This paper is concerned with the 2D elasticity solution for the buckling of a thick orthotropic ring under external hydrostatic pressure loading. The bifurcation buckling problem is first formulated using two methods, distinguished by the manner in which the external work done by the pressure loading during the buckling transition is treated. In doing so, the correct buckling equations and associated traction boundary conditions are derived. The resulting sets of equations and associated boundary conditions are then cast in a weak form, amenable to a numerical solution using the finite element method. The necessity of using the correct pairs of energetically conjugate stress and strain measures for the buckling problem is pointed out. Errors in using the incorrect traction boundary condition and terms that influence the buckling load and that have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. Results from the present two-dimensional analysis to predict the critical pressure are compared with previous theoretical results. The formulation and results presented here can be used as the correct benchmark solution to establish the accuracy in computing the buckling load of thick orthotropic composite structures, of contemporary interest, due to the increased use of thick-walled composite shell type structures in diverse engineering applications.

References

1.
Brush
,
D. O.
, and
Almroth
,
B. O.
,
1975
,
Buckling of Bars, Plates, and Shells.
McGraw-Hill
,
New York
.
2.
Carrier
,
G. F.
,
1947
, “
On Buckling of Elastic Rings
,”
J. Math. Phys.
,
26
(
2
), pp.
94
103
.
3.
Koiter
,
W. T.
,
1963
, “
Elastic Stability and Postbuckling Behavior
,”
Langer
,
R. E.
, ed.,
Proceedings of the Symposium on Nonlinear Problems
,
University of Wisconsin Press
, pp.
257
275
.
4.
Thompson
,
J. M. T.
,
1969
, “
A General Theory for the Equilibrium and Stability of Discrete Conservative Systems
,”
Z. Angew. Math. Phys.
,
20
(
6
), pp.
797
846
.10.1007/BF01592294
5.
Budiansky
,
B
.,
1974
, “
Theory of Buckling and Postbuckling Behavior of Elastic Structures
,”
Adv. Appl. Mech.
,
14
(
2
), pp.
1
65
.10.1016/S0065-2156(08)70030-9
6.
Kardomateas
,
G. A.
,
1993
, “
Buckling of Thick Orthotropic Cylindrical-Shells Under External Pressure
,”
ASME J. Appl. Mech.
,
60
(
1
), pp.
195
202
.10.1115/1.2900745
7.
Novozhilov
,
V. V.
,
1953
,
Foundations of the Nonlinear Theory of Elasticity
,
Graylock
,
Rochester, NY
.
8.
Fu
,
L.
, and
Waas
,
A. M.
,
1995
, “
Initial Postbuckling Behavior of Thick Rings Under Uniform External Hydrostatic Pressure
,”
ASME J. Appl. Mech.
,
62
(
2
), pp.
338
345
.10.1115/1.2895936
9.
Sewell
,
M. J.
,
1967
, “
On Configuration-Dependent Loading
,”
Arch. Ration. Mech. An.
,
23
(
5
), pp.
327
351
.10.1007/BF00276777
10.
Bodner
,
S. R.
,
1958
, “
On the Conservativeness of Various Distributed Force Systems
,”
J. Aeronaut. Sci.
,
25
(
2
), pp.
132
133
.
11.
Bažant
,
Z. P.
,
1971
, “
A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies
,”
ASME J. Appl. Mech.
,
38
(
4
), pp.
919
928
.10.1115/1.3408976
12.
Bažant
,
Z. P.
, and
Cedolin
,
L.
,
1991
,
Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories
,
Oxford University Press
,
New York
.
13.
Singer
,
J.
, and
Babcock
,
C. D.
,
1970
, “
On Buckling of Rings Under Constant Directional and Centrally Directed Pressure
,”
ASME J. Appl. Mech.
,
37
(
1
), pp.
215
218
.10.1115/1.3408445
14.
Bower
,
A. F.
,
2010
,
Applied Mechanics of Solids
,
CRC Press
,
Boca Raton
, FL.
15.
Ji
,
W.
, and
Waas
,
A. M.
,
2009
, “
2D Elastic Analysis of the Sandwich Panel Buckling Problem: Benchmark Solutions and Accurate Finite Element Formulations
,”
Z. Angew. Math. Phys.
,
61
(
5
), pp.
897
917
.10.1007/s00033-009-0041-z
16.
Ji
,
W.
, and
Waas
,
A. M.
,
2010
, “
Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs
,”
ASME J. Appl. Mech.
,
77
(
4
), p.
044504
.10.1115/1.4000916
17.
Ji
,
W.
, and
Waas
,
A. M.
,
2012
, “
Accurate Buckling Load Calculations of a Thick Orthotropic Sandwich Panel
,”
Compos. Sci. Tech.
,
72
(
10
), pp.
1134
1139
.10.1016/j.compscitech.2012.02.020
18.
Ji
,
W.
,
Waas
,
A. M.
, and
Bažant
,
Z. P.
,
2012
, “
On the Importance of Work-Conjugacy and Objective Stress Rates in Finite Deformation Incremental Finite Element Analysis
,”
ASME J. Appl. Mech.
(accepted for publication).
19.
Hibbitt
,
H. D.
,
1979
, “
Some Follower Forces and Load Stiffness
,”
Int. J. Numer. Meth. Eng.
,
14
(
6
), pp.
937
941
.10.1002/nme.1620140613
20.
Mang
,
H. A.
,
1980
, “
Symmetricability of Pressure Stiffness Matrices for Shells With Loaded Free Edges
,”
Int. J. Numer. Meth. Eng.
,
15
(
7
), pp.
981
990
.10.1002/nme.1620150703
21.
Schweizerhof
,
K.
, and
Ramm
,
E.
,
1984
, “
Displacement Dependent Pressure Loads in Nonlinear Finite-Element Analyses
,”
Comput. Struct.
,
18
(
6
), pp.
1099
1114
.10.1016/0045-7949(84)90154-8
22.
Simo
,
J. C.
,
Taylor
,
R. L.
, and
Wriggers
,
P.
,
1991
, “
A Note on Finite-Element Implementation of Pressure Boundary Loading
,”
Commun. Appl. Numer. M.
,
7
(
7
), pp.
513
525
.10.1002/cnm.1630070703
23.
Zienkiewicz
,
O. C.
, and
Taylor
,
R. L.
,
2000
,
The Finite Element Method
,
Butterworth-Heinemann
,
Oxford, Boston
.
24.
Kardomateas
,
G. A.
,
2000
, “
Effect of Normal Strains in Buckling of Thick Orthotropic Shells
,”
J. Aerosp. Eng.
,
13
(
3
), pp.
85
91
.10.1061/(ASCE)0893-1321(2000)13:3(85)
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