# Stress-Life (S-N) Approach

## S-N Curve

The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure. Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).
When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude ${S}_{a}$ or range ${S}_{R}$ versus cycles to failure $N$ , the relationship between $S$ and $N$ can be described by straight line segments. Normally, a one or two segment idealization is used. (1)
$S=S1{\left({N}_{f}\right)}^{b1}$

for segment 1

Where, $S$ is the nominal stress range, ${N}_{f}$ are the fatigue cycles to failure, ${b}_{l}$ is the first fatigue strength exponent, and $SI$ is the fatigue strength coefficient.

The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as high-cycle fatigue.

S-N curve data is provided for a given material on a MATFAT Bulk Data Entry. It is referenced through a Material ID (MID) which is shared by a structural material definition.

## Equivalent Nominal Stress

Since S-N theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the S-N curve.

Various stress combination types are available with the default being "Absolute maximum principle stress". "Absolute maximum principle stress" is recommended for brittle materials, while "Signed von Mises stress" is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value.

Parameters affecting stress combination may be defined on a FATPARM Bulk Data Entry. The appropriate FATPARM Bulk Data Entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information Entry.

## Mean Stress Correction

Generally, S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully-reversed, and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to take into account the effect of non-zero mean stresses.

The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:

Gerber:(2)
${S}_{e}=\frac{{S}_{r}}{\left(1-{\left(\frac{{S}_{m}}{{S}_{u}}\right)}^{2}\right)}$
Goodman:(3)
${S}_{e}=\frac{{S}_{r}}{\left(1-\frac{{S}_{m}}{{S}_{u}}\right)}$
Where,
${S}_{m}$
Mean stress given by ${S}_{m}=\left({S}_{max}+{S}_{min}\right)/2$
${S}_{r}$
Stress Range given by ${S}_{r}={S}_{max}-{S}_{min}$
${S}_{e}$
Stress range after mean stress correction (for a stress range ${S}_{r}$ and a mean stress ${S}_{m}$ )
${S}_{u}$
Ultimate strength

The Gerber method treats positive and negative mean stress correction in the same way that mean stress always accelerates fatigue failure, while the Goodman method ignores the negative means stress. Both methods give conservative result for compressive means stress. The Goodman method is recommended for brittle material while the Gerber method is recommended for ductile material. For the Goodman method, if the tensile means stress is greater than UTS, the damage will be greater than 1.0. For the Gerber method, if the mean stress is greater than UTS, the damage will be greater than 1.0, with either tensile or compressive.

A Haigh diagram characterizes different combinations of stress amplitude and mean stress for a given number of cycles to failure.

Parameters affecting mean stress influence may be defined on a FATPARM Bulk Data Entry. The appropriate FATPARM Bulk Data Entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information Entry.

FKm:

If only MSS2 field is specified for mean stress correction, the corresponding Mean Stress Sensitivity value ( $M$ ) for Mean Stress Correction is set equal to MSS2. Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( $R={S}_{\mathrm{min}}/{S}_{\mathrm{max}}$ ) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.
Note: The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( ${S}_{e}^{A}$ ). The actual value of stress used in the Damage calculations is the Corrected stress range (which is $2\cdot {S}_{e}^{A}$ ). These equations apply for SN curves input on the MATFAT entry (by default, any user-defined SN curve is expected to be input for a stress ratio of R=-1.0). For FKM equations applicable to spot weld analysis where the SN curve is input for a stress ratio of R=0.0, see the spot weld section below.

There are 2 available options for FKM correction in OptiStruct and are activated by setting UCORRECT to FKM/FKM2 or MCORRECT(MCi) fields to FKM on the FATPARM entry.

If only MSS2 is defined and if UCORRECT/MCORRECT(MCi) on FATPARM is set to FKM:
Regime 1 (R > 1.0)
${S}_{e}^{A}={S}_{a}\left(1-M\right)$
Regime 2 (-∞ ≤ R ≤ 0.0)
${S}_{e}^{A}={S}_{a}+M*{S}_{m}$
Regime 3 (0.0 < R < 0.5)
${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(M}{3}\right){S}_{m}}{1+M}{3}}$
Regime 4 (R ≥ 0.5)
${S}_{e}^{A}=\frac{3{S}_{a}{\left(1+M\right)}^{2}}{3+M}$
Where,
${S}_{e}^{A}$
Stress amplitude after mean stress correction (Endurance stress)
${S}_{m}$
Mean stress
${S}_{a}$
Stress amplitude
If only MSS2 is defined and if UCORRECT on FATPARM is set to FKM2:
Regime 1 (R > 1.0) and Regime 4 (R ≥ 0.5)
Mean stress correction is not applied $M=0.0$
Regime 2 (-∞ ≤ R ≤ 0.0)
${S}_{e}^{A}={S}_{a}+M*{S}_{m}$
Regime 3 (0.0 < R < 0.5)
${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(M}{3}\right){S}_{m}}{1+M}{3}}$
Where,
${S}_{e}^{A}$
Stress amplitude after mean stress correction (Endurance stress)
${S}_{m}$
Mean stress
${S}_{a}$
Stress amplitude
$M$
Equal to MSS2

