Page 108 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 83
Figure 3–10 cw x
( – )
Mohr’s circle diagram. y x y –
x y
2
F

y
B H
( , xy cw )
y
xy
E 2 D
O 2 y C x 1
xy
x – y

2 2 A 2 p
2 ccw )
x
( , xy
+ xy
x
+ G
ccw x y
2

Using the stress state of Fig. 3–8b, we plot Mohr’s circle, Fig. 3–10, by first look-
ing at the right surface of the element containing σ x to establish the sign of σ x and the
cw or ccw direction of the shear stress. The right face is called the x face where
◦
φ = 0 . If σ x is positive and the shear stress τ xy is ccw as shown in Fig. 3–8b, we can
establish point A with coordinates (σ x ,τ ccw ) in Fig. 3–10. Next, we look at the top y
xy
face, where φ = 90 , which contains σ y , and repeat the process to obtain point B with
◦
cw
coordinates (σ y ,τ ) as shown in Fig. 3–10. The two states of stress for the element
xy
◦
are φ = 90 from each other on the element so they will be 2 φ = 180 from each
◦
other on Mohr’s circle. Points A and B are the same vertical distance from the σ axis.
Thus, AB must be on the diameter of the circle, and the center of the circle C is where
AB intersects the σ axis. With points A and B on the circle, and center C, the complete
circle can then be drawn. Note that the extended ends of line AB are labeled x and y
as references to the normals to the surfaces for which points A and B represent the
stresses.
The entire Mohr’s circle represents the state of stress at a single point in a struc-
ture. Each point on the circle represents the stress state for a specific surface intersect-
ing the point in the structure. Each pair of points on the circle 180° apart represent the
state of stress on an element whose surfaces are 90° apart. Once the circle is drawn, the
states of stress can be visualized for various surfaces intersecting the point being ana-
lyzed. For example, the principal stresses σ 1 and σ 2 are points D and E, respectively,
and their values obviously agree with Eq. (3–13). We also see that the shear stresses
are zero on the surfaces containing σ 1 and σ 2 . The two extreme-value shear stresses, one
clockwise and one counterclockwise, occur at F and G with magnitudes equal to the
radius of the circle. The surfaces at F and G each also contain normal stresses of
(σ x + σ y )/2 as noted earlier in Eq. (3–12). Finally, the state of stress on an arbitrary
surface located at an angle φ counterclockwise from the x face is point H.