Second-order inelastic dynamic analysis of steel frames using fiber hinge method

doi:10.1016/j.jcsr.2011.03.022

Journal of Constructional Steel Research

Volume 67, Issue 10, October 2011, Pages 1485-1494

https://doi.org/10.1016/j.jcsr.2011.03.022 Get rights and content

Abstract

This paper presents a simple and effective numerical procedure for the nonlinear inelastic dynamic analysis of steel frames under dynamic loadings which considers both geometric and material nonlinearities. The geometric nonlinearities are included by using stability functions obtained from the exact stability solution of a beam–column subjected to axial force and bending moments. The spread of plasticity over the cross section and along the member length is captured by tracing the uniaxial stress–strain relations of each fiber on the cross-sections located at the integration points along the member length. A computer program utilizing the average acceleration method for the integration scheme is developed to numerically solve the equations of motion. The obtained results are compared with those generated by ABAQUS to illustrate the accuracy and the computational efficiency of the proposed procedure.

Research highlights

► A fiber element is proposed for nonlinear inelastic dynamic analysis of frames. ► A computer program is developed for the nonlinear dynamic analysis of frames. ► The proposed program is proved to be an efficient tool for frame analysis.

Introduction

With the recent advances in computer technology, more accurate and precise analysis techniques which consider both geometric and material nonlinearities of structures are developed to achieve a realistic insight into the response of structures under earthquake loadings. The geometric nonlinearities of structure can be captured by using either interpolation function or stability function. Since the interpolation function is derived based on an assumed cubic polynomial variation of transverse displacement along the element length, it is unable to capture accurately the effect of axial force acting through the lateral displacement of the element (P − δ effect) when one element per member is used [1]. Hence, it overestimates the strength of a member under significant axial force. Although the accuracy of this method can be improved by using several elements per member in the modeling, it is generally recognized to be computationally intensive because of a very refined discretization of the structures. Whereas, the stability function is derived from the exact stability function of a beam–column subjected to axial force and bending moments. It can capture accurately the P − δ effect of a beam–column member by using only one or two elements per member in the modeling, hence, to save computational time.

The material nonlinearities of structure can be included using either lumped plasticity model or distributed plasticity model. In the lumped plasticity model, the inelastic behavior of material is assumed to be concentrated at point hinges that are usually located at the ends of the member. The force–deformation relation at these hinges is based on force resultants. (Porter and Powell [2], Uzgider [3], Hilmy and Abel [4], Izzuddin and Elnashai [5], Al-Bermani and Zhu [6], Sekulovic and Nefovska-Danilovic [7], among others). The advantage of this model is that it is simple in formulation as well as implementation. However, the disadvantage of this model is that: (1) the force–deformation relation at the hinges is not always available and accurate for every section; and (2) the residual stress is not included directly in the analytical model. In the distributed plasticity model, the gradual yielding of a member is captured by dividing the member into several sections and each section is further subdivided into many fibers (Izzuddin and Elnashai [8], Mamaghani et al. [9], El-Tawil and Deierlein [10], [11], Asgarian et al. [12], among others). Although it is the most computationally intensive among others, it represents the inelastic behavior of material more accurately and rationally than concentrated plasticity models. Since the above-mentioned researches adopt the interpolation function in deriving stiffness matrix formulation, the members of structure need to be divided into many elements to capture accurately the P − δ effect, and, hence, the computational time is very long especially in the time-history analysis. Therefore, it is not recommended for practical use in analysis/design of steel structures.