If all four MSSi fields are specified for mean stress correction, the corresponding Mean Stress Sensitivity values are slopes for controlling all four regimes. Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( $R={S}_{\mathrm{min}}/{S}_{\mathrm{max}}$ ) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.

There are 2 available options for FKM correction in OptiStruct and are activated by setting UCORRECT to FKM/FKM2 and MCORRECT(MCi) fields to FKM on the FATPARM entry.

If all four MSSi are defined and if UCORRECT/MCORRECT(MCi) on FATPARM is set to FKM:
Regime 1 (R > 1.0)
${S}_{e}^{A}=\left({S}_{a}+{M}_{1}{S}_{m}\right)\left(\left(1-{M}_{2}\right)/\left(1-{M}_{1}\right)\right)$
Regime 2 (-∞ ≤ R ≤ 0.0)
${S}_{e}^{A}={S}_{a}+{M}_{2}{S}_{m}$
Regime 3 (0.0 < R < 0.5)
${S}_{e}^{A}=\left(1+{M}_{2}\right)\frac{{S}_{a}+{M}_{3}{S}_{m}}{1+{M}_{3}}$
Regime 4 (R ≥ 0.5)
${S}_{e}^{A}=\left({S}_{a}+{M}_{4}{S}_{m}\right)\left(\left(\left(1+3{M}_{3}\right)\left(1+{M}_{2}\right)\right)/\left(\left(1+3{M}_{4}\right)\left(1+{M}_{3}\right)\right)\right)$
Where,
${S}_{e}^{A}$
Stress amplitude after mean stress correction (Endurance stress)
${S}_{m}$
Mean stress
${S}_{a}$
Stress amplitude
${M}_{i}$
Equal to MSSi
If all four MSSi are defined and if UCORRECT on FATPARM is set to FKM2:
Regime 1 (R > 1.0) and Regime 4 (R ≥ 0.5)
Mean stress correction is not applied
Regime 2 (-∞ ≤ R ≤ 0.0)
${S}_{e}^{A}={S}_{a}+{M}_{2}{S}_{m}$
Regime 3 (0.0 < R < 0.5)
${S}_{e}^{A}=\left(1+{M}_{2}\right)\frac{{S}_{a}+{M}_{3}{S}_{m}}{1+{M}_{3}}$
Where,
${S}_{e}^{A}$
Stress amplitude after mean stress correction (Endurance stress)
${S}_{m}$
Mean stress
${S}_{a}$
Stress amplitude
${M}_{i}$
Equal to MSSi
For Spot Weld analysis, when the default S-N curves are used or if the field R on the SPWLD continuation line is set to 0.0 and UCORRECT is set to FKM, then the following FKM equations are used:
Regime 1 (R > 1.0)
${S}_{e}^{A}=\left({S}_{a}+{M}_{1}{S}_{m}\right)\left(\left(1-{M}_{2}\right)/\left(\left(1+{M}_{2}\right)\left(1-{M}_{1}\right)\right)\right)$
Regime 2 (-∞ ≤ R ≤ 0.0)
${S}_{e}^{A}=\left({S}_{a}+{M}_{2}{S}_{m}\right)/\left(1+{M}_{2}\right)$
Regime 3 (0.0 < R < 0.5)
${S}_{e}^{A}=\frac{{S}_{a}+{M}_{3}{S}_{m}}{1+{M}_{3}}$
Regime 4 (R ≥ 0.5)
${S}_{e}^{A}=\left({S}_{a}+{M}_{4}{S}_{m}\right)\left(\left(1+3{M}_{3}\right)/\left(\left(1+3{M}_{4}\right)\left(1+{M}_{3}\right)\right)\right)$

## Damage Accumulation Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:(4)
$\sum {D}_{i}=\sum \frac{{n}_{i}}{{N}_{if}}\ge 1.0$
Where,
${N}_{if}$
Materials fatigue life (number of cycles to failure) from its S-N curve at a combination of stress amplitude and means stress level $i$ .
${n}_{i}$
Number of stress cycles at load level $i$ .
${D}_{i}$
Cumulative damage under ${n}_{i}$ load cycle.

The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage, due to cyclic fatigue loading. However, it has been proved to work well for many applications.