# Mean Stress Correction in Low Cycle Fatigue Life of High Strength Steels

## Article information

## Abstract

Fatigue strength of welded joints varies depending on the shape, filler metal, welding procedure, and post-treatment method. In addition, mean stress affects the fatigue behavior in terms of stress ratio. Typically, it is well known that tensile mean stress tends to decrease fatigue life while compressive mean stress increase fatigue life. There are various methods to incorporate the mean stress effect in fatigue strength, but there are relatively few correction methods in compressive mean stress, especially in low cycle fatigue regime.

In this study, a load-based fatigue test was conducted to evaluate the effect of mean stress on the welded joints of High Yield strength steel(HY Steel) with T-joint shape. The effect of the mean stress on the welded part of HY Steel with a T-joint shape was investigated using stress-based mean stress correction methods and strain-based mean stress correction methods. When the effective mean stress approach, which can correct compressive mean stress and tensile mean stress, is applied to HY Steel with a T-joint shape, it shows accurate correction result than stress-based mean stress correction method and strain-based mean stress correction method. Sensitivity parameters to consider mean stress for HY steel are presented.

**Keywords:**High yield strength steel; T-joint; Mean stress correction; Effective mean stress approach

## 1. Introduction

HY80 and HY100 used in this study are classified as high-strength steels with yield strengths of 80 and 100 ksi or higher, respectively. They can contribute to weight reduction by meeting the required standard strength even with a lower thickness compared to typical steels. For high-strength steels, however, it is known that fatigue strength does not increase in proportion to the increase in static strength. Therefore, it is important to evaluate fatigue strength considering the thickness reduction.

In general, low cycle fatigue tests are conducted through strain control in accordance with ASTM E606, and the generated mean strain does not adversely affect fatigue life when it is not accompanied by mean stress1). It also does not adversely affect fatigue life when the mean stress is zero in the case of high cycle fatigue tests. It is known, however, that fatigue life decreases under tensile mean stress and it tends to increase under compressive mean stress. Among the methods to quantitatively correct the effect of mean stress on fatigue life, the commonly used traditional methods have limited applicability to sections under compressive mean stress.

In previous studies that reported that traditional correction methods derive inaccurate results, fatigue tests were conducted in stress ratio sections of R=-1, 0, and 0.08<R<0.6 for aluminum alloys 356-T6 and 356-T7, and the results corrected using the Goodman equation, Gerber equation, Soderberg equation, ASME-Elliptic equation, SWT equation, and Morrow equation were compared with the experiment results at R=-1. The error of SWT ranged from -23 to 44%, which was relatively close to the experimental values. The other five equations, however, exhibited an error rate of 134 to 2,362%, showing a tendency to overestimate fatigue life2).

Meanwhile, there is a lack of research on mean stress correction methods that can correct compressive and tensile mean stress considering the application of actual environmental factors or the purpose of use. As such, in this study, the results of applying various mean stress correction methods were compared and analyzed for HY80 and HY100. In particular, an attempt was made to propose a method for evaluating the fatigue strength by mean stress in the compression zone.

## 2. Mean Stress Correction Methods

### 2.1 Stress-based mean stress correction methods

The Soderberg equation, Walker equation, and Morrow equation among others were presented as traditional methods to consider the mean stress effect in stress- based fatigue strength assessment3-6). Among the traditional methods, the Goodman equation and Gerber equation, which are methods to correct mean stress using tensile strength as a coefficient, and the Soderberg equation and ASME-elliptic equation, which are methods to correct mean stress using yield strength as a coefficient, were used. They are expressed as equations (1) to (4), respectively6-9).

where *σ _{a,} σ_{ar}*,

*σ*and

_{m,}σ_{u}*σ*are the stress amplitude, equivalent stress amplitude, mean stress, tensile strength, and yield strength, respectively.

_{y}Equations (1) to (4) were expressed in Haigh diagrams in Fig. 1 and Fig. 2. *σ _{a}* tends to decrease under tensile mean stress in equation (1) to (4). Equations (1) and (3) neglected the effect of the mean stress when the compressive mean stress acts, and equation (2) and (4) cannot be used in the section where the compressive mean stress acts. The green area in Fig. 1 and Fig. 2 where the fitted curve overlaps with yield strength is defined as a safe area with a fatigue limit. The yellow area is defined as an area that does not have a fatigue limit due to the occurrence of failure.

