A New Load-Transfer Factor to the Slipping Analytical Formulation in Axially Loaded Piles

Abstract

The load-transfer factor ($\zeta$) in the concentric cylinder approach is often used in analytical formulation in axially loaded piles. The factor is a constant value (in a given pile slenderness ratio and soil condition) devised under the elastic and ‘pre-failure’ perfect pile–soil bonding conditions (a non-slip analytical model). Given most numerical methods have already considered the pile–soil slipping in the ‘pre-failure’ stage, the limitations of non-slip analytical models have recently been discussed, and slipping analytical models have been recommended. Therefore, this research aims to investigate the load-transfer factor in slipping analytical models. This paper reviews that the factor in slipping analytical models is only constant in linear elastic and some Gibson soil conditions. Beyond these conditions, the factor varies as the pile-head load increases in some cohesionless soils. Adopting the existing constant factor in slipping analytical models will deviate the load–displacement results, as supported by numerical results. Therefore, a new equation is proposed to the load-transfer factor, and a new analytical method is proposed by varying the load-transfer factor during loading for improvement. Results presented in this paper demonstrate improved load–displacement results.

1. Introduction

The concentric cylindrical approach was originally proposed by Randolph and Wroth [1] and is often adopted by researchers (such as Kraft et al. [2], Alawneh [3], Pando et al. [4], Wang et al. [5], Zhang et al. [6], Cheng et al. [7]) to determine the analytical load-response behavior of axially loaded piles. The approach relies on a load-transfer factor ($\zeta$), which is a function of the pile radius ($r_0$) and $r_m$ (Equation (1)). The $r_m$. value is a radial distance at which the shear stress becomes diminishingly small and negligible. In the past, the $\zeta$ factor was devised under elastic mediums in non-slip analytical models, as described in Lo et al. [8]. Recently, the limitations of non-slip analytical models have been discussed in Wang et al. [5], Lo et al. [8], Sheil and McCabe [9], Boonyatee and Lai [10], and slipping analytical models were recommended. In this study, the $\zeta$ factor in slipping analytical models is reviewed using finite element (FE) analyses of homogeneous and vertically inhomogeneous (Gibson soil, linear increase in the soil shear modulus ($G$) with depth) soils under the linear elastic and Mohr-Coulomb (MC, linear-elastic-perfectly-plastic) models. Results show that constant values are valid in elastic soils and soils with a constant ratio of shear modulus to the ultimate shear strength (${\tau }_{ult}$) in the Gibson condition. However, in certain cohesionless soils, $\zeta$ is found to vary as the pile-head load increases and a new equation for $\zeta$ is proposed to account for this. Furthermore, a new analytical method is proposed to determine the load–displacement curve. An automated procedure was employed by combining Python codes and VBA (Visual Basic Application) macro to control the FE analysis (PLAXIS 2D) and perform the analytical calculations.

$$\zeta =\ln \left(\frac{r_m}{r_o}\right)$$

(1)

2. Background

2.1. Load-Transfer Factor (ζ)

In the concentric cylindrical approach, the deformation of the soils surrounding the pile shaft is idealized as shearing of concentric cylinders [1]. Pando et al. [4] commented that the concentric cylinder approach provides a good approximation of the deformation patterns obtained using more rigorous analyses such as finite element analysis. Randolph and Wroth [1] demonstrated that an estimate of vertical displacement of soil at the pile shaft ($w_s$) is given by Equation (2), where ${\tau }_o$ is the shear stress at the pile shaft. Initially, Randolph and Wroth [1] evaluated $\zeta$ by comparing the settlement profile of the surrounding soil at the mid-depth level of the pile to that calculated from integral equation analysis, for a pile slenderness ratio of $L/r_o$ = 40 (where $L$ is the pile length). The $\zeta$ factor was further extended by Randolph [11] to improve the accuracy of predicting the load-displacement behavior of both short and long piles by comparing to Poulos and Davis [12] and Carter and Kulhawy [13]. Guo [14] then used a more rigorous approach using the finite difference software FLAC to evaluate $\zeta$ in an elastic medium. The approach involved iteratively adjusting $\zeta$ until the pile-head stiffness matched both the analytical closed-form solution and the finite different numerical results.

$$w_s=\frac{{\tau }_o{r}_o}{G}\zeta$$

(2)

The general form of $\zeta$ is provided in Equation (3), where ${\mathsf{\upsilon }}_s$ is the soil’s Poisson ratio, and ρ is equal to the ratio of the shear modulus at mid-depth of the pile to that at the base. The constants, A and B used in Equation (3) have been determined by various researchers and are summarized in

Table 1. Values for A and B (Equation (3)) under different literature.

