72 Mechanics of Materials 2 $3.8 经 Mox.B.M.-one hinge Max.B.M.-one hinge 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. Two points of "max."B.M. -two hinges B.M Diogram Fig.3.13. Collapse therefore occurs when MA =M8 =MFP,and thus two plastic hinges are required. (c)Built-in beam In this case there are three positions of local maximum B.M.,two of them being at the supports,and three plastic hinges are required for collapse (Fig.3.14). Three positions of local max.BM. -three hinges BM Diagram Fig.3.14.72 W i Mechanics of Materials 2 $3.8 rn Mox. B.M.- one hinp Mox.B.M.- one hinge "/ Fig. 3.12 (6) 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. Two ooints of "mxl' B.M. B.M L Diagram Fig. 3.13. Collapse therefore occurs when MA = MB = MFP, and thus two plastic hinges are required. (e) Built-in beam In this case there are three positions of local maximum B.M., two of them being at the supports, and three plastic hinges are required for collapse (Fig. 3.14). Three positms of local max BM -three hinges BY Diagram Fig. 3.14