Three-dimensional effects in slope stability for shallow ...

Three-dimensional effects in slope stability forshallow excavations

Analyses with the finite element program PLAXIS

Niclas Lindberg

Master of Science Program in civil engineering

Luleå University of Technology

Department of Civil, Environmental and Natural resources engineering

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PREFACE

This master thesis is the final part of my five year education in civil engineering at Luleåuniversity of technology. The investigation has been done on behalf of Luleå university withinspiration from Trafikverket.

I would like to thank my supervisor Hans Mattsson from Luleå University of technology forall the guidance and inspiration during both the courses and the time doing this investigation.I also want to thank for the possibility to work and learn more about numerical modelling. Aspecial thanks to Per Gunnvard at Luleå university for all the guidance and patience duringthe time of making this investigation.

Finally, thanks to all my friends and family during the years at Luleå university of technology.

Luleå, Mars 2018Niclas Lindberg

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ABSTRACT

The purpose with this study was to investigate the impact of three-dimensional effects in

slope stability for three-dimensional excavations and slopes with cohesive soils and compare

the results with the method provided by the Swedish commission of slope stability in 1995

regarding three-dimensional effects. Both the factor of safety and the shape of the slip surface

was compared between the methods but also the results from their equivalent two-

dimensional geometry.

The investigation was performed with models created in the finite element software PLAXIS

3D and the limit equilibrium software GeoStudio SLOPE/W. Three-dimensional excavations

with varying slope angles, external loads and slope lengths were tested for three different

geometry groups in PLAXIS 3D. The equivalent two-dimensional geometries were modeled

with SLOPE/W and recalculated with the three-dimensional effect method provided from the

Swedish commission of slope stability.

The results show that the methods match well for slopes with inclinations 1:2 and 1:1 when an

external load is present on the slope edge, and the factor of safety is greater and not close to

1,0. For an excavation with vertical walls or when no external load is present, the methods

match poorly. The results also show that for a long and unloaded slope, the factor of safety

approaches the value obtained from a simplified two-dimensional analysis.

The results imply that the recommendations from the Swedish commission of slope stability

are reliable for simple calculations of standard cohesive slopes.

Keywords: Slope stability; 3D-effects; FEM

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SAMMANFATTNING

Syftet med detta examensarbete var att undersöka tredimensionella effekters inverkan vid

släntstabilitet hos tredimensionella schakter och slänter av kohesiva jordar och jämföra

resultatet med den metod som svenska skredkommissionen rekommenderat år 1995 gällande

tredimensionella-effekter. Både säkerhetsfaktorn och formen hos den utbildade glidytan

jämfördes mellan metoderna samt resultatet från dess ekvivalenta tvådimensionella geometri.

Undersökningen utfördes med hjälp av modellering i det finita elementprogrammet PLAXIS

3D och gränslastanalysprogrammet GeoStudio SLOPE/W. Tredimensionella schakter med

varierande släntlutningar, externa laster och släntlängder testades hos tre olika

geometrigrupper i PLAXIS 3D. De ekvivalenta tvådimensionella geometrierna modellerades i

SLOPE/W och räknades sedan om tredimensionellt enligt den metod som svenska

skredkommissionen rekommenderat.

Resultatet visar att metoderna överensstämmer väl för schakter med släntlutningen 1:2 och

1:1 där en extern last finns närvarande på släntkrönet och säkerhetsfaktorn är större än och

inte nära 1,0. För schakter med vertikala schaktväggar eller schakter där ingen extern last

närvarar överensstämmer metoderna inte väl. Resultatet visar också att en långsträckt

obelastad slänt har en säkerhetsfaktor som stämmer väl överens med en simplifierad

tvådimensionell analys.

Resultatet föreslår att rekommendationerna från svenska skredkommissionen är tillförlitliga

för enklare beräkningar av normala släntstabilitetsproblem i kohesiva jordar.

Nyckelord: Släntstabilitet; 3D-effekter; FEM

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TABLE OF CONTENTS

1. INTRODUCTION .........................................................................................................1

1.1 Background ..............................................................................................................11.2 Purpose and objective ...............................................................................................21.3 Limitations ...............................................................................................................3

2. THEORETICAL BACKGROUND ................................................................................5

2.1 Finite element method ..............................................................................................52.2 Limit equilibrium method ....................................................................................... 102.3 Three-dimensional slope stability ........................................................................... 13

3. SOFTWARE ................................................................................................................ 19

3.1 PLAXIS ................................................................................................................. 193.2 GeoStudio, SLOPE/W ............................................................................................ 23

4. NUMERICAL MODELLING WITH PLAXIS 3D ....................................................... 25

4.1 Geometries and cases ............................................................................................. 254.2 Model specification ................................................................................................ 274.3 Phases .................................................................................................................... 294.4 Material parameters ................................................................................................ 294.5 Mesh and boundaries .............................................................................................. 30

5. SLOPE/W MODELS AND 3D-EFFECT CALCULATIONS ....................................... 31

5.1 SLOPE/W models .................................................................................................. 315.2 Calculation of three-dimensional effects ................................................................. 32

6. RESULTS AND ANALYSIS....................................................................................... 33

6.1 Two-dimensional comparison ................................................................................. 356.2 Three-dimensional analysis .................................................................................... 366.3 3D-effects method compared with PLAXIS 3D models .......................................... 396.4 Safety analysis in PLAXIS ..................................................................................... 43

7. DISCUSSION .............................................................................................................. 45

7.1 Suggestions on further studies ................................................................................ 46

REFERENCES ..................................................................................................................... 47

APPENDIX A1 –SFR/DEFORMATION-CURVES, MODEL A ..................................... 51APPENDIX A2 –SFR/DEFORMATION-CURVES, MODEL B ...................................... 55

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APPENDIX A3 –SFR/DEFORMATION-CURVES, MODEL C ...................................... 58APPENDIX B1 – TOTAL DEFORMATIONS, MODEL A ............................................. 59APPENDIX B2 – TOTAL DEFORMATIONS, MODEL B .............................................. 67APPENDIX B3 – TOTAL DEFORMATIONS, MODEL C .............................................. 74APPENDIX C1 –SLOPE/W SLIP SURFACE GROUP A ................................................ 78APPENDIX C2 –SLOPE/W SLIP SURFACE GROUP B ................................................ 79APPENDIX C3 –SLOPE/W SLIP SURFACE GROUP C ................................................ 80

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1 INTRODUCTION

1.1 Background

Slope stability analysis is a branch of geotechnical engineering and concerns the stability for

both natural and constructed soil slopes. Many construction projects require excavations in

soil to construct pipes and cables, and also foundations for buildings and bridges. Roads and

railways are usually built with soil embankments, where stability analysis must be performed

to satisfy the safety regulations. Several methods have been developed throughout history to

calculate the stability of slopes, which results in a factor of safety. The factor is defined as the

ratio of the shear strength available of the soil compared to the necessary strength to maintain

equilibrium (Bishop, 1955). The most common approach to calculate the stability of a slope is

with the limit equilibrium method (LEM), with an assumption of plane-strain conditions.

(Zhang, Guangqi, Zheng, Li, & Zhuang, 2013) With the software and computer power

available today, this method can obtain multiple failure surfaces with their factors of safety in

a very short time. The finite element method (FEM) is also a popular technique which is a

numerical method that discretizes a problem into elements and solves partial differential

equations numerically.

To establish safe and economical solutions for slopes and excavation stability, high accuracy

in the calculation models are required, even though several assumptions to simplify the real-

life problems are inevitable in most engineering models. A very common assumption in slope

stability analysis is the plane-strain condition, which is a simplifying assumption that means

that the value of the strain component perpendicular to the plane of interest is equal to zero.

This is usually valid when structures are very long in one dimension in comparison to the

others. With this assumption, no curvatures, corners or change of geometry in one dimension

can be accounted for at all. This means that the failure surface must exist perpendicular to the

plane of interest. (Zhang, Guangqi, Zheng, Li, & Zhuang, 2013) In the past, this assumption

was almost necessary because of the limitations in the three-dimensional slope stability

analysis methods. Location, shape and direction of the slip surface are usually unknown,

which makes the problem very complex to solve. (Cheng, Liu, Wei, & Au, 2005)

A three-dimensional slope stability analysis is important because the factor of safety is known

to be higher than in a two-dimensional analysis. This means that a more economic design is

possible. (Cheng, Liu, Wei, & Au, 2005) The Swedish commission of slope stability proposed

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a calculation method for the treatment of three-dimensional effects (Skredkommissionen 3:95,

1995). The calculations are valid for simple cases with an idealized slip surface in cohesive

soils and is based on the limit equilibrium approach. The treatment of three-dimensional

effects in slopes made of frictional soils were described as unknown. The Swedish transport

administration often perform shallow excavations for smaller construction works. If three-

dimensional effects are considered for slope stability, the total excavation cost could be

smaller. They are currently using the recommendations from 1995 but is interested in a

modern FE comparison to confirm or add knowledge to the subject.