Literature	A	B
(i) Randolph and Wroth [1]	2.5	0
(ii) Randolph [11]	2.5	5
(iii) Guo [14] A	2.1	1

^A An approximated value for deep homogenous layers with some loss of accuracy.

.

$$\zeta =\ln \left[A\mathsf{\rho }\left(1-{\upsilon }_s\right)\frac{L}{r_0}+B\right]$$

(3)

2.2. Pile–Soil Interface

As discussed by De Gennaro et al. [15], when considering pile-soil interaction, failure of the soil is characterized by the deformation of a thin soil lens (interface) orientated in the direction of the contract surface. Many researchers [15,16,17,18,19,20,21] have shown that assuming an interface along the pile shaft is essential in predicting pile behavior under loads. Sheil and McCabe [9] commented that most numerical methods have already included the pile–soil slipping behavior (pile–soil relative displacement) by introducing interface elements along the pile shaft in the numerical models. Coutinho et al. [18] emphasized that interface elements must allow for relative displacements between two bodies in contact or separated by a thin material layer. However, existing simplified analytical methods still assume ‘pre-failure’ conditions that imply perfect pile–soil bonding characteristics (non-slip analytical models).

This study adopted the formulation of interface elements in PLAXIS 2D to simulate the pile–soil interaction for use in the analytical method. Slipping is considered in terms of relative pile-soil displacement at the interface along the pile shaft. A virtual thickness is assigned to the interface, which is an imaginary dimension used in part to define its material properties. The coordinates of each pair of nodes at the pile–soil interface are identical, which means that the interface has a zero thickness [22]. The pile–soil slipping ($\Delta w$) is governed by

$$\Delta w=w_p-w_s=\frac{{\tau }_o{t}_i}{G_i}$$

(4)

where $w_p$ and $w_s$are the vertical displacement of pile and soil nodes at the interface, $G_i$ is the shear modulus of the interface (calculated by an interface factor $R_i^2$ multiplying the soil shear modulus $G$), and $t_i$ is the virtual thickness of the interface and calculated by a virtual thickness factor (default value = 0.1) times the global element size calculated from the model dimensions [22].

3. A Review of the Load-Transfer Factor by Finite Element Analysis

3.1. Model Verification

FE analyses using PLAXIS do not directly provide the $\zeta$ factor, however values for ${\tau }_o$ and $w_p$ which are used in Equation (5) to evaluate $\zeta$. Equation (5) is proposed in this study, which is a combination of Equations (2) and (4). Verification of this proposed method for evaluating $\zeta$ involved comparing values of $\zeta$ resulting from an FE model to those obtained from previously published values.

$$\zeta =\left(\frac{w_{p}G}{{\tau }_o{r}_o}-\frac{t_i}{R^2{r}_o}\right)$$

(5)

The first step in developing the FE model used in this study involved optimizing it in terms of mesh size and the model boundaries. The effect of the mesh size and total height ($H$) of the modeled soil on resulting values of $\zeta$ is shown in Figure 1 and Figure 2. From these figures, it can be seen that the effect is minimized when the $r_e$ and $H/L$ factors reach 0.35 and 6, respectively, for $L/r_0=40$ and $G=1923$ kPa. Similar results were obtained in other combinations among $L$ = 10 to 30 m, $L/r_0$ = 20 to 60, $H/L$ = 2 to 8, and $G$ = 1.92 to 11.54 GPa. Therefore, values of $r_e=0.35$ and $H/L=6$ were adopted for the FE model used in this study.

After determining the optimum mesh and boundary geometries, the FE model used in this study was verified by comparing the calculated $\zeta$ values using Equation (5) proposed in this study at the mid-depth level of the pile embedded in elastic soils, with values provided in Randolph [11]. Figure 3 demonstrates a high degree of correlation between $\zeta$ values calculated using Equation (5) and corresponding values provided by Randolph [11] (within approximately 2%).

