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

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ExAmpLE
m12.10 Coach Mohr’s Circle of Stress
Learn to construct and use Mohr’s circle to determine principal
stresses, including the proper orientation of the principal
stress planes.
ExAmpLE 12.7
y
5 ksi
6 ksi
x
9 ksi
Principal and Maximum in-Plane Shear Stresses
Consider a point in a structural member that is subjected to plane stress. Normal and shear
stresses acting on horizontal and vertical planes at the point are shown.
(a) Determine the principal stresses and the maximum in-plane shear stress acting at
the point.
(b) Show these stresses in an appropriate sketch.
(–5, 6 ccw) y
τ
(2, 0)
C
τ
x (9, 6cw)
σ
SolutioN
Begin with the normal and shear stresses acting on the x face of the
stress element. Since σ x = 9 ksi is a tension stress, the point on Mohr’s
circle will be plotted to the right of the τ axis. The shear stress acting on
the x face tends to rotate the stress element clockwise; therefore, point x
on Mohr’s circle is plotted above the σ axis.
On the y face, the normal stress σ y = −5 ksi will be plotted to the
left of the τ axis. The shear stress acting on the y face tends to rotate the
stress element counterclockwise; therefore, point y on Mohr’s circle is
plotted below the σ axis.
Note: Attaching either a positive or a negative sign to τ values at
points x and y does not add any useful information for the Mohr’s circle
stress analysis. Once the circle has been properly constructed, all computations
are based on the geometry of the circle, irrespective of any
signs. In this introductory example, the subscript “cw” has been added
to the shear stress at point x simply to emphasize that the shear stress on
the x face rotates the element clockwise. Similarly, the subscript “ccw”
is meant to emphasize that the shear stress on the y face rotates the element
counterclockwise.
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