Beam_Deflection_Formulae

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
MAXIMUM DEFLECTION
2
Pl
2EI
2
Px
6EI
θ= y = ( 3l−
x)
3
Pl
δ
max
=
3EI
2. Cantilever Beam – Concentrated load P at any point
2
Pa
θ=
2EI
3. Cantilever Beam – Uniformly distributed load ω (N/m)
2
Px
y = ( 3a− x)
for 0< x<
a
6EI
2
Pa
y = ( 3x− a)
for a< x<
l
6EI
2
Pa
δ
max
= 3 −
6EI
( l a)
3
ωl
6EI
2
ωx
24EI
2 2
θ= y = ( x + 6l − 4lx)
4
ωl
δ
max
=
8EI
4. Cantilever Beam – Uniformly varying load: Maximum intensity ω o (N/m)
ω l
24EI
ω x
120lEI
3
2
o
o 3 2 2 3
θ= y = ( 10l − 10l x+ 5lx − x )
4
ωol
δ
max
=
30EI
5. Cantilever Beam – Couple moment M at the free end
θ=
Ml
EI
y =
2
Mx
2EI
2
Ml
δ
max
=
2EI

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