Modified Wöhler Curve Method and multiaxial fatigue assessment of thin welded joints

The present paper is concerned with the use of the Modified Wöhler Curve Method to estimate fatigue lifetime of thin welded joints of both steel and aluminium subjected to in-phase and out-of-phase multiaxial fatigue loading. The Modified Wöhler Curve Method postulates that, in welded connections subjected to in-service complex time-variable loading, fatigue damage reaches its maximum value on that material plane experiencing the maximum range of the shear stress amplitude, such a stress quantity being calculated according to the Maximum Variance concept. The most important peculiarity of the above multiaxial fatigue criterion is that it can be applied by performing the stress analysis in terms of both nominal and local quantities, where in the latter case the relevant stress state at the assumed critical locations can be estimated according to either the reference radius concept or the Theory of Critical Distances. The accuracy and reliability of our multiaxial fatigue criterion was systematically checked through several experimental results taken from the literature and generated by testing, under in-phase and out-of-phase biaxial loading, welded joints of both steel and aluminium having thickness of the main tube lower than 5 mm. Such a systematic validation exercise allowed us to prove that the Modified Wöhler Curve Method is a powerful tool suitable for performing the fatigue assessment of thin welded joints, this holding true independently of the strategy adopted to perform the stress analysis. Finally, a microstructural motivation of the length scales included in the Theory of Critical Distances can be established by linking this technique to gradient mechanics, as we will argue.     

On the use of nominal stresses to predict the fatigue strength of welded joints under biaxial cyclic loading

In this paper, the modified Wöhler curve method proposed by Susmel and Lazzarin is employed to predict the fatigue life of welded connections subjected to biaxial cyclic loading. This criterion is reformulated here in order not to take into account the mean stress effect, as suggested by several design codes (at least when welded connections are not completely stress relieved). The accuracy of the proposed method in fatigue lifetime estimation was evaluated by using a number of data sets taken from the literature. The modified Wöhler curve method was applied in terms of nominal stresses and was calibrated using the uniaxial and torsional fatigue curve determined by reanalysing the experimental data, as well as using the standard fatigue curves of the Eurocode 3. The proposed approach was seen to be successful, giving multiaxial fatigue life predictions located within the widest scatter band related either to uniaxial or to torsional data, independently of both out‐of‐phase angle and load ratio value. Finally, the accuracy of the modified Wöhler curve method was compared to the one obtained by applying the procedure suggested by the Eurocode 3: the proposed criterion is demonstrated to be much more accurate and reliable than the standard one.     

A stress-based method to predict lifetime under multiaxial fatigue loadings

This paper extends to low/medium‐cycle fatigue a stress‐based method recently proposed by the same authors for high‐cycle multiaxial fatigue assessments. By considering the plane of maximum shear stress amplitude coincident with the microcrack initiation plane, the method requires the calculation both of the maximum shear stress amplitude and the maximum normal stress relative to the same plane. Multiaxial fatigue life estimates are made by means of bi‐parametric modified Wöhler curves, which take into account the mean stress effect, the influence of the out‐of‐phase angle and the presence of notches by using a generalization to multiaxial fatigue of the fatigue strength reduction factor K_f. Approximately 700 experimental data taken from the literature are used to demonstrate that the method is a useful tool to summarize fatigue strength data of both smooth and notched components, subjected to either in‐phase or out‐of‐phase loads. Finally, a simple practical rule for the calculation of the multiaxial fatigue strength reduction factor is proposed.     

An improved critical plane and cycle counting method to assess damage under variable amplitude multiaxial fatigue loading

The plane with the maximum variance of the resolved shear stress is taken as the critical plane. Two algorithms are used along with the maximum variance method (MVM) to determine the orientation of the critical plane. The maximum variance of the normal stress on the potential critical planes is calculated to determine the one experiencing the maximum extent of fatigue damage. A new multiaxial cycle counting method is proposed to count cycles on the critical plane. The modified Wöhler curve method is used to assess fatigue damage. About 200 experimental results were collected from the technical literature to validate the approaches being proposed. The results show that the improved design technique being proposed is successful in assessing fatigue damage under variable amplitude multiaxial cyclic loading.     

A bi-parametric Wöhler curve for high cycle multiaxial fatigue assessment

This paper presents a method for estimating high-cycle fatigue strength under multiaxial loading conditions. The physical interpretation of the fatigue damage is based on the theory of cyclic deformation in single crystals. Such a theory is also used to single out those stress components which can be considered significant for crack nucleation and growth in the so-called Stage I regime. Fatigue life estimates are carried out by means of a modified Wöhler curve which can be applied to both smooth and blunt notched components, subjected to either in-phase or out-of-phase loads. The modified Wöhler curve plots the fatigue strength in terms of the maximum macroscopic shear stress amplitudes, the reference plane − where such amplitudes have to be evaluated − being thought of as coincident with the fatigue microcrack initiation plane. The position of the fatigue strength curve also depends on the stress component normal to such a plane and the phase angle as well. About 450 experimental data taken from the literature are used to check the accuracy of the method under multiaxial fatigue conditions.     

Estimating fatigue damage under variable amplitude multiaxial fatigue loading

The present paper is concerned with the use of the modified Wöhler curve method (MWCM) to estimate both lifetime and high‐cycle fatigue strength of plain engineering materials subjected to complex load histories resulting, at critical locations, in variable amplitude (VA) multiaxial stress states. In more detail, when employed to address the constant amplitude (CA) problem, the MWCM postulates that fatigue damage reaches its maximum value on that material plane (i.e. the so‐called critical plane) experiencing the maximum shear stress amplitude, fatigue strength depending on the ratio between the normal and shear stress components relative to the critical plane itself. To extend the use of the above criterion to those situations involving VA loadings, the MWCM is suggested here as being applied by defining the critical plane through that direction experiencing the maximum variance of the resolved shear stress. Such a direction is used also to perform the cycle counting: because the resolved shear stress is a monodimensional quantity, stress cycles are directly counted by the classical rain‐flow method. The degree of multiaxiality and non‐proportionality of the time‐variable stress state at the assumed critical sites instead is suggested as being measured through a suitable stress ratio which accounts for the mean value and the variance of the stress perpendicular to the critical plane as well as for the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear stress. Accuracy and reliability of the proposed approach was checked by using several experimental results taken from the literature. The performed validation exercise seems to strongly support the idea that the approach formalized in the present paper is a powerful engineering tool suitable for estimating fatigue damage under VA multiaxial fatigue loading, and this holds true not only in the medium‐cycle, but also in the high‐cycle fatigue regime.     

Four stress analysis strategies to use the Modified Wöhler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading

The present paper investigates the different ways of using the Modified Wöhler Curve Method (MWCM) to perform the fatigue assessment of steel and aluminium welded joints subjected to in-service variable amplitude (VA) multiaxial load histories. Thanks to its specific features, the above critical plane approach can efficiently be applied in terms of both nominal, hot-spot, and local quantities, that is, by using any of the stress analysis strategies suggested by the Design Recommendations of the International Institute of Welding (IIW). The MWCM can efficiently be used also along with the so-called Theory of Critical Distances applied in the form of the Point Method (PM). The accuracy of the different formalisations of the MWCM investigated in the present paper was systematically checked against a large number of experimental results taken from the literature and generated by testing, under VA biaxial nominal loading, welded samples having different geometries. Such a systematic validation exercise allowed us to prove that our multiaxial fatigue criterion is successful in designing welded joints against VA multiaxial fatigue, this holding true independently from both definition adopted to calculate the necessary stress quantities and complexity of the assessed load history.     

