Validation of multiaxial fatigue strength criteria on specimens from structural steel in the high-cycle fatigue region

materials Article Validation of Multiaxial Fatigue Strength Criteria on Specimens from Structural Steel in the High-Cycle Fatigue Region František Fojtík1,* , Jan Papuga 2,3 , Martin Fusek 1and Radim Halama 1   Citation: Fojtík, F.; Papuga, J.; Fusek, M.; Halama, R. Validation of Multiaxial Fatigue Strength Criteria on Specimens from Structural Steel in the High-Cycle Fatigue Region. Materials 2021,14, 116. https://doi.org/10.3390/ma14010116 Received: 27 November 2020 Accepted: 23 December 2020 Published: 29 December 2020 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Faculty of Mechanical Engineering, Department of Applied Mechanics, VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 70800 Ostrava, Czech Republic; [email protected] (M.F.); [email protected] (R.H.) 2Department of Mechanics, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Technická4, 16607 Prague, Czech Republic; [email protected] 3Center of Advanced Aerospace Technology, Faculty of Mechanical Engineering, Department of Instrumentation and Control Engineering, Czech Technical University in Prague, Technická4, 16607 Prague, Czech Republic *Correspondence: [email protected] Abstract: The paper describes results of fatigue strength estimates by selected multiaxial fatigue strength criteria in the region of high-cycle fatigue, and compares them with own experimental results obtained on hollow specimens made from ˇ CSN 41 1523 structural steel. The specimens were loaded by various combinations of load channels comprising push–pull, torsion, bending and inner and outer pressures. The prediction methods were validated on fatigue strengths at seven different numbers of cycles spanning from 100,000 to 10,000,000 cycles. No substantial deviation of results based on the selected lifetime was observed. The PCRN method and the QCP method provide best results compared with other assessed methods. The results of the MMP criterion that allows users to evaluate the multiaxial fatigue loading quickly are also of interest because the method provides results only slightly worse than the two best performing solutions. Keywords: multiaxial fatigue; high-cycle fatigue; multiaxial fatigue experiments; S-N curve approximation 1. Introduction To validate the multiaxial fatigue strength criteria, the prediction results should be compared with experimental results. Any experimental campaign that would cover the mean normal stress effect, the mean shear stress effect, phase shift effect, etc., in various lifetimes for a single material (or more materials) presents a lengthy and costly process. The validation is thus often realized on experimental data retrieved from other sources: conference papers, papers in journals, books, PhD theses or technical reports. If such sources are used, they must be carefully verified, as to whether the data to be adopted are credible and usable for such validation. The prediction quality of multiaxial fatigue strength criteria is often assessed on experiments with various load combinations of push–pull and torsion [ 1 – 4 ]. By some experiments, axial loading of specimens can be induced by bending, which causes nonconstant stress distribution over the cross-section [ 5 – 9 ]. Relatively rarely, the inner pressure is applied on hollow specimens [ 10 – 12 ]. The stress gradient differs in this load mode—it is higher at the inner surface and lower at the outer surface. It is, therefore, important to know, which of those two surfaces is critical due to the other co-acting load channels, and both surfaces should be evaluated in some cases. The pressure is usually acting as a constant load. Such setup is simpler, because it allows the experimenter to run the desired load history quicker. If more load channels are superposed, the multiaxial stress state is likely to be induced. Such more complicated combinations are very interesting for validation purposes. Experimental sets comprising multiple various load cases on different Materials 2021,14, 116. https://doi.org/10.3390/ma14010116 https://www.mdpi.com/journal/materials Materials 2021,14, 116 2 of 19 specimen designs and sizes, that also provide a detailed information on material properties to tune the multiaxial fatigue strength criteria, are published only rarely. The experiments described in this paper cover various combinations of push–pull, bending, torsion, inner and outer pressures. The broad spectrum of various multiaxial load combinations with varying stress ratios on individual