2.5.1 Introduction and Purpose
The purpose of the structural analysis is twofold: (1) to relate IM (5%-damped, 1-sec spectral acceleration response) to EDPs (peak drifts, peak floor accelerations, and inelastic deformations of each reinforced concrete component) and (2) to estimate the collapse probability as a function of IM, i.e., to create a collapse fragility function. Although this second task can be seen as part of the damage analysis (depicted as such in Fig. 2.1), it is more naturally included with the structural analysis and so it is discussed here. The inputs and outputs of this step, for each structural design realization, are:
• Input
° Ground motion hazard curve (Section 4.1.1)
° Bin of ground motions for each of seven distinct hazard levels (Section 4.2) • Outputs
° EDPs for each ground motion record at each hazard-level bin
EDPs for mean structural model (all modeling variables set to their respective
expected values) (Section 5.11)
° Collapse capacity
Collapse capacity for mean structural model (all modeling variables set to their
respective expected values) (Section 5.12.2)
Total uncertainty in collapse capacity estimates, including uncertainties from
record-to-record variability and structural modeling uncertainties (Sections 5.12.2, E.5.2)
Probability of collapse at a given ground motion intensity level, denoted here by
P[C | im] (Section 5.12.2.2)
21 2.5.2 Structural Modeling
To produce accurate and dependable benchmark results that include the assessment of nonstructural damage, the structural model needs to accurately predict structural response from
low deformation levels (where cracking and tension-stiffening phenomena are important) up to
the extremely high deformation levels (where deterioration leading to collapse is important). To achieve these goals requires a structural model that accurately represents behavior from initial cracking up through collapse.
The structural analyses for this study are run using PEER’s Open System for Earthquake Engineering Simulation (OpenSees 2005). Based on our assessment of the available modeling features in OpenSees (as of 2005), we concluded that no single model would accurately capture the structural response over the full ranges of ground motion intensity. Models that could capture the initiation of cracking well did not accurately simulate strain softening at large deformations, and models that captured deterioration at large deformations did not simulate the initial loading stages well. Therefore, we used the following two different models to simulate the full range of response: (1) a fiber model for low-intensity levels (where cracking behavior governs) and (2) a lumped-plasticity model to deal with high intensities at which structural collapse can occur. Both models incorporate bond-slip (Section 5.5.2.4), element shear flexibility (Section 5.3.3.3), and the gravity system (Section 5.6) effects.
The fiber model is a force-based beam-column element implementation by Filippou (1999) that captures the cracking behavior of the concrete section using a uniaxial concrete constitutive law, and tracks the spread of plasticity through the element cross section and along the element length. This model is termed the “fiber-spring model” because the fiber elements include an uncoupled shear degree-of-freedom that is used in conjunction with rotational springs at the ends of each element to model bond-slip behavior. Section 5.3 discusses this model in more detail. The current fiber-element model in OpenSees (2005) cannot capture rebar buckling and low-cycle fatigue, and hence cannot accurately model side-sway system collapse of a ductile RC frame. (This is not an inherent limitation of the fiber-element formulation, but comes from an inability of the existing steel models to simulate rebar degradation.)
The lumped-plasticity model consists of elastic beam-column elements that are combined with concentrated hinge springs, which employ the peak-oriented material model (called “Clough” in OpenSees). The “Clough” spring implementation in OpenSees includes
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modifications introduced by Ibarra and Krawinkler (Ibarra 2003) to capture strain softening at large deformations. The model is composed of a trilinear backbone and is capable of modeling four types of cyclic deterioration of strength and stiffness (OpenSees 2005). Section 5.4 discusses this model in more detail.
