My guess would be oak because it is far superior for strength and with a beam in a barn that would be the most logical choice. Lots of oak available in that region. The small size of the growth rings on one piece shown in the link is indicator that it is more likely a hardwood.
I would like to point out that Chestnut is one of four American woods that are called open grain ring pourous woods. The others are Oak, Ash and Hickory. All have these open rings. I might have seen some rings in the far right piece of wood in the picture of three pieces. Hard to tell if those rings were present in the large beam.
I looked at all the pictures and the detail isn't there. Some fine sanding is needed to properly see the grain. Perhaps you can sand a few pieces down to 220 grit and show them again? Quench the wood with water and quickly take the pictures. Any finish that darkens the wood makes it harder to study.
