This is my first time here and I have looked at other questions regarding pergolas. However, my situation is unique. There will be no load on the beams. The pergola is 20 foot x 20 foot. It only contains 4 posts and 4 beams. There will be no rafters or lattices.
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For a beam that long deflection will control over strength. A 2x6 HF#2 would pass code with 0.6" sag in the middle if you are ok with that. Going with a 2x4 will result in a deflection of about 1-3/4" and it will want to roll over in the middle since there are no bracing members. If you go with cedar or pressure treated, it will be more.
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