In the design and manufacture of modern steel assemblies, permanent joining by welding is still common. The crane industry makes use of gas metal arc welding (GMAW), i.e., metal active gas welding, as a traditional and prevalent welding process, e.g., for the manufacture of truck and crawler cranes. Due to recent process developments, laser beam welding and the hybrid combination of laser beam and gas metal arc welding have appeared as expedient alternatives. However, for applications with highly stressed structures, as in the case of loading situations of the telescopic boom of truck cranes, for example, Fig. 1a, the service loads are related to the low cycle fatigue regime. Within such structures, butt joints are essential load-carrying welds (Fig. 1b). Therefore, the following points need to be considered in the fatigue estimation:

  • In highly stressed areas, welded joints are limiting factors in design, e.g., butt joints in the telescopic boom

  • Crane-specific load scenarios result in variable amplitude loading (VAL) and elastic-plastic stresses and strains (LCF)

  • A large number of load cases requires a method for fatigue life estimation that is easy to use

Fig. 1
figure 1

a Truck crane on the test site (photo from [1]) and b representation of a transverse butt weld in the tensile-stressed upper shell [2]

The fatigue assessment of welded joints according to recommendations and standards, Eurocode 3 [3], the crane design standard DIN EN 13001-3-1 [4], Hobbachers IIW recommendations [5], and the FKM guideline [6], is based on fatigue classes (FAT), i.e., characteristic stress ranges ΔσC at 2·106 cycles to failure. These FAT classes were derived from experimental results from specimens and components. Specimens for experimental fatigue investigations of steel structures with medium sheet thickness, i.e., 5 mm to approximately 20 mm, are usually of larger scale and tested under stress control to verify the fatigue assessment and FAT classes. To date, these fatigue recommendations are mostly limited to steels with a maximum yield strength of 960 MPa and the assessment by FAT classes does not include the low cycle fatigue (LCF) regime and elastic-plastic material behavior. Therefore, they are restricted to cycles to failure of N ≥ 1·104.

An overview of the fatigue design concepts, which are applied in the assessment of welded joints, is given in Fig. 2 and, in more detail, by Radaj et al. [8, 9]. The nominal stress (and strain) approach requires the definition of a nominal stress or strain, i.e., a reference cross-section, and assignment of a specific notch detail. FAT values for nominal stress and various notch details have been defined [3,4,5,6]. Even though FAT classes are defined for survival probabilities of 97.7%, analyses of fatigue results of butt welds, including data points for Nf < 5·104 from Olivier and Ritter [10], Leitner et al. [11], Berg and Stranghöner [12], and Möller [2] have found that the assessment by FAT 90 (or even FAT 80/FAT 71) tends to be unsafe at high loading levels in and close to the LCF regime.

Fig. 2
figure 2

Overview of fatigue design concepts for welded joints [7]

In the case of more complex structures, where it is not possible to define nominal stress, the structural stress (or strain) concept can be applied, as further described by the IIW recommendations of Hobbacher [5] and Fricke [13] as well as [8], where FAT classes for structural stresses can also be found. Due to the linear-elastic extrapolation, which is used in most of the concept variants, the existing stress concentration at the notch and notch support effects at this location are not captured. Furthermore, structural stress and strain concepts reveal their limitations when confronted with root notches.

As a consequence and in addition to the nominal and structural stress (or strain) concepts, local concepts (cf. [5, 7,8,9, 13]) are increasingly used for the design of welded joints. FAT classes for notch stresses result in fictitious (linear-elastic) stresses at the notch root. The notch stress approaches attracted wide interest and developed over recent decades, so that they are a useful assessment approach for welded joints, especially in the case of the reference radius concept (cf. [8, 13]). However, this approach cannot be applied in elastic-plastic stress-strain states, as is the case in the LCF regime or, rather, in the notch root of a weld for high loading applications. In order to cover this case, the notch strain concept (cf. [5, 7,8,9, 13]) is used, where the cyclic material behavior forms the basis for the simulation and assessment of the stress-strain state in the notch, in addition to the geometry of the joint, as introduced by Lawrence et al. [14]. The effort involved in such an assessment should not be underestimated and increases further, if transient effects and microstructural differences, as given by investigations on the effect of the microstructure of the different weld zones on the fatigue strength from various authors [15,16,17,18,19,20], are taken into account. Recent work of Saiprasertkit et al. [21,22,23,24] uses an analytical approximation for the description of the effective notch strain from the notch strain concept with a reference radius of rref = 1.00 mm in the case of load-carrying cruciform joints to show a conservative estimation of FAT 200 within the LCF and HCF regimes. On the contrary, this work aims at providing a fatigue life estimation by an integral treatment of butt welds under low to high constant and variable amplitude loading for elastic-plastic stress-strain conditions with moderate effort, as described in this paper.

