Three-Dimensional Effects in Slope Stability for Shallow Excavations Analyses with the Finite Element Program PLAXIS

Three-dimensional effects in slope stability for shallow 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

PREFACE

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

I would like to thank my supervisor Hans Mattsson from Luleå University of technology for all 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. A special thanks to Per Gunnvard at Luleå university for all the guidance and patience during the time of making this investigation.

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

Luleå, Mars 2018 Niclas Lindberg

i ii 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

iii iv 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

v vi TABLE OF CONTENTS

1. INTRODUCTION ...... 1

1.1 Background ...... 1 1.2 Purpose and objective ...... 2 1.3 Limitations ...... 3 2. THEORETICAL BACKGROUND ...... 5

2.1 Finite element method ...... 5 2.2 Limit equilibrium method ...... 10 2.3 Three-dimensional slope stability ...... 13 3. SOFTWARE ...... 19

3.1 PLAXIS ...... 19 3.2 GeoStudio, SLOPE/W ...... 23 4. NUMERICAL MODELLING WITH PLAXIS 3D...... 25

4.1 Geometries and cases ...... 25 4.2 Model specification ...... 27 4.3 Phases ...... 29 4.4 Material parameters ...... 29 4.5 Mesh and boundaries ...... 30 5. SLOPE/W MODELS AND 3D-EFFECT CALCULATIONS ...... 31

5.1 SLOPE/W models ...... 31 5.2 Calculation of three-dimensional effects ...... 32 6. RESULTS AND ANALYSIS...... 33

6.1 Two-dimensional comparison ...... 35 6.2 Three-dimensional analysis ...... 36 6.3 3D-effects method compared with PLAXIS 3D models ...... 39 6.4 Safety analysis in PLAXIS ...... 43 7. DISCUSSION ...... 45

7.1 Suggestions on further studies ...... 46 REFERENCES ...... 47

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

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

viii 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

1 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?

2 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.

3 4 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).

5 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:

00 0 0 0 0 + =0 (2.4) ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ 0 ⎥ 00 ⎢⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ This is called the strong formulation, which also can⎣ be written:⎦

+ =0 (2.5) ∇ 6 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

7 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 a3 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

8 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 constitutivė 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)

9 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).

10 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.

11 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).

12 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:

13 + 0.75 (2.24) , =, , − 1 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 + 2. For the same number of columns, it will result in 12 + 2 4 14 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

15 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)

16 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).

17 18 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

19 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.

20 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.

21 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

22 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-

23 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)

24 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.

25 Figure 12 Small excavations, group A

Figure 13 Large excavations, group A

Figure 14 Two-dimensional equivalence of model A

26 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 in Table 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, with symmetry 4 m 2 × 1 m 6 m 3 × 1 m 8 m 4 × 2 m

27 12 m 6 × 2 m 16 m 8 × 2 m 24 m 12 × 2 m 32 m 16 × 2 m

Table 2 Group A, model specifications

Group A, model specifications Model length, y, small/large 14 / 24 m Model width, x, small/large 12 / 12 m Model height, z, small/large 5/5 m Number of elements small/large 24523 / 24630 Element dimension small/large 0.6687 / 0.6824 Coarseness factor small/large 0.9 / 1 Boundary conditions y, x Normally fixed Boundary conditions z Fixed/free

Table 3 Group B, model specifications

Group B, model specifications Model length, y, small/large 12 / 22 m Model width, x, small/large 12 / 12 m Model height, z, small/large 4 / 4 m Number of elements small/large 19028 / 20303 Element dimension small/large 0.6102 / 0.6124 Coarseness factor small/large 1 / 1.05 Boundary conditions y, x Normally fixed Boundary conditions z Fixed/free

Table 4 Group C, model specifications

Group C, model specifications Model length, y, small/large 10 / 20 m Model width, x, small/large 12 / 12 m Model height, z, small/large 4 / 4 m Number of elements small/large 9109 / 17650 Element dimension small/large 0.6708 / 0.5477 Coarseness factor small/large 0.9 / 1.1 Boundary conditions y, x Normally fixed Boundary 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

