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Research ArticleNon-Gaussian Stochastic Equivalent LinearizationMethod for Inelastic Nonlinear Systems with SofteningBehaviour under Seismic Ground Motions
Francisco L Silva-Gonzaacutelez1 Sonia E Ruiz2 and Alejandro Rodriacuteguez-Castellanos1
1 Instituto Mexicano del Petroleo Eje Central Lazaro Cardenas 152 San Bartolo Atepehuacan07730 Del Gustavo A Madero DF Mexico
2 Coordinacion de Mecanica Aplicada Instituto de Ingenierıa Universidad Nacional Autonoma de MexicoCiudad Universitaria 04510 Del Coyoacan DF Mexico
Correspondence should be addressed to Francisco L Silva-Gonzalez flsilvaimpmx
Received 20 July 2014 Revised 6 October 2014 Accepted 20 October 2014 Published 25 November 2014
Academic Editor Salvatore Caddemi
Copyright copy 2014 Francisco L Silva-Gonzalez et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
A non-Gaussian stochastic equivalent linearization (NSEL) method for estimating the non-Gaussian response of inelastic non-linear structural systems subjected to seismic ground motions represented as nonstationary random processes is presentedBased on a model that represents the time evolution of the joint probability density function (PDF) of the structural responsemathematical expressions of equivalent linearization coefficients are derived The displacement and velocity are assumed jointlyGaussian and the marginal PDF of the hysteretic component of the displacement is modeled by a mixed PDF which is Gaussianwhen the structural behavior is linear and turns into a bimodal PDF when the structural behavior is hysteretic The proposedNSEL method is applied to calculate the response of hysteretic single-degree-of-freedom systems with different vibration periodsand different design displacement ductility values The results corresponding to the proposed method are compared with thosecalculated by means of Monte Carlo simulation as well as by a Gaussian equivalent linearization method It is verified that theNSEL approach proposed herein leads to maximum structural response standard deviations similar to those obtained with MonteCarlo technique In addition a brief discussion about the extension of the method tomuti-degree-of-freedom systems is presented
1 Introduction
The method of stochastic equivalent linearization (SEL) isone of the most common methods within the approximateapproaches for stochastic dynamic analysis of nonlinearsystems The SEL method has proven to be a versatile andefficient technique from the computation point of viewA great impulse to the method was given by Atalik andUtku [1] who demonstrated that for Gaussian responses thecalculation of linearization coefficients can be performed ina very simple way However an important deficiency of themethod has been found when the response of systems withnonlinear inelastic behavior subjected to intense excitationshas been considered as Gaussian [2ndash6] The authors [7]have found that when the hypothesis related to the response
of single-degree-of-freedom (SDOF) systems with hystereticbehavior has Gaussian distribution the standard deviationof the displacement response can be underestimated in theorder of 45 consequently the probability of failure widelydeviates from the Monte Carlo simulation results especiallyfor systems with high displacement ductility demandsThis ismainly due to the fact that a Gaussian probability distributionis assumed for all variables however the restoring forceshould lie on a finite region implying that its probabilitydensity function (PDF) is non-Gaussian Several attemptshave been made in the last years to overcome the problemmentioned above [8ndash10] For example Asano and Iwan [11]obtained alternate linearization results for a hysteretic systemby using equations that were identical to the usual form on allsubsets having a nonzero probability density in the nonlinear
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 539738 16 pageshttpdxdoiorg1011552014539738
2 Mathematical Problems in Engineering
system but differed in the zero probability subsets An amplereview related to applications of statistical and equivalentlinearization in the analysis of structure and mechanicalnonlinear stochastic dynamic systems is found in [12]
A non-Gaussian equivalent linearization criterion(NSEL) that uses mathematical expressions for the lineari-zation coefficients is presented herein In order to derive thenon-Gaussian equivalent linearization coefficients a modelappropriately representing the time evolution of the jointprobability density function of the structural response ofsoftening systems subjected to seismic ground motions isused The parameters of the joint PDF depend on a functionwhichmeasures the non-Gaussianity of the response process
The structural softening models are of particular interestin earthquake engineering because they are commonly usedto analyze buildings made with softening materials suchas steel reinforced concrete or masonry which presentsaturation of the restoring force when deformation increases[13] The nonlinearities considered in the present study areonly due to material behavior
It is noticed that theNSELmethodsmentioned in the firstparagraph of this section do not take into account the timeevolution character of the joint PDFof the structural responsein the way that is formulated here
Themathematical expressions corresponding to the non-Gaussian equivalent linearization coefficients derived hereare applied to calculate the response of hysteretic SDOFsystems with different vibration periods and design displace-ment ductility demands subjected to seismic groundmotionsthat are representative of a nonstationary ergodic randomprocess The comparison of the results with those obtainedbymeans ofMonte Carlo simulation shows that the proposedapproach predicts with acceptable accuracy the maximumstandard deviation of the structural response of hystereticSDOF systems
An extension of the proposedmethod tomulti-degree-of-freedom (MDOF) systems is outlined at the end of the paper
2 Equation of Motion
The equation of motion of hysteretic SDOF system studiedhere is
+ 2120585120596 + 120572
2120596
2119909 + (1 minus 120572
2) 120596
2119911 = minus119886 (119905)
(1)
where and 119909 represent the relative acceleration velocityand displacement respectively 120585 = 1198882119898120596 is the fraction ofviscous critical damping 120596 =
radic119896119898 is the circular frequency
of vibration of the system 119898 is the mass 119888 is the coefficientof viscous damping 119896 is the initial stiffness 120572
2is the ratio
between the postyielding stiffness and the initial stiffness 119886(119905)represents the ground acceleration 119905 is the time and 119911 is thehysteretic component of displacement represented here bytheBouc-Wenmodel [14 15]which is defined by the followingnonlinear differential equation
= ℎ ( 119911) = 119860 minus 120573119911 || |119911|
119899minus1minus 120574 |119911|
119899 (2)
where 119860 120573 120574 and 119899 are parameters that define the size andthe shape of the hysteretic cycles 119899 controls the smoothness
of the transition from elastic to plastic structural behavior insuch a way that the smaller 119899 the smoother the transitionand 119899 rarr infin corresponds to a bilinear model 120573 and 120574 definethe softening or the hardening of the system the first casecorresponds to120573 + 120574 gt 0while the second corresponds to120573 +
120574 le 0 Casciati and Faravelli [16] recommend considering120573 = 120574 for steel and 120573 = minus2120574 for reinforced concrete In thisstudy softening systems with smooth transition representedby 119899 = 1 are considered
3 Stochastic Model of the Seismic Excitation
The seismic motion is modeled as a nonstationary Gaussianrandom process generated from a filtered white noise modu-lated in amplitudeThe transfer function proposed by Cloughand Penzien [17] is used here therefore the power spectraldensity of the seismic excitation is given by
119878 (120596) = (
120596
4
119892+ 4120585
2
119892120596
2
119892120596
2
(120596
2
119892minus 120596
2)
2
+ 4120585
2
119892120596
2
119892120596
2
)
times(
120596
4
(120596
2
119891minus 120596
2)
2
+ 4120585
2
119891120596
2
119891120596
2
)119904
0
(3)
The square of the amplitude modulating function is [18]
119888
2(119905) = 119886
119905
119887
119889 + 119905
119890Exp (minus119896119905) (4)
In practical applications the amplitude of the power spectraldensity of the white noise (119904
0) and the filter parameters (120596
119892
120585
119892 120596119891 and 120585
119891) are determined with a nonlinear fitting of
the power spectral density estimated by means of the Fourierspectrum of the basic seismic ground motion representativeof the excitation process The parameters 119886 119887 119889 119890 and 119896
of the modulating function are determined with a nonlinearfitting of the Arias intensity curve of the basic accelerogram[19] In this study the excitation process is modulated onlyin amplitude to study the effect of the frequency modulationon the nonlinear systems response the reader is referred to[20ndash23]
In the analysis presented below the seismic recordobtained in Mexico City at the Ministry of Communicationsand Transportation (SCT) during the September 19 1985earthquake is used to obtain the parameters of the stochasticmodel of the seismic excitationThe amplitude of the spectraldensity of the white noise is 119904
0= 72776 times 10
minus4 cm2s3 thefilter parameters are 120596
119892= 31017 120585
119892= 00220 120596
119891= 22988
and 120585119891= 00492 and those corresponding to the modulating
function are 119886 = 58161 times 10
48 cm2s4 119887 = minus03388 119889 =
2166times10
47 119890 = 26461 and 119896 = minus01258The bimodal powerspectral densities of the seismic motion and its fitted modelare shown in Figure 1 The cumulative Arias intensity curveand its fitted model (119868
0) are in Figure 2
Mathematical Problems in Engineering 3
Power spectral density of SCTClough-Penzien model
0
1000
2000
3000
4000
5000
6000
7000
8000
0 1 2 3 4 5
S(120596
) (cm
2s3)
Circular frequency120596 (rads)
Figure 1 Bimodal power spectral density of the accelerogram andits corresponding fitted model
0
20000
40000
60000
80000
100000
120000
140000
160000
0 20 40 60 80 100 120 140 160
intc2(t)dtI 0
(t) (
cm2s3)
Time t (s)
I0(t)intc2(t)dt
Figure 2 Cumulative Arias intensity curve (1198680) and its correspond-
ing fitted model
4 Modeling the Joint Probability DensityFunction of the Structural Response
TheNSEL approach presented here is based on an appropriaterepresentation of the time evolution of the joint PDF of thestructural seismic response of softening systems with smoothtransition Results from Monte Carlo simulation analyseshave shown that the PDF of the displacement 119891
119883(119909) and
of the velocity 119891() of the structural system is Gaussian
and their evolution in time is as shown in Figures 3 and 4respectively However the probability density of the restoringforce and of the displacement hysteretic component 119891
119885(119911)
changes with time because it depends on the nonlinearityof the structural behavior At the beginning and at the endof the excitation when the structure is under low seismicintensities 119891
119885(119911) tends to be Gaussian because the structure
presents linear behavior however as the seismic intensity
minus1minus05
005
1
0
005
01
015
02
025
03
x (m)Time (s)0
2040
6080
100
fx
(x)
Figure 3 Time evolution of the probability density function of thedisplacement response
increases and the structural behavior becomes nonlinear(hysteretic) then 119891
119885(119911) becomes non-Gaussian as shown in
Figure 5 It can be seen that when the system is vibratingwithin the nonlinear range 119911 equals its maximum value 119911
119906
and its PDF presents concentrations at +119911119906and minus119911
119906 Figure 5
also shows that as the excitation intensity grows (up to 119905 =
59 s) and consequently the displacement ductility demandincreases 119891
119885(119911) turns into a bimodal PDF It is observed that
the support of 119891119885(119911) is minus119911
119906le 119911 le 119911
119906 where the maximum
value of 119911 is given by [13 19]
119911
119906= (
119860
120573 + 120574
)
1119899
(5)