Finally, fracture mechanics concepts, e.g., stress intensity or crack propagation approaches, assume a pre-existing crack-like defect for the life assessment, but will not be in the focus of this work.

Materials and welding process

Based on unalloyed structural steels [25] and normalized rolled fine-grained structural steels (N, NL) [26] with yield strengths ReH or Rp0.2 of 235 to 460 MPa, the steel designation for high-strength, fine-grained structural steels according to [27] is also based on the minimum yield strength. In a further subdivision, a distinction can be made between high-strength (Rp0.2 ≤ 960 MPa) and ultra-high-strength (Rp0.2 > 960 MPa) fine-grained structural steels. The yield strength specification is followed in the designation by information on the manufacturing process. A basic distinction is made between water quenched and tempered (Q, QL) [28] and thermomechanically treated (M, TM) [29] steels. In the context of this work, the S960QL (material no.: 1.8933), two thermomechanically rolled products of grade S960M (manufacturer “1” and “2”) and the ultra-high-strength S1100QL (material no.: 1.8942) fine-grained structural steel, suitable for welding, have been investigated. Starting from a conventional hot rolling above the recrystallization temperature of austenite, followed by normalizing annealing in normalized steels, water quenched and tempered steels are rapidly cooled in water, resulting in very good strength properties due to the bainitic and martensitic microstructure [30]. Microstructural transformations by subsequent tempering can, above all, improve their toughness properties. Thermomechanical rolling, i.e., a combined mechanical-thermal rolling process, which is partially carried out below the recrystallization temperature, comprises a large number of process variants (e.g., accelerated cooling), which, together with the alloying concept (microalloy), result in high toughness and strength [30].

The steel materials in this investigation are characterized by the mechanical properties listed in Table 1—yield strength Rp0.2, tensile strength Rm, elongation at fracture A5, and impact energy KV (at three different temperatures). The characteristic values meet the requirements according to [28]:

  • Minimum yield strength: Rp0.2 ≥ 960 MPa (S960), Rp0.2 ≥ 1100 MPa (S1100)

  • Tensile strength: 980 ≤ Rm ≤ 1150 MPa (S960), 1250 ≤ Rm ≤ 1550 MPa (S1100)

  • Minimum elongation at fracture: A5 ≥ 10% (S960)

  • Notched bar impact energy: KV ≥ 40 J (S960 at − 20 °C) and KV ≥ 30 J (S960 at − 40 °C)

Table 1 Mechanical properties of investigated steel grades [31]

The specifications for the impact energy are temperature-dependent and correspond to the test carried out by the steel manufacturer.

Butt joints were manufactured, using the 8-mm-thick S960QL, S960M, and S1100QL steel sheets described previously, by gas metal arc welding (GMAW), i.e., metal active gas welding, applying the process parameters given in Table 2, as this is the traditionally and commonly used welding process in the crane industry and related areas. A cooling time t8/5 of 5 to 10 s is an essential process requirement. For the ultra-high-strength, fine-grained structural steel S1100QL, the use of filler metal G 89 6 M Mn4Ni2CrMo with a minimum yield strength of Rp0.2 = 890 MPa results in a welded joint with weld areas of lower strength (undermatching), especially in the heat-affected zone (HAZ). The butt joints were created in a multi-layer weld with one weld root, one backing and one or two top layers. Two overlapping top layers have been chosen for manually welded S960M and S1100QL, while a single top layer is used for manually S960QL welds, in order to compare them with a partially automated joining of the steel sheets. Welds, manufactured by manual GMAW, were executed by qualified welders from the crane industry. In the past, efforts were made to introduce automated welding and laser beam welding (LBW) or a combined LBW-GMAW as a joining process, in order to reduce distorsions and increase the strength of joints, i.e., also the fatigue strength. Investigations on the fatigue behavior of these welds are still ongoing.

