Stress-strain And Failure Analysis Of Steel Pipe With Stress Concentration Factors (SCFs) Under Static 3 And 4 Point Bend Conditions.
SUPERVISOR DECLARATION
“I hereby declare that I have read this thesis and in my opinion this thesis is sufficient in terms of scope and quality for the award of the degree of
Bachelor of Mechanical Engineering (Material & structures)”
Signature : ……… Supervisor : ………
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STRESS-STRAIN AND FAILURE ANALYSES OF STEEL PIPE WITH STRESS CONCENTRATION FACTORS (SCFs) UNDER STATIC 3 AND 4
POINT BEND CONDITIONS
TIE SING ANN
This report is submitted in
partial fulfillment of requirement for degree of
Bachelor of Mechanical Engineering (Structure and Materials)
Universiti Teknikal Malaysia Melaka (UTeM)
Faculty of Mechanical Engineering University Teknikal Malaysia Melaka
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DECLARATION
“I hereby declare that the work in this report is my own except for summaries and quotation which have been duly acknowledged.”
Signature : ………
Author : ………
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Specially dedicated
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ACKNOWLEDGEMENT
It is a pleasure to thank those who have given their help to make this thesis possible. First of all, I would like to thank Universiti Teknikal Malaysia Melaka (UTeM) for giving me this opportunity to conduct this study as my final year project as a partial fulfillment of Bachelor in Mechanical Engineering (Material & Structures ).
I am heartily thankful to Pm. Abd. Salam Md Tahir for acting as my final year project supervisor. It is an honor for me to have such a great supervisor who willing to spend his precious time to give advices, guidance, support, and encouragement to me throughout the course of this Project Sarjana Muda.
I owe my deepest gratitude to my beloved parents who raise me up and supporting me this opportunity to study in UTeM. Their financial and mental support to make this project possible.
Lastly, I offer my regards and blessings to all of those who supported me either directly or indirectly in any aspect during the completion of the project. Without all of you, I believe that this thesis would not have been possible.
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ABSTRAK
Sistem dan struktur-struktur paip digunakan dengan meluas dalam banyak industri. Tiub dan paip digunakan terutamanya dalam bidang stuktur. Oleh itu, tekanan memainkan kriteria penting pada paip tersebut. Tetapi, kekuatan paip selalu dikira dengan menganggap paip tersebut dalam keadaan sempurna. Malah, semua paip mangandungi retak kecil yang bertindak sebagai penubuh tekanan atau penumbuan tegasan. Selain daripada retak kecil, lubang yang berlainan bentuk juga dihasilkan dengan sengaja pada paip demi kegunaannya. Penumbuan tekanan akan mengurangkan kekuatan asal sebatang paip. Penumbuan tekanan ini boleh dikirakan dengan faktor penumbuan tekanan, K. Faktor penumbuan tekanan bermaksud nisbah takanan maksima kepada tekanan nominal. Dengan itu, ujian lenturan digunakan untuk memastikan paip mempunyai kemuluran dan kekuatan yang cukup untuk menahan lenturan tanpa gagal. Tekanan lenturan berlaku apabila satu komponen itu dikenakan degan daya yang sebalik dari bercuba menghulurkan atau mengecut komponen tersebut. Ini akan menghasilkan gabungan daripada tekanan tegangan dan mampatan. Terdapt dua jenis ujian lenturan iaitu ujian liku tiga mata dan liku empat mata. Dalam laporan ini, paip keluli dianalisiskan dengan faktor penumbuan tekanan (SCFs) yang berlainan iaitu lubang bulat dan slot dengan kedua-dua ujian tersebut. Kedua-dua spesimen tersebut didilampirkan dengan lapan tolok terikan pada lokasi yang tertentu dari dekat ke jauh daripada penumbuan tekanan yang dihasilkan. Tolok terikan ini disambung kepada logger data untuk mendapatkan keputusan apabila tekanan dikenakan pada spesimen dengan mengunakan mesin ujian sejangat. Graf terikan terhadap jarak tolok terikan dari penubuh tekanan dilukis berasakan keputusan yang terdapat dan mengirakan faktor penumbuan tekanan secara eksperimen. Selain itu, faktor penumbuan tekanan juga dikirakan secara teori dan dibandingkan. Daripada keputusan projek ini, ditunjukkan dengan jelas bahawa bentuk yang berlainan akan mempunyai nilai faktor penumbuan tekanan yang berlainan. Tetapi, kedua-dua spesimen lubang bulat dan slot menujukkan bahawa
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apabila menjauhi dari penubuh tekanan, terikan akan berkurang. Selain itu, terikan adalah berkadar terus dengan daya yang dikenakan. Daya yang tinggi akan menghasilkan terikan yang tinggi.
