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Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 82

3. Metallic Bellows

The main element of an expansion joint, which consists of one or more

convolution, is called bellow. The performance of expansion joint is mainly

depends on the geometric features of bellow. Hence, type of raw material and its

properties, its geometric features, other influencing design factors, construction or

manufacturing method, and performance testing of bellows are necessary to

study. Study of theses parameters is helpful in achieving desirable performance of

expansion joints.

3.1 Construction of Bellows: There are mainly two types of bellows according to

manufacturing method. [20]

1. Formed bellows: The formed bellows are made from thin sheet metal. The

bellows are formed either hydraulically or mechanically, from a thin walled

tube. The tube contains longitudinal welds and exhibit significant flexibility as

the thickness is very less. Formed bellows are made in a single or multiple

plies according to requirement. The thickness of material is ranging from 0.20

to 2.5 mm and diameter of bellows from 20 mm to 3000mm. These bellows are

usually categorized according to convolution shape. Figure 3.1 shows formed

bellows and the initial thin wall tube of material.

Figure 3.1: Formed bellow


2. Fabricated bellows: Thin gauge diaphragms or discs are used in series and

joined by welding process. Fabricated bellows are made from heavier gauge

material than formed bellows. Hence fabricated bellows can withstand higher

amount of pressure. Figure 3.2 shows constructional arrangement of fabricated

bellows.

Figure 3.2: Fabricated bellow

Welded bellows can be fabricated from a greater variety of exotic metals and

alloys, whereas formed bellows are limited to alloys with good elongation. Welded

bellows are not fabricated from brass because of its fundamentally poor

weldability. Other advantages to welded bellows include compactness (higher

performance in a smaller package), ability to be compressed to solid height with

no damage, resistance to nicks and dents, and dramatically greater flexibility.

The welding of metal bellows is a microscopic welding process, typically

performed under laboratory conditions at high magnification.

The bellows convolutions are formed either hydraulically or mechanically, from a

thin walled tube. The forming method should be very precise so that material

thinning should be controlled, in order to maintain uniform thickness. The similar

size convolution shapes should be formed in a bellow.

3.2 Components of a bellow: The main components of bellows are convolutions,

crest, root and tangent part. The other important configurations of bellows are

pitch of convolutions, mean diameter of bellow, height of convolutions, convolution

depth, tangent part etc. Figure 3.3 shows various components of a bellow.


Figure 3.3: Components of a bellow

3.3 Geometry of a bellow: For the precise design of a bellow important

geometries should be defined. These are bellows mean diameter, height of

convolutions, pitch of convolutions, tangent length, collar length etc. These

parameters are shown in following figure 3.4.

Figure 3.4: Geometry of U shaped bellow

Db = Inside diameter of bellows, Do = Outside diameter of bellows

n = Number of plies, t = Thickness of material

w = Height of convolution, q = Pitch of convolutions

Lt = Tangent length of bellow, Lc = Collar length of bellows

N = Number of Convolutions, r = Radius of root & crest (U type)

COLLAR


3.4 Convolution shapes of bellows: Bellows can be made using different

shapes of convolutions. Performance behavior of bellows differs with reference to

each convolution shape and other parameters. Mostly U shape of convolutions is

preferred by designers because of its simplicity in design, manufacturing and also

permits more amount of deformation in axial direction. Other shapes are V type, S

shape, semi toroidal shape, toroidal shape, flat, stepped, sweep, ripple etc. Figure

3.5 shows various basic shapes of convolutions. As no standard machineries are

developed for forming of convolutions, simple hydraulic or mechanical press is

used in the industries. There is no standard dimensional sizes of convolutions are

determined, and customized approach is adopted for the design there are different

features amongst various manufacturers.

Figure 3.5: Various Convolution shapes of bellows 3.5 Bellows materials: Mainly cold rolled carbon sheets are used in the

manufacturing of bellows. Many times thin sheet of stainless steel or alloyed steel

are used for bellows to avoid corrosion. Formability is the main criteria for the

selection of material. Following are the material suggested as per the properties

required from ASM hand book. [B2] Classification of Cold rolled plain carbon steels

sheets are shown in table 3.1.


Table 3.1: Classification of Cold rolled plain carbon steels [B2]

Sr. No.

Material designation

Thickness (mm)

Width (mm)

Specification symbol / ASTM No.

1 Cold rolled sheet 0.35 – 2 50 – 300 A366, A619, A620,

A366M, A619 M, A 620 M

2 Cold rolled sheet > 0.35 > 300 A366, A619, A620,

A366M, A619 M, A 620 M

3 Cold rolled sheet < 5.839 610 - 1220 A506, A507

3.5.1 Mechanical properties of the material:

The relationship between formability and values of the strain hardening exponent,

n and the plastic strain ratio ‘r’ (determined in tensile testing) is important.

Plastic strain ratio (r) is the resistance of steel sheet to thinning during forming

operations. This is the ratio of true strain in the width direction (εw ) to the true

strain in the thickness direction (ε t ) of the plastically strained sheet metal.[B2]

Plastic strain ratio (r) = ε w / ε t (3.1)

This rate is related to the crystallographic orientation of low carbon steels. It can

be decided by standard tension test.

The strain hardening exponent (n) is the slope of the true stress – strain curve

when plotted on logarithmic co-ordinates. A significant portion of the curve is

nearly a straight line for many low carbon steels. The approximately value is 0.22.

Many times for manufacturing of bellows annealed sheets are used as raw

material.

Annealing is low temperature recrystallization annealing or process annealing can

be used to soften cold rolled low carbon steel. When done in batches process, this

type of annealing is known as box annealing.

---------------------------------------------------------------------------------------------------------------------- * B2 ASM Hand book (Formerly metal hand book), Volume 1; Properties and selection: iron, steels and high performance alloys; ASM International Hand Book Committee; USA; Seventh Print; December 2005.