The adopted FE model was further verified by comparing to the analytical closed-form solution under the suggested $\zeta$ factor from Guo [14] for 24 cases ($L$ = 10–30 m, $L/r_0$ = 40–60; ${\mathsf{\upsilon }}_{\mathrm{s}}$ = 0.2, $G$ = 1.92–7.70 GPa). Again, a high degree of correlation was obtained in all combinations. Figure 4 shows one of the typical results amongst the 24 cases analyzed. It compares the load–displacement curves at the pile head between the analytical formulation based on the load-transfer factor recommended from Guo [14] and the verified FE model in elastic soil.

3.2. A Review of the Load-Transfer Factor

An axially loaded pile was analyzed under tension and compression loads, using linear elastic (LE) and Mohr–Coulomb (MC) constitutive soil models in the FE model used in this study. Values for parameters used for modeling the pile comprised a diameter of 1.2 m and length of 15 m, with an elastic modulus ($E_p$) and Poisson ratio ($v_p$) of 30 GPa and 0.15 respectively. The $\zeta$ value at the mid-depth level of the pile was reviewed (similar to Randolph and Wroth [1]) by substituting the shear stress and vertical displacement of the FE results into Equation (5). The mid-depth level was chosen on the presumption that it is least affected by boundary edge effects and discontinuities at the pile head and toe. Five different FE models (

Table 2. Summary of soil and interface parameters in different FE models analyzed.

MODEL	1C/1T	2C/2T	3C/3T	4C/4T	5C/5T
Soil model	LE	LE	MC	MC	MC
$E_{ref}$ (kPa)	10,000	4000	4000	10,000	10,000
$E_{inc}$ (kPa/m)	0	400	400	0	0
$c_{ref}$ (kPa)	-	-	10	1	1
$c_{inc}$ (kPa/m)	-	-	1	5.2	-
$\varphi$ (°)	-	-	-	-	30

Note: ‘C’ and ‘T’ denote compression and tension loads; $c_{ref}$ and $c_{inc}$ are the shear strengths at the ground level and incremental value with depth, respectively for both the soil and interface; $E_{ref}$ and $E_{inc}$ are the stiffness at the ground level and incremental value with depth, respectively for both the soil and interface; $\varphi$ is the friction angle of the soil.

) were analyzed in both axial tension and compression:

• Model 1: Homogeneous, LE constitutive soil model.
• Model 2: Gibson (inhomogeneous), LE constitutive soil model.
• Model 3: Gibson (inhomogeneous), elastoplastic MC constitutive soil model with a constant $G/{\tau }_{ult}$ ratio where the entire pile–soil interface becomes plastic (reaching the ultimate shear strength, ${\tau }_{ult}$) along the total pile length simultaneously.
• Model 4: Homogenous (modulus only), MC constitutive soil model with a fixed strength profile along the interface. The shear strength profile is an input parameter converted from initial overburden stress based on Das [23] to simulate cohesionless soil. It is a simplified version of Model 5.
• Model 5: Homogenous (modulus only), MC constitutive soil model with a variable strength profile along the interface. Soil strength is a function of the friction angle (input parameter) and the rotation of principal stresses (updated automatically as pile-head load increases).

Figure 5 shows that the estimated $\zeta$ factor in FE Model 1 (3.85 compared to 3.89) and Models 2 and 3 (3.51 compared to 3.57) closely align with values suggested by Randolph [11]. It can also be seen from Figure 5 that estimated $\zeta$ values are constant as the pile-head load increases in the LE models (Models 1 and 2) and for a Gibson soil with a constant ratio of $G/{\tau }_{ult}$ (Model 3).

The constant value of $\zeta$ estimated in Model 3 (Figure 5) is a result of the interface behaving linearly-elastic as the pile load increases up until a certain load where it reaches a plastic state along its entire length simultaneously. It can be explained by the analytical theory supported by the FE results. Randolph and Wroth [1] commented that the distribution of shear stress along the pile shaft is similar to ${\tau }_i\left(z\right)=k_i\left(b+z\right)$ for soil shear modulus in the form of $G\left(z\right)=m\left(b+z\right)$ as shown in Figure 6, which can be expressed as

$$\frac{{\tau }_i\left(z\right)}{G\left(z\right)}=\frac{k_i\left(b+z\right)}{m\left(b+z\right)}=\frac{k_i}{m}$$

(6)

The term $k_i$ in Equation (6) is a constant for load step $i$, where PLAXIS divides each calculation phase into several calculation steps, and load is applied in small proportions called load steps [22]. In Model 3, if $m$, $b$, and$K$ are constants, the ratio of soil shear modulus to strength is constant and is equal to

$$\frac{G\left(z\right)}{{\tau }_{ult}\left(z\right)}=\frac{m\left(b+z\right)}{K\left(b+z\right)}=\frac{m}{K}$$

