Derivation of P-M interaction diagrams for axial+biaxially bended RC column 1

[JoeH78]

Dear All,

How am I supposed to derive a generic way or algorithm to extract a P-M diagram for arbitrary shaped RC columns, at least for primitive geometric shapes (paraleloid, L, I Shaped columns sections). Regarding the book REINFORCED CONCRETE Mechanics and Design ( by JAMES K. WIGHT and JAMES G. MACGREGOR) it instructs to find the centroid of shape and later on establish the equilibriium equations for forces and moments. Book also recommends to choose arbitrary neutral axis point(c) and iterate over it to find general P-M interaction diagram, for each c there is P-M pairs that represent a point on diagram.

Everything seems to be doable up to know from my point of view, the difficulty is if column is subjected to biaxially moments then neutral axis doesn't depend simply on neutral axis poisiton (c) but also the oblique angle it have in X-Y coordinate system. Having arbitrary values for angle + neutral axis location and combo them to find P-M may seem like costly thing even for nowaday computers and I'm also a bit dubious that I may hit the worse case scenerio.

I'll appreciate your guidance and comments,
Regards,

 

[KootK]

It sounds as though you've got a pretty good handle on the basics. I've attempted this myself and have had pretty good results using the goal seek functions in Excel and MathCAD. It was always a little buggy for complex shapes however. In the end, I gave up and went with approximations by Bressler and Parme etc.

There's a paper from the Journal of Computing in Civil Engineering called Interactive Column Design has suggests some interesting techniques such as annular rebar approximation. None of those worked much better than brute force methods for me however.

There is a member here that goes by the handle IDS. He's rather exceptional at programmatic stuff. Hopefully your thread will catch his eye and he can enlighten us as to how this is handled in commercial software.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[IDS]

I have a spreadsheet at:

https://newtonexcelbach.wordpress.c...nder-combined-axial-load-and-biaxial-bending/

that will find an interaction diagram under biaxial loads.

It works by iterating to find the NA direction and location for any given combination of axial load and moments.

The code is open source if you want to have a look, or let me know if you have any problems using it.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JoeH78]

Thank you all in advance,

@IDS, it seems that you put quite a lot effort on that, sure I'll study your code but that will require time, is it your proprietary way of solving diagram or did you refered to some articles/nomenclatures when doing that? If its refered to articles, do you mind sharing it which specific it was?

 

[WARose]

The 1991 edition of '246 Solved Structural Engineering Problems' (by Buckner) has such a problem solved. They find a few points that determine a conservative lower bound for the column.

 

[Denial]

There is a similar spreadsheet on my web site

http://rmniall.com

under Downloads>SoftwareDownloads.

I'm telling you this not for the spreadsheet itself but because the download package contains some detailed notes on the algorithm I used.

 

[IDS]

KootK - Thanks for the kind comments.

Joe - Yes, there is a lot of work that has gone into that spreadsheet over the years, and as a result the code is not as well documented as it could be. Much of the basics is just application of standard RC design theory, but there are several aspects that (as far as I know) are unique to my spreadsheets:

[ol 1]
[li]The basic section bending capacity routine uses a closed form solution, allowing for multiple reinforcement layers, any cross section, and prestress or axial load (or both). This greatly speeds up the iterative process to find the NA angle.[/li]
[li]The above routine requires the solution of cubic equations, so I have written a VBA routine for that. There is also a Quartic routine in the file, but it isn't needed for the ultimate strength calc.[/li]
[li]To apply the bending capacity routine to any cross section with any NA angle I needed to split any shape defined by a polyline into horizontal tapezoids, so that's what the routines HIP_2 and HIP2Trap do.[/li]
[li]The above routines allow the equilibrium bending moments to be found for any shape with an NA at any angle, and this can be used with the Excel solver to find the NA angle by iteration for any specified resultant moment combination. However the solver is very slow, and not always reliable, so I have also included my own solver routine (QuadBrent) which is much faster and seems to always work.[/li]
[li]The spreadsheet is also units aware, using my own VBA units functions.[/li]
[/ol]

There is some documentation to all of the above on my blog, but it's probably fairly hard to find the most relevant pages, so I will add a post in the next few days listing the best links for the routines listed above.

