Appendix E Evaluation of the Accidental Torsion Requirement in ASCE 7 by the FEMA P695 Methodology

Transcription

1 Appendix E Evaluation of the Accidental Requirement in ASCE 7 by the FEMA P695 Methodology E. Overview The purpose of this study is to evaluate the significance of the accidental torsion requirement in Section of ASCE 7-0 for buildings in SDC B. The accidental torsion provisions require application of a +/-5% offset of the center of mass in each of two orthogonal directions to compute a torsional moment, thereby increasing the design seismic base shear. The primary goal of this study is to quantitatively examine the possible elimination or revision of the accidental torsion requirement for SDC B buildings designed according to the newly proposed stand-alone code document. To this end, the study quantifies the effect of the accidental torsion design requirement in terms of building collapse capacity and collapse risk for a variety of SDC B buildings in order to determine the consequences, or lack thereof, of removing or revising the accidental torsion requirements. Seismic ground motions may induce torsional response in buildings. Some of this torsion is created by asymmetrical building geometry, hereafter referred to as inherent torsion. In contrast, accidental torsion is unexpected and may occur for a variety of reasons including: asymmetric distribution of lateral ground motions across the plan of the building, asymmetric stiffness contributions from the gravity system or nonstructural elements not accounted for in design, uneven live-load distribution, or changes in the center of rigidity due to nonlinear behavior. To account for all of these potential sources of accidental torsion, ASCE 7 defines an accidental torsional design moment to be considered in buildings with rigid diaphragms, in addition to any inherent torsion that may exist. The additional accidental torsional moment is equivalent to the torsion due to applying the seismic load at a distance from the center of mass equal to 5% of the building dimension perpendicular to the applied lateral load (Section of ASCE 7-0). These provisions have the effect of increasing the design base shear in the frames and walls that resist lateral forces. BSSC SDC B E: Accidental Studies E-

2 4.86% Draft E. Literature Review on Accidental Significant research on accidental and inherent torsion in buildings has been conducted in the past, producing varying results (Stathopoulos et al. 2009; De Stefano and Pintucchi 2007). However, those that have specifically addressed the issue of accidental torsion in design codes have all concluded that it is not needed for most regular buildings; regular loosely refers to buildings that are not particularly torsionally flexible or irregular, although the exact definition varies from study to study (Stathopoulos et al. 2005, Chang et. al 2009). Some of the key findings of the past research are outlined here. De Stefano and Pintucchi (2007) provide a more complete summary and assessment of recent research on torsion in buildings. This section summarizes selected past research of torsional seismic response of buildings, based on three main categories of models: ) linear models, 2) simplified single-story shearspring models and 3) lumped plasticity nonlinear frame models. Results from linear models have shown that design accidental torsion is not a significant factor for building performance under earthquake excitation for many buildings (Chopra 992. A strength of the linear models is that they represent realistic building geometries, such as multi-story space frames, very accurately. However, the linear models cannot accurately simulate post yielding behavior and collapse and, more recently, detailed nonlinear models have been used to evaluate torsional response. Some of the first nonlinear models used for studying torsion in buildings were models that represented the aggregated behavior of lateral force resisting systems with bilinear shear springs in a single-story. These models have the advantage of being able to simulate behavior beyond the linear range. Anagnostopoulos et al. (2009) showed that the procedures for calibrating such simplified models is crucial for obtaining accurate results. They demonstrated that by calibrating single story shear spring models to more high end lumped plasticity models using pushover analysis, they could obtain results that qualitatively agreed with the results from the more sophisticated nonlinear frame models with lumped plasticity elements. The results of the single-story simplified models even with agreed lumped plasticity models having more than one story (based on 3 and 5 story models). However, Anagnostopoulos et al. (2009) recommends that strong caution be taken when calibrating simplified shear spring models. For example, some researchers have scaled building strength of the simplified models based on design loads, without modifying the stiffness. Increasing strength E-2 E: Accidental Studies BSSC SDC B

3 independently of stiffness in bilinear models leads to inaccurate measures of initial stiffness, but also artificially increases the yield displacement. Since much of the early work on torsion with nonlinear models used ductility demand as a performance measure, artificially increasing the yield displacement of a lateral force resisting system artificially increased its performance; this phenomenon has been addressed quite specifically by Tso and Smith (999). Anagnostopoulos et al. (2009) showed that carefully calibrated simplified shear-spring models predict greater ductility demand on the flexible side of irregular buildings, which is in agreement with linear models and lumped plasticity models. However, increasing strength independently of stiffness in simplified models leads to the exact opposite prediction; ductility demand is increased in the stiff side elements. Since they used ductility demand as a main performance measure, the calibration of the simplified models made the difference between being qualitatively right or qualitatively wrong in their predictions. More recent work using nonlinear models of frame structures includes Stathopoulos and Anagnostopoulos (2009) and Chang et al. (2009). Stathopoulos and Anagnostopoulos (2009) used one, three and five story RC space frames with lumped plasticity models of beam and column elements to assess the importance of design accidental torsion, concluding that it is insignificant for the building heights and torsional rigidities studied and ought to be re-examined. A similar study by Chang et al. (2009) examined six and twenty-story steel space frames and reached the same conclusion; design accidental torsion requirements are not significantly beneficial, for the building types they studied. Both of these studies used ductility demand as the main measure of performance. This study will expand on past work by considering a wider variety of torsional flexibility and irregularity in buildings, focusing on collapse capacity as the primary performance metric. To our knowledge, no studies of accidental torsion have used collapse as a performance measure, as we do here, but instead have relied mostly on ductility demands to quantify the impacts of accidental torsion (De Stefano and Pintucchi 2007). The building designs considered here have base shear levels consistent with SDC B, in contrast to most previous studies that considered SDC D ground motion levels, and consider a broad range of torsional flexibilities. BSSC SDC B E: Accidental Studies E-3

