# A Numerical Study of the Behavior of Micropile Foundations under Cyclic Thermal Loading
## 2 Basis for Numerical Analyses
### Initial and Boundary Conditions and Modeling Sequence  
The analyses presented follow a four-stage sequence as described below:  
* The problem domain was initialized with an initial stress phase, based on a value of the at-rest earth pressure coefficient, K\({}_{0}\) of 0.429, and an initial temperature, T\({}_{0}\) of 14 \({}^{\circ}\)C;
* Recognizing that the near-surface region of the ground is influenced by seasonal climate variations, in this stage, a transient thermal analysis was undertaken in order to establish the extent of this region. To achieve this, a periodic temperature function, Equation (2) and Figure 3a, was applied at the ground surface over a period of 10 years, which was found to be sufficient to establish a dynamic thermal equilibrium with the imposed boundary condition, Figure 3b. \[\mathrm{T}=\mathrm{T}_{\mathrm{avg}}+\Delta\mathrm{T}\sin\left[2\pi\mathrm{t}/ \mathrm{P}\right]\] (2) where T\({}_{\mathrm{avg}}=\mathrm{T}_{0}=\mathrm{14}\)\({}^{\circ}\)C is the average annual air temperature; \(\Delta\mathrm{T}=\mathrm{11}\)\({}^{\circ}\)C is the temperature amplitude; P = 1 year is the period of the function, and t (years) is the time since the start of the calculation stage. In this phase of the calculation, adiabatic conditions were assumed \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Unit** & **Micropile** & **Soil** \\ \hline Density, \(\rho\) & kg/m\({}^{3}\) & 2500 & 1900 \\ Young’s modulus, E & MPa & 18,435 & Figure 1a \\ Poisson’s ratio, \(\
u\) & - & 0.25 & 0.3 \\ Angle of shearing resistance & degrees & n/a & 38 \\ Cohesion, c & kPa & n/a & 0.1 \\ Interface angle of shearing resistance, \(\delta\) & degrees & n/a & 32 \\ Interface friction coefficient, \(\mu=\tan\) & - & n/a & 0.624 \\ Thermal conductivity, k & W/m\(\cdot\)K & 2 & 2 \\ Specific heat, c & J/kg\(\cdot\)K & 940 & 800 \\ Linear coefficient of thermal expansion, \(\alpha\) & \(\mu\epsilon/\mathrm{K}\) & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties adopted in the analyses. along the side boundaries and a constant temperature (T\({}_{0}\) = 14 \({}^{\circ}\)C) was specified on the bottom boundary. * The pile was wished-in-place and then the allowable service load, F\({}_{\text{a}}\) (see Section 3.1.1) was applied;
* Operational thermal activation of the pile was then modeled for a duration of 10 years, through a \"coupled temperature-displacement (transient)\" procedure. This was achieved by applying a periodic temperature change over the length of the pile, on the centerline of the micropile. This approach was considered reasonable following the work of [29] which found that thermal load applied as a temperature at the pipe position in a micropile gave similar responses to that obtained by modeling the pipe flow using one-dimensional, forced-convection elements. The imposed temperature followed Equation (2) with T\({}_{\text{avg}}\) = 14 \({}^{\circ}\)C; \(\Delta\)T = 12 \({}^{\circ}\)C. Each simulation started from T = T\({}_{\text{avg}}\) and with a season for which there was a need for cooling (and, therefore, pile heating, for heat injection into the ground). It can be noted that analyses starting with a heating season (pile cooling) showed some small differences in the first few thermal cycles but converged to the same response as reported herein and thus, are not shown. This fourth stage was run with one of two surface temperature boundary conditions:  
* An \"outside\" condition, as described above was maintained to represent an outdoor environmental condition which would be consistent with, e.g., the running of a pile load test;
* An \"inside\" condition was modeled, again using Equation (2) but with T\({}_{\text{avg}}\) = 23 \({}^{\circ}\)C, \(\Delta\)T = 2 \({}^{\circ}\)C and P = 1, Figure 3a, which represents the indoor thermal comfort condition for a structure supported on the pile foundation.