On the application of the goodman equation to a multiaxial stress state

In summary, Whitebread explains how the modified goodman equation can be used to account for a multiaxial stress state and fatigue. He explains that uni-axial stress amplitude is replaced with von-mises stress amplitude, the fatigue limit at whatever number of cycles the designer is concerned with is replaced with the von-mises stress at that stress amplitude, mean stress is replaced with mean von-mises stress, and ultimate stress is replaced with (sqrt(2)/3)*sigma_u, or the von Mises stress at failure. He notes that equations for stress amplitude (tau_a) and mean stress (tau_m) are swapped, but the results are the same.
  • #1
Whitebread
23
0
I don't quite understand how the modified goodman equation can be applied to a multiaxial stress state. The explanation given in my stress analysis class has been quite confusing and verbose so I've come here to see if I can't get a better understanding.

First I'll lay out what I think to be true:

Utilizing the Von-Mises failure criterion in place of the uni-axial stresses in the goodman equation should be able to account for a multiaxial stress state AND fatigue. Here's how I think it should work

Uni-axial stress amplitude is replaced with von-mises stress amplitude (same equation, different stresses)

The fatigue limit at whatever number of cycles the designer is concerned with is replaced with the Von-Mises stress at that stress amplitude. Or: (Uni-Axial Fatigue limit at X cycles)*(1/3)=fully reversed stress amplitude (or SIGMAar in the good man equation).

Mean stress is replaced with mean von-mises stress (same equation, different stresses)

Ultimate stress is replaced with (sqrt(2)/3)*SIGMA(u) or the von mises stress at failure.

Equations:
Goodman.jpg

That last equation should be sqrt(2)/3. Made a mistake when writing the equations. Thanks.
Is this correct?
 
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  • #2
Actually you got the equations for stress amplitude (tau_a) and mean stress (tau_m) swapped, here.
 
  • #3
Whitebread wrote: "Utilizing the von Mises failure criterion in place of the uniaxial stresses in the [modified] Goodman equation should be able to account for a multiaxial stress state and fatigue. Here's how I think it should work. Uniaxial stress amplitude is replaced with von Mises stress amplitude."

Agreed.

Whitebread wrote: "The fatigue limit at whatever number of cycles the designer is concerned with is replaced with the von Mises stress at that stress amplitude. Or, (uniaxial fatigue limit at N cycles)*(1/3) = fully reversed stress amplitude (or sigma_ar in the [modified] Goodman equation)."

Disagree. Fatigue strength is a material property, not a von Mises stress. The fatigue strength should not be adjusted.

Whitebread wrote: "Mean stress is replaced with mean von Mises stress."

Agreed.

Whitebread wrote: "Ultimate stress is replaced with (sqrt(2)/3)*sigma_u, or the von Mises stress at failure."

Disagree. Tensile ultimate (mean) strength, Stu, is a material property, not a von Mises stress, and should not be adjusted.

A similar question is posted at thread https://www.physicsforums.com/showthread.php?t=304749" .
 
Last edited by a moderator:
  • #4
Well its been a while. Since I posted and the project I posted it for was long since been turned in. Thanks for the input and the link though. Its quite helpful since this information just doesn't seem to be recorded anywhere.
 

Related to On the application of the goodman equation to a multiaxial stress state

1. What is the Goodman equation and how is it used?

The Goodman equation is a fatigue analysis method used to predict the failure of a material under cyclic loading. It takes into account both the mean and alternating stresses in a multiaxial stress state to calculate the fatigue life of a material.

2. How is the Goodman equation different from other fatigue analysis methods?

The Goodman equation differs from other fatigue analysis methods because it considers both mean and alternating stresses, whereas other methods may only consider one or the other. This makes it more accurate for predicting the failure of materials under complex multiaxial stress states.

3. What are the limitations of the Goodman equation?

One limitation of the Goodman equation is that it assumes a linear relationship between stress and fatigue life. This may not always hold true for all materials and loading conditions. Additionally, it does not take into account any factors such as material defects or environmental conditions that may affect fatigue life.

4. How can the Goodman equation be applied in practical engineering situations?

The Goodman equation can be applied in practical engineering situations by using it to select appropriate materials and design components that will have a longer fatigue life under cyclic loading. It can also be used to determine the maximum stress levels that a material can withstand without failing under a given number of cycles.

5. Are there any alternative methods to the Goodman equation for fatigue analysis?

Yes, there are alternative methods to the Goodman equation for fatigue analysis, such as the S-N curve method and the Smith-Watson-Topper (SWT) method. Each method has its own advantages and limitations, and the most appropriate method should be chosen based on the specific material and loading conditions being analyzed.

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