Equation (1) is recommended for brittle materials, and it tends to be excessively conservative. Equation (2) is recommended for ductile materials, and it is not conservative. Equation (3) tends to be more conservative than equation (1), and it is recommended for brittle materials. Equation (4) may represent fatigue damage when the specimen is not actually damaged.

### 2.2 Strain-based mean stress correction methods

Hatanaka-Fujimitsu-Yamada, NK-curve, Iida and Fujii, Truchon, Sarma-Padmanabhan-Jaeger-Koethe have presented strain-based low cycle fatigue test results for various materials in Coffin-Manson form or Strain-N form. They are expressed as equations (5) to (9), respectively10-14).

where Δ*ε _{t,} N_{f,} σ_{u}, E*, and

*ε*are the total strain, fatigue life, tensile strength, elastic modulus, and failure ductility coefficient.

_{f}The Δ*ε _{t,-}N_{f,}* curve appears as a straight line for Hatanaka-Fujimitsu-Yamada and NK-curve and in the form of Coffin-Manson for Iida and Fujii, Truchon, and Sarma-Padmanabhan-Jaeger-Koethe.

In addition, mean stress correction methods for strain- based fatigue tests include the Morrow equation and Modified equation, which are expressed as equations (10) and (11)15).

where *ε _{a,} σ_{f}’, E, σ_{m,} N_{f,} b, ε_{f}*’and c are the strain amplitude, fatigue strength coefficient, elastic modulus, mean stress, fatigue life, fatigue strength exponent, fatigue ductility coefficient, and fatigue ductility exponent16).

### 2.3 Effective mean stress approach

Hensel performed an experiment under various stress ratios and a mean stress of 100 MPa for longitudinal reinforcement that used S355NL and S960QL, which are structural steels. Fig. 3 shows the geometry of the fatigue test specimen and fitting curve by Hensel are shown in Fig. 4 as a haigh diagram. The effective mean stress approach presented by Hensel estimates fatigue strength using effective mean stress, which is the sum of tensile/compressive mean stress and the residual stress, and sensitivity to mean stress that represents the sensitivity of fatigue strength to mean stress17,18).

The presented formulas were shown in equations (12) to (17) in order of the stabilized residual stress, effective mean stress, sensitivity to mean stress and effective mean stress approach.

In equation (12), *σ _{RS,N=}*

_{10,000},

*σ*,

_{RS,N=0}*σ*are the residual stress at a fatigue life of 10,000 cycles, residual stress at a fatigue life of 0 cycle, highest load stress, and yield strength. The highest load stress appears as the maximum stress under tensile residual stress and as the minimum stress under compressive residual stress. In equation (13),

_{LS,}, σ_{y}*σ*,

_{m,eff}*σ*and

_{m}*σ*are the effective mean stress, mean stress, and residual stress stabilized at a fatigue life of 10,000 cycles, respectively. In equations (14) to (16),

_{RS,stab}*m*

^{*},

*σ*, and

_{a,R(R=x)}*σ*are the sensitivity to mean stress, fatigue strength amplitude at a certain stress ratio, and mean stress. As the sensitivity to mean stress is closer to zero, fatigue strength is less affected by mean stress. As it is closer to 1, fatigue strength is more affected by mean stress. In equation (17),

_{m(R=x)}*σ*(

_{a,R}*σ*),

_{m}; σ_{RS}*σ*(

_{a,R}*σ*),

_{m}=0; σ_{RS}=0*m*

^{*,}

*σ*are the fatigue strength amplitude in consideration of mean stress and residual stress at a stress ratio other than R=-1, fatigue strength amplitude at R=-1, sensitivity to mean stress, and effective mean stress.

_{m,eff}The effective mean stress approach derives fatigue strength when considering the sensitivity to mean stress and effective mean stress which is the sum of the mean stress and residual. It can also estimate fatigue strength in sections under compressive and tensile mean stress.

## 3. Fatigue Test

### 3.1 Materials

The steels used in this study are HY80 and HY100, and T-joint shaped specimens were prepared using submerged arc welding (SAW). SAW has a high deposition rate, but it causes welding deformation and residual stress due to the welding heat input20,21). A fatigue test was conducted to derive fatigue life for the T-joint shaped weld zone subjected to SAW. Table 1 summarizes mechanical properties while Fig. 5 shows the specimen geometry and dimension.

