Settlement of a Pile Foundation Considering Linear and Rheological Properties of Soils

Abstract

Despite numerous studies of single piles and practical experience with their application, methods for calculating settlements of pile foundations remain limited. The existing objective need for specialized methods of pile foundation settlement calculation that take into account the rheological properties of the base soils is becoming more and more important, especially in the construction of unique objects in complex ground conditions. When predicting the stress–strain state of the pile–raft-surrounding soil mass system, it is allowed to consider not the entire pile foundation as a whole, but only a part of it—the computational cell. In the present work, we have solved the problems of determining the strains of the computational cell consisting of the pile, the raft and the surrounding soil according to the column pile scheme and hanging pile scheme, on the basis of the Kelvin–Voigt rheological model, which is a model of a viscoelastic body consisting of parallel connected elements: Hooke’s elastic spring and Newtonian fluid. According to our results, we obtained graphs of the dependence of strains of the computational cell on time at different pile spacing and different values of coefficients of viscosity of the surrounding soil, and a formula for calculating the reduced modulus of deformation of the pile. The results of the present study can significantly improve the accuracy of calculations during construction on clayey soils with pronounced rheological properties and, as a result, increase the reliability of pile structures in general.

1. Introduction

For many years, the pile foundation has been one of the most popular types of foundations used in practice due to its reliability, speed of construction and the possibility of construction in almost any engineering–geological and climatic conditions. At the same time, the widespread use of pile foundations is due to the increasing number of stories and height of buildings, as well as the increasing weight of technological equipment.

When an external load is applied to a pile foundation, in addition to the settlements of the piles themselves, there is settlement of the soil in the inter-pile space, causing the friction forces developed on the lateral surfaces of the piles to not fully develop, unlike in single piles. When increasing the spacing between piles to 3–4 pile diameters, the development of friction forces on the lateral surface will only be partially realized.

It should be noted that the stress–strain state of the active zone in pile foundations significantly differs from the stress–strain state of the soil mass around a single pile, as in pile foundations there is a mutual influence of piles on each other. Therefore, it is not permissible to use the results of static tests of single piles and analytical methods to determine their settlements when calculating the settlements of a pile foundation.

Despite a large number of scientific works devoted to the study of single piles (Luga A.A. [1,2], Ogranovich A.B. [3], Golubkov V.N. [4,5], etc.), and the availability of many years of experience in the use of piles in the construction of various buildings and structures, for quite a long time there was no single engineering method for calculating the settlement of pile foundations and pile groups.

In the 1960s, the method by Egorov K.E. began to be used to calculate the settlement of pile groups [6], based on the calculation of a rectangular foundation placed on a natural base. The specific feature of this calculation method was that the pile foundation was considered as a massive foundation, and the place of application of the calculated pressure on the soil under the foundation bottom was dependent on the ratio of the length of the piles to the width of the foundation. If this ratio was less than or equal to one, then the point of application of the calculated pressure was taken as the surface located at the tip of the pile. If the ratio of pile length to foundation width exceeded unity, the design pressure was distributed to the surface at a depth equal to the foundation width [7].

At present, the prediction of the rate of settlement of foundations over time, formed mainly by weak clay soils, is quite relevant in the design of various types of foundations, including pile foundations. Since most settlements are uneven and can develop over years and even decades, it is necessary to know the rate of development of these settlements over time to ensure the normal operation of buildings and structures erected on weak clay soils.

Foundation settlement over time can be predicted by taking into account the long-term complex physical processes that occur during load transfer to soils that have rheological properties (creep properties). To describe the stress–strain state of the base foundation system, taking into account the viscoelastic and viscoelastic–plastic properties of the base soils, it is necessary to take into account many parameters, the values of which may change during the operation of the structure [8]. Based on the above, it is important to note that the time factor must be taken into account when calculating foundations on bases formed by soils possessing rheological properties.

Numerous experimental and theoretical studies aimed at studying the rheological properties of clay soils, presented in the works of Tertsagi K. [9], Gersevanov N.M. [10], Florin V.A. [11,12], Cytovich N.A. [13], Meschyan S.R. [14], Vyalov S.S. [15], Maslov N.N. [16], Ter-Martirosyan Z.G. [17] and many others, allowed us to develop methods for calculating foundations on a natural base over time.

Many researchers have studied the interaction of piles with clay soils, including Vyalov S.S. [15], Bartolomei, A.A. [18], Dalmatov B.I. and Lapshin F.K. [19], Ter-Martirosyan Z.G. [17], Omelchak I.M. [8] and many others.