(7)

The ratio of shear stress (${\tau }_i\left(z\right)$) to shear strength (${\tau }_{ult}\left(z\right)$) along the shaft at load step$i$ is a constant, which is confirmed from the FE results presented in Figure 7 and is expressed as

$$\frac{{\tau }_i\left(z\right)}{{\tau }_{ult}\left(z\right)}\approx \frac{k_i}{K}=constant$$

(8)

For load steps with $k_i<K$, the shear stress${\tau }_i\left(z\right)$ at any pile depth is smaller than the ultimate value ${\tau }_{ult}\left(z\right)$ and the interface along the whole shaft remains in an elastic state. When $k_i=K$, ${\tau }_i\left(z\right)$ at any depth is equal to ${\tau }_{ult}\left(z\right)$ and the pile interface simultaneously reaches a plastic state along the entire pile length. Therefore, the estimated $\zeta$ factor is mainly the same as the linear elastic model. The near constant value for $\zeta$ can further be explained by the mathematical derivative of the shear strain ($\gamma$) with respect to $z$

$$\gamma {{}^{\prime }}_i\left(z\right)=\frac{\partial \left[\frac{{\tau }_i\left(z\right)}{G\left(z\right)}\right]}{\partial z}=\frac{m\left(b+z\right)\frac{\partial \left(k_i\left(b+z\right)\right)}{\partial z}}{m^2{\left(b+z\right)}^2}-\frac{k_i\left(b+z\right)\frac{\partial \left(m\left(b+z\right)\right)}{\partial z}}{m^2{\left(b+z\right)}^2}$$

(9)

When Equation (9) is simplified, the first and second terms simplify to $k_i/m\left(b+z\right)$ which results in$\gamma {}^{\prime }\left(z\right)=0$. This results in the shear strain ($\gamma {}_i$) in a load step $i$ being nearly constant along the pile shaft as shown in the FE results in Figure 8. The shear stress along the pile shaft in a load step$i$ is defined as

$${\tau }_i\left(z\right)=\gamma {}_i\left(z\right)\ast G\left(z\right)=\gamma {}_i\left(z\right)\ast m\left(b+z\right)\le K\left(b+z\right)={\tau }_{ult}\left(z\right)$$

(10)

Theoretically, the term $\gamma {}_i\left(z\right)\ast m$ in the above equation continuously increases with increasing pile head loads. Eventually, it reaches the ultimate value of $K,$ and the whole pile interface becomes plastic simultaneously.

Figure 9 demonstrates that the estimated value of $\zeta$ is no longer constant but changes as the pile-head load increases in the MC constitutive models (Models 4 and 5). This can be explained by rearranging Equation (2) ($w_s/{\tau }_0=\zeta r_0/G$) in which the ratio of $w_s/{\tau }_0$ is lesser than that in the elastic model (Model 1). It results from the reduction of soil movement adjacent to the pile shaft and the increase in ${\tau }_0$. When ${\tau }_0$ reaches ${\tau }_{ult}$ the interface will exhibit plastic behavior. That is, it will continuously displace at the ${\tau }_{ult}$ value (Figure 10). The pile–soil plastic slip commences at the upper portion of the pile shaft and progressively develops down the pile shaft in Models 4 and 5. This phenomenon is explained in Lo et al. [8] and adopted by many researchers to simulate cohesionless soils. Once the pile–soil plastic slip has occurred, the rate of vertical soil movement adjacent to the pile shaft around the slip portion (upper portion) reduces. This effect releases the downward ‘pushing’ effect from the upper portion onto the soil below (in compression piles), and the upper portion restrains the vertical soil movement below (in tension piles). Consequently, the vertical soil displacement ($w_s$) below the slip portion is lesser in Models 4 and Model 5 compared to the elastic model (Model 1), as shown in Figure 11. However, the pile keeps moving as the pile head load increases, which results in larger pile–soil relative displacement ($\Delta w=w_p-w_s$) and therefore larger mobilized shear stress (${\tau }_o$) based on Equation (4) ($w_p-w_s={\tau }_0{t}_i/G_i$).

