SIGN CONVENTIONS IN BENDING
The Problem…
Look at shear force in three-point bend.
L
W
L
W/2
W/2
M
free
body 1
Q
W/2
x
Shear force Q = W/2
M
free
body 2
x
W
Q
Q + W - W/2 = 0
Q = -W/2
Is Q positive or negative?
W/2
Now look at bending moment in same system
W
L
L
W/2
W/2
M
free
body 1
Q
W/2
x
Anticlockwise bending moment
M - xW/2=0
M=Wx/2
M
W
free
body 2
x
Q
W/2
M - (L - x)W + (2L - x)W/2 = 0
M = -Wx/2
Is M positive or negative?
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We resolve the problem by introducing the concept of the sign of a surface. The
surfaces concerned are those we expose when we create a free body. For beams the
horizontal x axis is in the conventional direction:
x
A surface is positive when its outward normal is along the x direction. It is negative
when the outward normal is in the direction opposite to the x axis. We also introduce
a y direction which is vertically downwards:
y
Thus, a shear force has two signs associated with it; one defined by its direction, and
one by the sign of the surface. If they are both positive or both negative, the shear
force is positive. If one sign is negative and the other positive, the shear force is
negative. Using this convention, the shear forces obtained for the two free bodies in
the shear force problem above are both the same, i.e. positive:
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We can do a similar thing for bending moment if we adopt the anticlockwise direction
as the positive rotation angle. Then, the moment has the sign associated with its
direction, and the sign of the relevant surface, associated with it. The bending
problem above is thus resolved:
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Note that whenever we create free bodies we expose surfaces at which the opposed
forces and moments are equal and opposite in direction. However, use of the sign
convention ensures that the shear forces and bending moments are equal. It therefore
makes sense to speak of positive and negative bending.
Now look again at the cantilever example – it is in negative bending.
Forces and moments are drawn as if they are positive. The sign which comes out
from the algebra is then the sign of the force or moment – both negative.
W
shear force
W
Q
Q + W = 0; Q = - W
M
bending moment
Wx + M = 0
M = - Wx
x
W
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EXAMPLES
(1)
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(2)
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SUPERPOSITION PRINCIPLE
Textbook diagram
Another example
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