Equilibrium Analysis of Linear Elastic Cable Networks Using Minimum Potential Energy

Cables have always been under consideration as a structural element because of important features such as large strength to weight ratios and long spans. Their equilibrium analysis is an important issue in this regard. But this analysis involves highly nonlinear equations arising from large deformations and material nonlinearity. Different methods have been under use for numerical analysis of such structures. Using the principle of minimum total potential energy is one of the common methods of analyzing the equilibrium configuration of structures. In this method, which is considered an alternative to direct solution of nonlinear equations of equilibrium analytically or numerically by finite element for example, the total potential energy of the structure is minimized using optimization techniques and forces and deformations corresponding to the equilibrium configuration are computed. In cable structures, which under ideal assumption cannot withstand compressive forces, the potential energy functional has discontinuous derivative and thus the classic methods of optimization, which make use of the derivative of the objective function, cannot be used in this case. Usually, the energy consideration is used as the basis of obtaining the equilibrium equations of the structure but rarely is it used as a function the numerical minimization of which gives the equilibrium configuration. In this paper a new method of solving nonlinear equations of elastic equilibrium of cable structures is presented. In this method, first the potential energy functional of the cable structure with large deformations is established. Then, the Powell algorithm of optimization, which doesn't depend on the derivatives of the objective function, is applied and the equilibrium configuration as the minimizer of the functional is obtained. The proposed method has the ability to determine the force in each cable and displacements of the cable junctions (nodes) and the slack cables (cables with no tension) with great speed and accuracy compared to the classic methods. This work consists of a brief explanation of different analyzing methods of cable structures currently in the market. Then, the potential energy for a single cable is obtained and is generalized for a cable network. After that, the Powell's minimization technique is explained. In the examples section, a very simple structure for which the analytical result is available is considered as a validating example. Following that, several illustrative examples are considered. The results are in good agreement with previous published ones.