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A three-node shell element based on the discrete shear gap and assumed natural deviatoric strain approaches

  • Gil Rama
  • Dragan Marinkovic
  • Manfred Zehn
Technical Paper

Abstract

Thin-walled structures are of enormous importance in the structural engineering world. Their successful design calls for numerically efficient, accurate and reliable numerical tools. A new three-node shell element with six degrees of freedom per node—three translations and three rotations—is presented in this paper. The discrete shear gap approach together with the cell smoothing technique is implemented for treatment of shear locking. The membrane behavior is resolved by means of the assumed natural deviatoric strains formulation with certain adjustments implemented to accommodate for shell behavior. Examples are given to demonstrate the applicability of the proposed element for modeling shell structures. The accuracy and convergence rate are tested on a chosen set of well-known challenging benchmark problems, and the results are compared with those yielded by the Abaqus S3 element.

Keywords

Triangular shell element Discrete shear gap Strain cell smoothing Assumed natural deviatoric strains Drilling degree of freedom 

References

  1. 1.
    Ahmad S, Irons BM, Zienkiewicz O (1970) Analysis of thick and thin shell structures by curved finite elements. Int J Numer Methods Eng 2(3):419–451CrossRefGoogle Scholar
  2. 2.
    Alvin K, Horacio M, Haugen B, Felippa CA (1992) Membrane triangles with corner drilling freedoms—I. The EFF element. Finite Elem Anal Des 12(3–4):163–187CrossRefMATHGoogle Scholar
  3. 3.
    Belytschko T, Stolarski H, Liu WK, Carpenter N, Ong JS (1985) Stress projection for membrane and shear locking in shell finite elements. Comput Methods Appl Mech Eng 51(1–3):221–258MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benjeddou A (2000) Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput Struct 76(1–3):347–363CrossRefGoogle Scholar
  5. 5.
    Benson D, Bazilevs Y, Hsu MC, Hughes T (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199(5–8):276–289MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bergan P, Nygård M (1984) Finite elements with increased freedom in choosing shape functions. Int J Numer Methods Eng 20(4):643–663CrossRefMATHGoogle Scholar
  7. 7.
    Berthelot JM (2012) Composite materials: mechanical behavior and structural analysis. Springer, BerlinGoogle Scholar
  8. 8.
    Bletzinger KU, Bischoff M, Ramm E (2000) A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput Struct 75(3):321–334CrossRefGoogle Scholar
  9. 9.
    Boukhari A, Atmane HA, Tounsi A, Adda B, Mahmoud S et al (2016) An efficient shear deformation theory for wave propagation of functionally graded material plates. Struct Eng Mech 57(5):837–859CrossRefGoogle Scholar
  10. 10.
    Carrera E, Pagani A, Valvano S (2017) Multilayered plate elements accounting for refined theories and node-dependent kinematics. Compos Part B Eng 114:189–210CrossRefGoogle Scholar
  11. 11.
    Cook RD (1993) Further development of a three-node triangular shell element. Int J Numer Methods Eng 36(8):1413–1425CrossRefMATHGoogle Scholar
  12. 12.
    Cook RD, Malkus DS, Plesha ME, Witt RJ (1974) Concepts and applications of finite element analysis, vol 4. Wiley, New YorkGoogle Scholar
  13. 13.
    Cui X, Liu GR, Gy Li, Zhang G, Zheng G (2010) Analysis of plates and shells using an edge-based smoothed finite element method. Comput Mech 45(2–3):141MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dvorkin EN, Bathe KJ (1984) A continuum mechanics based four-node shell element for general non-linear analysis. Eng Comput 1(1):77–88CrossRefGoogle Scholar
  15. 15.
    Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Felippa C, Militello C (1992) Membrane triangles with corner drilling freedoms—II. the ANDES element. Finite Elem Anal Des 12(3–4):189–201CrossRefMATHGoogle Scholar
  17. 17.
    Felippa CA (2003) A study of optimal membrane triangles with drilling freedoms. Comput Methods Appl Mech Eng 192(16):2125–2168CrossRefMATHGoogle Scholar
  18. 18.
    Felippa CA, Alexander S (1992) Membrane triangles with corner drilling freedoms—III. Implementation and performance evaluation. Finite Elem Anal Des 12(3–4):203–239CrossRefMATHGoogle Scholar
  19. 19.
    Filippi M, Carrera E (2016) Bending and vibrations analyses of laminated beams by using a zig-zag-layer-wise theory. Compos Part B Eng 98:269–280CrossRefGoogle Scholar
  20. 20.
    Hassen AA, Taheri H, Vaidya UK (2016) Non-destructive investigation of thermoplastic reinforced composites. Compos Part B Eng 97:244–254CrossRefGoogle Scholar
  21. 21.
    Iron BM (1976) The semiloof shell element. In: Gallagher RH, Ashwel DG (eds) Finite elements for thin shells and curved members, chap 11. Wiley, New York, pp 197–222Google Scholar
  22. 22.
    Jayasankar S, Mahesh S, Narayanan S, Padmanabhan C (2007) Dynamic analysis of layered composite shells using nine node degenerate shell elements. J Sound Vib 299(1–2):1–11CrossRefGoogle Scholar