The T-joint shaped specimens were subjected to the load-controlled fatigue test. To apply strain-based mean stress correction methods, a low cycle fatigue test was conducted at a strain ratio of -1 for the round bar specimens of HY80 and HY100 prepared in accordance with ASTM E606. Fig. 6 shows the specimen geometry while Table 2 lists the dimensions of the specimens.

### 3.2 Load-controlled fatigue test results

The fatigue test was conducted using Instron’s IST- 8800. The test was conducted 24 times (12 times at R=0.1 where tensile mean stress applies and 12 times at -1.2<R<-0.2). Fig. 7 and Fig. 8 show the test results of HY80 and HY100, respectively.

At R=0.1, a typical tendency that fatigue life decreases alongside the increase in *Δσ* was observed as expected. Results of R=-1.16 increased fatigue life than results of R=-0.98 and R=-0.8. Therefore, in the case of -1.2<R<-0.2, fatigue life tended to increase as the magnitude of tensile mean stress decreased and compressive mean stress was applied.

For T-joint shaped HY80 and HY100, the fatigue limit at 2 million cycles was derived using the experiment results at R=0.1. The results are shown in Fig. 9 and Fig. 10. Table 3 summarizes the derived fatigue limit.

### 3.3 Strain-controlled fatigue test results

The strain-controlled fatigue test was conducted using Istron’s IMT-8803 for the round bar specimens of HY80 and HY100. Table 4 summarizes the test conditions.

The fatigue test was conducted once for each specimen in the 0.6-1.2% strain range, and the results are shown in Fig. 11.

## 4. Results of Applying Mean Stress Correction Methods

### 4.1 Stress-based mean stress correction methods

At R=0.1, the fatigue limits of HY80 and HY100 at 2 million cycles were derived from the fatigue test results. Since the fatigue limits tested at -1<R<-0.2 could not be directly compared, the fatigue strength amplitude (*σ _{aR}*) at R=-1 was derived by applying equations (1) to (4) to the experimental data, and mean stress was corrected with R=0.1. The fatigue limit amplitude at R=0.1 calculated from the experiment results was compared with the fatigue limits derived using four traditional methods in the S-N curves in Fig. 12 and Fig. 13. The specific values are summarized in Tables 5 and 6.

After the test at a stress ratio of R=0.1, the fatigue limit of HY80 at 2 million cycles was derived to be 131.9 MPa. In equations (1) to (4), however, 121.4, 161.1, 99.3, and 166.3 MPa were derived, respectively. The value corrected using the Soderberg equation was very close to the test value, but the other three methods showed relatively large differences.

After the test at a stress ratio of R=0.1, the fatigue limit of HY100 at 2 million cycles was derived to be 106.6 MPa. 97.1, 132.9, 88.0, and 147.2 MPa were derived using equations (1) to (4), respectively. They were not close to the experimental value when traditional methods were applied to HY80 and HY100, only the Goodman equation of HY80 tended to be close. The traditional methods showed tendencies close to the experimental values when applied to T-joint shaped HY80 and HY100, but the correctionaccuracy varied depending on the steel and geometry compared to a previous study2). In addition, they cannot be applied to sections under compressive mean stress depending on the method.

### 4.2 Strain-based mean stress correction methods

To convert the results of the load-based fatigue test conducted in this study into *Δε* for low cycle fatigue life assessment, cyclic stress-strain curves were derived by applying the Ramberg-Osgood relation to the fatigue test results at strains of 0.6, 0.8, 1.0, and 1.2% for the round bar specimens of HY80 as shown in Fig. 14. *Δε* was derived by applying the Neuber’s rule to the derived cyclic stress-strain curves. The Ramberg-Osgood relation and the Neuber’s rule are expressed as equations (18) and (19).

where ε, σ, *E*, *H*, *n*’, *k _{t}*, and

*S*are the true strain, true stress, elastic modulus, cyclic strength coefficient, cycilc strain hardening exponent, stress concentration factor, and nominal stress. 2.4 was used as the stress concentration factor

*k*for T-joint based on DNV 30.722).

_{t}The cyclic stress-strain curves of HY80 and HY100 that applied equation (18) were summarized in equations (20) and (21).

