static testing of u-shaped formed metal bellows

Int. J. Pres. Ves. & Piping 41 (1990) 207-226

Static Testing of U-Shaped Formed Metal Bellows

Will iam P. Schonbergfl Phill ip A. Beasley, a Gerald R. G u i n n b & Alan J. Bean b

a Department of Mechanical Engineering, b Johnson Research Center, University of Alabama, Huntsville, Alabama 35899, USA

(Received 21 September 1989; accepted 6 December 1989)

A BSTRA CT

This paper summarizes the results of an investigation in which an innovative test facility was designed and constructed for the purpose of stress testing of thin-walled U-shaped formed metal bellows. Stress values calculated from strains obtained using commercially available strain gages for various modes o f deformation were compared with those predicted by currently available static stress equations. It was observed that the experimental test results are significantly different from the stress values obtained using the analytical static stress equations. Possible explanations for the discrepancies are presented and discussed. It is concluded that common, commercially available strain gages are inappropriate for measuring convolute strains in thin-walled formed metal bellows with small convolute radii.

NOTATION

b

Cd, Cf, Cp D Dm Do Db e

E

Int. J. Pres.

r+ Rp EJMA/Anderson correction factors 13 Bellows inner diameter Bellows mean diameter Bellows outer diameter

+ D o)/2 Equivalent axial elongation Young's modulus of elasticity

207 Ves. & Piping 0308-0161/90/$03-50 © 1990 Elsevier Science Publishers Ltd,

England. Printed in Great Britain

208 Will&m P. Sehonberg, Phillip A. Beasley, Gerald R. Guinn, Alan J. Bean

h Lb m Uc Up P P q

qi qo r Ri Rm Ro Rp s

Si So t

/eb tp W Z

C( 6

A

0 21 22 la Pc v

P O'BO

t

O'BO

w/2 q/No

12(1 - v 2 )

Number of convolutions Number of plies Internal pressure Total axial load Average bellows pitch length Interior bellows pitch length Exterior bellows pitch length Convolute radius Bellows inner radius Bellows mean radius Bellows outer radius Half-width of annular plate portion of convolute Average convolute width Interior convolute width Exterior convolute width Bellows ply thickness Npt(0"57 + w/2r) Ply thickness corrected for thinning, tp = tx /Di/D m Convolute height MSFC/Matheny correction factor 23'24

RpRm/r(R m + 0"7Rp) = p/(1 + 0 " 7 0 Total axial displacement Axial displacement per convolute Axial displacement per pitch length per convolute Lateral offset displacement Rp/ Rm Salzman's shape factor 23 End rotation t/r r/Rm 12(1 - v2)(r/Rm)(r/t) 12(1 - v2)br/(rc - 2)Rmt Poisson's ratio Rp/ Rm Rp/r Pr/rtRm tz 36BO Location of maximum equivalent stress

Static testing of U-shaped formed metal bellows 209

I N T R O D U C T I O N

Bellows are thin-walled corrugated tubes designed for high flexibility when subjected to a variety of load conditions. In predicting the life expectancy of formed metal bellows, two stress components are traditionally taken into account. These are the static and dynamic stresses that result from anticipated in-service loads, such as axial and lateral deformation, angulation, and internal pressure. Experimentally verified mathematical models have previously been developed to predict dynamic stresses that result from flow-induced vibrations. 1-2 However, very few attempts have been made to correlate analytical bellows models that predict static stresses with empirical stress data for extremely thin-walled U-shaped formed metal bellows. *'5 Such bellows are often found in aerospace liquid propulsion systems where low weight and high flexibility are important design considerations.

The objectives of this investigation were to design and construct a test facility that could be used for the testing of thin-waited formed metal bellows and to determine if common, commercially available strain gages can be used to accurately measure bellows strains. Experimental stress values calculated from measured strains are compared with those predicted by currently available static stress equations. The first section of this paper discusses the development, design and construction of the bellows static test facility. The test procedure is described in the second section and a description of the bellows test articles is presented in the third section. The correlation of experimental and analytically predicted stress values is discussed in the fourth section of this paper. The major conclusions of this investigation are presented in the final section along with recommendations for future investigations of static bellows stresses.

TEST FACILITY DESIGN AND C O N S T R U C T I O N

A test facility was designed and constructed to provide the static stress values that would be compared with the predictions of existing analytic static stress equations for U-shaped formed metal bellows. The facility consists of two six degree-of-freedom (six DOF) motion platforms, one fixed and one movable, as shown in Figs l(a) and l(b). The six DOF platform concept was derived from the so-called 'Stuart table' used to provide three- dimensional motion in flight simulators. 6 By differentially extending or retracting each of the six legs of a Stuart table, it is possible to provide, within limits, any desired position of the platform. This test facility is a significant departure from standard bellows test facilities which typically employ

210 William P. Schonberg, Phillip A. Beasley, Gerald R. Guinn, Alan J. Bean

Elevation [ ~ Weld flange Blind flange

tation ~ Hold-down T r a ~ I 1 I P-~".,, I \Moveable l I l "Fixed

Lvo I I l l ..... I ,a~,e l I I tab,e T~ ~rnbuckle i I ~ //n \ a / / I I I i ~,~d

(a)

T-slot (b)

Fig. 1. Facility design: (a) plan view; (b) top view.

modified tensile test machines as the test fixtures (see, for example, Refs 7 and 8). The facility developed for this investigation can be used to test any type of bellows (i.e. formed, welded, etc.) under any type of mot ion (i.e. axial, lateral, bending, etc.) and, with only a few modifications, can also be used to perform fatigue testing of formed and welded metal bellows.

