An assessment of refined hierarchical kinematic models for the bending and free vibration analyses of laminated and functionally graded sandwich cylindrical panels
Restricted accessResearch articleFirst published online September, 2021
An assessment of refined hierarchical kinematic models for the bending and free vibration analyses of laminated and functionally graded sandwich cylindrical panels
An effort is made in this study to systematically develop and evaluate two-dimensional (2D) displacement-based higher order theories for stress and vibration analyses of moderately thick laminated and functionally graded (FG) sandwich cylindrical shell panels. Comparison of various displacement models is presented in order to assess the contribution of thickness stretching effects in laminated and sandwich shells. The equivalent single layer (ESL) approach is followed and a refined thickness criterion is adopted to enhance the accuracy of present formulation for moderately thick to thick cylindrical shell panels. Contribution of membrane and bending deformations is examined using the hierarchical kinematic models. Governing set of equations is obtained using energy minimization based on a variational approach and solution is obtained subsequently with Navier’s method for cylindrical shells with all edges on diaphragm (simple) supports. Effective FG material properties are obtained using Voigt’s rule of mixture with power law gradation. Solutions are compared with the available three dimensional (3D) solutions and a comparative assessment is done for different higher order theories. The present study accentuates the necessity of including transverse normal strain in the theory for FG sandwich panels.
SureshSMortesnsenA.Fundamentals of functionally graded materials.
London:
IOM Publication, 1998.
2.
PaganoNJ.Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater1970;
4: 20–34.
3.
SrinivasSRaoAK.Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct1970;
6: 1463–1481.
BhimaraddiAChandrashekharaK.Three-dimensional elasticity solution for static response of simply supported orthotropic cylindrical shells. Compos Struct1992;
20: 227–235.
6.
YeJSoldatosKP.Three-dimensional stress analysis of orthotropic and cross-ply laminated hollow cylinders and cylindrical panels. Comput Meth Appl Mech Eng1994;
117: 331–351.
7.
VaradanTKBhaskarK.Bending of laminated orthotropic cylindrical shells – an elasticity approach. Compos Struct1991;
17: 141–156.
8.
FanJZhangJ.Analytical solutions for thick, doubly curved, laminated shells. J Eng Mech1992;
118: 1338–1356.
9.
Jia-RangFJu-YongZ.Exact solutions for thick laminated shells. Sci China (Series A)1992;
35: 1343–1355.
10.
AndersonTA.A 3-D elasticity solution for a sandwich composite with functionally graded core subjected to transverse loading by a rigid sphere. Compos Struct2003;
60: 265–274.
11.
KashtalyanMMenshykovaM.Three-dimensional elasticity solution for sandwich panels with a functionally graded core. Compos Struct2009;
87: 36–43.
12.
WoodwardBKashtalyanM.3D elasticity analysis of sandwich panels with graded core under distributed and concentrated loadings. Int J Mech Sci2011;
53: 872–885.
13.
AlibeiglooA.Three-dimensional thermo-elasticity solution of sandwich cylindrical panel with functionally graded core. Compos Struct2014;
107: 458–468.
14.
Brischetto S. Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells. Journal of Sandwich Structures & Materials2019;
21: 77–114.
15.
BrischettoS.an Exact 3D solution for free vibrations of multilayered cross-ply composite and sandwich plates and shells. Int J Appl Mechanics2014;
6: 1450076.
16.
BrischettoS.A general exact elastic shell solution for bending analysis of functionally graded structures. Compos Struct2017;
175: 70–85.
17.
ReddyJNLiuCF.A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci1985;
23: 319–330.
18.
YangPCNorrisCHStavskyY.Elastic wave propagation in heterogeneous plates. Int J Solids Struct1966;
2: 665–684.
19.
PuneraDKantT.Elastostatics of laminated and functionally graded sandwich cylindrical shells with two refined higher order models. Compos Struct2017;
182: 505–523.
20.
RohwerKRolfesRSparrH.High-order theories for thermal stresses in layered plates. Int J Solids Struct2001;
38: 3673–3687.
21.
CalladineCR.Theory of shell structures.
Cambridge:
Cambridge University Press, 1983.
22.
KoiterWT. A consistent first approximation in the general theory of thin elastic shells. In: Symposium on the theory of thin elastic shells, IUTAM. Amsterdam: North Holland Pub. Co., 1960, pp. 12–33.
23.
CarreraEBrischettoSCinefraM, et al.
Effects of thickness stretching in functionally graded plates and shells. Compos Part B Eng2011;
42: 123–133.
24.
ReddyJN.An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Compos Struct1993;
25: 21–35.
25.
CarreraE.Historical review of Zig-Zag theories for multilayered plates and shells. Appl Mech Rev2003;
56: 287.
26.
KapaniaRKRacitiS.Recent advances in analysis of laminated beams and plates, Part I: shear effects and buckling. AIAA J1989;
27: 935–946.
27.
AbrateSDi SciuvaM.Equivalent single layer theories for composite and sandwich structures: a review. Compos Struct2017;
179: 482–494.
