Shift of plastic neutral axis due to minor-axis bending & eccentric axial compression on channel

[faremusai]

I want to calculate the plastic section modulus and location of plastic neutral axis of a parallel flange channel subject to bending about the minor axis and eccentric axial compression load acting along the channel back surface for Eurocode 3 design check purpose.

I think this problem can be decomposed into 3 steps:
1) Fully-plastic distribution of stress due to bending about the geometric minor axis only. Location of plastic neutral axis can be determined easily.
2) Shift of plastic neutral axis due to the application of eccentric axial compression NOT taking into account its location of application (i.e. assume coincide with the plastic neutral axis. The amount of shift can be determined easily.
3) Additional shift of plastic neutral axis due to the moment formed between the actual location of axial compression force and section resistant force at plastic neutral axis.

How to find the additional shift of plastic neutral axis in step (3)?

 

[rb1957]

I don't know Eurocode so this may be of limited "help".

compression + bending = beam column. Beam column analysis (in lieu of superposition of the two loads) may be required, if the axial load is high enough.

I think I would determine the compression stress on the section due to the compression load then add this to the allowable bending stresses (reducing the compression allowable, increasing the tension allowable) and calc the plastic moment the section can carry.

Ideally, to calc a true MS, I'd do this for several axial loads to produce an allowable curve (based on the interaction of compression and axial).

another day in paradise, or is paradise one day closer ?

 

[BridgeSmith]

I think the plastic moment capacity can be determined most simply by subtracting the compression stress due to the axial load (P/A) from the yield stress (yield strength), and then multiplying by the plastic section modulus of the section. If the axial load is applied eccentrically, subtract the moment due to eccentricity (P*delta, with delta equal to the distance from the plastic neutral axis to center of the applied load) from the previously calculated plastic moment capacity, to get the remaining plastic moment capacity after application of the eccentric axial load.

 

[rb1957]

I took the problem to be that in plastic stress analysis you assume that the section everywhere is close to failure (as distinct from the familiar linear varying elastic stress).

Thinking of it that way maybe it's very simple ... under a plastic moment the stress distribution is fixed everywhere (well, near enough). so to add axial load "all" you can do is move the NA (I think as the OP suggested) in order to allow the section to react axial load. this will add a net force to the balanced stress field reacting the applied moment. In doing this have you changed the moment reacted by the section ? Possibly not ??

another day in paradise, or is paradise one day closer ?

 

[faremusai]

For the shift of plastic neutral axis in step (2), I used the method mentioned in the following document (

http://www.colincaprani.com/files/notes/SAIII/Plastic Analysis 1011.pdf

page 20 & 21) in which the value of "beta" is the shift due to axial compression force only. Design resistance for bending will be affected.

I wonder if it is possible to use a similar approach for step (3) by introducing a ficitionous load and find a corresponding value of "beta".

 

[dik]

Your plastic section has a 'neutral axis' through the centre of areas with the area above and below equal to the combined moment and compression/tension load.

Dik

 

[ukbridge]

Theres some helpful discussion about this in "Designer's Guide to EN 1993-2" by Hendy and Smith pg149.

This isn't very helpful to the question but in most cases I would just design it as a class 3 section, particularly if theres both major and minor axis moments as well as a compressive force, torsion etc.

The various interaction equations are much simpler as a result. I don't know anything about the project, but I'd find it hard to justify the small amount of economy offered by designing the beam plastically. The amount of time you spend researching exactly how to go about it, making a spreadsheet all for what is likely a one-off isn't worth it IMO.

 

[faremusai]

I have found a section properties table for parallel flange channels (PFC) in SteelConstruction.info (

https://www.steelforlifebluebook.co.uk/pfc/bs5950/section-properties-reduced-plastic-modulus/)

which provided values for calculating the reduced plastic modulus under axial load and moment induced stresses.

For minor axis bending, threes coefficient are involved i.e. K1, K2 and K3 which seems to depend on the change number "n". Anyone know where can we find the formulas for K1, K2 and K3?

 

[ukbridge]

I did a quick search in Evernote and couldn't find anything. To be fair I don't have much for buildings.

It might be in SCI P363 - a lot of the equations are derived from that document. Try some other SCI building documents or BS 5950 which is the superseded code in the UK.

IMO just design it as a class 3 member and move on with your life, unless for some reason something absolutely cannot be changed.

 

[faremusai]

ukbridge,

Thank you for the suggestion.

In SCI P363, only the value of plastic section moduli are tabulated. The related formulae are not provided.

In SCI P202 Section 3.2.6(c), only the formula for major axis bending are provided. Formula for minor axis bending are too complex and not quoted.

I am making an Excel worksheet to find the classification and plastic section modulus of sections under various combinations of load. So I need the formula for calculation.