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3 Lifetime Estimation of Welded Joints

4 Tadeusz Łagoda Lifetime Estimation of Welded Joints

5 Tadeusz Łagoda Politechnika Opolska Katedra Mechaniki I Podstaw Konstrukcji Maszyn OPOLE, ul. St. Mikolajszyka 5 Poland t.lagoda@po.opole.pl ISBN: e-isbn: Library of Congress Control Number: c 008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper springer.com

6 Preface In the paper the author attempts to assess the fatigue life of chosen welded joints. It focuses especially on chosen problems that accompany determination of the fatigue life of welded joints, taking into consideration the strain energy density parameter. Chapter describes the welded joint as a stress concentrator. The state of stress and strain in the notch are described and theoretical and fatigue coefficients are indicated. The fatigue coefficient of the notch effect is estimated on the basis of fictitious radius in the notch root. Chapter 3 presents a model of fatigue life assessment under uniaxial stress state with statistical handling of data presented. The new energy model of fatigue life assessment, which rests upon the analysis of stress and strain in the critical plane, is described in detail in chapter 4. The principle of such a description is presented in the uniaxial as well as in biaxial state of loading. Chapter 5 contains the analysis of tests of four materials subjected to different loadings: cyclic, variable-amplitude with Gaussian distribution, and variable amplitude with Gaussian distribution and overloading for symmetric and pulsating loading. The analysis is based on the determined fatigue characteristics for all the considered materials. Chapter 6 shows the application of the model in the fatigue life assessment in the complex state of loading (bending with torsion of flange-tube and tube-tube joints) based on fatigue research of steel and aluminum welded joints carried out in well-known German centres. Proportional and out-ofproportional cyclic research are carried out. Additionally, the influence of various bending and torsion frequencies and proportional and out-ofproportional variable amplitude loadings are analysed. Dealing with such a complicated problem as fatigue life of welded joints is requires a wide cooperation with other researchers and research centres. That is why I would like to express his gratitude to at least some of the people who contributed to the issue of this publication. This book is a result of my research work, considerations and discussions while my sixmonths stay at LBF Darmstadt, Germany, financed by NATO. Thus, I would like to thank all the workers of LBF, especially Prof. C.M. Sonsino [18] and Dr. M. Küppers for many discussions, access for their test results, laboratories and library while my visit, and for information sent in our correspondence. I was able to complete my book while my later two-week

7 V I Preface stay in Darmstadt, financed by DAAD. In this book, I also used some data obtained from Technical University of Clausthal, Germany, namely from Prof. H. Zenner. I want to thank Prof. Zenner, Prof. A.Esderts and Mr. A.Ahmadi for their help while my one-week visit at TU of Clausthal, financed by CESTI. I also used my experience obtained during my work with the postgraduate students at Opole University of Technology: Dr. Damian Kardas, Dr. Krzysztof Kluger, Ms Magorzata Kohut, Dr. Pawe Ogonowski, Dr. Jacek Sowik and Ms Karolina Walat. I must also thank my co-workers from Opole University of Technology: Dr. Adam Niesony, Dr. Aleksander Karolczuk, Dr. Roland Pawliczek, and especially Prof. Ewald Macha, also some other people not mentioned here. I would like to thank Ms Ewa Helleska for translation of this book and some previous papers into English. Finally, I want to thank Prof. M. Skorupa and Prof. K. Rosochowicz for their suggestions and advice. I wish to dedicate this book to my wife Boena for supporting me in my research work as well as for her constant understanding and care. Tadeusz agoda t.lagoda@po.opole.pl Department of Mechanics and Machine Design Faculty of Mechanical Engineering Opole University of Technology ul. Mikolajczyka Opole, Poland

8 Contents Notation... IX 1 Introduction...1 Welded Joints as the Stress Concentrator The Complex Stress State in the Notch...5. Theoretical Notch Coefficient The Fatigue Notch Coefficient The Fictitious Radius of the Welding Notch The Notch Coefficient with the Use of the Fictitious Notch Radius The Stress Model for the Assessment of Fatigue Life Under Uniaxial Loading Algorithm for the Assessment of Fatigue Life Under Uniaxial Loading State Statistic Evaluation The Energy Model of Fatigue Life Assessment The Energy Parameter Under Uniaxial Loading The Energy Parameter Under Multiaxial Loading The Generalized Criterion of the Parameter of Normal and Shear Strain Energy Density Parameter in the Critical Plane The Criterion of Maximum Parameter of Shear and Normal Strain Energy Density on the Critical Plane Determined by the Normal Strain Energy Density Parameter The Criterion of Maximum Parameter of Shear and Normal Strain Energy Density in the Critical Plane Determined by the Shear Strain Energy Density Parameter Algorithm for Fatigue Life Assessment...47

9 VIII Contents 5 An Example of Fatigue Life Evaluation Under Simple Loading Fatigue Tests Tests Under Constant-amplitude Loading Tests Under Variable-amplitude Loading Verification of the Results Obtained Under Variable-amplitude Loading An Example of Fatigue Life Evaluation Under Complex Loading States Fatigue Tests Verification of the Criteria Under Constant-amplitude Loading The Parameter of Shear and Normal Strain Energy Density on the Critical Plane Determined by the Parameter of Normal Strain Energy Density The Parameter of Shear and Normal Strain Energy Density in the Critical Plane Determined by the Shear Strain Energy Density Parameter The Influence of Different Frequencies of Bending and Torsion on Fatigue Life Verification Under Variable-amplitude Loading Conclusions...95 References...97 Summary...117

