Lower Bound Techniques – The Low Road
In practice, using hand calculations, lower bound solutions are generally obtained by the “strip” method (Hillerborg, 1975). In its simplest form, the strip method uses lines of assumed zero shear force to allow the determination of equilibrium moment distributions that provide a “safe” solution if used to design the slab’s reinforcement pattern. The basic methodology has been extended (Hillerborg, 1996) by a number of approximations and simplifications to allow the design of slabs with complex geometries, loadings, and boundary conditions and also to make it more convenient for practical application.
Lower bound solutions may also be generated by automated approaches, although these are closer to automated upper bound (yield line) techniques rather than to the strip system. As with the automated yield line system, triangulated nets are used, but, rather than taking the net edges as potential yield lines, prescribed distributions of bending moment (Krenk et al, 1994) within the triangular regions are presumed. If, as is common practice, the moment within each element is assumed to be a linear function of the values at its nodes, then a lower bound solution may be obtained by optimising the load subject to the nodal moment values not exceeding the slab’s yield capacity.