Higher order theory based Fourier analysis of cross-ply plates and doubly curved panels

Abstract

An analytical solution to the static analysis of a general cross-ply finite-dimensional doubly curved panel of rectangular plan-form, modeled using a higher order shear deformation theory, is presented. A solution methodology, based on boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the higher order-based laminated shell analysis, with the fully clamped type 4 (C4) boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the simply supported type 4 (SS4) constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of the deflection and moment of a cross-ply spherical panel, and also by comparison with the available results in literature. Additional comparisons were made with the finite element counterparts using commercially available software for distributed load. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to shell geometry (cylindrical and spherical), lamination, lamina material property, thickness and membrane effects as well as their interactions.

Keywords

Boundary discontinuous Fourier higher order theory cross-ply shells cross-ply plates fully clamped simply supported

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References

Swanson

. Introduction to design and analysis with advanced composite materials, Upper Saddle River, NJ: Prentice Hall Inc.

Jones

. Mechanics of composite materials, 2nd edn. London: Taylor & Francis Inc.

Hobson

. The theory of functions of a real variable 1950; Vol. II2nd edn. Washington: Harren Press.

Goldstein

. The stability of viscous fluid flow under pressure between parallel planes. Math Proc Cambridge Philos Soc 1936; 32: 40–55.

Goldstein

. The stability of viscous fluid flow between rotating cylinders. Math Proc Cambridge Philos Soc 1937; 33: 41–55.

Green

. Double Fourier series and boundary value problems. Math Proc Cambridge Philos Soc 1944; 40: 222–228.

Green

Hearmon

RFS

. The buckling of flat rectangular plywood plates. Philos Magazine 1945; 36: 659–687.

Whitney

. The effect of boundary conditions on the response of laminated composites. J Compos Mater 1970; 4: 192–203.

Whitney

Leissa

. Analysis of a simply supported laminated anisotropic rectangular plate. AIAA Journal 1970; 8: 28–33.

10.

Whitney

. Fourier analysis of clamped anisotropic plates. ASME J Appl Mech 1971; 38: 530–532.

11.

Whitney

Pagano

. Shear deformation in heterogeneous anisotropic plates. J Appl Mech 1970; 37: 1031–1036.

12.

Kan

Ito

. Shear deformation in heterogeneous anisotropic plates. J Compos Mater 1972; 6: 316–319.

13.

Sivakumaran

. Natural frequencies of symmetrically laminated rectangular plates with free edges. Compos Struct 1987; 7: 191–204.

14.

Malhotra

Ganesan

Veluswami

. Effect of fiber orientation and boundary conditions on the vibration behavior of orthotropic square plates. Compos Struct 1988; 9: 247–255.

15.

Barbero

Reddy

Teply

. General two-dimensional theory of laminated cylindrical shells. AIAA Journal 1990; 28(3): 544–553.

16.

Khedeir

. Free vibration of antisymmetric angle-ply laminated plates including various boundary conditions. J Sound Vibr 1988; 122(2): 377–388.

17.

Khedeir

. An exact approach to the elastic state of stress of shear deformable antisymmetric angle-ply laminated plates. Compos Struct 1989; 11: 245–258.

18.

Tauchert

. Thermal buckling of thick antisymmetric angle-ply laminates. J Therm Stresses 1987; 10: 113–124.

19.

Lim

Reddy

. On canonical bending relationships for plates. Int J Solid Struct 2003; 40: 3039–3067.

20.

Ferreira

AJM

Roque

CMC

Jorge

RMN

. Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Comput Struct 2005; 83: 2225–2237.

21.

Lin

Jen

. Analysis of a laminated anisotropic plate by Chebyshev collocation method. Composites: Part B 2005; 36: 155–169.

22.

Kim

Cho

. Enhanced first-order theory based on mixed formulation and transverse normal effect. Int J Solid Struct 2007; 44: 1256–1276.

23.

Akhras

. Spline finite strip analysis of composite plates based on higher-order zigzag composite plate theory. Compos Struct 2007; 78: 112–118.

24.

Bhaskar

Sivaram

. Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions. Compos Struct 2008; 83: 83–92.

25.

Chaudhuri

Kabir

HRH

. A boundary-continuous-displacement based Fourier analysis of laminated doubly-curved panels using classical shallow shell theories. Int J Eng Sci 1992; 30: 1647–1664.

26.

Chaudhuri

Kabir

HRH

. Static and dynamic analysis of finite general cross-ply doubly-curved panels using classical shallow shell theories. Compos Struct 1994; 28: 73–91.

27.

Chaudhuri

. On boundary-discontinuous double Fourier series solution to a system of completely coupled p.d.e.’s. Int J Eng Sci 1989; 27(9): 1005–1022.

28.

Chaudhuri

. On the roles of complementary and admissible boundary constraints in Fourier solutions to boundary-value problems of completely coupled rth order p.d.e.'s. J Sound Vibr 2002; 251: 261–313.

29.

Chaudhuri

Kabir

HRH

. Fourier solution to higher order theory based laminated shell boundary-value problem. AIAA Journal 1995; 33: 1681–88.

30.

Oktem

Chaudhuri

. Levy type analysis of cross-ply plates based on higher-order theory. Compos Struct 2007; 78: 243–253.

31.

Oktem

Chaudhuri

. Levy type Fourier analysis of thick cross-ply doubly-curved panels. Compos Struct 2007; 80: 475–488.

32.

Oktem

Chaudhuri

. Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels. Compos Struct 2007; 80: 489–503.

33.

Oktem

Chaudhuri

. Boundary discontinuous Fourier analysis of thick cross-ply clamped plates. Compos Struct 2008; 82: 539–548.

34.

Oktem

Chaudhuri

. Sensitivity of the response of thick doubly curved cross-ply panels to edge clamping. Compos Struct 2009; 87: 293–306.

35.

Oktem

Chaudhuri

. Effect of in-plane boundary constraints on the response of thick general (unsymmetric) cross-ply plates. Compos Struct 2008; 83: 1–12.

36.

Oktem

Chaudhuri

. Higher-order theory based boundary-discontinuous Fourier analysis of simply supported thick cross-ply doubly curved panels. Compos Struct 2009b; 89: 448–458.

37.

Oktem AS. The effect of boundary conditions on the response of laminated thick composite plates and shells. Ph.D. Dissertation, University of Utah, 2005.

38.

Reddy

Liu

. A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci 1985; 23: 319–330.

39.

Librescu

Khdeir

. Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory: part I – stress and displacement. Compos Struct 1988; 9: 189–213.

40.

Librescu

Khdeir

Frederick

. A shear deformable theory of laminated composite shallow shell type panels and their response, analysis I, free vibration and buckling. Acta Mechanica 1989; 76: 1–33.

41.

Reddy

. Theory and analysis of elastic plates and shells 2004, 2nd edn. Boca Raton: CRC Press.