Identification of tensile forces in tie rods with unknown boundary conditions

Tie rods are commonly used in arch and vault structures to counter the horizontal loads. Estimation of the tensile forces in these rods is essential for proper restoration of ancient and monumental buildings. The rods extend into the masonry and are anchored outside the main walls by means of mechanical devices while they are accessible from the inside of the structure. Due to their access, force-deflection methods and vibration based methods are applicable for identification of the rods tensile forces. However, the latter typically adopted due to the simpler experimental setup, consisting in one or few accelerometers and an instrumented hammer. The cross section and the clear length of the rod are easily measurable, although the extension length of the rod into the masonry is not always straightforward. The present paper investigates the accuracy of vibration based methods in identifying tensile force in presence of uncertain boundary conditions. The aim of the paper is the estimation of the tensile force of the tie rod when its total length (clear length plus the length of the portions inside the walls) is replaced by the clear length, with the addition of rotational springs as boundaries. In this approach, measured natural frequencies and the first mode shape will be used to estimate the tensile force and rotational stiffness at the boundaries. The analytical solution of the vibration problem and a Rayleigh-Ritz method are used to predict the analytical natural frequencies of the system as a function of axial load based on numerical simulations.