# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique
## 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, stiffness, 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 stiffness) and the lateral vibration of string (considering the axial effect). The boundary conditions are considered in the proposed solutions of differential equations. Figure 3a depicts small-amplitude free lateral vibration with a uniform cross-section of the beam with material density \(\rho\). 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**) differential segment of beam representing the positive orientation of bending moments, shear forces, and axial and inertial forces of mass. Shear forces (V\({}_{\rm Z}\)) are acting in a vertical direction of an element and are responsible for balancing a weight of segment that varies in time along with element (\(\rho\)dx\(\vartheta^{2}\)w/\(\vartheta^{2}\)t). The sum of vertical forces is equal to the product of the mass of the element and acceleration (Equation (1)):  
\[ \begin{array}{c}\sum{\rm z}=0\rightarrow{\rm V}_{\rm Z}-\rho{\rm dx}\frac{ \vartheta^{2}\rm w}{\vartheta^{2}}-\left({\rm V}_{\rm Z}+\frac{\vartheta{\rm V }_{\rm Z}}{\partial\kappa}{\rm dx}\right)=0,\\ \frac{\vartheta{\rm V}_{\rm Z}}{\partial\kappa}=-\rho\frac{\vartheta^{2}\rm w} {\vartheta^{2}}.\end{array}  \tag{1}\]  
If the moments are taken about point C of the element dx (Equation (2a)), the term contains vertical forces (V\({}_{\rm z}\) and P\({}_{\rm z}\)) and axial bending moment (M\({}_{\rm y}\)). Substituting the bending moment with flexural stiffness (EI) and taking the second derivative of the deflection of beam (M\({}_{\rm y}=-\)EI\(\vartheta^{2}\)w/\(\partial\kappa^{2}\)) and a component of axial force (P\({}_{\rm z}=\) Ptg\(\alpha\) = P\(\partial_{\rm w}\)/\(\partial_{\rm x}\), Equation (2b)) gives Equation (3)  
\[ \begin{array}{c}\sum{\rm M}_{\rm c}=0\rightarrow{\rm M}_{\rm y}+{\rm V}_{ \rm Z}\frac{\rm dx}{2}+{\rm P}_{\rm Z}\frac{\rm dx}{2}-\left({\rm M}_{\rm y}+ \frac{\partial{\rm M}_{\rm y}}{\partial\kappa}{\rm dx}\right)+\left({\rm V}_{ \rm Z}+\frac{\partial{\rm V}_{\rm Z}}{\partial\kappa}{\rm dx}\right)\frac{ \rm dx}{2}+{\rm P}_{\rm z}\frac{\rm dx}{2}=0,\\ 2{\rm V}_{\rm Z}\frac{\rm dx}{2}+2{\rm P}_{\rm z}\frac{\rm dx}{2}-\frac{ \partial{\rm M}_{\rm y}}{\partial\kappa}{\rm dx}=0,\end{array}  \tag{2a}\]  
\[{\rm V}_{\rm Z}=\frac{\partial{\rm M}_{\rm y}}{\partial\kappa}-{\rm P}\frac{ \partial{\rm w}}{\partial\kappa}, \tag{2b}\]  
\[\frac{\partial{\rm V}_{\rm z}}{\partial\kappa}=\frac{\partial^{2}{\rm M}_{\rm y }}{\partial\kappa^{2}}-{\rm P}\frac{\partial^{2}{\rm w}}{\partial\kappa^{2}}, \tag{2c}\]  
\[\frac{\partial{\rm V}_{\rm z}}{\partial\kappa}=-\rm EI\frac{\partial^{4}\rm w }{\partial\kappa^{4}}-{\rm P}\frac{\partial^{2}\rm w}{\partial\kappa^{2}}. \tag{3}\]  
Finally, substituting Equation (3) into Equation (1) provides the basic equation for lateral vibration of the beam with inner constant axial force (Equation (4)):  
\[\rm EI\frac{\partial^{4}\rm w}{\partial\kappa^{4}}+{\rm P}\frac{\partial^{2} \rm w}{\partial\kappa^{2}}-\rho\frac{\partial^{2}\rm w}{\partial t^{2}}=0. \tag{4}\]  
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}{c}{\rm w}={\rm A}(\cos\ \kappa+\cosh\kappa)+{\rm B}(\cos \kappa\kappa-\cosh\kappa)\\ +{\rm C}(\sin\kappa\kappa+\sinh\kappa)+{\rm D}(\sin\kappa-\sinh\kappa)\end{array} .