lecture 05 session 03

7/31/2019 Lecture 05 Session 03

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Lecture 05: Earthquake Excitation, Response and Design Spectra

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Lecture 05: Earthquake Excitation, Response and DesignSpectra

ObjectivesThe objectives of this lecture are: (i) to learn about the response of a SDOF structure to a

pulse excitation; (ii) to learn about the response quantities; (iii) to learn about the response

and design spectra. The response spectrum concept is central to earthquake engineering

together with the procedures to determine the peak response of a system directly from the

response spectrum and needs to be well understood.

Figure 5.1. An example of single pulse excitation stimulus.

Response to pulse excitations

In this section we consider an important class of excitations that consist of a single pulse

of force (see Figure 5.1). The response of a SDOF system to such pulses does not reach a

steady-state condition and the transient solution of equation of motion (4.10) is important

together with the knowledge of the initial conditions. If the excitation is a single pulse,

then the effect of damping can be relatively unimportant unless the system is highly

damped. In this case, the equation of motion becomes

( )gmu ku mu t + = (5.1)

with at-rest initial conditions, (0) (0) 0u u= = . In the case of the simplest type of pulse, a

rectangular pulse of duration dt , the equation to be solved is

0, 0

0, , 0

d

d

f t tmu ku

t t t

+ =

>


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Phase (i) Forced vibration phase. During this phase, the system is subjected to a step

force and the response of this system is given by equation (4.23) in which 0 , i.e.

( )

[ ]0

( ) 21 cos 1 cos , 0n d

st n

u tt t t t

u T

= =

(5.3)

where ( ) 00 /stu f k= is the static deformation due to the static force 0f .

Figure 5.3. The dynamic response of an undamped SDOF structure to a rectangular pulse force.

Phase (ii) free vibration phase. After the external force is removed at dt t= , the system

continues to vibrate freely according to the modified form of (4.17) ( 0 ), i.e.

( )( ) ( ) cos ( ) sin ( ) ,d

d n d n d d

n

u tu t u t t t t t t t

= + >

, (5.4)

where ( ) [ ]0

( ) 1 cosd st n d u t u t = and ( )0

( ) sind st n n d u t u t = . Expressing2

n

nT

= and

using trigonometric identities results in


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( )0

( ) 12 sin sin 2

2

d d

st n n n

t tu t t

u T T T

=

, dt t> . (5.5)

Equations (5.3) and (5.5) present the response of a SDOF system exposed to a rectangularpulse force of the duration

dt as a function of / nt T . These equations show that the

normalised response depends on the ratio of the maximum external force and the stiffness

of the SDOF system, 0 /f k, and on the ratio of the pulse duration and the natural period of

the SDOF system, /d n

t T . Figure 5.3 illustrates the dynamic response of an undamped

SDOF system to a rectangular pulse force (solid lines) for several ratios /d nt T . Here the

dashed lines correspond to the static solution.

In engineering applications it often required to determine the overall maximum response

of a SDOF system to a pulse force excitation, i.e.( )

0

( )maxst

u tu

. The number and

amplitude of local maxima (peaks) that develop in the forced vibration phase depends on

the /d nt T ratio (see Figure 5.3). One can expect that the longer the pulse duration, the

greater the number of peaks which can occur during the forced vibration phase. The first

peak with the maximum deformation ( )0 02 stu u= always occurs at / 2nt T= (see Figure

5.3). Therefore, the pulse duration,d

t , must be longer than / 2n

T for at least one peak to

develop during the forced vibration phase. If more than one peak develops during this

phase (Phase (i)) and the damping is negligibly small, then these peaks have similar value

and occur at / 2, 3 / 2, 5 / 2n n nt T T T = , etc. If / 2d nt T< , then no peak will develop duringthe forced vibration phase and the response simply builds up from zero to ( )du t .

In this way it is possible to define the maximum deformation during the forced vibration

phase using the deformation response factor

( )0

0

21 cos , / 1/ 2

2, / 1/ 2

dd n

d n

st

d n

tt Tu

R Tu

t T

= =

>

(5.6)

During the free vibration phase (Phase (ii)) the system oscillates in a simple harmonic

motion given by equation (5.5) with the amplitude

2

2

0

( )( ) dd

n

u tu u t

= +

(5.7)

and corresponding deformation response factorof

( )0

0

2 sin ddst n

u tR

u T

= = (5.8)


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which depends only on d

n

t

T. Finally, because the overall maximum deformation

(displacement) is given by the greatest of either the forced-response maximum (eq. (5.6))

or the free-response maximum (eq. (5.8)) we can define the overall deformation response

factor

( )0

0

2sin , / 1/ 2

2, / 1/ 2

dd n

nd

st

d n

tt Tu

TRu

t T

= = >

(5.9)

Figure 5.4. The deformation response factor as a function of d

n

t

T: (a) response during forced and

free vibration phases; (b) shock spectrum.