10 Notation A 5 contraction C coefficient containing circumferential stresses in the root of the notch E longitudinal modulus of elasticity (Young s modulus) f frequency G shear modulus k(n f ) ratio of allowable stresses for bending and torsion for a given number of cycles N f K f fatigue notch coefficient K t theoretical notch coefficient K ta theoretical notch coefficient for axial loading K tb theoretical notch coefficient for bending K tt theoretical notch coefficient for torsion N f number of stress cycles up to fracture m, m slope of fatigue S-N characteristic curve M moment N number of cycles P force r correlation coefficient R stress ratio R 0., R e yield point R m. tensile strength s coefficient of multiaxiality, standard deviation S(T o ) fatigue damage degree in observation time T o t time, thickness of sheet T N scatter band for life-time T mean scatter band for life-time N T o observation time W strain energy density parameter angle of the plane position shear strain normal strain

11 X Notation phase displacement angle Poisson ratio radius in the welding notch root normal stress shear stress 0xyz xyz co-ordinate system with the origin in the point 0 Indices a amplitude af fatigue limit b bending cal calculation e elastic eq equivalent exp experimental f fictitious l local m mean value max maximum value min minimum value n nominal p plastic t torsion x, y, z directions of axes of the co-ordinate system w weighed value Functions 1 for x 0 sgn( x ) 0 for x 0 1 for x 0 sgn(x) sgn(y) sgn(x, y)

12 1 Introduction The problem of determination of fatigue life of welded joints has been investigated for many years. As a result, there is a possibility to find solution of that issue in many publications. Typical handbooks concerning fundamentals of machine building are for example [33, 194], other books and monographs [45, 53, 199, 03, 04, 15, 16, 34 and others] or the latest work [44], to mention some. The problem has been also discussed in many journal publications and presented in conferences. Only few of publications have been cited in this paper. During the last 15 years, the in-depth analysis of the problem of fatigue life calculations has been presented in many books and other publications [40]. The author of this monograph refers to the most important of them [4, 5, 40, 73, 100, 166, 176, 33]. Correct design of welded joints seems to be very important, for example in transport facilities, including hoisting equipment [84, 167] where special safety regulations must be fulfilled, or in the structures with high pressure of a medium [186]. According to [176, 3], in order to define the fatigue life in welded joints, there are two basic approaches possible to determine calculation stresses: first on the basis of the nominal stresses, and second on the basis of the strictly local stresses determined in the potential point of crack initiation ( hot spot ). Analysis based on the nominal stresses is applicable in the situation where the considered element has been classified and when the stresses can be easily determined. In [33], Susmel and Tovo presented satisfactory results of many calculations of welded joints based on nominal stresses under constant-amplitude loading. The hot spot method is recommended for the cases where the strains can be measured near the joint [4, 176], or if the strains can be calculated with the finite element method. In [31] Dang Van et al., on the basis of the analysis of more than 00 fatigue tests of different steels (low- and high-strength) and different geometries of welded joints, founded that fatigue life of welded joints, calculated on the basis of stresses determined with the hot spot method [177, 178], is not strongly influenced by types of the materials joined. It is observable particularly for a number of cycles greater than In the case of a lower number of cycles, for

13 1 Introduction higher-strength materials, the permissible stresses are higher than for normal steels. It is important to draw attention to the fact that the notch coefficients for high-strength steels are greater than those for low-strength steels [4]. Thus, it is interesting to analyse relation between strength of normal steels and higher-strength steels in the local notation, i.e. including the theoretical notch coefficient. The safe fatigue life of the butt joints is higher than that of the fillet joints. Maddox [166] claims that good results can be obtained for nominal stresses but, in his opinion, the hot spot method should be developed in future. Principles of local stress determination according to the hot spot method are presented in Fig The local stresses can be defined from strains determined by extrapolation using two or three tensometers, or calculated with the finite element method. If the local approach for welded joints subjected to multiaxial loading is applied, it is necessary to know the stress concentration for bending and torsion (K tb, K tt ) at the fusion edge [165, 17, 0, 1]. Because of the fact that it is usually not possible to measure the actual radius of the fusion edge, in order to solve this problem, a suitable method is necessary. For welded joints subjected to uniaxial loading, the problem has been successfully solved owing to the application of so-called fictitious radius or, in other words, conventional radius [0, 08], based on the Neuber theory [173, 174]. Local methods for determination of fatigue life of welded joints under multiaxial fatigue were reviewed by Labesse-Jied [89]. The calculated fatigue lives of welded joints made of C45 steel and subjected to proportional and non-proportional random tension-compression with torsion loading were located in the scatter band of coefficient 4. The analysis was done on the basis of local stresses with plastic strains. The method using the conventional radius in the notch root can be applied for determination of the theoretical notch coefficient in ε hs ε 1 ε t Φ 0.4t t Fig Determination of strains with the hot-spot method

