        # How to Predict the Plastic Collapse of Structures Efficiently?

VITEZSLAV STEMBERA, TU Wien, Austria
How to Predict the Plastic Collapse of Structures Efficiently?

The prediction of the collapse load (limit load) of a structure made of a material exhibiting elastic plastic behavior is very often of practical interest. The standard approach to obtain such collapse loads is based on iterative calculation schemes using classical nonlinear finite element methods, as, e.g., can be found in .

However, as an alternative approach, the so-called finite-element-based limit analysis (FELA) can be applied. This approach is based on limit theorems, first formulated by A.A.Gvozdev in 1936. Thereby, the collapse load is obtained as the minimum of a certain optimization problem, either considering kinematically compatible velocity fields (upper bound approach) or statically admissible stress fields (lower bound approach) within the structure, at the time instant of collapse. Thus, the whole load history doesn’t need to be taken into account, resulting in a much more stable and numerically efficient approach compared to standard schemes based on classical finite element formulations. The two significant disadvantages of the FELA method, which are the assumption of geometrical linearity and ideal plasticity, can be overcome by the so-called sequential finite element limit analysis (SFELA), as, e.g., given in . Thereby, the FELA method is called repeatedly, where the geometry and the plastic strain is updated after each iteration.

In this talk, implementations of the FELA as well as SFELA method, for the case of shell and solid finite elements is presented. Two strength criteria are considered: an isotropic von Mises type strength criterion for shell elements (Ilyushin criterion) and the orthotropic Tsai-Wu strength criterion for solid elements. An incorporation of isotropic hardening and softening is possible and is demonstrated in two of the three presented examples: first, in plastic collapse of a steel frame and second, in plastic collapse of a S-rail tube. In the third example, the ability to analyze wooden structures is demonstrated, where the collapse of a three-layered pressure loaded cross laminated timber (CLT) plate is predicted. The numerically obtained collapse loads of the CLT plate for different wooden strength classes C18, C24, and C35 are compared with experimental data.
It is demonstrated that the FELA and SFELA methods are efficient numerical approaches for predicting the plastic collapse of shell and solid structures. Moreover it is shown, that hardening and softening can be incorporated within the framework of the SFELA method.

 A. Robertson, H. Li, D. Mackenzie, Plastic collapse of pipe bends under combined internal pressure and in-plane bending, Int. J. of Pressure Vessels and Piping, Vol.82, 2005, pp. 407-416.
 D. Kong, C. M. Martin, B. W. Byrne, Modelling large plastic deformations of cohesive soils using sequential limit analysis, Int. J. Numer. Anal. Meth. Geomech., Vol.41, 2017, pp. 1781–1806.

How to Predict the Plastic Collapse of Structures Efficiently?