Finite element analyses of slope stability problems using non-associated plasticity

Simon Oberhollenzer,Franz Tschuchnigg,Helmut F.Schweiger

Institute of Soil Mechanics,Foundation Engineering and Computational Geotechnics,Graz University of Technology,Graz,Austria

Keywords:Finite element limit analysis(FELA)Finite element method Slope stability Strength reduction technique Non-associated plasticity Adaptive mesh refinement Initial stresses

A B S T R A C T In recent years, finite element analyses have increasingly been utilized for slope stability problems.In comparison to limit equilibrium methods,numerical analyses do not require any definition of the failure mechanism a priori and enable the determination of the safety level more accurately.The paper compares the performances of strength reduction finite element analysis(SRFEA)with finite element limit analysis(FELA),whereby the focus is related to non-associated plasticity.Displacement-based finite element analyses using a strength reduction technique suffer from numerical instabilities when using non-associated plasticity,especially when dealing with high friction angles but moderate dilatancy angles.The FELA on the other hand provides rigorous upper and lower bounds of the factor of safety(FoS)but is restricted to associated flow rules.Suggestions to overcome this problem,proposed by Davis(1968),lead to conservative FoSs;therefore,an enhanced procedure has been investigated.When using the modified approach,both the SRFEA and the FELA provide very similar results.Further studies highlight the advantages of using an adaptive mesh refinement to determine FoSs.Additionally,it is shown that the initial stress field does not affect the FoS when using a Mohr-Coulomb failure criterion.

1.Introduction

In geotechnical engineering,no generally accepted definition of the factor of safety(FoS)exists.For many bearing capacity problems,the FoS is usually defined on the basis of the ultimate load bearing capacity.However,for slope stability analyses,it is more common that the FoS is related to the characteristic strength parameters of the soil.The limit equilibrium methods suggested by Janbu(1954),Bishop(1955)and Morgenstern and Price(1965)are based on the method of slices and have a wide tradition in slope stability analysis.Despite the long-lasting experience with limit equilibrium analysis(LEA),these methods have several disadvantages,e.g.assumptions regarding the shape of the failure plane and the forces acting between the slices lead to a non-unique definition of the FoS.Furthermore,it cannot be guaranteed that the failure mechanism is kinematically admissible.

Therefore,displacement-based finite element analyses in combination with a strength reduction technique became increasingly popular over the last decades.During the strength reduction standard procedure,the effective friction angle φ′and the effective cohesionc′are simultaneously reduced until no equilibrium can be achieved in the numerical procedure.The ratios of initial strength parameters to mobilized strength parameters define the FoS in terms of strength reduction finite element analysis(SRFEA).Further calculations are performed with an enhanced procedure at which the effective dilatancy angle ψ′is reduced simultaneously with the effective friction angle φ′and the effective cohesionc′from the beginning(see Section 2.1).Apart from that,the influence of initial stresses is investigated.It has been previously shown that LEA and SRFEA assuming associated plasticity yield similar results when employing a Mohr-Coulomb failure criterion.However,as shown by Tschuchnigg et al.(2015a),large differences between the effective friction angle φ′and the effective dilatancy angle ψ′maylead to numerical problems,making a clear definition of the FoS difficult.

Alternatively,rigorous upper and lower bounds of the FoS are obtained from finite element limit analysis(FELA).As FELA is restricted to associated plasticity,Davis(1968)proposed reduced strength parameters in combination with an associated flow rule to simulate non-associated soil behavior.In the case that the definition of safety is based on the strength parameters,the approach by Davis(1968)leads to(very)conservative results.Therefore,Tschuchnigg et al.(2015b)developed two modifications.These modified approaches are discussed briefly in the following.Furthermore,the paper tries to illustrate the significant advantage of adaptive mesh refinement,namely the significant reduction of the required degrees of freedom to obtain accurate FoSs.

