I'm building a 20 by 20 foot barn using the socket system. My questin is, will a 20 foot 8x12 pine beam be strong enough to act as center beam carrying second floor without post under the middle
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Yes but you have to modify it and so over engineer it but its better to be better than worse. Pine deflects naturally so you have to beef it up come what may. Your question "I'm building a 20 by 20 foot barn using the socket system. My questin is, will a 20 foot 8x12 pine beam" You need a steel strip 8" wide screwed at 4" intervals at center, staggered across the beams along the length of the beams on the tension side .... The lower (ceiling) side of the beams so in the end both beams are joined by a single strip of steel which can be clad afterward and this will prevent excess deflection and multiply the strength by 3.7%/foot. The steel is 1/4 (10mm) so as cheap as chips its just the drilling thats gonna take time but keep em tight else the beam will overload on areas that are excessively spaced You could reduce the width by 0.5" each side so as to fix battens hence reduce the steel width to 7" and be able to fix battens to fix cladding to cover the steel if you wanted to. Have a nice day bye