# Analyzing of thick plates with cutouts using the meshless (EFG) method based on higher order shear deformation theories for solving shear-locking issue

Document Type : Research

Authors

1 Ph.D. Student, Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

2 Professor, Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

3 Assistant Professor, Department of Civil Engineering, Quchan University of Technology, Quchan, Iran.

Abstract

Although the finite element method (FEM) is a well-established method for modelling the thick plates, in some cases FEM encounters some difficulties such as shear locking and decrease in the accuracy of results caused by stress concentration around the openings. In this paper for the first time, the EFG method based on the higher-order shear deformation theories is developed for analysis of thick plates with cutout to overcome these drawbacks. It should be mentioned that the EFG method does not need any mesh generation in problem domain and its boundaries. The Radial Point Interpolation method (RPIM) is used to discrete the problem domain. Several numerical examples are analyzed using proposed method and effects of aspect ratios, boundary conditions and location of cutout are discussed in details. Results show that by choosing the appropriate shape functions for the deflection and rotations, the presented EFG method has successfully overcome the shear-locking problem. Based on numerical results, the best position of circular cutout, which minimizes the maximum deflection is determined. The approximate equations for determination of maximum deflection are presented using the cubic polynomial method. Numerical implementations show that the presented method has high efficiency, good accuracy and easy implementation.

