Uplift capacity of the helical anchor installed in cohesionless soil using energy balance approach

Abstract

Helical anchors are being considered an efficient alternative in offshore applications due to their advantages of being rapid, easy, and inexpensive to install. Hydraulic motors apply torque to the upper end of the screw shaft during installation. Therefore, the most important parameter to be considered during installation is the applied torque. When installing helical anchors, the uplift capacity can be estimated by knowing the torque or vice versa. Because the uncertainty of helical anchor failure mechanisms makes estimating the uplift capacity of helical anchors difficult, few theoretical studies that directly correlate uplift capacity with installation torque have been published. The aim of this study is to develop a theoretical model that describes the relationship between the uplift capacity and installation torque of helical anchors installed in sand. This model is based on an energy balance between the external work applied to the anchor during the installation process and energy dissipation due to the combined effects of penetration and frictional resistance. The research demonstrates that final installation torque, helical anchor geometry, blade pitch, soil characteristics, and helical anchor displacement all have a major role in the uplift capacity. This theoretical model’s results agree well with several previous experimental and theoretical findings.

Keywords:

• helical anchors
• screw piles
• uplift capacity
• installation torque
• sand
• energy

1. Introduction

Anchor is a structural component that acts to transfer tensile forces from the main structure to the soil layers surrounding the anchor (Maming et al., 2022). Helical or screw piles, often known as anchors, are piled systems made up of one or more helix plates and a central steel shaft (Figure 1). Compared to various anchor alternatives, helical anchors are considered an efficient alternative due to their advantages of being rapid, easy, and inexpensive to install. Additionally, helical anchors have a high holding capacity-to-weight ratio. These elements do not produce waste and create minimal disturbance in the surrounding area of the anchor installation due to the pitch of the helical plate (Young, 2012). Moreover, helical anchors that used in anchored wall have more efficient seismic performance, especially in a higher ground acceleration up to 0.5 g (Olia et al., 2021). A helical anchor acts as an axially loaded end-bearing deep foundation when it is installed in the soil (Chance & Division of Hubbell Power Systems, Inc, 2006; Hoyt et al., 1989). Many offshore structures, such as submarine pipelines and vessels anchored to the sea floor, tension-leg platforms, tension-leg floating wind turbines, and wind turbines founded on tripod or jacket structures, require foundations–anchors with a significant tension capacity, converting moment at the base of the turbine into a compressive/tensile loading applied to the foundations, which are resisted by shaft friction and blade end bearing (Spagnoli & de Hollanda Cavalcanti Tsuha, 2020). However, helical anchors in offshore applications are mostly designed to provide uplifting forces. Despite being a common anchoring solution in offshore applications, helical anchors have not been extensively studied for their use in offshore floating structures, especially in cohesionless soil. For installation, hydraulic motors apply a torque and a downward force to the upper end of the helical screw shaft, which is transferred to the helical plate(s) (Al Hakeem, 2019); therefore, the most important parameter to be considered during helical pile installation is the torque, which is directly proportional to the uplift capacity of helical anchor. The installation effort of helical anchor, measured by torsional resistance, is utilized as a tool to measure the quality of the foundation (C. D. H. C. Tsuha & Aoki, 2010). The torsional resistance to penetration recorded at the end of installation of the helical anchor can be converted into the uplift capacity value. The maximum the torsional resistance during installation, the maximum the uplift capacity of the helical piles. Torsional resistance assessment is influenced by site conditions, pile geometry, and the design embedment depth assumed to achieve the required load capacity (Spagnoli, de Hollanda Cavalcanti Tsuha, Oreste, & Mauricio Mendez Solarte, 2018, 2019). In the last two decades, several researchers have proposed some theoretical methods to estimate the uplift capacity of helical anchors since the uncertainty of helical anchor failure mechanisms and some geometry factors is a problem for the designer when predicting the final uplift capacity (Niroumand et al., 2012). However, very few theoretical studies have been published that directly correlate uplift capacity with installation torque (A. M. Ghaly, 1995; A. Ghaly & Hanna, 1991; A. Ghaly et al., 1991; Perko, 2000). With the goal of determining the capacity/torque ratio of helical piles, Perko (2000) developed a theoretical model that takes into account the energy expended during installation. Perko (2000) presented an expression to predict pullout capacity of helical anchors in terms of variety of variables, including the final installation torque, the downward force, the total number of helical blades, and the pitch of blade. However, according to this model, above-mentioned factors have little effect on the capacity/torque ratio. The model predictions and previous field and laboratory measurements were highly correlated. C. D. H. C. Tsuha and Aoki (2010) presented a simple theoretical expression to calculate the uplift capacity of the helical pile in cohesionless soil depending on the torque resistance. The following equations express the relationship between the uplift capacity of the helical pile and the torque components T_h and T_s.