1.2 Purpose and objective

All real-life slopes are three-dimensional problems, and because the factor of safety is known

to be higher for slope calculations where three-dimensional effects are accounted for, more

knowledge is required to make more economic designs that are still considered safe. The

purpose of this master thesis is therefore to compare the difference in factor of safety and slip

surface shape between three-dimensional and two-dimensional slope stability analysis for

shallow excavations using the commercial software PLAXIS 3D and SLOPE/W. The

SLOPE/W results will then be extended to three-dimensional surfaces using the 3D-effects

method described by the Swedish commission of slope stability. Different slope geometries

and soil materials will be tested in the models along with external loads and excavations. The

objective is to compare the factor of safety and slip surface shapes for shallow excavations

obtained from the three-dimensional-effects method (3D-effects) and PLAXIS 3D models.

The following questions will be investigated:

· What difference in factor of safety is obtained from a stability analysis of a three-

dimensional excavation problem modeled in three-dimensions than in its two-

dimensional equivalence?

· Does the 3D-effects calculation method recommended by the Swedish commission of

slope stability match the results obtained from finite element 3D calculations?

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1.3 Limitations

This report is limited to only study undrained analyzes for one type of cohesive soil material.

The varied parameters are the slope angles, the excavation lengths and the magnitude of the

external load. The groundwater level, excavation depth and load area are kept the same in all

models.

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2 THEORETICAL BACKGROUND

2.1 Finite element method

Almost every physical phenomenon can be described mathematically using differential

equations. The problem with differential equations is that most of them are very hard or even

impossible to solve with analytical methods. The alternative way of solving them is with

numerical methods which gives approximate solutions, e.g. the finite element method. The

main feature with the finite element method is that the body or region is discretized into

smaller elements, on which the differential equations describe its behavior. Every element has

nodal points where the variables are assumed to be known. The nodal points are usually

located at the element boundaries. The elements are attached together to form an element

mesh, which can be seen in Figure 1. The more elements a body is discretized into, the more

nodal points it has. With more nodal points comes more unknowns, which in general produce

a solution with higher accuracy. The method is then carried out by solving for the unknowns

in the nodal points, and shape functions describe the behavior of the element in between

nodes. (Ottosen & Petersson, 1992)

Figure 1 Element mesh of a soil body in PLAXIS 3D

The finite element method was primarily developed to solve structural problems, but can be

used for several types of problems, such as heat transfer, electromagnetic problems and

groundwater flow, and is applicable to problems in one, two or three dimensions. (Ottosen &

Petersson, 1992).

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2.1.1 The mathematical background to the finite element method

For structural analysis, the differential equations describing the element behavior must be

derived from equilibrium equations. These equations can be obtained from an infinitesimal

stress cube. On the surface of the cube that can be seen in Figure 2, stress components are

acting, where indicates the plane on which the stress acts, and the direction of the stress.

Figure 2 An infinitesimal stress cube

A three-dimensional body have 9 stress components, where only 6 of them are independent

because of moment equilibrium. If there is force equilibrium in every cartesian direction, the

following three equilibrium equations can be obtained and expressed as:

+ + + = 0 (2.1)

+ + + = 0 (2.2)

+ + + = 0 (2.3)

In equation (1) – (3), b denotes the body forces in the three cartesian directions. These

equations written in matrix form, together with the body force vector gives the expression:

⎣⎢⎢⎢⎡ 0 0

0 0

0 0

0

0

0 ⎦⎥⎥⎥⎤

⎣⎢⎢⎢⎢⎡

⎦⎥⎥⎥⎥⎤

+ = 0 (2.4)

This is called the strong formulation, which also can be written:

∇ + = 0 (2.5)

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The operator ∇ is written in the transposed form because of its use in the non-transposed form

later. The finite element approach is based on the weak formulation of the differential

equations, which requires some mathematical manipulations from the strong formulation. The

equilibrium equations must be multiplied with an arbitrary weight function, in this case a

three-dimensional weight vector:

= (2.6)

The stresses acting on the surface of the body acts as boundary conditions, and can be

expressed with a traction vector:

= (2.7)

The cartesian components of must fulfill the boundary conditions:

= + += + += + +

(2.8)

The next step is to integrate over the volume and add all equilibrium equations together,

which gives the weak formulation the following expression:

∫ ∇ = ∫ + ∫ (2.9)

The weak formulation is necessary in the finite element method and gives some advantages.

The information in the strong form and the weak form is unchanged, but in the weak form the

approximate function can be one time less differentiable than in the strong form. This

property facilitates the approximation process. Also, the weak formulation can handle

discontinuities without change of the expression (Ottosen & Petersson, 1992). The greatest

advantage is that the boundary conditions are well sorted out directly in the expression, which

is the known part of the equation, the right expression in equation (2.9). This is not the case

for the strong formulation. (Ottosen & Petersson, 1992)

When the equilibrium equations are established in the weak formulation, the finite element

discretization can begin. This starts with expressing a displacement vector, ( , , ), that

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describes displacements in the three cartesian directions which forms a 3 1 matrix. Then a

global shape function matrix ( , , ) express how the behavior of each element in between

the nodes is acting by interpolation functions. This matrix forms a huge 3 3 matrix, where n

is the number of nodes in the element mesh. Last, a nodal displacement vector ( , , ) is

formed to give the deformation for each node in the element mesh, which results in a 3 1

vector. (Mattsson, 2017) Their relation is expressed as:

= (2.10)

This equation must be substituted into the weak formulation to discretize the problem. This

can be performed with the Galerkin method, where the first step is to define the arbitrary

weight vector as:

= (2.11)

In this expression, is an arbitrary 3 1 vector. The strain can be defined as:

= (2.12)

The matrix is defined as the operator multiplied with the global shape function matrix .

With these equations, the strain can also be expressed in the form:

= (2.13)

These equations can together form the expression:

= (2.14)

If expression (2.14) is substituted into the weak formulation (2.9) it can be expressed:

∫ − ∫ − ∫ = 0 (2.15)

From this expression, the parenthesis or the vector must be zero to fulfill the equation, and

because is arbitrary it must not be the zero vector. Therefore, it can be concluded that:

∫ − ∫ − ∫ = 0 (2.16)

This equation describes that internal and external forces must be equal to obtain equilibrium,

where is the inner stresses, is the outer stresses and is the body forces. It can be applied

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for all constitutive models. (Ottosen & Petersson, 1992) After this step, a constitutive relation

must be chosen and substituted into the internal part of the equation. The nodes in the

elements will deform in accordance with the chosen material model. The strains in the nodes

can be translated to a stress in the stress points, or Gauss points, located in each element. The

load in the solving procedure is increasing with small steps, called load increments. The

relation between the stress and the strain increments in stress point can be expressed as:

= (2.17)

where depends on the constitutive relation:

=

=

Because the constitutive relation is substituted into the FE-formulation, it means that a non-

linear elastoplastic material model gives a non-linear FE-formulation. For a linear elastic

material model that can be described with Hook´s law, the FE-formulation will be linear.

2.1.2 Slope stability analysis with the finite element method

Slope stability analysis can be performed with the finite element method as well. Several

approaches have been developed in order to obtain a factor of safety with the finite element

method, such as the gravity increase method, load increase method and strength reduction

method (Alkasawneh, Malkawi, Nusairat, & Albataineh, 2007). The most commonly used is

the strength reduction method, developed by (Matsui & San, 1992). Instead of comparing

resisting and driving moment or resisting and mobilized shear strength on a determined failure

body, like in LEM (explained in chapter 2.2), a strength reduction analysis continuously

decreases the strength parameters of the entire soil body until equilibrium cannot be

maintained. At this point, a failure occurs, and the factor of safety is defined as the initial

shear strength over the shear strength at failure. With the Mohr-Coulomb failure criterion, this

can be mathematically expressed as:

= = = (2.18)

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The continuous process is performed with a strength reduction factor (SRF), and can be

written:

∗ = (2.19)

∗ = tan (2.20)

In these expressions the ∗ and ∗ refers to the reduced cohesion and frictional angle.

