Academic Issues – Page 4 – vulcanhammer.net

My Response to Rodrigo Salgado’s “Forks in the road: decisions that have shaped and will shape the teaching and practice of geotechnical engineering” and an announcement

Posted on 9 April 202421 April 2024 by Don Warrington

I was intrigued by Purdue University’s Rodrigo Salgado’s “Forks in the road: decisions that have shaped and will shape the teaching and practice of geotechnical engineering.” It comes at a timely point in my own journey in geotechnical education: as I write this response, I am in the process of finishing up and putting a wrap on my last semester of teaching geotechnical courses at the University of Tennessee at Chattanooga. I will explain more about this at the end of the piece but it will probably be the last time I teach these topics “face to face.”

My effort in geotechnical education has never been strictly about the classes I teach at the University. This website has enabled me to disseminate information on the topic for many years, which is implicitly educational. In more recent times I have posted topics of interest, many of which have emerged from my teaching, and I will continue that practice.

With that introduction out of the way, I can proceed to respond to the various topics that Salgado brings up.

Empiricism, science and geotechnical design

As the scion of one of the oldest active families in the business–the Vulcan Iron Works–I feel I am in an interesting position to address this issue. The Warrington-Vulcan hammer was introduced in the late 1880’s into a world where wood piles–the same types of piles driven by the Lake Dwellers in Switzerland–were pretty much the norm. It was also the era when the Engineering News formula (itself an empirical project) came into being. In subsequent years many other types of piling were introduced, but as Salgado notes soil mechanics–with the exception of lateral earth pressures–were pretty much undiscovered territory. Pile dynamics suffered from a similar level of ignorance, one compounded by civil engineers’ historic aversion to things that move. Vulcan was aware of the problems with the dynamic formulae, but it wasn’t until after World War II that these problems were tackled with a more scientific approach, and even then one gets the impression that Smith’s choice of soil model was fortuitous as much as anything else. In the meanwhile Salgado’s statement that “There is a common misconception that all engineering done before the advent of science was conservatively done” was a misconception not shared by those who really understood the problems with the pile dynamics of the first half of the twentieth century.

The Original Sins of Mohr-Coulomb and the Associated Flow Rule

I have to confess that my thinking on these topics has been formed by two processes: 1) my years of teaching a discipline where the topics do not have as immediate an integration one to another as other branches of civil engineering, and 2) my dissertation, which wasn’t complete until 2016. The latter involved writing geotechnical finite element code from the ground up, which gives me something of a greater appreciation for the issues which Salgado brings up than many who teach this course on an undergraduate level.

Let’s start with Mohr-Coulomb itself: I used it in my FEA code with the following justification:

Based on this and the elasto-plastic discussion, the Mohr-Coulomb failure model was chosen. As Abbo et.al. (2011) point out: “The Mohr–Coulomb yield criterion provides a relatively simple model for simulating the plastic behavior of soil. Other more sophisticated constitutive models for predicting the behavior of soil have been developed over the past three decades, however the complexity of these models, as well as the additional testing required to determine the various soil parameters involved, minimizes their utility for practicing geotechnical engineers. The Mohr–Coulomb yield function is also of importance to finite element researchers and practitioners as it forms the basis of many analytical solutions. These analytical solutions serve as crucial benchmarks for validating numerical algorithms and software.” This observation is supported by McCarron (2013).

In spite of its limitations, Mohr-Coulomb still has its place in geotechnical engineering. Salgado focuses on two serious problems with Mohr-Coulomb: the sharp division between cohesionless and cohesive soils and the use of the associated flow rule, an issue which doesn’t emerge in many “closed form” solutions but is certainly an important one in finite element code.

To illustrate my own journey with the first problem, having come from a deep foundations background, I was taught the cohesionless/cohesive dichotomy which is built into the whole “alpha/beta” method business. That was challenged by John Burland a long time ago. The key to ditching alpha methods–and by doing this use beta methods–is to note that the cohesion of a soil is a function of its effective stress (and other variables.) It is for this reason why I have not taught alpha methods for driven piles for some time but prefer Fellenius’ Method, recognising that the student may find themselves forced to use another method once “in the wild.” (One of these days we’ll fix that dichotomy embedded in O”Neill and Reese’s method for drilled shafts.) Fellenius’ Method has its own problems (it requires the student to use some judgement, which they haven’t quite developed) but I think in the end beta methods will win out.

