# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
In the present study, only dynamic methods are addressed since they are generally more convenient in terms of time and costs. The fundamental idea behind dynamic methods is that in a dynamic test, the tensile force in a tie-rod can be related to its modal parameters, particularly its natural frequencies and mode shapes. As a consequence, a dynamic approach consists, first, in exciting and measuring the vibrations of the tie-rod and then in estimating the modal parameters by using a modal identification technique [20; 21; 22]. In the last decade, several research contributions on dynamic approaches for identifying the tensile force in tie-rods have appeared in the literature. Some of them require the determination of the modal shapes [23; 24; 25; 26], eventually by using model updating techniques based on the comparison between experimental and numerical results, the latter determined by FEM simulations [23; 26]. Other approaches use additional masses for perturbating the modal frequencies [27]; also, in this case, the identification could be supported by the comparison with FEM results and optimization methods based on genetic algorithms [28]. A relevant issue deepened in the literature is the influence of the constraints at the ends of the tie-rod on the dynamic behavior, and in particular, on the frequency of free vibrations. The more refined methods consider diffuse elastic constraints on the terminal portions of the tie-rod, represented by the Winkler model [29; 30]. The approach in [31] combines diffuse elastic end constraints with the possibility of having concentrated masses; moreover, the shear deformability of the tie-rod is taken into account by using a Timoshenko beam model. Even though dynamic methods are particularly suitable for the estimation of tensile force in tie-rods of historic constructions [32], there are still some ways for improvements of the experimental setup required for vibration measurement. Indeed, the vibrations of tie-rods are usually measured through accelerometers. The latter are very accurate instruments, but their installation is time-expensive, technically complex and requires full access to the tie-rod. These drawbacks can limit the diffusion of dynamic methods in practical applications. To overcome these drawbacks, some alternative measuring techniques have been studied; the first contributions in this direction concerned the similar, but somewhat different, case of the stay cables. In particular, the goal of contactless measurements of vibrations of cables has been pursued by using the radar interferometer [33; 34], the laser vibrometer [35], vision-based methods [36]. In [37], regarding the case of tie-rods of historic construction, acoustic recording by general-purpose microphones has been employed. Finally, in [38], both the laser vibrometer and the radar interferometer have been used for determining the first natural frequency of tie-rods belonging to monumental buildings, but no results in terms of tensile force were reported. In this context, the main purpose of the present study is to experimentally evaluate the effectiveness of an innovative experimental technique, based on the principle of radar interferometry, for the application represented by the estimation of tensile force in tie-rods. The radar interferometry allows for remote and quick measurement of structural displacements and vibrations [39; 40; 41; 42] overcoming the need to install a sensor on the structure. These features make this technique very appealing; nevertheless, also radar interferometry presents a major issue that should be considered. Indeed, from interferometric radar data are very difficult to estimate mode shapes unless special arrangements of the experimental setup are employed, but in this case, the advantages of the radar interferometry on accelerometric measurements substantially vanish. On the other hand, the natural frequencies of the first modes can be identified with good accuracy.