Extensible Beam Models in Large Deformation Under Distributed Loading: A Numerical Study on Multiplicity of Solutions

  • Francesco dell’Isola
  • Alessandro Della Corte
  • Antonio Battista
  • Emilio BarchiesiEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 120)


In this paper we present numerical solutions to a geometrically nonlinear version of the extensible Timoshenko beam model under distributed load. The particular cases in which: i) extensional stiffness is infinite (inextensible Timoshenko model), ii) shear stiffness is infinite (extensible Euler model) and iii) extensional and shear stiffnesses are infinite (inextensible Euler model) will be numerically explored. Parametric studies on the axial stiffness in both the Euler and Timoshenko cases will also be shown and discussed.


Timoshenko beam Large deformation of beams Extensional beam model Shooting technique 


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  1. Altenbach H, Eremeyev VA (2013) Cosserat-type shells. In: Altenbach H, Eremeyev VA (eds) Generalized Continua from the Theory to Engineering Applications, CISM International Centre for Mechanical Sciences, vol 541, Springer, Vienna, pp 131–178Google Scholar
  2. Altenbach H, Birsan M, Eremeyev VA (2013) Cosserat-type rods. In: Altenbach H, Eremeyev VA (eds) Generalized Continua from the Theory to Engineering Applications, CISM International Centre for Mechanical Sciences, vol 541, Springer, Vienna, pp 179–248Google Scholar
  3. Altenbach J, Altenbach H, Eremeyev VA (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics 80(1):73–92Google Scholar
  4. Andreaus U, Spagnuolo M, Lekszycki T, Eugster SR (2018) A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Continuum Mechanics and Thermodynamics 30(5):1103–1123Google Scholar
  5. Antman SS, Renardy M (1995) Nonlinear problems of elasticity. SIAM Review 37(4):637Google Scholar
  6. Atai AA, Steigmann DJ (1997) On the nonlinear mechanics of discrete networks. Archive of Applied Mechanics 67(5):303–319Google Scholar
  7. Ball JM, Mizel VJ (1987) One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation. In: Analysis and Thermomechanics, Springer, pp 285–348Google Scholar
  8. Balobanov V, Niiranen J (2018) Locking-free variational formulations and isogeometric analysis for the timoshenko beam models of strain gradient and classical elasticity. Computer Methods in Applied Mechanics and Engineering 339:137–159Google Scholar
  9. Barchiesi E, dell’Isola F, Laudato M, Placidi L, Seppecher P (2018) A 1D continuum model for beams with pantographic microstructure: Asymptotic micro-macro identification and numerical results. In: dell’Isola F, Eremeyev V, Porubov A (eds) Advances in Mechanics of Microstructured Media and Structures, Advanced Structured Materials, vol 87, Springer, Cham, pp 43–74Google Scholar
  10. Barchiesi E, Spagnuolo M, Placidi L (2019) Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids 24(1):212–234Google Scholar
  11. Battista A, Della Corte A, dell’Isola F, Seppecher P (2018) Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams. Zeitschrift für angewandte Mathematik und Physik 69(3):52Google Scholar
  12. Berezovski A, Yildizdag M, Scerrato D (2018) On the wave dispersion in microstructured solids. Continuum Mechanics and Thermodynamics pp 1–20, DOI 10.1007/s00161-018-0683-1Google Scholar
  13. Bernoulli D (1843) The 26th letter to Euler. Correspondence Mathématique et Physique 2Google Scholar
  14. Bernoulli J (1691) Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura. Die Werke von Jakob Bernoulli pp 223–227Google Scholar
  15. Birsan M, Altenbach H, Sadowski T, Eremeyev V, Pietras D (2012) Deformation analysis of functionally graded beams by the direct approach. Composites Part B: Engineering 43(3):1315– 328Google Scholar
  16. Boubaker BB, Haussy B, Ganghoffer J (2007) Discrete models of woven structures. macroscopic approach. Composites Part B: Engineering 38(4):498–505Google Scholar
  17. Boutin C, Giorgio I, Placidi L, et al (2017) Linear pantographic sheets: Asymptotic micro-macro models identification. Mathematics and Mechanics of Complex Systems 5(2):127–162Google Scholar
