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## Q. 14.6.2

A propped uniform cantilever is to be designed to support the loads in Fig. 14.6-6(a). Explain how the lower bound theorem may be used to select a value of the plastic moment of resistance, $M_{P}$, which will guarantee that the beam will not collapse under the loading.

## Verified Solution

ON Fig. 14.6-6(b), (c) and (d) show three bending moment diagrams which are in equilibrium with the external loads P and 2P. These are discussed in turn: Fig. 14.6-6(b): The bending moment diagram is obtained by superposition of the simple-beam moment diagram a1c2d2b on to the diagram a1a2b due to the redundant moment M at the fixed end A. The value of M actually acting is, of course, not known, but this presents no difficulties because the designer is at liberty to choose any value he considers appropriate. Suppose he sets M at a value represented to scale by the ordinate a1a2. The largest moment ordinate is then d1d2. Thus, provided the plastic moment of resistance $M_{p}$ exceeds d1d2 (= PL — M/4), the yield condition is satisfied and the lower bound theorem guarantees that the structure is safe.

Fig. 14.6-6(c): The bending moment diagram in Fig, 14.6-6(b) reduces to the simple- span diagram a1c2d2b in Fig. (c) when M is made equal to zero. Admittedly, there is little justification to suppose that M would be zero in the actual beam, but the lower bound theorem is specific: provided $M_{p} > PL$, the yield condition is satisfied and the safety of the beam is guaranteed.

Fig. 14.6-6(d): The moment diagram in Fig. 14.6-6(b) becomes that in Fig. (d) if the designer should (be so unreasonable as to) make M a sagging moment. Commonsense tells us that the loads P and 2P will never produce a sagging moment at the fixed end A of the beam, but the validity of the lower bound theorem is not affected by the ‘unreasonableness’ of the assumed bending moment distribution; as long as the designer provides a plastic moment of resistance $M_{p}$ exceeding PL + M/2 (See Fig. 14.6-6(d)), the safety of the structure is assured.