Theoretical t-z Curves for Axially Loaded Piles

Bateman, Abigail H.; Crispin, Jamie J.; Vardanega, Paul J.; Mylonakis, George E.

doi:10.1061/(ASCE)GT.1943-5606.0002753

Open access

Technical Papers

May 6, 2022

Theoretical $t - z$ Curves for Axially Loaded Piles

Authors: Abigail H. Bateman https://orcid.org/0000-0003-3454-1756, Jamie J. Crispin, Ph.D. https://orcid.org/0000-0003-3074-8493, Paul J. Vardanega, Ph.D., M.ASCE https://orcid.org/0000-0001-7177-7851, and George E. Mylonakis, Ph.D., M.ASCE https://orcid.org/0000-0002-8455-8946 [email protected]Author Affiliations

Publication: Journal of Geotechnical and Geoenvironmental Engineering

Volume 148, Issue 7

https://doi.org/10.1061/(ASCE)GT.1943-5606.0002753

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Abstract

Estimation of nonlinear pile settlement can be simplified using one-dimensional “

t - z

” curves that conveniently divide the soil into multiple horizontal “slices.” This simplification reduces the continuum analysis to a two-point boundary-value problem of the Winkler type, which can be tackled by standard numerical procedures. Theoretical “

t - z

” curves can be established using the “shearing-of-concentric-cylinders” theory of Cooke and Randolph-Wroth, which involves two main elements: (1) a constitutive model cast in flexibility form,

γ = γ (τ)

; and (2) an attenuation function of shear stress with radial distance from the pile,

τ = τ (r)

. Soil settlement can then be determined by integrating shear strains over the radial coordinate, which often leads to closed-form solutions. Despite the simplicity and physical appeal of the method, only a few theoretical “

t - z

” curves are available in the literature. This paper introduces three novel attenuation functions for shear stresses, inspired by continuum solutions, which are employed in conjunction with eight soil constitutive models leading to a set of 32 “

t - z

” curves. Illustrative examples of pile settlement calculation in two soil types are presented to demonstrate application of the method.

Introduction

Nonlinear settlement estimation of axially loaded piles requires continuum analysis using constitutive models in multiple dimensions, which might be difficult to implement in practice and are unavailable for many real soils. The problem can be simplified to a single dimension using the Winkler model of soil reaction through the introduction of pertinent stress-displacement curves (“

t - z

” curves) along the pile shaft (Seed and Reese 1957; Coyle and Reese 1966) and a stress-displacement curve (“

q - z

” curve) at the pile tip (Frank and Zhao 1982; Chow 1986; Armaleh and Desai 1987; API/ANSI 2011). The pile settlement can then be estimated by discretizing the pile, which is modeled as an elastic-compressible rod, into a number of elements separated by nodes. A “

t - z

” curve for each node and a “

q - z

” curve at the base are selected. A numerical integration approach of a two-point nonlinear boundary value problem is then applied to compute the settlement at the pile head due to a given load [e.g., Poulos and Davis (1980); Salgado (2008); Guo (2012); Poulos (2017)]. Relevant algorithms are easy to implement and available in different formats, including finite-differences (Ensoft TZPile, Reese et al. 2014), finite-elements (Oasys Pile, Oasys 2017), and alternative formulations (Scott 1981; Motta 1994; Seo et al. 2009; Crispin et al. 2018; Psaroudakis et al. 2019). The stress-displacement response at the pile tip (i.e., “

q - z

” curves) and group effects lie beyond the scope of this paper; therefore, the Illustrative Examples presented in the following work involve only end-bearing solitary piles.

Some empirical “

t - z

” curves are available in the literature [e.g., Coyle and Reese (1966); Coyle and Sulaiman (1967); Reese et al. (1969); Vijayvergiya (1977); Frank and Zhao (1982); Abchir et al. (2016)] but are limited to specific test configurations. In addition, the American Petroleum Institute and American National Standards Institute (API/ANSI) (2011) provides general recommendations to develop idealized “

t - z

” curves for axially loaded piles based on a database of field tests. Analogous (“

p - y

”) curves are available for laterally loaded piles [e.g., McClelland and Focht (1956); Matlock (1970); Lam and Martin (1986); Reese and Van Impe (2011); Khalili-Tehrani et al. (2014)].

Instead, theoretical “

t - z

” curves have been derived from the consideration of a (plane-strain) horizontal soil “slice.” Early solutions (Cooke 1974; Randolph and Wroth 1978; Baguelin and Frank 1979) assumed linear-elastic soil and employed the concentric cylinder model to describe the attenuation of shear stress with a radial distance from the pile. These idealizations have been used to establish linearly elastic springs for axially loaded piles which are widely used in practice [Scott (1981), Guo and Randolph (1998), Mylonakis and Gazetas (1998), Salgado (2008), Fleming et al. (2009), Viggiani et al. (2012), Guo (2012), Crispin et al. (2018)]. Following the same approach, nonlinear “

t - z

” curves were obtained by Vardanega et al. (2012), Kraft et al. (1981) and Chang and Zhu (1998) using power law, hyperbolic, and modified hyperbolic constitutive models, respectively. These curves have been employed in various studies by Chow (1986), Zhu and Chang (2002), Randolph (1994), Guo (2012), and Voyagaki et al. (2021).

This method is versatile because it can incorporate variable soil properties with radial distance from the pile, installation effects, interface action between soil and pile, variable soil properties with depth, negative skin friction, and dynamic phenomena (Akiyoshi 1982; El Naggar and Novak 1994; Michaelides et al. 1998; Crispin and Mylonakis 2021). In addition, the method provides information about the stress and displacement fields around a pile that can be used for developing solutions for pile groups, such as those by Randolph and Wroth (1979) and Mylonakis and Gazetas (1998). Moreover, the formulation can accommodate both total and effective stresses (by using appropriate constitutive relations); hence, it can handle both fine- and coarse-grained soil materials under drained and undrained conditions. Nevertheless, despite the elegance and practical appeal of this method, only a handful of theoretical “

t - z

” curves are reported in the literature.

This paper derives an extended set of “

t - z

” curves pertaining to different conditions based on the same horizontal soil slice model as the aforementioned theoretical solutions. A “toolbox” of “

t - z

” curves is provided to allow a designer to select an appropriate curve from a selected soil model and an easily fitted attenuation function. The unloading/reloading case is not investigated in this work (yet, no relevant restrictions are imposed by the derived “

t - z

” curves). Two illustrative calculation examples involving end-bearing piles in natural Pisa and remolded kaolinite are provided to demonstrate the simplicity of calibrating the “

t - z

” curves for application to a design problem. In these examples, the soil constitutive models and attenuation functions are fitted to available triaxial test data and numerical data, respectively. A preliminary version of some of the work presented in this paper is available in Bateman (2019).

Horizontal Soil Slice Model

Fig. 1 indicates the soil strip considered in the horizontal soil slice model, which is based on the following main assumptions.

1.

The soil around the pile is divided into a series of independent horizontal “slices” of infinitesimal thickness, Fig. 1, subjected to shearing on horizontal and vertical planes.

2.

Since the continuity of the soil medium with depth is not considered, the slices provide resistance only to shearing, Fig. 1. Additional resistance due to pile-induced normal stresses acting on the upper and lower faces of the slices is neglected.

3.

Shear strains depend solely on vertical soil displacements (i.e.,

γ ≅ \partial u / \partial r

). The effect of radial displacements on shear strains is small and can be neglected.

4.

The attenuation of shear stresses with radial distance from the pile,

τ = τ (r)

, can be established (often taken as proportional to

1 / r

by equilibrium considerations, as discussed in the following).

5.

Deformations are infinitesimal.

Note that the second assumption can be relaxed using an extended set of attenuation functions,

τ (r)

, to be discussed in the next section. To obtain explicit solutions, two additional assumptions (6 and 7) are needed:

6.

Soil stress-strain response in shear is cast in flexibility form, that is,

γ = γ (τ)

, where

τ

and

γ

are the shear stress and shear strain acting on a soil element, respectively.

7.

The composite function

γ = γ (τ (r))

is integrable over the radial coordinate,

r

.

Fig. 1. Horizontal soil slice model. [Reprinted from Bateman and Crispin (2020), under Creative Commons-BY-4.0 license (https://creativecommons.org/licenses/by/4.0/).]

Using these assumptions, shear strain can be integrated over the radial distance to provide the soil settlement at the pile-soil interface due to an applied shear stress,

τ_{0}

, at any given depth (Fig. 1)

u_{0} = \int_{d / 2}^{\infty} γ (r) d r = \int_{d / 2}^{\infty} γ [τ (r)] d r

(1)

where

u_{0}

= soil settlement at the pile circumference;

r

= radial distance from the pile centerline; and

d

= pile diameter. Since volumetric strains do not explicitly appear in Eq. (1), the analysis highlights how immediate settlement develops in nearly incompressible media, such as saturated clay. Regarding the upper integration limit in Eq. (1), this paper shows that it is not always feasible to extend to infinity under certain conditions.

After dividing the total side friction per unit pile length (

π d τ_{0}

) by Eq. (1), the nonlinear secant Winkler modulus,

k (τ_{0})

, of the soil slice due to the imposed shear stress

τ_{0}

can be readily obtained as

k (τ_{0}) \equiv τ_{0} \frac{π d}{u_{0}} = τ_{0} \frac{π d}{\int_{d / 2}^{\infty} γ [τ (r)] d r}

(2)

This stiffness parameter naturally carries units of stress and represents a nonlinear extension of the classical modulus of subgrade reaction for axially loaded piles. It is noted that no assumptions have been made, or restrictions imposed, in Eqs. (1) and (2) regarding the spatial distribution of soil properties, including the low strain shear modulus and strength, which might vary with the radial distance from the pile [e.g., observed in field tests by Kalinski et al. (2001) and O’Neill (2001)]. However, the explicit solutions obtained later in this paper assume radially homogenous soil. Corrections to the solutions provided to account for radial inhomogeneity are available [e.g., Kraft et al. (1981), Bateman and Crispin (2020)].

Eq. (1) indicates that two functions are required to obtain a theoretical “

t - z

” curve from the horizontal soil slice model describing, first, the attenuation of shear stresses,

τ (r)

, and, second, the soil constitutive relationship (in flexibility form),

γ (τ)

. Previous authors have utilized the concentric cylinder model to obtain an attenuation function, leading to a

1 / r

dependence (discussed later). This paper goes on to introduce three novel attenuation functions inspired by available continuum solutions [Mylonakis (2001a)], resulting in a combination of generalized power law and exponential decay functions. These attenuation functions are employed in conjunction with eight simplified soil constitutive models from the literature, leading to 32 families of theoretical “

t - z

” curves, the vast majority (28) of which are, to the authors’ knowledge, novel.

Attenuation Functions

To derive attenuation functions of shear stresses, the cylindrical pile depicted in Fig. 2 is considered. Taking the vertical equilibrium of an infinitesimal soil element in cylindrical coordinates and focusing on pile-induced stresses yields [Randolph and Wroth (1978), Mylonakis (2001b), Anoyatis and Mylonakis (2012)]

\frac{\partial (r τ)}{\partial r} + \frac{\partial σ}{\partial z} r = 0

(3)

where

σ

and

τ

= vertical normal and shear stresses, respectively; and

z

= depth below ground level.

Fig. 2. Equilibrium of a soil element in cylindrical coordinates. [Reprinted from Bateman and Crispin (2020), under Creative Commons-BY-4.0 license (https://creativecommons.org/licenses/by/4.0/).]

