Advertisement

Higher-Order Weak Formulation for Arbitrarily Shaped Doubly-Curved Shells

  • Francesco TornabeneEmail author
  • Michele Bacciocchi
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 110)

Abstract

The aim of this chapter is the development of an efficient and accurate higher-order formulation to solve the weak form of the governing equations that rule the mechanical behavior of doubly-curved shell structures made of composite materials, whose reference domain can be defined by arbitrary shapes. To this aim, a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) is introduced. It should be specified that the theoretical shell model is based on the Equivalent Single Layer (ESL) approach. In addition, the Generalized Integral Quadrature technique, that is a numerical tool which can guarantee high levels of accuracy with a low computational effort in the structural analysis of the considered shell elements, is introduced. The proposed technique is able to solve numerically the integrals by means of weighted sums of the values that a smooth function assumes in some discrete points placed within the reference domain.

References

  1. 1.
    Tornabene, F., Bacciocchi, M.: Anisotropic doubly-curved shells. Higher-Order Strong and Weak Formulations for Arbitrarily Shaped Shell Structures. Esculapio, Bologna (2018)Google Scholar
  2. 2.
    Kraus, H.: Thin Elastic Shells. Wiley, New York (1967)Google Scholar
  3. 3.
    Vinson, J.R.: The Behavior of Shells Composed of Isotropic and Composite Materials. Springer, Berlin (1993)CrossRefGoogle Scholar
  4. 4.
    Jones, R.M.: Mechanics of Composite Materials, 2nd edn. Taylor & Francis, London (1999)Google Scholar
  5. 5.
    Vasiliev, V.V., Morozov, E.V.: Mechanics and Analysis of Composite Materials. Elsevier, Oxford (2001)Google Scholar
  6. 6.
    Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells, 2nd edn. CRC Press, Boca Raton (2004)CrossRefGoogle Scholar
  7. 7.
    Barbero, E.J.: Introduction to Composite Materials Design. CRC Press, Boca Raton (2011)Google Scholar
  8. 8.
    Tornabene, F.: 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, 186–206 (2011)CrossRefGoogle Scholar
  9. 9.
    Tornabene, F.: Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method. Comput. Method. Appl. M 200, 931–952 (2011)CrossRefGoogle Scholar
  10. 10.
    Tornabene, F., Fantuzzi, N., Viola, E., Reddy, J.N.: Winkler-Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels. Compos. Part B Eng. 57, 269–296 (2014)CrossRefGoogle Scholar
  11. 11.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: the free vibration analysis. Compos. Struct. 116, 637–660 (2014)CrossRefGoogle Scholar
  12. 12.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E.: A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature. Compos. Struct. 131, 433–452 (2015)CrossRefGoogle Scholar
  13. 13.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E.: Higher-order theories for the free vibration of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method. Compos. Part B Eng. 81, 196–230 (2015)CrossRefGoogle Scholar
  14. 14.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Higher-order structural theories for the static analysis of doubly-curved laminated composite panels reinforced by curvilinear fibers. Thin Wall. Struct. 102, 222–245 (2016)CrossRefGoogle Scholar
  15. 15.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: a general formulation. Compos. Part B Eng. 92, 265–289 (2016)CrossRefGoogle Scholar
  16. 16.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Neves, A.M.A., Ferreira, A.J.M.: MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells. Compos. Part B Eng. 99, 30–47 (2016)CrossRefGoogle Scholar
  17. 17.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: On the mechanics of laminated doubly-curved shells subjected to point and line loads. Int. J. Eng. Sci. 109, 115–164 (2016)CrossRefGoogle Scholar
  18. 18.
    Bacciocchi, M., Eisenberger, M., Fantuzzi, N., Tornabene, F., Viola, E.: Vibration analysis of variable thickness plates and shells by the generalized differential quadrature method. Compos. Struct. 156, 218–237 (2016)CrossRefGoogle Scholar
  19. 19.
    Brischetto, S., Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Interpretation of boundary conditions in the analytical and numerical shell solutions for mode analysis of multilayered structures. Int. J. Mech. Sci. 122, 18–28 (2017)CrossRefGoogle Scholar
  20. 20.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Reddy, J.N.: A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation. Compos. Part B Eng. 126, 162–191 (2017)CrossRefGoogle Scholar
  21. 21.
