Page 106 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 81
Figure 3–9 y
n

x
dy dy
ds ds
xy dx
dx
x
xy
y
Differentiating Eq. (3–8) with respect to φ and setting the result equal to zero
maximizes σ and gives
2τ xy
tan 2φ p = (3–10)
σ x − σ y
Equation (3–10) defines two particular values for the angle 2φ p , one of which defines
the maximum normal stress σ 1 and the other, the minimum normal stress σ 2 . These two
stresses are called the principal stresses, and their corresponding directions, the princi-
pal directions. The angle between the two principal directions is 90°. It is important to
note that Eq. (3–10) can be written in the form
σ x − σ y
sin 2φ p − τ xy cos 2φ p = 0 (a)
2
Comparing this with Eq. (3–9), we see that τ = 0, meaning that the perpendicular sur-
faces containing principal stresses have zero shear stresses.
In a similar manner, we differentiate Eq. (3–9), set the result equal to zero, and obtain
σ x − σ y
tan 2φ s =− (3–11)
2τ xy
Equation (3–11) defines the two values of 2φ s at which the shear stress τ reaches an
extreme value. The angle between the two surfaces containing the maximum shear
stresses is 90°. Equation (3–11) can also be written as

σ x − σ y
cos 2φ p + τ xy sin 2φ p = 0 (b)
2
Substituting this into Eq. (3–8) yields

σ x + σ y
σ = (3–12)
2
Equation (3–12) tells us that the two surfaces containing the maximum shear stresses
also contain equal normal stresses of (σ x + σ y )/2.
Comparing Eqs. (3–10) and (3–11), we see that tan 2φ s is the negative reciprocal
of tan 2φ p . This means that 2φ s and 2φ p are angles 90° apart, and thus the angles
between the surfaces containing the maximum shear stresses and the surfaces contain-
ing the principal stresses are ±45 .
◦
Formulas for the two principal stresses can be obtained by substituting the
angle 2φ p from Eq. (3–10) in Eq. (3–8). The result is

2

σ x + σ y σ x − σ y
2
σ 1 ,σ 2 = ± + τ xy (3–13)
2 2