A simplified solution to this equation can be derived by assuming that variations in the vertical stress with a depth due to pile loading are negligible; accordingly, setting

\partial σ / \partial z = 0

and integrating over

r

yields the elementary solution [Cooke (1974), Randolph and Wroth (1978); Baguelin and Frank (1979)]

τ (r) = τ_{0} (\frac{d}{2 r})

(4)

This result is commonly known as the concentric cylinder (or plane strain) model, in which the soil is treated as a series of concentric cylinders with respective shear forces and no resistance by means of vertical normal stresses. This model provides a simple attenuation function that has been extensively employed in the literature (Scott 1981; Fleming et al. 2009; Guo 2012; Viggiani et al. 2012).

A difficulty associated with the use of this equation lies in the singular nature of the associated displacement field. Indeed, integrating shear stresses over the radial coordinate leads to a logarithmic solution for displacements that diverges with increasing

r

. To correct this problem, Randolph and Wroth (1978) suggested an empirical radius,

r_{m}

, beyond which the vertical soil settlement can be assumed negligible;

r_{m}

is usually taken as proportional to the pile length,

L

, and specific values have been suggested by Randolph and Wroth (1978), Guo and Randolph (1998), Fleming et al. (2009), and Guo (2012).

A more rigorous yet simple approach for tackling this problem is possible by first casting the governing equation [Eq. (3)] in displacement form through the introduction of the approximate stress-displacement relations

τ ≅ - G_{s} \partial u / \partial r

and

σ ≅ - M_{s} \partial u / \partial z

[as, for instance, done by Nogami and Novak (1976), Mylonakis (2001b), and Anoyatis et al. (2019)]

\frac{\partial}{\partial r} [r \frac{\partial u (r, z)}{\partial r}] + η^{2} r \frac{\partial^{2} u (r, z)}{\partial z^{2}} = 0

(5)

where

u

= vertical soil settlement;

η^{2} = M_{s} / G_{s}

is a dimensionless compressibility coefficient;

G_{s}

= elastic shear modulus;

M_{s}

= compressibility modulus of the soil material; and

η

= function of the Poisson’s ratio of the soil,

ν_{s}

. Suitable values of

η

(and, thus,

M_{s}

) are discussed in Appendix I and in Mylonakis (2001a, b).

An approximate solution to Eq. (5) can be obtained using a technique analogous to the one employed for spread footings by Vlasov and Leontiev (1966), previously applied to this problem by Vallabhan and Mustafa (1996), Lee and Xiao (1999), and Mylonakis (2000, 2001b) (similar approaches have been employed for laterally loaded piles, including Guo and Lee (2001), Basu and Salgado (2008), Shadlou and Bhattacharya (2014), and Bhattacharya (2019), among others). By separating the displacement function in Eq. (5),

u (r, z)

, into radial,

u (r)

, and vertical,

χ (z)

, components [

u (r, z) = u (r) χ (z)

], multiplying by a virtual displacement

χ (z)

and integrating over the vertical coordinate,

z

, these authors obtained an alternative form of Eq. (5) that is now independent of depth [modified from Mylonakis (2000)]

\frac{d^{2} u (r)}{d r^{2}} + (\frac{1}{r}) \frac{d u (r)}{d r} - {(\frac{2 q}{d})}^{2} u (r) = 0

(6)

where

q

= compressibility constant that results from the integration with depth and is proportional to

η

. The specific form of

q

is chosen to enable it to be employed as a dimensionless fitting parameter to simplify the problem.

Contrary to the elementary concentric cylinder model, Eq. (6) duly accounts for the continuity of the soil in the vertical direction. Eq. (6) is of the Bessel type and admits the following solutions for displacements and stresses (Mylonakis 2000; Olver et al. 2010)

u (r) = u_{0} \frac{K_{0} (2 q r / d)}{K_{0} (q)}

(7a)

τ (r) = τ_{0} \frac{K_{1} (2 q r / d)}{K_{1} (q)}

(7b)

where

K_{0}

and

K_{1}

= modified Bessel functions of the second kind and order 0 and 1, respectively. Interestingly, these solutions are identical to those of the related dynamic plane strain problem for axially loaded piles pioneered by Baranov (1967) and Novak (1974), and tend to zero with increasing

r

(thus rendering empirical corrections such as

r_{m}

unnecessary). A discussion of this remarkable similarity is provided in Mylonakis (2001a).

Due to its complexity, this expression does not result in closed-form “

t - z

” curves as desired. Concentrating on the solution for stresses in Eq. (7b), this can be simplified considering the asymptotic expansions of the associated Bessel function in different regimes. For instance, for small radial distances from the pile centerline, the asymptotic form of the modified Bessel function is the power law function

K_{1} (x) \sim (1 / 2) Γ (v) {(2 / x)}^{1}

, where

x

= independent variable and

Γ (v)

= Gamma function [Olver et al. (2010)]. Substituting this expression into Eq. (7b) yields Eq. (4), which implies that the first term in Eq. (3) governs the behavior of the solution in the vicinity of the pile, as assumed in the classical concentric cylinder model, regardless of soil compressibility.

For large radial distances from the pile centerline, the pertinent asymptotic expression is

K_{1} (x) \sim {(π / 2 x)}^{1 / 2} e^{- x}

[Olver et al. (2010)], which yields the approximate solution (referred in the ensuing as “power-exponential decay”)

τ (r) = τ_{0} {(\frac{2 r}{d})}^{- 1 / 2} \exp [- q (\frac{2 r}{d} - 1)]

(8)

Eqs. (4) and (8) are based on sound theoretical arguments that include some necessary simplifying assumptions. Therefore, to provide flexibility in the ensuing analyses, these can be generalized in the form shown in Eqs. (9) and (10). By applying the additional fitting parameters (

m

and

n

), a better fit can be achieved

τ (r) = τ_{0} {(\frac{2 r}{d})}^{- m}

(9)

τ (r) = τ_{0} {(\frac{2 r}{d})}^{- n} \exp [- q (\frac{2 r}{d} - 1)]

(10)

where

e x p (x)

= exponential function. The first [Eq. (9)] of these modified solutions is referred in the ensuing as “generalized concentric cylinder,” whereas the second [Eq. (10)] is referred to as “generalized power-exponential decay” model. The last model encompasses all others by using appropriate values for parameters

n

and

q

. In addition, these equations have previously been suggested as approximate attenuation functions for displacements and associated time derivatives in relevant dynamic problems involving frequency and damping (Mylonakis 1995; Gazetas et al. 1998; Su et al. 2019); here, they are used as semiempirical static solutions for stresses.

Eqs. (9) and (10) enable theoretical “

t - z

” curves to be analytically derived without reliance on the empirical radius

r_{m}

, which is employed in the concentric cylinder model [provided

m

is greater than 1 in Eq. (9)]. Instead, these equations require fitting the parameter

m

[Eq. (9)] or the parameters

q

and

n

[Eq. (10)] to field test data, numerical continuum solutions or more rigorous analytical models, similar to how

r_{m}

has been fitted by previous authors (Randolph and Wroth 1978; Guo 2012). An example of fitting these parameters to a more rigorous analytical solution (that is unsuitable for direct use in developing “

t - z

” curves and discussed in Appendix I) is detailed subsequently in the paper. The approximate range of the parameters from these examples, given in Table 2, are

0.1 < q < 0.3

,

1.0 < m < 1.25

, and

0.75 < n < 0.95

.

The attenuation functions in Eqs. (4), (8), (9), and (10) are indicated in Fig. 3 with the fitted parameters from illustrative Example B. These are compared with finite element method (FEM) results in PLAXIS 2D (Brinkgreve et al. 2019) for both linear and hyperbolic soil constitutive models (using the Mohr-Coulomb and HS Small soil models to match the two constitutive models used here). An elastostatic FEM analysis using axisymmetric conditions was carried out to model an end-bearing pile for each of the Illustrative Examples, which are discussed in the corresponding section. The pile (modeled using a linear-elastic material) has a fully fixed rigid layer at the pile base, and horizontal fixities are provided at the pile centerline and at a distance of

75 d

(Example A) or

50 d

(Example B). A mesh with 15-node triangular elements was generated by PLAXIS (total number of elements: 6026 for Example A and 3245 for Example B). The resulting attenuation functions for Example B are also plotted in Fig. 3. Further details on this example are provided later in this paper.

Fig. 3. Attenuation functions (parameters from Example B, Table 2): (a) linear; and (b) log scale.

Overall, the differences between the various curves are minor and start to manifest themselves at medium radial distances (

r > 5 d

), which agrees with the observations from the asymptotic solutions in Eqs. (4) and (8). These results suggest that the attenuation of shear stresses in the vicinity of the pile is governed by equilibrium, not stress-strain behavior. Soil compressibility starts becoming important at higher distances, where shear stresses and strains are generally lower than the corresponding normal stresses and volumetric strains. These traits justify using some elasticity arguments [used to derive Eqs. (8) and (10)] for tackling the stress distribution in the nonlinear problem at hand. Similar patterns are observed in other relevant problems, such as the Boussinesq and Mindlin solutions [e.g., Davis and Selvadurai (1996)].

Soil Constitutive Models

Eight soil constitutive models are considered in this paper. To obtain an explicit solution using Eq. (1), the selected constitutive relations must be cast in flexibility form, that is,

γ = γ (τ)

. Each soil constitutive model (illustrated in Fig. 4) is given as follows in both the (a) stiffness and (b) flexibility forms. A table summarizing the models is provided in Appendix III. Each model contains a number of parameters that should be fitted to site-specific laboratory data and/or in situ testing. An example of fitting these constitutive models to the soil test data is subsequently discussed.

Fig. 4. Illustration of constitutive models considered [Eqs. (11)–(18)].

The simplest constitutive relation considered is the linear stress-strain model defined by a single shear modulus value,

G

, given in stiffness and flexibility form by (for

τ < τ_{m a x}

)

τ = γ G

(11a)

γ = \frac{τ}{G}

(11b)

The bilinear model provides a simple adaption by splitting the linear model into two regions, the first having a stiffness

G_{1}

and the second having a reduced stiffness

G_{2}

, which is applicable after a predefined yield stress,

τ_{1}

, is reached. Additional stiffness changes could be introduced to more closely model nonlinear soil behavior. However, doing so would be at the expense of an increasingly complex function involving many fitting parameters. The bilinear model is given by (for

τ < τ_{m a x}

)

τ = {\begin{cases} γ G_{1} & γ \leq τ_{1} / G_{1} \\ G_{2} (γ - \frac{τ_{1}}{G_{1}}) + τ_{1} & γ \geq τ_{1} / G_{1} \end{cases}

(12a)

γ = {\begin{cases} \frac{τ}{G_{1}} & τ \leq τ_{1} \\ \frac{τ - τ_{1}}{G_{2}} + \frac{τ_{1}}{G_{1}} & τ \geq τ_{1} \end{cases}

(12b)

Alternatively, a simple power law relationship can be employed, in this case defined with the shear strain when 50% of the soil shear strength is mobilized,

γ_{50}

, and a soil nonlinearity exponent,

b

. This model was employed by Vardanega et al. (2012) to generate simple “

t - z

” curves [see also Williamson (2014), Vardanega (2015), Crispin et al. (2019)]. The power law relationship has the advantage that

r_{m}

is not required due to an infinite initial stiffness (

b < 1

). The power law model is given by (for

τ < τ_{m a x}

)

τ = \frac{τ_{m a x}}{2} {(\frac{γ}{γ_{50}})}^{b}

(13a)

