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# A simple load transfer method for energy pile groups

Huaibo Song, Huafu Pei, Jean-Michel Pereira, Anh Minh A.M. Tang, Chao Zhou

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## To cite this version:

Huaibo Song, Huafu Pei, Jean-Michel Pereira, Anh Minh A.M. Tang, Chao Zhou. A simple load transfer method for energy pile groups. Computers and Geotechnics, 2023, 159, pp.105483. 10.1016/j.compgeo.2023.105483. hal-04375003

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https://enpc.hal.science/hal-04375003v1

Submitted on 14 Jan 2024

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Distributed under a Creative Commons Attribution 4.0 International License
1 **A simple load transfer method for energy pile groups**

2 Huaibo Song¹, Huafu Pei²*, Jean-Michel Pereira³, Anh Minh Tang⁴, Chao Zhou⁵

3

4 ¹Ph.D. student, MSc, School of Civil Engineering, State Key Laboratory of Coastal and Offshore

5 Engineering, Dalian University of Technology, Dalian, China; Navier, Ecole des Ponts, Univ

6 Gustave Eiffel, CNRS, Marne-la-Vallée, France, E-mail: songhuaibo@mail.dlut.edu.cn;

7 huaibo song@enpc.fr, https://orcid.org/0000-0002-4592-9521

8 ²* Professor, Ph.D., School of Civil Engineering, State Key Laboratory of Coastal and Offshore

9 Engineering, Dalian University of Technology, Dalian, China (corresponding author).

10 E-mail: huafupei@dlut.edu.cn, https://orcid.org/0000-0003-3862-0865

11 ³Professor, Ph.D., Navier, Ecole des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vallée, France,

12 E-mail: jean-michel.pereira@enpc.fr, https://orcid.org/0000-0002-0290-5191

13 ⁴ Research Director, Ph.D., Navier, Ecole des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vallée,

14 France, E-mail: anh-minh.tang@enpc.fr, https://orcid.org/0000-0002-7149-8497

15 ⁵ Associate Professor, Ph.D., Department of Civil and Environmental Engineering, The Hong Kong

16 Polytechnic University, Hung Hom, Kowloon, HKSAR, China, E-mail: zhou@polyu.edu.hk

17 **Abstract**

18 Various analysis methods have been developed to simulate the performance of energy pile
19 groups subjected to thermal and mechanical loads. Although they considered the interactions
20 between piles in the group, the ‘sheltering-reinforcing’ effect was usually ignored during the
21 analysis, or the methods were not experimentally validated. Therefore, to address this gap, we
22 propose an experimentally validated load transfer model that considers the ‘sheltering-
23 reinforcing’ effect for the thermomechanical analysis of energy pile groups. The proposed
24 method modeled two well-documented full-scale field tests and two finite-element simulation
25 cases. The comparison results showed that the proposed method could capture several essential
26 aspects of the energy pile group, including thermally induced strain, stress, displacement, and
27 group effects. Furthermore, a parametric analysis was performed to evaluate the effects of
28 relevant parameters on the thermomechanical behavior of the energy pile group.

29 **Keywords:** load transfer method; energy pile group; thermomechanical behavior; geothermal
30 energy.

31 **1. Introduction**

32 Energy piles are environment-friendly technologies that have attracted increasing attention
because of their effective solicitation of shallow geothermal energy (Rotta Loria and Coulibaly 2021). Unlike conventional pile foundations, the operation of an energy pile involves complex coupling between thermal, mechanical, and other physical fields (Bourne-Webb and Bodas Freitas 2020). It is also necessary to consider pile-pile and pile-slab interactions under thermal-mechanical loading for energy pile groups (Moradshahi et al. 2020, Ravera et al. 2020a, 2020b, Fang et al. 2022). Recently, understanding this complex mechanism, seeking simple approaches to describe it, and then applying it to the engineering design of energy piles has become a hot topic in energy pile research.

Numerous in situ field tests (Bourne-Webb et al. 2009, Murphy et al. 2015, Faizal et al. 2019, Sutman et al. 2019a, Fang et al. 2020, 2022, Wu et al. 2020, 2021), small-scale laboratory model tests (Kalantidou et al. 2012, Yavari et al. 2016a, Yazdani et al. 2019b, Kong et al. 2020, Song et al. 2020), and centrifuge model tests (Ng et al. 2015, 2016, 2021, Ng and Ma 2019) have been conducted to investigate the thermomechanical behavior of energy pile and pile groups. It is impractical to use experimental methods to comprehensively investigate the effects of all the factors on the performance of energy piles considering the enormous time-consuming and financial expense. Thus, the development of experimentally verified numerical methods to supplement the experimental methods is necessary to thoroughly study the thermomechanical mechanisms of energy piles. Various numerical approaches with different complexities have been used in previous years, including finite element models (Yavari et al. 2014, Rammal et al. 2018, Garbellini and Laloui 2019a, Nguyen et al. 2019, Arzanfudi et al. 2020, Georgiadis et al. 2020, Sutman et al. 2020), finite difference models (Suryatriyastuti et al. 2012, 2014, Luo and Hu 2019, Najma and Sharma 2021, Liu et al. 2022), and other theoretical models (Rotta Loria and Laloui 2016, 2017a, Rotta Loria et al. 2018, 2020a, Garbellini and Laloui 2019a, Iodice et al. 2020, 2023, Ravera et al. 2020a, 2020b, Cui et al. 2021, Saeidi Rashk Olia et al. 2022, Liu et al. 2022, Song et al. 2022, Song and Pei 2022, Pei et al. 2022a). Among these, the load transfer method is commonly used for energy pile design because it overcomes the drawbacks of high time consumption, high computational cost, and high requirements of the complete
numerical simulation method. The load transfer method is very user-friendly and well-suited for the engineering design of energy piles.

The load transfer method was initially proposed by Seed and Reese (Seed and Reese 1957) and then gradually applied to investigate the thermomechanical response of an isolated energy pile. For example, Knellwolf et al. (2011) developed a load transfer method using the three-polyline load transfer curve proposed by Frank and Zhao (1982). In situ isolated energy piles were modeled, and their thermomechanical behavior was analyzed. The comparison results between the simulations and the measured data showed good agreement. Sutman et al. (2019b) developed a model by incorporating the unloading-reloading response into the load transfer curve and validated it through field test results. The above studies demonstrate the applicability of the load transfer method for studying the thermomechanical response of an isolated energy pile. As an essential part of the energy pile group, the successful application of the single pile response model lays the foundation for developing a pile group response model.

Recently, several studies (Ravera et al. 2020a, Fei et al. 2022, Liu et al. 2022) extended the load-transfer method to consider the thermomechanical interaction of energy piles in groups. Ravera et al. (2020a) modified the load transfer curve of an isolated energy pile in a group using a displacement factor that considered group effects. The field test results of an energy pile group consisting of four energy piles were used to validate the proposed method. A comparison between the simulated and measured results showed the suitability of the proposed method for analyzing the thermomechanical performance of the energy pile group. Nevertheless, it should be noted that the determination of the displacement factor is based on an empirical formula, which is proposed based on the fitting of the finite-difference modeling results of the 5×5 pile group (Comodromos et al. 2016). A significant difficulty in applying this empirical formula to the analysis of the energy pile group is that the effect of the temperature on displacement factor cannot be considered. Thus, additional experiments or numerical simulations are required for specific analysis cases to comprehensively determine the appropriate displacement factors. Besides, the load transfer method proposed by Ravera et al. (2020a) has some limitations in
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clarifying the thermomechanical behavior of energy pile groups because it cannot take into account the 'sheltering-reinforcing' effects of the energy pile group into consideration.

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The 'sheltering-reinforcing' effects characterize the interaction between piles in groups and are considered in previous studies of conventional pile groups (Mylonakis and Gazetas 1998, Wang et al. 2016). The results indicated that the calculated pile head displacement was overestimated without considering this effect, and the error increased significantly with increasing group size. For the energy pile group, the 'sheltering-reinforcing' effects act throughout thermomechanical loading, which can be observed in the field test results (Rotta Loria and Laloui 2017b, 2018, Fang et al. 2020). Liu et al. (2022) proposed a load transfer method to rapidly calculate the response of an energy pile group in which they considered the reinforcing effect of neighboring piles in the group. The applicability of this method to pile groups was verified by comparing it with finite element simulations that did not consider the nonlinear pile-soil interaction (Rotta Loria and Laloui 2016). Nevertheless, the presence of the slab and the pile-slab-soil interaction had not been considered in their study, which would limit the scope of application of the method due to the importance of the slab in the energy pile group. Besides, there is no adequate verification of whether the proposed model can accurately calculate the critical aspects of the thermomechanical responses of the energy pile group, including the pile axial strain, stress, and pile head displacement. Therefore, to address this gap, it is necessary to develop an experimentally validated model.

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This study aims to provide an experimentally validated simple load transfer model that considers the 'sheltering-reinforcing' effects for thermomechanical analysis of energy pile groups. Section 2 details the proposed model and the developed thermomechanical performance analysis algorithm for energy pile groups based on the proposed model. Section 3 verifies the results obtained by the proposed method using field tests and finite element simulations. A comprehensive sensitivity analysis of the parameters involved in the proposed model is presented in Section 4. Finally, conclusions are presented in Section 5.

