Mean Stress Corrections and Stress Ratios

Siemens Valued Contributor Siemens Valued Contributor
Siemens Valued Contributor

(view in My Videos)

 

When calculating fatigue life from a load time history, accounting for the mean stress of the loading is an important consideration.  Corrections must be made to account for mean stress.

 

Typically, more damage is accumulated when the product is under a tensile mean stress, while less damage is accumulated under compressive mean stress.

 

This article covers the Mean Stress Correction process using the Goodman diagram and Stress Ratio (R). 

 

Article index:

  1. What is Mean Stress?

  2. Mean Stress Correction Concept

  3. Mean Stress Correction with Goodman Diagram

  4. Stress Ratios

  5. Multi-Segment Mean Stress Correction

       5.1 Goodman

       5.2 Three Segment

       5.3 Four Segment

       5.4 Five Segment

       5.5 Example

  6. Mean Stress Corrections in Simcenter Testlab Neo

 

1. What is Mean Stress?

 

When a part that is subjected to repeated cyclic loads (above the endurance limit), it eventually fails.  These loads are often measured in units of stress.  Stress loads can have both an alternating component and a mean offset as shown in Figure 1.

 

cycles_with_and_without_mean.pngFigure 1: Left - A four cycle alternating stress time history with no mean stress. Right – A four cycle alternating stress time history with a mean stress offset.

Every time a cycle is applied, damage is accumulated (see knowledge base articles on Miner’s Rule and SN-Curves). While cycles are required for creating damage, what role does the mean offset play?  How significant is it?

 

When a product is used in the real world, the loads applied are complex, like those shown in Figure 2.  Some cycles have a mean, some may not.  The alternating stress amplitudes also vary.

 

load_history.pngFigure 2: Load time history with varying alternating stress and varying mean stress.

A typical SN-Curve is created by stress (S) cycling a material coupon with no mean stress present and counting the number of cycles (N) until it fails as shown in Figure 3.

 

sn_curve.pngFigure 3: A SN-Curve (green) relates the alternating stress level (S) to the number of cycles to failure (N). Projecting a stress level (via dashed blue line) to the SN-Curve the corresponding number of cycles to failure can be determined.

This curve can only be used to predict damage from cycles with no mean stress.  

 

The same SN-Curve test could also be performed under net tension or net compression as shown in Figure 4.

 

tension_vs_compression.pngFigure 4: A metal coupon being cycled under net tension (top) will have a crack/failure form before a metal coupon being cycled while in compression (bottom). The load history is shown in green, while the mean of the stress time history is shown in magenta.

Compression pushes the coupon or part together, helping prevent the formation of cracks.  Under net tension, the opposite happens.  The coupon or part will fail in fewer cycles as net tension tries to pull it apart.

 

In Figure 5, three different SN-Curves for the same material are shown:

  1. Zero Mean Stress - For a material with no mean stress is plotted (green curve).
  2. Tension - If the same test is performed under tension, the number of cycles until failure is reduced (magenta curve).
  3. Compression - If the same test is performed under compression, the number of cycles until failure is increased (yellow curve).

 

shifted_sn_curve.pngFigure 5: For the same material, compression increases the number of cycles to failure (yellow curve), while tension decreases the number of cycles to failure (magenta). A zero mean stress curve falls in between (green).

Performing multiple tests to get the SN-Curve for a material under different mean stress conditions takes time and effort.  The result would be multiple SN-Curves that would need to be used in a lookup table to determine the fatigue damage of a time history.  For many materials, multiple SN-Curves acquired at different stress ratio values do not exist.

 

Instead of shifting the SN-Curve to account for mean stresses, a more common procedure is to transform the rainflow matrix of the load time history.  The rainflow matrix contains all the stress cycle information for a given load time history.  See the knowledge base article “Rainflow Counting” for more information.

 

A process called mean stress correction is used to transform the original rainflow matrix to a new matrix with equivalent fatigue damage, but where every cycle has zero mean stress.

