Beam_Deflection_Formulae

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
2
Pl
θ
1
=θ
2
=
16EI
Px 2
⎛3
l 2 ⎞
y = ⎜ − x ⎟ for 0 < x<
l
12EI
⎝ 4 ⎠
2
3
Pl
δ
max
=
48EI
7. Beam Simply Supported at Ends – Concentrated load P at any point
Pbx 2 2 2
2 2
Pb( l − b )
y = ( l −x − b ) for 0 < x<
a
θ
1
=
6lEI
6lEI
Pb ⎡ l
3 2 2 3⎤
Pab(2 l − b)
y = ( x a) ( l b ) x x
θ
2
=
6lEI
⎢
− + − −
⎣b
⎥
⎦
6lEI
for a< x<
l
8. Beam Simply Supported at Ends – Uniformly distributed load ω (N/m)
2 2
Pb( l − b ) 32
δ
max
= at x= ( l 2 − b
2
)
3
9 3lEI
Pb
δ= ( 3l
2 −4b
2
) at the center, if a>
b
48EI
3
ωl
3 2 3
θ
1
=θ
2
= y = ( l − 2lx + x )
24EI
ωx
24EI
4
5ωl
δ
max
=
384EI
9. Beam Simply Supported at Ends – Couple moment M at the right end
θ =
1
θ =
2
Ml
6EI
Ml
3EI
Mlx ⎛ x
y = ⎜1−
6EI
⎝ l
2
2
⎞
⎟
⎠
2
Ml
δ
max
= at x =
9 3EI
2
Ml
δ= at the center
16EI
l
3
10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ω o (N/m)
3
7ωol
θ
1
=
360EI
ωox
4 2 2 4
y =
3
( 7l − 10l x + 3x
)
ωol
360lEI
θ
2
=
45EI
ω l
max
0.00652
EI
4
ωol
δ= 0.00651 at the center
EI
4
o
δ = at x = 0.519
l