Figure 9 demonstrates that during compression loading, the corresponding reduction of the estimated ζ factor in Model 5C ($c$ = 1 kPa and ϕ = 30°) is lesser than the simplified Model 4C ($c$ = 1 + 5.2 z and ϕ = 0°). In contrast, during tension loading, the reduction in Model 5T ($c$ = 1 kPa and $\varphi$ = 30°) is slightly higher than the simplified Model 4T ($c$ = 1 + 5.2 z and ϕ = 0°), but very similar. The difference in $\zeta$ is due to the increase in mobilized shear strength along the pile shaft in Model 5C under compression loads and decrease in Model 5T under tension loads, as observed in O’Neill [24] and De Nicola and Randolph [25]. It results in the depth of pile–soil plastic slip ($L-iL$) being lesser in Model 5C and larger in Model 5T compared to the simplified Model 4 (supported by the FE results in Figure 12). As a result, the reduction of the $\zeta$ factor in Model 5C is lesser than Model 4C and the reduction in Model 5T is larger than Model 4T.

Figure 9 shows that the estimated value of ζ in Model 4T (simplified model under tension loads) is very close to that in Model 5T. Therefore, Model 4T was adopted to simulate piles under tension loads in some cohesionless soils in the next section.

4. New Analytical Method and Improvement in the Load–Displacement Curve

This section considers the change in ζ when determining the load–displacement curve for some cohesionless soils. With the initial application of a tensile load to the pile, the whole interface remains elastic. As the tensile load increases, the plastic slip (Figure 10) of the soil–pile interface is first reached at the top of the pile. It progressively transcends downward to the toe (Figure 13b), where the whole interface becomes plastic. Many researchers (such as Misra et al. [26], Guo [14], and Meerdink et al. [27]) have adopted this elastoplastic form to model cohesionless soils.

4.1. Devising the Elastoplastic Analytical Solution

4.1.1. Elastic Component

The deformation of a pile under the pile elasticity theory and vertical force in equilibrium is shown in Equation (11). Where the $l$term in Equation (11) is the distance measured from the pile toe for piles under tension load, which is different from the $z$ term used in the previous sections, which is distance from the pile head. The terms $P\left(l\right)$, $w_p\left(l\right)$ and ${\tau }_0\left(l\right)$ are the pile axial load, pile axial displacement, and vertical shear stress at the pile shaft at a distance $l$ from the pile toe, respectively. Finally, the $\lambda$ term is equal to $E_p/G$.

$$\frac{{\partial }^2{w}_p\left(l\right)}{\partial l^2}=\frac{1}{\pi r_o^2\lambda G}\frac{\partial P\left(l\right)}{\partial l}=\frac{2}{r_0\lambda G}{\tau }_0\left(l\right)$$

(11)

If $w_s\left(l\right)$ is the vertical soil displacement at the pile shaft at a distance $l$ from the pile toe, the vertical soil displacement ($w_s$) at the pile shaft can be combined with the concentric cylindrical equation (Equation (2)) and Equation (11), which yields

$$\frac{{\partial }^2{w}_p\left(l\right)}{\partial l^2}=\frac{2w_s\left(l\right)}{r_o^2\lambda \zeta }$$

(12)

Combining the interface equation and concentric cylindrical equation, Equations (4) and (2), respectively, results in

$$w_s\left(l\right)=w_p\left(l\right)\frac{r_0\zeta R^2}{r_0\zeta R^2+t_i}$$

(13)

Combining Equations (12) and (13), yields the second-order differential equation

$$\frac{{\partial }^2{w}_p\left(l\right)}{\partial l^2}=\frac{2R^2}{r_0\lambda (r_0\zeta R^2+t_i)}{w}_p\left(l\right)$$

(14)

By solving the second-order differential equation under the boundary conditions of $P\left(0\right)=0$ and $w_p\left(0\right)=w_b$ at the pile toe ($l=0)$, the following equations for the axial displacement $w_p\left(l\right)$, axial load $P\left(l\right)$, and vertical shear stress at the pile shaft ${\tau }_0\left(l\right)$ at a distance $l$ from the pile toe are obtained

$$w_p\left(l\right)=w_{b}cosh\left(\mu l\right)$$

(15)

$$P\left(l\right)=w_b\mu \pi r_o^2\lambda Gsinh\left(\mu l\right)$$

(16)

$${\tau }_0\left(l\right)=\frac{w_b{\mu }^2{r}_o\lambda G}{2}\cosh \left(\mu l\right)$$

(17)

$$\frac{P\left(l\right)}{w_p\left(l\right)}=\mu \pi r_o^2\lambda Gtanh\left(\mu l\right)$$

(18)

where $\mu =\sqrt{\frac{2R^2}{r_0\lambda (r_0\zeta R^2+t_i)}}$.

