Second-order inelastic analysis of steel suspension bridges

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Abstract

The second-order inelastic analysis based on the refined plastic hinge model is proposed to predict the ultimate strength and behavior of steel suspension bridges. The cable members are modeled using the catenary cable element, while the girder and tower members are modeled using the beam-column element. The nonlinear equilibrium equations are solved using an incremental–iterative scheme based on the generalized displacement control method. This algorithm can accurately trace the equilibrium path of nonlinear problems with multiple limit points. Several numerical examples are presented. The obtained results are compared well with the existing results and those generated using the commercial finite element packages of SAP2000 and ABAQUS. It can be concluded that the proposed method is suitable for adoption in practice.

Introduction

Cable supported structures such as suspension bridges and cable-stayed bridges have been recognized as the most appealing structures due to their esthetic appearances as well as the structural advantages of cables. With the recent advances of structural analysis techniques and construction technologies, a suspension bridge with a main span length reaching almost 2,000 m emerges into reality. When the main span length of suspension bridge becomes longer, more accurate and precise analysis techniques are required to consider both geometric and material nonlinearities of structures. Material nonlinearity comes from the nonlinear stress–strain behavior of materials, whereas the geometric nonlinearities result from the cable sag effect, axial force-bending moment interaction in the girder and tower, and large displacement.

The analysis methods of suspension bridges can be classified into two categories of analytical and numerical types. The analytical method is based on elastic theory and deflection theory. The elastic theory does not consider the stiffening effect of the main cable under tension, and thus gives higher moments in the stiffening girder. The deflection theory accounts for the second-order effects of cable stiffness; hence, reduces the moments caused by the stiffening girder. A number of studies have been performed in the past based on the deflection theory [1], [2], [3], [4], [5], [6], [7]. Although the deflection theory provides appropriate results for the suspension bridge with the main span longer than 200 m, it is cumbersome for more complex models. Therefore, the numerical method based on advances in computer technology has been proposed to accurately predict the response of suspension bridges. In the numerical method, the tower and girder members are modeled using beam-column element, while the cable members are modeled using truss element or catenary cable element. The nonlinear inelastic analysis models for the beam-column element can be grouped into two categories of plastic zone and plastic hinge types based on the refinement used to represent yielding. The beam-column member in the former (i.e. the plastic zone method) is discretized into several finite elements along the member length, and the cross section of each finite element is further subdivided into many fibers, of which the uniaxial stress–strain relationships of material are monitored during the analysis process. Although the plastic zone solution is known as an “exact solution”, it is too computationally intensive because of a very refined discretization of the structure. Therefore, it is usually applicable for the research purpose. The beam-column member in the later (i.e. the plastic hinge method) is modeled by an appropriate way to eliminate its further subdivision, and the plastic hinges representing the inelastic behavior of material are assumed to be lumped at both ends of the member. The distinct advantages of this method are that it is simple in formulation as well as implementation and, more importantly, the least elements need to be used to model the member. Therefore, it is usually applicable for the practical design purpose. Although several studies on the nonlinear behavior of suspension bridges have been performed in the past using the numerical method [8], [9], [10], [11], [12], [13], these studies was only limited to the application of plastic zone method.

In this study, second-order inelastic analysis based on the refined plastic hinge model [14], [15], [16] is proposed to predict the ultimate strength and behavior of steel suspension bridges. The main cable and hanger are modeled using the catenary cable element, while the girder and tower are modeled using the plastic hinge beam-column element. To trace the descending branch of equilibrium path, the Generalized Displacement Control (GDC) method proposed by Yang and Shieh [17] is employed for solving the nonlinear equilibrium equations. This algorithm can accurately trace the equilibrium path of nonlinear problems with multiple limit points and snap-back points as demonstrated in the numerical example. A computer program is also developed. Several numerical examples are presented to prove the robustness of the proposed method in predicting the nonlinear inelastic response of space steel structures.

Section snippets

Catenary cable element

To accurately simulate the realistic behavior of cable members, the catenary cable element is employed to model the main cables as well as the hangers. It is assumed that the cable is perfectly flexible with the self-weight distributed along its length. Consider an elastic catenary cable as shown in Fig. 1, the projected lengths of the cable can be derived as follows (Refs. [18], [19])lx=F1L0EcAcF1w{ln[F12+F22+(wL0F3)2+wL0F3]ln(F12+F22+F32F3)},ly=F2L0EcAcF2w{ln[F12+F22+(wL0F3)2+wL0F3]

Stability functions accounting for second-order effects

Since the large displacements occur in the tower and girder members of suspension bridge under the combined effect of axial force and bending moments, the interaction between axial and flexural deformations in such members is significant and should be considered in the nonlinear analysis. This coupling effect can be considered using the stability functions. From Kim et al. [16], the incremental form of force–displacement relationship of space beam-column element can be expressed as{PMyAMyBMzAMzB

Nonlinear solution procedure

This section presents a numerical method for solving the nonlinear equations. Among several numerical methods, the GDC method proposed by Yang and Shieh [17] appears to be one of the most robust and effective method for solving the nonlinear problems with multiple critical points for its general numerical stability and efficiency. The incremental form of equilibrium equation can be rewritten for the jth iteration of the ith incremental step as[Kj1i]{ΔDji}=λji{Pˆ}+{Rj1i}where [Kj1i] is the

Verification study

A computer program is developed based on the above-mentioned algorithm to predict the nonlinear inelastic analysis of space steel structures. A flow chart of the proposed program is illustrated in Fig. 2. For the verification purpose, the obtained results are compared with the existing results, and those generated by SAP2000 and ABAQUS.

Case study

Fig. 11 shows the configuration of the bridge with its sectional properties. Young's modulus and yield stress of beam-column members are 200,000 and 248 MPa, respectively, whereas Young's modulus and yield stress of cable members are 165,500 and 1,103 MPa, respectively. The weight per unit volume of the cable and beam-column members is 60.5 and 76.82 kN/m3, respectively. The total loads of the example bridge consist of dead load and live load. The girder dead load is taken as 85 kN/m, which is

Conclusion

The nonlinear inelastic analysis based on the refined plastic hinge model has been proposed and implemented on a computer program to predict the ultimate strength and behavior of the steel suspension bridges. Both geometric and material nonlinearities are included in the analysis. The proposed method has been shown to be accurate in capturing the buckling load of columns with different end conditions using only one element per member. It can also accurately trace the equilibrium path of

Acknowledgement

This research has been supported by the Brain Korea 21 Project of the Korea Research Foundation.

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