The behaviour of the deformation response factor predicted by equation (5.9) is

graphically illustrated in Figure 5.4. This plot ( )d n

R T is called the response spectrum. In

the particular case of a rectangular pulse excitation this plot is also called the shock

spectrum. It is clear from equation (5.9) and Figure 5.4 that the maximum displacement

of a SDOF system exposed to a pulse force of amplitude0

f is up to two times as much as

the displacement of the same system exposed to a static force of the same amplitude.In

this way it is possible to define the equivalent static force which is required to cause

identical maximum deformation of the structure

0 0 0S df ku f R= = . (5.10)


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In practical calculations is can be easier to use the equivalent static force definition in the

static analysis of internal forces and stresses acting upon a structure exposed to an

earthquake excitation.

If the duration of a pulse force of arbitrary shape is shorter than / 2d nt T


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Since 2n

k m= , we can express the equivalent static force in the following form

2( ) ( )S nf t m u t= or ( ) ( )Sf t mA t= (5.12)

where ( )A t is the pseudo-acceleration which is a rather different quantity to that of the

ground acceleration, ( )gu t . The pseudo-acceleration of a structure can be calculated

directly using the displacement history, ( )u t , and knowing the structural mass, m and the

frequency of excitation,n

. Figure 5.6 presents the pseudo-acceleration of a SDOF

structure (with variable natural period, fixed damping ratio, 0.02 = ) exposed to an El

Centro 1940 ground motion. The pseudo-acceleration was calculated by multiplying the

pre-computed structural deformation, ( )u t , by the squared natural frequency of the

structure,

2

2 2n

nT

=

. The pseudo-acceleration data presented in Figure 5.6 can be used

to determine the internal forces in the frame at a selected instant, 0t , by static analysis of

the structure subjected to the equivalent static lateral force, 0( )Sf t . In this way, the

analysis of the structure would be necessary at each time instant when the structural

responses need to be known. In particular case of the frame shown in Figure 5.5, the base

shear force is ( ) ( ) ( )b SV t f t mA t = = and the overturning moment is

( ) ( ) ( )b SM t hf t hmA t= = .

Figure 5.6. The pseudo-acceleration response of SDOF structures to El Centro 1940 earthquake.

In order to summarise the response of all possible SDOF structures to a particular

component of ground motion it is convenient to introduce the concept of the response

spectrum. In this way it is possible to plot the peak value of a response quantity (e.g.displacement, velocity or acceleration) as a function of the natural vibration frequency,


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,n nf , or the natural period, nT . This plot is called the response spectrum for that

quantity. The response spectra are normally plotted for a range of damping ratios typical

in actual structures. Formally, the response spectra can be defined as following

- deformation response spectrum:0

( , ) max ( , , )n nt

u T u t T (5.13a)

- velocity response spectrum: 0 ( , ) max ( , , )n nt

u T u t T (5.13b)

- acceleration response spectrum: 0 ( , ) max ( , , )n nt

u T u t T . (5.13c)

In simple terms, the response spectrum is a plot of the peak or steady-state response

(displacement, velocity or acceleration) of a series of oscillators of varying natural

frequency, that are forced into motion by the same base vibration or shock.

The response spectrum for a given ground motion component, ( )gu t , can be determined

using the following procedure:

1. Numerically define the ground acceleration, ( )gu t

2. Select the natural vibration period,n

T , and damping ratio, , of a SDOF structure.

3. Compute the deformation response of this SDOF structure due to the ground

motion, ( )gu t , by a direct numerical method applied to eq. (4.10) or solving the

Duhamels integral (see also exp. (4.17) in Lecture 04)

( )

0

1( ) ( ) sin( ( ))D

t

t

g D

D

u t u e t d

=

4. For givenn

T and determine the peak value of the deformation, ( )u t , i.e.

0 max ( )T

D u u t= = .

5. For givenn

T and determine the peak values of acceleration,

2

2

n

A DT

=

, and

velocity,2

n

V DT

=

.

6. Repeat steps 2 to 5 to cover all possible values of nT and of engineering interest.7. Present the results of steps 2 to 6 graphically to produce three separate spectra like

those shown in Figure 5.7 for the ( )gu t recorded at the time of El Centro

earthquake (see Figure 4.8 in Lecture 04).

The deformation, velocity and acceleration response spectra contain by and large similar

information, i.e. knowing one of the spectra, the other two can be easily recalculated. One

reason why the three spectra are needed is that each of them provides a physically

meaningful quantity. The deformation spectrum provides the peak deformation of the

system. The pseudo-velocity is related to the peak strain energy stored in the system

during the earthquake. The pseudo-acceleration spectrum is related to the peak value ofthe equivalent static force and the base shear. The other reason is that it convenient to use


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the shapes of the three spectra can be used to derive an approximate spectrum illustrative

and useful for the design purposes. As a result, a combined type of spectral plot was

developed by Veletsos and Newmark in 1960 (see Figure 5.8). This integrated

presentation of the response spectra is possible because the three spectral quantities are

interrelated, i.e.

2

nV DT

= and2

nA VT

= . In this case, four-way logarithmic paper

can be used as explained in Appendix 5.1 to present the spectra as shown in Figure 5.8.

Figure 5.7. The D-V-A response spectra for El Centro 1940 earthquake ( 0.02 = ).

