# P-y curves on model piles: Uncertainty identification

Frédéric Rosquoët, Luc Thorel, Jacques Garnier, Meriam Khemakhem

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Frédéric Rosquoët, Luc Thorel, Jacques Garnier, Meriam Khemakhem. P-y curves on model piles: Uncertainty identification. 7th international Conference on Physical Modelling in Geotechnics, Jun 2010, Zürich, Switzerland. pp.997-1002. hal-05055108

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# P-y curves on model piles: uncertainty identification

F. Rosquoët

Laboratoire des Technologies Innovantes : Equipe Phénomènes de Transferts et Construction Durable, IUT d'Amiens, France

L. Thorel, J. Garnier, M. Khemakhem

Laboratoire Central des Ponts et Chaussées, Nantes, France

**ABSTRACT:** The effect of horizontal cyclic loading applied to a pile has been investigated on centrifuge models with the aim of determining the P-y reaction curves. The instrumented pile allows the bending moment profiles to be obtained. The uncertainty on the moment measurement is first quantified. The displacements are determined from a polynomial function fitted on the moment data and calculated by double-integrating the moment. An analytical integration type is used. The soil reaction is obtained through double derivation: the moment data are interpolated using quintic Splines, which are derived twice. The Spline functions are selected by studying the static equilibrium of the pile in force and moment. An analysis of the pile response through the obtained P-y curves is discussed including uncertainty of measurements. The principle of the analysis is to introduce an uncertainty on the experimental bending moment data, and to quantify its effect on the P-y curve.

## 1 INTRODUCTION

Lateral cyclic loading on vertical piles is generally produced by waves, wind, boat accosting and mooring to quays, variable overloads or thermal dilatations. Centrifuge modelling method enables a parametric study with realistic boundary limits and control of the testing conditions (Garnier, 1995). Series of works on pile in sand have been carried out in the L.C.P.C. centrifuge on laterally loaded piles: single pile (Mezazigh et al., 1994; Mezazigh, 1995), group of piles (Remaud et al., 1998; Remaud, 1999), single pile under impact (Bonab, 2003), single pile under cyclic loading (Rosquoët, 2004; Rosquoet et al., 2007) and seismic loading (Chenaf, 2007). All tests are carried out using the L.C.P.C. centrifuge (Corté & Garnier, 1986) on piles installed in sand.

The soil-structure interaction model, is classically used to design laterally loaded piles. This model, which requires the knowledge of the P-y curves, is based on the elastic beam theory to determine the pile lateral displacements y(z), and the bending moments M(z). A similar method is used for the interpretation of the experimental results. However, in this case the bending moments are measured whereas the pile lateral displacement, y(z), and the soil reaction profile, P(z), are calculated by double integration (Eq. 1) and double derivation (Eq. 2) of the bending moments, respectively.

$$M(z) = EI \frac{d^2y}{dz^2} \Rightarrow y(z) = \iint EI \frac{M(z)}{dz} dz^2 \quad (1)$$

$$\text{and } \frac{dM(z)}{dz} = -T(z) \Rightarrow P(z) = \frac{d^2M(z)}{dz^2} \quad (2)$$

If it is relatively easy to calculate the uncertainty on the pile head displacement (obtained with the uncertainty on displacement sensor) and the maximum bending moment (Rosquoët, 2008), it is more difficult to estimate the effect of this uncertainty on the double derivation and double integration.

The present study is conducted to examine the effects of uncertainty on the obtained P-y curves. Finally, it should be mentioned that the experimental conditions, in which all tests are carried out, correspond to real load service conditions (far from failure).

## 2 EXPERIMENTAL DEVICE

A 1:40-scale model pile (Figure 1) has been instrumented with 20 pairs of strain gauges facing each other every 15 mm (figure 2). Thanks to a preliminary calibration, the bending moment versus depth profile is achieved for twenty different depths.
<figure><img src="image_2.png" /><figcaption>Figure 1. Instrumented model pile</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. General view of the model pile</figcaption></figure>

Tested with an acceleration of 40g, the scale model represents a full size prototype pile with an external diameter of 0.72 m, a length of 12 m and a bending stiffness of 476 MN.m². Confirmed with former experiments (Remaud, 1999), the assumption that the pile, in dense to medium dense sand, is flexible (or long) is used to set the boundary condition M = 0 at the pile tip.

The ratio between the pile diameter B and the Fontainebleau sand d₅₀ is 18/0.2 = 90 larger than the minimum value of about 45 needed for eliminating any grain size effects (Garnier et al., 2007).

