Assessment of Experimental Data and Analytical Method of Helical Pile Capacity Under Tension and Compressive Loading in Dense Sand

Abstract

This study presents the results of axial tension (uplift) and compression tests evaluating the capacity of helical piles installed in Shahriyar dense sand using the UTM apparatus. Thirteen pile load experiments involving single-, double-, or triple-helix piles with shaft diameters of 13 mm were performed, including six compression tests and seven tension tests with different pitches (D_h =13, 20, and 25 mm). The tested helical piles with a helix diameter of 51 mm were considered, and the interhelix spacing approximately ranged between two and four times the helix diameter. Through laboratory testing techniques, the Shahriyar dense sand properties were identified. Alongside theoretical analyses of helical piles, the tensile and compressive pile load tests outcomes in dense sand with a relative density of 70% are presented. It was found that the maximum capacities of the compressive and tensile helical piles were up to six and eleven times that of the shaft capacity, respectively. With an increasing number of helices, the settlement reduced, and the bearing capacity increased. Consequently, helical piles can be manufactured in smaller sizes compared to steel piles. Overall, the compressive capacities of helical piles were higher than the tensile capacities under similar conditions. Single-helices piles with a pitch of 20 mm and double-helices piles with a pitch of 13 mm were more effective than others. Therefore, placing helices at the shallower depths and using smaller pitches result in better performance. In this study, when compared to values from the L1–L2 method, the theoretical method slightly underestimates the ultimate compression capacity and both overestimates and underestimates the uplift capacity for single- and double-helical piles, respectively, due to the individual bearing mode and cylindrical shear mode.

1. Introduction

Over the past two decades, numerous studies have investigated the axial performance of helical piles under compressive and tensile loads, focusing on load–displacement behavior, failure mechanisms, and load transfer characteristics in cohesionless, cohesive, and structured soils. These include experimental investigations [1,2,3,4,5], numerical simulations [6,7,8,9,10,11,12], and analytical modeling approaches [13,14,15,16]. While these works have advanced understanding of helical pile behavior, several limitations remain unresolved.

Helical piles have gained popularity as a reliable foundation solution in residential, commercial, and infrastructural applications, particularly where ease of installation, low environmental impact, and cost-effectiveness are critical factors [17,18,19].

Full-scale testing by Sakr [1,20] demonstrated the high load-bearing capacity of large-diameter helical piles in dense to very dense sands, with uplift capacities approaching 80% of their compressive capacity. However, such tests are expensive, and generalizing their results to different geometrical configurations or soil conditions is limited.

Elsherbiny and El Naggar [21] examined the compressive behavior of smaller helical piles in clay and sand and provided a method for their design based on the findings of the calibrated numerical model. A comparison between the compressive capacity gained from the numerical simulations and analytical methods was conducted. It was shown that the estimations of analytical solutions for piles in sand soil are significantly affected by the selection of the acceptable failure criteria and bearing capacity factors.

The uplift capability of multi-helix anchors in sand was investigated in the laboratory and on the ground by Mitsch and Clemence [22]. According to this research, a cylindrical failure surface formed under the top helical plate during pull-out, and the failure surface above the top plate was influenced by the depth of the anchor embedment. Based on the procedure, the bearing resistance on the top helical plate, frictional soil cylinder resistance, and friction on the shaft all contributed to the uplift capacity.

The pull-out capability of helical piles in a significant sample of sand was examined by Spagnoli et al. [23] using a series of experimental model on helical pile models. These studies considered the impact of the ratio of the helix (or helical plates) to shaft diameter and the helix diameter. According to the findings, the ideal design of helical pile has a helix-to-shaft diameter ratio of 1.5 to 2.0. Smaller helixes offered minimal resistance to vertical stresses, whereas larger helixes were substantially more difficult to install.

Recent advances in artificial intelligence (AI) and machine learning (ML) have demonstrated strong potential in modeling complex pile–soil interactions, as evidenced by Eslami et al. [24], who used data-driven techniques to predict ultimate load-bearing capacity (ULB). Nonetheless, these models often lack transparency in physical interpretation and are not yet widely integrated with mechanistic design frameworks or validated through controlled laboratory testing.

Although full scale testing is preferable a better approach for investigations, the cost of such testing is significant, and it may be difficult to interpret the loads and soil conditions, which restricts the number of piles that can be examined. The main benefit of numerical models is that they enable us to investigate in more detail, although a constitutive rule should be reliable. Therefore, small-scale models can be tested in many scenarios, which can consider the effects of various parameters with cost-effectiveness.

The present state of knowledge on helical piles is unsatisfactory, and the design techniques currently used to estimate the displacement and compression/uplift capability in dense sandy soils are inadequate. The current study provides and discusses the findings of studies relating to experimental testing modeling and theoretical predictions of helical piles in dense sandy soils in light of the necessity for more research on helical foundations.

2. Aim of Paper

The main objective of this research is to advance the current understanding of helical pile performance in cohesionless soils—particularly Shahriyar dense sand in Tehran, Iran—by addressing the analytical and empirical gaps identified in the existing studies. Several categories of helical piles were considered with various pitches ($P_h$), number of helices ($n$), and spacing between them ($S$) into Shahriar sand soil. The study adopts an integrated methodology involving experimental testing, analytical evaluation, and empirical modeling. The specific goals of the study are as follows:

• Quantify and compare axial compressive and tensile capacities of helical piles with different numbers of helices, helix spacings, and pitches and examine the associated displacement behaviors.
• Evaluate the accuracy of established theoretical models, such as the L1–L2 failure criterion and classical bearing capacity methods, by systematically comparing their predictions with experimental outcomes.
• Conduct a comparative analysis of failure mechanisms, particularly focusing on cylindrical shear mode (CSM) and individual bearing mode (IBM) across various pile configurations—an area underexplored in the existing literature.
• Identify optimal geometric configurations (e.g., pitch-to-diameter ratio, helix spacing) based on empirical trends, thereby offering design recommendations for practitioners.
• Develop new dimensionless empirical equations using nonlinear regression and dimensional analysis techniques to enhance the predictive capability for pile capacity estimation in practical engineering applications.

Through this framework, the study provides not only validated empirical models but also mechanistic insights that are directly applicable to foundation engineering in granular soils.

