# Prediction of Pile Axial Bearing Capacity Using Artificial Neural Network and Random Forest

^{*}

## Abstract

**:**

^{2}) were used to evaluate the performance of RF and ANN algorithms. In addition, the predicted results of pile load tests were compared with five empirical equations derived from the literature and with classical multi-variable regression. The results showed that RF outperformed ANN and other methods. Sensitivity analysis was conducted to reveal that the average SPT value and pile tip elevation were the most important factors in predicting the axial bearing capacity of piles.

## 1. Introduction

_{u}) is considered as the most important parameter in the design of pile foundation. Normally, the axial bearing capacity of piles can be determined by five approaches, namely the static analysis, dynamic analysis, dynamic testing, pile load testing, and in-situ testing [1]. Out of these methods, the pile load test is considered as the best method to determine the pile bearing capacity. However, such a method is time-consuming, and the costs are often difficult to justify for ordinary or small projects, whereas other methods have lower accuracy. As a result, several approaches have been developed to predict the axial bearing capacity of pile or to enhance the prediction accuracy. The nature of these methods included some simplifications, assumptions, or empirical approaches with respect to the soil stratigraphy, soil–pile structure interactions, and the distribution of soil resistance along the pile. In such studies, the test results were used as complementary elements to improve the prediction accuracy. Meanwhile, the European standard (Euro code 7) [2] recommends using the following ground field tests: DP (dynamic probing test), SS (press-in and screw-on probe test), SPT (standard penetration test), PMT (pressure meter tests), PLT (plate loading test), DMT (flat dilatometer test), FVT (field vane test), CPTu (cone penetration tests with the measurement of pore pressure). Among the above indicators, the SPT is commonly used to determine the bearing capacity of piles [3]. Several propositions relying on the results of SPT have been proposed to predict the bearing capacity of piles, including the empirical equations derived from the works of Meyerhof [4], Bazaraa and Kurkur [5], Robert [6], Shioi and Fukui [7], Shariatmadari et al. [8]. In general, the pile diameter, SPT blow counts along the pile shaft, and at the tip of the pile have been used in these equations. Besides, Lopes and Laprovitera [9] proposed a formula to estimate the pile bearing capacity for different types of soil, namely sand, and silt. Decort [10] has presented an empirical formula, including the adjustment factors for sandy and clay soils. Last but not least, the Architectural Institute of Japan (AIJ) [11] has a recommendation using an experimental formula, taking into account the effects of soil type or the use of SPT value for sandy soil and untrained shear strength of soil (C

_{u}) for clayey soil. Overall, traditional methods or empirical equations attempted to include a few key parameters to predict the pile strength. However, if the input parameters of pile geometry and soil properties increased, these methods were impossible to use.

^{2})—were applied to evaluate the prediction capability of the algorithms. In addition, 1000 simulations taking into account the random splitting of the dataset were conducted for each model in order to finely evaluate the accuracy of RF and ANN. Proved to outperform ANN, the RF algorithm was next used to compare with five empirical formulas in the literature as well as classical multivariable regression (MVR) in predicting the bearing capacity of piles. Finally, the feature importance analysis was proposed to find out the important factors affecting the prediction capability of the RF model.

## 2. Significance of the Research Study

## 3. Data Collection and Preparation

#### 3.1. Experimental Measurement of Bearing Capacity

- (i)
- (i) If the settlement of pile top at a given load level was 5 times higher than the settlement of pile top at the previous load level, or the settlement of the pile top at a given load level increased continuously while the load did not increase, the pile bearing capacity was determined based on that given failure load. The number of piles corresponded to this situation was 688, representing about 30% of the samples. An example of this situation is given in Appendix A (Figure A1).
- (ii)
- When the pile load capacity was too large to be able to test by the destructive load, the load curve (P)– settlement (S) was plotted in log(P)–log(S). The intersection point of two lines was considered as a result of failure and taken as the pile bearing capacity, according to De Beer (1968) [47]. The number of piles corresponded to this situation was 1225 piles (accounting for more than 50% of the samples). An example of this situation is given in Appendix A (Figure A2 and Figure A3).
- (iii)
- For the remaining samples, when the log(P)–log(S) relationship is linear, which could not find the intersection point compared with the previous case. The determination of the pile bearing capacity was taken at the load level when the settlement of the pile top exceeded 10% of the pile diameter.

