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Polar second moment of Area

Polar second moment of Area

Polar second moment of Area

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

RE: Polar second moment of Area

You should not be using the polar moment of inertia to calculate torsional stress or torsional rotation unless you are dealing with a circular cross section.  For other cross sections, there is warping and it is considerably more complicated.  Check membrane analogy in a book on theory of elasticity.


RE: Polar second moment of Area

Okay I understand that-
The polar moment of inertia cannot be used to analyze shapes with non-circular cross-sections. In such cases, the torsion constant can be substituted instead.
So how can I determine torsion constant of various cross sections without having to use a different formula for a every  cross sectional shape?

RE: Polar second moment of Area

I don't know where you come from or which code you are using.  In Canada, there are specific rules for designing for torsion in concrete sections.  I am pretty sure there are similar rules in the U.S.A. and probably the U.K. but where do you reside and which code do you use?


RE: Polar second moment of Area

I guess a few people dedicated half of their lifes to solve this problem and just a very few commercial softwares can calculate torsional constant for arbitrary section. One will need to mesh the section by finite element first, simple integration around the perimeter would not do.


RE: Polar second moment of Area

In other words, we do not know how to solve your problem, is that right, yakpol?


RE: Polar second moment of Area

Hmmm...So it seems that torsional constant is something that can not calculated for an arbitrary section very easily.
Does it involve masses of formula and a huge bunch of number crunching?
Yakpol - what does "mesh the section by finite element first"

RE: Polar second moment of Area

Quote (Russmuss):

Does it involve masses of formula and a huge bunch of number crunching?
Yes: it involves the solution of a Laplace equation (second order partial derivatives differential equation in 2D). With the background you have (based on your questions), you can't even figure out  what this problem looks like: you'll need to abandon it.

http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Polar second moment of Area

torsinal constants are very dependent on the section ... prizmatic, thin walled closed, thin walled open.  i don't think there is a general approach applicable to all, "one equation fits all".

RE: Polar second moment of Area

torsional constant for a rectangular section is not the polar second moment of area (like a circular cross-section) and it depends on the aspect ratio of the rectangle.

I believe without having a look at my textbooks that it is something like 0.33*x^3*y for square sections and 0.2*x^3*y for rectangular flat plate sections, where x is the shortest dimension (and therefore that face is subject to the greatest shear stress when a torque exists).

Torsion in reinforced or prestressed concrete sections can lead to torsional cracking if the cracking torque is exceeded and can reduce this stiffness by up to 90%, so I would be careful which the how much of the gross torsional stiffness you assume for your elements as not to attract to much torsion to them which will give you unrealistic behaviour of the structure.

For compatability torsion, I have seen a lot of engineers ignore the torsional constant all together, not saying that this is the correct approach to take.

RE: Polar second moment of Area

Thanks for the insides of this problem. Even the terminology of the solution sounds scary.
No, I don't know how to solve it. NASA does.


RE: Polar second moment of Area

Don't know who came up with this but it's nice to have it around when I'm feeling smug.

"Structural engineering is the art of molding materials we do not fully understand into shapes we cannot precisely analyze so as to withstand forces we cannot realistically assess in such a way that the community at large has no reason to suspect the extent of our ignorance."

The more I learn, the more questions I have.


RE: Polar second moment of Area

Let's assume that we're just talking about a "good enough" solution. After all, we're engineers not physicists.  

In that case look at it as a series of rectangles pieced together.  Torsional constant = summation of (b*t^3) / 3.  

If you want to get more sophisticated there are ways to make this more exact.  The reference I usually use is Formulas for Stress, Strain, and Structural Matrices by Walter Pilkey. In there, the equation for torsional constant of a rectangle is given as:
(b*t^3 / 3)*(1-0.63*t/b +0.052*t^5/b^5)

Keep in mind that this formula requires t to be smaller than b.

RE: Polar second moment of Area


I've tried to solve the same problem myself, via MathCAD.  I was on a real roll too.  I found coordinate formulas, similar to the criss-cross area calculation used in surveying, that addressed Area, Ix, Iy, and Ixy, yc, xc.  These were very easy to program.  

Ixy is what you're looking for, right?  I don't have my mechanics of materials book hand but I believe J = Ixy = Ix + Iy.  While the fancy torsional constants are required for the analysis of many members, Ixy is still a useful property in in its own right.  I say go ahead and calculate it.  To me, it sounds as though you're trying to automate some very tedious calculations.  It's up to the professional using that information (maybe you, I'm not sure) to determine whether or not they are valid or sufficient.

Anyhow, when I tried to calculate the torsional constants in my spreadsheet, I didn't get too far at all.  All the difficulties that arise with open/closed sections, multi-cellular sections etc made it intractable programatically.  I even went to a great deal of trouble to get what is considered the bible of torsion: Torsion in Structures by Kollbrunner.  It's a great book but didn't get me any closer to solving the problem.

My investigation concluded with the realization that the only practical, generalized method for finding shear centers, warping constants etc. was to using a finite element software package.  You can get one called Shape Builder from IES for $600 that will do everything you'll ever need.  I'm pretty sure that I wasted at least $600 in time away from my family trying to develop my own tool.  It was a very educational exercise however.

RE: Polar second moment of Area

Sorry, I meant to address Russmuss there, not Yakpol...

RE: Polar second moment of Area

I don't think Ixy is what we are looking for at all.

For very thin sections, the sum of bt^3/3 is adequate, but it does not work for thick sections.  I have not checked out JoshPlum's reference to Walter Pikey, so I won't comment on that.  Timoshenko gives factors for various b/t ratios.  If you were to include this in a computer program, you would need a table of values for various b/t ratios.


RE: Polar second moment of Area

Thanks everyone for your very interesting answers, they are very much appreciated.

 Prex sounds like he wants an apology from me - for posting the question on this forum. I do not know why he bothers to reply, when he writes patronizing comments such as:

"With your background you have (based on your questions) you
cant even figure out what the problem looks like: you will need to abandon it."

thats helpful to no one.

RE: Polar second moment of Area


The quote is from James Amrhein.

RE: Polar second moment of Area

I use R from Blodgett.

RE: Polar second moment of Area

one way of deoing it to idealise the general cross section to componenet rectangles and cover as much cross sectional area as much as possible.

The torsion inertiea can then be calculated as sum of 0.3*(b^3*d^3)/(b^2+d^2).

This is an approximation but is not very far from real value unless you are in a very tight design sitiation.

The other way is to use a coefficient depending upon width/depth ratio say K and the torsional inertiea in that cse would be K*b*d. This K value you can find in any standard structural engineering book like E.C.Hambly (Bridge deck behaviour) if you need to be more accurate.

For a thin wall section like steel box you can use this formula to calculate torsional inertia J = 4*A^2/(SUM(ds/dt).

Where A is the area enclosed by the centre line of the cross section, s  = width and t is the thickness of each individual section.

For thick hollow sections, calculate J for the outer section and for inner section and deduct from the bigger value to get the effective J.

RE: Polar second moment of Area

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.

RE: Polar second moment of Area

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.

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