(1) $\begin{array}{rl}& Q_u=\frac{2T_h}{d_c.t_g(\mathit{\theta }+{\delta }_r)}+\frac{2T_s}{\mathit{d}}\\ & \\ & d_c=\frac{2}{3}\left[\frac{D^3-d^3}{D^2-d^2}\right]\\ & \\ & \mathit{\theta }={tan}^{-1}\left(\frac{p}{\pi d_c}\right)\end{array}$(1)

Figure 1. Multi pitch helical anchor.

where T_h and T_s are the resisting moments from helical plate(s) and the shaft, respectively, d is the shaft external diameter, D is the helical plate external diameter, p is the helix pitch, and δ_r is the interface friction angle between helix material and surrounding sand. The evolving empirical relationship between helical pile capacity and torque is being supported by the development of a theoretical model that relates the uplift and bearing capacity of helical anchors to the installation torque.

In the present study, a theoretical model that relates the pullout and bearing capacity of helical anchor to the installation torque (the sum of T_h and T_s) is presented. This model is based on an energy balance between the external work applied to anchor and energy dissipation during the installation process. As a result, uplift and bearing capacity of helical anchors relates directly to installation torque T based on energy equivalency. The rotating helical pile penetrates a vertical distance equal to the pitch p. Therefore, the torque for every single revolution during installation can be measured, and during installation process of helix pile, these torque readings tell us how strong the soil is at the depth where the blade is passing. This theoretical approach can be used as a method of “field production control” to verify load capacity of helical anchors during installation. In this research, the capacity of helical pile installed in cohesionless soil will be determined taking into account the following parameters:

1. Applied downward force during installation F

2. Geometry of the helical blade (radius R, thickness t)

3. Shaft radius r

4. Blade pitch per revolution p

5. Soil properties φ

6. Helical anchor displacement d.

In the following section, the impact of each parameter will be explained with more details.

2. Model assumptions and derivation

For single revolution and soil local shear, penetration energy $E_{\mathit{Pent}.}$is proportional to the displaced soil volume multiplied by the distance that the volume of soil is moved. This is due to the characteristics of stress–displacement relationship and work done physical concept as shown in Figure 2.

Figure 2. Typical soil stress–displacement curve.

where C is the slope of stress–displacement function as shown in Figure 2. Penetration energy $E_{\mathit{Pent}.}$ is simply defined as in Equation 2.

(2) $E_{\mathit{pent}.}=\underset{0}{\overset{\delta }{\int }}\mathit{\sigma }\mathit{\hspace{0.17em}}\mathit{A}\mathit{\hspace{0.17em}}\mathit{d\delta }{ }^{\mathrm{\prime }}$(2)

where $\mathit{\delta }{ }^{\mathrm{\prime }}$: displacement, δ: final displacement, σ: penetration stress, and A: penetrator area. From soil stress–displacement curve (Figure 2), the penetration stress σ can be written as in Equation 3.(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$(3)

(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$(3)

The initial part of Equation 3(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$(3) is generally linear as shown in Figure 2. Substitute Equation 3(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$(3) in Equation 2(2) $E_{\mathit{pent}.}=\underset{0}{\overset{\delta }{\int }}\mathit{\sigma }\mathit{\hspace{0.17em}}\mathit{A}\mathit{\hspace{0.17em}}\mathit{d\delta }{ }^{\mathrm{\prime }}$(2) to obtain Equation 4(4) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathit{A}\mathit{\hspace{0.17em}}{\delta }^2$(4) :

