2018

Soil Mechanics I and Exercises

Final Exam

2019/1/29 (Tue.) 13:00-15:00 Kyotsu 2 Lecture room
Attention:
This exam consists of four questions for which you are provided with four answer sheets. Write down yourname and ID number on every answer sheet. Use one answer sheet per question and answer them in sequence,starting from [Question 1]. If the space provided in any answer sheet is insufficient, use the back of the pageafter clearly mentioning so (for example, “continues on the back”).

In addition to personal writing instruments, rulers and non-programmable calculators are permitted, butprogrammable calculators and all types of mobile phones are prohibited. Any attempts at cheating on the exam
will result in failed credit of the course and serious penalties.

Wherever necessary, specify the units in your answers.

[Question 1]
1) A soil mass on a mountain side is shown in its original state in Figure 1. Due to a landslide, this mass became thesliding soil mass shown in Figure 2. It is assumed that the soil mass is homogeneous both before and after thelandslide. A field survey based on topographic maps and slip surface depth revealed that the original volume of thesoil mass was 1000 m3, which increased to 1200 m3 after sliding. The water content of soil samples taken from thesliding soil mass immediately after a landslide was measured to be w = 25.0%. In addition, according to the resultsof a past soil survey, it was found that soil particle density of the original soil mass is ρs = 2.50 g/cm3 (or the specificgravity of soil particles Gs = 2.50) and dry density is ρd = 1.50 g/cm3. Gravitational acceleration is g = 9.81 m/s2.
Under these conditions, determine the following quantities by assuming that the water content remains unchanged
before and after a landslide. Show calculation processes and include units whenever necessary.
(1) Wet density of the original soil mass
(2) Void ratio of the original soil mass
(3) Total weight of the sliding soil mass
(4) Wet density of the sliding soil mass
(5) Degree of saturation of the original soil just before a landslide

Figure 1 Before landslide

Figure 2 After landslide

V = 1 000 m3，ρs = 2.50 g/cm3，ρd = 1.50 g/cm3

V = 1 200 m3，w = 25.0%

2) Explain the following terms.
(1) Liquid limit(2) Maximum dry density-1-

[Question 2]
1) A pumping test well was dug all through a homogeneous and uniform permeable layer, reaching the underlain
impermeable layer, as shown in Figure 3. This pump withdraws water at a constant rate of Q (volume/unit of time).
The figure shows the steady state when the water level measured at the pumping well and observation wells isconstant. The Dupuit assumption holds that the groundwater flow is assumed to be radial, towards the test well
(axisymmetric problem). Answer the following questions.
(1) The water head at a point located at an arbitrary distance r from the test well, in the radial direction, has aconstant value of h regardless of depth. Express the water gradient i, at this point, in differential form.
(2) The flow rate Q, flowing radially at an arbitrary distance r from the test well, can be calculated using Darcy'slaw (Q = kiA). Here, k is the permeability coefficient of the permeable layer (constant), and A is the area ofthe cylindrical inflow region. Using the expression for i that you found in (1), express Q in terms of r, h,and k.
(3) Integrate (2) by the variable separation method, applying the boundary conditions ℎ = ℎ1 at 𝑟𝑟 = 𝑟𝑟1, andℎ = ℎ2 at 𝑟𝑟 = 𝑟𝑟2, and solve for Q using 𝑘𝑘, 𝑟𝑟1 , 𝑟𝑟2 , ℎ1, and ℎ2.

(4) A pumping test well, with a radius of 10 cm, was installed in a soil where the initial groundwater table waslocated at a height of 8 m from the top of the impermeable layer, reaching a steady state flow of 30 L/min.
If the height of the water table at the test well is 5 m from the top of the impermeable layer, and the influence
radius is 1000 m, calculate the coefficient of permeability of the permeable layer.

Observation
Wells

Pumping Test Well

Q

Ground Surface

No.2

No.1

dh

Initial Groundwater Table
Groundwater
Table
ℎ1
𝑟𝑟1

𝑟𝑟

𝑟𝑟2

ℎdrℎ2

Permeable
Layer

Impermeable
Layer

Figure 32) Answer the following questions based on Figure 4, which shows three conditions (a, b, and c) for the same soil.
(1) Plot the distribution of the total head, elevation head, and pressure head, for the three cases, along thevertical elevation z = 0 m to 6 m through the soil sample, assuming z = 0 m as the reference datum.
′′′(2) Let 𝜎𝜎(𝑎𝑎), 𝜎𝜎(𝑏𝑏), and 𝜎𝜎(𝑐𝑐)be the values of effective stress at point B (located within the soil sample) for

samples (a), (b) and (c), respectively. Order them in descending order, based on their magnitudes, explaining
your reasoning.
-2-(a)6m(b)SoilSample
B(c)

