Question 14.1: Design diagram of pinned-clamped beam, which is subjected to......

From the elastic analysis of the pinned-clamped beam we know that the maximum moment occurs at the support B. Therefore here will be located the first plastic hinge; the corresponding plastic moment is M_y  (Fig. 14.6b).

Now we have simply supported beam, i.e., the appearance of the plastic hinge at B does not destroy the beam but led to the changing of design diagram; reaction of such beam is R_A=P / 2-M_y / l . For this beam we can increase the load P until the second plastic hinge appears at the point of application of the load. Thus we have three hinges (Fig. 14.6c), which are located on the one line, so this structure becomes instantaneously changeable.

Using superposition principle, the moment at the point of application of force is

M(P)+M\left(M_y\right)=\frac{P l}{4}-\frac{M_y}{2}

In the limit condition, the moment at the point of application of force equals to plastic moment:

\frac{P_{\lim } l}{4}-\frac{M_y}{2}=M_y       (a)

This equation leads to the limit load P

P_{\lim }=\frac{6 M_y}{l}

The procedure (a) may be presented graphically as shown in Fig. 14.6d.

Obviously, in the case of an arbitrary position \xi=u l  of the force P on the beam, the limit load is a function of position \xi .