Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free-vibration and buckling of Mindlin plates using SGN-FEM models and effects of parasitic shear in models performance

  • Leilson J. Araujo (Graduate Program in Civil Engineering, Federal University of Technology-Parana) ;
  • Joao E. Abdalla Filho (Graduate Program in Civil Engineering, Federal University of Technology-Parana)
  • Received : 2022.07.20
  • Accepted : 2023.07.29
  • Published : 2023.08.10
⟨ Previous Next ⟩

Abstract

Free-vibration and buckling analyses of plate problems are investigated with the aid of the strain gradient notation finite element method (SGN-FEM). As SGN-FEM employs physically interpretable polynomials in developing finite elements, parasitic shear sources, which are the cause of shear locking, can be precisely identified and subsequently eliminated. This allows two mutually complementary objectives to be defined in this work, namely, evaluate the efficiency of free-vibration and buckling results provided by corrected models, and study the severity of parasitic shear effects on plate models performance. Parasitic shear are flexural terms erroneously present in shear strain polynomials. It is reviewed here that six parasitic shear terms arise during the formulation of the four-node Mindlin plate element. Two parasitic shear terms have been identified in the in-plane shear strain polynomial while other two have been identified in each of the transverse shear strain polynomials. The element is corrected a-priori, i.e., during development, by simply removing the spurious terms from the shear strain polynomials. The computational implementation of the element in its two versions, namely, containing the parasitic shear terms (PS) and corrected for parasitic shear (SG), allows for assessments of the accuracy of results and of the deleterious effects of parasitic shear in free vibration and buckling analyses. This assessment of the parasitic shear effects is a novelty of this work. Validation of the SG model is done comparing its results with analytical results and results provided by other numerical procedures. Analyses are performed for square plates with different thickness-to-length ratios and boundary conditions. Results for thin plates provided by the PS model do not converge to the correct solutions, which indicates that parasitic shear must be eliminated. That is, analysts should not rely on refinement alone. For thick plates, PS model results can be considered acceptable as deleterious effects are really critical in thin plates. On the other hand, results provided by the SG model converge well for both thin and thick plates. The effectiveness of the SG model is established via high-accuracy results obtained in several examples. It is concluded that corrected SGN-FEM models are efficient alternatives for free-vibration and buckling analysis of Mindlin plate problems, and that precise elimination of parasitic shear is a requirement for sound analyses.

Keywords

Acknowledgement

The authors acknowledge Federal University of Technology-Parana and Graduate Program in Civil Engineering (PPGEC) for providing the environment for this research to be conducted. The first author acknowledges CAPES for providing a scholarship during his master's degree program. The second author acknowledges CNPQ (Brazilian Research Council) for partly supporting his research via a fellowship PQ-2 (Process: 310855/2019-5).