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ABSTRACT
Piping system and structures are widely used in many industries. Tube and pipe are mainly use in structural application. So, stress plays an important criterion on these pipes. But the strength of the pipes usually calculated by assumed the pipes are perfect. In fact, all pipes contain small cracks that act as stress raisers or stress concentration. Besides small cracks, some holes of different shape also purposely produced for its uses. Stress concentration will reduce the strength of the pipe. The stress concentration is measured by stress concentration factor, K. Stress concentration factor is defines as the ratio of the maximum stress to nominal stress. So, bending tests are carried out to ensure that the pipe has a sufficient ductility and strength to stand bending without fracture. Bending stress occurs when a component is loaded by force which instead of trying to stretch or shrink the component. It will generate a combination of tensile and compressive stress in the load-carrying components. There are two types of bending tests, which are three-point bend and four-point bend tests. In this report, steel pipes are analyzed with different types of stress concentration factors (SCFs) which are circular hole and slot under both bending tests. Each specimen was attached with eight strain gauges on desired location from near to far from the stress raisers. These strain gauges are connected to the data logger to take the reading when the load is applied by the universal testing machine. Graphs of strain against distance of the strain gauges from the stress raisers were plotted based on the data recorded to calculate the experimental stress concentration factor. Besides, the theoretical stress concentration factor also calculated by using analytical method. Both theoretical and experimental results were compared for validation. From the results, it is clearly shown that different shapes will have the different value of the stress concentration factor. But both circular hole and slot specimens showed that the strain will decrease as the distance from the stress raisers is increase. Besides, the strain is directly proportional to the applied load. Higher loads will contribute higher strain.
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TABLE OF CONTENT
CHAPTER CONTENT PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRAK v
ABSTRACT vii
TABLE OF CONTENT viii
LIST OF TABLES xii
LIST OF FIGURES xiv
LIST OF SYMBOLS xvi
LIST OF ABBREVIATIONS xvii
LIST OF APPENDIX xviii
CHAPTER 1 INTRODUCTION 1
1.1 Introduction 1
1.2 Problem Statement 2
1.3 Objective 3
1.4 Scope 3
CHAPTER 2 LITERATURE REVIEW 4
2.1 What Constututes Failure 4
2.1.1 Excessive Elastic Deformation 4 2.1.2Yielding, or Excessive Plastic Deformation 5
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CHAPTER CONTENT PAGE
2.1.3Fracture 6
2.1.4 Safety Factor 7
2.2 Stress Concentration Factor 8
2.2.1Selection of Nominal Stress 10
2.2.1.1 Tension Bar With a Hole 10 2.2.2 Accuracy of Stress Concentration Factors 14 2.2.3 Decay of Stress Away From The Peak Stress 14 2.2.4 Stress Concentration as a Two-dimensional
Problem 15
2.2.5 Stress Concentration as a Three-dimensional
Problem 17
2.2.6 Plane and Axisymmetric problems 19 2.3 Three Point Bend and Four Point Bend 20 2.3.1 Theory of three point and four point bending test 20
2.3.2 Hooke’s Law 21
2.3.3 Strain 21
2.3.4 Stress 22
2.3.5 Bending 22
2.3.6 Shear and Moment Diagrams 23
2.4 Application of Steel Pipes and Tubes 24 2.4.1 Industrial Steel Pipes and Tubes 24
2.4.2 Steel Water Pipes and Tubes 25
