When you’ve taught any course for as long as I taught Soil Mechanics and Foundation Design and Analysis, you’ll sometimes get inquiries from your students about topics that you either didn’t cover or they might have missed. (Many comforted themselves during the course with the thought that “I’ll never go into geotech,” only to have it be their first job.) This topic came from a student from Florida (which is, of course of special interest for a long list of reasons, esp. since I grew up there) who saw on a geotech report about bearing pressure on a shallow foundation (in this case, a pool) citing the net bearing pressure on the foundation.
Using the notation above, the pressure p is the gross bearing pressure on the foundation. The pressure p0 is the effective stress at the base of the foundation. The difference between the two is Δp, which is the net bearing pressure on the foundation.
So how is this relevant to foundation design? In the case of Schmertmann’s Method, it is a part of the method. For bearing capacity, the Soils and Foundations Reference Manual has the following to say:
The net ultimate bearing pressure is the difference between the gross ultimate bearing pressure and the pressure that existed due to the ground surcharge at the bearing depth before the footing was constructed, q (= γaDf). The net ultimate bearing pressure can thus be computed by subtracting the ground surcharge (q) from Equation 8-6:
The structural designer will typically include the self-weight of the concrete footing and the backfill over the footing (approximately equal to γaDf) in the loads that contribute to the applied bearing stress. Therefore, if the geotechnical engineer computes and reports a net ultimate bearing pressure, the effect of the surcharge directly over the footing area is counted twice. Reporting an allowable bearing capacity computed from a net ultimate bearing pressure is conservative and generally not recommended provided that a suitable factor of safety is maintained against bearing capacity failure. If the geotechnical engineer chooses to report an allowable bearing capacity computed from a net ultimate bearing pressure, this fact should be clearly stated in the foundation report.
In general, the ultimate capacity is the more conservative option to use. Combined with the difficulties associated with bearing capacity analysis, I decided not to emphasise net bearing capacity in my teaching. Additionally, for a small pool like the one we had in Palm Beach, bearing capacity failure is remotely possible. For a large (say, Olympic) size pool, we’re dealing with something approaching a mat foundation, and these generally fail in settlement, as noted in Foundation Design and Analysis: Shallow Foundations, Other Topics.
This post has been a long time coming. I’ve been teaching engineering for a quarter century now, and when not doing that developing sites like this so that engineers (and others) can learn more about design and construction of geotechnical structures (and all structures, to varying degrees, are geotechnical unless they float on the water, in the air or in space.)
I come from a long time of technically educated people, as visiting my sites vulcanhammer.info and Chet Aero Marine will show. My great-grandfather learned the basics from Smith’s Mechanic at the University of Illinois in the 1870’s and both he and his brother had successful careers as naval architects. So I come to this debate with a long family history in this profession and many years of experience in the design and application of construction equipment, which is why my teaching straddles both Civil and Mechanical engineering. Many of those who have visited this site are familiar with my Soil Mechanics, Soil Mechanics Laboratory and Foundations courses. After teaching these for a long time, it was evident that our students weren’t “getting it” on what I thought were fundamental concepts. Moving to Lee University and teaching Statics and Dynamics only confirmed those suspicions. Added to my experience with Fluid Mechanics Laboratory and later Fluid Mechanics and this post is the sum of my reflections on this experience.
The growth in the use of numerical methods while the curriculum still emphasizes the use of “hand calculations.” A lot of that is driven by testing; as my CFD professor said after a disastrous midterm (and it was a 500 level course!) testing isn’t perfect but it’s the best thing we have to evaluate whether students are learning the material. Conversely, the problem with simply relying on numerical methods without any recourse at all is that engineers tend to regard results that a computer produces come from Mt. Sinai (the “black box” phenomenon”) and this does not lead to good engineering practice.