channels are an important condition for any adequate validation of calculation methods used by different multiaxial fatigue strength criteria. One of the few papers, which looks for a general solution covering more different load modes in one campaign is the paper by Morel and Palin-Luc [ 13 ], in which they propose the use of the non-local model averaging the stress quantities over the critical volume. The paper by Papuga et al. [ 14 ] does not treat the same problem more generally and simply uses the axial load modes in the analysis according to assumed stress distribution. If multiaxial loading includes bending, the plane bending fatigue strengths are used in analyses. If it involves any other load case than bending, the fatigue strengths relevant to push–pull are used. This paper presents an extensive experimental campaign that was realized on hollow specimens made from ˇ CSN 41 1523 structural steel. There are results of 24 uniaxial and multiaxial load cases of very diverse setups. The paper validates several multiaxial fatigue strength criteria on fatigue strengths obtained at 750,000 cycles from the Kohout-Vˇechet approximations [ 15 ] for each load case. The new MMP method usable for multiaxial fatigue strength analysis is published in [ 14 ] for the first time. As MMP is an extension of the Manson–McKnight criterion (MMK) [ 16 , 17 ], its biggest advantage is the simplicity of the computational analysis. It can be easily run using a common spreadsheet program such as MS Excel. The newly introduced MMP criterion significantly improves the prediction quality found for the MMK solution to a point that the MMP criterion could reach the quality of the output comparable with much more complex multiaxial fatigue strength criteria. Papuga et al. used the same data set in [ 18 ] to describe the validation results by selected multiaxial fatigue strength criteria at three additional lifetimes (the analyses were run between 100,000 and 750,000 cycles). The current paper increases the scope of tested load cases from 24 to 34. This enlargement brings along some important load cases missing previously for some sizes of specimens—above all, the reversed torsion on smaller specimens or the repeated bending load case. These previously non-existent load cases had to be in some way substituted in the previous papers. Their inclusion into the computational scheme should result in a more consistent validation process. The recent paper by Karolczuk et al. [ 19 ] opened a question, whether the same material parameters weighting the effect of stress parameters in the multiaxial criteria could be used over bigger ranges of lifetimes. They proved that the actual material parameters valid for the given final lifetime should result in a superior output than some fixed constant parameters could provide. To confirm or to deny that finding, the validation campaign on all 34 load cases is performed in this paper on fatigue strengths derived at seven different lifetimes spanning from 100,000 cycles to 10,000,000 cycles. To also cover the high lifetime levels, a special approximation FF formula is adopted in this paper. With all these changes involved, the paper focuses on validating 11 different multiaxial fatigue strength criteria of various types of solutions in order to assess their credibility for an accurate fatigue strength evaluation. To reach this goal, the experimental campaign is first described in Section 2, including also the regression models used to derive the fatigue strengths from the S-N curves. The various multiaxial models processed in the validation are described in Section 3, together with the way the quality of the regression is assessed. Section 4discusses the obtained results and Section 5concludes the outcome of the presented paper. 2. Experiments and Processing of Their Results 2.1. Material and Specimens The specimens were manufactured from ˇ CSN 41 1523 structural steel (equivalent to S355JR or St52-3) delivered in bars retrieved from the single T31052 melt. The static Materials 2021,14, 116 3 of 19 material properties are provided in Table 1, and the chemical composition can be found in Table 2. Table 1. Static material parameters and chemical composition of the studied material. Designation Tensile Strength [MPa] Tensile Yield Stress [MPa] Elongation at Fracture [%] Reduction of Area at Fracture [%] True Fracture Strength in Torsion [MPa] ˇ CSN 41 1523 560 400 31.1 74.0 516.6 Table 2. Chemical composition of the studied material. Chemical Composition: C [%] Mn [%] Si [%] P [%] S [%] Cu [%] 0.18 1.38 0.4 0.018 0.006 0.05 Three different specimen types: S1, S2 and S3 have already been used in [ 14 ]. The diameters (D—outer diameter, d—inner diameter) in the critical cross-sections were: (S1) D= 11 mm and d= 8 mm, (S2 and S3) D= 20 mm and d= 18 mm—see Figure 1. The newly introduced specimen type S4 is also shown in the same figure with its cross-sectional parameters slightly modifying the S1 configuration: D= 12 mm and d= 8 mm. All specimens were polished on the outer surface, while the inner surface was reamed. Figure 1. Drawings of specimens used in this campaign: ( a ) S1—top left, ( b ) S2—top right, ( c ) S3—bottom left, ( d ) S4—bottom right. 