The Study of the Failure Envelopes of Hollow Shaft Single-Plate Helical Piles in Clay

Abstract

Helical piles are utilized worldwide as foundations for onshore infrastructure, providing fast installation and high capacity. Their unique advantages have led to their potential as an alternative to monopile foundations for offshore wind turbines. One challenge in their service life is dealing with combined loading caused by wind, waves, current, and the weight of the structure itself. While research has focused on helical piles’ capacity for uniaxial tensile loads, there is limited knowledge regarding their performance under combined loads. This study used FEM analysis to investigate the impact of aspects such as helix-to-shaft diameter ratios and helix position on the capacity and failure mechanisms of hollow shaft single-plate helical piles in clay. With 561 analysis cases under both uniaxial and combined loading, failure envelopes were evaluated for various helix-to-shaft diameter ratios and positions. The study revealed a linear positive correlation between helix-to-shaft diameter aspect ratios and load-bearing ability, while the effect of helix positioning on failure envelopes was more complex and nonlinear. The outcomes from this comprehensive analysis enabled the development of a formula to predict the bearing capacity of helical anchors under combined loading.

1. Introduction

Helical piles are widely used all over the world [1,2] to support or anchor light onshore structures, such as telecommunication towers [3] and low-rise buildings [4]. Normally, a typical helical pile consists of a steel central shaft and one or few helixes welded on it, as shown in Figure 1. Due to the existence of the helix, helical piles could be installed quickly and silently with torque and press, which causes less disturbance for marine mammals than conventional pile driving [5]. During service, helical piles can provide a larger capacity than normal piles because of the embedded helix [6,7,8]. Due to these advantages of helical piles mentioned above, it was treated as a kind of novel foundation for fixed offshore wind turbines [9,10,11].

For the design of foundations for fixed offshore wind turbines, the first step is to estimate the maximum loads on the foundation due to all possible design load cases and compare them with the capacity of the chosen foundation [12]. In the service phase, foundations for fixed offshore wind turbines are normally subjected to a combination of vertical loading (F_V) caused by the self-weight of the wind power structure, horizontal loading (F_H), and moment loading (M) caused by wind, wave, and current. In the ocean, kaolinite clay is a common marine sediment [13] and is especially prevalent in the South China Sea [14]. Various studies, including centrifuge [15,16] and 1 g model tests [17,18], in situ tests [19,20,21], and numerical analyses [22,23] have investigated the bearing capacity of helical piles in kaolinite clay. However, most existing studies have focused only on uniaxial tensile loads, and the understanding of their performance under combined loading is limited. More research is necessary to fully examine their bearing capacity for offshore wind turbines.

In recent decades, the ultimate capacity of foundations under combined loading were assessed by a failure envelope approach, first introduced by Roscoe and Schofield [24], which defines the n-dimensional combination of loads (n ≥ 1) that results in the ultimate limit state (or plastic failure) of a foundation [25]. This assessment has been widely adopted to analyze the foundation bearing capacity for different foundation types, such as the bearing capacity of bucket foundations [26,27,28,29,30], that of spudcan foundations [31], that of monopile foundations [32], and that of mudmat foundations [33,34]. For helical piles, the failure envelope approach can be also treated as an option to assess their bearing capacity under combined loading.

This study employs the finite element software (ABAQUS) to conduct a failure envelope approach in order to assess the bearing capacity of helical piles under combined loading. To achieve this goal, the research team conducted 51 cases of the uniaxial bearing capacity of hollow shaft single-plate helical piles to illustrate the variation trend of the uniaxial bearing capacity and relevant failure mechanisms. Additionally, 561 analysis cases are carried out to construct the failure envelopes and develop formulas for predicting the bearing capacity of hollow shaft single-plate helical piles under combined loading.

