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Research Papers: Design and Analysis

Semi-Analytical Approach in Buckling Analysis of Functionally Graded Shells of Revolution Subjected to Displacement Dependent Pressure

[+] Author and Article Information
Majid Khayat

Department of Civil Engineering,
Shahid Chamran University of Ahvaz,
Ahvaz 61357, Iran
e-mail: khayatmajid@yahoo.com

Davood Poorveis

Assistant Professor
Department of Civil Engineering,
Shahid Chamran University of Ahvaz,
Ahvaz 61357, Iran
e-mail: dpoorveis@scu.ac.ir

Shapour Moradi

Professor
Department of Mechanical Engineering,
Shahid Chamran University of Ahvaz,
Ahvaz 61357, Iran
e-mail: moradis@scu.ac.ir

1Corresponding author.

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received May 30, 2016; final manuscript received June 2, 2017; published online August 30, 2017. Assoc. Editor: Albert E. Segall.

J. Pressure Vessel Technol 139(6), 061202 (Aug 30, 2017) (19 pages) Paper No: PVT-16-1089; doi: 10.1115/1.4037042 History: Received May 30, 2016; Revised June 02, 2017

Linearized buckling analysis of functionally graded shells of revolution subjected to displacement-dependent pressure, which remains normal to the shell's middle surface throughout the deformation process, is described in this work. Material properties are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and a metal. The governing equations are derived based on the first-order shear deformation theory, which accounts for through the thickness shear flexibility with Sanders type of kinematic nonlinearity. Displacements and rotations in the shell's middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix, also known as the pressure stiffness matrix, which accounts for the variation of load direction, is derived for each strip and after assembling resulted in the global load stiffness matrix of the shell, which may be unsymmetric. The load stiffness matrix can be divided into two unsymmetric parts (i.e., load nonuniformity and unconstrained boundary effects) and a symmetric part. The main part of this research is to quantify the effects of these unsymmetries on the follower action of lateral pressure. A detailed numerical study is carried out to assess the influence of various parameters such as power law index of functionally graded material (FGM) and shell geometry interaction with load distribution, and shell boundary conditions on the follower buckling pressure reduction factor. The results indicate that, when applied individually, unconstrained boundary effect and longitudinal nonuniformity of lateral pressure have little effect on the follower buckling reduction factor, but when combined with each other and with circumferentially loading nonuniformity, intensify this effect.

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Figures

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Fig. 2

Geometry of shell of revolution

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Fig. 3

An FGM shell of revolution

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Fig. 4

Variation of volume fraction Vm through the thickness for various values of the power-law index, N

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Fig. 5

Force and moment resultants

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Fig. 7

Global and local coordinate systems for a shell of revolution

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Fig. 8

Deformation of an elemental area

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Fig. 9

Hemispherical shell under uniform lateral pressure

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Fig. 10

Influence of base boundary conditions of FG hemispherical shell on the follower buckling reduction factor, μ, (R/h = 25)

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Fig. 11

Influence of base boundary conditions of FG hemispherical shell on the follower buckling reduction factor, μ, (R/h = 50)

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Fig. 12

Influence of base boundary conditions of FG hemispherical shell on the follower buckling reduction factor, μ (R/h = 100)

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Fig. 13

Geometry of conical shell

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Fig. 14

Flutter and divergence dominant instability criteria regions for case “DB”

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Fig. 15

Flutter and divergence dominant instability criteria regions for case “DL”

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Fig. 16

Flutter and divergence dominant instability criteria regions for case “DC”

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Fig. 17

Flutter and divergence dominant instability criteria regions for case “DLB”

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Fig. 18

Flutter and divergence dominant instability criteria regions for case “DCB”

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Fig. 19

Flutter and divergence dominant instability criteria regions for case “DCL”

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Fig. 20

Flutter and divergence dominant instability criteria regions for case “DCLB”

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Fig. 21

Influence of domain, free boundary, and load nonuniformity of external virtual work integrals on the follower buckling reduction factor: (a) (γ  = 0 deg and h = 5 mm), (b) (γ  = 15 deg and h = 5 mm), (c) (γ  = 30 deg and h = 5 mm), and (d) (γ  = 45 deg and h = 5 mm)

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Fig. 22

Influence of domain, free boundary, and load nonuniformity of external virtual work integrals on the follower buckling reduction factor: (a) (γ  = 0 deg and h = 7.5 mm), (b) (γ  = 15 deg and h = 7.5 mm), (c) (γ  = 30 deg and h = 7.5 mm), and (d) (γ  = 45 deg and h = 7.5 mm)

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Fig. 23

Influence of domain, free boundary, and load nonuniformity of external virtual work integrals on the follower buckling reduction factor: (a) (γ  = 0 deg and h = 10 mm), (b) (γ = 15 deg and h = 10 mm), (c) (γ  = 30 deg and h = 10 mm), and (d) (γ  = 45 deg and h = 10 mm)

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Fig. 24

Catenary shell geometry: (a) meridian section and (b) circumferential section

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Fig. 25

Cycloid shell geometry: (a) meridian section and (b) circumferential section

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Fig. 26

Parabolic shell geometry: (a) meridian section and (b) circumferential section

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Fig. 27

Elliptic shell geometry: (a) meridian section and (b) circumferential section

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Fig. 28

Lateral buckling pressure of various FG shells of revolution for h = 1.25 mm

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Fig. 29

Lateral buckling pressure of various FG shells of revolution for h = 2.5 mm

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Fig. 30

Lateral buckling pressure of various FG shells of revolution for h = 5 mm

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Fig. 31

Variation of follower buckling reduction factor, μ  for different shells of revolution (h = 1.25 mm)

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Fig. 32

Variation of follower buckling reduction factor, μ for different shells of revolution (h = 2.5 mm)

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Fig. 33

Variation of follower buckling reduction factor, μ for different shells of revolution (h = 5 mm)

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