2.4.3 Sewage Pipes 25
2.4.4 Steel Scaffolding 26
2.4.5 Oil and Gas Pipes 26
2.4.6 Steel Casing Pipes and Tubes 26 2.4.7 Structural Steel Pipes and Tubing 27
2.4.8 Steel Tubes for Furniture 27
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CHAPTER CONTENT PAGE
2.5 Data acquisition 28
2.5.1 Strain gauges 28
2.5.2 Data logger 30
CHAPTER 3 METHODOLOGY 31
3.1 Experimental method 31
3.1.1 Three point bending bend tests 31
3.2 Material and specimens 32
3.2.1 Stress concentration factor of notched specimens 33
3.3 Equipments and Machine used 34
3.4 Symbol labeling 36
3.4.1 Symbol labeling for specimens without strain
gauges 36
3.4.2 Symbol labeling for specimens with strain gauges 37
3.5 Procedure 39
3.5.1 Three point bend test for specimens without strain gauges
3.5.2 Three point bend test for specimens with strain
gauges 41
3.5.3 Four point bend test for specimens with strain
gauges 42
CHAPTER 4 RESULT 43
4.1 Three point bend 43
4.1.1 Bending test results without strain gauges 43 4.1.1.1Bend test results for un-notched specimens 43 4.1.1.2 Bend test results for notched specimens of circular hole 45 4.1.1.3 Bend test results for notched specimen
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CHAPTER CONTENT PAGE
4.1.1.4 Bend test results for notched specimen
with strain gauges 49
4.2 Four point bend 60
4.2.1 Bend test results for notched specimen
of circular hole 60
4.2.2 Bend test results for notched specimen of slot 67
CHAPTER 5 ANALYSIS AND DISCUSSION 72
5.1 Bending test results for specimen without
stress concentration 72
5.2 Bending test results for specimen with
stress concentration 73
5.3 Discussion for results of specimens with strain
gauges reading 74
CHAPTER 6 CONCLUSION AND RECOMMENDATIONS 78
6.1.Conclusion 78
6.2 Recommnedations 79
REFERENCES 81
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LIST OF TABLES
NO. TITLE PAGE
2.1 Stress Concentration Factor as a Function of Poisson’s Ratio for a Shaft in Tension with a Groove 19
3.1 Symbol and meaning 36
3.2 Symbol labeling of specimens without strain gauges 36 3.3 Symbol labeling of specimens with strain gauges 37 3.4 Distance of strain gauges from center of circular hole 38 3.5 Distance of strain gauges from center of slot 39
4.1 Average results of three point bending tests 45
4.2 Average data of three point test on circular hole specimen 46 4.3 Average data of slot specimen under three point bend 48
4.4 Strain gauges data of SP-NHG 49
4.5 Average of strain gauges data of SP-NHG 50
4.6 Strain concentration factor for specimen with hole as
SCF under three point bend 54
4.7 Strain gauges data of SP-NSG 55
4.8 Average strain gauges data of SP-NSG 56
4.9 Strain concentration factor of slot specimen under three
point bend test 59
4.10 Strain gauges data of SP-NHG 60
4.11 Average strain gauges data of SP-NHG 61
4.12 Strain concentration factor of SP-NHG 66
4.13 Strain gauges data of SP-NSG 67
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NO. TITTLE PAGE
4.15 Strain concentration factor for specimen with slot as
SCF under four point bend test 71
5.1 Comparison of ultimate stress 73
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LIST OF FIGURES
NO. TITLE PAGE
2.1 (a) Tension bar with hole; (b) torsion bar with groove 11 2.2 Stress concentration factor Ktg and Ktn for the tension
of a finite width thin element with a circular hole
(Howland 1929-1930) 13
2.3 (a) Plane stress; (b) plane strain 15
2.4 Round bar with a circumferential groove and torsional
loading 17
2.5 Hyperbolic circumferential groove in a round bar 18 2.6 Shear and moment diagrams of three point bend test 23 2.7 Shear and moment diagrams of four point bend test 24
