DEGREE PROJECT IN ENGINEERING MECHANICS, SECOND CYCLE, 30 CREDITSSTOCKHOLM, SWEDEN 2021

Comparison of Fatigue Life Evaluation MethodsJämförelse av beräkningsmetoder för utmattning

LOUISE HEDBERG LUNDBLAD

ANNA LUND

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Abstract

The aim of this thesis is to investigate a selected set of fatigue life calculation methods andevaluate if they are suitable for fatigue life estimation of truck components at Scania. Failuredue to fatigue can be cause by road induced vibrations, which is an inevitable phenomenontrucks are exposed to. By estimating when and where these components will fail, they can bedesigned to reduce the amount of failure per vehicle.

Three types of fatigue life calculation methods, namely equivalent stress methods, critical planemethods and spectral methods, have been evaluated. These are methods for calculating fatiguelife in both the time domain and the frequency domain. The chosen calculation methods havebeen evaluated based on their sensitivity to input parameters, their accuracy on predictingfatigue life and their ability to find the critical areas where the components are most likely tofail. The methods have also been compared to a method already implemented at Scania. Toevaluate the methods, two different components were used. The first component was designedto give a multiaxial stress state and the other was a real truck component where fatigue datahad been collected from a shake rig test at Scania.

It was found that all investigated methods were successful in finding critical areas where failurewill occur. However, the resulting estimated fatigue life had a very low accuracy. To draw anyconclusions about the accuracy of the fatigue life estimations, a model that better reflects thedynamics of the real truck component is needed. Therefore, the conclusion is that the chosenmethods can be used for finding critical areas in a component but not to determine the absolutetime to failure for the model used. However, the method already implemented at Scania wasequally successful in finding the critical areas and it has a much shorter computational timethan the methods in the time domain. Since it is already implemented and used, the Scaniamethod is recommended for the purpose of finding the critical areas of a component.

A sensitivity study was conducted in order to investigate the influence of a variation of materialparameters on the fatigue life calculated with the different methods. This study showed thatthe SN-curve parameters are important for the resulting fatigue life of methods that considerthe endurance limit, and, therefore, that the choice of SN-curve is important. Since the roadinduced vibrations in this study caused load signals where the majority of the cycles were foundbelow the endurance limit, methods that account for the endurance limit have to be used forcalculations on components experiencing similar conditions. Furthermore, it was found thatthe resulting stress signal from the FE-analysis using input data from the shake rig test wasnon-Gaussian, this makes the results from all the chosen frequency domain methods invalid.To use these methods, they need to be extended to consider non-Gaussian signals.

Sammanfattning

Syftet med detta examensarbete ar att undersoka ett antal utvalda metoder for utmattnings-berakning och utvardera om dessa ar lampliga for att uppskatta livslangden pa lastbilskom-ponenter hos Scania. Haveri pa grund av utmattning kan orsakas av vibrationer fran vagen,ett fenomen som paverkar komponenter pa lastbilar. Genom att uppskatta nar och var dessakomponenter gar sonder kan de konstrueras for att minska antalet haverier.

Olika typer av metoder for utmattningsberakning i bade tidsdomanen och frekvensdomanenhar utvarderats. Dessa inkluderade ekvivalenta spannings-metoder, kritiska plan-metoder samtspektrala metoder. Metoderna har utvarderats med avseende pa deras kanslighet for variationi materialparametrar, hur den beraknade livslangden skiljer sig mot verkliga tester och hurbra de ar pa att hitta de kritiska omradena pa en lastbilskomponent. Detta har aven jamfortsmot en berakningsmetod som redan anvands pa Scania. Tva olika komponenter anvandesfor att utvardera metoderna. En av komponenterna var designad for att ge ett multiaxielltspanningstillstand och en var en riktig lastibilskomponent med data uppmatt fran ett skaktestpa Scania.

Alla studerade metoder fann de kritiska omradena dar utmattningsbrott riskerar att uppsta.Daremot visade det sig att berakningsmetoderna inte lyckades estimera livslangder som lag inarheten av de som uppmattes under testet i skakriggen. En mer verklighetsnara modell vilkenbattre motsvarar de dynamiska egenskaperna av systemet behovs for att kunna dra en slutsatsom modellernas traffsakerhet gallande estimeringen av livslangden.

For andamalet att hitta kritiska omraden rekommenderas metoden som redan anvands hosScania, eftersom denna var lika framgangsrik att hitta dessa, men gjorde det pa en avsevartkortare tid. Darutover identifierades att spanningssignalen fran FE-analysen, dar indata franskakriggen anvandes, inte var gaussisk. Detta innebar att signalen inte uppfyller kraven forde spektrala metoderna och darmed att resultaten fran berakningarna pa lastbilskomponenteninte gar att anvanda for att dra nagra slutsatser.

Kanslighetsanalysen visade att de metoder som tar hansyn till utmattningsgransen ar kansligafor andringar i SN-parametrar. Detta beror pa att manga cykler, for det studerade lastfallet, lagnara utmattningsgransen och att antalet cykler som ingick i berakningarna darfor paverkadesstort av SN-parametrarna. Eftersom de vibrationer som uppstar da lastbilar framfors pa vagarkan ge upphov till manga cykler med amplituder nara utmattningsgransen bor endast metodersom kan ta hansyn till utmattningsgransen anvandas vid dessa fall.

Acknowledgements

We would like to sincerely thank our supervisors at Scania, Dr. Ulrika Lagerblad and M.Sc ErikLundblad, for their guidance and endless patience throughout our master thesis work.

We are very grateful for the support and helpful advice from our supervisor Bo Alfredsson,Professor at the Department of Engineering Mechanics. We would also like to acknowledgeour supervisor and examiner Marten Olsson, Professor at the Department of EngineeringMechanics, for stepping in and helping us in our time of need.

We are also very thankful for all the support from the team RTLC at Scania, thanks for makingus feel welcome and taking your time to help us out when needed.

Nomenclature

αi Bandwidth parameter

G(f) PSD matrix of the stress components

Ml Multiaxial spectral moment

Q Constant matrix

γ Irregularity factor

Γ(•) Gamma function operator

σ(θ, φ) Stress in the θ, φ-plane

σe Endurance limit

σF Findley stress

σM Matake stress

σN Stress amplitude

σAMPS Absolute maximum principal stress

σa Stress amplitude

σeq,AMPS Equivalent absolute maximum principal stress

σeq,S Sines equivalent stress

σf,b Fatigue life under reversed bending

σf,tc Fatigue life under fully reversed tension-compression

σh Hydrostatic stress

σm Mean stress

σt,f Fatigue life under repeated tension

σvM von Mises stress∑(•) Summation operator

τ(θ, φ) Shear stress in the θ, φ-plane

τ vM Octahedral von Mises shear stress

τa Shear stress amplitude

τf,t Fatigue life under fully reversed torsion

ξ Correction factor

AMPS Absolute Maximum Principal Stress

BT Benasciutti-Tovo

BT.75 Benasciutti-Tovo α.75

Ca Basquin material constant

E[0+] Expected number of mean up-crossings

E[P ] Expected number of peaks

EVMS Equivalent von Mises Stress

f Frequency

fσa,i Dirlik’s probability density function for rainflow stress amplitudes

G(f) One-sided PSD

Geq(f) Equivalent one-sided PSD

kF Findley material parameter

kM Matake material parameter

kS Sines material parameter

m Fatigue slope exponent

Ml l :th spectral moment

Ml,eq l :th equivalent uniaxial spectral moment

n Number of applied cycles

OC Ortiz-Chen

PSD Power Spectral Density

RMS Root mean square

S(f) Two-sided PSD

SvM Signed von Mises

T Test duration

N Number of load reversals to failure

D Fatigue damage

E[•] Expected value operator

FE Finite Element

tr{•} Trace operator

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Signal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Time history methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Equivalent stress theories . . . . . . . . . . . . . . . . . . . . 52.2.2 Critical plane theories . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Dirlik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Benasciutti-Tovo . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Ortiz-Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Random response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 FEMFAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Equivalent stress methods . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Critical plane theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Random response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 FEMFAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8 Sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Fatigue life comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Sensitivity to material parameters . . . . . . . . . . . . . . . . . . . . . 31

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1 L-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Mudguard bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 FEMFAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Bibliography 40

1. INTRODUCTION

1 IntroductionThis thesis was conducted at Scania, with the aim to study fatigue calculation methods.Fatigue damage can be caused by road induced vibrations, and this is an inevitable phenomenonwhen it comes to vehicle components. By estimating when and where these components willfail, they can be designed to reduce the amount of failures per vehicle.

