Abstract

The formulation used by the most of studies on elastic torus are either Reissner’s mixed formulation or Novozhilov’s complex-form one; however, for vibration and some displacement boundary-related problem of torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for torus. In this article, we will carry on the first author’s previous work (Sun, 2010, “Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells,” J. Eng. Mech., 136, pp. 1281–1288.), and with the help of our own maple codes, we are able to simulate some typical problems of torus. The numerical results are verified by both finite element analysis and H. Reissner’s formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio. The analysis of a torus must be done by using the bending theory of a shell instead of membrane theory of shells, and also reveal that the inner torus is stronger than outer torus due to their Gaussian curvature. One of the most interesting discovery is that the crowns of a torus, the turning point of the Gaussian curvature at ϕ = 0, π, are the line where the mechanics response of inner and outer torus is almost separated.

References

1.
Zhang
,
W.
,
1944
, “
Der Spannungszustand in Kreisringschale und ähnlichen Schalen Mit Scheitelkreisringen Unter Drehsymmetrischer Belastung
,” Arbeitzur Erlangung des Grades eines Doctor-Ingenieurs der Technichen Hochschule, Berlin (Published in Scientific Report of National Tsinghua University, Ser. A, 1949), 289–349.
2.
Qian
,
W. Z.
, and
Liang
,
S. C.
,
1979
, “
Complex Form Equation and Asymptotic Solution
,”
J. Tsinghua University
,
19
(
1
), pp.
27
47
.
3.
Xia
,
Z. H.
, and
Zhang
,
W.
,
1986
, “
The General Solution for Thin-Walled Curved Tubes With Arbitrary Loadings and Various Boundary Conditions
,”
Int. J. Pressure Vessels Piping
,
26
(
2
), pp.
129
144
.
4.
Zhang
,
W.
,
Ren
,
W. M.
, and
Sun
,
B. H.
,
1990
, “
Toroidal Shells—History, Current Situation and Future
,”
Fifth Conference of Space Structures
,
Lanzhou, China
,
July 25–29
.
5.
Zhang
,
R. J.
, and
Zhang
,
W.
,
1991
, “
Turning Point Solutions for Thin Shell Vibration
,”
Int. J. Solids. Struct.
,
27
(
10
), pp.
1311
1326
.
6.
Zhang
,
R. J.
, and
Zhang
,
W.
,
1994
, “
Toroidal Shells Under Nonsymmetric Loading
,”
Int. J. Solids. Struct.
,
31
(
19
), pp.
2735
2750
.
7.
Audoly
,
B.
, and
Pomeau
,
Y.
,
2010
,
Elasticity and Geometry
,
University of Cambridge
,
Cambridge, UK
.
8.
Sun
,
B. H.
,
2010
, “
Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells
,”
J. Eng. Mech.
,
136
(
10
), pp.
1281
1288
.
9.
Sun
,
B. H.
,
2012
,
Toroidal Shells
,
Nova Novinka
,
New York
.
10.
Clark
,
R. A.
, and
Reissner
,
E.
,
1951
, “
Bending of Curved Tubes
,”
Adv. Appl. Mech.
,
2
, pp.
93
122
.
11.
Clark
,
R. A.
,
1950
, “
On the Theory of Thin Elastic Toroidal Shells
,”
J. Math. Phys.
,
29
(
1–4
), pp.
146
178
.
12.
Dahl
,
N. C.
,
1953
, “
Toroidal-Shell Expansion Joints
,”
ASME J. Appl. Mech.
,
20
(
4
), pp.
497
503
.
13.
Novozhilov
,
V. V.
,
1959
,
The Theory of Thin Shell
,
Noordhoff
,
Groningen
.
14.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
McGraw-Hill
