6. Performance Testing of Bellows - INFLIBNETshodhganga.inflibnet.ac.in/bitstream/10603/40881/15/15_chapter6.pdf · 6. Performance Testing of Bellows ... EJMA.[20] A minimum of ten

Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 190

6. Performance Testing of Bellows

Bellows are correlated with actual test results to demonstrate predictability of

design parameters like rupture pressure, meridional yielding, squirm and cycle life

for a consistent series of bellows of same basic design.[20] Minimum five

meridional yield rupture tests on bellows of varying sizes are recommended by

EJMA.[20] A minimum of ten squirm tests on bellows of varying diameters and

number of convolutions are required. A minimum of twenty five fatigue test on

bellows of varying diameters, thicknesses, convolution profiles are required to

construct a fatigue life versus combined stress plot. The test bellows must be

representative of typical bellows design and manufacturing process. Hence lot of

cost is incurred in testing facilities of bellows. Many times special purpose test rigs

are required to be prepared for experimental verification or testing of bellows.

Testing results can be used for the foolproof design of expansion joints. Frequent

testing is essential for the manufacturers as customized design approach.

6.1 Purpose of Testing:

To assure a purchaser (user) that the product has been properly designed and

manufactured; which requires some method of examination and testing of the

product. The user may specify the kind of test required in the acceptance criterion.

Type of testing may be depending upon individual application. To ensure that the

product has been precisely designed and carefully manufactured, certain tests are

required. To ensure that the product is totally defect free, some method of

examination of the product is also required. The testing can be categorized in to

two groups, destructive testing and non-destructive testing. All tests are not

required for bellows, but the required types of tests are selected for individual

application.

Some standard non-destructive examinations are mentioned below.

6.2 Non-destructive Testing:

1. Radiographic examination

2. Liquid penetration examination

3. Fluorescent penetrant examination


4. Magnetic particle examination

5. Ultrasonic examination

6. Halogen leak examination

7. Mass Spectrometer examination

8. Air jet leak examination

9. Pressure Testing

A. Hydrostatic testing

B. Pneumatic testing

10. Spring rate test

Pressure tests are useful for detecting leaks, and also way to test bellows squirm,

meridional yield and rupture

6.2.1 Radiographic Examination:

This method is based on the principle that extremely high frequency light waves,

usually x rays will penetrate solid materials and, when projected on to

photosensitive film, will reveal voids, areas of discontinuity, and lack of

homogeneity. This examination is widely used to evaluate the soundness of

welds. Unless required by the purchaser, radiographic examination of the

longitudinal seam of a bellow need not be specified.

6.2.2 Liquid Penetrant Test:

This method consists of cleaning a surface, coating it with a dye, wiping the dye

off and coating the surface with a developer which after sufficient time will draw

the dye from the cracks, pin holes, and make them apparent to the observer.

Liquid penetrant examination is limited in the scope to detecting the surface

defects.

6.2.3 Flourcent Penetrant Examination:

Flourcent penetrant examination is similar in purpose to the liquid penetrant

examination but is accomplished by the use of a dye which contains a flourcent

material and developer.

6.2.4 Magnetic Particle Examination:

Magnetic particle examination consists of coating a surface with finely powdered

iron and establishing a magnetic field in the material being examined. The


presence of discontinuities and irregularities in the magnetic field, as indicated by

the lines of powdered iron, will indicate surface and subsurface defects.

6.2.5 Ultrasonic Examination:

Ultrasonic examination used high frequency sound waves to detect flaws, and is

useful in determining thickness, depth and exact location of defects. Interpretation

of indications in sections of sharply varying thickness is difficult.

6.2.6 Halogen leak Examination:

Halogen leak examination utilizes a probe of suitable design which selectively

indicates the presence of halogen gases.

This examination is more sensitive than a hydrostatic test or air jet leak

examination but since it is done at low pressure, it can only determine the

presence of a leak and can not validate the structural integrity of the item being

examined.

6.2.7 Mass Spectrometer Examination:

Mass spectrometer examination is an extremely sensitive means of determining

the presence of a leak. The gas used is helium. This test is only recommended for

explosive service requirements.

6.3 Hydrostatic Pressure Testing:

The hydrostatic pressure testing is necessary to check the pressure withstanding

capability of bellow and detection of any leakage in the bellow. This test is carried

out in a suitable fixture as shown in the figure 6.1 or in case of large diameter; it

can be carried out without fixture with necessary fabrication. The bellows ends

must be closed and free length of the bellow should be made fixed with extra leg

support at three or four sides of the diameter.

This test involves filling of the expansion joint with a liquid, usually water. After

filling, it can be pressurized up to the test pressure. The test pressure is usually

1.5 times the design pressure at ambient temperature. Expansion joints placed in

high temperature service may require the pressure test to be performed at an

adjusted pressure. It is imperative that the test pressure does not produce any

membrane stress in excess of yield strength or cause permanent deformation or

instability of the bellows at test temperature. The observer has to take care about


pressure drop in the bellow, leakages if any in the bellow. Bellow should come to

its original shape after removal of pressure.

Figure 6.1: Set up diagram for Hydro test

6.4 Pneumatic Pressure Testing:

This test is having similar objectives as to check the pressure withstanding

capability of bellow and detection of leakage. This test involves filling of the

expansion joint with air or other gas. After filling, it can be pressurized up to the

test pressure. The test pressure is usually 1.1 times the design pressure at

ambient temperature. Expansion joints placed in high temperature service may

require the pressure test to be performed at an adjusted pressure. It is imperative

that the test pressure does not produce any membrane stress in excess of yield

strength or cause permanent deformation or instability of the bellows at test

temperature.

6.5 Spring Rate Test:

The force required to deflect (usually compress) a bellow is a function of the

dimensions of the bellows and the material from which it is made. It can be

measured as load per unit deflection. The curve of force versus deflection for most


bellows indicates motion extending into the plastic range, since material thickness

is taken very less in order to get higher flexibility.

Spring rate determination of a bellow becomes more critical as variation in

geometric parameters and bellow deforms in elastic range as well as plastic

range. Many times due to higher deflection taking place in piping length,

deformation stresses becomes very significant. But assuming the movement of

bellows deformation is within elastic limit, the axial theoretical spring rate can be

determined experimentally, which can be useful for the limit of axial deformations

of bellows.

Figure: 6.2 Setup diagram for spring rate test

A simple press type fixture is necessary for the spring rate test. Following

procedure can be used for test. The diagram is shown in figure 6.2.

1. The expansion bellow to be tested is placed in vertical position in the fixture

as shown in the figure 6.2. The bellow is held in place by means of

fastening clamps.

2. Measurement of the length of bellow should be feasible with fixed or free

scale at different loadings.


3. Measurement of force, which is created by operating screw or bringing ram

downwards, should be through electronic load cell.