One way of determining the fatigue life of a component is to measure the time to failureon a component in service, this however, is very time consuming and therefore ineffective.To increase efficiency, so called vibration shake rigs are used to simulate the road inducedvibrations on the relevant components, decreasing the duration of the measurements. Eventhough the shake rigs increases the efficiency of the tests, they are still very time consumingand expensive, which is why fatigue calculation methods are of interest. Furthermore, it isadvantageous to test new designs without creating prototypes. With fatigue calculations, itis possible to evaluate multiple design concepts simultaneously and, with the knowledge fromthe calculated results, only necessary prototypes and tests need to be made, saving both timeand money.

However, determining fatigue life using numerical methods is a challenging engineering task,which is why a variety of methods have been developed. These methods have different strengthsand weaknesses, for example in their ability to handle different types of stress signals. There-fore, a method that is good for one application might not be valid for another application.

On a truck, the mudguard bracket is the component that attaches the mudguard to the vehicle,and this is a component in which fatigue damage has been an issue. Therefore, shake rig testshave been performed on this component. These tests can be used when evaluating fatiguesimulations, which is why, together with the fact that fatigue failure had been recorded, thisspecific component is studied in this thesis.

Different methods of calculating fatigue life have been evaluated in this thesis. These werechosen from three different categories, namely equivalent stress methods, critical plane methodsand spectral methods. The aim is to compare these in terms of accuracy and sensitivity toinput parameters. The sensitivity is tested on a component designed to give a multiaxial stressstate from a uniaxial input signal. The accuracy is tested on a real truck component, thecalculated fatigue life is compared to data collected at Scania and, in addition, the point offailure in the simulation vs. physical tests is investigated.

1.1 Purpose

The purpose of this thesis work is to compare different types of methods for estimation of thefatigue life of a vehicle component, considering multiaxial random variable load cases. Themethods are evaluated with regard to the estimated fatigue life and their sensitivity consideringinput parameters. This is done to see which of the different methods are effective and accurateenough to be of interest for implementation at Scania.

1

1. INTRODUCTION

1.2 Scope and limitations

The subject of fatigue damage can be very complex, especially considering multiaxial stressstates and random load signals. Therefore, it is necessary to limit the scope of the project. Lowcycle fatigue is not applicable when studying the truck components chosen for this thesis, thus,this thesis is limited to high cycle fatigue. The load signals of interest might be non-stationaryduring long periods of time but can be considered stationary when looking at shorter periodsof the signals. The signals considered in this thesis is treated as stationary. The scope isalso limited to Gaussian, wide-banded signals and a plane stress state of the components isassumed. Furthermore, the simulations are set up as linear analyses.

2

2. THEORY

2 TheoryIn real life, many engineering components are subjected to vibrations which are of a randomnature and not possible to predict exactly, these are called random vibrations. A fatigueanalysis on a random vibration process is usually accomplished by either using time historymethods or spectral methods, both will be described later in this chapter.

2.1 Signal properties

Different fatigue prediction methods require the studied load process to have certain properties.The scope of this thesis limits the study to stationary processes. A stationary random processis defined as a process which statistical properties remain constant over time. For a processto be truly stationary, it would have to be infinite. This, however, does not occur in real lifesince all events studied will occur over a finite period of time. Therefore, stationery in thisthesis is defined as a process that is stationary during the entire load cycle. For example, as atruck is loaded or unloaded the process goes from one stationary state to another stationarystate. This means that the processes is non-stationary when looking at both load cases butstationary when looking at only one of them. The same can be said about the loads inducedby travelling on one type of road and switching to another type [1].

In order to use some of the spectral methods, a Gaussian process is required. A processis Gaussian if its values are normally distributed at all times. Furthermore, the knowledgeabout whether the studied process is wide-banded or narrow-banded also affect the selection ofmethod. A narrow-band process shows a clear spike around a well-defined frequency, whereasthe spectrum of a wide-band process is spread over a wider frequency range [1].

2.2 Time history methods

For the task of determining the fatigue life of a component subjected to a time varying load arainflow analysis is usually used. The rainflow analysis translates the varying stress into setsof simpler stress reversals, identified as load types or load cycles. These load cycles representthe original load spectrum. By identifying these cycles, the damage of each cycle can becalculated and the fatigue life determined. Rainflow counting is a well-established and widelyused method. Therefore, the theory behind it will not be described in this thesis. More indepth information can be found in [2, 3].

Uniaxial stress states are required to use rainflow counting. However, road induced vibrationscreate multiaxial stress states, which is a great challenge when it comes to fatigue calculations.Therefore, an equivalent uniaxial stress state needs to be created from the multiaxial stressstate. This can be done using different methods. In this work, two different types of timehistory methods are investigated in more detail, namely equivalent stress theories and criticalplane theories. For the equivalent stress theories, multiaxial stress histories are used to producean equivalent uniaxial stress history on which rainflow counting can be applied. The criticalplane theories evaluate all material planes to find the plane with the highest damage parameter.This plane is called the critical plane [4]. It is of interest to consider both theories, since

3

2. THEORY

they have different limitations. For example, the equivalent stress theories are only able tohandle proportional stressing. A stressing is considered proportional when the principal stressdirections and ratios remain constant, it is otherwise considered non-proportional. The criticalplane theories are able to handle both proportional and non-proportional stressing [4]. Theequivalent stress methods investigated are the Absolute Maximum Principal Stress method,the Signed von Mises method and Sines method. The critical plane theories studied are theFindley method and the Matake method.

The fatigue properties of a material can be described by a SN-curve, also called Wohler curve,shown in Figure 1. By conducting fatigue experiments, the number of cycles to failure isrecorded for a given load level. This is repeated for a number of different load levels, and theresult is presented as a SN-curve where the fatigue life is approximated for different number ofcycles. The endurance limit is the stress at which the material is not subjected to fatigue, in thisthesis it is defined as the stress at which the material survives 106 load cycles. The endurancelimit is marked as σe in Figure 1. The SN-curves usually represents a failure probability of50%.

Figure 1: Example of a SN-curve, where the endurance limit, σe, is marked. Values for load cyclesbelow 103 are not tested by Scania and, therefore, not included in the graph.

The SN-curve can be described according to Basquin’s equation

σN =

(N

Ca

)1/−m

, (1)

4

2. THEORY

where σN is the stress amplitude associated with the number of load reversals N to failure, Cais a constant and m is the fatigue slope exponent [5, 6]. At Scania, a value of m in the rangeof 3 - 6 is usually used for steel components whereas polished test specimens used in fatiguetests typically have a m-value of 10. An important detail to consider before fitting the data toBasquin’s equation is whether the number of cycles or the stress amplitude is the dependentvariable during the fatigue experiments. Equation (1) implies that the number of cycles is anindependent variable, however, these experiments are usually conducted at a selected stressamplitude recording the number of cycles until failure, making the number of cycles dependenton the stress amplitude. Thus, Basquin’s equation needs to be rewritten as

N = Caσ−mN . (2)

If the test conditions when determining the SN-curve and the conditions of the component ofinterest differs, the curve might have to be adjusted. This is done in order to consider factorsthat could have an impact on the fatigue life of the component, such as the surface roughness[7]. Furthermore, the chosen critical plane theories require an additional correction of theSN-curve when applied to cases with finite, high cycle fatigue [8]. More information about thiscorrection factor is given in Subsection 2.2.2.

The fatigue damage of every load cycle, calculated by the rainflow analysis and Basquin’sequation, is used to determine the total damage. This is done using the Palmgren-Minerdamage hypothesis which defines the damage parameter D as

D =k∑i=1

niNi

, (3)

where ni is the number of applied cycles of a certain stress level i, Ni is the fatigue limit forthe corresponding stress level, found in the material’s SN-curve, and D is the damage whichis equal to 1 at a failure probability 50%. By accumulating the damage from all types of loadcycles found by the rainflow counting, the fatigue life of an entire stress time history can becalculated using the equation

life =1

D, (4)

and the expected number of cycles until failure can be determined [7].

2.2.1 Equivalent stress theories

The equivalent stress theories are based on the concept of converting a multiaxial stress stateto a uniaxial one by finding an equivalent stress. The equivalent stress can be found usingdifferent criteria. The different criteria are relevant for different types of stressing. For example,the Absolute Maximum Principal Stress and the Signed von Mises methods cannot accountfor the effect of shear stresses on the fatigue life. However, this is not a problem in the Sinesmethod. This is why it is of interest to look at a variety of criteria.

The most basic criterion is called the Absolute Maximum Principal Stress (AMPS) criterionand uses the principal stresses to find the equivalent uniaxial stress history. The criterion is

5

2. THEORY

formulated asσeq,AMPS = sign · σAMPS, (5)

where σAMPS is the absolute maximum principal stress and sign is the sign that the absolutemaximum principal stress had before the absolute was calculated [4].

