where:

Nq: non-dimensional parameter

p0,tip: effective vertical pressure at the base of the pile.

The following charts collect the relationship needed to develop this formulation based on the soil relative density (Figure 6).

**Figure 6.** (**a**) Relationship between relative density and skin friction; (**b**) Relationship between relative density and resistance.

### *4.3. Pile Model*

Figure 7 shows the pile model used to solve the load-settlement relationship. It consists of splitting the pile in equal-length slices. The slices are united by an elastic spring (Ki). Each of them has a weight (Pi) and ultimate skin friction (Rfi). The deepest slice has additionally a base resistance (Rp). At pile top, it is applied a load (F0).

Each slice undergoes an absolute offset (ui), as a result of the application of the external load and weight. ui is positive downward. The following equation system solves the mathematical problem.

$$\mathbf{F}\_0 = \mathbf{R}\mathbf{f}\_1(\mathbf{u}\_1) + \mathbf{K}\_1(\mathbf{u}\_1 - \mathbf{u}\_2) - \mathbf{P}\_1 \tag{10}$$

$$0 = \mathbf{R}\xi\_2(\mathbf{u}\_2) + \mathbf{K}\_2(\mathbf{u}\_2 - \mathbf{u}\_3) - \mathbf{K}\_1(\mathbf{u}\_1 - \mathbf{u}\_2) - \mathbf{P}\_2 \tag{11}$$

$$0 = \mathbf{R} \mathbf{f} \mathbf{\hat{z}}(\mathbf{u} \mathbf{\hat{z}}) + \mathbf{K} \mathbf{\hat{z}}(\mathbf{u} \mathbf{\hat{z}} - \mathbf{u}\_4) - \mathbf{K} \mathbf{\hat{z}}(\mathbf{u} \mathbf{\hat{z}} - \mathbf{u} \mathbf{\hat{z}}) - \mathbf{P} \mathbf{\hat{z}} \tag{12}$$

$$0 = \mathbf{R}\mathbf{f}\_i(\mathbf{u}\_i) + \mathbf{K}\_i(\mathbf{u}\_i - \mathbf{u}\_{i+1}) - \mathbf{K}\_{i-1}(\mathbf{u}\_{i-1} - \mathbf{u}\_i) - \mathbf{P}\_i \tag{13}$$

$$0 = \mathbf{Rp}(\mathbf{u}\_{\mathrm{n}}) + \mathbf{Rf}\_{\mathrm{n}}(\mathbf{u}\_{\mathrm{n}}) + \mathbf{K}\_{\mathrm{n}}(\mathbf{u}\_{\mathrm{n}}) - \mathbf{K}\_{\mathrm{n}-1}(\mathbf{u}\_{\mathrm{n}-1} - \mathbf{u}\_{\mathrm{n}}) - \mathbf{P}\_{\mathrm{n}} \tag{14}$$

Kiis the pile stiffness that takes the value:

$$\mathbb{K}\_{\mathrm{i}} = \frac{\mathbb{E} \times \mathbf{A}\_{\mathrm{i}}}{\mathbb{L}\_{\mathrm{i}}}$$

where:

E: pile deformation modulus

Ai pile cross-section area.

Li: slice length.

The solution of the previous system provides the seeking relationship between applied external pressure F0 and displacement at the top pile (u1). Since skin friction Rfi(ui) and base resistance Rp(un) are functions of displacement, the system is nonlinear and it has to be used an iterative algorithm to reach the solution.

**Figure 7.** Soil–pile analytical model.

#### *4.4. t-z Curve and Q-z Curve*

Rfi functions are obtained from the t-z curve defined in standard ISO 19901-4:2003 (API RP 2GEO). Base resistance also follows the recommendations of this standard. The following Figure 8 shows the used functions.

**Figure 8.** Soil-pile deformation relationship (**a**): t-z curve; (**b**): Q-z curve.

### *4.5. Load Application*

The total load F0 is applied by steps as it is done in the large-scale load test. Positive load is downward. Model admits starting loads as tension (upwards) or compression (downwards).

The weight of the pile is considered through the density of the structural section. If the initial charge is zero the model provides only the settlement due to the weight of the pile.

#### **5. Pile Load Test Outputs**

There are sixty-two (62) load tests conducted on 0.8 m diameter concrete piles. The piles were executed by hollow-stemmed flight augering. Pile length ranged between 22 to 28 meters.

The difference in length between the piles is due to the depth where dense sand is located, which is reached by all the piles. Therefore, it can be said that all the piles are supported in a layer of similar mechanical properties.

The piles were tested in a single cycle of loading up to a value of 3.150 KN. Maximum load was reached on five load steps.

The following Figure 9 represents the length of the piles with respect to settlement. The following is observed:


**Figure 9.** Concrete pile load test outputs.

For the more frequent values, Figure 10 below shows the output distribution and their statistical parameters. Statistical parameters are shown in Table 2.

The following Figure 11 represents an example of some of the more-frequent-value load tests.

**Figure 10.** Settlement distribution of concrete load test.

**Table 2.** Statistical parameter of concrete load tests.


**Figure 11.** Concrete pile load test—zone 1.

#### **6. Fitting the Analytical Model and Pile Load Test**

Pile load tests have been grouped in eight zones and each zone is characterized by a borehole place at that zone. Therefore, eight different soil profiles have been used. Nevertheless, all of them were similar.