4.1.2. Plastic Component

It is assumed that the shear strength at the pile–shaft interface increases linearly with depth, as shown in Equation (19) and Figure 13a. The vertical shear stress (${\tau }_0\left(l\right)$) along the shaft has already reached the strength value $\left({\tau }_{ult}\left(l\right)\right)$.

$${\tau }_{ult}\left(l\right)=K\left(b+L-l\right)$$

(19)

where $K$ and $b$ are constants, and $L$ is the pile length as shown in Figure 13a.

Based on Equations (2), (13), (17)–(19), the axial displacement of the pile, $w_p\left(iL\right)$and corresponding load $P\left(iL\right)$at the transition point of $iL$ (the point measured from the pile toe at which the elastic and plastic portions separate; $i$ is the elastoplastic ratio of the remaining elastic portion which is equal to $L1/L$ as shown in Figure 13b) can be expressed as

$$P\left(iL\right)=\mu \pi r_o^2\lambda Gtanh\left(\mu iL\right)w_p\left(iL\right)$$

(20)

$$wp\left(iL\right)=\frac{2}{G{r}_0\lambda {\mu }^2}K\left[b+L\left(1―i\right)\right]$$

(21)

Within the plastic portion, the pile axial load $P\left(l\right)$ at a distance $l$ from the pile toe is

$$P\left(l\right)=P\left(iL\right)+{{\displaystyle \int }}_{iL}^{l}2\pi rK\left(b+L-l\right)dl$$

(22)

Within the plastic portion, the pile axial displacement $w_p\left(l\right)$ at a distance $l$ from the pile toe is

$$wp\left(l\right)=wp\left(iL\right)+{{\displaystyle \int }}_{iL}^l\frac{P\left(l\right)}{\pi r_o^2\lambda G}dl$$

(23)

Combining Equations (20) to (22) then (21) and (22) to (23), the pile axial load $P\left(L\right)$ and displacement $w_p\left(L\right)$ at the pile head are expressed as

$$P\left(L\right)=\frac{2\pi r_{0}tanh\left(\mu iL\right)}{\mu }K\left[b+L\left(1-i\right)\right]+{{\displaystyle \int }}_{iL}^{L}2\pi rK\left(b+L-l\right)dl$$

(24)

$$w_p\left(L\right)=\frac{2}{G{r}_0\lambda {\mu }^2}K\left[b+L\left(1-i\right)\right]+\frac{2tanh\left(\mu iL\right)}{\mu r_0\lambda G}K\left[b+L\left(1-i\right)\right]\left(L-iL\right)+{{\displaystyle \int }}_{iL}^L\frac{{{\displaystyle \int }}_{iL}^{l}2\pi rK\left(b+L-l\right)dl}{\pi r_o^2\lambda G}dl$$

(25)

An iterative process was developed using Python code combined with a VBA macro which is presented in Figure 14. This procedure evaluates the load $P\left(L\right)$ and displacement $w_p\left(L\right)$ at the pile head as head load increases.

4.2. Improvement in the Load–Displacement Curve at the Pile Head by the New Analytical Method

A total of 44 cases ($L$ = 10–30 m, $L/r_0$ = 40–60, $G$ = 3.8–7.7 MPa, ${\upsilon }_s$ = 0.3, ${\upsilon }_p$ = 0.15, $E_p$ = 30 GPa) were analyzed using the analytical method proposed in this paper. This new analytical method considers the change of the load-transfer factor (${\zeta }_{change}$) as the pile-head load increases using an automated process; referred to here as the new ${\zeta }_{change}$ analytical method. For comparison, an analysis of the same cases was undertaken using analytical methods that employ a traditional fixed load-transfer factor (${\zeta }_{fixed}$) (Section 2.1), which is referred to hereon as the traditional ${\zeta }_{fixed}$ analytical method. The resulting load–displacement curves at the pile head using the different analytical methods (the new ${\zeta }_{change}$ and traditional ${\zeta }_{fixed}$ analytical methods) were compared with FEM results. Figure 15 demonstrates that the new ${\zeta }_{change}$ analytical method (Case 1) yielded results markedly closer to those estimated using FEM than the traditional ${\zeta }_{fixed}$ analytical methods (Case 2). The deviation of the analytical results relative to the FEM results are summarized in

Table 3. Summary of percentage deviation from the FE results.

