Bending Moment Diagrams &
Shearing
Bending Moment Diagrams &
Shearing
Force Diagrams for Beams
Force Diagrams for Beams
Theory and Design of Structures I
Theory and Design of Structures I
Simply Supported Beam under UDL
Consider the simply supported beam AB carrying a
uniformly distributed load (UDL)
w.
The reactions at A and B are found by considering the
entire beam as a free body.
A
B
w
kN/m
wl
/2 kN
wl
/2 kN
x
l
m
Simply supported beam under UDL
• To evaluate the shearing force and bending moment
at a chosen section
x
from A, we consider the
equilibrium of the portion of the beam to the left of
this section, namely
wx
wl
V
x
−
=
2
2
2
2
wx
x
wl
M
x
−
=
A
B
w
kN/m
wl
/2 kN
wl
/2 kN
x
l
m
Simply supported beam under UDL
A
w
wl
/2
M
x
V
x
x
• If we consider the equilibrium of that portion of the
beam to the right of the section, we get
B
wl
/2
l

x
w
M
x
V
x
Free body diagram
()
2
wl
x
l
w
V
x
−
−
=
2
2
2
x
l
w
x
l
wl
M
x
−
−
−
=
A
B
w
kN/m
wl
/2 kN
wl
/2 kN
x
l
m
Simply supported beam under UDL
Simply Supported Beam under UDL
Simply Supported Beam under UDL
• Since the entire beam is in equilibrium, we must
obtain the same values for shear force and bending
moment at a chosen section by using the equilibrium
conditions for either portion of the beam
•B
y
v
a
r
y
i
n
g
x
from 0 to
l
, the values of
M
x
and
V
x
can
be obtained.
The bending moment diagram and the
shear force diagram can thus be constructed
accordingly
wl
/2 kN
wl
/2 kN
l
m
Simply supported beam under UDL
A
B
w
kN/m
+
Bending moment diagram
wl
2
/8
+
Shear force diagram

wl
/2

wl
/2
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View Full Document• Consider structure ABCD with a hinge at C as shown
A
20 kN/m
3 m
B
D
C
40 kN
40 kN/m
10 m
2 m
Frame ABCD with hinge at C
Structural Analysis of a Simple Frame
Structural Analysis of a Simple Frame
Reactions
Reactions
• Consider free body ABC.
Taking moment about C
gives
• Taking moment about B,
0
6
240
12
40
10
=
×
−
×
−
×
B
V
kN
192
=
B
V
0
2
40
4
240
10
=
×
−
×
+
×
−
C
V
kN
88
=
C
V
40
V
C
H
C
V
B
H
B
B
C
240
A
A
20 kN/m
B
D
C
40 k N
40 k N / m
10 m
2 m
Frame ABCD with hinge at C
• Then consider free body CD.
Taking moment about C
gives
• Consider the equilibrium of forces in the horizontal and
vertical directions (
x
&
y
directions).
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 Spring '10
 profchai
 Force, Shear Stress, shear force, Frame ABCD

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