>Beyond Bishop, why doesn’t there seem to be a more rational method of slices based on finite element analyses?

A method for doing this is described by Krahn in the 2002 R.M. Hardy Address (https://www.eoas.ubc.ca/courses/eosc433/lecture-ma...) and implemented in Slope/W / Sigma/w for ~20 years. You can use the stresses from SIgma/W in an LE analysis in Slope/W. Or alternatively you can just do strength reduction in FE. However, making things more complicated is not always better; predicting accurate in-situ stresses with an FE program is arguably just as dubious as the assumptions in conventional LE analysis.

Thanks for your reply, and I agree that making things more complicated isn't always better, but the method you refer to seems to involve an FE analysis of the specific slope. What I meant was a replacement of the X=E.λ.f(x) approach by a generic function obtained from different FE analyses. It's the dependence on the horizontal force E that I don't like as this could be zero or even negative. In the latter case this has the effect of reversing the shear forces, i.e. a negative X near the top of the slope and a non-zero (but positive) λ would imply that the ground at the top is tending to lift up the sloping ground below it.

The greater weight of earth higher up a slope obviously results in greater settlements which induce shear forces. These shouldn't change much as failure approaches if it's a circular slip as the slices don't move in relation to each other. (I'm assuming a decision has been made about tension cracks despite this being another can of worms.) So it seems that a typical profile of vertical shear forces could be arrived at depending on slice depth, soil density, slope angle, depth to bedrock or anything else that seems important. This function would be part of the software and would be adjusted using λ to arrive at equilibrium.

I accept this would only provide a rough estimate of internal forces, and it doesn't deal with non-circular slips which involve mobilising the soil's shear strength, but if one wants to go beyond Bishop yet not as far as a bespoke FE analysis then it seems a good idea to take some account of typical deformations.

Great resources geotechguy1 and totally agree in the way geostudio solves the problems I only would like to add the Flac software suite is common in industry practice for performing stress deformation and strength reduction modelling.

OK but these don't seem to provide a relatively simple method of slices that tries to directly take account of typical deformations.

I sort of see what you're saying A Smith but it seems like you want to fit a square peg into a round hole; LE analysis inherently don't deal with displacement / strains. If you want or need to deal with strains / displacements you can deal with this by moving fully to an FE analysis. Sigma/W, Plaxis, and RS2 etc are available. FLAC is also an option (although I thought this was an FD program not FE program? - but commonly used by bigger geotech firms).

Maybe I'm talking about square pegs and holes with rounded corners. But I still think there's a need for the relative simplicity of the method of slices which doesn't either ignore internal shear or else involve a choice of seemingly arbitrary functions tied to horizontal forces. Why not replace a half sine with a function that resembles the actual shear forces as estimated using FE? Or if a profile of typical shear forces can be obtained experimentally then so much the better.

It’s just occurred to me that the lack of a more rational method of slices may have arisen because Krahn and others talk about a “General limit equilibrium method”. This implies it covers all limit equilibrium approaches. If people think this is true they are less likely to think of other approaches. But the method isn’t general as it ties shear forces to horizontal forces. This constraint in the “general” method seems irrational as the idea of zero or negative horizontal forces in tension zones has been known since Rankine’s theory in 1856.

A Smith, take a look at Spencer's Method of slices. I don't recall off hand what method(s) it uses to determine the horizontal forces, but it is worth a look. I've used a limit equilibrium software called UTEXAS for over 30 years, different versions over the years. Very different software than Slope/w and I personally much prefer it, but each to their own.

As for the more general discussion, limit equilibrium methods were developed based on two over riding ideas. First they could be calculated by hand, there were no computers to do the analysis. Second, they are only valid for factors of safety greater than 1.0, i.e. everything is static and there is no movement.

My personal opinion is that limit equilibrium methods are adequate for the vast majority of projects. FE/FD methods have their place, but I'm not convinced that they add anything to most projects.

Thanks GeoPaveTraffic. Yes I've seen Spencer's method, and having looked more closely at Krahn’s splendidly clear paper it seems obvious that the General Limit Equilibrium Method should really be called the Generalised Spencer Method. It takes advantage of Spencer’s innovation of two factor of safety equations but it only generalises his (her?) assumption that each shear force equals the corresponding horizontal force. It doesn’t cover any other basis for estimating shear forces.

I also think Krahn and others should say more about the difference between predicted inter-slice forces and actual forces. The horizontal stresses from statics aren’t the same as a soil’s actual earth pressure at rest which depends on its stress history. Differences from reality could be important in non-circular slips which Krahn says can be analysed using the method of slices. If different parts of a slip surface meet an an angle then the shear force at this point obviously needs to overcome the shear strength so the necessary distortion can occur. But in general I don’t see how a half sine function for example can predict the actual shear strength at this point whilst not overestimating the shear elsewhere. Maybe someone can explain how this would happen. Otherwise I’ll continue to think that the “general” method is much less general than the misnomer implies.

