# Nonlinear static analysis

Nonlinear static analysis, widely known as pushover analysis, is a structural analysis method that applies an incrementally increasing lateral load to a structural model, accounting for material and geometric nonlinearity, until a target displacement is reached, in order to evaluate strength, stiffness, and seismic performance.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> Demand parameters at the target displacement, such as plastic hinge rotations and interstory forces, are then compared with acceptance criteria.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> Because of its simplicity, the profession has adopted the procedure described in FEMA-356 and ATC-40 broadly for seismic evaluation and retrofit.<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S014102960600068X)</sup>

| Key fact | Detail |
|---|---|
| Primary output | The pushover (capacity) curve: base shear versus control-point, usually roof, displacement, idealized with elastic, post-yield hardening, and softening branches<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> |
| Core assumption | One dominant mode and eigenvalue, assumed unchanged between elastic and inelastic response<sup>[3](https://www.dlubal.com/en/webfile/000912/543505/RF-DYNAMPro_PushOver_EN.pdf?hash=c750ecda36563ee849228017ef32987e2543a0ff)</sup> |
| Target displacement | In ASCE 41, the product of the elastic spectral displacement and three modification factors (coefficient method)<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> |
| Load patterns | FEMA 356 requires at least two patterns, typically uniform (floor mass) and triangular (fundamental period), to envelope the response<sup>[4](https://www.eolss.net/Sample-Chapters/C05/E6-139-14.pdf)</sup> |
| Codification | ATC-40 (1996), FEMA 273 (1997), and FEMA 356 (2000) made the procedure available to engineers; improvements followed in FEMA 440 (2005) and ASCE 41<sup>[5](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)</sup> |
| Accuracy against dynamic analysis | Reported deviations of deformation relative to nonlinear response history analysis range from more than 10% (maximum about 42%) in one study to up to 75% in another<sup>[6](https://www.sciencedirect.com/science/article/abs/pii/S2352710219306540)</sup><sup> • </sup><sup>[7](https://elar.urfu.ru/bitstream/10995/92345/1/10.1088-1757-899X-570-1-012108.pdf)</sup> |

## How it works

The model is subjected to an incremental lateral load whose distribution represents the inertia forces expected during ground shaking, applied until imposed displacements reach the target displacement.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> The static approximation of dynamic response rests on the assumption that the structure has one dominant eigenvalue and mode shape that remains the same during elastic and inelastic response; the pushover curve is then connected to the inelastic response spectrum to evaluate a performance point.<sup>[3](https://www.dlubal.com/en/webfile/000912/543505/RF-DYNAMPro_PushOver_EN.pdf?hash=c750ecda36563ee849228017ef32987e2543a0ff)</sup>

The resulting plot of base shear against control-point displacement can be idealized with elastic, post-yield hardening, and softening branches.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> For demand comparison, the curve is converted into spectral coordinates, spectral acceleration \( S_{a} \) versus spectral displacement \( S_{d} \), by means of dynamic modal participation factors, and plotted in Acceleration-Displacement Response Spectrum (ADRS) format, with periods represented by radial lines; \( S_{d} \) versus \( T \) and \( S_{a} \) versus \( T \) formats are also used.<sup>[8](https://engineering.purdue.edu/~ce573/Documents/SAF_Review%20of%20Capacity%20Spectrum%20Method.pdf)</sup><sup> • </sup><sup>[4](https://www.eolss.net/Sample-Chapters/C05/E6-139-14.pdf)</sup> Two ways of locating demand follow from the guidelines: the capacity spectrum method finds the performance point at the intersection of the capacity curve with a damped demand curve, while the coefficient method modifies an elastic prediction of displacement demand.<sup>[5](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)</sup>

