One of the main driving factors for structures’ evaluation is the foundation settlement. Measuring structures’ settlement in field is costly especially when heavy loads are applied. Settlement prediction models can be used to avoid the high cost of settlement field tests. Four advanced heuristic regression methods are developed and applied in this study to estimate raft foundations’ settlement, namely, multivariate adaptive regression splines (MARS), M5 model tree (M5Tree), generalized regression neural networks (GRNN), and support vector regression (SVR) techniques. Simulation of raft pile foundations is utilized to calculate the settlements of piles under the effect of static and dynamic loads. Previous studies are compared with the newly developed models. The results show that the four models can be used to accurately predict foundations’ settlements in the training stage. Also, the results reveal that the MARS and SVR models performed slightly better than the M5Tree and GRNN models in the testing stage and accordingly can be used to predict foundations’ settlement. The SVR model outperformed other models when few numbers of measurements are available.
Piles foundations are usually used to decrease the settlements of heavy loads structures. Monitoring or predicting piles foundations settlement of important structures (e.g., long-span bridges, and skyscrapers) is an essential task to ensure their safety. Understanding the behavior of foundations buried in soil is complex and not defined completely yet [
In the literature, many studies utilized different methods to study the behavior and predict the settlement of foundations subjected to different types of loads and with different foundations shapes [
In the literature, many numerical models are applied for the foundations settlement prediction under static and static-dynamic loads. Shahin [
On the other hand, regression computational methods are widely used to simplify the nonlinear prediction behavior for the engineering characteristics in many different areas [
This study aims, mainly, to investigate the ability of regression models to predict pile-raft foundations’ settlement under coupled static and dynamic loads. Four regression models, MARS, M5Tree, GRNN, and SVR, are developed, assessed, and evaluated using statistical analysis. In addition, the obtained results are discussed and compared with the results of a previous study by Ghorbani and Niavol [
Figure
(a) 9-pile foundation diagram and (b) affected dynamic load [
Input and output data used.
Table
Statistical analysis for the input and output parameters of the simulation model.
Variables | N | d (m) | s/d | l/d | P (kPa) | S (cm) |
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Max | 25.00 | 0.50 | 6.00 | 32.00 | 90.00 | 12.31 |
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Min | 9.00 | 0.30 | 4.00 | 16.00 | 60.00 | 2.83 |
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sd | 6.57 | 0.10 | 0.82 | 5.94 | 15.05 | 2.21 |
In this study, predicting settlement for pile-raft foundations’ models under static and dynamic loads using heuristic methods is developed. Five input parameters (N, d, s/d, l/d, and p) are used. A brief description of each model is presented in the following subsections.
The Multivariate adaptive regression spline (MARS) is a nonlinear nonparametric regression approach introduced by Friedman [
The MARS model for estimating pile-raft foundations’ settlement can be presented as follows:
where,
where d is a penalty for each BF included in the developed submodel, N is the number of training data, and e is the model error. The numerator represents the mean square error of the model in the training phase, penalized by the denominator which accounts for the variance increase if model complexity increases. The GCV penalizes both the number of BFs and the number of knots. More details for this method can be found in [
Quinlan [
M5Tree, splitting the input space into subspaces and the resultant diagram [
where T is the set of examples that reach the node,
Nonlinear prediction models using family of radial basis function neural networks (RBFNN) are used successfully in many engineering applications [
where h is the output of the hidden neurons (
The support vector regression (SVR) is introduced by Gunn [
where
with constraints
Based on Ghorbani and Niavol [
Statistical analysis of the training and testing data samples.
Training | N | d (m) | s/d | l/d | P (kPa) | S (cm) |
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Mean | 16.48 | 0.40 | 5.00 | 23.03 | 74.86 | 6.65 |
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Max | 25.00 | 0.50 | 6.00 | 32.00 | 90.00 | 12.31 |
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Min | 9.00 | 0.30 | 4.00 | 16.00 | 60.00 | 2.83 |
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sd | 5.98 | 0.10 | 0.82 | 5.96 | 15.07 | 2.31 |
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Testing | N | d (m) | s/d | l/d | P (kPa) | S (cm) |
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Mean | 17.22 | 0.40 | 5.00 | 22.92 | 75.41 | 6.72 |
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Max | 25.00 | 0.50 | 6.00 | 32.00 | 90.00 | 11.35 |
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Min | 9.00 | 0.30 | 4.00 | 16.00 | 60.00 | 3.62 |
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sd | 8.11 | 0.10 | 0.82 | 5.94 | 15.20 | 1.91 |
Figure
Models processing.
