Posted in Geotechnical Engineering

The Difference Between Flamant’s and Terzaghi’s Solution for Line Loads Behind Retaining Walls

Generally speaking, in Soil Mechanics courses elastic solutions on semi-infinite half spaces are presented to allow the geotechincal engineer to estimate the stresses induced in a soil by a load at the surface. Also presented in Soil Mechanics courses are charts and equations to estimate the lateral pressures on retaining walls induced by vertical loads on the surface behind the retaining walls. How either of these came into being is generally not explained; additionally, the fact that they are related is not explained either. The purpose of this article is to explain that relationship and, in the case of the retaining walls, how the original equations have been modified to suit experimental data. For simplicity’s sake, the discussed will be restricted to line loads.

The Original Equations: Flamant

As pointed out by Verruijt, the original equation for the stresses (vertical, horizontal and shear) were first set forth by Flamant in 1892. The equations are shown below.

Flamant Solution for a Line Load on a Semi-Infinite Elastic Medium, from Verruijt

The derivation of this equation can be found here or in Verruijt. Note carefully that the vertical load induces stresses in the horizontal direction (and in shear) as well as the vertical direction.

Verruijt then shows the following:

Mirror Flamant Loads on either side of an arbitrary dividing line and the computation of stress against a rigid wall.

Basically, by static equilibrium, if you replace the left half of the two mirrored loads (on the left) with a rigid wall, the horizontal stresses would be the same on the centre axis/wall as induced by two line loads. The resulting stresses and the resultant for the stress distribution are given below.

Flamant’s Equation for Horizontal Stresses Against a Rigid Wall, Including the resultants

The Modified Equations: Terzaghi

Generally, however, these equations are not presented in books such as DM 7, Sheet Pile Design by Pile Buck or others in this exact form. The following chart (frequently copied) comes from the Soils and Foundations Manual:

The line load is in the upper right hand corner. For values of m (the ratio of the distance from the line load to the wall over the height of the wall) greater than or equal to 0.4, the two results are the same. For those less than that, they are different. (In Verruijt’s notation, m = a/z.) The difference is because most books use the formulation of Terzaghi (1954). He explains the difference as follows:

However, the application of the line load tends to produce a lateral deflection of the vertical section, and the flexural rigidity of the bulkhead interferes with that deflection…However, for values smaller than (m=)0.4, the discrepancy between observed and computed values increases with decreasing values values of m…

From Terzaghi, K. (1954) “Anchored Bulkheads.” Transactions of the American Society of Civil Engineers, Vol. 119, Issue 1.

The whole issue of the flexibility of the retaining wall has been the chief complicating factor in this discussion, going back to Spangler’s tests in the 1930’s.

Comparing the Two Solutions

As an illustration, consider the pressure distribution situation when m=0.3:

Comparison of Flamant/Verruijt and Terzaghi Solutions for Line Load Pressures, m = 0.3

The pressures have been made dimensionless for generalisation. The Flamant solution comes to a higher peak nearer to the surface but falls off more rapidly down the wall. Terzaghi’s solution is more evenly distributed.

Now consider the situation at m=0.5:

Comparison of Flamant/Verruijt and Terzaghi Solutions for Line Load Pressures, m = 0.5

The two are identical in this range.

We can also consider the resultants as well:

Comparison of Flamant/Verruijt and Terzaghi Solutions for Line Load Resultants

The y-axis is made dimensionless by dividing the resultant by the vertical line load. For values of m less than or equal to 0.4, the results are different; for greater, they are the same.

In general, we can say that Flamant’s original formulation is more conservative. In the event that a deeper understanding of the interaction of surface loads with a retaining wall is desired, a finite element analysis needs to be done.

Posted in Geotechnical Engineering

G-I Geo Legends Series Interviews Harry Poulos

Most geotechnical engineers will recognize the name Harry Poulos. The Geo-Institute’s Geo-Legends series recently posted an interview with Professor Poulos of Coffey Engineering and the University of Sydney. He has worked on the foundations of some of the most well-known skyscrapers in the world in Dubai and elsewhere.

G-I Geo Legends Series Interviews Harry Poulos
Posted in Geotechnical Engineering, Soil Mechanics

Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” (or at least clarifying the problem)

Three years ago I posted Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results. Since that time he has been elected President of the American Society of Civil Engineers and I am in the process of retiring from full-time teaching, so our trajectories are a little different. (He’ll catch up, don’t worry.)

Nevertheless his Presidency would go unfinished if some explanation of the pet peeve wasn’t given. To remind my readers it is as follows:

The consolidation e versus log p’ curve is a stress-strain curve. Typically, stress-strain curves are plotted as stress on the vertical axis and strain on the horizontal axis. Both axes are on normal scales, not log scales. It’s my view that consolidation curves should be plotted in a similar fashion: effective vertical stresses on the vertical axis in arithmetic scale, and normal strain on the horizontal axis in arithmetic scale. When doing so, the steel ring confining the test specimen influences the the measurements and skews the stiffness data. Indeed the stress-strain curve, which usually has a downward curvature, has an upward curvature in such a plot.

This post won’t be very rigourous or mathematically detailed, but more of a qualitative statement of the problem. Perhaps a proper solution will solve this dilemma; I think it certainly needs it.

