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An analytical formulation to extract the capacity curve of steel structures

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Abstract

Among the methods for evaluating the nonlinear performance of structures, pushover analysis is an appropriate alternative instead of direct time history analysis. To accurately extract the capacity curve of a structure, according to the loading regulations/protocols such as FEMA-356 and ATC-40, lateral loads are incrementally applied to the structure in experimental tests until the structural failure occurs. Because of the cost and time-consuming nature of experimental tests, proposing mathematical/analytical methods could be the appropriate tools to predict the capacity curves of a system. The present study proposes a new method to find the capacity curves of cantilever steel beams based on mathematical formulations, structural analysis, and material properties. The reason to select a simple beam in this study is to shed more light on the unknown aspects of the system’s behavior. Therefore, in this research, the effect of axial load is ignored to clarify the nonlinear behavior of complicated systems such as frames. The strains, stresses, and other responses corresponding to large geometric deformations have been extracted in two cases with and without strain hardening by considering changes in the behavior of materials. The proposed method has been verified using the finite-element method with Abaqus software. The results indicate that the proposed method has acceptable accuracy and could be applied in the pushover analysis of steel structures.

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Abbreviations

\({L}_{u}\) :

The length of the beam between the point with \({M}_{y}\) and the support

\({\theta }_{u}\) :

Rotation of \({L}_{u}\)

\({\theta }_{y-B}\) :

The rotation of elastic length of the beam in failure

\({\theta }_{P}\) :

Rotation of LP

\({\theta }_{y-P}\) :

Rotation of the elastic length of the beam at the point of plastification of the support

\({L}_{P}\) :

The length of the beam between the point with My and the support for the status of plastification

\({\delta }_{y}\) :

Displacement of the free end of the beam due to deformation of the elastic length of the beam at the point of plastification of the support

\({\Delta }_{P}\) :

Displacement of the free end of the beam caused by \({P}_{P}\)

\({\Delta }_{y}\) :

Displacement of the free end of the beam when the support reaches the yielding moment

\({\Delta }_{H}\) :

Horizontal displacement of the free end of the beam

\(S\) :

Elastic section modulus

\(Z\) :

Plastic section modulus

\({\varepsilon }_{y}\) :

Yield strain

\({\varepsilon }_{U}\) :

Strain at the farthest fiber of the section corresponding to the flexural moment of the section failure

\({\mu }_{\theta }\) :

Rotational ductility of the beam

\(E\) :

Modulus of elasticity in the elastic region

\({\varepsilon }_{hs}\) :

Strain at the final stress \(\left({F}_{U}\right)\)

\({P}_{U}\) :

Collapse load

\({M}_{U}\) :

Collapse moment

\({M}_{P}=Z*{F}_{y}\) :

Plastic moment

\({F}_{U}\) :

The final stress of steel under tension

\({P}_{p}\) :

The load that causes plastic moment at the support

\({P}_{y}\) :

The load that causes yielding moment at the support

\({\Delta }_{u}\) :

Displacement of point B due to plastic deformation

\({\Delta }_{y-B}\) :

Displacement of the free end of the beam due to the deformation of the elastic length of the beam at the point of beam failure

\(\Delta\) :

Displacement of the free end of the beam under \(P\)

\({\delta }_{P}\) :

Displacement of the free end of the beam due to only the plastification of the beam at the support

\({\mathrm{A}}_{g}\) :

Area of the beam cross-section

\({M}_{y}=S*{F}_{y}\) :

Yielding moment

\(\uptheta\) :

Rotation of the whole beam length

\(d\) :

Beam cross-section depth

\({\varepsilon }_{P}\) :

Strain in the farthest section of fiber corresponding to the plastic flexural moment

\({\mu }_{\varnothing }\) :

Cross-sectional ductility based on curvature (material properties)

\({\mu }_{\Delta }\) :

Displacement ductility

\({\varepsilon }_{h}\) :

Strain at the end of the steel–plastic step

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Acknowledgements

The second author acknowledged the support of Malayer University when he was an assistant professor of civil engineering from September 2008 to June 2019.

Funding

For the preparation of this manuscript and conducting this research, no fund was received.

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Authors and Affiliations

  1. Malayer University, Malayer, Iran

    Abbas Shamivand

  2. Department of Civil Engineering, Bu Ali Sina University, Hamedan, Iran

    Jalal Akbari

  3. Department of Civil Engineering, Borujerd Branch, Islamic Azad University, Borujerd, Iran

    Parastoo Allahyari

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  1. Abbas Shamivand
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  2. Jalal Akbari
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  3. Parastoo Allahyari
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Contributions

The first and second authors proposed the formulation in this paper. The third author helped to write the manuscript and draw figures. The second author edited the manuscript and submitted it to the journal.

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Correspondence to Jalal Akbari.

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Shamivand, A., Akbari, J. & Allahyari, P. An analytical formulation to extract the capacity curve of steel structures. Asian J Civ Eng 23, 1183–1195 (2022). https://doi.org/10.1007/s42107-022-00472-6

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