Thin Extensional Beams under Large Deformations

Variational Principles, Global Bounds, Stability
  • H. Bufler
  • R. Lautenbach
  • H. Schneider
Conference paper


Under the usual kinematical assumptions and plane loading the basic equations for an extensional beam including shearing deformation are established for the case of a semilinear material. The corresponding mixed variational formulation allows independent variations of six field variables. If the force equilibrium is satisfied a priori, there results a mixed stationarity principle with only two field variables: The cross section rotation angle and the couple. From this one complementary variational principles are derived and conditions for the existence of global bounds are given. Generally the solution is not unique. The following cases are observed: (a) JI = min, JII = max (stable equilibrium, global bounds); (b) JI = min, JII = stat (stable equilibrium, no global bounds); (c) JI = stat, JII = stat (no stable equilibrium, no global bounds). Here JI means the total potential energy and JII the total complementary potential. Some illustrative examples are worked out numerically using finite elements.


Stable Equilibrium Total Potential Energy Stiffness Ratio Dead Weight Extremum Principle 
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  1. 1.
    Timoshenko, S.P; Gere, J.M.: Theory of elastic stability, 2nd ed. New York, Toronto, London: McGraw Hill 1961, p.134.Google Scholar
  2. 2.
    Reissner, E.: Some remarks on the problem of column buckling. Ing.-Arch. 52 (1982) 115–119.MATHCrossRefGoogle Scholar
  3. 3.
    Ziegler, H.: Arguments for and against Engesser’s buckling formulas. Ing.-Arch. 52 (1982) 105–113.MATHCrossRefGoogle Scholar
  4. 4.
    Reissner, E.: Formulation of variational theorems in geometrically nonlinear elasticity. J. of Engrg. Mech. 110 (1984) 1377–1390.CrossRefGoogle Scholar
  5. 5.
    Bufler, H.: The Biot stresses in nonlinear elasticity and the associated generalized variational principles. Ing.-Arch. 55 (1985) 450–462.MATHCrossRefGoogle Scholar
  6. 6.
    Sewell, M.J.: The governing equations and extremum principles of elasticity and plasticity generated from a single functional. J. Struct. Mech. 2 (1973) 1–32 and 135–158.CrossRefGoogle Scholar
  7. 7.
    Bufler, H.: Finite rotations and complementary variational principles. In: Finite rotations in structural mechanics, Proc. Euromech. Coll. No. 197 at Jablonna (ed. Pietraszkiewicz, W.), Springer-Verlag (1986).Google Scholar
  8. 8.
    Reeves, R.I.: Complementary variational principles for large deflections of a cantilever beam. Quart. Appl. Math. 33 (1975) 245–254.MathSciNetMATHGoogle Scholar
  9. 9.
    Noble, B.; Sewell, M.J.: On dual extremum principles in applied mathematics. J. Inst. Math. Appl. 9 (1972) 123–193.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Japan 1986

Authors and Affiliations

  • H. Bufler
    • 1
  • R. Lautenbach
    • 1
  • H. Schneider
    • 1
  1. 1.Institute of Mechanics (Civil Engineering)Stuttgart UniversityStuttgart 80Federal Republic of Germany

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