6 D2

This is the equation of a parabola with x = L/2 when d — 0 (i.e. fully plastic section)

Strains Beyond the Elastic Limit x' = L/3 when d = D (i.e. section elastic)

The yielded portion of the beam is thus as indicated in Fig. 3.10.

Yielded area

Fig. 3.10. Yielded area in beam carrying central point load.

Other beam support and loading cases may be treated similarly. That for a simply supported beam carrying a uniformly distributed load produces linear boundaries to the yielded areas as shown in Fig. 3.11.

Having determined the moment required to produce a plastic hinge for the shape of beam cross-section used in any design it is then necessary to decide from a knowledge of the support and loading conditions how many such hinges are required before complete collapse of the beam or structure takes place, and to calculate the corresponding load. Here it is necessary to consider a plastic hinge as a pin-joint and to decide how many pin-joints are required to convert the structure into a "mechanism". If there are a number of points of "local" maximum B.M., i.e. peaks in the B.M. diagram, the first plastic hinge will form at the numerical maximum; if further plastic hinges are required these will occur successively at the next highest value of maximum B.M. etc. It is assumed that when a plastic hinge has developed at any cross-section the moment of resistance at that point remains constant until collapse of the whole structure takes place owing to the formation of the required number of further plastic hinges.

Consider, therefore, the following loading cases.

(a) Simply supported beam or cantilever

Fig. 3.11. Yielded area in beam carrying uniformly distributed load.

Fig. 3.11. Yielded area in beam carrying uniformly distributed load.

3.8. Collapse loads - plastic limit design

Whatever the loading system there will only be one point of maximum B.M. and plastic collapse will occur with one plastic hinge at this point (Fig. 3.12).

Fig. 3.12.

(b) Propped cantilever

In the case of propped cantilevers, i.e. cantilevers carrying opposing loads, the B.M. diagram is as shown in Fig. 3.13. The maximum B.M. then occurs at the built-in support and a plastic hinge forms first at this position. Due to the support of the prop, however, the beam does not collapse at this stage but requires another plastic hinge before complete failure or collapse occurs. This is formed at the other local position of maximum B.M., i.e. at the prop position, the moments at the support remaining constant until that at the prop also reaches the value required to form a plastic hinge.

Collapse therefore occurs when Ma=Mb = Mfp, and thus two plastic hinges are required.

(c) Built-in beam

In this case there are three positions of local maximum BJVL, two of them being at the supports, and three plastic hinges are required for collapse (Fig. 3.!4).

Fig. 3.13.
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