estimation and impacts of model parameter correlation for...
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Estimation and Impacts of Model Parameter1
Correlation for Seismic Performance Assessment of2
Reinforced Concrete Structures3
B.U. Gokkayaa,⇤, J.W. Bakera, G.G. Deierleina4
aCivil & Environmental Engineering Department and John A. Blume Earthquake5
Engineering Center, Stanford University, Stanford, CA 943056
Abstract7
Consideration of uncertainties, including stochastic dependence among8
uncertain parameters, is known to be important for estimating seismic risk9
of structures. In this study, we characterize the dependence of modeling10
parameters that define the nonlinear response at a component level and the11
interactions of multiple components associated with a system’s response. We12
utilize random e↵ects regression models, and a component test database with13
multiple tests conducted by di↵ering research groups, to estimate correlations14
among parameters. Groups of tests that are conducted in similar conditions,15
and are investigating the impacts of particular properties of components that16
can e↵ectively represent di↵erent locations in a structure, are suitable for this17
estimation approach. Correlation coe�cients from these regression models,18
reflecting statistical dependency among properties of components tested by19
individual research groups, are assumed here to reflect correlations associ-20
ated with multiple components in a structure. To illustrate, correlations for21
reinforced concrete element parameters are estimated from a database of re-22
inforced concrete beam-column tests, and then used to assess the e↵ects of23
⇤Corresponding author
Email addresses: [email protected] (B.U. Gokkaya),
[email protected] (J.W. Baker), [email protected] (G.G. Deierlein)
Preprint submitted to Structural Safety March 22, 2017
correlations on dynamic response of a frame structure. Increased correla-24
tions are seen to increase dispersion in dynamic response and produce higher25
estimated probabilities of collapse. This work provides guidance for charac-26
terization of parameter correlations when propagating uncertainty in seismic27
response assessment of structures.28
Keywords: correlation, modeling uncertainty, random e↵ects regression,29
uncertainty propagation, collapse30
1. Introduction31
Performance-based earthquake engineering enables quantification and prop-32
agation of uncertainties in a probabilistic framework to make robust estima-33
tions of seismic risk and loss of structures. Quantification and propagation34
of ground motion uncertainties have received significant attention in the re-35
search community, but an important and somewhat less-explored topic is36
uncertainty in structural modeling (e.g., Bradley, 2013). The uncertainties37
related to use of idealized models and analysis methods, as well as uncertain-38
ties in a model’s parameters, influence assessments of the seismic reliability39
of a structure. Explicit quantification of uncertainties and characterization40
of dependence among the random model parameters are essential for propa-41
gating these uncertainties when assessing seismic performance.42
While quantification of model parameter uncertainties is relatively well43
studied, stochastic dependence among model parameters has received very44
little attention, in large part due to scarcity of appropriate calibration data.45
When it has been assessed or considered in assessments, it is typically in46
the form of correlation coe�cients. Where the random variables have a47
2
multivariate normal distribution, correlations provide a complete description48
of their dependence. They are also useful in first-order and other approximate49
reliability assessments.50
The current state-of-the-art in seismic reliability analysis is to use ex-51
pert judgment in quantifying the correlation structure of analysis model52
parameters. Haselton (2006) used judgment-based correlation coe�cients53
when considering model parameter uncertainty in assessing collapse risk of54
reinforced concrete structures, and showed that variability in collapse ca-55
pacity was strongly influenced by the correlation assumptions. Liel et al.56
(2009), Celarec and Dolsek (2013), Celik and Ellingwood (2010) and Pinto57
and Franchin (2014) all used assumed correlations among modeling parame-58
ters when propagating modeling uncertainty for seismic performance assess-59
ment of reinforced concrete structures.60
Although the e↵ects of correlations among random variables on system61
reliability are well known, few researchers have used observational data to62
quantify dependence. Idota et al. (2009) assessed the correlation of strength63
parameters for steel moment resisting frames using steel coupon tests from64
production lots. Vamvatsikos (2014) used those results to study the e↵ects of65
correlation of components at di↵erent locations in a building on its dynamic66
response. We are aware of no other studies that directly estimate correlations67
in component-level or phenomenological modeling parameters in order to68
study seismic collapse risk.69
In this study, we estimate the correlation structure of modeling param-70
eters that define a component’s nonlinear cyclic response, and study the71
interaction of di↵erent components on system-level dynamic response. Ran-72
3
dom e↵ects regression is used with a database of reinforced concrete column73
tests to infer correlation structure of parameters defining a concentrated plas-74
ticity model. The database is composed of reinforced concrete column tests75
performed by multiple research groups. Correlation coe�cients, representing76
statistical dependency among parameters within a set of tests conducted by a77
research group, are assumed to reflect dependency among parameters corre-78
sponding to components throughout a structural system. We then use the es-79
timated correlations to assess the e↵ects of correlations on dynamic response80
of a four-story reinforced concrete frame building, and to explore potential81
