Interactions between energy piles caused by temperature Open Access

$Arτ$

calculation parameter of temperature diffusion

C_pile

specific heat of the pile

C_soil

specific heat of the soil

c

specific heat capacity

c_soil

soil cohesion

E_pile

elastic modulus of the pile

E_soil

elastic modulus of the soil

G_bs

shear modulus of the soil at the pile end

H_pile

pile length

k

thermal conductivity

k_b

stiffness of the pile-end soil

k_s

initial stiffness of the shaft stress–relative displacement relationship at the pile–soil interface

L

pile length

l₁

zero point of displacement change

n

pile number of a pile group

r_m

pile limit radius

r_pile, r₀

pile radius

$Trτ$

temperature at the pile shaft at the middle point of the pile length

U_it

total additional displacement of pile i

$uijΔt$

displacement of pile i due to the increase in temperature at pile j

z

vertical position of the calculation point

α

thermal diffusivity

αl_pile

pile conductivity

αl_soil

soil conductivity

γ_pile

pile density

γ_soil

soil density

Δl

length of the computing unit

Δs_bΔT

pile base settlement by heating

Δs_m

changes in displacements for the mth computational unit of pile i

Δs_n

change in displacements for the nth computing unit

ΔT

temperature increase of the pile

Δt

increase in temperature at pile j

Δt_n

change in temperature at the top of the nth computing unit

Δt_n+1

change in temperature at the end of the nth computing unit

$ΔuiiΔt$

increase in displacement of pile i due to its own heating

λ_p

pile line expansion coefficient

λ_s

soil line expansion coefficient

ν_bs

Poisson’s ratio of the soil at the pile end

ν_pile

pile Poisson’s ratio

ν_soil

soil Poisson’s ratio

ρ

density

$∑j=1nuijΔt$

increased displacement of pile i caused by other piles due to heating

τ

time computing heat transfer

τ(z)

shaft shear stress at a given depth z

Φ_soil

friction angle of soil

In recent years, energy piles have been used increasingly due to their economic benefit and environment-friendly advantage. Energy piles, utilising shallow energy to transfer heat between the building foundations and the structure, couple the role of heat exchangers to that of structural foundations for buildings (Brandl, 2006; Laloui and Di Donna, 2013; Mimouni and Laloui, 2015). The advantage of energy pile application is that concrete as a heat-transfer medium has a higher thermal capacity. Therefore, as well as reducing land use and installation cost significantly, energy piles are able to increase thermal productivity while preserving environmental sustainability.

There are relatively few studies about pile bearing and deformation characteristics with thermo-mechanical coupling compared to studies about thermal performance (Bourne-Webb, 2013). Despite the widespread use of energy piles, many mechanistic questions need to be studied, such as additional settlements and pile axial force along pile depth during the heating and cooling cycle. Existing studies show that the heating and cooling cycle can change the adhesion forces at the pile–soil interface and produce additional stresses in the pile body (Murphy et al., 2014; Stewart and McCartney, 2014; Yavari et al., 2014).

Field measurements were carried out to investigate the changes in pile bearing characteristics due to temperature, such as the axial forces along pile depth, the displacements at pile head and the pile-side frictional resistance (Akrouch et al., 2014; Amatya et al., 2012; Hemmingway and Long, 2013; Laloui et al., 2006; Suryatriyastuti et al., 2012). The measured results showed that the temperature applied at the pile could lead to additional thermal stresses and displacements and even cause significant thermal deformation and stress in the pile. The test results also showed that additional thermal stresses mobilised in the pile during the heating and cooling processes had been subjected to restraint conditions (Batini et al., 2015; Murphy and McCartney, 2014; Nguyen et al., 2017; Rotta Loria and Laloui, 2016; Saggu and Chakraborty, 2015). While field measurements provide significant understanding of energy piles, such tests are site specific and it is difficult to analyse comprehensively the bearing characteristics of the energy pile due to many uncertainties in site conditions. In contrast, a laboratory experiment is able to isolate and study important variables while minimising uncertainties. Laboratory tests were carried out to examine the heat-transfer performance and bearing characteristics of energy piles (Kong et al., 2017; Wang et al., 2016), and the conclusion was that the heating–cooling cycle could lead to thermal strain and that stress in the pile shaft could occur. The finite-element method and numerical method were also used to simulate energy piles subjected to superstructure loads and seasonally cyclic thermal loads over several years (Di Donna and Laloui, 2015; Wang et al., 2019; Zarrella et al., 2017; Zhang et al., 2017). Besides field measurements and laboratory tests, the thermal behaviour of energy piles can be simulated using analytical models. Among various simulation tools for the heat transfer of energy piles, the finite-line-heat-source model, an analytical model derived by the Green function, has been widely used in commercial design programs. The finite-line-heat-source model is available to simulate the thermal behaviour of soils surrounding the energy pile by neglecting the radial dimension of the heat source and vertical heat flow (Li et al., 2014; Park et al., 2017).

