Mohr Circle and Strength Testing Chapter 11
CHAPTER 11
MOHR CIRCLE, FAILURE THEORIES, AND STRENGTH
TESTING OF SOIL AND ROCKS
11-1. Given an element with stresses as indicated in the figure, find: (a) The major and minor
principal stresses and the planes on which they act. (b) The stresses on a plane inclined at 30°
from the horizontal. (c) The max. shear stress and the inclination of the plane on which it acts.
SOLUTION:
20, 35
60, 53.15
49.69, 52.14
40
50
60
pole
Mohr Circle and Strength Testing Chapter 11
11-1 continued
xy xy
Solve using Mohr‘s circle and the pole method (see plot on next page).
20 kPa, 100 kPa, 35 kPa
   
yx
1
p1
1o
1
1
Pr incipal angle, cos
22R
1 100 20
cos 20.59 measured cw from the x-axis at .
2253.15


 



 


Mohr Circle and Strength Testing Chapter 11
11-2. Work Problem 11.1 with the element rotated 30° clockwise from the horizontal.
SOLUTION:
xy xy
Solve using Mohr‘s circle and the pole method (see plot on next page).
20 kPa, 100 kPa, 35 kPa
   
Mohr circle shown on next page
Mohr Circle and Strength Testing Chapter 11
11-2 continued
113.156.85
70, 52
60, 53.15
9.3, 16
pole
0
10
20
30
40
50
60
Mohr Circle and Strength Testing Chapter 11
11-3. With the element of Problem 11.1 rotated 40° clockwise from the horizontal, find the
magnitude and direction of the stresses on the vertical plane.
SOLUTION:
Solve using Mohr‘s circle and the pole method (see plot on next page).
From the Mohr circle: pole (98,37)
stresses on a vertical plane (98, 37)

100, 35
pole (98, 37)
30
40
50
60
Mohr Circle and Strength Testing Chapter 11
11-4. Work Example 11.3 with the element rotated 30° clockwise from the horizontal. In addition,
find the stresses (magnitude and direction) on the horizontal plane.
SOLUTION:
xyxy
Solve using Mohr‘s circle and the pole method (see plot on next page).
4 kPa, 6 kPa, 2 kPa
   
Mohr circle shown on next page
Mohr Circle and Strength Testing Chapter 11
11-4 continued
pole (0.6, 5.4)
4
5
6
7
8
9
Mohr Circle and Strength Testing Chapter 11
11-6. The state of plane stress in a body is described by the following stresses:
1 = 8500 kN/m2
compression,
3 = 1500 kN/m2 tension. Determine by means of the Mohr circle the normal stress
and shear stress on a plane inclined at 20° to the plane on which the minor principal stress acts.
Check the results analytically. (After A. Casagrande.)
SOLUTION:
13 xy
Solve using Mohr‘s circle and the pole method (see plot on next page).
8.5 MPa, 1.5 MPa, 0
  
(7.33, 3.21)
1
2
3
4
5
6
Mohr Circle and Strength Testing Chapter 11
11-7. At a certain critical point in a steel beam, on a vertical plane the compressive stress is 115
MPa and the shearing stress is 31.5 MPa. There is no normal stress on the longitudinal
(horizontal) plane. Find the stresses acting on the principal planes and the orientation of principal
planes with the horizontal. (After Taylor, 1948.)
SOLUTION:
xyxy
Solve using Mohr‘s circle and the pole method (see plot below).
115 MPa, 0, 31.5 MPa
   
40
60
80
Mohr Circle and Strength Testing Chapter 11
11-8. A soil sample is under a biaxial state of stress. On plane 1, the stresses are (13, 4), while
on plane 2, the stresses are (5.8, -2). Find the major and minor principal stresses.
SOLUTION:
Solve using Mohr‘s circle and the pole method (see plot below).
7
9
Mohr Circle and Strength Testing Chapter 11
11-9. For the element shown in the figure: (a) Find the magnitude of the unknown stresses
h
and
h on the horizontal plane. (b) Find the orientation of the principal stresses; clearly indicate
their orientation in a small sketch. (c) Show the orientation of the planes of maximum as well as
minimum shear.
SOLUTION:
Solve using Mohr‘s circle and the pole method using the given stresses.
Plot (2, -2) and (5, 3), and construct a perpendicular bisector.
(5, 3)
(4.4, 3.1)
2
3
4
Normal stress
Mohr Circle and Strength Testing Chapter 11
11-10. Given the element with stresses as shown in the figure: (a) Find the magnitude and
direction of
H and
H. (b) Find the magnitude and direction of
1 and
3.
SOLUTION:
Solve using Mohr‘s circle and the pole method using the given stresses.
Plot (4, 3) and (8, –2), and construct a perpendicular bisector.
4, 3
2
3
4
5
Mohr Circle and Strength Testing Chapter 11
11-11. Given the data of Example 11.5. (a) Find the magnitude of the stresses on the horizontal
plane. (b) Find the maximum shear stress, and determine the angle between the plane on which
it acts and the major principal plane.
SOLUTION:
xy
xyxy
Solve using Mohr‘s circle and the pole method (see plot below).
84
8 kPa, 4 kPa, 2 kPa, center 6 kPa
22

   
8, 2
pole (4, 2)
6, 2.83
1
2
3
4
5
Mohr Circle and Strength Testing Chapter 11
11-12. The state of stress on a small element is
v = 21 kPa,
h = 10 kPa, and the shear stress
on the horizontal plane is +3 kPa. (a) Find the magnitude and directions of the major and minor
principal stresses. (b) If the material is a loose sand, can you say whether the element is in a
state of failure? If it isn’t, how close is it? Why? State your assumptions clearly. (Assume
’ = 28o
for the loose sand.)
SOLUTION:
Solve using Mohr‘s circle and the pole method (see plot below).
(b) The element is not in a state of failure. The shear stress on the fail
ure plane is about
21, 3
Pole: (10, 3)
4
6
8
10
12