Arrivals and Departures
When using either an upper or a lower bound technique, whether one arrives at the true collapse load or departs significantly from it depends on the approach selected and the slab under investigation. A number of illustrative examples will therefore be studied to show the comparative abilities of the different methods.
Rectangular slab simply supported on three sides
An automated yield line procedure presented by Shoemaker (Shoemaker, 1989) followed the hand calculation approach of examining a number of predetermined collapse modes, the form of which depended on the particular slab under consideration. In the case of a rectangular slab, simply supported on three sides and free on the fourth, the two collapse modes shown in Figure 1a were examined and the critical values of x and y were determined by a grid based search procedure. For a particular example, the solution obtained by this procedure was as shown in Figure 1b.
Figure 1: a. Possible modes b. Shoemaker solution c. Hillerborg solution
Subsequently, Hillerborg (Hillerborg, 1991) showed that, in the absence of top reinforcement, the mechanism shown in Figure 1c is critical and that the mechanism employed by Shoemaker had overestimated the collapse loads by at least 50%. This led Hillerborg to conclude that “In my opinion, it is questionable whether yield line analysis is a suitable basis for the design of slabs, as the risk of getting an unsafe structure is too high.”
Figure 2: a. Fine mesh solution b. coarse mesh solution
The slab has also been analysed by the automated yield line procedure (Johnson, 1996). In this case, the “fine” net shown in Figure 2a, based on a half-slab model, was used to identify the nature of the collapse mode and the “coarse” net shown in Figure 2b was then used to optimise the geometry of the mode. As may be noted, this procedure produced a solution that corresponded closely to the one suggested by Hillerborg (Figure 1c). In this case, therefore, the automated yield line technique did successfully identify the appropriate mechanism. However, it might well be contended that the two stage solution is somewhat cumbersome since it is dependent on manual intervention to establish a coarse grid appropriate to the identified mode.