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

p2
p2
p2
σ p2
σ
p2
σ p2
σ
p1
σ
p1
σ
p1
p1
p1
p1
p3
p3
p3
(a) If both σ p1 and σ p2 are positive (b) If both σ p1 and σ p2 are negative (c) If σ p1 is positive and σ p2 is negative
FIGURE 12.13 Planes of absolute maximum shear stress for plane stress.
(b) If both σ p1 and σ p2 are negative, then
τ
absmax
σ p − σ p 0 − σ p σ p
=
= =−
2 2 2
3 2 2 2
(c) If σ p1 is positive and σ p2 is negative, then
τ
absmax
σ
=
− σ
p1 p2
2
These three possibilities are illustrated in Figure 12.13, in which one of the two orthogonal
planes on which the maximum shear stress acts is highlighted in each example. Note that
σ p3 = 0 in all three cases.
The direction of the absolute maximum shear stress can be determined by drawing a
wedge-shaped block with two sides parallel to the planes having the maximum and minimum
principal stresses, and with the third side at an angle of 45° with the other two sides.
The direction of the maximum shear stress must oppose the larger of the two principal
stresses.
Stress Invariance
A useful relationship between the principal stresses and the normal
stresses on the orthogonal planes, shown in Figure 12.14, is obtained by
adding the values for the two principal stresses as given in Equation
(12.12). The result is
σ
y
σ
p2
σ
p1
σ p1 + σ p2 = σx + σ y
(12.19)
y
x
τ
xy
σ
x
θ
p
In words, for plane stress, the sum of the normal stresses on any two
orthogonal planes through a point in a body is constant and independent
of the angle θ.
FIGURE 12.14
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