Simulated Collapse Modes

To the extent possible, index archetype models should directly simulate all significant deterioration modes that contribute to collapse behavior.

Typically, this is accomplished through structural component models that simulate stiffness, strength, and inelastic deformation under reverse cyclic loading. Research has demonstrated that the most significant factors

influencing collapse response are the strength at yield, F_y, maximum strength (at capping point), Fc, plastic deformation capacity, p, the post-capping tangent stiffness, Kpc, and the residual strength, Fr (Ibarra et al., 2005). These parameters can be used to define a component backbone curve, such as the one shown in Figure 5-4. Recently, such a curve has been designated a force-displacement capacity boundary (FEMA, 2009).

Cyclic deterioration, which reduces stiffness values and lowers the force-displacement capacity boundary established by the monotonic backbone curve, should also be included to the extent that it influences the collapse response in nonlinear dynamic analyses. An example of degrading hysteretic response is shown in Figure 5-5. Characterization of component backbone curves and hysteretic responses should represent the median response properties of structural components. While illustrated in an aggregate sense in Figure 5-4 and Figure 5-5, the behavior can be modeled through elements of varying degrees of sophistication using phenomenological or physics-based approaches.

F

 F_c

F_y

F_r

_y _c _r _u

K_e

_p pc

Effective yield strength and deformation (Fy and y) Effective elastic stiffness, Ke = Fy/_y

Strength cap and associated deformation for monotonic loading (Fc and _c) Pre-capping plastic deformation for monotonic loading, p

Effective post-yield tangent stiffness, Kp = (Fc-Fy)/_p Post-capping deformation range, pc

Effective post-capping tangent stiffness, Kpc = Fc/pc

Residual strength, Fr

Ultimate deformation, u

Figure 5-4 Parameters of an idealized component backbone curve

-1.5 -1 -0.5 0 0.5 1 1.5

-8 -6 -4 -2 0 2 4 6 8

Chord Rotation (radians) Normalized Moment (M/My)

Non-Deteriorated Backbone

Figure 5-5 Idealized inelastic hysteretic response of structural components with cyclic strength and stiffness degradation.

While of lesser importance than the definition of the maximum force and deformation at the capping point, the initial stiffness can have a significant effect on the ductility capacity. Element-level initial stiffness should reflect all important contributors to deformation (e.g., flexure, bond-slip, and shear), and should be validated against component and assembly test data. An effective initial stiffness defined as the secant stiffness from the origin

considered in phenomenological concentrated spring models. In continuum models, initial stiffness is usually modeled directly. Where results are sensitive to initial stiffness, attention should be given to effects related to initiation of cracking or yielding that may not be considered in the model, such as shrinkage cracking due to concrete curing and residual stresses due to fabrication.

Figure 5-4 and Figure 5-5 are intentionally portrayed in a generic sense, since critical response parameters will vary for each specific component and configuration. For example, in ductile reinforced concrete components (i.e., special moment frames), nonlinear response is typically associated with moment-rotation in the hinge regions where degradation occurs at large deformations through a combination of concrete crushing, confinement tie yielding/rupture, and longitudinal bar buckling. However, in less ductile reinforced concrete components (i.e., ordinary moment frames), nonlinear response may include shear failures and axial failure following shear failure.

Where the seismic-force-resisting system carries significant gravity load, characteristic force and deformation quantities may need to represent vertical deformation effects as well as horizontal response effects.

The development of analytical models is case specific, and no single model is universally applicable. For many steel, reinforced concrete, and wood components, the deterioration model proposed by Ibarra et al. (2005) satisfactorily matches experimental results and analytical predictions.

However, this model should be utilized for a proposed system only if it can be justified based on experimental evidence.

Referring to Figure 5-5, the backbone curve defines a boundary within which hysteresis loops are confined. The implication is that in the analytical model, the load-deformation response is not permitted to move outside this curve.

Such boundaries can be based on monotonic behavior, but ideally they should be based on series of tests including monotonic loading and cyclic loading with different loading protocols (FEMA, 2009). If such boundaries are fixed in the analytical model (i.e., cyclic deterioration is not incorporated explicitly), then estimates of the backbone curve parameters should account for average cyclic deterioration, to produce a modified backbone curve. If the initial stiffness is very different from the effective elastic stiffness, then it may affect the response close to collapse, and should become part of the modeling effort.

Figure 5-6 illustrates the effect of cyclic loading relative to a backbone curve obtained from monotonic loading. In almost all cases, the plastic

deformation capacity, p, is reduced by cyclic loading, and in many cases it is

reduced by a considerable amount from the monotonic loading case. A backbone curve is difficult to construct from a cyclic test (unless experience exists from other similar specimens) and often necessitates the execution of an additional monotonic test. If monotonic tests are not available, a curve enveloping the cyclic test (cyclic envelope) may be used as a conservative estimate of the modified backbone curve.

Figure 5-6 Comparison of monotonic and cyclic response, along with a cyclic envelope curve (adapted from Gatto and Uang 2002).

If the backbone curve is obtained from a monotonic test (or is deduced based on a cyclic deterioration model), then cyclic deterioration must be built into the analytical model representing component behavior. Most cyclic

deterioration models are energy based (e.g., Ibarra et al., 2005; Sivaselvan and Reinhorn, 2000). Validity of the component model must be

demonstrated through satisfactory matching of component, connection, or assembly test date from the experimental program.

Figure 5-6 also illustrates a simplified measure of performance, which is the deformation associated with a force value of 80% of the maximum strength measured in the test, F_c^c. In the figure, the deformation value, cc, which is obtained from the intersection of a horizontal line at 0.8Fcc with the cyclic envelope, can be viewed as a conservative estimate of the ultimate

deformation capacity of a component. In simplified analytical models it can be assumed that no deterioration occurs up to this value of deformation, provided that the strength of the component is assumed to drop to zero at deformations larger than this value. Both Fcc and cc may be different in the positive and negative directions.

The monotonic backbone curve of Figures 5-4 and 5-6 is similar but distinct

Seismic Rehabilitation of Existing Buildings, (ASCE, 2006b). In ASCE/SEI 41-06, generalized force-displacement curves utilize cyclic envelopes that incorporate some degree of cyclic degradation and, in most cases, result in conservative estimates of median response. In this Methodology, backbone curves are intended to represent median properties of monotonic loading response, where cyclic strength and stiffness degradation are directly modeled in the analysis, and statistical variations of the component response are explicitly accounted for in the assessment process.

The type of backbone curve and cyclic hysteretic model used will also impact the amount of equivalent viscous damping used in the model. Models that have backbone curves with a large initial elastic region (which do not dissipate energy under cyclic loading) will generally use higher equivalent viscous damping than models with small initial elastic regions (which do dissipate energy under small cycles).

While component models are expected to be rigorously calibrated to test data, available data may not be comprehensive enough to fully calibrate the models. Data are often particularly scarce for evaluating the capping point and post-capping behavior that occurs at large deformations in ductile components. In such cases, test data should be augmented by engineering analysis and judgment to establish the modeling parameters.

An example of the development and calibration of nonlinear component models for reinforced concrete moment frame systems is provided in Appendix E.