Limited Stress Surface Model for Bending and Torsion Fatigue Loading with the Mean Load Value

In this study, a linear model of the transformation of the stress amplitude due to the mean value was used. The coefficient of the material sensitivity to cycle asymmetry with consideration of the dependence of this coefficient on the number of fatigue loading cycles is also used. A three-parameter surface model of limited stresses is proposed in this paper. The model is verified using the results of fatigue tests for cyclic bending and torsion under mean loads. The tests are performed for two types of alloy steels—S355J0 and S355J2G1W. Comparison of the allowable stress amplitudes obtained experimentally with those predicted using the proposed model shows errors of no more than 18%, with the area of the surface with the largest error being relatively small.


Introduction
Structures and elements of machines can be subjected to time-varying loads, which cause fatigue processes in the materials. These loads are most often of a complex nature. Additionally, the mean load appears resulting from the self-weight of the structure or initial loads related to the functions fulfilled by the structure. The asymmetry of the fatigue load significantly influences the change of the allowable amplitudes; therefore, an understanding of the mutual relations between the average load and the maximum variable load is important when determining the durability of materials operating in such conditions. In the literature, mathematical models of the influence of the mean load on the fatigue life of construction materials are usually based on the basic strength properties of the material, such as the strength or yield point. In engineering practice, these properties are convenient to use because they do not require complex calculations or separate tests to determine the model parameters (material properties).
The commonly used transformation relationships based on changes in the mean stress values from zero are the Goodman, Gerber, and Marin models, which assess the mechanical properties of materials using static tensile tests [1][2][3][4]. In these cases, reservations are raised by the reference to the static properties of the material, which are not necessarily adequate for the fatigue phenomenon. On the other hand, the Morrow and Kliman models use material constants that characterize the fatigue properties of the material [5,6]. These are the parameters describing the Manson-Coffin-Basquin fatigue strain characteristics, namely the fatigue strength exponent and exponent of fatigue plastic strain. Another very popular model was presented by Smith, Watson, and Topper (SWT) [7]. This model is usually used in strain based fatigue analysis by combining the SWT parameter with the strain-life equation proposed by Manson [8] multiplied by the Basquin power law relation [9]. With the assumption that the plastic part of the strain amplitude is small and can be neglected during the fatigue life assessment, which is a typical practice in high-cycle fatigue, this model can only be expressed in terms of stress. A generalization of the SWT model, taking into account the mean value of the load, is the Walker model [10], where in special cases it is synonymous with the SWT model. Walker's model for low cycle loads proved to be better. The only disadvantage of the Walker model was the difficulty in determining one of the equation parameters in the experimental method. A set of 18 different materials [19,20] from both steel and aluminum alloy groups were analyzed. Ince and Glinka [21] indicated that in general, the SWT model expressed in stress equations reduces its application to high-cycle loading. Extending the SWT model to the range of low-cycle loading, the authors proposed the introduction of equivalent strains. The proposed model considers elastic and plastic strains and introduces a correction due to the mean value of the load. The proposed mean stress correction model was found to be better than the SWT model for Incoloy 901 superalloy and ASTM A723 steel, while both models provided equally good correlations with experimental data for 7075-T561 aluminum alloy and 1045 HRC 55 steel.
With respect to computational models based on stress-life (σ-N) or strain-life (ε-N) curves, Kwofie and Chandler [22] proposed a correction that allows the use of the SWT model for materials that show an increase in fatigue life when the average load is increased (e.g., copper). The so-called mean stress sensitivity index can be determined from experimental data. The effect of the increase in durability with an increase in the mean load itself is explained by the natural tendency of the mean tensile load to enlarge a fatigue crack. When the average load is very high, this could counteract the effect of the decreasing plastic strain amplitude. The second element is the effect of cyclic creep, which usually occurs for this type of material.