  23. 23.
    Klinkel S, Gruttmann F, Wagner W (2006) A robust non-linear solid shell element based on a mixed variational formulation. Comput Methods Appl Mech Eng 195(1–3):179–201CrossRefMATHGoogle Scholar
  24. 24.
    Knight N (1997) Raasch challenge for shell elements. AIAA J 35(2):375–381CrossRefMATHGoogle Scholar
  25. 25.
    Kulikov G, Plotnikova S, Carrera E (2017) A robust, four-node, quadrilateral element for stress analysis of functionally graded plates through higher-order theories. Mech Advan Mater Struct.  https://doi.org/10.1080/15376494.2017.1288994 Google Scholar
  26. 26.
    Kulikov GM, Plotnikova SV, Carrera E (2018) Hybrid-mixed solid-shell element for stress analysis of laminated piezoelectric shells through higher-order theories. In: Altenbach H, Carrera E, Kulikov G (eds.) Advanced structured materials: analysis and modelling of advanced structures and smart systems. Springer, Berlin, pp 45–68CrossRefGoogle Scholar
  27. 27.
    Li D (2016) Extended layerwise method of laminated composite shells. Compos Struct 136:313–344CrossRefGoogle Scholar
  28. 28.
    Li G, Xu F, Sun G, Li Q (2015) Crashworthiness study on functionally graded thin-walled structures. Int J Crashworthiness 20(3):280–300CrossRefGoogle Scholar
  29. 29.
    Liu GR, Trung NT (2016) Smoothed finite element methods. CRC Press, Boca RatonGoogle Scholar
  30. 30.
    Macneal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1(1):3–20CrossRefGoogle Scholar
  31. 31.
    Marinković D, Köppe H, Gabbert U (2006) Numerically efficient finite element formulation for modeling active composite laminates. Mech Adv Mater Struct 13(5):379–392CrossRefGoogle Scholar
  32. 32.
    Marinković D, Rama G (2017) Co-rotational shell element for numerical analysis of laminated piezoelectric composite structures. Compos Part B Eng 125:144–156CrossRefGoogle Scholar
  33. 33.
    Marinkovic D, Zehn M, Marinkovic Z (2012) Finite element formulations for effective computations of geometrically nonlinear deformations. Adv Eng Softw 50:3–11CrossRefGoogle Scholar
  34. 34.
    Milić P, Marinković D (2015) Isogeometric fe analysis of complex thin-walled structures. Trans FAMENA 39(1):15–26Google Scholar
  35. 35.
    Nguyen VA, Zehn M, Marinković D (2016) An efficient co-rotational fem formulation using a projector matrix. Facta Univ Ser Mech Eng 14(2):227–240Google Scholar
  36. 36.
    Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Xuan H, Ngo-Thanh P (2013) Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements. Comput Struct 125:100–113CrossRefGoogle Scholar
  37. 37.
    Parand AA, Alibeigloo A (2017) Static and vibration analysis of sandwich cylindrical shell with functionally graded core and viscoelastic interface using DQM. Compos Part B Eng 126:1–16CrossRefGoogle Scholar
  38. 38.
    Park K, Stanley G (1988) Strain interpolations for a 4-node ANS shell element. In: Atluri SN, Yagawa G (eds) Computational mechanics, vol 88. Springer, Berlin, pp 747–750Google Scholar
  39. 39.
    Rademacher T, Zehn M (2016) Modal triggered nonlinearities for damage localization in thin walled frc structures—a numerical study. Facta Univ Ser Mech Eng 14(1):21–36Google Scholar
  40. 40.
    Rama G (2017) A 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis of smart structures. Facta Univ Ser Mech Eng 15(1):31–44MathSciNetCrossRefGoogle Scholar
  41. 41.
    Rama G, Marinković D, Zehn M (2017) Efficient three-node finite shell element for linear and geometrically nonlinear analyses of piezoelectric laminated structures. J Intell Mater Syst Struct.  https://doi.org/10.1177/1045389X17705538
  42. 42.
    Rama G, Marinkovic DZ, Zehn MW (2017) Linear shell elements for active piezoelectric laminates. Smart Struct Syst 20(6):729–737Google Scholar
  43. 43.
    Robert Winkler DP (2016) A new shell finite element with drilling degrees of freedom and its relation to existing formulations. In: ECCOMAS Congress 2016. VII European congress on computational methods in applied sciences and engineering. Crete Island, pp 5–10Google Scholar
  44. 44.
    Rohwer K (2016) Models for intralaminar damage and failure of fiber composites—a review. Facta Univ Ser Mech Eng 14(1):1–19MathSciNetGoogle Scholar
  45. 45.
    Shin CM, Lee BC (2014) Development of a strain-smoothed three-node triangular flat shell element with drilling degrees of freedom. Finite Elem Anal Des 86:71–80MathSciNetCrossRefGoogle Scholar
  46. 46.
    Simo J, Fox D, Rifai M (1989) On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects. Comput Methods Appl Mech Eng 73(1):53–92CrossRefMATHGoogle Scholar
  47. 47.
    Valvano S, Carrera E (2017) Multilayered plate elements with node-dependent kinematics for the analysis of composite and sandwich structures. Facta Univ Ser Mech Eng 15(1):1–30CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.BerlinGermany

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