## How it is done

The workflow runs in a fixed order. First, the total gravity load is applied before the lateral load, so that gravity-induced forces and P-Δ effects (extra forces from gravity acting on a deformed, swaying structure) on component yielding and post-peak response are captured.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> Second, nonlinear behavior is assigned to members; for reinforced concrete moment frames, hinge models represent beam and column members with an elastic line element plus zero-length rotational springs at each end.<sup>[9](https://nvlpubs.nist.gov/nistpubs/gcr/2017/NIST.GCR.17-917-46v3.pdf)</sup> Third, lateral load patterns are selected: FEMA-356 recommends a uniform pattern proportional to floor mass and a triangular pattern dependent on the fundamental period, and at least two patterns to bound the inertia force distribution.<sup>[4](https://www.eolss.net/Sample-Chapters/C05/E6-139-14.pdf)</sup><sup> • </sup><sup>[3](https://www.dlubal.com/en/webfile/000912/543505/RF-DYNAMPro_PushOver_EN.pdf?hash=c750ecda36563ee849228017ef32987e2543a0ff)</sup>

The pushover itself may run under force control or displacement control; displacement control is required when the tangent stiffness matrix is not positive definite, which usually occurs in the later steps of the analysis.<sup>[10](http://www.ce.memphis.edu/7119/PDFs/FEAM_Notes/Topic15-5b-AdvancedAnalysisPart2Notes.pdf)</sup> Under ASCE 41-17 (section 7.4.3.2.4), the complex nonlinear base shear–control node displacement relationship must then be idealized into a manageable force-displacement curve for the Nonlinear Static Procedure.<sup>[11](https://protasoftware.com/wp-content/uploads/2025/05/ProtaStructure-Design-Guide-Assessment-to-ASCE-SEI-41.pdf)</sup>

## Origin

Nonlinear static procedures became available to engineers with the publication of ATC-40, *Seismic Evaluation and Retrofit of Concrete Buildings*, in 1996, FEMA 273, *Guidelines for the Seismic Rehabilitation of Buildings*, in 1997, and FEMA 356, which replaced FEMA 273, in 2000.<sup>[5](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)</sup> These were the first significant US guidelines on nonlinear analysis, all focused on pushover analysis, and they were carried forward into [ASCE 41](https://www.edgepedia.org/asce-41) (2007), with improvements proposed in FEMA 440 (2005) and FEMA P440A.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> The FEMA 440 study was proposed to FEMA in 2000 because engineers reported that the capacity spectrum and coefficient methods often gave different displacement demand estimates for the same building.<sup>[5](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)</sup> The conceptual basis is older: early work on pushover-type procedures developed the idea of determining the response of a multi-degree-of-freedom system from the dynamic response of an equivalent single-degree-of-freedom system, and the ADRS format plotting spectral acceleration against spectral displacement with periods as radial lines was proposed in the early 1990s.<sup>[12](http://eprints.usm.my/43424/1/Saber%20Taghavipour24.pdf)</sup>

## Variants

Published procedures group into three classes by lateral load pattern: invariant single load vectors (FEMA-356), invariant multi-mode vectors, and adaptive load vectors.<sup>[13](https://quakelogic.net/Pubs/45.pdf)</sup>