2. Numerical Modelling Aspects

2.1. Geometries of Hollow Shaft Helical Piles and Soil Properties

Considering that almost 80% of foundations of fixed offshore wind turbines are steel-pipe piles [35], it is also treated as a reference. In addition, the size of the steel-pipe pile is the same as that of the shaft of the helical pile, and just hollow shaft single-plate helical piles are investigated in the present study. The hollow shaft single-plate helical piles consist of a hollow shaft and one helix and are treated as a rigid body. Considering the typical ratio between pitch and diameter of the helix should be just 1/20, according to the design guidance [1] for helical piles; the influence of pitch on the bearing capacity could be ignored and the helix is simplified as a circular plate. The geometry of a hollow shaft single-plate helical pile is specified by the outer diameter of the hollow shaft (D_S), the diameter of the plate (D_H), the distance from the bottom of the shaft to the soil surface (L_S), the distance from the bottom of the plate to the soil surface (L_H), the wall thickness of shaft (t_S), and the thickness of plate (t_H), as shown in Figure 1, considering the length-to-diameter aspect ratio of steel-pipe piles for offshore wind turbines is typically in the scope of 4~8 [36,37,38]. In this study, the length-to-diameter aspect ratio for the steel-pipe pile used as a comparison is set as six, The length of the steel-pipe pile (L), the outer diameter (D_M) and the wall thickness (t_M) of the steel-pipe pile were kept constant as 6 m, 1 m and 50 mm, respectively. The dimension of the hollow shaft for the helical pile is the same as that of the steel-pipe pile.

The elastic perfectly-plastic Tresca material is used to model soil. A uniform undrained shear strength (s_u), which is set equal to 10 kPa, is considered. The submerged effective unit weight of soil (γ’) is set equal to 6 kN/m³. The friction angle and dilation angle are both set at 0°. The Young’s modulus (E) and Poisson’s ratio (ν) are set as 5000 s_u and 0.49, respectively (to model undrained conditions with minimal volume change but maintain numerical stability). The relevant parameters about piles and soil are collected and shown in

Table 1. Summary of the parameters about piles and soil.

Category	Parameter	Value
Steel-pipe pile	The outer diameter of monopile DM, (m)	1.0
	The length of monopile L, (m)	6.0
	The embedment ratio L/DM	6.0
	The wall thickness of the monopile tM, (m)	0.05
Helical pile	The outer diameter of hollow shaft DS, (m)	1.0
	The diameter of plate DH, (m)	1.5, 2.0, 2.5, 3.0
	The distance from the bottom of shaft to soil surface LS, (m)	6.0
	The distance from the bottom of helix to soil surface LH, (m)	6.0, 4.5, 3.0, 1.5
	The wall thickness of shaft tS, (m)	0.05
Soil	Soil strength su, (kPa)	10
	Young’s modulus E, (kPa)	5000
	The Poisson’s ratio of soil ν	0.49
	The submerged effective unit weight of soil γ′ (kN/m3)	6

.

2.2. Finite Element Model

Three-dimensional finite-element modelling is carried out by using the finite element package ABAQUS [39]. Considering the symmetry of the structure and the load, a half model is used to improve the computation efficiency. As one typical FE model shown in Figure 1, the soil domain, which is a half cylinder, is set as 20 D_S for diameter and 2 L_S for height to avoid boundary effect [40]. For soil seabed, the bottom plane is fully fixed, the top plane is free and the outer side of the cylinder is constrained in the x and y directions. For the plane of symmetry, the movement in the normal direction is fixed. Eight-node linear hybrid brick element C3D8H [39] was adopted for soil and an eight-node linear brick element with reduced integration C3D8R is adopted for both the helical pile and steel-pipe pile.

The interface between a single-plate helical pile (or steel-pipe pile) with soil is defined by the mechanical contact properties, including normal and tangential behaviours of the contacting surface [41]. No separation is permitted in the normal direction to consider the influence of suction, and the tangential behaviour is simulated by Coulomb friction.