2.Theory

2.1.Strength reduction method with displacement-based fi nite element method(strength reduction fi nite element analysis)

The displacement-based finite element code Plaxis(Brinkgreve et al.,2016)enables the definition of the FoS by means of effective friction angle φ′and effective cohesionc′.The method of strength reduction was first used by Zienkiewicz et al.(1975).This is achieved by simultaneously reducing tanφ′andc′until no equilibrium can be satisfied.A Mohr-Coulomb failure criterion is assumed and it is emphasized that this procedure only works for these types of(linear)failure criteria.The corresponding formulation of the FoS is shown in Eq.(1),where the subscript“mobilized”denotes the strength quantities at failure.For associated plasticity,both the effective friction angle φ′and the effective dilatancy angle ψ′are reduced at the same time.For a non-associated flow rule,the dilatancy angle ψ′is kept constant as long as the reduced effective friction angle φ′is larger than ψ′.At the point where the reduced friction angle φ′equals the effective dilatancy angle ψ′,both values are reduced simultaneously(Brinkgreve et al.,2016).

Additionally,calculations with a user-defined strength reduction procedure are performed.During this SRFEA,tanφ′,tanψ′andc′are simultaneously reduced,as shown in Eq.(2).As a consequence,the degree of non-associativity(Λ = φ′- ψ′)is also affected by the reduced value ofψ′.For the case whereψ′=0°as well as associated plasticity(ψ′= φ′),the standard and user-defined strength reduction procedures are the same.

2.2.Factor of safety obtained from finite element limit analysis

FELA has some tradition in geotechnical engineering,but only in recent years tools that are applicable in practice have been developed(Chen,2007).Based on the upper and lower bound theorems of plasticity,the prediction of various stability questions is possible by calculating the failure load from above and below(see e.g.Lyamin and Sloan,2002a,b).Thereby,the difference between the two bounds is an indication of the error in the solution.A limit analysis enables the definition of the FoS without performing an elasto-plastic analysis.The limit theorems of plasticity can be applied to any solid body if the material shows perfect plasticity and no hardening or softening is considered.The initial stresses and deformations do not affect the plastic limit(failure load).Furthermore,the yield surface is assumed to be convex and characterized by an associated flow rule(ψ′= φ′)(Chen,2007).

If the FoS is expressed as the ratio of collapse load to actual load,a single pair of upper and lower bound calculations leads to the required result.In slope stability analyses,however,the FoS is defined according to the strength parameters of the soil.In thispaper,a strength reduction procedure considering an adaptive mesh refinement is used as described in Sloan(2013).

Table 1 Comparison of different procedures(Tschuchnigg et al.,2015b).

2.3.Non-associated plasticity

A SRFEA performed with non-associated plasticity may lead to numerical instabilities without a clearly defined failure surface.This is particularly the case for materials with high friction angles(φ′＞ 40°)in combination with steep slopes(α ＞ 40°),where different failure mechanisms are kinematically possible leading to an alternating FoS(Tschuchnigg et al.,2015a).

FELA is restricted to associated plasticity(ψ′= φ′),since stress and velocity characteristics in a plastic field are equal only for the case of an associated flow rule(Davis,1968).Therefore,reduced effective strength parameters(c*,φ*)are used in combination with an associated flow rule to model non-associated behavior(ψ′≠ φ′).Davis(1968)suggested using reduced strength parameters(see Eqs.(3)and(4))based on the given effective strength parameters φ′,c′and ψ′(strength reduction factor β).Thereby,the amount of Eq.(5)diminishes with decreasing effective dilatancy angle ψ′and an increasing degree of non-associativity Λ = φ′- ψ′(Tschuchnigg et al.,2015b).Theβvalue given in Eq.(5)is denoted in the following as Davis procedure A.

Based on the fact that the original approach by Davis(1968)leads to very conservative results if the FoS is expressed by means of the strength parameters of soil,two modified procedures were developed(Tschuchnigg et al.,2015c).Both modified approaches are still conservative compared to the non-associated SRFEA,but yield a better agreement with SRFEA than the Davis procedure A(original approach).

The enhanced procedures Davis B and C require an iterative procedure for determining the reduced strength parametersc*and φ*.In Davis B,the β value is determined based on the “actual”effective strength parameters,because the degree of nonassociativity Λ = φ′- ψ′changes during the procedure.As shown in Eq.(6),the FoS of the previous iteration step reduces the “actual”dilatancy angle ψ′as well as the “actual”effective friction angle φ′simultaneously.Theβfailurevalue is calculated until no change in the FoS occurs.

The enhanced procedure C(Davis C)differs from Davis B,by assuming a constant dilatancy angle ψ′during the iterations,as shown in Eq.(7)(Tschuchnigg et al.,2015b).It is important to note that as long as the dilatancy angleψ′is zero,procedures B and C are the same.Table 1 underlines the most important differences between Davis A,B and C.

3.Numerical studies

3.1.Influence of initial stress field on strength reduction finite element analysis

In geotechnical engineering,it is a recurring question that stress field exists in situ.Related to that question,it is often necessary to make assumptions for numerical analyses.The study discussed in the following tries to clarify to what extent the initial stress distribution does affect the FoS when performing SRFEA considering a Mohr-Coulomb failure criterion.The finite element code Plaxis(Brinkgreve et al.,2016)is used for all displacement-based finite element analyses discussed in this section.

The example to be analyzed is a simple homogeneous slope shown in Fig.1,with a slope heightHequal to 5 m and a slope angle α of 26.6°(1:2).In SRFEA,the mesh refinement is performed manually with 15-noded triangular elements.The analyses consider drained conditions and a linear elastic-perfectly plastic constitutive model with a Mohr-Coulomb failure criterion.The homogeneous slope is characterized by a unit weight γunsat=16 kN/m3,an effective friction angle φ′=30°and an effective cohesionc′=2 kPa.The dilatancyangle ψ′is set tozero.To investigate the influence of initial stresses on the FoS,four cases are investigated.Both the initial geometry and the method(K0procedure or gravity loading)to compose the initial stresses are varied.In cases 1-3,the initial stresses are assumed as= γzandσ′h=whereby theK0values vary between 0.25 and 0.4(K0procedure).Referring to the geometry(see Fig.1),theK0procedure in case 1( fill)is performed only for the base layer(see Fig.1,section 1 is activated)followed by a construction phase(sections 1 and 2 are activated)and a φ′/c′reduction.For case 2(excavation),the initial stress distribution(K0procedure)is determined by activating sections 1,2 and 3.Subsequently,section 3 is deactivated(excavation)followed by a SRFEA.TheK0procedure of case 3 is performed on the final geometry(sections 1 and 2 are activated)followed by a plastic nil-step(i.e.a plastic calculation starts without defining any loads in order to guarantee continuous horizontal stresses)and a strength reduction.In the first calculation phase of case 4(sections 1 and 2 are activated),gravity loading is applied and subsequently a SRFEA is executed.

Fig.1.Finite element mesh used for SRFEA(ψ′=0°).

Table 2 FoS obtained with SRFEA(ψ′=0°).

Table 2 shows the evaluation of FoS for all four cases.It becomes obvious that the initial stress situation has no influence on the FoS(FoS=1.53)when using a Mohr-Coulomb failure criterion.The different stress paths of pointA(located inside the failure surface,see Fig.2)end up at the Mohr-Coulomb failure line of the same FoS although they have completely different origins(see Fig.3).

Fig.3.Stress paths of point A for case 1( fill),case 2(excavation),case 3(nil-step)and case 4(gravity loading).

3.2.Influence of adaptive mesh refinement on the factor of safety

The example studied considers again the stability of a homogeneous slope under drained conditions.The dimensions of the slope are identical to those of the previous study presented in Section 3.1.In all analyses,a linear elastic-perfectly plastic constitutive model with a Mohr-Coulomb failure criterion and the soil parameters γunsat=16 kN/m3,γsat=18 kN/m3,φ′=30°andc′=2 kPa are used.In the first calculation phase,gravity loading is applied and subsequently the strength reduction is performed.Strength reduction finite element analyses assuming associated plasticity(φ′= ψ′)are performed using Plaxis 2D and Optum G2(Krabbenh?ft et al.,2016)with 6-and 15-noded elements,thus quadratic shape functions and shape functions of 4th-order,respectively.Sensitivity analyses are performed by increasing the number of triangular elements with and without adaptive mesh refinement.A detailed description of the formulation used for the adaptive mesh refinement was given in Sloan(2013).

The failure surface passes through the toe of the slope and does not extend below this point(see also Fig.2).Fig.4 shows that the mesh refinement as well as the order and number of elements strongly affects the computed FoS.As is shown in Fig.4,both finite element codes,namely Plaxis 2D and Optum G2,are in good agreement when performing SRFEA without adaptive mesh refinement.The results obtained with SRFEA using 6-noded elements lead to significantly higher FoSs compared to those with higher-order elements.Furthermore,it is shown that the effect of adaptive mesh refinement is limited when using 15-noded elements(at least in combination with an associated flow rule).For the example considered,about 500 15-noded triangular elements without adaptive mesh refinement are required to give accurate estimates of the safety level(FoS=1.6).The advantage of the adaptive mesh refinement becomes clear when using 6-noded elements.Approximately 400 elements are needed to reach a value ofFoS=1.62.On the other hand,about 1500 elements are needed without adaptive mesh refinement to reach the same value(see Fig.4).It should be noted that approximately 1500 6-noded elements in combination with an adaptive mesh refinement are required to achieve a good agreement with elements using a shape function of 4th-order.

Fig.4.Factors of safety obtained from SRFEA(φ′= ψ′)with different numbers of elements(with and without adaptive mesh refinement).

It should be noted that both finite element programs(namely Plaxis 2D and Optum G2)lead to approximately the same computation time when performing SRFEA without adaptive mesh refinement.Thereby,analyses based on 15-noded elements take approximately 3-4 times longer compared to those with 6-noded elements.Furthermore,it is noticeable that SRFEA performed in Plaxis 2D using a shape function of 4th-order becomes slightly faster compared to Optum G2 with increasing number of elements(difference＜20%).If the number of elements is kept constant,an additional mesh refinement(with 3 iterative procedures)raises the computation time approximately by 60%-90%,while the precision of the solution increases(see Fig.4).When comparing the results of SRFEA(performed with and without adaptive mesh refinement)which lead to the same FoS,those calculations based on mesh adaptivity are approximately 30%-50%faster and the number of elements is reduced significantly(see Fig.4).

In the following,the same slope is used but several horizontal water tables with a vertical distanceLfrom the crest are considered(see Fig.5a).Thez-axis is defined positive downwards.The problem can be seen as an approximation of a slow drawdown process where the water table is initially defined above the crest(L/H=-0.2)and is lowered to the base(L/H=1).A similar study was performed by Griffiths and Lane(1999).The soil parameters remain unchanged,with the difference that a non-associated flow rule with ψ′=0°is assumed for all strength reduction finite element analyses.To study the effect of mesh density on the FoS,about 300 and 1500 15-noded elements are used(without adaptive mesh refinement).

Furthermore,FELA is performed to show the better agreement of Davis B,compared to Davis A,when dealing with non-associated SRFEA.It is also illustrated to what extent the adaptive mesh refinement influences the error margin between upper and lower bound calculations.FELA is carried out without an adaptive mesh refinement and 4000 elements as well as approximately 1000 triangular elements in combination with an adaptive mesh refinement.Examples of the meshes used in the FELA are shown in Fig.5a and b.

Fig.5.Finite element meshes for upper bound of FELA(a)with adaptive mesh refinement using approximately 1000 elements and(b)without adaptive mesh refinement using approximately 4000 elements.

Fig.6 shows the FoSs obtained from SRFEA,and it follows that the mesh density has a strong influence on the FoS.TheL/Hratio equal to 0 based on the coarse mesh(approximately 300 elements)yields a FoS of 1.87.If a higher mesh density of 1500 15-noded elements is assumed,the FoS decreases by approximately 3%to 1.81.

The results also confirm that Davis A gives a conservative estimate,whereas the results based on the enhanced procedure Davis B are in much better agreement with the ones obtained by SRFEA.As shown in Section 2.3,Davis B and C provide the same results for=0°.

