The `meanstress` module

Meanstress transformation

Meanstress transformation is used to take the influence of the meanstress of a load hysteresis into account. The fatigue of a cyclically loaded component is not only influenced by the amplitude but also by the mean stress. The usual way to take this into account is to compute a substitute amplitude that has the same fatigue result for the known mean stress. The mean stress sensitivity is usually defined by a haigh diagram, depending on the R-value regime.

In pyLife we have the class HaighDiagram that lets you define an arbitrary haigh diagram, i.e. a mean stress sensitivity for each interval of R. There are also convenience functions fkm_goodman and five_segment that lets you define the more common haigh diagrams more easily.

FKM Goodman
Five Segment Correction

class pylife.strength.meanstress.HaighDiagram(pandas_obj)[source]

Model for a Haigh diagram in order to perform meanstress transformations.

A Haigh diagram a set of meanstress sensitivity slopes $M$ that is changing with the R-values. The values of the `pd.Series represents that slopes $M$ and the pd.IntervalIndex represents the R-ranges.

classmethod five_segment(five_segment_haigh_diagram)[source]

Create a five segment slope Haigh diagram.

Parameters:: five_segment_haigh_diagram (pandas.Series or pandas.DataFrame) – The five segment meanstress slope data.

Notes

five_segment_hagih_diagram has to provide the following keys:

M0: the mean stress sensitivity between R==-inf and R==0
M1: the mean stress sensitivity between R==0 and R==R12
M2: the mean stress sensitivity betwenn R==R12 and R==R23
M3: the mean stress sensitivity between R==R23 and R==1
M4: the mean stress sensitivity beyond R==1
R12: R-value between M1 and M2
R23: R-value between M2 and M3

Examples

>>> haigh = HaighDiagram.five_segment(
...    pd.Series(
...        {"M0": 0.5, "M1": 0.25, "M2": 0.125, "M3": 1.0, "M4": -2.0, "R12": 0.2, "R23": 0.8}
...    )
... )
>>> haigh.to_pandas()
R
(1.0, inf]    -2.000
(-inf, 0.0]    0.500
(0.0, 0.2]     0.250
(0.2, 0.8]     0.125
(0.8, 1.0]     1.000
dtype: float64

>>> collective = pd.DataFrame(
...     {
...         "range": [600.0, 300.0, 500.0],
...         "mean": [400.0, -150.0, 0.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> haigh.transform(collective, 0.0)
        range        mean  cycles
0  640.000000  320.000000     1.0
1  100.000000   50.000000    10.0
2  333.333333  166.666667   100.0

classmethod fkm_goodman(haigh_fkm_goodman)[source]

Create a Haigh diagram according to FKM Goodman.

Parameters:: haigh_fkm_goodman (pd.Series or pd.DataFrame) – a series containing one or a dataframe containing multiple values for M and optionally M2.

Notes

The Haigh diagram according to FKM Goodman comes with the slope M which is valid between R==-inf and R==0. Beyond R==0 the slope is M2` if ``M2 is given or M/3 if not.

Examples

A FKM Goodman diagram with default M2

>>> HaighDiagram.fkm_goodman(pd.Series({"M": 0.5})).to_pandas()
R
(1.0, inf]     0.000000
(-inf, 0.0]    0.500000
(0.0, 1.0]     0.166667
dtype: float64

A FKM Goodman diagram with manual M2

>>> HaighDiagram.fkm_goodman(pd.Series({"M": 0.5, "M2": 0.2})).to_pandas()
R
(1.0, inf]     0.0
(-inf, 0.0]    0.5
(0.0, 1.0]     0.2
dtype: float64

>>> collective = pd.DataFrame(
...     {
...         "range": [600.0, 300.0, 500.0],
...         "mean": [100.0, 50.0, 80.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> HaighDiagram.fkm_goodman(pd.Series({"M": 0.5})).transform(collective, -1.0)
   range  mean  cycles
0  700.0   0.0     1.0
1  350.0   0.0    10.0
2  580.0   0.0   100.0

classmethod from_dict(segments_dict)[source]

Create a Haigh diagram from a dict.

Parameters:: segments_dict (dict) – dict resolving the R-value intervals to the meanstress slope

Example

>>> HaighDiagram.from_dict({
...    (1.0, np.inf): 0.0,
...    (-np.inf, 0.0): 0.5,
...    (0.0, 1.0): 0.167
... }).to_pandas()
R
(1.0, inf]     0.000
(-inf, 0.0]    0.500
(0.0, 1.0]     0.167
dtype: float64

sets up a FKM Goodman like Haigh diagram.

transform(collective, R_goal)[source]

Transform a load collective to defined R-value.

Parameters:

collective (pd.DataFrame) – The load collective data to transform containing either range and mean or from and to columns to describe the load cycles. All other columns e.g. cycles are copied to the result.
R_goal (float) – The target R-value for the transformation.

