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

and the normal stress acting on the planes of maximum shear
stress is
σ = C = −31.5 MPa
avg
(–31.50, 55.22)
S 2
τ
y (–16, 53)
A complete sketch showing the principal stresses, the maximum
in-plane shear stress, and the orientations of the respective
planes is shown.
Determine σ n , σ t , and τ nt
Next, the normal stresses σ n and σ t and the shear stress τ nt
acting on a stress element that is rotated 35° counterclockwise
from the x direction, as shown in the accompanying
sketch, must be determined.
P 2
R = 55.22
C (–31.5, 0) (23.72, 0)
(−86.72, 0) P σ
1
73.7°
σ t
(–47, 53) x
S 1 (–31.50, 55.22)
τ nt
σ n
τ
16 MPa
16 MPa
53 MPa
47 MPa
53 MPa
47 MPa
35°
x
x
36.85°
31.50 MPa
55.22 MPa
Begin at point x on Mohr’s circle. This statement may
seem obvious, but it is probably the most common mistake
made in solving problems of this type.
In the x–y coordinate system, the 35° angle is rotated
counterclockwise from the horizontal axis. As one transfers
this angular measure to Mohr’s circle, the natural tendency is
to draw a diameter that is rotated 2(35°) = 70° counterclockwise
from the horizontal axis. This is incorrect!
Remember that Mohr’s circle is a plot in terms of the
normal stress σ and shear stress τ. The horizontal axis in
Mohr’s circle does not necessarily correspond to the x face of
the stress element. On Mohr’s circle, the point labeled x is the
one that corresponds to the x face. (This fact explains why it
is so important to label the points as you construct Mohr’s
circle.)
To determine stresses on the plane that is rotated 35°
from the x face, a diameter that is rotated 2(35°) = 70° counterclockwise
from point x is drawn on Mohr’s circle. The
point 70° away from point x should be labeled point n. The
coordinates of this point are the normal and shear stresses
acting on the n face of the rotated stress element. The other
end of the diameter should be labeled point t, and its coordinates
are σ and τ, acting on the t face of the rotated stress
element.
23.72 MPa
τ
n
2(35°) = 70°
(–31.5, 0)
C
R = 55.22
t
τ
86.72 MPa
σ
523