a displacement-based seismic design for reinforced concrete structures-boulaouad-2011-isi

KSCE Journal of Civil Engineering (2011) 15(3):507-516DOI 10.1007/s12205-011-1009-z

507

www.springer.com/12205

Structural Engineering

A Displacement-Based Seismic Design for Reinforced Concrete Structures

Abderrachid Boulaouad* and Ahmed Amour**

Received September 27, 2009/Revised March 1, 2010/Accepted July 11, 2010

Abstract

The Displacement-Based Design method is presented for linear and non linear systems with some numerical applications on onestorey and multistory buildings using the spectra and formulae provided by the Algerian seismic code. Fundamentals and designprocedure of this method are given with implications and inherent problems. A brief review of the classic (force-based) method isalso given. Comparison between the two methods is made and the limits and advantages of each one are discussed. Furthermore, theimportance of target displacement and ductility level are outlined and more investigation is recommended to determine an accuraterelation between damping and ductility. The results of the analysis show that the Displacement-Based method is simple and efficientwith enough accuracy and confirm the idea, developed by many authors, that this method may be, in future, a good alternative to theForce-Based one providing that some problems can be resolved by further work such as the elaboration of appropriate design spectra.Keywords: force-based method, displacement-based method, target displacement, ductility, algerian seismic code

1. Introduction

In countries subjected to frequent severe earthquakes, such asAlgeria, attention must be focused on seismic design, andseismic codes must be frequently revised to be more improved.The Algerian seismic code (R.P.A.), amongst others, is based onthe conventional method known as force-based method (F.B.D.)which uses the acceleration spectra. Nevertheless, in many cases,the effect of externally applied loads is directly felt as defor-mation of structure members which may be easily related todamage. The relation is well illustrated by damage that occurredin the Olive View Hospital during the 1971 San Franciscoearthquake. It is known that deformation controlled design canbe achieved either by using the traditional force/strength baseddesign procedure together with a check on the displacement/driftlimit or by employing a direct displacement design procedure;but it seems to be rational to examine a seismic design methodwherein displacements are considered at the start of the designprocess with attention focused on deformations to provide astructure that meets the requirements for the several limit states(Medekhar and Kennedy, 1998). Such a procedure that designsthe structure for a given displacement profile, can control thepotential damages. Furthermore, recent seismic activities worl-dwide clearly proved that the seismic codes were generallysuccessful in achieving live safety goals, but there is still a lack ineconomical considerations. For this reason, deformation-baseddesign approaches have been developed to create a structure withcontrolled and predictable performance.

The Displacement Based Design (D.B.D.) is one of a numberof seismic design procedures recently developed and jointlytermed Performance Based Seismic Design. This alternativeseismic design philosophy differs in significant details from theconventional method (Priestley et al., 2007).

A performance objective consists of two major components: astated maximum level of expected damage (sometimes calledperformance level or limit state) and a level of seismic hazard. Ingeneral, performance objectives can be defined quantitatively orqualitatively. They may be expressed in a deterministic manner(FEMA-237) or in a reliability-based probabilistic approach(FEMA-350). Examples from FEMA-350 are:

2 probability of poorer performance than collapse preven-tion level in 50 years, or

50 probability of poorer performance than immediate occu-pancy level in 50 years.

Therefore, the design process involves association of a limitstates to a level of seismic hazard at a site for a given period oftime. The advantage of incorporating the performance objectivesin the design is that the owner of the building can requestdifferent levels of damage/functionality of the property.

Many researchers recommended the use of this new method inseismic codes instead of the conventional one. All the applica-tions made were conducted on steel structures only or on R.C.(i.e., reinforced concrete) structures but without any comparisonbetween the two methods (Panagiotakos and Fardis, 2001;Williams and Albermann, 2003) or without considering bothlinear and non linear cases (Borzi and Elnashai, 2000).

