Fatique in tubular structures: developing design tools that assist in analysing and optimising tubular structures that are subject to fatique

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Fatigue in tubular structures

Developing design tools that assist in analysing and optimising tubular structures that are subject to fatigue

by

C.F. Jilderda

A thesis submitted to the department of Design, Production & Management

University of Twente for the degree of Master of Science in Mechanical Engineering

to be defended on Tuesday the 8th of December, 2020 at 14:00.

Graduation committee:

Prof. dr. I. Gibson Ir. H. Tragter Dr. ir. S. Hoekstra

Prof. dr. ir. A.H. van den Boogaard Ir. S. Droste

Ir. J. Moen

Chairman Supervisor Supervisor External member Company mentor Company mentor

An electronic version of this thesis is available at https://essay.utwente.nl/.

Cover photo: steel structure of the Poort van Zuid-Limburg communication mast during the building phase of the project.

Photo by: RGB producties, Waalre

Photo used with written approval from RGB producties.

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Preface

This master thesis assignment is written as the concluding part of the Mechanical Engineering master programme. The research is conducted in cooperation with the Department of Design, Production & Management at University of Twente, and the consulting engineering company ABT bv.

The year 2020 was quite a strange one. I came back in February from an amazing exchange semester in Finland, to start right away with my thesis in March. Due to the Coronavirus pandemic I spent most of my time working on the thesis from home.

Despite the fact that contact with supervisors and colleagues was minimal and only via digital meetings, I learned many things in the process, both about the thesis topic and beyond.

First of all, I would like to sincerely thank my supervisors Hans Tragter and Sipke Hoekstra for their time and energy spent on guiding me. As supervisors they always kept me focused and sharp by triggering me with the right questions during meetings.

I would like to express my great appreciation to the colleagues from ABT: Stefan Droste for introducing me to ABT and arranging everything for me to work well, Matthijs van der Hulst, who thought of making this topic a graduation project and getting me started in the beginning and Rayaan Ajouz for helping me programming in Grasshopper.

I would like to offer special thanks to Joris Moen, who gave me loads of feedback, answers to my questions and always checked in on me every once in a while.

Finally, I would like thank my parents, family and my girlfriend for always being very supportive and encouraging.

Christiaan Jilderda Enschede, November 2020

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Management Summary

Fatigue is a material failure mechanism that might lead to problems when it is not considered in the design phase. There seems to be a trend that fatigue assessment is becoming more important in structural engineering. For tubular structures, fatigue assessment is generally done by using analytical formulas to determine the hot-spot stress. These formulas are complex and the design parameters are intertwined in multiple components of the fatigue assessment, which makes it hard for an engineer to estimate the impact of his design choices.

To gain insight in these parameters, one design tool is developed from scratch, and two existing software tools are programmed to perform fatigue calculations. The new design tool, called Fatigue Design& Assessment tool, is made in Excel. The two existing tools are Smart Synthesis Tool and Grasshopper. Two case studies are performed to analyse how the design tools can be used to optimise the fatigue characteristics of a design.

Each of the software tools provides insight in the design parameters, but all on a different level. The FDA tool can exactly tell how changing one certain parameter affects the outputs.

The SST provides insight by creating and comparing many solutions within the solution space.

The Grasshopper software offers an engineer insight in the fatigue performance while he is still creating the initial shape of the structure. A great advantage of Grasshopper is that no external software is needed to perform the structural analysis, which is input for the fatigue assessment.

The case studies have shown that with the insight provided by the design tools, an engineer is able to improve his design with regard to fatigue life, structure mass, weld volume or combined costs of materials and welding. In the first case study, one proposed concept shows a possible increase in fatigue life of 20%. A different concept shows that a mass reduction of 24% can be achieved, while maintaining the original fatigue life of 50 years.

In the second case study a concept is created in Grasshopper with a slightly different overall geometry. With this concept an improvement in fatigue life of 286% compared to the original design is achieved. The results from both case studies proof that the design tools are most effective when they are deployed early in the design process, or in a design where it is allowed to optimise many parts separately.

The research shows that by means of the analytical formulas, a hot-spot stress fatigue assessment can fairly easily be programmed in different software applications, and that it pays off to do so. The fact that the hot-spot stress method is not limited to tubular joints means that results similar to this research can be achieved for any type of welded joint.

In the end, the choice of fatigue assessment type is discussed. The hot-spot stress method with analytical formulas is an often used and accepted method, but also Finite Element Analysis could have been the basis for a parametric design tool. FE-based design tools are the only way to include any type of joint, since no analytical formulas exist for other than tubular joints.

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Abbreviations

ASTM American Society for Testing Materials CA Constant amplitude loading

CHS Circular hollow sections

CEN European Committee for Standardization

CIDECT International Committee for the Development and Study of Tubular Structures DNV GL Det Norske Veritas - Germanischer Lloyd

FDA Fatigue design & assessment FEA Finite element analysis HAZ Heat affected zone

IIW International Institute of Welding

ISO International Organization for Standardization NDI Non destructive inspection

NEN Nederlandse Norm

RHS Rectangular hollow sections

ROK Richtlijnen ontwerpen kunstwerken SBD Set-based design

SCF Stress concentration factor SHS Structural hollow sections SST Smart Synthesis Tool TIG Tungsten inert gas

VA Variable amplitude loading

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Symbols

A₀ cross-sectional area of chord A1 cross-sectional area of brace B_ref number of repititions to failure

