1. Introduction
In the realm of geotechnical engineering, the consideration of ultimate bearing capacity (UBC) and allowable settlement continues to be of paramount importance when designing shallow foundations [
1,
2,
3]. These aspects are crucial factors that dictate the structural integrity and performance of any construction project. UBC, a fundamental parameter, is deeply intertwined with the shear strength characteristics of the underlying soil, and its accurate estimation is essential for ensuring the stability of foundations. Theoretical formulations proposed by eminent geotechnical scholars such as Terzaghi [
4], Meyerhof [
5], Hansen [
6], and Vesic [
7] have provided a foundation for calculating UBC. However, the application of these theories has revealed that various UBC equations exhibit significant disparities, primarily due to the inherent simplifications and assumptions employed in these approaches [
8].
The pursuit of accurate UBC prediction has led to a proliferation of research studies that explore numerical and semi-empirical solutions. The works of Conte et al. [
9] and Achmus and Thieken [
10] stand as examples of such efforts, where researchers have sought to elucidate the complex bearing capacity phenomenon. These models are frequently evaluated through meticulously designed footing tests carried out on scale models. Yet, bridging the gap between laboratory-scale experiments and real-world, full-scale foundation scenarios remains a challenge. Scholars, cognizant of the potential scale effects, have undertaken initiatives to minimize discrepancies when extrapolating findings from controlled laboratory conditions to the complexities of real-world projects. This endeavor to account for scale effects, as seen in the works of de Beer [
11] and Yamaguchi et al. [
12], underscores the dedication within the geotechnical community to ensuring the accuracy and reliability of their predictions.
In a pursuit to unravel the intricate facets of bearing capacity, researchers have embarked on investigations that dissect the influence of factors such as particle size on UBC. The study carried out by Tatsuoka et al. [
13] exemplifies this endeavor, as it unravels the nuances of soil behavior during model-scale footing experiments. Interestingly, the results drawn from large-scale tests conducted on compacted sand have unveiled pronounced variations in shearing strains along the slip line. These findings have challenged conventional formulas that solely rely on the concept of maximum friction angle (ϕ
max) [
14]. This revelation underscores the critical need for a nuanced understanding of the underlying mechanics and limitations of various theoretical equations, ultimately driving the demand for innovative approaches to enhance the accuracy of UBC predictions in real-world scenarios.
Enter artificial intelligence (AI), a groundbreaking paradigm that has reshaped various scientific disciplines. The realm of geotechnical engineering, while somewhat reserved in its adoption of AI, has begun to witness the application of AI-based techniques in addressing complex challenges. AI methods such as artificial neural networks (ANNs), fuzzy inference systems (FISs), adaptive neuro-fuzzy inference systems (ANFISs), and others have shown remarkable potential in deciphering intricate relationships within complex datasets across diverse domains such as soil dynamics [
15,
16,
17,
18,
19,
20], deep foundations [
21,
22,
23,
24], soil cracking [
25,
26,
27], recycled materials [
28,
29,
30,
31,
32,
33,
34,
35,
36], soil mechanics [
37,
38], tunnelling and rock mechanics [
39,
40,
41] and other fields [
42,
43,
44,
45,
46,
47,
48,
49,
50,
51]. The beauty of these techniques lies in their capacity to capture nonlinear interactions between a myriad of variables, even when the underlying relationships are not fully understood. Recent forays into AI applications in geotechnical engineering have showcased the supremacy of AI over traditional analytical formulas when addressing UBC in shallow foundations [
14,
52,
53,
54,
55,
56]. This shift in approach offers a promising avenue for ushering in a new era of accurate and reliable UBC predictions.
Yet, despite the remarkable strides made by AI in various fields [
57,
58,
59], its penetration into the geotechnical engineering sphere remains somewhat limited, as evidenced by the existing literature. This research endeavor boldly pioneers the utilization of grey box AI methodologies and a hybrid AI approach to predict UBC in the context of shallow foundations on cohesionless soils.
This ambitious research journey encompasses a comprehensive ensemble of four distinct mathematical models, each contributing a unique facet to the overarching goal. The multiple linear regression (MLR) model forms the foundation as a statistical framework, shedding light on the significance of individual parameters. Concurrently, the integration of grey box AI models such as genetic programming (GP) and classification and regression tree (CART) methods introduces a novel dimension to UBC prediction, bridging the gap between traditional analytical approaches and cutting-edge AI methodologies. Then, the Genetic Algorithm-Emotional Neural Network (GA-ENN) method, a hybrid AI approach that harnesses the strengths of genetic algorithms and neural networks, was employed. This amalgamation of diverse approaches is harnessed for the very first time to unravel the enigma of predicting the ultimate bearing capacity of cohesionless soils beneath shallow foundations.