,
New York
.
15.
Flügge
,
W.
,
1973
,
Stresses in Shells
,
Springer-Verlag Berlin
,
Heidelberg
.
16.
Flügge
,
W.
,
1961
,
Theory of Elastic Thin Shells
,
Pergamon Press
,
New York
.
17.
Sun
,
B. H.
,
2013
, “
Centenary Studies of Toroidal Shells and in Memory of Prof. Zhang Wei
,”
Mech. Eng.
,
37
(
3
), pp.
94
97
.
18.
Föppl
,
L.
,
1907
,
Vorlesungen Über Technische Mechanik, Volume 5.B.G.
,
Teubner
,
Leipzig, Germany
.
19.
Weihs
,
G.
,
1911
,
Über Spannungs- Und Formänderungszustände in Dünnen
,
Hohlreifen
,
Halle
.
20.
Wissler
,
H.
,
1916
, “
Festigkeiberechung von ringsflachen
,”
Promotionarbeit, Zurich
.
21.
Kuznetsov
,
V. V.
, and
Levyakov
,
S. V.
,
2001
, “
Nonlinear Pure Bending of Toroidal Shells of Arbitrary Cross-Section
,”
Int. J. Solids. Struct.
,
38
(
40–41
), pp.
7343
7354
.
22.
Zingoni
,
A.
,
Enoma
,
N.
, and
Govender
,
N.
,
2015
, “
Equatorial Bending of an Elliptic Toroidal Shell
,”
Thin-Walled Struct.
,
96
, pp.
286
294
.
23.
Jiammeepreecha
,
W.
, and
Chucheepsakul
,
S.
,
2017
, “
Nonlinear Static Analysis of an Underwater Elastic Semi-Toroidal Shell
,”
Thin-Walled Struct.
,
116
, pp.
12
18
.
24.
Enoma
,
N.
, and
Zingoni
,
A.
,
2020
, “
Analytical Formulation and Numerical Modelling for Multi-Shell Toroidal Pressure Vessels
,”
Computers Struct.
,
232
, p.
105811
.
25.
Reissner
,
H.
,
1912
, “
Spannungen in kugelschalen (kuppeln)
,”
Festschrift Heinrich Müller-Breslau (A. Kröner, Leipzig)
, pp.
181
193
.
26.
Meissner
,
E.
,
1915
, “
Über und elastizitat festigkeit dunner schalen
,”
Viertelschr. D. nature. Ges, Bd.60, Zurich.
27.
Tölke
,
F.
,
1938
, “
Zur Integration Der Differentialgleichungen Der Drehsymmetrisch Belasteten Rotationsschale Bei Beliebiger Wandstärke
,”
Ingenieur-Archiv
,
9
(
4
), pp.
282
288
.
28.
Reissner
,
E.
,
1949
, “
On Bending of Curved Thin-Walled Tubes
,”
Proc. Natl. Acad. Sci. USA
,
35
(
4
), pp.
204
208
.
29.
Tao
,
L. N.
,
1959
, “
On Toroidal Shells
,”
J. Math. Phys.
,
38
(
1–4
), pp.
130
134
.
30.
Steele
,
C. R.
,
1965
, “
Toroidal Pressure Vessels
,”
J. Spacecraft Rockets
,
2
(
6
), pp.
937
943
.
31.
Sun
,
B. H.
,
2016
, “
Exact Solution of Qian’s Equation of Slender Toroidal Shells
,”
Mech. Eng.
,
38
(
5
), pp.
567
569
.
32.
Sun
,
B. H.
,
2021
, “
Small Symmetrical Deformation of Thin Torus With Circular Cross Section
,”
Thin-Walled Struct.
,
163
, p.
107680
.
33.
Sun
,
B. H.
,
2021
, “
Geometry-Induced Rigidity in Elastic Torus From Circular to Oblique Elliptic Cross-Section
,”
Int. J. Non-Linear Mech.
,
135
, p.
103754
.
34.
Sun
,
B. H.
,
2022
, “
Gol’denveizer’s Problem of Elastic Torus
,”
Thin-Walled Struct.
,
171
, p.
108718
.
35.
Sun
,
B. H.
,
2022
, “
Nonlinear Elastic Deformation of Mindlin Torus
,”
Commun. Nonlinear Sci. Numerical Simul.
,
114
, p.
106698
.
36.
Song
,
G. K.
, and
Sun
,
B. H.
,
2022
, “
Nonlinear Investigation of Gol’denveizer’s Problem of a Circular and Elliptic Elastic Torus
,”
Thin-Walled Struct.
,
180
, p.
109862
.
37.
Reddy
,
J. N.
,
2007
,
Theory and Analysis of Elastic Plates and Shells
,
CRC Press
,
Boca Raton, FL
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