M bending moment

N number of cycles

N_f number of cycles to failure N_g number of wind gust loads

P load

R stress ratio S nominal stress

∆S nominal stress range S_a nominal stress amplitude S_k static extreme wind load

W₀ elastic section modulus of the chord W₁ elastic section modulus of the brace

a crack length

b₀ chord width (RHS) b₁ brace width (RHS) d₀ chord diameter (CHS) d₁ brace diameter(CHS) da/dN crack growth rate e eccentricity

f frequency

f_mc multiplanar correction factor

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Symbols

f_skew skew joint correction factor f_y yield strength

g gap

g⁰ ratio of gap length to chord wall thickness h0 chord height (RHS)

h₁ brace height (RHS) km stress magnification factor k_t stress concentration factor

l chord length between simple supports or contraflexure points m slope of the fatigue strength curve

r weld toe notch radius

t thickness

t0 chord wall thickness t₁ brace wall thickness

α relative chord length β width ratio

γ chord slenderness

γ_Mf partial factor for fatigue strength δ distance to the weld toe

θ brace to chord angle ρ air density

σ local stress in material

∆σ stress range σ_a stress amplitude σ_b bending stress

∆σ_C reference value of the fatigue strength at N = 2 million cycles

∆σ_D fatigue limit for constant amplitude loading σ_e endurance limit

∆σ_E,2 equivalent constant amplitude stress range related to 2 million cycles σ_hs hot-spot stress

∆σ_hs hot spot stress range

∆σ_L cut-off limit for variable amplitude loading

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Symbols

σ_m mean stress σ_m membrane stress

σmax maximum stress in stress history σ_min minimum stress in stress history σ_nlp non-linear peak stress

τ wall thickness ratio

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1 — Introduction

1.1 Fatigue in structures

Fatigue is a material failure mechanism that is induced by cyclic loading. In the fatigue process microscopic damage accumulates with continued cycling of stresses, which can be far below the ultimate material strength. This eventually develops into a crack or other macroscopic damage.

Failure occurs when the remaining cross-section is no longer capable to bear the loads, in that case sudden fracture occurs.

Fatigue has been researched for more than 150 years, with most commonly known the railway axles tests by W¨ohler. The fatigue phenomenon is often encountered in mechanical engineering appliances due to the dynamic nature of machines and vehicles.

In structural engineering, mainly static loads are considered, with the exemption of some structures like bridges and off-shore structures. Residential- and commercial structures or buildings usually are only checked on static loads like dead loads, imposed loads, wind loads and snow loads. While these loads are not truly static, they are neither of such dynamic magnitude that they will lead to fatigue.

Now there seems to be a trend where fatigue is playing a more important role in the structural assessment of residential-, commercial- and industrial buildings. There are several reasons for this:

Optimising weight of structures: This usually goes along with greater static and dynamic stresses. These high stress amplitudes might induce fatigue.

The use of high strength steels: High quality steels will have a greater yield and tensile strength, but the fatigue strength of the material might not increase in the same ratio.

Slender design: Structures that are slender are generally more flexible, and will therefore be more sensitive to vibrations.

Optimising welds: Welding is still a costly process, and so it pays off to not only optimise the structural members, but also the welds. Since fatigue generally initiates at a weld, it is very important to understand the fatigue phenomenon before optimising the welds.

Increasing lifespan: In the context of durability the lifespan of existing structures might be increased to use them longer than they were initially designed for. Fatigue may be one of the criteria that determines the lifespan of a structure.

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Chapter 1 Introduction

1.2 Research scope

This research is conducted as graduation assignment for the master Mechanical Engineering, in the Department of Design, Production & Management at the University of Twente. The topic of this research is created by ABT bv, a consulting engineering company in the Netherlands.

All the work is performed within, and with assistance of the company ABT bv.

The research focusses on structures that consists of tubular members. Only circular hollow sections will be considered, rectangular hollow sections are out of the scope. Out of all the fatigue assessment methods, only the hot-spot stress method is used for the calculations and case studies in this research.

The complete process of assessing how external loads act on a structure is not researched in detail in this project. This is a complete discipline on its own in structural engineering. For this research the most important aspect is what the dynamic behaviour of the loads look like.

In this research, there is a project that is often used as example for a fatigue assessment. This project is a communication mast, of which the structural assessment is performed by ABT in 2013. In the latter part of the research, this project is used as a case study. This tubular structure goes by the name ‘Poort van Zuid-Limburg’ and is shown in figure 1.1.

Figure 1.1: The ‘Poort van Zuid-Limburg’ communication mast at the A2 highway in The Netherlands.

This image shows the mast before it is wrapped in a special cloth. Due to this cloth, the mast catches a lot of wind and is expected to be prone to fatigue. (Photograph by: RGB Producties.)

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1.3 Problem statement

There are numerous methods to calculate the risk that a structure will fail due to fatigue. In structural engineering, standards and regulations are important aspects. The Eurocodes are the ten European standards specifying how structural design should be conducted within the European Union (EU).

The norm EN 1993-1-9 gives methods for the assessment of fatigue resistance of members, connections and joints subjected to fatigue loading. This norm focusses mainly on standard components or connections that are used in structures which are made up of beams and plates.

Each type of component is classified into a level of how well it is resistant to fatigue. This method is therefore often referred to as the classification-, or nominal stress-method.

Tubular structures and joints are often not suitable to be assessed with the classification method.

The many design parameters that belong to a tubular joint and affect fatigue life make it hard to put them into standard levels. Because of this, usually the hot-spot stress method is used to assess fatigue in tubular structures.

