Limit state analysis: Adaptive finite element upper and lower bound approach to the evaluation of the limit load of a Von Mises rigid-plastic material body in plane stress. - Raymundo Cordero (2005)
A new approach to the computation of the Limit load of a Von Mises rigid-plastic material structure modelled in plane stress is assessed. Most international design codes require the engineer to establish the safety of a structure for a given set of design loads under the so-called limit state conditions. The limit state represents the failure point at which the structure begins to exhibit unbounded deformations. Under limit state conditions, the deformation of the solids tend to concentrate on thin failure bands, known as slip-lines. This makes the finite element analysis a challenging task as the mesh needs to be adapted to capture these bands accurately. In order to achieve this, an adaptive technique is required whereby the error produced in each finite element is measured and if required the element is subdivided automatically. In order to measure this error both an upper and lower bound of the exact solution need to be evaluated. In this thesis, a novel technology to obtain the lower bound is derived and implemented together with mesh adaptivity technology. A lower bound is found from a state of stresses in equilibrium with the external forces. The proposed technique obtains such equilibrated state using the stresses obtained during the upper bound evaluation. These stresses, although not strictly in equilibrium, can be balanced using procedures available in the literature. The present aim of the research project is to develop numerical technology based on the finite element method to calculate the limit state of two-dimensional solids in plane stress. The upper bound theorem of limit analysis is implemented by means of a Lagrangian optimization technique solved by the Newton-Raphson method with Line Search. A control parameter to deal with the singularity of the tangent stiffness matrix due to the yielding condition is used along the range of admissible rate of deformations for a rigid-plastic material. The lower bound theorem is then applied by performing a technique to equilibrate the interelement tractions, kinematically solving a sequence of local problems using the equilibrated tractions as an updated load input, which lets us determine the elementwise contribution to both the upper and lower bounds. An adaptive technique is then implemented, based on the elemental contributions to the difference between the upper bound and the lower bound of the collapse multiplier. Both non-adaptive and adaptive results are evaluated. Results show a good performance of the solution technique, both in comparison with well known plane stress bound values and also in the graphical output obtained in the form of refined regions which describe the occurrence of slip-line patterns and/or localized yielding regions.