Weapon Damage by Hit Quality.

Maldoon the Mighty closes in on the dragon Hearthwrack ithas been a long and grueling combat and had had it been for the magics and missiles provided by his now suffering companions Maldoon would not be ready to strike this final blow
Player: oh yeah a 20 !
DM: and it's a hit of course
Player rolls his 1d12 and it's a 1...whoopie 1 point of damage

...Has this ever happened in your game?

A common solution to this heartbreak of unsatisfying dice rolls is the critical hit. A doubling or tripling of the damage roll that is just as prone to be unsatisfying. Possibly slow the game down with extra die rolls (not that drama isn't worth focus) or big charts of results.

Why not have the weapon damage tied to the quality of the hit roll? How about only requiring the hit roll?

Here is a brief suggestion to supply fixed damage scores where damage varies with the quality of the weapon blow.

I'm going to suggest 3 magnitudes of a hit; there shall be a graze/glancing blow, a sold blow and a devastating blow.

On a hit roll of the score required to hit a target and up to 2 points higher a glancing blow is inflicted.
On a hit roll 3 points higher to 9 points higher a solid blow is struck.
On a hit roll of 10 or more higher then the required score a devastating blow is inflicted.

Damage scores are needed for each weapon that reflect the glancing blow/the solid blow and the devastating blow. If coming from a game that uses variable damage dice by weapon type it's not to hard to work up some numbers. We need the minimum, average and maximum damage for each weapon.
To find damage from a glancing blow take the minimum and aveage damage scores and divide this score by 3. Round down any score of with a fraction of .5.
To find damage from a solid blow take the average and use that score. If the average has a fraction round up to the next whole number.
To find damage for a Devastating Blow double the maximum (if you want to have a critical hit degree of wepon dmage as I do, otherwise use the maximum).

Factoring in STR modifiers to weapon damage. As STR has an impact on the chance to hit it already improves the damage score, this must be accounted for (unless of course STR doesn't modify hit rolls in yuor game).
Always apply a penalty to a hit roll but the penalty can't drop a glancing or solid blow to less then 1, nor can a devastating blow be dropped to less then 2.
When one has a Damage bonus from this bonus it isn't applied to a glancing blow. A STR bonus (as indicated by your rules) is applied to solid blows and devestating blow. I recommend agaisnt double the damage bonus for devastating blows as STR typically improves the quality of the weapon blow already.
Magic weapons get a big oomph as they add the damage bonus to all grades of damage so even a glancing blow from a +3 weapon will be a big deal.

Pre-calcualted glancing/solid and devastating blow damage for some damage scores form die roll based damage:
1-4 ..... 1/3/8
1-6 .... 1/4/12
1-8 .... 2/5/16
1-10.... 2/6/20
2-12.... 3/7/24
2-16.... 4/9/32
3-18.... 4/11/36
4-24.... 6/14/48

Here's a sample combat table that precalculates glancing/solid/devastating Blows.









(Click to enlarge and get a clear view)

There you go now the scenario above could have ended differently:

Player: oh yeah a 20, that means Maldoon the Mighty does 21 points of damage!!!
DM: The dragon is slain.

(note in the example here the player already recorded weapon damage with properly factored modifiers, good luck getting the other players to all do that)

So there we go, a presentation for a possible method to alter how damage to fit the hit rolls could be applied to a host of games (and maybe even speed up combat).