## 4 Numerical Results and Discussion  
### General Response  
The general heat transfer response of the pile foundation is discussed in this section using the case with \(k\) = 2 W/m-\({}^{\circ}\)C, \(c\) = 940 J/kg-\({}^{\circ}\)C, and \(d_{i}\) = 200 mm as an example. Figure 7 shows the temperature distribution contours in the first loading cycle at two different times for the concrete and the surrounding soil. The temperature profile in the pile foundation and the surrounding soil is uniform along the circumferential direction because of the symmetry of geometry and loading conditions. Therefore, the rest of the paper only presents temperature distributions along the radius direction (\(r\)), along which the temperature changes. Figure 6: The 2D plane strain model. Figure 8 shows the temperature change in the first 24 h loading cycle for a couple of locations in the concrete and nearby soil. As seen in Figure 8a, as an input loading, the temperature at the inner surface (\(r\) = 100 mm) of the pile foundation follows the one in the compressed air (refer to Figure 4a): increasing from 7:00 to 19:00 but decreasing from 19:00 to 7:00. As time goes by, the heat is gradually transferred from the inner to the outer surface and causes a temperature change at other places. The temperature change becomes smaller and slower from the inner to the outer surface. The temperature near the outer surface (\(d_{i}\) = 500 mm) does not even decrease during the entire loading cycle. As observed in Figure 8b, the heat also penetrates the soil and increases the temperature in the nearby soil. This penetration becomes smaller as moving further from the pile foundation and stops at \(r\) = 1100 mm, where the initial temperature (12 \({}^{\circ}\)C) remains. Figure 9 shows the temperature distribution in the first 24 h loading cycle for various times. From the 7:00 to 19:00 (see Figure 9a), similar temperature distributions are observed in the concrete section though the magnitude increases with the time. On the other hand, from the 19:00 to 7:00 (see Figure 9b), different temperature distributions are observed in the concrete section because the temperature near the inner surface drops, while that near the outer surface still increases. At the end of the first loading cycle, the significant residual temperature is observed in the concrete section. The temperature distribution in the soil is similar for the entire loading cycle, with the magnitude increasing gradually (see Figure 9b). Sudden temperature drops between the concrete surface and the soil surface are observed because of the interface heat transfer effect. Figure 8: Temperature change in the first loading cycle: (**a**) concrete; (**b**) soil. Figure 7: Temperature profile contour in the 1 st loading cycle at: (**a**) 19:00; (**b**) 7:00 (end of the loading cycle). The temperature in subsequent cycles is expected to be affected by the residual temperature at the end of the first loading cycle. Figure 10 shows the temperature history for a couple of locations in the concrete and the soil during the first 36 loading cycles. In general, except for the inner surface, which follows the input temperature change, the temperature in both concrete and soil gradually increases with cycles because of the cumulative effect of the residual temperature. The increase rate is initially high but starts to slow down as the increase of the loading cycle. Moving from the inner surface to the outer, the temperature variation within each cycle becomes smaller. This observation indicates that the effect of the temperature change in the compressed air becomes lower when moving further from the center of the pile foundation. For a distance of 10.1 m from the center of the pile foundation, the soil temperature is not affected (see Figure 10b). Figure 10: Temperature history for the first 36 loading cycles in the: (**a**) concrete; (**b**) soil. Figure 9: Temperature distribution in the 1 st loading cycle: (**a**) 7:00 to 19:00; (**b**) 19:00 to 7:00.