Table 2 Main process parameters for GMAW of butt joints made of high-strength, fine-grained structural steels

Stress-based fatigue life derived from force-controlled testing

Fatigue tests on butt joints have been performed on resonance and servohydraulic fatigue test rigs under force control at a load ratio of RF = 0.1 [31, 33, 34]. The specimen was designed to have a free length between the clamping areas of 250 mm. Low test frequencies in the range of 0.1 Hz ≤ f ≤ 7.0 Hz were defined for tests based on the “Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculations” [35] with a focus on low cycle fatigue. The test frequency was reduced with increasing force amplitude to a minimum of 0.1 Hz and was increased to 35 Hz for fatigue testing in the high cycle fatigue (HCF) regime. “Total rupture of the specimen” was defined as the final failure criterion.

The evaluation of the fatigue test results in the HCF regime, both under constant amplitude loading (Wöhler curve) and variable amplitude loading (Gaßner curve) with a random load time history derived from a Gaussian-like load spectrum [34, 36], has been carried out according to the maximum likelihood estimation (MLE) [37], Fig. 3. Both Wöhler and Gaßner curves for butt-welded S960QL, S960M, and S1100QL fine-grained steels are evaluated for all steel grades together. In Fig. 3, the lines are represented for survival probabilities of PS = 10, 50, and 90%. The Wöhler line is characterized by the slope k = 3.1, the nominal stress amplitude σa,n (Nr = 1·106,PS = 50%) = 60 MPa, and the scatter Tσ = 1:1.98—the Gaßner line is, on the other hand, described by k = 4.2, σa,n (Nr = 1·106,PS = 50%) = 184 MPa, and Tσ = 1:1.56. Wöhler and Gaßner lines are in good agreement with those of other evaluation methods shown in [31, 33, 34]. This is due to the fact that the evaluation is limited to the HCF and that there are no differences in applying the method of the least squares and the maximum likelihood method including just one run-out. Differentiation between the various test series (combination of base material and welding process) shows a dependence of the fatigue strength on the weld quality and welding process [31, 33, 34]. It has been found that partially automated welding results in a reduced misalignment and an increase in fatigue strength. In the LCF regime, force-controlled fatigue testing is sensitive to plasticity and flow, so that small changes in force can have a considerable influence on the fatigue life of the welded joint [19]. This restriction can be counteracted by strain-controlled fatigue testing.

Fig. 3
figure 3

a Fatigue test setup. b Load time history. c Load spectrum. d Fatigue test results

Integral treatment and cyclic behavior of butt welds

Integral treatment of welded joints

The complexity of and effort in the application of local fatigue assessment approaches increases drastically, if the elastic-plastic material model of welded joints is supplemented by a local differentiation of the material behavior and the consideration of transient effects. Therefore, an integral treatment of a butt joint is introduced as a basis for the fatigue life assessment. This integral treatment uses the cyclic material behavior, evaluated from strain-controlled fatigue testing, but characterizes butt welds by the cyclic stress-strain behaviors and strain-life curves with an integral approach from the base material across the weld to the base material (Fig. 4).

Fig. 4
figure 4

Schematic presentation of the integral treatment of butt joints and resulting stress-strain states under strain-controlled fatigue testing

Cyclic behavior of butt welds

A servohydraulic test rig with a maximum load of 100 kN has been used for strain-controlled fatigue tests of the small-scale flat specimens (Fig. 5). Although the greater sheet thickness of 8 mm does not fall within the range of thin sheets, the tests were carried out in accordance with the specifications of SEP1240 [35]. As a result of the strain ratio of Rε = − 1, an anti-buckling device is used. Depending on the strain level, the test frequencies were selected as 0.1 Hz (εa,t > 0.4%), 0.5 Hz (0.4% ≥ εa,t > 0.2%), and 4.0 Hz (εa,t ≤ 0.2%). From the test results, strain-life curves and cyclic stress-strain curves are derived, which describe the cyclic material behavior (base material) and the cyclic behavior of the butt weld. For this purpose, an extensometer with a 25 mm measuring length was used in the strain-controlled tests. In the case of the butt weld, this extensometer contains the entire weld area (weld metal, HAZ, and base metal on both sides).

