Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells

[+] Author and Article Information
Erasmo Carrera

Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; carrera@polito.it

Appl. Mech. Rev 54(4), 301-329 (Jul 01, 2001) (29 pages) doi:10.1115/1.1385512 History:
Copyright © 2001 by ASME
Your Session has timed out. Please sign back in to continue.


Reissner  E (1984), On a certain mixed variational theory and a proposed application, Int. J. Numer. Methods Eng. 20, 1366–1368.
Reissner  E (1986a), On a mixed variational theorem and on a shear deformable plate theory, Int. J. Numer. Methods Eng. 23, 193–198.
Reissner E (1986b) On a certain mixed variational theorem and on laminated elastic shell theory, Proc of Euromech-Colloquium219 , 17–27.
Hildebrand FB, Reissner E, and Thomas GB (1938), Notes on the foundations of the theory of small displacements of orthotropic shells, NACA TN-1833, Washington DC.
Reissner  E (1952), Stress strain relation in the theory of thin elastic shell, J. Math. Phys. 31, 109–119.
Reissner  E (1964), On the form of variationally derived shell equations, ASME J. Appl. Mech. 31, 233–328.
Carrera E (1995), A class of two-dimensional theories for anisotropic multilayered plates analysis, Accademia delle Scienze di Torino, Memorie Scienze Fisiche, 19–20 (1995–1996), 1–39.
Carrera  E (1997c), Cz0 Requirements-Models for the two dimensional analysis of multilayered structures, Compos. Struct. 37, 373–384.
Koiter WT (1959), A consistent first approximations in the general theory of thin elastic shells, Proc. of Symp. on the Theory of Thin Elastic Shells, Aug, North-Holland, Amsterdam, 12–23.
Cauchy  AL (1828), Sur l’equilibre et le mouvement d’une plaque solide, Exercises de Matematique 3, 328–355.
Poisson  SD (1829), Memoire sur l’equilibre et le mouvement des corps elastique, Mem. Acad. Sci. 8, 357.
Kirchhoff  G (1850), Über das Gleichgewicht und die Bewegung einer elastishen Scheibe, J. Angew Math. 40, 51–88.
Love AEH (1927), The Mathematical Theory of Elasticity, 4th Edition, Cambridge Univ Press, Cambridge.
Cicala  P (1959), Sulla teoria elastica della parete sottile, Giornale del Genio Civile, fascicoli 6e, 9.
Cicala P (1965), Systematic Approach to Linear Shell Theory, Levrotto and Bella, Torino.
Goldenveizer AL (1961), Theory of Thin Elastic Shells, Int. Series of Monograph in Aeronautics and Astronautics, Pergamon Press, New York.
Jones RM (1975), Mechanics of Composite Materials, Mc Graw-Hill, New York.
Reissner  E (1945), The effect of transverse shear deformation on the bending of elastic plates, ASME J. Appl. Mech. 12, 69–76.
Mindlin  RD (1951), Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates, ASME J. Appl. Mech. 18, 1031–1036.
Whitney  JM (1969), The effects of transverse shear deformation on the bending of laminated plates, J. Compos. Mater. 3, 534–547.
Reddy JN (1997), Mechanics of Laminated Composite Plates, Theory and Analysis, CRC Press.
Sun  CT and Whitney  JM (1973), On the theories for the dynamic response of laminated plates, AIAA J. 11, 372–398.
Lo  KH, Christensen  RM, and Wu  EM (1977), A higher-order theory of plate deformation, Part 2: Laminated plates, ASME J. Appl. Mech. 44, 669–676.
Librescu L (1975), Elasto-Statics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures, Noordhoff Int, Leyden, Netherland.
Vlasov  BF (1957), On the equations of Bending of plates, Dokl. Akad. Nauk Arm. SSR , 3, 955–979.
Reddy  JN (1984b), A simple higher order theories for laminated composites plates, ASME J. Appl. Mech. 52, 745–742.
Reddy  JN and Phan  ND (1985), Stability and vibration of isotropic, orthotropic, and laminated plates according to a higher order shear deformation theory, J. Sound Vib. 98, 157–170.
Srinivas  S (1973), A refined analysis of composite laminates, J. Sound Vib. 30, 495–507.
Cho  KN, Bert  CW, and Striz  AG (1991), Free vibrations of laminated rectangular plates analyzed by higher-order individual-Layer theory, J. Sound Vib. 145, 429–442.