Liu J. et al. [20] proposed a tri-linear softening model to describe the interaction between a single pile and the surrounding soil, and a tri-linear plastic model that describes the interaction between the pile and the underlying soil. Based on the mechanical properties of soils and the theory of elasticity, the authors developed equations that describe the dependence of the settlement of a single pile on the vertical load in layered soils. In performing the calculations, the pile was considered as a flexible shaft with a constant cross-section. The authors of the study have demonstrated the effectiveness of the proposed calculation method for a single pile and provided recommendations on using the proposed models in different soil layers for a correct analysis of the relationship between the pile settlement and the applied vertical load.

The study [21] presents an analytical solution for determining the stress–strain state of a soil mass interacting with underground structures, including with piles.

The interaction of a single long pile with a multilayer soil mass, taking into account the elastic and rheological properties of the soil mass, was investigated in [22,23]. In [22], the solution of the problem by analytical method taking into account linear and nonlinear properties of soils is presented. For describing nonlinear shear deformations, the elasto-plastic Timoshenko S.P. model was used. The authors performed a comparison of the results of the analytical solution to the elastic problem with the results of numerical modeling in the Plaxis 3D software package, and also obtained an expression for determining the reduced shear modulus for a multilayer soil mass. The study [23] considers the solution to the problem of determining the displacements of a long pile in a surrounding multilayer base. It was found that the rate of stress change under the pile heel depends on the viscosity of the soil, while the rheological hardening coefficient significantly influences the time of pressure stabilization under the pile heel and the time of pile settlement stabilization.

Based on the modified Burgers model, in which the spring is replaced by a hyperbolic model, a nonlinear approach to calculate the settlement over time of a vertically loaded single pile as well as a group of piles in a layered soft soil was developed in [24]. In this work, parametric studies were carried out to investigate the influence of the parameters included in the calculation on the time-dependent settlement of a single pile. Based on the conducted research, it was shown that the theoretical results agree well with the measurement results, indicating a sufficiently high accuracy of the approach proposed in the work.

For calculating the long-term settlement of a single pile embedded in viscoelastic base soils, a three-dimensional viscoelastic model with a fractional-order derivative was proposed by Li, Zhang and Liu to describe the rheological behavior of the soils around the pile [25].

A simple approach to analyzing the elasto-plastic behavior of a single pile in layered soils, which allows for a quick assessment of the settlement of a single pile as well as a group of piles, is presented in [26]. The proposed approach is based on the application of two models, one of which describes the nonlinear relationship between the shear stresses occurring on the lateral surface of the pile and the displacement of the soil around the pile. The second model is based on a bilinear hardening model to simulate the relationship between the settlement of the pile base and the resistance of the pile tip.

In [27], a model was proposed for determining the settlement of a vertically loaded single hanging pile embedded in layered base soils.

To evaluate the behavior of a single vertical pile over time, embedded in soil with elasto-plastic properties, the authors of [28] conducted an analysis using the finite element method in a two-dimensional setting. For modeling the soil around the pile, a linear elastic perfectly plastic Mohr–Coulomb model was used, and for modeling the pile material, a linear elastic model was employed. Based on the calculations performed, it was found that a pile in non-cohesive soil has greater resistance under rapid loading than under long-term loading, whereas the opposite results were obtained for a pile in cohesive soil. The geotechnical model presented in this work allows the determination of lateral deformations as well as lateral soil stress and its variation over time.

An analytical method for evaluating the behavior of a single tapered pile and a group of piles from applied static axial compressive loads was proposed in [29]. Based on the proposed method, an iterative computer program was developed to calculate the settlement and bearing capacity of a single tapered pile.

An analytical approach for predicting the settlement of vertically loaded pile and pile groups was proposed by Xia and Zou [30]. A piecewise function was applied to investigate the nonlinear relationship between surface friction (friction on the side surface of the pile) and relative displacement of the pile–soil system. In this paper, a comparative analysis of the results obtained using the proposed calculation methodology with field test data and results obtained by other researchers has been carried out.

In the study by Long Li and Yousheng Deng [31], an algorithm is proposed to calculate the settlement of a pile foundation in a soil base having linear–viscous–elastic properties. The calculation algorithm presented in this paper is based on the analysis of the change in linear viscoelastic settlement of the soil depending on the geometric parameters of the pile (pile spacing, pile length-to-diameter ratio) and the deformation characteristics of the soil (elastic modulus and Poisson’s ratio). The authors of the work performed a comparative analysis of the obtained results of the study with the results of field tests of piles.

