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Scientific Reports volume 15, Article number: 38195 (2025) Cite this article
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Several studies have extensively explored the axial (uplift and compressive) behavior of helical piles, while lateral behavior—an equally critical aspect in the design of foundation systems— remains relatively uncharted. In this study, key geometric features with significant influence on lateral behavior were explored through machine learning to derive optimization insights. The lateral response of the piles was modelled using the modified p-y springs method, with installation effects accounted for through empirically adjusted geotechnical parameters from field measurements. The results indicate that the optimal helix-to-pile diameter ratio (dH/dP) is dependent on the helix-to-pile embedment ratio (zH/zP) and is strongly influenced by the applied-to-failure load ratio (aL/fL) of each unique configuration. Response surface analyses further reveal that the contribution of the helix to lateral resistance diminishes beyond a zH/zP ratio of 0.2 and becomes negligible past 0.6 — at which point the pile behaves similarly to a conventional shaft without helical reinforcement. The geometric features identified at Test Site I were then applied to guide the pile configurations for Test Sites II and III, with lateral behavior validated through comparison between simulated profiles and strain gauge-derived field measurements, using site-specific geotechnical data. Notably, the derived geometric features demonstrated a degree of independence from localized soil conditions, suggesting their potential applicability across a wide range of ground profiles. Overall, the findings of this study offer both a practical and comprehensive framework for the design of laterally loaded helical piles, enabling practitioners to achieve an optimized balance between geo-structural performance and material efficiency.
Helical piles (see Fig. 1) are considered one of the rising alternatives to concrete pads and deep foundations for various small to mid-scale infrastructure projects. Traditional concrete spread foundations are susceptible to erosion and differential settlement, particularly in loose silty sand soils during heavy rainfall. Moreover, the demolition of concrete structures generates substantial waste, contributing to environmental challenges when disposed in landfills. In contrast, helical piles present a more sustainable and environmentally friendly solution. They can be installed and removed quickly, and their recyclability allows them to be repurposed for future projects, reducing waste and environmental impact.
Distinction between traditional and helical pile foundation. (a) concrete spread, (b) helical pile.
Helical piles offer several advantages, including increased axial compression and uplift capacity^1,2,3, ease of installation and eventual decommissioning, and the ability to mitigate environmental concerns, such as noise during driving⁴. Notably, their performance in soft and liquefiable soils, where differential settlement is a concern, has been highlighted by⁵. These attributes make helical piles particularly well-suited for sustainable development initiatives, especially for renewable energy infrastructures. Such structures are often installed in open fields with minimal obstructions, leaving them vulnerable to extreme lateral forces and overturning moments caused by variable wind loads.
Schematic illustration of solar farm testing sites at Saemangeum, Republic of Korea.
This study encompasses three test sites with distinct soil profiles located in Saemangeum, Republic of Korea, as illustrated in Fig. 2. Test Site I, characterized predominantly by silty sand (SM), serves as the basis for a machine learning model to derive insights into the factors that influence the optimal configuration of a laterally loaded helical pile. Test Site II, comprising a silty sand (SM) layer overlying soft clay (CL), and Test Site III, composed primarily of soft clay (CL), are used to implement the insights obtained from Test Site I by comparing simulated responses using site-specific soil properties and field-measured data.
Research on the lateral capacity of helical piles has progressed significantly over the years, with pioneering and modern studies contributing to the understanding of their behavior under lateral loading. Early investigators, such as⁶, developed a mathematical model that provided reliable predictions of lateral capacity for helical piles with multiple helix plate configurations. Reference⁷ demonstrated that the lateral capacity of helical piles is approximately 20–50% higher than that of conventional straight-shaft piles, with capacity increasing in relation to soil shear strength and embedment depth. Reference⁸ explored the role of structural stiffness, emphasizing its impact on the lateral performance of helical piles and concluded that shaft resistance predominantly governs the lateral behavior of helical piles. Reference⁹ conducted experiments on tapered central shafts with single and double helix configurations, reporting an 8–42% improvement in lateral resistance depending on embedment depth. Reference¹⁰ developed a mathematical model and empirical equations to predict the ultimate lateral capacity of helical piles. Reference¹¹ performed full-scale tests comparing plain helical piles with grout-reinforced piles, finding that grout reinforcement significantly enhanced lateral resistance by increasing shaft stiffness. Reference¹² conducted laboratory-scale physical modeling and concluded that the inclusion of helices generally improves lateral capacity by mobilizing additional soil resistance. These studies collectively highlight the critical influence of pile geometry, soil properties, and structural modifications on the lateral performance of helical piles, forming a strong basis for advancing their application in modern geotechnical engineering practices.
Recent advancements in helical pile research have increasingly incorporated computational approaches, such as the finite difference and finite element methods, to model lateral behavior, often validated through field test data. For instance¹³, conducted full-scale lateral load tests on large-diameter helical piles and analyzed their behavior using a p-y curve model, demonstrating that helical piles can develop substantial lateral resistance, primarily influenced by shaft diameter. Reference¹⁴ performed a small-scale parametric study on the lateral capacity of helical piles, identifying embedment ratio and soil properties as key factors affecting lateral capacity. Reference¹⁵ extended this line of inquiry by investigating the effects of helix pitch and diameter through small-scale model tests. Their findings revealed that lateral capacity increases with larger helix diameters but decreases as the helix pitch increases. These studies underscore the importance of both geometric parameters and soil-structure interactions in governing the lateral performance of helical piles, while highlighting the role of advanced computational methods in enhancing predictive accuracy and design efficiency.
