2023 Soil Mechanics I and Exercises [Final Exam]
January 30, 2024 (Tue.) 13:15–15:15
Notes:
The examination consists of four questions for which you are provided with four answer sheets andone diagram.

Write down your name and student ID number on every sheet including the diagram sheet. Notethat any sheets without your name may not be graded.

Use one answer sheet per question and answer them in sequence, starting from [Question 1]. If
the front page of an answer sheet is insufficient to complete your answer, continue your answer onthe back side of the same sheet of paper. Be careful not to spread the answer to a single question
across multiple sheets of paper. Note that answers given over multiple sheets of paper may not begraded.

Wherever necessary, specify the units in your answers.

In addition to personal writing instruments, use of non-programmable calculators and rulers arepermitted. However, programmable calculators and calculator functions of mobile phones areprohibited.

Any attempt at cheating on the exam will result in failed credit of the course and serious penalties.

- 1 -[Question 1] Answer the following questions.
(1) Consider the construction of an embankment by collecting soil from a nearby excavation site for residential landdevelopment. Assuming that the density of water is 1.00 Mg/m3, answer the following questions.
1) To determine the physical properties of the soil, a soil sample with a volume of 4.00 m3 is collected from theexcavation site, and its mass is measured to be 8.00 Mg. The water content of the soil is 18.0%, and the soil
particle density is 2.50 Mg/m3. Find the wet density, dry density, void ratio, and degree of saturation of thesoil in its natural state at the site.
2) An embankment with a height of 2.00 m and an area of 2.00×104 m2 is constructed by compacting the soil
collected from the excavation site while adding water by sprinkling. If the target dry density is 1.50 Mg/m3,
find the volume of soil in its natural state that should be taken from the excavation site.
3) Find the mass of water to be added by sprinkling to make the degree of saturation of the embankment 80.0%
after compaction. Also, find the water content and air content (i.e., air volume / total volume) of theembankment at this degree of saturation.
4) State whether it is possible to compact the soil to a dry density of 1.80 Mg/m3 while maintaining the watercontent in 3) and give reasons for this. You may use a diagram if necessary.
5) Attempting embankment construction under conditions 2) and 3), it has been found that the dry density ofthe embankment is not uniform in the vertical direction; the dry density of the upper layer above a certain
height is 1.42 Mg/m3, while the dry density of the lower layer below the height is 1.55 Mg/m3. Find theheight of the boundary between the upper and lower layers, as well as the water content and degree ofsaturation in each layer. Assume that the mass of water contained in the upper and lower layers is equal.

(2) Briefly explain the following terms, giving definitions with mathematical expressions.
1) Relative density2) Coefficient of uniformity
3) Plasticity index- 2 -[Question 2] Answer the following questions about the seepage flow in the soil.
(1) A permeability test was conducted in the horizontal direction on a specimen consisting of two layers of different
soils: Soil 1 (thickness H1 [cm], coefficient of permeability k1 [cm/s]) and Soil 2 (thickness H2 [cm], coefficientof permeability k2 [cm/s]) (see Figure 2-1). Answer the following questions. Note that the boundary between thelayers is horizontal, and unit depth (perpendicular to the page) is considered in all questions.
1) Explain Darcy’s law.
2) With H1 = 10 cm and H2 = 20 cm, a permeability test was conducted on the specimen by applying 15 cm
head difference between the left and right sides. During the test, 27.0 cm3 of water flowed out from thedownstream side in 10 minutes. Calculate the value of the equivalent horizontal coefficient of permeability
kh, considering Soil 1 and Soil 2 as a single layer.
3) Next, with H1 = 25 cm and H2 = 5 cm, a permeability test was conducted on the specimen by applying a 20cm head difference between the left and right pipes. During the test, 54.0 cm3 of water flowed out from thedownstream side in 10 minutes. Together with the results of the permeability test in question 2), calculate
the values of the coefficient of permeability k1 and k2 for Soil 1 and Soil 2, respectively.

hQ

Soil 1
k1

H1

Soil 2k2

H230 cm

Figure 2-1(2) Consider an earth dam of a reservoir constructed on a horizontal impermeable layer as shown in Figure 2-2.
Assume that the water level of the reservoir reaches up to 12 m height from the impermeable layer, and a drainage
system is installed at the foot of the downstream side of the dam, which is in a steady state. Answer the followingquestions.
1) In Figure 2-2, the phreatic surface and streamlines (flow lines) from the reservoir to the drainage system aredepicted. Explain the criteria that should be met when drawing a flow net. Then, complete the flow net byadding equipotential lines (draw them on the diagram provided).
- 3 -2) Given that the daily leakage rate per unit depth (perpendicular to the page) from the inner side of the dam
was measured to be 0.58 m3 and the coefficient of permeability of the soil can be approximated by Hazen's
formula k = 70D2 [cm/s] (D: effective grain diameter [cm]), calculate the effective grain diameter of the fillmaterial constituting the earth dam. Then, based on this, determine which of the grain size distribution curves(a), (b), or (c) in Figure 2-3 is expected to represent the fill material. Note that the reference plane forelevation head is the top surface of the impermeable layer, and the total (piezometric) head is assumed to be
zero along the boundary between the fill and the drainage system.

