Summary of doctor thesis of engineerin: Research of earth embankment stability on natural ground

MINISTRY OF EDUCATION & TRAINING
UNIVERSITY OF TRANSPORT AND COMMUNICATIONS





Do Thang



RESEARCH OF EARTH EMBANKMENT STABILITY
ON NATURAL GROUND

Major: Highway Engineering
Code: 62.58.02.05.01



SUMMARY OF DOCTOR THESIS OF ENGINEERING

Hanoi - 2014
WORK TO BE COMPLETED IN
UNIVERSITY OF TRANSPORT AND COMMUNICATIONS


SUPERVISORS:
1: Prof. Dr Ha Huy Cuong
(Military University of Science and Technology)
2: PhD Vu Duc Sy
(University of Transport and Communications)


Reviewers 1: Prof. Dr Nguyen Xuan Truc
(National University of Civil Engineering)
Reviewers 2: Prof. Dr Nguyen Van Quang
(Hanoi Architectural University)
Reviewers 3: Prof. PhD Nguyen Truong Tien
(Vietnam Society Soil Mechnical and
Geotechnical Engineering)

The thesis is defended to the Council assessing doctoral dissertation
at the case level: University of Transport and Communications
at ….h…' ……, 2014.

Thesis can be found in the library:
1. National Library VietNam
2. University Library of Transport and Communications

LIST OF PUBLISHED WORKS

1. Do Thang (2013). “Stress field in soil is obtained by using the theory
of elastic and the theory of min (
max
)”. Vietnam Bridge and
Road Magazine. 10/2013. pp. 30 - 33.
2. Do Thang (2013). “Research of stability for vertical slope by limit
analysis method”. Review of Ministry of Construction of VietNam.
11/ 2013. pp. 103 - 104.
3. Do Thang (2014). “New method research earth embankment stability
on natural ground”. Review of Ministry of Construction of
VietNam. 6/2014.



1
INTRODUCTION
1. The reason of selecting project
Subgrade is an important part of highway. To ensure the stability of
subgrade is a prerequisite to ensure the stability of the pavement structure.

Research methodology stable subgrade is widely used in today's
designs is limited equilibrium methods. Basic equations of this method
consists of two balance equations (plane stress problem) and Mohr-
Coulomb yield condition.
However, the limit equilibrium method is not consider the
phenomenon volume change of soil when using the yield condition Mohr-
Coulomb. On the other hand, the basic equations above do not allow
determining the stress state in the yield imperfections, ie not consider the
stress state of the entire soil mass. Therefore, in the thesis "research of
earth embankment stability on natural ground" is presented below, using
theoretical min (
max
), author can apply directly limit theorem to study the
stability of the overall soil mass and stability of the embankment on natural
ground.
2. Purposes of the research
Building a new method (method directly applicable limit theorem)
evaluate soil stabilization in accordance with the actual working of the soil
environment, the study contributes to the development of stable subgrade .
Applying the above method to build a computer program, set the
tables and nomogram helps engineers quickly determine the height and
slope of embankment limited. In addition, using the lower limit theorems
of the theory of limit analysis tells us that the stress distribution in the soil
mass before ruin and slip surfaces occur in soil mass, freely given
variables appropriate enhance soil stabilization when necessary.
3. Subjects and scope of the research
Research subjects: The earth embankment on natural ground.
Scope of the research: The research of the stability problem of earth
embankment on natural background consider in the case of plane
problems.

2
4. Scientific meanings and pratice of the project
Soil is not so elastic material in plane problems, two balance
equations are not sufficient to determine the three components of stress.
Author used more conditions min (
max
) to have enough equations
determine the stress state in the entire soil mass and directly applicable
limit theorem for stability studies and natural embankment and foundation.
In the thesis presents the various stability problems: limited intensity
of the ground under load horizontally forward hard (Prandtl problem),
block slope of dry sand , steep tomorrow so natural on under Business Use
of the external ear and self-weight, trapezoidal embankment on so natural
under the effect of self-weight. Since the study was able to draw
conclusions and explanations and quantitative following:
- The yield condition Mohr- Coulomb said materials with internal
friction greater the bigger the load capacity. However, for embankment
construction materials such as soil, sand, shred lascivious the material
has a large capacity unit new headquarters is the material guarantee a
better slope stabilization. Practices embankment construction in our
country attest to that.
- Slip surface appears on the sliding surface slope and embankment
surface when external load effects.
- When study the stability of embankment only consider self weight
of the soil does not appear on the slip surface on slope and embankment
surface.
- Depending on the intensity (c, ) patch material to natural ground
which happens all cases Disruptive packing material intensity the greater
the height of the dam increasingly limited so large, increasingly large talus
slope . When embankment intensity (c, ) equal to or less than the natural

