# Practical design example

 Figure 3: Part-plan of floor slab for block of flats and trial yield line patterns (Kennedy et al, 2003)

The part-plan of one storey of a seven-storey block of flats is shown in Figure 3 (Kennedy et al, 2003). The floor slab was initially analysed by hand using no less than ten possible yield line patterns, which are also shown in Figure 3. On the basis of this analysis, a required moment of resistance of 43.3 kNm/m was established for the slab, the critical mechanism being Pattern 2.

The analysis was subsequently checked by the use of the automated yield procedure, which produced the complex yield line pattern shown in Figure 4a, and an ultimate moment requirement of

47.2 kNm/m. The pattern demonstrates a basic problem with the use of the automated yield line method, namely that the form of the collapse mechanism is not readily obvious from Figure 4a. In these circumstances, reference to an associated contour plot of the collapse mode (Figure 4b) is often useful and, in this case, demonstrates that the collapse mode is of the “folded plate” nature. This collapse mode approximates to a combination of the Patterns 1 and 10 considered in the hand analysis (Figure 3) and these two patterns were, in fact, identified as the third and second most critical cases, respectively, by the hand analysis.

Figure 4: a. Yield line pattern b. collapse mode contour plot

To obtain a bound on the true collapse load, an automated lower bound solution was also obtained and produced the collapse mode shown in Figure 5. As may be seen from Figure 5, the collapse mode is a variation on that given by the upper bound solution (Figure 4b) and the associated ultimate moment requirement of 64.1 kNm/m is a 36% increase on the 47.2 kNm/m obtained from the upper bound analysis. A subsequent upper bound analysis, based on a finer, more restricted net, gave an ultimate moment of 53.8 kNm/m, which corresponded with a value of 55.8 kNm/m derived from a hand analysis (Kennedy et al, 2003). These revised results, however, are still an underestimate of 15% from the lower bound prediction, suggesting that the safer lower bound should generally be preferred, unless it can be shown to produce unduly conservative results.