(4) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathit{A}\mathit{\hspace{0.17em}}{\delta }^2$(4)

Volume of soil displaced due to single revolution in the field $\mathrm{\Delta V}=\mathit{A}\times \mathit{\delta }$ will be substituted in Equation 4(4) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathit{A}\mathit{\hspace{0.17em}}{\delta }^2$(4) to obtain Equation 5.(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5)

(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5)

where C is the constitutive parameter for small displacements typical local shear as in Figure 2. The procedure to derive the helical anchor capacity Q_u installed in sand relating to installation torque T can be divided into two main steps as follows:

Step 1: Determine the value of constitutive parameter c for each single revolution

Installation energy of helical anchor must match the energy needed to break through the soil (penetration resistance) plus the energy loss due to friction (friction resistance). Friction is exerted on the helical anchor (helix plate(s) and along the shaft) and the surrounding soil. The following section derives the constitutive parameter C, utilizing E _installation in Equation 6.(6) $E_{\mathit{installation}}=E_{\mathit{penetration}}+E_{\mathit{losses}}$(6) E _penetration and E _losses will be presented in more details.

(6) $E_{\mathit{installation}}=E_{\mathit{penetration}}+E_{\mathit{losses}}$(6)

2.1. Installation energy E _installation

The machine, which provides the installation energy, comprises two components: rotation energy (installation torque T) provided by the torque motor and downward force F. Typically, during installation, it is a standard procedure to apply a downward force while rotating the anchor pile into the ground. The downward force F overcomes penetration resistance but only at the start of the installation or when the lead helix passes from weak to hard soil (Chance & Division of Hubbell Power Systems, Inc, 2006). A downward force F releases potential energy proportional to the product of the force and the distance, which is equal to the pitch of the blade p. The amount of energy that must be applied to rotate a shaft through one complete revolution is equal to the torque multiplied by the angle of rotation, which is equal to $2\mathit{\pi }$ as in Equation 7.(7) $E_{\mathit{installation}}=2\mathit{\pi T}+\mathit{Fp}$(7)

(7) $E_{\mathit{installation}}=2\mathit{\pi T}+\mathit{Fp}$(7)

where T is the total final installation torque, F is the applied axial downward force, and p is the pitch distance as shown in Figure 3. Return to Equation 5(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5) , $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$

Figure 3. Helical pile installation.

where $\mathrm{\Delta V}$ is the soil volume displaced by the helix pile per one revolution. This volume equals the sum of volumes of all individual blades and the shaft by moving downward for the distance p.

(8) $\mathrm{\Delta V}=\mathrm{\Delta }{V}_{\mathit{helix}}+\mathrm{\Delta }{V}_{\mathit{shaft}}$(8)

(9) $\mathrm{\Delta }{V}_{\mathit{helix}}=\sum _{\mathit{i}=1}^n({R_i}^2-r^2)\mathit{\pi }\mathit{\hspace{0.17em}}{t}_i$(9)

(10) $\mathrm{\Delta }{V}_{\mathit{shaft}}=r^2\mathit{\pi p}$(10)

where R is the radius of helix, r is the shaft radius, t is the helix thickness, n is the number of helices, and p = pitch as shown in Figure 3. Since p is tiny, helical plate volume is almost the same as circular plate volume with same radius. Regarding $\mathit{\delta }$ in Equation 5(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5) , the soil is split and pushed to the sides as a blade passes through it, as seen in Figure 4a. The displacement of soil δ on average is equivalent to roughly half of the helix thickness t. Consequently, an increase in thickness results in a corresponding increase in the amount of work or energy required to push the helix through the soil, meaning that a thick helix requires more soil displacement than a thin helix (Figure 4 a).

Figure 4. Displacement distance for one revolution.

