2022

Soil Mechanics II and Exercises

2022/6/8 (Wed.)

8:45-10:15

Midterm Exam

Kyotsu 4 lecture room

Attention:
The exam consists of three questions for which you are provided with three answer sheets. Write down
your name and ID number on every answer sheet. Use one answer sheet per question and answer them insequence, starting from [Question 1]. If the front page of an answer sheet is insufficient to complete youranswer, use the back page of the same answer sheet after clearly indicating your intent.

Scores for each question are equally weighted.

In addition to personal writing instruments, non-programmable calculators are permitted. However,programmable calculators and calculator functions of mobile phones are prohibited. Any attempt atcheating on the exam will result in failed credit of the course and serious penalties.

Wherever necessary, specify the units in your answers.

[Question 1] Answer all the following questions(1) A saturated clay layer of thickness 𝐻 lies above an impermeable rock layer. Assume that the groundwaterlevel coincides with the ground surface. After the rapid application of a load over the clay layer, theconsolidation process can be analyzed based on Terzaghi’s one dimensional consolidation equation.
(1-1)

Derive Terzaghi’s one dimensional consolidation equation from the following four equations, andanswer the relationship between coefficient of consolidation 𝑐 and other parameters:
Relation between effective stress and strain:𝑑𝜀𝑚 𝑑𝜎Constant total stress condition:

𝜕𝜎
𝜕𝑡

𝜕𝜎
𝜕𝑡

Darcy’s law:𝑣𝑘Continuity equation for water:

𝜕𝑣
𝜕𝑧

𝜕𝜀
𝜕𝑡where, 𝜀 is soil strain, 𝑚Eq. (1)𝜕𝑢𝜕𝑡𝜕ℎ𝜕𝑧

0

Eq. (2)𝑘 𝜕𝑢𝛾 𝜕𝑧

Eq. (3)Eq. (4)is coefficient of volume compressibility, 𝜎 is effective stress, 𝜎 istotal stress, 𝑢 is excess pore water pressure, 𝑡 is time，𝑣 is flow velocity, 𝑘 is coefficient ofpermeability, ℎ is total head, 𝑧 is the position coordinate and 𝛾

is unit weight of water.

(1-2)

Write down the boundary condition for the upper surface of the clay layer (ground surface, 𝑧(1-3)

Write down the boundary condition for the lower surface of the clay layer (in contact with theimpermeable rock layer, 𝑧(1-4)

0).

𝐻).

The solution of the consolidation equation can be expressed using the time factor 𝑇 . Assuming thatthe initial pore water pressure 𝑢 is constant regardless of depth, explain the changes in the porewater pressure distribution with the time factor 𝑇 by drawing a schematic diagram of the solution- 1 -of consolidation equation. Make the horizontal axis represent the ratio of the excess pore waterpressure 𝑢 to 𝑢 (𝑢⁄𝑢 ), and the vertical one the ratio of the depth 𝑧 to the thickness of clay layer
𝐻 (𝑧⁄𝐻 ).
(2) The thickness of the clay layer in (1) is 𝐻 = 3 m. When a soil sample was taken from this clay layer anda consolidation test was conducted, it took 20 minutes for the 2-cm-thick specimen to reach 90%
consolidation. Find the time required for this clay layer to reach 90% consolidation.
(3) When a uniformly distributed rectangular (length of each side: 𝑎 and 𝑏) load of 𝑞 shown in Figure 1 isapplied on the surface of a clayey ground, the vertical stress 𝜎 at a depth of 𝑧 under the corner of therectangular can be given by the Eq. (5) in terms of the arguments 𝑎⁄𝑧 and 𝑏⁄𝑧 of the function 𝐼 . By
using this function, describe the vertical stress 𝜎 at point A shown in Figure 2. Herein, the groundbehavior is assumed to be linearly elastic.