3. Dimensional Analysis and Scaling

3.1. Dimensional Analysis

In this part, we look for a collection of normalized parameters that, regardless of the size or scope of the physical issue, may be utilized for a thorough interpretation of test results. This may be accomplished by means of dimensional analysis; see, for instance, Westine et al. [25] and Sedran et al. [26]. Buckingham $\pi$ theorem [27], as one of the methods for nondimensionalization and dimensional analysis, was applied. The performance of helical piles is indicated by the effective parameters in Figure 1. The function, $f\left(q_i\right)$, is formally given by

$$f\left(q_i\right)=f\left(\phi , n, d, H, H_e, S, t_p, L_a^{\prime }, D_h, t_h, P_h, E_p, E_s, G_p, Q, d_s\right)=0,$$

(1)

where $q_i\left(i=1,\dots ,N\right)$ are effective parameters including, friction angle φ, number of helices $n$, shaft diameter $d$, total length of pile $H$, embedded length of pile $H_e$, spacing between helices $S$, shaft thickness $t_p$, spacing between the pile tip and bottom helix $L_a^{\prime }$, diameter of helix $D_h$, helix thickness $t_h$, pitch $P_h$, elastic modules of pile $E_p$, elastic modules of soil $E_s$, shear modules of pile $G_p$, axial load $Q$, and pile displacement $d_s$. The dimension of the effective parameters in Equation (1) are

$$\begin{array}{l}\dim \left(\phi ,n\right)=1\\ \dim \left(d, H, H_e, S, t_p, L_a^{\prime }, D_h, t_h, P_h, d_s\right)=\left[L\right]\\ \dim \left(E_p, E_s,G_p\right)=\left[M{L}^{-1}{T}^{-2}\right]\\ \dim \left(Q\right)=\left[ML{T}^{-2}\right]\end{array}.$$

(2)

A set of dimensionless parameters ${\pi }_j\left(j=1,\dots ,p\right)$, where p represents the total number of dimensionless parameters constructed from the effective parameters, $q_i$, can typically be interpreted as the ratio of two quantities with the same dimension, as follows:

$$\begin{array}{l}{\pi }_1=\phi , {\pi }_2=n, {\pi }_3={\displaystyle \frac{H}{d}}, {\pi }_4={\displaystyle \frac{H_e}{d}}, {\pi }_5={\displaystyle \frac{S}{d}}, {\pi }_6={\displaystyle \frac{t_p}{d}}, {\pi }_7={\displaystyle \frac{L_a^{\prime }}{d}}, {\pi }_8={\displaystyle \frac{D_h}{d}}, {\pi }_9={\displaystyle \frac{t_h}{d}}, {\pi }_{10}={\displaystyle \frac{P_h}{d}},\\ {\pi }_{11}={\displaystyle \frac{E_s}{E_p}}, {\pi }_{12}={\displaystyle \frac{G_p}{E_p}}, {\pi }_{13}={\displaystyle \frac{d_s}{d}},\end{array}$$

(3)

The remaining quantities are $Q$, $d$, and $E_p$. We need only the $k=N-p=1$ ($k$: maximal dimensionally independent) dimensionless parameter, denoted by ${\pi }_{14}$, and the model can be re-expressed as

$${\pi }_{14}=Q\times E_p^{\alpha }\times d^{\beta }=ML{T}^{-2}\times M^{\alpha }{L}^{-\alpha }{T}^{-2\alpha }\times L^{\beta }=M^{\alpha +1}{L}^{1-\alpha +\beta }{T}^{-2-2\alpha }=1.$$

(4)

Equating the coefficient of each power of fundamental units ($M$, $L$, and $T$) to zero, the exponents $\alpha$ and $\beta$ are obtained equal to −1 and −2, respectively. Hence, we have

$${\pi }_{14}={\displaystyle \frac{Q}{E_p\times d^2}}.$$

(5)

Applying dimensional analysis, and selecting two parameters $E_p$ and $d$ as repeating variables, Equation (1) can be restated as

$$g\left(\phi , n, {\displaystyle \frac{H}{d}}, {\displaystyle \frac{H_e}{d}}, {\displaystyle \frac{S}{d}}, {\displaystyle \frac{t_p}{d}}, {\displaystyle \frac{L_a^{\prime }}{d}}, {\displaystyle \frac{D_h}{d}}, {\displaystyle \frac{t_h}{d}}, {\displaystyle \frac{P_h}{d}}, {\displaystyle \frac{E_s}{E_p}}, {\displaystyle \frac{G_p}{E_p}}, {\displaystyle \frac{Q}{E_p\times d^2}}, {\displaystyle \frac{d_s}{d}}\right)=0.$$

(6)

Most of the parameters in Equation (6) are associated with the pile geometry. The bearing capacity for helical piles with different numbers of helices and installation spacing is measured. Regarding the similar geometry of piles and soil properties in the present experiments, some of the effective dimensionless parameters in Equation (6) are constant and can be neglected. Therefore, Equation (6) can be reduced to Equation (7).

$$Q_d={\displaystyle \frac{Q}{E_p\times d^2}}=ħ(n, {\displaystyle \frac{S}{d}}, {\displaystyle \frac{P_h}{d}}, {\displaystyle \frac{d_s}{d}})$$

(7)

The analysis assumes homogeneous, isotropic, and dry granular soil conditions, with uniform axial loading and negligible effects of pile installation disturbance. These simplifications were necessary for dimensional consistency and to maintain analytical tractability. However, it should be acknowledged that under real field conditions, factors such as installation torque, soil layering, and anisotropy may influence behavior.

3.2. Scaling

Full-scale testing provides the most accurate representation of foundation behavior across various ground conditions—particularly in sandy soils. However, such tests are both expensive and time-consuming. To address these constraints, geotechnical researchers often employ small-scale physical models, which offer a cost-effective and time-efficient alternative for experimental investigation.

Although small-scale models can accurately capture critical response data, translating these results to full-scale conditions requires careful extrapolation. This process is complicated by the many interacting factors that govern the performance of soil–helical pile systems [28].

Scaling considerations for helical piles embedded in sand typically fall into three categories: (1) pile geometry, (2) structural properties of the pile, and (3) soil characteristics. Ideally, similitude laws should be applied to maintain the physical consistency between the model and prototype. However, due to practical laboratory constraints, it is often not feasible to satisfy all scaling laws simultaneously, which may limit the generalizability of the results.

