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

in the x–y coordinate system is rotated 18.43° clockwise from the x face of the stress
element.
The angle between point x and point S 1 is 53.14°; therefore, this plane of maximum
in-plane shear stress in the x–y coordinate system is rotated 26.57° counterclockwise
from the x face of the stress element.
To determine the direction of the shear stress arrow acting on this face, note that
point S 1 is on the lower half of the circle, below the σ axis. Consequently, the shear stress
acting on the x face rotates the stress element counterclockwise.
A complete sketch showing the principal stresses, the maximum in-plane shear
stress, and the orientations of the respective planes is given.
ExAmpLE 12.9
53 MPa
16 MPa
47 MPa
35°
x
τ
y (– 16, 53)
R = 55.22
σ
t
τ
nt
σ
n
Stresses on an inclined Plane
The stresses shown act at a point on the free surface of a stressed
body.
(a) Determine the principal stresses and the maximum in-plane
shear stress acting at the point.
(b) Show these stresses in an appropriate sketch.
(c) Determine the normal stresses σ n and σ t and the shear stress
τ nt that act on the rotated stress element.
SolutioN
Construct Mohr’s Circle
From the normal and shear stresses acting on the x and y faces of
the stress element, Mohr’s circle is constructed as shown.
The center of Mohr’s circle is located at
C
47 ( 16)
= − + − =−31.5 MPa
2
The radius R is found from the hypotenuse of the shaded triangle:
2 2
R = 15.5 + 53 = 55.22 MPa
15.5
73.7°
53
C (–31.5, 0)
σ
The angle between the x–y diameter and the σ axis is 2θ p , and it can
be computed as follows:
tan2θ
p
53
= ∴ 2θp
= 73.7 ° (cw)
15.5
(– 47, 53) x
τ
Principal and Maximum Shear Stress
The principal stresses (points P 1 and P 2 ) are determined from the
location of the center C of the circle and the radius R:
σ
σ
p1
p2
= C + R =− 31.5 + 55.22 = 23.72 MPa
= C − R =−31.5 − 55.22 = −86.72 MPa
The maximum in-plane shear stress corresponds to points S 1 and S 2
on Mohr’s circle. The maximum in-plane shear stress magnitude is
τ = R = 55.22 MPa
max
522