## Example 214

Frame subjected to distributed load

When a frame is subjected to distributed loads, the maximum moment and hence the plastic hinge location are not known in advance. The exact location of the plastic hinge may be determined by writing the work equation in terms of the unknown distance and then maximizing the plastic moment by formal differentiation.

Consider the frame shown in Figure 2.101a. The side sway collapse mode in Figure 2.101b leads to the following work equation:

which gives

The beam mechanism of Figure 2.101c gives which gives

In fact the correct mechanism is shown in Figure 2.101d, in which the distance Z from the plastic hinge location is unknown. The work equation is

which gives

To maximize Mp, the derivative of Mp is set to zero, that is,

In practice, uniform load is often approximated by applying several equivalent point loads to the member under consideration. Plastic hinges thus can be assumed to form only at the concentrated load points, and the calculations become simpler when the structural system becomes more complex.

1.6 kip/ft

24 kip

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