  18. Cazzani A, Malagù M, Turco E (2016) Isogeometric analysis of plane-curved beams. Mathematics and Mechanics of Solids 21(5):562–577Google Scholar
  19. Challamel N (2013) Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Composite Structures 105:351–368Google Scholar
  20. Challamel N, Zhang Z, Wang C (2013) Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams. Journal of Nanomechanics and Micromechanics 5(1):A4014,004Google Scholar
  21. Chebyshev P (1878) Sur la coupe des vetements. Complete works by PL Chebyshev 5:165–170Google Scholar
  22. Chróścielewski J, Schmidt R, Eremeyev VA (2019) Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Continuum Mechanics and Thermodynamics 31(1):147–188Google Scholar
  23. Cosserat E, Cosserat F (1909) Théorie des corps déformables. A Hermann et filsGoogle Scholar
  24. Della Corte A, dell’Isola F, Esposito R, Pulvirenti M (2016) Equilibria of a clamped Euler beam (Elastica) with distributed load: Large deformations. Mathematical Models and Methods in Applied Sciences pp 1–31Google Scholar
  25. Della Corte A, Battista A, dell’Isola F, Seppecher P (2019) Large deformations of Timoshenko and Euler beams under distributed load. Zeitschrift für angewandte Mathematik und Physik 70(52),
  26. dell’Isola F, Giorgio I, Pawlikowski M, Rizzi N (2016a) Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472(2185):20150,790Google Scholar
  27. dell’Isola F, Steigmann D, Della Corte A (2016b) Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Applied Mechanics Reviews 67(6):060,804–060,804–21Google Scholar
  28. dell’Isola F, Seppecher P, Alibert JJ, Lekszycki T, Grygoruk R, Pawlikowski M, Steigmann D, Giorgio I, Andreaus U, Turco E, Golaszewski M, Rizzi N, Boutin C, Eremeyev VA, Misra A, Placidi L, Barchiesi E, Greco L, Cuomo M, Cazzani A, Corte AD, Battista A, Scerrato D, Eremeeva IZ, Rahali Y, Ganghoffer JF, Müller W, Ganzosch G, Spagnuolo M, Pfaff A, Barcz K, Hoschke K, Neggers J, Hild F (2018) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics 31(4):851–884Google Scholar
  29. Diyaroglu C, Oterkus E, Oterkus S, Madenci E (2015) Peridynamics for bending of beams and plates with transverse shear deformation. International Journal of Solids and Structures 69:152–168Google Scholar
  30. Diyaroglu C, Oterkus E, Oterkus S (2017) An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework. Mathematics and Mechanics of Solids 24(2):361–376Google Scholar
  31. Dortdivanlioglu B, Javili A, Linder C (2017) Computational aspects of morphological instabilities using isogeometric analysis. Computer Methods in Applied Mechanics and Engineering 316:261–279Google Scholar
  32. Dos Reis F, Ganghoffer J (2012) Construction of micropolar continua from the asymptotic homogenization of beam lattices. Computers & Structures 112:354–363Google Scholar
  33. Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Mathematics and Mechanics of Complex Systems 3(1):43–82Google Scholar
  34. Engelbrecht J, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philosophical Magazine 85(33-35):4127–4141Google Scholar
  35. Eremeyev VA (2017) On characterization of an elastic network within the six-parameter shell theory. In: Pietraszkiewicz W, Witkowski W (eds) Shell Structures: Theory and Applications Volume 4: Proceedings of the 11th International Conference in Shell Structures: Theory and Applications, SSTA 2017, CRC Press, pp 81–84Google Scholar
  36. Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Mathematics and Mechanics of Solids 21(2):210–221Google Scholar
  37. Eugster S, Hesch C, Betsch P, Glocker C (2014) Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. International Journal for Numerical Methods in Engineering 97(2):111–129Google Scholar
  38. Eugster SR (2015) Geometric Continuum Mechanics and Induced Beam Theories, Lecture Notes in Applied and Computational Mechanics, vol 75. SpringerGoogle Scholar
  39. Euler L (1952) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti (ed. by C. Carathéodory), Opera mathematica, vol 1. Birkhäuser, BaselGoogle Scholar
  40. Fertis DG (2006) Nonlinear Structural Engineering. SpringerGoogle Scholar
