An exact closed-form solution for vibration analysis of beams based on third-order shear deformation theory

In this paper, an exact analytical solution is presented to study the vibration of isotropic single-span and multi-span beams based on Levinson’s third-order shear deformation theory. The displacement field of the beam, which accounts for the variation of shear stress along the beam thickness, is defined. Based on Hamilton’s principle, the governing equations and boundary conditions of the problem are derived. The dynamics of beam vibration is captured by two coupled partial differential equations. These equations are solved by first separating the time and space response and then decoupling the two equations to obtain a single, sixth-order differential equation for beam deflection. An analytical solution is presented for this equation, which is able to exactly satisfy the beam’s boundary conditions. The natural frequencies and mode shapes are calculated for different boundary conditions and geometries of the beam and are compared with the results of Euler-Bernoulli, Rayleigh, and Timoshenko beam theories. The effectiveness of this solution for analysis of multi-span beams has been demonstrated by applying it to a two-span beam with various support arrangements. Finally, based on the results obtained, some conclusions are presented.