From what I've seen, you may have trouble making a bear stand still when it's dead. Chest shots do work, eventually, but not like it would on a deer. To make a bear die dead you need to shoot it in the head or spine. It doesn't really seem that it matters what you shoot it with; if you don't shoot it in the spine or head, it'll run away, two or three hundred yards, up a mountain, with three shots through the chest, over slash, up a tree, then die. (.308 Win)

Or it'll not know you're there when you shoot and run towards you, necessitating an African moment as two or three people start shooting a running bear thirty feet away. (30-06)

Or it'll run headlong into a deep ravine and die in a creek at the bottom. (7 mag)

Or through the neck, where it just dies. (7 RUM)

That was two weekends, five guns, in British Columbia. FWIW I was the gun that didn't get one!