An Improvement of the Load Transfer Method for Energy Piles Under Thermo-Mechanical Loads

Abstract

The energy pile integrates shallow geothermal energy extraction with underground structural engineering, thereby expanding the functional applications and scope of pile foundations. Due to its widespread adoption, research on energy pile analysis theory has advanced significantly. Among existing analytical methods, the load transfer method is widely employed owing to its computational simplicity and readily obtainable parameters. However, current load transfer models for energy piles remain imperfect, primarily because their results often fail to accurately reflect real-world loading conditions. This study investigates the underlying causes of this discrepancy and proposes an iterative method to eliminate unbalanced forces at the pile head, based on the displacement coordination algorithm for energy pile load transfer. The calculated results at the pile head show an 18% reduction in error compared to previous studies. The average error compared with field test results is within 20%, with consistent trend patterns, confirming the feasibility of the proposed method. Computational results demonstrate that the proposed method effectively captures the combined effects of mechanical load and temperature variations on the bearing behavior of energy piles. It should be noted that this paper focuses specifically on improving the temperature-dependent load transfer method for energy piles. Consequently, the conventional load transfer method and results under purely mechanical loading are not discussed herein.

1. Introduction

Geothermal energy, as a low-carbon and energy-saving renewable energy, can be developed and utilized in various fields all over the world. (Abas et al. [1], 2015; De Moel et al. [2], 2010). Energy piles represent an innovative technology that integrates ground-source heat pump systems with building pile foundations. By embedding heat exchange pipes within the pile foundation and circulating a heat transfer medium, this system utilizes the stable thermal properties of subsurface soil to provide building heating and cooling. Structurally, energy piles primarily consist of load-bearing piles, embedded heat exchange pipes, and a heat pump connection system. During operation, heat transfer occurs through the pipes into the surrounding soil, leading to temperature variations in the adjacent geomaterials. Additionally, the restraint imposed by the surrounding soil on pile deformation induces thermal stresses within the energy pile. This complicates the load-transfer mechanism between the pile and soil, ultimately altering the bearing behavior of energy piles (Brandl H. et al. [3], 2006; Jiang Qiangqiang et al. [4], 2019; YANG et al. [5,6], 2023). Therefore, it is necessary to study the influence of the coupling effect of temperature and force on the working characteristics of energy piles to ensure safe and sustainable operation.

At present, many researchers are devoted to the study of thermo-mechanical coupling characteristics of energy piles, and some scholars have conducted relevant field tests to investigate the effect of temperature on structural and load-bearing characteristics (Lu Hongwei [7], 2017. Laloui L. [8], 2005. Bourne-Webb P. [9], 2009. Wang [10], 2015). The field test results show that the change of temperature will lead to the change of pile shaft force, lateral resistance, and pile top settlement. In other words, the effect of temperature on energy piles is not negligible. Although the load test in the field presents a reliable method for observing and determining the thermal-mechanical behaviors of energy piles, it is manifestly both time-consuming and costly and requires numerical modeling approaches as more efficient alternatives. Some scholars have carried out research on the operating characteristics of energy piles by means of numerical simulation and theoretical analysis. (Wang et al. [11], 2024; RAVERA et al. [12], 2020; NAJMA et al. [13], 2021; NAJMA et al. [14], 2019).

Among the many methods, the load transfer method has received wide attention from scholars because of its simplicity and practicality of calculation and because the method requires only fundamental mechanical parameters (e.g., density and internal friction angle) that are routinely obtained through standard geotechnical investigations and geotechnical testing.

Knellowlf [15] (2011) considered the effect of temperature on pile-soil action, and the first energy pile load transfer displacement coordination algorithm was established by using the static equilibrium condition at the neutral surface. The load-bearing characteristics of energy piles under the coupling effect of temperature and force were calculated by using a segmental linear load transfer model, and the reasonableness of the method was verified by comparing it with the field-measured data. Although the method achieves good computational results, it still has the following shortcomings. (1) When the top of the pile is not restrained, that is, the top of the pile is free, the pile top axial force calculation result is not zero; thus, it is considered that this method cannot accurately reflect the constraint at the top of the pile. (2) It is necessary to assume the location of the neutral point and calculate the relatively accurate location of the neutral point by iteration, and the calculation results may not necessarily converge. (3) The segmented linear model cannot accurately reflect the progressivity of the soil deformation trend. Plaseied [16] (2012) continues to use Knellwolf’s method and improves the load transfer model so that the progressive nature of the deformation trend at the interface between pile and soil is taken into account. He makes an in-depth study of the thermal-force-coupled load-bearing characteristics of energy piles, but the calculation results of such a method still do not reflect the pile top constraint. Yinpei Huang [17] (2019) used a load transfer model in the form of an exponential function to simulate the interaction between the pile and the surrounding soil and improved Knellwolf’s method to make the method perform better in terms of convergence, but still the calculation results do not correctly reflect the constraint at the top of the pile.

