Beam formulas

Beam formulas

1. BEAM DEFLECTION FORMULAE

BEAM TYP
E

SLOPE AT FREE END
DEFLE
C
TION AT
ANY SECTION IN TERMS
OF
x


M
A
X
I
M
U
M

D
E
F
L
E
C
T
I
O
N
1. Cantilever Beam
– Concentrated load
P
at the free e
nd



2
2Pl
E
I
θ=

()
2
3
6Px
yl
x
EI
=
−

3
max
3Pl
E
I
δ=

2. Cantilever Beam
– Concentrated load
P
at any poi
nt




2
2Pa
E
I
θ=

()
2
3f
o
r
0
6Px
ya
x
x
a
EI
=
−<
<
()
2
3f
o
r
6Pa
yx
a
a
x
l
EI
=
−<
<

()
2
max
3
6Pa
l
a
EI
δ
=−

3. Cantilever Beam
– Uniform
l
y
distributed load
ω

(N/
m
)




3
6
lE
I
ω
θ=

()
2
22
6
4
24
x
yx
l
l
x
EI
ω
=+
−

4
max
8
lE
I
ω
δ=

4. Cantilever Beam
– Uniform
l
y
var
y
ing load: Maxi
m
u
m intensity
ω
o

(N/
m
)



3
o
24
lE
I
ω
θ=

()
2
32
2
3
o
10
10
5
120
x
yl
l
x
l
x
x
lEI
ω
=−
+
−

4
o
ma
x
30
lE
I
ω
δ=

5. Cantilever Beam
– Couple
m
o
m
e
nt
M
at the free en
d



M
l
E
I
θ=

2
2M
x
y
E
I
=

2
ma
x
2M
l
E
I
δ=

2. BEAM DEFLECTION FORMULAS

BEAM TYP
E

SLOPE A
T
ENDS
DEFLE
C
TI
ON AT
ANY
SECTI
O
N I
N
TERMS
OF
x

MAX
I
MUM
AND C
E
NT
ER
DEFLE
C
TI
ON
6. Beam
Simpl
y
Su
pporte
d at Ends – C
oncentrated load
P
at the center



2
12
16
Pl
E
I
θ=
θ
=

2
2
3
for
0
12
4
2
Px
l
l
yx
x
EI
⎛⎞
=
−<
<
⎜⎟⎝⎠

3
ma
x
48
Pl
E
I
δ=

7. Beam
Simpl
y
Su
pporte
d at Ends – C
oncentrated load
P
at an
y
point

22
1
() 6
Pb
l
b
lEI
−
θ=

2
(2
)
6
Pab
l
b
lE
I
−
θ=

()
22
2
for
0
6Pb
x
yl
x
b
x
a
lEI
=
−−
<
<

()
()
3
22
3
6
for
Pb
l
yx
a
l
b
x
x
lE
I
b
ax
l
⎡
⎤
=−
+
−
−
⎢
⎥
⎣
⎦
<
<

(
)
32
22
ma
x
93
Pb
l
b
lE
I
−
δ=
at
()
22
3
xl
b
=−

(
)
2
2
at the center, if

3
4
48
Pb
l
b
EI
δ=
−
ab
>
8. Beam
Simpl
y
Su
pporte
d at Ends – U
n
iform
l
y
distr
i
buted load
ω

(N/
m
)



3
12
24
lE
I
ω
θ=
θ
=

()
32
3
2
24
x
yl
l
x
x
EI
ω
=−
+

4
ma
x
5
384
lE
I
ω
δ=

9. Beam
Simpl
y
Su
pporte
d at Ends – C
ouple m
o
m
e
n
t
M
at the right end

1
6M
l
E
I
θ=

2
3M
l
E
I
θ=

2 2
1
6M
lx
x
y
E
Il
⎛⎞
=−
⎜⎟⎝⎠

2
ma
x
93
M
l
EI
δ=
at
3l
x
=

2
16
M
lE
I
δ=
at the center


10. Beam
Sim
p
ly
Supp
ort
e
d at Ends –
Uniform
l
y
vary
i
ng l
o
ad: M
a
xim
u
m
intensity
ω
o

(N/
m
)


3
o
1
7360
lE
I
ω
θ=

3
o
2
45
lE
I
ω
θ=

()
42
2
4
o
71
0
3
360
x
yl
l
x
x
lEI
ω
=−
+

4
o
ma
x
0.00652
l
E
I
ω
δ=
at
0.51
9
x
l
=

4
o
0.0
065
1
l
E
I
ω
δ=
at the center

3. 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