Graph-Analytical Method for Calculating Settlement of a Single Pile Taking into Account Soil Slippage

Abstract

Most of the existing methods of pile settlement calculation, including normative methods, do not fully take into account the processes occurring in the soil when loads are transferred to them and the changes in the properties of the contact zone soils. This leads to underutilisation of the bearing capacity of the soil, and the calculated settlement value may differ several times from the real values. In this paper, a graph-analytical solution to the problem of interaction of a single pile with a three-layer soil foundation is proposed to determine the settlement, taking into account the complex nature of the pile operation and the processes occurring in the soil when loads are transferred to them. The proposed method allows to use the non-linear behaviour of the soil on the lateral surface and under the tip of the pile, the possibility of its detachment and slippage after reaching the ultimate strength of the soil, changes in the properties of the contact zone soils, and the load distribution on the pile between its lateral surface and the tip. To verify the proposed graph-analytical solution, a comparative analysis was performed with the numerical method in the Plaxis 2d software (version 21.00.01.7) and with the results of static tests of piles at the construction site. To determine the strength reduction factor at the contact of soils with concrete, laboratory tests were carried out on a direct shear apparatus. Based on the results of the performed calculations, graphs of the dependence of settlements on loads were plotted, conclusions were drawn about the possibility of using the graph-analytical method, and prospects for further development and improvement of the graph-analytical method were proposed.

1. Introduction

Experimental studies of piles using field methods show that piles are designed with a large reserve for both bearing capacity and settlement. First of all, this is due to the fact that, according to the regulatory documentation in force in the Russian Federation, piles are designed in the elastic stage of the soil [1,2,3,4]. In the second place, some studies suggest that the underutilisation of bearing capacity may also be due to the fact that the soils under the tip of long piles have a higher bearing capacity than assumed in the design [1,5]. Despite this, the authors in [5,6,7] have proposed methods for calculating pile settlement that take into account the non-linear behaviour of soils only along the lateral surface of the pile, while the authors in [8,9], on the contrary, take into account the non-linear behaviour of soils only under the pile tip. However, no calculation method has been proposed in engineering practice that takes into account the non-linear behaviour of the soil both along the lateral surface of the pile and under the tip of the pile. Exceptions are numerical calculation methods, e.g., [10], which allow elastic, non-linear, viscous behaviour of the soil to be taken into account, but the final result and its interpretation will depend on the user.

Secondly, the design of pile foundations does not take into account the load distribution on the pile between its lateral surface and the tip. However, this process has a significant influence on the formation of the stress–strain state of the soil mass and on the pile settlement. Based on a study of the experimental results, it can be noted that initially, most of the load is transferred to the soil through the side surface of the pile and the rest of the load is transferred through the tip of the pile. With increasing loads on the pile head, there is a gradual depletion in the bearing capacity: first, at the level of the pile head and at the contact point with the weak soil layers; then, along the entire lateral surface; and finally, failure under the pile tip, which leads to total failure [11,12]. For example, in [13], at a load of 100 kN, almost the entire load on the soil is transmitted through the lateral surface of the pile (Figure 1). A further increase in the loads occurs mainly only on the lateral surface, and at the maximum applied load of 1100 kN, there is an increase in the pile tip load, but it does not exceed 30%. Similar results can be seen for different pile lengths with different cross-sectional shapes [14,15,16,17,18]. However, this factor was only considered by the authors in [5], and they considered the kinematic inclusion of the pile in that work. When loads are applied, the lateral surface area at the level of the head is included in the work, and then, gradually deeper soil layers are involved in the work, gradually reaching the tip of the pile and involving the soil beneath it in the work. This load distribution is an approximation as it will also depend on the stiffness of the soil.

Thirdly, most of the theoretical and numerical solutions are based on the laws of continuum mechanics, which leads to their joint deformation, i.e., the settlement of the pile is equal to the settlement of the adjacent soil. But in reality, when the bearing capacity limit of the soil is reached, failure occurs, which leads to pile failure. The works by Grigoryan A.A. [19,20] considered the possibility of pile detachment from the ground using the mechanics of discrete bodies; however, such methods are more voluminous and difficult to calculate than methods based on the mechanics of continuous bodies, which is undoubtedly their disadvantage, especially in engineering practice.