simplified approaches for representing parameter correlations. Although the82
reported correlation results are for reinforced concrete model parameters, the83
presented framework can be applied for other types of materials or models.84
2. Probabilistic Seismic Performance Assessment85
We use the probabilistic performance-based earthquake engineering method-86
ology to assess structural performance (e.g., Krawinkler and Miranda, 2004;87
Deierlein, 2004). Nonlinear structural analyses are run using a suite of ground88
motions to propagate uncertainties related to ground motion variability and89
seismic hazard. The results from structural analyses are then related to the90
risk of collapse and other damage states of interest.91
The concentrated plasticity model by Ibarra et al. (2005), which has92
been frequently used to simulate sidesway collapse in frame structures (e.g.,93
Zareian and Krawinkler, 2007; Eads et al., 2013), is used in this study to94
model component response. Specific attention is given to the correlation of95
model parameters used to define plastic hinges in seismic resisting moment96
4
frames. Phenomenological concentrated plasticity models are well-suited for97
modeling collapse of structures (Deierlein et al., 2010). However, model pa-98
rameters that define concentrated plasticity models are generally related to99
physical engineering parameters by empirical relationships. Modeling uncer-100
tainty becomes more pronounced for collapse response simulations than for101
elastic or mildly nonlinear simulations, due to both the relatively limited102
knowledge of parameter values and the highly nonlinear behavior associated103
with collapse.104
The concentrated plasticity model has a trilinear ”backbone curve,” shown105
in Figure 1, defined by five parameters: capping plastic rotation (✓cap,pl), ef-106
fective sti↵ness (secant sti↵ness up to 40% of the component yield moment107
EIstf ), yield moment (My), capping moment (Mc), and post-capping rota-108
tion (✓pc). A sixth parameter, �, defines the normalized energy dissipation109
capacity, which controls the rate of deterioration (under cyclic loading) of110
basic strength, post-capping strength, unloading sti↵ness, and accelerated111
reloading sti↵ness.112
The uncertainty in these modeling parameters is large, as estimated by113
a predictive model for these parameter values that will be discussed further114
below (Haselton et al., 2008). For a given column design, the parameters115
associated with elastic and peak strengths are moderately uncertain: the116
EIstf/EIg, My and Mc/My parameters have logarithmic standard deviations117
of 0.28, 0.3 and 0.1, respectively. The parameters associated with more non-118
linear displacement capacities and cyclic deterioration, however, are highly119
uncertain: the ✓pc, ✓cap,pl and � parameters have logarithmic standard devi-120
ations of 0.73, 0.59 and 0.51, respectively.121
5
My
Mc
Chord Rotation
Mo
me
nt
θpc
θcap,pl
EIstf
Figure 1: Ibarra et al. (2005) model for moment versus rotation of a plastic hinge
in a structure. The model parameters of interest are labeled.
In this study, we aim to characterize correlation of these model pa-122
rameters in a structure. Parameter correlations are grouped into within-123
component and between-component correlations. The former refers to cor-124
relations among modeling parameters that define response of a single com-125
ponent, whereas the latter refers to correlations among parameters from dif-126
fering components, as illustrated Figure 2. This distinction is useful because127
within-component correlations can be estimated from tests of individual com-128
ponents, while between-component correlations require more e↵ort to esti-129
mate. Between-component correlations are caused by similarities throughout130
a structure in the properties of structural materials, and member geometries131
and details. If these similarities are not captured in estimating mean val-132
ues for model parameters, they will result in stochastic dependence of the133
resulting component model parameters.134
Incremental dynamic analyses (IDA) involves performing nonlinear dy-135
6
Component i: { θcap,pl , My , ... }
Component j: { θcap,pl , My , ... }
Within-component
Between-component
Figure 2: Illustration of correlation within a component and correlation between
components in a structure.
namic analysis using multiple ground motions scaled to particular ground136
motion intensity measure (IM) levels (Vamvatsikos and Cornell, 2002). Seis-137
mic capacity is the IM value causing dynamic instability in the structure.138
This capacity is random due to the uncertain nature of a ground motion with139
a given IM level and uncertainties associated with structural performance.140
Its distribution is quantified by a collapse fragility function defining the prob-141
ability of collapse (C) at a given IM level, im (P (C|IM = im)). Below, the142
fragility function will be estimated either by an empirical distribution or by143
7
fitting a lognormal cumulative distribution function:144
P (C|IM = im) = �
✓ln (im/✓)
�
◆(1)
where � () is a standard normal cumulative distribution function, ✓ is the145
median and � is the logarithmic standard deviation (or “dispersion”) of the146
distribution. Values for ✓ and � will be estimated and reported below.147
The mean annual frequency of collapse (�c) is obtained by integrating a148
collapse fragility function with a ground motion hazard curve (Ibarra and149
Krawinkler, 2005), as given in Equation 2.150
�c =
Z 1
0
P (C|IM = im)
����d�IM(im)
d(im)
���� d(im) (2)
where �IM(im) is the mean annual rate of exceeding the ground motion im151
and d�IM (im)d(im)
defines the slope of the hazard curve at im.152
Fragility functions corresponding to alternative limit states, such as ex-153
ceeding a particular story drift ratio, sdr, can be also obtained from incre-154
mental dynamic analysis results. Using IDA, ground motions are scaled until155
the structure displays a story drift ratio of sdr and the fragility function is156
obtained. This function can be integrated with the seismic hazard curve157