All LE methods overpredict and underpredict shear forces at different points of the slip surface; it's only really the average driving force vs average resisting force over the entire surface that is somewhat credible. If you want - or need - to consider soil behaviour that is something more complicated than that you need to use FEM or FDM or do something very careful in your LE analysis. You can go down a rabbit hole of making things more complicated (Shouldn't you use a 3D model, since the real world is 3D, and for many practical landslide sizes that occur in the real world FOS3D could be 1.5-2x FOS2D plane strain? Shouldn't you model it in FE with critical state soil mechanics? Actually - soil is a continuum of particles, so shouldn't you actually toss all of this FE and LE stuff in the trash and use a discrete element model run on a super computer?).

You could also have mentioned the problems of soil variability, sample disturbance, testing relevance, matric suction, tension cracks, climate change and vegetation unknowns. We seem to be in agreement about the risk of trying to be too clever about analysing stresses.

Bishop’s method has the merits of simplicity and honesty: the internal stresses can’t be determined from statics so we’ll just ignore them and use a healthy safety factor - which is also needed for all the other unknowns. This approach of course is generally a bit over-conservative unless the soil is weaker lower down. In this case Spencer’s method should be safer, although if there’s a weak lower stratum and a composite slip surface then a wedge analysis seems more sensible.

Krahn’s paper doesn’t explain why a half-sine function is better than Spencer’s (and other functions are of course available). Some people may be attracted to a half-sine because it seems more sophisticated, but Spencer’s approach seems more plausible to me as well as having the advantage of simplicity. As I’ve said, I don’t know why the shear to horizontal force ratio would approach zero at the ends of the slip surface. The ratio seems likely to be particularly high in the zone of tension cracking. I also think elastic settlements would produce shear stresses beyond a slope’s crest and toe whatever the slip surface position. So to return to my original question, when it’s appropriate to use the relative simplicity of method of slices software, why not use a typical shear stress profile derived from generic FE analyses instead of seemingly arbitrary functions based on horizontal forces? Why not try to take advantage of insights from FE over the last few decades since the methods of Spencer and others were introduced?

>You could also have mentioned the problems of soil variability, sample disturbance, testing relevance, matric suction, tension cracks, climate change and vegetation unknowns. We seem to be in agreement about the risk of trying to be too clever about analysing stresses.

Yes, and all models require simplifications :). Many of these issues can be dealt with using current software but practically the problem is now getting the inputs / lab data for them.

>Bishop’s method has the merits of simplicity and honesty: the internal stresses can’t be determined from statics so we’ll just ignore them and use a healthy safety factor - which is also needed for all the other unknowns. This approach of course is generally a bit over-conservative unless the soil is weaker lower down. In this case Spencer’s method should be safer, although if there’s a weak lower stratum and a composite slip surface then a wedge analysis seems more sensible.

I have heard some people argue that we should switch back to simpler methods that can be checked by hand if an LE analysis is going to be used; They might have a case.

>In this case Spencer’s method should be safer, although if there’s a weak lower stratum and a composite slip surface then a wedge analysis seems more sensible.

Maybe but then, despite the points you've raised Spencer and Morgenstern-Price give essentially the exact same answer, or at least close enough that you shouldn't need to worry about it if you have any understanding of the extent of all of the other simplifying assumptions in the model.

>Krahn’s paper doesn’t explain why a half-sine function is better than Spencer’s (and other functions are of course available). Some people may be attracted to a half-sine because it seems more sophisticated, but Spencer’s approach seems more plausible to me as well as having the advantage of simplicity

Practically speaking the methods return essentially the same answer and with modern computer software it really makes no difference, you can run all of the avaialable methods in Slope/W on tens of thousands of slip surfaces in a couple of minutes.

>So to return to my original question, when it’s appropriate to use the relative simplicity of method of slices software, why not use a typical shear stress profile derived from generic FE analyses instead of seemingly arbitrary functions based on horizontal forces? Why not try to take advantage of insights from FE over the last few decades since the methods of Spencer and others were introduced?

Because you can just run an FE analysis and uses the stresses in the LE analysis, which is what Krahn proposed 20 years ago? If you need something more complicated than that, use FE; if you need something less, current LE methods are adequate. From what I understand there aren't really many practical cases where the difference between FE, LE with FE derived stresses, and LE makes a difference. Sure I guess you could try coming up with a revised LE method (I haven't done a literature search, but maybe someone has already done this). Thing is, the odds of getting any serious uptake of it are pretty close to nil, and there aren't really any practical advantages of doing so. If you want a better understanding of the stress distribution, use an FE model (they are widely available and similarly easy to run now).

