This should be a straightforward question for anyone who remembers
their strength of materials class from engineering school.

We have a boom made from square steel tubing 3" x 3" x 11 gage (about
1/8" wall)and 20 feet long.  The boom is freely hinged at one end and
supported by a diagonal chain attached 12 feet from the hinged end
with the other end about 3 feet above the hinged end.  The boom does
not carry very much weight, but it is mounted on an agricultural
tractor and drivers tend to whack it into obstructions such as fence
posts.

The question currently being debated is whether it is better to have
the tube mounted "square" (flat faces horizontal and vertical) or
rotated 45 degrees in a "diamond" orientation.

Mandatory question:  Which orientation of the tube is stronger and
stiffer, and by approximately how much?  Why?

Optional question:  Which orientation will tolerate fence post impacts better? Why?

Hello kann, you are correct in your thinking. The section modulus,
which is used to calculate bending strength, does depend on
orientation in the case of a square shape. Whether the section is a
piece of square tubing or a solid square bar the same ratio holds
true. In the first case where your boom is mounted "square", the
technical term is "axis of moments through center". In the second
"diamond" mounting, it would be said to have "axis of moments on
diagonal". The formulas for the two cases are as follows:

Axis of moments through center:

Section modulus = d^3 / 6 = d^3 x 0.167

Axis of moments on diagonal:

Section modulus = d^3 / (6 x sqrt 2) = d^3 x 0.118

In both formulas d is the length of a side or 3 inches in your case.

You can see by this that the section modulus is larger when the boom
is mounted "square" by the ratio of 0.167/0.118 or 1.42. This means
that your boom is 1.42 times stronger when it is mounted as you have
it now in the "square" orientation. However, this is only true if you
are using the boom to pick up a load in a straight vertical pull. If
you were to pull at 45 degrees, the "diamond" mounting would be
stronger.

Now to the problem of fence post impacts. If the boom impacts the post
and the load applied is in a purely horizontal plane, then the boom is
still stronger by exactly the same ratio. The only thing that would
change this is if the post were to hit the boom at an odd angle.

I think I have answered all the points of your question. However, if I
have left something out or there is anything you don't understand,
please ask for a clarification and I will do my best to answer.

Hope this wins your bet, Redhoss

A good clear answer.  Thank you very much, Redhoss.