Saturday, August 11, 2012

Mech: Understanding strength and deflection of truss structures

Have you ever wondered whether a truss with a single cross bar is better than a truss with many of them? How many cross bars exactly do we need?

Sure, we see a lot of things with truss bars at 45-deg angles so we can assume smart engineers determined this was best. But was it best for reducing deflection or was it best for strength to weight? If they were doing it for total strength to weight but not for deflection to weight, I might want to do it differently.

Let's start with strength calculations of the single cross-brace truss. If we assume the weight of the steel in each beam is 1 lb/ft and the truss is 3' then we end up with 11.16lbs for the truss.

What's the strength of that truss? Let's assume the steel will not buckle or compress, and it has a pull strength of 10,000 lb/beam. So if a beam was hung vertically and we put 9,999lbs on it, it'd be fine. If we put 10,001lbs on it, it'd break. If we put a weight on the end of this truss we see that center beam will break at a weight of 3,160 pounds on the end.

What about if we had that same 3' truss with 3 spans? Now the weight is 14.24, a lot heavier. What's the strength? This is harder to calculate. The way to do it is one truss section at a time. Let's start with the last truss section, it will take 7,070 pounds to break that center beam. What about the one before it? Well the last truss section only adds a few pounds of weight so this one will also break at around 7,070 lbs. Same with the section before this. So this truss is actually the strength of any one section, assuming the weight of the truss is negligible.

If we divide the weight a truss can hold by it's own weight, the multi section truss gets a ratio of 496 while the single section truss gets only 283. So the multi section truss is 75% stronger for it's weight.

What about deflection? Does the single truss deflect less? Could that be an advantage?

I made a model of this on my nifty new structure simulating I've been using for the mech. I allowed for 20% stretch in the sections that would be experiencing it; both the top piece and the cross bar of each section. I modeled trusses with 1 section, 3 sections, and 6 sections. The 3-section truss puts each cross bar at exactly 45 degree angles.

It's easy to see from the model the 3 section truss is deflecting less. It deflects 1.8' while the 1 section truss deflects 2.2', 22% more. Then again, the 3 section truss weighs 28% more. Here though I don't think we can just take the weight ratio. The weight ratio was appropriate before because we could just thicken the truss bars to increase weight and strength linearly. The bar thickness will not have a linear relationship with deflection. For starters we haven't defined what the stretch algorithm is, we just set it to 20% arbitrarily.

How are we going to get a handle on this? For starters we know that stretch is usually linear with strain in steel. And that mild steel will top out around 0.2% stretch before it breaks. So if we made our bars in the 1 section truss 28% thicker, we'd reduce the stretch about 28%; to 0.1448% stretch. Now our two sections are of equal weight but the 1 section truss deflects 20% less.

Unfortunately, to get here I had to make an assumption about the stretch on the top of the truss, that we're going to have all truss sections of equal thickness, and that buckling never happens. But in the end it's clear the many-section truss is shit, the truss with 45-deg cross bracing is strongest by a good margin, and the 1 cross brace truss probably has the least deflection but only by about 20% of minimum possible deflection.