Beşinci Ulusal Deprem Mühendisliği Konferansı, 26

TIME HISTORIES FOR SEISMIC ANALYSIS OF STRUCTURES PROS AND CONS OF AVAILABLE METHODS
Nicolas HUMBERT1, Irmela ZENTNER2, Paola TRAVERSA3, Frédéric ALLAIN4
ABSTRACT
Due to increased potentialities of computers, it is nowadays possible to perform dynamic nonlinear computation of structures to evaluate their ultimate behaviour under seismic loads using refined
finite element models. Nevertheless, one key parameter for such complex computations is the input
motion, which may lead to important discrepancies in the results and therefore difficulties to deal with
for engineering purpose.
Like many codes worldwide, Eurocode 8 (EC8), allows to perform dynamic computation for
seismic analysis of structures with the use of time histories. The method to get these accelerograms is
not clearly defined by EC8, possible alternatives being (i) time histories selection in a waveform
database, (ii) numerical waveform generation or (iii) a composite method using selection of recording
with a scaling or a filtering step.
However these methods are frequently presented in contradiction with each other: natural
motions should give a less biased estimation of seismic load, or generated motions should not catch all
the complexity of natural ones.
The objective of this paper to evaluate these assertions using statistics based on heuristic cases
(i.e. test case).
For this purpose, each of the 10548 accelerograms in the NGA database is used as an input for a
test case. For each test case, a time history is generated that matches several parameters of the natural
one: response spectrum, strong motion duration or cumulative absolute velocity (CAV). Then a natural
accelerogram is selected that matches magnitude, distance, soil condition, response spectrum (..) of the
target.
The numerical codes used in this work are Powerspec and Code Aster with an overall process
that guaranties the representativeness of the generated motions. The selection process is a classical one
based on macroseismic data, response spectrum and strong motion duration.
The statistical evaluation of the NGA database compared to the new databases generated in this
study allows us to give some recommendations on how to obtain input motion for the dynamic
analysis. These recommendations are in the conclusion of this paper.
1
Engineer, EDF, Hydro Engineering Centre, France, [email protected]
R&D Engineer , EDF R&D, France, [email protected]
3
Seismologist, EDF – TEGG, France, [email protected]
4
Engineer, EDF, SEPTEN, France, frederic-1.allain @edf.fr
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INTRODUCTION
This study uses the NGA database as input data for a large number of heuristic cases: each
motion of the database is reproduced with the methods evaluated. The validity of a given method is
then evaluated by its capacity to reproduce the NGA motion with an equivalent level of demand, for
different types of mechanical models.
Firstly, we present the input data (NGA database) and the methods used to generate or select the
accelerograms. Secondly the evaluation method based on usual statistics and a Student Fisher
estimator is described. Finally the results and the recommendations are presented.
NGA DATABASE
The NGA (Next Generation Ground-Motion Attenuation Models) project was a
multidisciplinary research program coordinated by the Pacific Earthquake Engineering Research
Center (PEER), in partnership with the U.S. Geological Survey and the Southern California. The main
goal of the NGA project was to develop new ground-motion prediction models for shallow crustal
earthquakes in the western United States and similar active tectonic regions. To attain this goal, a
common ground motion database was implemented (Power et al., 2008). It consists of 3551 multicomponent records from 173 shallow crustal earthquakes, ranging in magnitude from 4.2 to 7.9. The
NGA data collection includes acceleration time series of the multi-component recordings, pseudoabsolute-spectral acceleration at multiple damping levels and supporting information about the
earthquakes and recordings. In particular, metadata consist of information about the record, the
earthquake source, the strong-motion station, and the propagation path. NGA data were systematically
checked and reviewed by experts and NGA developers (Chiou et al., 2008).
For these reasons, the NGA database is a robust and statistically representative input motion for
this study.
Figure 1. Epicenter distribution of the 173 earthquakes of the NGA database (Chiou et al., 2008).
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Figure 2. Magnitude and distance distribution of strong-motion records in the NGA database (Chiou et al.,
2008).