Fig. 5
figure 5

Strain-controlled fatigue testing and investigated flat specimen geometries

Two approaches are used to evaluate the strain-life curve: the classical description of the Basquin-Coffin-Manson-Morrow (BCMM) [38,39,40,41] strain-life curve and the description by the tri-linear strain-life curve [42]. In the latter description, continuation of the slope of the elastic line from range 2 into 3 (b2 = b3) is selected, since no results are available for N > 1·106. The comparison of the BCMM strain-life curves with the tri-linear strain-life curves for the S1100QL butt welds (Fig. 6a) shows differences between the two estimation methods—with a more accurate description by the tri-linear strain-life curve and cyclic stress-strain curve derived from compatibility equations. Therefore, the tri-linear strain-life curve is preferred in describing the strain-life relation and will be used to estimate the fatigue life. There are approx. 1–2 decades in number of cycles to crack initiation between the base material and butt weld. In [19], it has also been demonstrated that a fatigue life reduction has been found for machined specimens compared with the base material state. Thus, it can be seen that the strain-life curves determined on base material specimens cannot be directly transferred to other material states, such as weld seams.

Fig. 6
figure 6

Comparison of the BCMM evaluation and tri-linear approach of S1100QL butt welds for a strain-life curves and b cyclic stress-strain curves

Cyclic stress-strain curves are described according to Ramberg-Osgood [43]. In addition to the direct regression of the test results, cyclic stress-strain curves have been derived from compatibility with the strain-life curve in the form of BCMM and the tri-linear strain-life curve. The three descriptions are compared for the S1100QL butt welds in Fig. 6b. The stress-strain curves from the compatibility with the BCMM strain-life curves tend to underestimate the results for lower plastic strains (0.2% < εa,t < 0.5%) and to overestimate the results for large plastic strains (εa,t > 0.6%). In contrast, the direct regression and the cyclic stress-strain curve describe the tri-linear strain-life curve more accurately. This also applies in a similar way for the curves of the other base materials. All evaluated cyclic stress-strain curves are based on the results for the cyclically stabilized state and thus do not reflect any transient effects. Cyclic material parameters of the investigated high-strength, fine-grained steels and butt-welded joints have been documented and analyzed in more detail in [2, 19].

Cyclic transient behavior

A first impression regarding transient effects is given by a comparison of the initial loading with the cyclic stabilized state and, especially the cyclic stress-strain curve. Distinctive cyclic softening from initial loading to cyclic stabilization can be observed for high-strength steel base materials as well as butt joints (cf. Fig. 4 (right)). However, deeper insight into the cyclic transient behavior is gained from changes in the hysteresis loops from cycle to cycle. The transient behavior is shown more clearly in the evolution of the cyclic yield strength R’p0.2 or the cyclic hardening coefficient K′ and the cyclic hardening exponent n′. An overview is provided by the representation of K′ and n′ in relation to the normalized number of cycles to crack initiation N/Ni for S960QL in Fig. 7, from which a drop in the cyclic hardening coefficient K′ can be derived as a result of softening and crack initiation. The cyclic hardening exponent n′ also shows a decreasing trend, although for S960QL steel and butt welds, it remains almost constant with increasing N/Ni, apart from a few minor changes. For both cyclic parameters, a linearization is performed for continuous evolution, so that (linear) functional relations for K′ = f (N/Ni) and n′ = f (N/Ni) are found. Due to the change in the characteristic values resulting from cyclic softening and crack initiation, the normalized number of cycles from strain-controlled fatigue tests can be directly interpreted as total damage, so that D = N/Ni follows. This definition results in the linearized descriptions for K′ and n′ depending on the damage according to Eqs. 1 and 2, where the index “0” denotes the initial state. More complex, non-linear relationships are also conceivable, but for the following investigation, this approximation is sufficient for the description of the cyclic transient behavior.

