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Chapter 14

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.

fig14.6-6a

Step-by-Step

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.

COMMENTS
The lower bound or safe theorem of plastic collapse is, whether he knows it or not, the fundamental tool of the conventional elastic steelwork designer. In the elastic design of steel frames, it is usual to design floor beams as simply supported even when the drawing clearly shows that they have fairly rigid bolted joints at the ends. However, the lower bound theorem guarantees that the beams so designed are safe; if a beam can be shown to be safe in the pin-ended state, then it cannot be unsafe if the ends are in fact rigid jointed or semi-rigid jointed.
Also, the elastic designer proportions the members of, say, a structural framework on the basis of an elastic analysis. However, unless the structure happens to be statically determinate, the bending moments calculated from an elastic analysis might bear so little resemblance to the actual bending moments as to be quite meaningless. For example, an elastic analysis of the propped cantilever AB in Fig. 14.6-6(a) would indicate a moment of 0.844 PL at the built-in support; but this analysis is based on the assumptions of no differential settlement of the supports and of full encastre effect at A.

In the final paragraph of Section 9.5, it was stated with reference to Fig. 9.5-3 that a so-called built-in end must be very stiff indeed to give anything approaching a full encastre effect. Thus the full encastre effect assumed in the analysis is unlikely to be achieved in practice; this fact, coupled with some small differential settlement of the supports which must inevitably occur in practice, means that the calculated moment of 0.844 PL at support A may be much larger or smaller than the actual moment. Thus the bending moment diagrams which are used in design and which the elastic designer believes to be correct might possibly be quite incorrect. They merely represent bending moment distributions which are in equilibrium with the applied loads; however, because of the lower bound theorem of plastic collapse, the mere fact that the structure has been properly proportioned on the basis of an equilibrium distribution of bending moments is sufficient to guarantee that the strength is adequate to support the applied loading. Whether the bending moments have been obtained by an elastic analysis or by intelligent guess (or, indeed by unintelligent guesses, as in Fig. 14.6-6(c) and (d)) does not matter.

fig14.6-6
fig9.5-3