4. The initial position or free length of bellow is measured on the scale or

using separate scale.

5. Force is gradually applied in the steps by rotation of the screw or

pressurizing the ram.

6. Measure the load readings with reference to bellow length at various

intervals.

7. Plot the curve of force vs deflection, which is the spring rate of the bellow.

8. Compare the spring rate with theoretical designed value.

6.6 Destructive Testing:

1. Squirm testing

2. Meridional yield- rupture testing

3. Fatigue life testing

6.6.1 Squirm Testing:

Main objective of the test is to determine the internal pressure which will cause a

bellows to become unstable. Squirm is defined on the basis of change in pitch of

the bellows convolutions under internal pressure.

Test Procedure:

1. Expansion joint should be placed in a suitable fixture, with bellows in

straight position.

2. Bellows should be effectively sealed at the ends during pressurization.

3. End movements must be prevented.

4. Bellows can be tested either in horizontal or vertical position. Horizontal is

preferred.

5. Testing medium should be water for safety.

6. No restrictions to convolutions.


7. Use two dial gauges in perpendicular direction (on outer surface of

convolution), to observe deflection of end centers of a bellow.

8. Change in pitch of all convolutions.

9. Pressurize the specimen in steps without releasing the pressure between

steps.

10. Each interval should not exceed 10% of the final anticipated instability

pressure, although smaller intervals are preferred.

11. Instability of axially aligned bellows is generally characterized by a sudden

acceleration of either the change in resultant lateral deflection and/or

change in convolution pitch.

6.6.2 Meridional yield rupture testing:

Rupture test is to determine the internal pressure which will cause yielding and

rupture of bellows. Place the expansion joint in any suitable fixture, with the

bellows fixed in the straight position which will effectively seal the ends during

pressurization, and most importantly will prevent any movement of the ends

during testing. Test medium should be limited to water as safety precautions.

Pressurize the specimen in steps, retaining to zero pressure after each step, up to

at least twice the yield pressure. Instrumentation should be arranged such as

pressure – time recorder, strain gauges etc.

6.6.3 Fatigue Life Testing:

This test must be on proto type bellows. Fatigue life testing is a verification of the

ability of a bellow to withstand a given number of flexing cycles. With all other

shape factors remaining constant, cycle life will generally increase with diameter.

But for prototype testing it may be acceptable to cycle test the smaller size of

expansion joint being furnished for a given series of identical service condition.

Figure 6.3 shows the arrangement required for cycle life test of a bellow.

Test Procedure:

1. Place the bellow element in the suitable fixture as shown in the figure 6.3.

2. Bellows should be effectively sealed at the ends during pressurization.

Apply pressure gradually till it reaches design pressure.


3. Set axial movement of bellow as per designed permissible limit. Check

whether the bellow test is as per free length or extended length. Set limits

according to these values.

4. Pressurize the bellow with water.

5. Start fatigue testing at room temperature, keeping it pressurized at design

pressure. The fatigue life frequency shall be kept constant as far as

possible.

Figure 6.3: Set up diagram for cycle life test

6. Continue cycle life testing till it reaches 10,000 cycles.

7. Check for any leakage in the bellow.

8. Carry out Die penetrant examination for any surface cracks after the fatigue

test.


Fatigue may be performed at constant pressure or varying pressure condition. It is

also acceptable to cycle test at room temperature any expansion joint which will

be furnished for operating temperatures up to the active creep range. For

expansion joints operating above this range, consideration should be given to

testing at elevated temperature.

6.7 Experimental Work

All activities associated with the development of any product for the full

satisfaction of the customer requires extensive planning and as well as conducting

various research studies so that the optimum value of all decision variables can

be achieved. The quality engineering techniques are very much helpful in

producing robust design of the products. [B13] Many more quality engineering

principles are helpful for making robust product. They are system design through

innovations, parameter design, tolerance design, product design optimization,

process design optimization, statistical quality control etc.

6.8 Design of Experiments

In the present business scenario of globalization a revolution is taking place due

to customers’ higher expectations and breakneck technical changes are taking

place which are causing yesterday’s realities as tomorrow’s irrelevancies. Quality,

reliability and durability are the primary factors in the customer’s buying decisions

in the present overall business revolution. The robust design of products is the

fundamental requirement of the customers.

Robust design means that the performance of the system is always acceptably

close to the ideal function of the system. A systematic and efficient way to meet

the challenge of developing a robust product is the statistical approach to the

optimization of the product and process design which was originally developed by

Sir Ronal A Fisher and later adapted by Genichi Taguchi for industrial products.

This work is an attempt towards optimization of various geometric parameters for

the spring rate of bellows. After that some tests are carried out and using results

some meaningful conclusions are drawn.

6.8.1 Taguchi’s philosophy:

All products have characteristics that describe their performance relative to

customer requirements or expectations. The quality of product is measured in


terms of these characteristics. Quality is related to the loss to society caused by a

product during its life cycle. A truly high quality product will have a minimum loss

to the society as it goes through this life cycle.

6.8.2 Purpose of Experimentation:

The purpose of product or process development is to improve the performance

characteristics of the product or process relative to customer needs and

expectations. The purpose of experimentation should be to understand how to

reduce and control variation of a product or process; subsequently, decisions

must be made concerning which parameters affect the performance of a product

or process. By adjusting the average and reducing variation, the product or

process losses can be minimized.

6.8.3 Basis of experimentation:

The basis of experimentation should be based on the use of orthogonal arrays to

conduct small, highly fractional factorial experiments up to larger, full factorial

experiments. The use of orthogonal arrays is just a methodology to design an

experiment, but probably the most flexible in accommodating a variety of

situations and yet easy for industry people to execute on a practical basis.

6.8.4 Introduction to Design of Experiments (DOE):

Design of Experiments (DOE) is a statistical technique introduced by R A Fisher in

England. This technique is useful for simultaneously study of multiple variables on

the any parameter or outcomes. Dr. Taguchi has carried out significant research

with DOE techniques in the field of electronics. This technique has many

advantages over classical experimentation procedure. DOE is helpful in

addressing quality of the product issue in the design phases of products. DOE is

helpful in finding influence of individual parameters, determination of relative

influence of individual factors and it leads to optimum design of the product or

process.

A designed experiment is the simultaneously evaluation of two or more factors

(design parameters) for their ability to affect the resultant average or variability of

particular product or process characteristics. To accomplish this in an effective

and statistically proper fashion, the levels of the factors are varied in a strategic

manner. The results of a particular test conditions are observed and a complete


set of results are analyzed to determine the influential factors and preferred levels,

and weather increases or decreases of those levels will potentially leads to further

improvement. Basically this is an iterative process. Later on experiments typically

involve few factors at more than two levels to determine conditions of further

improvement.

6.8.5 The process of Design of Experiments DOE: [B13]

Following are the steps suggested for design of experiments by Taguchi

philosophy.