We all too frequently lose sight of the fact that our first purpose is to teach people how to think, not just how to make computations or apply formulae. The latter is especially tempting in geotechnical engineering because so many of our formulae are empirical to varying degrees, but we’re not the only people with this problem.
AI, for all of its potential and actual benefits, encourages mental laziness. There, I said it. That’s the root problem with AI, and everything else only leads to that. We don’t need mental laziness in this profession or any other for that matter.
Students’ ability to visualise problems has deteriorated with the growth in computer graphics, from CAD to all kinds of 3D modeling. One thing that vanished before I got into this was students’ ability to draw, but we need to make CAD a solution rather than a problem.
Engineering curricula suffer from being squeezed into a smaller and smaller portion of the course of study engineers are required to take. State school people will recognize the fight with “GenEd” people but it’s not only a problem with state schools.
This last point is not only fueled by the amour-propre of non-STEM faculty; it comes from something that I’ve noticed over the years on Positive Infinity: there is a deep-seated fear that engineers and other scientifically trained people, without the benefit of a liberal arts education, will take over society and enforce a cold, uncultured ethic on everyone else. But that exposes one of the main weaknesses of the American educational system: we warehouse people for over a decade before expecting colleges and universities to impart to them “culture” and “critical thinking skills.” Both of these should be formed long before they step into the halls of “Old Ivy” or “Old Kudzu” (the latter is becoming more important these days.) Our biggest problem is that we cannot agree on a “culture” to teach our young people let alone whether an educational institution of any kind can or should impart such things.
Having said all that, let’s get to the concrete suggestions.
Dial Back or Lose Vector Analysis in Solid Mechanics
One of the advantages of living in the internet era is that we can easily look up books from the past, especially if they’re out of copyright. Engineering education, in a sense, is moving from the oldest knowledge to the newest, the oldest coming first in the early years and as one progresses one learns the newer stuff until, hopefully, the student meets the latest when they get the “terminal degree.” (Geotech warps this process because its transition into true scientific territory is later than other disciplines.)
Now that we have archived textbooks, we can see this process in the books that are readily available. From my great-grandfather’s Smith’s Mechanic to Analytical Mechanics for Engineers to Statics and Dynamics of a Particle, both the content and the pedagogy advance. We also have the Soviet books of a newer vintage; they had a poor economic system but an excellent educational one, and produced texts such as Theoretical Mechanics and Theoretical Mechanics: A Short Course. Other disciplines show the same trend. In all cases engineers trained under these methods went on to produce excellent designs even with the lack of computational power because they were forced to develop serious engineering judgement.
Around World War II we saw in this country a shift to a more precise and analytical approach to teaching solid mechanics. A large part of that was the application of vector analysis, especially for three dimensional problems. When I was taking these courses in the early 1970’s, one could expect to use these in practice. Today that expectation is gone; all problems but the simplest are subject to some kind of numerical method. We would be better off going back to what I call “old coot statics” and dynamics (two-dimensional analysis without vectors) and leave the three-dimensional problems to the numerical methods. Students today, especially with their visualisation problems, need to concentrate on developing their thinking skills and not get bogged down in vector analysis, which in turn is similar enough to linear algebra to develop confusion of both.
An example of the contrast between old coot statics and vector statics can be seen in the example Vector Statics and “Old Coot” Statics: An Example. In the example the two computational methods are set out side by side; the vector analysis is considerably more complicated.
Vector analysis also obscures the fact that a moment is the product of the force and the perpendicular moment arm from the point of application to the line of action of the force. This is illustrated in An Example of 2D Moment Computation. In fact that’s the key problem to vector analysis: the student gets bogged down in setting up the problem rather than understanding its nature.
Problems–especially three-dimensional ones–are best left to numerical methods and emphasis on two-dimensional problems for basic understanding would be a better approach. The last example I’d like to give is An Example of 3D Vector Statics With a Simple Truss. Here we have a problem where (as you can see in the example itself) the vector solution is just too much for the problem; we could either a) use a numerical method or b) split the problem into two two-dimensional problems, one in the x-y plane and the other in the 2-3-4 plane.
