# Thermally Induced Displacements and Rotations of Pillars for Precise Geodetic Measurements
## 2 Experimental Setting and Computational Model  
The behavior of a typical pillar used for forced centering of geodetic instruments was experimentally analyzed under controlled laboratory conditions (see Figure 2). The pillar was made of reinforced concrete and rigidly fixed into a solid concrete block. The concrete was placed using a plastic tube with an inner diameter of 0.217 m, an outer diameter of 0.250 m, and a height of 1.510 m. A standard pre-prepared dry mix of class C25/30 was used for the concrete. Four vertical steel bars, each with a diameter of 12 mm, were installed before pouring. The plastic tube was later removed to allow for the installation of temperature sensors on the pillar's surface. Thermal loading was applied using several heaters mounted on a vertical bar on one side of the pillar, while the opposite, cooler side was partially insulated (Figure 2a). This insulation prevented the back side of the pillar from heating due to radiation, allowing it to warm up solely through internal heat transfer. Additionally, it reduced temperature effects on the measuring equipment. Studies, such as [26], have examined the effect of temperature on inclinometers. In our research, we were aware of this influence, so we thermally insulated the inclinometers. As a result, the heating of the column had only a minimal impact on their operation. Several temperature sensors were installed on both the front heated side of the pillar (Figure 2a) and the back cooler side (Figure 2b), with measurements recorded every 30 s. The installed sensors enabled preliminary studies that led to an optimized vertical arrangement of the heaters, ensuring a nearly uniform temperature distribution along the height of the pillar. This distribution was not only simple but also aligned with the numerical model presented later, making it crucial for establishing a robust relationship between rotations and displacements. A displacement sensor mount was installed behind the pillar, measuring displacement at the top to capture its actual movement due to thermal loading (Figure 2c). The displacements of the pillar were measured with a standard Novotechnik TR25 potentiometric position transducer with \(\pm\)0.2% linearity. The sensor's repeatability is 0.002 mm, and it provides virtually infinite resolution due to its potentiometric design. Measurements were  
Figure 2: Experimental setting: (**a**) heated side of the pillar; (**b**) cooler side of the pillar; (**c**) displacement sensor and two inclinometers. taken every 30 s, and heating was stopped once the top sensor registered a displacement of 1 mm. Two inclinometers were mounted on the pillar (Figure 2c). The first (Leica Geosystems Nivel 210--top inclinometer) was placed at the very top of the pillar on an extruded polystyrene (XPS) insulation board to prevent direct heat transfer from the pillar to the inclinometer. The second (Leica Geosystems Nivel 210--side inclinometer) was mounted laterally on a right-angle bracket attached to the pillar. Measurements were taken every 30 s. The inclinometers used in the experiment were Leica Geosystems Nivel210 sensors, which have been specially developed for high-precision monitoring. These biaxial sensors offer a resolution of 0.01 mrad and a measurement range of \(\pm\)3.00 mrad and accuracy \(\pm\) 0.47. A more detailed positioning of the sensors, including dimensions and distances, is shown in Figure 3. Rotations and displacements strongly depend on the vertical position relative to the column. Since the mounting plate is placed at the top, the displacement sensor and inclinometers are positioned nearby. However, due to geometrical constraints, they cannot measure the exact same point. The key geometrical data are as follows: height of the pillar \(h=1.510\) m; position of the top inclinometer \(x_{R2}=1.572\) m; position of the side inclinometer \(x_{R1}=1.537\) m; and position of displacement sensor \(x_{D}=1.470\) m.  
Based on the experimental setting, we now present the model.