Abstract:
Although the Timoshenko beam theory is termed as the two generalized displacement beam theory, there is an interrelationship between the displacement and rotation, which are dependent. To derive the analytical functions of mode shape in Timoshenko continuous beam, and determine the relations among the displacement, rotation, moment and shear force modes, the initial parameter solutions to the free vibration of Timoshenko beam are employed to establish the transfer matrix method and the transfer matrix formulation of state vector passing through the rigid node. Consequently, the frequency equation and the analytical functions of shape mode are constructed. Based on the properties of the solutions to the frequency equation of a two–span Timoshenko continuous beam, the simplified functions of shape mode is demonstrated, and the peak value ratios of vertical displacement, bending moment and shear force modes to the rotation mode are derived. At the same time, the mode shapes of a two–span Timoshenko continuous beam are calculated by the finite element method and the transfer matrix method. The vibration figures of the first four mode shapes are achieved. Furthermore, the mode shapes of a three–span Timoshenko continuous beam is analyzed, the mode differences of a two–span continuous beam are captured by the Timoshenko beam theory and the Euler–Bernoulli beam theory in the present paper. Results demonstrate that the shape modes of the analytical solutions presented by the transfer matrix method agree with the eigenvectors of the finite element method. Shape modes of vertical displacement and rotation are interdependent. The analytical solutions of peak values of shape modes are consistent with the numerical values. Some conclusions are drawn that the transfer matrix method can be used to derive the exact mode shapes of Timoshenko continuous beam; the traditional method is not rational to simultaneously normalize the shape modes of the vertical displacement and rotation.