I used to work for Delta and have flown quite a few times with a longbow in a pvc tube. I've had it broken two or three times and I use the heavy scedule 40 stuff for my tube. Most times it broke it was on the end like it fell and landed on it. The pvc is thick but it must be a little brittle. It also costs about $40 or $50 bucks to build a tube out of the heavy pvc so when I had mine broken I went ahead and filed a claim and got money to make a new tube. I've watched my bow tube roll out of a bag cart when being taken out to the plane and saw it go rolling across the ramp.
I've had it get stuck in the conveyor system and cause a bag jam. I've had it get lost or deliberately left off of a flight because it was a large item. It always turned up again but still... I've also stood at the baggage carousell and had it not show up at all only to find that it was either hand carried up to the baggage office or sent up a different conveyor. Dallas is notorious for this. They have a couple straight conveyor systems for bulky or long items like golf bags, skiis and bow or gun cases that are nowhere near the regular bag returns.
I got to where I'd usually ask in advance if it would be hand carried or sent up a different conveyor at the destination airport. Hand carried is the way to go but sometimes you have to pay extra for it.
Also, try to modify your tube so it does not roll. I've used a couple pieces of wood glued on to give it a triangle shape where it will set flat. That helps reduce the chances that it will roll to the side on a baggage conveyor and get jammed or roll out of a bag cart and get run over on the ramp.
Oh yea, it's been a couple years since I've flown with my bow but after 9/11 the security got changed. You couldn't lock the case or they would cut the lock. If you use a tube, be sure to mark which end gets opened and also be sure to not screw the cap on so tight that they can't unscrew it by hand.
Also, I used to take a piece of foam like from an old sofa cushion and cut two "donuts" about twice the diameter of the tube with just a little hole in the middle so that a longbow in a bow sock will just fit through.
Put a piece of foam in the bottom of the tube.
Put your bow in the sock.
Put a foam donut between mid limb and the tip on each end and start the bow into the case.
Squish the foam down tight so it will go into the tube.
When the bow is all the way in the tube put another piece of foam in to protect the tip.
This way, your bow never even touches the inside of the tube and if the tube gets broken on the end like mine did, the tight fitting foam keeps it from sliding out of the tube.