# Thermally Induced Displacements and Rotations of Pillars for Precise Geodetic Measurements
## 2 Experimental Setting and Computational Model
Given the relatively small dimensions of the cross-section, it is reasonable to assume a linear temperature distribution over the volume. This can be expressed as follows:  
\[\Delta T(x,y,z)=\Delta T_{x}+\Delta T_{z}(x)\;z\]  
Here, \(\Delta T_{x}\) represents the average incremental temperature along the reference axis, while \(\Delta T_{z}(x)\) denotes the temperature gradient of a cross-section at position \(x\), where \(x\) is the arc-length parameter along the centroidal axis of the pillar, and \(z\) denotes the axis orthogonal to the height of the pillar and parallel to the direction of heating. In our model, the cross-sectional temperature gradients are estimated by subtracting the  
Figure 3: Details and geometrical data of the setting: (**a**) more detailed positioning; (**b**) dimensions and distances. measured temperatures on the cooler (non-heated) side from those on the heated side at several discrete points along the height of the pillar. Temperature gradients measured by sensors at three distinct heights are depicted in Figure 4. The sensor positions are as follows: bottom, \(x^{(I)}=0.150\) m; middle, \(x^{(III)}=0.750\) m; and top, \(x^{(V)}=1.350\) m. The gradients were estimated as  
\[\Delta T_{z}^{(i)}=\Big{(}T_{w}^{(i)}-T_{c}^{(i)}\Big{)}/d, \tag{1}\]  
where \(T_{c}^{(i)}\) and \(T_{w}^{(i)}\) represent the temperatures on the cooler and heated sides, respectively, at position \(i\), and \(d\) denotes the diameter of the pillar. For the present study, we have \(d=0.217\) m. From Figure 4, we can see that the temperature gradients are almost evenly distributed over the height. This was achieved by carefully planning the experiment to realistically model the effect of solar radiation on such structures. The results allow us to assume a constant \(\Delta T_{z}(x)\) with respect to \(x\), while its temporal distribution in our experiment can be seen in Figure 4.  
The relationship between lateral displacements and thermal loads in beam-like structures, in the special case where no mechanical loads are present, is governed by a simple ordinary differential equation [27]:  
\[\frac{d^{2}}{dx^{2}}w(x)=-\alpha_{T}\Delta T_{z}. \tag{2}\]  
Integrating once gives the slope as follows:  
\[\frac{d}{dx}w(x)=-\alpha_{T}\Delta T_{z}\;x+C_{1}=-\varphi(x), \tag{3}\]  
and integrating again results in the lateral deflection as follows:  
\[w(x)=-\alpha_{T}\Delta T_{z}\;\frac{x^{2}}{2}+C_{1}x+C_{2}. \tag{4}\]  
Here \(w(x)\) denotes the lateral deflections of a reference axis, and \(\alpha_{T}=1.25\times 10^{-5}/^{\circ}\)C is the temperature coefficient of the concrete forming the pillar. The integration constants are determined from the boundary conditions at the clamped end, leading to \(C_{1}=C_{2}=0\). Note that the locations of measured displacements and rotations generally do not coincide. Figure 3.0 Temperature gradients at three different heights of the pillar. However, for a constant cross-sectional temperature gradient, \(\Delta T_{z}(x)=\text{const}\), a simple relationship between displacements and rotations can be derived:  
Figure 3.1  
which leads to  
Figure 3.2  
In the above equations, \(x_{D}\) denotes the position of displacement sensor, while \(x_{R}\) denotes the position of an inclinometer, and \(f_{G}\) is the correlation coefficient between displacements and rotations. For such a simple temperature distribution, uniform along the height, the coefficient relating displacement and rotation is independent of temperature. This allows for straightforward transformations between these quantities, regardless of the measurement locations. In the present experiment, displacements were measured at \(x_{D}=1.470\) m, while inclinometers were positioned at \(x_{R1}=1.537\) m and \(x_{R2}=1.572\) m. This yields the coefficients used for postprocessing the measured data as follows:  
Figure 3.3  
for the rotations at the top and  
Figure 3.4  
for the side measurements.