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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.

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    $\begingroup$ Does this answer your question? What are the governing equations solved in FEA for structural mechanics? $\endgroup$
    – Solar Mike
    Commented Sep 28, 2021 at 10:15
  • $\begingroup$ 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. $\endgroup$
    – Rameez Ul Haq
    Commented Sep 28, 2021 at 11:57
  • $\begingroup$ Then check out books by Timoshenko - that will outline the theory for you. $\endgroup$
    – Solar Mike
    Commented Sep 28, 2021 at 13:53
  • $\begingroup$ 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). $\endgroup$
    – alephzero
    Commented Sep 28, 2021 at 13:55

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FEM cuts a structure into several elements (pieces of the structure). -Then reconnects elements at “nodes” as if nodes were pins or drops of glue that hold elements together (boundary conditions). -This process results in a set of simultaneous algebraic equations in the form of - [K]{u} = {F}; for structural mechanics, these terms represent [stiffness matrix]{displacement} = {force}.

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http://web.mit.edu/16.810/www/16.810_L4_CAE.pdf

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  • $\begingroup$ Actually, I already have this document and I have read it as well. If you look at the first page of this document, you will see ''Governing Equations" (which are actually partial differential equations). These governing equations can vary depending upon which field of physics or engineering we are trying to use FEM for. For example, the governing equations in Solid Mechanics are going to be different than in Acoustics or CFD. Even in Solid Mechanics, they can be different depending upon what are we using it for, like for Static Structural, Transient Analysis or Rigid Body Dynamics. $\endgroup$
    – Rameez Ul Haq
    Commented Sep 28, 2021 at 12:21
  • $\begingroup$ In this document, it has described the process that how does FEM proceeds. I actually want to know what governing equations are actually used in Solid Mechanics, and more specifically, in Static Structural module, Transient Analysis module, Rigid Body Dynamics module etc. $\endgroup$
    – Rameez Ul Haq
    Commented Sep 28, 2021 at 12:22
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    $\begingroup$ You need to buy a textbook then. $\endgroup$
    – r13
    Commented Sep 28, 2021 at 12:49
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    $\begingroup$ The Finite Element method can be used to solve any system of partial differential equations. Your question is really about continuum mechanics, not about FEA. $\endgroup$
    – alephzero
    Commented Sep 28, 2021 at 13:58

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