response spectrum of linear elastic system … response spectrum of linear elastic system for 2001...

RESPONSE SPECTRUM OF LINEAR ELASTIC SYSTEM

FOR 2001 BHUJ EARTHQUAKE

Bhumika B. Mehta M. E. CIVIL – CASAD.

B-2, Kalindi Flats, Opp. Kadwa Patidar Boarding, C. G. Road, Ahmedabad – 380 006

Ph. No. – (079) 6561093 [email protected]

[email protected]

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CONTENTS

1. CONCEPT OF RESPONSE SPECTRUM

2. RESPONSE QUANTITIES

3. NUMERICAL EVALUATION OF RESPONSE SPECTRUM

4. DEFORMATION, VELOCITY AND ACCELERATION RESPONSE

SPECTRUM

4.1 Deformation response spectrum

4.2 Pseudo velocity response spectrum


4.4 True/Relative velocity response spectrum

4.5 True/Relative acceleration response spectrum

5. PROCEDURE TO CONSTRUCT RESPONSE SPECTRUM OF THE

EARTHQUAKE

6. PEAK STRUCTURAL RESPONSE FROM THE RESPONSE

SPECTRUM

7. TRIPARTITE RESPONSE SPECTRUM

7.1 Procedure to construct tripartite plot

8. FACTORS AFFECTING RESPONSE SPECTRUM

9. RESPONSE SPECTRUM CHARACTERISTICS

10. APPLICATIONS OF RESPONSE SPECTRUM

11. DESIGN SPECTRUM

11.1 Procedure to construct design spectrum



RESPONSE SPECTRUM OF LINEAR ELASTIC

SYSTEM FOR 2001 BHUJ EARTHQUAKE

One of the central problem of engineering seismology is the calculation of the

behaviour of a structure subjected to a given ground motion. An exact solution of this

problem in transient dynamics is rarely possible because of the great complexity of the

ground motions associated with earthquakes, and because of the complicated nature of

many of the structures of interest to the engineer. One of the attempts to simplify this

problem has involved the introduction of the “ response spectrum”.

1. CONCEPT OF RESPONSE SPECTRUM

Response spectrum of any earthquake ground motion is simply a plot of the

maximum response during that earthquake of a single degree of freedom (SDF)

system as a function of its natural vibration period Tn or frequency (ωn or fn), for a

given value of damping. Maximum response may be displacement, velocity or

acceleration. Each such plot has a fixed damping ratio ξ, and several such plots for

different values of ξ are included to cover the range of damping values encountered in

actual structures.

Now a central concept in earthquake engineering, the response spectrum

provides a convenient means to summarize the peak response of all possible linear

single degree of freedom systems to a particular component of earthquake ground

motion.

2. RESPONSE QUANTITIES

As a structural engineer, mostly we concerned with the deformation of the

structural system or displacement u(t) of the mass relative to the moving ground, to

which the internal forces are linearly varied.

This response quantity is deformation or displacement, velocity or

acceleration. These are the bending moments and shear forces in beams and columns



of single degree of freedom system (SDF) or the spring force in the system or strain

energy stored in the system. The deformation spectrum provides the peak deformation

umax of a system. The velocity response spectrum is directly related to the total strain

energy stored in the structural system during the earthquake. The acceleration

response spectrum is directly related to the peak value of the equivalent static force

and base shear of the structural system.

Knowing the total displacement u(t) of the structure would be useful in

providing enough separation between adjacent structures to prevent their pounding

against each other during earthquake. Similarly, the total acceleration )(tu&& of the mass

would be needed if the structure is supporting sensitive equipment and the motion

imparted to the equipment is to be determined.

3. NUMERICAL EVALUATION OF RESPONSE SPECTRUM

The equation of motion of a single degree of freedom system subjected to the

earthquake ground motion is:

For undamped vibration condition

gumkuum &&&& −=+

)(2 tauu =+ ω&&

Elastic response spectrum is obtained by integrating the equation of motion for

a Single Degree of Freedom system.

)(2 tauu =+ ω&& (1)

Ground acceleration co-ordinate is denoted by a(t) for convenience. A number

of efficient methods are available for integrating equation (1). In this, equation (2) is

solved analytically within each successive time step assuming the ground acceleration

varies linearly between designated points.