Table 3.2 : Compositions of Cold rolled plain carbon steels[B2]

ASTM Specifications Type of material C Mn P S

A 611 CR, SQ Grades A, B, C, E 0.2 0.6 0.04 0.04

A 366 CR, SQ Commercial quality 0.15 0.6 0.035 0.04

A 619 CR, SQ Drawing quality 0.1 0.5 0.025 0.035

A 414

Pressure Vessel Grade A Grade B Grade C Grade D Grade E Grade F Grade G

0.15 0.22 0.25 0.25 0.27 0.31 0.31

0.9 0.9 0.9 1.2 1.2 1.2 1.35

0.035 0.035 0.035 0.035 0.035 0.035 0.035

0.04 0.04 0.04 0.04 0.04 0.04 0.04

CR = Cold Rolled, SQ = Structural Quality

Following commercial named materials are used for manufacturing metallic

bellows.

Table 3.3 : Bellow materials according to temperature range

Bellows material Temperature range 0F (ASME Sec. VII)

304 Stainless steel -300 to 750




Nickle 200 300 to 600

Monel 400 -250 to 900

Inconel 600 -250 to 1200

Inconel 625 -250 to 1200

Inconel 800 -250 to 1500

Incol 825 -250 to 800

304 Stainless Steel

Stainless steel 304 is an austenitic grade that can be severely deep drawn. This

property has resulted in 304 being the dominant grade used in applications like

sinks and saucepans.


304L Stainless Steel

Type 304L is the low carbon version of Stainless steel 304. It is used in heavy

gauge components for improved weldability. Some products such as plate and

pipe may be available as “dual certified” material that meets the criteria for both

304 and 304L. 304H Stainless Steel

304H, a high carbon content variant, is also available for use at high

temperatures. Property data given in this document is typical for flat rolled

products covered by ASTM A240/A240M. ASTM, EN or other standards may

cover products sold by ‘Aalco’. It is reasonable to expect specifications in these

standards to be similar but not necessarily identical to those given in this

datasheet.

Table 3.4: Composition of SS 304[W4]

Material C Mn Si P S Cr Ni N

S S 304 0.08 max. 2.0 0.75 0.045 0.03 18-20 10.5 0.1

SS 304L 0.03 max. 2.0 0.75 0.045 0.03 18-20 12 0.1

SS 304H 0.1 max. 2.0 0.75 0.045 0.03 18-20 10.5 -

Table 3.5: Mechanical Properties of Stainless steel sheets[W4]

Material Tensile strength (MPa)

Compression strength (MPa)

Proof stress 0.2% (MPa)

Elongation A 5 (%)

Hardness Rockwell

B S S 304 520 210 210 45 92

SS 304L 500 210 200 45 92

SS 304H 520 210 210 45 92

Table 3.6: Physical Properties of Stainless steel sheets – SS 304[W4]

Property Value

Density 8.00 g/cm3 Melting point 1400-1450°C Modulus of elasticity 193 000 MPa Thermal conductivity 16.2 W/m.K at 100°C Thermal expansion 17.2x10-6 /K at 100°C


Table 3.7: Physical Properties of Inconel sheets – inconel 600

Property Value

Density 8.47 g/cm3 Melting point 1355 - 1413°C Modulus of elasticity 207 000 MPa Thermal conductivity 14.9 W/m.K Thermal expansion 13.3 µm/mK

Inconel 600:

Inconel 600 is a nickel- chromium alloy with good oxidation resistance at high

temperatures and resistance to chloride ion stress corrosion cracking, corrosion

by high purity water, and caustic corrosion. It is used for furnace components, in

chemical and food processing, in nuclear engineering and for sparking electrodes.

Inconel 800:

A Ni-Cr-Fe alloy resists the high temperature oxidation. This alloy is a first choice

for an upgrade from the 300 series stainless steels when improved performance

or strength at temperature is required. For higher ASME Boiler and Pressure

Code design values, consider Alloy 800HT.

3.5.2 Properties of Inconel alloys: [W4]

1. Inconel alloys are oxidation and corrosion resistant materials well suited for

service in extreme environment.

2. When heated or at elevated temperature, inconel forms a thick, stable,

passivating oxide layer protecting the surface from further attack.

3. Inconel retains strength over a wide temperature range, attractive for high

temperature applications.

4. Inconel’s high temperature strength is developed by solid solution

strengthening or precipitation strengthening, depending on the alloy.

5. Inconel is difficult metal to shape and machine using traditional techniques

due to rapid work hardening.

6. Welding of inconel alloys is difficult due to cracking and microstructural

segregation of alloying elements in the heat affected zone. However some

alloys are designed for welding to overcome this problem.


3.5.3 General applications of inconel sheets:

Iconel sheets are often used in extreme environments. It is common in gas turbine

blades, seals, combustors, turbocharger rotors and seals, pressure vessels, heat

exchanger tubing, etc.

3.6 Manufacturing of bellows: Hydraulic forming or mechanical forming process

is used for forming convolutions. A welded cylinder is placed in the center of a

stack of split dies, which are machined to determine the final convolution shape.

Internal pressure and controlled axial compression is applied. A high pressure is

used in forming process, which thus imposes a leak test on the final bellows;

however, because this pressure is applied against external rings, the structural

strength and stability of the bellows are not proven in the forming process. Initially

sheet metal is welded in longitudinal direction. The weld efficiency is tested during

convolution forming process. If welding is not effective, then material will fail from

welding during forming process. Manufacturing methods are varying with different

manufacturers, as special purpose machinery is not developed for the forming of

convolutions.

3.7 Single or multi-ply material: [20] Bellows are made from thin sheet metal in

order to get higher flexibility. But these bellows can not withstand higher amount

of pressure. Hence to reduce the risk of sudden failure or complete failure,

multiple plies are used for high pressure applications. The inner ply is high

corrosion resistant material and outer ply is less costly higher strength material for

load resistance. Also if thick material is used, its fatigue life is reduced. So its

overall life is also reduced. For multi-ply, fatigue resistance is limited ply.

