BeamFormulas

BEAM DEFLECTION

BEAM TYPE SLOPE AT FREE END DEFLECTION

1. Cantilever Beam – Concentrated load P at the free end

θ = Pl 2
2EI

2. Cantilever Beam – Concentrated load P at any point

θ = Pa2 y=6
2EI y=

3. Cantilever Beam – Uniformly distributed load ω (N/m)

θ = ωl 3 y
6EI

4. Cantilever Beam – Uniformly varying load: Maximum intens

θ = ωol 3 y = ω
24EI 12

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

θ = Ml
EI

N FORMULAE MAXIMUM DEFLECTION

N AT ANY SECTION IN TERMS OF x

y = Px2 (3l − x) δmax = Pl 3
6EI 3EI

Px2 (3a − x) for 0< x<a Pa2
6EI 6EI
δmax = (3l − a)
Pa2
6EI (3x − a ) for a< x<l

( )yωx2 ωl 4
= 24EI x2 + 6l2 − 4lx δmax = 8EI

sity ωo (N/m)

ωo x2 ωol 4
30EI
( )20lEI
10l3 −10l 2 x + 5lx2 − x3 δmax =

y = Mx2 δmax = Ml 2
2EI 2EI

BEAM DEFLECTION

BEAM TYPE SLOPE AT ENDS DEFLECTION

6. Beam Simply Supported at Ends – Concentrated load P at the

θ1 = θ2 = Pl 2 y = P
16EI 12

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

Pb(l 2 − b2 ) y = Pb
6lEI 6lE
θ1 =

θ2 = Pab(2l − b) y = Pb
6lEI 6lEI

8. Beam Simply Supported at Ends – Uniformly distributed load

θ1 = θ2 = ωl 3 y
24EI

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

θ1 = Ml
6EI

θ2 = Ml
3EI

10. Beam Simply Supported at Ends – Uniformly varying load:

θ1 = 7ωol 3
360EI
y=
ωol 3
θ2 = 45EI

N FORMULAS

N AT ANY SECTION IN TERMS OF x MAXIMUM AND CENTER
e center DEFLECTION

Px ⎛ 3l 2 − x2 ⎞ for 0 < x < l δmax = Pl 3
2EI ⎜ 4 ⎟ 2 48EI
⎝ ⎠

y point

( )bx l2 − x2 − b2 for 0 < x < a ( )Pb l2 − b2 3 2
( )δmax = 9 3 lEI
EI at x = l2 − b2 3

b⎡l a )3 x3 ⎤
⎣⎢ b ⎥⎦
( )I
( x − + l2 − b2 x − Pb
48EI
( )δ
for a < x < l = 3l 2 − 4b2 at the center, if a > b

d ω (N/m)

( )y ωx 5ωl 4
= 24EI l3 − 2lx2 + x3 δmax = 384EI

right end

δmax = Ml 2 at x = l
9 3 EI 3
Mlx ⎛⎜1 − x2 ⎞
y = 6EI ⎝ l2 ⎟ Ml 2
⎠ 16EI
δ = at the center

Maximum intensity ωo (N/m)

ωo x δmax = 0.00652 ωol 4 at x = 0.519l
EI
( )360lEI
7l 4 −10l2 x2 + 3x4 δ = 0.00651 ωol4 at the center
EI

http://www.advancepipeliner.com/Resources/Others/Beams/Beam_Deflection_Formulae.pdf

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