Iai [29] proposed scaling relations for soil–pile–structure interaction under single-gravity conditions, addressing parameters such as length, density, strain, mass, time, and acceleration. In accordance with this framework, the present study assumes identical soil properties for both the model and prototype, setting the density and strain scaling factors to unity. Since all tests were performed under 1 g conditions, the acceleration scaling factor is also unity. Consequently, only geometric scaling (η) was applied to relate the model to the prototype.

It is important to note that one of the primary challenges in geotechnical model testing is accurately replicating the gravitational loading effects, which significantly influence the interaction between the pile and surrounding soil.

In practical applications, helical piles are generally embedded at relatively shallow depths. For example, Perko [17] compiled a database of over 300 load tests, noting that 80% of piles were installed at depths between 2 and 10 m. Based on this, geometric scaling factors (η) in the range of approximately 3 to 15 are considered acceptable. Therefore, a 1 g laboratory model of a 0.6–0.65 m long helical pile can produce reasonably representative results for typical field applications.

4. Experimental Setup

4.1. Pile Configuration and Soil Tests

The experiments were performed at the structural laboratory of Shahid Rajaee Teacher Training University. The criteria suggested by previous studies [30,31] are used to identify the properties of used helices. Thus, the properties of helices, including their diameter ($D_h$), thickness ($t_h$), pitch ($P_h$), and spacing ($S$) are calculated due to the diameter of the pile model. Regarding the cylindrical shear mode (CSM) and individual bearing mode (IBM) performances, the spacing of helices is identified as follows:

$$\begin{array}{l}{D}_h=\left(2\sim 4\right)d=3.9\times 13\simeq 51 \mathrm{mm}\\ {\displaystyle \frac{t_h}{d}}={\displaystyle \frac{1}{5}} \to t_h={\displaystyle \frac{13}{5}} \to t_h\simeq 2.5 \mathrm{mm}\\ {\displaystyle \frac{P_h}{d}}=1, 1.5, \mathrm{and} 2 \to P_h\simeq 13, 20, \mathrm{and} 25 \mathrm{mm}\end{array}.$$

(8)

Helical piles were made of a steel tube with a diameter of 13 mm, and the helical plate pitch was constant (13 mm) for the compression case; while in the tension test, the helical plate pitches of 25, 20, and 13 mm were considered to evaluate the pitch effects on the displacement and pile capacities.

$$\begin{array}{l}{S}_{IBM} >3D_h \Rightarrow S_{IBM} =4D_h\simeq 200 \mathrm{mm} \mathrm{for} \mathrm{compression}\\ S_{CSM} =\left(1.5\sim 2\right)D_h \Rightarrow S_{CSM} =2D_h\simeq 100 \mathrm{mm} \mathrm{for} \mathrm{compression}\\ S_{CSM} =\left(2\sim 3\right)D_h \Rightarrow S_{CSM} =2.3D_h\simeq 115 \mathrm{mm} \mathrm{for} \mathrm{tension}\end{array}$$

(9)

In addition, the pile length was limited to 650 mm and 600 mm for the compression and tension tests, respectively due to the laboratory limitations. Consequently, helices with dimensions of 2.5 mm in thickness and 51 mm in diameter were used. TIG welding was used to secure these helices to the steel tube at their desired locations. Figure 1 displays the configurations for various piles, taking into account the helical pile load test program. Figure 2 shows the grain size distribution curve for Shahryar sand, based on testing conducted in accordance with the ASTM D422standard [32]. In addition, the physical and mechanical properties of Shahriar double-washed sand are presented in

Table 1. The results of initial tests on “Shahriar sand soil”.

Parameter	Value	Descriptions	Standard Code
Soil type	SW	Unified soil classification standard (USCS)	ASTM D422 [32]
$D_{15}\left(\mathrm{mm}\right)$	0.38	Diameter corresponding to 15% fines
$D_{50}\left(\mathrm{mm}\right)$	1.76	Diameter corresponding to 50% fines
$C_C$	1.26	Coefficient of gradation
$C_u$	7.6	Uniformity coefficient
$G_s$	2.46	Specific gravity	ASTM D854 [33]
$\phi$(deg)	38	Friction angle	ASTM D3080 [34]
${\gamma }_{d\min }\left(\mathrm{kN}/{\mathrm{m}}^3\right)$	17.46	Minimum dry density	ASTM D4254 [35]
${\gamma }_{d\max }\left(\mathrm{kN}/{\mathrm{m}}^3\right)$	19.42	Maximum dry density	ASTM D4253 [36]
$\omega \left(\%\right)$	0.48	Moisture content	ASTM D2216 [37]

.

4.2. Experimental Setup for Axial Compressive and Tensile Load Tests

The Universal Testing Machine (UTM), Zwick/Roell Z150 apparatus, was employed for compressive and tensile load tests, as shown in Figure 3. The test box with dimensions of 700 mm × 500 mm × 1000 mm was made of steel plates with a thickness of 5 mm, considering the restrictions of the helical pile and UTM dimensions, as well as the pile’s pressure isobars. The test box was filled with Shahriyar sand with relative density, $D_r$, equal to 70%. For this purpose, at first, the box was divided into 100 mm sections. The required soil weight per layer was calculated based on the known relative density and volume. Having determined the weight of the required soil, the sand of each layer was poured into the box and tapped with a plastic hammer to reach the intended level. Furthermore, the density of soil-in-place test by the rubber balloon method (ASTM Designation D-2167 [38]) was conducted to ensure the validity of the relative density reaching 70%.

A drilling machine was used to simulate the real conditions of the helical pile installation in the field. The pile was attached to the machine and vertically installed at the central point of the box so that 100–170 mm of its head remained out of the soil allowing for load application. Afterward, the test box was transferred below the UTM’s jaw through the rails, embedded under the box. The computer communicated with a data recorder using Test Xpert11 V3.2 software. Then, making software adjustments, the UTM’s jaw was closed by a lever and was prepared to apply the load.

According to Stanier et al. [39], the test speed was set at 0.2 mm/s, and the graph’s endpoint represented 70% of the maximal force. According to ASTM standards D 1143 [40] and D 3689 [41], respectively, the axial compression and tensile stress tests were performed. Since the primary goal of the load testing was to ascertain the helical pile’s displacement and maximum bearing capacity, thirteen pile load tests consisting of seven tension and six compression tests were completed. Three repeated tests were conducted to confirm the validity of the outcomes from the two tests.