Substitute Equations 9 and(9) $\mathrm{\Delta }{V}_{\mathit{helix}}=\sum _{\mathit{i}=1}^n({R_i}^2-r^2)\mathit{\pi }\mathit{\hspace{0.17em}}{t}_i$(9) Equation 10(10) $\mathrm{\Delta }{V}_{\mathit{shaft}}=r^2\mathit{\pi p}$(10) into Equation 8 and(8) $\mathrm{\Delta V}=\mathrm{\Delta }{V}_{\mathit{helix}}+\mathrm{\Delta }{V}_{\mathit{shaft}}$(8) Equation 5(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5) ,$E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$, where δ= t_i /2 for helical plate penetration and δ = r for shaft penetration as mentioned earlier.

$E_{\mathit{pent}.}=E_{\mathit{pent})\mathit{helix}}+E_{\mathit{pent})\mathit{shaft}}$

$E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}[\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2)\mathit{\hspace{0.17em}}\mathit{\pi }{t}_i\mathit{\hspace{0.17em}}\mathit{\delta }]+\mathit{\hspace{0.17em}}\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}[r^2\mathit{\pi p\delta }]$

(11) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathit{\pi }\mathit{\hspace{0.17em}}[\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2)\mathit{\hspace{0.17em}}\frac{{t_i}^2\mathit{\hspace{0.17em}}}{2}]+\mathit{\hspace{0.17em}}\frac{1}{2}\mathit{C\pi }\mathit{\hspace{0.17em}}[r^2\mathit{p\delta }]$(11)

$E_{\mathit{pent}.}=\frac{\pi }{4}\mathit{C}\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2)\mathit{\hspace{0.17em}}{t_i}^2\mathit{\hspace{0.17em}}+\frac{\pi }{2}\mathit{C}\mathit{\hspace{0.17em}}{r}^3\mathit{p}\mathit{\hspace{0.17em}}$

It can be noticed from Equation 11(11) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathit{\pi }\mathit{\hspace{0.17em}}[\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2)\mathit{\hspace{0.17em}}\frac{{t_i}^2\mathit{\hspace{0.17em}}}{2}]+\mathit{\hspace{0.17em}}\frac{1}{2}\mathit{C\pi }\mathit{\hspace{0.17em}}[r^2\mathit{p\delta }]$(11) that the penetration resistance E_pent increases with helical anchor size (r, R, and t). Additionally, a higher pitch p leads the helix to travel a greater distance every revolution, requiring more work (more E_pent).

2.2. Energy losses E _losses

Frictional energy losses can be evaluated by transforming shear stress to torque and multiplying by $2\mathit{\pi }$, which is the angle of rotation (Perko, 2000). When the helix pile rotates, shear stress along the shaft and blade will be mobilized (Figure 5).

(12) $\mathit{\tau }=\mathit{\alpha }\mathit{\hspace{0.17em}}\mathit{\sigma }\mathit{\hspace{0.17em}}$(12)

Figure 5. Soil side friction due to penetration.

where τ is the shear stress mobilized, and σ is the penetration stress. If the friction angle between the soil and the steel is ranging from 25° to 30°, the friction along the blade and hub is equivalent to 0.5$\mathit{\sigma }$ as shown in Figure 5. In this study, and for comparison purposes with C. H. C. Tsuha et al. (2007) and C. D. H. C. Tsuha and Aoki (2010) as shown in Table 1, the average interface friction angle φ' between the soil and the helical anchor was taken 10.6 and 15.1 for relative densities D_r = 56% and 85%, respectively.

(13) $\mathit{\tau }=\mathit{\sigma }\mathit{\hspace{0.17em}}tan\mathit{\varphi }{ }^{\mathrm{\prime }}=\mathit{\sigma }\mathit{\hspace{0.17em}}tan\mathit{\hspace{0.17em}}(15.1)\mathit{\hspace{0.17em}}\approx 0.27\mathit{\sigma }$(13)

Table 1. Comparison of the present study to C. H. C. Tsuha et al. (2007) experimental results and C. D. H. C. Tsuha and Aoki (2010) analytical solution

