This is a sample sieve analysis result. For the lab (especially when sample washing is involved) it can get this complex, and the procedure is described in Materials Testing. To look at this more schematically (and this is common for most “textbook” problems) the following simplifications are done:
There are no losses in the process of sieve analysis. This is obviously unrealistic but these should be kept to a minimum. For textbook type problems this means that Blocks 8 and 23 are the same.
The tare (sieve or pan) weight is removed from consideration. This means that typically Column 15 is the first column in the problem, given the soil weights retained on each sieve. The weight that “hits the pan” is in Block 22.
Washing losses and errors are not considered, which means that Blocks 21 and 24 are set to zero.
With these out of the way, we can proceed to the analysis. The top sieve in the stack (the 2″ sieve) retained no soil (it doesn’t always happen this way) so it has zero soil retained and all passing. The next sieve (the 1 1/2″ sieve) retained 83.7 g (Column 15.) For the cumulative weight retained (Column 16) we add the value of Column 15 just above it to this one, thus 0 + 83.7 = 83.7 g. The next sieve (the 3/4″ sieve) retained 161 g, so for the cumulative weight retained for this sieve we add in the same way, thus 83.7 + 161 = 244.7 g. We keep going in the same way “down the stack” until we have all of the cumulative weights retained for each sieve computed.
This is terrific, but what we really need is the percentages of the total sample each cumulative retained value represents. This is what goes in Column 17. These values are obtained by dividing each result in Column 16 by the total sample weight in Block 23 and multiplying them by 100 to obtain a percentage. In this way, the cumulative percent retained for the 2″ sieve is obviously zero, for the 1 1/2″ sieve 83.7/4381.4 x 100 = 1.9%, for the 3/4″ sieve 244.7/4381.4 x 100 = 3.7%, and so on.
We can also compute the percent passing. An easy way to do this is to start by noting that 100% passes the whole sieve stack. We can than successively subtract the cumulative retained percentage as we go down. Thus the percentage of the 2″ sieve is again obviously 100 – 0 = 100%, for the 1 1/2″ sieve 100-1.9 = 98.1%, for the 3/4″ sieve 98.1-3.7 = 94.4% (note that we use the percentage passing from the previous sieve each time) and so on.
We then plot the results on the sieve curve as follows:
It’s a temptation these days to use a spreadsheet and its semi-logarithmic plotting features. I would avoid this: using a dedicated form like this has two advantages:
It lines up the grain sizes and sieve designations for you.
It can be plotted either from the percent passing size (left) or percent retained side (right.)
It’s also possible to use a tool such as the Spears Lab Spreadsheet, but this takes a lot of practice and is a little tricky to use, especially for “textbook” type problems where the tare is not considered. It’s easy to get a really stupid looking result; I have seen quite a few over the years.
Most retaining walls are designed with active or passive earth pressures derived from Rankine, Coulomb or Log-Spiral theories. One notable exception to that are braced cuts. The development of the earth pressure distributions is attributable to Karl Terzaghi and Ralph Peck. In the process of developing those, the way the wall is modelled was simplified to avoid statically indeterminate structures. Although this is not the problem that it was in their day, the method is still dependent upon those statically determinate structures.
The example below is a simple example in that the supports are symmetrically placed and there is no sheeting toe penetrating the bottom of the excavation. It’s primarily intended to illustrate the concepts, both geotechnical and structural, of the design of these structures.
Overview of the Example
Let us consider a braced cut excavation which is 45′ deep and which has supports at a depth of 5′, 17′, 28′ and 40′. The soil behind the wall is uniform with c = 1100 psf and γ = 110 pcf. The water table is at the bottom of the excavation and does not enter into our calculations. To show how this lays out we’ll use Pile Buck’s SPW 911 sheet pile software. We’ll assume PZ-27 sheeting is being used, and that there is no surcharge on the wall.