γ = γ_{50} {(\frac{2 τ}{τ_{m a x}})}^{\frac{1}{b}}

(13b)

However, since infinite initial stiffness is not physically realizable, a linear model can be employed at low stress (

τ \leq τ_{i}

), with an initial stiffness

G_{i}

, resulting in the linear-power law soil constitutive model (for

τ < τ_{m a x}

)

τ = {\begin{cases} γ G_{i} & γ \leq τ_{i} / G_{i} \\ \frac{τ_{m a x}}{2} {(\frac{γ}{γ_{50}})}^{b} & γ \geq τ_{i} / G_{i} \end{cases}

(14a)

γ = {\begin{cases} \frac{τ}{G_{i}} & τ \leq τ_{i} \\ γ_{50} {(\frac{2 τ}{τ_{m a x}})}^{\frac{1}{b}} & τ \geq τ_{i} \end{cases}

(14b)

where

τ_{i} = (\frac{τ_{m a x}}{2}) {(\frac{2 G_{i} γ_{50}}{τ_{m a x}})}^{\frac{b}{b - 1}}

(14c)

A combination of the linear and power law models was suggested by Ramberg and Osgood (1943), which is defined here with a reference shear strain,

γ_{r}

, and two fitting constants,

c_{1}

and

c_{2}

(Ishihara 1996). This model is available in only the flexibility form (for

τ < τ_{m a x}

)

γ = γ_{r} [\frac{τ}{τ_{m a x}} + {(c_{1} \frac{τ}{τ_{m a x}})}^{c_{2}}]

(15)

The five constitutive relationships previously discussed do not approach a limiting stress; therefore, a shear strength cap,

τ_{m a x}

, is introduced. For undrained conditions, this ultimate strength should be taken as equal to the undrained shear strength,

τ_{m a x} = c_{u}

, which, in turn, depends on the shear mode of the soil test [e.g., Mayne (1985), Beesley and Vardanega (2020)]. Nevertheless, a generic parameter

τ_{m a x}

is employed because the “

t - z

” curves are not linked to a particular test or limited to undrained conditions. In practice, the shear strength of the resultant “

t - z

” curves might be capped at a lower value to represent slip at the pile-soil interface. To this end, empirical factors such as the familiar adhesion parameter

α

(Tomlinson 1957; Skempton 1959; Meyerhof 1976; Chakraborty et al. 2013) are available for undrained problems, which can be used to limit the maximum shear stress. Note that in the interest of space, simple names such as “Bilinear model” are adopted here over more accurate alternatives such as “capped Bilinear model” or “Bilinear-fully plastic model.”

Instead, the hyperbolic model, which approaches a single value at large strain, can be employed. Different forms of this model are available in the literature [e.g., Kondner (1963)], and it was utilized by Kraft et al. (1981) to develop a “

t - z

” curve using the concentric cylinder model. The form of the hyperbolic model adopted by Kraft et al. (1981) is employed with a fitting parameter,

R_{f}

, which acts as a factor to

τ_{m a x}

that alters the location of the asymptote. This enables a better fit of this model over a desired stress range, although this results in the model performing poorly at high stresses. In the classical model,

R_{f}

is expected to be less than 1 and requires a shear strength cap similar to the previous five models. However, in this paper, it has been employed as a fitting parameter and allowed to vary outside this range. The hyperbolic model is given by

τ = \frac{τ_{m a x}}{R_{f}} {[\frac{τ_{m a x}}{R_{f} G_{i} γ} + 1]}^{- 1}

(16a)

γ = \frac{τ_{m a x}}{R_{f} G_{i}} {[\frac{τ_{m a x}}{R_{f} τ} - 1]}^{- 1}

(16b)

In addition, a modified hyperbolic function is employed with an exponent,

c_{3}

, to give more control over the stiffness degradation, as discussed by Fahey and Carter (1993). This model has previously been utilized to develop a “

t - z

” curve in conjunction with the concentric cylinder model by Chang and Zhu (1998). The modified hyperbolic model is given only in the flexibility form

γ = \frac{τ}{G_{i}} {[1 - {(\frac{R_{f} τ}{τ_{m a x}})}^{c_{3}}]}^{- 1}

(17)

A similar asymptotic property can be obtained using an exponential relationship resulting in the exponential model indicated with a similar fitting parameter,

R_{f}

, employed as before

τ = \frac{τ_{m a x}}{R_{f}} {1 - \exp [- γ (\frac{R_{f} G_{i}}{τ_{m a x}})]}

(18a)

γ = - \frac{τ_{m a x}}{R_{f} G_{i}} \ln (1 - \frac{R_{f} τ}{τ_{m a x}})

(18b)

Note that some of the formulations include an initial (low-strain) shear modulus,

G_{i}

, which is employed as a model parameter to enable a better fit of the constitutive relation to the available data over the strain range of interest. Alternatively, when small strains are important, this modulus might be set equal to the maximum shear modulus,

G_{m a x}

, from high-quality experimental testing (e.g., seismic cone penetration testing).

“ $t - z$ ” Curves

By substituting the four attenuation functions [Eqs. (4), (8), (9), and (10)] into each constitutive model [Eqs. (11)–(18)], theoretical “

t - z

” curves can be derived analytically using Eq. (1). As an example, substituting the concentric cylinder model [Eq. (4)] into the linear soil constitutive model Eq. (11b) results in a function,

γ (r)

. By inputting this directly into Eq. (1), infinite settlement would be predicted when integrating over an infinite distance; thus, the upper integration limit is replaced with

r_{m}

, and the following equation is derived. This is valid until slip occurs at the pile-soil interface (Randolph and Wroth 1978; Fleming et al. 2009)

\frac{u_{0}}{d} = \frac{τ_{0}}{2 G} \int_{d / 2}^{r_{m}} \frac{1}{r} d r = \frac{τ_{0}}{2 G} \ln (\frac{2 r_{m}}{d})

(19)

Theoretical “

t - z

” curves derived using the concentric cylinder model Eq. (4) are given as follows and are summarized in Appendix IV.

Bilinear [Eq. (12)]

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{1}} \ln (\frac{2 r_{m}}{d}) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [\frac{G_{2}}{G_{1} - G_{2}} \ln (\frac{2 r_{m}}{d} \frac{τ_{1}}{τ_{0}}) + \frac{G_{1}}{G_{1} - G_{2}} \ln (\frac{τ_{0}}{τ_{1}}) + (\frac{τ_{1}}{τ_{0}} - 1)] & τ_{0} \geq τ_{1} \end{cases}

(20)

Power Law, for

b < 1

(Vardanega et al. 2012; Williamson 2014; Vardanega 2015; Crispin et al. 2019) [Eq. (13)]

\frac{u_{0}}{d} = \frac{γ_{50} b}{2 (1 - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}}

(21)

Linear-Power Law [Eq. (14)]

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{i}} \ln (\frac{2 r_{m}}{d}) & τ_{0} \leq τ_{i} \\ \frac{γ_{50} b}{2 (1 - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} [1 - {(\frac{2 r_{i}}{d})}^{\frac{b - 1}{b}}] + \frac{τ_{0}}{2 G_{i}} \ln (\frac{r_{m}}{r_{i}}) & τ_{0} \geq τ_{i} \end{cases}

(22a)

where

\frac{2 r_{i}}{d} = \frac{2 τ_{0}}{τ_{m a x}} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{b}{1 - b}}

(22b)

Ramberg-Osgood [Eq. (15)]

\frac{u_{0}}{d} = \frac{τ_{0} γ_{r}}{2 τ_{m a x}} \ln (\frac{2 r_{m}}{d}) + \frac{γ_{r}}{2 (c_{2} - 1)} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} [1 - {(\frac{2 r_{m}}{d})}^{1 - c_{2}}]

(23)

Hyperbolic [Kraft et al. (1981)] [Eq. (16)]

\frac{u_{0}}{d} = \frac{τ_{0}}{2 G_{i}} [\ln (\frac{2 r_{m}}{d} - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}})]

(24)

Modified Hyperbolic [Chang and Zhu (1998)] [Eq. (17)]

\frac{u_{0}}{d} = \frac{τ_{0}}{2 G_{i} c_{3}} [\ln ({(\frac{2 r_{m}}{d})}^{c_{3}} - {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{c_{3}}) - \ln {(1 - (\frac{R_{f} τ_{0}}{τ_{m a x}})^{c_{3}})]}^{}

(25)

Exponential [Eq. (18)]

\frac{u_{0}}{d} = \frac{τ_{m a x}}{2 R_{f} G_{i}} [\ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \frac{2 r_{m}}{d} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} (\frac{d}{2 r_{m}})) + \frac{R_{f} τ_{0}}{τ_{m a x}} [\ln (\frac{2 r_{m}}{d} - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}})]]

(26)

Theoretical “

t - z

” curves derived using the generalized concentric cylinder model [Eq. (9)] are given in Eqs. (27)–(34) and are summarized in Appendix IV. The

r_{m}

value has been included in these solutions; however, when

m > 1

, the relevant terms vanish and the solutions can be further simplified.

Linear

\frac{u_{0}}{d} = \frac{τ_{0}}{2 G (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}]

(27)

Bilinear

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{1} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} {\frac{1}{m - 1} [\frac{G_{1}}{G_{1} - G_{2}} - \frac{G_{2}}{G_{1} - G_{2}} {(\frac{2 r_{m}}{d})}^{1 - m} - {(\frac{τ_{0}}{τ_{1}})}^{\frac{1 - m}{m}}] + [\frac{τ_{1}}{τ_{0}} - {(\frac{τ_{0}}{τ_{1}})}^{\frac{1 - m}{m}}]} & τ_{0} \geq τ_{1} \end{cases}

(28)

Power Law, for

b < m

\frac{u_{0}}{d} = \frac{γ_{50} b}{2 (m - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}}

(29)

Linear-Power Law

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{i} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \leq τ_{i} \\ \frac{γ_{50} b}{2 (m - b)} {(\frac{2 τ_{0}}{τ_{\max}})}^{\frac{1}{b}} [1 - {(\frac{2 r_{i}}{d})}^{\frac{b - m}{b}}] + \frac{τ_{0}}{2 G_{i} (m - 1)} [{(\frac{2 r_{i}}{d})}^{1 - m} - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \geq τ_{i} \end{cases}

(30a)

where

\frac{2 r_{i}}{d} = {[\frac{2 τ_{0}}{τ_{\max}} {(\frac{2 γ_{50} G_{i}}{τ_{\max}})}^{\frac{b}{1 - b}}]}^{\frac{1}{m}}

(30b)

Ramberg-Osgood

\frac{u_{0}}{d} = \frac{τ_{0} γ_{r}}{2 τ_{m a x} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] + \frac{γ_{r}}{2 (c_{2} m - 1)} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} [1 - {(\frac{2 r_{m}}{d})}^{1 - c_{2} m}]

(31)

Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{m a x}}{2 R_{f} m G_{i}} {(\frac{R_{f} τ_{0}}{τ_{\max}})}^{\frac{1}{m}} [B (\frac{τ_{\max}}{R_{f} τ_{0}} {(\frac{2 r_{m}}{d})}^{m}; \frac{1}{m}, 0) - B (\frac{τ_{\max}}{R_{f} τ_{0}}; \frac{1}{m}, 0)]

(32)

where

B (x; a, b)

= incomplete beta function (see Appendix II).