In order to obtain the ultimate bearing capacity and the relevant failure envelopes of the helical pile and steel-pipe pile, the displacement-control ‘probe test’ approach [28] was adopted by applying a given displacement at the reference point (RP) in this study. Two batches of probe tests are conducted in this study. The first batch is conducted to obtain the uniaxial ultimate bearing capacity of the steel-pipe pile and helical piles with different configurations by applying a displacement in the uniaxial direction without restricting the displacement in the other two directions. For the vertical, horizontal, and moment loads F_V, F_H, and M, and the corresponding displacement w, u, and θ, they are illustrated in Figure 1. The uniaxial bearing capacity factors N_FV, N_FH, and N_M are expressed as the ultimate loadings F_Vult, F_Hult, and M_ult normalized by the pile dimension (D_S·L_S or D_S·L_S·L_S) and the soil shear strength, s_u, i.e., N_FV = F_Vult/(D_S·L_S·s_u), N_FH = F_Hult/(D_S·L_S·s_u), and N_M = M_ult/(D_S·L_S·L_S·s_u). The second batch is conducted under different displacement ratios (w/u, w/θ, and u/θ) to obtain a series of failure points to characterize the failure envelope. To guarantee the accuracy and efficiency, the mesh sensitivity was conducted and the mesh used was found to represent a good balance between accuracy and computational efficiency. Fine meshes were used around the target pile to improve the accuracy of the analysis, where the minimum element size was D_S/50, D_S/5, and D_S/10 in the target pile’s radial, embedment, and circumferential directions, respectively.

3. Results and Discussion

3.1. Model Validation

Considering the lack of a study about the bearing capacity of a hollow helical pile in clay, the validation of the model used in the present study is conducted by the comparison between current research results of steel-pipe pile with that of existing studies.

Five existing studies are used in the validation part and the bearing capacity factors in vertical and horizontal directions (N_FV and N_FH) are used as the standard of comparison. The length of the steel-pipe pile (L), the outer diameter (D_M), and the wall thickness (t_M) of the steel-pipe pile used for validation are constant at 6 m, 1 m, and 50 mm, respectively. The corresponding results are collected and shown in

Table 2. Comparison with current research results.

	The Current Results	Existing Results			Deviation (%)
NFV	3.90	4.06 (a)	4.26 (b)	3.82 (c)	−3.94	−8.45	2.09
NFH	4.10	4.20 (a)	4.23 (d)	4.37 (e)	−2.38	−3.07	−6.18

(a): FE results [32]; (b): FE results [42]; (c): Results obtained from API [43]; (d): Upper bound results [44]; (e): Upper bound results [45].

.

It can be seen from Table 2 that all deviations of the current results from the existing investigation do not exceed ±8.45%. Consequently, both numerical models of the helical pile and the steel-pipe pile are viable and will be used afterwards.

3.2. Uniaxial Ultimate Bearing Capacity

Parametric analysis is conducted in this section to investigate the effects of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the position of the helix (L_H/L_S) on the uniaxial ultimate bearing capacity of helical piles and compared with that of monopile. A total of 51 cases of uniaxial bearing capacity are carried out, and the corresponding results are collected and shown in

Table 3. Analysis cases for uniaxial ultimate bearing.

No.	Case Identifier	DH/DS	LH/LS	NFV	NFH	NM
1	E-0	0	0	3.9	4.1	2.8
2	E-1	1.5	1	5.5	4.3	3.1
3	E-2	2.0	1	8.1	4.6	3.5
4	E-3	2.5	1	11.5	4.9	3.9
5	E-4	3.0	1	15.6	5.2	4.4
6	E-5	1.5	0.75	5.6	4.1	2.9
7	E-6	2.0	0.75	8.2	4.1	2.9
8	E-7	2.5	0.75	11.4	4.2	3.0
9	E-8	3.0	0.75	15.6	4.4	3.1
10	E-9	1.5	0.5	5.6	4.2	2.8
11	E-10	2.0	0.5	8.1	4.4	2.9
12	E-11	2.5	0.5	11.3	4.6	2.9
13	E-12	3.0	0.5	14.8	4.9	3.0
14	E-13	1.5	0.25	5.8	4.3	2.9
15	E-14	2.0	0.25	7.8	4.7	3.1
16	E-15	2.5	0.25	10.2	5.3	3.6
17	E-16	3.0	0.25	13.0	6.0	3.7