Relative density Dr (%)	Number of helical plates	Geometry of the prototype helical pile C. D. H. C. Tsuha and Aoki (2010)				Torque T (kN.m)	Predicted uplift capacity (kN) present study	Measured uplift capacity (kN) Tusha et al., 2007	Predicted uplift capacity (kN) Tusha & Aoki, 2010
		Shaft diameter (m)	Helical diameter (m)	Pitch of helical p (m)	Shaft length λ
56	1	0.0642	0.214	0.0643	3.1	0.9	13.86	14	30.6
56	2	0.0977	0.326	0.0695	4.6	3.9	80.4	83	110
56	2	0.132	0.44	0.077	6.2	7.2	134.6	113	150.9
85	2	0.0977	0.326	0.0695	4.6	14.4	252.5	270	326.2
85	3	0.0642	0.214	0.0643	3.1	4.9	145.3	122	153.2
85	3	0.0977	0.326	0.0695	4.6	12.6	304.6	302	284.8

Energy losses in the helical pile due to friction resistance can be calculated as the energy loss in each helix blade and shaft as in the following equations:

(14) $E_{\mathit{losses}}=E_{\mathit{losses}\mathit{\hspace{0.17em}}in\mathit{\hspace{0.17em}}\mathrm{blade}}+E_{\mathit{losses}\mathit{\hspace{0.17em}}in\mathit{\hspace{0.17em}}\mathrm{shaft}}$(14)

Frictional resistance along the steel shaft and on the helical plate(s) can be converted to torques as follows:

2.3. Shear stress in blade convert to torque

Since the surface area of the helix plate in contact with the soil increases with the square of the radius R, frictional resistance rises with helix size. Torque due to shear along the blade can be calculated by $\mathit{\alpha }\mathit{\hspace{0.17em}}\mathit{\sigma }\mathit{\hspace{0.17em}}\times \mathrm{surfacearea}\times \mathrm{momentarm}$. Two-third of R is the moment arm for resultant force from the centroid.

2.4. Shear stress in shaft convert to torque

The applied torque from the machine is transmitted to the helix plate(s) by the shaft. The minimum the shaft radius, the minimum the friction during installation. In this study, the variation of shaft resistance along the pile is considered constant. Torque due to shear stress along the shaft can be calculated by multiplying $\mathit{\alpha }\mathit{\hspace{0.17em}}\mathit{\sigma }\times \mathrm{surfaceareaoftheshaft}\times \mathrm{momentarm}(\mathit{r})$.

For circular shaft, which is used in this study, the length of the mobilized shaft experiencing friction is $\mathit{\lambda }$ (effective hub length). Therefore, the calculated E _losses will be as follows:

$E_{\mathit{losses}}=E_{\mathit{loss})\mathit{helix}}+E_{\mathit{loss})\mathit{shaft}}$

(15) $E_{\mathit{losses}}=2\mathit{\pi }\left[\left(2\mathit{\alpha \sigma }\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\pi }\times \frac{2}{3}{R}_i\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2\mathit{\pi }\times \frac{2}{3}\mathit{r})\right)+2\mathit{\pi }{r}^2\mathit{\alpha \sigma }\mathit{\hspace{0.17em}}\mathit{\lambda }\right]$(15)

(16) $E_{\mathit{losses}}=\left[\left(\frac{8}{3}{\pi }^2\mathit{\alpha \sigma }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\right)+4{\pi }^2\mathit{\alpha \sigma }\mathit{\hspace{0.17em}}{r}^2\mathit{\lambda }\right]$(16)

Return to Equation (3)(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$(3) $\mathit{\sigma }=\mathit{C}\mathit{\hspace{0.17em}}\mathit{\delta }$ and substitute $\mathit{\sigma }$ into Equation (16)(16) $E_{\mathit{losses}}=\left[\left(\frac{8}{3}{\pi }^2\mathit{\alpha \sigma }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\right)+4{\pi }^2\mathit{\alpha \sigma }\mathit{\hspace{0.17em}}{r}^2\mathit{\lambda }\right]$(16)

(17) $E_{\mathit{losses}}=\left[\left(\frac{8}{3}{\pi }^2\mathit{\alpha C\delta }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\right)+4{\pi }^2\mathit{\alpha C\delta }\mathit{\hspace{0.17em}}{r}^2\mathit{\lambda }\right]$(17)

