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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 which may consider 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 which represents elastic behavior. Response will then advance in a nonlinear pattern, possibly increasing to an ultimate point before degrading to a residual strength value. An F-D relationship may also be referred to as a backbone curve.
Material nonlinearity is captured through either of two relationship types, including the following:
Monotonic curve
Static-pushover analysis utilizes monotonic loading. Here, a component or system is subjected to a load condition where the independent variable, a deformation parameter, increases from zero to an ultimate value. The corresponding dependent variable, a force-based parameter, is plotted against deformation to produce a nonlinear, monotonic curve. Some examples of monotonic force-deformation relationships (and their mechanism) may include stress-strain (axial), moment-curvature (flexure), plastic-hinging (rotation), etc.
Figure 2 presents a static-pushover curve. Under monotonic loading, the force-deformation relationship begins linear-elastically, following the initial-stiffness tangent to a yield point. Inelastic behavior then begins, advancing through a series of limit states until an ultimate condition is achieved. Any strength-gain represents hardening, and strength-loss represents softening. After softening, a residual value may be achieved, which may sustain through unrealistically large displacements before reaching an ultimate condition. This nonlinear force-deformation relationship may then be simplified with little compromise to analysis accuracy through idealization as a series of linear segments, as shown in Figure 3. Please notice that immediate-occupancy (IO), life-safety (LS), and collapse-prevention (CP) limit states are denoted on the curve. While these parameters relate to structural serviceability, limit states may also be specific to plastic thresholds, as shown in Figure 4:
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Figure 3 - Idealized monotonic backbone curve
Limit states may be incorporated into the nonlinear material relationship.
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Figure 4 - Serviceability limit states
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 5 - 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 6 - 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.