The application of supervised machine learning (SML) has garnered significant attention in recent years, emerging as a promising alternative modeling approach in geotechnical engineering, particularly for predicting the performance and design optimization of helical piles. Numerous studies have explored various SML techniques to address primarily the axial bearing and pullout resistance, settlement profile and installation torques, leading to the development of optimized design strategies for helical piles. This literature review consolidates findings from recent studies to evaluate the feasibility and efficiency of SML-based models in addressing key behaviors of helical piles.
References^16,17 demonstrated the effectiveness of combining artificial neural networks (ANN) with optimization techniques to improve the prediction of helical pile pullout resistance. Reference¹⁶ utilized an ANN model optimized with the Imperialist Competitive Algorithm (ICA) and found improved accuracy using 36 experimental observations. Similarly¹⁷, employed an ANN model optimized with Particle Swarm Optimization (PSO) and reported significantly better performance compared to conventional ANN models. Both studies highlight the importance of integrating optimization methods to enhance predictive accuracy and better capture nonlinear soil-pile interactions under pullout loads.
Reference¹⁸ developed a model integrating Gradient Boosting Decision Trees (GBDT) with Particle Swarm Optimization (PSO) to predict the uplift behavior of helical piles in dense sand. Data from centrifuge tests were utilized to train and validate the model. The study identified the embedment ratio as the most critical variable influencing uplift resistance. The GBDT-PSO hybrid model achieved high predictive accuracy, demonstrating the potential of gradient boosting methods in geotechnical engineering applications, particularly for renewable energy projects.
Reference¹⁹ employed metaheuristic algorithms—Grey Wolf Optimization (GWO), Differential Evolution (DE), and Ant Colony Optimization (ACO)— to tune an adaptive neuro-fuzzy inference system (ANFIS) for predicting helical pile pullout resistance. While GWO-ANFIS excelled during training, DE-ANFIS and ACO-ANFIS produced smaller errors in testing. The study concluded that metaheuristic-enabled models provide robust and accurate predictions, particularly for varying soil conditions, and hold potential for real-world geotechnical applications.
Reference²⁰ explored various SML models to predict the installation torque of helical piles using data from 707 installation reports, including Standard Penetration Test (SPT) results. Among the eight evaluated SML algorithms (ANN-Artificial Neural Networks, DT-Decision Trees, KNN-k Nearest Neighbors, RF-Random Forests, SVM-Support Vector Machines, and Cubist), Cubist outperformed others in accuracy and reliability, whereas DT was the least effective. As Cubist is a model tree algorithm that extends traditional DT, it offers a better representation of the nonlinear relationships in the dataset. Their study demonstrated the potential of SML in leveraging routine geotechnical data, like SPT results and in-situ reports, for efficient pile design and analysis.
Reference²¹ investigated the use of SML models to predict the settlement of large-diameter helical piles in c-ϕ soils. They compared four SML models: Decision Trees (DT), Random Forest (RF), AdaBoost, and Artificial Neural Networks (ANN). Supplementing limited field test data with numerical simulations from finite element models (FEM), the authors evaluated various soil conditions and pile geometries. Their findings demonstrated the practicality of SML in designing large diameter helical piles for axial compressive loads, highlighting ANN as an efficient tool for this purpose. Their study underscores the potential of combining field and numerical data for SML model training in settlement prediction.
Reference²² addressed the compression capacity of screw piles in clean sand by training SML models on data derived from 1667 finite element models (FEM). They evaluated several SML algorithms, including linear regression (LR), artificial neural networks (ANN), Support Vector Machines (SVM), and Gaussian Process Regression (GPR). The GPR model achieved the best performance, offering excellent predictions of axial compression capacity. Their findings underscore the advantages of combing FEM simulations with SML algorithms to overcome limitations of empirical design methods.
To date, no studies on the optimization of helical piles for lateral design have been identified in the existing body of literature. While numerous investigations have extensively explored the axial behavior of helical piles, lateral behavior—an equally critical aspect in the design of foundation systems— remains largely uncharted. This research aims to address this gap by presenting a novel approach to understanding and optimizing the lateral behavior profile of helical piles, specifically through the application of supervised machine learning (SML) techniques. Building upon the work of²³, which modeled the nonlinear lateral behavior of helical piles by introducing the novel approach of modified p-y springs, this study advances the methodology by incorporating optimization strategies. This study also provides new insights into the mechanics of helical pile-soil interaction and highlights their practical implications for improved foundation design.
The project area is situated near the Saemangeum industrial complex in South Korea. The soil’s mechanical properties were assessed using standard in-situ tests, including the Standard Penetration Test (SPT), Cone Penetration Test (CPT), and Swedish Weight Sounding Test (SWST). Additionally, routine laboratory analyses were performed on both disturbed and partially disturbed soil samples to supplement the field data.
Distribution of collected soil data from across three test sites.
