Abstract

The post-buckling and nonlinear dynamic response of shallow spherical caps subjected to external pressure is analyzed. The Novozhilov's nonlinear thin shell theory is used to express the strain–displacement relations. Following the Rayleigh-Ritz method, the displacement fields are expanded using a mixed series: Legendre polynomials in the meridional direction, harmonic functions in the circumferential direction. Once the linear analysis is completed, the displacement fields are re-expanded and the nonlinear dynamic model is obtained by using the Lagrange equations. The response of clamped caps, made of isotropic and homogeneous material, is investigated. The bifurcation analyses of equilibrium points and periodic orbits are presented by using continuation techniques. Benchmark results are provided in terms of natural frequencies and critical buckling loads. The dynamic effects due to the interaction between static and dynamic pressure are investigated. Numerical results pointed out that, under particular load conditions, dynamic bifurcation results in nonnegligible asymmetric states activation in the response of the structure.

References

References
1.
Budiansky
,
B.
,
1959
, “
Buckling of Clamped Shallow Spherical Shells
,”
Proceedings of IUTAM Symposium on the Theory of Thin Elastic Shells
,
Delft, Netherlands
,
Aug. 24–28
, pp.
64
94
.
2.
Budiansky
,
B.
, and
Hutchinson
,
J. W.
,
1966
, “
A Survey of Some Buckling Problems
,”
AIAA J.
,
4
(
9
), pp.
1505
1510
.10.2514/3.3727
3.
Hutchinson
,
J. W.
,
1967
, “
Imperfection Sensitivity of Externally Pressurized Spherical Shells
,”
ASME J. Appl. Mech.
,
34
(
1
), pp.
49
55
.10.1115/1.3607667
4.
Fitch
,
J. R.
, and
Budiansky
,
B.
,
1970
, “
Buckling and Postbuckling Behavior of Spherical Caps Under Axisymmetric Load
,”
AIAA J.
,
8
(
4
), pp.
686
693
.10.2514/3.5742
5.
Huang
,
N.
,
1964
, “
Unsymmetrical Buckling of Thin Shallow Spherical Shells
,”
ASME J. Appl. Mech.
,
31
(
3
), pp.
447
457
.10.1115/1.3629662
6.
Krenzke
,
M. A.
, and
Kiernan
,
T. J.
,
1963
, “
Elastic Stability of Near-Perfect Shallow Spherical Shells
,”
AIAA J.
,
1
(
12
), pp.
2855
2857
.10.2514/3.2187
7.
Lock
,
M. H.
,
Okubo
,
S.
, and
Whittier
,
S. J.
,
1968
, “
Experiments on the Snapping of a Shallow Dome Under Step Pressure Load
,”
AIAA J.
,
6
(
7
), pp.
1320
1326
.10.2514/3.4742
8.
Ball
,
R. E.
, and
Burt
,
J. A.
,
1973
, “
Dynamic Buckling of Shallow Spherical Shells
,”
ASME J. Appl. Mech.
,
40
(
2
), pp.
411
416
.10.1115/1.3422998
9.
Akkas
,
N.
,
1976
, “
Bifurcation and Snap-Through Phenomena in Asymmetric Dynamic Analysis of Shallow Spherical Shells
,”
Comput. Struct.
,
6
(
3
), pp.
241
251
.10.1016/0045-7949(76)90035-3
10.
Evensen
,
H. A.
, and
Evan-Iwanowski
,
R. M.
,
1967
, “
Dynamic Response and Stability of Shallow Spherical Shells Subject to Time-Dependent Loading
,”
AIAA J.
,
5
(
5
), pp.
969
976
.10.2514/3.4110
11.
Yasuda
,
K.
, and
Kushida
,
G.
,
1984
, “
Nonlinear Forced Oscillations of a Shallow Spherical Shell
,”
Bull. JSME
,
27
(
232
), pp.
2233
2240
.10.1299/jsme1958.27.2233
12.
Gonçalves
,
P. B.
,
1993
, “
Jump Phenomenon, Bifurcations, and Chaos in a Pressure Loaded Spherical Cap Under Harmonic Excitation
,”
Appl. Mech. Rev.
,
46
(
11S
), pp.
S279
S288
.10.1115/1.3122646
13.
Grossman
,
P. L.
,
Koplik
,
B.
, and
Yu
,
Y. Y.
,
1969
, “
Nonlinear Vibrations of Shallow Spherical Shells
,”
ASME J. Appl. Mech.
,
36
(
3
), pp.
451
458
.10.1115/1.3564701
14.
Yu
,
Y. Y.
,
1964
, “
Generalized Hamilton's Principle and Variational Equation of Motion in Nonlinear Elasticity Theory, With Application to Plate Theory
