# An Improvement of the Load Transfer Method for Energy Piles Under Thermo-Mechanical Loads
## 2 Materials and Methods
### 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,  
Figure 1: Schematic diagram of the simplified calculation of the load transfer method. (**a**) Equivalent calculation model of pile-soil interaction; (**b**) Illustration of the force on the energy pile. 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. \[\sum Q_{T}^{NP}=\sum_{i=1}^{NP}Q_{s,T}^{i}+Q_{h,T}^{1}+\sum_{i=NP+1}^{n}Q_{s,T} ^{i}+Q_{base,T}=0 \tag{1}\]  
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 ([PERSON] et al. (2011), 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) ([PERSON] et al. (2011), 2011). \[(\sum Q_{T}^{NP})_{\text{min}}=(\sum_{i=1}^{NP}Q_{s,T}^{i}+Q_{h,T}^{1}+\sum_{i =NP+1}^{n}Q_{s,T}^{i}+Q_{base,T})_{\text{min}} \tag{2}\]  
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.