Question 18.8: Determine the value of the load W required to cause collapse...
Determine the value of the load W required to cause collapse of the frame shown in Fig. 18.16(a) if the plastic moment of all members of the frame is 200 \mathrm{kN} \mathrm{m}. Calculate also the support reactions at collapse.
We note that the frame and loading are unsymmetrical so that sway occurs. The bending moment diagram for the frame takes the form shown in Fig. 18.16(b) so that there are three possible collapse mechanisms as shown in Fig. 18.17.
In Fig. 18.17(a) the horizontal member BCD has collapsed with plastic hinges forming at B, C and D; this is termed a beam mechanism. In Fig. 18.17(b) the frame has swayed with hinges forming at A, B, D and E; this, for obvious reasons, is called a sway mechanism. Fig. 18.17(c) shows a combined mechanism which incorporates both the beam and sway mechanisms. However, in this case, the moments at \mathrm{B} due to the vertical load at \mathrm{C} and the horizontal load at \mathrm{B} oppose each other so that the moment at \mathrm{B} will be the smallest of the five peak moments and plastic hinges will form at the other locations. We say, therefore, that there is a hinge cancellation at \mathrm{B}; the angle \mathrm{ABC} then remains a right angle. We shall now examine each mechanism in turn to determine the value of W required to cause collapse. We shall designate the plastic moment of the frame as M_{\mathrm{P}}.
