Nonlinear Truss Analysis by One Matrix Inversion

A new method for nonlinear structural analysis has been developed. The novelty of the method is that only one stiffness matrix inversion is required without the need for updating and reinverting the matrix at every load increment. This stiffness matrix is not necessarily the real stiffness matrix of the structure. Instead any stiffness matrix compatible with the geometry and the constraints of the truss can be used. The advantage of this option is that if the design of some members is revised the already inverted and stored matrix is used for the analysis of the revised structure. Nonlinearities due to strain hardening, strain softening, buckling, breaking, and stiffness degradation are handled by iterations involving only multiplications of the banded matrix with a transformed force vector. The inversion of the half-banded original stiffness matrix is done using Gauss elimination performed on the half-banded matrix without destroying the bandedness, and the inverted matrix replaces the original without the need for additional storage. The coefficients for the transformation of the force vector are stored permanently in a new matrix of size equal to the size of the half-banded original. Thus, the total storage needed is equal to the storage for the banded original stiffness. Because, after the Gauss elimination, only multiplications of a matrix with a vector are involved, the method is computationally efficient. The method is not a step-by-step procedure. Any load increment can be applied, therefore, proportional, nonproportional, and cyclic loads are treated in a unified way. The energy dissipation and the residual stresses and strains after one or more cycles are readily available, and thus the method can be used in quasi-dynamic analysis (e.g. pushover) for an evaluation of the dynamic parameters of the structure.