ground intensity it takes only appear at the foot sliding embankment
slopes, embankments When intense than natural background is ingrained
into the sliding surface nature.
- The calculation and comparison shows embankment height limit
under the author’s method approximates the heights rebate under sliding
surface methods (using a safety factor greater than 1). This is explained by
the method of sliding surface gives the upper limit of the height of the
embankment .
3
Author has developed a computer program, set the tables and
nomogram helps engineers quickly determine the height and slope of
embankment limited . Also, from the graph contours plastic flow rate will
determine the net slip surfaces should be able to come up with measures
for reinforcing appropriate place to raise the roadbed stability when needed
5. Layout of thesis
The thesis includes the following sections and chapters:
- Introduction
- Chapter 1: Overview of research of earth embankment stability on
natural ground
- Chapter 2: Facility theory to research stabilize earth embankment
stability on natural ground
- Chapter 3: Fundamental problem about limit load and slope
stability
- Chapter 4: Research of stability of soil mass with vertical slope
- Chapter 5: New method to research stability problem of the earth
embankment on natural ground
- Conclusions and Recommendations
- The appendix
6. New contributions of the thesis
1- Different from traditional methods of soil mechanics, author uses

theory min (
max
) to be able to directly apply the method limit analysis to
research earth embankment stability on natural ground (not given stress
state or shape of slip line). Use the lower bound theorem of the theory of
limit analysis gives us the stress distribution in the soil before failure and
found slip line field, that we can take appropriate measures to improve
stability when needed.
2- Different from traditional methods is research methods separate
slope and bearing capacity of the natural, author built overall stability
problem of the embankment so natural to be able to study the impact
between them.
3- The soil stability problems presented in the thesis is correct on
mechanics, mathematics and strict new. In terms of mathematical is that
the non-linear programming problem because constraining is the yield
4
condition Mohr- Coulomb. The solution method is a method of finite
difference and to optimize the use of available content, author programmed
on Matlab's software to solve. Difference schemes for the solution of the
thesis results with high accuracy, such as Flamant problem with some,
limited slope angle of internal friction materials that do not intend to use
ice internal friction angle of the material, load within the limits of medium
to steep tomorrow theoretical formula (this result is also new), etc
4- Research methods stability problem of the earth embankment on
natural ground is presented in the thesis in a new method. Author has
developed a computer program, set the tables and nomogram helps
engineers quickly determine the height and slope of embankment limited.
Also, from the diagram contours plastic flow capabilities will determine
the slip line field should be given the appropriate measures for reinforcing
the right position to raise the embankment stability when required.

Chapter 1
OVERVIEW OF RESEARCH OF EARTH EMBANKMENT
STABILITY ON NATURAL GROUND
This chapter presents the research of earth embankment stability on
natural ground has been applied in Vietnam and countries around the
world. Next, the authors analyze strengths and weaknesses and the
existence of that approach. Finally presentation objectives and content of
the research thesis.
1.1. Analysis of related research in the country and abroad
1.1.1. The unstable form embankment on natural ground
According roads design standards TCVN 4054-2005 [7] and its
foundation to ensure stability, maintaining the geometric size, have
sufficient strength to withstand the impact of vehicle and load factors
nature during use. Therefore, the embankment must not be phenomena
such as slope sliding, sliding part up on the slopes, slip surfacing,
embankment subsidence on soft soil
1.1.2. Research methodology stable subgrade
Soil material is complicated, we do not know the full range of
mechanical and physical characteristics of it. However, soil samples
studied in the laboratory experiments and pressure plate at the scene
5
showed that the land can be considered as ideal materials comply wizened
the yield condition Mohr- Coulomb [34] to be able to use limit equilibrium
method or the more general theorem of limit analysis to study the stability
of the soil mass. So, in this section, before the introduction of the research
methodology stable ground, the author presents the basic contact ideal
plastic materials.
1.1.2.1. The basic contact of elastic perfectly plastic material
There are many different mathematical models to establish the
relationship between stress and deformation of plastic material. So far, the

researchers have agreed to use the model determines the speed plastic
deformation according to the following equation [35], [36],[40], [41]:
(1.9)
where: is a scalar proportionality factor;
≥ 0 if f = k and
'
f
= 0 (k is a yield limit);
= 0 if f < k or f = k and
'
f

