**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.

**Figure 1.** Cathedral of St. James in Šibenik (Croatia). Tie rods are supporting the walls and arches.

Several uncertainties exist when determining the forces in tie rods, including complex boundary conditions [1–3] 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.

**Figure 2.** Examples of complex (unclear) boundary conditions.

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 [5–7]. 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 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 [9–11].

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 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 sti ffness 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 sti ffness on each edge [15]. Besides the sti ffness 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. Although a mixed approach showed good results in laboratory conditions, measurement errors can cause a significant deviation in results. If the number of unknown parameters is larger, the probability of error is also greater. Although there are two such parameters in this case, they are enough to cause a significant deviation in the results [16–18].

In this study, a combination of experimental and numerical research was used for the determination of forces in tie rods based on the natural frequencies (fn), flexural sti ffness (EI), mass (m), and boundary conditions. The method was verified on a historical building case study. One characteristic tie rod in this building was analyzed in detail. Based on these analyses of the characteristics of a single tie rod and the experimentally-determined values of the natural frequencies on the other tie rods in the building, the force in all the tie rods was determined reliably.

The remainder of this paper is structured as follows: Section 2 presents an analytical solution for lateral vibration of a tie rod and provides an equation for the determination of the tensile force, Section 3 describes the proposed methodology in detail, and Section 4 presents the application of the proposed methodology on a real historical building.

#### **2. Analytical Solution for Lateral Tie Rod Vibration**

As discussed above, the natural frequencies of a tie rod depend on three basic assumptions—axial load, sti ffness, and boundary conditions. The lateral vibrations of the tie rod are assumed as a superposition of two solutions—the lateral vibrations of the Euler-Bernoulli beam (considering sti ffness) and the lateral vibration of string (considering the axial e ffect). The boundary conditions are considered in the proposed solutions of di fferential equations.

Figure 3a depicts small-amplitude free lateral vibration with a uniform cross-section of the beam with material density ρ. In the cross section, dx is the acting internal forces (P) with a positive orientation, including the weight of beam caused by vibrations (Figure 3b).

**Figure 3.** (**a**) Beam under lateral vibration and axial loading and (**b**) di fferential segmen<sup>t</sup> of beam representing the positive orientation of bending moments, shear forces, and axial and inertial forces of mass.

Shear forces (Vz) are acting in a vertical direction of an element and are responsible for balancing a weight of segmen<sup>t</sup> that varies in time along with element (ρdx∂2w/∂2t). The sum of vertical forces is equal to the product of the mass of the element and acceleration (Equation (1)):

$$\begin{split} \nabla \mathbf{z} = 0 \rightarrow \mathbf{V}\_{\mathbf{Z}} - \rho \mathbf{dx} \frac{\partial^2 \mathbf{w}}{\partial \mathbf{t}^2} - \left( \mathbf{V}\_{\mathbf{Z}} + \frac{\partial \mathbf{V}\_{\mathbf{Z}}}{\partial \mathbf{x}} \mathbf{dx} \right) = \mathbf{0}, \\ \frac{\partial \mathbf{V}\_{\mathbf{Z}}}{\partial \mathbf{x}} = -\rho \frac{\partial^2 \mathbf{w}}{\partial \mathbf{t}^2}. \end{split} \tag{1}$$

If the moments are taken about point C of the element dx (Equation (2a)), the term contains vertical forces (Vz and Pz) and axial bending moment (My). Substituting the bending moment with flexural stiffness (EI) and taking the second derivative of the deflection of beam (My = −EI∂2w/∂x2) and a component of axial force (Pz = Ptgα = P∂w/∂x, Equation (2b)) gives Equation (3)