References

  1. Abdalla Filho, J.E. (1992), "Qualitative and discretization error analysis of laminated composite plate models", Ph.D. Dissertation, University of Colorado, Boulder.
  2. Abdalla Filho, J.E. and Dow, J.O. (1994), "An error analysis approach for laminated composite plate finite element model", Comput. Struct., 52, 611-616. https://doi.org/10.1016/0045-7949(94)90343-3.
  3. Abdalla Filho, J.E., Belo, I.M. and Dow, J.O. (2016), "A serendipity plate element free of modeling deficiencies for the analysis of laminated composites", Compos. Struct., 154, 150-171. https://doi.org/10.1016/j.compstruct.2016.07.042.
  4. Abdalla Filho, J.E., Belo, I.M. and Pereira, M.S. (2008), "A laminated composite plate finite element a-priori corrected for locking", Struct. Eng. Mech., 28(5), 603. https://doi.org/10.12989/sem.2008.28.5.603.
  5. Abdalla Filho, J.E., Dow, J.O. and Belo, I.M. (2020), "Deficiencies in the eight-node Mindlin plate finite element physically explained", J. Eng. Mech., 146, 04019131. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001715.
  6. Abdalla Filho, J.E., Fagundes, F.A. and Machado, R.D. (2006), "Identification and elimination of parasitic shear in a laminated composite beam finite element", Adv. Eng. Softw., 37, 522-532. https://doi.org/10.1016/j.advengsoft.2005.11.001.
  7. Abdalla, J.E., Belo, I.M. and Dow, J.O. (2017), "On a four-node quadrilateral plate for laminated composites", Lat. Am. J. Solid. Struct., 14, 2177-2197. https://doi.org/10.1590/1679-78253663.
  8. Al Janabi, B.S., Hinton, E. and Vuksanovic, D.J. (1989), "Free vibrations of Mindlin plates using the finite element method: Part 1. Square plates with various edge conditions", Eng. Comput., 6, 90-96. https://doi.org/10.1108/eb023763.
  9. Baier-Saip, J.A., Baier, P.A., de Faria, A.R., Oliveira, J.C. and Baier, H. (2020), "Shear locking in one-dimensional finite element methods", Eur. J. Mech.-A/Solid., 79, 103871. https://doi.org/10.1016/j.euromechsol.2019.103871.
  10. Belounar, A., Benmebarek, S., Houhou, M.N. and Belounar, L. (2020), "Free vibration with Mindlin plate finite element based on the strain approach", J. Inst. Eng. (India): Ser. C, 101, 331-346. https://doi.org/10.1007/s40032-020-00555-w.
  11. Bletzinger, K.U., Bischoff, M. and Ramm, E. (2020), "A unified approach for shear-locking-free triangular and rectangular shell finite elements", Compu. Struct., 75, 321-334. https://doi.org/10.1016/S0045-7949(99)00140-6.
  12. Boussem, F., Belounar, A. and Belounar, L. (2021), "Assumed strain finite element for natural frequencies of bending plates", World J. Eng., 19(5), 620-631. https://doi.org/10.1108/WJE-02-2021-0114.
  13. Byrd, D.E. (1988), "Identification and elimination of errors in finite element analyses", Doctoral Thesis, University of Colorado, Boulder.
  14. Dawe, D.J. and Roufaeil, O.L. (1980), "Rayleigh-Ritz vibration analysis of Mindlin plates", J. Sound Vib., 69, 345-359. https://doi.org/10.1016/0022-460X(80)90477-0.
  15. Dawe, D.J. and Roufaeil, O.L. (1982), "Buckling of rectangular Mindlin plates", Comput. Struct., 15, 461-471. https://doi.org/10.1016/0045-7949(82)90081-5.
  16. Dow, J.O. and Byrd, D.E. (1990), "Error estimation procedure for plate bending elements", AIAA J., 28, 685-693. https://doi.org/10.2514/3.10447.
  17. Dow, J.O. (1999), A Unified Approach to the Finite Element Method and Error Analysis Procedures, 1st Edition, Academic Press, San Diego.
  18. Dow, J.O. and Abdalla Filho, J.E. (1994), "Qualitative errors in laminated composite plate models", Int. J. Numer. Meth. Eng., 37, 1215-1230. https://doi.org/10.1002/nme.1620370707.
  19. Dow, J.O. and Byrd, D.E. (1988), "The identification and elimination of artificial stiffening errors in finite elements", Int. J. Numer. Meth. Eng., 26, 743-762. https://doi.org/10.1002/nme.1620260316.
  20. Dow, J.O., Cabiness, H.D. and Ho, T.H. (1986), "Linear strain element with curved edges", J. Struct. Eng., 112, 692-708. https://doi.org/10.1061/(ASCE)0733-9445(1986)112:4(692).
