STRESS AND BURST PRESSURE DETERMINATION OF SHRINK FITTED COMPOUND CYLINDRICAL PRESSURE VESSEL

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STRESS AND BURST PRESSURE DETERMINATION

OF SHRINK FITTED COMPOUND CYLINDRICAL

PRESSURE VESSEL

Thesis is submitted in thepartial fulfillment of therequirement for thedegreeof

MASTER OF MECHANICAL ENGINEERING

By :

HARERAM LOHAR

Examination Roll No. : M4MEC13-24

Registration No. : 117131 of 2011-2012

Department of Mechanical Engineering

Jadavpur University

Under the Guidance of :

DR. SUSENJIT SARKAR

&

DR. SAMAR CHANDRA MONDAL

Faculty of Engineering and Technology

Department of Mechanical Engineering

Jadavpur University

Kolkata – 700 032

May, 2013



ii

FACULTY OF ENGINEERING AND TECHNOLOGY

DEPARTMENT OF MECHANICAL ENGINEERING

JADAVPUR UNIVERSITY

KOLKATA – 700 032

DECLARATION OF ORIGINALITY AND

COMPLIANCE OF ACADEMIC ETHICS

I hereby declare that this thesis contains literature survey and original research work by the

undersigned candidate, as part of his Master of Mechanical Engineering (MachineDesign)

studies.

All information in this document havebeen obtained and presented in accordancewith academic rules

and ethical conduct.

I also declare that, as required by these rules and conduct, I have fully cited and referred all

material and results that arenot original to this work.

Name : Hareram Lohar

Examination Roll Number :

Registration Number :

Thesis Title : Stress Analysis and Burst Pressure

Determination Of Shrink fitted

Compound Cylindrical Pressure Vessel

Signature :

Date :



iii


JADAVPUR UNIVERSITY

CERTIFICATE OF APPROVAL *

This foregoing thesis is hereby approved as a credible study of an engineering subject carried

out and presented in a manner satisfactory to warrant its acceptanceas a prerequisiteto the

degreefor which it has been submitted. It is understood that by this approval the undersigned do

not endorse or approve any statement made, opinion expressed or conclusion drawn therein but

approvethethesis only for thepurposefor which it has been submitted.

Committee Signature_______________________________

on Date_______________________________

Final Examination for Seal

Evaluation of theThesis

Signature_________________________________

Date_________________________________

Seal

* Only in casethethesis is approved



iv


JADAVPUR UNIVERSITY

CERTIFICATE OF SUPERVISION

We hereby recommend that the thesis presented under our supervision by Mr. HareramLohar

entitled “STRESS AND BURST PRESSURE DETERMINATION OF SHRINK FITTED

COMPOUND CYLINDRICAL PRESSURE VESSEL” beaccepted in partial fulfillment of the

requirements for thedegreeof Master of Mechanical Engineering.

1)

________________________________

2)

________________________________

(Signatures of theThesis Adviser)

Countersigned by : Date:

Seal :

_______________________________________________________

( Signatureof theHead of theDepartment, Mechanical Engineering )

Date:

Seal :

______________________________________________________

( Signatureof theDean of Faculty of Engineering and Technology )

Date:

Seal



v

ACKNOWLEDGEMENT

In every work, there have been so many heads behind the screen without whomthe

work might not reach that far as it should be. Very likely, I can remember many names who helped

me a lot during my project work. In this regard I pay my well gratitude to themwhosesupport

and cooperation inspired memorally in my good and bad days during thecourseof work.

At thevery outset, I acknowledgethegenerous and arduous assistanceand inspiration

provided by my guide, Dr. Susenjit Sarkar and my co. guideDr. Samar Chandra Mondal. Without

whosekind attention and tremendous support my thesis could not havereached this stage in proper

way. Words of thanks fall short to express my gratitude to them in proper dimensions for

their kind cooperation and methodical guidance in theentirespan of theprocess.

I would like to convey my regards to the Laboratory-in-Charge of ‘MachineElements

Laboratory’ and all other Professors who helped meduring thework.

I am greatly indebted to my friends and seniors of machine design specialization who

havehelped meconceiving certain ideas related to my project and also gavemesubstantial support in

thesoftwarepart.

Lastly, it is likely to mention that without the suggestions and hospitality of my family

and well wishers, it could havebeen difficult to carry out theproject work and completethethesis in

time. Hence, I would liketo express my gratitudeto themtoo.

Date:

______________________________________

( HARERAM LOHAR )

Examination Roll No.: M4MEC13-24

Registration No. : 117131 of 2011-2012



vi

This thesis is solely dedicated to my

P arents



vii

CONTENTS

SUBJECTS PAGENO.

DECLARATION ii

CERTIFICATE OF APPROVAL iii

CERTIFICATE OF SUPERVISION iv

ACKNOWLEDGEMENT v

DEDICATION vi

CONTENTS vii

LIST OF FIGURES ix

LIST OF TABLES xi

NOMENCLATURE xii

1. INTRODUCTION 1-11

1.1

Fundamental Idea of Pressure Vessel 2

1.2 Compound Cylinder 2

1.3 Burst Pressure 5

1.4 FEM Analysis 6

1.5

History and Context of the Present Work 8

1.6

Objective and Methodology 9

1.7 Summary of the Thesis 10

2. LITARATURE REVIEW 12-19

2.1 Compound Cylinder 13

2.2 Burst Pressure 16

2.3

FEM Analysis 18

3. MATHEMATICAL FORMULATION 20-43

3.1

Stress Analysis 21-38

3.1.1

Stress Analysis of Single Layer Cylinder 21

3.1.2 Stress Analysis of Two Layer Compound Cylinder 26



viii

3.1.3

Stress Analysis of Three Layer Compound Cylinder 29

3.2

Burst Pressure Determination 39-43

3.2.1

Burst Pressure of Single Layer Compound Cylinder 39

3.2.2

Burst Pressure of Two Layer Compound Cylinder 41

3.2.3

Burst Pressure of Three Layer Compound Cylinder 42

4. ANALYTICAL MODEL 44-49

4.1 Stress Analysis 45-47

4.1.1 Stress Analysis of Single Layer Cylinder 45

4.1.2

Stress Analysis of Two Layer Compound Cylinder 45

4.1.3 stress analysis of three layer compound cylinder 46

4.2

Burst Pressure Determination 47-49

4.2.1

Burst Pressure of Single Layer 47

4.2.2

Burst Pressure of Two Layer Compound Cylinder 47

4.2.3


5. FEM ANALYSIS 50-56

5.1

Stress Analysis 51-

5.1.1

Stress Analysis of Single Layer Cylinder 51

5.1.2

Stress Analysis of Two Layer Compound Cylinder 51

5.1.3 Stress Analysis of Three Layer Compound Cylinder 53

5.2 Burst Pressure Determination 53-56

5.2.1 Burst Pressure of Single Layer 53

5.2.2 Burst Pressure of Two Layer Compound Cylinder 54

5.2.3


6. DISCUSSION 57-58

7. CONCLUSION 59-61

8. REFERENCES 62-64



ix

LIST OF FIGURES

Fig. 1.1 Compound Cylinder

Fig. 1.2 Internal Pressure Vs Hoop Stress Distribution

Fig. 1.3 Two Layer Compound CylinderFig. 1.4 Three Layer Compound Cylinder

Fig. 3.1 Stress in a Thick Wall Cylinder

Fig. 3.2 A Three Layer Compound Cylinder Vessel

Fig. 3.3 Radial and Hoop Stress Distribution in Three Separate Cylinders of Three Layer

Compound Cylinder

Fig. 3.4 Interference between Cylinder 1 & 2 of Three Layer Compound Cylinder

Fig. 3.5 Interference between Cylinder 2 & 3 of Three Layer Compound Cylinder

Fig. 3.6 Superposition of Hoop Stress due toi

P & Residual Stress due to12s

P in Cylinder

1 of Three Layer Compound Cylinder

Fig. 3.7 Superposition of Hoop Stress due toiP & Residual Stress due to

12sP &23sP in

Cylinder 2 of Three Layer Compound Cylinder


23sP in Cylinder

3 of Three Layer Compound Cylinder


12sP &23sP in

all Cylinder of Three Layer Compound Cylinder

Fig. 5.1 Maximum Principal Stress due to 35 Mpa Internal Pressure in Thick Cylinder inANSYS

Fig. 5.2 Maximum Principal Stress due to 35 Mpa Internal Pressure in Two Layer

Compound Cylinder in ANSYS

Fig. 5.3 Maximum Principal Stress due to 35 Mpa Internal Pressure in Inner Tube in

ANSYS of Two Layer Compound Cylinder

Fig. 5.4 Maximum Principal Stress due to 35 Mpa Internal Pressure in Jacket in ANSYS

of Two Layer Compound Cylinder

Fig. 5.5 Maximum Principal Stress due to 35 Mpa Internal Pressure in Three Layer


Fig. 5.6 Maximum Principal Stress due to Elastic Break-Down Pressure of Single

Cylinder Layer in ANSYS

Fig. 5.7 Maximum Principal Stress due to Burst Pressure of Single Layer Cylinder in

ANSYS



x

Fig. 5.8 Maximum Principal Stress due to Burst Pressure of Two Layer Cylinder in

ANSYS

Fig. 5.9 Maximum Principal Stress due to Elastic Break-Down Pressure of Three Layer


Fig.5.10 Maximum Principal Stress due to Burst Pressure of Three Layer Compound

Cylinder in ANSYS



xi

LIST OF TABLES

Table 4.1 Stresses due to Internal Pressure of Two Layer Compound Cylinder

Table 4.2 Stresses due to Shrinkage Pressure for Inner Tube & Jacket of Two Layer