$$\begin{aligned} \boldsymbol{\Sigma} \cdot \mathbf{M}\_{\mathbf{c}} = 0 & \to \mathbf{M}\_{\mathbf{y}} + \mathbf{V}\_{\mathbf{Z}} \, \frac{d\mathbf{x}}{2} + \mathbf{P}\_{\mathbf{Z}} \, \frac{d\mathbf{x}}{2} - \left( \mathbf{M}\_{\mathbf{y}} + \frac{\partial \mathbf{M}\_{\mathbf{y}}}{\partial \mathbf{x}} \, \mathbf{dx} \right) + \left( \mathbf{V}\_{\mathbf{Z}} + \frac{\partial \mathbf{V}\_{\mathbf{Z}}}{\partial \mathbf{x}} \, \mathbf{dx} \right) \overset{\text{div}}{2} + \mathbf{P}\_{\mathbf{z}} \, \frac{d\mathbf{x}}{2} = 0, \\ & \mathbf{2V}\_{\mathbf{Z}} \frac{d\mathbf{x}}{2} + 2 \mathbf{P}\_{\mathbf{z}} \frac{d\mathbf{x}}{2} - \frac{\partial \mathbf{M}\_{\mathbf{y}}}{\partial \mathbf{x}} \, \mathbf{dx} = 0, \end{aligned} \tag{2a}$$

$$\mathbf{V}\_{\mathbf{Z}} = \frac{\partial \mathbf{M}\_{\mathbf{y}}}{\partial \mathbf{x}} - \mathbf{P} \frac{\partial \mathbf{w}}{\partial \mathbf{x}}\,\mathrm{\,}\tag{2b}$$

$$\frac{\partial \mathbf{V}\_x}{\partial \mathbf{x}} = \frac{\partial^2 \mathbf{M}\_\mathbf{y}}{\partial \mathbf{x}^2} - \mathbf{P} \frac{\partial^2 \mathbf{w}}{\partial \mathbf{x}^2} \tag{2c}$$

$$\frac{\partial \mathbf{V}\_{\mathbf{z}}}{\partial \mathbf{x}} = -\mathrm{EI}\frac{\partial^4 \mathbf{w}}{\partial \mathbf{x}^4} - \mathrm{P}\frac{\partial^2 \mathbf{w}}{\partial \mathbf{x}^2}. \tag{3}$$

Finally, substituting Equation (3) into Equation (1) provides the basic equation for lateral vibration of the beam with inner constant axial force (Equation (4)):

$$\mathrm{EI}\frac{\partial^4 \mathbf{w}}{\partial \mathbf{x}^4} + \mathrm{P}\frac{\partial^2 \mathbf{w}}{\partial \mathbf{x}^2} - \rho \frac{\partial^2 \mathbf{w}}{\partial \mathbf{t}^2} = 0. \tag{4}$$

The solution of this equation can be expressed as a form of harmonic functions of w(x,t) (Equation (5)) or in terms of exponential functions. For the selected function, constants (A, B, C, and D) should be found considering various boundary conditions with applying known conditions to the deflection, slope, bending moment, and shear forces:

$$\begin{array}{l} \mathbf{w} = \mathbf{A}(\cos \kappa \mathbf{x} + \cosh \kappa \mathbf{x}) + \mathbf{B}(\cos \kappa \mathbf{x} - \cosh \kappa \mathbf{x}) \\ + \mathbf{C}(\sin \kappa \mathbf{x} + \sinh \kappa \mathbf{x}) + \mathbf{D}(\sin \kappa \mathbf{x} - \sinh \kappa \mathbf{x}) \end{array} \tag{5}$$

Ultimately, a natural frequency for the nth mode shape in tie rods can be determined according to [19,20] as

$$\mathbf{f\_n} = \frac{\kappa^2}{2\pi\mathbf{l}^2} \sqrt{\frac{\mathbf{E I}}{\mathbf{m}'}} \sqrt{\mathbf{1} + \frac{\mathbf{P}\mathbf{l}^2}{\mathbf{E}\mathbf{I}\pi^2\mathbf{n}'}} \tag{6}$$

where n is the mode shape number, fn is the nth natural frequency, l is a span of tie rod, m is mass per unit length (m'= ρbh, b—width, and h—height of cross section) and κ is a boundary condition parameter, as presented in Table 1. By rearranging the previous equation, the boundary conditions (κ) can be determined:

$$\kappa = \sqrt{\frac{\text{f}\_{\text{n}} 2\pi \text{l}^2}{\sqrt{\frac{\text{EI}}{\text{m}^\circ} + \frac{\text{Pl}^2}{\text{m}^\circ \pi^2 \text{n}^2}}}} \tag{7}$$


**Table 1.** Value of boundary condition parameter κ for the first two [19] and nth [20] natural frequencies having various boundary conditions.

