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COLLAPSE PROBABILITY OF FRAME STRUCTURES WITH DETERIORATING
PROPERTIES
Helmut Krawinkler1 and Luis Ibarra1
1Department of Civil & Environmental Engineering, Stanford
University, Stanford, CA, USA Email: [email protected]
SUMMARY
This paper addresses the collapse potential of deteriorating
systems when subjected to seismic excitations. A global collapse
assessment approach, which is based on a hysteretic model that
includes history dependent strength and stiffness deterioration,
and considers the uncertainty in the ground motion frequency
content, is illustrated on hand of SDOF systems and MDOF frame
structures. The deterioration model is energy-based and traces
deterioration as a function of past loading history and the energy
dissipation capacity of the components of the structural system. In
the proposed approach global collapse is described by a relative
intensity measure, defined as the ratio of ground motion intensity
(measured by the spectral acceleration at the first mode period) to
a structure strength parameter (base shear coefficient). The
relative intensity at which collapse occurs is called the collapse
capacity. A parametric study is performed using SDOF systems and
generic MDOF frames subjected to a set of Californian ground
motions. The frames include nonlinear behavior by means of
concentrated plasticity. The nonlinear springs at the end of the
elements include hysteretic models with strength and stiffness
deterioration characteristics. To obtain the collapse capacity, the
relative intensity is increased in small increments and dynamic
analyses are carried out at each increment. Global collapse occurs
when the relative intensity-EDP curve becomes flat. In MDOF
systems, a flat slope implies that in a specific story the gravity
induced P-delta effects have overcome the deteriorating story
lateral resistance. The collapse capacity data are used to generate
fragility curves for a given MDOF system. The mean annual frequency
of collapse can be obtained by integrating the collapse fragility
curve for a given MDOF system over the spectral acceleration hazard
curve pertaining to a specific site.
1. INTRODUCTION Protection against collapse has always been a
major objective of seismic design. Collapse refers to the loss of
ability of a structural system, or any part thereof, to resist
gravity loads. Local collapse may occur, for instance, if a
vertical load carrying component fails in compression, or if shear
transfer is lost between horizontal and vertical components (e.g.,
shear failure between a flat slab and a column). Such local
collapse issues are not discussed in this paper. Global (or at
least story) collapse will occur if local collapses propagate
(cascading collapse) or if an individual story displaces
sufficiently so that the second order P-delta effects fully offset
the first order story shear resistance and instability occurs
(incremental collapse). Deterioration in strength and stiffness of
individual components plays a critical role in the incremental
collapse mode. Therefore, assessment of collapse safety
necessitates the capability to predict the dynamic response of
deteriorating systems, particularly for existing older construction
in which deterioration commences at relatively small deformations.
To this date, the system collapse issue was seldom addressed
because of the lack of hysteretic models capable of simulating
deterioration behavior, and collapse is
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usually associated with an acceptable story drift or the
attainment of a limit value of deformation in individual components
of the structure. This approach does not permit a redistribution of
damage and does not account for the ability of the system to
sustain significantly larger deformations before collapse than
those associated with first attainment of a limiting deformation in
a component. These shortcomings are overcome in the procedure
proposed in this paper The approach presented in this paper is
based on hysteretic models of structural elements that account for
history-dependent strength deterioration and stiffness degradation.
The cyclic deterioration model is energy based and traces
deterioration as a function of past loading history and the energy
dissipation capacity of each component. System collapse can then be
evaluated in a reliability format that considers the uncertainties
in the intensity and frequency content of the earthquake ground
motions as well as the deterioration characteristics of each
structural element. Research on SDOF systems and MDOF frame
structures utilizing these component deterioration models has been
performed. Parameter studies are carried out in which the period
(number of stories) of the structural system and the deterioration
properties of the component models are varied. Salient findings of
this study on deteriorating SDOF systems and MDOF frame structures
are the subject of this paper. 2. DETERIORATING HYSTERESIS MODEL
2.1 Basic Component Models Three basic component models are widely
used for a generic representation of hysteretic characteristics,
i.e., the bilinear model, the peak-oriented (Clough) model, and the
pinching model. If no deterioration exists, these three models can
be described by a small number of parameters; i.e., the elastic
stiffness Ke, the yield strength Fy, the strain hardening stiffness
Ks = sKe, an unloading stiffness, Ku, if different from the elastic
stiffness, and two more parameters to define the pinching effect
(for the pinching model only). In this paper all of the presented
results are for the peak-oriented model. For an assessment of the
effects of basic hysteresis properties on collapse the reader is
referred to [Ibarra, 2003]. 2.2 Component Models with Deterioration
in Strength and Stiffness Replication of collapse necessitates
modeling of deterioration characteristics of structural components.
