[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Formulation and evaluation of a new four-node quadrilateral element for analysis of the shell structures

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Shell structures are lightweight constructions which are extensively used by engineering. Due to this reason presenting an appropriate shell element for analysis of these structures has become an interesting issue in recent decades. This study presents a new rectangular flat shell element called ACM-SQ4 obtained by combining bending and membrane elements. The bending element is a well-known plate bending element called ACM which is based on the classical thin-plate theory and the membrane element is an unsymmetric quadrilateral element called US-Q4θ, the test function of this element is improved by the Allman-type drilling DOFs and a rational stress field is used as the element’s trial function. Finally, some numerical benchmark problems are used to evaluate the performance of the proposed flat shell element. The obtained results show that despite its simple formulation, the proposed element has reasonable accuracy and acceptable convergence in comparison with other shell elements.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Chapelle D, Bathe KJ (2010) The finite element analysis of shells-fundamentals, 2nd edn. Springer, Science & Business Media, New York

    MATH  Google Scholar 

  2. Carrera E, Petrolo M (2012) Refined beam elements with only displacement variables and plate/shell capabilities. Meccanica 47:537–556

    MathSciNet  MATH  Google Scholar 

  3. Nguyen-Hoang S, Phung-Van P, Natarajan S, Kim HG (2016) A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells. Eng Comput 32:267–284

    Google Scholar 

  4. Hernández E, Spa C, Surriba S (2018) A non-standard finite element method for dynamical behavior of cylindrical classical shell model. Meccanica 53:1037–1048

    MathSciNet  MATH  Google Scholar 

  5. Yuqi L, Jincheng W, Ping H (2002) A finite element analysis of the flange earrings of strong anisotropic sheet metals in deep-drawing processes. Acta Mech Sin 18:82–91

    Google Scholar 

  6. Zienkiewicz OC, Taylor RL (1977) The finite element method, vol 36. McGraw-Hill, London

    Google Scholar 

  7. Stolarski H, Belytschko T, Carpenter N, Kennedy JM (1984) A simple triangular curved shell element. Eng Comput 1:210–218

    Google Scholar 

  8. Surana KS (1982) Geometrically nonlinear formulation for the axisymmetric shell elements. Int J Numer Methods Eng 18:477–502

    MathSciNet  MATH  Google Scholar 

  9. Li LM, Li DY, Peng YH (2011) The simulation of sheet metal forming processes via integrating solid-shell element with explicit finite element method. Eng Comput 27:273–284

    Google Scholar 

  10. Gallagher RH (1976) Problems and progresses in thin shell finite element analysis. In: Ashwell DG, Gallagher RH (eds) Finite element for thin shells and curved members. Wiley, New York

    Google Scholar 

  11. Batoz JL, Hammadi F, Zheng C, Zhong W (2000) On the linear analysis of plates and shells using a new-16 degrees of freedom flat shell element. Comput Struct 78:11–20

    Google Scholar 

  12. Batoz JL, Zheng CL, Hammadi F (2001) Formulation and evaluation of new triangular, quadrilateral, pentagonal and hexagonal discrete Kirchhoff plate/shell elements. Int J Numer Methods Eng 52:615–630

    MathSciNet  MATH  Google Scholar 

  13. Zengjie G, Wanji C (2003) Refined triangular discrete Mindlin flat shell elements. Comput Mech 33:52–60

    MATH  Google Scholar 

  14. Sabourin F, Carbonniere J, Brunet M (2009) A new quadrilateral shell element using 16 degrees of freedom. Eng Comput 26:500–540

    MATH  Google Scholar 

  15. Wang Z, Sun Q (2014) Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness. Acta Mech Sin 30:418–429

    MathSciNet  MATH  Google Scholar 

  16. Zhang Y, Zhou H, Li J, Feng W, Li D (2011) A 3-node flat triangular shell element with corner drilling freedoms and transverse shear correction. Int J Numer Meth Eng 86:1413–1434

    MathSciNet  MATH  Google Scholar 

  17. Hamadi D, Ayoub A, Abdelhafid O (2015) A new flat shell finite element for the linear analysis of thin shell structures. Eur J Comput Mech 24:232–255

    Google Scholar 

  18. Shang Y, Cen S, Li CF (2016) A 4-node quadrilateral flat shell element formulated by the shape-free HDF plate and HSF membrane elements. Eng Comput 33:713–741

    Google Scholar 

  19. Allman DJ (1984) A compatible triangular element including vertex rotations for plane elasticity analysis. Comput Struct 19:1–8

    MATH  Google Scholar 

  20. Providas E, Kattis MA (2000) An assessment of two fundamental flat triangular shell elements with drilling rotations. Comput Struct 77:129–139