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3.2.1. Failure Mechanisms under Uniaxial Load

The effects of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the position of the helix (L_H/L_S) on the failure mechanisms of helical piles under different uniaxial loads are discussed in this section. The typical results of the incremental displacement vectors and relevant contour plots of helical piles and pipe piles at failure under different uniaxial loads are illustrated in Figure 2 and Figure 3.

As shown in Figure 2, the failure mechanism of the steel-pipe pile under vertical load (E-0) is shaped as a downward-moving cylinder, the vertical displacement of soil located inside and bottom of the pipe pile is obvious, that of soil located outside of the pipe pile is unapparent. Under horizontal and moment loads, the steel-pipe pile (E-0) has the similar failure mechanism reported by Murff and Hamilton [46], which shows that the steel-pipe pile rotates along a rotation centre located on the steel-pipe pile, and two wedges, one behind the pile moving direction because of tension and one in front of a pile moving direction because of pushing, form near the soil surface. The difference between the mechanism of a steel-pipe pile under horizontal load and that under moment load is the position of the rotation centre and the following changes in the shape of the mechanism. The position of the rotation centre (~1/2 L_S) under moment load is closer to the soil surface than that (~3/4 L_S) under horizontal load.

The failure mechanisms of helical piles (E-1~E-4) with different helix-to-shaft diameter aspect ratios (D_H/D_S) vary from those of steel-pipe piles because of the presence of the plate. Under vertical load, the most distinctive difference between the failure mechanism of a steel-pipe pile with that of a helical pile is the additional interaction between soil and plate. The influence area gradually enlarges with the increase of the helix-to-shaft diameter aspect ratio (D_H/D_S), especially the compressive zone under the plate and the tensile region upside the plate, which extends to the position ~1/4 L_S away from the soil surface. Under horizontal and moment loads, the shape of the influence area and the changing trend of the helical piles are similar to that of the steel-pipe pile. With the existence of the plate, more soil around the plate is motivated. This causes the enlarged influence area and the rotation centre movement towards the plate. The influence area enlarges and the position of the rotation centre moves downward with the increase of the helix-to-shaft diameter aspect ratio (D_H/D_S). For the influence of the position of the helix (L_H/L_S), some typical results are shown in Figure 3.

As illustrated in Figure 3, the position of the helix (L_H/L_S) also has a significant effect on the failure mechanisms of helical piles under different uniaxial loads. Under vertical load, the soil above the upper surface of the plate gradually moves down with the decrease of L_H/L_S, and the compressive zone under the plate gradually shrinks to the shaft. Under horizontal load, the shape of the influence area is similar to that of the steel-pipe pile but the position of the plate changes the position of the rotation centre and the influence area. With the decrease of L_H/L_S, the distance between the rotation centre and the plate decreases first, as shown in E-4 and E-8, and then increases, as shown in E-8, E-12, and E-16. The size of the influence area also has a similar changing trend and relation with the position of the rotation centre, it also decreases firstly, as shown in E-4 and E-8, and then increases, as shown in E-8, E-12, and E-16. The closer between the plate and the rotation centre, the smaller the influence area. Under moment load, the small influence area is also corresponding to the short distance between the plate and the rotation centre. Furthermore, the rotation center moves upward along with the plate.

3.2.2. Effect of the Helix-to-Shaft Diameter Aspect Ratio on Uniaxial Ultimate Bearing Capacity

The effect of the helix-to-shaft diameter aspect ratio (D_H/D_S) on pure uniaxial ultimate bearing capacity is analyzed in this section. The relevant load-displacement curves of helical piles and the steel-pipe pile under different uniaxial loads with different helix-to-shaft diameter aspect ratios (D_H/D_S) are shown in Figure 4.