But there is a more compelling factor at work: the most important single factor in the behaviour of a soil is its grain size distribution. With this soils don’t neatly divide themselves into two categories; it’s a continuum from coarse-grained to fine-grained soils. Even the classification system doesn’t quite do this justice; it’s hard to argue that a soil with 51% fines is that much different (all other things being equal) to one with 49% fines, even though they will have different classifications and the former will be considered “cohesive” and the latter “cohesionless.”

Looking at things this way, “purely” cohesive or cohesionless soils–each at the extreme of grain size distribution–are rare exceptions rather than the rule. This means that, from a Mohr-Coulomb standpoint, virtually all soils have a non-zero $c$ and $\phi$ , something that Salgado notes engineers figured out a long time ago (erroneously, in his opinion.) But that’s just the beginning of the problem, we must now deal with the other quantity that Salgado brings up: dilatancy. And that in turn leads us to the business of the associated flow rule.

Once again my position on that topic is in my dissertation:

Except for purely cohesive soils, a purely associated flow rule is to be avoided for soil materials in the model. Such a rule is acceptable for many engineering materials but does not realistically model the dilation of soils, especially cohesionless ones. The downside to this is that the elasto-plastic constitutive matrix is non-symmetric, which, strictly speaking, will result in a non-symmetric stiffness matrix, increasing the cost of the problem solution.

With an associated flow rule $\phi = \psi$ and the dilatancy is “locked” into the value of the friction angle. With a non-associated rule $\phi \neq \psi$ and we must determine the value of $\psi$ . The literature on meaningful values of $\psi$ isn’t as copious as one would like but the situation is improving. Including the dilatancy solves many of the problems that Salgado raises with the Mohr-Coulomb failure criterion, especially the sole reliance on the Mohr-Coulomb failure function. (An implementation of such a rule is here.) His application of a non-associated flow rule to the bearing capacity problem is intriguing, and would be more intriguing except for the fact that generally settlement is a more important failure mode to consider than bearing capacity. (Application of this to Coulomb and log-spiral failure of retaining walls is another matter altogether.)

At this point we run into the serious problem: how to implement the solutions properly in the class setting. The problem we have in geotechnical engineering is that, in order to consider the effects of elasticity (assuming that exists, which is problematic,) plasticity and dilatancy in “one shot,” you really need finite element code, which can move from elastic to plastic deformation continuously. Most of the “hand” solutions we’ve used up to now have assumed either an elastic state or a plastic one, and the latter with an associated flow rule, as Salgado observes. I think that, moving forward, we need to consider numerical methods as the best way to solve problems in this profession. But it is dangerous to rely on these solely; they tend to become “black boxes” in the hands of many practitioners, whose results are taken as “final” even when they are erroneous. (Those of us involved in pile dynamics are all too aware of this problem.) The hand solutions in these cases need to be ancillary to make sure we are not doing something really stupid with the numerical ones, and I think our education needs to reflect that.

Shear Strain Localisation

The way I have dealt with this issue is to use the lower and upper bound concept, one which I have borrowed from Verruijt. By showing that there is a beginning of failure in the “surface” (a term Salgado objects to) and an end we can bracket the beginning and the end of plastic failure, which shows a truth students need to get used to in soil mechanics: our designs inherently have a degree of failure; the question that remains is how much is acceptable. Some of Verruijt’s examples of lower and upper bound are lame but that can be fixed. It’s also interesting that Salgado picked as a primary example a driven pile. In spite of being the driven pile person I am, I would not, but perhaps that’s because we get to slope stability, lateral earth pressures and bearing capacity before deep foundations. But a more comprehensive solution to the problem would be to pitch the “bearing capacity in piles” concept altogether and base design on load-settlement, which forces us to consider the progressive failure of the shaft surface from the head down and finally the failure of the toe itself.

Since he brought up the subject of driven piles and the “groundbreaking” studies of Randolph and Wroth: in that study they included varying the modulus of elasticity as one got away from the shaft surface. It is possible to use a hyperbolic stress-strain model of the soil to vary this with the strain continuously, as I demonstrate here.

Particulate Modelling and Artificial Intelligence

One thing I’m thankful that my dissertation committee (or one member) talked me out of was particulate modelling. I think that the non-homogeneity of soils and particle shapes will make this a challenging pursuit; continuum modelling should suffice for the foreseeable future.

Salgado is spot on about using any kind of artificial intelligence and machine learning without inclusion of physical modelling: it will not work. Reducing our analysis to a statistical pursuit with no regard for the basic physics of the problem–ours or anyone else’s–will lead to disaster. I have commented on this in other fora and hope our community will resist the temptation of yet another type of “magical thinking” that permeates our society.