Each six D O F simulator consists of a 25in diameter rotary table top suppor ted by six legs. Each leg of the movable table consists of 0.5 in threaded rods connected to rod end bearings, which allows the table top to be moved to any six D O F position. The legs of the movable pla t form also contain turnbuckles that enable each leg to be extended or retracted, and a


load cell, which is used to determine if any of the table motions are out of a vertical movement plane. Out-of-plane motions were not considered in this investigation in order to avoid the complications arising from torsional deformat ion of formed metal bellows. The legs of the articulating table contain a Linear Variable Differential Transducer (LVDT) which indicates the axial movement of the leg. The general construction of both leg types can be seen in Fig. 2. Before installation, the load cells and LVDTs were calibrated to insure the accuracy of the required movements. A control program was written to calculate the leg extensions required to achieve a specified platform position based on the three-dimensional equations of mot ion for a Stuart Table. 6

l/2;nch threaded rod

LVDT Load cel I /

~"-~-I~11 H lll~" ~,,,.ii.iiiiiliiiii,, llO,i,i,liiii.i"iliiiii ~llllln~

g-- ll I 1 Turnbuckle Rod end bearing

Moveable table

Fig. 2. Fixed and movable Stuart table legs.

Once the platforms were assembled and the control program was verified, transducers were installed and preliminary movements were conducted to verify the accuracy of the tables. The initial stage of the checkout procedure insured that the two table tops were level and aligned with respect to each other. This was accomplished through the use of bubble levels and an aligning device. Once the tables were correctly positioned, the neutral height o f the articulating table was established and entered into the control program.

The accuracy with which the articulating table could be placed relative to a specified location was established by moving the centroid of the table to a position which would have resulted in a 25 mm extension of a bellows test article. It was found that the table could be moved to within ___0-3% of the desired position. The tables were also checked to see if the movement was in a single vertically oriented plane by determining whether or not the tables were level in inertial space, in the same plane, and did not rotate in the common plane. Once the positioning accuracy of the tables was verified, 90 °


flanged elbow end fixtures were mounted to the table tops. The end fixtures were positioned by insuring that the distance from one flange to the other was the same for any point on the flange circumference. A blind flange was then mounted on the inside of each of the end fixtures so that the internal pressure tests could be performed.

The bellows test articles were mounted between the two platforms, with one end attached to the fixed table and the other to the movable table. The desired deflection of the bellows was achieved by extension or retraction of the legs supporting the movable table. Movement of the platform required that only two legs needed to be adjusted simultaneously, thus ensuring the feasibility of manual positioning by a single operator. The test fixture was heavily instrumented to provide the strain data necessary for the calculation of the bellows convolute stresses. It is noted that a significant amount of warping and droop was observed when some of the bellows were placed in the test fixture. This was attributed to the extremely thin wall thickness and extensive free lengths of some of the test articles.

The strain data were acquired through the use of Hott inger-Baldwin 90 ° biaxial foil strain gages mounted to the crowns of selected convolutes. The strain gages were mounted on the bellows test articles using standard preparation and mounting procedures. The positions of these strain gages are shown in Fig. 3. The resistance changes of the strain gages were measured

O0

I0~01 ® 4: ®

_ _ , , ° ,

1 2:3 I/: N-2N-1 N

lCOOC,ICOOM (~ 2COOC,2COOM ( ~ 3COOC,3COOM (~ 1C01C,1C01M (~) 2C01C,2C01M (~ 3C01C,3C01M

1C10C,1C10M ~) 2C10C,2C10M ~) 3C10C,3C10M

C = Circumferential M = Meridianal Fig. 3. Strain gage locations.

7A

- - - 4 - - -


using Wheatstone bridge circuitry incorporated in a Metrobyte Das-16 analog board data acquisition system used in conjunction with an IBM PC/XT. A Labtech Notebook was used as the controlling program for the data acquisition process. The data were obtained from the instrumentation with the use of several control boards.

TESTING PROCEDURE

Before each test, the alignment of the six DOF tables was checked by leveling the tables and bringing them in line with each other. The articulating table was then moved to a position corresponding to a 25mm horizontal extension to check the accuracy of the test movements. The table was considered to be properly aligned when the centroid was in _+0"3% of the desired position. The distance separating the end fixtures was also determined to ensure that the ends of the bellows test articles would be in alignment.

Once the alignment was verified, each bellows test article was subjected to extension, lateral offset, angulation and internal pressure loadings. A test was considered complete when any of the stresses induced in the test article reached or exceeded 80% of the yield strength of the bellows material. Displacement testing (i.e. extension, lateral offset and angulation) was performed using the leg movements calculated by the control program based on a specified final displacement value. The internal pressure tests were performed using a HEISE 730 Calibrator as the measuring apparatus. This calibrator has a gage pressure range of 0-10 s N/m 2 with an accuracy of 0.05% full scale traceable to the National Bureau of Standards. Nitrogen gas was used as the flow medium for the internal pressurization tests.