As shown in Figure 4, the displacement under different loading conditions normalized by the outer diameter of the steel-pipe pile (D), and soil stiffness ratio (E/s_u) [32,47] is the horizontal axis. For different uniaxial loads, they are characterized by the corresponding uniaxial bearing capacity factors N_FV, N_FH, and N_M as the vertical axis, and the uniaxial bearing capacity is obtained when the gradient of the load-displacement curve approaches zero. For the steel-pipe pile, N_FV, N_FH, and N_M are 3.9, 4.1, and 2.8, respectively. For helical piles, N_FV are 5.5, 8.1, 11.5, and 15.6 when D_H/D_S = 1.5, 2, 2.5, and 3; N_FH are 4.3, 4.6, 4.9, and 5.2 when D_H/D_S = 1.5, 2, 2.5 and 3; N_M are 3.1, 3.5, 3.9 and 4.4 when D_H/D_S = 1.5, 2, 2.5 and 3. These results show that the uniaxial load-carrying properties of helical piles are better than that of the steel-pipe pile. To quantitatively denote the effect of the helix-to-shaft diameter aspect ratio (D_H/D_S) on the uniaxial bearing capacity of piles, the normalized capacity factors, (η_FV, the normalized vertical bearing capacity factor; η_FH, the normalized horizontal bearing capacity factor; η_M, the normalized moment bearing capacity factor), which are the ratio between the uniaxial bearing capacity factors of helical piles and that of the steel-pipe pile, are adopted. The corresponding results are collected and shown in Figure 5.

It can be seen from Figure 5, the normalized capacity factors in different loading conditions all increase with the rise of D_H/D_S when L_H/L_S is fixed. However, the effect degree of D_H/D_S is different on different uniaxial ultimate bearing capacities, the degree of influence could be expressed by the sequence vertical > moment > horizontal. For example, when L_H/L_S = 1 and D_H/D_S changed from 0 to 3, the normalized capacity factors η_FV, η_FH, and η_M are changed from 1 to 4.0, 1.3, and 1.6, respectively.

3.2.3. Effect of the Position of Helix on Uniaxial Ultimate Bearing Capacity

To analyze the effect of the position of the helix (L_H/L_S) on uniaxial ultimate bearing capacity, the relevant normalized load-displacement curves of helical piles under different uniaxial loads with different positions of the helix (L_H/L_S) and those of monopile are collected and shown in Figure 6.

As shown in Figure 6, the uniaxial load-carrying properties of helical piles are still much better than that of the steel-pipe pile when the plate is located in a different position. However, there is no obvious linear relationship between the corresponding uniaxial bearing capacity factors with L_H/L_S. When the D_H/D_S is fixed equal to 3, N_FV are 15.6, 15.6, 14.8, and 13.0 with L_H/L_S = 1.0, 0.75, 0.5, and 0.25; N_FH are 5.2, 4.4, 4.9, and 6.0 with L_H/L_S = 1.0, 0.75, 0.5, and 0.25; N_M are 4.4, 3.1, 3.0; and 3.7 with L_H/L_S = 1.0, 0.75, 0.5 and 0.25. To quantitatively denote the effect of the position of the helix (L_H/L_S) on the uniaxial bearing capacity of piles, the corresponding normalized capacity factors, η_FV, η_FH, and η_M, are collected and shown in Figure 7.

It can be seen from Figure 7 that the relationship between the size of L_H/L_S and the changing amplitude of the normalized capacity factors η_FV, η_FH, and η_M is different from that shown in Figure 5. For η_FV, it has a trend that it gradually reduces with the decrease of L_H/L_S. This is because of the different failure mechanisms with different L_H/L_S and the corresponding different influence areas, as shown in Figure 3. For η_FH and η_M, as shown in Figure 3b,c, the nonlinear relationship between the size of L_H/L_S with them is obvious; the position of L_H/L_S = 0.75 could be treated as a critical point to distinguish positive and negative on η_FH and η_M. This behaviour is also corresponding to the failure mechanism shown in Figure 3, the relevant influence area under position L_H/L_S = 0.75 is smaller than that of the others. In addition, the position of the plate is nearest to the position of the rotation centre.