< 0.
Relation (1.9) shows the dimensions of the plastic deformation of the
surface normals coincide with the construction of flexible plastic surface in
stress coordinates.
So the formula (1.9) is called the normal rules, also known as
associated flow rule, regarded dimensional plastic deformation rate
coincides with the gradient of the plastic flow function.
It is possible that the plastic problem is complex because flexible
nonlinear properties. However, designers are often interested in power
limit, or limit load of the structure, ie the capacity to cause structural
damage. In that case use "limit analysis" is a simple method that designers
are interested in [25], [33], [34], [48]. The foundation of this approach are
two definitions and the following theorem:
Definition 1: An equilibrium system, or a statically admissible field
of stresses is a distribution of stresses that satisfies the following
conditions:
a. It satisfies the conditions of equilibrium in each point of the body,
b. It satisfies the boundary conditions for the stresses,

c. The yield condition is not exceeded in any point of the body.
ij
ij
p
ij
)(f





6
Lower bound theorem: The true failure load is larger than the load
corresponding to an equilibrium system.
Definition 2: A mechanism, or a kinematically admissible field of
displacements is a distribution of displacements and deformations that
satisfies the following conditions:
a. The displacement field is compatible, i.e. no gaps or overlaps are
produced in the body (sliding of one part along another part is allowed),
b. It satisfies the boundary conditions for the displacements,
c. Wherever deformations occur the stresses satisfy the yield
condition.
Remark: From the definition 2 we can see the structure or hard state,
or plastic (hard plastic systems).
Upper bound theorem: The true failure load is smaller than the load
corresponding to a mechanism, if that load is determined using the virtual
work principle.
From the definitions and theorem of limit analysis we see: lower
bound - the stress balance; upper bound - only determine the stresses in the
yield point. The upper bound indicates the same flow as ranges or chute

should be to determine the load limit, the upper bound can not be used
separately but must use the lower bound. Correct answers when the upper
bound equal the lower bound.
1.1.2.2. Method of subgrade stability research
Method of subgrade stability research (Bearing capacity of natural
ground and slope stability) in plane stress problem is the method to
solution following system of equations:



(1.14)



where: 
x
, 
y
, 
xy
,


yx
is the stress state at a point in the soil;
 is the angle of internal friction;
c is the cohesion.

























cos.csin
2
0
xy
0
yx
yx
max
xyy

yx
x
7
Third equation of the system (1.14) is the yield condition Mohr-
Coulomb written as stress component.
1.1.2.3. Bearing capacity of natural ground
Prandtl (1920) was the first to solve the equations analytically for
the case of problem on the ice when the foundation could not land at the
weight. The load limit is determined from the lower limit theorems and
limit theorems on the results can be considered equal to the Prandtl
solution is the correct solution of the limit analysis method.
Novotortsev (1938 ) address the general problem when the work load
versus vertical oblique angle.
In addition, there are several methods of calculating load limits other
sliding surface is determined from the limit equilibrium methods such as
Terzaghi method, Berezansev, Vesic, Ebdokimov, Meyerhof, Hansen,…
Exact mathematical solution to the problem is an important
consideration volumetric weight of the ground is very complex. Therefore ,
many methods approximate solutions have been developed . Sokolovski
(1965 ) provide numerical solution methods based on approximation by
finite difference .
Actual construction and experimental models have shown that the
soil mass destruction , the soil mass of the state of damage at the same time
that place is still in stable equilibrium [24].
1.1.2.4. Research methodology slope stability
a. Method assumes the slip surface
Actually common method used classic W.Fellenius fragmentation
and auditing methods to Bishop slope stability assumption soil on the slope
instability will slip round cylindrical slip surface. Bishop method takes into
account the effect of horizontal thrust from both sides of the sliding slice