The hot-spot stress can be computed with Finite Element analyses or by measuring it experi- mentally. This is possible for any type of design. Now for tubular joints, there is another way to calculate the hot-spot stress; based on analytical formulas. These formulas are the result of years of numerical and experimental research to fatigue in tubular joints.

The big advantage of these formulas is that results can be found without performing a very time consuming FE analysis. Besides that, it allows for quicker iterations. The disadvantage is that there are many design parameters involved in the hot-spot stress method and the analytical formulas are complex. This all makes it difficult for a structural engineer to estimate how his design decisions will impact the fatigue life of a structure.

1.4 Research objective

The main objective is to provide insight in the fatigue behaviour of a structure with regard the design decisions. In order to achieve this, different design tools will be developed or set-up such that they can perform a fatigue assessment. With these design tools, it should be possible to give an engineer information about how his choices affect different parts of the fatigue assessment.

In two case studies it will be researched if it is possible to optimise the fatigue life of a structure without compromising other performances of the structure.

1.5 Research questions

1.5.1 Main research question

How to make it easier for a structural engineer to estimate the impact of changing certain design parameters in order to optimise a tubular structure that is subjected to fatigue?

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1.5.2 Sub research questions

What is fatigue of materials? (Ch. 2)

To what extent does the fabrication process affect fatigue behaviour? (Ch. 3)

What are the methods to assess fatigue in structures? (Ch. 4)

How do the design parameters affect fatigue behaviour? (Ch. 5 and 7)

How can software be used to gain insight in the relation between design parameters and fatigue? (Ch. 7, 8, 9 and 10)

How can a structure be optimised on its fatigue performance, while keeping other performances in consideration? (Ch. 11 and 12)

1.6 Reading guide

This thesis consists of a total of six parts. Each part is divided into a number of chapters.

Part I, Theoretical framework: this part provides the theoretical background and concepts related to the topic. The first chapter discusses the general fatigue theory. The next chapter focuses on the specific fatigue aspects in welded structures. Following is a chapter that describes in detail how a fatigue life assessment can be made. The second to last chapter of the theoretical framework discusses the relevant characteristics of hollow sections as structural members. The last chapter presents the summarised answers to the first three sub research questions.

Part II, Development of design tools: this part starts off with describing how a new design tool was developed for this research, and how it can be used. The next chapter discusses the Smart Synthesis Tool, and how it can be set-up for a fatigue assessment. The following chapter covers the development of how the visual programming software Grasshopper can be used to perform a fatigue assessment. The last chapter of Part II is about a Finite Element Analysis, in which the results are compared to the analytical formulas.

Part III, Case studies: the last results of the research are gained by two case studies. In the first case study, the self made design tool is compared with the SST tool, on how the best tube geometry can be chosen for a structure that is subject to fatigue. The second, and last chapter is about the fatigue assessment in the Grasshopper software, and how it can be used to optimise the total layout of a structure with regard to fatigue.

Part IV, Conclusion and recommendations: this part starts off with the answer to the main research question, followed by the answers to the last three sub research questions.

After that, a discussion of the results is presented, followed by recommendations for future research.

Part V, lists with all the references, figures and tables.

Part VI Appendices: information that is too elaborate to put in the main part of the report is added to the appendices.

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Part I

Theoretical framework

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2 — General fatigue theory

For humans, fatigue is a term used to describe an overall feeling of tiredness or lack of energy (O’Connell, 2019). There are various reasons for humans to become fatigued, but it generally builds up over time. When one is fatigued, it is hard to do the things one normally used to do.

The same goes more or less for fatigue of materials. Even at stresses far below the ultimate strength of a material, microscopic damage will occur. If this takes place over and over again, due to a cyclic load, the damage can accumulate. As explained by Dowling (2012), the damage develops into a crack or other macroscopic damage that might lead to failure of the component.

In case of failure the material is not capable of delivering the same performance as before fatigue occurred. Fatigue of materials is a process that is irreversible.

2.1 Phases of the fatigue process

The fatigue life of a technical product is often split up into two distinctive phases: a crack initiation period and a crack growth period. As Schijve (2009) describes: the initiation period is supposed to include some microcrack growth, but the fatigue cracks are still too small to be visible. In the second period, the crack is growing until complete failure.

2.1.1 Crack initiation

To understand crack initiation, we have to think at microscopic material level. In metals, the crystalline structure allows sliding of planes of atoms over one another, by dislocation movements. This phenomenon called slip is caused by shear stresses at crystal level. Slip is a plastic deformation of the crystal lattice.

Schijve (2009) explains elaborately how slip (a plastic deformation) is possible with low stresses:

“Fatigue occurs at stress amplitudes below the yield stress. At such a low stress level, plastic deformation is limited to a small number of grains of the material. This microplasticity preferably occurs in grains at the material surface because of lower constraint on slip. At the free surface of a material, the surrounding material is present at one side only. The other side is the environment, usually a gaseous environment (e.g. air) or a liquid (e.g. sea water). As a consequence, plastic deformation in surface grains is less constrained by neighbouring grains than in subsurface grains; it can occur at a lower stress level.”

Now with cyclic loads, there will be cyclic slip. Cyclic slip has the crucial characteristic that it is irreversible. There are mainly two reasons why this is the case, which is best explained with visual support.

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Chapter 2 General fatigue theory

Figure 2.1: Cyclic slip leads to crack nucleation. Image by: Schijve (2009).