ACCELEROGRAMS GENERATION
Different methods for the generation of accelerograms and numerous ground motion models are
available in literature. One particular class of stochastic methods is based on the identification of a
response-spectrum-compatible Power Spectral density (PSD). Vanmarcke was among the first to
implement this approachfor engineering design (Simqke – Gasparini, 1976). Following models
introduced non stationary features e.g. Preumont (1980 & 1985) and Mertens (1997). More recently, a
module for the generation of artificial ground motion has been introduced in Code_Aster that
addresses the above objectives.
The two methods for the generation of accelerograms developed in this publication are based on
POWERSPEC and Code_Aster. However these two codes are included in a selection process based on
the recommendations of both, EC8 and ASN (Autorité de Sûreté Nucléaire, the French nuclear
authority) guide 2.01: the generation of accelerograms is done using other parameter than spectral
shape as target. For this reason, iterative generation process is implemented for both codes in order to
fit parameters such as strong motion duration (Tstr), maximum velocity (Vmax), maximum
displacement (Dmax), Arias intensity (Iarias) or Cumulative Absolute Velocity (CAV).
THE NON STATIONARY MODEL IMPLEMENTED IN POWERSPEC
POWERSPEC is a software developed by Westinghouse based on the work of Preumont and
Mertens. The non separable PSD model implemented in Powerspec is presented in equation (1) :
S , t   A(, t ) G( )
2
A, t   t  e  ( )t
(1)
The α function has a polynomial evolution:      0   1   2 2 with α0, α1 and α2
determined with the strong motion duration target and two parameters xp and x0 , giving the evolution
of the spectral content with the time.
xp and x0 parameters represent respectively the decreasing rate of the number of cycles and the
decreasing rate of the number of peaks. This parameter usually varies between 0. and 0.075. The way
to determine these two parameters is developed by Saragoni (1974): the evolution of the central
frequency is described by the exponential function (2) :
C t   2 02 e  X t
(2)
0
POWERSPEC adds a new term to the Vanmarcke formula to identify PSD coherent with the
target spectrum (Preumont 1985). Then iteration on the spectral content of the PSD is implemented to
improve the adjustment with the target spectrum.
Finally a polynomial correction (or «baseline correction ») is applied using equation (3) to avoid
the drift on the displacement, obtained integrating twice the accelerogram.
a   a0  a1t  a 2 t 2
(3)
THE NON STATIONARY MODEL IMPLEMENTED IN CODE ASTER
The detailed methodology for Code_Aster non-stationary model is described in the Code_Aster
documentation (Zentner, 2013). Hereafter a quick overview of the process is given.
The spectrum-compatible, non-stationary ground motion model based on the concept of
evolutionary PSD (Priestley M B, 1965) implemented in Code_Aster uses a general formulation where
ground motion is expressed as filtered white noise, close to the classical Kanai-Tajimi PSD model
(Kanai, 1957 and Tajimi, 1960). Time dependency of the central frequency is introduced by
considering the central frequency of the filter as a function of time, yielding an evolutionary PSD
model, Eq. (4)
S R  , t  

R12  R22 2
2
 0 (t ) 2

2
 2 2 20 (t ) 2
(4)
The parameters of this model, R1,R2,w0 and  are identified by minimising the distance to the
spectrum compatible PSD G (w).
The variation of the amplitude is introduced by a deterministic modulating function h(t). The
Gamma function and the Jennings&Housner model are proposed by Code_Aster.
The obtained evolutionary PSD model will yield time histories whose median spectrum is close
to the target. However, since the PSD is a parametrical model, exact matching is in general not
possible. The equal energy criterion (Preumont 1985) allows then to conciliate the spectrumcompatible model h(t)2G(ω) with the evolutionary PSD h(t)2S(ω,t)S0(ω), where S0(ω) is the corrective
term:
0 ht  S R , t S0  dt  0 ht  G dt
T
T
2
2
(5)
Moreover, a high pass filter allows to remove the low frequency components of synthetic
ground motion (ω² decrease), yielding non diverging behaviour of the integrated velocity and
displacement time histories. The filtered PSD reads (Clough & Penzien 1975):
S  , t  
2
 2f   2  2i f  f 
2
S R  , t 
(6)
The filter frequency ωf is also known as the corner frequency in the literature (Rezaeian and Der
Kiureghian, 2008).