4 4.86% Draft E..2 Methodology In this study, we consider the effects of accidental torsion code provisions on a set of archetype nonlinear building models, which include torsionally stiff and torsionally flexible structures, and have ductility, mass, and strength characteristics of SDC B buildings. Since a main purpose of seismic codes is to reduce the likelihood of earthquake-induced collapse, accidental torsion requirements are evaluated with regard to seismic collapse risk. FEMA P695 proposes a methodology for systemically evaluating the seismic design provisions of new seismic resisting lateral systems on the basis of ensuring an acceptably low probability of collapse. The method uses building collapse capacity as a metric for determining appropriate response coefficients R, C d, and Ω 0 for newly proposed systems. The process for implementing the FEMA P695 methodology is illustrated in Figure E-. Figure E- Flow chart schematic of FEMA P695 methodology (FEMA 2009) E-4 E: Accidental Studies BSSC SDC B

5 In this study, the FEMA P695 method has been adapted to evaluate a particular code provision, namely the 5% offset requirement to account for accidental torsion, rather than a specific Seismic Force Resisting System (SFRS), but the main concepts have not changed. Rather than focus on a specific system, the method has been used to evaluate the collapse performance of a set of typical SDC B buildings designed with and without the accidental torsion requirement. To this end, an archetype design space is developed, analytical models created and analyzed, and their collapse performance is evaluated. The difference in collapse risk with and without accidental torsion provides quantitative information as to the importance of including accidental torsion requirements in SDC B. Each of these steps is documented in detail in the following sections. E.2 Archetype Design Space The objective of this study is to quantify the effect of the accidental torsion requirement on the design and safety of buildings in SDC B. Therefore, it is important to identify a range of archetype designs that encompass as many SDC B buildings as possible, with special emphasis on those buildings that may be most affected by accidental torsion requirements. This section discusses building characteristics that may affect the influence of accidental torsion requirements in design and how these characteristics were considered in developing a representative set of buildings. Table E-3 and BSSC SDC B E: Accidental Studies E-5

6 4.86% Draft Table E-4, at the end of this section, summarize the suite of archetype designs that are analyzed. Every archetype building is designed in two versions: one with and one without the accidental torsion design requirement considered, to provide a direct assessment of the impacts of accidental torsion design requirements on building collapse performance. The archetype design models were created by calibrating their linear and nonlinear properties to a set subset of baseline high end OMF frame models. E.2. Seismic Force Resisting System (SFRS) Building systems most commonly used in SDC B are less ductile than those used in higher seismic design categories. In fact, most have values of R, the response modification coefficient, of around 3. Due to the infeasibility of analyzing every available SFRS for SDC B, the models in this study are based on the design and behavior of reinforced concrete Ordinary Moment Frame (OMF) models. The choice of OMFs to represent SDC B buildings more generally is justified by this study s focus on measuring the effect of designing for accidental torsion on collapse capacity and collapse risk, not comparing specific systems. Reinforced concrete OMFs are used because they are non-ductile, their nonlinear behavior is fairly well documented and modelable, and they are commonly used in SDC B. In addition, the most important properties pertaining to collapse capacity such as ductility, overstrength, and deformation capacity are fairly similar to many other systems used in SDC B. E.2.2 Building Height Three different building heights are used in this study in order to capture the effects of designing for accidental torsion:, 4, and 0 stories. The height of 0 stories (32 ft.) was chosen as the tallest archetype structure because it is tall enough to adequately capture the effects of higher modes in tall buildings. Past studies by Chang et al. (2009) and Stathopoulos and Anagnostopoulos (2009) have suggested that accidental torsion requirements are less beneficial for taller buildings (5, 6, and 20 stories) than single story buildings. E.2.3 Building Weight Since gravity loads can play a major role in the design of SDC B buildings, a range of building weights are considered. The low and high gravity scenarios in this study are 00 psf and 200 psf of un-factored dead weight, respectively, for all stories except the roof level. Low and high roof weights are 80 psf and 60 psf, respectively, and are used for the single-story buildings. These values are intended to represent a reasonable range of E-6 E: Accidental Studies BSSC SDC B

