BEAM DEFLECTION FORMULAE
BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF
 x
 MAXIMUM DEFLECTION
1. Cantilever Beam – Concentrated load
P
 at the free end
2
2
Pl EI 
θ =
 
( )
2
36
Px y l x EI 
=
 
3max
3
Pl EI 
δ =
 2. Cantilever Beam – Concentrated load
P
 at any point
2
2
Pa EI 
θ =
 
( )
2
3for06
Px y a x x a EI 
= < <
( )
2
3fo6
Pa y x a a x l EI 
= < <
 
( )
2max
36
Pal a EI 
δ =
 3. Cantilever Beam – Uniformly distributed load
ω
 
(N/m)
3
6
l EI 
ωθ =
 
( )
222
6424
 x y x l lx EI 
ω= +
 
4max
8
l EI 
ωδ =
 4. Cantilever Beam – Uniformly varying load: Maximum intensity
ω
o
 
(N/m)
3o
24
l EI 
ωθ =
 
( )
23223o
10105120
 x y l l x lx xlEI 
ω= +
 
4omax
30
l EI 
ωδ =
 5. Cantilever Beam – Couple moment
 M 
 at the free end
 Ml EI 
θ =
 
2
2
 Mx y EI 
=
 
2max
2
 Ml EI 
δ =
 
 
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
212
16
Pl EI 
θ = θ =
 
22
3for01242
Px l l y x x EI 
= < <
 
3max
48
Pl EI 
δ =
 7. Beam Simply Supported at Ends – Concentrated load
P
 at any point
221
()6
Pb l blEI 
θ =
 
2
(2)6
Pab l blEI 
θ =
 
( )
222
for06
Pbx y l x b x alEI 
= < <
 
( )
 ( )
3223
6for 
Pb l y x a l b x xlEI ba x l
= + < <
 
( )
3222max
93
Pb l blEI 
δ =
 at
( )
22
3
 x l b
=
 
( )
22
 at the center, if
3448
Pbl b EI 
δ =
 a b
>
8. Beam Simply Supported at Ends – Uniformly distributed load
ω
 
(N/m)
312
24
l EI 
ωθ = θ =
 
( )
323
224
 x y l lx x EI 
ω= +
 
4max
5384
l EI 
ωδ =
 9. Beam Simply Supported at Ends – Couple moment
 M 
 at the right end
1
6
 Ml EI 
θ =
 
2
3
 Ml EI 
θ =
 
22
16
 Mlx x y EI l
=
 
2max
93
 Ml EI 
δ =
 at
3
l x
 =
 
2
16
 Ml EI 
δ =
 at the center 10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity
ω
o
 
(N/m)
3o1
7360
l EI 
ωθ =
 
3o2
45
l EI 
ωθ =
 
( )
4224o
7103360
 x y l l x xlEI 
ω= +
 
4omax
0.00652
 l EI 
ωδ =
 at
0.519
 x l
=
 
4o
0.00651
 l EI 
ωδ =
 at the center
 
file:///G|/BACKUP/Courses_and_seminars/0MAE4770S12/url%20for%20beam%20formulas.txt[1/23/2012 12:15:35 PM]
http://www.advancepipeliner.com/Resources/Others/Beams/Beam_Deflection_Formulae.pdf 
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