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Optimized earthquake simulation for high-reliability aseismic design
Nuclear Engineering and Design 71 (1982) 299-300 North-Holland Publishing Company
299
Short contribution
OPTIMIZED EARTHQUAKE SIMULATION FOR HIGH-RELIABILITY
ASEISMIC
DESIGN
Alessandro BARATTA lstituto di Costruzioni - Facolta' di Architettura, University of Naples, 1-80134 Naples, Italy
The chance to couple random-search optimization with procedures to generate earthquake-type functions is outlined in the paper. The purpose is to produce simulated earthquakes of the kind that are the most dangerous for the structure under examination, thus contributing to reduce the uncertainty in the calculations.
tion
1. Basic ideas Most of the procedures employed to generate synthetic earthquakes make use of a large number of basic parameters x i ( i - - 1 . . . . . n) strictly related to the ordinates of the accelerogram, that are determined by iterated sampling of a random variables ~ with given statistics. Thus, it can be assessed in general that synthesized earthquakes depend on the basic vector x, the sample of realizations of ~, and on a local vector h which represents constants conditioning the properties of the accelerograms that must be generated (e.g. the peak acceleration, duration, main frequencies, etc.). After h has been fixed, the accelerogram a depends only on the initial vector x
a----a(x).
(1)
Hence, it can be assessed that, under the condition that the procedure adopted for simulation fully reflects ground motion at the site, the manifold of x vectors corresponds to all possible earthquakes exhibiting properties compatible with the parameters in h. Assume that a given component R of the structure response must be evaluated. This component can be viewed as a function of the accelerogram a at the foundations, i.e. it is a function R(x, h) of the basic vector x and of the local vector h. It is clear therefore that, given h, the structure strength can be investigated by looking at the behaviour of R(x, h) while x spans the range Sx. In order to define the range SX, consider first that x is usually a sample of ~ of very large size n (say 1000). Statistical properties of ~ should then be reflected in x. In particular, if rn x and s~ are the expected value and the variance of ~ and P(x) is its PDF, the following relations should be verified to a very good approxima0029-5493/82/0000-0000/$02.75
Xi=rtm x , i-I
( x , - rag) 2 = ns2,
(2)
i=l
~(a,b)=n[P(b)-P(a)]
V ( a , b ) ~ X 2,
where ~(a,b) is the number of the components of x falling in the interval (a, b), and X is the range of values of .~. The above equalities play the role of constraints on the variables xi, while the response function R(x, h) can be viewed as the objective function of an optimization problem R* = R * ( h ) = s u p R ( x , h ) .
(3)
x
From the practical point of view, the constraints in eq. (2) would make the analysis of the optimum R* very cumbersome by classical methods. On the contrary, such difficulties are readily overcome by resorting to random-search procedures (e.g. Rao [1]), that, by their intrinsic nature, make constraints verifiable almost spontaneously.
2. Numerical results and conclusions In order to evaluate the consequences of the above considerations, the writer has attempted to build up elastic response spectra for a given seismic site, namely the town of Tolmezzo in Friuli (Italy). The method adopted for simulation is the one reported by Ruiz and Penzien [2] that yields artificial
© 1982 N o r t h - H o l l a n d
300
A. Baratta i" Optimiced earthquake simulation SfO(cm
400
~
,)
The values found are
,*, ,*,
o:,~ - 2 6 . 2 s
i ,~_0.52,
t0-0,c=0.66,a
300" f
~ * .....
*-- @--'~
¢ . - - * ' - - @-- @
200" 100
o.5
1.o
1.5
2.0
0-0.36g,
T0 - 2 0 s .
I)
2.5
To (see)
Fig. 1. The maximum possible ordinates of the velocity spectrum of the one-degree-of-freedom oscillator without damping: 1) simulated, 2) maximum values obtained by looking at four earthquake samples, normalized to the same peak acceleration. earthquakes by first modulating and then by filtering a white-noise process. The local vector h contains the following components: ~0, '~ = the pulsation and the damping ratio of the filter; t 0, c = parameters of the modulating function; and a o, To= expected peak acceleration and duration. The above parameters were calibrated by the analysis of four digital accelerograms recorded at the site.
The procedure outlined in the previous section has been applied, using the above values, to evaluate the maximum possible ordinates of the velocity spectrum of the one-degree-of-freedom oscillator without damping. Fig. 1 shows that the obtained spectral ordinates (line 1) are larger than the maximum values that can be obtained by looking at the four sample earthquakes (line 2) after normalization to the same peak acceleration. with a ratio varying from 2.5 to 7. Obviously, such results should be evaluated in connection with an overall view of structural safety, that takes into account the reduction of the uncertainty in the loading allowed by the procedure.
References
[1] S.S. Rao, Optimization: theory and applications (Wiley E.L., New Delhi, 1978). [2] P. Ruiz and J. Penzien, Stochastic Seismic Response of Structures, Proc. ASCE, Vol. 97, ST4, (1971).
