Abstract: A semi-analytical method and a two-direction expansion method , have been presented and used to find thenatural vibration characteristics of two-dimension beam with arbitrary boundary conditions. In these two methodsthere are no any assumptions on the transverse and the longitudinal displacements along x -axis and z -axis directions , i . e. , the beam is considered to be in a state of plane stress , and the elements of structure stiffness and massmatrices can be given explicitly. The geometrical properties of mode shapes are investigated , and the simply supported beam is taken as an example to discuss the accuracy and efficiency of semi-analytical method and two-direction expansion method of comparing the results of the two methods and that of the analytical method.

Key words: beam, plane problem, semi-analytical method, inherent characteristics, two-direction expansion method