To consider the elastic linear behaviour of the pile material, there is a relationship between the load applied Q in the extremity of the beam and the electric data uᵢ measured by each pairs of strain gauges. At each level zᵢ, according to the bending stiffness of

the pile and the electric signal by a deformation of strain gauges the bending moment is:

$$M(z)_i = 4 \frac{EI}{B} \frac{u_i}{u\Omega} 10^{-3} \quad (3)$$

With E the Young's modulus, I the bending stiffness, uᵢ the electric tension, u the supply electric tension, Ω the gauge factor and B the pile diameter.

The pile is driven into the sand sample at 1g (earth gravity) before rotating the centrifuge.

The force controlled load is applied with a servo jack and transmitted to the pile through a cable (Figure 3) 4 cm above soil surface.

<figure><img src="image_4.png" /><figcaption>Figure 3. One way loading device</figcaption></figure>

# 3 HYPOTHESES

## 3.1 P-y curve constructions

Practically, displacements are determined from a polynomial function fitted on the moment data. Displacement is calculated by double-integrating the moment using two integration constants achieved from the boundary conditions at the pile cap. An analytical integration type is used, with two integration constants (boundary conditions on head and tip of the pile) to "rise" to the displacement. The solution for P(z) is obtained through double derivation carried out by the Slivalic-LCPC piece of software (Degny, 1985). The moment data are interpolated using quintic Splines, which are derived twice. The Spline functions are selected by studying the static equilibrium of the pile in force and moment.

## 3.2 Uncertainty

It is assumed that the measurements are only subjected to random uncertainties and that the limiting distribution generated by a great number of measured values x go to only one maximum X, while the other values are regularly distributed here and there of this maximum value thus following a normal law (or Gauss function) (Taylor, 1982). The probability that a value x is located between an interval [a,b] is
the equation (4) to introduce the standard deviation σ.

$$P(x \in [a,b]) = (1/\sigma\sqrt{2\pi}) \int_a^b e^{\frac{(x-X)^2}{2\sigma^2}} dx \quad (4)$$

### 3.3 Uncertainty on bending moment measurement

It is possible to characterize three types of uncertainty (NF E11-062, 1985) during the measurement of the bending moments:

- uncertainty related to measurements;

- uncertainty due to the positioning of the gauges;

- uncertainty due to the installation.

Uncertainty of measurement can be given starting from the data obtained during the calibration. The assembly carried out during this operation, is cantilever beam type. The bending moment is calculated by equation (5).

$$M_x = EI_x \frac{\partial^2 Y}{\partial z^2} \quad (5)$$

If we consider that the behaviour of the material is elastic linear, a relation exists between the load P applied to the extremity of the beam and the electric signal uᵢ measured by each pair of gauges.

If uncertainties are independent and random, equation (6), the propagation of uncertainties allows us to write:

$$\frac{\delta Mm(z)_i}{M(z)_i} = \sqrt{\left(\frac{\delta E}{E}\right)^2 + \left(\frac{\delta I}{I}\right)^2 + \left(\frac{\delta B}{B}\right)^2 + \left(\frac{\delta \Phi}{\Phi}\right)^2 + \left(\frac{\delta k_i}{k_i}\right)^2 + \left(\frac{\delta Q}{Q}\right)^2} \quad (6)$$

with the following uncertainties (NF ENV 13005, 1999):

* $\partial E$ : on Young modulus of aluminum (±0.7%);

* $\partial I$ : on moment of inertia (±1.8%);

* $\partial B$ : on diameter (±0.6%);

* $\partial \Phi$ : on the power tension of the strain gauge (±0.01%);

* $\partial k_i$ : uncertainty of strain gauge coefficient (negligible);

* $\partial Q$ : on calibration mass (±0.04%).

Finally uncertainty related to moment measurements is

$$\partial M(z)_i = \pm 3.1\% \times M(z)_i \quad (7)$$

Uncertainty due to the position of the strain gauge can be calculated starting from equation (8):

$$\left| \frac{\partial M(z)_i}{M(z)_i} \right| \le \left| \frac{\partial a}{(a - z_i)} \right| + \left| \frac{\partial z_i}{(a - z_i)} \right| + \left| \frac{\partial P}{P} \right| \quad (8)$$

We understand the strain gauge position with uncertainty of 0.1 mm (δzi). The distance between embedding and the load application point (a = 407

mm) is known with uncertainty of 0.5 mm. Uncertainty of calibration mass is equal to the precision of the digital scale: 0.1 g (δP). Uncertainty due to the position of the strain gauge is ±0.6% for the extreme position (zᵢ = 307 mm) of the strain gauge and for a calibration mass of 257 g.

Uncertainty due to the installation of the pile in the soil (rotation during the driven) is negligible. This effect has been estimated using finite element calculation with CESAR-L.C.P.C. (Rosquoët, 2004).

