Mean stress effect | FatigueToolbox.org

The effect of mean stresses can be handled in different ways depending on the choices made under Setup SN curve as described in the following. In all cases the knee point stress range of the SN curve is shifted by some amount depending on the mean stress of the cycle being treated. A selection of implemented mean stress corrections is shown in Fig. 1.

Fig. 1: Some of the mean stress corrections implemented in Fatlab.

For compressive mean stresses the user is presented with the option to extrapolate the correction line or not, as illustrated by the dashed lines in the figure. It is of course more conservative not to do this, however many experiments have shown it is safe [2].

All calculations in Fatlab are carried out using stress ranges $\Delta\sigma$ , however, in the particular case of mean stress handling, the matter is better explained in terms of stress amplitudes $\sigma$ .

The interested reader/user may refer to mean_stress.m for the implementation of mean stress handling in Fatlab.

In the following $\sigma_{R,-1}$ refers to the fatigue strength amplitude at the knee point at fully reversed loading, i.e. $R=-1$ . Then, $\sigma_{R,m}$ is the corrected value at a given mean stress and $\sigma_u$ is the ultimate tensile stress (Rm) and $\sigma_y$ is the yield stress (Re).

The mean stress associated with experimental data is usually given through the stress ratio $R$

$\displaystyle R = \frac{\sigma_{min}}{\sigma_{max}}$

Some mean stress corrections also operate on $R$ . However, for calculations in Fatlab, it was found expedient to express all mean stress corrections in terms of mean stress $\sigma_m$ . The equations below are therefore converted in some cases.

No mean stress correction

Selecting this (default) option will leave the SN curve unaffected by the mean stress level. This is recommended in case of e.g. welded or bolted joints, for which the effect of mean stresses is already included in the SN curve.

Linear

The linear mean stress correction reduces the allowable stress range depending on a single parameter called the mean stress sensitivity.

$\sigma_{R,m} = \sigma_{R,-1} -M\sigma_m$

The mean stress sensitivity $M$ is determined from experiments and is defined as

$\displaystyle M = \frac{\sigma_{R,-1}}{\sigma_{R,0}}-1$

Where $\sigma_{R,0}$ is the fatigue strength amplitude at $R=0$ . The value of $M$ is typically in the range of $0.2$ to $0.4$ .

Bilinear

The bilinear setting is identical to the linear for $R<0$ . For $R>0$ , on the other hand it follows a straight line down to $Rm$ similar the to Goodman concept. This is the principle used in [1].

$\displaystyle \sigma_{R,m} = \left\{ \begin{matrix} \sigma_{R,-1} -M\sigma_m & for & \sigma_m \leq \sigma_{R,0}\\ \displaystyle\frac{\sigma_{R,0}(\sigma_u-\sigma_m)}{\sigma_u-\sigma_{R,0}} & for & \sigma_m > \sigma_{R,0}\\ \end{matrix} \right.$

where $\displaystyle \sigma_{R,0} = \frac{\sigma_{R,-1}}{(M+1)}$

Modified Goodman

The Modified Goodman correction is equivalent to the Linear mean stress correction in that the reduction of the allowable stress range is linear. Indeed, $M$ can be selected in such a way, that the two corrections are identical.

$\displaystyle\frac{\sigma_{R,m}}{\sigma_{R,-1}} + \frac{\sigma_m}{\sigma_u} = 1$

The fatigue strength at $\sigma_m$ is isolated

$\displaystyle \sigma_{R,m} = \sigma_{R,-1}\left(1-\frac{\sigma_m}{\sigma_u}\right)$

This option is recommended for high-strength/low-ductility materials in [2].

Soderberg

The Soderberg equation is given as

$\displaystyle\frac{\sigma_{R,m}}{\sigma_{R,-1}} + \frac{\sigma_m}{\sigma_y} = 1$

It describes as straight line between the fully reversed fatigue strength on the secondary axis and the yield strength on the primary axis. it can be rewritten as follows to express the fatigue strength at a given mean stress

$\displaystyle \sigma_{R,m} =\sigma_{R,-1} \left(1-\frac{\sigma_m}{\sigma_y}\right)$

Gerber parabola

The Gerber correction is mathematically similar to the Modified Goodman correction, except that the last term is squared, causing it to describe a parabola in the Haigh diagram. It is thus less conservative compared to the Modified Goodman correction in the tensile region.