#### 3.2. Data Preparation

_{s}) and pile tip (N

_{t}) were calculated. It is worth mentioning that, in order to obtain the average SPT (N

_{t}) value around the pile tip, Meyerhof’s recommendation (1976) [4] was considered. The average SPT (N

_{t}) value for 8D above and 3D below the pile tip was obtained, where D represented the pile diameter.

_{1}); (iii) length of second pile segment (L

_{2}); (iv) length of pile top segment (L

_{3}); (v) the natural ground elevation (E

_{g}); (vi) pile top elevation (E

_{p}); (vii) guide pile segment stop driving elevation (E

_{t}); (viii) pile tip elevation (Z

_{m}); (ix) the average SPT blow along the embedded length of the pile (N

_{s}) and (x) the average SPT blow at the tip of the pile (N

_{t}). The bearing capacity was the single output variable in this study (P

_{u}).

_{1}) ranged from 3 m to 8.4 m. The length of the second pile segment (L

_{2}) ranged from 1.47 m to 8 m. The length of pile top segment (L

_{3}) ranged from 0 m to 3.95 m, where a 0 value means that segment did not exist. The natural ground elevation (E

_{g}) varied from −1.6 m to 3.4 m. The pile top elevation (E

_{p}) varied from 2.05 m to 4.13 m. The guide piles stop driving elevation (E

_{t}) varied from −1.6 m to 8.4m. The pile tip elevation (Z

_{m}) varied from 8.27 m to 18.35 m. The average SPT blow counts along the embedded length of the pile (N

_{s}) ranged from 5.57 to 19.2. The average SPT blow counts at the tip of the pile (N

_{t}) ranged from 4.35 to 8.47. The bearing capacity load (P

_{u}), ranged from 384 kN to 1860 kN with a mean value of 1164.5 kN and a standard deviation of 268.6 kN.

## 4. Machine Learning Methods

#### 4.1. Random Forest (RF)

#### 4.2. Artificial Neural Network (ANN)

#### 4.3. Performance Evaluation

_{u}were used, namely the mean absolute error (MAE), root mean square error (RMSE), squared correlation coefficient, or the coefficient of determination (R

^{2}). The R

^{2}measured the squared correlation between the predicted and actual P

_{u}values, having values in the range of [0, 1]. Low RMSE and MAE values showed better accuracy of the proposed ML algorithms. On the other hand, RMSE calculated the squared root average difference, whereas MAE calculated the difference between the predicted and actual P

_{u}values. These values could be calculated using the following equations [64,65,66,67,68]:

## 5. Results and Discussion

#### 5.1. Comparison of RF and ANN

^{2}= 0.969, RMSE = 47.333, and MAE = 2.178. The ANN model produced an intermediate accuracy (R

^{2}= 0.818, RMSE = 114.882, and MAE = 1.050) for the training data. This was also confirmed by the values of the standard deviation of error (denoted as StDerror in Table 5). For the training part, RF yielded a lower value of StDerror compared to ANN (i.e., StDerror = 10.605% and 4.223% for ANN and RF, respectively).

^{2}, RMSE, MAE, the mean of error m

_{error}, and StDerror (i.e., R

^{2}= 0.866, 0.809; RMSE = 98.161, 116.366; MAE = 2.924, 3.190; m

_{error}= 0.573%, 1.202%; StD

_{erro}r =9.461%, 10.786% using RF and ANN, respectively). The MAE value of RF was slightly higher in the training part but much lower in the testing one compared to ANN because the performance of such a model might be influenced by choice of the selected index of the training data.

^{2}values of these simulations are plotted in Figure 8a,c, and the corresponding histograms are plotted in Figure 8b,d. It was observed that the proposed RF model gave satisfactory R

^{2}values within the range of 0.83 to 0.87. The most frequent R

^{2}obtained over 1000 simulations was R

^{2}= 0.855 with a frequency of about 170. Besides, ANN algorithm showed a lower accuracy when R

^{2}values ranged from 0.78 to 0.84 with the most frequent values of R

^{2}= 0.82 (frequency of about 260). Summarized values of the accuracy corresponded to the two models for the testing part is presented in Table 6.

^{2}= 0.861 compare to ANN (average R

^{2}= 0.811). In conclusion, from the statistical analysis and prediction errors, RF algorithm was the better model to predict the bearing capacity of pile.

#### 5.2. Comparison with Empirical Equations and Multi-Variable Regression

_{b}) and unit shaft (Q

_{s}) resistance are summarized in Table 7.

