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Material nonlinearity is associated with the inelastic behavior of a component or system. Inelastic behavior is characterized by a force-deformation (F-D) relationship. This relationship may consider either translational or rotational displacement. A general F-D relationship is shown in Figure 1. As seen in this figure, once a structure achieves its yield strength, additional loading will cause response to deviate from the initial tangent stiffness representative of elastic behavior. Response will then advance in a nonlinear pattern, possibly increasing to an ultimate point (hardening) before degrading to a residual strength value (softening). An F-D relationship may also be referred to as a backbone curve.
Material nonlinearity is captured through either of two relationship types which include the following:
Monotonic curve
A monotonic curve is produced when a load pattern is placed on a component or system such that the deformation parameter (independent variable) increases continuously from zero to an ultimate condition. The corresponding force-based parameter (dependent variable) is then plotted across this range, indicating the pattern of material nonlinearity. Static-pushover analysis is a static-nonlinear method where structural performance is indicated through a monotonic curve. Some examples of monotonic F-D relationships (and their associated mechanism) include stress-strain (axial), moment-curvature (flexure), and plastic-hinging (rotation).
To simplify the expression of a monotonic F-D relationship, and to provide for numerically efficient structural analysis, the nonlinear curve should be idealized as a series of linear segments. Figure 2 presents one such model. When Figures 1 and 2 are compared, it is evident that an exact formulation (1) may be simplified (2) with little compromise to accuracy.
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Figure 2 - Idealized monotonic backbone curve
Serviceability parameters may then be superimposed onto the nonlinear F-D relationship to provide insight into structural performance. Property owners and the general public are very much interested in performance measures which relate to daily use. Therefore it may be useful to introduce such limit states as immediate-occupancy (IO), life-safety (LS), and collapse-prevention (CP), which indicate the correlation between material nonlinearity and deterministic projections for structural damage sustained. Figure 3 depicts the serviceability limit states of a F-D relationship.
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Figure 3 - Serviceability limit states
Limit states may also be specific to inelastic behavioral thresholds. For example, under static pushover, a confined [reinforced-concrete] column may experience 1. yielding of longitudinal steel; 2. spalling of cover concrete; 3. crushing of core concrete; 4. fracture of transverse reinforcement; and 5. fracture of longitudinal steel.
Hysteretic cycle
Dynamic time-history analysis tracks the hysteretic behavior of a component or system subjected to cyclic loading. Here, material nonlinearity is plotted in a series of hysteresis loops. Rather than following a single monotonic curve to an ultimate condition, hysteresis repeatedly reverses the orientation of loading. Once some degree of inelasticity is achieved, behavior will begin to deviate from that of the monotonic curve with each unloading and reloading in the opposite direction. As shown in Figure 5, both stiffness and strength will deviate from their initial relationships as hysteretic cycles progress. Stiffness typically degrades, which is indicated by a decrease in slope upon load reversal. Strength levels may increase initially, but typically also degrade with cyclic behavior. A ductile system succeeds in maintaining its post-peak strength through hysteretic behavior and increasing levels of deformation.
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Figure 4 - Hysteresis loop
Characterizing the development of strength and stiffness relationships, as they progress through dynamic time-history analysis, is an important feature of accurate nonlinear modeling. PERFORM-3D is a computational tool which provides this capability.
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Figure 5 - Hysteresis loop types
Depending on structural configuration and material, a hysteretic cycle may be one of many different types. Figures 6-10 illustrate some of the possible behaviors.
While accurate prediction of structural behavior is desirable, analysis models can only idealize the performance of real structures. Those using software tools should note that exact prediction of behavior is not possible. The objective of structural analysis is to generate information useful to the design decision-making process. Nonlinear methods enable greater insight into dynamic and inelastic structural behavior.