2.2. Load Cases The original experimental campaign from [ 14 ] is extended in this paper by 10 further new load cases. The summary of all load cases imposed in the total of 34 configurations is provided in Figures 2and 3. The detailed description of individual tests sets, of their types, of applied test frequencies and of geometric parameters of used specimens can be found in Table 3. The new experimental load cases concern: •Load case FF041—repeated plane bending. • Load cases FF048-FF051—repeated bending with constant inner pressure imposed in the mode of a pressure vessel. Materials 2021,14, 116 4 of 19 •Load cases FF062-FF063—fully reversed push–pull combined with inner pressure. •Load cases FF064-FF065—repeated push–pull combined with inner pressure. •Load case FF092—fully reversed torsion on S4 specimen (see Figure 1). The S4 specimens were not manufactured on purpose after publishing [ 14 ]—they were prepared before the whole test campaign in the moment, when the search for an optimum geometry of specimens was targeted. As other specimen types were later selected for testing, the results of S4 specimens tested in fully reversed torsion were put aside. They were again uncovered, when the lack of the S-N curve for fully reversed torsion on smaller specimen type, S1, became obvious. Figure 2provides schematic drawings explaining the way the individual specimens were loaded for specific load cases marked FFXXX, where XXX is replaced by a unique ID number for each load case. All load cases were run under the load control. The ranges of forces, moments and pressures applied to individual load cases are provided in Table 4, and the information on the phase shift between the axial load channel and the torsion load channels accompanies them there. Figure 2. Overview of various setups of experiments. F—push/pull, Mb—bending moment, Mk—torque and P—pressure. Materials 2021,14, 116 5 of 19 Figure 3. Setup of the experiments for load cases of torsion and bending on the torque controlled FF044 (right) and bending load case FF040 (left), both using S2 specimens. Table 3. Summary of all load cases. Abbreviations used: Ten—tension, To—torsion, RP—pressurized, PV—pressure vessel mode, PB—plane bending, A—amplitude, M—mean value, D—outer diameter, d—inner diameter. Mark Specimen Type Machine No. (Frequency [Hz]) Load Combination Measured Diameters Input Load Channels D[mm] d[mm] Ten PB RP PV To FF001 S1 1 (10) Ten + To 10.95 8.02 A A FF002 S1 1 (10) Ten + To 10.95 8.02 A A FF003 S1 1 (10) Ten + To 10.95 8.02 A A FF004 S1 1 (10) Ten + To 10.95 8.02 A A FF005 S1 1 (10) Ten + To 10.95 8.02 A, M A, M FF030 S1 1 (10) Ten 10.95 8.02 A FF031 S1 1 (10) Ten 10.95 8.02 A, M FF033 S1 1 (10) To 10.95 8.02 A, M FF040 S2 2 (25) PB 19.99 18.05 A FF041 S2 3 (10) PB 19.96 18.06 A, M FF042 S2 2 (25) To 19.99 18.05 A FF044 S2 2 (25) PB + To 19.99 18.05 A A FF045 S2 2 (25) PB + To 19.99 18.05 A A FF046 S2 2 (25) PB + RP 20.00 18.03 A M FF047 S2 2 (25) PB + RP 20.02 18.02 A M FF048 S2 4 (20) PB + PV 19.93 18.05 A, M M FF049 S2 4 (20) PB + PV 19.94 18.05 A, M M FF050 S2 4 (20) PB + PV 19.94 18.05 A, M M FF051 S2 4 (20) PB + PV 19.92 18.05 A, M M FF054 S2 2 (25) To + PV 19.97 18.05 M A FF055 S2 2 (25) To + PV 19.94 18.09 M A FF056 S2 2 (25) To + PV 19.94 18.07 M A FF057 S2 2 (25) To + Ten 19.99 18.05 M A FF058 S2 2 (25) To + Ten 19.99 18.05 M A FF059 S2 2 (25) To + Ten + RP 20.04 18.25 M M A FF060 S2 2 (25) To + Ten + RP 19.96 18.29 M A FF061 S2 3 (4) To + Ten 19.99 18.05 A A FF062 S2 4 (20) Ten + RP 20.00 18.02 A M FF063 S2 4 (20) Ten + RP 20.01 18.02 A M FF064 S2 4 (20) Ten + RP 20.01 18.02 A, M M FF065 S2 4 (20) Ten + RP 20.03 18.02 A, M M FF074 S3 2 (25) To + RP 20.00 18.05 M A FF075 S3 2 (25) To + RP 20.00 18.05 M A FF092 S4 2 (25) To 12.00 8.00 A Materials 2021,14, 116 6 of 19 Table 4. Summary of all load cases as regards to the range of applied forces, moments and pressures; also, the phase shift ϕat between the axial and torsion load signals