The culmination of this research is not merely a prediction tool; it represents a paradigm shift that opens new vistas for accurate and reliable foundation design. This shift comes with a rigorous evaluation of the significance of input parameters, accompanied by a comprehensive sensitivity analysis to ensure the robustness and applicability of the predictive models in real-world scenarios. As we peer into the future, the impact of this pioneering effort is poised to extend beyond the boundaries of geotechnical engineering, ushering in a new era of precision and innovation across the construction industry.
2. Database Collection and Processing
The collected database contains a total of 97 datasets comprising results from tests on square, rectangular, and strip footings of various sizes conducted in sand beds with different unit weights (dry unit weight, γd or effective unit weight, γ′). To improve the model’s performance, the data are evenly distributed, ensuring an equal number of samples for large and small-sized footings.
Out of the 97 datasets [
14,
60], 47 are derived from load tests on large-scale footings from different references [
14,
60,
61,
62]. The remaining 50 datasets are from load tests on smaller-sized model footings, as reported by Gandhi [
63].
For the large-scale tests conducted at the DEGEBO test area in Berlin, submerged unit weights (γ
d or γ′) are utilized due to the submerged conditions during testing [
14,
60]. The angle of shearing resistance reported by the respective authors is adopted for the analysis, despite some differences in the mobilized angle of shearing resistance at failure between axisymmetric and plain strain conditions. Nevertheless, these differences do not exceed 10% [
14,
60].
In the case of laboratory model tests, the internal friction angles used are obtained from direct shear tests conducted at very low normal stresses, which also includes the effect of dilation.
Regarding the determination of ultimate load, for large-scale footings, it is defined as the load corresponding to the point where the slope of the load settlement curve is at its minimum. For smaller-sized model footings, it is defined as the load corresponding to the point of break of the load settlement curve in a log–log plot.
Figure 1 shows the distribution of the database.
Figure 2 illustrates the influence of the foundation depth on the foundation load. From these diagrams, it can be observed that the bearing capacity of the shallow foundation increases with the increasing foundation depth.
Prior to utilizing the existing database, a thorough check for inappropriate data was conducted. To accomplish this, a box plot for the output parameter was generated, and
Figure 3a displays the results. Following the examination, 22 datasets were identified as inappropriate and subsequently removed from the overall database. The refined database consists of 75 suitable datasets, and their corresponding box plot is depicted in
Figure 3b.
Table 1 presents the distribution of the utilized database after the exclusion of outliers. The data reveal valuable insights into the statistical characteristics of each variable. The ultimate bearing capacity of the foundation, represented by q
u, varies significantly across the datasets, ranging from 58.500 kPa to 464.000 kPa, with a mean value of 211.274 kPa and a standard deviation of 108.959 kPa. On the other hand, the foundation width (B) displays a relatively tighter distribution, ranging from 0.059 m to 0.600 m, with a mean of 0.250 m and a standard deviation of 0.195 m. Similarly, the foundation depth (D) exhibits a narrow range, varying from 0.000 m to 0.500 m, with a mean value of 0.098 m and a standard deviation of 0.106 m. The aspect ratio (L/B), calculated as the ratio of foundation length to width, shows wider variability, ranging from 1.000 to 6.000, with a mean of 3.409 and a standard deviation of 2.269. The unit weight of the soil (γ
d or γ′) displays variations across the datasets, ranging from 9.850 kN/m
3 to 17.100 kN/m
3, with a mean value of 14.615 kN/m
3 and a standard deviation of 2.625 kN/m
3. Similarly, the internal friction angle of the soil (ϕ) exhibits moderate variability, ranging from 34.000° to 42.500°, with a mean value of 38.363° and a standard deviation of 2.691°. The information provided in
Table 1 serves as a basis for further analysis, offering crucial insights into the key factors influencing surface foundation load behavior and supporting improved foundation design and performance assessment. Width of footing (B), depth of footing (D), footing geometry (L/B), unit weight of sand (γ
d or γ′), and internal friction angle (ϕ). Furthermore,
Figure 4 shows histograms for the output (q
u).
5. Discussion
5.1. Comparison of Different Proposed Models
Table 11 provides a comprehensive comparison of the predictive performance of different models for estimating the ultimate bearing capacity of shallow foundations. The evaluation is conducted on both the training and testing databases, and various performance metrics are utilized to assess the accuracy and reliability of each model.
For the training database, the Mean Absolute Error (MAE) values range from 33.938 for the Multiple Linear Regression (MLR) model to an impressive 6.634 for the Genetic Algorithm-Emotional Neural Network (GA-ENN) model. This indicates that the GA-ENN model exhibits the lowest average absolute deviation between its predictions and the actual values, showcasing its superior accuracy in predicting the bearing capacity of shallow foundations during training. The Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) also demonstrate the same trend, where the GA-ENN model outperforms all other models, yielding the smallest errors.
