1
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
A beam is a structural member or machine component that is designed to
support primarily forces acting perpendicular to the axis of the
member.
Types of Beams
Simply
Supported
(One pin, one roller)
Overhanging
(One pin, one roller)
Cantilever
(One fixed end)
Propped
(One fixed end
and one roller)
Continuous
(Several pins and
rollers)
Builtin
(Both ends fixed)
Statically Determinate
Statically Indeterminate
2
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
Types of Supports
Types of Loads
Roller:
one unknown
Pin:
two unknowns
Fixed:
three unknowns
Concentrated
loads
Distributed
loads
Concentrated
moments
3
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
1.
Determine the reactions, if necessary,
using the freebody diagram of the
overall beam.
2.
Cut the beam at cross section where
the shear force and bending moment
are to be determined. Draw the free
body diagram.
3.
Set up equilibrium equations for the
freebody diagram and use them to
determine the shear force and
bending moment at the cross section.
4.
Repeat steps 2 through 4 for as many
cross sections as needed.
Procedure for determining shear forces and bending moments
()
a
x
P
wx
Rx
M
M
P
wx
R
V
F
r
a
a
r
y
−
−
−
=
⇒
=
−
−
=
⇒
=
∑
∑
−
1
2
1
2
0
0
4
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
P
a
L
A
B
()
(
)
P
L
a
B
a
P
L
B
M
y
y
A
⎟
⎠
⎞
⎜
⎝
⎛
=
⇒
=
−
=
∑
0
y
A
x
A
x
V
x
M
P
L
a
A
V
V
A
F
y
x
x
y
y
⎟
⎠
⎞
⎜
⎝
⎛
−
=
=
⇒
=
−
=
∑
1
0
Px
L
a
x
A
M
M
x
A
M
y
x
x
y
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⋅
=
⇒
=
+
⋅
−
=
∑
1
0
P
y
A
a
x
A
x
V
x
M
P
L
a
V
V
P
A
F
x
x
y
y
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⇒
=
−
−
=
∑
0
Pa
L
x
M
M
a
x
P
x
A
M
x
x
y
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⇒
=
+
−
+
⋅
−
=
∑
1
0
)
(
a
x
≤
≤
0
For
L
x
a
≤
≤
For
Reactions
P
y
A
y
B
a
L
A
B
x
A
P
L
a
A
P
B
A
F
y
y
y
y
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⇒
=
−
+
=
∑
1
0
0
=
=
∑
x
x
A
F
5
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
q
L
A
B
Reactions
()
()
()
2
0
2
qL
B
L
L
q
L
B
M
y
y
A
=
⇒
=
−
=
∑
q
L
A
B
y
A
y
B
x
A
2
0
qL
A
qL
B
A
F
y
y
y
y
=
⇒
=
−
+
=
∑
0
=
=
∑
x
x
A
F
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⇒
=
−
−
=
∑
x
L
q
V
V
qx
A
F
x
x
y
y
2
0
()
x
L
qx
M
M
qx
x
A
M
x
x
y
−
=
⇒
=
+
+
⋅
−
=
∑
2
0
2
2
y
A
x
A
x
V
x
M
6
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
Sign Convention
7
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
A
R
B
R
(
)
()
()
0
2
2
1000
6000
3
2000
0
2
1000
6000
2000
2
=
+
−
+
−
+
=
=
−
−
−
+
−
=
∑
∑
−
M
x
x
x
M
V
x
F
a
a
y
()
(
)
()
()
()
0
2
2000
5
6
1000
13
2000
10
=
+
+
+
−
=
∑
A
B
R
M
lb
6000
=
A
R
lb

ft
8000
6000
500
lb
6000
1000
2
−
+
−
=
+
−
=
x
x
M
x
V
8
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
Relations among w, V, and M
()
x
w
L
A
B
x
dx
(
)
wdx
dV
dV
V
wdx
V
Fy
=
=
+
−
+
=
∑
or
0
w
dx
dV
=
1
2
1
2
2
1
2
1
2
1
V
wdx
V
V
V
dV
wdx
x
x
V
V
x
x
+
=
−
=
=
∫
∫
∫
9
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
Relations among w, V, and M
(
)
()
()
2
or
0
2
2
2
dx
w
Vdx
dM
dM
M
dx
w
Vdx
M
M
O
+
=
=
+
+
−
−
−
=
∑
V
dx
dM
=
()
x
w
L
A
B
x
dx
1
2
1
2
2
1
2
1
2
1
M
Vdx
M
M
M
dM
Vdx
x
x
M
M
x
x
+
=
−
=
=
∫
∫
∫
10
MEM202 Engineering Mechanics Statics
MEM
Shear Forces and Bending Moments in Beams
q
L
A
B
q
L
A
B
2
qL
2
qL
0
2
0
0
=
=
−
=
M
qL
V
q
w
2
qL
x
A
x
V
x
M
⎟
⎠
⎞
⎜
⎝
⎛
−
=
x
L
q
Vx
2
()
0
0
0
V
V
qx
dx
q
wdx
x
x
x
−
=
−
=
−
=
∫
∫
0
2
0
0
2
2
2
M
M
x
Lx
q
dx
x
L
q
Vdx
x
x
x
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
∫
∫
()
x
L
qx
M
x
−
=
2
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