1. State the problem or area of concern:

A statement of problem should be critically framed so that will make clear and

concise description of the problem. Expansion joints are manufactured with

customized approach for individual application. The performance is mainly

depending on precise design and manufacturing methodology. The expansion

joint must perform expected flexibility while working. The flexibility of bellow is

depending on its material property and selection geometric parameters. The

initial theoretical axial spring rate can be evaluated from the parameters. The

spring rate of bellow must maintain consistently within limits, so performance is

assured. This testing will also help to reduce variation in manufacturing

procedure and quality will improve. The statement of the problem is framed as

“Optimization of parameter design of expansion joints for the desired or

expected value of initial axial spring rate using Design of Experiment (DOE)

technique”.

2. State the objectives of the experiment.

The objectives of the experiment are

(a) Effect of various geometric parameters on axial spring rate,

(b) Study of percentage influence of each parameter,

(c) To check expected performance of expansion joint,

(d) To verify existing design procedure.

3. Select the quality characteristics and measurement system.


The initial axial spring rate of the bellow can be measured by spring rate test

as suggested by EJMA. This test is basically non-destructive test. Here the

spring rate is measured by movements of bellows at various pressure values.

The convolution movement can be measured by variations of pitch of the

convolutions. This movement can be measure by vernier caliper. The both

ends must be welded with flanges and their movements should be restricted

by a fixture.

4. Select the factors that may influence the selected quality characteristics.

Here the list of factors to be evaluated in the experiment for their effect on the

selected quality characteristics should be determined. The initial axial spring

rate of bellows depends on following factors

(a) Selection of material and its modulus of elasticity

(b) Selection material thickness,

(c) Design parameter - height of convolutions,

(d) Design parameter - pitch of convolutions

(e) Design parameter – Mean diameter of bellow

(f) Design parameter – number of plies of material.

5. Identify control and noise factors. (Taguchi-specific)

Control factors are those factors that a manufacturer can control the design of

a product, the design of a process, or during a process.

(a) Control on variations in thickness while forming convolutions.

(b) Control on precise dimension of height of convolutions.

(c) Control on pitch of convolutions.

Noise factors are those things that a manufacturer can not or wishes not to

control for cost reasons.

1. Very high precision level of dimensions

6. Select the levels for the factors.

Basically the spring rate of bellows mainly depends on two parameters for a

particular material. First parameter is thickness and number of plies. As higher


the thickness, spring rate is increases and for lower thickness the spring rate

will be reduces. These two parameters must be considered as a common

parameter named as “total thickness”. This will simplify the understanding as a

common parameter.

So, Total thickness = Material thickness x number of plies.

Second important parameter is height of convolutions, as the height of

convolution increases, spring rate reduces and for lower height of convolution

spring rate is always higher.

Here, two levels of parameters can be selected for the total thickness and

height of convolution parameters for the design of experiment.

7. Select the appropriate orthogonal array. [B13, B4]

The determination of appropriate orthogonal array for the experiment is major

criteria for the experiment. Since two parameters are selected for two levels,

following orthogonal array may be selected for the experimentation.

Parameter A = Thickness of bellow material

Parameter B = Height of convolutions

Table 6.1: Experimental parameters for Axial Spring Rate

Thickness of bellow material t (cm)

Height of convolutions w (cm)

A1 B1

A1 B2

A2 B1

A2 B2

8. Select interactions that may influence the selected quality characteristics,

or go to step 4. (iterative process)

9. Assign factors to orthogonal array and locate interactions.

10. Conduct tests described by trials in orthogonal arrays.

11. Analyze and interpret results of the experimental trials.

12. Conduct confirmation experiment.

Steps 8 to 12 are performed in the experiment as per experimental results.


6.9 Applying DOE on Spring Rate of Bellows:

The force required to deflect a bellow is a function of the geometric parameters of

the bellows and the material from which it is made. The curve of force versus

deflection for bellows may be represented by straight line, based on Hook’s law of

elasticity (within elastic region). Here lower material thickness will permit higher

flexibility and leads to higher spring rate.

Spring rate will not be consistent because of large variations in geometric

parameters. U shape of convolutions permits higher spring rate, while toroidal

shape of convolutions will not. Hence for expected movement of expansion joint,

determination of spring rate of bellow becomes essential.

The influencing parameters for the spring rate of bellows are mean diameter,

thickness of material, number of plies, height of convolution, number of

convolutions and elastic modulus of the material. If the spring rate of bellows is

evaluated in force required per unit deflection, per convolution, than number of

convolution parameter can be reduced from the analysis.

The initial theoretical spring rate of bellows can be evaluated analytically using

following relationship which is suggested by EJMA. [20]

Theoretical initial spring rate, f

pbm

CwntED

fiu 3

3

7.1 (6.1)

where, Dm = Mean diameter of the bellow, 193.88 cm.

Eb = Elasticity of the bellow at room temperature, 19728608 N/cm2

tp = Thickness of bellow material, 0.058 cm

n = Number of plies, 3

w = Height of convolutions, 3.8cm.

Cf = Constant based on inside dia. and pitch of convolutions, 1.7

6.9.1 Axial Spring Rate of Metallic Bellows

The factors, which can influence axial spring rate, are thickness of material (t),

number of ply (n), convolution height (w), mean diameter of bellow (Dm), elastic

property of material, and constants. Table 6.1 shows the factors considered and

its corresponding levels along with the interactions. The inner array along with the

experimental results is given in Table 6.2. Experimental results are taken by

manual measurement. The movement of convolution is measured by distance

between two end flanges using Vernier calliper. For the analysis “smaller is better”


quality characteristic is selected. Due to economic reasons, one test result for

each trial is used in this investigation only the interactions between two factors are

considered and all other interactions are ignored.

Table 6.2 Factors and Levels No. Factors Level 1 Level 2 1 t x n 0.058 0.065 2 w 3.5 3.8 3 t x w INTERACTION

Table 6.3 Experimental Results

Sr. No.

t (cm)

w (cm)

Axial Spring Rate (N/cm) Experimental data

Axial Spring Rate (N/cm)

1 0.058 3.5 3100 3210 2 0.058 3.8 2408 2260 3 0.065 3.5 4390 4240 4 0.065 3.8 3420 3580

Table 6.4: Experimental Results (In two level formats)

Parameter (A1)

Parameter (A2)

Total

Parameter (B1)

3100, 3210 4390, 4240 14940

Parameter (B2)

2408, 2260 3420, 3580 11668

10978 15630 Total = 26608

6.9.2 Analytical Approach:

Sum of Squares (SS):

SST = NTy

N

ii

2

1

2

(6.2)

= N

T 222222222 35803420424043902260240832103100

= 92582664 – 88498208

= 4084456

Variations due to thickness (t):

SSA =

kA

i i

i

NT

nAA

1

22

(6.3)

