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.

References

References
1.
Datta
,
P. K.
, and
Biswas
,
S.
,
2011
, “
Aeroelastic Behaviour of Aerospace Structural Elements With Follower Force: A Review
,”
J. Aeronaut. Space Sci.
,
12
(
2
), pp.
134
148
.
2.
Argyris
,
J. H.
, and
Symeonidis
,
S.
,
1981
, “
Nonlinear Finite Element Analysis of Elastic System Under Nonconservative Loading–Natural Formulation—Part 1: Quasistatic Problems
,”
Comput. Methods Appl. Mech. Eng.
,
26
(
1
), pp.
75
123
.
3.
Cohen
,
G. A.
,
1966
, “
Conservative of a Normal Pressure Field Acting on a Shell
,”
AIAA J.
,
4
(
10
), p.
1886
.
4.
Romano
,
G.
,
1971
, “
Potential Operators and Conservative Systems
,”
14th Polish Solid Mechanics Conference
, Kroscienko, Poland, Sept. 11, pp. 141–146.
5.
Sheinman
,
I.
, and
Tene
,
Y.
,
1973
, “
Potential Energy of a Normal Pressure Field Acting on an Arbitrary Shell
,”
AIAA J.
,
11
(
8
), p.
1216
.
6.
Loganathan
,
K.
,
Chan
,
S. C.
,
Gallagher
,
R. H.
, and
Abel
,
J. F.
,
1979
, “
Finite Element Representation and Pressure Stiffness in Shell Stability Analysis
,”
Int. J. Numer. Methods Eng.
,
14
(
9
), pp.
1413
1429
.
7.
Mang
,
H. A.
,
1980
, “
Symmetricability Pressure Stiffness Matrices for Shells With Loads Free Edges
,”
Int. J. Numer. Methods Eng.
,
15
(
7
), pp.
981
990
.
8.
Iwata
,
K.
,
Tsukimor
,
K.
, and
Kubo
,
F.
,
1991
, “
A Symmetric Load-Stiffness Matrix for Buckling Analysis of Shell Structures Under Pressure Loads
,”
Int. J. Pressure Vessels Piping
,
45
(
1
), pp.
101
120
.
9.
Bolotin
,
V. V.
,
1963
,
Nonconservative Problems of the Theory of Elastic Stability
,
Pergamon Press
,
New York
, pp.
53
55
.
10.
Hibbitt
,
H. D.
,
1979
, “
Some Follower Forces and Load Stiffness
,”
Int. J. Numer. Methods Eng.
,
14
(
6
), pp.
937
941
.
11.
Schweizerhof
,
K.
, and
Ramm
,
E.
,
1984
, “
Displacement Dependent Pressure Loads in Nonlinear Finite Element Analysis
,”
Comput. Struct.
,
18
(
6
), pp.
1099
1114
.
12.
Visy
,
D.
,
2012
, “
Elastic and Geometric Stiffness Matrices for Semi Analytical Finite Strip Method
,”
Conference of Junior Researchers in Civil Engineering
, Budapest, Hungary, June 19–20, pp. 272–281.
13.
Tornabene
,
F.
, and
Viola
,
E.
,
2008
, “
2-D Solution for Free Vibrations of Parabolic Shells Using Generalized Differential Quadrature Method
,”
Eur. J. Mech. A
,
27
(
6
), pp.
1001
1025
.
14.
Teng
,
J. G.
, and
Hong
,
T.
,
1998
, “
Nonlinear Thin Shell Theories for Numerical Buckling Predictions
,”
Thin-Walled Struct.
,
31
(
1–3
), pp.
89
115
.
15.
Hong
,
T.
, and
Teng
,
J. G.
,
2002
, “
Non-Linear Analysis of Shell of Revolution Under Arbitrary Loads
,”
Compos. Struct.
,
80
(
18–19
), pp.
1547
1568
.
16.
Singh
,
S.
,
Patel
,
B. P.
, and
Nath
,
Y.
,
2009
, “
Postbuckling of Angle-Ply Laminated Cylindrical Shells With Meridional Curvature
,”
Thin-Walled Struct.
,
47
(
3
), pp.
359
364
.
17.
Ovesy
,
H. R.
, and
Fazilati
,
J.
,
2009
, “
Stability Analysis of Composite Laminated Plate and Cylindrical Shell Structures Using Semi-Analytical Finite Strip Method
,”
Compos. Struct.
,
89
(
3
), pp.
467
474
.
18.
Tornabene
,
F.
,
2011
, “
2-D GDQ Solution for Free Vibrations of Anisotropic Doubly-Curved Shells and Panels of Revolution
,”
Compos. Struct.
,
93
(
7
), pp.
1854
1876
.
19.
