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

8. Several points on Mohr’s circle are of particular interest. The principal
stresses are the extreme values of the normal stress that exist in the
stressed body, given the specific set of stresses σ x , σ y , and τ xy that act in
the x and y directions. From Mohr’s circle, the extreme values of σ are
observed to occur at the two points where the circle crosses the σ axis.
The more positive point (in an algebraic sense) is σ p1 , and the more negative
point is σ p2 .
Notice that the shear stress τ at both points is zero. As discussed previously,
the shear stress τ is always zero on planes where the normal stress σ
has a maximum or a minimum value.
9. The geometry of Mohr’s circle can be used to determine the orientation of
the principal planes. From the geometry of the circle, the angle between
point x and one of the principal stress points can be determined. The angle
between point x and one of the principal stress points on the circle is 2θ p .
In addition to the magnitude of 2θ p , the sense of the angle (either clockwise
or counterclockwise) can be determined from the circle by inspection.
The rotation of 2θ p from point x to the principal stress point should
be determined.
In the x–y coordinate system of the stress element, the angle between the
x face of the stress element and a principal plane is θ p , where θ p rotates in the
same sense (either clockwise or counterclockwise) in the x–y coordinate system
as 2θ p does in Mohr’s circle.
τ
τ
τ
y
y
(σ p2 , 0)
R
R
517
MOHR’S CIRCLE FOR
PLANE STRESS
C
C
x
2θ
p
x
σ
(σ p1 , 0)
σ
10. Two additional points of interest on Mohr’s circle are the extreme shear
stress values. The largest shear stress magnitudes will occur at points located
at the top and at the bottom of the circle. Since the center C of the circle is
always located on the σ axis, the largest possible value of τ is simply the
radius R. Note that these two points occur directly above and directly below
the center C. In contrast to the principal planes, which always have zero
shear stress, the planes of maximum shear stress generally do have a normal
stress. The magnitude of this normal stress is identical to the σ coordinate
of the center C of the circle.
τ
τ
y
R
(σ avg , τ max )
C
σ
11. Notice that the angle between the principal stress points and the maximum
shear stress points on Mohr’s circle is 90°. Since angles in Mohr’s circle are
doubled, the actual angle between the principal planes and the maximum
shear stress planes will always be 45°.
τ
(σ avg , τ max )
2θ
s
x
The stress transformation equations presented in Sections 12.7 and 12.8
and the Mohr’s circle construction presented here are two methods for attaining
the same result. The advantage offered by Mohr’s circle is that it provides a concise
visual summary of all stress combinations possible at any point in a stressed
body. Since all stress calculations can be performed with the geometry of the
circle and basic trigonometry, Mohr’s circle provides an easy-to-remember tool
for stress analysis. While developing mastery of stress analysis, the student may
find it less confusing to avoid mixing the stress transformation equations presented
in Sections 12.7 and 12.8 with Mohr’s circle construction. Take advantage
of Mohr’s circle by using the geometry of the circle to compute all desired quantities
rather than trying to merge the stress transformation equations into the
Mohr’s circle analysis.
τ
τ
(σ avg , τ max )
y
R
C
( σ p2 , 0)
90°
x
(σ avg , τ max )
( σ p1 , 0)
σ