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

p12.25–p12.26 The stresses shown in Figures P12.25 and
P12.26 act at a point on the free surface of a machine component.
Determine the normal stresses σ x and σ y and the shear stress τ xy at
the point.
σ
y
y
x
τ
xy
σ
x
30°
48 MPa
82 MPa
y
σ
y
x
τ
xy
σ
x
13.8 ksi
24°
4.7 ksi
19.1 ksi
59 MPa
FIGURE p12.25
FIGURE p12.26
12.8 principal Stresses and maximum Shear Stress
The transformation equations for plane stress [Equations (12.3), (12.4), (12.5), and (12.6)]
provide a means for determining the normal stress σ n and the shear stress τ nt acting on any
plane through a point in a stressed body. For design purposes, the critical stresses at a
point are often the maximum and minimum normal stresses and the maximum shear
stress. The stress transformation equations can be used to develop additional relationships
that indicate
(a) the orientations of planes where maximum and minimum normal stresses occur,
(b) the magnitudes of maximum and minimum normal stresses,
(c) the magnitudes of maximum shear stresses, and
(d) the orientations of planes where maximum shear stresses occur.
The transformation equations for plane stress were developed in Section 12.7. Equations
(12.3) and (12.4), for normal stress and shear stress are, respectively, as follows:
2 2
σ = σ cos θ + σ sin θ + 2τ sinθcosθ
n x y xy
2 2
τ =−( σ − σ )sinθcos θ + τ (cos θ − sin θ)
nt x y xy
These same equations can also be expressed in terms of double-angle trigonometric functions
as Equations (12.5) and (12.6):
σ
n
σx + σ y σx − σ y
= + cos2θ + τxy
sin2θ
2 2
τ
nt
σx
=−
− σ y
sin2θ + τxy
cos2θ
2
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