Modified Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{m a x}}{2 R_{f} G_{i} c_{3} m} {(\frac{R_{f} τ_{0}}{τ_{\max}})}^{\frac{1}{m}} [B ({(\frac{R_{f} τ_{0}}{τ_{\max}})}^{c_{3}}; \frac{m - 1}{c_{3} m}, 0) - B ({(\frac{R_{f} τ_{0}}{τ_{\max}} {(\frac{d}{2 r_{m}})}^{m})}^{c_{3}}; \frac{m - 1}{c_{3} m}, 0)]

(33)

Exponential

\frac{u_{0}}{d} = \frac{τ_{m a x}}{2 R_{f} G_{i}} [\ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \frac{2 r_{m}}{d} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} {(\frac{d}{2 r_{m}})}^{m}) + {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{\frac{1}{m}} [B (\frac{τ_{m a x}}{R_{f} τ_{0}}; \frac{1}{m}, 0) - B (\frac{τ_{m a x}}{R_{f} τ_{0}} {(\frac{2 r_{m}}{d})}^{m}; \frac{1}{m}, 0)]]

(34)

Theoretical “

t - z

” curves derived using the power-exponential decay [Eq. (8)] are given as follows and summarized in Appendix V.

Linear

\frac{u_{0}}{d} = \frac{τ_{0} e^{q} \sqrt{π}}{2 G \sqrt{q}} erfc (\sqrt{q})

(35)

where

e r f c (x)

= complementary error function (see Appendix II).

Bilinear

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0} e^{q} \sqrt{π}}{2 G_{1} \sqrt{q}} erfc (\sqrt{q}) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [\frac{e^{q} \sqrt{π}}{\sqrt{q}} [\frac{G_{1}}{G_{1} - G_{2}} erfc (\sqrt{q}) - erfc (\sqrt{\frac{2 q r_{1}}{d}})] - \frac{τ_{1}}{τ_{0}} (\frac{2 r_{1}}{d} - 1)] & τ_{0} \geq τ_{1} \end{cases}

(36a)

where

\frac{2 r_{1}}{d} = \frac{1}{2 q} W (2 q e^{2 q} {(\frac{τ_{0}}{τ_{1}})}^{2})

(36b)

where

W (x)

denotes the Lambert W-Function (see Appendix II).

Power Law

\frac{u_{0}}{d} = \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{2 b - 1}{2 b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ (\frac{2 b - 1}{2 b}, \frac{q}{b})

(37)

where

Γ (s, x)

is the upper incomplete gamma function (see Appendix II).

Linear-Power Law

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0} e^{q} \sqrt{π}}{2 G_{i} \sqrt{q}} erfc (\sqrt{q}) & τ_{0} \leq τ_{i} \\ \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{2 b - 1}{2 b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} [Γ (\frac{2 b - 1}{2 b}, \frac{q}{b}) - Γ (\frac{2 b - 1}{2 b}, \frac{2 r_{i}}{d} \frac{q}{b})] + \frac{τ_{0} e^{q}}{2 G_{i} \sqrt{q}} Γ (\frac{1}{2}, q \frac{2 r_{i}}{d}) & τ_{0} \geq τ_{i} \end{cases}

(38a)

where

\frac{2 r_{i}}{d} = \frac{1}{2 q} W (2 q e^{2 q} {(\frac{2 τ_{0}}{τ_{m a x}})}^{2} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{2 b}{1 - b}})

(38b)

Ramberg-Osgood

\frac{u_{0}}{d} = \frac{τ_{0} γ_{r} e^{q}}{2 τ_{m a x} \sqrt{q}} Γ (\frac{1}{2}, q) + \frac{γ_{r}}{2} e^{c_{2} q} {(q c_{2})}^{\frac{c_{2} - 2}{2}} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} Γ (\frac{2 - c_{2}}{2}, q c_{2})

(39)

Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{m a x}}{R_{f} d G_{i}} \int_{d / 2}^{\infty} {\frac{τ_{m a x}}{R_{f} τ_{0}} \sqrt{\frac{2 r}{d}} \exp (q (\frac{2 r}{d} - 1)) - 1}^{- 1} d r

(40)

Modified Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{0}}{G_{i} d} \int_{d / 2}^{\infty} \sqrt{\frac{d}{2 r}} \exp (- q (\frac{2 r}{d} - 1)) {[1 - {(\frac{R_{f} τ_{0}}{τ_{\max}} \sqrt{\frac{d}{2 r}} \exp (- q (\frac{2 r}{d} - 1)))}^{c_{3}}]}^{- 1} d r

(41)

Exponential

\frac{u_{0}}{d} = - \frac{τ_{m a x}}{R_{f} G_{i} d} \int_{d / 2}^{\infty} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} \sqrt{\frac{d}{2 r}} \exp (- q (\frac{2 r}{d} - 1))) d r

(42)

Theoretical “

t - z

” curves derived using the generalized power-exponential decay [Eq. (10)] are given as follows and summarized in Appendix V.

Linear

\frac{u_{0}}{d} = \frac{τ_{0}}{2 G} e^{q} q^{n - 1} Γ (1 - n, q)

(43)

Bilinear

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{1}} e^{q} q^{n - 1} Γ (1 - n, q) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [e^{q} q^{n - 1} [\frac{G_{1}}{G_{1} - G_{2}} Γ (1 - n, q) - Γ (1 - n, q \frac{2 r_{1}}{d})] - \frac{τ_{1}}{τ_{0}} (\frac{2 r_{1}}{d} - 1)] & τ_{0} \geq τ_{1} \end{cases}

(44a)

where

\frac{2 r_{1}}{d} = \frac{n}{q} W (\frac{q}{n} e^{\frac{q}{n}} {(\frac{τ_{0}}{τ_{1}})}^{\frac{1}{n}})

(44b)

Power Law

\frac{u_{0}}{d} = \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{b - n}{b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ (\frac{b - n}{b}, \frac{q}{b})

(45)

Linear-Power Law

\frac{u_{0}}{d} = {\begin{cases} \frac{τ_{0}}{2 G_{i}} e^{q} q^{n - 1} Γ (1 - n, q) & τ_{0} \leq τ_{i} \\ \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{b - n}{b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} [Γ (\frac{b - n}{b}, \frac{q}{b}) - Γ (\frac{b - n}{b}, \frac{2 r_{i}}{d} \frac{q}{b})] + \frac{τ_{0}}{2 G_{i}} e^{q} q^{n - 1} Γ (1 - n, q \frac{2 r_{i}}{d}) & τ_{0} \geq τ_{i} \end{cases}

(46a)

\frac{2 r_{i}}{d} = \frac{n}{q} W (\frac{q}{n} e^{\frac{q}{n}} [\frac{2 τ_{0}}{τ_{m a x}} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{b}{1 - b}}]^{\frac{1}{n}})

(46b)

Ramberg-Osgood

\frac{u_{0}}{d} = \frac{τ_{0} γ_{r}}{2 τ_{m a x}} e^{q} q^{n - 1} Γ (1 - n, q) + \frac{γ_{r} e^{c_{2} q}}{2} {(q c_{2})}^{c_{2} n - 1} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} Γ (1 - c_{2} n, q c_{2})

(47)

Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{m a x}}{R_{f} d G_{i}} \int_{d / 2}^{\infty} {\frac{τ_{m a x}}{R_{f} τ_{0}} {(\frac{2 r}{d})}^{n} \exp [q (\frac{2 r}{d} - 1)] - 1}^{- 1} d r

(48)

Modified Hyperbolic

\frac{u_{0}}{d} = \frac{τ_{0}}{G_{i} d} \int_{d / 2}^{\infty} {(\frac{d}{2 r})}^{n} e^{- q (\frac{2 r}{d} - 1)} {[1 - {(\frac{R_{f} τ_{0}}{τ_{\max}} {(\frac{d}{2 r})}^{n} e^{- q (\frac{2 r}{d} - 1)})}^{c_{3}}]}^{- 1} d r

(49)

Exponential

\frac{u_{0}}{d} = - \frac{τ_{m a x}}{R_{f} G_{i} d} \int_{d / 2}^{\infty} \ln {1 - \frac{R_{f} τ_{0}}{τ_{m a x}} {(\frac{d}{2 r})}^{n} \exp [- q (\frac{2 r}{d} - 1)]} d r

(50)

The derived “

t - z

” curves are all presented in closed form with the exception of the hyperbolic, modified hyperbolic, and exponential constitutive models in Eqs. (8) and (10) [Eqs. (40), (41), (42), (48), (49), and (50)], for which explicit solutions were not found. The “

t - z

” curves [Eqs. (19)–(50)] are given normalized by the pile diameter,

d

, which indicates that the normalized settlement,

u_{0} / d

, is independent of the pile diameter for a given pile slenderness,

L / d

. The parameters dependent on the pile slenderness come from the attenuation functions (

r_{m}

,

m

,

q

,

n

). Therefore, for a given pile slenderness, the settlement at the pile–soil interface is proportional to the pile diameter independent of the constitutive model selected.

These “

t - z

” curves act as a toolbox, which a designer might easily employ to calculate the pile settlement (at any depth). This requires only the parameters for both the soil constitutive model and the attenuation functions to be determined. First, the soil constitutive parameters can be fit directly to a chosen stress range from experimental test data (e.g., a standard triaxial test from a representative soil sample), similar to how input parameters for a standard numerical analysis are determined. Second, the attenuation function can also be directly fitted to a numerical analysis or a more rigorous analytical solution accounting for material nonlinearity in the soil and the pile-soil interface [e.g., Akiyoshi (1982)], if available.

The “

t - z

” curves developed in this work can be used to establish the total head settlement. The pile is discretized, and a “

t - z

” curve is selected for each section (with fitted parameters as previously discussed). Various numerical integration techniques for two-point boundary-value problems are then available to determine the total head settlement, such as the Runge-Kutta methods. Alternatively, numerical analysis software packages [e.g., Oasys Pile, Oasys (2017)] often enable user-defined “

t - z

” curves to be inputted. Two illustrative examples involving end-bearing single piles are provided to demonstrate the simplicity of determining these parameters in practice.

Illustrative Examples

Two example scenarios, indicated in Fig. 5, are examined using two representative soils to determine “

t - z

” curves at the mid-depth of each example pile and demonstrate the application of the method described in this paper. These two examples are: (A) an end-bearing concrete pile embedded in Pisa clay, and (B) an end-bearing steel pile embedded in kaolinite. Note that Example B is treated as an equivalent solid pile with the same

E_{p} A

, where

A

is the cross-sectional area of the pile. The example uses end-bearing piles to only consider the response of the pile shaft (“

t - z

” curves) and remove the effect of the base response (a “

q - z

” curve), which is out of the scope of this paper. In practice, this can be repeated for multiple depths to obtain the total pile settlement.

Fig. 5. Examples of end-bearing piles in: Pisa clay (Example A) and kaolinite (Example B).

To analyze these examples, representative soil samples are selected using soil test data from Soga (1994) (note that this is an example analysis and not intended to provide site-specific design values or comparisons to load test data). Example A is represented by an undisturbed sample of a high plasticity Pisa clay taken from a depth of 10 m (

z / L = 0.5

). Example B is represented by a flocculated kaolinite which was remolded and subsequently tested at a confining pressure of 98 kPa to represent a sample at a depth of approximately 5 m (

z / L = 0.5

). Full details of the material tested can be found in Soga (1994). For both soils, Soga (1994) undertook an isotropic undrained triaxial compression test for a strain rate of 0.5% per min to obtain stress-strain results (

τ - γ

) for each material, plotted in Fig. 6. From this digitized data,

c_{u}

was interpreted as the maximum shear stress reached [

c_{u} = 45 kPa

for Pisa clay and

c_{u} = 29 kPa

for kaolinite; see Figs. 6(a and b), respectively].

