Question 14.8.2: Fig. 14.8-3(a) shows a propped cantilever supporting a unifo...
Fig. 14.8-3(a) shows a propped cantilever supporting a uniformly distributed load w per unit length. If the collapse load factor is to be A, determine the required plastic moment of resistance M_{P}.
Let the unknown location of the sagging plastic hinge be defined by x, as in Fig. 14.8-3(b). The work equation is
\qquad \quad \lambda wL\left(\frac{x\phi }{2} \right) =M_{P}\left[\phi +\frac{L\phi }{L-x} \right]or \qquad \qquad M_{P}=\lambda wL\frac{x(L-x)}{2L-x}
By the unsafe theorem, the value of M_{P} corresponding to an arbitrarily assigned value of x is less than, or at best equal to, the plastic moment required to withstand the load λw. Hence,M_{P} is maximised with respect to x:
\hspace{5em} \frac{dM_P}{dx} =0 \ gives \ x=\underline{0.586L} \\ \hspace{5em} so \ that \qquad M_P=\underline{0.0858 \ wL^2}COMMENTS
The above solution is based on the unsafe or upper bound theorem. An alternative solution, based on the safe or lower bound theorem is given in Example 14.8-3.
