# Uniaxial Fatigue Analysis

Uniaxial Fatigue Analysis, using S-N (stress-life) and E-N (strain-life) approaches for predicting the life (number of loading cycles) of a structure under cyclical loading may be performed by using OptiStruct.

The stress-life method works well in predicting fatigue life when the stress level in the structure falls mostly in the elastic range. Under such cyclical loading conditions, the structure typically can withstand a large number of loading cycles; this is known as high-cycle fatigue. When the cyclical strains extend into plastic strain range, the fatigue endurance of the structure typically decreases significantly; this is characterized as low-cycle fatigue. The generally accepted transition point between high-cycle and low-cycle fatigue is around 10,000 loading cycles. For low-cycle fatigue prediction, the strain-life (E-N) method is applied, with plastic strains being considered as an important factor in the damage calculation.

Sections of a model on which fatigue analysis is to be performed must be identified on a FATDEF Bulk Data Entry. The appropriate FATDEF Bulk Data Entry may be referenced from a fatigue subcase definition through the FATDEF Subcase Information Entry.

## Stress-Life (S-N) Approach

### S-N Curve

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:

- ${S}_{m}$
- 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.

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:

`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.

`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.

`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)
- ${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right){S}_{m}}{1+\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$
- Regime 4 (R ≥ 0.5)
- ${S}_{e}^{A}=\frac{3{S}_{a}{\left(1+M\right)}^{2}}{3+M}$

- ${S}_{e}^{A}$
- Stress amplitude after mean stress correction (Endurance stress)
- ${S}_{m}$
- Mean stress
- ${S}_{a}$
- Stress amplitude

`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)
- ${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right){S}_{m}}{1+\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

- ${S}_{e}^{A}$
- 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.

`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)
- $${S}_{e}^{A}=\left({S}_{a}+{M}_{4}{S}_{m}\right)\left(\left(\left(1+3{M}_{3}\right)\left(1+{M}_{2}\right)\right)/\left(\left(1+3{M}_{4}\right)\left(1+{M}_{3}\right)\right)\right)$$

- ${S}_{e}^{A}$
- Stress amplitude after mean stress correction (Endurance stress)
- ${S}_{m}$
- Mean stress
- ${S}_{a}$
- Stress amplitude
- ${M}_{i}$
- Equal to
`MSSi`

`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)
- ${S}_{e}^{A}=\left(1+{M}_{2}\right)\frac{{S}_{a}+{M}_{3}{S}_{m}}{1+{M}_{3}}$

- ${S}_{e}^{A}$
- Stress amplitude after mean stress correction (Endurance stress)
- ${S}_{m}$
- Mean stress
- ${S}_{a}$
- Stress amplitude
- ${M}_{i}$
- Equal to
`MSSi`

`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

- $${N}_{if}$$
- 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.

## Strain-Life (E-N) Approach

Strain-life analysis is based on the fact that many critical locations such as notch roots have stress concentration, which will have obvious plastic deformation during the cyclic loading before fatigue failure. Thus, the elastic-plastic strain results are essential for performing strain-life analysis.

### Neuber Correction

Neuber correction is the most popular practice to correct elastic analysis results into elastic-plastic results.

Where, $${\sigma}_{e}$$ , $${\epsilon}_{e}$$ is locally elastic stress and locally elastic strain obtained from elastic analysis, $\sigma $, $\text{\epsilon}$ the stress and strain at the presence of plastic strain. Both $\sigma $ and $\text{\epsilon}$ can be calculated from Equation 9 together with the equations for the cyclic stress-strain curve and hysteresis loop.

### Monotonic Stress-Strain Behavior

Where, $$A$$ is the current cross-section area, $$l$$ is the current objects length, $${l}_{0}$$ is the initial objects length, and $\sigma $ and $\text{\epsilon}$ are the true stress and strain, respectively, Figure 5 shows the monotonic stress-strain curve in true stress-strain space. In the whole process, the stress continues increasing to a large value until the object fails at C.

### Cyclic Stress-Strain Curve

- Stable state
- Cyclically hardening
- Cyclically softening
- Softening or hardening depending on strain range

- $${K}^{\text{'}}$$
- Cyclic strength coefficient
- $${n}^{\text{'}}$$
- Strain cyclic hardening exponent

### Hysteresis Loop Shape

### Mean Stress Correction

The fatigue experiments carried out in the laboratory are always fully reversed, whereas in practice, the mean stress is inevitable, thus the fatigue law established by the fully reversed experiments must be corrected before applied to engineering problems.

Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain.

The SWT method will predict that no damage will occur when the maximum stress is zero or negative, which is not consistent with the reality.

When comparing the two methods, the SWT method predicted conservative life for loads predominantly tensile, whereas, the Morrow approach provides more realistic results when the load is predominantly compressive.

### Damage Accumulation Model

In the E-N approach, use the same damage accumulation model as the S-N approach, which is Palmgren-Miner's linear damage summation rule.