Archive of Applied Mechanics

, Volume 84, Issue 2, pp 263–275 | Cite as

A simple method for determining large deflection states of arbitrarily curved planar elastica



The paper discusses a relatively simple method for determining large deflection states of arbitrarily curved planar elastica, which is modeled by a finite set of initially straight flexible segments. The basic equations are built using Euler–Bernoulli and large displacement theory. The problem is solved numerically using Runge–Kutta–Fehlberg integration method and Newton method for solving systems of nonlinear equations. This solution technique is tested on several numerical examples. From a comparison of the results obtained and those found in the literature, it can be concluded that the developed method is efficient and gives accurate results. The solution scheme displayed can serve as reference tool to test results obtained via more complex algorithms.


Large deflections Arbitrarily curved elastica Geometrical nonlinearity Plane deflection Elastica problem 


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  1. 1.
    Brojan, M., Sitar, M., Kosel, F.: On static stability of nonlinearly elastic Euler’s columns obeying the modified Ludwick’s Law. Int. J. Struct. Stab. Dyn. 12(6), 1250077 (1) –1250077 (19) (2012)Google Scholar
  2. 2.
    Campanile L.F., Hasse A.: A simple and effective solution of the elastica problem. J. Mech. Eng. Sci. 222(12), 2513–2516 (2008)CrossRefGoogle Scholar
  3. 3.
    Chen L.: An integral approach for large deflection cantilever beams. Int. J. Non-Linear Mech. 45, 301–305 (2010)CrossRefGoogle Scholar
  4. 4.
    Dado M., Al-Sadder S.: A new technique for large deflection analysis of non-prismatic cantilever beams. Mech. Res. Commun. 32(6), 692–703 (2005)CrossRefMATHGoogle Scholar
  5. 5.
    Holden J.T.: On the finite deflections of thin beams. Int. J. Solid Struct. 8(8), 1051–1055 (1972)CrossRefMATHGoogle Scholar
  6. 6.
    Levyakov S.V., Kuznetsov V.V.: Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads. Acta Mech. 211(1–2), 73–87 (2010)CrossRefMATHGoogle Scholar
  7. 7.
    Vaz M.A., Silva D.F.C.: Post-buckling analysis of slender elastic rods subjected to terminal forces. Int. J. Non-Linear Mech. 38(4), 483–492 (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Wang C.Y.: Post-buckling of a clamped-simply supported elastica. Int. J. Non-Linear Mech. 32(6), 1115–1122 (1997)CrossRefMATHGoogle Scholar
  9. 9.
    Bunce J.W., Brown E.H.: Non-linear bending of thin, ideally elastic rods. Int. J. Mech. Sci. 18(9–10), 435–441 (1976)CrossRefGoogle Scholar
  10. 10.
    De Bona F., Zelenika S.: A generalized elastica-type approach to the analysis of large displacements of spring-strips. J. Mech. Eng. Sci. 211(7), 509–517 (1997)CrossRefGoogle Scholar
  11. 11.
    Nallathambi A.K., Rao C.L., Srinivasan S.M.: Large deflection of constant curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445 (2010)CrossRefGoogle Scholar
  12. 12.
    Shinohara A.: Large deflection of a circular C-shaped spring. Int. J. Mech. Sci. 21, 179–186 (1979)CrossRefGoogle Scholar
  13. 13.
    Somervaille I.: Quadrature matrices and elastica problems. Comp. Method App. Mech. Eng. 69(3), 345–354 (1988)CrossRefMATHGoogle Scholar
  14. 14.
    Srpčič S., Saje M.: Large deformations of thin curved plane beam of constant initial curvature. Int. J. Mech. Sci. 28(5), 275–287 (1986)CrossRefMATHGoogle Scholar
  15. 15.
    Wang C.Y., Watson L.T.: On the large deformations of C-shaped springs. Int. J. Mech. Sci. 22, 395–400 (1980)CrossRefMATHGoogle Scholar
  16. 16.
    Watson L.T., Wang C.Y.: A homotopy method applied to elastica problems. Int. J. Solids Struct. 17(1), 29–37 (1981)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Dado M., Al-Saddar S.: The elastic spring behavior of a rhombus frame constructed from non-prismatic beams under large deflection. Int. J. Mech. Sci. 48, 958–968 (2006)CrossRefMATHGoogle Scholar
  18. 18.
    Faulkner M.G., Lipsett A.W., Tam V.: On the use of a segmental shooting technique for multiple solutions of planar elastica problems. Comp. Methods App. Mech. Eng. 110(3–4), 221–236 (1993)CrossRefMATHGoogle Scholar
  19. 19.
    Lee S.L., Manuel F.S., Rossow E.C.: Large deflections and stability of elastic frames. J. Eng. Mech. Div. 94(2), 521–548 (1968)Google Scholar
  20. 20.
    Manuel F.S., Lee S.L.: Flexible bars subjected to arbitrary discrete loads and boundary conditions. J. Franklin Inst. 285(6), 452–474 (1968)CrossRefGoogle Scholar
  21. 21.
    Mattiasson K.: Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals. Int. J. Num. Methods Eng. 17(1), 145–153 (1981)CrossRefMATHGoogle Scholar
  22. 22.
    Phungpaingam B., Chucheepsakul S.: Postbuckling of elastic beam subjected to a concentrated moment within span length of beam. Acta Mech. 23(3), 287–296 (2007)CrossRefMATHGoogle Scholar
  23. 23.
    Saje M.: Finite element formulation of finite planar deformation of curved elastic beams. Comput. Struct. 39(3–4), 327–337 (1991)CrossRefMATHGoogle Scholar
  24. 24.
    Thacker W.I., Wang C.Y., Watson L.T.: Effect of flexible joints on the stability and large deflections of a triangular frame. Acta Mech. 200(1–2), 11–24 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    Brojan M., Cebron M., Kosel F.: Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end. Acta Mech. Sin. 28(3), 863–869 (2012)CrossRefGoogle Scholar
  26. 26.
    Brojan M., Kosel F.: Approximative formula for post-buckling analysis of nonlinearly elastic columns with superellipsoidal cross-sections. J. Reinf. Plast. Comp. 30(5), 409–415 (2011)CrossRefGoogle Scholar
  27. 27.
    Brojan M., Videnic T., Kosel F.: Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law. Meccanica 44, 733–739 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Burden R.L., Faires J.D.: Numerical Analysis, 9th ed. Brooks/Cole, Boston (2010)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratory for Nonlinear Mechanics, Faculty of Mechanical EngineeringUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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