14 1 Introduction 3 the case when the notch radius in the welded joint is small and tends to zero. In this paper, this method is considered for complex loading. Similar to the Neuber s method based on the fictitious radius in the notch root, is the method proposed by Lawrence et al. (for example in [98]). In this model, determination of the maximum fatigue coefficient of stress concentration K fmax is suggested. That value is determined for the critical radius in the notch root, equal to the critical value a* dependent on the material. It took the values from about 0.1 mm for welds made of highalloy steels to 0.5 for low-alloy steels. Another method of determination of geometric stresses, next applied for fatigue life calculations, was presented by Xiao and Yamada [46]. It was proposed by them to perform calculations with the use of stresses occurring 1 mm from the point of contact on the surface of the joined materials. In [5] and [38, 47], Sonsino et al. claim that in practice damages should be accumulated on the assumption that the sum of damages according to the Palmgren-Miner hypothesis is D = 0.5. In [5], on the basis of the results of the tests under non-sinusoidal variable loading of the welded joints in bridges, at the drilling platforms or steel chimneys, it was observed that the sum of fatigue damages was D < 1. In [50], rough steel welded joints were subjected to variable-amplitude loading and it was found that damage accumulation in the considered joints varied about D = 1, and for the machined welded joints D = In [96], Lahti found that the damage sum for variable-amplitude loading was less than 1. In [168], Mayer et al. stated that the experimental life was usually 3.5 times less than the life calculated according to the Palmgren-Miner fatigue damage accumulation hypothesis. Thus, it can be assumed that the damage sum is included in (0.33 1) according to various test results. Standard recommendations referred to calculations of welded joints can be found in Eurocode 3 [38, 07], or the standards of the International Institute of Welding (IIW) [48] (for steels), and Eurocode 9 [39] (aluminium alloys). In [1], typical fatigue diagrams for welded joints under axial loading (or bending) and torsion with constant inclination coefficients are presented (see Fig. 1.). In present paper two models of fatigue life estimation, based on stresses and the strain energy density parameter are presented. For the uniaxial loading state (bending or axial loading), the model using local stresses was discussed. This model includes a value of the theoretical notch coefficient. As stated before, on the basis of [177, 178], it can be said that fatigue life of steel welded joints does not depend on a kind of material. In Chap. 3, there is a model of fatigue life assessment under uniaxial stress state with statistical handling of data presented. Chapter 5 contains the analysis of tests of four materials subjected to different

15 4 1 Introduction σ a τ a m σ = 3 m σ = 5 m τ = 5 σ af τ af 5* N f cycle Fig. 1.. Standard slopes of stress fatigue graphs for welded joints ( a normal stress, a shear stress) loading: cyclic, variable-amplitude with Gaussian distribution, and variable amplitude with Gaussian distribution and overloading for symmetric and pulsating loading. The analysis was based on the determined fatigue characteristics for all the considered materials. Another approach is applied for the complex stress state. When the stress and strain tensors are determined for the welded joint, it is necessary to reduce the multiaxial loading state to the equivalent uniaxial state. For this purpose the fatigue effort criteria based on stress, strain, or the strain energy density parameter [115, 13, 136, 164] referred to the critical plane can be used. In Chap. 4, the energy model using the strain energy density parameter for complex loading was presented. This model includes both stresses and strains occurring in the material. In [156, 190, 193] it has been proved that in the case of great number of cycles the stress and energy models are the most appropriate for fatigue description, and for low numbers of cycles the strain and energy models are good. Thus, the energy model seems to be universal and it was verified many times in many papers concerning uniaxial loading [13, 14, 55, 59, 64, 65, 66, 83, 10, 103, 105, 106, 1, 136, 154, 179, 180] and complex loading [57, 83, 116, 118, 10, 14, 15, 130, 135, 140, 141, 14, 149, 150, 151, 15, 155, 187, 193]. In this paper, known results obtained for tube-tube and flange-tube joints under pure bending and torsion and their combination, in- or out-of-phase, and also for chosen steel welded joints under variable-amplitude loading [165, 17, 0, 1, 40, 41, 4, 43] and aluminum joints [86, 87, 88, 6] were evaluated. For analysis, some selected criteria based on the energy parameter for multiaxial fatigue were applied [3, 115, 13, 136, 164, 165, 14, 17, 0, 1].

16 Welded Joints as the Stress Concentrator.1 The Complex Stress State in the Notch Complex stress concentration characterizes welded joints in which both geometrical and structural notches can be distinguished. In the case of geometrical notches under simple loading states, e.g. bending or axial loading, on the surface of the element in the notch root the plane stress state occurs. In round elements, apart from the nominal stress x, the additional circumferential stress is also observed along the element. It can expressed by a formula y C x, (.1) where 0 C. For simplification it can be written as 0 C 0 for for for K t 1 K t 1,, (.) K t where K t is the stress concentration ratio. Analysing the results of calculations performed in LBF Darmstadt by Sonsino et al., the following equation can be formulated [193, 9] C K t 1. (.3) K t The coefficient C, defining the values of circumferential stresses depending on the stress concentration ratio is shown in Fig..1. From the analysis of the figure it appears that the value of the coefficient C tends to the Poisson ratio,, for K t close to (according to (.)).

17 6 Welded Joints as the Stress Concentrator It should be also noted that in the case of a sharp notch, the plane stress state is accompanied by the plane strain state ( x, y = 0, z ). It results from the adoption of the elastic body model, the generalized Hooke s law, and y x according to (.1) and (.). For sharp notches, stress distributions for tension, bending and torsion are shown in Figs..,.3 and.4. v Value of coefficient C K tb 3.5 Fig..1. Coefficient C versus theoretical stress concentration coefficient (a) P (b) P axial stresses σ max nominal stresses σ max axial stresses circumferentia l stresses transverse stresses radial stresses P P Fig... Stress distributions in elements with sharp notches under tension: (a) flat element, (b) cylindrical element

18 .1 The Complex Stress State in the Notch 7 (a) M g (b) M g axial stresses σ max nominal stresses σ max radial stresses circumferential stresses axial stresses M g M g Fig..3. Stress distributions in elements with sharp notches under bending: (a) flat element, (b) cylindrical element M s local stresses τ max nominal stresses M s Fig..4. Stress and strain distributions in the element with a sharp notch under torsion Under tension and bending, stress distributions have been shown for flat and cylindrical elements. In both cases, on the notch root surface plane stress state is observable. In the case of flat elements, inside the material, the plane stress state occurs, and in cylindrical elements the spatial stress state is observed.

