The development of sheet piling and the Warrington-Vulcan hammers was about the same time (along with concrete piles, other types of steel piles, and the Engineering News Formula.) The years leading up to World War I were ones of rapid development in the marine construction and deep foundation industries, and Vulcan was in the middle […]Vulcan and Sheet Piling — vulcanhammer.info
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.
Verruijt then shows the following:
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.
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:
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:
The two are identical in this range.
We can also consider the resultants as well:
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.
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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
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. 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.
- is constant for a given soil type.
- Effective stress for a given sample (that’s just about a given for triaxial tests in any event)
- Other constants, such as and also remain constant.
That leaves the variable 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:
Substituting that into the equation before it yields
Let’s consider the case of . 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.
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.