$$ {K}_j^{\prime}\left({D}_j\right)=\Delta {K}^{\prime}\cdotp {D}_j+{K}_0^{\prime } $$
$$ {\mathrm{n}}_j^{\prime}\left({D}_j\right)=\Delta {n}^{\prime}\cdotp {D}_j+{n}_0^{\prime } $$
Fig. 7
figure 7

Description of the cyclic transient behavior by (linear) functional relations between Ramberg-Osgood parameters and total damage for a the cyclic hardening coefficient K′ and b the cyclic hardening exponent n′

Fatigue life estimation approach

For the fatigue life estimation from the low cycle to the high cycle fatigue regime, a strain-based approach, using damage parameters to assess simulated stress-strain paths, is used. The approach is based on the integral treatment of the butt weld already examined and its characterization by the cyclic (transient) behavior (Fig. 8). This approach includes the following steps:

  1. 1.

    The starting points of the evaluation are the butt-welded specimens described before, which provide “integral characteristic values of the cyclic behavior of transverse loaded butt joints” as a result of strain-controlled fatigue tests. For the initial loading, stress-strain curves based on the initial parameters K′0 and n′0 are applied. Likewise, additional stresses (and strains) resulting from an angular misalignment are applied in the first load step.

  2. 2.

    In the second step, the cyclic transient behavior of these “integral weld seams” is directly used in the analytical simulation of stress-strain paths, considering memory and Masing behavior without explicitly determining a fictitious notch stress or strain. Hysteresis loops are related to the load history. Different loading histories therefore need to be simulated computationally with this procedure. According to the force-controlled tests of the butt welds with RF = 0, constant and variable amplitude loading is simulated in order to verify this method. The application of this method allows the inclusion of transient effects, such as cyclic softening, in the fatigue assessment with moderate effort. Based on the description of the cyclic transient behavior, the first load with stress relief is calculated using K0′ and n0′ from the initial Ramberg-Osgood equation, while the following stress hystereses are calculated by damage-dependent linearized parameters of K′ and n′. The Ramberg-Osgood parameters K′ and n′ are modified cycle by cycle on the basis of the functional relationship with the damage content.

  3. 3.

    The damage of every resulting hysteresis is derived by common damage parameters PSWT by Smith-Watson-Topper [44] (Eq. 3), Pε by Sonsino-Werner [45, 46] (Eq. 4), modified PHL,mod by Haibach-Lehrke [47] (Eq. 5), and PJ by Vormwald [48] or the evolved PRAJ by Fiedler et al. [49] (Eq. 6), which are finally compared. As a result of the analysis of these damage parameters, a generalized damage parameter Pm (Eq. 7) is introduced, which includes additional factors for mean stresses and mean strains as a combination of Pε and the damage parameter of Bergmann PB [50].

Fig. 8
figure 8

Overview of the fatigue life estimation procedure for butt welds using cyclic transient behavior

$$ {P}_{\mathrm{SWT}}=\sqrt{\sigma_{\mathrm{m}\mathrm{ax}}\kern.15em {\varepsilon}_a\kern.15em E}=\sqrt{\left({\sigma}_{\mathrm{a}}+{\sigma}_{\mathrm{m}}\right)\kern.15em {\varepsilon}_{\mathrm{a}}\kern.15em E} $$
$$ {P}_{\upvarepsilon}=\sqrt{\kern.15em {\varepsilon}_{\mathrm{m}\mathrm{ax}}\kern.15em {\sigma}_{\mathrm{a}}\kern.15em E}=\sqrt{\left(\kern.15em {\varepsilon}_{\mathrm{a}}+{\varepsilon}_{\mathrm{m}}\right){\sigma}_{\mathrm{a}}\kern.15em E}\kern1em bzw.\kern0.75em {P}_{\upvarepsilon}=\sqrt{\left(\kern.15em {\varepsilon}_{\mathrm{a}}+k\kern.15em {\varepsilon}_{\mathrm{m}}\right){\sigma}_{\mathrm{a}}\kern.15em E} $$
$$ {P}_{\mathtt{HL},\operatorname{mod}}=\sqrt{{\Delta \sigma}_{\mathrm{eff}}\kern.15em {\Delta \varepsilon}_{\mathrm{eff}}\kern.15em E} $$
$$ {P}_{\mathrm{J}}={P}_{\mathrm{RAJ}}=\mathtt{1.24}\frac{{\left({\Delta \sigma}_{\mathrm{eff}}\right)}^{\mathtt{2}}}{E}+\frac{\mathtt{1.02}}{\sqrt{n^{\prime }}}{\Delta \sigma}_{\mathrm{eff}}\left({\Delta \varepsilon}_{\mathrm{eff}}-\frac{{\Delta \sigma}_{\mathrm{eff}}}{E}\right) $$
$$ {P}_{\mathrm{m}}=\sqrt{\left({m}_{\upsigma}\cdot {\sigma}_{\mathrm{m}}+{\sigma}_{\mathrm{a}}\right)\cdot \left({m}_{\upvarepsilon}\cdot {\upvarepsilon}_{\mathrm{m}}+{\varepsilon}_{\mathrm{a}}\right)\cdot E} $$
  1. 4.