The literature on this subject is extensive, but few simple
deterioration models exist, and little systematic research on the
effects of component deterioration on the collapse potential has
been performed in the past. The reader is referred to the following
references, which are representative examples of important work in
this area: [Kunnath et al., 1997], [Sivaselvan and Reinhorn, 2000],
and [Song and Pincheira, 2000]. Rigorous evaluation of collapse
safety requires more emphasis on deterioration models. Utilization
of the PEER framework equation for prediction of the collapse
probability [Krawinkler, 2002] is feasible only if modeling of
history dependent deterioration is incorporated in the response
prediction. Refined component models that incorporate deterioration
characteristics are being developed as part of the PEER OpenSees
effort. These models are detail-specific and cannot be employed for
general sensitivity studies. Thus, a general deterioration model
had to be developed as part of one of the PEER demand studies
[Ibarra, 2003]. This hysteresis model attempts to model all
important modes of deterioration that are observed in experimental
studies. An example of a monotonic load-displacement response and a
superimposed cyclic response of identical plywood shear wall panels
is illustrated in Figure 1. The monotonic test result shows that
strength is capped and is followed by a negative tangent stiffness
(which often degrades gradually, a phenomenon that is ignored in
this model). The cyclic hysteresis response indicates that the
strength in large cycles deteriorates with the number and amplitude
of cycles, even if the displacement associated with the strength
cap has not been reached. It also indicates that similar strength
deterioration occurs in the post-capping range, and that the
unloading stiffness may also deteriorate. Furthermore, it is
observed that the reloading stiffness may deteriorate at an
accelerated rate if the hysteresis response is of a pinched nature
(as in this example).
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Thus, the hysteresis model should incorporate a backbone curve
that represents the monotonic response, and deterioration rules
that permit modeling of all important deterioration modes and
should be applicable to any of the three basic hysteresis models
summarized previously. Thus, as a minimum, the backbone curve has
to be trilinear in order to include strength capping and post-cap
strength deterioration. The strength cap Fc is associated with the
cap deformation, c, and is followed by a post capping tangent
stiffness Kc = cKe, which is either zero or negative. The branches
of the backbone curve for monotonic loading are shown in Figure 2.
As seen, the ratio c/y may be viewed as the ductility capacity, but
deformations larger than c can also be tolerated.
UCSD Test PWD East Wall
-10
-5
0
5
10
-6 -4 -2 0 2 4 6 8Displacement (in)
Loa
d (k
ips)
Monotonic
ISO
UCSD Test PWD East Wall
-10
-5
0
5
10
-6 -4 -2 0 2 4 6 8Displacement (in)
Loa
d (k
ips)
Monotonic
ISO
c
Ke
s
cKeFy
Fc
y c
sKe
min. strength
c
Ke
s
cKeFy
Fc
y c
sKe
min. strength
Figure 1 Experimental results from plywood Figure 2 Backbone
curve and its movement shear wall tests; modes of deterioration
with deterioration (Ibarra et al., 2003) The strain hardening and
post capping branches may remain stationary or may deteriorate
(i.e., translate towards the origin) in accordance with a
relatively simple energy-based deterioration model [Rahnama and
Krawinkler, 1993] defined by a deterioration parameter of the
type
c
i
jjt
ii
EE
E
=
=1
(1)
in which i = parameter defining the deterioration in excursion i
Ei = hysteretic energy dissipated in excursion i Et = hysteretic
energy dissipation capacity, expressed as a multiple of Fyy, i.e.,
Et = Fyy Ej = hysteretic energy dissipated in all previous
excursions c = exponent defining the rate of deterioration
This deterioration parameter can be applied to one or all of the
following four deterioration modes: 1. Basic strength
deterioration, defined by translating the strain hardening branch
towards the origin by
an amount equivalent to reducing the yield strength to
( ) 11 = iisi FF (2) where = deteriorated yield strength after
excursion i iF = deteriorated yield strength before excursion i 1iF
is = given by Eq. 1, employing an appropriate value to model
strength deterioration, i.e., s.
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In addition, the slope of the strain hardening branch is
continuously rotated by an angle equivalent to the amount of
strength deterioration, such that the strain hardening stiffness is
equal to zero when the yield strength has deteriorated to zero.
2. Post-cap strength deterioration, defined by translating the
post-capping branch towards the origin by an amount equivalent to
reducing the cap strength to
( ) 11 = iici FF (3) using the same definitions as in the
strength deterioration case but employing ic associated with an
appropriate value to model cap deterioration, i.e., c.
3. Unloading stiffness deterioration, defined by reducing the
unloading stiffness Ku in accordance with
111 == i,uui,uii,u KK)(K (4) using the same definitions as in
the strength deterioration case but employing ik associated with an
appropriate value to model unloading stiffness degradation, i.e.,
k.
4. Accelerated reloading stiffness deterioration, defined by
moving the target deformation t,i (which defines the point targeted
in the reloading branch of the peak-oriented and pinching model)
along the backbone curve to a value of
111 =+= i,tki,tii,t )( (5) employing ia associated with an
appropriate value to model accelerated reloading stiffness
degradation, i.e., a.
In addition to these deterioration modes, a residual strength of
Fy can be assigned to the model. When such a residual strength is
specified, the backbone curve is supplemented by a horizontal line
with ordinate Fy, and the strength will not drop below this value.
