Estimation of Bearing Capacity of Piles in Cohesionless Soil Using Optimised Machine Learning Approaches

Abstract

Accurate estimation of the bearing capacity of piles requires complex modelling techniques which are not justified by timeframe, budget, or scope of the projects. In this study, six advanced machine learning algorithms including decision tree, k-nearest neighbour, multilayer perceptron artificial neural network, random forest, support vector regressor and extremely gradient boosting are employed to model the bearing capacity of piles in cohesionless soil, and the particle swarm optimisation algorithm is used to optimate the hyper-parameters of machine learning algorithms. A dataset comprising of 59 cases is employed and the R-squared value, root mean square error and variance accounted for are used as performance metrics to compare the performance of optimised machine learning methods. The comparison reveals that the optimised machine learning methods have great potential to estimate bearing capacity of piles and the particle swarm optimisation algorithm is efficient in the hyper-parameter tuning. The results show that R-squared values of six optimised machine learning approaches on the testing set vary from 0.731 to 0.9615. Also, the optimised extremely gradient boosting (R-squared value = 0.9615) shows the best performance compared with other algorithms. Furthermore, the relative importance of influential variable is investigated, which shows that effective stress is the most influential variable for bearing capacity of piles with an importance score of 30.9%. In addition, the results by the optimised machine learning method are compared to the β-method which is a popular empirical method. It is revealed the prominent performance of optimised machine learning approaches.

Appendices

Appendix 1: Dataset of the Multinational Pile Cases

See Table 5.

Table 5 Dataset of the multinational pile cases

Appendix 2: Calculating the Unit Shaft Resistance

In an effective stress analysis, the unit shaft resistance is calculated from the following expression:

$$Q_{s} = \int_{0}^{L} {C(\beta \sigma_{v}^{{\prime }} + c_{a} )dz}$$

(16)

where $\beta = K^{\prime}{ \tan }\left( \delta \right)$, $\sigma^{\prime}_{\text{v}}$ is the effective vertical stress and C is the pile circumference. Usually $c_{a}$ is set to zero for driven piles but may be non-zero for cast-in-place piles. Fellenius (1991) tells us that typical values for $\beta$ depend on soil gradation, mineral composition, density and soil strength within a fairly narrow range. Some empirical ranges for $\beta$ coefficients in different soil types are,

Meyerhof (1976) has proposed values of $K^{\prime}{ \tan }\left( \delta \right)$ for driven piles. Note that the shaft resistance values reflect the likely changes of stress state in the soil due to the method of installation. In using Table 6, the undisturbed value is used in all cases.

Table 6 Approximate range of β and $N_{t}$ coefficients (Fellenius 1991)

The unit base resistance is calculated from:

$$Q_{p} = A_{p} N_{t} \sigma^{\prime}_{\text{v}}$$

(17)

where $N_{t}$ = base bearing capacity coefficient, $A_{p}$ = cross-sectional area and $\sigma^{\prime}_{\text{v}}$ = effective stress.

Recommended ranges of β and $N_{t}$ coefficients as a function of soil type and φ_s angle from Fellenius (1991) are presented in Table 6. Fellenius notes that factors affecting the β and $N_{t}$, coefficients consist of the soil composition including the grain size distribution angularity and mineralogical origin of the soil grains, the original soil density and density due to the pile installation technique, the soil strength, as well as other factors. Even so, β coefficients are generally within the ranges provided and seldom exceed 1.0.

For sedimentary cohesionless deposits, Fellenius states $N_{t}$, ranges from about 30 to a high of 120. In very dense non-sedimentary deposits such as glacial tills,$N_{t}$ can be much higher, but can also approach the lower bound value of 30. In clays, Fellenius notes that the toe resistance calculated using a $N_{t}$ of 3 that is similar to the toe resistance calculated from a traditional analysis using undrained shear strength. Therefore, the use of a relatively low $N_{t}$, coefficient in clays is recommended unless local correlations suggest higher values are appropriate.

Graphs of the ranges in β and $N_{t}$ coefficients versus the range in φ_s angle as suggested by Fellenius. These graphs may be helpful in the selection of β or $N_{t}$. The inexperienced user should select conservative β and $N_{t}$ coefficients. As with any design method, the user should also confirm the appropriateness of a selected β or $N_{t}$, coefficient in a given soil condition with local correlations between static capacity calculations and static load tests results.

It should be noted that the effective stress method places no limiting values on either the shaft or base resistance.

Step 1 Delineate the soil profile into layers and determine φ_s angle for each layer.

• a. Construct $\sigma^{\prime}_{\text{v}}$ diagram along the depth of the pile.

• b. Divide soil profile throughout the pile penetration depth into layers and determine the effective stress,$\sigma^{\prime}_{\text{v}}$, at the midpoint of each layer.

• c. Determine the φ_s angle for each soil layer from laboratory or in-situ test data.

• d. In the absence of laboratory or in-situ data for cohesionless layers, determine the average corrected N_cor value for each layer and estimate φ_s angle from Schmertmanns (1975) SPT correlation for soils.

Step 2 Select the β coefficient for each soil layer.

• a. Utilise local experience to select β coefficient for each layer.

• b. In the absence of local experience, use Meyerhof’s Pile Factors to estimate β coefficient from φ_s angle for each layer. NB $\beta = K^{\prime}{ \tan }\left( \delta \right)$

Step 3 For each soil layer calculate the shaft resistance $Q_{s}$ (kPa) from Eq. (1). The total shaft resistance is simply the sum of the shaft resistance from each soil layer.

Step 4 Calculate the ultimate toe resistance, $Q_{m}$ (kPa).

• a. Utilise local experience to select $N_{t}$ coefficient.

• b. In the absence of local experience, estimate $N_{t}$, from Table 6 based on φ_s angle.

• c. Calculate the effective stress at the pile base, $\sigma^{\prime}_{\text{v}}$.

Step 5 Calculate the ultimate pile capacity, $Q_{m} = Q_{s} \left( {KN} \right) + Q_{p}$.