Question 18.9: In the portal frame of Ex. 18.8 the plastic moment of the me...

Since the vertical members are the weaker members plastic hinges will form at B in \mathrm{AB} and at \mathrm{D} in \mathrm{ED} as shown, for all three possible collapse mechanisms, in Fig. 18.19. This has implications for the virtual work equation because in Fig. 18.19(a) the plastic moment at \mathrm{B} and \mathrm{D} is M_{\mathrm{P}} while that at \mathrm{C} is 2 M_{\mathrm{P}}. The virtual work equation then becomes

W 2 \theta=M_{\mathrm{P}} \theta+2 M_{\mathrm{P}} 2 \theta+M_{\mathrm{P}} \theta

which gives

W=3 M_{\mathrm{P}}

For the sway mechanism

W 4 \theta=M_{\mathrm{P}} \theta+M_{\mathrm{P}} \theta+M_{\mathrm{P}} 2 \theta+M_{\mathrm{P}} 2 \theta

so that

W=\frac{3}{2} M_{\mathrm{P}}

and for the combined mechanism

W 4 \theta+W 2 \theta=M_{\mathrm{P}} \theta+2 M_{\mathrm{P}} 2 \theta+M_{\mathrm{P}} 3 \theta+M_{\mathrm{P}} 2 \theta

from which

W=\frac{5}{3} M_{\mathrm{P}}

Here we see that the minimum value of W which would cause collapse is 3 M_{\mathrm{P}} / 2 and that the sway mechanism is the critical mechanism.

We shall now examine a portal frame having a pitched roof in which the determination of displacements is more complicated.