Depending on the wall thickness and convolution size, single wall thin bellows

may be limited by stress or stability to lower pressure application. To overcome

this limitation, multi-ply bellows can be made by telescoping two or more cylinders

and forming together. Multi-ply bellows may be advantageous for reducing the risk

of sudden and complete failure. Also, in case of multi-ply the inner ply highly

corrosion resistance material is used and as outer ply less costly high strength

material can be used. Here, the fatigue resistance is limited by the inner ply.

The multi-ply can be used in many applications. It is important to understand the

functional characteristics of each type of constructions.


A. Multi-ply construction with the same total thickness as a single ply

construction (tt=spt)

B. Multi-ply construction with the same thickness for each ply as a single ply

construction (tt=n x spt)

C. Multi-ply construction with greater thickness for each ply than for single ply

construction (tt > n x spt)

Table 3.8 : Behavior of multi-ply bellows

Parameter Multi-ply construction characteristics

Design feature tt = spt tt = n x spt tt > n x spt

Circumferential stress same decreases Decreases

Longitudinal bending

stress

increases decreases Decreases

Fatigue life increases little change Decreases

3.8 Reinforcement of bellows: [20] Sometimes reinforcing or equalizing rings are

added while the bellow material is very thin. Reinforcing rings resist any distortion

of the convolution root and are easily fitted to bellows that are formed

hydraulically. Equalizing rings can be of cast or fabricated construction, generally

in two halves bolted together. These rings also prevent convolution root distortion

but additionally limit the compressive axial deflection taken by each element. Both

types of rings are claimed to improve fatigue life. Figure 3.6 shows the

arrangement of equalizing rings in thin wall bellows.

Figure 3.6: Reinforcing rings and equalizing rings details


3.9 Internal Sleeve: [20] Bellows can be sleeved for various reasons. First is to

reduce turbulence and thus pressure drop, to minimize erosion on the walls and to

restrict entry of foreign material. Sleeves should be designed with the minimum

practical clearance to restrict entry of foreign material. There should be sufficient

overlap at the free end to ensure that with all possible movements, especially if

lateral movement is involved, there is no chance of the sleeve end fouling the

convolutions.

It is wrong to assume that a sleeve can completely prevent deposition of solid

material in the convolutions, since back eddies can easily result in sedimentation

behind the sleeve. In fact, a sleeve can frequently help to trap solid material

against the bellows, where it might otherwise have been carried away in the

turbulent flow. The most practical way to prevent solid from getting into the

bellows/sleeve space is by use of purge medium continuously supplied to this

space. The draining of this space of any corrosive products must be considered.

3.10 Criteria affecting Bellows Design: The designer is having freedom in

deciding the geometric parameters of bellows, but he has to take care about the

cumulative effect of these parameters on the various performance criteria. They

are internal pressure capacity, squirm failure, stability of bellow, fatigue life etc.

Each criterion affects on the performance of expansion joint. They are elaborated

as following.

3.10.1 Internal Pressure Capacity: Excessive hoop stress in the straight

cylindrical end tangents of a bellow will cause circumferential yielding. This stress

is calculated by a modification of the Barlow’s equation. For un-reinforced bellows

straight tangents can be reinforced by collars.

Excessive hoop stress in the convoluted section of the bellows can produce

circumferential yielding and possible rupture. As in any cylindrical shell, this stress

is inversely proportional to the cross sectional areas and material properties.

Excessive longitudinal pressure stress in the convoluted section of a U shaped

bellows will produce bulging of the side wall. Any gross change in the convolution

shape will decrease the space between convolutions, and the ability of the bellows

to absorb movement. Such change in shape will also affect the fatigue life.


Deflection stresses are produced in the convoluted section due to deflection.

Typical stress range values are very high. These values are not true stresses,

since they exceed the elastic limit of the material. They are useful for the

prediction of fatigue life.

3.10.2 Fatigue life Expectancy: The fatigue life expectancy can be defined as

the total number of complete cycles which can be expected from the expansion

joint based on data tabulated from tests performed at room temperature under

simulated conditions. A cycle is defined as one complete movement from initial

positioning the piping system to the operating position and back to initial position.

Fatigue life is dependent upon the maximum stress range which the bellows is

subjected, the maximum stress amplitude being the far less significant factor. The

fatigue life expectancy of an expansion joint is affected by various factors such as

operating pressure, operating temperature, material of bellows, movement per

convolution, the convolution pitch, the depth and shape of the convolutions and

bellows heat treatment. Any change in these factors will result in a change of

fatigue life of the expansion joint.

The fatigue life expectancy can be evaluated from the total number of complete

cycles which can be expected from the expansion joint based on data tabulated

from tests performed at room temperature under simulated conditions.

The fatigue life expectancy of an expansion joint is affected by various factors

such as operating pressure, operating temperature, material of bellows,

movement per convolution, the convolution pitch, the depth and shape of the

convolutions and bellows heat treatment. Any change in these factors will result in

a change of fatigue life of the expansion joint. The work hardening of austenitic

stainless steel, induced during the forming of convolutions, generally improves the

fatigue life of an expansion joint.

The fatigue life of a bellows is a function of the sum of the meridional pressure

stress range and the total meridional deflection stress range. The number of

cycles to failure may be evaluated using total stress range (St) versus number of

cycles (Nc) to failure from actual fatigue tests of a series of bellows of similar

materials at room temperature. In actual practice bellows are subjected to

varieties of stress cycles during its operating life. Hence, EJMA suggests Miner’s


hypothesis for predicting the effect of cumulative fatigue based on different stress

cycles. The relation is mentioned as follows.

Cumulative usage factor, U = U1 + U2 + U3 + U4 + …. + Un (3.2)

.....4

4

3

3

2

2

1

1 Nn

Nn

Nn

NnU (3.3)

This factor should not exceed 1.

Where,

Stress cycle = St1, St2, St3 …Variations in stresses (absolute values)

n1, n2, n3 ... = Number of each stress cycles

N1, N2, N3 …. = Maximum number of stress cycles which would be

allowable if this type of cycle were acting alone.