is stated. Load Case Axial Force [kN] Torque [Nm] Bending Moment [Nm] Phase Shift [◦]ϕat Pressure [MPa] FaFmMkaMkmMbaMbm From–To From–To From–To From–To From–To From–To Pm FF001 4.34 0 34–25 0 0 0 0 0 FF002 8.5 0 24.5–16 0 0 0 0 0 FF003 4.34 0 34–26.15 0 0 0 90 0 FF004 8.5 0 33–23 0 0 0 90 0 FF005 5 5 24.1–17.5 24.1–17.5 0 0 0 0 FF030 12.85–10.4 0 0 0 0 0 0 0 FF031 9.5–8.2 9.5–8.2 0 0 0 0 0 0 FF033 0 0 37.5–27.7 37.5–27.7 0 0 0 0 FF040 0 0 0 0 99.6–81.6 0 0 0 FF041 0 0 0 0 73.3–64.7 73.3–64.7 0 0 FF042 0 0 94.5–83.9 0 0 0 0 0 FF044 0 0 81.1–63.7 0 60.5–47.5 0 0 0 FF045 0 0 42.2–33 0 94.4–73.7 0 0 0 FF046 0 0 0 0 90.1–76.5 0 0 23.3 FF047 0 0 0 0 88.0–74.0 0 0 36.0 FF048 0 0 0 0 65.6–59.6 65.6–59.6 0 10.5 FF049 0 0 0 0 67.7–57.6 67.7–57.6 0 20.0 FF050 0 0 0 0 68.5–56.5 68.5–56.5 0 30.0 FF051 0 0 0 0 66.7–54.1 66.7–54.1 0 40.0 FF054 0 0 93.0–76.2 0 0 0 0 15.0 FF055 0 0 95.0–78.0 0 0 0 0 10.0 FF056 0 0 75.5–62.8 0 0 0 0 20.2 FF057 0 14 98.8–71.6 0 0 0 0 0 FF058 0 10.2 100.6–76.6 0 0 0 0 0 FF059 0 0 91.0–77.5 0 0 0 0 40.0 FF060 0 5.9 84.7–77.0 0 0 0 0 40.0 FF061 16.6–12.3 0 27.1–20.1 0 0 0 0 0 FF062 17.0–14.6 0 0 0 0 0 0 20.0 FF063 16.5–13.0 0 0 0 0 0 0 40.0 FF064 7.2–6.3 7.2–6.3 0 0 0 0 0 20.0 FF065 7.3–6.4 7.3–6.4 0 0 0 0 0 40.0 FF074 0 0 87.6–74.9 0 0 0 0 13.0 FF075 0 0 85.7–48.2 0 0 0 0 27.0 FF092 0 0 55.7–45.2 0 0 0 0 0 To obtain the described load cases, various experimental machines had to be used as noted in Table 3, and different special fixtures to impose the desired load configurations had to be applied. Figure 3, left, depicts the example of the fixture model with the test specimen in grips as used for the load case of reversed bending marked as FF040. The combined bending and torsion loading as applied in FF044 test series is depicted in Figure 3, right. The FF046 load case combining the reversed bending with constant internal pressure can be found in Figure 4, left. The test specimens loaded by reversed torsion with inner and outer pressure in the FF059 test case can be seen in Figure 4, right. These tests were performed on the reconstructed and modernized biaxial testing machine Schenck type PWXN, which is originally equipped by the control of torque. It is extended by the possibility to apply additional axial constant force. Table 3refers to this machine type by number 2. Number 1 in Table 3corresponds to the biaxial servohydraulic pulsator INSTRON 8802. Another used testing machine is the biaxial servohydraulic pulsator LABCONTROL 100 kN/1000 Nm, which is equipped by a combined hydraulic actuator able to impose push–pull and torque. This machine was derived during the reconstruction of the original INOVA ZUZ 200 machine. It is marked by number 3 in Table 3. Number 4 in Table 3concerns the uniaxial hydraulic pulsator INOVA FU-63-930-V1. Materials 2021,14, 116 7 of 19 Figure 4. The load case of bending and pressurizing with the pressure chamber and the pressure sensor—FF046 load case (left). The setup of the test case with the pressure chamber inducing inner and outer pressure—FF059 load case (right). The necessary input into all multiaxial fatigue strength criteria are local stresses. The purely elastic material response is assumed to derive them. To locate the hot-spot on more complicated testing specimens for various superposed load channels’ acting, the finite element (FE) solution is necessary in order to deliver stress tensor components induced by individual load channels. All test cases and specimens were modeled and computed within the Ansys FE-solver. The stress tensors were obtained for unit loads acting on individual load channels, and the obtained stress components were then multiplied by the factor related to the ratio between the actual load and the unit load. Experiments FF062-FF065 are special, because the pressure causes higher (tangential) stress on the inner surface, while the stress response to axial loading induces more or less uniform axial stress distribution over the cross-section. For these specimens, thus, inner and outer surfaces were evaluated in the stress analysis and also in the subsequent fatigue analysis. During the fatigue tests, responses to individual load channels were measured by certified sensors, which are regularly checked by the Czech Metrology Institute. To set up the load parameters by individual tests, strain gages were installed on chosen tested specimens; see the examples in Figure 5. The strain gages provided the information on strains and stresses attained on the surface of test specimens for individual load cases. These data items then could be compared with results of the finite element analyses to verify the applied boundary conditions. Figure 5. Examples of the load setup validations taken to ensure the applied stress are conforming to the expectations. 2.3. Regression Analyses In the study by [ 14 ], experimental results were processed to obtain the regression S-N curves either by the linear Basquin model or by the Kohout-Vˇechet [ 15 ] non-linear model. Each load case was covered by at least 5 finished experiments on different load levels, and by one run-out test, which was left unfinished at 10 million cycles. Some experimental test cases were not described well above 750,000 cycles by any of the two mentioned models—see Figure 12 in [ 14 ] as regards to the FF033 test case (or see Figure 6hereafter). This was the main reason why the fatigue strength analysis was covered only at 750,000 cycles in [ 14 ]. The newer paper by [ 16 ] documents a similar analysis of chosen multiaxial fatigue strength criteria in the limited lifetime region; more precisely at 100,000, 200,000, 500,000 and 750,000 cycles. The tests were performed in compliance with the valid ˇ CSN 42 0362 standard [ 20 ]. Every experimental load case is completed by at least one run-out test, for which the specimen did not break even at 10 million cycles. The difference between Materials 2021,14, 116 8 of 19 amplitudes of the dominant stress channel for the run-out specimen and for the last broken specimen with the longest lifetime does not exceed 10 MPa for most cases. The ˇ CSN 42 0363 standard [ 20 ] in its paragraph No. 49 recommends choosing this difference in applied stress levels in dependency on the expected fatigue limit of the evaluated tests’ case. The basic number of cycles to determine the fatigue limit is set to 10 million by ˇ CSN 42 0363 for steels. Figure 6. Comparison of the three approximation models for three different load cases. The value of 10 million cycles is therefore set as the limit of the possible approximation domain. This paper compares results of three approximation methods for the S-N curve data. The first, and the most commonly used method is the Basquin approximation, which should be optimally used only within the experiments with limited lifetime. It is formulated by the following [21]: σ=σ0f·(2N)b(1) where σ0f is the coefficient of fatigue strength, and b is the exponent of fatigue strength. Though this model is more than 100 years old, it presents the part of standards for steel structures (e.g., Eurocode 3, ISO12107). If the experimental points are selected to be included in the approximation, two tendencies cause some bias. The first one concerns the requirement to include as many experimental points into the regression analysis, as only possible. This requirement ensures that the curve will really correspond to the behavior of material and will not describe the response of only several experimental points. The second tendency to bear in mind are the attempts to omit those data points, which do not conform to the expected material model in the limited lifetime region. For the Basquin model, such data points then can be a part of the S-N curve transition into the quasi-static domain or to the fatigue limit domain, which cannot be approximated by it reasonably well. A suitable approximation, which can integrate into the model most of experimental data from all those domains, is the Kohout-Vˇechet regression model ([15], also, K&V hereafter): σ=a·C·N+B N+Cβ(2) The non-linear regression analysis demands a certain setup for initial estimates of material parameters a , β , B , C . They can be set based on the previously obtained Basquin regression curve: a=2b·σ0f,β=b,B=10[maxi(logσi)−loga]/b,C=10[mini(logσi)−loga]/b(3) Thanks to the additional two parameters available in this model in comparison with the Basquin formula, the curve can follow the trend of the S-N data with two bends—one Materials 2021,14, 116 9 of 19 in the transition to the horizontal line at the quasi-static domain, and the other in the transition to the horizontal line at the fatigue limit region. To define better the quasi-static response, the tensile strength at 1 4 or 1 2 cycles can be taken as an additional regression input to other experimental data points. Such inclusion affects, above all, the shape of the curve in the domain of low-cycle fatigue and in the quasi-static domain. However, the Kohout-Vˇechet model is not suitable for regression of multiple experiments per the outer load levels (load level tests), because the outermost points substantially affect the trend of the curve and its transition to horizontal lines. To