Shock and Vibration

Volume 2015, Article ID 890474, 6 pages

http://dx.doi.org/10.1155/2015/890474

## Determination of the Strain-Free Configuration of Multispan Cable

^{1}State Key Lab for Mechanical Structural Strength and Vibration, Xi’an Jiaotong University, Xi’an, Shannxi 710049, China^{2}Shanghai Electric Power Generation Equipment Co., Ltd. Shanghai Turbine Plant, Shanghai 200240, China

Received 16 March 2015; Revised 25 May 2015; Accepted 26 May 2015

Academic Editor: Tai Thai

Copyright © 2015 Chuancai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For building a reasonable finite element geometric model, a method is proposed to determine the strain-free configuration of the multispan cable. The geometric conditions (the end conditions and the unstretched length conditions) as constraints for the configuration of multispan cable are given. Additionally, asymptotic static equilibrium conditions are given for determining the asymptotic shape of the multispan cable. By solving these constraint equations, a set of parameters are determined and then the strain-free configuration of multispan cable is determined. The method reported in this paper provides a technique for building reasonable FEA geometric model of multispan cables. Finally, a three-span cable is taken as example to illustrate the effectiveness of the method, and the computed results are validated via the software ADINA.

#### 1. Introduction

The cable structures are widely used in the engineering, such as bridge, building, and power transmission system [1, 2]. In [1], a new force method is proposed for analysing the dynamic behaviour of flat-sag cable structures. The accepted dynamic model of such cables reduces to a partial differential equation and an integral equation. The support reaction forces are considered as excitations, allowing D’Alembert’s solution to be used. In this way, single or multispan cables have been developed in the form of a single-degree-of-freedom system in terms of the additional dynamic tension. Finally, two examples are presented to illustrate the accuracy of the proposed force method for single and multispan cable systems subjected to harmonic forces. In [2], a three-dimensional modeling procedure is proposed for cable-stayed bridges with rubber, steel, and lead energy dissipation devices. In this study, the cable structure can be simplified as a four-node isoparametric cable element. The multispan cables and towers are the basic structures in the power transmission system. Under the wind load, the multispan cable is apt to generate large displacements because of its high flexibility and large span. So the multispan cable exhibits strong geometrical nonlinearity. Usually, the dynamic behavior of such cable structures is analyzed on the platform of the finite element analysis (FEA) software. To build a reasonable FEA model, two alternative configurations of the cable structures need to be determined in the modeling process; one is the initial static equilibrium configuration and the other is the assumed strain-free configuration. As a prestressed state, the static equilibrium configuration of a cable is the starting point of the subsequent dynamic analysis, whereas the strain-free configuration of a cable is only an assumed state which does not exist in the real world due to the ubiquitous gravity and it is the starting point for static and dynamic analyses. If the FEA geometric model is built based upon the static equilibrium configuration of the cable, the prestrain or prestress in each element caused by gravity should be specified one element by one element; this is always cumbersome. The gravity load is included in the equilibrium configuration. The FEA geometric model contains two gravity loads in this way, so that the model does not match the actual cable structure. The gravity load is not included in the strain-free configuration. Alternately, if the geometric model is built based on the strain-free configuration, the prestrain or prestress in each element needs not to be specified in advance and they are set up naturally in the nonlinear static analysis via the FEA. After that, the gravity load applied in the model; the system achieves equilibrium. Therefore, the assumed strain-free configuration is more preferable to the static equilibrium configuration for building the FEA geometric model of multispan cable.

For the static and dynamic analysis of cable structures, Irvine [3] systematically summarized the classical achievements reached up to 1981 and the static equilibrium shape of a cable is described as an elastic catenary. For the modeling of cable structures, in [4], a method for modeling cable supported bridges for nonlinear finite element analysis is presented in this paper. A two-node catenary cable element, derived using exact analytical expressions for the elastic catenary, is proposed for the modeling of cables. Based upon the elastic catenary, Such et al. [5] proposed a method for determining the static equilibrium configuration of arbitrary three-dimensional cable structures subject to gravity and point loads. Srinil et al. [6, 7] formulated a system of nonlinear partial differential equations describing the large amplitude three-dimensional free vibrations of inclined sagged elastic cables. Ai and Imai [8] proposed an iterative scheme to find the static equilibrium shape of beam-cable mixed system. In [9], based on exact analytical expressions of elastic catenary, Thai and Kim presented a catenary cable element for the nonlinear static and dynamic analysis of cable structures. In [10], Vu et al. presented a spatial catenary cable element for the nonlinear analysis of cable-supported structures and proposed an algorithm for form-finding of cable-supported structures. Rienstra [11] derived the partial differential equations and boundary conditions satisfied by the static equilibrium configuration of multiple spans coupled via suspension strings. In [12], Impollonia et al. obtained the deformed shape of the elastic cable in closed form for the cases of uniformly distributed load and multiple-point forces. Greco and Cuomo [13] proposed a method for obtaining an exact configuration of slack cable nets by using the exact expressions of the equilibrium derived from the equation of the catenary.

All the above papers focused on the determination of static equilibrium configuration of the cable structures. However, the present paper concentrates on determining the strain-free or unstretched configuration of multispan cable. Firstly, based upon the elastic catenary theory, a set of geometric boundary conditions, cable length condition, and static equilibrium conditions are derived for determining the configuration of the multispan cables when they are approaching asymptotically to the strain-free state. These constraint conditions constitute a system of nonlinear algebraic or transcendental function equations. Secondly, these nonlinear equations are solved and one finds a set of parameters which determine the unstretched configuration of the multispan cable. Finally, an example of three-span cable is given to illustrate the method.

#### 2. The Governing Equations of Multispan Cable

First we assume that (1) the bending stiffness of cable is negligible; (2) the material of cable is of Hookian type; that is, its constitutive relation is linear elastic; and (3) only the self-weight as the static force is applied to the cable structure. Under these assumptions, the governing equation for the cable in its static equilibrium state is as follows [6]:Here, the “” denotes the derivative with respect to , denotes the tensional strain in the cable when it is in static equilibrium state, is Young’s modulus, is the sectional area of the cable and assumed to be constant, represents the static equilibrium shape of the cable, and is the cable weight per unit length in the unstretched state.

The tensional strain is related to the horizontal tensional force as follows:According to (1), the final static equilibrium governing equation of the cable can be rewritten in the following form:This is the so-called elastic catenary equation. Being different from the traditional catenary equation, it takes account of the extensibility of the cable. Under the dynamic wind load, the multispan cable usually vibrates nonlinearly due to its large displacement, so the finite element analysis is preferable to investigate the static and dynamic behavior. To build the FEA model, firstly the geometric model of multispan cable is needed. Therefore, one of the initial configurations, the strain-free state or the static equilibrium state, should be determined. As stated in the Introduction, in the following, we only discuss how to determine the strain-free configuration of the multispan cable.

#### 3. Determination of the Strain-Free Configuration

Determining the strain-free configuration of the cable based on the data of static equilibrium configuration can be viewed as an inverse problem in nonlinear structural mechanics. The unstretched state of cable is an assumed state and it can be viewed as an asymptotic state that corresponds to (which means the cable is inextensible), and (note that is negative), so the equation for cable in unstretched state is as follows:This is the rigid catenary equation. Its solution can be found as

For the three-span cable, as illustrated in Figure 1, the shape of the left span cable in its unstretched state can be expressed asSimilarly, the shape of the middle span cable can be expressed asand the shape of the right span cable is expressed as