70 Mechanics of Materials 2 $3.7 W/2 一L/2一儿/2 w/2 B.M.Diagram WL/4 Fig.3.9. the centre.If loading is increased,yielding will commence therefore at the central section when (WL/4)=(BD2/6)oy and will gradually penetrate from the outside fibres towards the N.A.As this proceeds with further increase in loads,the B.M.at points away from the centre will also increase,and in some other positions near the centre it will also reach the value required to produce the initial yielding,namely BD2/6.Thus,when full plasticity is achieved at the central section with a load Wp,there will be some other positions on either side of the centre,distance x from the supports,where yielding has just commenced at the outer fibres;between these two positions the beam will be in some elastic-plastic state.Now at distance x from the supports: 2 2WpL B.M.=w,2=MFP=54 L x=3 The central third of the beam span will be affected therefore by plastic yielding to some depth.At any general section within this part of the beam distance x'from the supports the B.M.will be given by xBa (3D2-d2] BM.=W2=12 (1) BD WpL Now since 0,=w4 0y= BD2 Therefore substituting in (1), x' w,2= 23D2-4 pL BD2 (3D-)L 6D2 L d21 术= 2 3D2 This is the equation of a parabola with x'=L/2 when d =0 (i.e.fully plastic section)70 Mechanics of Materials 2 03.7 w/2 F-% L/2 & L/2 w/2 Fig. 3.9. the centre. If loading is increased, yielding will commence therefore at the central section when (WL/4) = (BD2/6)o, and will gradually penetrate from the outside fibres towards the N.A. As this proceeds with further increase in loads, the B.M. at points away from the centre will also increase, and in some other positions near the centre it will also reach the value required to produce the initial yielding, namely BD20,/6. Thus, when full plasticity is achieved at the central section with a load W,, there will be some other positions on either side of the centre, distance x from the supports, where yielding has just commenced at the outer fibres; between these two positions the beam will be in some elastic-plastic state. Now at distance x from the supports: x2 2 W,L B.M. = W,- = -MFp = -- 23 34 L 3 .. x=- The central third of the beam span will be affected therefore by plastic yielding to some depth. At any general section within this part of the beam distance x’ from the supports the B .M. will be given by Now since XI Bo B.M. = W,- = ‘[3D2 - d2] 2 12 L WPL --(Tv=wp- -(T -- BD~ 4 4 ’- BD2 Therefore substituting in (l), W,,- x’ B WPL = -[3D2 - d2]- 2 12 BD2 L I (3D2 -d2) 6D2 x= 2 This is the equation of a parabola with XI = L/2 when d = 0 (i.e. fully plastic section)