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

12.11 General State of Stress at a point
σ z
z
FIGURE 12.17
y
σ y
τ
τ yx
yz
τ xy
τ zy
τ zx σ x
τ xz
x
The general three-dimensional state of stress at a point was previously introduced in
Section 12.2. This state of stress has three normal stress components and six shear stress
components, as illustrated in Figure 12.17. The shear stress components are not all independent,
however, since moment equilibrium requires that
τ = τ τ = τ and τ = τ
yx xy yz zy xz zx
The stresses are all positive, according to the normal and shear stress sign conventions
outlined in Section 12.2.
Normal and Shear Stresses
Expressions for the stresses on any oblique plane through a point, in terms of stresses on
the reference x, y, and z planes, can be developed with the aid of the free-body diagram
shown in Figure 12.18a. The n axis is normal to the oblique (shaded) face. The orientation
of the n axis can be defined by three angles α, β, and γ as shown in Figure 12.18b. The area
of the oblique face of the tetrahedral element is defined to be dA. Areas of the x, y, and z
faces are thus dA cos α, dA cos β, and dA cos γ , respectively. 1 The resultant force F on the
oblique face is S dA, where S is the resultant stress on the area. The resultant stress S is
related to the normal and shear stress components on the oblique face by the expression
S σn 2 τnt
2
= + (12.24)
y
n
y
dA x σ x
dz
dA x τ xz
dA x τ xy
dy
dA y τ yx
dA z τ zx
dn
dA z τ zy
dA y τ yz
σ z dA z
dx
F = S dA
x
z
γ
n axis
β
α
l
m
n
x
z
σ y dA y
(a) Tetrahedron free-body diagram (b) Orientation of n axis defined
FIGURE 12.18 Tetrahedron for deriving principal stresses on an oblique plane.
1
These relationships can be established by considering the volume of the tetrahedron in Figure 12.18a. The
volume of the tetrahedron can be expressed as V = 1/3 dn dA = 1/3 dx dA x = 1/3 dy dA y = 1/3 dz dA z . However,
the distance dn from the origin to the center of the oblique face can also be expressed as dn = dx cos α = dy cos β =
dz cos γ . Thus, the areas of the tetrahedron faces can be expressed as dA x = dA cos α, dA y = dA cos β, and
dA z = dA cos γ .
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