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By Helmut F Schweiger & Franz Tschuchnigg, Graz University of Technology
Limit equilibrium analyses (eg by Janbu 1954, Bishop 1955, Morgenstern & Price 1965 and Spencer 1967) are currently the preferred method in practical geotechnical engineering to calculate factors of safety, particularly in slope stability analysis. However, alternative methods such as the strength reduction technique in combination with the displacement based finite element method (eg Brinkgreve & Bakker 1991, Griffiths & Lane 1999, Dawson et al, 1999) have been proposed as an alternative approach with significant advantages over limit equilibrium methods, albeit with somewhat increased computational costs. It has been shown that limit equilibrium and finite element methods can produce similar factors of safety for slope stability analysis (eg Cheng et al, 2007). However, as is shown in this paper, significant differences may occur under certain circumstances. Consequently, the question has to be answered as to which of the methods leads to more realistic results. An attempt to demonstrate that strength reduction methods produce reliable results has been made by Tschuchnigg et al. (2015a, 2015b) by comparison with finite element limit analysis (Fela), which provides rigorous upper and lower bounds on the factor of safety (eg Sloan 1988, Sloan 1989, Sloan & Kleeman 1995, Lyamin & Sloan 2002a, Lyamin & Sloan 2002b, Krabbenhoft et al, 2005).
It should be mentioned that although the concept of factor of safety is well established in geotechnical engineering, there is no unique definition for it. The concept of partial factors of safety applied to loads, strength and resistances as introduced in Eurocode 7 is not considered here.
Indeed, in bearing capacity problems it is common practice to define the factor of safety in terms of the load capacity, whereas in slope stability problems the safety factor is usually defined with respect to the soil strength. The latter definition is used throughout this paper.
A common method for calculating factors of safety of natural and cut slopes is the limit equilibrium method. Besides some global methods, the method of slices is generally adopted (eg Janbu,1954, Bishop, 1955, Morgenstern & Price, 1965, Spencer, 1967). Various method of slices approaches are available; they differ in the assumptions made for the interslice forces and whether force or moment equilibrium, or both, is considered. A key limitation, which does not hold for the finite element methods, is the a priori assumption of the shape of the failure mechanism and the fact that these methods in general produce neither rigorous upper nor lower bounds in the sense of plasticity theory. Thus, it is not guaranteed that the results are conservative, as is shown later in this paper. A summary of the limits of limit equilibrium methods (Lem) can be found eg in Duncan (1996) and Krahn (2003).
The concept of limit analysis is based on the theorems of plasticity developed by Drucker et al (1951, 1952), namely the lower and upper bound theorems. The assumptions made are perfectly plastic material behaviour, associated flow rule, and effects of geometry changes neglected. The approach considers the stress equilibrium equations, the stress-strain relationship and kinematical compatibility throughout the whole soil body. Hence, the approach calculates “true” collapse loads, or in most cases, upper and lower bounds of the collapse load. If a safety factor based on the loads is desired, which is defined as the actual load over the limit load, the solution can be obtained from a single pair of upper and lower bound analyses. However, if the safety factor needs to be expressed in terms of the material strength, which is defined as the actual material strength over the mobilised material strength, a strength reduction process must be performed as described in Sloan (2013). This involves several upper and lower bound analyses, each with different strength parameters (the standard case for slope stability analysis).
The lower bound analysis looks at equilibrium and the yield criterion. A stress distribution is considered where; (i) the equilibrium equations, (ii) stress boundary conditions and (iii) the yield criterion are satisfied. Any loads which produce a stress distribution that fulfils these requirements is a lower bound and equal to, or lower than, the actual collapse load. The upper bound analysis on the other hand considers velocities and energy dissipation.
The external work, resulting from the applied loads and deformations, is equated to the internal energy dissipation. A velocity field is sought that satisfies (i) the velocity boundary conditions, (ii) the strain and velocity compatibility conditions, and (iii) the flow rule. Any loads which produce a velocity field that satisfies these requirements is an upper bound (ie equal to or greater than the actual collapse load). By maximising the lower bound and minimising the upper bound the collapse load can be bracketed in between the lower and upper bound (Chen, 2008).