    With the help of the relation between damage parameter P and cycles to crack initiation Ni, the partial damage of the jth cycle dj is derived and added to the damage sum of previous cycles. The accumulation is ongoing as long as the damage sum is below the theoretical damage sum of Dth = 1.

  2. 5.

    The failure in the form of the crack initiation from the strain-controlled testing is achieved and the calculated number of cycles Ncalc is returned, when the theoretical damage sum of Dth = 1 is reached.

A comparison between experimentally derived and simulated stress-strain paths, for a constant amplitude stress-controlled loading situation with RF = − 1 until the theoretical damage sum of Dth = 1 is reached, is shown in Fig. 9. Experimentally derived, simulated stress-strain paths and resulting hysteresis loops are in good agreement. Smaller differences occur at the minimum stress under compression, where cyclic creep might have an influence on the turning point. However, fatigue lives have been calculated by application of introduced damage parameters. Except for a short calculated fatigue life using PHL,mod, small variations can be found for most of the damage parameters and in comparison with the experimentally determined value, in this specific example. In any case, the previously defined Pm seems to be a good choice. In other loading situations, advantages and disadvantages become clearer, as can be seen from [2].

Fig. 9
figure 9

Comparison between experimentally derived and simulated stress-strain paths for a constant amplitude stress-controlled loading situation with RF = − 1 as well as estimated fatigue lives depending on the choice of damage parameter

This procedure offers the advantage of a simplified fatigue life estimation, based on consideration of elastic-plastic stresses. Using this approach, a detailed and complex finite element simulation for the assessment of notch stresses or strains, which should additionally take transient effects into account, is avoided.

Fatigue life estimation

Constant amplitude loading (CAL)

According to the procedure described before, a fatigue life estimation under constant amplitude loading for the integral treatment of butt welds considering the cyclic transient behavior using damage parameters as defined in Eqs. 3 to 7 has been performed, where Pm is the introduced generalized damage parameter including mean stresses and mean strains. The results, applying the damage parameter Pm, are presented in Fig. 10, since Pm can be adjusted to the LCF regime (mean strains) and HCF regime (mean stresses) by choice of the corresponding factors in addition to the consideration of stress and strain amplitudes. In the nominal stress system, fatigue lives estimated with the help of Pm are compared with experimentally derived cycles to failure and fatigue classes FAT 50/71 (dot-dash-lines corresponding to a survival probability of PS = 97.7%) derived for the different test series, which are characterized by the combination of base material and welding process execution. Figure 10a shows that the estimation results in too long fatigue lives for S960QL butt welds (green triangles), where specimens include a comparably high angular misalignment of α = 3.9° on average, if this angular misalignment is not taken into account in the fatigue life simulation(gray curve for PS = 50%). Including an average angular misalignment α = 3.9° in the estimation procedure, it can be seen, from the course of the estimated fatigue life curve (cf. black curve) corresponding to a survival probability of PS = 50%, that the accuracy of the fatigue life estimation is increased. Therefore, calculated fatigue lives of the other test series in Fig. 10b, c, and d already include the angular misalignment (black curves for PS = 50%). It can be seen that the estimated fatigue lives coincide very well with the load-controlled experimental results for S960QL (Fig. 10a) and S1100QL (Fig. 10b), while more conservative fatigue lives are found for S960M (supplier 2, Fig. 10d). S960M (supplier 1) shows a good agreement at higher load levels and tends to be non-conservative at lower load amplitudes (Fig. 10c).

