Lab 8: Stress and Mohr Circles

Skills: Use the Mohr’s Circle construction to predict stresses on surfaces, construct theCoulomb failure envelope.

Bring: Calculator or computer, stereonet, tracing paper, graph paper, drawing compass

Recall that in 2D:

σN =σ1 + σ3

2+σ1 − σ3

2∗ cos(2θ) (1)

σS =σ1 − σ3

2∗ sin(2θ) (2)

Constructing the Failure Envelope (25%)

A rock mechanics laboratory performed a series of failure tests oncylinders cut from the same marble. The confining stress (σ3) was set,and the axial stress (σ1) was increased until the rock fractured.

σ1 (MPa) σ3 (MPa)164 28111 1070 0

1. Plot the Mohr Circles for each failure test. (5%)

The equation for a circle is:

(x− h)2 + (y − k)2 = r2 (3)

where (x, y) are points on the circle, (h, k) is the center of the circle, and r is the radius of thecircle.

2. Define x, y, h, k, and r in terms of Equations 1 and 2. (5%)

3. The overlapping circles encompass a region of the graph. The outer edges of this regiondefine a failure envelope which can be described by two lines tangent to the circles. Find theequations of these lines in the form σS = µσN + C0. The intersect of the failure envelopewith the shear stress axis, b, is called cohesion (C0) and the slope m is called the coefficientof friction (µ). (10%)

4. In each circle, draw a straight line connecting the center of the circle to the failure envelope.What is the value of θ for each case? (5%)

Stress calculations (30%)

Refer to Figure 1 to answer the following questions. Plane 1 is horizontal. The principalstresses σ1 and σ2 are horizontal and σ3 is vertical.

5. Using Equations 1 and 2, what are the normal and shear stresses resolved on Plane 3, if σ1 is120 MPa and σ3 is 25 MPa? (5%)

6. If the material in the box shown in Figure 1 is made of the same marble from the previoussection, will slip occur on a fault parallel to Plane 3? (5%)

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7. Plot a stereonet showing each of the 6 planes, and the principal stresses as lines. Theazimuth (compass orientation) of the long side of the rectangular area is shown; the strike ofthe planes is perpendicular to this. (10%)

8. Elevated fluid pressure within the rock has the effect of pressing outward, against thecompressive stresses, reducing the effective stress. This reduces the effective values of σ1 andσ3 by the relation:

σ∗ = σ − Pf (4)

How much fluid pressure increase is required to cause Plane 4 to fail? (10%)

Plane 1

Plane 2σ1

σ3

σ2

Plane 6Plane 5

Plane 4Plane 3

45° 45° 30°

60°

120°

Figure 1: Six possible fault planes within a single stress field

Applications (45%)

A deep mine in the Val d’Or area extends to 2 km below ground surface. The density of thegreenstone in the area is 2800 kg/m3. From borehole experiments, the engineers were able todetermine the orientation of two planes of maximum shear stress: 150/45 SW and 150/45NE. Themaximum shear stress was found to be 12 MPa with a normal sense of shear.

9. What is the lithostatic stress in the mine? Is the lithostatic stress the maximum,intermediate, or minimum principal stress? (5%)

10. What are the magnitudes and orientations of all three principal stresses? (5%)

11. Rock mechanics experiments have shown that greenstone has a cohesion of 18 MPa and acoefficient of friction of 0.6. Assuming the rock is homogeneous with no fractures, use aMohr diagram to illustrate the current state of stress in the rock. (5%)

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12. A new horizontal tunnel is proposed which would cross an ancient fault zone. Fluid flowingthrough the fault in the distant past has altered the fault rock material to talc, which has acohesion of 10 MPa and a coefficient of friction of 0.2. If the fault slips, the mine wouldbecome dangerously unstable. Add these conditions to your plot. (5%)

13. Is the talc-bearing fault close to failure at the present time? (5%)

14. If the tunnel is constructed, this will relieve the horizontal stress on the fault (horizontalstress will drop to zero). Will this destabilize the fault and cause it to slip? Discuss therange of possible effects, referring to the Mohr diagram to illustrate your answer. (10%)

15. A cone-shaped tailings pile of greenstone dust is sitting on the mine property. Unlike solidrock, the dust pile has no cohesion. If the angle of repose (φ) is defined as tan−1(µ), howhigh can the pile be built to sit within a circular area 500 m wide? (5%)

16. A bulldozer pushes tailings into the large pile. As it does so, thrust faults develop in the pilein response to the horizontal compression. What dip would you expect these thrust faults tohave? (5%)

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