in a similar fashion to Equation 2 to estimate the mean annual frequency158
(�SDR�sdr) of exceeding a given limit state.159
3. Assessing Correlations of Model Parameters160
The correlation assessment procedure described in this section requires161
two inputs: (1) a set of observed parameter values from test data where there162
are groups of components analogous to a set of components in a building,163
8
and (2) predictive equations that estimate means and standard deviations of164
those parameters values based on component properties such as dimensions165
and material strengths. With those two inputs, a mixed e↵ects analysis166
can be performed to estimate correlations as described here. It would also167
be natural to start with only the observed parameter values, and fit the168
predictive equations at the same time as the correlations are estimated. To169
illustrate, we consider the case of concrete beam-columns with the lumped-170
plasticity component model described above; in this case predictive equations171
for means and standard deviations are already available, so we adopt those172
equations and focus only on the estimation of correlations.173
3.1. Observed Parameter Values174
The six parameters illustrated in Figure 1 are treated here as random175
variables. Haselton et al. (2008) estimated values for these parameters for176
255 column tests from the Pacific Earthquake Engineering Research Center177
Structural Performance Database (Berry et al., 2004). This database pro-178
vides force-displacement histories from cyclic and lateral-load tests, along179
with information related to reinforcement, column geometry, test configura-180
tion, axial load, and failure type for each column.181
Haselton et al. (2008) considered rectangular column tests whose failure182
modes were either flexure or combined flexure and shear. The model param-183
eters were calibrated so that a cantilever column, with an elastic element184
and a concentrated plastic hinge at the base, has behavior that matches the185
corresponding experimental force-displacement data. The study authors fil-186
tered the data to remove outliers and parameters whose values could not187
be estimated for a given test (typically these were parameters characteriz-188
9
ing post-peak cyclic deterioration response, in cases where a test did not189
induce this behavior). The total number of estimated parameter values are190
232, 197, 255, 233, 65 and 223 for ✓cap,pl, EIstf/EIg, My, Mc/My, ✓pc and �,191
respectively.192
The 255 column tests used for the calibration were conducted by 42 di↵er-193
ent research laboratories, referred to here as “test groups”. The test groups194
are listed in Table 6, along with information for each regarding the varia-195
tion among tests in member dimensions, concrete strength (f 0c), longitudinal196
yield strength (fy), axial load ratio, and area of longitudinal and transverse197
reinforcement.198
3.2. Evidence of Parameter Correlations199
Using the observed parameter values described above, we compute pre-200
diction residuals by comparing the observations to model predictions:201
ln�y
kij
�= ln
�y
kij
�+ "
kij (3)
where subscripts i and j represent the test group and test number, respec-202
tively, and the superscript k indicates the random variable of interest. Ran-203
dom variable k from the test specified by i and j is associated with observed204
value y
kij, predicted value y
kij, and residual "kij.205
Predicted values are obtained in this study from the empirical equations of206
Haselton et al. (2008) and Panagiotakos and Fardis (2001). These equations207
relate column design details to the six model parameters using equations that208
are based on regression analysis of observed data and judgment on expected209
behavior. Haselton et al. (2008) provide a full and a simplified equation for210
10
some of the model parameters; we use the full equations if both are provided.211
The predictive model studies found that parameter values are generally log-212
normal, so a logarithmic transformation is used in equation 3 (and in the213
original predictive models) to obtain normally distributed residuals.214
For each model parameter, the residuals from each group of tests are215
plotted against each other, and a subset of the data are shown for illustra-216
tion in Figure 3. Each test group is denoted by a specific symbol and color.217
Grouping of these residuals by test group implies the presence of correlated218
residuals; this is most evident for My in Figure 3c. Here it is observed that219
tests from group 1 (TG1) have negative residuals, implying that the My val-220
ues of the tests conducted in that test group are consistently overestimated221
by the predictive equation. Conversely, tests from group 3 (TG3) have posi-222
tive residuals indicating an underestimation of observations by the predictive223
equation.224
The grouping of residuals within test groups is not surprising, considering225
that the tests have common features whose e↵ects are not captured by the226
predictive equations. Reviewing Table 6 in the Appendix, we observe that: 1)227
The majority of groups have tests with similar specimen dimensions. 2) Steel228
yield strength and area ratio of longitudinal reinforcing steel are constant in229
approximately three-quarters of the test groups. 3) The major di↵erences230
among the tests within each group are the level of axial load and transverse231
reinforcement. While not explicitly documented, we also expect that tests232
from a single laboratory would have similarities in environmental conditions,233
workmanship, and other factors that might influence the component behav-234
ior. These features within each test group are similar to features we would235
11
−1 0 1
−1
0
1
ε k
εk
θcap,pl
(a)
−0.5 0 0.5
−0.6
0
0.6
ε k
εk
EIstf/EI
g
(b)
−1 0 1
−1
0
1
ε k
εk
My
(c)
TG1
TG3
Figure 3: Example model parameter residuals, ("k), plotted against the residuals
of other tests within the test group to which they belong for a) ✓cap,pl, b) EIstf/EIg,
c) My. A subset of data is shown for illustrative purposes.