*Assistant Professor, Dept. of Civil Engineering, M.A.C.C., University of Batna, 500, Algeria (Corresponding Author, E-mail: [email protected])**Assistant Professor, Dept. of Civil Engineering, University of M'sila, 2800, Algeria (E-mail: Amourahmed @yahoo.fr)

Abderrachid Boulaouad and Ahmed Amour

508 KSCE Journal of Civil Engineering

This paper gives fundamentals of both conventional andD.B.D. method on the one hand and outlines the advantages andlimitations of each one on the other hand, with numericalexamples based on the formulae and spectra provided by theAlgerian seismic code.

2. Historical Considerations

The reason that seismic design is currently based on force (oracceleration) rather than displacement is based largely onhistorical considerations. The occurrence of the 1994 Northridgeearthquake (Mw 6.7) was a keystone in development of seismicregulations. Prior to this event, building structures were designedprimarily with one performance objective in mind: prevention ofloss of lives.

The idea of displacement based design was introduced about40 years ago. Gulkan and Sozen (1974) developed the concept ofsubstitute structure to estimate the non linear structure responsethrough an equivalent elastic model assuming a linear behaviorand a viscous damping equivalent to the non-linear response.This idea has been adopted recently by Kowalsky et al. (1994)for a direct displacement design of SDOF (i.e., single degree offreedom) R.C. structures and by Priestley et al. (1996) for bothSDOF and MDOF (i.e., multi degree of freedom) bridges andbuildings starting from a target peak displacement. Anotherdisplacement-based procedure for MDOF bridge structures,particularly suitable for symmetric bridges has been proposed byCalvi and Kingsley (1995). Qi and Moehle (1991) proposed adisplacement-based procedure for MDOF systems with therequirement of preliminary design and further modification ofthe design according to the displacement or drift limit. WhileWallace (1995), Sasani and Anderson (1996) and Bachman andDazio (1997) focus on wall systems, Panagiotakos and Fardis(1999) implemented an overall performance-based deformationcontrolled design of MDOF R.C. structures subjected to bothseismic and non seismic actions. Another direct displacement-based design approach was proposed by Fajfar (1999) based onthe capacity spectrum method, where the capacity curve is ob-tained from a non linear static pushover analysis and representedby a bilinear force-displacement model. Qiang (2001), amongstothers, presented a D.B.D. procedure derived from the capacityspectrum method using Newmark-Hall reduction factors for theinelastic demand spectrum.

3. A Brief Review of Force-Based Seismic Design(F.B.D.)

According to Priestley et al. (2007), the sequence of operationsrequired in F.B.D. can be summarized as follows:1. The structural geometry, including member sizes, is estimated.2. Member elastic stiffnesses are estimated, based on preliminary

estimates of member sizes.3.Based on the assumed member stiffnesses, the fundamental

period T is calculated, for a SDOF representation, by:

(1)

where Me is the effective seismic mass (normally taken as thetotal mass) and K the stiffness of the structure.In some building codes, a height-dependent fundamentalperiod is specified, given by Eq. (2):

(2)

where Ct depends on the structural system and Hn is thebuilding height.

4. The elastic design base shear VBE for the structure is given byan equation of the form:

(3)

where CT is the basic seismic coefficient dependent on seismicintensity, soil conditions and period T, and I is a factor reflect-ing different levels of acceptable risk for different structures,and g is the acceleration of gravity.

5. After selecting the appropriate force-reduction factor R, thedesign base shear force (VBR) is found from:

(4)

The base shear force is then distributed to different parts of thestructure. For building structures, the distribution is typicallyproportional to the product of the height and mass at differentlevels; and the total seismic force is distributed between dif-ferent lateral force-resisting elements in proportion to theirelastic stiffness.

6. The structure is then analyzed under the vector of lateralseismic design forces, and the required moment capacities atpotential locations of inelastic action (plastic hinges) is deter-mined.