As explained before in this section, slip is most likely to happen at a free surface. In figure 2.1a it can be seen that a loading causes a slip step in the material. After this slip step, a new fresh surface is exposed to the environment. In most types of environments, an oxide layer will cover the new rim of material. The oxidation appears as monolayers, which strongly adhere to the material surface and are not easily removed (Schijve, 2009).

The second aspect is that in the slip band strain hardening occurs during the increase of the load. The consequence is that a larger shear stress will be present on the same slip band upon unloading (figure 2.1b). This shear stress is felt in the opposite direction compared to the loading step. Reversed slip will therefore most likely take place in the same slip band.

The result of both characteristics make it such that reversed slip, although occurring in the same slip band, will not occur in the same slip plane, but in a plane adjacent and parallel to it. Because of the cyclic loading, these steps are repeated as can be seen in figure 2.1c and 2.1d. The back-and-forth movement of the slip bands leads to the formation of intrusions and extrusions at the surface (Campbell, 2008). Figure 2.2 shows schematically the development of extrusions and intrusions upon cyclic loading.

Figure 2.2: Development of extrusions and intrusions during fatigue. Image by: Higgins (1993).

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An intrusion can be classified as a microcrack, and many of these arise in the crack initiation period. In this period, fatigue is a material surface phenomenon. It is of great importance to make a distinction between the two phases of the fatigue process. According to Schijve (2009) there are several surface conditions that affect the initiation period, but have a negligible influence on the crack growth period.

2.1.2 Crack growth

Once a microcrack or multiple microcracks have been formed in the crack initiation phase, and the cycling continues, the phase of crack growth commences. In this phase, as explained by Stephens, Fatemi, Stephens, and Fuchs (2000), fatigue cracks tend to coalesce and grow along the plane of maximum tensile stress range. The crack growth phase can be divided into two stages: “stage I” (shear mode) and “stage II” (tensile mode).

In stage I, the microcrack at the surface grows further across several grains, in a direction that maximises the local shear stresses and shear strains. In stage II, the crack grows in a zigzag manner essentially perpendicular to, and controlled primarly by, the maximum tensile stress range (Stephens et al., 2000). The crack growth of both stages is visualised in Figure 2.3.

Stage

I Stage II

Free surface

Loading direction

Figure 2.3: Schematic of stages I (shear mode) and II (tensile mode) transcrystalline microscopic fatigue crack growth. Image by: Stephens et al. (2000).

Most of the time, fatigue cracks are growing across grain boundaries, which is referred to as transcrystalline or transgranular. The fatigue cracks prefer to grow within grains because of the fact that there is a lower restraint on slip, compared to the grain boundaries. In this phase of the fatigue process, the crack growth resistance depends on the bulk property of the material.

It is therefore no longer considered a surface phenomenon (Schijve, 2009). During this crack growth phase, so called striations are formed in the material. For the sake of brevity, the more detailed discussion about the mechanisms involved in crack growth is placed in Appendix A, section A.1.

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Chapter 2 General fatigue theory 2.1.3 Fatigue failure

Figure 2.4: Fracture surfaces for fatigue and final brittle fracture in an 18 Mn steel member.

Image by: A. Madeyski, Westinghouse Science and Technology.

A fatigue failure is the result of ongoing crack growing in the material. Every time the crack grows, the remaining cross-section decreases. This is possible until the remaining material is no longer able to bear the stresses or strains. At this point, sudden fracture occurs, which means that the component or structure is separated into two or more parts (Stephens et al., 2000).

The fracture surface of a broken part tells a lot about the load history, material properties and us- age conditions. The fracture surface shows clear- ly two different parts: the fatigue damage of the growing crack, and the sudden fracture surface.

In the fatigue part, there is no macroplastic deformation visible, since the stresses are far below the yield strength of the material. Both sides of the fatigue part usually fit perfectly together after failure. Because of this characteristic, cracks are usually very difficult to see before end of life. Ap- pendix section A.2 offers an overview of all characteristic fatigue failures that can be found on a fracture surface.

2.2 Stress concentrations

Figure 2.5: The crowding and bending of flow lines near obstructions helps to visualise the concentration of stresses and strains near notches. The large section and the small section are the same in both cases, but the tran- sitions are different. Image by: Stephens et al.

(2000).

A technical product often consists of some geometric discontinuities in the form of holes, chamfers, fillets or grooves. Geometric discontinuities are the cause of a local increase of stress, and are therefore also called stress raisers or notches. A designer should always pay special attention to notches, because the high stresses might cause fatigue cracks to initiate at these positions. The presence of notches therefore reduces the fatigue resistance of a component (Dowling, 2012).

In order to understand the severity of different types of geometric discontinuities, the (imperfect) analogy between stresses or strains and liquid flow can be made. Restrictions or enlargements inside a pipe will affect the the flow in a way that is somewhat similar to the stress raising caused by the notches in a design. A positive result is often obtained by ‘streamlining’ the discontinuity, as as indicated in Fig. 2.5.

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The mathematical, numerical and experimental methods to determine a stress concentration factor are discussed in detail in appendix A, section A.3. It depends on the fatigue assessment method how the effects of stress raisers are taken into consideration.

In welded components, the welds will always lead to stress concentrations. This is due to geometric discontinuities in the structure, and the local geometry of the weld, which act as a notch. A stress concentration is always found at the weld toe or the weld root. The fatigue behaviour of welded components is further discussed in chapter 3.

2.3 Constant amplitude loading

Every component in a technical product or structure has its own load history. When the loads are purely static, there will be no risk of failure due to fatigue, but in many situations, the load behaviour varies throughout time. The load spectrum can be rather simple and repetitive, but it can also be the opposite: completely random.

