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Open Access 21-12-2023 | Original Paper

Elastic–Plastic Analysis of Rigid Passive Piles in Two-Layered Soils

Author: Ivo Bellezza

Published in: Geotechnical and Geological Engineering | Issue 4/2024

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Abstract

The article delves into the elastic-plastic analysis of rigid passive piles in two-layered soils, focusing on their application in slope stability. It discusses the various methods used to evaluate the performance and design of reinforcing piles, highlighting the lack of a widely accepted design procedure. The study introduces a new method that considers the variation of soil movement, pile deformation, and soil properties with depth, providing a more realistic analysis of pile response. The results are expressed in terms of dimensionless parameters, offering a valuable tool for engineers to assess the contribution of piles to slope stability at both ultimate and intermediate states.

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A, B; C, Δ, X, Y

Auxiliary functions of λ, R_U and ρ

Depth of negative soil plasticization in the unstable soil; b_n = b/L₁

Depth of positive soil plasticization in the unstable soil; c_n = c/L₁

Pile diameter

E _P

Young’s modulus of pile

E _S1 (E _S2)

Modulus of subgrade reaction in the unstable (stable) layer

Final depth of negative soil plasticization in the stable soil; f_n = f/L₁

f _Tys f _yT, f _MT

Functions of λ and R_E

Initial depth of positive soil plasticization in the stable soil; g_n = g/L₁

J _P

Moment of inertia of pile

Pile length (= L₁ + L₂); L_n = normalized pile length = L/L₁ = 1 + λ

L ₁ (L ₂)

Pile length in the unstable (stable) layer

m ₁ (m ₂)

Gradient of P_u in the unstable (stable) layer

M(z)

Bending moment at depth z; M_n = M/m₁L

M _max

Maximum bending moment; M_maxn = normalized bending moment (= M_max/(m₁L₁³)

Gradient of E_S1 in the unstable layer

P _e

Soil reaction per unit length in elastic conditions

P _u1 (P _u2)

Ultimate force per unit length in the unstable (stable) layer

R _E

Subgrade modulus ratio at the layer interface = E_S2/(nL₁)

R _U

Strength ratio at the layer interface = P_u20/(m₁L₁)

T(z)

Shear force at depth z; T_n = T/m₁L₁²

T _s

Shear force at the sliding depth z = L₁; T_sn = T_s/(m₁L₁²)

T _sne

T_sn at the elastic threshold

T _snp

T_sn at the plastic threshold

T _snR

T_sn requested to stabilize a slope

y ₀

Pile head deflection; y_0n = normalized pile head deflection = y₀ E_s2/m₁L₁

y _p (z)

Pile deflection at depth z

y _s

Soil movement; y_sn = normalized soil movement = y_s E_s2/m₁L₁

y _s0

Soil movement at z = 0; y_s0n = normalized soil movement at z = 0 = y_s0 E_s2/m₁L₁

y _s0nA

Normalized limiting soil movement relevant to first soil plasticization above the sliding surface

y _s0nB

Normalized limiting soil movement relevant to first soil plasticization below the sliding surface

y _s0nH

Normalized limiting soil movement relevant to first soil plasticization at the pile head

y _s0ne

Normalized soil movement at the elastic threshold

y _s0np

Normalized soil movement at the plastic threshold

Generic depth; z_n = z/L₁

z _m1 (z _m2)

Depth of maximum bending moment above (below) the sliding surface; z_m1n = z_m1/L₁; z_m2n = z_m2/L₁

Embedment ratio = L₂/L₁

Rotation angle of rigid pile; ω_n = normalised rotation angle = tanω E_s2/m₁

m₂ : m₁

1 Introduction

Slope stability is often improved by using passive piles. In recent decades there have been several reports in the literature on successful use of piles to stabilize a slope (e.g. Sommer 1977; Fukuoka 1977; Reese et al. 1992; Poulos 1995; Smethrust and Powrie 2007).

Different methods have been proposed to evaluate the performance and design of reinforcing piles in slopes, as well as to evaluate the safety factor of a reinforced slope (Ito et al. 1979, 1981, 1982; Chow 1996, Hassiotis et al. 1997, Cai and Ugai 2000, 2011; Ausilio et al. 2001; Jeong et al. 2003, Won et al. 2005, Wei and Cheng 2009, Ellis et al. 2010, Yamin and Liang 2010; Kourkoulis et al. 2011, 2012; Ashour and Ardalan 2012; Galli et al. 2017; Di Laora and Fioravante 2018). However a widely accepted design procedure is still lacking (Di Laora et al. 2017). As an example, the effect of stabilizing piles on slope stability is considered somewhere as an additional resistance (e.g. Poulos 1995) and elsewhere as a negative action (e.g.Yamin and Liang 2010). Moreover a clear distinction should be made between piles installed to arrest an active landslide and piles used as a preventive measure in stable slopes. In the former case the pile design can be reasonably based on the assumption that the critical slip surface, on which residual strength is mobilized, does not change after the pile installation. In the latter case the pile response must be evaluated for different locations of the potential slip surface and it is likely that the critical slip surface varies respect to the unreinforced slope (Hassiotis et al. 1997).

Although numerical three-dimensional analyses are in principle the most rigorous approach to analyze the problem, they are computationally intensive and time-consuming. Therefore in the practice the so-called decoupled methods are widely used, in which the behavior of slope and piles is analyzed separately (Viggiani 1981; Poulos 1995). This design approach may be simplified in three main steps: (a) computing the lateral force needed to increase the factor of safety of the slope to the target value using the traditional limit equilibrium methods; (b) evaluating the shear force that each pile can offer as a consequence of soil sliding; (c) selecting a pile configuration able to provide this required force without structural damage. The available approaches in the literature (Viggiani 1981; Di Laora et al. 2017; Bellezza and Caferri 2018) are mainly based on the assumption that soil strength is fully mobilized along the Soil–pile interface and, at varying the depth of sliding respect to pile length, a set of analytical expressions is provided for the shear and moment which have to be included in the computation of the safety factor of the reinforced slope (Lee et al. 1995). These procedures refer to the ultimate state only, giving no indication on pile response at intermediate states prior to the ultimate state and on pile and soil movement required to achieve the ultimate state. Alternatively, some analytical methods assume a soil response fully elastic (e.g. Chen and Poulos 1997; Cai and Ugai 2003, 2011; Guo 2014) whereas the elastic–plastic condition is rarely investigated. To overcome these drawbacks, Poulos (1995) proposed a valuable displacement-based design procedure for a passive pile embedded on a continuum elastic, but even for simplified soil profiles the solution is not given in closed-form and the application to real cases requires the use of a specific computer program.

Nowadays the general trend in engineering practice is to install stabilizing piles with a center to center spacing of three/four pile diameter which is the most effective solution to assure the development of soil arching (Ellis et al. 2010). To increase the structural capacity, the use of large pile diameters with a high reinforcement ratio is recommended (Kourkoulis et al. 2011). Therefore, in certain circumstances, the assumption of rigid deformation for the pile can be reasonable.

Bellezza et al. (2017) presented an example of elastic–plastic analysis of rigid passive piles embedded in a single layer with modulus of subgrade reaction and strength linearly increasing with depth providing design charts for pile displacement and maximum bending moment at varying the shear force at the sliding surface. More recently, Lei et al. (2022) proposed an analysis for flexible piles embedded in cohesive layered soils assuming both strength and modulus of subgrade reaction constant with depth.

In this paper a similar analysis is extended to a more realistic two-layer soil profile with soil strength linearly increasing with depth in both layers. The purpose of this study is to provide an effective tool to analyze the pile response at varying the soil movement, so that the pile contribution in terms of stability can be evaluated not only at the ultimate state but also at intermediate states.

2 Method of Analyis

Figure 1 shows a passive pile of length L and diameter D embedded for a length L₁ a layer subjected to a lateral soil movement y_s and for a length L₂ in a stable layer.

Fig. 1

Basic assumptions of the proposed method: a soil movement; b rigid pile deformation; c variation of P_u with depth and d variation of E_S with depth

Full size image

The rigid displacement of the pile, y_p, at any depth z can be written as

$$y_{p} = y_{0} - \tan \omega \cdot z$$

(1)

where y₀ is the pile head displacement and ω is the rotation angle.

Similarly to previous studies (e.g. Hassiotis et al. 1997; Cai and Ugai 2000; Gou 2014) the elastic soil reaction [FL⁻¹] is calculated by modeling the soil as a series of independent springs:

$$P_{e} = - E_{s} \left( {y_{p} - y_{s} } \right)$$

(2)

where E_s is the modulus of subgrade reaction [FL⁻²].

According to Poulos (1995) a uniform distribution of soil movement is assumed in the unstable layer:

$$y_{s} = \left\{ {\begin{array}{*{20}c} {y_{s0} } & {0 \le z \le L_{1} } \\ 0 & {L_{1} < z \le L} \\ \end{array} } \right.$$

(3)

In the unstable layer the modulus of horizontal subgrade reaction E_S1 and the ultimate lateral soil resistance P_u1 [FL⁻¹] vary linearly with depth:

$$E_{S1} = n \cdot z$$

(4)

$$P_{u1} = m_{1} \cdot z$$

(5)

where n [FL⁻³] and m₁ [FL⁻²] are the gradient of E_s and P_u1, respectively.

In the stable layer the horizontal subgrade reaction is assumed to be constant (E_S2), whereas the ultimate soil strength is assumed to linearly increase with depth (Fig. 1d–e):

$$P_{u2} = P_{u20} + m_{2} \left( {z - L_{1} } \right)$$

(6)

where P_u20 is the ultimate soil resistance at the top of the stable layer and m₂ is the gradient of P_u2.

The assumption of a uniform E_s is generally acceptable for overconsolidated clays (Terzaghi 1955; Viggiani et al. 2012; Zhang & Ahmari 2013).

The appropriate selection of the horizontal modulus of subgrade and limiting lateral soil pressure, although fundamental for the analysis, will not be discussed in this paper. Some suggestions for the choice of P_u values for both isolated pile and pile group can be found in Ito and Matsui 1975; De Beer and Carpentier 1977; Poulos (1995), Georgiadis et al. (2013). A comprehensive discussion on the selection of E_s for isolated piles is given by Zhang (2009) and Zhang and Ahmari (2013).

The method does not consider the formation of plastic hinge along the pile shaft; the absence of plastic hinges can be achieved by a proper design of the pile section and reinforcement based on the maximum moment provided by the analysis.

Finally, the assumption of a rigid deformation of the pile holds for particular combinations of pile flexural stiffness, mechanical properties of both unstable and stable layers and relative embedment lengths (L₁ and L₂). However, for practical purpose, the following condition can be used for a rough, but conservative, check of pile rigidity:

$$L < 2 \cdot \sqrt[4]{{\frac{{E_{P} J_{P} }}{{E_{s2} }}}}$$

(7)

where E_p is the pile elastic modulus of pile and J_P is the moment of inertia of pile cross sectional area.

Numerical analyses performed by the author confirm that when the condition (7) is met the pile behaves as a rigid one.

2.1 Dimensionless Parameters

The results of the present study are expressed in terms of dimensionless parameters.