All the above scholars have improved the load transfer method of energy piles established by Knellwolf and have performed analytical studies on their load-carrying performance. However, it is found that this method cannot accurately reflect the constraint of the pile top, especially when the pile top is free; the pile top axial force obtained by this method is not zero, i.e., additional temperature stress is generated at the free pile top, and such calculation results cannot be explained. Therefore, this paper analyzes the reasons for this phenomenon and further improves the load transfer method for energy piles, which is called the iterative algorithm for the elimination of unbalanced forces at the pile top, so that the calculation results can more accurately reflect the load transfer mechanism of energy piles. Moreover, the improved method proposed in this study is universally applicable to energy piles of any configuration.

2. Materials and Methods

The load transfer method requires the pile body to be divided into several elastic units, and the contact between the pile side, the pile end, and the surrounding soil is replaced by nonlinear springs. Specifically, the relative displacement s of the pile-side and pile-tip resistances with respect to the surrounding soil can be represented by a load–transfer model. (He Huibin et al. [18], 2008; Li Yonghui et al. [19], 2011; Chen Min et al. [20], 2016; SONG et al. [21], 2023), as shown in Figure 1a. To simplify the calculations, the following assumptions were made: First of all, the positive direction of displacement and force is made as follows: displacement downward is positive, strain and stress direction are the same, both are positive by pressure, and the positive direction of frictional force is upward. Secondly, it is assumed that the modulus of elasticity E, thermal expansion coefficient α, and pile-soil interaction relationship are not affected by temperature. Lastly, considering that the radial dimension of the pile is much smaller than the axial dimension, the effect of temperature on the axial deformation of the pile is ignored.

When using the load transfer method of energy piles, it is necessary to determine the location of the neutral surface of the energy pile at first, so the definition of the neutral surface and the method of determining the location of the neutral surface are described next.

2.1. Neutral Surface Definition and Determination Method

According to the assumption that the energy pile can be regarded as a one-dimensional linear heat source. The change of pile temperature will lead to tensile or compressive deformation in the axial direction. Therefore, there must be a cross-section along the pile direction that does not generate strain; the cross-section is called the neutral surface, noted as NP. When the pile is not restrained, the temperature change will cause the pile to deform freely, and the neutral surface is located at the midpoint of the pile without additional temperature stress. In practical engineering applications, however, energy piles are embedded in soil masses and are constrained by both the surrounding soil and the pile-end bearing stratum, resulting in variations in the neutral plane position. As shown in Figure 1b, when only the effect of temperature is considered, the energy pile is compressed and has the maximum axial force at the neutral surface when the temperature rises, and the upper part of the pile generates negative frictional resistance; when the temperature decreases, the energy pile is pulled and has the maximum axial force at the neutral surface, and the lower part of the pile generates negative frictional resistance.

Where $P_{load}$ is the mechanical load acting on the top of the pile; $f_s$ and $f_b$ represent the lateral resistance and end resistance of the pile respectively; $Q_t$ is the axial force of the pile; $\Delta T$ is the variable of the temperature; and H is the depth of pile.

Firstly, the location of the neutral plane needs to be determined when exploring the thermomechanical coupling properties of energy piles using the load transfer method. It is worth noting that the mechanical equilibrium conditions for the neutral plane should satisfy the following equation.

$${\displaystyle \sum Q_T^{NP}}={\displaystyle \sum _{i=1}^{NP}{Q}_{s,T}^i}+Q_{h,T}^1+{\displaystyle \sum _{i=NP+1}^n{Q}_{s,T}^i}+Q_{base,T}=0$$

(1)

where $Q_{s,T}^i$ is the lateral frictional resistance of unit i of the pile; $Q_{base,T}$ is the end resistance; and $Q_{h,T}^1$ represents the vertical structural load (Knellwolf et al. [15], 2011).