Compound CylinderTable 4.3 Resultant Stresses of Two Layer Compound Cylinder

Table 4.4 Data for Modeling Stress Analysis due to internal pressure 35 Mpa of Three

Layer Compound Cylinder

Table 5.1 Data for Modeling Stress Analysis in ANSYS for Thick Single Layer Cylinder

Table 5.2 Data for Modeling Stress Analysis in ANSYS for of Two Layer Compound

Cylinder

Table 5.3 Data for Modeling in ANSYS for Stress Analysis of Three Layer Compound

Cylinder

Table 5.4 Data for Modeling Elastic Break-down Pressure and Burst Pressure of Single

Layer Cylinder

Table 5.5 Data for Modeling Burst Pressure in ANSYS of Two Layer Compound Cylinder

Table 5.6 Data for Modeling Elastic-Breakdown Pressure in ANSYS of Three Layer

Compound Cylinder

Table 5.7 Data for Modeling Burst pressure in ANSYS of Three Layer Compound Cylinder



xii

NOMENCLATURE

SINGLE LAYER CYLINDER

a Inner radius of Cylinder

b Outer radius of Cylinder

wP Working Pressure on the Inner Surface

t Hoop Stress in Cylinder

r Radial stress in the cylinder

a Axial Stress in Cylinder

Poisson’s ratio

t Hoop Strain in Cylinder

r Radial Strain in Cylinder

a Axial Strain in Cylinder

TWO LAYER COMPOUND CYLINDER

j

Increase in inner diameter of jacket

c Decrease in outer diameter of cylinder

= total Interference

t j Tangential Strain for Jacket

t c Tangential Strain for Inner Cylinder

1 D Inner Diameter of the Inner Cylinder

2 D Outer Diameter of the Inner Cylinder & Outer Diameter of the Jacket

3 D Outer Diameter of the Jacket

iP Internal Pressure

P Shrinkage Pressure between Inner Cylinder and Jacket

t Hoop Stress in Cylinder

THREE LAYER COMOUND CYLINDER

iP Internal pressure acting on the cylinder 1



xiii

Hoop stress in the cylinder

r Radial stress in the cylinder

1r Inner radius of cylinder 1

2r Outer radius of cylinder 1 and Inner radius of cylinder 2

3r Outer radius of cylinder 2 and Inner radius of cylinder 3

4r Outer radius of cylinder 3

12sP Contact pressure between cylinder 1 and 2

23sP Contact pressure between cylinder 2 and 3

12 Total interference at the contact between cylinder 1 and 2

23 Total interference at the contact between cylinder 2 and 3

Poisson’s ratio

1t 2

1

r r

2t 3

2

r

r

3t 4

3

r

r



xiv

ABSTRACT

PressureVessels, storepressurized fluids in different phase, pressureand temperature

conditions with great importanceand safety precautions, arewidely used in household

and industrial purpose. Multilayer pressurevessel is designed to work under high-

pressure condition. This thesis introduces the stress analysis and burst pressure

calculation of a multi-layer shrink fitted pressurevessel. In theshrink-fitting problems,

considering long hollow cylinders, the plane strain hypothesis can be regarded as

more natural. Generally hoops stress distribution is non-linear and sharply reduced

toward theouter surface. By shrink fitting concentric shells towards theinner shells are

placed in residual compression so that theinitial compressive hoop stress must be

relieved by internal pressurebefore hoop tensile stress are developed. Therefore the

maximumhoop stress will bereduced, resulting moreburst pressure. Theanalytical

results of stress distribution and burst pressureis calculated and validated by ANSYS

Workbench results.



Chapter – 1

Introduction

This chapter named as Introduction provides a general

introduction to the entire work. It gives an overall idea of

pressure vessel and burst pressure context of the

present work, objective and planning of the work and

brief summary of the entire thesis

sequentially.



Introduction

2

1.1

INTRODUCTION TO PRESSURE VESSELS

Pressure vessels are used to contain fluid (liquid or gas) under pressure and

temperature. The pressure acting on vessel may sometimes be from outside also. So the

pressure situation may be internal or external.

Vessels which carry fluid under pressure and temperature are also subjected to the

action of steady or dynamic support loading, piping reactions and thermal shock which require an

overall knowledge of the stresses imposed by these conditions.

Pressure vessels commonly have the form of spheres, cylinders, ellipsoids, or some

composite of these. In practice, vessels are usually composed of two or more pressure containing

shells together with flange, rings and fastening devices for connecting and securing mating parts.

The main difference of thick shell pressure vessel from the thin shell is the

stresses developed over the section thickness cannot be assumed to be uniformly distributed

(as in the thin shells have). Only the axial stress is found to be uniformly distributed over

the wall thickness. Both the tangential and radial stresses are dependent on the radius of the

geometry under consideration.

1.2

COMPOUND CYLINDER

One of the common method of pre-stressing is to use compound (or composite) cylinders

- two or more cylinders which are assembled with an interference fit. The analysis which follows

is elastic since the method does not generally involve yielding - and may be applied to sleeves

pressed onto shafts, etc.

Two open cylinders only are considered. They are shown in exaggerated fashion both

before assembly (individually completely free), and after assembly and pressurising (ie. after all

load components have been applied). The bore of the outer cylinder and the outside diameter of

the inner cylinder are made to the same nominal common diameter Dc - however there is a known

small diametral interference, Δ << Dc. The cylinders are then assembled concentrically using heat

or force, and loaded by pressures internal and external to the assembly, pi and po respectively.



Introduction

3

Before safety of either cylinder can be addressed, the common interface pressure pc must

be known. But the problem is statically indeterminate since statics reveals only that if pcexists as a

contact pressure internal to the outer cylinder, then the inner cylinder is pressurised externally by

the same amount. Geometric compatibility and the cylinders' constitutive laws must be invoked.

If the common diameter Dc increases by δi measured on the inner cylinder as sketched, and by

δo measured.

On the outer the compatibility requires that:

Δ = δo - δi

Fig.1.1 Compound Cylinder

In thick walled cylinders subjected to internal pressure only, it can be seen from the

equation of the hoop stress that the maximum stresses occur at the inside radius and this can be

given by:

This means that as pi increases σt may exceed yield stress even when p i < σyield .

Furthermore, it can be shown that for large internal pressures in thick walled cylinders the wall

thickness is required to be very large. This is shown schematically in figure

Fig.1.2 Internal pressure vs. hoop stress distribution



Introduction

4

This means that the material near the outer edge of the cylinder is not effectively used

since the stresses near the outer edge gradually reduce.

In order to make thick-walled cylinders that resist elastically large internal pressure

and make effective use of material at the outer portion of the cylinder the following

methods of pre-stressing are used:

• Shrinking a hollow cylinder over the main cylinder. (Compound cylinders)

• Multilayered or laminated cylinders.

• Autofrettage or self hooping.

An outer cylinder (jacket) with the internal diameter slightly smaller than the outer

diameter of the main cylinder is heated and fitted onto the main cylinder. When the

assembly cools down to room temperature, a compound cylinder is obtained. In this process the

main cylinder is subjected to an external pressure leading to radial compressive stresses at the

interface (Pc) as shown in figure 3.

The outer cylinder is subjected to an internal pressure leading to tensile circumferential

stresses at the interface (Pc) as shown in figure 4. Under these conditions as the internal pressure

increases, the compression in the internal cylinder is first released and then only the cylinder

begins to act in tension.

Fig.1.3 Two layer compound cylinder



Introduction

5

Fig.1.4 Three layer compound cylinder

1.3

BURST PRESSURE

On subjecting a thick-walled vessel to increasing internal pressure, stresses are induced in

the shell which are maximum at the bore, as predicted by the various criteria for failure based

upon theory of elasticity. Contrary what might be expected, failure of a shell of ductile material

usually does not begin at the fibers along the bore but the fibers along the outside surface of the

shell. On stressing beyond the yield point, most metals pass through a region of plastic flow in

which elongation progresses without an increase in resisting stress. This condition is firstly

reached in the inner part of the cylinder. However, the strain of the inner zone is limited by the

outer zone, which is not stressed beyond the yield point; thus the inner fibers are incapable of

rapture. The inner fibers of an over-stressed vessel often show evidence of slip where failure

began but halted because of the restraint offered by the outer fibers. The inner fibers therefore

prevented from failing, provided that the outer fibers offer sufficient restraint. There is no such

protection of the outer fibers by the inner fibers.

Manning has discussed the rapture of thick-walled cylinders of ductile metal. The

pressure necessary for the yield point of the metal fibers in the bore to be reached is known as the

“elastic-breakdown” pressure. At this pressure the maximum fiber stress is the tangential stress at



Introduction

6

the inner surface. The radial stress also has its maximum value at the bore, and this stress is equal

to the internal pressure. As the pressure is raised, the region of plastic flow, termed overstrain

moves radially outward and causes the tangential stress to decrease in the inner layers and to

increase rapidly in the outer layers. Progressive increase in pressure moves the elasto-plastic

interference radially outward until the interference reaches the outer radius and no elastic zone

remains. At this situation maximum hoop stress is at the outer surface. Manning has reported that

from the beginning of overstrain for a vessel in which the outer diameter to inside diameter ratio

was 2:1 and the pressure was 12750 psi, the tangential stress at the inner radius was 21000 psi and

at the outer radius 8000 psi. for the same vessel. for the same pressure vessel with 100%

overstrain (plastic-elastic interface at r o) at a pressure of 27,620 psi, the tangential stress at bore

was 16000 psi, and at the outer surface 34000 psi. thus it is apparent that the tangential stress

distribution is totally different in the 100% plastic state than in the complete elastic state. On the

other hand, the radial stress pattern has similar shapes for the complete elastic and complete

plastic state.