Fourthly, a new stress–strain state appears at the contact point of the structures with the soil, and the properties of the soils in the contact zone will differ from the properties of the same soils in their natural mass. The conducted studies and the accumulated practical experience have shown that the change in mechanical properties of soils at the contact point occurs under the influence of the technology of pile construction, which is called the “installation effect” [21,22]; under the influence of pile installation technology [21,22]; and under the influence of friction reduction at the contact point. Numerous results of laboratory shear tests of soils on solid materials such as concrete, metal, and polymer confirm the fact of that the strength of soils reduces when interacting with foundations of buildings and other elements of structures erected in soil mass [23,24,25,26]. The reduction in strength of the contact zone soil is influenced by the type of soil, the roughness of the material, moisture, and density; however, the greatest influence is exerted by the relative roughness R_n, as evidenced by its large number of laboratory studies [27,28]. However, the proposed analytical calculation methods do not allow for taking into account the change in the mechanical characteristics of soils at the contact point.

Recently, interface models designed to describe the behaviour of the soil layer involved with a structure have become very popular and can be implemented in various design software packages. As a rule, these models are based on laboratory shear tests of soil on the material and, depending on the number of input parameters, can predict the behaviour of the soil at the contact point with varying degrees of accuracy. For example, De Gennaro and Frank [29] proposed a simple interface model to describe the behaviour of sandy soils based on the Mohr–Coulomb strength criterion. Hu and Pu [30,31] proposed a contact interface model with 10 input parameters based on the performed shear tests of sandy soils over different materials. The plots of shear stresses versus shear strains obtained numerically using the proposed model agreed well with the results of laboratory tests. A similar model was proposed by Huabei Liu et al. [32], which was extended to unify the interface behaviour with different roughness, and also, Saberi, Annan, and Konrad [33,34] proposed a model with 10 input parameters that takes into account density, roughness, and de-strengthening at the contact point but was not designed to describe the behaviour of water-saturated and clayey soils.

Due to the fact that in reality, soil properties are random, some authors have presented a different approach to pile design, taking into account reliability calculations, in order to design the most economical structure. By analysing a number of parameters such as the pile diameter, spacing and location, and their degree of influence on slope reinforcement with piles, the authors in [35] proposed an approach using the multi-objective optimisation (MOO) design system, which allows for choosing the most optimal solution for these parameters to reduce the involved construction cost while retaining the same reliability level. Similar studies on calculating the stability of slopes taking into account the soil spatial variability were also carried out by other authors [36,37].

Thus, the interaction between the pile and the soil mass is complex and multifactorial. At the same time, the existing calculation methods do not fully reflect the actual operation of the pile and the processes occurring in the soil when loads are transferred to them. In addition, piles are designed in the elastic stage of the soil, which leads to underutilisation of the bearing capacity of the pile. The combination of these factors leads to increased construction costs and increased complexity of construction and installation work due to the increase in the length and diameter of the pile, and the need for more powerful pile driving equipment. The purpose of this work is to improve the graph-analytical method of calculating the settlement of a single pile, the novelty of which lies in the fact that this method takes into account the non-linear behaviour of the soil both on the lateral surface and under the tip of the pile; the possibility of its detachment and slippage after reaching the ultimate strength of the soil, based on the laws of continuum mechanics; the distribution of loads between the lateral surface and the tip of the pile; and the changes in the properties of soils of the contact zone.

2. Materials and Methods

The graph-analytical method consists of two parts: analytical and graphical parts. In the analytical part of the method, a system of several equations is solved to determine the unknown components of the stress–strain state (SSS) of the soil mass around the pile: the shear stress τ on the lateral surface of the pile due to friction forces and the vertical stress σ under the tip of the pile. The analytical solution is elastic; hence, there is no failure of the soil around the pile and no exhaustion of the bearing capacity.