The solution in ANSYS (I know this is an old version, we run on a low budget on this site) is below.
I’m aware that vector analysis is well embedded in the teaching of mechanics, but I think it’s time to take a serious look at the problem.
Bring Back Graphical Methods
One hallmark of engineering practice in the past was the use of graphical methods to resolve forces and perform other tasks which would be computationally expensive. The advent of the calculator and later the computer into the profession gave the impression that graphical methods were a thing of the past.
With CAD that’s not the case: we can solve problems with the same precision in CAD we do with either vector analysis or “old coot” statics. An example of this comes from the very first example shown; the graphical solution is given in the article Stiff Leg Derrick Part II: Truss Analysis, and can be seen below. The magnitudes were measured in CAD and the directions were done using the basic derrick layout in CAD.
Graphical methods will assist visualisation and give students a better feel for the problem, both of which they need in their development as engineers.
Distributed Loads and Concentrated Resultants
This comes out of my geotech teaching but also Fluid Mechanics as well: my students struggled with the transition from distributed loads to concentrated resultants. And the distributions were the usual ones we see in geotech and Fluid Mechanics: linear, either uniform, triangular, or trapezoidal. I’m not sure what the solution is but there are two things we need to consider: a) the aforementioned vector statics and b) the inclusion of algebraically complex loadings with polynomial or other higher order equations. As a practical matter it’s unlikely that a complex loading encountered in practice will obey these equations but will more likely require a stepped type of loading of some kind, which in turn begs a numerical solution.
Keep Basic Fluid Mechanics Basic
Fluid Mechanics has experienced much of the same transition as solid mechanics. From books like Hydraulics and Fluid Mechanics to those we usually teach from now, an emphasis on practical fluid mechanics has been lost. In my own career I have found that many of the problems I had would have been much easier to see if the curriculum I was taught under had stuck with a more basic approach (and books which emphasise that do exist, as you can see from my own course.) What we need is to do the following: a) save much of the more theoretical treatment for an advanced course, paralleling that of mechanics of materials, and b) integrating that type of material into basic Computational Fluid Dynamics, which is a necessity for many fluid mechanics problems.
One other thing worth mentioning is that, in the past, solid and fluid mechanics were more integrated, as you can see in books from Smith’s Mechanic to Mechanics by S.P. Strelkov. We might consider some of this if we decide to seriously rearrange how we teach these subjects.
Putting a Wrap
This is not meant to be an all-inclusive list, and I realise that some of these suggestions will be controversial. Today it’s fashionable in higher education, as it has been down the line for many years, to emphasise the pedagogy methods as opposed to the content. But a simpler, more straightforward approach to subject would lend itself to easier pedagogy; the two aren’t unrelated. The tendency to push practitioners out out teaching–one driven in part by our accreditation process–has left many with an unclear understanding of what is required of engineers once they graduate. There are many issues we face; this piece is only meant to start the conversation, not finish it.
We start with an existing technology: low-strain integrity testing of piles. A simple example of this is shown above, it’s the Pilewave program from Piletest. (Yes, I’m aware that it’s the Windows 3.1 version, if you’re interesting in running DOS and Windows 3.1 programs to save on the expense of “new” engineering software, you can visit Partying Like It’s 1987: Running WEAP87 and SPILE (and other programs) on DOSBox.)
With that distraction out of the way, note that, as the stress wave goes down and back up the pile, there is attenuation due to the interaction with the soil. In the simple demo of Pilewave, the soil resistance is constant along the shaft. But…if we could determine that the pile didn’t have defects which reflected waves, could we use information from the soil attenuation to determine the type of soil surrounding the pile at any given elevation? The answer in principle is “yes” and this paper, although not unique, it is an interesting step forward.