Keywords

#### References

1. M.A. Puso, J.S. Chen, E. Zywicz, W. Elmer. Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering. 74 (2008) 416-446. [DOI:10.1002/nme.2181] [DOI:10.1002/nme.2181]
2. G.R. Liu. Meshfree methods: moving beyond the finite element method, Taylor & Francis, 2009.
3. G.R. Liu, Y.T. Gu. An introduction to meshfree methods and their programming. Springer Science & Business Media, 2005.
4. V.M. Sreehari, D.K. Maiti. Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading. In Structures. 6 (2016) 9-19. [DOI:10.1016/j.istruc.2016.01.002] [DOI:10.1016/j.istruc.2016.01.002]
5. G.R. Liu, G.Y. Zhang, Y. Gu, Y.Y. Wang. A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Computational Mechanics, 36 (2005) 421-430. [DOI:10.1007/s00466-005-0657-6] [DOI:10.1007/s00466-005-0657-6]
6. A. Hussain, Y.P. Liu, S.L. Chan. Finite element modeling and design of single angle member under bi-axial bending, In Structures. 16 (2018) 373-389. [DOI:10.1016/j.istruc.2018.11.001] [DOI:10.1016/j.istruc.2018.11.001]
7. J. Akl, F. Alladkani, P. Dumond. Comparing and optimizing analytical, numerical and experimental vibration models for a simply-supported ribbed plate, In Structures. 23 (2002) 690-701. [DOI:10.1016/j.istruc.2019.12.003] [DOI:10.1016/j.istruc.2019.12.003]
8. D.L. Christy, T.M. Pillai, P. Nagarajan. Thin plate element for applied element method, In Structures. 22 (2019) 1-12. [DOI:10.1016/j.istruc.2019.07.010] [DOI:10.1016/j.istruc.2019.07.010]
9. T. Liszka, J. Orkisz. The finite difference method at arbitrary irregular grids and its application in applied mechanics. Computers & Structures, 11 (1980) 83-95. [DOI:10.1016/0045-7949(80)90149-2] [DOI:10.1016/0045-7949(80)90149-2]
10. L.B. Lucy. A numerical approach to the testing of the fission hypothesis, The astronomical journal, 82 (1977) 1013-1024. [DOI:10.1086/112164] [DOI:10.1086/112164]
11. N. R. Aluru, A point collocation method based on reproducing kernel approximations, International Journal for Numerical Methods in Engineering, 47 (2000) 1083-1121. 10.1002/(SICI)1097-0207(20000228)47:6<1083::AID-NME816>3.0.CO;2-N [DOI:10.1002/(SICI)1097-0207(20000228)47:63.0.CO;2-N] 10.1002/(SICI)1097-0207(20000228)47:6<1083::AID-NME816>3.0.CO;2-N []
12. T. Belytschko, Y.Y. Lu, L. Gu, Element‐free Galerkin methods, International journal for numerical methods in engineering, 37 (1994) 229-256. [DOI:10.1002/nme.1620370205] [DOI:10.1002/nme.1620370205]
13. J.S. Chen, D. Wang, A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, International Journal for Numerical Methods in Engineering, 68 (2006) 151-172. [DOI:10.1002/nme.1701] [DOI:10.1002/nme.1701]
14. S. Tanaka, H. Suzuki, S. Sadamoto, M. Imachi, T.Q. Bui, Analysis of cracked shear deformable plates by an effective meshfree plate formulation, Engineering Fracture Mechanics. 144 (2015) 142-157. [DOI:10.1016/j.engfracmech.2015.06.084] [DOI:10.1016/j.engfracmech.2015.06.084]
15. S.N. Atluri, T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational mechanics, 22 (1998) 117-127. [DOI:10.1007/s004660050346] [DOI:10.1007/s004660050346]
16. G.R. Liu, Y.T. Gu, A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids, Journal of Sound and vibration, 246 (2001) 29-46. [DOI:10.1006/jsvi.2000.3626] [DOI:10.1006/jsvi.2000.3626]
17. G.R. Liu, Y.T. Gu, A local point interpolation method for stress analysis of two-dimensional solids, Structural Engineering and Mechanics, 11 (2001) 221-236. [DOI:10.12989/sem.2001.11.2.221] [DOI:10.12989/sem.2001.11.2.221]
18. L. Gu, Moving kriging interpolation and element‐free Galerkin method, International journal for numerical methods in engineering, 56 (2003) 1-11. [DOI:10.1002/nme.553] [DOI:10.1002/nme.553]
19. Y.T. Gu, G. R. Liu, A boundary radial point interpolation method (BRPIM) for 2-D structural analyses, Structural engineering and mechanics, 15 (2003) 535-550. [DOI:10.12989/sem.2003.15.5.535] [DOI:10.12989/sem.2003.15.5.535]
20. G.R. Liu, Y.L. Wu, H. Ding, Meshfree weak-strong (MWS) form method and its application to incompressible flow problems, International journal for numerical methods in fluids, 46 (2004) 1025-1047. [DOI:10.1002/fld.785] [DOI:10.1002/fld.785]
21. J. Chen, W. Tang, P. Huang, L. Xu, A mesh-free analysis method of structural elements of engineering structures based on B-spline wavelet basis function, Structural Engineering and Mechanics, 57 (2016) 281-294. [DOI:10.12989/sem.2016.57.2.281] [DOI:10.12989/sem.2016.57.2.281]
22. S. Beissel, T. Belytschko, Nodal integration of the element-free Galerkin method, Computer methods in applied mechanics and engineering, 139 (1996) 49-74. [DOI:10.1016/S0045-7825(96)01079-1] [DOI:10.1016/S0045-7825(96)01079-1]
23. G.R. Liu, L. Yan, J.G. Wang, Y.T. Gu, Point interpolation method based on local residual formulation using radial basis functions. Structural Engineering and Mechanics, 14 (2002) 713-732. [DOI:10.12989/sem.2002.14.6.713] [DOI:10.12989/sem.2002.14.6.713]
24. T. Most, C. Bucher, A moving least squares weighting function for the element-free Galerkin method which almost fulfills essential boundary conditions, Structural Engineering and Mechanics, 21 (2005) 315-332. [DOI:10.12989/sem.2005.21.3.315] [DOI:10.12989/sem.2005.21.3.315]
25. Y. Liu, T. Belytschko, A new support integration scheme for the weak form in mesh‐free methods, International journal for numerical methods in engineering, 82 (2010) 699-715. [DOI:10.1002/nme.2780] [DOI:10.1002/nme.2780]