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## 2. Development of the load transfer model for the energy pile group

Some assumptions of the load transfer mechanism for energy piles are employed in this study: (1) the load-displacement relationship (i.e., t-z model) and pile and soil properties are assumed to be independent of temperature variations, based on the fact that previous studies suggested that the effect of temperature within the range of 5 - 40 °C on soil and concrete properties is insignificant (Yavari et al. 2016b, Yazdani et al. 2019a, Rotta Loria and Coulibaly 2021); (2) the strain and stress analysis of the energy pile in the group is achieved by considering the pile-to-pile interaction in the pile group; (3) the interaction between the slab, energy piles in the group, and soil can be considered through assuming an additional spring connecting with the pile heads; (4) The influence of the axial deformation of the piles on the thermomechanical behavior of the piles is mainly considered in the study to simplify the calculation(Ravera et al. 2020a). The last assumption is based on the fact that a numerical study of energy piles (Olgun et al. 2014) and full-scale field studies (Faizal et al. 2019) reported that the normal contact stress of the piles did not change significantly owing to their radial thermal expansion of the energy piles. Other studies on the load transfer analysis (Knellwolf et al. 2011; Chen and McCartney 2016) have also confirmed that the influence of radial deformation on the pile-soil axial interaction is negligible. Therefore, the present study did not consider the influence of the radial deformation of the energy pile.

### 2.1 Load transfer function at the shaft of the isolated energy pile

According to a previous study (Lee and Xiao 2001), the nonlinear performance of the pile load displacement is divided into (i) the nonlinear relationship between the pile shaft shear stress and pile-soil relative displacement and (ii) the nonlinear stress-strain relationship of the surrounding soil induced by the pile shaft shear stress. Caputo (1984) and Trochanis et al. (1991) divided surrounding soil into disturbed and undisturbed zones. The disturbed zone is a thin layer of soil on the soil/pile interface. The nonlinearity of the pile-soil interaction was concentrated in this interface. The area outside the interface is the undisturbed zone. Because this zone is subjected to a relatively low stress change, the stress-strain behavior of the soil is
mainly elastic. The nonlinear response of pile groups primarily depends on the nonlinear characteristics of single piles, and the pile-pile interaction is elastic. The total shaft displacement at a certain depth (i.e., $u$) can be divided into pile-soil relative displacement (i.e., $p$) and the elastic displacement of the surrounding soil (i.e., $s$). Recently, these concept has been introduced into the thermomechanical analysis of energy pile group (Fei et al. 2020, Liu et al. 2022). As suggested by Lee and Xiao (2001), the thickness of the interface was assumed to be zero, which can provide a simplified mathematical processing method for simulating the pile-soil relative displacement and elastic displacement of the soil around the pile. Thus, the total pile shaft displacement can be expressed as follows:

$$u = p + s \qquad (1)$$

The elastic soil displacement induced by pile shaft stress can be calculated using the expression proposed by Randolph and Worth (1978):

$$s = \frac{\tau r_0}{G} \ln \frac{r_m}{r_0} = \psi \tau \qquad (2)$$

where $G$ is the shear modulus of the soil, $r_0$ is the pile radius, $r_m$ is the shear influence radius.

In this study, an exponential model was employed to calculate the pile-soil relative displacement of the pile-soil interface because it can better simulate the field test results without adopting complex fitting parameters (Kezdi. 1957). The exponential model is expressed as follows:

$$\tau = \tau_u \left[ 1 - \exp \left( - \frac{G_{s,max}}{\tau_u} p \right) \right] \qquad (3)$$

where $\tau$ is the pile shaft shear stress, $\tau_u$ is the ultimate shear stress, $G_{s,max}$ is the initial stiffness of the pile-soil interface, the unit of which is kPa/m.

It should be noted that the deformation mechanisms of energy piles and conventional piles differ. For the energy pile, when the pile body is heated up, it expands, the upper part of the pile side friction decreases, and it is in the unloading stage. In contrast, the shaft shear stress at the
lower end of the pile side increases and is in a loading state. When the pile body is cooled, it shrinks, and the situation is the opposite of that during heating. Therefore, Eq. (3) should be improved for the thermomechanical analysis of the energy pile. We developed a model by employing the improved Masing rule to simulate the loading-unloading-reloading path of the pile-soil interaction, and its expression can be written as Eq. (4). Further details can be found in (Pei et al. 2022b).

$$ \tau = \tau_0 + \eta(\tau_u - \eta\tau_0) \left[ 1 - \exp\left(-\eta \frac{G_{s,max}}{\tau_u - \eta\tau_0}(p - p_0)\right) \right] \quad (4) $$

where $\eta$ denotes the loading and unloading branching judgment factor, $\eta = -1$ and $\eta = 1$ represent unloading and reloading conditions, respectively, $p_0$ denotes the relative displacement of the pile and soil corresponding to the stress reverse point, $\tau_0$ is the pile shaft stress corresponding to the stress reversal point. When $p_0$ and $\tau_0$ are equal to zero, Eq.(4) degenerates to the expression of the initial loading curve.

Substituting Eqs. (2) and (4) into Eq. (1), the total pile shaft displacement at a given depth can be expressed as Eq. (5):

$$ u = p + s = p_0 + \frac{\eta\tau_0 - \tau_u}{\eta G_{s,max}} \ln\left(1 - \frac{\tau - \tau_0}{\eta(\tau_u - \eta\tau_0)}\right) + \psi\tau \quad (5) $$

where $p_0 = u_0 - s_0 = u_0 - \psi\tau_0$, and $u_0$ denotes the total shaft displacement of the pile corresponding to the shaft shear stress reversal point. Thus, Eq. (5) can be expressed using Eq. (4), which is the load transfer function used to simulate the relationship between the pile shaft resistance and the relative pile-soil displacement.

$$ u = u_0 + \frac{\eta\tau_0 - \tau_u}{\eta G_{s,max}} \ln\left(1 - \frac{\tau - \tau_0}{\eta(\tau_u - \eta\tau_0)}\right) + \psi(\tau - \tau_0) \quad (6) $$

## 2.2 Group interaction for energy piles considering ‘sheltering-reinforcing’ effect

As discussed in the previous section, the pile-soil-pile interaction in the pile group can be considered as linearly elastic and suitable for the principle of superposition. According to previous studies (Randolph and Wroth 1979), for the two energy piles system (i and j)
subjected to thermomechanical loading, the vertical elastic displacement of the pile i can be divided into three parts: (i) the displacement ($s_{i0}$) induced by its own load; (ii) the displacement ($\Delta s_{i0}$) induced by the ‘sheltering-reinforcing’ effect of pile j; (iii) the additional displacement ($s_{ij}$) caused by j pile under its own load. If the ‘sheltering-reinforcing’ effect is not considered, the total elastic displacement of i pile can be written as follows:

$$s_i = s_{i0} + s_{ij} \qquad (7)$$

In the proposed method, we consider the ‘sheltering-reinforcing’ effect. Therefore, the total elastic displacement of i pile can be written as follows:

$$s_i = s_{i0} - \Delta s_{i0} + s_{ij} \qquad (8)$$

According to the formulation proposed by Randolph and Worth (1978), $s_{i0}$ can be calculated as follows:

$$s_{i0} = \frac{\tau_{si} r_0}{G} \ln \frac{r_m}{r_0} \qquad (9)$$

The shear stress of pile i transferred to the side of pile j can be expressed as follows:

$$\tau_{sji} = \frac{\tau_{sij} r_0}{s_{aij}}, (r_0 < s_{aij} < r_m) \qquad (10)$$

where $s_{aij}$ denotes the distance between the axes of two piles. Considering the ‘sheltering-reinforcing’ effects of pile j on pile i, $\Delta s_{i0}$ can be obtained by:

$$\Delta s_{i0} = \frac{\tau_{sji} r_0}{G} \ln \frac{r_m}{s_{aij}} = \frac{\tau_{sij} r_0}{G} \ln \frac{r_0}{s_{aij}} \qquad (11)$$

If the existence of pile j is not considered, then the displacement at the position of pile j caused by pile i under its own load is:

$$s_{ji0} = \frac{\tau_{sji} r_0}{G} \ln \frac{r_m}{s_{aij}}, (r_0 < s_{aij} < r_m) \qquad (12)$$

In addition, the shear stress transferred from pile i to the side of pile j (i.e., $\tau_{sij}$) reduces the displacement of pile j as follows:
$$211 \qquad \Delta s_{ji0} = \frac{\tau_{sji} r_0}{G} \ln \frac{r_m}{r_0} = \frac{\tau_{sji} r_0}{s_{aij}} \frac{r_0}{G} \ln \frac{r_m}{r_0} \qquad (13)$$

Therefore, the additional displacement of pile j induced by the load of pile i can be expressed as follows:

$$214 \qquad s_{ji} = s_{ji0} - \Delta s_{ji0} = \frac{\tau_{sji} r_0}{G} \left( \ln \frac{r_m}{s_{aij}} - \frac{r_0}{s_{aij}} \ln \frac{r_m}{r_0} \right) \qquad (14)$$

According to the assumption that $\tau_{sij} = \tau_{sji} = \tau_{si}$ (Zhang et al. 2016), the additional elastic displacement of pile i caused by the load of pile j is equal to the additional elastic displacement of pile j caused by the load of pile i, that is:

$$218 \qquad s_{ij} = s_{ji} = \frac{\tau_{si} r_0}{G} \left( \ln \frac{r_m}{s_{aij}} - \frac{r_0}{s_{aij}} \ln \frac{r_m}{r_0} \right) \qquad (15)$$

