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

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