On the estimation of the material fatigue properties required to perform the multiaxial fatigue assessment

To accurately perform the fatigue assessment of engineering components subjected to in‐service multiaxial fatigue loading, the adopted design criterion must properly be calibrated, the used information usually being the fatigue strength under both pure uniaxial and pure torsional fatigue loading. Because of the complex fatigue response of metallic materials to multiaxial loading paths, the only reliable way to generate the necessary pieces of calibration information is by running appropriate experiments. Unfortunately, because of a lack of both time and resources, very often, structural engineers are requested to perform the multiaxial fatigue assessment by guessing the necessary fatigue properties. In this complex scenario, initially, the available empirical rules suitable for estimating fatigue strength under both pure axial and pure torsional fatigue loading are reviewed in detail. Subsequently, several experimental results taken from the literature and generated by testing metallic materials under a variety of proportional and non‐proportional multiaxial loading paths are used to investigate the way such empirical rules affect the accuracy in estimating fatigue strength, the damage extent being evaluated according to the modified Wöhler curve method. Such a systematic validation exercise allowed us to prove that under proportional loading (with both zero and non‐zero mean stresses), an adequate margin of safety can be reached even when the necessary calibration information is directly estimated from the material ultimate tensile strength. On the contrary, in the presence of non‐proportional loading, the use of the empirical rules reviewed in the present paper can result, under particular circumstances, in a non‐conservative fatigue design.     

Two methods for predicting the multiaxial fatigue limits of sharp notches

In this paper the problem of the multiaxial fatigue limit estimation of sharply notched components has been addressed using two different methods: a critical distance method and a method involving modified Wöhler curves. These two methods had been previously developed by the authors, but required modification for use in conjunction with finite element stress analysis of sharply notched specimens subjected to multiaxial loadings. Initially, it was demonstrated mathematically that these methods are equivalent in terms of multiaxial stresses near the notch tip. Subsequently, by employing some well‐known uniaxial notch fatigue concepts, some assumptions have been made in order to extend the use of these methods to in‐phase multiaxial notch fatigue situations. Experimental data were obtained from tests conducted on V‐notched specimens subjected to in‐phase mixed Mode I and Mode II loadings. Both methods were successful in giving fatigue limit predictions with an error usually less than 15%. This is interesting because the two methods make quite different assumptions about the nature of fatigue crack growth in the vicinity of the notch.     

Local and structural multiaxial stress states in weldedjoints under fatigue loading

A cylindrical stiffener on a plate and a tube through a hollowed plate, both circularly welded, were tested under uniaxial fatigue loading. Their fatigue cracking behaviour was seen to be really complex due to the fact that crack initiation sites changed their position as the number of cycles to failure increased. To investigate this anomalous behaviour, an accurate numerical investigation was carried out to study the distribution of both local and structural linear elastic stresses along the weld toe circumferences. The numerical results proved that the weld beads were subjected to complex stress states, even though the applied nominal load was uniaxial. By the light of this evidence, the fatigue behaviour of the investigated welded joints was then re-interpreted from a multiaxial fatigue point of view by applying the Modified Wöhler Curve Method in terms of hot-spot stresses. The proposed approach was seen to be successful allowing us to estimate both crack initiation sites and fatigue lifetime with a high precision level. This fact is very interesting because it strongly supports the idea that our method can be used to assess real welded components subjected to multiaxial fatigue loading by simply post-processing linear-elastic finite-element results.     

Critical plane approach for the assessment of the fatigue behaviour of welded aluminium under multiaxial loading

ABSTRACT Multiaxial stress states occur in many welded constructions like chemical plants, railway carriages and frames of trucks. Those stresses can have constant and changing principal stress directions, depending on the loading mode. Latest research results on welded steel joints show a loss of fatigue life for changing principal stress directions simulated by out‐of‐phase bending and torsion compared to constant directions given by in‐phase loading. However, aluminium welds reveal no influence of changing principal directions on fatigue life compared to multiaxial loading with constant principal stress directions. This behaviour is not predictable by any conventional hypothesis. A hypothesis on the basis of local normal and shear stresses in the critical plane has been developed and applied to aluminium weldings.     