 

key1_mean_stress.png

 

This way, only one SN-Curve acquired with zero mean stress is needed to calculate fatigue damage. The mean stress correction process is described in the next section.

 

2. Mean Stress Correction Concept

 

A mean stress correction is used to transform a stress cycle to an equivalent stress cycle with zero mean stress (Figure 6).  The transformed stress cycle must still contain the same fatigue damage potential.

 

mean_stress_correction.pngFigure 6: Multiple stress cycles with a tensile mean stress (left) transformed to equivalent cycles with the same fatigue damage and zero mean stress (right).

In Figure 6, stress cycles with a positive mean are transformed to a higher amplitude alternating stress cycles with a zero mean.  If using alternating stress only (and ignoring the mean stress), the cycles on the right side of the figure accumulate more damage, properly reflecting that load history is in tension.

 

For a given load time history, the damage accumulation can be very different if mean stress correction is performed (Figure 7).

 

accumulated_damage_no_mean_correction_vs_mean_correction.pngFigure 7: Stress level (Y-axis) versus normalized accumulated damage (X-axis) from the same load time history with (red) and without (green) mean stress correction.

For a load history in net tension, the accumulated damage with mean stress correction (red) is much larger than without mean stress correction (green). Of course, if the all cycles in a load history have zero mean stress, there would be no difference between the calculated damage with and without mean stress correction.

 

To get to the correct equivalent fatigue damage, the amplitude of the stress cycle is changed using a Goodman-Haigh diagram and Stress Ratio (R) information as described in the next section. 

 

3. Mean Stress Correction with Goodman Diagram

 

To calculate an equivalent stress cycles with zero mean, the Goodman-Haigh diagram is used.

 

The alternating amplitude of the stress cycle and mean stress amplitude are plotted on a Goodman-Haigh diagram as shown in Figure 8.

 

stress_cycle_goodman_haigh.pngFigure 8: A stress cycle (upper left) is plotted on the Goodman-Haigh Diagram. The stress cycle consists of an alternating stress and a mean stress which is marked with an X on the diagram.

In the Goodman-Haigh diagram, the alternating stress (the cyclic part) is plotted on the Y axis, while the mean stress is plotted on the X axis. In this case, the cycle is in net tension, and falls on the right side of the Goodman-Haigh diagram.

 

Next, a line that connects the endurance limit (se) and the ultimate strength (su) of the material is plotted on the diagram (solid blue line in Figure 9). 

 

endurance_ultimate_goodman.pngFigure 9: A line that connects the endurance limit (e) to the ultimate strength (u) of the material is plotted on the diagram (solid blue line).

The endurance limit and ultimate strength are described in the Knowledge Base article “Stress and Strain”.

 

Then a line with the same slope (dotted blue) as the endurance-ultimate line is projected through the cycle as shown in Figure 10.

 

projection_for_tension.pngFigure 10: A line of the same slope (dotted blue) as the endurance-ultimate line (solid blue) is projected through the cycle. The intersection (green dot) on the alternating stress axis, where the mean stress is zero, is used as the new cycle amplitude.

The amplitude (green dot) where the projected line (dotted blue) intercepts the alternating stress axis is used as the stress in the new time history.  At this intersection, the mean stress value is zero.

 

key1_tension.png

 

For a cycle in tension like this, the alternating stress is higher in amplitude than the original cycles.  This compensates for the fact that the mean stress of the newly adjusted time history is zero.

 

Consider a different situation, with net compression as shown in Figure 11.

 

projection_for_compression.pngFigure 11: Stress cycles with a mean stress in compression yield a smaller alternating stress.

Here, when projecting from a cycle in compression, the new load time history would have a lower alternating stress.

 

key1_compression.png

  

This does not always work as desired.  If the mean stress is high enough, then the projection would yield a negative alternating stress as shown in Figure 12.

 

negative_projection.pngFigure 12: When projecting from a compressive cycle with a high mean stress, the alternating stress amplitude is negative, which is not possible.