A methodology based on the use of stress coefficients is presented by Gendy [32] to determine the settlement of hanging piles located in clay soil. The stress coefficients were obtained from the Mindlin equation by eliminating the Poisson’s ratio from this equation. Based on the results of the conducted research, equations were derived that allow predicting the settlement of a single pile, a pile group and a pile–raft foundation on clay soils.

It should be emphasized that taking into account the rheological properties of the base soils is important not only in static calculations, but also in dynamic calculations of foundations. This statement is supported by a number of studies and scientific papers that demonstrate the influence of rheological characteristics on the behavior of foundations subjected to dynamic loads [33,34]. The correct interpretation of these properties allows for a more accurate assessment of the stability and durability of structures, especially under variable and dynamic loads such as seismic impacts or vibrations induced by machinery.

Based on the analysis of the scientific and technical literature on the research topic, the following main conclusions can be drawn:

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Most scientific papers focus on determining the settlement and bearing capacity of a single pile, rather than the pile foundation as a whole;

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When determining the settlement of pile foundations, most scientific works do not take into account the rheological properties of the base soils;

-

The existing scientific papers present limited data on the influence of one of the most important rheological parameters—the coefficient of soil viscosity—on the settlement of pile foundations in general;

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A number of papers present calculations that take into account the rheological properties of base soils in relation to dynamic calculations.

Based on the above, the purpose of this paper is to determine the settlement of a pile foundation, taking into account the linear and rheological properties of the base soils based on the Kelvin–Voigt viscoelastic rheological model of the viscoelastic body according to the column pile and hanging pile schemes.

2. Materials and Methods

It is known that when a uniformly distributed load is applied to a pile foundation, it will be distributed among the piles, the raft and the surrounding soil mass located in the inter-pile space. Moreover, this distribution significantly depends on the diameter, length and spacing of the piles, as well as on the physical–mechanical and rheological properties of the soils and the pile material. With such a load transfer mechanism (using piles and a raft), a complex, heterogeneous stress–strain state is formed in the soil mass, which can change in space and time, especially when considering the rheological properties of the soils.

The important and determining design parameters when calculating a pile foundation are the settlement and bearing capacity of both the pile foundation as a whole and the individual piles within it (in the case of uneven distribution of piles in the pile foundation).

When predicting the stress–strain state of the pile–raft-surrounding soil mass system, it is allowed to consider not the entire pile foundation as a whole, but only a part of it—the computational cell.

The computational cell is a thick-walled soil cylinder of limited dimensions $\left(L, 2b_1\right)$, accommodating the pile, the raft and the surrounding soil mass. The geometric dimensions of the computational cell are chosen based on the load on the foundation and the engineering–geological conditions of the construction site.

Figure 1 shows the design scheme of the pile foundation with the delineation of the boundaries of the computational cell $\left(L, 2b_1\right)$ (dashed line).

When calculating piles as part of a pile foundation, the main thing is the quantitative assessment of the settlement of the pile foundation as a whole, which is largely determined by the magnitude of the load on the raft and the correct assessment of the stress–strain state of the pile–raft-surrounding soil mass system.

Depending on the ratio of the stiffness of the surrounding soil ($G_1$) and the underlying soil ($G_2$), the calculation of the stress–strain state of the pile foundation should be carried out either according to the column pile scheme when $G_2\gg G_1$, or according to the hanging pile scheme when $G_2>G_1$.

According to SP 24.13330.2021 [35], a column pile is a pile resting on rocky or weakly deformed soils and transmitting the load to the base mainly through the heel.

A hanging pile is a pile that transmits the load to the base through the lateral surface and the heel [35].

Figure 2 shows the design scheme of the interaction of the pile as part of the pile foundation with the surrounding and underlying soils and the raft according to two schemes.

Since, in the following, all main calculations will be based on the Kelvin–Voigt rheological model, the main features of this model are presented below for a better understanding of the results obtained.

The Kelvin–Voigt model is a rheological model of a viscoelastic body consisting of parallel connected elements: an elastic Hooke spring (H) and a Newtonian fluid (N) (Figure 3). When two elements are connected in parallel, the strain of each element will be the same and the total stress will be defined as the sum of the stresses on each element.

The dependence of strains ($\epsilon$) on stresses ($\sigma$) according to the Kelvin–Voigt rheological model will be as follows:

$$\sigma =\epsilon ·E+\dot{\epsilon }·\eta$$

(1)

where $E$ is the modulus of deformation; $\dot{\epsilon }$ is the strain rate; and $\eta$ is the coefficient of soil viscosity.

