A bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axially symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze the bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis program is developed to analyze various bellows. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. A shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function with weighting factors. The stiffness, strength, and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the natural frequencies, the fatigue limit, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is utilized to solve the problem.

Issue Section:
Research Papers
Topics:
Bellows (Equipment), Optimization, Shapes, Shells, Design, Finite element analysis, Stiffness, Algorithms, Deformation, Euler-Bernoulli beam theory, Fatigue life, Fatigue limit, Geometry, Manufacturing, Piping systems, Quadratic programming, Scalar functions
1.
Arora, J. S., and Tseng, C. H., 1986, IDESIGN User’s Manual Version 3.5, Optimal Design Laboratory, University of Iowa.
2.
Arora, J. S., 1989, Introduction to Optimum Design, McGraw-Hill Book Company, New York, NY.
3.
ASME Boiler and Pressure Vessel Code, 1969, Design by Analysis in Sections III and VIII, Division 2.
4.
Clark, R. A., 1964, Asymptotic Solutions of Elastic Shell Problem, Asymptotic Solutions of Differential Equations and Their Applications, John Wiley & Sons, Inc., New York, NY.
5.
Expansion Joint Manufacturers Association, Inc., 1993, Standards of the Expansion Joint Manufacturers Association, Inc. Sixth Edition.
6.
Koh, B. K., Park, G. J., and Lee, W. I., 1991, “An Optimal Design of the Bellows in the Automobile Exhaust System,” The 6th International Pacific Conference on Automotive Engineering, Vol. 1, pp. 401–411.
7.
Koh
B. K.
,
Ha
S. K.
, and
Lee
W. I.
,
1992
, “
A Study on the Design Standard of Automobile Bellows
” (in Korean),
Journal of the Research Institute of Industrial Sciences
, Hanyang University, Korea, Vol.
35
, pp.
247
–
263
.
8.
Koh
B. K.
,
Suh
Y. J.
, and
Park
G. J.
,
1994
, “
A Finite Element Analysis and Shape Optimal Design with Specified Stiffness for U-Typed Bellows
” (in Korean),
Transactions of Korea Society of Automotive Engineers
, Vol.
3
, No.
6
, pp.
96
–
111
.
9.
Koh, B. K., 1995, “A Top-Down Approach Design Methodology for Mechanical Systems with Bellows” (in Korean), Ph.D. thesis, Hanyang University, Seoul, Korea.
10.
Koh
B. K.
,
Park
G. J.
, and
Lee
W. I.
,
1995
a, “
Development of a Bellows Finite Element for the Analysis of Piping System
” (in Korean),
Transactions of Korea Society of Mechanical Engineers
, Vol.
19
, No.
6
, pp.
1439
–
1450
.
11.
Koh
B. K.
,
Park
G. J.
, and
Lee
W. I.
,
1995
b, “
Mechanical Behavior of U-Shaped Bellows and Shape Optimal Design Using Multiple Objective Optimization Method
,”
Transactions of Korea Society of Mechanical Engineers
, Vol.
9
, No.
1
, pp.
91
–
101
.
12.
Koh
B. K.
, and
Park
G. J.
,
1996
, “
Design of an Automobile Exhaust System Using a Top-Down Approach Design Methodology
,”
International Journal of Vehicle Design
, Vol.
17
, No.
3
, pp.
276
–
294
.
13.
Laupa
A.
, and
Weil
N. A.
,
1962
, “
Analysis of U Shaped Expansion Joint
,”
ASME Journal of Applied Mechanics
, Vol.
29
, pp.
115
–
123
.
14.
Lee
W. I.
, and
Kim
T. W.
,
1989
, “
A Study on the Modified Energy Theory for the Bellows Problem
” (in Korean),
Transactions of Korea Society of Automotive Engineers
, Vol.
11
, No.
2
, pp.
55
–
65
.
15.
Osyczka, A., 1984, Multicriterion Optimization in Engineering with FORTRAN Programs, John Wiley & Sons Inc., New York, NY.
16.
Ota
T.
, and
Hamada
M.
,
1963
, “
On the Strength of Toroidal Shells (1st Report, The Proposition on the Solutions)
,”
Bulletin of Japan Society of Mechanical Engineering
, Vol.
6
, No.
24
, pp.
639
–
654
.
17.
Richards, T. H., 1977, Energy Methods in Stress Analysis with an Introduction to Finite Element Techniques, Halsted Press, a division of John Wiley & Sons Inc., New York, NY.
18.
Ross, C. T. F., 1991, Finite Element Programs for Structural Vibrations, Springer-Verlag London Limited, U.K.
19.
Swanson, J., 1994, ANSYS Engineering Analysis System User’s Manual Revision 5.0a, Swanson Analysis Systems, Inc., Houston, TX.
20.
Tsukimori
K.
, et al.,
1992
, “
Fatigue and Creep-Fatigue Life Evaluation of U-Shaped Bellows
,”
ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY
, Vol.
114
, pp.
280
–
291
.
21.
Zienkiewicz, O. C., 1977, The Finite Element Method, McGraw-Hill, New York, NY.
This content is only available via PDF.
Copyright © 1998
 by The American Society of Mechanical Engineers
You do not currently have access to this content.