𝜎𝐼𝑎 𝑏, ∙𝑞𝑧 𝑧

Eq. (5)aabb

Uniformly distributed load: qzz

z

Uniformly distributed load: q
AFigure 11b2

1 a2

z

Figure 2[Question 2] Answer the following questions.
(1) An unconfined compression test using a clay sample was conducted. The unconfined compressive strength𝑞was 120 kN/m2. An undrained triaxial compression test was also conducted using the same clay sample

at a confining pressure of 150 kN/m2.
(1-1)

In the unconfined compression test, the effective stress at the sampled depth was maintained. Give thereason why the effective stress is maintained.

(1-2)

Find the undrained shear strength 𝑐(1-3)

Draw Mohr's circles at failure for the above unconfined compression test and undrained triaxialof this clay.

compression test, respectively.

- 2 -(2) A clay sample was isotropically consolidated with a confining pressure of 150 kN/m2 and then subjected
to a consolidated undrained triaxial compression test. The deviator stress 𝑞 was 126 kN/m2 and the excess
pore pressure 𝑢 was 72 kN/m2 at failure.
(2-1)

Find the mean effective stress 𝑝′ at failure, Skempton's pore pressure coefficient 𝐴 , and the failure
stress ratio 𝑀 .

(2-2)

Find the internal friction angle 𝜙′ when the cohesion 𝑐′ can be assumed to be zero.

(2-3)

Draw the total stress path in the 𝑝𝑞 plane and the expected effective stress path in the 𝑝′𝑞plane for this test.
(2-4)

If this specimen is isotropically consolidated with a confining pressure of 150 kN/m2, followed by aconsolidated-drained triaxial compression test, determine the deviator stress 𝑞 and the effective mean𝑝′ at failure.

(3) Copy Figure 3 in your answer sheet and sketch the expected experimental results for: (a) the consolidateddrained (CD) triaxial compression tests of dense and loose sands; and (b) the consolidated undrained (CU)
triaxial compression tests on dense and loose sand samples. Assume that compression is positive. Herein,
𝜀 is the axial strain and 𝜀 is the volumetric strain.

(a) CD test(b) CU test

Figure 3- 3 -[Question 3] The back surface of a retaining wall is not necessarily vertical, but may be inclined. For ahorizontal cohesionless backfill (c = 0 kPa), answer the following questions regarding the active earth pressure
on an inclined retaining wall (α > 0º) as shown in Figure 4, where point x is the base of the wall, point y is thetop of the wall and point z is the intersection of the vertical line through x and the surface of the backfill.

yz

Virtual surface

ΔW Backfill
HRetainingwall

θαPH

Horizontal cohesionless backfill
Unit weight γ = 16 kN/m3
Friction angle ϕ = 30°
Cohesion c = 0 kPa
Retaining wall
Height of wall H = 5 m
Inclined angle of wall α = 15°
Wall friction angle δ = 20 °PAxFigure 4(1) Explain why Rankine’s earth pressure theory cannot take the inclination of the back surface of retaining
wall into account but Coulomb’s earth pressure theory can.
(2) If the vertical plane xz through the heel is taken to be the virtual surface where no shear stresses act on thissurface, the active Rankine’s earth pressure theory can be applied. Determine the horizontal active force
per unit width (PH) on the virtual surface in the backfill of height H = 5 m having the unit weight γ = 16
kN/m3 and the friction angle ϕ = 30º.
(3) Calculate the weight per unit width of triangle xyz (ΔW) for which α = 15º. There must be equilibrium
among weight ΔW and the forces across the virtual plane and the back of the wall. Obtain the magnitudeof the active resultant force on the wall (PA) and its tilt angle (θ) from the horizontal plane by finding thesum of force vectors PH and ΔW.
(4) According to Coulomb’s earth pressure theory, the resultant force PA on a retaining wall is designated by
the following equations. Herein, the wall friction angle δ is 20º.
PA 12

 H 2 K A where K A cos 2    
sin  sin     cos  cos     1 
cos  cos     22

Based on Coulomb’s earth pressure theory, determine θ and PA on an inclined retaining wall using theparameters listed in Figure 4.
(5) Compare PA and θ obtained from (3) and (4), and discuss about the benefit and limitation of the assumptionof the virtual surface.
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