Note that the Kelvin–Voigt model is designed to analyze the mechanical properties of continuous media. In the context of our study, the ground can be classified as a continuous medium, which allows us to use this model to describe its behavior. This is confirmed by the ability of the model to take into account both elastic and viscous properties of the material, which makes it convenient for analyzing the stress–strain state of the soil mass under various loads.

A more detailed description of the Kelvin–Voigt rheological model can be found in the following references: [17,36,37].

3. Results

3.1. Solution of the Problem in a Linear Formulation According to the Column Pile Scheme

Let a long pile with a given diameter of $2a_1$ and length $l_1$ in a pile foundation be embedded in the soil mass, with its lower end resting on an underlying relatively dense soil layer ($G_2\gg G_1$). The computational domain of the massif containing the pile and the foundation is a thick-walled soil cylinder of limited dimensions $\left(L,2b_1\right)$.

When a static uniformly distributed load (${\sigma }_N=const$) is applied to the raft, it will be distributed between the piles and the surrounding soil within each computational cell. Considering the compressive deformation of the pile shaft and the surrounding soil mass (in accordance with the problem conditions), we can write the following equilibrium equation:

$${\sigma }_N={\sigma }_p·\omega +{\sigma }_s·(1-\omega ),$$

(2)

where ${\sigma }_N$ is the uniformly distributed load applied to the raft; ${\sigma }_p$ is the stress acting in the pile shaft at the level of the pile head; ${\sigma }_s$ is the stress acting in the soil mass at the contact with the raft; $\omega$ is the dimensionless coefficient equal to the ratio of the cross-sectional area of the pile shaft to the cross-sectional area of the entire computational cell, i.e., ${\omega =a}_1^2/b_1^2$; and $a_1$ and $b_1$ are the radius of the pile and the radius of the computational cell, respectively.

Let us also write down the condition of equality of settlements of the raft, the pile and the surrounding soil mass:

$$S_N=S_p=S_s$$

(3)

Equation (3) can be represented in the following form:

$$m_r·{\sigma }_N=m_p·{\sigma }_p=m_s·{\sigma }_s$$

(4)

where $m_p$ and $m_s$ are the coefficients of relative compressibility of the pile and the soil, respectively.

The joint solution of Equations (2) and (4) will determine ${\sigma }_p$ and ${\sigma }_s$, respectively:

$${\sigma }_p={\sigma }_N·{\displaystyle \frac{E_p}{E_p·\omega +E_s·\left(1-\omega \right)}}; {\sigma }_s={\sigma }_N·{\displaystyle \frac{E_s}{E_p·\omega +E_s·\left(1-\omega \right)}}$$

(5)

The reduced modulus of deformation of the computational soil cell can be found from condition (2), i.e.:

$$E_r=E_p·\omega +E_s·(1-\omega )$$

(6)

From Formula (6), it follows that as $\omega$ increases, $E_r$ also increases, and conversely. Consequently, other things being equal, by changing the ratio ${\omega =a}_1^2/b_1^2$, it is possible to adjust both the value of the reduced modulus of deformation of the computational cell $E_r$ and the calculated value of the settlement of the cell as a whole.

Based on the above, the settlement of the pile foundation can be determined according to the following formula:

$$S_{pr}={\displaystyle \frac{{\sigma }_N}{E_r}}·0.8·l_1$$

(7)

where $l_1$ is the length of the pile.

When calculating by Formula (7), the settlement of the underlying layer should be neglected.

Based on the known dependency $S=\epsilon ·l$, the solution to the settlement problem of a pile foundation cell can be obtained when the length ($l$) and strain ($\epsilon$) are known. Since length is an input parameter, the following problems will be considered in order to determine the strains of the computational cell.

3.2. The Stress–Strain State of the Cell in an Elasto-Viscous Formulation According to the Column Pile Scheme Based on the Kelvin–Voigt Model

Let us write down the equilibrium condition characterizing the process of development of settlement of the raft over time, assuming in the first approximation that it coincides with the equilibrium Equation (2), then we obtain:

$${\sigma }_N={\sigma }_p\left(t\right)·\omega +{\sigma }_s\left(t\right)·(1-\omega )$$

(8)

The stresses acting in the pile shaft at the level of the pile head (${\sigma }_p$) and the stresses acting in the soil mass at the contact with the raft (${\sigma }_s$) can be represented by the following relationships:

$${\sigma }_p\left(t\right)={\epsilon }_p·E_p; {\sigma }_s\left(t\right)={\epsilon }_s·E_s+\dot{{\epsilon }_s}·{\eta }_s$$

(9)

where ${\epsilon }_p$ is the strain of the pile; $E_p$ is the modulus of deformation of the pile; ${\epsilon }_s$ is the strain of the surrounding soil; $E_s$ is the modulus of deformation of the surrounding soil; $\dot{{\epsilon }_s}$ is the rate of development of strains of the surrounding soil; and ${\eta }_s$ is the coefficient of viscosity of the surrounding soil.

