Geology for Engineers
Fall 2003
Assignment: PS-3 (200 points)
Due:
All questions from this
problem set come from your textbook (West, 1995, chapter 6, questions 2, 3, 5,
6, 8, 9, 10, and 11). Each question is
worth 25 points. The questions are
written out below.
1. (6-2) Two triaxial compression tests were performed on two prepared
samples of the same limestone. At a
confining pressure of 2000 psi, the first sample
failed at 9000 psi.
Then at a confining pressure of 5000 psi, the
other sample failed at 21,000 psi. a)
Plot the two test results on a shear stress-normal stress diagram and
draw the Mohr envelope. b) determine the phi value for the limestone. c) what is the So
value? d) What was the shear stress on the failure
plane at failure for both tests?
2. (6-3) Limestone similar
to that tested in problem 1 exists in a mountainous region at a depth of 600
feet. a)
If the vertical stress is the confining stress of the rock at that
depth, what is this value in psf and psi? Assume a unit
weight of 160 lb/ft3 for the rock.
b) At this confining pressure,
what horizontal stress would be required to cause a shear failure in the
rock? Which is s1 and
which s3? Use the plot constructed in problem 1 to
obtain the answer. What is the shear
stress on the failure plane?
3. (6-5) A core sample of
basalt, 4 inches long and 2 inches in diameter, was tested in unconfined
compression. The basalt is known to have
a modulus of elasticity = 9.1 x 106 psi. a) At
a pressure of 10,000 psi, what would be the reduction
in length of the sample? b) At this pressure, what would be the vertical
load on the sample? c) At 10,000 psi
stress the core showed an increase in diameter of 5.5 x 10-4
inches. What is the Poisson’s ratio for
this sample?
4. (6-6) A concrete gravity
dam with a trapezoidal shape as show on page 99 of your text will be founded on
sedimentary rock. The concrete has a
unit weight of 150 lb/ft3.
a) What is the average pressure
exerted on the rock foundation in lb/ft2? What is the maximum pressure? Explain.
b) Refer to the different
sedimentary rocks listed in table 6.1 of your text (p. 83). Using average compressive strength and shear
strength values for these sedimentary rocks, compare this to the average
pressure calculated in part a above. Are any so low as to be of concern? Explain.
c) Assume that the pressure from
the weight of the dam is dissipated within 100 feet below the rock surface and
the full pressure acts over this distance.
How much settlement will occur in the center of the dam because of the
gravity force for the different sedimentary rocks listed in Table 6.1, using
average values.
5. (6-8) An 1100-foot long,
horseshoe-shaped tunnel is to be driven through massive granite with a unit
weight of 170 lb/ft3. The
dimensions of the tunnel cross section are shown on page 99 of your text. The maximum rock cover above the tunnel in
the middle of the mountain is 1200 feet.
a) What is the total overburden
stress at the midway point in the tunnel in psi and psf? b)Because of
arching effects that transfer load around the tunnel opening, the roof or crown
of the tunnel does not commonly support the full overburden stress, but instead
is typically much less. If the roof
stress was found to be 2500 psf, how many feet of
rock above the tunnel does this represent?
How do you suppose this load would be supported in the tunnel? c) If
the tunnel excavation was completed in 80 working days with two shifts per day,
how many lineal feet of tunnel were excaved per
day? How many feet per shift? What was the average volume of rock removed
per day, per shift? (Hint: Calculate the area of the corss
setion).
d) The tunnel project was bid at
$1,878,000. What would be the cost per
lineal foot of the tunnel?
6. (6-9) An arch dam
transfers much of the water load onto the abutments (the rock mass on the sides
of the dam) [see drawing on page 99 of your text for details]. The E value for the concrete is 6 x 106
psi. a) What would occur if the abutments were shale
with an E of 3 x 106 psi? b)
What if the abutments were diabase with an E
of 14 x 106 psi? Which of the two rocks would be preferred,
the shale or the diabase? Explain why.
c) What would be the effect if
the thrust (force direction from the dam) were parallel to the slaty cleavage direction in the abutments? What could be a conclusion of this
arrangement? d) What type of force is preferred within the
arch dam, compression or tension?
Explain why. Which type of force
would develop if the abutments had a high modulus of elasticity?
7. (6-10) For room and pillar
underground mines, the pillars must carry the total weight of the
overburden. a) Why does the arching effect that acts in
tunnels not contribute any support for these mines? b) If
a coal mine is 500 feet deep, can 40% of the coal be removed and still yield a
factor of safety [FS] of 3 relative to compressive strength of the coal in the
pillars? Assume an unconfined
compressive strength of 3000 psi for the coal and use
the equation FS = [rock strength (1-fraction removed0/rock load]. Hint:
Use a unit weight of rock = 160 lbs/ft3. What is the FS? c)
What percent of the coal can be removed at a depth of 800 feet using an
FS = 3 and the same compressive strength as in part b above?
8. (6-11) A massive hillside
of rhyolite contains joint planes that dip at an
angle of 40o toward a stream valley.
A rectangular-shaped rock bounded by other joint planes rests on the
dipping joint plane. The block measures
5 feet high, 15 feet wide, and 10 feet long.
The unit weight of the rhyolite is 172 lb/ft3. Assume a dry slope. a)
What is the weight of the rock block?
b) Assuming that only friction
holds the block in place (no cohesion along the joint plane), what is the
minimum frictional force needed to resist sliding? C)
What coefficient of friction is needed to produce this minimum
force? d) What would be an actual coefficient
of friction for this rock block, that is, rhyolite
sliding against rhyolite? A friction angle is commonly used, the
coefficient of friction being equal to the tangent of that angle. Determine that angle of friction.