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*Nonlinear* methods are best applied when [inelastic material behaviorstructural behavior may be associated with either geometric or material response. These sources of nonlinearity are described as follows:

* *Geometric* nonlinearity concerns the [P-Delta|kb:P-Delta effect] effects which are associated with the application of external loading upon the displaced configuration of a structure.

* *Material* nonlinearity concerns [inelastic|kb:Material nonlinearity] isstructural consideredresponse in structural modeling and analysis. Nonlinear methods include static-pushover and dynamic [time-history|kb:Time-history analysis] analyses which the behavior of a component, system, or connection deviates from the initial stiffness tangent characteristic of linear-elastic behavior.


h1. Linear vs. nonlinear analysis

*Nonlinear* analysis methods are best applied when either [geometric|kb:P-Delta effect] or [material|kb:Material nonlinearity] nonlinearity is considered during structural modeling and analysis. If only elastic material behavior is to be considered, linear analysis methods should be suitable for designsuffice, though P-Delta formulation may still be applied. Linear and nonlinear methods include strength-based and [response-spectrum|kb:Response-spectrum analysis] analyses. Thesemay be static or dynamic. A few of the traditional analysis methods, and the relations between their attributes, are presented in Figure 1:

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!Analysis type.png|align=center,border=1!

{center-text}Figure 1 - Analysis methods{center-text}

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Each of these analysis methods has benefits and limitations. An overview of each method is as follows:

* *Strength-based* analysis is a static-linear procedure wherein which structural components are selectedspecified such that their capacityelastic capacities exceedsexceed the demanddemands fromof loading conditions. Strength-based demand-capacity (D-C) ratio indicate the adequacy of each component. IfSince capacityonly doesthe notelastic meetstiffness demand,properties redesignare isapplied necessary.to Modelingthe andanalytical analysis is fairly simple and straight-forward in that only component elastic-stiffness properties are considered. Strengthmodel, strength-based analysis is also thethe most simplified and least time-consuming analysis method.

* *ResponseStatic-spectrumpushover* analysis is a dynamicstatic-linearnonlinear procedure methodin wherewhich maximuma structural responsesystem is plottedsubjected asto a functionmonotonic ofload structuralwhich periodincreases and [damping|kb:Damping]. Response may be the peak [acceleration|kb:acceleration], velocity, or displacement which results from a given time-history record. Response-spectrum analysis is a linear application dependent upon the superposition of gravity and lateral effectsiteratively, through an ultimate condition, to indicate a range of elastic and inelastic performance. As a result, structures must remain essentially elastic during response-spectrum analysis. A seriesfunction of time-historyboth records may be analyzed and enveloped to form a smooth design spectrum.

* *Static-pushover* analysis is a static-nonlinear procedure which indicates the elastic and inelastic performance of a component or system subjected to monotonic loading which continually increases through an ultimate condition. This method produces a strength and deformation, the resultant nonlinear force-deformation (F-D) relationship which provides insight into the ductility and limit-state behavior of a structure. Deformation parameters may be translational or rotational. Response is a function of both strength and deformation.

* *Time-history[Response-spectrum|kb:Response-spectrum analysis]* analysis is a dynamic-nonlinearlinear method in which characterizesmaximum structural theresponse dynamicis responseplotted andas inelastica behaviorfunction of astructural structureperiod subjectedfor toa thegiven time-history acceleration record ofand alevel ground motion (earthquake)of [damping|kb:Damping]. The [nonlinear material properties|kb:Material nonlinearity] of ductile components, designed to yield under substantial loading, are modeled such that step-by-step integration may capture inelastic effect. Simultaneously, [P-Delta effect|P-Delta effect] take into account the influence of gravity loading on deformed structural configuration. Output may provide data, plots, and animations indicating structural response and component behavior.

Engineers may use any of these four analysis methods to suit the following purposes:

* To characterize and gain insight into structural behavior; and

* To generate information useful to the design decision-making process.


h1. Sources of nonlinearity

There are two sources of nonlinearity which include the following:

* *Geometric*, which involves the effect of gravity load on displaced configuration, as discussed further in the [Geometric nonlinearitylinear superposition of SDOF systems for a set of structural [mode shapes|kb:Modal analysis] and corresponding natural frequencies represent response. Response measures may be in terms of peak [acceleration|kb:acceleration], velocity, or displacement relative to the ground or the structure. Structures must remain essentially elastic since response-spectrum analysis is dependent upon the superposition of gravity and lateral effects. Results may be enveloped to form a smooth design spectrum. [Modal|kb:Modal analysis] and [FNA|kb:Fast Nonlinear Analysis (FNA)] are additional methods which utilize structural mode shapes.

* *[Time-history|kb:Time-history analysis]* analysis is a dynamic-nonlinear method in which the equations of motion are integrated at a series of time steps to characterize the dynamic response and inelastic behavior of a structural system. Loading may be taken from either a ground-motion record or another dynamic condition. In addition to material nonlinearity, time-history analysis may include [P-Delta|P-Delta effect] article.

* *Material*, which is associated with the inelastic behavior of a structure, as discussed further in the [Material nonlinearity|kb:Material nonlinearity] articleeffects.


h1. Analysis objective

Engineers may use any of these analysis methods to:

* Characterize and gain insight into structural behavior.

* Generate information useful to the design decision-making process.


h1. Capacity Design

Nonlinear modeling and analysis is fundamental to *Capacity Design*. Please visit the [Capacity Design|kb:Capacity Design] article to read more on this topic.


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