    Fantuzzi, N., Tornabene, F., Bacciocchi, M., Neves, A.M.A., Ferreira, A.J.M.: Stability and accuracy of three fourier expansion-based strong form finite elements for the free vibration analysis of laminated composite plates. Int. J. Numer. Meth. Eng. 111, 354–382 (2017)CrossRefGoogle Scholar
  22. 22.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Linear static behavior of damaged laminated composite plates and shells. Materials 10(811), 1–52 (2017)Google Scholar
  23. 23.
    Jouneghani, F.Z., Mohammadi Dashtaki, P., Dimitri, R., Bacciocchi, M., Tornabene, F.: First-order shear deformation theory for orthotropic doubly-curved shells based on a modified couple stress elasticity. Aerosp. Sci. Technol. 73, 129–147 (2018)CrossRefGoogle Scholar
  24. 24.
    Tornabene, F., Dimitri, R.: A numerical study of the seismic response of arched and vaulted structures made of isotropic or composite materials. Eng. Struct. 159, 332–366 (2018)CrossRefGoogle Scholar
  25. 25.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E.: Mechanical behavior of damaged laminated composites plates and shells: higher-order shear deformation theories. Compos. Struct. 189, 304–329 (2018)CrossRefGoogle Scholar
  26. 26.
    Tornabene, F., Bacciocchi, M.: Effect of curvilinear reinforcing fibers on the linear static behavior of soft-core sandwich structures. J. Compos. Sci. 2(14), 1–43 (2018)Google Scholar
  27. 27.
    Brischetto, S., Tornabene, F.: Advanced GDQ models and 3D stress recovery in multi-layered plates, spherical and double-curved panels subjected to transverse shear loads. Compos. Part B Eng. 146, 244–269 (2018)CrossRefGoogle Scholar
  28. 28.
    Tornabene, F., Brischetto, S.: 3D capability of refined GDQ models for the bending analysis of composite and sandwich plates, spherical and doubly-curved shells. Thin Wall. Struct. 129, 94–124 (2018)CrossRefGoogle Scholar
  29. 29.
    Tornabene, F., Bacciocchi, M.: Dynamic stability of doubly-curved multilayered shells subjected to arbitrarily oriented angular velocities: numerical evaluation of the critical speed. Compos. Struct. 201, 1031–1055 (2018)CrossRefGoogle Scholar
  30. 30.
    Tornabene, F., Fantuzzi, F., Bacciocchi, M.: Foam core composite sandwich plates and shells with variable stiffness: effect of curvilinear fiber path on the modal response. J. Sandw. Struct. Mater. 21, 320–365 (2019)CrossRefGoogle Scholar
  31. 31.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Refined shear deformation theories for laminated composite arches and beams with variable thickness: natural frequency analysis. Eng. Anal Bound Elem.100, 24–47 (2019)CrossRefGoogle Scholar
  32. 32.
    Tornabene, F., Viola, E.: Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 44, 255–281 (2009)CrossRefGoogle Scholar
  33. 33.
    Tornabene, F., Viola, E., Inman, D.J.: 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. J. Sound Vib. 328, 259–290 (2009)CrossRefGoogle Scholar
  34. 34.
    Tornabene, F.: Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput. Method Appl. M 198, 2911–2935 (2009)CrossRefGoogle Scholar
  35. 35.
    Tornabene, F., Viola, E.: Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution. Eur. J. Mech. A Solid 28, 991–1013 (2009)CrossRefGoogle Scholar
  36. 36.
    Viola, E., Tornabene, F.: Free vibrations of three parameter functionally graded parabolic panels of revolution. Mech. Res. Commun. 36, 587–594 (2009)CrossRefGoogle Scholar
  37. 37.
    Tornabene, F., Viola, E.: Static analysis of functionally graded doubly-curved shells and panels of revolution. Meccanica 48, 901–930 (2013)CrossRefGoogle Scholar
  38. 38.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos. Part B Eng. 67, 490–509 (2014)CrossRefGoogle Scholar
  39. 39.
    Tornabene, F., Fantuzzi, N., Viola, E., Batra, R.C.: Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory. Compos. Struct. 119, 67–89 (2015)CrossRefGoogle Scholar
  40. 40.
    Fantuzzi, N., Tornabene, F., Viola, E.: Four-parameter functionally graded cracked plates of arbitrary shape: a GDQFEM solution for free vibrations. Mech. Adv. Mater. Struct. 23, 89–107 (2016)CrossRefGoogle Scholar
  41. 41.
