8th ACSE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-119
Structural Seismic Demand Analysis: Consideration of
“Collapse”
Nilesh Shome
Stanford University, Stanford, CA 94305.
nilesh@stanfordalumni.org
C. Allin Cornell, M. ASCE
Stanford University, Stanford, CA 94305.
cornell@ce.stanford.edu
Abstract
It is well known that a structure generally responds well beyond its elastic capacity at high seismic loading;
nonlinear time-history analysis is therefore the most accurate procedure for the seismic demand
calculations. Such analyses are being used systematically, for example, in the current SAC steel momentresisting frame project. There is apparently no methodology currently available to characterize the seismicdemand calculations when we observe that in some cases structures, especially tall structures, “collapse”
because of strength degradation and/or significant P-Δ effects. In this paper we will propose a novel way to
calculate the seismic demand by considering a three-parameter probability distribution model. The
parameters of the distribution at any ground-motion intensity level can be calculated easily from the
response results when the ground-motion records are scaled to that intensity level. More generally, the
parameters can be estimated economically from (1) a conventional-regression analysis of the
deformations/drifts of a structure versus the intensity of ground motion conditioned on no collapse and (2)
a binary-regression analysis of observations of “collapse” or “no collapse” versus intensity level. We
illustrate the seismic-demand calculations through an example problem of a 20-story steel structure that
collapses in a subset of the nonlinear time-history analyses under high-intensity ground motions. These
regression results can be used with the conventional seismic hazard analysis to calculate the annual
response-exceedance probabilities.
Introduction
We will illustrate in this study the estimation of seismic demand calculation versus
intensity of ground motion for a 20-story steel moment resisting frame (MRF) at a site in
California. As a structure is designed to go beyond its elastic capacity under severe
ground motion, we will carry out nonlinear time-history analysis to predict the seismic
demand. The 20-story building was designed for the SAC Steel project. See Gupta (1999)
for a detailed description of this structure. We will consider in this study a simple
centerline representation of the bare moment-resisting perimeter frames. We have not
considered the additional strength and stiffness of the frames for (1) the panel zones, (2)
the interior simple frames, and (3) the composite action of floor slabs and girders. See
Gupta (1999) for the effect of the consideration of these additional stiffnesses on
structural response of the 20-story structure. In this study we have used the DRAIN-2DX
(1993) program for carrying out the nonlinear time-history analysis. The first-mode and
the second-mode period of the structure are 4 sec and 1.4 sec. The first-mode and the
second-mode mass-participation factors are 80% and 12%.
We assume here that the main threat to the structure is from earthquakes of magnitude
(M) 7.0 at a distance (R) 20km from a site in California. Note that in general we get this
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information from the disaggregation of seismic hazard results for a particular site (see
Bazzurro and Cornell, 1999, for details). We will first estimate the demand at a high
intensity of ground motion. We will find that because of the strong P-Δ effect the
structure may “collapse” at this high intensity, i.e., the numerical algorithm does not
converge. We can not characterize the responses by a typical two-parameter probability
distribution model when we have results that contain collapse of the structure; we will
introduce a three-parameter model for this purpose. Ina more thorough performance
evaluation we need to estimate the demand at several intensities. In this case the
estimation of demand by regression analysis becomes convenient. We will demonstrate
this demand estimation procedure..
Selection of Records
We select accelerograms for nonlinear analysis from the magnitude range 6.7 to 7.3 and
distance range 10 to 30 km. These accelerograms are recorded in California at stiff soil
site (soil type S2 as in UBC, 1994). Because the period of the structure is high (4 sec), we
need to pay special attention to check whether a record has any power at the low
frequency range that excites the structure. In general a large fraction of records do not
have a significant signal-to-noise ratio at very low periods. Hence while processing these
records, a low-cut filter was used to remove the low frequency power from these records.
For the purpose of analysis of this structure we consider only those records in the
catalogue (Silva, 1995) whose low-corner filter frequency is equal to or less than 0.10
Hz. Note that as a structure undergoes nonlinear deformations, the “effective” frequency
of that structure becomes lower than that of its elastic counterpart. Hence while selecting
the records we considered a low-corner frequency that is significantly lower than the
lowest elastic frequency of the 20-story structure (0.25 Hz). In this process we have
selected 44 records from the catalogue for nonlinear time-history analysis.
Dynamic Analysis Results
It has been observed (Shome et al., 1998) that scaling ground-motion records to the target
spectral acceleration at the first-mode period of a structure is an efficient way to estimate
the seismic demand at that intensity. Here by efficient we mean the estimation of the
unbiased value of seismic demand from a smallest possible number of records. In the
present study we will first estimate the seismic demand of the 20-story building at 0.27g
spectral acceleration at 4 sec period and 5% damping. This spectral acceleration is
approximately the 5000-year intensity of ground motion at a central Los Angeles site.
Subsequently we will estimate the demand at several other intensity levels by regression
analysis. In this paper we will show the results of the seismic demand for the maximum
interstory drift only (see Shome and Cornell, 1999, for demand results from other damage
measures).