Our World In Stress

I usually reserve things like this to another site but since he brought it up:

Our fertility rate globally is dropping. The biggest problem we have with this is that our democracies are not prepared on dealing with funding the consequences of that change. So we will not have so many people who are consuming.
I find it difficult to understand why, since the central problem is the emission of carbon dioxide, nuclear power is not included in the solutions. The anti-nuclear movement had its genesis in the 1960’s and 1970’s, a time when the mood among many was decidedly anti-technological. It’s time for a change.
It is ironic that the methods used by the oft-demonised oil industry are now key to offshore wind.

The second point, in a sense, leads me to why I am departing my 20+ years of teaching geotechnical engineering in a university setting. This fall, Lord willing, I will begin teaching at Lee University, my church’s institution of higher learning. They are beginning an engineering program which has not developed to the point of being discipline-specific. I think, however, that Lee needs an engineering program, its students need the option of one, and that the liberal arts centred educational system, one we inherited from the British and have had since before the Republic began, has failed us, not only in producing a leadership class totally incapable of informed decision on technological issues, but also bereft of what Antoine Arnauld and Pierre Nicole called Logic, or the Art of Thinking. Much of this is an outcome of the same era as the anti-nuclear movement; we have replaced a pursuit of understanding with a pursuit of raw power, and the end result will be that we will end up with neither.

In summary, I think that Salgado’s idea is basically a good one. In addition to what I have observed above, I think that his program will be easier to implement in a well-funded and deep-research institute such as Purdue than in places (and there are many) where geotechnical engineering is an afterthought.

Academic Issues, Geotechnical Engineering, Soil Mechanics

Getting to the Legacy of B.K. Hough and his Settlement Method

Posted on 14 March 202413 June 2025 by Don Warrington

Note: since I first wrote this, more information has come to light (for me at least,) which is contained in the post Closing the Loop (or at least trying to) on Hough’s Settlement Method.

Last year I posted The Sorry State of Compression Coefficients where I a) gave a brief summary of earlier posts on consolidation settlement and b)showed that there was more than one way to express them. An example of the “alternative” (for geotechs in some countries it’s the accepted way) compression coefficient is “Hough’s Method” which is featured in both the Soils and Foundations Reference Manual and the Shallow Foundations manual. Hough’s Method, however, is for cohesionless soils. Why, you ask, can methods usually associated with cohesive soils be applied to cohesionless ones? Because consolidation settlement methods using the logarithmic difference of pressure reflect the fact that the elastic (or shear) modulus of a soil increases as the void ratio/porosity of the material decreases, which I discuss in From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”.

In this post I will attempt to do two things:

The method as presented in the above references has been described as too conservative, i.e., the settlements predicted are too large. I will attempt to explain this and perhaps offer a solution based on Hough’s own works.
Discuss the whole business of bearing capacity vs. settlement failure in shallow foundations, which was perhaps the greatest legacy of Hough’s work and remains an important issue in geotechnical engineering today.

Bearing Capacity vs. Settlement

Terzaghi’s solution (or more accurately his adaptation of Prandtl’s punching shear theory) of the bearing capacity problems was one in a number of solutions that became “reference standard” in geotechnical engineering.

We were regaled with photos of Canadian grain elevators on their side to show that bearing capacity failure was the first thing we should look for in shallow foundation design. Terzaghi’s formula was so highly regarded that for many years it was fashionable for introductory geotechnical courses to require students to learn both Terzaghi’s method and the subsequent improvements/extensions of that method by researchers such as Meyerhof, Vesic, Brinch Hansen, etc..

Up until that time shallow foundations were generally designed using what we call “presumptive bearing capacities” based on soil types and foundation configurations. These were enshrined in the building codes of the day. They were generally purely empirical in nature, as was most of geotech in the era before Terzaghi and his contemporaries. They had one advantage however: because they were derived from actual performance, be it ever so crude, they included the effects of soil settlement under load.

Like any other engineering material, only on a larger basis (because their elastic/shear moduli were several orders of magnitude lower than more conventional materials) soil is deformable under load. That deformation not only allows the foundation to deflect under load, it also affects the failure surfaces as they develop. The latter reality became apparent and so we have the modification of the bearing capacity for punching and local shear. It should be noted that Terzaghi and Peck were well aware of the problem of settlement, and included provision for it in their classic 1948 textbook Soil Mechanics in Engineering Practice.