3.3. F_V-F_H-M Failure Envelopes

In this section, the effects of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the position of the helix (L_H/L_S) on both the size and shape of the failure envelopes are systematically investigated. A total of 561 analysis cases, as collected and listed in

Table 4. The summary of analysis cases.

Analysis Batch	Analysis Case	FV-FH-M	DH/DS	LH/LS	w/u	w/θ	u/θ	Number of Cases
1	1–17	Pure FV	0, 1.5, 2.0, 2.5,3.0	0, 0.25, 0.5, 0.75, 1	+∞, 100, 10, 2, 1, 0.5, 0.33, 0.2, 0.1, 0	+∞, 100, 10, 2, 1, 0.5, 0.33, 0.2, 0.1, 0	+∞, 5.5, 5, 4.5, 4, 3.5, 2, 1, 0.1, 0	17
	18–34	Pure FH						17
	35–51	Pure M						17
2	52–221	Varying FV-FH						170
	222–391	Varying FV-M						170
	392–561	Varying FH-M						170

, were conducted to obtain the failure points used to construct the failure envelopes. The corresponding results are collected and shown in Figure 8, Figure 9, Figure 10 and Figure 11.

3.3.1. Effect of the Helix-to-Shaft Diameter Aspect Ratio on Failure Envelopes

In order to investigate the effect of the helix-to-shaft diameter aspect ratio (D_H/D_S) on the load-carrying property of helical piles and the steel-pipe pile under combined loading, the corresponding dimensionless failure envelopes are illustrated in Figure 8.

As shown in Figure 8, the sizes of envelopes for helical piles under different loading combinations all increase with the increase of D_H/D_S. For the horizontal and moment capacities of helical piles and the steel-pipe pile, both decrease obviously with the increase of vertical force, as shown in Figure 8a,b. For the F_H-M failure envelopes of helical piles and the steel-pipe pile shown in Figure 8c, they have a similar unsymmetrical shape; this is because of the adverse coupling effect of F_H and M in different directions. For further investigation about the effect of D_H/D_S on the shape of failure envelopes, the corresponding failure envelopes are normalized and shown in Figure 9.

It can be seen from Figure 9, the normalized F_V-F_H failure envelope shrinks inwards as the helix-to-shaft diameter aspect ratio (D_H/D_S) increases, and the shape of the normalized F_V-F_H, F_H-M, and F_H-M failure envelopes almost keeps constant.

3.3.2. Effect of the Position of Helix on Failure Envelopes

In order to investigate the effect of the position of the helix (L_H/L_S) on the load-carrying property of helical piles and steel-pipe pile under combined loading, the dimensionless failure envelopes are illustrated in Figure 10.

As shown in Figure 10, the effect of the position of the helix (L_H/L_S) on failure envelopes is more complex than that of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the corresponding failure envelopes under different loading combinations all have the overlapping condition. This is because of the nonlinear relationship between the size of L_H/L_S with η_FH and η_M, as shown in Figure 3b,c, and the failure mechanism under combined loading should be more complex. When the amount of steel used for the manufacturing of a helical pile (under constant D_H/D_S with fixed shaft) is constant, the position of the plate (L_H/L_S) could adjust the uniaxial ultimate bearing capacity under different directions and the ratio between each other. It is worth noting that the F_H-M failure envelope expands outwards with the increase of L_H/L_S and the behaviour of the helical pile with a plate on the bottom of the shaft (E-4) is best under F_V-M and F_H-M combined loading conditions. For further investigation about the effect of the position of the plate (L_H/L_S) on the shape of failure envelopes, the corresponding failure envelopes are normalized and shown in Figure 11.