7. Structural design of the member sections at plastic hinge loca-tions is carried out, and the displacements under the seismicaction are estimated and then, compared with code-specifieddisplacement limits

8. If the calculated displacements exceed the code limits, re-design is required

9. If the displacements are satisfactory, the final step of thedesign is to determine the required strength.

4. Limitations of Force-based Seismic DesignMethod

Medekhar and Kennedy (1998) and Priestley et al. (2007)pointed out the following problems with F.B.D. method:

The fundamental period required to start the design is deter-mined using empirical expressions

The values specified for the modification factor, R, by seismiccodes, appear to be somewhat arbitrary

Displacements are checked at the end of the design processonly. There appears to be a lack of concern about the implied

T 2 MeK------=

T Ct Hn( )3 4=

VBE CTI gMe( )=

VBRVBER

--------=


Vol. 15, No. 3 / March 2011 509

inelastic displacements which may be excessive and contri-bute to the instability of the structure.

5. Fundamentals of Displacement-Based DesignMethod (D.B.D.)

5.1 IntroductionThe D.B.D method is based on the Substitute Structure ap-

proach pioneered by Sozen and co-workers (Gulkan and Sozen,1974; Shibata and Sozen, 1976) and developed into a designapproach in Priestley et al. (2007).

According to this latter, the design approach attempts todesign a structure which would achieve, rather than be boundedby, a given performance limit state under a given seismic in-tensity, essentially resulting in uniform-risk structures, which isphilosophically compatible with the uniform-risk seismic spectraincorporated in most design codes.

5.2 Basic Formulation of the MethodThe design method is illustrated with reference to Fig. 1, which

considers a SDOF representation of a frame building (Fig. 1(a)).The bilinear envelope of the lateral force-displacement responseof the SDOF representation is shown in Fig. 1(b). An initialelastic stiffness Ki is followed by a post yield stiffness of rKi.

While F.B.D. characterizes a structure in terms of elastic, pre-yield properties, D.B.D. characterizes it by secant stiffness Ke atmaximum displacement d (Fig. 1(b)), and a level of equivalentviscous damping .

In Fig. 1(d), the displacement is, actually, the spectral displace-

ment. In Fig. 1(c), the displacement ductility, or structure ductility(), considers the behavior of the whole structure and is definedas the ratio of the maximum structure displacement in theinelastic range to the displacement corresponding to the yieldingpoint. The quantity of displacement ductility is selected on thebasis of many parameters (type of material and earthquake,distance from source, site conditions, etc.); the methods availableto the designer are either monotonic static nonlinear analyses(push-over type) or dynamic time history analyses.

With the design displacement at maximum response determin-ed and the corresponding damping estimated from the selectedductility demand (Fig. 1(c)), the effective period Te at maximumdisplacement response can be read from a set of displacementspectra for different levels of damping, as shown in Fig. 1(d).The effective stiffness Ke of the equivalent SDOF system atmaximum displacement can be found by Eq. (5a):

(5a)

where me is the effective mass of the structure participating in thefundamental mode of vibration.

From Fig. 2(b), the design lateral force, which is also thedesign base shear force, is thus:

(5b)

6. Some Implications of D.B.D.

Priestley et al. (2007) showed that:

Ke42me

Te2--------------=

F Vb Ked= =

Fig. 1. Fundamentals of Displacement-Based Design: (a) SDOF Simulation, (b) Effective Stiffness Ke, (c) Equivalent Damping vs Ductil-ity, (d) Design Displacement Spectra



The required base-shear design force is not sensitive to theseismic intensity, under F.B.D. procedure, whereas it is pro-portional to the square of the seismic intensity, under D.B.D.procedure.

The design base shear strength is independent of the numberof storey under D.B.D. procedure, whereas it is dependentunder F.B.D. procedure.

