# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique
## 5 Conclusions  
Tie rods are used in historical buildings to prevent horizontal displacement and resulting structural damage to elements like walls, buttresses, arches, and vaults. In their usual application, tie rods transfer axial tensile loads. Due to the importance of preventing damage to historical buildings, determining these loads and the corresponding stress levels is crucial, and can be achieved using several approaches based on static, dynamic, and mixed methods. This paper presented a methodology for the determination of axial forces in tie rods of historical buildings based on the model-updating technique. This methodology is composed of three stages. The first stage involves an on-site experimental study. In addition to the determination of the geometrical and material properties of the selected tie rod, this stage results in the determination of its natural frequencies and mode shapes. In the second stage, an initial numerical model is developed based on the experimentally-determined geometrical and material properties and the assumed boundary conditions. The mode shapes from the numerical model and the experiment are then compared by using the root-mean-square error. A minimum value of error indicates that the numerical and experimental findings overlap and the boundary conditions are adequately updated. Axial forces are then tuned in the numerical model to match the experimentally-determined natural frequencies from the first stage. In the third stage, the geometrical and material properties are combined with the tuned axial force into an analytical solution, resulting in the determination of the boundary conditions coefficient \(\kappa\). This coefficient can then be applied to other tie rods with similar geometrical and material characteristics in their respective historical buildings without repeating the second stage of the methodology. The outlined methodology was verified using a historical building case study. Notably, the boundary coefficient \(\kappa\) indicated in this paper is valid only for the analyzed building. Therefore, the values obtained for this coefficient are not universal and the tie rods of each building should be investigated individually based on the proposed methodology. The importance of the proposed methodology lies in its nondestructive nature, a very important feature in case of historical buildings. Also, the proposed method is relatively simple and quick to implement on site. We considered two mode shapes in the determination of tie rod axial load. To investigate the possibility of axial load determination from higher-order mode shapes, we recommend applying a denser disposition of acceleration measurement positions.  
Figure 14: Stress levels measured on the tie rods in the Cathedral of St. James in Sibenik at level R4.  
**Author Contributions:** Conceptualization, I.D., D.D., and M.B.; methodology, I.D and D.D.; formal analysis, S.E. and I.D.; investigation, I.D. and D.D.; writing--original draft preparation, S.E.; writing--review and editing, S.E. and M.B.; visualization, I.D.; supervision, I.D. and D.D.; project administration, I.D.; funding acquisition, I.D., D.D., and M.B. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the European Union through the European Regional Development Fund's Competitiveness and Cohesion Operational Program, grant number KK.01.1.1.04.0041, project \"Autonomous System for Assessment and Prediction of infrastructure integrity (ASAP).\" The APC was funded by the authors' affiliation.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.