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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 may be captured through a variety of F-D relationship types which include the following:

Monotonic curve

A monotonic curve is produced when a load pattern is progressively applied to a component or system such that the deformation parameter (independent variable) continuously increases 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 which indicates structural behavior through a monotonic curve. Some examples of monotonic F-D relationships (and their associated physical 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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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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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

Another relationship type which indicates material nonlinearity is the hysteretic cycle. When the F-D relationship is developed for a component or system subjected to cyclic loading, hysteresis loops are produced. 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. Hysteretic response may be developed during dynamic time-history analysis, where a structural system is evaluated under the cyclic loading induced by the application of an acceleration time-history record to its supports.

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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.

As hysteresis loops develop, the profile of peak values forms the cyclic envelope. The backbone curve produced by the cyclic envelope will be less than the monotonic curve which would result from the same structure being subjected to monotonic loading. This may be attributed to strength and stiffness degradation. An important provision of nonlinear modeling is the accurate characterization of strength and stiffness relationships as a structure progresses through hysteretic behavior. PERFORM-3D is a computational tool which offers this capability.

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:

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Interaction surface

An interaction surface is developed for a structural element when the combined relationship between various strength parameters is plotted. Von Mises, Mroz, or another such plasticity theory may be used to develop a 2D or 3D surface which represents a performance envelope for a given limit state. Behavior exceeds the limit state when the performance measure is outside the envelope. An example may be a 3D P-M-M interaction surface describing the yielding of a column under combined axial, strong-axis, and weak-axis bending. These three performance measures interact in a way which may be plotted to create a 3D ellipse. A response measure outside of the P-M-M envelope would indicate that the column has yielded. Interaction surfaces become increasingly complicated with the degree of sophistication.