  41. Forest S (2005) Mechanics of Cosserat media - an introduction. Ecole des Mines de Paris, Paris pp 1–20Google Scholar
  42. Franciosi P, Spagnuolo M, Salman OU (2019) Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mechanics and Thermodynamics 31(1):101–132Google Scholar
  43. Giorgio I, Del Vescovo D (2018) Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics 7(4):60Google Scholar
  44. Giorgio I, Rizzi N, Turco E (2017) Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473(2207):20170,636Google Scholar
  45. Golaszewski M, Grygoruk R, Giorgio I, Laudato M, Di Cosmo F (2019) Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Continuum Mechanics and Thermodynamics 31(4):1015–1034Google Scholar
  46. Greco L, Cuomo M, Contrafatto L, Gazzo S (2017) An effcient blended mixed b-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Computer Methods in Applied Mechanics and Engineering 324:476–511Google Scholar
  47. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol 42. SpringerGoogle Scholar
  48. Javili A, dell’Isola F, Steinmann P (2013a) Geometrically nonlinear higher-gradient elasticity with energetic boundaries. Journal of the Mechanics and Physics of Solids 61(12):2381–2401Google Scholar
  49. Javili A, McBride A, Steinmann P (2013b) Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. a unifying review. Applied Mechanics Reviews 65(1):010,802Google Scholar
  50. Javili A, McBride A, Steinmann P, Reddy B (2014) A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology. Computational Mechanics 54(3):745–762Google Scholar
  51. Javili A, Dortdivanlioglu B, Kuhl E, Linder C (2015) Computational aspects of growth-induced instabilities through eigenvalue analysis. Computational Mechanics 56(3):405–420Google Scholar
  52. Jawed MK, Novelia A, O’Reilly OM (2018) A Primer on the Kinematics of Discrete Elastic Rods. SpringerGoogle Scholar
  53. Ladevèze P (2012) Nonlinear Computational Structural Mechanics: New Approaches and Nonincremental Methods of Calculation. Springer Science & Business MediaGoogle Scholar
  54. Luongo A, D’Annibale F (2013) Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. International Journal of Non-Linear Mechanics 55:128–139Google Scholar
  55. Luongo A, Zulli D (2013) Mathematical Models of Beams and Cables. John Wiley & Sons Google Scholar
  56. Milton G, Briane M, Harutyunyan D (2017) On the possible effective elasticity tensors of 2- dimensional and 3-dimensional printed materials. Mathematics and Mechanics of Complex Systems 5(1):41–94Google Scholar
  57. Misra A, Placidi L, Scerrato D (2016) A review of presentations and discussions of the workshop computational mechanics of generalized continua and applications to materials with microstructure that was held in Catania 29–31 October 2015. Mathematics and Mechanics of Solids 22(9):1891–1904Google Scholar
  58. Misra A, Lekszycki T, Giorgio I, Ganzosch G, Müller WH, dell’Isola F (2018) Pantographic metamaterials show atypical poynting effect reversal. Mechanics Research Communications 89:6–10Google Scholar
  59. Niiranen J, Balobanov V, Kiendl J, Hosseini S (2017) Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Mathematics and Mechanics of Solids 24(1):312–335Google Scholar
  60. Pepe G, Carcaterra A, Giorgio I, Del Vescovo D (2016) Variational feedback control for a nonlinear beam under an earthquake excitation. Mathematics and Mechanics of Solids 21(10):1234–1246Google Scholar
  61. Piccardo G, Pagnini LC, Tubino F (2015a) Some research perspectives in galloping phenomena: critical conditions and post-critical behavior. Continuum Mechanics and Thermodynamics 27(1-2):261–285Google Scholar
  62. Piccardo G, Tubino F, Luongo A (2015b) A shear–shear torsional beam model for nonlinear aeroelastic analysis of tower buildings. Zeitschrift für angewandte Mathematik und Physik 66(4):1895–1913Google Scholar
  63. Placidi L, Rosi G, Giorgio I, Madeo A (2014) Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Mathematics and Mechanics of Solids 19(5):555–578Google Scholar
