Analysis of statically indeterminate non-uniform bar problem in post elastic domain by an iterative variational method

The present study investigates the growth of elastic-plastic front of a statically indeterminate non-uniform bar in post-elastic regime. The solutions of statically indeterminate bar problems are critical in general, because they are not amenable to a ready analytical solution. A clamped axially loaded bar problem becomes indeterminate when the load is concentric, and it result in a singularity point in the domain. In the present bar problem more such singularity points arise when the bar is in post-elastic state, at higher magnitude of concentrated load and the other points come from the yield front location. The computational domain is divided into sub-domains based on the location of singularity points. The formulation is based on von-Mises yield criterion and for linear strain hardening type material behavior. The governing equation is derived through an extension of a variational method in elasto-plastic regime and solution is obtained by using Galerkin's approximation principle. The approximate solution further needs an iterative method to locate the growth in the yield front. The solution algorithm is implemented with the help of MATLAB® computational simulation software and validation of the formulation is carried out successfully for some reduced problems. The effect of geometry parameters like aspect ratio, slenderness ratio and the type of taperness on the post-elastic performance of the bar is investigated and the relevant results are obtained in dimensionless form. The term bar used in this paper is in generic sense and hence the formulation is applicable for all one dimensional elements, e.g., rods, pipes, truss members, etc.