5. Failure Modes

The failure mechanisms governing the behavior of helical piles under axial compressive and tensile loads in sandy soils are complex and influenced by various factors, including pile geometry, helix configuration, helix pitch, helix spacing, embedment depth of helix and pile, and soil confinement. Accurately identifying these modes is critical for understanding load transfer and ensuring reliable capacity predictions.

According to Figure 4, under axial compression, two dominant failure mechanisms are typically observed: cylindrical shear mode (CSM), which generally occurs when the helix spacing $S\le \left(1.5\sim 2\right)D_h$ [42], and individual bearing mode (IBM), which is more likely to develop when $S>3D_h$ [43].

In CSM [22,44,45], a continuous cylindrical failure surface envelops all helices and extends along the pile shaft. This mode mobilizes both shaft friction and shear resistance along the failure surface and is commonly associated with piles that have closely spaced helices and sufficient embedment depth. In contrast, IBM occurs when each helix operates independently, transferring load primarily through local bearing resistance. This mode is often observed in piles with larger helix spacing installation depths. The failure surface takes on a conical–cylindrical shape, with its maximum radial extension measuring 1.5 times the diameter of the helix at the top helix level. It features a conical tensile zone above and exhibits substantial bearing resistance—exceeding 2 diameters—below the bottom helix. In this region, the tip resistance plays a significant role in contributing to the overall capacity. The results further suggest that reducing the helix pitch or increasing the number of helices promotes a shift from IBM to CSM, though excessive spacing may revert the mechanism back to IBM due to reduced helix interaction.

Under tensile (uplift) loading, the geometry of the failure surface changes significantly [22,46]. For shallowly embedded helices, where the embedment ratio $H_1/D_h$ is less than a critical value ${\left(H_1/D_h\right)}_{cr}$, the failure surface typically extends to the ground surface in the form of a truncated cone. This results in reduced uplift resistance due to limited overburden and confinement. In contrast, deeply embedded helices, with $H_1/D_h>{\left(H_1/D_h\right)}_{cr}$, tend to develop cylindrical failure surfaces that remain entirely confined within the soil. This enhances uplift capacity by increasing confinement and enabling more effective mobilization of shaft friction and helix–soil interface resistance. In sandy soils, the critical embedment ratio ${\left(H_1/D_h\right)}_{cr}$ is influenced by the internal friction angle of the soil [43].

Additionally, a transitional failure mode (TFM) may occur between the idealized CSM and IBM, especially in double-helix piles with intermediate spacing. This intermediate mechanism features partial helix interaction and a non-idealized evolving failure surface. Factors such as helix pitch, embedment ratio, soil density, and confinement play a critical role in influencing this transitional behavior.

6. Test Results

This part provides an analysis of the test findings, specifically focusing on the axial load versus displacement curves, the various failure modes seen, and the prediction methods of bearing capacities.

6.1. Axial Compressive Load Test Results

The load–displacement behavior for piles with varying numbers of helices (n = 0, 1, 2, and 3) is illustrated in Figure 5a. For the conventional pile (PC1), the minimum load required to initiate displacement is approximately 0.4 kN. As the pile penetrates deeper soil layers, the applied load increases gradually, reaching approximately 0.70 kN at a displacement of about 10 mm. Beyond this point, a slight increase in load induces a substantial displacement. The ultimate bearing capacity of PC1 is around 1.2 kN, corresponding to a settlement of nearly 67 mm.

Similarly, Figure 5a shows that the single-helix pile (PC2) also initiates displacement at around 0.4 kN. Following a brief settlement phase, a minor concavity appears at approximately 1 kN, after which, the load–displacement curve peaks. Beyond this point, the presence of the helix becomes significant, enhancing resistance to further settlement as the load increases. The bearing capacity in this case exceeds 9 kN, attributable to the interlocking mechanism between the helix and surrounding soil.

These results indicate that helical piles significantly reduce settlement and enhance bearing capacity due to improved soil–pile interaction. The load–displacement trends for other helical piles (PC3–PC6) follow a similar pattern to that of PC2. As shown in Figure 5b, the double-helix pile (PC4) initially exhibits a sudden drop in the load–displacement curve at 7 kN, likely due to a loss of soil integrity and rapid slippage. However, the load resistance subsequently recovers, reaching a higher capacity as a result of helix engagement.

The maximum capacities of PC3 and PC4 are approximately seven times greater than that of the conventional pile (PC1). Furthermore, as depicted in Figure 5c, PC5 and PC6—each with three helices—exhibit maximum capacities approximately eleven times greater than PC1. At peak load, the head displacement of PC1 is about three times higher than that of PC3, PC4, and PC6. From Figure 5, the ultimate bearing capacities of helical piles with one and two helices are in the range of 8–9 kN, while the triple-helix configuration achieves a capacity of approximately 13 kN. According to Chen et al. [47], when the maximum bearing capacity is reached, the lowest helix is particularly susceptible to failure, which may lead to a decline in the load-bearing performance. The observed settlements for helical piles range from roughly 20 mm to 65 mm. As illustrated in Figure 5a, the two-helix pile exhibits about one-third of the settlement observed in the single-helix pile at ultimate load. The addition of a third helix increases the bearing capacity by approximately 50%, while reducing the settlement to nearly half that of the double-helix pile under comparable loading. This confirms that increasing the number of helices enhances the bearing capacity while mitigating the settlement.

Additionally, Figure 5b,c indicates that piles installed using the impact-based method (IBM), such as PC4 and PC6, display greater initial stiffness and reduced settlement, particularly in dense sands, due to improved interaction with the surrounding soil and minimized helix interference. In contrast, piles installed using the continuous screw method (CSM), such as PC3 and PC5, experience larger displacements prior to reaching ultimate capacity. Nevertheless, they still demonstrate substantial load-carrying capacities, underscoring the influence of mobilized soil mass. Notably, for double-helix piles, IBM-induced settlement is generally lower than that resulting from CSM installation. The triple-helix pile shows more in terms of this distinction. In comparison to the CSM, the settlement of the helical pile with three helices is reduced about 60% in the IBM.