(Carrión, Vargas, Velloso, & Farfan, 2017) Even though the finite element analysis with the

strength reduction method search for the factor of safety differently than with limit

equilibrium, it is defined in the same way with available strength over strength at failure. This

makes the results from the two different methods directly comparable. (Camargo, Velloso,

Euripedes, & Vargas, 2016) The finite element approach with strength reduction have both

advantages and drawbacks compared to the limit equilibrium method. One big advantage with

this approach, especially with a three-dimensional analysis, is the ability to find the most

critical slip surface directly without making several computations. The slip surface can have

practically any shape and is not bound to circular or linear shapes. On the other hand, the

weakness with this method is that it can only find one failure surface at the time. This means

that theoretically a small local failure may be the most unstable, while a larger much more

dangerous slip surface with a factor of safety almost as low is not found. If a certain failure

surface of interest must be studied, the strength reduction method is not recommended.

2.2 Limit equilibrium method

Slope stability analysis with the limit equilibrium method is known to be used for the first

time in Sweden 1916, by the engineer Sven Hultin, after a landslide in Gothenburg. The

method he used was a very simplified case of the limit equilibrium method which describes

equilibrium equations for a slope with an assumed or known failure surface. This method was

further developed by Wollmar Fellenius and was later known as the Swedish method

(Johansson, 1991). Several similar approaches were later developed to calculate the stability

of slopes. This became the most popular approach due to the simplicity when dealing with

complex geometries and different pore water pressures (Terzaghi & Peck, 1967).

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2.2.1 The method of slices

The method of slices is an approach where the slip body is divided into vertical slices, which

is then individually calculated as free bodies using equilibrium. The pressures acting on every

slice is translated to equivalent forces with a point of application. A free body diagram of a

slice from the rigorous Spencers method can be seen in Figure 3. These acting slice forces are:

· Effective base normal force

· Shear force on the base of the slice

· Effective normal force between the slices

· Shear forces between the slices.

Together with the unknown factor of safety and the points of application of the forces, they

produce 6n-2 unknowns for the number of slices n. The equilibrium equations that can be

obtained are:

· Vertical equilibrium

· Horizontal equilibrium

· Moment equilibrium

· Shear strength of the material.

With these equilibrium equations, 4n equations can be obtained for n slices. The problem is

that 2n-2 equations are missing to form a statically determinate system, which implies that

this system is statically indeterminate. (Johansson, 1991). When equilibrium equations

together with shear strength are insufficient to create a statically determinate system, further

information about the force distribution or assumptions must be added to solve for a factor of

safety. (D.G Fredlund, 1977).

Several methods exist to solve slope stability problems, and what separate them from one

another are the necessary assumptions about the normal interslice forces. Not every method

has all equilibrium conditions satisfied. The ones that fulfill all equilibrium conditions are

called rigorous methods e.g. Morgenstern-Price’s method and Spencer’s method. This means

that both global (entire slip body) and local (slice) force and moment equilibrium are satisfied.

(Johansson, 1991). The problem remains statically indeterminate.

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Figure 3 Interslice forces from the Spencers method. (Spencer, 1967)

In order to obtain equilibrium equations in the limit equilibrium method, a slip surface must

first be defined. For simplicity, a circular slip surface is usually assumed. The solving

procedure is then performed in three steps. Step one is to determine the normal force of every

slice base. Step two is the establishment of the force and moment equilibrium around the

vertical axis. The third and last step is to solve for the factor of safety by using the following

equation along the slip surface:

= (2.21)

This is the general expression of the factor of safety. An alternative way to define the factor of

safety, with an assumed circular slip surface, is with moment equilibrium around a rotation

center (Johansson, 1991). This assumption defines the resisting moment from the shear

strength of the soil and the impelled moment from the self-weight and loads and is expressed:

= (2.22)

Further knowledge about the classical two-dimensional methods to solve slope stability

problems with the limit equilibrium method and its applications can be found in (Alkasawneh,

Malkawi, Nusairat, & Albataineh, 2007), (Bishop, 1955), (Spencer, 1967), (Johansson, 1991),

(Mattsson, 2017).

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2.3 Three-dimensional slope stability

2.3.1 Two-dimensional analysis with 3D-effects

Calculations of how three-dimensional effects could be treated for slope stability analysis has

been suggested since the 1950’s, and many reports have been published since then. The

earliest methods were extensions of the two-dimensional limit equilibrium approaches with

additional end surface contributions. This gives a cylindrical failure surface with a finite

length and with parallel planes as end surfaces. This method was also the approach given

from the Swedish commission of slope stability (Skredkommissionen 3:95, 1995). The factor

of safety in this method is initially calculated two-dimensionally with the plane strain

assumption. Then the end surface contribution is added to the equation, which yields the

following expression:

= , (2.23)

where , is the resisting moment from the soil body surface over the slope length .

refers to the resisting moment from the end surfaces and the mobilized shear

strength.

Figure 4 Idealized three-dimensional slip surface with plane end surfaces. (Skredkommissionen 3:95, 1995)

This expression assumes that the end surfaces consists of two parallel planes which is

calculated to contribute with a constant or weighted average shear strength.

(Skredkommissionen 3:95, 1995) The value gives the factor of safety for a cylindrical

surface with plane parallel ends, see Figure 4, but the most critical end surface has a slightly

curved shape, see Figure 5 (Gens, Hutchinsson, & Cavounidis, 1988). This shape is

empirically corrected by inserting the value of into the following equation:

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, = , + 0.75,

− 1 (2.24)

where , is the 2-dimensional factor of safety and , is the 3-dimensional factor of

safety.

Figure 5 Failure surface with curved shape. (Skredkommissionen 3:95, 1995)

In equation (2.24) it can be seen that , will always be greater than , . This is because

the factor will always be greater than , , since it contains contribution from the end

surfaces. This makes the parenthesis is equation (2.24) positive, and thus making ,

greater than , .

The disadvantage with this method is that it is only applicable for cohesive soils. If frictional

soils exist within the end surfaces, it is regarded as not contributing at all.

(Skredkommissionen 3:95, 1995) This method will not consider a real three-dimensional slip

surface shape, because it is extended from the least stable 2-dimensional failure surface.

2.3.2 Method of columns

Like the method of slices for two-dimensional problems, an equivalent three-dimensional

technique was developed by Hovland (1977) called the method of columns. This method is an

extension from the two-dimensional case with similar assumptions about the inter column

forces. The three-dimensional equivalence of the ordinary method for example means that all

the inter column forces are ignored, and the shear- and normal force at the bottom surface are

functions of the column weight. (Lam & Fredlund, 1993) The method of columns is like the

method of slices a statically indeterminate problem. With number of columns in the x-

direction and in the y-direction, the number of knowns from equilibrium equations and a

failure criterion is 4 + 2. For the same number of columns, it will result in 12 + 2

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unknowns. (Lam & Fredlund, 1993) This is the reason that the system requires assumptions or

neglections to obtain a solution. More suggestions of this method were later developed with

assumptions equivalent to Bishops method and Spencers method.

In order to obtain equilibrium equations from the method of columns, an assumed failure

surface is discretized into vertical columns, see Figure 6. The bottom plane acts as the failure

surface.

Figure 6 Column discretized three-dimensional failure surface. (Chen, Mi, Zhang, & Wang, 2003)

Every column surface except for the top has three acting forces, one that is perpendicular to

the surface and two parallel shear forces, see Figure 7.

Figure 7 Forces acting on the planes of a column. (Chen, Mi, Zhang, & Wang, 2003)

The different assumptions give different simplified free body diagrams for the columns.

(Chen, Mi, Zhang, & Wang, 2003) proposed a simplified method in 2003, with inter column

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force assumptions shown in Figure 8. All inter column shear forces are neglected, and

moment equilibrium is only fully satisfied in the z-direction. Force equilibrium is satisfied in

all coordinate directions.

Figure 8 The Assumed forces acting on the column planes. (Chen, Mi, Zhang, & Wang, 2003)

The solving procedure is performed in three steps, in a general limit equilibrium manner. Step

one is the determination of the normal force of every column base. Step two is the

establishment of force equilibrium in all the remaining directions, and moment equilibrium

around the z-direction. The last step is to solve for the factor of safety. (Chen, Mi, Zhang, &

Wang, 2003)

2.3.3 Direction of sliding and shape of failure body

A new feature in the three-dimensional analysis compared with its two-dimensional

equivalence is the direction of sliding, introduced by Huang & Tsai (2000). The direction of

sliding is defined as the main route the sliding mass is moving during a failure in the

horizontal plane. In the two-dimensional plane strain analysis, the direction of sliding is

obviously limited to the plane of interest. However, in a three-dimensional analysis the

direction can be assumed or calculated, depending on method. Because the factor of safety

will have a minimum value in one direction for a given failure body, an assumed direction of

sliding might give an inaccurate solution. (Kalatehjari, Arefnia, Rashid, Ali, & Hajihassani,

2015) The direction of sliding is only relevant to consider for methods where the failure

surface is predetermined, like in the limit equilibrium method. In methods like the strength

reduction method or similar, the direction of sliding and the shape of the failure body is a

result of the method itself. (Camargo, Velloso, Euripedes, & Vargas, 2016)

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When the limit equilibrium method is used to determine the factor of safety for a slope, an

iterative process is usually performed to test different failure surfaces in the search for the

least stable one. For three-dimensional applications, this process might be very time-

consuming and difficult because the method requires qualified assumptions. A few

suggestions can be found in the literature about optimization and determination of the failure

surface with different methods. Kalatehhari, Arefnia, Rashid, Ali & Hajihassani (2015)

proposed a method using particle swarm optimization to determine the shape of three-

dimensional failure surfaces as a complement to the method of columns, with good results.