The pile length L is defined by the embedment ratio λ:

$$\lambda = {{L_{2} } \mathord{\left/ {\vphantom {{L_{2} } {L_{1} }}} \right. \kern-0pt} {L_{1} }}$$

(8a)

$$L_{n} = {L \mathord{\left/ {\vphantom {L {L_{1} }}} \right. \kern-0pt} {L_{1} }} = 1 + \lambda$$

(8b)

The distributions of E_s and P_u with depth are described by R_E, R_U and ρ, defined as:

$$R_{E} = {{E_{S2} } \mathord{\left/ {\vphantom {{E_{S2} } {nL_{1} }}} \right. \kern-0pt} {nL_{1} }}$$

(9a)

$$R_{U} = {{P_{u20} } \mathord{\left/ {\vphantom {{P_{u20} } {m_{1} L_{1} }}} \right. \kern-0pt} {m_{1} L_{1} }}$$

(9b)

$$\rho = {{m_{2} } \mathord{\left/ {\vphantom {{m_{2} } {m_{1} }}} \right. \kern-0pt} {m_{1} }}$$

(9c)

As shown in Fig. 1c-d, the parameters R_U and R_E are the ratio of E_s and P_u at the layer interface, respectively.

The shear force mobilized at the sliding depth (which essentially gives the main contribution of a row of passive piles in slope stability) is expressed by:

$$T_{sn} = \frac{{T_{s} }}{{m_{1} L_{1}^{2} }}$$

(10)

In such a way T_sn = 0.5 when the horizontal soil resistance of the unstable layer is fully mobilized.

Similarly, the maximum bending moment along the pile shaft (useful for structural design of passive piles) is normalized as:

$$M_{\max n} = \frac{{M_{\max } }}{{m_{1} L_{1}^{3} }}$$

(11)

Finally, the soil movement, pile head displacement and pile rotation (useful for serviceability limit state analysis) are conveniently expressed by:

$$y_{s0n} = y_{s0} \frac{{R_{E} n}}{{m_{1} }} = y_{s0} \frac{{E_{s2} }}{{m_{1} L_{1} }}$$

(12)

$$y_{0n} = y_{0} \frac{{R_{E} n}}{{m_{1} }} = y_{0} \frac{{E_{s2} }}{{m_{1} L_{1} }}$$

(13)

$$\omega_{n} = \tan \omega \frac{{R_{E} nL_{1} }}{{m_{1} }} = \tan \omega \frac{{E_{s2} }}{{m_{1} }}$$

(14)

3 Results and Discussion

A passive pile is gradually loaded for increase of the soil movement. Referring to Soil–pile interaction three different conditions can be generally distinguished: (1) initially, for relatively small soil displacement an elastic condition is achieved for the entire length; (2) then, for soil movement exceeding a first threshold value, y_s0ne, an elastic–plastic condition is achieved with the soil at the ultimate state only in limited zones whose extent increases at increasing soil movements; (3) finally, for soil movement exceeding a second threshold value, y_s0np, a plastic condition can be achieved corresponding to the full plasticization of soil above and/or below the sliding depth. For the assumed distributions of soil movement and soil properties (Fig. 1), the extent of the elastic and elastic–plastic zones (i.e. the values of y_s0ne and y_s0np) depends on λ, R_E, R_U and ρ.

In the following paragraphs the above-mentioned cases (elastic, elastic–plastic and plastic) are analyzed separately with the aim to obtain, whenever possible, closed-form expressions useful for design purposes.

3.1 Elastic Case

On the basis of (1)-(4), the normalized elastic soil reaction versus depth can be expressed as:

$$P_{en} = \frac{{P_{e} }}{{m_{1} L_{1} }} = \left\{ {\begin{array}{*{20}l} {{{\left( {\omega_{n} z_{n}^{2} - \Delta y_{0n} \cdot z_{n} } \right)} \mathord{\left/ {\vphantom {{\left( {\omega_{n} z_{n}^{2} - \Delta y_{0n} \cdot z_{n} } \right)} {R_{E} }}} \right. \kern-0pt} {R_{E} }}} \hfill & {0 \le z_{n} \le 1} \hfill \\ {\omega_{n} z_{n} - y_{0n} } \hfill & {1 \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$$

(15)

where $\Delta y_{0n} = \left( {y_{0n} - y_{s0n} } \right)$ and z_n = z/L₁.

Imposing the horizontal force equilibrium and the moment equilibrium about the pile head, a linear system of two variables is obtained:

$$\frac{1}{2}\left( {y_{0n} - y_{s0n} } \right) - \frac{1}{3}\omega_{n} + y_{0n} \left( {L_{n} - 1} \right)R_{E} - \frac{1}{2}\omega_{n} \left( {L_{n}^{2} - 1} \right)R_{E} = 0$$

(16)

$$\frac{1}{3}\left( {y_{0n} - y_{s0n} } \right) - \frac{1}{4}\omega_{n} + \frac{1}{2}y_{0n} \left( {L_{n}^{2} - 1} \right)R_{E} - \frac{1}{3}\omega_{n} \left( {L_{n}^{3} - 1} \right)R_{E} = 0$$

(17)

The analytical solution in dimensionless form is found to be:

$$y_{0n} = \frac{{1 + 12R_{E} \lambda \left( {1 + \lambda } \right)^{2} }}{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}y_{s0n}$$

(18)

$$\omega_{n} = \frac{{6R_{E} \lambda \left( {2 + 3\lambda } \right)}}{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}y_{s0n}$$

(19)

Once y_0n and ω_n have been obtained, the internal forces at any depth can be calculated by the expressions listed in Table 1. An example of the distribution of the soil reaction, shear force and bending moment is shown in Fig. 2.

Table 1

Internal forces for the elastic case

range	Normalised shear force T and bending moment M
\(0 \le z_{n} \le 1\)	\(T_{n} = \frac{T}{{m_{1} L_{1}^{2} }} = - \frac{{\Delta y_{0n} }}{{2R_{E} }}z_{n}^{2} + \frac{{\omega_{n} }}{{3R_{E} }}z_{n}^{3}\) \(\Delta y_{0n} = \left( {y_{0} - y_{s} } \right)\frac{{nR_{E} }}{{m_{1} }}\) \(z_{n} = {z \mathord{\left/ {\vphantom {z {L_{1} }}} \right. \kern-0pt} {L_{1} }}\) \(M_{n} = \frac{M}{{m_{1} L_{1}^{3} }} = - \frac{{\Delta y_{0n} }}{{6R_{E} }}z_{n}^{3} + \frac{{\omega_{n} }}{{12R_{E} }}z_{n}^{4}\) M_max at \(\frac{{z_{m1} }}{{L_{1} }} = \frac{3}{2}\frac{{\Delta y_{0n} }}{{\omega_{n} }}\) provided that 0 < \(\frac{{z_{m1} }}{{L_{1} }}\) < 1
\(1 \le z_{n} \le L_{n}\)	\(T_{n} = - y_{0n} \left( {z_{n} - L_{n} } \right) + \frac{1}{2}\omega_{n} \left( {z_{n}^{2} - L_{n}^{2} } \right)\)\(M_{n} = - \frac{1}{2}y_{0n} \left( {z_{n} - L_{n} } \right)^{2} + \frac{1}{6}\omega_{n} \left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)\)M_max at \(\frac{{z_{m2} }}{{L_{1} }} = \frac{{2y_{0n} }}{{\omega_{n} }} - L_{n}\) provided that 1 < \(\frac{{z_{m2} }}{{L_{1} }}\) < L_n

$0 \le z_{n} \le 1$

$T_{n} = \frac{T}{{m_{1} L_{1}^{2} }} = - \frac{{\Delta y_{0n} }}{{2R_{E} }}z_{n}^{2} + \frac{{\omega_{n} }}{{3R_{E} }}z_{n}^{3}$ $\Delta y_{0n} = \left( {y_{0} - y_{s} } \right)\frac{{nR_{E} }}{{m_{1} }}$ $z_{n} = {z \mathord{\left/ {\vphantom {z {L_{1} }}} \right. \kern-0pt} {L_{1} }}$ $M_{n} = \frac{M}{{m_{1} L_{1}^{3} }} = - \frac{{\Delta y_{0n} }}{{6R_{E} }}z_{n}^{3} + \frac{{\omega_{n} }}{{12R_{E} }}z_{n}^{4}$ M_max at $\frac{{z_{m1} }}{{L_{1} }} = \frac{3}{2}\frac{{\Delta y_{0n} }}{{\omega_{n} }}$ provided that 0 < $\frac{{z_{m1} }}{{L_{1} }}$ < 1

$1 \le z_{n} \le L_{n}$

$T_{n} = - y_{0n} \left( {z_{n} - L_{n} } \right) + \frac{1}{2}\omega_{n} \left( {z_{n}^{2} - L_{n}^{2} } \right)$$M_{n} = - \frac{1}{2}y_{0n} \left( {z_{n} - L_{n} } \right)^{2} + \frac{1}{6}\omega_{n} \left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)$M_max at $\frac{{z_{m2} }}{{L_{1} }} = \frac{{2y_{0n} }}{{\omega_{n} }} - L_{n}$ provided that 1 < $\frac{{z_{m2} }}{{L_{1} }}$ < L_n

Fig. 2

Example of the elastic response of a rigid passive pile subjected to a uniform soil movement: a pile and soil displacement; b soil reaction; c shear force d bending moment

Full size image

In the presence of a uniform soil movement the shear force (Fig. 2c) achieves a relative maximum at the depth of sliding (z_n = 1) and it can be calculated as:

$$T_{sn} = y_{0n} \lambda - \frac{1}{2}\omega_{n} \lambda \left( {2 + \lambda } \right)$$

(20)

After rearranging the terms, taking into account Eqs. (18)-(19), the following linear relationships can be derived:

$$T_{sn} = f_{Tys} \cdot y_{s0n}$$

(21)

$$y_{0n}^{{}} = f_{yT} \cdot T_{sn}^{{}}$$

(22)

where $f_{Tys} = \frac{{\lambda + 3R_{E} \lambda^{4} }}{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}$ and $f_{yT} = \frac{{1 + 12R_{E} \lambda \left( {1 + \lambda } \right)^{2} }}{{\lambda + 3R_{E} \lambda^{4} }}$.

The bending moment can peak both above and below the sliding surface depending on the combinations of the values of λ and R_E. Then, the maximum bending moment is obtained as

$$M_{\max ,n} = \max \left\{ {\left| {M_{1n} } \right|;M_{2n} } \right\} = \max \left\{ {\frac{{9\Delta y_{0n}^{4} }}{{64R_{E} \omega_{n}^{3} }};\frac{{2\omega_{n} }}{3}\left( {L_{n} - \frac{{y_{0n} }}{{\omega_{n} }}} \right)^{3} } \right\}$$

(23)

where M_1n and M_2n are the maximum bending moment above and below the sliding surface, respectively (Fig. 2d).

For values of the embedment ratio of practical interest the maximum moment is always achieved in the stable zone (see Fig. 2d). Then, rearranging (23) taking into account of (18), (19) and (20), a linear correlation between M_maxn and T_sn can be obtained:

$$M_{\max n} = f_{MT} \cdot T_{sn}$$

(24)

where $f_{MT} = \frac{{\left[ {1 + \lambda - \left( {6R_{E} \lambda^{2} } \right)^{ - 1} } \right]^{3} }}{{\left( {1 + 1.5\lambda } \right)^{2} \left[ {3 + \left( {R_{E} \lambda^{3} } \right)^{ - 1} } \right]}}$

3.2 Elastic–Plastic Cases

By combining different values of y_s, λ, R_E, R_U and ρ various elastic–plastic (EP) cases can occur. The simplest cases are those in which soil reaction reaches its ultimate value only in one zone (Fig. 3a, b). For increasing soil movement more complex cases can develop, where two, three or four zones are simultaneously in plastic state (Fig. 3c, d, e).