2.8 Wheatstone bridge 29
2.9 Strain gauges 29
2.10 Data logger 30
3.1 Isometric view of circular hole specimen 32
3.2 Isometric view of slot specimen 32
3.3 Instron Universal Test Machine 35
3.4 Schematics of (a) three-point bend test apparatus;
(b) four point bend test apparatus 35 3.5 Location of strain gauges on circular hole specimen 37
3.6 Location of strain gauges on slot specimen 38
3.7 Bluehill software 39
3.8 Selection of method 40
4.1 Graph of un-notched specimen (SP-UN1) 44
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NO. TITTLE PAGE
4.3 Graph of circular hole specimen (SP-NH1) 45
4.4 Graph of circular hole specimen (SP-NH2) 46
4.5 Graph of slot specimen (SP-NS1) 47
4.6 Graph of slot specimen (SP-NS2) 47
4.7 Location of strain gauges of circular hole specimen 50
4.8 Graph of applied load vs strain of SP-NHG 51
4.9 Graph of strain vs distance of strain gauges of SP-NHG 52 4.10 Graph of strain vs distance of strain gauges of Sp-NHG 53
4.11 Location of strain gauges of slot specimen 56
4.12 Graph of applied load vs strain of Sp-NSG 57
4.13 Graph of strain vs distance of strain gauges of SP-NSG 58
4.14 Graph of applied load vs strain of SP-NHG 62
4.15 Graph of applied load vs strain of circular hole specimen 63 4.16 Graph of strain vs distance of strain gauges of SP-NHG 64 4.17 Graph of strain vs distance of strain gauges of Sp-NHG 65
4.18 Graph of applied load vs strain of SP-NSG 69
4.19 Graph of strain vs distance of SP-NSG 70
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LIST OF SYMBOLS
σw = working stress
σ0 = yield strength σu = tensile strength
N0 = factor of safety based on yield strength Nu = factor of safety based on tensile strength
σmax = Maximum stress
σnom = Nominal stress
P = Total Load
H = Height
h = Notch height
Ktg = Gross stress concentration factor Ktn = Net stress concentration factor
Ke = Experimental stress concentration factor
E = Modulus of elasticity
I = Moment of inertia
L = Span length of bending member
M = Maximum bending moment
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LIST OF ABBREVIATIONS
SCFS = Stress Concentration Factors
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LIST OF APPENDICES
NO. TITLE PAGE
A Gantt Chart For Projek Sarjana Muda(PSM) 1 85
B Gantt Chart For Projek Sarjana Muda(PSM) 2 86
C Chart of stress concentration factor of circular hole 87
D Chart of stress concentration factor of slot 88
E Detailed drawing if circular hole specimen 89
F Detailed drawing of slot specimen 90
G (i) Circular hole specimen with strain gauges 91
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1 CHAPTER 1
INTRODUCTION
1.1 Introduction
A pipe or tube is a tubular section or hollow cylinder. It is usually, but not necessarily of circular cross-section. It is mainly used to convey substances which can flow such as liquid and gas (fluids), slurries, powder, masses of small solids. Besides, it also can be used for structural applications such as for trusses, offshore structures and aerospace members where weight reduction is a prime concerned. This is because hollow pipe is far stronger per unit weight than solid members. With the same diameter and the same material, a solid rod is stronger than a hollow pipe. However, with the same weight and material, the hollow pipe is stronger than a solid rod. Because a hollow pipe is lighter than a solid rod for the same outside diameter and the hollow pipe would have a larger diameter in order to match weights with the solid rod and a larger diameter rod or pipe is stronger than a smaller diameter one.
Pipe or tube is generally specified by a nominal diameter with a constant outside diameter (OD and the thickness) according to a certain standard. It generally manufactured to one or several international or national industry standards. While similar standards exist for specific industry tubing, tube is often made to custom sizes and a broader range of diameters and tolerances. Many industrial and government standards exist for the production of pipe and tubing.
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Bending tests are carried out to ensure that a metal pipe or tube has a sufficient ductility and strength to stand bending without failure. Bending stress occurs when a component is loaded by force which, also trying to stretch or shrink the component. Those bending forces generate a combination of tensile and compressive stresses in the load-carrying components, known as bending stresses. In bending tests, the load is applied vertically or transversely at either one or two points. As a result, these tests are referred to as three-point and four point bending, respectively.
In this project, steel pipe or tube contained stress concentration factors (SCFs) was studied or analyzed. Stress concentration factor is defined as the ratio of the greatest stress in the area of a notch or other stress raiser to the corresponding nominal stress. It is a theoretical indication of the effect of stress concentrators on mechanical behavior. Stress concentration factor usually is higher than the empirical fatigue notch factor or strength reduction ratio, because it does not take into account stress relief due to local plastic deformation. The examples of shapes or discontinuities that cause these stress concentrations are cracks, sharp corners, holes, and, changes in the cross-section area of the object.
1.2 Problem Statement
An object is strongest when force that applied on it is evenly distributed over its area, so a reduction in area for example caused by stress raisers, will results in a localized increase in stress. A material can fail, via a propagating crack, when a concentrated stress exceeds the material’s fracture strength. The real fracture strength of a material is always lower than the theoretical value because most of the materials contain stress raisers that cause stress concentrations. Failures always start at stress concentration areas, so removing such defects will increase the fatigue strength of the structures or components. To prevent the materials fail, the real stress-strain distributions should be investigated with different stress concentration factors (SCFs) which subjected to static 3 and 4 point bend loads.
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3 1.3 Objective
The objective of this project is to investigate and define stress-strain distributions and failure analysis of steel pipe or tube with different stress concentration factors (SCFs) subjected to static 3 and 4 point bend loads.
1.4 Scope
The scopes of this project involve the following:
• Conduct the literature study on stress-strain and failure analysis of structure in general. This includes the effect of different type of stress concentration factors (SCFs) such as circular or elliptical holes and notches.
• Design the test specimens from the steel tube that contains the proposed SCFs. • Determination of theoretical SCFs (kt) by using analytical or Finite Element
(FE) method.
• By using the strain gages, measure the strains at selected and critical locations to investigate and validate the theoretical analysis and predictions.
• The use of finite element (FE) method may also be considered to verify or validate the measured and analytical stress-strain data.
• Define the actual failure stress / collapsed load of the proposed design by conducting the real 3 and 4 bend until the specimen fail.
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4 CHAPTER 2
LITERATURE REVIEW
2.1 What constitutes failure?
Structural members and machine elements can fail to perform their intended functions in three general ways:
i. Excessive elastic deformation
ii. Yielding, or excessive plastic deformation iii. Fracture
i.
An understanding of the common types of failure is important in good design because it is always necessary to relate the loads and dimensions of the member to some significant material parameters which limits the load-carrying capacity of the member.
2.1.1 Excessive elastic deformation
Two general types of excessive elastic deformation may occur: (1) excessive deflection under condition of stable equilibrium, such as the deflection of beam under gradually applied loads; (2) sudden deflection, or buckling, under conditions of unstable equilibrium.