Another method proposed by Cheng, Lui, Wei & Au (2005) used NURBS functions to

describe failure surfaces. It was concluded that an assumed symmetric elliptical failure

surface was sufficient for normal problems, and NURBS functions were recommended when

the surface was highly irregular.

2.3.4 Three-dimensional slope stability with the finite element method

Slope stability analysis with the finite element method can be performed in three dimensions

as well as two. Like in the two-dimensional equivalence the shape of the failure body is a

consequence of the method and can therefore be obtained as a result instead of an input like in

the limit equilibrium method. This is a huge advantage, because of the difficulty to assume a

reasonable slip surface, especially in three dimensions. A unique feature with a three-

dimensional finite element analysis is the direction of sliding of a slope failure, which also is a

result of the method itself (Camargo, Velloso, Euripedes, & Vargas, 2016). These advantages

make the finite element method very powerful to collect insight in the behavior of slope

failures. The drawback with the three-dimensional analysis using finite element is the huge

computational time, especially with the fact that this procedure only results in one slip

surface. Another drawback is the high license costs, which is why three-dimensional finite

element software is today seldom used by geotechnical engineers for slope stability analysis.

Further knowledge on the topic can be found in Alkasawneh, Malkawi, Nusairat & Albataineh

(2007), Camargo, Velloso, Euripedes & Vargas (2016), Kalatehhari, Arefnia, Rashid, Ali &

Hajihassani (2015).

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3 SOFTWARE

3.1 PLAXIS

PLAXIS is a software developed to study geotechnical problems and is commonly used by

engineers and researchers around the world. It was first developed in 1987 for the analysis of

deformation, stability and groundwater flow with the finite element method, which makes it

possible to model most geotechnical problems. Initially PLAXIS was a 2D application only,

and in 2010 a 3D application was released. (PLAXIS, 2016)

3.1.1 Elements and mesh

Like the normal finite element modeling procedure, PLAXIS discretize the studied region into

finite elements that together form a mesh. The elements contain nodes with a number of

degrees of freedom, which are the unknowns to be solved in an analysis. For a deformation

analysis, the unknowns correspond to displacements of the nodes (PLAXIS, 2016). The

elements also contain Gauss points (stress points), where the constitutive relation is applied to

define the relation between stresses and strains. In the 2D program the default elements

contain 15 nodes and 12 Gauss points. Elements with fewer nodes and Gauss points can be

chosen as well, which contains less information because of fewer unknowns. This choice will

reduce the calculation time but results in an inferior solution. (Mattsson, 2017) In PLAXIS

3D, the basic element have 10 nodes and 4 Gauss points with a tetrahedral shape (PLAXIS,

2016). The 2D-elements can be seen in Figure 9 as alternative a and b. The 3D-element can be

seen in alternative c.

Figure 9 Elements used in the PLAXIS-program. Reference: www.plaxis.nl

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A sufficient fine mesh is necessary in PLAXIS to obtain accurate solutions. (PLAXIS, 2016)

The model must be tested with finer mesh until next refinement change the results

insignificantly.

3.1.2 Model size and Boundary conditions

The boundary surfaces and their influence on the result are important factors to consider in

every PLAXIS model. If the space between the expected event and the boundaries during a

phase is large, the boundary influence will become small. The disadvantage with a large

model is the greater number of elements required, which will lead to longer calculation times.

A suitable compromise must be found for effective modeling, where the calculation time is

held as short as possible without significant influences from the model boundaries.

How the elements behave at the boundaries can be prescribed with boundary conditions. They

can be manually adjusted for every surface with five different settings, free, normally fixed,

horizontally fixed, vertically fixed or fully fixed. The free setting gives the surface the ability

to move freely in all cartesian directions, while the fixed alternative means that they are

locked in all directions. The normally fixed alternative locks the elements at the boundary

from moving in the normal direction. Movements in the two remaining directions parallel to

the surface are free. The horizontal or vertical fixed option are similar to the normally fixed

alternative but is also fixed in the horizontal or vertical direction. (PLAXIS, 2016)

3.1.3 Constitutive models

A constitutive relation must be defined to describe the material behavior in the PLAXIS

model, which can be chosen with different degrees of accuracy. The simplest constitutive

model uses a linear elastic relation between stresses and strains. The relation is based on the

Hooke´s law of isotropic elasticity and is suitable for the modeling of stiff materials like steel,

concrete or rock that is not expected to plasticize. (PLAXIS, 2016) The most commonly used

model for soil materials is the Mohr-Coulomb model, which describes the stress-strain

relationship as linear elastic perfectly plastic. With this model, deformations will develop

linearly in proportion to the effective stresses up to a certain point where the stresses cannot

be increased any further. At this point, plastic deformations occur with a constant rate, which

is illustrated in Figure 10.

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Figure 10 Linear elastic perfectly plastic material model

The Mohr-Coulomb material model requires five input parameters, E (young´s modulus),

(poisons ratio), (friction angle), c (cohesion) and (the dilatancy angle). The parameters E

and describes the elasticity, and the strength and the dilatancy. More advanced

material models are selectable as well, which contains more parameters to e.g. define the

stiffness. In the hardening soil and soft soil material models, the stiffness is stress dependent.

(PLAXIS, 2016) However, what most models have in common is the Mohr-Coulomb failure

criterion, which is defined with six equations, that together forms six planes in the , and

-effective principal stress space, called a yield surface. An illustration of the Mohr-

Coulomb yield surface is shown in Figure 11.

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Figure 11 Mohr-Coulomb failure criterion

3.1.4 Initial conditions: Gravity loading and 0-procedure

A PLAXIS model is usually calculated in phases, and the first one is called the initial phase.

This is where the initial stresses of the soil are generated. In PLAXIS, this can be modeled in

two ways, -procedure or gravity loading. The -procedure generate the initial effective

vertical stresses to obtain equilibrium with the self-weight of the soil and then calculates the

effective horizontal pressures using the -value. (PLAXIS, 2016) This is expressed with the

following equation:

′ = ′ (3.1)

The value of in PLAXIS is by default calculated with Jaky’s empirical formula with

drained parameters as:

= 1 − sin (3.2)

For an undrained analysis the default value of is therefore one but can be manually

adjusted. To ensure that equilibrium is fulfilled, only horizontal layers and surfaces are

recommended to use in the -procedure. If the initial geometry contains non-horizontal

23

layers or surfaces, the gravity loading option is recommended instead. The initial stresses in

the gravity loading is strongly dependent on the value of the Poisson’s ratio. The value of

is calculated with the following equation:

= (3.3)

With this definition, there is a problem with generating -values larger than 1, because is

restricted to be smaller than 0.5. This requires the user to model a historic pre-consolidation

phase with different values of for loading and unloading. (PLAXIS, 2016)

3.1.5 Drained and undrained analysis

It is possible to perform both drained and undrained analyses with PLAXIS. The drained

option calculates the behavior of soil where no excess pore pressures remain, which is suitable

for coarse grained soils and long-term analysis. Undrained soil behavior is possible to model

in three ways, termed Undrained A, B and C. What separates them is their definitions of

strength and stiffness parameters. Undrained A use effective strength and stiffness parameters,

which makes the undrained shear strength a consequence of the chosen effective parameters

and stresses rather than an input. The undrained B option use an input parameter for the shear

strength and an effective parameter for the stiffness. The last one, undrained C defines both

the shear strength and the stiffness with undrained input parameters. The most suitable model

depends on the type of analysis. (PLAXIS, 2016)

3.1.6 Safety calculation

Safety calculation in PLAXIS refers to the step by step reduction of strength parameters

explained in 2.1.2. To make sure that the obtained factor of safety from an analysis is valid,

the output program must be inspected to ensure that a reasonable failure surface is formed.