Pile Integrity Testing is a low-strain technique. That’s in contrast to the high-strain methods we’re used to in pile driving analysis. This one takes a leaf from the seismic refraction method (which will be featured as before in Soils in Construction, Seventh Edition) which is also a low-strain technique, as it is a geophysical method. The idea is that the pile acts as a probe into the soil; the response to exitation can be inversely analysed to determine the types of soils around the pile. As the paper notes, if you divide up the pile into enough “layers” the actual soil layering itself (based on the properties returned to you by the method) will basically emerge from the data.
As is generally the case with inverse methods, the solution is complex; it is described in the paper. There are a few comments that I would like to make as follows:
His governing equations are similar to the Telegrapher’s Equation used in Closed Form Solution of the Wave Equation for Piles and include a strain term but lack a damping term. Usually a damping term is necessary to model the energy dissapation into the soil; whether that applies to this problem remains to be seen.
Driven piles are subject to compaction and disturbance at the soil-pile interface; how this affects the results remains to be seen. The difference in soil response based on rate effects also will need to be addressed.
I hope that this research continues; I think it has potential.
I recently received in inquiry from an organisation which has proposed a shallow foundation of an embankment. They used (wisely IMHO) an FEA analysis to estimate the settlement. The owner’s response was that, since their result was a little greater than Hough’s Method, and Hough’s method reputedly overestimates the settlements by a factor of 2, that the FEA analysis overestimated the settlements. They referred this person to my posts Getting to the Legacy of B.K. Hough and his Settlement Method and Closing the Loop (or at least trying to) on Hough’s Settlement Method, which is evidently about the only ongoing discussion of the topic around these days.
Both of these posts have two objectives: a) they attempt to trace the development of the method, both by Hough and those who came after, and b) to begin the journey to a resolution of the accuracy of the method. The problem with both of these is that the problem is simple to state but, because of the nature of the evidence, difficult to resolve. I’ll start with a brief review of these two objectives and then set forth a worked example (something that is admittedly lacking in my first two posts) to see how things work out. I’ll end with some thoughts on how to more accurately determine the values of C’, which is the core issue with this method.
The Method and Its Development: A Review
“The SPT is a dynamic test, while soil bearing capacity is a matter of statics, interpreting one in terms of the other is analogous to determining the bearing capacity of piles from pile driving formulas. Consequently, it is felt that attempts to present correlations between blow counts and bearing capacities of soils would be an oversimplification of a much too complex subject.” From Fletcher (1965)
“Hough’s Method” is not univocal; he presented it in two forms in Hough (1959) and Hough (1969). The governing equation is the same for both:
(1)
This equation is identical to Equation (3) of my post The Sorry State of Compression Coefficients except for the form of the variables. In some places equations like this are used for fine-grained soils; this is explained in Verruijt.
The basic problem is determining C’ and there are two difficulties with this:
Hough changed the SPT N vs. C’ curves in the intervening decade between the two forms.
We’re not informed what type of SPT hammer Hough used, or if/how he corrected them as we do now (there’s no evidence that he did.)
Let’s start with the first problem: the curves reproduced from the 1959 version (from the FHWA’s Soils and Foundations Manual) are here:
Figure 1 Bearing capacity index (C’) values used in Modified Hough method for computing immediate settlements of embankments (from FHWA (2006))
We’ll deal with the business of N160 shortly. There is no evidence that Hough meant to restrict his method to embankments.
The chart from the 1969 version is reproduced below (my reproduction):
Figure 2 Hough’s Method Relationship between N Values and C’ Values (redrawn from Hough (1969))
One of the more thoughtless things the FHWA has done in publishing this method is never presenting any equations for these curves, which are easily obtained using linear regression. I have done this and you can see them in Getting to the Legacy of B.K. Hough and his Settlement Method.
Obviously these sets of curves are not identical; the soil classifications he uses aren’t either, and there are five (5) curves in the 1959 version while there are seven (7) in the 1969 one.