One possible solution was to use elastic theory for the initial settlement. The implementation of that is discussed in Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads. It is even possible to develop a lower bound solution for the bearing capacity problem, as was discussed in Lower and Upper Bound Solutions for Bearing Capacity. The problem with this is twofold. The first is that the lower bound solution assumes that the footing is a purely flexible foundation, which is not really true with this type of foundation. The second is that, if we went to the other extreme and assumed a purely rigid foundation, by elastic theory the stresses at the corners is infinite for any load. (This is conventionally attributed to foundations in cohesive soil, but it can be shown to be true by elastic theory.)

Hough’s Settlement Method

Hough presented his settlement method in his 1959 ASCE paper. He starts by presenting a graph similar to the following, which illustrates the transition from small, elastic displacement to large inelastic ones:

The solution is in the form (similar to that presented in Verruijt) of

$S=\frac{H_{o}}{C'} \log_{10} \frac{\sigma'_f}{\sigma'_o}$

The coefficient $C'$ is determined using a chart which is below.

The chart itself is basically the same as the one in Hough (1959,) but now things get complicated.

In Shallow Foundations we are informed of the following:

Cheney and Chassie (2000) recommend that the SPT blowcounts be corrected for overburden pressure before correlating the N-values to the bearing capacity index,
C′. An overburden correction by Bazaraa (1967) was recommended by Cheney and Chassie (2000). Since that time, many researchers have studied the effect of overburden stress on the SPT N-value, largely in support of liquefaction hazard assessment procedures. Recent consensus by the 1996 and 1998 National Center for Earthquake Engineering Research (NCEER) (Youd et al., 2001) concluded that the correction proposed by Liao & Whitman (1986) (shown in Figure 5-18) could be used for routine engineering applications. Therefore, the correction by Liao & Whitman is included here as part of the Hough procedure, in particular because it is easy to calculate and can be used without charts in simple computation spreadsheets.

The basic problem with this is that Hough himself never mentions any kind of correction for the variations of the SPT tests, overburden or otherwise, and that in modifying Hough’s Method we run the risk of making some assumptions that may not be applicable.

The idea of using the SPT tests, even with shallow foundations in cohesionless soils such as the ones Hough is concerned about, is admirable on the face because undisturbed samples of these soils are hard to obtain in normal soil testing. The problem with SPT tests–which were coming into acceptance when Hough presented his original method–is that their variations in both configuration and those induced by the overburden are significant. It wasn’t until the 1980’s that this was “formally” sorted out with the correction system that we have today.

With our current method we have two stages of correction. The first stage is for variations in the configuration of the split spoon, the effect of the rod and borehole, and most importantly the efficiency of the hammer itself. The second is for the overburden. In both his original paper and the presentation of the method in the Second Edition of his textbook Basic Soils Engineering (1969,) Hough does not delve into either of these.

The whole point of the N₆₀ correction (the first series) was to harmonise the results to a single mechanical standard, and one that was generally attained “back in the day” so that empirical methods such as Hough’s could be used today. We could assume that Hough’s value are at least N₆₀ values, although we’re not guaranteed of that due to the lack of supporting data.

The business of overburden correction brings up Bazaraa (1967.) At this point credit needs to be given where credit is due: Bazaraa had the thankless task of sorting out the settlement methods which were current in his day, and that task included dealing with the variations of the SPT method. Since his overburden correction method was originally used with Hough’s Method and then changed, let us compare the two. We start be defining

$p_{norm} = \frac{\sigma'_o}{p_{atm}}$

The two correction factors $C_N$ for overburden are shown below.

Except for the region where Liao and Whitman is “flat topped” the two are reasonably close, and so substituting Liao and Whitman’s correction is legitimate, assuming it should be used at all.

It should be noted, however, that Bazaraa’s objective is not to correct Hough’s method, which he did not do: it was to provide a new method of estimating settlements in cohesionless soils, and to advance Terzaghi and Peck’s 1948 method. So we are still “up in the air” about how to apply all of this to Hough’s Method.

If the overburden stress is less than the standard atmospheric pressure (approx. 2 ksf) then the N value used will be increased, the C’ values will likewise increase, and the estimated settlement will decrease, which may be more accurate but moves away from conservatism. If the opposite is true then the result will be more conservative. For shallow foundations such as Hough’s Method the lower overburden stresses will be more of a factor (depending upon the depth D.) At this point it is not clear to me (at least) that including an overburden correction is really significant given the other unsolved problems that exist with this method.