The number of steps can be manually changed and is set to 100 by default. For some cases

even up to 10000 steps are required. (PLAXIS, 2016) The safety curve is supposed to be

horizontal before a valid factor can be obtained.

3.2 GeoStudio, SLOPE/W

GeoStudio is a compound of several commercial softwares that are using different methods to

solve geotechnical problems. SLOPE/W is the most commonly used software to solve two-

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dimensional slope stability problems and is applying the limit equilibrium method with the

method of slices to determine the factor of safety. SLOPE/W can solve for the factor of safety

of thousands of slip surfaces fast and then present them in increasing order. The most

common methods like the ordinary, Bishop’s simplified, Janbu’s simplified, Spencers and

Morgenstern-Price can all be used simultaneously. (GEO-SLOPE, 2012)

3.2.1 Slip surface definition with enter and exit or grid and radius

The failure surface input can be selected by choosing “enter and exit” zones as well as a “grid

and radius”. The enter and exit zones describes the spans over which the failure surface can

start and exit the slope. Several failure surfaces are then tested within the selected spans. The

grid and radius option make the user define the moment centers and radiuses of the circular

failure surfaces. All combinations of the chosen moment centers and radiuses are then

calculated. (GEO-SLOPE, 2012)

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4 NUMERICAL MODELLING WITH PLAXIS 3D

In this chapter the studied geometries will be shown along with the chosen parameters used in

PLAXIS 3D.

4.1 Geometries and cases

The studied geometries were divided into three groups, A, B and C. The groups were defined

with respect to their slope angles. Group A models simulate excavations with slopes inclined

1:2, group B models with slopes inclined 1:1 and group C were vertical walls. All models

simulate three-dimensional excavations with a depth of 2 meters. Within each group, the

excavation length was varied along with an external load that is acting on a 4×4 meter area, 1

meter from the excavation edge in the center of the slope length, see Figure 12. The

excavation lengths were divided into 7 categories, 4, 6, 8, 12, 16, 24 and 32 meters, measured

along the excavation bottom. For group A and B, the load magnitude started at 5 kN/m2 and

was increased in steps of 5 kN/m2 until failure was reached. For group C the same procedure

was repeated but was initially started from 2.5 kN/m2 and with 2.5 kN/m2 increments. A non-

loaded case was performed for all groups and lengths as well. The groundwater surface was

defined to follow the ground surface in all calculated phases. This means that the water level

was located at the soil surface boundary in the initial phase and then adjusted in the

excavation phase to follow the excavated soil surface. The excavated volumes were defined as

dry. The three groups that simulate three-dimensional excavations were also modeled as their

two-dimensional equivalence in PLAXIS 3D. This means that a slice with finite length of the

3D excavation was modeled with normally fixed boundary conditions. The geometry can be

seen in Figure 14. All conditions and parameters remain the same for all models.

To make the models more efficient regarding the calculation time, symmetry was utilized,

meaning that only a quarter of the total excavation geometry was modeled. To keep the

number of elements as small as possible, the models were calculated in size groups, one small

and one large. The excavation sizes 4 - 12 meters were in the small group, and 16 - 32 were in

the large. The two size groups and two-dimensional model for group A is presented in Figure

12, Figure 13 and Figure 14 respectively. The remaining group geometries are found in

Appendix B.

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Figure 12 Small excavations, group A

Figure 13 Large excavations, group A

Figure 14 Two-dimensional equivalence of model A

27

4.2 Model specification

All models were constructed equally except for the different slope angles. To make the

models more efficient regarding the calculation time, a rectangular excavation bottom was

used for all groups. This makes one of the slopes longer than the other, and therefore making

one of the slopes more likely to collapse during the safety calculation (see chapter 3.1.6). The

excavation bottom size in the models for every corresponding excavation bottom length can

be found in

Table 1. How the symmetry is considered in the models is illustrated in Figure 15. The model

specifications for the different groups are found inTable 2,

Table 3 and

Table 4.

Figure 15 Example of how model symmetry is considered for a small A-group model

Table 1 Excavation bottom size for every modeled excavation slope length in PLAXIS 3D

Excavation bottom length, no symmetry Model excavation bottom size, withsymmetry

4 m 2 × 1 m6 m 3 × 1 m8 m 4 × 2 m

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12 m 6 × 2 m16 m 8 × 2 m24 m 12 × 2 m32 m 16 × 2 m

Table 2 Group A, model specifications

Group A, model specificationsModel length, y, small/large 14 / 24 mModel width, x, small/large 12 / 12 mModel height, z, small/large 5/5 mNumber of elements small/large 24523 / 24630Element dimension small/large 0.6687 / 0.6824Coarseness factor small/large 0.9 / 1Boundary conditions y, x Normally fixedBoundary conditions z Fixed/free

Table 3 Group B, model specifications

Group B, model specificationsModel length, y, small/large 12 / 22 mModel width, x, small/large 12 / 12 mModel height, z, small/large 4 / 4 mNumber of elements small/large 19028 / 20303Element dimension small/large 0.6102 / 0.6124Coarseness factor small/large 1 / 1.05Boundary conditions y, x Normally fixedBoundary conditions z Fixed/free

Table 4 Group C, model specifications

Group C, model specificationsModel length, y, small/large 10 / 20 mModel width, x, small/large 12 / 12 mModel height, z, small/large 4 / 4 mNumber of elements small/large 9109 / 17650Element dimension small/large 0.6708 / 0.5477Coarseness factor small/large 0.9 / 1.1Boundary conditions y, x Normally fixedBoundary conditions z Fixed/free

To be able to create a safety factor curve with respect to deformation from the safety

calculations in PLAXIS, a node located at the slope edge in the symmetry line was selected.

The curve is then generated by plotting the SFR-value with the total deformation of the node

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which gives information about the factor of safety and the convergence of the solution. The

location of the selected node for a group A model is shown in Figure 16. The location of the

node is selected equivalently for the group B- and C models.

Figure 16 Location of the selected node

4.3 Phases

All cases were calculated using three phases, starting with the initial phase where the K0-

procedure was used to generate the initial state stresses. The second phase was a plastic phase,

where the excavation was performed, and the load was applied. The last phase was the safety

calculation, using the strength reduction method. More specific phase information can be

found in Table 5.

Table 5 Phase settings in the PLAXIS-models

Phase settingsInitial phase

Initial condition -procedurePlastic phase

Tolerated error 0.001Safety phase

Steps 100 - 250Tolerated error 0.001

4.4 Material parameters

The same material parameters are used for all models and can be found in Table 6.

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Table 6 Material parameters

Material parametersMaterial model Mohr-Coulomb (Linear elastic – perfectly

plastic)Drainage type Undrained B

5 kN/m2

3 kN/m2/m′ 1000 kN/m2

0,350,516 kN/m3

17 kN/m3

4.5 Mesh and boundaries

The element mesh in the models were initially coarse in order to determine if a result was

possible to obtain with a specific geometry and set of parameters. The mesh was then refined

stepwise until another refinement gave insignificantly different results. This element size is

then chosen for all similar models. An example of how the safety factor depends on the

number of elements for a group A model is presented in Figure 17. The mesh was only

globally refined using the element distribution category and the coarseness factor. This type

of check was performed for all the different group geometries.

Figure 17 The diagram shows how the factor of safety change with the increasing number of elements

1

1,1

1,2

1,3

1,4

1,5

1,6

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Fact

orof

safe

ty

Number of elements

Factor of safety and number of elements

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5 SLOPE/W MODELS AND 3D-EFFECT CALCULATIONS

The SLOPE/W models were divided into equivalent groups as the PLAXIS models. The same

number of load increment cases was performed as well.

5.1 SLOPE/W models

In order to generate an equal shear strength profile in the SLOPE/W models as in the PLAXIS

models, the strength relative to a specified level was used as the strength function. This was

used because the initial stresses in the PLAXIS models were generated with K0-procedure

and a horizontal ground surface. This makes the shear strength defined as a function of depth

in the initial phase, giving the excavation bottom a higher strength even after the excavation

phase. The porewater-pressures were generated by drawing a piezometric line along the

ground surface after the excavation.

The slip surface shape was defined with the grid and radius alternative, which means that the

shape was assumed to be circular. The grid and radius locations from a group A model can be

seen in Figure 18.

Figure 18 Grid and radius selection of a group A model

The grid was defined with 20 x 20 points and the radius with 10 lines. If the most critical slip

surface resulted with a moment center near the edge of the grid or at the top or the bottom of

the radius, their locations was moved or extended. Further specific model information can be

found in Table 7.