At Test Site I, the subsurface profile consists of a loose, saturated silty sand layer extending to a depth of 3.8 m, underlain by a thick deposit of loose to very loose saturated silty clay. Test Site II features a silty sand layer from the surface to a depth of 1.5 m, transitioning into predominantly soft clay strata. Meanwhile, Test Site III is characterized by a soft clay layer of low to medium plasticity extending to 5.8 m, underlain by interbedded thin layers of silty sand and low-plasticity silt. The distribution of field test results across the three sites is presented in Fig. 3. A summary of the soil properties adopted for the analyses is presented in Tables 1, 2 and 3, corresponding to each respective test site. The rationale behind the selection of these parameters — particularly with regard to installation effects — will be elaborated in the following sections.
As the helical pile undergoes lateral displacement, the faces of the helix plate oriented relative to the loading direction compress against the surrounding soil, contributing to resistance against movement. It is hypothesized that the equilibrium of forces acting on the helix plates provides additional pile stiffness, significantly reducing the lateral deflection of a helical pile compared to a conventional pile. To determine whether the helix mobilizes soil resistance or merely slices through the soil stratum, an instrumented helical pile equipped with strain gauges on the first helix plate was tested and the raw strain readings are presented in the Supplementary Information (SI) section.
An idea was conceptualized wherein the helix plate is envisioned as a series of finite cantilevered beams, with their fixed ends attached to the central pile shaft (see Fig. 4). It was hypothesized that the soil resistance acting on the helix plate behaves as a linearly distributed force, reaching its maximum value near the fixed end (attached to the pile shaft) and gradually decreasing to zero at the free end. To validate these assumptions, the authors explored a novel approach by combining one-dimensional springs with repurposed p-multipliers to model the lateral behavior, specifically tailored to helical piles.
Concept of modified p-y springs method: (a) Theoretical concept of soil resistance distribution, (b) p-y spring distribution between a regular shafted pile vs. a helical pile shaft.
One-dimensional springs used in lateral analysis are called p-y springs which model the nonlinear relationship of the soil resistance (p) and the lateral displacement of the pile (y) that increases in value as a function of depth (z) for a single uniform layer soil. They are commonly derived from site test data or finite element calibration procedures that are specific to a particular pile-soil configuration²⁴. The concept of p-multipliers is conventionally used as strength adjustment factors for laterally loaded pile groups as mentioned in studies such as^25,26,27. The p-multipliers applied to this study serve as reinforcement factors to account for the stiffness provided by the helix plates. Together, the modified p-y springs method is formed.
A zone of influence is established to define the boundary within which the modified p-y springs are affected by the increased stiffness due to the presence of the helical plate. Finite element studies^3,28,29,30, and³¹ on the failure mechanism of helical piles subjected to axial and lateral loads display noticeable stress bulbs around the helical plates. Estimation of the diameter of these stress bulbs (as a function of the helix plate diameter) vary differently across multiple studies hinting its empirical nature. The zone of influence defined by the authors behave similarly to the stress bulbs developed around the helix plate. To determine the zone of influence, data from full-scale lateral field test of twelve helical piles (Test Site I) with varying geometries were analyzed. The zone of influence was then determined to be 1.75 times the diameter of the helix plate from a series of fitting and regression analyses (see²³ for details).
The soil resistance along the shaft length is modeled using a series of uncoupled nonlinear springs, formulated based on the differential equation proposed by Hetenyi (1946) (see Eq. (1)). The soil resistance value is computed as the minimum of the values obtained from Eq. (5) and Eq. (6) (see Eq. (2)). To account for the presence of helices, the soil resistance is modified using repurposed p-multipliers introduced in Eq. (5). The p_mult is expressed as a ratio of effective contact surface area of the helix diameter (D_h) to pile shaft diameter (D_p) multiplied by the general fitting coefficient (Ω) as defined in Eq. (6). A larger ratio signifies a greater contribution of the helix plate to the overall stiffness. The Ω coefficient was empirically fitted through linear regression from observing the behavior of the field test pile configurations²³. The calculated p_mult values are directly applied only to the discretized soil springs within the helix’s zone of influence. Within this zone, p_mult reaches its maximum value at the helix location and decreases linearly, approaching a minimum value of 1.0 at the zone’s boundary. A p_mult value of 1.0, referred to as p_normal, indicates that the original spring stiffness remains unaltered.
where:y is the deflection at any point along the length of the pile, R is the flexural rigidity (EI), P_x is the constant axial load, q is the uniformly distributed vertical load, and p is the soil resisting pressure
where: (:gamma:) is the soil unit weight, (:z) is the depth, (:b) is the pile diameter, (:phi:) is the friction angle of soil, (:alpha:=phi:/2), (:beta:=45+left(phi:/2right)), (:{K}_{0}=1-sinphi:), (:{text{K}}_{text{a}}={text{tan}}^{2}left[45-left(phi:/2right)right])
where: (:varOmega::)– coefficient, (:{D}_{h}) is the helix diameter, (:{D}_{p}) is the pile shaft diameter
where: (:{c}_{u}) = undrained shear strength, (:b) = pile shaft diameter, (:gamma:) = unit weight of soil, (:z) = depth relative to the ground line, and (:J) = Matlock empirical constant for clay soil (0.5 for soft clays, while 0.25 for soft to medium clays).
