New first order shear deformation beam theory with in-plane shear influence

•In Timoshenko beam theory bending and shear deflection cannot be determined uniquely.•An alternative formulation with total and bending deflection gives unique results.•The known theories give incorrect natural frequencies in overthreshold domain.•The problem is overcome by coupling flexural and in-plane shear vibrations.•The theory is based on Hamilton’s principle and it is variationally consistent.

In the conventional formulation of the Timoshenko beam theory bending and shear deflection cannot be determined uniquely. Recently, an alternative formulation which deals with total deflection and bending deflection has been developed with unique results. In the present paper a new beam theory, taking coupling between flexural and in-plane shear vibrations into account, is derived by employing Hamilton’s principle. First, uncoupled flexural and in-plane shear vibrations are considered. Then, coupling of vibrations, manifested in the case of geometric boundary conditions, is realized in a physically transparent way. Accuracy is confirmed by 2D FEM vibration analysis of illustrative examples. The proposed formulation is superior to the known first order shear deformation beam theories.