Thus, the deterioration model has two parameters defining the
capping phenomenon (c [or Fc] and c), up to four deterioration
parameters (s, c, k, a) [presuming that the exponent in Eq. 1 is
equal to 1.0, which is the only case considered so far], and a
residual strength parameter . This model was tested on
force-deformation data obtained from experiments on steel,
reinforced concrete, and wood components. Adequate simulations were
obtained in all cases by tuning the model parameters to the
experimental data. Examples illustrating the effects of cyclic
deterioration on the time history response of an SDOF system are
shown in Figure 3. The backbone curve parameters are indicated in
the figure, and values of 100 and 25, respectively, are used for
the four modes of deterioration. The NR94hol ground motion recorded
in the 94 Northridge earthquake is used as input. The small values
(25 as compared to 100) lead to pronounced cyclic deterioration,
which is reflected in the decrease in strength and stiffness
evident in Figure 3(b), which in turn increases the maximum
displacement by about 50% compared to the case with slow cyclic
deterioration, but does not yet lead to collapse.
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HYSTERETIC BEHAVIOR W/CYCLIC DET.Peak Oriented Model, NR94hol
Record , =5%,
P-=0, s=0.03, c=-0.10, c/y=4, s,c,k,a=100
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-4 0 4 8Normalized Displacement, /y
Nor
mal
ized
For
ce, F
/Fy
Initial Backbone
HYSTERETIC BEHAVIOR W/CYCLIC DET.Peak Oriented Model, NR94hol
Record , =5%,
P-=0, s=0.03, c=-0.10, c/y=4, s,c,k,a=25
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-4 0 4 8Normalized Displacement, /y
Nor
mal
ized
For
ce, F
/Fy
Initial Backbone
(a) (b)
Figure 3 Effect of Cyclic Deterioration in Time History Analysis
(a) s,c,k,a =100, (b) s,c,k,a = 25 3. STRUCTURAL SYSTEMS AND GROUND
MOTIONS USED IN THIS STUDY 3.1 Structural Systems Both SDOF systems
and MDOF frames are investigated in this study. Even though the
deterioration model described in Section 2 is a component model, in
the SDOF study it is assumed that the system response follows the
same hysteresis and deterioration rules as a representative
component. Clearly this is an approximation, as it is idealistic to
assume that all components of a structural system have the same
deterioration properties and it is unrealistic to assume that all
components yield and deteriorate simultaneously. But such
assumptions are made often when conceptual SDOF studies are
performed in which the SDOF system is intended to represent a MDOF
structure. The yield level of the SDOF system is defined by the
parameter = Fy/W, with Fy being the yield strength and W being the
seismically effective weight. The MDOF systems are single-bay
moment resisting frames with number of stories, N, equal to 3, 6,
9, 12, 15, and 18, and a fundamental period, T1, of 0.1N and 0.2N.
Note that there are overlaps at T1 = 0.6 s., 1.2 s. and 1.8 s.,
which allows an assessment of the effects of N in the response of
the frames given T1. The main characteristics of this family of
frames are as follows: The same mass is used at all floor levels
Centerline dimensions are used for beam and column elements
Relative stiffnesses are tuned so that the first mode is a straight
line Plastic hinges can occur only at the end of the beams and the
bottom of the first story columns (no weak
stories permitted) Frames are designed so that simultaneous
yielding is attained under a parabolic (NEHRP, k = 2) load
pattern The global shear strength of the frame is defined by the
parameter = Vy/W, with Vy being the base
shear yield strength and W being the seismically effective
weight of the full frame (i.e., is equivalent to of the SDOF
system)
Moment-rotation hysteretic behavior is modeled by using
rotational springs with the appropriate hysteresis and
deterioration properties. In all cases, 3% strain hardening is
assumed.
The effect of gravity load moments on plastic hinge formation is
not included Global (structure) P-Delta is included For the
nonlinear time history analyses, 5% Rayleigh damping is assigned to
the first mode and the
mode at which the cumulative mass participation exceeds 95%.
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3.2 Ground Motions A set of 40 ordinary ground motions (denoted
as LMSR-N set) is used to carry out the time history analysis. The
ground motions are from Californian earthquakes of moment magnitude
between 6.5 and 6.9 and a closest distance to the fault rupture
between 13 km and 40 km (i.e., near-fault effects are not
considered). These ground motions were recorded on NEHRP site class
D [FEMA368, 2000]. Qualitatively, conclusions drawn from the
collapse evaluation using this set of ground motions are expected
to hold true also for stiffer soils and rock. The records are
selected from the PEER (Pacific Earthquake Engineering Research)
Center Ground Motion Database (http://peer.berkeley.edu/smcat/). A
comprehensive documentation of the properties of the LMSR-N ground
motion set is presented in [Medina, 2003]. 4. DETERMINATION OF
COLLAPSE CAPACITY Since only bending elements are utilized in the
frame models (i.e., shear and axial failures are not modeled),
collapse implies that the interstory drift in a specific story
grows without bounds (incremental collapse). The basic parameter
used to drive the structure to collapse is the relative intensity
[Sa(T1)/g]/, with Sa(T1), the spectral acceleration at the first
mode period, being a measure of the intensity of the ground motion,
and being a measure of the strength of the structure. [For SDOF
system the equivalent parameter is (Sa/g)/.] The parameter
[Sa(T1)/g]/ represents the ductility dependent response
modification factor (often denoted as R), which, in the context of
present codes, is equal to the conventional R-factor if no
overstrength is present. Unless gravity moments are a major portion
of the plastic moment capacity of the beams, and there are
considerable changes in column axial forces due to overturning
moments as compared to the gravity axial forces in columns, the use
of [Sa(T1)/g]/ as a relative intensity measure can be viewed two
ways; either keeping the ground motion intensity constant while
decreasing the base shear strength of the structure (the R-factor
perspective), or keeping the base shear strength constant while
increasing the intensity of the ground motion (the Incremental
Dynamic Analysis, IDA. perspective [Vamvatsikos and Cornell 2002]).
