Material nonlinearity is associated with the inelastic behavior of a component or system. Inelastic behavior may be characterized by a force-deformation (F-D) relationship, also known as a backbone curve, which measures strength against translational or rotational deformation. The general F-D relationship shown to the right indicates that once a structure achieves its yield strength, 
additional loading will cause response to deviate from the initial tangent stiffness (elastic behavior). Nonlinear response may then increase (hardening) to an ultimate point before degrading (softening) to a residual strength value.
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To simplify the expression of a monotonic F-D relationship, and to provide for numerically-efficient formulation, the nonlinear curve may be idealized as a series of linear segments. Figure 2 presents one such model. When the general curve is compared with the idealized, it is evident that an exact formulation may be simplified with minimal compromise to accuracy.
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.
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.
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Figure 4 illustrates hysteretic behavior. Again, translational or rotational deformation is the independent variable. As the orientation of loading continually reverses, a strength-based parameter is plotted against the physical oscillation of the system. Hysteresis is useful for characterizing dynamic response under application of a time-history record.
Figure 4 - Hysteresis loop
As seen in Figure 4, both stiffness and strength deviate from their initial relationship once yielding occurs. This behavior advances with additional hysteretic cycles, and becomes more pronounced with greater inelastic deformations. Initially, strength may increase through hardening behavior, though ultimately, stiffness and strength will both degrade through softening behavior. Whereas strength gain or loss is indicated by the strength level achieved, the decrease in slope upon load reversal indicates degradation of stiffness. Ductility describes the ability of a system to maintain post-peak strength levels during hysteretic behavior and increasing levels of deformation.
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Depending on structural geometry and materials, a hysteretic cycle may follow one of many different possible patterns. Four possible hysteretic-behavior types are illustrated in Figure 5:
Figure 5 - Hysteresis loop types
Information on plotting hysteresis loops is available in the Plotting link hysteresis article.
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