3.10.3 Bellows Stability: Excessive internal pressure may cause a bellow to

become unstable and squirm. Squirm is determining parameter to bellows

performance in that it can greatly reduce both fatigue life and pressure capacity.

This phenomenon is similar to buckling of long columns. The buckling of bellows

is called squirm. Squirm harmful to the performance of bellows as it can reduce

both pressure capacity and fatigue life. The two most common type of There are

two basic types of squirm, column squirm and in-plane squirm.

Figure 3. 7 : Column Squirm Figure 3.8 : In-plane squirm

Column squirm is defined as a gross lateral shift of the middle section of the

bellow. It results in a curvature of the bellows centerline as shown in figure 3.7.

This type of squirm is associated with length to diameter ratio. According to this

ratio, bellows can be categorized in long or short columns. Failure of column is


depends on the kind of column. Squirm is similar to buckling of column under

compressive load. Buckling failure consists of an elastic and in-elastic region.

Since bellows are made from thin sheet metal, deformation of bellows can be in

elastic and plastic mode. Hence determination of stresses is much more difficult.

Figure 3.9: Force vs. Deflection curve

Figure 3.9 shows a graph which indicates critical column squirm pressure for

series of bellows having same diameter, thickness and convolution shapes. As the

number of convolution is increases, the curve passes through a transition from

inelastic to elastic behavior. The other condition which is related to column squirm

is end condition. Usually expansion joint is rigidly supported (fixed) at both the

ends. The equations suggested by EJMA for buckling pressure to avoid column

squirm is

Buckling Pressure = Psc = qN

fC iu2

34.0 when zb

b CDL

(3.4)

Buckling Pressure = Psc =

bDz

b

b

yc

CL

qDSA 73.0

187.0

when zb

b CDL

(3.5)

In-plane squirm is defined as deflection occurred in individual convolutions,

parallel to the surface of bellow materials. It looks like warping of perpendicular

faces of convolutions. This deflection is associated with high meridional bending

stress and the formation of plastic hinges at root and crest of convolutions. It is

more likely to occur in small length to diameter ratio bellows. For the estimation of

critical pressure to avoid in-plane squirm EJMA has given following relation

Critical pressure, Psi = 2

51.0K

S i (3.6)


Squirm failure also depends on end conditions of the bellows. Normally bellows

ends are welded to collars and they are further welded to flanges of pipes.

Generally both ends rigidly fixed condition is considered. This may vary for other

application. Bellows when subjected to internal pressure is acted upon by an

unbalance pressure force or couple which, is sufficiently large, could result in

distortion of the bellows. The magnitude of the unbalance pressure force or couple

is proportional to the internal pressure and the displacement of the convolutions, a

reduction in either of these values will improve the stability of expansion joint.

3.10.4 Spring Rate of bellows: The force required to deflect a bellows axially is a

function of the dimensions of the bellows and the material from which it is made.

The flexibility of bellows is measured by spring rate of bellows. This is also helpful

for expected movement of piping for the design purpose. The curve of force

versus deflection for most bellows indicates motion extending into the plastic

range. Initially the bellow is deflected through elastic range. But as bellows

continuous and extends into plastic range, the force versus 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 to bellows to its initial position, a restoring force must be

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

Figure 3.10: General curve of Bellows Force vs. Deflection

The use of the initial elastic spring rate in place of the working spring rate for a

bellows whose deflection extends into the plastic range predicts forces which can


be considerably higher than actual. Line B, drawn from the origin to the point of

maximum force and deflection, is used as the bellows working spring rate, fw. But

this has a disadvantage of underestimating the actual force over the full range.

Line C drawn from the point of maximum force and deflection to the value of the

restoring force required to return the bellows to zero deflection, becomes line C

when transferred to the origin. A working spring rate based on line C can be used.

This reduces the discrepancy between the indicated and true values although the

difference can still be significant. A relation to estimate the bellows theoretical

axial elastic spring rate suggested by EJMA is as follows.

Bellows theoretical axial elastic spring rate = f

pbm

CwntED

fiu 3

3

7.1 (3.7)

3.10.5 Cold Springing of bellows: Actually cold springing is defined as the pre-

straining of the elements of a piping system at the time of installation, so that the

thermal stresses in the piping in the operating positions are appreciably reduced.

Foe expansion joints, cold springing is defined as the lateral or angular offset of

the ends of an expansion joint when installed and should not be considered as

axial pre-compressing or pre-extending. Where expansion joint is used to relieve

loading on sensitive equipment, or anchor structures are limited to extremely small

loads, cold springing the expansion joint at installation will effect a reduction in the

maximum deflection force value of as much as 50%. In other cases, 100% cold

spring may be used to provide minimum lateral deflection forces at the operating

position.

3.10.6 Vibration in bellows: [20]

The metallic bellow component will have its own natural frequency. Metallic

bellows are used in the applications where there are low amplitudes and high

frequencies. Expansion joints should not be used to absorb vibrations created by

reciprocating machines or pumps. There will be two kinds of vibrations. The

vibration will depend on number of convolutions of bellows. Since both ends of

bellows will be rigidly connected with pipe ends, vibration area will be between

first and last convolution of bellows. The vibration will develop in axial direction

and lateral direction. This natural frequency of metallic bellows can be measured

using following mathematical relation.


Axial vibration, WKCf sr

nn Hertz (3.8)

Where, Ksr = Overall bellow spring rate, (kg/cm)

W = weight of bellows including reinforcement, flanges, liquid, kg.

Cn = Constant used for calculation of frequencies.

Where, n = 1, 2, 3, 4, 5….

Number of convolution C1 (first mode)

1 8.84

2 9.51

3, 4 9.75

5, 6, 7, 8, 9, 10 etc. 9.81

Lateral vibrations: Vibration induced in the perpendicular direction of bellows axis

is called lateral vibrations. It is also known as beam mode of vibration. It can be

calculated using following relationship.