7 weights of buildings, but are not linked to any particular floor system or occupancy. Live load was taken to be 20 psf at the roof level and 50 psf for all other stories, and live load reductions were made according to section 4.7 of ASCE 7-0 Past research has shown that gravity load levels can significantly affect system ductility, overstrength, and collapse performance (FEMA 2009). Only the high gravity load level was used for the 0-story archetype designs because the -story and 4-story archetypes showed that high gravity buildings performed worse overall and had more significant improvements from design accidental torsion than their low gravity counterparts. E.2.4 Building Plan Layout Most of the archetype building layouts considered in this study are symmetric (rectangular layouts). Past research (Llera and Chopra 995, Stathopoulos and Anognostopoulos 2005), have shown that accidental torsion requirements have a larger effect on the performance of symmetric buildings than asymmetric buildings, because the relative increase in torsional design forces due to accidental torsion increases as inherent torsion decreases. In addition, we consider buildings with different torsional rigidities because the torsional period or frequency affects response to earthquake excitation. The rectangular building plans follow the schematic in Figure E-2Error! t a valid bookmark self-reference., with overall building dimensions 200 ft. x 00 ft. and relative frame spacing of S/L=S /L =S 2 /L 2 =.0,,, and. This configuration is used for all of the rectangular buildings in this study. (In the context of this study, the term frame refers to any frame or wall line that is part of the lateral force resisting system). The extent to which designing for accidental torsion increases the design base shear in frame lines depends on the relative torsional stiffness of a structure and its frame-line spacing. This effect is illustrated using the building plan that is illustrated in Figure E-2. The building has plan dimensions L and L 2 and frames are spaced at distances S and S 2 apart. All frames are considered to have equal stiffness k. Taking a normalized design base shear of in each frame and then computing the additional shear due to accidental torsion produces the results shown in Table E-. In general, as relative frame spacing decreases, torsional rigidity decreases and the contribution of accidental torsion to the design base shear in frames increases. Frame 3 S BSSC SDC B E: Accidental 2 Studies E-7 L 2 =00' Frame 4 S =2S 2 Frame 2

8 4.86% Draft Figure E-2 Plan view of a symmetric archetype structure with a rectangular frame layout Table E- E Increase in Base Shear Due to the 5% Offset Accidental Requirement for Building Layout shown in Figure E-2 E S/L (Perimeter) Design Base Shear (rmalized) Frames &3 Frames 2&4 Total Design Base Shear, Accounting for Accidental Design Base Shear (rmalized) Total Design Base Shear, Accounting for Accidental In addition to the rectangular frame layout that was used most for most of the archetypes analyzed, a subset of archetypes with an I-shaped frame layout was also analyzed. I-shaped or similar frame layouts are common in parking garages and other structures. L 2 =00' E-8 E: Accidental Studies BSSC SDC B S

9 Figure E-3 I-shaped frame layout E.2.5 Building Plans with Inherent A few selected archetype buildings were analyzed with asymmetric building plan layouts, as depicted in Figure E-4. Two different inherent torsion plan layouts were used: one with S/L=S /L =S 2 /L 2 = and one with S/L=S /L =S 2 /L 2 =. For each of these layouts, two of the frames are located at the building s edge and the other two are inset according to the prescribed relative frame spacing. Eccentricities in each direction are labeled as e and e 2. Both of the archetype building geometries used to represent buildings with inherent torsion are classified as having horizontal irregularity type b (extreme torsional irregularity) according to ASCE 7-0. e L 2 =00' S =2S 2 S 2 CR X X CM e 2 L =2L 2 =200' Figure E-4 Inherent torsion frame layout E.2.6 Natural Accidental We use the term natural accidental torsion to describe the effective offset between center of mass and center of stiffness, accounting for the many sources of accidental torsion that may exist. Levels of natural accidental torsion were systematically introduced in the model, but not the design, by offsetting the center of mass (CM) of the models from the design CM along BSSC SDC B E: Accidental Studies E-9

10 4.86% Draft the diagonal of the building. Center of mass offset distances of 0%, 5%, and 0% of the total diagonal length of the building were used. E.2.8 Design Assumptions and Methodology for OMF Models A subset of the archetype buildings was designed as reinforced concrete OMF s according to ASCE 7-0 and ACI 38-0 and are listed in Table E-2. Each archetype building was designed for dead, live, and seismic loads using all applicable load combinations; additional loading from snow and wind were not considered. The design short period and one-second spectral accelerations were taken as the maximum allowable values for SDC B: S DS =0.33(g) and S D =0.33(g). The buildings were designed as space frames with 2-way slabs, having spans of 30 ft., and story heights of 5 ft. and 3 ft. in the first story and all other stories, respectively. For design, they were modeled as 2D portal frames with SAP2000, using the Equivalent Force Procedure (ELFP) to determine design loads, story forces and drifts. The design of all members was force controlled, with the exception of the ten story archetypes whose lowest six stories were governed by the stability (P- ) requirements of Section in ASCE 7. Columns of the one-story buildings were modeled pinned at the base, whereas all other designs used a fixed foundation assumption for design, to be consistent with common design practice. Each OMF design depended on the number of stories and gravity loads and had two versions, which are summarized in Table E-2. The first version was designed as a space frame with 30 ft. bays and 30 ft. of tributary width and an equivalent tributary seismic mass. Space frame OMFs were selected because they are common and have nonlinear behavior that we believe is representative of many SDC B type buildings. This design ignores accidental torsion effects, i.e. it is designed only for the base shear calculated according to the equivalent lateral force method. These are later referred to as the low base shear models because they have the lowest design base shear of all designs for their particular height and gravity load levels. In Table E-2, the low base shear designs are the odd numbered designs. The second version was designed with the same geometry and loads, except with larger design base shear due to the consideration of 5% accidental torsion (later referred to as the high base shear models). For symmetric archetypes, the increase was 32%, which was due to the base shear increase from accidental torsion when relative wall spacing (S/L) is in a building with the geometry shown in Figure E-2. Frames with extreme values of design base shear were selected for design (even numbered designs in Table E-2), because simplified models are later calibrated by interpolation of properties between high end OMF E-0 E: Accidental Studies BSSC SDC B