Generally, this demonstrates the importance of the initial stiffness of the load–displacement curve of the helical piles controlled by helical spacing. The bearing capabilities in the two modes are practically identical. Finally, based on the nonlinear regression analysis, an empirical polynomial equation (Equation (10)), derived from Equation (7), has been developed to estimate the capability of various helical pile configurations. The equation is presented in a dimensionless form, allowing its application to full-scale (prototype) conditions:

$$\begin{array}{l}{Q}_d={\displaystyle \frac{Q}{E_p\times d^2}}=a_1+a_{2}n+a_3\left({\displaystyle \frac{S}{d}}\right)+a_4\left({\displaystyle \frac{d_s}{d}}\right)+a_5{n}^2+a_6{\left({\displaystyle \frac{S}{d}}\right)}^2+a_7{\left({\displaystyle \frac{d_s}{d}}\right)}^2\\ +a_{8}n\left({\displaystyle \frac{S}{d}}\right)+a_{9}n\left({\displaystyle \frac{d_s}{d}}\right)+a_{10}\left({\displaystyle \frac{S}{d}}\right)\left({\displaystyle \frac{d_s}{d}}\right)+a_{11}{\left({\displaystyle \frac{d_s}{d}}\right)}^3,\end{array}$$

(10)

where $a_i, (i=1, 2, \dots , 11)$ are dimensionless regression constants. Among several nonlinear regression functions evaluated, the model demonstrating the highest coefficient of determination (R²) was selected, and its corresponding parameters were calibrated.

Table 2. The values of constants in Equation (10) for different pier conditions.

Variable	Estimate	Standard Error	t-Statistic	p-Value	Interpretation
a1	1.91 × 10−6	3.76 × 10−6	0.508128	0.6116	Not statistically significant
a2	−2.85 × 10−5	2.68 × 10−6	−10.6106	5.77 × 10−24	Highly significant
a3	−8.98 × 10−6	1.13 × 10−6	−7.96189	1.05 × 10−14	Highly significant
a4	1.93 × 10−4	7.10 × 10−6	27.1235	5.33 × 10−102	Highly significant
a5	−1.56 × 10−5	2.17 × 10−6	−7.16558	2.63 × 10−12	Highly significant
a6	−5.85 × 10−7	8.29 × 10−8	−7.05416	5.50 × 10−12	Highly significant
a7	−6.96 × 10−5	3.82 × 10−6	−18.2059	8.53 × 10−58	Highly significant
a8	1.20 × 10−5	6.87 × 10−7	17.4566	3.62 × 10−54	Highly significant
a9	2.23 × 10−5	1.96 × 10−6	11.3685	6.25 × 10−27	Highly significant
a10	1.40 × 10−7	3.67 × 10−7	0.381391	0.7031	Not statistically significant
a11	7.74 × 10−6	5.34 × 10−7	14.4957	2.64 × 10−40	Highly significant

presents the calibrated values of constants for a helical pile, along with their influence on the model response. The predictive accuracy of the equation is illustrated in Figure 6, which compares the experimental results with the model predictions. As shown, most data points align closely with the line of best fit, confirming the high reliability and precision of the proposed empirical model.

Notable discrepancies persist for non-helical piles, particularly specimen PC1. To enhance the reliability of the predictive model, data from non-helical piles were excluded from the regression analysis, as Equation (7) was specifically formulated for piles incorporating helices. The divergence observed in the PC1 results can be attributed to the following factors:

• PC1, being a plain shaft pile without helices, relies primarily on shaft friction for load transfer. In contrast, helical piles mobilize both shaft friction and bearing resistance through their helix–soil interaction. As a result, the current model, which assumes enhanced resistance due to helices, underestimates the settlement of PC1, especially under large deformations.
• The presence of helices in helical piles provides additional anchorage and confinement, leading to greater stiffness and reduced displacement. In PC1, the lack of this anchorage permits more significant axial movement and progressive slippage along the shaft. The empirical model, which does not account for this condition, fails to capture this behavior accurately.

6.2. Axial Tensile (Uplift) Load Test Results

In this section, the tensile capacity and displacement behavior of piles equipped with one helix (PT2 to PT4) and two helices (PT5 to PT7) are compared to that of a shaft pile without helices (PT1). Figure 7 presents the applied load versus displacement responses for various pile configurations, including no-helix, single-helix, and double-helix piles, with differing helix pitch values (P_h = 13, 20, and 25 mm). The influence of the helix pitch on the maximum tensile capacity is clearly demonstrated in Figure 7. As observed in Figure 7a–c, the maximum displacement for helical piles ranges approximately from 14 to 30 mm. The maximum tensile capacity reaches 1.267 kN for the double-helical pile PT5, whereas the shaft pile (PT1) consistently exhibits the lowest load-bearing capacity, achieving peak loads near 0.2 kN with minimal displacement (about 2.44 mm), indicative of limited soil–pile interaction and mechanical anchorage. The double-helical piles (PT5–PT7) also exhibit a notably stiffer initial response compared to the PT1.

The helical piles equipped with one and two helices (PT2–PT7) exhibit approximately the same maximum bearing capacity, ranging from 1.0 to 1.2 kN, although the displacement at maximum load for double-helix piles is generally less than that of single-helix piles. Notably, the load–displacement curve for double-helix piles shows an abrupt drop after reaching the peak capacity, whereas the single-helix piles experience a gradual decrease. The failure modes differ correspondingly: double-helix piles exhibit nearly cylindrical failure, while single-helix piles show no cylindrical failure, indicating that single-helix piles perform better at higher displacement levels. Figure 7d,e compare the load–displacement behavior of single- and double-helix piles with different pitch-to-diameter ratios (P_h/d = 1, 1.5, and 2), respectively. When comparing the plain shaft pile PT1 with the single-helix pile PT2 (P_h = 13 mm), the maximum tensile capacity increases from 0.2 kN to 1.12 kN, representing an approximate 450% improvement. This enhancement is attributed to the contribution of the twisting blades, which increase friction and mobilize the weight of the soil above the helix during uplift.

Similar to PT2, the trend in the changes in the curves for other helical piles with different pitches follows the same pattern. The results showed that the maximum uplift capacity of PT3 and PT4 in comparison with PT1 increased by about 520%, and 435%, respectively.

Figure 7d shows spikes on the load–settlement curves. These spikes are due to slight slippage between the jaw of the UTM machine and the middle axis of the pile, despite the measures that were taken for the connection between the pile axis and the machine jaw (such as the working thread of the axis).

