I am trying to do some beam calculations..

I am building a pergola in the back yard, and need to calculate what size of square pipe steel to buy.... The span is 24', and the load will be a vine. This load weight has been approximated to a couple hundred pounds. The span will have a 3" contact point end. For cross beams i am using 5"X3"X1/4" angle iron, with the 5" side angled down.

Right now i am thinking of using 1/8" square pipe steel for the 24' runs, with a width and height of 4"X6" and a length of 288". I have calculated that this choice can handel 14.57lb/in.. However, i am unable to determine the amount of deflection, this is due to my lack of conceptual understanding... This is the part that i am stuck on.

Any help would be greatly appreciated.

You can compute deflection with beam tables, but this is generally only if your particular scenario is in the table. You need two properties of the beam and material itself, the modulus of elasticity (E), and the moment of inertia about the axis of bending (I). I can be computed based on the dimensions if not tabulated somewhere, and E is based on the material. A list of deflections equations (Delta x max) is at the available link.

http://www.awc.org/pdf/DA6-BeamFormulas.pdf

However, it may be difficult to find your particular configuration, you need to decide if the ends are fixed or pinned, the number of spans, loading conditions etc.

May not produce desired results.

Other alternative methods are double integration, work energy methods, and software. The beam tables are the easiest if you have the correct scenario.

However- if you have a 3-dimensional structure things may be completely different depending on how it is all bolted, welded or connected.

That's probably the only somewhat helpful information I can really provide as I don't really have a structural engineer mindset.

Maybe someone with construction experience with the particular materials and end goal can provide better help.