Fig. 6. Chosen fitting (this work) of each constitutive model to undrained triaxial compression test data for (a) Pisa clay; and (b) kaolinite fitted over the range $0.2 c_{u} \leq τ \leq 0.8 c_{u}$ (fitted parameters given in Table 1). $N$ denotes number of data points considered in each range. (Data from Soga 1994.)

Parameters for each soil constitutive model [Eqs. (11)–(18)] are determined using a nonlinear least squares fit to the triaxial test data. In some cases, model parameters can be directly interpreted from the test results instead of being employed as a fitting parameter. In addition to the undrained triaxial tests, Soga (1994) undertook cyclic torsional shear tests at 0.5 Hz on the same samples that allow a direct interpretation of

G_{m a x}

[29 MPa for Pisa clay and 78 MPa for kaolinite; Soga (1994)]. Both combinations of fitted (i.e.,

G_{i}

as a free parameter) and interpreted (i.e.,

G_{i} = G_{m a x}

) stiffnesses were considered to ascertain which provides the best results for each model. It should be noted that to ensure a reasonable initial stiffness for the modified hyperbolic model,

G_{i} = G_{m a x}

is selected.

Moreover, a desired stress range of

0.2 c_{u} \leq τ \leq 0.8 c_{u}

was selected for both examples. Thus, only data points within this region were considered during the fitting stage to optimize the results. This stress range was chosen to better reflect the response at moderate strains and to limit the effect of the scatter at the onset of plastic behavior at the expense of accurately modeling the small strain response [this approach was also employed in Vardanega and Bolton (2011) when calibrating a power-law using collected soil stress-strain data]. Although the two examples in this paper are assessed within a specific stress range (

0.2 c_{u} \leq τ \leq 0.8 c_{u}

), an appropriate stress range should be selected that is specific to each problem. This should be decided based on the expected loading conditions of the pile, which might affect which soil constitutive model is chosen.

Since multiple fittings for each soil constitutive model are considered, a method to select the most appropriate fitted line is required. To assess each fitting, the measured shear stress from the triaxial test data can be compared with that predicted by each model and an error bound obtained based on a selected proportion of the data (95% is used here). The fitting with the smallest error bound is selected (or an alternative method can be applied). Note that this method does not bias the result toward conservative predictions. Full details and results of this process are given in the Supplemental Materials. A summary of selected parameters and error bounds for each constitutive model are given in Table 1 and plotted in Fig. 6.

Table 1. Parameters from the selected fitting for each soil constitutive model and error bounds (%) on Predicted versus Measured shear stress plots (Fig. 6) that 95% of data (within

0.2 c_{u} - 0.8 c_{u}

) fall between.

c_{u}

and

G_{m a x}

are interpreted values from Soga (1994). Corresponding figure numbers in Supplemental Materials are indicated in brackets next to soil constitutive model name

Soil type	Soil constitutive model	Parameters ( $τ_{m a x} = c_{u}$ )	Error bounds (%)
Pisa	Linear [Fig. S1(a)]	$G = G_{m a x}$	244
	Bilinear [Fig. S2(a)]	$G_{1} = 96.5 MPa$ , $G_{2} = 1.1 MPa$ , $τ_{1} = 12.6 kPa$	28
	Power law [Fig. S3(a)]	$γ_{50} = 0.0079$ , $b = 0.41$	2.6
	Linear-Power law [Fig. S4(a)]	$G_{i} = G_{m a x}$ , $γ_{50} = 0.0079$ , $b = 0.41$	2.6
	Ramberg-Osgood [Fig. S5(a)]	$γ_{r} = 0.00053$ , $c_{1} = 5.7$ , $c_{2} = 2.5$	2.3
	Hyperbolic [Fig. S6(a)]	$G_{i} = 7.6 MPa$ , $R_{f} = 1.12$	37
	Modified Hyperbolic [Fig. S7(a)]	$G_{i} = G_{m a x}$ , $R_{f} = 1$ , $c_{3} = 0.17$	22
	Exponential [Fig. S8(a)]	$G_{i} = 5.8 MPa$ , $R_{f} = 1.39$	63
Kaolinite	Linear [Fig. S1(b)]	$G = 6.4 MPa$	114
	Bilinear [Fig. S2(b)]	$G_{1} = 12.2 MPa$ , $G_{2} = 0.4 MPa$ , $τ_{1} = 15.4 kPa$	18
	Power law [Fig. S3(b)]	$γ_{50} = 0.0028$ , $b = 0.24$	36
	Linear-Power law [Fig. S4(b)]	$G_{i} = G_{m a x}$ , $γ_{50} = 0.0028$ , $b = 0.24$	36
	Ramberg-Osgood [Fig. S5(b)]	$γ_{r} = 0.0021$ , $c_{1} = 1.8$ , $c_{2} = 6.8$	2.6
	Hyperbolic [Fig. S6(b)]	$G_{i} = 20 MPa$ , $R_{f} = 1.26$	6.8
	Modified Hyperbolic [Fig. S7(b)]	$G_{i} = G_{m a x}$ , $R_{f} = 1.21$ , $c_{3} = 0.18$	8.6
	Exponential [Fig. S8(b)]	$G_{i} = 14.2 MPa$ , $R_{f} = 1.40$	17

For Pisa clay in Example A, the smallest error bound is observed with the Ramberg-Osgood model. However, the error bounds of both the power law model and the linear-power law model are only slightly larger (2.6% instead of 2.3%). In this case, the power law model has been chosen for Pisa clay because it provides similarly accurate results as the other two models and only requires two model parameters to be fitted.

For the kaolinite, the Ramberg-Osgood model also has the smallest error bound. In addition, the second best-performing model—the hyperbolic model—employs the fitting parameter,

R_{f}

, which has the effect of scaling

c_{u}

to provide a better fit within the desired stress range at the expense of an accurate fit at high stresses. This is evident from the kaolinite triaxial test data for which the model does not perform well beyond

0.8 c_{u}

. Although not the particular region of interest, the Ramberg-Osgood model also provides a reasonable fit in this range and is, thus, selected for the kaolinite.

If the small strain response is of interest in a problem, an experimentally determined

G_{m a x}

should be employed (e.g., in the linear-power law model). If the stress is unlikely to exceed the small strain region, the linear model might be suitable (Leung et al. 2010). In addition, these simplified constitutive models are likely to break down at higher stresses, particularly in softer clays, so more advanced soil models (with specific treatment of yielding and/or plastic behavior) may be more appropriate.

The parameters for the attenuation functions were then determined by fitting the simplified functions to a more rigorous analytical solution of Eq. (5) developed by Mylonakis (2001b) for perfectly end-bearing piles in linear-elastic material. This continuum solution allows for a closed-form, depth-dependent attenuation function to be derived in the form of trigonometric series (Appendix I), which is indicated to depend on depth

z / L

, pile slenderness

L / d

, and pile-soil stiffness ratio

E_{p} / E_{s}

. An

E_{s}

value is selected based on the fitted

G_{i}

of the chosen soil constitutive model (except for the power law model for which

G_{m a x}

must be selected) because much of the soil is under relatively low strain.

With this in mind, the parameters of the simplified attenuation functions in Eqs. (4), (8), (9), and (10) can be fitted. Both the concentric cylinder model [Eq. (4)] and the power-exponential decay [Eq. (8)] require only one fitting parameter,

r_{m}

and

q

, respectively, whereas two parameters are required for the generalized concentric cylinder model [Eq. (9)] (

m

,

r_{m}

), and the generalized power-exponential decay model [Eq. (10)]

(q, n)

.

These parameters, with the exception of

r_{m}

, are fitted to linearly spaced data points along the more rigorous attenuation function using the method of nonlinear least squares.

r_{m}

is calculated using the following expression (Randolph and Wroth 1978):

r_{m} \approx 2 L (1 - v_{s})

(51)

where

ν_{s}

= Poisson’s ratio of the soil, taken here as 0.5.

This expression is assumed to be an appropriate estimate for

r_{m}

for an end-bearing pile; although further work should be completed to verify its suitability in this method for both the concentric cylinder and the generalized concentric cylinder model. To assess each simplified attenuation function, the discrepancy (as a percentage difference) between the fitted functions and the rigorous solution is calculated (Appendix I). This discrepancy is obtained from the area under the “

t - z

” curve from the linear constitutive model [

\int_{0}^{τ_{m a x}} τ_{0} (u) d τ_{0}

, proportional to the energy stored by the Winkler spring] obtained with each fitted function compared with the rigorous solution. The fitted parameters and the calculated discrepancy are given in Table 2 for both example piles and plotted alongside the rigorous attenuation function in Fig. 7.

Table 2. Parameters from the fitting of each simplified attenuation function and the discrepancy compared with continuum solution by Mylonakis (2001b)

Example	Attenuation function	$z / L$	0.25	0.5	0.75	0.25	0.5	0.75
Example	Attenuation function		Parameter			Percentage discrepancy from rigorous solution
Example A ( $E_{p} / E_{s} = 287$ , $L / d = 50$ )	Concentric cylinder model [Eq. (4)]	$2 r_{m} / d$	100	100	100	18.3	2.4	5.7
	Generalized concentric cylinder model [Eq. (9)]	$m$	1.04	1.00	0.98	8.2	2.5	0.9
	Generalized concentric cylinder model [Eq. (9)]	$2 r_{m} / d$	100	100	100	8.2	2.5	0.9
	Power-exponential decay [Eq. (9)]	$q$	0.16	0.14	0.13	22.3	26.2	28.5
	Generalized power-exponential model [Eq. (10)]	$q$	0.02	0.01	0.01	2.2	4.6	5.8
	Generalized power-exponential model [Eq. (10)]	$n$	0.92	0.91	0.91	2.2	4.6	5.8
Example B ( $E_{p} / E_{s} = 96$ , $L / d = 10$ )	Concentric cylinder model [Eq. (4)]	$2 r_{m} / d$	20	20	20	36.2	22.3	16.5
	Generalized concentric cylinder model [Eq. (9)]	$m$	1.23	1.17	1.15	0.9	4.1	5.8
	Generalized concentric cylinder model [Eq. (9)]	$2 r_{m} / d$	20	20	20	0.9	4.1	5.8
	Power-exponential decay [Eq. (8)]	$q$	0.27	0.22	0.21	4.9	3.7	3.3
	Generalized power- exponential decay [Eq. (10)]	$q$	0.15	0.12	0.11	0.2	1.7	2.4
	Generalized power- exponential decay [Eq. (10)]	$n$	0.78	0.76	0.77	0.2	1.7	2.4

Fig. 7. Fitting of attenuation functions at depth $z / L = 0.5$ of (a) Example A; and (b) Example B (fitted parameters given in Table 2).

Due to the two fitting parameters, the generalized power-exponential decay model [Eq. (10)] is observed to fit best in most situations. However, the simpler concentric cylinder model [Eq. (4)] provides a better fit for Example A at mid-depth. The generalized concentric cylinder model [Eq. (9)] provides a very similar result as

m \approx 1

, reducing this equation to the concentric cylinder model. For Example B, the power-exponential decay model [Eq. (8)] performs similarly to the generalized power-exponential decay [Eq. (10)], while requiring only one parameter. The results in Fig. 7 indicate that the power-exponential decay model [Eq. (8)] might be a better fit for compressible piles (Example B, low

E_{p} / E_{s}

), whereas the concentric cylinder model [Eq. (4)] might be a better fit for stiff piles (Example A, high

E_{p} / E_{s}

).

