Jesus never carried the material cross, and he never fell either!
"Then he delivered Him to them to be crucified. So they took Jesus and led Him away. And He, bearing His cross, went out to a place called the Place of a Skull, which is called in Hebrew, Golgotha" (John 19:16-17).
John's account is clear that Jesus left the place of judgment bearing his own cross. Somewhere along the way, the soldiers compelled a bystander named Simon to carry his cross.
"And when they had mocked Him, they took the robe off Him, put His own clothes on Him, and led Him away to be crucified. Now as they came out, they found a man of Cyrene, Simon by name. Him they compelled to bear His cross" (Matthew 27:31-32)
"And when they had mocked Him, they took the purple off Him, put His own clothes on Him, and led Him out to crucify Him. Then they compelled a certain man, Simon a Cyrenian, the father of Alexander and Rufus, as he was coming out of the country and passing by, to bear His cross" (Mark 15:20-21).
"Now as they led Him away, they laid hold of a certain man, Simon a Cyrenian, who was coming from the country, and on him they laid the cross that he might bear it after Jesus" (Luke 23:36).
We are not told what led to the shifting the cross from Jesus to Simon. The most logical supposition is that Jesus was not able to bear the weight that distance because of his prior scourging. That would also imply that Jesus collapsed while attempting to bear the cross, so the soldiers to keep things moving pulled a nearby bystander to carry the cross to the crucifixion site.