In general, it can be concluded that it is not possible to indicate one universal calculation method due to the mean load, as both these factors affect the efficiency of calculations. Numerous effects and parameters that are indicated as important elements for estimation relate to material properties and loading conditions. Simple computational models (e.g., Goodman and Gerber models) do not guarantee good results for durability estimation, which is important from the point of view of structural safety. A number of algorithms indicate the necessity of basing such estimations on the results of experimental studies. The purpose of this study is to learn the behavior of specimens made of S355J0 and S355J2G1W steels under conditions of cyclic bending and torsional loading with the participation of various mean values. In this paper, using a model of the transformation of stress amplitudes due to the mean value, a general three-parameter limiting stress surface is defined in the form of a relationship between the stress amplitude, mean stress value and the number of fatigue cycles.

Transformation Model for the Mean Load Value in Relation to the Number of Fatigue Cycles
The previously presented calculation models contain material properties that are constant values. Meanwhile, some research results indicate that the fatigue properties of the material may change with the number of cycles, and it is postulated that this effect should be taken into account in calculation algorithms [14,16,23,24]. Some proposals extend the applicability of the Goodman, Gerber, and other mentioned models in the range of lowcycle fatigue by taking into account the current number of failure cycles in transformation models [25,26].
In this paper, the influence of the mean load value on the limit amplitude value can be described by the cycle asymmetry sensitivity ratio ψ, which is defined according to the scheme of the two-parametric characteristic σ a = f(σ m ). A simplified bilinear model of this characteristic is presented in Figure 1. The cycle asymmetry sensitivity ratio ψ can be expressed as: where σ −1 is the stress amplitude for fatigue tests results under purely alternating loads (R = −1), σ 0 is the stress amplitude for fatigue tests results under purely pulsating loads (R = 0), and R is the stress ratio R = σmin/σmax. It is assumed that the factor ψ may change with the number of cycles N. Figure 2 shows the scheme used for determining the value of factor ψ depending on the number of N cycles. If the results of fatigue tests for purely alternating loads (R = −1) are described with line 1 in Figure 2, and for pulsating loads with line 2 in Figure 2, the material's sensitivity coefficient to cycle asymmetry for the life of N cycles can be calculated as: where σ −1 (N) is the stress amplitude at the N cycle life level for loads with the cycle asymmetry factor R = -1 and σ 0 (N) is the stress amplitude at the durability level of N cycles for loads with the cycle asymmetry factor R = 0. Equation (2) is explained in [27] for normal and shear stresses with these formulas. The cycle asymmetry sensitivity ratio ψ can be expressed as: where σ −1 is the stress amplitude for fatigue tests results under purely alternating loads (R = −1), σ 0 is the stress amplitude for fatigue tests results under purely pulsating loads (R = 0), and R is the stress ratio R = σ min /σ max . It is assumed that the factor ψ may change with the number of cycles N. Figure 2 shows the scheme used for determining the value of factor ψ depending on the number of N cycles. The cycle asymmetry sensitivity ratio ψ can be expressed as: where σ −1 is the stress amplitude for fatigue tests results under purely alternating loads (R = −1), σ 0 is the stress amplitude for fatigue tests results under purely pulsating loads (R = 0), and R is the stress ratio R = σmin/σmax. It is assumed that the factor ψ may change with the number of cycles N. Figure 2 shows the scheme used for determining the value of factor ψ depending on the number of N cycles. If the results of fatigue tests for purely alternating loads (R = −1) are described with line 1 in Figure 2, and for pulsating loads with line 2 in Figure 2, the material's sensitivity coefficient to cycle asymmetry for the life of N cycles can be calculated as: where σ −1 (N) is the stress amplitude at the N cycle life level for loads with the cycle asymmetry factor R = -1 and σ 0 (N) is the stress amplitude at the durability level of N cycles for loads with the cycle asymmetry factor R = 0. Equation (2) is explained in [27] for normal and shear stresses with these formulas. If the results of fatigue tests for purely alternating loads (R = −1) are described with line 1 in Figure 2, and for pulsating loads with line 2 in Figure 2, the material's sensitivity coefficient to cycle asymmetry for the life of N cycles can be calculated as: where σ −1 (N) is the stress amplitude at the N cycle life level for loads with the cycle asymmetry factor R = -1 and σ 0 (N) is the stress amplitude at the durability level of N cycles for loads with the cycle asymmetry factor R = 0. Equation (2) is explained in [27] for normal and shear stresses with these formulas. Based on the fatigue characteristics for the tests at R = −1 and R = 0, it is possible to determine changes in the material sensitivity coefficient to cycle asymmetry.