Iterations on the frequency content of the accelerogram are performed to adjust the spectral
match.
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METHODOLOGY FOR THE SELECTION OF STRONG MOTIONS
The selection process is a usual two-step method (Buratti, 2008) based on (i) macroseismic
parameters and (ii) accelerometric parameters.
First, the 20 motions that best fit the macroseismic score presented in the following equation are
selected:
K1  k 0 M  M TARGET
2  k1 logD / DTARGET 2  k 2 logVs30 / Vs30TARGET 2
,
(7)
with M the magnitude, D the hypocentral distance, Vs30 the mean velocity of shear waves in
the last 30 meters, and k0, k1 and k2 fitted to minimize the distance between spectra (for this paper
k0=0.3, k1=1, k2=1)
Secondly, the 20 resulting motions are classified using a score on spectrum and strong motion
duration.
30Hz
K2  k
 Sp f   Sp
'
0
TARGET
 f 2 df
 k 0' TSTR  TSTR _ CIBLE  ,
2
(8)
1Hz
with Sp the spectral acceleration, TSTR, the strong motion, and k’0 and k’1 fitted in order to
represent the sensitivity of the model to each parameter, the for this study k’0=1.5 and k’1=1.
Moreover, the selected motion cannot be linked to the same station, the same earthquake or an
aftershock/main shock of the same earthquake that the accelerogram we try to reproduce.
For the selection of natural motions we do not use any scaling factor or frequency filter. This
type of composite method is not developed in this paper but gives results very close to the generation
methods.
EVALUATION OF THE GENERATION AND SELECTION METHODS
The comparison of the level of seismic demand from natural and artificial time histories will be
made on the 10 548 time histories of the NGA database and on each of the 6 fragility models. Based
on this 52 740 comparisons, this paper will conclude on the eventual over-conservatism or bias of
artificial accelerograms.
Models
NGA data
Record 1
Record 2
...
Record 10548
Conclusions
Statistical Analysis
Figure 3. Overview of the process
The three methods studied in this paper to obtain accelerograms (two methods of generation and
one of selection) are evaluated with respect to the level of demand they give in mechanical models.
Two issues are evaluated: first they have to be unbiased; secondly they have to be as precise as
possible.
These two issues are addressed by performing a statistical comparison between the design value
obtained for each accelerogram of the NGA dataset and the generated or selected one.
Bias is evaluated by the global distance between the “x=y” line and the plot. The
Bias 
1
N
 ln( D(acc
AccNGA
NGA
))  ln( D(acc  copy NGA )) ,
(9)
Efficiency of the method is evaluated with the standard deviation of the “copy” data around the
respective NGA data (with a natural logarithm as Eq. (10)).

1
N
ln( D(acc
AccNGA
NGA
2
))  ln( D(acc  copy NGA )) ,
(10)
For purposes of clarity this paper gives an estimation of the number of computation needed to
obtain a 30% precision on the mean result with a 95% confidence. The evaluation of this number is
based on the Student-Fisher estimator (Viallet et al. 2005 & 2007). The main characteristics of this
estimator are described in the next paragraph.
Figure 4. Illustration of the validation process: left figure: OK, no bias, low SD / middle figure: NOK, bias (here
over-conservatism) / right figure: No bias, but high SD
DESCRIPTION OF THE STUDENT FISHER ESTIMATOR
Student-Fisher law SN (N being its degree of freedom) is commonly used in statistics. Its main
characteristic is to tend to a Normal law when N tends to infinity. This law is particularly appropriate
to predict the average value of a population D based on the average value of the sample DAV and its
standard deviation SD, for a desired confidence level and depending on the size of the sample N,
considering the following expression.
Where T0.05;N-1 is the Student-Fisher parameter taken for N-1 degrees of freedom and for the
appropriate probability (0.05 unilateral for 95% confidence level in our case). See next table to get
characteristic values of Student-Fisher parameter.
Table 1. Student-Fisher Parameters for a confidence level of 95%.