    Google Scholar 

  21. Pimpinelli G (2004) An assumed strain quadrilateral element with drilling degrees of freedom. Finite Elem Anal Des 41:267–283

    Google Scholar 

  22. Madeo A, Zagari G, Casciaro R (2012) An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes. Finite Elem Anal Des 50:21–32

    Google Scholar 

  23. Choi N, Choo YS, Lee BC (2006) A hybrid Trefftz plane elasticity element with drilling degrees of freedom. Comput Methods Appl Mech Eng 195:4095–4105

    MATH  Google Scholar 

  24. Rojas F, Anderson JC, Massone LM (2016) A nonlinear quadrilateral layered membrane element with drilling degrees of freedom for the modeling of reinforced concrete walls. Eng Struct 124:521–538

    Google Scholar 

  25. Nestorović T, Marinković D, Shabadi S, Trajkov M (2014) User defined finite element for modeling and analysis of active piezoelectric shell structures. Meccanica 49:1763–1774

    MathSciNet  MATH  Google Scholar 

  26. Areias P, de Sá JC, Cardoso R (2015) A simple assumed-strain quadrilateral shell element for finite strains and fracture. Eng Comput 31:691–709

    Google Scholar 

  27. Kim KD, Liu GZ, Han SC (2005) A resultant 8-node solid-shell element for geometrically nonlinear analysis. Comput Mech 35:315–331

    MATH  Google Scholar 

  28. Li ZX, Izzuddin BA, Vu-Quoc L (2008) A 9-node co-rotational quadrilateral shell element. Comput Mech 42:873

    MATH  Google Scholar 

  29. Li Z, Xiang Y, Izzuddin BA, Vu-Quoc L, Zhuo X, Zhang C (2015) A 6-node co-rotational triangular elasto-plastic shell element. Comput Mech 55:837–859

    MathSciNet  MATH  Google Scholar 

  30. Shang Y, Ouyang W (2018) 4-node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh-distortion. Int J Numer Methods Eng 113:1589–1606

    MathSciNet  Google Scholar 

  31. Adini A, Clough RW (1961) Analysis of plate bending by the finite element method. Report to the National Science Foundation, G 7337, Arlington

  32. Cook RD, Malkus DS, Plesha ME, Witt RJ (1974) Concepts and applications of finite element analysis, vol 4. Wiley, New York

    Google Scholar 

  33. Timoshenko SP, Woinowsky-Krieger S (1969) Theory of plates and shells, 2nd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  34. Melosh RJ (1963) Basis for derivation of matrices for the direct stiffness method. AIAA J 1:1631–1637

    Google Scholar 

  35. Lee Y, Lee PS, Bathe KJ (2014) The MITC3+ shell element and its performance. Comput Struct 138:12–23

    Google Scholar 

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

    Google Scholar 

  37. Ko Y, Lee PS, Bathe KJ (2017) A new 4-node MITC element for analysis of two-dimensional solids and its formulation in a shell element. Comput Struct 192:34–49

    Google Scholar 

  38. Ibrahimbegović A, Frey F (1994) Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: linearized kinematics. Int J Numer Methods Eng 37:3659–3683

    MATH  Google Scholar 

  39. Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects. Comput Methods Appl Mech Eng 73:53–92

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  41. Belytschko T, Leviathan I (1994) Physical stabilization of the 4-node shell element with one point quadrature. Comput Methods Appl Mech Eng 113:321–350

    MATH  Google Scholar 

  42. Alves de Sousa RJ, Cardoso RP, Fontes Valente RA, Yoon JW, Grácio JJ, Natal Jorge RM (2005) A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: part I—geometrically linear applications. Int J Numer Meth Eng 62:952–977

    MATH  Google Scholar 

  43. Wang C, Hu P, Xia Y (2012) A 4-node quasi-conforming Reissner–Mindlin shell element by using Timoshenko’s beam function. Finite Elem Anal Des 61:12–22

    MathSciNet  Google Scholar 

  44. Norachan P, Suthasupradit S, Kim KD (2012) A co-rotational 8-node degenerated thin-walled element with assumed natural strain and enhanced assumed strain. Finite Elem Anal Des 50:70–85

    MathSciNet  Google Scholar 

  45. Alves de Sousa RJ, Natal Jorge RM, Fontes Valente RA, César de Sá JMA (2003) A new volumetric and shear locking-free 3D enhanced strain element. Eng Comput 20:896–925

    MATH  Google Scholar 

  46. César de Sá JM, Natal Jorge RM, Fontes Valente RA, Almeida Areias PM (2002) Development of shear locking-free shell elements using an enhanced assumed strain formulation. Int J Numer Methods Eng 53:1721–1750

