Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 79–99 | Cite as

Structural response of existing spatial truss roof construction based on Cosserat rod theory

  • Mikołaj MiśkiewiczEmail author
Open Access
Original Article


Paper presents the application of the Cosserat rod theory and newly developed associated finite elements code as the tools that support in the expert-designing engineering practice. Mechanical principles of the 3D spatially curved rods, dynamics (statics) laws, principle of virtual work are discussed. Corresponding FEM approach with interpolation and accumulation techniques of state variables are shown that enable the formulation of the \(C^{0}\) Lagrangian rod elements with 6-degrees of freedom per node. Two test examples are shown proving the correctness and suitability of the proposed formulation. Next, the developed FEM code is applied to assess the structural response of the spatial truss roof of the “Olivia” Sports Arena Gdansk, Poland. The numerical results are compared with load test results. It is shown that the proposed FEM approach yields correct results.


Cosserat rods Structural response SO(3) SHM 



The research reported in this paper was partially supported by the National Centre of Science of Poland with the Grant DEC—2012/05/D/ST8/02298.


  1. 1.
    Wilde, K., Miśkiewicz, M., Chróścielewski, J.: SHM system of the roof structure of sports Arena “Olivia”. In: Structural Health Monitoring, Tom II. Pennsylvania 17602, pp. 1745–1752. DEStech Publications Inc., USA (2013)Google Scholar
  2. 2.
    Miśkiewicz, M.: Nonlinear FEM Analysis and the String-Rod Structures Structural Health Monitoring (in Polish). Wydawnictwo Politechniki, Gdańskiej (2016)Google Scholar
  3. 3.
    Klikowicz, P., Salamak, M., Poprawa, G.: Structural health monitoring of urban structure. Procedia Eng. 161, 958–962 (2016)CrossRefGoogle Scholar
  4. 4.
    Miśkiewicz, M., Pyrzowski, Ł., Chróścielewski, J., Wilde, K.: Structural health monitoring of composite shell footbridge for its design validation. In: Baltic Geodetic Congress (Geomatics), vol. 2016, pp. 228–233 (2016).
  5. 5.
    Kaminski, W., Makowska, K., Miśkiewicz, M., Szulwic, J., Wilde, K.: System of monitoring of the Forest Opera in Sopot structure and roofing. In: 15th International Multidisciplinary Scientific GeoConference SGEM, Book 2, vol. 2, pp. 471–482 (2015).
  6. 6.
    Miśkiewicz M., Pyrzowski Ł., Wilde K., Mitrosz O.: Technical monitoring system for a new part of Gdańsk Deepwater Container Terminal. In: Polish Maritime Research Special Issue 2017 S1 (93) 2017, vol. 24, pp. 149–155 (2017).
  7. 7.
    Miśkiewicz M., Mitrosz O., Brzozowski, T.: Preliminary field tests and long-term monitoring as a method of design risk mitigation: a case study of Gdańsk Deepwater Container Terminal. In: Polish Maritime Research, No 3(95), vol. 24, pp. 106-114 (2017).
  8. 8.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67(4), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Greco, L., Cuomo, M., Contrafatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 324, 476–511 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions. The plane case: part I. ASME J. Appl. Mech. 53, 849–54 (1986)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions. The plane case: part II. ASME J. Appl. Mech. 53, 855–63 (1986)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II. Computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions. A geometrically exact approach. Comput. Methods. Appl. Mech. Eng. 66, 125–61 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Smoleński, W.M.: Statically and kinematically exact nonlinear theory of rods and its numerical verification. Comput. Methods. Appl. Mech. Eng. 178, 89–113 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Green, A.E., Laws, N.: A general theory of rods. Proc. R. Soc. Lond. A 293, 145–55 (1966)ADSCrossRefGoogle Scholar
  16. 16.
    Rubin, M.B.: Cosserat Theories: Shells, Rods, and Points. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Antman, S.S.: Nonlinear Problems of Elasticity. Series Applied Mathematical Sciences, vol. 107. Springer, New York, NY (1995)CrossRefGoogle Scholar
  18. 18.
    Sander, O., Schiela, A.: Variational analysis of the coupling between a geometrically exact Cosserat rod and an elastic continuum. Z. Angew. Math. Phys. 65, 1261–88 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: On a thermodynamic theory of rods with two temperature fields. Acta Mech. 223(8), 1583–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. B Eng. 43(3), 1315–28 (2012)CrossRefGoogle Scholar
  21. 21.
    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rod. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications, vol. 541, pp. 179–248. CISM International Centre for Mechanical Sciences, Udine (2013)CrossRefGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Alibert, J.-J., Della, C.A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–70 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218, 1239–1262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 20150790 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. B Eng. 118, 1–14 (2017)CrossRefGoogle Scholar
  28. 28.
    De Masi, A., Ferrari, P.A.: Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process. J. Stat. Phys. 107(3–4), 677–683 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angew. Math. Phys. 67, 114 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an under-estimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887–928 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Buttà, P., De Masi, A., Rosatelli, E.: Slow motion and metastability for a nonlocal evolution equation. J. Stat. Phys. 112(3–4), 709–764 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRefGoogle Scholar
