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

The choice of either Equation (12.3) or Equation (12.5) to calculate the normal stress acting
on the inclined plane is a matter of personal preference.
The shear stress τ nt acting on the n face in the t direction can be computed from
Equation (12.4):
2 2
τ =−( σ − σ )sinθcos θ + τ (cos θ − sin θ)
nt x y xy
2 2
=−[(16 MPa) − (42 MPa)]sin58° cos58 °+ ( − 50MPa)[cos 58° − sin 58] °
= 33.6 MPa
Alternatively, Equation (12.6) may be used:
τ
nt
σx
− σ y
=− sin2θ + τxy
cos2θ
2
(16MPa) − (42 MPa)
=−
sin2(58 ° ) + ( − 50MPa) cos2(58 ° )
2
= 33.6 MPa
n
To complete the problem, the stresses acting on the inclined plane are shown in a
sketch. Since σ n is negative, the normal stress acting in the n direction is shown as a
compressive stress. The positive value of τ nt indicates that the stress arrow points in
the positive t direction on the positive n face. The arrows are labeled with the stress
magnitude (i.e., absolute value). The signs associated with the stresses are indicated
by the directions of the arrows.
t
58°
10.24 MPa
33.6 MPa
ExAmpLE 12.4
The stresses shown 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.
Plan the Solution
The stress transformation equations are written in terms of σ x , σ y ,
and τ xy ; however, the x and y directions do not necessarily have to be
the horizontal and vertical directions, respectively. Any two orthogonal
directions can be taken as x and y, as long as they define a righthand
coordinate system. To solve this problem, we will redefine the x
and y axes, aligning them with the rotated element. The faces of the
unrotated element will be redefined as the n and t faces.
σy
τxy
σx
20°
28 ksi
42 ksi
66 ksi
SolutioN
Redefine the x and y directions, aligning them with the rotated element.
The axes of the unrotated element will be defined as the n and
t directions.
Accordingly, the stresses acting on the rotated element are now
defined as
σ
σ
τ
x
y
xy
= 66 ksi
=−28 ksi
= 42 ksi
t
σ y
τ xy
n
σ x
20°
28 ksi
42 ksi
y
x
66 ksi
495