To start, let’s pick up where we left off, with the E vs. \epsilon plot below:

It was noted at the time that the apparent elastic modulus increased more or less (that’s about as good as it gets with most geotechnical phenomena) linearly with strain.

From this, it can be noted that the shear modulus can be estimated for a soil (excluding strain-softening effects) as follows:


where the notation is shown in the source. Let’s make some assumptions:

  • Poisson’s ratio remains constant, thus the relationship between strain and elastic modulus is constant.
  • S is constant for a given soil type.
  • Effective stress \sigma_o for a given sample (that’s just about a given for triaxial tests in any event)
  • Other constants, such as p_{atm} and \bar{n} also remain constant.

That leaves the variable F(e) to change. The shear (and by extension the elastic) modulus of a material is a function of the void ratio. From the same source,


We can convert this to strain by noting the following relationship, which is written so that compressive strain is positive:

\epsilon={\frac {{\it e_0}-{\it e_1}}{1+{\it e_0}}}

Substituting that into the equation before it yields

F(e) = -{\frac {1}{\left (1+{\it e0}\right )^{3}\left (-1+\epsilon\right )^{3}}}

Let’s consider the case of e_0 = 1 . Substituting that into the previous equation and plotting it yields the following result:

It’s not perfect, but it’s close to a linear relationship, at least in the strains under consideration. And, of course, it shows an increasing shear modulus with increasing strain (or decreasing void ratio.)

Verruijt makes two important observations that should be noted. The first is his commentary on the image below, in the caption.

When performing the test, it is observed, as expected, that the increase of vertical stress caused by a loading from say 10 kPa to 20 kPa leads to a larger deformation than a loading from 20 kPa to 30 kPa. The sample becomes gradually stiffer, when the load increases. Often it is observed that an increase from 20 kPa to 40 kPa leads to the same incremental deformation as an increase from 10 kPa to 20 kPa. And increasing the load from 40 kPa to 80 kPa gives the same additional deformation. Each doubling of the load has about the same effect. This suggests to plot the data on a semi-logarithmic scale. In this figure log(σ/σ0 ) has been plotted against ε, where σ0 denotes the initial stress. The test results appear to form a straight line, approximately, on this scale. The logarithmic relation between vertical stress and strain has been found first by Terzaghi, around 1930.

The second is his comment on the use of strain vs. void ratio:

It is of course unfortunate that different coefficients are being used to describe the same phenomenon. This can only be explained by the historical developments in different parts of the world. It is especially inconvenient that in both formulas the constant is denoted by the character C, but in one form it appears in the numerator, and in the other one in the denominator.

The need to treat compression due to settlement completely differently than that of elastic (or elasto-plastic) settlement is one of the anomalies of geotechnical engineering. The observation that the elastic modulus decreases with void ratio (or increases with strain) is a start in putting the two together and presenting a more or less unified theory of soil deformation. Coupled with agreement on using strain in consolidation tests, this would bring us a long way to solving the dilemma of Jean-Louis Briaud’s–and some of the rest of our–pet peeves.

Posted in Academic Issues, Geotechnical Engineering

An Anniversary, An Announcement and Looking Ahead

Today is an anniversary I’ve commemorated before: it’s the anniversary this web site/blog (take your pick) got its start as the Wave Equation Page for Piling. It’s been twenty-four years since I put the first pages on GeoCities, and it’s been going (with spin-offs) ever since. It’s time for a little looking back, and some looking forward too.

The year 2020 was traumatic for just about everyone but it was a good year for this site. It was even a better year in that most of the traffic to the site came from outside the United States (that trend has continued into 2021.) This is in spite of the fact that my students at the University of Tennessee at Chattanooga mostly access it from within the country, having no small part in the visits/page views for the site. (I say mostly; a few actually did so from outside the country, as they were forced to continue their coursework from overseas due to COVID.)

One of the long-term goals of this site has been to disseminate knowledge about geotechnical engineering to where it’s needed most: to developing countries which need to build their infrastructure and bring a better life to their citizens. In the first decade of this millennium, it tended to dominate the field, but realistically this is no longer the case. Nevertheless it remains an important resource in a shifting internet, and in a field where social media cannot (or at least has not) replaced the open internet.

One thing that has helped this change–and the long-term value of this site–has been the growing educational component of the site in Soil Mechanics, Soil Mechanics Laboratory and Foundations classes. I have taught consistently at UTC since 2009 and have put up most of my educational material for these courses on this site. The COVID pandemic only accelerated that; I taught this past academic year completely online, which necessitated putting the lectures onto YouTube. This means that one can take entire courses (except for the homework and tests) on this site, or use this material as a facilitator for online courses.

That leads to the next announcement: I am retiring from full-time teaching at the end of the month. There’s a lot of academic “inside baseball” in that, but I will revert to adjunct teaching after that time, as I did before my full-time appointment in 2019. I will continue, Lord willing, to teach in the immediate future, and also plan to continue to build this site with new educational materials of all kinds, both for the courses and for the documents that have been a hallmark of this site from its earliest times.

As always, thanks for your support, or as I say at the end of all my videos, thanks for watching and God bless.