Similarly, in the testing dataset, the GA-ENN model attains the lowest MAE (6.018), MSE (65.397), and RMSE (8.087) values, indicating its remarkable predictive accuracy on unseen data. Additionally, the GA-ENN model shows the lowest Mean Squared Logarithmic Error (MSLE) and Root Mean Squared Logarithmic Error (RMSLE) values, indicating its ability to accurately predict both small and large errors on the testing data.
Furthermore, the coefficient of determination (R2) values for the GA-ENN model are the highest in both the training (0.994) and the testing (0.995) datasets, illustrating the model’s excellent fit to the data and its capability to explain a significant proportion of the variance in the target variable.
In
Figure 10, the outcomes of various statistical and AI models for predicting the bearing capacity of shallow foundations are displayed. Among these models, it is evident that the Genetic Programming (GP), Classification and Regression Trees (CART), and Genetic Algorithm-Emotional Neural Network (GA-ENN) models have exhibited the most promising performance.
5.2. Comparison with Previous Studies
To delve deeper into the investigation,
Table 12 displays the outcomes of this study alongside numerical models from the historical record, as well as the Gaussian Process Regression Approach introduced by Ahmad et al. [
60]. To facilitate a more comprehensive comparison, an additional parameter referred to as Nash–Sutcliffe Efficiency (NSE), as defined in Equation (9), was employed alongside the
R2. The findings clearly highlight the superiority of the suggested models in contrast to the pre-existing historical models.
In the equations, T is the number of data points, and and are the actual and predicted output of ith sample of the data, respectively; is the averaged actual output of the data.
5.3. Sensitivity Analysis
In the domain of artificial intelligence methodologies, the evaluation of the importance of input parameters holds a pivotal role. This assessment follows a structured procedure: each individual input parameter is deliberately adjusted both upwards and downwards by 100%. Subsequently, the resulting discrepancies within the models are meticulously monitored. This thorough analysis functions as a mechanism for assessing the sensitivity of each model to specific parameters. Elevated error values indicate heightened model sensitivity to those distinct parameters, while lower error values suggest that the scrutinized parameter has a relatively modest impact on the overall model efficacy. This approach enables us to precisely identify the input parameters that wield substantial influence over the model’s outcomes, facilitating the refinement of the model for superior performance.
To visually illustrate this analysis of parameter significance across diverse models for predicting q
u,
Figure 11 is presented. These visuals provide valuable insights into the models’ responses to variations in various input parameters and aid in the identification of pivotal factors that shape the predictive proficiency of the models for each property.
Table 13 presents a hierarchy of input parameter importance across various models employed for the anticipation of qu. Within this ranking, a rank of 1 corresponds to the utmost importance, signifying the paramount parameter, while a rank of 5 designates the least significance.
Analyzing the outcomes of
Table 13, it becomes evident that the parameters foundation depth, internal friction angle of soil and foundation width emerge as the most pivotal elements in predicting ultimate bearing capacity of shallow foundations (q
u). These parameters are instrumental in determining the foundation’s ability to withstand load and stress distribution. Conversely, the parameters unit weight of soil (γ
d or γ′) and L/B exhibit minimal relevance in this predictive context. The unit weight of soil (γ
d or γ′) primarily influences the overall density of the foundation materials, which might not be as critical a factor in the prediction of ultimate bearing capacity as compared to parameters directly related to the geometry and material properties of the foundation. Similarly, the parameter L/B, representing the ratio of length to width, showcases minimal importance in this prediction. This could be attributed to the fact that other parameters like foundation depth, internal friction angle, and foundation width have more direct and substantial impacts on the foundation’s load-bearing capability.
5.4. Implications and Applications
This study presents significant implications for the field of geotechnical engineering by exploring the potential of soft computing techniques in predicting the ultimate bearing capacity (UBC) of shallow foundations on cohesionless soils. The inclusion of multiple linear regression (MLR), Genetic Programming (GP), Classification and Regression Trees (CART), and GA-ENN (Genetic Algorithm-Emotional Neuron Network) models in this research represents a novel approach, as they are used for the first time as grey box AI models in the literature for UBC prediction. The application of these advanced AI models expands the range of predictive tools available to engineers and practitioners, offering diverse options for accurate and reliable UBC estimation.