Reservoir

Downstream side
Phreatic surface

Drainage system
Impermeable layer

Streamlines

Percentage
finer by weight
通過質量百分率[%] [%]Figure 2-21009080706050403020
1000.001(a)(b)

0.010.1

1

Grainsize[mm]
粒 径[mm]Figure 2-3- 4 -(c)10100[Question 3] Answer the following questions.
(1) As shown in Figure 3-1, there is a ground consisting of a 6 m thick sand layer and a 10 m thick clay layer on animpermeable bedrock. Here, the saturated unit weight of the sand layer below the groundwater table, sat, is 20kN /m3, and the wet unit weight of the sand layer in the area where the groundwater table drops, t, is assumed
to be 19 kN /m3, which is the same as that of the soil above the groundwater table. In addition, the saturated unit
weight of the clay layer is 17 kN / m3, the initial void ratio, e0, is 1.5, the coefficient of compressibility, mv, is 8.0
× 10-4 m2/kN, and the coefficient of consolidation, cv, is 50 cm2/day. Furthermore, the unit weight of water is w= 10 kN/m3.
1) Find the effective stress at the bottom of the sand layer when the groundwater level is 1m from the groundsurface.
2) Pumping up groundwater lowered the groundwater level to 4m from the ground surface. At this time, findthe effective stress at the bottom of the sand layer.
3) The clay layer will consolidate and settle due to the above groundwater pumping. Find the volumetric strain
and final settlement due to this consolidation.
4) Find the time it takes for the degree of consolidation to reach 90 %. Note that the time factor is 0.85 when
the degree of consolidation is 90 %.
5) After that, when pumping of groundwater was stopped, the groundwater level rose to the ground surface. At
this time, find the overconsolidation ratio at the top surface of the clay layer. The overconsolidation ratio is
the ratio of the consolidation yield stress (the maximum stress that has been applied to the clay in the past)
to the effective overburden pressure (the stress that the clay is currently experiencing).

Figure 3-1(2) Answer the following questions regarding consolidation of clay.

- 5 -1) Answer whether the following statement is true or false as a prerequisite for one-dimensional consolidation
theory.
(a) The soil is assumed to be fully saturated, and the soil pores are completely filled with water.
(b) It is assumed that consolidation is also important in the horizontal direction, and horizontal deformationis also taken into account.
(c) The soil particles themselves are assumed to be compressed, and consolidation occurs throughcompression of both soil particles and pores.
(d) The hydraulic conductivity is assumed to be constant throughout the consolidation process.
(e) The flow of pore water is assumed to follow Darcy's law, and the flow of water is laminar.
(f) The stress-strain relationship in soil is nonlinear, and changes in stress are not proportional to changes instrain.
(g) The soil is assumed to be homogeneous and isotropic, and the physical and mechanical properties areconstant within the clay layer and independent of direction.
2) Derive Terzaghi's consolidation theory using the following four equations. Also, express the relationshipbetween the consolidation coefficient Cv and other soil parameters by equation.
𝑣=−𝑘 𝜕𝑢𝛾𝑤 𝜕𝑧𝜕𝑣 𝜕𝜀=𝜕𝑧 𝜕𝑡𝑑𝜀 = 𝑚𝑣 𝑑𝜎 ′𝜕𝜎 𝜕𝜎 ′ 𝜕𝑢=+=0𝜕𝑡𝜕𝑡𝜕𝑡

In above equations, v is the flow rate of pore water, k is the hydraulic conductivity of the target clay layer,

w is the unit weight of water, u is the pore water pressure, z is the depth from the ground surface, t is thetime, is the vertical strain, mv is the coefficient of compressibility, and ' is the effective stress.

- 6 -[Question 4] Answer the following questions.
(1) The stress state at a certain point in the ground is given in Figure 4-1.
1) Draw a Mohr’s stress circle and find the maximum and minimum principal stresses.
2) Find the direction of the major principal plane (i.e., the plane where the maximum principal stress acts).
3) Find the normal and shear stresses acting on plane A-A'.
240

A'60
8060

Unit: kN/m2
80

4560

oA60
240

Figure 4-1̅̅̅̅) triaxial compression test was conducted on a saturated normally consolidated(2) A consolidated undrained (CUclay under lateral pressure 3 = 200 kN/m2. Failure occurred when axial pressure 1 = 500 kN/m2 and pore waterpressure uw = 50 kN/m2 were reached.
1) Draw the Mohr’s stress circles for total stress and effective stress at failure.
2) By assuming that the Mohr-Coulomb failure criterion is satisfied, determine the internal friction angle ' ofthis clay based on the effective stress. Note that cohesion c' = 0 kN/m2 can be adopted for normallyconsolidated clays.

(3) Answer the following questions regarding shear tests for sand and clay. Graphics and equations may be used ifnecessary.
1) Explain how to determine the internal friction angle  and cohesion c of a soil from the direct shear test.
̅̅̅̅) triaxial compression tests were conducted on a2) Consolidated drained (CD) and consolidated undrained (CUsaturated dense sandy soil, with the latter having greater shear strength. Discuss the reasons for this, focusing
on the dilatancy of the soil.

3) Explain why consolidated drained (CD) triaxial compression tests are generally not conducted on clays.
4) Explain the sensitivity ratio obtained by the unconfined compression test.

- 7 -