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.
It’s been a while since the last post on this subject; this has slowed things down. But in the course of getting started again a little “side trip” shows a good illustration of how sometimes determining the engineering properties of a structural element–in this case a driven concrete pile–can be challenging.
The study involved the installation and testing of three different types of piles, as shown below. We’ll concentrate on the prestressed concrete pile on the left.
The prestressed concrete pile was a 610 mm square solid pile. This means that the cross-sectional area is . The pile was 18 m long, as shown below.
Stress-strain curves were developed for the three materials, and these are shown below.
From the stress-strain curve for the concrete alone (and we usually assume that the concrete governs the pile elastic properties for compression at least) the curve would indicate that the modulus of elasticity is somewhere around . The diagram below, however, indicates that those involved in the project determined the modulus of elasticity to be around .
The interest from the STADYN standpoint is to obtain a force-time and velocity-time curve from the Pile Driving Analyzer, and this is certainly forthcoming:
The value of was probably determined from the two force peaks. The first force peak is the impact of the hammer on the pile and the second is the reflection of that impact from the toe. Both are compressive and the second is strong, which indicates a high level of toe resistance.
As is typical with PDA output, the force and the velocity (multiplied by the impedance) are plotted together. Unfortunately the document does not give the impedance for this case, so it’s necessary to back compute the impedance. Since we have a reasonably good idea of from the PDA, and the impedance Z is
we can determine the impedance. Solving for c from ,
We need to pause at this point and note that other values of acoustic speed are implied in the data. For example, the following table states an acoustic or wave speed of 3800 m/sec.
Before and after the test, PIT (Pile Integrity Tests) were run on the pile. The results are below.
Converted to SI units, the acoustic or wave speed becomes 4037 m/sec, which is fairly close to the PDA tests. The PDA results will be used for the remainder of this piece.
In any case, using the EA values from the earliest part of the test, the impedance is
The data was extracted from the PDA results. The force values could be used “as is.” The velocities were in reality the product of the velocity and the impedance, so the dashed line values were divided by the impedance just obtained to yield a velocity. Unfortunately, when this was put into STADYN, the velocities that resulted–even in the early stages of impact, where semi-infinite pile conditions predominate–the velocities of the program varied from the velocities extracted from the data by a factor of two. Checks in the program did not show any change in the way the program executed the algorithm from earlier runs, but the impedance values the program was yielding were considerably different from the one above.
In an attempt to sort things out, it is good to start by noting that the acoustic speed is computed by the equation
The report states that the pile was poured to normal Virginia DOT specifications. A fair assumption is that the density or unit weight of the concrete is close to normal, or . That being the case, the computed acoustic speed from the values of Young’s modulus E (which is necessary to put into Pa for unit consistency) and the density assumed yields
Something is clearly wrong here, and the most probable culprit is the modulus of elasticity of the concrete. A common way to estimate the modulus of elasticity of concrete in MPa is to use the formula
where is the 28-day compressive strength of the concrete. The report gives this to us at the time of the load tests as 55 MPa, which yields a modulus of elasticity of
This is considerably higher than the earlier data would indicate. It’s worth noting the the specifications for the pile set a minimum value for as 35 MPa; this indicates that the values of Young’s Modulus for concrete in piles can vary widely.
Another–and given the data probably a stronger–approach to compute the value of Young’s Modulus is to back compute it from the acoustic speed (which is known within reasonable values) and the density (see assumption above.) Solving the basic equation for acoustic speed for Young’s Modulus yields
Substituting our values yields
The impedance from this would be
Applying values along this line and recomputing the velocities, the results of the STADYN program and the actual PDA results were much closer.
The reason for the discrepancy in Young’s Modulus–and thus the pile impedance–is unclear. It may be due to rate effects on the elastic response to concrete, or it may be due to other factors.
Wave equation analysis are typically run according to “standard” material properties. Those who run these should be aware that, with concrete and wood, those properties may not reflect the properties of what actually gets driven into the ground.
Any force- and velocity-time data such as are produced by the PDA should have their axes labelled properly (with both force and velocity) or with the impedance reported.
Even with controlled research projects, discrepancies can arise in the data which can impede (pun somewhat intentional) the use of the data, and careful analysis is necessary to avoid problems such as was seen in this situation.
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