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Load and Stress Analysis 87
Figure 3–12

Mohr’s circles for three- 1/3
dimensional stress.

1/2
2/3
1/2

3 2 1
1

2

(a) (b)

stress components σ x ,σ y ,σ z ,τ xy ,τ yz , and τ zx , involves finding the roots of the cubic
equation 1
2
3
2
2
2

σ − (σ x + σ y + σ z )σ + σ x σ y + σ x σ z + σ y σ z − τ xy − τ − τ zx σ
yz
2 2 2
− σ x σ y σ z + 2τ xy τ yz τ zx − σ x τ − σ y τ − σ z τ xy = 0 (3–15)
zx
yz
In plotting Mohr’s circles for three-dimensional stress, the principal normal
stresses are ordered so that σ 1 ≥ σ 2 ≥ σ 3 . Then the result appears as in Fig. 3–12a. The
stress coordinates σ, τ for any arbitrarily located plane will always lie on the bound-
aries or within the shaded area.
Figure 3–12a also shows the three principal shear stresses τ 1/2 , τ 2/3 , and τ 1/3 . 2
Each of these occurs on the two planes, one of which is shown in Fig. 3–12b. The fig-
ure shows that the principal shear stresses are given by the equations
σ 1 − σ 2 σ 2 − σ 3 σ 1 − σ 3
τ 1/2 = τ 2/3 = τ 1/3 = (3–16)
2 2 2
Of course, τ max = τ 1/3 when the normal principal stresses are ordered (σ 1 >σ 2 >σ 3 ),
so always order your principal stresses. Do this in any computer code you generate and
you’ll always generate τ max .

3–8 Elastic Strain
Normal strain is defined and discussed in Sec. 2–1 for the tensile specimen and is
given by Eq. (2–2) as = δ/l, where δ is the total elongation of the bar within the
length l. Hooke’s law for the tensile specimen is given by Eq. (2–3) as
σ = E (3–17)

where the constant E is called Young’s modulus or the modulus of elasticity.

1 For development of this equation and further elaboration of three-dimensional stress transformations see:
Richard G. Budynas, Advanced Strength and Applied Stress Analysis, 2nd ed., McGraw-Hill, New York,
1999, pp. 46–78.
2 Note the difference between this notation and that for a shear stress, say, τ xy . The use of the shilling mark is
not accepted practice, but it is used here to emphasize the distinction.