4.3.2. Updated Model

When updating the numerical model of tie rods, rotational springs were added at each boundary, starting from the initial hinged model. Spring stiffness was determined by an iterative procedure. Their values were changed until the mode shapes of the numerical model entirely corresponded to the mode shapes obtained by on-site measurements. During the iteration, the change in the RMSE value was monitored depending on the change of the spring stiffness (k) (Figure 12). Through several iteration steps, due to the minimum value of RMSE, it was possible to select the appropriate spring stiffness for each mode shape. In this way, the finite element (FE) model was updated to overlap with the experimental model considering only the boundary conditions.

**Figure 12.** The values of the RMSE for the observed mode shapes and different spring stiffness values.

Table 4 shows the natural frequencies computed by SAP2000 (fnum n ) and the relative error between these frequencies and the frequency values determined by on-site measurement (fexp n ). The error varied between 20% and 30% for the first two frequencies, which confirmed the assumptions about the axial force in the tie rods.



Therefore, the next step was to adjust the natural frequencies of the numerical model to correspond to those obtained from on-site measurements. The natural frequencies were adjusted by changing the tension in the FE model of the tie rod. In addition, the procedure of tuning natural frequencies is iterative, and it was implemented for the first two mode shapes. The values of experimentally-measured natural frequencies (fexp n ) were compared to the numerical model (fnum n ) and are presented in Figure 13. After tuning the tension for both mode shapes (P1, P2) in the FE model, the results of natural frequencies were in good agreemen<sup>t</sup> with those measured for both mode shapes (Figure 13). The tuning of dynamic properties of the FE model enabled us to determine the actual tension in the tie rod (Table 4).

**Figure 13.** Change of force values depending on the ratio of numerical and experimental frequency values for two mode shapes (P1, P2).

Based on the proposed methodology, we continued to the final stage (stage 3) of determination of axial force and coefficient κ. With known axial force (from the updated numerical model) and geometrical and material properties (from on-site observations), coefficient κ could be determined by using Equation (7). As expected, the values of coefficient κ (Table 5) obtained for both mode shapes were within the theoretical boundaries of the hinge–hinge and clamp–clamp conditions previously presented in Table 1. Based on the force values obtained from two mode shapes indicated in Table 5, one can observe that there is a difference of around 12% between these values. It can be concluded that the force value varies depending on the number of the observed mode shapes as indicated in [27].


**Table 5.** The force values read for the first two mode shapes and the value of coefficient κ determined by the calculation of the updated model.

We assumed that the coefficient κ found by using this procedure could be applied to all remaining tie rods. The stated assumption was based on the on-site visual inspection and the observed geometrical and material properties.

Based on the defined coefficient κ, experimentally-determined natural frequencies, and analytical equations, the values of the axial forces and the stress levels (Table 6) were determined for the observed iron tie rods in the Cathedral of St. James in Šibenik, Croatia.

**Table 6.** The value of forces Pn (kN) and stress levels σn (MPa) for the observed mode shapes in the tie rods observed in the Cathedral of St. James in Šibenik at level R4.


The shown stress levels were determined as the arithmetic values of the stress determined for the first two mode shapes (Table 6). Based on Figure 14, we concluded that the stress levels in the observed tie rods at level R4 in the Cathedral of St. James are far below the yield strength. Similar values were presented in study [28].

**Figure 14.** Stress levels measured on the tie rods in the Cathedral of St. James in Šibenik at level R4.
