Geol 542: Advanced Structural Geology
Fall 2011
Problem Set #7: Coulomb Failure Criterion / Stress Tensors Applied to Planes
The Coulomb criterion for failure of intact rock may be written in terms of the Coulomb Stress S as follows:
S = |s| - in = So (at failure)
where s is shear stress, i is the coefficient of internal friction, and n is the normal stress. Using the Mohr
equations, this expression may be rewritten as:
S
1
1   3 sin 2  i cos2   i 1   3 
2
where  is the angle between the maximum compressive stress (1) and n.

1. The maximum value of the Coulomb stress Smax will occur when the rate of change of S with respect to  is
zero:
dS
0
d
Calculus alert!! Use this relationship to show that this critical value of  (called f), satisfies the condition:
 cos2 f  i sin 2 f  0 .
(5)
2. Although you don’t have to prove it, from the above result, it can be calculated that:

f 
 1 
1
 tan1  where i ≥ 0.
2 2
i 

Use the above equation to find f where i= 0.6.
(2)

3. Show that the maximum Coulomb Stress Smax at a depth of 3 km in a sandstone section having i= 0.6 and
density  = 2300 kg/m3 is -6.4 MPa. Assume that g = 9.8 ms-2. In calculating the stresses at this depth,
assume zero pore pressure and a tectonic stress component of 120 MPa that causes the maximum
compressive stress, 1, to be horizontal and 3 to be vertical. Show all calculations!
(8)
4. If the sandstone has an inherent shear strength of So = 8 MPa, why is a fault not likely to form in the
sandstone for the stress field described in question 3? How much more Coulomb stress will be needed?
(3)
5. Draw a Mohr circle to represent the conditions described in questions 3 and 4. Include the Coulomb
criterion frictional failure line for the sandstone. Use an entire sheet of graph paper in landscape format.
(8)
6. Based on your Mohr circle diagram, write a sentence or two to corroborate your answer in question 4.
(2)
Geol 542: Advanced Structural Geology
Fall 2011
7. On the same Mohr diagram, draw the frictional failure line for a pre-existing fault that has zero cohesion
and a coefficient of friction  = 0.5. Use your diagram to predict which range of fault orientations (in terms
of ) are prone to failure in the existing stress field (i.e., the range of  defined by the portion of the Mohr
circle crossing the lower failure line).
(6)
8. Will a pre-existing fault with a dip of 50° be prone to slip? Explain your answer.
(4)
9. Write out the stress tensor for the case where 11, 22 and 33 are 1, 4 and 5 MPa, respectively, 12 is -3 MPa,
13 is 3 MPa, and 23 is 2 MPa.
(2)
10. Assuming a choice of coordinate system such that the principal stresses act along the coordinate axis
directions, write out stress tensors (using the terms 1, 2, 3, and p) that would be representative of the state
of stress:
a) in a uniaxial loading laboratory test
b) in a triaxial loading laboratory test
c) for a hydrostatic stress state, p
(2)
(2)
(2)
Note: the last question can only be attempted after the Tues. Dec. 6th lecture.
11. The stress at some point in a body of rock is given by:
MPa.
Use the following sequence of questions to determine the normal and shear tractions acting on a plane defined by
a normal vector v, where [vi] = [1 3 5] m.
a) What is the unit normal vector, n = [n1 n2 n3] for this plane?
(3)
b) What are the three components of the traction vector t = [t1 t2 t3] acting across this plane?
(4)
c) Calculate the value of the normal traction, tn, using the vector dot product t • n.
(3)
d) Determine tn in its vector notation, tn where tni = [tn1 tn2 tn3], using (t • n) • n.
(3)
e) Use the vector subtraction ts = t - tn to determine the shear traction vector.
(3)
f) Now calculate the scalar length of vector ts to determine the shear stress on the plane.
(3)
g) What is the largest coefficient of friction, , allowable in order for this fault to fail?
(3)
[68]