Computational Mechanics

, Volume 52, Issue 6, pp 1313–1330 | Cite as

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

  • Shota Sadamoto
  • Satoyuki TanakaEmail author
  • Shigenobu Okazawa
Original Paper


A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.


Meshfree method Reproducing kernel approximation Large deflection analysis Multiple point constraint 



This research was partially supported by JKA through its promotion funds from KEIRIN RACE. A part of the present research conducted by Satoyuki Tanaka was financially supported by The Research Council of Norway (RCN) through the Yggdrasil mobility programme.


  1. 1.
    Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318CrossRefzbMATHGoogle Scholar
  2. 2.
    Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Method Eng 37:229–256MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Method Fluids 20:1081–1106MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Durate CA, Oden JT (1996) An h-p adaptive method using clouds. Comput Method Appl Mech Eng 139:237–262CrossRefGoogle Scholar
  5. 5.
    Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Belytschko T, Kronggauz Y, Organ D, Fleming M (1996) Meshless methods: an overview and recent developments. Comput Method Appl Mech Eng 139:3–47CrossRefzbMATHGoogle Scholar
  7. 7.
    Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34CrossRefGoogle Scholar
  8. 8.
    Babuška I, Banerjee U, Osborn JE (2003) Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 12:1–125MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li S, Liu WK (2004) Meshfree particle methods. Springer, BerlinzbMATHGoogle Scholar
  10. 10.
    Liu GR (2009) Mesh free methods: moving beyond the finite element method, 2nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  11. 11.
    Li S, Hao W, Liu WK (2000) Numerical simulations of large deformation of thin shell structures using meshfree methods. Comput Mech 25:102–116CrossRefzbMATHGoogle Scholar
  12. 12.
    Qian D, Eason T, Li S, Liu WK (2008) Meshfree simulation of failure modes in thin cylinders subjected to combined loads of internal pressure and localized heat. Int J Numer Method Eng 76:1159–1184CrossRefzbMATHGoogle Scholar
  13. 13.
    Gato C (2010) Meshfree analysis of dynamic fracture in thin-walled structures. Thin Walled Struct 48:215–222CrossRefGoogle Scholar
  14. 14.
    Krysl P, Belytschko T (1996) Analysis of thin plates by the element-free Galerkin method. Comput Mech 17:26–35MathSciNetCrossRefGoogle Scholar
  15. 15.
    Krysl P, Belytschko T (1996) Analysis of thin shells by the element-free Galerkin method. Int J Solids Struct 33:3057–3080CrossRefzbMATHGoogle Scholar
  16. 16.
    Long SY, Atluri SN (2002) A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate. Comput Model Eng Sci 3:53–63zbMATHGoogle Scholar
  17. 17.
    Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput Method Appl Mech Eng 193:1065–1083CrossRefzbMATHGoogle Scholar
  18. 18.
    Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. Int J Numer Method Eng 74:368–390CrossRefzbMATHGoogle Scholar
  19. 19.
    Wang D, Lin Z (2010) Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration. Comput Mech 46:703–719MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang D, Lin Z (2011) Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures. Comput Mech 48:47–63MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen JS, Wang D (2006) A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. Int J Numer Method Eng 68:151–172CrossRefzbMATHGoogle Scholar
  22. 22.
    Liu Y, Hon YC, Liew KM (2006) A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems. Int J Numer Method Eng 66:1153–1178CrossRefzbMATHGoogle Scholar
  23. 23.
    Noguchi H, Kawashima T, Miyamura T (2000) Element free analyses of shell and spatial structures. Int J Numer Method Eng 47:1215–1240CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang D, Sun Y (2011) A galerkin mashfree method with stabilized conforming nodal integration for geometrically nonlinear analysis of shear deformable plates. Int J Comput Method 8:685–703CrossRefzbMATHGoogle Scholar