  21. Fakher, M. and Hosseini-Hashemi, S. (2020), "Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution", Eng. Comput., 1-15. https://doi.org/10.1007/s00366-020-01058-z.
  22. Fang, J.S. and Zhou, D. (2017), "Free vibration analysis of rotating Mindlin plates with variable thickness", Int. J. Struct. Stab. Dyn., 17, 1750046. https://doi.org/10.1142/S0219455417500468.
  23. Hinton, E. (1988), Numerical Methods and Software for Dynamic Analysis of Plates and Shells, 1st Edition, Pineridge Press, Swansea.
  24. Horta, T.P. and Abdalla Filho, J.E. (2021), "Free vibration of laminated composites beams using strain gradient notation finite element models", Mater. Res., 24, e20210394. https://doi.org/10.1590/1980-5373-MR-2021-0394.
  25. Hosseini-Hashemi, S., Khorshidi, K. and Amabili, M. (2008), "Exact solution for linear buckling of rectangular Mindlin plates", J. Sound Vib., 315, 318-342. https://doi.org/10.1016/j.jsv.2008.01.059.
  26. Hughes, T.J.R. and Cohen, M. (1978), "The ''heterosis" finite element for plate bending", Comput. Struct., 9, 445-450. https://doi.org/10.1016/0045-7949(78)90041-X.
  27. Hughes, T.J.R. and Tezduyar, T.E. (1981), "Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element", J. Appl. Mech., 48, 587-596. https://doi.org/10.1115/1.3157679.
  28. Hughes, T.J.R., Cohen, M. and Haroun, M. (1978), "Reduced and selective integration techniques in finite element analysis of plates", Nucl. Eng. Des., 46, 203-222. https://doi.org/10.1016/0029-5493(78)90184-X.
  29. Hughes, T.J.R., Taylor, R.L. and Kanoknukulchai, W. (1977), "A simple and efficient finite element for plate bending", Int. J. Numer. Meth. Eng., 11, 1529-1543. https://doi.org/10.1002/nme.1620111005.
  30. Jinsong, T., Linfang, Q. and Guangsong, C. (2021), "A smoothed GFEM based on taylor expansion and constrained MLS for analysis of Reissner-Mindlin plate", Int. J. Comput. Meth., 18(10), 2150048. https://doi.org/10.1142/S0219876221500481.
  31. Jung, J., Jun, H. and Lee, P.S. (2021), "Self-updated four-node finite element using deep learning", Comput. Mech., 69(1), 23-44. https://doi.org/10.1007/s00466-021-02081-7.
  32. Le, C.I., Tran, Q.D., Pham, V.N. and Nguyen, D.K. (2021), "Free vibration and buckling of bidirectional functionally graded sandwich plates using an efficient Q9 element", Viet. J. Mech., 43, 277-295. https://doi.org/10.15625/0866-7136/15981.
  33. Lee, S.J. (2004), "Free vibration analysis of plates by using a four-node finite element formulated with assumed natural transverse shear strain", J. Sound Vib., 278, 657-684. https://doi.org/10.1016/j.jsv.2003.10.018.
  34. Li, G., Xing, Y. and Wang, Z. (2021), "Closed-form solutions for free vibration of rectangular nonlocal Mindlin plates with arbitrary homogeneous boundary conditions", Compos. Part C: Open Access, 6, 100193. https://doi.org/10.1016/j.jcomc.2021.100193.
  35. Liew, K.M., Wang, J., Ng, T. and Tan, M.J. (2004), "Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method", J. Sound Vib., 276, 997-1017. https://doi.org/10.1016/j.jsv.2003.08.026.
  36. Mirsalehi, M. Azhari, M. and Amoushahi, H. (2017), "Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method", Eur. J. Mech.-A/Solid., 61, 1-13. https://doi.org/10.1016/j.euromechsol.2016.08.008.
  37. Mizusawa, T. (1993), "Buckling of rectangular Midlin plates with tapered thickness by the spline strip method", Int. J. Solid. Struct., 30, 1663-1677. https://doi.org/10.1016/0020-7683(93)90196-E.
  38. Mohammadi Nia, M., Shojaee, S. and Hamzehei-Javaran, S. (2020), "Utilizing new spherical Hankel shape functions to reformulate the deflection, free vibration, and buckling analysis of Mindlin plates based on finite element method", Scientia Iranica, 27, 2209-2229.
  39. Mohammadi, M., Saidi, A.R. and Jomehzadeh, E. (2010), "A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply supported opposite edges", J. Mech. Eng. Sci., 224, 1831-1841. https://doi.org/10.1243/09544062JMES1804.