Equation (1) to Eq. (4) are standard equations widely used in one-dimensional spring analyses found in^32,33. Since the soil layer classified at Test Site I is sand, the shape of the p-y functions is computed using Eq. (2) to Eq. (4), which are based on the procedure of Reese et al., (1974). In contrast, Test Sites II and III indicates soft clay layers, for which the procedure proposed by Matlock (1970) is used. The soil resistance (see Eq. (7)) is calculated as the minimum value between Eq. (8) and Eq. (9). The validity of the spring distribution was confirmed by comparing simulated profiles—bending moment, shear, soil resistance, and pile displacement—with those back-calculated from strain gauge measurements of full-scale instrumented helical piles.
The authors assembled lines of code based in finite difference form in MATLAB to streamline the process of implementing the modified p-y springs approach. To ensure the reproducibility of this study, an LPile procedure—a widely used commercial software that also employs the finite difference method—is provided in the Supplementary Information (SI) section of this manuscript.
The data utilized for machine learning was derived from full-scale lateral load tests conducted at the 300 MW Onshore Solar Farm Project (Test Site I) in Saemangeum, South Korea. Twelve helical piles with varying geometrical configurations were fabricated from high-strength steel with a yield strength of 355 MPa. All helical pile configurations were installed using the pitch-matched method (AR = 1), with careful field monitoring throughout the process. This installation approach aligns with the recommendations from studies conducted by^34,35. Recent studies also recommend a pitch-matched method, as it induces limited particle displacement, thereby reducing soil disturbance. This observation is corroborated by the studies conducted^36,37, and³⁸. The twelve test piles were subjected to static lateral load tests after 14 days to allow the soil to adjust to any soil disturbance caused during the installation. Combined loading conditions were not attempted during the field test; axial (ASTM D3689; ASTM D1143) and lateral (ASTM D3966) load tests were conducted separately.
Refeence³⁹ proposed accounting for installation effects by applying a reduction factor ((:{D}_{f})) to the initial modulus of subgrade reaction (k). Their study recommends implementing a value of 0.2 for piles tested over a short-term duration (i.e. after 14days from installation) to the undisturbed soil parameters. Plotted in Fig. 5 shows the comparison of the load-displacement curves from several field test piles and corresponding calculated results using different (:{D}_{f}-:)values.
Comparison of load-displacement curves between field test data and calculated results using different values of adjusted initial modulus of subgrade reaction (k); shaft diameters: (a) 76.3 mm, (b) 89.1 mm, (c) 101.6 mm, and (d) 114.3 mm.
As shown, applying the suggested 0.2 value resulted in significantly underestimated responses. To better align with the actual field test data, the authors adjusted to 0.5, improving the agreement between simulated and observed behavior. This is due to their study focused on highly over consolidated clayey glacial deposits, which are prone to significant shear strength reduction after soil disturbance. The driving process alters the soil’s microstructural composition by breaking capillary tension, which plays a crucial role in maintaining shear strength. This strength is largely dependent on a specific particle arrangement and pore spacing that sustain cohesive forces. In contrast, silty sand primarily relies on the mechanical interlocking of larger, more angular granular particles. Since frictional resistance in these particles is more easily mobilized, silty sand is less affected by remolding disturbances compared to clay.
Hypothesized factors influencing the optimal design were utilized to generate a comprehensive dataset for the machine learning model. These include (in millimeters for dimensions; kilonewtons for loading): pile shaft diameter (dP), pile shaft thicknesses (tP), pile head offset from the ground line (eP), depth of the first helix from the ground line (dH₁), location of the first helix (zH₁), applied pile head load (P), depth of the second helix from the ground line (dH₂), and location of the second helix (zH₂). The helix diameters (dH) are determined as ratios of the pile shaft diameter (dP) with values of 3.50dP, 4.75dP, and 6.0dP. Additionally, the applied lateral load (aL) is expressed as ratios of the failure load (fL) for the helical pile configuration, with values of 0.25 fL, 0.50 fL, 0.75 fL, and 1.00 fL [see Supplementary Information (SI) section, Table C-1].
A MATLAB script was used to generate a dataset consisting of 3456 samples. Since one helical pile model is subjected to four magnitudes of loading, the dataset essentially contains 864 helical pile configurations in total. The dataset was randomly split into training and testing batches based on the number of configurations, not the number of samples generated. 80% of the generated data was allocated for training, comprising of 690 configurations (a total of 2670 samples), while the remaining 20% was reserved for testing, consisting of 174 configurations (a total of 696 samples). The field test results of the pile configurations installed at Saemangeum are used for independent validation for the final model. These configurations were not included in the SML dataset to ensure non-biased validation.
The input data consists of eight features used to predict the output variable, pile head displacement (δ). All variables were normalized using the min-max scaling method to standardize the features onto a consistent scale, promoting faster convergence and minimizing the risk of numerical instability during model training. The normalized pile head displacement underwent further transformation to a logarithmic scale to approximate a normal distribution, thereby reducing variance and improving detection against outliers.
Supervised machine learning (SML) is often criticized for its dependence on subjective engineering judgment and experience, particularly in the field of geotechnical engineering. To address these challenges, sensitivity analyses are frequently integrated to identify the optimal model configuration, ensuring both accuracy and interpretability for practitioners.
A comparison of several supervised machine learning (SML) algorithms, each subjected to different optimization methods, was conducted to identify the most suitable model for the study. Each SML algorithm was initialized using their default hyperparameter search range and executed five times for each optimization method to determine the optimal hyperparameters. During each run, performance metrics (RSME, MAE, MSE, and R²) were recorded along with their corresponding hyperparameters. In the Supplementary Information (SI) section, Table D-1 tabulates the machine learning algorithms and their respective hyperparameter search ranges and Table D-2 presents the best performance metrics achieved by the model across multiple runs.
