trajectory neutral beam axis with variable section not symmetrically

[Vaik]

Hello.
having searched for a long time on the web and on the forum itself regarding what is reported in the title, without finding any exhaustive answer, I write these lines. I will refer to the scheme I attach. also specific that when I write "an", I will mean "neutral axis".
My question is:
how can I determine the position of the neutral axis with which to calculate the second-order surface inertia module (i [mm^4] ) in a beam with carvings, section variations ?

My reasoning is as follows:
example a:
I start from the simple example of a constant section beam (for example rectangular bxh), fig1, stuck to an extreme and loaded with concentrated force to the extreme opposite. the neutral axis to which to calculate the moment of inertia izz, in this configuration, will coincide with the axis of symmetry z-z, fig6. No problem.

example b:
If I now add a slit as in fig9, removing material, I get, in correspondence of the cut I have operated, a section bxh', with h'<h. I have to think of the position of the neutral axis in this section I would be spontaneous to place it also in half-try and thus coincide with z'-z' (fig9 ).

considering now the whole beam of fig9 along its larger size and thus displaying the trajectory of the successive positions of the neutral axis for each section that compose it, I would get a broken line with a pattern similar to the lower edge of the same beam of fig9 which passes exactly for the half-points of each section leaving so much traction material above, as material in compression below (I did not prepare an image in the diagram... ).

d1) is this a correct and truthful result?
d2) or the above trajectory is, instead, that of fig.10 and fig11 ? or the trajectory of the neutral axis undergoes a δy variation, compared to the position obtained by calculating according to the example a, which places the neutral axis of the bxh section ( near the gap ) to a "half way" between the position of the neutral axis of example a and example b.

d3) if the question is affirmed d2, then the "throughway" how can one quantify? Could you approach the average position between those obtained by example a and example b?

I tried to give myself an answer by reproducing examples with the inventor fem, and I also looked for autocad plug-ins that calculated the position of the neutral axis, but nothing satisfactory.
If there are any engineers who can also only refer to a link (also sites in English) that explains exactly what they asked, I would be grateful!
in the calculation of the position of the an I have thought of the balance that the sections that stand above and below the neutral axis must respect, but still do not return anything.

 

[meccanicamg]

Unfortunately all the theory made on books takes into account that the section remains constant along the beam. Therefore in terms of section variations, an average section of the profile must be considered, valid from beginning to end. Accidentality, i.e. sectional variations should be evaluated locally with carving coefficients and what else and not as a real calculation template.
what can be very close to reality will be an accurate fem. Only in this way can you validate the retouched media theory locally.
Unfortunately, there are no other paths that can be traced to reality.
therefore the neutral axis is the average beam as if it had no carvings.

 

[Vaik]

Unfortunately all the theory made on books takes into account that the section remains constant along the beam. Therefore in terms of section variations, an average section of the profile must be considered, valid from beginning to end. Accidentality, i.e. sectional variations should be evaluated locally with carving coefficients and what else and not as a real calculation template.
what can be very close to reality will be an accurate fem. Only in this way can you validate the retouched media theory locally.
Unfortunately, there are no other paths that can be traced to reality.
therefore the neutral axis is the average beam as if it had no carvings.

suggest that you consider the largest all-resistant hazelnut section. a little like you do when sizing a tree that has a tab. in case of sizing, considering safety coefficients and oversize is correct and lawful.
However, in case you want to compare two sections, both with forms that do not share the same main axes of inertia, of which I can still calculate the moment of inertia and identify the position of the main axes of inertia (my case for which I solved the integral and then verified with autocad the veracity of results). I should refer both sections to a chosen axis arbitrarily, but nevertheless parallel to the main one, and then recalculate the moment of inertia of the sections taken into account by comparing it? Can it be correct or doesn't make sense?

 

[meccanicamg]

in case you have two beams with different section, you must consider the average geometry of both sections and evaluate them for both the main axes of inertia.
the method is just like the tree with a tab.

 

[Vaik]

Thank you for the prompt intervention you responded to!
I will make some further calculation as an example because this is very curious to me.
I will post these examples in case to conclude the discussion and draw conclusions and to verify that I have understood well what you mean.

If someone else wants to add something, go ahead. even any suggestions on software or methods to locate the an would be well accepted.

 

[meccanicamg]

Thank you for the prompt intervention you responded to!
I will make some further calculation as an example because this is very curious to me.
I will post these examples in case to conclude the discussion and draw conclusions and to verify that I have understood well what you mean.

If someone else wants to add something, go ahead. even any suggestions on software or methods to locate the an would be well accepted.

So how did the calculation of the simplistic hypothesis end?