The graphical part of the method is used to describe the non-linear behaviour of the soil. It consists in the construction of two shear stress diagrams along the lateral surface of the pile. The first is a mobilised (effective) shear stress diagram obtained on the basis of analytical equations, while the second is a limit (resource) diagram based on the selected soil strength criterion. When overlaying the diagrams on each other, there are characteristic areas where there is exhaustion of the bearing capacity, i.e., where the mobilised diagram exceeds the ultimate diagram, and, vice a versa, where the ultimate diagram is greater than the mobilised diagram, which indicates the remaining unexhausted reserve of bearing capacity of the soil. The load that exceeds the soil strength limit is redistributed laterally and downwards until the bearing capacity is completely exhausted and pile failure occurs. At the same time, the graph-analytical method allows for controlling the distribution of loads on the pile and changing the strength characteristics of the contact zone soils. The section of the pile where the bearing capacity of the soil has been exhausted is removed from further work, which gives the pile the opportunity to move relative to the soil.

The analytical solution of the settlement problem of a single pile interacting with a three-layer base in axisymmetric formulation is presented below. The calculation scheme is a «cell» consisting of a cylinder-shaped soil volume with a pile immersed in it. The engineering geological element number is designated as EGE. When a load N is applied to the pile head, friction forces along the lateral surface of the pile T₁, T₂, and T₃ and earth back pressure R under the pile tip occur. The origin is located at the level of the pile tip, with the z-axis pointing vertically upwards and the r-axis pointing horizontally. The pile radius is denoted as a and the cell radius as b. The width of the cell was iteratively selected so that the condition of zero settlement at the lateral boundaries was fulfilled, i.e., at r = b, the soil settlement S = 0. Figure 2 shows the calculation scheme of the problem with the indication of the acting forces on the pile and with the indication of the arising stresses from the action of the applied load N.

As a result, the following stress components need to be determined: the reactive stress σ₀ under the tip pile and the four components of the shear stresses τ₀, τ_l1, τ_l2, and τ_l3 acting along its lateral surface. The distribution of shear stresses along the length of the pile is assumed to follow the “trapezoidal” law, linearly decreasing with depth. For the bottom soil layer, this law has the following form:

$$\tau \left(z\right)={\tau }_0+{\displaystyle \frac{{\tau }_{l1}-{\tau }_0}{l_1}}·z$$

(1)

where τ₀ is the value of shear stress at the level of the pile tip; τ_l1 is the value of shear stress at the boundary of the bottom and middle layers of soil; l₁ is the length of the pile located in the bottom layer of soil or the thickness of the bottom layer of soil.

In the middle layer of soil, the law of shear stress distribution along the depth has the following form:

$$\tau \left(z\right)={\tau }_{l1}+{\displaystyle \frac{{\tau }_{l2}-{\tau }_{l1}}{l_2}}·\left(z-l_1\right)$$

(2)

where τ_l2 is the value of the shear stress at the boundary of the middle and top soil layers; l₂ is the length of the pile located in the middle soil layer or the thickness of the middle soil layer.

In the upper soil layer, the law of distribution of tangential stresses along the depth takes the following form:

$$\tau \left(z\right)={\tau }_{l2}+{\displaystyle \frac{{\tau }_{l3}-{\tau }_{l2}}{l_3}}·\left(z-l_2-l_1\right)$$

(3)

where τ_l3 is the value of shear stress at the level of the pile head; l₃ is the length of the pile located in the top soil layer or the thickness of the top soil layer.

To solve the problem and determine the unknown components of the stress state, it is necessary to compose a system of equations that includes one equation of equilibrium and four equations of equality of settlement of the pile and the adjacent soil at the characteristic boundaries, i.e., at the level of the pile tip and at the boundaries of the layers:

$$\left\{\begin{array}{c}N=R+T_1+T_2+T_3\\ S_{pile}\left(0\right)=S_{soil}\left(0\right)\\ S_{pile}\left(l_1\right)=S_{soil}\left(l_1\right)\\ S_{pile}\left(l_1+l_2\right)=S_{soil}\left(l_1+l_2\right)\\ S_{pile}\left(l_1+l_2+l_3\right)=S_{soil}\left(l_1+l_2+l_3\right)\end{array}\right.$$