When failure occurs in the shell of a ductile metal as the result of progressive increase in stress, it

usually follows the path of a continuous helix from the outer surface onward.

Prager and Hodge have defined the internal pressure in a cylindrical vessel that is

required to place the elasto-plastic interference on the outer surface of the vessel. this is the

pressure required to place all the vessel wall beyond the yield point. In deriving the relationship it

is necessary to establish the condition under which plastic flow is initiated.

1.4 FEM ANALYSIS

The Finite element Analysis (FEA) method, originally introduced by Turner (1956), is a

powerful computational technique for approximate solutions to a variety of “real-world”

engineering problems having complex domains subjected to general boundary conditions. FEA

has become an essential step in the design or modeling of physical phenomenon in various

engineering disciplines. A physical phenomenon usually occurs in a continuum of matter (solid,



Introduction

7

liquid, gas) involving several field variables. The field variables vary point to point, these

processing infinite solutions in the domain.

In this method, the body or structure is divided into on equivalent system of smaller

bodies or units. The assemblage of such a units that represents the original bodies. Instead of

solving the problem for the entire body in one operation, the solutions are formulated for each

constituents unit and combine the solution for the original body or structure. This approach is

known as going from part to whole.

The method can be symmetrically programmed to accommodate such complex and

difficult problems as non-homogeneous materials, non-linear stress-strain behavior and

complicated boundary conditions.

The basic theory of Finite Element Method is representation of a body or a structure by an

assemblage of small sub-divisions called finite elements. These elements are considered inter-

connected at joints, which are nodes or nodal points. The elements are super-imposed on to a co-

ordinate guided system, where nodal points are preferred with respect to a co-ordinate system.

The position and elastic properties of elements are defined by the matrices, so that the

displacement of each elements can be related to the forces on the element. Finally a composite

matrix of the system of every elements of the structure is are formed which relates to the

displacements of the nodal point of each element to the external force on the structure. Once the

displacement field is determined, the strains can be evaluated by using the strain-displacement

relationship and finally, the stress can be evaluated by stress-strain relations.

The finite element analysis method requires the following steps:

Discretization of the domain into a finite number of sub domains (elements).

Selection of interpolation function (the value of field variables at specific points

refers to as nodes).

Development of the element matrix for the sub domain (elements).

Assembly of the element matrices for each sub domain to obtain the global

matrix for the entire domain.



Introduction

8

Imposition of the boundary conditions.

Solution of equations.

Additional computation (if desired).

1.5

HISTORY AND CONTEXT OF THE PRESENT WORK

Pressure vessel is an age old term, with the advancement of modern civilization

the use of pressure vessel also increased. It is used in everywhere from household goods to large

scale industries with same importance. The basic conception was invented long ago in about

15th

century related to the history of canons, but its significance to the practical life started

increasing since 19th century when Lamè proposed the eq.(s) for thick walled cylinder.

Relative to this plasticity also attracted the researcher’s attention in 19th century when

Tresca published a preliminary account of experiments on punching and extrusion, which

led him to state that a metal yielded plastically when the maximum shear stress attained a critical

value.

The idea of determining burst pressure of a compound pressure vessel is a very new

approach. A full understanding of the behavior of a pressurized thick walled cylinder was not

possible without an understanding of the post-yield behavior of metals under a biaxial stress

state. This was provided by the maximum shear stress theory of ‘‘failure’’ as proposed by

Tresca. Saint-Venant published an analysis of the plastic deformation of a pressurized

thick-walled cylinder which introduced the concept of Autofrettage.

Many researchers have focused on methods to extend lifetimes of vessels. Majzoobi et

al. have proposed the optimization of bi-metal compound cylinders and minimized the

weight of compound cylinder for a specific pressure [6]. The variables were shrinkage radius

and shrinkage tolerance. Patil S. A. has introduced optimum design of two layer compound

cylinder and optimized intermediate, outer diameter and shrinkage tolerance to get minimum

volume of two layer compound cylinders [7-8]. Niranjan et at.have proposed the optimization of

shell thickness in multilayer pressure vessels and also investigated the effect of no. of shell on



Introduction

9

maximum hoops stress[4]. At the same time Ayub A. Miraje and Sunil A. Patil have proposed the

optimization of three layer shrink fitted cylinder for uniform stress distribution[5].

To incorporate this new movement into the determination of burst pressure of a

compound pressure vessel this entire work is carried out, hope we will have some fruitful results.

1.6 OBJECTIVE AND METHODOLOGY

In a sentence Increased Strength -to- weight Ratio and Extended Burst Pressure are the

main objectives of the optimal Autofrettaging. The process of compounding itself ensure

uniform stress distribution and increased burst pressure. The process involves cost so optimum

uniform distribution is very important and also give better result.

The objective of this present work is to formulate the stress distribution equations

in the pressurize condition for the each layer then to find out the shrinkage pressure in between

the two layer by equating the inner side hoop stress of the each layer. From this shrinkage

pressure the shrinkage interference in between two layers will be determined. As the busting of a

compound cylinder occur at the outer layer, so the pressure required at the inner side of the outer

layer to bust the outer layer have to calculate. Then have to equate this pressure with the sum of

shrinkage pressure and the interference pressure to get the burst pressure of the whole compound

cylinder.

The problem is defined now and to evaluate we need to plan our methodology as

follows :

1)

Mathematically formulate the problem for the stress distribution using Lame’s equations

for the each layer.

2)

After finding hoop stresses at all the radii, the principle of superposition is applied, i.e.

the various stresses are then combined algebraically to produce the resultant hoop stresses

in the compound cylinder subjected to both shrinkage pressures and internal pressure.



Introduction

10

3) For optimum stress distribution it is considered that the resultant stress in the inner

surface of each layer is equal and the optimum contact shrinkage pressure and the

optimum shrinkage interferences are calculated.

4)

For formulating the burst pressure of compound cylinder it is considered that the outer

layer will burst first and the whole compound cylinder.

5)

After mathematical deduction is done, an analytical model is deduced by taking a certain

problem.

6)

After that we validated our results by using FEM software ANSYS Workbench version

v11.

7)

Finally we compared results, analyzed and got to some conclusions.

The entire work can be divided into three phases they are:

I. Mathematical Formulation (step 1 – step 4, chapter 3),

II. Analytical Model (step 5, chapter 4), and

III. FEM Analysis and Validation (step 6 – step 7, chapter 5).

1.7

SUMMARY OF THE THESIS

The thesis is based on the work discussed above. It constitutes of mainly five chapters.

The first chapter named as Introduction provides a general introduction to the entire

work. It gives an overall idea of pressure vessel, compound cylinder, burst pressure and FEM

Analysis, context of the present work objective and planning of the work and brief summary of

the entire thesis sequentially.

The second chapter is titled as Literature Review, where different works of several

researchers up to present time are surveyed, historical background is nurtured and context of the

present work are planned.



Introduction

11

The third chapter named as Mathematical Formulation gives the through deduction of

stress distributions for single layer thick cylinder, two layer compound cylinder and three layer

compound cylinder. Also the burst pressure of single layer thick cylinder, two layer compound

cylinder and three layer compound cylinder is determined at the end of the chapter.

The fourth chapter named as Analytical Model where a certain problem is considered

and stress analysis and burst pressure determination is done for corresponding single layer thick

cylinder, two layer compound cylinder and three layer compound cylinder.

The fifth chapter named as FEM Analysis where a finite element analysis is done using

ANSYS Workbench v11 software and the stresses and burst pressure of single layer thick

cylinder, two layer compound cylinder and three layer compound cylinder is validated.

The sixth chapter is the most important chapter named as Discussion. Here the results of

analytical model and FEM analysis are compared and error is also computed and shown that the

errors are in acceptable limits.

The seventh chapter is the general Conclusion, where merits and demerits of the present

work are dealt and the readers of the thesis are directed towards the future scope of work or

further continuation or modification of the present work.



Chapter – 2

Literature Review

This chapter is titled as Literature Review, where different

works of several researchers upto present time are surveyed,

historical background is nurtured and context of the present

work are planned.



Literature Review

13

It is customary and quite meaningful to introduce any subject through its historical

development; hence the review work may start mentioning some earlier studies in the area of

proposed work constructing the context of the same.

The history of high-pressure technology began in the early 14th century. The earliest

pioneer in the field of Autofrettage was a German monk named Berthold Schwarz, who invented

the first known “cannon”. A canon is a thick walled pressure vessel open at one end, intended to

contain the high pressure and temperature gaseous products of an explosive detonation while they

accelerate projectile through its muzzle. For several centuries, the gun was the only significant

application of high pressure in which the pressure is contained. Gradually as the time passed

away, range of high-pressure technology got broad. Then it covers pipes, boilers, reactors etc.

2.1. COMPOUND CYLINDER

Due to the ever-increasing industrial demand for axisymmetric pressure vessels which

have had wide applications in chemical, nuclear, fluid transmitting plants, power plants, gas

storages [1,2] and military equipments, the attention of designers has been concentrated on this

particular branch of engineering. On the other hand, the increasingly scarcity of materials and

higher costs have led researchers not to confine themselves to the customary elastic regime but

attracted their attention to the elastic-plastic along with optimization approach which offer more

efficient use of materials. Basically there are two basic different elastic-plastic techniques to

increase the pressure capacity of thick-walled cylinders. In the first, the cylinder is subjected to

internal pressure so that its wall becomes partially plastic. The pressure is then released and the

resulting residual stresses increase the pressure capacity of the cylinder in the next loading stage.

This procedure is called 'autofrettage' [3,4]. The analysis of residual stresses and deformation in

an autofrettaged thick-walled cylinder has been given by Chen [5] and Franklin and Morrison [6].

