This is a little over-simplified, but makes the main points.
The 2 types of forces that are important for this type of rope failure are compression (pushing) and tensile (pulling).
When you bend something, the inside radius is squeezed, the outside radius is pulled. ﾃつ The classic example of this is to bend a deck of cards. ﾃつ If you look at the cards on the inside of the bend they seem to stick out in relation to the other cards. ﾃつ Conversely, the top cards are "shorter." ﾃつ Now imagine the cards are glued together and cannot move in relation to each other. ﾃつ If you are successful in bending the deck, the cards on the inside of the center are being squeezed. ﾃつ The farther out you go from the center of the deck (neutral), the higher the "squeezing force." ﾃつ There is a similar set of "pulling" forces on the outside of the deck. ﾃつ The tighter the bend, the higher the forces.
Thus when considering making equipment for climbing, and you are looking at the forces on a rope during a fall they include:
1) Force applied to the system (force of the fall)
2) Characteristics of what is being bent (rope)
3) Radius of the curve that the rope wraps around during a fall(Rope Bearing Radius)
When you're making carabiners, the only one of these you have control over is the rope bearing radius and thus you want to make that as big as possible - thus reducing stress on the rope. ﾃつ Unfortunately, big clunky carabiners, don't sell, so you want the carabiners to be as small and light as possible. ﾃつ
You now have conflicting design requirements and need to decide what direction you want to go in (safer vs lighter).
One other interesting note is that tensile and compressive forces tend to cancel each other out. ﾃつ When you pull on a rope (no bends) all forces are tensile (pretty intuitive). ﾃつ
Thus, during a fall, the highest forces on a rope going over a carabiner are on the outside of the rope. ﾃつ They are being doubly pulled because 1) they are pulled by virtue of being on the outside of the bend and, 2) being pulled because the rope is being stretched by the fall.
The compressive forces on the inside of the bend are actually relieved to some extent by the fall.
This issues comes up time and time again in all aspects of mechanical design. ﾃつ Here are some pictures from the same issues for kayaks:http://www.oneoceankayaks.com/Sandcore.htm
All of this, of course, depends on the rope actually coming in contact with the rope bearing radius of the carabiner during a fall - which is not a sure bet. ﾃつ But that's a different topic.