28 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 settings Initial phase Initial condition -procedure Plastic phase Tolerated error 0.001 Safety phase Steps 100 - 250 Tolerated error 0.001

4.4 Material parameters

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

29 Table 6 Material parameters

Material parameters Material model Mohr-Coulomb (Linear elastic – perfectly plastic) Drainage type Undrained B 5 kN/m2 3 kN/m2/m 1000 kN/m2 0,35 ′ 0,5 16 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.

Factor of safety and number of elements 1,6

1,5 y

t 1,4 e f a s f

o 1,3 r o t c a

F 1,2

1,1

1 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 Number of elements

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

30 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.

31 Table 7 Model specific information from SLOPE/W

Model specifications Analysis type Morgenstern-price method Number of slices 30 Side function Half-sine function PWP-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.

32 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.

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

Group A Length 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 1,17 6 m 2,41 2,24 2,05 1,84 1,61 1,38 1,17 8 m 2,36 2,21 2,03 1,83 1,61 1,37 1,16 12 m 2,29 2,17 2,02 1,83 1,60 1,37 1,16 16 m 2,26 2,18 2,04 1,85 1,62 1,40 1,20 24 m 2,23 2,16 2,04 1,85 1,62 1,40 1,20 32 m 2,21 2,15 2,04 1,85 1,62 1,40 1,20 2D 2,17 1,92 1,69 1,48 1,31 1,16 1,03 SLOPE/W 2,08 1,84 1,62 1,42 1,26 1,14 1,03 3D-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 B Length 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 1.12 6 m 1.94 1.78 1.60 1.42 1.26 1.12 8 m 1.89 1.76 1.59 1.42 1.26 1.12 12 m 1.86 1.74 1.59 1.42 1.26 1.12 16 m 1.86 1.74 1.59 1.42 1.26 1.11 24 m 1.83 1.72 1.59 1.42 1.26 1.11 32 m 1.82 1.72 1.59 1.42 1.26 1.11 2D 1.77 1.56 1.35 1.18 1.04 Fail SLOPE/W 1.62 1.45 1.25 1.10 0.98 0.88 3D-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 C Length No load 2,5 kN/m2 5 kN/m2 4 m 1.32 1.25 1.18 6 m 1.22 1.16 1.09 8 m 1.20 1.13 1.06 12 m 1.15 1.10 Fail 16 m 1.13 1.09 Fail 24 m 1.12 1.09 Fail 32 m 1.12 1.09 Fail 2D 1.13 Fail Fail SLOPE/W 0.97 0.89 0.81 3D-effects Table 11 1.22 1.15

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

No load, 3D-effect Length Model A Model B Model C 4 m 2.35 1.89 1.30 6 m 2.26 1.80 1.19 8 m 2.22 1.75 1.13 12 m 2.17 1.71 1.08 16 m 2.15 1.69 1.05 24 m 2.13 1.67 1.02 32 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

35 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 the group 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.

36 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-excavation and its 2D-equivalence

37 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

38 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.

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

Group A Length No load 5 kN/m2 10 15 20 25 30 kN/m2 kN/m2 kN/m2 kN/m2 kN/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 method proposed by the Swedish commission of slope stability for group B

Group B Length 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 method proposed by the Swedish commission of slope stability for group C

Group C Length 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% -

40 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.

41 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.

42 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.

43 44 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.

45 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.

46 REFERENCES

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48 Zhang, Y., Guangqi, C., Zheng, L., Li, Y., & Zhuang, X. (2013). Effects of geometries en three-dimensional slope stability. Canadian Geotechnical Journal, 233-249.

49 50 APPENDIX A1 –SFR/DEFORMATION-CURVES, MODEL A

Safetyfactor Model A, No load 2,6

2,4