)(2i

i

ii tt

t

aauu −

∆∆

+=+ ω&& , 1+≤≤ ii ttt (2)



Where, the ground acceleration, a(t) has been replaced by its piecewise-linear

approximation. The response u(t) over the time interval itt ∆≤≤0 is the sum of three

parts:

§ Free vibration due to initial displacement ui and velocity iu& at t = 0,

§ Response to step force ai with zero initial conditions,

§ Response to ramp force (∆ai/∆ti)(t-ti) with zero initial conditions.

The solutions for the relative displacement u and velocity u& are:

∆

−∆

∆+−++=

in

n

i

in

in

n

ini t

t

t

t

k

at

k

at

ututu

ωω

ωωω

ωsin

)cos1(sincos)(&

(3)

−

∆∆

+++−= )cos1(sincossin)( ttk

at

k

at

ututu n

in

in

in

n

inin ω

ωωω

ωωω

&& (4)

Evaluating these equations at t = ∆ti gives the relative displacement ui+1 and velocity

1+iu& at time i+1.

{ } { })sin()cos(1)sin()cos(1 ininin

iin

iin

n

iinii tt

tk

at

k

at

utuu ∆−∆

∆∆

+∆−+∆+∆=+ ωωω

ωωω

ω&

{ }

∆−

∆∆

+∆+∆+∆−=+ )cos(1)sin()cos()sin(1 inin

iin

iin

n

iinini t

tk

at

k

at

utuu ω

ωωω

ωωω

&&

These can be rewritten for damped vibration system

11 ++ +++= iiiii DaCauBAuu & (5)

11 '''' ++ +++= iiiii aDaCuBuAu && (6)

Where,



∆+∆

−= ∆− )cos()sin(

1 2tteA DD

tn ωωξ

ξξω

D

Dt teB n

ωωξω )sin( ∆

= ∆−

∆

∆

+−∆

−−

∆−

+∆

= ∆− )cos(2

1)sin(1

21212

2

tt

tt

etk

C Dn

DD

t

n

n ωω

ξω

ξ

ξω

ξω

ξ ξω

∆

∆+∆

∆−

+∆

−= ∆− )cos(2

)sin(122

11 2

tt

tt

etk

D Dn

DD

t

n

n ωω

ξω

ωξ

ωξ ξω

∆

−= ∆− )sin(

1'

2teA D

ntn ωξ

ωξω

∆

−−∆= ∆− )sin(

1)cos('

2tteB DD

tn ωξ

ξωξω

∆∆

+∆

−∆+

−+

∆−= ∆−

t

tt

te

tkC D

Dntn

)cos()sin(

11

11'

22

ωω

ξ

ξ

ξ

ωξω

∆+∆

−−

∆= ∆− )cos()sin(

11

1'

2tte

tkD DD

tn ωωξ

ξξω

Where,

21 ξωω −= nD

If the record is digitized at equal time intervals, the coefficients of A, B, C and

D are constant for a given frequency. Hence, given the initial conditions for a single

degree of freedom system, response computations proceed rapidly by applying the

recursion relationships defined in equation (5) and (6). The calculations are repeated

for a family of frequencies for each selected damping value. Thereby an entire set of

elastic response spectra is developed for the given earthquake record.

The time step used in the response computations is selected as the smaller of

the digitized interval of the earthquake accelerogram or some fraction of the period of



free vibration, for example T/10. For systems whose natural period governs the

selection of ∆ti, i.e. for high frequencies, ∆ti must be chosen so that an integral number

of time steps comprise the digitized interval of the accelerogram. This restriction on

∆ti preserves uniform time intervals and guarantees that response quantities will be

computed at times corresponding to those of the given earthquake record.

4. DEFORMATION, VELOCITY AND ACCELERATION RESPONSE

SPECTRUM

4.1 Deformation response spectrum

Deformation response spectrum is a plot of the maximum displacement that a

SDF structure is experienced during the earthquake as a function of its natural

vibration period for a given value of damping.

The deformation response spectrum provides all the necessary information to

compute peak values of deformation SD = umax and internal forces. Here the spectrum

is developed for a longitudinal component of Bhuj earthquake of 2001 at station

Ahmedabad. For each system the peak value is of deformation SD = umax is

determined from the deformation history. Usually the peak occurs during ground

shaking, however, for lightly damped systems with very long natural periods the peak

response may occur during the free vibration phase after the ground shaking has

stopped. The spectrum is complete when all possible values of damping are plotted.

