求解比例加载下结构负刚度问题的虚加弹簧法

Assumed-Spring Method for Analysis of Structures with Negative Stiffness under Proportional Loading

针对恒载和比例变载共同作用下软化结构按位移法求解的负刚度问题,提出了一种虚加弹簧法。在各比例加载点沿荷载方向布置刚度适当的弹簧,得到没有负刚度问题的硬化结构,对虚加弹簧的结构进行逐级加载,在每级加载时从总荷载中扣除弹簧所承担的荷载,即得到结构所承担的荷载。将后者中不符合比例加载要求的非比例成分反向施加在虚加弹簧的结构上重新求解,将得到新的非比例成分。重复此过程进行迭代运算,直至非比例成分可略,就得到本级比例加载下结构负刚度问题的解,重复此过程可得到各级加载下的解。本方法同时也适合结构硬化阶段的求解。该方法力学概念直观,编程简明,可以用于非线性结构加卸载变形全过程的数值分析。运用该方法对多种工况进行了计算,结果表明该方法可行并具有良好的收敛性。

The assumed -spring method is developed to solve the negative stiffness problem of softening structures under dead and live proportional loads. At each point carrying a component of the proportional loads, a spring with appropriate stiffness is installed along the load direction, and then a structure without negative stiffness is constructed. Loading the structure with springs step by step and deducting the loads exerted on the springs, one can find the loads acting on the original structure, Get the non-proportional part which does not conforms to proportional loading and put it reversely on the structure with springs. Then resolving this new case results in a new non-proportional part. Repeat this process until the non- proportional part is negligible, thus the solution of the proportional loading of the current loading step is obtained. Repeating these procedures can provide the solutions of each loading steps. This method can also apply to handle hardening structures. This scheme has direct mechanical meaning and can be programmed concisely. It is applicable in the numerical analysis of the whole deformation process of nonlinear structures. A few cases are studied with this approach, which show good feasibility and convergence of this method.