Studies have shown that temperature would lead to additional pile displacements and axial forces along pile depth and there is a point at which the temperature-induced displacement is zero (Wang et al., 2019). In practical engineering, it is very important to determine that position and estimate the additional pile force. Meanwhile, studies about bearing characteristics of energy pile groups are also important, and simplified analytical methods for energy pile groups are badly needed due to deficiency relating to engineering application.

In this study, the perfect line elastic model is adopted to describe the relationship between the pile base settlement and the pile base load, and the response between the pile shaft and surrounding soils is also simulated by the line elastic model. Based on the assumption that the pile shaft and end resistance are in the elastic stage, a new simplified method is proposed to estimate the zero point of displacement variation, and the additional pile axial force was calculated by using the proposed method. The finite-line-source model was adopted to estimate the temperature distribution among surrounding soils, and the factors affecting the distribution were also analysed. The temperature distribution obtained by the proposed method was compared with that obtained from the finite-element model, and the analysis results show that the proposed method can be used to estimate the temperature distribution accurately. A method for estimating interactions caused by heating between energy piles is proposed, and a comparison between the results obtained by the proposed method and those obtained using Abaqus software on energy pile was done to verify the accuracy of the proposed method. The analysed results show that reasonable predictions can be made without expensive and time-consuming analyses by means of the method proposed in this paper.

Even though there are many existing methods for analysing the settlement of pile foundations and the load-transfer mechanism of a single pile, these methods are not very applicable to energy piles due to the temperature load. The behaviour between the pile shaft and surrounding soils in this paper is described by a simple bilinear model, as shown in Figure 1(a). From the bilinear relationship shown in Figure 1(a), it can be seen that the pile-side friction increases linearly with gradually increasing relative displacement between the pile shaft and surrounding soils. When the pile–soil relative displacement reaches the value s_u1, the shear stress at the interface between the pile side and surrounding soils changes. This relationship can be expressed in the following equation

where τ (z) is the shaft shear stress at a given depth z, Δs_z is the pile–soil relative displacement developed in the pile–soil interface at a given depth z and k_s are empirical coefficients determined by laboratory test or by back-analysis of field test results.

As Figure 1(b) shows, the relationship between the pile-end loads and settlements until the ultimate bearing capacity can also be expressed by the bilinear model based on the existing research results (Xie et al., 2013).

where τ_b is the pile-end resistance and s_b is the pile-end settlements, and the relationship between the pile-end settlements and loads is shown in Figure 1(b). When the displacement of the pile-end soil is within s_ub1, the stiffness of the pile-end soil is k_b. The coefficient k_s, as the initial stiffness of the shaft stress–relative displacement relationship at the pile–soil interface, can be obtained as (Randolph and Wroth, 1978; Wang et al., 2012; Zhang et al., 2010, 2016)

The initial soil stiffness at the pile base, k_b is written as the following expression

where G_bs and ν_bs are the shear modulus and Poisson’s ratio, respectively, of the soil at the pile end.

As per the conclusion expressed by Wang et al. (2019), the applied load at the energy pile head cannot be heavier than the value which will lead to non-linear settlement at the pile top. The relative displacement between the pile shaft and surrounding soils is less than the value of s_u1 as Figure 1(a) shows; the zero point of displacement change is considered as the origin, and the length from the zero point to the pile head has the value of l₁, as shown in Figure 2. Assuming that the settlement of soil at the pile end caused by heating was negligible and the pile base settlement, Δs_bΔT, due to heating is written as the expression

l₁, the zero point of displacement change due to heating from the top of the pile, can be written by the equilibrium expression

where λ_p is the coefficient of linear expansion for the pile, λ_s is the coefficient of linear expansion for soils surrounding the pile and ΔT is the temperature increase of the pile body. Thus, the zero point of displacement change l₁ can be obtained by using the following equation

From Equation 7, it can be seen that the position of the zero point of displacement change of the pile top has no direct relationship to the temperature change of the pile body. The main factors that influence the position of the zero point of displacement change are the initial soil stiffness at the pile base, the shear stiffness of soils surrounding the pile and coefficients of linear expansion for the pile and surrounding soils. In addition, the additional axial pile shaft force from the pile head to the zero point can be obtained by the following equation, while the relative displacement between the pile shaft and surrounding soils is less than s_ub1.