  64. Placidi L, Greco L, Bucci S, Turco E, Rizzi NL (2016) A second gradient formulation for a 2D fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5):114Google Scholar
  65. Placidi L, Andreaus U, Giorgio I (2017) Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model. Journal of Engineering Mathematics 103(1):1–21Google Scholar
  66. Ravari MRK, Kadkhodaei M (2015) A computationally effcient modeling approach for predicting mechanical behavior of cellular lattice structures. Journal of Materials Engineering and Performance 24(1):245–252Google Scholar
  67. Ravari MRK, Kadkhodaei M, Badrossamay M, Rezaei R (2014) Numerical investigation on mechanical properties of cellular lattice structures fabricated by fused deposition modeling. International Journal of Mechanical Sciences 88:154–161Google Scholar
  68. Ravari MRK, Kadkhodaei M, Ghaei A (2016) Effects of asymmetric material response on the mechanical behavior of porous shape memory alloys. Journal of Intelligent Material Systems and Structures 27(12):1687–1701Google Scholar
  69. Reda H, Rahali Y, Ganghoffer JF, Lakiss H (2016) Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models. Composite Structures 141:328–345Google Scholar
  70. Rezaei DAH, Kadkhodaei M, Nahvi H (2012) Analysis of nonlinear free vibration and damping of a clamped–clamped beam with embedded prestrained shape memory alloy wires. Journal of Intelligent Material Systems and Structures 23(10):1107–1117Google Scholar
  71. Romano G, Rosati L, Ferro G (1992) Shear deformability of thin-walled beams with arbitrary cross sections. International Journal for Numerical Methods in Engineering 35(2):283–306Google Scholar
  72. Scerrato D, Giorgio I, Rizzi NL (2016) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3):53Google Scholar
  73. Serpieri R, Rosati L (2014) A frame-independent solution to Saint-Venant’s flexure problem. Journal of Elasticity 116(2):161–187Google Scholar
  74. Spagnuolo M, Barcz K, Pfaff A, dell’Isola F, Franciosi P (2017) Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mechanics Research Communications 83:47–52Google Scholar
  75. Steigmann D, Faulkner M (1993) Variational theory for spatial rods. Journal of Elasticity 33(1):1–26Google Scholar
  76. Steigmann DJ (2017) Finite Elasticity Theory. Oxford University Press Google Scholar
  77. Taig G, Ranzi G, D’annibale F (2015) An unconstrained dynamic approach for the generalised beam theory. Continuum Mechanics and Thermodynamics 27(4-5):879–904Google Scholar
  78. Timoshenko SP (1921) Lxvi. on the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41(245):744–746Google Scholar
  79. Timoshenko SP (1922) X. on the transverse vibrations of bars of uniform cross-section. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43(253):125–131Google Scholar
  80. Turco E (2018) Discrete is it enough? the revival of Piola–Hencky keynotes to analyze threedimensional Elastica. Continuum Mechanics and Thermodynamics 30(5):1039–1057Google Scholar
  81. Turco E, Golaszewski M, Cazzani A, Rizzi NL (2016) Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete lagrangian model. Mechanics Research Communications 76:51–56Google Scholar
  82. Turco E, Golaszewski M, Giorgio I, D’Annibale F (2017) Pantographic lattices with nonorthogonal fibres: Experiments and their numerical simulations. Composites Part B: Engineering 118:1–14Google Scholar
  83. Turco E, Misra A, Sarikaya R, Lekszycki T (2019) Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Continuum Mechanics and Thermodynamics 31(1):209–223Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Francesco dell’Isola
    • 1
    • 2
    • 3
  • Alessandro Della Corte
    • 1
    • 2
  • Antonio Battista
    • 1
    • 4
  • Emilio Barchiesi
    • 1
    • 2
    Email author
  1. 1.International Research Center for the Mathematics and Mechanics of Complex Systems-M&MoCS, Università dell’AquilaL’AquilaItaly
  2. 2.Department of Structural and Geotechnical EngineeringUniversità di Roma La SapienzaRomeItaly
  3. 3.Research Institute for Mechanics, Nizhny Novgorod Lobachevsky State UniversityNizhny Novgorod,Russia
  4. 4.LaSIE, Université de La RochelleLa RochelleFrance

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