When the stresses alternate between maximum and minimum levels that are constant, this is called constant amplitude loading (CA). The definitions and nomenclature in this section are derived directly from the book by Dowling (2012).

The stress range, ∆σ = σmax− σmin, is the difference between the maximum and minimum values. Averaging the maximum and minimum values gives the mean stress, σ_m. The mean stress may be zero, as in Fig. 2.6(a), but often it is not, as in (b). Half the range is called the stress amplitude σa, which is the variation about the mean. In appendix A, section A.4 more definitions related to cyclic loading are discussed.

Section 9.2 Deﬁnitions and Concepts 419

0

one cycle (a)

time σ_max σa

σa Δσ

σ_min

σmax

σmin σ

0

(b) σ

0

σmax σa

σa

σ =Δσ

(c)

a

a σ

σ Δσ σm

Figure 9.2 Constant amplitude cycling and the associated nomenclature. Case (a) is completely reversed stressing,σm= 0; (b) has a nonzero mean stress σm; and

(c) is zero-to-tension stressing,σmin= 0.

amplitude, σa, which is the variation about the mean. Mathematical expressions for these basic deﬁnitions are

σa = σ

2 = σmax − σmin

2 , σm = σmax + σmin

2 (a, b) (9.1)

The term alternating stress is used by some authors and has the same meaning as stress amplitude.

It is also useful to note that

σmax = σm + σa, σmin = σm − σa (9.2) The signs ofσaandσ are always positive, since σmax > σmin, where tension is considered positive.

The quantitiesσmax,σmin, andσm can be either positive or negative.

The following ratios of two of these variables are sometimes used:

R = σmin

σmax, A = σa

σm

(9.3) where R is called the stress ratio and A the amplitude ratio. Some additional relationships derived from the preceding equations are also useful:

σa = σ

2 = σmax

2 (1 − R), σm = σmax

2 (1 + R) (a, b)

= 1− A

, = 1− R (9.4)

Figure 2.6: Constant amplitude cycling and the associated nomenclature. Case (a) is completely reversed stressing, σm= 0; (b) has a nonzero mean stress σm. Image by: Dowling (2012).

In real life engineering appliances, it is very rare to have this perfect constant amplitude loading on the same mean stress level. The sinusoidal load shape is often found in rotating components, for example in train axles or aeroplane engines. However, in these components, the stress amplitude and mean stress still varies throughout time, depending on factors such as passenger loading or engine mode. The constant amplitude loading is generally used in fatigue tests, as it makes it easier for analysing the numerous parameters.

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2.4 Fatigue testing

Fatigue testing is very important as it contributes to the knowledge about fatigue behaviour in various materials and applications. The extensive literature on fatigue problems illustrates the large variety of purposes of fatigue investigations. Some categories are:

Collecting data on material fatigue properties for material selection by the designer.

Investigations on effects of different surface finishes and production techniques.

Investigations on joints and other structural elements.

Investigations on environmental effects.

Investigations on crack nucleation and crack propagation.

Verification of fatigue prediction models.

2.4.1 Test specimens

In order to avoid scatter in the test results, the test specimens have to be manufactured and treated with care. Especially in low amplitude stress testing, the variability in manufacturing quality can lead to variance in results. However, in some cases it is not desired that the test specimens are of lab quality. Quantitative tests can be done, in which the test specimens are made by the standards of every day work. Welded joints, for example, will contain imperfections, in the weld material but possibly also misalignment between parts. If these characteristics are present in the test, the results might be a better representation of reality.

There is a clear distinction between notched and unnotched specimens. It is obvious that fatigue tests on unnotched specimens cannot give an indication of the material notch sensitivity, a property of relevant engineering significance. However, unnotched specimens can be advantageous for problems related to the quality of the surface finish. A variety of specimens containing stress raisers, called notched specimens, are also used. These permit the evaluation of materials under conditions more closely approaching those in an actual component (Dowling, 2012). Compara- tive experiments on candidate materials should preferably be performed on notched specimens.

In special cases, fatigue tests are performed on full size structures. These tests require more specialized equipment.

2.4.2 Fatigue test machines

One of the most used fatigue testing machines is the rotating bending machine. The device is designed such that the test specimen is loaded with a uniform, pure bending moment which is present over the entire testing length. These type of test machines are often referred to as constant amplitude machines, because the load amplitudes do not change regardless of a change in material properties or crack growth. Constant amplitude fatigue tests are useful to compare different types of materials, surface effects, or small geometry effects like notch diameter.

Hollow section joints are usually tested in big testing rigs, that are equipped with one or more hydraulic cylinders (see figure 2.7). The test specimens for hollow sections are relatively big compared to the fatigue tests discussed before, and therefore also the applied forces are much greater. The testing rig should be very stiff compared to the joint to be sure that the results 11

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are accurate. These tests are performed to research various phases of the fatigue process.

The members are equipped with strain gauges to calculate the stress concentration factor and determine the location of crack initiation. Subsequently the crack growth can be measured. This is done by a special probe that goes into the crack. These type of tests are often performed to verify theories that are created with numerical methods. A fatigue test may also be performed in order to demonstrate the fatigue strength of a certain joint, and use the experimental data instead of the guidelines in norms and regulations.