$\displaystyle\frac{\sigma_{R,m}}{\sigma_{R,-1}} + \left(\frac{\sigma_m}{\sigma_u}\right)^2 = 1$

Again, the mean stress corrected fatigue strength is isolated

$\displaystyle \sigma_{R,m} = \sigma_{R,-1}\left(1-\frac{\sigma_m^2}{\sigma_u^2}\right)$

This option is recommended for “reasonably ductile” materials [2].

Smith-Watson-Topper

Smith, Watson and Topper defined an equivalent stress amplitude as [3]

$\sigma_{a,eq} = \sqrt{\sigma_{max}\sigma_a} = \sqrt{(\sigma_m+\sigma_a)\sigma_a}$

Inserting fatigue strength and rewriting to find the mean stress corrected fatigue strength

$\sigma_{R,-1} = \sqrt{(\sigma_m+\sigma_{R,m})\sigma_{R,m}}$

we get

$\displaystyle \sigma_{R,m} = -\frac{1}{2}\left( \sigma_m - \sqrt{\sigma_m^2+4\sigma_{R,-1}^2} \right)$

This correction only works in the tensile region.

60% compression rule

Some codes, e.g. EC3 allows using only part of the compressive stresses under the condition, that the component under investigation is not affected by detrimental tensile residual stresses, e.g. stress relieved welded joints.

$\Delta\sigma_{EC3} = |\sigma_{max}| + 0.6\cdot |\sigma_{min}|$

In Fatlab, this principle is incorporated as a modification to the SN curve instead of a modification on the stress range, i.e. by scaling up the SN curve under compressive mean stresses.

$\sigma_{R,m} = \left\{ \begin{matrix} \ \sigma_{R,c} & for & \sigma_m \leq -\sigma_{R,c}\\ \ 1.25\sigma_{R,-1}-0.5\sigma_m & for & -\sigma_{R,c} < \sigma_m < \sigma_{R,0}\\ \ \sigma_{R,-1} & for & \sigma_m > \sigma_{R,0}\\ \end{matrix} \right.$

Where $\sigma_{R,c}=\sigma_{R,-1}/0.6$ is the maximum fatigue strength obtainable under fully compressive loading. This option is of course recommended when a code calls for it.

IIW medium/low residual stresses (RS)

The IIW recommendations [4] has a slightly different approach compared to EC3. It offers an enhancement factor f(R) which scales up the fatigue strength under partially compressive loads. The enhancement factor depends on the following catergories:

I: low residual stress: unwelded base material, stress relieved welded joints
f(R) up to 1.6
II: medium residual stress: small scale, thin welded joints with no constraints in assembly
f(R) up to 1.3
III: high residual stress: normal welded joints
f(R) = 1.0

FKM guideline

Mean stress correction according to the FKM guideline [5] is divided into 4 zones.

$\displaystyle \sigma_{R,m} = \left\{ \begin{matrix} \sigma_{R,-\infty} & for & \sigma_m \leq -\sigma_{R,-\infty}\\ \sigma_{R,-1} - M\sigma_m & for & -\sigma_{R,-\infty} < \sigma_m \leq \sigma_{R,0}\\ \sigma_{R,0} (1+\frac{M}{3}) - \frac{M}{3} \sigma_m & for & \sigma_{R,0} < \sigma_m \leq \sigma_{R,0.5}\\ \sigma_{R,0.5}/3; & for & \sigma_m > \sigma_{R,0.5}\\ \end{matrix} \right.$

Where $\displaystyle \sigma_{R,-\infty} = \frac{\sigma_R}{(1-M)}$ and $\displaystyle \sigma_{R,0.5} = \sigma_{R,0} \frac{(M+3)}{M+1}$ .

References

[1] Gudehus & Zenner, Leitfaden für eine Betriebsfestigkeitsrechnung, 4th ed, 2004.

[2] Schijve J., Fatigue of structures and materials, 2nd ed., Springer, 2009.

[3] Dowling, Mean Stress Effect in Stress-Life and Strain-Life Fatigue. In: SAE Paper No. 2004-01-2227, 2004.

[4] Hobbacher A., IIW recommendations for fatigue design of welded joints and
components, IIW-doc. XIII-2460-13, 2013.

[5] FKM-Richtlinie: Rechnerischer Festigkeitsnachweis für Maschinenbauteile aus Stahl, Eisenguß- und Aluminiumwerkstoffe, 4. Ausgabe. VDMA-Verlag, Frankfurt/M. 2002