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!General monotonic curve, Figure 2.PNG|align=right,border=0!

*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 ismay be captured through a eithervariety of two relationship types which include the following:

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h5. Monotonic curve

A *monotonic curve* is produced when a [load pattern|kb:Load pattern] is placedprogressively onapplied a component or system such that the deformation parameter (independent variable) continuously 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 which whereindicates structural performance isbehavior indicated 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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!Idealized general curves.png|align=center,border=1,width=600px600pxpxpxpx!

{center-text}Figure 2 - Idealized monotonic backbone curve{center-text}

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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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!Serviceability curves.png|align=center,border=1,width=600px600pxpxpxpx!

{center-text}Figure 3 - Serviceability limit states{center-text}

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Limit states may also be specific to inelastic behavioral thresholds. For example, under static pushover, a confined [reinforced-concrete|kb: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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h5. Hysteretic cycle

Dynamic time-history analysis tracksAnother relationship type which indicates material nonlinearity is the *hysteretic behavior of cycle*. When the F-D relationship is developed for a component or system subjected to cyclic loading, hysteresis loops are produced. Here, material nonlinearity 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 inagainst the aphysical seriesoscillation of hysteresisthe loopssystem. 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 Hysteretic response may be developed during dynamic [time-history|kb:Time-history analysis] 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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!Hysteresis.png|align=center,border=1,width=700px!

{center-text}Figure 4 - Hysteresis loop{center-text}

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As seen in Figure 4, both stiffness and strength will deviate from their initial relationships asrelationship once yielding occurs. This behavior advances with additional hysteretic cycles progress. Stiffness typically degrades, which, 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 a the strength level achieved, the decrease in slope upon load reversal. Strengthindicates levelsdegradation mayof increase initially, but typically also degrade with cyclic behavior. A ductile system succeeds in maintaining its stiffness. *Ductility* describes the ability of a system to maintain post-peak strength levels throughduring hysteretic behavior and increasing levels of deformation.

As hysteresis loops develop, 
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!Hysteresis.png|align=center,border=1,width=700px!

{center-text}Figure 4 - Hysteresis loop{center-text}

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Characterizing the development ofthe 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 relationships,degradation. asAn theyimportant progressprovision throughof dynamicnonlinear time-history analysis, modeling is anthe importantaccurate featurecharacterization of accurate nonlinear modeling. PERFORM-3D strength and stiffness relationships as a structure progresses through hysteretic behavior. [PERFORM-3D|perform:Home] is a computational tool which providesoffers 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 6:

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!Hysteresis types.PNG|align=center,border=1!

{center-text}Figure 5 - Hysteresis loop types{center-text}

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h5. Interaction surface

An *interaction 
Depending on structural configuration and material, a hysteretic cyclesurface* 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 one of many different types. Figures 6-10 illustrate some of the possible behaviors.

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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.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.