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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 whichrelationship 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 whichrepresentative representsof 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 includinginclude the following:

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h5. Monotonic curve

Static-pushover analysis utilizes monotonic loading. Here,A *monotonic curve* is produced when a component or system[load pattern|kb:Load pattern] is subjectedplaced toon a loadcomponent conditionor wheresystem thesuch independent variable, athat the deformation parameter, (independent variable) increases continuously from zero to an ultimate valuecondition. The corresponding force-based parameter (dependent variable) is then plotted across this range, indicating the pattern of material nonlinearity. Static-pushover analysis is a forcestatic-basednonlinear parameter,method iswhere plottedstructural againstperformance deformationis toindicated producethrough a nonlinear, monotonic curve. Some examples of monotonic forceF-deformationD relationships (and their associated mechanism) may include stress-strain (axial), moment-curvature (flexure), and plastic-hinging (rotation), etc.

FigureTo 2simplify presentsthe aexpression 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 of a monotonic F-D relationship, and to provide for numerically efficient structural analysis, the nonlinear curve should be idealized as a series of limitlinear statessegments. untilFigure an2 ultimatepresents conditionone issuch achieved. Any strength-gain represents hardening, and strength-loss represents softening. After softening, a residual valuemodel. When Figures 1 and 2 are compared, it is evident that an exact formulation (1) may be achieved, which may sustain through unrealistically large displacements before reaching an ultimate condition. This nonlinear force-deformation relationshipsimplified (2) with little compromise to accuracy. 

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

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*Serviceability* parameters may then be superimposed simplifiedonto the withnonlinear littleF-D compromiserelationship to analysis accuracy through idealization as a series of linear segments, as shown in Figure 3. Please notice thatprovide 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), limitwhich statesindicate arethe denotedcorrelation onbetween thematerial curve.nonlinearity Whileand thesedeterministic parametersprojections relatefor to structural serviceability,damage limitsustained. statesFigure may3 alsodepicts bethe specificserviceability tolimit plasticstates thresholds,of asa shown in Figure 4:F-D relationship.

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!IdealizedServiceability general curves.png|align=center,border=1,width=600pxpxpxpxpxpxpxpx600px!

{center-text}Figure 3 - IdealizedServiceability monotoniclimit backbone curvestates{center-text}

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Limit states may also be incorporated into the nonlinear material relationship.

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

{center-text}Figure 4 - Serviceability limit states{center-text} 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 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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!Hysteresis.png|align=center,border=1,width=700pxpxpxpxpxpxpxpx700px!

{center-text}Figure 54 - Hysteresis loop{center-text}

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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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{center-text}Figure 65 - Hysteresis loop types{center-text}

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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.

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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.