Skip to main content
SpringerLink
Log in

Large Oscillations Around Curled Equilibrium Configurations of Uniformly Loaded Euler–Bernoulli Beams: Numerical and Experimental Evidences

  • Chapter
  • First Online:
Recent Developments in the Theory of Shells

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 110))

  • 562 Accesses

Abstract

In this paper, we show that equilibrium configurations of a clamped beam under distributed load, resembling a curled pending wire—whose existence has been mathematically established—can be obtained experimentally using ‘soft’ beams, i.e. beams for which the ratio between amplitude of the load and bending stiffness is large enough. Moreover, we introduce a Hencky-type discrete model, i.e. a finite dimensional Lagrangian model, for the ‘soft’ Elastica and build a numerical code for determining its motion, in the most general nonlinear regime. This code is able to qualitatively describe observed nonlinear dynamical behavior.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It has to be remarked that an outline of Hencky’s idea can also be found in the work of Gabrio Piola almost one century before, see [20, 23].

References

  1. Abali, B.E., Wu, C.C., Müller, W.H.: An energy-based method to determine material constants in nonlinear rheology with applications. Contin. Mech. Thermodyn. 28(5), 1221–1246 (2016)

    Article  CAS  Google Scholar 

  2. Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

    Article  Google Scholar 

  3. Alibert, J.J., Della Corte, A., Giorgio, I., Battista, A.: Extensional elastica in large deformation as \(\varGamma \)-limit of a discrete 1D mechanical system. Zeitschrift für angewandte Mathematik und Physik 68(2), 19 (2017)

    Google Scholar 

  4. Alibert, J.J., Della Corte, A., Seppecher, P.: Convergence of Hencky-type discrete beam model to Euler inextensible elastica in large deformation: rigorous proof. In: Mathematical Modelling in Solid Mechanics, pp. 1–12. Springer (2017)

    Google Scholar 

  5. Altenbach, H., Eremeyev, V.A.: Generalized Continua-from the Theory to Engineering Applications, vol. 541. Springer (2012)

    Google Scholar 

  6. Altenbach, H., Eremeyev, V.A.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)

    Article  Google Scholar 

  7. Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Generalized Continua from the Theory to Engineering Applications, pp. 179–248. Springer (2013)

    Google Scholar 

  8. Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Contin. Mech. Thermodyn., 1–21 (2018)

    Google Scholar 

  9. Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Q. Appl. Math. 32(3), 221–240 (1974)

    Article  Google Scholar 

  10. Barchiesi, E., Dell’Isola, F., Laudato, M., Placidi, L., Seppecher, P.: A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results. In: Advances in Mechanics of Microstructured Media and Structures, pp. 43–74. Springer (2018)

    Google Scholar 

  11. Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn., 1–13 (2018)

    Google Scholar 

  12. Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212–234 (2019)

    Article  Google Scholar 

  13. Battista, A., Della Corte, A., dell’Isola, F., Seppecher, P.: Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams. Zeitschrift für angewandte Mathematik und Physik 69, 22 (2018)

    Google Scholar 

  14. Berezovski, A., Yildizdag, M.E., Scerrato, D.: On the wave dispersion in microstructured solids. Contin. Mech. Thermodyn., 1–20 (2018)

    Google Scholar 

  15. Bernoulli, D.: Letter from Daniel Bernoulli to Euler, 20 Oct 1742. http://eulerarchive.maa.org/correspondence/letters/OO0147.pdf

  16. Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B: Eng. 43(3), 1315–1328 (2012)

    Article  Google Scholar 

  17. Born, M.: Untersuchungen über die stabilität der elastischen linie in ebene und raum, unter verschiedenen grenzbedingungen. Ph.D. thesis, University of University of Gottingen (1906)

    Google Scholar 

  18. Cuomo, M.: Forms of the dissipation function for a class of viscoplastic models. Math. Mech. Complex Syst. 5(3), 217–237 (2017)

    Article  Google Scholar 

  19. Della Corte, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations. Math. Model. Methods Appl. Sci. 27(08), 1391–1421 (2017)

    Article  Google Scholar 

  20. dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S.: The complete works of Gabrio Piola: Volume I Commented English Translation. Springer Publishing Company, Incorporated (2014)

    Google Scholar 

  21. dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)

    Article  Google Scholar 

  22. dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 23 (2016)

    Article  Google Scholar 

  23. dell’Isola, F., Andreaus, U., Cazzani, A., Esposito, R., Placidi, L., Perego, U., Maier, G., Seppecher, P.: The Complete works of Gabrio Piola: Volume II Commented English Translation. Springer Publishing Company, Incorporated (2019)

    Google Scholar 

  24. Duncan, W.J.: A critical examination of the representation of massive and elastic bodies by systems of rigid masses elastically connected. Q. J. Mech. Appl. Math. 5(1), 97–108 (1952)

    Article  Google Scholar 

  25. Dupac, M., Marghitu, D.B.: Nonlinear dynamics of a flexible mechanism with impact. J. Sound Vib. 289(4–5), 952–966 (2006)

    Article  Google Scholar 

  26. Eugster, S.R., Hesch, C., Betsch, P., Glocker, C.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng. 97(2), 111–129 (2014)