= 8

266084

156304

10978 222

= 2705138


SSA’ = N

AA 221 =

81563010978 2 = 2705138 (6.4)

Variations due to convolution height (w) :

SSB = N

BB 221 =

81166814940 2 = 1338248 (6.5)

Variations due to combined effect (t and w):

A1B1 = (A x B)1 = 6310

A2B2 = (A x B)2 = 8630

A1B2 = (A x B)3 = 4668

A2B2 = (A x B)4 = 7000

SS (AxB) = BA

C

iSSSS

NT

inAxBiAxB

2

1

22

)()( (6.6)

= 133824827051388

266082

70002

46682

86302

6310 22222

= 18

SS (AxB) = N

AxBAxB 221 = 18

SST = SSA + SSB + SS (AxB) + SSe 6.7

4084456 = 2705138 + 1338248 + 18 + SSe

SSe = 41052

Degree of freedom:

VT = N – 1 = 8 – 1 = 7

VT = VA + VB + V AxB + Ve

VA = kA – 1 = 2 – 1 = 1

VB = kB – 1 = 2 – 1 = 1

VAxB = VA x VB = 1 x 1 = 1

Ve = VT – VA – VB – VAxB = 7 – 1 - 1 – 1 = 4


6.9.3 Results and discussions:

The results are evaluated by Taguchi method and the methodology is shown

earlier. The results are tabulated in the ANOVA table 6.5. The influence of each

parameters can be observed easily by referring last column. Figure 6.4 and 6.5

shows effect of both geometric parameters by line graph.

Table 6.5: ANOVA Table Factor Sum of squares

(S S) Degree of

freedom (v) Variance

(V) F-Ratio

(F) Percent contribution

( % ) t 2705138 1 2705138 263.582 65.98 w 1338248 1 138248 130.395 32.51

t x w 18 1 18 0.002 -0.25 Error 41052 4 10263 1.76 Total 4084456 7 100

Variance = SS / degree of freedom

Factor F = Variance / Variance (error)

SSA’ = SSA – (Ve) vA = 2705138 – 10263 = 2694875

Percent contribution = (SS’A / SST ) x 100

= (2694875 / 4084456) x 100 = 65.98

Effect of t

0

2000

4000

6000

0.058 0.065

t, cm

ASR

, N/c

m

Figure 6.4 Influence of thickness of material


Effect of w

01000200030004000

3.5 3.8

w , cm

ASR

, N/c

m

Figure 6.5 Influence of height of convolution

6.9.4 Observations:

1. The experiment investigation and the subsequent analysis bring out the

influence of dominancy of selected geometric parameters (thickness and

height of convolution) for the axial spring rate of bellows.

2. The thickness is being the most significant parameter (65.98%), followed

by convolution height plays influence of (32.51%) and the combination of

these two parameters affects very negligible (-0.36%) for achievement of

axial spring rate in metallic bellows.

3. The factors are predominant for a confidence level of 95%, since error part

is very negligible, the results may consider reliable.

4. The most significant parameter is thickness of material (t) for desired axial

spring rate.

6.9.5 Limitation:

Spring rate measurement is carried out on four bellows and results are

extrapolated for L-8 orthogonal array. Further if all eight experimental data are

available, influence can be evaluated more precisely.


6.10 Spring Rate Test:

The force required to deflect a bellow is a function of the dimensions of the

bellows and the material from which it is made. The curve of force versus

deflection for most bellows indicates motion extending into the plastic range, since

material thickness is taken very less in order to get higher flexibility. Spring rate

determination of a bellow becomes more difficult as variation in geometric

parameters and bellow deforms in elastic range as well as plastic range. Many

times due to higher deflection taking place in piping length, deformation stresses

becomes very significant.

Bellows are loaded by internal pressure, which may cause a bellow to become

deflect axially, laterally or angularly. Bellows performance is depending on critical

pressure and temperature and their fluctuations. The bellow convolution may get

expand or contract axially and laterally.

Figure 6.6: General curve of Bellows Force vs Deflection

The curve shown in figure 6.6 shows the curve of force vs deflection for most

bellows indicates motion extending into the plastic range. The first portion of the

curve is a straight line as the bellows is deflected through its elastic range

(Hooke’s law). As bellows deflection continues and extends into plastic range, the


force vs deflection relationship becomes non-linear until the point of maximum

deflection is reached.

When the restraining force is released, the curve again becomes linear until the

applied force is zero at which point the residual deflection of the bellows still has a

positive value. To return the bellows to its initial position, a restoring force must be

applied in the opposite direction as shown by the curve below the abscissa. This

phenomenon is similar to hystersis loop behavior of materials while supplying

electric or magnetic energy.

Line A represents theoretical initial elastic spring rate, which can be determined

analytically with reasonable accuracy. This equation is mentioned in analytical

approach and for U shape convolution as shown in figure 6.7.

6.10.1 Experimental method to check Spring Rate:

An experimental set up requires a bellow with both ends blind. This bellow is

mounted between two end plates with fixed lungs. The bellow can not expand but

due to pressure force, it can contract. The whole set up is made vertical and at the

top, pressure gauge is mounted. Same side one opening is kept through water

pump. The water is filled till it overflows, and then the hole is closed.

Now using water pump inside pressure can be increased at different values and

the movements of convolutions can be observed and measured. The length

Figure 6.7: Convolutions of bellows


variations can be measured with reference to convolution tip. It can also be

termed as pitch. The pitch variation is taken at 4 sides, 900 to each other, named

as A, B, C and D.

6.10.2 Assumptions of Analysis:

1. Bellow material is having uniform thickness.

2. Bellow material is homogeneous.

3. Pitch measurement is carried out with vernier caliper, but due to manual

approach, measurement error can be approximately ± 0.2 mm.

Geometric Dimensions of a bellow:

A bellow with following dimensions is taken for experiment for the spring rate

measurement. Table 6.6: Geometric dimensions of a bellow

Db

(cm)

Dm

(cm)

t

(cm)

w

(cm)

q

(cm)

N n E

(N/cm2)

190 193.68 0.06 3.5 5 10 3 19728608

Mean diameter, Dm = Db + w + (n x t) = 190 + 3.5 + (3x0.06) = 193.68 cm

Cylinder bore = 10 cm.

Figure 6.8: Measurement indication of a bellow


6.10.3 Experimental Results: Table 6.7: Experimental Results

Change in length between convolutions, cm Sr. No.

Pressure N/cm2 A B C D

Average change,

cm 1 0 45.10 45.20 45.30 45.50 45.27

2 20 44.85 45.00 45.00 45.30 45.04

3 40 44.60 44.90 44.80 45.00 44.82

4 60 44.35 44.60 44.50 44.80 44.56

5 80 44.10 44.25 44.20 44.55 44.28

6 100 43.80 44.00 44.00 44.25 44.02

7 0 45.10 45.20 45.30 45.50 45.27

Maximum variations between 20-100

N/cm2. 1.05 1.00 1.00 1.05 1.025

6.10.4 Result Analysis:

Table 6.8: Computation of Axial Spring Rate

Sr. No.