Barbero  EJ, Reddy  JN, and Teply  JL (1990), General two-dimensional theory of laminated cylindrical shells, AIAA J. 28, 544–553.
Nosier  A, Kapania  RK, and Reddy  JN (1993), Free vibtation analysis of laminated plates using a layer-wise theory, AIAA J. 31, 2335–2346.
Pagano  NJ (1969), Exact solutions for composite laminates in cylindrical bending, J. Compos. Mater. 3, 398–411.
Lekhnitskii  SG (1935), Strength calculation of composite beams, Vestn. inzhen. i tekhnikov 9,540-544.
Lekhnitskii SG (1968), Anisotropic Plates, 2nd Edition, Translated from the 2nd Russian Edition by SW Tsai, Cheron, Bordon, and Breach.
Ambartsumian SA (1969), Theory of Anisotropic Plates, Translated from Russian by T Cheron and Edited by JE Ashton Tech. Pub. Co.
Grigolyuk  EI and Kulikov  GM (1988), General directions of the development of theory of shells, Mekh. Kompoz. Mater. 24, 287–298.
Ren  JG (1987), Exact solutions for laminated cylindrical shells in cylindrical bending, Compos. Sci. Technol. 29, 169–187.
Rath  BK and Das  YC (1973), Vibration of layered shells, J. Sound Vib. 28, 737–757.
Aitharaju  VR and Averill  RC (1999), C0 zig-zag kinematic displacement models for the analysis of laminated composites, Mech. Compos. Mater. Struct. 6, 31–56.
Reddy JN (1984a), Energy and Variational Methods in Applied Mechanics, John Wiley, NY.
Atluri SN, Tong P, and Murakawa H (1983), Recent studies in hybrid and mixed finite element methods in mechanics, in Hybrid and Mixed Finite Element Methods, Edited by SN Atluri RH Gallagher, and O Zienkiewicz, John Wiley and Sons, 51–71.
Murakami  H (1984), A laminated beam theory with interlayer slip, ASME J. Appl. Mech. 51, 551–559.
Murakami  H (1985), Laminated composite plate theory with improved in-plane responses, ASME Proc. of PVP Conf, New Orleans 98-2, 257–263.
Murakami  H (1986), Laminated composite plate theory with improved in-plane responses, ASME J. Appl. Mech. 53, 661–666.
Toledano  A and Murakami  H (1987), A high-order laminated plate theory with improved in-plane responses, Int. J. Solids Struct. 23, 111–131.
Toledano  A and Murakami  H (1987b), A composite plate theory for arbitrary laminate configurations, ASME J. Appl. Mech. 54, 181–189.
Librescu L and Reddy JN (1986), A critical review and generalization of transverse shear deformable anisotropic plates, Euromech Colloquium 219, Kassel, Sept, 1986 Refined Dynamical Theories of Beams, Plates and Shells and their Applications, Elishakoff and Irretier (eds), Springer Verlag, Berlin, 32–43.
Kapania  RK (1989), A review on the analysis of laminated shells, J. Pressure Vessel Technol. 111, 88–96.
Kapania  RK and Raciti  S (1989), Recent advances in analysis of laminated beams and plates, AIAA J. 27, 923–946.
Noor  AK and Burton  WS (1989b), Assessment of shear deformation theories for multilayered composite plates, Appl. Mech. Rev. 42 (1), 1–13.
Noor  AK and Burton  WS (1990), Assessment of computational models for multilayered composite shells, Appl. Mech. Rev. 43(4), 67–97.
Reddy  JN and Robbins  DH (1994), Theories and computational models for composite laminates, Appl. Mech. Rev. 47 (6), 147–165.
Soldatos  KP and Timarch  T (1993), A unified formulation of laminated composites, shear deformable, five-degrees-of-freedom cylindrical shell theories, Composite Structures 25, 165–171.
Noor  AK, Burton  S, and Bert  CW (1996), Computational model for sandwich panels and shells, Appl. Mech. Rev. 49(3), 155–199.
Washizu K (1968), Variational Method in Elasticity and Plasticity, Oxford, Pergamon Press.
Antona E (1991), Mathematical model and their use in engineering, Applied Mathematics in the Aerospace Science/Engineering, by A Miele and A Salvetti (eds), vol 44, 395–433.