11 models. In addition, for select archetypes with inherent torsion, additional high end OMF frames were designed and modeled considering more extreme changes in design base shear. The high-end OMF models are designed as space frames with 30 ft. of tributary width, but the 3D frame layouts have just two frame lines in each direction and plan dimensions of 200 ft. x 00 ft., as shown in Figure E-2. te that these simplified models have only two frame lines in each orthogonal direction to more easily capture a wide range of torsional flexibilities, creating a discrepancy with the original OMF space frame design. The discrepancy between the two building plans was reconciled by adjusting the mass and weight of the 3D models to reflect the correct building mass and weight tributary to just two of the OMF frames. Since gravity loads contribute significantly to the frame element design moments and forces, much care was taken to design the two versions of each frame consistently. For each OMF, the lower base shear version was designed first. Columns were designed to be as small as possible while keeping the longitudinal reinforcement ratio below about 4.5%. Beams were designed as T-beams, but with smaller longitudinal reinforcement ratios (2.5%-3%) than columns. The beam longitudinal reinforcement ratios were often governed by maximum reinforcement requirements (which limit reinforcement and promote steel yielding before concrete crushing). Transverse shear reinforcement was designed with bar sizes ranging from #3 to #5, and bar size was kept consistent for all columns and for all beams throughout each building; in every case, rebar size was determined such that the maximum allowable spacing could be used for all or the elements of the frame, reflecting common engineering practice. After designing the frame with the lower base shear, the high base shear version of the same frame was designed. Starting with the first design, element sizes and reinforcement were increased to accommodate the larger loads. We aimed to keep reinforcement ratios as similar as possible by increasing the reinforcement and element sizes concurrently. Table E-2 E Design # Building Height (stories) Matrix of OMF Designs (Baseline Models) System Gravity (Story Weight) Relative Frame Spacing (S/L) Inherent torsion Design Accidental Concrete * ne 80 psf ne 2 OMF 5% BSSC SDC B E: Accidental Studies E-

12 4.86% Draft Design # Building Height (stories) System Gravity (Story Weight) Relative Frame Spacing (S/L) Inherent torsion Design Accidental 3 * ne 60 psf 4 5% 5 * ne 00 psf 6 4 5% 7 * ne 200 psf 8 5% 9 * ne psf 0 5% *Frame spacing does not matter if the building is symmetric and accidental torsion is not considered E.2.8 Design Assumptions and Methodology for Simplified Frame Models Simplified models have been constructed such that the design lateral earthquake force in each frame, without considering accidental torsion, is exactly the same as the baseline case for the corresponding high end OMF model, so that their nonlinear properties can be matched directly. For simplified archetypes designed for accidental torsion, the earthquake forces are increased and frame properties are obtained by interpolation between the low and high base shear versions of the high end OMF frames. This process is described in more detail in section E.4. E.2.9 Archetype Design Space Tables Table E-3 summarizes key properties of the archetype design space that has been used for this study. E-2 E: Accidental Studies BSSC SDC B

13 Table E-4 lists all the buildings and design properties considered in the study, including a total of 96 archetypical models. Table E-3 E Summary of Archetype Design Space Design # Building Height (stories) System Gravity (Story Weight) Relative Frame Spacing (S/L) Configuration Inherent Design Accidental Natural Accidental 96 Total Archetypes 4 0 Concrete OMF Low High *0.45 *0.4 *0.35 *0.3 Rectangular Frame Layout *I-Shaped Frame Layout ne (ally Symmetric) *25% 0% 5% 0% 5% 0% *Properties only represented by selected subgroups of the archetype design space BSSC SDC B E: Accidental Studies E-3

14 4.86% Draft Table E-4 E Full Archetype Design Space Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) 80 psf Rectangular Frame Layout 60 psf Rectangular Frame Layout ne (Symmetric) ne (Symmetric) 0 *0 5% 0% 0 E-4 E: Accidental Studies BSSC SDC B

15 Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) 60 psf Rectangular Frame Layout ne (Symmetric) 0 *0 5% BSSC SDC B E: Accidental Studies E-5

