Designing Cantilever Sheet Pile Walls Using a Chart

Sheet pile walls are deceptively simple retaining walls to analyse. Because a) the loads are generally distributed and b) the layering of the soil is complex, a simple problem can turn into a complicated one very rapidly. An outline of the hand solution method for cantilever walls is shown below.

That’s why today software such as SPW911 is very commonly used to solve these problems. In some cases, a “back of the envelope” solution can be useful. For cantilever walls, some of these are as follows:

Equations for dry walls and cohesionless soils, for both cantilever and anchored walls. For the former I cover this in my Foundations course (both slides and video) in the post Foundation Design and Analysis: Retaining Walls, Cantilever Sheet Pile Walls. The theory behind this is discussed in Basic Configuration of Sheet Pile Design Analysis.
Hand solutions, such as are shown above.
Graphical solutions. These were very popular before serious computational power came into the hands of engineers. It is interesting to note that, with CAD software, the limited numerical precision problem is solved. (Whether this contributes to actual better results in a geotechnical problem is debatable.) Some discussion of this is in the US Steel Sheet Pile Design Manual.
Chart solutions. They’re limited in the soil, water and wall configurations they’ll cover, but they’re handy and relatively easy to use.

Because of its focus on temporary works, Soils in Construction focuses on cantilever walls, and features the last solution type. There are two cases considered: cantilever walls a) in a purely cohesionless soils and b) in a cohesive soil with cohesionless backfill. The former has a worked example in Soils in Construction, the latter does not. The charts shown below come from NAVFAC DM 7.02 and originally was presented in US Steel Sheet Piling Design Manual and were used in Sheet Pile Design by Pile Buck.

Cantilever Example, Purely Cohesionless Soil

We’ll start with a cantilever example driven into a cohesionless soil , which is similar to the one shown in Basic Configuration of Sheet Pile Design Analysis except that we’re going to add water to the design. The chart is shown below.

The soil and excavation configuration is as follows:

Dry cantilever wall, uniform cohesionless soil, level backfill
φ = 30°, γ = 109.2 pcf
Rankine conditions. Soils in Construction only discusses Rankine conditions with a vertical wall and level backfill. A great deal of actual sheet piling design is done using Coulomb or log-spiral theory for lateral earth pressures and forces. For the purposes of Soils in Construction Rankine will suffice. It is conservative and easy to apply. It is worth noting that a full soil mechanics textbook like Tsytovich restricts itself to Rankine conditions. Rankine lateral earth coefficients for conditions other than level backfill are discussed in the post Coulomb and Rankine Earth Pressure Coefficients for Vertical Walls. Using equations from that post, the active and passive earth pressure coefficients for level backfill are as follows:
- K_p = 3
- K_a = 1/3
h = 10’ (distance from the top of the wall to the dredge line)
Passive reduction factor of safety = 1.5. The simplest way of applying a factor of safety to sheet pile wall design is to reduce the passive earth pressure coefficient(s) by dividing them by a passive reduction factor of safety. That being the case, for this problem K’_p = 3/1.5 = 2. Once you do this you don’t need to apply the factor of safety shown in the chart.
Depth of the water table (on both sides, thus no unbalanced hydrostatic forces are an assumption of the chart) from the surface = 5′, thus the ratio of the two $\alpha = \frac{5}{10} = 0.5$ .
The chart assumes that the submerged unit weight of the soil below the water table is half of the unit weight above it. Since only one unit weight was given, γ‘ = 109.2/2 = 54.6 pcf.

From this, we can first compute the value of the x-axis of the chart, thus

$\frac{K'_p}{K_a} = \frac{2}{\frac{1}{3}} = 6$ (1)

There are two sets of curves to consider: the depth ratio curves, which reference the right hand y-axis, and the moment ratio curves, which reference the left hand y-axis. In both cases we choose the one we use based on the value of $\alpha$ we have computed, in this case $\alpha = 0.5$ .

The depth ratio for the values given is approximately 1.6. The values increase as you go up the axis.
The moment ratio for the values given is approximately 1.1. Note carefully that the values increase as you go down the axis.

Applying the depth ratio is relatively simple: you multiply the height H by the depth ratio to obtain the depth D. Since H = 10′, D = (10)(1.8) = 18′. Since the passive earth pressure coefficient has been reduced, it is unnecessary to increase D further.

The moment ratio is a little trickier: it is equal to

$M_{rat} = \frac{M_{max}}{\gamma' K_a H^3}$ (2)

Note carefully that the unit weight given in Equation (2) is the submerged unit weight, while the unit weight in the chart is the saturated unit weight. The chart has an error here: this is because, when DM 7 was written the chart shown above (which was taken from the US Steel Sheet Piling Design Manual) dropped the prime, and unfortunately this carried over into the worked example in Soils in Construction. Solving for the maximum moment,

$M_{max} = M_{rat} \gamma' K_a H^3$ (3)

Substituting, M_max = (1.1)(54.6)(1/3)(10³) = 20,020 ft-lbs/ft of wall.

Results from the SPW 911 software are given below. The maximum moment is very close; the penetration depth is longer in the software and may reflect some additional conservatism in the software.

Cantilever Example: Cohesive Soil with Cohesionless Backfill

The second chart given in Soils in Construction is for a cohesive base soil with cohesionless backfill above the dredge line. The chart is shown below.

NAVFAC DM 7.02 gives a worked example to go along with the chart; it is shown below.

We note the following about this:

The active earth pressure coefficient of the cohesionless soil was probably computed using Coulomb earth pressure theory. We left it as is; had we converted it to Rankine, it would have had the same value as the first example.
The x-axis quantity is no longer the passive to active ratio of the first example but a more complex quantity, which is computed above.
The results of the software are shown below. They are close to the hand solution above. The penetration depth of the software is closest to the D_calculated of the hand solution; the hand solution used a multiplier method to increase the penetration depth. It is also possible to divide the cohesion by a factor to reduce it as a method of applying a factor of safety.