# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 6 Conclusions  
Monitoring tensile force in tie-rods over time is of great importance for the structural safety of historical masonry constructions. Among the experimental methods available, the dynamic methods appear as the most suitable. However, dynamic tests may be time-expensive, technically complex and require full access to the tie-rod. To overcome these drawbacks, here, the effectiveness of the interferometric radar technique for vibration measurements was studied by performing laboratory tests on a tie-rod under known tensile forces. Indeed, radar interferometry is a quick and easy to install technique allowing for remote displacement and vibration measurements. Nevertheless, at the present level of development with radar interferometry is very difficult to determine mode shapes. As a consequence, this measurement technique should be complemented by a dynamic identification approach for the tensile force in tie-rod based only on frequency data. In particular, the approach by [PERSON] in [PERSON] [PERSON] (2014) was considered. Laboratory tests indicated that for obtaining good quality data in interferometric radar measurements, it was necessary to install a corner reflector on the tie-rod. To take into account the not-negligible mass of the corner reflector, the theoretical approach in [PERSON] and [PERSON] (2014) was accordingly modified. From the results reported in Section 5.3, the following considerations can be drawn. First, Tables 4-6 clearly indicate that provided a corner reflector is installed of the tie-rod, interferometric radar data yield to a determination of the natural frequencies practically coincident with that evaluated by employing three accelerometers and the more sophisticated FDD approach. Therefore, the radar interferometer is actually capable of gathering the vibration data needed for applying tensile force identification approaches based only on the knowledge of the natural frequency of the tie-rod. Second, for what concerns the method for determining the tensile force starting from frequency data, the simple natural frequency-based Method, where the flexural stiffness of the tie-rod is neglected, leads to too large error for the demands of practical applications (see Table 7). On the other hand, by employing the modified version of the approach by [PERSON] and [PERSON], here proposed, very accurate results can be obtained, with a mean value of the error in the determination of the tensile force of 1.85%, with and a maximum value 2.42% (see Table 8). In addition, the obtained results show that the above-recalled dynamic identification approach for the tensile force is suitable for practical applications. To this aim, it is worth noting that to date, the sampling frequency of standard radar interferometers is limited to 200 Hz: this, in view of [PERSON]'s theorem [PERSON] (1952), obstructs the possibility of measuring vibration frequencies higher than 100 Hz since the aliasing phenomena. Anyway, in practical applications, the bending stiffness, the length, and the tensile force of tie-rods are such that the above-recalled limitation is not relevant (only for thick tie-rods with high tensile forces such high frequencies are expected). In conclusion, these considerations suggest that the modified version of the approach by [PERSON] and [PERSON] applied to interferometric radar data acquired in the presence of a corner reflector installed to the tie-rods leads to accurate and reliable results, matching the needs of practical applications. Anyway, the need for the corner reflector in some sense obstructs the advantages of using radar interferometry since it is necessary to access the tie-rod, even if the installation of the corner reflector is faster and easier than that of accelerometers and of the required