It is easy to show that if the slope is the same for straight lines 1 and 2 in Figure 2, the material's sensitivity factor to cycle asymmetry is constant over the entire range of fatigue life. If the lines shown in Figure 1 clearly differ in their slope, then the value of the coefficient ψ changes with the number of fatigue cycles.
The changes of the ψ factor with the number of cycles N can be determined experimentally using the equation: where η, λ are parameters, the values of which are determined on the basis of fatigue tests results under purely alternating (R = −1) and purely pulsating (R = 0) loads. Taking the logarithm of Equation (3), the following expression is obtained: The relationship (4) describes the equation of a straight line in a bilogarithmic coordinates system ( Figure 3). The slope λ in Equation (4) is related to the angle ξ by the following equation: Based on the fatigue characteristics for the tests at R = −1 and R = 0, it is possible to determine changes in the material sensitivity coefficient to cycle asymmetry.
It is easy to show that if the slope is the same for straight lines 1 and 2 in Figure 2, the material's sensitivity factor to cycle asymmetry is constant over the entire range of fatigue life. If the lines shown in Figure 1 clearly differ in their slope, then the value of the coefficient ψ changes with the number of fatigue cycles.
The changes of the ψ factor with the number of cycles N can be determined experimentally using the equation: where η, λ are parameters, the values of which are determined on the basis of fatigue tests results under purely alternating (R = −1) and purely pulsating (R = 0) loads.
Taking the logarithm of Equation (3), the following expression is obtained: The relationship (4) describes the equation of a straight line in a bilogarithmic coordinates system ( Figure 3). The slope λ in Equation (4) is related to the angle ξ by the following equation: (5) Figure 3. Material cycle asymmetry sensitivity factor in a bilogarithmic coordinate system.
The values of the cycle stress amplitude for purely alternating and purely pulsating loads can be determined from the equations describing the results of fatigue tests.
For purely alternating loads: For purely pulsating loads: Considering Equations (6) and (7), Equation (2) takes the following form: The values of the cycle stress amplitude for purely alternating and purely pulsating loads can be determined from the equations describing the results of fatigue tests.
For purely alternating loads: For purely pulsating loads: Considering Equations (6) and (7), Equation (2) takes the following form: where a = 1 (5) for the calculation of the coefficient λ takes the following form: According to Equation (3) and considering Equation (8), the coefficient η can be determined as: Equations (9) and (10) allow us to determine the values of the parameters λ and η in Equation (3) describing the function of the change in the value of the coefficient of material sensitivity to cycle asymmetry with the number of cycles N.

Limiting Stress Amplitude Surface Model
During the design of machine elements or structures due to the phenomenon of fatigue, loading conditions are determined for the expected life of the elements. In the case of cyclic loads with the contribution of the mean stress, the limiting stress surface is represented in the σ a -σ m -N coordinate system as shown in Figure 4. (5) for the calculation of the coefficient λ takes the following form: According to Equation (3) and considering Equation (8), the coefficient η can be determined as: (10) Equations (9) and (10) allow us to determine the values of the parameters λ and  in Equation (3) describing the function of the change in the value of the coefficient of material sensitivity to cycle asymmetry with the number of cycles N.

