Keywords: cable structures, nonlinear analysis, dynamic relaxation, force density method.

This paper presents a comparison of computational methods used in the static analysis of cable structures. Two different methods are used, the dynamic relaxation and force density methods. The dynamic relaxation method is presented in more detail. The sample design will be compared for accuracy, the computation time and the conditions and speed of convergence of the methods used.

A single cable example should be analysed before proceeding to analyse an entire cable structures. There are basically two approaches to this issue. The first approach approximates a single cable perfectly flexible cable element [1]. The second approach consists in replacing the cables of several bar elements that carry only tension. In this paper only the first approach is considered. Several methods exist to solve cable structures. In this work will be compared two selected methods: the dynamic relaxation method [2] and the force density method [1].

The dynamic relaxation method [2] is not used for dynamic analysis of structures, but uses a dynamic solution for a fictitious damped structure to achieve a static solution. When calculating the response of the structure the assembled stiffness matrix is not used and therefore the dynamic relaxation is suitable for large scale nonlinear cases. The method is based on Newton's second laws of motion. During the static analysis fictitious damping is used. The basic unknowns are the nodal velocities, from which the nodal displacements are calculated. The speed of the solution is dependent on a suitable distribution of mass structure in each node.

The basic equations in the force density method [1] form nodal equilibrium equations of the free joints. The basic unknowns are the nodal coordinates of the unsupported nodes. The state of unknown nodal coordinates must be determined, so that all internal forces acting on the node are in equilibrium with the external nodal forces. The basic governing equations can be solved by employing an iterative algorithm based on Newton's method for solving nonlinear problems.

To compare both methods an example given in paper [1] is selected. The method of dynamic relaxation in this structure has a smaller number of iterations and faster time of solution at the same accuracy as the force density method. It is clear that the calculation time can be affected by the use of scripts that must be written effectively. However a lower number of iterations, shows that the method of dynamic relaxation has considerable potential for numerical calculations of cable structures.

1

A.S.K. Kwan, "A new approach to geometric nonlinearity of cable structures", Computers and Structures, 67, 243-252, 1998. doi:10.1016/S0045-7949(98)00052-2

2

B.H.V. Topping, P. Iványi, "Computer Aided Design of Cable Membrane Structures", Saxe-Coburg Publications, Stirlingshire, UK, 2008.

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