Pressure N/cm2

Force = Pr. x Area

(N)

Unit Deflection (Reference : 45.27)

(cm)

Bellow Spring Rate =Force/Unit deflection

(N/cm ) 1 20 1570 0.23 6826

2 40 3140 0.45 6978

3 60 4710 0.71 6633

4 80 6280 0.99 6344

5 100 7850 1.25 6280

Average: = 6612.2

6.10.5 Sample calculations:

Force at 100 N/cm2 pressure = Pr.xArea =100x 2

4 boreD =100 x 2)10(4 = 7850 N

Unit deflection at 100 N/cm2 pressure measured = 1.25 cm

Spring rate = Force / unit deflection = 7850 / 1.25 = 6280 N/cm


Graphs:

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Deflection (cm)

Forc

e (N

)

Figure 6.9: Force Vs Deflection curve

6.10.6 Analytical Approach:

EJMA suggests an analytical approach to check the initial spring rate of bellows. It

also suggests that; since there is no standard manufacturing methodology at

various industries, frequent experimental testing must be carried out to validate

the design methodology.

Bellows initial Axial Elastic Spring Ratef

pbm

CwntED

fiu 3

3

7.1 (4)

= 8.15.3

306.01972860868.1937.13

3

xxxxx = 54540 N/cm/convolution.

Axial spring rate of bellow = 545.4 / 10 = 5454 N/cm.


010002000

3000400050006000

70008000

0 20 40 60 80 100 120

Pressure, N/cm2

Sprin

g ra

te, N

/cm

/con

v

Experimental Analytical Average Exp

Figure 6.10 : Comparison of Spring Rate Results

6.10.7 Comparison of Results: Table 6.9: Comparison of Results

Average Experimental

Spring rate

(N/cm)

Analytical

Spring rate

(N/cm)

Deviation in

spring rate

(N/cm)

% deviation

6610 5454 1156 21.19

6.10.8 Observations:

1. Experimental results shows that the spring rate of bellows vary with respect to

internal pressure load. Hence, the average movement of convolution is

considered for various pressure loadings in elastic range. As the pressure

increases towards designed value, the spring rate also approaches to

expected value. In the present study maximum deviation is up to 21.19%.

2. Bellows with lower value of spring rate are flexible, while bellows with higher

spring rate value are stiffer. We desire more flexibility from expansion joints.

3. Stiffness of bellow is directly proportional to mean diameter of bellow,

thickness of material, number of plies of bellow, while inversely proportional to

height of convolutions.


6.11 Squirm Test:

The purpose of squirm test is to check critical buckling pressure of a bellow. The

objectives of the test are to find actual factor of safety, validation of the design

procedure and confirmation of manufacturing process. The bellow is said to be

squirmed on the basis of major (sudden) change in pitch of the bellows

convolutions under internal pressure. The test will also be helpful to determine the

critical internal pressure at which it will become unstable.

6.11.1 Geometric dimensions of a bellow:

Material type = SS 304 Table 6.10: Geometric dimensions of a bellow

Db

(cm)

Dm

(cm)

t

(cm)

w

(cm)

q

(cm)

N n E

(N/cm2)

16.90 20.48 0.04 3.5 2.6 7 2 19728608

Length of a bellow = N x q = 18.20 cm

Design Pressure = 50 N/cm2

6.11.2 Estimation of Critical pressure:

Ratio of Length to diameter of bellow = (18.20 / 16.90) = 1.0769

Transition Point factor, Cz = cby

ui

ADSqf 272.4

= 2.25

Since Lb/Db ratio is less than transition point factor (1.0769<2.25), it is short

column.

Where, fw = Theoretical spring rate = 1998 N/cm/convolution

Sy = Yield strength of material = 20310 N/cm2

Ac = Cross section metal area of one bellow = 0.365 cm2

Critical Pressure (In-plane) = 72 N/cm2

Critical Pressure (Column) Psc =

bz

b

b

yc

DCL

qDSA 73.0

187.0

= 95.4 N/cm2


6.11.3 Experimental Readings: Table 6.11: Experimental Readings of Pitch dimensions

1 2 3 4 5 6 25.7 25.7 26.3 26.7 26.4 27.1 25.9 26.1 26.1 26.1 25.8 26.4 26.7 26.1 26.3 26 26.3 26.8 25.9 26.7 26.1 26.8 26.7 26.7 0

26.05 26.15 26.2 26.4 26.3 26.750 26.30

28.1 26.1 26.5 26.7 27.3 27 25.8 26.1 26.5 27 26.7 26.7 26.5 27.1 26.3 26 26.5 26.5 26 26.5 26.7 27.1 23.7 26.4 1

26.6 26.45 26.5 26.7 26.05 26.650 26.492

26.1 26.5 26.5 26.9 26.9 26.9 25.8 26.3 26.4 26.9 26.5 26.3 27 26.8 26.3 26.7 26.7 26.7 26 25.8 26.5 26.5 26.5 26.5 2

26.225 26.35 26.425 26.75 26.65 26.600 26.50

26.2 26.6 27 27 26.6 26.6 26.3 26.7 25.7 26.7 26.2 27 26.6 26.6 25.7 26.3 26.3 26.2 25.5 25.7 25.8 27.1 26 26 6

26.15 26.4 26.05 26.775 26.275 26.450 26.35

25.5 25.9 26.3 26.3 26.3 26.3 26 26 26 26 26 26

25.7 26.4 25.6 26.3 26.7 26.1 25.7 26.2 27 26.5 26.7 26 9

25.725 26.125 26.225 26.275 26.425 26.100 26.15

25.9 25.9 26.4 26.6 26.9 26.3 25.9 26.3 26 26.4 26.3 26.3 26.2 26.8 26.3 26.4 26.3 26.2 25.5 25.6 27 26.8 26.4 26.3 12

25.875 26.15 26.425 26.55 26.475 26.275 26.30

26.2 26.2 26.3 26.5 26.5 26.5 26.7 26.5 26.5 26.5 26.4 26.2 26.5 26.5 26.1 26.2 26.1 26.1 26.1 26.1 26.9 26.7 26.7 26.5 15

26.375 26.325 26.45 26.475 26.425 26.325 26.40

1 2 3 4 5 6 26.2 26.2 26.3 27.2 26.6 26.4 26.1 26.1 25.9 26 26.9 26.5 26.2 26.2 26.2 26.2 26.2 26.2 26.9 26.4 26.4 26.4 26.4 26.4

26.35 26.225 26.2 26.45 26.525 26.375 26.35

26.1 26.5 26.9 26.5 26.3 26.2 26.1 26.1 26.1 26.1 26.1 26.1 26.2 26.7 26.1 26.2 26.6 26.2 26.3 26.3 26.7 26.7 26.7 26.4