(not to the point of interest of the two horizontal forces put it).
In addition to the above two methods is very much the way
fragmentation methods such as: Janbu method, Morgenstern-Price,
Spencer, American engineers association, or methods based on general
limit equilibrium theory GLE, the method takes into account the forces
between the pieces to reflect most closely the actual interaction between
the slice
8
b. Method assumed stress field
To determine the height limit of a vertical slope according to the
lower bound theorem, WF Chen [33], [34] has assumed stress field in three
regions correspond to two equilibrium equations.
Conduct Mohr circle for each area and get points to reach the foot of
the steep yield limit first plastic (Mohr circle tangent to the Coulomb)
when increment height of the vertical slope H.
1.2. Shortcoming issues in the research of earth embankment stability
on natural ground
Research methodology stable roadbed is widely used in today's
designs is limit equilibrium method or methods of solving equations (1.14)
includes two balance equations and yield condition Mohr-Coulomb (plane
stress problem). Solve the system under stress on users to limit theorem
under the assumption of stress states in each region soil mass balance
equation satisfied and Mohr - Coulomb condition, so here 's how indirect.
Solve the system used on the skating track on the limit theorem by writing
equations in polar coordinates .
However, the slope of the applied solution is very difficult to have
assumed before sliding surface . Methods commonly used method today is
fragmented classical and Bishop method assuming circular slip surface is
cylindrical . W. F. Chen used a logarithmic spiral slip surface to calculate .
Limit equilibrium method with the above two solutions , as WF Chen

commented [34], is not a proper application of limit analysis method of the
above theory - ideal plasticity by for not considering the volume of soil
mass phenomenon altered flow situations using the yield condition Mohr-
Coulomb. On the other hand, the basic equations above do not allow
determining the stress state in the plastic flow imperfections, which is not
considered a state of stress of the whole block of land because land is not
so elastic material with two balance equation which has three hidden, so
can not determine the state of stress in the soil.
1.3. Objectives and contents of the thesis research
Ngo Thi Thanh Huong when researchers calculated stresses in the
ground transportation works [19], under the guidance of Prof. Dr Ha Huy
Cuong combined max shearing stress conditions to achieve the greatest
9
minimum value (min (
max
)) with two balance equations in plane stress
problem to be system of equations:

 

























0
xy
0
yx
0
xyy
yx
x
yx
2
(1.47)
with 
2
denotes the Laplace operator.
System ( 1.47 ) has three equations to find three hidden is unknown

x

, 
y
and 
xy
so the problem is defined. Therefore, using this system of
equations we can determine the stress state in the entire soil mass .
PhD. Ngo Thi Thanh Huong in his thesis on applied theory to solve
the following problem :
- Status subcritical stress natural soil under the effect of self-weight .
- Angle of slope of the critical mass of dry sand .
- Load capacity of the ground under the foundation tape not consider
myself weight .
PhD. Nguyen Minh Khoa in his thesis was developed to solve
theoretical limit stress in the natural ground under the weight of the
embankment effects and counter pressure pad .
However, load embankment and breaks the rules applied load is
distributed, ie not research of the simultaneous embankment and natural
ground.
Therefore, the author based on theoretical min (
max
) can be directly
applied to limit theorem subgrade stability (research of the simultaneous
stability of embankment and natural ground). Author should use the lower
bound theorem without upper bound theorem by assuming that all points
are capable of plastic flow. For plane problems, we have:

 


V

2
max
mindV)x(f
G
1
Z (1.48)
where: 



 cos.csin
2
)x(f
yx
;
10
G is the shear modulus of soil;
In brackets [ ] is the yield condition Mohr-Coulomb written as
stress component.
Chapter 2
FACILITY THEORY TO RESEARCH STABILIZE
EARTH EMBANKMENT STABILITY ON NATURAL GROUND
This chapter presents the theoretical min (
max
) and differentiate with
elastic theory, followed by presenting the problem constructively
determine stress field in the earth. Finally, the method presented in
accordance finite difference solution and some results to show properties
can use this theory to research of earth embankment stability on natural
ground.

2.1. Theory min (
max
)
Soil is the product of weathering processes on the same layer of the
earth's crust, which formed the sediments. In natural conditions, soil is
multi-phase materials: solid phase (particles), liquid and gas phases. The
mechanical properties of the soil are complex, depend directly on the three
phases interact with each other. However, in the process of sediment due to
self weight over time more and more land is "stable".
To distinguish theory min (
max
) with elastic theory, author study
stress field in the soil based on two theories.
2.1.1. Elastic stress field in soil
If soil is considered elastic material, the elastic stress field in the
earth can be determined through displacement field, its deformation. In the
plane problem, using stress is unknown, the stress field can be determined
by the minimum potential energy problem (2.1).



