*Article* **Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique**

#### **Ivan Duvnjak, Suzana Ereiz, Domagoj Damjanovi´c and Marko Bartolac \***

Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia; ivan.duvnjak@grad.unizg.hr (I.D.); suzana.ereiz@grad.unizg.hr (S.E.); domagoj.damjanovic@grad.unizg.hr (D.D.)

**\*** Correspondence: marko.bartolac@grad.unizg.hr

Received: 7 August 2020; Accepted: 28 August 2020; Published: 31 August 2020

**Abstract:** Tie rods are structural elements that transfer axial tensile loads and are typically used on walls, vaults, arches, and buttresses in historical buildings. To verify their load-bearing capacity and identify possible structural damage risks, the forces transferred by tie rods and the corresponding stresses must be determined. However, this is often a challenging task due to the lack of project documentation for historical buildings. Uncertainties like complex boundary conditions or unknown material and geometrical properties make it hard to assess the tie rods' load level. This paper presents a methodology for the determination of axial forces in tie rods that combines on-site experimental research and a numerical model-updating technique. Along with the common approach based on a determination of the natural frequency of tie rods, this paper presents an approach based on tie rods' mode shapes. Special emphasis is placed on the boundary conditions coefficient, which is a crucial parameter in the analytical solution for axial forces determination based on the conducted on-site experiments. The method is applied in a historical building case study.

**Keywords:** tie rod; structural health monitoring (SHM); natural frequencies; mode shapes; root-mean-square error (RMSE)