Considering the concentration of values of 119911 in the vicinityof its maximum value 119911
119906and the shape that such PDF adopts
(see Figure 5 for 119905 = 59 s) the following mathematical modelis used
119891
119885(119911) =
119886120593
119885(119911) (1 minus 2119901) +
119887120593
119885(119911 minus 119911
119911) 119901
+
119887120593
119885(119911 + 119911
119911) 119901 0 le 119901 le 05
(6)
where119886120593
119885is a Gaussian density functionwith zeromean and
standard deviation 120590
119911119886119887120593
119885is a Gaussian density function
with mean 119911
119911(or minus119911
119911) and standard deviation 120590
119911119887 and 119901 is a
weighting factorThe idea is that when the nonlinear behaviorbecomes important the weighting of
119887120593
119885 functions should
increase while that corresponding to119886120593
119885should decrease It
is noticed that (6) uses the Gaussian density functions119886120593
119885
and119887120593
119885and not Diracrsquos pulses as has been proposed by
other authors [9] The advantage of using Gaussian densityfunctions instead of Diracrsquos pulses is that it is possible tomodelmore appropriately the time evolution of the joint PDFand as a consequence theNSEL approach leads to results thatbetter approximate the Monte Carlo results Furthermore itis possible to get mathematical expressions of the equivalentlinearization coefficients
Using the marginal PDFs 119891119883(119909) 119891() and the informa-
tion about the correlation structure of the random variablesit is possible to compute the joint PDF [24 25] Here the joint
4 Mathematical Problems in Engineering
Figure 4 Time evolution of the probability density function of thevelocity response
PDF is obtained using the conditional PDF of 119909 and given119911 ℎ(119909 | 119911) as follows
119891
119883119885(119909 119911) = ℎ (119909 | 119911) 119891
119885(119911) (7)
Based on the simulation results it is reasonable to assumethat ℎ(119909 | 119911) is jointly Gaussian The parameters of ℎ(119909 |119911) that is conditional means and variances are estimatedassuming that the former are linear functions of 119911 andthe latter do not depend on 119911 as it happens in a meansquare estimation of normal variables [26] Additionally it isconsidered that the response process has zero mean
Based on (6) and (7) the following expression for the jointPDF is obtained
119891
119883119885(119909 119911)
=
1 minus 2119901
(2120587)
32120590
119909120590
119909120590
119911119886120581
12
times Exp[minus 1
2120581
(
119911
2
120590
2
119911119886
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
119911
2
120590
2
119911119886
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 minus 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 minus 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 + 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 + 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
(8)
where 120590
119909 120590119909 and 120590
119911are the standard deviations of 119909
and 119911 respectively 120588119909119911 120588119909119911 and 120588
119909 119909are the corresponding
correlation coefficients and 120581 = 1 minus 120588
2
119909119911minus 120588
2
119909119911minus 120588
2
119909 119909+
2120588
119909 119909120588
119909119911120588
119909119911
41 Evaluating the Parameters 120590119911119886 120590119911119887 119901 and 119911
119911 The four
parameters 120590119911119886 120590119911119887 119901 and 119911
119911of the joint PDF given by (8)
Mathematical Problems in Engineering 5
020
4060
80100
minus01minus005
0005
01
0
02
04
06
08
z (m)Time (s) minuszu
+zu
fz(z
)
Figure 5 Time evolution of the probability density function of thehysteretic component of displacement response
should satisfy that the area under the PDF given by (6) is closeto unity
int
119911119906
minus119911119906
119891
119885(119911) 119889119911 = (1 minus 2119901)Erf(
119911
119911
radic2120590
119911119886
)
+ 119901[Erf(119911
119906minus 119911
119911
radic2120590
119911119887
)
+ Erf(119911
119906+ 119911
119911
radic2120590
119911119887
)] asymp 1
(9)
where Erf(sdot) is the error function given by
Erf (119906) = 2
radic120587
int
119906
0
Exp (minus1199052) 119889119905 (10)
The above mentioned condition is necessary because (6) isnot a bounded function If (9) is satisfied the secondmoment(in this case equal to the variance 1205902
119911) can be approximated
as
119864 [119911
2] = ∭
infin
minusinfin
119911
2119891
119883119885(119909 119911) 119889119909 119889 119889119911
= int
infin
minusinfin
119911
2119891
119885(119911) 119889119911 = (1 minus 2119901) 120590
2
119911119886
+ 2119901 (119911
2
119911+ 120590
2
119911119887) = 120590
2
119911
(11)
The four parameters 120590119911119886 120590119911119887 119901 and 119911
119911could be computed
by means of their first four moments however here theyare determined bymeans of nondimensional coefficients thatdepend on the system response non-Gaussianity level whichis represented by [13]
120582 (119905) = int
minus119911119906
minusinfin
1
radic2120587120590
119911(119905)
Exp(minus 119911
2
2120590
2
119911(119905)
) 119889119911
=
1
2
(1 minus Erf(119911
119906
radic2120590
119911(119905)
))
(12)
From (12) it is observed that 120582 is close to zero when theresponse is Gaussian that is when the system strength is
high (where 119911119906is high) and the excitation or response is low
(and as a consequence 120590119911is low) On the other hand if the
response is non-Gaussian that is if the system strength is low(where 119911
119906is low) and the excitation or response is high (and
consequently 120590119911is high) then 120582 is high
Here the following mathematical expressions for thenormalized parameters 119911
119911119911
119911119906 120590119911119886120590
119911 120590119911119887120590
119911 and 119901 are
proposed
119911
119911(119905)
119911
119906
= 119886
1Ln (1198862120582 (119905) + 119886
3) + 119886
4
119901 (119905) = 119887
1Ln (1198872120582 (119905) + 119887
3) + 119887
4
120590
119911119886(119905)
120590
119911(119905)
= 119888
1Ln (1198882120582 (119905) + 119888
3) + 119888
4
120590
119911119887(119905)
120590
119911(119905)
= 119889
1Ln (1198892120582 (119905) + 119889
3) + 119889
4
(13)
where 119886
119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 are constants to
be determined The general behavior of (13) is presentedin Figures 6(a)ndash6(d) where it can be seen that 119911
119911and 119901
increase with 120582 (Figures 6(a) and 6(b)) meaning that whenthe nonlinearity of the response increases the function
119887120593
119885
moves far from 119911 = 0 and its weight increases hence 119891119885(119911)
(given by (6)) becomes a bimodal PDF On the other handwhen the structural behavior is linear120582 is close to zero and theweight of
119887120593
119885vanishes therefore 119891
119885(119911) becomes Gaussian
It can also be seen (Figures 6(c) and 6(d)) that119886120593
119885120590
119911and
119887120593
119885120590
119911decrease with 120582 but it should be noticed that as 120590
119911
increases 120582 grows as wellEquations (13) show that the parameters 120590
119911119886 120590119911119887 119901 and
119911
119911are time dependent thus the proposed approach considers
the time evolutionary character of the joint PDF of theresponse while other methods proposed in the literature donot take it into account Figure 7 shows an example of thetime evolution of 119891
119885(119911) given by (6)
411 Case of Narrow-Band Seismic Inputs Parameters 119886
119894
119887
119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 of (13) corresponding to
the response of single-degree-of-freedom systems (SDOF)systems subjected to the action of narrow-band seismicinputs were calculated by means of Monte Carlo simulationanalysis The critical case where the natural vibration periodof SDOF system is equal to the dominant period of seismicinput (in this case 119879 = 21 s see Figure 1) and the SDOFsystem has a high design ductility demand 120578 = 4 wasconsidered The design ductility factor 120578 is defined hereas the ratio between the expected maximum and the yielddisplacement of the system In this study the first one is foundby means of a stationary Gaussian equivalent linearizationanalysis and the latter is estimated by means of the initialstiffness and the yield force of the system
The SDOF system was subjected to the action of 50000artificial accelerograms based on the narrow-band motionrecorded in SCT on September 19 1985 (see Section 3) Forthis case it was found that 120582 lies within the interval 0 lt 120582 le
01 and that 120590119911119886
gt 120590
119911119887leads to good results The parameter
values obtained from the analysis are shown in Table 1
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
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2 Mathematical Problems in Engineering
system but differed in the zero probability subsets An amplereview related to applications of statistical and equivalentlinearization in the analysis of structure and mechanicalnonlinear stochastic dynamic systems is found in [12]
A non-Gaussian equivalent linearization criterion(NSEL) that uses mathematical expressions for the lineari-zation coefficients is presented herein In order to derive thenon-Gaussian equivalent linearization coefficients a modelappropriately representing the time evolution of the jointprobability density function of the structural response ofsoftening systems subjected to seismic ground motions isused The parameters of the joint PDF depend on a functionwhichmeasures the non-Gaussianity of the response process
The structural softening models are of particular interestin earthquake engineering because they are commonly usedto analyze buildings made with softening materials suchas steel reinforced concrete or masonry which presentsaturation of the restoring force when deformation increases[13] The nonlinearities considered in the present study areonly due to material behavior
It is noticed that theNSELmethodsmentioned in the firstparagraph of this section do not take into account the timeevolution character of the joint PDFof the structural responsein the way that is formulated here
Themathematical expressions corresponding to the non-Gaussian equivalent linearization coefficients derived hereare applied to calculate the response of hysteretic SDOFsystems with different vibration periods and design displace-ment ductility demands subjected to seismic groundmotionsthat are representative of a nonstationary ergodic randomprocess The comparison of the results with those obtainedbymeans ofMonte Carlo simulation shows that the proposedapproach predicts with acceptable accuracy the maximumstandard deviation of the structural response of hystereticSDOF systems
An extension of the proposedmethod tomulti-degree-of-freedom (MDOF) systems is outlined at the end of the paper
2 Equation of Motion
The equation of motion of hysteretic SDOF system studiedhere is
+ 2120585120596 + 120572
2120596
2119909 + (1 minus 120572
2) 120596
2119911 = minus119886 (119905)
(1)
where and 119909 represent the relative acceleration velocityand displacement respectively 120585 = 1198882119898120596 is the fraction ofviscous critical damping 120596 =
radic119896119898 is the circular frequency
of vibration of the system 119898 is the mass 119888 is the coefficientof viscous damping 119896 is the initial stiffness 120572
2is the ratio
between the postyielding stiffness and the initial stiffness 119886(119905)represents the ground acceleration 119905 is the time and 119911 is thehysteretic component of displacement represented here bytheBouc-Wenmodel [14 15]which is defined by the followingnonlinear differential equation
= ℎ ( 119911) = 119860 minus 120573119911 || |119911|
119899minus1minus 120574 |119911|
119899 (2)
where 119860 120573 120574 and 119899 are parameters that define the size andthe shape of the hysteretic cycles 119899 controls the smoothness
of the transition from elastic to plastic structural behavior insuch a way that the smaller 119899 the smoother the transitionand 119899 rarr infin corresponds to a bilinear model 120573 and 120574 definethe softening or the hardening of the system the first casecorresponds to120573 + 120574 gt 0while the second corresponds to120573 +