The peak deformation SD = 46.917mm for Tn = 1 sec and ξ = 5%.






Figure 1

N 78 E Component of Bhuj Earthquake

Tn = 1 sec

ξ = 5%



Tn = 5 sec

ξ = 5%

Tn = 10 sec

ξ = 5%

Tn = 15 sec

ξ = 5%

Figure 2

Deformation Response of Four SDOF Systems



Displacement Response Spectrum

237.35

98.516

117.9

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14

Natural period Tn in Sec

Sp

ectr

al D

isp

lace

men

t S

D in

mm

Figure 3

Displacement Response Spectrum for ξ = 5%


Pseudo velocity response spectrum is a plot of pseudo spectral velocity PSV

and natural period Tn that an SDF structure is experienced during the earthquake for a

given value of damping.

)(2

)(ω SDT

SDPSVn

nπ==

PSV is pseudo spectral velocity. It is pseudo because it is not velocity but it has

units of velocity. It is related to the peak value of strain energy Emax stored in the SDF

system during the earthquake.

2222maxmax )(

2

1)(

2

1)(

2

1

2

1PSVmSDmSDkkuE ==== ω

The peak pseudo spectral velocity PSV for a system with natural period Tn and

spectral displacement SD can be determined from the equation.

For Tn = 1 sec, ξ = 5% and D = 46.917 mm, PSV = 294.79 mm/sec.






4.3 Pseudo acceleration response spectrum

Pseudo acceleration response spectrum is a plot of pseudo spectral acceleration

PSA and natural period Tn that an SDF structure is experienced during the earthquake

for a given value of damping.

)(2

)(2

2 SDT

SDPSAn

n

==

πω

PSA is pseudo spectral acceleration. It is pseudo because it is not acceleration

but it has units of acceleration. It is related to the peak value of base shear VBmax or the

peak value of equivalent static force fs.

)()()( 2maxmax PSAmSDmSDkkuV nB ==== ω

Rewriting above equation

Wg

PSAVB =max

Where,

W is the seismic weight of the structure

g is the gravitational acceleration.

Here, (PSA/g) may be interpreted as the Base shear Coefficient normally used in

design codes.

The peak pseudo spectral acceleration PSA for a system with natural period Tn

and spectral displacement SD can be determined from the equation.

For Tn = 1 sec, ξ = 5% and D = 46.917 mm, PSA = 1852.21 mm/sec2.






Displacement Response Spectrum

237.35

98.516

0

50

100

150

200

250

300

0 5 10 15

Natural period Tn in Sec

Spec

tral

Dis

plac

emen

t SD

in m

m

Figure 4

Spectral Displacement Response Spectrum

Velocity Response Spectrum

123.80149.13

0

100

200

300

400

0 5 10 15

Natural Period Tn in sec

Spec

tral

Vel

ocit

y SV

in

mm

/sec

Figure 5

Pseudo Spectral Velocity Response Spectrum

Acceleration Response Spectrum

0

1000

2000

3000

4000

0 5 10 15Natural Period Tn in sec

Spec

tral

Acc

eler

atio

n SA

in m

m/s

2

Figure 6

Pseudo Spectral Acceleration Response Spectrum



4.4 True/Relative velocity response spectrum

The true or relative velocity response spectrum )(tu& can be determined from

the equation (6) of section 3. The difference between relative velocity spectra and

pseudo velocity spectra depend on the natural period of the system. For long period

systems, PSV is less than maxu& and differences between the two are large. This can be

understood by recognizing that as Tn becomes very long, the mass of the system stays

still while the ground underneath moves. For short period systems PSV exceeds maxu& ,

with the differences increasing as Tn becomes shorter. For medium period systems, the

differences between PSV and maxu& are small over a wide range of Tn.

Velocity Response Spectrum

0

50

100

150

200

250

300

350

400

0 5 10 15


Sp

ectr

al V

eloc

ity

SV

in m

m/s

ec

Pseudo velocity

Relative velocity

Figure 7

Comparison of True and Pseudo Velocity Response Spectrum

4.5 True/Relative acceleration response spectrum

The true or relative acceleration response spectrum )(tu&& can be determined

from the following equation.