expect to see among components located throughout a real-world building.236
When modeling seismic performance of a real structure, we would use the237
same predictive equations discussed above to predict parameter values for a238
numerical model. Because those predictive models rely on the limited set of239
column properties, we would expect components in a real building to also240
behave in a correlated (but not perfectly dependent) manner. By assuming241
that a group of components in a single laboratory’s tests corresponds to a242
group of components in a building, we can utilize statistical analysis of this243
test data to quantitatively estimate parameter correlations for components244
within a real building. While the correspondence between test groups and245
real-world structures is not strictly true, the authors believe it is reasonable,246
and this assumption provides a unique opportunity to estimate correlations247
that are otherwise nearly impossible to observe. We will keep in mind the248
12
approximate nature of this correspondence when evaluating the numerical249
results below.250
3.3. Random E↵ects Regression251
The observed clustering of residuals within a test group motivates the252
use of random e↵ects regression to study the correlations among model pa-253
rameters. Random e↵ects models are used when at least one of the response254
variables is categorical. The discrete levels for the categorical variable are255
termed the “e↵ects” in the model, and the qualifier random implies that the256
observed levels represent a random sample from a population and do not257
contain all possible levels (Searle et al., 2009; Pinheiro and Bates, 2010).258
A one-way random e↵ects model is applied to residuals from Equation259
3 to assess the correlation structure of the model parameters, using the R260
software package (Team, 2014). The test groups are treated as a random261
e↵ect, and logarithmic residuals of each random variable, "kij, are considered262
without any further transformation, leading to the following equation:263
ln�y
kij
�� ln
�y
kij
�= "
kij
= µ
k + ↵
ki + "
kij
(4)
where where µ
k is an intercept indicating the mean of the data, and ↵
k and264
"
k represent between- and within-test-group variability, respectively. The265
↵
k and "
k terms are independent random variables with zero means and266
variances �
2k and ⌧
2k , respectively. These variances are estimated from the267
regression procedure. From Equation 4 and the above definitions, it follows268
that the variance of a model parameter, k, is �2k + ⌧
2k .269
13
From Equation 4, the covariance of the logarithms of the model parame-270
ters k and k
0 within a component j is given by:271
cov
⇣ln�y
kij
�, ln
⇣y
k0
ij
⌘⌘= cov("kij, "
k0
ij )
= cov(µk + ↵
ki + "
kij, µ
k0 + ↵
k0
i + "
k0
ij )
= corr(↵ki ,↵
k0
i )�k�k0 + corr("kij, "k0
ij )⌧k⌧k0
(5)
where cov(·, ·) and corr(·, ·) refer to covariance and correlation, and by def-272
inition cov(↵ki ,↵
k0j ) = corr(↵k
i ,↵k0i )�k�k0 and cov("ki , "
k0j ) = corr("ki , "
k0i )⌧k⌧k0 .273
Correlation of model parameters within a component is then given by:274
corr(ln�y
kij
�, ln
⇣y
k0
ij
⌘) =
corr
�↵
ki ,↵
k0i
��k�k0 + corr
�"
kij, "
k0ij
�⌧k⌧k0p
�
2k + ⌧
2k
p�
2k0 + ⌧
2k0
(6)
The covariance of the logarithms of the model parameters k and k
0 between275
components j and j
0 is given by:276
cov(ln�y
kij
�, ln
⇣y
k0
ij0
⌘) = cov(µk+↵
ki +"
kij, µ
k0+↵
k0
i +"
k0
ij0) = corr(↵ki ,↵
k0
i )�k�k0
(7)
where the cov("kij, "k0ij0) = 0 since "
kij and "
k0ij0 are independent. Correlation of277
model parameters between components can then be shown to equal278
corr(ln�y
kij
�, ln
⇣y
k0
ij0
⌘) =
corr
�↵
ki ,↵
k0i
��k�k0p
�
2k + ⌧
2k
p�
2k0 + ⌧
2k0
(8)
When assessing the correlation of the same model parameter between279
components, corr(↵ki ,↵
ki ) = 1 and Equation 8 simplifies to280
corr(ln�y
kij
�, ln
�y
kij0�) =
�
2k
�
2k + ⌧
2k
(9)
14
Table 1: Between and within test group standard deviations obtained from ran-
dom e↵ects regression.
�k ⌧k
p�
2k + ⌧
2k
✓cap,pl 0.41 0.44 0.59EIstfEIg
0.20 0.20 0.28
My 0.26 0.10 0.30
McMy
0.07 0.08 0.10
✓pc 0.24 0.69 0.73
� 0.20 0.46 0.51
3.4. Regression Results281
Estimated between-component (�k) and within-component (⌧k) standard282
deviations for the example database are shown in Table 1. Table 2 shows283
the correlation coe�cients obtained using Equations 5 to 9. Table 3 shows284
the same correlation coe�cients obtained after rounding to one significant285
figure, reflecting the approximate nature of the way in which we are using286
these data and the finite sized database used here. The correlations shown287
in Table 3 are used in the rest of the paper.288
15
Table
2:Initialcorrelationcoe�cientsobtainedfrom
random
e↵ectsregression.
Componenti
Componentj
✓ cap,pl i
⇣EI s
tf
EI g
⌘ iM
yi
⇣M
cM
y
⌘ i✓ p
c i� i
✓ cap,pl j
⇣EI s
tf
EI g
⌘ jM
yj
⇣M
cM
y
⌘ j✓ p
c j� j
Componenti
✓ cap,pl i
1.0000
-0.0183
0.0578
0.2538
0.2083
-0.0260
0.6839
0.0106
0.0277
0.0975
0.0533
-0.0202
⇣EI s
tf
EI g
⌘ i1.0000
0.1354
-0.1018
0.0375
0.0799
0.6853
0.0612
-0.0950
-0.0305
0.0379
Myi
1.0000
0.2838
0.1067
0.0722
0.9263
0.2482
0.0951
0.0549
⇣M
cM
y
⌘ j1.0000
0.0077
0.1681
0.6728
0.0415
0.0192
✓ pc i
(sym.)