Fig. 3

Soil reaction in some elastic–plastic cases: a one plastic zone above the sliding surface; b one plastic zone below the sliding surface; c two plastic zones above and below the sliding surface; d three plastic zones above and below the sliding surface and on tip; e generalized case with four plastic zones

Full size image

A generalized EP-case is here comprehensively analyzed assuming a combination of the input parameters y_s, λ, R_E, R_U and ρ for which the soil reaches the ultimate value in four zones, as shown in Fig. 3e.

Horizontal and rotational equilibrium are assured when:

$$\begin{gathered} - \frac{1}{2}b_{n}^{2} - \frac{1}{{2R_{E} }}\left( {y_{0n} - y_{s0n} } \right)\left( {c_{n}^{2} - b_{n}^{2} } \right) + \frac{1}{{3R_{E} }}\omega_{n} \left( {c_{n}^{3} - b_{n}^{3} } \right) + \frac{1}{2}\left( {1 - c_{n}^{2} } \right) - R_{U} \left( {f_{n} - 1} \right) - \frac{1}{2}\rho \left( {f_{n} - 1} \right)^{2} \hfill \\ - y_{0n} \left( {g_{n} - f_{n} } \right) + \frac{1}{2}\omega_{n} \left( {g_{n}^{2} - f_{n}^{2} } \right) + R_{U} \left( {L_{n} - g_{n} } \right) + \frac{1}{2}\rho \left( {L_{n} - 1} \right)^{2} - \frac{1}{2}\rho \left( {g_{n} - 1} \right)^{2} = 0 \hfill \\ \end{gathered}$$

(25)

$$\begin{gathered} - \frac{1}{3}b_{n}^{3} - \frac{1}{{3R_{E} }}\left( {y_{0n} - y_{s0n} } \right)\left( {c_{n}^{3} - b_{n}^{3} } \right) + \frac{1}{{4R_{E} }}\omega_{n} \left( {c_{n}^{4} - b_{n}^{4} } \right) + \frac{1}{3}\left( {1 - c_{n}^{3} } \right) - \frac{1}{2}R_{U} \left( {f_{n}^{2} - 1} \right) - \frac{1}{3}\rho \left( {f_{n}^{3} - 1} \right) \hfill \\ + \frac{1}{2}\rho \left( {f_{n}^{2} - 1} \right) - \frac{1}{2}y_{0n} \left( {g_{n}^{2} - f_{n}^{2} } \right) + \frac{1}{3}\omega_{n} \left( {g_{n}^{3} - f_{n}^{3} } \right) + \frac{1}{2}R_{U} \left( {L_{n}^{2} - g_{n}^{2} } \right) + \frac{1}{3}\rho \left( {L_{n}^{3} - g_{n}^{3} } \right) - \frac{1}{2}\rho \left( {L_{n}^{2} - g_{n}^{2} } \right) = 0 \hfill \\ \end{gathered}$$

(26)

where b_n, c_n, f_n and g_n are the normalized depths that define the extent of the plastic zones (Fig. 3e).

By equalizing elastic soil reaction (P_e) and ultimate resistance per unit length (P_u) the following expressions can be derived:

$$b_{n} = \frac{b}{{L_{1} }} = \frac{{y_{0n} - y_{s0n} - R_{E} }}{{\omega_{n} }}$$

(27)

$$c_{n} = \frac{c}{{L_{1} }} = \frac{{y_{0n} - y_{s0n} + R_{E} }}{{\omega_{n} }}$$

(28)

$$f_{n} = \frac{f}{{L_{1} }} = \frac{{y_{0n} - R_{U} + \rho }}{{\omega_{n} + \rho }}$$

(29)

$$g_{n} = \frac{g}{{L_{1} }} = \frac{{y_{0n} + R_{U} - \rho }}{{\omega_{n} - \rho }}$$

(30)

Equations (25–30) represent a nonlinear system of 6 equations in 6 variables (y_0n, ω_n, b_n, c_n, f_n and g_n). The solution of this system cannot be obtained in closed form, but it can be readily accomplished by simple spreadsheet software (e.g. the tool Solver of Microsoft Excel).

On the basis of y_0n and ω_n, the shear force and bending moment can be computed by the expressions listed in Table 2. The normalized shear force at the sliding depth can be obtained as:

$$T_{sn} = R_{U} \left( {f_{n} + g_{n} - \lambda - 2} \right) - \frac{1}{2}\rho \left[ {\lambda^{2} - \left( {g_{n} - 1} \right)^{2} - \left( {f_{n} - 1} \right)^{2} } \right] + y_{0n} \left( {g_{n} - f_{n} } \right) - \frac{1}{2}\omega_{n} \left( {g_{n}^{2} - f_{n}^{2} } \right)$$

(31)

Table 2

Internal forces for the generalized elastic–plastic case

Range	Normalized shear force T and bending moment M
0 ≤ z ≤ b_n	$T_{n} = \frac{T}{{mL_{1}^{2} }} = - \frac{1}{2}z_{n}^{2}$ $M_{n} = \frac{M}{{mL_{1}^{3} }} = - \frac{1}{6}z_{n}^{3}$
b_n ≤ z_n ≤ c_n	$T_{n} = - \frac{1}{2}b_{n}^{2} - \frac{{\Delta y_{0n} }}{{2R_{E} }}\left( {z_{n}^{2} - b_{n}^{2} } \right) + \frac{{\omega_{n} }}{{3R_{E} }}\left( {z_{n}^{3} - b_{n}^{3} } \right)$ $\Delta y_{0n} = y_{0n} - y_{s0n}$ $M_{n} = - \frac{1}{6}b_{n}^{2} \left( {3z_{n} - 2b_{n} } \right) - \frac{{\Delta y_{0n} }}{{6R_{E} }}\left( {z_{n}^{3} - 3b_{n}^{2} z_{n} + 2b_{n}^{3} } \right) + \frac{{\omega_{n} }}{{12R_{E} }}\left( {z_{n}^{4} - 4b_{n}^{3} z_{n} + 3b_{n}^{4} } \right)$ The depth M_max at. z_m1 is the solution of a 3-order equation: $z_{m1n}^{3} - 1.5\frac{{\Delta y_{0n} }}{{\omega_{n} }}z_{m1n}^{2} + \left\{ {1.5b_{n}^{2} \frac{{\Delta y_{0n} }}{{\omega_{n} }} - b_{n}^{3} - \frac{{1.5R_{E} b_{n}^{2} }}{{\omega_{n} }}} \right\} = 0$ provided that b_n ≤ $\frac{{z_{m1} }}{{L_{1} }}$ ≤ c_n
c_n ≤ z_n ≤ 1	$T_{n} = \frac{1}{2}\left( {z_{n}^{2} - c_{n}^{2} - b_{n}^{2} } \right) - \frac{{\Delta y_{0n} }}{{2R_{E} }}\left( {c_{n}^{2} - b_{n}^{2} } \right) + \frac{{\omega_{n} }}{{3R_{E} }}\left( {c_{n}^{3} - b_{n}^{3} } \right)$ $\begin{gathered} M_{n} = \frac{1}{6}\left\{ {z_{n}^{3} - 3\left( {b_{n}^{2} + c_{n}^{2} } \right)z_{n} + 2\left( {c_{n}^{3} + b_{n}^{3} } \right)} \right\} - \frac{{\Delta y_{0n} }}{{6R_{E} }}\left\{ {3\left( {c_{n}^{2} - b_{n}^{2} } \right)z_{n} - 2\left( {c_{n}^{3} - b_{n}^{3} } \right)} \right\} \hfill \\ + \frac{{\omega_{n} }}{{12R_{E} }}\left\{ {4\left( {c_{n}^{3} - b_{n}^{3} } \right)z_{n} - 3\left( {c_{n}^{4} - b_{n}^{4} } \right)} \right\} \hfill \\ \end{gathered}$ M_max at depth $\frac{{z_{m1} }}{{L_{1} }} = \sqrt {c_{n}^{2} + b_{n}^{2} + \frac{{\Delta y_{0n} }}{{R_{E} }}\left( {c_{n}^{2} - b_{n}^{2} } \right) - \frac{2}{3}\frac{{\omega_{n} }}{{R_{E} }}\left( {c_{n}^{3} - b_{n}^{3} } \right)}$ provided that c_n ≤ $\frac{{z_{m1} }}{{L_{1} }}$ ≤ 1
1 ≤ z_n ≤ f_n	$T_{n} = - R_{U} \left( {z_{n} - f_{n} - g_{n} + L_{n} } \right) + y_{0n} \left( {g_{n} - f_{n} } \right) - \frac{1}{2}\omega_{n} \left( {g_{n}^{2} - f_{n}^{2} } \right) - \frac{1}{2}\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \left( {g_{n} - 1} \right)^{2} - \left( {f_{n} - 1} \right)^{2} + \lambda^{2} } \right]$$\begin{gathered} M_{n} = - \frac{1}{2}R_{U} \left[ {z_{n}^{2} + f_{n}^{2} + g_{n}^{2} - L_{n}^{2} - 2\left( {f_{n} + g_{n} - L_{n} } \right)z_{n} } \right] - \frac{1}{2}y_{0n} \left[ {g_{n}^{2} - f_{n}^{2} - 2\left( {g_{n} - f_{n} } \right)z_{n} } \right] \hfill \\ + \omega_{n} \left[ {\frac{1}{3}\left( {g_{n}^{3} - f_{n}^{3} } \right) - \frac{1}{2}\left( {g_{n}^{2} - f_{n}^{2} } \right)z_{n} } \right] - \frac{1}{2}\rho \left\{ \begin{gathered} \frac{1}{3}\left[ {\left( {z_{n} - 1} \right)^{3} - \left( {f_{n} - 1} \right)^{3} - \left( {g_{n} - 1} \right)^{3} + \lambda^{3} } \right] + \lambda^{2} \left( {z_{n} - L_{n} } \right) \hfill \\ - \left( {g_{n} - 1} \right)^{2} \left( {z_{n} - g_{n} } \right) - \left( {f_{n} - 1} \right)^{2} \left( {z_{n} - f_{n} } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered}$ M_max at $\frac{{z_{m2} }}{{L_{1} }} = \frac{{\rho - R_{U} + \sqrt {\left( {R_{U} - \rho } \right)^{2} - \rho C_{P} } }}{\rho } \le f_{n}$$C_{P} = \left\{ \begin{gathered} \left( {g_{n}^{2} - f_{n}^{2} } \right)\omega_{n} - 2\left( {g_{n} - f_{n} } \right)y_{0n} + 2R_{U} \left( {L_{n} - f_{n} - g_{n} } \right) \hfill \\ - \rho \left[ {\left( {g_{n} - 1} \right)^{2} + \left( {f_{n} - 1} \right)^{2} - \lambda^{2} - 1} \right] \hfill \\ \end{gathered} \right\}$
f _n ≤ z_n ≤ g_n	$T_{n} = R_{U} \left( {g_{n} - L_{n} } \right) - y_{0n} \left( {z_{n} - g_{n} } \right) + \frac{1}{2}\omega_{n} \left( {z_{n}^{2} - g_{n}^{2} } \right) + \frac{1}{2}\rho \left[ {\left( {g_{n} - 1} \right)^{2} - \lambda^{2} } \right]$$\begin{gathered} M_{n} = \frac{1}{2}R_{U} \left( {L_{n} - g_{n} } \right)\left( {L_{n} + g_{n} - 2z_{n} } \right) - \frac{1}{2}y_{0n} \left( {z_{n} - g_{n} } \right)^{2} + \frac{1}{6}\omega_{n} \left( {z_{n}^{3} - 3g_{n}^{2} z_{n} + 2g_{n}^{3} } \right) \hfill \\ + \frac{1}{2}\rho \left\{ {\left( {g_{n} - 1} \right)^{2} \left( {z_{n} - g_{n} } \right) + \lambda^{2} \left( {L_{n} - z_{n} } \right) + \frac{1}{3}\left[ {\left( {g_{n} - 1} \right)^{3} - \lambda^{3} } \right]} \right\} \hfill \\ \end{gathered}$ M_max at $\frac{{z_{m2} }}{{L_{1} }} = \frac{{y_{0n} - \sqrt {y_{0n}^{2} - \omega_{n} C_{E} } }}{{\omega_{n} }} \ge f_{n}$$C_{E} = 2R_{U} \left( {g_{n} - L_{n} } \right) + 2y_{0n} g_{n} - \omega_{n} g_{n}^{2} + \rho \left[ {\left( {g_{n} - 1} \right)^{2} - \lambda^{2} } \right]$
g_n ≤ z_n ≤ L_n	$T_{n} = R_{U} \left( {z_{n} - L_{n} } \right) + 0.5\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \lambda^{2} } \right]$$M_{n} = \frac{1}{2}R_{U} \left( {z_{n} - L_{n} } \right)^{2} + \frac{1}{6}\rho \left[ {\left( {z_{n} - 1} \right)^{3} - \lambda^{3} } \right] - \frac{1}{2}\rho \lambda^{2} \left( {z_{n} - L_{n} } \right)$

