I'm looking for official rules for Shadowrun 5th edition (4th would be close enough) for damaging and losing limbs. Specifically I'd like to know how some augmentation would help a character survive and how do you determine what limbs are prone to damage.
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Try 4th Edition's Augmentation
Augmentation may be for Fourth Edition, so you will need to do some minor conversions, but it does explicitly have a section on Advanced Wounds (much as 3rd Edition's Man and Machine did).
Note that there are specific situations for incoming damage which lead to this in-game, but the mechanic is pretty solid; you count as being bleeding out and you take more wound penalties.
Furthermore, there are rules for other nasty injuries in Augmentation, if you're interested. Of course, 5th Edition's combat guide is due out soon, so I don't know if I'd jump to the 4th Edition rules for injury right now (especially since there is some slight difference in the way damage works), but if you absolutely have to you can.
SR4 Core doesn't have any rule for localized damage.
Arsenal details how pieces of armor work together, and each piece has its own armor ratings that applies to the whole body. Like trousers give 1/1, jacket gives 3/1, so wearing both gives 4/2. There is no "trousers only prevent damage to the legs."
About augmentations, SR4A p.343 :
As I understand it, Damage Resistance tests use the average Body and Armor ratings of all limbs.
EDIT: As Kyle said, Augmentations have optional rules for severe wounds.
Assuming you don't have access to 4th Ed. Augmentation, or don't like the optional rules in there for severe wounds (p.120), I'd be prone to following something along these lines.
If a character loses a limb, apply the usual modifiers... If you just shot a guy's weapon-bearing arm with a shotgun loaded with HE rounds, and it exceeds the armor rating for that zone, assume he had lost use of that limb (you get to make up the details) and he is now forced to deal with wound modifiers as well as off-hand (no pun intended) modifiers if he continues to fight. Adjust this for the drama of the situation, whether it was cyberware that got blown off (depending on where hit and the extensiveness of the cyberware, they may not bleed out, per se, but the biofeedback shock to the system could still be fairly staggering for at least one initiative segment), the importance of the character (redshirt NPC vs. pivotal NPC vs. PC), and how gritty you've made your game.
Tin Foil Hat moment:
I remember reading a long time ago (around 2nd or 3rd ed.) that the DR tests and damage system was built to simulate not only gross system damage (i.e. how much meat is now lying on the ground) but also the harder to judge side effects of blood loss and shock. I know there have been systems that tried to simulate on a more granular level (I seem to recall Twilight 2k), but Core SR errs in the side of storytelling in this case, and, I believe, expects players and GMs to apply some common sense.