SEISMIC DEMANDS AND CAPACITIES OF SINGLE-STORY AND LOW- RISE MULTI-STORY WOODFRAME STRUCTURES

Transcription

1 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -, Paper No. SEISMIC DEMANDS AND CAPACITIES OF SINGLE-STORY AND LOW- RISE MULTI-STORY WOODFRAME STRUCTURES Farzin ZAREIAN, Luis IBARRA, and Helmut KRAWINKLER SUMMARY This paper summarizes a systematic process for the evaluation of seismic demands imposed by ground motions on single-story and low-rise woodframe structures. This process involves development of representative sets of ground motions and scaling them to specific hazard levels, development of analytical component models that closely simulate experimental results (including monotonic and cyclic strength deterioration), and utilization of analytical models in the development of a new representation of seismic demands called the performance spectrum. Such a spectrum provides information on deformation demands as well as on the collapse capacity of a structural system. The proposed process is exercised on SDOF systems representing plywood and OSB wall assemblies of single story woodframe structures. The importance of various structural parameters on seismic demands and collapse capacities is evaluated. The study is extended to the MDOF domain, and it is demonstrated that the comprehensive demand results developed for SDOF systems also can be applied with confidence to low-rise MDOF systems through the use of an equivalent SDOF system. INTRODUCTION The main objective of the study summarized here is to introduce a systematic process for seismic demand and collapse capacity assessment of structural systems that are representative of single-story and low-rise multi-story woodframe structures. The emphasis is on the evaluation of story displacements and of the weight to strength ratio that causes collapse of the structural system. The results of this study are based on extensive simulations, using a wide range of important structural system parameters and a set of forty ground motions to generate representative seismic demands through nonlinear time history analyses and statistical evaluation of response data. The demand assessment process consists of the following components: Development of representative sets of ground motions Consistent scaling of ground motions to specific hazard levels Evaluation of experimental results for the purpose of improving analytical models Graduate Student, Dept. of CEE, Stanford Univ., Stanford, CA 9-, USA, farzin@stanford.edu Senior Research Engineer, Southwest Research Institute, USA, luis.ibarraolivas@swri.org Professor, Dept. of CEE, Stanford University, Stanford, CA 9-, CA, krawinkler@stanford.edu

2 Development of analytical component models that closely simulate experimental results, including monotonic and cyclic strength deterioration Utilization of these models to develop demand information in a new representation scheme, called the performance spectrum. Utilization of these models for assessing the collapse capacity of structural systems. The information developed here can be used to evaluate the adequacy of current code provisions and to provide the background for a rational performance-based seismic design procedure. SEISMIC HAZARD AND GROUND MOTION SELECTION Seismic Hazard The purpose of this demand study is to provide information that assists in performance assessment and seismic design. Thus, the need exists to tie in the demands with well established hazard or design levels. Hazard levels quoted most widely in recent codes and guidelines are /, /, and / hazards, with return periods of 7, 7, and 7 years, respectively. In view of the emphasis of this study on California conditions, the UBC 97 ground motion spectrum for seismic zone and soil type D (and without near-fault effects) is chosen as the primary ground motion spectrum for demand assessment. This spectrum can be considered as a / hazard spectrum. The argument for equating the UBC 97 ground motion spectrum with a USGS/FEMA / hazard spectrum is based on a comparison with spectral values used in the SAC steel project (Somerville []) for the same soil type. The points for the / hazard match well with the UBC 97 spectrum, justifying the use of the latter as a representative / spectrum for soil type D. Ground Motion Selection and Scaling Considering the large variation in the intensity and frequency content of ground motions, there is no option but using statistical means to predict seismic demands. The selection of representative ground motions is a critical aspect of any demand study. A necessary hypothesis is that within the range of applicability of the results, the ground motion frequency content is not sensitive to earthquake magnitude, M, and source to site distance R. Clearly, the intensity strongly depends on M and R, but this is accounted for in the hazard analysis that results in a uniform hazard spectrum. The insensitivity to M and R does not hold true for near-fault ground motions, and for this reason near fault ground motions are not considered here. The following discussion is concerned with the selection and scaling of a sufficiently large set of ordinary ground motions, where ordinary implies that the records do not contain near-fault forward directivity characteristics. For this study a set of ground motions records, denoted as LMSR-N, which fall into a bin with boundaries of. < M < 7 and km < R < km are chosen to represent large magnitude small distance records (Medina []). These ground motions do not include soft soil and basin effects, and therefore the computed demands are not directly applicable to these conditions. In order to make selected ground motion records compatible with a specified hazard level, it will be necessary to scale the records to a common intensity measure. The most widely used technique is to scale each record so that its spectral value at the first mode period of the structure coincides with the spectral value of the targeted hazard spectrum. Such a scaling technique is illustrated in Fig. for T =. second. It is seen that all spectra go through the same point at T =. second. This scaling technique has the great advantage of simplicity and it assures that at least the elastic first mode response is always properly represented. However, the disadvantage of this technique is the dispersion in the spectral values at all other periods but the period of scaling, see Fig.. The consequence is that there will be a