It is worth to note that all possible elastic–plastic cases can be viewed as special cases of that described above. For example, the case with two yielded zones above and below L₁ (Fig. 3c) can be analyzed simplifying (25) and (26), as well as the expressions of Table 2, by putting b_n = 0 and g_n = L_n and solving a nonlinear system with 4 variables (y_0n, ω_n, c_n, and f_n).

3.3 Elastic Thresholds

For the assumed distributions of P_u, E_s and y_s the zone of first plasticization is found to generally occur immediately above (Fig. 3a) or below (Fig. 3b); only for very low embedment ratios the first plasticization can occur at the pile head. The relevant limiting soil displacement (y_s0nA, y_s0nB and y_s0nH) can be obtained by imposing c_n = 1 in (28), f_n = 1 in (29) and b_n = 0 in (27), respectively. Taking into account of (18) and (19), the following values are computed:

$$y_{s0n,A} = \frac{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}{{6R_{E} \lambda^{4} + 6\lambda \left( {1 + \lambda } \right)}}$$

(32a)

$$y_{s0n,B} = \frac{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}{{1 + 6R_{E} \lambda^{2} \left( {1 + 2\lambda } \right)}}R_{U}$$

(32b)

$$y_{s0n,H} = \frac{{1 + 6R_{E}^{2} \lambda^{4} + 6R_{E} \lambda \left( {1 + 2\lambda + 2\lambda^{2} } \right)}}{{6\lambda \left( {1 + 2\lambda } \right) - 6R_{E} \lambda^{4} }}$$

(32c)

The elastic threshold is taken as the minimum (positive) among the above values.

$$y_{s0ne} = \min \left\{ {y_{s0nA} ;y_{s0nB} ;y_{s0nH} } \right\}$$

(33)

The pile head deflection, shear force at the sliding surface T_sne and the maximum bending moment M_maxne relevant to the elastic threshold can be obtained by substituting y_s0ne into (18), (20) and (24), respectively.

In Fig. 4 the values of the normalized shear force at the elastic threshold T_sne are plotted against the embedment ratio for various combinations of R_E and R_U. It is evident that T_sne monotonically increases with λ regardless of the value of R_E and R_U. For an assigned value of R_U a limiting value of λ (= λ^*) exists beyond which T_sne depends only on R_E because the elastic threshold is governed by (32a) that does not include R_U. The value of λ^* decreases for increasing R_U. For λ > λ^* T_sne increases with R_E (solid lines in Fig. 6); as an example, for R_E = 5 and λ = 1.6, T_sne is high as 0.45, i.e. the pile response is fully elastic until it reaches 90% of its maximum response. On the other hand for λ < λ* the elastic threshold is governed by (32b) and different curves of T_sne versus λ can be drawn at varying R_U (dashed lines in Fig. 4).

Fig. 4

Influence of λ, R_E and R_U on the shear force at the sliding surface at the elastic threshold

Full size image

3.4 Plastic Cases

It is well recognized that, for a free head pile, all elastic–plastic cases converge in one of three failure mechanisms indicates as: short-pile mode or mode A, intermediate mode or mode B and flow mode or mode C (e.g. Viggiani 1981; Poulos 1995; Kanagasabai et al. 2011; Di Laora et al. 2017; Bellezza and Caferri 2018).

In mode A the slide is relatively deep (Fig. 5a) and there is full mobilization of strength in the stable soil (i.e. f_n = g_n = L_n in eqs. 22 and 23), so that T_snp = R_U λ +0.5 ρλ². Mode A is of little practical interest in design and therefore it is marginally investigated in this paper.

Fig. 5

Soil reaction distribution for plastic cases: a mode A; b mode B; c mode C1; d mode C2; e mode C3

Full size image

In mode B the soil strength is fully mobilized both in the stable and unstable soil (Fig. 5b). This case can be analytically treated as a special case of the generalized EP case putting b_n = c_n and g_n = f_n in Eq. (25) and (26). The computation of the maximum shear force (T_snp = 0.5—c_n²) is not straightforward, as the values of c_n and f_n must be numerically obtained by imposing horizontal and rotational equilibrium. The numerical solution has a practical meaning only when 0 < b_n < 1 and 1 < f_n < L_n and these conditions are met only for particular combinations of λ, R_U and ρ as detailed in Table 3.

Table 3

Analytical equations for failure mode B

Conditions of existence
$\Delta \le 0$ where $\Delta = B^{2} - AC$ $A = 4R_{U}^{2} + 2\rho X$ $B = R_{U} X - \rho Y$ $C = X^{2} + 2R_{U} Y$ $X = 1 + 2R_{U} \lambda + \rho \lambda^{2}$$Y = 1 - 3R_{U} \lambda^{2} - 2\rho \lambda^{3}$
Equilibrium equations
$- \frac{1}{2}c_{n}^{2} + \frac{1}{2}\left( {1 - c_{n}^{2} } \right) - R_{U} \left( {f_{n} - 1} \right) - \frac{1}{2}\rho \left( {f_{n} - 1} \right)^{2} + R_{U} \left( {L_{n} - f_{n} } \right) + \frac{1}{2}\rho \left( {L_{n} - 1} \right)^{2} - \frac{1}{2}\rho \left( {f_{n} - 1} \right)^{2} = 0$$\begin{gathered} - \frac{1}{3}c_{n}^{3} + \frac{1}{3}\left( {1 - c_{n}^{3} } \right) - \frac{1}{2}R_{U} \left( {f_{n}^{2} - 1} \right) - \frac{1}{3}\rho \left( {f_{n}^{3} - 1} \right) + \frac{1}{2}\rho \left( {f_{n}^{2} - 1} \right) + \frac{1}{2}R_{U} \left( {L_{n}^{2} - f_{n}^{2} } \right) + \frac{1}{3}\rho \left( {L_{n}^{3} - f_{n}^{3} } \right) \hfill \\ - \frac{1}{2}\rho \left( {L_{n}^{2} - f_{n}^{2} } \right) = 0 \hfill \\ \end{gathered}$
Internal forces
$T_{n} = \left\{ {\begin{array}{{20}l} { - 0.5z_{n}^{2} } \hfill & {0 \le z_{n} \le c_{n} } \hfill \\ {0.5z_{n}^{2} - c_{n}^{2} } \hfill & {c_{n} \le z_{n} \le 1} \hfill \\ { - R_{U} \left( {z_{n} - 2f_{n} + L_{n} } \right) - 0.5\rho \left[ {\left( {z_{n} - 1} \right)^{2} - 2\left( {f_{n} - 1} \right)^{2} + \lambda^{2} } \right]} \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ {R_{U} \left( {z_{n} - L_{n} } \right) + 0.5\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \lambda^{2} } \right]} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$ \(M_{n} = \left\{ {\begin{array}{{20}l} {{{ - z_{n}^{3} } \mathord{\left/ {\vphantom {{ - z_{n}^{3} } 6}} \right. \kern-0pt} 6}} \hfill & {0 \le z_{n} \le c_{n} } \hfill \\ {{{\left( {z_{n}^{3} - 6c_{n}^{2} z_{n} + 4c_{n}^{3} } \right)} \mathord{\left/ {\vphantom {{\left( {z_{n}^{3} - 6c_{n}^{2} z_{n} + 4c_{n}^{3} } \right)} 6}} \right. \kern-0pt} 6}} \hfill & {c_{n} \le z_{n} \le 1} \hfill \\ {\left\{ \begin{gathered} - \frac{1}{2}R_{U} \left[ {z_{n}^{2} + 2f_{n}^{2} - L_{n}^{2} - 2\left( {2f_{n} - L_{n} } \right)z_{n} } \right] - \frac{1}{6}\rho \left[ {\left( {z_{n} - 1} \right)^{3} - 2\left( {f_{n} - 1} \right)^{3} + \lambda^{3} } \right] \hfill \\ - \frac{1}{2}\rho \left[ {\lambda^{2} \left( {z_{n} - L_{n} } \right) + 2\left( {f_{n} - 1} \right)^{2} \left( {z_{n} - f_{n} } \right)} \right] \hfill \\ \end{gathered} \right\}} \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ {\frac{1}{2}R_{U} \left( {z_{n} - L_{n} } \right)^{2} - \frac{1}{2}\rho \lambda^{2} \left( {z_{n} - L_{n} } \right) + \frac{1}{6}\rho \left[ {\left( {z_{n} - 1} \right)^{3} - \lambda^{3} } \right]} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.\) M_max at depth $\frac{{z_{m2} }}{{L_{1} }} = \frac{{\rho - R_{U} + \sqrt {\left( {R_{U} - \rho } \right)^{2} + \rho C_{P} } }}{\rho }$$C_{P} = \left[ \begin{gathered} 2R_{U} \left( {2f_{n} - L_{n} } \right) - \rho \hfill \\ + 2\rho \left( {f_{n} - 1} \right)^{2} - \rho \lambda^{2} \hfill \\ \end{gathered} \right]$

Finally, in mode C the slide depth is relatively shallow and there is a full strength mobilization of the soil in the unstable layer, so that T_snp = 0.5 (Fig. 5c, d, e). The equilibrium equations are obtained by putting b_n = c_n = 0 in (22) and (23).

For an assigned combination of λ, R_U and ρ the normalized shear force at the sliding surface, T_sn, is the minimum value among those relevant to mode A, mode B and mode C.

$$T_{snp} = \frac{{T_{s} }}{{m_{1} L_{1}^{2} }} = \min \left\{ {R_{U} \lambda + 0.5\rho \lambda^{2} ;0.5 - c_{n}^{2} ;0.5} \right\}$$

(34)

Figure 6 shows the trend of the normalized shear T_snp as a function of the embedment ratio λ for different values of R_U and ρ. For low values of λ mode A develops and T_snp increases linearly with λ; then, for λ greater than a first threshold value (λ_AB), mode B starts to govern the problem and the increase of T_snp is no more linear; finally, a second threshold value of λ (λ_BC) exists beyond which T_snp becomes independent of λ and R_U (mode C). The values of λ_AB and λ_BC are found to decrease with increasing R_U and ρ. The effect of ρ is appreciable only to low values of R_U.