Lateral vibrations, WK

LDCf sr

b

mnn Hertz (3.9)

Where, C1 = 24.8 (For first mode)

The predicted amplitude and frequency of external mechanical vibrations to be

imposed on the bellows, such as caused by reciprocating or pulsating machinery

(kind of pump) shall be specified. The expansion joint must be designed to avoid

the resonant vibration of the bellows to prevent the possibility of sudden fatigue

failure. Many times layout and anchor position, alteration may be done in order to

control the vibration amplitudes.

3.11 Design approach: Every individual application of bellows is unique

considering type of internal fluid, its temperature variations, its pressure, pipe

diameter, fluctuations in pressure, corrosion, pipe length and many others. Hence

expansion joints design and manufacturing prefers customized approach. For a

specific application it is designed, than individually manufactured and non

destructive testing is carried out. Here high degree of understanding is required

between manufacturer and user in order to assure a safe and reliable installation.


The user is asked to give basic technical information about the requirements,

pressure, temperature, maximum possible axial movement, maximum lateral

movement etc. Then according to this requirement, the manufacturer suggests the

technical design of expansion joint, which includes the dimensions and its

technical capabilities. If the user is satisfied with this design, then only commercial

aspect or rates are quoted. This approach is suggested by Expansion Joint

Manufacturing Association.

3.12 Design procedure: The design of a bellows is complex in that it involves an

evaluation of pressure capacity, stress due to deflection, fatigue life, spring forces

and column instability. The determination of a suitable design is further

complicated by the numerous variables involved such as diameter, material

thickness, pitch, height, number of plies, method of reinforcement, manufacturing

technique, material type and heat treatment. In many cases, the design for a

particular application will involve a compromise of conflicting requirements.

EJMA has developed theoretical stress analysis of bellows. The analysis is based

on certain assumptions. These assumptions are idealized bellow configuration, a

uniform thickness, a homogeneous and isotropic material and elastic behavior.

These assumptions are not precisely correct for most applications. A bellows

usually operates in the elastic and plastic stress region and cold work, due to

forming, alters the mechanical properties of the material.

Few investigators have employed computerized analysis technique to more

accurately consider the effect of thickness and shape variations as well as

plasticity. This procedure is obviously more complex than a simple elastic analysis

and yet does not fully solve the design problem in the absence of experimental

verification. Also a bellow design should be based on the actual bellows metal

temperature expected during operation.

Design of bellows includes evaluation of major stresses in the circumferential

membrane and longitudinal membrane and bending stress with reference to

pressure and deflection. It also requires estimating spring forces and fatiguing life

of bellows. The detailed theoretical design is elaborated at later stage.

3.13 Testing of bellows:[20] Bellows are correlated with actual test results to

demonstrate predictability of rupture pressure, meridional yielding, squirm and


cycle life for a consistent series of bellows of same basic design. Usually, five

meridional yield rupture tests on bellows of varying sizes with not less than three

convolutions are required. 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 needs to be prepared for experimental verification or testing

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

joints. The testing is necessary to for the verification of the design procedure.

After manufacturing bellows are necessary to test or specific inspection procedure

is decided and which is followed. This testing is required to assure the user about

the satisfactory design and performance verification.

Usually following non-destructive examinations are recommended for the

inspection and testing after manufacturing.

1. Radiographic examination

2. Liquid penetration examination

3. Flourscent penetrant examination

4. Magnetic particle examination

5. Ultrasonic examination

6. Halogen leak examination

7. Mass Spectrometer examination

8. Air jet leak examination

Following non-destructive tests are also recommended depending upon the

application.

1. Pressure Testing

(a) Hydrostatic test

(b) Pneumatic test


Following destructive tests are also recommended depending upon the

application.

1. Squirm testing

2. Meridional yield rupture testing

3. Fatigue life testing

3.14 Failure of Bellows:

Bellows are loaded with combined tensile and compressive loadings during its

service life. Bellows may fail due following reasons during its application.

1. Stress corrosion: Stress-corrosion which is evidenced by cracking of the

material as the result of a combination of stress and corrosive environment.

This is occurring because of chlorides of austenitic stainless steel.

Corrosion can significantly reduce the service life of expansion joints.

2. Fatigue failure: Bellows undergoes low cycle fatigue during its service life.

Bellows may fail due to fatigue because of its randomly occurring (different

stress ranges) thermal expansion and compression movements. The

fatigue life may be estimated based on its expected stresses due to

deflection. The bellows should be designed for finite number of life cycles.

3. Carbide Precipitation: Bellow material becomes unstable at elevated

temperature and due to vibrations occurring in the bellows. The designer

has to insure that vibrations loads will not be detrimental to the function of

the bellows. Vibrations should be controlled by providing external damping

devices or system mass adjustment.

4. Squirm Failure: Excessive internal pressure may cause a bellow to

become unstable and squirm. The buckling of bellows is called squirm. This

phenomenon is similar to buckling of long columns. Squirm reduces

pressure capacity and fatigue life. The two most common type of There are

two basic types of squirm, column squirm and in-plane squirm. This failure

can be avoided by suitable geometric parameters pitch, height of

convolution and material properties.

5. Rupture failure: Bellows may yield (shear cracks) due to excessive

internal pressure, is called rupture failure. This failure is normally


successive failure after squirm failure. This failure can be avoided by over

pressurization and material properties.

3.15 General Applications of Bellows:

Convoluted (formed) bellows are used in a large number of industrial applications

other than piping. Some applications are mentioned below.

1. Load cells: A load cell deforms if a certain load in the form of a pressure or a

strain is imposed on it. This deformation is then detected by a strain gauge

through which a low voltage direct current is flowing. The change in voltage is

detected and made visible on a control panel. To protect this strain gauge from

outside damages or weather influences a bellow is mounted over the gauge to

protect it from outside influences.

2. Vacuum interrupters: For the switching of very high voltages in transformer

stations sparks should be avoided. To prevent any danger that the surrounding

atmosphere will explode, oxygen has to be removed in the area where the sparks

occur. This can be done by sealing the spark area completely. Bellows are used

to seal this confined area and the inside of the bellow is vacuumized or an inert

gas is filled into the bellow.

