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Discrete-Mindlin finite element for nonlinear geometrical analysis of shell structures

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Abstract

A finite degenerated shell element based on Mindlin discrete approach is presented for nonlinear geometric analysis in large displacements and small deformations. The element, called DMQS (Discrete Mindlin Quadrilateral Shell) with four-nodes and 6 dof/node, includes a constant transverse displacement and rich quadratic rotations. Since the global kinematics of shell is dominated by a rigid body rotation and motion involving small displacements and rotations, an Updated Lagrangian Formulation at each Iteration is used. The tangent stiffness matrix has been treated as a sum of two parts: linear part related to the degenerated shell model (DMQS) as well as the initial stress part associated with the well-known membrane standard element. Some standard nonlinear benchmarks are presented herein, where very good results have been obtained and show better accuracy than similar quadrilateral elements.

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References

  • Ahmad S, Irons BM, Zienkiewicz OC (1970) Analysis of thick and thin shell structures by curved finite elements. Int J Numer Methods Eng 2(3):419–451

    Article  Google Scholar 

  • Altenbach H, Eremeyev V (2014) Actual developments in the nonlinear shell theory-state of the art and new applications of the six-parameter shell theory. Shell Struct Theory Appl 3:3–12

    Google Scholar 

  • Ammar S, Dhatt G, Fafard M (1996) Exact stability model of space frames. Comput Struct 60(1):59–71

    Article  MATH  Google Scholar 

  • Ayad R (2002) Contribution to the numerical modeling for the analysis of solids and structures, and for the forming of non-newtonian fluids. application to packaging materials. HDR Thesis, University of Reims Champagne-Ardenne (in French)

  • Bathe KJ (1982) Finite Element Procedures in Engineering Analysis. In: Prentice-Hall civil engineering and engineering mechanics. Prentice-Hall, Englewood Cliffs

  • Bathe KJ, Bolourchi S (1980) A geometric and material nonlinear plate and shell element. Comput Struct 11(1):23–48

    Article  MATH  Google Scholar 

  • Bathe KJ, Dvorkin EN (1985) A four-node plate bending element based on mindlin/reissner plate theory and a mixed interpolation. Int J Numer Methods Eng 21(2):367–383

    Article  MATH  Google Scholar 

  • Bathe KJ, Dvorkin EN (1986) A formulation of general shell elementsthe use of mixed interpolation of tensorial components. Int J Numer Methods Eng 22(3):697–722

    Article  MATH  Google Scholar 

  • Batoz JL, Cantin G (1985) Geometrically nonlinear analysis of shell structures using flat DKT shell elements. Progress Report, 1 October 1984-30 September 1985, Naval Postgraduate School, Monterey, California

  • Batoz JL, Dhatt G (1986) Plate and shell finite elements: linear and nonlinear analysis: [conférences] 2–4 juin 1986. Inst Adv Eng Sci (in French)

  • Batoz JL, Jaamei S (1987) Study of different Lagrangian formulations for analysis of beams and thin elastic shells in large rotations. In: Tech. rep., University of Technology of Compiègne, France (in French)

  • Batoz JL, Guo Y, Shakourzadeh H (1998) Nonlinear analysis of elasto-plastic thin shells with DKT12 element. Eur J Comput Mech 7(1–3):223–239 (in French)

    MATH  Google Scholar 

  • Belytschko T, Wong BL, Stolarski H (1989) Assumed strain stabilization procedure for the 9-node Lagrange shell element. Int J Numer Methods Eng 28(2):385–414

    Article  MATH  Google Scholar 

  • Bognet B, Bordeu F, Chinesta F, Leygue A, Poitou A (2012) Advanced simulation of models defined in plate geometries: 3d solutions with 2d computational complexity. Comput Methods Appl Mech Eng 201:1–12

    Article  MATH  Google Scholar 

  • Bognet B, Leygue A, Chinesta F (2014) Separated representations of 3d elastic solutions in shell geometries. Adv Model Simul Eng Sci 1(1):1–34

    Article  MATH  Google Scholar 

  • Brendel B, Ramm E (1980) Linear and nonlinear stability analysis of cylindrical shells. Comput Struct 12(4):549–558

    Article  MATH  Google Scholar 

  • Bucalem ML, Bathe KJ (1993) Higher-order MITC general shell elements. Int J Numer Methods Eng 36(21):3729–3754