_{b}and Q

_{s}, following the well-known formula

_{i}denotes the thickness and Q

_{s(i)}indicates the value for unit shaft resistance of the i

^{th}soil layer which piles penetrated through. In addition, the use of classical MVR to predict the bearing capacity of pile was also applied. Multi-variable regression technique is commonly used in several studies, such as Egbe et al. [70] and Silva et al. [71] to predict the properties and chemical composition of soil. The regression coefficient results are shown in Table 8.

^{2}, RMSE, MAE (i.e., R

^{2}= 0.866, 0.702, 0.467, 0.485, 0.334, 0.391, 0.611; MAE = 2.924, 94.267, 77.830, 280.727, 312.297, 340.668, 205.72; RMSE = 98.161, 183.046, 224.5, 340.606, 483.335, 400.470, 265.53 using RF and Meyerhof (1976), Shioi and Fukui (1982), Decourt (1995), Shariatmadari (2008), and AIJ (2004), respectively). The results also showed that the MVR was less accurate than the RF model, but both models gave better accuracy than traditional formulas. It is worth noticing that these formulas were developed for granular soil. However, the effect of soil type in estimating the bearing capacity of piles was neglected in this study, knowing that such bearing capacity depends on soil types. The main purpose of this section was to demonstrate the higher prediction capacity of ML approach compared with empirical equations even without information related to soil types.

#### 5.3. Feature Importance Analysis Using RF

_{s}) was the most important variable, as an increase in MSE of 33% was noticed (Figure 10). Indeed, N

_{s}is an important indicator of predicting pile-bearing capacity, and it is related to the ultimate friction along the pile shaft. The pile tip elevation Z

_{m}was the second important variable, confirmed by an increase in MSE of 25%. From a soil mechanic point of view, it meant that with a little change in the soil properties, the pile tip elevation played an important role in the bearing capacity of pile; such a variable is involved in the pile tip resistance. The variables L

_{2}, L

_{1}, and E

_{t}were ranked as the third to the fifth important predictors, with an increase in MSE, ranging from 5% to 19% (Figure 10). Other predictor variables included in the model (Z

_{g}, N

_{t}, Z

_{p}, L

_{3}, and D) had lower than 5% of the increase in MSE. This observation was also in good agreement with MVR results, where the coefficients of input variables L

_{3}and Z

_{p}were equal to 0, while D had a relatively small coefficient of 2.33 (see Table 8).

## 6. Conclusions

^{2}= 0.866, RMSE = 98.161 kN, MAE = 2.924 kN using RF compared with R

^{2}= 0.809, RMSE = 116.366 kN, MAE =3.190 kN using ANN). Moreover, the results of this study indicated that the RF algorithm was more accurate in predicting the pile bearing capacity than those obtained from traditional approaches, namely the formulas or empirical equations from the work of Meyerhof (1976), Shioi and Fukui (1982), Decourt (1995), Shariatmadari (2008), AIJ (2004), and a classical MVR. In addition, a sensitivity analysis using RF indicated that the average SPT value along pile shaft N

_{s}, pile tip elevation, and L

_{2}, L

_{1}, E

_{t}had the most significant effect on the predicted bearing capacity of piles.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 1.**(

**a**) Experimental location; (

**b**) pre-cast square-section concrete pile; (

**c**) hydraulic pile presses machine; (

**d**) experimental layout.

**Figure 3.**Histograms of the dataset in this study: (

**a**) D; (

**b**) L

_{1}; (

**c**) L

_{2}; (

**d**) L

_{3}; (

**e**) E

_{g}; (

**f**) E

_{p}; (

**g**) E

_{t}; (

**h**)Z

_{m}; (

**i**) N

_{s}; (

**j**) N

_{t}; (

**k**) P

_{u}.