At Head	Numerical	Proposed (Case 1)	Case 2—i	Case 2—ii	Case 2—iii
$P\left(L\right)$	8746 kN	+0.5%	−10.2 %	−11.0%	−8.3%
$w_p\left(L\right)$	54.6 mm	−0.8%	+18.4%	+20.0%	+14.7%

Literature (i), (ii), and (iii) refer to Table 1.

.

When applying a factor of safety of 1.5 to the ultimate capacity of the FE result (allowable capacity = 8746 kN), the corresponding pile head displacement is approximately 54.6 mm. Using this displacement as a reference point, the deviation of the head load ($P\left(L\right)$) in the analytical solution comparing to the FE analysis estimates is less than +0.5% in Case 1 and a maximum of −11% in Case 2 (using Randolph [11]). To achieve the allowable capacity of 8746 kN, the head displacement deviated −0.8 per cent from the FE analysis result in Case 1, and a maximum of +20.0 per cent in Case 2 (using Randolph [11]). The coefficient of determination ($R^2$) for all 44 cases is summarized in Figure 16, which shows the new ${\zeta }_{change}$ analytical method has at least a 0.99 correlation with results using FE analysis. In comparison, the traditional ${\zeta }_{fixed}$ analytical method provides a correlation of between 0.91 and 0.95 (using Randolph [11]) with FE analysis results.

Figure 17, Figure 18, Figure 19 and Figure 20 show the approximate change of the $\zeta$ factors as pile-head load increases in the new ${\zeta }_{change}$ analytical method. The $\zeta$ factors versus the pile length are plotted on the right $y-$axis, with $r_m$ values also provided on the left $y-$axis for reference. The $\zeta$ factors start from an initial maximum value at the purely elastic condition (${\zeta }_{elastic}$ can be calculated from the existing literature, Equation (3)). Plastic slip (Figure 10) occurs from the top of the interface when the pile-head load increases and progressively develops down the pile (Figure 13b). The $\zeta$ factors reduce and reach a minimum value at the fully plastic condition (${\zeta }_{plastic}$). The $\zeta$ values at purely elastic and plastic conditions are the upper and lower bounds, respectively. In this study, the maximum reduction in the $\zeta$ factors are approximately 40% (80% in $r_m$ values), as pile-head load increases. Figure 17, Figure 18, Figure 19 and Figure 20 show fluctuations in the minimum values (${\zeta }_{plastic}$), which is due to a small deviation of the calculated head load just before the fully plastic condition under the iteration procedure (Case 1 against the FE analysis results in Figure 15). When the pile-head load in the analytical solution is manually adjusted to match the same as the FE results by refining values for $\zeta$, the ${\zeta }_{plastic}$ values are then straight lines against the pile length (the horizontal dotted lines in Figure 17, Figure 18, Figure 19 and Figure 20). These lines show that ${\zeta }_{plastic}$ (the minimum load-transfer factor at the fully plastic condition) is independent of pile length (the dotted lines are horizontal) and independent of the soil shear modulus (the $y$ intercepts of the dotted lines are the same in different soil modulus, i.e., Figure 17 vs. Figure 19), but dependent on the $L/r_0$ratio (the $y$ intercepts are different in different $L/r_0$ ratio, i.e., Figure 17 versus Figure 18). Additionally, the ${\zeta }_{plastic}$ values increase with the $L/r_0$ ratio, and this pattern agrees with the ${\zeta }_{elastic}$ formula proposed under the literature in Equation (3).

The above scenarios show that load-transfer factor values change with pile head loads (in turn of the elastoplastic ratio, $i$). Therefore, information on the change and lower bound value (${\zeta }_{plastic}$) is necessary to construct the analytical solution. Equations (26) and (27) were developed in a parametric study of over 300 combinations (summarized in

Table 4. Combination of pile geometries and parameters.