  25. 25.
    Liew KM, Chen XL, Reddy JN (2004) Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrary shaped shear deformable plates. Comput Method Appl Mech Eng 193:205–224CrossRefzbMATHGoogle Scholar
  26. 26.
    Liew KM, Peng LX, Kitipornchai S (2006) Buckling analysis of corrugated plates using a mesh-free Galerkin method based on the first-order shear deformation theory. Comput Mech 38: 61–75Google Scholar
  27. 27.
    Bui TQ, Nguyen MN, Zhang Ch (2011) Buckling analysis of Reissner-Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng Anal Boundary Elem 35:1038–1053MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liew KM, Peng LX, Kitipornchai S (2007) Geometric non-linear analysis of folded plate structures by the spline strip kernel particle method. Int J Numer Method Eng 71:1102–1133CrossRefzbMATHGoogle Scholar
  29. 29.
    Lu H, Cheng HS, Cao J, Liu WK (2005) Adaptive enrichment meshfree simulation and experiment on buckling and post-buckling analysis in sheet metal forming. Comput Method Appl Mech Eng 194:2569–2590CrossRefzbMATHGoogle Scholar
  30. 30.
    Rabczuk T, Areias PMA, Belytschko T (2007) A meshfree thin shell method for non-linear dynamic fracture. Int J Numer Method Eng 72:524–548MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
  32. 32.
    Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Method Eng 50:435–466CrossRefzbMATHGoogle Scholar
  33. 33.
    Chen JS, Yoon S, Wu CT (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Method Eng 53:2587–2615CrossRefzbMATHGoogle Scholar
  34. 34.
    Cho JY, Song YM, Choi YH (2008) Boundary locking induced by penalty enforcement of essential boundary condition in mesh-free methods. Comput Method Appl Mech Eng 197:1167–1183MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Belyschko T, Organ D, Krongauz Y (1995) A coupled finite element—element free Galerkin method. Comput Mech 17: 186–195Google Scholar
  36. 36.
    Krongauz Y, Belyschko T (1996) Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput Method Appl Mech Eng 131:133–145CrossRefzbMATHGoogle Scholar
  37. 37.
    Hegen D (1996) Element-free Galerkin methods in combination with finite element approaches. Comput Method Appl Mech Eng 135:143–166CrossRefzbMATHGoogle Scholar
  38. 38.
    Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Comput Method Appl Mech Eng 139:195–227MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Chen JS, Wang HP (2000) New boundary condition treatments in meshfree computation of contact problems. Comput Method Appl Mech Eng 187:441–468CrossRefzbMATHGoogle Scholar
  40. 40.
    Nagashima T (2000) Development of a CAE system based on the node-by-node meshless method. Comput Method Appl Mech Eng 187:1–34CrossRefzbMATHGoogle Scholar
  41. 41.
    Wu CT, Koishi M, Skinner G, Shimamoto H (2008) A meshfree procedure for the microscopic simulation and design of rubber compounds and its application to multi-scale simulation of tires. Proceedings of WCCM8 and ECCOMAS 2008, pp. 1–2Google Scholar
  42. 42.
    Noguchi H, Zhang Z (2007) Analysis of large deformation of rubber-filler structures under periodic boundary conditions using an enhanced meshfree method. Proc Comput Mech Conf 20: 661–662Google Scholar
  43. 43.
    Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Method Eng 38:1655–1679 Google Scholar
  44. 44.
    Chui CK, Wang GZ (1991) A cardinal spline approach to wavelet. Proc Am Math Soc 113:785–793MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    MSC.Marc 2005r3, User’s GuideGoogle Scholar
  46. 46.
    Fujikubo M, Yao T, Khedmati MR, Harada M, Yanagihara D (2005) Estimation of ultimate strength of continuous stiffened panel under combined transverse thrust and lateral pressure Part 1: Continuous Plate. Marine Struct 18:383–410CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shota Sadamoto
    • 1
  • Satoyuki Tanaka
    • 1
    Email author
  • Shigenobu Okazawa
    • 1
  1. 1.Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

Personalised recommendations