Three optimization methods were explored. In Bayesian search method, iterations were set to 30, maximum training time was limited to 300 s and the expected improvement per second plus (EI+) was selected as the acquisition function. This function calculates the degree of improvement that a point can achieve when exploring the vicinity of the current optimum value. If the improvement of the function value is less that the expected value after the search method is executed, then the current optimal value may be the local optimal solution, and the search method will find the optimum value point in other positions of the domain⁴⁰. In the Grid search method, maximum running time was limited to 300 s and the number of grid divisions were set to 10. Finally, for the Random search method, the iterations were set to 30 and the maximum training time limited to 300 s.
The final model, a Bayesian-optimized neural network, was selected as it achieved the best performance metrics compared to the other machine learning algorithms and optimization methods. Its outperformance can be attributed to several key factors highlighted in the literature:
Bayesian optimized neural networks outperforms other methods for hyperparameter optimization and learning algorithms in general machine learning field^41,42.
Bayesian optimization generally outperforms random search and grid search as it employs an informed search strategy rather than exploring all possible combinations randomly. By leveraging prior knowledge and iteratively refining the search space, it systematically reduces deviations and identifies optimal solutions more efficiently. This has been supported by studies conducted by^42,43,44,45, and⁴⁶.
The Bayesian network method addresses uncertainty inference and complex nonlinear problems based on probabilistic reasoning and graph theories^47,48.
Hyperparameter refinement focused on the most influential parameters identified through Bayesian optimization: network depth, node allocation per layer, activation function and regularization strength (λ).
The Sigmoid logistic function was selected as the activation function because its asymptotic characteristics parallel the nonlinear stress-strain responses in p-y soil curves. It also proved to be the most effective, being chosen three out of five times during the hyperparameter optimization. It is also widely accepted for most machine learning applications⁴⁹.
In MATLAB’s hyperparameter search, the regularization strength (λ) was explored within a range of 0.000001/n to 100,000/n, where n is the number of observations in the training observations. The optimization yielded λ = 0.00000066, but this was conservatively adjusted to 0.00001 to reduce overfitting risk and improve generalization to independent validation cases. Extremely small λ values tend to overfit, while excessively large values promote underfitting. Prior studies^42,50 recommend a practical range of [0.01 ≥ λ ≥ 0.0000001]; sensitivity testing within this range showed negligible performance variation, justifying adoption of the median value in this study.
A sensitivity analysis of 18 neural networks models with varying later configurations was performed to resolve inconsistencies observed in the initial hyperparameter search. Incremental increases in layer depth and node count showed diminishing returns beyond two hidden layers. The optimal configuration consisted of two hidden layers with 10 and 20 nodes, respectively, which provided the best balance between predictive accuracy and model simplicity. Further increases offered no meaningful improvement, and in some cases, degraded performance due to overfitting and unnecessary complexity. This architecture reflects the principle of parsimony in model design, where compact yet representative models are favored for both computational efficiency and interpretability. Detailed results of the sensitivity analysis are presented in the Supplementary Information (Tables D-3 to D-5).
Underfitting and overfitting occur when a machine learning model fails to generalize effectively, often performing well on the training dataset but significantly worse on the testing dataset or during validation. To address overfitting in this study, the authors implemented the following measures:
Data pre-processing – The data was standardized to approximate a normal distribution, which helps reduce model complexity and improve generalization performance^40,51.
Hyperparameter sensitivity – Careful oversight of hyperparameter selection for the final model was performed, as improper settings can result in underfitting or overfitting, thereby restricting the model’s learning capability⁵¹.
Cross-validation – this study adopted a cross validation scheme using k = 10 folds to ensure the model performs well across different subsets of data.
The model was evaluated on a reserved testing set (20% of the total data) separate from the training data (80% of the total data).
By default, MATLAB has no training iteration limit. In this study, the iteration limit is set to 100. With unlimited training time, risk of overfitting is increased.
Machine learning application: (a) and (b) scatter plot of predicted vs. true response for training and testing, respectively.
The Bayesian-optimized neural network scatter plots of predicted versus true responses in the training and testing phase is presented in Fig. 6a and b. This configuration yielded the lowest values for MAE, RMSE, and MSE, along with the highest R² value, indicating superior performance and a better model fit. This model is used to predict the lateral displacements of the field data for verification purposes, as shown in Fig. 7. The configurations used for verification were not included in the training and testing pool, ensuring their independence for assessment of the final machine learning model. The helical piles used for verification exhibit varying geometrical configurations (e.g., dP, dH, zH) all except for the embedment length of 3.0 m, with 0.2 m of the pile exposed above the ground surface. Their first helix is placed 0.2 m below the ground line, and subsequent helices are spaced 1.0 m apart. The predicted load-displacement by machine learning (this study) show a reliable estimation of the lateral profile for each helical pile configuration compared to study published by²³.
Predicted values of helical pile displacement using the trained model vs. the field test data (Kim et al., 2022): (a) test pile 76.3dP, (b) test pile 89.1dP, (c) test pile 101.6dP, and (d) test pile 114.3dP.