Turning to the second problem, in neither of Hough’s original monographs is any kind of correction–mechanical or overburden–are mentioned. The FHWA has consistently added overburden correction. As far as mechanical correction is concerned, in Design and Construction of Driven Pile Foundations, 2016 Edition the FHWA has assumed (not unreasonably) that Hough obtained data from a donut hammer and their correction (which also includes overburden correction) looks like this:
Figure 3 Values of the Compression Index C’ for granular soil (from FHWA (2016))
In the same vein I shifted the x-axis of Figure 2 for N60 values as shown below.
Figure 4 Relationship of N60 Values to C’ for Hough (1969) Method Assuming Original Donut Hammer
With that out of the way, the best way to illustrate the use of Hough’s Method is using a worked example, in this case a retaining wall foundation from A Simplified Method to Design Cantilever Gravity Walls. The diagram at the top of the page shows the foundation; the settlement calculations for a variety of methods (using the U.S. Army Corps of Engineers’ CSANDSET program) are given there. Let’s begin by reproducing those results below.
This program includes a fairly broad selection of methods, from elastic/theoretical ones to purely empirical ones. These methods are described in the program manual. While some of them may not be really applicable to this type of foundation, they show the wide variations of these methods, which suggests that there is not a consensus on computing these values.
Hough’s Method is not included. The detailed solution to the problem is contained in this spreadsheet. We assumed that the soil was well-graded fine to medium sand. There are four variations to the results, which are shown below:
As has been documented widely, the results of Hough’s Method are generally above most of the methods used in CSANDSET, although in the case of Schmertmann’s Method (which has been widely disseminated) the difference is not so great. The largest of the Hough’s Method variations is the Closing the Loop (or at least trying to) on Hough’s Settlement Method proposal, so I ran this with the N1(60) values, which resulted in settlements between the two FHWA methods.
One thing I would caution about using an “academic” problem as an illustration is that the parameters–many of which are taken from “typical” values–may not be representative of what actually occurs in the field, and may yield less than satisfactory results, especially for methods with a strong empirical basis. I ran into this problem with Driven Pile Design: Three Methods of Analysis. On the other hand field results are specific to their location and may not be representative of soils that the geotechnical engineer can expect to encounter.
New Values of C’
I’m not sure how much progress has been really made in this discussion. First I summarised my last two posts on Hough’s Method and how it comes up with the value of the compression constant C’, which is an alternative method of using consolidation settlement techniques to estimate one-dimensional settlement. Then I applied this to an example. Both of these have some value but they don’t get to the heart of the issue: we need more reliable (or at least values of which we understand the source) of the compression constant C’.
One hallmark of many of the fixes for this method is the invocation of overburden correction, which (as we saw above) reduces the resulting settlement. Doing this reminds me of something my Computational Fluid Dynamics I professor put in his notes many years ago:
Also, a few words need to be said about how one should interpret results ensuing from a computational simulation. There are a couple of anecdotal-based observations that are often used to describe how to approach a calculated result: (1) Computed results are guilty until proven innocent, and (2) There’s nothing more dangerous than answers that look about right. These observations are related but have slightly different interpretations. The first says that newly computed results should always be viewed with aggressive skepticism. In other words, a CFD practitioner should never accept a computed result as “truth” or representative of Mother Nature until exhaustive means have been taken to ensure that the result is a “reasonable” approximation to reality. The second observation simply means that if a calculation gives results that are orders of magnitude different from those intuitively expected, then the results can usually be quickly judged as erroneous and there is work to do to find out why. The difficult part comes when a calculation gives results that are “close” to what was expected. Such an outcome often lulls the researcher and/or practitioner into thinking that “all is well” and there is no reason to continue scrutinizing the results. However, it is very possible that a “good” answer was obtained for the wrong reason.
