Polar second moment of Area 6

[Russmuss]

Hi I have written a program in visual lisp (part of Autocad)
that determines the cross sectional properties of a composite concrete beam section. Composite materials are either beam concrete, topping concrete, strands and rebar. It gives the moment of inertia, centroid and section modulus, using the transformed areas method.
Now I want to go one step further and find the 'polar second moment of area' (ability to resist torsion).
As I can understand the formula changes depending on the shape of the cross section. Can the polar second moment of area be determined from section modulus or other means?
I would rather not have a different formula for every different shaped cross section, this makes programing difficult.
I am not a structural engineer - so I hope my questions don't sound naive.
thanks if any one can help me russ

 

[KootK]

Polar moment of inertia is what Russmuss specifically asked for in his post.

It's important to remember that all the bt^3 methods only apply to thin walled, open sections. If what russmuss wants is a general, programmable method, that isn't it.

 

[petertor]

ShapeBuilder is hard to beat for finding section properties, although notice it just calculates Ipolar by adding Ix + Iy.

 

[Teguci]

I've been working on a torsional problem (crane girder with a rail offset) over the last week and I found this post very interesting. Just to follow with a couple of my findings:

Torsion is resisted by two different but related principals: Twisting and warping. The twisting of the beam causes the whole beam to rotate and induces shear stresses in the shell. Warping can best be described as bending in opposite ended elements (think of an I-beam where the top flange bends in one direction while the bottom flange bends in the other direction - This moment couple resists torsion)

Twisting (aka pure torsion or Saint-Venant's Torsion) is related to the polar moment of inertia (aka torsional constant J) and to the shear modulus of elasticity G.

Warping is related to the warping torsional constant Cw and the modulus of elasticity E.

For the modified wide flange girder that I have been working on, both of these elements play a role. The girder rotates a small amount and induces some pure torsion resistance. However, most of the torsional resistance comes from the opposingly bending top and bottom flanges (about 87% of the resistance).

Now to make it hopelessly complicated(smile):
Both of these torsional resistances are related to the rotation (theta) of the beam

Pure torsion Ms = G J x dTheta/dx
Warping torsion Mw = -E Cw x d3Theta/dx3
Ms + Mw = Torsion

the homogenous solution involves hyperbolic functions and it all depends on the loading of the beam (point load, distributed etc..) and thats as far as I'm going with this.

It is not as simple (smile) as saying the shape is closed or open and applying the J factor or the Cw factor.