To identify the factors that most significantly influence the optimal configuration of helical piles, a feature importance plot was generated to visualize the contribution of each feature to the output value. As illustrated in Fig. 8, the applied lateral load (P) is identified as the most influential factor, followed by the pile head offset from the ground line (eP), the diameter of the first helix from the ground line (dH₁) and position of first helix from the ground line (zH₁). In contrast, the least influential features are the pile shaft thickness (tP), the pile shaft diameter (dP), diameter of the second helix (dH₂), and the position of the second helix from the ground line (zH₂).
Feature importance plot of the parameters used in the machine learning process.
As expected, pile head displacement (δ) increases with applied lateral load (aL) once local shear strength is exceeded and the soil microstructure yields. A similar trend is observed with pile head offset (eP), where elevating the pile head amplifies displacement due to a larger bending moment generated by the increased lever arm. Both features intensify deflection near the ground surface, where bending stresses are concentrated, while also reducing lateral restraint from surrounding soil. The diameter of the first helix (dH₁) significantly influences mobilized soil stiffness, as resistance is proportional to the helix surface area in contact with the soil. The helix plates act as rotational inhibitors, partially restricting the pile shaft from movement. Likewise, the depth of the first helix (zH₁) governs the distribution of soil resistance by determining which strata are engaged. Shallow placement provides immediate resistance but may be insufficient under higher loads, whereas deeper placement mobilizes stiffer soils, optimizing resistance distribution along the shaft. Therefore, a well-positioned first helix can optimize the distribution of lateral resistance, balancing the forces along the pile’s embedment depth, as discussed in the parameter sensitivity analysis.
In contrast, the pile shaft thickness (tP) exhibits minimal influence due to scaling effects, as its contribution to flexural capacity is secondary to the influence of helix geometry. Similarly, the pile shaft diameter (dP) primarily provides structural stability but contributes less directly to lateral resistance compared to the helices. The second helix diameter (dH₂) and its depth (zH₂) rank low in importance, since the first helix near the ground surface predominantly governs soil-pile interaction. At greater depths, the second helix exerts limited influence due to its distance from the load application point and the dominance of upper-layer soil resistance.
Incorporating the findings from the previous section, the optimal positioning of the first helix (zH₁) was evaluated using the data extracted from the machine learning model, with eP fixed at 200 mm, dH₁ at 400 mm, and tP at 3.2 mm. Piles were incrementally loaded to the failure load (fL), defined as the load producing a bending moment equal to the plastic moment capacity of the pile section [see Supplementary Information, Table C-1]. Response surface plots (Fig. 9) illustrate displacement behavior as a function of the helix-to-pile embedment ratio (zH/zP) and applied-to-failure load ratio (aL/fL).
Two key trends emerge. First, piles without helices (zH/zP = 0) exhibit substantially greater displacements than helical configurations. Second, the optimal zH/zP ratio consistently lies between 0.1 and 0.2 across load levels, as reflected by the concave response surface in this range. The helix effect diminishes beyond zH/zP = 0.2 and becomes negligible past 0.6, where the pile behaves similarly to a plain shaft. The underlying mechanics of this observation can be attributed to the load-transfer response in soil-structure interaction.
Each finite strip of the soil layer exhibits a maximum soil resistance (p) that can be mobilized for displacement (y). Once this capacity is exceeded, the excess external forces are transferred deeper into the soil until equilibrium is achieved. At lower load levels, where the soil remains in its elastic state, the zH/zP is within the range of 0 ≤ x ≤ 0.1, indicating that the shear strength of the near surface region has not been exceeded and can still provide sufficient lateral resistance. As loading approaches failure, the zH/zP shifts deeper to 0.1 ≤ x ≤ 0.2, signifying the upper layers have reached their shear capacity. Placement of the helices at greater depths yields limited benefits, as the near -surface layers already provide sufficient resistance, rendering deeper helices less effective in enhancing lateral capacity.
Response surface plots indicating the displacement behavior of the helical piles under different zH/zP (helix to pile embedment depth ratio) and aL/fL (applied load to failure load ratio) values: (a) 76.3 mm, (b) 89.1 mm, (c) 101.6 mm, and (d) 114.3 mm.
The findings of this study are utilized to optimize the geometric configuration of helical piles for lateral design applications in renewable energy infrastructure, particularly in solar farms. These infrastructures are often installed in open fields with minimal obstructions, making them particularly vulnerable to extreme lateral forces and overturning moments induced by variable transient wind loads.
This case examines the construction of a solar farm unit, where the pile embedment depth (zP) is restricted to 2.5 m due to an underlying rocky substratum, preventing further penetration. The project requires that pile head displacement remain within 15 mm under the design load (aL) of 5.30 kN. Figure 10a presents the extracted datapoints from the machine learning model that meet the displacement criteria within [10 ≤ x ≤ 20], visualized in three-dimensional space. These datapoints also have an eP value set to 150 mm above ground, a tP of 3.2 mm, and for lateral design considerations, only the first helix from the ground line (zH₁) is taken into account. The displacement is normalized and expressed as a function of the dH/dP ratio, zH/zP ratio, and aL. Here, the set value represents the idealized 15 mm displacement, while the response value corresponds to the actual configuration’s displacement. Positive values indicate displacements below the threshold, whereas negative values signify displacements exceeding the set value limit. Values close to 0 are considered optimal, representing the ideal configuration.