The Signed von Mises (SvM) criterion is based on the von Mises yield criterion and is givenby

σeq,SvM = sign · σvM , (6)

where σvM is the von Mises stress. There are different ideas of how to define sign in thiscriterion, here the sign is the sign of the principal stress that has the absolute maximum value[4].

One of the most commonly known criterion is the Sines criterion. This criterion is based onthe octahedral von Mises shear stress amplitude, τ vMa , given by

τ vMa =√σ2x,a + σ2

y,a + σ2z,a − σx,aσy,a − σy,aσz,a − σx,aσz,a + 3(τ 2xy,a + τ 2yz,a + τ 2xz,a) (7)

and the Sines equivalent stress is given by

σeq,S = τ vMa + kSσh,m, (8)

where σh,m is the mean hydrostatic stress of the entire time history. The mean stress parameter,kS, is a material parameter given by

kS =2√

2(σf,tc − σf,t

2

)σf,t

, (9)

where σf,tc and σf,t are the fatigue limit under fully reversed tension-compression and repeatedtension, respectively [8–12]. The value of kS ranges from 0.05 - 0.32 for studied steels in[13].

2.2.2 Critical plane theories

The critical plane theories are based on the evaluation of the stress states projected ontodifferent material planes, to find the plane where the damage parameter, D, is maximized. Bycalculating and combining the normal and shear stresses to find the critical plane, a multiaxialstress state can be reduced to an equivalent uniaxial stress state [4, 14]. The normal and shearstresses are found by projecting the time varying multiaxial stress onto each plane accordingto

τA(θ, ϕ) = (τxycos2θ + 0.5 · (σy − σx)sin2θ) sinϕ

τB(θ, ϕ) = 0.5 · (σxcos2θ + σysin2θ + τxysin2θ)sin2ϕ

σN(θ, ϕ) = (σxcos2θ + σysin

2θ + τxysin2θ)sin2ϕ,

(10)

6

2. THEORY

for each time step, where σN is the stress normal to the plane and τA and τB are the shearstresses in the y”- and x”-direction respectively, illustrated in Figure 2. The parameters τxy,σx and σy are the stress components caused by the varying load history and ϕ and θ are theangles describing the normal directions of the plane. These equations are valid when assumingplane stress, no surface pressure and that the failure is due to a single dominant crack, whichis true for most metallic alloys [15].

Figure 2: Illustration of the directions of the projected stresses and angles defining the plane.

Depending on the stress state of the process, either τA or τB will be more dominant towardscrack initiation on different planes. By rainflow counting the dominant shear stress for allangles, (θ, ϕ), the shear stress amplitude history can be found for each plane. The shear stressamplitude history and the normal stress history are used to find the critical plane. However,the different critical plane theories are separated by the criterion on which the critical plane isdefined.

The Findley and Matake criteria are both widely used, which is why they have been chosenfor comparison in this thesis. The first criterion studied is the Findley criterion. Findleydefined the critical plane as the material plane that maximizes the Findley stress, σF , givingthe criterion

σF = maxθ,ϕ

(τa + kFσn,max), (11)

where τa is the shear stress amplitude and σn,max the maximum normal stress [4, 9]. Thematerial parameter kF is given by

kF =2− r−1

2√r−1 − 1

(12)

where r−1 = σf,b/τf,t. The material parameters τf,t and σf,b are the fully reversed torsion andthe fully reversed bending endurance limit respectively. These can be obtained from uniaxialfatigue testing. The value of kF is typically in the range of 0.2 - 0.3 for ductile material [4, 8].However, in [13], the studied steels had kF -values ranging from 0.05 - 0.3.

7

2. THEORY

The other criterion studied is the Matake criterion. It is similar to the Findley criterion exceptfor how the critical plane is defined. In the Matake criterion, the critical plane is defined as thematerial plane that maximizes the shear stress amplitude [4]. This definition gives the Matakestress

σM = maxθ,ϕ

(τa) + kMσn,max, (13)

where the material parameter kM is given by

kM =2− r−1r−1

. (14)

To apply Basquin’s Equation (2) to the Findley stress, the Basquin material parameter, Ca,needs to be multiplied with a correction factor, ξF [8]. This correction factor is given by

ξF =

√1 + kF

3. (15)

No information about a correction factor could be found for the Matake criterion. Due to thesimilarity between the two criteria, one could suggest that the same correction factor could beused for both criteria, changing the parameter k depending on which criterion is used.

2.3 Spectral methods

Random loading in the time domain is complicated to evaluate and explicit time domain cal-culations can require long computational time [16]. This can be avoided by instead consideringthe problem in the frequency domain, where the computational time can be reduced. This canbe done by introducing the Power Spectral Density (PSD), here denoted S(f), which is a rep-resentation of the random process on the frequency spectrum [1, 16]. Methods for computingthe PSD along with more information of its characteristics are well documented, and can befound in for example [17–19].

The function S(f) is a two-sided PSD valid for frequencies that are both positive and negative.A one-sided PSD limited to the positive frequency range is introduced as

G(f) =

{2S(f), 0 < f <∞S(0), f = 0.

(16)

This format is the most commonly used for this type of application [1, 20].

The geometrical shape of the PSD can be described by

Ml =

∫ ∞0

|f |lG(f)df (17)

where Ml are called spectral moments [1, 20–23]. The moments M0,M1,M2 and M4 have beenfound to sufficiently describe the characteristics of the random process which are necessary

8

2. THEORY

for fatigue analysis [1, 21]. Some important statistical properties of the process itself can bedescribed using the spectral moments, such as the zero-order moment which is the variance ofthe process, M0 = σ2

X . [1, 22].

There exists a large amount of different spectral methods to approximate the fatigue life of astructure. The methods that are included in the scope of this project are Dirlik, Benasciutti-Tovo, Benasciutti-Tovo α.75 and Ortiz-Chen.

All of the chosen spectral methods are so called uniaxial spectral methods and cannot handlemultiaxial stress states directly. To apply these methods on a multiaxial process, a uniaxialequivalent stress needs to be found. There exists several different methods to find the uniaxialequivalent stress, in this thesis the Equivalent von Mises Stress method (EVMS) has been usedfor all spectral methods. The EVMS is, due to its simplicity, one of the more frequently usedmethods for frequency domain equivalent stress calculations. This method was chosen since itis widely used and well documented, which makes it easy to reproduce and compare to otherstudies. The method is based on the well-known von Mises stress, σvM , which, in the timedomain, is defined as

σ2vM = σ2

x + σ2y + σ2

z − σxσy − σxσz − σyσz + 3(σ2xy + σ2

xz + σ2yz). (18)

By writing σvM in Voigt notation as σvM = (σx, σy, σz, σxy, σxz, σyz), the expected value of thevon Mises stress can be expressed as

E[σ2vM ] = tr{QE[σvMσvM

T ]}, (19)

where Q is the constant matrix

Q =

1 −1/2 −1/2 0 0 0−1/2 1 −1/2 0 0 0−1/2 −1/2 1 0 0 0

0 0 0 3 0 00 0 0 0 3 00 0 0 0 0 3

. (20)

The expected value E[σvMσvMT ], in Equation (19), is in fact equal to the expected value of

the PSD matrix of the stress components, G(f). This equality means that the expected valueof the von Mises stress calculated using the time-domain formulation and the frequency-domainformulation is also equal [24, 25], which leads to the definition of the EVMS as

Geq(f) = Tr{QG(f)}. (21)

As the spectral moments are a function of the PSD, the EVMS can be applied directly ontoEquation (17), giving the following expression

Ml,eq = Tr{QM l}, (22)

where Ml,eq is the equivalent uniaxial spectral moment and M l the corresponding multiaxialspectral moment.

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2. THEORY

The EVMS has some limitations to be aware of before applying the method. Since the methodis based on the von Mises stress, it is limited to materials for which the slope of the SN-curveis equal for tension-compression and torsion [26]. This is fulfilled for many isotropic materials.To use the EVMS, the process is also assumed to be a Gaussian zero-mean process [25].

It is important to remember that the choice of definition for the uniaxial equivalent stress hasan impact on the results of the spectral methods. By using the same definition for all chosenmethods, the influence of this definition is assumed to be reduced [16].

2.3.1 Dirlik

Dirlik’s method is one of the most commonly used spectral methods for fatigue damage cal-culations. It is a method that is of interest to Scania, and it is therefore relevant to includeit in this thesis. The method was developed using Monte Carlo simulations, which is why itcould be argued that it lacks theoretical background and other methods should be consideredinstead. However, Dirlik’s method has proven to be one of the most accurate for rainflowfatigue life prediction [20].