When the pile is discretized into a number of elasticities, the neutral surface can only exist on the face where the unit nodes are located. Therefore, the nodes of the pile are assumed to be the neutral surface in turn from top to bottom, and the combined force at the neutral surface is calculated according to Equation (1). The hypothetical neutral surface that makes the combined force minimum is approximated as the actual neutral surface, i.e., the mechanical equilibrium condition at the neutral surface when using the load transfer method for energy piles should satisfy Equation (2) (Knellwolf et al. [15], 2011).

$$({\displaystyle \sum Q_T^{NP}{)}_{\min }}=({\displaystyle \sum _{i=1}^{NP}{Q}_{s,T}^i}+Q_{h,T}^1+{\displaystyle \sum _{i=NP+1}^n{Q}_{s,T}^i}+Q_{base,T}{)}_{\min }$$

(2)

Then, the deformation due to temperature change at the location of the neutral surface is zero as a known condition, and the axial force and strain values of the pile due to temperature are calculated from the node location of the neutral surface to both sides, respectively. Therefore, when using the load transfer method to explore the thermal-mechanical coupling bearing characteristics of the energy pile, it is first necessary to determine the displacement where the neutral surface of the energy pile is located. The definition and determination method of the neutral surface have been explained in detail above, and the flow chart of the algorithm for determining the neutral surface is shown below. The meaning of the equations designed into the flow chart (Figure 2) is shown in Figure 3, which shows the process of determining the neutral surface more graphically through the form of a schematic diagram.

2.2. Load Transfer Methods for Energy Piles

The location where the longitudinal deformation of the energy pile is zero after the temperature influence on the pile, i.e., the location of the neutral surface, can be found by following the method shown in Figure 2. This is also a prerequisite for using the load transfer method. Once the location of the neutral surface is found, the second module of the load transfer method, i.e., the module for calculating the performance of the energy pile, can be applied, and the corresponding calculation process is shown in Figure 4.

2.3. Improvement of Load Transfer Method for Energy Piles

2.3.1. Find the Problem

Figure 2, Figure 3 and Figure 4 show the algorithm of the coupled thermo-mechanical analysis method for the energy piles established by Knellwolf [15] (2011), and now the London energy pile in-situ test is investigated using the adapted method. The pile used in this test has a diameter that is 0.6 m, a length that is 22.5 m, a modulus that is 40 GPa, and a coefficient of thermal expansion that is 8.5 × 10⁻⁶ °C⁻¹. The parameters of the physical and mechanical indexes of the soil layer in this test are shown in

Table 1. Soil physical and mechanical parameters of the London test.

Soil Properties	River-Stage Sedimentary Soils	London Clay	London Clay	London Clay
Depth, h (m)	0~6.5	6.5~10.5	10.5~16.5	16.5~22.5
Density, $\rho$ (kg/m3)	2300	2000	1870	1870
Cohesion, c (kPa)	25	15	15	10
Friction angle, φ (°)	15	18	18	18
Elastic modulus, E (MPa)	72	45	45	45
Poisson’s ratio, v	0.2	0.5	0.5	0.5
End-resistance τb (kPa)	—	—	—	460

. While the static load test is in effect, the energy pile heat transfer test is conducted through the heat transfer tube, and the force on the top of the energy pile is controlled to be constant, i.e., the load is maintained at 1200 kN during the test so as to obtain the load transfer characteristics of the energy pile under the coupled thermo-mechanical action, and the field test and calculation results are shown below. A comparison of the calculated results with the field test results is shown in Figure 5. It can be found that the calculated top loading of the pile is 1591.46 kN, while the actual loading is 1200 kN, a difference of 391.46 kN, i.e., the error between the calculated and actual loading is 32%, so it can be considered that the load transfer characteristics of the energy pile under the coupled thermo-mechanical action cannot be accurately calculated by the above method.

2.3.2. Analyze the Cause

The analysis shows that both the size of the cells and the equilibrium condition of the neutral surface may lead to the phenomenon that the calculated axial force does not match the actual loading condition at the top of the pile. This is due to the mismatch between the unit size and the calculation tolerance error. Therefore, the matching degree of the cell’s size and the calculation tolerance of the error will lead to the non-convergence of the calculation results, so the matching problem between the size of the cell and the calculation tolerance of the error in the load transfer method is also the difficulty of the method. Therefore, the problem of matching the cell size and the computational tolerance of the error in the load transfer method is also a difficult point in the promotion of the method, which will not be discussed too much in this paper. In the following, the reasons for the mismatch between the axial force calculated from the neutral plane equilibrium condition and the actual load at the top of the pile will be discussed, the iterative method to eliminate the unbalanced force at the top of the pile will be used, and an iterative algorithm to eliminate the unbalanced force at the top of the pile will be proposed so that the load transfer displacement coordination method can accurately reflect the load transfer characteristics of energy piles coupled under thermodynamic action.

