# The Effect of Mean Load for S355J0 Steel with Increased Strength

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analyzed Fatigue Models

_{a}(τ

_{a}) and mean stress value σ

_{m}(τ

_{m}). Figure 1 shows the algorithm for calculating fatigue life under bending and torsion for courses with different values of the stress ratio R. In the algorithm, the equivalent stress amplitude was calculated using the Huber–Misess–Hencky hypothesis.

_{ag}, σ

_{mg}—bending stress amplitude and stress mean value respectively, τ

_{as}, τ

_{ms}—torsional stress amplitude and stress mean value respectively, M

_{a}, M

_{m}—amplitude of the moment and mean value of the load moment, respectively, W

_{x}—bending indicator, W

_{0}—torsion indicator, α—angle between bending and torsion moments.

_{m}—ultimate strength, σ

_{m}—mean stress, σ

_{a}—stress cycle amplitude with a mean value σ

_{m}≠ 0, σ

_{aT}—fatigue equivalent stress cycle amplitude (about mean value σ

_{mT}= 0).

_{f}—fatigue strength coefficient, b, c—fatigue strength and ductility exponent, respectively.

_{a}= f(σ

_{m}) and τ

_{a}= f(τ

_{m}), their limit for stress level corresponding to the unlimited life (fatigue limit) is indicated. It was indicated that material sensitivity on mean loading is not a material constant and depends on the number of the cycles corresponding to the failure of an element. The amplitude of the corresponding normal and shear stress components is calculated as follows:

_{−1}—fatigue strength and ductility exponent, respectively fatigue limit at the oscillatory cycle, ψ

_{σ}, ψ

_{τ}—material sensitivity factor for the asymmetry of cycle.

_{σ}for bending is defined as

_{0}, τ

_{0}—fatigue limit at one-sided cycle for bending and torsion.

_{σ}and ψ

_{τ}determined by the relationships (8) and (9) is correct for the fatigue limit. Determining the function of changing the value of the material sensitivity factor for the asymmetry of cycle depending on the number of cycles N allows the assessment of the effect of the mean stress value depending on the material life. By making the factor ψ

_{σ}and ψ

_{τ}dependent on the number of cycles until failure N, in Formulas (8) and (9), stress amplitudes will appear depending on the number of cycles:

_{−1}(N), τ

_{−1}(N)—stress amplitude at oscillatory bending and torsion for a fixed number of N cycles, σ

_{0}(N), τ

_{0}(N)—one-sided cycle stress amplitude under bending and torsion for a fixed number of N cycles.

_{a}and τ

_{a}and mean values of stress σ

_{m}and τ

_{m}with the assumed number of cycles N

_{i}coefficient is adopted, which allows to refer the case of multiaxial loads to uniaxial tension-compression

_{rc}—fatigue limit under cyclic tension-compression, Z

_{i}—fatigue limit expressed by the equivalent stress amplitude for bending, torsion or a combination of bending and torsion for zero mean load, determined using the Huber-Misess-Hencky hypothesis.

## 3. Material and Methods

_{g}

_{α}(t) = M(t)—the specimen is subjected to bending. When α = 90°, we have M

_{g}

_{α}(t) = M(t)—the specimen is subjected to torsion. The combined bending with torsion is applied to specimen for every value of the angle in range 0 < α < 90°. For combined bending with torsion, torsion moment M

_{g}

_{α}(t) is proportional to bending moment M

_{g}

_{α}(t), where proportion is equal to tgα.

α = 0 (rad), (0°) (bending), |

α = 1.107 (rad), (63.5°) (combined bending with torsion), |

${M}_{s\alpha}=2{M}_{g\alpha};{\sigma}_{\alpha}\left(t\right)={\tau}_{\alpha}\left(t\right),$ |

α = π/2 (rad), (90°) (torsion). |

_{ag}and the mean moment value M

_{mg}for bending and M

_{as}and M

_{ms}respectively for torsion. The tests were performed for 4–5 levels of stress amplitudes, for each combination of load at least two or three specimens were used. As a result, for the given load levels, the corresponding fatigue life was obtained. Obtaining fracture of specimen was adopted as the failure criterion for the sample.

_{rc}= 204 MPa and correspond her N

_{0}= 1.12 × 10

^{6}cycles.

## 4. Results and Discussion

_{a})

_{a}(logτ

_{a}), while Table 4 contains the fatigue limits for individual load combinations.