(2.1)










































0
xy
0
yx
mindV
2
)1(
2E
1
Z
xyy
yx
x
2
yx
2
xy
yx
2
y
2
x
V
11

where: Z is the elastic strain potential energy in the plane stress
problem [1];


x
, 
y
, 
xy
,


yx
is the stress state at a point in the soil;
E,  is the elastic modulus and Poisson's ratio of soil;
By variational calculus problem leads to extreme on the basic
equations of the elastic theory.
2.1.2. Stress field based on theory min (

max
)
The plane stress problem to determining stress field in soil based on
theory min (
max
) as follows:




































0
xy
0
yx

min
2
xyy
yx
x
2
xy
2
yx
max
(2.10)
where:  is the volume weight of the soil.
The problem gives enough the equation to determine the stress state
in the soil. In addition, we also received a volumetric strain of 0. This is an
important factor to be applied strictly limited analytical methods for soil
that yield condition Mohr-Coulomb.
Now, we have the stress field in the soil is static determinacy field
enough equation to solve the equation. Therefore, the problem is soil
mechanics problems identified, we can use to solve the problem of
different stress states (such as external load).
2.2. Establish problem to identify stress field in soil
After obtaining these results, the problem identified stress field in the
soil of roads, houses, dikes, dams entirely possible. In the need to further
examine the problem of constraint conditions. For clearer presentation, we
consider the problem to the stress state of embankment on natural ground
due to self weight and external load (Figure 2.4).

12
O
x

y
n
0
m
1
m
2
n
2
p
1
c ,
11

1

c ,
00

0

m'
1
m'
2



n
1

n
3
n
4

Figure 2.4. Diagram trapezoidal embankment
Stress boundary conditions
+ On horizontal surface n
2
-n
0
:

y
= 0; 
xy
= 0 while only consider self weight (2.16)

y
≠ 0; 
xy
= 0 within the sphere of external load. (2.17)
+ On inclined surface (slope):
)n,ycos().n,xcos( 2)n,y(cos)n,x(cos
xy
2
y
2
xn
 (2.18)

+ On horizontal surface m
1
-n
1
:

y
= 0; 
xy
= 0 when not surcharge (2.19)
+ On boundary m
1
-m
2
:








min)(
min)(
2)m,2(
xy
)m,1(
xy
2)m,2(

x
)m,1(
x
(2.20)
+ At bottom:
The more depth of soil, the more the stress state of the soil nearly to
each other. By means of least squares as we have:










min)(
min)y.(
2)n,1
2
m(
xy
)n,
2
m(
xy
2)n,1
2
m(

y
)n,
2
m(
y
(2.21)
Soil conditions inability tensile
The compressive stress satisfies the following conditions:
0
x


and
0
y


. (2.22)
13
Yield condition Mohr-Coulomb
Stress state in soil must satisfy the yield condition Mohr-Coulomb
follows:
0cos.csin
2
yx
max





 (2.23)
Condition every node is likely to yield

mindxdy)cos.csin
2
(
G
1
2
V
yx
max






(2.24)
where: G is the shear modulus of soil.
2.3. Finite difference method to solve the problem
Direct solution problem is very difficult, especially when considering
the volumetric weight of the soil. Therefore, the authors solve the problem
by finite difference method [15], [22].
Divide the soil mass into square blocks, each node has three
unknown stresses, except for the nodes on the boundary mentioned above.
In general there are 3 hidden in each node is 
x
,


y
,
xy
.
Balance equations and the objective function is written for center
points of the finite difference grid
Problem has form squares objective function, constraints are linear
and nonlinear. There are many methods of solving nonlinear programming
problem [29], but to take advantage of the extreme function is available
[37], author programmed on Matlab's software to solve.
2.4. Flamant problem solution by numerical method
To verify the correctness of the solution method and computer
program, author solve Flamant problem by finite difference method, then
compared with analytical solutions.
Author writes program Dothang1 and Dothang1a to solve the
problem.
Results calculated vertical normal stress 
y
at the position between
the strip load by the finite difference method for results approximation
with analytical solutions (less than 5% difference). The difference is due to
the number of mesh elements difference is not large enough.
14
2.5. Solution of the plane problem by theory min (
max
)
To compare stress field based on the theory min (
max
) with elastic

theory, we solve the problem to determine the stress field in the soil caused
by distributed load evenly on the homogeneous soil surface is limited by
horizontal plane by theory min (
max
).
Author writes program Dothang2 and Dothang2a to solve the
problem.
We see the stress distribution
y
follow horizontal and depth in case
soil is considered elastic broader and deeper based on theory min (
max
).
2.6. Results and discussion
1- The problem determined stress field in the soil is essential.
However, today the stress field problem is not determined .
2- If soil is considered elastic material, use two balance equation
combined with the minimum potential energy conditions. By variational
calculus problem leads to extreme on the basic equations of the elastic
theory.
3- Given the condition min (
max
), combined with two balance
equations , we can build stress field in the soil .
4- To get solution by numerical method, author use the finite
difference method. The balance equations and the objective function is
written for center points; constraints conditions (2.16), (2.17), (2.18),
(2.19), (2.20), (2.21), (2.22), (2.23), (2.24).
5 - To check the convergence of the finite difference method, author
writes program Dothang1 and Dothang1a for evenly distributed loads on

the horizontal plane and compared with the Flamant solution for small
difference results than 5% .
6- To compare stress field based on the theory min (
max
) with elastic
theory, author writes program Dothang2a and Dothang2 for the
distributed load evenly on the on the horizontal plane . The results show
that the stress distribution based on theory min (
max
) in accordance with
the nature of the soil than in case soil is considered elastic .
15
Chapter 3
FUNDAMENTAL PROBLEM ABOUT LIMIT LOAD
AND SLOPE STABILITY
This chapter first presents the basic problem is a natural stress state
of the ground in infinite half space to determine the coefficient of
horizontal earth pressure. Next, using the theory min (
max
) and limit
analysis method to solve the problem of Prandtl about limit load and the
problem about limit steep angle of dry sand blocks.
3.1. Natural stress state of the ground in half infinite space
To determine the important parameters in geotechnical is coefficient
of horizontal earth pressure, author study problem natural stress state of
the ground in infinite half space because of self weight.
The problem determine the stress state in the nature ground is the
problem (2:10) with the constraints (2.16), (2.19), (2.20), (2.21). Author
writes program Dothang3 to solve the problem.
Calculation results showed that compressive stress value 

x
, 
y
in the
soil column are equal, increases linearly by depth with rule 
x
= 
y
= .y.
values of shear stress 
xy
at nodes approximately zero and the coefficient of
static earth pressure calculation
1K
0

.
3.2. Problem Prandtl
Determining the load limits of natural ground due to the effect of
uniformly distributed load on the foundation put ice on the ground , then
compare the analytic solution of Prandtl to verify the correctness of the
theory min (
max
) and directly applying the limit theorem of limit analysis
methods to the problem of limit load of the ground.
Author writes program Dtlim4, Dtlim4a and Dtlim4b to solve
problem.
Calculation results show that the limit load, the yield node
developing and connecting extending to the surface. Meanwhile, the soil
can see has formed a failure mechanism. That is the load capacity or load

limit of the ground.
In addition, we also get the yield deformation zone and hard soil
wedge below foundation similar solution of Prandtl.
16
Limit load of ground approximately with a solution of Prandtl, p
gh
=
5,14c (difference 2,8%). This difference is due to the solution of Prandtl
consider only the stress state of yield deformation zone limited to a certain
range below the foundation , the author's solution allows us to identify the
stress state of the entire soil mass.
3.3. Problem about limit steep angle of dry sand blocks
Sand is the material being used in most road construction in our
country today. However, according to the Highway-Specifications for
design TCVN 4054-2005 [7] and Highway embankment and cuttings -
Construction and quality control TCVN 9436-2012 [9] sand embankment
must be cover by clay side slope and the upper layer of subgrade to prevent
erosion.
The author writes program Dtlim5, Dtlim5a and Dtlim5b to solve the
problem .
Results calculated slope angle limit in the case showed that the
critical angle 
gh
slope of dry sand equal internal friction angle of sand .
We found that limited research steep angle of dry sand blocks by the
way of PhD. Ngo Thi Thanh Huong [19] gives us the full status of the
entire mass of sand, while previously only solution is balanced review of
the counterclaim on the slope .
From solving the problem of limited angle of sand, the author
received a stable shape of the sand mass. So, we can see the outer