3. Mechanical Seals: These are mostly used to close the inside of a pump from

the outside world to prevent leakage. For that purpose, a mechanical seal is

mounted on the pump shaft. As the pump shaft is turning, there has to a rotating

sealing element consisting of a stationary and a rotating ring. To enforce sufficient

pressure on the two rings one is fitted with a spring. This spring can also have the

form of a diaphragm (welded) bellow.

4. Pressure gauges: If the pressure of aggressive fluids or gases has to be

measured, the gauge has to be isolated from the flow. For critical applications a

diaphragm sealing is used instead of a bourdon tube in the gauge. This gives

more security that the aggressive media cannot leak. The diaphragm is a self

contained sensor, transmitting the displacement to the measuring device.

5. Sensors: In this application diaphragm or convoluted bellows are completely

sealed and filled with a certain gas. Two electrical poles are penetrating the inside

of the bellow. By variation the current of those two poles the temperature inside


the bellow can be regulated. The expansion or contraction of the bellow is used as

an actuator to control a certain movement.

6. Valve Sealing: A bellow is used between the housing and the rising stem to

seal the inside completely from the outside world. In Europe this is of particular

importance as regulations such as TA Luft prohibit any leakage.

7. Couplings for stepper motors and servomotors: The flexible part, capable of

compensating for misalignment is made by a bellow. It ensures that there is no

angular positioning difference between the two coupling halves. This is essential if

the positioning accuracy should be extremely precise.

8. Exhaust pipe expansion joints: Running engines cause self vibration. To

compensate for those movements and temperature differences resulting in

thermal expansion, bellows are used to connect the exhaust gas pipes to the

funnel.

Metal bellows are also used other products and marketplaces, including medical

applications like implantable drug pumps, to industrial actuators, to aerospace

applications such as altitude sensors and fluid management devices

(accumulators, surge arresters, volume compensators, and fluid storage). Metal

bellows are also found in space applications, providing reservoirs with potable

water as well as accumulators to collect wastewater.

3.16 Characteristics of Metallic Bellows used in Instrumentations:

1. Absolute leak tightness – zero permeation to mass spectrometer

sensitivity

2. High reliability

3. Compatibility with many environments

4. High humidity

5. Salt spray

6. Corrosive fluids

7. Liquid or gas applications

8. Wide temperature extremes


9. Long predictable life at operating conditions

10. No degradation in performance after long storage periods

11. Maintenance free service

12. Contaminant free operation

3.17 Conventional Design of Bellows: (As suggested by EJMA)

Design for strength is an essential criterion for any mechanical system. The

objective of this design is to avoid failure at minimum cross section areas for the

required loading conditions. Design of bellows, since they are made from thin

sheets, the design for thin cylinders methodology is useful. For thin cylindrical

objects with some distinguish geometric features can be designed with reference

to Barlow’s equation.

3.17.1 Design methodology single expansion joint :

Data: Design pressure (P) = 5 kg/cm2 = 50 N/cm2

Design temperature (T) = 500C

Modulus of elasticity at room temperature (Eb) = 19897350 N/cm2

Modulus of Elasticity at 500 C (Eb) = 19728610 N/cm2

Yield strength of material (Sy)= 20300 N/cm2

Allowable stress (Sab) = 12730 N/cm2

Thickness of material (t) = 0.08 cm

Number of plies (n) = 1 no.

Number of convolutions (N) = 15 nos.

Inside diameter (Db) = 40.60 cm

Height of convolutions (h) = 2.30 cm

Pitch of convolutions (p) = 2.26 cm

Tangent length (Lt) = 2.5 cm

Collar length (Lc) = 2.5 cm

Collar thickness (tc) = 0.16 cm


Length of a below (Lb) = (N x p) = 33.90 cm (3.10)

Mean diameter of bellow (Dm) = Db + h + ( n x t ) = 42.98 cm (3.11)

Collar diameter (Dc) = Db + (2 x tc) = 40.92 cm (3.12)

Thickness after thinning (tp) = t DmDb = 0.078 cm (3.13)

Cross section area of a convolution (Ac) =(0.571 x pitch) + (2 x h) x tp x n (3.14)

= 0.459 cm2

Stiffening factor (k) = txDb

Lt5.1

= 0.925

Values taken from Graph, Cp = 0.65, Cd = 1.75, Cf = 1.70

3.17.2 Design calculations:[20]

Bellows tangent circumferential membrane stress (S1)

S1 = DcLcEcktcLtDbLtEbtn

kEbLttnDbP

2

2

= 4110 N/cm2 (3.15)

Collar circumferential membrane stress (S11)

S11 = DcLcEcktctnDbLtEbtnkEcLtDcP

2

2

= 4160 N/cm2 (3.16)

Bellows circumferential membrane stress (S2)

S2 =

qwKr

tpnDmP

/2571.02= 7137 N/cm2 (3.17)

Here, S1 & S2 < Sab * Cw; (3.18)

Where, Cw = Longitudinal weld efficiency factor, may be taken as 1.

Bellows meridional membrane stress due to pressure (S3)

S3 = tpnwP

2= 740 N/cm2 (3.19)

Bellows meridional bending stress due to pressure (S4)


S4 = Cptpw

nP

2

2

= 14220 N/cm2 (3.20)

Here, (S3 + S4) should be < Sab * Cm (3.21)

Where, Cm = Material strength factor,

Cm = 1.5 for annealed condition and 3 for as formed condition.

Bellows meridional membrane stress due to deflection (S5)

S5 = Cfw

etpEb3

2

2 = 0 (As e = 0) (3.22)

Bellows meridional bending stress due to deflection (S6)

S6 = Cdw

etpEb23

5 = 0 (As e = 0) (3.23)

For bellow to be designed for 5 cm axial motion.

