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A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates

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Abstract

A Hermite reproducing kernel Galerkin meshfree approach is proposed for buckling analysis of thin plates. This approach employs the Hermite reproducing kernel meshfree approximation that incorporates both the deflectional and rotational nodal variables into the approximation of the plate deflection and the C 1 continuous approximation requirement for the Galerkin analysis of thin plates can be easily achieved herein. The strain smoothing operation is consistently introduced to construct the smoothed rotation and curvature fields which appear in the weak form governing the thin plate buckling. The domain integration of the weak form is carried out by the method of sub-domain stabilized conforming integration with the smoothed measures of rotation and curvature, as leads to an efficient discrete meshfree formulation for the eigenvalue problem of thin plate buckling. A series of benchmark buckling problems are presented to assess the proposed algorithm and the results uniformly demonstrate the present approach is very effective and it performs superiorly compared to the conventional Galerkin meshfree formulations whose domain integration are performed by Gauss quadrature rules.

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References

  1. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover publications, Mineola

    MATH  Google Scholar 

  2. Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New York

    MATH  Google Scholar 

  3. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: volume 2, solid mechanics, 6th edn. Butterworth & Heinemann, Oxford

    Google Scholar 

  4. Belytschko T, Lu YY, Gu L (1994) Element-free Gakerkin methods. Int J Numer Methods Eng 37: 229–256

    Article  MathSciNet  MATH  Google Scholar 

  5. Belytschko T, Kronggauz Y, Organ D, Fleming M (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139: 3–47

    Article  MATH  Google Scholar 

  6. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20: 1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu WK, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (I) methodology and convergence. Comput Methods Appl Mech Eng 143: 113–154

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Comput Methods Appl Mech Eng 139: 195–227

    Article  MathSciNet  MATH  Google Scholar 

  9. Babuška I, Banerjee U, Osborn JE (2003) Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 12: 1–125

    Article  MathSciNet  MATH  Google Scholar 

  10. Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55: 1–34

    Article  Google Scholar 

  11. Huerta A, Belytschko T, Fernández-Méndez S, Rabczuk T (2004) Meshfree methods. Encycl Comput Mech 1: 279–309

    Google Scholar 

  12. Atluri SN, Shen SP (2002) The meshless local Petrov-Galerkin method. Tech Science Press, Encino

    MATH  Google Scholar 

  13. Zhang X, Liu Y (2004) Meshless methods. Tsinghua University Press/Springer, Beijing/Berlin

    Google Scholar 

  14. Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin

    MATH  Google Scholar 

  15. Liu GR (2009) Mesh free methods: moving beyond the finite element method, 2nd edn. CRC Press, London

    Book  Google Scholar 

  16. Krysl P, Belytschko T (1995) Analysis of thin plates by the element-free Galerkin method. Comput Mech 16: 1–10

    Article  MathSciNet  Google Scholar 

  17. Rabczuk T, Areias PMA, Belytschko T (2007) A meshfree thin shell method for nonlinear dynamic fracture. Int J Numer Methods Eng 72: 524–548

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method: Part I. Theoretical formulation. Comput Methods Appl Mech Eng 193: 933–951

    Article  MathSciNet  MATH  Google Scholar 

  19. Li S, Lu H, Han W, Liu WK, Simkins DC (2004) Reproducing kernel element method, Part II. Global conforming I m/C n hierarchy. Comput Methods Appl Mech Eng 193: 953–987

    Article  MathSciNet  MATH  Google Scholar 

  20. Lu H, Li S, Simkins DC, Liu WK, Cao J (2004) Reproducing kernel element method Part III, Generalized enrichment and applications. Comput Methods Appl Mech Eng 193: 989–1011

    Article  MATH  Google Scholar 

  21. Simkins DC, Li S, Lu H, Liu WK (2004) Reproducing kernel element method Part IV. Globally compatible C n(n ≥ 1) triangular hierarchy. Comput Methods Appl Mech Eng 193: 1013–1034

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang YM, Chen SM, Wu CP (2010) A meshless collocation method based on the differential reproducing kernel interpolation. Comput Mech 45: 585–606

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen SM, Wu CP, Wang YM (2011) A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli-Euler beams and Kirchhoff-Love plates. Comput Mech 47: 425–453

    Article  MathSciNet  MATH  Google Scholar 

  24. 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: 61–75

    Article  MATH  Google Scholar 

  25. Liew KM, Peng LX, Kitipornchai S (2006) Buckling of folded plate structures subjected to partial in-plane edge loads by the FSDT meshfree Galerkin method. Int J Numer Methods Eng 65: 1495–1526