Li
,
Z. M.
,
Lin
,
Z. Q.
, and
Chen
,
G. L.
,
2009
, “
Nonlinear Buckling and Postbuckling Behavior of 3D Braided Composite Cylindrical Shells Under External Pressure Loads in Thermal Environments
,”
ASME J. Pressure Vessel Technol.
,
131
(
6
), p.
061206
.
20.
Khayat
,
M.
,
Poorveis
,
D.
,
Moradi
,
S.
, and
Hemmati
,
M.
,
2016
, “
Buckling of Thick Deep Laminated Composite Shell of Revolution Under Follower Force
,”
Struct. Eng. Mech.
,
58
(
1
), pp.
59
91
.
21.
Carnoy
,
E. G.
,
Guennoun
,
N.
, and
Sander
,
G.
,
1984
, “
Static Buckling Analysis of Shell Submitted to Follower Force by Finite Element Method
,”
Comput. Struct.
,
19
(
1–2
), pp.
41
49
.
22.
Poorveis
,
D.
, and
Kabir
,
M. Z.
,
2006
, “
Buckling of Discretely Stringer-Stiffened Composite Cylindrical Shells Under Combined Axial Compression and External Pressure
,”
Sci. Iran.
,
13
(2), pp.
113
123
.
23.
Cagdas
,
I. U.
, and
Adali
,
S.
,
2011
, “
Buckling of Cross-Ply Cylinders Under Hydrostatic Pressure Considering Pressure Stiffness
,”
Ocean Eng.
,
38
(
4
), pp.
559
569
.
24.
Altman
,
W.
, and
Oliveira
,
M. G. D.
,
1988
, “
Vibration and Stability Cantilevered Cylindrical Shell Panels Under Follower Forces
,”
J. Sound Vib.
,
122
(
2
), pp.
291
298
.
25.
Altman
,
W.
, and
Oliveira
,
M. G. D.
,
1990
, “
Vibration and Stability Shell Panels With Slight Internal Damping Under Follower Forces
,”
J. Sound Vib.
,
136
(
1
), pp.
45
50
.
26.
Altman
,
W.
, and
Oliveira
,
M. G. D.
,
1987
, “
Stability of Cylindrical Shell Panels Subjected to Follower Forces Based on a Mixed Finite Element Formulation
,”
Comput. Struct.
,
27
(
3
), pp.
367
372
.
27.
Mahamood
,
R. M.
, and
Akinlabi
,
E. T.
,
2012
, “
Functionally Graded Material: An Overview
,” World Congress on Engineering (
WCE
), London, July 4–6, pp. 1593–1597.
28.
Shen
,
H. S.
,
1998
, “
Postbuckling Analysis of Stiffened Laminated Cylindrical Shells Under Combined External Liquid Pressure and Axial Compression
,”
Eng. Struct.
,
20
(
8
), pp.
738
751
.
29.
Zuo
,
Q. H.
, and
Schreyer
,
H. L.
,
1996
, “
Flutter and Divergence Instability of Nonconservative Beams and Plates
,”
Int. J. Solids Struct.
,
33
(
9
), pp.
1355
1367
.
30.
Lanhe
,
W.
,
2004
, “
Thermal Buckling of a Simply Supported Moderately Thick Rectangular FGM Plate
,”
Compos. Struct.
,
64
(
2
), pp.
211
218
.
31.
Chi
,
S.-H.
, and
Chung
,
Y.-L.
,
2006
, “
Mechanical Behavior of Functionally Graded Material
,”
Int. J. Solids Struct.
,
43
(
13
), pp.
3657
3674
.
32.
Santos
,
H.
,
Soares
,
C. M. M.
,
Soares
,
C. A. M.
, and
Reddy
,
J. N.
,
2008
, “
A Semi-Analytical Finite Element Model for the Analysis of Cylindrical Shells Made of Functionally Graded Materials Under Thermal Shock
,”
Compos. Struct.
,
86
(
1–3
), pp.
10
21
.
33.
Reddy
,
J. N.
, and
Chin
,
C. D.
,
2007
, “
Thermo Mechanical Analysis of Functionally Graded Cylinders and Plates
,”
J. Therm. Stresses
,
21
(
6
), pp.
593
626
.
34.
Tornabene
,
F.
,
Fantuzzi
,
N.
, and
Bacciocchi
,
M.
,
2014
, “
Free Vibrations of Free-Form Doubly Curved Shells Made of Functionally Graded Materials Using Higher Order Equivalent Single Later Theories
,”
Composites, Part B
,
67
, pp.
490
509
.
35.
Dung
,
D. V.
, and
Hoa
,
L. K.
,
2015
, “
Nonlinear Torsional Buckling and Postbuckling of Eccentrically Stiffened FGM Cylindrical Shells in Thermal Environment
,”
Composites, Part B
,
69
, pp.