3 significant dispersion in the seismic demands once a structure enters the inelastic range (period elongation) or if higher mode effects are important. Also, the median spectrum of the ground motions may not (and usually does not) match the uniform hazard spectrum in the period range of interest for the specific structure. As a consequence, there may be a systematic error associated with this difference in spectral shapes. This is illustrated in Fig., which shows the median spectra of the LMSR-N record set scaled to T =. and T =. seconds. The shape of the median spectra is the same, but the intensity is different because of the scaling to different periods. The difference in the median spectra is significant, but the error introduced for each specific structure may be small because the primary period range of interest is in the neighborhood of the first mode period. Sa/g... Elastic Response Spectra LMSR-N, Scaled to T =. sec., ξ=% Individual ground motion spectrum Median UBC Uniform Hazard Spectrum (%/) Sa/g.. Elastic Response Spectra LMSR-N, Different Scaling, ξ=% UBC Uniform Hazard Spectrum (%/) T Scaling (T=. sec.) T Scaling (T=. sec.)..... Period (sec.) Figure. Median Spectra of Records Scaled to Common S a at. Second.... Period (sec.) Figure. Median Spectra of Records Scaled to Common S a at. and. Seconds. EXPERIMENTAL EVIDENCE AND ANALYTICAL MODELING OF LATERAL LOAD RESISTING COMPONENTS OF WOODFRAME STRUCTURES Observations from Experimental Studies In the process of building analytical models and identifying their feasible range of parameters, it was decided to evaluate the full range of plywood and OSB wall behavior, considering that such walls likely will make up the major portion of the reliable lateral load resisting system of wood frame structures. Other relatively deformable elements, such as stiff framing elements, sheeting, gypsum board walls, etc., will make the structural system stronger but likely will not radically change its deformation response to one that falls way outside the range of behavior of wood panels. Presence of stucco walls may radically increase the initial stiffness and the elastic strength but because of their brittle nature, sooner or later they will crack and the strength will drop rather rapidly. Thus, the presence of very stiff and strong stucco walls are not considered in this study. For this study, the experimental studies reported in Uang [] and SEAOSC [], which are concerned with the monotonic and cyclic lateral load-displacement behavior of plywood and OSB wall panels of different configurations, thickness, and nailing patterns are evaluated. Based on experimental results, the following observations can be made, see Fig. : The monotonic and cyclic load-displacement responses are nonlinear even at very small displacements. The monotonic response can be described by a continuously decreasing tangent stiffness, which gradually approaches a zero value and takes on a negative value after maximum strength is attained. The negative value of the tangent stiffness varies considerably, depending on specimen

4 details and boundary conditions. In some cases the strength decrease after maximum strength is gradual, in other cases it is rather abrupt and sudden. The cyclic response is characterized by pinched hysteresis loops, i.e., it consists of a large unloading stiffness, followed by a rapid decrease in stiffness (due to slip at nail holes or anchors), followed by a rather sudden increase in stiffness (due to re-establishment of bearing between nails and wood), and followed by a decrease in stiffness due to nail yielding, nail pullout, or wood crushing. The cyclic response deteriorates as the number and amplitude of the cycles increases. The deterioration depends on the loading history applied to the specimen. The following four modes of cyclic deterioration can be identified, see Fig. : basic strength deterioration (), post-capping strength deterioration (), unloading stiffness deterioration (), and accelerated reloading stiffness deterioration (). The identification of a definite elastic stiffness and yield strength is not possible. Any representation of the response by straight lines requires much judgment. In all test results the unloading stiffness, K u, determined from judgment, is clearly larger than the effective initial stiffness, K i, determined from judgment. Load (kips) - Monotonic ISO Force F F c Fmax F y κ sfmax Ke δ y α s Ke C κ d δ perm δ perm δ max Displacement α cke Displacement (in) B F y - Figure. Monotonic and Cyclic Experimental Response and Modes of Cyclic Deterioration Figure. Pinched Hysteresis Model Used in this Study Analytical Modeling Recognizing the need for a simple analytical model that contains conventional design parameters, a trilinear backbone (monotonic envelope) curve of the type illustrated in Fig. provides a reasonable representation of the load deformation characteristics of wood panels. The backbone curve is matched to the monotonic load-displacement response curve. The following notation, as shown schematically in Fig., is used to define the backbone curve throughout this paper: F y = yield strength F c = strength cap (maximum monotonic strength) δ y = yield displacement = displacement associated with F c (cap displacement) K e = elastic stiffness K i = initial stiffness, is the same as the elastic stiffness, but is used in conjunction with K u K u = unloading stiffness K s = strain hardening stiffness = α s K e K c = post-cap stiffness = α c K e α s = strain hardening ratio K s /K e α c = post-cap softening ratio K c /K e

