# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique
## 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, f\({}_{n}^{\mathrm{exp}}\), 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):  
\[\mathrm{RMSE}_{n}=\frac{\sqrt{\frac{1}{k}\sum_{j=1}^{k}\left(\Phi_{nj}^{\mathrm{ num}}-\Phi_{nj}^{\mathrm{exp}}\right)^{2}}}{\max(\Phi_{n})}, \tag{9}\]  
where n is the mode shape number, \(\Phi_{nj}^{\mathrm{num}}\) is the numerically-obtained normalized mode shape vector, and \(\Phi_{nj}^{\mathrm{exp}}\) 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 f\({}_{n}^{\mathrm{exp}}\), f\({}_{n}^{\mathrm{exp}}\), and a previously-updated numerical model, the axial force was tuned to a numerical model to match the natural frequency, f\({}_{n}^{\mathrm{num}}\), with that from an experiment, f\({}_{n}^{\mathrm{exp}}\) (Figure 4, stage 2--updated model (BC + f\({}_{n}^{\mathrm{num}}\)). Using this procedure, considering experimental measurements, and updating the numerical model, the axial force (P) was determined (Figure 4, end of stage 2).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Boundary**} & \multicolumn{1}{c}{**Static System**} & \multicolumn{3}{c}{**Coefficient \(\kappa\)**} \\ \cline{3-5} \multicolumn{1}{c}{**Condition**} & & **1 st Mode** & **2 nd Mode** & **nth Mode** \\ \hline \multirow{2}{*}{Hinge-hinge} & & & 3.142 & 6.283 & \(\cong\) n\(\pi\) \\ \hline \multirow{2}{*}{Clamp-clamp} & & & 4.730 & 7.853 & \(\cong\)\(\frac{(2n+1)}{2}\pi\) \\ \hline \multirow{2}{*}{Clamp-hinge} & & & 3.927 & 7.069 & \(\cong\)\(\frac{(4n+1)}{4}\pi\) \\ \hline \multirow{2}{*}{Clamp–free} & & & 1.875 & 4.694 & \(\cong\)\(\frac{(2n-1)}{2}\pi\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Value of boundary condition parameter \(\kappa\) for the first two [19] and nth [20] natural frequencies having various boundary conditions.  
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 \(\kappa\), 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 \(\kappa\).