    Brischetto, S., Tornabene, F., Fantuzzi, N., Viola, E.: 3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders. Meccanica 51, 2059–2098 (2016)CrossRefGoogle Scholar
  42. 42.
    Fantuzzi, N., Brischetto, S., Tornabene, F., Viola, E.: 2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels. Compos. Struct. 154, 573–590 (2016)CrossRefGoogle Scholar
  43. 43.
    Tornabene, F., Brischetto, S., Fantuzzi, N., Bacciocchi, M.: Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures. Shock Vib. 2373862, 1–17 (2016)Google Scholar
  44. 44.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E., Reddy, J.N.: A numerical investigation on the natural frequencies of FGM sandwich shells with variable thickness by the local generalized differential quadrature method. Appl. Sci. 7(131), 1–39 (2017)Google Scholar
  45. 45.
    Jouneghani, F.Z., Dimitri, R., Bacciocchi, M., Tornabene, F.: Free vibration analysis of functionally graded porous doubly-curved shells based on the first-order shear deformation theory. Appl. Sci. 7(1252), 1–20 (2017)Google Scholar
  46. 46.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E.: Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos. Part B Eng. 89, 187–218 (2016)CrossRefGoogle Scholar
  47. 47.
    Kamarian, S., Salim, M., Dimitri, R., Tornabene, F.: Free vibration analysis of conical shells reinforced with agglomerated carbon nanotubes. Int. J. Mech. Sci. 108–109, 157–165 (2016)CrossRefGoogle Scholar
  48. 48.
    Fantuzzi, N., Tornabene, F., Bacciocchi, M., Dimitri, R.: Free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates. Compos. Part B Eng. 115, 384–408 (2017)CrossRefGoogle Scholar
  49. 49.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Compos. Part B Eng. 115, 449–476 (2017)CrossRefGoogle Scholar
  50. 50.
    Nejati, M., Asanjarani, A., Dimitri, R., Tornabene, F.: Static and free vibration analysis of functionally graded conical shells reinforced by carbon nanotubes. Int. J. Mech. Sci. 130, 383–398 (2017)CrossRefGoogle Scholar
  51. 51.
    Nejati, M., Dimitri, R., Tornabene, F., Yas, M.H.: Thermal buckling of nanocomposite stiffened cylindrical shells reinforced by functionally graded wavy carbon nano-tubes with temperature-dependent properties. Appl. Sci. 7(1223), 1–24 (2017)Google Scholar
  52. 52.
    Banić, D., Bacciocchi, M., Tornabene, F., Ferreira, A.J.M.: Influence of Winkler-Pasternak foundation on the vibrational behavior of plates and shells reinforced by agglomerated carbon nanotubes. Appl. Sci. 7(1228), 1–55 (2017)Google Scholar
  53. 53.
    Tornabene, F., Bacciocchi, M., Fantuzzi, N., Reddy, J.N.: Multiscale approach for three-phase CNT/polymer/fiber laminated nanocomposite structures. Polym. Compos. 40, E102–E126 (2019)CrossRefGoogle Scholar
  54. 54.
    Whitney, J.M., Pagano, N.J.: Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. T ASME 37, 1031–1036 (1970)CrossRefGoogle Scholar
  55. 55.
    Whitney, J.M., Sun, C.T.: A higher order theory for extensional motion of laminated composites. J. Sound Vib. 30, 85–97 (1973)CrossRefGoogle Scholar
  56. 56.
    Reissner, E.: On transverse bending of plates, including the effect of transverse shear deformation. Int. J. Solids Struct. 11, 569–573 (1975)CrossRefGoogle Scholar
  57. 57.
    Green, A.E., Naghdi, P.M.: A theory of composite laminated plates. IMA J. Appl. Math. 29, 1–23 (1982)CrossRefGoogle Scholar
  58. 58.
    Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. T ASME 51, 745–752 (1984)CrossRefGoogle Scholar
  59. 59.
    Bert, C.W.: A critical evaluation of new plate theories applied to laminated composites. Compos. Struct. 2, 329–347 (1984)CrossRefGoogle Scholar
  60. 60.
    Reddy, J.N., Liu, C.F.: A higher-order shear deformation theory for laminated elastic shells. Int. J. Eng. Sci. 23, 319–330 (1985)CrossRefGoogle Scholar
  61. 61.