5 The hysteresis model used for modeling wood panel hysteretic behavior during cyclic loading is the pinching model shown in Fig.. This model is based on the backbone curve described above and specific hysteresis rules. Briefly, during reloading it always targets the point associated with the maximum displacement in the previous history via two branches separated by a break point as shown by the character C and defined by two parameters κ s and κ d. Detailed description of the Pinching Hysteretic Model is available in Ibarra []. The cyclic deterioration of system properties is implemented in the analytical model using the cyclic deterioration model proposed firstly by Rahnama [] and modified and expanded by Ibarra []. The four modes of cyclic deterioration shown in Fig. are captured by a model defined by a deterioration parameter of the type: where: β I E i E t ΣE j c Ei βi = i Et E j j= c = parameter defining the deterioration in excursion i = hysteretic energy dissipated in excursion i = hysteretic energy dissipation capacity, expressed as a multiple γ of a reference energy term such as F y δ y or F c /Κ u, i.e., E t = γf y δ y, or E t = γ F c /Κ u = hysteretic energy dissipated in all previous excursions = exponent defining the rate of deterioration (in this study c =. is used) () This deterioration parameter can be applied to any or all of the four deterioration modes using a generic equation of the type: DP ( β ) = DP () i i, DP i where: DP i = deteriorated property (i.e. yield strength, unloading stiffness) after excursion i DP i = deteriorated property (i.e. yield strength, unloading stiffness) before excursion i β i,dp = given by Eq., employing an appropriate γ value to model the deteriorated properties, i.e., γ s for basic strength deterioration, γ c for post-cap strength deterioration, γ k for unloading stiffness degradation, and γ a for accelerated reloading stiffness degradation. Range of System Parameters Considered in This Study The range of structural parameters considered in this study mirrors the full range of plywood and OSB wall behavior. For such components, the experiments carried out within the CUREE woodframe program by Uang [] and the CoLA program, SEAOSC [], are utilized to establish the range of parameters of interest. The load-displacement curves of monotonic tests performed at UCI [], together with two monotonic tests of the CUREE test series performed at UCSD [], are presented in Fig.. For these tests, the displacement associated with the strength cap F c is safely bracketed between and, and the yield displacement appears to be safely bracketed between. and., as indicated by vertical lines. These limits, together with a thorough evaluation of all monotonic and cyclic behavior parameters led to the establishment of a matrix of structural systems that became the focus of the analytical studies. The basic parameters of the backbone curves of this system matrix are summarized in Fig.. The strength

6 cap, F c, and the associated displacement,, are used as anchor points, rather than the yield strength and yield displacement. For the cap displacement,, values of,,, and are used. This provides a rather dense matrix, which covers the realistic range of displacement capacities without the need for much interpolation in. The yield displacement δ y is obtained by using predetermined ductility capacity values /δ y of,, and 8. The yield strength F y is taken as a fraction,.,.7, and.8, of the strength cap F c. For the post-cap softening ratio, α c, it was decided to use two values, both referenced to the anchor point (F c, ). As shown in Fig., values of -. times and -. times F c / are used for K c. The first value is used to represent a slow decrease in strength, and the second value is used to represent a rapid decrease. In addition to variations in the backbone parameters, variations in the cyclic deterioration parameters have to be considered. Values for γ s,c,a of,, and are selected for this purpose, with γ k being twice the selected value. This amounts to a total of (cap displacements) times ( /δ y ratios) times (F y /F c ratios) times (α c values) times (γ values) = structural systems. To limit the number of systems to this value, only a single κ value of. is used to describe pinching, and zero residual strength is assumed in all cases. A limited parameter study was performed also on the unloading stiffness, K u, which in many cases is significantly larger that the initial loading stiffness, K i. For selected subsets of the structural systems, the effects of values of K i /K u equal to /, /., and / are evaluated. 8 UCI UCSD F c Range of Backbone Curves α s =. α c =.F c / ("small").8f c Load (lb.) 8 Force.7F c.f c α s =. α c =.F c / ("large") δ y(min) δ y(max) (min) (max) Displacement (inch) 8 Displacement Figure. Monotonic Test Results from UCI and UCSD Wood Panel Testing Programs Figure. Range of System Backbone Curves Used in Demand Study SDOF DEMANDS AND COLLAPSE CAPACITIES This section is concerned with the evaluation of demands imposed by earthquake ground motions on typical single-story woodframe houses. Two parameters are of specific interest: the story displacement, and the weight to strength ratio that causes collapse of the building. Both can be obtained from SDOF studies, provided the hysteretic behavior of the lateral load resisting system is known and can be described with the tools discussed in earlier sections. For the types of structural systems identified in the previous section, and using the aforementioned set of LMSR-N records, comprehensive data on displacement demands and collapse capacities have been derived and are documented in Krawinkler [7]. Here, only a few pertinent observations are made, and a new representation scheme for demand data is introduced.

