摘要：
In order to study the internal forces and displacement of circular arch under complex stress state, the axial deformation of circular arch was considered, the equilibrium equations, geometric equations and physical equations were analyzed, and the governing equations for the axial and radial displacement of the arch were developed.The circular arch structure under three typical loads was studied, and the analytical solution of axial and radial displacement expressed by primary function vector and integration constant were derived.Then the displacement coefficient was obtained according to the displacement boundary condition, and the stiffness equilibrium equation in matrix form was established in line with the equations of internal forces.After the matrix transformation, the stiffness matrix model and equivalent nodal force vector of circular arch were obtained.And, by comparing the results of the model, analytical solutions and the classic straight beam element in the tradition finite element method, the accuracy and efficiency of the unit were verified.The results show that, by using the stiffness matrix model and equivalent nodal force vector, the numerical static analysis of all kinds of circular arch can be obtained, as well as the accurate solutions of internal forces and displacement.%为了研究圆拱在复杂受力状态下的内力及位移大小,考虑了圆拱的轴向变形,对其平衡方程、几何方程、物理方程进行了分析,建立了圆拱轴向位移和径向位移的控制方程;并对三类典型荷载作用下的圆拱结构分别进行了研究,求得了用基函数向量以及积分常数向量表达的圆拱轴向位移和径向位移的解析解;根据位移边界条件,得到了位移系数;根据圆拱内力方程,建立了以矩阵形式表达的刚度平衡方程;经矩阵变换得到了圆拱分析的刚度矩阵模型以及等效节点力向量.同时,通过算例,与理论解析解和传统有限元模型中的经典直梁单元进行对比与分析,验证了单元的精度及效率.研究结果表明:通过直接刚度法求得的圆拱刚度矩阵及等效节点力模型,可用于各类圆拱结构的静力数值分析,得到圆拱内力及位移的精确解.