The strain values were computed from the differential resistances of the strain gages as recorded by the acquisition system. These values were computed using the equation

S t r a i n - AR, (1) RgGF

where AR, is the differential strain gage resistance, R, is the nominal strain gage resistance, and G F is the gage factor (equal to 1"95). Stresses were computed from strains according to Hooke's Law:

E Merid ional stress: am - (1 -- v 2) (Em "q- V/3c) (2a)

E Circumferent ia l stress: ac - (1 - v 2) (ec + win) (2b)


where a i and ei are the calculated stresses and the measured strains, respectively.

The test bench was checked against faulty readings by attaching extra strain gages to two bellows test articles and then checking the consistency and repeatability of the readings obtained. The extra strain gages were the same type (90 ° biaxial), but were obtained from a different manufacturer (Micromeasurements, Inc.). The gages were mounted next to gage positions 2C00C and 2C00M on the same convolute as indicated in Fig. 3. Based on the consistent readings obtained from the additional strain gages, it was concluded that the strain data generated by the test facility were reproducible.

It is noted that although the static stress testing was performed using U- shaped formed metal bellows, any type of formed or welded bellows can be tested in the facility with only slight modifications to the existing hardware and instrumentation. Additionally, the test facility can be easily automated to perform fatigue testing of formed or welded metal bellows.

BELLOWS TEST SPECIMENS

The specimens used in the test program were single-ply 12.7 cm and 20.3 cm diameter U-shaped formed bellows made from 321 Stainless Steel (E = 193 GPa, v = 0.32). The geometry of each test specimen was measured to determine its uniformity and conformity to the fabrication specifications provided to the manufacturer. These measurements included the inside and outside diameters, the convolution height, the thickness of the material, and the internal and external width and pitch of each convolute. The average geometric parameters of the test specimens are summarized in Table 1; the variations of convolute geometry within each bellows are given in Table 2. As can be seen in Table 2, convolute dimensions varied considerably in

T A B L E 1 Test Article Geometr ies

Article No. D i s w t N,.

88635 12.7 0"353 1"107 0"015 2 33 88637 12.7 0.376 1'151 0"025 4 43 88639 12.7 0-394 1'128 0-035 6 33 88641 20"3 0.450 1-471 0-030 5 30 88642 20"3 0"668 1.715 0"050 8 25

All measurements in cm.

Static testing o f U-shaped formed metal bellows

TABLE 2 Variation of Bellows Convolute Geometry

215

Test article So qo si qi

88635 Average 0-352 519 0.715 683 0.503 872 0-778 391 Std dev. 0.020470 0.034775 0.045 113 0-047381

88637 Average 0'376 215 0-691 363 0.498 323 0.743 008 Std dev. 0.011 631 0.021 562 0.037 526 0.041 1 ! 8

88639 Average 0-393 469 0'695 325 0.488 315 0.729 298 Std dev. 0.015 611 0-034 201 0.033 025 0.036 830

88641 Average 0.448 648 0.874 766 0.637 276 0.919 785 Std dev. 0.009 784 0"051 262 0.039 367 0-046 378

88642 Average 0.667 918 1.188 402 0-653 308 1.220 892 Std dev. 0.018448 0-023 198 0.021 600 0.246736

All measurements in cm.

several of the bellows test articles. The effects of these irregularities on the stresses induced by the applied loads are discussed in subsequent sections.

C O R R E L A T I O N OF E X P E R I M E N T A L RESULTS AND A N A L Y T I C A L PREDICTIONS

An exhaustive literature survey was performed to obtain analytical equations currently being used to calculate static bellows stresses. All available equations were evaluated for their applicability to U-shaped formed metal bellows and for the ease with which they can be used. A list of static stress equations and axial spring-rate equations for U-shaped bellows is provided in Appendixes A and B, respectively.

A review of the bellows stress analysis literature revealed a definite lack of terminological and parametric definition consistency among the equations. It was often unclear, for example, as to whether the convolution radius was measured to the inner surface, the outer surface, or the mid-surface of the convolute. An unwise assumption was found to lead to a stress value differing from all others by 10% or more. Likewise, equations were often presented for 'bulging stress' and 'bending stress' without a clear definition of the direction of the stress. It was also unclear as to whether the stress in question referred to a total quanti ty or to a component of a particular stress.