7 A fundamental problem is that for plywood and OSB wall panels the elastic stiffness and associated yield strength F y, and yield displacement δ y, are ill defined. This is illustrated in Fig. 7, which shows a typical monotonic load-displacement test result for a plywood panel (the curved line somewhat hidden behind three simulations). It is evident that there is no best bilinear elastic strain hardening fit to this curve, whereas values for F c,, and K c are rather well defined. Three judgmental bilinear diagrams are placed between the origin and the point (F c, ), each one of them defensible, but with elastic stiffnesses that vary by a factor as much as. Depending on which stiffness is selected for analysis, the demand predictions for periods that differ by a factor as large as. have to be consulted. It is difficult to say what the outcome will be, because the ratios /δ y and F y /F c also vary between the three cases if the loaddeformation response is to go through the common point (F c, ). The consequence is that conventional means of demand evaluation, which usually are period and yield level based, will provide ambiguous predictions. This is particularly important for short period structures, such as single story woodframe buildings, because of the sensitivity of the inelastic response to period and yield level, which is illustrated in Fig. 8. Moreover, it can be shown that large R-factors (strength reduction factors F e /F y ) in the very short period range are unrealistic, and therefore, graphs of the type shown in Fig. 8 may be misleading. For instance, tests on plywood and OSB panels rarely will exhibit an effective yield displacement of less than. inch (7. mm), whatever judgmental definition of yield displacement is used. Using inches as units and the definition R = (S a /g)/(f y /W), the elastic period becomes T = π W /( gk) =. ( W / Fy ) δ y =. [ R /( Sa / g)] δ y () which means that for T =. sec. and δ y =. inch the spectral acceleration would have to be.g in order to result in a R-factor of. Thus, an R-factor on the order of or larger rarely ever will be realistic. Load (kips) F c 8 K i UCSD Test PWD East Wall Pinching Model, κ=., F c =8. Kips, =.7" /δ y =var., F y /F c =var., α s =var., α c =var., K i /K u =var., λ= K c K u 8 Drift (in) C C m m = = δ in Ξ in / /Ξ δ el el.. Displacement Modification Factor, C R Bilinear Model, LMSR-N, ξ=%, No P-δ, α s=. R=. R= R= R= R= Period (sec.) Figure 7. Matching of Backbone Curves to Result of a Monotonic Test Figure 8. Period Dependence of Ratio δ in /δ el for Specified Values of R-factor Moreover, providing demand data in terms of T requires a commitment to T in design. This may not be the best choice either. Up front, the period is not known (even if a clear elastic stiffness exists) because the structural system has first to be selected. In general, this may require an iteration process, particularly if the seismic demands are sensitive to the period range of interest. These arguments, and others, point towards the need for an alternative demand representation, which focuses explicitly on the stable point of available load-deformation diagrams (F c, ), and de-emphasizes

8 the importance of yield strength and yield displacement. Such a representation, called the performance spectrum, is discussed next. Performance Spectrum The performance spectrum is a different way of representing the peak deformation demand for inelastic SDOF and MDOF systems. It can be postulated that structural systems have a bracketed range of deformation characteristics which can be characterized by the family of simplified models shown in Fig.. The stable anchor point for strength, in general, is either the strength cap F c, or the yield strength F y. For a given type of shear panel, the deformation behavior (defined by δ y and ) is given, but the strength can take on any value. Let us use the case of shear walls (of either RC, masonry, or wood) as an example. Somewhat simplified, the yield displacement of such walls is a material property (presuming that shear deformations control) and does not depend significantly on the length, thickness, or number of walls. But the strength is, more or less, linearly proportional to length, thickness and number of walls. Thus, providing sufficient lateral resistance or stiffness, often means adding walls of similar displacement characteristics. Thus, it is reasonable to assume that in the design process δ y and are given (or targeted) values, but F y or F c (the number or length or thickness of walls) is tuned to fit the performance target and depends on the mass (weight) the system has to support. If δ y is assumed to be constant, then, as Eq. () shows, the period of the system varies in proportion to W / F ). ( y A performance spectrum is a representation that illustrates the relationship between a system strength parameter (either W/F y or W/F c ) and an engineering demand parameter, EDP (for an SDOF system the basic EDP is the system displacement), given that the shape of the load-deformation response of the system is fixed (in terms of a reference displacement such as δ y or, the shape of the backbone curve, and appropriate hysteretic rules). The use of the system strength parameter W/F y (or W/F c ) implies that either the weight (mass) or the yield strength (or strength cap) of the system is variable. It makes no difference which of the two is varied because both have the same effect on the period (since δ y remains constant). In the performance spectrum context, the response of SDOF systems to ground motions is determined by subjecting the structure to a set of representative ground motion records, while keeping the yield and cap displacements constant and varying the ratio W/F y or W/F c (i.e., the strength parameter F y or F c or the weight W). This approach is fundamentally different from conventional demand assessment approaches, as the requirement of constant δ y means that the period of the SDOF system varies continuously as the ratio W/F y or W/F c is changed, whereas in conventional approaches the period T is kept constant and the yield displacement is changed. Unfortunately, this also means that the response becomes dependent on the hazard level, i.e., the parameter W/F y (or W/F c ) can no longer be normalized to S a at the period T associated with the elastic system. The reason is that the period, as given by Eq. (), varies with W/F y and, consequentially, the W/F y versus displacement relationship becomes non-proportional to the intensity of the ground motion. Thus, performance spectra always are associated with a specific hazard level (which is assumed to be described by a uniform hazard spectrum). The basic strength parameter for performance spectra could be W/F y or W/F c. In concept, it makes no difference because the two parameters are related by the ratio F y /F c, which is one of the parameters belonging to the specific structural system for which the demands are being evaluated. For plywood or OSB panel systems, F c is a stable value whereas the determination of F y requires much judgment. Therefore, it is decided to use W/F c as the basic strength parameter. If the results presented in this section are to be viewed in the W/F y domain, all that is needed is to multiply the presented W/F c values by the ratio F c /F y.