In the second technique, two or more cylinders are shrunk into each other with different

diametral interferences to form a compound cylinder. The shrinkage produces a residual stress



Literature Review

14

distribution within the walls of the cylinders, which improves the cylinder behavior against the

working pressure.

Many researchers contributed in the field of Compound Cylinder. Some of them

discussed various ways to find out stress distribution in pressure vessels considering the pressure

condition and sometimes thermal loading as well.

The basic governing mathematical formulation for stress generation in thick walled

pressure vessel was proposed by Lamè (1795-1870) and his colleagues in 1833 [1].

Gun designers during the mid-1800s recognised the poor utilisation of material (uneven

stress distribution) within thick-walled guns, applying a practical upper limit on the range of wall

ratios that could be used. It was realised that if some inwards force could be applied to a barrel,

the effect of poor material utilisation could be mitigated. To this end, barrels were “hooped” to

pre-compress the inner surface of the barrel – for example, William Armstrong [2] assembled

compound cylinders from wrought iron tubes. Additionally, it was recognised by Rodman [3] that

if the cooling of cast cannon were controlled, the sequence of crystallisation from liquid could

influence the residual stress distribution and hardness of the tube material at the bore.

The relationships which follow in compound cylinder were first presented by H.L. Cox in

1936. The theory as developed is based upon the assumption that the maximum combined stresses

(hoop-pressure stress plus hoop-shrinkage stress) existing at the inner surface of each of the

several shells will attain a certain identical value. It is assumed that both the internal and external

diameters are known and that the number of shells is to be a minimum. It is also assumed that the

combined cylinder is fabricated by shrinking on each successive shell from the inside outwards

and that after each shell is shrunk on, the outside diameter is machined to size before the next

cylinder is shrunk on to the inner shell or shells.

In 2005 G.H. Majzoobi, A. Ghomi proposed Optimisation of two layer compound

pressure cylinders [13].The purpose of this paper was optimization of the weight of compound



Literature Review

15

cylinder for a specific pressure where the variables were shrinkage radius and shrinkage tolerance.

SEQ technique for optimization, the finite element code ANSYS for numerical simulation is

employed to predict the optimized conditions. They investigated that the weight of a compound

cylinder could reduce by 60% with respect to a single steel cylinder. The reduction is more

significant at higher working pressures. While the reduction of weight is negligible for k<2.5, it

increases markedly for 2.5<k<5.5. The stress at the internal radii of the outer and inner cylinders

become equal to the yield stresses of the materials used for compound cylinders.

In 2011 investigated on optimization of shell thickness in multilayer pressure vessel and

study on effect of number of shells on maximum hoop stress [11]. In this paper, optimization of

thickness of each layer in multilayer vessel was carried out by Genetic Algorithm and then stress

distribution was analyzed under optimum shrink-fit condition. The fatigue life was calculated for

shrink-fit multilayer vessel. Thickness of each vessel was considered as design variable and

objective function was maximum hoop stress through-out the thickness at the given working

pressure. Multilayer vessel was assumed to be constructed by insertion of different vessels with

zero interference and zero clearance such that interface pressure at the mating surfaces was equal

to the pressure generated at the same surface due to interference fit. Effect of number of shells on

the maximum value of hoop stress was analyzed. Apart from this, effect of overall thickness of

pressure vessel on the effectiveness of multi-layering was brought into focus. Stress distribution

and fatigue life for the obtained thickness of each vessel from Genetic Algorithm was nearer to

that obtained from Lagrange’s multiplier method.

In 2011he (Niranjan Kumar et al.) extended his work to optimum autofrettage pressure

and shrink-fit combination for minimum stress in multilayer pressure vessel. In this paper,

optimization of thickness of each layer in multilayer vessel was carried out by Genetic Algorithm

and then stress distribution was analyzed under optimum shrink-fit condition. The fatigue life was

calculated for shrink-fit multilayer vessel. Thickness of each vessel was considered as design



Literature Review

16

variable and objective function was maximum hoop stress through-out the thickness at the given

working pressure.

At the same time (2012) Ayub A. Miraje and Sunil A. Patil introduced the optimum

design for minimization of thickness of three-layer shrink-fitted compound cylinder to get equal

maximum hoop stresses in all the cylinders [12]. In this paper effort is made to find optimum

minimum thicknesses of three cylinders so that material volume is reduced and hoop stress is

equal in all the cylinders. The analytical results of optimum design calculated with computer

programming are validated in comparison with Finite Element Analysis in ANSYS Workbench.

2.2. BURST PRESSURE

Many researchers contributed in the field of burst pressure.

In 2009 burst tests and volume expansions of vehicle toroidal LPG fuel tanks was

proposed by kisioglu [18]. This study addressed the prediction of the burst pressures and

permanent volume expansions of the vehicle toroidal LPG fuel tanks using both experimental and

finite element analysis (FEA) approaches. The experimental burst test investigations were carried

out hydrostatically in which the cylinders were internally pressurized with water. The LPG tanks

were subjected to incremental internal uniform pressure in the FEA modeling. 2D nonlinear plane

models were developed and evaluated under non-uniform and axisysmmetric boundary

conditions. For the analysis, the required actual shell properties including weld zone and thickness

variations were investigated. Therefore, the results of the burst pressures and volume expansions

were predicted and compared to experimental one.

Kaptan and Kisioglu determined the BP of the cylindrical LPG fuel tanks using both

experimental and finite element method _FEM_ approaches in 2007 [19]. The primary aim of

this study was to determine the burst pressures (BP) and the failure locations of LPG cylinders

whose service pressure (SP) and test pressures (TP) are known. The purpose of this work was to

investigate the BP and failure locations of LPG fuel tanks using both experimental burst tests and



Literature Review

17

FEA. To predict the BP and the failure location using computer modeling, the actual shell and

weld zone material properties (MPs) including thickness variations were investigated from the

LPG tanks. These properties were used in the FEA to approximate the BP values obtained

experimentally. Two different types of two-dimensional (2D) nonlinear FEA models, plane and

shell, were developed under axi-symmetric boundary conditions.

Aksoley et al. compared the BP of the LPG tank, used for home-kitchen applications,

employing the experimental and FEM techniques in 2008 [20].

Mirzaei analyzed the failure of the exploded cylinders containing and transporting the

hydrogen gas using the analytical methods in 2008 [21]. This paper reported the major activities

carried out during the failure analysis of an exploded cylinder containing hydrogen. The general

cracking pattern of the cylinder, the fractographic features, and the stress analysis results were all

indicative of an internal gaseous detonation. Accordingly, several specific characteristics of

detonation-driven fracture of closed-end cylindrical tubes were identified. These characteristics

were analyzed through detailed examinations of the fracture surfaces, cracking patterns, and

dynamic stress analysis of the cylinder using a transient analytical model. Based on the size and

location of special markings found on the shear lips, and using the time duration of flexural

waves, the crack growth increments and speed were computed. Consequently, the basic features

of the gaseous detonation and the composition of the original gas mixture were identified. The

results indicated that the detonation of a low-pressure oxygen-rich mixture of hydrogen and

oxygen was the cause of this failure. The presence of oxygen was attributed to an improper

usageof an oxygen cylinder for hydrogen storage.

Xue et al. determined the influence of geometrical parameters on the burst pressure of a

cylindrical shell intersection using analytical method in 2010 [22]. In this paper, the FEA

simulation procedure was validated by comparison with the results from experiments employing

test vessels, which were hydraulically pressurized to burst. Based on the agreement between the

numerical and experimental results, a parametric analysis was here carried out to determine the



Literature Review

18

influence of geometric parameters (diameter ratio, thickness ratio, and diameter to thickness ratio)

on the burst pressure of radial intersections. An empirical formula for this pressure was then

developed based on the parametric analysis results.

Finite element analysis of burst pressure of composite hydrogen storage vessels by P. Xu

a, J.Y. Zheng b, P.F. Liu b in 2009 [23]. In this research, a 3D parametric finite element model

was proposed to predict the damage evolution and failure strength of the composite hydrogen

storage vessels, in which a solution algorithm was proposed to investigate the progressive damage

and failure properties of composite structures with increasing internal pressure. The maximum

stress, Hoffman, Tsai–Hill and Tsai–Wu failure criteria which awere employed respectively to

determine the failure properties of composite vessels were incorporated into the numerical method

as individual subroutines. The birth-to-death element technique in the finite element analysis was

used to describe the mechanical properties of carbon fiber/epoxy composite elements. Parametric

studies in terms of the effects of different failure criteria were performed and the calculated

failure strengths of composite vessels were also compared with the experimental results.

Research on bursting pressure formula of mild steel pressure vessel by ZHENG Chuan-

xiang, LEI Shao-hui in 2006 [24]. Of several formulas for calculating bursting pressure of mild

steel vessel, the Faupel formula is the most famous one. In fact, Faupel formula is conservative in

calculating mild steel pressure. Based on hundreds of bursting experiments on mild steel pressure

vessels such as Q235(Gr.D), 20R(1020) and, after statistically analyzing data on bursting

pressure, it was found that the Faupel formula had some errors in calculation. The authors derived

a more approximate modified formula from the data, which proved more general after examining

the data on other mild steel pressure vessels with different diameters and shell thickness.

2.3. FEM ANALYSIS

The Finite element Analysis (FEA) method, originally introduced by Turner (1956), is a

powerful computational technique for approximate solutions to a variety of “real-world”

engineering problems having complex domains subjected to general boundary conditions. FEA



Literature Review

19

has become an essential step in the design or modeling of physical phenomenon in various

engineering disciplines.

The Finite Element Method and Application in Engineering Using ANSYS by Erdogan

Madenci and Ibrahim Guven [25]. In this book mathematical formulation of finite element

method and a variety of analysis tutorials are given.