The load–deflection curves for piles with a pitch ratio of two (PT4 and PT7) show sharper drops post-peak capacity than those with lower pitch ratios (PT2/PT3 and PT5/PT6), particularly when the helix is located close to the soil surface. The optimal uplift performance under tension is observed when P_h/d ranges between 1 and 1.5, corroborating the findings in prior research. The single-helix piles with P_h/d ratio 1.5, PT3, have a better performance than PT2 and PT4, while the double-helices piles with P_h/d equal to 1, PT5, show better performances than PT6 and PT7. Therefore, when helices are placed at shallower depths, a smaller pitch results in better performance. For the single-and double-helical piles with a pitch of 25 mm, PT4 and PT7, the performance improvement rate is lower compared to the previous cases. The higher pitch causes a higher slope of the blade, which decreases the effect of the weight of the soil above the helices, subsequently decreasing the maximum capacities.

Similarly, based on the nonlinear regression analysis, an empirical polynomial equation (Equation (11)), derived from Equation (7), was developed to estimate the capability of various helical pile configurations under tension load. The following equation is in a dimensionless form that can be used for real prototype cases:

$$\begin{array}{r}{Q}_d = {\displaystyle \frac{Q}{E_p\times d^2}} = a_1+a_{2}n+a_3\left({\displaystyle \frac{P_h}{d}}\right)+a_4\left({\displaystyle \frac{d_s}{d}}\right)+a_5{n}^2+a_6{\left({\displaystyle \frac{P_h}{d}}\right)}^2+a_7{\left({\displaystyle \frac{d_s}{d}}\right)}^2\\ +a_{8}n\left({\displaystyle \frac{P_h}{d}}\right)+a_{9}n\left({\displaystyle \frac{d_s}{d}}\right)+a_{10}\left({\displaystyle \frac{P_h}{d}}\right)\left({\displaystyle \frac{d_s}{d}}\right)+a_{11}{\left({\displaystyle \frac{d_s}{d}}\right)}^3+a_{12}{\left({\displaystyle \frac{d_s}{d}}\right)}^4,\end{array}$$

(11)

where $a_i, (i=1, 2, \dots , 12)$ are dimensionless regression constants.

Table 3. The values of constants in Equation (11) for different piers under tension tests.

Variable	Estimate	Standard Error	t-Statistic	p-Value	Interpretation
a1	−4.0 × 10−5	1.05 × 10−6	−37.8611	1.1 × 10−230	Highly significant
a2	−8.6 × 10−6	5.96 × 10−7	−14.4382	9.02 × 10−45	Highly significant
a3	5.6 × 10−5	1.8 × 10−6	31.0741	9 × 10−170	Highly significant
a4	7.65 × 10−5	1.08 × 10−6	70.7934	0	Highly significant
a5	8.1 × 10−6	3.82 × 10−7	21.2236	2.62 × 10−89	Highly significant
a6	−1.7 × 10−5	5.7 × 10−7	−29.4859	5.7 × 10−156	Highly significant
a7	−4.0 × 10−5	7.5 × 10−7	−53.7552	0	Highly significant
a8	−5.1 × 10−6	4.62 × 10−7	−11.1181	8.22 × 10−28	Highly significant
a9	−9.5 × 10−6	2.64 × 10−7	−36.0566	2.9 × 10−214	Highly significant
a10	−1.2 × 10−6	1.98 × 10−7	−5.86939	5.2 × 10−9	Highly significant
a11	9.77 × 10−6	2.29 × 10−7	42.5791	7.2 × 10−274	Highly significant
a12	−8.2 × 10−7	2.31 × 10−8	−35.4876	4.1 × 10−209	Highly significant

presents the calibrated values of constants for a helical pile. The predictive accuracy of the equation is illustrated in Figure 8, which compares the experimental results with the model predictions. As shown, most data points align closely with the line of best fit, confirming the high reliability and precision of the proposed empirical model.

6.3. Comparison Between Measured and Estimated Pile Capacities

To assess the robustness and accuracy of the proposed empirical model, the predicted compressive and tensile capacities of helical screw piles were systematically compared with those obtained from classical analytical methods—specifically the L1–L2 failure criterion and theoretical formulations [48,49] based on Meyerhof’s breakout theory and Vesić’s bearing resistance model. The L1–L2 criterion [50] estimates the interpreted ultimate load by applying linear extrapolation between offset points along the load–displacement curve. In contrast, the classical theoretical model (TM) is derived from limit equilibrium analysis and assumes that axial capacity is governed by the combination of shaft friction and individual helix bearing resistance. The ultimate capacities of the piles in this study were calculated using both the L1–L2 criterion and TM across different helical configurations, as presented in

Table 4. Theoretical estimation of axial capacity of shaft and helical piles for sand.

Pile Type	Axial Compressive Capacities	Additional Information
PC1	$Q_c={\displaystyle \frac{\pi d^2}{4}}\left(\gamma H_e{N}_q^{*}+{\displaystyle \frac{1}{2}}\gamma D_h{N}_{\gamma }^{*}\right)+{\displaystyle {\int }_0^{He}{p}_s{k}_s}\gamma z\tan \delta dz$	$k_s=2\left(1-\sin \phi \right)$
PC3 and PC5	$\begin{array}{l}{Q}_c={\displaystyle \frac{\pi d^2}{4}}\left(\gamma H_e{N}_q^{*}+{\displaystyle \frac{1}{2}}\gamma D_h{N}_{\gamma }^{*}\right)+{\displaystyle {\int }_{H_t}^{H_b}\pi D_h{k}_s}\gamma \tan \phi dz\\ +{\displaystyle \int p_s}{k}_s\gamma z\tan \delta dz\end{array}$
PC3, PC4, and PC5	$Q_c={\displaystyle {\sum }_i^n\frac{\pi D_h^2}{4}}\left(\gamma H_i{N}_{qi}^{*}+{\displaystyle \frac{1}{2}}\gamma D_h{N}_{\gamma i}^{*}\right)+{\displaystyle \int p_s{k}_s\gamma z\tan \delta dz}$
Pile Type	Axial Tension Capacities
PT1	$Q_T=Q_s={\displaystyle \underset{0}{\overset{H_e}{\int }}\pi d{k}_u\gamma z\tan \delta dz}$	$\begin{array}{l}{F}_q={\left({\displaystyle \frac{H_1}{D_h}}\right)}^2{k}_u\tan \phi \left[{\cos }^2\left({\displaystyle \frac{\phi }{2}}\right)\right]\left[{\displaystyle \frac{2D_h}{H_1}}+{\displaystyle \frac{4}{3}}{\tan }^2\left({\displaystyle \frac{\phi }{2}}\right)\right]\\ +2{\left({\displaystyle \frac{H_1}{D_h}}\right)}^2\tan \left({\displaystyle \frac{\phi }{2}}\right)\end{array}$
PT2, PT3, and PT4	$Q_{hI}+Q_s+W_p={\displaystyle \frac{\pi }{4}}{F}_q{D}_h^2{H}_1+{\displaystyle {\int }_0^{H_1-H_{cr}}\pi d{k}_u\gamma z\tan \delta dz}$	$H_{cr}=6.144D_h$ $W_p=$ Effective self-weight of the pile, supposed to neglect
PT5, PT6, and PT7	${\displaystyle \frac{\pi }{4}}{F}_q{D}_h^2{H}_1+{\displaystyle \underset{H_1}{\overset{H_2}{\int }}\pi D_h}{k}_u\gamma z\tan \delta dz$	$k_u=0.6+m\left({\displaystyle \frac{H_1}{D_h}}\right)\le k_{u\max }, k_{u\max }=2, m=0.226$ For sand $\phi =38 (\mathrm{deg})$
		$Q_s={\displaystyle {\int }_0^{H_1-H_{cr}}\pi d{k}_u}\gamma z\tan \delta dz,H_1>H_{cr}$$; Q_s=0,H_1\le H_{cr}$

