索结构找形分析的精确单元方法

索找形问题是索结构分析中所要解决的首要问题,在给定边界条件下,所施加的预张力和外部荷载通过调节索的 形状来平衡。本文研究索结构初始形状确定的精确有限单元法,对于常见的荷载形式,构造了线性和非线性两大类共5种 单元,适用于一般的索结构找形计算,并且可以给出精确的解答。本文通过将水平方向和竖直方向的平衡方程解藕,得出 了索单元的精确描述格式,并保证了索结构形状的唯一性。文中以索结构内部结点坐标作为基本未知量,以结点平衡方程 为基本方程,通过直接求解单元的平衡微分方程得到单元刚度矩阵的解析表达式。对于由线性单元组成的索结构,其基本 方程为线性,可直接求解;对于含有非线性单元的索结构,其基本方程为非线性,需通过迭代求解,文中构造了相应的 Newton法迭代格式。本文方法所需单元数目少,计算量小,所得到的解答为数值精确解。数值算例表明本法稳定可靠。

Exact element method for form-finding analysis of cable structures

The problem of form-finding is the first task for analysis of cable structures, that is to obtain the equilibrium of the applied prestension and external loads by adjusting the form of cable under the given boundary condition. This paper presents an exact finite element method for initial form-finding analysis of cable structures by constructing exact element models for commonly-used loads. This method can be used for form-finding calculations of general cable structures and exact results can be obtained. In the proposed method, the horizontal and vertical equilibrium equations are uncoupled, the exactly described type of elements are achieved as well as the sole form of cable structure are proved. The basic unknowns are chosen to be the joint coordinates, and the equilibrium equations at these joints are formed. The element stiffness matrices are analytically given by solving the corresponding governing differential equations. The elements are classified into linear and nonlinear types according to different load types. For linear type of elements, the basic equations are linear and can be solved directly, while for nonlinear type of elements, the basic equations are nonlinear and can only be solved by iteration method. A Newton type of iteration method is proposed to solve the nonlinear problem in an efficient way. Numerical examples are given to show the exactness, efficiency and applicability of the proposed method for general space cable structures.