ANSYS Workbench Tutorial (ANSYS Release 10) by Kent L. Lawrence [26]. In this

book a variety of tutorials are given which help to understand the ANSYS Workbench Software.



Chapter-3

Mathematical Formulation

This chapter named as Mathematical Formulation gives the through deduction of stress

distributions for single layer thick cylinder, two layer compound cylinder and three layer

compound cylinder. Also the burst pressure of single layer thick cylinder, two layer compound

cylinder and three layer compound cylinder is determined at the end of the chapter.




21

3.1 STRESS ANALYSIS

3.1.1

STRESS ANALYSIS OF SINGLE LAYER CYLINDER

From the geometry given below, let ‘r’ be any radius, ‘dr’ is any elementary

radius after r, ‘l’ is any length in axial direction. Now, ifr and

t denote the radial and

tangential stresses,

Fig.3.1 Stress in a thick walled cylinder

then from the consideration of equilibrium under vertical forces :

. . . . 2 . . .sin 02

r r r t

d d l r dr d l rd i dr

For small values of ‘ d ’, sin2

d

2

d ; and the above equation becomes :

. . 0r r t dr rd ld dr l d

neglecting the product of small quantities, this implies :

0r r t dr rd dr




22

or, r t r

d r

dr

(1)

Eq. (1) is known as the equilibrium equation of the cylinder.

From the deformation point of view there is a congruence of elongations and they are

related to each other as follows :

Let us consider a circular ring of inner and outer radii ‘ x ’ and ‘ y ’ respectively, with the

thickness ‘ dr ’. now because of the circumferential straint

, the radius x of inner circle has an

elongation dx =t r and also the radius y of the outer circle will have an elongation

t dy d r dr . To impose the congruence, the difference between these two

elongations must corresponds to the increment of the thickness of the ring element,r dt dr i.e.

where,r is the radial strain.

Therefore, in other words, dy dx dt .

neglecting the products of small quantities, this implies :

t r t

d r

dr

(2)

Eq. (2) is known as the equation of congruence of the cylinder

Now, let us remember

1

r r t a

E

1

t t r a E

(3)

1

a a t r E




23

Where, E is the Modulus of Elasticity and ‘ν’ is the Poisson’s ratio of the thick cylinder material.

Now applying eq. (3), eq. (2) becomes :

0ar r t r t r d d d r r r

dr dr dr

This can also be written as :

1 0t ar r t r

d d d d r r r r

dr dr dr dr

Comparing this eq. with eq. (1) we find :

t ar d d d

dr dr dr

Now, if we differentiate 1

a a t r E

1a a t r d d d d

dr E dr dr dr

Now, comparing last two eq. (s) we have :

21a ad d

dr E dr

As thea

is uniform throughout radial direction, i.e. transverse section remains plane, and hence

the axial strain ‘a ’ is constant, therefore :

0t r d d

dr dr

Now, if we differentiate eq. (1) w.r.t ‘r’, we have :




24

2

20t ar r

d d d d r

dr dr dr dr

From above two eq.(s) we find :

2

2

3r r

d d

dr r dr

(4)

Finally solving this differential eq. (4) we obtain

2r r

(5)

And from eq. (1) and eq. (5), we found :

2t r

(6)

Now, for Cylinder subjected to internal pressure ‘wP ’;

Where, inner radius r = a, andr =

wP and outer radius r = b, andr = 0

Substituting these values in eq. (E), we get,

2

2 2w

aP

b a

2 2

2 2w

a bP

b a

Now substituting these values of ‘α’ and ‘β’ in eq. (5) and eq. (6), we get :

2 2 2

22 2 2 2

w

r w

Pa a bP

r b a b a

or,

2 2

2 2 21w

r

P a b

b a r




25

2 2 2

22 2 2 2

w

t w

Pa a bP

r b a b a

or,

2 2

2 2 21w

t

P a b

b a r

As far asa is concern, it can be calculated considering the equilibrium of the forces along

axial direction i.e. mainly the thrust on the heads. It will come out to be constant

2 2 2

w aP a b a or,

2

2 2

wa

P a

b a

To summarize, the three principal stresses are as follows :

2 2

2 2 2

1w

r

P a b

b a r

2 2

2 2 21w

t

P a b

b a r

(7)

2

2 2

wa

P a

b a

The equations shown in eq. (7) are known as Lamè’s equations.

The Lamè’s equations are based on two basic assumptions. 1) Cylinder material is Brittle

in nature, and 2) Cylinder has closed end conditions.

The entire analysis in this thesis is based on Lamè’s equations and assumptions. Now we

will see what happens if we pre stress the same cylinder with an enough high amplitude internal

pressure. One thing should be cleared, that though Lamè’s equations and assumptions deals with

brittle material, but in practice it has been seen that ductile materials are also responds well in

overloading Autofrettaging as those material passes through a strain – hardening process in the

post elastic region.




26

Now we will discuss about the formulation of stress distribution of the Autofrettaged

thick cylinder. Most of the assumptions and calculations are same for both considering and not

considering Bauschinger Effect. Only the difference comes when we formulate the unloading

stress distribution. For both the cases, for vessels to be subjected to internal pressure it is required

to pre-stress the vessel in the same direction as the working pressure do, but in a enough larger

magnitude. This will develop a compressive residual stress upon release.

From the elementary study on pressure vessel it is clear that mainly Circumferential or

Tangential Stress is more prone to failure, and hence it is taken for designing pressure vessels

with suitable factor of safety.

3.1.2

STRESS ANALYSIS OF TWO LAYER COMPOUND CYLINDER

Let us consider a compound cylinder, consisting of a cylinder and a jacket as shown in

fig. The inner diameter of the jacket is slightly smaller than the outer diameter of the cylinder.

When the jacket is heated, it expands sufficiently to move over the cylinder. As the jacket cools, it

tends to contract onto the inner cylinder, which induces residual compressive stresses. There is a

shrinkage pressure p between the cylinder and jacket. The pressure p tends to contract the cylinder

and expand the jacket .

3.1.2.1 TOTAL INTERFERENCE

Let,

j increase in inner diameter of jacket

c decrease in outer diameter of cylinder

The tangential strain (t

) for the jacket is given by,

t j change in circumference / original circumference

2

j

D

(8)

The tangential strain t c for the cylinder is given by,




27

t c

2

c

D

(9)

Also, 1

t t r j E (10)

From (1), 2 j t r

D

E

(11)

Where,

2 2

3 2

2 2

3 2

i

t

P D D

D D

and

r iP

Substituting the above values in Eq. (11),

2 2

3 22

2 2

3 2

i j

D D D P

E D D

Similarly, t c

1t r

E

From (2), 2c t r

D

E (12)

Where,

2 2

2 1

2 2

2 1

i

t

P D D

D D

and

r iP

Substituting the above values in Eq. (12),

2 2

2 12

2 2

2 1

ic

D D D P

E D D

The negative sign indicates contraction. Neglecting the positive and negative signs and

considering only magnitudes, the total deformation ( ) is given by,

2 2 2

2 3 12

2 2 2 2

3 2 2 1

2i

D D DPD

E D D D D

(13)

3.1.2.2 STRESS DUE TO INTERNAL PRESSURE

Stress due to internal pressure at1

D




28

1

2 2

1 3

22 213 1

1i

t atD

P D D

D D D

(14)

At2

D

2

2 2

1 3

22 223 1

1i

t atD

PD D

D D D

(15)

At3 D

3

2

1

2 2

3 1

2 i

t atD

PD

D D

(16)

3.1.2.3

STRESS DUE TO SHRINKAGE PRESSURE

3.1.2.3.1 JACKET

Let P is the shrinkage pressure

So stress due to shrinkage pressure P at2 D of the jacket

2

22

32

22 223 2

1t atD

DPD

D D D

(17)

At 3 D

3

2

2

2 2

3 2

2t atD

PD

D D

(18)

3.1.2.3.2

INNER TUBE

So stress due to shrinkage pressure P at1 D of inner tube

1

2

2

2 2

2 1

2t atD

PD

D D

(19)

At2

D

2

2 2

2 1

22 222 1

1t atD

PD D

D D D

(20)




29

3.1.2.4 SHRINKAGE PRESSURE

Equating stresses at the inner surfaces of tube and jacket. From equation (14), (15), (19) & (17).

2 2 2 2 22 21 3 1 3 32 2

2 2 22 2 2 2 2 2 2 21 2 23 1 2 1 3 1 3 2

21 1 1i iPD D P D D DPD PD

D D D D D D D D D D D

Or,2 2 2 2 2 22 2

3 1 3 2 3 21 2

2 2 2 2 2 2 2 2 2

3 1 2 3 1 2 1 3 2

2/

i

D D D D D D D DP P

D D D D D D D D D

(21)

3.1.3

STRESS ANALYSIS OF THREE LAYER COMPOUND CYLINDER

Consider three cylinders have same material. The method of solution for compound

cylinders constructed from similar materials is to break the problem down into four separate

effects:

i) shrinkage pressure12sP only on the cylinder 1

ii) shrinkage pressure12s

P and23s

P only on the cylinder 2

iii) shrinkage pressure23s

P only on the cylinder 3

iv) internal pressureiP only on the complete cylinder

Thus for each condition the hoop and radial stresses at any radius can be evaluated.