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Table 7 Model specific information from SLOPE/W

Model specificationsAnalysis type Morgenstern-price methodNumber of slices 30Side function Half-sine functionPWP-conditions Piezometric line

5 kN/m2

3 kN/m2/m

5.2 Calculation of three-dimensional effects

The result obtained from the SLOPE/W models were used as the 2D slip surface for every

load case. The calculations of the three-dimensional factor of safety were performed with the

equations given in chapter 2.3.1 with the input values from the SLOPE/W result tab. The part

of the equation that refers to the end surface planes were calculated approximately from the

slip surface results. The slip surface shape was divided into three or four triangles and

rectangles, and a weighted average shear strength was calculated. The distance from the

rotation center to the center of gravity for the entire slip body was calculated from the results

tab data in SLOPE/W and approximations from the resulting slip body geometry.

The excavation length (L) in equation 2.23 was defined as the length of the surface load.

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6 RESULTS AND ANALYSIS

The models calculated in PLAXIS 3D resulted in a load magnitude of maximum 30 kN/m2 for

group A, 25 kN/m2 for group B and 5 kN/m2 for group C. Further load increase resulted in

large deformations and the initiation of a failure mechanism already in the plastic calculation

phase.

The factor of safety for all models was evaluated from the selected node in the generated

SFR/deformation-curve. To ensure that this curve is representative, the PLAXIS output

program was always inspected to verify that the developed failure occurred at the location of

the node. This is important because the SFR-factor is plotted against the total deformation of

this particular node. A total deformation of 0.2 meters in the plastic- and safety calculation

phase added together was used as the failure criterion, given that the model was deforming

reasonably. A problem with the evaluation of the safety factor from the strength reduction

method, especially in 3D, is the continuously rising safety factor with larger deformations.

The geotechnical support engineer at PLAXIS, S. Papavasileiou (E-mail communication, 15

Feb 2018) explained that in some cases, there is a stress redistribution when a failure is

occurring which results in an increasing safety factor. This means that the soil can behave

differently after the redistribution and therefore develop higher strengths. The

recommendation was therefore to always inspect the failure mechanism in the PLAXIS output

program and evaluate the factor at a reasonable deformation. The recommendations were

followed for every evaluation.

The evaluated factors of safety for the models in group A are found in Table 8, group B in

Table 9 and group C in Table 10. The obtained results from the SLOPE/W calculations and

the calculated three-dimensional effects from the Swedish commission of slope stability can

be seen in the same tables for their corresponding group. Because the case with no load has no

reason to be calculated for a failure width matching the load, this case is calculated with the

3D-effects-width of the excavation bottom instead. The result can be seen in Table 11.

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Table 8 The evaluated factor of safety from the models in group A

Group ALength No load 5 kN/m2 10 kN/m2 15 kN/m2 20 kN/m2 25 kN/m2 30 kN/m2

4 m 2,52 2,32 2,09 1,86 1,62 1,38 1,176 m 2,41 2,24 2,05 1,84 1,61 1,38 1,178 m 2,36 2,21 2,03 1,83 1,61 1,37 1,1612 m 2,29 2,17 2,02 1,83 1,60 1,37 1,1616 m 2,26 2,18 2,04 1,85 1,62 1,40 1,2024 m 2,23 2,16 2,04 1,85 1,62 1,40 1,2032 m 2,21 2,15 2,04 1,85 1,62 1,40 1,202D 2,17 1,92 1,69 1,48 1,31 1,16 1,03SLOPE/W 2,08 1,84 1,62 1,42 1,26 1,14 1,033D-effects Table 11 2,14 1,96 1,76 1,60 1,48 1,37

Table 9 The evaluated factor of safety from the models in group B

Group BLength No load 5 kN/m2 10 kN/m2 15 kN/m2 20 kN/m2 25 kN/m2

4 m 2.03 1.84 1.63 1.44 1.27 1.126 m 1.94 1.78 1.60 1.42 1.26 1.128 m 1.89 1.76 1.59 1.42 1.26 1.1212 m 1.86 1.74 1.59 1.42 1.26 1.1216 m 1.86 1.74 1.59 1.42 1.26 1.1124 m 1.83 1.72 1.59 1.42 1.26 1.1132 m 1.82 1.72 1.59 1.42 1.26 1.112D 1.77 1.56 1.35 1.18 1.04 FailSLOPE/W 1.62 1.45 1.25 1.10 0.98 0.883D-effects Table 11 1.75 1.54 1.39 1.27 1.18

Table 10 The evaluated factor of safety from the models in group C

Group CLength No load 2,5 kN/m2 5 kN/m2

4 m 1.32 1.25 1.186 m 1.22 1.16 1.098 m 1.20 1.13 1.0612 m 1.15 1.10 Fail16 m 1.13 1.09 Fail24 m 1.12 1.09 Fail32 m 1.12 1.09 Fail2D 1.13 Fail FailSLOPE/W 0.97 0.89 0.813D-effects Table 11 1.22 1.15

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Table 11 The calculated 3D-effect for a failure body withdifferent lengths and no load for group A-C

No load, 3D-effectLength Model A Model B Model C4 m 2.35 1.89 1.306 m 2.26 1.80 1.198 m 2.22 1.75 1.1312 m 2.17 1.71 1.0816 m 2.15 1.69 1.0524 m 2.13 1.67 1.0232 m 2.11 1.65 1.01

6.1 Two-dimensional comparison

The evaluated factors of safety for the 2D cases in PLAXIS match well with the SLOPE/W

values for group A and B. The reason behind the slightly higher factor of safety-value

obtained in the PLAXIS models are unknown. A general explanation regarding different

results is the slip surface shape. In the analytical SLOPE/W-analyses the slip surface shape is

defined and by the user, which in this case is circular. In the PLAXIS-models on the other

hand, the slip surface shape is a consequence of the method itself. This might be one reason

for the different safety factors between the methods, but reasonably this should explain lower

safety factors for the PLAXIS-analyses, not higher. For the group A models, the difference

varies between 0 and 4,3% between the methods. For group B the difference varies between

6,1 and 9,3%, and for group C, 16,9%. The big difference between the results in group C

might be explained by the factor of safety value being very close to 1,0. The stress

redistribution effect explained earlier might be more palpable for low values because the

methods are compared relatively to each other. If the relative shear stress in the plastic phase

is studied in the PLAXIS output program for the unloaded 2D-case of group C (upper picture

in Figure 19), the shear strength of an entire slip surface is not fully mobilized, which should

mean that the factor of safety is greater than 1,0, unless a local failure is occurring in the

slope. For the 2,5 kN/m2 case (bottom picture in Figure 19), the red-colored area matches a

possible slip surface region where the mobilized shear stress is very close or equal to the

maximum shear strength. Even though a soil body collapse was not obtained for the 2,5

kN/m2 model, the factor of safety was evaluated to be smaller than 1,0. This was also the case

for the 5 kN/m2 model in the same group. A soil body collapse was not obtained even though

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deformations of nearly 0,35 m were developed already in the plastic phase. The reason why a

soil body collapse was not obtained is unknown.

Figure 19 The relative shear stress in the plastic phase (PLAXIS),comparison between the non-loaded and the 2,5 kN/m2 loaded case of thegroup C model in 2D

6.2 Three-dimensional analysis

It can be seen in Table 8, Table 9 and Table 10 that the factor of safety generally becomes

lower or remain constant as the excavation length is increased, especially for the unloaded

cases. As the load increases, the factor of safety difference between the excavation length

cases becomes smaller. For group A in the unloaded case, the factor of safety difference

between a 4-meter- and a 32-meter excavation is about 14,0%. Already at a load magnitude of

10 kN/m2 for the same models, the factor of safety difference is just 2,5%. Similar

phenomenon can be observed to happen in all groups. In the total deformation plots

(Appendix B), it is seen that the failure surface shape changes less as the load becomes higher

when the excavation length is increased. If the factor of safety result and the total deformation

plot is both compared, a load magnitude of about 15 kN/m2 or more for group A and B seems

to be the limit where the failure surface and stability does not change at all with excavation

length any more. This effect cannot be seen in group C because of the low loads required for

the vertical wall to collapse.

37

In contrast to previous observations, the factor of safety is slightly higher as the excavation

length is increased for the high load cases in group A. This is most likely because of model

specific reasons and not an actual phenomenon. The reason that the large models (16 – 32 m)

result in a slightly higher factor of safety is probably a consequence of the element

distribution. PLAXIS is automatically generating finer mesh around the surface load area and

might have generated smaller elements over that region in the large model than in the small. A

finer element mesh will decrease the safety factor (see 4.5 Mesh and boundaries).