- **Capacity spectrum method (CSM).** The primary pushover in ATC-40 (1996) is based on equivalent linearization using a capacity spectrum.<sup>[14](https://www.mdpi.com/2076-3417/14/1/151)</sup><sup> • </sup><sup>[5](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)</sup>
- **Displacement coefficient method (DCM).** Used in FEMA 273 and incorporated in FEMA 356, it estimates peak inelastic displacement directly with modifying factors; A revised \( C_{1} \) equation exists.<sup>[14](https://www.mdpi.com/2076-3417/14/1/151)</sup>
- **N2 method and improved CSM.** The N2 method, recommended in Eurocode 8, combines pushover analysis of an MDOF model with response spectrum analysis; the improved capacity spectrum method updates the original CSM with constant-ductility inelastic design spectra per FEMA-440 and requires iteration.<sup>[14](https://www.mdpi.com/2076-3417/14/1/151)</sup>
- **Modal pushover analysis (MPA).** Reported by Anil K. Chopra and Rakesh K. Goel in *Earthquake Engineering & Structural Dynamics* in 2001, MPA determines the demand from each term of the modal expansion of effective earthquake forces by a separate pushover with that mode's inertia force distribution, then combines modal demands by SRSS or CQC rules.<sup>[15](https://doi.org/10.1002/eqe.144)</sup><sup> • </sup><sup>[16](http://www.civil.ist.utl.pt/~rbento/tmp/3DISP/Structure&InfrastructureEng_2014.pdf)</sup>
- **Upper-bound pushover analysis (UBPA).** Reported by Tysh Shang Jan, Ming Wei Liu, and Ying Chieh Kao in *Engineering Structures* in 2003 for high-rise buildings.<sup>[17](https://doi.org/10.1016/j.engstruct.2003.09.003)</sup>
- **Adaptive procedures.** Displacement-based adaptive pushover (DAP), reported by S. Antoniou and R. Pinho in the *Journal of Earthquake Engineering* in 2004, updates load vectors at each pushover step using instantaneous mode shapes; the adaptive capacity spectrum method (ACSM) builds on DAP and accounts for stiffness degradation, period elongation, progressive damage, and higher-mode effects.<sup>[18](https://doi.org/10.1080/13632460409350504)</sup><sup> • </sup><sup>[16](http://www.civil.ist.utl.pt/~rbento/tmp/3DISP/Structure&InfrastructureEng_2014.pdf)</sup><sup> • </sup><sup>[4](https://www.eolss.net/Sample-Chapters/C05/E6-139-14.pdf)</sup>
- **Energy-based pushover.** These procedures integrate inertial properties and kinetic energy into force or displacement patterns, computing an energy term \( \Pi_{x} = \tfrac{1}{2} F_{y} \delta_{y} (2\mu^{*} - 1) \) with ductility ratio \( \mu^{*} = \delta_{\max}^{*}/\delta_{y}^{*} \); they have limitations for braced structures.<sup>[14](https://www.mdpi.com/2076-3417/14/1/151)</sup>

## Applications

Under the 2017 NIST guidelines, seismic effects can be modeled either through a nonlinear static (pushover) analysis or a nonlinear dynamic analysis, conducted under appropriate gravity loads, other non-seismic loads, and seismic load effects.<sup>[19](https://nvlpubs.nist.gov/nistpubs/gcr/2017/NIST.GCR.17-917-46v1.pdf)</sup> Commercial implementations differ mainly in hinge defaults and control options: [SAP2000](https://www.edgepedia.org/sap2000) includes built-in default hinge properties based on average values from ATC-40 for concrete members and FEMA-273 for steel members, with user-defined properties recommended for final analyses.<sup>[20](https://psfeg.com/wp-content/uploads/2014/01/Ashraf-Pushover-paper.pdf)</sup>

## Limitations and alternatives

The nonlinear static procedure is applicable to low-rise regular buildings whose response is dominated by the fundamental sway mode, and is less suitable for taller, slender, or irregular buildings where multiple modes affect behavior.<sup>[1](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)</sup> Invariant load patterns are inadequate when modes higher than the first contribute and inelasticity alters the height-wise distribution of inertia forces.<sup>[21](https://www.iitk.ac.in/nicee/wcee/article/WCEE2012_0351.pdf)</sup> The good performance of single-mode methods for low-rise buildings rapidly deteriorates as the number of stories increases; multi-modal techniques extend applicability at additional computational cost without guaranteeing reliability.<sup>[22](http://users.ntua.gr/divamva/docs/Journals/frag-vamva-aschheim-prES2013_NSPAssessmentRCFrames.pdf)</sup>

