## Parallel Axis Theorem for Z ?

## Parallel Axis Theorem for Z ?

(OP)

I need to calculate the plastic section modulus for some moderately complicated shapes.

If I were computing the elastic section modulus (via the moment of inertia), I'd have a parallel axis theorem to help me.

Does something like a parallel axis theorem exist for plastic section modulus calcs? I've never seen something like this and I looked through my books already.

A reference to something like this would be greatly appreciated.

e

If I were computing the elastic section modulus (via the moment of inertia), I'd have a parallel axis theorem to help me.

Does something like a parallel axis theorem exist for plastic section modulus calcs? I've never seen something like this and I looked through my books already.

A reference to something like this would be greatly appreciated.

e

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

Based on that find, the moment due to the tension/compression couple (similar to elastic design) which is your Mp and you know Mp=Z*SigmaYield; Z=Mp/SigmaYield

I think programs like SAP2000 let you input custom section dimensions, then it calculates the I, Z, S etc... I could be mistaken though.

## RE: Parallel Axis Theorem for Z ?

Thanks guys.

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

BTW, I've calculated Z for years and understand exactly what Z and S are!

Save your keystrokes with regard to the basic stuff, LOL.

## RE: Parallel Axis Theorem for Z ?

this is reasonable and conservative (which is a good thing). if you need to squeeze a little more out of something "Cozzone" gives you a methodology for allowing the remote fibers to be at a higher stress (exceed yield).

## RE: Parallel Axis Theorem for Z ?

It is an interesting idea and I'd like to research it. My analysis books don't have anything on it, so I'll have to google it or (gasp) use the library! Do you have a paper on the subject that I could download?

The yield plateau for mild steel is very long, so I don't know if we get enough enough rotation in the plastic hinge to get into strain hardening before eventual local buckling.

Our b/t limits for I-shapes are not stringent enough to actually allow squashing of the flanges. They're set to preclude local buckling up to some rotation. AISC calls it a "rotation capacity of 3" but I forgot how that's defined.

## RE: Parallel Axis Theorem for Z ?

I-beams don't gain very much from this approach (the whole idea is that the material near the neutral axis can absorb more load after the remote fibers yield); if there isn't much material there (as in an I beam) there isn't much benefit. i'd stick with your assumption of constant yield stress; much simpler and conservative.

## RE: Parallel Axis Theorem for Z ?

If you consider a simple (2) column with (1) beam framing between them framing system - assume shearwalls so no lateral loads present. Once the plastic hinge forms in the center of the beam, the framing has failed. You can not get additional capacity out of it. The amount of deformation necessary to get to that strain hardening will cause the beam to act like (2) cables just pulling the columns over.

It would certainly make for interesting reading, but I would not ever consider using it for normal building applications.

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

You are exactly right. Plastic bending is simply Zx*Fy. rb1957 is suggesting using a value greater than Fy at the extreme fibers. That only occurs with strain hardening - which requires a great deal of deformation.

## RE: Parallel Axis Theorem for Z ?

This is not a problem I have ever given much thought to before, but what happens if the cross section is not symmetrical? You can no longer assume that the neutral axis will be parallel to the axis about which the applied moment is acting, so there is an infinite number of lines which divide the cross section into equal areas above and below that line.

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Let's say monosymmetric and bent about one of the principal axes to keep it simple.

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

I wouldn't recommend buying this book though, I dont think it's all that good.

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?

there is an approach for plastic bending of non-symmetric section in the MSFC Stress Analysis Manual (at http://euler9.tripod.com/analysis/asm.html).

this also covers the more typical Cozzone analysis, which is a bit more complicated than swivel63's suggestion.

but i agree with swivel's final comment !

## RE: Parallel Axis Theorem for Z ?

## RE: Parallel Axis Theorem for Z ?