Advances in aircraft and spacecraft science

  • 제1권2호
  • /
  • Pages.145-175
  • /
  • 2014
  • /
  • 2287-528X(pISSN)
  • /
  • 2287-5271(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Strong formulation finite element method for arbitrarily shaped laminated plates - Part II. Numerical analysis

  • Fantuzzi, Nicholas (Department of Civil, Chemical, Environmental and Materials - DICAM, University of Bologna) ;
  • Tornabene, Francesco (Department of Civil, Chemical, Environmental and Materials - DICAM, University of Bologna)
  • 투고 : 2013.12.17
  • 심사 : 2014.01.17
  • 발행 : 2014.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

The results of a series of numerical experiments are presented to verify some of the important developments made in the first part of this paper. Firstly, the static solution of an algebraic system obtained through Strong Formulation Finite Element Method (SFEM) is presented. Secondly, the stress and strain recovery procedure is descripted for the present technique. It will be clear that the present approach is suitable for any strong formulation finite element methodology, due to the presented general approach based on the unknown displacements and on the elasticity equations. Thirdly, the numerical solutions for some classical and other numerical results found in literature are exposed. Finally, an arbitrarily shaped composite plate is solved and good agreement is observed for all the presented cases.

키워드

참고문헌

  1. Bellman, R.E. and Casti, J. (1971), "Differential quadrature and long-term integration", J. Math. Anal. Appl., 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  2. Bellman, R.E., Kashef, B.G. and Casti, J. (1972), "Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations", J. Comput. Phys., 10(1), 40-52. https://doi.org/10.1016/0021-9991(72)90089-7
  3. Boyd, J.P. (2001), Chebyshev and Fourier spectral methods, Dover Publications.
  4. Chen, C.N. (1999a), "The development of irregular elements for differential quadrature element method steady-state heat conduction analysis", Comp. Meth. Appl. Mech. Eng., 170(1-2), 1-14. https://doi.org/10.1016/S0045-7825(98)00185-6
  5. Chen, C.N. (1999b), "The differential quadrature element method irregular element torsion analysis model", Appl. Math. Model, 23(4), 309-328. https://doi.org/10.1016/S0307-904X(98)10083-5
  6. Chen, C.N. (2000a), "A generalized differential quadrature element method", Comp. Meth. Appl. Mech. Eng., 188(1-3), 553-566. https://doi.org/10.1016/S0045-7825(99)00283-2
  7. Chen, C.N. (2000b), "Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integrated numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes", Appl. Math. Model., 24(8-9), 637-655. https://doi.org/10.1016/S0307-904X(00)00007-X
  8. Chen, W.L., Striz, A.G. and Bert, C.W. (1997), "A new approach to the differential quadrature method for fourth-order equations", Int. J. Numer. Methods Eng., 40(11), 1941-1956. https://doi.org/10.1002/(SICI)1097-0207(19970615)40:11<1941::AID-NME145>3.0.CO;2-V
  9. Chen, W.L., Striz, A.G. and Bert, C.W. (2000), "High accuracy plane stress and plate element in the quadrature element method", Int. J. Solids Struct., 37(4), 627-647. https://doi.org/10.1016/S0020-7683(99)00028-1
  10. Fantuzzi, N. (2013), "Generalized differential quadrature finite element method applied to advanced structural mechanics", Ph.D., Thesis, University of Bologna.
  11. Fantuzzi, N. and Tornabene, F. (2014), "Strong formulation finite element method for arbitrarily shaped plates - I. Theoretical", Adv. Aircraft Spacecraft Sci., 1(2), 124-142.
  12. Fantuzzi, N., Tornabene, F. and Viola, E. (2014), "Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes", Int. J. Mech. Sci., 79, 216-251. https://doi.org/10.1016/j.ijmecsci.2013.12.008
  13. Ferreira, A.J.M., Viola, E., Tornabene, F., Fantuzzi, N. and Zenkour, A.M. (2013), "Analysis of sandwich plates by generalized differential quadrature method", Math. Probl. Eng., 2013, 1-12, Article ID 964367, http://dx.doi.org/10.1155/2013/964367.
  14. Ferreira, A.J.M., Carrera, E., Cinefra, M., Viola, E., Tornabene, F., Fantuzzi, N. and Zenkour, A.M. (2014), "Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a unified formulation", Compos. Part B-Eng., 58(1), 544-552. https://doi.org/10.1016/j.compositesb.2013.10.088
  15. Gottlieb, D. and Orszag, S.A. (1977), Numerical analysis of spectral methods: theory and applications, CBMSNSF, SIAM.
  16. Jang, S., Bert, C.W. and Striz, A.G. (1989), "Application of differential quadrature to static analysis of structural components", Int. J. Numer. Meth. Eng., 28(3), 561-577. https://doi.org/10.1002/nme.1620280306