Thus, the process of determining the collapse capacity of a
structural system consists of subjecting the structure to a set of
ground motions, and for each ground motion incrementing the
relative intensity until dynamic instability occurs. This implies
that the curve relating the relative intensity, [Sa(T1)/g]/, and a
relevant engineering demand parameter, EDP, (e.g., roof drift,
maximum story drift, maximum story ductility) becomes flat
(horizontal), as illustrated in Figure 4, because the EDP increases
indefinitely for a minute increase in relative intensity. Thus, the
relative intensity associated with the last point of each
[Sa(T1)/g]/ - EDP curve (the maximum story ductility is used as EDP
in Figure 4) can be viewed as the collapse capacity of the
structural system, denoted here as [Sa,c(T1)/g]/ . Thus, if 40
ground motions are used, up to 40 data points for the collapse
capacity are obtained (Figure 4 illustrates an example for only 20
ground motions). It is evident that the collapse capacity has a
very large scatter, and for several ground motions it may be
attained at very high [Sa(T1)/g]/ values that are outside the range
of interest. Thus, the collapse capacity can only be evaluated
statistically, and often from an incomplete data set. For good
reasons [Ibarra, 2003] it is assumed that the distribution of the
collapse capacity data is lognormal, and counted statistics is
employed because of the sometimes occurring incompleteness of the
date set (i.e., for 40 records the average of the 20th and 21st
sorted value is taken as the median, the 6th sorted value is taken
as the 16th percentile, and the 34th sorted value is taken as the
84th percentile). If the data set is complete, a distribution can
be fitted to the data as shown in Figure 4. The meaning of this
distribution is discussed further in Section 6.. In the following
discussion the emphasis is on median values of collapse
capacity.
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MAX. STORY DUCTILITY vs. NORM . STRENGTHN=9, T1=0.9, =0.05, K 1,
S1, BH, =0.015, Peak-Oriented M odel,
s=0.05, c/y=4, c=-0.10, s=8 , c=8 , k=8 , a=8 , =0, LM SR
0
2
4
6
8
10
0 10 20M aximum Story Ductility Over the Height, s,max
[Sa(
T 1)/
g]/
30
Individual responsesMedian
MAX. STORY DUCTILITY vs. NORM . STRENGTHN=9, T1=0.9, =0.05, K 1,
S1, BH, =0.015, Peak-Oriented M odel,
s=0.05, c/y=4, c=-0.10, s=8 , c=8 , k=8 , a=8 , =0, LM SR
0
2
4
6
8
10
0 10 20M aximum Story Ductility Over the Height, s,max
[Sa(
T 1)/
g]/
30
Individual responsesMedianMedianMedianMedian
Figure 4 Statistical determination of the collapse capacity
The median collapse capacity can be determined for different
periods and different system parameters, which permits an
evaluation of the effects of deterioration parameters. The collapse
capacity is a property of the selected structural system and the
selected ground motion set. For a system of given strength ( or ),
it represents the median Sa value leading to collapse, and for a
given Sa value (hazard level), it represents the median strength
leading to collapse. As mentioned before, in the latter context
[Sa(T1)/g]/ represents the response modification factor (R-factor)
without overstrength, and therefore, the median [Sa(T1)/g]/ value
at collapse is equivalent to the median R-factor causing collapse.
In the following discussion the median [Sa(T1)/g]/ value at
collapse is being used to assess the sensitivity of the collapse
capacity to the period and deterioration properties of the
structural system. An example of the dependence of the collapse
capacity on the system period is presented in Figure 5, showing
data points for individual records as well as median and 16th
percentile values. The data are for an SDOF system that is defined
by a c/y value of 4.0, a cyclic deterioration parameter of s,c,k,a
= 100, and a post-capping slope of c = -0.1. The results are
obtained by performing collapse analysis for structural systems
whose period is varied in closely spaced intervals. It can be
observed that the statistical measures for the collapse capacity
vary only slightly with period, except in the short period range (T
< 0.6 sec.) in which they decrease considerably. This is also
the range in which many past studies have shown that even for
nondeteriorating systems the R-factor for constant ductility
demands decreases rapidly with a decrease in period. The dependence
of the SDOF median collapse capacity on two system parameters (c/y
and c) is illustrated in Figure 6. If a flat post-capping slope (c
= -0.1) exists, it permits a significant increase in (Sa/g)/ after
c is reached but before the relatively large collapse displacement
is attained. Thus, the effect of c/y on collapse values of (Sa/g)/
is not very large unless a steep post-capping slope exists, in
which case collapse occurs soon after c has been reached (see curve
for c/y = 2 and c = -0.3 vs c = -0.1).