)(2)()( 2 tututu nn&&& ξωω += (7)

Where,

u(t) = Spectral Displacement



)(tu& = Relative velocity

The pseudo acceleration and relative acceleration response spectra are identical

for systems without damping. This is clear from equation (7).

PSAtutu n == )()( 2ω&&

For systems with damping, it is possible only when )(tu& = 0. In this case u(t)

attains its peak umax. at this instant )(2 tunω represent the true acceleration of the mass.

Equation (7) suggests that the differences between PSA and )(tu&& are expected to

increases as the damping increases.

The difference between two spectra is small for short period systems and is of

some significance only for long period systems with large values of damping. As the

natural period Tn of a system approaches infinity, the mass of the system stays still

while the ground moves. Thus deformation reaches to peak ground displacement and

PSA reaches zero.

Acceleration Response Spectrum

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15


Sp

ectr

al A

ccel

erat

ion

SA

in m

m/s

2

Pseudo acceleration

Relat ive acceleration

Figure 8

Comparison of True and Pseudo Acceleration Response Spectrum



5. PROCEDURE TO CONSTRUCT RESPONSE SPECTRUM OF THE

EARTHQUAKE

a) Take the ground motion data of an earthquake. This ground acceleration data

gu&& is available from the accelerogram, which is earthquake-measuring

instrument. This ground motion ordinates are defined at suitable interval.

Generally it is at 0.02 sec.

b) For the particular value of damping ξ take the different value of natural period

Tn of a single degree of freedom system (SDF).

c) Compute the deformation response u(t) of this SDF system due to the ground

motion )(tug&& by method developed in section 3.

d) Determine umax, the peak value of u(t).

e) The spectral ordinates are SD = umax, PSV = (2π/Tn)SD and PSA = (2π/Tn)2

SD. here SD is spectral displacement, PSV is pseudo spectral velocity and PSA

is pseudo spectral acceleration.

f) Repeat above steps except first for a different range of Tn and ξ values

covering all possible systems of engineering interest.

g) Draw the results graphically to produce three separate spectra.

h) Above all figure shows the graphical representation of construction of

displacement response spectrum, velocity response spectrum and acceleration

response spectrum of Bhuj earthquake of 2001 at station Ahmedabad.

6. PEAK STRUCTURAL RESPONSE FROM THE RESPONSE

SPECTRUM

If the response spectrum for a given ground motion component is available, the

peak value of deformation or of an internal force in any linear SDF system can be

determined readily. Corresponding to the natural vibration period Tn and damping

ratio ξ of the system, the values of SD, PSV, PSA are read from the spectrum. Now all

response quantities of interest can be expressed in terms of SD, PSV, PSA and the

mass or stiffness properties of the system. The peak deformation of the system is



PSAT

PSVT

SDunn

2

max22

===

ππ

And the peak value of the equivalent static force fsmax is

)(maxmax PSAmkufs ==

Static analysis of the one story frame subjected to lateral force fsmax provides the

internal forces (e.g. shear force s and bending moments in columns and beams). The

peak values of shear force and bending moment at the base of the one story structure

are:

)(maxmax PSAmkuVB ==

maxmax BB hVM =

h

fsmax

VBmax

MBmax

Figure 9

Peak Value of Equivalent Static Force

Only one of the three spectra – deformation, pseudo velocity and pseudo

acceleration is sufficient for computing the peak deformations and forces required in

structural design.

7. TRIPARTITE RESPONSE SPECTRUM

Each of the three spectra, SD, PSV and PSA, give exactly the same

information. No new information is given by any one of them. The three spectra are

simply different ways of presenting the same information on structural response. From

one of them we can get the other spectra. Usually, these spectra are together drawn on

a single log plot.



If all the three spectra provide same information than why do we need three

different spectra and what can be obtained from any one of the other two? One of the

reasons is that each spectrum directly provides a physically meaningful quantity. The

deformation response spectrum provides the peak deformation or displacement of a

system. The pseudo velocity response spectrum is directly related to the peak strain

energy stored in the system during the earthquake. The pseudo acceleration response

spectrum is directly related to the peak value of the equivalent static force and base

shear. The second reason is the shape of the spectrum can be approximated more

readily for design purposes with the support of all three spectral quantities rather than

any one of them alone (elastic design spectrum)

The three spectral quantities are interrelated by relation

)(SDPSVPSA

n

n

ωω

==

SDT

PSVPSAT

n

n ππ

2

2==

The 4-way plot is a compact representation of the three – displacement, pseudo

velocity and pseudo acceleration response spectra. A response spectrum should cover

wide range of natural vibration periods and several damping values so that it provides

peak response of all possible structures.

