Correct me if I am wrong but the armor of a shield is like armor on a person, it decreases the damage done to the shield, however when a weapon has a bonus against shields it ignores the armor of the shield, therefore low hit point shields break faster to things like axes, but high hitpoint shields with less armor take more damage from things like arrows because of the reduced armor. Personally you might want to consider the triple heirloomed fur covered shield, nothing says barbarian like a 700 hit point shield covered in fur.
Wrong. I'll explain...
Bonus to shield is a 2x Multiplier to damage when striking shields.
That is all -- However that is more than you think it is just by hearing it.
Mathematically speaking though, since the damage is raised it will negate the armor at a Higher percentage of its damage dealt. This can make axes MORE useful for higher armor shields than lower armor shields -- But still super useful always. Always.
Example 1: Hit a fur covered shield for 50 damage with no bonus, let's say 8 damage is negated by armor and 42 is absorbed by shield HP. With a bonus to shield multiplier that is now 100 damage, 8 still negated by armor and 92 damage absorbed by the shield. Since it has such High HP it will still last a long long time, but it will last roughly half as long as normal when struck by a bonus to shield weapon.
Example 2: Hit a steel shield which negates 50 damage. You hit it for 60 damage with a 2 handed sword and do 10 damage to it's HP only. Now say you hit for 60 damage from a shield bonus weapon for a total of 120 damage. 50 negated damage leaves 70 damage still dealt to the shield's HP. Since it has a low amount of HP, it won't last very long to a high damage axe. In this example, an axe does 7 times more damage than a non shield bonus weapon, due to having more damage negate the armor's effectiveness.
No, This is not the exact math, nor the amount of damage negated by the armor presented. There are lots of variables on armor effectiveness, damage, etc in a normal game that I am not going to bother adding. Many of which are randomly generated effectivenesses per strike and stuff.
*due to heirlooming effects, I'd loom a steel shield, huscarl, or knightly heater since it raises shield armor on them even higher. Shield armor has exponential increasing effectiveness, not a simple linear increase to effectiveness which HP provides.