It is well known that Euler}Bernoulli beam theory neglects the e!ect of transverse shear strain on the bending solutions because the assumption of plane cross-sections perpendicular to the axis of the beam remaining plane and perpendicular to the axis after deformation. This simple beam theory can give excellent solutions to the vibration analysis of slender beams. However, it cannot present accurate values for the modes of thick beams or sandwich beams.
Timoshenko [1,2] was evidently the "rst to study thick beams taking into consideration the in#uences of transverse shear deformation. In the Timoshenko beam theory, plane cross-sections remain plane but not necessarily normal to the neutral axis after deformation, thus admitting a non-zero transverse shear strain. The study on vibration of multi-span Euler}Bernoulli beams has been carried out by various methods such as graphical network method [3], "nite element method [4], integral equation method [5] and U-transformation method [6, 7], etc. Huang [8] derived the exact solutions of eigenfrequencies and modes for a one-span Timoshenko beam under various boundary conditions. He and Huang [9] used the dynamic sti!ness method to analyze the free vibration of continuous Timoshenko beam. Moreover, Chen and Cai [10] used the U-transformation method to analyze the static deformation of the Timoshenko beams with period supports.
In this paper, the free vibration of multi-span Timoshenko beams is studied by the Rayleigh}Ritz method. The static Timoshenko beam functions, which are composed of a set of transverse de#ection functions and a set of rotational angle functions, are developed as the trial functions. These transverse de#ection functions and rotation-angle functions are the complete solutions of a multi-span Timoshenko beam under a series of static sinusoidal loads distributed along the length of the beam. Each of the trial functions is a sine or cosine function plus a polynomial function of no more than the third order. A uni"ed program can easily be provided because the change of boundary conditions of the beam and the number and locations of internal point supports only results in a corresponding change of coe$cients of the low order polynomials.
2. EIGENFREQUENCY EQUATION
Consider a straight multi-span beam with the length l, the cross-sectional area A and the area moment of inertia I, as shown in Figure 1. The beam has J internal point supports,