Finite element linear programming approach to foundation shakedown

Aboustit, Baher Labeeb (1979) Finite element linear programming approach to foundation shakedown. Masters thesis, Memorial University of Newfoundland.

[English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (19MB) [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

Abstract

Shakedown load factors are evaluated for certain plane stress and plane strain problems using the kinematic shakedown theorem. The numerical procedure is based on coupling of the finite element method with linear programming. -- For foundations subjected to loads varying in time in a nonproportional manner within prescribed limits, the classical limit theorems can give unsafe estimates of the collapse loads, as failure can occur at loads well below the static collapse values. Shakedown theorems, which are generalizations of the limit theorems, can provide appropriate safe bounds for complex loading programmes. A few applications of the static shakedown theorem to plane stress problems are available in the literature, but plane strain applications have not yet been reported. -- The nonlinear mathematical formulations for shakedown analysis of continuum problems can be adapted to linear programming by piecewise linearization of the yield surfaces. The continuum is discretised into a finite number of constant strain triangular elements. The elastic-perfectly plastic piecewise linear constitutive law for each element is defined by the yield condition and the associated flow rule. The equilibrium and compatibility conditions are derived from the requirements of the static and the kinematic shakedown theorems respectively. The plastic dissipation energy is minimized subject to compatibility and maximum positive external work conditions. -- A computer programme is developed for application of the kinematic shakedown theorem. The software is applied to (1) shakedown analysis of a square plate, with a circular central hole, subjected to biaxial, variable repeated loading for the plane stress condition, and (2) limit analysis of a strip footing subjected to uniformly distributed loading for the plane strain conditions. The results are in excellent agreement with available analytical (Case 1) and numerical (Cases 1 and 2) solutions. The code is then applied to the problem of a footing subjected to inclined, eccentric and variable repeated loading for the plane strain case. It is observed that the shakedown load varies almost linearly with the uniaxial compressive strength of the underlying soil for the particular case of tension cut-off. -- Although the analysis is restricted to dry soil, the work can be extended to include the effect of cyclic loading on pore pressure. Nonassociative behaviour, work hardening/softening properties, and inertia and damping effects for a prescribed loading history can also be considered. The use of the hybrid finite element model and the sparse matrix technique in linear programming will give better estimates of the shakedown loads with minimum computation time.

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