Demand at a Specific Intensity
The unusual response characteristic of the 20-story structure at the 0.27g intensity is that
13 of the 44records introduce “fatal” numerical instability in the solution scheme. This
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numerical instability can be attributed to the non-convergence at a high level of
nonlinearity. We refer to this numerical instability as “collapse” of the structure. The
drifts at collapse can be considered to be infinite. Hence we can not estimate the central
value of the responses by taking the average of the sample of results. An alternative isto
estimate the median from the middle of the ordered response results. We can call this
estimation the “counted” median ( θ̂ st, count ). We find that the counted median drift is
7.5%. The median is calculated as the average of the 22nd and 23rd ordered responses out
of 44 results. Similarly the 16th-percentile response can be calculated from the 7th ordered
response. With an implicit lognornal assumption an estimate of an “equivalent
dispersion” of the responses (analogous to the conventional standard deviation of the
natural logs) can be defined by the following:
ˆ

(1)
 eqst  ln  st ,16count
% 
  st 
The 16th percentile drift response (  st16% ) at 0.27g intensity is 2.4%. So the equivalent
dispersion is 1.1. One use of this dispersion is to estimate a measure of the accuracy of
the median (  ˆ    st / n , where n is the sample size).
st
Three-Parameter Distribution Model
If we are interested only the estimation of only the median seismic demand at a specific
intensity, the calculation of counted median may be sufficient. The consideration of
record-to-record variability in the demand is, however, also important for the evaluation
of performance of structures especially when the dispersion is high. We have found here
that the dispersion (  eqst ) at high intensity is very high (>1.1). The thorough way of
dealing with the uncertainty in the demand is to calculate the annual probability of
exceedance of any allowable demand (formally this may be called the seismic demand
hazard calculation). This calculation involves integration of the product of the usual
seismic intensity curve and the conditional probability of demand exceedance given
intensity (Bazzurro et al, 1998, Shome and Cornell, 1999). Therefore this procedure calls
for the characterization of the complete distribution of the response results given
intensity. A typical frequency distribution of the damage results at a high intensity of
ground motion is shown in Figure 1. The figure shows that the conventional
characterization of the distribution of the results by a lognormal distribution and the mean
and the standard deviation on the logs is no longer possible in the case of collapses as
these moments cannot be estimated. We choose instead three parameters to characterize
the distribution. These parameters are the probability of no collapse (PNC), and the
median and the dispersion conditioned on no-collapse. (For a given intensity) the
probability of exceedance of a level of damage y for any damage measure Y can be
calculated from the following:
GY ( y )  GY |C ( y )  PC  GY |NC ( y )  PNC
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 PC  GY |NC ( y )  PNC
(2)
where GY |NC ( y ) is the complementary cumulative distribution function of Y conditioned
on no collapse, PNC is the probability of no collapse, and PC is the probability of collapse
(=1- PNC ). If we assume that the distribution of no-collapse damage results is lognormal,
then Equation 2 simplifies to
 ln( y / yˆ ) 
GY ( y )  PC  PNC   YC|NC 
(3)

 

where  C () is the standardized complementary cumulative normal distribution. Note
that at the low-intensity level PNC is zero and the distribution is then simply lognormal.
The parameters of the distribution yˆ ,  and PNC can be easily calculated from the
response results. For example, the 20-story structure collapsed in 13 cases out of 44 at the
0.27g intensity. Hence PNC is 0.71. The median and dispersion of the 31 non-collapsing
results are 5.7% and 0.84. With this information we can characterize the response results
quite accurately (see Shome and Cornell, 1999, for a discussion on the verification of the
fitting of this model). From Eq. 3 we can also determine the median (7.9%), the 16th
percentile (1.9%), and (if desired) an equivalent dispersion as per Eq. 1 (1.4).. These
estimates are less sensitive to the sample-to-sample variability of the response results.
Demand at Several Intensities
For performance based design or for annual drift hazard estimation we need to estimate
the demand at several or a continuum of intensities. In this case, instead of conducting
nonlinear analyses for a set of records scaled to each of several intensities, it is more
economical to carry out regression analysis of response results on spectral acceleration at
the first mode ( S a ) for a (smaller) number of records. The records should be selected (or
scaled) in such a way that the intensities are spread over the range of required intensities.
A conventional regression analysis of the non-collapsing results gives the median ( ŷ ) as
function of S a and the conditional dispersion of Y given Sa (  ). Here regression analysis
of maximum interstory drifts gives  st  0.16  ( S a ) 0.95 and   0.44 . See Shome and
Cornell (1999) for details of the records, the regression model and a discussion of the
regression results. The homoskedastic (constant  ) assumption may need modification
in some cases.
Binary Regression Analysis
More challenging is the calculation of PNC as a function of S a to estimate the seismic
demand at several intensities. This is accomplished by binary regression analysis of the
response data. In this case the response data are represented by a binary variable, NC.