9 A conceptual illustration of a performance spectrum is shown in Fig. 9. It belongs to a structural system whose backbone curve is defined by a yield displacement δ y and a cap displacement, and whose hysteretic behavior follows specific rules. It is obtained from the response to a set of ground motions that represent a specific hazard level. Statistical values are determined as discussed later. For each ground motion, the ratio W/F c is incremented in small steps, and for each increment the maximum displacement is obtained from time history analysis. The elastic period of the system changes according to Eq. () for each new value of W/F c. Thus, one can place a period axis on the graph, as is done on the right vertical axis in Fig. 9. The EDP of primary interest is the maximum displacement of the SDOF system. There is no reason to normalize this quantity, because its relation to the system characteristics is uniquely defined for the specific structural system selected for the analysis. There are different ways to interpret a performance spectrum. For the specified ground motion hazard, it provides a complete picture of the displacement response of a given system as a function of W/F c, from elastic behavior up to the point of collapse. Thus, it provides all needed information for performance assessment and for design for the hazard level for which the spectrum has been computed. Within limitations, the performance spectrum also can be scaled to other hazard levels by multiplying the W/F c ratio by the inverse of the ratio of short period spectral accelerations (S as ) of the uniform hazard spectra. These limitations are discussed in Krawinkler [7]. The performance spectrum can be used for different purposes. It can be used to assess performance at the given hazard level, using any performance level of interest, ranging from drift based damage control to collapse. It can be used for design (this is probably its primary function), by deciding on the strength F c, (or F y ) given the seismically effective weight and given desired performance in terms of a displacement limit (such as δ t in Fig. 9) or an acceptable probability of collapse (median values of the collapse capacity at the given hazard level are shown in Fig. ). It also can be used to derive R-factors for design, using either the conventional definition R = (S a /g)/(f y /W) or the modified definition R = (S a /g)/(f c /W). Appropriate values for F c /W (or F y /W) can be obtained from the performance spectrum, and the associated S a needs to be computed from the hazard spectrum, using the period given by Eq. (). In all cases the option exists to use the strength cap F c (as presented directly in the graphs) or the yield strength F y as a strength parameter. The advantage is that the elastic period does not show up explicitly as a rigid design parameter; rather it is the deformation behavior of the structural system that is explicitly represented in the performance spectrum. W/Fc δ y δ t median 8th percentile Displacement Period T Weight / Strength W / F c Individual Records Median 8 Percentile C ollapse Point.... Displacement δ max (inch) Figure 9. Illustration of Performance Spectrum Figure. Statistical Evaluation of δ max Given (W/F c ), and of Collapse Capacity

