Ultimate limit state design of sheet pile walls by finite elements and nonlinear programming Kristian Krabbenhoft a , Lars Damkilde b, * , Sven Krabbenhoft b a Department of Civil Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark b Institute of Chemistry and Applied Engineering Science, Section for Structural Mechanics, Aalborg University Esbjerg, Niels Bhor Vej 8, DK-6700 Esbjerg, Denmark Received 16 December 2002; accepted 9 August 2004 Abstract The design of sheet pile walls by lower bound limit analysis is considered. The design problem involves the determi- nation of the necessary yield moment of the wall, the wall depth and the anchor force such that the structure is able to sustain the given loads. This problem is formulated as a nonlinear programming problem where the yield moment of the wall is minimized subject to equilibrium and yield conditions. The finite element discretization used enables exact ful- fillment of these conditions and thus, according to the lower bound theorem, the solutions are safe. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Sheet pile walls; Plasticity; Limit analysis; Material optimization; Finite elements; Nonlinear programming 1. Introduction 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 con- sists 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 nec- essary extrusion pressure in metal forming problems, when evaluating the bearing capacity of reinforced con- crete slabs or the stability of slopes, and generally, when- ever 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 dimen- sions, 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 following we consider the latter of these prob- lems with particular reference to the design of sheet pile walls. The example problem which we will return to in Section 7 is sketched in Fig. 1. Given the weight of the soil and the relevant strength parameters, for the Mohr–Coulomb criterion the cohesion and the angle of friction, the task is to determine the necessary yield 0045-7949/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.08.016 * Corresponding author. Tel.: +45 7912 7648; fax: +45 7545 3643. E-mail address: ld@aaue.dk (L. Damkilde). Computers and Structures 83 (2005) 383–393 www.elsevier.com/locate/compstruc