299
Short contribution
OPTIMIZED EARTHQUAKE SIMULATION FOR HIGH-RELIABILITY
ASEISMIC
DESIGN
Alessandro BARATTA lstituto di Costruzioni - Facolta' di Architettura, University of Naples, 1-80134 Naples, Italy
The chance to couple random-search optimization with procedures to generate earthquake-type functions is outlined in the paper. The purpose is to produce simulated earthquakes of the kind that are the most dangerous for the structure under examination, thus contributing to reduce the uncertainty in the calculations.
tion
1. Basic ideas Most of the procedures employed to generate synthetic earthquakes make use of a large number of basic parameters x i ( i - - 1 . . . . . n) strictly related to the ordinates of the accelerogram, that are determined by iterated sampling of a random variables ~ with given statistics. Thus, it can be assessed in general that synthesized earthquakes depend on the basic vector x, the sample of realizations of ~, and on a local vector h which represents constants conditioning the properties of the accelerograms that must be generated (e.g. the peak acceleration, duration, main frequencies, etc.). After h has been fixed, the accelerogram a depends only on the initial vector x
a----a(x).
(1)
Hence, it can be assessed that, under the condition that the procedure adopted for simulation fully reflects ground motion at the site, the manifold of x vectors corresponds to all possible earthquakes exhibiting properties compatible with the parameters in h. Assume that a given component R of the structure response must be evaluated. This component can be viewed as a function of the accelerogram a at the foundations, i.e. it is a function R(x, h) of the basic vector x and of the local vector h. It is clear therefore that, given h, the structure strength can be investigated by looking at the behaviour of R(x, h) while x spans the range Sx. In order to define the range SX, consider first that x is usually a sample of ~ of very large size n (say 1000). Statistical properties of ~ should then be reflected in x. In particular, if rn x and s~ are the expected value and the variance of ~ and P(x) is its PDF, the following relations should be verified to a very good approxima0029-5493/82/0000-0000/$02.75
Xi=rtm x , i-I
( x , - rag) 2 = ns2,
(2)
i=l
~(a,b)=n[P(b)-P(a)]
V ( a , b ) ~ X 2,
where ~(a,b) is the number of the components of x falling in the interval (a, b), and X is the range of values of .~. The above equalities play the role of constraints on the variables xi, while the response function R(x, h) can be viewed as the objective function of an optimization problem R* = R * ( h ) = s u p R ( x , h ) .
(3)
x
From the practical point of view, the constraints in eq. (2) would make the analysis of the optimum R* very cumbersome by classical methods. On the contrary, such difficulties are readily overcome by resorting to random-search procedures (e.g. Rao [1]), that, by their intrinsic nature, make constraints verifiable almost spontaneously.
2. Numerical results and conclusions In order to evaluate the consequences of the above considerations, the writer has attempted to build up elastic response spectra for a given seismic site, namely the town of Tolmezzo in Friuli (Italy). The method adopted for simulation is the one reported by Ruiz and Penzien [2] that yields artificial
© 1982 N o r t h - H o l l a n d
300
A. Baratta i" Optimiced earthquake simulation SfO(cm
400
~
,)
The values found are
,*, ,*,
o:,~ - 2 6 . 2 s
i ,~_0.52,
t0-0,c=0.66,a
300" f
~ * .....
*-- @--'~
¢ . - - * ' - - @-- @
200" 100
o.5
1.o
1.5
2.0
0-0.36g,
T0 - 2 0 s .
I)
2.5
To (see)
Fig. 1. The maximum possible ordinates of the velocity spectrum of the one-degree-of-freedom oscillator without damping: 1) simulated, 2) maximum values obtained by looking at four earthquake samples, normalized to the same peak acceleration. earthquakes by first modulating and then by filtering a white-noise process. The local vector h contains the following components: ~0, '~ = the pulsation and the damping ratio of the filter; t 0, c = parameters of the modulating function; and a o, To= expected peak acceleration and duration. The above parameters were calibrated by the analysis of four digital accelerograms recorded at the site.
The procedure outlined in the previous section has been applied, using the above values, to evaluate the maximum possible ordinates of the velocity spectrum of the one-degree-of-freedom oscillator without damping. Fig. 1 shows that the obtained spectral ordinates (line 1) are larger than the maximum values that can be obtained by looking at the four sample earthquakes (line 2) after normalization to the same peak acceleration. with a ratio varying from 2.5 to 7. Obviously, such results should be evaluated in connection with an overall view of structural safety, that takes into account the reduction of the uncertainty in the loading allowed by the procedure.
References
[1] S.S. Rao, Optimization: theory and applications (Wiley E.L., New Delhi, 1978). [2] P. Ruiz and J. Penzien, Stochastic Seismic Response of Structures, Proc. ASCE, Vol. 97, ST4, (1971).