19 8 Welded Joints as the Stress Concentrator Stress and strain distributions in smooth and notched elements have been already analysed in [67, 108, 110, 135, 139, 153, 00, 01, 10, 1, 13]. They were used mainly in non-local methods of fatigue life assessment.. Theoretical Notch Coefficient The theoretical notch coefficient is defined as K t e xx. (.4) xxn According to the Neuber rule, this coefficient can be expressed as the geometric mean from the stress and strain concentration coefficients K t K K, (.5) which are defined as and ep xx K (.6) xxn ep xx K, (.7) xxn ep ep where xx and xx are the elastic-plastic strain and the stress in direction of x axis, respectively. The notch coefficients can be determined in a numerical way, for example with the finite element method, appropriate monograms or suitable formulas, more or less complicated [171, 181, 183, 184 and 185]. There are also other models joining nominal and actual stresses [54], but they are not considered in this book. Up to the yield point, the following relation takes place K t K K. (.8)

20 For greater stresses, the known relation is valid [0, 4]. Theoretical Notch Coefficient 9 K K t K, (.9) ε (see Fig..5). Xiao and Yamada [46] point that the theoretical notch coefficient K t for welded joints can be determined as a product of the weld geometry action K w and the influence of structure change in the weld K s, which can be expressed as K t = K w K s. (.10) Influence of K s changes in a welded joint was considered by Chen et al. in [8], and Cheng et al. [9], who tested specimens made of 1Cr 18Ni 9Ti steel under pure tension, pure torsion, and non-proportional tension with torsion. In the specimens tested there was no change of geometry in the joint, i.e. K w = 1. The result scatters for welded joints were greater than those for the native metal. Under pure tension compression and pure torsion, the fatigue strength of the weld material was less than that of the native material. Such a change was not observed under non-proportional tension with torsion. Thus, the coefficients K s, including structure changes usually are not separately calculated, and it is assumed that K t = K w. (.11) Whereas, the influence of the K s coefficient is taken into account during the determination of fatigue notch performance coefficient. The theoretical notch performance coefficient K t can be used for transformation of stress K t K σ K ε K σ K t K ε R e σ Fig..5. Relation between theoretical notch coefficients and stress value

21 10 Welded Joints as the Stress Concentrator σ a,n, σ a 1000 MPa local values 500 K t nominal values N f cycles Fig..6. Transformation of stress amplitudes from nominal to local system amplitude values from nominal to local system, shown in Fig. (.6) according to (.4)..3 The Fatigue Notch Coefficient The fatigue notch coefficient K f [37, 19] is determined by comparison of stresses in smooth, sm and notched not elements K f sm. (.1) not Interpretation of the coefficient K f is shown in Fig..7 [1,, 3, 119, 11, 135, 189, 190]. The fatigue notch coefficient is usually determined for 10 6 cycles, i.e. [97]

22 .3 The Fatigue Notch Coefficient 11 σ an smooth element notched element K f (N f ) N f Fig..7. Comparison of nominal stresses an for smooth and notched elements K f 6 sm 10 cycles. (.13) 6 not 10 cycles From Fig..8 it appears that the fatigue notch coefficient increases as a number of cycles rises. Generally speaking, it can be stated that it is dependent on a number of cycles [0, 43, 197] K f sm (Nf ). (.14) not (Nf ) Let us derive a relationship [11] K f log Kf Nf (N f ). (.15) 3 10 According to this relationship, on the assumption that N f = 10 3 cycles, the fatigue notch coefficient K f = 1. However, it is usually given as a constant value for N f = 10 6 cycles (.13) (see Fig..8). It can be written as K f log Kf Nf (N f ) K f (10 ). (.16) 6 10

23 1 Welded Joints as the Stress Concentrator K f 1 N f cycles Fig..8. Fatigue notch coefficient K f versus number of cycles N f There are many models joining theoretical and fatigue notch coefficients in the following general form K f = f(k t ). (.17) Many papers, among these [175, 47] present relationships among different forms of (.17) for various materials, types of notches and plastic strains occurring in the notch bottom..4 The Fictitious Radius of the Welding Notch For determination of notch coefficients the fictitious (conventional) radius in the notch root is applied. Determination of this radius results from stress averaging according to the Neuber s proposal [174]. It is assumed that the crack initiation is controlled by stress in the notch root averaged in a small volume of the material, in the point of the maximum stress occurrence. A suitable material parameter is a substitute microstructural length *. Stresses in the notch root must be averaged in the interval * in the direction normal to the surface along this length normal to the notch surface. Taking into account an actual radius in the notch root and the coefficient of multiaxiality s, the expression for the fictitious radius in the notch root is obtained f = +s *. (.18a)

24 .4 The Fictitious Radius of the Welding Notch 13 For the weld, the worst case can be assumed, i.e. the radius = 0, which corresponds to the crack. Then the calculated fictitious radius is expressed as f = s *. (.18b) If the radius is known, it is possible to calculate the notch coefficient, components of the local stress tensor and the corresponding strains. The fictitious radius also depends on geometry of the specimen and a loading mode [159, 173, 174, 0, 08] they should be taken into account under biaxial bending and torsion (see Table.1). The fictitious notch coefficient f depends on the actual notch coefficient, the substitute microstructural length * and the coefficient of multiaxiality s from Table.1 (according to the Neuber s proposal) resulting from the stress state multiaxiality in the notch root. In [9, 30] and some other papers, the authors proposed to determine the substitute microstructural length * according to the following equation K t 1 K f 1 * 1 s (.19) (see Fig..9). As it was said above, the zero notch radius, = 0, is often assumed with * = 0.4 mm for welded steels, * = 0.1 mm for aluminium alloys [173, 174, 0, 08] and s =.5 for plane specimens, when the Huber-Mises-Hencky criterion is used. Then, the fictitious radius f = 1 mm for welded steels and f = 0.5 mm for aluminium is obtained, on the basis of which the Table.1. Coefficients of multiaxiality s according to Neuber [173, 174] Loading axial or bending shearing or torsion Specimen plane Round - Criterion Huber-Mises-Hencky Tresca 1 1 maximum normal stresses 1 Beltrami - 1 1

25 14 Welded Joints as the Stress Concentrator 0.5 ρ*, mm 0.4 Cast steels and welded joints 0.3 Austenitic steels Al, Cu, Mg alloys steel R p0., MPa Fig..9. Substitute lengths of microstructure * for chosen materials and different yield points determination of the fatigue notch performance coefficient is possible. However, in the case of round specimens subjected to bending, on the assumption that the Poisson s number =0.3, for welded steels the following formula is obtained fb = mm = 1.16 mm (.0) under torsion the following expression is obtained ft = mm = 0.4 mm. (.1) for aluminium subjected to bending the following formula is obtained fb = 0.1 and for torsion 5 mm = 0.9 mm (.) ft = mm = 0.1 mm. (.3)

26 .5 The Notch Coefficient with the Use of the Fictitious Notch Radius 15.5 The Notch Coefficient with the Use of the Fictitious Notch Radius Huther et al. [51] considered fillet joints and analysed influence of the angle of weld face inclination within (30º 55º), and the radius in the notch root within (0.5 3) mm on the fatigue limit. Geometry of such a joint is shown in Fig..10. When the angle rises under stresses determined according to the nominal system, the fatigue limit decreases. The fatigue limit decreases also as the radius in the notch bottom increases. In [17], influence of the weld face inclination (0º 90º) and the transfer radius ( ) mm on the theoretical notch coefficient was considered. Greater notch coefficients are obtained for smaller notch radii. For angles (45º 75º) stabilization of the notch coefficient is observed. In the weld penetration zone the existence of the fictitious radius in the notch root can be assumed, keeping the same weld face inclination, and then it is possible to determine the theoretical notch coefficient K t. Thus, in order to calculate K tb and K tt from the fictitious radius in the notch root f in round specimens, it is necessary to chose separately the fictitious radii for welded elements subjected to bending ( fb = 1.16 mm) and torsion ( ft = 0.4 mm), and for elements made of aluminium alloy under bending ( fb = 0.9 mm) and torsion ( ft = 0.1 mm) when the constant angle is kept. Θ ρ Fig..10. Weld joint with the marked angle of weld face inclination and the radius in the notch root

27 3 The Stress Model for the Assessment of Fatigue Life Under Uniaxial Loading 3.1 Algorithm for the Assessment of Fatigue Life Under Uniaxial Loading State Fatigue failure of machine and structure elements caused by service loading often occurs under random stress state. In such a situation, fatigue life is usually calculated with analytic methods or cycle counting methods. The analytic methods use spectral analysis of stochastic processes, and the cycle counting methods are based on numerical algorithms of cycle and halfcycle counting from histories of stress, strain or the energy parameter. The cycle counting methods include schematization of random loading histories, damage accumulation and then fatigue life calculation. Schematization of random histories includes counting of amplitudes and mean values of cycles and half-cycles occurring in the loading history. In order to define fatigue life under random stress states, determination of the basic fatigue characteristic of the considered material is necessary. It is defined on the basis of cyclic fatigue tests. The basic characteristics for great number of cycles are the stress characteristics in the system a N f, the so-called S N characteristics. The first characteristic was elaborated by Wöhler [44] in 1860 in a single logarithmic system log N f a b. (3.1) a In 1910, Basquin [16] proposed a characteristic that can be written in a double logarithmic system as log N f a b log. (3.) a It is necessary to point out that many authors meaning the Basquin characteristic call it the Wöhler curve. In 1914, Stromeyer [31] presented another proposal including the fatigue limit

28 18 3 Stress Model for the Assessment of Fatigue Life log N f a b log a af. (3.3) The next proposals were formulated by Corson in 1955 (see [195]) Nf a exp c a af a (3.4) af and Bastenaire in 1974 (see [174, 195] c a a af N f exp. (3.5) a af b Other models were discussed in papers by Palmgren [196], Weibull (1949), Stüssi (1955) and Bastenaire (1963) (see [173]), Kohout, Vchet (001) [81]. However, the most frequently applied is the Basquin model expressed by (3.), the so-called S N fatigue curve. Under uniaxial loading, fatigue life is calculated according to the algorithm shown in Fig. 3.1 and the stress model [17, 18, 94, 95, 17, 143, 144, 145, 148, 160, 164]. A similar algorithm for uniaxial loading has been proposed by Goo [43]. Stage 1 The input data for fatigue life calculations are strain (t) or stress (t) histories, which can be obtained from: measurements of actual strains [41] or forces (strain gauges, extensometers, force gauges). Under uniaxial tension and on the assumption of a 1 Measurement, calculation or generation of σ(t) Determination of extrema σ max,loc and σ min,loc 3 Cycle counting σ ai, σ mi 4 Transformation of amplitudes in relation to mean values σ ati 5 Calculation of damage degree S(T o ) 6 Calculation of fatigue life T cal Fig Algorithm for determination of fatigue life under uniaxial random loading

29 3.1 Algorithm for the Assessment of Fatigue Life 19 perfectly elastic body, the relationship between the stress and strain histories can be written as (t) = E(t), (3.6) previous numerical calculations [85] (FEM finite element method, BEM boundary element method, FDM finite difference method), computer generation of random sequences with shaped probabilistic characteristics corresponding to service conditions or the predicted states. Standard programs elaborated in some research centers can be used for this purpose. Some well-known standards are: WASH1 for loading simulation in drilling platforms [49, 05], Broad64 and MMMOD64 [6] (see also [5]) for drilling platforms, too, CARLOS [06] for car wheel loading, wind load [5]. Other possibilities of generation of signals have been presented, among others, in [6, 5, 85, 170, 198, 11, 36, 45, 48]. Stage At this stage, extrema of the stress history are defined. Under random history, values of successive extrema are determined. This process includes observation of the derivative from the history and search of its monotonic changes, see Fig. 3.. Figure 3. shows some exemplary determined local minima (, 4, 6, 8) and extrema (1, 3, 5, 7, 9) in a course fragment. σ(t) MPa t 4 6 Fig. 3.. Determination of local extrema 8

30 0 3 Stress Model for the Assessment of Fatigue Life Stage 3 In the case of random histories with wide frequency bands, several cycle counting methods can be applied [80, 38]. The full cycle method in which half-cycles are not included, and the obtained life is usually overestimated. However, with the use of three other methods, i.e. the methods of range pairs, hysteresis loop and rain flow [10, 34, 35], the cycles, half-cycles and their mean values can be determinated. These three methods usually give similar results. In practice, the rain flow method (so-called envelope method) is most often applied. Its scheme is shown in Fig Envelopes are drawn from each local extremum (maximum or minimum). If the envelope has its beginning at the local minimum, it ends at the local maximum located opposite the local minimum, the value of which is lower than the initial minimum. The envelope beginning at the local minimum 0 ends at the local maximum 3 opposite the local minimum 4. The same procedure is applied when the envelope begins at the local maximum. Then it ends at the local minimum opposite the local maximum, the value of which is higher than the initial maximum. The envelope beginning at the local maximum 1 ends at the point located opposite the local maximum 3. Half-cycles should be isolated from the determined cycles. The half-cycle of the largest span is included between the global maximum and global minimum (3 and 8). If the local minimum occurs as the first local minimum, the half-cycle is determined between this local minimum and the global maximum (0 and 3), and between this global maximum and the preceding local minimum (3 and 8). One more half-cycle is obtained between the global minimum and the last local maximum (8 and 9). The same procedure is applied when the beginning of the cycle is in the local maximum. σ(t) MPa t halfcycle cycle halfcycle cycle cycle halfcycle Fig Cycle counting with the rain-flow method [7, 34, 43, 80]

31 3.1 Algorithm for the Assessment of Fatigue Life 1 As it was mentioned above, the rain flow method, so-called envelope method, allows to define both cycles and half-cycles, which are determined by suitable envelopes (see Fig. 3.3). This method has been programmed and cycles are counted by the computer program. The amplitude ai and the mean value mi of a cycle or a half-cycle are determined each time. Stage 4 At this stage, transformation of cycle amplitudes ai takes place in relation to the occurring mean values mi according to the general equation for the transformed amplitude, analysed in some previous papers [74, 75, 76, 113, 146, 147], ati = f( ai, mi ). (3.7) There are many models that take into account the influence of mean values. In this paper, the above transformations have not been widely presented because of the fact that in welded joints high residual stresses are often observed and then probable loading with the mean value occurring while a cycle do not influence the fatigue life. Stage 5 There are many hypotheses of fatigue damage accumulation (stage 4) [35] (linear and nonlinear). The linear hypotheses proposed by Palmgren Miner [169, 196], Haibach [47] and Serensen-Kogayev [09], Corten- Dolan [30], Liu-Zenner [99] are most frequently applied. Damages can be accumulated according to the Palmgren Miner hypothesis [169, 196], including amplitudes below the fatigue limit and the coefficient a 1 j ni for ai a m af i1 S af PM To N, (3.8) o ai 0 for ai a af

32 3 Stress Model for the Assessment of Fatigue Life Haibach hypothesis [46] j ni for m ai af i1 af No ai S H To, (3.9) k ni for mp ai af i j af No ai where [3]: p =1 for steels and aluminium alloys, p = for casts and sintered steels, Serensen-Kogayev hypothesis [09] j ni for ai a m af i1 S af SK To bn, (3.10) o ai 0 for ai a af where: b k i1 ai t a max i a a af af for b > 0.1, (3.11) is the Serensen-Kogayev coefficient, connected with a history character, and n i t i k (3.1) n i1 i is frequency of occurrence of particular levels ai in observation time T 0, and af is the general fatigue limit. The relationship (3.11) is valid if the a max 1 k following condition is satisfied 1and ait i af a max i1

33 3.1 Algorithm for the Assessment of Fatigue Life 3 Corten-Dolan hypothesis [30] j ni for m' ai af i1 S a max CD To N1 ai 0 for ai af, (3.13) where m = ( ) m, m af N 1 N o (3.14) a max Liu-Zenner hypothesis [99] j ni for ai a m' af i1 S a max LZ To N1 ai 0 for ai a af, (3.15) where m mi m', m af N 1 N o, (3.16) a max and m i is a slope of the fatigue curve S N for fatigue crack initiation. It is important to note that the models (3.1) and (3.15) act in a similar way as the Liu-Zenner model, however the Liu-Zenner model explains a new slope of the fatigue curve. A model similar to the Serensen-Kogayev proposal was formulated for alloys of non-ferrous metals [58]. From calculations, b less or greater than 1 is obtained, depending on the mean-square weighed amplitude [107, 157]