120574 le 0 Casciati and Faravelli [16] recommend considering120573 = 120574 for steel and 120573 = minus2120574 for reinforced concrete In thisstudy softening systems with smooth transition representedby 119899 = 1 are considered
3 Stochastic Model of the Seismic Excitation
The seismic motion is modeled as a nonstationary Gaussianrandom process generated from a filtered white noise modu-lated in amplitudeThe transfer function proposed by Cloughand Penzien [17] is used here therefore the power spectraldensity of the seismic excitation is given by
119878 (120596) = (
120596
4
119892+ 4120585
2
119892120596
2
119892120596
2
(120596
2
119892minus 120596
2)
2
+ 4120585
2
119892120596
2
119892120596
2
)
times(
120596
4
(120596
2
119891minus 120596
2)
2
+ 4120585
2
119891120596
2
119891120596
2
)119904
0
(3)
The square of the amplitude modulating function is [18]
119888
2(119905) = 119886
119905
119887
119889 + 119905
119890Exp (minus119896119905) (4)
In practical applications the amplitude of the power spectraldensity of the white noise (119904
0) and the filter parameters (120596
119892
120585
119892 120596119891 and 120585
119891) are determined with a nonlinear fitting of
the power spectral density estimated by means of the Fourierspectrum of the basic seismic ground motion representativeof the excitation process The parameters 119886 119887 119889 119890 and 119896
of the modulating function are determined with a nonlinearfitting of the Arias intensity curve of the basic accelerogram[19] In this study the excitation process is modulated onlyin amplitude to study the effect of the frequency modulationon the nonlinear systems response the reader is referred to[20ndash23]
In the analysis presented below the seismic recordobtained in Mexico City at the Ministry of Communicationsand Transportation (SCT) during the September 19 1985earthquake is used to obtain the parameters of the stochasticmodel of the seismic excitationThe amplitude of the spectraldensity of the white noise is 119904
0= 72776 times 10
minus4 cm2s3 thefilter parameters are 120596
119892= 31017 120585
119892= 00220 120596
119891= 22988
and 120585119891= 00492 and those corresponding to the modulating
function are 119886 = 58161 times 10
48 cm2s4 119887 = minus03388 119889 =
2166times10
47 119890 = 26461 and 119896 = minus01258The bimodal powerspectral densities of the seismic motion and its fitted modelare shown in Figure 1 The cumulative Arias intensity curveand its fitted model (119868
0) are in Figure 2
Mathematical Problems in Engineering 3
Power spectral density of SCTClough-Penzien model
0
1000
2000
3000
4000
5000
6000
7000
8000
0 1 2 3 4 5
S(120596
) (cm
2s3)
Circular frequency120596 (rads)
Figure 1 Bimodal power spectral density of the accelerogram andits corresponding fitted model
0
20000
40000
60000
80000
100000
120000
140000
160000
0 20 40 60 80 100 120 140 160
intc2(t)dtI 0
(t) (
cm2s3)
Time t (s)
I0(t)intc2(t)dt
Figure 2 Cumulative Arias intensity curve (1198680) and its correspond-
ing fitted model
4 Modeling the Joint Probability DensityFunction of the Structural Response
TheNSEL approach presented here is based on an appropriaterepresentation of the time evolution of the joint PDF of thestructural seismic response of softening systems with smoothtransition Results from Monte Carlo simulation analyseshave shown that the PDF of the displacement 119891
119883(119909) and
of the velocity 119891() of the structural system is Gaussian
and their evolution in time is as shown in Figures 3 and 4respectively However the probability density of the restoringforce and of the displacement hysteretic component 119891
119885(119911)
changes with time because it depends on the nonlinearityof the structural behavior At the beginning and at the endof the excitation when the structure is under low seismicintensities 119891
119885(119911) tends to be Gaussian because the structure
presents linear behavior however as the seismic intensity
minus1minus05
005
1
0
005
01
015
02
025
03
x (m)Time (s)0
2040
6080
100
fx
(x)
Figure 3 Time evolution of the probability density function of thedisplacement response
increases and the structural behavior becomes nonlinear(hysteretic) then 119891
119885(119911) becomes non-Gaussian as shown in
Figure 5 It can be seen that when the system is vibratingwithin the nonlinear range 119911 equals its maximum value 119911
119906
and its PDF presents concentrations at +119911119906and minus119911
119906 Figure 5
also shows that as the excitation intensity grows (up to 119905 =
59 s) and consequently the displacement ductility demandincreases 119891
119885(119911) turns into a bimodal PDF It is observed that
the support of 119891119885(119911) is minus119911
119906le 119911 le 119911
119906 where the maximum
value of 119911 is given by [13 19]
119911
119906= (
119860
120573 + 120574
)
1119899
(5)
Considering the concentration of values of 119911 in the vicinityof its maximum value 119911
119906and the shape that such PDF adopts
(see Figure 5 for 119905 = 59 s) the following mathematical modelis used
119891
119885(119911) =
119886120593
119885(119911) (1 minus 2119901) +
119887120593
119885(119911 minus 119911
119911) 119901
+
119887120593
119885(119911 + 119911
119911) 119901 0 le 119901 le 05
(6)
where119886120593
119885is a Gaussian density functionwith zeromean and
standard deviation 120590
119911119886119887120593
119885is a Gaussian density function
with mean 119911
119911(or minus119911
119911) and standard deviation 120590
119911119887 and 119901 is a
weighting factorThe idea is that when the nonlinear behaviorbecomes important the weighting of
119887120593
119885 functions should
increase while that corresponding to119886120593
119885should decrease It
is noticed that (6) uses the Gaussian density functions119886120593
119885
and119887120593
119885and not Diracrsquos pulses as has been proposed by
other authors [9] The advantage of using Gaussian densityfunctions instead of Diracrsquos pulses is that it is possible tomodelmore appropriately the time evolution of the joint PDFand as a consequence theNSEL approach leads to results thatbetter approximate the Monte Carlo results Furthermore itis possible to get mathematical expressions of the equivalentlinearization coefficients
Using the marginal PDFs 119891119883(119909) 119891() and the informa-
tion about the correlation structure of the random variablesit is possible to compute the joint PDF [24 25] Here the joint
4 Mathematical Problems in Engineering
Figure 4 Time evolution of the probability density function of thevelocity response
PDF is obtained using the conditional PDF of 119909 and given119911 ℎ(119909 | 119911) as follows
119891
119883119885(119909 119911) = ℎ (119909 | 119911) 119891
119885(119911) (7)
Based on the simulation results it is reasonable to assumethat ℎ(119909 | 119911) is jointly Gaussian The parameters of ℎ(119909 |119911) that is conditional means and variances are estimatedassuming that the former are linear functions of 119911 andthe latter do not depend on 119911 as it happens in a meansquare estimation of normal variables [26] Additionally it isconsidered that the response process has zero mean
Based on (6) and (7) the following expression for the jointPDF is obtained
119891
119883119885(119909 119911)
=
1 minus 2119901
(2120587)
32120590
119909120590
119909120590
119911119886120581
12
times Exp[minus 1
2120581
(
119911
2
120590
2
119911119886
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
119911
2
120590
2
119911119886
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 minus 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 minus 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 + 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 + 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
(8)
where 120590
119909 120590119909 and 120590
119911are the standard deviations of 119909
and 119911 respectively 120588119909119911 120588119909119911 and 120588
119909 119909are the corresponding
correlation coefficients and 120581 = 1 minus 120588
2
119909119911minus 120588
2
119909119911minus 120588
2
119909 119909+
2120588
119909 119909120588
119909119911120588
119909119911
41 Evaluating the Parameters 120590119911119886 120590119911119887 119901 and 119911
119911 The four
parameters 120590119911119886 120590119911119887 119901 and 119911
119911of the joint PDF given by (8)
Mathematical Problems in Engineering 5
020
4060
80100
minus01minus005
0005
01
0
02
04
06
08
z (m)Time (s) minuszu
+zu
fz(z
)
Figure 5 Time evolution of the probability density function of thehysteretic component of displacement response
should satisfy that the area under the PDF given by (6) is closeto unity
int
119911119906
minus119911119906
119891
119885(119911) 119889119911 = (1 minus 2119901)Erf(
119911
119911
radic2120590
119911119886
)
+ 119901[Erf(119911
119906minus 119911
119911
radic2120590
119911119887
)
+ Erf(119911
119906+ 119911
119911
radic2120590
119911119887
)] asymp 1
(9)
where Erf(sdot) is the error function given by
Erf (119906) = 2
radic120587
int
119906
0
Exp (minus1199052) 119889119905 (10)
The above mentioned condition is necessary because (6) isnot a bounded function If (9) is satisfied the secondmoment(in this case equal to the variance 1205902
119911) can be approximated
as
119864 [119911
2] = ∭
infin
minusinfin
119911
2119891
119883119885(119909 119911) 119889119909 119889 119889119911
= int
infin
minusinfin
119911
2119891
119885(119911) 119889119911 = (1 minus 2119901) 120590
2
119911119886
+ 2119901 (119911
2
119911+ 120590
2
119911119887) = 120590
2
119911
(11)
The four parameters 120590119911119886 120590119911119887 119901 and 119911
119911could be computed
by means of their first four moments however here theyare determined bymeans of nondimensional coefficients thatdepend on the system response non-Gaussianity level whichis represented by [13]
120582 (119905) = int
minus119911119906
minusinfin
1
radic2120587120590
119911(119905)
Exp(minus 119911
2
2120590
2
119911(119905)
) 119889119911
=
1
2
(1 minus Erf(119911
119906
radic2120590
119911(119905)
))
(12)
From (12) it is observed that 120582 is close to zero when theresponse is Gaussian that is when the system strength is
high (where 119911119906is high) and the excitation or response is low
(and as a consequence 120590119911is low) On the other hand if the
response is non-Gaussian that is if the system strength is low(where 119911
119906is low) and the excitation or response is high (and
consequently 120590119911is high) then 120582 is high
Here the following mathematical expressions for thenormalized parameters 119911
119911119911
119911119906 120590119911119886120590
119911 120590119911119887120590
119911 and 119901 are
proposed
119911
119911(119905)
119911
119906
= 119886
1Ln (1198862120582 (119905) + 119886
3) + 119886
4
119901 (119905) = 119887
1Ln (1198872120582 (119905) + 119887
3) + 119887
4
120590
119911119886(119905)
120590
119911(119905)
= 119888
1Ln (1198882120582 (119905) + 119888
3) + 119888
4
120590
119911119887(119905)
120590
119911(119905)
= 119889
1Ln (1198892120582 (119905) + 119889
3) + 119889
4
(13)
where 119886
119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 are constants to
be determined The general behavior of (13) is presentedin Figures 6(a)ndash6(d) where it can be seen that 119911
119911and 119901
increase with 120582 (Figures 6(a) and 6(b)) meaning that whenthe nonlinearity of the response increases the function
119887120593
119885
moves far from 119911 = 0 and its weight increases hence 119891119885(119911)
(given by (6)) becomes a bimodal PDF On the other handwhen the structural behavior is linear120582 is close to zero and theweight of
119887120593
119885vanishes therefore 119891
119885(119911) becomes Gaussian
It can also be seen (Figures 6(c) and 6(d)) that119886120593
119885120590
119911and
119887120593
119885120590
119911decrease with 120582 but it should be noticed that as 120590
119911
increases 120582 grows as wellEquations (13) show that the parameters 120590
119911119886 120590119911119887 119901 and
119911
119911are time dependent thus the proposed approach considers
the time evolutionary character of the joint PDF of theresponse while other methods proposed in the literature donot take it into account Figure 7 shows an example of thetime evolution of 119891
119885(119911) given by (6)