30

Fig.3.2 A three layer compound pressure vessel

Fig.3.3 Radial and hoop stress distribution in three separate cylinders

Fig.3.4 Interference between cylinder 1 & 2 Fig.3.5 Interference between 2 & 3




31

3.1.3.1 Radial and Hoop stress in Cylinder 1

Ifi

P i.e. no internal pressure, radial stress in cylinder 1 is given by using Lame’s equation

2 2

2 1

12 2 2 22 1

1r s

r r P

r r r

(22)

r is maximum at outer radius2r of cylinder 1 . Using equation (22)

2 12max sr atr P (23)

Hoop stress in cylinder 1 is given by using Lame’s equation

2 2

2 112 2 2 2

2 1

1s

r r P

r r r

(24)

Hoop stress at outer radius2

r is

2

2 2

2 112 2 2

2 1

satr

r r P

r r

(25)

While hoop stress at inner radius1

r is

1

2

12 2

max 2 2

2 1

2 s

atr

P r

r r

(26)

In the shrink-fitting problems, considering long hollow cylinders, the plane strain hypothesis

(in general, 0 z ) can be regarded as more natural. Hence as per the relation

z r

the expression for the hoop strain is given by

21 1 11r z r r r

E E E

Using equations (23) and (25), assuming plane strain condition the hoop strain at the outer wall2r

of cylinder 1 is




32

2 2

2 11 12 122 2

2 1 2

1 11 1 rlo

o r s s

U r r P P

E E r r r

(27)

Radial displacement1r oU is

2 212 2 2 1

1 12 2 2

2 1

1s

r o s

P r r r U P

E r r

(28)

3.1.3.2

Radial and Hoop stress in Cylinder 2

Contact pressure12sP is acting as internal pressure and contact pressure

23sP is acting as external

pressure on cylinder 2.

Using Lame’s equation, radial stress in the cylinder 2 at inner radius2

r is given by

212sr atr

P (29)

While radial stress in the cylinder 2 at outer radius3

r is given by

3 23sr atr

P (30)

Hoop stress in the cylinder 2 at inner radius2

r is given by

2

2 2 2

3 2 23 312max 2 2 2 2

3 2 3 2

2 ssatr

r r P r P

r r r r

(31)

While hoop stress in the cylinder 2 at outer radius3r is given by

3

2 2 2

12 2 3 2232 2 2 2

3 2 3 2

2 ssatr

P r r r P

r r r r

(32)

Using equations (29) and (31), assuming plane strain condition the hoop strain at the inner wall2

r

of cylinder 2 is




33

2 2 2

3 2 23 3 22 12 122 2 2 2

3 2 3 2 2

21 11 1 s r i

i r s s

r r P r U P P

E E r r r r r

(33)

Radial displacement2r i

U

2 2 22 3 2 23 3

2 12 2 2 2 2

3 2 3 2

1 21 1 s

r i s

r r r P r U P

E r r r r

(34)

Referring figure 3 and using equations (28) and (34), total interference12

at the contact between

cylinder 1 and 2 is

12 2 1r i r oU U

2 2 2 2 22 12 23 2 23 3 2 112 122 2 2 2 2 2

3 2 3 2 2 1

1 121 1 ss

s sr P r r r P r r r P P

E r r r r E r r

2 2 22 22 3 2 32 1

12 232 2 2 2 2 2

3 2 2 1 3 2

12s s

r r r r r r P P

E r r r r r r

(35)

Using equations (30) and (32), hoop strain in the outer wall3r of cylinder 2 is given by

2 22

23 3 212 2 22 232 2 2 2

3 2 3 2 3

21 11 1

ss r oo r s

P r r P r U P

E E r r r r r

(36)

Hence radial displacement2r oU

2 2 2 212 23 3 2

2 232 2 2 2

3 2 3 2

2 11 1

s

r o s

P r r r r U P

E r r r r

(37)

3.1.3.3

Radial and Hoop stress in Cylinder 3

Contact pressure23sP is acting as internal pressure on cylinder 3 and external pressure

oP is zero.

Radial stress in the cylinder 3 at inner radius3

r is given by

3 23sr atr P (38)




34

Hoop stress in the cylinder 3 at inner radius3r is given by

3

2 2

4 323max 2 2

4 3

satr

r r P

r r

(39)

While hoop stress in the cylinder 3 at outer radius4

r is given by

4

2

23 3

2 2

4 3

2 s

atr

P r

r r

(40)

Using equations (38) and (39), hoop strain at inner wall3

r of cylinder 3 is given by

2 2

4 3 33 23 232 2

4 3 2

1 11 1 r i

i r s s

r r U P P

E E r r r

(41)

Radial displacement3r iU

2 223 3 4 3

3 2 2

4 3

11

s

r i

P r r r U

E r r

(42)

Referring figure 4 and using equations (37) and (42), total interference23

at the contact between

cylinder 2 and 3

23 3 2r i r oU U

2 22 2 2 212 223 3 4 3 3 3 2

232 2 2 2 2 2

4 3 3 2 3 2

2 111 1 1

ss

s

P r P r r r r r r P

E r r E r r r r

2 2 2 2 23 4 3 3 2 2

23 122 2 2 2 2 2

4 3 3 2 3 2

12s s

r r r r r r P P

E r r r r r r

(43)

Hoop stress at any radius r in compound cylinder due to internal pressure only is given by

2 2

1 4

2 2 2

4 1

1i

r r P

r r r

(44)




35

3.1.3.4 Principle of superposition

After finding hoop stresses at all the radii, the principle of superposition is applied, i.e. the various

stresses are then combined algebraically to produce the resultant hoop stresses in the compound

cylinder subjected to both shrinkage pressures and internal pressure iP .

3.1.3.4.1

Resultant hoop stress in cylinder 1

Using equations (44) and (26), maximum hoop stress at the inner surfaces of cylinder 1 at1

r

2 2 2

4 1 21 122 2 2 2

4 1 2 1

2i s

r r r P P

r r r r

(45)

Fig.3.6 Superposition of hoop stress due toi

P & residual stress due to12s

P in cylinder 1

3.1.3.4.2 Resultant hoop stress in cylinder 2

Using equations (44) and (31), maximum hoop stress at the inner surfaces of cylinder 2 at2r

2 2 2 22 2

1 12 3 2 23 34 22 2 2 2 2 2

2 4 1 3 2

( ) 2i s s

Pr P r r P r r r

r r r r r

(46)




36

Fig.3.7

Superposition of hoop stress due toiP & residual stress due to

12sP &23sP in cylinder 2

3.1.3.4.3

Resultant hoop stress in cylinder 3

Using equations (39) and (44), maximum hoop stress at the inner surfaces of cylinder 3 at3r

2 2 2 2 21 4 3 4 3

3 232 2 2 2 2

3 4 1 4 3

is

Pr r r r r P

r r r r r

(47)



P in cylinder 3

3.1.3.4.4

Resultant hoop stress in compound cylinder



P &23s

P in all cylinder




37

3.1.3.5 Methodology of optimum compounder cylinder

To obtain optimum values of the contact (shrinkage) pressures12s

P and23s

P which will produce

equal hoop (tensile) stresses in all the three cylinders, maximum hoop stresses given by the

equations (45), (46) and (47) have been equated.

Equating equations (45) and (46) i. e.1 2

and rearranging,

2 2 2

4 1 2122 2 2 2

4 1 2 1

2i s

r r r P P

r r r r

=

2 2 2 22 2

1 12 3 2 23 34 2

2 2 2 2 2

2 4 1 3 2

( ) 2i s s

Pr P r r P r r r

r r r r r

2 2 22 2 2 2 2 2

3 2 32 4 1 1 4 112 232 2 2 2 2 2 2 2 2 2 2

2 1 3 2 4 1 2 4 1 3 2

22s i s

r r r r r r r r r P P P

r r r r r r r r r r r

(48)

Let the ratios1t = 2

1

r

r ,

2t = 3

2

r

r ,

3t = 4

3

r

r (49)

Let

2 2 2 2

2 4 1 21 2 2 2 2 2

2 1 4 1 2

21

21

12

1

2

1

r r r t k

r r r r t

t

t

(50

2 2 2 2 22 2 2 2 2

1 2 3 2 34 1 1 4 22 2 2 2 2 2 2 2 2 2 2 2

4 1 2 4 1 1 2 3 1 2 3

1 1

1 1

t t t t t r r r r r k

r r r r r t t t t t t

(51)

2 2

3 23 2 2 2

3 2 2

22

1

r t k

r r t

(52)

Hence equation (48) becomes

3212 23

1 1

s i s

k k P P P

k k

(53)

Equating equations (46) and (47) i. e. 2 3 and rearranging,

2 2 2 2 2 2 2 2 22 2

1 12 3 2 23 3 1 4 3 4 34 2232 2 2 2 2 2 2 2 2 2

2 4 1 3 2 3 4 1 4 3

( ) 2i s s i

s

Pr P r r P r Pr r r r r r r P

r r r r r r r r r r




38

2 2 2 2 2 2 22 2 2 2

3 2 4 3 4 3 31 1 4 212 232 2 2 2 2 2 2 2 2 2 2 2

3 2 3 4 1 2 4 1 4 3 3 2

2s i s

r r r r r r r r r r r P P P

r r r r r r r r r r r r

(54)

Let

2 2 2

4 2 2

4 2 2 24 1 2

1

1

r r t

k r r t

(55)

2 2 22 2 2 2 2 2

3 2 31 4 2 1 4 25 2 2 2 2 2 2 2 2 2 2 2 2

3 4 1 2 4 1 1 2 3 1 2 3

1 1

1 1

t t t r r r r r r k

r r r r r r t t t t t t

(56)

2 2 2 2

3 4 3 36 2 2 2 2 2

3 2 4 3 3

22

22

12

1

2

1

r r r t k

r r r r t

t

t

(57)

Hence equation (54) becomes

5 612 23

4 4

s i s

k k P P P

k k

(58)

Equations (32) and (36) have been solved to get12s

P and23s

P in terms ofi

P as follows,

5 6 2 3

12

4 6 1 3

/ /

/ /s i

k k k k P P

k k k k

(59)

5 4 2 1

23

3 1 6 4

/ /

/ /s i

k k k k P P

k k k k

(60)

Putting the values of1

t ,2

t and3

t , the equations (35) and (43) can be written as

2 2 2 22 2 1 2

12 12 232 2 2

2 1 1

1 1 12

1 1 1s s

r t t t P P

E t t t

(61)

2 2 23 3 122

23 23 2 2 2

3 2 2

1 1 21

1 1 1

ss

r t Pt P

E t t t

(62)




39

3.2 BURST PRESSURE DETERMINATION

3.2.1 BURST PRESSURE OF SINGLE LAYER CYLINDER

A widely used yield criterion is that of Von Mises. This criterion can be expressed by the

following relationship:

. . 3 y p s y

(63)

Where. . y p = yield-point stress of the material in the simple tension, pound per square inch

.s y =yield limit in simple shear, pound per square inch

We know that the maximum shear stress in three component system is:

max2

t r

s

(64)

Combining the above relationships with eq (63) gives:

.