The other thing to consider in all of this is: What are you trying to accomplish with your analysis of the slope?

If your goal is to show that the proposed slope meets accepted criteria, then you need to run your analysis using the same or at least similar methods as were used to develop the criteria. For example, in dam and levee design, the criteria were developed using LE slope stability methods. The required Factors of Safety are based on LE methods and experience using those methods on real projects over the course of decades. So if you use a different method to analyses the slope, what criteria are you going to compare your results to? Will the approving agency and/or owner accept your analysis?

I'm all for progressing the profession, but every change in the analysis method results in other changes that need to be made. It is a long and tedious process not to be undertaken lightly. So you need to ask why make changes? Are the results really better or just different? Do we really know the inputs required for a FE/FD analysis to a degree of accuracy such that the results are really better than a LE analysis? In reality (at least in the US), most slope stability analyses are performed by engineers with a BS degree. They have had ONE geotechnical theory class. Are these engineers really capable of running a good FE/FD analysis?

Just food for thought.

Thanks geotechguy1 for your comprehensive reply.

I expect Krahn and Fredlund have done well financially from their allegedly general method of limit equilibrium, but even in the developed word not all engineers can afford software like Slope/W and not all problems warrant investing much time. I know geotechnical firms are not going to adopt a method having a rational basis for inter-slice forces but a student in a third world country might think it’s a worthwhile idea for a thesis - except that if a general method is believed to already exist then this is unlikely.

It seems a good idea for engineers to understand the principles of how their software works, so I wonder what goes through their minds when they use the half-sine function. As this seems to be the default option in Slope/W I guess the answer is not a lot. And why would anyone bother to think about it when the geotechnical community seems to accept this is part of the general solution.

The half-sine function means the shear stress is reduced to zero at the ends of the slip surface. Yet if one tries reducing the slip radius then the stresses become zero at the ends of the smaller slip where previously they weren’t zero. I personally can’t defend the logic of this when alternative assumptions exist, but I realise my view can be dismissed as pedantic given that any method that achieves static equilibrium is likely to give similar answers. This seems to depend on how important one thinks rational principles should be in engineering. As society in general has abandoned the age of reason I suppose the answer is not very.

I think you make some good points GeoPaveTraffic.

I’d rather use a simpler method that I know makes sense and can be directly checked than something that takes longer and can’t realistically be checked personally. One could use two completely different FE programs but that’s more time and cost, and for engineers who only need FE occasionally I think they would still have the feeling that FE is a black box, albeit two boxes not one. Designing something when you aren’t confident you understand what you’re doing isn’t a good feeling.

Engineers who routinely use FE are obviously happy with it, but many civil engineers work in small organisations where they can be expected to tackle a fairly wide range of work. For them FE will often not be the best choice.

Great points by GeoPaveTraffic

>It seems a good idea for engineers to understand the principles of how their software works, so I wonder what goes through their minds when they use the half-sine function.

Well, what goes through their mind if they use Spencer and get a FoS of 1.312 instead of 1.311? One reason engineers don't worry about it is the interslice force function actually doesn't make much of a difference to the result.

Thanks geotechgy1.

Side forces can make a difference in the calculated FOS and should always be examined to ensure that they are reasonable. As for you example, reporting reasonable degrees of precision is one of my pet peeves.

A Smith, side forces going to zero at the top and bottom of a slide plane makes perfect sense to me.

If we think about the top of a slide plane, the only way to generate side forces is for the soil to be in tension. If you consider the soil saturated, more in a minute, then very little if any tensile force can be generated. If you are considering unsaturated soil mechanics, then some amount of soil tension could be possible. My problem with unsaturated soil mechanics is: How do you know that the soil will always be unsaturated? So to me, while unsaturated soil mechanics may be more accurate in some situations; it must be used very carefully.

If we think about the bottom of a slide plane, side forces become harder and harder to resist as the size and weight of the slices becomes smaller. So significant side forces are very difficult to resist due to the small weight of the slice and since by definition nothing is moving in LE, the side forces must be very small.

It's been interesting reading the various comments from colleagues for which I have a lot of respect! I wonder about FEA when a 4th order theoretical analysis is using 1st order soil data . . . I also wonder whatever happened to the use of the various slope stability charts. Michalowski states: "While computational tools have made most graphical methods and charts obsolete, stability charts for slopes are still routinely used in practice."

I was on a job in Indonesia and the mine had a slide during the open pit cut. They asked me to have a look at it; I did. They asked their own geotechnical department to have a look at it; they did. I spent 30 minutes on it and most was to get the presented graph right. They spent 2 days with Slope/W. We ended up with the same answer. BTW, I used charts.