Quantified accuracy against nonlinear [response history analysis](https://www.edgepedia.org/response-history-analysis) (NRHA) varies by study. Chopra and Goel showed, as cited in later work, that the conventional procedure can underestimate deformation responses with deviations of up to 75%, while another comparison of NSP with nonlinear time-history analysis found maximum horizontal displacements differing by more than 10%, with a maximum difference of about 42%, and differences increasing as building height decreases.<sup>[6](https://www.sciencedirect.com/science/article/abs/pii/S2352710219306540)</sup><sup> • </sup><sup>[7](https://elar.urfu.ru/bitstream/10995/92345/1/10.1088-1757-899X-570-1-012108.pdf)</sup> The ATC-40 capacity spectrum method underestimates the actual response of low-ductility systems and overestimates that of high-ductility systems, the opposite of the equal displacement approximation.<sup>[23](https://onlinelibrary.wiley.com/doi/10.1002/eqe.597)</sup> MPA gives good estimates of floor displacements and story drifts and identifies locations of most plastic hinges, but is not acceptable for buildings deforming far into the inelastic range.<sup>[6](https://www.sciencedirect.com/science/article/abs/pii/S2352710219306540)</sup> For torsionally susceptible buildings, agreement with nonlinear RHA is best at the center of mass and deteriorates at building edges; the N2 method gives slightly better results than FEMA at the center of mass and stiff side but overly conservative results at the flexible side due to large amplification factors.<sup>[24](https://www.iitk.ac.in/nicee/wcee/article/14_05-01-0195.PDF)</sup>

Against these limits, NRHA with simple [hysteretic models](https://www.edgepedia.org/hysteretic-model) outperforms pushover for quantifying engineering demand parameters, except for low-rise, first-mode-controlled structures without significant torsion; pushover retains value for visualizing behavior characteristics that a demand/capacity-focused NRHA does not explore, and the recommended practice is to employ a combination of both.<sup>[25](https://ascelibrary.org/doi/10.1061/41171%28401%29193)</sup> Until a reliable, simple pushover method for irregular 3-D structures is devised, nonlinear dynamic RHA with a good selection of earthquake records remains the method of choice for special and important buildings.<sup>[24](https://www.iitk.ac.in/nicee/wcee/article/14_05-01-0195.PDF)</sup>

