## 3 Experimental Measurement
### Realization of Experimental Measurement  
#### 3.2.1 Test Measurement at Unit Load  
Experimental measurements were performed on three different roofs of the same car model. Two roofs were painted and one was unpainted (Figure 7). It should be noted that these roofs are not the same. Marking A--painted roof, in selected places material was taken for other mechanical tests. Marking B--unpainted roof. For this reason, roofs A and B were used for preparatory experimental measurements; therefore, only four strain gauges were applied. The measured results were not compared with numerical modelling, as the model with the collected material was not used in the FEM (Finite Element Analysis) analysis. The third roof marked C corresponds in shape to the numerical model, so that the measured values were compared with FEM analysis. In this case, strain gauges were applied at six locations. In order to ensure the same operating conditions, one person used the same type of antenna and the same tightening procedure in all cases. Figure 8 shows a view of the applied strain gauges on the analyzed roofs. Figure 8: Roof loading at the mounting hole. Figure 6: Strain gauges’ locations. Figure 7: View of applied strain gauges: (**a**) painted roof A; (**b**) unpainted roof B; (**b**) painted roof C.  
The following presents the results of tensometric measurements performed on roofs B and C. The aim was to compare the values of relative deformations \(\varepsilon_{a}\), \(\varepsilon_{b}\), and \(\varepsilon_{c}\) in locations BL1, BL2, BP1, BP2, CL1, CL2, CP1, and CP2 at unit load caused at the location of the antenna mounting hole (Figure 8). Five repeated measurements were performed. The time courses of the measured relative deformations are shown in Figure 9.  
From the measured strain \(\varepsilon_{a}\), \(\varepsilon_{b}\), and \(\varepsilon_{c}\), the values of the principal stresses were calculated according to Equation (1)  
\[\sigma_{1/2}=\frac{E}{(1-\mu)}\frac{\varepsilon_{a}+\varepsilon_{c}}{2}\pm \frac{E}{\sqrt{2(1+\mu)}}\sqrt{\left(\varepsilon_{c}-\varepsilon_{b}\right)^{2 }+\left(\varepsilon_{a}-\varepsilon_{b}\right)^{2}}, \tag{1}\]  
where \(E\) is [PERSON]'s modulus and \(\mu\) is [PERSON]'s ratio, and subsequently, using Equation (2), the values of equivalent stresses according to [PERSON] theory were determined  
\[\sigma_{nd}^{HMH}=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}-\left( \sigma_{1}\sigma_{1}+\sigma_{2}\sigma_{3}+\sigma_{1}\sigma_{3}\right)}. \tag{2}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are principal stresses. Figure 9: Time records of strains for repeated loading at the hole: (**a**) roof B; (**b**) roof C.  
The values of the main normal stresses and reduced stresses according to the von Mises theory are given in Table 3 to compare the stress states on roofs B and C under the same load. From the above values it follows that:  
In place 1, in both cases (roofs B and C), the values are significantly lower compared to values in place 2, while on the painted roof (roof C) there are larger percentage differences; in the places on the left side, the measured values are lower than on the right side in the case of both painted and unpainted roofs, which is most likely closely related to the shape of the roof (see Figure 6--displacements determined by 3D scanning). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measuring Place** & \(\sigma_{\text{max}}\) **(MPa)** & \(\sigma_{\text{min}}\) **(MPa)** & \(\sigma_{\text{red}}\) **(MPa)** \\ \hline BL1 & 1.83 & \(-0.51\) & 2.13 \\ BL2 & 2.55 & \(-2.21\) & 4.12 \\ BP1 & 2.12 & \(-0.91\) & 2.69 \\ BP2 & 2.65 & \(-2.50\) & 4.46 \\ CL1 & 1.72 & \(-0.55\) & 2.05 \\ CL2 & 5.22 & 0.26 & 5.09 \\ CP1 & 1.61 & \(-1.16\) & 2.41 \\ CP2 & 5.77 & 0.92 & 5.36 \\ \hline \hline \end{tabular}
\end{table}