7. Problems with D.B.D.

According to Medekhar and Kennedy (1998), the use ofD.B.D. introduces the following issues:

Selection of an appropriate maximum displacement of theSDOF

Effect of axial column deformations on the displaced shape Greater cumulative P- effect on the building Influence of higher modes on inter storey drift

8. Design Procedure of D.B.D.

8.1 Single Degree of Freedom SystemThe central concept of the method is that the structure is

designed for a specified target displacement.The method is illustrated by reference to a single storey, single

bay building that may be modeled as a SDOF shear building(Fig. 2). According to Paz (1985), a shear building may bedefined as a structure with no rotation of a horizontal section atthe level of the floors. In this respect, the deflect building willhave many of the features of a cantilever beam that is deflectedby shear forces only; hence the name shear building. For this, itmust be assumed that:1. The total mass of the structure is concentrated at the levels of

the floors2. The girders on the floors are infinitely rigid as compared to the

columns.3. The deformation of the structure is independent of the axial

forces present in the columns.A set of elastic displacement spectra for different levels of

equivalent viscous damping is required (Fig. 1d).The design procedure is as follows:

1. Estimate the yield displacement of the structure y as afunction of the geometry and the material properties.

2. Select an appropriate maximum inelastic displacement inwhich depends on the ductility level of the elements

3. Calculate the maximum displacement max as the sum of thetwo previous displacements.

4. Select an appropriate value of effective structural damping effaccording to the ductility level implied in step 2.

5. The effective period (Teff) corresponding to max and eff isobtained from spectrum

6. The effective stiffness of the system is:

(6)

7. The base shear capacity required is: (7) The structure is dimensioned to give an effective period.

8. If the effective period is not sufficiently close to the requiredperiod, repeat the process.

8.2 Multi Degree of Freedom SystemThe D.B.D. method is applied to the 3 degrees of freedom

system as shown in Fig. 3, by transforming it into an equivalentSDOF system in exchange for some assumptions, and applyingthe method described previously for a SDOF system. Then wereturn to the MDOF system.

According to Medekhar and Kennedy (1998), the designprocedure is as follows:1. Select an initial desired displaced share for the structure, i2. Select the effective damping, eff, which depends on the ducti-

lity implied with i3. Determine the effective displacement:

(8)

4. Obtain the normalized profile:

(9)

5. Obtain the effective mass: (10) 6. Determine the effective period Teff from spectrum, correspond-ing to eff and eff.

7. Obtain Keff from Eq. (6)8. Obtain:

(11) 9. Obtain the lateral forces acting on MDOF system:

(12)

9. Numerical Application

As all the applications are based on the Algerian seismic code,Keff

42MeffTeff2

-----------------=

Vb Keffmax=

eff mii2

mii----------------=

Ciieff-------=

Meff miCi=

Vb Keffeff=

Fimiimjj---------------Vb=

Fig. 2. Structure Modeled as a SDOF System


Vol. 15, No. 3 / March 2011 511

it may be appropriate to state briefly the most important informa-tion about this code, with a focus on the seismic design spectraand the seismic map.

9.1 About the Algerian Seismic Code (R.P.A.)

9.1.1 Seismic Design SpectraThe Algerian seismic code proposes three approaches to

calculate the seismic forces The first, known as equivalent staticmethod, consists to replace the real dynamic forces acting onthe structure by a system of fictive static forces which aresupposed to have the same effects. In the second method, knownas spectral modal analysis method, the seismic forces arecalculated for the most important vibration modes and combinedto obtain the maximum structure response using one of theknown combinations such as the square root of the sum of thesquares. The third method calculates the response by integratingthe equation of movement using seismic records as input.Although the last method is the most accurate, it is rarely usedbecause it is relatively complex as it requires adequate analysisand appropriate real or simulated earthquakes. The first andsecond methods are based on the concept of seismic designspectrum.

In the equivalent static method, the base shear Vb is given byEq. (13a).

