I can provide you with the answer that comes is the result of college statics and dynamics classes.
What I can not do is provide you with an answer that carries the weight that would be provided by the stamp of a public engineer licensed to practice in the state of Florida, or your state of residence.
In all sincerity, I would ask that you not accept the word anyone who has not already approved residential construction issues in the City or County of your residence. And even then, do not take their word as accurate until you have the data in writing.
Hurricanes, slow or no insurance payments, and political inaction seem to happen at the same time.
My in-laws lived between I-95 and the Ocean, in Palm Beach County. I think the biggest difference between their house and yours, is the pool edging not being flush with the deck, and the house next door was a different color.
Realize I am guessing about everything from ASTM rating of the steel that would be used, to the hurricane wind design rating required in your zip code.
I would not blink if you were told that it needs a heavy 4 or a standard 5 inch beam. The 5 inch beam is less than 1/4” larger on each side. So we will go with that one.
I know nothing about the most recent building code and large pieces of steel around residential pools. But I will bet that 32’ bits of it weighing about 325 pounds and raised in the air, on steel, are not allowed within 6 feet of the water.
I am sure that there is now something about needing enough space to walk all the way around the perimeter of a pool.
There is likely also something about footings, saturated soil, flying debris, and the term ‘grounding’ will be used at least twice. If we add 6 feet to each side, we are well inside the bounds of reasonable.
We have now hit a 44’ clear span.
It gets a little tough to tell distance, but I am (again) guessing that this will put 1 post over your pool pump water works, and the other in front of the bathroom window. You are not going to be able to move the post closer to the pool, but
I am open your thoughts.
The question is now, what are you going to do about the beam that will now overlap with your eaves. The easiest answer it likely to wrap a hollow fiberglass post around the steel one. Then paint it blue and hope it is not noticed.
Since you are changing the patio roof so much, from the materials used, the distance between the supports, the additional length, and joists now being attached to steel instead of wood. I would expect the patio roof to be seen as new construction and not repair. This will require the patio roof to be self supported. This is much better for you than the alternative of having the patio roof seen as a part of your house.
It appears from your pictures that the house side of the patio roof requires the outside wall of your house, and is even attached to it with hurricane straps. The 2x on the wall, but connected to nothing else seems to be there as an attachment point should you want to put in a flat ceiling.
To recap up to this point, removing the 2 center posts and replace it with a single steel beam spanning the entire distance, you will need to build a 4 post, self standing, cabana type roof.
It can be against your house, and the roof can be integrated into your house, but it must be able to stand on its own.
In reality the posts closest to your house will need to be far enough away from your house that they will not disturbing you house’s slab. (But you might be able to offset the footing enough that the post can be right against the house. Way too much math for right now.)
Ok, at this point, it makes no sense to build anything but a rectangle of steel, and put it up on steel legs. You will need ’notches’ to allow for both ends of the new roof, to make room for where your house becomes wider.
You will need to decide what you want the roof to look like, the notched portions will be the hardest. Thanks to this now being all steel, you can add gussets, strengthening web where needed, this will help to keep the beam size down.
We still need to deal with the new joists, roof and shingles, there are some other decisions to be made first
The costs at this point are likely in excess of the quality of life you gain in having the 2 wooden posts removed.
It would be informative to call planing and zoning, and ask if they can think of any issues with you installing about 32’ of steel, where the posts are within 6 feet of a pool. The answer would allow a few facts to get involved here.
And this is how we truly figure out what size beam you need.
Best Answer
Some physics. Assuming the same material, length and loading, the bending strength is limited by the Second Moment of Inertia (I) of the cross sectional area. The math is complex, and even given a calculator, it's a bit confusing.
Assuming a vertical loading, You can estimate the relative change in the I value by cubing the scale change. (This estimate is accurate only for solid rectangular beams) This suggests that multiplying the vertical dimension by 1.5 would give a resulting increase in strength of 3.375 times.
If you're increasing the horizontal thickness, then you simply multiply by the scale. So increasing thickness by 1.5 would result in a strength increase of 1.5
I ran some numbers in an I calculator, and assuming a 3mm thickness, I got a value 3 times higher for 1.5" than for 1" if loaded against the longer axis, which shows that the estimator is reasonable.
For a 3mm tube of one inch, I got values of .8 and 2.75 for 1.5".
All things being equal, a hollow square is better than a hollow circle, and a hollow rectangle is better than both -- if you load it in the right direction.