To achieve a good fit with these tests, it is necessary to modify some parameters of the model. It was found that the influencing parameters are skin friction and Zpeak value.

Other parameters with less influence are: the base resistance, pile stiffness and pile diameter. The following Figure 12 represents one of the adjustments made for load tests conducted in zone 2. In this case, the adjustment is achieved improving skin friction by 2 and using a Zpeak value of 0.17%D.

**Figure 12.** Numerical model and pile load test fitting in zone 1 Concrete pile.

It is represented also for this adjustment the distribution of forces and displacement against depth in Figure 13.

**Figure 13.** Numerical models outputs for each load step (**a**) skin friction versus depth (**b**) displacement distribution versus depth.

The following Figure 14 shows the mobilized base resistance. It is shown that the maximum load reached at the base is less than 400 KN, which means that only 10% of the total load of the test is supported by the pile tip.

**Figure 14.** Base resistance. Concrete piles.

The sensitivity of the model against the two main parameters has also been analyzed. The outputs are presented in the following Figure 15. It is noted that for a proper fit, the skin friction has to be multiplied by a factor that is between 2.0 to 2.5.

**Figure 15.** Sensitivity analysis of Zpeak values for concrete piles.

This result is reasonable considering that the basic value is for metal-soil surface and that the tested piles are casted in-situ. The different roughness and bounding of both surfaces is obvious.

In addition to skin friction, the Zpeak value has also to be reduced. This value gives the slope of the t-z curve; so that a low value means a rapid mobilization of skin friction, and therefore, for the same level of load, there is less deformation.

### **7. Probabilistic Approach**

The model presented above was tested by a probabilistic approach in order to quantify the uncertainty of the model. The applied methodology is as follows:


This procedure and its outputs are described below:

#### *7.1. Artificial Soil Resistance Profiles Generation*

Based on the expected mean and standard deviation of soil resistance, artificial profiles are generated. A total of 50 profiles have been used. They are represented in Figure 5b.

### *7.2. Pile Bearing Capacity*

The model requires working out the total bearing capacity in order to define the t-z and base-displacement functions. Since the inputs are statistical functions, the outputs are also given in the same way. The following Figures 16 and 17 show separately the point and shaft resistance of the piles.

**Figure 16.** Generated distribution for point resistance.

**Figure 17.** Generated distribution for shaft resistance.

It is noted that:


### *7.3. Pile Settlement Calculation*

The model provides the pile settlement. The settlement is also a probability distribution function. Figure 18 below compares it with the pile load test distribution.

**Figure 18.** Density functions for generated and pile load test settlement.

So far, the formulated model is a deterministic one in which the same input yields always the same outputs. It is noted that the settlement distribution generated in this way has far less dispersion that the actual pile load test, even though the input data takes into account the soil strength dispersion. A better match is achieved if a probabilistic model is used.

### *7.4. Model Uncertainty*

The fact that the model does not include all the possible variables that intervene in settlement generation but only the main ones can be taken into account by introducing a probabilistic function in the model itself.

The calculation model requires the introduction of a statistical parameter to represent the dispersion of results that is observed in the field.

This statistical parameter is the Zpeak factor, which controls the slope of the elastic run on the t-z curves.

The used probabilistic function is the Uniform Density Function whose value is:

$$\text{iz}\_{\text{peak}} = 0.02 \times \text{Random} + 0.0001 \tag{15}$$

where "Random" is an aleatory number. It varies between 0 and 1. In this way, the Zpeak parameter varies uniformly between 0.0001 and 0.02.

The randomness of this parameter reflects the uncertainty of the model in settlement prediction. Its value has been calibrated with usual working load for piles, which on the other hand is where deformations are most interesting.

### **8. Discussion and Conclusions**

This paper develops a numerical model for axially loaded pile which is able to predict its deformation and gives insight about the ultimate skin friction and its distribution along the pile, base resistance and how it is mobilized as loading.

The model is calibrated against sixty two (62) pile load tests carried out on a fairly homogenous soil conditions.

Pile material and piling method are relevant to defined maximum skin friction. In particular, concrete set-in-place shaft resistance has proven to be in the order of 2.0 to 2.5 times greater than metallic piles.

Shear deformation modulus has also been assessed. On the spring model, the shear deformation modulus is related to the slope of the curve t-z, controlled by the Zpeak parameter. For concrete piles, Zpeak values ranging from 0.25% to 0.17% pile diameter show a good fitting with pile load tests.

In addition, a probabilistic approach helps to understand the uncertainties of the method. Uncertainties due to the measure equipment and soil strength variability have been quantify and represented by their probabilistic functions.

The stochastic variation applied to the input parameters is not enough to simulate the dispersion observed on the pile load test. Therefore, a probabilistic function is also introduced in the model to account for the model uncertainty. Eventually, the generated density function matches in mean and deviation to the pile load test distribution, which means that the model is able to accurately predict settlement and confidence intervals.

**Author Contributions:** Conceptualization, methodology and investigation, M.B.A., F.E.S.; soft-ware, M.B.A.; review and supervision, E.S.P.; editing, supervision and formal analysis, M.B.A., F.E.S. and E.S.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the authors. The data are not publicly available due to they have been gathered and treated for the authors. **Acknowledgments:** The authors of the article appreciate the support given in time and resources by Geointec company and the Geology and Research group Geología Aplicada a la Ingeniería Civil of Universidad Politécnica de Madrid.

**Conflicts of Interest:** The authors declare no conflict of interest.