16 4.86% Draft Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) 5% 60 psf Rectangular Frame Layout ne (Symmetric) 0% * psf I- shape ne (symmetric) % E-6 E: Accidental Studies BSSC SDC B

17 Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) *0 +5% 60 psf Rectangular High Inherent (Extremely Asymmetric) +0% -5% -0% * psf Rectangular ne (symmetric) 5% BSSC SDC B E: Accidental Studies E-7

18 4.86% Draft Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) 4 00 psf Rectangular ne (symmetric) 0% * psf Rectangular ne (symmetric) 5% 0% psf Rectangular ne (symmetric) *0 E-8 E: Accidental Studies BSSC SDC B

19 Building Height (stories) Gravity (Story Weight) Relative Frame spacing (S/L) LRFS Configuration Inherent Design for Accidental Natural Eccentricity (Offset of CM) * psf Rectangular ne (symmetric) 5% 0% * The natural eccentricity is zero, but small amounts of torsion are introduced due to the nature of the simplified frame models (this occurs for any kind of frame in 3 dimensions) 2. E3 Analysis Procedure E3. Ground Motions This study uses a set of 22 pairs of far-field strong ground motions selected by the FEMA P695 project. These motions are recorded from large magnitude events at moderate fault rupture distances. Although there are no ground motions in the far-field set from SDC B-like environments, the FEMA P695 strong ground motion set is used without modification because it: () provides a consistent ground motion record set through which to examine relative changes in collapse capacity due to accidental torsion requirements, and (2) contains broadband frequency content, which is BSSC SDC B E: Accidental Studies E-9

20 4.86% Draft important for obtaining unbiased results for multiple buildings with varying lateral and torsional periods. In incremental dynamic analysis of the two-dimensional models, each component of each of the 22 ground motions was applied, leading to a total of 44 records scaled until collapse occurs. Ground motions were applied bidirectionally and simultaneously to the three-dimensional models. Each analysis was repeated twice for each of the 22 pairs of ground motions: once with the north-south (NS) component acting along the x-axis of the building and the east-west (EW) component acting along the y-axis, then again with the components switched so that the NS and EW components acted along the y-axis and x-axis, respectively. All of the results from the 44 cases were used for computing collapse statistics, per FEMA P695. E3.2 Incremental Dynamic Analysis Ground motions are scaled to increasing intensities until collapse occurs for incremental dynamic analysis. In this study, ground motion scaling is based on the geometric mean of the spectral acceleration of the two components at a specific building period, i.e. Sa(T ). The fundamental period of the model, obtained from eigenvalue analysis, was used for scaling ground motions for all two-dimensional models. Periods of the three-dimensional designs and models vary slightly (0% or less) depending on how much the design base shear is increased to account for accidental torsion; however, it is desirable to use the same period for scaling ground motions such that results can be directly compared to one another. Therefore, one representative period has been selected to scale ground motions for each combination of height and gravity load level that is used. Once incremental dynamic analysis is performed, two statistical measures of collapse performance are used: the Adjusted Collapse Margin Ratio (ACMR) and probability of collapse given the maximum considered earthquake (MCE) ground motion intensity level, denoted P(Collapse MCE). The maximum considered earthquake ground motion intensity (MCE) in ASCE 7-0 is based on a target risk of % probability of collapse in 50 years. At many locations, the risk-targeted MCE is similar to a ground motion intensity whose likelihood of occurrence corresponds to a 2% probability of occurring in a 50 year time period (approximately a 2500 year return period) at a site. This scaling procedure is slightly different than the FEMA P695 method, which scales a set of pre-normalized records together, but the end result of either method, in terms of the assessed margin against earthquake-induced collapse, is expected to be indistinguishable from the other (FEMA P695). E-20 E: Accidental Studies BSSC SDC B

21 To compute the ACMR of a building, the Collapse Margin Ratio (CMR) must be computed first, based on the ratio of the median collapse capacity, or spectral acceleration causing collapse in incremental dynamic analysis, to the MCE spectral acceleration at the site of interest as in: CMR = Sa collapse,median (T )/Sa MCE (T ) (E.) In addition, Baker and Cornell (2006) have shown that rare ground motions tend to have a different spectral shape than the ASCE code-defined design spectrum; in fact, the spectra tend to have peaks at the period of interest. Therefore, analysis using broadband sets of ground motions, such as the FEMA P695 far-field set, which do not have the expected peaks and valleys in the response spectra, yield conservative estimates of median ground motion intensity at which collapse occurs. To account for the frequency content of the ground motion set, the FEMA P695 methodology uses a spectral shape factor (SSF) to adjust the CMR. The spectral shape factor is based on the site hazard of interest and a building s period and ductility and ranges between. and.2 for the SDC B structures in this study. These factors have been calibrated to adjust the CMR to the value that would be obtained if ground motions with the appropriate spectral shape were selected specifically for the building, rather than using a general set. The equation for ACMR of 3-dimensional buildings is: ACMR =.2 x SSF x CMR (E.2) Tables of SSF values and a more detailed description of how to compute SSF and ACMR can be found in Chapter 7 of FEMA P695. The.2 factor adjusts three-dimensional model results to a two-dimensional equivalent collapse capacity, as described in FEMA P695. Since ACMR corresponds to a median collapse value that is scaled by MCE, a collapse cumulative distribution can be constructed if the dispersion in the spectral intensity at which collapse occurs is known. Chapter 7 of the FEMA P695 report gives a detailed explanation of important factors such as uncertainty in design and modeling properties that contribute to total collapse dispersion, as well as how to combine them to obtain total collapse dispersion (β TOT ), quantified by the logarithmic standard deviation. Several tables of pre-computed dispersion values for different combinations of model quality, quality of design requirements, and quality of system test data are also presented in FEMA P695, Chapter 7. Values of β TOT can vary from to 0.95, but are mostly between 0.45 and 0.7. For this study, a typical value of the total dispersion β TOT was assumed to be 0.65, based on the tables in chapter 7 of FEMA P695. It should be noted, however, that factors such BSSC SDC B E: Accidental Studies E-2