(Sa/g) / at COLLAPSE vs PERIODPeak Oriented Model, LMSR-N, =5%,
P-='0.1N',
s=0.03, c=-0.10, c/y=4, s,c,k,a=100
0
5
10
15
20
25
30
0 1 2 3 4Period (sec)
(Sa/
g) /
Median84th
EFFECT OF c/y ON MEDIAN (Sa/g)/ AT COLLAPSEPeak Oriented Model,
LMSR-N, =5%, P-='0.1N',
s=0.03, c=Var, c/y=Var, s,c,k,a=100
0
2
4
6
8
10
0 1 2 3 4Period (sec)
(Sa/
g) /
c/y=6, c=-0.1c/y=4, c=-0.1c/y=2, c=-0.1c/y=2, c=-0.3
Figure 5 Variation of SDOF collapse capacity Figure 6 Variation
of SDOF median collapse with period; one specific system, all data
points capacity with period, various systems
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5. MEDIAN COLLAPSE CAPACITY FOR FRAME STRUCTURES In the MDOF
study it is assumed that every plastic hinge in the structure can
be described by the same hysteresis and deterioration model. This
simplifying assumption permits a consistent evaluation of the
collapse data and an assessment of the effect the various
deterioration parameters have on the collapse capacity. The
parameters of primary interest in the following discussion are:
The ductility capacityc/y. Values of 2, 4, and 6 are used. The
post-capping tangent stiffness ratio c. Values of 0.1, -0.3, and
0.5 are used. The cyclic deterioration parameter = Et/Fyy. Value
for s = c = k = a of (no cyclic
deterioration), 100, 50, and 25 are used. The first observation
to be made is that none of these parameters can be evaluated
independently of the others. The simplest example is the one
illustrated in Figure 6. It appears that the ductility capacity
does not have an overriding effect on the collapse capacity. But
this holds true only if the post-capping stiffness is flat (e.g., c
= -0.1) because then the component strength capacity deteriorates
slowly after the cap displacement c is reached. If the post-capping
stiffness is steep (e.g., c = -0.3), collapse occurs soon after c
has been reached. The interdependence of the various parameters
must be kept in mind when the collapse capacity results are
interpreted. Moreover, there is a clear dependence of the parameter
effects on the first mode period and the number of stories in the
frame structure. 5.1 Effect of Ductility Capacity and Post-Capping
Stiffness The effect of ductility capacity (c/y) on the median
collapse capacity is illustrated in Figure 7 for frame structures
with T1 = 0.1N and T1 = 0.2N, with N = 3, 6, 9, 12, 15, and 18. The
effect clearly is larger if the post-capping tangent stiffness is
steep, but even then the collapse capacity does not increase in the
same proportion as the ductility capacity. The effect of the
post-capping stiffness is isolated in Figure 8, which shows median
collapse capacities for frames with c/y = 4 but different c values.
There is a large difference between the collapse capacities for c =
0.1 and 0.3, but little difference between the capacities for c =
0.3 and 0.5. The reason for the latter is that c = 0.3 corresponds
already a steep slope. There is some, but not much, sensitivity to
the number of stories for a given T1. Perhaps most striking is the
strong dependence of the collapse capacity on the first mode period
T1. This is expected for the short period structure with T1 = 0.3
sec., where the collapse capacity is much smaller than that for T1
= 0.6 sec. But the large decrease in collapse capacity for long
period structures is striking, indicating that the period
independent R-factor concept is way off. The reason is the P-delta
effect, which is much more important than might be expected. This
paper does not address P-delta effects in detail, but it must be
emphasized that this effect is severely underestimated in present
practice. It turns out that for long period structures the elastic
story stability coefficient ( = P/hV) severely underestimates the
P-delta effect in the inelastic range. In most practical cases the
lower stories experience large drifts when the structure undergoes
large inelastic deformations, and the story stability coefficient
increases correspondingly. The fact that the elastic story
stability coefficient is a poor measure of the inelastic P-delta
effect is illustrated in Figure 9, in which the collapse capacities
of SDOF and MDOF systems are compared, utilizing the MDOF first
story elastic stability coefficient (for T1 = 0.2N) in the analysis
of the SDOF systems. It can be seen that the results are very close
to each other for systems for which the P-delta effects are not
dominating. But for long period structures the results deviate
considerably, with the SDOF system predicting much too large a
collapse capacity.