Fig. 10
figure 10

Fatigue life estimation under constant amplitude loading applying the damage parameter Pm compared with fatigue classes (FAT) based on the cyclic transient behavior of the butt-welded steels a S960QL, b S1100QL, c S960M (supplier 1) and d S960M (supplier 2)   

Variable amplitude loading (VAL)

Under variable amplitude loading, the simulated stress-strain path is more complex than the one for constant amplitude loading. Again, the fatigue life estimation is carried out for the integral treatment of butt welds with the described procedure considering cyclic transient behavior. Due to the intersection of the black curves for PS = 50% with the green symbols, Fig. 11a, b, and c show that the highest stress amplitudes of load-controlled tests have been assessed by the approach applying the damage parameter Pm. For manually welded butt joints made of S960M (supplier 1, cf. Fig. 11c), there is a very good agreement between estimated and experimentally determined fatigue lives under variable amplitude loading. In the other cases, S960QL (Fig. 11a), S1100QL (Fig. 11b), and S960M (supplier 2, Fig. 11d), there is a tendency to be more conservative, when the stress amplitude decreases. Compared with the linear damage accumulation for PS = 50% (here: using the allowable damage sum acc. to [5] of Dall = 0.5), this approach is not based on Wöhler (SN) lines derived under constant amplitude loading and is therefore independent of the slope of the Wöhler line in the HCF regime. As can be seen from Fig. 11, the transition from low cycle to high cycle fatigue is represented in a good way using this method, so that the calculated Gaßner (or Wöhler) curve is not purely linear (in double-logarithmic scale). However, in the HCF regime, the approximation of a straight line for the Gaßner curve becomes obvious and is reasonable. At lower load levels, the estimation from the integral treatment of butt joints shows a good agreement for S960QL (cf. Fig. 11a) and S960M (supplier 1, cf. Fig. 11c), while the linear damage accumulation is closer to the experimental data for S1100QL (Fig. 11b) and S960M (supplier 2, Fig. 11d).

Fig. 11
figure 11

Fatigue life estimation under variable amplitude loading using the damage parameter Pm compared with the linear damage accumulation with Dall = 0.5 based on the cyclic transient behavior of the butt-welded steels a S960QL, b S1100QL, c S960M (supplier 1) and d S960M (supplier 2)

Discussion of the fatigue life estimation

Estimated (calculated) fatigue lives Ncalc for the integral treatment of butt welds, considering the cyclic transient behavior using damage parameter Pm, are directly compared with experimentally derived fatigue lives Nexp for both constant and variable amplitude loading. Resulting data in the range of 1:4 from the optimum Ncalc = Nexp (factor of 4) show a good agreement within the overall scatter of welded joints. In the case where Ncalc is larger by more than a factor of 4 compared with Nexp, the estimation is on the unsafe side, while it is defined to be too safe, if Ncalc = 0.25 · Nexp.

In Fig. 12, the direct comparison between calculated and experimentally determined fatigue lives for constant amplitude loading, applying the damage parameter Pm, confirms the good agreement which Fig. 10 indicates, if the angular distortion is taken into account—apart from the still very conservative estimate for the S960M (supplier 2). It can be seen that most results for S960QL (cf. Fig. 10a), S1100QL (cf. Fig. 10b), and S960M (supplier 1, Fig. 10c) are in the 1:4 range. In addition to the results of S960M (supplier 2), some of the other results at lower load amplitudes tend to be too safe. However, test results at low load levels are rare, since the focus of this investigation is set on the regime from LCF to higher load levels of the HCF. Therefore, the significance of the fatigue life estimation for very low load levels is limited, based on the results of this investigation. On the other hand, a very good agreement between estimated and experimentally derived fatigue lives at high load levels is found, at least for S960QL, S1100QL, and S960M (supplier 1) butt welds.