In order to use Dirlik’s method, the considered process needs to be Gaussian. In a Gaussianprocess, the spectral moments can give the expected number of mean up-crossings (i.e. therate of crossings of the mean value with a positive slope) defined as E[0+] =

√M2/M0 as well

as the expected rate of peaks E[P ] =√M4/M2. The spectral moments can also be combined

to estimate the spectral width of the PSD, through the bandwidth parameters αi, where themost commonly used are given by

α1 =M1√M0M2

α2 =M2√M0M4

.

(23)

Since the bandwidth parameters describes the PSD, they are crucial for many spectral meth-ods. For a Gaussian process, as is considered when using Dirlik’s method, α2 is equal to theirregularity factor γ = E[0+]/E[P ], which gives the number of expected peaks enclosed by twomean up-crossings. For a wide-banded process, the number of peaks is much greater than thenumber of up-crossings and the irregularity factor approaches zero [1, 22].

Dirlik’s formula for fatigue damage is

DDirlik =E[P ]T

∑(σa,i)

mfσa,i(σa,i)∆σa

Ca, (24)

where E[P ] is the expected rate of peaks, T is the test duration, ∆σa is the change in stressamplitude and σa,i is the stress amplitude for every increment, m is the fatigue slope exponentand Ca is the Basquin material constant [22]. The increment i denotes each time step in thestress history. The function fσa,i is Dirlik’s probability density function for rainflow stressamplitudes, defined by the spectral moments as

fσa,i =D1√M0Q

e−Zσa,iQ +

D2Zσa,i√M0R2

e−Z2σ2a,i

2R2 +D3Zσa,i√

M0

e−Z2σ2a,i

2 , (25)

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2. THEORY

where the parameters are

Z = 1√M0

Xm =(M1

M0

)√M2

M4

D1 = 2(Xm−γ2)1+γ2

R =γ−Xm−D2

1

1−γ−D1+D21

D2 =1−γ−D1+D2

1

1−R Q = 1.25(γ−D3−D3R)D1

D3 = 1−D1 −D2.

(26)

As well as a requirement for the process to be Gaussian, the process also needs to be stationaryand wide-banded to use Dirlik’s method. Furthermore, the method does not allow for calcu-lation of mean stress influence due to the statistical properties of fσa,i . For more informationabout this limitation, the reader is referred to [1].

2.3.2 Benasciutti-Tovo

The information that is presented in this subsection is found in [22].

The Benasciutti-Tovo (BT) method is very different from Dirlik’s method, as it is based on theso called narrow-band approach. The narrow-band approach is a method for approximatingthe damage parameter of a narrow-banded signal. In the narrow-band approach, the damageparameter, DNB, is defined as

DNB =E[0+]T (

√2M0)

m

CaΓ(

1

2m+ 1), (27)

where E[0+] is the expected rate of positive mean up-crossings, T is test duration, M0 is thevariance of the process, m is the fatigue slope exponent, Ca is the Basquin material constantand Γ() is the gamma function. To utilize the narrow-band approach for wide-banded signals,a correction factor, ξ, is needed. The correction factor is multiplied with Equation (27) inorder to calculate the fatigue damage. The damage according to the Benasciutti-Tovo methodis defined as

DBT = ξBTDNB, (28)

whereξBT = b+ (1− b)αm−12 (29)

in which m is the fatigue slope exponent and b is defined by the bandwidth parameters as

b =(α1 − α2) [1.112(1 + α1α2 − (α1 + α2))e

2.11α2 + (α1 − α2)]

(α1 − 1)2, (30)

where α1 and α2 are calculated according to Equation (23).

After Benasciutti and Tovo observed that the correction factor is dependent on the bandwidthparameter α.75, they developed another method called Benasciutti-Tovo α.75 (BT.75), whereα.75 refers to

α.75 =M(.75)√M0M(1.5)

. (31)

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2. THEORY

The correction factor ξBT is then replaced by ξBT.75 = α2.75 resulting in

DBT.75 = ξBT.75DNB, (32)

To use both Benasciutti-Tovo and Benasciutti-Tovo α.75 the considered process needs to be,as for Dirlik’s method, stationary and wide-banded. However, the Benasciutti-Tovo can beextended to consider non-Gaussian processes [1].

2.3.3 Ortiz-Chen

Another method for calculating fatigue damage based on the narrow-band approach using acorrection factor is the Ortiz-Chen (OC) method

DOC = ξOCDNB. (33)

The damage using the narrow-band approach, DNB, is defined in Equation (27) and the cor-rection factor is given by

ξOC =1

α2

(√M2Mj

M0Mj+2

)m

(34)

with α2 as defined in Equation (23), j = 2/m, where m is the fatigue slope exponent, and Mj

is the spectral moment given by Equation (17) [22].

As Dirlik’s method, the Ortiz-Chen method requires the considered process to be Gaussian,stationary and wide-banded.

2.4 Random response

Random response is a method for finding critical areas that is already implemented at Scania.It is a procedure which can be used when a system is subjected to excitations that are non-deterministic and continuous, such as random vibrations. The random response analysis pre-dicts the response of the system by using the system’s eigenmodes to calculate the correspond-ing PSD of nodal and element response variables, such as stresses and strains. In this thesis,the random response analysis was conducted in Abaqus/CAE.

Due to the excitations being nondeterministic, they can only be described using statistics.Therefore, a few assumptions need to be made to successfully describe the loading. Firstly,the process must be stationary. Secondly, the process needs to be ergodic, which means thatthe average of any random samples of the process are the same. Thirdly, all variables in theanalysis are assumed to be real [27].

For more information about the theory behind the random response analysis, the reader isreferred to [17, 27–29].

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2. THEORY

2.5 FEMFAT

FEMFAT is a software frequently used at Scania. The software is used to calculate fatigue life ordamage in a component and can successfully be used considering both static and dynamic loads.By using the stresses calculated by finite element analysis, FEMFAT utilizes a combination ofclassical methods, such as nominal stress and stress-strain concepts and calculation standards,as well as newer methods. The newer methods include the mean stress influence and stressgradients, using critical plane theory and other methods for determination of the equivalentstresses [30].

For more information about FEMFAT, the reader is referred to their official website [30].

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3 MethodologyTo evaluate the chosen methods, two components were used; an L-bracket and a mudguardbracket. The aim of the L-bracket simulations was to test all the methods and their sensitivityto changes in input parameters. For the mudguard bracket, a shake rig test had already beenperformed by Scania. Signals and results from the shake rig test will be referred to as the”measured” results. The aim of the mudguard bracket simulations was to see if they showedthe same critical areas as in the shake rig test and also compare the measured fatigue life tothe calculated fatigue life.

To get an understanding of the methodology of this thesis, an overview of the project ispresented in Figure 3. This methodology was applied on both the L-bracket and the mudguardbracket. Each process will be described in more detail later in this chapter.

Figure 3: Flowchart that illustrates the simplified step-by-step procedure of this thesis project.The purple boxes illustrate processes performed in Abaqus/CAE while the orange boxes representsprocesses performed using MATLAB.

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3. METHODOLOGY

As illustrated in Figure 3, the first steps were to define the load signal and the model. Forthe L-bracket, these had to be created while the load signal and the model of the mudguardbracket were provided by Scania. More information about the load signals and the models isfound in Section 3.1 and 3.2 respectively.

The load signals were applied to the models in Abaqus/CAE, were the boundary conditions ofthe components were set. For a more visual description, information on the boundary condi-tions can be found in Section 3.1. In Abaqus/CAE, an eigenfrequency analysis was performedto find the component’s eigenfrequencies. Thereafter, the next step differed depending on whattype of analysis was performed.

For the random response analysis, the next step utilized the random response tool in Abaqus/CAE.Thereafter, the Scania plug-in ”random response analyzer” was used to view the stress stateand find the critical element. For the other methods, the procedure followed the left path inFigure 3. A modal dynamics analysis was performed, giving an input file used in both FEM-FAT and the data compilation. In order to make the data from the Finite Element-analysis(FE-analysis) compatible with MATLAB, the data was compiled using python. The FEM-FAT methodology is described in Section 3.7 and the compiled data was used in the timehistory methods and the spectral methods as described in Sections 3.3, 3.4 and 3.5. Fromthese methods, the fatigue life was computed.

All chosen methods for calculation of the fatigue life are dependent on the Basquin parametersm and Ca. The slope exponent m was set in accordance with material tests performed atScania. Using the corresponding SN-curve, the endurance limit, σe, was recorded and thematerial parameter Ca was determined by rearranging Equation (2). The components of themudguard bracket were made of different types of steel. For the fatigue calculations, thematerial parameters presented in Table 1 were used for the mudguard bracket and for the L-bracket. The fatigue parameters were found by looking at steels which material’s characteristicscorrelated well with the material tests from Scania and the chosen SN-curve.

