Fatigue of bellows, a new design approach

International Journal of Pressure Vessels and Piping 77 (2000) 843±850

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Fatigue of bellows, a new design approach C. Becht IV* Becht Engineering Co. Inc., 22 Church Street, P.O. Box 300, Liberty Corner, NJ 07938, USA

Abstract Consideration of fatigue is generally an important aspect of the design of metallic bellows expansion joints. These components are subject to displacement loading which frequently results in cyclic strains well beyond the proportional limit for the material. At these high-strain levels, plastic strain concentration occurs. Current design practice relies on use of empirical fatigue curves based on bellows testing. Prediction of fatigue behavior based on the combination of analysis and polished bar fatigue data is not considered to be reliable. One of the reasons for the unreliability is plastic strain concentration. It is shown that the difference between bellows and polished bar fatigue behavior, as well as the difference between reinforced and unreinforced bellows, can be largely attributed to this strain concentration. Further, it is shown that fatigue life of bellows can be better predicted by partitioning the bellows fatigue data based on a geometry parameter. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Bellows; Expansion joints

1. Background on bellows A bellows is a convoluted shell consisting of a series of toroidal shells, usually connected with annular plates that are called the sidewalls. The shape and terms used are shown in Fig. 1. A bellows with no sidewalls is a special case, called a semitoroidal bellows. Bellows are used to provide additional ¯exibility in shell structures such as piping and heat exchanger shells. They must withstand the forces due to internal pressure, while at the same time providing ¯exibility, the ability to absorb de¯ections. These de¯ections are typically in the form of axial and bending de¯ection, and in some cases lateral displacement of the ends. Bellows are often much thinner walled than the cylindrical shells to which they are attached, but are provided with suf®cient metal area to resist the circumferential pressure forces by folding the wall into convolutions. Where this does not provide suf®cient metal wall area, reinforcing rings are added, usually on the outside of the bellows in the roots, to provide additional resistance to burst type failures. Because of a variety of reasons, discussed in more detail in Refs. [1,2], including plastic strain concentration, the approach that has been taken in industry is to use empirical fatigue curves based on bellows testing for bellows design. To use a polished bar fatigue curve would require evaluation of plastic strain concentration effects. The actual bellows * Tel.: 11-908-580-1119; fax: 11-908-580-9260. E-mail address: [email protected] (C. Becht IV).

fatigue test data inherently includes typical strain concentration as well as other effects. Fatigue testing of bellows is both material speci®c and expensive. The intent of the present work is to lead to a better understanding of plastic strain concentration effects in bellows in order to lead towards a reduction or elimination of the present requirements for conducting fatigue tests of actual bellows for bellows design. If nothing else, an improved understanding could lead to greater con®dence in assigning fatigue curves based on analogy to materials for which no bellows fatigue tests were performed. 2. Bellows response to de¯ection loading De¯ection of the bellows can be envisioned as an axial displacement of the crown relative to the root of the bellows. This creates a bending distribution, with maximum bending stresses in the root and the crown [3,4]. The maximum bending stress is at the outermost and innermost part of the convolution for bellows with shallower convolutions (higher QW) and/or thicker walls and/ or smaller pitch relative to the diameter (lower QDT). For example, see Fig. 2. Otherwise, the location of highest bending stress shifts towards the junction between the toroid and the sidewall. For example, see Fig. 3 [5]. In the latter case, a countermoment from circumferential stresses in the toroid offsets the meridional bending moment. See the circumferential stress distribution in Fig. 3. Stresses in bellows are predicted in industry using

