2018

Soil Mechanics II and Exercises Midterm Exam

2018/5/30 (Wed.) 8:45-10:15

Kyotsu 4 and Kyotsu 155 lecture rooms

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, givenbelow. Here, u is the excess pore water pressure (= pore water pressure – hydrostatic pore water pressure),
Cv is the coefficient of consolidation, t is time, and z is the coordinate along the vertical direction.

u
 2u
 Cv 2tz(1)(a) Derive equation (1) from the following assumptions and conditions:v: flow velocityk uv  ki  ① Darcy’s law
k: coefficient of permeability w zγw: unit weight of water②

Continuity equation forwaterv 
z t

ε: soil strain③

Relation between
effective stress and straind   mv d  

mv: coefficient of volumetric compression
σ’: effective stress④

Constant total stresscondition   u

0
t
t
t

σ: total stress(b) Write down the boundary condition for the upper surface of the clay layer (ground surface, z = 0).
(c) Write down the boundary condition for the lower surface of the clay layer (in contact with theimpermeable rock layer, z = H).
(d) Assuming that the initial pore water pressure uo is constant regardless of depth, the solution of theconsolidation equation is given by the following equation. Explain the changes in the pore water
distribution with time by drawing a schematic diagram of the following equation. Make the horizontal
axis represent the excess pore water pressure u, and the vertical one the depth z.

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 2uM u    0 sin  z  exp   M 2Tv  H m0  MMCt2m  1
 , Tv  v2H2(2)(2) A uniformly distributed load of 100 kN/m2 is acting on a rectangular foundation of dimensions 8 m × 4 m.
Assuming the ground as a linear elastic body, find the vertical stress at a depth of 2 m just under the centerof the foundation. The vertical stress under the corner of a rectangular distributed load q can be obtainedby the following equation. In addition, the value of the influence factor Iq may be obtained from Figure 1.

 z  qI q  m, n  mn  m 2  n 2  2 q mn m a sin 1 
22z2   m 2  1 n 2  1 m 2  n 2  1
 m  1 n  1  
Figure 1. Influence Factor Iq
2/5nbz(3)[Question 2]
Unconfined compression test of a saturated normally consolidated clay sample taken from the ground layer
was carried out. The states of stress (total and effective stress) and pore water pressure at the time of failure
are exhibited in Figure 2 by Mohr’s stress circles. The symbols used in this figure are defined as follows:τ

Shear stressσ, σʹNormal stress (total, effective)σʹf, τfNormal effective stress and shear stress acting on the failure plane

σ1f, σʹ1f

Major principal stress at failure (total, effective)

σ3f, σʹ3f

Minor principal stress at failure (total, effective)ϕʹ, cʹ

Internal friction angle, cohesion

quUnconfined compressive strength

uf, Δuf

Total pore water pressure, excess pore water pressure at failureuo

Pore water pressure before shearing

ττf
uf
c′=0
σ3f=0ufϕ′σ′3f σ′fσ1f=qu∆σ1f=quuo<0Pore water pressurein the sample
before shearing+∆uf>0

σ′1fσ, σ′σ1f=σ′1f +uf=qu=

Excess pore waterpressure developed due
to unconfinedcompression test

uf=uo+∆uf σ3f=σ′3f +uf=0

Total pore water pressure
developed due tounconfined compression
test at failure

Figure 2
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The change in pore water pressure, which cannot be regularly measured in an unconfined compression test,
is considered in this problem. Assuming that the Mohr-Coulomb failure criterion is valid, answer thefollowing questions.
(1) The unconfined compressive strength qu = 160 kPa was obtained. Calculate the total pore water pressure
at failure, uf, provided that cʹ = 0, ϕʹ = 30º were obtained from a sample.
(2) Find normal effective stress, σʹf, normal total stress, σf, and shear stress, τf, acting on the failure plane atthat time.
(3) Total pore water pressure uf is considered to be a combination of pore water pressure uo before shearing
and excess pore water pressure Δuf at failure, i.e., uf = uo + Δuf. Calculate uo when Skempton’s pore pressure
parameter Af ＝ Δuf / (σ1f - σ3f ) ＝ 0.7 at the time of failure.
(4) To avoid negative pore water pressure, uo, before shearing, unconsolidated undrained (UU) triaxial
compression was conducted on the same sample under a constant confining pressure of 200 kPa. Determine
the undrained shear strength, cu.
(5) Explain why the pore water pressure before shearing, uo, is negative in the initial state of (1).

[Question 3]
Consider the retaining wall shown in Figure 3. Let the cohesion 𝑐′ of the backfill soil be zero. Assume thatthe retaining wall is smooth without friction, and that the groundwater level is sufficiently deep so that porewater pressure in the backfill soil is assumed to be zero. Answer the following questions.
(1) Briefly describe:(a) An empirical relationship between the coefficient of earth pressure at-rest and the internal friction
angle 𝜙′.
(b) Three criteria taken into account when considering stability of a retaining wall(2) Graphically show the three representative coefficients of earth pressure acting on the retaining wall. Use
the displacement 𝛿 of the retaining wall as the horizontal axis, and the coefficient of earth pressure as thevertical axis. Use as reference the direction of displacement from Figure 3.
(3) Determine the earth pressure acting on the retaining wall based on Coulomb’s earth pressure theory. In4/5

Figure 3, the weight of the soil wedge is W，the resultant force acting against the retaining wall is P, thereaction force normal to the assumed failure plane is N, and the frictional resistance tangent to the same
assumed failure plane is T. Show the force polygon diagram consisting of the forces that act on the soilwedge under the following two cases as well as describing the expression of the resultant force P in termsof W，𝜃，and 𝜙′.
(a) When the retaining wall displaces to the left side (𝛿 ) direction of the figure(b) When the retaining wall displaces to the right (𝛿 ) direction of the figure(4) Briefly explain the procedure of determining the earth thrust acting on the retaining wall from the value ofP obtained in (3), for each case (a) and (b).
(5) After experimentally confirming the ground conditions of the backfill, it was found that the cohesion, 𝑐′,was not zero. Based on Rankine’s earth pressure theory, obtain the earth pressure that acts on the retaining
wall when it moves to the left (𝛿 ) direction of the figure under this new condition.
(a) Obtain the earth pressure acting on the wall at the depth z from the surface of the backfill soil.
(b) Find the direction of the failure plane(c) Clearly show the distribution of earth pressure acting on the wall

Movements of retaining wallBackfill soil , , cHFigure 3

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