I don't think it is common or normal once you get used to the bike, but I know that at the beginning of the season, especially when I start climbing on my bike rides, if I haven't been doing my ab work and lower back extensions, I get a sore back. Then, as I get stronger with more climbing, ab work, and back extensions in the gym, my back pain disappears for the season.
I don't have any history of back injury or chronic soreness, and I can do a lot of lifting, moving and so forth and consider my back pretty strong. But...
Abs and lower back muscles are among those muscles that we need for bike riding, but that bike riding does not develop. I don't know you or your history, but for me, doing abs and back extensions in the gym takes care of back pain on the bike. If I don't do them, my back can start hurting. It is predictable for me.
"The best rides are the ones where you bite off much more than you can chew, and live through it." ~ Doug Bradbury