The response spectrum has proven so useful in earthquake engineering that

spectra for virtually all ground motions string enough to be of engineering interest are

now computed and published soon after they are recorded. Enough of them have been

obtained to give us a reasonable idea of the kind of the motion that is likely to occur in

future earthquakes, and how response spectra are affected by distance to the causative

fault, local soil conditions, and regional geology.

7.1 Procedure to construct tripartite plot

a) Y-axis is the log(PSV) and X – axis is the log(Tn).

b) Now relation between PSV, PSA and Tn is PSVPSATn

=π2

Taking log scale on both side



PSVPSATn log2logloglog =−+ π

now, log Tn = x, log PSV = y and log PSA is constant.

∴ π2loglog −+= PSAxy

∴y = x + constant

1=dx

dy

This is the equation of a straight line with slope = +1. the intercept on

Y – axis depends on PSA. Hence, constant PSA lines have a slope pf +1 means

they make an angle of +45° with X - axis.

c) Similarly, relation between PSV, SD and Tn is SDT

PSVn

π2=

Taking log scale on both sides

nTSDPSV loglog2loglog −+= π

Now, log Tn = x, log PSV = y and log SD is constant.

∴y = log 2π + log SD - x

∴y = -x + constant

1−=dx

dy

This represents a straight line with slope of – 1 means the constant SD

lines make an angle of -45° with X – axis.

d) Hence, SD and PSA values can be read in a log(PSV) versus log(T) plot on the

logarithmic scales oriented at 45° to the period scale. This 4-way plot is a

compact representation of the three response spectra.



1000

1000

0

SD mm

1000SA

mm

/sec2

0.1

0.001

0.01

0.01

10

1

SV

mm

/sec

100

100.1

10

0.001

0.01

Tn sec

0.1 1

0.0000

1

100

1000

1

0.0001

0.01

0.10.001

1000

10

100

1100

8

22.55

25.13

Figure 10

Combined D-V-A or Tripartite Plot

From the figure, for a pseudo spectral velocity PSV = 8 mm/sec and Tn = 2 sec,

PSA = 25.13 mm/sec2 and SD = 2.55 mm.

8. FACTORS AFFECTING RESPONSE SPECTRUM

The actual shape of the response spectrum is affected by

§ Fault length and size of shock,

§ Distance from fault

§ Local geology and geography.

For example, at large distances from fault, the high frequencies get filtered and

only low frequency waves spread. Thus, a site nearer to the fault will have more high

frequencies than farther ones. This is significant while designing structures. For



structures with high natural periods, a larger fault lying at greater distance from the

site may be critical, while for structures with low natural periods, a smaller fault lying

close to the structure may be more significant. This is also a reason why design

spectrum is at times arrived at by combining expected response spectra from different

faults for different frequency range.

9. RESPONSE SPECTRUM CHARACTERISITCS

For systems with very short natural period (stiff system), the peak pseudo

acceleration approaches maxgu&& and SD is very small. For example, a fixed mass and

very short period system is extremely stiff or essentially rigid. Such a system would be

expected to undergo very little deformation and its mass would move rigidly with the

ground; its peak acceleration should be approximately equal to maxgu&& and maximum

relative displacement SD and maximum relative velocity SV are small. Thus the

acceleration spectrum gives the maximum ground acceleration corresponding to the

acceleration value.

For systems with very long natural period (flexible system), the maximum

displacement approaches maxgu and pseudo acceleration is very small. Thus the forces

in the structure, which are related to m*PSA, would be very small. For example, a

fixed mass and very long period system is extremely flexible. In this type of system

mass would be expected to remain essentially stationary while the ground below

moves. Thus maximum relative displacement and maximum relative velocity

approaches maxgu and maxgu& . An increase in damping smoothens the spectrum curves

and lowers the average amplitude.

For intermediate systems with medium natural period, maximum velocity

approaches maxgu& . Over this period range, PSV may be idealized as constant at a value

equal to peak ground velocity.



