OPTI 222
Mechanical Design in Optical Engineering
43
Shear Forces and Bending Moments in Beams
Bending Stress:
My
I
σ
=
Moment of Inertia:
I
x
=
∫
A
y
2
dA
I
y
=
∫
A
x
2
dA
Parallel Axis Theorem:
I
x
= I
xc
+ Ad
2
I
y
= I
yc
+ Ad
2
Beam Classifications:
Beams are also classified according to the shape of their cross sections.
Shear Force and Bending Moment:
When a beam is loaded by forces or couples, internal stresses and strains are created.
To determine these stresses and strains, we first must find the internal forces and
couples that act on cross sections of the beam.
Consider the following example.
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View Full DocumentOPTI 222
Mechanical Design in Optical Engineering
44
It is convenient to reduce the resultant to a shear force, V, and a bending moment, M.
Because shear forces and bending moments are the resultants of stresses distributed
over the cross section, they are known as
stress resultants
and in statically
determinate beams can be calculated from the equations of static equilibrium.
Deformation Sign Conventions:
As can be seen in the previous diagram (left hand section vs. right hand section), we
recognize that the algebraic sign does not depend on its direction in space, such as
upward or downward or clockwise or counterclockwise.
The sign depends on the
direction of the stress resultant with respect to the material against which it acts.
Shear and Moment Sign Convention
Deformations highly exaggerated
Positive shear forces always deform right hand face downward with respect to the left
hand face.
Positive bending moments always elongate the lower section of the beam.
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 Fall '11
 STAFF
 Force, Shear Stress, Shear, Second moment of area, shear force

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