34 4 3 Stress Model for the Assessment of Fatigue Life j ni for ai a m af i1 S af K To b' N, (3.17) o ai 0 for ai a af where a coefficient including a damage degree (D1) is determined. This coefficient characterizes the history and takes the form (N f ) b', (3.18) aw where aw stress amplitude for the given number of cycles, expressed by the following equation: 1 ai aw i i ni ai stress amplitude, n i a number of stress cycles with amplitude ai. for ai > a af, (3.19) where: aw stress amplitude for the given number of cycles. In the discussed calculations, it was assumed that a number of cycles N f was equal to N 0, so: ( N f ). (3.0) af Thus, when aw aw aw (Nf ) (Nf ) (Nf ) then then then b' 1 b' 1. b' 1 The damage degree changes depending on a level and a number of stress amplitudes. It decreases as the amplitudes increase. Figure 3.4 shows a

35 3.1 Algorithm for the Assessment of Fatigue Life 5 log(σ a ) b =1, σ aw = σ af σ amax b >1, σ aw < σ af σ af b <1, σ aw > σ af aσ af Fig Changes of the coefficient b for ( N f ) N o af log(n) scheme of changes of the coefficient b depending on the weighed amplitudes for the S N curve. The hypotheses (3.8, 3.9, 3.10, 3.13, 3.15, 3.17) can be written as one expression: j ni dla ai a m af i1 b * N * ( af / ai ) S(To ), (3.1) k ni h dla (m p) ai a af i j1 N * ( af / ai ) where: S(T o ) material damage degree at time T o according to (3.8, 3.9, 3.10, 3.13, 3.15) or (3.17), n i a number of cycles with amplitudes ai int o, T o observation time (for analysis of loadings with variable amplitudes a number of cycles in one block, N bloc is assumed), m exponent of the S N fatigue curve, m modified slope coefficient for the S N fatigue curve for Corten-Dolan (3.13) and Liu-Zenner (3.17) hypotheses, in another case m =m, N o a number of cycles corresponding to the fatigue limit af, N* = N 1 for Corten Dolan (3.13) and Liu Zenner (3.17), in another case N* = N 1, k a number of class intervals of the amplitude histogram (j < k), a coefficient allowing to include amplitudes below af in the damage accumulation process, (for Haibach (3.9) and Corten-Dolan (3.13) a = 1),

36 6 3 Stress Model for the Assessment of Fatigue Life b* coefficient including history character; for Serensen-Kogayev (3.10) b* = b, for Kardas-agoda [57] (3.17) b*= b, in other cases b* = 1, p coefficient modifying the fatigue curve according to Haibach for amplitudes below the fatigue limit, h coefficient for the Haibach hypothesis (3.9) h =1 (for other hypotheses h = 0). General forms are shown in Fig. 3.5, and method of damage accumulation according to the Palmgren Miner rule is presented in Fig As it was stated before, the assumption of linear summation of fatigue damage with modifications was proved many times during experiments log(σ a ) σ amax m m b,b b σ af m-p N 1 N o aσ af log(n f ) Fig Original Basquin fatigue curve and its modifications for fatigue damage accumulation log(σ a ) σ a1 n n i (σ ai ) n i1 (σ a1 ) n i (σ a ) = i = 1 N i (σ ai ) N 1 (σ a1 ) N (σ a ) σ a log(n f ) N f1 N f Fig A way of fatigue damage accumulation

37 3. Statistic Evaluation 7 under uniaxial loading being a stationary stochastic process of normal probability distribution. Stage 6 After determination of a damage degree during observation time T o according to a general form (3.1), fatigue life is determined T cal To. (3.) S(T ) o After determination of the damage degree S(N block ) for a number of cycles N block in a loading block according to the general formula (3.1), fatigue life is calculated according to the following equation N block N cal S(N block ). (3.3) 3. Statistic Evaluation From references, for example from [96], it appears that for fatigue tests large scatters are typical. There are scatters of life under a given loading or scatters of loading (stresses, strains, the strain energy density parameter) under the given life. From the paper by Lahti et al. [96] it appears that in the case of life of welded joints test results are included in the scatter band with the coefficient about 4 with probability 95%. It is defined as life scatters, i.e. T N = N cal /N exp, (3.4) or inverse of (3.4) T N =1/T N = N exp /N cal. (3.5) Ratios (3.) and (3.5) can be called the scatter band with coefficient T N. This scatter band varies depending on materials, loading level and mode, and it is included within the range from 1.5 for a low number of cycles, steel and notched specimens to 5 for a level close to the fatigue limit, cast iron or welded joints. The least scatters are obtained for tension where all the section area is equally subject to cracking (Fig. 3.7). A little greater

38 8 3 Stress Model for the Assessment of Fatigue Life tension torsion bending Fig The most loaded parts of the section under different simple loadings scatters are often obtained under torsion (in the case of notches they are much greater), where the greatest stresses occur at the perimeter. The greatest scatters can be observed for bending, where only extreme fibres are loaded to a highest degree. In [90, 91, 3], the error of life determination was related to the experimental life according to the following formula E = N N cal N exp exp. (3.6) In [1] a standard deviation of logarithms of the calculated lives related to the experimental ones is presented s N n i1 log N cal log N exp. (3.7) n 1 In [1], a scatter of the calculation results related to the experimental results is considered N TN10% TN90% T, (3.8) where T N is defined by (3.8), for 10% and 90% of probability of damage, respectively. Other authors analyse the stress scatters. Bellet et al. [19] compare calculated and experimental lives trying to assess efficiency of the models according to the following formula

39 3. Statistic Evaluation 9 E = exp cal. (3.9) Another possibility of stress comparing is proposed by Sonsino and coauthors in many papers, for example [36, 19, 0,, 39] who apply the following equation: T = P 10% P 90% for N f = const. (3.30) Such a formula includes only 80% calculating points. Equation (3.30) for life can be written as: T N = T m, (3.31) and in this case T N is defined according to (3.31) as: T N = N N f f P 10%. (3.3) P 90% There are also models that include scatters of damage degrees analysis (see [7]), which is similar to (3.3) Dcal Dexp E. (3.33) D exp During the analysis of life scatters the life ratios according to (3.4) or (3.5), or logarithms of life are usually used, according to E = log N N exp cal (3.34) (see [11]). The mean value of the considered quantity can be defined as n 1 E E i, (3.35) n i1