A comparison of the predicted meridional and circumferential stresses and the actual stresses experienced by the five test specimens under the various loading conditions is presented in Tables 3-7. In these tables, the first eleven rows contain static stress values calculated using the equations


TABLE 3 (a) T o t a l C r o w n M e r i d i o n a l St resses ( x 105 N / m 2) Bel lows No. 88635

6 (0"254 cm) p (27579 N/m 2) 0 (0"1 °) A (0"254 em)

Blair & Wel l s 17 x 728.1" x x

D a n i e l s 2° 224"6 a 728.1 ~ 10.6" x

J o h n s o n 22 184"6" 728.1" x x

A n d e r s o n 14"15 215"1" 571"1" x x

E J M A 13 192"8 577-2 9"2 358"0

J a n z e n 25 180.1 554-9 x x

A n d reeva 16 292.9 - - 13"5 x

M S F C z3 183.3 a 562.0" 8.7" 340-3"

C h a n d & G a r g ~8 --- x x x

C l a r k 19 x x x

H a m a d a & T a k e z o n o 21 167.3" x x x

Avg. 776.7 510.0 368.2 259.6

Exp. Max . 1 278.6 625.2 1 453.1 1 370"3

M a x . / A v g . 1.65 1.23 3.91 5-28

(b) T o t a l C r o w n C i r c u m f e r e n t i a l St resses ( x 105 N/m 2) Bel lows No. 88635

6 (0"254cm) p (27579N/m 2) 0 (0-1 °) A (0254cm)

Blai r & Wel l s I 7 x 32.5 x x

D a n i e l s 2° x x x x

J o h n s o n 22 x x x x

A n d e r s o n 14.15 x x x x

E J M A 18 x 33.9 x x

J a n z e n 25 103-4 14.1 x x

A n d r e e v a 16 93.8 - - 4.3 x

M S F C 23 x 35.0 x x

C h a n d & G a r f 8 - - x x x

C l a r k ~ 9 x x x

H a m a d a & T a k e z o n o 2~ 46.7 h x x x

Avg. 194.8 1 182-4 525.1 253.0

Exp. Max . 1 675.7 1 413.3 1 214.5 1 192.4

M a x . / A v g . 8'60 1.19 2.31 4.71

x , St ress e q u a t i o n no t p r o v i d e d for m o d e o f d e f o r m a t i o n .

, E q u a t i o n a v a i l a b l e for m o d e o f d e f o r m a t i o n , but g e o m e t r i c p a r a m e t e r s o f the be l l ows

u n d e r c o n s i d e r a t i o n lie ou t s i de the l imi t s o f app l i cab i l i t y .

" Bend ing s t ress c o m p o n e n t value.

b M e m b r a n e c o m p o n e n t value.


TABLE 4 (a) Tota l C r o w n Mer id iona l Stresses ( x 105 N / m 2) Bellows No. 88637

t~ (0.508 cm) p (20 684 N/m 2) 0 (0.2 °) A (0"254 cm)

Blair & Wells 17 x 211.8" x x

Danie ls 2° 511-0" 211.8" 24-1" x

J o h n s o n 22 437.5" 211.8" x x

Anderson~, .~ 5 509'2" 176.2" x x

E J M A 13 484.6 180.4 22.9 662'6

Janzen 25 449-4 158"4 x x

Andreeva 16 692.6 - - 31 "8 x

M S F C 23 416.8" 158.4" 19.6" 570.0" C h a n d & G a r g 18 - - x x x

Clark 19 - - x x x

H a m a d a & T a k e z o n o 21 427.7 a x x x

Avg. 1 077.7 1 145.3 615.6 894.0

Exp. Max. 1 197.0 1 292-2 759.4 1217.2

Max. /Avg. 1.11 1.28 1.23 1.36

(b) Tota l C r o w n Circumferent ia l Stresses ( x 105 N / m 2) Bellows No. 88637

6 (0"508 cm) p (20 684 N/m 2) 0 (0.2 °) A (0"254 cm)

Blair & Wells 17 x 14-5 x x

Danie ls 2° x x x x


Anderson14, t 5 x x x x

E J M A 13 x 15.1 x x

Janzen 2 s 190"4 8.1 x x

Andreeva t 6 221.6 - - 10' 1 x

M S F C 23 x 15.7 x x

C h a n d & G a r g 18 - - x x x Clark 19 - - x x x

H a m a d a & T a k e z o n o 2~ 758.4 b x x x

Avg. 637'8 1 331.0 862.3 789.0

Exp. Max. 1 174.0 1 465.7 1 429.5 892.3

Max. /Avg. 1-85 1.10 1.66 1.13

× , Stress equa t ion no t p rov ided for m o d e o f de fo rmat ion .

- - , E q u a t i o n avai lable for m o d e o f de fo rma t ion , bu t geometr ic p a r a m e t e r s o f the bel lows unde r cons ide ra t ion lie outs ide the limits o f applicabili ty.

" Bending stress c o m p o n e n t value.



TABLE 5 (a) Tota l C r o w n Merid ional Stresses ( x 105 N / m 2) Bellows No. 88639

6 (0"381 cm) p (34 474 N/m 2) 0 (0"2 °) A (0"254 cm)

Blair & Wells I 7 × 173'4 a × x

Daniels 2° 694.2" 173.4" 43-6" x

J o h n s o n 22 623.1" 173.4" x x

A n d e r s o n 14,15 763.2" 144.8 a x × E J M A 13 685.0 149.6 43.0 795"9

Janzen 25 658.6 135'5 x ×

Andreeva t 6 987.4 60.4 ×

M S F C 23 566"5" 142'7" 35"6" 658"2"

