The Föppl–Hencky nonlinear membrane theory is employed to study the axisymmetric deformation of annular elastic membranes. The general solutions for displacements and stresses are established for arbitrary edge boundary conditions. New exact solution results are developed for central loading and edge forcing conditions. Both positive and negative radial stress solutions are found. Comparisons are made for special cases to previously known solutions with excellent agreement.

1.
Föppl
,
A.
, 1907,
Vorlesungen uder Technische Mechanic
, Vol.
5
,
Teubner
,
Leipzig
.
2.
Karman
,
V.
, 1910, “
Festigkeitsprobleme in Maschinenbau
,”
Encyklopadie Der Mathematischen Wissenschaften
, Teubner, IV/4, pp. 311–385.
3.
Way
,
S.
, 1934, “
Bending of Circular Plates With Large Deflection
,”
Trans. ASME
0097-6822,
56
, pp.
627
633
.
4.
Hencky
,
H.
, 1915, “
Uber den Spannungszustand in Kreisrunden Platten Mit
,”
Zeitschrift fur Mathematik und Physik
,
63
, pp.
311
317
.
5.
Reissner
,
E.
, 1948, “
Note on Membrane Theory of Shells of Revolution
,”
J. Math. Phys.
,
26
, pp.
290
293
. 0097-1421
6.
Reissner
,
E.
, 1950, “
On Axisymmetrical Deformation of Thin Shells of Revolution
,”
Proc. Symp. Appl. Math.
0160-7634,
3
, pp.
27
52
.
7.
Weinitschke
,
H.
, 1980, “
On Axisymmetric Deformations of Nonlinear Elastic Membranes
,”
Mechanics Today
,
S.
Nemat-Nasser
, ed.,
Pergamon
, Oxford, pp.
523
541
.
8.
Dickey
,
R. W.
, 1983, “
The Nonlinear Circular Membrane Under a Vertical Force
,”
Q. Appl. Math.
,
41
, pp.
331
338
. 0033-569X
9.
Dickey
,
R. W.
, 1967, “
The Plane Circular Elastic Surface Under Normal Pressure
,”
Arch. Ration. Mech. Anal.
,
26
, pp.
219
236
. 0003-9527
10.
Weinitschke
,
H.
, 1987, “
On Finite Displacements of Circular Elastic Membranes
,”
Math. Methods Appl. Sci.
,
9
, pp.
76
98
. 0170-4214
11.
Callegari
,
A. J.
, and
Reiss
,
E. L.
, 1968, “
Non-Linear Boundary Value Problems for the Circular Membrane
,”
Arch. Ration. Mech. Anal.
,
31
, pp.
390
400
. 0003-9527
12.
Kelkar
,
A.
,
Elber
,
W.
, and
Raju
,
I. S.
, 1985, “
Large Deflections of Circular Isotropic Membranes Subjected to Arbitrary Axisymmetric Loading
,”
Comput. Struct.
,
21
, pp.
413
421
. 0045-7949
13.
Grabmüller
,
H.
, and
Novak
,
E.
, 1987, “
Nonlinear Boundary Value Problems for the Annular Membranes: A Note on Uniqueness of Positive Solutions
,”
J. Elast.
,
17
, pp.
279
284
. 0374-3535
14.
Schwerin
,
E.
, 1929, “
Uber Spannungen und Formanderungen Kreisringformige Membranen
,”
Z. Angew. Math. Mech.
0044-2267,
12
, pp.
651
659
.
15.
Yang
,
W.
, and
Hsu
,
K. H.
, 1971, “
Indentation of a Circular Membrane
,”
ASME J. Appl. Mech.
,
38
, pp.
227
230
. 0021-8936
16.
Chen
,
D.
, and
Cheng
,
S.
, 1996, “
Non-Linear Analysis of Prestretched Circular Membrane and a Modified Iteration Technique
,”
Int. J. Solids Struct.
,
33
, pp.
545
553
. 0020-7683
17.
Dienes
,
J. K.
, and
Miles
,
J. W.
, 1977, “
A Membrane Model for the Response of Thin Plates to Ballistic Impact
,”
J. Mech. Phys. Solids
,
25
, pp.
237
256
. 0022-5096
18.
Wierzbicki
,
T.
, and
Nurick
,
G. N.
, 1996, “
Large Deflection of Thin Plates Under Localized Impulsive Loading
,”
Int. J. Impact Eng.
,
18
, pp.
899
918
. 0734-743X
19.
Mori
,
D.
,
David
,
G.
,
Humphrey
,
J. D.
, and
Moores
,
J. E.
, 2005, “
Stress Distribution in a Circular Membrane With a Central Fixation
,”
ASME J. Biomech. Eng.
0148-0731,
127
, pp.
549
553
.
20.
David
,
G.
, and
Humphrey
,
J.
, 2004, “
Redistribution of Stress Due to a Circular Hole in a Nonlinear Anisotropic Membrane
,”
J. Biomech.
0021-9290,
37
, pp.
1197
1203
.
21.
Scott
,
O. N.
,
Begley
,
M. R.
,
Komaragiri
,
U.
, and
Mackin
,
T. J.
, 2004, “
Indentation of Freestanding Circular Elastomer Films Using Spherical Indenters
,”
Acta Mater.
1359-6454,
52
, pp.
4877
4885
.
22.
Pelesko
,
J.
, and
Chen
,
X.
, 2003, “
Electrostatic Deflections of Circular Elastic Membranes
,”
J. Electrost.
,
57
, pp.
1
12
. 0304-3886
23.
Begley
,
M. R.
, and
Mackin
,
T. J.
, 2004, “
Spherical Indentation of Freestanding Circular Thin Films in the Membrane Regime
,”
J. Mech. Phys. Solids
0022-5096,
52
, pp.
2005
2023
.
24.
Tuan
,
C. Y.
, 1998, “
Ponding on Circular Membranes
,”
Int. J. Solids Struct.
,
35
, pp.
269
283
. 0020-7683
25.
Beck
,
A.
, and
Grabmüller
,
H.
, 1992, “
Wrinkle-Free Solutions of Circular Membrane Problems
,”
Z. Angew. Math. Phys.
,
43
, pp.
481
504
. 0044-2275
26.
Callegari
,
A. J.
,
Reiss
,
E. L.
, and
Keller
,
H. B.
, 1971, “
Membrane Buckling: A Study of Solution Multiplicity
,”
Commun. Pure Appl. Math.
0010-3640,
24
, pp.
499
516
.
27.
MATLAB, 2006, The MathWorks Inc.
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