embankment mission against surface erosion also another important task is
to stabilize the roadbed slope by slope angle slopes often greater internal
friction angle of sand. In addition to clay earthen embankment how we can
use geotextile to stabilize the slope .
3.4. Results and discussion
From these studies showed that the theoretical correctness of min
(
max
) and directly applying the limit theorem of limit analysis methods.
17
Chapter 4
RESEARCH OF STABILITY
OF SOIL MASS WITH VERTICAL SLOPE
In this chapter, using the theory min (
max
) and limit analysis method
for soil stability study has a vertical slope in the case due to the effects of
external loads and cases due to self weight.
4.1. Research vertical slope stability due to external load
Consider a vertical slope of weightlessness (= 0), external load as
Figure 4.1.
c ,




n
1
n
0

m
1
m
2
p
c ,
0 0
1
1
1
gh
H
O
x
y

Figure 4.1. Diagram of calculation vertical slope stability
due to external load
We see that, when the external load increases, the stress state in the
soil and increase the load reaches the value of soil mass began failure
mechanism called limit load p
gh
. Load P is the unknown of problems .
The objective function of vertically slope stability problems is
written as follows:
18

minpdV
2G
1

dVcos.csin
22G
1
Z
gh
2
xy
2
yx
V
2
V
yx
2
xy
2
yx
1














































(4.1)
The objective function (4.1) must satisfy two balance equations and
the constraints follows:
- Soil conditions not likely to be pulled (2.22);
- Yield condition Mohr- Coulomb (2.23);
- Boundary condition (2.17), (2.18), (2.19), (2.20), (2.21).
Author writes program Dtlim6, Dtlim6a and Dtlim6b to solve the
problem.
Next, author conducted survey of the various cases with physical and
mechanical characteristics of embankment and natural ground, the
placement of load to get remark.
4.2. Research vertical slope stability due to self weight
Consider a vertical slope in Figure 4.9.
H
O
x
y
c ,




n
1
n
0
m
1

m
2
c ,
0 0
1 1

1

0
x
(b)

y
(a)

Figure 4.9. Diagram of calculation vertical slope stability
due to self weight
Soil mass is divide into finite difference grid as Figure 4.9a . Each
node has three unknown stresses 
x
, 
y
, 
xy
. Splitting a rectangular
difference grid (Figure 4.9b), the horizontal size and vertical size is Δx,
19
Δy. Fixed Δx, Δy rise to the height of the vertical slope H = (m
1
-1)Δy will

increase. When the height of the slope at a value that soil mass begins to
form failure mechanism called critical height. Therefore, the height of
slope H is unknown of the problem. This is new way. Because the normal
way, they have reduced soil shear strength to the soil ruined by dividing a
coefficient K
min
stable or decreasing elastic modulus E for horizontal
displacement slope to a limiting value, ie not determine directly the critical
height.
The objective function of vertically slope stability problems due to
self weight is written as follows:

minHdV
2G
1
dVcos.csin
22G
1
Z
2
xy
2
yx
V
2
V
yx
2
xy
2

yx
1













































(4.2)
The objective function (4.2) must satisfy two balance equations and
the constraints follows:
- Soil conditions not likely to be pulled (2.22);
- Yield condition Mohr- Coulomb (2.23);
- Boundary condition (2.16), (2.18), (2.19), (2.20), (2.21).
Author writes program Dtlim7, Dtlim7a và Dtlim7b to solve the
problem.
Then, author conducted survey of the various cases with physical and
mechanical characteristics of embankment and natural ground to get
remark.
4.3. Results and discussion
Research of stability of soil mass with vertical slope in the case of
external load as well as for self weight get remark following :

1 - When the external load form slip surface and if the load placed
back into, the slip surface will start from the toe to point beginning to set
load. If layer above has greater intensity than layer below the limit load
increases, then slip surface deepening into the natural ground .
20
2 - Limit load of embankment stability finding equal 2c.tg(45
0
+/2),
consistent with other authors [33], [34], [47] .
3 - When considering the self weight, author has not get slip surface
eaten up on and the result of critical height as )2/45(tg
c3,2
H
0
gh


 .
4 - The directly determine critical height H
gh
is new way compared to
the usual way to calculate indirectly through K
min
stability or displacement
limits.
There are these results are due the accurately diagram and directly
applicable limit theorem of limit analysis methods.
Chapter 5
NEW METHOD TO RESEARCH STABILITY PROBLEM
OF THE EARTH EMBANKMENT ON NATURAL GROUND