10 The writers do not take credit for having invented the type of representation given in the performance spectra. It has been used in New Zealand for rating of structural components (Dean [8]). Figure presents an example of the data and process involved in arriving at statistical values for performance spectra and for collapse capacity. The thin grey lines represent performance spectra for the forty individual records. The traces vary widely, and individual curves may have several returns (decrease in displacement for an increase in W/F c ). These returns are in the nature of the beast, and acknowledge the great variability in the frequency content of the ground motions. Statistics could be performed horizontally (to obtain statistical measures of displacement, given specific values of W/F c ), or it could be performed vertically (to obtain statistical measures of W/F c, given specific values of displacement. The two statistical measures will be similar but not identical because of the returns in the individual spectra. In order to avoid the problem of double counting curves that contain returns, it was decided to obtain statistical values (median and 8 th percentile) for the performance spectrum through horizontal statistics. The counting process is used, rather than assuming a distribution, to avoid problems when the data set becomes incomplete because of collapses (individual curves turning horizontal). Tests were performed on assessing the differences between counted statistical measures and computed values from a lognormal distribution, using cases in which the data set is complete, and the differences were found to be small. Most individual curves become horizontal sooner or later, i.e., the W/F c parameter no longer can be increased, which constitutes collapse of the system. The last stable value of W/F c, which is marked on each curve in Fig., is denoted as the collapse capacity of the system for the ground motion for which the curve is determined. Thus, statistical values can be obtained for the collapse capacity of the given system at the given hazard level. This can be achieved only through vertical statistics, using either a lognormal distribution (as shown in the figure) or by counting of collapse points. The collapse capacity issue is summarized later. Typical median and 8 th percentile performance spectra are shown in Fig.. They belong to SDOF systems whose load-displacement behavior can be described by the trilinear backbone curve shown on the figure ( =, /δ y = 8, F y /F c =.), when subjected to ground motions that represent the / hazard spectrum shown in Fig.. Spectra are shown for values of the cyclic deterioration parameter γ of infinite,, and, indicating that cyclic deterioration is only important when the displacement demands approach the capping displacement. The figure shows, for instance, the expected (median) displacement the system will experience as a function of W/F c, and it shows, approximately, what base shear coefficient F c /W should be used in design if the displacement is to be limited to a specific value. It also shows that there is a significant probability of collapse if the ratio W/F c approaches.. Many figures of this kind are compiled in Krawinkler [7]. Sensitivity of Performance Spectra to Different Models of the Same Structure One strong argument for the use of performance spectra is that they no longer depend on a commitment to an elastic stiffness and yield strength. The problem inherent in defining these quantities has been pointed out previously and is illustrated in Fig. 7. Three different backbone curves are illustrated, but all models have the anchor point (F c, ) as well as the post-capping stiffness K c and the unloading stiffness K u in common. The ratio of elastic, or better, initial stiffness K i to the unloading stiffness K u is taken as /, /., and / in the three models. Median performance spectra for the three models are shown in Fig.. The observation is that the three spectra vary insignificantly. Thus, regardless of which model is used, the performance evaluation for this structural system would come out about the same. What the selected models have in common is the

11 anchor point (F c, ), which largely controls the response. This supports the argument for using the point (F c, ) as the control point of the backbone curve and to use performance spectra rather than conventional (constant period) approaches for demand evaluation of woodframe structures. Weight / StrengthΞ W / F c Displacement δ max (inch) gama=inf. (median) gama=inf. (8 Percentile) gama= (median) gama= (8 Percentile) gama= (median) gama= (8 Percentile) Figure. Performance Spectra for a System with Different Cyclic Deterioration Rates Weight / StrengthΞ W / F c Model Model Model 8 Displacement δ max, Median (inch) Figure. Median Performance Spectra for Three Models of System Shown in Figure 7 It is noteworthy that even the initial branch of the performance spectra is not very different for the three models, even though the initial stiffness varies by a factor as large as.. The reason for the relatively small difference is that for the flexible model, i.e., K i /K u = /, the fact that K u is twice K i causes energy dissipation from the beginning, i.e., there is no elastic range of behavior. This early energy dissipation reduces the displacement compared to an elastic system. Sensitivity of Performance Spectra to Cap Displacement and Ductility Capacity /δ y The range of performance spectra, as a function of (ranging from to ) is illustrated in Fig. for systems with a ductility capacity /δ y =. In the elastic range (up to W/F c =.) the displacement demands for a given W/F c value are inversely proportional to the relative elastic stiffnesses, as expected. This proportionality is approximately maintained rather far into the inelastic range. All systems show reserve capacity beyond the point defined by and its associated W/F c value. This reserve capacity is larger for stiff systems (small ) and smaller for flexible systems because of P-delta effects that become important for systems with large displacements. The sensitivity of performance spectra to the ductility capacity /δ y is illustrated in Fig. for systems with =. The very large differences in the initial slopes of the spectra are only a consequence of the use of a constant and a variable /δ y. The important effect to notice is that for these flexible systems there is little reserve capacity beyond because of P-delta effects, regardless of the ductility capacity. The benefit of larger ductility is evident primarily on the vertical axis, as the W/F c values associated with displacements approaching and exceeding are clearly larger for more ductile systems. But the increase in W/F c beyond is not overwhelming. Thus, for systems with a rapid post-cap decrease in strength, the ductility capacity (defined by /δ y ) alone does not make a drastic difference in the W/F c values near and near collapse. The reason is that for the same the difference between /δ y = and 8 diminishes as the large displacement region in reached.