Thus, for a two-pile system, the total elastic displacement of pile i can be obtained as follows:

$$221 \qquad s_i = s_{i0} - \Delta s_{i0} + s_{ij} = \frac{\tau_{si} r_0}{G} \left[ \ln \frac{r_m}{r_0} - \frac{r_0}{s_{aij}} \ln \frac{r_m}{s_{aij}} + \ln \frac{r_m}{s_{aij}} - \frac{r_0}{s_{aij}} \ln \frac{r_m}{r_0} \right] \qquad (16)$$

For a group of n piles, the total elastic displacement of pile i can be calculated as follows:

$$224 \qquad s_i = s_{i0} - \sum_{j=1, j \neq i}^{n} (\Delta s_{i0} - s_{ij}) = \frac{\tau_{si} r_0}{G} \left[ \ln \frac{r_m}{r_0} - \sum_{j=1, j \neq i}^{n} \left( \frac{r_0}{s_{aij}} \ln \frac{r_m}{s_{aij}} - \ln \frac{r_m}{s_{aij}} + \frac{r_0}{s_{aij}} \ln \frac{r_m}{r_0} \right) \right] \qquad (17)$$

Therefore, the total shaft displacement of the energy pile i is:

$$226 \qquad u = p + s = u_0 + \frac{\eta \tau_0 - \tau_u}{\eta G_{s \max}} \ln \left( 1 - \frac{\tau_{si} - \tau_0}{\eta(\tau_u - \eta \tau_0)} \right) + \psi'(\tau_{si} - \tau_0) \qquad (18)$$

where $\psi'$ is the pile-pile interaction parameter, which can be written as follows:

$$228 \qquad \psi' = \frac{r_0}{G} \left[ \ln \frac{r_m}{r_0} - \sum_{j=1, j \neq i}^{n} \left( \frac{r_0}{s_{aij}} \ln \frac{r_m}{s_{aij}} - \ln \frac{r_m}{s_{aij}} + \frac{r_0}{s_{aij}} \ln \frac{r_m}{r_0} \right) \right] \qquad (19)$$
## 2.3 Load transfer function at the base of energy pile group

Similar to the definition of the pile-side displacement, the displacement at the pile end is the sum of the pile-soil relative displacement (i.e., $p_p$) and the elastic displacement of the soil at the pile tip (i.e., $s_p$). It can be written as

$$u_p = p_p + s_p \qquad (20)$$

The pile-soil relative displacement can be simulated using an exponential model (Liu et al., 2022), which can be calculated as follows:

$$\sigma_p = \sigma_{up} \left[ 1 - \exp \left( - \frac{G_{p \max}}{\sigma_{up}} p_p \right) \right] \qquad (21)$$

where $\sigma_p$ is the pile tip resistance, $G_{pmax}$ is the initial stiffness of the pile tip-soil interface; $\sigma_{up}$ is the ultimate pile tip resistance.

According to the improved Masing rule (Pei et al. 2022), the load transfer function for simulate the loading-unloading-reloading path of the pile-soil interaction at the pile tip can be written as

$$\sigma_p = \sigma_{p0} + \kappa (\sigma_{up} - \eta \sigma_{p0}) \left[ 1 - \exp \left( -\kappa \frac{G_{p \max}}{\sigma_{up} - \kappa \sigma_{p0}} (p_p - p_{p0}) \right) \right] \qquad (22)$$

Thus, the pile-soil relative displacement of the energy pile tip can be written as

$$p_p = p_{p0} + \frac{\kappa \sigma_{p0} - \sigma_{pu}}{\kappa G_{p \max}} \ln \left( 1 - \frac{\sigma_p - \sigma_{p0}}{\kappa (\sigma_{pu} - \kappa \sigma_{p0})} \right) \qquad (23)$$

where $\kappa$ denotes the loading and unloading branching judgment factor, $\kappa = -1$ and $\kappa = 1$ represent unloading and reloading conditions, respectively, $p_{p0}$ denotes the relative displacement of the pile tip and soil corresponding to the stress reversal point, $\sigma_{p0}$ is the pile tip resistance corresponding to the stress reversal point. When $p_{p0}$ and $\sigma_{p0}$ equal zero, Eq. (22) degenerates to the expression for the initial loading curve.

According to Randolph and Wroth (1979), the elastic vertical displacement of the soil induced by the pile tip resistance can be obtained as
$$s_p = \frac{\pi r_0 (1 - \nu)}{4G_{\text{soil}}} \sigma_p = \lambda \sigma_p \quad (24)$$

where $\lambda$ is the model parameter. According to the studies in (Lin and Dai 2014), the pile-pile interaction at the pile tip is insignificant. Therefore, the total displacement of the pile tip can be expressed as follows:

$$u_p = p_p + s_p = p_{p0} + \frac{\eta\sigma_{p0} - \sigma_{pu}}{\eta G_{p\max}} \ln\left(1 - \frac{\sigma_p - \sigma_{p0}}{\eta(\sigma_{pu} - \eta\sigma_{p0})}\right) + \lambda\sigma_p \quad (25)$$

where $p_{p0} = u_{p0} - s_{p0} = u_{p0} - \lambda\sigma_{p0}$, and $u_{p0}$ denotes the total displacement of the pile tip corresponding to the pile tip resistance reversal point. Thus, Eq.(25) can be expressed as Eq.(26), which is the load-transfer function used to simulate the relationship between the pile tip resistance and the relative pile tip-soil displacement.

$$u_p = u_{p0} + \frac{\eta\sigma_{p0} - \sigma_{pu}}{\eta G_{p\max}} \ln\left(1 - \frac{\sigma_p - \sigma_{p0}}{\eta(\sigma_{pu} - \eta\sigma_{p0})}\right) + \lambda(\sigma_p - \sigma_{p0}) \quad (26)$$

## 2.4 Pile-slab-soil interaction for the energy pile group

The pile-slab-soil interaction of the energy pile group is simulated using a spring linked to the pile head during the load transfer method, as adopted by Knellwolf et al. (2011) and Ravera et al. (2020a). The spring stiffness is calculated by (Randolph 1994, Selvadurai 2013)

$$k_h = \frac{E_s \sqrt{B_{slab} L_{slab}}}{(1 - v_s^2) \rho_0} \quad (27)$$

where $E_s$ is Young's modulus of the soil; $v_s$ is Poisson's ratio of the ssoil; $L_{slab}$ and $B_{slab}$ are the length and width of the slab, respectively; and $\rho_0$ is a coefficient, which can be evaluated from the dimensions of the slab (Ravera et al. 2020a).

## 2.5 Parameters determination

As shown in Fig. 1, the proposed model has four parameters that are calculated as follows:

(1) Ultimate shaft resistance $\tau_u$

The ultimate shear stress is the stress corresponding to the asymptote of the stress-strain
curve, which is obtained by laboratory interface shear tests. If the test data are not available,
the ultimate shear stress can be calculated as follows:

$$
276 \qquad \tau_u = (1 - \sin \varphi_s) \sigma'_z \tan \delta_s \qquad (28)
$$

277 where $\sigma'_z$ is the effective overburden pressure, $\sigma'_z = \gamma'z$ for the homogeneous soil and
278 $\sigma'_z = \sum_{i=1}^{n} \gamma'_i z_i$ for the layered soil, where $\gamma'$ is the effective weight of the soil. $\delta_s$ is the
279 friction angle of the pile-soil interface, which can be expressed by the soil friction angle ($\varphi_s$)
280 as $\delta_s = \tan^{-1} \frac{\sin \varphi_s \cos \varphi_s}{1 + \sin^2 \varphi_s}$ (Pei et al. 2022b).

281 (2) Initial stiffness of the pile-soil interface $G_{smax}$

282 According to Randolph and Worth (1978), $G_{smax}$ can be obtained as follows:

$$
283 \qquad G_{smax} = \frac{2G_s}{D \ln \frac{2r_m}{D}} \qquad (29)
$$

284 where $G_s$ is the soil shear modulus and $r_m$ is the shear influence radius, which is related to
285 the pile length and soil layer distribution. The value of the shear influence radius increases with
286 increasing soil depth. It can be expressed by:

$$
287 \qquad r_m = 2\delta(1-\nu_s)L \qquad (30)
$$

288 where $\delta$ is the soil shear modulus ratio, $\nu_s$ is the Poisson's ratio of the soil, $L$ is the pile
289 length. For homogeneous soils, the soil shear modulus ratio can be calculated as $\delta = \frac{G_L}{G_L}$,
290 where $G_L$ and $G_L$ are the soil shear modulus at the middle and tip of the pile, respectively.
291 For multilayered nonhomogeneous soils, the soil shear modulus ratio can be calculated as $\delta =$
292 $\int G_{iz}dz/G_L L$ according to Lee et al. (1993), in which $G_{iz}$ is the shear modulus of the soil
293 layer and $z$ is the soil depth. According to Wong and Teh (1995), the shear influence radius
294 should adopt a value in the range of 0.5-2.5L when the length-to-diameter ratio of the pile falls
295 within the range of 20-100.

