Response spectra

1
2. Elastic Earthquake Response Spectra
• Definition
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Chapter 4 – Seismic Analysis
25
• Definition
There are different types of response spectra:
SD = |x| max = relative displacement response spectrum (spectral|| max p p p ( p
displacement)
SV = |x| max
 = relative velocity response spectrum (spectral velocity)
SA = |x| max
 = relative acceleration response spectrum
SDa= |x+x| s max = absolute displacement response spectrum
SVa= |x+x| s max = absolute velocity response spectrum
SA = |x+x| s max = absolute acceleration response spectrum (spectral
acceleration)
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)
The earthquake spectra that are most useful in earthquake engineering are SD, SV and
SA.

2
• Definition
Northridge-Rinaldi
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• Properties of Response Spectra
Response spectra have the following properties :
1 they give the maximum response values of a SDOF system subjected to a given1. they give the maximum response values of a SDOF system subjected to a given
earthquake accelerogram;
2. they give the maximum response values in each mode of a MDOF system
subjected to a given earthquake accelerogram; This result will be discussed
further in this chapter.
3. they indicate the frequency distribution of the seismic energy of a given
earthquake accelerogram, meaning that the response of a SDOF system is
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q g , g p y
amplified when the seismic energy is close to its natural frequency.

3
• Exact Response Spectra
The relative displacement response spectrum is obtained directly by Duhamel’s integral
given by equation 4.69 :
|d)-(te)(x
1
-|=|x|=S d
)-(t-
s
t
0d
D maxmax sin 



The acceleration and the relative velocity are as follows :
dt
dx(t)
=(t)x
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dt
x(t)d
=(t)x 2
2

By convolution, if a time function, F(t), is given by :
(t)
then its time derivative becomes :
 d)(t,f=F(t)
(t)u
(t)u
1
0

t)(t),u(f
dt
(t)du
-t)(t),u(f
dt
(t)du
+d
t
)(t,f
=
dt
dF(t)
0
0
1
1
(t)u
(t)u
1
0





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(t)u0
the above is known also as Leibnitz derivative of an integrated function

4
Applying convolution to Duhamel’s integral given by equation 4.69, we have :
)-(te)(x
1
-=)(t,f d
)-(t-
s
d


 
sin



d)-(te)(x
1
-=x(t) d
)-(t-
s
t
0d
sin
t=(t)u
0=(t)u
1
0
The relative velocity is then :
 
d)-(te)(x-=(t)x d
)-(t-
s
t
0
cos 
(4 100)
t)(t),u(f
dt
(t)du
-t)(t),u(f
dt
(t)du
+d
t
)(t,f
=
dt
dF(t)
0
0
1
1
(t)u
(t)u
1
0





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The relative velocity response spectrum is given by :


 
d)-(te)(x
-1
+ d
)-(t-
s
t
0
2
sin
|(t)x|=SV max

(4.100)
Similarly, the relative acceleration is obtained by differentiating equation 4.100 with
respect to time :
(t)x-d)-(te)(x
-1
)2-(1
+
d)-(te)(x2=(t)x
sd
)-(t-
s
t
0
2
2
d
)-(t-
t
0








sin
cos


(4.102)
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The absolute acceleration response spectrum is then :
|(t)x+(t)x|=S sA max

5
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Datafile
• Pseudo Response Spectra
Usually, a civil engineering structure has low damping (lower than 20% critical). The
following hypotheses can then be made :g yp
)(t-sinbyreplacedbecancos d 


)-(t
0,
d
d
2



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6
• Pseudo Response Spectra
With these assumptions, equation 4.100 becomes:
x(t)=d)-(tex-(t)x d
)-(t-
s
t

sin 
 
d)-(te)(x-=(t)x d
)-(t-
s
t
0
cos 
(4.106)
The pseudo relative velocity response spectrum is then:
With the same assumptions, equation 4.102 is written:
( ))(ex( ) ds
0

S=S DV 
2)(t
t




 
d)-(te)(x
-1
+ d
)-(t-
s
t
0
2
sin
d)-(te)(x2=(t)x d
)-(t-
t
0
  
cos



d)-(te)(x
1
-=x(t) d
)-(t-
s
t
0d
sin
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The pseudo absolute acceleration response spectrum becomes:
x(t)=d)-(texx+x 2
d
)-(t-
s
0
s  
sin 
S=S=S VD
2
A 
(t)x-d)-(te)(x
-1
)2-(1
+ sd
)-(t-
s
t
0
2
2
 

 
sin
(4.108)
• Comparison Between Exact and
Pseudo Response Spectra
– Comparing exact response spectra
(equations 4.100, 4.102) with pseudo
response spectra (equations. 4.106,
4.108) for different accelerograms,
yields the following tendencies:yields the following tendencies:
• in a system with zero damping,
results are essentially identical for
natural periods less than one second
(T < 1 s);
• when damping increases to 20 %
critical, differences are within 20 %
but without any observable bias;
• pseudo acceleration response
spectrum more precise than pseudo
velocity response spectrum.
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– Variations within acceptable range
expected from seismic analysis.
– Pseudo response spectra must not be
used for highly damped systems ( >
20 % critical) or for systems with
long natural periods (T >> 1 s).