Any specific questions, please ask.

I can recommend a look at Denial's site as well. He has a different approach to mine, which works for an elastic stress distribution (if I recall correctly). Mine only works with a rectangular stress block at the moment.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[RFreund]

I've been wondering what the difference is between the IDS and Denial approach. Can you guys comment on this at all?

EIT

http://www.HowToEngineer.com

 

[Denial]

RFreund.

I have not looked into Doug's approach in any depth, but I'll stick my neck out (and invite Doug to wield the axe if necessary).[ ] In so doing I will assume that by "approach" you are referring to what the spreadsheets do rather than how they do it.

Code accommodation.[ ] Doug's is code-aware, allowing the explicit incorporation of load factors and capacity reduction factors.[ ] Mine is completely code-unaware, applying no factors to the raw problem-description numbers that the user enters.

Prestress accommodation.[ ] Doug's allows for prestress in a way that is much more intuitive, easy and flexible than mine.[ ] I added prestress as a bit of an afterthought, years after I had developed the main capabilities of the spreadsheet, and it shows.

Generality of material stress-strain models.[ ] Doug's is limited to a "rectangular stress block" for concrete, and straight elastoplastic for steel.[ ] Mine can handle pretty much any stress-strain curve that can be approximated in a piecewise linear manner, for both the concrete and the steel.

Units.[ ] Doug's allows the user to specify which units are used for which quantities.[ ] Mine offers absolutely no choice of units.

Output.[ ] Doug's can produce an interaction diagram.[ ] Mine is limited to tabulating stress and strain values for a (small) number of unrelated load cases.

Column curvature.[ ] Doug's does not seem to calculate curvature.[ ] Mine calculates it for each load case.

Sign convention.[ ] Doug uses the standard RC sign convention of compression positive for concrete and tension positive for steel.[ ] I use tension positive for all materials (just to make life difficult for structural engineers ).

 

[IDS]

Good summary.

Regarding how the programs work, the big difference is that mine uses a closed form solution to find the NA depth for any given angle, which speeds things up. The drawback is that it will only work with a rectangular stress block (or potentially linear elastic), whereas Denial's can use any defined stress-strain relationship.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[RFreund]

Thanks for the explanation!

A couple of questions:

1. So Denial's does not iterate through different locations of the N.A.? Rather it solves to match section "capacity" at a given (applied force) P, M input?
2. How does the column curvature effect the solution?
3. Do they yield different solutions when run with the same input parameters?


EIT

http://www.HowToEngineer.com

 

[Denial]

And a couple(!) of answers.

1.[ ] My solution does iterate, but not to find the neutral axis location directly.[ ] That would be a problem in two unknowns.[ ] Instead it iterates to find the equation that defines the strain variation over the cross-section,
E(X,Y) = a.X + b.Y + c
a problem in three unknowns (a,b,c).[ ] Once it has found this equation, the location of the neutral axis falls out as
a.X + b.Y + c = 0
although the method never actually uses this.[ ]

For each load case the method assumes an initial (a,b,c), and then integrates over the cross-section to calculate what set of external forces (P',M'x,M'y) would cause this strain distribution.[ ] Comparing (P',M'x,M'y) with the actual applied forces (P,Mx,My) gives a measure of the error at that iteration, and applying a correction (P-P',Mx-M'x,My-M'y) to the cross-section at its marginal stiffness leads to an improved estimate of (a,b,c).

2. Column curvature does not effect the solution.[ ] It falls out from the solution, because the curvature is given by Sqrt(a²+b²).

3. I believe that if they were fed identical problems the two approaches would come up with identical answers insofar as their answers overlap.[ ] This is because both methods are arithmetically exact.[ ] (Actually a purist would probably not call my method "exact" because it uses Gaussian integration, but, hey, we're engineers here.)[ ] Give it a go.[ ] Feed Doug's a problem, then "transcribe" the appropriate values into mine.

 

[Denial]

Inspired by this thread, I have finally got off my dot and modified the way my spreadsheet requires its users to specify any prestress.[ ] I believe the new approach is a bit easier, more intuitive, and more flexible.[ ] The new version of the spreadsheet is now on my web site.