The math behind shield bashes

[Rull]

The vanguard tree got me excited. But I always wondered, how much do those bashes really proc?
As I understand them, they are hard-capped to proc at most once a second. Now if you have a proc chance of 20% and attack not too fast, this is not likely to restrict you. On the other hand, if you attack 3x per second and always proc, you only get an actual bash on 1 out of 3 attacks.

One very important thing to realise is that the '1 bash per second' does not behave like like a cutoff threshold. Because ?fter that second, you still have to score your proc.
If you proc approximatly once a second, it takes on average a second to score a proc after the second of 'downtime', so you bash once per 2 seconds!

I ran the numbers for 10%-60% proc rate, on various attack speeds, and here are the results:



So with 3 attacks per second (I don't think this is reachable on sword and board), you still only get the 0,64 bashes per second even with 60% proc rate. The 1 bash per second is not nearly approached.




The same math also applies to stacks of destruction and similar effects

edit: corrected for time between attacks

[Rull]

Edit: fixed calculation error in accounting for the time between attacks

(with 1 attack per second or less your bash chance should of course equal your proc chance, and the same idea effects carries over in higher attack speeds)

Formula is now 1/(MAX(0;S-1)/S+((1/P)/S)) bashes per second, and (1/(MAX(0;$S-1)/S+((1/P)/S)))/S bashes per attack.

[Rull]

Oh, and I want to thank ArcaneArcher52689.

Okay, so I just ran some approximations with a computer.
Assuming 3 attacks per second
...
Out of the maximum 60 shield bashes you can make in a minute, you'd make
10% 15 bashes
20% 25 bashes
30% 33 bashes
40% 40 bashes
50% 45 bashes
60% 49 bashes
70% 52 bashes
80% 55 bashes
90% 58 bashes
100% 60 bashes

It's good to see those numbers. What I get from the general formula was
10%: 15
20%: 25,71
30%: 33,75
40%: 40
50%: 45
60%: 49,09
70%: 52,5
80%: 55,38
90%: 57,85
100%: 60

For now i'll assume that if it matches this closely for 3 attacks per second my calculations for other attack speeds might be accurate as well

I'll be adding 70%-100% bash chance asap, didn't know they were possible.

[Xianio]

The highest you can reach is 95% with 12 fighter levels, a shield that gives +20%, the shield bash feat and EK shield striking. Not quite 100% but very close.

[Grailhawk]

So with 3 attacks per second (I don't think this is reachable on sword and board), you still only get the 0,64 bashes per second even with 60% proc rate. The 1 bash per second is not nearly approached.

S&B with 35% Alacrity (capstone + haste) should be about 2.08 APS, Haste Boost brings it up to 2.60. to get base 3.00 APS you need ~45% double strike (assuming you can bash on a double strike)

45% double strike should be do able on a S&B.

[ArcaneArcher52689]

S&B with 35% Alacrity (capstone + haste) should be about 2.08 APS, Haste Boost brings it up to 2.60. to get base 3.00 APS you need ~45% double strike (assuming you can bash on a double strike)

45% double strike should be do able on a S&B.

Here's my program using 2APS



bash %actual bashes/60
0% 0% 0
10% 18% 10.6
20% 33% 20
30% 46% 27.5
40% 57% 34.2
50% 66% 40
60% 75% 45
70% 82% 49.2
80% 89% 53.4
90% 95% 57
100% 100% 60

[Scrabbler]

Here's a chart I made of how much adding passive shield bash proc improves your actual rate of getting passive shield bashes. The attack rate is 2.07 per second (realistic) and the bash cooldown is 1 second. First column is nominal bash chance, found by adding whatever you have of Vanguard cores + Vanguard t5 + Imp Bash feat + Bashing shield item (only goes up to 80 unless you're an ill-advised multiclass). Second column is what percent of your attacks actually get a bash (you can think of that in comparison to the 80% offhand rate a TWF guy has). Third shows how much your real bash rate improved by moving up 10% from the previous line.

Nominal   Actual  Gained
 10%       9.07%   9.07% 
 20%      14.73%   5.66%  
 30%      19.10%   4.38%   
 40%      22.61%   3.51% 
 50%      25.21%   2.60% 
 60%      27.50%   2.29% 
 70%      29.33%   1.83% 
 80%      30.92%   1.59% 
 90%      32.28%   1.36% 
100%      33.33%   1.05%

The key thing to learn is that anyone who is a Vanguard build will already have 30% bash rate from the level 3-6-12 cores, and if you care at all about bash DPS your shield will have 10% or 20% bash as well. That's an effective bash rate of 23-25% almost for free, and investing a feat and tier5 AP into more bashing only produces a very small increase to 30%. So not likely to be worth trying to raise.

This was assuming doublestrike won't help you bash more; if that is incorrect, then raising nominal bash rate is even less useful.

[Xianio]

45% double strike should be do able on a S&B.

You'll be able to hit 45% doublestrike fairly easily if that's the goal. With a fully built out toon you'll have 34% without any active skills, points in other trees, or spells. If you want to maximize for it you'll be looking a 70-75%+ in the vanguard tree with decent items.

[Grailhawk]

You'll be able to hit 45% doublestrike fairly easily if that's the goal. With a fully built out toon you'll have 34% without any active skills, points in other trees, or spells. If you want to maximize for it you'll be looking a 70-75%+ in the vanguard tree with decent items.

Agreed. The interesting question is if a double strike will proc a bash if so there should be no problem getting what ever the max actual bash per second is.

[Scrabbler]

Another thing to remember about bash proc rate:

Although normally the 1-second cooldown means that bash proc rate has steeply diminishing returns, there is one key situation where the cooldown doesn't apply: When you start a fight with a powerful active attack which can proc on shield bashses. That could mean something like Exalted Smite, but not Smite Evil. Getting double use out of an Exalted Smite is pretty nice, and a high bash chance will let that happen if your Smite is the first attack of a battle. After the first attack, your bash is probably on cooldown.