Excessive elastic deformation of a machine part can mean failure of the machine just as much as if the part completely fractured. For example, a shaft which is too
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flexible can cause rapid wear of bearing, or the excessive deflection of closely mating parts can result in interference and damage to the parts. The sudden buckling type of failure may occur in a slender column when the axial load exceeds the Euler critical load or when the external pressure acting against a thin-walled shell exceeds a critical value. Failures due to excessive elastic deformation are controlled by the modulus of elasticity, not by the strength of the material. Generally, little metallurgical control can be exercised over the elastic modulus. The most effective way to increase the stiffness of a member is usually by charging its shape and increasing the dimensions of its cross section.
2.1.2 Yielding, or excessive plastic deformation
Yielding, or excessive plastic deformation, occurs when the elastic limit of the metal has been exceeded. Yielding produces permanent change of shape, which may prevent the part from functioning properly any longer. In a ductile metal under conditions of static loading at room temperature yielding rarely results in fracture, because the metal strain hardens as it deforms, and an increased stress is required to produce further deformation. Failure by excessive plastic deformation is controlled by the yield strength of the metal for a uniaxial condition of loading. For more complex loading conditions the yield strength is still significant parameter, but it must be used with a suitable failure criterion. At temperatures significantly greater than room temperature metals no longer exhibit strain hardening. Instead, metals can continuously deform at constant stress in a time-dependent yielding known as creep. The failure criterion under creep conditions is complicated by the fact that stress is not proportional to strain and the further fact that the mechanical properties of the material may change appreciably during service.
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LIST OF APPENDICES
NO. TITLE PAGE
A Gantt Chart For Projek Sarjana Muda(PSM) 1 85
B Gantt Chart For Projek Sarjana Muda(PSM) 2 86
C Chart of stress concentration factor of circular hole 87
D Chart of stress concentration factor of slot 88
E Detailed drawing if circular hole specimen 89
F Detailed drawing of slot specimen 90
G (i) Circular hole specimen with strain gauges 91
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1 CHAPTER 1
INTRODUCTION
1.1 Introduction
A pipe or tube is a tubular section or hollow cylinder. It is usually, but not necessarily of circular cross-section. It is mainly used to convey substances which can flow such as liquid and gas (fluids), slurries, powder, masses of small solids. Besides, it also can be used for structural applications such as for trusses, offshore structures and aerospace members where weight reduction is a prime concerned. This is because hollow pipe is far stronger per unit weight than solid members. With the same diameter and the same material, a solid rod is stronger than a hollow pipe. However, with the same weight and material, the hollow pipe is stronger than a solid rod. Because a hollow pipe is lighter than a solid rod for the same outside diameter and the hollow pipe would have a larger diameter in order to match weights with the solid rod and a larger diameter rod or pipe is stronger than a smaller diameter one.
Pipe or tube is generally specified by a nominal diameter with a constant outside diameter (OD and the thickness) according to a certain standard. It generally manufactured to one or several international or national industry standards. While similar standards exist for specific industry tubing, tube is often made to custom sizes and a broader range of diameters and tolerances. Many industrial and government standards exist for the production of pipe and tubing.
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Bending tests are carried out to ensure that a metal pipe or tube has a sufficient ductility and strength to stand bending without failure. Bending stress occurs when a component is loaded by force which, also trying to stretch or shrink the component. Those bending forces generate a combination of tensile and compressive stresses in the load-carrying components, known as bending stresses. In bending tests, the load is applied vertically or transversely at either one or two points. As a result, these tests are referred to as three-point and four point bending, respectively.
In this project, steel pipe or tube contained stress concentration factors (SCFs) was studied or analyzed. Stress concentration factor is defined as the ratio of the greatest stress in the area of a notch or other stress raiser to the corresponding nominal stress. It is a theoretical indication of the effect of stress concentrators on mechanical behavior. Stress concentration factor usually is higher than the empirical fatigue notch factor or strength reduction ratio, because it does not take into account stress relief due to local plastic deformation. The examples of shapes or discontinuities that cause these stress concentrations are cracks, sharp corners, holes, and, changes in the cross-section area of the object.