## References

1. [Nonlinear Structural Analysis For Seismic Design (NIST GCR 10-917-5, ATC-76-8)](https://nehrp.gov/pdf/nistgcr10-917-5.pdf)
2. [Effects of plastic hinge properties in nonlinear analysis of reinforced concrete buildings (Engineering Structures)](https://www.sciencedirect.com/science/article/abs/pii/S014102960600068X)
3. [Evaluate the Capacity Curve for Pushover Analysis (Dlubal RFEM tutorial)](https://www.dlubal.com/en/webfile/000912/543505/RF-DYNAMPro_PushOver_EN.pdf?hash=c750ecda36563ee849228017ef32987e2543a0ff)
4. [Pushover Analysis of Building Structures (EOLSS chapter)](https://www.eolss.net/Sample-Chapters/C05/E6-139-14.pdf)
5. [Improvement of Nonlinear Static Seismic Analysis Procedures (FEMA 440)](https://nehrpsearch.nist.gov/static/files/FEMA/PB2008108249.pdf)
6. [Assessment of modal pushover analysis for mid-rise concrete buildings with and without viscous dampers](https://www.sciencedirect.com/science/article/abs/pii/S2352710219306540)
7. [Performance-based seismic evaluation methods for the estimation of inelastic deformation demands (IOP, 2019)](https://elar.urfu.ru/bitstream/10995/92345/1/10.1088-1757-899X-570-1-012108.pdf)
8. [Review of the Development of the Capacity Spectrum Method](https://engineering.purdue.edu/~ce573/Documents/SAF_Review%20of%20Capacity%20Spectrum%20Method.pdf)
9. [Guidelines for Nonlinear Structural Analysis and Design of Buildings. Part IIb – Reinforced Concrete Moment Frames](https://nvlpubs.nist.gov/nistpubs/gcr/2017/NIST.GCR.17-917-46v3.pdf)
10. [FEAM Notes: Topic 15-5b Advanced Analysis Part 2 (event-to-event pushover)](http://www.ce.memphis.edu/7119/PDFs/FEAM_Notes/Topic15-5b-AdvancedAnalysisPart2Notes.pdf)
11. [ProtaStructure Design Guide - Assessment and PBD to ASCE/SEI 41-17](https://protasoftware.com/wp-content/uploads/2025/05/ProtaStructure-Design-Guide-Assessment-to-ASCE-SEI-41.pdf)
12. [Assessment of Nonlinear Static (Pushover) Procedures for Seismic Evaluation of Reinforced Concrete Structures (PhD thesis)](http://eprints.usm.my/43424/1/Saber%20Taghavipour24.pdf)
13. [Enhanced pushover procedures (Kalkan and Kunnath, Engineering Structures, 2006)](https://quakelogic.net/Pubs/45.pdf)
14. [Pushover Analysis in Seismic Engineering: A Detailed Chronology and Review of Techniques for Structural Assessment](https://www.mdpi.com/2076-3417/14/1/151)
15. [Anil K. Chopra, Rakesh K. Goel (2001). A modal pushover analysis procedure for estimating seismic demands for buildings. Earthquake Engineering & Structural Dynamics.](https://doi.org/10.1002/eqe.144)
16. [An overview/comparison of nonlinear static procedures (Structure & Infrastructure Engineering, 2014)](http://www.civil.ist.utl.pt/~rbento/tmp/3DISP/Structure&InfrastructureEng_2014.pdf)
17. [Tysh Shang Jan, Ming Wei Liu, Ying Chieh Kao (2003). An upper-bound pushover analysis procedure for estimating the seismic demands of high-rise buildings. Engineering Structures.](https://doi.org/10.1016/j.engstruct.2003.09.003)
18. [S. ANTONIOU, R. PINHO (2004). DEVELOPMENT AND VERIFICATION OF A DISPLACEMENT-BASED ADAPTIVE PUSHOVER PROCEDURE. Journal of Earthquake Engineering.](https://doi.org/10.1080/13632460409350504)
19. [Guidelines for Nonlinear Structural Analysis for Design of Buildings. Part I - General (NIST GCR 17-917-46)](https://nvlpubs.nist.gov/nistpubs/gcr/2017/NIST.GCR.17-917-46v1.pdf)
20. [Practical Three Dimensional Nonlinear Static Pushover Analysis (SAP2000)](https://psfeg.com/wp-content/uploads/2014/01/Ashraf-Pushover-paper.pdf)
21. [Assessment of current nonlinear static procedures for seismic evaluation of special moment-resisting frames (16WCEE)](https://www.iitk.ac.in/nicee/wcee/article/WCEE2012_0351.pdf)
22. [Application of Nonlinear Static Procedures for the Seismic Assessment of Regular RC Moment Frame Buildings](http://users.ntua.gr/divamva/docs/Journals/frag-vamva-aschheim-prES2013_NSPAssessmentRCFrames.pdf)
23. [Evaluation of the structural response under seismic actions using non-linear static methods](https://onlinelibrary.wiley.com/doi/10.1002/eqe.597)
24. [An Overview of Pushover Procedures for the Analysis of Buildings Susceptible to Torsional Behavior (14WCEE)](https://www.iitk.ac.in/nicee/wcee/article/14_05-01-0195.PDF)
25. [Prediction of Nonlinear Response, Pushover Analysis versus Simplified Nonlinear Response History Analysis](https://ascelibrary.org/doi/10.1061/41171%28401%29193)

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*Topic: Encyclopedia › Technology and the built world › Architecture, buildings, and civil works*

*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*

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