THE EFFECT OF SHEAR DEFORMATION ON THE VIBRATIONS OF ELASTIC THIN WALLED BEAMS

In this paper, a theory is proposed for the vibrations of elastic thin walled beams of open cross sections, where the effect of shear deformation is approximately taken into account. The theory is based on the following assumptions:1) The transverse cross section of the beam is rigid. Therefore its motion may be defined by the deflections ξ(z,t), η(z,t) and the angle of torsion θ(z,t) about the center of twist.(2) The longitudinal displacement ω(z,8,t)and the longitudiual normal stress σ(z,8,t) have the following four torms expression:3) The mean value of the shcaring stress across the thickness of beam has the following three terms expression:4) The effect of non-uniform shearing stress distribution may be counted for by introducing a twisting moment:Then the stress deformation relations, equations of motion, and boundary conditions are derived from a variational principle of reference 4, The final equations are Eqs. (24), (26), (21).In the case of beams having two planes of symmetr...