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

ExAmpLE 12.10
Stresses on an inclined Plane
The stresses shown act at a point on the free surface of a stressed body. Determine the
normal stress σ n and the shear stress τ nt that act on the indicated plane surface.
SolutioN
From the normal and shear stresses acting on the x and y faces of the stress element,
Mohr’s circle is constructed as shown.
How is the orientation of the inclined Plane Determined?
We must find the angle between the normal to the x face (i.e.,
the x axis) and the normal to the inclined plane (i.e., the n axis).
The angle between the x and n axes is 50°; consequently, the
inclined plane is oriented 50° clockwise from the x face.
On Mohr’s circle, point n is located 100° clockwise from
point x.
Using the coordinates of point x and the center C of the circle,
we find the angle between point x and the σ axis to be 67.38°.
Consequently, the angle between point n and the σ axis
must be 32.68°.
The horizontal component of the line between C and point
n is
∆ σ = Rcos32.62 °= (71.5 MPa)cos32.62°=
60.22 MPa
and the vertical component is
∆ τ = Rsin32.62 °= (71.5 MPa)sin32.62°=
38.54 MPa
The normal stress on the n face of the rotated stress element can
be computed by using the coordinates of the center C and ∆σ:
σ =− 11.0 MPa + 60.22 MPa = 49.22 MPa
n
The shear stress is computed similarly:
τ = 0 + 38.54 MPa = 38.54 MPa
nt
Since point n on Mohr’s circle is located below the σ axis, the shear stress acting on the
n face tends to rotate the stress element counterclockwise.
t
(–38.5, 66.0) y
(–11, 0) C
R = 71.5
τ
τ
66.0 MPa
y
40°
x (16.5, 66.0)
67.38°
32.62°
66.0 MPa
16.5 MPa
40°
n
y
38.5 MPa
σ
n
t
38.5 MPa
x
16.5 MPa
100° (cw)
38.54 MPa
49.22 MPa
n
t
mecmovies
ExAmpLE
m12.19 Mohr’s Circle learning tool
Illustrates the proper usage of Mohr’s circle to determine
stresses acting on a specified plane, principal stresses, and
the maximum in-plane shear stress state for stress values
specified by the user. Detailed “how-to” instructions.
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