1 2. Elastic Earthquake Response Spectra • Definition CIE 619 Chapter 4 – Seismic Analysis 25 2. Elastic Earthquake Response Spectra • Definition There are different types of response spectra: SD = | x | max = relative displacement response spectrum (spectral max 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) CIE 619 Chapter 4 – Seismic Analysis 26 The earthquake spectra that are most useful in earthquake engineering are SD, SV and SA.
Transcript
1
2. Elastic Earthquake Response Spectra
• Definition
CIE 619 Chapter 4 – Seismic Analysis
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2. Elastic Earthquake Response Spectra
• Definition
There are different types of response spectra:
SD = |x| max = relative displacement response spectrum (spectral || max p p p ( p
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
2. Elastic Earthquake Response Spectra
• Definition
Northridge-Rinaldi
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2. Elastic Earthquake Response Spectra
• 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 discussedfurther 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 yamplified when the seismic energy is close to its natural frequency.
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2. Elastic Earthquake Response Spectra
• Exact Response SpectraThe relative displacement response spectrum is obtained directly by Duhamel’s integral given by equation 4.69 :
| d )-(t e )(x 1
-| = | x | = S d)-(t-
s
t
0dD 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)x2
2
2. Elastic Earthquake Response Spectra
• Exact Response Spectra
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( fdt
(t)du - t) (t),u( f dt
(t)du + d t
)(t, f =
dt
dF(t)0
01
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
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2. Elastic Earthquake Response Spectra
• Exact Response SpectraApplying convolution to Duhamel’s integral given by equation 4.69, we have :
)-(t e )(x 1
- = )(t, f d)-(t-
sd
sin
d )-(t e )(x
1- = x(t) d
)-(t -s
t
0d
sin
t = (t)u
0 = (t)u
1
0
The relative velocity is then :
d )-(t e )(x - = (t)x d)-(t-
s
t
0
cos
(4 100)
t) (t),u( fdt
(t)du - t) (t),u( f dt
(t)du + d t
)(t, f =
dt
dF(t)0
01
1(t)u
(t)u
1
0
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The relative velocity response spectrum is given by :
d )-(t e )(x -1
+ d)-(t-
s
t
02
sin
|(t)x | = SV max
(4.100)
2. Elastic Earthquake Response Spectra
• Exact Response SpectraSimilarly, the relative acceleration is obtained by differentiating equation 4.100 with respect to time :
(t)x - d )-(t e )(x - 1
)2 - (1 +
d )-(t e )(x 2 = (t)x
sd)-(t-
s
t
02
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
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2. Elastic Earthquake Response Spectra
• Exact Response Spectra
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Datafile
2. Elastic Earthquake Response Spectra
• 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- sin by replaced becan cos d
)-(t
0 ,
d
d
2
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2. Elastic Earthquake Response Spectra
• Pseudo Response SpectraWith these assumptions, equation 4.100 becomes:
x(t)=d)-(tex-(t)x d)-(t-
s
t
sin
d )-(t e )(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)(tt
d )-(t e )(x -1
+
d)-(t-
s
t
02
sin
d )-(t e )(x 2 = (t)x d)-(t-
t
0
cos
d )-(t e )(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 2d
)-(t-s
0
s sin
S = S = S VD2
A
(t)x - d )-(t e )(x - 1
)2 - (1 +
sd)-(t-
s
t
02
2
sin
(4.108)
2. Elastic Earthquake Response Spectra• Comparison Between Exact and
(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).
This result indicates that a line at -45º on a tripartite graph represents a constant relative displacement spectrum, SD.
1- =
T d
S d
10
V10
log
log
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2. Elastic Earthquake Response Spectra• 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.
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2. Elastic Earthquake Response Spectra
• 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.
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:
• 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.
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 %.
• 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:
• 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 41, start reducing linearly acceleration branch of spectrum until reaching limit of peak ground acceleration for a frequency of 101.
– Theoretically, displacement branch should also be modified for low
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y, pfrequencies (lower than 0.1 Hz). But, as low frequencies have little impact on Civil Engineering structures, modification can be omitted.
– 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 the g p p p gequipment 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).