Compression constant typical values aren’t exactly plentiful. This table, from Verruijt, is one I have put in my course materials for many years (for log10 formulations):
Type of Soil
C’
Sand
20-200
Silt
10-50
Clay
4-40
Peat
1-10
Another tabulation comes from this source, converted to log10 values:
Soil
Minimum C’
Maximum C’
Loess silt
6.5
19.6
Clay
13.0
52.2
Silts
26.1
65.2
Medium dense and dense sands
65.2
87.0
Sand with gravel
108.7
None
Hough (1969) himself suggests another way forward. Referring to his table of compression coefficient parameters reproduced in Getting to the Legacy of B.K. Hough and his Settlement Method, we start by noting that he computes the values of Cc using the following equation:
(2)
Since the compression coefficient and constant are related in this way
(3)
we can combine these equations and compute the compression constant thus
(4)
Doing this for Hough’s values of a and b (and one should be aware of the caveats he puts on values of b) for a range of void ratios yields the following tabular result:
Hough’s Coefficients
Initial Void Ratio e0 (first row) Values of C’ (rows that follow)
Soil Type
a
b
1.1
1
0.9
0.8
0.7
Clean Gravel
0.05
0.5
70.0
80.0
95.0
120.0
170.0
Coarse Sand
0.06
0.5
58.3
66.7
79.2
100.0
141.7
Medium Sand
0.07
0.5
50.0
57.1
67.9
85.7
121.4
Fine Sand
0.08
0.5
43.8
50.0
59.4
75.0
106.3
Inorganic Silt
0.1
0.5
35.0
40.0
47.5
60.0
85.0
Silty sand and gravel
0.09
0.2
25.9
27.8
30.2
33.3
37.8
Clean, coarse to fine sand
0.12
0.35
23.3
25.6
28.8
33.3
40.5
Coarse to fine silty sand
0.15
0.25
16.5
17.8
19.5
21.8
25.2
Sandy silt (inorganic)
0.18
0.25
13.7
14.8
16.2
18.2
21.0
Silt, some clay; silty clay; clay
0.29
0.27
8.7
9.4
10.4
11.7
13.6
Organic silt, little clay
0.35
0.5
10.0
11.4
13.6
17.1
24.3
Graphically this is what it looks like:
Although I would be reluctant to reconstruct the method based on this, it shows one important thing: there’s more than one way to get to these constants. If we want to have a method for consolidation settlement type solutions for cohesionless soils, we need to pursue all of the following:
SPT correlations, based on current practice for correcting and applying the results.
CPT correlations. Although not appropriate in all stratigraphies (what method is?) CPT is very useful and more consistent than the SPT in those stratigraphies where it can be applied successfully.
Basic soil properties such as void ratio, relative density and unit weight. This suggests lab tests on undisturbed samples; the problem here is that getting undisturbed samples of cohesionless materials into a consolidation testing machine is easier said than done.
It’s also possible to use tests such as the pressuremeter and dilatometer, but these would only be meaningful in places where they are commonly used.
Doing all of these things would advance our understanding of the settlement of shallow foundations and give us more meaningful comparison with finite element methods.
Unlinked References
Fletcher, G.F.A. (1965) “Standard Penetration Test: Its Uses and Abuses.” Journal of the Soil Mechanics and Foundations Division : Proceedings of the American Society of Civil Engineers. Vol. 94 No. 4, pp. 67-75. It is interesting to note that Fletcher cites Hough’s First Edition of Basic Soils Engineering, while Hough (1969) cites Fletcher (1965).
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.
One of the reasons I was interested in teaching Statics at Lee University was because I was continually disappointed at my students’ memory of their statics. Statics is crucial in the design and analysis of geotechnical structures, and most of the problems–at the undergraduate level at least–aren’t that involved, or at least I thought they weren’t. A great deal of the problem is that geotechnical statics usually involves converting distributed loads into resultants, which Statics–and Mechanics of Materials for that matter–generally associate this with beam problems, not always the case with geotechnical problems.