A novel engineering method based on the critical plane concept to estimate the lifetime of weldments subjected to variable amplitude multiaxial fatigue loading

This paper summarizes an attempt at proposing a new engineering method suitable for estimating the fatigue lifetime of steel‐ and aluminium‐welded connections subjected to variable amplitude multiaxial fatigue loading. In particular, the proposed approach is based on the use of the so‐called Modified Wöhler Curve Method (MWCM), i.e. a bi‐parametrical critical plane approach, whose accuracy has been checked so far solely in addressing the constant amplitude multiaxial fatigue problem. In order to extend the use of our criterion to variable amplitude situations, the critical plane is suggested here as being determined by taking full advantage of the maximum variance concept, that is, such a plane is assumed to be the one containing the direction along which the variance of the resolved shear stress reaches its maximum value. The main advantage of such a strategy is that the cycle counting can directly be performed by considering the shear stress resolved along the maximum variance direction: by so doing, the problem is greatly simplified, allowing those well‐established cycle counting methods specifically devised to address the uniaxial variable amplitude problem to be extended to those situations involving multiaxial fatigue loading. The validity of the proposed methodology was checked by using two different datasets taken from the literature and generated by testing both steel and aluminium tube‐to‐plate welded connections subjected to in‐phase and 90° out‐of‐phase variable amplitude bending and torsion. This new fatigue life assessment technique was seen to be highly accurate allowing the estimates to fall within the calibration scatter bands not only when the constants in the governing equations were calculated by using the experimental uniaxial and torsional fully reversed fatigue curves, but also when they were determined by using the reference curves supplied, for the investigated geometry, by the available standard codes. These results seem to strongly support the idea that, thanks to its peculiar features, our method can be considered as an effective engineering approach capable of performing multiaxial fatigue assessment under variable amplitude loading which fully complies with the recommendations of the available standard codes.     

Multiaxial fatigue assessment of welded joints – Recommendations for design codes

In the assessment of welded joints submitted to multiaxial loading the calculations method applied, independently of the concept (nominal, structural, hot-spot or local), must consider primarily the materials ductility. While proportional loading can be assessed by von Mises, the principal stress hypothesis, the Findley method or the Gough–Pollard relationship, using any of the mentioned concepts, difficulties occur when the loading is non-proportional, i.e. the principal stress (strain) direction changes. This causes a significant fatigue life reduction for ductile steel welds, but an indifferent behaviour for semi-ductile aluminium welds. This different response to non-proportional loading can be assessed when ductility related mechanisms of fatigue failures, i.e. the mean value of plane oriented shear stresses for ductile materials and a combination of shear and normal stresses for semi-ductile materials, are properly considered.However, as these methods require a good expertise in multiaxial fatigue, for design codes used by non-fatigue experts, simpler but sound calculation methodologies are required. The evaluation of known fatigue data obtained with multiaxial constant and variable amplitude (spectrum) loading in the range N > 10⁴ cycles suggests the application of the modified interaction algorithm of Gough–Pollard. In the case of variable amplitude loading, constant normal and shear stresses are replaced by modified reference normal and shear stresses of the particular spectrum. The modification of the reference stresses is based on the consideration of the real Palmgren–Miner damage sum of D_PM = 0.5 (for spectra with constant mean loads) and the modification of the Gough–Pollard algorithm by consideration of the multiaxial damage parameter D_MA = 1.0 or 0.5, which is dependent on the material’s ductility and on whether the multiaxial loading is proportional or non-proportional. This method is already part of the IIW-recommendations for the fatigue design of welded joints and can also be applied by using hot-spot or local stresses.     

Fatigue assessment of laserbeam‐welded aluminium joints under multiaxial loading

This paper presents the results and evaluation of the multiaxial fatigue behaviour of laserbeam‐welded overlapped tubular joints made from the artificially hardened aluminium alloy AlSi1MgMn T6 (EN AW 6082 T6) under multiaxial loadings with constant and variable amplitudes. Several fatigue test series under pure axial and pure torsional loadings as well as combined axial and torsional proportional and non‐proportional loadings have been carried out in the range of 2·10⁴ to 2·10⁷ cycles. The assessment of the investigated thin‐walled joints is based on a local notch stress concept. In this concept the fatigue critical area of the weld root is substituted by a fictitious notch radius r_ref = 0.05 mm. The equivalent stresses in the notch, considering especially the fatigue life reducing influence of non‐proportional loading in comparison to proportional loading, were calculated by a recently developed hypothesis, which is called the Stress Space Curve Hypothesis (SSCH). This hypothesis is based on the time evolution of the stress state during one load cycle. In addition, the fatigue strength evaluation of multiaxial spectrum loading was carried out using a modified Gough‐Pollard algorithm.     

