On the Plastic Collapse of Structures and Linear Programming

This paper is concerned with the solution of various problems in the plastic collapse of plane structures.

In Chapter I the basic problems and theorems of limit analysis are reviewed and formulated in a convenient notation.

A pair of superposition principles are developed for limit analysis of structures in Chapter II. These principles lead to upper and lower bounds to the safety factor for a superimposed load system in terms of bounds to the safety factors for the individual loads. In addition several special problems are posed and solved in the second chapter. These include a minimax problem in which a safety factor which is valid for all load systems in a given range is found. Finally an iterative method is given for obtaining bounds to the safety factor for the proportional loading of frames when axial forces as well as bending moments are to be considered. Examples are included at the end of the chapter.

Chapter III reviews three basic methods of solution for linear programming problems. The problem of the plastic collapse of structures is reduced to forms suitable for the application of these three methods. A collapse problem is solved by the several linear programming methods in Chapter IV for demonstration and comparison.

A method for obtaining an initial feasible solution for Lemke's dual method of solving the linear programming problem is given in Appendix C. This method is analogous to a procedure developed by Dantzig for the simplex method.