Fig. 8.10 Chord shear model

Fig. 8.11 Chord preload N_op

Fig. 8.12 Test rig for isolated joint tests θ1

θ2

N2 N1

N0

A

V N0 gap

θ1

θ2

N2 N1

A

N0P

Ν₀ = ΣN_1,2cosθ1,2 + N0P

N0

Ν0,gap = N1 cosθ1 + N0P

"Copy free of charge - for educational purposes only"

Figure 2.7: Testing rig for hollow section joints. Image by: Wardenier (2001).

2.5 Stress based approach

As explained in the section before, fatigue tests are often performed with a constant amplitude loading. The specimen is subjected to a certain nominal stress amplitude S_a, which is repeated until the specimen fails due to fracture at a certain number of cycles N_f. Another test is performed at a higher stress level, which leads to a lower number of cycles to failure. When this process is repeated for a range of different stress amplitudes, the results can be plotted to obtain a stress-life-curve, often abbreviated as S-N curve (Dowling, 2012).

A typical S-N curve can be seen in Figure 2.8. The Y-axis plots the nominal stress amplitude S_a , and the X-axis plots the number of cycles to failure N_f. A characteristic feature of fatigue is that the number of cycles to failure changes rapidly with stress level and the results may be spread over several orders of magnitude (Dowling, 2012). In order to keep S-N curves readable, the number of cycles to failure is generally plotted on a logarithmic scale. A logarithmic scale might also be used for the stress axis. Figure 2.8 shows the advantage of using the logarithmic scale by plotting the same data in two different ways.

Throughout years of fatigue research it is proven that for some materials, fatigue does not lead to failure when the load is below a distinct stress level. This property appears most notably in plain-carbon and low-alloy steels, according to Dowling (2012). This feature can be observed in the S-N curve, as it goes flat after a certain number of cycles to failure. The point where the curve flattens is often referred to as the knee.

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Chapter 2422 Chapter 9 Fatigue of Materials: Introduction and Stress-Based ApproachGeneral fatigue theory

(a)

7075-T6 Al S =0 k =1^m_t 300

200

100

0 10 20 30 40x10⁶

Sa, Stress Amplitude, MPa

10⁴ 10⁵ 10⁶ 10⁷ 10⁸

300

200

100

50

40

30

20

10 (b)

S

ksi 7075-T6 Al

S =0 k =1^m_t

a

N_f, Cycles to Failure N_f, Cycles to Failure

Figure 9.4 Stress versus life (S-N) curves from rotating bending tests of unnotched specimens of an aluminum alloy. Identical linear stress scales are used, but the cycle numbers are plotted on a linear scale in (a), and on a logarithmic one in (b).

(Data from [MacGregor 52].)

Figure 9.5 Rotating bending S-N curve for unnotched specimens of a steel with a distinct fatigue limit. (Adapted from [Brockenbrough 81]; used with permission.)

The results of such tests from a number of different stress levels may be plotted to obtain a stress–life curve, also called an S-N curve. The amplitude of stress or nominal stress,σa or S_a, is commonly plotted versus the number of cycles to failure Nf, as shown in Figs. 9.4 and 9.5.

A group of such fatigue tests giving an S-N curve may be run all at zero mean stress, or all at some speciﬁc nonzero mean stress,σm. Also common are S-N curves for a constant value of the

Figure 2.8: Stress versus life (S-N ) curves from rotating bending tests of unnotched specimens of an aluminum alloy. Identical linear stress scales are used, but the cycle numbers are plotted on a linear scale in (a), and on a logarithmic one in (b). (Data from: MacGregor and Grossman (1952)).

This phenomenon, often called fatigue limit is generally treated as a material property, and listed amongst other material properties. According to Stephens et al. (2000) the knee in the S-N curve is only found in a few materials (notably the low- and medium-strength steels) between 10⁶ and 10⁷ cycles under non-corrosive conditions.

The fatigue limit is generally denoted as S_f, and the definition according to ASTM (2000) is that it is “the limiting value of stress at which failure occurs as N_f becomes very large”.

A typical S-N curve does not separate crack nucleation from growth, and only the total life to fracture is given. The number of cycles to form a small crack in smooth, unnotched or notched fatigue specimens and components can range from a small percentage to almost the entire life.

Stephens et al. (2000) visualised this nicely as can be seen in Fig. 2.9, where applied stress amplitude is plotted versus number of cycles to failure (fracture) and number of cycles to crack nucleation. Linear scales are implied so as not to disguise axis compression due to log-log scales.

The conclusion that can be drawn is that at high stress levels a large fraction of life consists of crack growth, while at lower stress levels, most of the lifespan is spent on crack nucleation.

Fatigue crack growth region

Final fracture

Fatigue crack nucleation

N S

a

Figure 2.9: S-N schematic of fatigue crack nucleation, growth, and final fracture. Image and explanation by: Stephens et al. (2000).

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2.6 Variable amplitude loading

Most of the fatigue tests and resulting stress-life curves are performed with a constant amplitude loading. However, in real life, a product or structure will in most cases be subject to stress amplitudes that change in an irregular manner. An example can be seen in figure 2.10. This is a time-stress diagram that is measured by ABT for the fatigue assessment of a bridge that bears road and railway traffic. The explanations of executing the Palmgren-Miner rule and the rainflow method are retrieved from the book by Dowling (2012).

