Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Abstract

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.