N
5
10
15
20
30
50
Confidence level: 95% (unilateral)
t(0.05; N-1)
t(0.05;N-1)/N1/2
2.13
0.95
1.83
0.58
1.76
0.45
1.73
0.39
1.70
0.31
1.68
0.24
This parameter is inverted in order to obtain the number of computations needed to obtain a
given precision on the mean value. For this study the precision needed is taken equal to 30% and the
confidence level equal to 95%.
DESCRIPTION OF FRAGILITY MODELS SELECTED
The structural damages produced by real earthquake and artificial time histories are evaluated
on several simplified models, from the most linear, to the most non linear:
Mono modal linear model: this model represents a structure or a material with one controlling
mode which represent the majority of the mass.
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Multimodal linear model: this model is a finite differences computation of a double mode
structure or material. This model gives information of the capacity of the generation or selection
method to represent a combination of solicitations.
Model with hardening behaviour: this model has a bilinear behaviour. For this study it
represents the first step to non linearity. (The equivalent linear models are not presented here but for
reasonable level of distortion the result is close to the linear models)
Force
1. 50E+02
1. 00E+02
5. 00E+01
Displacment
0. 00E+00
- 3. 00E- 0
1
- 2. 00E- 0
1
- 1. 00E- 0
1
0. 00E+00
1. 00E- 01
2. 00E- 01
3. 00E- 01
- 5. 00E+01
- 1. 00E+02
- 1. 50E+02
Figure 5. Schematic loops “force .vs. displacement” Hardening behaviour.
Takeda behaviour: elastic-plastic model taking into account an hysteretic behaviour
characteristic of structures under earthquakes, and finally, a simplified model based on the frequency
degradation as a function of a damage index.
Force
1. 50E+02
1. 00E+02
5. 00E+01
Displacment
0. 00E+00
- 3. 00E- 0
1
- 2. 00E- 0
1
- 1. 00E- 0
1
0. 00E+00
1. 00E- 01
2. 00E- 01
3. 00E- 01
4. 00E- 01
- 5. 00E+01
- 1. 00E+02
- 1. 50E+02
Figure 6. Schematic loops “force .vs. displacement” Takeda bilinear model.
Stiffness degradation with the number of cycles: represents a soil behaviour
Force
150
100
50
Displacment
0
- 1. 5
-1
- 0. 5
0
0. 5
1
1. 5
- 50
- 100
- 150
Figure 7. Schematic loops “force .vs. displacement” Stiffness degradation model.
Elastoplastic model: sliding structure – cliff edge effect. The demand parameter for this model
is the maximum displacement.
Acceleration
0. 15
0. 1
0. 05
Displacment
0
- 0. 000006
- 0. 000005
- 0. 000004
- 0. 000003
- 0. 000002
- 0. 000001
0
0. 000001
- 0. 05
- 0. 1
Figure 8. Elasto-plastic model left: Schematic loops “force .vs. displacement”. Right: Acceleration vs.
displacement
- 0. 15
RESULTS FOR THE GENERATION WITH CODE ASTER
Mono-modal behaviour
Takeda behaviour
Multi-modal behaviour
Stiffness degradation with the number of cycles
Hardening behaviour
Elastoplastic
Figure 9. Analysis of the validity of the generation process with Code Aster
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RESULTS FOR THE GENERATION WITH POWERSPEC
Mono-modal behaviour
Takeda behaviour
Multi-modal behaviour
Stiffness degradation with the number of cycles
Hardening behaviour
Elastoplastic
Figure 10. Analysis of the validity of the generation process with POWERSPEC
RESULT FOR THE SELECTION OF NATURAL ACELEROGRAMS
Mono-modal behaviour
Takeda behaviour
Multi-modal behaviour
Stifness degradation with the number of cycles
Hardening behaviour
Elastoplastic
Figure 11. Analysis of the validity of the generation process with the selection of natural motions
ANALYSIS OF RESULTS
The following table summarizes the results of the analysis. The three methods of
generation/selection tested present no bias and are validated for linear and non linear analysis. The
generation of accelerograms gives a higher precision than the selection of natural motions. This is
mainly due to the fact that the sample of selected natural accelerograms includes variability on
magnitude / distance and soil condition. These uncertainties are carried over to the result variability.
Table 2. Analysis of the validity of results.