.

The corresponding displacements and axial capacities for both shaft and helical piles are summarized in

Table 5. Comparison between measured and estimated axial compressive and uplift capacities.

Pile Type	$\mathit{n}$	$\mathit{S}\left(\mathbf{mm}\right)$	${\mathit{P}}_{\mathit{h}}\left(\mathbf{mm}\right)$	${\mathit{Q}}_{\mathbf{max}}$	${\mathit{d}}_{\mathit{s}\mathbf{max}}$	${\mathit{Q}}_{\mathit{L}1-\mathit{L}2}$	${\mathit{d}}_{\mathit{s}\mathit{L}1-\mathit{L}2}$	${\mathit{Q}}_{\mathit{T}\mathit{M}}$
PC1	0	N.A.	N.A.	1.18	67.4	0.705	10.67	0.64
PC2	1	N.A.	13	9.09	66.1	5.54	26.67	4.04
PC3	2	100	13	8.25	21.9	5.97	13.3	4.1
PC4	2	200	13	8.65	22.7	6.83	12.8	5.90
PC5	3	100	13	13.1	49	6.57	15.86	4.15
PC6	3	200	13	13.3	20.3	7.622	9.41	6.33
PT1	0	N.A.	N.A.	0.20	2.44	0.18	1.24	0.111
PT2	1	N.A.	13	1.12	28.05	0.93	9.95	1.52
PT3	1	115	20	1.26	27.08	1.17	11.58	1.52
PT4	1	115	25	1.09	29.82	0.89	10.19	1.52
PT5	2	115	13	1.27	14.56	1.18	10.06	0.68
PT6	2	115	20	1.17	15.08	1.07	8.3	0.68
PT7	2	115	25	0.99	14.3	0.88	9.11	0.68

Note: $Q_{\max }\left(\mathrm{kN}\right)$: maximum capacity of pile, $Q_{L1-L2}\left(\mathrm{kN}\right)$: ultimate capacity of pile, $Q_{TM}\left(\mathrm{kN}\right)$: theoretical capacity of pile according Table 4, $d_{s\max }\left(\mathrm{mm}\right)$: displacement corresponding to $Q_{\max }$, $d_{sL1-L2}\left(\mathrm{mm}\right)$: displacement corresponding to $Q_{L1-L2}$.

. The results highlight a clear distinction between compressive and tensile behavior: while the compressive capacities are generally higher, both the theoretical and experimental uplift (tensile) capacities were significantly lower—consistent with previous findings in sandy soils.

Notably, the compressive capacities predicted by TM were consistently underestimated relative to L1–L2 estimates. This underprediction is likely due to the model’s inability to account for the interaction between closely spaced helices and enhanced shaft resistance, particularly in dense granular soils. In tensile behavior, the discrepancies were more pronounced. Theoretical predictions overestimated the capacity of single-helix piles—governed by the individual bearing mode (IBM)—by approximately 41% and underestimated the capacity of double-helix piles—dominated by cylindrical shear mode (CSM)—by about 42%. These deviations can be attributed to several key factors not captured in the classical TM formulation. These include the transitional failure mechanism (TFM), which bridges IBM and CSM; the effect of the pitch-to-diameter ratio on the geometry of failure surfaces; and the compound interaction between the shaft and helical plates. Such effects play a critical role in the mobilization of uplift resistance, particularly under varying embedment depths and confining pressures.

While deeply embedded helices typically exhibit similar breakout and bearing factors under tensile and compressive loading, shallow installations often result in reduced uplift resistance due to insufficient overburden confinement. Moreover, the theoretical model neglects the influence of helix pitch—an important parameter affecting soil arching and the mobilization of overlying soil mass, especially under tensile conditions.

Unlike TM, the proposed empirical model incorporates dimensional analysis and has been calibrated against experimental data. This integration allows it to inherently capture the effects of geometry, spacing, and interaction mechanisms through nonlinear regression fitting. As a result, it demonstrates superior predictive accuracy across a range of helical pile configurations.

In summary, while classical analytical methods remain useful for theoretical benchmarking, the empirical model developed in this study provides a more robust, accurate, and practical tool for design purposes—particularly when validated by laboratory testing and future field-scale applications. These findings align with previous studies (e.g., Sakr [1] and Chen et al. [51]) and further emphasize the limitations of classical theory in modeling complex soil–pile interactions, especially under uplift loading conditions.

7. Practical Applications

This study provides practical guidance for designing helical piles under axial compression and uplift loads in dense cohesionless soils, specifically Shahriyar sand. Helical piles offer a cost-effective alternative to conventional deep foundation systems, particularly in areas with limited access or environmental constraints. They can achieve axial capacities up to 11 times higher than plain shafts, making them a practical substitute for larger and more expensive traditional piles. The experimental results demonstrate that the inclusion of helical plates significantly improves the axial load capacity and reduces the settlement when compared to shaft piles. For example, triple-helix piles exhibited compressive capacities up to eleven times higher than non-helical piles, while also showing a substantial reduction in displacement under load. The findings highlight that pile configurations with reduced pitch and helices placed at shallower depths perform more efficiently. Specifically, single- and double-helix piles with pitch-to-diameter ratios of 1.0–1.5 yielded optimal uplift capacity. In contrast, larger pitches were associated with reduced performance due to increased blade inclination and a decrease in soil confinement above the helix.