Limiting Stress Amplitude Surface Model
During the design of machine elements or structures due to the phenomenon of fatigue, loading conditions are determined for the expected life of the elements. In the case of cyclic loads with the contribution of the mean stress, the limiting stress surface is represented in the σ a -σ m -N coordinate system as shown in Figure 4. The mathematical description of the surface is a function f(σ a, N, σ m ) that relates the stress amplitude, mean value, and number of failure cycles. The search for a surface model based on fatigue test results is a complex and complicated issue [28]. The surface shown in Figure 4 carries information about the mutual relations of stress amplitude σa, mean stress value σm (curve 1 in Figure 4), stress amplitude σa, and number of failure cycles N (curve 2 in Figure 4). Assuming a certain function σ a = f(σ m ) to describe curve 1 and σ a = f(N) to describe curve 2, it is possible to determine the mathematical form of the The mathematical description of the surface is a function f(σ a , N, σ m ) that relates the stress amplitude, mean value, and number of failure cycles. The search for a surface model based on fatigue test results is a complex and complicated issue [28]. The surface shown in Figure 4 carries information about the mutual relations of stress amplitude σ a , mean stress value σ m (curve 1 in Figure 4), stress amplitude σ a , and number of failure cycles N (curve 2 in Figure 4). Assuming a certain function σ a = f(σ m ) to describe curve 1 and σ a = f(N) to describe curve 2, it is possible to determine the mathematical form of the limiting stress surface. The accuracy of the mathematical model of the surface depends on the accuracy of the adopted mathematical models describing curves 1 and 2.
Taking into account the coefficient of the material's sensitivity to cycle asymmetry and its interpretation presented in Figure 1, the relationship between the allowable stress amplitude and the mean stress value can be described by the following formula: where σ a , σ m are the amplitude and mean value of the asymmetric load (R = −1) and σ −1 is the stress amplitude of the symmetric (purely alternating) load (R = −1) equivalent in terms of fatigue to the asymmetric load. Assuming that the effects of the mean value of the material are different for different numbers of cycles N applied to the material and using Equation (3) to describe the change the coefficient of material sensitivity to cycle asymmetry with the number of cycles to failure, Equation (11) can be written in the following form: Experimental fatigue characteristics of the material for purely alternating loads (R = −1, curve 2 in Figure 4) are described by the following equation: where m is the slope of the Wöhler curve, σ f is the fatigue limit, and N 0 is the limit (edge) number of load cycles. Determining the stress amplitude σ −1 (N) from Equation (13) and inserting it into relation (12), one obtains the limiting stress amplitude surface equation in the following form: Equation (14) represents a mathematical model of the limiting stress amplitude surface determined from fatigue test results for purely alternating and purely pulsating loads, assuming that the effect of the mean value on the stress amplitude is a linear function.

Fatigue Tests
The subjects of the study are S355J2G1W steel (European Standard EN 10155:1993, Low-alloy rust-resistant steel, The European Committee for Standardization) and S355J0 steel (European Standard EN 10025:2019, Hot rolled products of structural steels, The European Committee for Standardization) [12,27,29]. The strength properties are presented in Table 1.  Figure 5 shows the shape and dimensions of the specimen used in the tests. The tests were carried out on the MZGS-100 fatigue test stand (Opole University of Technology, Opole, Poland) [30]. The stand allows testing for fatigue bending or torsion and additionally proportional bending with torsion (not applied in this research). The tests of cyclic bending and torsion were performed within the low and high range numbers of cycles where the moment amplitude was controlled. The loads present sinusoidal waveforms under a frequency range of 25-29 Hz. The nominal stress amplitude and nominal mean stress value were used in calculations.
The results of fatigue tests were approximated by the following regression equation: The tests were carried out on the MZGS-100 fatigue test stand (Opole University of Technology, Opole, Poland) [30]. The stand allows testing for fatigue bending or torsion and additionally proportional bending with torsion (not applied in this research). The tests of cyclic bending and torsion were performed within the low and high range numbers of cycles where the moment amplitude was controlled. The loads present sinusoidal waveforms under a frequency range of 25-29 Hz. The nominal stress amplitude and nominal mean stress value were used in calculations.