411 Case of Narrow-Band Seismic Inputs Parameters 119886
119894
119887
119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 of (13) corresponding to
the response of single-degree-of-freedom systems (SDOF)systems subjected to the action of narrow-band seismicinputs were calculated by means of Monte Carlo simulationanalysis The critical case where the natural vibration periodof SDOF system is equal to the dominant period of seismicinput (in this case 119879 = 21 s see Figure 1) and the SDOFsystem has a high design ductility demand 120578 = 4 wasconsidered The design ductility factor 120578 is defined hereas the ratio between the expected maximum and the yielddisplacement of the system In this study the first one is foundby means of a stationary Gaussian equivalent linearizationanalysis and the latter is estimated by means of the initialstiffness and the yield force of the system
The SDOF system was subjected to the action of 50000artificial accelerograms based on the narrow-band motionrecorded in SCT on September 19 1985 (see Section 3) Forthis case it was found that 120582 lies within the interval 0 lt 120582 le
01 and that 120590119911119886
gt 120590
119911119887leads to good results The parameter
values obtained from the analysis are shown in Table 1
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Power spectral density of SCTClough-Penzien model
0
1000
2000
3000
4000
5000
6000
7000
8000
0 1 2 3 4 5
S(120596
) (cm
2s3)
Circular frequency120596 (rads)
Figure 1 Bimodal power spectral density of the accelerogram andits corresponding fitted model
0
20000
40000
60000
80000
100000
120000
140000
160000
0 20 40 60 80 100 120 140 160
intc2(t)dtI 0
(t) (
cm2s3)
Time t (s)
I0(t)intc2(t)dt
Figure 2 Cumulative Arias intensity curve (1198680) and its correspond-
ing fitted model
4 Modeling the Joint Probability DensityFunction of the Structural Response
TheNSEL approach presented here is based on an appropriaterepresentation of the time evolution of the joint PDF of thestructural seismic response of softening systems with smoothtransition Results from Monte Carlo simulation analyseshave shown that the PDF of the displacement 119891
119883(119909) and
of the velocity 119891() of the structural system is Gaussian
and their evolution in time is as shown in Figures 3 and 4respectively However the probability density of the restoringforce and of the displacement hysteretic component 119891
119885(119911)
changes with time because it depends on the nonlinearityof the structural behavior At the beginning and at the endof the excitation when the structure is under low seismicintensities 119891
119885(119911) tends to be Gaussian because the structure
presents linear behavior however as the seismic intensity
minus1minus05
005
1
0
005
01
015
02
025
03
x (m)Time (s)0
2040
6080
100
fx
(x)
Figure 3 Time evolution of the probability density function of thedisplacement response
increases and the structural behavior becomes nonlinear(hysteretic) then 119891
119885(119911) becomes non-Gaussian as shown in
Figure 5 It can be seen that when the system is vibratingwithin the nonlinear range 119911 equals its maximum value 119911
119906
and its PDF presents concentrations at +119911119906and minus119911
119906 Figure 5
also shows that as the excitation intensity grows (up to 119905 =
59 s) and consequently the displacement ductility demandincreases 119891
119885(119911) turns into a bimodal PDF It is observed that
the support of 119891119885(119911) is minus119911
119906le 119911 le 119911
119906 where the maximum
value of 119911 is given by [13 19]
119911
119906= (
119860
120573 + 120574
)
1119899
(5)
Considering the concentration of values of 119911 in the vicinityof its maximum value 119911
119906and the shape that such PDF adopts
(see Figure 5 for 119905 = 59 s) the following mathematical modelis used
119891
119885(119911) =
119886120593
119885(119911) (1 minus 2119901) +
119887120593
119885(119911 minus 119911
119911) 119901
+
119887120593
119885(119911 + 119911
119911) 119901 0 le 119901 le 05
(6)
where119886120593
119885is a Gaussian density functionwith zeromean and
standard deviation 120590
119911119886119887120593
119885is a Gaussian density function
with mean 119911
119911(or minus119911
119911) and standard deviation 120590
119911119887 and 119901 is a
weighting factorThe idea is that when the nonlinear behaviorbecomes important the weighting of
119887120593
119885 functions should
increase while that corresponding to119886120593
119885should decrease It
is noticed that (6) uses the Gaussian density functions119886120593
119885
and119887120593
119885and not Diracrsquos pulses as has been proposed by
other authors [9] The advantage of using Gaussian densityfunctions instead of Diracrsquos pulses is that it is possible tomodelmore appropriately the time evolution of the joint PDFand as a consequence theNSEL approach leads to results thatbetter approximate the Monte Carlo results Furthermore itis possible to get mathematical expressions of the equivalentlinearization coefficients
Using the marginal PDFs 119891119883(119909) 119891() and the informa-
tion about the correlation structure of the random variablesit is possible to compute the joint PDF [24 25] Here the joint
4 Mathematical Problems in Engineering
Figure 4 Time evolution of the probability density function of thevelocity response
PDF is obtained using the conditional PDF of 119909 and given119911 ℎ(119909 | 119911) as follows
119891
119883119885(119909 119911) = ℎ (119909 | 119911) 119891
119885(119911) (7)
Based on the simulation results it is reasonable to assumethat ℎ(119909 | 119911) is jointly Gaussian The parameters of ℎ(119909 |119911) that is conditional means and variances are estimatedassuming that the former are linear functions of 119911 andthe latter do not depend on 119911 as it happens in a meansquare estimation of normal variables [26] Additionally it isconsidered that the response process has zero mean
Based on (6) and (7) the following expression for the jointPDF is obtained
119891
119883119885(119909 119911)
=
1 minus 2119901
(2120587)
32120590
119909120590
119909120590
119911119886120581
12
times Exp[minus 1
2120581
(
119911
2
120590
2
119911119886
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
119911
2
120590
2
119911119886
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 minus 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 minus 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 + 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 + 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
(8)
where 120590
119909 120590119909 and 120590
119911are the standard deviations of 119909
and 119911 respectively 120588119909119911 120588119909119911 and 120588
119909 119909are the corresponding
correlation coefficients and 120581 = 1 minus 120588
2
119909119911minus 120588
2
119909119911minus 120588
2
119909 119909+
2120588
119909 119909120588
119909119911120588
119909119911
41 Evaluating the Parameters 120590119911119886 120590119911119887 119901 and 119911
119911 The four
parameters 120590119911119886 120590119911119887 119901 and 119911
119911of the joint PDF given by (8)
Mathematical Problems in Engineering 5
020
4060
80100
minus01minus005
0005
01
0
02
04
06
08
z (m)Time (s) minuszu
+zu
fz(z
)
Figure 5 Time evolution of the probability density function of thehysteretic component of displacement response
should satisfy that the area under the PDF given by (6) is closeto unity
int
119911119906
minus119911119906
119891
119885(119911) 119889119911 = (1 minus 2119901)Erf(
119911
119911
radic2120590
119911119886
)
+ 119901[Erf(119911
119906minus 119911
119911
radic2120590
119911119887
)
+ Erf(119911
119906+ 119911
119911
radic2120590
119911119887
)] asymp 1
(9)
where Erf(sdot) is the error function given by
Erf (119906) = 2
radic120587
int
119906
0
Exp (minus1199052) 119889119905 (10)
The above mentioned condition is necessary because (6) isnot a bounded function If (9) is satisfied the secondmoment(in this case equal to the variance 1205902
119911) can be approximated
as
119864 [119911
2] = ∭
infin
minusinfin
119911
2119891
119883119885(119909 119911) 119889119909 119889 119889119911
= int
infin
minusinfin
119911
2119891
119885(119911) 119889119911 = (1 minus 2119901) 120590
2
119911119886
+ 2119901 (119911
2
119911+ 120590
2
119911119887) = 120590
2
119911
(11)
The four parameters 120590119911119886 120590119911119887 119901 and 119911
119911could be computed
by means of their first four moments however here theyare determined bymeans of nondimensional coefficients thatdepend on the system response non-Gaussianity level whichis represented by [13]
120582 (119905) = int
minus119911119906
minusinfin
1
radic2120587120590
119911(119905)
Exp(minus 119911
2
2120590
2
119911(119905)
) 119889119911
=
1
2
(1 minus Erf(119911
119906
radic2120590
119911(119905)
))
(12)
From (12) it is observed that 120582 is close to zero when theresponse is Gaussian that is when the system strength is
high (where 119911119906is high) and the excitation or response is low
(and as a consequence 120590119911is low) On the other hand if the
response is non-Gaussian that is if the system strength is low(where 119911
119906is low) and the excitation or response is high (and
consequently 120590119911is high) then 120582 is high
Here the following mathematical expressions for thenormalized parameters 119911
119911119911
119911119906 120590119911119886120590
119911 120590119911119887120590
119911 and 119901 are
proposed
119911
119911(119905)
119911
119906
= 119886
1Ln (1198862120582 (119905) + 119886
3) + 119886
4
119901 (119905) = 119887
1Ln (1198872120582 (119905) + 119887
3) + 119887
4
120590
119911119886(119905)
120590
119911(119905)
= 119888
1Ln (1198882120582 (119905) + 119888
3) + 119888
4
120590
119911119887(119905)
120590
119911(119905)
= 119889
1Ln (1198892120582 (119905) + 119889
3) + 119889
4
(13)
where 119886
119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 are constants to
be determined The general behavior of (13) is presentedin Figures 6(a)ndash6(d) where it can be seen that 119911
119911and 119901
increase with 120582 (Figures 6(a) and 6(b)) meaning that whenthe nonlinearity of the response increases the function
119887120593
119885
moves far from 119911 = 0 and its weight increases hence 119891119885(119911)
(given by (6)) becomes a bimodal PDF On the other handwhen the structural behavior is linear120582 is close to zero and theweight of
119887120593
119885vanishes therefore 119891
119885(119911) becomes Gaussian
It can also be seen (Figures 6(c) and 6(d)) that119886120593
119885120590
119911and
119887120593
119885120590
119911decrease with 120582 but it should be noticed that as 120590
119911
increases 120582 grows as wellEquations (13) show that the parameters 120590
119911119886 120590119911119887 119901 and
119911
119911are time dependent thus the proposed approach considers
the time evolutionary character of the joint PDF of theresponse while other methods proposed in the literature donot take it into account Figure 7 shows an example of thetime evolution of 119891
119885(119911) given by (6)
411 Case of Narrow-Band Seismic Inputs Parameters 119886
119894
119887
119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 of (13) corresponding to
the response of single-degree-of-freedom systems (SDOF)systems subjected to the action of narrow-band seismicinputs were calculated by means of Monte Carlo simulationanalysis The critical case where the natural vibration periodof SDOF system is equal to the dominant period of seismicinput (in this case 119879 = 21 s see Figure 1) and the SDOFsystem has a high design ductility demand 120578 = 4 wasconsidered The design ductility factor 120578 is defined hereas the ratio between the expected maximum and the yielddisplacement of the system In this study the first one is foundby means of a stationary Gaussian equivalent linearizationanalysis and the latter is estimated by means of the initialstiffness and the yield force of the system
The SDOF system was subjected to the action of 50000artificial accelerograms based on the narrow-band motionrecorded in SCT on September 19 1985 (see Section 3) Forthis case it was found that 120582 lies within the interval 0 lt 120582 le
01 and that 120590119911119886
gt 120590
