# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique
## 4 Case Study Using the Proposed Methodology
### Experimental Identification of Dynamic Properties of Tie Rods  
As a part of stage 1 of the proposed methodology, dynamic properties of tie rods were experimentally observed by using the operational modal analysis (OMA) method. Unlike the classical experimental modal analysis (EMA), the OMA method does not require known excitation. The reference [26] reports negligible differences between the dynamic parameters of simple structures obtained by these two methods. The obtained data (natural frequency, mode shapes, and damping coefficient) were used to improve the numerical model of the tie rods to estimate the tensile forces based on natural frequency. During the experiment, irregularities in the geometry of tie rods, their lengths, and the length of the anchoring to the wall were observed in the north and south galleries (Figure 2). Figure 8 shows the reference tie rod, 6B-C at level R4, with the geometric properties and arrangement of measuring points. As such, a total of nine measurement points were obtained for each tie rod, and the two endpoints were assumed to be fixed. For the purpose of determining the boundary conditions, the measuring points at the ends of the tie rods were more densely distributed.  
Figure 6: Cathedral of St. James in Sibenik, Croatia. (**a**) West view of cathedral and (**b**) iron tie rods inside the cathedral.  
Figure 8: Reference tie rod (6B-C, R4) with measuring points and dimensions.  
Figure 7: Cathedral of St. James in Sibenik. (**a**) Ground floor and (**b**) cross-section.  
For simplicity, tie rod 6B-C was taken as the reference, and a more detailed analysis was performed on it, as follows: The on-site measurements were performed using five piezoelectric accelerometers (Bruell & Kjaer, Denmark, type 4508-B, nominal sensitivity 100 mV/g). The OMA was conducted by roving four accelerometers through two measuring stages, using one as a referent. The tie rods were excited by randomly using a rubber impact hammer. Frequency domain decomposition (FDD) and enhanced frequency domain decomposition (EFDD) were used for the estimation of modal parameters (Figure 9). The values of the experimentally-obtained frequencies for the concerned mode shapes were read from the characteristic record (Figure 10).  
The natural frequencies of the remaining tie rods were measured using only one accelerometer placed at the node 4 (Figure 8). Geometric properties (L--length, b--width, and h--height) and the first two experimentally-obtained natural frequencies (f\({}_{\text{n}}^{\text{exp}}\), n = 1, 2) for the observed tie rods are presented in Table 2. Please note that the cross-section dimensions present an average value of five measurements performed in quarters of tie rods' length.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Tie** & **L** & **h** & **b** & **f\({}_{\text{1}}^{\text{exp}}\)** & **f\({}_{\text{2}}^{\text{exp}}\)** \\
**Rod** & **(m)** & **(mm)** & **(mm)** & **(Hz)** & **(Hz)** \\ \hline
2B-C & 6.84 & 55 & 55 & 7.25 & 17.94 \\
3B-C & 6.71 & 64 & 64 & 7.56 & 19.00 \\
4B-C & 6.81 & 60 & 60 & 7.31 & 18.69 \\
5B-C & 6.87 & 68 & 68 & 7.31 & 19.56 \\
6B-C & 6.90 & 61 & 61 & 6.94 & 17.50 \\
7B-C & 6.95 & 56 & 56 & 8.25 & 18.88 \\
7-8B & 6.97 & 56 & 56 & 8.13 & 19.38 \\
7-8C & 6.98 & 60 & 60 & 8.63 & 19.38 \\ \hline \end{tabular}
\end{table}
Table 2: Values of the first two frequencies of experimentally-observed tie rods in the Cathedral of St. James in Sibenik at level R4.  
Figure 10: Characteristic record of frequency domain decomposition (FDD) for the determination of natural frequencies on reference iron tie rod 6B-C.  
Figure 9: The first two experimentally-obtained mode shapes of the reference iron tie rod 6B-C.