The Effects of Either Height of Bellows Ends on the Stress Distribution according to Rotation Angles

As analysis research, the effect of either height of bellows ends on the stress distribution has been proposed in the study. Rotation angle only is considered as a boundary condition. FEM solution for a u-shaped flexible tube under the action of angle of rotation is obtained. The design factor, convolution height of bellows ends, is considered for the simulation. The analysis is performed using the finite element analysis program. The maximum von-Mises stress and its reduction rate according to the height of bellows ends is compared respectively.


Introduction
Numerous papers have dealt with various aspects of bellows; Broman et al. [1] have determined dynamic characteristics of bellows by manipulating certain parameters of the beam finite elements, Jakubauskas and Werner [2] have considered the transverse vibrations of fluid-filled double-bellows expansion joints.Jha et al. [3] have investigated the stress corrosion cracking of stainless steel bellows of satellite launch vehicle propellant tank assembly.Zhu et al. [4] have investigated the effect of environmental medium on fatigue life for ushaped bellows expansion joints.In addition, Becht [5] has investigated the fatigue behavior of expansion joint bellows.
Usage of the bellows has been increased due to its inherent flexibility and excellent anti vibration properties.It has been used for a long time in many engineering applications, therefore, numerous papers dealt with bellows have found in the literatures.Many design formulae of the bellows can be found in ASME code [6].And the most comprehensive and widely accepted text on bellows design is the Standards of Expansion Joint Manufactures Association, EJMA [7].A comparison of the ASME code and the EJMA standards is given by Hanna [8], concluding that the two conform quite well in most aspects.In addition, the EJMA standards were compared with finite element and experimental analyses in some papers.But simplified equations are contained in EJMA, so it includes excessive safety factor and design errors.In order to compensate errors, service conditions in the design are considered in some studies [9].
Flexible tubes consist of a thin walled shell of revolution with a corrugated meridian in order to provide the flexibility needed to absorb mechanical movements.As the geometry of flexible tube is complex, it is difficult to analyze the mechanical behavior of flexible tubes.In some cases, the axi-symmetrical deformation problems of flexible tubes have been studied [10], and these problems were investigated by the finite difference method [11].
Proper dimensioning requires deep understanding of the characteristics of the flexible tubes and their interaction with the rest of the exhaust system.Off the shelf products seldom fit a specific application, which was experienced when flexible tubes were introduced into exhaust systems.Failure as shown in Figure 1,2 took place after rather short operation times and substitution of stronger and much more expensive material did not solve the problem [12].In order to prevent these failures, it is necessary to design an optimum shape reducing maximum stress.The design factors, mountain height, length, thickness, and the number of convolutions should be considered and the proper values should be chosen for optimum design.As maximum stress is initiated at both ends of bellows, it is necessary to analysis the effects of height of bellows ends on the maximum stress.The aim of this work is to find a relationship between the height of bellows ends and the rotation angle upon maximum von_Mises stress and stress reduction rate.

Analysis Model
To obtain the flexible tube profile, it was modeled with the finite element code.The flexible tube was meshed with 8 node shell elements and elastic -plastic non-linear analysis was performed.
Material properties and dimensions used in the analysis are described in Table 1.The yield stress for the model used in the study is 353MPa and ANSYS was used as FE-solver for stress analysis.Analysis model is shown in Fig. 1.Fig. 1 displays the geometry profile for the analysis model.The height of convolution from both end caps of the flexible tube varies as shown in Fig. 1.The height of 1st convolution varies from 4mm to 10mm.The pitch of convolution is 6.8mm.The mesh consists of 112,800 elements.One side is fixed and lateral displacement of 0mm and angle of rotation from 0.01 to 0.21 degrees was applied on the other side for boundary condition.SRR is maximum at the rotation angle 0.05 degree.If the rotation angle is less than 0.09 degree (elastic range), SSR decreases steadily and the maximum von-Mises stress increases as the height of both ends increases.In case of 0.09 degree rotation, SSR becomes nearly 0 and maximum von-Mises does not change as the height of either ends varies.If the rotation angle is more than 0.09 degree (plastic range), SSR increases steadily as the height of either ends increases.

Conclusions
The results for the effects of either height of bellows ends on the stress distribution at the changes of rotation angle can be summarized as follows.
(1) The maximum von-Mises stress is affected not only by the angle of rotation, but also the height of bellows ends.
(2) Stress reduction rate according to the height of bellows ends varies from -10% to 30% for the model.SSR decreases steadily and the maximum von-Mises stress increases as the height of both ends increases in the

Figure 1 .
Figure 1.Analysis Model and Boundary Condition

Figure 2 .Figure 3 , 4 MATECFigure 4 ,Figure 5 ,
Figure 2. von-Mises stress with 0mm deflection and 0.01 degree rotation range.On the contrary, SSR increases steadily as the height of either ends increases in the plastic range

Table 1 ,
Material properties and geometric dimensions