(13a)

where A, D, Q, R and W are, respectively, the accelerationcoefficient of zone, the dynamic factor of amplification, thequality factor, the global behavior factor and the total structureweight. All these factors are given by appropriate tables accord-ing to some parameters as: the seismic zone, the site, the geo-metry, the lateral loading resisting system.

In the spectral modal analysis method, the base shear Vb isgiven by Eq. (13b).

(13b)

where M is the total mass and g the acceleration of gravity (Sa/g)which stands for the normalized acceleration spectrum, is givenby the following equations:

(14a)

(14b)

(14c)

(14d)

where T is the fundamental period of the structure given by theempirical equation:

(15)

Ct is a coefficient depending on the type of lateral loadingresisting system and the filling type, T1 and T2 are the charac-teristic periods depending on site type (i.e., rock, firm, soft orvery soft soil). is a correction factor of the damping, given, in terms of the

damping ratio , by Eq. (16):(16)

9.1.2 Seismic MapAt the beginning, the Algerian territory was subdivided into

four zones on the seismic map: Zone 0 (i.e., of neglected seismicity) covering the great South

of the country called Sahara Zone I (i.e., of low seismicity) covering a strip beyond the

previous zone of about 400 kilometers of width Zone II (i.e., of medium seismicity) situated between the

previous zone and the Mediterranean Sea Zone III (i.e., of high seismicity) covering some regions of the

VbADQW

R-----------------=

Vb M.g.Sag----=

Sgg----

1.25A 1 TT1----- 2.5QR---- 1 + if 0 T T1

3.125AQR---- if T1 T T2

3.125A QR---- T2T-----

23---

if T2 T 3.0s

3.125A QR---- T23-----

23---

3T---

53---

if T 3.0s

=

T Ct Hn( )3 4=

7 2 +( ) 0.7=

Fig. 3. System with 3 Degrees of Freedom and Equivalent SDOF System



previous zone such as Chlef and BoumerdesRecently, the zone II has also been subdivided into two zones

namely zone IIa and zone IIb with seismicity in zone IIb higherthan in zone IIa. Thus, there are now five zones on the Algerianseismic map (Fig. 4).

9.2 Applied ModelsSince the aim of this paper is to make comparison between two

analytical procedures, the geometrical and mechanical modelsapplied have been taken as simple as possible. The simplest modelavailable in structural dynamics is the shear building definedabove which is based on the assumption of lumped masses at thefloor levels. Furthermore, the model of the curve giving the load-deformation relationship has been idealized to the elastic-per-fectly plastic model. Other models may be used to describe thebehavior of the structure which is really with distributed massand an infinite number of degrees of freedom. These modelswhich are often based on finite element method or derived methodsare more realistic and, consequently, more accurate but they aremore complex and may be required only in special cases as highdynamic loading (blast for example) and large deformations.Among these recent and sophistical models, those based on theso called mesh-free methods are well suited to give explicitrepresentation of the crack evolution in 2 and 3 dimensions forboth cases of static

9.3 Numerical DataFor simplicity, numerical applications have been conducted on

regular reinforced concrete shear buildings (Fig. 5) with a LateralLoad Resisting System (L.L.R.S.) which consists of 4 frames ineach direction and a maximum number of storey taken equal tothree, following the R.P.A. advices for zone IIa. The mass perfloor has been taken constant and equal to 20 t and the flexibilityof the roof diaphragm neglected. Although the spectra based on aset of specifically selected records are more convenient, the

displacement spectra used are those derived from the accelera-tion spectra given by the R.P.A. for firm soil and zone of mediumseismicity. The design procedure is similar to that used byMedekhar (1998).

Ductility demand: the greatest value that may be agreed for aR.C. structure is =2. (Edjtemai, 1981).

Drift to yield: Referring to the R.P.A., inter storey drift is limit-ed to 1 of the building height. It should be noted that this limitcorresponds to the immediate occupancy performance level(Djebbar et al., 2007) and dynamic loading (Oller et al., 1990;Rabczuk and Belytschko, 2007 and 2008; Rabczuk and Eibl,2006; Belytschko et al., 2000; Saatci and Vecchio, 2009).

