Upper Bound Techniques – The High Road
The traditional hand calculation approach to yield line analysis (Johansen, 1964) requires the initial specification of a potential mechanism for which a collapse load is calculated. This estimate will be an upper bound on the true collapse load. Subsequent trial mechanisms may then be investigated and the lowest collapse load found is taken as the exact value. A refinement is to investigate the effect of geometric variation of a given yield line arrangement, whilst maintaining the same basic topology. This refinement normally results in less significant reductions in the collapse load than is possible from the detection of a more critical mechanism and is therefore sometimes approximated by applying a 10-15% reduction in the collapse load. This reduction is assumed to cover this geometric refinement effect, together with subsidiary variations in collapse mechanisms, such as the presence of “corner levers” (Kennedy et al, 2003).
The yield line approach may be automated by casting it as a linear programming formulation (Munro et al, 1978). The slab system to be analysed is subdivided by a triangulated grid and the optimisation process then evaluates a critical mechanism, taking the edges of the triangles as potential yield line sections. The process does not necessarily identify either the correct form of mechanism or the exact geometric positioning of the yield lines, since it is limited by the stipulation that yield should only occur along the edges of the specified grid. To improve the accuracy of the estimated collapse load, a two part process has been suggested (Johnson, 1994), in which a “fine” grid is used to identify the likely critical mechanism and a “coarse” grid is then used with geometric optimisation to obtain an improved approximation to the positioning of the yield line pattern.