Variational principle of partial-interaction composite beams using Timoshenkoʹs beam theory

Based on the kinematic assumptions of Timoshenkoʹs beam theory, this paper formulates the principle of virtual work and reciprocal theorem of work for the partial-interaction composite beams. Then the principle of minimum potential energy and minimum complementary energy are derived and proved. The variational principles for the frequency of free vibration and critical load of buckling are also deduced afterward as well as the mixed variational principle with two types of variables. These variational formulae are all rendered in terms of shearing force, bending moment and axial force as well as corresponding deflection, rotation angle and interlayer slip, which can be applied conveniently for analyzing of composite beams. According to the proposed variational principles, the governing equations of static bending, free vibration and buckling can be obtained for the partial-interaction composite members as well as the corresponding boundary conditions. Finally, some numerical examples are presented and compared with the other solutions available in literatures to demonstrate the present theory.