26.175 26.4 26.45 26.375 26.425 26.225 26.34

26.8 26.8 26.8 27.1 26.9 26.8 27.2 26.8 26.9 26.9 26.9 26.9 27 27 26.7 26.7 26.7 26.7

27.2 27.2 27 27 26.8 26.5 27.05 26.95 26.85 26.925 26.825 26.725 26.88

27.9 26.6 27.2 27.3 26.9 26.9 28 27.2 26.6 26.6 26.8 26.9

28.3 27.6 27 27 27.3 26.9 27.5 27.5 26.8 27 27.6 27.6

27.925 27.225 26.9 26.975 27.15 27.075 27.20

33 20.4 27.1 32.8 26.8 24.3 21 33.5 26.9 21.7 28.6 31.8 33 20.6 26.9 32.7 27.5 23.4

20.9 24.7 26.6 22.1 28.2 31.1 26.975 24.8 26.875 27.325 27.775 27.650 26.90

6.11.4 Summary of Results:

Table 6.12: Summary of Pitch Dimensions Pressure, N/cm2 Average pitch, mm

0 26.30 10 26.50 20 26.50 60 26.35 90 26.15

120 26.30 150 26.40 180 26.35 210 26.35 250 26.90 290 27.20 350 26.90

25.625.8

2626.226.426.626.8

2727.227.4

0 10 20 60 90 120 150 180 210 250 290 350

PRESSURE, N/cm2

AVG

. PIT

CH

, mm

Figure 6.11: Graph showing pitch variations



1. Experimental results of squirm test of bellows shows variations of pitch well

within elastic limits up to the pressure 210 N/cm2. But beyond that pressure,

deformation exceeds continuously till squirm failure.

2. The pitch variation suddenly increases from 250 N/cm2, i.e. because of drastic

deformation of bellows beyond elastic limits. Here the pitch disturbs

permanently even after releasing pressure. This is called squirm failure.

3. Bellow should be loaded well within the limits of critical pressure to avoid

squirm failure.

4. In case of short column bellows, it is observed that the bellow initially failed by

in-plane squirm, than subsequently by column squirm.

5. Critical value of pressure suggested by EJMA involves following safety factor.

Factor of safety in in-plane squirm failure = (250 / 72) = 3.47

Factor of safety in column squirm failure = (290 / 95.4) = 3.04


6.12 In-Plane Stability Tests:

In actual practice, piping are operates at various temperatures in a specific range.

Like for a particular application of expansion joint, the piping operates between

250 F to 3750 F (-30 C to 1900 C). Now the installation of expansion joint should be

carried out at minimum design temperature, and at minimum temperature, the

bellow will be under compression mode. To facilitate the installation for such

cases, bellows are initially pre-compressed axially by certain amount and than

installation is carried out. The amount of pre-compression is calculated based on

coefficient of thermal expansion at various temperatures.

Pre-compression of bellows creates very high longitudinal bending stresses due to

deflection. If the compression amount is higher, than permanent deformation of

material takes place in the convolution area. The maximum stresses are develops

at roots of convolutions.

To analyze the amount of longitudinal stresses developed in the bellows, following

calculations are made using EJMA relations.

Table 6.13: Evaluation of longitudinal stresses under pre-compression

Pre-compression (cms)

Bellows meridional membrane stress due to deflection

N/cm2

Bellows meridional bending stress

due to deflection N/cm2

Total meridional stresses N/cm2

0 0 0 0

0.25 800 69080 69880

0.5 1600 138160 139760

0.667 2140 184300 186440

1.0 3200 276320 279520

2.0 6400 552640 584600

3.0 9600 828960 838560

5.0 16000 1381590 1397590

7.5 24010 2072390 2096400

10.0 32010 2763190 2795200


6.12.1 Bellow subjected to compressive deformation:

Bellows are used in free length installation, compression length installations or

expanded length installations. These conditions are selected for space availability

and for special requirements. This test is conducted on a bellow after 10 cm initial

compression. The geometric dimensions of bellow are as following. The objectives

of the test are to study convolution movement, in-plane behavior, and squirm

failure (critical pressure) of bellows in compression mode.


Material type = SS 304

Design Pressure = 0.5 Pascal Table 6.14: Geometric dimensions of a bellow

Db

(cm)

Dm

(cm)

t

(cm)

w

(cm)

q

(cm) N n

E

(N/cm2)

40.60 42.98 0.08 2.3 2.26 15 1 19728608

Length of a bellow = N x q = 33.90 cm

Initial compression of bellow = 10 cm

6.12.3 Evaluation of Critical pressure:



w

ADSqf 272.4 = 1.451

Since Lb/Db ratio is less than transition point factor, it is short column.




Db = Inside diameter of bellow = 42.98 cm,

Limiting critical pressure (in-plane) (Psi) = 53.3 N/cm2

Buckling pressure, (column) Psc =

bz

b

b

yc

DCL

qDSA 73.0

187.0

= 51.1 N/cm2


6.12.4 Experimental Readings: Table 6.15: Experimental Readings of Pitch Dimensions

Pr. Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 0 A 18 17.2 16.4 16.8 15.5 15.8 15.8 15.8 15.6 15.6 16.1 15 16.8 B 19 16.6 16.7 15.1 15 15.6 15 14.8 16.6 15.4 17.5 14.9 17.7 C 15.5 15.7 15.2 15.5 15.3 14.6 16.3 16 15.6 16.5 16.4 16.1 19.9 D 18.5 16.1 18.2 15.3 15.6 15.3 15.4 16.7 16.7 16.3 17.3 15 18 66.5 62.7 66.5 62.7 61.4 61.3 62.5 63.3 64.5 63.8 67.3 61 72.4 64.3 2 A 17 17 17 17 16 16 16 17 16 16 16 16 18 B 18 16 18 15 16 15 15 17 17 16 17 14.5 18 C 16 15.5 17 15 15.5 15 14 16 16.5 15.5 16.5 15.5 20 D 17.5 17 17 16.5 15 16 15 15 16.5 16 18 15 17 68.5 65.5 69 63.5 62.5 62 60 65 66 63.5 67.5 61 73 65.15 6 A 18.5 17 17 17 16.5 16.5 16 16.5 16 16 16 15 17.5 B 18 16.5 17.5 15.5 16 15 15 16.5 16.5 15.5 17 15 17 C 15.5 15.5 17.5 15 15.5 15 14 16.5 16 15 16 15 20.5 D 18.5 16 17 16.5 15 16 15 15 17 17 18.5 15 18 70.5 65 69 64 63 62.5 60 64.5 65.5 63.5 67.5 60 73 65.23 9 A 18 16.5 17.5 17 16 16 16 16 16 16 16 15.5 17 B 18 16.5 18 15 16 15 15 16.5 17 15.5 17 15 18 C 16 15.5 17.5 15 16 14.5 14 16 16 15.5 16.5 15 20.5 D 18.5 16.5 17 17 15 16 15.5 15 16.5 16.5 18.5 15 18 70.5 65 70 64 63 61.5 60.5 63.5 65.5 63.5 68 60.5 73.5 65.30