Fig. 6

Influence of λ, R_U and ρ on the normalized shear force at the sliding depth in plastic conditions

Full size image

Bellezza (2020) showed that within mode C three distinct sub-cases can be identified (C1, C2 and C3) which differ for the distribution of soil reaction in the stable layer. In mode C1 the soil is in the plastic state both below the sliding surface and near the tip (Fig. 6c); in mode C2 there is only a plastic zone below the sliding surface (Fig. 6d), whereas in mode C3 the soil remains elastic in the stable zone (Fig. 6e).

For each case (C1, C2 or C3) the governing equations can be algebraically manipulated to obtain closed-form expressions for normalized pile head displacement (y_0n), rotation (ω_n), limiting soil movement (y_s0np) and maximum bending moment (M_maxn), as well as the extent of the eventual plastic zone below the sliding surface (f_n and g_n). Tables 4–6 contain the governing expressions for mode C1, C2 and C3, respectively.

Table 4

Analytical equations and closed-form solutions for failure mode C1

Conditions of Existence
$0 \le \Delta \le \left( {A\lambda - B} \right)^{2}$ where $\Delta = B^{2} - AC$ $A = 4R_{U}^{2} + 2\rho X$ $B = R_{U} X - \rho Y$ $C = X^{2} + 2R_{U} Y$ $X = 1 + 2R_{U} \lambda + \rho \lambda^{2}$$Y = 1 - 3R_{U} \lambda^{2} - 2\rho \lambda^{3}$
Governing Equations
$\frac{1}{2} - R_{U} \left( {f_{n} - 1} \right) - \frac{1}{2}\rho \left( {f_{n} - 1} \right)^{2} - \left( {g_{n} - f_{n} } \right)y_{0n} + \frac{1}{2}\left( {g_{n}^{2} - f_{n}^{2} } \right)\omega_{n} + R_{U} \left( {L_{n} - g_{n} } \right) + \frac{1}{2}\rho \left( {L_{n} - 1} \right)^{2} - \frac{1}{2}\rho \left( {g_{n} - 1} \right)^{2} = 0$$\begin{gathered} \frac{1}{3} - \frac{1}{2}R_{U} \left( {f_{n}^{2} - 1} \right) + \frac{1}{2}\rho \left( {f_{n}^{2} - 1} \right) - \frac{1}{3}\rho \left( {f_{n}^{3} - 1} \right) - \frac{1}{2}\left( {g_{n}^{2} - f_{n}^{2} } \right)y_{0n} + \frac{1}{3}\left( {g_{n}^{3} - f_{n}^{3} } \right)\omega_{n} \hfill \\ + \frac{1}{2}R_{U} \left( {L_{n}^{2} - g_{n}^{2} } \right) - \frac{1}{2}\rho \left( {L_{n}^{2} - g_{n}^{2} } \right) + \frac{1}{3}\rho \left( {L_{n}^{3} - g_{n}^{3} } \right) = 0 \hfill \\ \end{gathered}$ $f_{n} = \frac{f}{{L_{1} }} = \frac{{y_{0n} - R_{U} + \rho }}{{\omega_{n} + \rho }}$$g_{n} = \frac{g}{{L_{1} }} = \frac{{y_{0n} + R_{U} - \rho }}{{\omega_{n} - \rho }}$
Solutions
$y_{0n} = \frac{{R_{U} \left( {A + B} \right) + \rho \left( {B + C} \right)}}{\sqrt \Delta }$$\omega_{n} = \frac{{R_{U} A + \rho B}}{\sqrt \Delta }$ $f_{n} = 1 + \frac{B - \sqrt \Delta }{A}$ $g_{n} = 1 + \frac{B + \sqrt \Delta }{A}$ $y_{s0np} = y_{0n} + R_{E}$
Internal forces
$T_{n} = \left\{ {\begin{array}{{20}l} {0.5z_{n}^{2} } \hfill & {0 \le z_{n} \le 1} \hfill \\ {0.5 - R_{U} \left( {z_{n} - 1} \right) - 0.5\rho \left( {z_{n} - 1} \right)^{2} } \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ { - y_{0n} \left( {z_{n} - g_{n} } \right) + 0.5\omega_{n} \left( {z_{n}^{2} - g_{n}^{2} } \right) + R_{U} \left( {g_{n} - L_{n} } \right) + 0.5\rho \left[ {\left( {g_{n} - 1} \right)^{2} - \lambda^{2} } \right]} \hfill & {f_{n} \le z_{n} \le g_{n} } \hfill \\ {R_{U} \left( {z_{n} - L_{n} } \right) + 0.5\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \lambda^{2} } \right]} \hfill & {g_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$\(M_{n} = \left\{ {\begin{array}{{20}l} {{{z_{n}^{3} } \mathord{\left/ {\vphantom {{z_{n}^{3} } 6}} \right. \kern-0pt} 6}} \hfill & {0 \le z_{n} \le 1} \hfill \\ {0.5\left( {z_{n} - 1} \right) - 0.5R_{U} \left( {z_{n} - 1} \right)^{2} - {{\rho \left( {z_{n} - 1} \right)^{3} } \mathord{\left/ {\vphantom {{\rho \left( {z_{n} - 1} \right)^{3} } 6}} \right. \kern-0pt} 6} + {1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6}} \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ {\left\{ \begin{gathered} - \frac{{y_{0n} }}{2}\left( {z_{n} - g_{n} } \right)^{2} + \frac{{\omega_{n} }}{6}\left( {z_{n}^{3} - 3g_{n}^{2} z_{n} + 2g_{n}^{3} } \right) + \frac{{R_{U} }}{2}\left( {g_{n} - L_{n} } \right)\left( {2z_{n} - g_{n} - L_{n} } \right) \hfill \\ + \frac{\rho }{6}\left[ {\left( {g_{n} - 1} \right)^{3} - \lambda^{3} } \right] + \frac{\rho }{2}\left[ {\left( {g_{n} - 1} \right)^{2} \left( {z_{n} - g_{n} } \right) - \lambda^{2} \left( {z_{n} - L_{n} } \right)} \right] \hfill \\ \end{gathered} \right\}} \hfill & {f_{n} \le z_{n} \le g_{n} } \hfill \\ {0.5R_{U} \left( {z_{n} - L_{n} } \right)^{2} + \frac{\rho }{6}\left[ {\left( {z_{n} - 1} \right)^{3} - \lambda^{3} } \right] - \frac{\rho }{2}\lambda^{2} \left( {z_{n} - L_{n} } \right)} \hfill & {g_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.\) $M_{\max n} = \frac{1}{6} + \frac{{2\left( {R_{U}^{2} + \rho } \right)^{1.5} - 3\rho R_{U} - 2R_{U}^{3} }}{{6\rho^{2} }}$ at depth $\frac{{z_{m2} }}{{L_{1} }} = 1 + \frac{{\sqrt {R_{U}^{2} + \rho } - R_{U} }}{\rho } \le f_{n}$

3.5 Thresholds Values of the Embedment Ratio

For the investigated soil profile, the transition value between mode A and mode B cannot be derived in closed form, but the value of λ_AB for ρ = 0 can be obtained by interpolating the numerical results as:

$$\lambda_{AB} \cong 0.211 \cdot R_{U}^{ - 0.918}$$

(35)

For ρ > 0 the values of λ_AB slightly decrease.

Also the ranges of occurrence of case C1 and C2 for ρ > 0 can be obtained only numerically on the basis of the expressions listed at the top of Tables 4–5. Only for ρ = 0 it is possible to obtain a set of closed-form expressions of the threshold values of λ for a given value of R_U (Bellezza 2020):

$$\lambda_{C1} = \lambda_{BC} = \frac{{3 + \sqrt {18 + 24R_{U} } }}{{6R_{U} }}$$

(36a)

$$\lambda_{C2} = \frac{{1 + \sqrt {3 + 4R_{U} } }}{{2R_{U} }}$$

(36b)

$$\lambda_{C3} = \frac{{1 + \sqrt {1 + R_{U} } }}{{R_{U} }}$$

(36c)

Table 5

Analytical equations and closed-form solutions for failure mode C2

Conditions of existence
$R_{U} \lambda^{2} - 2\lambda - 1 \le 0$ and $A\lambda^{2} - 2B\lambda + C \le 0$ where $A = 4R_{U}^{2} + 2\rho X$ $B = R_{U} X - \rho Y$ $C = X^{2} + 2R_{U} Y$ $X = 1 + 2R_{U} \lambda + \rho \lambda^{2}$$Y = 1 - 3R_{U} \lambda^{2} - 2\rho \lambda^{3}$
Equations
$\frac{1}{2} - R_{U} \left( {f_{n} - 1} \right) - \frac{1}{2}\rho \left( {f_{n} - 1} \right)^{2} - \left( {L_{n} - f_{n} } \right)y_{0n} + \frac{1}{2}\left( {L_{n}^{2} - f_{n}^{2} } \right)\omega_{n} = 0$$\frac{1}{3} - \frac{1}{2}\left( {R_{U} - \rho } \right)\left( {f_{n}^{2} - 1} \right) - \frac{1}{3}\rho \left( {f_{n}^{3} - 1} \right) - \frac{1}{2}\left( {L_{n}^{2} - f_{n}^{2} } \right)y_{0n} + \frac{1}{3}\left( {L_{n}^{3} - f_{n}^{3} } \right)\omega_{n} = 0$ $f_{n} = \frac{f}{{L_{1} }} = \frac{{y_{0n} - R_{U} + \rho }}{{\omega_{n} + \rho }}$
Solutions
$y_{0n} = \frac{{\left[ {2\left( {R_{U} + 1} \right)\lambda - \left( {R_{U} - \rho } \right)\lambda^{2} } \right]\left( {\rho \lambda^{2} + 2R_{U} \lambda - 1} \right)^{2} }}{{\left( {\rho \lambda^{3} + 3R_{U} \lambda^{2} - 3\lambda - 1} \right)^{2} }} + R_{U} - \rho$ $y_{s0np} = y_{0n} + R_{E}$ $\omega_{n} = \frac{{\left( {\rho \lambda^{2} + 2R_{U} \lambda - 1} \right)^{3} }}{{\left( {\rho \lambda^{3} + 3R_{U} \lambda^{2} - 3\lambda - 1} \right)^{2} }} - \rho$$f_{n} = \frac{{2\left( {R_{U} + 1} \right)\lambda - \left( {R_{U} - \rho } \right)\lambda^{2} }}{{\rho \lambda^{2} + 2R_{U} \lambda - 1}}$
Internal forces
$T_{n} = \left\{ {\begin{array}{{20}l} {0.5z_{n}^{2} } \hfill & {0 \le z_{n} \le 1} \hfill \\ {0.5 - R_{U} \left( {z_{n} - 1} \right) - 0.5\rho \left( {z_{n} - 1} \right)^{2} } \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ { - y_{0n} \left( {z_{n} - L_{n} } \right) + 0.5\omega_{n} \left( {z_{n}^{2} - L_{n}^{2} } \right)} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$$M_{n} = \left\{ {\begin{array}{{20}l} {{{z_{n}^{3} } \mathord{\left/ {\vphantom {{z_{n}^{3} } 6}} \right. \kern-0pt} 6}} \hfill & {0 \le z_{n} \le 1} \hfill \\ {0.5\left( {z_{n} - 1} \right) - 0.5R_{U} \left( {z_{n} - 1} \right)^{2} - \frac{1}{6}\rho \left( {z_{n} - 1} \right)^{3} + {1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6}} \hfill & {1 \le z_{n} \le f_{n} } \hfill \\ { - 0.5y_{0n} \left( {z_{n} - L_{n} } \right)^{2} + \omega_{n} {{\left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)} \mathord{\left/ {\vphantom {{\left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)} 6}} \right. \kern-0pt} 6}} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$ $M_{\max n} = \frac{1}{6} + \frac{{2\left( {R_{U}^{2} + \rho } \right)^{1.5} - 3\rho R_{U} - 2R_{U}^{3} }}{{6\rho^{2} }}$ at depth $z_{mn} = \frac{{z_{m2} }}{{L_{1} }} = 1 + \frac{{\sqrt {R_{U}^{2} + \rho } - R_{U} }}{\rho } \le f_{n}$ Otherwise $M_{\max n} = \frac{1}{6}\left( {z_{mn} - L_{n} } \right)^{2} \left[ {\omega_{n} \left( {z_{mn} + 2L_{n} } \right) - 3y_{0n} } \right]$ at depth $z_{mn} = \left( {\frac{{2y_{0n} }}{{\omega_{n} }} - L_{n} } \right) \ge f_{n}$

The last threshold value (λ_C3) is found to be independent of ρ, so that (36c) is valid also for ρ > 0.

