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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

Pl 2 Px 2 Pl 3
θ= y= ( 3l − x ) δ max =
2 EI 6 EI 3EI

2. Cantilever Beam – Concentrated load P at any point

Px 2
y= ( 3a − x ) for 0 < x < a
Pa 2 6 EI Pa 2
θ= δ max = ( 3l − a )
2 EI Pa 2 6 EI
y= ( 3x − a ) for a < x < l
6 EI
3. Cantilever Beam – Uniformly distributed load ω (N/m)

ωl 3 ωx 2 ωl 4
θ=
6 EI
y=
24 EI
( x 2 + 6l 2 − 4lx ) δ max =
8 EI

4. Cantilever Beam – Uniformly varying load: Maximum intensity ωo (N/m)

ωol 3 ωo x 2 ωo l 4
θ=
24 EI
y=
120lEI
(10l 3 − 10l 2 x + 5lx2 − x3 ) δ max =
30 EI

5. Cantilever Beam – Couple moment M at the free end

Ml Mx 2 Ml 2
θ= y= δ max =
EI 2 EI 2 EI
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 Px ⎛ 3l 2 ⎞ l Pl 3
θ1 = θ2 = y= ⎜ − x 2 ⎟ for 0 < x < δ max =
16 EI 12 EI ⎝ 4 ⎠ 2 48 EI

7. Beam Simply Supported at Ends – Concentrated load P at any point

Pb(l 2 − b 2 ) y=
Pbx 2
( l − x 2 − b 2 ) for 0 < x < a Pb ( l 2 − b 2 )
32

θ1 =
6lEI
6lEI δ max =
9 3 lEI
at x = (l 2
− b2 ) 3
Pb ⎡ l
( x − a ) + ( l 2 − b2 ) x − x3 ⎤⎥
3
Pab(2l − b) y= ⎢
θ2 =
6lEI
6lEI ⎣ b ⎦ δ=
Pb
48 EI
( 3l 2 − 4b2 ) at the center, if a > b
for a < x < l
8. Beam Simply Supported at Ends – Uniformly distributed load ω (N/m)

ωl 3 ωx 3 5ωl 4
θ1 = θ2 =
24 EI
y=
24 EI
( l − 2lx 2 + x3 ) δmax =
384 EI

9. Beam Simply Supported at Ends – Couple moment M at the right end

Ml 2 l
Ml δmax = at x =
θ1 = 9 3 EI 3
6 EI Mlx ⎛ x 2 ⎞
y= ⎜1 − ⎟
Ml 6 EI ⎝ l 2 ⎠ Ml 2
θ2 = δ= at the center
3EI 16 EI

10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ωo (N/m)
7ωol 3 ωo l 4
θ1 = δ max = 0.00652 at x = 0.519 l
ωo x
360 EI
ω l3
y=
360lEI
( 7l 4 − 10l 2 x 2 + 3x4 ) ωol 4
EI
θ2 = o δ = 0.00651 at the center
45 EI EI
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