9.4 Force Based Method

9.4.1 One-storey Building (SDOF system)a) Base shear: It is obtained using Eq. (13b) with the following

data: T1 =0.15 s, T2 =0.40 s, Ct=0.05, Hn= 3 m; then T=0.11 s; =7, =0.88, A= 0.15, Q=1.1 and R=5As: 0 < T < T1, substituting in Eq. (14a) leads to: Sa/g=0.076;then, from Eq. (13b), Vb =14.9 kN.If =5, =1 and Sa/g=0.083; then: Vb =16.3 kN, and if =10 : Sa/g=0.07, then Vb =13.14 kN

b) Check of displacement: It is done for the displacementcorresponding to the greatest value of Vb (i.e., 16.3 kN): The force acting upon a column is 1.06 kN and the minimumcolumn cross section is 25 cm25 cm. The correspondingstiffness and displacement are, respectively, K=1249.28 kN/m and = 0.84 mm.The maximum displacement is max =R. =4.2 mm. This latter must be less or equal to 1 of the building height(11). Indeed, 1 of 3 m equals 3 cm.

c) P- effect: According to the R.P.A., the P- effect is nottaken into account if: Pk k/Hk 0.1 Vk, where:Pk is the total structure weight over level k; Hk and Vk arerespectively the height and force shear at level k.M.g. max/H=200009.810.001/3 < 0.113700 N. So the P- effect is not taken into account.

9.4.2 Two-storey Buildinga) Base shear: T=0.19 s. For =5, Vb =39 kN; for =7, Vb =35

Fig. 4. Algerian Earthquake Map

Fig. 5. Symmetric Layout of 3-storey Building


Vol. 15, No. 3 / March 2011 513

kN and for =10, Vb =31. kNb) Distribution of strength over height: Vb =Ft+ Fi, but since

T< 0.7 s, then Ft =0. and Vb =FiFi =Vb.Wi.Hi/ WjHj, with: Wi: portion of weight allocated tofloor (i) and Hi: height to floor (i)F1 =13.1 kN and F2 =26.16 kN

c) Horizontal distribution of strength: it is done proportionally tothe stiffnessThe force acting upon a column is: F1i =F1 /16=818.7 N (firststorey) and F2i =F2 /16=1635. N (second storey)

d) Check of displacements: According to the R.P.A. the interstorey drift k = k - k-1 must be 0.01 Hk with: k =R. ek, where ek is the floor displacement due toseismic forces and R the behavior factor.e1=0.65 mm and e2=1.31 mm. Then: 1 =3.25 mm, 2=6.56mm and 1 =3.25 mm, 2 =3.31 mm1 of Hk =30. mm. So, the condition is verified.

e) P- effect: P1. 1/H1 =17.33 N < 1047 N (= 0.1 F1) and P2.2/H2 =17.67 N < 2093 N (= 0.1 F2).So the P- effect is not taken into account.

9.4.3 Three-storey Buildinga) Base shear: T=0.05(9)3/4 =0.26 s.

For =5, Vb =59. kN; for =7, Vb =53. kN and for =10,Vb =47. kN

b) Distribution of strength over height: F1 =9.81 kN, F2 =19.62kN and F3 =29.43 kN

c) Horizontal distribution of strength: F1i =F1/16=613.12 N,F2i =1226.25 N and F3i =1839.37 N

d) Check of displacements: condition of R.P.A. satisfiede) P- effect: it is not important enough to be taken into account.9.5 Displacement Method

9.5.1 One-storey Building (SDOF system)a) Ductility Level =2. with y =1 cm

y =1 cm and in =1. y =1 cm; thus: max =y + in =2. y =2cm According to Qiang (2001), the damping model presented byIwan and Gates (Eq. (17)) gives the most accurate results forstructures with ductility factor 4:

(17)

For =2, Eq. (17) gives eff =0.108, then eff =10 (commonvalue used for damping).