In Figure 4, the calculated model of the interaction of a pile in a pile foundation and the surrounding soil mass under compressive deformation is presented based on the Kelvin–Voigt model.

Substituting into the equilibrium condition (8) the dependences obtained in (9), we obtain:

$${\sigma }_N={\epsilon }_p·E_p·\omega +({\epsilon }_s·E_s+\dot{{\epsilon }_s}·{\eta }_s)·(1-\omega )$$

(10)

Based on the condition of equality of strains of the pile and the surrounding soil ${\epsilon }_p={\epsilon }_s=\epsilon$, we obtain:

$${\sigma }_N=\epsilon ·(E_p·\omega +E_s·\left(1-\omega \right))+{\dot{\epsilon }·\eta }_s·(1-\omega )$$

(11)

where $\epsilon ={\epsilon }_p={\epsilon }_s$, $\dot{\epsilon }=\dot{{\epsilon }_s}$.

Let us represent the differential Equation (11) in the following form:

$$\dot{\epsilon }+\epsilon ·P=Q$$

(12)

$$P={\displaystyle \frac{E_p·\omega +E_s·\left(1-\omega \right)}{{\eta }_s·(1-\omega )}};Q={\displaystyle \frac{{\sigma }_N}{{\eta }_s·\left(1-\omega \right)}}$$

(13)

The solution of the differential Equation (12) is known [38] and has the following form:

$$\epsilon (t)=e^{-\int Pdt}[\int Q·e^{\int Pdt}dt+C]$$

(14)

By performing the corresponding transformations in (14), we obtain the following equation:

$$\epsilon (t)={\displaystyle \frac{Q}{P}}+C·e^{-P·t}$$

(15)

We determine the integration constant from the initial condition when $\epsilon \left(0\right)=0$ at $t=0$:

$$C=-{\displaystyle \frac{Q}{P}}$$

(16)

Substituting the integration constant $C$ rom Equation (16) into Equation (15), we obtain:

$$\epsilon (t)={\displaystyle \frac{Q}{P}}·(1-e^{-P·t})$$

(17)

It follows from Equation (17) that:

-

at $t\to 0$ $\epsilon \left(0\right)=0$;

-

at $t\to \infty$ $\epsilon \left(\infty \right)={\displaystyle \frac{Q}{P}}={\displaystyle \frac{{\sigma }_N}{E_p·\omega +E_s·\left(1-\omega \right)}}$.

Table 1. Characteristics of the pile and surrounding soil (according to the column pile scheme).

Name of the Parameter	Symbol	Value	Units
Characteristics of the pile
Modulus of deformation of the pile	$E_p$	3 × 107	kPa
Radius of the pile	$a_1$	0.30	m
Pile spacing *	${2b}_1$	1.80	m
	${2b}_2$	2.40	m
	${2b}_3$	3.00	m
	$2b_4$	3.60	m
Characteristics of the surrounding soil
Modulus of deformation of the surrounding soil	$E_s$	2 × 104	kPa
Coefficient of viscosity of the surrounding soil	${\eta }_{s_1}$	1 × 109	Poise
	${\eta }_{s_2}$	1 × 1011	Poise
	${\eta }_{s_3}$	1 × 1013	Poise

*—in the present study, the pile spacing is equal to the diameter of the computational cell (Figure 1).

shows the characteristics of the pile and surrounding soil used in the calculation using Formula (17).

$E_p$$a_1$${2b}_1$${2b}_2$${2b}_3$$2b_4$$E_s$${\eta }_{s_1}$${\eta }_{s_2}$${\eta }_{s_3}$Figure 5, Figure 6 and Figure 7 show the graphs of the dependence of strains of the computational cell on time $\left(\epsilon -t\right)$, obtained with different pile spacing $\left({2b}_1<{2b}_2<{2b}_3<{2b}_4\right)$ and with different coefficients of viscosity of the surrounding soil $\left({\eta }_{{\mathrm{r}}_1}<{\eta }_{{\mathrm{r}}_2}<{\eta }_{{\mathrm{r}}_3}\right)$ based on Formula (17).

Figure 8 shows the graphs of the dependence of strains of the computational cell on time $\left(\epsilon -t\right)$, obtained with different coefficients of viscosity of the surrounding soil $\left({\eta }_{s1}<{\eta }_{s2}<{\eta }_{s3}\right)$ and with pile spacing ${2b}_2$, based on Formula (17).