We obviously have a clay soil, thus our selection will be either (b) or (c). Whether the soil is soft to medium or stiff depends upon the stability number , which is computed as follows:
This is between (b) and (c), we are thus supposed to use the “larger” of the two diagrams. The earth pressure coefficient for (b) is
Assuming m = 1,
If we turn to Case (c) and assume that
this is obviously “larger” than Case (b), so we will use Case (c), even when using a “medium” case between the two extreme pressure profiles.
We thus have a pressure distribution that can be described as follows:
Beginning at the top, it linearly rises from zero to the maximum value of 1485 psf at a point a quarter down the wall, or 45/4 = 11.25′.
From that point until a quarter from the bottom of the wall, or 0.75 * 45 = 33.75′, it is a constant pressure of 1485 psf.
From that point until the bottom of the wall, it linearly decreases to a value of zero at the bottom of the wall.
Guidelines for Structural Analysis of Wall
Turning to the structural aspects of the wall, the guidelines for dividing the wall up are as follows:
If the wall is cantilevered at either end, then the endmost support and the one next to it form a simply supported beam with a cantilever at one end and a distributed load.
Segments in the middle are analysed as simply supported beams with a distributed load.
If there’s a support at the top or the bottom of the wall, the beam at that location is analyzed as a simply supported beam.
Reactions are computed for each beam. For supports where two segments meet, you simply add the two reactions from each beam for a total reaction for the support.
Maximum moments are computed for each beam; the largest of these maximum moments is the maximum moment of the system and the one used to size the sheeting.
This was Terzaghi and Peck’s attempt to make the calculations simple. If the distributions are simple, then “handbook” type formulas can be used. The trout in the milk takes place (as it does here) when the break points in the distribution don’t coincide with the supports, in which case you end up with a more complicated distribution. There are two ways of dealing with this problem.
The first is to reduce the distributed loads to point load resultants. This is a favourite tactic among geotechnical engineers and is used extensively with shallow foundations. For purely hand calculations, it makes sense. The moments will be higher (which is conservative) but the reactions will be identical, assuming the concentration of the moments went off according to plan.
The first thing we need to do is to specify the distributed loads. CFRAME, like most finite element programs, considers the beam between each support (and the beams from the outermost supports to the cantilever element) as one element. So there are six elements. CFRAME asks us to specify the distributed load (constant or linearly varying) for each element, and requires us to specify the constant loads and the varying loads separately.
But here we run into something that trips up students. Sheet piles are analysed as beams, but they’re “infinite” beams; we analyse them in terms of moment of inertia per length of wall, section modulus per length of wall, load per unit length of wall, etc. The good news is that, for distributed loads, the pressure at any point is the load per unit length! Pressure is expressed, in this case, as lb/ft^2 of wall, when in reality it’s lb/ft/ft of wall. That makes things simpler; as long as we enter the moment of inertia and cross sectional area in terms of “per foot of wall” (which any US unit section should furnish us) then we’re good. In this case for PZ-27 the moment of inertia is 184.2 in^4/ft of wall and the cross-sectional area is 7.94 in^2/ft of wall, and these are entered directly into CFRAME.
With that technicality out of the way, for are areas of constant earth pressure (the middle) we’re also good; it’s just 1485 psf, and we enter this directly into CFRAME. With the ramped portions, they increase from the top and bottom of the wall at a rate of 1485/11.25 = 132 psf/ft from the end. Looking at the topmost element, which we enter into CFRAME as (surprise!) element 1, the pressure at the topmost support is 132 * 5 = 660 psf, which we enter as the maximum pressure for the “triangle load” on the top element.
For element 2, we have two loads. The first is a continuation of the ramped load from 660 psf at the top end of the beam to 1485 psf at a point 11.25′ from the top of the wall or 11.25′ – 5′ = 6.25′ from the top end of the beam. The second load is simply a constant load to the bottom end of the beam.
The middle element 3 has a constant distribution across its entire length. The bottom two elements are mirror images of the top two elements.
Results from CFRAME
We entered the data into CFRAME via a small text file. First we present the model itself.