In this chapter, using the theory min (
max
) and limit analysis method
to research stability problem of the earth embankment on natural ground.
5.1. Research stability problem of the earth embankment on natural
ground
The problem arises: For width of the embankment and slope
gradient, the physical properties of embankment and natural ground; asked
to identify the critical height to ensure a stable embankment
H
H
gh
1:m
1:m
c ,
11

1

c ,
00

0




B
nÒn

Figure 5.1. Diagram determined the critical height of embankment


21
The author's solution is the hypothesis an embankment height
initially small, then increase the height to the embankment in the limit
state, then we have a critical height H
gh
of embankment (Figure 5.1).
Embankment and natural ground slopes with a slope given is divide
into finite difference grid as Figure 5.2a . Each node has three unknown
stresses 
x
, 
y
, 
xy
. Splitting a rectangular difference grid (Figure 5.2b), the
horizontal size and vertical size is Δx, Δy. Fixed Δx, Δy rise to the height
of the vertical slope H = (m
1
-1)Δy will increase. When the height of the
slope at a value that soil mass begins to form failure mechanism called
critical height. Therefore, the height of slope H is unknown of the problem.
n
3
n
4
O
x

y
n
1
n
0
n
5
m
1
m
2
n
2
c ,
11

1

c ,
00

0

m'
1
m'
2




x
(b)

y

(a)

Figure 5.2. Diagram of finite difference grid used to calculate
the critical height of embankment
The objective function of problems determined the critical height of
embankment due to self weight similar (4.2) as follows:

minHdV
2G
1
dVcos.csin
22G
1
Z
2
xy
2
yx
V
2
V
yx
2
xy
2

yx
1













































(5.1)
The objective function (5.1) must satisfy two balance equations and
the constraints follows:
- Soil conditions not likely to be pulled (2.22);
22
- Yield condition Mohr- Coulomb (2.23);
- Boundary condition (2.16), (2.18), (2.19), (2.20), (2.21).
Author writes program Dtlim8, Dtlim8a và Dtlim8b to solve the
problem.
Next, author conducted survey of the various cases about the
geometric structure of the embankment, physical and mechanical
characteristics of embankment and natural ground to get remark.

Figure 5.4. Chart of contours yield ability

(Line has value equal 0 is the line running through node
that yield limit is reached)
To clarify the suitability of the analysis methods used in study about
embankment stabilization, author compare the calculated limit equilibrium
methods are commonly used today as the ordinary method W . Fellenius,
Bishop and WF Chen in many different cases.
To facilitate the designer can quickly determine the limit level of the
red line to ensure a stable embankment, author tabulated lookup table ratio
H
gh
*/c
0
to determine critical height of embankment in many different
cases. The results summarized in Table 5.8.
23
Table 5.8. The relationship between the ratio H
gh
*

/c
0
with angle of internal friction and the ratio of cohesion
Slope
Angle of
friction
(Degree)
Ratio c
1
/c
0

1 1.5 2 3
1/1
0 4,76 5,25 5,31 5,33
5 5,42 6,25 6,61 6,13
10 6,06 7,46 8,23 9,32
15 6,81 8,92 10,3 12,08
20 7,61 10,71 12,94 15,53
25 9,12 12,94 16,39 20,61
30 11,73 15,75 20,97 27,70
1/1.25
0 5,06 5,34 5,42 5,47
5 5,82 6,41 6,80 7,26
10 6,69 7,71 8,54 9,61
15 7,70 9,30 10,77 12,54
20 8,89 11,26 13,66 16,17
25 10,64 13,74 17,47 21,48
30 13,19 16,90 22,56 29,57
1/1.5
0 5,20 5,37 5,47 5,55
5 6,17 6,51 6,93 7,34
10 7,27 7,90 8,78 9,73
15 8,54 9,62 11,18 12,95
20 10,11 11,76 14,32 16,83
25 12,09 14,48 18,48 22,49
30 14,59 17,98 24,10 32,19
1/1.75
0 5,28 5,41 5,53 5,64
5 6,71 6,69 7,16 7,66
10 7,90 8,39 9,33 10,13

15 9,57 10,52 12,11 13,73
20 11,78 13,20 15,57 18,45
25 14,38 16,72 20,44 24,86