Zienkiewicz OC (1986), The Finite Element Method, Mc Graw-Hill, London.
Carrera  E (1998d), Layer-wise mixed models for acurate vibration analysis of multilayered plates, ASME J. Appl. Mech. 65, 820–828.
Carrera  E (1999a), Multilayered shell theories that account for a layer-wise mixed description, Part I. Governing equations, AIAA J. 37(9), 1107–1116.
Carrera  E (1999b), Multilayered shell theories that account for a layer-wise mixed description, Part II. Numerical evaluations, AIAA J. 37(9), 1117–1124.
Carrera  E (1999c), A Reissner’s mixed variational theorem applied to vibration analysis of multilayered shells, ASME J. Appl. Mech. 66, 69–78.
Rao  KM and Meyer-Piening  HR (1990), Analysis of thick laminated anisotropic composites plates by the finite element method, Compos. Struct. 15, 185–213.
Messina  A (2001), Two generalized higher order theories in free vibration studies of multilayered plates, J. Sound Vib. 242, 125–150.
Carrera  E (1996), C0 Reissner-Mindlin multilayered plate elements including zig-zag and interlaminar stresses continuity, Int. J. Numer. Methods Eng. 39, 1797–1820.
Murakami  H and Yamakawa  J (1996), Dynamic response of plane anisotropic beams with shear deformation, ASCE J. Eng. Mech. 123, 1268–1275.
Murakami  H, Reissner  E, and Yamakawa  J (1996), Anisotropic beam theories with shear deformation, ASME J. Appl. Mech. 63, 660–668.
Soldatos KP (1987), Cylindrical bending of Cross-ply Laminated Plates: Refined 2D Plate theories in comparison with the Exact 3D elasticity solution, Tech Report No. 140, Dept. of Math., University of Ioannina, Greece.
Carrera  E (1998a), A refined multilayered finite element model applied to linear and nonlinear analysis of sandwich structures, Compos. Sci. Technol. 58, 1553–1569.
Carrera  E (1998b), Mixed layer-wise models for multilayered plates analysis, Compos. Struct. 43, 57–70.
Carrera  E (1998c), Evaluation of layer-wise mixed theories for laminated plates analysis, AIAA J. 26, 830–839.
Carrera  E (1999d), Transverse normal stress effects in multilayered plates, ASME J. Appl. Mech. 66, 1004–1012.
Carrera  E (1999e), A study of transverse normal stress effects on vibration of multilayered plates and shells, J. Sound Vib. 225, 803–829.
Carrera  E (2000a), Single-layer vs multi-layers plate modelings on the basis of Reissner’s mixed theorem, AIAA J. 38, 342–343.
Carrera  E (2000b), A priori vs a posteriori evaluation of transverse stresses in multilayered orthotropic plates, Compos. Struct, 48, 245–260.
Carrera  E (2000c), An assessment of mixed and classical theories for thermal stress analysis of orthotropic plates, J. Therm. Stresses 23, 797–831.
Bhaskar  K and Varadan  TK (1991), A higher-order theory for bending analysis of laminated shells of revolution, Comput. Struct. 40, 815–819.
Jing H and Tzeng KG (1993a), On two mixed variational principles for thick laminated composite plates, Compos. Struct.
Jing  H and Tzeng  KG (1993b), Refined shear deformation theory of laminated shells, AIAA J. 31, 765–773.
Jing  H and Liao  ML (1989), Partial hybrid stress element for the analysis of thick laminate composite plates, Int. J. Numer. Methods Eng. 28, 2813–2827.
Toledano  A and Murakami  H (1987c), A high-order mixture model for periodic particulate composites Int. J. Solids Struct. 23, 989–1002.
Carrera  E and Kröplin  B (1997), Zig-Zag and interlaminar equilibria effects in large deflection and postbuckling analysis of multilayered plates, Mech. Compos. Mater. Struct. 4, 69–94.
Carrera  E and Krause  H (1998b), An investigation on nonlinear dynamics of multilayered plates accounting for Cz0 requirements, Comput. Struct. 69, 463–486.
Carrera  E (1997b), An improved Reissner-Mindlin-Type model for the electromechanical analysis of multilayered plates including piezo-layers, J. Intell. Mater. Syst. Struct. 8, 232–248.
Carrera E and Niglia F (1998), A refined multilayered FEM model applied to sandwich structures, Mechanics of Sandwich Structures, A Vautrin (ed), Kluwer Academic Publ, 61–69.
