Stress analysis of a cantilever beam using FDM

[rajaditya]

I am a CSE researcher with a not so in depth background of physics. As a part of my research in object modelling, I am trying to computationally figure out the stress for various objects by using Finite Difference Method (not FEM which is the defacto standard for stress analysis). The reason for this has to do with the other part of my research.

Anyhow I have written and solved the 3D elastostatic Lame equations and calculated the Von Mises stress. This is the problem I am facing :

When I am doing this for a 2D beam, the stress is calculated correctly. The Stress gradually decreases from the hinge point to the other parts. But in 3D the calculations are not correct as I am seeing the maximum Von Mises stress only in the middle portion of the beam instead of at the starting.

I am only considering the weight of the grid cell at each grid point for the external force. (in the Z direction). The displacements at the hinge points are all 0.

My question is : What changes in the stress computation of a cantilever beam between a 2D and a 3D formulation. Or am I ignoring some reactive forces which are of importance in 3D that is likely causing the maximum mises stress to shift to the center of the beam. (Of course it is entirely possible that my code is wrong but in that case it should have given haywire results for 2D case also).

If my question is not clear, I can supply as many details as required. Any inputs will be greatly appreciated.

 

[eljose79]

Dear researcher,

Thank you for sharing your current research and the specific problem you are facing. It is commendable that you have taken the initiative to solve the 3D elastostatic Lame equations and calculate the Von Mises stress, even with a limited background in physics.

Firstly, I would like to clarify that the Finite Difference Method (FDM) and Finite Element Method (FEM) are both widely used numerical methods for solving stress analysis problems. FEM may be considered as the "defacto standard" due to its versatility and ability to handle complex geometries, but FDM is also a valid approach.

In regards to your question about the differences in stress computation between 2D and 3D formulations, the main factor to consider is the additional degree of freedom in the third dimension. In 2D, the beam is only able to deform in the plane of the beam, but in 3D, it can also deform in the transverse direction (perpendicular to the plane of the beam). This additional degree of freedom allows for more complex stress distributions, which may explain why the maximum Von Mises stress is not occurring at the starting point in the 3D case.

Additionally, it is important to consider the boundary conditions and external forces applied to the beam. In 3D, there may be reactions at the end of the beam that are not present in the 2D case, which could affect the stress distribution. It is also possible that there are errors in your code that are causing the discrepancy in results.

I would suggest carefully reviewing your code and boundary conditions to ensure they are accurately representing the 3D problem. You may also want to compare your results to analytical solutions or results from other numerical methods to confirm the accuracy of your calculations.

I hope this helps in addressing your question. If you require further clarification or assistance, please do not hesitate to provide more details and I will do my best to help. Good luck with your research!