_{m}= 0) and for R = 0 they are 35% lower. During torsion for R = −0.5, the allowable stress amplitudes are about 19% smaller compared to the stress amplitudes for R = −1, while for R = 0 this decrease is from 40% for a small number of cycles and up to 27% for a large number of cycles. For the combination bending with torsion under R = −0.5 the stress amplitudes are reduced by 35% for a small number of cycles and to 27% for a large number of cycles, while for R = 0 the allowable stress amplitudes are reduced for a small and a large number of cycles respectively 37% and 32%.

_{m}and torsional stress τ

_{m}on the change in allowable stress amplitudes σ

_{a}and τ

_{a}. Lines in Figure 5 present the interpolation of the data on the graph.

^{4}cycles, an increase in the mean stress value σ

_{m}in relation to σ

_{m}= 0 causes a decrease in the stress amplitude σ

_{a}to varying degrees, e.g., for R = 0.5 by 22% and for R = 0 by 39%. For durability N = 10

^{5}cycles, the decreases in σ

_{a}are 20% and 30% for R = −0.5 and 0, respectively. These differences for all R values are no longer as large as for N = 10

^{4}cycles. However, in the case of durability at the level of 10

^{6}cycles, there is a decrease in the value of stress amplitude σ

_{a}by an average of 23% in relation to σ

_{m}= 0, while for R = −0.5 and 0 the value of allowable stress amplitude practically does not depend on the value of the mean stress.

^{4}cycles, a significant decrease in stress τ

_{a}due to the increase in value τ

_{m}is also visible, e.g., in relation to loads τ

_{m}= 0 the decrease is 17% and for R = −0.5 and about 40% for R = 0. At durability level N = 10

^{5}cycles these decreases are 20% and 34% respectively. For a large number of cycles, this difference virtually disappears and the decrease in the permissible amplitude values is very similar (23% and 27%).

^{4}cycles, it can be seen that with an increase in average stress, there is a large decrease in the allowable stress amplitudes (about 35%) for both R = −0.5 and R = 0. Similarly, for N = 10

^{5}cycles- the decrease in the allowable stress amplitudes are 31% and 34% for R = −0.5 and R = 0, respectively. For N = 10

^{6}cycles, an increase in the mean stress value reduces the stress amplitude by 32% for R = −0.5 and 27% for R = 0 compared to the case σ

_{m}= τ

_{m}= 0. The small influence of the mean stress value from the durability level N = 10

^{5}cycles is characteristic. A further increase in mean stress practically does not cause major changes in the values of allowable stress amplitudes.

^{4}− 2.5 × 10

^{6}) cycles. The change of the value of the factor ψ depending on the number of cycles N is shown in Figure 6. For bending, this relationship was described by the function ψ

_{σ}(N) = 3.1621N

^{−0.164}. There is a visible decrease in the value of the factor in terms of durability N = (5⋅× 10

^{4}− 7.5⋅× 10

^{5}) cycles. In the case of torsion, the function has the form ψ

_{τ}(N) = 2.897N

^{−0.131}, while the nature of the curve is similar to that for bending. In the case of a combination of bending and torsion, the function takes the form ψ

_{τ}(N) = ψ

_{σ}(N) = 0.854N

^{−0.044}. From this function one can notice a milder course than for bending and torsion, with the largest decreases in the coefficient value also occurring in the durability range N = (5⋅× 10

^{4}− 7.5⋅× 10

^{5}) cycles.

_{i}(16) coefficients were determined assuming the fatigue limits of individual types of loads at σ

_{m}= τ

_{m}= 0 (Table 4) respectively:

- for bending$${K}_{g}=\frac{{Z}_{rc}}{{\sigma}_{-1}};\sigma {-}_{1}=271\mathrm{MPa},$$
- for torsion$${K}_{s}=\frac{{Z}_{rc}}{\sqrt{3\cdot}{\tau}_{-1}};\tau {-}_{1}=152\mathrm{MPa},$$
- for bending and torsion$${K}_{gs}=\frac{{Z}_{rc}}{\sqrt{{\sigma}_{-1}^{2}+3\cdot {\tau}_{-1}^{2}}};\sigma {-}_{1}=\tau {-}_{1}=175\mathrm{MPa}.$$

_{aTi}were determined from Equation (17), computational stability N

_{cal}according to Equation (18), which is the fatigue characteristics of the tested steel under uniaxial tension-compression (where there is an even distribution of stress across the specimen cross-section).