Based on known natural frequencies (fn), properties (EI, m), and boundary conditions (κ), we can determine the axial force of a tie rod:

$$\mathbf{P} = \frac{\pi^4 \mathbf{n}^2}{\kappa^4} 4 \mathbf{f}\_\mathbf{n}^2 \mathbf{m}' \mathbf{l}^2 - \pi^2 \mathbf{n}^2 \frac{\mathbf{E}\mathbf{l}}{\mathbf{l}^2}. \tag{8}$$

Although the previous equation is simple, it only considers ideal boundary conditions. Generally, in real life, the boundary conditions are quite complicated and very often nonsymmetrical problems. This is why we assessed the axial forces in tie rods using analytical solutions, experimental research, and numerical analysis.

#### **3. Methodology for Boundary Conditions and Axial Load Identification**

The proposed methodology is composed of three stages divided into experimental research on-site and numerical optimization, considering an analytical solution for the determination of axial force (Figure 4). The experimental research (stage 1) involved vibration-based measurement [21] of natural frequencies, fexp n , and mode shapes determination by using operational modal analysis (OMA) [22,23]. Based on the assumed boundary conditions, known materials, and geometric properties, an initial numerical model of a tie rod was developed (Figure 4, stage 2). Mode shapes from the numerical simulation and experiments were compared using normalized root-mean-square error (RMSE) [24], as presented in Equation (9):

$$\text{RMSE}\_{\text{n}} = \frac{\sqrt{\frac{1}{k} \sum\_{j=1}^{k} \left(\Phi\_{\text{n},j}^{\text{num}} - \Phi\_{\text{n},j}^{\text{exp}}\right)^2}}{\max(\Phi\_{\text{n}})},\tag{9}$$

where n is the mode shape number, Φnum n,j is the numerically-obtained normalized mode shape vector, and Φexp n,j is the experimentally-obtained operating deflection shape at the jth point on the tie rod. Based on the RMSE values, the numerical model was updated by adapting the boundary conditions (BC). When the RMSE reached its minimum value, it indicated that the experimental and numerical mode shapes were overlapping, which ultimately meant that boundary conditions were updated adequately (Figure 4, stage 2—updated model BC). Finally, based on known natural fexp n , fexp n , and a previously-updated numerical model, the axial force was tuned to a numerical model to match the natural frequency, fnum n , with that from an experiment, fexp n (Figure 4, stage 2—updated model (BC + fnum n ). Using this procedure, considering experimental measurements, and updating the numerical model, the axial force (P) was determined (Figure 4, end of stage 2).

**Figure 4.** Methodology for boundary conditions determination.

Generally, the simulated boundary conditions in the numerical model are complicated and do not coincide with the basic support conditions given in Table 1. Hence, by applying known geometrical and material properties with the resulting axial force in an analytical solution (Equation (7)), we can determine the coefficient κ, which is associated with the boundary conditions (Figure 4, stage 3). Using this methodology on one characteristic tie rod, a boundary coefficient was determined and this value can then be applied to other tie rods with the same boundary conditions (Figure 5). Therefore, it is sufficient to perform on-site measurements of the natural frequencies in each tie rod to determine the axial force (Figure 5, end of stage 3). In case of varying boundary conditions for tie rods in the building, the numerical model updating should be applied to each tie rod (stages 1 and 2) without determination of coefficient κ.

**Figure 5.** Methodology for axial load identification.

#### **4. Case Study Using the Proposed Methodology**

The previously-described methodology (Figures 4 and 5) was applied to a historical building case study. The Cathedral of St. James in Šibenik (Figure 6), Croatia, is a good example of a historical building with defined dynamic tie rod parameters (frequencies and mode shapes). For the sake of simplicity, one of the tested iron tie rods was taken as a reference for which we conducted a detailed analysis.

**Figure 6.** Cathedral of St. James in Šibenik, Croatia. (**a**) West view of cathedral and (**b**) iron tie rods inside the cathedral.