    Article  MathSciNet  MATH  Google Scholar 

  • Bucalem M, Bathe K (1997) Finite element analysis of shell structures. Arch Comput Methods Eng 4(1):3–61

    Article  MathSciNet  Google Scholar 

  • Buechter N, Ramm E (1992) Shell theory versus degeneration-a comparison in large rotation finite element analysis. Int J Numer Methods Eng 34(1):39–59

    Article  MATH  Google Scholar 

  • Chapelle D, Bathe K (1998) Fundamental considerations for the finite element analysis of shell structures. Comput Struct 66(1):19–36

    Article  MATH  Google Scholar 

  • Cook R, Malkus D, Plesha M (1989) Concepts and applications of finite element analysis. Wiley, New York

    MATH  Google Scholar 

  • Crisfield A (1986) Finite elements and solution procedures for structural analysis V. 1: linear analysis. Pineridge Press, Swansea

  • Dvorkin EN, Bathe KJ (1984) A continuum mechanics based four-node shell element for general non-linear analysis. Eng Comput 1(1):77–88

    Article  Google Scholar 

  • Fafard M (1987) Automatic computation of pre- and post-buckling configurations in nonlinear analysis of structures. Ph.D. thesis, University of Laval, Ottawa (in French)

  • Fafard M, Dhatt G, Batoz J (1989) A new discrete kirchhoff plate/shell element with updated procedures. Comput Struct 31(4):591–606

    Article  Google Scholar 

  • Gal E, Levy R (2006) Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element. Arch Comput Methods Eng 13(3):331–388

    Article  MATH  Google Scholar 

  • Hammadi F (1998) Formulation and evaluation of finite elements with \(C ^0\) geometrical continuity for linear and nonlinear analysis of shells. Ph.D. thesis, University of Technology of Compiègne

  • Huang H, Hinton E (1986) A new nine node degenerated shell element with enhanced membrane and shear interpolation. Int J Numer Methods Eng 22(1):73–92

    Article  MathSciNet  MATH  Google Scholar 

  • Hughes T (2000) The finite element method: linear static and dynamic finite element analysis. In: Dover civil and mechanical engineering. Dover Publications, New York

  • Irons BM (1976) The semiloof shell element. In: Ashwell DG, Gallagher RH (eds) Finite elements for thin shells and curved members. Wiley, London, pp 97–222

  • Jaamei S (1986) Study of different Lagrangian formulations for nonlinear analysis of elasto-plastic thin plates and shells in large displacements and large rotations. Ph.D. thesis, University of Technologie of Compiègne (in French)

  • Katili I (1993) A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fieldspart ii: an extended DKQ element for thick-plate bending analysis. Int J Numer Methods Eng 36(11):1885–1908

    Article  MATH  Google Scholar 

  • Kratzig W (1993) Best transverse shearing and stretching shell theory for nonlinear finite element simulations. Comput Methods Appl Mech Eng 103(12):135–160

    Article  MATH  Google Scholar 

  • Liu WK, Law E, Lam D, Belytschko T (1986) Resultant-stress degenerated-shell element. Comput Methods Appl Mech Eng 55(3):259–300

    Article  MATH  Google Scholar 

  • Milford R, Schnobrich W (1986) Degenerated isoparametric finite elements using explicit integration. Int J Numer Methods Eng 23(1):133–154

    Article  MathSciNet  MATH  Google Scholar 

  • Oliver J, Onate E (1984) A total Lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. part i. two-dimensional problems: shell and plate structures. Int J Numer Methods Eng 20(12):2253–2281

    Article  MATH  Google Scholar 

  • Onate E, Hinton E, Glover N (1978) Techniques for improving the performance of Ahmad shell elements. In: Proceedings of the international conference on applied numerical modelling, Madrid. Pentech Press, London

  • Parisch H (1978) Geometrical nonlinear analysis of shells. Comput Methods Appl Mech Eng 14(2):159–178

    Article  MATH  Google Scholar 

  • Pietraszkiewicz W (1984) Lagrangian description and incremental formulation in the non-linear theory of thin shells. Int J Non-linear Mech 19(2):115–140

    Article  MATH  Google Scholar 

  • Pol P (1992) Modeling of elasoplastic behavior of thin shells with finite elements. Ph.D. thesis, University of Technologie of Compiègne