    Article  MATH  Google Scholar 

  26. Peng LX, Liew KM, Kitipornchai S (2006) Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method. J Sound Vib 289: 421–449

    Article  Google Scholar 

  27. Liew KM, Chen XL (2004) Mesh-free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in-plane point loads. Int J Numer Methods Eng 60: 1861–1877

    Article  MATH  Google Scholar 

  28. Liu L, Chua LP, Ghista DN (2007) Mesh-free radial basis function method for static, free vibration and buckling analysis of shear deformable composite laminates. Compos Struct 78: 58–69

    Article  Google Scholar 

  29. 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 Bound Elem 35: 1038–1053

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu GR, Chen XL, Reddy JN (2002) Buckling of symmetrically laminated composite plates using the element-free Galerkin method. Int J Struct Stab Dyn 2: 281–294

    Article  MATH  Google Scholar 

  31. Bui TQ, Nguyen MN (2011) A novel meshfree model for buckling and vibration analysis of rectangular orthotropic plates. Struct Eng Mech 39: 579–598

    Google Scholar 

  32. Wang D (2006) A stabilized conforming integration procedure for Galerkin meshfree analysis of thin beam and plate. In: Proceeding of the 10th enhancement and promotion of computational methods in engineering and science, Sanya, China, August 21–23

  33. Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. Int J Numer Methods Eng 74: 368–390

    Article  MATH  Google Scholar 

  34. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Methods Eng 50: 435–466

    Article  MATH  Google Scholar 

  35. Chen JS, Yoon S, Wu CT (2002) Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Methods Eng 53: 2587–2615

    Article  MATH  Google Scholar 

  36. Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput Methods Appl Mech Eng 193: 1065–1083

    Article  MATH  Google Scholar 

  37. Wang D, Chen JS (2006) A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration. Comput Mech 39: 83–90

    Article  MATH  Google Scholar 

  38. Wang D, Dong SB, Chen JS (2006) Extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry. Int J Solids Struct 43: 144–171

    Article  MATH  Google Scholar 

  39. Chen JS, Wang D (2006) A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. Int J Numer Methods Eng 68: 151–172

    Article  MATH  Google Scholar 

  40. 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: 703–719

    Article  MathSciNet  Google Scholar 

  41. 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: 47–63

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang D, Lin Z (2012) A comparative study on the dispersion properties of HRK and RK meshfree approximations for Kirchhoff plate problem. Int J Comput Methods 9: 1240015

    Article  MathSciNet  Google Scholar 

  43. Chen JS, Wang HP (2000) New boundary condition treatments for meshless computation of contact problems. Comp Methods Appl Mech Eng 187: 441–468

    Article  MATH  Google Scholar 

  44. Xiong YB, Long SY (2005) Analysis of buckling for an anisotropic plate by the meshless local Petrov-Galerkin (MLPG) method. Mech Eng 27: 50–53

    Google Scholar 

  45. Yao LQ, Chen FJ, Liu X (2010) HRBF meshfree method for buckling analysis of piezoelectric laminate plates. In: Symposium on piezoelectricity, acoustic waves and device applications

  46. Chen JS, Wang HP (2000) New boundary condition treatments for meshless computation of contact problems. Comput Methods Appl Mech Eng 187: 441–468

    Article  MATH  Google Scholar 

  47. Chen W, Cheung YK (1998) Refined triangular discrete Kirchhoff plate elment for thin plate bending, vibration and buckling analysis. Int J Numer Methods Eng 41: 1507–1525

    Article  MATH  Google Scholar 

  48. Timoshenko SP, Gere JM (1988) Theory of elastic stability, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  49. Leissa AW, Ayoub EF (1988) Vibration and buckling of a simply supported rectangular plate subjected to a pair of in-plane concentrated forces. J Sound Vib 127: 155–171

    Article  MATH  Google Scholar 

  50. Liew KM, Chen XL (2004) Buckling of rectangular Mindlin plates subjected to partial in-plane edge loads using the radial point interpolation method. Int J Solids Struct 41: 1677–1695

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Xiamen University, Xiamen, Fujian, 361005, China

    Dongdong Wang & Huikai Peng

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  1. Dongdong Wang
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  2. Huikai Peng
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Correspondence to Dongdong Wang.

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Wang, D., Peng, H. A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Comput Mech 51, 1013–1029 (2013). https://doi.org/10.1007/s00466-012-0784-9

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  • DOI: https://doi.org/10.1007/s00466-012-0784-9

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