Figure 7 shows the threshold values of λ plotted against R_U for two different values of ρ. It can be observed that λ_C1 significantly decreases with R_U and consequently the range of existence of mode C greatly increases. As an example, for R_U = 2.5 and ρ = 0, λ_AB = 0.091, λ_C1 = λ_BC ≈ 0.789, λ_C2 ≈ 0.921 and λ_C3 ≈ 1.148. As expected, at increasing ρ, λ_AB, λ_C1 and λ_C2 decrease: for R_U = 2.5 and ρ = 1 λ_AB ≈0.090, λ_C1 ≈ 0.732 and λ_C2 ≈ 0.822, whereas λ_C3 remains unchanged, as the first plasticization occurs immediately below the sliding surface.

Fig. 7

Ranges of existence of different soil failure modes

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In Fig. 8 the values of the normalized pile head displacement y_0n are plotted as a function of the embedment ratio λ for different values of the strength ratio R_U and ρ = 0. For combinations of λ and R_U which implies the development of mode B, y_0n does not have a finite value and therefore all curves have a vertical asymptote in correspondence of λ_C1. Then, for λ > λ_BC, y_0n starts to decrease with λ. The rate of decrease of y_0n versus λ is initially very high and decreases at increasing λ according to the expression of Table 4; in the range of occurrence of mode C1 and C2, y_0n depends on λ, R_U and ρ (Tables 4–5). Finally, for λ > λ_C3 the normalized pile head displacement becomes independent of the plastic parameters R_U and ρ (see Table 6).

Fig. 8

Effect of λ and R_U on the normalized pile head deflection for failure mode C

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Table 6

Analytical equations and closed-form solutions for failure mode C3

Condition of existence
$\lambda^{2} R_{U} - 2\lambda - 1 > 0$
Equilibrium equations
$\frac{1}{2} - \left( {L_{n} - 1} \right)y_{0n} + \frac{1}{2}\left( {L_{n}^{2} - 1} \right)\omega_{n} = 0$$\frac{1}{3} - \frac{1}{2}\left( {L_{n}^{2} - 1} \right)y_{0n} + \frac{1}{3}\left( {L_{n}^{3} - 1} \right)\omega_{n} = 0$
Solutions
$y_{0n} = \frac{{2\left( {1 + \lambda } \right)^{2} }}{{\lambda^{3} }}$$\omega_{n} = \frac{2 + 3\lambda }{{\lambda^{3} }}$$y_{s0n} = y_{0n} + R_{E}$
Internal forces
$T_{n} = \left\{ {\begin{array}{{20}l} {0.5z_{n}^{2} } \hfill & {0 \le z_{n} \le 1} \hfill \\ { - y_{0n} \left( {z_{n} - L_{n} } \right) + 0.5\omega_{n} \left( {z_{n}^{2} - L_{n}^{2} } \right)} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$$M_{n} = \left\{ {\begin{array}{{20}l} {{{z_{n}^{3} } \mathord{\left/ {\vphantom {{z_{n}^{3} } 6}} \right. \kern-0pt} 6}} \hfill & {0 \le z_{n} \le 1} \hfill \\ { - 0.5y_{0n} \left( {z_{n} - L_{n} } \right)^{2} + \omega_{n} {{\left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)} \mathord{\left/ {\vphantom {{\left( {z_{n}^{3} - 3L_{n}^{2} z_{n} + 2L_{n}^{3} } \right)} 6}} \right. \kern-0pt} 6}} \hfill & {f_{n} \le z_{n} \le L_{n} } \hfill \\ \end{array} } \right.$$M_{\max n} = \frac{{2\left( {1 + \lambda } \right)^{3} }}{{3\left( {2 + 3\lambda } \right)^{2} }}$ at depth $\frac{{z_{m2} }}{{L_{1} }} = 1 + \frac{{\lambda^{2} }}{2 + 3\lambda }$

For a uniform distribution of y_s, the limiting soil movement required to reach the plastic case C can be obtained considering c_n = 0 in (25), i.e.

$$y_{s0np} = y_{0n} + R_{E}$$

(37)

Obviously, the value of y_0n in (37) is that pertaining to the case of occurrence (C1, C2 or C3).

Figure 9 shows the values of the normalized maximum bending moment M_maxn as a function of the embedment ratio λ for different value of R_U. In the investigated range of λ, for a given value of R_U, M_maxn first significantly increases with λ for mode B (dashed curves in Fig. 9); then, there is a small range of in which M_maxn is constant (mode C1) and finally M_maxn starts again to increase with λ at the occurrence of mode C2 and C3. For mode C3 the law of variation of M_maxn versus λ is the same, regardless of the value of R_U (see Table 6 for details).

Fig. 9

Effect of λ and R_U on the normalized maximum bending moment

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4 Mobilization Curves

On the basis of the analysis developed in the previous sections, the shear force at the sliding depth T_sn, the maximum bending moment, M_maxn, and the pile head displacement, y_0n, can be calculated for any value of the soil movement, obtaining the so-called mobilization curves generally subdivided in elastic, elastic–plastic and plastic part. Figure 10 shows typical examples of mobilization curves relevant to three different combinations of λ, R_U and ρ which implies the final development of mode A, mode B and mode C, respectively. Note that for mode A (Fig. 10a) the plastic threshold of soil movement, y_s0np, exists; it is possible to demonstrate that for y_s0n > y_s0np, T_sn and M_maxn remain constant, but y_0n continues to increase with y_s0n, maintaining the same difference between y_0n and y_s0n. For mode B the plastic threshold does not exist; T_sn and M_maxn tend asymptotically to the plastic values of Table 3, whereas y_0n continue to increase monotonically with y_s0n (Fig. 10b). The final part of the curves can be obtained by numerically solving the generalized EP case (Fig. 5e). For mode C the pile head displacement and maximum bending moment remain constant after the plastic threshold and the pile response in terms of shear force at the sliding surface reaches its maximum (Fig. 10c). In such a case closed-form expressions are available also to evaluate both y_0n and M_maxn at the plastic threshold (Tables 4–6).

Fig. 10

Effect of the soil movement on the normalized shear force at the sliding depth (T_sn), maximum bending moment (M_maxn) and pile head deflection (y_0n) for three different combinations of λ, R_E,R_U and ρ: a λ = 0.12, R_U = R_E = 1.5, ρ = 0; b λ = 0.7 R_U = R_E = 2, ρ = 0; c λ = 1.2 R_U = R_E = 2 ρ = 0

Full size image

In design procedure the passive piles are requested to provide a stabilizing force needed to increase the factor of safety to the target value (Viggiani 1981; Poulos 1995). When the requested force for a single pile is less than the maximum one (i.e. T_snR < 0.5), pile response can fall in the elastic or in the elastic–plastic range and pile head deflection and the maximum bending moment are obviously less than those calculated in plastic conditions by the expressions obtained for mode C. For a fully elastic pile response (i.e. T_snR < T_sne) the closed-form expressions derived in this paper can be used for ready and accurate evaluations of the y_0n and M_maxn values. Conversely, in the elastic–plastic range (i.e. T_sne < T_snR < 0.5), closed-form expressions are not available and y_0n and M_maxn can be determined on the basis of the mobilization curves similar to those of Fig. 10. Considering that in design process more attention is focused on pile displacement and internal forces instead of soil movement, the mobilization curves of Fig. 11 can be conveniently drawn only in terms of y_0n, T_sn and M_maxn.

Fig. 11

Relationship between normalized shear force and pile head deflection for different combinations of λ, R_U R_E and ρ

Full size image

Figure 11 shows the values of T_sn as a function of y_0n for different combinations of λ and R_U with R_E = R_U and ρ = 0.

It is evident that the mobilization curve is mainly influenced by the embedment ratio λ and slightly by R_U. The effect of R_U is appreciable only for low embedment ratios and when T_sn approaches its maximum value. For λ = 1 the curve for R_U = 2 is distinct from other ones; the difference is due to the different failure mode, i.e. for R_U = 2 mode C1 occurs, whereas for R_U > 2.5 mode C3 develops, at which a smaller pile head displacement occurs (see Table 4 and Table 6 for details). A similar trend is observed for λ = 1.2, but in this case the difference in the final values of y_0n is smaller because for R_U = 2 mode C2 develops (see Fig. 8). For higher embedment ratios (λ = 1.5 or λ = 2) mode C3 is always activated and all curves converge to the same final value of $y_{0n}$ for all the investigated values of R_U. For higher embedment ratios the mobilization curves are practically superimposed, although the elastic thresholds are slightly different. On the other hand, for increasing embedment ratios it is always more difficult to satisfy the condition of pile rigidity.

Figure 12 shows the values of M_maxn as a function of T_sn for different combinations of λ and R_U with R_E = R_U and ρ = 0.

Fig. 12

Relationship between normalized shear force and maximum bending moment for different combinations of λ, R_U, R_E and ρ

Full size image

The trend of the curves is similar: in the elastic range M_maxn linearly increases with T_sn (Eq. 24), whereas beyond the elastic threshold M_maxn increases nonlinearly, at increasing rates, with increasing T_sn. The slope of the curves in the elastic range, given by (24), significantly increases with the embedment ratio, whereas the effect of R_E is appreciable only for low values of λ. In the elastic–plastic range a crossover is evident for λ = 1 because the maximum bending moment for R_E = R_U = 2 is higher than that for R_U ≥ 3 owing the different plastic mechanism (C1 and C3, respectively). On the contrary, for λ = 1.5 and λ = 2 all curves converge to the same final value of M_maxn relevant to case C3.

To obtain more accurate numerical values, Table 7 and Table 8 list the values of the normalized pile head deflection y_0n and maximum bending moment M_maxn calculated for some intermediate values of the normalized shear force (T_sn < 0.5) for different combinations of λ, R_E, R_U and ρ. As expected, for increasing values of the embedment ratio λ, y_0n decreases whereas M_maxn increases. At increasing λ the effect of R_E and R_U tends to be negligible.