22 4.86% Draft as model quality and quality of design requirements are subjective, and therefore, our selection of β TOT =0.65 was somewhat subjective as well. The probability of collapse given MCE is computed from cumulative distribution function that is defined by the adjusted collapse margin ratio (ACMR) and the total logarithmic dispersion (β TOT ) as follows: P(Collaspe MCE)=LognormalCDF(,ACMR, β TOT ) (E.3) E.4 nlinear Modeling E.4. Overview of Modeling Approach The majority of the analysis for this study of accidental torsion relies on simplified models, which have been calibrated to the fully designed OMF buildings and models. The following steps outline the general method used for building simplified models: ) Build and analyze high end OMF 2D models of archetypes in Table E-2, 2) Calibrate simplified models to match the 2D OMF behavior, and 3) Build simplified 3D models for all archetypes in Table E-4 using the 2D frames. Each of these steps is discussed in more detail in the following sections. E.4.2 High End OMF Models Each of the fully designed OMFs (listed in Table E-2) was modeled as a moment frame in OpenSEES (Open Source Earthquake Engineering Software). Columns and beams were modeled using a lumped plasticity approach, with plastic hinge properties of beams and columns computed according to empirical relationships developed by Haselton et al. (2008). These relationships are based on the design properties of the beams and columns (i.e. concrete compression strength, element dimensions, axial load ratio, and reinforcement detailing) and are therefore capable of representing the influence of changes in design on the element modeling. Plastic hinges were modeled using the Ibarra Material in OpenSEES developed by Ibarra et. al (2005). The Ibarra hinge materials have tri-linear monotonic backbones and incorporate cyclic and in-cycle deterioration, which are important for modeling collapse. Shear failure is not modeled directly in the high end models. However, shear failure has been accounted for by means of a non-simulated collapse mechanism. The non-simulated collapse mechanism is triggered by postprocessing of dynamic analysis results and depends on the column deflection. Physically, the non-simulated collapse mode represents the loss of vertical load carrying capacity in at least one column due to shear failure. nsimulated collapse modes are described in more detail in section E.4.5. E-22 E: Accidental Studies BSSC SDC B

23 In addition to plastic hinges in the beams and columns, nonlinear joint behavior was modeled using 2D shear panels with an Ibarra pinching material. nlinear joint properties were obtained from Lowes and Altoontash (Altoontash 2004; Lowes et al. 2004). The primary factors affecting joint strength and/or ductility are confinement, joint area, and column axial load ratio. Many of the outer joints of the high end models failed during analysis, but failure of the interior joints was prevented, which is what we expect in interior space frames, due to the high level of confinement of interior joints. Distributed gravity loads were applied to the beams, and all remaining dead loads were applied to P- columns, connected to the frame by rigid truss elements. Building mass was lumped at the joints and foundation connectivity was modeled as pinned in the -story models and fixed for the others. (Since 4-story fixed and grade-beam foundation models resulted nearly identical computed CMRs, these foundation fixities were judged to be reasonable.) The high end OMFs were analyzed using Incremental Dynamic Analysis (IDA) and static pushover analysis, and the results of each were used to calibrate simplified models. Beam/Column P- Truss Beam/Column Plastic Hinge nlinear Joint Figure E-5 Schematic of a four-story OMF model BSSC SDC B E: Accidental Studies E-23

