Vibration-based estimation of bolt tension in non-slender bolts using Timoshenko beam theory

Many industrial applications apply non-slender bolts, from small bolts in machinery to large bolts in offshore structures. Ensuring the correct tension in such bolts is a significant problem. Recent work suggests a vibration-based approach for estimating bolt tension. The idea is to assume the bolt is an Euler–Bernoulli beam and measure the bending natural frequencies. When tightening the bolt, the frequencies increase. For non-slender bolts, the Euler–Bernoulli assumption is no longer valid. Therefore, a tensioned Timoshenko beam model with flexible boundary conditions is derived in this work. Derivation and investigation of a tensioned Timoshenko beam with boundary mass, inertia, and flexible boundary conditions is not well described in the literature. Besides the purpose of estimating tension, the investigation provides a fundamental understanding of how boundary conditions influence natural frequencies in the Timoshenko formulation, offering novel insights that may be useful in other applications. The Timoshenko model is incorporated into a previously applied parameter estimation method and validated by testing numerical scenarios of tightened bolts. Despite finding that non-slender bolts’ natural frequencies depend relatively less on tension than slender bolts, it is still possible to make estimations with an average deviation of less than 2%. Finally, to test that the Timoshenko model is a valid assumption, experiments are performed on a non-slender M72 bolt.