Page 107 - Shigley's Mechanical Engineering Design

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82 Mechanical Engineering Design
In a similar manner the two extreme-value shear stresses are found to be

2

σ x − σ y
2
τ 1 ,τ 2 =± + τ xy (3–14)
2
Your particular attention is called to the fact that an extreme value of the shear stress
may not be the same as the actual maximum value. See Sec. 3–7.
It is important to note that the equations given to this point are quite sufficient for
performing any plane stress transformation. However, extreme care must be exercised
when applying them. For example, say you are attempting to determine the principal
state of stress for a problem where σ x = 14 MPa,σ y =−10 MPa, and τ xy =−16 MPa.
Equation (3–10) yields φ p =−26.57 and 63.43°, which locate the principal stress sur-
◦
faces, whereas, Eq. (3–13) gives σ 1 = 22 MPa and σ 2 =−18 MPa for the principal
stresses. If all we wanted was the principal stresses, we would be finished. However,
what if we wanted to draw the element containing the principal stresses properly ori-
ented relative to the x, y axes? Well, we have two values of φ p and two values for the
principal stresses. How do we know which value of φ p corresponds to which value of
the principal stress? To clear this up we would need to substitute one of the values of
φ p into Eq. (3–8) to determine the normal stress corresponding to that angle.
A graphical method for expressing the relations developed in this section, called
Mohr’s circle diagram, is a very effective means of visualizing the stress state at a point
and keeping track of the directions of the various components associated with plane stress.
Equations (3–8) and (3–9) can be shown to be a set of parametric equations for σ and τ,
where the parameter is 2φ. The parametric relationship between σ and τ is that of a cir-
cle plotted in the σ, τ plane, where the center of the circle is located at C = (σ, τ) =

2
2
[(σ x + σ y )/2, 0] and has a radius of R = [(σ x − σ y )/2] + τ . A problem arises in
xy
the sign of the shear stress. The transformation equations are based on a positive φ
being counterclockwise, as shown in Fig. 3–9. If a positive τ were plotted above the
σ axis, points would rotate clockwise on the circle 2φ in the opposite direction of
rotation on the element. It would be convenient if the rotations were in the same
direction. One could solve the problem easily by plotting positive τ below the axis.
However, the classical approach to Mohr’s circle uses a different convention for the
shear stress.

Mohr’s Circle Shear Convention
This convention is followed in drawing Mohr’s circle:
• Shear stresses tending to rotate the element clockwise (cw) are plotted above the
σ axis.
• Shear stresses tending to rotate the element counterclockwise (ccw) are plotted below
the σ axis.
For example, consider the right face of the element in Fig. 3–8b. By Mohr’s circle con-
vention the shear stress shown is plotted below the σ axis because it tends to rotate the
element counterclockwise. The shear stress on the top face of the element is plotted
above the σ axis because it tends to rotate the element clockwise.
In Fig. 3–10 we create a coordinate system with normal stresses plotted along the
abscissa and shear stresses plotted as the ordinates. On the abscissa, tensile (positive)
normal stresses are plotted to the right of the origin O and compressive (negative) nor-
mal stresses to the left. On the ordinate, clockwise (cw) shear stresses are plotted up;
counterclockwise (ccw) shear stresses are plotted down.