    MATH  Google Scholar 

  47. Argyris JH, Papadrakakis M, Apostolopoulou C, Koutsourelakis S (2000) The TRIC shell element: theoretical and numerical investigation. Comput Methods Appl Mech Eng 182:217–245

    MATH  Google Scholar 

  48. Abed-Meraim F, Combescure A (2007) A physically stabilized and locking-free formulation of the (SHB8PS) solid-shell element. Eur J Comput Mech 16:1037–1072

    MATH  Google Scholar 

  49. Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas SP (2008) A smoothed finite element method for shell analysis. Comput Methods Appl Mech Eng 198:165–177

    MATH  Google Scholar 

  50. Moreira RAS, Rodrigues JD (2011) A non-conforming plate facet-shell finite element with drilling stiffness. Finite Elem Anal Des 47:973–981

    Google Scholar 

  51. Cook RD (1993) Further development of a three-node triangular shell element. Int J Numer Methods Eng 36:1413–1425

    MATH  Google Scholar 

  52. Felippa CA (2003) A study of optimal membrane triangles with drilling freedoms. Comput Methods Appl Mech Eng 192:2125–2168

    MATH  Google Scholar 

  53. 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 86:71–80

    MathSciNet  Google Scholar 

  54. Timoshenko S, Goodier JN (1979) Theory of elasticity, 3rd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  55. Flügge W (1973) Stresses in shells. Springer, New York

    MATH  Google Scholar 

  56. Macneal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1:3–20

    Google Scholar 

  57. Ko Y, Lee Y, Lee PS, Bathe KJ (2017) Performance of the MITC3+ and MITC4+ shell elements in widely-used benchmark problems. Comput Struct 193:187–206

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Civil Engineering Department, University of Sistan and Baluchestan, Zahedan, 98167-45845, Iran

    Hosein Sangtarash, Hamed Ghohani Arab, Mohammad Reza Sohrabi & Mohammad Reza Ghasemi

Authors
  1. Hosein Sangtarash
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamed Ghohani Arab
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammad Reza Sohrabi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohammad Reza Ghasemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamed Ghohani Arab.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A

Appendix A

Natural coordinates for the plate bending element are shown in Fig. 14.

Fig. 14
figure 14

Geometry of the plate bending element in Local coordinate system

Full size image

The shape function of the ith degree of freedom in the natural coordinate system is

$$ \begin{aligned} N_{1} (\xi ,\eta ) = (1/8)(1 - \xi )(1 - \eta )(2 - \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{2} (\xi ,\eta ) = (b/8)(1 - \xi )(1 - \eta )( - (1 - \eta^{2} )) \hfill \\ N_{3} (\xi ,\eta ) = (a/8)(1 - \xi )(1 - \eta )(1 - \xi^{2} ) \hfill \\ N_{4} (\xi ,\eta ) = (1/8)(1 + \xi )(1 - \eta )(2 + \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{5} (\xi ,\eta ) = (b/8)(1 + \xi )(1 - \eta )( - (1 - \eta^{2} )) \hfill \\ N_{6} (\xi ,\eta ) = (a/8)(1 - \xi )(1 - \eta )( - (1 - \xi^{2} )) \hfill \\ N_{7} (\xi ,\eta ) = (1/8)(1 + \xi )(1 + \eta )(2 + \xi + \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{8} (\xi ,\eta ) = (b/8)(1 + \xi )(1 + \eta )(1 - \eta^{2} ) \hfill \\ N_{9} (\xi ,\eta ) = (a/8)(1 + \xi )(1 + \eta )( - (1 - \xi^{2} )) \hfill \\ N_{10} (\xi ,\eta ) = (1/8)(1 - \xi )(1 + \eta )(2 - \xi + \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{11} (\xi ,\eta ) = (b/8)(1 - \xi )(1 + \eta )(1 - \eta^{2} ) \hfill \\ N_{12} (\xi ,\eta ) = (a/8)(1 - \xi )(1 + \eta )(1 - \xi^{2} ). \hfill \\ \end{aligned} $$
(27)

The element stiffness matrix in the natural coordinate system is calculated by Eq. (28)

$$ {\mathbf{K}} = \int\limits_{V} {{\mathbf{B}}^{\text{T}} {\mathbf{DB}}{\text{d}}V} , $$
(28)

where \( {\text{d}}V \) is the differential volume and its value in the natural coordinates of the element is equal to \( abtd\xi d\eta \) and B is a \( 3 \times 12 \) matrix shown in Eq. (29).