Another culprit is that Statics, in the U.S. at least, is a vector proposition from the start. At the University of Tennessee at Chattanooga where I taught, it was called “Vector Statics,” which gives the game away early. (At Lee we use the same book and teach the same material, but simply title it “Statics.”)
But what if we applied a vector approach to a simple geotechnical problem? That’s what we’re going to do here with a concrete gravity wall. I will use the method outlined in my post A Simplified Method to Design Cantilever Gravity Walls. You can refer to the theory there, I will try to keep it to a minimum. The wall is pictured at the top of the post, I will reproduce it below.
Analysing Overturning
We have three forces acting on the wall:
The weight of the gravity wall itself, Wconc
The weight of the soil trapped by the heel of the wall, Wsoil
The lateral force of the soil on the wall, Fh
Forces 1 and 2 are determined by computing the cross-sectional area of the concrete and soil and multiplying each by the unit weight as shown above, and then converting the result to a vector force and placing it at the centroid of the area (another Statics topic.) Instead of the “manual” approach in A Simplified Method to Design Cantilever Gravity Walls, the was was drawn in CAD and both the areas and centroids were determined automatically. You can see the magnitudes and locations of those resultants above.
The lateral force of the soil is computed using Rankine’s theory. The first thing is to determine the working internal friction angle of the soil by applying the Shear Mobilisation Factor SMF. Assuming an SMF of 2/3, that friction angle changes from the 30 degree one shown above to a 21.05 degree one, which is applied to the formula for Rankine active pressures for level backfill,
(1)
Doing that results in a kh = .471. The force on the wall is then determined by the formula
(2)
The division by two reflects the fact that soil effective stress (and thus lateral earth pressure) increases linearly with depth (like a fluid,) creating a triangular distribution (yet another concept from Statics.)
At this point there the resisting forces R and T are not defined. The forces themselves are easily computed by summing forces in the x and y directions. Doing this, we have
(3a) (3b)
The location of F–along the surface of the footing–is evident. The location of R is not; it is some distance x from the toe (Point “A”) of the footing. We can obtain x by summing moments around Point “A,” and with a vector method that means taking cross products of the moment arms with the forces.
Converting both the moment arms r and the forces to vector notation yields the following:
Concrete Weight: r = 2.358 i + 4.049 j, Wcon = −4.35 j
Soil Weight: r = 5.576 i + 8.264 j, Wsoil = −6.30 j
Lateral Earth Pressure: r = 8 i + 4 j, Fh = −4.073261616 i
Vertical Footing Force: r = x i, R = 10.65 j (Equation (3a))
The force T does not enter into this because its line of action runs through Point “A,” thus its moment is zero as its moment arm is zero.
The cross product moments around the toe (Point “A” in the drawing) are as follows:
Concrete Weight:
Soil Weight:
Lateral Earth Pressure:
Vertical Footing Force:
Summing these moments,
(4)
Solving yields x = 2.732′. At this point we need to determine whether this is an acceptable location or not for the force. The goal is for the pressure to be positive (downward) along the entire surface of the footing. There are two ways of determining this:
We will do the latter. The middle third of this foundation falls between 2.67′ < x < 5.33′, so the vertical footing force is within the middle third (barely.) As I noted in A Simplified Method to Design Cantilever Gravity Walls, “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.”
Analysing Sliding
With the lack of keys or deep foundations, the only lateral resistance to sliding is the friction force T. We computed that force based on Equation (3b,) but in reality that force cannot exceed–and there should be a factor of safety in that inequality–the frictional force possible, which is defined by the equation
(5)
in which case
(6)
Equation (5) is written in “mechanical engineers format.” Geotechnical engineers understand all too well the concept of a friction angle. In my post Explaining the Relationship Between the Coefficient and the Angle of Friction I relate the two from a non-geotech standpoint; we can turn Equation (5) into a more “geotech-friendly” form by noting that
(7)
Let us assume that the value of is the same under the wall as next to the wall, and let us also assume that the friction angle between the base and the soil is the same as the friction angle of the soil overall, as was done in A Simplified Method to Design Cantilever Gravity Walls. That being the case, . Substituting into Equation (5,) . The factor of safety from Equation (6) is thus , which is barely over the minimum criterion for usual loads given in A Simplified Method to Design Cantilever Gravity Walls.
