## 2 Materials and Methods
### Temperature Loading
#### 2.1.2 Boundary Conditions  
The temperature boundary condition \(\mathrm{T}=\mathrm{f}(\varphi,\mathrm{t})\) (i.e., \(\mathrm{T_{1}},\mathrm{T_{2}}\ldots,\mathrm{T_{24}}=\mathrm{f}(\varphi, \mathrm{t})\), as shown in Figures 2-4), depends on time \(\mathrm{t}\) and the angular dimension of the Earth and acts on the outer surface of the crust. This boundary condition simulates the cyclic temperature changes based on the daily cycle of the Earth's rotation and the annual cycle of the planet's orbit around the Sun. An example of \(\mathrm{T_{2}}\) dependence loading for 2 years and 5 days is shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Section** & **Angle \(\varphi\) /deg/** & \begin{tabular}{c} **Young’s** \\ **Modulus E/Pa/** \\ \end{tabular} & \begin{tabular}{c} **Poisson Number** \\ **/I** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **/kg\(\cdot\)m\({}^{-3}\)/** \\ \end{tabular} & \begin{tabular}{c} **Conductivity** \\ **/W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)/** \\ \end{tabular} & \
\begin{tabular}{c} **Specific Heat** \\ **/J\(\cdot\)K\({}^{-1}\)/** \\ \end{tabular} &
\begin{tabular}{c} **Thermal Expansion** \\ **/K\({}^{-1}\)/** \\ \end{tabular} \\ \hline
1 & 345–360 & 3.00 \(\times\) 10\({}^{10}\) & 0.30 & 2760 & 3 & 1100 & 3.50 \(\times\) 10\({}^{-5}\) \\ \hline
2 & 0–15 & 3.50 \(\times\) 10\({}^{10}\)  
For the calculation, surface temperatures from the ALA monitoring system [10; 11] (in addition to surface temperature and air temperature) were taken from stations located between latitudes of approximately 40 and 50 degrees north for the period October 2007 to August 2009, at 15 min increments.  
The basic characteristic values of the statistical data of temperature measurements \(\rm T_{1},T_{2}\ldots,T_{24}=f(\varphi,t)\) are:  
* Minimum temperature = 263.65 K,
* Mean temperature = 282.20 K,
* Median temperature = 282.15 K,
* Standard deviation of temperature = 7.68 K,
* Maximum temperature = 303.35 K.  
The data were then processed for importing into the MSC.Marc Mentat 2006 software [42]. The resulting time-dependent temperature series contained 35,424 values, with 110 real temperature anomalies where the temperature continuity check (large variability between two measurements) was not met. These were mainly large fluctuations in air temperature, which could have been caused by the outbreak of a severe storm, etc. The surface temperature followed the air temperature pattern. In this case, the above-mentioned anomalies occurred only in the spring and summer periods, from 3 April 2008 to 30 August 2008. Of course, these anomalies, which are real in nature, were also used in the solution.  
Figure 5: Earthcylinder and its initial temperature distribution \(\rm T_{initial}\).  
of 1 February 2008 01:00:00 and, for each subsequent section, the time series is shifted by another hour, as already mentioned.  
The different temperature boundary conditions in each section represent a virtual rotation of the model, i.e., the change in temperature simulates the rotation of the Earth around the Sun.  
The temperature time series, however, had to be reduced for importing into the MSC.Marc Mentat software due to the large amount of data, so that the original time step of 15 min had to be increased to a time step of 3519 s (i.e., 58 min 39 s). This leads to the faster solution by FEM.  
Since the initial surface temperature of the model is constant around the perimeter and the boundary temperature conditions are angle dependent, it is necessary to let the model settle at the beginning of the analysis. According to the results obtained, the model is considered to be steady after three days of loading, i.e., the values from 4 February 2008 00:48:37 are taken for processing the results. More information regarding the boundary conditions is given in [9].