Imagine a cantilever beam modelled with solid elements in FEA. Now, I want to apply loads to it in X, Y and Z axis. First option is to apply the load altogether, and then observe the results after it reaches equilibrium. Second option is that I can first apply the complete load along X axis, body reaches equilibrium, then I apply the complete load in Y axis, again body reaches equilibrium and then finally I apply the load in Z axis.

Can just anyone tell me that should the displacements, strains, stresses etc should be the same for the first case as compare to the second case?

I talked with some people and few of them say that it should be the same because ultimately, the total load that you are applying at the end of final equilibrium is the same, so it shouldn't make a difference. Furthermore, the method of superposition directly alludes that the results arising from one loading condition can be added on top of arising from other loading conditions.

However, others say that it shouldn't be the same because the stiffness of the structure changes while you apply the load. If you apply the complete X axis load first, then the final stiffness of the structure will be different, onto which you now will apply the load in Y direction, and subsequently in the Z direction. So the final results such as deformations, strains and stresses shouldn't be the same as compare to the case where you were applying the load together at the same time.

Which one is true? A comprhensive elaboration will be extremely appreciated.

  • $\begingroup$ Why not try both and see what happens? $\endgroup$ Commented Jan 4, 2022 at 14:32

1 Answer 1


Short answer - no equations answer

  • for a linear analysis ( small displacement) you should get the same results. (this is essentially the principle of superposition, or linear combination).

  • For a non linear analysis (second order/large displacement) in the general case the order of the load can affect the outcome. Simplest example an analysis with compression load to induce buckling in the z axis. If you apply first a load in X axis then the z axis and finally Y you'd get totally different results than when you apply Y-Z- X

  • $\begingroup$ But what about applying the loads altogether at the same time for non-linear analysis? Will it still generate results different than, say, if I apply the same loads one by one, in the same non-linear analysis? $\endgroup$ Commented Jan 3, 2022 at 20:48
  • $\begingroup$ What you are asking I interpret it as: If A is not equal to B, is it possible for C to be equal to A and C equal to B at the same time? It might be to one but it to both. $\endgroup$
    – NMech
    Commented Jan 3, 2022 at 20:53
  • $\begingroup$ You know why I asked that again? Because I was hoping someone would reply or maybe answer with a detailed reasoning behind that lol $\endgroup$ Commented Jan 3, 2022 at 21:47
  • $\begingroup$ Because applying the load altogether concurrently will change the stiffness of the body in a different manner as compare stiffness change which we will get if we apply the load one by one, which in turn changes the results. Yes, I know this. But this is just something which is known, and people believe it. I couldn't visualize it or imagine it, unfortunately (sorry for my bad imagination skills). $\endgroup$ Commented Jan 3, 2022 at 21:47
  • $\begingroup$ I think the non-linear here means the non-linearity of the force, not the geometry or material properties. Within the linear-elastic range, how stiffness can change due to load, or load sequence? Provide an example, please. $\endgroup$
    – r13
    Commented Jan 4, 2022 at 0:06

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