296 (3) Ultimate base resistance $\sigma_{up}$

297 $\sigma_{up}$ can be calculated as $\sigma_{up} = c' N_c + \sigma'_{bb} N_q$, where $c'$ is the effective cohesion of the
soil, $\sigma'_{vb}$ is the effective vertical stress at the pile tip; $N_q$ and $N_c$ are the load capacity coefficients, which can be calculated by $N_q = e^{\pi t \tan \varphi'} \tan^2(\frac{\pi+2\varphi'}{4})$, $N_c = (N_q - 1) \cot \varphi'$, where $\varphi'$ is the effective friction angle of the pile tip soil.

(4) Initial shear stiffness of the pile tip $G_{pmax}$

The initial shear stiffness of the pile tip, which can be calculated as $G_{pmax} = \frac{4G_s}{\pi r_0 (1 - v_s)}$

<figure><img src="image_4.png" /><figcaption>Fig. 1 Steps of calculation for the determination of model parameters</figcaption></figure>

## 2.6 Implementation for thermomechanical analysis of an energy pile group

Approaches such as the one-dimensional element and finite difference methods can implement the proposed load transfer model. Song and Pei (2022) improved the finite-difference method proposed by Chen and McCartney (2016), which was divided into mechanical and thermomechanical analyses, as shown in Fig.2. During the mechanical analysis, the temperature change is set to zero and the pile is subjected to the mechanical load only. Besides, no constraint is applied to the pile head since the pile only moves downwards caused by the mechanical load. The mechanical response could be obtained through an iterative
314 procedure.

315 When the energy pile is subjected to heating/cooling, the pile expands or contracts, a null
316 point (NP) is defined as the position below and above which the pile has opposite displacements.

317 In the thermomechanical analysis stage, the position of the NP should be assumed first, and
318 then the element number above and below the NP should be determined. The displacement
319 calculation of the element below the NP starts from the NP+1 element. Since the displacement
320 of NP caused by the thermal load is zero, the initial displacement of this element is the
321 displacement under the mechanical load. Taking this as the initial element, one iteratively
322 calculates the displacement of all pile elements below the NP and calculates the stress of each
323 element from the pile bottom element upward based on the displacement. The displacement
324 and stress calculation methods for elements above the NP are similar to those below the NP.

325 The difference is that the stress is calculated sequentially from the downward pile head element.

326 The stress at the top of the pile head element is determined by the mechanical load, restraint
327 conditions, and element displacement. The convergence condition of each element under the
328 thermal-mechanical coupling is that the thermal deformation tends to converge (i.e., |Δ<sup>i</sup><sub>T,new</sub> - Δ<sup>i</sup><sub>T</sub>|<10<sup>-6</sup>). At the same time, the position of the assumed NP is adjusted by observing the stress
329 balance of the upper and lower elements on the NP until the positions of the new and old NP
330 reach the convergence condition and the thermomechanical response of all elements of the pile
331 body can be obtained. It should be noted that the load transfer functions of the pile shaft
332 resistance and tip reaction presented in Section 2 are employed during the thermomechanical
333 analysis of the energy pile group in the present study.
334
<figure><img src="image_5.png" /></figure>

**Note:** ΔT=temperature change; K_h=contact stiffness; P=mechanical load; n=thermal cycles; N=number of pile elements; NP=neutral plane of the pile subjected to thermal loading; Δtᵢ=free thermal deformation of element; αᵢ=thermal expansion coefficient of the pile; ΔT=temperature change of the pile; Q_b,MT=pile tip reaction; Qᵢ^s,MT=side friction of the element subjected to thermal and mechanical loading; Qᵢ^t,MT=thermomechanical top axial stress of the element; Qᵢ^b,MT=thermomechanical base axial stress of the element; σᵢ=axial stress of the element; Δtᵢ^new=calculated thermal elongation or contraction of the element; ΣQᵢ^NP=unbalanced force between the pile above and below the neutral plane.

<caption>**Fig. 2** Flow chart for computational thermo-mechanical analysis of energy piles (after Song and Pei 2022)</caption>

### 3. Validation of the present model

#### 3.1 Tests of an isolated energy pile

Guo et al. (2018) reported an energy pile field test conducted at Southeast University, Nanjing City, China. The groundwater of the site is located at a depth of 1m. The test pile with a length of 24 m and a diameter of 0.5 m was made of high-strength concrete. The pile was placed in five soil layers, as shown in Fig.3. The top layer of the soil deposit is clay, reaching a depth of 4.3 m. The following layers were silty clay with a thickness of 7.2 m. Below the silty
345 clay, two clay layers were found and reached depths of 16 and 17.4 m, respectively. The last 346 layer is silty clay with a thickness of 6.6 m, which provides pile tip resistance. Relevant 347 parameters for the present study were extracted from Guo et al. (2018) and are shown in Table 348 349 1. The pile was continuously heated for 168 h at an average power of 3.44 kW. In this study, 350 the heating durations of 24, 72, 96, and 168 h are selected to be modeled, corresponding to the 351 pile temperature variations of 11.1, 13.2, 16.6, and 19.3 °C. No mechanical load was imposed 352 on the pile head during testing, thus the change in pile stress was caused by the thermal load 353 only. In addition, the contact stiffness between the pile head and superstructure was not 354 considered in this case study because there were no structural constraints on the top of the pile. 355

<figure><img src="image_6.png" /><figcaption>Fig. 3 Schematic of the soil and pile referring to the study of Guo et al. (2018)</figcaption></figure>

<table><caption>Table 1 Soil parameters considered for modelling the field tests of Guo et al. (2018)</caption><thead><tr><th>Variable</th><th>Clay [0-4.3]</th><th>Silty clay [4.3-11.5]</th><th>Clay [11.5-13.5]</th><th>Clay [13.5-16]</th><th>Silty clay [16-24]</th><th>Pile tip [24, -]</th><th>Pile</th></tr></thead><tbody><tr><td>Unit weight, γ<sub>sat</sub> [kN/m<sup>3</sup>]</td><td>18.9</td><td>17.4</td><td>19.7</td><td>19.5</td><td>19.9</td><td>-</td><td>24.9</td></tr><tr><td>Ultimate skin friction, τ<sub>u</sub> [kPa]</td><td>2.6</td><td>6.5</td><td>10.3</td><td>22.9</td><td>31.9</td><td>-</td><td>-</td></tr><tr><td>Young's modulus, E [MPa]</td><td>4.6</td><td>3.6</td><td>7.5</td><td>6.6</td><td>7.3</td><td>-</td><td>45700</td></tr><tr><td>Shear modulus, G<sub>soil</sub> [MPa]</td><td>17.1</td><td>13.9</td><td>28.5</td><td>25.2</td><td>15.1</td><td>-</td><td>-</td></tr><tr><td>Friction angle, φ [°]</td><td>10</td><td>6.6</td><td>6.8</td><td>18.4</td><td>17.3</td><td>-</td><td>-</td></tr><tr><td>Initial stiffness of skin friction, G<sub>max</sub> [MPa/m]</td><td>13.1</td><td>11.1</td><td>22.9</td><td>20.3</td><td>11.8</td><td>-</td><td>-</td></tr><tr><td>Ultimate tip resistance, q<sub>b</sub> [MPa]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>50</td><td>-</td></tr><tr><td>Initial stiffness of tip reaction,</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1200</td><td>-</td></tr></tbody></table>
<table><tr><td>G<sub>pmax</sub>[MPa/m]</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Thermal dilation coefficient, α<sub>s</sub> (με/°C)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>11.6</td></tr></table>

Fig. 4 shows the thermally induced pile axial stresses obtained from the experiments and the model used in this study. A close agreement between the measured and computed values for all the heating steps is observed. The computed thermal stress meaningfully yields a value of zero at the pile head, which reflects the absence of restraint provided by any structural elements. The thermal stress increases with the pile depth and decreases again towards the pile tip. Differences between the simulations and measurements along the pile in the first two soil layers can be found. The reason may be attributed to the significant variations of the ultimate skin friction of the first three soil layers, i.e., 37.82, 12.4 and 106.4 kPa for the first, second and third soil layers, respectively. As the temperature change increases, the second layer of soil needs to mobilize more friction than under lower temperature changes to resist the significantly increased relative pile-soil displacement. Even the third soil layer may mobilize more friction when the second soil layer is incapable of providing enough friction to reach the limit equilibrium, which would induce the stress redistribution along the pile depth and thus explains the variation of the thermal stress of the pile body in the second layer of soil. A similar phenomenon can also be found in previous studies (Rotta Loria and Laloui 2018, Fang et al. 2020).
<figure><img src="image_7.png" /><figcaption>**Fig. 4** Experimental and modeled thermally induced pile axial stresses referring to the tests of Guo et al. (2018). (a) $\Delta T=11.1 \,^\circ C$; (b) $\Delta T=13.2 \,^\circ C$; (c) $\Delta T=16.6 \,^\circ C$; (d) $\Delta T=19.3 \,^\circ C$</figcaption></figure>

Fig. 5 shows the measured and modeled thermal stress at different pile depths versus the average temperature variations of the pile. The computed results capture the overall evolution of the measurements. It can be seen that the computed thermal stress at each depth along the pile has an approximately linear relationship with the temperature change. The maximum thermal stress is at a depth of about 16.3m. The relationship with the average temperature change is $\sigma = 236\Delta T$ kPa.
<figure><img src="image_8.png" /><figcaption>Fig. 5 Experimental and modeled thermal stresses at the different pile depths during thermal loading referring to Guo et al. (2018)</figcaption></figure>

