Unified Free Vibration Solution for Three Versions of Timoshenko Beam Theory Based on Dynamic Stiffness Matrix Method

In this paper, we introduce 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). With the presented unified method, the free vibration analysis of these three beam theories can be performed by only one equation, which avoids separate modeling and solution with different beam theories. Firstly, the comparison of three beam theories is carried out with the 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 with the frequency-dependent mass and stiffness matrices are derived in a unified and analytical form by reducing the order of inverse matrix operation from 2n to n. The dynamic properties of three beam theories can be characterized with only a single model. The discrepancy of the three beam theories is discussed as well in terms of prediction accuracy under different boundary conditions and slenderness ratio.