The results of fatigue tests were approximated by the following regression equation: where σ a is the bending stress amplitude (for torsion accordingly τ a ), N is the number of cycles until failure, and A and B are the parameters of the regression equation.
The parameters in Equations (3), (14) and (15) were determined based on fatigue test data from Pawliczek and Rozumek [27,29]. Fatigue tests were performed under cyclic bending and cyclic torsion with the mean value of the load at fixed stress ratios R = −1, R = −0.5 and R = 0 for S355J0 steel and for fixed mean stress values σ m (τ m ) 0, 75, 150 and 225 for S355J2G1W steel. For each load level, the tests were carried out with a minimum of 3 specimens. A set of loading conditions for fatigue tests is presented in Table 2.  Table 3 shows the values of coefficients A and B of regression Equation (15) for bending and torsion and the stress ratios R = −1 and R = 0, respectively. Analyzing the parameters contained in Table 3, it can be concluded that the values of the A coefficient for bending and the presented materials are different in all cases. During torsion, for R = −1, the values of the coefficient A show smaller differences than for R = 0, where greater differences between their values are noticed.
Based on the data in Table 3, the coefficients η and λ of Equation (3) were determined. The results of the calculations are shown in Table 4 and Figure 6 shows the functions for the tested steels.  The values of  and λ coefficients for bending and torsion show similar effects of the mean stress value on the allowable stress amplitudes for both steels. Figure 6 also indicates that for the durability range of 5 × 10 4 to about 7.5 × 10 5 cycles, the value of the coefficient ψ decreases significantly, while for the high-cycle durability range, the value of the material's sensitivity factor to cycle asymmetry changes slightly.
In the case of S355J0 steel bending, this relationship was described by the function ψ(N) = 3.16N −0.164 . There is a visible decrease in the value of the factor in terms of durability at N = (5 × 10 4 -7.5 × 10 5 ) cycles. In the case of torsion, the function has the form ψ(N) = 2.89N −0.131 , while the nature of the curve is similar to that for bending.
In a similar way for S355J2G1W steel, we have ψ(N) = 1.01N −0.072 functions for bending and ψ(N) = 0.82N −0.113 for torsion, respectively. The nature of the functions is similar to that of S355J0 steel. However, it can be noted that the range of changes in ψ for the maximum value at N = 5 × 10 4 and minimum value at N = 2.5 × 10 6 for S355J0 steel is about 2 times larger than for the second material. This suggests a greater sensitivity of the material to cycle asymmetry, especially in the low cycle load range.

Analysis of the Limiting Stress Amplitude Surface Model
This paper presents a mathematical model of the surface description of allowable stress amplitudes depending on the value of the mean load and the number of cycles to failure of the material. It was assumed that the influence of the mean load value on the allowable stress amplitudes depends on the material's sensitivity to cycle asymmetry. A function was proposed to describe the change in the value of the material's sensitivity coefficient to cycle asymmetry, and its parameters were determined from fatigue tests of specimens under purely alternating (R = −1) and purely pulsating (R = 0) loading conditions.
The analysis of the limiting stress amplitude surface model was based on the data shown in Tables 3 and 4. The experimental surfaces results for tests with stress ratios R = −1 and 0 are presented in [27,29]. The experimental surfaces of the limiting stress amplitudes were determined using Kriging multiparameter approximation [31], then the mathematical formula based on Equation (14) for the surfaces describing the different loading cases for both steels was determined. The values of η and λ coefficients for bending and torsion show similar effects of the mean stress value on the allowable stress amplitudes for both steels. Figure 6 also indicates that for the durability range of 5 × 10 4 to about 7.5 × 10 5 cycles, the value of the coefficient ψ decreases significantly, while for the high-cycle durability range, the value of the material's sensitivity factor to cycle asymmetry changes slightly.