Finite element formulations of upper and lower bound theorems of plasticity have been developed to a stage that they can be readily applied to a wide variety of geotechnical problems (eg Sloan, 2013). Finite element limit analysis is particularly powerful when both upper and lower bound estimates are calculated so that the true collapse load for the idealised material is bracketed. Differences between upper and lower bound values are usually in the order of a few percent; these solutions therefore provide good estimates of the exact solution. The narrowing of the gap between lower and upper bound solutions has been made possible by means of adaptive mesh refinement techniques. The formulations used in this paper stem from the methods originally developed by Sloan (1988, 1989) and Sloan & Kleeman (1995), and further improved by Lyamin & Sloan (2002a, 2002b) and Krabbenhoft et al. (2005, 2007).
A disadvantage of limit analysis is the implicit assumption of an associated flow rule (φ’ = ψ’) which is in general not appropriate for geotechnical problems. Davis (1968) suggested that one can use reduced strength values φ* and c* with an associated flow rule, if the dilatancy of the considered material is less than implied by an associated flow rule (Equation 1 to 3):
where the reduction factor β is given as
and c’ is the effective cohesion, φ’ is the effective friction angle, and ψ’ is the dilatancy angle.
Davis argued that the flow rule will not have a significant influence on the computed limit load unless the problem is kinematically constrained, and only for these situations would his approach need to be applied. As discussed by Sloan (2013), however, it is not straightforward to define the “degree of constraint” for practical problems.
It should be mentioned that Davis did not use his approach in the context of strength reduction techniques, but for the analysis of collapse loads.
When using the approach suggested by Davis (1968) in combination with strength reduction techniques different possibilities exist to evaluate the reduced strength parameters φ* and c* as outlined below.
Procedure A: Based on the given (characteristic) effective parameters c´, φ´ and ψ´ the factor β is calculated once and kept constant throughout the strength reduction procedure, ie β at failure (βfailure) is the same as at initial conditions (β0).
Procedure B: β is calculated based on the effective strength parameters applied in the current iteration of the strength reduction procedure, thus β changes during the strength reduction process (Eq. (5)) because tanφ’ and tanψ’ change (βinitial ≠ βfailure). In this case tanψ’ is also reduced simultaneously with c’ and tanφ’.
Procedure C: Similar to Procedure B, except that tanψ´ is not reduced, but is kept constant throughout the strength reduction procedure. The difference between Procedure C and B in terms of calculated factor of safety is typically small for common values of friction and dilatancy angles.
However, it can be stated that Procedure A provides the most conservative result and Procedure C is slightly unconservative. More details and comparisons related to Procedures A, B and C and displacement based finite elements can be found in Tschuchnigg et al (2015b).
In displacement based finite element codes the factor of safety (FoS) is obtained by means of the strength reduction method (SRM), ie an analysis is performed with characteristic strength properties for the friction angle φ’ and the cohesion c’, followed by an incremental decrease of tanφ’ and c’ (for the case of a Mohr-Coulomb failure criterion). This procedure eventually produces stress states which violate the strength criterion, which are then resolved in an iterative manner using the same stress point algorithm employed for a standard elastic-plastic analysis. This leads to a stress redistribution in the system until equilibrium can no longer be established and failure is reached. It should be noted that this procedure works only for linear failure criteria such as the Mohr-Coulomb criterion. It is well known that element type, mesh discretisation and convergence tolerances have a pronounced influence on the factor of safety obtained from the displacement finite element method. Therefore, the influence of these parameters has to be minimised by using high order elements, fine meshes and stringent convergence tolerances. The factor of safety obtained from this procedure is defined by:
Alternatively, the factor of safety can be obtained by performing a series of analyses reducing the strength parameters for each of the analyses at the beginning. At least for simple geometries this will lead to the same result.
It is emphasised that this procedure is not equivalent to modelling strain softening behaviour, where a defined stress-strain relationship from peak to residual strength is followed by employing an appropriate, advanced constitutive model. In addition, such an analysis would require an enhancement of the finite element formulation by means of a so-called regularisation method, eg a non-local formulation (eg Galavi & Schweiger, 2010, Summersgill et al, 2017) in order to avoid severe mesh dependency of the results.
This implies that careful considerations have to be given to the input of strength parameters on which the strength reduction procedure is applied, including the dilatancy angle, see eg Bolton, 1986, Rowe 1962, Nanda & Patra, 2015.
An issue which must be addressed in displacement finite element analysis of failure is the definition of the flow rule. If a non-associated flow rule with a dilatancy angle ψ’ smaller than the friction angle φ’ is employed, which could be considered as the standard case in geotechnics, numerical instabilities may occur leading to strong oscillations of the resulting factor of safety during the strength reduction procedure. The consequence is a non-unique failure mechanism making it difficult or in some cases impossible to define a unique value for the factor of safety. To overcome this problem, the approach suggested in Tschuchnigg et al. (2015b) and described in the previous section as “Procedure B” could be employed for the strength reduction procedure as well. Although there seems to be agreement in the literature that in slope stability analysis the flow rule does not have a significant influence on the computed factor of safety (eg Cheng et al, 2007), it will be shown later that this cannot be generalised, in particular for steep slopes of materials with high friction angles (>35°), a situation regularly found in mountainous regions.
