# Finite Element Method for Spherical Objects

I am familiar with the procedure of the FEM for simple planar objects like truss and etc. but my question would be whether it would be possible to use this method to calculate the stiffness or displacement of spherical (or ellipsoidal) objects under the pressure or load.

I guess if I approximate the round and closed object with a cells-shaped connected trusses (similar to soccer ball) and apply Hooke's law I might be able to calculate stiffness matrix by solving the algebraic equations that are derived as the result of applying Hooke's law to each side of the underlying grid.

Or I am totally wrong?

• After the answer of aaquib I wanted to remind that I meant 3d trusses (if it is a correct expression) Oct 2, 2023 at 13:08

This is certainly possible, variation of this is axisymmetry, which is available in any decent FEA software. If you are interested only in basic linear elasticity and small deformations, you could use Lamé solution as the basis of 1D element capable only of radial displacements. This is also a good starting point for something more advanced.

To validate your solution, you can use axisymmetric analysis in any FEA software with additional symmetry to a plane perpendicular to the axisymmetry axis, so you will need to model just 1/4 of a circle (which is the section of the sphere).

• I am thinking of doing it manually for now and the file you have attached is very helpful. My model is almost spheric though not totally a sphere. Oct 6, 2023 at 5:53
• Lamé solution was immensely helpful and I was totally unaware of it. thank you so much. May I please if I have a hollow sphere opposite to the example of the attached file (zero pressure inside and P outside), then will I be able to measure the displacement? will I simply plug P (rather than -P) in eq.11 and zero for eq-12? Oct 18, 2023 at 11:07
• @farhad_hep_81 You would put -P to eq. 11, because stress convention is positive for tension and negative for compression. Pressure acting from the outside will lead to compressive radial stress, which must be in equilibrium with the pressure at the outer surface. Oct 18, 2023 at 15:54
• Thanks for clarification. I googled a bit and it seems that lame solution is for cylinder rather than spheres. Am I missing something? By the way do you have a short numerical example of using it? Oct 19, 2023 at 6:34
• @farhad_hep_81 It is for both, the derivations are very similar. Cylinder solution is just used more in practice and you can also find the derivation in many sources. Oct 19, 2023 at 14:11

One of the advantages of the finite element method is that it can be very easily used with a multitude of geometries. Trusses/Frames represent only a tiny fraction of the geometries which can be analyzed. In your case, you are looking for 3D solid elements. Take a look at any good FE book and find the finite element procedure for 2D/3D linear elasticity. You will have your answer. Trusses/Frames are just 1D applications

Edit: OP clarified that he meant 3D trusses. In that case, this exercise should be straightforward since you will still be using 1D elements, and applying the appropriate rotations. This is just like how we solve 2D trusses, just that the rotation equations will now be $$3 \times 3$$

• Yes I was asking about 3d trusses. Oct 2, 2023 at 13:07
• Thanks for clarification, Being a physics graduated and not an engineer I am thinking of thoroughly reading the FEM book (e.g. By Moaveni or etc.). Do you have a better suggestion please? Oct 6, 2023 at 6:02
• This is an area where every other person you ask will have a different suggestion. I would ask you to pick a decent book and stick with it. Oct 6, 2023 at 14:04