1
BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x 1. Cantilever Beam – Concentrated load P at the free end
Pl 2 2 EI
θ=
y=
MAXIMUM DEFLECTION
Px 2 ( 3l − x ) 6 EI
δ max =
Pl 3 3EI
2. Cantilever Beam – Concentrated load P at any point
Px 2 ( 3a − x ) for 0 < x < a 6 EI Pa 2 y= ( 3x − a ) for a < x < l 6 EI
y=
Pa 2 θ= 2 EI
δ max
Pa 2 = ( 3l − a ) 6 EI
3. Cantilever Beam – Uniformly distributed load ω (N/m)
ωl 3 6 EI
θ=
δ max =
ωl 4 8 EI
ωo x 2 (10l 3 − 10l 2 x + 5lx2 − x3 ) 120lEI
δ max =
ωo l 4 30 EI
Mx 2 2 EI
δ max =
Ml 2 2 EI
y=
ωx 2 x 2 + 6l 2 − 4lx ) ( 24 EI
4. Cantilever Beam – Uniformly varying load: Maximum intensity ωo (N/m)
θ=
ωol 3 24 EI
y=
5. Cantilever Beam – Couple moment M at the free end
θ=
Ml EI
y=
MEADinfo
2
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
Pl 2 θ1 = θ2 = 16 EI
⎞ Px ⎛ 3l 2 l y= − x 2 ⎟ for 0 < x < ⎜ 12 EI ⎝ 4 2 ⎠
δ max =
Pl 3 48 EI
7. Beam Simply Supported at Ends – Concentrated load P at any point
Pb(l 2 − b 2 ) θ1 = 6lEI Pab(2l − b) θ2 = 6lEI
Pbx 2 l − x 2 − b 2 ) for 0 < x < a ( 6lEI Pb ⎡ l 3 y= ( x − a ) + ( l 2 − b2 ) x − x3 ⎤⎥ ⎢ 6lEI ⎣ b ⎦ for a < x < l y=
δ max = δ=
Pb ( l 2 − b 2 )
32
9 3 lEI
at x =
(l
2
− b2 ) 3
Pb ( 3l 2 − 4b2 ) at the center, if a > b 48 EI
8. Beam Simply Supported at Ends – Uniformly distributed load ω (N/m)
θ1 = θ2 =
ωl 3 24 EI
y=
ωx 3 l − 2lx 2 + x3 ) ( 24 EI
δmax =
5ωl 4 384 EI
9. Beam Simply Supported at Ends – Couple moment M at the right end
Ml θ1 = 6 EI Ml θ2 = 3EI
y=
Mlx ⎛ x 2 ⎞ ⎜1 − ⎟ 6 EI ⎝ l 2 ⎠
δmax = δ=
Ml 2 l at x = 9 3 EI 3
Ml 2 at the center 16 EI
10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ωo (N/m)
7ωol 3 360 EI ω l3 θ2 = o 45 EI
θ1 =
y=
ωo x ( 7l 4 − 10l 2 x 2 + 3x4 ) 360lEI
δ max = 0.00652 δ = 0.00651
ωo l 4 at x = 0.519 l EI
ωol 4 at the center EI MEADinfo