The way the dilatancy angle is taken into account in the strength reduction technique depends on the numerical code used. To the authors knowledge most codes reduce tanφ’ and c’, but keep tanψ’ constant as long as φ’reduced > ψ’ as illustrated in Figure 1. As discussed in section 2.2 this produces generally conservative results which could be avoided by reducing tanψ’ stepwise in the same way as tanφ’ and c’.
In this section a comparison of SRFEA and Fela is made to provide confidence in the strength reduction procedure when performed with conventional finite element codes. The geometry of the slope is given in Figure 2. The slope height HS is 10m and the slope angle αS varies between 15° and 45° in 5° intervals. A typical finite element mesh and the resulting failure mechanism is shown in Figure 3 for SRFEA and Figure 4 for Fela. In Fela the mesh refinement is performed adaptively as part of the analysis, whereas in the SRFEA the mesh is fixed and a sensitivity analysis is required to ensure that it is sufficiently fine. Here about 6,500, 15-noded triangular elements are used to ensure high accuracy of the results. In a first step, gravity loading is applied and subsequently the strength reduction is performed in the finite element analysis. The material sets include two different soil types (Table 1), a purely frictional material (no cohesion) and a cohesive-frictional material. Four different materials are specified to allow for an associated flow rule (ψ’ = φ’) and a non-associated flow rule with zero plastic volume change (ψ’ = 0). Drained conditions are adopted and the groundwater table is assumed to be below the slope.
The results are summarised in Figure 5. The factors of safety for both methods, SRFEA assuming an associated flow rule and Fela, are almost identical, which is expected because in Fela the flow rule is inherently associated. For the non-associated cases, the data differ as expected. In SRFEA the influence of the flow rule on the factor of safety is marginal for material set 2, but for material set 1 the differences are no longer negligible, at least not for slope inclinations >25°. Fela employing the Davis approach Procedure A (as described in the previous section) provides very conservative results, ie significantly lower factors of safety for both material sets, whereas the Davis approach Procedure B is close to the non-associated SRFEA. So, it is clear that the flow rule has some influence on the results, and as expected, a non-associated flow rule leads to lower factors of safety. In turn it can be concluded that the assumption of an associated flow rule, which is not realistic for soils, is on the unsafe side. It should be mentioned that the Davis approach (A and B) can be also used with SRFEA, but results are not shown here because they agree well with Fela results. The results presented demonstrate that the strength reduction technique employed with conventional finite element codes is a reliable method for assessing the factor of safety for slopes. In this study the code Plaxis (Brinkgreve et al, 2016) was used, but the procedure can be adopted in any fe-code and to the author’s knowledge most commercially available codes have this feature implemented.
Having shown that SRFEA is a well suited method to calculate factors of safety of slopes, comparison with more commonly used limit equilibrium methods (here Morgenstern & Price, 1965) are presented. Again, a homogeneous slope, illustrated in Figure 2 is considered and slope inclination αs and effective friction angle φ’ are varied in this study. A constant value for the effective cohesion of c’ = 5kPa is assumed for all investigated cases and the height of the slope Hs = 20m in this case.
The results are summarised in Table 2, and the following can be observed: the results obtained from Lem and SRFEA agree for all cases extremely well when an associated flow rule (φ’ = ψ’) is used in SRFEA. If the more realistic case of a non-associated flow rule is adopted in SRFEA, differences appear and become more significant for steeper slopes for a given value of φ’, ie when the factor of safety reduces, or for increasing values of φ’ for a given slope inclination because the larger the difference between φ’ and ψ’ the larger the difference in FoS compared to the associated case.
As an example, Lem (and associated SRFEA) yields a factor of safety of 1.27 for a slope of 30° and a friction angle of 30° (cohesion = 5kPa), whereas for the non-associated case SRFEA gives 1.17. For a very steep slope of 40° and a friction angle of 40° (again cohesion = 5kPa) the results are 1.28 (Lem) and 1.08 (SRFEA non-associated). In some cases, Lem indicates a stable slope FoS > 1.0 whereas the non-associated SRFEA indicates failure.