One thing I noticed - as almost no one has done Bishops or Janbu by hand - many don't have a good feel as to whether a potential slip surface is reasonable or not. I have seen a Slope/W analaysis of a roadway cut into deep into the ground - similar slopes on each side and one of the slip surfaces they came up with was a full half circle starting near the crest of one of the slopes, going under the road and then up the other side! Really. Would never have happened in the field.

Thanks BigH for pointing out the usefulness of slope stability charts. So the consensus in this thread isn’t in favour of mathematical sophistication - the main sources of error lie within the physics.

GeoPaveTraffic mentioned side forces being zero at the ends of the slip surface. The horizontal force within the slip at these two points certainly is zero because the vertical height of the slice is zero here. In my previous post I was talking about stresses not forces, and the stresses in the soil are independent of the chosen slice positions and depths. Similarly there will be a positive horizontal force in the toe region below the end of the slip surface.

As for horizontal forces near the crest, these could be positive in a stable slope but, as GeoPaveTraffic says, negative at the limit of stability. However forces of cohesion and suction can exist in saturated as well as unsaturated clay. About 10 years ago I submerged lumps of clay inside a sealed glass jar and they haven’t fallen apart yet, so cohesion can persist even when the matric suction is zero. (Apologies for this not being quite as impressive as BigH’s mining anecdote.)

I subscribe to the view that matric suction is one of the most important factors engineers should consider in slope stability even in cases where it is practically impossible to quantify reliably - which is most of them.

Responding to the comment that the Spencer and half-sine methods would change the FoS by about 1 part in a 1000, it would seem to make no sense for Slope/W to allow for different guesses at the shear force function if these made no significant difference. I expect such differences can arise when soil strata within a slip circle have different strengths. However my point was about the use of a seemingly counter-intuitive shear function.

Krahn quotes an example of FoSs of 1.318 and 1.145 (a difference of about 15%) for methods that take ground movements into account via FE and those that don’t. He says “... different normal stresses in the toe area result from the shear stress concentration in this part of the section. Localized shear stress concentrations are, of course, not captured in a limit equilibrium formation ... This is one of the limitations of the limit equilibrium method.” But this isn’t true of an LE method that’s outside the “general” LE method. A typical FE shear profile having greater stresses at the toe can be used in a LE method of slices that doesn’t itself need the complication of FE software.

Suppose it was possible to construct a slope using separate blocks of soil of different heights. These would be allowed to settle and bulge, then their sides would be trimmed so they could all be fitted together and fused to form a single soil mass. It would then be valid to ignore the previous ground settlements, so any of the “general” methods would be sensible ways to analyse this slope. But of course this isn’t the way slopes are formed either naturally or artificially, so it’s no wonder that the “general” methods can give consistent yet misleading results. These methods are solving a different problem.

The example Krahn quotes is specifically for a slip surface exiting at the toe (where there would be a shear stress concentration), which is a case where there is a bigger difference (and the LE model was conservative in that case anyway).

>Suppose it was possible to construct a slope using separate blocks of soil of different heights. These would be allowed to settle and bulge, then their sides would be trimmed so they could all be fitted together and fused to form a single soil mass. It would then be valid to ignore the previous ground settlements, so any of the “general” methods would be sensible ways to analyse this slope. But of course this isn’t the way slopes are formed either naturally or artificially, so it’s no wonder that the “general” methods can give consistent yet misleading results. These methods are solving a different problem.

Back to the supercomputer DEM model to run a simulation of a few million years of geological history based on your preferred geological story, then?

You should chat with an engineering geologist or geomorphologist sometime - they'll tell you that all of this slope stability modeling civil engineers turned geotech engineers do is a load of total rubbish anyway (at least for natural slopes); perhaps the point about using bishops method / and or slope stability charts isn't so bad.

There is nothing unusual about slip circles exiting at the toe, and it’s important to know how different assumptions can affect the resulting FoS.

I thought we’d agreed that mathematical sophistication is often unnecessary along with the need for a supercomputer.

Nor do I have a preferred geological history going back millions of years. When an embankment is constructed, the higher part settles more and this induces vertical shear stresses. Such stresses also occur when overburden is removed to form a cutting. It seems obvious these stresses should be taken into account when engineers use a method of slices requiring vertical shear stresses to be estimated.

Artificial slopes are the province of the engineer and natural slopes are more the province of geologists. Where a watercourse has made a cutting through alluvium this seems akin to an artificial cutting, but where the soil on a slope has weathered in-situ from rock I think your reference to engineering geologists is instructive. The weathering process probably destroys earlier shear stresses and a FE analysis that treated the soil as alluvium or artificial fill would probably be invalid. So I agree that Bishop or any other time-honoured approach wouldn’t be at all bad in these circumstances.

A rational method of slices would therefore need to distinguish between different types of soil origin. But FE or other software that’s even more sophisticated should also take account of geological considerations and hence be acceptable to geologists. Why doesn’t this seem to be the case?