3.3. The Stress–Strain State of the Cell in an Elasto-Viscous Formulation According to the Hanging Pile Scheme Based on the Kelvin–Voigt Model

The analysis of the solution of the problem in the elastic–viscous formulation according to the column pile scheme showed that if we replace the modulus of deformation of the pile with the reduced modulus of deformation, taking into account the local settlement under the pile heel, we can find the solution of the problem according to the hanging pile scheme. For this purpose, it is sufficient to change the stiffness of the pile in such a way that the following condition is satisfied:

$${\displaystyle \frac{{\sigma }_p}{E_p^{\prime }}}·l_1={\displaystyle \frac{{\sigma }_p}{E_p}}·l_1+{\displaystyle \frac{{\sigma }_p·\pi ·a_1·\left(1-{\nu }_2^2\right)}{2·E_2}}$$

(18)

$$E_p^{\prime }={\displaystyle \frac{E_p}{1+\frac{E_p·\pi ·a_1·\left(1-{\nu }_2^2\right)}{2·E_2·l_1}}}$$

(19)

where $E_p^{\prime }$ is the reduced modulus of deformation of the pile; $E_p$ is the modulus of deformation of the pile; $E_2$ is the modulus of deformation of the underlying soil mass; ${\nu }_2$ is the Poisson’s ratio of the underlying soil mass; $a_1$ is the radius of the pile; and $l_1$ is the length of the pile.

Table 2. Characteristics of the pile and surrounding soil.

Name of the Parameter	Symbol	Value	Units
Characteristics of the pile
Modulus of deformation of the pile	$E_p$	3 × 107	kPa
Radius of the pile	$a_1$	0.30	m
Pile spacing *	${2b}_2$	2.40	m
Length of the pile	$l_1$	30	m
Characteristics of the soil
Modulus of deformation of the surrounding soil	$E_s$	2 × 104	kPa
Modulus of deformation of the underlying soil mass	$E_2$	3 × 104	kPa
Poisson’s ratio of the underlying soil mass	${\nu }_2$	0.36	-
Coefficient of viscosity of the surrounding soil	${\eta }_s$	1 × 1012	Poise

*—in the present study, the pile spacing is equal to the diameter of the computational cell (Figure 1).

shows the characteristics of the pile and surrounding soil used in the calculation using Formula (17), taking into account Formula (19).

Based on the above, we can conclude that by replacing the modulus of deformation of the pile $E_p$ with the reduced modulus of deformation $E_p^{\prime }$, in the rheological Equation (11), we obtain the stress–strain state of the cell according to the hanging pile scheme (Figure 9a).

As a result, we can predict the strains of the computational cell both using the column pile scheme $\left({\epsilon }_1-t\right)$ and the hanging pile scheme $\left({\epsilon }_2^{\prime }-t\right)$ (Figure 10) based on Formula (17), taking into account (19).

4. Discussion

When the load is transferred to the pile foundation, in addition to the settlement of the piles themselves, the settlement of the surrounding soil located in the inter-pile space occurs. When considering the stress–strain state of the active zone in pile foundations, it should be noted that it differs significantly from the stress–strain state of the surrounding soil mass around a single pile, because in pile foundations there is a mutual influence of the piles on each other, as well as a time distribution of the external load between the raft, piles and the surrounding soil. However, despite this, numerous studies [20,21,22,23,25,26,27,28,29] have focused on determining the settlement of a single pile rather than the pile foundation as a whole.

The formation of a complex and heterogeneous stress–strain state of the system pile–raft-surrounding soil mass system depends on:

-

geometric characteristics (cross-sectional and longitudinal profile shape, length) and pile material, as well as pile spacing;

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engineering–geological and hydrogeological conditions of the construction site;

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mechanical characteristics of the surrounding soil, which include deformation $(E,\nu )$, strength $(c,\phi )$ and rheological $\left(\eta \right)$ properties.

Indeed, the factors presented above have a significant influence on the formation of the stress–strain state of the active zone in the calculation of the pile foundation as a whole, which has been reviewed and noted by Long Li and Yousheng Deng in [31]. This study investigated the change in linear viscoelastic settlement of soil depending on the pile spacing, the ratio of pile length to diameter, soil elasticity modulus and Poisson’s ratio. However, unfortunately, this study did not pay sufficient attention to the influence of the rheological properties of clay soils (including soil viscosity) on the obtained values of settlement of the pile foundation over time.

Taking into account the rheological properties of clay soils is necessary in order to predict over time the final settlements of pile foundations constructed on clay soils, to prevent in advance the negative phenomena associated with creep of clay soils and, if necessary, to optimize the design solutions of foundations.