C h a n d & G a r g 18 x x x

Clark 19 x x x

H a m a d a & T a k e z o n o 21 616.3" x × x

Avg. 1 137.6 527-7 810.5 390-0

Exp. Max. 1 682'0 723-5 1 508'6 1 199"0

Max. /Avg. 1.48 1.37 1.86 3.07

(b) Tota l C r o w n Circumferent ia l Stresses ( × 105 N/m 2) Bellows No. 88639

6 (0.381 cm) p (34 474 N/m 2) 0 (0"2 °) A (0"254 cm)

Blair & Wells 17 × 17"9 × ×

Daniels 2° × × × ×

J o h n s o n 22 × × × x

A n d e r s o n 14,1 s × x × x

E J M A 13 x 18"6 x x

Janzen 25 239.3 12.1 x ×

And reeva L 6 315.9 - - 19"3 ×

M S F C z3 x 19"3 × ×

C h a n d & G a r g is x × x

Clark 19 - - X X X

H a m a d a & T a k e z o n o 21 80"8 h x x x

Avg. 1 027.4 1 148.3 984.4 551-6

Exp. Max. 1 654.6 1 234.0 1 645.8 1 278.1

Max. /Avg. 1.61 1.07 1.67 2.32

x , Stress equa t ion not p rovided for m o d e o f de fo rmat ion . - - , E q u a t i o n available for m o d e o f de fo rma t ion , bu t geometr ic p a r a m e t e r s o f the bel lows

under cons idera t ion lie outs ide the limits o f applicability.

Bending stress c o m p o n e n t value.

h M e m b r a n e c o m p o n e n t value.


TABLE 6 (a) Tota l C r o w n Mer id iona l Stresses ( x 105 N / m 2) Bellows No. 88641

6 (0"381 cm) p ( 13 789 N/m 2) 0 (0"2 °) A (0"254 cm)

Blair & Wells iv x 160-5" x x

Danie ls 2° 403-5 a 160.5" 40.1 a M

J o h n s o n 2z 345'6" 160"5" x x

Anderson14.15 429.2" 135"0" x x

E J M A 13 393'2 137-8 39.1 1 017.6

Janzen25 375.1 126-7 x x

Andreeva 16 552.5 - - 53.6 x

M S F C 23 329.3" 133"8" 32'7" 852.32"

C h a n d & G a r g 18 x x x

Cla rk 19 - - x x x

H a m a d a & T a k e z o n o 2~ 353"0" x x x

Avg. 914.8 558"9 570.2 337.3

Exp. Max. 1 277.1 676-1 592.0 431.5

Max. /Avg. 1.40 1'21 1.04 1.28

(b) Tota l C r o w n Circumferent ia l Stresses ( x 105 N / m 2) Bellows No. 88641

6 (0.381 cm) p (13 789 N/m 2) 0 (0.2 °) A (0.254 cm)

Blair & Wells ~ × 12.1 × x

Danie ls 2° × x x x


A n d e r s o n 14.15 x x x x

E J M A 13 x 12.4 x x

Janzen 25 143.5 6.6 x x

Andreeva 16 176.8 - - 17.1 x

M S F C 23 x 12.8 x x

C h a n d & G a r g ~s - - x x x

Clark19 x x x

H a m a d a & T a k e z o n o 21 47.1 b X X X

Avg. 773"2 1 196"8 1 342"3 703"8

Exp. Max. 1 000"1 1 831"4 1 407"9 1 658"2

Max. /Avg. 1.29 1.53 1.05 2.36

× , Stress equa t ion no t p rov ided for m o d e o f de fo rma t ion .

- - , E q u a t i o n avai lable for m o d e o f de fo rma t ion , bu t geometr ic p a r a m e t e r s o f the bel lows

under cons ide ra t ion lie outs ide the limits o f applicabili ty.




T A B L E 7 (a) Tota l C r o w n Merid ional Stresses ( x 105 N / m 2) Bellows No. 88642

6 (0"254cm) p (27579N/m 2) 0 (0"3 °) A (0"381 cm)

Blair & Wells z7 x 157-2" x x

Daniels 2° 488.4" 157-2" 74.6" x

J o h n s o n 22 508.4" 157.2" x x

A n d e r s o n ~4"15 586'1" 124"4" x x

E J M A 13 534"0 128"5 80"6 1 153-9

Janzen 25 501.7 114.7 x x

Andreeva 16 808.3 - - 120.2 x

M S F C 23 398-5" 122.6" 60"9" 860-9"

C h a n d & G a r g 18 - - x x x

Clark 19 - - X X X

H a m a d a & T a k e z o n o zl 471.9" x x x

Avg. 835.0 921.5 203.2 131.3

Exp. Max. 1 134.6 1 316.7 273.0 398.9

Max. /Avg. 1.35 1.43 1.34 3.04

(b) Tota l C r o w n Circumferent ia l Stresses ( x 105 N / m a) Bellows No. 88642

(0.381 cm) p (27"579 N/m 2) 0 (0"3 °) A (0"381 cm)

Blair & Wells 17 x 18.2 x x

Daniels z° x × x x


Anderson14"1 s x x x x

E J M A ~3 x 18"8 x x

Janzen z5 202.6 11.3 x x

Andreeva t 6 258.6 - - 38-4 x

M S F C 23 x 19.3 x x

C h a n d & G a r g TM - - x x x

Clark ~9 x x x

H a m a d a & T a k e z o n o 2~ 79-6 ~ x x x

Avg. 1 165"1 1 105"9 472"8 87'0

Exp. Max. 1 450.5 1 633.1 1 120.2 102-1

Max. /Avg. 1.25 1.48 2.36 1-17

x , Stress equa t ion no t p rovided for m o d e o f de fo rmat ion .