Carrera  E and Parisch  H (1998a), Evaluation of geometrical nonlinear effects of thin and moderately thick multilayered composite shells, Compos. Struct. 40, 11–24.
Brank  B and Carrera  E (2000a), A family of shear-deformable shell finite elements for composite structures, Comput. Struct. 76, 297–297.
Brank  B and Carrera  E (2000b), Multilayered shell finite element with interlaminar continuous shear stresses: A refinement of the Reissner-Mindlin formulation, Int. J. Numer. Methods Eng. 48, 843–874.
Carrera E and Demasi L, Multilayered finite plate element based on Reissner Mixed Variational Theorem. Part I: Theory; Part II: Numerical analysis, to appear.
Carrera E and Demasi L (2000b), An assessment of multilayered finite plate element in view of the fulfillment of the Cz0-Requirements, AIMETA GIMC Conf, Brescia, Nov, 340–348
Carrera E and Demasi L (2000c), Sandwich plate analysis by finite plate element and Reissner Mixed Theorem, V Int. Conf. on Sandwich Construction, Zurich, Sept, vol I, 301–312
Latham  C and Toledano  A and Murakami  H and Seible  F (1988), A shear deformable two-layer plate element with interlayer slip, Int. J. Numer. Methods Eng. 26, 1769–1789.
Toledano  A and Murakami  H (1988), A shear-deformable two-layer plate theory with interlayer slip, ASCE Journal of Engineering Mechanics 114(4), 604–623.
Hegemier  GA, Murakami  H, and Hageman  LJ (1985), On tension stiffening in reinforced concrete, Mech. Mater. 4(2), 161–179.
Murakami  H and Hegemier  GA (1986a), On simulating steel-concrete interaction in reinforced concrete, Part I: Theoretical development, Mech. Mater. 5, 171–185.
Murakami  H and Hegemier  GA (1986b), A mixture model for unidirectionally fiber-reinforced composites, ASME J. Appl. Mech. 53(4), 765–773.
Murakami  H and Hegemier  GA (1989), Development of a nonlinear continuum model for wave propagation in jointed media: theory for single joint set, Mech. Mater. 8, 199–218.
Murakami  H and Toledano  A (1990), A high-order mixture homogenization of bi-laminated composites, ASME J. Appl. Mech. 57, 388–397.
Toledano  A and Murakami  H (1991), High-order mixture homogenization of fiber-reinforced composites, ASME J. Energy Resour. Technol. 113, 254–263.
Murakami  H, Impelluso  TJ, and Hegemier  GA (1991), A continuum finite element for single-set jointed media Int. J. Numer. Methods Eng. 31, 1169–1194.
Impelluso  TJ and Murakami  H (1995), A homogenized continuum model for fiber-reinforced composites, Z. Angew. Math. Mech. 75 (3), 171–188.
Noor  AK and Burton  WS (1989a), Stress and free vibration analyses of multilayered composite plates, Compos. Structu. 11, 183–204.
Cho  M and Parmerter  RR (1993), Efficient higher order composite plate theory for general lamination configurations, AIAA J. 31, 1299–1305.
Idlbi  A, Karama  M, and Touratier  M (1997), Comparison of various laminated plate theories, Compos. Struct. 37, 173–184.
Ren  JG (1986), A new theory for laminated plates, Compos. Sci. Technol. 26, 225–239.
Bhaskar  K, Varadan  TK, and Ali  JSM (1996), Thermoelastic solution for orthotropic and anisotropic composites laminates, Composites, Part B 27B, 415–420.
Varadan  TK and Bhaskar  K (1991), Bending of Laminated Orthotropic Cylindrical Shells - An Elasticity Approach, Compos. Struct. 17, 141–156.
Dennis  ST and Palazotto  AN (1991), Laminated shell in cylindrical bending, two-dimensional approach vs exact, AIAA J. 29, 647–650.
Ye  JQ and Soldatos  KP (1994), Three-dimensional vibration of laminated cylinders and cylindrical panels with symmetric or antisymmetric cross-ply lay-up, Composites Eng. 4, 429–444.
Timarci  T and Soldatos  KP (1995), Comparative dynamic studies for symmetric cross-ply circular cylindrical shells on the basis of a unified shear deformable shell theory, J. Sound Vib. 187, 609–624.
