A Lateral-Response Analyzed Model for Piles in Sloping Ground Considering Vertical Resistance Based on Transfer Matrix Method

Abstract

In order to analyze the lateral response of piles subjected to combinational loads located at sloping ground, a simplified physical model is proposed and developed based on the Timoshenko beam theory and tri-parameter elastic foundation beam model. In this model, the effects of vertical resistance, shear deformation and initial inclination of pile shafts are taken into account, and the semi-analytical solution to the lateral behavior of pile is obtained by utilizing the transfer matrix approach. Specifically, the relative transfer matrix coefficients in the free, passive, and active segment of the pile are analytically determined through Laplace transformation. In addition, the validation of this model is conducted by comparing with field observations and predictions obtained from other existing methods. Moreover, the influences of initial inclination angle, vertical resistance in passive segments and the vertical load on the lateral response of the pile are also discussed in depth. The results indicate that, although there is a significant reduction effect emerged in both the lateral displacement and bending moment with the increase in the vertical resistance, a growth trend for those responses can be obtained with an increase in the initial inclination angle and vertical load.

1. Introduction

Piles buried in sloping ground are frequently encountered in practice engineering to meet the requirement of environment protection and road direction control, especially in the western mountain areas of China. Rather than at the flat ground, the loading state for pile shafts constructed in sloping ground is complicated, since they are usually under coupling effects due to several types of loads, which chiefly includes axial and lateral load transmitted by upper structure, the lateral thrust caused by rainfall or self-weight of soil and construction etc.” [1,2,3]. Therefore, in view of such complexity and their crucial role in safety for engineering practices, analysis of deformation and internal force have become a topic of everlasting interest to researchers.

To date, much presented study effort has been directed towards the exploration of relevant topics and a number of celebrated works can be seen. Specifically, studies focused on characterizing the behavior of flexible pile subjected to both eccentric and inclined loads showed that pile could be significantly affected by the secondary deflection induced by eccentric vertical loads [4,5,6,7,8]. Moreover, several laboratory experiments for mircro-piles under large lateral loads in stiff clay and sand showed that lateral displacement was apparently affected by the magnitude of the constant axial loads [9,10]. In more recent developments, the numerical modelling treated as an efficient tool for evaluating the performance of laterally loaded piles has attracted significant attention in the academic field. In detail, Karthigeyan et al. [11,12,13] investigated the P-Δ effect on the response of inclined loads piles in sand and clay soil by using the finite element method (FEM), and proposed a numerical model to quantify the influence of inclined-loads piles in sand [14]. Lassaad et al. [15] successfully conducted a series of three-dimensional finite differences analysis for investigating the influence of vertical loads on the lateral performance of pile. In addition to numerical methods, some theoretical methods [16,17,18,19,20,21,22,23] have also been used to facilitate designs.

Although the above-mentioned studies have been applied widely, the slope effect (which refers to the effect of soil resistance on a pile shaft) has not been taken into account. In detail, the slope effect mainly consists of three aspects of resultants: (i) the reduction in the soil horizontal resistance in front of piles; (ii) the earth pressure caused by potential sliding soils behind piles; and (iii) the initial inclination of piles resulting from the thrusting of sliding soils. Reflection on reduction in the soil horizontal resistance has mainly been performed by experimental investigation [24,25,26,27,28,29], numerical simulation [30,31,32,33] and theoretical methods [34,35,36,37]. In addition, the earth pressure behind piles caused by potential sliding soils can be approximatively predicted by landslide thrust [38]. In further developments, studies about the behavior of a pile with complicated loads in sloping ground have been performed by experimental investigation [39], and field test research [40]. Theoretical approaches to the combined effect of vertical and lateral loads on sloping ground are still insufficient [41,42], especially for combinatorial loaded large diameter pile in sloping ground.

In engineering practice, it is noteworthy that the diameter D for most piles in usage is greater than 0.8 m, a number of studies [43,44,45,46,47,48] indicated that, for such piles, the moment of resistance induced by vertical friction could affect the behaviors of both the lateral deformation and internal force of a pile shaft, and this effect will be enhanced with an increasing in diameter D or the rigidity of the surrounding soils. Mcvay et al. [45] conducted a centrifugal model test and developed relevant theoretical and numerical calculation models for estimating the moment of resistance for the vertical friction of the pile shaft. Zhu et al. [47] proposed a simplified theoretical calculation model for evaluating the above-mentioned moment of resistance by adopting friction enhancement theory and several vertical friction models, and further conducted lateral horizontal deformation analysis for pile shafts based on the transfer-matrix method. However, up to now, the previous studies were mainly focusing on the analysis of the lateral response for pile shafts under horizontal load; the studies on the lateral deformation of the pile shaft in a slope by considering coupling effects, including the vertical friction, are extremely limited.

Therefore, based on the above assertions, the main objective of the current research is to develop an analytical model for evaluating the lateral deformation of a pile shaft in a slope under the aforementioned combined loads by considering vertical friction. With this objective, the remainder of this work is organized as follows. Firstly, the governing equations for mimicking the lateral deformation of different characteristic segments, including the passive, active and free segment of the pile shaft, are determined. Then, the semi analytical solutions for the lateral deformation and internal force of the pile shaft are obtained by adopting the Laplace transformation and the transfer-matrix method; the effectiveness and the feasibility of the proposed model is also verified against experimental results and one existing model. Finally, sensitivity analysis by considering the factors of vertical friction in the passive segment, and initial inclination angle and P-Δ effect of vertical load on the lateral response of the pile shaft, is carried out.

2. Theoretical Analysis

2.1. Problem Definition

The problem addressed in this study is schematically described in Figure 1. The slightly inclined pile is embedded into a slope-ground multilayered soil deposit. It is divided into three characteristic segments by the slope surface and potential sliding surface. The pile length above the slope surface is $L_f$, above and beneath the potential sliding surface is $L_p$ and $L_a$, respectively. The segment of the pile is divided into free, passive and active segments by the similarities and differences between the lateral displacement of pile and soil and the loaded main body. Taking into account the difference in the subjected horizontal loads over those segments, corresponding local coordinate systems were developed by treating the vertexes of those characteristic segments as the origins, respectively.$z_f$, $z_p$ and $z_a$ denote the distances between certain points to the vertexes for the free, passive and the active segment, respectively; $y_f$, $y_p$ and $y_a$ are the horizontal deformation of pile shaft for the free, passive and the active segment, respectively; ${\phi }_{IN}$ is the initial inclination of the pile shaft. This pile is assumed to have a horizontal force $Q_t$, a moment $M_t$, and an axial load $N_t$ acting on the pile head. The distributed load $q_f\left(z_f\right)$ acting on the pile above the slope surface is also considered to simulate the extra external loadings, such as wind and wave. The distributed load $q_p\left(z_p\right)$ acting on the pile above the potential sliding surface is also considered to simulate the extra external loadings, such as soil lateral pressure caused by the landslide.

In this study, the interaction between soil and pile is simulated as a linear spring in series with a slider distributed along the pile shaft. The sign conventions for shear force, moment, horizontal deformation, and rotation angle of pile shaft follows: the positive directions for the horizontal deformation and the rotation angle are assumed as the right direction; for that of the bending moment, it is presumed that the left side of the pile shaft is subjected to tension; and for the shear force, the clockwise direction is assumed as the positive direction.

2.2. Load Transfer Model of Soil–Pile Interaction

In the current paper, the horizontal resistance of the pile shaft yielded by the soil in the active zone is characterized by a series of discrete springs. Based on the three-parameter foundation resistance model, regarding the horizontal soil resistance and the displacement for the pile shaft per unit meter, the soil–pile interaction models can be described in a general form as follows:

$$\{\begin{array}{l}{p}_p=k_p\left(z_p\right)y_p={\delta }_{p}m{\left(z_0+z_p\right)}^n{y}_p\\ p_a=k_a\left(z_a\right)y_a=m{\left(z_0+L_p+z_a\right)}^n{y}_a\end{array}$$

(1)

In which the notations $p_p$ and $p_a$ mean the soil resistance for the passive and active segments, respectively; the notations $k_p\left(z_p\right)$ and $k_a\left(z_a\right)$ mean the horizontal subgrade reaction coefficients for the passive and active segments, respectively; ${\delta }_p$ is the weaken coefficient of the soil resistance in front of the pile shaft, which can be determined from [48]; $m$ denotes the proportion coefficient for the foundation resistance; and $n$ and $z_0$ are the index of the depth and the equivalent depth at ground surface.

In the current paper, the distribution of vertical soil friction and the resistance moment for pile shaft are graphically illustrated in Figure 2, in which the vertical soil friction is simplified and evaluated as linear elastic distribution. By integrating the area of the cross-section A-A’, the vertical friction resistance moment for the pile shaft in passive and active segments are

$$\{\begin{array}{ll}{M}_{\mathit{vp}}\left(z_p\right)& =-4{\displaystyle {\int }_0^{z_p}{\displaystyle {\int }_0^{\pi /2}{\tau }_{\mathit{vp}}\left(z_p\right)r^2\cos \psi d\psi d{z}_p}}\\ & =-4{\displaystyle {\int }_0^{z_p}{\displaystyle {\int }_0^{\pi /2}{k}_{\mathit{vp}}\left(z_p\right)r^3\theta {\cos }^2\psi d\psi d{z}_p}}=-{\displaystyle {\int }_0^{z_p}{R}_{\mathit{vp}}\left(z_p\right)\theta d{z}_p}\\ M_{\mathit{va}}\left(z_a\right)& =-2{\displaystyle {\int }_0^{z_a}{\displaystyle {\int }_0^{\pi /2}{\tau }_{\mathit{va}}\left(z_a\right)r^2\cos \psi d\psi d{z}_a}}\\ & =-2{\displaystyle {\int }_0^{z_a}{\displaystyle {\int }_0^{\pi /2}{k}_{\mathit{va}}\left(z_a\right)r^3\theta {\cos }^2\psi d\psi d{z}_a}}=-{\displaystyle {\int }_0^{z_a}{R}_{\mathit{va}}\left(z_a\right)\theta d{z}_a}\end{array}$$

(2)

in which the $M_{\mathit{vp}}\left(z_p\right)$ and $M_{\mathit{va}}\left(z_a\right)$, ${\tau }_{\mathit{vp}}\left(z_p\right)$ and ${\tau }_{\mathit{va}}\left(z_a\right)$, $k_{\mathit{vp}}\left(z_p\right)$ and $k_{\mathit{va}}\left(z_a\right)$, $R_{\mathit{vp}}\left(z_p\right)$ and $R_{\mathit{va}}\left(z_a\right)$ are the vertical friction resistance moment, vertical friction, vertical subgrade coefficient, vertical friction stiffness coefficient for the pile shaft in passive and active segments, respectively; $R_{\mathit{vp}}\left(z_p\right)=\frac{\pi D^3{k}_{\mathit{vp}}\left(z_p\right)}{8}$, $R_{\mathit{va}}\left(z_a\right)=\frac{\pi D^3{k}_{\mathit{va}}\left(z_a\right)}{16}$, $\theta$ denote the cross-section rotation angle, $D$ and $r$ denote the diameter and radius of the cross-section for the pile shaft, respectively.

2.3. Determination of Axial Force along the Pile Shaft and Pile-Shaft Model

The pile-unit shaft resistance is mobilized under vertical loads and will affect the distribution of the axial load along the pile. It is important to estimate the distribution of the pile axial force along the pile shaft to accurately evaluate the pile response. By taking the assumption that the axial force of the pile shaft is varied linearly along the depth, for the pile shaft in the free segment above the ground surface, the axial force $N_f\left(z_f\right)$ can be formulated by

$$N_f\left(z_f\right)=N_t+f_0{z}_f$$

(3)

where $f_0=A_c{\gamma }_c$ is the coefficient that describes the increase in the axial force of the per-unit-depth pile shaft in the free segment, and where $A_c$ and ${\gamma }_c$ represent the cross-sectional area and unit weight of the pile.

For piles below the ground surface, the axial forces $N_p\left(z_p\right)$ and $N_a\left(z_a\right)$, which denote the passive and active segments, respectively, can be evaluated by

$$N_p\left(z_p\right)=N_t+f_0{L}_f+f_1{z}_p$$

(4)

$$N_a\left(z_a\right)=N_t+f_0{L}_f+f_1\left(L_p+z_a\right)$$

(5)

where $f_1$ represent the coefficients of the axial force growth for the pile shaft below the ground surface.

$$f_1=A_c{\gamma }_c-\frac{\mu \tau }{2}$$

(6)

where $\mu$ is perimeter of pile shaft; and $\tau$ represents the ultimate shear friction at the pile–soil interface.

In view of the influence of shear deformation on the whole deformation for the pile shaft, in the current paper, Timoshenko beam theory [49] is adopted to establish associated differential equations for later analysis for simplicity, as illustrated graphically in Figure 3. In detail, in this theory, the plane section hypothesis is used, viz., the cross-section of the pile shaft remains as a plane after bending, whereas the cross-section is no longer perpendicular to the original neutral axis.

Hence, according to Timoshenko beam theory, the relationship between the deformation and the internal forces is:

$$Q=\kappa A_c{G}_c\left(\frac{dy}{dz}-\theta \right)$$

(7)

$$M=EI\frac{d\theta }{dz}$$

(8)

where $EI$ is the bending stiffness of the pile shaft; $\kappa A_c{G}_c$ is the equivalent shear stiffness for the pile foundation; $\kappa$ is the equivalent shear stiffness; and $G_c$ and $\theta$ denote the shear modulus, cross-section rotation angle the for the pile shaft, respectively.

2.4. Governing Differential Equations and Transfer-Matrix Coefficient for Pile in the Free Segment

For the pile in the free segment above the ground surface, an independent local coordinate system on the ith discrete segment with subscript f (Figure 4a) is established. By discretizing the pile length of free segment $L_f$ into $n_f$ parts, the pile length of each discrete segment can be determined as $h_f\left(=L_f/n_f\right)$, the average axial force ${\overline{N}}_{fi}$ of the ith discrete segment (equal to the axial force emerged in both the upper and lower cross-section) can be formulated by

$${\overline{N}}_{fi}=N_t+f_0\left(2i-1\right)h_f/2$$

(9)

Assuming that the average lateral distributed load ${\overline{q}}_{fi}$ acting on the ith discrete segment is constant and equals to the mean value of that acting on the corresponding upper and lower cross-section yields

$${\overline{q}}_{fi}=\left[q_f\left(i{h}_f-h_f\right)+q_f\left(i{h}_f\right)\right]/2$$

(10)

The flexural differential equations for the ith discrete free segment of pile shaft can be readily found based on Figure 4b, which are found to be

$$\frac{d^4{y}_{\mathit{fi}}}{d{z}_{\mathit{fi}}^4}+\frac{1}{E{I}_f}\frac{d}{d{z}_{\mathit{fi}}}\left[{\overline{N}}_{\mathit{fi}}\frac{d{y}_f}{d{z}_{\mathit{fi}}}\right]-\frac{{\phi }_{IN}}{E{I}_f}\frac{d}{d{z}_{\mathit{fi}}}\left[{\overline{N}}_{\mathit{fi}}\frac{d{y}_{\mathit{fi}}}{d{z}_{\mathit{fi}}}\right]=\frac{b_1}{E{I}_f}{q}_{\mathit{fi}}$$

(11)

where the notation f indicates the free segment for the pile shaft. Furthermore, $b_1$, $E$, $I_f$ and $E{I}_f$ are the calculated width, material’s elastic modulus, moment of inertia and bending stiffness of the pile shaft in the free segment, respectively.

Thus, in view of the response continuity for independent discrete segment, the transfer matrix equation for the ith discrete free segment of the pile shaft can be expressed as

$$S_{\mathit{fi}}=U_{\mathit{fi}}{S}_{f\left(i-1\right)}$$

(12)

where $S_{\mathit{fi}}={\left[\begin{array}{ccccc}{y}_{\mathit{fi}}& {\phi }_{\mathit{fi}}& M_{\mathit{fi}}& Q_{\mathit{fi}}& 1\end{array}\right]}^T$ and $S_{f\left(i-1\right)}={\left[\begin{array}{ccccc}{y}_{f\left(i-1\right)}& {\phi }_{f\left(i-1\right)}& M_{f\left(i-1\right)}& Q_{f\left(i-1\right)}& 1\end{array}\right]}^T$ are all 5-by-1 transposed matrices and represent the pile horizontal deformation, axis rotation angle, bending moment and shear force for the ith upper and i − 1th lower discrete free segment for the pile shaft, respectively. In which ${\mathit{U}}_{\mathit{fi}}$ is the transfer matrix coefficient of the ith discrete free segment for the pile shaft.

Let ${\alpha }_{\mathit{fi}}^2={\overline{N}}_{\mathit{fi}}/E{I}_f$, the transfer coefficient matrix $U_{\mathit{fi}}$ of the ith discrete free segment for the pile shaft can be obtained as

$$U_{\mathit{fi}}=\left[\begin{array}{ccccc}1& \frac{\sin ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\alpha }_{\mathit{fi}}}& \frac{1-\cos ({\alpha }_{\mathit{fi}}{z}_{\mathit{gi}})}{{\overline{N}}_{\mathit{fi}}}& \frac{{\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}-\sin ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\overline{N}}_{\mathit{fi}}{\alpha }_{\mathit{fi}}}& {\chi }_{1,fi}\\ 0& \cos ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})& \frac{{\alpha }_{\mathit{fi}}\sin ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\overline{N}}_{\mathit{fi}}}& \frac{1-\cos ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\overline{N}}_{\mathit{fi}}}& {\chi }_{2,fi}\\ 0& \frac{-{\overline{N}}_{\mathit{fi}}\sin ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\alpha }_{\mathit{fi}}}& \cos ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})& \frac{\sin ({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}})}{{\alpha }_{\mathit{fi}}}& {\chi }_{3,fi}\\ 0& 0& 0& 1& {\chi }_{4,fi}\\ 0& 0& 0& 0& 1\end{array}\right]$$

(13)

With

$$\begin{array}{l}{\chi }_{1,fi}=\frac{b_1{\overline{q}}_{\mathit{fi}}\left\{{\alpha }_{\mathit{fi}}^2{z}_{\mathit{fi}}^2+\left[2\cos \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)-2\right]\left(-\frac{{\overline{N}}_{\mathit{fi}}}{\kappa A_c{G}_c}+1\right)\right\}+2{\overline{N}}_{fi}{\phi }_{\mathit{IN}}{\alpha }_{\mathit{fi}}\left[{\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}-\sin \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)\right]}{2{\overline{N}}_{\mathit{fi}}{\alpha }_{\mathit{fi}}^2}\\ {\chi }_{2,fi}= \frac{b_1{\overline{q}}_{\mathit{fi}}\left[{\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}-\sin \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)\right]\left(-\frac{{\overline{N}}_{\mathit{fi}}}{\kappa A_c{G}_c}+1\right)+{\overline{N}}_{\mathit{fi}}{\phi }_{\mathit{IN}}{\alpha }_{\mathit{fi}}\left[1-\cos \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)\right]}{{\overline{N}}_{\mathit{fi}}{\alpha }_{\mathit{fi}}}\\ {\chi }_{3,fi}= \frac{b_1{\overline{q}}_{\mathit{fi}}\left[\cos \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)-1\right]\left(\frac{{\overline{N}}_{\mathit{fi}}}{\kappa A_c{G}_c}-1\right)+{\overline{N}}_{\mathit{fi}}{\phi }_{\mathit{IN}}{\alpha }_{\mathit{fi}}\sin \left({\alpha }_{\mathit{fi}}{z}_{\mathit{fi}}\right)}{{\alpha }_{\mathit{fi}}^2}\\ {\chi }_{4,fi}=b_1{\overline{q}}_{\mathit{fi}}{z}_{\mathit{fi}}\end{array}$$

(14)

As a result, the transfer matrix equation of the lateral response for the free segment in the pile shaft can be evaluated by

$$S_{f{n}_f}=U_{f{n}_f}{U}_{f\left(n_f-1\right)}\cdots U_{f2}{U}_{f1}{S}_{f0}=U_f{S}_{f0}$$

(15)

in which the notation $U_f$ is the total transfer coefficient matrix of the free segment of the pile shaft; $S_{f0}={\left[\begin{array}{ccccc}{y}_{f0}& {\phi }_{f0}& M_{f0}& Q_{f0}& 1\end{array}\right]}^T$ is the mechanical deformation parameter matrix at the head of the pile shaft and $S_{f{n}_f}={\left[\begin{array}{ccccc}{y}_{f{n}_f}& {\phi }_{f{n}_f}& M_{f{n}_f}& Q_{f{n}_f}& 1\end{array}\right]}^T$ is that between the free and passive segments for the pile shaft.

2.5. Governing Differential Equations and Transfer-Matrix Coefficient for Pile in the Passive Segment

Similarly, for the pile in the passive segment of the pile shaft, an independent local coordinate system on the ith discrete segment with subscript p (Figure 5a) is established and, discretizing it into $n_p$ parts with length $h_p=\left(L_p/n_p\right)$, the average axial force ${\overline{N}}_{pi}$ of each discrete segment can be evaluated by

$${\overline{N}}_{pi}=N_t+f_0{L}_0+f_1\left(2i-1\right)h_p/2$$

(16)

Presuming that the average passive load ${\overline{q}}_{pi}$ and the average vertical friction stiffness coefficients ${\overline{R}}_{vpi}$ on the ith discrete segment are constant and equal to the mean value of those on the corresponding upper and lower cross-section, namely

$${\overline{q}}_{pi}=\left[q_p\left(i{h}_p-h_p\right)+q_p\left(i{h}_p\right)\right]/2$$

(17)

$${\overline{R}}_{vpi}=\left[R_{vp}\left(i{h}_p-h_p\right)+R_{vp}\left(i{h}_p\right)\right]/2$$

(18)

Moreover, by taking into account that the average coefficient of horizontal foundation resistance ${\overline{k}}_{pi}$ surrounding the ith pile segment is constant, the determination of corresponding value is performed on the basis of the mean value theorem of integrals:

$${\overline{k}}_{pi}=m{\displaystyle {\int }_{\left(i-1\right)h_p}^{i{h}_p}{\left(z_0+z_p\right)}^n}d{z}_p/h_p=\frac{m\left[{\left(z_0+i{h}_p\right)}^{n+1}-{\left(z_0+\left(i-1\right)h_p\right)}^{n+1}\right]}{\left(n+1\right)h_p}$$

(19)

For the ith pile segment in the passive segment shown in Figure 5b, the differential equation is expressed as

$$\begin{array}{l}\frac{d^4{y}_{pi}}{d{z}_{pi}^4}+\frac{1}{E{I}_p}\frac{d}{d{z}_{pi}}\left[{\overline{N}}_{pi}\frac{d{y}_{pi}}{d{z}_{pi}}\right]-\frac{{\phi }_{IN}}{E{I}_p}\frac{d}{d{z}_{pi}}\left[{\overline{N}}_{pi}\frac{d{y}_{pi}}{d{z}_{pi}}\right]\\ +2{\overline{R}}_{vpi}\frac{d^2{y}_{pi}}{d{z}_{pi}{}^2}-\frac{\left(2{\overline{R}}_{vpi}E{I}_p+\kappa A_c{G}_c\right)}{\kappa A_c{G}_{c}E{I}_p}{b}_1\left({\overline{q}}_{pi}-{\overline{k}}_{pi}{y}_{pi}\right)=0\end{array}$$

(20)

where the notation of p indicates the passive segment for the pile shaft.

Similarly, for the piles in the passive segment, the transfer-matrix equation for the ith discrete passive segment of the pile shaft can be formulated as (Appendix B)

$$S_{pi}=U_{pi}{S}_{p\left(i-1\right)}$$

(21)

where $S_{pi}={\left[\begin{array}{ccccc}{y}_{pi}& {\phi }_{pi}& M_{pi}& Q_{pi}& 1\end{array}\right]}^T$ and $S_{p\left(i-1\right)}={\left[\begin{array}{ccccc}{y}_{p\left(i-1\right)}& {\phi }_{p\left(i-1\right)}& M_{p\left(i-1\right)}& Q_{p\left(i-1\right)}& 1\end{array}\right]}^T$ are all 5-by-1 transposed matrices and represent the pile horizontal deformation, axis rotation angle, bending moment and shear force for the ith upper and i − 1th lower discrete passive segment for the pile shaft, respectively. In addition, in which ${\mathit{U}}_{pi}$ is the transfer matrix coefficient of the ith pile slice of the pile in the passive segment.

By setting $T_{pi}={\overline{N}}_{pi}+2{\overline{R}}_{vpi}^{}$, ${\lambda }_{pi}=D_{pi}/4E{I}_p$, $D_{pi}=T_{pi}+\left(b_1{\overline{k}}_{pi}E{I}_p/\kappa A_c{G}_c\right)$, $J_{pi}=1+\left(2{\overline{R}}_{vpi}^{}/\kappa A_c{G}_c\right)$, ${\eta }_{pi}=b_1{\overline{k}}_{pi}/4E{I}_p$, ${\beta }_{pi}^2={\left(b_1{\overline{k}}_{pi}/4E{I}_p\right)}^{1/2}$, ${\kappa }_{pi}={\left({\beta }_{pi}^2-{\lambda }_{pi}\right)}^{1/2}$, ${\xi }_{pi}={\left({\beta }_{pi}^2+{\lambda }_{pi}\right)}^{1/2}$, and ${\omega }_{x,pi}=\pm \left({\kappa }_{pi}\pm {\xi }_{pi}\mathrm{i}\right)$ (where x equals 1, 2, 3, 4, respectively, and the symbol “i” refers to imaginary, ${\kappa }_{pi}$ and ${\xi }_{pi}$ are real part and imaginary part), the transfer coefficient matrix for the ith passive segment $U_{pi}$ can be generated by incorporating with the Laplace transformation

$$U_{pi}=\left[\begin{array}{ccccc}{\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,pi}^2+4{\lambda }_{pi}-\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\omega }_{x,pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\lambda }_{pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{D_{pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{pi}{\lambda }_{pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{D_{pi}{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\zeta }_{1,pi}\\ {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{pi}\left(\frac{{\omega }_{x,pi}^2}{\kappa A_c{G}_c}+\frac{J_{pi}}{E{I}_p}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\omega }_{x,pi}^2{e}^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\lambda }_{pi}{\omega }_{x,pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{D_{pi}\left({\omega }_{x,epi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{pi}{\lambda }_{pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{D_{pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\zeta }_{2,pi}\\ {\displaystyle \sum _{x=1}^4\frac{b_1{\overline{k}}_{pi}\left(\frac{T_{pi}}{\kappa A_c{G}_c}-J_{pi}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-\left(T_{pi}{\omega }_{x,pi}^2+b_1{\overline{k}}_{pi}{J}_{pi}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,pi}^2+\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{pi}\left({\omega }_{x,pi}^2+\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\zeta }_{3,pi}\\ {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{pi}\left({\omega }_{x,pi}^2+4{\lambda }_{pi}-\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-{\eta }_{pi}{e}^{{\omega }_{x,pi}{z}_{pi}}}{{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,pi}^2+4{\lambda }_{pi}\right)e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}& {\zeta }_{4,pi}\\ 0& 0& 0& 0& 1\end{array}\right]$$

(22)

With

$$\begin{array}{l}{\chi }_{1,pi}=b_1{\overline{q}}_{pi}\left\{\frac{1}{b_1{\overline{k}}_{pi}}-{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}\left[\frac{\left({\omega }_{x,pi}^2+4{\lambda }_{pi}\right)}{b_1{\overline{k}}_{pi}}-\frac{1}{\kappa A_c{G}_c}\right]}\right\}+\frac{4{\lambda }_{pi}{\overline{N}}_{pi}{\phi }_{IN}}{D_{pi}}{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}\\ {\chi }_{2,pi}=\left[ b_1{\overline{q}}_{pi} \left(\frac{4{\lambda }_{pi}}{D_{pi}}{J}_{pi}+\frac{{\omega }_{x,pi}^2}{\kappa A_c{G}_c}\right)+\frac{4{\lambda }_{pi}{\overline{N}}_{pi}{\phi }_{IN}{\omega }_{x,pi}}{D_{pi}}\right]{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}\\ {\chi }_{3,pi}=b_1{\overline{q}}_{pi}\left(-\frac{T_{pi}}{\kappa A_c{G}_c}+J_{pi}\right){\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}+{\overline{N}}_{pi}{\phi }_{IN}\left({\omega }_{x,pi}^2+\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right){\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}}\\ {\chi }_{4,pi}=b_1{\overline{q}}_{pi}\left[\left({\omega }_{x,pi}^2+4{\lambda }_{pi}\right)-\frac{b_1{\overline{k}}_{pi}}{\kappa A_c{G}_c}\right]{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4{\omega }_{x,pi}\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}+{\overline{N}}_{pi}{\phi }_{IN}\left[-1+\left({\omega }_{x,pi}^2+4{\lambda }_{pi}\right){\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,pi}{z}_{pi}}}{4\left({\omega }_{x,pi}^2+2{\lambda }_{pi}\right)}}\right]}\end{array}$$

(23)

Finally, the transfer matrix equation of the lateral response in the passive segment for the pile shaft can be expressed as

$$S_{p{n}_p}=U_{p{n}_p}{U}_{p\left(n_p-1\right)}\cdots U_{p2}{U}_{p1}{S}_{p0}=U_p{S}_{p0}$$

(24)

In which the notation $U_p$ is the total transfer coefficient matrix in the passive segment; $S_{p0}={\left[\begin{array}{ccccc}{y}_{f{n}_f}& {\phi }_{f{n}_f}& M_{f{n}_f}& Q_{f{n}_f}& 1\end{array}\right]}^T$ is the mechanicals deformation parameter matrix of the interface between free and passive segments, and $S_{p{n}_P}={\left[\begin{array}{ccccc}{y}_{p{n}_p}& {\phi }_{p{n}_p}& M_{p{n}_p}& Q_{p{n}_p}& 1\end{array}\right]}^T$ is that between passive and active segments for the pile shaft.

2.6. Governing Differential Equations and Transfer-Matrix Coefficient for Pile in the Active Segment

Analogously to the foregoing approaches, for the lateral response in the active segment of the pile shaft, an independent local coordinate system on the ith discrete segment with subscript a (Figure 6a) is established and, discretizing it into $n_a$ parts with length $h_a=\left(L_a/n_a\right)$, the average axial force ${\overline{N}}_{ai}$ for ith segment can obtained with:

$${\overline{N}}_{ai}=N_t+f_0{L}_f+f_1{L}_p+f_1\left(2i-1\right)h_a/2$$

(25)

Similarly, the average vertical friction stiffness coefficient ${\overline{R}}_{vai}$ and the average horizontal foundation resistance coefficient ${\overline{k}}_{ai}$ surrounding it for ith active segment of pile shaft are

$${\overline{R}}_{vai}=\left[R_{va}\left(i{h}_a-h_a\right)+R_{va}\left(i{h}_a\right)\right]/2$$

(26)

$${\overline{k}}_{ai}=\frac{m\left[{\left(z_0+L_p+i{h}_a\right)}^{n+1}-{\left(z_0+L_p+\left(i-1\right)h_a\right)}^{n+1}\right]}{\left(n+1\right)h_a}$$

(27)

For the ith pile segment in the passive segment shown in Figure 6b, the differential equation is expressed as

$$\begin{array}{l}\frac{d^4{y}_{ai}}{d{z}_{ai}^4}+\frac{1}{E{I}_a}\frac{d}{d{z}_{ai}}\left[{\overline{N}}_{ai}\frac{d{y}_{ai}}{d{z}_{ai}}\right]-\frac{{\phi }_{IN}}{E{I}_a}\frac{d}{d{z}_{ai}}\left[{\overline{N}}_{ai}\frac{d{y}_{ai}}{d{z}_{ai}}\right]\\ +{\overline{R}}_{vai}\frac{d^2{y}_{ai}}{d{z}_{ai}{}^2}+\frac{\left(2{\overline{R}}_{vai}E{I}_a+\kappa A_c{G}_c\right)}{\kappa A_c{G}_{c}E{I}_a}{b}_1{\overline{k}}_{pi}{y}_{ai}=0\end{array}$$

(28)

where the notations of a indicate the active segment for the pile shaft.

Similarly, the transfer-matrix equation for the ith discrete active segment of the pile shaft can be formulated as (Appendix C).

$$S_{ai}=U_{ai}{S}_{a\left(i-1\right)}$$

(29)

where $S_{ai}={\left[\begin{array}{ccccc}{y}_{ai}& {\phi }_{ai}& M_{ai}& Q_{ai}& 1\end{array}\right]}^T$ and $S_{a\left(i-1\right)}={\left[\begin{array}{ccccc}{y}_{a\left(i-1\right)}& {\phi }_{a\left(i-1\right)}& M_{a\left(i-1\right)}& Q_{a\left(i-1\right)}& 1\end{array}\right]}^T$ are all 5-by-1 transposed matrices and represent the pile horizontal deformation, axis rotation angle, bending moment and shear force for the ith upper and i − 1th lower discrete active segment for the pile shaft, respectively. In which $U_{ai}$ is the transfer matrix coefficient of the ith pile slice of the pile in the active segment, by setting $T_{ai}={\overline{N}}_{ai}+{\overline{R}}_{vai}^{}$, $D_{ai}=T_{ai}+\left(b_1{\overline{k}}_{ai}E{I}_a/\kappa A_c{G}_c\right)$, ${\lambda }_{ai}=D_{ai}/4E{I}_a$, $J_{ai}=1+\left({\overline{R}}_{vai}^{}/\kappa A_c{G}_c\right)$${\eta }_{ai}=b_1{\overline{k}}_{ai}/4E{I}_a$, ${\beta }_{ai}^2={\left(b_1{\overline{k}}_{ai}/4E{I}_a\right)}^{1/2}$, ${\kappa }_{ai}={\left({\beta }_{ai}^2-{\lambda }_{ai}\right)}^{1/2}$, ${\xi }_{ai}={\left({\beta }_{ai}^2+{\lambda }_{ai}\right)}^{1/2}$, and ${\omega }_{x,ai}=\pm \left({\kappa }_{ai}\pm {\xi }_{ai}\mathrm{i}\right)$ (where x equals 1, 2, 3, 4, respectively, and the symbol “i” refers to imaginary,${\kappa }_{ai}$ and ${\xi }_{ai}$ are real part and imaginary part), the transfer-matrix equation $U_{ai}$ for ith active segment is capable of evaluating in following form:

$$U_{ai}=\left[\begin{array}{ccccc}{\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,ai}^2+4{\lambda }_{ai}-\frac{b_1{\overline{k}}_{ai}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\omega }_{x,ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\lambda }_{ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{D_{ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{ai}{\lambda }_{ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{D_{ai}{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\chi }_{1,ai}\\ {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{ai}\left(\frac{{\omega }_{x,ai}^2}{\kappa A_c{G}_c}+\frac{1}{E{I}_a}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\omega }_{x,ai}^2{e}^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{{\lambda }_{ai}{\omega }_{x,ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{D_{ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{ai}{\lambda }_{ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{D_{ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\chi }_{2,ai}\\ {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{ai}\left(-\frac{T_{ai}}{\kappa A_c{G}_c}+J_{ai}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-\left(T_{ai}{\omega }_{x,ai}^2+b_1{\overline{k}}_{ai}{J}_{ai}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,ai}^2+\frac{b_1{\overline{k}}_{ai}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{J_{ai}\left({\omega }_{x,ai}^2+\frac{b_1{\overline{k}}_{ai}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\chi }_{3,ai}\\ {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{ai}\left({\omega }_{x,ai}^2+4{\lambda }_{ai}-\frac{b_1{\overline{k}}_{ai}}{\kappa A_c{G}_c}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-b_1{\overline{k}}_{ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{-{\eta }_{ai}{e}^{{\omega }_{x,ai}{z}_{ai}}}{{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\displaystyle \sum _{x=1}^4\frac{\left({\omega }_{x,ai}^2+4{\lambda }_{ai}\right)e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}& {\chi }_{4,ai}\\ 0& 0& 0& 0& 1\end{array}\right]$$

(30)

With

$$\begin{array}{l}{\chi }_{1,ai}=\frac{4{\lambda }_{ai}{\overline{N}}_{ai}{\phi }_{IN}}{D_{ai}}{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}\\ {\chi }_{2,ai}=\frac{4{\lambda }_{ai}{\overline{N}}_{ai}{\phi }_{IN}}{D_{ai}}{\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}\\ {\chi }_{3,ai}={\overline{N}}_{ai}{\phi }_{IN}\left({\omega }_{x,ai}^2+\frac{b_1{\overline{k}}_{ai}}{\kappa A_c{G}_c}\right){\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,ai}{z}_{ai}}}{4{\omega }_{x,ai}\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}\\ {\chi }_{4,ai}={\overline{N}}_{ai}{\phi }_{IN}\left[-1+\left({\omega }_{x,ai}^2+4{\lambda }_{ai}\right){\displaystyle \sum _{x=1}^4\frac{e^{{\omega }_{x,ai}{z}_{ai}}}{4\left({\omega }_{x,ai}^2+2{\lambda }_{ai}\right)}}\right]\end{array}$$

(31)

Then, in view of the response continuity for independent discrete segment, the following relationship holds true:

$$S_{a{n}_a}=U_{a{n}_a}{U}_{a\left(n_a-1\right)}\cdot \cdot \cdot U_{a2}{U}_{a1}{S}_{a0}=U_a{S}_{a0}$$

(32)

in which the notation $U_a$ is the total transfer coefficient matrix in the active segment; $S_{a0}={\left[\begin{array}{ccccc}{y}_{p{n}_p}& {\phi }_{p{n}_p}& M_{p{n}_p}& Q_{p{n}_p}& 1\end{array}\right]}^T$ is the mechanical deformation parameter matrix of the interface between passive and active segments, and $S_{a{n}_a}={\left[\begin{array}{ccccc}{y}_{a{n}_a}& {\phi }_{a{n}_a}& M_{a{n}_a}& Q_{a{n}_a}& 1\end{array}\right]}^T$ is that at the bottom of the pile shaft.

2.7. General Solutions for the Pile Responses under Combined Loads

Considering that the continuity for values of the lateral responses for the active, passive and free segment of the pile shaft emerged equally at their interfaces, the following relationships hold true:

$$\begin{array}{l}{S}_{f{n}_f}=S_{p0}\\ S_{p{n}_p}=S_{a0}\end{array}$$

(33)

Hence, the transfer-matrix equation for the total pile shaft is derived by combining Equations (15), (24), (32) and (33):

$$S_{a{n}_a}=U_a{U}_p{U}_f{S}_{f0}=U{S}_{f0}$$

(34)

$U_a$ and $U_p$ denote the transfer-matrix coefficient, which is calculated by Equation (32) for piles in the active segment and by Equation (28) for piles in the passive segment. $U_f$ denotes the transfer matrix coefficient of the pile in the free segment as calculated by in Equation (15), $U$ denotes total transfer coefficient matrix of the pile shaft, which is the product of the transfer-matrix coefficient of each pile slice in the sequence from the pile bottom to the pile top.

Equation (34) is composed of four equations with eight variables, including the pile deflection, slope, bending moment, and sheer force of the pile bottom and pile head. The boundary conditions of the pile toe and pile head must be specified to solve this equation. Specifically, in the current paper, the boundary conditions can be categorized by two major divisions:

(1) Constraints in the top of the pile shaft

For the case of free constraint, the relative boundary conditions need to be satisfied:

$$M_{f0}=M_t;Q_{f0}=Q_t$$

(35)

For the case of fixed constraint, the relative boundary conditions need to be satisfied:

$$y_{f0}=0;{\phi }_{f0}=0$$

(36)

(2) Constraints in bottom of the pile shaft

For the case of free constraint, the relative boundary conditions need to be satisfied:

$$M_{a{n}_a}=0;Q_{a{n}_a}=0$$

(37)

For the case of fixed constraint, the relative boundary conditions need to be satisfied:

$$y_{a{n}_a}=0;{\phi }_{a{n}_a}=0$$

(38)

Substituting one of the pile-top boundary conditions and one of the pile-bottom boundary conditions into Equation (15), the rest of the variables of the pile top could be obtained. Thus, based on the response of the pile top $S_{f0}$, the solution of the pile in the free segment at any depth is expressed as

$$S_{fi}=U_{fi}{U}_{f\left(i-1\right)}\cdots U_{f2}{U}_{f1}{S}_{f0}$$

(39)

Similarly, the responses of pile in the passive segment and active segment at any depth are expressed as

$$S_{pi}=U_{pi}{U}_{p\left(i-1\right)}\cdots U_{p2}{U}_{p1}{U}_f{S}_{f0}$$

(40)

$$S_{ai}=U_{ai}{U}_{a\left(i-1\right)}\cdots U_{a2}{U}_{a1}{U}_p{U}_f{S}_{f0}$$

(41)

The algorithm to calculate $S_{fi}$, $S_{pi}$ and $S_{ai}$ is summarized in Figure 7.

3. Validation

3.1. Case Study 1

For the purpose of verifying the effectiveness of the proposed method in this work, an engineering practice of a bridge pile in a steep slope illustrated in work of Peng et al. [34] was adopted for relevant analysis herein.

Specifically, in reference to this example, the basic parameters are listed as below, in

Table 1. Detailed parameters in work of Peng et al. [34].

Parameters	Value
Type of soil stress distribution	Parabola
Diameter of pile shaft (m)	2.0
Length of pile shaft (m)	25.0
Elastic modulus (Gpa)	29.6
Subjected axial force at top of pile shaft (kN)	7312.0
Subjected bending moment at top of pile shaft (kN × m)	520.0
Subjected horizontal force at top of pile shaft (kN)	50.0

:

Additionally, the angle for the slope is approximately 40°, the distributions of soil deposits along with the pile shaft from top to the bottom are planting soil, gravelly silty clay, strongly weathered dolomitic limestone and moderately weathered dolomitic limestone, with thicknesses of 1.8 m, 9.7 m, 6.1 m and 7.4 m, respectively.

Therefore, according to the national code [48], in this paper, the proportional coefficients of horizontal resistance for the foundation against soil layer m₁, m₂ and the coefficients of horizontal resistance against rock layer C₃, C₄ can be determined as 1.5 MN/m⁴, m₂ = 4 MN/m⁴, C₃ = 80 MN/m³ and C₄ = 420 MN/m³, respectively. By noticing the fact that, for the pile shaft, the passive segment occurs in the soil layers, the horizontal resistance reduction coefficient for foundation ${\delta }_p$, the vertical subgrade coefficients for the foundation in both the planting soil layer and the gravelly silty-clay soil layer are able to be obtained as 0.8 and 40 MN/m³ [48], respectively.

Hence, the predictions of the bending moment for each segment of the pile shaft yielded by the proposed approach in this work is illustrated diagrammatically in Figure 8, along with the site measured results reported by Peng et al. [34]:

In Figure 8, it is found that the predictions highly coincide as satisfactorily as possible with the site measured results. The error between predicted results and measured value in the maximum bending moment of pile shaft is about 6.19%. In detail, although there are some underestimations for the predicted results compared with the measured results, especially around the locations near the interface between the layers of the highly weathered lime stone and the moderately weathered limestone, from the viewpoint of applicability, the majority of the predicted curve is capable of indeed delineating the bending moments for the pile shaft in real engineering practice.

3.2. Case Study 2

In order to further verify the rationality and applicability of the approach proposed in this work, a comparison between the predictions and the results generated by ZHAO et al. [41] in perspective of the lateral response for pile shaft was also conducted; the relative parameters of the pile shaft for examining are listed as below, in

Table 2. Detailed parameters in work of ZHAO et al. [41].

Parameters	Value
Type of soil stress distribution	Rectangle
Diameter of pile shaft (m)	2.0
Length of pile shaft (m)	30.0
Elastic modulus (Gpa)	18.0
Concrete density for the pile shaft (kN/m3)	25.0
Subjected axial force at top of pile shaft (kN)	9100.0
Subjected bending moment at top of pile shaft (kN × m)	1000.0
Subjected horizontal force at top of pile shaft (kN)	179.0

:

Specifically, in this case, the pile shaft is located in the rock slope, the corresponding rock layers for the passive and active segments are the strongly weathered rock and slightly weathered rock, with the thickness of 10 m and 20 m, respectively. Therefore, according to the national code [48], the horizontal resistance proportional coefficients for those two types of rock layers are determined as m_p = 5 MN/m⁴ and m_a = 50 MN/m⁴, respectively. In this case, the influence of the vertical friction in the passive segment of the pile shaft was taken into account. By assuming that the other parameters are constant and ignoring the slope effects on the vertical resistances in the passive segment of the pile shaft, the vertical subgrade coefficients of the foundation can be determined as k_vp = 3 × 10⁵ kN/m³ [48]. Hence, the comparison results are obtained as illustrated in Figure 9. In detail, in Figure 9, it can be observed that the predictions generated by the proposed method are extremely closed to that of ZHAO et al.’s [41] work for both the lateral deformation (Figure 9a) and the bending moment (Figure 9b). The errors between predicted results and ZHAO et al.’s [41] work of maximum lateral displacement and maximum ending moment of pile shaft is about 4.99% and 7.52%, respectively. It is also understood from Figure 9 that there are marked reduction effects on the lateral response of the pile shaft due to the vertical friction in the passive segment. It is illustrated that the horizontal deformation for the pile-top is reduced by 25.36% and the maximum bending moment for the pile shaft is reduced by 19.40% by dint of the effects caused by the vertical friction, respectively.

4. Discussion

In this section, for the purpose of investigating the influences caused by vertical friction of the passive segment, initial inclination angle, shear deformation and the P-Δ effect of the vertical load, a deep analysis on an environmental remediation utilization project of an abandoned mine is conducted. the schematic diagram of engineering case is described in Figure 10. The detailed information for this example is listed in the following,

Table 3. Detailed parameters.

Parameters	Value
Type of soil stress distribution	Rectangle
Diameter of pile shaft (m)	2.0
Length of free segment for pile shaft $L_f$ (m)	5.0
Length of passive segment for pile shaft $L_p$ (m)	10.0
Length of active segment for pile shaft $L_a$ (m)	20.0
Elastic modulus (Gpa)	18.0
Concrete density for the pile shaft (kN/m3)	25.0
Subjected axial force at top of pile shaft (kN)	6000.0
Subjected bending moment at top of pile shaft (kN × m)	400.0
Subjected horizontal force at top of pile shaft (kN)	400.0
Resultant earth pressure (kN)	600.0
Foundation vertical resistance coefficient of passive segment kvp (kN/m3)	1 × 104
Foundation vertical resistance coefficient of active segment kva (kN/m3)	1 × 105
Foundation horizontal resistance coefficient of passive segment mp (kN/m3)	7.5 × 103
Foundation horizontal resistance coefficient of active segment ma (kN/m3)	6.0 × 104

:

In detail, the pile shaft is located on the pit wall of the abandoned mine pit, the horizontal resistance reduction coefficient of the foundation ${\delta }_p$ below the slope within the depth range L_p is determined as 0.8. The boundary conditions of pile top and pile bottom are free and fixed, respectively. The detailed schematic diagram is showed in Figure 10, as follows:

4.1. Influence of Vertical Resistance in Passive Segment

In order to analyze the influence of the vertical resistance in the passive segment of the pile shaft, the vertical subgrade coefficients $k_{vp}$ for the pile passive segment is treated as the main reflex index. The corresponding results are demonstrated in Figure 11, in which the values of $k_{vp}$ varied from 0, 1 × 10⁴ kN × m³, 2 × 10⁴ kN × m³, 4 × 10⁴ kN × m³ and 6 × 10⁴ kN × m³ and other parameters are remaining constant.

It can be seen from Figure 11 that, during the increasing of the vertical subgrade coefficients $k_{vp}$ of the foundation, the horizontal deformation and the bending moment at the pile-top showed a unceasingly decreasing tendency. In detail, the horizontal deformations and the maximum bending moment are diminishing by 14.29%, 24.14%, 36.88%, 50.42% and 9.42%, 14.47%, 19.27%, 22.77%, respectively, when the $k_{vp}$ is assigned the values of 0, 1 × 10⁴ kN × m³, 2 × 10⁴ kN × m³, 4 × 10⁴ kN × m³ and 6 × 10⁴ kN × m³, respectively. Meanwhile, it is worth stressing that the distribution of the pile shaft bending moment along the pile length changes from a single peak mode to a double peak mode gradually; the positions of the maximum bending moment are slowly moving to the surface, which reveals that the load bearing ratio of the passive segment for the pile shaft is enlarged, and its horizontal bearing ability is distinctly improved.

In order to further demonstrate the ability to delineate the influences evoked by the vertical friction of the passive segment over the lateral responses of the pile shaft, it will be of interest to examine several parameters, including the relative ratio of fixed base of the maximum horizontal deformation ${\eta }_{y\max ,k_{vp}}$, that of the maximum bending moment ${\eta }_{M\max ,k_{vp}}$, the sequential variation ratio of maximum horizontal deformation $\Delta y_{\max ,k_{vp}}$ and that of the maximum bending moment $\Delta M_{\max ,k_{vp}}$ during the changing in the vertical subgrade coefficients $k_{vp}$ in which the corresponding parameters are, respectively, defined as:

$$\begin{array}{l}{\eta }_{y\max ,k_{vp}}=\frac{y_{\max }}{y_{\max ,k_{vp}=0}}\\ {\eta }_{M\max ,k_{vp}}=\frac{M_{\max }}{M_{\max ,k_{vp}=0}}\end{array}$$

(42)

$$\begin{array}{l}\Delta y_{\max ,k_{vp}}=\frac{y_{\max ,k_{vp,i}}-y_{\max ,k_{vp,i-1}}}{y_{\max ,k_{vp,i-1}}}\\ \Delta M_{\max ,k_{vp}}\frac{M_{\max ,k_{vp,i}}-M_{\max ,k_{vp,i-1}}}{M_{\max ,k_{vp,i-1}}}\end{array}$$

(43)

As shown in Figure 12, the variation tendencies of each of the above-mentioned parameters under different vertical subgrade coefficients $k_{vp}$ are exhibited lucidly. Specifically, in Figure 12, it is shown that the relationships of ${\eta }_{y\max ,k_{vp}}$ and ${\eta }_{M\max ,k_{vp}}$ with $k_{vp}$ emerge as linearly and non-linearly reduction trends, respectively, which indicates that, during the growth in $k_{vp}$, the reduction effects of vertical foundation friction in the passive segment over both the maximum horizontal deformation and the bending moment of the pile shaft are gradually decaying. Meanwhile, it can also observed that, in Figure 12, the $\Delta y_{\max ,k_{vp}}$ and $\Delta M_{\max ,k_{vp}}$ are initially diminishing dramatically and then followed increasing gently, the total amount of the drop for the $\Delta y_{\max ,k_{vp}}$ is larger than that of $\Delta M_{\max ,k_{vp}}$. Hence, it is then revealed that, while the vertical friction of the passive segment is going up, there is a critical value (denoted as $k_{vp,cr}$) for the vertical foundation resistance coefficient $k_{vp}$ existing where, when the value of $k_{vp}$ is larger than the $k_{vp,cr}$, the influence resulting from this coefficient over the lateral response of the pile shaft can be ignored.

4.2. Influence of Initial Inclination Angle of Pile Shaft

In this subsection, the influence of the initial inclination angle ${\phi }_{IN}$ for the pile shaft is analyzed by setting up its value as 0, 5 mrad, 10 mrad and 15 mrad, respectively. Assuming that the downhill direction is positive in direction, the relative results are illustrated in Figure 13. It is clear that when the ${\phi }_{IN}$ is varying from 0 to 15 mrad, the maximum horizontal deformation for the pile shaft is increasing by 4.91%, 9.74% and14.66%, respectively; the maximum bending moment for the pile shaft is increasing by 6.24%, 12.48%, and 18.71%, respectively. Therefore, this indicates that there is an obvious enlargement effect caused by the increase in ${\phi }_{IN}$ on the lateral response of the pile shaft when the length of the free segment is adequate.

Similarly, in order to further delineate the influences evoked by the initial inclination angle over the lateral responses of the pile shaft, it will be of interest to examine several parameters, including fixed base of the maximum lateral displacement ${\eta }_{y\max ,{\phi }_{IN}}$, that of the maximum bending moment ${\eta }_{M\max ,{\phi }_{IN}}$, the sequential variation ratio of maximum horizontal deformation $\Delta y_{\max ,{\phi }_{IN}}$ and that of the maximum bending moment $\Delta M_{\max ,{\phi }_{IN}}$, in which the corresponding parameters are, respectively, defined as:

$$\begin{array}{l}{\eta }_{y\max ,{\phi }_{IN}}=\frac{y_{\max }}{y_{\max ,{\phi }_{IN}=0}}\\ {\eta }_{M\max ,{\phi }_{IN}}=\frac{M_{\max }}{M_{\max ,{\phi }_{IN}=0}}\end{array}$$

(44)

$$\begin{array}{l}\Delta y_{\max ,{\phi }_{IN}}=\frac{y_{\max ,{\phi }_{IN,i}}-y_{\max ,{\phi }_{IN,i-1}}}{y_{\max ,{\phi }_{IN,i-1}}}\\ \Delta M_{\max ,{\phi }_{IN}}=\frac{M_{\max ,{\phi }_{IN,i}}-M_{\max ,{\phi }_{IN,i-1}}}{M_{\max ,{\phi }_{IN,i-1}}}\end{array}$$

(45)

The analysis results involved in the above definitions and the ${\phi }_{IN}$ are listed in Figure 14. It is shown that there is a linear increase relationship between the ${\phi }_{IN}$ and both the ${\eta }_{y\max ,{\phi }_{IN}}$ and ${\eta }_{M\max ,{\phi }_{IN}}$, which indicates that the increase in ${\phi }_{IN}$ has an amplification effect on those parameters. Moreover, the gaps of the increments between ${\eta }_{y\max ,{\phi }_{IN}}$ and ${\eta }_{M\max ,{\phi }_{IN}}$ generated by changing ${\phi }_{IN}$ are gradually distinguished; in addition, compared with those of the ${\eta }_{y\max ,{\phi }_{IN}}$ and ${\eta }_{M\max ,{\phi }_{IN}}$, the relationships between the ${\phi }_{IN}$ and $\Delta y_{\max ,{\phi }_{IN}}$ or $\Delta M_{\max ,{\phi }_{IN}}$ initially experienced a non-linearly decreasing trend and then flattened, which revealed that, as the ${\phi }_{IN}$ is increasing, there is an attenuation effect for the influence of the $\Delta y_{\max ,{\phi }_{IN}}$ and $\Delta M_{\max ,{\phi }_{IN}}$ over the lateral response of the pile shaft, and such effect is unceasingly diminishing.

4.3. Influence of Vertical Loading

In view of the fact that the vertical load is an essential factor which can influence the lateral response of the pile shaft, such influences are investigated by varying the ratio of vertical load and passive load for the pile shaft N/q as 0, 5, 10, 20 and 40. Figure 15 compares the relationships between lateral deformation and depth with varied N/q when the passive load q is equal to 400 kN. It is revealed that both the horizontal deformation and bending moment of the pile shaft are increasing as the vertical load is increasing. There is an apparent amplified action for the lateral response of the pile shaft resulting from the P-Δ effect of the vertical load.

In order to further explore the influence on the lateral response of the pile shaft caused by the vertical load, the relationship between passive load q and the lateral response of the pile shaft under different N/q are obtained and illustrated in Figure 16. It is noteworthy that, in Figure 16, when q is larger than 300 kN, as the value of q is increasing, the differences among maximum horizontal deformations and bending moments under varied N/q reveal an increasing trend, while the amplified action caused by the P-Δ effect of the vertical load on the lateral response of the pile shaft is significant. However, when q is less than 300 kN, such influence can be ignored. Moreover, it is also found that, when N/q is less than 10, as the value of q increases, the maximum horizontal deformation and the bending moment of the pile shaft increases linearly, while a non-linear growth trend can be observed when N/q is greater than 10.

5. Conclusions

In this work, a model for obtaining the semi-analytical solution to the lateral responses of a pile shaft in a high-steep slope including deformation and internal force was proposed based on the transfer-matrix method. Precisely, based on the tri-parameters pile–soil interaction model, the solution for pile lateral responses induced by combined loads in which the influence of vertical friction and initial inclination angle was considered can be formulated by utilizing the Laplace transformation. Conclusions can be drawn as follows:

(1) In the proposed model, by considering the influences of vertical friction, shear deformation and initial inclination angle for the pile shaft, the flexural differential governing equations for the piles in a high-steep slope were firstly established based on the tri-parameters pile–soil interaction model, and the semi-analytical solution for the lateral response of the pile shaft was obtained by applying the transfer-matrix method. The predictions are in close agreement with field observations and the results generated by previous methods.

(2) Through sensitivity analysis of each parameter in the proposed model, the lateral responses of the pile shaft were relatively sensitive to the vertical friction in the passive segment, initial inclination angle and P-Δ effect of vertical load; specifically, it was found that a reduction effect occurred for the horizontal deformation and bending moment of the pile shaft when the vertical friction of the passive segment is increasing. Subsequently, as the value of the vertical friction was increasing, the location of the maximum bending moment was gradually closing toward the shallow stratum, the relative bearing ratio of the load in passive segment of the pile shaft was enlarged, which improved its horizontal bearing characteristics.

(3) The lateral deformation and the internal force of the pile shaft were both increasing linearly due to the growth in the initial inclination angle with a decreasing amplitude. There were significant influences yielded by both the vertical and passive load on the lateral deformation and internal force of the pile shaft: when the ratio of vertical load N and passive load q was less than 10, the lateral deformation and internal force were linearly increasing as the q was largening, while a non-linearly increasing trend was found when the ratio is larger than 10. In addition, greater influence was found during the N was rising up when the q is constant.