Table 7

Values of the normalized pile head displacement y_0n × 10² for intermediate values of T_sn

λ	R_E = R_U = 2				R_E = R_U = 3				R_E = R_U = 4				R_E = R_U = 5
	T_sn				T_sn				T_sn				T_sn
	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45
	ρ = 0
0.7	758	1116	-⁽¹⁾	-	791	968	1207	1666	825	991	1189	1419	854	1011	1200	1422
0.8	599	768	1139	-	632	757	903	1099	658⁽²⁾	775	914	1075	676	790	922	1078
0.9	493	596	764	1146	520	612	720	845	537	627	729	848	547	638	735	851
1.0	416	493	589	754	435	508	591	687	445	520	598	689	452	527	604	692
1.1	356	418	489	583	369	431	496	572	376	438	502	574	380	443	507	576
1.2	309	361	418	484	317	370	424	486	322	375	429	488	325	379	433	490
1.3	270	315	363	418	276	322	368	420	279	326	373	421	281	328	375	423
1.4	239	278	319	366	243	283	324	367	245	286	328	369	246	288	329	370
1.5	213	248	284	324	216	252	288	325	217	254	290	326	218	255	291	328
1.6	191	223	255	289	193	226	258	291	194	227	260	292	195	228	260	293
1.7	173	202	230	261	175	204	233	262	175	205	234	263	176	205	235	264
1.8	157	184	210	237	159	185	211	238	159	186	213	239	160	186	213	240
1.9	144	168	192	217	145	169	193	218	146	170	195	219	146	170	194	219
2.0	133	155	177	199	133	156	178	200	134	156	179	201	134	156	179	201
	ρ = 1
0.7	756	1071	1893	-	791	968	1202	1616	825	991	1189	1419	854	1011	1200	1422
0.8	599	760	1057	1763	632	757	903	1095	658	775	914	1075	676	790	922	1078
0.9	493	595	749	1021	520	612	720	845	537	627	729	848	547	638	735	851
1.0	416	493	588	732	435	508	591	687	445	520	598	689	452	527	604	692
1.1	356	418	489	579	369	431	496	572	376	438	502	574	380	443	507	576
1.2	309	361	418	484	317	370	424	486	322	375	429	488	325	379	433	490

⁽¹⁾ the specified value of T_sn cannot be reached as mode B develops; ⁽²⁾ values in bold refer to the elastic soil response where closed-form expressions are available

Table 8

Values of the normalized maximum bending moment × 10³ for intermediate values of T_sn

λ	R_E = R_U = 2				R_E = R_U = 3				R_E = R_U = 4				R_E = R_U = 5
	T_sn				T_sn				T_sn				T_sn
	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45	0.30	0.35	0.40	0.45
	ρ = 0
0.7	67	94	- ⁽¹⁾	-	75	96	124	161	81	100	125	156	87	104	127	157
0.8	77	101	134	-	86	105	131	163	93⁽²⁾	110	133	163	98	114	136	164
0.9	87	108.0	138	178	97	114.6	139	169	103	120	142	170	107	124	144	171
1.0	97	117	143	180	106	124	146	175	111	129	150	177	114	133	153	178
1.1	106	125	150	183	114	133	154	182	118	138	158	184	121	141	161	185
1.2	114	134	157	188	121	141	162	189	125	145	166	191	127	148	169	192
1.3	122	142	165	195	127	149	170	196	130	152	174	198	132	154	176	199
1.4	129	150	173	201	133	156	178	204	136	159	181	205	137	160	183	207
1.5	135	157	180	208	139	162	185	211	141	165	188	213	142	166	190	214
1.6	141	164	198	215	144	168	192	218	146	171	195	220	147	172	196	221
1.7	146	171	195	222	149	174	199	225	151	176	202	227	152	177	203	228
1.8	152	177	202	230	154	180	206	232	156	182	208	234	157	183	209	235
1.9	157	183	209	237	159	186	212	239	161	187	215	241	161	188	215	242
2.0	162	0.189	216	244	164	191	219	246	165	193	220	248	166	194	221	249
	ρ = 1
0.7	67	93	129	-	75	96	123	161	81	100	125	156	87	104	127	157
0.8	77	100	133	175	86	105	131	163	93	110	135	163	98	114	136	164
0.9	87	108	137	177	97	115	139	169	103	120	142	170	107	124	144	171
1.0	97	117	143	179	106	124	146	175	111	129	150	177	114	133	153	178
1.1	106	125	150	182	114	133	154	182	118	138	158	184	121	141	161	185
1.2	114	134	157	188	121	141	162	189	125	145	166	191	127	148	169	192

⁽¹⁾ the specified value of T_sn cannot be reached as mode B develops; ⁽²⁾ values in bold refer to the elastic soil response where closed-form expressions are available

Finally, it can be observed that the effect of ρ is slightly appreciable only for the lower investigated values of λ and R_U and for high value of T_sn when soil plasticization occurs also below the sliding surface (see Fig. 3c, d, e); otherwise, when soil plasticization occurs only above the sliding surface (Fig. 3a) the values of y_0n and M_maxn obtained for ρ = 1 are perfectly coincident with those obtained for ρ = 0. Therefore the values of y_0n and M_maxn calculated for ρ = 1 and λ > 1.2 are not included in Tables 7–8.

4.1 Effect of R_U

Tables 7, 8 and Figs. 11, 12 are obtained by assuming R_E = R_U. While the effect of R_U is null below the elastic threshold, it can be potentially appreciable in the elastic–plastic zone. As an example, Fig. 13 shows the curves T_sn-y_0n and M_maxn-T_sn for λ = 1.2, R_E = 3 and ρ = 0 at varying R_U. As expected, the curves are perfectly superimposed in the elastic range. Note that for R_U > 1.68 the elastic threshold is independent of R_U, as the soil plasticization first occurs above the sliding surface and Eq. 32a—which does not include R_U—determines the beginning of the elastic–plastic part of the mobilization curve. For R_U = 1.5 the elastic threshold is slightly lower, as for this combination of R_E and R_U the elastic threshold is governed by the soil plasticization below the sliding surface (Eq. 32b). Only for T_sn greater than about 0.4 the curves are clearly distinct as the final values of y_0n and M_maxn depend on which plastic mode is reached (C1, C2 or C3) and this on turn depends on R_U and ρ. For λ = 1.2 mode C3 always develops when R_U > 2.36 and the final values of y_0n and M_maxn become independent of R_U. This implies that the mobilization curve relevant to R_U = 3—or in general for R_U > 2.37—is perfectly coincident with that for R_U = 2.37.

Fig. 13

Effect of R_U on the mobilization curve for λ = 1.2, R_E = 3 and ρ = 0: a T_sn vs y_0n b M_maxn vs T_sn

Full size image

4.2 Effect of R_E

Figure 14 shows different mobilization curves T_sn-y_0n and M_maxn-T_sn obtained by varying R_E for an assigned combination of λ, /R_U and ρ (λ =1; R_U = 2; ρ = 0). In this case the final values of y_0n and M_maxn are the same, as they depend only on R_U and ρ. The elastic thresholds and the extent of the elastic–plastic zone depends on R_E. For a given value of T_sn (< 0.5) both y_0n and M_maxn slightly increase with R_E. The effect of R_E becomes negligible approaching the plastic condition.

Fig. 14

Effect of R_E on the mobilization curve for λ = 1 R_U = 2 and ρ = 0: a T_sn vs y_0n b M_maxn vs T_sn

Full size image

5 Numerical Example

The procedure for designing a stabilizing pile is illustrated by a numerical example.

The soil profile includes an unstable cohesionless layer (γ = 18 kN/m³; ϕ = 30°; n = 2000 kN/m³) of thickness 3.75 m overlying a stable layer of OC clay. Let us assume that a traditional two-dimensional stability analysis requires an additional resistant force of 245 kN/m to achieve the desired safety factor.

Consider a row of stabilizing bored concrete piles (E_p = 32 GPa) with D = 1.5 m and L = 8.4 m, so that the condition of rigidity of Eq. (7) is satisfied. Moreover consider a center to center spacing of 6 m (i.e. 4D), which implies that each pile must support 1470 kN. For this spacing it is reasonable to calculate the value of m₁ using the relationship suggested by Fleming et al. (2009) for isolated piles in sand, i.e. $m_{1} = DK_{P}^{2} \;\gamma$ = 243 kN/m². Assuming for the stable layer the lateral resistance per unit length, P_u2, equal to 1950 kN/m and subgrade modulus E_s2 = 20 MPa, the dimensionless parameters T_snR, λ, R_E, R_U and ρ can be computed as 0.43, 1.24, 2.67, 2.14 and 0, respectively.

With this input the normalized shear force at the elastic threshold T_sne is computed as 0.37; then, for the required resistant force, the pile response falls in the elastic–plastic range.

Using the values of Table 7 a first linear interpolation is made to compute y_0n for λ = 1.24 and R_U = 2, obtaining 3.966 and 4.582 for T_sn = 0.40 and T_sn = 0.45, respectively. Repeating the interpolation for R_U = 3 gives y_0n = 4.026 for T_sn = 0.40 and y_0n = 4.602 for T_sn = 0.45.

Starting from these values, a second interpolation is performed to calculate the values of y_0n relevant to T_sn = 0.43 which are equal to 4.3356 for R_U = 2 and 4.3716 for R_U = 3. Finally, a third linear interpolation is required to compute the value of y_0n for R_U = 2.14 obtaining y_0n = 4.34, which corresponds to y₀ = 4.34∙ m₁/(nR_E) = 19.8 cm.

The same procedure can be applied to evaluate the maximum bending moment by the data of Table 8. Specifically, by triple linear interpolation a value of M_maxn = 0.1817 is obtained, i.e. M_max = 0.1817∙m₁L₁³ = 2328 kNm.

The pile head deflection and the maximum bending moment are 82% and 79%, respectively, of those calculated at the plastic threshold using the closed-form equations of case C2 listed in Table 5 which gives y_0n = 5.284 and M_maxn = 0.2295.

Note that the previous elastic–plastic computations are strictly valid for R_E = R_U = 2.14; by performing a specific numerical analysis with R_E = 2.67 and R_U = 2.14 the normalized pile head deflection and the maximum bending moment are computed as 4.325 and 0.1797, with a difference percentage of about 1% in comparison with the values obtained by the simplified procedure based on Tables 7, 8.

6 Conclusions

An approach based on the modulus of subgrade reaction has been proposed to analyze the response of a rigid unrestrained passive pile subjected to a uniform horizontal soil movement in two-layered soil. The investigated soil profile includes an unstable layer with the modulus of subgrade reaction and the ultimate strength that vary linearly with depth, whereas both are assumed to be constant in the stable layer. In the investigated soil profile the ultimate strength vary linearly with depth in both unstable and stable layer, and the modulus of subgrade reaction is assumed to increase with depth in the unstable layer, whereas is assumed to be constant in the stable layer.

Using dimensionless parameters, the pile response has been analyzed in terms of the shear force developed at the sliding surface, maximum bending moment and pile head deflection.

Unlike some previous studies, the analysis has been focused not only to the soil ultimate state but also to the intermediate soil response, when soil reaction is fully elastic or locally plastic along the pile shaft. The analysis described in this paper allows obtaining the mobilization curves, i.e. the relationships between soil movement, pile head deflection, maximum bending moment and shear force at the sliding depth. For the elastic part of the mobilization curves, analytical expressions have been derived to calculate internal forces, pile deformation and limiting soil movement beyond which the soil response ceases to be elastic. For usual pile embedment the extent of the elastic zone increases with the pile embedment (λ) and with the strength and modulus ratio at the layer interface (R_U and R_E) and in certain circumstances the pile response remains fully elastic until it reaches 90% of the maximum shear force at the sliding surface.

For the elastic–plastic part of the mobilization curves, a general case has been discussed and the governing equations have been also provided. In such a case the numerical solution of a nonlinear system is needed and closed-form solutions are not available. All the mobilization curves reach one of the possible three failure mechanisms already known in the literature (short pile, intermediate and flow mode). For the investigated soil profile, the occurrence of such failure modes is found to depend on the combined values of the embedment ratio (λ) and the strength ratio at the soil interface (R_U), as well as the ratio of the gradients of soil strength (ρ).

