A numerical strategy for computing the stability boundaries for multi-loading systems by using generalized inverse and continuation method

A strategy for computing the stability boundaries for structural problems subjected to two loading parameters has been proposed. In this strategy, the incremental solution to a known point lying on the stability boundary is first expressed as a linear combination of the homogeneous and particular solutions for the linear incremental equilibrium equations by using the generalized inverse. Next by imposing a linear constraint on the two loading parameters and then carrying out iteration at a constant load ratio, a point lying in the vicinity of the next target point on the stability boundary is obtained. Using this point as the initial point and keeping the same linear constraint on the two loading parameters active, the next target point lying on the stability boundary is then obtained by tracing the equilibrium curve and monitoring the occurrence of the critical point simultaneously. During equilibrium curve tracing, the original two-parameter problem is reduced to a single-parameter problem by virtue of the linear constraint imposed on the two loading parameters. Numerical analysis has been carried out on three numerical examples. From the obtained results it can be concluded that the proposed strategy is worthy of further development.