- - , E q u a t i o n available for m o d e o f de fo rma t ion , but geometr ic p a r a m e t e r s o f the bel lows

unde r cons ide ra t ion lie outs ide the limits o f applicability.




developed by the investigator named in the first column. The final three rows in each table contain average and maximum experimental static stress values as well as ratios of maximum to average stress values in a particular bellows for each mode of deformation. A review of these tables reveals several interesting features.

First, it can be seen that, for the geometric parameters considered in this study, stress values calculated using the analytical equations had very little correlation with the experimental values calculated from the strain values obtained using commercially available strain gages. The differences between the experimental results and the theoretical predictions can be attributed to several factors.

(1) The use of flat strain gages to measure strains on sharply curved surfaces may have introduced a significant error in the strain measurements. 9'1° Additionally, because of the sharp radii of curvature in the circumferential and meridional directions, it was extremely difficult to conform the flat strain gages to the curved surfaces of the convolute.

(2) The thicknesses of the strain gages were no longer negligible when compared to the bellows ply thickness. The resulting error may have been increased by the removal of some of the already thin convolute walls during the preparation of the convolutes for strain gage application. Together, these effects contributed to a local stiffening of the bellows material, which most likely resulted in an increased e r r o r . 11,12

(3) The analytical stress equations were all developed based on the assumption of a uniform and symmetric bellows. As noted previously, the manufactured test specimens revealed many asymmetries in the circumferential and meridional directions as well as varying degrees of geometric non-uniformity.

These factors indicate that common, commercially available strain gages are inadequate for measuring convolute strains in thin-walled formed metal bellows with very small convolute radii.

Second, although there is an abundance of equations that predict static stresses due to axial deformation and internal pressure, there is a dearth of equations for stresses due to angulation and lateral offset. The authors are currently developing a series of static stress equations for each of the four modes of deformation considered in this investigation in an effort to rectify this situation.

Third, from the experimental stress ratio values, it can be seen that there was often a wide range of stress levels within a particular test specimen. The variations in stress levels within a bellows were most likely due to the


inherent asymmetry and geometric non-uniformity of the bellows test articles.

CONCLUSIONS A N D R E C O M M E N D A T I O N S

An innovative experimental facility was developed and constructed to provide static stress data for U-shaped formed metal bellows. The facility consists of two six degree-of-freedom platforms which proved to be quite versatile in the static testing of formed metal bellows. The facility design can be easily modified to perform static testing of welded bellows as well as fatigue testing of formed and welded metal bellows.

An extensive survey of existing static stress equations for U-shaped formed metal bellows was performed. Analytical equations currently being used to calculate static bellows stresses were evaluated for their applicability to the study of stresses in U-shaped metal bellows. It was found that the experimental test results were significantly different from the stress values obtained using these analytical equations. The large differences observed were most likely due to the use of flat strain gages mounted on thin, sharply curved surfaces to measure convolute deformation and the non-uniformity and asymmetry of the bellows test articles. As such, it was concluded that commercially available strain gages should not be used to measure convolute strains in thin-walled metal bellows with very small convolute radii.

Based on the observations made during the course of this investigation, it is recommended that a study be performed to analyze the limits of the applicability of a strain gage to measure bellows deformation. A hybrid photo-elastic/strain gage analysis of a series of convolutes with decreasing radii is suggested as a means of developing a correction factor for combined ply thinness and convolute curvature effects. Additionally, it is recommended that the effect of geometric asymmetry and non-uniformity on bellows stress be thoroughly investigated. An in-depth study of the effects of variations in convolute width, height and pitch on bellows response should be performed in order to determine if a manufactured bellows with deviations from a requested geometry can perform adequately under anticipated loading conditions.

R E F E R E N C E S

1. Gerlach, C. R., Bass, R. L., Holster, J. L. & Schroeder, E. C., Flow induced vibration of bellows with internal cryogenic fluid flows. Interim Report No. 2, NAS-21133, Southwest Research Institute, San Antonio, TX, August 1970.

2. Gerlach, C. R., Bass, R. L., Holster, J. L. & Schroeder, E. C., Bellows flow- induced vibrations and pressure loss. Final Report, NAS8-21133, Southwest Research Institute, San Antonio, TX, April 1973.


3. Tygielski, P. J., Smyly, H. M. & Gerlach, C. R., Bellows flow-induced vibrations. NASA TM-83556, Washington, DC, October 1983.

4. Wilson, J. F., Mechanics of bellows: A critical survey. Int. J. Mech. Sci., 26 (1984) 593-605.

5. Hulbert, L. E., Keith, R. E. & Trainer, T. M., State-of-the-art survey of metallic bellows and diaphragms for aerospace applications. Report No. AFRPL-TR- 65-215, Edwards Air Force Base, CA, November 1965.

