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

P 2
(–7.22, 0)
(–5, 6 ccw) y
τ
(2, 9.22 cw)
S 2
49.40°
R = 9.22
40.60°
(2, 0)
C
R = 9.22
x (9, 6cw)
(11.22, 0)
P σ
1
shear stress magnitude at these points is simply equal to the
radius R of the circle. Notice that the normal stress at points
S 1 and S 2 is not zero. Rather, the normal stress σ at these
points is equal to the stress at the center C of the circle.
The angle between points P 1 and S 2 is 90°. Since the
angle between point x and point P 1 was found to be 40.60°,
the angle between point x and point S 2 must be 49.40°.
This angle rotates in a counterclockwise direction.
One plane subjected to the maximum in-plane shear
stress is oriented 24.70° counterclockwise from the x face.
The magnitude of this shear stress is equal to the radius
of the circle:
Mohr's circle point S 1
corresponds to this face.
6 ksi
5 ksi
9 ksi
τ
2 ksi
24.7°
x
S 1
(2, 9.22ccw)
S
2 ksi
9.22 ksi
Mohr’s circle point
S2 corresponds
to this face.
6 ksi
τ = R = 9.22 ksi
max
To determine the direction of the shear stress arrow acting on
this face, note that point S 2 is on the upper half of the circle,
above the σ axis. Consequently, the shear stress acting on
this face rotates the stress element clockwise. Once the shear
stress direction on one face has been determined, the shear
stress directions on the other three faces are known.
A complete sketch showing the principal stresses,
the maximum in-plane shear stress, and the orientations
of the respective planes can now be prepared.
Mohr’s circle point S 1
corresponds to this face.
5 ksi
9 ksi
2 ksi
24.7°
x
20.3°
Mohr’s circle point P 2
corresponds to this face.
S
P
2 ksi
9.22 ksi
7.22 ksi
Mohr’s circle point
S2 corresponds
to this face.
11.22 ksi
Mohr’s circle
point P1
corresponds
to this face.
ExAmpLE 12.8
y
12 MPa
27 MPa
x
60 MPa
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.
SolutioN
Begin with the normal and shear stresses acting on the x face of the stress element. The
normal stress is σ x = 60 MPa (C), and the shear stress τ acting on the x face rotates the
520