Teff =1.55 s, for eff =0.1 and max =2 cm. Keff =328.33 kN/m(Eq. (6)). Then, Vb =6.57 kN (Eq. (7)).The portion of base shear on a frame is Vb1 =Vb/4=1642 Nand on a column Vb2 =Vb1/4=411 N.The deflection at the free end of a cantilever beam actedupon by a static force F at the free end is given by:

(18)

where, l, E and I are respectively the length, modulus ofelasticity and modulus of inertia of the beam.Consequently, the corresponding stiffness coefficient is:

(19)

Then, the required stiffness is Kr =19250 N/m; and conse-quently, the required period is Tr =2.13 s.The required period is not close enough to the effective one.Then, the process is repeated from the beginning with a newvalue of the yield displacement y =3.5 mm. Then, max =7mm and Vb =11.3 kN.

b) Ductility level=1.1 with y =5 mmmax =1.1 y =5.5 mm, then: eff =0.07 and Teff =0.55 s. Thus:Vb =14.34 kN; Tr =0.73 s and T1 =0.18 s.

c) Ductility level=1. with y =5 mmmax =y =5 mm, then: eff =0.05 and Teff =0.5 s. Thus: Vb =15.8 kN; Tr =0.67 s and T1 =0.17 s.

9.5.2 Two-storey Buildinga) =2. in the first storey and 1. in the second storey, with

r1 =3.5 mm and r2 =5 mm eff =10.16 mm and Meff =37.4 t=93 Mt, with Mt =total mass.So, Teff =0.92 s for eff =0.1 and eff =10.16 cm Then:Keff =1743. kN/m and Vb =18. kN=base shear withoutincluding P- effect.Lateral force at each level, Fi: F1 =14018/380=6.63 kN andF2 =24018/380=11.37 kNLateral forces equivalent to P- effect: F1=M1.g 1/H1 =0.46kN and F2=M2.g. 2/H2 =0.40 kNTotal lateral forces including P- effect: F1t =F1 + F1=7.10kN and F2t =F2 + F2=11.77 kNTr =1.19 s and T1 =0.27 s. Table 1 summarizes the most im-portant data and results.

b) =1.1 over height with r1 =3.5 mm and r2 =5 mm eff =7.75 mm, eff =7%, Teff =0.7 s, Vb =21.2 kN

eff 0.05 0.0587 1( )0.371+=

Fl3

3EI---------=

K F---3EIl3

---------= =

Table 1. Lateral Forces Including P- Effect, for =2. and 1, with r1 =3.5 mm and r2 =10 mm

FloorHeightto floorHi (m)

MassMi (t)

Ductility level

Drift to yield ri (mm)

Assumedshapei (mm)

Mi i (tmm)

Mi ( i)2 (tmm2)

Profile of shape

Ci

MiCi(t)

Fi(kN)

Fi(kN)

Fit(kN)

1 3. 20. 2. 3.5 7. 140. 980. 0.69 13.8 6.63 0.46 7.1

2 6. 20. 1. 5. 12. 240. 2880. 1.18 23.6 11.37 0.4 11.77

40. 380. 3860. 37.4 18. 18.87



c) =1. over height, with: r1 =5 mm= r2 eff =8.33 mm, eff =5%, Teff =0.63 s, Vb =30. kN

9.5.3 Three-storey Buildinga) =1.5 over height, with r1 = r2 = r3 =3.5 mm

eff =12.25 mm, eff =10, Teff =1.05 s, Vb =23. kN b) =1.1 over height, with r1 = r2 = r3 =3.5 mm

eff =8.98 mm and eff =7, Teff =0.78 s, Vb =30 kN

c) =1. over height, with r1 = r2 = r3 =3.5 mm eff =0.05 and eff =9.12 mm, Teff =0.7 s, Vb =36.3 kN

9.5.4 Summary of the Main ResultsIn order to make comparison between the two methods, the

main results are grouped in a single table and the values of thedifference between both the base shear forces and the periods areplotted in different charts.