The intersection of the 5.30kN plane in Fig. 10b highlights two dP configurations that closely align with the ideal displacement of 15 mm (normalized deviation value of 0). Figure 10c and d present the response contour plots for pile diameters (dP) of 89.1 and 101.6, respectively, sliced at the 5.30kN plane, while Fig. 10e displays the corresponding steel weights for each configuration. Based on the response plots, the optimized configuration for a helical pile with a design capacity (aL) of 5.30kN is a pile diameter of 89.1, with a dH/dP ratio of 3.50 and a zH/zP ratio of 0.15. This configuration effectively balances both performance and economic considerations, ensuring an efficient and practical design solution.
Case study to determine the optimal configuration of a single helical pile: (a) extracted datapoints from the machine learning model, (b) isolated possible configurations; displacement response contour of the isolated configurations under the design load (aL) of 5.30kN for pile diameters (dP): (c) 89.1 and (d) 101.6, respectively, and (e) configuration total steel weight.
At Test Site II, the solar farm units are limited to a pile embedment depth of 2.5 m and are required to withstand a design load of 5.46 kN, with a maximum allowable pile head displacement of 15 mm. Insights derived from the machine learning model identified the optimal configuration with the following parameters: dP = 89.1 mm, tP = 3.2 mm, eP = 150 mm, dH₁ ≥ 380 mm, and zH₁ = 0.5 m below ground. The implemented helical pile design adopted the suggested values for dP, tP, eP, and zH₁, while increasing dH₁ to 400 mm and incorporating two additional helices (both 200 mm) at depths of 1.4 m and 2.3 m to enhance axial capacity. A full-scale instrumented pile test was performed to evaluate the lateral response behavior of the helical pile configuration, aimed at assessing its suitability for design applications. Pairs of electrical resistance strain gauges, mounted diametrically on the central shaft perpendicular to the load direction, were installed to monitor strain during the pile test. Table 2 summarizes the soil properties used in the analysis for Test Site II. Detailed information on the subsurface investigation, pile instrumentation, and loading protocol is available in the earlier study by⁵². Figure 11 presents a comparison between the simulated bending moment, shear distribution, soil resistance, and pile displacement profiles with the converted strain gauge measurements. The results demonstrate a satisfactory agreement, validating the reliability of the simulation model. The findings of this investigation validate the adequacy and reliability of the helical pile configuration for its intended application in the solar farm project.
The solar farm units at Test Site III are subject to a relatively lower design load of 3.96 kN compared to Test Sites I and II; however, the maximum allowable pile head displacement remains constrained to 15 mm. The optimal configuration was identified to be dP = 76.3 mm, tP = 3.2 mm, eP = 150 mm, dH₁ ≥ 360 mm, and zH₁ = 0.4 m below the ground line. The total embedment depth zP was set to 2.7 m. To enhance axial capacity, particularly in soft ground conditions prone to settlement, two additional helices (each 200 mm in diameter) were incorporated at depths of 1.3 m and 2.3 m. This configuration yields a zH/zP ratio of 0.15, aligning with the optimal range established in the previous section. The manufactured helical pile configuration follows the recommended design parameters, with the exception of the first helix diameter (dH₁), which was set to 400 mm—slightly larger than the optimized value—to maintain uniformity across the other test sites. A full-scale instrumented pile test was performed on the proposed configuration to verify its performance and suitability.
The lateral behavior was simulated using the modified p–y springs approach, in which soil resistance is calculated as the minimum value between Eq. (7) and Eq. (8), following the p–y curve formulation for soft clay originally proposed by Matlock (1970). Table 3 summarizes the soil properties used in the analysis for Test Site III. The strain at one-half the maximum principal stresses of clay specimens (:left({varepsilon:}_{50}right)) was empirically adjusted to account for the effects of soil disturbance resulting from pile installation. This adjustment was necessary due to the noticeable discrepancies observed between the simulated profiles against the converted strain gauge measurements. In clay soils, the stiffness of the p-y curves is highly dependent on the (:{varepsilon:}_{50}) value, similar to how the initial modulus of subgrade reaction k influences the shape of the p-y curve behavior in sandy soils. A comprehensive and articulate discussion on the mechanics of selecting representative values for soil stiffness can be found in Chap. 3 of³².
Figure 12 presents a comparison between the simulated bending moment, shear distribution, soil resistance, and pile displacement profiles with the converted strain gauge measurements. The results exhibit a generally satisfactory agreement, with the exception of the field test data (FTD) of the pile head displacements. The discrepancy is most likely attributed to instrumentation effects, as the refence block for measuring pile head displacement was not rigidly fixed but merely placed on the ground surface. This setup may have been influenced by localized ground movements, which in turn affected the accuracy of the recorded pile head displacement values.
From an engineering standpoint, the discrepancies observed at the pile head do not undermine the adequacy of the results. While measurement uncertainty exists at the surface, the deeper structural response indicators (bending moment, shear, soil resistance) reliably captured the fundamental soil-structure interaction.
Lateral response plots for a helical pile with shaft diameter of 89.1 mm and three helix configuration (400-200-200; all in mm) at Test Site II showing the comparison between the simulated profiles and the corresponding converted strain gauge readings (SGR): (a) bending moment (SBM), (b) shear distribution (SSD), (c) soil resistance (SSR), and (d) pile displacement (SPD) alongside field test data (FTD).