This variable is defined to be 1 when the structure does not collapse and is 0 when it
collapses. The independent variable is the record’s first-mode spectral acceleration ( S a ).
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Because NC is an “indicator variable”, the probability that NC=1 for a given S a is also
the expected value of NC given S a , i.e., P(nc | sa ) is the regression of NC on S a . We
assume the following model of P(nc | sa ) (see Shome and Cornell, 1999, for details):
C
s 
P(nc | sa )   a 
for sa  sa 0 and where 0  P(nc | sa )  1
(4)
 sa 0 
where s a 0 and  C are the regression coefficients. By the method of maximum likelihood
we estimate sa 0  0.22 g and  C = -2.8. When spectral acceleration is less than 0.22g, the
likelihood of collapse of the structure is zero.
The median drift versus S a as calculated from Equation 4 is shown in Figure 2. This plot
can be called the seismic demand curve of the 20-story structure. Following one popular
definition of performance-based assessment one might evaluate the performance of the
structure by comparing this (median) drift of the structure at each of several different
intensities with the allowable drift at that intensity. For example, at the site under
consideration the 2% in 50 year or 2500-year intensity of ground motion is 0.19 g at 4 sec
period. The seismic drift demand at that intensity from Figure 2 is 3.5% which is less
than the FEMA-273 (1996) 5% allowable drift for seismic rehabilitation of steel moment
frames. Hence this performance objective of the structure is deemed to be satisfactory.
Note that the consideration of the likelihood of collapse is enters only when the required
intensity is greater than s a 0 ; otherwise we can estimate the seismic demand from the
regression results conditioned on no collapse. We find in Figure 2 that when sa is greater
than s a 0 the median drift increases rapidly with small increase in S a , indicating a severe
”softening” of the structure. Figure 2 thus demonstrates the potential importance of
consideration of likelihood of collapse in seismic demand calculations. In this regard
more realistic models of such tall steel frames should include not only the stiffening
benefits of interior gravity frames but also the less-than-perfectly- ductile behavior of the
connections (e.g., Luco and Cornell, 2000).
Conclusion
In this study we have shown the demand calculation procedure for structures when the
likelihood of collapse is significant. The conventional procedure of the estimation of a
central value of responses does not work in the case of collapse. Although the simple
counted median can be used for this purpose, we can make a better estimation by using
the three-parameter probability distribution model introduced in this study. The
additional advantage of this model is that it simplifies the demand calculation procedure
for multi-level performance evaluation. This model also facilitates the annual demand
hazard calculation.
References:
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Bazzurro, P., Cornell, C.A., Shome, N., and Carballo, J.E., “Three Proposals for Characterizing MDOF
Nonlinear Seismic Response@, Journal of Structural Engineering, ASCE, Vol. 124, No. 11, pp.
1281-1289, November, 1998.
Bazzurro, P., and Cornell, C.A., AOn Disaggregation of Seismic Hazard@, Bulletin of Seismological Society
of America, B.S.S.A.., Vol. 89, No.2, pp. 501-520, April, 1999.
DRAIN-2DX 1993), “DRAIN-2DX: Basic Program Description and User Guide”, Report No
UCB/SEMM-93/ 17, by Prakash, V., Powell, G. H., and Campbell, S., University California,
Berkeley.
Gupta, A. (1999), “Seismic Demands for Performance Evaluation of Steel Moment Resisting Frame
Structures”, Ph.D. Thesis, Stanford University, Stanford.
Luco, Nicolas and Cornell, C.Allin, “Effects of Connection Fractures on SMRF Seismic Drift Demands”,
ASCE Journal of Structural Engineering, Vol. 126, No. 1, pp.127-136, January, 2000.
Shome, N., Cornell, C. A., Bazzurro, P., and Carballo, J. (1998), “Earthquake, Records, and Nonlinear
MDOF Responses”, Earthquake Spectra, 14(3), 469-500.
Shome, N. and Cornell, C. A. (1999), “Probabilistic Seismic Demand Analysis of Nonlinear Structures”,
RMS Report-35, Reliability of Marine Structures Group, Stanford University, Stanford.
Silva, W. J. (1995), “Strong Motion Catalogue”, Pacific Engineering and Analysis, El Cerrito, CA.
UBC (1994), :Uniform Building Code”, International Conference of Building Officials, Whittier, CA.
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40%
Probability (%)
35%
30%
25%
20%
15%
10%
5%
0%
0-2
2-4
4-6
6-8
8-10
10-12 12-14
coll
Drift (%)
Figure 1. Frequency distribution of maximum story
drift from the scaled results at the 0.27g intensity
level.
0.3
0.25
Sa (g)
0.2
0.15
0.1
0.05
Neglecting the Likelihood of Collapse
Considering the Likelihood of Collapse
0
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
Maximum Interstory Drift (%)
Figure 2. Variation of the median maximum
interstory drift with the first-mode spectral
acceleration.
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