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DUCTILITY CAPACITY EFFECT ON [Sa,c(T1)/g]/ N=Var, T1=Var, BH,
Peak Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=-0.10, s,c,k,a=Inf, =0
0
2
4
6
8
10
0 1 2 3 4Period (sec)
[Sa,
c(T1)/
g]/
c/y = 6 c/y = 4c/y = 2
T1 = 0.1 NT1 = 0.2 N
DUCTILITY CAPACITY EFFECT ON [Sa,c(T1)/g]/ N=Var, T1=Var, BH,
Peak Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=-0.30, s,c,k,a=Inf, =0
0
2
4
6
8
10
0 1 2 3 4Period (sec)
[Sa,
c(T1)/
g]/
c/y = 6 c/y = 4c/y = 2
T1 = 0.1 NT1 = 0.2 N
(a) (b) Figure 7 Effect of ductility capacity c/y on median
collapse capacity of frames (a) flat (c = -0.1),
and (b) steep (c = -0.3) post-capping tangent stiffness
POST-CAPPING STIFFNESS EFFECT ON [Sa,c(T1)/g]/ N=Var, T1=Var,
BH, Peak Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, c=Var, s,c,k,a=Inf, =0
0
2
4
6
8
10
0 1 2 3 4Period (sec)
[Sa,
c(T1)/
g]/
c = -0.1c = -0.3c = -0.5
T1 = 0.1 NT1 = 0.2 N
[Sa,c(T1)/g]/ & (Sa,c/g)/ vs PERIODN=Var, T1=0.2N, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, cap=-0.10, s,c,k,a=Inf, =0
0
2
4
6
8
10
0 1 2 3Period (sec)
[Sa,
c(T1)
/g]/
& (S
a,c/g
)/
4
SDOF, P-D=0.2N
MDOF, T=0.2N
Figure 8 Effect of post-capping stiffness on Figure 9 Collapse
capacities of SDOF and MDOF collapse capacity of frames systems
with same elastic stability coefficient 5.2 Effect of Cyclic
Deterioration on Collapse Capacity The effect of cyclic
deterioration on the collapse capacity of frames with c/y = 4 and c
= 0.1 is illustrated in Figure 10. The effect is evident, although
not overpowering, which indicates that the combination of ductility
capacity and post-capping stiffness is in general equally or more
important than the effect of cyclic deterioration. The effect
diminishes for long period structures because of the dominant
importance of P-delta effects. It should be said that the ground
motion set used in this study is comprised of records with
relatively short strong motion duration. A parallel study utilizing
records with long strong motion duration did show a somewhat but
not much larger effect of cyclic deterioration. Thus, cyclic
deterioration appears to be an important but not dominant issue for
collapse evaluation unless the energy dissipation capacity of the
structural components is very small ( = 25). The results shown so
far represent median collapse capacities. To assess the reliability
of structures, the measure of dispersion of the collapse capacities
is equally important. Since it is assumed that the distribution of
collapse capacities is lognormal, the appropriate measure of
dispersion is the standard deviation of the natural log of the
data. Typical data for this measure of dispersion are shown in
Figure 11. It is noted that the measure of dispersion is rather
large but is not sensitive to the period of the structural system.
This is an important observation in view of the probability of
collapse issue discussed in the next section.
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CYCLIC DETERIORATION EFFECT ON [Sa,c(T1)/g]/ N=Var, T1=Var, BH,
Peak Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, c=-0.10, s,c,k,a=Var, =0
0
2
4
6
8
10
0 1 2 3Period (sec)
[Sa,
c(T1)/
g]/
4
s,c,k,a= Infs,c,k,a=100s,c,k,a=50s,c,k,a=25
T1 = 0.1 NT1 = 0.2 N
DISPERSION OF [Sa,c(T1)/g]/ vs PERIODN=Var, T1=Var, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, c=-0.10, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 1 2 3Period (sec)
Stan
dard
Dev
. ln[
S a,c(
T 1)/g
]/
4
T1=0.1N
T1=0.2N
Figure 10 Effect of cyclic deterioration on Figure 11 Dispersion
of collapse capacity of collapse capacity of frames frames 6.
COLLAPSE FRAGILITY CURVES AND MEAN ANNUAL FREQUENCY OF COLLAPSE In
the context of seismic performance assessment, collapse constitutes
one of several limit states of interest. It can be argued that it
is not a fundamental limit state. On one hand it contributes to the
cost of damage, if monetary losses or downtime are performance
targets. In this context collapse could be viewed as a damage
measure for which it is useful to develop fragility curves. On the
other hand, collapse contributes to (but is not solely responsible
for) casualties and loss of lives. Thus, for the performance target
of casualties and loss of lives, collapse is an intermittent
decision variable that could be described by means of a Mean Annual
Frequency (MAF) of exceedance. Both, fragility curves and MAFs can
be derived from the collapse capacity data as illustrated next. 6.1
Collapse Fragility Curves Data of the type shown in Figure 4 can be
utilized to develop fragility curves, which describe the
probability of failure (in this case failure implies collapse),
given the value of [Sa(T1)/g]/ (or (Sa/g)/ in the case of SDOF
systems). Such fragility curves are obtained from the CDF of the
last point of each of the curves shown in Figure 4 (the collapse
point). Typical results of fragility curves are shown in Figure 12
for SDOF systems of various periods and a set of specific
structural parameters, which may be viewed as baseline properties
that are used in many of the graphs to follow. The baseline
properties are
c/y = 4 c = -0.1 No cyclic deterioration (s = c = k = a = ) No
residual strength ( = 0)
Figure 12,, which is for SDOF systems, shows ragged lines that
are obtained from the ordered data points, as well as smooth curves
that are obtained from fitting a lognormal distribution to the
data. The general observation is that a lognormal distribution fits
the data rather well, and for this reason in the subsequent graphs
for MDOF frames only the fitted distributions are shown. The
dependence of the fragility curves on the system period is evident
in Figure 12(a), with the curve for T = 0.3 sec. indicating a much
higher probability of collapse for a short period system. Systems
with T = 0.6 sec. to 3.6 sec. show only small period sensitivity,
which is no surprise for SDOF systems (see also Figure 9 for median
values). But this observation is not valid for MDOF frames as is
discussed in the next section.