Lateral response plots for a helical pile with shaft diameter of 76.3 mm and three helix configuration (400-200-200; all in mm) at Test Site III showing the comparison between the simulated profiles and the corresponding converted strain gauge readings (SGR): (a) bending moment (SBM), (b) shear distribution (SSD), (c) soil resistance (SSR), and (d) pile displacement (SPD) alongside field test data (FTD).
As no perfect set of functions can comprehensively encapsulate all conceivable irregularities and deviations, there are limitations to the presented methodology that should be accounted for.
In this study, combined axial-lateral loading conditions were not investigated. Instead, axial load tests (ASTM D1143; ASTM D3689) and lateral load tests (ASTM D3966) were performed independently, in accordance with standard practice that prescribes distinct procedures, instrumentation and interpretation methods. The modified p-y springs method was therefore calibrated exclusively from lateral load test data, under the assumption of independent soil-pile interaction. As a result, the current methodology does not capture potential coupling effects between axial and lateral responses. Although⁵³ has shown that coupled analysis can lead to more reliable foundation design, extending the framework to incorporate combined loading would require significant methodological modifications and the development of appropriate verification datasets. This limitation is acknowledged, and combined loading is identified as an important direction for future research.
Although the modified p-y springs method can be applied to various soil layers (including adjustments to other p-y curve functions, such as those for soft clay), as demonstrated in the results of Test Site II and Test Site III, further research is needed to determine a generalized value for the pile coefficient (Ω).
While this study focused on displacement control as the primary structural design consideration, it is acknowledged that durability aspects, such as corrosion resistance and life cycle costs, are also critical for solar farm infrastructure. Although these factors were not explicitly explored due to the scope of this work, related studies comparing foundation alternatives and incorporating sustainable design concepts in geotechnical engineering^54,55 provide valuable perspectives for extending this research.
This study investigated the potential of supervised machine learning in gaining valuable insights into the optimal configuration of laterally loaded helical piles. Despite the inherent variability of the soil layers across the three test sites, the key features identified by the machine learning model were effectively utilized for the initial assessments and selection of helical pile geometry. These insights contributed significantly to determining optimal configurations. Based on the research findings, the following conclusions can be drawn.
A Bayesian-optimized neural network outperformed other supervised machine learning algorithms and optimization methods explored in this study.
The first helix (the nearest relative to the ground line) provides the most beneficial soil resistance, significantly reducing lateral deflection and its optimal position is sensitive to the applied lateral loading.
An increase in helical plate diameter decreases the pile deflection due to an increase in surface area in-contact with the soil. While the addition of succeeding helix plates provides minimal to almost negligible contribution to the overall lateral resistance.
The diminishing lateral resistance offered by the helix plate results from the further positioning away from the ground line, which can be observed from a helix-to-pile embedment ratio of 0.2 until a ratio of 0.6 where the presence of the helical plate is no longer significant.
The applied lateral load (aL) is the most influential feature in predicting the lateral displacement of a helical pile, followed by the pile head offset from the ground line (eP), the diameter of the first helix from the ground line (dH₁) and position of first helix from the ground line (zH₁).
The least influential features are the pile shaft thickness (tP), the pile shaft diameter (dP), diameter of the second helix (dH₂), and the position of the second helix from the ground line (zH₂).
The derived geometric features of the helical pile configuration demonstrated a level of independence from site-specific geotechnical properties, suggesting that these features can be used to guide helical pile design across varying soil types. Nevertheless, site-specific geotechnical characterization remains essential for accurately capturing the pile–soil behavior especially accounting installation effects.
The authors assembled lines of code based in finite difference form in MATLAB to streamline the process of implementing the modified p-y springs approach. To ensure the reproducibility of this study, an LPile procedure—a widely used commercial software that also employs the finite difference method—is provided in the Supplementary Information (SI) section of this manuscript. Specific details and configurations can be shared upon reasonable request to the corresponding author.
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (RS-2021-NR060134). Additionally, this work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry & Energy (No. RS-2021-KP002506).
Department of Civil Engineering, Kunsan National University, 558 Daehak-ro, Miryong-dong, Gunsan, 54150, Republic of Korea
Hyeong-Joo Kim
Department of Civil and Environmental Engineering, Kunsan National University, 558 Daehak-ro, Miryong-dong, Gunsan, 54150, Republic of Korea
James Vincent Reyes, Hyeong-Soo Kim & Tae-Eon Kim
Renewable Energy Research Institute, Kunsan National University, 558 Daehak- ro, Miryong-dong, Gunsan, 54150, Republic of Korea
Tae-Woong Park & Peter Rey Dinoy
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H.J.K: Resources, Funding acquisition, Supervision. J.V.R: Conceptualization, Methodology, Data curation, Formal analysis, Validation, Writing — original draft and editing. P.R.D: Data curation, Formal analysis, Validation. H.S.K: Project administration, Investigation, Funding acquisition. T.W.P: Project administration, Investigation, Funding acquisition. T.E.K: Investigation, Project administration.
Correspondence to James Vincent Reyes.
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Kim, HJ., Reyes, J.V., Kim, HS. et al. Optimization and field validation of laterally loaded helical piles for solar farm infrastructure projects. Sci Rep 15, 38195 (2025). https://doi.org/10.1038/s41598-025-22106-y
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DOI: https://doi.org/10.1038/s41598-025-22106-y
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