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.
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.
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.
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.
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:
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.
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.
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).
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.
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 (s_{e}) and the ultimate strength (s_{u}) of the material is plotted on the diagram (solid blue line in Figure 9).
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.
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.
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.
Here, when projecting from a cycle in compression, the new load time history would have a lower alternating stress.
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.
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.
In the region highlighted in blue, the projection is broken into two parts:
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.
Three different stress ratios are shown:
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.
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.
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
.
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.
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.
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.
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).
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.
The Simcenter Testlab Neo Process Designer has the following sections:
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.
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.
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.
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).
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.
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.
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