It is obvious that the upper and lower bound analyses with and without adaptive mesh refinement provide approximately the same mean values(see Fig.6).Nevertheless,it should be noted that the adaptive mesh refinement with approximately 1000 elements leads to much smaller differences between upper and lower bounds compared with those analyses based on approximately 4000 elements without adaptive mesh refinement.Table 3 summarizes the FELA results obtained forL/H=0.The differences of upper and lower bounds using no adaptive mesh refinement and about 4000 elements range between 4.6%and 5.1%.The gap between the upper and lower bounds can be reduced to approximately 1.5%by using an adaptive mesh refinement.These results show clearly the significant advantage of using an adaptive mesh refinement.

Table 3 Factors of safety obtained from FELA(with and without adaptive mesh refinement)for L/H=0.

Table 4 Soil properties:Comparison of studies 1 and 2.

Fig.6.Factors of safety obtained from SRFEA(ψ′=0°)and FELA(with and without adaptive mesh refinement)for different L/H ratios.

3.3.Comparison of standard and user-defined strength reduction finite element analyses with Davis procedures A,B and C

The following studies are performed to show how the effective friction angle φ′,the degree of non-associativity Λ = φ′- ψ′and the effective cohesionc′influence the different Davis approaches and non-associated SRFEA.SRFEA is performed in Plaxis 2D,while Optum G2 is used for FELA(Davis A,B and C).Emphasis is put on the comparison of Davis B and C.Davis A and C(mean values of upper and lower bounds)will be compared with the standard strength reduction as implemented in Plaxis 2D(Brinkgreve et al.,2016),where the dilatancy angle ψ′is kept constant until the reduced effective friction angle φ′equals ψ′.Since in a user-defined SRFEA,the effective friction angle φ′and the dilatancy angle ψ′are reduced simultaneously from the beginning,Davis B is compared with this enhanced SRFEA.

The studies are performed using the slope shown in Fig.1.Again,the drained conditions and linear elastic-perfectly plastic material behavior with a Mohr-Coulomb failure criterion are considered.The results presented in Section 3.2 prove that about 1000 15-noded elements are needed to give accurate estimates of the FoS.In the first calculation phase,gravity loading is applied followed by a plastic nil-step and a strength reduction.On the other hand,in FELA(Davis A,B and C),the mesh refinement is performed adaptively as part of the analysis with approximately 1000 elements.

Two different studies are performed.The homogeneous slope of study 1 is characterized by a unit weight ofγunsat=16 kN/m3and an effective cohesionc′of 2 kPa.The dilatancy angle ψ′is reduced from the associated case(ψ′= φ′)at intervals of 5°,until zero.This is done for the effective friction angles=25°,=30°,=35°and=40°.On the other hand,study 2 investigates how the cohesion influences the difference between Davis calculations and non-associated SRFEA.For this purpose,the effective friction angle φ′=30°is kept constant,while the effective cohesion varies between 0 and 10 kPa(see Table 4).

The calculations for study 1 show that standard and user defined strength reduction finite element analyses as well as Davis B and C give the same results for a dilatancy angle ofψ′=0°(see Fig.7a).Furthermore,it was found that,with an increasing degree of non-associativity(Λ=φ′-ψ′),the differences between all Davis approaches and SRFEA(standard and user-defined)become larger.Fig.7a clearly indicates that with increasing effective friction angle φ′,those differences become significantly larger for Davis A,while differences for Davis B and C do not increase considerably.A general conclusion from Fig.7a is that the difference between the original Davis approach(Davis A)and the enhanced procedures(Davis B and C)increase with increasing φ′and Λ.

Fig.7.Study 1:(a)Standard SRFEA,user-defined SRFEA,and Davis A,B and C results for different values of Λ and φ′;and(b)Differences between Davis A and standard SRFEA,Davis B and user-defined SRFEA as well as Davis C and standard SRFEA for different values of Λ and φ′.

