The shear strain influence on the dynamics of thin-walled beams

Abstract The influence of middle-surface shear strains on the dynamics of thin-walled beams is analyzed. By means of a suitable choice of the axial displacement field, shear lag effects in bending and torsion are taken into account for open and closed cross-sections. The equations of motion are obtained via Hamilton's principle. The flexural-torsional natural frequencies in the presence of warping which varies along the beam axis are compared with those given by Timoshenko-Vlasov models. The solution to the problem is pursued by means of the most natural choice, i.e. the classical trigonometric series expansion. A suitable algorithm is developed to solve the resulting eigenvalue problem which turns out to be strongly ill-conditioned.

[1]  A. Gjelsvik,et al.  The Theory of Thin-Walled Bars , 1981 .

[2]  T. Kármán.  Methods of Analysis for Torsion with Variable Twist , 1944 .

[3]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[4]  T. Huang,et al.  The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions , 1961 .

[5]  A. Heidebrecht,et al.  Frequency Analysis of Thin-Walled Shear Walls , 1971 .

[6]  E. Reissner.  The effect of transverse shear deformation on the bending of elastic plates , 1945 .

[7]  B. Downs,et al.  Transverse Vibration of a Uniform, Simply Supported Timoshenko Beam Without Transverse Deflection , 1976 .

[8]  G. Cowper,et al.  On the Accuracy of Timoshenko's Beam Theory , 1968 .

[9]  V. Vlasov.  Thin-walled elastic beams , 1961 .

[10]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[11]  Seng-Lip Lee,et al.  Free vibration of asymmetric shear wall-frame buildings , 1984 .

[12]  Alexander Coull,et al.  Free vibrations of open-section shear walls , 1977 .

[13]  F. Laudiero,et al.  A consistent approach to linear stability of thin-walled beams of open section , 1988 .

[14]  Enrico Volterra,et al.  Second approximation of method of internal constraints and its applications , 1961 .

[15]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[16]  Raymond D. Mindlin,et al.  TIMOSHENKO'S SHEAR COEFFICIENT FOR FLEXURAL VIBRATIONS OF BEAMS , 1953 .

[17]  C. Dolph.  On the Timoshenko theory of transverse beam vibrations , 1954 .

[18]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .

[19]  M. Savoia,et al.  Shear strain effects in flexure and torsion of thin-walled beams with open or closed cross-section , 1990 .