Because of the possibility of negative alternating cycles, the projection is performed differently when cycles fall into certain areas of the Goodman-Haigh diagram as shown in Figure 13.

 

safe_projection.pngFigure 13: When cycles fall into the area marked in blue, the projection is done in two parts: zero slope projection to edge of blue area, then a projection based on the ultimate-endurance line to determine the alternating stress with zero mean.

In the region highlighted in blue, the projection is broken into two parts:

  1. Using a slope of zero, the amplitude of the cycle is projected to the line at the edge of the blue area.
  2. Then a projection using the slope from endurance-ultimate line to define the alternating stress where the mean stress is equal to zero.

 

Doing the projection in this manner results in a lower alternating stress, but without a negative alternating stress value being possible.

 

What defines the blue region?  The Goodman-Haigh Diagram can be broken into multiple regions or segments based on stress ratio (R).  Stress ratios are discussed in the next section.

 

4. Stress Ratios

 

The “stress ratio” or “R-Ratio” in short is the ratio of the lower value of a cycle divided by the upper value of a cycle as shown in Figure 14. Each cycle shown has a unique stress ratio.

 

stress_ratio.pngFigure 14: Stress Ratio (R) for different alternating stress and mean stress time histories.

Three different stress ratios are shown:

  • R = -1: Zero mean stress, the upper and lower alternating stress value are the same magnitude, opposite sign.
  • R = 0: The lower alternating stress value is zero, the alternating and mean stress magnitudes are equal. These cycles are in tension.
  • R = -∞: The upper alternating stress value is zero, the mean stress and alternating stress have equal magnitude, but the mean stress is negative. These cycles are in compression.

 

The R-ratio is often used to describe how the stress cycles for a S-N curve were applied to the material coupon. The R-ratio can be specified in lieu of reporting the mean stress, lower alternating stress, and higher alternating stress.

 

These stress ratios can be plotted on a Goodman-Haigh Diagram as shown in Figure 15.

 

goodman_segments.pngFigure 15: Goodman-Haigh diagram divided into segments by different stress ratios (R).

These three stress ratios (R) define different segments of the Goodman-Haigh diagram.  Different mean stress correction schemes can be used for each different segment depending on the correction method selected. The different methods are described in the next section.

 

5. Multi-Segment Mean Stress Correction

 

Based on these segments, different correction schemes can be defined.  They range from the Goodman mean stress correction, which is conservative, to multi-segment mean stress corrections.

 

The available mean stress corrections options available in Simcenter Testlab Neo are explained next.  In each section, the menu selections, with their default settings, are shown in the upper right.

 

5.1 Goodman Correction

 

The Goodman correction is shown in Figure 16. There is a single slope (M) defined in this correction.

 

goodman_correction.pngFigure 16: The Goodman Correction (Simcenter Testlab settings shown in upper right) has two regions. In the first region, the correction is the slope of a line (1/M) from the endurance limit to the ultimate strength. The default settings of Simcenter Testlab Neo listed in upper right.

The Goodman default slope is 0.3 (1/M) outside of the R = -∞ segment.

 

Some consider the Goodman mean stress correction to overestimate the amount of damage. 

 

5.1 Three Segment

 

A three segment correction is shown in Figure 17. Three different slopes (M1, M, and M3) can be specified

.

three_segment.pngFigure 17: Three Segment Correction, default settings of Simcenter Testlab Neo in upper right.

Schutz (1967) proposed a more gradual slope in the region M3. If using the default settings, this slope ensures that smaller alternating amplitude cycles in the segment above R-ratio equal to zero do not translate to as high of an alternating cycle as the Goodman method.

 

5.2 Four segment correction

 

The Four Segment Correction is shown in Figure 18.

 

four_segment.pngFigure 18: Four Segment Correction, default settings of Simcenter Testlab Neo in upper right.

The parameter LimitR is not a slope, but rather allows the user to arbitrarily define an additional segment in the Goodman-Haigh diagram. 