Total axial motion x = 5 cm

Axial motion per convolution, ex = 155

NX = 0.34 cm (3.24)

Axial force = Fa = Spring rate x movement / convolution (3.25)

= 33070 x 0.334 = 11045 N.

Axial spring rate = 5

11045

deflectionAxialforceAxial = 2210 N/cm (3.26)


S5 = Cfw

etpEb3

2

2=

70.13.2234.0078.019728610

3

2

xxxx = 987 N/cm2


S6 = Cdw

etpEb23

5 = 75.13.23

34.0078.01972861052 xx

xxx = 94194 N/cm2

For bellow to be designed for 5 cm axial motion and 2 cm lateral motion.

Axial motion, X = 5 cm

Lateral motion, Y = 2 cm


Axial motion per convolution , ex = 155

NX = 0.34 m

Lateral motion/convolution, ey= XLbN

YDm

3 = 59.3315298.423

xx =0.454 cm/con. (3.27)

Vertical lateral force = Vl = XLbeyDmfw

2 = 59.332

454.098.4233080

xx = 8296 N. (3.28)

Equivalent movement, ee = ey + et + [ex] = 0.454+0+0.34=0.794 cm (3.29)

Equivalent movement, ec = ey + et - [ex] = 0.454+0–0.34=0.114 cm (3.30)


S5 = Cfw

etpEb3

2

2=

70.13.22794.0078.019728610

3

2

xxxx = 2303 N/cm2


S6 = Cdw

etpEb23

5 = 75.13.23

794.0078.01972861052 xx

xxx = 219972 N/cm2

3.17.3 Thermal Considerations in design:

Metallic bellow movement occurs because of temperature and pressure

variations in the piping. The bellow deformation depends on piping layout

and position of anchors.

Figure 3.11: Lay out of piping

Figure 3.11 shows one layout of piping with a bellow. Bellow will be

fluctuating along X direction (towards anchor B) as the flow of fluid is in this

direction, but the expansion effect will be developed due to region between


bellow and anchor B. The temperature of fluid will increase the temperature

of pipe materials as well as bellow. Due thermal expansion of the pipe

material, its length will be increased. Hence, thermal aspect is important in

the design of bellow. The pipe and bellow materials approach the

temperature equivalent to fluid temperature. The elastic modulus of the

bellow material is decreases at elevated temperature. Hence, an elastic

modulus should be considered at particular temperature during the design.

For the higher temperature applications, as the elastic modulus reduces, its

yield stress reduces, and finally the permissible stress limit is reduces.

Hence, the designer should control the developed stresses corresponding

to permissible stresses at designed temperature.

3.18 Estimation of Stresses as per EJMA:

A program is prepared in excel worksheet using EJMA relations to evaluate the

stresses, transition parameter, spring rate, critical pressure of bellow considering

column buckling and in-plane squirm etc. First part is data sheet, all data related

to bellows and its requirements is required to feed as input. Part 2 is an

evaluation.


Part 1: Data input:

Estimation of Stresses developed in Matallic Bellows as per EJMA

Pressure, P 50 N/cm2 q/2w 0.491 Inside Dia, Db 40.6 cm q/2.2(Dm*tp)1/2 0.562 No. of Ply, n 1 nos.

Yield stress, Ys 20310 N/cm2 Read from Graphs Thickness, t 0.08 cm Cd 1.75

Reduced thickness, tp 0.078 cm Cp 0.65 Height of convolution, w 2.3 cm Cf 1.7 Elasticity (bellow), Eb 19728608 N/cm2

Tangent length, Lt 2.5 cm Material Properties data Collar Thickness, tc 0.16 cm Sy 20310 Elasticity (Collar), Ec 19728608 N/cm2 Sab 12730 Length of collar, Lc 2.5 m Eb 19728608 Stiffening Factor, k 0.925

Collar. Dia, Dc 40.92 cm Mean Dia., Dm 42.98 cm

Pitch, q 2.26 cm No. of Convolution, N 15 Area of convo. , Ac 0.4580037 cm2 Axial deflection, ex 0.667 cm

Lateral deflection, ey 0 m Factor, Kr 1.295

Part 2: Calculation of Stresses S1 S2

Num 3.77406E+12 Num1 2149 D1 160511954.7 Num2 0.496905 D2 298629563.4 Den 0.155507

Den = 2(D1+D2) 918283036.2

S1=Num/Den 4109.91 S2 =

Num/Den 6866.89

S3 S4 Num 115 Num1 25 Den 0.155506947 Num2 875.0162

S3 = Num/Den 739.52 Cp 0.65 S4 14219.01

S5 S6 Num 79553.97 Num 5115783 Den 41.3678 Den 27.7725 S5 1923.089 S6 184203.2

S1 + S2 10976.80 Sab = 12738

S3 + S4 14958.53 Sab = 38214

(cold formed bellow) S5 + S6 186126.26 Total stress, St 196597.23


3.19 Design of Components of Expansion Joints:

The basic unit of every expansion joint is the bellows. By adding additional

components expansion joints of increasing complexity and capability are created

which are suitable for wide range of applications. These components are limit

rods, lugs, hinge plates, clevis plates, flanges, collar, cover, etc. Fundamental

design rules should refer to design these components. For a specific application,

these component logical design methodology is developed as follows.

Material = Carbon Steel

Permissible tensile stress = 8500 N/cm2

Permissible shear stress = 0.8 x 8500 = 6800 N/cm2

Permissible average shear stress = 0.6 x 8500 = 5100 N/cm2

Permissible crushing stress = 1.5 x 8500 N/cm2

Inside Pressure = 30 N/cm2

Mean diameter of Bellow = 32.4 cm.

Thrust force = Pressure x Area (3.31)

= 30 x 4 [32.4]2 = 24740 N

3.19.1 Tie rod or Limit rod:

Limit rods are used to limit the maximum movements of expansion joint as per

design. Bellow is not supposed to take up the additional expansion or contraction

movement.