,”
J. Acoust. Soc. Am.
,
36
(
1
), pp.
111
120
.10.1121/1.1918921
15.
Amabili
,
M.
, and
Breslavsky
,
I. D.
,
2015
, “
Displacement Dependent Pressure Load for Finite Deflection of Doubly-Curved Thick Shells and Plates
,”
Int. J. Non-Linear Mech.
,
77
, pp.
265
273
.10.1016/j.ijnonlinmec.2015.09.007
16.
Novozhilov
,
V. V.
,
1953
,
Foundations of the Nonlinear Theory of Elasticity
,
Graylock Press
,
Rochester, NY
.
17.
Leissa
,
A. W.
,
1973
,
Vibration of Shells, NASA SP-288
,
Government Printing Office, Now Available from the Acoustical Society of America
,
Washington, DC
.
18.
Amabili
,
M.
,
2005
, “
Non-Linear Vibrations of Doubly Curved Shallow Shells
,”
Int. J. Non-Linear Mech.
,
40
(
5
), pp.
683
710
.10.1016/j.ijnonlinmec.2004.08.007
19.
De Souza
,
V. C. M.
, and
Croll
,
J. G. A.
,
1980
, “
An Energy Analysis of the Free Vibrations of Isotropic Spherical Shells
,”
J. Sound Vib.
,
73
(
3
), pp.
379
404
.10.1016/0022-460X(80)90522-2
20.
Strozzi
,
M.
, and
Pellicano
,
F.
,
2013
, “
Nonlinear Vibrations of Functionally Graded Cylindrical Shells
,”
Thin-Walled Struct.
,
67
, pp.
63
77
.10.1016/j.tws.2013.01.009
21.
Pellicano
,
F.
,
2007
, “
Vibrations of Circular Cylindrical Shells: Theory and Experiments
,”
J. Sound Vib.
,
303
(
1–2
), pp.
154
170
.10.1016/j.jsv.2007.01.022
22.
Evensen
,
D. A.
,
1966
, “
Nonlinear Flexural Vibration of Thin Circular Rings
,”
ASME J. Appl. Mech.
,
33
(
3
), pp.
553
560
.10.1115/1.3625121
23.
Zoelly
,
R.
,
1915
, “
Ueber Ein Knickungsproblem an Der Kuegelschale
,” Ph.D. thesis, ETH Zurich, Zürich, Switzerland.
24.
Zarghamee
,
M. S.
, and
Robinson
,
A. R.
,
1967
, “
A Numerical Method for Analysis of Free Vibration of Spherical Shells
,”
AIAA J.
,
5
(
7
), pp.
1256
1261
.10.2514/3.4180
25.
Narasimhan
,
M. C.
, and
Alwar
,
R. S.
,
1992
, “
Free Vibration Analysis of Laminated Orthotropic Spherical Shells
,”
J. Sound Vib.
,
154
(
3
), pp.
515
529
.10.1016/0022-460X(92)90783-T
26.
Thomas
,
O.
,
Touzé
,
C.
, and
Chaigne
,
A.
,
2005
, “
Non-Linear Vibrations of Free-Edge Thin Spherical Shells: Modal Interaction Rules and 1: 1: 2 Internal Resonance
,”
Int. J. Solids Struct.
,
42
(
11–12
), pp.
3339
3373
.10.1016/j.ijsolstr.2004.10.028
27.
Avramov
,
K. V.
, and
Malyshev
,
S. E.
,
2018
, “
Periodic, Quasi-Periodic, and Chaotic Geometrically Nonlinear Forced Vibrations of a Shallow Cantilever Shell
,”
Acta Mech.
,
229
(
4
), pp.
1579
1595
.10.1007/s00707-017-2087-x
28.
Doedel
,
E. J.
,
Champneys
,
A. R.
,
Fairgrieve
,
T. F.
,
Kuznetsov
,
Y. A.
,
Sandstede
,
B.
, and
Wang
,
X.
,
1997
, “
AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (With HomCont)
,” Concordia University, Montreal, QC, Canada.
29.
Yamada
,
S.
,
Uchiyama
,
K.
, and
Yamada
,
M.
,
1983
, “
Experimental Investigation of the Buckling of Shallow Spherical Shells
,”
Int. J. Non-Linear Mech.
,
18
(
1
), pp.
37
54
.10.1016/0020-7462(83)90017-3
30.
Grigolyuk
,
E. I.
, and
Lopanitsyn
,
Y. A.
,
2003
, “
The Non-Axisymmetric Postbuckling Behaviour of Shallow Spherical Domes
,”
J. Appl. Math. Mech.
,
67
(
6
), pp.
809
818
.10.1016/S0021-8928(03)90501-6
31.
Marcinowski
,
J.
,
2007
, “
Stability of Relatively Deep Segments of Spherical Shells Loaded by External Pressure
,”
Thin-Walled Struct.
,
45
(
10–11
), pp.
906
910
.10.1016/j.tws.2007.08.034
32.
Silva
,
F. M. A.
,
Soares
,
R. M.
,
del Prado
,
Z. G. N.
, and
Gonçalves
,
P. B.
,
2020
, “
Intra-Well and Cross-Well Chaos in Membranes and Shells Liable to Buckling
,”
Nonlinear Dyn.
, 102(2), pp.
877
906
.10.1007/s11071-020-05661-z
You do not currently have access to this content.