Substitute $\mathit{\delta }$ = $\frac{1}{2}\mathit{t}$ for helix and $\mathit{\delta }$ = $\mathit{r}$ for shaft into Equation. 17(17) $E_{\mathit{losses}}=\left[\left(\frac{8}{3}{\pi }^2\mathit{\alpha C\delta }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\right)+4{\pi }^2\mathit{\alpha C\delta }\mathit{\hspace{0.17em}}{r}^2\mathit{\lambda }\right]$(17) to determine E_losses as in Equation. 18.(18) $E_{\mathit{losses}}=\left[\left(\frac{8}{6}{\pi }^2\mathit{\alpha C}\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\mathit{\hspace{0.17em}}{t}_i\right)+4{\pi }^2\mathit{\alpha C}\mathit{\hspace{0.17em}}{r}^3\mathit{\lambda }\right]$(18)

(18) $E_{\mathit{losses}}=\left[\left(\frac{8}{6}{\pi }^2\mathit{\alpha C}\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)\mathit{\hspace{0.17em}}{t}_i\right)+4{\pi }^2\mathit{\alpha C}\mathit{\hspace{0.17em}}{r}^3\mathit{\lambda }\right]$(18)

The magnitude of C for the helical pile can be divided to C_h for helix and C_s for shaft. To find C_h, substitute E _penetration and E _losses values for helix part in Equation. 19.(19) $E_{\mathit{installation})\mathit{\hspace{0.17em}}\mathrm{helix}}=E_{\mathit{penetration})\mathit{\hspace{0.17em}}\mathrm{helix}}+E_{\mathit{losses})\mathit{\hspace{0.17em}}\mathrm{helix}}$(19) Similarly, C_s value can be calculated by Equation 22–Equation 24.(24) $C_s=\frac{2\pi T_s+\mathit{Fp}}{\frac{\pi }{2}{r}^3\mathit{p}+4{\pi }^2\mathit{\alpha }{r}^3\mathit{\lambda }}$(24) (22) $E_{\mathit{installation})\mathit{shaft}}=E_{\mathit{penetration})\mathit{shaft}}+E_{\mathit{losses})\mathit{shaft}}$(22) Then, the total value of C can be found by adding C_h and C_s as in Equation. 25.(25) $\mathit{C}=C_h+C_s$(25)

(19) $E_{\mathit{installation})\mathit{\hspace{0.17em}}\mathrm{helix}}=E_{\mathit{penetration})\mathit{\hspace{0.17em}}\mathrm{helix}}+E_{\mathit{losses})\mathit{\hspace{0.17em}}\mathrm{helix}}$(19)

(20) $2\mathit{\pi }{T}_h+\mathit{Fp}=\frac{\pi }{4}{C}_h\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2){t_i}^2+\frac{8}{6}{\pi }^2\mathit{\alpha }\mathit{\hspace{0.17em}}{C}_h\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)t_i$(20)

(21) $C_h=\frac{2\pi T_h+\mathit{Fp}}{\frac{\pi }{4}\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2){t_i}^2+\frac{8}{6}{\pi }^2\mathit{\alpha }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)t_i}$(21)

(22) $E_{\mathit{installation})\mathit{shaft}}=E_{\mathit{penetration})\mathit{shaft}}+E_{\mathit{losses})\mathit{shaft}}$(22)

(23) $2\mathit{\pi }{T}_s+\mathit{Fp}=\frac{\pi }{2}\mathit{C}\mathit{\hspace{0.17em}}{r}^3\mathit{p}+4{\pi }^2\mathit{C\alpha }{r}^3\mathit{\lambda }$(23)

(24) $C_s=\frac{2\pi T_s+\mathit{Fp}}{\frac{\pi }{2}{r}^3\mathit{p}+4{\pi }^2\mathit{\alpha }{r}^3\mathit{\lambda }}$(24)

(25) $\mathit{C}=C_h+C_s$(25)

Step 2: Determination of uplift capacity of helical anchor Q_u equation in terms of C

In practice, the permissible movement of a helical anchor is usually limited to modest displacements. And that is because the mobilization of shear strength and general bearing capacity failure may require significant displacements to satisfy limit state conditions. Determining the capacity of small displacements requires a balancing between the energy applied during loading with the suitable energy of the supporting blade, and this is done using the penetration energy $E_{\mathit{penteration}}$ hypothesis.

