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Today for serious projects, if one wants to estimate the settlement of a deep foundation, one uses a “t-z” method of some kind. These methods involve analyzing the pile-soil interaction (and the internal flexibility of the pile) to develop a load-settlement curve similar to one obtained with a static load test. There are several types of algorithms available for this purpose, and they work in different ways.

But what if one wants a “back of the envelope” estimate of the settlement of a deep foundation, or possibly a preliminary load-settlement curve? As is frequently the case with geotechnical engineering, there is more than one way to do this, but probably the simplest–and well represented in practice–is Vesić’s Method, named after Aleksandar Vesić. The method is presented here as shown in NAVFAC DM 7.02, but it appears in other publications and textbooks.

First, we’ll present the method itself:

Let’s break this down: first, it’s necessary to identify some parameters.

Ultimate static capacity of the shaft $Q'_s$ . It isn’t necessary to split up the shaft resistance further along the length; that problem will be handled by other factors. Note that this is a primed variable, and not directly entered into the equations. This will be explained shortly. The method handles toe resistance differently; it is necessary to compute the ultimate unit toe resistance $q_0$ , which is usually a result of whatever method you choose to compute the ultimate static capacity of the pile.
Pile geometry, including length $L$ , cross-sectional area $A$ , pile “diameter” $B$ , length of the pile embedded into the soil $D$ and modulus of elasticity of the pile $E_s$ . The method assumes a uniform cross-section and material for the entire pile length.
The factor $\alpha$ , which considers the distribution of soil resistance along the shaft. For a uniform stratum, conventional wisdom has been that sandy soils will have a triangular distribution (due to the dependence of shaft resistance from effective stress) and clay soils a uniform distribution, as the resistance is based on the soil cohesion. However, since Burland, that notion has been challenged, and has created something of a controversy. The best resolution, as shown by the current API method and Kolk and van der Velde, is that the adhesion itself varies with effective stress; it’s not really an either/or, but a both/and situation. Unless the soil data indicates otherwise, the simplest way to resolve this is to say that the default value of $\alpha = \frac{2}{3}$ .
The variable $C_p$ , given in the table above.

The simplest way to illustrate how this method works is through an example. Consider the case of a 16″ square concrete pile, 125′ long, driven into loose sands. We want to estimate the settlement of the pile under two axial loads: 400 kips and 600 kips. The water table is at the surface. From the TAMWAVE program, we learn the following:

Ultimate shaft resistance $Q'_s = 527.95\,kips$ .
Unit toe resistance $q_0 = 238.0\,ksf$ .
Pile length $L = 125'$ .
Pile cross-sectional area $A = 1.778\,ft^2$ .
Pile “diameter” $B = 1.333\,ft$ .
Embedment of pile into soil $D = 125'$ (TAMWAVE only analyses groundline piles.)
Modulus of elasticity of pile $E_s = 719,910\,ksf$ .
Shaft distribution factor $\alpha = 0.667$ .
Settlement coefficient $C_p = 0.04$ .

With that out of the way, we can proceed; however, at this point we get to the tricky part of this method, namely computing $Q_s$ and $Q_p$ . This is where students get tripped up, but the root problem is how we look at the “capacity” of deep foundations.

Static capacity methods imply that there is “a number” which represents the static capacity of a pile. They also imply that this capacity is divided neatly between shaft and toe resistance. The truth is that neither is the case; a deep foundation is loaded progressively and the resistance is mobilised progressively, and you end up with a load-displacement relationship from which you determine the maximum permissible load of the foundation.

Vesić’s Method replicates that by the way that $Q_s$ and $Q_p$ are computed, and it is as follows:

$Q_s$ is either a) equal to the applied load $Q_{applied}$ if $Q'_s > Q_{applied}$ or b) equal to $Q'_s$ if $Q'_s > Q_{applied}$ .
$Q_p$ is either a) equal to zero, if $Q'_s$ if $Q'_s > Q_{applied}$ or b) equal to $Q_{applied} - Q'_s$ if $Q'_s > Q_{applied}$ .

With those definitions, the first thing we do is to compute the coefficient $C_s$ . By substitution of the given variables into the equation above, $C_s = 0.64$ .

Then for the case of $Q_{applied} = 400\,kips$ , since $Q'_s > Q_{applied}$ , $Q_s = 400\,kips$ and $Q_p = 0$ . Substitution of these values along with those given or computed above yields the following:

$W_s = 0.026'$
$W_{pp} = 0$
$W_{ps} = 0.009'$
$W_o = 0.026 + 0 + 0.009 = 0.035' = 0.416"$

Turning now to the case of $Q_{applied} = 600\,kips$ , since $Q'_s < Q_{applied}$ , $Q_s = 527.95\,kips$ and $Q_p = 600 - 527.95 = 72.05\,kips$ . Substitution of these values along with those given or computed above yields the following:

$W_s = 0.041'$
$W_{pp} = 0.009'$
$W_{ps} = 0.011'$
$W_o = 0.041 + 0.009 + 0.011 = 0.062' = 0.742"$

It should be evident that, if these calculations are repeated for a range of $Q_{applied}$ , a load-settlement curve could be developed, and in fact this has been done and it is documented on this spreadsheet. In addition to Vesić’s Method being shown, the results of TAMWAVE (which uses a numerical t-z method to develop the load-settlement curve) are also included, and the results are shown below.

With this the following should be noted:

The results in the range which the two methods share are similar, especially with the “break point” where shaft resistance is maximised and toe resistance starts to kick in.
The t-z method which TAMWAVE uses assumes a purely plastic response once the ultimate static capacity is exceeded, i.e. plunging failure. Vesić’s Method does not, and in many cases this is more realistic.
The TAMWAVE method has an unloading curve which is not shown here, which Vesić’s Method does not replicate.
It is possible to apply a static load test criterion (such as Davisson’s Method) to either or both of these results. It should be expected that the ultimate load derived in this way is different from the ultimate load computed by the static capacity method.

From this we can see that Vesić’s Method is a useful tool to find an initial estimate of the load-settlement characteristics of either driven or bored piles. The difficult part is grasping the method by which Vesić’s Method models progressive mobilisation of the shaft and toe resistance. Doing this, however, can give us a better understand of the whole concept of resistance mobilisation, which is crucial in understand how deep foundations resist and transfer loads.

Our newest research item is this one, which is an expansion of the work with steel piles earlier this year. Abstract is as follows:

The application of semi-infinite pile theory to the behaviour of driven piles has been studied since Parola (1970). Most of the effort, however, has been concentrated on piles which do not require a cushion between the pile head and the pile driving accessory, such as steel piles. Concrete piles, on the other hand, are generally driven with this additional cushion. In this paper the same type of semi-infinite type of analysis is applied to this problem. Both the case of a rigid pile head and a pile head which responds without reflection from the pile are studied using both closed-form and numerical solutions. Two case histories are included which illustrate the application of the method, along with parametric studies of both pile head conditions.

You can download the paper here, or go to Researchgate to do so.