  17. Marzani, A., Tornabene, F. and Viola, E. (2008), "Nonconservative stability problems via generalized differential quadrature method", J. Sound Vib., 315(1-2), 176-196. https://doi.org/10.1016/j.jsv.2008.01.056
  18. Natarajan, S., Ferreira, A.J.M., Bordas, S.P.A., Carrera, E. and Cinefra, M. (2013), "Analysis of composite plates by a unified formulation-cell based smoothed finite element method and field consistent elements", Compos. Struct., 105, 75-81. https://doi.org/10.1016/j.compstruct.2013.04.040
  19. Orszag, S.A. (1969), "Numerical methods for the simulation of turbulence", Phys. Fluids Suppl. II, 12(12), 250-257. https://doi.org/10.1063/1.1692279
  20. Orszag, S.A. (1980), "Spectral methods for problems in complex geometries", J. Comput. Phys., 37(1), 70-92. https://doi.org/10.1016/0021-9991(80)90005-4
  21. Reddy, J.N. (2003), Mechanics of laminated composite plates and shells, CRC Press.
  22. Shu, C., Chew, Y.T., Khoo, B.C. and Yeo, K.S. (1995), "Application of GDQ scheme to simulate incompressible viscous flows around complex geometries", Mech. Res. Commun., 22(4), 319-325. https://doi.org/10.1016/0093-6413(95)00031-L
  23. Shu, C. and Chew, Y.T. (1997), "Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems", Commun. Numer. Meth. En., 13, 643-653. https://doi.org/10.1002/(SICI)1099-0887(199708)13:8<643::AID-CNM92>3.0.CO;2-F
  24. Shu, C. and Chew, Y.T. (1999), "Application of multi-domain GDQ method to analysis of waveguides with rectangular boundaries", Prog. Electromagn. Res., 21, 1-19. https://doi.org/10.2528/PIER98052601
  25. Shu, C. (2000), Differential quadrature and its application in engineering, Springer.
  26. Tornabene, F. and Viola, E. (2007), "Vibration analysis of spherical structural elements using the GDQ method", Comp. Math. Appl., 53(10), 1538-1560. https://doi.org/10.1016/j.camwa.2006.03.039
  27. Tornabene, F. and Viola, E. (2008), "2-D solution for free vibrations of parabolic shells using generalized differential quadrature method", Eur. J. Mech. A-Solid., 27(6), 1001-1025. https://doi.org/10.1016/j.euromechsol.2007.12.007
  28. Tornabene, F. (2009), "Vibration analysis of functionally graded conical, cylindrical and annular shell structures with a four-parameter power-law distribution", Comp. Meth. Appl. Mech. Eng., 198(37-40), 2911-2935. https://doi.org/10.1016/j.cma.2009.04.011
  29. Tornabene, F. and Viola, E. (2009a), "Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution", Eur. J. Mech. A-Solid., 28(5), 991-1013. https://doi.org/10.1016/j.euromechsol.2009.04.005
  30. Tornabene, F. and Viola, E. (2009b), "Free vibration analysis of functionally graded panels and shells of revolution", Meccanica, 44(3), 255-281. https://doi.org/10.1007/s11012-008-9167-x
  31. Tornabene, F., Viola, E. and Inman, D.J. (2009), "2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures", J. Sound Vib., 328, 259-290. https://doi.org/10.1016/j.jsv.2009.07.031
  32. Tornabene, F., Marzani, A., Viola, E. and Elishakoff, I. (2010), "Critical flow speeds of pipes conveying fluid by the generalized differential quadrature method", Adv. Theor. Appl. Mech., 3(3), 121-138.
  33. Tornabene, F. (2011a), "2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution", Compos. Struct., 93(7), 1854-1876. https://doi.org/10.1016/j.compstruct.2011.02.006
  34. Tornabene, F. (2011b), "Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations", Compos. Struct., 94(1), 186-206. https://doi.org/10.1016/j.compstruct.2011.07.002
  35. Tornabene, F. (2011c), "Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method", Comp. Meth. Appl. Mech. Eng., 200(9-12), 931-952. https://doi.org/10.1016/j.cma.2010.11.017
  36. Tornabene, F., Liverani, A. and Caligiana, G. (2011), "FGM and laminated doubly-curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations", Int. J. Mech. Sci., 53(6), 446-470. https://doi.org/10.1016/j.ijmecsci.2011.03.007
  37. Tornabene, F. (2012), Meccanica delle Strutture a Guscio in Materiale Composito, Esculapio, Bologna.
  38. Tornabene, F., Liverani, A. and Caligiana, G. (2012a), "General anisotropic doubly-curved shell theory: a differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian", J. Sound Vib., 331(22), 4848-4869. https://doi.org/10.1016/j.jsv.2012.05.036
  39. Tornabene, F., Liverani, A. and Caligiana, G. (2012b), "Laminated composite rectangular and annular plates: a GDQ solution for static analysis with a posteriori shear and normal stress recovery", Compos. Part B-Eng., 43(4), 1847-1872. https://doi.org/10.1016/j.compositesb.2012.01.065