∆ε was derived after obtaining the difference between the maximum strain and the minimum strain by applying the Neuber’s rule to the cyclic stress-strain curves of the round bar specimens of HY80 and HY100 as shown in Fig. 15. The results of deriving ∆ε and *R _{eff}*for T-joint shaped HY80 and HY100 are shown in Fig. 16 and Tables 7 and 8.

HY80 and HY100 tested at R=0.1 have a strain ratio of 0< *R _{ε}*<0.03, which is larger than

*R*=-1 for equation (5)~(9), when stress is converted into strain using the Neuber’s rule and Ramberg-Osgood relation. Therefore, the results of correcting the mean stress by applying the Morrow equation and Modified Morrow equation were compared with the round bar data for HY80 and HY100 as shown in Fig. 17 and Fig. 18.

_{ε}For both HY80 and HY100, the correction results of the Modified Morrow method were higher than those of the Morrow method, and they tended to appear in the upper part of the *ε _{a}* -N curve. This appears to be because the effect of plastic strain on mean stress was removed in the Modified Morrow method. When the curves of the round bar specimens of HY80 and HY100 were compared with the corrected data of T-joint shaped HY80 and HY100, it was found that the Modified Morrow method was closer to the round bar data than the Morrow method.

Fig. 19 and Fig. 20 compare the equation (5)~(9) with T-joint shaped HY80 and HY100 on the *ε _{a}* -N curve.

The coefficient and exponent of the Coffin-Manson equation for T-joint shaped HY80 and HY100 are summarized in Tables 9 and 10.

In the case of the curves of HY80 and HY100 that applied the Morrow method, the transition life defined as the point of intersection between elastic strain and plastic strain occurred at 1,244 and 70 cycles, respectively. When the Modified Morrow method was applied, the fatigue transition life occurred at 124 and 8 cycles, respectively. HY100 tended to show a shorter fatigue transition life than HY80.

The Morrow method in Fig. 19 and Fig. 20 considered both the mean stresses of the elastic and plastic regions, but the Modified Morrow method considered only the mean stress of the elastic region. Therefore, the Modified Morrow method had an increased proportion of the elastic strain region, and it was located higher than the Morrow method.

When the Morrow method and Modified Morrow method, which are strain-based mean stress correction methods, were compared with equation (5)~(9), conservative curves of HY80 and HY100 were derived in *N _{f}* <10

^{4}, which is generally defined as a low cycle region.

### 4.3 Effective mean stress approach

In this study, it was assumed that the effect of residual stress was included in mean stress to consider only the effect of mean stress. Therefore, the condition that effective mean stress is identical to mean stress was derived. The fatigue limit and mean stress at 2 million cycles derived by testing HY80 and HY100 at R=1.0 are summarized in Table 11.

When the above *σ _{a,R}* and

*σ*were expressed together with the curve presented by Hensel, similar tendencies were observed as shown in Fig. 21 and Fig. 22.

_{m}HY steel has lower mechanical properties than S960QL used by Hensel. When the presented sensitivity to mean stress was applied, the fatigue limit data of HY80 and HY100 tended to be close to the curve presented by Hensel in Fig. 21 and Fig. 22. This indicates that the effective mean stress approach presented by Hensel can be used.

*σ _{a,R}* was estimated to be 82.1 MPa for HY80 and 66.3 MPa for HY100 at R=-1 by substituting the fatigue limits of HY80 and HY100 tested at R=0.1 and the sensitivity to mean stress

*m** of 0.2 into equation (17). Since sensitivity to mean stress is zero in the

*R*>0.5section, the impact of sensitivity to mean stress in the -∞≤

_{eff}*R*≤0.5 section was observed. A sensitivity to mean stress of 0.4 was used in the

_{eff}*R*< -1 section and a sensitivity to mean stress of 0.2 in -1≤

_{eff}*R*≤0.5, and the tendency in the -300 MPa ≤

_{eff}*σ*≤ 120 MPa section is shown in Fig. 23 and Fig. 24. It was found that the data of T-joint shaped HY80 and HY100 tested in this study were not consistent with the sensitivity to mean stress presented by Hensel.

_{m}In Fig. 23 and Fig. 24, the tendency that fatigue strength increases in R<-1 where compressive mean stress is applied and decreases in 0<R≤0.5 where tensile mean stress is applied was observed.