Table 1: Material parameters used.

Parameter Material 1 Material 2 Material 3 Descriptionm 7.6 6.8 8.2 Fatigue slope exponent

σe [MPa] 250 250 170 Endurance limit

σf,tc [MPa] ± 200 ± 200 ± 110Fatigue limit, fully reversedtension-compression

σf,t [MPa] 180 ± 180 180 ± 180 110 ± 110Fatigue limit, repeated ten-sion

σf,b [MPa] ± 270 ± 270 ± 170Fatigue limit, fully reversedbending

τf,t [MPa] ± 150 ± 150 ± 100Fatigue limit, fully reversedtorsion

The chosen SN-curves were found using bending tests. For it to be fully accurate for themultiaxial stress state in the L-bracket and the mudguard bracket, it should be adjusted. Tosuccessfully adjust the chosen SN-curve for each method, a full analysis of the stress state

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3. METHODOLOGY

and the dominating fracture modes would have had to be conducted. Due to time limitationsof this thesis and the extent of such an analysis, it was decided to use the chosen SN-curveswithout conducting any adjustments. This, of course, had an impact on the results given andwas kept in mind when drawing any conclusions. Furthermore, the tests were conducted onnon-perfect specimens resulting in already partly reduced SN-curves.

3.1 Models

The shape of the L-bracket is designed to give a multiaxial stress state. This design was presen-ted in [25]. The model of the L-bracket was created in HyperMesh using 2D-shell elements.The dimensions of the model are shown in Figure 4.

Figure 4: Geometry and dimensions of the L-bracket.

The model was meshed and a set of 26 elements were chosen for the fatigue life analyses. Theseelements were positioned in various places of interest where large stresses had been observed inthe random response analysis and a test run of the modal dynamics analysis, using a shorterversion of the time signal. These elements are highlighted in Figure 5. The material assignedto the model was a steel with a density of 7880 g/m3, a Young’s modulus of 207 GPa and aPoisson’s ratio of 0.3. A modal damping of 0.03 was applied for all modes of the model. For thefatigue calculations, the material parameters of Material 1, found in Table 1, were used.

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3. METHODOLOGY

Figure 5: Display of the set of elements that were chosen for the fatigue analyses on the L-bracket.

The boundary condition on the L-bracket was set in Abaqus/CAE as fixed at the lower bound-ary, as shown in Figure 6. The load signal was applied as base excitation at the same bound-ary.

Figure 6: Illustration of the boundary condition applied to the L-bracket.

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3. METHODOLOGY

The model of the mudguard bracket provided by Scania is presented in Figure 7. A lampand a mudflap were missing from this model, these were therefore modelled as point masses.However, they are not shown in the figure. The three coloured parts have been given names inthis report since they were of interest, the reason for this will be described later in this section.The orange component will hereafter be referred to as the rubber housing, the blue componentas the frame bracket and the green component as the circular beam. The entire model wasmeshed and all component were assigned material properties. Furthermore, all componentsexcept two were assigned a steel with density of 7880 g/m3, a Young’s modulus of 207 GPaand a Poisson’s ratio of 0.3 in HyperMesh.

Figure 7: An overview of the studied mudguard bracket. The orange component is the rubberhousing, the blue component is the frame bracket and the green component is the circular beam. Thedark pink edges indicates where the boundary conditions and the load signal was applied.

The non-steel components were a rubber component, located between the circular beam andthe rubber housing, and a plastic component indicated with an arrow in Figure 7. The rubbercomponent was connected to its surrounding components using TIE-connections and it wasmodelled with an isotropic, incompressible rubber material. The modal damping of the modelwas given from the shake rig measurements and used in the Abaqus/CAE analyses, the dampingfor the different modes are presented in Table 2. During the calculations, the fatigue parametersof Material 1 were used for the rubber housing, the parameters of Material 2 for the framebracket and the parameters of Material 3 for the circular beam, see Table 1.

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3. METHODOLOGY

Table 2: Modal damping used for the mudguard bracket.

Mode number Damping1 0.012 0.0373 0.0464 0.038

Remaining 0.01

The frame, which the frame bracket is attached to, was set as fixed on the edges marked darkpink in Figure 7. The load signal was applied at the same boundaries. An eigenfrequencyanalysis was performed to see if the model showed properties corresponding to the shake rigtest. Thereafter, a random response analysis was conducted in order to get an understandingof the stress state in the mudguard bracket.

To validate the model, the areas that showed high stress in the random response analysis werecompared to the shake rig test. This showed that failure in the shake rig test occurred in thesame location as the high stress areas from the random response analysis. Therefore, onlythese areas were investigated in the modal dynamics evaluation. They were found on the threecoloured components, which is why these components were of interest (for a closer view seeFigure 8). The chosen elements of each high stress area are marked with white in the figure.At the location of SG1 and SG2, shown in the same figure, the strain was measured. Thesepositions were chosen since strain gauges were placed there during the shake rig test. Thesesstrains were used to validate the model by comparing the PSDs of the measured strains fromthe shake rig test to the ones from the ones measured in the model.

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3. METHODOLOGY

Figure 8: The elements chosen for the three high stress areas are marked with white elements.There were 234 elements chosen for the circular beam, the dashed circle means that the componentof interest is found underneath other components in the figure. On the rubber housing, 28 elementswere chosen and 150 elements were chosen for the frame bracket. The locations where strains weremeasured in the simulations and the shake rig tests are indicated with SG1 and SG2.

3.2 Signal

The load signal used for the simulations on the L-bracket was created using MATLAB. The loadsignal amplitude was defined as an acceleration, with unit m/s2, exciting the lower boundaryof the L-bracket, shown in Figure 6. An eigenfrequency analysis was conducted to find thefirst six eigenfrequencies (4.53, 11.5, 52.6, 61.6, 102 and 167 Hz) of the L-bracket. The timesignal was 50 seconds long and was created using a normally distributed random functionto simulate random road-induced vibrations. A sampling frequency of 20 times 250 Hz wasused, and a 9th order lowpass butterworth filter was applied to get a signal including the firstsix eigenfrequencies. The amplitude was chosen to simulate a stress state with high enoughstresses to contribute to fatigue, i.e. stresses above the endurance limit. A too high amplitude,however, would result in stresses outside the valid range of the SN-curve, this limited how largethe amplitude was allowed to be. The amplitude was controlled by multiplying an amplificationfactor to the normally distributed random variables, giving the signal in Figure 9.

The created time signal was used to estimate a PSD through the implementation of Welch’smethod in MATLAB. This PSD was used in the random response simulation of the L-bracket.The PSD created from the time signal is presented in Figure 10.

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3. METHODOLOGY

Figure 9: The load signal used in the FE-analysis on the L-bracket.

Figure 10: The PSD, corresponding to the load signal in Figure 9, used in the random responseanalysis on the L-bracket.

The time signals used for the simulations on the mudguard bracket were measured using threeaccelerometers placed on the frame, close to the frame bracket, during the shake rig test. The

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3. METHODOLOGY

signals were 60 seconds long and sampled with a frequency of 410 Hz. The acceleration wasmeasured in three different directions, namely the x -, y-, and z -direction. These are presentedin Figure 11. All three measured acceleration signals were used as input acceleration on themudguard bracket during the simulations.

Using the signals of the measured acceleration, three PSDs were created. One for each dir-ection. These PSDs were used as input for the random response analysis of the mudguardbracket.

Figure 11: The acceleration signals used as input in the FE-analysis on the mudguard bracket.

3.3 Equivalent stress methods

In order to use the AMPS method, the principal stress history was extracted from Abaqus/CAE.For each time step, the principal stresses were compared to each other. The largest absolutevalue was stored together with its sign in order to create the equivalent stress history accordingto Equation (5). Rainflow counting was performed on the equivalent stress and the fatiguelife was calculated using the Basquin, Palmgren-Miner and life equations (Equations (2)-(4)).This was done for all elements in the chosen element set and the minimum fatigue life wasfound.

The SvM method and the Sines method were conducted using the same procedure as the AMPSmethod. In order to use the SvM criterion, the von Mises stress, σvM , was extracted fromAbaqus/CAE and entered into Equation (6) together with the sign calculated in the AMPSmethod. For the Sines criterion, the relevant stresses (σx, σy, σz, τxy, τxz, τyz), were extractedfrom Abaqus/CAE from which the mean and amplitude values were calculated. The meanstress parameter kS = 0.16 was calculated using Equation (9) with the material parameters

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3. METHODOLOGY

found in Table 1. With this data, the equivalent stress, according to the Sines criterion, wasevaluated with Equation (8). After the equivalent stress was calculated according to thesetwo methods, rainflow counting was performed on the equivalent stress and fatigue life wasdetermined using Equations (2) - (4). After repeating this process for all elements in the chosenelement set, the minimum fatigue life was recorded.

