Background <p>FG anisotropic beams are widely used as basic structures in various engineering fields. However, it is difficult to obtain exact solutions due to material inhomogeneity, tension-shear coupling and other effects. Thus, a simple and universal general solution suitable for arbitrary loads and constraints is necessary for mechanical behavior analysis.&#xa0;</p> Methods <p>Firstly, the stress function method combined with the semi-inverse method is used to analytically solve the boundary value problem of partial differential equations. Then, the exact solution is reasonably truncated, and an approximate model for bending and vibration analysis is proposed. Finally, the effectiveness is verified by comparison with finite element results and existing literature results.</p> Results <p>The general solution for the plane problem of FG anisotropic beams is provided, and the explicit particular solution of a specific example is obtained.</p> Conclusions <p>The proposed analytical formulas are universal and effective. FG anisotropic beams have complex and obvious shear deformation behaviors due to inhomogeneity and tension-shear coupling.</p>