In the case of S355J0 steel bending, this relationship was described by the function ψ(N) = 3.16N −0.164 . There is a visible decrease in the value of the factor in terms of durability at N = (5 × 10 4 -7.5 × 10 5 ) cycles. In the case of torsion, the function has the form ψ(N) = 2.89N −0.131 , while the nature of the curve is similar to that for bending.
In a similar way for S355J2G1W steel, we have ψ(N) = 1.01N −0.072 functions for bending and ψ(N) = 0.82N −0.113 for torsion, respectively. The nature of the functions is similar to that of S355J0 steel. However, it can be noted that the range of changes in ψ for the maximum value at N = 5 × 10 4 and minimum value at N = 2.5 × 10 6 for S355J0 steel is about 2 times larger than for the second material. This suggests a greater sensitivity of the material to cycle asymmetry, especially in the low cycle load range.

Analysis of the Limiting Stress Amplitude Surface Model
This paper presents a mathematical model of the surface description of allowable stress amplitudes depending on the value of the mean load and the number of cycles to failure of the material. It was assumed that the influence of the mean load value on the allowable stress amplitudes depends on the material's sensitivity to cycle asymmetry. A function was proposed to describe the change in the value of the material's sensitivity coefficient to cycle asymmetry, and its parameters were determined from fatigue tests of specimens under purely alternating (R = −1) and purely pulsating (R = 0) loading conditions. The analysis of the limiting stress amplitude surface model was based on the data shown in Tables 3 and 4. The experimental surfaces results for tests with stress ratios R = −1 and 0 are presented in [27,29]. The experimental surfaces of the limiting stress amplitudes were determined using Kriging multiparameter approximation [31], then the mathematical formula based on Equation (14) for the surfaces describing the different loading cases for both steels was determined.   To determine the effectiveness of the results from predicting allowable stress amplitudes σa using the proposed model, the relative error of the stress amplitudes calculated from the theoretical model versus the experimental results was determined: (16) Figure 11 shows the distribution of relative error as a function of mean stress and number of cycles N for specimens made of S355J0 steel. The maximum error for bending is 14% at σm = 80 MPa and life N > 1 × 10 6 cycles. For torsion, the error is 16% for τm = 4060 MPa and life N > 1 × 10 6 cycles. Figure 12 shows the relative error areas of the calculation results versus the experimental results for bending and torsion of S355J2G1W steel specimens. To determine the effectiveness of the results from predicting allowable stress amplitudes σ a using the proposed model, the relative error of the stress amplitudes calculated from the theoretical model versus the experimental results was determined: Figure 11 shows the distribution of relative error as a function of mean stress and number of cycles N for specimens made of S355J0 steel. To determine the effectiveness of the results from predicting allowable stress amplitudes σa using the proposed model, the relative error of the stress amplitudes calculated from the theoretical model versus the experimental results was determined: (16) Figure 11 shows the distribution of relative error as a function of mean stress and number of cycles N for specimens made of S355J0 steel. The maximum error for bending is 14% at σm = 80 MPa and life N > 1 × 10 6 cycles. For torsion, the error is 16% for τm = 4060 MPa and life N > 1 × 10 6 cycles. Figure 12 shows the relative error areas of the calculation results versus the experimental results for bending and torsion of S355J2G1W steel specimens. The maximum error for bending is 14% at σ m = 80 MPa and life N > 1 × 10 6 cycles. For torsion, the error is 16% for τ m = 40 ÷ 60 MPa and life N > 1 × 10 6 cycles. Figure 12 shows the relative error areas of the calculation results versus the experimental results for bending and torsion of S355J2G1W steel specimens. The maximum error for bending is 14% at σm = 75 MPa and life N > 1.5 × 10 6 cycles. For torsion, the maximum error is 18% for τm = 6090 MPa and life N > 1.5 × 10 6 cycles. For durability N > 2.3 × 10 6 and τm = 75 MPa, the error reaches 20%, although this error covers an insignificant region of the plane of limit stresses.