Fig.8.Study 2:(a)Standard SRFEA,user-defined SRFEA,and Davis A,B and C results for different values of Λ and c′;and(b)Differences between Davis A and standard SRFEA,Davis B and user-defined SRFEA as well as Davis C and standard SRFEA for different values of Λ and c′.

Fig.9.Reinforced embankment:Cross-section(unit:m).

The differences between Davis B and the user-defined SRFEA as well as the differences between Davis C and the standard SRFEA are in the same order of magnitude.This is illustrated in Fig.7b,where the blue and green dashed lines show a good match.The black dashed lines(reference)represent the standard and user-defined SRFEA results,from which the differences of Davis A(=standard SRFEA-Davis A),Davis B(=user-defined SRFEA-Davis B)and Davis C(=standard SRFEA-Davis C)are subtracted.To give a better overview of the results,different reference values for the calculations(based on an effective friction angle φ′equal to 25°,30°,35°and 40°)are chosen but do not represent FoSs.Those differences between SRFEA and Davis approaches are illustrated exemplarily by arrows in Fig.7a and b.

Study 2 shows similar trends as study 1.Keeping the effective friction angle φ′constant and varying the effective cohesionc′between 0 and 10 kPa lead to the conclusion that with increasing cohesion,Davis A becomes more conservative compared to the standard strength reduction.It can be seen in Fig.8a that Davis B and Care not strongly affected by the variation of the cohesion.Again,the differences between Davis B and user-defined SRFEA as well as the differences between Davis C and standard strength reduction areapproximately the same.Despite the change of non-associativity Λ,the green and blue dashed lines in Fig.8b do match well.

Table 5 Reinforced embankment:Soil properties for SRFEA and FELA.

Fig.10.Reinforced embankment-safety analysis:(a)Incremental shear strains obtained with SRFEA;(b)Shear dissipation obtained with FELA-upper bound(Davis B);and(c)Failure mechanism obtained with LEA.

Fig.11.Reinforced embankment:Factors of safety obtained from FELA with different element numbers.

4.Applications

4.1.Reinforced embankment

4.1.1.Problem description

This section deals with the application of the original and modified Davis approaches to an embankment,which is reinforced by 6 horizontal geotextile layers at the toe.As shown in Fig.9,the four soil layers,namely back fill material,sandy top layer,gravel layer and clay layer,define the soil layering.A horizontal water table is defined on the top of the sandy layer.The three underlying layers are fully saturated and are assumed drained.

For SRFEA and FELA,a 35.5 m long and 20.6 m high model,with an embankment height of 5.6 m,is defined.The cross-section is characterized by an embankment inclination of aboutα =35°.The underlying geogrid layers are 5 m long with a vertical distance of about 0.3 m.Consequently,the height of the reinforced area is 1.5 m.

The hardening soil model(Schanz,1998)and the hardening Mohr-Coulomb model(Doherty and Muir Wood,2013),both assuming a Mohr-Coulomb failure criterion,are used for SRFEA in Plaxis 2D and FELA using Optum G2,respectively.The material parameters used are listed in Table 5.The reinforcements are modeled by means of linear-elastic geogrid elements with a stiffnessEAof 10,000 kN/m.This study excludes any external load,thus only self-weight is acting.

In SREFA,the discretization of the domain is carried out manually using 15-noded elements.Since the failure plane develops behind the reinforcements,that area shows a higher mesh density(see Fig.10a).The final model consists of 10,913 elements.It should be noted that for all strength reduction finite element analyses carried out in Plaxis 2D,the initial stresses are calculated with theK0procedure,where no embankment is present.In the following,the construction of the embankment is modeled in one step,followed by the φ′/c′reduction.The analyses are performed for both associated(ψ′= φ′)and non-associated(ψ′=0°) flow rule.In FELA,safety analyses are performed with and without adaptive mesh refinement for different numbers of elements.As shown in Fig.11,the difference between the upper and lower bounds reduces significantly once the adaptive mesh refinement is used.Therefore,all calculations discussed in Section 4.1.2 consider an adaptive mesh refinement procedure in combination with approximately 1000 triangular elements.Fig.10b shows exemplarily the result of the automatic mesh adaptivity procedure for an upper bound analysis.In addition to the standard upper and lower bounds,FELA(using ψ′=φ′),Davis A withψ′=0°and Davis B withψ′=0°are calculated.Additionally,LEA using the Morgenstern and Price(1965)method is also performed and compared with the results obtained with SRFEA and FELA.

4.1.2.Comparison of the results

The SRFEA of the reinforced embankment confirms again that the flow rule has a significant influence on the FoS,where associated plasticity(ψ′= φ′)leads to a FoS value of about 1.66 and the zero dilatancy case(ψ′=0°)reaches a FoS value of 1.55(seeTable 6).On the other hand,it can be seen that theFoSMeanvalue((FoSLB+FoSUB)/2)of the FELA with adaptive mesh refinement matches remarkably well with the SRFEA(ψ′= φ′)results.Comparing Fig.10a and b,one can see that also the failure planes of SRFEA and FELA are in good agreement.The very fine mesh(see Fig.10b)in the region of the slip surface(when using FELA)is a consequence of the adaptive mesh refinement.On the other hand,for all strength reduction finite element analyses,approximately 10,000 triangular elements with a shape function of 4th-order are used to reach a comparable FoS value.Due to the inherent assumptions of LEA,this method yields a different failure mechanism(see Fig.10c)and a much higher FoS equal to 1.72.By comparing the FoSs of Davis A and B with that of the SRFEA(ψ′=0°),it is obvious that both approaches are conservative,but Davis B predicts FoS values in better agreement with SRFEA(see Table 6).This is also highlighted in Table 7 where one can see that Davis A deviates about 14.8%compared to the SRFEA,and Davis B on the other hand shows only a difference of about 4%.

Table 7 Reinforced embankment:Comparison of Davis A and B with SRFEA(ψ′=0°).

Table 6 Reinforced embankment:FoS obtained with LEA,SRFEA and FELA.

Fig.12.Upstream slope:Cross-section(unit:m).

Table 8 Upstream slope:Soil properties for SRFEA and FELA.

4.2.Upstream slope

4.2.1.Problem description

Fig.13.Upstream slope:Finite elements meshes used for(a)SRFEA and(b)FELA.

Fig.14.Upstream slope:Failure mechanism obtained in(a)SRFEA and(b)FELA.

Table 9 Upstream slope:FoS obtained with SRFEA and FELA.

Table 10 Upstream slope:Comparison of Davis A and B with SRFEA(ψ′=0°).

This boundary value problem deals with a reinforced upstream slope next to a reservoir.The section investigated,with a total height of approximately 33m,can be divided into an area below the berm disposed about 26.6°(1:2)towards the horizontal,and an area above the berm with a slope angle of 30°.The slope below the berm is lined with a sealing foil,thus loaded additionally by water in the reservoir.At the berm,a pile trestle consisting of two thread bars(GEWI system)connected with a ridgepole is installed.As shown in Fig.12,the upper section is separated by an intermediate berm.The soil stratigraphy is characterized by three layers(see Fig.12).The top layer(light blue)represents moraine material.Underneath,marked in light green and brown,fractured and intact rock layers are present,respectively.The water level can be found primarily in the fractured rock layer.In the slope,an inclined water table is defined,as shown in Fig.12.

In this study,the moraine layer as well as the fractured rock layer is described by the hardening soil model.For the deeper intact rock layer,a Mohr-Coulomb model is utilized.Table8 gives an over view of the input parameters.The calculations assume drained conditions and ignore any external loads.Furthermore,the reservoir is assumed to be emptied in order to generate the worst-case scenario.Instead of modeling the two GEWI piles with structural elements,the area between the piles is defined with an increased effective cohesionc′equal to 18.5 kPa.All the other soil properties in the region of the pile trestle correspond to those of the fractured rock layer.

The finite element code Plaxis 2D is used for all strength reduction finite element analyses discussed in this section.About 8932 15-noded elements are used to compute FoSs.Referring to Fig.13a,the cut slope and the region of the pile trestle are object of local mesh refinements.For all finite element limit analyses,the mesh refinement is performed adaptively using approximately 1000 3-noded elements in Optum G2(see Fig.13b).Strength reduction finite element analyses assuming associated(ψ′= φ′)and non-associated plasticity(ψ′=0°)are compared with FELA as well as Davis procedures A and B(Davis B equals Davis C forψ′=0°)(see Fig.14).

