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

t
y
n
θ
x
σn dA
Figure 12.10b is a free-body diagram of a wedge-shaped element in which the
areas of the faces are dA for the inclined face (plane A–A), dA cos θ for the vertical
face (i.e., the x face), and dA sin θ for the horizontal face (i.e., the y face). The equilibrium
equation for the sum of forces in the n direction gives
σx dA cos θ
τxy dA cos θ
θ
θ
τnt dA
τyx dA sin θ
∑ F = σ dA − τ ( dA sin θ)cos θ − τ ( dA cos θ)sinθ
n n yx xy
− σ ( dA cos θ)cos θ − σ ( dA sin θ)sinθ
= 0
x
Since τ yx = τ xy , this equation can be simplified to give the following expression for
the normal stress acting on the n face of the wedge element:
y
σy dA sin θ
FIGURE 12.10b
2 2
σ = σ cos θ + σ sin θ + 2τ sinθcosθ
(12.3)
n x y xy
From the free-body diagram in Figure 12.10b, the equilibrium equation for the sum of
forces in the t direction gives
∑ F = τ dA − τ ( dAcos θ)cos θ + τ ( dAsin θ)sinθ
t nt xy yx
+ σ ( dAcos θ)sin θ − σ ( dAsin θ)cosθ
= 0
x
Again from τ yx = τ xy , this equation can be simplified to give the following expression for
the shear stress acting in the t direction on the n face of the wedge element:
y
2 2
τ =−( σ − σ )sinθcos θ + τ (cos θ − sin θ)
(12.4)
nt x y xy
The equations just derived for the normal stress and the shear stress can be written in an
equivalent form by substituting the following double-angle identities from trigonometry:
2
1
cos θ = (1 + cos2 θ)
2
2
1
sin θ = (1 − cos2 θ)
2
2sinθcosθ = sin 2θ
Using these double-angle identities, we can write Equation (12.3) as
σ
n
σx + σ y σx − σ y
= + cos2θ + τxy
sin2θ
(12.5)
2 2
mecmovies 12.6 presents an
animated derivation of the plane
stress transformation equations.
and Equation (12.4) as
τ
nt
σx
− σ
=−
2
y
sin2θ + τxy
cos2θ
(12.6)
Equations (12.3), (12.4), (12.5), and (12.6) are called the plane stress transformation
equations. They provide a means for determining normal and shear stresses on any plane
whose outward normal is
(a) perpendicular to the z axis (i.e., the out-of-plane axis), and
(b) oriented at an angle θ with respect to the reference x axis.
Since the transformation equations were derived solely from equilibrium considerations,
they are applicable to stresses in any kind of material, whether it is linear or nonlinear,
elastic or inelastic.
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