119911119887leads to good results The parameter
values obtained from the analysis are shown in Table 1
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Figure 4 Time evolution of the probability density function of thevelocity response
PDF is obtained using the conditional PDF of 119909 and given119911 ℎ(119909 | 119911) as follows
119891
119883119885(119909 119911) = ℎ (119909 | 119911) 119891
119885(119911) (7)
Based on the simulation results it is reasonable to assumethat ℎ(119909 | 119911) is jointly Gaussian The parameters of ℎ(119909 |119911) that is conditional means and variances are estimatedassuming that the former are linear functions of 119911 andthe latter do not depend on 119911 as it happens in a meansquare estimation of normal variables [26] Additionally it isconsidered that the response process has zero mean
Based on (6) and (7) the following expression for the jointPDF is obtained
119891
119883119885(119909 119911)
=
1 minus 2119901
(2120587)
32120590
119909120590
119909120590
119911119886120581
12
times Exp[minus 1
2120581
(
119911
2
120590
2
119911119886
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
119911
2
120590
2
119911119886
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 minus 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 minus 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
+
119901
(2120587)
32120590
119909120590
119909120590
119911119887120581
12
times Exp[minus 1
2120581
(
(119911 + 119911
119911)
2
120590
2
119911119887
(1 minus 120588
2
119909 119909)
+ (
119911
2
120590
2
119911
minus
(119911 + 119911
119911)
2
120590
2
119911119887
)
times (120588
2
119909119911+ 120588
2
119909119911minus 2120588
119909 119909120588
119909119911120588
119909119911)
+
119909
2
120590
2
119909
(1 minus 120588
2
119909119911) +
2
120590
2
119909
(1 minus 120588
2
119909119911)
minus 2
119909119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119911
120590
119909120590
119911
(120588
119909119911minus 120588
119909 119909120588
119909119911)
minus 2
119909
120590
119909120590
119909
(120588
119909 119909minus 120588
119909119911120588
119909119911))]
(8)
where 120590
119909 120590119909 and 120590
119911are the standard deviations of 119909
and 119911 respectively 120588119909119911 120588119909119911 and 120588
119909 119909are the corresponding
correlation coefficients and 120581 = 1 minus 120588
2
119909119911minus 120588
2
119909119911minus 120588
2
119909 119909+
2120588
119909 119909120588
119909119911120588
119909119911
41 Evaluating the Parameters 120590119911119886 120590119911119887 119901 and 119911
119911 The four
parameters 120590119911119886 120590119911119887 119901 and 119911
119911of the joint PDF given by (8)
Mathematical Problems in Engineering 5
020
4060
80100
minus01minus005
0005
01
0
02
04
06
08
z (m)Time (s) minuszu
+zu
fz(z
)
Figure 5 Time evolution of the probability density function of thehysteretic component of displacement response
should satisfy that the area under the PDF given by (6) is closeto unity
int
119911119906
minus119911119906
119891
119885(119911) 119889119911 = (1 minus 2119901)Erf(
119911
119911
radic2120590
119911119886
)
+ 119901[Erf(119911
119906minus 119911
119911
radic2120590
119911119887
)
+ Erf(119911
119906+ 119911
119911
radic2120590
119911119887
)] asymp 1
(9)
where Erf(sdot) is the error function given by
Erf (119906) = 2
radic120587
int
119906
0
Exp (minus1199052) 119889119905 (10)
The above mentioned condition is necessary because (6) isnot a bounded function If (9) is satisfied the secondmoment(in this case equal to the variance 1205902
119911) can be approximated
as
119864 [119911
2] = ∭
infin
minusinfin
119911
2119891
119883119885(119909 119911) 119889119909 119889 119889119911
= int
infin
minusinfin
119911
2119891
119885(119911) 119889119911 = (1 minus 2119901) 120590
2
119911119886
+ 2119901 (119911
2
119911+ 120590
2
119911119887) = 120590
2
119911
(11)
The four parameters 120590119911119886 120590119911119887 119901 and 119911
119911could be computed
by means of their first four moments however here theyare determined bymeans of nondimensional coefficients thatdepend on the system response non-Gaussianity level whichis represented by [13]
120582 (119905) = int
minus119911119906
minusinfin
1
radic2120587120590
119911(119905)
Exp(minus 119911
2
2120590
2
119911(119905)
) 119889119911
=
1
2
(1 minus Erf(119911
119906
radic2120590
119911(119905)
))
(12)
From (12) it is observed that 120582 is close to zero when theresponse is Gaussian that is when the system strength is
high (where 119911119906is high) and the excitation or response is low
(and as a consequence 120590119911is low) On the other hand if the
response is non-Gaussian that is if the system strength is low(where 119911
119906is low) and the excitation or response is high (and
consequently 120590119911is high) then 120582 is high
Here the following mathematical expressions for thenormalized parameters 119911
119911119911
119911119906 120590119911119886120590
119911 120590119911119887120590
119911 and 119901 are
proposed
119911
119911(119905)
119911
119906
= 119886
1Ln (1198862120582 (119905) + 119886
3) + 119886
4
119901 (119905) = 119887
1Ln (1198872120582 (119905) + 119887
3) + 119887
4
120590
119911119886(119905)
120590
119911(119905)
= 119888
1Ln (1198882120582 (119905) + 119888
3) + 119888
4
120590
119911119887(119905)
120590
119911(119905)
= 119889
1Ln (1198892120582 (119905) + 119889
3) + 119889
4
(13)
where 119886
119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 are constants to
be determined The general behavior of (13) is presentedin Figures 6(a)ndash6(d) where it can be seen that 119911
119911and 119901
increase with 120582 (Figures 6(a) and 6(b)) meaning that whenthe nonlinearity of the response increases the function
119887120593
119885
moves far from 119911 = 0 and its weight increases hence 119891119885(119911)
(given by (6)) becomes a bimodal PDF On the other handwhen the structural behavior is linear120582 is close to zero and theweight of
119887120593
119885vanishes therefore 119891
119885(119911) becomes Gaussian
It can also be seen (Figures 6(c) and 6(d)) that119886120593
119885120590
119911and
119887120593
119885120590
119911decrease with 120582 but it should be noticed that as 120590
119911
increases 120582 grows as wellEquations (13) show that the parameters 120590
119911119886 120590119911119887 119901 and
119911
119911are time dependent thus the proposed approach considers
the time evolutionary character of the joint PDF of theresponse while other methods proposed in the literature donot take it into account Figure 7 shows an example of thetime evolution of 119891
119885(119911) given by (6)
411 Case of Narrow-Band Seismic Inputs Parameters 119886
119894
119887
119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 of (13) corresponding to
the response of single-degree-of-freedom systems (SDOF)systems subjected to the action of narrow-band seismicinputs were calculated by means of Monte Carlo simulationanalysis The critical case where the natural vibration periodof SDOF system is equal to the dominant period of seismicinput (in this case 119879 = 21 s see Figure 1) and the SDOFsystem has a high design ductility demand 120578 = 4 wasconsidered The design ductility factor 120578 is defined hereas the ratio between the expected maximum and the yielddisplacement of the system In this study the first one is foundby means of a stationary Gaussian equivalent linearizationanalysis and the latter is estimated by means of the initialstiffness and the yield force of the system
The SDOF system was subjected to the action of 50000artificial accelerograms based on the narrow-band motionrecorded in SCT on September 19 1985 (see Section 3) Forthis case it was found that 120582 lies within the interval 0 lt 120582 le
01 and that 120590119911119886
gt 120590
119911119887leads to good results The parameter
values obtained from the analysis are shown in Table 1
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
020
4060
80100
minus01minus005
0005
01
0
02
04
06
08
z (m)Time (s) minuszu
+zu
fz(z
)
Figure 5 Time evolution of the probability density function of thehysteretic component of displacement response
should satisfy that the area under the PDF given by (6) is closeto unity
int
119911119906
minus119911119906
119891
119885(119911) 119889119911 = (1 minus 2119901)Erf(
119911
119911
radic2120590
119911119886
)
+ 119901[Erf(119911
119906minus 119911
119911
radic2120590
119911119887
)
+ Erf(119911
119906+ 119911
119911
radic2120590
119911119887
)] asymp 1
(9)
where Erf(sdot) is the error function given by
Erf (119906) = 2
radic120587
int
119906
0
Exp (minus1199052) 119889119905 (10)
The above mentioned condition is necessary because (6) isnot a bounded function If (9) is satisfied the secondmoment(in this case equal to the variance 1205902
119911) can be approximated
as
119864 [119911
2] = ∭
infin
minusinfin
119911
2119891
119883119885(119909 119911) 119889119909 119889 119889119911
= int
infin
minusinfin
119911
2119891
119885(119911) 119889119911 = (1 minus 2119901) 120590
2
119911119886
+ 2119901 (119911
2
119911+ 120590
2
119911119887) = 120590
2
119911
(11)
The four parameters 120590119911119886 120590119911119887 119901 and 119911
119911could be computed
by means of their first four moments however here theyare determined bymeans of nondimensional coefficients thatdepend on the system response non-Gaussianity level whichis represented by [13]
120582 (119905) = int
minus119911119906
minusinfin
1
radic2120587120590
119911(119905)
Exp(minus 119911
2
2120590
2
119911(119905)
) 119889119911
=
1
2
(1 minus Erf(119911
119906
radic2120590
119911(119905)
))
(12)
From (12) it is observed that 120582 is close to zero when theresponse is Gaussian that is when the system strength is
high (where 119911119906is high) and the excitation or response is low
(and as a consequence 120590119911is low) On the other hand if the
response is non-Gaussian that is if the system strength is low(where 119911
119906is low) and the excitation or response is high (and
consequently 120590119911is high) then 120582 is high
Here the following mathematical expressions for thenormalized parameters 119911
119911119911
119911119906 120590119911119886120590
119911 120590119911119887120590
119911 and 119901 are
proposed
119911
119911(119905)
119911
119906
= 119886
1Ln (1198862120582 (119905) + 119886
3) + 119886
4
119901 (119905) = 119887
1Ln (1198872120582 (119905) + 119887
3) + 119887
4
120590
119911119886(119905)
120590
119911(119905)
= 119888
1Ln (1198882120582 (119905) + 119888
3) + 119888
4
120590
119911119887(119905)
120590
119911(119905)
= 119889
1Ln (1198892120582 (119905) + 119889
3) + 119889
4
(13)
where 119886
119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 are constants to
be determined The general behavior of (13) is presentedin Figures 6(a)ndash6(d) where it can be seen that 119911
119911and 119901
increase with 120582 (Figures 6(a) and 6(b)) meaning that whenthe nonlinearity of the response increases the function
119887120593
119885
moves far from 119911 = 0 and its weight increases hence 119891119885(119911)
(given by (6)) becomes a bimodal PDF On the other handwhen the structural behavior is linear120582 is close to zero and theweight of
119887120593
119885vanishes therefore 119891
119885(119911) becomes Gaussian
It can also be seen (Figures 6(c) and 6(d)) that119886120593
119885120590
119911and
119887120593
119885120590
119911decrease with 120582 but it should be noticed that as 120590
119911
increases 120582 grows as wellEquations (13) show that the parameters 120590
119911119886 120590119911119887 119901 and
119911
119911are time dependent thus the proposed approach considers
the time evolutionary character of the joint PDF of theresponse while other methods proposed in the literature donot take it into account Figure 7 shows an example of thetime evolution of 119891
119885(119911) given by (6)
411 Case of Narrow-Band Seismic Inputs Parameters 119886
119894
119887
119894 119888119894 and 119889
119894 119894 = 1 2 3 and 4 of (13) corresponding to
