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.

home_dwarring_PDF-job_43 — Basic layout of braced cut example, using Pile Buck’s SPW-911 software

The options for earth pressure distribution behind braced cuts are shown below, from NAVFAC DM 7.2. or Sheet Pile Design by Pile Buck.

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 $N_o$ , which is computed as follows:

$N_o = \frac{\gamma H}{c} = \frac{110 \times 45}{1100} = 4.5$

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

$K_a= 1 - m \frac {4c}{\gamma H}$

Assuming m = 1,

$K_a = 1 - m \frac {4c}{\gamma H} = 1 - \frac{4 \times 1100}{110 \times 45} = 0.11$

and thus

$\sigma_h = K_a \gamma H = 0.11 \times 110 \times 45 = 550\,psf$

If we turn to Case (c) and assume that

$\sigma_h = 0.3 \gamma H = 0.3 \times 110 \times 45 = 1485\,psf$

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 second is to employ beam software to analyse each segment. Although there’s a lot of beam software out there, being the old coots we are, we’ll use CFRAME, a DOS program for two-dimensional structures. It gets the job done and is fairly easy to use. (Note: because of some bad interaction between CFRAME and DOSBox, we ran it on a Windows XP installation. The manual for CFRAME: Computer Program with Interactive Graphics of Plane Frame Structures is here.)

Implementation in CFRAME

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.

Screenshot_20191031_142020 — The basic layout of the model. The model is simply supported at all braces. Additionally–and this is one reason we wanted to use CFRAME–the central element 3 was additionally pinned at the ends to simulate Terzaghi and Peck’s original intent for the method.

Screenshot_20191031_142045 — The pressure distribution on the model, replicating that Terzaghi and Peck distribution for stiff clays.

Now we show the results.

Screenshot_20191031_142158 — The deflections of the model. The pinned (discontinuous) nature of the middle element 3 can be easily seen.

Screenshot_20191031_142214 — The shear diagram for the model. As is the case for all of these diagrams, the maximum value can be seen in the lower left corner of the image.

Screenshot_20191031_142321 — The moment diagram for the braced excavation. Although it is not explicit, the values for shear and moment are per foot of wall. As with the deflection, the effects of pinning the ends of element 3 are easily seen.

The individual element results are shown below. The tabular results of the program are here.

Analysing the Results

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

$M_{max} = \frac{wl^2}{8}$

Substituting the values for Element 3, we have

$M_{max} = \frac{wl^2}{8} = \frac {1485 \times 11^2}{8} = 22,461\,\frac{ft-lb}{ft}$

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.