(4)

By assuming that the pile acts as a circular rigid die at some depth, the settlement under the tip can be determined using the following formula [38,39]:

$$S_{pile}\left(0\right)={\displaystyle \frac{{\sigma }_0·\pi ·a·\left(1-{\nu }_1\right)·K}{4G_1}}$$

(5)

where σ₀ is the vertical stresses acting under the tip of the pile; a is the radius of the pile; ν₁ is the Poisson’s ratio of the soil under the pile tip; G₁ is the shear modulus of the soil under the pile tip; K is the coefficient depending on the depth of application of the load on the die, which is assumed to be 0.6…0.8 for piles.

After a number of transformations, the system of five equations for determining the unknown components of SSS will take the following form:

$$\left\{\begin{array}{c}\pi {·a}^2·{\sigma }_l=\pi ·a^2·{\sigma }_0+\pi ·a·l_1·\left({\tau }_0+{\tau }_{l1}\right)+\pi ·a·l_2·\left({\tau }_{l1}+{\tau }_{l2}\right)+\pi ·a·l_3·\left({\tau }_{l2}+{\tau }_{l3}\right)\\ {\displaystyle \frac{{\sigma }_0·\pi ·a·\left(1-{\nu }_1\right)·K}{4G_1}}={\displaystyle \frac{{\tau }_0·a}{G_1}}·ln\left({\displaystyle \frac{b}{a}}\right)\\ S_{l1}+{\displaystyle \frac{{\sigma }_0·l_1·{\beta }_c}{E_{pile}}}+{\displaystyle \frac{{\sigma }_0·\pi ·a·\left(1-{\nu }_1\right)·K}{4G_1}}={\displaystyle \frac{{\tau }_{l1}·a}{G_1}}·ln\left({\displaystyle \frac{b}{a}}\right)\\ S_{l1}+S_{l2}+{\displaystyle \frac{{\sigma }_0·\left(l_1+l_2\right)·{\beta }_c}{E_{pile}}}+{\displaystyle \frac{{\sigma }_0·\pi ·a·\left(1-{\nu }_1\right)·K}{4G_1}}={\displaystyle \frac{{\tau }_{l2}·a}{G_2}}·ln\left({\displaystyle \frac{b}{a}}\right)\\ S_{l1}+S_{l2}+S_{l3}+{\displaystyle \frac{{\tau }_0·l·{\beta }_c}{E_{pile}}}+{\displaystyle \frac{{\sigma }_0·\pi ·a·\left(1-{\nu }_1\right)·K}{4G_1}}={\displaystyle \frac{{\tau }_{l3}·a}{G_3}}·ln\left({\displaystyle \frac{b}{a}}\right)\end{array}\right.$$

(6)

where σ_l is the vertical stresses acting on the pile head; b is the cell radius; βc is the coefficient of impossibility of lateral expansion of the pile material; E_pile is the modulus of elasticity of the pile; l is the pile length; S_l1 is the settlement of the bottom layer of soil with thickness l₁ on the lateral surface of the pile; S_l2 is the settlement of the middle layer of soil with thickness l₂ on the lateral surface of the pile; S_l3 is the settlement of the top layer of soil with thickness l₃ on the lateral surface of the pile.

The elastic settlement of a pile in a three-layer soil mass will be the sum of the settlement under the pile tip, the settlement adjacent to the lateral surface of the pile, and the settlement due to compression of the pile shaft. Thus, the formula for determining the elastic settlement is as follows:

$$S_{el}=S_{l1}+S_{l2}+S_{l3}+S_{compress}+S_0$$

(7)

where S_compress is the settlement due to compression of the pile shaft; S₀ is the soil settlement under the pile tip, determined by Formula (5).