Figure 20 Comparison of group A, total deformations in PLAXIS between a cross section of the 6-meter-long 3D-excavationand its 2D-equivalence

38

In Figure 20 a cross section comparison between the slip surface shapes in 3D and its 2D-

equivalence can be seen for the 6-meter long group A models. The slip surface in the 3D-

model with no load is very similar to the 2D-model regarding both size and shape. When the

load is increasing, the size of the slip surface is decreasing considerably for the 3D-model but

is relatively maintained for the 2D-model. The similar 3D-models with a top view of the total

deformation can be seen in Figure 21. As the load is increasing, the width of the failure body

is decreasing. When the load is higher, the failure surface is more local. This result might be

explained by the fact that PLAXIS is using the strength reduction method, which is a global

searching method. For the unloaded case, the entire slope may be close to collapse when the

first failure mechanism is initiated. This results in a large collapse when one part of the slope

is deforming. When the load is locally high, and the strength is decreased incrementally, only

a small part of the slope is close to failure when the first failure mechanism is initiated.

Figure 21 Top view of the total deformation of the 6-meter-long group A models in 3D

39

6.3 3D-effects method compared with PLAXIS 3D models

In Table 12, Table 13 and Table 14 the factor of safety obtained from the PLAXIS 3D models

are compared with the results from the 3D-method proposed by the Swedish commission of

slope stability. A positive value means that the PLAXIS result is higher.

40

Table 12 The difference between the factor of safety obtained from PLAXIS 3D and the calculated 3D-factor from the methodproposed by the Swedish commission of slope stability for group A

Group ALength No load 5 kN/m2 10

kN/m215kN/m2

20kN/m2

25kN/m2

30kN/m2

4 m 7,2% 8,4% 6,6% 5,7% 1,3% -6,8% -14,6%6 m 6,6% 4,7% 4,6% 4,5% 0,6% -6,8% -14,6%8 m 6,5% 3,3% 3,6% 4,0% 0,6% -7,4% -15,3%12 m 5,5% 1,4% 3,1% 4,0% 0,0% -7,4% -15,3%16 m 5,2% 1,9% 4,1% 5,1% 1,3% -5,4% -12,4%24 m 4,9% 0,9% 4,1% 5,1% 1,3% -5,4% -12,4%32 m 4,5% 0,5% 4,1% 5,1% 1,3% -5,4% -12,4%

Table 13 The difference between the factor of safety obtained from PLAXIS 3D and the calculated 3D-factor from the methodproposed by the Swedish commission of slope stability for group B

Group BLength No load 5 kN/m2 10 kN/m2 15 kN/m2 20 kN/m2 25 kN/m2

4 m 7,6% 5,1% 5,8% 3,6% 0,0% -5,1%6 m 7,9% 1,7% 3,9% 2,2% -0,8% -5,1%8 m 7,7% 0,6% 3,2% 2,2% -0,8% -5,1%12 m 8,8% -0,6% 3,2% 2,2% -0,8% -5,1%16 m 10,2% -0,6% 3,2% 2,2% -0,8% -5,9%24 m 9,8% -1,7% 3,2% 2,2% -0,8% -5,9%32 m 10,0% -1,7% 3,2% 2,2% -0,8% -5,9%

Table 14 The difference between the factor of safety obtained from PLAXIS 3D and the calculated 3D-factor from the methodproposed by the Swedish commission of slope stability for group C

Group CLength No load 2,5 kN/m2 5 kN/m2

4 m 1,5% 2,5% 2,6%6 m 2,6% -4,9% -5,2%8 m 5,8% -7,4% -7,8%12 m 6,7% -9,8% -16 m 7,6% -10,7% -24 m 9,5% -10,7% -32 m 11,1% -10,7% -

41

In Table 12 the difference is found to be very small for group A between the 3D-effects

method and PLAXIS 3D for several cases. The exceptions are the high load cases, no load

case and the 4-meter-long excavation cases. For loads smaller than 20 kN/m2, the 3D-effects

method results in a lower safety factor than the PLAXIS models. At 20 kN/m2, the match

between the methods is almost perfect and when the load is higher than 20 kN/m2, the 3D-

effects method results in a higher safety factor instead. This seems to be the case for group B

as well (Table 13), with the exception that a very good match is also found with a 5 kN/m2

load. Like for group A, the group B comparison also show small differences for several cases,

especially the intermediate load cases. The cases with no load, a high load or a 4-meter

excavation gives the largest differences between the 3D-effects method and the PLAXIS

models. For group C (Table 14), no good matches are found. For the no load case, the result

difference is increasing with the slope length, and the PLAXIS result is the highest for all

lengths in this load class. The 2,5 and 5 kN/m2 models show large differences as well, but the

PLAXIS result is higher, with the exception for the 4-meter case.

The results suggest that for conventional undrained slope stability calculations when there is a

surface load defining the failure boundaries, the 3D-effects method generally give safety

factors close to what PLAXIS 3D will give. Because the general 3D safety factor is higher

than the equivalent 2D safety factor, the empirical correction equation in the 3D-effects

method is probably not suited for 3D-effects without a load that is defining a certain failure

surface. Vertical walls are found to correlate bad as well between the methods, which might

depend on that the 3D-effects equation is empirically corrected for typical circular failure

surfaces. The vertical excavation walls (group C) give almost plane failure surfaces in both

PLAXIS 3D and SLOPE/W (Appendix C3).

The reason why high load cases does not match well between the methods is most certainly

because a totally different failure surface is generated in PLAXIS 3D than in SLOPE/W

(Figure 20 and Appendix C3). In the 3D-effects calculation, the SLOPE/W result is used as

the failure body cross-section that is extended in the third dimension by defining the length.

The cross-section input from a slope stability calculation in 2D does not match with the cross-

section of the most critical 3D-surface. Hence, it is reasonable that the safety factors do not

match well either. For all excavation lengths in group A and B, the 3D-effects calculation

creates a more conservative design than PLAXIS 3D when the load is larger than 20 kPa.

42

In the 4-meter excavation case, the surface load and the excavation bottom are equally long.

This means that the perpendicular excavation walls could potentially create some support if

the most critical failure body is wider than the load. This contribution of support is accounted

for in the PLAXIS 3D-models, but not definable in the 3D-effects method. This could explain

why the 4-meter excavation models for some cases have a greater difference between the

methods than the other length cases. This seems to be the case when the surface load is

relatively low. In Table 12 and Table 13 it can be seen that the factor of safety for the higher

load cases is almost identical between the 4-meter excavation and the longer ones. This is

probably the case because the surface load is high enough to create a more local failure, where

the supporting perpendicular walls does not matter. With other words, when the surface load

is high enough, a longer excavation will not decrease the factor of safety.

43

6.4 Safety analysis in PLAXIS

In appendix A, the safety factor diagrams plotted against the total deformation of the selected

node is found. Note that the y-axis scale is different between the groups. The x-axis in the

figures are spanning up to a total node deformation of 0,2 meters, which corresponds to the

decided failure criterion. The strength increase effect explained in chapter 6 can be seen in the

figures, especially for the group C models in appendix A3. If the number of steps is increased

in the safety analysis, the value of ∑ is increased as well. Therefore, a realistic

deformation is necessary to determine a realistic factor of safety.

The results from Table 8, Table 9 and Table 10 was confirmed with the safety factor plots in

Appendix A, regarding the constant safety factor for varying lengths of the slope with a high

load. With other words, the deformation plots between the different length cases are very

similar for all high load cases. They all follow the same failure path, which suggests that the

failure no matter the length of the slope, is similar.

44

45

7 DISCUSSION

The results from this study was partly expected. The fact that the three-dimensional safety

factor was higher than its two-dimensional equivalence was confirmed. This difference is of

course larger with a local load present compared to the unloaded case. The reason is because

the 2D model represents an infinite long slope with an infinite long load acting as well, which

means that the total load compared to the resisting soil volume is greater. What was

unexpected was that the factor of safety for a 32-meter long and 2-meter deep excavation with

no load applied already had the same value as a the equivalent 2D-calculated slope. This

means that long unloaded excavation slopes, like cable excavations, can be calculated two-

dimensionally without making the design too conservative. In order to optimize the

excavation design, the excavation could be calculated two dimensionally as long as no loads

are acting on the slope top, and three-dimensionally when a local load is present.

The results from this study suggests that the recommendations given in 1995 from the

Swedish commission of slope stability regarding 3D-effects gives satisfying results for normal

cases of undrained slope stability calculations. For slope stability calculations where the

defined 3D-length (e.g. surface load) and the excavation width are similar, if the slope is very

steep (steeper than 1:1), or the 2D-safety factor is very low due to a high local load, the

scenario should be calculated with a real 3D-method instead, like FE-modelling. The results

suggest that both conservative and unconservative designs can be obtained with the 3D-

effects calculation compared to a FE-analysis. If an error of less than approximately 10-15%

is very important, a real 3D calculation should be performed instead.

