Exact vibration solution for three versions of Timoshenko beam theory: A unified dynamic stiffness matrix method

This paper introduces a unified and exact method for the vibration solution of three versions of Timoshenko beam theory, namely the classical Timoshenko beam theory (TBT), truncated Timoshenko beam theory (T-TBT), and slope inertia Timoshenko beam theory (S-TBT). The proposed unified method enables the free vibration analysis of these three beam theories performed by only one equation, which avoids separate modeling and solution with different beam theories. Firstly, the comparison of three beam theories is conducted with a special focus on deriving the governing differential equation of T-TBT based on Hamilton's principle. Then, two parameters are introduced to unify the three governing differential equations. The dynamic stiffness matrix is derived in a unified form by reducing the order of inverse matrix operation from 2n to n, allowing the dynamic properties of three beam theories to be characterized with only a single model. Additionally, this paper also derives the unified frequency-dependent mass and stiffness matrices based on the unified dynamic stiffness matrix, which eliminates the limitation of DSM in studying the independent influence of the structure's mass and stiffness characteristics. Finally, the discrepancies of the three beam theories are discussed under different boundary conditions and aspect ratios.