 

Cycles that fall above LimitR are projected with zero slope to the line defined by LimitR, and then projected along the M3 slope to the alternating stress axis.

 

5.3 Five segment correction

 

In the five segment method, there is an additional parameter defined by “LimitR2” as shown in Figure 19.

 

five_segment.pngFigure 19: Five Segment Correction, default settings of Simcenter Testlab Neo in upper right.

Both “LimitR” and “LimitR2” are R-ratio values and not slopes.  Five different slopes can be defined for a given material.

 

5.3 Example

 

The accumulated damage from a load history in net tension are shown in Figure 20.  Default settings were used.

 

accumulated_damage_different_corrections.pngFigure 20: Stress level (Y-axis) versus Normalized Accumulated Damage (X-axis) from the same load time history with Goodman (red), three segment (green), and four segment (blue) mean stress corrections.

Using the default settings, the Goodman damage estimate is the most conservative of the methods.  To get meaningful results, it is important that the slopes used in the corrections accurately reflect the material of the product.

 

6. Mean Stress Correction in Simcenter Testlab Neo

 

When using Simcenter Testlab Neo Durability processing, a mean stress correction can be used when doing damage calculations.

 

6.1 Getting Started

 

Start with opening the Simcenter Testlab Neo Desktop.  This is located in “Start -> Programs -> Testlab -> Testlab Neo General Processing -> Testlab Desktop”.

 

Once the Desktop is started, go to “File -> Add-ins” and turn on Process Designer, Load Data Analysis, and Fatigue Life Analysis (Figure 21).  

 

addins.pngFigure 21: Add-ins to do fatigue damage calculations with mean stress corrections.

This requires 69 tokens. Switch to the Processing tab on the bottom of the screen and select the Universal view.  The Process Designer has several sections as shown in Figure 22.

 

neo_interface.pngFigure 22: The Simcenter Testlab Neo Process Designer has several sections: data selection, process area, display, and action buttons.

The Simcenter Testlab Neo Process Designer has the following sections:

  • Data Selection: This area is used to select data for processing and to select analysis results for viewing.
  • Process Area: Methods can be selected and combined to define a data analysis process.
  • Display: When data is selected, it will automatically display here.
  • Action Area: Has action buttons. Press the “Run” button to start the analysis process, and press “Accept” to store the results of an analysis run to the current project.

 

6.2 Select Data for Processing

 

In the Data Selection area, navigate to the time histories of interest.  Right click on them and select “Add to Input Basket” to make them available for processing as shown in Figure 23.

 

input_basket.pngFigure 23: Right click on data and choose “Add to Input Basket”.

Data can be selected from different directories, etc.  The input basket is links to the data to be processed.

 

6.3 Create Process

 

Choose methods from the Method Library and insert them in the process area.  Inserting can be done by double clicking on the methods in the library. Select methods as shown in Figure 24

 

method_library.pngFigure 24: To calculate fatigue damage, the methods should be listed and connected as shown.

Click and drag to make connections between the methods.

 

6.4 Set Mean Stress Correction Method

 

Click once and highlight the Mean Stress Correction method as shown in Figure 25

 

properties.pngFigure 25: Highlight the Mean Stress Correction method and select the correction method.

After highlighting the method, the settings of the method can be changed in the Properties pane.

 

6.5 Run Process, View Results, and Save

 

To analyze the time data, press the “Run” button in the lower left (Figure 26).

 

action_buttons.pngFigure 26: The action buttons “Run” and “Accept”.

After pressing “Run” and the calculation finishes, the results are stored in the “Active Analysis”.  They can be viewed by clicking on them in the “Active Analysis” folder as shown in Figure 27.

 

final_results.pngFigure 27: View analysis results by clicking in the Data selection area.

If the results are acceptable, the button “Accept” can be pushed and results are stored in the active project.

 

Questions?  Email john.hiatt@siemens.com, post a reply, or contact Siemens PLM GTAC support.

 

Related Durability Links: