Keywords: sheet pile walls, plasticity, limit analysis, material optimization, finite elements, nonlinear programming.

Limit analysis has been used for decades in civil and mechanical engineering practice as a means of analyzing structures of materials which with reasonable accuracy can be described as being rigid-perfectly plastic. Such materials include steel, concrete and soils. Traditionally, most attention has been given to the problem which consists of determining the ultimate magnitude of a given set of loads acting on a structure with a given geometry. This problem is relevant when determining e.g. the necessary extrusion pressure in metal forming problems, when evaluating the bearing capacity of reinforced concrete slabs or the stability of slopes, and generally, whenever all information about the structure, except for the ultimate magnitude of the load set, is known. However, in the design of structures the situation is the opposite. Here the loads are known whereas the necessary dimensions, boundary conditions, material strengths, etc. must be determined in such a way that the structure is able to sustain the given loads. Thus, limit analysis embraces two different scenarios, one where everything except the maximal permissible load intensity is known, and one where all that is known is the load intensity.
In the paper we consider the latter of these problems with particular reference to the design of sheet pile walls, a typical example of which is shown in Figure 56.1. Given the weight of the soil and the relevant strength parameters the task is to determine the necessary yield moment of the wall, the anchor force, and the depth of the wall below level 0.00m.
If the wall depth is known it is possible, by means of the lower bound theorem, to formulate a material optimization problem in which the necessary yield moment is minimized subject to equilibrium and yield conditions. The finite element discretization is performed such that all equilibrium and yield conditions are fulfilled, Poulsen and Damkilde [1], and thus, according to the lower bound theorem the solutions are safe.
In the case where the wall depth is not know a two-phase strategy is proposed. In the phase one problem each wall segment is assigned an independent design parameter and the sum of yield moments then minimized. This gives a good indication of the necessary depth since yield moments approximately equal to zero will be chosen below the necessary depth. The phase two problem then consists of determining the necessary yield moment with all wall segments being assigned a common design parameter.
For the soil the Mohr-Coulomb yield criterion is used and the resulting optimization problem is thus nonlinear. The problem is solved by means of an algorithm previously used for load optimization, Krabbenhoft and Damkilde [2].
In the paper examples of the design of both rough and smooth cantilever and anchored walls are given. The results are compared to what is obtained by the method of Brinch Hansen [3] and good agreement is found.

1

P. N. Poulsen and L. Damkilde. Limit analysis of reinforced concrete plates subjected to in-plane forces. International Journal of Solids and Structures, 37:6011-6029, 2000. doi:10.1016/S0020-7683(99)00254-1

2

K. Krabbenhoft and L. Damkilde. Lower-bound limit analysis with a nonlinear interior point method. In Z. Waszczyszyn and J. Pamin, editors, Proceedings of the Second European Conference on Computational Mechanics. Cracow University of Technology, Cracow, Poland, 2001. Paper no. 181 in CD-ROM proceedings.

3

J. Brinch Hansen. Earth Pressure Calculation. The Danish Technical Press, Copenhagen, Denmark, 1953.

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