Conclusions
On the basis of the obtained results of experimental research and the conducted analyses, the following conclusions were formulated: 1. The effect of the mean load value on the allowable stress amplitudes was taken into account by means of a linear relationship, in which the variation of the material's sensitivity coefficient to cycle asymmetry with the number of cycles to failure was taken into account; 2. A mathematical model of limiting stress amplitude surfaces was determined, which was verified by the results of fatigue tests for S355J0 and S355J2G1W steel specimens subjected to cyclic bending and torsional loading with different values of the mean load. For both S355J0 and S355J2GW1 steels in bending, a maximum error of 14% appeared for fatigue life N values greater than 1 × 10 6 cycles and a mean stress value of about 80 MPa. For torsion, similar trends were observed for both materials, with an error of 18%. Compared to bending for torsional loading, the estimation error of more than 14% covered a wider range of the average load in the range of about 40-90 MPa; 3. For both tested materials, it was observed that the greatest error in estimating allowable stress amplitudes occurred for a life of N = 1 × 10 6 cycles, which may have been due to the largest scatter of experimental tests for loads in the high cycle fatigue range; 4. The proposed model proved to be effective in estimating the maximum stress amplitude values for low-cycle fatigue over a wide range of mean loads, where the estimation error was less than 10%.
The algorithm proposed in this paper for determining the values of permissible stress amplitudes provided consistent results with those of the experimental tests. The model can be applied to components subjected to bending as well as torsional loads. This requires that the characteristics for purely alternating and purely pulsating loads are appropriate for the type of loading.
Author Contributions: Conceptualization, R.P. and D.R.; methodology, R.P. and D.R.; formal analysis, R.P. and D.R.; investigation, R.P. and D.R.; writing-original draft preparation, R.P. and D.R. All authors have read and agreed to the published version of the manuscript. The maximum error for bending is 14% at σ m = 75 MPa and life N > 1.5 × 10 6 cycles. For torsion, the maximum error is 18% for τ m = 60 ÷ 90 MPa and life N > 1.5 × 10 6 cycles. For durability N > 2.3 × 10 6 and τ m = 75 MPa, the error reaches 20%, although this error covers an insignificant region of the plane of limit stresses.

Conclusions
On the basis of the obtained results of experimental research and the conducted analyses, the following conclusions were formulated: 1.
The effect of the mean load value on the allowable stress amplitudes was taken into account by means of a linear relationship, in which the variation of the material's sensitivity coefficient to cycle asymmetry with the number of cycles to failure was taken into account; 2.
A mathematical model of limiting stress amplitude surfaces was determined, which was verified by the results of fatigue tests for S355J0 and S355J2G1W steel specimens subjected to cyclic bending and torsional loading with different values of the mean load. For both S355J0 and S355J2GW1 steels in bending, a maximum error of 14% appeared for fatigue life N values greater than 1 × 10 6 cycles and a mean stress value of about 80 MPa. For torsion, similar trends were observed for both materials, with an error of 18%. Compared to bending for torsional loading, the estimation error of more than 14% covered a wider range of the average load in the range of about 40-90 MPa; 3.
For both tested materials, it was observed that the greatest error in estimating allowable stress amplitudes occurred for a life of N = 1 × 10 6 cycles, which may have been due to the largest scatter of experimental tests for loads in the high cycle fatigue range; 4.
The proposed model proved to be effective in estimating the maximum stress amplitude values for low-cycle fatigue over a wide range of mean loads, where the estimation error was less than 10%.
The algorithm proposed in this paper for determining the values of permissible stress amplitudes provided consistent results with those of the experimental tests. The model can be applied to components subjected to bending as well as torsional loads. This requires that the characteristics for purely alternating and purely pulsating loads are appropriate for the type of loading.
Author Contributions: Conceptualization, R.P. and D.R.; methodology, R.P. and D.R.; formal analysis, R.P. and D.R.; investigation, R.P. and D.R.; writing-original draft preparation, R.P. and D.R. All authors have read and agreed to the published version of the manuscript.