Composite materials are increasingly used in several types of structural applications. Advantages that motivate their use are high strength and sti!ness, low weight, improved fatigue life, etc. Many structural components made of composites have the form of beams. Accordingly, some re"ned theories were developed in order to consider the special structural properties of composite beams such as the important role of the shear deformability and the e!ects of anisotropy.
A theory of shear deformable orthotropic beams was developed by Nowinsky [1]. On the other hand, Dharmarajan and McCutchen [2] have discussed a method for obtaining shear correction factors for these types of beams. An orthotropic beam theory including normal deformability along with the shear e!ect was presented by Soldatos and Elishako! [3]. Recently, Murakami and Yamakawa developed an anisotropic beam theory of the Timoshenko type from a mixed variational principle [4]. The model was used to calculate vibration frequencies of cantilever and simply supported beams.
In this article, a two-dimensional vibration analysis for anisotropic beams is presented in order to verify the Murakami}Yamakawa beam theory. The beam is modelled by means of a plane state of stress corresponding to an anisotropic elastic body. Natural frequencies are determined by means of the "nite element system FLEXPDE [5]. The results are compared against the values obtained in reference [4]. Additional results are given for clamped beams.
2. GOVERNING EQUATIONS
An anisotropic beam with a narrow rectangular cross-section is considered (see Figure 1). According to the elasticity theory the problem is governed by means of the following equations:
* V *x # * VW *y " *u *t , (1a,b) * VW *x # * W *y " *v *t , V " *u *x , W " *v *y , 2 VW " *u *y # *v *x , (2a}c)