**Figure 6.**Graphs of regression results between measured P

_{u}versus predicted Pu for the training part using (

**a**) ANN, (

**b**) RF, (

**c**) mean error ANN, (

**d**) mean error RF, (

**e**) standard deviation ANN, and (

**f**) standard deviation RF.

**Figure 7.**Graphs of regression results between measured P

_{u}versus predicted P

_{u}for the testing part using (

**a**) ANN, (

**b**) RF, (

**c**) mean error ANN, (

**d**) mean error RF, (

**e**) standard deviation ANN, and (

**f**) standard deviation RF.

**Figure 8.**Scatter plot of R

^{2}values for 1.000 simulations for (

**a**) RF, (

**c**) ANN and the corresponding histograms of R

^{2}values, (

**b**) RF, (

**d**) ANN.

N° | D | L_{1} | L_{2} | L_{3} | E_{g} | E_{p} | E_{t} | Z_{m} | N_{s} | N_{t} | P_{u} |
---|---|---|---|---|---|---|---|---|---|---|---|

Unit | mm | m | m | m | m | m | m | m | - | - | kN |

1 | 400 | 3.45 | 8 | 0.07 | 2.95 | 3.42 | 2.95 | 14.47 | 11.52 | 7.44 | 1163 |

2 | 400 | 3.4 | 7.29 | 0 | 3.4 | 3.49 | 3.4 | 14.09 | 10.69 | 7.27 | 1240 |

3 | 400 | 4.35 | 8 | 1.2 | 2.05 | 3.4 | 5.8 | 15.6 | 13.55 | 7.74 | 1297.8 |

. | . | . | . | . | . | . | . | . | . | . | . |

. | . | . | . | . | . | . | . | . | . | . | . |

. | . | . | . | . | . | . | . | . | . | . | . |

2312 | 400 | 5.72 | 8 | 1.67 | 0.68 | 4.13 | 1.06 | 16.07 | 15.39 | 7.50 | 1344 |

2313 | 400 | 4.1 | 2.19 | 0 | 2.7 | 3.72 | 2.73 | 8.99 | 6.29 | 4.94 | 480 |

2314 | 400 | 4.05 | 8 | 0.7 | 2.35 | 3.5 | 2.4 | 15.1 | 12.75 | 7.58 | 1318 |

Min | 300 | 3.00 | 1.47 | 0.00 | −1.60 | 2.05 | −1.60 | 8.27 | 5.57 | 4.35 | 384 |

Average | 393.3 | 4.02 | 7.27 | 0.49 | 2.53 | 3.52 | 2.70 | 14.30 | 11.78 | 7.24 | 1164.5 |

Max | 400 | 8.40 | 8.00 | 3.95 | 3.40 | 4.13 | 8.40 | 18.35 | 19.20 | 8.47 | 1860 |

SD | 24.85 | 0.55 | 1.53 | 0.50 | 0.63 | 0.09 | 0.75 | 1.68 | 2.02 | 0.71 | 268.63 |

D | L_{1} | L_{2} | L_{3} | E_{g} | E_{p} | E_{t} | Z_{m} | N_{s} | N_{t} | |
---|---|---|---|---|---|---|---|---|---|---|

Min | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 | −1.0 |

Average | 0.868 | 0.845 | −0.983 | −0.999 | 0.651 | 0.410 | −0.139 | 0.197 | 0.401 | 0.284 |

Max | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |

SD | 0.497 | 0.413 | 0.172 | 0.042 | 0.253 | 0.084 | 0.150 | 0.333 | 0.342 | 0.959 |

Parameters | Value and Description |
---|---|

Number of neurons in the input layer | 10 |

Number of hidden layers | 1 |

Number of neurons in the hidden layer | 7 |

Number of neurons in the output layer | 1 |

Activation function for the hidden layer | Logistic |

Activation function for the output layer | Linear |

Training algorithm | Quasi-Newton methods |

Cost function | Mean Square Error (MSE) |

Parameters | Value and Description |
---|---|

Number of trees | 15 |

Number of features to consider when looking for the best split | 10 |

Minimum number of samples required to split an internal node | 2 |

Minimum number of samples required to be at a leaf node | 1e-7 |

Cost function | Mean Square Error (MSE) |

Maximum depth of the tree | None |

Part | Method | R^{2} | MAE (kN) | RMSE (kN) | m_{error} (%) | StD_{error} (%) |
---|---|---|---|---|---|---|

Training | ANN | 0.818 | 1.050 | 114.882 | 0.884% | 10.605% |

RF | 0.969 | 2.178 | 47.333 | 0.069% | 4.223% | |

Testing | ANN | 0.809 | 3.190 | 116.366 | 1.202% | 10.786% |

RF | 0.866 | 2.924 | 98.161 | 0.573% | 9.461% |

Part | Method | Avr. R^{2} | StD. R^{2} |
---|---|---|---|

Testing | ANN | 0.811 | 0.318 |

RF | 0.861 | 0.277 |

Ground Type | Sandy Ground | Clayey Ground |
---|---|---|

Meyerhof (1976) [4] | ${\mathrm{Q}}_{\mathrm{b}}\left(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2}\right)=40{\mathrm{N}}_{\mathrm{t}}\left(\frac{\mathrm{L}}{\mathrm{D}}\right)\le 400{\mathrm{N}}_{\mathrm{t}}$ | |