3.4 Critical plane theories

The stress components σx, σy and τxy were extracted from the Abaqus/CAE simulation resultsand projected onto a material plane (θ, ϕ) for each time step according to Equation (10). Theshear stress amplitude, τa, together with the corresponding number of cycles, n, was foundby rainflow counting the projected shear stress. Since there was a plane stress state in theL-Bracket and the mudguard bracket, the plane ϕ = 90◦ gave the largest damage. Thus, therainflow counting was conducted on τA for all θ (due to τB = 0 for ϕ = 90◦ according toEquation (10)). The maximum normal stress, σn,max, was found for each plane θ. Hereafter,the methodology for the critical plane methods differs due to their difference in the definitionof the critical plane, and will be described further in separate.

In Findley’s method, the shear stress amplitude and the maximum normal stress for each timestep was calculated according to Equation (11). The value of kF = 0.11 was calculated usingEquation (12) and the material parameters found in Table 1. Using Equation (2), modifiedwith the Findley correction factor from Equation (15), the number of cycles to failure, Nf ,was calculated. Thereafter, Equation (3) was used to calculate the damage on each planeθ. The plane with the largest accumulated damage was defined as the critical plane. Thisprocess was repeated for each studied element to find the element with the largest accumulateddamage. The fatigue life of this element in the critical plane was then calculated using Equation(4).

In Matake’s method, the single maximum shear stress amplitude on each plane θ was found andcompared, and the plane θ with the single largest maximum shear stress amplitude was chosenas the critical plane. Thereafter, the Matake stress was calculated according to Equation (13),with kM = 0.11 from Equation (14) and the material parameters found in Table 1. Equation(2), modified using the Findley correction factor from Equation (15), was used to determinethe number of cycles to failure. The damage in the critical plane was calculated using Equation(3). This was repeated for each studied element where the fatigue life of the element with thelargest accumulated damage was found using Equation (4).

3.5 Spectral Methods

All chosen spectral methods used the stresses σx, σy and τxy to determine the PSD matrixof the stress components G(f). Thereafter, the EVMS method was used to calculate theequivalent uniaxial spectral moments according to Equation (22), as explained in Section 2.3.After this step, the different spectral methods differ.

In order to determine the fatigue life using Dirlik’s formula, the equivalent uniaxial spectralmoments were used to determine the expected rate of peaks, E[P ], the bandwidth parameter,

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3. METHODOLOGY

α2, and the input parameters presented in Equations (25) and (26). The cycles below theendurance limit were removed and the parameters were entered into Equation (24) giving thedamage according to Dirlik’s formula. Finally, the fatigue life was determined using Equation(4).

For the other spectral methods, the equivalent uniaxial spectral moments were used to calculatethe expected rate of mean up-crossings, E[0+]. After this, the damage according to the narrow-band approach was computed using Equation (27). To calculate the correction factor in theBT method, b was determined by entering the bandwidth parameters into Equation (30).Thereafter, the correction factor was calculated with Equation (29) and the damage DBT ,from Equation (32), was used to determine the fatigue life for every element in the chosenelement set.

Both the BT.75 and the OC methods utilizes the narrow-band approach, which had alreadybeen computed in the BT calculations. Therefore, the only thing needed to calculate thedamage according to these methods was the correction factors. For the BT.75 method, α.75was calculated according to Equation (31) which was then squared to give the correctionfactor ξα. The damage was then determined using Equation (32). For the OC method, thespectral momentsMj andMj+2 were calculated in addition to the previously calculated spectralmoments. These were entered into Equation (34), giving the correction factor ξOC which wasused to calculate the fatigue damage, according to Equation (33). The fatigue life of the chosenelements was computed using Equation (4) for both the BT.75 and the OC method.

3.6 Random response

The results from the random response simulation in Abaqus/CAE were imported into AbaqusViewer where the Scania implemented plug-in ”random response analyzer” was used. This ana-lyser calculated the accumulated von Mises stress in every element. The relationship betweenthe stresses in every element was shown as a colour plot. The element with the highest accu-mulated stress was deemed the most critical element.

3.7 FEMFAT

FEMFAT has a number of different analyses which are suitable for different types of loads. Inthis thesis, the so called transMAX analysis was used. To conduct the transMAX analysis, astep-by-step guide provided by Scania was followed. The settings used was chosen by suggestionof this guide.

To set up the analysis, the results of the Abaqus/CAE FE-analysis was imported and theelement set of interest was chosen. By using the material parameters σf,tc and σf,t of thecomponent’s material, found in Table 1, a material with corresponding SN-curve was generatedin FEMFAT. This material was applied to the chosen element set. A number of influence factorsfor the analysis was set. These included stress gradients, mean stress, surface roughness, mean(and amplitude) stress rearrangement, modified Haigh Diagram (Ultimate Tensile Strength)as well as statistical and rotating principal stresses influence. Finally, additional analysisparameters were set, such as a survivability of 50% and an absolute stress limit analysis filter

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3. METHODOLOGY

of 1 MPa. Furthermore, the elementary Palmgren-Miner approach was chosen for damagecalculations, which means that the endurance limit is not considered.

Unfortunately, the computational time of the analysis was extremely long and the requiredlicenses to shorten this time were not available. Therefore, the analyses could not be finishedin time and no results were found for this method.

3.8 Sensitivity study

A sensitivity study was performed on the L-bracket to investigate how much the calculatedfatigue life differed with variation of the input parameters. When gathering data for the SN-curves, the tests are performed under certain conditions. However, the conditions that the realcomponents are exposed to will not truly match the test-conditions, which is why the SN-curvemight not fully describe the fatigue behaviour of the studied component. Therefore, it is ofinterest to see how much the computed fatigue life from the different methods vary when thematerial parameters m and Ca are altered, i.e. when different SN-curves are used. This wasdone by varying the value of the fatigue slope exponent m while keeping the same top valuefor every curve. The material parameter Ca was then recalculated for every m. This resultedin the endurance limit varying for every SN-curve. It was decided to change m in incrementsof 10% from -50% to +100%.

In addition to the parameter m, a sensitivity study of the material parameter k, found inthe Sines, Findley and Matake method, was conducted. For these methods, the parameterskS, kF and kM depend on various material parameters, found in Table 1. These are gatheredfrom material fatigue tests. Depending on what material is used, the exact values of theseparameters might be difficult to find and the certainty of k will therefore vary. The influenceof this uncertainty on the resulting fatigue life was investigated by varying the parameters kfor each method with -50%, -10%, 0%, +10%, +50%, +100% and +500%. These values werechosen to include both small and large uncertainties of the parameters.

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4. RESULTS

4 ResultsIn this chapter, all results from the studies on both the L-bracket and the Mudguard will bepresented.

4.1 Model evaluation

The result from the random response analysis of the L-bracket is shown in Figure 12. Thecolours represent the accumulated von Mises stress. The area with the highest accumulatedstress is found in the notch of the L-bracket.

Figure 12: Random response result of the L-bracket.

The random response analysis of the mudguard bracket was used to validate the model andit is presented in Figure 13 together with pictures taken off the real life components after theshake rig test was performed. The model fulfills the requirement that the critical zones shouldbe found on the same place in the random response results as where fracture was found inthe shake rig test. The high stress areas are indicated with pink circles in the figure. The redareas found around the holes are neglected since the stress values are affected by singularitiescaused by the modelling approach.

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4. RESULTS

Figure 13: Random response result of the mudguard bracket. The pink circles show where highstress areas are found. The rubber housing has been cut to show the circular beam inside it. Thearrows points to the same location on the real life components after the shake rig test.

The PSD of the strains measured during the test in the shake rig is presented in Figure 14,together with the PSD of the strains measured in Abaqus/CAE. The left hand side graphrepresents the PSDs of the strains at location SG1 while the right hand side graph representsstrain at SG2 (see Figure 8). The strains were extracted at the same location in the modelas where the strains were measured in the tests. To further evaluate the model, the rootmean square (RMS) values from the strain time signals of the model and the real system werecalculated. The RMS value of the strain signal measured at location SG1 was 209 for themodelled system and 144 for the real system. At location SG2, the RMS of the strains fromthe modelled system was 66 and the RMS of the strains measured in the shake rig test was34.

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4. RESULTS

Figure 14: PSD of the strains at location SG1 and SG2 from the shake rig test and the FEsimulations.