    Reddy, J.N.: A generalization of the two-dimensional theories of laminated composite plates. Commun. Appl. Numer. M 3, 173–180 (1987)CrossRefGoogle Scholar
  62. 62.
    Librescu, L., Reddy, J.N.: A few remarks concerning several refined theories of anisotropic composite laminated plates. Int. J. Eng. Sci. 27, 515–527 (1989)CrossRefGoogle Scholar
  63. 63.
    Reddy, J.N.: On refined theories of composite laminates. Meccanica 25, 230–238 (1990)CrossRefGoogle Scholar
  64. 64.
    Robbins, D.H., Reddy, J.N.: Modeling of thick composites using a layer-wise laminate theory. Int. J. Numer. Meth. Eng. 36, 655–677 (1993)CrossRefGoogle Scholar
  65. 65.
    Carrera, E.: A refined multi-layered finite-element model applied to linear and non-linear analysis of sandwich plates. Compos. Sci. Technol. 58, 1553–1569 (1998)CrossRefGoogle Scholar
  66. 66.
    Carrera, E.: Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch. Comput. Methods Eng. 9, 87–140 (2002)CrossRefGoogle Scholar
  67. 67.
    Carrera, E.: Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Methods Eng. 10, 215–296 (2003)CrossRefGoogle Scholar
  68. 68.
    Carrera, E.: Historical review of zig-zag theories for multilayered plates and shells. Appl. Mech. Rev. 56, 287–308 (2003)CrossRefGoogle Scholar
  69. 69.
    Carrera, E.: On the use of the Murakami’s zig-zag function in the modeling of layered plates and shells. Comput. Struct. 82, 541–554 (2004)CrossRefGoogle Scholar
  70. 70.
    Dozio, L.: A hierarchical formulation of the state-space Levy’s method for vibration analysis of thin and thick multilayered shells. Compos. Part B Eng. 98, 97–107 (2016)CrossRefGoogle Scholar
  71. 71.
    Dozio, L., Alimonti, L.: Variable kinematic finite element models of multilayered composite plates coupled with acoustic fluid. Mech. Adv. Mater. Struct. 23, 981–996 (2016)CrossRefGoogle Scholar
  72. 72.
    Vescovini, R., Dozio, L.: A variable-kinematic model for variable stiffness plates: vibration and buckling analysis. Compos. Struct. 142, 15–26 (2016)CrossRefGoogle Scholar
  73. 73.
    Wenzel, C., D’Ottavio, M., Polit, O., Vidal, P.: Assessment of free-edge singularities in composite laminates using higher-order plate elements. Mech. Adv. Mat. Struct. 23, 948–959 (2016)CrossRefGoogle Scholar
  74. 74.
    Demasi, L.: \(\infty ^3\) hierarchy plate theories for thick and thin composite plates: the generalized unified formulation. Compos. Struct. 84, 256–270 (2008)CrossRefGoogle Scholar
  75. 75.
    D’Ottavio, M.: A sublaminate generalized unified formulation for the analysis of composite structures. Compos. Struct. 142, 187–199 (2016)CrossRefGoogle Scholar
  76. 76.
    Viola, E., Tornabene, F., Fantuzzi, N.: Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories. Compos. Struct. 101, 59–93 (2013)CrossRefGoogle Scholar
  77. 77.
    Viola, E., Tornabene, F., Fantuzzi, N.: General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels. Compos. Struct. 95, 639–666 (2013)CrossRefGoogle Scholar
  78. 78.
    Tornabene, F., Viola, E., Fantuzzi, N.: General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels. Compos. Struct. 104, 94–117 (2013)CrossRefGoogle Scholar
  79. 79.
    Tornabene, F., Fantuzzi, N., Viola, E., Ferreira, A.J.M.: Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer formulation. Compos. Part B Eng. 55, 642–659 (2013)CrossRefGoogle Scholar
  80. 80.
    Tornabene, F., Fantuzzi, N., Viola, E., Carrera, E.: Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method. Compos. Struct. 107, 675–697 (2014)CrossRefGoogle Scholar
  81. 81.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E.: Accurate inter-laminar recovery for plates and doubly-curved shells with variable radii of curvature using layer-wise theories. Compos. Struct. 124, 368–393 (2015)CrossRefGoogle Scholar
  82. 82.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Dimitri, R.: Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials. Compos. Struct. 133, 278–299 (2015)CrossRefGoogle Scholar
  83. 83.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Dimitri, R.: Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method. Thin Wall. Struct. 97, 114–129 (2015)CrossRefGoogle Scholar
  84. 84.
