Dynamic stiffness model for vertical vibration of hybrid cable-stayed suspension bridges

Dynamic characteristic analysis (DCA) of cable-supported bridges typically begins with finite element (FE) modelling of an initial static equilibrium, with considerable element numbers. Achieving this equilibrium is not straightforward, but requires a preceding nonlinear form-finding iteration procedure (NFIP) given a set of structural parameters. This process can be particularly tedious for the preliminary dynamic design of a hybrid cable-stayed suspension bridge (HCSB), especially when extensive parameter selection is involved. The continuum model can bypass the NFIP but fails to capture the spanwise variation in structural stiffnesses, limiting the precision of DCA. The semi-analytical methods such as the dynamic stiffness method might serve as a promising solution, but have rarely been used for complex cable structures such as HCSBs. This is mostly due to the lack of an accurate dynamic stiffness matrix of cable element that fulfills additional horizontal force equilibrium under vibration. For this issue, this study proposes a dynamic stiffness model for the vertical vibration of HCSBs based on the dynamic stiffness theory. This model enables a semi-analytical approach with fewer divisions for a numerical DCA at an element scale, while avoiding NFIP. The dynamic stiffness matrices of the main cable considering accurate additional horizontal force were firstly derived from the element dynamic differential equations. Along with the establishment of the stiffness impact matrices for the stayed cables and hangers, the model equation was formulated. The frequencies and mode shapes of vertical vibrations were determined by solving a nonlinear eigenvalue problem. The model’s effectiveness was validated through comparative and parametric studies, making it a valuable tool for HCSB dynamic analysis.