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Axial and Bending - Find Neutral Axis Routine

Axial and Bending - Find Neutral Axis Routine

I'm curious to know how others handle this -

When you have a linear elastic material member subject to axial and bending forces, say a concrete or masonry section (anything though really); how do you typically find the neutral axis when creating a calculation?

'Typically' you have 3 unknowns - axial force of section (not of load), moment of section (again not load) and the location of the neutral axis. I believe you may also have 3 equations to solve. 1.) sum of the forces, 2.) sum the moments and 3.) the ratio of the stress (or maybe strain) and distance to the neutral axis. This 3rd equation may not be correct. In any case it seems as though the neutral axis is normally found by iteration. Basically move the NA until the eccentricity of the section matches that of the applied load (or the equilibrium equations are satisfied). So my questions are as follows:

1.) Do you use iteration to find the NA when programming a calculation?
2.) If yes, How. Meaning what does the loop syntax look like or do you do this in excel?
3.) Do you run into problems with extremely large applied compressive forces (causes the distance to the NA to become very large) or very large bending moments?
4. If yes, how do you handle this?

Thanks in advance!


RE: Axial and Bending - Find Neutral Axis Routine

For a material that is linear-elastic you can find the stresses due to bending and due to axial load separately, then add them together. You don't need to know the position of the NA, but if you want it it's easy to calculate from the known stresses on the top and bottom face.

For a material like reinforced concrete which is designed on the basis of zero tensile strength the behaviour is non-linear, so that method does not work. In this case you can find the NA by iteration by finding the eccentricity of the resultant reaction force, but it's reasonably straightforward to develop a cubic equation for the depth of the NA, which can be solved analytically or by iteration, as you prefer.

More details and sample spreadsheets here:
and following posts (click on the "ping-back" at the bottom of the post to go to the next in the series)

Doug Jenkins
Interactive Design Services

RE: Axial and Bending - Find Neutral Axis Routine


Thanks. I had myself all twisted around yesterday. I spent 2 hours convincing myself that P/A +- M/s would give me the same result as using the equations of equilibrium for a linear elastic material. I should have held off on posting this as I don't think I fully understood my problem.
Just in-case anyone else comes across this mental battle here is my work:

I was creating another calculation for masonry for a member subject to axial/bending when I was started thinking more about using iteration to find the NA. At first I wondered if it were possible to somehow use a P/A + M/S approach for ASD or Strength Design masonry, which I don't think it is as you really need to find the strains which cannot be directly solved for with by only knowing the external forces. Basically thinking can I take the strain due to axial + strain due to bending and add these. The only problem is that you usually are assuming a maximum allowable strain and working from there. Anyway I will have a look at your blog, I actually hope read more of it, I just haven't had the time and my math skills need to improve.

Is it reasonable to solve for the NA if you have multiply layers of reinforcement?


RE: Axial and Bending - Find Neutral Axis Routine

I think I may have confused you. If the resultant stress on the tension face is less than the flexural tensile strength then the behaviour will be linear elastic and you can combine the stresses due to the axial force and bending moment to find the stresses at the top and bottom face, and hence the position of the NA, as you have done in your spreadsheet.

It only becomes more complicated if you have a reinforced section, and the resultant stress at the tensile face exceeds the flexural tensile strength. In this case you can derive an equation for the eccentricity of the reaction force, which depends only on the position of the NA (assuming the steel is fully linear elastic and the concrete is linear-elastic in compression). The same method can be applied with multiple layers of reinforcemnt and non-rectengular sections.

Doug Jenkins
Interactive Design Services

RE: Axial and Bending - Find Neutral Axis Routine

Many algorithms have been proposed to perform cross sectional calculations, many of the utilize fiber elements, or finite/boundary elements, or even semi-analytic solutions, with each of them having different Pros and Cons. I would suggest to have a look at the documentation of "Cross Section Analysis & Design" by ENGISSOL, which seems very comprehensive and reports the basics of such calculation routines.

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