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Lower bound limit analysis using finite elements and linear programming. (English) Zbl 0626.73117

The paper describes a technique for computing lower bound limit loads in soil mechanics under conditions of plane strain. In order to invoke the lower bound theorem of classical plasticity theory, a perfectly plastic soil model is assumed, which may be either purely cohesive or cohesive- frictional, together with an associated flow rule. Using a suitable linear approximation of the yield surface, the procedure computes a statically admissible stress field via finite elements and linear programming. The stress field is modelled using linear 3-noded triangles and statically admissible stress discontinuities may occur at the edges of each triangle. Imposition of the stress-boundary, equilibrium and yield condition leads to an expression for the collapse load which is maximized subject to a set of linear constraints on the nodal stresses. Since all of the requirements for a statically admissible solution are satisfied exactly (except for small round-off errors in the optimization computations), the solution obtained is a strict lower bound on the true collapse load and is therefore ‘safe’.
A major drawback of the technique, as first described by J. Lysmer [J. Soil Mech. Found. Div., A.S.C.E. 96(SM 4), 1311-1334 (1970)] is the large amount of computer time required to solve the linear programming problem. This paper shows that this limitation may be avoided by using an active set algorithm, rather than the traditional simplex or revised simplex strategies, to solve the resulting optimization problem. This is due to the nature of the constraint matrix, which is always very sparse and typically has many more rows than columns. It also proved that the procedure can, without modification, be used to derive strict lower bounds for a purely cohesive soil which has incresing strength with depth. This important class of problem is difficult to tackle using conventional methods. A number of examples are given to illustrate the effectiveness of the procedure.

MSC:

74L10 Soil and rock mechanics
74R20 Anelastic fracture and damage
74S99 Numerical and other methods in solid mechanics
Full Text: DOI

References:

[1] Lysmer, J. Soil Mech. Found. Div., A. S. C. E. 96 pp 1311– (1970)
[2] ’Theories of plasticity and the failure of soil masses’, in Soil Mechanics–Selected Topics (Ed. ), Chap. 6, Butterworths, London, 1968.
[3] Limit Analysis and Soil Plasticity, Elsevier, Amsterdam, 1975. · Zbl 0354.73033
[4] Anderheggen, Int. J. Solids Struct. 8 pp 1413– (1972)
[5] ’Application de l’analyse Limite a l’etude de la Stabilitie des Pentes et des Talus’, Thesis, Institute of Mechanics, Grenoble (1976).
[6] Bottero, Comp. Meth. Appl. Mech. Eng. 22 pp 131– (1980)
[7] and , Linear Programming: Active Set Analysis and Computer Programs, Prentice-Hall, New Jersey, 1985.
[8] and , Practical Optimisation, Academic Press, London, 1981.
[9] Davis, Geotechnique 23 pp 551– (1973)
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[12] ’FORTRAN subroutines for handling sparse linear programming bases’, Harwell Report, A. E. R. E., R8269 (1976).
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