Returns:

DataFrame with columns: - ‘range’: transformed amplitude range - ‘mean’: transformed mean value - ‘cycles’: copied unchanged from the input if present

Return type:

pd.DataFrame

Examples

>>> collective = pd.DataFrame(
...     {
...         "from": [300.0, -150.0, -250.0],
...         "to": [-300.0, 150.0, 250.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> HaighDiagram.from_dict({(-np.inf, np.inf): 0.5}).transform(collective, 0.0)
        range        mean  cycles
0  400.000000  200.000000     1.0
1  200.000000  100.000000    10.0
2  333.333333  166.666667   100.0

>>> collective = pd.DataFrame(
...     {
...         "range": [600.0, 300.0, 500.0],
...         "mean": [100.0, 50.0, 80.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> HaighDiagram.from_dict({(-np.inf, np.inf): 0.5}).transform(collective, -1.0)
   range  mean  cycles
0  700.0   0.0     1.0
1  350.0   0.0    10.0
2  580.0   0.0   100.0

class pylife.strength.meanstress.MeanstressTransformCollective(pandas_obj)[source]

Meanstress transformer class for a load collective.

five_segment(five_segment, R_goal)[source]

Perform a Five segment transformation on a load collective

Parameters:

five_segment (pd.Series or pd.DataFrame) – The meanstress sensitivities and R transition values (see HaighDiagram.five_segment())
R_goal (float) – The R-value to transform to

Returns:

transformed_collective – The transformed load collective. After the meanstress transformation, the resulting (range, mean) interval bins may no longer be continuous.

Return type:

LoadCollective

Examples

>>> collective = pd.DataFrame(
...     {
...         "range": [600.0, 300.0, 500.0],
...         "mean": [400.0, -150.0, 0.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> haigh = pd.Series(
...    {"M0": 0.5, "M1": 0.25, "M2": 0.125, "M3": 1.0, "M4": -2.0, "R12": 0.2, "R23": 0.8}
... )
>>> transformed = collective.meanstress_transform.five_segment(haigh, 0.0)
>>> transformed.amplitude
0    320.000000
1     50.000000
2    166.666667
Name: amplitude, dtype: float64

>>> transformed.meanstress
0    320.000000
1     50.000000
2    166.666667
Name: meanstress, dtype: float64

fkm_goodman(goodman, R_goal)[source]

Perform a FKM Goodman transformation on a load collective

Parameters:

goodman (pd.Series or pd.DataFrame) – The meanstress sensitivity data needs M and optionally M2
R_goal (float) – The R-value to transform to

Returns:

transformed_collective – The transformed load collective

Return type:

LoadCollective

Examples

>>> collective = pd.DataFrame(
...     {
...         "from": [300.0, -150.0, -250.0],
...         "to": [-300.0, 150.0, 250.0],
...         "cycles": [1.0, 10.0, 100.0],
...     }
... )
>>> collective.meanstress_transform.fkm_goodman(pd.Series({"M": 0.5}), 0.0).amplitude
0    200.000000
1    100.000000
2    166.666667
Name: amplitude, dtype: float64

class pylife.strength.meanstress.MeanstressTransformMatrix(pandas_obj)[source]

Meanstress transformer class for a load histogram.

five_segment(five_segment, R_goal)[source]

Perform a Five segment transformation on a load histogram

Parameters:

five_segment (pd.Series or pd.DataFrame) – The meanstress sensitivities and R transition values (see HaighDiagram.five_segment())
R_goal (float) – The R-value to transform to

Returns:

transformed_histogram – The transformed load histogram. After the meanstress transformation, the resulting (range, mean) interval bins may no longer be continuous.

Return type:

LoadHistogram

Notes

If continuous bins are required afterwards, e.g. for visualization, the transformed histogram can optionally be rebinned. Be aware that such a rebinning is a post-processing step for presentation purposes and may reduce the accuracy of the transformed histogram.

Examples

>>> histogram = pd.Series(
...     [10, 90, 900, 9000, 90000, 900000],
...     index=pd.MultiIndex.from_arrays(
...         [
...             pd.IntervalIndex.from_arrays(
...                 [650, 550, 450, 350, 250, 150], [750, 650, 550, 450, 350, 250]
...             ),
...             pd.IntervalIndex.from_arrays([0, 0, 0, 0 ,0, 0], [0, 0, 0, 0, 0, 0]),
...         ],
...         names=["range", "mean"],
...     ),
...     name="cycles"
... )
>>> five_segments = pd.Series(
...     {
...         "M0": 0.5,
...         "M1": 0.2,
...         "M2": 0.1,
...         "M3": 1.0,
...         "M4": 2.0,
...         "R12": 0.2,
...         "R23": 0.8,
...     }
... )
>>> transformed = histogram.meanstress_transform.five_segment(five_segments, 0)
>>> transformed.amplitude
range                                     mean
(433.3333333333333, 500.0]                (216.66666666666666, 250.0]                 233.333333
(366.6666666666667, 433.3333333333333]    (183.33333333333334, 216.66666666666666]    200.000000
(300.0, 366.6666666666667]                (150.0, 183.33333333333334]                 166.666667
(233.33333333333334, 300.0]               (116.66666666666667, 150.0]                 133.333333
(166.66666666666669, 233.33333333333334]  (83.33333333333334, 116.66666666666667]     100.000000
(100.0, 166.66666666666669]               (50.0, 83.33333333333334]                    66.666667
Name: amplitude, dtype: float64

