Nonlinear static and dynamic analysis of cable structures

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Abstract

This paper presents a catenary cable element for the nonlinear analysis of cable structures subjected to static and dynamic loadings. The element stiffness matrix and element nodal forces, which account for self-weight and pretension effects, are derived based on exact analytical expressions of elastic catenary. Cables encountered in cable networks as well as cable-supported bridges can be modeled using the proposed element. An incremental-iterative solution based on the Newmark direct integration method and the Newton–Raphson method is adopted for solving the nonlinear equation of motion. The accuracy and reliability of the present element are verified by comparing the predictions with those generated by commercial finite element package SAP2000, and the results given by other authors using different analytical or numerical approaches.

Introduction

In recent decades, cable element has been widely used in tension structures such as cable-supported bridges and roofs of structures covering large unobstructed areas due to their aesthetic appearances as well as the structural advantages of cables. Since the highly nonlinear behavior exhibits in this element, the effects of flexibility and large deflection in the cable should be considered in establishing the equilibrium equations. In general, the cable member can be modeled using two different approaches: (1) the finite element approach based on the polynomial interpolation functions and (2) the analytical approach based on analytical expressions of elastic catenary.

In the first method, the interpolation functions are adopted to represent the nonlinear effects of the cable. This method has been employed to formulate two-node element, multi-node element, and curved element with rotational degrees of freedom. The two-node element is the most common element used in the modeling of cables, and was adopted by several researches [1], [2], [3]. This element is only suitable for modeling the cables with high pretension. To account for the sag effect, the elastic modulus is modified by the equivalent modulus proposed by Ernst [4]. Several researchers have adopted the equivalent modulus for modeling the cables, which have been proved to be sufficiently accurate for the cases of cable under relatively high stress and small length [5]. For cables with large sag, a series of straight elements is used to model the curved geometry of cables. The multi-node element was developed instead of using many two-node elements. The multi-node element is based on the higher order polynomials for the interpolation functions [6], [7], [8]. Since the formulations of this element are complex expressions, the tangent stiffness matrix, and nodal force vector are obtained using the isoparametric formulation. These elements give accurate results for cables with small sag. When the large sag cable is modeled by several elements, the continuity of slopes is violated. The continuity of the slopes can be enforced by adding rotational degrees of freedom to the nodes. Such an element was developed by Gambhir and Batchelor [9]. In general, the polynomial based elements are only appropriate to model the cable with small sag. For cable element with large sag, it is necessary to use a large number of elements to model the curved geometry of cable. Therefore it causes computational costs.

In the second approach, exact analytical expressions of elastic catenary are used to describe the realistic behavior of cables. This method was originally proposed by O’Brien and Francis [10] and later developed by Jayaraman and Knudson [11], Wang et al. [12], Andreu et al. [13], Yang and Tsay [14], and Such et al. [15]. In this method, the curved cables are modeled by a single two-node catenary element without internal joints. This element can be used to model the small sag cables in cable-stayed bridges as well as large sag cables in suspension bridges. Compared to the finite element method, the analytical approach has some advantages such as requiring fewer number of degree of freedom and exactly considering the nonlinear effects of the cable. The catenary cable element presented in this study is derived based on the second approach.

The purpose of this paper is to develop a spatial two-node catenary cable element for the nonlinear analysis of cable structures subjected to static and dynamic loadings. The tangent stiffness matrix and internal force vector of the element are derived explicitly based on the exact analytical expressions of elastic catenary. Self-weight of the cables can be directly considered without any approximations. The effect of pre-tension of cable is also included in the element formulation. It should be noted that most of the finite element package still lack suitable cable element; therefore the proposed element is also implemented in a computer program for practical use in design. An incremental-iterative solution based on the Newmark direct integration method and the Newton–Raphson method is adopted for solving the nonlinear equation of motion. Several numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed element in predicting the static and dynamic responses of cable structures.

Section snippets

Catenary cable element

To accurately simulate the realistic behavior of cable structures, the cable element presented in this paper is derived based on the exact analytical expressions of the elastic catenary element. It is assumed that the cable is perfectly flexible with the self-weight distributed along its length, and the cross-sectional area of the cable is kept constant. Fig. 1 shows the cable suspended between two points I and J which have the Cartesian coordinates (0, 0, 0) and (lx, ly, lz), respectively. The

Procedure for computing the stiffness matrix

The tangent stiffness matrix and internal force vector of cable element are evaluated using an iteration procedure. This procedure requires the initial values of end forces (F1, F2, F3). Based on the well-known catenary expressions, the initial values of end forces are obtained as follows [11]:F1=wlx2λ0F2=wly2λ0F3=w2(lzcoshλ0sinhλ0+L0)in whichλ0={106if(lx2+ly2)=00.2ifL02lx2+ly2+lz23(L02lz2lx2+ly21)ifL02>lx2+ly2+lz2

The iteration procedure for obtaining tangent stiffness matrix and internal

Solution algorithm

For the nonlinear static analysis, the residual forces in each load increment can be dissipated using the Newton–Raphson method. For the nonlinear time–history analysis, an incremental-iterative solution based on the Newmark direct integration method and the Newton–Raphson method is employed to solve the nonlinear equation of motion. The incremental equation of motion of a structure can be written as[M]{ΔD̈}+[C]{ΔḊ}+[K]{ΔD}={ΔF}where [ΔD̈], [ΔḊ], and [ΔD] are the vectors of incremental

Numerical verifications

A computer program is developed based on the above-mentioned algorithm. The flow chart of the proposed program for the application of the Newmark method and the Newton–Raphson method is illustrated in Fig. 3. Two earthquake records of the El Centro and the Loma Prieta as shown in Fig. 4 are used as ground excitation in the dynamic analysis. Their peak ground accelerations and time steps are listed in Table 1. In the dynamic time–history analysis, the mass- and stiffness-proportional damping

Conclusion

An accurate and effective catenary cable element is presented for the nonlinear analysis of cable structures subjected to static and dynamic loadings. The explicit form of tangent stiffness matrix and corresponding internal force vector are presented. With the present element, each cable member in cable nets can be modeled by using only one element. The computer program developed for this research is verified for accuracy and computational efficiency through several numerical examples. The good

Acknowledgement

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

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