1.0000
0.2195
(sym.)
0.3466
0.0357
� i1.000
0.4102
16
Table
3:Finalcorrelationcoe�cientsareobtainedafterroundingtoonesignificantfigure.
Com
pon
enti
Com
pon
entj
✓
cap,pl i
⇣EI s
tf
EI g
⌘ iM
y i
⇣M
cM
y
⌘ i✓
pc i
�
i✓
cap,pl j
⇣EI s
tf
EI g
⌘ jM
y j
⇣M
cM
y
⌘ j✓
pc j
�
j
Componenti
✓
cap,pl i
1.0
0.0
0.1
0.3
0.2
0.0
0.7
0.0
0.0
0.1
0.1
0.0
⇣EI s
tf
EI g
⌘ i1.0
0.1
-0.1
0.0
0.1
0.7
0.1
-0.1
0.0
0.0
M
y i1.0
0.3
0.1
0.1
0.9
0.2
0.1
0.1
⇣M
cM
y
⌘ j1.0
0.0
0.2
0.7
0.0
0.0
✓
pc i
(sym
.)1.0
0.2
(sym
.)0.3
0.0
�
i1.0
0.4
17
We note that rounding of the correlation coe�cients, and estimation of289
correlations from data with missing values, can result in a correlation matrix290
without the required positive semi-definiteness property. Although Table 3291
produces positive definite matrices, in our initial calculations some violation292
of positive semi-definiteness was observed. In such cases, minor changes293
can be made to transform the correlation matrix into a positive definite one294
(Jackel, 2001).295
We observe that within a component (i.e., the left half of Table 3), corre-296
lations of model parameters are rather small; the largest coe�cient being 0.3,297
which corresponds to the correlations between Mc/My and My, and Mc/My298
and ✓cap,pl. These values suggest moderate interactions between strength pa-299
rameters and hardening behavior. Small interactions are observed between300
the parameters defining post-capping cyclic behavior within a component301
(e.g., � with Mc/My and ✓pc).302
Between components, like parameters have larger correlation coe�cients303
(i.e., the diagonal components on the right half of Table 3). We see that My,304
✓cap,pl, EIstf/EIg and Mc/My have correlations of 0.7 or greater. In the set-305
ting of a structure, this implies that values of these parameters across compo-306
nents will tend to take similar values; note that Haselton and Deierlein (2007)307
and Liel et al. (2009) assumed perfect correlation between like-parameters308
across components.309
We also observe that correlations of di↵erent model parameters between310
components are small (i.e., the o↵-diagonal terms in the right half of the311
table). This is expected, given that the correlations of these parameters312
within a component are also small. There is even a small negative correlation313
18
between EIstf/EIg and Mc/My. This is likely to be a numerical artifact, as314
there is no clear physical reason why such a correlation would be negative.315
This artifact also motivates the decision to retain only one significant figure316
in the correlation estimates.317
4. Impacts of Parameter Correlations on Dynamic Structural Re-318
sponse319
4.1. Case Study Structure320
A reinforced concrete special moment frame structure is considered here,321
to demonstrate the impact of parameter correlations and evaluate potential322
model simplifications for structures with many uncertain parameters. The323
building was designed by Haselton (2006) for a high seismicity site in Cali-324
fornia in accordance with 2003 IBC and ASCE 7-02 provisions (IBC, 2003;325
American Society of Civil Engineers, 2002). Here we assume the building to326
be located at the same Los Angeles site as considered in its original design.327
One three-bay four-story frame of the building is modeled, with a total of328
12 beam and 16 column elements. The structural system is modeled using329
the concentrated plasticity approach described above, in which elements of330
the frame are modeled using elastic elements with rotational springs at the331
ends. A Rayleigh damping of 3% is defined at the first and third mode pe-332
riods of the structure, and P-� e↵ects are modeled using a leaning column.333
The fundamental period of the structure is 0.94 s. The Open System for334
Earthquake Engineering Simulation platform is used to analyze the struc-335
ture (OpenSEES, 2015). The FEMA-P695 far-field set of 44 ground motion336
components is used for structural response simulations (FEMA, 2009).337
19
Monte Carlo simulation is used for propagating uncertainties related to338
modeling and ground motion variability (Kalos and Whitlock, 2009). The339
six parameters mentioned previously are treated as random, with marginal340
means and standard deviations as predicted by Haselton et al. (2008), and341
correlations as defined below. Further, equivalent viscous damping and col-342
umn footing rotational sti↵ness are assumed to be random, with logarithmic343
standard deviation values of 0.6 and 0.3, respectively (Haselton, 2006; Hart344
and Vasudevan, 1975; Porter et al., 2002), and to be independent of the345
other parameters. A multivariate normal distribution is assumed for the346
logarithms of all parameters except Mc/My. Since, by definition Mc/My is347
always greater than 1, we use a one-sided truncated normal distribution for348
this parameter. Because only one frame of the structure is modeled, it is349
implicitly assumed that the parameters for frames in a given direction are350
fully correlated.351
Table 4 lists four correlation models considered in the following analyses.352
As the name implies, the No Correlation model assumes all parameters in353
the building to be uncorrelated. This model has 170 random variables (six354
parameters for each of 28 elements, plus damping and foundation sti↵ness355
parameters). The Partial Correlation A model uses correlation coe�cients356
from Table 3 for all within- and between-component correlations, and also357
has 170 random variables. In Partial Correlation B, Table 3 is used to define358
correlations within a component and correlations of beam-to-column com-359
ponents. Column-to-column and beam-to-beam parameters are assumed to360
be fully correlated (e.g., all column components for a given model realiza-361
tion have the same parameters). These assumed full correlations reduce the362
20
Table 4: Correlation models used with Monte Carlo simulations. “0”, “P” and “1”
refer to the cases of No Correlation, Partial Correlation and Perfect Correlation,
respectively.