The response spectrum should cover a wide range of natural vibration periods and several

damping values so that it provides the peak response for all possible structures. Practicalvalues of damping ratio and natural vibration periods can cover the range of 0 0.2

and 0.02 50nT seconds, respectively. This procedure is also repeated for the

perpendicular component of the ground acceleration. The above procedure requires a

considerable computational effort and is conducted using a specialised software package,

e.g. SeismoSignal.

The response spectrum has proven so useful in earthquake engineering that spectra for

virtually all ground motions strong enough to be of engineering interest are now

computed and published soon after they are recorded. There are enough spectra in the

database to give a reasonable idea of the kind of motion that is likely to occur in future

earthquakes and how the response spectra are affected by distance to the earthquakeepicentre, local soil conditions and regional geology. If the response spectrum for a given


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ground acceleration can be available readily for any SDOF structure with givenn

T and .

In this way all the response quantities of interest can be expressed in terms of D , V , A ,mass of the system, m, or its stiffness, k. Therefore, the peak value of the equivalent static

force can be determined from

0Sf kD mA= = . (5.14)

Figure 5.8. The combined D-V-A spectrum for El Centro 1940 earthquake ( 0;0.02;0.05;0.20 = ).

Design spectrum

The design spectrum is intended for the design of new structures within a seismic zone or

for the evaluation of existing structures to resist future earthquakes. For this purpose the

response spectra for the ground motion recorded during the past earthquakes are

inadequate because of their considerable statistical variation. Figure 5.9 presents theresponse spectra for the north-south component of ground acceleration recorded at the

Imperial Valley Irrigation district substation in El Centro, California, USA. This figure

suggests that it is difficult or even impossible to describe accurately the jagged response

spectrum in all its details for a ground motion which can occur in the future. Therefore, it

would be convenient if the design spectrum could consist of a set of smooth curves or a

series of straight lines with one curve for each level of damping. This spectrum should be

representative of ground motions recorded at the site during the past earthquakes and

include some statistical information. If none of the response spectra are available for a

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

site under similar conditions. These conditions should relate to: (i) the magnitude of the

earthquake; (ii) epicentral distance; (iii) geology of the travel path for seismic waves; and(iv) local soil conditions at the site. Statistical analysis of the past response spectra can


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provide the probability distribution for the spectral quantity, its mean value and its

standard deviation at each period nT . Connecting all mean values for all possible natural

vibration periods should give the mean response spectrum. Similar procedure can be used

to obtain the mean plus one standard deviation response spectrum (see Figure 5.10).

Figure 5.9. The pseudo-acceleration response spectra calculated for earthquakes which wererecorded at an El Centro site in California during 1940, 1956 and 1968.

Figure 5.10. Statistical response spectra and idealised design spectrum.

One result from the application of such procedure to numerical data assembled by Riddell

and Newmark is illustrated in Figure 5.10. Here the mean spectrum is approximated by

four straight lines which intersect at the recommended period values 1/33a

T = seconds,

1/ 8bT = seconds, 10eT = seconds and 33fT = seconds. In these regions amplification


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factors can be applied to these lines to guarantee non-existence probabilities (the

probability that the structural response spectrum does not exceed the design spectrum) and

to satisfy several values of damping.

The design spectrum differs from the response spectrum in two important ways. Firstly,

the uneven response spectrum is a plot of the peak response of all possible SDOF systemsand hence it is a description of a particular ground motion. On the other hand, the smooth

design spectrum is a specification of the level of seismic design force or deformation as a

function of natural vibration period and damping ratio. It is common that the shapes of

these spectra differ considerably. If the design spectrum is determined by statistical

analysis of several comparable response spectra, then the shapes of the two spectra can be

similar. Secondly, for some sites a design spectrum is the envelope of two different

design spectra (see Figure 5.11).

Figure 5.11. Combined pseudo-acceleration spectrum based on the design spectra for two diffirent sites.

GlossaryDeformation response factor the ratio of the amplitude of the dynamic (vibratory) deformation tothe static deformation.Design spectrum an adapted response spectrum used for the design of new structures within aseismic zone or for the evaluation of existing structures to resist future earthquakes.

Equivalent static force the product of the deformation response factor and dynamic force.1 kip(kilopounds force) = 4448 NPseudo-acceleration the ratio of the equivalent static force and the structural massResponse spectrum- the plot of the peak value of a response quantity (e.g. displacement, velocity

or acceleration) as a function of the natural vibration frequency, ,n nf , or the natural period, nT .

Shock spectrum the response spectrum of a structure exposed to a rectangular pulse force.

References

K. Chopra,Dynamics of Structures: Theory and Applications to Earthquake Engineering.

Chapters 4 and 6. 3rd

Edition, Pearson Education, Inc. 2007.

K. V. Horoshenkov, Earthquake Engineering Module: ENG4075M. Lecture 04. School ofEngineering, Design and Technology, University of Bradford, 2008.


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Appendix 5.1: Four-way graph paper.

(see also section 3.2.4 in Chopra, 2007 for the relations between dynamic response factors)

lecture 05 session 03

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