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

Since points x and y are always the same distance
above or below the σ axis, the center of Mohr’s circle
can be found by averaging the normal stresses acting
on the x and y faces:
σx
+ σ
C =
2
y
9ksi + ( −5 ksi)
=
= 2ksi
2
The center of Mohr’s circle always lies on the σ axis.
The geometry of the circle is used to calculate
the radius. The (σ, τ) coordinates of point x and center
C are known. Use these coordinates with the Pythagorean
theorem to calculate the hypotenuse of the shaded
triangle:
R = (9 ksi − 2 ksi) + (6 ksi − 0)
2 2
(–5, 6 ccw) y
τ
(2, 0)
τ
C
R = 9.22
x (9, 6cw)
6
40.60°
7 σ
2 2
= 7 + 6 = 9.22 ksi
The angle between the x–y diameter and the σ axis is
2θ p , and it can be computed by means of the tangent
function:
τ
x (9, 6cw)
tan2θ
p
6
= ∴ 2θp
= 40.60°
7
Note that this angle turns clockwise from point x to
the σ axis.
The maximum value of σ (i.e., the most positive
value algebraically) occurs at point P 1 , where Mohr’s
circle crosses the σ axis. From the geometry of the circle,
P 2
(–7.22, 0)
(–5, 6 ccw) y
40.60°
(2, 0)
C
R = 9.22
R = 9.22
(11.22, 0)
P σ 1
σ = C + R = 2ksi + 9.22 ksi = 11.22 ksi
p1
τ
The minimum value of σ (i.e., the most negative value
algebraically) occurs at point P 2 . Again, from the
geometry of the circle,
σ = C − R = 2ksi − 9.22 ksi =−7.22 ksi
p2
The angle between point x and point P 1 was calculated as
2θ p = 40.60°; however, angles in Mohr’s circle are double
angles, so, to determine the orientation of the principal
planes in the x–y coordinate system, divide 40.60° by 2.
Therefore, the principal stress σ p1 acts on a plane rotated
20.30° from the x face of the stress element. The 20.30°
angle in the x–y coordinate system rotates in the same
sense as 2θ p in Mohr’s circle. In this example, the 20.30°
angle rotates clockwise from the x axis.
The principal stresses, as well as the orientation of
the principal planes, are shown in the sketch.
The maximum values of τ occur at points S 1 and S 2 ,
located at the bottom and at the top of Mohr’s circle. The
6 ksi
5 ksi
9 ksi
x
20.3°
Mohr’s circle point P 2
corresponds to this face.
P
7.22 ksi
11.22 ksi
Mohr’s circle
point P 1
corresponds
to this face.
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