When engine warms (2-3 hours of driving) accelerating causes sputtering and shaking

No warning lights are on. This summer when we were driving 6 hours and it was raining all the way, about 4 hours into the trip we couldn't accelerate. If we pushed the gas, the car shook violently and wouldn't go any faster. We were able to maintain the current speed as long as we didn't go up hill. Finally we stopped for supper and then could continue. We took the car to a local repair shop, they could duplicate the problem, but don't know how to fix. They said it would be a bent axle.