the response of single-degree-of-freedom systems (SDOF)systems subjected to the action of narrow-band seismicinputs were calculated by means of Monte Carlo simulationanalysis The critical case where the natural vibration periodof SDOF system is equal to the dominant period of seismicinput (in this case 119879 = 21 s see Figure 1) and the SDOFsystem has a high design ductility demand 120578 = 4 wasconsidered The design ductility factor 120578 is defined hereas the ratio between the expected maximum and the yielddisplacement of the system In this study the first one is foundby means of a stationary Gaussian equivalent linearizationanalysis and the latter is estimated by means of the initialstiffness and the yield force of the system
The SDOF system was subjected to the action of 50000artificial accelerograms based on the narrow-band motionrecorded in SCT on September 19 1985 (see Section 3) Forthis case it was found that 120582 lies within the interval 0 lt 120582 le
01 and that 120590119911119886
gt 120590
119911119887leads to good results The parameter
values obtained from the analysis are shown in Table 1
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
minus01
01
03
05
07
09
000 002 004 006 008
120582
zz
zu
12
c3
c4
c5
c6
(a) 119911119911119911119906 versus 120582
000 002 004 006 008
120582
045
035
025
015
005
minus005
p
12
c6
c5c4c3
(b) 119901 versus 120582
120582
000 002 004 006 008
11
10
09
08
07
06
05
04
120590za120590z
12
c6
c5
c4
c3
(c) 120590119911119886120590119911 versus 120582
minus01
01
03
05
07
09
120582
000 002 004 006 008
120590zb120590z
12
c6
c5
c4c3
(d) 120590119911119887120590119911 versus 120582
Figure 6 Behavior of the functions 119911119911119911
119911119906 119901 120590119911119886120590
119911 and 120590
119911119887120590
119911versus 120582
minus10minus8minus6 minus4
minus2 0 2 4 6 8 10
020
4060
80100
120140
1601800
0102030405
z (cm)
Time (s)
fz(z
)
Figure 7 Example of time evolution of 119891119885(119911)
Points 1198881ndash1198886in Figures 6(a)ndash6(d) represent the results of
Monte Carlo simulation analysesIt is noticed that parameters in Table 1 are proposed for
narrow-band seismic inputs and are independent of any otherparameter such as structural vibration period and designductility factor
Table 1 Values of the parameters in (13)
119886
1119886
2119886
3119886
4
0187 31990 0042 0590119887
1119887
2119887
3119887
4
0043 455126 0003 0240119888
1119888
2119888
3119888
4
minus0269 8247 0125 0442119889
1119889
2119889
3119889
4
minus0239 1408 0007 minus0391
5 Non-Gaussian EquivalentStochastic Linearization
In the equivalent linearization method (2) is replaced by alinear differential equation as follows [5 27]
= 119904
119890119909 + 119888
119890 + 119896
119890119911 (14)
where 119904119890 119888119890 and 119896
119890are linearization coefficients In order to
minimize the expected value of the square of the difference of(2) and (14) the linearization coefficients must satisfy [5 2829]
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (15)
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
where 119876 is the response vector 119876 = [119909 119911]
119879 119867119890is the
vector of equivalent linearization coefficients given by 119867119890=
[119904119890119888
119890119896
119890]
119879 and ℎ is given by (2)When the joint PDF proposed in this paper (see (8))
is substituted in (15) mathematical expressions for the lin-earization coefficients are derived (see Appendix A) In par-ticular for 119899 = 1 which corresponds to SDOF systems withsoftening behavior the equivalent linearization coefficientsare as follows
119904
119890= 0
119888
119890= (1 minus 2119901)119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886
+ 2119901119860 minus 120573
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)
119896
119890= (1 minus 2119901)119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
+ 2119901119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+3120590
2
119911119887minus120590
2
119911)Erf(
119911
119911120588
119911
radic2120576
119887
)
+ 120574
120588
119909119911120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)Erf(
119911
119911
radic2120590
119911119887
)
(16)
where 120576
119886= (1 minus 120588
2
119909119911)120590
2
119911+ 120588
2
119909119911120590
2
119911119886and 120576
119887= (1 minus 120588
2
119909119911)120590
2
119911+
120588
2
119909119911120590
2
119911119887
6 Covariance of the Response ofthe Hysteretic System
The covariance matrix of the response of the hystereticSDOF system Σ(119905) = 119864[119906119906
119879] where 119906(119905) =
[119909 119911 119909
119891119909
119892
119891
119892]
119879 is calculated by solving thefollowing equation [28 29]
119889Σ (119905)
119889119905
= 119867 (119905) Σ (119905) + Σ (119905)119867
119879(119905) + 2120587119878
119865
(17)
where 119867 is a matrix depending on the mechanical andgeometrical properties of the system the linearization coef-ficients the modulating function coefficients and the filterparameters In this study119867 is given by
119867(119905) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0 0 0 0
minus120572
2120596
2minus2120585120596 minus (1 minus 120572
2) 120596
2120596
2
119891119888 (119905) minus120596
2
119892119888 (119905) 2120585
119891120596
119891119888 (119905) minus2120585
119892120596
119892119888 (119905)
119904
119890119888
119890119896
1198900 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 minus120596
2
119891120596
2
119892minus2120585
119891120596
1198912120585
119892120596
119892
0 0 0 0 minus120596
2
1198920 minus2120585
119892120596
119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(18)
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)
Time (s)
Monte Carlo
Proposedmethod
Gaussian SEL
Figure 8 Standard deviation of displacement119879 = 21 s and 120578 = 15
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160 180
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Time (s)
Proposed method
Gaussian SEL
Monte Carlo
Figure 9 Standard deviation of velocity 119879 = 21 s and 120578 = 15
119878
119865is a 7times7matrix of zeros except the element 119878
119865(7 7)which
is equal to the amplitude of the white noise spectral density(119904
0)In order to compute Σ(119905) it is essential to start the
integration process of (17) when the Clough-Penzien filterresponse has reached the stationary stage Thus the initialcondition for Σ(119905) should correspond to the covariance of thestationary response of the Clough-Penzien filter as describedin Appendix B [30]
7 Verification of the Accuracy ofthe Proposed NSEL Approach
In this section the standard deviation of the response ofseveral hysteretic SDOF systems is calculated bymeans of theproposedNSEL criterion using themathematical expressionsobtained above ((13) and (16) using values of Table 1) TheSDOF systems have vibration periods 119879 = 21 s and 119879 = 40 sand design ductility factors 120578 = 15 and 120578 = 4 andthey are subjected to a nonstationary random process withthe statistical properties of the seismic motion recorded
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Time (s)
GaussianSEL
MonteCarlo Proposed method
Figure 10 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
)Monte Carlo
Gaussian SEL
Proposed method
Figure 11 Standard deviation of displacement 119879 = 21 s and 120578 = 4
in Mexico City at the SCT during the September 19 1985earthquake (see Section 3)
The response of the SDOF hysteretic systems is calculatedbymeans of SEL and alternatively byMonte Carlo simulationusing 50000 simulated accelerograms Two SEL criteria areused (a) assuming Gaussian response [1] and (b) assumingnon-Gaussian response using the criterion and the mathe-matical expressions proposed herein
The statistical response of the SDOF systemwith119879 = 21 sand 120578 = 15 is shown in Figures 8 9 and 10 and thatcorresponding to the system with 119879 = 21 s and 120578 = 40 is inFigures 11 12 and 13 Figures 8 to 13 show that the proposedapproach predicts with reasonable accuracy the maximumresponse however as well as theGaussianmethod it does notpredict the permanent drift of the hysteretic system especiallyfor high values of the design ductility (see Figures 8 and 11)The statistical response of the SDOF system with 119879 = 40 sand 120578 = 15 is shown in Figures 14 15 and 16 and thatcorresponding to the system with 119879 = 40 s and 120578 = 40 isin Figures 17 18 and 19 As in the case of SDOF system with119879 = 21 s the proposed approach predicts with reasonable
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
accuracy the maximum response of the SDOF system with119879 = 40 s however it does not predict with good accuracy thepermanent drift
The relative error (120576) in the calculation of the maximumstandard deviation of the response (120590
119909 120590119909and 120590119911) is obtained
as follows
120576 () =
response obtained from linearization minus response obtained from simulationresponse obtained from simulation
times 100 (19)
The relative errors (120576) of themaximum response correspond-ing to all the cases analyzed are shown in Tables 2 and 3corresponding to structural vibration periods of 21 and 40 srespectively A negative value of 120576 indicates that the equivalentlinearization underestimates the response Tables 2 and 3show that the proposed NSEL method leads to results closeto those obtained by means of Monte Carlo simulation
8 Extension to Multiple-Degree-of-FreedomSystems
In this section an extension of the proposed method tomultiple-degree-of-freedom (MDOF) systems is presented
Consider a nonlinear dynamic MDOF structural systemsubjected to a vector input P(119905) with an 119898-dimensionalresponse q(119905) satisfying the nonlinear differential equation
Φ (q (119905) q (119905) q (119905)) = P (119905) (20)
Here bold letters are related to MDOF systems The equiva-lent linear system of (20) is defined as
Meq (119905) + Ceq (119905) + Keq (119905) = P (119905) (21)
where Me Ce and Ke are defined as the equivalent massviscous damping and stiffness matrices of 119898 times 119898 They aredetermined minimizing the mean square error 119890 = 119864[120576120576
119879]
where 120576 is the119898-dimensional vector
120576 = Φ (q q q) minus (Meq + Ceq + Keq) (22)
Denoting by Q and He the 3119898-dimensional response vectorand the 3119898 times 119898 equivalent structural matrix given respec-tively by
Q = [q q q]119879 He = [MeCeKe]119879 (23)
it can be shown thatHe can be obtained from [5 28 29]
He = 119864 [QQ119879]minus1
119864 [QΦ
119879(Q)]
(24)
Herein the random excitation P(119905) is modelled as a linearcombination of the responses of linear filters to white or shotnoiseWith the aim of calculating the statistical second-orderresponse of the MDOF structure it is convenient to express(21) in state space form
119889Z (119905)
119889119905
= AeZ (119905) + BZF (119905) (25)
where Z(119905) = [q(119905) q(119905)]119879 is the 2119898 times 1 state vector and Aethe time-dependent system matrix given by
Ae (119905) = [
0 IminusMminus1e Ke minusMminus1e Ce
] (26)
and the matrix B contains the coefficients of the linearcombination of the response ZF of the filters The dynamicsof the latter is governed by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (27)
where the matrixD contains the coefficients of the filters andthe vector W
1015840
(119905) the white or shot noise Equations (25) and(27) can be rewritten as
119889u119889119905
= H (119905) u +W(28)
where
u (119905) =
Z (119905)
ZF (119905) (29)
H (119905) = [
A B0 D] (30)
Matrix W has the same structure as W1015840
but with higherdimensionThus (28) controls the response of an augmentedsystem consisting of the linear filters and the structuralsystem in series [5] Finally the covariance matrix of theresponse of the MDOF system is calculated by solving
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (31)
where SF is a matrix of zeros except the last element whichis equal to the amplitude of the white noise spectral density(119904
0)The mathematical expressions given in this section cor-
respond to a MDOF structural system with 119898-dimensionalresponse while those given in the previous sections corre-spond to the special case of a SDOF with 2-dimensionalresponse (q(119905) = [119909 119911]
119879) For example (20) took the
particular form of (1) and (2) (21) took the particular formof (14) (24) was reduced to (15) while (30) took the particularform of (18)
9 Conclusions
A model appropriately representing the time evolution ofthe joint PDF of the structural response of softening systems
