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 HyperLife.
Multiaxial Fatigue Analysis, using S-N (stress-life), E-N (strain-life), and Dang Van Criterion (Factor
of Safety) approaches for predicting the life (number of loading cycles) of a structure under cyclical
loading may be performed by using HyperLife.
Seam Weld Fatigue analysis is available to facilitate Fatigue analysis for seam welded structures. It allows you to
simulate the Fatigue failure at the seam weld joints to assess the corresponding fatigue failure characteristics like
Damage and Life.
The study of fatigue life of structures under Random Loading.
Power Spectral Density (PSD) results from the Random Response Analysis are used to calculate
Moments () that are used to generate the probability density
function for the number of cycles versus the stress range.
There are two ways to perform Random Response Fatigue Analysis in
HyperLife:
by using PSD Stresses which are already available for Random Fatigue
by using PSD Stresses generated from Frequency Response Analysis and Power
Spectral Density Functions Loading versus Frequency in HyperLife
PSD Stresses Are Already Available for Random Fatigue
The PSD Moments are calculated based on the Stress PSD generated from the Random
Response Analysis, as shown below.
Input
Calculates Random Response Fatigue.
Power Spectral Density (PSD) Moments
PSD Moments () are calculated based on
the Stress PSD generated from the Random Response Analysis as:
The moments are calculated based on:(1)
Where,
Frequency value
PSD response value at frequency
The stability of can be checked by setting PARAM, CHKM0, YES. A warning is printed if
the frequency interval must be further refined.
Calculate Probability of Stress Range Occurrence
Calculation of the Probability of occurrence of a stress range between the initial
and final stress range values within each bin section are user-defined.
The probability of a stress range occuring between and is .
Probability Density Function (probability density of number of cycles versus
stress range)
PSD Moments calculated as shown above are used in the generation of a Probability
Density Function for the stress range. The function is based on the specified
damage model in the Random Response Fatigue section of the SN Fatigue
module. Currently, DIRLIK, LALANNE,
NARROW, and THREE options are available to
define the damage model.
DIRLIK (Default Damage Model):
DIRLIK postulated a closed form solution to the
determination of the Probability Density Function as:(2)
Where,
Irregularity Factor
Stress range
LALANNE:
The LALANNE Random Fatigue
Damage model depicts the probability density function as: (3)
Where,
Irregularity factor
Stress range
NARROW:
The Narrow Band Random Fatigue Damage model uses
the following probability functions:(4)
Where, is the stress range.
In the NARROW band
model, by default, HyperLife uses number of zero crossings () instead of number of peaks (), because the numerical calculations involving may lead to unstable numerical
behavior.
THREE:
The Steinberg 3-Band Random Fatigue Damage model
uses the following probability function.
Unlike the other damage models,
for THREE band, the following values are probability (and not probability
density). This is also evident in the use of upper case compared to the lower case for the other damage models. For the THREE
damage model, the following probabilities are directly used to calculate the
number of cycles by multiplying with the total number of zero-crossings in the
entire time history (for other damage models (except THREE), the probability
density values are first multiplied by DS (bin size) to get the
probability).(5)
Where, is the stress range.
The probability density function can be adjusted based on the following
parameters defined in the Random
Response Fatigue section of the SN Fatigue module:
Upper Stress Range (Calculated)
Calculates the upper limit of the stress range. This is calculated
as: upper limit of the stress range = 2*RMS Stress*Upper
Stress Range (Calculated) input. The RMS Stress is
output from Random Response Subcase. The stress ranges of interest
are limited by the upper limit of the stress range. Any stresses
beyond the upper limit are not considered in Random Fatigue Damage
calculations.
Width: Stress Range (Calculated)
Calculates the width of the stress ranges (DS = ) for which the probability is
calculated (Figure 2). The default is 100 and the first bin starts
from 0.0 to . The width of the stress range is
calculated as DS = upper limit of the stress
range / Width: Stress Range (Calculated).
Calculate Probability of Stress Range Occurrence
Calculation of the Probability of occurrence of a stress range between the initial
and final stress range values within each bin section are based on the damage
models.
DIRLIK, LALANNE, and
NARROW Damage Models
The probability of a stress range occurring between and is .
THREE Damage Model
The probability is directly defined
using the probability function defined above. It is being repeated here for
clarity.(6)
Where, is stress range.
For the THREE damage model, there are only three bins. The number of
cycles at each stress range (2*RMS, 4*RMS, and 6*RMS) are calculated by directly
multiplying the corresponding probabilities with the total number of zero-crossings
(refer to section below regarding calculation of number of zero-crossings).
Select Damage Models
DIRLIK, LALANNE, NARROW, and
THREE are available for selection in the Random
Response Fatigue section of the SN Fatigue module. The following
information may provide additional understanding to help choose the damage model for
aHyperLife run.
