# What are the fundamental and governing equations used for structural/solid mechanics in FEA?

Can anyone please provide me a list of governing equations of structural/solid mechanics which are used by FEA softwares during solving. And how are these fundamental equations implemented onto each and every element in FEA? (And also, can these equations be different for different element types)?

I am already aware that FEA softwares basically makes use of these governing equations of structural/solid mechanics and apply its own Finite Element procedure in order to compute and approximate the results (such as displacements, stresses, energy etc) for each of the nodes and elements. But what fundamental equations and how are they incorporated for each of the elements, I couldn't understand it. These governing equations are essentially partial differential equations (whose solution is basically approximated by the FEA solver) but which ones, don't exactly know.

Just talking about simple static structural analysis here, with structure being in elastic range all the time.

• Does this answer your question? What are the governing equations solved in FEA for structural mechanics? Commented Sep 28, 2021 at 10:15
• I already checked out the thread that you have linked before opening this thread. Unfortunately, only the navier-stokes equation is mentioned overall, which is useful for the CFD but not for solid mechanics. The replier tried to make some analogy with the solid mechanics equations but none of them were provided. Thats why I specifically opened this thread where (a person already aware of all the fundamental and governing equations for solid mechanics used in FEA) they can listed clearly. Commented Sep 28, 2021 at 11:57
• Then check out books by Timoshenko - that will outline the theory for you. Commented Sep 28, 2021 at 13:53
• To answer this, you need a (large) textbook for the first part, and a (very large) collection of research papers for the second part. (FWIW Timoshenko is an excellent summary of the state of the art if FEA 50 years ago, if that is enough to get you started). Commented Sep 28, 2021 at 13:55