# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 2 Tensile Force Dynamic Identification in Tie-Rods: Theoretical Approaches  
Several vibration-based tensile force identification approaches have been proposed in the literature, mainly differing in the structural model of the tie-rod and in the modal parameters required for estimating the tensile force. In principle, the more advanced is the structural model, the more reliable is the tensile force value estimated. However, this rule admits exceptions; moreover, not all methods are suitable for reprocessing dynamic data acquired by interferometric radar measurements. In particular, since it is generally challenging to identify modal shapes by interferometric radar measurements acquired in closed spaces like that spanned by tie-rods in historical buildings, tensile force dynamic identification approaches based on the determination of modal shapes are not suitable for purposes of the present study. For the above consideration, a literature review aimed at selecting an appropriate method for reprocessing interferometric radar data was performed. Particular attention was devoted to the simplicity of the experimental setup needed. In general, the structural model considered in most approaches (Figure 1), as well as in the present paper, is composed of a one-dimensional linear elastic beam element representing the tie-rod, with elastic translational and rotational restraints at both ends; \(k_{1}\) and \(k_{2}\) are the stiffness of the rotational elastic constraints, and \(k_{3}\) and \(k_{4}\) are the stiffness of the translational elastic constraints. The tie-rod is characterized by mass per unit length \(\rho(x)\), cross-section \(A(x)\), moment of inertia of the cross-section about an axis orthogonal to the plane of figure \(I(x)\), Young's modulus \(E(x)\), shear modulus \(G(x)\), in general, all variable on the element according to the abscissa \(x\). The length of the element is \(L\), and the tensile force is represented by \(N\). The simplest approach is the natural frequency-based method, developed starting from the taut string theory [33; 35]: the tensile force \(N\) can be calculated from the n-th experimentally measured natural frequency \(f_{n}\) by:  
\[N=4\rho L^{2}\bigg{(}\frac{f_{n}}{n}\bigg{)}^{2}. \tag{1}\]  
This method requires the knowledge of only one natural frequency, and, therefore, it appears suitable for reprocessing interferometric radar dynamic measurements. However, it is worth noting that it is based on the assumption that the tie-rod can be assimilated to a taut string, i.e., that the bending stiffness is negligible compared to the axial stiffness, and that the influence of the eventual sag is negligible. Consequently, the stiffness of the rotational springs at the ends is negligible on the tensile force determination: in short, the tie-rod is considered a simply supported beam with no bending stiffness. This strong simplification, of course, may yield substantial differences between the actual value of the tensile force and the value determined by (1). A more advanced approach using only modal frequencies was proposed by [PERSON] and [PERSON] in [14] expressly for the determination of the tensile force in the tie-rods of historical constructions. There, the considered structural model employed for the tie-rod is an Euler-Bernoulli beam subjected to a constant axial tensile force. Moreover, the beam is assumed to be spring-hinged at both ends with rotational elastic-springs having the same flexural stiffness \(k\), i.e., \(k_{1}=k_{2}=k\); both the translational constraints at the extremities are assumed to be perfect (see Figure 1). By the separation of variables method, the governing partial differential equation describing the free transverse vibrations of the beam subject to the axial load \(N\) can be separated into two ordinary differential equations. A linear algebraic system can be obtained imposing the appropriate static and kinematic boundary conditions to the general solution of one of these two ordinary differential equations.