(26) $E_{\mathit{loading}}=E_{\mathit{penetration}}$(26)

Equation. 26(26) $E_{\mathit{loading}}=E_{\mathit{penetration}}$(26) assumed that shaft friction losses are negligible because only a fraction of the shear strength is mobilized for tiny displacements, and this is consistent with Perko (2000). Considering an energy balance between the energy expended during loading and the required energy of each of the helix plates to penetrate, one realizes that any installation energy not specifically related to helix penetration is wasted (Chance & Division of Hubbell Power Systems, Inc, 2006). Furthermore, the uplift is approximately equal to the bearing capacity since small displacement in either direction should be influenced primarily by the effective mean stress surrounding the blade. To reduce the disturbance of the surrounding soil during installation, d as in Figure 6 should be equal to the helical anchor pitch p. The determination of energy during loading can be achieved by integrating the force applied over a specific displacement of the helix. For linear force displacement, function E_loading can be written as in Equation. 27(27) $E_{\mathit{loading}}=\frac{1}{2}{Q}_u\mathit{\hspace{0.17em}}\mathit{p}$(27) :

(27) $E_{\mathit{loading}}=\frac{1}{2}{Q}_u\mathit{\hspace{0.17em}}\mathit{p}$(27)

Figure 6. Penetration displacement during loading.

where Q_u = uplift capacity and d = vertical displacement.

$E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta }\mathit{V}\mathit{\hspace{0.17em}}\mathit{\delta }$ as in Equation (5(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5)

Substitute Equations. 21(21) $C_h=\frac{2\pi T_h+\mathit{Fp}}{\frac{\pi }{4}\sum _{\mathit{i}=1}^n({R_i}^2\mathit{\hspace{0.17em}}-\mathit{\hspace{0.17em}}{r}^2){t_i}^2+\frac{8}{6}{\pi }^2\mathit{\alpha }\sum _{\mathit{i}=1}^n({R_i}^3-\mathit{\hspace{0.17em}}{r}^3)t_i}$(21) , 24(24) $C_s=\frac{2\pi T_s+\mathit{Fp}}{\frac{\pi }{2}{r}^3\mathit{p}+4{\pi }^2\mathit{\alpha }{r}^3\mathit{\lambda }}$(24) , and 5(5) $E_{\mathit{pent}.}=\frac{1}{2}\mathit{C}\mathit{\hspace{0.17em}}\mathrm{\Delta V}\mathit{\hspace{0.17em}}\mathit{\delta }$(5) into Equation. 27(27) $E_{\mathit{loading}}=\frac{1}{2}{Q}_u\mathit{\hspace{0.17em}}\mathit{p}$(27) (for helix and shaft) provides an expression for the helical anchor capacity (Q_h or Q_s) in terms of installation torque (T_h or T_s), applied downward force F, geometry of the helix R, and displacement d as in Equations. 29(29) $Q_h=C_h\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^2-r^2)\mathit{\pi d}$(29) and Equation 30.(30) $Q_s=C_s\mathit{\hspace{0.17em}}{r}^2\mathit{\pi d}$(30) Then, the total uplift capacity of helical anchor Q_u can be determined by Equation. 31.(31) $Q_u=Q_h+Q_s$(31)

(28) $\frac{1}{2}{Q}_h\mathit{\hspace{0.17em}}\mathit{d}=\frac{1}{2}{C}_h[({R_i}^2-r^2)\mathit{\pi d}\times \mathit{d}]$(28)

(29) $Q_h=C_h\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^2-r^2)\mathit{\pi d}$(29)

$\frac{1}{2}{Q}_s\mathit{\hspace{0.17em}}\mathit{d}=\frac{1}{2}{C}_s\mathit{\hspace{0.17em}}{r}^2\mathit{\pi d}\times \mathit{d}$