Within- Between-component E↵ective
Model Name component Column-
to-Column
Beam-to-
Beam
Beam-to-
Column
# of
R.V.s
No Correlation 0 0 0 0 170
Partial Correlation A P P P P 170
Partial Correlation B P 1 1 P 16
Full Correlation 1 1 1 1 3
e↵ective number of random variables for this model to 14 (six beam param-363
eters, six column parameters, damping and foundation sti↵ness). In the Full364
Correlation model all of the element parameters are assumed to have per-365
fect correlation, such that there are e↵ectively three random variables (one366
component parameter, damping and foundation sti↵ness).367
For each correlation model, we simulate 4400 realizations of model pa-368
rameters from their joint distribution, each of which are randomly matched369
with one ground motion. Incremental dynamic analysis is then conducted370
to scale each ground motion up until structural collapse is observed for the371
given model realization. A maximum story drift ratio (SDR) � 0.1 is as-372
sumed to indicate structural collapse. Ground motion IM values are defined373
as 5%-damped first-mode spectral acceleration, Sa(0.94s).374
21
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
SDR
Me
dia
n S
a (
0.9
4 s
) (g
)
(a)(a)(a)(a)
No CorrelationPartial Correlation AFull CorrelationMedian Model
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
SDR
(b)(b)(b)
Partial Correlation APartial Correlation B
0 0.02 0.04 0.06 0.08 0.10
0.1
0.2
0.3
0.4
0.5
0.6
SDR
Dis
pe
rsio
n
Me
dia
n S
a (
0.9
4 s
) (g
)D
isp
ers
ion
(c)(c)
No CorrelationPartial Correlation AFull CorrelationMedian Model
0 0.02 0.04 0.06 0.08 0.10
0.1
0.2
0.3
0.4
0.5
0.6
SDR
(d)
Partial Correlation APartial Correlation B
Figure 4: IDA results obtained using di↵erent correlation models a) median
IDA response for varying correlation levels, b) median IDA response for Partial
Correlation models, c) dispersion in IDA curves for varying correlation levels, and
d) dispersion in IDA curves for Partial Correlation models.
4.2. Results375
Figures 4.a and 4.b show the median IDA curves from the four correlation376
models, along with results for a structure with median model parameters (i.e.,377
no parameter uncertainty). There are remarkably small di↵erences among378
22
the median IDA curves. A small di↵erence between the No Correlation and379
Full Correlation cases is observed for SDR � 0.03, with a di↵erence of 7%380
at SDR = 0.1.381
Figures 4.c and 4.d show the dispersions in the IDA curves, and these vary382
more significantly. The Partial Correlation A and No Correlation models383
yield similar variability for SDR< 0.03, and at SDR= 0.1, the di↵erence in384
dispersion values for these two cases is 12%. At SDR= 0.1, the di↵erence in385
dispersion between No Correlation and Full Correlation models is 47%. The386
Median model consistently underestimates dispersion, where for SDR= 0.1,387
a di↵erence of 17% is observed between dispersion values of Median and388
Partial Correlation models. The Partial Correlation A and B models have389
very similar medians and dispersions.390
The mean annual frequency of collapse, �c, is obtained by integrating391
the empirical collapse fragility curves with the seismic hazard curve of the392
Los Angeles site using equation 2. Fragility functions and corresponding and393
�SDR�sdr values are also obtained for alternative values of sdr. Figure 5394
shows �SDR�sdr with respect to sdr using the assumed correlation models,395
where the �SDR�sdr di↵er for SDR values greater than approximately 0.03.396
The plots show, for example, that the No Correlation and Partial Correlation397
cases produce nearly identical �SDR�sdr. On the other hand, the �SDR�sdr398
for the Full Correlation case is 30% to 110% higher than the No Correlation399
model for drift values of 0.05 and 0.1, respectively.400
Figure 6 shows empirical collapse cumulative distribution functions for401
the structure obtained using the considered correlation models. At smaller402
Sa(0.94s)) levels, as the correlations among parameters increase, the struc-403
23
0 0.02 0.04 0.06 0.08 0.110
−4
10−3
10−2
sdr
λS
DR
≥ s
dr
(a)(a) No CorrelationPartial Correlation AFull CorrelationMedian Model
0 0.02 0.04 0.06 0.08 0.110
−4
10−3
10−2
sdr
λS
DR
≥ s
dr
(b) Partial Correlation APartial Correlation B
Figure 5: Mean annual frequency of exceedance of maximum story drift ratio
using the considered correlation models.