Finally, uncertainty on maximum bending moments is equal to 3.7%.

## 4 P-y CURVES

### 4.1 Constructions

The theory of beams makes it possible to obtain the displacement y(z) and the soil reaction profile P(z) by respectively a double integration and a double derivation of the bending moments. Displacements are determined by a polynomial function fitted on the moment data. An analytic integration type is used, with two integration constants (boundary conditions on head and tip of the pile).

However the double derivation is much more difficult. In mathematics, the result of the double derivation is the curvature of the curve. The recorded values of bending moment vs. depth must first be properly fitted by a function that will then be derived twice. King (King, 1994) has demonstrated that a single polynomial function is not satisfactory and that it would be better to use a quintic spline. Cubic splines have indeed been used successfully to fit the bending moment profiles.

In the present analysis, the double derivation is carried out using the software Slivalic-LCPC (Degny, 1985). The selection of the Spline function is done by minimizing the elastic energy stored into the bent pile and by controlling the static equilibrium of the pile in force and moment.

However, before analyzing the obtained P-y reaction curves and studying the cyclic loading effects, these experimental P-y curves are first validated by back analyzing the response of the pile as described below.

### 4.2 Validation

The obtained P-y static curves are validated by back analysis, thanks to the Pilate-LCPC software (Rosmagny, 1985). This software is commonly used in practice to design horizontally loaded piles from P-y curves derived from in situ pressuremeter test (M.E.L.T. French code of practice Fascicule 62). The experimental P-y curves and the boundary conditions are introduced into Pilate. The obtained results (bending moments, shearing loads, pile displacements and soil reactions versus depth) are
compared with experimental data for each load increment. A good agreement is observed between results calculated by Pilate and experimental data. At the maximal static applied load F (960 kN prototype value), the difference is always less than 10 % as also demonstrated by Remaud (Remaud, 1999).

# 5 EFFECT OF THE UNCERTAINTY ON P-Y CURVES

It is very difficult to directly calculate the consequences on reaction P and displacement y of uncertainty on the bending moment measurement. In fact the classical methods of uncertainty based on uncertainty propagation are unsuitable for such a problem. We suggest to compute, from an example, the influence that an uncertainty on the bending moment could have on the soil reaction and pile displacement.

The uncertainty on the bending moment measurements (3.7 % generated by uncertainty related to measurements, uncertainty due to the positioning of the gauges and uncertainty due to the installation) is in the first step put systematically on all recorded values of moment and then in a second step included randomly.

To simplify this study of sensibility, only the case of a pile under the monotonous maximum load is considered.

## 5.1 Systematic uncertainty

We generate a 3.7 % systematic uncertainty on the bending moment measurement and we observe the effect on the calculated displacement and the soil reaction after the double derivation and double integration procedure. The uncertainty on the displacement and the soil reaction is the relationship between the values calculated from the experimental measures of moment (considered as being the "true" values) and the values obtained by modifying the moments of +3.7 % and -3.7 % (Figures 4, 5 and 6 and Table 1).

<figure><img src="image_5.png" /><figcaption>Figure 4. Systematic uncertainty of +3.7 % and -3.7 % on experimental value of bending moment</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. Effect on displacement profile (y) generated by systematic +3.7 % and -3.7 % uncertainty on bending moment</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. Effect on soil reaction (P) generated by systematic +3.7 % and -3.7 % uncertainty on bending moment</figcaption></figure>

<table><caption>Table 1. Uncertainty on pile displacement (y) and soil reaction (P) generated by +3.7 % and -3.7 % systematic uncertainty on values of bending moment.</caption><thead><tr><th>Depth (m)</th><th>y values (mm)</th><th>Uncertainty on y (%)</th><th>P value (kN/m)</th><th>Uncertainty on P (%)</th></tr></thead><tbody><tr><td>0</td><td>96.76</td><td>±1.51</td><td>133.36</td><td>±3.70</td></tr><tr><td>1.2</td><td>59.22</td><td>±3.96</td><td>472.37</td><td>±3.70</td></tr><tr><td>2.4</td><td>29.83</td><td>±9.83</td><td>537.54</td><td>±3.70</td></tr><tr><td>3.6</td><td>9.76</td><td>±32.48</td><td>266.99</td><td>±3.70</td></tr><tr><td>4.8</td><td>-1.84</td><td>±167.90</td><td>9.91</td><td>±3.70</td></tr><tr><td>6.0</td><td>-7.03</td><td>±39.59</td><td>-76.11</td><td>±3.70</td></tr><tr><td>7.2</td><td>-8.06</td><td>±28.73</td><td>-138.70</td><td>±3.70</td></tr><tr><td>8.4</td><td>-6.84</td><td>±25.83</td><td>-152.76</td><td>±3.70</td></tr><tr><td>9.6</td><td>-4.68</td><td>±25.25</td><td>-116.61</td><td>±3.70</td></tr><tr><td>10.8</td><td>-2.33</td><td>±25.34</td><td>-56.71</td><td>±3.70</td></tr></tbody></table>

It may be seen that a systematic uncertainty of 3.7 % on bending moment will be the same magnitude as the uncertainty on the soil reaction P. The double derivation is not affected by a systematic uncertainty. This result seems coherent. Indeed the curvature calculated during the double derivation remains the same in spite of the translation of the curve of the uncertainty factor.

