In these page we want to analyze and compare the technical characteristics of three different elements often used in steel constructions:
Square hollow section
Circular hollow section
Wide flanges beams (HEA-HEB)

These elements have similar applications and are often used as equivalent alternatives, depending on the conditions of the application, stress, joint.
All the three considered elements, in the range H = 100 - 300 mm, have a similar square shape H x B or D x D and can have the same cross-sectional area.
In the following figure the three elements are placed side by side, with same dimensions and cross sectional area. In the same figure are also indicated the typical dimensions:
High, width, diameter
Thickness
Fillet radius, corner radius

Now we want to analyze their resistance characteristics, in details:
Momentum of inertia according axis x-x
Momentum of inertia according axis y-y
Torsional momentum of inertia, axis z

For the exact geometry, dimensions and tolerances, we refer to the relevant European standards:
UNI-EN 10210 - 10219
EURONORM 53-62

The mentioned resistance characteristics can be summarised in the following formulas:

Now let's try to imagine the criteria to be used to compare the three different profiles here considered, referring to the general criteria that may draw their choice into a project.

In details, we propose to use the following conditions to compare them:
Same dimensions
Same weight

This means to find the ratio between the momentum of inertia of the three different sections, having same dimensions ( H, B, D ) and same weight or cross sectional area.

The complexity of the formulas written in the previous pages is not giving a possibility to write the mentioned ratio, between momentum of inertia of different profiles, in a simple and synthetic form, clear and evident for everybody, with no need of computers and equation solving software. We must find a different way to come to a simple and synthetic solution.
We have processed the mentioned formulas in deep, with the intention to find a simple and synthetic form, even if not complete and rigorous as the previous ones, possibly approximate but useful for our goal.

We have made a sensitivity analysis of the formulas versus the different variables, to evaluate their effective influence on the final results, then we have tried different strategies of approximation, to find homogeneous expressions, easy to be compared. The result of this operation has been very interesting and has given a different formulation that we submit to Your opinion and to Your check. We came to the conclusion that all the momentum of inertia, for the considered profiles, in the range H = 100-300 mm, can be written in the following form:

Obviously, they are approximate expressions, using approximate constant factors K.
But the approximation, as we can see in the following table, is in the same range of the production tolerances

Now, having these simplified formulas, we can easily calculate the ratios between the momentum of inertia of the different profiles, having same dimensions and cross-sectional area, being these ratios simply the ratios between the K factors.

In the following graphs we can see that in the range, H= 100-300 mm, where the profiles can be compared having same dimensions and weight, the K factors are represented with pretty flat curves. This is confirming that, in the considered range, they can be considered constant factors with very good approximation.

Same in these following graphs, we can see that the ratios between the K factors of different profiles, in the considered range H=100-300 mm, are represented with pretty flat curves, confirming that also these ratios can be considered constant factors with very good approximation.

At the end
of this work we have reached our goal successfully.
We have found a simple way to compare the resistance characteristics ( momentum of inertia ) of the three considered profiles, that apparently seems very different between them.

The previous tables and graphs show this direct comparison, that can be summarised as follows:

Flection according axis x-x:
Wide flanges beam HEA-HEB is in average
20% superior to square hollow section
56% superior to circular hollow section

Flection according axis y-y-:
Square hollow section is in average
130% superior to wide flange beam HEA-HEB
30% superior to circular hollow section

Torsion :
Square hollow section is in average
9% superior to circular hollow section
250 times superior to wide flanges beam HEA-HEB

To come to the final conclusion we had to find a different form of analythical representation of the momentum of inertia that, always considering the limits of validity of the approximation, can be considered itself a significant result.

We hope our work might be considered a useful reference for steel constructions designer.