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 2 Relative errors of the maximum response of SDOF system with 119879 = 21 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus560 minus1359 minus567 minus929 minus708 minus1942Proposed method minus104 272 minus172 650 minus421 minus031
Table 3 Relative errors of the maximum response of SDOF system with 119879 = 40 s
Method Error in max[120590119909] Error in max[120590
119909] Error in max[120590
119911]
120578 = 15 120578 = 4 120578 = 15 120578 = 4 120578 = 15 120578 = 4
Gaussian minus499 minus1302 minus199 minus383 minus492 minus1795Proposed method minus392 minus822 minus121 086 minus407 minus148
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 12 Standard deviation of velocity 119879 = 21 s and 120578 = 4
7
6
5
4
3
2
1
00 20 40 60 80 100 120 140 160 180
Time (s)
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed methodMonteCarlo
GaussianSEL
Figure 13 Standard deviation of hysteretic variable 119879 = 21 s and120578 = 4
subjected to seismic ground motions was proposed Thedisplacement and velocity was assumed jointly Gaussianand the marginal PDF of the hysteretic component ofthe displacement was modeled by a mixed PDF whichis Gaussian when the structural behavior is linear andturns into a bimodal PDF when the structural behavior ishysteretic
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
18
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 14 Standard deviation of displacement 119879 = 40 s and 120578 =
15
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
50
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s) Monte Carlo
Gaussian SEL
Proposed method
Figure 15 Standard deviation of the velocity 119879 = 40 s and 120578 = 15
TheNSELmethod proposed in this study usesmathemat-ical expressions for the equivalent linearization coefficients(16) corresponding to the non-Gaussian response of inelasticnonlinear systems The mathematical expressions depend onthe parameters of the joint PDF of the structural responseand they take into account the time evolutionary characterof the PDF It is noticed that other NSEL methods found
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 180
Time (s)
14
12
10
8
6
4
2
0
Std
dev
of h
yste
retic
com
pone
nt (c
m) Proposed methodMonte Carlo
Gaussian SEL
Figure 16 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 15
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
10
12
14
16
Time (s)
Stan
dard
dev
iatio
n of
disp
lace
men
t (cm
) Monte Carlo
Gaussian SEL
Proposed method
Figure 17 Standard deviation of displacement 119879 = 40 s and 120578 =
40
in the literature have proposed mathematical expressions forthe equivalent linearization coefficients however they do notconsider the time evolution of the joint PDF
The mathematical expressions derived here (16) can beapplied to obtain the non-Gaussian response of any kind ofSDOF softening system with smooth transition from elasticto plastic behavior (119899 = 1) It is possible to derive expressionsof the linearization coefficients for other values of 119899
The NSEL method proposed was applied to calculate thestatistical response of SDOF systems with different vibrationperiods and different design ductility factors The resultsshow that for the SDOF systems analyzed the proposedcriterion estimates with good accuracy the time evolutionof the structural response however it does not predict thepermanent drift exhibited by hysteretic systems with highdesign ductility values
It is concluded that the proposed NSEL approach isuseful to estimate the maximum standard deviation of thestructural response of softening systems subjected to seismicground motions The relative errors in the calculation of themaximum standard deviation of the displacement velocity
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
Time (s)
Stan
dard
dev
iatio
n of
vel
ocity
(cm
s)
Monte Carlo
Gaussian SEL
Proposed method
Figure 18 Standard deviation of the velocity 119879 = 40 s and 120578 = 40
0 20 40 60 80 100 120 140 160 180
Time (s)
7
6
5
4
3
2
1
0
Std
dev
of h
yste
retic
com
pone
nt (c
m)
Proposed method
Monte Carlo
Gaussian SEL
Figure 19 Standard deviation of the hysteretic variable 119879 = 40 sand 120578 = 40
and hysteretic component of displacement as compared withthose obtained by means of Monte Carlo technique are lowand in general they are smaller than those obtained by theGaussian equivalent linearization method
The parameters 119886119894 119887119894 119888119894 and 119889
119894 119894 = 1 2 3 4 in (13) can be
found for other types of seismic excitation (eg wide-bandground motions)
Appendices
A Linearization Coefficients
This appendix presents the derivation of the equivalentlinearization coefficients given by (16)
The linearization coefficients are obtained from
119867
119890= 119864 [119876119876
119879]
minus1
119864 [ℎ119876] (A1)
where 119876 is the response vector and 119867
119890is the vector of
linearization coefficients given by 119876 = [119909 119911]
119879 and 119867
119890=
[119904119890119888
119890119896
119890]
119879
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
The inverse of the covariance matrix is
Σ
minus1
119876=
1
120590
2
119909120590
2
119909120590
2
119911120581
[
[
[
[
(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909119911120588
119909119911minus 120588
119909 119909) 120590
119909120590
119909120590
2
119911(120588
119909119911120588
119909 119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911
(120588
119909119911120588
119909119911minus 120588
119909) 120590
119909120590
119909120590
2
119911(1 minus 120588
2
119909119911) 120590
2
119909120590
2
119911(120588
119909 119909120588
119909119911minus 120588
119909119911) 120590
119909120590
2
119909120590
119911
(120588
119909119911120588
119909minus 120588
119909119911) 120590
2
119909120590
119909120590
119911(120588
119909 119909120588
119909119911minus 120588
119911) 120590
119909120590
2
119909120590
119911(1 minus 120588
2
119909 119909) 120590
2
119909120590
2
119911
]
]
]
]
(A2)
where 120581 = 1minus120588
2
119909119911minus120588
2
119909119911minus120588
2
119909 119909+2120588
119909 119909120588
119909119911120588
119909119911 Substituting (A2)
in the second factor of (A1) the following is obtained
119864 [ℎ119876] =
[
[
[
119864 [119909ℎ ( 119911)]
119864 [ℎ ( 119911)]
119864 [119911ℎ ( 119911)]
]
]
]
=
[
[
[
119860119865
1minus 120573119865
2minus 120574119865
3
119860119865
4minus 120573119865
5minus 120574119865
6
119860119865
7minus 120573119865
8minus 120574119865
9
]
]
]
(A3)
where
119865
1= 119864 [119909] = ∬
infin
minusinfin
119909119891
119883119889119909 119889
119865
2= 119864 [119909119911 || |119911|
119899minus1]
= ∭
infin
minusinfin
119909119911 || |119911|
119899minus1119891
119883119885119889119909 119889 119889119911
119865
3= 119864 [119909 |119911|
119899] = ∭
infin
minusinfin
119909 |119911|
119899119891
119883119885119889119909 119889 119889119911
119865
4= 119864 [
2] = int
infin
minusinfin
2119891
119889
119865
5= 119864 [119911 || |119911|
119899minus1] = ∬
infin
minusinfin
119911 || |119911|
119899minus1119891
119885119889 119889119911
119865
6= 119864 [
2|119911|
119899] = ∬
infin
minusinfin
2|119911|
119899119891
119885119889 119889119911
119865
7= 119864 [119911] = ∬
infin
minusinfin
119911119891
119885119889119889119911
119865
8= 119864 [119911
2|| |119911|
119899minus1] = ∬
infin
minusinfin
119911
2|| |119911|
119899minus1119891
119885119889 119889119911
119865
9= 119864 [119911 |119911|
119899] = ∬
infin
minusinfin
119911 |119911|
119899119891
119885119889 119889119911
(A4)
Considering that 119891119883119885
is modeled by (8) 119891119885
119891119883
and119891
can be easily obtained thus making it possible to obtain
mathematical expressions for the linearization coefficientsFor the case of 119899 = 1 (A4) are as follows
119865
1= (1 minus 2119901) 119865
1119886+ 2119901119865
1119887
119865
2= (1 minus 2119901) 119865
2119886+ 2119901119865
2119887
119865
3= (1 minus 2119901) 119865
3119886+ 2119901119865
3119887
119865
4= (1 minus 2119901) 119865
4119886+ 2119901119865
4119887
119865
5= (1 minus 2119901) 119865
5119886+ 2119901119865
5119887
119865
6= (1 minus 2119901) 119865
6119886+ 2119901119865
6119887
119865
7= (1 minus 2119901) 119865
7119886+ 2119901119865
7119887
119865
8= (1 minus 2119901) 119865
8119886+ 2119901119865
8119887
119865
9= (1 minus 2119901) 119865
9119886+ 2119901119865
9119887
(A5)
where
119865
1119886=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911120590
2
119911119886]
119865
1119887=
120590
119909120590
119909
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911) + 120588
119909119911120588
119909119911(119911
2
119911+ 120590
2
119911119887)]
119865
2119886=
radic
2
120587
120590
119909120590
119909120590
2
119911119886
120590
2
119911radic120576
119886
[120588
119909119911120576
119886
+ 120588
119909119911120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119886]
119865
2119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
]
times 120588
119909119911(119911
2
119911+ 120590
2
119911119887) 120576
119887
+ 120590
2
119911119887120588
119909119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911120590
2
119911119887)
+
radic
2120587119911
119911(120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887))
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
3119886=
radic
2
120587
120590
119909120590
119909120590
119911119886
120590
2
119911
[120590
2
119911(120588
119909 119909minus 120588
119909119911120588
119909119911)
+ 2120588
119909119911120588
119909119911120590
2
119911119886]
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
119865
3119887
=
120590
119909120590
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 2120590
2
119911119887)]
+
radic
2120587119911
119911Erf[
119911
119911
radic2120590
119911119887
]
times [120590
2
119911(120588
119909minus 120588
119909119911120588
119909119911)
+ 120588
119909119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)]]
119865
4119886=
120590
2
119909
120590
2
119911
120576
119886
119865
4119887=
120590
2
119909
120590
2
119911
(120576
119887+ 120588
2
119909119911119911
2
119911)
119865
5119886=
radic
8
120587
120590
2
119909120590
2
119911119886
120590
2
119911
120588
119909119911radic120576
119886
119865
5119887
=
120590
2
119909
radic2120587120590
2
119911
[2120588
119909119911(119911
2
119911+ 2120590
2
119911119887)radic120576
119887Exp[minus
119911
2
119911120588
2
119909119911
2120576
119887
]
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911120588
119909119911
radic2120576
119887
]]
119865
6119886=
radic
2
120587
120590
2
119909120590
119911119886
120590
2
119911
(120576
119886+ 120588
2
119909119911120590
2
119911119886)
119865
6119887
=
120590
2
119909
radic2120587120590
2
119911
[2120590
119911119887Exp[minus
119911
2
119911
2120590
2
119911119887
] 120576
119887+ 120588
2
119909119911(119911
2
119911+ 120590
2
119911119887)
+
radic
2120587119911
119911120576
119887+ 120588
2
119909119911(119911
2
119911+ 2120590
2
119911119887)
times Erf[119911
119911
radic2120590
119911119887
]]
119865
7119886=
120590
119909120588
119909119911120590
2
119911119886
120590
119911
119865
7119887=
120590
119909120588
119909119911
120590
119911
(119911
2
119911+ 120590
2
119911119887)
119865
8119886=
radic
2
120587
120590
119909120590
2
119911119886
120590
119911radic120576
119886
(120576
119886+ 120590
2
119911119886120588
2
119909119911)
119865
8119887
=
120590
119909
radic2120587120590
119911
[
2
radic120576
119887
Exp[minus119911
2
119911120588
2
119909119911
2120576
119887
] 120576
119887(119911
2
119911+ 120590
2
119911119887) + 120590
4
119911119887120588
2
119909119911
+
radic
2120587119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911120588
119909119911
radic2120576
119887
]]
119865
9119886=
radic
8
120587
120590
119909120590
3
119911119886120588
119911
120590
119911
119865
9119887
=
120590
119909120588
119909119911
radic2120587120590
119911
[2120590
119911119887(119911
2
119911+ 2120590
2
119911119887)Exp[minus
119911
2
119911
2120590
2
119911119887
]
+
radic
2120587119911
119911(119911
2
119911+ 3120590
2
119911119887)Erf[
119911
119911
radic2120590
119911119887
]]
(A6)
where 120576119886= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119886 120576119887= (1minus120588