Taking the parameters from the fitting of each soil constitutive model and the simplified attenuation functions (Tables 1 and 2), “

t - z

” curves can be generated using Eqs. (19)–(50). These are plotted in Fig. 8 for both example piles and have been compared with numerical “

t - z

” curves obtained from PLAXIS 2D for the linear and hyperbolic constitutive models, as discussed for the similar comparison in Fig. 3. The error in the “

t - z

” approach is obtained by comparing the measured settlement from the FEM results to those predicted using the “

t - z

” approach and calculating the error bound within which 95% of the data points lie. Considering the stress range

0.2 c_{u} \leq τ \leq 0.8 c_{u}

(as employed for fitting the constitutive models), the linear and hyperbolic constitutive models resulted in error bounds of 21% and 15%, respectively, for Example A and 4.0% and 3.8%, respectively, for Example B. Note that all of these error bounds are smaller than those obtained from fitting the constitutive models [Eqs. (11)–(18)], indicating that the additional error introduced by the “

t - z

” assumption is very small than compared with the assumptions in soil parameters for modeling both a single element and the spatial distribution.

Fig. 8. “ $t - z$ ” curves for each illustrative example (parameters from Tables 1 and 2): (a) linear; (b) bilinear; (c) power law; (d) linear-power law; (e) Ramberg-Osgood; (f) exponential; (g) hyperbolic; (h) modified hyperbolic.

In addition, Fig. 8 indicates that a strong variation exists in the normalized settlement (

u_{0} / d

) at one-half of the ultimate load, ranging approximately between 0.1% and 1%. Ultimate loads are attained at settlements between 0.25% to more than 2%. These results are problem-specific and should not be viewed as design recommendations.

Summary and Conclusions

A theoretical method for deriving “

t - z

” curves, employed to predict nonlinear pile settlement, has been presented. These curves should be used in conjunction with a pressure-displacement relation at the pile tip (“

q - z

” curve) to model the axial response of floating piles. The “

t - z

” curves were obtained by combining an attenuation function

τ

=

τ (r)

and a soil constitutive model

γ

=

γ (τ)

and integrating over the radial coordinate. The curves can be employed through hand or spreadsheet calculations or in available commercial software such as Oasys Pile [Oasys (2017)] or Ensoft TZPile [Reese et al. (2014)]. Use of these functions enables simple estimations of foundation performance without reliance on numerical continuum methods in multiple dimensions, which are complex, time consuming, and often unavailable for real soils. In addition, the method provides a much better physical insight into the mechanics of the problem. Nevertheless, despite the elegance and practical appeal of the method at hand, only a handful of such “

t - z

” curves are available to date—hence, the motivation for this study.

To this end:

1.

Three novel stress attenuation functions with radial distance

r

[Eqs. (8), (9), and (10)] were developed, inspired by an available continuum solution by the corresponding author [Mylonakis (2001a)].

2.

The new attenuation functions, in addition to the well-known concentric cylinder model (Eq. 4), were employed in conjunction with eight soil constitutive models [Eqs. (11)–(18)] to derive 32 theoretical “

t - z

” curves [Eqs. (19)–(50)], 28 of which are, to the authors’ knowledge, novel, and 26 are presented in closed form.

3.

In principle, the generalized power-exponential decay model [Eq. (10)] can describe all other attenuation functions by using appropriate values for parameters

n

and

q

. However, the corresponding solutions in Eqs. (43)–(50) cannot describe all “

t - z

” curves derived in this paper; these do not employ the empirical radius

r_{m}

required due to the singular nature of the power law models. Moreover, simpler solutions are obtained with the power law attenuation functions, whereas three of the constitutive models are not integrable using Eq. (10).

4.

Two illustrative example problems involving end-bearing piles have been provided to demonstrate the process of employing these “

t - z

” curves by selecting the relevant model parameters. To this end, the attenuation functions were fitted to a more rigorous continuum solution for two pile configurations, and the soil constitutive models were fitted to triaxial test data for two materials. The most appropriate attenuation function and constitutive model for each example was selected, which enabled the most suitable “

t - z

” curves to be calculated. All 32 “

t - z

” curves were plotted using the fitted parameters for both examples (Fig. 8).

5.

The results from the illustrative examples indicate that the power-exponential decay model in Eq. (8) might be a better fit for compressible piles (low

E_{p} / E_{s}

), whereas the concentric cylinder model in Eq. (4) might be a better fit for stiff piles (high

E_{p} / E_{s}

).

6.

A strong variation exists in the settlement at 50% of the ultimate load among the various curves indicated in Fig. 8, ranging between approximately 0.1% and 1% of pile diameter

d

. Ultimate shaft loads are attained at values varying from 0.25% to more than 2% of

d

. These results are clearly problem specific and should not be viewed as universal.

The “

t - z

” curves derived in this paper can be used to produce design-orientated stress-displacement curves informed by experimental results for specific pile configurations. It should be noted that the range of parameters obtained from the illustrative example presented in this paper should not be viewed as generic design recommendations but are provided as an example of obtaining parameters for the method shown. Additionally, the method could be experimentally verified by comparing against “

t - z

” curves from in situ pile tests. However, for a specific site, the ideal “

t - z

” curve depends on the ground conditions, the pile properties, and the installation method. A systematic comparison, including relevant statistics, against a database of field data [e.g., the DINGO database; Vardanega et al. (2021); Voyagaki et al. (2021)] is a formidable task and lies beyond the scope of this work.

As a final remark, it is fair to mention that no attempt has been made here to identify the most suitable element test for fitting a theoretical

τ - γ

curve to experimental data, especially the shear strength parameter

τ_{m a x}

(or

c_{u}

). As pile loading induces rotations of principal stresses in the surrounding soil, no elementary shear test can faithfully reproduce the pertinent stress state. Evidently, the fitting process requires judgment and should be conducted on a case-by-case basis.

Notation

The following symbols are used in this paper:

$B_{m}$: integration constant;
$b$: soil nonlinearity exponent;
$c_{u}$: soil undrained shear strength;
$c_{1}$ , $c_{2}$ , $c_{3}$: fitting constants from constitutive models;
$d$: pile diameter;
$E_{p}$ , $E_{s}$: stiffness of pile and soil, respectively;
$F (z)$: distributed axial force in pile;
$G$ , $G_{i}$: linear shear modulus, initial (low strain) shear modulus in constitutive models;
$G_{s}$ , $G_{m a x}$: soil shear modulus, measured maximum (low strain) soil shear modulus;
$G_{1}$ , $G_{2}$: low-strain shear modulus and reduced shear modulus in bilinear constitutive model;
$K_{v}$: modified Bessel function of second kind with order $v$ ;
$k (τ_{0})$: nonlinear secant Winkler modulus;
$L$: pile length;
$M_{s}$: soil compressibility modulus;
$m$: stress attenuation fitting parameter;
$n$: stress attenuation fitting parameter;
N: number of data points;
$P$: axial force acting on pile head;
$q$: shear attenuation fitting parameter;
$R_{f}$: fitting constant to scale $τ_{m a x}$ in constitutive models;
$r$ , $r_{i}$ , $r_{1}$: radial distance from pile centerline, radius at which $τ_{i}$ occurs, radius at which $τ_{1}$ occurs;
$r_{m}$: empirical radius beyond which vertical settlement is assumed negligible;
$u$ , $u_{0}$: vertical displacement, vertical displacement at pile-soil interface, respectively;
$x$: real variable;
$z$: depth below ground level;
$α$: empirical adhesion factor;
$α_{m}$: eigenvalue parameter;
$γ$ , $γ_{r}$ , $γ_{50}$: soil shear strain, reference shear strain, strain when ½ the peak shear stress is mobilized, respectively;
$η$: compressibility coefficient;
$v_{s}$: Poisson’s ratio of soil;
$σ$: vertical normal stress acting on soil slice;
$τ$ , $τ_{0}$: shear stress, shear stress at pile-soil interface, respectively;
$τ_{i}$ , $τ_{1}$: shear stress at change in stiffness in linear-power law and bilinear models, respectively; and
$τ_{m a x}$: shear strength of soil.

Supplemental Materials

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Appendix I. Continuum Solution for Attenuation of Shear Stress

A more rigorous solution to Eq. (6) can be derived by introducing the separation of variables and applying the boundary conditions of zero displacement at large radial distances, zero vertical stress at the surface, and zero displacement at depth

z = L

, which corresponds to a perfectly end-bearing pile. Thus, the vertical displacements in the soil medium can be written as an infinite trigonometric series [Mylonakis (2001b)]

u (r, z) = \sum_{m = 0}^{\infty} B_{m} K_{0} (α_{m} η r) \cos (α_{m} z)

(52)

where

B_{m}

= coefficient to be determined;

K_{0}

= modified Bessel function of the second kind with order 0; and

α_{m}

is obtained by imposing the requirement of zero displacement at the base of the pile, as follows:

\cos (α_{m} L) = 0, α_{m} = \frac{π}{2 L} (2 m + 1), m = 0, 1 \dots

(53)

An attenuation function is obtained by substituting Eq. (52) into the stress-strain relation

τ ≅ - G_{s} \partial u / \partial r

τ (r) ≅ {- G}_{s} \frac{\partial u}{\partial r} = - G_{s} \sum_{m = 0}^{\infty} α_{m} η B_{m} K_{1} (α_{m} η r) \cos (α_{m} z)

(54)

Considering vertical equilibrium of the pile, assuming both a linear-elastic pile material and perfect bonding at the pile/soil interface yields the following boundary condition:

- E_{p} (\frac{π}{4} d^{2}) \frac{\partial^{2} u (d / 2, z)}{\partial z^{2}} = π d τ (d / 2, z) + F (z)

(55)

where

F (z)

= distributed axial force in the pile, assumed to be uniformly distributed within the pile cross-section. This is determined by resolving the force at the pile head into equivalent body forces using a cosine series

F (z) = \sum_{m = 0}^{\infty} \frac{2 P}{L} \cos (α_{m} z)

(56)

where

P

= the axial force at the pile head. Eqs. (52), (54), and (56) can be substituted into Eq. (55) and the cosine transformation applied to obtain a solution for

B_{m}

B_{m} = \frac{2 P}{π L d η G_{s}} {[\frac{d}{4 η} \frac{E_{p}}{G_{s}} α_{m}^{2} K_{0} (\frac{α_{m} η d}{2}) + α_{m} K_{1} (\frac{α_{m} η d}{2})]}^{- 1}

(57)

Substituting this back into Eq. (54) yields an analytical attenuation function for perfectly end bearing piles

\frac{τ (r)}{τ_{0}} = \frac{τ_{r z} (r, z)}{τ_{r z} (d / 2, z)} = \sum_{m = 0}^{\infty} \frac{\frac{K_{1} (α_{m} η r)}{K_{1} (α_{m} η d / 2)} \cos (α_{m} z)}{α_{m} (\frac{d}{4 η}) (\frac{E_{p}}{G_{s}}) \frac{K_{0} (α_{m} η d / 2)}{K_{1} (α_{m} η d / 2)} + 1} / \sum_{m = 0}^{\infty} \frac{\cos (α_{m} z)}{α_{m} (\frac{d}{4 η}) (\frac{E_{p}}{G_{s}}) \frac{K_{0} (α_{m} η d / 2)}{K_{1} (α_{m} η d / 2)} + 1}

(58)

To select a value of

η

, specific assumptions regarding the radial and tangential soil stresses and displacements must be chosen, which are discussed in Mylonakis (2001b). The assumption of zero radial stresses and zero tangential strains in the soil medium requires

η^{2} = \frac{2}{1 - ν_{s}}

(59)

Appendix II. Special Functions

The lesser known mathematical functions used in this study are detailed as follows (Olver et al. 2010). First, the upper incomplete gamma function

Γ (s, x) = \int_{x}^{\infty} e^{- t} t^{s - 1} d t

(60)

This equation simplifies to the ordinary gamma function when x = 0. In addition, if

s = 1 / 2

, this can be simplified using the complementary error function,

e r f c (x)

, which is related to the standard error function,

e r f (x)

as indicated

Γ (\frac{1}{2}, x) = \sqrt{π} erfc (\sqrt{x}) = \sqrt{π} [1 - \erf (x)]

(61)

The incomplete beta function is defined as

B (x; a, b) = \int_{0}^{x} t^{a - 1} {(1 - t)}^{b - 1} d t

(62)

Variations of the functions previously discussed are available in common spreadsheet software (although often in the form of statistical distribution functions).