ture has a higher probability of collapse. As expected, the median model404
provides smaller probabilities of collapse, especially for smaller ground mo-405
tion intensities, leading to unconservative estimates of collapse risk. Note406
that since median collapse capacity is higher for the median model, it would407
result in slightly larger collapse margin ratio (i.e., the ratio of the median408
collapse capacity to Maximum Considered Event (MCE) intensity) (FEMA,409
2009), which can be misleading from a collapse safety point of view. The No410
Correlation and Partial Correlation models have similar lower tail behavior411
and only di↵er at higher IM levels.412
Table 5 summarizes the counted median and logarithmic standard devia-413
tion (�) of collapse capacities obtained using alternative correlation models,414
along with the associated collapse rates. Any di↵erences observed at the415
lower tail of the fragility curve due to increasing levels of correlation trans-416
late into pronounced di↵erences in �c estimates. For example, there is a417
24
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Pco
llap
se
(a)(a)
No CorrelationPartial Correlation AFull CorrelationMedian Model
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Sa (0.94 s) (g)Sa (0.94 s) (g)
Pco
llap
se
(b)
Partial Correlation APartial Correlation B
Figure 6: Empirical cumulative distribution functions obtained using di↵erent
correlation models
factor of 2.4 di↵erence between �c estimates obtained using the No Cor-418
relation and Full Correlation models. The similarity in lower tail collapse419
fragilities for No Correlation and Partial Correlation models leads to similar420
�c estimates using these models, since the lower tail of the collapse fragility421
function contributes the most to �c (Eads et al., 2013).422
Given the above results, we can make a few observations about the im-423
pact of these models, taking the Partial Correlation A case as a benchmark424
because it fully utilizes the previously estimated parameter correlations. The425
Partial Correlation B model is appealing, as it produces comparable results426
to the Partial Correlation A case, but reduces the number of modeled ran-427
dom variables; this is helpful when using reliability assessment procedures428
that scale in e↵ort with the number of random variables. The Full Cor-429
relation case further reduces the number of random variables, but with an430
apparent loss in accuracy for this case. The Median Model and No Correla-431
25
Table 5: Counted median and logarithmic standard deviation (�ln) of collapse
capacity, and mean annual frequency of collapse (�c), obtained using alternative
models.
Model Name Median � �c (⇤10�4)
Median Model 1.69 0.39 1.81
No Correlation 1.57 0.42 2.65
Partial Correlation A 1.67 0.48 2.75
Partial Correlation B 1.64 0.49 3.06
Full Correlation 1.69 0.67 6.39
tions cases are also simplified representations of the model, but they produce432
unconservative estimates of seismic collapse risk and so should be used with433
caution.434
The structure considered here was designed to have a regular strength and435
sti↵ness distribution over its height, and so the typical collapse mechanisms436
were not notably altered when considering No Correlation and Partial Corre-437
lation models. Although we did not investigate the influence of ductility and438
strength irregularities in detail, we expect that di↵erent results are likely to439
be obtained for buildings with strength irregularities, since presence of even440
partial correlations may enable the triggering of alternate modes of failure441
(e.g., creation of a story mechanism by simulation of weak column-strength442
parameters).443
26
5. Conclusions444
We have considered model parameter uncertainty in seismic performance445
assessment of structures, both in estimating parameter correlations and in446
quantifying the impacts of these correlations on building performance. We447
have characterized the dependence of modeling parameters that define cyclic448
inelastic response at a component level and the interactions of multiple com-449
ponents associated with a system’s response. Parameter correlations were450
estimated from component tests using random e↵ects regression on grouped451
tests of structural components. Variation in parameter values within and452
between test groups were incorporated as random e↵ects in the regression453
model, and statistical dependency between the estimated parameters were454
assessed.455
Dependence in the parameters defining a lumped-plasticity model for con-456
crete columns were estimated using a database of reinforced concrete beam-457
column tests. Correlation coe�cients from these regression models, reflecting458
statistical dependency among properties of components tested by individ-459
ual research groups, are assumed to reflect correlations among components460
within a given structure. The random treatment of research groups, com-461
bined with the aforementioned observations in the data set (i.e., similarity of462
column dimensions and di↵erences in axial load and transverse reinforcement463
in the tests), justified this assumption. We found that correlations between464
di↵ering parameters (both within and between components) have low corre-465
lation (correlation coe�cients from -0.1 to 0.3), while like parameters across466
components have higher correlations of as large as 0.9.467
The impact of these estimated parameter correlations on dynamic re-468
27
sponse of a four story reinforced concrete frame structure was then assessed,469
by performing Incremental Dynamic Analysis of the structure using Monte470
Carlo realizations of uncertain model parameters. Variations in correlation471
assumptions did not strongly influence median response, even for large drifts.472
Variability in correlation assumptions did, however significantly influence dis-473