  40. Tornabene, F., Liverani, A. and Caligiana, G. (2012c), "Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method", Int. J. Mech. Sci., 61(1), 71-87. https://doi.org/10.1016/j.ijmecsci.2012.05.007
  41. Tornabene, F. and Ceruti, A. (2013a), "Free-form laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations: a 2-D GDQ solution for static and free vibration analysis", World J. Mechanics., 3(1), 1-25.
  42. Tornabene, F. and Ceruti, A. (2013b), "Mixed static and dynamic optimization of four-parameter functionally graded completely doubly-curved and degenerate shells and panels using GDQ method", Math. Probl. Eng., 2013, 1-33, Article ID 867089, http://dx.doi.org/10.1155/2013/867079.
  43. Tornabene, F., Fantuzzi, N., Viola, E. and Ferreira, A.J.M. (2013a), "Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer theory", Compos. Part B-Eng., 55(1), 642-659. https://doi.org/10.1016/j.compositesb.2013.07.026
  44. Tornabene, F. and Reddy, J.N. (2013), "FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundation: a GDQ solution for static analysis with a posteriori stress and strain recovery", J. Indian Inst. Sci., 93(4), 635-688.
  45. Tornabene, F. and Viola, E. (2013), "Static analysis of functionally graded doubly-curved shells and panels of revolution", Meccanica, 48(4), 901-930. https://doi.org/10.1007/s11012-012-9643-1
  46. Tornabene, F., Viola, E. and Fantuzzi, N. (2013b), "General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels", Compos. Struct., 104, 94-117. https://doi.org/10.1016/j.compstruct.2013.04.009
  47. Tornabene, F. and Fantuzzi, N. (2014), Mechanics of laminated composite doubly-curved shell structures, Esculapio, Bologna.
  48. Tornabene, F., Fantuzzi, N., Viola, E. and Carrera, E. (2014a), "Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method", Compos. Struct., 107(1), 675-697. https://doi.org/10.1016/j.compstruct.2013.08.038
  49. Tornabene, F., Fantuzzi, N., Viola, E. and Reddy, J.N. (2014b), "Winkler-Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels", Compos. Part B-Eng., 57(1), 269-296. https://doi.org/10.1016/j.compositesb.2013.06.020
  50. Viola, E. and Tornabene, F. (2005), "Vibration analysis of damaged circular arches with varying cross-section", Struct. Integr. Durab. (SID-SDHM), 1(2), 155-169.
  51. Viola, E. and Tornabene, F. (2006), "Vibration analysis of conical shell structures using GDQ method", Far East J. Appl. Math., 25(1), 23-39.
  52. Viola, E., Dilena, M. and Tornabene, F. (2007), "Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches", J. Sound Vib., 299(1-2), 143-163. https://doi.org/10.1016/j.jsv.2006.07.001
  53. Viola, E. and Tornabene, F. (2009), "Free vibrations of three parameter functionally graded parabolic panels of revolution", Mech. Res. Commun., 36(5), 587- 594. https://doi.org/10.1016/j.mechrescom.2009.02.001
  54. Viola, E., Rossetti, L. and Fantuzzi, N. (2012), "Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery", Compos. Struct., 94(12), 3736-3758. https://doi.org/10.1016/j.compstruct.2012.05.034
  55. Viola, E., Tornabene, F. and Fantuzzi, N. (2013a), "Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape", Compos. Struct., 106(1), 815-834. https://doi.org/10.1016/j.compstruct.2013.07.034
  56. Viola, E., Tornabene, F. and Fantuzzi, N. (2013b), "Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories", Compos. Struct., 101, 59-93. https://doi.org/10.1016/j.compstruct.2013.01.002
  57. Viola, E., Tornabene, F. and Fantuzzi, N. (2013c), "General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels", Compos. Struct., 95, 639-666. https://doi.org/10.1016/j.compstruct.2012.08.005
  58. Viola, E., Tornabene, F., Ferretti, E. and Fantuzzi, N. (2013d), "Soft core plane state structures under static loads using GDQFEM and cell method", CMES-Comp. Model. Eng., 94(4), 301-329.
  59. Viola, E., Tornabene, F., Ferretti, E. and Fantuzzi, N. (2013e), "GDQFEM numerical simulations of continuous media with cracks and discontinuities", CMES-Comp. Model. Eng., 94(4), 331-369.
  60. Viola, E., Tornabene, F., Ferretti, E. and Fantuzzi, N. (2013f), "On static analysis of composite plane state structures via GDQFEM and cell method", CMES-Comp. Model. Eng., 94(5), 421-458.
  61. Xing, Y. and Liu, B. (2009), "High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain", Int. J. Numer. Meth. Eng., 80(13), 1718-1742. https://doi.org/10.1002/nme.2685
  62. Xing, Y., Liu, B. and Liu, G. (2010), "A differential quadrature finite element method", Int. J. Appl. Mechanics, 2(1), 1-20. https://doi.org/10.1142/S1758825110000433
  63. Zong, Z. and Zhang, Y. (2009), Advanced differential quadrature methods, CRC Press.