  40. Nguyen, N.T., Hui, D., Lee, J. and Nguyen-Xuan, H. (2015), "An efficient computational approach for size-dependent analysis of functionally graded nanoplates", Comput. Meth. Appl. Mech. Eng., 297, 191-218. https://doi.org/10.1016/j.cma.2015.07.021.
  41. Nguyen, T.N., Ngo, T.D. and Nguyen-Xuan, H. (2017), "A novel three-variable shear deformation plate formulation: Theory and Isogeometric implementation", Comput. Meth. Appl. Mech. Eng., 326, 376-401. https://doi.org/10.1016/j.cma.2017.07.024.
  42. Nguyen, T.N., Thai, C.H., Luu, A.T., Nguyen-Xuan, H. and Lee, J. (2019), "NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells", Comput. Meth. Appl. Mech. Eng., 347, 983-1003. https://doi.org/10.1016/j.cma.2019.01.011.
  43. Pham, Q.H., Nguyen, P.C., Tran, T.T. and Nguyen-Thoi, T. (2021), "Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory", Eng. Comput., 1-19. https://doi.org/10.1007/s00366-021-01531-3.
  44. Reddy, J. (1997), "On locking-free shear deformable beam finite elements", Comput. Meth. Appl. Mech. Eng., 149, 113-132. https://doi.org/10.1016/S0045-7825(97)00075-3.
  45. Ruocco, E., Mallardo, V., Minutolo, V. and Di Giacinto, D. (2017), "Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions", Appl. Math. Model., 50, 497-508. https://doi.org/10.1016/j.apm.2017.05.052.
  46. Shojaee, S., Valizadeh, N., Izadpanah, E., Bui, T. and Vu, T.V. (2012), "Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method", Compos. Struct., 94, 1677-1693. https://doi.org/10.1016/j.compstruct.2012.01.012.
  47. Sun, E.Q. (2006), "Shear locking and hourglassing in MSC nastran, ABAQUS, and ANSYS", Msc Software Users Meeting, 1-9.
  48. Tabarraei, A. and Sukumar, N. (2006), "Application of polygonal finite elements in linear elasticity", Int. J. Comput. Math., 3, 503-520. https://doi.org/10.1142/S021987620600117X.
  49. Taczala, M., Ryszard, B. and Michal, K. (2021), "Elastic-plastic buckling and postbuckling finite element analysis of plates using higher-order theory", Int. J. Struct. Stab. Dyn., 21(07), 2150095. https://doi.org/10.1142/S0219455421500954.
  50. Thai, C.H., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T.H., Nguyen-Thoi, T. and Rabczuk, T. (2012), "Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach", Int. J. Numer. Meth. Eng., 91, 571-603. https://doi.org/10.1002/nme.4282.
  51. Tran, T.T., Pham, Q.H. and Nguyen-Thoi, T. (2021), "Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method", Defence Technol., 17, 971-986. https://doi.org/10.1016/j.dt.2020.06.001.
  52. Verma, G., Sengupta, S., Mammen, S. and Bhattacharya, S. (2021), "Free vibration analysis of Reissner-Mindlin plates using FEniCS", Recent Advances in Structural Engineering: Select Proceedings of NCRASE 2020, Springer Singapore.
  53. Vu-Huu, T., Phung-Van, P., Nguyen-Xuan, H. and Wahab, M.A. (2018), "A polytree-based adaptive polygonal finite element method for topology optimization of fluid-submerged breakwater interaction", Comput. Math. Appl., 76(5), 1198-1218. https://doi.org/10.1016/j.camwa.2018.06.008.
  54. Wang, B., Lu, C., Fan, C. and Zhao, M. (2021), "A meshfree method with gradient smoothing for free vibration and buckling analysis of a strain gradient thin plate", Eng. Anal. Bound. Elem., 132, 159-167. https://doi.org/10.1016/j.enganabound.2021.07.014.
  55. Yang, Z., Chen, X., Zhang, X. and He, Z. (2013), "Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element", Appl. Math. Model., 37, 3449-3466. https://doi.org/10.1016/j.apm.2012.07.055.
  56. Zienkiewicz, O.C., Taylor, R.L. and Too, J.M. (1971), "Reduced integration technique in general analysis of plates and shells", Int. J. Numer. Meth. Eng., 3(2), 275-290. https://doi.org/10.1002/nme.1620030211.
  57. Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. (2013), The Finite Element Method: Its Basis and Fundamentals, 7th Edition, Butterworth-Heinemann, Elsevier, Amsterdam.