The Goodman-Haigh Diagram for Infinite Life
Infinite life is often used in designing critical components of products with demanding use. Examples include crankshafts of an engines, vehicles for public transportation, spacecraft, etc.
What is meant by infinite life? Ferrous materials have an ‘infinite life’ region defined by an ‘endurance limit’. The endurance limit is a specific stress level for a material, where stress cycles below a certain amplitude and mean will not accumulate fatigue damage.
The Goodman-Haigh diagram is used to check if a cyclic stress time history is within the infinite life region for a product made of a given material (Figure 1).
It is important that none of the stress cycles in a load history exceed the infinite life endurance limit. If they do, the material will behave as if the infinite life region does not exist, and failure will occur given enough additional cycles, even if they are below the endurance limit.
Goodman published his original diagram in 1899. Haigh added alternating and mean stress in 1917. The combination of these two is referred to as the ‘Goodman-Haigh Diagram’.
Two major pieces of information are needed to use a Goodman-Haigh diagram:
The material information is used to define an infinite life region. The stress cycles are plotted against this region to see if they are contained within it.
A stress time history can be broken down into individual cycles. A cycle has an alternating component as shown in Figure 2.
A stress cycle can also have a mean stress. This mean stress puts the part in either net compression or tension as shown in Figure 3.
The mean stress is very important factor in governing the fatigue life. Net tension on a part tries to pull it apart, which significantly reduces its life. Net compression pushes a part together, which is not as damaging.
In the Haigh diagram, the alternating and mean stress of the cycles will be plotted against each other as shown in Figure 4.
The alternating stress level is plotted on the Y axis. The mean stress level is plotted on the X axis. Negative mean stress is compression, and positive mean stress is tension.
Using a static stress-strain test on a material, the following material properties can be determined:
These material properties are determined via applying static loads to the material and plotting the relationship of stress and strain as shown in Figure 5.
The Yield strength and Ultimate strength are plotted on the Goodman-Haigh diagram as shown in Figure 6.
A yield envelope is created by connecting the yield strength points. However, this yield envelope is symmetric around the Y-axis, and does not distinguish between compression and tension.
Additional material information is needed from a dynamic/cyclic stress test. The result of a dynamic stress test can be found in a SN-curve as shown as shown in Figure 7.
The endurance limit is determined from the SN-Curve. The endurance limit is then plotted on the Goodman-Haigh diagram as shown in Figure 8.
An infinite life region can then be created by:
This infinite life region defined by these connections and projections are shown in Figure 9.
This infinite life region has a smaller region for tension versus compression, as would be expected. A stress time history can then be evaluated against the infinite life region.
Using the Goodman-Haigh diagram
The mean and alternating stress of a stress time history is plotted on the Goodman-Haigh diagram as shown in Figure 10.
This is done for each cycle in the time history. Each cycle is evaluated as to whether it falls in the infinite life region. In Figure 10, the stress cycles are contained entirely in the infinite life region.
Any stress time history, no matter how complicated, can be broken into individual cycles via the rainflow counting process. These cycles produced by the rainflow counting process include a mean and alternating stress.
Projecting from the origin to the cycle versus the region, a factor of safety can be calculated (Figure 11).
In this case, the factor of safety is approximately two: the ratio of the magenta and green lines. In many engineering applications, a factor of safety of three or higher is often desired. This would ensure that the part would survive with three times higher than expected loads.
To check infinite life with a Goodman-Haigh diagram in LMS Virtual.Lab, right click on ‘Local Tensor History Solution’ and choose ‘New Display’ as shown in Figure 12.
In the resulting dialog box, select ‘Haigh Diagram (Signed Von Mises)’ as shown in Figure 13.
A Goodman-Haigh diagram will be created, showing the results of each cycle from the stress time history superimposed on the diagram as shown in Figure 14.
In the diagram shown in Figure 14, it can be clearly seen if some cycles are outside the infinite life region.
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