The results of a parametric study shows that the mobilization curves is strongly influenced by the embedment ratio (λ) while the effect of the soil properties (R_E, R_U and ρ) is minor.

Tabulated values of the normalized pile head deflection and maximum bending moment as a function of the required stabilizing force are provided for different combination of λ, R_E, R_U and ρ for a quick assessment of pile response. A simplified procedure has been described by a numerical example, to check the pile performance even in the elastic–plastic range without the need to solve the nonlinear system governing the problem. The accuracy of this design methodology was demonstrated to be very high, especially if compared with the intrinsic uncertainties involved in the choice of deformability and strength parameters of the soil layers.

Declarations

Competing interests

The authors have not disclosed any competing interests.

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Title: Elastic–Plastic Analysis of Rigid Passive Piles in Two-Layered Soils
Author: Ivo Bellezza
Publication date: 21-12-2023
Publisher: Springer International Publishing
Published in: Geotechnical and Geological Engineering / Issue 4/2024
Print ISSN: 0960-3182
Electronic ISSN: 1573-1529
DOI: https://doi.org/10.1007/s10706-023-02673-5

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Lateral Performance of Single and Grouped of Hollow Bar Micropiles in Cohesionless Soil

Original Paper

Springer Professional

Elastic–Plastic Analysis of Rigid Passive Piles in Two-Layered Soils

Abstract

Abstract

Publisher's Note

1 Introduction

2 Method of Analyis

2.1 Dimensionless Parameters

3 Results and Discussion

3.1 Elastic Case

3.2 Elastic–Plastic Cases

3.3 Elastic Thresholds

3.4 Plastic Cases

3.5 Thresholds Values of the Embedment Ratio

4 Mobilization Curves

4.1 Effect of R_U

4.2 Effect of R_E

5 Numerical Example

6 Conclusions

Declarations

Competing interests

Publisher's Note

Other articles of this Issue 4/2024

Comparing 1D Regression and Evolutionary Polynomial Analyses for Predicting Brazilian Tensile Strength of Limestone in Dry and Saturated Conditions

Using Limestone Mining Waste to Stabilize Expansive Soils in Ceará, Brazil

Initiation and Propagation of Interface Cracks in Layered Composite Rocks: An Investigation Using True Triaxial Experiments

Brittle Failure Modes and Mechanisms in Foliated Rock Under Uniaxial Compression: Laboratory Testing and Particle Flow Modeling

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Range	Normalized shear force T and bending moment M
0 ≤ z ≤ b_n	\(T_{n} = \frac{T}{{mL_{1}^{2} }} = - \frac{1}{2}z_{n}^{2}\) \(M_{n} = \frac{M}{{mL_{1}^{3} }} = - \frac{1}{6}z_{n}^{3}\)
b_n ≤ z_n ≤ c_n	\(T_{n} = - \frac{1}{2}b_{n}^{2} - \frac{{\Delta y_{0n} }}{{2R_{E} }}\left( {z_{n}^{2} - b_{n}^{2} } \right) + \frac{{\omega_{n} }}{{3R_{E} }}\left( {z_{n}^{3} - b_{n}^{3} } \right)\) \(\Delta y_{0n} = y_{0n} - y_{s0n}\) \(M_{n} = - \frac{1}{6}b_{n}^{2} \left( {3z_{n} - 2b_{n} } \right) - \frac{{\Delta y_{0n} }}{{6R_{E} }}\left( {z_{n}^{3} - 3b_{n}^{2} z_{n} + 2b_{n}^{3} } \right) + \frac{{\omega_{n} }}{{12R_{E} }}\left( {z_{n}^{4} - 4b_{n}^{3} z_{n} + 3b_{n}^{4} } \right)\) The depth M_max at. z_m1 is the solution of a 3-order equation: \(z_{m1n}^{3} - 1.5\frac{{\Delta y_{0n} }}{{\omega_{n} }}z_{m1n}^{2} + \left\{ {1.5b_{n}^{2} \frac{{\Delta y_{0n} }}{{\omega_{n} }} - b_{n}^{3} - \frac{{1.5R_{E} b_{n}^{2} }}{{\omega_{n} }}} \right\} = 0\) provided that b_n ≤ \(\frac{{z_{m1} }}{{L_{1} }}\) ≤ c_n
c_n ≤ z_n ≤ 1	\(T_{n} = \frac{1}{2}\left( {z_{n}^{2} - c_{n}^{2} - b_{n}^{2} } \right) - \frac{{\Delta y_{0n} }}{{2R_{E} }}\left( {c_{n}^{2} - b_{n}^{2} } \right) + \frac{{\omega_{n} }}{{3R_{E} }}\left( {c_{n}^{3} - b_{n}^{3} } \right)\) \(\begin{gathered} M_{n} = \frac{1}{6}\left\{ {z_{n}^{3} - 3\left( {b_{n}^{2} + c_{n}^{2} } \right)z_{n} + 2\left( {c_{n}^{3} + b_{n}^{3} } \right)} \right\} - \frac{{\Delta y_{0n} }}{{6R_{E} }}\left\{ {3\left( {c_{n}^{2} - b_{n}^{2} } \right)z_{n} - 2\left( {c_{n}^{3} - b_{n}^{3} } \right)} \right\} \hfill \\ + \frac{{\omega_{n} }}{{12R_{E} }}\left\{ {4\left( {c_{n}^{3} - b_{n}^{3} } \right)z_{n} - 3\left( {c_{n}^{4} - b_{n}^{4} } \right)} \right\} \hfill \\ \end{gathered}\) M_max at depth \(\frac{{z_{m1} }}{{L_{1} }} = \sqrt {c_{n}^{2} + b_{n}^{2} + \frac{{\Delta y_{0n} }}{{R_{E} }}\left( {c_{n}^{2} - b_{n}^{2} } \right) - \frac{2}{3}\frac{{\omega_{n} }}{{R_{E} }}\left( {c_{n}^{3} - b_{n}^{3} } \right)}\) provided that c_n ≤ \(\frac{{z_{m1} }}{{L_{1} }}\) ≤ 1
1 ≤ z_n ≤ f_n	\(T_{n} = - R_{U} \left( {z_{n} - f_{n} - g_{n} + L_{n} } \right) + y_{0n} \left( {g_{n} - f_{n} } \right) - \frac{1}{2}\omega_{n} \left( {g_{n}^{2} - f_{n}^{2} } \right) - \frac{1}{2}\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \left( {g_{n} - 1} \right)^{2} - \left( {f_{n} - 1} \right)^{2} + \lambda^{2} } \right]\)\(\begin{gathered} M_{n} = - \frac{1}{2}R_{U} \left[ {z_{n}^{2} + f_{n}^{2} + g_{n}^{2} - L_{n}^{2} - 2\left( {f_{n} + g_{n} - L_{n} } \right)z_{n} } \right] - \frac{1}{2}y_{0n} \left[ {g_{n}^{2} - f_{n}^{2} - 2\left( {g_{n} - f_{n} } \right)z_{n} } \right] \hfill \\ + \omega_{n} \left[ {\frac{1}{3}\left( {g_{n}^{3} - f_{n}^{3} } \right) - \frac{1}{2}\left( {g_{n}^{2} - f_{n}^{2} } \right)z_{n} } \right] - \frac{1}{2}\rho \left\{ \begin{gathered} \frac{1}{3}\left[ {\left( {z_{n} - 1} \right)^{3} - \left( {f_{n} - 1} \right)^{3} - \left( {g_{n} - 1} \right)^{3} + \lambda^{3} } \right] + \lambda^{2} \left( {z_{n} - L_{n} } \right) \hfill \\ - \left( {g_{n} - 1} \right)^{2} \left( {z_{n} - g_{n} } \right) - \left( {f_{n} - 1} \right)^{2} \left( {z_{n} - f_{n} } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered}\) M_max at \(\frac{{z_{m2} }}{{L_{1} }} = \frac{{\rho - R_{U} + \sqrt {\left( {R_{U} - \rho } \right)^{2} - \rho C_{P} } }}{\rho } \le f_{n}\)\(C_{P} = \left\{ \begin{gathered} \left( {g_{n}^{2} - f_{n}^{2} } \right)\omega_{n} - 2\left( {g_{n} - f_{n} } \right)y_{0n} + 2R_{U} \left( {L_{n} - f_{n} - g_{n} } \right) \hfill \\ - \rho \left[ {\left( {g_{n} - 1} \right)^{2} + \left( {f_{n} - 1} \right)^{2} - \lambda^{2} - 1} \right] \hfill \\ \end{gathered} \right\}\)
f _n ≤ z_n ≤ g_n	\(T_{n} = R_{U} \left( {g_{n} - L_{n} } \right) - y_{0n} \left( {z_{n} - g_{n} } \right) + \frac{1}{2}\omega_{n} \left( {z_{n}^{2} - g_{n}^{2} } \right) + \frac{1}{2}\rho \left[ {\left( {g_{n} - 1} \right)^{2} - \lambda^{2} } \right]\)\(\begin{gathered} M_{n} = \frac{1}{2}R_{U} \left( {L_{n} - g_{n} } \right)\left( {L_{n} + g_{n} - 2z_{n} } \right) - \frac{1}{2}y_{0n} \left( {z_{n} - g_{n} } \right)^{2} + \frac{1}{6}\omega_{n} \left( {z_{n}^{3} - 3g_{n}^{2} z_{n} + 2g_{n}^{3} } \right) \hfill \\ + \frac{1}{2}\rho \left\{ {\left( {g_{n} - 1} \right)^{2} \left( {z_{n} - g_{n} } \right) + \lambda^{2} \left( {L_{n} - z_{n} } \right) + \frac{1}{3}\left[ {\left( {g_{n} - 1} \right)^{3} - \lambda^{3} } \right]} \right\} \hfill \\ \end{gathered}\) M_max at \(\frac{{z_{m2} }}{{L_{1} }} = \frac{{y_{0n} - \sqrt {y_{0n}^{2} - \omega_{n} C_{E} } }}{{\omega_{n} }} \ge f_{n}\)\(C_{E} = 2R_{U} \left( {g_{n} - L_{n} } \right) + 2y_{0n} g_{n} - \omega_{n} g_{n}^{2} + \rho \left[ {\left( {g_{n} - 1} \right)^{2} - \lambda^{2} } \right]\)
g_n ≤ z_n ≤ L_n	\(T_{n} = R_{U} \left( {z_{n} - L_{n} } \right) + 0.5\rho \left[ {\left( {z_{n} - 1} \right)^{2} - \lambda^{2} } \right]\)\(M_{n} = \frac{1}{2}R_{U} \left( {z_{n} - L_{n} } \right)^{2} + \frac{1}{6}\rho \left[ {\left( {z_{n} - 1} \right)^{3} - \lambda^{3} } \right] - \frac{1}{2}\rho \lambda^{2} \left( {z_{n} - L_{n} } \right)\)

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Abstract

Publisher's Note

1 Introduction

2 Method of Analyis

2.1 Dimensionless Parameters

3 Results and Discussion

3.1 Elastic Case

3.2 Elastic–Plastic Cases

3.3 Elastic Thresholds

3.4 Plastic Cases

3.5 Thresholds Values of the Embedment Ratio

4 Mobilization Curves

4.1 Effect of RU

4.2 Effect of RE

5 Numerical Example

6 Conclusions

Declarations

Competing interests

Publisher's Note

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4.1 Effect of R_U

4.2 Effect of R_E