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

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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.

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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 LIST OF PUBLISHED WORKS UNIVERSITY OF TRANSPORT AND COMMUNICATIONS 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. SUPERVISORS: 2. Do Thang (2013). “Research of stability for vertical slope by limit 1: Prof. Dr Ha Huy Cuong analysis method”. Review of Ministry of Construction of VietNam. (Military University of Science and Technology) 11/ 2013. pp. 103 - 104. 2: PhD Vu Duc Sy 3. Do Thang (2014). “New method research earth embankment stability (University of Transport and Communications) on natural ground”. Review of Ministry of Construction of VietNam. 6/2014. 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
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]: f ( ij ) p  ij   (1.9)  ij 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.
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:   x  yx  x  y  0    y  xy (1.14)    0  y x  x  y  max  sin   c. cos   2 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.
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:   2      0 x y    x  yx    0 (1.47)  x y   y  xy    0  y  x 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: 1 Z    max  f ( x ) dV  min 2 (1.48) VG x  y where: f (x)  sin   c. cos  ; 2
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 2 1  x  y 2  2  xy yx  Z   . x . y  (1   )  dV  min  V E 2 2      x  yx   0  (2.1) x y   y  xy   0  y x  
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:  x  y  2  2  max   2    xy  min       x  yx   0  (2.10) x y   y  xy    0  y x   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 x p1 n2 n0 n3 O y  c1,  1 1 n1 n4 m1 m' 1  c0,  0 0   m2 m' 2  Figure 2.4. Diagram trapezoidal embankment Stress boundary conditions + On horizontal surface n2-n0: 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   x cos 2 ( x , n )   y cos 2 ( y, n )  2. xy . cos( x, n ). cos( y, n ) (2.18) + On horizontal surface m1-n1: y = 0; xy = 0 when not surcharge (2.19) + On boundary m1-m2: ( (x1, m )   (x2,m ) ) 2  min   (1,m ) ( 2, m ) 2 (2.20) ( xy   xy )  min  + 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: ( (ym2 ,n )   (ym2 1,n )  .y) 2  min   ( m 2 ,n ) (2.21) ( xy    (xy 2 1,n ) ) 2  min m Soil conditions inability tensile The compressive stress satisfies the following conditions:  x  0 and  y  0 . (2.22)
13 Yield condition Mohr-Coulomb Stress state in soil must satisfy the yield condition Mohr-Coulomb follows: x  y  max  sin   c. cos   0 (2.23) 2 Condition every node is likely to yield 1 x  y G ( max  sin   c. cos ) 2 dxdy  min (2.24) V 2 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 K 0  1. 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, pgh = 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. pgh O x n1 n0 c1 ,1 y H 1 m1  c0,0  m2  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 pgh. Load P is the unknown of problems . The objective function of vertically slope stability problems is written as follows:
18 2 1  2  x  y  x  y Z1        2  sin   c. cos  dV  2  xy VG    2    (4.1) 2 1   x   y         2  dV  p gh  min 2  xy V G      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. O x n1 n0 y H c1 ,1 y 1 x (b) m1  c0, 0 0  m2 (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,