approximate the experimental data, the FF approximation first published in [ 22 ] was used. The proposed approximation function is: σ=σ0−(σ0−σC)·sinnπ 2·[log(4·N0)/log(4·NC)]a2o. (4) This approximation is here used as a one-parametric, where a2 is the only fitted parameter and σ0 is tensile strength. The σC parameter corresponds to the highest stress level, at which the specimen did not break until the lifetime NC= 10 7 cycles. This stress level is in accordance with [ 19 ], assumed to correspond to the fatigue limit. Thanks to the use of the sinus function, the approximations are twice bent, which enables to follow the S-N curve trends in both transitions to the horizontal lines. Materials and specimens leading to S-N curve data items that show such S-like trend can thus be suitably modeled by the FF function. On the other hand, the same function limits the use of this formula for lifetimes longer than NC=107cycles, where the function would start to increase again to higher stresses, which is unlikely for any material. The FF approximation is also not suitable for load level tests. The examples of approximations by the three mentioned formulas can be compared in Figure 6. The functions of the first two formulas (Basquin—Equation (1), Kohout-Vˇechet—Equation (2)) are shown also outside the interpolation domain of analyzed data, so that their general trends were clearer. Due to the mentioned character of the sinus function, the FF approximation is shown only until NC=107cycles, and not at higher lifetimes. The shape of the Kohout-Vˇechet curve, e.g., for FF033 experiment in repeated torsion (Figure 6), is caused by the model properties, where the transition to the quasi-static region best follows the experimental data items, and it leads to the smallest coefficient of determination R2 . As only data items related to broken specimens are used for this regression model, the obtained regression curves for most evaluated load cases are limited in their interpolation region only to the lifetimes up to 2 million cycles. Some test cases show the limitation even more stringently, as e.g., the FF033 test case that is usable only up to approx. 900,000 cycles. To get the reasonably set fatigue strengths for each load case, another approximation rather than the Kohout-Vˇechet model should be applied. To increase the multiaxial fatigue strength analyses reported hereafter, also to the lifetimes of 1, 2, 5 and 10 million cycles, the FF approximation was chosen. The quality of each of the approximations described in Equations (1), (2) and (4) is compared in Figure 7. The chosen characteristics shown for each load case is the coefficient of determination R 2 . In this comparison, the Kohout-Vˇechet model clearly attains the best results, and the Basquin model and FF model are comparable one to another, but are weaker than the Kohout-Vˇechet approximation. It should be anyhow reminded, that R 2 parameters are computed on different sets of experimental points—the Basquin curve is regressed only on points in the inclined part of the S-N curve, the Kohout-Vˇechet curve excludes all run-outs, and only the FF model covers all data points. Logically, its results can be comparably worse to the Kohout-Vˇechet model in R 2 parameters due to the largest scope of regression inputs. The parameters of the FF model for each tested load case are summarized in Table 5. To also show the limitation on the scope of usable lifetimes that should be imposed when dealing with the regression curves, the shortest fatigue life obtained experimentally for each load case is documented in Table 5as Nmin parameter. Materials 2021,14, 116 16 of 19 Figure 8. Histograms of ∆ FI errors for all 34 test cases, 7 evaluated lifetimes and all 11 multiaxial fatigue strength criteria validated according to Table 6. 5. Conclusions The analysis presented in this paper focuses on validating 11 different multiaxial fatigue strength criteria on the own test set composed of 34 S-N curves describing the fatigue response for different multiaxial load cases. The S-N curves were approximated on the experimental data based on the FF regression function. All test cases relate to hollow specimens manufactured from a single melt of the ˇ CSN 41 1523 structural steel equivalent to S355JR. Eleven chosen fatigue strength criteria comprise commonly used criteria and some new criteria, which were recently proven to provide good prediction results. For each load case, the validation is realized on fatigue strengths retrieved from the FF approximation at seven different lifetimes between 0.1 and 10 million