Cross section of tie rod = circular

Type of Loading = Tension or compression

Maximum tensile stress = rodstieofnoxdx

loadThrust.785.0 2 (3.32)

Diameter of tie rod = 2785.08500

24740xx

= 1.36 cm. = 15 mm


3.19.2 Lugs:

Lugs are provided for the support of tie rods. Normally square cross section plates

are used here. Basically it acts as a base of the tie rod assembly.

Cross section of lugs = rectangle

Type of loading = Membrane and bending

Number of lugs = 4

Height of lugs, distance from outer diameter = 10 cm.

Bending moment = 24740 x 10 = 247400 N-cm.

Section modulus = (b x t2 / 6)

Cross section (b x t2) = 485006247400

xx = 45 cm.

If we take width =15 cm, thickness of plate should be 3 = 1.71 cm = 17.1 mm

3.19.3 Hinge Plates:

Hinge assembly is provided to get lateral movement of expansion joints.

Rectangle cross section is used for hinge plates. Plates are joined by rivets at the

middle of bellow. Figure 3.11 shows schematic arrangement.

Number of hinges = 2

Hinge plates (2 nos.) Clevis plate (1 no.)

Figure 3.12 : Hinge plate assembly

Thrust load = 24740 N

Cross section b x t = 28500

24740x

= 1.5 cm2

If width 3 cm is selected, thickness required is 0.5 cm.


3.19.4 Design of Pin

Pins are made from structural steel and its function is permit lateral and angular

motion to ends of the pipe.

Cross section of Pin = circular

Type of loading = double shear

Pin diameter = 26800785.02

24740xxx

= 1.25 cms. = 12.5 mm

3.19.5 Clevis Plates:

It is also a part of hinge assembly. Cross section dimensions will be similar to

hinge plates, but crushing failure will be required to check.

Crushing stress = platesofnoxtxX

loadThrust (3.33)

Distance x = 1275025.0

24740xx

= 1.96 = 2 cms. = 20 mm

Figure 3.13 : Clevis plate

Figure 3.12 shows schematic arrangement of clevis plate, designed for distance x.

3.19.6 Hinge Support Plates:

Hinge plates are fixed at the top of support plates. The plate is under tension as

well as bending moment. Figure 3.14 shows schematic arrangement of hinge

support plates.

x


Figure 3.14 : Hinge support plates A and B

Bending moment = Thrust force x height of plate (h)

24740 x 10 cms = 247400 N-cm.

Section modulus = (b x t2 / 6)

Cross Section (b x t2) = 4212750

6247400xx

x (considering 2 plates opposite sides)

= 15 cm2 We can take width = 15 cm and thickness = 1 cm.

3.19.7 Gimbal ring:

Gimbal ring is floated over the bellow with the support of four pins. Gimbal plates

may be square loop or circular section.

Cross section : Rectangular plate (b x t)

Loading: Tension plus bending

Cross section of gimbal plate = (b x t2 /6 )= pinsofnoxstressePermissibl

loadThrust.

= 48500

61024740x

xx = 45 cm2 .

We can take width as 30 cms, thickness = 1.25 cms = 12.5 mm

3.19.8 Design of Pin:

Type of cross section = Circular

Type of loading = double shear

Pin diameter = 46800785.02

24740xxx

= 0.76 cms. = 8 mm

A B h

b


Check for crushing:

Crushing stress = platesofnoxtxX

loadThrust

Distance x = 1275045.0

24740xx

= 0.97 = 1 cm. = 10 mm

3.19.9 Design of Pantograph linkages:

Figure 3.15: Links arrangements for hinged expansion joint

Figure 3.15 shows the arrangement of pantograph linkages in the hinge type

expansion joints. The type of loading is axial due to thrust force. In one bellow,

there will be four linkages are joined by pins.

Thrust force = 25000 N.

Maximum load on the each linkage will depend on the angular position, which is

achieved after expansion effects. Assuming that the angle is 450 as shown in

figure 3.16.

Figure 3.16: Load distribution in the links

Sin 450 = BC / 20

Dimension BC = Sin 450 x 20 = 14.142 cm.

Fa

Fa450

Actual link 20 cm

X cm

B

C A


Force on link AB = 25000 x sin 450 = 17678 N

Load on each link = 17678 / 4 = 4420 N

Assuming factor of safety as 4

Critical load = 4420 x 4 = 17678 N

Taking thickness of link = t and width of link = b; and assuming b = 5 t

Cross section area of the link = b x t = 5 t x t = 5t2

Moment of inertia of cross section of link = (1/12) (b) (t3) = (1/12) (t) (5t) 3

= 10.416 (t4)

Radius of gyration; k = AI = 2

4

5416.10t

t = 1.44 t (3.34)

Checking the cross section for buckling,

Considering both ends hinged, L = l = 200 mm

Using Rankine’s relation

Critical load = 2

1

kla

Af c = 2

2

44.1200

750011

5100

t

tx (3.35)

17678 = 2

2

57.21

500

t

t

36 = 57.22

4

tt

t4 – 36 t2 – 92.52 = 0

t2 =

252.9243636 2

(Taking positive sign)

hence; t = 5.76 = 10 mm; b = 5 t = 50 mm

Considering perpendicular direction

I = (1/12) (b) (t3) = (1/12) (5 t) (t3) = 0.416 t4

Cross section area = t x b = 5 t2


Radius of gyration, k = AI = 2

4

5416.0

tt = 0.29 t

Considering both ends fixed, L = l/2 = (200/2) = 100 mm

Critical load = 2

1

kla

Af c = 2

2

29.0100

750011

5100

t

tx

17678 = 2

2

85.151

500

t

t

36 = 85.152

4

tt

t4 – 36 t2 – 570.6 = 0

t2 =

26.57043636 2

(Taking positive sign)

hence; t = 5.80 = 10 mm; b =5 t = 50 mm

Considering both axis the dimensions: width = 50 mm and thickness = 10 mm.

These are safe in buckling from both axes.

Note:

The designer has to keep in mind that the bellows are used for expansion and

contraction of length of pipes. No extra load or force should be transmitted on it.

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