# A Numerical Study of the Behavior of Micropile Foundations under Cyclic Thermal Loading
## 2 Basis for Numerical Analyses
### Material Thermal and Mechanical Properties  
The medium supporting the pile was assumed to be frictional and was assigned parameters typical for a dense sand, as per [40]. It was modeled as linear elastic-perfectly plastic, with a Mohr-Coulomb failure criteria and assuming a non-associated flow rule, with the properties outlined in Table 1. In using this soil model, it was recognized that it did not reproduce a number of the known aspects of granular soil behavior including stress path dependency, and stress and strain dependency of the soil stiffness [41]. It can, however, provide reasonable estimates of failure for boundary value problems involving granular soils and interfaces, capturing the basic idea of shear strength increasing with effective stress. While more complex models for granular soil behavior are available [42], the model has been successfully employed in the back-analysis of TA pile tests [43; 44; 45], and it is considered that the model is, therefore, sufficient, and its simplicity allows the focus of this work to be maintained. The Young's modulus for the soil approximates Equation (1), with a stepwise variation that increases with depth. As seen in Figure 1a, Equation (1) captures the stress dependency of soil stiffness which allows the increase in stiffness with depth to be approximated. This yields an average soil Young's modulus over a depth equal to the length, L, of the piles, of 129 and 91.3 MPa for MP-30 and MP-15, respectively.  
\[\mathrm{E}=1.2\mathrm{p_{a}}\left(\frac{\sigma_{h}^{\prime}}{p_{a}}\right)^{0.5} \tag{1}\]  
where \(p_{a}=\) atmospheric pressure = 101.3 kPa, and \(\sigma_{h}^{\prime}=\) horizontal effective stress.  
Figure 2: FE 2D model with the details of the adopted structured spatial discretization.  
The micropile was modeled as a linear elastic material ([PERSON] et al., 2023) [46], with the properties detailed in Table 1. The effective stiffness of the micropile was determined based on a composite section in which it was assumed that a 140 mm diameter steel tube with a wall thickness of 10 mm extended the full length of the micropile, and the grout comprising the rest of the pile body had a stiffness of 12 GPa [47; 48]. Concerning the tangential behavior, the pile-soil interface was modeled by attributing a \"penalty\" friction formulation, where the contact between the two materials is imposed by means of a shearing resistance that is proportional to the normal stress, i.e., \(\tau=\sigma_{\mathrm{n}}.\mu\), where \(\mu\) is a friction coefficient (Table 1) [49; 50]. The friction coefficient \(\mu=\tan\varphi_{\mathrm{cs}}\)' assumes that a critical state angle of shearing resistance, \(\varphi_{\mathrm{cs}}\)' = 32\({}^{\circ}\), is applicable on the contact with a value of 0.624 ([PERSON] et al., 2009) [51]. Initial analyses of the piles under axial mechanical load were run in order to define the parameter representing the relative displacement at which the contact shearing resistance was fully mobilized, and a value of 0.002 m was chosen. Regarding the normal behavior at the interface, a pressure-overclosure of \"hard contact type\" was selected.  
In granular media, the permeability tends to be sufficiently large that load and thermally induced excess pore water pressures are not likely to be significant [52; 53]. The analyses undertaken assumed the soil was dry, which would alter (increase) the absolute ultimate pile resistance values relative to, e.g., saturated conditions, but it was expected that the relative effects and the conclusions reported in this study will be similar.  
The thermal properties for the micropile and soil are largely generic, however, they were selected as being representative of concrete [54] and soil [55].