6. Dieudonne, J. E., An actuator extension transformation for a motion simulator and an inverse transformation applying Newton-Raphson's method. NASA TN-D-7067, Washington, DC, 1972.

7. Feely, F. J. & Goryl, W. M., Stress studies on piping expansion bellows. J. Appl. Mech., 17 (1950) 13541.

8. Samoiloff, A. A., Evaluation of expansion joint behavior. Power, 105 (1961) 57-9.

9. Gerdeen, J. C., Effects of pressure on small foil strain gages. Exp. Mech. (March 1963) 73-80.

10. Milligen, R. V., The effects of high pressure on foil strain gages on convex and concave surfaces. Exp. Mech. (February 1965) 59-64.

11. Beatty, M. F. & Chewning, S. W., Numerical analysis of the reinforcement effect of a strain gage applied to a soft material. Int. J. Engng Sci., 17 (1979) 907 15.

12. Perry, C. C., The resistance strain gage revisited. Engng Mech., (December 1984) 286-99.

13. Standards of the Expansion Joint Manufacturers Association, Inc. (5th edn), White Plains, NY, 1980.

14. Anderson, W. F., Analyis of stress in bellows, Part I: Design criteria and test results. Atomics International Report No. NAA-SR-4527, Canoga Park, CA, 1964.

15. Anderson, W. F., Analysis of stresses in bellows, Part II: Mathematical. Atomics International, Report No. NAA-SR-4527, Canoga Park, CA, 1965.

16. Andreeva, L. E., Elastic Elements of Instruments. Gos. Nau.-Teck. lzdat., Moscow, 1962.

17. Blair, R. R. & Wells, J. D., Mechanical Design of Stainless Steel Bellows. Stainless Steel Products, Inc., Burbank, CA, December 1963.

18. Chand, S. & Garg, S. B. L., Design formulas for expansion bellows. J. Mech. Des., Trans. ASME, 103 (1981) 881-91.

19. Clark, R. A., An expansion bellows problem. J. Appl. Mech., 37 (1970) 61-9. 20. Danieis, C. M., Predicting fatigue life of metal bellows. NASA Tech. Brief 68-

11026, Technical Support Package, 1968. 21. Hamada, M. & Takezono, S., Strength of U-shaped bellows. Bull JSME, 8

(1965) 525-31. 22. Johnson, J. E., Deffenbaugh, D. M., Astlefort, W. J. & Gerlach, G. R., Bellows

flow-induced vibrations. Final Report, NAS8-31994, Southwest Research Institute, San Antonio, TX, October 1979.

23. Assessment of flexible lines for flow induced vibration. Preliminary Document, Marshall Space Flight Center, Huntsville, AL, May 1973.

24. Matheny, J. D., Bellows spring rate for seven typical convolution shapes. Mach. Des. (4 January 1962) 137-9.

25. Janzen, P., Formulae and graphs of elastic stresses for design and analysis of U- shaped bellows. Int. J. Pres. Ves. & Piping, 7 (1979) 407-27.

2 2 4 W i l l i a m P. S c h o n b e r g , Ph i l l i p ,4. B e a s l e y , G e r a l d R . G u i n n , A l a n J. B e a n

A P P E N D I X A: S T A T I C S T R E S S E Q U A T I O N S

E J M A 13

Axia l d i s p l a c e m e n t

M e r i d i o n a l m e m b r a n e : M e r i d i o n a l b e n d i n g :

I n t e r n a l p re s su re

C i r c u m f e r e n t i a l m e m b r a n e :

M e r i d i o n a l m e m b r a n e : M e r i d i o n a l b e n d i n g :

O'mm = Et2e/2w3Cf , e = 6/N¢ trmb = 5Etpe/3W2Cd, e = 6/Nc

0"cm = PDm/2Nptp(0"571 + 2w/q) O'mm ~- pw/2Nptp O'mb = p w 2 Cp/2Npt 2

A n g u l a t i o n - - u s e axial d i s p l a c e m e n t e q u a t i o n s wi th e = DmO/2N ¢ L a t e r a l o f f s e t - - u s e axial d i s p l a c e m e n t e q u a t i o n s wi th e = 3 D m A / q N ~

A n d e r s o n 1 4 , 1 5

Axia l d i s p l a c e m e n t

B e n d i n g stress: I n t e r n a l p ressu re

M e m b r a n e stress:

B e n d i n g stress:

A n d r e e v a 16

Axia l d e f o r m a t i o n M e r i d i o n a l stress:

C i r c u m f e r e n t i a l stress:

A n g u l a t i o n M e r i d i o n a l stress:

C i r c u m f e r e n t i a l stress:

I n t e r n a l p re s su re

M e r i d i o n a l stress: C i r c u m f e r e n t i a l stress:

Bla i r & Wel ls iv

I n t e r n a l p re s su re H o o p stress:

Bu lge stress:

abd = 0 .412EtRm6/h2N¢CdRi

O'mp = p h / N p t ~rbp = 2Cpph2 Rm/t2 NpRi

G m = BQEtfi/2NcR2o

G c ~ Y(7 m

B 0 = 2c2(c 2 - 1 - 2 In c)/ [(17 2 - - 1 ) 2 - - 4C 2 In 2 c](1 -- v 2)

c = Ro /R i

t7 m = B4,ckEt/2NcR o O'c ~ P r i m

B e = 3(c z -- 1)/rc(c 2 + 1)A~ A e = (3/re)(1 - v 2)

x [In c - (c 2 - 1)/(c 2 + 1)]

