Earth Surface Processes, GEOL 4880 Humphrey, fall 2022, Homework #2, due next Thursday
Here are 4 questions to work on. Question 4 is for you to think about, do not feel bad if you can’t answer it, but with clear thinking it should be understandable.
Please be careful… the homework gets posted just before it is assigned. Links to homework further on in the course lead to old homework. I change some questions, and also change some of the data each year, so you will be doing the wrong homework. The due date has to be this years for the homework to be valid.
Question 1 is a standard slope stability question of the sort that some parts of might appear on an exam (hint). Questions 2 and 3 require short calculations, question 4 only requires careful thinking. Clear thinking will hugely reduce the amount of work that you need do. Here is a summary table of most of the equations used so far, please use it and compare with your notes (Note this table is also available on the bottom right of the main web page).
1 Time to do some slope stability
calculations for which you will need our stability equations (see the web
pages). We will look at a small slope failure, modelled on this failure above Centenial, WY.
Here are data for the
slump. Slope is 0.47, internal angle of
friction for the slope is 30 degrees. Dry soil density is 1600kg/m3,
density of water 1000kg/m3, and the porosity is 0.4. The bulk
cohesion is very little on this lightly vegetated slope, at about 103Pa. The soil is about 2m deep and the potential
slide surface is at the soil/bedrock interface.
a) Calculate the factor of safety for
the dry slope?
b) After the spring snow melt, and a
heavy rainstorm, the water table in the soil on the slope rises to about 1m
depth. Find the water pressure at the bedrock/soil interface (in Pascals)?
c) Calculate the effective normal
stress controlling the friction at the slide surface.
d) Now calculate the frictional
resistance on the slide plane, along with the driving stress, and calculate the
factor of safety for this slope to see why the slump occurred.
2
(This may be hard for some of
you). In class, I claimed that for a dry
slope failure, the H/L ratio for a sliding block (or for a landslide) is the
tangent of the angle of repose (Φ), independent of the failure angle
of the initial slide. Use a little algebra, and show that for any
constant slope (α) dumping onto a flat runout plane, that a sliding block
will have a similar H/L ratio. Remember, H is the vertical height of the block
above the plane, and L is the horizontal distance from directly below the
block, to where it stops (L is not the distance from the toe of the slope, see
the diagram). This is true, independent of the complexity of the path
taken by the block.
You
can do this problem 2 different ways.
The elegant way is set up the acceleration and deacceleration equations
with α and Φ for straight slopes, but without
putting in a value for α.
Then a little algebra shows that everything cancels and H/L is equal to tanΦ. You can assume no energy is lost
transitioning from the steep slope to the flat. (Note that curved slide paths
also end up with an H/L ratio of tangent of Φ, although this requires
calculus to prove.)
As an alternative, If you can’t get the algebra to work (as above), chose an angle for α more than Φ and show that the H/L for this randomly chosen slope is tan Φ.
3 Turbidity, or the opacity of water is often used as a rough estimate of the amount of sediment in ocean, lake or river water. If the sediment size is fairly constant with time and locality this works well, with some initial calibration (check out a Secci disk on Wikipedia). However the turbidity depends heavily on grain size and if the grain size varies, estimates of suspended sediment based on turbidity can have large errors. Investigate this problem by looking at the turbidity created by two different crushed rock samples mixed into water. We have actually basically done this problem in class, when we calculated how much smoke you breath in when the visibility is bad. This is attacked the same way.
Calculate the amount (volume) of crushed rock that is needed to be suspended in water to make a 1 meter cube of water 10% opaque, (in other words: so that 10% of the light hits a particle while traveling through the 1 meter cube of water). Use two different grain sizes, fine silt, and coarse sand. Hint: it is x-section area of each grain that blocks light, but the mass is due to the volume. Make the simplifying assumptions that no rock particles hide behind each other and that rock particles are cubes. Report your answer in volume of crushed rock per cubic meter to get 10% opacity for the two different sizes.
4 (part d is very difficult to do correctly, but everybody should try part a,b,c: HINT, it does not need any factor of safety or other complex calculation(!), clear thinking is required, but very little work) At Vedauwoo, a sheet of exfoliating granite 1 meter in downslope length lies on an exposed granite slope, with the slope less than the friction angle (let’s say a slope angle 20 degrees, friction angle 30 degrees). The granite slab is only a few centimeters thick and warms up and cools down with the diurnal (daily) cycle. The underlying granite is such a large thermal mass that it hardly heats or cools at all. Assume the granite slab goes through a steady 10 degree C peak to peak sinusoidal diurnal cycle. Also assume the slab is basically a rectangular block that is touching the underlying rock only at its upper and lower ends. Coefficient of thermal expansion of granite is 0.00001 per degree C (ie. dimensionally a strain per degree)
a) How much does the block lengthen each day (and contract each night)?
b) Calculate the rate of motion, in meters per year of the slab, resulting from the diurnal thermal cycle. Although this example of thermal creep is extreme, in fact, most loose materials (soils, rock debris etc.) creep downhill under the action of a large variety of thermal, water, freeze-thaw, biological disturbances etc. The various types of slow downslope motion are important erosion/transport processes and we will examine in detail later.
c) How does the angle of the slope affect the motion?
d) (VERY hard) Assume the slab touches the underlying rock along its entire length, not just at the ends, now how fast does it move? A complete answer would give a definite speed, however, a description of the details of motion is probably sufficient, but such a description should include a comment on the speed dependence on the slope angle, which is different from the answer to part c.