Finally, the lambert W function is defined as the solution to the equation

W e^{W} = x

(63)

For the application in this paper,

x \geq 0

, the principle branch is employed. This function is available in various numerical computing packages.

Appendix III. Soil constitutive models considered ( $τ \leq τ_{\max}$ ).

Constitutive model	Eq. Number	Number of parameters	Stiffness form, $τ$	Flexibility form, $γ$
Linear	Eq. (11)	2	$γ G$	$\frac{τ}{G}$
Bilinear	Eq. (12)	4	${\begin{cases} γ G_{1} & γ \leq τ_{1} / G_{1} \\ G_{2} (γ - \frac{τ_{1}}{G_{1}}) + τ_{1} & γ \geq τ_{1} / G_{1} \end{cases}$	${\begin{cases} \frac{τ}{G_{1}} & τ \leq τ_{1} \\ \frac{τ - τ_{1}}{G_{2}} + \frac{τ_{1}}{G_{1}} & τ \geq τ_{1} \end{cases}$
Power Law	Eq. (13)	3	$\frac{τ_{\max}}{2} {(\frac{γ}{γ_{50}})}^{b}$	$γ_{50} {(\frac{2 τ}{τ_{\max}})}^{\frac{1}{b}}$
Linear-Power Law	Eq. (14)	4	${\begin{cases} γ G_{i} & γ \leq τ_{i} / G_{i} \\ \frac{τ_{\max}}{2} {(\frac{γ}{γ_{50}})}^{b} & γ \geq τ_{i} / G_{i} \end{cases}$	${\begin{cases} \frac{τ}{G_{i}} & τ \leq τ_{i} \\ γ_{50} {(\frac{2 τ}{τ_{\max}})}^{\frac{1}{b}} & τ \geq τ_{i} \end{cases}$
Ramberg-Osgood	Eq. (15)	4	—	$γ_{r} [\frac{τ}{τ_{\max}} + {(c_{1} \frac{τ}{τ_{\max}})}^{c_{2}}]$
Hyperbolic	Eq. (16)	3	$\frac{τ_{\max}}{R_{f}} {[\frac{τ_{\max}}{R_{f} G_{i} γ} + 1]}^{- 1}$	$\frac{τ_{\max}}{R_{f} G_{i}} {[\frac{τ_{\max}}{R_{f} τ} - 1]}^{- 1}$
Modified Hyperbolic	Eq. (17)	4	—	$\frac{τ}{G_{i}} {[1 - {(\frac{R_{f} τ}{τ_{\max}})}^{c_{3}}]}^{- 1}$
Exponential	Eq. (18)	3	$\frac{τ_{\max}}{R_{f}} [1 - \exp (- γ (\frac{R_{f} G_{i}}{τ_{\max}}))]$	$- \frac{τ_{\max}}{R_{f} G_{i}} \ln (1 - \frac{R_{f} τ}{τ_{\max}})$

Note: [ $τ_{i} = (τ_{m a x} / 2) {(2 G_{i} γ_{50} / τ_{m a x})}^{b / (b - 1)}$ ] [Eq. (14c)].

Appendix IV. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (4) and (9) ( $τ \leq τ_{\max}$ )

Soil constitutive model	Concentric cylinder model [Eq. (4)]^a $[τ = τ_{0} (d / 2 r)]$	Generalized concentric cylinder model [Eq. (9)]^b $[τ = τ_{0} (d / 2 r)^{m}]$
Linear [Eq. (11)]	$\frac{τ_{0}}{2 G} \ln (\frac{2 r_{m}}{d})$	$\frac{τ_{0}}{2 G (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}]$
Bilinear [Eq. (12)]	${\begin{cases} \frac{τ_{0}}{2 G_{1}} \ln (\frac{2 r_{m}}{d}) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [\frac{G_{2}}{G_{1} - G_{2}} \ln (\frac{2 r_{m}}{d} \frac{τ_{1}}{τ_{0}}) + \frac{G_{1}}{G_{1} - G_{2}} \ln (\frac{τ_{0}}{τ_{1}}) + (\frac{τ_{1}}{τ_{0}} - 1)] & τ_{0} \geq τ_{1} \end{cases}$	${\begin{cases} \frac{τ_{0}}{2 G_{1} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [\frac{1}{m - 1} [\frac{G_{1}}{G_{1} - G_{2}} - \frac{G_{2}}{G_{1} - G_{2}} {(\frac{2 r_{m}}{d})}^{1 - m} - {(\frac{τ_{0}}{τ_{1}})}^{\frac{1 - m}{m}}] + (\frac{τ_{1}}{τ_{0}} - {(\frac{τ_{0}}{τ_{1}})}^{\frac{1 - m}{m}})] & τ_{0} \geq τ_{1} \end{cases}$
Power Law [(Eq. (13)]	$\frac{γ_{50} b}{2 (1 - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}}$ where $b < 1$	$\frac{γ_{50} b}{2 (m - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}}$ where $b < m$
Linear-Power Law [Eq. (14)]	${\begin{cases} \frac{τ_{0}}{2 G_{i}} \ln (\frac{2 r_{m}}{d}) & τ_{0} \leq τ_{i} \\ \frac{γ_{50} b}{2 (1 - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} [1 - {(\frac{2 r_{i}}{d})}^{\frac{b - 1}{b}}] + \frac{τ_{0}}{2 G_{i}} \ln (\frac{r_{m}}{r_{i}}) & τ_{0} \geq τ_{i} \end{cases}$	${\begin{cases} \frac{τ_{0}}{2 G_{i} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \leq τ_{i} \\ \frac{γ_{50} b}{2 (m - b)} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} [1 - {(\frac{2 r_{i}}{d})}^{\frac{b - m}{b}}] + \frac{τ_{0}}{2 G_{i} (m - 1)} [{(\frac{2 r_{i}}{d})}^{1 - m} - {(\frac{2 r_{m}}{d})}^{1 - m}] & τ_{0} \geq τ_{i} \end{cases}$
Linear-Power Law [Eq. (14)]	where $\frac{2 r_{i}}{d} = \frac{2 τ_{0}}{τ_{m a x}} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{b}{1 - b}}$	where $\frac{2 r_{i}}{d} = {[\frac{2 τ_{0}}{τ_{m a x}} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{b}{1 - b}}]}^{1 / m}$
Ramberg-Osgood [Eq. (15)]	$\frac{τ_{0} γ_{r}}{2 τ_{m a x}} \ln (\frac{2 r_{m}}{d}) + \frac{γ_{r}}{2 (c_{2} - 1)} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} [1 - {(\frac{2 r_{m}}{d})}^{1 - c_{2}}]$	$\frac{τ_{0} γ_{r}}{2 τ_{m a x} (m - 1)} [1 - {(\frac{2 r_{m}}{d})}^{1 - m}] + \frac{γ_{r}}{2 (c_{2} m - 1)} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} [1 - {(\frac{2 r_{m}}{d})}^{1 - c_{2} m}]$
Hyperbolic [Eq. (16)]	$\frac{τ_{0}}{2 G_{i}} [\ln (\frac{2 r_{m}}{d} - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}})]$	$\frac{τ_{m a x}}{2 R_{f} m G_{i}} {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{\frac{1}{m}} {B (\frac{τ_{\max}}{R_{f} τ_{0}} {(\frac{2 r_{m}}{d})}^{m}; \frac{1}{m}, 0) - B (\frac{τ_{\max}}{R_{f} τ_{0}}; \frac{1}{m}, 0)}$
Modified hyperbolic [Eq. (17)]	$\frac{τ_{0}}{2 G_{i} c_{3}} [\ln ({(\frac{2 r_{m}}{d})}^{c_{3}} - {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{c_{3}}) - \ln (1 - {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{c_{3}})]$	$\frac{τ_{m a x}}{2 R_{f} G_{i} c_{3} m} {(\frac{R_{f} τ_{0}}{τ_{\max}})}^{\frac{1}{m}} [B ({(\frac{R_{f} τ_{0}}{τ_{\max}})}^{c_{3}}; \frac{m - 1}{c_{3} m}, 0) - B ({(\frac{R_{f} τ_{0}}{τ_{\max}} {(\frac{d}{2 r_{m}})}^{m})}^{c_{3}}; \frac{m - 1}{c_{3} m}, 0)]$
Exponential [Eq. (18)]	$\frac{τ_{m a x}}{2 R_{f} G_{i}} [\ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \frac{2 r_{m}}{d} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} (\frac{d}{2 r_{m}})) + \frac{R_{f} τ_{0}}{τ_{m a x}} [\ln (\frac{2 r_{m}}{d} - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}})]]$	$\frac{τ_{m a x}}{2 R_{f} G_{i}} [\ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}}) - \frac{2 r_{m}}{d} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} {(\frac{d}{2 r_{m}})}^{m}) + {(\frac{R_{f} τ_{0}}{τ_{m a x}})}^{\frac{1}{m}} [B (\frac{τ_{m a x}}{R_{f} τ_{0}}; \frac{1}{m}, 0) - B (\frac{τ_{m a x}}{R_{f} τ_{0}} {(\frac{2 r_{m}}{d})}^{m}; \frac{1}{m}, 0)]]$

Note: Constitutive model parameters are given in Eqs. (11)–(18); $r_{m}$ = empirical radius [Randolph and Wroth (1978)]; and $B (x; a, b)$ = incomplete Beta function (see Appendix II).

a

“

t - z

” curves available in the literature with the concentric cylinder model [Linear: Randolph and Wroth (1978), Power Law: Vardanega et al. (2012); Williamson (2014); Vardanega (2015); Crispin et al. (2019); Hyperbolic: Kraft et al. (1981); Modified Hyperbolic: Chang and Zhu (1998)].

b

For

m > 1

, all terms

{(2 r_{m} / d)}^{1 - m}

in the generalized concentric cylinder model (right column) vanish when

r_{m}

approaches infinity, which simplifies the solutions.

