CFT: Moment of Inertia? 1

[ElliottJames]

I'm working on a design that includes a concrete-filled steel tube. Might anyone have equations handy for the moments of inertia when the neutral axis does not go through the center (see attached diagram)?

Referring to the diagram, I'm looking for the MoI of the concrete in compression (area Cc), steel in compression (partial arc Sc) and the steel in tension (partial arc St). I guess I could derive it by integration (gasp!), but before I reinvent the wheel, I'm hoping these equations might be tabulated somewhere. Thanks...

 

[Agent666]

Eurocode 4 has a design procedure where the effective Stiffness is calculated.

Procedure is outlined in this pdf

Link

Be aware though this document is based on the previous 1990's revision of EN 1994: (Eurocode 4) Design of composite steel and concrete structures, but I think the stiffness part is the same in the latest revision. The procedure for working out the strength is slightly different in the most recent revision.

 

[ElliottJames]

@ Agent666: thanks, I understand the concept of effective stiffness. This document doesn't provide what I'm looking for, which is a general geometric equation for the moment of inertia of a chordal area and of a chordal portion of a hollow tube.

 

[PMR06]

Do you have access to Roarks? Table A.1 section 19&20 has the properties for circle segment you describe. If you google enough you might be able to find a "bootleg" copy. I suggest you buy the book if you don't have it though. Its a very handy reference to have around for just these situations.

 

[ElliottJames]

@ PRM06: Thanks, that's exactly what I was looking for. I'll take your advice and add it to my bookshelf.

 

[Agent666]

If you would work it out from first principles it's not really relevant to use directly in design. The effective stiffness method is more appropriate for deflection checks, as it takes into account tension stiffening and other confinement effects specific to concrete filled tubes and has been verified against real world tests. Similar to working out the effective stiffness of a reinforced concrete member, the moment of inertia isn't simply the moment of inertia at the location of a tension crack extending to the neutral axis as you have drawn.

I guess the way you are wanting to calculate the moment of inertia would result in a lower value so it would possibly be conservative if you are simply looking at deflections of an isolated member. But if the distribution of loads in your structure are affected by the stiffness of various elements in say a moment frame for example, then you might be missing out on some load in the tube as the member would be stiffer than your calculated value (which possibly be unconservative, can't say for sure as don't know what you are designing).

As long as you know the geometric properties about one axis/edge you can transform it back to the center of the pipe via the parallel axis theorem if required. Don't forget to transform the properties to one material if you are wanting to work with one modulus of elasticity.