MonoMultimodal Model with
Takeda
Stiffness Elastoplastic
modal linear
linear
hardening
behaviour degradation
model
model
model
behaviour
POWERSPEC
generation
ASTER generation
Sigma
0.11
0.15
0.19
0.24
0.33
1.1
Number of
computations*
3
3
4
5
7
52
Sigma
0.07
0.09
0.24
0.25
0.35
0.72
Number of
computations*
2
2
5
5
7
23
0.48
0.56
4.22
11
15
>100
SIgma
0.36
0.37
0.43
SELECTION of
natural
Number of
7
8
10
accelerograms computations*
* number of computations needed to match a 30% precision on the result
Codes and guides usually recommend using a minimum of 3 computations for linear models
and 5 for non linear model. Based on this study, we propose to limit this recommendation using
synthetic accelerograms.
In the case of natural accelerogram selection, this number should be raised to 8 for linear
models and 11 for non linear models.
For models with stiffness degradation, we recommend a minimum of 7 accelerograms in the
case of synthetic accelerograms and 15 in the case of natural accelerograms selection.
For highly non linear models (such as sliding structures) the number of accelerograms required
has to be discussed case by case.
In any case, the number of computations has to be discussed depending on the type of model,
the type of generation of accelerograms and the needed precision for the results.
CONCLUSION
The three methods studied in this paper are validated: the generation with either, Code_Aster, or
Powerspec, as well as the selection of natural accelerograms from strong motion databases, do not
show any bias for any degree of non linearity of the studied structure or material.
The whole variability of the ground motions included in the NGA database can be reproduced
using a numerical code with a sufficient degree of complexity (spectrum matching for a given
damping / realistic evolution of amplitude / non stationary model). Moreover the accelerograms
generation methods are not over-conservative and well reproduce the intensity of natural strong
motion.
However a computation based on a selection of natural accelerograms need a higher number of
computations to reach the same level of precision on the mean result.
Furthermore the main issue to obtain a dynamic analysis without any bias is not a software or a
natural vs. artificial question. The quality of the study is based on the attention given to the interfaces
between the seismic hazard assessment team, the dynamic computation team and the team in charge of
the accelerograms generation.
For this reason, we recommend the teams that realize dynamic study to follow these five steps:
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
Evaluation of the dynamic model retained to represent the behaviour of the structure,
evaluation of the (structural?) parameters of influence for the model.
 Assessment by the seismic hazard study of the target for the parameters of influence
determined by step 1 coherent with the level of hazard needed (cf. table 3).
 Generation or selection of the times histories that match the target (in mean value or
each one).
 Dynamic assessment;
 Evaluation of the mean value with a Student Fisher coefficient. (e.g. mean+0.95
Standard-deviation for 5 computations and 95% confidence level)
In case of a result judged over-conservative, go to step 3 and raise the number of time histories.
The feedback proves that in most cases this is the pre-study (phase 1) that is missing. We
recommend assessing systematically a pre-study of the structure or material before the seismic hazard
study and then communicate to the hazard assessment team the needed parameters (damping value /
sensitivity to duration, strong motion / effect of vertical or horizontal excitation, etc.).
Table 3 gives in detail the main recommendations for the generation of time histories.
Table 3. Recommendations for the generation of time histories
Type of dynamic model
Elastic + 1 main mode
Elastic + multimodal
several dampings
Non linear
Highly non linear
Needed parameter for the time history generation
Only spectrum with the damping of the structure
or
Spectrum for the minimal damping of the structure +
strong motion duration (Tstr) or IARIAS or CAV
Add a first computation to evaluate an equivalent
damping for the structure and use this damping for the target
spectrum. If this step is not possible, prefer a low damping for
the target spectrum.
Other parameters: at least TSTR (but the following
parameters are recommended too: IARIAS, CAV, VMAX, DMAX
or AMAX/VMAX + specific parameter for the study: e.g. De
Biasio 2014.
The chronological time evolution of the frequency
content seems to have only a second order effect on the
noxiousness of the time history, but this evolution can
nevertheless be taken into account, especially if the eigen
frequencies decrease with the number or the amplitude of the
cycles.
Idem non linear +
Anticipate a large number of computations (more than
5) + prefer generation than selection of natural accelerograms
in order to avoid a too large number of computations
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