    Tornabene, F., Fantuzzi, N., Viola, E.: Inter-laminar stress recovery procedure for doubly-curved, singly-curved, revolution shells with variable radii of curvature and plates using generalized higher-order theories and the local GDQ method. Mech. Adv. Mater. Struct. 23, 1019–1045 (2016)CrossRefGoogle Scholar
  85. 85.
    Tornabene, F.: General higher order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels. Mech. Adv. Mater. Struct. 23, 1046–1067 (2016)CrossRefGoogle Scholar
  86. 86.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M., Reddy, J.N.: An equivalent layer-wise approach for the free vibration analysis of thick and thin laminated sandwich shells. Appl. Sci. 7(17), 1–34 (2017)Google Scholar
  87. 87.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Strong and weak formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells: convergence and accuracy. Eng. Anal. Bound Elem. 92, 3–37 (2018)CrossRefGoogle Scholar
  88. 88.
    Tornabene, F., Fantuzzi, N., Ubertini, F., Viola, E.: Strong formulation finite element method based on differential quadrature: a survey. Appl. Mech. Rev. 67, 020801 (2015)CrossRefGoogle Scholar
  89. 89.
    Shu, C.: Differential Quadrature and Its Application in Engineering. Springer, Berlin (2000)CrossRefGoogle Scholar
  90. 90.
    Fantuzzi, N., Tornabene, F., Viola, E., Ferreira, A.J.M.: 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. Meccanica 49, 2503–2542 (2014)CrossRefGoogle Scholar
  91. 91.
    Fantuzzi, N.: New insights into the strong formulation finite element method for solving elastostatic and elastodynamic problems. Curved Layer Struct. 1, 93–126 (2014)Google Scholar
  92. 92.
    Fantuzzi, N., Bacciocchi, M., Tornabene, F., Viola, E., Ferreira, A.J.M.: Radial basis functions based on differential quadrature method for the free vibration of laminated composite arbitrary shaped plates. Compos. Part B Eng. 78, 65–78 (2015)CrossRefGoogle Scholar
  93. 93.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: Mechanical behaviour of composite cosserat solids in elastic problems with holes and discontinuities. Compos. Struct. 179, 468–481 (2017)CrossRefGoogle Scholar
  94. 94.
    Fantuzzi, N., Leonetti, L., Trovalusci, P., Tornabene, F.: Some novel numerical applications of cosserat continua. Int. J. Comp. Meth. 15, 1850054 (2018)CrossRefGoogle Scholar
  95. 95.
    Fantuzzi, N., Tornabene, F., Bacciocchi, M., Ferreira, A.J.M.: On the convergence of laminated composite plates of arbitrary shape through finite element models. J. Compos. Sci. 2(16), 1–50 (2018)Google Scholar
  96. 96.
    Fantuzzi, N., Tornabene, F.: Strong formulation isogeometric analysis (SFIGA) for laminated composite arbitrarily shaped plates. Compos. Part B Eng. 96, 173–203 (2016)CrossRefGoogle Scholar
  97. 97.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: The GDQ method for the free vibration analysis of arbitrarily shaped laminated composite shells using a NURBS-based isogeometric approach. Compos. Struct. 154, 190–218 (2016)CrossRefGoogle Scholar
  98. 98.
    Fantuzzi, N., Puppa, G.D., Tornabene, F., Trautz, M.: Strong formulation isogeometric analysis for the vibration of thin membranes of general shape. Int. J. Mech. Sci. 120, 322–340 (2017)CrossRefGoogle Scholar
  99. 99.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: A new doubly-curved shell element for the free vibrations of arbitrarily shaped laminated structures based on weak formulation isogeometric analysis. Compos. Struct. 171, 429–461 (2017)CrossRefGoogle Scholar
  100. 100.
    Tornabene, F., Fantuzzi, N., Bacciocchi, M.: DiQuMASPAB: Differential Quadrature for Mechanics of Anisotropic Shells, Plates, Arches and Beams. User Manual. Esculapio, Bologna. https://DiQuMASPAB.editrice-esculapio.com (2018)
  101. 101.
    Liew, K.M., Wang, C.M., Xiang, Y., Kitipornchai, S.: Vibration of Mindlin Plates. Elsevier, London (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of SalentoLecceItaly
  2. 2.University of BolognaBolognaItaly

Personalised recommendations