It is illustrated in Figure 11 that the position of the helix (L_H/L_S) has a significant effect on the shape of failure envelopes. For the normalized F_V-F_H failure envelope, it gradually shrinks with the decrease of L_H/L_S. However, the law of variation for the normalized F_V-M and F_H-M failure envelopes are relatively vague.

3.3.3. The Establishment of Failure Envelopes

To quantitatively evaluate the effect of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the position of the helix (L_H/L_S) on both the size and shape of the failure envelopes and predict the bearing capacity of hollow shaft single-plate helical piles under combined loading, an equation modified from that in Cassidy et al. [48] and Zhang et al. [31] is proposed below.

$${\left(\frac{\left|F_{\mathrm{V}}\right|}{F_{\mathrm{V},\mathrm{ult}}}\right)}^{{\alpha }_{\mathrm{V}}}+{\left[{\left(\frac{\left|F_{\mathrm{H}}\right|}{F_{\mathrm{H},\mathrm{ult}}}\right)}^{{\alpha }_{\mathrm{H}}}+{\left(\frac{\left|M\right|}{M_{\mathrm{ult}}}\right)}^{{\alpha }_{\mathrm{M}}}-e\frac{F_{\mathrm{H}}}{F_{\mathrm{H},\mathrm{ul}t}}\frac{M_{\mathrm{H}}}{M_{\mathrm{ult}}}\right]}^{{\alpha }_{\mathrm{HM}}}-1=0$$

(1)

where ${\alpha }_{\mathrm{V}},\begin{array}{c}\end{array}{\alpha }_{\mathrm{H}},\begin{array}{c}\end{array}{\alpha }_{\mathrm{M}},\begin{array}{c}\end{array}{\alpha }_{\mathrm{HM}}\begin{array}{c}\end{array}\mathrm{and}\begin{array}{c}\end{array}e$ are fitting parameters used to determine the shape and size of the failure envelope and the corresponding values of these parameters under different conditions are collected and shown in

Table 5. Summary of the parameters used to define failure envelopes.

Case Identifier	DH/DS	LH/LS	αV	αH	αM	αHM	e
E-0	0	0	3.75	3.91	2.24	1.14	3.48
E-1	1.5	1	3.35	3.57	2.19	1.41	−2.70
E-2	2.0	1	2.61	2.79	2.08	2.11	−6.62
E-3	2.5	1	2.52	1.49	1.18	3.56	−0.71
E-4	3.0	1	2.53	1.17	0.97	3.86	−13.28
E-5	1.5	0.75	3.94	0.66	0.72	2.66	−4.14
E-6	2.0	0.75	2.89	1.72	1.21	2.01	−4.33
E-7	2.5	0.75	2.07	6.08	5.54	0.66	−5.26
E-8	3.0	0.75	2.83	1.25	1.63	1.14	−0.99
E-9	1.5	0.5	2.70	9.64	9.32	0.36	−7.70
E-10	2.0	0.5	2.24	1.97	1.71	1.61	−1.82
E-11	2.5	0.5	2.01	2.63	2.89	1.02	−3.60
E-12	3.0	0.5	1.63	1.95	2.03	1.73	−1.81
E-13	1.5	0.25	2.75	3.58	2.84	0.74	−6.57
E-14	2.0	0.25	2.31	1.62	1.65	1.54	−1.30
E-15	2.5	0.25	2.16	1.24	1.74	1.78	−0.89
E-16	3.0	0.25	2.18	1.28	1.81	1.83	−0.92

.

