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 or range versus cycles to failure , the relationship between and can be described by straight line segments.
Normally, a one or two segment idealization is used.
(1)
for segment 1
Where, is the nominal stress range, are the fatigue cycles to failure, is the first fatigue strength exponent, and 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)
Goodman:(3)
Where,
Mean stress given by
Stress Range given by
Stress range after mean stress correction (for a stress range and a mean stress )
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 () 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 () 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 (). The actual value of stress used in the Damage
calculations is the Corrected stress range (which is ). 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)
Regime 2 (-∞ ≤ R ≤ 0.0)
Regime 3 (0.0 < R < 0.5)
Regime 4 (R ≥ 0.5)
Where,
Stress amplitude after mean stress correction (Endurance stress)
Mean stress
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
Regime 2 (-∞ ≤ R ≤ 0.0)
Regime 3 (0.0 < R < 0.5)
Where,
Stress amplitude after mean stress correction (Endurance stress)
Mean stress
Stress amplitude
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 () 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)
Regime 2 (-∞ ≤ R ≤ 0.0)
Regime 3 (0.0 < R < 0.5)
Regime 4 (R ≥ 0.5)
Where,
Stress amplitude after mean stress correction (Endurance stress)
Mean stress
Stress amplitude
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)
Regime 3 (0.0 < R < 0.5)
Where,
Stress amplitude after mean stress correction (Endurance stress)
Mean stress
Stress amplitude
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)
Regime 2 (-∞ ≤ R ≤ 0.0)
Regime 3 (0.0 < R < 0.5)
Regime 4 (R ≥ 0.5)
Damage Accumulation Model
Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:(4)
Where,
Materials fatigue life (number of cycles to failure) from its S-N curve
at a combination of stress amplitude and means stress level .
Number of stress cycles at load level .
Cumulative damage under 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.