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

The angle θ from the redefined x axis to the n axis is 20° in a clockwise sense; therefore,
θ = −20°.
The normal stress on the vertical face of the unrotated element can be computed
from Equation (12.3):
2 2
σ = σ cos θ + σ sin θ + 2τ sinθcosθ
n x y xy
2 2
= (66 ksi)cos ( − 20) ° + ( −28ksi)sin ( − 20) ° + 2(42 ksi)sin( − 20 ° )cos( − 20) °
= 28.0 ksi
The normal stress on the horizontal face of the unrotated element can be computed
from Equation (12.3) if the angle θ is changed to θ = −20° + 90° = 70°:
9.99 ksi
62.4 ksi
28.0 ksi
20°
28 ksi
42 ksi
2 2
σ = σ cos θ + σ sin θ + 2τ sinθcosθ
t x y xy
2 2
= (66 ksi)cos 70 ° + ( − 28ksi)sin 70°+ 2(42 ksi)sin70° cos70°
= 9.99 ksi
66 ksi
The shear stress on the unrotated element can be computed
from Equation (12.4):
2 2
τ =−( σ − σ )sinθcos θ + τ (cos θ − sin θ)
nt x y xy
=−[(66 ksi) − ( −28ksi)]sin( − 20 ° )cos( − 20) °
2 2
+ (42 ksi)[cos ( − 20) ° − sin (20 − ° )]
= 62.4 ksi
The stresses acting on the horizontal and vertical planes are shown in
the sketch.
mecmovies
ExAmpLES
m12.7 Determine the normal and shear stress acting on a
specified plane surface.
m12.8 Determine the normal and shear stress acting on a
specified plane surface in a wooden object.
496