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Mechanical behavior of U-shaped bellows and shape optimal design using multiple objective optimization method

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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 :

Corrugation height

l :

Length of annular plate

\(\bar F\) :

3(1−ν 2)a 3/Et 3, dimensionless pressure

q :

Corrugation pitch

r :

Arbitary circumferential radius

t :

Material thickness

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


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Correspondence to Byoung-Gab Ko.

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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).

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Key Words

  • Bellows Expansion Joint
  • Quadrant-Toroidal Shell
  • Multiple Objective Optimization Method
  • Weighting Objective Method
  • EJMA Standard
  • Corrugation