  33. 33.
    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. Z. Angew. Math. Phys. 66(6), 3473–98 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.J.: Buckling modes in pantographic lattices. C. R. Méc. 344, 487–501 (2016)CrossRefGoogle Scholar
  35. 35.
    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, 171153 (2017). CrossRefGoogle Scholar
  36. 36.
    Campello, E.M.B., Pimenta, P.M., Wriggers, P.: An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells. Comput. Mech. 48, 195-21 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Chróścielewski, J., Witkowski, W.: Discrepancies of energy values in dynamics of three intersecting plates. Int. J. Numer. Methods Biomed. Eng. 26, 1188–1202 (2010)CrossRefzbMATHGoogle Scholar
  38. 38.
    Simmonds, J.G.: The nonlinear thermodynamical theory of shells: descent from 3-dimensions without thickness expansions. In: Axelrad, E.L., Emmerling, F.A. (eds.) Flexible Shells, Theory and Applications, pp. 1–11. Springer, Berlin (1984)Google Scholar
  39. 39.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  40. 40.
    Chróścielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int. J. Numer. Methods Eng. 35, 63–94 (1992)CrossRefzbMATHGoogle Scholar
  41. 41.
    Chróścielewski, J., Witkowski, W.: Four-node semi-EAS element in six-field nonlinear theory of shells. J. Numer. Methods Eng. 68, 1137–79 (2006)CrossRefzbMATHGoogle Scholar
  42. 42.
    Eremeyev, V., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–52 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pietraszkiewicz, W., Eremeyev, V.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46, 774–87 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Konopińska, V., Pietraszkiewicz, W.: Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. Int. J. Solids Struct. 44, 352–69 (2007)CrossRefzbMATHGoogle Scholar
  45. 45.
    Eremeyev, A.V., Lebedev, L.P., Cloud, M.J.: The Rayleigh and Courant variational principles in the six-parameter shell theory. Math. Mech. Solids 20(7), 806–822 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Cardona, A., Geradin, M.: Flexible Multibody Dynamics. A Finite Element Approach. Wiley, Chichester (2001)Google Scholar
  47. 47.
    Chróścielewski, J.: Family of C\(^{0}\) finite elements in six parameter nonlinear theory of shells (in Polish). In: Proceeding of Gdansk Technical University, Civil Eng series 540(LIII), pp. 1–291 (1996)Google Scholar
  48. 48.
    Crisfield, M.A.G.: Objectivity of strain measures in geometrically exact 3D beam theory and its finite element implementation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455, 1125–47 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Chróścielewski, J., Kreja, I., Sabik, A., Witkowski, W.: Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mech. Adv. Mater. Struct. 18, 403–19 (2011)CrossRefGoogle Scholar
  50. 50.
    Ibrahimbegović, A.: On the choice of finite rotation parameters. Comp. Methods Appl. Mech. Eng. 149, 49–71 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Pietraszkiewicz, W., Badur, J.: Finite rotations in the description of continuum deformation. Int. J. Eng. Sci. 21, 1097–115 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Argyris, J.H., Dunne, P.C., Malejannakis, G., Scharpf, D.W.: On large dis-placement-small strain analysis of structures with rotational degrees of freedom. Comput. Methods Appl. Mech. Eng. 15, 99–135 (1978)ADSCrossRefzbMATHGoogle Scholar
  53. 53.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs (1982)Google Scholar
  54. 54.
    Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory. Comput. Methods Appl. Mech. Eng. 79, 21–70 (1990)ADSCrossRefzbMATHGoogle Scholar
  55. 55.
    Wagner, W., Gruttmann, F.: A robust non-linear mixed hybrid quadrilateral shell element. Int. J. Numer. Methods Eng. 64, 635–66 (2005)CrossRefzbMATHGoogle Scholar
  56. 56.
    Witkowski, W.: 4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom. Comput. Mech. 43, 307–19 (2009)CrossRefzbMATHGoogle Scholar
  57. 57.
    Bathe, K.J., Bolourchi, S.: Large displacement analysis of three-dimensional beam structures. Int. J. Numer. Methods Eng. 14, 961–86 (1979)CrossRefzbMATHGoogle Scholar
  58. 58.
    Cardona, A., Geradin, M.: A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Eng. 26, 2403–38 (1988)CrossRefzbMATHGoogle Scholar
  59. 59.
    Crisfield, M.A.: A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements. Comput. Methods Appl. Mech. Eng. 81, 131–50 (1990)ADSCrossRefzbMATHGoogle Scholar
  60. 60.
    Crivelli, L.A., Felippa, C.A.: A three-dimensional non-linear Timoshenko beam based on the core-congruential formulation. Int. J. Numer. Methods Eng. 36, 3647–73 (1993)CrossRefzbMATHGoogle Scholar
  61. 61.
    SOFiSTiK. Oberschleißheim: SOFiSTiK AG (2006)Google Scholar
  62. 62.
    Chróścielewski, J., Miśkiewicz, M., Pyrzowski, Ł.: The Introduction to FEM Analysis in SOFiSTiK ( in Polish). Wydawnictwo Politechniki Gdańskiej, Gdańsk (2016)Google Scholar
  63. 63.
    Mariak, A., Miśkiewicz, M., Meronk, B., Pyrzowski, Ł., Wilde, K.: Reference FEM model for SHM system of cable-stayed bridge in Rzeszów. In: Kleiber, et al. (eds.) Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues, pp. 383–387. Taylor & Francis Group, London (2016). CrossRefGoogle Scholar
  64. 64.
    Miśkiewicz, M., Pyrzowski, Ł., Wilde, K.: Structural health monitoring system for suspension footbridge. In: Baltic Geodetic Congress (BGC Geomatics), vol. 2017, pp. 321–325 (2017).
  65. 65.
    Hou, J., Jankowski, L., Ou, J.: An online substructure identification method for local structural health monitoring. Smart Mater. Struct. 22(9), article id 095017. (2013)

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdańskPoland

Personalised recommendations