z=6 m

z=6 m

z=4 m

z=4 m

SoilSample

4m

B

z=0 m

z=6 m
8m

z=0 m

SoilSample
B

z=4 m

z=0 m

Figure 4[Question 3]
Due to Japan’s rapid development in the 1950s and 1960s, many large cities, including nearby Osaka, increased theamounts of water pumped from the groundwater to satisfy their industrial needs. This overpumping caused an overdrawn of the water table, which dropped up to 30 meters in different areas of the city.
After it was recognized that the overpumping was causing subsidence problems, Osaka city expanded its waterworks
and groundwater users were required to stop using groundwater, which helped with the recovery of the water table.
Although it would be impossible to travel back in time, assume that you somehow managed to get hold of an old,undisturbed, soil sample, taken from the middle of the normally consolidated clay layer described in Figure 5, froman unspecified location in Osaka city. You run a double-drained consolidation test of the cylindrical soil sample(diameter ϕ = 6.00 cm, height h = 2.50 cm) using a laboratory oedometer, and find that it takes 2 minutes and 3
seconds for it to reach 50% average degree of consolidation. The obtained coefficients of compression and swelling
1were 𝐶𝐶𝐶𝐶 = 1.20, and 𝐶𝐶𝑆𝑆 = 𝐶𝐶𝐶𝐶 . The initial void ratio was e0 = 2.00. Unit weight of water is γw = 9.81 kN/m3.
67m
33 m

Sand 150 m

Clay

Unsaturated Sand 1:
Gs=2.65, e = 0.70, Sr = 0.15 (= 15%)
Saturated Sand 1:
Gs=2.65, e = 0.70, Sr =1.00 (= 100%)Clay:
Gs=2.70, e0 = 2.00

Sand 2

Sand 2:Gs=2.67, e = 0.70
Figure 5-3-With this information in mind, answer the following questions:(1) From the laboratory test, calculate the coefficient of consolidation of the clay layer, cv.
(2) If, due to the overpumping discussed earlier, the water table dropped 25 m from the original condition
shown in Figure 5, calculate the total consolidation settlement of the clay layer. Assume for this problem
that the drop of the water table is instantaneous. For calculating consolidation, you may consider thestress and strain at the middle of the clay layer as representative of the full layer.
(3) Assume that the development of the city meant an additional distributed overburden of 100 kN/m2 over
the whole area, in addition to the one you calculated in (2). What would have been the consolidation
settlement due to the combined effects of the water table drop and the development of the city? Assume
for this problem that the drop of the water table and the application of the additional overburden are
simultaneous and instantaneous.
(4) Calculate the average degree of consolidation reached 10 years and 45 years after the beginning of theconsolidation process described in (3).
The following assumptions are valid for the whole problem:i.

All soils below the water table are fully saturatedii.

Both sand layers are incompressible and provide full drainageiii. When using Table 1, average degree of consolidation vs. time factor, if necessary, you may round thevalues of the time factor to the nearest second decimal positioniv.

Terzaghi’s theory of one-dimensional consolidation is fully applicable

Table 1

U (%)

Tv

U (%)

Tv

U (%)

Tv

00.000350.096700.403

50.002400.12675

0.477

100.008450.15980

0.567

150.018

50

0.197

85

0.684

200.031

55

0.239

90

0.848

250.049

60

0.286

95

1.129

30

0.071

65

0.340

100

∞-4-[Question 4]
Answer the following questions.
1) The stress tensor for which the state of stress of the soil element in the ground described in Figure 6 can be
expressed in matrix form byand③①. The invariants λ (Eigenvalues) obtained from the stress tensor are②. Therefore, the major principal stress iskN/m2 and the minor principal stress iskN/m2. As the major and minor principal stresses are known, Mohr’s stress circle whose center is ata radius⑤④②③with

can be drawn. This stress circle implies that the major principal plane on which the major⑥principal stress acts in Figure 6 is rotated-wise from plane B by an angle of150
Plane B

6060

⑦degrees.

Plane A60
6060
y60
150

x

Unit: kN/m2

Figure 6(1) Fill in the blank spaces no.① to no.⑤．
(2) Draw the Mohr’s stress circle corresponding to the given state of stress. In Mohr’s space, clearly indicate
the coordinate of the center, the radius, and the intersecting points between the circle and the axis σ(3) In regard to the stress circle obtained in (2), describe the state of stress (traction) acting on planes A and B
specified in the physical space.
(4) Fill in the blank space no.⑥ by using appropriate terms and determine the angle α for the blank space
no.⑦ by imposing it on the Mohr’s stress circle obtained in (2).
(5) Use the pole method and explain how to obtain the principal stresses and their principal planes (the planeson which the principal stresses act) by using graphics.

2) A consolidated undrained triaxial test was conducted on a specimen of normally-consolidated clay under saturated
condition (regarded as c’ = 0). After isotropic consolidation by applying the confining pressure σ3 equal to 100
kN/m2, the axial stress was increased in the triaxial test. At failure, the axial stress σ1 was 200 kN/m2 and the porewater pressure u was 20 kN/m2.
(1) Draw the Mohr’s stress circle for effective stress at failure (indicate the coordinate of the center, the radius
and the intersecting points between a circle and the axis σ) and determine the internal friction angle φ’.
(2) By assuming that Mohr-Coulomb failure criterion is valid, determine the state of stress (σ, τ) acting on thefailure plane of a specimen as well as the orientation of this failure plane.

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