Applied Mathematics and Mechanics

, Volume 14, Issue 5, pp 417–428 | Cite as

The applications of generalized variational principles in nonlinear structural analysis

  • Cheng Xiang-sheng


This paper discusses the generalized variational principles founded by the technique of Lagrangian multipliers in structural mechanics and analyzes the nonlinear statically indeterminate structures. It is assumed that the stress-strain relationship of the materials of structures has the form ofσ=Bε 1/m orτ=Cγ 1/m ; namely, the physical equations of structures have the shape of exponential functions. Several examples are given to illustrate the statically indeterminate structures such as the trusses, beams, frames and torsional bars.

Key words

generalized variational principle nonlinear structures statically indeterminate structures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Chien Wei-zang,On Generalized Variational Principles in Elasticity with Applications in Problems of Plates and Shells, (1964), (in Chinese)Google Scholar
  2. [2]
    Chien Wei-zang,Mechanics and Practice,1, 1 (1979) 16–24;1, 2 (1979) 18–27. (in Chinese)Google Scholar
  3. [3]
    Chien Wei-zang,Calculus of Variation and the Finite Element Method, Vol. I, Science Publishing House, (1980), (in Chinese)Google Scholar
  4. [4]
    Chien Wei-zang, Method of Lagrangian multipliers, method of high-order Lagrangian multipliers and generalized variational principles of elasticity with more general forms of functionals,Appl. Math. and Mech. 4, 2 (1983).Google Scholar
  5. [5]
    Chien Wei-zang,The Generalized Variatinal Principles, Knowledge Publishing House, (1985). (in Chinese)Google Scholar
  6. [6]
    Cheng Xiang-sheng, The method of undetermined multipliers applied to finding problems of statically indeterminate structures of rod systems, Papers read at the meeting of Society of Mechanics in Kiangsu Province, (1964), (in Chinese)Google Scholar
  7. [7]
    Cheng Xiang-sheng,Mechanics and Practice,1, 1 (1979) 46–47. (in Chinese)Google Scholar
  8. [8]
    Cheng Xiang-sheng, The generalized variational principles with their applications in structural analysis,Appl. Math. and Mech. 6, 7 (1985) 639–646.Google Scholar
  9. [9]
    Washizu, K.,Variational Methods in Elasticity and Plasticity, Pergamon, London, (1968).Google Scholar
  10. [10]
    Washizu, K.,Introduction of the Theory of Energy, PeiFeng Publ. House, (1970). (in Japanese)Google Scholar
  11. [11]
    Pian, T. H., and Tong Pin., Finite element method in continuum mechanics,Adv. in App. Mech.,12 (1972) 1–53.MATHCrossRefGoogle Scholar
  12. [12]
    Hsuch Dah-wei,Science Information,20, 4 (1975) 81.Google Scholar
  13. [13]
    Fikhtengolets, G.M.,Course of Differential and Integral Calculus, Vol. I, National Technical Publishing House, Moscow-Leningrad, (1949). (in Russian)Google Scholar
  14. [14]
    Timoshenko, S. and J. Gere,Mechanics of Materials, Van Nostrand Reinhold Company, (1972)Google Scholar
  15. [15]
    Engesser, F., On statically indeterminate frame with any low of deformation and on theorem of minimum complementary energy,Jour. of Arch and Engin. Union, Hannover,35 (1889) 733–744. (in German)Google Scholar
  16. [16]
    Westergaard, H. M., On the method of complementary energy and its applications to structures stressed beyond the proportional limit, to buckling and vibrations to suspension bridges,Proc. of ASCE,67, 2 (1941) 199.Google Scholar
  17. [17]
    Westergaard, H. M., On the method of complementary energy,Trans. of ASCE,107, (1942) 765–793.Google Scholar
  18. [18]
    Chien Ling-xi, Theory of complementary energy,Scientia Sinica,1 (1950) 449.Google Scholar
  19. [19]
    Charlton, T. M., Analysis of statically-indeterminate structures by the complementary energy method,Engineering,174 (1952) 389–391.Google Scholar
  20. [20]
    Brown, E. H., The energy theorems of structural analysis,Engineering,179 (1955) 305–308, 339–342, 400–403.Google Scholar
  21. [21]
    Hoff, N. J.,The Analysis of Structures, John Wiley and Sons, Inc., New York, (1956) 493.MATHGoogle Scholar
  22. [22]
    Argyris, J. J. and S. Kelsey,Energy Theorems and Structural Analysis, Butterworth and Co., Ltd., London, (1960) 85.Google Scholar
  23. [23]
    Langhaar, H. L.,Energy Methods in Applied Mechanics, John Wiley and Sons, Inc., New York, (1962) 350.Google Scholar
  24. [24]
    Au, T.,Elementary Structural Mechanics, (1963) 521.Google Scholar
  25. [25]
    Libove, C., Complementary energy method for finite deformations,Pro. of ASCE, Jour. of Engi. Mech. Divi. 90, EM6 (1964) 49–71.Google Scholar
  26. [26]
    Oran, C., Complementary energy method for buckling,Pro. of ASCE, Jour. of EMD.,93, EM1 (1967) 57–75.Google Scholar
  27. [27]
    Oden, J. T.Mechanics of Elastic Structures, (1967) 381.Google Scholar
  28. [28]
    Oran, C., Complementary energy concept for large deformations,Proc. of ASCE, Jour. of Struc. Divi. 93, ST1 (1967) 57–75.Google Scholar

Copyright information

© Shanghai University of Technology (SUT) 1993

Authors and Affiliations

  • Cheng Xiang-sheng
    • 1
  1. 1.Tongji UniversityShanghai

Personalised recommendations