Based on these observations response spectrum is divided into three parts with

low, medium and high natural periods. The long natural period region is called

displacement sensitive region. The short natural period region is called acceleration

sensitive region. And intermediate system having medium natural period is called

velocity sensitive region.

10. APPLICATIONS OF RESPONSE SPECTRUM

The use of the response spectrum has frequently been criticized of the grounds

that it is not possible to represent a complex structure by the very simplified model

containing only a single mass, spring and dashpot. It is important that it be clearly

understood that no exact equivalence between the response spectrum and the

behaviour of an actual structure is necessarily assumed nor implied in the response

spectrum technique. The practical usefulness of the response spectrum method is

based on the following consideration.

b) For complex structures in which system response must be calculated by

considering motions in a number of modes of vibration simultaneously, the

response spectrum may be used directly to get approximate solutions by the

principle of superposition. The response spectrum is concerned with maximum

values only, loses some information concerning the phases of the motions in

various modes. A superposition of the response values from the response

spectrum corresponding to the periods of the modes of vibration does,

however, give an approximate value of total system response. This

approximation is always on the safe side, since the assumption that the

maximum responses in the various modes will always occur at the same time,

will give a total value somewhat higher than the actual maximum response.

c) Many structures may, in spite of their complexity, behave under some

circumstances essentially as single degree of freedom systems, and for these

situations the response spectrum can be applied directly to give system

behaviour. Because of the way in which motion is excited, even very

complicated structures may move in essentially one mode of vibration. Ground

accelerations, for example, since they are equivalent to a distributed set of



inertia forces acting always in the same direction at all points in a structure,

tend to excite motion in which the fundamental mode of vibration

predominates.

d) A realistic evaluation of seismic coefficients or lateral force coefficients cannot

be obtained without the use of the response spectrum. The maximum

accelerations to be expected in a structure are not those which are recorded by

the ground motion accelerometer, since dynamic amplification effects can

occur which may take the structural accelerations considerably larger that the

ground accelerations. From the response spectrum, the maximum value of the

total base shear force is directly obtained.

e) The velocity spectrum gives directly the energy input into the system. By

equating this energy input to the sum of the various energy dissipation, which

occurs in the structure, a determination of overall system behaviour can be

made. Such energy methods, used in connection with a limit design technique,

will permit the establishment of rational dynamic strength criteria for many

structures subjected to earthquake loading.

f) Response spectrum gives the maximum response of a linear SDF system to a

given ground motion. Many real life structures can be idealized as single

degree of freedom systems. Hence, such structures can be directly analyzed

using the response spectrum.

11. DESIGN SPECTRUM

The earthquake design spectrum for elastic systems should satisfy certain

requirements because it is intended for the design of new structures or the seismic

safety evaluation of existing structures, to resist future earthquakes. For this purpose

the response spectrum for a ground motion recorded is inappropriate. The unevenness

in the response spectrum is characteristic of one excitation. The response spectrum for

another ground motion recorded at the same site during different earthquake is also

uneven, but the peaks and valleys are not necessarily at the same period. Similarly, it

is not possible to predict the uneven response spectrum in all its detail for a ground



motion that may occur in the future. Thus the design spectrum should consist of a set

of smooth curves or a series of straight lines with one curve for each level of damping.

The design spectrum should, in a general sense, be representative of ground

motions recorded at the site during past earthquakes. If none have been recorded at the

site, the design spectrum should be based on ground motions recorded at other sites

under similar conditions.

The design spectrum is based on statistical analysis of the response spectra for

the group of ground motions. Statistical analysis of these groups of data provides the

probability distribution for the spectral ordinate, its mean value, and its standard

deviation at each period Tn. The probability distributions are shown schematically at

three selected Tn values, indicating that the coefficient of variation (= standard

deviation ÷ mean value) varies with Tn. connecting all mean values gives the mean

response spectrum. Similarly connecting all the mean-plus-one-standard-deviation

values gives the mean-plus-one-standard-deviation response spectrum. These two

response spectra are much smoother than the response spectrum for an individual

ground motion.

The recommended period values Ta = 1/33 sec, Tb = 1/8 sec, Te = 10 sec and

Tf = 33 sec and the amplification factors αA, αV, and αD for the three spectral

regions were developed from the group of earthquake data. The amplification factor

for two different nonexceedance probabilities, 50% and 84.1% are given in table

4.1.for several values of damping and table 4.2 as a function of damping ratio. The

50% nonexceedance probability represents the median value of the spectral ordinates

and the 84.1% approximates the mean-plus-one-standard-deviation value assuming

lognormal probability distribution for the spectral ordinates.