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(Sa,c/g)/ vs PROBABILITY OF COLLAPSEPeak Oriented Model, LMSR-N,
=5%, P-='0.1N'
s=0.03, c=-0.10, c/y=4, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 4 8 12(Sa,c/g)/
Prob
abili
ty o
f Col
laps
e
16
T = 0.3 secT = 0.9 secT = 1.8 sec
(Sa,c/g)/ vs PROBABILITY OF COLLAPSEPeak Oriented Model, LMSR-N,
=5%, P-='0.1N'
s=0.03, c=-0.10, c/y=4, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 4 8 12(Sa,c/g)/
Prob
abili
ty o
f Col
laps
e
16
T = 0.6 secT = 1.8 secT = 3.6 sec
(a) (b)
Figure 12 Fragility curves for SDOF systems of various periods
6.2 Collapse Fragility Curves for MDOF Frames with Parameter
Variations Figure 13 presents MDOF fragility curves that can be
compared directly to the SDOF fragility curves of Figure 12. They
are for baseline structural properties. The four digit code
identified for each frame the number of stories and the first mode
period, i.e., 0918 means a 9-story frame with T1 = 1.8 sec. There
is a clear pattern, equivalent to that exhibited in the median in
Figure 9, of high fragility (small collapse capacity) for short
period structures (T1 = 0.3 sec.), a large decrease in the
fragility for medium period structures (T1 = 0.6 and 0.9 sec.), and
then again an increase in fragility for long period structures (T1
= 1.8 and 3.6 sec.) because of the predominance of P-delta
effects.
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=Var, T1=Var, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, cap=-0.10, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
030309091818
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=Var, T1=Var, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=4, cap=-0.10, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
0306
0918
1836
(a) (b) Figure 13 Fragility curves for frame structures with
baseline properties; (a) three frames with T1 =
0.1N, (b) three frames with T1 = 0.2N The effects of ductility
capacity and post-capping tangent stiffness are illustrated in
Figure 14 for four frames with the first mode period varying from
0.3 sec. to 3.6 sec. An increase in the ductility capacity shifts
the fragility curves to the right, but not by an amount
proportional to the increase in ductility capacity. An increase in
the slope of the post-capping tangent stiffness (from flat to
steep) has a very detrimental effect on the fragility. In concept,
all observations that have been made previously for median collapse
capacities hold true also for the fragility curves. The value of
these curves lies in their probabilistic nature that permits
probabilistic expressions of performance and design decisions. For
instance, if for a given long return period hazard (e.g., 2/50
hazard) a 10% probability of collapse could be tolerated, then the
intersections of a horizontal line at a probability of 0.1 with the
individual fragility curves provides targets for the R-factor that
should be employed in design in conjunction with the spectral
acceleration associated with this hazard. If such
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horizontal lines are drawn in the graphs of Figure 14, it can be
conjectured that the indicated R values are low, even for rather
ductile systems. The second value of the fragility curves lies in
the opportunity they provide for a rigorous computation of the mean
annual frequency of collapse, as is discussed in the next
section.
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=3, T1=0.3, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=Var, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
c/y=6, c=-0.10c/y=4, c=-0.10 c/y=2, c=-0.10 c/y=4, c=-0.30
c/y=2, c=-0.30
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=9, T1=0.9, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=Var, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
c/y=6, c=-0.10c/y=4, c=-0.10 c/y=2, c=-0.10 c/y=4, c=-0.30
c/y=2, c=-0.30
(a) (b)
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=18, T1=1.8, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=Var, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
c/y=6, c=-0.10c/y=4, c=-0.10 c/y=2, c=-0.10 c/y=4, c=-0.30
c/y=2, c=-0.30
[Sa,c(T1)/g]/ vs PROBABILITY OF COLLAPSEN=18, T1=3.6, BH, Peak
Oriented Model, LMSR-N, =5%,
s=0.03, c/y=Var, c=Var, s,c,k,a=Inf, =0
0
0.2
0.4
0.6
0.8
1
0 5 10 15[Sa,c(T1)/g]/
Prob
abili
ty o
f Col
laps
e
c/y=6, c=-0.10c/y=4, c=-0.10 c/y=2, c=-0.10 c/y=4, c=-0.30
c/y=2, c=-0.30
(c) (d)
Figure 14 Fragility curves for frame structures with parameter
variations;
(a) 3-story, T1 = 0.3 sec.; (b) 9-story, T1 = 0.9 sec.; (c)
18-story, T1 = 1.8 sec.; (d) 18-story, T1 = 3.6 sec 6.3 Mean Annual
Frequency of Collapse If the collapse fragility curve for a given
system has been determined, probabilistic collapse assessment can
be carried out according to the following equation:
|)(|)( xdxFaSa SCf = (6)
where f = mean annual frequency of collapse = probability of the
S)(xF
SaC a capacity, Sa,c, (for a given or value) exceeding x )(xSa =
mean annual frequency of Sa exceeding x (ground motion hazard)
Thus, given the Sa hazard curve and fragility curves of the type
shown in Figures 12 to 14, it is a matter of numerical integration
to compute the mean annual frequency of collapse. The process of
integrating Equation (6) is illustrated in Figure 15. corresponds
to the specific fragility curve of interest. In
this context, the structure strength parameter ( or ) is kept
constant, i.e., the individual curves shown in Figure 15 represent
IDAs.