(30) $Q_s=C_s\mathit{\hspace{0.17em}}{r}^2\mathit{\pi d}$(30)

(31) $Q_u=Q_h+Q_s$(31)

3. Strengths and limitations of the research

Theoretical model using the energy balance method has been developed in this study to describe the relationship between the uplift capacity and installation torque of a helical anchor installed in cohesionless soil. This research makes a significant contribution by deriving two important physical correlations. The first one is between Q_h and T_h as in Equation 29(29) $Q_h=C_h\mathit{\hspace{0.17em}}\sum _{\mathit{i}=1}^n({R_i}^2-r^2)\mathit{\pi d}$(29) and the second between Q_s and T_s as in Equation 30(30) $Q_s=C_s\mathit{\hspace{0.17em}}{r}^2\mathit{\pi d}$(30) . The energy equivalency principles utilized in this research can be extended to predict the pile capacity that uses a combination of downward force and torque to install, which is called the Gentle Driving of Pile (GDP) method.

In this research, the uplift capacity of a helical anchor was calculated by adding the capacities of the individual helices to that of the anchor’s shaft. Given that this method is suggested for deep helical piles (C. D. H. C. Tsuha & Aoki, 2010). Also, the current model postulates that failure occurs above each individual helix, as examined by Chance & Division of Hubbell Power Systems, Inc (2006). According to Adams and Klym (1972), helical plates can be assumed to act independently of one another if the vertical distance between them is at least twice the diameter of the plates. Therefore, this study did not take the effect of distance between plates into account.

Additionally, friction constitutes a significantly greater proportion of torque resistance when compared to the other factor. However, this research assumed that the energy losses expended in the shaft friction during loading (when displacement d > p) are negligible because only a fraction of the shear strength is mobilized for small displacements as in Equations 26(26) $E_{\mathit{loading}}=E_{\mathit{penetration}}$(26) and 27(27) $E_{\mathit{loading}}=\frac{1}{2}{Q}_u\mathit{\hspace{0.17em}}\mathit{p}$(27) .

4. Comparison the current theoretical model and previous experimental results

The uplift capacity of helical anchor for different relative densities (D_r = 56% and 85%) based on the aforementioned theoretical model was compared with theoretical and experimental results obtained by C. H. C. Tsuha et al. (2007) and C. D. H. C. Tsuha and Aoki (2010). The predictions of the present theoretical model were in good agreement to the experimental results of C. H. C. Tsuha et al. (2007) as shown in Table 1. It can also be noticed from Table1 and Figure 7 that the present theoretical model underestimates with the theoretical predictions of the uplift capacity of true scale helical piles obtained by C. D. H. C. Tsuha and Aoki (2010).

Figure 7. Comparison of the uplift capacity to C. D. H. C. Tsuha and Aoki (2010) analytical solution for different torque and D_r values.

5. Conclusions

A study was conducted to evaluate the physical relationship between the uplift capacity and installation torque of helical anchors in cohesionless soil. The findings in this study support the following conclusions:

1. A theoretical expression was developed to predict the uplift and bearing capacity of a helical anchor utilizing the balance of energy during installation.

2. It was found that the behavior of helical anchors during the application of an installation torque depends on the applied downward force during installation, geometry of the helical anchor, soil properties, blade pitch per revolution, and displacement.

3. The results according to the present theoretical study correlate well with some previous field measurements and other theoretical models.

4. Two important physical correlations have been derived in this research. The first one is between Q_h and T_h and the second between Q_s and T_s.

5. It would be highly recommended to perform experimental tests and numerical analyses to verify this theoretical model. Also, scarcity of previous research studies on this topic presents the need for further development in this field of research.

6. This study can be extended to other domains in offshore applications, such as monopile foundations, which is most commonly utilized for offshore wind turbines. The energy equivalency principles can be adopted to predict the pile capacity that was installed using the GDP method by applying a combination of downward force and torque to the pile.

Disclosure statement

No potential conflict of interest was reported by the author(s).