Practical formula to calculate tension of vertical cable with hinged-fixed conditions based on vibration method

Vertical cables are widely used in the tied-arch bridges and suspension bridges as the vital components to transfer load. It is very important to accurately estimate the cable tensions in the cable supported bridges during both construction and in-service stages. Vibration method is the most widely used method for in-situ measurement of cable tensions. But for the cables with hinged-fixed boundary conditions, no analytical formulas can be used to describe the relationship between the frequencies and the cable tension. According to the general solution of the vibration equation and based on its numerical computational results, practical formula to calculate tensions of vertical cables by multiple natural frequencies satisfying hinged-fixed boundary conditions is proposed in this paper. The expression of the practical formula is the same as the solution derived from an axially loaded beam with simple supported ends and can use the first 10 order frequencies to calculate the cable tension conveniently and accurately. Error analysis showed that when using the fundamental frequency to estimate cable force, the estimated tension errors of the cables with its dimensionless parameter ξ≥ 2.8 are less than 2 %. It contained nearly all of the vertical cables used in bridge engineering. In addition, with multiple natural frequencies being measured, bending stiffness of the cable can be identified by using the formulas presented in this paper with an iterative method. At last, the practical formula in this paper is verified to have high precision with several numerical examples, and can be conveniently applied to field test for cable-supported bridges.