$$ B = \left[ {\begin{array}{*{20}c} {B_{1,1} } & {B_{1,2} } & {B_{1,3} } & {B_{1,4} } & {B_{1,5} } & {B_{1,6} } & {B_{1,7} } & {B_{1,8} } & {B_{1,9} } & {B_{1,10} } & {B_{1,11} } & {B_{1,12} } \\ {B_{2,1} } & {B_{2,2} } & {B_{2,3} } & {B_{2,4} } & {B_{2,5} } & {B_{2,6} } & {B_{2,7} } & {B_{2,8} } & {B_{2,9} } & {B_{2,10} } & {B_{2,11} } & {B_{2,12} } \\ {B_{3,1} } & {B_{3,2} } & {B_{3,3} } & {B_{3,4} } & {B_{3,5} } & {B_{3,6} } & {B_{3,7} } & {B_{3,8} } & {B_{3,9} } & {B_{3,10} } & {B_{3,11} } & {B_{3,12} } \\ \end{array} } \right]. $$
(29)

Matrix B elements in the natural coordinate system are defined as

$$ \begin{array}{*{20}c} \begin{aligned} B_{1,1} = - \frac{3}{{4a^{2} }}z\xi (1 - \eta ) \hfill \\ B_{1,2} = 0 \hfill \\ B_{1,3} = - \frac{1}{4a}z(\eta - 1)(1 - 3\xi ) \hfill \\ B_{1,4} = - \frac{3}{{4a^{2} }}z\xi (\eta - 1) \hfill \\ \end{aligned} & \begin{aligned} B_{2,1} = - \frac{3}{{4b^{2} }}z\eta (1 - \xi ) \hfill \\ B_{2,2} = \frac{1}{4b}z(\xi - 1)(1 - 3\eta ) \hfill \\ B_{2,3} = 0 \hfill \\ B_{2,4} = - \frac{3}{{4b^{2} }}z\eta (1 + \xi ) \hfill \\ \end{aligned} & \begin{aligned} B_{3,1} = \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,2} = \frac{2}{8a}z( - 3\eta^{2} + 2\eta + 1) \hfill \\ B_{3,3} = - \frac{2}{8b}z( - 3\xi^{2} + 2\xi + 1) \hfill \\ B_{3,4} = - \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ \end{aligned} \\ \begin{aligned} B_{1,5} = 0 \hfill \\ B_{1,6} = - \frac{1}{4a}z(1 - \eta )(1 + 3\xi ) \hfill \\ B_{1,7} = \frac{3}{{4a^{2} }}z\xi (\eta + 1) \hfill \\ B_{1,8} = 0 \hfill \\ B_{1,9} = - \frac{1}{4a}z(1 + \eta )(1 + 3\xi ) \hfill \\ B_{1,10} = \frac{3}{{4a^{2} }}z\xi (\eta + 1) \hfill \\ B_{1,11} = 0 \hfill \\ B_{1,12} = \frac{1}{4a}z(1 + \eta )(1 - 3\xi ) \hfill \\ \end{aligned} & \begin{aligned} B_{2,5} = - \frac{1}{4b}z(\xi + 1)(1 - 3\eta ) \hfill \\ B_{2,6} = 0 \hfill \\ B_{2,7} = \frac{3}{{4b^{2} }}z\eta (1 + \xi ) \hfill \\ B_{2,8} = \frac{1}{4b}z(\xi + 1)(1 + 3\eta ) \hfill \\ B_{2,9} = 0 \hfill \\ B_{2,10} = \frac{3}{{4b^{2} }}z\eta (1 - \xi ) \hfill \\ B_{2,11} = - \frac{1}{4b}z(\xi - 1)(1 + 3\eta ) \hfill \\ B_{2,12} = 0 \hfill \\ \end{aligned} & \begin{aligned} B_{3,5} = - \frac{2}{8a}z( - 3\eta^{2} + 2\eta + 1) \hfill \\ B_{3,6} = \frac{2}{8b}z(3\xi^{2} + 2\xi - 1) \hfill \\ B_{3,7} = \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,8} = \frac{2}{8a}z(3\eta^{2} + 2\eta - 1) \hfill \\ B_{3,9} = - \frac{2}{8b}z(3\xi^{2} + 2\xi - 1) \hfill \\ B_{3,10} = - \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,11} = - \frac{2}{8a}z(3\eta^{2} + 2\eta - 1) \hfill \\ B_{3,12} = \frac{2}{8b}z( - 3\xi^{2} + 2\xi + 1). \hfill \\ \end{aligned} \\ {} & {} & {} \\ \end{array} $$
(30)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sangtarash, H., Arab, H.G., Sohrabi, M.R. et al. Formulation and evaluation of a new four-node quadrilateral element for analysis of the shell structures. Engineering with Computers 36, 1289–1303 (2020). https://doi.org/10.1007/s00366-019-00763-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-019-00763-8

Keywords

Search

Navigation