28.
GulatiSTEssenburgF.Effects of anisotropy in axisymmetric cylindrical shells. J Appl Mech1967;
34: 659.
BerchaFGGlocknerPG.Thick shell stress resultants. J Eng Mech Div ASCE1972;
98: 757–762.
31.
BerchaFGGlocknerPG.Thick shell and oriented surface theories. J Eng Mech Div1972;
98: 823–833.
32.
WhitneyJMSunC-T.A refined theory for laminated anisotropic, cylindrical shells. J Appl Mech1974;
41: 471.
33.
KantT.Thick shells of revolution-some studies.
India:
Indian Institute of Technology Bombay, 1976.
34.
KantTRameshCK. Analysis of thick orthotropic shells. In: Proceedings of the IASS World Congress on Space Enclosures. Montereal, Canada, 4-9 July 1976, pp. 401–409.
35.
BhimaraddiA.A higher order theory for free vibration analysis of circular cylindrical shells. Int J Solids Struct1984;
20: 623–630.
36.
KantTSwaminathanK.Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Compos Struct2002;
56: 329–344.
37.
KantTKhareR.A higher-order facet quadrilateral composite shell element. Int J Numer Meth Engng1997;
40: 4477–4499.
38.
KhareRKKantTGargAK.Closed-form thermo-mechanical solutions of higher-order theories of cross-ply laminated shallow shells. Compos Struct2003;
59: 313–340.
39.
DasMBarutAMadenciE, et al.
A triangular plate element for thermo-elastic analysis of sandwich panels with a functionally graded core. Int J Numer Meth Engng2006;
68: 940–966.
40.
CarreraE.A class of two-dimensional theories for anisotropic multilayered plates analysis. 19th–20th ed.
Torino:
Accademia delle Scienze, 1995.
41.
CarreraE.Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng2003;
10: 215–296.
42.
FazzolariFACarreraE.Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core. J Sound Vib2014;
333: 1485–1508.
43.
CarreraECinefraMFazzolariFA.Some results on thermal stress of layered plates and shells by using unified formulation. J Therm Stress2013;
36: 589–625.
44.
BrischettoS.Hygrothermoelastic analysis of multilayered composite and sandwich shells. J Sandw Struct Mater2013;
15: 168–202.
45.
GroverNSinghBNMaitiDK.New nonpolynomial shear-deformation theories for structural behavior of laminated-composite and sandwich plates. AIAA J2013;
51: 1861–1871.
46.
MantariJLBonillaEMGuedes SoaresC.A new tangential-exponential higher order shear deformation theory for advanced composite plates. Compos Part B Eng2014;
60: 319–328.
47.
MantariJLGuedes SoaresC.A trigonometric plate theory with 5-unknowns and stretching effect for advanced composite plates. Compos Struct2014;
107: 396–405.
48.
El MeicheNTounsiAZianeN, et al.
A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int J Mech Sci2011;
53: 237–247.
49.
NguyenV-HNguyenT-KThaiH-T, et al.
A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos Part B Eng2014;
66: 233–246.
50.
ZenkourAM.Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J Sandw Struct Mater2013;
15: 629–656.
51.
FerreiraAJMCarreraECinefraM, et al.
Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. Compos Part B Eng2011;
42: 1276–1284.
52.
MantariJLOktemASGuedes SoaresC.Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Compos Struct2011;
94: 37–49.
53.
PuneraDKantT.A critical review of stress and vibration analyses of functionally graded shell structures. Compos Struct2019;
210: 787–809.
54.
TornabeneFFantuzziNViolaE, et al.
Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory. Compos Struct2015;
119: 67–89.
55.
ShahPHBatraRC.Through-the-thickness stress distributions near edges of composite laminates using stress recovery scheme and third order shear and normal deformable theory. Compos Struct2015;
131: 397–413.
56.
ShahPHBatraRC.Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory. Mech Adv Mater Struct2018;
25: 1276–1296.
57.
KalninsA.Effect of bending on vibrations of spherical shells. J Acoust Soc Am1964;
36: 74–81.
58.
KantTManjunathaBS.An unsymmetric FRC laminate C° finite element model with 12 degrees of freedom per node. Eng Comput1988;
5: 300–308.
59.
KantTPuneraD.A refined higher order theory for statics and dynamics of doubly curved shells. Proc Indian Natl Sci Acad2017;
83: 611–630.
60.
PuneraDKantT.Free vibration of functionally graded open cylindrical shells based on several refined higher order displacement models. Thin-Walled Struct2017;
119: 707–726.
61.
HilderbrandFBReissnerEThomasGB.Notes on the foundations of the theory of small displacements of orthotropic shells.
Washington, DC:
NACA-TN-1833, 1949.
62.
ReddyJN.Mechanics of laminated composite plates and shells : theory and analysis.
Boca Raton, FL:
CRC Press, 2004.
63.
KrausH.Thin elastic shells: an introduction to the theoretical foundations and the analysis of their static and dynamic behavior. 1st ed.
New York:
John Wiley & Sons, Ltd., 1967.