First let’s look at the reactions at the supports, which come from the element results. They are as follows:
Support 1 (Node 2): The reaction/shear at that point from element 1 is 1650 lb/ft of wall and from element 2 7009 lb/ft of wall, summing it comes to 8659 lb/ft of wall.
Support 2 (Node 3): The reaction/shear at that point from element 2 is 8233 lb/ft and from element 3 8168 lb/ft, summing it comes to 16401 lb/ft.
Support 3 (Node 4) is the same as Node 3 by symmetry.
Support 4 (Node 5) is the same as Node 2 by symmetry.
Thus the maximum brace load is on Supports 2 and 3, 16401 lb/ft. We have for convenience ignored the sign conventions and simply added the reactions, since they’re all in the same direction.
The maximum moment is actually in Element 2 (or 4,) and is 273,900 in-lb/ft of wall. Since the elastic section modulus for PZ-27 is 30.2 in^3/ft of wall, the maximum bending stress is 273,900/30.2 = 9070 psi, which is well within most allowable specifications. A lighter section can probably be employed, depending upon the allowable deflection and other requirements.
As a quick check, for a uniformly distributed load on a simply supported beam, the maximum moment is given by the equation
Substituting the values for Element 3, we have
Now we compare these with SPW 911, whose output is as follows:
The differences are minor (SPW 911 and the hand calculation report the maximum moment in ft-lb/ft of wall, not in-lb/in of wall.) Some discussion of eliminating the additional pins in the simply supported spans is given in Sheet Pile Design by Pile Buck.
Practitioners who design gravity retaining walls are familiar with the existence of Terzaghi’s “low walls” curves to estimate the equivalent fluid pressure on the wall, horizontal and vertical. The basic chart comes in several versions but the one (for straight backfill) from NAVFAC DM 7.02 is above. The explanation for it (important when one is using it) is below.
In the “slide rule” days, reading charts like this was routine. A better way now, with spreadsheets abounding, is to have a formula available. Some least squares curve fitting correlations are shown below. The variable is the slope angle as shown in the figure. As is the case with the chart, the formulas return and in units of of wall length.
Many of you (and especially those who are familiar with the companion site vulcanhammer.info) are aware that I’ve spent much of my career in geotechnical engineering and deep foundations dealing with contractors. As such I am both sympathetic with their situation and also aware that they need good information to make decisions that can “make or break” a job or their company. It was for that reason and more that I was very happy to be invited to co-author the Fifth Edition fifteen years ago and to help prepare this revision as well.
Soils in Construction is designed to teach the basics of soil mechanics and foundation design to construction management students and to be a reference for those “in the field.” It takes a practical approach to the subject, and it also deals with “temporary works” such as dewatering and cantilever sheet pile walls that many design engineers are unfamiliar with.
There will be more resources for this book available, both from Waveland Press and on this site as well; I’ll keep you posted. Waveland has been great to work with and I appreciate the effort they have put into the book. But the one person I want to say special thanks to is Lee Schroeder, Professor Emeritus (and former Interim Athletic Director) at Oregon State University. Eminent in his own right outside of the book (he’s the Schroeder of the Schroeder-Maitland method for cellular cofferdams,) his vision for the original work, his practical and experienced implementation of same, and his graciousness and support in allowing me to be a part of this project are deeply appreciated.
Van der Merwe’s method, which was first introduced in the 1960’s, is a simple method for estimating the vertical movement of expansive soils. The method at its most basic is described in publications such as Foundations in Expansive Soils, and basically looks like this:
The problem with this presentation is that it is entirely in “Imperial” units, which were the standard in the South Africa of van der Merwe’s day. We need to convert this to make it usable in SI units as well. First we present the potential expansion chart, from Soil Mechanics:
Then we reconstruct van der Merwe’s equation to make it applicable for SI units as well as US (or Imperial) units:
Note that the swell potential is now in the denominator. That’s a convenience to make the PE values of a reasonable order of magnitude. But now it’s dimensionless. Also note that the depth reduction factor has a different exponent for SI units than it does for US units.