Abstract
In this paper, buckling analysis of a functionally graded thick cylindrical shell with variable thickness subjected to combined external pressure and axial compression is carried out. Moreover, the effect of an axisymmetric imperfection on the buckling load of the shell is investigated. It is assumed that material roperties of the shell vary smoothly through the thickness according to a power law distribution of the volume fraction of constituent materials, while the Poisson’s ratio is assumed to be constant. The shell is considered to be simply supported at both ends. The governing differential equations are obtained based on the second Piola–Kirchhoff stress tensor and are then reduced to a homogenous linear system of equations using differential quadrature method. Effects of several parameters of the shell including the volume fraction of constituents, geometric ratios, thickness variation amplitude factor, imperfection parameter and loading conditions on the buckling behavior of the functionally graded thick cylindrical shell are investigated. The results obtained by the present method are compared with results reported in the literature.
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Abbreviations
- R 1,R 2 :
-
Shell inner, outer radius, respectively
- L :
-
Length of the shell
- a(x):
-
Mid-surface radius of the shell
- h(x):
-
Thickness of the shell
- r, θ, x :
-
Radial, circumferential, longitudinal coordinate, respectively
- z :
-
Radial coordinate at the mid-surface
- h 0 :
-
Thickness of a perfect cylindrical shell
- \({\eta}\) :
-
Thickness variation amplitude
- \({\varepsilon}\) :
-
Non-dimensional parameter of the imperfection
- \({\alpha, \beta}\) :
-
Constant coefficients
- k :
-
Volume fraction index
- E m, E c :
-
Elastic modulus of metal, ceramic, respectively
- V m, V c :
-
Volume fraction of the metal, ceramic, respectively
- \({\vec{t}}\) :
-
Traction vector
- \({\bar{{F}}}\) :
-
Deformation gradient
- \({\vec{V}}\) :
-
Displacement vector
- I :
-
Unit tensor
- F :
-
Axial compressive load
- P :
-
Uniform lateral pressure
- \({\sigma_{ij},\varepsilon_{ij}}\) :
-
Stress, strain tensor components, respectively
- w, v, u :
-
Displacement fields in \({r, \theta, x}\) directions, respectively
- m :
-
Buckling mode number in the circumferential direction
- \({G, \lambda}\) :
-
Lame coefficients
- N :
-
Number of grid points in the r direction
- Q :
-
Number of grid points in the x direction
- \({w^{(n)}_{ij}}\) :
-
Weighting coefficients of the nth order derivative
- \({a^{(n)}_{ij}, b^{(n)}_{ij}}\) :
-
Weighting coefficients of the nth order derivative in the r, x-direction, respectively
- P cr :
-
Critical buckling pressure
- \({\lambda}\) :
-
Ratio of the buckling pressure of the imperfect to the perfect shell
- \({\lambda^{\prime}}\) :
-
Ratio of the buckling axial stress of the imperfect to the perfect shell
- K :
-
Structural stiffness matrix
- \({\phi_i}\) :
-
Eigenvector
- \({\lambda_i}\) :
-
Eigenvalue
- S :
-
Stress stiffness matrix
References
Koiter, W.T.: Buckling of cylindrical shells under axial compression and external pressure: thin shell theory new trends and applications. In: Olzak, W. (ed.) CISM Courses and Lectures, vol. 40, pp. 77– 87. Springer, New York (1980)
Elishakoff I., Li Y., Starners J.H.: Non-Classical Problem in Theory of Elastic Stability, pp. 43–98. Cambridge University Press, Cambridge (2001)
Gusic G., Combescure A., Jullien J.F.: The influence of circumferential thickness variations on the buckling of cylindrical shells under lateral pressure. Comput. Struct. 74, 461–477 (2000)
Nguyen T.H.L., Thach S.S.H.: Stability of cylindrical panel with variable thickness. Vietnam J. Mech. VAST. 28(1), 56–65 (2006)
Nguyen, T.H.L.; Thach, S.S.H.: Influence of the thickness variation and initial geometric imperfection on the buckling of cylindrical panel. In: Proceeding of the 8th Vietnamese Conference on Mechanics of Solids, Thai Nguyen, pp. 491–499 (2006)
Nguyen T.H.L., Elishakoff I., Nguyen T.V.: Buckling under external pressure of cylindrical shell with variable thickness. Int. J. Solids Struct. 46, 4163–4168 (2009)
Elishakoff, I.; Li, Y.W.; Starnes, J.H.: The combined effect of the thickness variation and axisymmetric initial imperfection on the buckling of the isotropic cylindrical shell under axial compression. Preliminary Report, Florida Atlantic University (1992)
Koiter W.T.: The effect of axisymmetric imperfection on the buckling of cylindrical shells under axial compression. Akademie van Wetenschappen-Amsterdam, Ser. B 66, 265–279 (1963)
Akbari Alashti, R.; Ahmadi, S.A.: Buckling of imperfect thick cylindrical shells and curved panels with different boundary conditions under external pressure. J. Theor. Appl. Mech. 52(1), 25–36 (2014)
Koiter, W.T.; Elishakoff, I.; Li, Y.W.; Starnes, J.H.: Buckling of axially compression imperfect cylindrical shells of variable thickness. In: Proceedings of the 35th (AIAA/ASME/ASCE/AHS/ASC) Structural Dynamics and Materials Conferences, Hilton Head, pp. 277–289 (1994)
Sofiyev A.H.: The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure. J. Compos. Part B 34, 27–233 (2003)
Civalek O.: A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution. J. Thin-Walled Struct. 45, 692–998 (2007)
Mirfakhraei P., Redekop D.: Buckling of circular cylindrical shells by the differential quadrature method. J. Press. Vessel. Pip. 75, 347–353 (1998)
Civaleka O.: Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. J. Eng. Struct. 26, 171–186 (2004)
Koizumi M.: The concept of FGM. Ceramic Transactions. Funct. Gradient Mater. 34, 3–10 (1993)
Lai W.M., Rubin D., Krempl E.: Introduction to Continuum Mechanics, 3rd edn. Butterworth-Heinemann, Massachusetts (1996)
Ciarlet P.G.: Mathematical Elasticity. Three Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988)
Shu C.: Differential Quadrature and Its Application in Engineering. Springer, London (2000)
Kardomateas G.A.: Benchmark three-dimensional elasticity solution for the buckling of thick orthotropic cylindrical shells. J. Appl. Mech. (ASME) 5, 569 (1996)
Timoshenko S.P., Gere J.M.: Theory of Elastic Stability. McGraw Hill, New York (1961)
Flugge W.: Stresses in Shells. Springer, Berlin (1960)
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Akbari Alashti, R., Ahmadi, S.A. Buckling Analysis of Functionally Graded Thick Cylindrical Shells with Variable Thickness Using DQM. Arab J Sci Eng 39, 8121–8133 (2014). https://doi.org/10.1007/s13369-014-1356-4
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DOI: https://doi.org/10.1007/s13369-014-1356-4