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

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.