4.2 Fatigue life comparison

The shortest calculated fatigue life, the critical element and critical planes were calculatedin MATLAB. The critical element is the element with the shortest fatigue life in the chosenelement set.

The critical element in the L-bracket, found from the different methods, is highlighted in Figure15. The results of the different methods for the L-bracket is presented in Table 3. The fatiguelife is shown in number of completed cycles. The Findley and Matake method found differentcritical planes. The Findley method found the plane (φ = 90◦, θ = 109◦) while the Matakemethod found the plane (φ = 90◦, θ = 113◦).

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4. RESULTS

Figure 15: The critical element of the L-bracket for all methods is marked with red.

Table 3: Fatigue life, critical element and critical plane of the L-bracket, calculated using the differentmethods.

Method Fatigue life [cycles]AMPS 533SvM 252Sines 253

Findley 199Matake 213

Dirlik 215BT 7

BT.75 131OC 48

The critical elements in the three mudguard bracket components, found from the differentmethods, are highlighted in Figure 16. The different methods found the same, or adjacent,elements to be the critical one. They all matched the results from the shake rig test (see Figure13). The calculated fatigue life is presented in Table 4.

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Figure 16: The critical elements found on the three investigated components of the mudguardbracket are marked as red.

Table 4: The fatigue life of the three components of the mudguard bracket, calculated using thedifferent methods, is presented. The fatigue lives recorded from the shake rig test are presented inthe first row of the table.

Rubber housing Frame bracket Circular beamMethod Fatigue life (cycles) Fatigue life (cycles) Fatigue life (cycles)Measured 1407 3644 - 4572 1501

AMPS 492 18698 <1SvM 1076 21058 <1Sines 1076 21058 <1

Findley 1941 25480 <1Matake 2428 26986 <1

Dirlik 15652 805777 13BT 59 657 <1

BT.75 9301 87204 <1OC 9211 86415 <1

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4. RESULTS

The fatigue life of the elements in the chosen element set in the notch of the L-bracket ispresented in Figure 17. The other elements are not shown here since they did not show fatiguedamage. The element number on the x-axis represent the element number found in HyperMesh.and the order of the elements is the same in the plot as in Figure 15. Every curve representsone method.

Figure 17: The calculated fatigue life from all methods for part of the chosen element set on theL-bracket. Only the fatigue life of the elements in the notch are plotted due to the other not beingaffected by fatigue. The numbers on the x -axis represent the element number from HyperMesh.

4.3 Sensitivity to material parameters

When calculating fatigue life, it is of interest to know how sensitive the different methods are toa variations in the material parameters since the material parameters are often not known withgreat certainty for each specific component. The influence of the value of the slope exponent mon the calculated fatigue life for the different methods was investigated and is plotted in Figure18 for evaluation and comparison. The influence of a variation of m on the SN-curve was alsocompared with the stress amplitude distribution in the most critical element from the rainflowcounting conducted on the von Mises stress. This is presented in Figure 19. The von Misesstress was chosen since this was the largest stress in the L-bracket. The stress distributionwas of interest to see how many of the stress amplitudes fell below the endurance limit for thedifferent SN-curves.

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4. RESULTS

Figure 18: Sensitivity study of m, the grey area are values of m outside the typical range for steels.The AMPS curve continues to a value of 5124 for m = 15.2 but was cut out for a plot that is easierto interpret.

Figure 19: The left plot is a visualization of the von Mises stress amplitude distribution on theL-bracket. The plot shows that the majority of the stress amplitudes are below the endurance limit.The plot to the right shows the resulting SN-curve with a varying m-value and how the endurancelimit varies with m.

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4. RESULTS

The influence of variation of the material parameters kF , kM and kS, used in the Findley,Matake and Sines method, was studied. The results of these studies are presented in Figure20. The ”x”s in the figures indicates the calculated fatigue life for the chosen k-values.

Figure 20: The influence on the fatigue life of variations in the material parameters kF , kM and kSused in the Findley, Matake and Sines method, respectively. The left plot shows the results for theFindley and Matake method and the right plot shows the results for the Sines method.

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5. DISCUSSION

5 DiscussionIn this chapter, a discussion on all results from the studies on both the L-bracket and themudguard bracket is given.

5.1 L-bracket

Fatigue life

Considering the results of the fatigue life calculations for the L-bracket, presented in Table3, one can see that five of the methods give similar life estimations. Namely the SvM, Sines,Findley, Matake and Dirlik method. The AMPS method showed a fatigue life roughly twiceas big as the above mentioned methods. The AMPS method is calculated solely based on theprincipal stresses while the other methods are based on the stresses in x - and y-direction. Theprincipal stresses from the L-bracket simulation were lower than the von Mises which mightexplain this outcome.

The life of the L-bracket calculated with the spectral methods show that the results differ alot. The Dirlik method show a longer expected life, which can be explained by the fact thatthis method considers the endurance limit. This is possible due to the method containinga probability density function for rainflow stress amplitudes, which enables it to estimateand remove cycles that fall below the endurance limit. The other spectral methods do notcontain such probability function and are, therefore, not able to consider the endurance limit.Since many cycles in the studied signal lied below the endurance limit, see Figure 19, it isexpected that the methods not accounting for this show a shorter life. All time history methodsaccounted for the endurance limit which explains why the calculated fatigue life from Dirlik’smethod is closer to the results of time history methods than the other spectral methods.

Among the deviating spectral methods, the BT.75 method shows a much longer fatigue lifethan the BT and OC method. The fact that it shows such a long life without considering theendurance limit makes the method questionable. Due to the amount of cycles lying below theendurance limit, the BT.75’s fatigue life would increase considerably if these cycles were to beremoved and most likely result in a longer fatigue life than reasonable compared to the othermethods. This raises questions regarding the reliability of the results from this method.

Critical plane methods

The Findley and Matake fatigue life calculations of the L-bracket showed that the two methodsfound different critical planes. This can be explained by the fact that the methods’ definitions ofthe critical plane are different. In Findley’s criterion the maximum normal stress is consideredwhen finding the critical plane, while only the maximum shear stress amplitude is consideredin Matake’s criterion. Due to Matake’s neglect of the normal stress when finding the criticalplane, it is reasonable that the Matake fatigue life is longer than the Findley fatigue life, evenif this difference is quite small.

The way the critical plane was chosen by Matake’s criterion is also important to discuss. Since

34

5. DISCUSSION

the critical plane was evaluated as the plane with the single maximum shear stress amplitudeand not the plane with the largest Matake stress, there might be other planes with a largerdamage which are not considered with this method. This results in a higher fatigue life andraises questions about the reliability of the method for the load case studied. Another sourceof error in Matake’s method is the lack of a correction factor exclusive for this method. It isunknown how much influence the use of Findley’s correction factor has on the resulting fatiguelife.

Critical element

Figure 17 shows that, even though the absolute fatigue life calculated with the methods differ,the trend of the results show the same behaviour. This means that all methods calculated asimilar relative fatigue life between all elements. The spectral methods, especially BT and OC,are not as obvious in this trend as the other methods, this could be due to their absolute fatiguelife results not varying that much between the different elements. The results are also closerin absolute fatigue life between the different methods for the most critical elements, element18676 and 18293 in the figure. The most critical element, element number 18676, is shown inFigure 15 and corresponds to the results from the random response, see Figure 12.

This, along with what is discussed above, shows that all methods are successful in findingthe critical elements even though they might not find the correct fatigue life. However, sincethe random response analysis result is equally successful in finding the critical elements andhas significantly shorter computational time, it is advantageous for the purpose of finding thecritical areas. The shorter computational time of the random response analysis is a result ofit only working in the frequency domain. The downside of this method is that fatigue life cannot be predicted since it only computes the accumulated von Mises stress.

5.2 Mudguard bracket

Fatigue life

When comparing the calculated fatigue lives in Table 4 to the measured results, they show awide range in accuracy. For the rubber housing, the time history methods are fairly close tothe measurements. However, the calculated fatigue life of the frame bracket is much longerand the circular beam much shorter, compared to the measurements. This raises the questionif the rubber housing results are trustworthy or if the similarity to the measured result is onlya coincidence.

The method used to convert the multiaxial spectral moments into equivalent uniaxial spectralmoments have an impact on the outcome. Therefore, the EVMS method was chosen for allthe spectral methods which is why the converting method should not be the reason for thespread in the results. However, it was not investigated if different definitions of the uniaxialequivalent stress were favourable for different spectral methods. This investigation is left forfuture work.

It was also observed that the results were very sensitive to the SN-curve used in the calculations.By using a slightly lower endurance limit, the calculated fatigue life of the methods which

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5. DISCUSSION

considers the endurance limit, differed substantially. This was due to the majority of thestress amplitudes being close to the endurance limit.

