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力密度法最初是求解膜结构找形问题的方法,经发展可用于计算桁架结构的几何非线性问题。本文应用力密度法建立结构变形后的非线性平衡方程及相应的雅可比矩阵,用于迭代求解;从能量原理出发,推导出杆单元应变能、外荷载势能、结构总势能在每次迭代位移方向上关于步长λ的显式列式。相对于固定步长的牛顿法,本文将最优迭代步长λ引入求解,使结构在每次迭代位移方向上均达到总势能最小。经桁架算例验证,表明该方法可加快计算收敛进程。
Force density method is the first method to solve the problem of finding the shape of the membrane structure, which can be used to calculate the geometric nonlinearity of the truss structure. In this paper, the nonlinear equilibrium equations of structural deformation and corresponding Jacobian matrixes are established by force density method for iterative solution. Based on the energy principle, the rod unit strain energy, the external load potential energy and the total potential energy of the structure are deduced at each iteration The explicit formulation of the step size λ in the direction. Compared with the Newton method with fixed steps, this paper introduces the optimal iteration step λ into the solution to make the structure reach the minimum total potential energy in every iteration. The truss example shows that this method can speed up the process of convergence.