2
119909119911)120590
2
119911+120588
2
119909119911120590
2
119911119887 Exp[sdot]
is the exponential function and Erf[sdot] is the error functiongiven by
Erf [119906] = 2
radic120587
int
119906
0
Exp [minus1199052] 119889119905 (A7)
Substituting ((A5)-(A6)) in (A3) the value of 119864[ℎ119876] for119899 = 1 is obtained This result together with the inverse ofthe covariance matrix (A2) is substituted in (A1) in order toobtain the linearization coefficients corresponding to 119899 = 1Consider
119904
119890= 0
119888
119890= (1 minus 2119901)
times [119860 minus
radic
2
120587
120573
120588
119909119911120590
2
119911119886
radic120576
119886
minus
radic
2
120587
120574120590
119911119886]
+ 2119901[119860 minus 120573
times
radic
2
120587
120588
119909119911120590
2
119911119887
radic120576
119887
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
+ 119911
119911Erf(
119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
radic
2
120587
120590
119911119887Exp(minus
119911
2
119911
2120590
2
119911119887
)
+ 119911
119911Erf(
119911
119911
radic2120590
119911119887
)]
119896
119890= (1 minus 2119901) [119860
120588
119909119911120590
119909(120590
2
119911119886minus 120590
2
119911)
120590
3
119911
minus
radic
2
120587
120573
120590
119909120590
2
119911119886
120590
3
119911
(120576
119886+ 120588
2
119909119911(120590
2
119911119886minus 120590
2
119911))
radic120576
119886
minus
radic
2
120587
120574
120588
119909119911120590
119909120590
119911119886(2120590
2
119911119886minus 120590
2
119911)
120590
3
119911
]
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
+ 2119901[119860
120588
119909119911120590
119909(119911
2
119911+ 120590
2
119911119887minus 120590
2
119911)
120590
3
119911
minus 120573
120590
119909
120590
3
119911
times
radic
2
120587
Exp(minus119911
2
119911120588
2
119909119911
2120576
119887
)
times
119911
2
119911120576
119887+ 120590
2
119911119887(120576
119887+ 120588
2
119909119911(120590
2
119911119887minus 120590
2
119911))
radic120576
119887
+ 119911
119911120588
119909119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911120588
119909119911
radic2120576
119887
)
minus 120574
120588
119909119911120590
119909
120590
3
119911
radic
2
120587
Exp(minus119911
2
119911
2120590
2
119911119887
)
times 120590
119911119887(119911
2
119911+ 2120590
2
119911119887minus 120590
2
119911)
+ 119911
119911(119911
2
119911+ 3120590
2
119911119887minus 120590
2
119911)
times Erf(119911
119911
radic2120590
119911119887
)]
(A8)
B Initial Condition of the CovarianceMatrix of the Response
Following the state space approach the state equation of alinear system subjected to a nonwhite excitation modeled asa filtered white noise is [29]
119889Z (119905)
119889119905
= AZ (119905) + BZF (119905) (B1)
where Z(119905) is the state vector the matrix A is the systemmatrix and thematrixB contains the coefficients of the linearcombination of the response ZF of the filter The response ofthe filters is controlled by
119889ZF (119905)
119889119905
= DZF (119905) +W1015840 (119905) (B2)
where the matrixD contains the coefficients of the filters andW1015840
(119905) is a vector with zeros except the last element whichcorrespond to white noise (B1) and (B2) can be rewrittenas
119889u119889119905
= H (119905) u +W(B3)
where
u (119905) =
Z (119905)
ZF (119905) (B4)
H (119905) = [
A B0 D
] (B5)
Matrix W has the same structure as W1015840
but with a higherdimension To avoid introducing the effect arising from thetransient response of the filter the output of the filter ZFmust be allowed to reach the stationary phase before it ismultiplied by the modulating function [5] Analytically it isreached by selecting the suitable initial conditions for thecovariance matrix Σ of the overall state variable vector Zwhich is governed by the following differential equation
119889Σ (119905)
119889119905
= H (119905)Σ (119905) + Σ (119905)H119879 (119905) + 2120587SF (B6)
where SF is a matrix of zeros except the last element which isequal to the amplitude of the white noise spectral density (119904
0)
It is convenient to introduce a partition in Σ in agreementwith (B4) Thus
Σ = [
Σ119885119885Σ119885119885119865
Σ119879
119885119885119865
Σ119885119865119885119865
] (B7)
where Σ119885119885
= 119864[ZZ119879] Σ119885119885119865
= 119864[ZZ119879F ] and Σ119885119865119885119865 =
119864[ZFZ119879F ] If the original system is at rest at 119905 = 0 then theresponse will be zero and all the elements of Σ
119885119885and Σ
119885119885119865
will be zero however the elements of Σ119885119865119885119865
at 119905 = 0 willcorrespond to the stationary response of the filter which isgoverned by
DΣ119904119885119865119885119865
+ Σ119904
119885119865119885119865
D119879 + P1015840 = 0 (B8)
where P1015840 has the same form as P but with a lower dimensionFor the case when the Clough-Penzien filter is used thesolution of (B8) is
Σ119904
119885119865119885119865
=
[
[
[
[
[
[
]11
]12
]13
]14
]21
]22
]23
]24
]31
]32
]33
]34
]41
]42
]43
]44
]
]
]
]
]
]
(B9)
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
where
]11
= (120587119904
0120596
119892(4120585
2
119891120585
119892120596
2
119891120596
119892+ 120585
119892120596
119892(4120585
2
119892120596
2
119891+ 120596
2
119892)
+ 120585
119891((1 + 4120585
2
119892) 120596
3
119891+ 4120585
2
119892120596
119891120596
2
119892)))
times (2120585
119891120585
119892120596
3
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]12
= ]21
= (120587119904
0(120596
2
119891+ 8120585
119891120585
119892120596
119891120596
119892+ (minus1 + 8120585
2
119892) 120596
2
119892))
times (2120585
119892120596
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]13
= ]31
= 0
]14
= ]41
= (120587119904
0(minus120585
119891120596
119891120596
119892+ 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]22
=
120587119904
0
2120585
119892120596
3
119892
]23
= ]32
= (120587119904
0(120585
119891120596
119891120596
119892minus 120585
119892(120596
2
119891minus 2120596
2
119892)))
times (120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]24
= ]42
= 0
]33
= (120587119904
0120596
2
119892(4120585
3
119892120596
2
119891+ 120585
119891120596
119891120596
119892
+ 4120585
119891120585
2
119892120596
119891120596
119892+ 120585
119892120596
2
119891))
times (2120585
119892120585
119892120596
119891(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]34
= ]43
= (120587119904
0120596
119892((1 + 4120585
2
119892) 120596
2
119891+ 4120585
119891120585
119892120596
119891120596
119892minus 120596
2
119892))
times (2120585
119892(120596
4
119891+ 4120585
119891120585
119892120596
3
119891120596
119892
+ 2 (minus1 + 2120585
2
119891+ 2120585
2
119892) 120596
2
119891120596
2
119892
+ 4120585
119891120585
119892120596
119891120596
3
119892+ 120596
4
119892))
minus1
]44
=
120587119904
0
2120585
119892120596
119892
(B10)
Thus (B6) should be integrated numerically with the follow-ing initial condition for Σ
Σ0= [
0 00 Σ119904119885119865119885119865
] (B11)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thanks are given to DGAPA-UNAM for the support givenunder Project IN102114
References
[1] T S Atalik and S Utku ldquoStochastic linearization of multi-degree-of-freedom non-linear systemsrdquo Earthquake Engineer-ing amp Structural Dynamics vol 4 no 4 pp 411ndash420 1976
[2] J J Beaman ldquoAccuracy of statistical linearizationrdquo in NewApproaches to Nonlinear Problems in Dynamics P J HolmesEd pp 195ndash207 SIAM Philadelphia Pa USA 1980
[3] P D Spanos ldquoStochastic linearization in structural dynamicsrdquoApplied Mechanics Reviews vol 34 no 1 pp 1ndash8 1981
[4] F Fan and G Ahmadi ldquoLoss of accuracy and nonunique-ness of solutions generated by equivalent linearization andcumulant-neglected methodsrdquo Tech Rep MIE-168 Depart-ment of Mechanical and Industrial Engineering ClarksonUniversity Potsdam NY USA 1988
[5] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Chichester UK 1990
[6] G I Schueller H J Pradlwarter and C G Bucher ldquoEfficientcomputational procedures for reliability estimates of MDOF-systemsrdquo International Journal of Non-LinearMechanics vol 26no 6 pp 961ndash974 1991
[7] F L Silva and S E Ruiz ldquoCalibration of the equivalentlinearization gaussian approach applied to simple hystereticsystems subjected to narrow band seismic motionsrdquo StructuralSafety vol 22 no 3 pp 211ndash231 2000
[8] G Falsone ldquoAn extension of the Kazakov relationship fornon-Gaussian random variables and its use in the non-linearstochastic dynamicsrdquo Probabilistic Engineering Mechanics vol20 no 1 pp 45ndash56 2005
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[9] K Kimura H Yasumuro and M Sakata ldquoNon-Gaussianequivalent linearization for non-stationary random vibration ofhysteretic systemrdquo Probabilistic Engineering Mechanics vol 9no 1-2 pp 15ndash22 1994
[10] G Ricciardi ldquoA non-Gaussian stochastic linearizationmethodrdquoProbabilistic Engineering Mechanics vol 22 no 1 pp 1ndash11 2007
[11] K Asano and W D Iwan ldquoAn alternative approach to therandom response of bilinear hysteretic systemsrdquo EarthquakeEngineering amp Structural Dynamics vol 12 no 2 pp 229ndash2361984
[12] L Socha ldquoLinearization in analysis of nonlinear stochastic sys-tems recent resultsmdashpart II applicationsrdquo Applied MechanicsReviews vol 58 no 5 pp 303ndash314 2005
[13] J E Hurtado and A H Barbat ldquoImproved stochastic lineariza-tion method using mixed distributionsrdquo Structural Safety vol18 no 1 pp 49ndash62 1996
[14] R Bouc ldquoForced vibration of mechanical systems with hys-teresisrdquo in Proceedings of the 4th Conference on NonlinearOscillation Academia Prague Czech Republic 1967
[15] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980
[16] F Casciati and L Faravelli Fragility Analysis of ComplexStructural Systems Research Studies Press Taunton UK 1991
[17] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 2nd edition 1993
[18] C H Yeh and Y K WenModeling of nonstationary earthquakeground motion and biaxial and torsional response of inelasticstructures [dissertation] Civil Engineering Studies StructuralResearch Series no 546 University of Illinois 1989
[19] F L Silva Calibracion del metodo de linealizacion equivalenteestocastica para sistemas histereticos simples [MS thesis] Engi-neering Graduate School Universidad Nacional Autonoma deMexico UNAM 1998 (Spanish)
[20] M Grigoriu S E Ruiz and E Rosenblueth ldquoMexico earth-quake of September 19 1985mdashnonstationary models of seismicground accelerationrdquo Earthquake Spectra vol 4 no 3 pp 551ndash568 1988
[21] F L Silva J L Rivera S E Ruiz and J E Hurtado ldquoInfluenceof the mathematical modeling of the seismic input on the non-Gaussian response of non-linear systemsrdquo in Proceedings of the8th International Conference on Structural Safety and ReliabilityAA Balkema Publishers Newport Beach Calif USA June2001
[22] G M Atkinson and K Goda ldquoInelastic seismic demand ofreal versus simulated ground-motion records for Cascadiasubduction earthquakesrdquo Bulletin of the Seismological Society ofAmerica vol 100 no 1 pp 102ndash115 2010
[23] F de Luca I Iervolino and E Cosenza ldquoCompared seismicresponse of degrading systems to artificial and real recordsrdquoin Proceedings of the 14th European Conference on EarthquakeEngineering Ohrid Republic of Macedonia August-Sepember2010
[24] P-L Liu and A der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986
[25] N L Johnson S Kotz andN BalakrishanContinuous Univari-ate Distributions John Wiley and Sons New York NY USA2nd edition 1994
[26] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Singapore 3rd edition 1991
[27] T K Caughey ldquoEquivalent linearization techniquesrdquo Journal ofthe Acoustical Society of America vol 35 pp 1706ndash1711 1963
[28] F Casciati and L Faravelli ldquoMethods of non-linear stochasticdynamics for the assessment of structural fragilityrdquo NuclearEngineering and Design vol 90 no 3 pp 341ndash356 1985
[29] T T Soong and M Grigoriu Random Vibration of Mechanicaland Structural Systems Prentice Hall Englewood Cliffs NJUSA 1993
[30] F L Silva Respuesta estocastica de estructuras histereticassujetas a sismos [PhD thesis] Engineering Graduate Schoolof Universidad Nacional Autonoma de Mexico UNAM 2002(Spanish)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of