Engineers can use the empirical polynomial models developed in this study to predict pile displacement under both compressive and tensile loading. These models are based on dimensionless parameters and regression analysis, offering a useful design tool for preliminary assessments without the need for extensive field testing. However, it is important to note that the predictive accuracy of these models is limited to helical piles and does not extend reliably to shaft piles.

The study also clarifies the different failure mechanisms associated with various pile types and loading conditions. Recognizing the dominant failure mode based on geometric and soil conditions is essential for optimizing the performance of helical pile foundations, particularly in sandy soils. Effective design strategies should consider the interaction between pile geometry and soil behavior: closely spaced helices are recommended in dense sands to promote cylindrical shear mechanisms (CSM), enhancing shaft friction, where beneficial. Conversely, applications requiring high initial stiffness and minimal displacement benefit from larger helices with greater embedment to activate individual bearing mechanisms (IBM). Additionally, accurately assessing the critical embedment ratio is vital to ensure sufficient soil confinement under uplift loads, thereby maximizing resistance. Incorporating predictive models that account for transitional behaviors—such as the transitional failure mechanism (TFM)—further improves design accuracy. These considerations emphasize the importance of site-specific design, moving beyond simplistic assumptions to ensure the safety and efficiency of helical pile systems in sandy conditions.

8. Limitations and Research Prospects

This study provides valuable insights into the axial behavior of helical piles in dense cohesionless soils; however, several limitations should be acknowledged, and future research directions are proposed accordingly.

First, the experiments were conducted at a small scale due to laboratory constraints. While dimensional analysis was employed to develop generalized dimensionless relationships, true similitude principles—such as those used in geotechnical centrifuge testing—were not fully applied. Therefore, direct extrapolation of the results to full-scale applications should be approached with caution. Differences in gravity effects, stress scaling, and boundary conditions between laboratory and field settings may influence pile–soil interaction mechanisms and failure responses.

Second, the developed empirical models are specifically calibrated for helical piles and may not be directly applicable to conventional shaft piles. Furthermore, the relatively limited sample size introduces statistical variability, which may affect model reliability. To address this, future research should incorporate probabilistic approaches, such as Monte Carlo simulations [52], to quantify the uncertainty associated with model predictions and improve robustness.

Third, the experimental setup constrained the ability to capture visual documentation of failure mechanisms, particularly in sandy soils where failure surfaces are often indistinct. Although theoretical interpretations of failure modes—namely cylindrical shear mode (CSM), individual bearing mode (IBM), and transitional failure mode (TFM)—were discussed, the integration of advanced imaging techniques, such as transparent soil models or high-resolution digital image correlation, could significantly enhance understanding of soil–pile interaction in future work.

Additionally, the study did not utilize modern data-driven methods such as artificial intelligence (AI) and machine learning (ML) [24], which have shown increasing relevance in geotechnical analysis and prediction tasks. Incorporating these techniques could improve the accuracy and adaptability of predictive models, especially when large-scale datasets become available. For instance, Fang et al. [53] demonstrated the power of generative adversarial networks (GANs) for 3D data reconstruction in complex environments, highlighting their potential for visualizing subsurface interactions and spatial deformation patterns in geotechnical systems. Similarly, Hu et al. [54] employed ML-based optimization of finite element meshes to enhance modeling and prediction of excavation-induced settlement, a methodology that could be extended to simulate helical pile behavior in layered or heterogeneous soils.

Looking ahead, future work will focus on field-scale validation, geotechnical centrifuge modeling, and the incorporation of advanced numerical simulations to improve scalability and applicability. Expanding the experimental dataset and integrating AI/ML frameworks will support the development of more comprehensive and generalizable tools for the analysis, design, and optimization of helical piles in varied geotechnical contexts.

9. Conclusions

This study investigated the axial performance of helical piles under compression and tension loads through a series of small-scale laboratory experiments supported by theoretical analysis. The findings provide a deeper understanding of how geometric and installation parameters influence pile behavior in dense cohesionless soils. Some important features are as follows:

(1)

Increasing the number of helices significantly improves the load-bearing performance. Triple-helix piles exhibited compressive capacities up to 11 times higher than non-helical piles, while also reducing the settlement by over 60% compared to single-helix piles. This improvement is attributed to enhanced soil–helix interaction and increased resistance.

(2)

Load–displacement curves showed that piles operating in IBM generally exhibited stiffer responses and smaller settlements than those in CSM, particularly for double- and triple-helix piles. The distinction between IBM and CSM is governed by the helix spacing and depth, emphasizing the importance of geometric configuration in design.

(3)

Compressive capacities of helical piles were consistently higher than their tensile (uplift) counterparts. The maximum uplift capacity increased by more than 500% in piles with helices compared to shafts, mainly due to additional frictional resistance and overburden confinement mobilized during pull-out.

(4)

Optimal performance was observed in piles with pitch-to-diameter ratios between 1.0 and 1.5. Shallower helix placement also contributed to better load transfer and reduced displacements. Piles with larger pitch values showed a sharper drop in load-bearing after peak capacity, indicating reduced confinement and effectiveness.

(5)

Empirical equations based on dimensionless parameters were developed to estimate pile capacities. These models showed high predictive accuracy but are limited to helical pile configurations. Comparisons between the theoretical and experimental results revealed that the theoretical method (TM) underestimates the compressive capacity and inconsistently predicts the uplift performance due to its simplified assumptions about failure modes (IBM, CSM, and TFM).

(6)

The outcomes of this research have direct implications for the design and optimization of helical pile foundations. Understanding the interaction between helix geometry, failure mode, and loading type can lead to more efficient foundation systems in dense sandy soils. The results also highlight the need for more advanced modeling approaches that integrate transitional failure mechanisms and account for nonlinearity in soil–pile behavior.

It could be mentioned that these findings are based on model-scale experiments in Shahriyar dense sand and should be applied cautiously to field-scale scenarios without proper scaling or validation.