Table 1

Amplification Factors for Elastic Design Spectra

Median (50th percentile) One sigma (84.1th percentile) Damping

ξ% αA αV αD αA αV αD

1 3.21 2.31 1.82 4.38 3.38 2.73

2 2.74 2.03 1.63 3.66 2.92 2.42

5 2.12 1.65 1.39 2.71 2.30 2.01

10 1.64 1.37 1.20 1.99 1.84 1.69

20 1.17 1.08 1.01 1.26 1.37 1.38

Table 2

Amplification Factors as a Function of Damping Ratio

Median (50th percentile) One sigma (84.1th percentile)

αA 3.21 – 0.68 ln ξ 4.38 – 1.04 ln ξ

αV 2.31 – 0.41 ln ξ 3.38 – 0.67 ln ξ

αD 1.82 – 0.27 ln ξ 2.73 – 0.45 ln ξ

11.1 Procedure to construct design spectrum

a) Plot three dashed lines corresponding to the peak values of ground acceleration

maxgu&& , velocity maxgu& and displacement maxgu for the design ground motion.

b) Take the appropriate value of Aα , Vα and Dα for the damping ratio ξ.

c) Multiply the peak value of ground acceleration maxgu&& by amplification factor

Aα to obtain the straight-line b-c representing the constant value of pseudo

acceleration PSA.

d) Multiply the peak value of ground velocity maxgu& by amplification factor Vα to

obtain the straight-line c-d representing the constant value pseudo velocity

PSV.



e) Multiply the peak ground displacement maxgu by the amplification factor Dα

to obtain the straight-line d-e representing the constant value of spectral

displacement SD.

f) Draw the line maxguPSA &&= for periods shorter than Ta and the line maxguSD =

for periods longer than Tf.

g) The transition lines a-b and e-f complete the spectrum.

133 1

8 33

Pseu

do S

pect

ral V

eloc

ity

PSV

mm

/sec

(lo

g sc

ale)

Natural Period Tn sec

c d

e

f

b

a

10

Figure 11

Construction of Elastic Design Spectrum



1000

SD mm

1

0.1

0.001

0.01 10.1

0.01

SV m

m/s

ec

100010

1

100

10

SA m

m/sec2100

0.110

1000

0.001

Tn sec

10 100

0.001

0.01

0.01

0.1

0.0001

1

1000

0

100

1000

1

0.107

0.085

0.25

2.71

0.171

Figure 12

Construction of Elastic Design Spectrum (84.1th percentile)

Longitudinal component of 2001 Bhuj Earthquake

The design spectrums of different component of 26 January Bhuj earthquake

are given below.



Design Spectrum for Longitudinal Component

0.5

73

4.3

39

0

500

1000

1500

2000

2500

3000

0 1 2 3 4 5Natuaral Period Tn in sec

PSA

mm

/sec

2

Figure 13

Elastic Design Spectrum for Longitudinal Component

Design Spectrum for Tansverse Component

1.8

720

.86

2

0

500

1000

1500

2000

2500

0 1 2 3 4 5

Natuaral Period Tn in sec

PSA

mm

/sec

2

Figure 14

Elastic Design Spectrum for Transverse Component



Design Spectrum for Vertical Component

0.3

26

4.2

93

0

500

1000

1500

2000

0 1 2 3 4 5

Natuaral Period Tn in sec

PSA

mm

/sec

2

Figure 15

Elastic Design Spectrum for Vertical Component



REFERENCES

Chopra, A. K., Dynamics of Structures- Theory and Applications to Earthquake

Engineering, 2001, Earthquake Engineering Research Institute, Berkeley, California

94704.

Hudson, D. E., Response Spectrum Techniques in Engineering Seismology,

Proceedings of the first world conference in earthquake engineering, Berkeley,

California 1956, pp 4-1 to 4-12.

Newmark, N. M., Hall W. J., Earthquake Spectra and Design, earthquake

engineering research institute, Berkeley, California, 1982, pp 29 - 37



response spectrum of linear elastic system … response spectrum of linear elastic system for 2001...

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