You can see from the previous sections, that the PSD moments of stress are
used to calculated corresponding moments, which are used to determine the
probability density function for the stress-range.
DIRLIK and LALANNE models generate
probabilities across a wider distribution of the stress-range spectrum.
Therefore, these models should be used when the input random signal consists
of a variety of stress-ranges across multiple frequencies. Therefore, the
information in the probability density function better captures the wider
range in stress-range distribution if DIRLIK and
LALANNE are used.
The NARROW model is intended for random signals for which
the stress range is expected to be closely associated with a high
probability of particular certain stress range distribution. Therefore, if
you know that the input random data does not have a wide range of
stress-range distribution, and that the distribution is mainly concentrated
about a particular stress range, then you should select
NARROW, since it expects the highest probability of
stress-ranges to lie at or around this particular stress range.
The THREE model is like the NARROW model,
except that it expects the distribution of the random signal to contain, in
addition to the association with 1*RMS, associations (albeit smaller) with
2*RMS, and 3*RMS. Therefore, if your input random data is mainly clustered
around stress ranges in 1*RMS, and to a lesser extent, 2*RMS, and 3*RMS,
then you should select THREE.
Number of Peaks and Zero Crossings
NARROW and THREE Damage Models
The
number of zero crossings per second in the original time-domain random loading
(from which the Frequency based Random PSD load is generated) is determined
as:(7)
DIRLIK and LALANNE Damage Models
The
number of peaks per second in the original time-domain random loading (from
which the Frequency based Random PSD load is generated) is determined
as:(8)
The total number of cycles for THREE band and
NARROW band is calculated as:(9)
The total number of cycles for DIRLIK,
LALANNE, and NARROWis calculated as:(10)
Where, is Total exposure time given by the Repeats/Time
values when creating events in the Load Map dialog.
Total Number of Cycles of a Particular
Stress Range
The total number of cycles with Stress range is calculated as:(11)
Fatigue Life and Damage
Fatigue Life (maximum number of cycles of a particular stress range
for the material prior to failure)
is calculated based on the Material SN curve as:(12)
Total Fatigue Damage as a result of the applied Random Loading is calculated based
on:(13)
To account for the mean stress correction with any loading that leads to a mean
stress different from zero, you can define a static subcase that consists of such
loading (typically gravity loads). This static subcase can be referenced within the fatigue
event created in the Load Map dialog.
Setup
After loading an FE result file that contains a Random subcase, Random Response
Fatigue is activated by setting the Type of Loading field to Random (PSD
Stresses). The supported result datatypes for Random Fatigue are
Segalman Von mises PSD stresses (Stress Life only) or Absolute Max Principal PSD
Stresses (Stress Life and Strain Life).
PSD Stresses to be Generated from Frequency Response Analysis and Power Spectral Density
Functions Loading vs Frequency in HyperLife for Random
Fatigue
The PSD stresses are calculated internally using the FRF stresses and the input PSDs.
The fatigue event created first calculates the PSD stress (Random Response Analysis), which
are then used to calculate the PSD moments.
Different Load Cases (a and b)
If and are the complex frequency responses (stresses) of the th degree of freedom, due to Frequency
Response Analysis load cases and respectively, the power spectral density of the
response of the th degree of freedom, is:(14)
Where, is the cross power spectral density of two
(different, ) sources, where the individual source is the excited load case and is the applied load case. This value can possibly be
a complex number.
Same Load Case (a)
If is the spectral density of the individual source
(load case ), the power spectral density of the response of th degree of freedom due to the load case will be:(15)
Combination of Different (a,b) and Same (a,a) Load Cases in a Single Random
Response Analysis
If there is a combination of load cases for Random Response Analysis, the total power
spectral density of the response will be the summation of the power spectral density
of responses due to all individual (same) load cases as well as all cross
(different) load cases.
Setup
After loading an FE result file that contains a Frequency Response Analysis subcase,
Random Response Fatigue is activated by setting the Type of Loading field to
Random (Input PSD with FRF).
If the power spectral density of the loading versus frequency is of Magnitude and
Phase, it is converted to real and imaginary.
Loadmap:
Input PSD is based on the Channel Type selection.
Input PSD: Real and Imaginary
Input PSD: Magnitude and Phase
The event is created with the correlations based on the number of Frequency Response
Subcases selected. The imported channels (input power spectral density of the
Loading vs Frequency) are then assigned to each correlation.
Excitation1 and Excitation2 refer to the correlations of the FRF subcases. The Input
PSDs are used to scale the complex stresses. If the Input PSDs are of the Phase and
Magnitude form, they are internally converted to Real and Imaginary.
PSD Stresses calculated from the above event are then utilized for fatigue
calculations, like the case where PSD stresses are already available (described in
the beginning of this chapter).
Note: RMS stress value over the entire frequency range is output along with damage and
life.