12 A 18.5 17 17 17 16.5 16.5 16.5 17 16.5 16.5 16.5 16 17.5 B 18 16 18 15 16 15 15 16 16.5 16 17.5 15 18 C 17 15.5 17.5 14.5 15.5 15.5 14.5 16 16 15 16.5 15.5 20.5 D 18 17 17 17 16 16 15.5 15.5 17 16.5 18 15 17.5 71.5 65.5 69.5 63.5 64 63 61.5 64.5 66 64 68.5 61.5 73.5 65.88

13 A 15 15 16 18 21 21 20 20 19 16 19 14 15 B 22 19 20 13 10 10 10 12 16 16 18 15 20 C 20 18 20 14 12 10 10 10 11 15 21 22 24 D 17 16 17 17 18 18 17 19 20 18 19 11 14 74 68 73 62 61 59 57 61 66 65 77 62 73 66.00

6.12.5 Summary of Results:

Table 6.16: Summary of Pitch Dimensions Pressure, N/cm2 Average pitch, mm

0 64.3 20 65.2 60 65.2 90 65.3

120 65.9 130 66


63

63.5

64

64.5

65

65.5

66

66.5

0 0.2 0.6 0.9 1.2 1.25

Pressure, N/cm2

Mov

emen

t of c

onvo

lutio

ns, m

m

Movement

Figure 6.12: Movement of convolutions 6.12.6 Observations:

1. In this experiment the pressure intervals are comparatively larger than

previous column squirm test.

2. Experimental results of squirm test of bellows shows average pitch as 16.5

mm instead of 22.6 mm as bellow is compressed by 10 cm.

3. The pitch variations are almost negligible at all pressure values (64.3 to 66.00

for the pressure 0 to 130 N/cm2 ). This is because all convolutions do not have

space or room for the movement as the bellow is compressed.

4. Bellow squirm occurs at 130 N/cm2 pressure.

5. In case of short column bellows, it is observed that the bellow initially failed by

in-plane squirm, than subsequently by column squirm.

6. Critical value of pressure suggested by EJMA includes factor of safety of as

following. Factor of safety in in-plane squirm failure = (130 / 49.2) = 2.64 and

Factor of safety in column squirm failure = (130 / 5.11) = 2.54.

7. The factor of safety is less compared to normal bellow in earlier experiment.


6.12.7 In-Stability Test of Bellow Subjected To Tensile Mode

Bellows are used in free length installation, compression length installations or

expanded length installations. These conditions are selected for space

management and for special requirements. The squirm test in pre-extension mode

is conducted on an expansion joint of following dimensions. The objective of the

test is to study convolution movement, and squirm failure of bellows in extension

mode.


Material type = SS 304

Design Pressure = 0.5 Pascal Table 6.17: Geometric dimensions of a bellow

Db

(cm)

Dm

(cm)

t

(cm)

w

(cm)

q

(cm) N n

E

(N/cm2)

40.60 193.68 0.08 2.3 2.26 15 1 19728608

Length of a bellow = 33.90 cm

Initial extension of bellow = 5 cm

6.12.9 Evaluation of Critical pressure:



w

ADSqf 272.4 = 1.451

Since Lb/Db ratio is less than transition point factor, it is short column.




Db = Inside diameter of bellow = 42.98 cm,

Limiting critical pressure (in-plane) (Psi) = 50 N/cm2

Buckling pressure, (column) Psc =

bz

b

b

yc

DCL

qDSA 73.0

187.0

= 51.1 N/cm2


6.12.10 Experimental Readings: Table 6.18: Experimental Readings of Pitch Dimensions

Pres 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 A 26 25 25 26 25 24 25 25 26 24 25.5 24 24.5 25.5 B 24.5 25.5 25 24.5 24 24 24 24.5 25 25 25 25 25.5 25.5 C 24 25 25 24 24 24 24 25 24 25.5 25.5 25 25.5 25.5 D 24.5 25.5 24.5 25.5 24 23.5 25 25 24.5 23.5 24 23.5 24 25 99 101 99.5 100 97 95.5 98 99.5 99.5 98 100 97.5 99.5 101.5 5 A 25 25.5 25 25.5 24 24 24 24 25.5 23.5 25 23.5 24.5 25.5 B 25 25 25.5 24.5 23.5 25.5 24 23.5 25 24.5 25 25 25 26 C 24 25.5 25.5 24 25 24.5 25 25 23.5 25 25 25 24.5 25 D 25 25.5 24.5 25 24 23.5 25 25.5 24.5 24 25 24.5 23.5 25 99 102 100.5 99 96.5 97.5 98 98 98.5 97 100 98 97.5 101.5 7 A 25 25.5 25.5 25 24 23.5 24.5 24.5 25.5 23.5 25 23.5 24 25 B 24.5 24.5 25.5 25.5 23.5 25 24.5 24.5 25.5 25 25 24 25 24.5 C 24 25 25 23.5 24.5 25 24.5 24.5 24 25.5 25.5 25 24.5 25.5 D 25.5 25.5 25 25.5 24 23 25 25 24 23.5 25 25.5 24.5 25.5 99 101 101 99.5 96 96.5 98.5 98.5 99 97.5 101 98 98 100.5

10.5 A 25.5 25 25 26 24.5 24 24.5 24 25.5 23.5 26 23 24.5 25 B 24.5 24.5 25 25 23.5 25.5 24.5 24.5 26 25 25 25 24.5 24.5 C 23 24.5 25.5 23.5 24 24.5 24 25 24 25.5 25 25 25.5 25 D 26.5 25.5 24 25.5 25 23 24.5 25.5 23.5 25 25 25 24.5 26 99.5 99.5 99.5 100 97 97 97.5 99 99 99 101 98 99 100.5

13 A 22.5 26.5 26 27 23.5 22.5 22 19.5 23.5 23.5 27 25.5 22.5 28.5 B 19 22 23 24 24 30 35 35 30 26 24 22.5 22 16.5 C 19 23 26 25.5 26.5 28.5 26.5 26.5 25.5 26.5 25.5 25.5 24.5 21 D 32 29.5 24.5 25.5 21 18.5 21 22 18 22.5 22 24 26.5 35.5 92.5 101 99.5 102 95 99.5 105 103 97 98.5 98.5 97.5 95.5 101.5

6.12.11 Summary of Results: Table 6.19: Summary of Pitch Dimensions

Pressure, N/cm2 Average pitch, mm 0 98.96

50 98.75 70 98.79

105 98.96 135 98.96


98.6

98.65

98.7

98.75

98.8

98.85

98.9

98.95

99

0 50 70 105 135

Pressure, N/cm2

Mov

emen

t of c

onvo

lutio

ns, m

m

Movement

Figure 6.13: Movement of convolutions 6.12.12 Observations:

1. In this experiment the pressure intervals are comparatively larger than

previous column squirm test.