To take advantage of computational efficiency of the stability function and overcome the above-mentioned weakness of the lumped plasticity model, a new fiber hinge beam–column element is proposed herein for the nonlinear inelastic dynamic analysis of steel frames. The geometric nonlinearities are accurately captured by using stability functions derived from the closed-form solution of a beam–column subjected to end forces, while the material nonlinearities are included by tracing the uniaxial stress–strain relations of each fiber on the cross-sections located at the selected integration points along the member length. The Gauss–Lobatto integration rule is adopted herein for evaluating numerically element stiffness matrix instead of the classical Gauss integration rule because it always includes the end sections of the integration field. Since inelastic behavior in beam elements often concentrates at the ends of member, monitoring the end sections of the element results in improved accuracy and numerical stability [13]. Warping torsion is ignored. A computer program utilizing the average acceleration method for the integration scheme is developed to numerically solve the nonlinear equations of motion of framed structure. Although the programs DRAIN-3DX [14], OpenSees [15], and FRAME3D [16] used the fiber hinge method in representing the material nonlinearity in the analysis of space steel frames, the P − δ effect was not considered in these programs. Therefore, they overestimate the strength of a member subjected to significant axial force if only one or few elements per member are used in the modeling.

Section snippets

Geometric nonlinear P − δ effect

To capture the effect of axial force acting through the lateral displacement of the beam–column element (P − δ effect), the stability functions reported by Chen and Lui [17] are used to minimize modeling and solution time. Generally only one element per member is needed to accurately capture the P − δ effect. From Kim et al. [18], the incremental force–displacement equation of space beam–column element can be expressed as $\{\begin{array}{c} Δ P \\ Δ M_{yA} \\ Δ M_{yB} \\ Δ M_{zA} \\ Δ M_{zB} \\ Δ T \end{array}\} = [\begin{array}{c} \frac{E A}{L} & 0 & 0 & 0 & 0 & 0 \\ 0 & S_{1 y} \frac{E I_{y}}{L} & S_{2 y} \frac{E I_{y}}{L} & 0 & 0 & 0 \\ 0 & S_{2 y} \frac{E I_{y}}{L} & S_{1 y} \frac{E I_{y}}{L} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{1 z} \frac{E I_{z}}{L} & S_{2 z} E \end{array}]$

Nonlinear solution procedure

The Newmark's method has been chosen for the numerical integration of the equation of motion because of its simplicity. The residual forces in each time step can be eliminated by using the Newton–Raphson iterative procedure. The incremental equation of motion of a structure can be written as $[M] \{Δ \ddot{D}\} + [C] \{Δ \dot{D}\} + [K] \{Δ D\} = \{Δ F\}$ where $\{Δ \ddot{D}\}$ , $\{Δ \dot{D}\}$ , and {ΔD} are the vectors of incremental acceleration, velocity, and displacement, respectively; [M], [C], and [K] are the mass, damping, and tangent stiffness matrices,

Verifications

A computer program written in FORTRAN is developed based on the above-mentioned formulations to predict vibration behavior of steel framed structures as well as its nonlinear response under earthquake loadings. It is verified for accuracy and efficiency by the comparison of the predictions with those generated by ABAQUS through two numerical examples. Four earthquake records of the El Centro 1940, the Loma Prieta 1989, the Northridge 1994, and the San Fernando 1971 as presented in Fig. 4, are

Conclusions

A simple and effective numerical procedure for the nonlinear inelastic dynamic analysis of steel frames considering both geometric and material nonlinearities has been presented. The geometric nonlinearities are considered by using the stability functions, while the material nonlinearities are included by tracing the uniaxial stress–strain relations of each fiber on the cross-sections. The computer program developed for this research is verified for accuracy and computational efficiency through

Notations

The following notations are used in the paper throughout.

A_i

area of fiber i

axial stiffness of element

e_i

uniaxial strain of fiber i

E_i

tangent modulus of fiber i

EI_y

bending stiffness of element with respect to y axis

EI_z

bending stiffness of element with respect to z axis

external force vector

f_y

yield stress of material

shear modulus of material

torsional stiffness of element

total number of monitored sections (Gauss point)

length of element

total number of fiber on each section

M_y

bending moment at

Acknowledgments

This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (no. 20104010100520).

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Second-order inelastic dynamic analysis of steel frames using fiber hinge method

Abstract

Research highlights

Introduction

Section snippets

Geometric nonlinear P − δ effect

Nonlinear solution procedure

Verifications

Conclusions

Notations

Acknowledgments

Eng Struct

Comput Struct

Comput Struct

Eng Struct

Eng Struct

Eng Struct

Comput Struct

Eng Struct

Static and dynamic analysis of inelastic frame structures