    Article  Google Scholar 

  27. Euler, L.: De curvis elasticis, Additamentum I to his Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne and Geneva (1744)

    Google Scholar 

  28. Franciosi, P., Spagnuolo, M., Salman, O.U.: Mean green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Contin. Mech. Thermodyn., 1–32 (2018)

    Google Scholar 

  29. Greco, L., Cuomo, M.: Consistent tangent operator for an exact Kirchhoff rod model. Contin. Mech. Thermodyn. 27(4–5), 861–877 (2015)

    Article  Google Scholar 

  30. Hencky, H.: Über die angenäherte lösung von stabilitätsproblemen im raum mittels der elastischen gelenkkette. Der Eisenbau 11, 437–452 (1920)

    Google Scholar 

  31. Leckie, F.A., Lindberg, G.M.: The effect of lumped parameters on beam frequencies. Aeronaut. Q. 14(3), 224–240 (1963)

    Article  Google Scholar 

  32. Maurin, F., Greco, F., Desmet, W.: Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models. Contin. Mech. Thermodyn., 1–14 (2018)

    Google Scholar 

  33. Milton, G.W., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)

    Article  Google Scholar 

  34. Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)

    Article  Google Scholar 

  35. Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67(5), 121 (2016)

    Article  Google Scholar 

  36. Placidi, L., Barchiesi, E., Battista, A.: An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations. In: Mathematical Modelling in Solid Mechanics, pp. 193–210. Springer (2017)

    Google Scholar 

  37. Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Zeitschrift für angewandte Mathematik und Physik ZAMP 23(5), 795–804 (1972)

    Article  Google Scholar 

  38. Rosi, G., Placidi, L., Auffray, N.: On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure. Eur. J. Mech.-A/Solids 69, 179–191 (2018)

    Article  Google Scholar 

  39. Rubinstein, D.: Dynamics of a flexible beam and a system of rigid rods, with fully inverse (one-sided) boundary conditions. Comput. Methods Appl. Mech. Eng. 175(1–2), 87–97 (1999)

    Article  Google Scholar 

  40. Ruocco, E., Zhang, H., Wang, C.M.: Hencky bar-chain model for buckling analysis of non-uniform columns. Structures 6, 73–84 (2016)

    Article  Google Scholar 

  41. Šalinić, S.: An improved variant of Hencky bar-chain model for buckling and bending vibration of beams with end masses and springs. Mech. Syst. Signal Process. 90, 30–43 (2017)

    Article  Google Scholar 

  42. Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 96(11), 1268–1279 (2016)

    Article  Google Scholar 

  43. Spagnuolo, M., Andreaus, U.: A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Math. Mech. Solids 24(1), 258–280 (2019)

    Article  Google Scholar 

  44. Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)

    Article  Google Scholar 

  45. Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33(1), 1–26 (1993)

    Article  Google Scholar 

  46. Turco, E.: Discrete is it enough? The revival of Piola-Hencky keynotes to analyze three-dimensional Elastica. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0656-4:1-19

  47. Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 28 (2016). https://doi.org/10.1007/s00033-016-0681-8

  48. Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.H., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016b)

    Article  Google Scholar 

  49. Turco, E., Golaszewski, M., Cazzani, A., Rizzi, N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete lagrangian model. Mech. Res. Commun. 76, 51–56 (2016c)

    Article  Google Scholar 

  50. Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R. Soc. Open Sci. 4(10), 20 (2017)

    Article  Google Scholar 

  51. Wang, C.M., Zhang, H., Gao, R.P., Duan, W.H., Challamel, N.: Hencky bar-chain model for buckling and vibration of beams with elastic end restraints. Int. J. Struct. Stab. Dyn. 15(07), 16 (2015)

    Google Scholar 

  52. Wang, Y., Huston, R.L.: A lumped parameter method in the nonlinear analysis of flexible multibody systems. Comput. Struct. 50(3), 421–432 (1994)

    Article  Google Scholar 

  53. Zhang, H., Wang, C.M., Challamel, N.: Buckling and vibration of Hencky bar-chain with internal elastic springs. Int. J. Mech. Sci. 119, 383–395 (2016)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aalto University School of Engineering, Aalto University, 00076, Aalto, Finland

    D. Baroudi

  2. Department of Mechanical and Aerospace Engineering, SAPIENZA Università di Roma, 00184, Rome, Italy

    I. Giorgio

  3. International Research Center for the Mathematics and Mechanics of Complex Systems, Università degli studi dell’Aquila, 67100, L’Aquila, Italy

    I. Giorgio & E. Turco

  4. Department of Architecture, Design and Urban Planning, University of Sassari, 07041, Alghero, Italy

    E. Turco

Authors
  1. D. Baroudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Giorgio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Turco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Giorgio .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Jacek Chróścielewski

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  4. Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

    Krzysztof Wiśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Baroudi, D., Giorgio, I., Turco, E. (2019). Large Oscillations Around Curled Equilibrium Configurations of Uniformly Loaded Euler–Bernoulli Beams: Numerical and Experimental Evidences. In: Altenbach, H., Chróścielewski, J., Eremeyev, V., Wiśniewski, K. (eds) Recent Developments in the Theory of Shells . Advanced Structured Materials, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-030-17747-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation