Available online at www.sciencedirect.com

ScienceDirect

Procedia Engmeering 2 (2010) 2027-2035

Procedia Engineering

www.elsevier.com/locate/procedia

Fatigue 2212

A comparative study on fatigue damage assessment of welded joints under uniaxial loading based on energy methods

F. Pakandam, A. Varvani-Farahani*

Department of Mechanical & Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario M5B 2K3 Canada

Received 28 February 2212; revised 12 March 2212; accepted 15 March 2212

Abstract

The present study intends to evaluate fatigue damage of different welded joints under uniaxial loading condition and its response on fatigue lifetime. The main variables influencing the fatigue life of a welded joint are: applied stress amplitude, material properties, geometrical stress concentration factor. nnergy approaches were employed to evaluate the fatigue damage of various weld joints under uniaxial loading conditions. Energy-fatigue life (W-N) curves were further discussed and comported for their capabilities in assessing fatigue life of various joints through different parameters including curve slope, life data scatter, and how readily coefficients/constants are determined and employed in the energy methods. The critical plane/energy approach was found to be the most suitable energy-based approach for fatigue damage and life assessment of welded joints by offering sharper W-N curves and less life scatter. This approach also allowed employing readily available material coefficients/properties as compared with the notch stress-intensity energy approach. © 2010 Published by Elsevier Ltd.

Keywords: Fatigue damage; Welded joints; Energy approaches; W-N curves; Critical plane/energy model

1. Introduction

Many structures and components that include welded joints are often subjected to cyclic loads leading to deformation. At these stressed sites, the local stress and strain will exceed the elastic limit. Such situations are very common; resulting in early failures in parts designed for long life applications. Fatigue failures in welded joints happen mostly close to the welds rather than in the base metal far from the weld [1]. Fatigue of welded joints is very much dependant on local stress/strain components acting at the joint. Fatigue assessment of welded joints should address material microstructure, elastic-plastic response of the material under cyclic deformation, and modes of failure at the joints. Other variables influencing the fatigue life of a welded joint consist of the magnitudes of the stresses at the weld toe (/IS), strain-controlled fatigue properties, and the concentration of stress and strain at the weld toe [2,3].

Generally, fatigue damage approaches in welded joints are categorized either as 'global approaches' or as 'local approaches'. Strength assessments are named 'global approaches' if they result from external forces or nominal

* Corresponding author. Tel.: +1-416-979-5222 ; fax: +1-416-979-5265. E-mail address: avarvani@ryerson.ca.

1877-7058 © 2010 Published by Elsevier Ltd. doi:10.1016/j.proeng.2010.03.218

stresses in the critical cross-section under constant stress distribution assumption, also named 'nominal stress approach' [1]. Energy-based approaches are also considered 'local approaches' and incorporate both stress and strain in fatigue damage assessment of materials.

2. Energy methods in fatigue damage assessment of welded joints

Stress and strain-based criteria lack in comprehensively addressing materials response required for fatigue damage assessment. During fatigue cycles, both the elastic and plastic strain components and their corresponding stress values are involved to describe fatigue damage phenomena on the tested materials appropriately. To modify damage approaches and to construct precise continuum mechanics fundamentals, the fatigue approach should include both the stress and the strain components. Energy-based approaches were introduced to analyze the damage accumulation of notched/welded components [4]. It has been confirmed that in the case of large numbers of cycles, the stress and energy models are the best for explaining fatigue, and for a small number of cycles the strain and energy models are superior [5].

In each category of welded joints, varying from butt joint to cruciform joint, the weld toe geometry is mainly comparable to an open notch. The energy approaches describe the simultaneous stress-strain situation in the area near to the weld toe, concerned by the fatigue damage progress. In this regard, in welded joints subjected to cyclic loading, the highly stressed regions where cracks initiate and spread are normally located at weld toes. The stressstrain response near the crack initiation locations is required to describe direct relations between energy values, fatigue strength, and factors influencing fatigue of the structural welded components [6].

In the present study, energy-based approaches of (i) hysteresis-loop method, (ii) fracture mechanics approach (known as notch stress-intensity method), and (iii) critical plane/energy have been used to assess the fatigue of welded joints of various types [7-11].

2.1. Hysteresis loop energy method

The calculation of notch stresses and strains is based on the stabilized cyclic stress-strain curve generated by the Ramberg-Osgood equation. The stabilized cyclic stress-strain hysteresis loop diagram is estimated by Masing's hypothesis by doubling the amplitudes of the cyclic stress-strain curve for each cycle. The area within the loop corresponds to the dissipated plastic energy per cycle, per unit volume of the material, representing a measure of the plastic deformation work/energy done on the material. The hysteresis-loop energy method describes the plastic strain energy equation for Masing type metallic material (equation 1) [12]. The plastic energy range, AWp, is calculated based on components of the stress range (Ao) and the plastic strain range (Aep) extracted from stabilized stress-strain hysteresis loops as [7]:

A W p = ( i—ni) A aA £ p (1

1 — n'

Substituting for the plastic strain range and the stress range from the Coffin-Manson equation, equation 1 is rewritten as:

A W p = a', £' f (2 N , )b—c

p 1 — n' f fV '' (2)

where Nf corresponds to the number of cycles to failure, and ef are the axial fatigue strength and ductility

coefficients, and b and c are the fatigue strength and ductility exponents. Term n' is the cyclic hardening exponent.

2.2. Notch stress-intensity energy method

The weld toe is an important location for fatigue crack initiation and propagation in which high stresses are present. The local stresses at the crack tip are based on notch stress intensity factors. Based on the Linear Elastic Fracture Mechanics (LEFM) method, the fatigue life of welded joints is taken as crack propagation at the toe of the weld seam. LEFM relates the applied stress to the crack propagating on the welded joints as the number of cycles progresses. The goal is to find an averaged value for the strain energy density in the welded joints of structural steels and aluminium alloys with a V notch at the weld toe, where stresses are highly concentrated [13].

Lazzarin et al. [13] have introduced a strain energy density approach to estimate fatigue failure of welded joints. Based on this approach, fatigue failure occurs when the average value of the total or plastic strain energy density reaches a critical value in a cylindrical volumetric region around the notch tip with a radius Rc, independent of the loading mode. The total deviatoric strain energy density averaged over a circular sector with its center at the weld toe and radius Rc shown in figure 1 turns out to be [2,13]:

-( K i)2( Rc)

2 (¿1 -1)

^ d 2 E

(K2)2 (Rc )

2(¿2 -1)

( K 3)2( Rc )

2 (¿3 -1)

where E is the elastic modulus, ed1, ed2, ed3 are angular function integrals, X1, X2, X3 are the eigenvalues, and K1, K2, K3 are the notch stress intensity values for mode I, II, and III stresses, respectively.

This model is based on an averaged value for the strain energy density at the weld toe. The total energy value is found through the summation of mode I, II and Ill notch stress intensity factor energies. For a cyclic uniaxial applied load, the second and third energy terms corresponding to the mode II and mode III stresses are equal to zero leaving the mode I stress energy as the only present energy expression.

The nominal stress range, AS, or maximum stress, Smax, applied to each welded joint, and the fatigue notch stress concentration factor, Kf, are used to calculate the local stress, a, at the weld toe of the welded joints. All the required cyclic stress-strain properties of the base metals are used for some of the parameters needed for the calculations. The geometry factor, k1, required for the notch stress-intensity factor, K1, is calculated. The related notch loading stress modes; normal tension stress, an, transverse shear stress, zn(per), and longitudinal shear stress, zn(par), used to calculate the notch stress intensity factors, K1, K2 and K3, at .the weld toe. The fatigue notch factor, Kf, was calculated from the elastic notch stress-concentration factor, Kt.

Fig. 1. Presentation of the weld toe geometry [14].

2.3. Critical plane-energy method

In the critical plane-energy fatigue damage model, both the normal and the shear energies are computed on the most damaging plane of materials, referred to as the critical plane [11]. In this fatigue damage approach, the critical plane is defined by the largest shear strain and stress Mohr's circles during the reversals of a cycle, and the model consists of tensorial stress and strain range components acting on this critical plane. The range of maximum shear stress, ATmax, and shear strain, A(ymax/2), found from the largest stress and strain Mohr's circles for loading and unloading during the first and the second reversals of a loading cycle and the related normal stress range, Aon, and the normal strain range, Asn, on that plane are the components of this approach, shown in figure 2.

Fig. 2. Strain and stress Mohr's circles presenting stress and strain components acting on the critical plane-energy approach [11].

In Varvani's fatigue damage model both the normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively, and it is expressed as [11]:

(a'f e'f )

(AanAen ) +

(T f Y' f )

(At A (

v max v

where t f and y f are the shear fatigue strength and ductility coefficients, respectively.

The axial fatigue strength coefficient, of, and axial fatigue ductility coefficient, ef, are found from the cyclic stress-strain properties. The shear fatigue strength and ductility coefficients Tf and Yf are estimated as:

Y' f = f/3" x e' f

The ranges of maximum shear stress, Azmax, normal stress range, Act n, and shear strain, A(ymax/2), and the normal strain range, As n, are calculated as:

At = (°1 - 03 -, - (°1 - 03 x

max V n. ' loading V n. ' unloading

Ao = (oi + o3 x - (°1 + °3 x

^ U n ~ V ) loading V )

unloading

(6) (7)

A (± max ) = (_1_— _

2 2 loading 2 unloading

e + e e + e

Ae =( 1 3 ) -( 1 3 )

n v 2 ' loading \ 2 'unloading

(8) (9)

t ', =

where a1 and a3 are the maximum and minimum principal stresses and e1 and e3 are the maximum and minimum principal strains calculated from loading (90°) and unloading (270°) reversals of a cycle.

The critical plane/energy model of Varvani (equation 4) integrates shear energies defined on the critical plane. The model incorporates the normal and shear stress and strain components, through their summation, in the damage assessment of materials. Fatigue coefficients are used to calibrate the damage model for local stress and strain components at the weld toe. The local stresses are calculated from nominal stresses and the fatigue notch stress concentration factor, Kf.

To evaluate the energy based fatigue damage approaches, fatigue data of different types of seam-welded joints have been extracted from the available literature [14-17]. The fatigue tests on the welded specimens were performed under transverse uniaxial loading conditions. The S-N data obtained from these experiments were extended over low and high cycle fatigue regimes. The nominal applied stresses are used to calculate the local stresses/strains, based on

Neuber's notch analysis required in energy approaches to assess damage of welded joints based on energy approaches. It must be noted that the joints selected from the literature are ideal weldments in which the fatigue crack is assumed to initiate at the weld toe, in the weld metal [3]. Figure 3 presents various types of welded joints fatigue tested in references [14-17] and examined in the present study.

Fig. 3. Fatigue tested welded joints with different geometries [14-17].

4. Results and discussion

Figures 4-6 plot the energy-fatigue life diagrams generated based on energy approaches of hysteresis loop (HL), Lazzarin's notch stress-intensity (NS), and Varvani's critical plane/energy (CP). These figures compare fatigue W-N curves based on energy models for butt, cruciform (double fillet), and butt-strap welded joints. Fatigue life assessments of butt versus cruciform welded joints for the same material were evaluated based on energy approaches and are presented in figure 4. Fatigue life assessments of butt-welded joints were evaluated based on energy-based approaches for three sets of different weld cap height (same material), and are presented in figure 5. Fatigue life assessments of butt-strap fillet versus butt-ground welded joints for aluminium alloys (magnesium-manganese-aluminium alloy) were evaluated based on energy-based approaches and are presented in figure 6.

While the notch stress-intensity energy model shows slightly better scatter convergence for butt and cruciform joints presented in figures 4 and 5 as compared with the critical plane/energy model, the critical plane/energy model shows a considerably higher curve slope. This indicates that for any given energy value, the fatigue life range resulting from the diagram scatter band, is significantly smaller. This enables a designer to predict fatigue life of welded joints more accurately with less scatter.

Figure 4 presents W-N diagrams for butt joints versus cruciform joints made from low carbon steel calculated based on the energy approaches. In figure 4 the butt-welded joint of Chapetti possesses a fatigue notch factor of Kj=2.0 while Kf for the cruciform welded joint of Lazzarin (13x10 mm) is 25% higher in magnitude. The higher value of Kf shortens the fatigue life of the cruciform welded joint. The reduction in life based on the hysteresis-loop energy approach exceeds 4 times. This reduction in life, based on the critical plane/energy approach is up to 10 times. The notch stress-intensity approach however shows no reduction in life (figure 4b). The hysteresis loop

energy approach in figure 4a fails to introduce the weaker cruciform joint in terms of curve slope (curve slope=0.61) compared to the stronger butt joint (curve slope=0.67).

Comparison of W-N diagrams for the low carbon steel butt and cruciform welded joints in figure 4 shows a lower curve slope and higher scatter of energy-life for the butt joint as compared with the cruciform joint of the same material, in the notch stress-intensity and critical plane/energy models. This clearly designates a stronger joint and lower fatigue notch factor, Kf, for the butt joint in comparison with the cruciform joint, indicating that the scatter in energy values is inversely associated with Kf.

0.1 (a)

- butt joint of Chapetti et al. data

- fillet joint of Lazzarin data (steel-13x10)

: -0.31, R= 0. : -0.48, R= 0.

- ♦ butt joint of Chapetti et al. data — fillet joint of Lazzarin data (steel13x10)

N. ♦

: ^ _ \ ♦ ;

(c) -slope: -0.89, R= 0.96 --slope: -1.04, R= 0.98 J J 1 I L L ' ' ^ A

N [cycles]

N [cycles]

N [cycles]

Fig. 4. Energy versus fatigue life diagrams for low carbon steel welded joint types of butt and cruciform, based on (a) hysteresis- loop energy, (b) notch stress-intensity energy , and (c) critical plane/energy.

The butt joints of Reemsnyder (figure 5) with weld cap heights of 1.5 mm (Kf =2.10) and 2.3 mm (Kf =2.30) show similar response, with the lowest slope values compared to the other butt joint with weld cap height of 3.8 mm (Kf =2.35), in the notch stress-intensity and critical plane/energy models. This shows that welded joints with greater fatigue resistance have W-N diagrams that are more nearly horizontal. This obviously designates higher fatigue strengths for the butt joints having weld cap heights of 1.5 and 2.3 mm, with relatively small fatigue notch factors. The butt- joint with weld cap height of 3.8 mm also has a relatively small fatigue notch factor (Kf =2.35) and is also categorized in the same group of joints with weld cap heights of 1.5 and 2.3 mm in terms of fatigue notch factor, but shows higher curve slope in the energy models. In joints with the same geometry and similar fatigue notch factors, it appears that the joints with smaller weld cap heights demonstrate greater fatigue strengths. This may suggest that in

weld caps up to 2.3 mm height, a similar response can be expected in the W-N curves, while the weakest butt joint of the three compared joints, belongs to the joint with h=3.8 mm, possibly due to its large weld cap.

A comparison of W-N diagrams for the aluminium alloys butt-strap fillet and butt-ground welded joints in figure 6 shows a lower curve slope and higher scatter of energy-life for Lazzarin's butt-joint as compared with Webber's butt-strap fillet joint in the notch stress-intensity and critical plane/energy models. This clearly designates a stronger joint and lower fatigue notch factor, Kf, for butt-ground joint type of Lazzarin (Kf =1) as compared with butt-strap fillet joint type of Webber (Kf =1.95), for aluminium alloys, once again indicating that the scatter in energy values is inversely associated with Kf.

: ' ! -slope: -0.59, R= 0.90

\ ! --slope: -0.45, R= 0.62

❖ --slope: -0.78, R= 0.99

. X ♦♦ ♦ ♦ ! . O ! A v.« * X.V............i...........................

XN ? ! ♦ "

(a) O v. ¡V \ i - V\

o Reemsnyder data-h=1.5 -♦— Reemsnyder data-h=2.3 -A — Reemsnyder data-h=3.8 ^ 1 ° oi ♦ : ♦

slope: -1.50, R= 0.98 slope: -1.45, R= 0.94 --slope: -1.97, R= 0.90

N [cycles]

N [cycles]

N [cycles]

Fig. 5. Energy versus fatigue life diagrams for butt joints with different weld cap heights [16] developed based on (a) hysteresis loop energy, (b) notch stress-intensity energy, and (c) critical plane/energy.

In evaluating the W-N diagrams, it was found that the hysteresis-loop energy model fails to differentiate the stronger welded joints from the weaker welded joints using either the scatter index or the curve slope criteria (see figures 4a, 5a and 6a). The critical plane/energy model not only differentiates the stronger welded joints from the weaker welded joints using either the scatter and the curve slope index, but also recognizes a higher energy level for the stronger joint (with smaller fatigue notch factor). This could be seen in the comparison of the joints with different geometries; butt joint versus cruciform joint (figure 4c), and butt joints with different weld cap heights (figure 5c). In the case of butt-strap fillet joint versus butt-ground joint of similar alloys, both having relatively small values of fatigue notch factor, indicating strong joints, the critical plane/energy model differentiates the stronger welded joint from the weaker welded joint and shows energy levels with close proximity (figure 6c).

In assessing fatigue damage of welded joints through the energy models, another important comparison that must be made is how readily the coefficients/constants are determined and employed in the energy parameters. In the

hysteresis-loop energy model, no local stress analysis and fatigue notch factor were required. The fatigue assessment was determined from the nominal stress versus fatigue life data and base metal material coefficients; n', b, c, of, ef In the notch stress-intensity energy model, variables of ed and X are determined from diagrams [2]. Constants such as Rc and E are properties of the base metal. In order to obtain the stress intensity factor, K, the geometry coefficient, k, is obtained from a diagram through geometric specifications of the joint, also used to determine plate thickness [2]. The eigenvalue, X, is obtained from a diagram, and for normal tension-stress ont, the local stress amplitude is calculated [2]. Coefficients in the critical plane/energy approach are fatigue properties and are determined readily from strain-life data. Reference [19] provides details of coefficients/constants determined from diagrams and tables.

o o .......'1 ' ....... --slope: -0.84, R= 1 o° °

♦ ° % o - - ° C 0 ;

(a) •N

........1 1 1 1 1 III

- - O - - butt-strap joint of Webber data -♦— butt joint of Lazzarin data (al. ref.) ..............l--l.-JL.'-.l.UH.

1 d"d.' '!-1—.......I

slope: -0.50, R= 0.98 slope: -0.32, R= 0.97

- - O - - butt-strap joint of Webber data 1 ^^ -♦— butt joint of Lazzarin data (al. ref.)

o 'O bo. o • slope: -0.50, R= 0.98 - slope: -0.33, R= 0.97

(c) ♦ ' - ° 'o

- -O - - butt-strap joint of Webber data -♦— butt joint of Lazzarin data (al. ref.)

N [cycles]

N [cycles]

N [cycles]

Fig. 6. Energy versus fatigue life diagrams for aluminum welded joint types of butt and butt-strap developed based on (a) hysteresis loop energy, (b) notch stress-intensity energy, and (c) critical plane/energy.

5. Conclusions

Three energy-based methods were employed to assess fatigue of welded joints. These methods consist of the hysteresis-loop method, the notch stress-intensity method, and the critical plane/energy method. Calculated energy values based on the methods were plotted versus fatigue life data (W-N curves) for different joint types; (i) butt versus cruciform joint for the same base metal, (ii) butt joints with different weld cap heights, and (iii) butt versus butt-strap fillet joint of aluminium alloys. Energy-based fatigue models of the hysteresis-loop energy method, notch stress-intensity energy method, and the critical plane/energy method have corresponded differently as they were plotted versus fatigue lives. Of these models, the critical plane/energy model possessed a steeper W-N curve slope. This suggests that the critical plane/energy approach has correlated the fatigue data for the joints with less scatter in

fatigue life range. Both approaches of the notch stress-intensity and the critical plane/energy methods are dependant on Kf factor and they incorporate the effect of weld joint types in the fatigue analysis, while the hysteresis-loop energy approach fails to correspond to various joint types as the Kf factor is not used in the analysis.