To evaluate the models, three statistical measures are utilized: coefficient of determination (R2), route mean square error (RMSE), and mean absolute error (MAE):
where
For the MARS model, the number of BFs should be selected first. The mean square error (MSE) and GCM are calculated to estimate the BFs numbers for the training data set. Figure
Basis functions and corresponding equations of MARS model for settlement.
BF | Equation | BF | Equation |
---|---|---|---|
BF1 | | BF8 | |
BF2 | | BF9 | |
BF3 | | BF10 | |
BF4 | | BF11 | |
BF5 | | BF12 | |
BF6 | | BF13 | |
BF7 | | BF14 | |
BFs number and coefficient for the settlement prediction: (a) BFs numbers, (b) BFs coefficients.
To design the M5Tree model, the regression tree and model tree, combined regression with linear regression function at the leaves, are utilized. In this study, the two models of tree are examined. For the model tree, the number of trees should be selected first to improve the nonlinearity performance of the settlement prediction. Figure
Regression tree obtained from M5Tree for settlement modeling.
if x4 <= 22 | else | else | else |
M5Tree model tree selection.
In the GRNN model design, the spread or
The effect of spread constant on GRNN model.
Finally, the design of the SVR model for foundations settlement prediction depends mainly on the
Lagrange multiplier vector for the SVR model.
As per the results presented, the four designed models outperformed the neural network and the genetic algorithm for the Ghorbani and Niavol [
Statistical performance of the designed settlement prediction models.
Model | Training | Testing | ||||
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R2 | RMSE (cm) | MAE (cm) | R2 | RMSE (cm) | MAE (cm) | |
MARS | 0.993 | 0.179 | 0.133 | 0.983 | 0.245 | 0.195 |
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M5Tree | 0.968 | 0.413 | 0.275 | 0.869 | 0.681 | 0.529 |
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GRNN | 0.999 | 4.67e-06 | 2.89e-06 | 0.962 | 0.366 | 0.316 |
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SVR | 0.995 | 0.096 | 0.095 | 0.994 | 0.148 | 0.131 |
Measured and predicted settlement for the (a) MARS, (b) M5Tree, (c) GRNN, and (d) SVR models.
The performance of the MARS model is almost constant during both the training and testing stages. The slopes of the linear regression for the training and testing cases are found to be 0.99, and 0.94, respectively. In addition, the R2, RMSE, and MAE are 0.01, 0.066 cm, and 0.062 cm, respectively (Table
Finally, the results of the developed models are compared with Ghorbani and Niavol [
Figure
Measured and predicted values for the SVR and MARS models for the (a) training and (b) testing sets.
This study investigates the use of soft computing models based on heuristic regression methods to predict the settlements of pile-raft foundations. Four models, namely, MARS, M5Tree, GRNN, and SVR, are introduced, designed, evaluated, and compared with Ghorbani and Niavol [
The heuristic regression tools can be used effectively for predicting the settlements of pile-raft foundations under static and dynamic loads. The evaluation of the four models in the training stage shows that the GRNN model outperformed the other models, with the M5Tree model performing the worst. Comparing the results of the four models with previous study shows that the four models can be used to predict the foundations settlement.
The results in the testing stage show a change in the performance of the four designed models. The MARS and SVR models performed better than the GRNN model. Moreover, the SVR model outperformed all the other models and can be used to accurately predict foundations settlement. The results, also, show that the SVR performance is high when low numbers of data sets are used while it is recommended to use the MARS model with higher number of data sets.
The settlement data presented and used to support the findings of this study have been included within the article in Figures
The authors declare that they have no conflicts of interest.
This work was supported by Post-Doctor Research Program in 2018 through the Incheon National University (INU), Incheon, Republic of Korea. Also, this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2017R1A2B2010120).