One further complicating factor is that, in the aforementioned Second Edition of Basic Soils Engineering, Hough presents a new chart (again with no explanation of correction of any kind) which is redrawn as follows:

Some trend line work results in a correlation in this form:

$C = A \exp^{BN}$

where the coefficients A and B are below.

Soil Type	A	B
Organic Silt, Little Clay	7.22	0.0229
Inorganic Sandy Silt	18.28	0.021
Very Well Graded Fine to Coarse Sand	22.86	0.0203
Well Graded Clean Fine to Coarse Sand	28.22	0.0216
Well Graded Silty Sand and Gravel	32.85	0.0216
Uniform Clean Inorganic Silt	37.02	0.0221
Very Uniform Clean Medium Sand (Similar to Std. Ottawa)	58.66	0.0225

It should be noted that, where the curves are directly comparable with Hough (1959), they tend to be lower, i.e., the values of C are less. This would increase settlement and thus conservatism. At this point perhaps we should consider Hough’s Methods in the plural rather than the singular.

Beyond Hough’s Method

To start this part consider this, from NAVAC DM 7.1:

We have the conventional C_c based on either liquid limit, initial void ratio or water content. Hough is represented here as well; however, these correlations are for cohesive soils. Hough himself had a broader application for these.

In the Second Edition of Basic Soils Engineering, he presents a function to compute C_c as follows:

$C_c = a(e_0 - b)$

He also establishes a relationship between correlations based on liquid limit and those on void ratio, but getting to that is for another time.

In any case he presents values for this equation which are tabulated below, and include both cohesionless and cohesive soils.

	a	b*
Uniform cohesionless Material (C_u < 2)
Clean Gravel	0.05	0.5
Coarse Sand	0.06
Medium Sand	0.07
Fine Sand	0.08
Inorganic Silt	0.1
Well-graded, cohesionless soil
Silty sand and gravel	0.09	0.2
Clean, coarse to fine sand	0.12	0.35
Coarse to fine silty sand	0.15	0.25
Sandy silt (inorganic)	0.18	0.25
Inorganic, cohesive soil
Silt, some clay; silty clay; clay	0.29	0.27
Organic, fine-grained soil
Organic silt, little clay	0.35	0.5

*The value of the constant b should be taken as e_min whenever the latter is known or can conveniently be determined. Otherwise, use tabulated values as a rough estimate.

From here we can use the well-tried “C_c” formulae (which would include preconsolidated soils) to estimate settlement. This opens up a new vista for using conventional consolidation settlement theory to estimate the settlements of cohesionless soils.

To return to our original objectives, settlement and bearing capacity are, in the process of failure, two sides of the same thing. It is also worth noting that most foundations fail in settlement, although the failure isn’t as “spectacular” as bearing capacity.

The only way to treat them together is to use a method that can combines both phenomena, and finite element methods are capable of doing that. However our profession has been uncomfortable with “black box” methods such as these, although they really apply the same laws we use to formulate closed form solutions. In that respect they have value, and to apply methods such as Hough’s to cohesionless soils with better calibration of the constants would be a good step forward.

References

Bazaraa, A.R.S. (1967). “Use of the Standard Penetration Test for Estimating Settlements of Shallow Foundations on Sand,” Ph.D. Thesis presented to University of Illinois, Urbana.
Hough, B.K. (1959). “Compressibilty as the Basis for Soil Bearing Value,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 85, Part 2.
Hough, B.K. (1969). Basic Soils Engineering. Second Edition. New York: Ronald Press Company.
Kimmerling, R.E. (2002). Shallow Foundations: Geotechnical Engineering Circular No. 6. FHWA-SA-02-054. Washington, DC: Federal Highway Administration.
Terzaghi, K. and Peck, R. (1948). Soil Mechanics in Engineering, 1st Ed., John Wiley & Sons.

Academic Issues, Geotechnical Engineering

Bengt Broms RIP

Posted on 12 March 2024 by Don Warrington

We may be a little late to the announcement but we regret to inform you of the following:

Professor Bengt Baltzar Broms, Stockholm, Sweden, passed away at the age of 95. He is survived by his wife Carina and children Erik, Karin, and Peter.

Bengt Broms was a leading figure in Swedish and International Geotechnology and Foundation Engineering. He is mourned by many colleagues around the world, especially his former doctoral students, whom he mentored with exceptional expertise and generous advice.

For this site, in addition to his research work his contribution has been the numerous drawings that he posted online back in the late 1990’s/early 2000’s, in the early years of both this site and of my teaching at the University of Tennessee at Chattanooga. I still find them very useful and certainly better than my own “chicken scratchings.”