1.2 Problem Statement
An object is strongest when force that applied on it is evenly distributed over its area, so a reduction in area for example caused by stress raisers, will results in a localized increase in stress. A material can fail, via a propagating crack, when a concentrated stress exceeds the material’s fracture strength. The real fracture strength of a material is always lower than the theoretical value because most of the materials contain stress raisers that cause stress concentrations. Failures always start at stress concentration areas, so removing such defects will increase the fatigue strength of the structures or components. To prevent the materials fail, the real stress-strain distributions should be investigated with different stress concentration factors (SCFs) which subjected to static 3 and 4 point bend loads.
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3 1.3 Objective
The objective of this project is to investigate and define stress-strain distributions and failure analysis of steel pipe or tube with different stress concentration factors (SCFs) subjected to static 3 and 4 point bend loads.
1.4 Scope
The scopes of this project involve the following:
• Conduct the literature study on stress-strain and failure analysis of structure in general. This includes the effect of different type of stress concentration factors (SCFs) such as circular or elliptical holes and notches.
• Design the test specimens from the steel tube that contains the proposed SCFs. • Determination of theoretical SCFs (kt) by using analytical or Finite Element
(FE) method.
• By using the strain gages, measure the strains at selected and critical locations to investigate and validate the theoretical analysis and predictions.
• The use of finite element (FE) method may also be considered to verify or validate the measured and analytical stress-strain data.
• Define the actual failure stress / collapsed load of the proposed design by conducting the real 3 and 4 bend until the specimen fail.
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4 CHAPTER 2
LITERATURE REVIEW
2.1 What constitutes failure?
Structural members and machine elements can fail to perform their intended functions in three general ways:
i. Excessive elastic deformation
ii. Yielding, or excessive plastic deformation iii. Fracture
i.
An understanding of the common types of failure is important in good design because it is always necessary to relate the loads and dimensions of the member to some significant material parameters which limits the load-carrying capacity of the member.
2.1.1 Excessive elastic deformation
Two general types of excessive elastic deformation may occur: (1) excessive deflection under condition of stable equilibrium, such as the deflection of beam under gradually applied loads; (2) sudden deflection, or buckling, under conditions of unstable equilibrium.
Excessive elastic deformation of a machine part can mean failure of the machine just as much as if the part completely fractured. For example, a shaft which is too
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5
flexible can cause rapid wear of bearing, or the excessive deflection of closely mating parts can result in interference and damage to the parts. The sudden buckling type of failure may occur in a slender column when the axial load exceeds the Euler critical load or when the external pressure acting against a thin-walled shell exceeds a critical value. Failures due to excessive elastic deformation are controlled by the modulus of elasticity, not by the strength of the material. Generally, little metallurgical control can be exercised over the elastic modulus. The most effective way to increase the stiffness of a member is usually by charging its shape and increasing the dimensions of its cross section.
2.1.2 Yielding, or excessive plastic deformation
Yielding, or excessive plastic deformation, occurs when the elastic limit of the metal has been exceeded. Yielding produces permanent change of shape, which may prevent the part from functioning properly any longer. In a ductile metal under conditions of static loading at room temperature yielding rarely results in fracture, because the metal strain hardens as it deforms, and an increased stress is required to produce further deformation. Failure by excessive plastic deformation is controlled by the yield strength of the metal for a uniaxial condition of loading. For more complex loading conditions the yield strength is still significant parameter, but it must be used with a suitable failure criterion. At temperatures significantly greater than room temperature metals no longer exhibit strain hardening. Instead, metals can continuously deform at constant stress in a time-dependent yielding known as creep. The failure criterion under creep conditions is complicated by the fact that stress is not proportional to strain and the further fact that the mechanical properties of the material may change appreciably during service.