2.6 Variable amplitude loading

Most of the fatigue tests and resulting stress-life curves are performed with a constant amplitude loading. However, in real life, a product or structure will in most cases be subject to stress amplitudes that change in an irregular manner. iets verder introductie over verschillende load spectra The explanations of executing the Palmgren-Miner rule and the rainflow method are retrieved from the book by Dowling [2012]:

2.6.1 Palmgren-Miner rule

Consider a situation of variable amplitude loading, as illustrated in Fig. 2.16. A certain stress amplitude σa1is applied for a number of cycles N1, where the number of cycles to failure from the (S-N ) curve for σa1 is Nf 1. The fraction of the life used is then N1/Nf 1. Now let another stress amplitude σa2, corresponding to Nf 2 on the S-N curve, be applied for N2 cycles. An additional fraction of the life N2/Nf 2is then used. The Palmgren-Miner rule simply states that fatigue failure is expected when such life fractions sum to unity− that is, when 100% of life is exhausted:

N1

Nf 1

+ N2

Nf 2

+ N3

Nf 3

+ ... =X Nj

Nf j

= 1 (2.7)

This simple rule was employed by A. Palmgren in Sweden in the 1920s for predicting the life of ball bearings, and then it was applied in a more general context by B. F. Langer in 1937.

However, the rule was not widely known or used until its appearance in 1945 in a paper by M. A. Miner. A particular sequence of loading may be repeatedly applied to an engineering component, or, for a continually varying load history, a typical sample may be available. Under these circumstances, it is convenient to sum cycle ratios over one repetition of a given load sequence and then multiply the result by the number of repetitions required for the summation to reach unity:

B_fX N_j Nf j

one rep.

= 1 (2.8)

Here, Bf is the number of repetitions to failure. The application of this equation is illustrated in Fig. 2.16.

468 Chapter 9 Fatigue of Materials: Introduction and Stress-Based Approach

9.9 VARIABLE AMPLITUDE LOADING

As discussed earlier in this chapter, fatigue loadings in practical applications usually involve stress amplitudes that change in an irregular manner. We will now consider methods of making life estimates for such loadings.

9.9.1 The Palmgren–Miner Rule

Consider a situation of variable amplitude loading, as illustrated in Fig. 9.43. A certain stress amplitudeσa1 is applied for a number of cycles N₁, where the number of cycles to failure from the S-N curve forσa1is N_{f 1}. The fraction of the life used is then N₁/Nf 1. Now let another stress amplitudeσa2, corresponding to Nf 2 on the S-N curve, be applied for N2 cycles. An additional fraction of the life N₂/Nf 2is then used. The Palmgren–Miner rule simply states that fatigue failure is expected when such life fractions sum to unity—that is, when 100% of the life is exhausted:

N₁ N_{f 1} + N₂

N_{f 2} + N₃

N_{f 3} + · · · = N_j

Nf j = 1 (9.33)

This simple rule was employed by A. Palmgren in Sweden in the 1920s for predicting the life of ball bearings, and then it was applied in a more general context by B. F. Langer in 1937. However, the rule was not widely known or used until its appearance in 1945 in a paper by M. A. Miner.

A particular sequence of loading may be repeatedly applied to an engineering component, or, for a continually varying load history, a typical sample may be available. Under these circumstances, it is convenient to sum cycle ratios over one repetition of a given load sequence and then multiply the result by the number of repetitions required for the summation to reach unity:

Bf

Nj

Nf j

one rep.= 1 (9.34)

σ σ

σ σ σ

N_f3 N_f1 N_f2 a3

a1

a2 a

N N

N

N . . . = 1

+ +

1 f1

2 f2 f3

3+ σa2 σa1

σa3

N₁ N₂ N₃cycles

time

N , Cycles to Failuref

Figure 9.43 Use of the Palmgren–Miner rule for life prediction for variable amplitude loading which is completely reversed.

Figure 2.16: Use of the Palmgren-Miner rule for life prediction for variable amplitude loading which is completely reversed [Dowling, 2012].

Figure 2.10: Example of a real life stress history plot.

2.6.1 Palmgren-Miner rule

Consider a situation of variable amplitude loading, as illustrated in Fig. 2.11. A certain stress amplitude σ_a1 is applied for a number of cycles N₁, where the number of cycles to failure from the (S-N ) curve for σa1 is N_f1. The fraction of the life used is then N1/N_f1. Now let another stress amplitude σ_a2, corresponding to N_f2 on the S-N curve, be applied for N₂ cycles. An additional fraction of the life N₂/N_f2 is then used. The Palmgren-Miner rule simply states that fatigue failure is expected when such life fractions sum to unity− that is, when 100% of life is exhausted:

N₁ N_f1 + N₂

N_f2 + N₃

N_f3 + ... =X N_j

N_{f j} = 1 (2.1)

This simple rule was employed by A. Palmgren in Sweden in the 1920s for predicting the life of ball bearings, and then it was applied in a more general context by B. F. Langer in 1937.

However, the rule was not widely known or used until its appearance in 1945 in a paper by M. A. Miner. A particular sequence of loading may be repeatedly applied to an engineering component, or, for a continually varying load history, a typical sample may be available. Under these circumstances, it is convenient to sum cycle ratios over one repetition of a given load sequence and then multiply the result by the number of repetitions required for the summation to reach unity:

B_fX Nj

N_{f j}

one rep.

= 1 (2.2)

Here, B_f is the number of repetitions to failure. Figure 2.11 shows how the equation can be used.

In many cases, there are mean stresses involved in a variable amplitude loading history. Usually, the effect of the mean stress can be taken into account by calculating the equivalent completely reversed stress, before applying a S-N curve. However, for fatigue in welded joints, there 14

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Chapter 2 General fatigue theory 468 Chapter 9 Fatigue of Materials: Introduction and Stress-Based Approach

9.9 VARIABLE AMPLITUDE LOADING

As discussed earlier in this chapter, fatigue loadings in practical applications usually involve stress amplitudes that change in an irregular manner. We will now consider methods of making life estimates for such loadings.