0308-0161/00/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(00)00078-8

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Nomenclature Dm ˆ Db 1 w 1 t Mean diameter of convolutions Db Bellows inside diameter QW ˆ q=2w QDT ˆ q=‰2:2…Dm tp †1=2 Š q Bellows pitch t Nominal thickness of one ply tp ˆ t…D=Dm †1=2 Bellows material thickness for one ply, corrected for thinning during forming w Convolution depth equations provided in the Standards of the Expansion Joint Manufacturer's Association [6], originally developed by Anderson [7,8] and Winborne [9], described by Broyles [10], and now used widely in international standards [11]. These equations are based on equilibrium considerations (for membrane stress) and parametric nondimensional shell analysis (for bending stress and stiffness). The nondimensionalized bending stress and stiffness are plotted versus two parameters, q/2w, (termed QW herein) and q/ [2.2(Dm tp) 1/2] (termed QDT herein), where q is bellows pitch, w the convolution height, tp the bellows thickness including an approximate adjustment for thinning due to forming, and Dm the mean bellows diameter. These were the parameters used by Anderson to nondimensionalize the shell analysis of the toroidal portions of the bellows. The effect of these parameters on bellows geometry is illustrated in Fig. 4. Note that the ®gure is based on constant thickness and mean diameter. Plasticity complicates bellows response, even under de¯ection loading. Tanaka [12], following Hamada and

Tanaka [13,14], found that strain concentration occurs at the high-stress zones after yielding under axial displacement loading. Plastic strain concentration is the ratio of elastic±plastic strain due to displacement, from an elastic±plastic analysis, to the elastic strain, based on an elastic analysis. It can have a signi®cant impact on design for fatigue since the fatigue life prediction is typically based on an elastic stress calculation. The work of Hamada and Tanaka was very signi®cant in terms of identi®cation of the plastic strain concentration effects. However, as shown by Becht [15], it does not provide a correct understanding of the phenomenon. Becht found that the strain concentration in unreinforced bellows was highly dependant on the parameter QW, which characterizes the shape of the convolution. The effects of the parameters QW and QDT on plastic strain concentration are illustrated in Fig. 5 from Becht [15]. Fig. 5 is based on parametric inelastic analyses of unreinforced bellows. As shown in Fig. 5, the strain concentration is not very signi®cant for values of QW greater than about 0.45. For values of QW less than 0.45, strain concentration can be very signi®cant and depends signi®cantly on QW and QDT. The present paper extends this work to reinforced bellows and draws comparisons between the response of reinforced and unreinforced bellows. 3. Inelastic analyses approach A number of elastic±plastic large displacement ®nite element analyses of reinforced and unreinforced bellows

Fig. 1. Reinforced bellow geometry (EJMA, 1998).

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Fig. 2. Surface stress due to displacement …QW ˆ 0:5; QDT ˆ 0:4†:

were performed to evaluate the strain range due to displacement loading. The evaluations of unreinforced bellows had previously been reported by Becht [15]. The model used a single element wide wedge of one half of a convolution. Most of the analyses were based on mean bellows diameter, Dm, of 610 mm and thickness of 0.51 mm. The thickness was varied in some analyses to evaluate the sensitivity of strain concentration to the parameters QDT and QW. A bilinear stress±strain curve was assumed, with a yield strength of 207 MPa. The slope beyond yield strength was assumed to be 10% of the elastic slope and kinematic hardening was assumed. All of the analyses were performed using the COSMOS/M ®nite element program. The model

was checked for unreinforced bellows against strain gage results, as reported by Becht [15]. The models were subject to one and a half cycles of compressive displacement. A typical chart showing straindisplacement behavior for an unreinforced bellows is provided in Fig. 6. The strain range was taken from the last half cycle of the analysis. This provides for the shifting of the yield surface that occurs on the initial displacement which self springs the bellows. After the ®rst cycle, the elastic stress range can reach twice yield. To permit comparison of the effects of the geometric parameters, independent of the degree of plasticity, analyses were run with different bending stress levels. They were run

1.50E+04

Outside Meridional

Inside Meridional

1.00E+04

Inside Circumferential Stress (psi)

5.00E+03

0.00E+00

-5.00E+03

Outside Circumferential -1.00E+04

-1.50E+04 0

100

200

300

400

500

Element

Surface Stress Du

ent (QW=0.2, QDT=1.2)

Fig. 3. Surface stress due to displacement …QW ˆ 0:2; QDT ˆ 1:2†:

600

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Fig. 4. Bellows geometries.

at half, three, six and twelve times the displacement that would result in an elastic meridional bending stress equal to the yield strength of the material. As should be expected, the degree of strain concentration depended on the magnitude of displacement. All the results in Fig. 5 are based on an imposed displacement of six times the elastic displacement.