Based on the results in Table 5, the relevant formulas for evaluating these fitting parameters are developed and written as Equation (2).

$$\begin{array}{l}{\alpha }_{\mathrm{V}}=3.85-1.63\cdot D_{\mathrm{H}}/D_{\mathrm{S}}+2.61\cdot L_{\mathrm{H}}/L_{\mathrm{S}}\\ \begin{array}{cc}& \end{array}+0.32\cdot {\left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)}^2-0.38\cdot {\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)}^2-0.57\cdot \left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)\\ {\alpha }_{\mathrm{H}}=4.12-1.75\cdot D_{\mathrm{H}}/D_{\mathrm{S}}+9.86\cdot L_{\mathrm{H}}/L_{\mathrm{S}}\\ \begin{array}{cc}& \end{array}-0.12\cdot {\left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)}^2-10.22\cdot {\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)}^2+1.23\cdot \left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)\\ {\alpha }_{\mathrm{M}}=2.46-0.98\cdot D_{\mathrm{H}}/D_{\mathrm{S}}+11.11\cdot L_{\mathrm{H}}/L_{\mathrm{S}}\\ \begin{array}{cc}& \end{array}-0.15\cdot {\left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)}^2-11.67\cdot {\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)}^2+1.07\cdot \left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)\\ {\alpha }_{\mathrm{HM}}=1.73+0.94\cdot D_{\mathrm{H}}/D_{\mathrm{S}}+4.02\cdot L_{\mathrm{H}}/L_{\mathrm{S}}\\ \begin{array}{cc}& \end{array}-0.64\cdot {\left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)}^2-6.41\cdot {\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)}^2+1.87\cdot \left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)\\ e=2.24-6.98\cdot D_{\mathrm{H}}/D_{\mathrm{S}}+6.31\cdot L_{\mathrm{H}}/L_{\mathrm{S}}\\ \begin{array}{cc}& \end{array}+2.51\cdot {\left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)}^2+2.87\cdot {\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)}^2-6.48\cdot \left(D_{\mathrm{H}}/D_{\mathrm{S}}\right)\left(L_{\mathrm{H}}/L_{\mathrm{S}}\right)\end{array}$$

(2)

Based on the equation of the failure envelope, and that of corresponding fitting parameters, the typical results about E-1 and E-12 are shown in Figure 12. As shown in Figure 12, the agreement between the original bearing capacity points between the failure envelopes expressed by Equations (1) and (2) are good. The maximum error is not more than 4%.

4. Conclusions

In order to predict the bearing capacity of hollow shaft single-plate helical piles under combined loading, a total of 561 cases under different loading conditions were conducted through 3D FEM to investigate the relevant failure envelopes, considering the effect of the helix-to-shaft diameter aspect ratio (D_H/D_S) and the position of the helix (L_H/L_S). The relevant results were also compared with that of a steel-pipe pile. Based on the present study, it could be found that:

Helical piles have greater uniaxial ultimate bearing capacity compared with a steel-pipe pile which has the same size as the shaft of a helical pile. This phenomenon is caused by the larger influence area due to the existence of the helix. The uniaxial ultimate bearing capacity of a helical pile in different directions will increase as D_H/D_S increases, while L_H/L_S remains unchanged. When the helix-to-shaft diameter aspect ratio (D_H/D_S) remains unchanged, the position of the helix (L_H/L_S) has a significant effect on the corresponding failure mode and the uniaxial ultimate bearing capacity of the helical pile has a nonlinear relationship with the size of L_H/L_S.

For the effect of D_H/D_S and L_H/L_S on the envelope, they show a significant difference. For the shape of the normalized failure envelopes, D_H/D_S is almost no influence on them; on the contrary, the effect of L_H/L_S on them is obvious. For the size of the dimensionless failure envelope, the effect of D_H/D_S is a linear positive effect; however, L_H/L_S causes the nonlinear positive effect on failure envelopes. Based on the systematic analysis of the load-carrying properties of helical piles, a formula was proposed to predict corresponding failure envelopes and the accuracy of this formula is also tested. The bearing capacity of hollow shaft single-plate helical piles under combined loading could be evaluated by the expressions of failure envelopes. Further, due to the specificity of the marine environment, the cyclic loading condition caused by wind, wave, and current also needs to be considered in the future study.