Beam_Deflection_Formulae

BEAM DEFLECTION FORMULAE BEAM TYP E SLOPE AT FREE END DEFLE CTION AT ANY SECTION IN TERMS OF x MAXIMUM DEFLECTION 1. Cantilever Beam – Concentrated load P at the free e nd 2 2Pl EIθ= ()2 36Pxyl xEI=− 3 max3Pl EIδ= 2. Cantilever Beam – Concentrated load P at any poi nt 2 2Pa EIθ= ()2 3f or06Pxya x xaEI= −< < ()2 3f or6Payx a axlEI= −< < ()2 max 36PalaEIδ=− 3. Cantilever Beam – Uniform ly distributed load ω (N/m) 3 6l EIωθ= ()2 226 424xyx l lxEIω=+ − 4 max8l EIωδ= 4. Cantilever Beam – Uniform ly varying load: Maxi mum intensity ωo (N/m) 3 o 24l EIωθ= ()2 32 2 3 o10 10 5120xyl lxlxxlEIω=− +− 4 o max30l EIωδ= 5. Cantilever Beam – Couple moment M at the free en d Ml EIθ= 2 2MxyEI= 2 max2Ml EIδ= BEAM DEFLECTION FORMULAS BEAM TYP E SLOPE A T ENDS DEFLE CTION AT ANY SECTI ON IN TERMS OF x MAX IMUM AND C ENTER DEFLE CTION 6. Beam Simpl y Supporte d at Ends – C oncentrated load P at the center 2 1216Pl EIθ=θ= 2 2 3for012 4 2Px l lyx xEI⎛⎞= −< < ⎜⎟⎝⎠ 3 max48Pl EIδ= 7. Beam Simpl y Supporte d at Ends – C oncentrated load P at an y point 22 1() 6Pblb lEI−θ= 2(2 ) 6Pab lb lEI−θ= ()22 2for06Pbxyl xb xalEI= −− << ()()3 22 3 6 forPb lyx a lbxxlEIb ax l⎡ ⎤=− +−−⎢ ⎥⎣ ⎦ << ()3222 max93Pblb lEI− δ= at ()223 xl b=− ()2 2 at the center, if 3 448Pbl bEIδ= − ab> 8. Beam Simpl y Supporte d at Ends – U niform ly distr ibuted load ω (N/m) 3 1224l EIωθ=θ= ()32 3224xyl lxxEIω=− + 4 max5 384l EIωδ= 9. Beam Simpl y Supporte d at Ends – C ouple m oment M at the right end 16Ml EIθ= 23Ml EIθ= 2 216Mlx xyEIl⎛⎞=−⎜⎟⎝⎠ 2 max93Ml EIδ= at 3lx= 2 16Ml EIδ= at the center 10. Beam Simply Supp orted at Ends – Uniform ly vary ing load: M aximum intensity ωo (N/m) 3 o 17 360l EIωθ= 3 o 245l EIωθ= ()42 2 4 o71 0 3360xyl lx xlEIω=− + 4 o max0.00652l EIωδ= at 0.519 x l= 4 o0.00651l EIωδ= at the center