detailed nonlinear elastic (large displacement with gap elements) investigations of reinforced bellows. A reinforced bellows model with QW ˆ 0:5 is shown in Fig. 7. With respect to displacement, a signi®cant consideration is that the elastically calculated meridional bending stress in a reinforced bellows signi®cantly differs between when the bellows is compressed and when it is stretched. The reinforcing ring enhances the stress more in the compression mode due to the greater interference between the ring and the bellows convolution. Fig. 8 shows the elastically calculated meridional bending stress for the bellows illustrated in Fig. 7 with the bellows compressed and extended. Also

4. De¯ection response of reinforced bellows Some salient points of behavior are illustrated by a

9.0

Strain Concentration

8.0 7.0 QDT = 0.4

6.0 QDT = 0.6

5.0 4.0

QDT = 1

3.0

QDT = 1.2

2.0

QDT = 2

1.0 0.0 0

0.2

0.4

0.6

0.8

QW Fig. 5. Strain concentration vs. QW (all data).

1

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847

0.0050

0.0047

0.0045 0.0040

Bending Strain

0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0009 0.0005 0.0000 0

2

4

6

8

10

12

14

16

18

20

Displacement (mm) Fig. 6. Bending strain vs. displacement …QW ˆ 0:5; QDT ˆ 2:0†:

plotted, for comparison purposes, is the stress in reinforced and unreinforced bellows calculated in accordance with the EJMA equations. Considering inelastic behavior, and the fact that bellows are usually cycled well beyond yield, compression would cause permanent deformation, wrapping the convolution around the ring. When the bellows is returned to the original, neutral position, the sidewalls can be pulled off of the reinforcing ring. If this occurs, the strain range due to displacement cycling after the ®rst cycle could be more analogous to the tensile displacement loading case. This hypothesis was investigated using elastic±plastic ®nite element analysis. The model was run through a de¯ection cycle from 0 to 7.6 mm compression (for the 1/2 convolution), back to zero, and again to 7.6 mm compression. In the analysis, the sidewalls were in fact pulled away from the reinforcing ring when the bellows was returned to a zero end displacement

Fig. 7. Finite element model.

condition. Fig. 9 plots the meridional bending strain versus displacement for this analysis. The strain range is dramatically reduced after the ®rst cycle due to the permanent distortion of the convolution. The effect of the reinforcing ring in restricting bellows de¯ection signi®cantly decreases after the initial cycle(s). Strain concentration was calculated for a variety of reinforced bellows convolution geometries and is shown in Fig. 10. Note that a number of single points are provided for various values of QDT at QW ˆ 0:4: The curves are for QDT ˆ 1:0: All of the cases shown are for a displacement that is six times the displacement that will cause the meridional bending stress to reach yield. For the closing case, elastic±plastic strain range at the

Fig. 8. Meridional bending stress vs. displacement, elastic …QW ˆ 0:5; QDT ˆ 1:0†:

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Fig. 9. Meridional bending strain vs. displacement, elastic±plastic …QW ˆ 0:5; QDT ˆ 1:0†:

root and crown, divided by maximum elastic strain range, is charted. For the opening case, the maximum elastic±plastic strain range in the bellows, divided by maximum elastic strain range, is charted. There are quite a few complexities in reinforced bellows response, beyond what can be reported in this paper. However, a general observation can be made that strain concentration is not very signi®cant for reinforced bellows with QW greater than about 0.4.

5. Effect of internal pressure on de¯ection stress Internal pressure is considered in the EJMA equations to increase the stress due to de¯ection in reinforced bellows. It is considered to increase the interaction between the convolution and the reinforcing ring. While space does not

permit discussion, pressure was not found to have a signi®cant affect on de¯ection stress in ring-reinforced bellows. 6. Effect of convolution pro®le on stress distribution and strain concentration As shown in Ref. [5], the effect of QW and QDT on plastic strain concentration is due to their effect on the stress distribution. When QW is low, bellows plastic strain concentration increases dramatically with QDT. In these bellows, the peak bending stress shifts along the toroid towards the sidewall and becomes much more localized as QDT increases. The region undergoing plasticity and plastic strain concentration becomes much less ¯exible relative to the remainder of the bellows. Thus, higher plastic strain concentration occurs.

Fig. 10. Strain concentration in reinforced bellows.