fkm_goodman(goodman, R_goal)[source]

Perform a FKM Goodman transformation on a load histogram

Parameters:

goodman (pd.Series or pd.DataFrame) – The meanstress sensitivity data needs M and optionally M2.
R_goal (float) – The R-value to transform to

Returns:

transformed_histogram – The transformed load histogram. After the meanstress transformation, the resulting (range, mean) interval bins may no longer be continuous.

Return type:

LoadHistogram

Notes

If continuous bins are required afterwards, e.g. for visualization, the transformed histogram can optionally be rebinned. Be aware that such a rebinning is a post-processing step for presentation purposes and may reduce the accuracy of the transformed histogram.

Examples

>>> histogram = pd.Series(
...     [10, 90, 900, 9000, 90000, 900000],
...     index=pd.MultiIndex.from_arrays(
...         [
...             pd.IntervalIndex.from_arrays(
...                 [650, 550, 450, 350, 250, 150], [750, 650, 550, 450, 350, 250]
...             ),
...             pd.IntervalIndex.from_arrays([0, 0, 0, 0 ,0, 0], [0, 0, 0, 0, 0, 0]),
...         ],
...         names=["range", "mean"],
...     ),
...     name="cycles"
... )
>>> transformed = histogram.meanstress_transform.fkm_goodman(pd.Series({"M": 0.0}), 0)
>>> transformed.amplitude
range           mean
(650.0, 750.0]  (325.0, 375.0]    350.0
(550.0, 650.0]  (275.0, 325.0]    300.0
(450.0, 550.0]  (225.0, 275.0]    250.0
(350.0, 450.0]  (175.0, 225.0]    200.0
(250.0, 350.0]  (125.0, 175.0]    150.0
(150.0, 250.0]  (75.0, 125.0]     100.0
Name: amplitude, dtype: float64

pylife.strength.meanstress.experimental_mean_stress_sensitivity(sn_curve_R0, sn_curve_Rn1, N_c=inf)[source]

Estimate the mean stress sensitivity from two FiniteLifeCurve objects for the same amount of cycles N_c.

The formula for calculation is taken from: “Betriebsfestigkeit”, Haibach, 3. Auflage 2006

Formula (2.1-24):

\[M_{\sigma} = {S_a}^{R=-1}(N_c) / {S_a}^{R=0}(N_c) - 1\]

Alternatively the mean stress sensitivity is calculated based on both SD values (if N_c is not given).

Parameters:

sn_curve_R0 (pylife.strength.sn_curve.FiniteLifeCurve) – Instance of FiniteLifeCurve for R == 0
sn_curve_Rn1 (pylife.strength.sn_curve.FiniteLifeCurve) – Instance of FiniteLifeCurve for R == -1
N_c (float, (default=np.inf)) – Amount of cycles where the amplitudes should be compared. If N_c is higher than a fatigue transition point (ND) for the SN-Curves, SD is taken. If N_c is None, SD values are taken as stress amplitudes instead.

Returns:

Mean stress sensitivity M_sigma

Return type:

float

Raises:

ValueError – if the resulting M_sigma doesn’t lie in the range from 0 to 1 a ValueError is raised, as this value would suggest higher strength with additional loads.

pylife.strength.meanstress.five_segment_correction(amplitude, meanstress, M0, M1, M2, M3, M4, R12, R23, R_goal)[source]

Performs a mean stress transformation to R_goal according to the: Five Segment Mean Stress Correction

Parameters:

Sa – the stress amplitude
Sm – the mean stress
Rgoal – the R-value to transform to
M – the mean stress sensitivity between R=-inf and R=0
M1 – the mean stress sensitivity between R=0 and R=R12
M2 – the mean stress sensitivity betwenn R=R12 and R=R23
M3 – the mean stress sensitivity between R=R23 and R=1
M4 – the mean stress sensitivity beyond R=1
R12 – R-value between M1 and M2
R23 – R-value between M2 and M3

Returns:

the transformed stress range

pylife.strength.meanstress.fkm_goodman(amplitude, meanstress, M, M2, R_goal)[source]

The meanstress module

Meanstress transformation

The `meanstress` module