Abstract
The buckling and postbuckling of beams is revisited taking into account both the influence of axial compressibility and shear deformation. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adopted, in which the stress resultants depend linearly on the generalized strain measures. The equilibrium equation is derived in a general form that holds for the statically determinate and indeterminate combinations of boundary conditions representing the four fundamental buckling cases. The eigenvalue problem is recovered by consistent linearization of the governing equations, the critical loads at which the trivial solution bifurcates are determined, and the influence of shear on the buckling behavior is investigated. By a series of transformations, the equilibrium equation is rearranged such that it allows a representation of the solution in terms of elliptic integrals. Additionally, closed-form relations are provided for the displacement of the axis, from which buckled shapes are eventually obtained. Even for slender beams, for which shear deformation can usually be neglected, both the buckling and the postbuckling behavior turn out to be affected by shear not only quantitatively, but also qualitatively.
Similar content being viewed by others
References
Euler L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti. Marc-Michel Bousquet et Soc, Lausanne (1744)
Truesdell, C.: The Rational Mechanics of Flexible or Elastic Bodies, 1638—1788: Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei Secundae. O. Füssli (1960)
Reissner E.: On one-dimensional finite-strain beam theory: the plane problem. Z. Angew. Math. Phys. 23(5), 795–804 (1972)
Antman S.: The theory of rods. In: Truesdell, C. Handbuch der Physik, Vol. VIa/2, Springer, Berlin (1972)
Timoshenko S.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43(6), 125–131 (1922)
Timoshenko S., Gere J.: Theory of Elastic Stability. McGraw-Hill, New York (1961)
Simitses G., Hodges D.: Fundamentals of Structural Stability. Butterworth-Heinemann, Oxford (2006)
Kirchhoff, G.: Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. Reine Angew. Math. 56, 285–343 (1859)
Love A.: A Treatise on the Mathematical Theory of Elasticity. Dover Publications Inc., New York (1944)
Saalschütz, L.: Der belastete Stab unter Einwirkung einer seitlichen Kraft: Auf Grundlage des strengen Ausdrucks für den Krümmungsradius. Teubner (1880)
Bisshopp K., Drucker D.: Large deflection of cantilever beams. Q. Appl. Math. 3(3), 272–275 (1945)
Frisch-Fay R.: Flexible Bars. Butterworths, London (1962)
Stern J.: Der Gelenkstab bei großen elastischen Verformungen. Arch. Appl. Mech. 48(3), 173–184 (1979)
Stern J.: Der Gelenkstab mit Vorverformung. Arch. Appl. Mech. 54(2), 152–160 (1984)
Domokos G., Holmes P., Royce B.: Constrained Euler buckling. J. Nonlinear Sci. 7(3), 281–314 (1997)
Mikata Y.: Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube. Acta Mech. 190(1), 133–150 (2007)
Bažant Z.: A correlation study of formulations of incremental deformation and stability of continuous bodies. J. Appl. Mech. 38(197), 9–19 (1971)
Bažant Z., Cedolin L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. Oxford University Press, New York (1991)
Engesser F.: Die Knickfestigkeit gerader Stäbe. Centralblatt der Bauverwaltung 11(49), 483– (1891)
Haringx, J.: On the buckling and lateral rigidity of helical springs. I, II. Proc. Konink. Ned. Akad. Wetenschappen 45, 533–539, 650–654 (1942)
Antman S., Rosenfeld G.: Global behavior of buckled states of nonlinearly elastic rods. Siam Rev. 20(3), 513–566 (1978)
Pflüger A.: Stabilitätsprobleme der Elastostatik. Springer, Berlin (1964)
Goto Y., Yamashita T., Matsuura S.: Elliptic integral solutions for extensional elastica with constant initial curvature. Struct. Engng./Earthquake Engng. 4(2), 299–309 (1987)
Goto Y., Yoshimitsu T., Obata M.: Elliptic integral solutions of plane elastica with axial and shear deformations. Int. J. Solids Struct. 26(4), 375–390 (1990)
Humer A.: Elliptic integral solution of the extensible elastica with a variable length under a concentrated force. Acta Mech. 222(3–4), 209–223 (2011)
Magnusson A., Ristinmaa M., Ljung C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. 38(46–47), 8441–8457 (2001)
Wang C.: Post-buckling of a clamped-simply supported elastica. Int. J. Non-Linear Mech. 32(6), 1115–1122 (1997)
Vaz M., Silva D.: Post-buckling analysis of slender elastic rods subjected to terminal forces. Int. J. Non-Linear Mech. 38(4), 483–492 (2003)
Mazzilli C.: Buckling and post-buckling of extensible rods revisited: a multiple-scale solution. Int. J. Non-Linear Mech. 44(2), 200–208 (2009)
Nayfeh A., Mook D.: Nonlinear Oscillations. Wiley, New York (1979)
Simo J., Vu-Quoc L.: On the dynamics of flexible beams under large overall motionsthe plane case: part I. J. Appl. Mech. 53(4), 849–854 (1986)
Simo J., Vu-Quoc L.: On the dynamics of flexible beams under large overall motionsthe plane case: part II. J. Appl. Mech. 53(4), 855–863 (1986)
Gerstmayr J., Matikainen M., Mikkola A.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20(4), 359–384 (2008)
Nachbagauer K., Pechstein A., Irschik H., Gerstmayr J.: A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 26(3), 245–263 (2011)
Irschik H., Gerstmayr J.: A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli-Euler beams. Acta Mech. 206(1-2), 1–21 (2009)
Irschik H., Gerstmayr J.: A continuum-mechanics interpretation of Reissner’s non-linear shear-deformable beam theory. Math. Comput. Model. Dyn. Syst. 17(1), 19–29 (2011)
Reissner E.: Some remarks on the problem of column buckling. Arch. Appl. Mech. 52(1-2), 115–119 (1982)
Lakes R.: Foam structures with a negative poisson’s ratio. Science 235, 1038–1040 (1987)
Prall D., Lakes R.: Properties of a chiral honeycomb with a poisson’s ratio of −1. Int. J. Mech. Sci. 39(3), 305–314 (1997)
Attard M., Hunt G.: Column buckling with shear deformations—a hyperelastic formulation. Int. J. Solids Struct. 45(14), 4322–4339 (2008)
Abramowitz M., Stegun I.: Handbook of Mathematical Functions. Dover Publications Inc., New York (1972)
Kuznetsov V., Levyakov S.: Complete solution of the stability problem for elastica of Euler’s column. Int. J. Non-Linear Mech. 37(6), 1003–1009 (2002)
Levyakov S., Kuznetsov V.: Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads. Acta Mech. 211(1), 73–87 (2010)
Gröbner W., Hofreiter N.: Integraltafel. 1. Teil: Unbestimmte Integrale. Springer, Wien (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Humer, A. Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mech 224, 1493–1525 (2013). https://doi.org/10.1007/s00707-013-0818-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-013-0818-1