2. Experimental results of squirm test of bellows shows average pitch as 25.93

mm instead of 22.6 mm as bellow is extended by 5 cm.

3. The pitch variations are almost negligible at all pressure values (98.75 to 98.96

for the pressure 0 to 105 N/cm2). This is because of bellow is extended by 5

cm. and convolution movement is constrained by tension force.

4. The convolutions became unstable at around 135 N/cm2 pressure. This is

called in-plane squirm. The critical pressure value for instability is 51.1 N/cm2

as per EJMA.

5. Critical value of pressure suggested by EJMA involves following safety factor.

Factor of safety in in-plane squirm failure = (135 / 50.1) = 2.69

Factor of safety in column squirm failure = (135 / 51.1) = 2.64

6. The factor of safety is lowest, while bellow is in compression mode, and

highest when it is in normal mode.


6.12.13 Comparison of Results

Table 6.20: Comparison of results of in-plane stability tests

Initial condition

S5+S6 (Theo.) N/cm2

In-plane critical

pressure, N/cm2

(Theoretical)

Column Critical

Pressure, N/cm2

(Theoretical)

Actual squirm

pressure, N/cm2

Acting f o s

(in-plane)

Acting f o s

(column)

Compression by 10 cm

280564 50.1 51.1 130 2.59 2.54

Extension by 5 mm

142324 50.1 51.1 135 2.69 2.64

6.13 Squirm failure mechanism:

While performing above experiments, following observations are made about

squirm phenomena in case of short column. The squirm failure for short column

bellow may be explained in to three stages.

1. Bulging of flanks: The pressure inside the bellow is gradually increases at

periodic intervals. The convolutions of bellow will remain in elastic limit up

to the pressure for which it is designed. However, when the pressure is

further increases, convolution flanks becomes inclined between their root

and crest part of each convolution. The convolutions are bulging from

flanks. At this stage the maximum axial force is developed at root and crest

part of convolutions. These two sections provide strength to the

convolutions of bellows.

Figure 6.14: Bulging of flanks

2. Deformation along plane: The further increase in internal pressure will

develop very high amount of force at root sections. This will create very

high membrane and bending stresses in the convolution flanks. Practically,

not all convolutions will have similar precise wall thickness and diameters,

hence in-plane deformations occurs in weak areas. This deformation

initiates in elastic region and may continue until plastic region. This


deformation will be non-uniform in convolutions only. This stage is in-plane

squirm.

Figure 6.15: In-plane deformation

3. Deformation along longitudinal axis: When further internal pressure is

increases in the bellow, gradually the whole structure becomes unstable

along its longitudinal axis. The end centers are slowly disturbs from its

coinciding axis. An individual convolution may come closer to each other at

one side and becomes spreader from opposite side. This stage is column

squirm.

Figure 6.16: Axial displacement

Above three stages of failure of bellows are snapped and shown below.

Initial condition of

convolutions

Convolutions planes are deformed within plane

Convolutions deformed

laterally - Column squirm Figure 6.17: Photographic images: In-plane stability test



1. Actual squirm failure occurs at minimum 2.5 times the designed critical

pressure.[4] This can be visualized by comparing values of actual critical

pressure and design critical pressure. Hence, this may be considered as

the factor of safety provided in the design procedure.

2. Short bellows having Lb/Db less than transition point factor; the in-plane

critical pressure is always less than column squirm critical pressure. This

observation are agreed and verified with the analytical approach of EJMA.

3. By experimental observation we may conclude that the short bellows

(Lb/Db<Cz), initially deformed by in-plane squirm and subsequently

deformed by column squirm.

4. Bellows may fail by column squirm, if number of convolutions and pitch of

bellows are selected on higher side. As, these two parameters are directly

proportional to the length of bellow. In addition, these bellows will be

amongst the long column bellows.


6.14 Dynamic Analysis:

Every individual metallic bellows are different in dimensions, and unique for the

applications. The natural frequency of expansion joints must be evaluated

analytically and designer should take care to avoid similar/near by natural

frequency of expansion joint and frequency of vibration, because of pumping

machinery in the piping. Overlapping of both frequencies will leads to resonant

condition and very heavy vibration amplitudes may be created.

A metallic bellow with following dimensions is selected for the analysis.

6.14.1 Geometric dimension of Flanges bellow:

Material: SS 304

Table 6.21: Geometric dimensions of a bellow

Db

(cm)

Dm

(cm)

t

(cm)

w

(cm)

q

(cm) N n

E

(N/cm2)

20 21.62 0.04 1.5 1.3 8 3 19728608

Initial spring rate, fi = 282.5 N/cm/convolution

Overall spring rate of bellow, Ksr = fi / N = 282.5 / 8 = 35.3 N/cm

Axial vibration, WKCf sr

nn

= 9.81 857.1

3.35 = 42 Hertz

The frequency of vibration can be measured with FFT analyzer. In this experiment

FFT analyzer (make: “Pruftechnique”, Germany) is used to measure natural

frequency. The vibrations are created with rubber coated hammer with manual

hammering on the expansion joint. Total three sets of readings are taken to check

the repeatability of the experiment. The readings are mentioned in table 6.22. The

objectives of an experiment are to measure natural frequency of vibration of

expansion joint. The detailed specifications of FFT analyzer is mentioned in

appendix E.


6.14.2 Experimental results: Table 6.22: Experimental results of FFT Analyzer

Reading Number First peak (Hz) Second Peak (Hz)

1 33 76

2 33 77

3 33 77

6.14.3 FEA Approach:

The metallic bellow which is tested for measurement of natural frequency earlier,

same is modeled in the ANSYS software and analyzed for dynamic analysis. The

model is shown in figure 6.17. The results are mentioned in table 6.23. The result

of natural frequency from FEA is shown in figure 6.18.

Figure 6.18: Axi-symmetry FEA model (Full view and close view)

6.14.4 FEA results:

Table 6.23: FEA Results

Reading Number First peak (Hz) Second Peak (Hz)

1 38 73


Figure 6.19 : FEA Results


1. Analytically natural frequency of expansion joint is 42 cycles/sec, actual

measurement with experiment shows that natural frequency of vibration

is 33 cycles/sec and FEA results shows frequency as 38 cycles/sec.

2. It should be noted that, there are three layers of bellow in the expansion

joint. Hence, the natural frequency vibration is the cumulative natural

frequency of all three layers. Since, the transducer is attached at flange

part. This is an example of parallel mode of frequency.

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