The settlement of the bottom layer of soil with thickness l₁ on the lateral surface of the pile is determined by the following formula:

$$S_{l1}={\displaystyle \frac{2·{\tau }_0·l_1^2+{\tau }_{l1}·l_1^2}{3·E_{pile}·a}}$$

(8)

The settlement of the middle layer of soil with thickness l₂ on the lateral surface of the pile is determined by the following formula:

$$S_{l2}={\displaystyle \frac{{\tau }_{l1}·{\left(l_1+l_2\right)}^2}{E_{pile}·a}}+{\displaystyle \frac{\left({{\tau }_{l2}-\tau }_{l1}\right)·\left(\frac{{\left(l_1+l_2\right)}^3}{3}-l_1·{\left(l_1+l_2\right)}^2\right)}{E_{pile}·a·l_2}}$$

(9)

The settlement of the top soil layer with thickness l₃ on the side of the pile is determined by the following formula:

$$S_{l3}={\displaystyle \frac{{\tau }_{l2}·l^2}{E_{pile}·a}}+{\displaystyle \frac{\left({{\tau }_{l3}-\tau }_{l2}\right)·\left(\frac{l^3}{3}-l_2·l^2-l_1·l^2\right)}{E_{pile}·a·l_3}}$$

(10)

In the graphical part of the method, the mobilised tangential stress diagram is plotted using the shear stresses obtained on the basis of the analytical part of the method, and the ultimate shear stress diagram is determined on the basis of the Coulomb–Mohr strength criterion. The characteristic diagram of mobilised shear stresses and ultimate shear stresses is shown in Figure 3.

The superposition of the mobilised and ultimate shear stresses results in two main sections. The first section occurs when the mobilised shear stress profile exceeds the ultimate shear stress profile, i.e., the bearing capacity is exhausted at this section of the pile length and the pile is shut down. The load that can no longer be carried by this section is denoted by ΔT_n1. The second section characterises the remaining reserve of the bearing capacity of the soil ΔT_n2, i.e., where the ultimate shear stress diagram exceeds the mobilised diagram. The load ΔT_n1 is redistributed between the lateral surface of the pile, where reserve ΔT_n2 remains, and the pile tip. Figure 4 shows the calculation scheme of the graph-analytical method for determining the pile settlement for a three-layer soil mass.

The non-linear settlement of the soil under the pile tip is determined by the following formula [8]:

$$S_{pl}=S_{el}·{\displaystyle \frac{p_{cr}}{p_{cr}-p_0}}$$

(11)

where S_el is the elastic component of the settlement; p_cr is the ultimate (critical) load; p₀ is the acting load on the pile.

The elastic component of the settlement S_el is determined from the analytical solution. The ultimate load p_cr is determined using the formula proposed by Brinch Hansen [40,41]:

$$p_{cr}=\omega ·B_{\omega }·L_{\omega }·\left(s_c·c·N_c+s_q·\gamma ·d·N_q+\mathrm{0,5}·s_{\gamma }·\gamma ·B_{\omega }·N_{\gamma }\right)$$

(12)

where ω is the shape factor of the foundation’s cross-section, for the transition from a square to circular cross-section; Bω and Lω are given values of length and width of the pile cross-section; s_c, s_q, and s_γ are shape factors of the foundation’s cross-section; Nc, Nq, and Nγ are dimensionless constants depending on the angle of internal friction of the soil; c is the cohesion of the soil; γ is the unit weight of the soil within the limits of a possible soil bulge prism; d is the depth of foundation embedment, equal to the pile length l.

Thus, we can distinguish the following steps for using the graph-analytical method at each stage of load application N:

1. Solving the system of Equation (6) to determine the unknown components of the stress–strain state (τ_n) and solving Equation (7) to determine the elastic settlement S_el.

2. Construction of the mobilised shear stress epuple obtained in the previous step (Figure 3a).

3. Constructing the ultimate tangential stress diagram based on the Mohr–Coulomb strength criterion (Figure 3b).

4. Superimposition of the mobilised and ultimate tangential stresses on each other (Figure 4).

5. Determination of the areas where the mobilised epyre exceeds the ultimate epyre, and calculation of the delta ΔT_n1 for each soil layer along the lateral surface of the pile.

6. Determination of the areas where the ultimate epyre exceeds the mobilised epyre, and calculation of the delta ΔT_n2 for each soil layer along the lateral surface of the pile.