Dischiuva  M and Carrera  E (1992), Elasto-dynamic behavior of relatively thick, symmetrically laminated, anisotropic circular cylindrical shells, ASME J. Appl. Mech. 59, 222–223.
Liou  WJ and Sun  CT (1987), A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates, Comput. Struct. 25, 241–249.
Moriya  K (1986), Laminated plate and shell elements for finite element analysis of advanced fiber reinforced composite structure, Laminated Composite Plates, in Japanese, Trans. Jpn. Soc. Mech. Eng. 52, 1600–1607.
Bathe  KJ and Dvorkin  EN (1985), A four node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation, Int. J. Numer. Methods Eng. 21, 367–383.
Kraus H (1967) Thin Elastic Shells, John Wiley, NY.
Bhaskar  K and Varadan  TK (1992), Reissner’s new mixed variational principle applied to laminated cylindrical shells, ASME J. Pressure Vessel Technol. 114, 115–119.
Dischiuva  M (1993), A general quadrilater, multilayered plate element with continuous interlaminar stresses, Compos. Struct. 47, 91–105.
Kant  T and Kommineni  JR (1989), Large amplitude free vibration analysis of cross-ply composite and sandwich laminates with a refined theory and C0 finite elements, Comput. Struct. 50, 123–134.
Pagano  NJ and Hatfield  SJ (1972), Elastic behavior of multilayered bidirectional composites, AIAA J. 10, 931–933.
Pagano  NJ (1970), Exact solutions for rectangular bidirection composites and sandwich plates, J. Compos. Mater. 4, 20–34.


Grahic Jump Location
Examples of multilayer structures: Plates (upper part) made of layers of different materials (left) and by unidirectional fibers (right) and sandwich shell (lower part)
Grahic Jump Location
Geometry and notations used for multilayered shells
Grahic Jump Location
Typical through the thickness stress (in-plane and transverse components) fields in a three layer plate: Cz0-Requirements
Grahic Jump Location
Details and notations of stresses at the interface
Grahic Jump Location
Geometrical meaning of zig-zag function, Linear case
Grahic Jump Location
Meanings of the used acronyms
Grahic Jump Location
Examples of assumed fields in the thickness plate direction in a four layer plate
Grahic Jump Location
Amplitude of in-plane displacement Uz×ET/PztNlh vs z/h. Comparison of present and other ESLM results to 3D-elasticity 32. Antisymmetric 4-layer case, a/h=4. Mechanical data of the lamina: EL=25×106 psi,ET=1×106 psi,GLT=0.5×106 psi,GTT=0.2×106 psi,νLTTT=.25
Grahic Jump Location
Transverse shear stress amplitude Sxz×1/PxiNl vs z/h; Comparison of present and other ESLM to Elasticity 32; Antisymmetric 4 layer case, a/h=4; Same data as in Fig 8
Grahic Jump Location
Amplitude of in-plane displacements Uβ×100ETh2/PztNlRβ3 vs z; Ren’s cylindrical panel with Rβ/h=4; Five layer case
Grahic Jump Location
Amplitude of transverse shear stress |Sαz×10h/Pzb1Rβ| vs z; Varadan and Bhaskar’s cylindrical shells with Rβ/h=4; Five layer case
Grahic Jump Location
Amplitude of in-plane displacemnets Uβ×10ELh2/Pzb1Rβ3 vs z; Varadan and Bhaskar’s cylindrical shells with Rβ/h=4; Ten layer case
Grahic Jump Location
Postbuckling of a compressed plate: Load parameter vs plate defection at the center; Comparisons among RMZC and classical theories. a=10,h=1; mesh 4×4. 0°/90°/0°/90°/0°, EL/ET=40,GLT/ET=.5,GTT/ET=.35,νLTTT=.3.
Grahic Jump Location
Phase diagram corresponding to the steady-state-solution: u̇zvsuz; Comparison between linear and nonlinear analysis for RMZC, FSDT and CLT results. a=10,h=1, mesh 4×4; 45°/−45°/45°/−45°/45°; Mechanical data as Fig. 13; Pz(5,5)=1;ωe=0.2;w1=0.535,w2=0.534
Grahic Jump Location
Shell panel with Rβ/h=4(h=2.5); Transverse shear stress σyz×h/PztNlRβ
Grahic Jump Location
An example showing how Cz0-Requirements are imposed




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In