On the other hand, the same systematic uncertainty seems to have much more effects on the pile displacement. The uncertainties vary between 1.5 and 170 %. The double integration being a surface calculation, a small modification of the moment values.
especially in the zone of the largest moment. can significantly modify the result.

However, it is important to notice that the most important uncertainty (170 %) is obtained at a depth of 4.8 m which corresponds to the pile rotation point and to extremely small soil reaction. The pile displacement errors in this zone will not have perceptible effects on the study of the pile response to lateral loads.

## 5.2 Random uncertainty

The calculation of the effects of a systematic uncertainty on the double derivation and double integration procedures shown in the previous paragraph gives some information on the phenomena. However, uncertainties are generally random. To investigate this other aspect, we allocated randomly a 3.7 % change to some measures of moment (between 5 and 10). We present (figures 7. 8 and 9 and table 2) the effect of this random uncertainty on pile displacement and soil reaction.

<figure><img src="image_8.png" /><figcaption>Figure 7. Random uncertainty of ±3.7 % on experimental values of bending moment</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8. Effect on displacement profile (y) generated by random ±3.7 % uncertainty on bending moment</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9. Effect on soil reaction (P) generated by random ±3.7 % uncertainty on bending moment</figcaption></figure>

<table><caption>Table 2. Uncertainty on pile displacement (y) and soil reaction (P) generated by a ±3.7 % random uncertainty on experimental values of bending moment.</caption><thead><tr><td>Depth (m)</td><td>Uncertainty on y (%)</td><td>Uncertainty on P (%)</td></tr></thead><tbody><tr><td>0.0</td><td>0.38</td><td>34.39</td></tr><tr><td>1.2</td><td>1.05</td><td>0.97</td></tr><tr><td>2.4</td><td>2.86</td><td>-8.92</td></tr><tr><td>3.6</td><td>9.67</td><td>-8.71</td></tr><tr><td>4.8</td><td>-48.95</td><td>298.05</td></tr><tr><td>6.0</td><td>-11.30</td><td>10.60</td></tr><tr><td>7.2</td><td>-8.18</td><td>-1.62</td></tr><tr><td>8.4</td><td>-7.39</td><td>-0.71</td></tr><tr><td>9.6</td><td>-7.18</td><td>-0.70</td></tr><tr><td>10.8</td><td>-7.18</td><td>-1.54</td></tr></tbody></table>

A random uncertainty modifies the curvature of the bending moment versus depth and the results obtained after the double derivation. The smallest soil reaction are more affected by uncertainties on bending moment (layer near to the surface or at depths close to the pile rotation point).

The pile displacement seems less sensitive to random uncertainty. Indeed, the calculation of the surface is less affected when some points are only perturbed by the uncertainty. We can nevertheless notice that the systematic uncertainty will always limit the random uncertainty on displacement.

## 6 CONCLUSION

It is rather difficult to directly calculate the impact on pile displacement and on soil reaction of the uncertainties on recorded bending moments during either true scale or centrifuge model tests. Nevertheless, it is essential to quantify the errors that may exist on the P-y reaction curves before any analysis and use of these experimental P-y curves.

The present study is carried out on an example of centrifuge pile lateral loading test. The prototype pile installed into a medium dense sand is 12m long and 0.72m in diameter.
The uncertainty on bending moment measured through 20 pairs of strain gages data placed on the pile model is small and about ±3.7%.

The study shows how this uncertainty propagates
towards the pile displacement (y) and the soil reac-
tion (P). Only one random distribution of error has
been considered so to say that systematic error is al-
ways worst has only been proved for the specific
case considered

In shallow or deep layers. the resulting uncertain-
ties are variable but always smaller than 35 %. At in-
termediate depths. close to the pile rotation point
(deep of 4.8m). errors on the pile displacement and
soil reaction may be much larger. However at such
depths pile displacement and soil reaction are very
small and do not carry weight on the pile equilibrium
and pile response to lateral loads.

It should be mentioned that the results of the pre-
sent study are only valid when the uncertainty (or er-
ror) on the measured bending moment are small (a
few per cent).

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