2 3

y pt r (65)

By making allowance for sign convention with the compressive stressr

, negative, it may be

written as:

r t r

d r

dr

(66)

Substituting fort

in eq (66) by eq (65) gives:

.2

3

y p

r

dr d

r

(67)

Integration of eq (E) gives:




40

. .2ln

3

y p o

i

r p

r

(68)

Where p = internal pressure required to stress the outer surface to the yield point, pound per

square inch

. . y p = yield point of the shell material in single tention, pound per square inch

or =outside radius of the vessel

ir = inside radius of the vessel

Equation (68) was derived for a ideal plastic solid. For this ideal condition the yield strength and

tensile strength have the same value; therefore the bursting strength would be that predicted by

either eq (68) or eq (69).

. .2

ln3

t s o

i

r p

r

(69)

Where. .t s = ultimate tensile strength, pound per square inch

In the actual case for ordinary metals the ultimate tensile stress is appreciably higher the

yield strength, and the stress at bursting will be lie between the yield and ultimate strengths.

Faupel has proposed that eqs. (68) and (69) be modified as follows:

. . . .. . . .

. . . .

2 2ln 1 ln

3 3

y p y pt s o t s o

t s i t s i

r r p

r r

(70)

Equation (70) reduces to:

. . . .

. .

2ln 2

3

y p y po

i t s

r p

r

(71)




41

Equations (70) and (71) proportionally weight the stress values to their ratios; thus, when

the ratio…… is equal to 0.25, Eq (68) contributes 75% and Eq (69) contributes 25% to the

bursting pressure. Faupel reported the tests on the rapture of nearly 100 thick-walled cylinders

fabricated from a variety of high strength steels and Equation (71) was reliable within +-15% for

predicting the observed rapture pressure on a 90% certainty.

3.2.2

BURST PRESSURE OF TWO LAYER COMPOUND CYLINDER

Stress generated in interference between inner tube and jacket due to only for the internal

pressurei

P (excluding the shrinkage pressure) is

2

2 21 3

22 223 1

1i

t atD

PD D

D D D

To generate 2t atD

stress, pressure required at the interference (int er

p )

2

2 2

int 2 3

22 223 2

1er

t atD

P D D

D D D

Or, 2

2 2

3 2int 2 2

3 2

er t atD

D DP

D D

Total interference pressure in the inner surface of the jacket

int inter er total p p P

But pressure required at the inner surface of jacket to burst the jacket of the compound cylinder

from equation (1)

. . . .

. .

2ln 2

3

y p y pob

i t s

r p

r




42

So to calculate the burst pressure of the two layer compound cylinder int er total p will be equal to

b p .

2

2 2. . . . 3 2

2 2

. . 3 2

2 ln 23

y p y po

t atD

i t s

r D D Pr D D

Or,

2 2 2 2. . . . 1 3 3 2

2 2 22 2. . 2 3 23 1

2ln 2 1

3

y p y po i

i t s

r P D D D DP

r D D D D D

2 2 2 2 2 2 2 2 2 22 2. . . . 1 3 3 2 3 1 3 2 3 21 2

2 2 2 2 2 2 2 2 2 2 2 22 2. . 2 3 2 3 1 2 3 1 2 1 3 23 1

2 2ln 2 1 /

3

y p y po ii

i t s

r PD D D D D D D D D D D DP

r D D D D D D D D D D D D D D

From this equation we can find out the burst pressure of the compound cylinderiP as the all other

parameters are known.

3.2.3 BURST PRESSURE OF THREE LAYER COMPOUND CYLINDER

Stress generated in interference between layer 2 and 3 due to only for the internal

pressurei

P (excluding the shrinkage pressure) is

2 2

1 423 2 2 2

4 1 3

1iPr r

r r r

To generate23 stress, pressure required at the interference (

int er p )

So,

2 2

int 3 423 2 2 2

4 3 3

1er p r r

r r r

Or,

2 2

4 3int 23 2 2

4 3

er

r r p

r r




43

Total interference pressure in the inner surface of the outer layer

int er total p =

inter p +

23sP

But pressure required at the inner surface of outer layer to burst the outer layer of the compound

cylinder from equation (1)

. . . .

. .

2ln 2

3

y p y pob

i t s

r p

r

So to calculate the burst pressure of the three layer compound cylinder int er total p will be equal to

b p .

2 2. . . . 5 4 2 14 3

23 2 2

. . 4 3 3 1 6 4

2 / /ln 2

/ /3

y p y poi

i t s

k k k k r r r P

r r r k k k k

Or,

2 2 22. . . . 5 4 2 11 4 34

2 2 2 2 2

. . 4 1 3 4 3 3 1 6 4

2 / /ln 2 1

/ /3

y p y po ii

i t s

k k k k r Pr r r r P

r r r r r r k k k k

From this equation we can find out the burst pressure of the compound cylinder iP as the all other

parameters are known.



Chapter- 4

Analytical Model

This chapter named as Analytical Model where a certain problem is considered and

stress analysis and burst pressure determination is done for corresponding single layer thick

cylinder, two layer compound cylinder and three layer compound cylinder.



Analytical Model

45

4.1 STRESS ANALYSIS

4.1.1 STRESS ANALYSIS OF SINGLE LAYER CYLINDER

Let us consider a thick cylinder with= 25 mm and = 50 mm subjected to an internal

pressure 35 Mpa.

So hoop stress in the inner surface from Lame’s Equation

i = 58.333 Mpa

Hoop stress in the outer surface from Lame’s Equation

0 = 23.333 Mpa

4.1.2

STRESS ANALYSIS OF TWO LAYER COMPOUND CYLINDER

Let us consider a two layer compound cylinder with1r =25mm or

1 D = 50mm,

2r

=37.50mm or2

D =75mm and3

r =50mm or3

D =100mm subjected to an internal pressure 35

Mpa. The material for all three layer is structural steel with E = 210 Gpa, = 0.3,. . y p f =250

Mpa,. .t s

f =460 Mpa.

4.1.2.1

STRESSES DUE TO INTERNAL PRESSURE (i

P ):

Substituting the proper values in equations (14), (15), (16), we got the tangential stresses

at radius1r ,

2r and

3r are:

Table 4.1. Stresses due to internal pressure:

r 25 37.5 50

t 58.35 32.42 23.34

4.1.2.2

STRESSES DUE TO SHRINKAGE PRESSURE ( P ):

Substituting the proper values in equations (17), (18), (19) and (20), we got the tangential

stresses for the shrinkage pressure ( P ) for jacket and inner tube at radius1r , 2r and

3r are:



Analytical Model

46

Table 4.2. Stresses due to shrinkage pressure:

For jacket

r 37.5 50

t 3.57 P 2.57 P

For inner tube

r 25 37.5

t -3.6 P -2.6 P

4.1.2.3

TOTAL SHRINKAGE INTERFERENCE ( ):

Substituting the proper value in Eq (13), we got the shrinkage interference ( ) = 0.008

mm.

The inner diameter of the jacket should be (75 – 0.008) or 74.992 mm.

Table 4.3. Resultant stresses:

Inner tube Jacket

r = 25 r =37.5 r = 37.5 r = 50

Stresses due to P b

(P b=332.78Mpa)

555.74 309.49 309.49 222.96

Stresses due to P

(P= 34.34 Mpa)

-123.4 -89.28 122.59 88.25

Resultant stresses 432.34 220.21 432.08 311.21

4.1.3

STRESS ANALYSIS OF THREE LAYER COMPOUND CYLINDER

Let us consider a three layer compound cylinder with1r =25mm,

2r =33.33mm,

3r

=41.67mm and4r =50mm subjected to an internal pressure 35 Mpa . The material for all three

layer is structural steel with E = 210 Gpa, = 0.3,. . y p f =250 Mpa,

. .t s f =460 Mpa.



Analytical Model

47

Substituting these value in the equation (5) and (6) we will get,

12sP = 0.1048iP

23sP = 0.0673 iP

Substituting the value of12sP and

23sP in the equation (2), (3) and (4) we will get,

1 =

2 =

3 = 1.188

iP

Also substituting in the equation (7) and (8) we will get,

12 = 0.000083957

iP

23 =0.00005547iP

Table 4.4. Parameter due to internal pressure 35 Mpa

1

2

3

12sP 23sP

12

23

41.58 41.58 41.58 3.668 2.3555 0.00294 0.00194

4.2

BURST PRESSURE DETERMINATION

4.2.1

BURST PRESSURE OF SINGLE LAYER

From Faupel Equation (71) burst pressure for the thick cylinder is

b p = 291.44 Mpa

Stress generated due to burst pressure in the inner side

i = 485 Mpa

Stress generated due to burst pressure in the out side

0 = 194 Mpa

4.2.2

BURST PRESSURE OF TWO LAYER COMPOUND CYLINDER

Let assume pressure required in the inner surface to burst the compound cylinder = P b



Analytical Model

48

Here shrinkage pressure P = 0.1032 P b

Interference pressure due to P b:

22

int

2 2

*75 500.93 1

100 75 37.5

er b

PP

Or,int 0.2604

er bP P

So, total interference pressure

inter totalP = (Shrinkage pressure (P) + interference pressure due to P b)

Or, inter totalP = 0.1032 bP + 0.2604 bP

Or, inter totalP = 0.3636

bP

Let assume, pressure required to burst the jacket =bj

P

From Faupel formula,

bjP =

. . . .