7. Sum of the deltas ΔT_n1 obtained from point 5, and distribution between the deltas ΔT_n2 obtained in point 6 and the pile tip. The percentage of the exceeded load distribution depends on the stiffness of the soil and the geometric parameters of the pile and is determined from the system of Equation (6).

8. Calculation of the critical force on the pile tip using the Brinch Hansen formula (Equation (12)).

9. Determination of the settlement using Equation (7).

10. Increasing load N, and repeating steps 1–9.

The proposed graph-analytical method of settlement calculation was verified with the numerical calculation method in the Plaxis 2d software package in order to evaluate the correctness of the graph-analytical method and to correct it if necessary. In order to ensure the identity of the initial conditions, in the numerical modelling, the problem was solved in axisymmetric formulation using the Mohr–Coulomb soil model. The adopted soil properties are presented in

Table 1. Physical and mechanical characteristics of soils and pile material for verification with numerical solution.

Soil	Unit Weight γ, kN/m3	Modulus of Elasticity/Deformation E, MPa	Poisson’s Ratio ν	Cohesion c, kPa	Internal Friction Angle φ, deg
EGE-3 Loam	19.5	16.0	0.36	22	20
EGE-2 Sandy loam	19.6	13.0	0.36	16	15
EGE-1 Sand	18.0	21.1	0.30	1	30
Concrete	24.0	30 · 103	0.20	-	-

.

The length of the pile was assumed to be l = 15 m, the radius of the pile was a = 0.5 m, and the radius of the model was b = 10 m; thus, in the numerical solution, the settlement S at r = b is zero, similar to the analytical formulation of the problem. The cross-sections of the analytical and numerical models are shown in Figure 5a,b, respectively.

However, despite the high convergence with numerical methods, any proposed solution should be verified with the results of in situ pile tests to assess the confidence level of the results. Therefore, to verify the graph-analytical solution, the results of the static tests of a 20 m long bored pile with a diameter of Ø1200 mm were used. The scheme of the test setup is presented in Figure 6.

The soils at the construction site are characterised by complex heterogeneous layering of soils. The physical and mechanical properties of the soils are given in

Table 2. Physical and mechanical characteristics of soils for verification with static pile test.

№ EGE	Soil	Unit Weight γ, kN/m3	Modulus of Elasticity/Deformation E, MPa	Cohesion c, kPa	Internal Friction Angle φ, deg
4, 10	Soft-firm and very soft-firm loam	19.9	9	7	20
13	Fine sand	20.5	28	3	33
14	Sandy silt	20.3	30	4	31
16	Stiff loam	19.9	28	29	20

. Figure 7 shows the engineering–geological section with pile foundation planting. Due to the fact that the piles interact with several soil layers, some of which have similar characteristics, they were combined into one layer. For example, EGE-4 and EGE-10, which are loams, were combined into one layer because they have identical characteristics, and EGE-13 and EGE-14, which are sandy soils, were combined and the properties for this layer were assumed to be those of EGE-14.

To investigate the strength of the soils of the contact zone, laboratory experiments were conducted on a direct shear apparatus, as presented in Figure 8a. The soil samples were prepared and tested in accordance with GOST 12248.1-2020 [42]. Very stiff clays and loams were used for the tests, and the concrete plates shown in Figure 8b were used to define the strength reduction factor. The soils were cut from monolithic cores of natural moisture content with the following dimensions: sample diameter—71.4 mm and height—35 mm. Before the test, the samples were pre-compacted with a vertical load equal to 100 kPa for eight hours to restore the natural density of the soil. After pre-compaction, soil shearing was carried out with a constant vertical load and at a constant rate according to the consolidated, drained (slow) shear scheme until the value of shear strain reached 15%. During the test with a concrete sample, it was placed in the lower part of the cage of the shear device and a soil sample cut from the core was placed in the upper part of the cage. It was assumed that the contact zone fell into the the resulting gap. The main physical properties of the soils and the shear rate at which the tests were conducted are given in

Table 3. Physical characteristics of the soils and shear rate.