To derive the sensitivity to mean stress suitable for HY80 and HY100, the sensitivity to mean stress closest to the curve presented by Hensel was selected by substituting the fatigue limit amplitude *σ _{a,R}* and the mean stress

*σ*at R=0.1 into equation (17) and changing the sensitivity to mean stress. Consequently, 0.51 was derived in R<-1 and 0.25 in -1≤R≤0.5 for HY80 and HY100 as shown in Table 12. The curves that used the sensitivity to mean stress suitable for HY80 and HY100 are shown in Fig. 25 and Fig. 26.

_{m}To present sensitivity to mean stress for high-strength steels, the fatigue limit amplitude *σ _{a,R}* and the mean stress

*σ*were derived at 2 million cycles by integrating HY80 and HY100 as shown in Table 13. In addition, the S-N curve is shown in Fig. 27.

_{m}Fig. 28 shows the results of applying the sensitivity to mean stress presented by Hensel after substituting the fatigue limit amplitude and mean stress into equation (17). When compared with the curve presented by Hensel, inconsistent curve were obtained.

Therefore, in this study, the sensitivity to mean stress for high-strength steels was presented by integrating HY80 and HY100 subjected to the load-controlled fatigue test at R=0.1 as shown in Table 14. Fig. 29 shows the results of applying the presented sensitivity to mean stress to HY steel.

In this study, the sensitivity to mean stress for T-joint shaped high-strength steels was presented as 0.51 in R<-1, 0.25 in -1≤R≤0.5, and 0 in 0.5<R. In all sections, *m*^{*} of high-strength steels was higher than *m*^{*} presented by Hensel. The T-joint shaped high-strength steels used in this study are more affected by mean stress than S355NL and S960QL used by Hensel in all sections.

To compare mean stress correction methods that are close to the curve presented by Hensel, the fatigue limits corrected with R= -1 after applying the traditional methods and effective mean stress approach to HY80 and HY100 are shown in Fig. 30 and Fig. 31. Each value is summarized in Tables 15 and 16.

For HY80 and HY100, a fatigue limit amplitude of approximately 70.0 MPa was derived. When the traditional methods were applied, however, 98.0 to 106.0 MPa was derived for HY80 and 87.0 to 94.0 MPa for HY100.

The effective mean stress approach results were closer to the curve presented by Hensel than the traditional methods. Therefore, effective mean stress can derive more accurate correction results than the traditional methods.

In Fig. 32 and Fig. 33, the compressive mean stress section of -10 to -300 MPa was corrected using the traditional methods for HY80 and HY100, and the corrected results were compared with the effective mean stress approach results.

The Goodman equation and Soderberg equation exhibited very conservative correction results. For the Gerber equation and ASME-Elliptic equation, the fatigue strength amplitude tended to decrease in the section under compressive mean stress, indicating that the equations cannot be applied in the section under compressive mean stress.

Based on this, it was found that the effective mean stress approach derived more accurate mean stress correction results than the traditional methods. In particular, it exhibited more accurate correction results in the compression area.

## 5. Conclusions

In this study, load-based fatigue tests were conducted after preparing T-joint shaped specimens using high- strength steels of HY80 and HY100, and traditional mean stress correction methods, strain-based mean stress correction methods, and the mean stress correction method presented by Hensel were applied. The following conclusions could be drawn.

1) When traditional mean stress correction methods applicable in the section under tensile mean stress were applied for HY80 and HY100, tendencies close to the experimental fatigue limit were observed. Considering the results of a previous study2), however, the corrected values of the traditional methods were not close to the experimental values depending on the material and geometry.

2) The Morrow method considers both the mean stresses of the elastic and plastic regions, but the Modified Morrow method considers only the mean stress of the elastic region. Therefore, when strain-based mean stress was applied for T-joint shaped HY80 and HY100, the Modified Morrow method curve was located higher than the Morrow method curve. In addition, both methods exhibited a more conservative tendency than equation (5)~(9)

3) The sensitivity to mean stress of HY80, HY100, HY80&HY100 was presented using the effective mean stress approach. Its application can derive more accurate correction results than the traditional methods. In particular, it can correct mean stress in the compression area.

## Acknowledgment

This work was supported by the basic research support program (2 years) of Pusan National University.