. .

2ln 2

3

y p y po

i t s

f f r

r f

Or,2*250 50 250

ln 237.5 4603

bjP

Here, for structural steel. . y p

f =250Mpa and. .t s f =460Mpa

Or,bjP = 121 Mpa.

At the time of bursting, shrinkage pressure (P) will be equal tobj

P .

Hence, P =bj

P

Or 121= 0.3636b

P

Or,b

P = 332.78 Mpa

Therefore, burst pressure of the compound cylinder is 332.78 Mpa



Analytical Model

49

4.2.3 BURST PRESSURE OF THREE LAYER COMPOUND CYLINDER

Stress generated in interference between layer 2 and 3 due to only for the internal pressureiP

(excluding the shrinkage pressure) is

2 2

1 423 2 2 2

4 1 3

1iPr r

r r r

=>23 = 0.8133

iP

To generate23

stress, pressure required at the interference (int er p )

So,

2 2

int 3 423 2 2 2

4 3 3

1er p r r

r r r

=>inter

p = 0.1466i

P

Total interference pressure in the inner surface of the outer layer

inter total p =

inter p +

23sP

=> inter total p = 0.2139

iP

But pressure required at the inner surface of outer layer to burst the outer layer of the compound

cylinder from equation (1)

b p =76.626 Mpa.

So to calculate the burst pressure of the three layer compound cylinder inter total p will be equal to

b p So,

iP = 358.23 Mpa

Therefore, burst pressure of the three layer compound cylinder will be 358.23 Mpa.



Chapter-5

FEM Analysis

This chapter named as FEM Analysis where a finite element analysis is done using ANSYS

Workbench v11 software and the stresses and burst pressure of single layer thick cylinder, two

layer compound cylinder and three layer compound cylinder is validated.



FEM Analysis

51

5.1 STRESS ANALYSIS

5.1.1. STRESS ANALYSIS OF A SINGLE LAYER CYLINDER

Table 5.1. Data for modeling in ANSYS

iP 1

D 2 D

35 50 100

Fig.5.1 Maximum principal stress in thick cylinder

5.1.2. STRESS ANALYSIS OF TWO LAYER COMPOUND CYLINDER

Table 5.2. Data for modeling in ANSYS

iP 1

D 2 D 2i

D 3 D

35 50 75 74.992 100 0.008

Where,

D1 ,D2 = inner and outer diameter of the tube

D2i, D3 = inner and outer diameter of the jacket

Shrink fit is applied between tube and jacket in ANSYS Workbench. Contact between tube and

jacket is applied using contact tool in ANSYS Workbench.



FEM Analysis

52

Fig.5.2 Maximum principal stress in two layer compound cylinder

Fig.5.3 Maximum principal stress in inner tube

Fig.5.4 Maximum principal stress in jacket



FEM Analysis

53

5.1.3. STRESS ANALYSIS OF THREE LAYER COMPOUND CYLINDER

Table 5.3 Data for modeling in ANSYS

iP 1 D 2 D 2i D 3 D 3i D 4 D 12 23

35 50 66.66 66.60 83.34 83.30 100 0.00294 0.00194

Fig.5.5 Maximum principal stress in three layer compound cylinder

5.2. BURST PRESSURE DETERMINATION

5.2.1. BURST PRESSURE OF SINGLE LAYER

Table5.4: Data for modeling Elastic break-down pressure and burst pressure

1 D

2 D

50 100



FEM Analysis

54

Fig.5.6 Maximum principal stress due to Elastic break-down pressure

Fig.5.7 Maximum principal stress due to burst pressure

5.2.2. BURST PRESSURE OF TWO LAYER COMPOUND CYLINDER

Table 5.5 Data for modeling in ANSYS

1 D 2

D 2i D 3

D

50 75 74.924 100 0.076



FEM Analysis

55

Fig.5.8 Maximum principal stress due to burst pressure

Using Design of Experiment in ANSYS Workbench, the pressure required to generate maximum

principle stress 332.34Mpa is 327Mpa.

5.2.3. BURST PRESSURE OF THREE LAYER COMPOUND CYLINDER

Table 5.6 Data for modeling Elastic-breakdown pressure in ANSYS

1 D

2 D

2i D

3 D

3i D

4 D

12

23

50 66.66 66.625 83.34 83.317 100 0.0177 0.0116

Where

1 D ,

2 D = inner & outer diameters of cylinder 1 respectively.

2i D ,

3 D = inner & outer diameters of cylinder 2 respectively for shrink fit.

3i D ,

4 D = inner & outer diameters of cylinder 3 respectively for shrink fit.



FEM Analysis

56

Fig. 5.9 Maximum principle stress due to Elastic break-down Pressure


principle stress 250 Mpa is 203 Mpa.

Table 5.7. Data for modeling Burst pressure in ANSYS

1 D

2 D

2i D

3 D

3i D

4 D

12 23

50 66.66 66.60 83.34 83.30 100 0.0301 0.0199

Fig. 5.10 Maximum principle stress due to Burst Pressure


principle stress 425. 57 Mpa (1.188 * 358.23) on the inner surface of the outer layer is 354 Mpa.



Chapter-6

Discussion

The sixth chapter is the most important chapter named as Discussion. Here the results of

analytical model and FEM analysis are compared and error is also computed and shown that the

errors are in acceptable limits.



Discussion

58

6.1. SINGLE LAYER COMPOUND CYLINDER

Table6.1 . Comparison of burst pressure

Burst pressure(P b)

Analytical 291.44

ANSYS Workbench 291

%Error Less than 1%

6.2. TWO LAYER COMPOUND CYLINDER

Table 6.2. Comparison of burst pressure

Burst pressure(P b)

Analytical 332.78

ANSYS Workbench 327

%Error 1.74

6.3. THREE LAYER COMPOUND CYLINDER

Table 6.3. Comparison of Burst Pressure

Burst Pressure

Analytical 358.23

ANSYS Workbench 354

%Error 1.18



Chapter-7

Conclusions

This chapter is the generalConclusion , where merits and demerits of the present work

are dealt and the readers of the thesis are directed towards the future scope of work or further

continuation or modification of the present work.



Conclusions

60

CLOSING REMARK

The combination of advanced materials and techniques of multi-layering processes are

being employed to develop pressure vessels that have improved pressure capacity, fatigue life,

and reduced weight that enables pressure vessels to ensure higher performance and more

reliability.

In this thesis of shrink fitted compound cylinder, the stress analysis is studied and at the

same time the burst pressure of the compound cylinder is determined. From the discussion it is

clear that the difference in analytical and ANSYS Software results are within acceptable

limits. This difference is due to numerical techniques of Finite Element Method in ANSYS. Since

analytical results are validated by FEM calculations, the design methodology proposed in this

paper can be successfully applied into the real-world mechanical applications for the stress

analysis and to determine the burst pressure of multi-layered compound cylinders to assure best

utilization of material.

From this work, following conclusions are drawn:

I.

Maximum hoop stress at the innermost surface decreases due to compounding. Thus

compound cylinder with same dimension of monobloc cylinder can withstand high

pressure without actual failure.

II.

Compounding of vessel decreases the difference between maximum and minimum hoop

stress. Thus material is safer compared to monobloc or autofrettaged vessel as there is

approximate uniform stress dustribution.

III. Though hoop stress decreases with increase in number of layers, but decrease in hoop

stress is more effective up to n=3. Hence, number of layers used in pressure vessel

depends on requirement.

IV. The burst pressure of compound cylinder is higher than the monobloc cylinder upto three

layer. For the given problem in the thesis the burst pressure of monobloc cylinder is 291

Mpa where of two layer and three layer compound cylinder with same dimension and

same material properties is 332.78 Mpa and 358.23 Mpa.



Conclusions

61

V.

As the outer layer of compound cylinder is subjected to burst first. So provision can be

made to increase the burst pressure using harder material for the outer layer.

7.2 FUTURE SCOPE OF WORK

In this thesis we solely dedicate our focus to the stress analysis and burst pressure

determination of multi layer shrink fitted compound cylindrical pressure vessel. we selected a

common same material for the each layer.

In this present work we did not focus our attention on the fatigue life of the vessel, though

it is well aware that compounding increases fatigue life. There should be a scope remains.

Other than compounding there are other techniques that help increase the pressure

capacity and reliability – like autofrettage technology. Autofrettaging and shrink fitting in

different combination can do the trick. In that case, the percentage of Autofrettage and degree of

interference fit serve the purpose.

The main motto of the study of pressure vessel is to increase pressure capacity and

reliability with adequate safety at a time. And also at the same time it should be more economical.

For these reasons, we need to optimise all the facts related to its performance. Hence optimization

has a crucial role and future prospect in the study of multi layered pressure vessel technology.

This can be done extensively using different material data, different material model,

selecting different types of strain hardening and different yield criteria in different combinations.

In the present time many research are going on functionally graded material. So we also

can incorporate FGM to analyze the same thing.



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62

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Documents

STRESS AND BURST PRESSURE DETERMINATION OF SHRINK FITTED COMPOUND CYLINDRICAL PRESSURE VESSEL