persion in response estimates, especially at large drift levels associated with474
severe nonlinearity and collapse. Models considering uncorrelated and par-475
tially correlated parameters had similar collapse fragility functions at the476
critical lower tail, resulting in similar mean annual frequencies of collapse.477
Models assuming perfectly correlated parameters, however, had higher prob-478
abilities of collapse for low-intensity shaking; the perfectly correlated model479
had a mean annual frequency of collapse that was 2.4 times the frequency480
of collapse of the fully uncorrelated model (even though the parameters had481
the same marginal distributions in both cases). A slightly simplified model482
representation, with full correlation among beam-to-beam and column-to-483
column parameters (and partially correlated beam-to-column parameters),484
produced nearly identical results to the benchmark model with partial corre-485
lations in all parameters. This simplified model has significantly fewer unique486
random variables, and so is a promising approach for considering parameter487
correlations while also managing computational expense. In aggregate, these488
results provide further evidence that parameter correlations are an important489
consideration in seismic collapse safety assessments.490
The results presented here were for reinforced concrete components, but491
the framework allows these evaluations to be performed on any model with492
uncertain parameters that are estimated from experimental data. The cor-493
28
relation estimation approach requires a set of component tests with multiple494
tests that can be grouped and considered as having commonalities consis-495
tent with those among components in a given structure. Tests that are496
conducted in similar conditions, and are investigating the impacts of partic-497
ular properties of components, are most suitable for this approach. While498
the appropriateness of considering groups of tests to represent components499
throughout a structure will need to be evaluated on a case-by-case basis, this500
proposed approach o↵ers a unique solution to the otherwise vexing problem501
of estimating parameter correlations for studying the seismic reliability of502
buildings.503
6. Data and Resources504
The data for the reinforced concrete column tests are obtained from the505
PEER Structural Performance Database (http://nisee.berkeley.edu/spd/)506
and Professor Curt Haselton’s Reinforced Concrete Element Calibration Database507
(http://www.csuchico.edu/structural/researchdatabases/reinforced_508
concrete_element_calibration_database.shtml).509
7. Acknowledgements510
We thank Dr. Shrey Shahi and Professor Art Owen for providing valu-511
able feedback on mixed e↵ects modeling. This work was supported by the512
National Science Foundation under NSF grant number CMMI-1031722. Any513
opinions, findings and conclusions or recommendations expressed in this ma-514
terial are those of the authors and do not necessarily reflect the views of the515
National Science Foundation.516
29
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A. Appendix: Information on Test Groups Conducting Reinforced727
Concrete Column Tests728
This appendix provides summary data of the test groups of concrete com-729
ponent tests considered here. The variables in the table are defined as follows:730
“Dim.” is member dimensions, f 0c is concrete compressive strength, fy is re-731
inforcing steel yield strength, ALR is Axial Load Ratio, LRS is the ratio of732
Longitudinal Reinforcing Steel to section area, TRS is the ratio of Transverse733
Reinforcing Steel to section area.734
Table 6: Test groups conducting reinforced concrete column tests. “Y” and “-”
indicate when the tests within the test group have similar or di↵erent properties,
respectively. “N/A” is used when the group has only one test.
Are the properties similar among tests?
Test Group Reference # Tests Dim. f 0c fy ALR LRS TRS
1 Galeota et al. (1996) 24 Y Y Y - - -
2 Bayrak and Sheikh (2001) 16 - - - - - -
3 Pujol (2002) 14 Y - Y - Y -
4 Wight and Sozen (1973) 13 Y - Y - Y -
5 Matamoros (1999) 12 Y - - - - -
6 Thomson and Wallace
(1994)
11 Y - - - Y -
7 Atalay and Penzien (1975) 10 Y - - - Y -
8 Saatcioglu and Grira
(1999)
10 Y Y - - - -
9 Mo and Wang (2000) 9 Y - Y - Y -
10 Bayrak and Sheikh (1996) 8 Y - Y - Y -
11 Muguruma et al. (1989) 8 Y - Y - Y Y
12 Tanaka (1990) 8 - - - - - -
13 Sakai (1990) 7 Y Y - Y - -
14 Kanda et al. (1988) 6 Y - Y - Y Y
15 Legeron and Paultre
(2000)
6 Y - - - Y -
16 Paultre et al. (2001) 6 Y - Y - Y -
40
Table 6: Test groups conducting reinforced concrete column tests. “Y” and “-”
indicate when the tests within the test group have similar or di↵erent properties,
respectively. “N/A” is used when the group has only one test.
Are the properties similar among tests?
Test Group Reference # Tests Dim. f 0c fy ALR LRS TRS
17 Saatcioglu and Ozcebe
(1989)
6 Y - - - Y -
18 Takemura and Kawashima
(1997)
6 Y - Y Y Y Y
19 Xiao and Yun (2002) 6 Y - Y - Y -
20 Xiao and Martirossyan
(1998)
6 Y - Y - - -
21 Zhou et al. (1987) 6 Y - Y - Y -
22 Bechtoula (1985) 5 - - - - - -
23 Sugano (1996) 5 Y Y Y - Y -
24 Watson (1989) 5 Y - Y - Y -
25 Esaki (1996) 4 Y - Y - Y -
26 Gill (1979) 4 Y - Y - Y -
27 Soesianawati (1986) 4 Y - Y - Y -
28 Wehbe et al. (1999) 4 Y - Y - Y -
29 Ohno and Nishioka (1984) 3 Y Y Y Y Y Y
30 Sezen and Moehle (2002) 3 Y - Y - Y Y
31 Ang (1981) 2 Y - Y - Y -
32 Azizinamini et al. (1988) 2 Y - Y - Y -
33 Lynn et al. (1996) 2 Y - Y - - -
34 Lynn (2001) 2 Y - Y - Y Y
35 Ohue et al. (1985) 2 Y - - - - Y
36 Ono et al. (1989) 2 Y Y Y - Y Y
37 Zahn (1985) 2 Y - Y - Y -
38 Zhou et al. (1985) 2 Y - Y - Y Y
39 Amitsu et al. (1991) 1 N/A N/A N/A N/A N/A N/A
40 Arakawa et al. (1982) 1 N/A N/A N/A N/A N/A N/A
41 Nagasaka (1982) 1 N/A N/A N/A N/A N/A N/A
42 Park and Paulay (1990) 1 N/A N/A N/A N/A N/A N/A
41