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

538
STRESS TRANSFORMATIONS
y
σ
p2
Arbitrary plane
τ
abs max
x
σ
p1
z
O
σ
p3
σ
p2
σ
p1
σ
σ
p3
Assume σ p1 > σp2 > σp3 > 0
(a) Principal stress element
FIGURE 12.20
τ
( σ, τ)
Stress combination on arbitrary plane
(b) Mohr’s circle
Application of mohr’s circle to Three-Dimensional Stress Analysis
In Figure 12.20a, the principal stresses σ p1 , σ p2 , and σ p3 at a point are shown on a stress
element. We will assume that the principal stresses have been ordered such that σ p1 > σ p2 >
σ p3 > 0. Furthermore, observe that the principal planes represented by the stress element are
rotated with respect to the x–y–z axes. From the three principal stresses, Mohr’s circle
can be plotted to visually represent the various stress combinations possible at the point
(Figure 12.20b). Stress combinations for all possible planes plot either on one of the circles
or in the shaded area. From Mohr’s circle, the absolute maximum shear stress magnitude
given by Equation (12.30) is evident.
pRoBLEmS
p12.64 At a point in a solid body subjected to plane stress, σ x =
130 MPa and σ y = 48 MPa, acting as shown in Figure P12.64a. For
plane n shown in Figure P12.64b, determine
(a) the resultant stress S.
(b) the normal stress σ n and the shear stress τ nt .
p12.65 At a point in a solid body subjected to plane stress, σ x =
9.2 ksi, σ y = 35.8 ksi, and τ xy = 6.4 ksi, acting as shown in Figure
P12.65a. For plane n shown in Figure P12.65b, determine
(a) the resultant stress S.
(b) the normal stress σ n and the shear stress τ nt .
y
y
y
y
z
σ x
x
σ y n
x
45°
45°
σ y
τ xy
z
z
σ x
x
z 45° 45°
x
n
FIGURE p12.64a
FIGURE p12.64b
FIGURE p12.65a
FIGURE p12.65b