In the present study, the problem of the interaction between the pile, the raft and the surrounding soil mass was solved in a linear formulation using the column pile scheme. As a result, Formula (6) was obtained for calculating the reduced deformation modulus of the computational soil cell, as well as Formula (7) for calculating the settlement of the pile foundation.

To take into account the rheological properties of the base soils, a problem was solved using an elastic–viscous formulation in the column pile scheme and in the hanging pile scheme based on the well-known Kelvin–Voigt rheological model. This model allows us to describe creep of clay soils over time quite well because it includes a viscosity parameter.

Let us consider in more detail the main advantages of this model for a better understanding of the results obtained in this study.

• The structure of the model: as mentioned earlier in this paper, the Kelvin–Voigt model consists of an elastic element (spring) and a viscous element (Newtonian fluid) connected in parallel. This allows this model to describe the processes of elastic deformation and viscous flow, which is characteristic of the behavior of clayey soils under the action of a constant load. We also note that the Kelvin–Voigt model takes into account both the rate and magnitude of deformation, describing the behavior of the soil under the action of constant or temporary loads. For example, when the load is suddenly increased, the elastic deformation will occur immediately, while the viscous part will determine how the soil will continue to deform over time.
• The ability to take creep into account: in contrast to simple elastic models, the Kelvin–Voigt model allows us to take into account the creep phenomenon inherent in clayey soils. This means that when a constant load is applied, deformations will slowly continue to develop in the clayey soil over a long period of time. These deformations can be quantified using this model, which represents an important aspect in the design of various foundations and enclosing structures, since the final settlement can significantly affect the project and operation of the structure.
• Taking into account the time factor: the Kelvin–Voigt model makes it possible to estimate deformations of soil with rheological properties both in the short-term (a few days) and long-term (several decades) time range, which is especially important in the design of foundations and various enclosing structures.
• Flexibility and adaptability: the Kelvin–Voigt model can be used to calculate both single-layer and multilayer bases.
• Ease of use: the mathematical description of the model is relatively simple, which will allow its future implementation in numerical calculation methods.
• Empirical validity: the model agrees well with experimental data obtained from tests of clayey soils. This makes it a reliable tool for engineering analysis.

Based on the above, we conclude that the Kelvin–Voigt model is a powerful tool for calculating and predicting long-term deformations of soil bases with rheological properties. The model’s ability to take into account both elastic and viscous components allows engineers to more accurately estimate long-term deformations and make informed decisions in the design of various foundations and enclosing structures. Thus, the application of this model provides an opportunity for more reliable and efficient design and minimization of potential risks for the objects under construction.

Based on the solution to the problem using the column pile scheme based on the Kelvin–Voigt model, the following results were obtained:

-

It is shown that the vertical strains of the computational cell also increase with increasing pile spacing $\left(2b_1<{2b}_2<{2b}_3<{2b}_4\right)$ (Figure 5, Figure 6 and Figure 7), because with increasing pile spacing, the load applied to the raft starts to be supported more by the surrounding soil mass located in the inter-pile space, which has a modulus of deformation much lower than that of the pile itself;

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At low values of the coefficient of viscosity of the surrounding soil (${\eta }_{s_1}={10}^9$ Poise) (Figure 5), comparable to the liquid medium, there is a peak value of vertical strains exceeding the residual vertical strains, which is not observed in more viscous media (${\eta }_{s_2}={10}^{11}$ Poise, ${\eta }_{s_3}={10}^{13}$ Poise) (Figure 6 and Figure 7). This phenomenon indicates significant differences in soil behavior depending on its rheological properties;

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In accordance with the character of location of the graphs of dependence of strains of the computational cell on time $\left(\epsilon -t\right)$ presented in Figure 8, it can be seen that the time of stabilization of vertical strains of the computational cell is directly proportional to the value of the coefficient of viscosity of the surrounding soil, i.e., the higher the value of the coefficient of viscosity, the more time will be needed in order to stabilize the vertical strains of the computational cell from the applied load. However, it should be noted that changing the coefficient of viscosity of the surrounding soil, both increasing and decreasing, will affect only the time of stabilization of vertical strains of the computational cell, and will not change their value. The results obtained confirm the importance of considering time factors in assessing the stability and durability of pile structures, especially under variable loads.

