# What could be the reason for these sections not agreeing?

I have two results that don't seem to match up and I can't figure out why.

I built a hollow square section (HSS) using two different structural analysis softwares (they both agreed). However, they don't seem to agree with the AISC steel table despite the dimensions being identical. It seems to be the Torsion Constant (J) which is slightly off in the table.

What could be the possible reason for this as I am totally stumped?

Simple formulas for torsion constants of this type of hollow section are all approximate. The AISC tables may be using a different formula from the calculator and making some allowance for the radii at the corners of the section. Or, the AISC values might be derived from a detailed three-dimensional finite element analysis, which will converge to the "correct" torsion constant value.

For "serious" work, use the values from the data sheets, even though for your section the difference is too small to be important.

• Yes i agree the difference in the values is probably negligible. However it is interesting how the values differ and the softwares produce a slightly different result (~2-3%). I would have thought the softwares would be using a more exact method of calculation over the table - possibly the table doesn't factor in the radii. Sep 8 '15 at 15:49
• In practice, this precise of a value is plenty. When you take into account small differences in the composition of particular pieces of material you use, imperfect fabrication methods, and several other factors, +/- 5% is a decent workable tolerance. One of many reasons we incorporate a factor of safety into important calculations. Sep 9 '15 at 19:14
• @pauloz1890 the OP's calculator can't factor in the radii, because they are not specified. The AISC table might factor them in (but I don't know if it does). Also, you have to be careful what you mean by "exact" here. For example the cross section will not remain plane under torsion, but the change in stiffness caused by warping out of plane depends on the length of the section, and on the way the ends are fixed. Sep 9 '15 at 19:29
• I said a "more exact" method of calculation -- closer to the true solution if the section was manufactured according to specification perfectly and obviously it is under no load. Of course the software takes into account the radii - it has a field for it and I've used the software before - that field changes the values. By the way for a HSS section the radius does equal the wall thickness - the OP has done this correctly. Sep 9 '15 at 22:53
• @alephzero In fact you can see on this page using the same values that the OP used with a radius of 0 that the software is taking the radius into account. Sep 9 '15 at 23:12

If you look at the "book" values and the "calculated" value of many shapes, you will often find that there are differences. This doesn't only happen with hollow sections.

The differences are usually small enough that it doesn't matter (and there is as much variation in the material properties).

As @alephzero mentions, most of the variation is from the fillet or radius at the corners.

The fun happens when a given shape changes properties between editions of the manual without changes in the dimensions. Sometimes this is errata, sometimes it is "re-calculating" the table properties to match the "as-rolled" properties. The rolling mill has tolerances, so it may just be that one side of the tolerance is more common than the other. This would skew the average properties.

The approximate equation for the torsional constant, $J$ of a thin-walled square tube is: $$J = tb^3$$ Where t is the wall thickness and b is the width between the wall centerlines.

Using this equation you can verify that SkyCiv is ignoring the corner radius when calculating J.

There's no simple closed-form solution for $J$ with rounded corners, but we can observe that accounting for the corner radii increases $J$. Where the rectilinear corners produced stress concentrations, the curved corners produce a more 'uniform' torsional stress flow, and therefore a torsionally stiffer section. (As least, I think that's a valid mental model.)

You can read the full AISC discussion of torsion in Design Guide 9.

This paper by Darwish and Johnston has a fairly extensive discussion of the effect of corner radii on torsional constants.