A bellows is frequently applied in piping systems for absorbing mechanical movement. Its geometry is an axial symmetric shell, which is composed of two toroidal shells and one annular plate. The mechanical behavior of U-shaped bellows under axial force and internal pressure is estimated by changing the dimensions of the geometric parameters. The changing ranges of the geometric dimensions is so selected as to invest the results with practical environments in many fields. The minimization of strength and spring rate is considered simultaneously as a multiple objective function. The weighting objective method is implemented, in which a vector function is transformed to a scalar function. The structure is analyzed by the energy method for toroidal sections. Optimization is carried out by the Recursive Quadratic Programming algorithm.
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- a 1,a 2 :
Outer and inner torus radius, respectively
- C n ,C p :
Factors from EJMA
- D :
Et 3/12(I−ν 2), shell stiffness
- 91-2 :
Fa/4πD, dimensionless total axial forc
- h :
- l :
Length of annular plate
- \(\bar F\) :
3(1−ν 2)a 3/Et 3, dimensionless pressure
- q :
- r :
Arbitary circumferential radius
- t :
- v, w :
Meridional and normal displacement, respectively
- y, z :
Axial and radial displacement, respectively
A/r, radius ratio
- ϕ, gq:
Meridional and circumferential angle, respectively
- σ t :
Total stress from EJMA
3(1−ν 2)a 4/y 2 t 2, shell parameter, dimensionless
- ϕ, gq:
Meridional and circumferential direction, respectively
- 1, 2:
Outer and inner torus, respectively
Arora, J. S. and Tseng, C. H., 1986,IDESIGN User’s Manual Version 3.5, Optimal Design Laboratory, University of Iowa, Iowa City, IA.
Bhavikatti, S. S. et al., 1979, “Optimum Design of Flanged and Flued Expansion Joints”,Engineering Optimization, Vol. 4, pp. 121–128.
Calladine, C. R., 1974, “Flexibility of Axially Symmetric Bellows Under Axial Loading”,Int. J. Mech. Sci., Vol. 16, pp. 843–853.
Chakraverti, G., 1976, “Optimum Design of Expansion Bellows for Piping Systems,” M. Thesis, Dept. of Applied Mechanics, Ilt, Delhi.
Clark, R. A., 1970, “An Expansion Bellows Problems,”J. Appl. Mech., pp. 61–69.
Hamada, M., 1973, “On the Optimum Shapes of Some Axisymmetric Shells”,IUTAM Symp. on Optimization in Structural Design, Warsaw/Poland, Springer Verlag, pp. 248–262.
Hamada, M. et al., 1976, “Design Diagrams and Formulae for U-Shaped Bellows”,Int. J. Pres. Ves. & Piping (4), pp. 315–328.
Janzen, P., 1979, “Formulae and Graphs of Elastic Stresses for Design and Analysis of U-Shaped Bellows”,Int. J. Pres. Ves. & Piping 0308-0161, pp. 407–423.
Kellogg, M. W., 1957,Design and Piping System, John-Willey & Sons Kraus, H., 1967,Thin Elastic Shells, John-Willey & Sons.
Laupa, A. and Weil, N. A., 1962, “Analysis of U Shaped Expansion Join t,”J. Appl. Mech., 29(1), pp. 115–123.
Lee, W. I., Ko, B. G. and Park, G. J., 1991a, “An Optimal Design of the Bellows in the Auto-mobile Exhaust System”,The 6th International Pacific Conference on Automotive Engineering, Vol. 1, pp. 401–411.
Lee, W. I., Ko, B. G. and Park, G. J., 1991b, “Shape Optimal Design of Bellows,”KSEA Spring Conference, pp. 327–336.
Lee, W. I., Ha, S. K. and Ko, B. G., 1992, “A Study on the Design Standards of the Automobile Bellows,”J. of the Research Institute of Industrial Sciences, Vol. 35, Hanyang Univ. Korea, pp. 247–263.
Osyczka, A., 1984,Multicriterion Optimization in Engineering with FORTRAN Programs, Ellis Horwood Limited.
Robotshaw Catalog, 1992,Bellows and Bellows Assembly for Pressure or Temperature Sensing, Robotshaw LTD.
Standards of the Expansion Joint Manufacturers Association (EJMA), 1993, Inc, sixth edition.
Takezono, S., 1971, “Stress Analysis of Expansion Joints of Pressure Vessel Under Internal Pressure”,Bul. JSME, Vol. 14, pp. 673–685.
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Ko, B., Park, G. & Lee, W. Mechanical behavior of U-shaped bellows and shape optimal design using multiple objective optimization method. KSME Journal 9, 91–101 (1995). https://doi.org/10.1007/BF02954357
- Bellows Expansion Joint
- Quadrant-Toroidal Shell
- Multiple Objective Optimization Method
- Weighting Objective Method
- EJMA Standard