40 30 3 Stress Model for the Assessment of Fatigue Life and the mean error of the mean value can be defined as s SE, (3.36) n where n a number of measurements, s mean standard deviation. For determination of the variance the following formula should used n s 1 n 1, (3.37) E i E i1 and the standard deviation should be determined from variance (3.34) s s. (3.38) In the case of material fatigue, the significance level is usually assumed at the minimum level = 5% or 10%, sometimes even 0%. Thus, the mean value should be included within the range or (SE) E t (SE) (3.39) t ( n1), / (n1), / (s) E t (s), (3.40) t ( n1), / (n1), / where t (n 1), / constant from the t-student s distribution for the mean value error SE (3.36) or the population error s (3.38). Constant t (n 1), / from the t-student s distribution is determined for a half of the significance level / because of section of the normal distribution edges (see Fig. 3.8). The mean scatter is determined from the following relationship E TN 10 (3.41)

41 3. Statistic Evaluation p(z) α/ α/ random variable, z Fig Normal distribution with sections at significance level in the scatter band with the scatter coefficient T N expressed as T N t / s n (3.4) For the significance level / =.5% and n = 60 (often 0 30 measurements) the scatter band for all population is obtained. It is equal to two standard deviations (s) (or maximum.s for 0 measurements), which is often applied in tests [166] and corresponds to the scatter band with the coefficient 3. The scatters 3s correspond to a large significance level tending to zero but they are not applied in fatigue tests [9].

42 4 The Energy Model of Fatigue Life Assessment For the complex loading state the energy model is being proposed. The model is based on the strain energy density parameter (SEDP) and analyses changes of stress (normal and shear) and strain (normal and shear) in the critical plane. It also distinguishes tension and compression. 4.1 The Energy Parameter Under Uniaxial Loading The change of strain energy density, widely used in theory of plasticity, is also proposed as a parameter of the multiaxial fatigue analysis. Suitability of this parameter for description of fatigue processes seems to be promising, especially while formulation of thermal-elastic-plastic models of strain in the materials subjected to random thermomechanical loading. The models do not include a division of strain energy density into elastic and plastic parts, like in case of the parameters proposed by Smith Watson Topper (SWT) [14], Hoffman and Seeger [48], Bergman and Seeger []. In the elastic range, energy can be calculated from W 1. (4.1) In the time domain, energy density can be expressed as 1 W (t) (t) (t). (4.) When the equation is connected with the damage parameter where P a E, (4.3) SWT max m max, (4.4) a

43 34 4 The Energy Model of Fatigue Life Assessment and when the mean value of stress is equal to zero max = a, (4.5) the strain energy density takes the form P SWT W. (4.6) E It should be pointed that parameter P SWT has a dimension of stress. However, it is expressed as energy per a volume unit, i.e. MJ/m 3. Further modification of the considered parameter can be written as [] P k E (4.7) m a a or for shear stresses and shear strains [48] P a max G. (4.8) In order to distinguish tension and compression in a fatigue cycle, functions sgn[(t)] and sgn[(t)] should be substituted to (4.): W(t) (t) (t) (t) (t) sgn[ (t)] 1 4 (t) (t) sgn[ (t)] sgn[ (t)] sgn[ (t)] 1 sgn[ (t)] sgn[ (t)] (t) (t). (4.9) A two-argument logical function is sensitive to the signs of variables, and it is defined as sgn(x) sgn(y) sgn(x, y) when when when when when sgn(x) sgn(y) 1 (x 0 and sgn(y) 1) sgn(x) sgn(y) (x 0 and sgn(y) 1) sgn(x) sgn(y) 1 or or (y 0 (y 0 and and sgn(x) 1) sgn(x) 1), (4.10) where 1 when x 0 sgn( x) 0 when x 0. (4.11) 1 when x 0

44 4.1 The Energy Parameter Under Uniaxial Loading 35 After the substitution sgn(x,y) to (4.9) the following formula is obtained 1 W(t) (t) (t)sgn[ (t), (t)]. (4.1) Equation (4.1) expresses positive and negative values of the strain energy density parameter in a fatigue cycle and it allows to separate energy (work) under tension from energy (work) under compression. If the parameter is positive, it means that the material is subjected to tension. If the parameter is negative, the material is subjected to compression with energy equal to this parameter for the absolute value. Equation (4.1) has another advantage: a course of the strain energy density parameter has the zero mean value, and the cyclic stress and strain have also the expected zero value, i.e. R = 1. Moreover, when stress or strain reaches zero, (4.9) is equal to zero, so sgn( xy, ) 0.5 does not occur. Thus, (4.10) can be written in the reduced form sgn(x, y) 1 when sgn(x) sgn(y) 1 sgn( x) sgn(y) (4.13) 0 when sgn(x) sgn(y) 1 when sgn(x) sgn(y) 1. Figure 4.1a shows a course of the energy parameter according to (4.6), and the strain energy density parameter, taking into account signs of both stress and strain for an elastic body. If the signs of stresses and strains are not taken into account (Fig. 4.1b), a number of cycles with small ranges of energy parameters (Fig. 4.1b) is doubled and a non-zero mean value is obtained. This model is valid for R = 1. If cyclic stresses and strains reach their maximum values, a and a, then the amplitude of maximum strain energy density parameter according to (4.9) is W a (4.14) a a