)(xFSaC
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(Sa/g) Fragility
CurvePDF for
Sa,c/g
x
EDP
Sa/g
Sa
Hazard CurvedSa(x)
Probability of Sa,c/g exceeding xgiven Sa/g FC,Sa(x)
Figure 15 Illustration of process used to compute the mean
annual frequency of collapse The results obtained will depend
strongly on the selected hazard curves and collapse fragility
curve. The former is site dependent and the latter is site (ground
motion) and structure dependent. To provide an illustration of
typical results, hazard curves for various periods are derived
(approximately) from the equal hazard spectra employed in PEER
studies for a Los Angeles building, see Figure 16, and fragility
curves for SDOF systems with specific structural properties are
utilized to develop curves for the MAF of collapse for various
periods and selected strength levels .. These curves are shown in
Figure 17. They are for illustration only, as they are rather site
and structure system specific. But they illustrate general trends
and are the product of a rigorous process for computing the mean
annual frequency of collapse.
Equal Hazard Spectra, Van Nuys, CA.
0
0.5
1
1.5
2
0 0.5 1 1.5Period, T (sec)
Spec
tral
Acc
eler
atio
n
2
50% in 50 years10% in 50 years2% in 50 years
MEAN ANNUAL PROB. OF COLLAPSE, Van Nuys, CA.Peak Oriented Model,
LMSR-N, =5%, P-='0.1N', HC-LR
s=0.03, c=-0.10, c/y=6, s,c,a=Inf, k=Inf, =0
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 0.5 1 1.5Period, T (sec)
Mea
n A
nnua
l Pro
b C
olla
pse
2
= 1.0 = 0.5 = 0.2 = 0.1
Figure 16. Equal hazard spectra used to derive Figure 17. Mean
annual frequency of collapse hazard curves for specific periods for
SDOF systems of given strength = Fy/W 7. SUMMARY AND PRELIMINARY
CONCLUSIONS The study summarized here demonstrates that collapse
assessment needs to account for several
deterioration phenomena of the inelastic cyclic response
characteristics of the important components of the structural
system.
The level of displacement at which a monotonically loaded
component attains its maximum strength (defined by c/y), as well as
the stiffness after attainment of this displacement, are important
parameters that can be accounted for in the backbone curve of the
hysteretic rules that describe the system (component) behavior.
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338
The slope of the post-capping stiffness has a significant effect
on the collapse capacity. If this slope is small (flat), the
strength of the component (system) decreases only slowly and
deformations much larger than c can be attained before collapse
occurs.
The rate of cyclic deterioration, which is assumed to be
controlled by energy dissipation demands and capacity, is another
important parameter. It can be accounted for by developing rules
for history dependent cyclic deterioration. Everything else equal,
cyclic deterioration is of somewhat greater importance for short
period systems because of the larger number of inelastic cycles to
which such systems are subjected.
Deterioration in unloading and reloading stiffnesses is in
general of smaller consequence than the deterioration
characteristics of the aforementioned parameters.
P- effects, which depend on the period of the structural system
and on the relationship between period and the number of stories,
require much attention. They will accelerate collapse of
deteriorating systems, and they may be the primary source of
collapse for flexible but very ductile structural systems.
Collapse fragility curves derived from equivalent SDOF systems
for long period frame structures may provide misleading information
unless the large inelastic P-delta effect is accounted for in the
equivalent SDOF system. The elastic story stability coefficient
will not do the job.
8. ACKNOWLEDGEMENTS This research is supported by the Pacific
Earthquake Engineering Research (PEER) Center, an Engineering
Research Center sponsored by the US National Science Foundation.
This support is much appreciated. 9. REFERENCES FEMA 368 (2000).
NEHRP recommended provisions for seismic regulations for new
buildings and other
structures, Building seismic safety council, Washington D.C.
Ibarra, L.F. (2003). "Global collapse of frame structures under
seismic excitations," Ph.D. Dissertation,
Department of Civil and Environmental Engineering, Stanford
University, Stanford, CA. Krawinkler, H. (2002). A general approach
to seismic performance assessment, Proceedings,
International Conference on Advances and New Challenges in
Earthquake Engineering Research, ICANCEER 2002, Hong Kong, August
19-20.
Kunnath, S.K., Mander, J.B. and Lee, F. (1997). Parameter
identification for degrading and pinched hysteretic structural
concrete systems. Engineering Structures, 19 (3), 224-232.
Medina, R. A. (2003). Seismic demands for nondeteriorating frame
structures and their dependence on ground motions, Ph.D.
Dissertation, Department of Civil and Environmental Engineering,
Stanford University, Stanford, CA.
Rahnama, M. and Krawinkler, H. (1993). "Effects of soft soils
and hysteresis model on seismic demands." John A. Blume Earthquake
Engineering Center Report No. 108, Department of Civil Engineering,
Stanford University.
Sivaselvan, M.V., and Reinhorn, A.M. (2000). Hysteretic models
for deteriorating inelastic structures. Journal of Engineering
Mechanics, 126(6), 633-640.
Song, J-K., and Pincheira, J.A. (2000). Spectral displacement
demands of stiffness- and strength-degrading systems. Earthquake
Spectra, EERI, 16(4), 817-851.
Vamvatsikos, D., and Cornell, C. A. (2002). Incremental dynamic
analysis, Earthquake Engineering & Structural Dynamics, 31(3),
491-514.