What to keep in mind when 3D-stability is calculated is that extra effort should be put in the

evaluation of the reasonability of the results, but also in the input definition. Usually in

geotechnical engineering when a 2D-slope stability calculation is performed, the entire slope

length has the same weighted average shear strength, and a critical section is then chosen.

This gives in general a very conservative design. In a 3D-analysis, the slope failure will be

more local. Therefore, a more local definition of the shear strength parameters should be

considered as well, and because all 3D-slope stability calculations give a more aggressive

design due to the increasing factor of safety, an evaluation of the reasonability of the result is

more important.

46

Another important aspect to keep in mind when the FE-method is used to evaluate 3D-slope

stability is the element mesh size. PLAXIS is using an automatic element size generation from

the chosen coarseness factor, where a refinement zone is automatically performed around

loads or other objects. If results are to be compared between different models, the element

size around the failure zone should be compared and maybe adjusted so that the models have

similar numerical conditions. As seen in Figure 17 the factor of safety can differ up to 20%

depending on element size!

7.1 Suggestions on further studies

Further study on the topic could be to investigate another set up of strength parameters, in

order to see if the 3D-effects calculation correlates similar with low strength and high strength

cohesive soils.

Another investigation could include frictional soils in the analysis. Especially because there is

no good suggestion for the treatment of the three-dimensional effects with frictional soils

today. Analysis of drained behavior for cohesive soils in long term could be investigated as

well.

47

REFERENCES

Alkasawneh, W., Malkawi, A. H., Nusairat, J. H., & Albataineh, N. (2007). A comperative

study of various commercially available programs in slope stability analysis.

Computers and geotechnics, 428-435.

Bishop, A. W. (1955). The use of the slip circle in the stability analysis of slopes.

Géotechnique, 7-17.

Camargo, J., Velloso, R. Q., Euripedes, A., & Vargas, J. (2016). Numerical limit analysis of

three-dimensional slope stability problems in catchment areas. Acta Geotechnica,

1369-1383.

Carrión, M., Vargas, E. A., Velloso, R. Q., & Farfan, A. D. (2017). Slope stability analysis in

3D using numerical limit analysis (NLA) and elasto-plastic analysis (EPA).

Geomechanics and Geoengineering, 250-265.

Chen, Z., Mi, H., Zhang, F., & Wang, X. (2003). A simplified method for 3D slope stability

analysis. Canadian Geotechnical Journal, 675-683.

Cheng, Y., Liu, H., Wei, W., & Au, S. (2005). Location of critical three-dimensional non-

spherical failure surface by NURBS functions and ellipsoid with applications to

highway slopes. Amsterdam: Elsevier.

Cheng, Y., Lui, H., Wei, W., & Au, S. (2005). Location of critical three-dimensional non-

spherical failure surface by NURBS functions and ellipsoid with applications to

highway slopes. Computers and geotechnics, 387-399.

D.G Fredlund, J. K. (1977). Comparison of slope stability methods of analysis. Canadian

geotechnical journal, 429-439.

Gens, A., Hutchinsson, J., & Cavounidis, J. (1988). Three-Dimensional Analysis of Slides in

Cohesive Soils. Géotechnique 38, 1-23.

GEO-SLOPE. (2012, 11 05). Stability Modeling with SLOPE/W. Retrieved from geo-slope:

https://www.geoslope.com/support/downloads/whatsnew

Hovland, H. (1977). Three dimensional slope stability analysis method. Geotechnical

Engineering Division, 971-986.

48

Huang, C.-C., & Tsai, C.-C. (2000). New method for 3D and asymmetrical slope stability

analysis. Journal of Geotechnical and Geoenviromental Engineering, 917-927.

Johansson, A. (1991). Släntstabilitetsberäkningar med klassiska beräkningsmetoder. Luleå:

Skredkommisionen. (Reference in swedish)

Johansson, L., & Axelsson, K. (1991). Släntstabilitetsberäkningar med klassiska

beräkningsmetoder. Luleå: SGI. (Reference in swedish)

Kalatehjari, R., Arefnia, A., Rashid, A. S., Ali, N., & Hajihassani, M. (2015). Determination

of three-dimensional shape of failure surface in soil slopes. Canadian Geotechnical

Journal, 1283-1301.

Lam, L., & Fredlund, D. (1993). A general limit equilibrium model for three-dimensional

slope stability analysis. Canadian Geotechnical Journal, 905-919.

Matsui, T., & San, K.-C. (1992). Finite element slope stability analysis by shear strength

reduction technique. Japanese Society of Soil Mechanics and Foundation Engineering,

59-70.

Mattsson, H. (2017, 11 20). LTU.SE. Retrieved from Luleå tekniska universitet:

https://ltu.instructure.com

Ottosen, N. S., & Petersson, H. (1992). Introduction to the Finite Element Method. Lund:

Prentice hall.

PLAXIS. (2016, 10 25). General Information. Netherlands: PLAXIS.

PLAXIS. (2016). Material models manual. Netherlands: PLAXIS.

PLAXIS. (2016). Reference manual 2016. Netherlands: PLAXIS.

Skredkommissionen 3:95. (1995). Anvisningar för släntstabilitetsutredningar. Linköping:

Ingenjörsvetenskapsakademien. (Reference in swedish)

Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel

inter-slice forces. Géotechnique, 11-26.

Terzaghi, K., & Peck, R. (1967). Soil mechanics in engineering practice. New York, NY:

John Wiley and Sons, inc.

49

Zhang, Y., Guangqi, C., Zheng, L., Li, Y., & Zhuang, X. (2013). Effects of geometries en

three-dimensional slope stability. Canadian Geotechnical Journal, 233-249.

50

51

APPENDIX A1 –SFR/DEFORMATION-CURVES, MODEL A

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, No load

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 5 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

52

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 10 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 15 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

53

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 20 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 25 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

54

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model A, 30 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

55

APPENDIX A2 –SFR/DEFORMATION-CURVES, MODEL B

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, No load

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, 5 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

56

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, 10 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, 15 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

57

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, 20 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

1,0

1,2

1,4

1,6

1,8

2,0

2,2

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model B, 25 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

58

1,0

1,1

1,2

1,3

1,4

1,5

1,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model C, 2,5 kN/m2

4 m

6 m

8 m

12 m

16 m

24 m

32 m

APPENDIX A3 –SFR/DEFORMATION-CURVES, MODEL C

1,0

1,1

1,2

1,3

1,4

1,5

1,6

0,00 0,05 0,10 0,15 0,20

ΣMsf

Deformation [m]

Safetyfactor Model C, No load

4 m

6 m

8 m

12 m

16 m

24 m

32 m

2D

59

APPENDIX B1 – TOTAL DEFORMATIONS, MODEL A

No load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

60

5 kN/m2 Load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

61

10 kN/m2 load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

62

15 kN/m2 load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

63

20 kN/m2 load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

64

25 kN/m2 load

32 meter

16 meter 24 meter

8 meter 12 meter

4 meter 6 meter

65

30 kN/m2 load

32 meter

24 meter16 meter

12 meter8 meter

4 meter 6 meter

66

2D

30 kN/m2

20 kN/m2 25 kN/m2

15 kN/m210 kN/m2

5 kN/m2No load

67

APPENDIX B2 – TOTAL DEFORMATIONS, MODEL B

No load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

68

5 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

69

10 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

70

15 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

71

20 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

72

25 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

73

2D

No load 5 kN/m2

10 kN/m2 15 kN/m2

20 kN/m2

74

APPENDIX B3 – TOTAL DEFORMATIONS, MODEL C

No load4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

75

2,5 kN/m2 load

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

32 meter

76

5 kN/m2 load

32 meter

4 meter 6 meter

8 meter 12 meter

16 meter 24 meter

77

2D

No load 2,5 kN/m2

5 kN/m2

78

APPENDIX C1 –SLOPE/W SLIP SURFACE GROUP A

No load 5 kN/m2

10 kN/m2 15 kN/m2

20 kN/m2 25 kN/m2

30 kN/m2

79

APPENDIX C2 –SLOPE/W SLIP SURFACE GROUP B

No load 5 kN/m2

10 kN/m2 15 kN/m2

20 kN/m2 25 kN/m2

80

APPENDIX C3 –SLOPE/W SLIP SURFACE GROUP C

5 kN/m2

No load 2,5 kN/m2