12 Weight / StrengthΞ W / F c 7 PERFORMANCE SPECTRA - DISPLACEMENTS =var, /δ y =, F y /F c =., α s =., α c =-.(L), K i /K u =/, γ s,c,a =, γ k =, λ= 8 Displacement δ max, Median (inch) dc=" dc=" =" dc=" =" δdc=" c Weight / StrengthΞ W / F c 7 PERFORMANCE SPECTRA - DISPLACEMENTS =", /δ y =var., F y /F c =.8, α s =var., α c =var.(l), K i /K u =/, γ s,c,a =, γ k =, λ= 8 Displacement δ max, Median (inch) dc/dy=8 /δ y =8 dc/dy= /δ y = dc/dy= /δ y = Figure. Effect of Cap Displacement on Median Performance Spectra, /δ y = Figure. Effect of Ductility Ratio /δ y on Median Performance Spectra, = Collapse Capacity Derived from Performance Spectra As illustrated in Fig., performance spectra for individual ground motions permit tracking of the system response till collapse takes place (the slope in the spectrum becomes zero). Performing vertical statistics on the last point of the spectra permits evaluation of the W/F c value causing collapse. This value is called the collapse capacity of the system associated with the seismic hazard for which the performance spectra are computed (in this case the / hazard represented by the equal hazard spectrum shown in Fig. ). Collapse may occur because of P-delta effects, post-capping strength deterioration (defined by α c ), cyclic deterioration (defined by γ s,c,a ), or combinations thereof. Figure shows the dependence of the median collapse capacity on various combinations of α c and γ s,c,a, for the full range of values and for ductility ratios of and 8. Interpretation is left to the reader, as the benefits or detriments of the individual parameters and their combinations can be judged from the graphs. In design for life safety, these collapse capacities take on considerable importance. 7 DEPENDENCE OF COLLAPSE ON α c & γ =var, /δ y =8, F y /F c =.8, α s =.7, α c =var., K i /K u =/, γ s,c,a =var., γ k =var., λ= 7 DEPENDENCE OF COLLAPSE ON α c & γ =var, /δ y =, F y /F c =., α s =., α c =var., K i /K u =/, γ s,c,a =var., γ k =var., λ= W/F Collapse W/F Collapse Cap Displacement (inch) Cap Displacement (inch) γ s,c,a =, γ k = γ s,c,a =, γ k = γ s,c,a=, γ k= γ s,c,a =, γ k = gama=inf. α c="small" Ac=s gama= α c="small" Ac=s gama=inf. α Ac=l c="large" gama= α c="large" Ac=l γ s,c,a=, γ k= γ s,c,a=, γ k= γ s,c,a=, γ k= γs,c,a=, γk= gama=inf. α Ac=s c="small" gama= α Ac=s c ="small" gama=inf. α c="large" Ac=l gama= α c ="large" Ac=l Figure. Median Collapse Capacities for Various Values of, α c and γ s,c,a ; (a) /δ y = 8, (b) /δ y =

13 MDOF DEMANDS AND COLLAPSE CAPACITIES The dynamic response of low-rise ( to stories) woodframe structures is dominated by their fundamental mode of vibration. The extent to which the seismic demands and collapse capacities of these structures can be derived from the results of an equivalent SDOF system is assessed and briefly discussed in this section. A series of -story MDOF structures, which cover the full range of -story woodframe structures from a uniform strength and stiffness distribution over the height to severe weak first story conditions, and which deform in a pure shear mode in each story, are evaluated for this purpose. These structures are modeled as frames with rigid beams and flexible columns, with the latter being capable of developing plastic hinges at the ends, and with the properties tuned such that the desired story loaddisplacement behavior is obtained. Individual performance spectra for the ground motions of the LMSR-N set are obtained by increasing the mass of the structure, which is distributed equally to the three floors, in small steps and performing time history analysis to compute maximum roof and story drifts. Median spectra are used to establish a correlation between MDOF results and results obtained from an equivalent SDOF system, ESDOF. Detailed description of the MDOF and ESDOF models and the demand evaluation process can be found in Krawinkler [7]. The use of an ESDOF system in deriving MDOF response characteristics has been studied by many. In this work the formulation discussed in Seneviratna [9] is utilized. The assumption is that the deflected shape of the MDOF system can be represented by a shape vector, Φ that remains constant during the excitation, regardless of the level of deformation. The shape vector and the force-deformation characteristics of the ESDOF system are determined from the results of a pushover analysis of the MDOF system. In this study, the base shear roof displacement relationship of the -story MDOF structures is idealized into a trilinear backbone curve of the type shown in Fig. and transformed into the ESDOF domain. The inelastic stability coefficient, as recommended by Bernal [], is incorporated in order to model the P-delta effects. In order to evaluate the response sensitivity to the shape vector, three shape vectors are utilized corresponding to a roof displacement equal to the yield displacement δ y, two times the yield displacement, δ y, and the cap displacement,. For each MDOF system, three different ESDOF systems are derived and performance spectra are generated in the same fashion as described in the previous sections. A comparison of median performance spectra and collapse capacities of MDOF systems and spectra derived from their corresponding ESDOF systems shows good correlation with relatively small differences on the order of %, regardless of the choice of shape vector. A typical result is presented in Fig., which shows first story drift performance spectra for a three story structure, derived from MDOF analysis and from its three ESDOF systems with different shape vectors. The presented results are for a -story structure with a soft first story in which the lateral stiffness and strength is / th of the stiffness and strength of stories and. For -story structures without a soft story, the prediction of performance spectra from ESDOF systems is also rather accurate, particularly if shape vectors corresponding to a roof displacement of δ y or are employed. The results lead to the conclusion that performance spectra of ESDOF systems can be used with confidence to predict performance spectra for MDOF structures of the type discussed in this paper.