24 4.86% Draft E.4.3 Simplified Model Calibration Procedure For each high end OMF model, a simplified 2D model was made that matched its properties as exactly as possible. The simplified models are single bay x-braced frames with nonlinear braces, as shown in Figure E-6. The braces are truss elements with hysteretic material properties defined by the nonlinear Ibarra material. Like the nonlinear hinge materials in the high end models, the brace materials are characterized by a tri-linear monotonic backbone and different modes of cyclic and in-cycle deterioration properties. The properties of the tri-linear backbones were calibrated to the high end models, as described in the following paragraphs. The columns of the simplified models are rigid beam/columns; multi-story simplified models have elastoplastic hinges in columns between the stories to allow for storystory interaction to occur as it would in a moment frame structure. P- loads for the 2D simplified models were applied directly to the columns. Rigid Beam/Column Element Rigid Truss Element nlinear Truss Element Elastoplastic Hinge Figure E-6 Schematic of a four-story simplified model The first step for calibrating the simplified 2D models was to match the static pushover properties of the corresponding high end 2D models, with P- effects included in the analysis. This calibration was achieved by modifying the brace properties, specifically initial stiffness, strength, hardening stiffness, capping displacement and negative post-capping slope, until the pushover analysis results of each story of the simplified and high end OMF models matched as nearly as possible. After matching the story by story pushover analysis results, the pushover results of the building as a whole, as well as modal periods, were checked to ensure that the overall static behavior of the simplified models matched the behavior of the high end OMF models as closely as possible. Figure E-7 illustrates the pushover calibration E-24 E: Accidental Studies BSSC SDC B

25 comparison for the 2D, 4-story, high gravity archetype designed without accidental torsion. All of the simplified model properties except for cyclic deterioration parameters were calibrated using static pushover. Lastly, the cyclic deterioration properties of the simplified models were adjusted until the IDA results matched the IDA results of the corresponding high end model. Table E-5 illustrates the IDA comparison between the two models. One difficulty with calibrating simplified braced frame models to represent the high end OMF models was the inherent lack of story-to-story interaction in the simplified models. If all column and beam elements are modeled as truss elements, each story of the simplified braced frame assemblies behaves independently of the stories above and below. Two major problems arise from this behavior: higher mode periods are much different for the simplified models than the high end models, and damage concentrates in just one story during pushover and dynamic analysis, rather than distributing to multiple stories. This problem has been remedied by making the columns flexurally rigid and adding plastic hinges between stories to simulate the story-to-story interaction that occurs in the OMF frames. Plastic hinge properties in the simplified models are based on beam and column properties in the corresponding OMF frames. As a result, higher modes of the simplified models matched those of the high end models and earthquake damage was distributed to multiple stories in a similar manner as well. Table E-5 shows a comparison of the first 3 modal periods for the high end and simplified versions of the 4-story high gravity OMF archetype. BSSC SDC B E: Accidental Studies E-25

26 4.86% Draft 250 OMF Simplified 200 Total Base Shear (kips) Roof Displacement (in) Figure E-7 Table E-5 E Static pushover results for the 2D, 4-story high gravity model designed without accidental torsion and analyzed using a triangular loading pattern with P- effects considered IDA Results for the 2D, 4-story High Gravity Archetypical Model Designed without Accidental Measure OMF Simplified Difference Period (sec) % Median Sa collapse (g) % β total NA CMR % ACMR % P(Collapse MCE) % Table E-6 E Modal Periods of the 4-story 4 High Gravity OMF Archetype without Accidental Considered Period (s) Mode Difference 'High end' Simplified % E-26 E: Accidental Studies BSSC SDC B

27 Mode Period (s) 'High end' Simplified Difference % % % Once the 2D behavior of the simplified models was calibrated to the high end 2D OMF models, 3D simplified models were created. These models reflect the design plan dimensions of 200ft. x 00ft. There are two frame lines in each orthogonal direction of the simplified models and one leaning column in the center of each quadrant of the building to transmit P- forces to the rigid diaphragm. The P- columns in the 3D models are not a part of the frames like they are in the simplified 2D models; the reason for this difference is because real buildings typically have gravity carrying elements that are distributied fairly evenly throughout the building, not just in the lateral system. Therefore, P- columns have been placed at the center of each quadrant in order for P- forces to have an appropriate lever arm for impacting torsional response. The thick black lines in Figure E-8 represent the frame lines of a sample 3D model (each frame is modeled as shown in Figure E-6, except that they no longer carry P- loads) and the squares indicate P- columns. L 2 =00' X CM +0% X CM, CR L =2L 2 =200' Figure E-8 Plan layout of a 3D simplified model Determination of the 3D brace frame properties was based on the design base shear of the structure. For cases where the frames in the 3D models had exactly the same design base shear as the frames in the 2D model, the modeled frames were identical. For cases where the design base shear due to accidental torsion was different, because of the building of interest did not fall in the subset of archetypes fully designed as 2D frames OMFs, the properties of the braces (and plastic hinge elements between stories for multi- BSSC SDC B E: Accidental Studies E-27