2.3.3. Improve the Method

It has been mentioned above that the calculated results of the load transfer method established by Knellwolf [15] (2011) applied to the study of energy piles showed large deviations between the calculated axial force values at the top of the pile and the actual loading. After analysis, it is known that this is caused by using ${({\displaystyle \sum Q_T^{NP}})}_{\min }$ instead of $({\displaystyle \sum Q_T^{NP}})=0$ in determining the neutral surface. Although this method can determine the approximate location of the neutral plane, it also brings a large error to the calculation results of the pile top, which can be recorded as the pile top unbalance force. The iterative algorithm for pile top unbalance force elimination established in this paper can well solve the pile top error brought by the above problem, and the specific steps are as follows.

Step 1: The approximate location of the neutral surface is determined according to the algorithm shown in Figure 2.

Step 2: The distribution of the axial force, strain, and lateral resistance of the energy pile along the pile is determined according to the algorithm shown in Figure 4.

Step 3: The value of the loading at the top of the pile is known and is denoted as P. The calculated axial force at the top of the pile is denoted as ${\displaystyle \sum Q_{\mathrm{top},T}^1}$. The difference between them is calculated as the unbalanced force at the top of the pile.

Step 4: The unbalanced force at the top of the pile obtained from the calculation is added back to the top of the pile, and the axial force, strain, and displacement due to the unbalanced force are obtained using the load transfer method.

Step 5: The axial forces, strains, and displacements due to unbalanced forces are superimposed on the results calculated by the algorithms shown in Figure 2 and Figure 4, thus updating the results of the axial force, strain, and displacement distribution of the pile.

The flow chart of the iterative algorithm for the unbalanced force at the top of the pile is shown below.

The test conditions shown in Figure 6 were calculated and analyzed according to the iterative algorithm for the elimination of the unbalanced forces at the top of the pile. The calculated results were also compared with the results of the load transfer method established by Knellwolf, the test results, and the matrix displacement method constructed by Ouyang et al. [22], as shown in Figure 7. It can be found that the axial force at the top of the pile calculated by the improved method is consistent with the value of the actual loading at the top of the pile, and the results of the axial force and lateral resistance distribution along the pile body obtained by the method are consistent with the pattern of the experimental values, and the results are similar, which can better reflect the effect of thermo- mechanical coupling on the performance of energy piles.

3. Results and Discussion

In order to verify the energy pile load transfer method established in this work that can accurately reflect the loading at the top of the pile, a numerical simulation was conducted on the Kunshan test (performed preliminarily by the authors), and the calculated results were compared with the test results.

3.1. Description of the Project and the Test Setup

Kunshan tests of energy piles were preliminarily carried out by the authors in the city of Kunshan in southeast China for both driven and bored energy piles. A total of 94 bored piles and 43 driven piles were selected as the foundation of the building in the construction project, and this paper focuses on the field tests and numerical simulations of bored energy pile tests. Soil information of the site is shown in

Table 2. The physical and mechanical properties of soil and piles.

(a) Properties of Soil
Soil Properties	Fill	Silty Clay 1	Silty Clay 2	Silty Clay 3	Silt with Sand	Silty Sand
Depth, h (m)	0~3.8	3.8~4.8	4.8~6.8	6.8~10.6	10.6~15.6	15.6~21.8
Density, $\rho$ (kg/m3)	1800	1830	1770	1940	1840	1860
Cohesion, c (kPa)	—	20.5	12.0	40.5	7.5	5.9
Friction angle, φ (°)	11.6	12.8	10.0	15.0	25.1	27.1
Elastic modulus, E (MPa)	28	25	15	80	100	140
CPT tip resistance (MPa)	0.79	0.77	0.48	2.38	5.11	7.73
(b) Properties of Piles
Numbering of Piles	Type of Piles	Pile Length (m)	Pile Diameter (mm)	Strength Grade	Form of Buried Pipe
F1	Bored piles	17	600	C30	Double Spiral

a. Table 2b shows the properties of the bored piles, and it can be seen from Figure 8 that the energy piles were fully instrumented, including temperature sensors and rebar stress gauges. The stiffness of the pile at the top when the energy pile is heated can be taken as 0.26 GPa/m by inversion. More details of the project setup and the testing procedures can be referred to Lu et al. [7] (2017), Jiang et al. [23] (2021), and Jiang et al. [24] (2022).