피인용 문헌

  1. Optimization of flutter boundaries of cantilevered trapezoidal functionally graded sandwich plates 2017, https://doi.org/10.1177/1099636217697492
  2. Vibration and flutter analyses of cantilever trapezoidal honeycomb sandwich plates 2017, https://doi.org/10.1177/1099636217728746
  3. Stability and accuracy of three Fourier expansion-based strong form finite elements for the free vibration analysis of laminated composite plates vol.111, pp.4, 2017, https://doi.org/10.1002/nme.5468
  4. A strong formulation finite element method (SFEM) based on RBF and GDQ techniques for the static and dynamic analyses of laminated plates of arbitrary shape vol.49, pp.10, 2014, https://doi.org/10.1007/s11012-014-0014-y
  5. A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners vol.7, pp.2, 2016, https://doi.org/10.1016/j.asej.2015.05.016
  6. Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations vol.23, pp.1, 2016, https://doi.org/10.1080/15376494.2014.933992
  7. Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method vol.52, 2015, https://doi.org/10.1016/j.euromechsol.2015.02.004
  8. A new component mode synthesis for dynamic mixed thin plate finite element models vol.86, pp.5, 2016, https://doi.org/10.1007/s00419-015-1072-x
  9. Optimization for flutter boundaries of cantilevered trapezoidal thick plates vol.39, pp.5, 2017, https://doi.org/10.1007/s40430-016-0688-2
  10. Static analysis of higher order sandwich beams by weak form quadrature element method vol.116, 2014, https://doi.org/10.1016/j.compstruct.2014.06.015
  11. Various double component mode synthesis and sub-structuring methods for dynamic mixed FEM vol.53, 2015, https://doi.org/10.1016/j.euromechsol.2015.04.005
  12. Strong Formulation IsoGeometric Analysis for the vibration of thin membranes of general shape vol.120, 2017, https://doi.org/10.1016/j.ijmecsci.2016.10.033
  13. Nonlinear dynamic analysis and vibration of shear deformable piezoelectric FGM double curved shallow shells under damping-thermo-electro-mechanical loads vol.125, 2015, https://doi.org/10.1016/j.compstruct.2015.01.041