**1. Introduction**

Pile foundations are structural elements that are mainly used when the surface soil is weak and there is an urgent need to transfer the structural load to the further layers of the soil, or when soil settlement is an essential concern in the designing process. In terms of the pile's role in load transmission, calculating the precise ultimate bearing capacity of pile foundations is an important topic for geotechnical engineers. Besides this, some scholars have indicated that pile bearing capacity can be considered as a time-dependent parameter, exhibiting an increasing trend after a specific period [1–3]. Pile setup is a geotechnical phenomenon referring to a time-dependent increase in the ultimate bearing capacity of pile foundations. It is assumed that pile setup occurs due to the dissipation of the excess pore water pressure (EPWP) generated as a result of pile installation [4].

Furthermore, it is widely accepted that this phenomenon develops by incorporating three main stages, including the non-uniform dissipation of EPWP, the uniform dissipation of EPWP, and aging [5]. Results of different studies indicate that setup considerably affects the side resistance, while when it comes to the tip resistance, it has exhibited less change or a decrease owing to relaxation [1,6–10]. Predicting the time-dependent bearing capacity of pile foundations has always been an interesting topic for researchers. Moreover, considering the pile setup, the design process of piles can be more economical.

Many studies have been presented in which analytical or numerical models were developed to forecast the pile setup [11–13]. One of the most well-known investigations

**Citation:** Khanmohammadi, M.; Armaghani, D.J.; Sabri Sabri, M.M. Prediction and Optimization of Pile Bearing Capacity Considering Effects of Time. *Mathematics* **2022**, *10*, 3563. https://doi.org/10.3390/ math10193563

Academic Editors: Camelia Petrescu and Valeriu David

Received: 16 July 2022 Accepted: 13 September 2022 Published: 29 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

in this area is a study conducted by Skov and Denver [14] to find an equation to estimate the pile setup. The setup equation was revised using different geotechnical properties to achieve this goal. Finally, a semi-empirical equation was proposed by them introducing a practical variable called setup parameter (A). This pioneering study was a starting point for other researchers. For example, Haque and Abu-Farsakh [6] published a paper in which the application of a nonlinear multivariable regression model in the prediction of pile setup was investigated. Although the studies conducted using this group of techniques were able to create an effective equation, pile setup is a complex issue considering the complicated soil–pile interaction. Therefore, analytical methods and regression analysis do not seem to be powerful enough for prediction purposes [15].

In recent years, several studies have presented the successful usage of intelligent algorithms to simulate complex problems in civil and geotechnical engineering [16–29]. Several scholars have highlighted the applicability of these techniques in predicting pilerelated issues, e.g., pile capacity, settlement, lateral deflection [30–33]. In a study conducted by Lee and Lee [34], the application of artificial neural networks (ANNs) in the prediction of pile bearing capacity was investigated. The results of the model and in situ pile load tests were utilized to verify the developed model. Finally, it was concluded that the error back-propagation neural network used in this study had good performance since the maximum error in the prediction process did not exceed 25%. Shahin [35] utilized intelligent computing to model the axial capacity of pile foundations. For this purpose, an ANN technique was employed to predict the axial capacity of driven piles and drilled shafts using a total of 174 data points. Furthermore, a comparison was made between CPT-based methods and the ANN to evaluate their performances in the prediction area. The results indicated that ANN with a correlation coefficient of 0.85 and 0.97 for driven and drilled shaft validation datasets showed acceptable performance. Samui [36] investigated the application of the support vector machine as a powerful machine learning technique to estimate the pile bearing capacity. Three inputs, including penetration depth ratio, mean normal stress, and the number of bowls, were considered for this aim. Eventually, using evaluation criteria such as coefficient of correlation, the developed model predicted the pile bearing capacity with sufficient accuracy. In another study, Momeni et al. [37] used the results from 50 dynamic load tests to predict the bearing capacity of piles using an ANN-based predictive model optimized with a genetic algorithm. The final data indicated that the developed model, with a correlation coefficient of 0.99, successfully predicted the target very close to its actual value.

Other studies tried to improve the performance of the base intelligent models using optimization algorithms. For instance, Dehghanbanadaki et al. [38] used the gray wolf optimization (GWO) algorithm to enhance the performance of the adaptive neuro-fuzzy inference system (ANFIS) for estimating the ultimate bearing capacity of single driven piles. The results showed that the actual values of pile bearing capacity had been successfully estimated using the GWO-ANFIS model, and their results improved upon the ANFIS model. In another study implemented by Armaghani et al. [33], a combination of ANFIS and group data handling methods optimized with a competitive imperialism algorithm (ICA) was utilized to forecast the pile bearing capacity. Based on the data and the evaluation criteria, the proposed model could be considered a powerful technique regarding pile foundations' design process.

Previous works did not include a time component in their input parameters, and their input parameters were mostly pile geometry-related. However, this study includes a separate input directly related to time, which is the main difference between this study and those published previously. Another contribution in this study is related to the optimization phase. An intelligent equation has been developed to predict pile capacity using the genetic programming (GP) technique. Then, the proposed GP equation is used in two optimization techniques, namely artificial bee colony (ABC) and GWO to maximize pile capacity. A database containing information about 256 data samples has been considered to achieve these goals. The models mentioned above and their results are discussed and compared to introduce a new procedure for predicting pile capacity.

The rest of this paper is organized as follows:

Section 2 describes the methodology background of the used models in predicting and optimizing pile capacity. Section 3 gives the needed information regarding the database used for modeling. Section 4 discusses the process of prediction models to develop a GP model and its evaluation. The optimization process regarding two algorithms, i.e., ABC and GWO, is given in Section 5. Section 6 discusses both the prediction and optimization phases. Sections 7 and 8 describe limitations, future works, and concluding remarks of this study.