The analytic solutions to Equation 8 can be expressed as the following equation

The additional axial pile shaft force from the pile end to the zero point can be obtained by using the following equation

The analytic solutions to Equation 8 can be expressed in the following equation

In multilayered soils, assuming that the zero point of displacement change is in the ith layer, the length from the ith layer soil heat to the zero point is l₁, as shown in Figure 3. The zero point of displacement change can be obtained by using the following equation

All the parameters in the preceding equation are shown in Figure 3. The additional axial pile shaft force from the pile head to the zero point can be obtained by the following equation, assuming that the relative displacements between the pile shaft and surrounding soils are in the first linear stage in all the multilayered soils

The additional axial pile shaft force from the pile top to the zero point can be obtained by using the following equation

In order to determine the specific position of layers in Equation 14, the following procedure can be adopted for calculation.

• Confirm the values of k_si according to the relative displacements between the pile shafts and surrounding soils obtained in the preceding step.

• Assume that the zero point is in the lowest layer – that is, the nth layer, shown in Figure 3.

• Calculate to obtain the solution of l₁.

• If the solution of l₁ is negative, then assume that the zero point is in the (n − 1)th layer, as shown in Figure 3.

• Continue to calculate the solution to Equation 12 until a positive value of l₁ is obtained.

• Then, calculate the additional axial pile shaft force using Equations 13 and 14 along the pile shaft in multilayered soils.

The two-dimensional model was initially studied by Eskilson (1987), in which a line heat source has a finite length from the ground surface to a certain depth h along the z-axis, as shown in Figure 4. This model has been widely used and developed to simulate temperature conduction (Man et al., 2010, 2011). The initial temperature uniformity in a semi-infinite medium is t ₀. At a given time, the finite-length uniform line heat source perpendicular to the boundary surface with a strong degree q ₁ (W/m) begins to give off exothermic (or endothermic) heat. Using the principle of the virtual heat source method, a virtual line heat source at the symmetrical position of the boundary surface to the line heat source is assigned, as shown in Figure 4. An analytical solution has been derived by the method to ensure the boundary condition of constant temperature on the surface, which takes the form of Equation 15 (Man et al., 2010). In this paper, the finite-line-heat-source model is adopted to simulate the thermal expansion in the surrounding soils around the pile.

where H is the length of the energy pile; z is the vertical position of the calculation point; τ is the time computing heat transfer; $r=[(x−x′)2+(y−y′)2]1/2$⁠; erfc(z) is an error function and has the characteristic of $erfc(z)=1−(2/π1/2)∫0zexp(−μ2)dμ=(2/π1/2)∫z∞exp(−μ2)dμ$⁠; k is thermal conductivity (W/(m K)); and α = λ/ρc (m²/s) is thermal diffusivity, where ρ (kg/m³) is density and c (J/(kg K)) is specific heat capacity.

Around the surrounding soils, the thermal diffusions can be approximately estimated by the following expression, derived from Equation 15

$Trτ$ is the temperature at the pile shaft at the middle point of the pile length and $Arτ$ is the value at the middle point of pile length when r = r _pile; $Arτ$ can be expressed in the following equation

The thermal diffusions around the surrounding soils obtained from Equations 16 and 17 were identical to that calculated by the finite-element method, as shown in Figure 5. The parameters in the analysis adopted the values from Table 1, for both the finite-element method (using Abaqus software) and the proposed method.

In order to estimate the vertical displacements of energy pile groups subjected to thermal loads, the interaction method of conventional pile groups could also be performed to analyse the interaction between energy piles. The pile displacement along the pile depth caused by the other piles enduring heating could be considered as the composition of a series of units of the pile shaft. The pile could be divided to obtain enough computational units that could satisfy the computing requirement. The changes in displacements for every computational unit could be estimated by the following expression

Δs _n is the change in displacements for the nth computing unit, Δl is the length of the corresponding unit, Δt _n is the change in temperature at the top of the nth computing unit and Δt _n+1 is the change in temperature at the end of the nth computing unit; Δt _n and Δt _n+1 could be estimated by Equation 16. The total change in displacements for the pile could be obtained by summing up changes in displacements of all computing units, as shown in the following equation