## 3.2 Tests of an energy pile group

The analysis case was a full-scale field test of a pile group consisting of four energy piles, as reported by Mimouni and Laloui (2015) and Rotta Loria and Laloui (2017, 2018). The energy piles (EP) have a diameter of 0.9 m and a length of 28 m; their arrangement is shown in Fig. 6a. The pile group joint has a slab of 26 m long, 0.9 m thick, and 10 m wide. The piles were placed in five soil layers as shown in Fig 6b. The top layer of the soil deposit was alluvial, reaching a depth of 8.6 m. The subsequent layer is a sandy-gravelly moraine with a thickness of 8 m. Below it, a thin bottom moraine layer is found and reaches a depth of 20.1 m. The last layer is molasses, reaching a depth of 28.9 m. The soil deposit was considered to be fully saturated with water since the groundwater table was located at the surface. The relevant parameters used in this study were extracted from existing studies or calculated according to the determination methods described in Section 2, as shown in Table 2. It is noted that Young's modulus of the Molasse layer provided by Ravera et al. (2020a) was in the range of 3000-7000 MPa, based on which the calculated shear modulus fell in the range of 1153.8 - 2692.3 MPa. The initial stiffness of skin friction and tip reaction of the load transfer in the Molasse layer were calculated according to the equations mentioned in section 2.5 and fell in the range of
318.2-742.5 MPa/m and 2176.5-5078.5 MPa/m, respectively. Therefore, this study conducted two kinds of simulations using the maximum and minimum values of the initial stiffness of skin friction and tip reaction estimated from the shear modulus. The four energy piles were subjected to a heating-passive cooling cycle during the test. In this study, the monitoring results of EP1 pile (see Fig.6a) during the heating phase were used to validate the proposed model. The temperature changes were 5, 10, 15, and 20 °C, respectively. Additionally, an initial vertical mechanical load of 495 kN was calculated for the single pile tested by Rotta Loria and Laloui (2017). A contact stiffness calculated as 4450 MPa/m was considered. Besides, thermal stress means pile vertical stress induced by temperature changes only.

<figure><img src="image_9.png" /></figure>

410

<caption>Fig.6 Schematic of (a) the energy pile foundation; (b) the soil stratigraphy (modified from Rotta Loria and Laloui 2017a, 2018a)</caption>

<table><caption>Table 2 Soil and pile parameters considered for modelling the field tests referring to (Rotta Loria and Laloui 2017b, Ravera et al. 2020a)</caption><thead><tr><th>Variable</th><th>Alluvial soil [0.0, 8.6]</th><th>Sandy-gravelly moraine [8.6, 16.6]</th><th>Bottom moraine [16.6, 20.1]</th><th>Molasse [20.1, 28.9]</th><th>Pile tip [24, -]</th><th>Pile</th></tr></thead><tbody><tr><th>Unit weight, γ<sub>sat</sub> [kN/m<sup>3</sup>]</th><td>19.8</td><td>19</td><td>19.8</td><td>25.5</td><td>-</td><td>25</td></tr></tbody></table>
<table><tr><td>Ultimate skin friction, τ<sub>u</sub> [kPa]</td><td>7</td><td>17</td><td>28.9</td><td>50.8</td><td>-</td><td>-</td></tr><tr><td>Young's modulus, E [MPa]</td><td>5</td><td>84</td><td>90</td><td>3000-7000</td><td>-</td><td>2800</td></tr><tr><td>Shear modulus, G<sub>soil</sub> [MPa]</td><td>2</td><td>30</td><td>32.1</td><td>1153.8 - 2692.3</td><td>-</td><td>-</td></tr><tr><td>Friction angle, φ [°]</td><td>30</td><td>23</td><td>27</td><td>35</td><td>-</td><td>-</td></tr><tr><td>Initial stiffness of skin friction, G<sub>max</sub> [MPa/m]</td><td>0.6</td><td>8.3</td><td>8.9</td><td>318.2 - 742.5</td><td>-</td><td>-</td></tr><tr><td>Ultimate tip resistance, q<sub>b</sub> [MPa]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>15.8</td><td>-</td></tr><tr><td>Initial stiffness of tip reaction, G<sub>pmax</sub> [MPa/m]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>2176.5-5078.5</td><td>-</td></tr><tr><td>Thermal dilation coefficient, α<sub>s</sub> (με/°C)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>10</td></tr></table>

Fig. 7 shows the comparison between experimentally observed and computed thermal vertical stress variations along the pile depth. It is noted that only the measured thermal stress along the pile depth of 4-28 m is presented for comparison because the heat exchange pipes were thermally insulated to a depth of 4 m below the pile heads to eliminate the effects of the climatic conditions on the test results and the temperature changes imposed to the piles in the present simulations were the mean values of the piles at a depth of 4-28 m recorded during the thermal operation (Rotta Loria and Laloui 2017b). It can be seen that the simulated results match well with the experimental results, especially in the lower temperature change operation ($\Delta T = 5$ and $10$ °C). An increasing difference between the experimental and simulated values of the pile in the molasse layer is found under the higher thermal loads of $\Delta T = 15$ and $20$ °C. The reason may be attributed to the fact that in the molasse layer, the present method is incapable of capturing the stress redistributions of the pile well under the condition that thermally induced deformation of the soil is more pronounced than that of the piles because the load transfer method used in this study is suitable for the case where the thermally induced deformations of piles govern the performance of energy piles. Similar phenomena and explanations can also be found in studies of Ravera et al. (2020a) and Rotta Loria et al. (2020b). Besides, the simulated results of the model proposed by Ravera et al. (2020a) are also shown for comparison purposes. It can be found that the present model does not need to use experiments or numerical simulations to determine the calculation parameters such as displacement factors and can achieve the same good calculation results as the model of Ravera et al. (2020a). The excellent
agreement between the simulated and experimental values of the current model and the
calculated values of other models shows the accuracy of the current model and its promising
potential for simulating thermomechanical responses of an energy pile group.

<figure><img src="image_10.png" /><figcaption>438
439 Fig. 7 Comparison between experimentally observed and computed thermal vertical stresses
440 variations. (a) ΔT=5°C; (b) ΔT=10°C; (c) ΔT=15°C; (d) ΔT=20°C</figcaption></figure>

441 3.3 FE simulations of energy pile groups

442 Saggu and Chakraborty (2016) conducted numerical studies of energy pile groups with
spacings of 2.5 and 4 m using the finite element software Abaqus. In these studies, the pile group size was 3×2, and the slab was considered to be composed of the same concrete as the pile. The slabs were 15.5 m long, 1.2 m thick, and 12.9 m wide. All piles, 20 m long and 1m in diameter, were wished into a single layer of dry Ottawa sand. During the simulations, an axial load of 6000 kN was continuously applied to the slab, and all the piles were first heated and then recovered. The temperature changes were 21 and 2.8 °C at the heating and recovery ends, respectively. The parameters used in this study are listed in Table 3. A contact stiffness calculated as 401.7 MPa/m was considered in the study.

<table><caption>**Table 3** Soil and pile parameters considered in the present study referring to Saggu & Chakraborty (2016)</caption><thead><tr><th>Variable</th><th>Soil layer, z [m]</th><th>Ottawa sand [0, 20]</th><th>Pile tip [24, -]</th><th>Pile</th></tr></thead><tbody><tr><td>Unit weight, γ [kN/m³]</td><td></td><td>18</td><td>-</td><td>25</td></tr><tr><td>Ultimate skin friction, τ<sub>u</sub> [kPa]</td><td></td><td>31.6</td><td>-</td><td>-</td></tr><tr><td>Shear modulus, G<sub>soil</sub> [MPa]</td><td></td><td>32.6</td><td>-</td><td>-</td></tr><tr><td>Friction angle, φ [°]</td><td></td><td>32</td><td>-</td><td>-</td></tr><tr><td>Initial stiffness of skin friction, G<sub>max</sub> [MPa/m]</td><td></td><td>16.2</td><td>-</td><td>-</td></tr><tr><td>Ultimate tip resistance, q<sub>b</sub> [MPa]</td><td></td><td>-</td><td>8.3</td><td>-</td></tr><tr><td>Initial stiffness of tip reaction, G<sub>pmax</sub> [MPa/m]</td><td></td><td>-</td><td>118.6</td><td>-</td></tr><tr><td>Young's modulus, E<sub>p</sub> (GPa)</td><td></td><td>-</td><td>-</td><td>33.7</td></tr><tr><td>Thermal dilation coefficient, α<sub>s</sub> (με/°C)</td><td></td><td>-</td><td>-</td><td>10</td></tr></tbody></table>

Fig.8 shows the FE simulated and modeled pile axial stresses, referring to Saggu and Chakraborty (2016). Fig.9(a), (b), and (c) show the axial stress in piles with 2.5 m center-to-center spacing. Close agreement was observed between the FE simulations and the results of the present model. Under mechanical loading, the pile stress decreases with depth. The axial stress of the pile significantly increased when the pile temperature change reached 21 °C. The vertical stress at the end of the recovery phase was slightly higher than that under mechanical loading, because the pile did not return to the initial temperature. An interesting phenomenon is that the corner piles show a more significant thermal stress than the central piles because of the interaction effects, which is in agreement with previous studies (Jeong et al. 2014, Suryatriyastuti et al. 2016). A similar phenomenon can be observed in Fig.9(d), (e), and (f) for the piles with 4.0 m center-to-center spacing.
<figure><img src="image_11.png" /><figcaption>**Fig. 8** FE simulated and modeled pile axial stresses referring to Saggu & Chakraborty(2016). (a), (b), and (c) are the axial stresses in pile with 2.5 m center-to-center spacing; (d), (e), and (f) are the axial stresses in pile with 4.0 m center-to-center spacing</figcaption></figure>