7
• Tripartite Representation
of Pseudo Response
Spectra
– In practice, responsep , p
spectra represented by a
graph with multiple
logarithmic scales called a
tripartite graph.
– Tripartite graph display on
same curve the following
information :
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information :
• exact relative displacement
response spectrum;
• pseudo relative velocity
response spectrum;
• pseudo absolute acceleration
response spectrum.
• Tripartite Representation of Pseudo Response Spectra
– To understand tripartite graph, consider variation of
log10Sv with log10T for constant values of SA or SD.
a) SA = constant = C1
If the pseudo acceleration response spectrum is equal to a constant, C1, it can be
written:
operating with the log10 on this equation, it yields:
  S
T
2
=S=S=C=S VDD
2
1A


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or
S+T-2=C V101010110 loglogloglog 
 2-C+T=S 1011010V10 loglogloglog

8
 2-C+T=S 1011010V10 loglogloglog
then
This result indicates that a line at + 45º on the tripartite graph represents a constant
spectral acceleration, SA .
 
 
1+=
Td
Sd
10
V10
log
log
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b) SD = constant = C2
If the relative displacement response spectrum is equal to a constant, C2, it can bep p p q
written:
S
2
T
=
S
=C=S V
V
2D

operating with the log10 on this equation, it yields :
S+2-T=C V101010210 loglogloglog 
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or
 2+C+T-=S 1021010V10 loglogloglog

9
then
 2+C+T-=S 1021010V10 loglogloglog
then
This result indicates that a line at -45º on a tripartite graph represents a constant
relative displacement spectrum, SD.
 
 
1-=
Td
Sd
10
V10
log
log
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• Tripartite Representation of Pseudo
Response Spectra
– Figure illustrates, for different damping
values, response spectra of El Centro
earthquake (1940 05 18, comp. S00E).
– Response spectrum of an earthquake
er irreg larvery irregular.
– Spectrum has a general trapezoidal
shape (shape of a tent) characteristic of
earthquake response spectra and has
the following physical explanation :
• for long natural periods, maximum
relative displacement equal to
maximum ground displacement and
maximum absolute acceleration tends
toward zero;
• for intermediate natural periods
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for intermediate natural periods,
relative displacement, relative velocity
and absolute acceleration amplified;
• for short natural periods, maximum
absolute acceleration equal to
maximum ground acceleration and
maximum relative displacement tends
toward zero.

10
• Simplified Design Response Spectra
– Motivation
• For practical seismic design of structures, simplified response
spectra are used.
• Different regions of simplified spectra represented by straight
lines.
• Position of these lines (amplitude) function of seismic hazard
of the region.
• Many simplified design response spectra have been proposed.
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y p g p p p p
• Most common are described in the following sections.
– Housner’s Response Spectra (1959)
Relying on response spectra obtained for four historical earthquakes of Southern
California (El Centro, 1934, M=6.5; El Centro, 1940, M=6.7 and Tehachapi, 1952,
M=7.7) and one from Washington (Olympia, 1949, M=7.1)., G. Housner (1959)
proposed, for the first time, an “average design spectrum”. This spectrum was
calibrated for a maximum ground acceleration of 0,20g and for a probability of
exceedence of 50 %, in other words, for the average of the historical spectral values.
The values obtained from this spectrum are to be multiplied by a scale factor to take
into account the seismic hazard of the region. For example, if the design earthquake of
a given site is 0,15g, then the spectral acceleration will be:
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HousnerofSof
20
15
equals A





SA

11
– Housner’s Response Spectra (1959)
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– Newmark and Hall’s Response Spectra (1969)
• Developed for nuclear industry
• Simplified spectra based on standard ground motion
parameters:
– maximum ground acceleration : 0,50 g;
– maximum ground velocity : 61 cm/s (24 in/s);
– maximum ground displacement : 46 cm (18 in).
– Relying on the study of 28 earthquake records, values represent
l i b diff d i i
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average relation between different ground seismic parameters.
– For a given site, values are scaled directly to the maximum design
acceleration, which is function of seismic hazard of the region.
– Simplified spectrum obtained by multiplying each branch of ground
parameters by an amplification factor which depends on damping
coefficient of the structure.