Appendix V. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (8) and (10) ( $τ \leq τ_{\max}$ )

Soil constitutive model	Power-exponential decay [Eq. (8)] $[τ = τ_{0} (2 r / d)^{- 1 / 2} \exp (- q (2 r / d - 1))]$	Generalized power-exponential decay [Eq. (10)] $[τ = τ_{0} (2 r / d)^{- n} \exp (- q (2 r / d - 1))]$
Linear [Eq. (11)]	$\frac{τ_{0} e^{q} \sqrt{π}}{2 G \sqrt{q}} erfc (\sqrt{q})$	$\frac{τ_{0}}{2 G} e^{q} q^{n - 1} Γ (1 - n, q)$
Bilinear [Eq. (12)]	${\begin{cases} \frac{τ_{0} e^{q} \sqrt{π}}{2 G_{1} \sqrt{q}} erfc (\sqrt{q}) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [\frac{e^{q} \sqrt{π}}{\sqrt{q}} X_{1} - \frac{τ_{1}}{τ_{0}} (\frac{2 r_{1}}{d} - 1)] & τ_{0} \geq τ_{1} \end{cases}$	${\begin{cases} \frac{τ_{0}}{2 G_{1}} e^{q} q^{n - 1} Γ (1 - n, q) & τ_{0} \leq τ_{1} \\ \frac{τ_{0} (G_{1} - G_{2})}{2 G_{1} G_{2}} [e^{q} q^{n - 1} X_{1} - \frac{τ_{1}}{τ_{0}} (\frac{2 r_{1}}{d} - 1)] & τ \geq τ_{1} \end{cases}$
	where $X_{1} = \frac{G_{1}}{G_{1} - G_{2}} erfc (\sqrt{q}) - erfc (\sqrt{\frac{2 q r_{1}}{d}})$	where $X_{1} = \frac{G_{1}}{G_{1} - G_{2}} Γ (1 - n, q) - Γ (1 - n, q \frac{2 r_{1}}{d})$
	$\frac{2 r_{1}}{d} = \frac{1}{2 q} W (2 q e^{2 q} {(\frac{τ_{0}}{τ_{1}})}^{2})$	$\frac{2 r_{1}}{d} = \frac{n}{q} W (\frac{q}{n} e^{\frac{q}{n}} {(\frac{τ_{0}}{τ_{1}})}^{\frac{1}{n}})$
Power Law [Eq. (13)]	$\frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{2 b - 1}{2 b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ (\frac{2 b - 1}{2 b}, \frac{q}{b})$	$\frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{b - n}{b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ (\frac{b - n}{b}, \frac{q}{b})$
Linear-Power Law [Eq. (14)]	${\begin{cases} \frac{τ_{0} e^{q} \sqrt{π}}{2 G_{i} \sqrt{q}} erfc (\sqrt{q}) & τ_{0} \leq τ_{i} \\ \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{2 b - 1}{2 b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ_{g} + \frac{τ_{0} e^{q}}{2 G_{i} \sqrt{q}} Γ (\frac{1}{2}, q \frac{2 r_{i}}{d}) & τ_{0} \geq τ_{i} \end{cases}$	${\begin{cases} \frac{τ_{0}}{2 G_{i}} e^{q} q^{n - 1} Γ (1 - n, q) & τ \leq τ_{i} \\ \frac{γ_{50}}{2} e^{\frac{q}{b}} {(\frac{b}{q})}^{\frac{b - n}{b}} {(\frac{2 τ_{0}}{τ_{m a x}})}^{\frac{1}{b}} Γ_{g} + \frac{τ_{0}}{2 G_{i}} e^{q} q^{n - 1} Γ (1 - n, q \frac{2 r_{i}}{d}) & τ \geq τ_{i} \end{cases}$
	where $Γ_{g} = Γ (\frac{2 b - 1}{2 b}, \frac{q}{b}) - Γ (\frac{2 b - 1}{2 b}, \frac{2 r_{i}}{d} \frac{q}{b})$	where $Γ_{g} = Γ (\frac{b - n}{b}, \frac{q}{b}) - Γ (\frac{b - n}{b}, \frac{2 r_{i}}{d} \frac{q}{b})$
	$\frac{2 r_{i}}{d} = \frac{1}{2 q} W (2 q e^{2 q} {(\frac{2 τ_{0}}{τ_{m a x}})}^{2} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{2 b}{1 - b}})$	$\frac{2 r_{i}}{d} = \frac{n}{q} W (\frac{q}{n} e^{\frac{q}{n}} {(\frac{2 τ_{0}}{τ_{m a x}} {(\frac{2 γ_{50} G_{i}}{τ_{m a x}})}^{\frac{b}{1 - b}})}^{\frac{1}{n}})$
Ramberg-Osgood [Eq. (15)]	$\frac{τ_{0} γ_{r} e^{q}}{2 τ_{m a x} \sqrt{q}} Γ (\frac{1}{2}, q) + \frac{γ_{r}}{2} e^{c_{2} q} {(q c_{2})}^{\frac{c_{2} - 2}{2}} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} Γ (\frac{2 - c_{2}}{2}, q c_{2})$	$\frac{τ_{0} γ_{r}}{2 τ_{m a x}} e^{q} q^{n - 1} Γ (1 - n, q) + \frac{γ_{r} e^{c_{2} q}}{2} {(q c_{2})}^{c_{2} n - 1} {(\frac{c_{1} τ_{0}}{τ_{m a x}})}^{c_{2}} Γ (1 - c_{2} n, q c_{2})$
Hyperbolic [Eq. (16)]	$\frac{τ_{m a x}}{R_{f} d G_{i}} \int_{d / 2}^{\infty} {[\frac{τ_{m a x}}{R_{f} τ_{0}} \sqrt{\frac{2 r}{d}} \exp (q (\frac{2 r}{d} - 1)) - 1]}^{- 1} d r$	$\frac{τ_{m a x}}{R_{f} d G_{i}} \int_{d / 2}^{\infty} {[\frac{τ_{m a x}}{R_{f} τ_{0}} {(\frac{2 r}{d})}^{n} \exp (q (\frac{2 r}{d} - 1)) - 1]}^{- 1} d r$
Modified hyperbolic [Eq. (17)]	$\frac{τ_{0}}{G_{i} d} \int_{d / 2}^{\infty} \frac{\sqrt{\frac{d}{2 r}} \exp (- q (\frac{2 r}{d} - 1))}{1 - [\frac{R_{f} τ_{0}}{τ_{m a x}} \sqrt{\frac{d}{2 r}} {\exp (- q (\frac{2 r}{d} - 1))]}^{c_{3}}} d r$	$\frac{τ_{0}}{G_{i} d} \int_{d / 2}^{\infty} \frac{{(\frac{d}{2 r})}^{n} \exp (- q (\frac{2 r}{d} - 1))}{1 - {[\frac{R_{f} τ_{0}}{τ_{m a x}} {(\frac{d}{2 r})}^{n} \exp (- q (\frac{2 r}{d} - 1))]}^{c_{3}}} d r$
Exponential [Eq. (18)]	$- \frac{τ_{m a x}}{R_{f} G_{i} d} \int_{d / 2}^{\infty} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} \sqrt{\frac{d}{2 r}} \exp (- q (\frac{2 r}{d} - 1))) d r$	$- \frac{τ_{m a x}}{R_{f} G_{i} d} \int_{d / 2}^{\infty} \ln (1 - \frac{R_{f} τ_{0}}{τ_{m a x}} {(\frac{d}{2 r})}^{n} \exp (- q (\frac{2 r}{d} - 1))) d r$

Note: Constitutive model parameters are given in Eqs. (11)–(18); $e r f c (x)$ = complementary error function; $Γ (s, x)$ = upper incomplete gamma function; and $W (x)$ = Lambert W-Function (see Appendix II). $X_{1}$ and $Γ_{g}$ are parameters defined for convenience.

Data Availability Statement

All data and models generated or used during the study appear in the published article. All of the code that supports the findings of this study is available from the corresponding author on reasonable request.

Acknowledgments

The first and second authors would like to thank the Engineering and Physical Sciences Research Council for their support (Grant Nos. EP/T517872/1 and EP/N509619/1, respectively). Partial funding was received by EU/H2020 under grant agreement number 730900 (SERA) with George Mylonakis as Principal Investigator for University of Bristol.

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Journal of Geotechnical and Geoenvironmental Engineering

Volume 148 • Issue 7 • July 2022

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

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Received: Apr 28, 2021

Accepted: Nov 5, 2021

Published online: May 6, 2022

Published in print: Jul 1, 2022

Discussion open until: Oct 6, 2022

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Abigail H. Bateman https://orcid.org/0000-0003-3454-1756

Ph.D. Student, Dept. of Civil Engineering, Univ. of Bristol, Bristol BS8 1TR, UK. ORCID: https://orcid.org/0000-0003-3454-1756

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Jamie J. Crispin, Ph.D. https://orcid.org/0000-0003-3074-8493

Formerly, Ph.D. Student, Dept. of Civil Engineering, Univ. of Bristol, Bristol BS8 1TR, UK. ORCID: https://orcid.org/0000-0003-3074-8493

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Paul J. Vardanega, Ph.D., M.ASCE https://orcid.org/0000-0001-7177-7851

Associate Professor in Civil Engineering, Dept. of Civil Engineering, Univ. of Bristol, Bristol BS8 1TR, UK. ORCID: https://orcid.org/0000-0001-7177-7851

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George E. Mylonakis, Ph.D., M.ASCE https://orcid.org/0000-0002-8455-8946 [email protected]

P.E.

Professor, Dept. of Civil Infrastructure and Environmental Engineering, Khalifa Univ., Abu Dhabi, United Arab Emirates; University Chair in Geotechnics and Soil-Structure Interaction, Dept. of Civil Engineering, Univ. of Bristol, Bristol BS8 1TR, UK; Adjunct Professor, Dept. of Civil and Environmental Engineering, Univ. of California Los Angeles, CA 90024 (corresponding author). ORCID: https://orcid.org/0000-0002-8455-8946. Email: [email protected]

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Elia Voyagaki, Jamie J. Crispin, Charlotte E. L. Gilder, Konstantina Ntassiou, Nick O’Riordan, Paul Nowak, Tarek Sadek, Dinesh Patel, George Mylonakis, Paul J. Vardanega, The DINGO database of axial pile load tests for the UK: settlement prediction in fine-grained soils, Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 10.1080/17499518.2021.1971249, 16, 4, (640-661), (2021).
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Theoretical $t - z$ Curves for Axially Loaded Piles

Abstract

Introduction

Horizontal Soil Slice Model

Attenuation Functions

Soil Constitutive Models

“ $t - z$ ” Curves

Illustrative Examples

Summary and Conclusions

Notation

Supplemental Materials

Appendix I. Continuum Solution for Attenuation of Shear Stress

Appendix II. Special Functions

Appendix III. Soil constitutive models considered ( $τ \leq τ_{\max}$ ).

Appendix IV. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (4) and (9) ( $τ \leq τ_{\max}$ )

Appendix V. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (8) and (10) ( $τ \leq τ_{\max}$ )

Data Availability Statement

Acknowledgments

References

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Abstract

Introduction

Horizontal Soil Slice Model

Attenuation Functions

Soil Constitutive Models

“t-z” Curves

Illustrative Examples

Summary and Conclusions

Notation

Supplemental Materials

Appendix I. Continuum Solution for Attenuation of Shear Stress

Appendix II. Special Functions

Appendix III. Soil constitutive models considered (τ≤τmax).

Appendix IV. Normalized Displacement, u0/d, for Each Constitutive Model Obtained Using Eqs. (4) and (9) (τ≤τmax)

Appendix V. Normalized Displacement, u0/d, for Each Constitutive Model Obtained Using Eqs. (8) and (10) (τ≤τmax)

Data Availability Statement

Acknowledgments

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“ $t - z$ ” Curves

Appendix III. Soil constitutive models considered ( $τ \leq τ_{\max}$ ).

Appendix IV. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (4) and (9) ( $τ \leq τ_{\max}$ )

Appendix V. Normalized Displacement, $u_{0} / d$ , for Each Constitutive Model Obtained Using Eqs. (8) and (10) ( $τ \leq τ_{\max}$ )