Furthermore, finite element simulations of Rotta Loria and Laloui (2016) were employed for comparison with the simulations of the proposed model in this study to investigate the 'sheltering-reinforcing' effect in an energy pile group. The piles in the group had a length of 10 m and a diameter of 1 m. Young's modulus and Poisson's ratio of the piles were 30 GPa and 0.25, respectively. The thermal dilation coefficient of the piles was $10 \mu\epsilon/°C$. The piles were sited into homogeneous sand with a shear modulus of 30 MPa and a Poisson's ratio of 0.3. The temperature change is 10 °C during the simulations. The contact stiffness was considered to be zero since the pile head could move freely.
476 Rotta Loria and Laloui (2016) defined the displacement ratio to analyze the displacement
477 characterizing energy pile group. The displacement ratio is expressed as:

$$
478 \qquad R_d = \frac{\text{Average displacement of group}}{\text{Displacement of single pile under the same average load}} \qquad (31)
$$

<figure><img src="image_12.png" /><figcaption>Fig. 9 Displacement ratio under the different pile spacings</figcaption></figure>

481 Fig.9 shows the displacement ratio for various pile groups under different pile spacings. It
482 can be seen that there is good agreement between the results of the finite element simulation
483 and the results obtained by the present method with and without the ‘sheltering-reinforcing’
484 effect in the group. The difference between the results without and with consideration of the
485 ‘sheltering-reinforcing’ effect tend to disappear when s/D is larger and the convergence is
486 delayed if the number of piles increases. The displacement ratio obtained by the finite element
487 simulation is the largest, and the results without and with consideration of the ‘sheltering-
488 reinforcing’ effect are smaller and smallest, respectively. The possible reasons, similar to those
489 discussed by Liu et al. (2022), are as follows:

490 (i) The nonlinear interaction of the pile-soil interface is not considered in the finite
491 element simulations of Rotta Loria and Laloui (2016). Thus, when the pile is
492 subjected to the thermal load, the pile and soil will deform together, and there is
493 no relative slip between them. However, the nonlinear pile-soil interaction is
494 considered in the proposed model, which means there will be a relative slip
495 between the pile and soil. The shaft resistance induced by surrounding soil will
496 hinder the upward expansion of the pile. Therefore, the displacement ratio
497 obtained by the finite element simulation is greater than that obtained by the model
498 proposed in this study.

499 (ii) Even though the nonlinear mechanical properties of the interface have been
500 considered, the calculation without considering the ‘sheltering-reinforcing’ effect
501 would overestimate the displacement ratio of the pile.

502 Besides, the displacement ratio obtained in the free displacement field increases significantly
503 when s/D=3. The reason may be that the shear displacement of the surrounding soil in the free
504 displacement field varies according to a non-linear decreasing function with pile spacing and
505 increases rapidly at smaller pile spacings (Wang et al., 2016). The comparison results show the
506 importance of considering the ‘sheltering-reinforcing’ effect in estimating the energy pile group
507 response and the relevance of the model proposed in the present study.

508 4. Parametric study

509 This section aims to study the influence of the pile-slab-soil interaction, pile spacing, pile
510 position and number in a group using the present load transfer model. To this end, a series of
511 simulations was conducted using the parameters listed in Table 4.

<table><caption>512 Table 4 Soil and pile parameters considered in the present study</caption><thead><tr><td>Variable</td><td>Value</td></tr></thead><tbody><tr><td>Length, L[m]</td><td>25</td></tr><tr><td>Pile diameter, D[m]</td><td>1</td></tr><tr><td>Young's modulus of piles, E<sub>p</sub> (GPa)</td><td>30</td></tr><tr><td>Unit weight, γ [kN/m<sup>3</sup>]</td><td>19.8</td></tr><tr><td>Friction angle, φ [°]</td><td>33</td></tr><tr><td>Thermal dilation coefficient of piles, α<sub>s</sub> (με/°C)</td><td>10</td></tr><tr><td>Mechanical load, P [kN]</td><td>3000</td></tr><tr><td>Ultimate skin friction, τ<sub>u</sub> [kPa]</td><td>40.1</td></tr><tr><td>Initial stiffness of skin friction, G<sub>s,max</sub> [MPa/m]</td><td>30</td></tr><tr><td>Ultimate tip resistance, σ<sub>up</sub> [MPa]</td><td>10</td></tr><tr><td>Initial stiffness of tip reaction, G<sub>pmax</sub> [MPa/m]</td><td>1500</td></tr></tbody></table>
## 4.1 Effect of the pile-slab-soil interaction

To study the effect of the pile-slab-soil interaction on the thermomechanical behavior of an energy pile group, a 2×2 pile group and an isolated pile were modeled. The pile-to-pile spacing of the group was three times the pile diameter. The parameters used in simulations are shown in Table 4. The piles were subjected to temperature changes of 10 and 30 °C. Contact stiffness values of 0, 1, 3, 5, and 7 GPa/m were used in simulations. Fig.10 shows the effect of the contact stiffness on the thermal vertical stress of the pile. It can be observed that more thermal stress was induced as the contact stiffness increased. The thermal stress generated at the pile head of the pile group was slightly larger owing to the pile-slab-soil interaction. Fig.11 shows the effect of the contact stiffness on the maximum thermal vertical stress in the pile. It can be found that the maximum thermal stress in the pile in the group under the same contact stiffness was smaller than that of the isolated pile. This phenomenon has also been observed in previous studies (Suryatriyastuti et al. 2016, Ravera et al. 2020a).

<figure><img src="image_13.png" /><figcaption>**Fig. 10** Effect of contact stiffness on the thermal vertical stress of the pile. (a) and (b) are the thermal vertical stress under a temperature change of 10 °C of an isolated pile and a pile in a 2×2 group, respectively; (c) and (d) are the thermal vertical stress under a temperature change of 30 °C of an isolated pile and a pile in a 2×2 group</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11** Effect of contact stiffness on maximum thermal vertical stress of the pile</figcaption></figure>

The thermally induced downward displacement of the pile under different levels of pile-slab-soil interaction is compared in Fig. 12. According to the study of Najma and Sharma (2021), the location of the NP (i.e., thermally induced displacement equal to zero) approaches the pile head with greater pile head constraint, which explains why the position of NP gradually moves upward as the contact stiffness increases. It is also observed that the absolute values of the pile head and end displacements decreased and increased, respectively, as the contact stiffness increased. Additionally, the NP of the pile group was lower than that of the isolated pile under the same constraint conditions. Fig. 13 presents the effect of the contact stiffness on the absolute value of the maximum thermal vertical displacement of the pile. Compared with the isolated piles, the pile group had larger thermally induced displacements under the same constraints; however, this difference became insignificant as the temperature change increased.
<figure><img src="image_15.png" /><figcaption>Fig. 12 Effect of the contact stiffness on thermal vertical displacement of the pile. (a) and (b) are the thermal vertical displacement under a temperature change of 10 °C of an isolated pile and a pile in a 2×2 group; (c) and (d) are the thermal vertical displacement under a temperature change of 30 °C of an isolated pile and a pile in a 2×2 group</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13 Effect of the contact stiffness on the absolute value of maximum thermal vertical displacement of the pile</figcaption></figure>

## 4.2 Effect of the pile spacing

Section 3.3 investigated the effect of pile spacing on a group with 2, 3 and 4 piles. In this section, the study was extended to a 5×5 pile group. The parameters used in the simulation are shown in Table 4. The pile-to-pile spacing of the group was three times the pile diameter. During the simulation, all the group piles were energy piles and subjected to a temperature change of 30 °C. Besides, contact stiffness equal to zero was considered. Fig. 14 shows the vertical thermal
displacement of the pile head subjected to different spacings. It can be observed that the thermal displacements decreased with increasing pile spacing, which is consistent with the results of Jeong et al. (2014) and Rotta Loria et al. (2016). This may be because an increase in pile spacing weakens the pile interactions within the group. The mechanism was discussed in detail in a study by Rotta Loria et al. (2016) on the interaction factor of energy-pile groups. In addition, the comparison results with and without considering the ‘sheltering-reinforcing’ effect, shown in Fig.14, further indicate the advantage of the present method in the analysis of the response of the energy pile group.

<figure><img src="image_17.png" /><figcaption>Fig. 14 Thermal displacement of the pile head under different pile spacing</figcaption></figure>

### 4.3 Effect of pile position in a group

This section aims to study the effect of the pile position in a 3×3 group on the thermomechanical response. The pile-to-pile spacing of the group was three times the pile diameter. During the simulation, all the group piles were energy piles and subjected to a temperature change of 10 and 30 °C. Besides, contact stiffness equal to zero was considered. Fig.15 shows a comparison of the profiles of the vertical thermal stress and displacement for the corner, edge, and center energy piles. For the sake of clarity, the effect of pile position in the group on the maximum thermal vertical stress and displacement of the pile under
temperature changes of 10 and 30 °C is summarized in Fig.16. It can be observed that the
maximum and minimum thermal stresses occur at the corner and center piles, respectively,
whereas the minimum and maximum thermal displacements occur at the corner and center piles,
respectively. The position of the NP for the corner and center piles are the highest and lowest,
respectively, which means that the corner piles are more constrained than the center piles in the
group. The results of the energy pile group subjected to thermomechanical loads were in
accordance with those of the energy pile group conducted by Ravera et al. (2020a) and the
conventional pile group performed by Comodromos et al.(2016).