$uijΔt$ is the displacement of pile i due to the increase in temperature at pile j, Δt is the increase in temperature at pile j, Δs _m is the changes in displacements for the mth computational unit of pile i. The total increase in displacements for pile i in the energy pile group caused by other piles enduing heating can be estimated approximately by the following equation

As described earlier, the displacement of pile i in the pile group can be expressed as the sum of the self-additional displacement caused by its own heating and the additional displacements caused by the other piles at its position. Assuming that the number of piles in the pile group is n, the additional displacement of pile i can be expressed as

In this formula, U _it is the total additional displacement of pile i, n is the pile number of the group, $ΔuiiΔt$ is the increase in displacement of pile i due to its own heating and $∑j=1nuijΔt$ is the increased displacement of pile i caused by other piles due to heating.

The zero point of displacement change can be obtained by using Equation 7, and the displacements at the pile top can be obtained by using Equations 7 and 17.

The behaviour of pile groups has been studied by many scholars in the past decades. However, researches about the interactions between energy piles are quite limited. Many tests have been carried out to study the responses of piles due to change in temperature, and the tests’ results provided some great insight into the thermo-mechanical response of energy piles subjected to heating. However, existing tests mainly focused on single piles. However, mechanical interactions between the piles may occur when only a part of the foundation is thermally activated piles – for example, the case described by Mimouni and Laloui (2015). As a result, different displacements between energy piles and conventional piles could develop and induce potential damage to the supported structure. Due to the lack of experimental data about the interaction between energy piles, the finite-element method was adopted in this paper to analyse the interaction between energy piles. The parameters used in the calculation model were consistent with that shown in Table 1, and the layout of the 3 × 3 pile group is shown in Figure 6; the distance between two closely adjacent piles was 3 m. The extension of the simulated domain was chosen to be large enough to avoid boundary effects, as shown in Figure 6. The finite-element analysis adopted the sequential thermo-mechanical coupling method. The implication of this method is that stress does not affect temperature distribution, but temperature causes stress changes. In the process of heat-conduction analysis, the heat-conduction elements were adopted and the three-dimensional stress elements were adopted in the process of coupled thermo-mechanical analysis. The pile adopted the elastic model and the soils adopted the Mohr–Coulomb elastoplasticity model in the process of coupled thermo-mechanical analysis.

The proposed method is an approximate calculation method, and the thermal parameters of soil and pile are regarded as identical, excepting the value of the expansion coefficient. The existing tests mainly focused on a single pile to study the responses of piles due to change in temperature, and experimental data about the interaction between energy piles were rare. Therefore, the finite-element method is adopted to compare with the results obtained by the proposed method. The zero point of displacement change in this case study is estimated to be 18 m below the pile head, and the case study applied a temperature increase of 20°C. The parameters in the analysis using Abaqus adopted the values from Table 1. The parameters used in this case for the finite-line-source model are also listed in Table 1. Comparisons between the results form Abaqus and those computed by the proposed approach are shown in Figure 7. The results show that the pile-head displacements caused by heating calculated by the proposed method are generally consistent with the finite-method results.

On the assumption that the pile shaft and end resistance are in the elastic stage – that is, the applied pile head load should not be heavier than the value from which non-linear settlement occurred on the load–settlement curve – the following conclusions can be obtained.

• A new simplified method is proposed to estimate the zero point of additional displacement along the pile depth. The position of the zero point has no obvious relationship to the temperature change in the pile body. The main factors that influence the point are the initial soil stiffness at the pile base, the shear stiffness of soils surrounding the pile and the coefficients of linear expansion for the pile and surrounding soils.

• A simplified method for calculating the additional pile axial force caused by heating is also proposed.

• The finite-line-source model can be adopted to estimate the temperature distribution among surrounding soils. The temperature distribution obtained by the finite-line-source model has good agreement with that obtained using Abaqus software, and the analysed results show that the proposed method can be used to estimate the temperature distribution accurately among surrounding soils.

• A method fort estimating the interaction caused by heating between energy piles was proposed. The calculated results obtained by the proposed method have good agreement with those obtained using Abaqus software. The analysed results show that the proposed method can be used to estimate the interaction between energy piles with accuracy.

This research presented in this paper was supported by the National Natural Science Foundation of China (research grant numbers 51708496 and 51579217) and the Zhejiang Provincial Natural Science Foundation (research grant number LY16E080010). These financial supports are gratefully acknowledged.