Questions 1
Questions 1
Questions 1
5:
introduction
Q 1 .4 Why do you think that the techniques used in rock mechanics for
site characterization, analysis and modelling are not the same as those
used in soil mechanics?
Q2. l The picture in Fig. Q2.1 shows a limestone slope above a highway
in Spain. Comment briefly on the geological factors that could influence
rock slope stability at this location.
Q2.2 The picture in Fig. Q2.2 shows the surface of a fault in a hard rock
aggregate quarry on which a rock slide has occurred. Explain
(a) why the existence of this fault could indicate that other similar
features will be encountered as quarrying continues, and
(b) why encountering an adverse geological feature such as this is likely
to be less significant in a quarry than in a road cutting.
Q2.3 The picture in Fig. Q2.3 shows tooth marks from the bucket of a
mechanical excavator in the Carboniferous rocks of a near-surface slope
figure Q2.1
404 Questions 2. 1-2. 10: geological setting
Figure Q2.2
Figure Q2.3
Figure Q2.4
406 Questions 2. 1-2. 10: geological setting
Q3. 1 Show how the stress state in a solid can be described via the stress
components (normal and shear) on an elemental cube of rock. Also,
show how these components are listed in the stress matrix. What do the
components in a row of the stress matrix have in common? What do the
components in a column of the stress matrix have in common?
Q3.3 Explain the differences between scalar, vector and tensor quantit
ies. Why is stress a tensor quantity?
Q3.4 How are normal and shear stress components plotted on Mohr's
circle?
Q3.6 What are the following stress states: uniaxial stress, biaxial stress,
triaxial stress, polyaxial stress, pure shear stress, hydrostatic stress?
Q3.7 Show how to add two tensors and hence how to calculate the
mean of n stress states. How would you calculate the mean of n different
stress states which were specified by their principal stresses and the
associated principal stress directions?
Q3.8 What are the first, second and third stress invariants?
to find a cube orientation such that the shear stresses disappear on all
faces and only normal stresses (the principal normal stresses) remain.
Is it possible to find a complementary orientation such that the normal
stresses disappear on all faces and only shear stresses (i.e. principal shear
stresses) remain? Explain the reason for your answer.
Questions 4.1-4.10:
in situ rock stress
o 30v0,
Q4.2 Add the following 2-D rock stress states, and find the principal
stresses and directions of the resultant stress state.
30 I
15MPa
f,
/ SMPa 20MPa
/\
\\
2MPa
10MPa 20MPa
jacks were cut normal to the wall of the tunnel, and were oriented
relative to the tunnel axis as shown.
The cancellation pressure for each of the flatjacks A, B and C was
7.56 MPa, 6.72 MPa and 7.50 MPa, respectively. Compute the principal
stresses and their directions, and ascertain whether they accord with
worldwide trends.
Q4.5 Two further flatjack measurements have been made in the wall of
the tunnel considered in Q4.4. These dip at 20 and 90 relative to the
tunnel axis, and produced cancellation pressures of 7.38 MPa and 7.86
MPa, respectively. Compute the best estimate of the principal stresses.
Q4.6 The stress in a granitic rock mass has been measured by the
hydraulic fracturing technique. Two tests were conducted in a vertical
borehole: one test at a depth of 500 m, and the other test at a depth of
1000 m. The results were as follows:
Depth Breakdown pressure, P8 Shut-in pressure, r
(rn) (MPa) (MPa)
500 14.0 8.0
1000 24.5 16.0
Given that the tensile strength, a1, of the rock is 10 MPa, estimate and list
the values of a1, a2 and a3 at the two depths.
State all of the assumptions you have to make in order to produce
these estimates. Are any of them doubtful? State whether the two sets
of results are consistent with each other, and justify your reasons for the
statement. Are the results in agreement with trends exhibited by collated
worldwide data?
Q4.8 Suppose that we have measured the pre-existing stress state in the
ground by some means and that the results are as follows:
a1, magnitude 15 MPa, plunges 35 towards 085;
a2, magnitude 10 MPa, plunges 43 towards 217";
os. magnitude 8 MPa, plunges 27" towards 335.
Find the 3-D stress tensor in the right-handed x y z
Right-
co-ordinate system with
handed
x, horizontal to the east, co-ordinate
y, horizontal to the north, system
z, vertically upwards.
Q4.9 A fault is present in the same rock (continuing from Q4.8) with
an orientation of 295/50. Determine the stress components in a local
co-ordinate system aligned with the fault. Assume for this question that
the presence of the fault does not affect the stress field.
Questions 4. 1-4. 10: in situ rock stress 411
Q4.10 The plan below shows a horizontal section through a rock mass.
Stress measurements were made from the gallery along the borehole
line Ox using the Japanese CCBO technique. The measured principal
stresses in the horizontal plane are plotted on the plan. What are the
main conclusions that you can draw from the stress variations?
Fault Ill
20 MPa
Stress scale
O 10 m
Questions 5.1-5.10:
strain and the theory of
elasticity
QS.1 What is the meaning of the first stress invariant and the first strain
invariant?
QS.3 Draw a Mohr circle for strain, indicating what quantities are on
the two axes, how to plot a 2-D strain state, and the location of the
principal strains, c1 and c2
QS.4 Show why the shear modulus, Young's modulus and Poisson's
ratio are related as G = E /2(1 + v) for an isotropic material. This
equation holds for an isotropic material but not for an anisotropic
material - why? Hence explain why five elastic constants are required
for a transversely isotropic material rather than six.
QS.5 (a) How can the strain in a particular direction be found from the
strain matrix components and hence how can a strain gauge rosette be
used to estimate the state of strain at a point, and hence the state of stress
at a point?
(b) Assume that strains measured by a strain gauge rosette are
cp = 43.0 x 10-6, CQ = 7.8 x 10-6 and CR = 17.0 x 10-6, and that the
gauges make the following angles to the x-direction: f)p = 20, f)Q = 80
and f)R = 140. Determine the principal strains and their orientations and
then, using values for the elastic constants of E = 150 GPa and v = 0.30,
determine the principal stresses and their orientations.
414 Questions 5. 1-5. 10: strain and the theory of elasticity
QS.6 Explain clearly why an isotropic rock has two independent elastic
constants, a transversely isotropic rock has five independent constants
and an orthotropic rock has nine independent constants (compared to
the general anisotropic case where there are twenty-one independent
elastic constants).
QS.8 (a) At the time of writing this book, most elastic analyses that have
been conducted for rock engineering design purposes have assumed that
the rock is perfectly isotropic with two elastic constants. Why do you
suppose that is, given that most rock masses are clearly not isotropic?
(b) Conversely, no elastic analysis for rock mechanics has been con
ducted assuming that the rock mass is fully anisotropic with 21 elastic
constants? Why is that?
(c) In this context, what do you think will happen in future analyses?