<figure><img src="image_18.png" /><figcaption>**Fig. 15** Effect of pile arrangement in a group. Thermal vertical stress of the pile under (a) $\Delta T=10$ °C and (b) $\Delta T=30$ °C; thermal vertical displacement of the pile under (c) $\Delta T=10$ °C; (d) $\Delta T=30$ °C</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 16 Effect of pile position in the group on the maximum thermal vertical stress and the absolute value of the displacement of the pile under temperature changes of 10 and 30 °C</figcaption></figure>

4.4 Effect of the number of piles in the group

This section aims to study the effect of the number of piles in the group on the thermomechanical response. The pile-to-pile spacing of the group was three times the pile diameter. During the simulation, all the group piles were energy piles and subjected to a temperature change of 10 and 30 °C. Besides, contact stiffness equal to zero was considered in the present case. Fig.17 presents the effect of the number of piles in the group, in which the thermally induced stress and displacement of an isolated pile (1×1) and 2×2, 3×3, 4×4, and 5×5 pile groups were modeled and compared. For the sake of clarity, the effect of the number of piles in the group on the maximum thermal vertical stress and the absolute values of the displacement of the pile under temperature change of 10 and 30 °C is summarized in Fig.18.

The difference between thermal stresses in the pile group and the isolated pile decreased as the temperature increased. However, the pile head displacement increased with the number of piles.

These results are consistent with those found by experimental and numerical studies (Rotta Loria and Laloui 2017b, Ravera et al. 2020a, Fang et al. 2022) and validate the applicability of the present method to simulate the interactions and group effects of energy piles.
<figure><img src="image_20.png" /><figcaption>Fig. 17 Effect of the number of piles in the group. Thermal vertical stress of the pile under (a) $\Delta T=10\,^\circ C$ and (b) $\Delta T=30\,^\circ C$; thermal vertical displacement of the pile under (c) $\Delta T=10\,^\circ C$; (d) $\Delta T=30\,^\circ C$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18 Effect of the number of the pile in the group on the maximum thermal vertical stress and the absolute value of the displacement of the pile under temperature changes of 10 and $30\,^\circ C$.</figcaption></figure>

## 5. Discussion

In this study, we proposed an experimentally validated load transfer model that considers
the ‘sheltering-reinforcing’ effect for the thermomechanical analysis of energy pile groups. The proposed method modeled two well-documented full-scale field tests and two finite-element simulation cases. The comparison results showed that the proposed method could capture several essential aspects of the energy pile group, including thermally induced strain, stress, displacement, and group effects. Also, we conducted a parametric analysis to evaluate the effects of relevant parameters on the thermomechanical behavior of the energy pile group.

Recently, several methods considering pile-pile interactions have been developed for the thermomechanical analysis of energy pile groups. For example, Rotta Loria and Laloui (2018) proposed an analytical model to analyze the vertical displacement and the increased deformation of energy pile groups subjected to thermal loads more comprehensively by defining a pile-soil-pile interaction factor. Iodice et al. (2021) provided a unified methodology involving a novel explicit formula for evaluating the maximum thermally induced axial load by establishing the effect of the actual restraint condition at the pile head as a function of the stiffness ratio between the inactive and the active pile subgroups. Nevertheless, the present study is different from the above studies. First, the present study aims to provide an experimentally validated simple load transfer model that considers the ‘sheltering-reinforcing’ effects for thermomechanical analysis of energy pile groups. The model can estimate all the thermomechanical behavior of energy pile groups involving vertical displacement, strain, stress, side friction, and group effects. This aim is very different from Rotta Loria and Laloui (2018) study, which mainly focuses on estimating the vertical displacement of the energy pile group. Furthermore, this study developed the load transfer method for the single energy pile under thermomechanical loading by considering the pile-pile interaction in the group to make it suitable for the thermomechanical behavior of energy pile groups. This is different from the
study of Iodice et al. (2021) because they fulfilled their aim by analyzing numerical simulation
results and proposed the design procedure by using analytical solutions, which can be worked
out to obtain explicit formulae for the direct computation of axial force in a multilayer soil.
Their practical method is more geared toward calculating the thermomechanical responses of
single energy piles. Second, The present study considers both nonlinear pile-soil interaction
and elastic deformation of the surrounding soil. The surrounding soil is divided into disturbed
and undisturbed zones. The soil in the disturbed zone is a skinny layer of soil on the soil/pile
interface. The nonlinearity of the pile-soil interaction was concentrated in this interface. The
area outside the interface is the undisturbed zone. Because this zone is subjected to a relatively
low stress change, the stress-strain behavior of the soil is mainly elastic. The pile-soil-pile
interaction in the pile group is considered linearly elastic and suitable for the principle of
superposition. The advantage of this approach is that it can consider the relative slip between
piles and soil for the single pile and the interaction between piles-soil-pile in pile groups. Thus,
the consideration of pile-soil-pile interaction in the present study is very different from the two
studies(Rotta Loria & Laloui, 2018; Iodice et al., 2021).

One of basic assumptions of the simple load transfer method is that the mechanical behavior
of the energy pile is governed by thermally induced pile axial deformation. Thus, the radial
deformation induced by thermal loading is not considered in the present study. There is
controversy about whether the influence of radial deformation of energy piles on the response
of piles can be ignored in the study of simplified analysis methods. Wang et al. (2015) and
Faizal et al.(2018) showed that axial thermal strains were more restricted to thermal expansion
than radial thermal strains, which indicated that the radial thermal strains are closer to that of
the pile in free thermal expansion. On the contrary, the study by Mimouni and Laloui (2015)
showed that the radial thermal strains of the pile were much lower than the free thermal strains,
indicating that stiffer soils provided higher restriction to radial thermal strains than the case reported by Wang et al. (2015) and Faizal et al. (2018). It suggests that building loads could affect the development of radial thermal strains along the pile length. Saggu et al. (2015) observed that the radial strains induced in a pile due to thermo-mechanical loading are higher than that induced due to only mechanical loading. Elzeiny & Suleimam (2021) pointed out that the horizontal pressure decreased by 2~5 kPa when colling the pile, which corresponds quite well with the average stress associated with the measured reduction in pull-out capacity. Bourne-Webb et al. (2022) conducted numerical simulations and comprehensively investigated the problem. They reported that the horizontal stress changes induced by thermal loading have significant potential effects on the thermomechanical behavior of energy piles. The effects depend on the margin concerning the initial mobilization of the shaft resistance by the mechanical load. In summary, the studies mentioned above suggested that the horizontal thermally induced contact stresses would significantly influence the thermomechanical behavior of energy piles because of the radial expansion and contraction of the pile under heating and cooling.

However, these require a confirmation from field tests under building loads representing actual boundary conditions. It is noted that the existing studies have different opinions. Olgun et al. (2014) reported that radial contact pressures typically increase less than 15 kPa, which cannot fully explain the shaft resistance observed in heating tests. They thought that the increase of heat exchanger pile capacity in response to heating, observed in several small-scale laboratory studies, cannot be directly attributed to the increase of contact pressure at the soil/pile interface. Faizal et al. (2019) conducted a field test of the energy pile and found that the magnitudes of the axial thermal strains were more constrained than the radial thermal strains at all depths, leading to the development of axial and radial thermal stresses of up to -4.5 and -0.015 MPa, respectively, for a change in average pile temperature of 24.1 °C. The magnitudes of the radial thermal stresses with changes in pile temperature were significantly lower than the axial thermal stresses at all depths of the pile, indicating that the radial thermal expansion had
690 negligible effects on the development of axial thermal strains and stresses.

691 In some studies (Garbellini and Laloui 2019b, Iodice et al. 2020, 2023, Liu et al. 2022),
692 radial thermal effects are commonly considered relatively small compared to axial thermal
693 effects when predicting the axial thermal response of energy piles due to the low diameter-to-
694 length ratio of the pile. Some load transfer analysis methods have consistently validated and
695 predicted the axial thermal response of field- and centrifuge-scale energy piles by neglecting
696 the radial thermal effects (Knellwolf et al. 2011, Suryatriyastuti et al. 2014, Chen and
697 McCartney 2016, Luo and Hu 2019, Ravera et al. 2020a, Rotta Loria et al. 2020a, Hu and Luo
698 2020, Najma and Sharma 2021, Fei et al. 2022). Some numerical studies have also noted that
699 radial thermal stresses in the energy pile are significantly low compared to the axial thermal
700 stresses along the length of the pile (Ozudogru et al. 2015, Gawecka et al. 2017). Besides,
701 preliminary numerical and analytical studies on energy piles (Olgun et al. 2014, Zhou et al.
702 2016), along with a field study on an energy pile without end restraints (Faizal et al. 2018) have
703 used cavity expansion analyses to confirm that no significant changes in pile-soil contact
704 stresses are expected from the radial thermal expansion of the pile. However, these also need
705 validation at various depths of a field-scale energy pile under building loads.