$\mathit{L}$ (m)	$\mathit{L}/{\mathit{r}}_0$	${\mathit{\upsilon }}_{\mathit{s}}$	${\mathit{c}}_{\mathit{i}\mathit{n}\mathit{c}}(\mathbf{kPa}/\mathbf{m})$	G (MPa)
10, 20, 30	20, 40, 60, 80	0.2, 0.3, 0.4	5, 10, 20	2, 6, 15

) to match the load–displacement curve between the analytical and FEM results. The same automation using Python code and a VBA macro described above and in Figure 14 were used to undertake the analyses.

$${\zeta }_{plastic}=\ln \left[0.368\left(\frac{L}{r_0}\right)·\left(1-0.890{\upsilon }_s\right)+3.619\right]$$

(26)

$$\zeta =\ln \left[\frac{0.368\left(\frac{L}{r_o}\right)·\left(1-0.890{\upsilon }_s\right)+3.619}{2.1\frac{L}{r_o}\left(1-{\upsilon }_s\right)+1}\right]·\left(1-i\right)+\ln \left[2.1\frac{L}{r_o}\left(1-{\upsilon }_s\right)+1\right]$$

(27)

4.3. Application to a Case Study

A case history is stimulated by adopting the new equations (Equations (24)–(27)) in the new ${\zeta }_{change}$ analytical method. A full-scale test (P1) was carried out by Sowa [28] on a reinforced concrete pile (0.53 m in diameter and 12 m in length) embedded mainly in fine sand with a unit weight of 18.4 $\mathrm{kN}/{\mathrm{m}}^3$. Parameters suggested by Zhang et al. [29] were adopted to facilitate later comparison. A soil shear modulus of 2 MPa (average SPT N value of 11) and pile elastic modulus of 30 GPa were adopted, as reported by Zhang et al. [29]. In our new ${\zeta }_{change}$ analytical method, the shear strength profile of the interface (solid line) shown in Figure 21 is adopted such that the ultimate pull-out capacity of 401 kN is achieved. It is the same capacity obtained under Zhang et al. [29] who divides the pile into 12 segments (each approximately 1 m height) with average strength in each segment (dotted line).

The results presented in Figure 22 show a high degree of correlation between the new ${\zeta }_{change}$ analytical method and numerical method (MC model in PLAXIS 2D) for load–displacement estimates. The new ${\zeta }_{change}$ analytical method that considers the change of the load-transfer factor, better agrees with the FEM results compared to the results from the traditional fixed factor (${\zeta }_{fixed}$). Furthermore, it shows an improvement from Zhang et al. (2015) when their suggested parameters are adopted. However, the proposed analytical method slightly underestimates the initial load–displacement curve and overestimates the portion just before the interface becomes fully plastic (compared to the measured field data). This is a disadvantage of the linear-elastic-perfectly-plastic behavior in the Mohr–Coulomb (MC) model. To address this, a non-linear stress-dependent modulus was used in the hyperbolic form of

$$G=G_0\left(1-\frac{{\tau }_0}{{\tau }_{ult}}\right)$$

(28)

where $G_0$ is the initial shear modulus, $G_0=G_{50}/\left(1-R_f/2\right)$, $R_f$ is the ratio of the maximum allowable shear stress to the asymptotic value in the hyperbolic form, and $G_{50}$ is assumed the same as the soil shear modulus adopted in the MC model. Using this hyperbolic function improved these two portions of the load-displacement curve as shown in Figure 23.

5. Conclusions

Many researchers have considered the pile–soil slipping theory in the analytical method by introducing an interface in the analytical formulation that allows relative displacement between the pile and soil surfaces. It is similar to many rigorous numerical methods that adopt interface elements to simulate the pile–soil slipping behaviors to achieve more accurate and realistic results. This study investigated the use of the existing load-transfer factor (a constant value during pile loading and devised under the elastic soils) in slipping analytical models and found deviation in the load–displacement results as explained by mathematical theories and numerical results. It draws attention to the use of the existing load-transfer factor in the slipping analytical theory. The study shows a constant $\zeta$ factor is valid in elastic soils and soils with a constant ratio of shear modulus to strength in the Gibson condition for axially loaded piles. Some cohesionless soils are characterized by the pile–soil plastic slip commencing at the upper portion of the pile shaft and progressively developing down the pile. It is found that the $\zeta$ factor changes as pile-head load increases in these soils. To address this, a new equation for the load-transfer factor and a new analytical method to develop the load–displacement curve are proposed to improve the slipping analytical method in these soils. Using the new equation and analytical method, the results demonstrate that the load–displacement curve is improved and correlates well with corresponding Mohr–Coulomb models in the FEM. The study results suggested that the change of the $\zeta$ factor is independent of soil modulus, and therefore, it is expected to apply to nonlinear soils. Therefore, further research and verification based on this fundamental study expanding to nonlinear models are recommended for continuous improvement.