Multi-mode beam theory with constrained cross-sectional warping

The paper presents a generalized model of the bending of beams (with arbitrary compact cross-sections) due to the effect of constrained cross-sectional warping. The longitudinal displacement is given in the form of a finite series of power functions, satisfying the appropriate boundary conditions imposed on the free surface of a beam. The assumption of functional relationship between the bending angle, shearing angles, and slope of the deformed centroid of the beam has been abandoned. Taking into consideration the influence of non-uniform and constrained shearing, self-balanced shearing forces are derived. Using the example of some elementary problems, it was possible to investigate the effect of the assumptions made on free vibration frequencies and phase velocities of wave propagation. The assumed model enables frequency branches of a simply supported beam, and a finite number of phase velocity modes to be determined. Some more accurate numerical values of the shear thickness mode parameter have also been calculated.