${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=2{\mathrm{N}}_{\mathrm{s}}$ | ||

Shioi and Fukui (1982) [7] | ${\mathrm{Q}}_{\mathrm{b}}\left(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2}\right)=300{\mathrm{N}}_{\mathrm{t}}$ | |

${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=2{\mathrm{N}}_{\mathrm{s}}$ | ||

Decourt (1995) [10] | ${\mathrm{Q}}_{\mathrm{b}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=325{\mathrm{N}}_{\mathrm{t}}$ | ${\mathrm{Q}}_{\mathrm{b}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=100{\mathrm{N}}_{\mathrm{t}}$ |

${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=(0.5\xf70.6)(2.8{\mathrm{N}}_{\mathrm{s}}+10)$ | ${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=2.8{\mathrm{N}}_{\mathrm{s}}+10$ | |

Shariatmadari (2008) [8] | ${\mathrm{Q}}_{\mathrm{b}}\left(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2}\right)=385{\mathrm{N}}_{\mathrm{t}}$ | |

${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=3.65{\mathrm{N}}_{\mathrm{s}}$ | ||

AIJ (2004) [11] | ${\mathrm{Q}}_{\mathrm{b}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=300{\mathrm{N}}_{\mathrm{t}}$ | ${\mathrm{Q}}_{\mathrm{b}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=8{\mathrm{C}}_{\mathrm{u}}=50.{\mathrm{N}}_{\mathrm{t}}$ |

${\mathrm{Q}}_{\mathrm{s}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})=2{\mathrm{N}}_{\mathrm{s}}$ | ${\mathrm{Q}}_{\mathrm{b}}(\mathrm{k}\mathrm{N}/{\mathrm{m}}^{2})={\mathrm{C}}_{\mathrm{u}}=6.25{\mathrm{N}}_{\mathrm{s}}$ |

Variable | Intercept | D | L_{1} | L_{2} | L_{3} | Z_{p} | Z_{g} | E_{t} | Z_{m} | N_{s} | N_{t} |
---|---|---|---|---|---|---|---|---|---|---|---|

Coefficients | −611.39 | 2.33 | 86.29 | 142.11 | 0 | 0 | −54.05 | 62.09 | 31.85 | 23.16 | −169.43 |

RF Model | MVR | Meyerhof (1976) | Shioi and Fukui (1982) | Decourt (1995) | Shariatmadari (2008) | AIJ (2004) | |
---|---|---|---|---|---|---|---|

R^{2} | 0.866 | 0.702 | 0.467 | 0.485 | 0.334 | 0.391 | 0.611 |

MAE (kN) | 2.924 | 94.267 | 77.830 | 280.727 | 312.297 | 340.668 | 205.721 |

RMSE(kN) | 98.161 | 183.046 | 224.500 | 340.606 | 483.335 | 400.470 | 265.531 |

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## Share and Cite

**MDPI and ACS Style**

Pham, T.A.; Ly, H.-B.; Tran, V.Q.; Giap, L.V.; Vu, H.-L.T.; Duong, H.-A.T.
Prediction of Pile Axial Bearing Capacity Using Artificial Neural Network and Random Forest. *Appl. Sci.* **2020**, *10*, 1871.
https://doi.org/10.3390/app10051871

**AMA Style**

Pham TA, Ly H-B, Tran VQ, Giap LV, Vu H-LT, Duong H-AT.
Prediction of Pile Axial Bearing Capacity Using Artificial Neural Network and Random Forest. *Applied Sciences*. 2020; 10(5):1871.
https://doi.org/10.3390/app10051871

**Chicago/Turabian Style**

Pham, Tuan Anh, Hai-Bang Ly, Van Quan Tran, Loi Van Giap, Huong-Lan Thi Vu, and Hong-Anh Thi Duong.
2020. "Prediction of Pile Axial Bearing Capacity Using Artificial Neural Network and Random Forest" *Applied Sciences* 10, no. 5: 1871.
https://doi.org/10.3390/app10051871