Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

ExAmpLE 12.11
y
(–14, 0)
y
(–14, 0)
14 ksi
14 ksi
50 ksi
A B
τ
50 ksi
Element A
R = 32
x σ
σ p2
C
σ p1 (50, 0)
τ
τ
Element A
R = 32
R = 25
z
x σ
σ p2 σ p3 C
σ p1 (50, 0)
τ
Absolute Maximum Shear Stress
Two elements subjected to plane stress are shown. Determine the
absolute maximum shear stress for each element.
SolutioN
In Section 12.8, it was shown that shear stress does not exist on
planes where the maximum and minimum normal stresses occur.
Furthermore, the following statement must also be true:
If the shear stress on a plane is zero, then
that plane must be a principal plane.
Since there is no shear stress acting on the x and y faces for both
element A and element B, one can conclude that the stresses acting
on these elements are principal stresses.
The Mohr’s circle for element A is constructed as shown.
Notice that point x is the principal stress σ p1 and point y is the
principal stress σ p2 . This circle shows all possible combinations
of σ and τ that occur in the x–y plane.
What is meant by the term x–y plane? This term refers to
plane surfaces whose normals are perpendicular to the z axis.
The maximum in-plane shear stress for element A is simply
equal to the radius of Mohr’s circle; therefore, τ max = 32 ksi.
In the statement of the problem, we are told that element A is
a point subjected to plane stress. From Section 12.4, we know that
the term plane stress means that there are no stresses on the outof-plane
face of the stress element. In other words, there is no
stress on the z face; hence, σ z = 0, τ zx = 0, and τ zy = 0. We also
know that a plane with no shear stress is, by definition, a principal
plane. Therefore, the z face of the stress element is a principal
plane, and the principal stress acting on this surface is the third
principal stress: σ z = σ p3 = 0.
The state of stress on the z face can be plotted on Mohr’s
circle, and two additional circles can be constructed:
• The circle defined by σ p1 and σ p3 depicts all combinations of
σ–τ that are possible on surfaces in the x–z plane (meaning
plane surfaces whose normal is perpendicular to the y axis).
• The circle connecting σ p2 and σ p3 depicts all combinations of
σ–τ that are possible on surfaces in the y–z plane (meaning
plane surfaces whose normal is perpendicular to the x axis).
The maximum shear stress in the x–z plane is given by the
radius of the Mohr’s circle connecting points x and z, and the maximum
shear stress in the y–z plane is given by the radius of the
circle connecting points y and z. By inspection, both of these circles
are smaller than the x–y circle. Consequently, the absolute
maximum shear stress—that is, the largest shear stress that can
occur on any possible plane—is equal to the maximum in-plane
shear stress for element A.
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