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bellows that were tested, QW varied between 0.27 and 0.61 and QDT varied between 0.46 and 1.74. For reinforced bellows, QW varied between 0.41 and 0.48 and QDT varied between 0.90 and 1.57. The stress range was calculated in accordance with the EJMA standards with the following modi®cations.

Fig. 11. Unreinforced bellows fatigue data with QW . 0:45:

For bellows with higher QW, plastic strain concentration decreases with increasing QDT, although it is low in any case. This is because, with higher QDT, the radius of the toroid increases, so with the same convolution height (same QW), the length of the sidewall decreases. Because the ¯exibility of the remainder that shifts its strain to the local region of plastic strain concentration is decreasing (due to shorter sidewall), plastic strain concentration decreases. 7. Evaluation of bellows fatigue data Reinforced and unreinforced bellows fatigue data that were provided by the Expansion Joint Manufacturer's Association for the development of the design rules for bellows in ASME Section VIII, Div 1 and ASME B31.3 were used to evaluate tentative modi®cations to the EJMA equations for prediction of stress in reinforced bellows. For unreinforced

² For unreinforced bellows, only bellows with QW . 0:45 were included. These had exhibited relatively low strain concentration in the inelastic analyses. ² The stress range was multiplied by a factor of 1.4. ² Pressure was assumed to have no effect on the de¯ection stress in reinforced bellows. ² Stress due to internal pressure was not added to the de¯ection stress in calculating the stress range because the pressure was non-cyclic. The unreinforced bellows fatigue test data are plotted in Fig. 11. The reinforced bellows fatigue data are plotted in Fig. 12. The data are compared to the raw (no design margin) polished bar fatigue curve for austenitic stainless steel that was used in the development of Section VIII, Div 2, as provided in the Criteria Document [16]. The polished bar fatigue curve is shown as a curve in Figs. 11 and 12. The data calculated with the modi®ed equation for calculating de¯ection stress range is quite well characterized by the raw material data. The factor of 1.4 difference can be attributed to surface ®nish, some degree of plastic strain concentration, and other differences between actual component and polished bar fatigue tests. It is well within typical differences between component and polished bar fatigue tests. Note that the difference between polished bar and plate fatigue tests is reported to be 1.43 in Ref. [17]. The data in Figs. 11 and 12 indicate that reinforced and unreinforced bellows could possibly use a single fatigue curve, and could possibly be compared to polished bar

Fig. 12. Fatigue data for reinforced bellows.

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greater accuracy in design of bellows for fatigue. Further, improved understanding of bellows fatigue behavior may lead to a reduction in the requirements for bellows fatigue testing to develop design fatigue curves. References

Fig. 13. Unreinforced bellow fatigue data with QW , 0:45:

fatigue data as long as deep convolution bellows …QW , 0:45† are excluded. This would also result in a higher fatigue curve for these bellows than is used, for example, in ASME B31.3 Appendix X. Fatigue data for the deep convolution unreinforced bellows …QW , 0:45† are plotted in Fig. 13, with the same 1.4 factor applied. Note that these bellows exhibit signi®cantly greater scatter and generally fall lower than the bellows with shallower convolutions. This is to be expected due to the substantial and widely varied strain concentration that occurs in these deep convolution bellows. Further adjustments could be made to the equations to re¯ect the sensitivity of strain range to QW, QDT and displacement range for these bellows, or a lower bound curve could be developed for these deep convolution bellows. While all the reinforced data, with the factor of 1.4, follows the polished bar fatigue curve reasonably well, it should be noted that no bellows that are subject to signi®cant plastic strain concentration were included in the database. Reinforced bellows with QW , 0:4 can be subject to plastic strain concentration; the present fatigue curve may be unconservative for these bellows. The prior observed difference between unreinforced and reinforced bellows behavior appears to be largely due to the difference in bellows geometries that were tested. The unreinforced bellows database included deep convolution bellows that were subject to signi®cant strain concentration, whereas the reinforced bellows that were tested did not have deep convolutions. When the unreinforced bellows fatigue test is partitioned between deep and shallower convolution bellows, the fatigue performance of reinforced and unreinforced bellows (with QW . 0:45† was found to be very similar. 8. Conclusions The results contained herein point to a new approach to designing bellows for fatigue. Consideration of strain concentration clari®es the fatigue data and will permit

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