# BEAM DEFLECTION FORMULAS

```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 &lt; x &lt; a
6 EI
Pa 2
y=
( 3x − a ) for a &lt; x &lt; 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=
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
− x 2 ⎟ for 0 &lt; x &lt;
y=
⎜
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 &lt; x &lt; a
(
6lEI
Pb ⎡ l
3
y=
( x − a ) + ( l 2 − b2 ) x − x3 ⎤⎥
⎢
6lEI ⎣ b
⎦
for a &lt; x &lt; 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 &gt; 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