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Technically 1 point of strength adds to Attack, so yes, I would say my 126 strength should be making an impact on my overall melee damage...but it rarely does
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Have you tried parsing any of this?
Also is your weapon skill maxed for whatever weapon you are testing this with?
For the purpose a proper test I would limit myself to a 2-handed weapon (if you are a duel wielding class) just to make it easier to see exactly what's happening without the offhand getting in the way.
If you can get a good haste that will also help produce the numbers you need to look at more quickly.
Out of curiosity what is the strength range you were testing with? You mention 126 being your current STR but what is the minimum/maximum strength you are testing with? You should at least try to get a 50 strength difference and see if you notice a difference.
If your str range is only differing like 5-15 points then the range probably isn't good enough to see past the R.N.G. hit range.