Table 2. Main Results of the Two Methods

Dampingratio (%) Number of storey Floor

F.B.D. method D.B.D. methodRatioVbF/FtT

(kN)Base shearVbF (kN)

Ductilitylevel Teff(s) Base shearVbD (kN)

Total force Ft (kN)

Required period Tr (s)

5

1 1 0.11 16.3 1. 0.50 15.8 16.13 0.67 1.01

21

0.19 39.1.

0.63 30. 30.66 0.83 1.32 1.

3

1

0.26 59.

1.

0.71 36.3 37.15 0.92 1.592 1.

3 1.

7

1 1 0.11 14.9 1.1 0.55 14.3 14.70 0.73 1.02

21

0.19 35.1.1

0.7 21.2 21.75 0.96 1.612 1.1

3

1

0.26 53.

1.1

0.78 30. 30.75 1.03 1.722 1.1

3 1.1

10

1 1 0.11 13.14 2. 0.70 11.3 11.3 0.93 1.16

21

0.19 31.2.

0.92 18. 18.87 1.19 1.642 1.

3

1

0.26 47.

1.5

1.05 23. 24.02 1.35 1.962 1.5

3 1.5

Fig. 6. Effect of the Storey Number and the Rate Meff/Mt on the Base Shear Forces: (a) Effect of the Storey Number, (b) Effect of the RateMeff/Mt


Vol. 15, No. 3 / March 2011 515

10. Analysis of the Results, Comparison and Com-mentaries

This study, based on the Algerian seismic code, has beenachieved on R.C. regular buildings with one, two and threestorey. More complex structures can be studied with the sameprocedures.

The following results can be outlined: The base shear under the F.B.D. method is greater than that

under the D.B.D. one (1-2 times). The difference between theresults of the two methods increases with the number ofstorey and with the value of the damping ratio. This may beattributed to the deficiency in defining the substitute structureparameters, especially the maximum displacement and theequivalent viscous damping.

The effective mass used in the D.B.D. method compared tothe total mass may give an idea of the accuracy of thismethod: more the rate Meff/Mtot is near from 100 , more theresult is realistic.

In the D.B.D. method, the values of base shear correspondingto a damping ratio of 5 seem to be more realistic. Thisaccounts for the fact that this value is commonly used inseismic design.

This method is an efficient design procedure with no need toempirical equation or arbitrary coefficient.

It enables to take directly into account the P- effect, whichmay be important for slender structures.

It also enables to take into consideration the deformations ofsecondary elements.

It is simpler to apply and better suited to incorporation indesign codes.

System-level ductility is an essential factor for earthquakeresistant design of structures.

The target displacement is a main factor in the D.B.D method.Some authors proposed its estimation by the Pushovermethod (Williams and Albermann, 2003).

More investigation must be done to determine an accuraterelation between damping and ductility.

11. Conclusions

This analysis confirms the idea, developed by many authors,that the Displacement-based method may be, in future, a goodalternative to the Force-based method providing that someproblems can be resolved by further work in a number of areas toassess the following issues:

Estimation of the target displacement and the effective damp-ing

System ductility and its relation to the damping Damage thresholds for common non structural elements Appropriate displacement spectra for design purposes Application of this method to other systems such as: asym-

metric structures, systems with shear walls

Acknowledgments

The work on this paper is based on research work supported bythe Algerian Ministry of High Education and Scientific Researchwith the contribution of both University of M'sila and Universityof Batna. The valuable orientations of Dr Mohamed Benchikhfrom M'sila University and Dr Kamel Abdelkader Tayebi, Pre-sident of Novel Technologies Solutions, Ontario, are also grate-fully acknowledged.

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