Three and one half hours for this supplemental examination.
You may use a Chinese-English dictionary.
1a)
For the case of an element of clay (soil) in one dimensional strain only (vertical
consolidation) show how the linear one-dimensional consolidation equation is
derived. Expressed in differential form the equation is:
where k z = coefficien
of vertical
permeability,
kz
u
 2t u
where c v =
= t of consolidation,
c=v coefficien
2
t
mv w  z
b)
Estimate the total vertical consolidation settlement of the clay layer in metres if a
vertical surcharge (v) is applied as shown
v = 25 kPa
sand
clay
sand
The sand has a total unit weight of 20 kN/m3. The average void ratio of the clay is
1.1, it has a total saturated unit weight of 18.2 kN/m3, and the compressibility
characteristics are plotted on the next page.
c)
What would be the effect on the time required for consolidation of the clay if the
base layer were impermeable shale instead of permeable sand?
3
2.
A retaining wall in dry sand is shown after the wall has moved outwards, in the
direction shown, causing the sand behind the wall to be in the Rankine active earth
pressure state (Ka) and the sand in front of the toe to be in the passive state (Kp).
a)
Given that the friction angle, ', of the sand is 30, calculate the horizontal stresses
on the wall because of the sand behind and in front. Assume that the dry density
of the sand is 15.5 kN/m3.
b)
Sketch a distributed force diagram for the stresses applied to the front and to the
back of the wall, assuming now that there is water to the top of the sand behind
the wall, and assuming that the interface between the sand and the wall is very
smooth (i.e. no wall friction).
c)
What steps would you take as an engineer to increase the factor of safety of the
wall if there is concern as to its stability in sliding along the base as well as the
stability with respect to overturning?
4
3.
In a rock mass with a major fault plane, the total stresses are as shown in the
diagram.
1 = 40 MPa
1 = major principal total stress
3 = 20 MPa
3 = minor principal total stress
 = shear stress along plane
n = normal stress along plane
a)
Calculate the shear stress  and the total normal stress to the plane n.
b)
Plot this stress state on a Mohr Circle, clearly showing where 1, 3, n,  are found
on this circle.
c)
Assume that the fault has a slip criterion (yield or failure criterion) of
f = 1.0 MPa + 0.5 n'
Where n' = n - u = effective normal stress on the plane and f = shear stress at
slip initiation (yield point). Plot this failure criterion on your Mohr Circle Plot.
d)
For the conditions shown in the figure, what pore pressure (u) will be necessary
just to initiate slip? (You may solve this graphically or analytically.)
5
4a)
Given the following data plot the total and effective vertical (v, v') and lateral (h,
h') stresses versus depth for this column of rock. In the Table, Ko' = h'/v' is the
effective lateral earth pressure coefficient.
------------------------------------------------------------------------------------------------------Depth & Thickness
Rock Type Unit Weight of Rock
Ko'
m
kN/m3
------------------------------------------------------------------------------------------------------0-100
100-500
500-600
600-1200
1200- 1600
1600- 2000
2000- 2100
sandstone
shale
limestone
sandstone
shale
sandstone
salt
21
26
26
23
27
24
21
1.7
1.3
1.6
1.2
1.0
0.8
1.0
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Depth
Measurement of Pore Pressure
m
MPa
----------------------------------------------------------------------------------100
600
1000
1500
0.5
5
10
18
------------------------------------------------------------------------------------
Answer the following questions, based on this plot and based on your knowledge of
rocks and mechanics.
b) Is it possible to define a pore pressure in salt (halite) which has a porosity of 0.1% (i.e.
almost zero porosity)?
c) At what depths would you expect induced horizontal fractures to be vertical?
horizontal?
d) Sandstones are permeable, shales are not. What value of
dp
= pore pressure gradient with depth
dz
would you expect in a sandstone? Would you expect this in a shale?
6
5a)
In deciding, as an engineer, whether to use an elastic analysis or an elastoplastic
analysis for a deep underground structure in a jointed rock mass (for example, a
tunnel underneath a mountain pass), what factors would you consider? (For
example: if temperature is low, rock is more likely to behave elastically than if
temperature is high. Thus, depending on rock type, if temperature is high, I may
have to use an elastoplastic analysis.)
Consider all aspects of this question including:
- initial state in the rock mass
- geometry of opening
- rock type, structure
- process to be implemented in the structure
5b)
Define, using appropriate sketches with the axes correctly labelled, the following
behaviour modes:
- brittle and ductile behaviour
- strain-weakening behaviour
- stress-dependent stiffness behaviour
- dilatancy suppression with increased normal stress
- a plasticity surface in a stress-space diagram
- the normality rule for associated plastic strain increments
- viscous behaviour
- ideal elastic followed by ideal plastic behaviour
7
6. Using the Limiting Equilibrium approach for the forces on the potential slip plane,
derive an expression for the Factor of Safety for this unit width slope in a rock mass:
 = slip plane angle
 = slope angle
 = height of slope
m = total unit weight of rock material
w = unit weight of water (9.8 kN/m3)
Pore pressure at D = w_H/2
Pore pressure at A and C = 0.0, and otherwise, the pressure is distributed linearly
between A-D & C-D (i.e.: a straight-line pressure distribution).
Slip (Yield) Criterion for the slip plane shown
Sf = c'L + N' tan'
where Sf = shear force at failure
c' = unit cohesion along slip plane
L = length of slip plane (A-D-C)
N' = effective normal force on slip plane
' = friction angle on slip plane
8
7. Hooke's Law for linear thermoelasticity can be written as:
i =
where
1
(  i  - v(   j  +  k  )) - 
E
i = compressive strain in the i direction.
i' = stress change (in effective stress terms!)
 = temperature change
i,j,k = x,y,z cyclic
E,,= elastic parameters
- Young's Modulus
- Poisson's Ratio
- Coefficient of thermal linear expansion
Given the case sketched below for a deep, laterally extensive and relatively thin
reservoir, the following boundary conditions can be assumed:
x, y, z = principal stresses
x, y = 0 (no lateral strains allowed)
z = 0 (for z = constant, i.e. total vertical stress remains the same).
Also, from the theory of porous media, we can assume that pore pressure is a scalar,
and that the effective stress law can be written in the form of stress and pressure
changes:
i = i' + u (or [] = ['] + [u]
where u = pore pressure change
i' = change in principal effective stress in direction i
i
= change in principal total stress in direction i
a) Derive an expression that predicts vertical strain as function of change in pore
pressure u and change in temperature .
b) Assuming u = 0 (pore pressure constant in the reservoir), derive an expression for
x' and y' (they are always equal) if there is a change in temperature .
c) Discuss what might happen to a reservoir if you change the temperature dramatically,
by injecting cold water, or by injecting steam (very hot!).