26. J.S. Chen, C.T. Wu, S. Yoon, Y. You, A stabilized conforming nodal integration for Galerkin mesh‐free methods, International journal for numerical methods in engineering, 50 (2001) 435-466. 10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A [DOI:10.1002/1097-0207(20010120)50:23.0.CO;2-A] 10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A []
27. J. Sladek, V. Sladek, H. A. Mang, Meshless formulations for simply supported and clamped plate problems, International Journal for Numerical Methods in Engineering, 55 (2002) 359-375. [DOI:10.1002/nme.503] [DOI:10.1002/nme.503]
28. J. Alihemmati, Y.T. Beni, Developing three-dimensional mesh-free Galerkin method for structural analysis of general polygonal geometries, Engineering with Computers, 1 (2019) 1-10.
29. Y. Cao, L. Yao, Y. Yin, New treatment of essential boundary conditions in EFG method by coupling with RPIM, Acta Mechanica Solida Sinica, 26 (2013) 302-316. [DOI:10.1016/S0894-9166(13)60028-2] [DOI:10.1016/S0894-9166(13)60028-2]
30. Y. Choi, S. Kim, Analysis of Mindlin plate by the element free Galerkin method applying penalty technique, In 40th Structures, Structural Dynamics, and Materials Conference and Exhibit, 1999. [DOI:10.2514/6.1999-1238] [DOI:10.2514/6.1999-1238]
31. B.M. Donning, W.K. Liu, Meshless methods for shear-deformable beams and plates, Computer Methods in Applied Mechanics and Engineering, 152 (1998) 47-71. [DOI:10.1016/S0045-7825(97)00181-3] [DOI:10.1016/S0045-7825(97)00181-3]
32. V. Gulizzi, I. Benedetti, A. Milazzo, An implicit mesh discontinuous Galerkin formulation for higher-order plate theories, Mechanics of Advanced Materials and Structures, 1 (2019) 1-15. [DOI:10.1080/15376494.2018.1516258]
33. M. Khezri, M. Gharib, K.J.R. Rasmussen, A unified approach to meshless analysis of thin to moderately thick plates based on a shear-locking-free Mindlin theory formulation, Thin-Walled Structures, 124 (2018) 161-179. [DOI:10.1016/j.tws.2017.12.004] [DOI:10.1016/j.tws.2017.12.004]
34. D.H. Konda, J.A.F. Santiago, J.C.F. Telles, J.P.F. Mello, E.G.A. Costa, A meshless Reissner plate bending procedure using local radial point interpolation with an efficient integration scheme, Engineering Analysis with Boundary Elements, 99 (2019) 46-59. [DOI:10.1016/j.enganabound.2018.11.004] [DOI:10.1016/j.enganabound.2018.11.004]
35. Y. Li, G.R. Liu, An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems, Computers & Mathematics with Applications, 77 (2019) 441-465. [DOI:10.1016/j.camwa.2018.09.047] [DOI:10.1016/j.camwa.2018.09.047]
36. X. Liu, G.R. Liu, J. Wang, Y. Zhou, A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment, Computational Mechanics, 64 (2019) 989-1016. [DOI:10.1007/s00466-019-01691-6] [DOI:10.1007/s00466-019-01691-6]
37. M.H.G. Rad, F. Shahabian, S.M. Hosseini, A meshless local Petrov-Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping, Acta Mechanica, 226 (2015) 1497-1513. [DOI:10.1007/s00707-014-1266-2] [DOI:10.1007/s00707-014-1266-2]
38. C.H. Thai, H. Nguyen-Xuan, A moving kriging interpolation meshfree method based on naturally stabilized nodal integration scheme for plate analysis, International Journal of Computational Methods, 16 (2019) 1850100. [DOI:10.1142/S0219876218501001] [DOI:10.1142/S0219876218501001]
39. Thai, H. T., & Choi, D. H. (2013). Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates. Applied Mathematical Modelling, 37(18-19), 8310-8323. [DOI:10.1016/j.apm.2013.03.038] [DOI:10.1016/j.apm.2013.03.038]
40. A.C.A. Ferreira, P.M.V. Ribeiro, Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method, Latin American Journal of Solids and Structures, 16 (2019) e140. [DOI:10.1590/1679-78255191] [DOI:10.1590/1679-78255191]
41. Tash, F. Y., & Neya, B. N. (2020). An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness. Applied Mathematical Modelling, 77, 1582-1602. [DOI:10.1016/j.apm.2019.08.017] [DOI:10.1016/j.apm.2019.08.017]
42. J.N. Reddy, Theory and analysis of elastic plates and shells. CRC press. 1999.
43. E. Reissner, On the theory of transverse bending of elastic plates, International Journal of Solids and Structures, 12 (1976) 545-554. [DOI:10.1016/0020-7683(76)90001-9] [DOI:10.1016/0020-7683(76)90001-9]
44. Dorduncu, M., Kaya, K., & Ergin, O. F. (2020). Peridynamic Analysis of Laminated Composite Plates Based on First-Order Shear Deformation Theory. International Journal of Applied Mechanics, 2050031. [DOI:10.1142/S1758825120500313] [DOI:10.1142/S1758825120500313]
45. Cui, X. Y., Liu, G. R., & Li, G. Y. (2010). Analysis of Mindlin-Reissner plates using cell-based smoothed radial point interpolation method. International Journal of Applied Mechanics, 2(03), 653-680. [DOI:10.1142/S1758825110000706] [DOI:10.1142/S1758825110000706]
46. M. Hu, X. Shi, T. Wang, F. Liu, A note on cubic polynomial interpolation, Computers & Mathematics with Applications, 56 (2008) 1358-1363. [DOI:10.1016/j.camwa.2008.02.032] [DOI:10.1016/j.camwa.2008.02.032]
47. E. Meijering, M. Unser, A note on cubic convolution interpolation, IEEE Transactions on Image processing, 12 (2003) 477-479. [DOI:10.1109/TIP.2003.811493] [DOI:10.1109/TIP.2003.811493]
48. M.H. Kharrazi, Rational method for analysis and design of steel plate walls. Ph.D. Dssertation, University of British Columbia, Canada, 2005.
49. Ghugal, Y. M., & Sayyad, A. S. (2013). Stress analysis of thick laminated plates using trigonometric shear deformation theory. International Journal of Applied Mechanics, 5(01), 1350003. [DOI:10.1142/S1758825113500038] [DOI:10.1142/S1758825113500038]
50. N.J. Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of composite materials, 4 (1970) 20-34. [DOI:10.1177/002199837000400102] [DOI:10.1177/002199837000400102]

### History

• Receive Date: 26 February 2021
• Revise Date: 06 April 2021
• Accept Date: 07 May 2021