# experiment

Topics: Beam, Deflection, Cantilever Pages: 8 (353 words) Published: November 24, 2013
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 < x < a
6 EI
Pa 2
y=
( 3x − a ) for a < x < 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 < x <
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 − x2 − b2 ) for 0 < x < a
6lEI
Pb ⎡ l
3
2
2
3⎤
y=
⎢ b ( x − a ) + (l − b ) x − x ⎥
6lEI ⎣

for a < x < 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 > 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 − 2lx2 + 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

file:///G|/BACKUP/Courses_and_seminars/0MAE4770S12/url%20for%20beam%20formulas.txt[1/23/2012 12:15:35 PM]