28 4.86% Draft story buildings) were computed using linear interpolation between the high and low base shear versions of the 2D frames. Model strength, stiffness and cyclic deterioration parameters were interpolated based on the design base shear of the frames. Such interpolations were only performed between frames that had the same gravity load and number of stories. Using interpolation to compute the frame properties meant that several archetype buildings could be modeled in 3-D using only two fully designed baseline archetypes for each combination of height and gravity load level. It should be noted that the capping displacement of the calibrated 2D simplified models was always determined such that no interpolation would be needed to compute capping displacement for intermediate models. In other words, the capping displacement of the high base shear version of a given archetype was kept the same as the capping displacement of the low base shear version. The reason that capping displacement was kept constant for each archetype is because we believe that system ductility should be independent of design base shear. Therefore, linking capping displacement to design base shear would introduce error into the experiment by calibrating intermediate models to design idiosyncrasies, rather than meaningful system properties. Additionally, the capping displacemt for the high and low base shear versions of each high end OMF frame in this study were extremely similar (consistantly less than0% different), which confirmed our decision to keep it constant during calibration. An example of the interpolation of simplified frame properties is shown below in Figure E-9. The interpolation of cyclic deterioration properties is not presented in the figure, but is based on design base shear just as the monotonic backbone properties have been. Story Shear (kips) Figure E Displacement (in) Low Base Shear (factor of.0) High Base Shear (factor of.32) Interpolated Intermediate Model (base shear factor of.6) Example interpolation of nonlinear monotonic backbone properties for the second story of the 4- story, high gravity archetype (P- effects not included) E-28 E: Accidental Studies BSSC SDC B

29 E.4.4 n-simulated Collapse Modes Collapse is defined in a number of different ways for this study. For IDA, a building is considered to collapse when the maximum interstory drift ratio begins to increase rapidly, without any significant increase in ground motion intensity (side-sway collapse). However, two other forms of collapse are considered in addition to sideway collapse: ) Failure of the (unmodeled) gravity system and 2) Loss of vertical load carrying capacity of the lateral system, due to shear failure of a column and its subsequent loss of ability to carry gravity loads. Neither shear failure modes nor gravity system failure are simulated by the simplified or OMF frame models, so these failure modes are assessed through non-simulated methods. These failure modes are of interest because both result in structural members no longer having the capacity to withstand vertical loads, which can lead to building collapse. gravity systems are design or modeled in this study, but it is still important to acknowledge the fact that collapse in real buildings can result due to failure of gravity elements, even if the lateral system is still in tact. Assessing non-simulated collapse due to failure of the gravity system is achieved in this study by setting a threshold interstory drift, beyond which the gravity system is assumed to fail. If the maximum interstory drift in any story of a building exceeds that threshold, then the building is assumed to collapse. Thresholds of 3% and 6% were used for assessing non-simulated collapse due to failure of the gravity system. These thresholds were chosen to represent the range in ductility in gravity-load bearing systems possible in SDC B. Design standards for OMF s do not require capacity design, so, as a result, transverse reinforcement may be inadequate for carrying loads associated with plastic hinging of the columns, resulting in brittle shear failure. This specific type of brittle failure only applies to SDC B reinforced concrete columns, but it is still relevant to include when we are trying to use OMF s to represent a SDC B lateral systems in general, because several other systems with low R-factors are prone to brittle failure as well (joint shear failure and weld failure in steel frames for example). Column shear failure has been shown to depend on a combination of displacement demand and shear force demand (Aslani 2005, and Elwood 2004). Therefore, the second non-simulated collapse mode, loss of vertical load carrying capacity, is also assessed using interstory drift thresholds. However, the drift thresholds are story specific, because the expected column drift for which shear failure occurs depends on multiple parameters such as BSSC SDC B E: Accidental Studies E-29

30 4.86% Draft column dimensions, axial load ratio, and reinforcement detailing. Using those parameters, Aslani (2005) and Elwood (2004) have developed empirical methods for predicting the probabilities of shear failure and subsequent loss of gravity-load bearing capacity in reinforced concrete columns. In this study, column drifts corresponding to a 50% probability of loss vertical load carrying capacity are computed according to the methods of Aslani (2005) and Elwood (2004) and interpreted as non-simulated collapse related to column shear failure. These column drifts are then mapped to total interstory drifts using results from static pushover analysis, accounting for drift contributions from column, beam, and joint rotations. Collapse occurs if the drift in any column exceeds the collapse interstory drift threshold. For example, the loss of vertical load carrying capacity drift threshold for a second story interior column in the 4-story, high gravity, archetype is.80%, but the interstory drift threshold for non-simulated collapse for the second story is taken to be 2.35%, due to the portion of the drift resulting from beam and joint rotations. The adjusted collapse margin ratio for each archetype varies significantly with varying methods of assessing non-simulated collapse, however, the relative improvement gained from designing for accidental torsion in this study is mostly independent of which, if any, non-simulated collapse mechanism is implemented. Therefore, all of the results figures combine the results from each of the non-simulated collapse modes considered in addition to the results obtained without non-simulated collapse, unless otherwise specified. Complete results are provided in the subsequent section, E.5. E.5 Sensitivity of Collapse Risk Assessments to Designing for Accidental Results of the assessments, in terms of the change in collapse risk due to designing SDC B buildings with and without accidental torsion, and the absolute collapse risk (ACMR or probability of collapse), are presented in this section. The following figures and paragraphs describe the main trends observed in this study. These trends include: Trends specifically relevant to the scope of the study: ally flexible buildings benefit more from being designed for accidental torsion than torsionally stiff buildings. As a result, the relative frame spacing parameter (S/L) is an excellent predictor of the effectiveness of designing accidental torsion for all building E-30 E: Accidental Studies BSSC SDC B