$5.6 Slope and Deflection of Beams 107 APPLIED LOAD SYSTEM EQUIVALENT LOAD SYSTEM Applied loading w/metre Applied w Continuation -0- R R Compensating Fig.5.14. BMRax一罗子+wt空吗 Compensating' X 3w Applied w -0 R: Fig5.15. MR2wluap] Second Continuation compensating Applied w X R X First compensating Fig5.16. BMRax-wt2%+w空21-wf空) st compensating 2w Fig5.17. M2管，及a-1+w学，…g2四 compensating Figs.5.14.5.15,5.16 and 5.17.Typical equivalent load systems for Macaulay method together with appropriate B.M.expressions. Ra B C M Mo B.M diagram ML=a】 L Fig.5.18.Beam subjected to applied couple or moment M
$5.6 Slope and Depection of Beams 107 APPLIED LOAD SYSTEM EQUIVALENT LOAD SYSTEM Applied loading w/metre Continuation Applied w lx / ++-a4 RA Compensating EM -H2+w[('a'? 2 2 RA E%-lRB Fig 5 14 +Compensating' I T RA RE RE 2w Fig. 5.15. 8 M., =RAx -e2- 2 2w [??I Second RA First compensating Fig 5 16. EMxx ;R,X-W [I&& + w ['?*]-W[(?)~] is' compensating 2w compensatlng 2 2"d BM,,=-2wL2 t RA [(a-a)] +w [(X-b"] +w['x-c' 1 2 2 2 Fig. 5 17. Figs 5 14,5 15,5 16 and 5.17. Typical equivalent load systems for Macaulay method together with appropnate B M. expressions A n B M diagram MIL-a1 L Fig. 5.18. Beam subjected to applied couple or moment M
108 Mechanics of Materials §5.7 For sections between and C the B.M.is M MX -M. For sections between Cand B the B.M.is The additional(-M)term which enters the B.M.expression for points beyond C can be adequately catered for by the Macaulay method if written in the form M[(x-a)] This term can then be treated in precisely the same way as any other Macaulay term, integration being carried out with respect to(x-a)and the term being neglected when x is less than a.The full B.M.equation for the beam is therefore Mx-E装-2--a] (5.17) Then BE史-M2-M[K-]+Acc 1=2L 5.7.Mohr's“area-moment'”method In applications where the slope or deflection of beams or cantilevers is required at only one position the determination of the complete equations for slope and deflection at all points as obtained by Macaulay's method is rather laborious.In such cases,and in particular where loading systems are relatively simple,the Mohr moment-area method provides a rapid solution. B.M.diagram Fig.5.19. Figure 5.19 shows the deflected shape of part of a beam ED under the action of a B.M. which varies as shown in the B.M.diagram.Between any two points B and C the B.M. diagram has an area A and centroid distance x from E.The tangents at the points B and C give an intercept of xoi on the vertical through E,where oi is the angle between the tangents. Now ds=Rδi
108 Mechanics of Materials 45.7 M L For sections between A and C the B.M. is -x. Mx L For sections between C and B the B.M. is ~ - M The additional (- M) term which enters the B.M. expression for points beyond C can be adequately catered for by the Macaulay method if written in the form M[I(x-a)Ol This term can then be treated in precisely the same way as any other Macaulay term, integration being carried out with respect to (x - a) and the term being neglected when x is less than a. The full B.M. equation for the beam is therefore d2y Mx dx2 L M,,=EI-=--M[(x-a)0] (5.17) Then dy Mx2 dx 2L El- = - - M[(x-a)]+A, etc. 5.7. Mohr’s “area-moment” method In applications where the slope or deflection of beams or cantilevers is required at only one position the determination of the complete equations for slope and deflection at all points as obtained by Macaulay’s method is rather laborious. In such cases, and in particular where loading systems are relatively simple, the Mohr moment-area method provides a rapid solution. \ ‘- B.M. diagram I I I 17 I/ I I I I I Fig. 5.19. Figure 5.19 shows the deflected shape of part of a beam ED under the action of a B.M. which varies as shown in the B.M. diagram. Between any two points B and C the B.M. diagram has an area A and centroid distance X from E. The tangents at the points B and C give an intercept of xSi on the vertical through E, where Si is the angle between the tangents. 6s = R6i Now