2.5.3 Structural Analysis Methodology
We employed incremental dynamic analysis (IDA) (Vamvatsikos 2002) with the lumped- plasticity model to estimate collapse probability as a function of IM, denoted here by P[C | im]. In an IDA, a set of ground motions is systematically scaled and applied to the structural model over a broad range of IM levels. At each IM level, the input ground motions are scaled to the desired value of IM, and the resulting EDPs from the structural analysis are recorded. For the purpose of assessing collapse, the ground motion record set corresponding to largest earthquake magnitude bin was applied over the IDA. On the other hand, for evaluating the onset of damage at lower-intensity values, we employed a related method to IDA called stripe analysis with the fiber-element model, where separate ground-motion-record bins are applied for each IM level. This is in contrast to the conventional IDA, where records from one ground motion bin are scaled over the full range of intensities, the rationale being that there may be differences in the record bins that should be reflected in the various hazard intensity levels.
Stripe analysis. Figure 2.3 illustrates the results of the stripe analysis for one of the fiber models of the benchmark building. This figure shows the individual drift responses for each earthquake record with lines for the median, 16th and 84th percentile responses, assuming that the drift is lognormally distributed for a given Sa level (Krawinkler and Miranda 2004). This shows the uncertainty in drift response due to the variability between ground motion records, conditioned on the Sa intensity. Since different ground motion bins are used at each stripe level, the variation in response reflects both the record-to-record variability within a bin (at a given strip) and variations between record sets at the various intensity stripes.
23 0 0.01 0.02 0.03 0.04 0 0.2 0.4 0.6 0.8 1 1.2 Sa g. m . (T = 1 s e c ) [g ] Roof Drift
Fig. 2.3 IDA results using fiber-spring model (Design A).
IDA. As noted above, the stripe analysis is used to estimate losses prior to collapse, whereas IDA is used to model collapse. There are several potential local and global collapse mechanisms, notably (a) loss of gravity-load-carrying capacity of a column, (b) local collapse of a gravity slab system, and (c) global side-sway collapse caused by dynamic instability in one or more stories. We assume that the detailing provisions of the current building code (ICC 2003) will effectively prevent local collapse modes (a) and (b), so this study focuses on global side- sway collapse.
Figure 2.4 shows the collapse behavior of the structural model when subjected to each of the 36 earthquake records. None of the 36 records cause collapse near the 2%-in-50-years hazard level (0.82g), thus demonstrating that the simulation model predicts high collapse capacities, with a corresponding low associated probability of collapse at the 2%-in-50-years hazard level. Section 5.12 discusses how information of the type shown in Figure 2.4 is used to evaluate the collapse probability as a function of IM and the mean annual collapse frequency.
0.82g is 2%-in-50- years ground motion 0.55g is 10%-in-50- years ground motion
24 0 0.05 0.1 0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 Sa g. m . (T=1 .0 s )[ g ]
Maximum Interstory Drift Ratio
Fig. 2.4 Collapse IDA using lumped-plasticity element model (Design A, record Bins 4A and 4C, controlling horizontal direction).
2.5.4 Effects of Structural Uncertainties
Despite the apparently low collapse probability in the 2%-in-50-years event suggested by Figure 2.6, consideration of structural and modeling uncertainties can increase the collapse probability by an order of magnitude. Uncertainties in the structural behavior and modeling include those in element plastic-rotation capacities, cyclic deterioration modeling parameters, and other variables. These uncertainties affect both the structural behavior before collapse and the collapse capacity, but we have considered only their effect on the more sensitive collapse capacity.
Details of the modeling uncertainty analysis are presented in Appendix E of this report. Briefly, the process involved (1) quantifying uncertainties in structural modeling parameters, using both previous research and calibration to experimental data (Table E.4, Appendix C) and (2) applying the first-order second-moment (FOSM) approximation (Baker and Cornell 2003) to integrate the modeling uncertainties with the record-to-record uncertainties to evaluate the collapse capacity (Section E.5.1). A significant observation from this exercise was that the degree of correlation between variables strongly affects the final uncertainty in collapse capacity.
0.82g is 2% in 50 year motion (b) Capacity Stats.: Median = 2.1g σLN = 0.29
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