14 Total Weight / V y.... MDOF Results (Median) and Prediction from Eq. SDOF Structure S,8, st Story Displacement MDOF_st Eq. SDOF_dc_st Eq. SDOF_dy_st Eq. SDOF_dy_st 8 Maximum Displacement (inch) Figure. Performance Spectra for First Story Displacement of -Story Structure, Derived From MDOF Analysis and From ESDOF Systems CONCLUSIONS A systematic approach for the evaluation of seismic demands and collapse capacities for single-story and low-rise woodframe structures is summarized. A new representation scheme is introduced, called the performance spectrum. The advantage of the performance spectrum representation is that it deemphasizes the importance of the elastic stiffness and yield strength, which are difficult to define in woodframe structures, and that it follows a displacement-based approach, insofar that the resulting spectra provide direct insight into the displacement demand of the selected system at all performance levels of interest for a given hazard level. A detailed explanation of the performance spectrum along with its applications for design and assessment of structural systems is provided. The main conclusions drawn from this study are as follows: A relatively large set of ground motion records is needed in order to evaluate seismic demands with sufficient confidence. The median value and the measure of dispersion are sensitive to the size of the record set. The variability in the frequency content of the ground motions introduces a large dispersion in the seismic demands of inelastic systems. This study is concerned only with the response to ordinary ground motions recorded on soil type D, and the results cannot be extrapolated for sites with very different soil conditions, especially soft soil, or for sites with near-fault forward directivity effects. The load-displacement behavior of plywood and OSB wall panels can be described adequately by a tri-linear backbone curve, a pinched hysteresis model, and a deterioration model that accounts for four modes of cyclic deterioration. The stable anchor point for describing the loaddisplacement behavior of plywood and OSB wall panels is the maximum strength (strength cap) and the associated displacement. Thus, it is desirable to describe the seismic demands in terms of these parameters. The sensitivity of seismic demands and collapse capacities to variations in important structural parameters is evaluated. The results show that effect of these parameters on the displacement response depends on the performance regime of the structure. For instance, variation in cap displacement,, and ductility capacity, /δ y, have a significant effect on displacement demands at all performance levels. However, variations in post-cap strength deterioration, α c, and cyclic deterioration, γ s,c,a, affect primarily the system behavior close to collapse. It is also found that the effect of any one structural parameter cannot be evaluated independently of the others because of the interaction effects of the parameters.

15 It is demonstrated that the demand results for SDOF systems can be applied with confidence to low-rise MDOF systems through the use of an equivalent SDOF system. ACKNOWLEDGMENT Funding of the research summarized in this paper war provided by the Consortium of Universities for Research in Earthquake Engineering (CUREE) as part of the CUREE-Caltech Woodframe Project ( Earthquake Hazard Mitigation of Woodframe Construction ), under a grant administered by the California Office of Emergency services and funded by the Federal Emergency Management Agency (FEMA). Additional funding for the performance spectra study was provided by the NSF sponsored Pacific Earthquake Engineering Research (PEER) Center. REFERENCES. Somerville, P., et al., 997. Development of Ground Motion Time Histories for Phase of the FEMA/SAC Steel Project, SAC Background Document, Report No. SAC/BD-97/.. Medina, et. al., Seismic Demands for Nondeteriorating Frame Structures and Their Dependence on Ground Motions, Ph.D. thesis, Stanford University, Stanford. Uang, C.M.,. Loading Protocol and Rate of Loading Effects, Report on CUREE/Caltech Woodframe Project Task.., CUREE Publication W-.. SEAOSC,. "Report of a Testing Program of Light-Framed Walls with Wood-Sheathed Shear Panels", City of Los Angeles Department of Building and Safety.. Ibarra, et. al., Global Collapse of Frame Structures Under Seismic Excitations, Ph.D. thesis, Stanford University, Stanford. Rahnama, M. and Krawinkler, H., 99. Effects of Soft Soil and Hysteresis Model on Seismic Demands, John A. Blume Earthquake Engineering Center Report No. 8, Department of Civil Engineering, Stanford University. 7. Krawinkler, H., Zareian, F., Ibarra, L., Medina, R., and Lee, S.,. Seismic Demands for Single and Multi-Story Wood Buildings, Report on CUREE/Caltech Woodframe Project Task.., CUREE Publication W- 8. Dean, B.L.,. Rating Seismic Bracing Elements for Timber Buildings, Proceedings of WCEE, New Zealand. 9. Seneviratna, G.D.P.K., and Krawinkler, H., 997. Evaluation of Inelastic MDOF Effects for Seismic Design. John A. Blume Earthquake Engineering Center Report No., Department of Civil Engineering, Stanford University.. Bernal, D., 99. "Instability of Buildings Subjected to Earthquakes," Journal of Structural Engineering, ASCE, 8(8), pp. 9-.