  • Ramm E (1977) A plate/shell element for large deflections and rotations. In: US-Germany symposium on formulations and computational algorithms in finite element analysis. MIT-Press, Cambridge pp 264–293

  • Roelandt JM, Batoz JL (1992) Shell finite element for deep drawing problems: computational aspects and results. In: Besdo D, Stein E (eds) Finite inelastic deformations theory and applications, International union of theoretical and applied mechanics, Springer, Berlin, pp 423–430

  • Sakami S (2008) Numerical modeling of multilayer composite structures using a discrete approach within the meaninig of Mindlin. The DDM model ( Discrete Mindlin Displacement). Ph.D. thesis, Universit of Reims Champagne- Ardenne (in French)

  • Sakami S, Sabhi H, Ayad R, Talbi N (2008) Formulation and evaluation of a finite element model, discrete within the meaning of Mindlin for the analysis of isotropic structures. Eur J Comput Mech 17(4):529–552

    MATH  Google Scholar 

  • Sedira L (2013) Contribution to the modeling of 2D/3D composites using special finite elements. Ph.D. thesis, Universities of Biskra and Reims Champagne-Ardenne (in French)

  • Sedira L, Ayad R, Sabhi H, Hecini M, Sakami S (2012) An enhanced discrete Mindlin finite element model using a zigzag function. Eur J Comput Mech 21(1–2):122–140

    Google Scholar 

  • Stanley G (1985) Continuum-based shell elements. Ph.D. thesis, Applied mechanics division. Stanford University, CA

  • Stolarski H, Belytschko T (1983) Shear and membrane locking in curved \( {C}^0\) elements. Comput Methods Appl Mech Eng 41(3):279–296

    Article  MATH  Google Scholar 

  • Surana KS (1983) Geometrically nonlinear formulation for the curved shell elements. Int J Numer Methods Eng 19(4):581–615

    Article  MathSciNet  MATH  Google Scholar 

  • Sze K, Liu X, Lo S (2004) Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des. 40(11):1551–1569

    Article  Google Scholar 

  • Sze K, Zheng SJ (1999) A hybrid stress nine-node degenerated shell element for geometric nonlinear analysis. Comput Mech 23(5–6):448–456

    Article  MATH  Google Scholar 

  • Wall WA, Gee M, Ramm E (2000) The challenge of a three-dimensional shell formulation-the conditioning problem. In: Proceedings of IASS-IACM 2000-fourth international colloquium on computation for shells & spatial structures, Chania-Crete, Greece

  • Yang HT, Saigal S, Liaw D (1990) Advances of thin shell finite elements and some applications–version i. Comput Struct 35(4):481–504

    Article  MATH  Google Scholar 

  • Yang HT, Saigal S, Masud A, Kapania R (2000) A survey of recent shell finite elements. Int J Numer Methods Eng 47(1–3):101–127

    Article  MathSciNet  MATH  Google Scholar 

  • Zienkiewicz OC, Taylor R, Too J (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng 3(2):275–290

    Article  MATH  Google Scholar 

  • Zienkiewicz OC, Taylor RL (2000) The finite element method: solid mechanics, vol 2. Butterworth-heinemann, London

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratory of Mechanical Engineering (LGM), University of Biskra, rue Sidi-Okba, 07000, Biskra, Algeria

    Lakhdar Sedira & Mabrouk Hecini

  2. Laboratory of Mechanics: Modeling and Experimentation (L2ME), University of Bechar, BP 417, 08000, Bechar, Algeria

    Fodil Hammadi

  3. LISM, EA 4695, University of Reims Champagne-Ardenne IUT de Troyes, 9 rue de Québec, 10026, Troyes, France

    Rezak Ayad

  4. Laboratoire de Génie Energétique et Matériaux (LGEM), University of Biskra, 07000, Biskra, Algeria

    Kamel Meftah

Authors
  1. Lakhdar Sedira
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  2. Fodil Hammadi
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  3. Rezak Ayad
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  4. Kamel Meftah
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  5. Mabrouk Hecini
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Correspondence to Lakhdar Sedira.

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Communicated by Taoufik BOUKHAROUBA.

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Sedira, L., Hammadi, F., Ayad, R. et al. Discrete-Mindlin finite element for nonlinear geometrical analysis of shell structures. Comp. Appl. Math. 35, 951–975 (2016). https://doi.org/10.1007/s40314-015-0279-3

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  • DOI: https://doi.org/10.1007/s40314-015-0279-3

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