4.2.2.Comparison of the results

The results show that SRFEA(ψ′= φ′)and FELA fail in an analogous manner and present a good agreement in the FoS(see Table 9).The SREFA assuming associated plasticity yields a FoS equal to 1.38,while the corresponding FELA computes a lower boundFoSLBof 1.37 and an upper boundFoSUBof 1.4.Referring to Table 9,the mean value of the upper and lower bounds leads to aFoSMeanof 1.39.Again,it becomes clear that by using an adaptive mesh refinement with approximately 1000 3-noded elements,accurate estimates on the FoS are possible.

By comparing the FoSs of Davis A and B with a non-associated SRFEA,it is obvious that both methods are conservative,but Davis B offers a better agreement with the calculation of Plaxis 2D(see Table 10).Whereas Davis procedure A deviates about 7.6%compared to the SRFEA(ψ′=0°),Davis B shows only a difference of about 1.6%.

5.Conclusions

The results presented in this paper confirm that the LEA,FELA and SRFEA assuming associated plasticity are in good agreement.SRFEA using non-associated plasticity(ψ′≠ φ′)yields lower FoS values compared to LEA.It is also shown that determination of the initial stress field does not influence the computed FoS when using a Mohr-Coulomb failure criterion.FELA presented confirms that,by using an adaptive mesh refinement,it is possible to remarkably reduce the number of elements.For all considered finite element limit analyses including applications of boundary value problems,about 1000 elements in combination with an adaptive mesh refinement lead to accurate FoSs.Detailed numerical studies have been performed with SRFEA and FELA and have proven that an increasing effective friction angle φ′,effective cohesionc′and degree of non-associativity Λ lead to larger differences between Davis A and the standard SRFEA.Since the dilatancy angle is kept constant in a standard strength reduction technique,a user-defined SRFEA has been used,where the friction angle and dilatancy angle are reduced simultaneously from the beginning.Due to the fact that the difference between the friction angle φ′and the dilatancy angle ψ′defines the amount of non-associativity,the latter is considered to be more appropriate.It is suggested to use the enhanced procedure in displacement-based finite element analyses.It should be noted that the differences between Davis B and the user-defined SRFEA as well as between Davis C and the standard SRFEA(constant ψ′)are approximately the same.The results presented highlight that in Davis B and C,the degree of nonassociativity Λ has a noticeable influence on the computed FoSs,whereas changes in the effective cohesionc′show negligible effect when using the enhanced Davis procedures.

An important outcome of the studies is that the proposed modified Davis procedure B in combination with the FELA seems to compute more realistic FoS than the original Davis(1968)approach,but is still slightly conservative.However,the enhanced Davis procedure is also suitable in combination with SRFEA to avoid numerical instabilities when dealing with non-associated plasticity.

Conflicts of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Nomenclatures

α Slope angle

β Strength factor according to Davis(1968)

β0Strength factor according to Davis(1968)at initial conditions

βfailureStrength factor according to Davis(1968)at failure

γunsatBulk unit weight

γsatSaturated unit weight

φ′Effective friction angle

ψ′Dilatancy angle

Λ Amount of non-associativity,Λ = φ′- ψ′

c′Effective cohesion

Mobilized effective cohesion during SRFEA

mPower for stress dependency of stiffness

Reference secant modulus from triaxial test

Reference tangential modulus from oedometer test

Reference unloading/loading modulus

EA Extensional stiffness

FELA Finite element limit analysis

HSlope height

K0Coefficient of lateral earth pressure

LEA Limit equilibrium analysis

FoSFactor of safety

FoSLBFactor of safety obtained with lower bound analysis

FoSMeanMean factor of safety obtained with lower and upper bound analysis

FoSUBFactor of safety obtained with upper bound analysis

SRFEA Strength reduction finite element analysis