Lebensdauerermittlung bei mehrachsigen wechselnden Beanspruchungen im niedrigen und hohen Temperaturbereich

Life time assessment on multiaxial cyclic loadings at low and high temperatures For the calculation of fatigue strength of components made out of ductile materials under complex cyclic load different assessments are present. As typical representatives of stress theories the shear stress intensity hypothesis (SIH) as well as the method of critical plane approach (MKS) are considered and compared for rigid and non rigid principle stress directions. Furthermore for synchronous loads the calculation methods are compared with Bach's method. The calculation method becomes more complex, if time dependent material properties at corresponding high temperatures have to be taken into account. In this case the application of viscoplastic material models is necessary, which allows the consideration of combination of creep and fatigue. As an example a modified material model by Chaboche / Nouailhas is used in order to present the calculation of multiaxial creep fatigue tests.     

Der Gradient der Wöhlerkurve

The Gradient of the S, N-curve. The classical two-parameter basic equations of Wöhler and Basquin and also the four-parameter equation of Palmgren/Weibull approximate the Wöhler curve in certain ranges only. They fail in particular when one attempts to use them to convert a service life increase into the appropriate gain in strength. For a more exacting analysis these approximations must be replaced by models of the Wöhler curve which are nearer to actual reality.     

On the interaction of normal and shear stresses in multiaxial fatigue damage

One of the important issues in assessing multiaxial fatigue damage is interactions between different components of stress such as normal and shear stresses. The present study investigated this interaction effect on the fatigue behavior of materials with shear failure mode when subjected to multiaxial loading conditions. A method is introduced to model this interaction based on the idea that two types of influence are caused by the normal stress acting on the critical plane orientation. These two types of influence are affecting roughness induced closure, as well as fluctuating normal stress which affects the growth of small cracks in mode II. Shear‐based critical plane damage models which use normal stress as a secondary input, such as FS damage model, could then use the summation of these terms. In order to investigate the effect of the method, constant amplitude load paths with different levels of interaction between the normal and shear stresses, as well as variable amplitude tests with histories both taken from service loading conditions and generated using random numbers were designed for an experimental program. The proposed method was observed to result in improved fatigue life estimations where significant interactions between normal and shear stresses exist.     

Multiaxial fatigue life assessment of welded structures

Welded structures, such as welded pressure vessel components subjected to multiaxial cyclic loading, are particularly susceptible to fatigue damage. In this paper, a new path-length-based effective stress range is proposed to assess the fatigue life of weld joints under multiaxial fatigue loading. The path-length measure, a function of both normal and shear components on a critical crack plane, has a solid root in classic fracture mechanics and its application is validated by correlating nominal fatigue data including pure-bending, pure-torsion, in-phase, and out-of-phase loading. Path-Dependent Maximum Range (PDMR), a unique general-purpose fatigue life assessment package for multiaxial variable-amplitude loading, is introduced in this paper. Finally, the application of PDMR to multiaxial fatigue life assessment of complex loading cases is also discussed.     

Strain-based multiaxial fatigue damage modelling

A new multiaxial fatigue damage model named characteristic plane approach is proposed in this paper, in which the strain components are used to correlate with the fatigue damage. The characteristic plane is defined as a material plane on which the complex three‐dimensional (3D) fatigue problem can be approximated using the plane strain components. Compared with most available critical plane‐based models for multiaxial fatigue problem, the physical basis of the characteristic plane does not rely on the observations of the fatigue crack in the proposed model. The cracking information is not required for multiaxial fatigue analysis, and the proposed model can automatically adapt for different failure modes, such as shear or tensile‐dominated failure. Mean stress effect is also included in the proposed model by a correction factor. The life predictions of the proposed fatigue damage model under constant amplitude loading are compared with a wide range of metal fatigue results in the literature.