krawinkler paper otani

1-1: Krawinkler Otani Symposium 2003

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

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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

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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=25

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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

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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

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[Sa(

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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

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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

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(Sa/

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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%,

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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=-0.30, s,c,k,a=Inf, =0

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(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%,

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[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

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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

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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

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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



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


(a) (b)



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




0

0.2

0.4

0.6

0.8

1

0 5 10 15[Sa,c(T1)/g]/

Prob

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f Col

laps

e


(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.

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