12
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47
Note!
Table 4.2 Amplification factors for Newmark and Hall’s design response spectra
(From Newmark et Hall, 1982).
Damping ratio
% of critical
Probability of exceedence
of 16 %
(mean + 1 standard deviation)
Probability of exceedence of 50 %
(mean value)
SA SV SD SA SV SD
0,5 5,10 3,84 3,04 3,68 2,59 2,01
1 4,38 3,38 2,73 3,21 2,31 1,82
2 3,66 2,92 2,42 2,74 2,03 1,63
3 3,24 2,64 2,24 2,46 1,86 1,52
5 2,71 2,30 2,01 2,12 1,65 1,39
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7 2,36 2,08 1,85 1,89 1,51 1,29
10 1,99 1,84 1,69 1,64 1,37 1,20
20 1,26 1,37 1,38 1,17 1,08 1,01
For the seismic design of nuclear plants, Newmark and Hall recommended the use of
amplification factors corresponding to a probability of exceedence of 16 %.

13
• Procedure to plot Newmark’s simplified spectra
– Step 1 - Plot of the ground motion parameters
» Limits of ground motion parameters linked by straight lines:
maximum horizontal acceleration, maximum horizontal velocity
and maximum horizontal displacement.
» If maximum horizontal acceleration is only known parameter at
site, standard ground motion parameters can be used with
maximum design acceleration, e.g. if maximum design
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acceleration of 0,33g, at site is only known parameter:
 
 
  cm30,36=cm46
0,50
0,33
=ntdisplacemeground
cm/s40,26=cm/s61
0,50
0,33
=velocityground
g0,33=g0,50
0,50
0,33
=onacceleratiground


















• Simplified Design Response
Spectra
– Newmark and Hall’s
Response Spectra (1969)
Table 4.3 Recommended Damping Values.
Strain Level
Types of structures and conditions % of Critical
Damping
Response Spectra (1969)
• Procedure to plot Newmark’s
simplified spectra
– Step 2 - Plot different regions
of spectrum
» Amplification factors,
shown in Table 4.2, used
to plot different regions
welded steel, prestressed concrete,
reinforced concrete with light
cracking 2 to 3
reinforced concrete with heavy
cracking 3 to 5
less than 50 % of
the elastic limit
bolted or rivetted steel, nailed or
bolted timber 5 to 7
welded steel, prestressed concrete
without complete loss of
prestressing 5 to 7
prestressed concrete with
prestressing loss 7 to 10
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p g
of simplified spectrum.
» Table 4.3 shows
recommended damping
values to be used.
prestressing loss 7 to 10
reinforced concrete 7 to 10
bolted or rivetted steel, bolted
timber 10 to 15
close to or over the
elastic limit nailed timber 15 to 20

14
• Procedure to plot Newmark’s simplified spectra
– Step 3 - Modification of spectral limits for high frequencies
– Find corner frequency, 1, which links velocity branch to acceleration
branch.
– At frequency of about 41, start reducing linearly acceleration branch
of spectrum until reaching limit of peak ground acceleration for a
frequency of 101.
– Theoretically, displacement branch should also be modified for low
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51
y, p
frequencies (lower than 0.1 Hz). But, as low frequencies have little
impact on Civil Engineering structures, modification can be omitted.
– Design Response Spectrum of ASCE 7-05
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15
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16
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17
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18
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19
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61
– Consider a seismic zone in Southern California on Site
Class B (rock)
 
 
H )f(
g.
T
g.g.S
g.g.S
.FF
g.Sg;.S
D
Ds
va
s
338120
60
20
6090
3
2
0151
3
2
01
9051
1
1








0.4
0.6
0.8
1
1.2
ctralAcceleration(g)
ASCE 7-05 Design Spectrum
Newmark-Hall Spectrum, 5%damping, PGA=0.40 g
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Hz).f(orT
Hz).f(or.
g.
g.
T
Hz).f(or.
g.
g
.T
LL
os
oo
0830sec12
671sec600
01
60
338sec120
01
20













0
0.2
0 1 2 3 4 5
Period (sec)
Spec

1
• Floor Response Spectra
– Response spectra, discussed in previous sections, used to determine maximum
response of SDOF structure subjected to base motion.
– Similarly, maximum response of equipment, located in a building, can be
obtained using response spectrum corresponding to the floor where theg p p p g
equipment is located.
– Vibration of a complex building varies from storey to storey, creating,
therefore, a variation in the response spectra of the various floors.
– Traditional technique used to generate a floor response spectrum is, first, to
calculate historical horizontal acceleration of a floor and then use this
accelerogram to construct a response spectrum.
– If a simplified design response spectrum of a floor is to be constructed, the
procedure starts with an ensemble of accelerograms at the base and the
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procedure starts with an ensemble of accelerograms at the base and the
resulting spectra are smoothened.
– Because of the large quantity of calculations required to generate a floor
response spectrum, approximate methods have been proposed (Singh, 1975,
Biggs and Roesset, 1970).
Generation of
Floor Motion Ensembles
Generation of
Floor Motion Ensembles
Floor Response Spectra
• Floor Response Spectra
Dynamic
Analysis
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Period (sec)
SpectralAcceleration(g)
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Period (sec)
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65
Ground Motion Ensembles
(Ground Response Spectra)
Ground Motion Ensembles
Ground Response Spectra
Building
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Period (sec)

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