Fig. 12
figure 12

Comparison between calculated and experimentally determined fatigue lives for constant amplitude loading applying the damage parameter Pm

Figure 13 shows the comparison between calculated and experimentally determined fatigue lives for variable amplitude loading, applying the damage parameter Pm, again taking the angular distortion into account. For VAL, the estimation is less accurate than for CAL and leads to some results on the unsafe side—at least in a few cases of high loads for S1100QL and one result for S960M (supplier 1). For many results in the LCF regime, when elastic-plastic behavior and mean strains come into play, but also at low stress amplitudes of the HCF regime, where linear-elastic behavior can be assumed, the experimentally determined fatigue lives exceed the calculated ones, which gives a conservative estimation beyond the factor of 4. However, again, the same applies as for low load levels from constant amplitude loading: the significance of the estimation is low due to few test results in this regime.

Fig. 13
figure 13

Comparison between calculated and experimentally determined fatigue lives for variable amplitude loading applying the damage parameter Pm

Based on the integral treatment of butt welds, the cyclic behavior of the seam weld is determined and successfully used for a fatigue life estimation of corresponding joints under CAL and VAL. The estimation for CAL is in good agreement with the experiment showing some results that are too safe, while the VAL estimation tends to be very conservative, but still having some unsafe results. In both cases, consideration of the transient behavior for the fatigue life estimation is beneficial, especially at high load levels at the border to the LCF regime. The conservative estimations in the LCF regime using this method is still superior compared with the linear damage accumulation, which just gives a constant slope of the calculated Gaßner line depending on the slope of the Wöhler curve. A limitation in the LCF regime can then just be constituted by knowledge about the yield or tensile strength of the joint. Therefore, further work on this approach is expedient, in order to improve the fatigue life estimation in the regime from HCF to LCF.


A method for fatigue life estimation based on an integral treatment of transversely loaded butt joints has been introduced. The following conclusions can be drawn from this investigation and the developed fatigue life estimation using an integral treatment of welds:

  1. (a)

    A description of the substructure from base material to base material by its cyclic behavior (integral treatment) is the basis. The characterization by an accumulated set of parameters results in a unification and simplification, compared with locally detailed modeling of the weld. As might be seen from the hardness distribution of a section cut from the weld seam, huge variations in micro-hardness from one measurement point to another exist (cf. [2, 51, 52]). By implementation of a distribution of local characterization parameters in local fatigue concepts by FE modeling and simulation, calculation time increases drastically.

  2. (b)

    For the assessment from the low cycle to the high cycle fatigue regime, the cyclic elastic-plastic behavior was considered. The cyclic transient behavior shows a distinctive cyclic softening of the base materials and butt welds, which is described by a damage-dependent definition of the cyclic characteristic values K′ and n′ of the Ramberg-Osgood equation. This is the main proposal to describe transient effects within this approach. Other transient effects, mean stress relaxation and cyclic creep, were not explicitly considered. In particular, cyclic creep requires further strain-rate-dependent investigations.

  3. (c)

    Additional influences on the fatigue performance result from global geometrical factors, such as axial and angular misalignments. Axial misalignments are small and can be neglected. The angular distortion depends on the execution of the weld with respect to the boundary conditions. Furthermore, the heat input of the welding process influences the angular misalignment or residual stresses for fixed connections to the entire structure. While LBW creates a smaller angular misalignment, the increased heat input of GMAW induces a larger distortion. Therefore, the angular misalignment is considered with the help of additional stresses (and strains), which improves the fatigue life estimation.

  4. (d)

    The choice of damage parameter—PSWT, Pε, PHL,mod, PJ, and PRAJ were evaluated—influences the fatigue life estimation. Finally, a generalized damage parameter Pm (combination of PB and Pε) has been introduced, which can be adjusted to the regime, where damage is dominated by mean stresses (HCF) or mean strains (LCF) in addition to stress and strain amplitudes. This has been achieved by additional factors for the mean stress and mean strain in the mathematical formulation.

  5. (e)

    The application of the integral life estimation method has been illustrated by lifetime estimations for constant and variable amplitude loading. It has been found that the estimation is in good agreement with experimental stress-controlled results under constant amplitude loading, while the estimated life under variable amplitude loading tends to be on the safe side for lower loading situations. In general, the (double-logarithmic) non-linear transition from LCF to HCF can be estimated with this method.