The use of input parameters such as the width of footing (B), depth of footing (D), footing geometry (L/B), unit weight of sand (γd or γ′), and internal friction angle (f) further enhances the versatility and practicality of the developed AI models. Engineers can leverage these models to perform UBC predictions for a wide range of shallow foundation designs on cohesionless soils, optimizing foundation designs and ensuring structural stability. The comparison of the AI models with existing theoretical approaches demonstrates their superiority in UBC prediction, underscoring their potential to outperform traditional methods. Consequently, the adoption of AI models like GP, CART, and GA-ENN can lead to more accurate and efficient foundation design practices, resulting in cost-effective and safer construction projects. Moreover, the sensitivity analysis conducted in this study empowers engineers to gain a deeper understanding of the influence of individual input parameters on UBC predictions, facilitating the identification of critical factors that impact the foundation’s load-bearing capacity. Overall, the successful integration of AI models in UBC prediction for shallow foundations on cohesionless soil contributes to the advancement of geotechnical engineering practices and promotes more reliable and informed decision making during the design and construction phases.
5.5. Geotechnical Challenges and Remediation Strategies
The prediction and assessment of the ultimate bearing capacity (UBC) for shallow foundations on cohesionless soils pose several geotechnical challenges that need to be addressed for reliable and safe foundation design. One of the primary challenges is the complex and heterogeneous nature of cohesionless soils, which can exhibit significant spatial variability in properties such as grain size distribution, void ratio, and shear strength. This variability can lead to unpredictable bearing capacity behavior, making it challenging to accurately estimate UBC using traditional methods. Moreover, the presence of soft spots, loose zones, or weak layers within the soil profile can further complicate the foundation response and load-carrying capacity. Dealing with these uncertainties and spatial variations is crucial to ensure the integrity and stability of shallow foundations.
To overcome these geotechnical challenges, the implementation of advanced soft computing techniques, such as Genetic Programming (GP), Classification and Regression Trees (CART), and Genetic Algorithm-Emotional Neuron Network (GA-ENN), as demonstrated in this study, offers promising remediation strategies. These AI models possess the capability to learn and adapt to complex patterns and non-linear relationships within the data, enabling more accurate and robust predictions of UBC for shallow foundations on cohesionless soils. By utilizing these AI models, engineers can account for the spatial variability and uncertainties in soil properties, leading to more reliable and safer foundation designs.
Another significant geotechnical challenge is the consideration of various influential parameters, such as the width of footing (B), depth of footing (D), footing geometry (L/B), unit weight of sand (γd or γ′), and internal friction angle (f), when estimating UBC. The interplay of these parameters requires a comprehensive understanding of their individual and combined effects on the foundation’s load-bearing capacity. Soft computing techniques, as used in this study, facilitate sensitivity analysis to assess the impact of each input parameter on the UBC prediction, aiding engineers in identifying critical factors and optimizing foundation design.
In addition to AI-based approaches, thorough site investigations, including detailed soil exploration and testing, play a vital role in addressing geotechnical challenges. The collection of high-quality data enables engineers to characterize the soil properties accurately and capture the spatial variations, providing a solid foundation for the AI models’ training and validation.
5.6. Limitations and Future Research
Despite the promising results and contributions of this study in predicting the ultimate bearing capacity (UBC) of shallow foundations on cohesionless soils using advanced soft computing techniques, there are certain limitations that should be acknowledged. One limitation is the reliance on historical data for model development and validation. The accuracy and performance of the AI models heavily depend on the quality and representativeness of the available data. Therefore, future research should focus on expanding the dataset and including data from a broader range of soil conditions and foundation geometries to enhance the models’ generalization and applicability to diverse engineering scenarios.
Another limitation lies in the assumption of certain soil behaviors and conditions. The AI models developed in this study may not capture certain complex geotechnical phenomena, such as soil liquefaction or soil–structure interaction effects, which can significantly influence the bearing capacity of shallow foundations. Incorporating additional factors and advanced modeling techniques to address these complexities can further improve the accuracy and reliability of UBC predictions in more challenging geotechnical conditions.
Furthermore, the present study only considers cohesionless soils, while, in practice, engineers encounter various types of soils with different characteristics. Extending the AI models to account for cohesive soils and investigating their performance in mixed soil conditions can enrich the predictive capabilities of the models and enable a more comprehensive assessment of foundation bearing capacity.
Additionally, it is essential to recognize that the predictive accuracy of the AI models is subject to the quality and availability of input parameters. In practice, obtaining precise measurements of certain parameters, such as internal friction angle, might be challenging due to site-specific constraints. Integrating advanced geophysical and non-destructive testing techniques to estimate soil properties can contribute to more accurate and reliable data inputs for the AI models.
Moreover, while the sensitivity analysis conducted in this study provides valuable insights into the influence of individual input parameters, exploring the effects of other potential factors, such as groundwater table variations, temperature, and time-dependent soil behavior, can further enhance the models’ predictive capabilities and broaden their applications.