When studying Table 4, it is noted that the calculated fatigue life of the circular beam isbelow 1 for almost all methods. This is due to the fact that the stresses in this area exceedthe ultimate tensile strength and, therefore, the component fail before completing even onecycle. Dirlik’s method does not show the same result. This can be explained by the fact thatthe Dirlik method estimates the amount of cycles there will be at each stress level based ona normally distribution within the stress range of the SN-curve. Longer life for the Dirlikmethod is also observed in the other components. When looking at all the fatigue lives inTable 4 one can see that the results from the Dirlik method show the highest life. After astudy of the signal measured in the shake rig test, it was found that it was non-Gaussian.This was the signal used as an input acceleration on the modelled mudguard bracket andbecause of the linear model this resulted in non-Gaussian output stress signals. Since Dirlik’smethod estimates a stress amplitude distribution from a normal distribution it will not givea correct representation of the stress signals. The method estimates more cycles with lowerstress levels than the true signals which was leptokurtic. This explains why Dirlik’s methodgenerally give higher results than all other methods for the mudguard bracket. Since the stresssignals obtained from the mudguard bracket simulation is not Gaussian, the requirements forthe EVMS and the spectral methods are not fulfilled. Therefore, none of the results from thespectral methods can be trusted for the mudguard bracket and no conclusions about the lifeestimations can be drawn.

Critical element

The conclusions that can be drawn about the critical elements when looking at the results ofthe mudguard bracket are similar to those of the L-bracket. Figure 16 shows that all methodspredicts failure in the same, or adjacent, elements in places which matches the results of theshake rig test seen in Figure 13.

For the circular beam, one element is a bit further away. This is the critical element foundfrom the Dirlik method. However, they all follow the same line of failure as the one seen in theshake rig test and are all placed in the critical area from the random response. This impliesthat the methods are successful in finding the correct failure points. This is most obvious forthe frame bracket, where the two critical elements in Figure 16 are found at the same point asthe crack initiation seen in the shake rig test. As for the L-bracket, the random response resultsof the mudguard bracket in Figure 13 show that the random response analysis also finds wherethe failure will occur. Due to the random response method’s shorter computational time, thismethod is recommended for the purpose of finding the failure points.

Strain validation

Though the general pattern of the PSD of the measured strains and the strains from the modelin Figure 14 are similar, it does not give any information about the strains in the critical zones.We know that the critical zones appear at the correct positions, but it is difficult to validateif the stress levels are correct in said zones.

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5. DISCUSSION

Another source of error that is seen in Figure 14 is that the eigenfrequencies of the realmudguard bracket are significantly lower than the ones for the modelled mudguard bracket forboth strain gauges. For example, the left plot shows that the dominating eigenfrequency forthe real system is found at 9 Hz while the dominating eigenfrequency of the modelled systemis found at 16 Hz.

In an effort to correct this, the mass of the model was increased by increasing the mass ofthe only component that we had estimated the mass of, the lamp. However, this gave nosignificant change in eigenfrequencies, but rather strains that were unreasonably high. Since ashorter part of the frame has been modelled, the modelled system does not reflect the stiffnessof the real system correctly. This is probably why the eigenfrequencies end up higher for themodelled system. It is difficult to know how, and if, this difference in the modelled systemwill affect the fatigue life estimations. Modelling the complete system for a closer match tothe real system is recommended as future work to be able to successfully compare the fatiguecalculation methods to shake rig tests. Furthermore, the RMS values indicates that the strainsare higher in the model than the strains from the measured in the real system. This alsoindicated that the energy is too high in the modelled system.

Due to the uncertainties of the study of the mudguard bracket discussed above, it is difficultto draw any conclusion about the calculated fatigue life results. A larger study would have tobe conducted to investigate which uncertainties have to be minimized to successfully use anyor some of these methods.

5.3 FEMFAT

The FEMFAT analyses could not be finished due to the extremely long computational timerequired. Therefore, results were not found for this method. One reason for the long computa-tional time is believed to be the large number of increments used in the Abaqus/CAE modaldynamics analysis, which is used as input to the FEMFAT analysis. For successful use of thismethod, an FE-analysis using a smaller number of increments would be recommended. Fur-thermore, there might, of course, be other factors which contributes to the long computationaltime, such as the model used. However, the influence of such factors were not investigatedfurther. Since no results were found for the FEMFAT method, no other conclusions can bedrawn for this method.

5.4 Sensitivity study

In the sensitivity study of the fatigue slope exponent, m, some values above the typical rangewere used. This was done in order to capture the behaviour of the results for large variationsof m, even though these are less likely to be used. The results in Figure 18 show that thefatigue life computed from the time history methods and Dirlik are noticeably affected by thevalue of m, while the fatigue life computed using the spectral methods BT, BT.75 and OC arenot. This is explained by looking at the uniaxial stress amplitude distribution in Figure 19.The majority (99.7%) of the stress amplitudes are below the endurance limit when m = 7.6.However, the amount of stress amplitudes below the endurance limit will vary with m. Forthe lowest m-value (m = 3.8), 93.1% of the stress amplitudes are below the endurance limit

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5. DISCUSSION

while 100% of the stress amplitudes fall below the endurance limit for the largest value of m(m = 15.2). The BT, BT.75 and OC methods does not consider the endurance limit and aretherefore not as affected by this variation of m. However, as the result is highly dependent onthe stress amplitude distribution, the influence of m would most likely be different for otherratios of high and low stress amplitudes.

Furthermore, the results in Figure 18 indicates that the Dirlik method shows potential sinceit behaves like the time history methods which represent the real event. For example, thetime history methods considers the order in witch the stress cycles appear. However, in orderto further evaluate if the Dirlik method will be usable, a correction for non-Gaussian cases isneeded.

When studying the sensitivity plot for kF and kM , found in the left plot in Figure 20, it canbe seen that they behave similarly. This is expected due to the similar definition of the twocriteria. The Findley method is shown to be a bit more sensitive to variations in kF , wherethe fatigue life varies with about 150% in the whole range of kF . The Matake fatigue lifevaries around 125% in the same range of kM . The two curves can be divided into two partswith different slopes. For values of k above 0.2, the variation shows a linear behaviour ofapproximately 10% per 0.1 change in k. Below 0.2 the slope is much steeper and the influenceof k is approximately 50% per 0.1 change. This suggests that the uncertainty of k is more severefor lower values of k. However, for most ductile materials, the value of kF is in the range 0.2 -0.3, in which the fatigue life does not vary more than 10%. It is important to point out thatthese results are valid for the specific stress state used. For Findley’s and Matake’s criteria,the material parameter k acts as a weight factor to the maximum normal stress. Therefore,the influence of k is directly correlated to the size of the normal stress.

The right plot in Figure 20 shows that the Sines fatigue life vary less than 2% within thechosen range of kS. From these results, one could argue that the results are not sensitiveregarding the material parameter kS. However, because of how the criterion is defined, the kSbecomes a weight factor to the mean hydrostatic stress, σh,m. This means that the influence ofkS will vary with σh,m. The stress state in the L-bracket resulted in a low hydrostatic stress.It is possible that a more dominant σh,m would result in a bigger sensitivity to changes in kS.Therefore, no conclusions could be drawn from the stress state investigated.

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6. CONCLUSIONS

6 Conclusions• All studied methods are successful in finding the critical areas in a component. However,

the computational time for the random response method is significantly lower than thesimulation time of the FE-analysis using modal dynamics in the time domain. Sincethe random response method is already implemented and used at Scania, we do notrecommend changing method for the purpose of finding the critical areas.

• In cases with many cycles below the endurance limit, such as stresses from the roadinduced vibrations simulated in this study, it is of importance to use methods that canaccount for the endurance limit.

• The SN-parameters have a significant impact when calculating the fatigue life with theinvestigated methods.

• In this study, the level of non-Gaussianity in the stress signals obtained from the mudguardbracket simulations gives unreliable results from the studied spectral methods. For thistype of load, the studied spectral methods are, therefore, not recommended without usinga correction for non-Gaussianity.

• Changes in the Findley and Matake material parameters, kF and kM , within the ductilespan does not significantly affect the calculated fatigue life. However, it is difficult toconclude if this is the case for other stress states.

• Matake’s method is not as reliable as Findley’s method in finding the critical plane for thestudied component and load case. Therefore, it will also give a longer fatigue life. Thelack of documentation regarding a correction factor for Matake’s method also increasesthe uncertainty of the method. Therefore, Findley’s method is the recommended criticalplane method.

• For the model used, no method will give a reliable estimation of the fatigue life, sincethe model itself is not accurate enough. A model that better reflects the dynamics of thesystem is needed.

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