Memory eternal.

Academic Issues, Geotechnical Engineering

A Simplified Method to Design Cantilever Gravity Walls

Posted on 6 October 202316 June 2025 by Don Warrington

On this site we feature the U.S. Army Corps of Engineers publication Retaining and Flood Walls, which details the design of several types of retaining walls. As it was published a good while back, it details the design of these walls using hand calculations. Sometimes these can get tedious, especially when the aptly named “trial wedge” solutions are employed.

These days it’s more likely that a computer solution–be it a true finite element analysis or simply the automation of those tedious hand calculations–is used to finalise the design of a retaining wall. An example of such an analysis is shown above, and addresses in particular an issue that gets the short shrift in classical retaining wall analysis of any kind: global stability failure. An idea of a hand solution to this problem is shown at the right. Global stability failures still happen and are generally disastrous; beyond a conventional slope their analysis is fairly complex.

But in the meanwhile, what do we do to verify that we’ve got it right with a computer solution? Or what can we do to start with a reasonable design that we can refine with numerical analysis? Back in the “slide rule era” we used quick, “back of the envelope” methods to design things, and we can use them today both to get started and to gain a basic understanding of the elements of the design.

In this case we’re going to discuss the design of a cantilever retaining wall, an example of which is shown below.

The methodology is based on the aforementioned Retaining and Flood Walls and the example (which is a little clearer description of same) comes from Appendix A of Seismic Analysis of Cantilever Retaining Walls, Phase I. I’ve modified it in a couple of spots and will detail those modifications as we go.

The example we’ll look at is below. We need to design this wall to prevent failure against three events: sliding, overturning and bearing capacity failure.

Defining the Geometry, Weights and Centres of Gravity

Our datum/coordinate origin is at the toe, which is convenient since we assume that the overturning moment is computed around the toe. Since this is a cantilever/gravity wall, the weight of the wall is part of the resistance force and moment. Included with that is the backfill that is trapped behind the wall (shown above.)

We start by dividing up the wall into sections, each of which has a weight and a centre of gravity. The first section we take up is the backfill, which is a simple rectangle shown above.

At this point we need to make one correction to the Corps’ work: the weights and moments are in pounds per unit length of wall and ft-pounds per unit length of wall, respectively. Leaving those per unit length designations is a common shortcut among practitioners but is confusing for students, who frequently find the unit length concept difficult to grasp at first. The weight of the backfill is properly 18,000 lbs/ft of wall and the moment around the toe ((clockwise) is 162,000 ft-lbs/ft of wall.

Computing the weights and moments of the various sections of the wall itself yields the following results.

Taking all of this information and processing it yields the total weight, total moment around the toe, and moment arm around the toe:

Since most engineers have learned their statics via vectors, a review of Vector Statics and “Old Coot” Statics: An Example and What is a Resultant in Geotechnical Engineering? may be in order.

The Shear Mobilisation Factor

Now comes the tricky part: in retaining wall design, we traditionally define the factor of safety as

$FS = \frac{F_{resisting}}{F_{driving}}$ (1)

Where the “F” values can be forces or moments and FS is the factor of safety. Sometimes this is not the optimal way of applying factors to account for uncertainties, especially when we get to LRFD. Another approach is either to increase the driving force or reduce the resisting force. We do the latter with sheet piling (where there there are earth forces to resist) but here we’ll to the former. To make this happen we first define a shear mobilisation factor (SMF) thus

$SMF = \frac{\tan \phi'_{mob}}{\tan \phi'}$ (2)

The value of $\phi'_{mob}$ can be computed thus:

$\phi'_{mob} = \tan^{-1}(SMF tan \phi')$ (3)

For this problem, assuming SMF = 2/3 and φ’ = 35°, by substitution φ’_mob = 25°. We will discuss the effect of cohesion later.

Now we turn to computing the force of the soil behind the wall on the wall.

We note the following:

Rankine theory is used. Methods shown in both Retaining and Flood Walls and the Soils and Foundations Reference Manual use Coulomb and/or log-spiral for these computations. The problem with this is that the value of wall friction δ is sometimes difficult to determine without data or experience, and in reality most of this pressure bears on soil, not the wall. So Rankine is easier to start with, and is more conservative.
The backfill is level, the formula for Ka only applies in that case. We will discuss sloping backfill below.
Note that φ’_mob (not φ’ ) is being used to compute the soil force.