cycles. The results are assessed based on the defined fatigue index error ∆ FI and its statistical processing over all evaluated lifetimes and checked load cases. The mean value of ∆ FI, the sum of its squares, sample standard deviation or its variation range are assessed. Based on the comparison for various evaluated multiaxial criteria, for the tested material and for tested load cases, it can be concluded: • PCRN, QCP and MMP methods result in the best prediction quality, and their sample standard deviation is multiple times lower than the sample standard deviation of the Dang Van method, though the Dang Van method is commonly used in the engineering practice. • In the range between 0.1 and 10 million cycles, the prediction quality is stable, and the variability of ∆ FI over this interval is negligible if compared with the overall prediction scatter. This holds true only then, when the complete calculation of the equivalent Materials 2021,14, 116 17 of 19 stress amplitude and of the material parameters specific to each multiaxial fatigue strength criterion, as described in Tables 6and 7, is done at the same fatigue life. Author Contributions: Conceptualization, J.P. and F.F.; methodology, J.P.; software, J.P. and F.F.; validation, J.P. and F.F.; formal analysis, J.P. and F.F.; investigation, J.P. and F.F.; data curation, F.F.; writing—original draft preparation, J.P. and F.F.; writing—review and editing, J.P. and F.F visualization, J.P., F.F, M.F. and R.H; supervision, M.F. and R.H.; project administration, F.F., M.F. and J.P. All authors have read and agreed to the published version of the manuscript. Funding: The research by František Fojtík, Martin Fusek and Radim Halama was supported by 19-03282S project of the Czech Science Foundation, and by the specific research SP2020/23 project, supported by the Ministry of Education, Youth and Sports of the Czech Republic. Jan Papuga acknowledges support from the ESIF, EU Operational Programme Research, Development and Education, from the Center of Advanced Aerospace Technology (CZ.02.1.01/0.0/0.0/16_019/0000826); Faculty of Mechanical Engineering, Czech Technical University in Prague, and from the Grant Agency of the Czech Technical University in Prague, grant number SGS20/158/OHK2/3T/12. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to their complexity. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Nomenclature a2[-] parameter for FF approximation a,b,c,d[-] material parameters for various multiaxial fatigue limit estimation methods a,β,B,C[-] material parameters of Kohout-Vˇechet regression b−1[MPa] fatigue strength in fully reversed bending loading b[MPa] exponent of fatigue strength β[-] mean stress coefficient C[MPa] shear stress on an examined plane d[mm] inner diameter D[mm] outer diameter ∆FI [%] fatigue index error F[N] axial force ϕ,θ[◦] Euler angles defining the orientation of the examined plane ϕat [◦] phase shift between signals on axial and torsion load channels J2[MPa] second invariant of the deviatoric stress tensor κ[-] ratio of fatigue strengths in fully reversed loadings κ=s−1/t−1 κ0[-] ratio of fatigue strengths in repeated loadings κ0=s0/t0 Mb [Nmm] bending moment Mk [Nmm] torque N[MPa] normal stress on an examined plane Nx[-] number of cycles NC[-] number of cycles at σc p−1[MPa] fatigue strength in fully reversed push–pull P[MPa] pressure R2[-] coefficient of determination s−1[MPa] fatigue limit in fully reversed axial loading (stress amplitude of the cycle) s0[MPa] fatigue limit in repeated axial loading (maximum stress of the cycle) Su[MPa] tensile strength σ1[MPa] maximum principal stress σ3[MPa] minimum principal stress σc[MPa] the highest stress level, at which the specimen did not break until Nc= 107cycles Materials 2021,14, 116 18 of 19 σeq,a [MPa] equivalent stress amplitude σ’f[MPa] coefficient of fatigue strength σH[MPa] hydrostatic stress t0[MPa] fatigue limit in repeated torsion (maximum stress of the cycle) t−1[MPa] fatigue limit in fully reversed torsion (stress amplitude of the cycle) T[MPa] resolved shear stress (shear stress projection into a given direction) Xm[-] shear stress weight coefficient in the MMP method w[-] Walker exponent Abbreviations K&V Kohout-Vˇechet regression curve PB plane bending PV pressure vessel mode RP pressure loading Ten tension-compression (push–pull) loading To torsion loading Indexes a,Aamplitude m,Mmean value max maximum value 0 referring to repeated loading (from zero to maximum value) −1 referring to fully reversed loading (from −x to +x) References 1. 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