Based on the solution to the problem using the hanging pile scheme based on the Kelvin–Voigt model, the following results were obtained:

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Formula (19) for calculating the reduced modulus of deformation of the pile ($E_p^{\prime }$) was obtained, which makes it possible to take into account the strains of the underlying soil layer and considerably simplify the procedure for calculating the stress–strain state of the computational cell according to the hanging pile scheme. The resulting equation takes into account both the stiffness of the pile itself ($E_p$) and the stiffness of the underlying soil mass under the pile heel ($E_2$);

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Building graphs of dependence of vertical strains of the computational cell over time according to the column pile scheme $\left({\epsilon }_1-t\right)$ and according to the hanging pile scheme $\left({\epsilon }_2^{\prime }-t\right)$, presented in Figure 10, showed that they are significantly different. This difference shows that taking into account the local settlement under the pile heel in the design diagram leads to an increase in settlement, which corresponds to the “hanging pile” design diagram (Figure 9a);

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When the modulus of deformation of the surrounding soil ($E_s$), the modulus of deformation of the underlying soil mass ($E_2$) and the Poisson’s ratio of the underlying soil mass (${\nu }_2$) decrease, the vertical strains of the computational cell in the hanging pile scheme will increase, and conversely, when these parameters increase, the vertical strains of the computational cell will decrease;

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Change in the coefficient of viscosity of the surrounding soil (${\eta }_s$) does not influence the value of vertical strains of the computational cell both for the column pile and hanging pile schemes. However, this parameter significantly influences the time of stabilization of vertical strains, which can be important in the design of foundations, especially under conditions of dynamic loads on building structures.

The analysis of the obtained results allows us to conclude that when designing pile foundations, it is necessary to take an individual approach to each specific construction case, taking into account the type of soil, its rheological properties and its mechanical behavior. The use of the proposed Kelvin–Voigt viscoelastic body model and the obtained formula for calculating the reduced modulus of deformation can significantly increase the accuracy of calculations and, consequently, the reliability of pile structures in general. Importantly, further research in this area will help to optimize existing design standards, leading to safer and more efficient solutions in engineering practice.

5. Conclusions

The acceleration of scientific and technical progress in foundation engineering is inextricably linked to the development of new calculation methods that allow for a more reliable description of the interaction between different types of foundations and base soils during the transfer of loads from buildings, structures and technological equipment [8]. Numerous observations of the settlement of various buildings and structures over the years have shown that a promising direction of research into the interaction of various foundations and soil bases is to take into account the rheological properties of soils (consideration of viscous deformation under the action of constant and variable loads).

Taking into account the fact that many buildings and structures are constructed on bases composed of weak clay soils, the settlement of which, as a rule, is uneven and can develop over years and even decades, it is necessary to consider the rheological properties of clay soils when calculating and designing various types of foundations, including pile foundations. The widespread use of pile foundations is due to their reliability, speed of construction and possibility of construction in almost any engineering–geological and climatic conditions.

The following main results were obtained, meeting the purpose of this study as outlined in the introduction:

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The dependence (17) was obtained, which allows us to determine the vertical strains of the computational cell (and, at known pile length, it is possible to obtain the settlement of the pile foundation as a whole) both for the column pile scheme and for the hanging pile scheme;

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The Formula (19) for calculation of the reduced modulus of deformation of the pile ($E_p^{\prime }$) was obtained, which allows us to take into account the local settlement of the underlying soil under the heel of the pile and thus to pass from the solution of the problem according to the column pile scheme to the solution of the problem according to the hanging pile scheme (Figure 10 clearly shows the difference in the values of vertical strains of the computational cell obtained when solving the problem using the column pile scheme $\left({\epsilon }_1-t\right)$ and the hanging pile scheme, taking into account the reduced modulus of deformation of the pile $\left({\epsilon }_2^{\prime }-t\right)$);

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It was found that the coefficient of viscosity of the surrounding soil has no influence on the values of vertical strains of the computational cell. This parameter only influences the stabilization time of the vertical strains, i.e., how quickly the vertical strains stabilize. Figure 8 effectively demonstrates how the stabilization time of vertical strains of the computational cell varies significantly with different values of the coefficients of viscosity of the surrounding soil $\left({\eta }_{s1}<{\eta }_{s2}<{\eta }_{s3}\right)$.

The practical significance of this study consists in solving the actual problem of determining the settlement of a pile foundation, taking into account the linear and rheological properties of soils under the column pile scheme and under the hanging pile scheme on the basis of the known viscoelastic Kelvin–Voigt model.

As future research, it is planned to further develop methods of quantitative assessment of the stress–strain state of the base-foundation system, taking into account rheological properties of the clay soil in order to exclude excessive development of settlements, tilts and displacements of various buildings and structures, as well as to optimize the adopted design solutions of foundations.