706 In the present study, a simple load transfer method was proposed for analyzing the
707 thermomechanical behavior of energy pile groups. Therefore, the same assumption adopted by
708 some studies, including the load transfer method for the thermomechanical analysis of energy
709 pile and energy pile groups (Knellwolf et al. 2011, Suryatriyastuti et al. 2014, Chen and
710 McCartney 2016, Luo and Hu 2019, Ravera et al. 2020a, Rotta Loria et al. 2020a, Hu and Luo
711 2020, Najma and Sharma 2021, Fei et al. 2022) is employed to reduce the complexity of coupled
712 thermomechanical analysis.

713 During the development of the simple load transfer model, the shear radius proposed by
714 Randolph and Wroth (1978) is employed to calculate the initial stiffness of the pile-soil
715 interface, as shown in Eq.(30). The shear radius is affected by pile length, shear modulus and
716 Poisson's ratio of the soil. For the pile under pure mechanical loads, the shear radius has been
found to be almost constant with pile depth and more significant than the pile length. However, when the pile was subjected to thermal loads, Rotta Loria and Laloui (2018) conducted finite element simulations and found that the shear radius was not constant with the pile depth and smaller than the pile length. In this study, the calculation of the shear radius is considered in a simplified method. For the multi-layered soil, the shear modulus and Poisson's ratio of each soil layer are different, which means there will be different values of shear radius for each soil layer. Thus, the calculated shear radius varies with pile depth but remains unchanged within the same soil layer through this simplified method. The simplified method mentioned above for the homogeneous soil will fail to obtain a shear radius varying with depth. The calculated shear radius remains unchanged within all the pile depths because the shear modulus and Poisson's ratio of the soil keep constant, which may adversely affect the application of the proposed simple load transfer method for energy pile groups in the homogeneous soil. Furthermore, the shear radius is greater than the pile length under mechanical loads and generally smaller than the pile length under thermal loads (Randolph & Wroth, 1978; Rotta Loria & Laloui, 2018).

The proposed simple load transfer method for energy pile groups in the presented study can consider the combined thermal and mechanical loads. During the thermomechanical analysis, it is necessary to consider the shear radius induced by both mechanical and thermal loads. Therefore, the shear radius induced by mechanical loads is employed for the thermomechanical analysis because the radius induced by mechanical loads is much greater than that induced by thermal loads.

## 6. Conclusions

We proposed a simple load transfer method for the thermomechanical analysis of an energy pile group subjected to thermomechanical loads. The proposed method was validated using full-scale field tests and finite element numerical simulations. In addition, a numerical analysis was performed to evaluate the effects of relevant parameters on the thermomechanical behavior of energy pile groups. The main conclusions are as follows:
743 (1) The results obtained with the proposed method were compared with those from full-scale field tests and FE simulations. The comparison indicated that the proposed model could capture the thermal strain, stress, and displacement of the energy piles in the group under thermal and mechanical loads.

747 (2) Under the same constraints and thermal loads, the energy piles in the group exhibited less thermal stress and larger thermal displacement than the isolated piles. The difference in the thermal stress and displacement between the single energy pile and the energy pile in groups increases with the temperature change.

751 (3) The maximum and minimum thermal stresses occurred at the corner and center piles, whereas the minimum and maximum thermal displacements occurred at the corner and center piles, respectively. In addition, the thermal displacement decreased with increasing pile spacing.

754 (4) The thermal stress decreased as the number of energy piles in the group increased under the same thermal load. The difference between thermal stresses in the pile group and the isolated pile decreased as the temperature change increased.

757 Acknowledgments

758 This research was supported by the National Natural Science Foundation of China (Grant Nos. 52122805 and 52078103) and the China Scholarship Council (CSC) (Grant No. 202106060093).

761 List of notations

762 $B_{\text{slab}}$ width of the slab

763 $c'$ effective cohesion of the soil

764 $D$ diameter of the pile

765 $E_P$ Young's modulus of the pile

766 $E_{\text{soil}}$ Young's modulus of the soil

767 $G$ shear modulus of the soil

768 $G_{s,max}$ initial stiffness of the pile-soil interface

769 $G_{pmax}$ initial stiffness of the pile tip-soil interface

770 $G_s$ soil shear modulus

771 $G_{L/2}$ soil shear modulus at the middle of the pile

772 $G_L$ soil shear modulus at the tip of the pile
<table>
  <tr>
    <td>773</td>
    <td>K<sub>h</sub></td>
    <td>contact stiffness between the pile head and slab</td>
  </tr>
  <tr>
    <td>774</td>
    <td>L</td>
    <td>length of the pile</td>
  </tr>
  <tr>
    <td>775</td>
    <td>L<sub>slab</sub></td>
    <td>length of the slab</td>
  </tr>
  <tr>
    <td>776</td>
    <td>N<sub>q</sub></td>
    <td>load capacity coefficients</td>
  </tr>
  <tr>
    <td>777</td>
    <td>N<sub>c</sub></td>
    <td>load capacity coefficients</td>
  </tr>
  <tr>
    <td>778</td>
    <td>P</td>
    <td>mechanical load applied to the pile head</td>
  </tr>
  <tr>
    <td>779</td>
    <td>p</td>
    <td>pile-soil relative displacement</td>
  </tr>
  <tr>
    <td>780</td>
    <td>p<sub>0</sub></td>
    <td>relative pile-soil displacement corresponding to the stress reversal point</td>
  </tr>
  <tr>
    <td>781</td>
    <td>p<sub>p</sub></td>
    <td>pile-soil relative displacement of the pile tip</td>
  </tr>
  <tr>
    <td>782</td>
    <td>p<sub>po</sub></td>
    <td>relative displacement of the pile tip and soil corresponding to the stress reverse point</td>
  </tr>
  <tr>
    <td>783</td>
    <td>r<sub>0</sub></td>
    <td>pile radius</td>
  </tr>
  <tr>
    <td>784</td>
    <td>r<sub>m</sub></td>
    <td>shear influence radius</td>
  </tr>
  <tr>
    <td>785</td>
    <td>s</td>
    <td>elastic displacement of the surrounding soil</td>
  </tr>
  <tr>
    <td>786</td>
    <td>s<sub>i0</sub></td>
    <td>displacement of pile i induced by its own load</td>
  </tr>
  <tr>
    <td>787</td>
    <td>s<sub>ij</sub></td>
    <td>additional displacement caused by j pile under its own load</td>
  </tr>
  <tr>
    <td>788</td>
    <td>s<sub>aij</sub></td>
    <td>distance between the axes of two piles</td>
  </tr>
  <tr>
    <td>789</td>
    <td>s<sub>p</sub></td>
    <td>elastic displacement of the soil at the pile tip</td>
  </tr>
  <tr>
    <td>790</td>
    <td>u</td>
    <td>total shaft displacement at a certain pile depth</td>
  </tr>
  <tr>
    <td>791</td>
    <td>u<sub>0</sub></td>
    <td>total pile shaft displacement corresponding to the shaft shear stress reversal point</td>
  </tr>
  <tr>
    <td>792</td>
    <td>α<sub>s</sub></td>
    <td>thermal dilation coefficient of the pile</td>
  </tr>
  <tr>
    <td>793</td>
    <td>Δs<sub>i0</sub></td>
    <td>displacement induced by the ‘sheltering-reinforcing’ effects of pile j</td>
  </tr>
  <tr>
    <td>794</td>
    <td>δ</td>
    <td>soil shear modulus ratio</td>
  </tr>
  <tr>
    <td>795</td>
    <td>δ<sub>s</sub></td>
    <td>friction angle of the pile-soil interface</td>
  </tr>
  <tr>
    <td>796</td>
    <td>η</td>
    <td>loading and unloading branching judgment factor</td>
  </tr>
  <tr>
    <td>797</td>
    <td>κ</td>
    <td>loading and unloading branching judgment factor</td>
  </tr>
  <tr>
    <td>798</td>
    <td>ν<sub>soil</sub></td>
    <td>Poisson’s ratio of the soil</td>
  </tr>
  <tr>
    <td>799</td>
    <td>σ<sub>p</sub></td>
    <td>pile tip resistance</td>
  </tr>
  <tr>
    <td>800</td>
    <td>σ<sub>up</sub></td>
    <td>ultimate pile tip resistance</td>
  </tr>
  <tr>
    <td>801</td>
    <td>σ<sub>po</sub></td>
    <td>pile tip resistance corresponding to the stress reversal point</td>
  </tr>
  <tr>
    <td>802</td>
    <td>σ'<sub>vb</sub></td>
    <td>effective vertical stress at the pile tip</td>
  </tr>
  <tr>
    <td>803</td>
    <td>σ'<sub>z</sub></td>
    <td>effective overburden pressure</td>
  </tr>
  <tr>
    <td>804</td>
    <td>τ</td>
    <td>pile shaft shear stress</td>
  </tr>
  <tr>
    <td>805</td>
    <td>τ<sub>u</sub></td>
    <td>ultimate shear stress</td>
  </tr>
  <tr>
    <td>806</td>
    <td>τ<sub>0</sub></td>
    <td>pile shaft stress corresponding to the stress reverse point</td>
  </tr>
  <tr>
    <td>807</td>
    <td>φ<sub>s</sub></td>
    <td>friction angle of the soil</td>
  </tr>
  <tr>
    <td>808</td>
    <td>φ'</td>
    <td>effective friction angle of the pile tip soil</td>
  </tr>
</table>

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