Soil	Plastic Limit WP, %	Liquid Limit WL, %	Plasticity Index IP	Shear Rate V, mm/min
Very stiff clay	45	90	0.45	0.005
Very stiff loam	28	44	0.16	0.05

.

3. Results and Discussion

Based on laboratory tests results, the graphs of dependence of shear stresses on linear displacement τ = f(d) were plotted. Figure 9 shows the results of testing of very stiff clay according to the schemes “soil–soil” and “soil–concrete”. The ultimate shear resistance values obtained in the “soil–soil” test change from 100 to 165 kPa, while in the “soil–concrete” test, the ultimate shear resistance values range from 55 kPa to 80 kPa.

During the testing of very stiff loams on the scheme “soil–soil”, the ultimate shear stress ranged between 100 and 160 kPa; during the testing with concrete plates, the values of shear stresses ranged between 50 and 80 kPa. Graphs of the dependence of shear stresses on linear displacement τ = f(d) for very stiff loams are presented in Figure 10.

The strength reduction at the soil–structure contact can be characterised by the friction coefficient or the strength reduction factor R_inter, which is defined as the ratio of the strength at the contact point to the strength of the soil. The shear resistance, angle of internal friction, or specific cohesion can be taken as the strength. The obtained average values of the strength reduction factor at the contact point of the soil with concrete are equal to the following:

-

For clays, 0.53;

-

For loams, 0.44.

The greatest influence on the convergence of the graph-analytical solution with the numerical solution, as well as on the value of settlement, was the influence of the load distribution along the length of the pile. Experimental studies of piles [14,15,16] have shown that the pile tip accounts for 0–30% of the applied load, and the lateral surface of the pile about 70–100%. However, often, the pile is embedded in several layers of soil, whose strength and stiffness are not equal; accordingly, the load will not be distributed uniformly on the lateral surface. In the graph-analytical method, it is assumed that the pile tip accounts for 10–15% of the applied load and the lateral surface for 85–90%, with the load distribution along the lateral surface distributed according to the stiffness of the soil. Such relations allowed for good convergence to be obtained with the numerical solution.

Figure 11 shows the characteristics of the appearance and distribution of zones of plastic failure of soil during growth of a vertical load on a pile, which, in general, confirm the accepted assumptions and preconditions for solving the problem using the graph-analytical method. At first, plastic zones appeared in the head zone; then, in the weak middle layer; and further, gradually until full exhaustion of the bearing capacity of the soil occurs along the entire length of the pile. However, it should be noted that the active growth of plastic points under the pile tip occurs, among other things, due to the presence of stress concentrators in the corner zone of the pile.

Figure 12 shows the calculation diagram of the problem for calculating the settlement of the pile using the graph-analytical method at the stage when a load equal to 2200 kN is applied. Exhaustion of the bearing capacity of the soil occurs in two sections: in the upper layer of the soil (ΔT₃₁), with a length of 3.05 m, and in the middle layer of soil (ΔT₂₁), with a length of 1.97 m. The total excess load N_add is distributed between the pile tip and the sections of the lateral surface of the pile where there is a reserve of bearing capacity. According to the results of the analytical solution, the share of load on the tip of the pile is 0.13 and, on the lateral surface, 0.87, which agrees with the data from experimental studies of piles. At the same time, in the top layer of the soil (EGE-3), the load share is 0.29; in the middle weak layer (EGE-2), 0.23; and in the bottom layer (EGE-1), 0.35, i.e., the lower the soil stiffness, the less load falls on this section.

According to the results of the calculations, the graph-analytical and numerical methods were used to construct the graphs of the dependence of settlement on load with the non-linear properties of soils, and the graph of the dependence of settlement on load, obtained via the analytical method and using Formula (7) in the linear formulation. The obtained results are presented in Figure 13 and show good convergence. The maximum difference in settlement values was 12.9%, which does not exceed the engineering accuracy limit of 15%. In addition, there is a similar characteristic of deformation in the load–deposition graphs, and pile failure occurs at the same value for pile load—2600 kN.