_{cal}and the experimental N

_{exp}, while the dashed lines represent the scatter bands of results with the coefficients N

_{cal}/N

_{exp}= 3 (1/3).

_{cal}/N

_{exp}= 3 (1/3) coefficient for all three cases. It should be noted, however, that this model underestimates computational durability compared to experimental. The remaining transformation formulas clearly underestimate the calculated fatigue life; the calculation results are outside the scatter band N

_{ca}

_{l}/N

_{exp}= 3 (1/3).

## 5. Conclusions

- In the case when R = −1 satisfactory results of the fatigue life assessment for bending, torsion and a combination of bending and torsion were obtained using the K
_{i}coefficient, which is the ratio of the fatigue limit at cyclic tension-compression to the fatigue limit, calculated in accordance with the Huber–Misess–Hencky hypothesis for analyzed loads. - For loads with a non-zero mean value, it was found that:
- satisfactory results of the fatigue life assessment for bending were obtained using the Gerber transformation dependence and the CASF model described by Equation (14),
- in the case of torsion, the best compatibility of computational fatigue life with experimental life is obtained by using Equation (15) and Morrow’s formula,
- for the combination of bending and torsion for both R = −0.5 and R = 0, the CASF model gives the best results of fatigue life assessment.

- The CASF model described by Equations (14) and (15), including the effect of the mean stress value on fatigue life using the material sensitivity factor for the asymmetry of cycle, gives good results for predicting fatigue life for all tested load cases.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**The dependence of the stress amplitude on the mean stress for: (

**a**) bending, (

**b**) torsion, (

**c**) bending with torsion.

**Figure 8.**Comparison of the computational life N

_{cal}with the experimental N

_{exp}under bending for: (

**a**) R = −0.5, (

**b**) R = −0.

**Figure 9.**Comparison of the computational life N

_{cal}with the experimental N

_{exp}under torsion for: (

**a**) R = −0.5, (

**b**) R = −0.

**Figure 10.**Comparison of the computational life N

_{cal}with the experimental N

_{exp}under bending and torsion for: (

**a**) R = −0.5, (

**b**) R = −0.

C | Mn | Si | P | S | Cr | Ni | Cu | Fe |
---|---|---|---|---|---|---|---|---|

0.21 | 1.46 | 0.42 | 0.019 | 0.046 | 0.09 | 0.04 | 0.17 | Bal. |

R_{e},MPa | R_{m},MPa | A_{10},% | Z, % | E, GPa | v | σ′_{f},MPa | b | c | n′ | K′ MPa |
---|---|---|---|---|---|---|---|---|---|---|

357 | 535 | 21 | 50 | 210 | 0.30 | 782 | −0.118 | −0.410 | 0.287 | 869 |

R | Bending | Torsion | Bending and Torsion | ||||||
---|---|---|---|---|---|---|---|---|---|

B | A | r | B | A | r | B | A | r | |

−1 | 23.93 | −7.19 | −0.97 | 32.81 | −11.82 | −0.94 | 29.73 | −10.62 | −0.98 |

−0.5 | 23.71 | −7.40 | −0.96 | 27.60 | −10.01 | −0.93 | 36.73 | −14.67 | −0.99 |

0 | 31.40 | −10.73 | −0.96 | 59.76 | −25.25 | −0.95 | 31.62 | −12.42 | −0.94 |

Moment and Stress | Bending | Bending and Torsion | Torsion |
---|---|---|---|

α = 0° | α = 63.5° | α = 90° | |

M_{a} (N⋅m) | 13.59 | 16.82 | 17.53 |

σ_{−1} (τ_{−1}) (MPa) | σ_{−1} = 271 | σ_{−1} = τ_{−1} = 152 | τ_{−1} = 175 |

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**MDPI and ACS Style**

Pawliczek, R.; Rozumek, D.
The Effect of Mean Load for S355J0 Steel with Increased Strength. *Metals* **2020**, *10*, 209.
https://doi.org/10.3390/met10020209

**AMA Style**

Pawliczek R, Rozumek D.
The Effect of Mean Load for S355J0 Steel with Increased Strength. *Metals*. 2020; 10(2):209.
https://doi.org/10.3390/met10020209

**Chicago/Turabian Style**

Pawliczek, Roland, and Dariusz Rozumek.
2020. "The Effect of Mean Load for S355J0 Steel with Increased Strength" *Metals* 10, no. 2: 209.
https://doi.org/10.3390/met10020209