9.9.1 The Palmgren–Miner Rule

Consider a situation of variable amplitude loading, as illustrated in Fig. 9.43. A certain stress amplitude σa1 is applied for a number of cycles N₁, where the number of cycles to failure from the S-N curve for σa1 is Nf 1. The fraction of the life used is then N1/Nf 1. Now let another stress amplitude σa2, corresponding to N_{f 2} on the S-N curve, be applied for N₂ cycles. An additional fraction of the life N2/Nf 2is then used. The Palmgren–Miner rule simply states that fatigue failure is expected when such life fractions sum to unity—that is, when 100% of the life is exhausted:

N₁

N_{f 1} + N₂

N_{f 2} + N₃

N_{f 3} + · · · = Nj

N_{f j} = 1 (9.33)

This simple rule was employed by A. Palmgren in Sweden in the 1920s for predicting the life of ball bearings, and then it was applied in a more general context by B. F. Langer in 1937. However, the rule was not widely known or used until its appearance in 1945 in a paper by M. A. Miner.

A particular sequence of loading may be repeatedly applied to an engineering component, or, for a continually varying load history, a typical sample may be available. Under these circumstances, it is convenient to sum cycle ratios over one repetition of a given load sequence and then multiply the result by the number of repetitions required for the summation to reach unity:

B_f N_j Nf j

one rep. = 1 (9.34)

σ σ

σ σ σ

N_f3 N_f1 N_f2

a3

a1

a2 a

N N

N

N . . . = 1

+ +

1 f1

2 f2 f3

3+ σa2

σa1

σa3

N₁ N₂ N₃cycles

time

N , Cycles to Failuref

Figure 9.43 Use of the Palmgren–Miner rule for life prediction for variable amplitude loading which is completely reversed.

Figure 2.11: Use of the Palmgren-Miner rule for life prediction for variable amplitude loading which is completely reversed. Image by: Dowling (2012).

is a consensus that mean stress does not have a notable effect. This will be substantiated in chapter 3. For above-mentioned reason the detailed equations of calculating equivalent completely reversed stress are not further discussed.

2.7 Strain based approach

Compared to the stress-based approach, the strain-based approach differs in the sense that it emphasizes on local stresses and strains, rather then the nominal (average) stresses. In the localised analysis both elastic and elastic strain is taken into consideration. Since the strain based approach allows detailed consideration of fatigue where local yielding is involved, it is of added value in situations of relatively short live and ductile materials (Dowling, 2012). However, the theory behind this approach also applies in situations where there is little plasticity at long fatigue lives, which makes it suitable to be used also instead of the stress-based approach.

2.8 Fracture mechanics

This approach is fundamentally different from the stress-based an the strain-based approaches.

Fracture mechanics does not make a life estimation based on experimental data or design codes, but it calculates how a crack propagates in the material until failure.

When the crack with length a propagates due to a loading cycle N , the increase in crack length is called ∆a. Now the rate of crack growth can be written as ∆a/∆N or, for small intervals, by the derivative da/dN. A value of fatigue crack growth rate, da/dN ,is the slope at a point on an a versus N curve (Dowling, 2012).

Crack growth analysis needs information about the initial crack as input. If the initial crack can not be observed or measured, assumptions can be made. Crack growth analysis is for this reason generally not used in the design phase of products and structures. It can however be useful in various other applications. Examples are: structural integrity analysis for planes and brigdes, often combined with NDI, or forensic engineering.

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3 — Fatigue in welded structures

Welding is one of the most used production techniques to connect two or more metal parts to each other. Due to the nature of the welding process (rapid local changes in the micro-structure due to heat) there are various problems and challenges encountered that are characteristic for welding only. Welding has become a discipline on its own, which is reflected by the extensive research and literature in this subject, as well as the numerous standards and design codes, certified by welding institutes and organizations. Not only welding as a production technique is a discipline on its own, also the subject of fatigue in welded members differs on many points from the general fatigue theory. This chapter covers the various aspects that characterize fatigue in welded structures.

3.1 Weld imperfections

In unnotched specimens generally a substantial part of the total life is spent on crack initiation.

In welded structures, there are many factors that effectively reduce the first phase of the fatigue process. As discussed by Wahab and Alam (2004) these are weld imperfections such as slag inclusions at weld toes, under-cut, residual stresses, lack of penetration or misalignment.

Figure 3.1: Some terms and defects of a butt welded joint. Image by: Schijve (2009).

Fatique in tubular structures: developing design tools that assist in analysing and optimising tubular structures that are subject to fatique

Fatigue in tubular structures

Developing design tools that assist in analysing and optimising tubular structures that are subject to fatigue

C.F. Jilderda

Preface

Management Summary

Contents

Abbreviations

Symbols

1 — Introduction

1.1 Fatigue in structures

1.2 Research scope

1.3 Problem statement

1.4 Research objective

1.5 Research questions

1.6 Reading guide

Part I

Theoretical framework

2 — General fatigue theory

2.1 Phases of the fatigue process

2.2 Stress concentrations

2.3 Constant amplitude loading

2.4 Fatigue testing

2.5 Stress based approach

Fatigue crack growth region

Final fracture

Fatigue crack nucleation

N S

2.6 Variable amplitude loading

2.7 Strain based approach

2.8 Fracture mechanics

3 — Fatigue in welded structures

3.1 Weld imperfections