378
388
.
36.
Bich
,
D. H.
,
Ninh
,
D. G.
, and
Tran
,
I. T.
,
2016
, “
Non-Linear Buckling Analysis of FGM Toroidal Shell Segments Filled Inside by an Elastic Medium Under External Pressure Loads Including Temperature Effects
,”
Composites, Part B
,
87
, pp.
75
91
.
37.
Duc
,
N. D.
,
Thang
,
P. T.
,
Dao
,
N. T.
, and
Tac
,
H. V.
,
2015
, “
Nonlinear Buckling of Higher Deformable S-FGM Thick Circular Cylindrical Shells With Metal–Ceramic–Metal Layers Surrounded on Elastic Foundations in Thermal Environment
,”
Compos. Struct.
,
121
, pp.
134
141
.
38.
Zhang
,
Y.
,
Huang
,
H.
, and
Han
,
Q.
,
2015
, “
Buckling of Elastoplastic Functionally Graded Cylindrical Shells Under Combined Compression and Pressure
,”
Composites, Part B
,
69
, pp.
120
126
.
39.
Shen
,
H. S.
,
2002
, “
Postbuckling Analysis of Axially-Loaded Functionally Graded Cylindrical Shells in Thermal Environments
,”
Compos. Sci. Technol.
,
62
(
7–8
), pp.
977
998
.
40.
Na
,
K. S.
, and
Kim
,
J. H.
,
2004
, “
Three-Dimensional Thermal Buckling Analysis of Functionally Graded Materials
,”
Composites, Part B
,
35
(
5
), pp.
429
437
.
41.
Khazaeinejad
,
P.
,
Najafizadeh
,
M. M.
, and
Jenabi
,
J.
,
2010
, “
On the Buckling of Functionally Graded Cylindrical Shells Under Combined External Pressure and Axial Compression
,”
ASME J. Pressure Vessel Technol.
,
132
(
6
), p.
064501
.
42.
Wang
,
Z. W.
,
Zhang
,
Q.
,
Xia
,
L. Z.
,
Wu
,
J. T.
, and
Liu
,
P. Q.
,
2015
, “
Thermo-Mechanical Analysis of Pressure Vessels With Functionally Graded Material Coating
,”
ASME J. Pressure Vessel Technol.
,
138
(
1
), p.
011205
.
43.
Kar
,
V. R.
, and
Panda
,
S. K.
,
2016
, “
Nonlinear Thermo-Mechanical Behavior of Functionally Graded Material Cylindrical/Hyperbolic/Elliptical Shell Panel With Temperature-Dependent and Temperature-Independent Properties
,”
ASME J. Pressure Vessel Technol.
,
138
(
6
), p.
061202
.
44.
Ganapathi
,
M.
,
2007
, “
Dynamic Stability Characteristics of Functionally Graded Materials Shallow Spherical Shells
,”
Compos. Struct.
,
79
(
3
), pp.
338
343
.
45.
Sofiyev
,
A. H.
,
2010
, “
The Buckling of FGM Truncated Conical Shells Subjected to Combined Axial Tension and Hydrostatic Pressure
,”
Compos. Struct.
,
92
(
2
), pp.
488
498
.
46.
Torki
,
M. E.
,
Kazemi
,
M. T.
,
Haddadpour
,
H.
, and
Mahmoudkhani
,
S.
,
2014
, “
Dynamic Stability of Cantilevered Functionally Graded Cylindrical Shells Under Axial Follower Forces
,”
Thin-Walled Struct.
,
79
, pp.
138
146
.
47.
Nguyen
,
T. K.
,
Sab
,
K.
, and
Bonnet
,
G.
,
2007
, “
Shear Correction Factors for Functionally Graded Plates
,”
Mech. Adv. Mater. Struct.
,
14
(
8
), pp.
567
575
.
48.
Goyal
,
V. K.
, and
Kapania
,
R. K.
,
2008
, “
Dynamic Stability of Laminated Beams Subjected to Nonconservative Loading
,”
Thin-Walled Struct.
,
46
(
12
), pp.
1359
1369
.
49.
Hibbitt, Karlsson & Sorensen
,
1998
, “
ABAQUS/Standard User's Manual Vols. I–III, Version 5.8
,” Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI.
50.
Khayat
,
M.
,
Poorveis
,
D.
, and
Moradi
,
S.
,
2016
, “
Buckling Analysis of Laminated Composite Cylindrical Shell Subjected to Lateral Displacement-Dependent Pressure Using Semi-Analytical Finite Strip Method
,”
Steel Compos. Struct.
,
22
(
2
), pp.
301
321
.
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