Authors:Longkai Chen, Chao Zhang
Abstract
Arches are often connected to other structural members, introducing elastic restraints that enhance out-of-plane vibration performance. While existing literature primarily focuses on translational and torsional restraints, warping and minor axis rotational restraints also play a role in practice, yet their influence on vibrations remains underexplored. This paper employs the Laplace transform method (LTM) to derive exact closed-form solutions for the free vibration of discretely restrained arches (DRAs). The study accounts for arbitrary restraint positions, four types of restraints (translational, torsional, warping, and minor axis rotational), and general boundary conditions. The LTM is applied to derive general solutions for the mode shapes and frequencies of DRAs across the entire domain, eliminating the need for beam partitioning, Green's functions, or Lagrange multipliers. Numerical calculations, validated against results from existing literature and finite element methods, confirm the effectiveness of the proposed solution. The study thoroughly examines the influence of restraint parameters (type, stiffness, position, and number) on the modal characteristics of a DRA, with Courant's maximum-minimum principle being demonstrated in the curved beam/arch structures. An exact method is employed to efficiently determine the critical stiffness that maximizes the fundamental frequency in arches with multiple discrete lateral translational and torsional restraints. The findings are significant for optimizing restraint and dynamic analysis of DRA structures.