Analysing Sliding and the Location of the Resultant/Overturning

At this point we can compute the forces on the base (both the forces T’ and N’ and the location of the resultant x_N’,) which are shown below.

The calculations are shown below.

Turning to the sliding problem, the driving force is the earth pressure force and the resisting force is the maximum Coulomb friction of the base/soil interface. In this case the value of δ is equated to the unmodified value of φ’, although that isn’t always the case. (The basis for this is that there is a thin layer of backfill sand under the wall, under which is a different foundation soil.) We then apply Equation (1) and determine the factor of safety against sliding, which checks out against the Corps criteria.

Knowing the location and magnitude of the resultant, we compute the maximum and minimum pressures on the base. Since both pressures are compressive, the resultant is in the middle third, and thus we can proceed with the base design.

One thing that is missing from this analysis is a specific analysis for overturning. In this case we make a common assumption that, as long as the resultant force of the wall is within the kern and there are no negative pressures on the base, overturning will not be experienced. It is certainly possible to do an explicit overturning analysis to check this result.

Bearing Capacity Analysis

With the wall’s sliding and overturning established, we turn to the bearing capacity analysis of the base. The complete bearing capacity equation, from the Soils and Foundations Reference Manual (with modification,) is

$q_{ult}=s_{c}b_{c}l_{c}cN_{c}+s_{q}b_{q}l_{q}C_{wq}q_{o}N_{q}+\frac{1}{2}s_{\gamma}b_{\gamma}l_{\gamma}C_{w\gamma}\gamma B_{f}N_{\gamma}$ (4)

where

q_ult = ultimate unit upper bound bearing capacity
s_c, s_γ and s_q = shape correction factors. These are unity in this case since it is a continuous foundation (generally the case with retaining walls)
b_c , b_γ and b_q = base inclination correction factors (unity in this case since the foundation is level)
C_wγ and C_wq are groundwater correction factors (unity in this case since groundwater isn’t an issue, a rare event with retaining and especially flood walls)
I_c, I_γ and I_q = load inclination factors, discussed below
N_c , N_q and N_γ are bearing capacity factors that are a function of the friction angle of the soil. N_c , N_q and N_γ are shown in the table below. These are handled differently when a slope is present. They are given in the Soils and Foundations Reference Manual. There is a general consensus for N_c and N_q but not N_γ. In this case we will use Vesić’s values for it, following AASHTO/FHWA practice. For this case the base soil φ’ = 40° we have N_c = 75.3, N_q = 64.2 and N_γ = 109.4
B_f = Base width of the foundation. In this case, with an eccentrically loaded foundation, this must be reduced to the equivalent foundation width by the formula B’_f = B_f – 2e = 13 – (2)(1.27) = 10.46’.
q₀ = overburden pressure on the base from the dredge (low) side of the wall. For walls such as this we neglect all effects of this, both any potential passive lateral pressure and overburden pressure.
c = cohesion of the soil = 0
- Because of this and the previous point, we can neglect the first two terms of Equation (4) and only concern ourselves with the last one.

Load inclination is the result of two perpendicular loads acting on the base of the foundation. It is illustrated in the sketch at the left.

The load inclination factors are given as follows:

$l_{c}=l_{q}=\left(1-\frac{\delta'}{90\textdegree}\right)^{2}$ (5a)

$l_{\gamma}=\left(1-\frac{\delta'}{\phi'}\right)^{2}$ (5b)

where the load inclination angle is given as follows

$\delta' = \tan^{-1}\frac{T}{N'}$ (6)

Substituting yields 𝞭’ = tan^-1 (10,137.5 lbs/26,625 lbs) = 20.8°. The friction angle of the base soil proper is 𝟇 = 40°. Substituting into Equation (5b) yields l_𝞬 = (1-20.8/40)² = 0.23. We can neglect the factors for Equation (5a) as those terms do not apply to this situation, but for completeness l_c = l_q = (1-20.8/90)² = 0.591.

Making all relevant substitutions:

q_ult = (0.5)(1)(1)(0.23)(1)(10.46)(125)(109.4) = 16,450 psf
Q_ult = q_ult B_f = (16,450)(10.46) = 172,067 lbs. = 172 kips
N = 26,625 lbs
FS = Q_ult/N = 172067/26625 = 6.46 (acceptable)

Settlement Analysis

Settlement of retaining wall is an important topic, as settlement of walls and levees has led to overtopping (as we found out the hard way during Hurricane Katrina.) Instead of picking a method and doing it “by hand,” we will use the USACOE software package CSANDSET, developed by Virginia Knowles. To accomplish this we need to do the following:

The foundation with we use is the reduced foundation width, or B’_f = B_f – 2e = 13 – (2)(1.27) = 10.46’.
The normal load is N = 26,625 lbs.
The unit load on the foundation is thus 26625/(2000*10.46)=1.27 tsf (the units used by the program.)
We assume a length of 50′; this puts the L/B > 10, which is an assumption for continuous foundations.

