Improved engineering theory for uniform beams

Summary

A small static symmetric bending deformation of isotropic linear elastic beams under arbitrary transverse loading varying slowly with the axial coordinate is considered. An asymptotic analysis of the three-dimensional variational equation — in which the small parameter is the ratio of maximum cross-sectional dimension to beam length — gives Timoshenko type governing equations, corresponding boundary conditions, improved formulae for the displacements, and, unlike known beam theories, for all stresses a plane problem in the cross-sectional domain to be solved. Predictions of the theory for beams of narrow rectangular and circular cross-sections are compared with explicit elasticity solutions.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Bresse, M.: Cours de mecanique appliquee. Paris: Mallet-Bacheher 1859.

    Google Scholar 

  2. [2]

    Lord Rayleigh: On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. London Math. Soc.10, 225–234 (1889).

    Google Scholar 

  3. [3]

    Timoshenko, S. P.: On the correction for shear of the differential equation for the transverse vibrations of prismatic bars. Phil. Mag.41, 744–746 (1921).

    Google Scholar 

  4. [4]

    Rehfield, L. W., Murthy, P. L. N.: Toward a new engineering theory of bending: fundamentals. AIAA J.20, 693–699 (1982).

    Google Scholar 

  5. [5]

    Levinson, M.: On Bickford's consistent higher order beam theory. Mech. Res. Comm.12, 1–9 (1985).

    Google Scholar 

  6. [6]

    Rychter, Z.: A simple and accurate beam theory. Acta Mech.75, 57–62 (1988).

    Google Scholar 

  7. [7]

    Fan, H., Widera, G. E. O.: Refined engineering beam theory based on the asymptotic expansion approach. AIAA J.29, 444–449 (1991).

    Google Scholar 

  8. [8]

    Renton, J. D.: Generalized beam theory applied to shear stiffness. Int. J. Solids Struct.27, 1955–1967 (1991).

    Google Scholar 

  9. [9]

    Docmeci, M. C.: A general theory of elastic beams. Int. J. Solids Struct.8, 1205–1222 (1972).

    Google Scholar 

  10. [10]

    Love, A. E. H.: A treatise on the mathematical theory of elasticity, 4th ed., chapters 15, 16, Cambridge: Cambridge University Press 1952.

    Google Scholar 

  11. [11]

    Timoshenko, S. P., Goodier, J. N.: Theory of elasticity, 3rd ed., §§22, 23, New York: McGraw-Hill 1970.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kathnelson, A.N. Improved engineering theory for uniform beams. Acta Mechanica 114, 225–229 (1996). https://doi.org/10.1007/BF01170406

Download citation

Keywords

  • Fluid Dynamics
  • Governing Equation
  • Small Parameter
  • Plane Problem
  • Axial Coordinate