The critical plane/energy approach is more readily used compared with other energy-based approaches. In addition, it is not based on geometrical specifications of the welded joint, thus reducing the error probability due to dimensional measurements.

Acknowledgement

Authors wish to thank the financial support of Natural Science and Engineering Research Council (NSERC) of Canada.

References

[1] W. Fricke, Fatigue analysis of welded joints: state of development, Marine Structures, 2003, 185-200.

[2] D. Radaj, C. M. Sonsino, W. Fricke, Fatigue assessment of welded joints by local approaches, 2nd Ed., WoodheadPublishing Limited and CRC Press LLC, 2006, Abington Cambridge.

[3] F. V. Lawrence, S. D. Dimitrakis, W. H. Munse, Factors influencing weldment fatigue, ASM Handbook, Volume 19, 1996, 274-286.

[4] W. Cui, A state-of-the-art review on fatigue life prediction methods for metal structures, Journal of Marine Science and Technology (7), 2002, 43-56.

[5] T. Lagoda, Lifetime estimation of welded joints, Springer Publications, 2008, Springer-Verlag Berlin Heidelberg.

[6] P. Lazzarin, R. Tovo, A notch stress intensity factor approach to the stress analysis of welds, Fatigue & Fracture of Engineering Materials& Structures 21, 1998, 1089-1103.

[7] J. Dziubinskyi, Fatigue failure criterion based on plastic strain energy density applied to welds, Int J Fatigue, 1991, 13 (3), 223-226.

[8] P. Lazzarin, R. Tovo, A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches, International Journal of Fracture 78, 1996, 3-19.

[9] B. Atzori, P. Lazzarin, R. Tovo, From a local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints, Fatigue Fracture Engineering Mater Struct 22, 1999, 369-381.

[10] P. Lazzarin, T. Lassen, P. Livieri, A notch stress intensity approach applied to fatigue life prediction of welded joints with different local toe geometry, Blackwell Publishing Ltd. Fatigue Fracture Engineering Material Struct. 26, 2003, 49-58.

[11] A. Varvani-Farahani, A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to inphase and out-of-phase multiaxial fatigue loading conditions, International Journal of Fatigue 22, 2000, 295-305.

[12] D. Lefebvre, F. Ellyin, Cyclic response and inelastic strain energy in low cycle fatigue, International Journal of Fatigue 6, 1984, 5-15.

[13] P. Lazzarin, P. Livieri, F. Berto, M. Zappalorto, Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading, Engineering Fracture Mechanics 75, 2008, 1875-1889.

[14] P. Livieri, P. Lazzarin, Fatigue strength of steel and aluminium welded joints based on generalized stress intensity factors and local strain energy values, International Journal of Fracture 133, 2005, 247-276.

[15] M. D. Chapetti, J. Belmonte, T. Tagawa, T. Miyata, Integrated fracture mechanics approach to analyze fatigue behaviour of welded joints, Science and Technology of Welding and Joining, 2004, Vol. 9, No. 5, 430-438.

[16] H. S. Reemsnyder, Development and application of fatigue data for structural steel weldments, Fatigue Testing of Weldments, ASTM STP 648, D. W. Hoeppner, Ed., American Society for Testing and Materials, 1978, 3-21.

[17] D. Webber, An evaluation of possible improvement methods for aluminium alloy fillet welded joints, Improving the Fatigue Performance of Welded Joints (TWI), 1983, Abington Cambridge.

[18] S. J. Maddox, D. Webber, Fatigue crack propagation in aluminum-zinc-magnesium alloy fillet-welded joints, Fatigue Testing of Weldments, ASTM STP 648, D. W. Hoeppner, Ed., American Society for Testing and Materials, 1978, 159-184.

[19] F. Pakandam, Masters Thesis on: Fatigue Damage and Life Assessment of Welded Joints Based on Energy Methods, Ryerson University, Toronto, Canada, 2009.