However, any proposed problem-solving method should be verified with field pile tests to evaluate the accuracy of the proposed method. Verification of the graph-analytical method was performed based on the results of static pile tests at a construction site. A 20 m long pile with a diameter of 1.2 m was buried in a multilayer foundation represented by dusty sands and loams of different consistencies. The maximum load on the pile was 9600 kN, and the settlement of the pile was 290 mm, i.e., the test was carried out until loss of bearing capacity of the pile.

Figure 14 shows the graphs of the dependence of settlement on load for a single pile constructed using the graph-analytical and numerical methods, as well as based on the results of the static test of the pile. Figure 14a shows the results of calculations without the use of friction coefficients at the pile–soil contact point. The deformation characteristic of the curve obtained via the graph-analytical method is similar to the real curve, but the values are smaller than those obtained from the results of the static test of the pile. The numerical calculation showed that at the initial section of deformation, the curve is close to the graph-analytical curve, but after reaching the ultimate load, it quickly goes to failure.

To approximate the results of the calculations to the real deformation curve, friction coefficients were taken into account in the graph-analytical and numerical calculations: for sands, the coefficient was taken equal to 1.0; for loams, 0.9. The results presented in Figure 14b show that when friction coefficients are taken into account, the graph-analytical curve approaches the real deformation curve. Thus, based on laboratory tests of soils or field tests of piles, it is possible to obtain values of friction coefficients close to the real ones and relevant to the given conditions of the construction site, which will allow for a more accurate description of the pile deformations under load. In numerical modelling, the use of special interface elements and a reduction in soil strength at the contact point does not always lead to positive results. In this case, it resulted in pile failure at a lower load and very rapid accumulation of plastic deformations of soil after failure.

4. Conclusions

• The interaction of a pile with a soil mass is complex and multifactorial, which is reflected in the graph-analytical method. At the same time, the normative methods of calculation do not fully reflect the actual operation of the pile and the processes occurring in the soil when loads are transferred to them, as well as changes in the properties of the contact zone soils. In addition, piles are designed at the initial stage of non-linear soil behaviour, which leads to underutilisation of the bearing capacity of the pile. The combination of these factors leads to higher construction costs and increased complexity of construction and installation work due to the increased length and diameter of the pile.
• Laboratory tests on the schemes “soil–soil” and “soil–concrete” on the direct shear device showed that the strength of soils in contact with the concrete plate is lower than the strength of the same soil in its natural mass. To the greatest extent, the reduction in strength of the soils of the contact zone is influenced by the technology of construction of structures in soil mass, the type of soil, and the roughness of the solid material. According to the results of the performed laboratory tests, the strength reduction factor at the contact point with concrete was equal to 0.53 for clays and 0.44 for loams.
• In the paper, an improved graph-analytical method of calculating a single pile settlement is proposed to utilise the non-linear behaviour of soil on the lateral surface and under the tip of the pile, the possibility of its detachment and slippage after reaching the ultimate strength of the soil, the change in the properties of the contact zone soils, and the load distribution on the pile between its lateral surface and the tip.
• A comparison of the graphs of the dependence of settlement on load obtained using the graph-analytical and numerical methods showed good convergence of the results. The maximum discrepancy of the settlement values was 12.9%, which does not exceed the engineering accuracy limit of 15%, indicating that this method can be used for calculations. In addition, a similar characteristic of deformation of the graphs is observed, and pile failure occurs at the same load value—2600 kN.
• Verification of the proposed graph-analytical method with the results of static pile testing showed that without the use of the strength reduction factors that reduce the strength of the contact zone soils, the graphs have a similar deformation character, but the calculated settlement values are less than the actual ones. The use of the strength reduction factors of 0.9 for loams and 1.0 for sands brought the calculated settlement values obtained by the graph-analytical method closer to the actual values. This indicates that these factors should be taken into account in the calculations.
• For a more accurate prediction of the settlement of a single pile calculated using the graph-analytical method, it is recommended to determine the strength reduction factors experimentally, either with laboratory tests or static tests of pile analogues, in similar engineering–geological conditions. In addition, in order to use the graph-analytical method of calculation in engineering practice, it is necessary to perform a larger number of calculations and compare it with the results of field tests of piles.