The input data is shown in the screenshot below.

The SPT and CPT are taken from “typical” values as they were not given in the problem statement. The option data is generated by the program. The horizontal (at-rest) earth pressure is Jaky’s Equation for normally consolidated soils.

The solutions the program gives are as follows:

The various methods are described in the program documentation. Schmertmann’s Method is given a full description in Foundation Design and Analysis: Shallow Foundations, Settlement. Elastic methods are treated in Soil Mechanics: Elastic Solutions to Soil Deflections and Stresses and related posts. It is interesting to note the wide variance in results; this is typical of geotechnical methods in a state of flux, and can also be applied to bearing capacity of driven piles.

The supplemental data generated by the program is at the end of the post.

Dealing with Sloping Backfill

The example problem above has a level backfill. Sloping backfills–usually positive (up from the wall,) occasionally negative, are common with retaining walls. The problem of the sloping backfill is illustrated at the right.

Without going into the actual solution of the problem using a sloping backfill, the following changes must be made in order to accommodate the effects of this condition:

The Rankine equation for sloping backfill needs to be applied. This can be found in the post Rankine and Coulomb Earth Pressure Coefficients. The coefficient computed is actually Coulomb theory applied without wall friction; the differences between this and “extended Rankine” theory for sloping backfill are discussed in the post on the coefficients.
An additional region needs to be defined with the soil below the sloping backfill (assuming it’s positive) as shown in the diagram below.
The length a of this region is already defined. The length b is given by the equation b = a tan (β). That length is important in a) computing the weight of the region and b) computing the additional length of the height the soil bears on the wall, or H + b.
The backfill force must be separated into horizontal and vertical components, as shown in the diagram above. The vertical component actually increases both the weight on the foundation and the resisting moment of the weight.

Assuming a β = 10° and applying φ’_mob = 25°, with the geometry shown we note the following:

With the two angles shown, K_a = 0.462, which is higher than the level backfill value.
The value of b = (8)(tan(10°))=1.41′. This means that the total height is 20 + 1.41 = 21.41′.
The lateral pressure at the base is (0.462)(21.41)(125) = 1236.5 psf.
The total lateral force on the wall F_h is (1236.4)(21.41)/2 = 13,236 lb/ft of wall. The resultant of that force is 21.41/3 = 7.14′ above the base of the foundation.
That lateral force has a horizontal component of (13236)(cos((10°)) = 13,035 lb/ft and a vertical component of (13236)(sin(10°)) = 2298 lb/ft. The latter has a moment arm of 13′ from the toe.
The area added by the sloping backfill has a weight W_ba = (125)(8)(1.41)/2 = 705 lb/ft. Its centroid is located 5 + (2)(8)/3 = 10.3′ from the toe.

We will leave working out the effects of this backfill slope to the reader.

The Shear Mobilisation Factor (SMF) and Cohesive Soils

If we have soils with cohesion in the backfill, the cohesion should be modified in a similar way to the friction angle thus:

$c' = c \times SMF$ (7)

A more complete treatment of the SMF is given in Retaining and Flood Walls.

Academic Issues

My View of Student Evaluations

Posted on 21 September 2023 by Don Warrington

One thing any academic does (except those at grand institutions where they get to have someone else do it for them) is evaluate students through grading. Most institutions afford students the opportunity to retaliate through the faculty evaluation system in place. That usually takes places at the end of the term, although my institution now had mid-term evaluations in place.

Student evaluations of faculty always take me back to this incident in my own undergraduate saga:

One of the things that the Mechanical Engineering department required* its majors to take was Logic, which was offered by the Philosophy Department. Most of the engineers did pretty well in this course, which was doubtless a source of secret frustration to liberal arts’ professors.

One day I went up to pick up a test from the professor. The professor looked at the grade, noted that I had nearly aced it, looked at me, and exclaimed, “You’re not as dumb as you look!”

The purpose of student evaluations of me is to determine whether they agree with my Logic teacher’s opinion or not.

*Originally posted here. Since then I discovered that there was another option available, but my advisor at the time did not avail me of that choice.