In the test, the settlement was monitored using four micrometers installed at the pile head. The load was applied to the pile using a hydraulic jack, and the magnitude of the load was measured by a load cell placed between the hydraulic jack and the pile head. The arrangement of temperature sensors and reinforcement stress gauges is shown in Figure 8, where temperature sensors are uniformly arranged at 3, 10, and 16 m below the top of the pile, while reinforcement stress gauges are installed in six groups at 0.5, 3.7, 9.0, 11.1, 13.9, and 16.5 m below the top of the pile according to the soil distribution characteristics in the site.

The results of the heating recovery test of the bored energy pile (F1) and the comparison of the results with the numerical calculation method used in this work are presented and discussed in the following subsections.

3.2. Results and Discussion of Kunshan Test

The unbalanced force elimination iterative algorithm at the top of the pile is used to calculate and analyze the bored pile (F1) to study the effect of thermo-mechanical coupling on the load transfer characteristics and performance of the energy pile.

3.2.1. Case 1: 25% of the Ultimate Load and Coupled Heating

A comparison of the calculated results for the load of 420 kN and simultaneous heating shows that the results of the iterative method of unbalanced force elimination are in general agreement with the results of the test. After the superimposed temperature effect with the load, the axial force and lateral resistance of the pile are redistributed, and it can be seen from Figure 9b that the temperature change has less effect on the lateral resistance of the upper part of the pile, but it can still be found that the negative resistance of the upper part of the pile due to the temperature is offset by the positive resistance of the upper load, so that the lateral resistance of the upper part of the pile gradually decreases, while the lower part of the pile is the opposite. The results of the iterative method of unbalanced force elimination also reflect this phenomenon well, and both the test results and the calculated results of the lateral resistance distribution curve (Figure 9b) can be found to show that the neutral surface of the pile is about 11.1 m. Therefore, the above analysis shows that the iterative method of unbalanced force removal at the top of the pile for energy piles established in this paper can be used to study the effect of thermo-mechanical coupling on its load transfer characteristics and performance.

3.2.2. Case 2: 50% of the Ultimate Load and Coupled Heating

As shown in Figure 10, the results of axial force and lateral resistance distribution along the pile body obtained by the iterative algorithm of unbalanced forces at the top of the pile are more consistent with the measured results when the structural load of 840 kN is combined with heating. Similarly, the temperature change caused the change in pile-bearing performance. As can be seen from Figure 10a, heating will cause an increase in the axial force at all points of the pile, and such an effect is greatest at the top of the pile and decreases gradually with increasing depth. Figure 10b shows the distribution curve of the lateral resistance along the pile. The calculation results show that the temperature change has less effect on the upper lateral resistance of the pile; the lower lateral resistance of the pile changes more obviously by the temperature, i.e., the temperature increase causes the pile to expand, which leads to the positive resistance of the lower part of the pile making the lateral resistance of the pile increase. From the test results, it can be seen that the increase in temperature weakens the lateral resistance of the upper part of the pile, which is also caused by the expansion of the pile. Although there is a large error between the calculated value and the test value in the upper lateral resistance, it can still reflect the effect of temperature change on the pile bearing performance, so the method established in this paper is still applicable.

4. Conclusions

The improved energy pile load transfer method eliminates the effect of unbalanced forces at the top of the pile caused by the inaccurate determination of the neutral surface. The calculation results of the improved load transfer method were compared with those of the load transfer method established by Knellwolf [15]. The error between Knellwolf’s method and the test results at the pile head was approximately 20%, whereas the error using the proposed method was only 2%. Additionally, the average error between the proposed method and Ouyang’s matrix displacement method was only 4.5%. In comparison with field test data from Kunshan, the average error of axial force was 9.6%, and the average error of shaft resistance was 17.3%, with both showing consistent trends. Thus, it shows that the method in this paper has better applicability. The results of both calculations and tests show that the coupled load-temperature effect of the energy pile will cause changes in load transfer characteristics such as axial force and lateral resistance, indicating that the temperature effect at the pile-soil interface has a non-negligible effect on the performance of the energy pile. It is also found that the performance of energy piles is closely related to the temperature variation, end restraint, and value of loading through the study of this paper. Therefore, the temperature variation range should be controlled according to the loading at the top of the pile and the end restraint in order to ensure the safety of energy pile work in practical engineering.