# BeamFormulas.pdf

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BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF  x  MAXIMUM DEFLECTION 1. Cantilever Beam – Concentrated load P  at the free end 2 2 Pl EI  θ =   ( ) 2 36 Px y l x EI  = −   3max 3 Pl EI  δ =  2. Cantilever Beam – Concentrated load P  at any point 2 2 Pa EI  θ =   ( ) 2 3for06 Px y a x x a EI  = − < < ( ) 2 3for 6 Pa y x a a x l EI  = − < <   ( ) 2max 36 Pal a EI  δ = −  3. Cantilever Beam – Uniformly distributed load ω   (N/m) 3 6 l EI  ωθ =   ( ) 222 6424  x y x l lx EI  ω= + −   4max 8 l EI  ωδ =  4. Cantilever Beam – Uniformly varying load: Maximum intensity ω o   (N/m) 3o 24 l EI  ωθ =   ( ) 23223o 10105120  x y l l x lx xlEI  ω= − + −   4omax 30 l EI  ωδ =  5. Cantilever Beam – Couple moment  M   at the free end  Ml EI  θ =   2 2  Mx y EI  =   2max 2  Ml EI  δ =    BEAM DEFLECTION FORMULAS BEAM TYPE SLOPE AT ENDS DEFLECTION AT ANY SECTION IN TERMS OF  x  MAXIMUM AND CENTER DEFLECTION 6. Beam Simply Supported at Ends – Concentrated load P  at the center 212 16 Pl EI  θ = θ =   22 3for01242 Px l l y x x EI  ⎛ ⎞= − < <⎜ ⎟⎝ ⎠   3max 48 Pl EI  δ =  7. Beam Simply Supported at Ends – Concentrated load P  at any point 221 ()6 Pb l blEI  −θ =   2 (2)6 Pab l blEI  −θ =   ( ) 222 for06 Pbx y l x b x alEI  = − − < <   ( )  ( ) 3223 6for  Pb l y x a l b x xlEI ba x l ⎡ ⎤= − + − −⎢ ⎥⎣ ⎦< <   ( ) 3222max 93 Pb l blEI  −δ =  at ( ) 22 3  x l b = −   ( ) 22  at the center, if 3448 Pbl b EI  δ = −  a b > 8. Beam Simply Supported at Ends – Uniformly distributed load ω   (N/m) 312 24 l EI  ωθ = θ =   ( ) 323 224  x y l lx x EI  ω= − +   4max 5384 l EI  ωδ =  9. Beam Simply Supported at Ends – Couple moment  M   at the right end 1 6  Ml EI  θ =   2 3  Ml EI  θ =   22 16  Mlx x y EI l ⎛ ⎞= −⎜ ⎟⎝ ⎠   2max 93  Ml EI  δ =  at 3 l x  =   2 16  Ml EI  δ =  at the center 10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ω o   (N/m) 3o1 7360 l EI  ωθ =   3o2 45 l EI  ωθ =   ( ) 4224o 7103360  x y l l x xlEI  ω= − +   4omax 0.00652  l EI  ωδ =  at 0.519  x l =   4o 0.00651  l EI  ωδ =  at the center  file:///G|/BACKUP/Courses_and_seminars/0MAE4770S12/url%20for%20beam%20formulas.txt[1/23/2012 12:15:35 PM] http://www.advancepipeliner.com/Resources/Others/Beams/Beam_Deflection_Formulae.pdf

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