Observations
The use of vectors for this problem is overkill from a computational standpoint. It also requires locating the centroid/CG of the two regions in both the x- and y-directions, although with using CAD this is trivial. On the other hand doing it using vectors is more “bullet proof” in that the student is not required to “think” but just “plug and chug” without having to identify lines of action and perpendicular moment arms.
The fact that the word “barely” appears in both analyses should inspire some additional conservatism in the design. The simplest way to improve the situation would be to move the heel to the right, which would shift the resisting forces away from the toe (and thus increase their resisting moment) and also put the footing force resultant deeper into the middle third.
Both bearing capacity and settlement of the wall’s foundation, the methodology for which are discussed in A Simplified Method to Design Cantilever Gravity Walls, are beyond the scope of this post. Also beyond the scope of this post is the structural design of the wall and of course the global stability of the wall as well.
(1)
(2)
(3)
(4)
(1)
(2)
(3a)
(3b)![\left [\begin {array}{ccc} i&j&k\\{\medskip} 2.358& 4.049&0 \\{\medskip}0&- 4.35&0\end {array}\right ] = -10.257 k](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin+%7Barray%7D%7Bccc%7D+i%26j%26k%5C%5C%7B%5Cmedskip%7D+2.358%26+4.049%260+%5C%5C%7B%5Cmedskip%7D0%26-+4.35%260%5Cend+%7Barray%7D%5Cright+%5D++%3D+-10.257+k+&bg=ffffff&fg=777777&s=0&c=20201002)
![\left [\begin {array}{ccc} i&j&k\\{\medskip} 5.576& 8.264&0\\{\medskip}0&- 6.3&0\end {array}\right ] = -35.129k](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin+%7Barray%7D%7Bccc%7D+i%26j%26k%5C%5C%7B%5Cmedskip%7D+5.576%26+8.264%260%5C%5C%7B%5Cmedskip%7D0%26-+6.3%260%5Cend+%7Barray%7D%5Cright+%5D+%3D+-35.129k+&bg=ffffff&fg=777777&s=0&c=20201002)
![\left [\begin {array}{ccc} i&j&k\\{\medskip}8&4&0\\{\medskip}- 4.073&0&0\end {array}\right ] = 16.293k](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin+%7Barray%7D%7Bccc%7D+i%26j%26k%5C%5C%7B%5Cmedskip%7D8%264%260%5C%5C%7B%5Cmedskip%7D-+4.073%260%260%5Cend+%7Barray%7D%5Cright+%5D+%3D+16.293k&bg=ffffff&fg=777777&s=0&c=20201002)
![\left [\begin {array}{ccc} i&j&k\\{\medskip}x&0&0\\{\medskip}0& 10.65&0\end {array}\right ] = 10.65 x k](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin+%7Barray%7D%7Bccc%7D+i%26j%26k%5C%5C%7B%5Cmedskip%7Dx%260%260%5C%5C%7B%5Cmedskip%7D0%26+10.65%260%5Cend+%7Barray%7D%5Cright+%5D+%3D+10.65+x+k+&bg=ffffff&fg=777777&s=0&c=20201002)
(4)
(5)
(6)
(7)
is the same under the wall as next to the wall, and let us also assume that the friction angle between the base and the soil is the same as the friction angle of the soil overall, as was done in 
. Substituting into Equation (5,) 
. The factor of safety from Equation (6) is thus 
, which is barely over the minimum criterion for usual loads given in 
