# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique
## 1 Introduction  
Displacements that occur with historical buildings can be arrested using metal beams, or tie rods, which support the masonry walls, buttresses, arches, and vaults in the plane of bending out. Tie rods are subjected to axial tension and are an essential element in the control of horizontal forces (displacements) produced by static and dynamic loads related to seismic actions. In extreme cases, a tie rod can reach its maximum bearing capacity due to high stress or the pulling out of its anchor point. Both scenarios can lead to a loss of structural integrity. Therefore, the value of the internal tensile force in tie rods is a frequent subject of discussion. Figure 1 represents a typical tie rod in historically-important buildings such as cathedrals, churches, or castles. Several uncertainties exist when determining the forces in tie rods, including complex boundary conditions [1; 2; 3] and geometrical and material properties [3; 4]. Boundary conditions can vary, ranging from theoretically-fixed and pinned conditions to those that should be considered with spring elements (Figure 2). In practice, the length of the anchoring of the tie rod is associated with geometrical problems. Due to the limitations of inspection, material properties such as the Young's modulus are often unknown. Evaluation of tensile forces in tie rods can be achieved using static, dynamic, or mixed approaches. Two static approaches can be used for the determination of tensile forces in tie rods. The first involves loading of the tie rod in several stages and then measuring deflections and strains in representative locations [5; 6; 7]. The second approach is known as the residual stress method [8]. This involves attaching strain gauge rosettes to the surface, drilling a hole at the center of the gauge, and measuring the residual strain caused by the relaxation of the material surrounding the drilled hole. The several disadvantages to static approaches include the following:  
* The tie rods are generally located at elevated positions;
* High-accuracy displacement sensors should be used due to small vertical displacements. These should be placed on a previously-determined referent location;
* The strain gauge installation can be complicated at elevated positions. The aforementioned approaches require considerable time for application and they are slightly destructive. Based on the mentioned disadvantages, many authors have researched a dynamic approach based on numerical or analytical methods [9; 10; 11]. Vibration-based methods (dynamic approach), as nondestructive methods, are widely applied for the evaluation of axial forces in tensile elements of structures. These methods are mainly based  
Figure 1: Cathedral of St. James in Sibenik (Croatia). Tie rods are supporting the walls and arches. Figure 2: Examples of complex (unclear) boundary conditions. on tie rods' natural frequencies, which are later used for the determination of forces. A distinction must be made between transverse oscillations of strings and beams. String theory is widely used for the determination of cable forces in structures such as bridges [12] due to the high ratio of length to cross-section dimensions. Unlike beam theory, string theory does not consider the flexural stiffness or boundary conditions [13; 14]. Tensile forces in tie rods can also be evaluated using a mixed approach. This approach is a combination of previously-mentioned approaches--static- and vibration-based. In this case, tie rods are modeled as simply supported Euler beams with rotational springs with unknown stiffness on each edge [15]. Besides the stiffness of rotational springs, the second unknown parameter is force. These unknown parameters are obtained by solving a system of equations composed of static equations for deflection and dynamic equations for natural frequencies. This method requires data from two separate experiments. The first experiment involves measuring the deflection in representative locations on a beam and the second involves measuring the natural frequency.