The influence of the dilatancy angle in the SRFEA data is significant from a practical point of view when a certain FoS is required for a particular project to comply with standards and guidelines. The results demonstrate that Lem data may significantly overestimate the factor of safety. Please note that the dilatancy angle ψ’ does not explicitly enter the calculation when using the method of slices. It should be mentioned that, at least for higher friction angles, the assumption of ψ’ = 0° is conservative; for moderate values for ψ’ differences remain, but are slightly lower. However, this is accounted for when using SRFEA, because an appropriate dilatancy angle can be introduced in the analysis. As described in section 2.2, the dilatancy angle (if not 0) should be reduced as well as the friction angle in the strength reduction procedure although the effect is usually small as shown in Tschuchnigg et al (2015b).
Based on the observation that SRFEA provides very similar results as rigorous limit analyses, it can be concluded from the results presented in the previous section that SRFEA has advantages over LEM in practical geotechnical design because it avoids possible overestimation of factors of safety by employing non-associated plasticity. However, this advantage may turn into a disadvantage when very high friction angles (in the order of 40°) and low dilatancy angles have to be considered. This is of practical relevance in alpine environments where steep slopes with an inclination of 40° or higher are quite common. For these cases, where the factor of safety of the natural slope is usually low, any disturbance of the slope due to construction activities has to be carefully examined and the assessment of a reliable factor of safety becomes crucial. A slope with a height Hs of 10m and an inclination αS of 45° (see Figure 1) is considered here, with effective strength parameters of φ’ = 45° and c’ = 6 kPa. The dilatancy angle in this study is varied between ψ’ = 0° and 45°, intentionally covering a range of extremes, with ψ ≈ 10° to 15° being probably a realistic value.
Figure 6 shows the factor of safety from the SRFEA, and it is immediately clear, as expected, that the flow rule has a significant influence on the factor of safety, with the associated case giving a value of about 1.54 and the zero dilatancy case (ψ’ = 0°) giving a value of approximately 1.32. It is also apparent that the non-associated case for ψ’ = 0° gives highly erratic results, which makes it difficult to infer a precise value for the factor of safety. This type of behaviour is well known and is a consequence of the large difference between the friction angle and the dilatancy angle and is difficult to avoid (see eg also Nordal, 2008). This problem could be overcome however by employing the modification of the Davis approach as briefly described in section 2.2.
Limit equilibrium method (method of slices, LEM)
Finite element limit analysis (Fela)
Strength reduction finite element displacement based analysis (SRFEA)
In summary it can be concluded that the SRFEA is the most general and reliable method of obtaining factors of safety for natural and cut slopes. However, it is noted that this method in the simplest form as described here is only valid for linear failure criteria such as the Mohr-Coulomb failure criterion. If constitutive models are used where strength is a function of an internal variable, such as the void ratio, more elaborate procedures have to be adopted, see Potts and Zdravkovic (2012). This also implies that for undrained conditions the analysis has to be performed in total stresses employing the Tresca failure criterion. Because advanced constitutive models are required when performing undrained analyses in terms of effective stresses the strength reduction approach is not readily applicable to these conditions.
It should be mentioned at this stage that a strength reduction procedure does not necessarily have to be taken to failure. If it has only to be proven that the factor of safety is at least as high as the required partial factor of safety on soil strength as required eg in EC7, the strength reduction procedure could be stopped, once this partial factor on characteristic soil strength has been achieved. This has the advantage that the numerical model does not need to be driven to failure, making the analysis less vulnerable to convergence and error tolerances.
The finite element method is routinely applied in geotechnical engineering to calculate displacements and stresses under working load conditions. For ultimate limit state analysis, limit equilibrium methods are often preferred to calculate factors of safety of natural and cut slopes. It has been shown in this paper that the finite element method provides an alternative way to obtain factors of safety for slopes with a number of advantages compared with the commonly applied method of slices. It has been proven by comparison with rigorous finite element limit analysis that the strength reduction technique applied with a displacement-based finite element method can be safely applied in practice, provided a linear failure criterion such as the Mohr-Coulomb criterion is employed. However, for non-associated plasticity numerical instabilities may occur, in particular for high friction angles and small dilatancy angles. Furthermore, it is shown that limit equilibrium methods may produce non conservative factors of safety.
It should be mentioned that displacements resulting from in SRFEA have no meaning and are not the purpose of the analysis. Therefore, a strength criterion such as Mohr-Coulomb is sufficient for most practical problems.
Some results from the finite element limit analyses presented in this paper have been provided by Christoph Schmüdderich from Ruhr University Bochum.
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