~r m = BppRZo/t z O- c = VO" m

Bp = 3(c / - - 1)/4c z

tr H = PDb/2teb tr B = PDmW2/2Npt 2


C h a n d & G a r g ts

Axial load Equiva len t stress:

< 0 .45 . - . tro/trso = 0"52(~ + sin ~bm)/(1 - 0"66~x °'72) t ~ m = 95(1 - 0"75~°'86)/# °'Sz

0"45 < ~ < 2"80"" tre/a8o = 0"88~°'~°(~ + sin q~m) (~m = 40/°c°'35# °'45

Clark 19

Axial load Mer id iona l bend ing stress:

O'¢B,max = 0"222(1 - v2)-o.5.~¢o.75n/...r/rt Circumferen t ia l m e m b r a n e stress:

a¢O,ma x = 0.0972#o. 75P/rt

Daniels 2°

Axial d e f o r m a t i o n Bending stress: aB = Eft~2(1 - v2)w2~wN c

In te rna l pressure Bulging stress: tr b = pw2/2t2Np

A n g u l a t i o n - - u s e axial d e f o r m a t i o n equa t ion with 6 -- DmO/2

H a m a d a & T a k e z o n o 21

Axial load Ci rcumferen t ia l m e m b r a n e stress: ato = ahop{2.7538 x 10-1 - - 1"9387 x 10- 5122 + 4"0752 x 10- 7#4

+ ~[3"143 x 10 - l -- 1"5508 x 1 0 - 3 p 2 + 5"8730 × 10-7/ t4]}/ (1-0+2"0111 x 1 0 - ~ # 2 + 8"1170 x 1 0 -4 p 4)

Mer id iona l bend ing stress: trb~ = t r h o { 1 . 0 - 2 . 4 9 3 0 x 1 0 - 2 # 2 + 1"6318 x 1 0 - s p 4

+ ~[1-0 - 6"235 x 1 0 - 2 p 2 + 6"9164 x 10 -5#4]} /

(1 -0+2-0111 x 10 -1p 2 + 8"1170 x 1 0 -4 p 4)

J o h n s o n 22

Axial d i sp lacement C o m p r e s s i o n stress:

In te rna l pressure Pressure stress:

M S F C d o c u m e n t 23

Axial d i sp lacement Bend ing stress:

In te rna l pressure H o o p stress: Bulge stress:

ac = EtA/wZN~

ap = pwZ/2t2 Np

an = Et6/2"2Zw2Nc

an = pDoq/4Npt(w -- 2r + 7tr) aB = pCpwZ Dm/2NptZ Dl


A n g u l a t i o n - - u s e axial d i sp lacement equa t ion with 6 = DmO°/114"6 Latera l o f fse t - -use axial d isp lacement equa t ion with 6 = 3DmA/N¢q

A P P E N D I X B: A X I A L S P R I N G R A T E E Q U A T I O N S

Samoi lof f 8

PlY' = rCEDmt3/24w3[O'083 + 0"287(r/w) - O" 144(r/w) 2 + O'046(r/w) 3]

E J M A 13

K = l '7DmEtpNp/w3Cf

A n d e r s o n 14

K = 0"431RmEt3Np/N¢h3Cf

C h a n d & Ga r g 18

< 0 " 4 5 " " P / 6 " = 9-5rtEr222(1 -- 0.68~°'5% ° 1 z 5,a)/12(1 - v2)#3(/;/)~ 2 + sin ¢km) 3,

- 4"2:~ A = e

0.45 < ~ < 2 .80 . . .P /6 " = 3"75rcEr222(e °'46s~ + 0"092~25#)/12(1 - vZ)#3(e/2e + sin qSm) 3

Clark 19

P/cY = rEhZ/O.646x/12(1 - v z,)#co.25-

H a m a d a & T a k e z o n o 21

P/6" = 2zcEr23/mZ(422Ac + ~Ao)

A c = [7"8540 x 10-1 + 1"0684 x 10-2#z + 1"3996 x 10-5/24 + 0¢(1.0 + 2-8699 x lO-Z# z + 5"4164 x 10-5#4) ] / (1 -0+ 2-0111 x 10-1#2 + 8"1171 x 10-4#4)

Ap = 2(2 + v~){1.0 + 2"8690 x 10-2# 2 + 5"4185 x 1 0 -5 # 4 + ~[1"5708 + 1-1383 x 1 0 - 2 # 2 + 3"1311 x 10 -5#4]} / ( 1 " 0 + 2 . 0 1 1 1 x 10 1 # 2 + 8 " 1 1 7 1 x 1 0 -4 # 4 ) + 2(1 + ~/3)c¢p -- (2 -- ~)p2/3

M S F C 23

K = DmENpt3/weNc

M a t h e n y 24

K = 0"575Rmt3Er/h3qN~Z

static testing of u-shaped formed metal bellows

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