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Some days ago, I created for someone a barbarian dwarf character sheet.
Everything gone well, and the sheet is complete.

As I'm very new to D&D, I decided to read through the barbarian class to get what could be nice this future player to have, as I know pretty much know what she's looking for for this character.

What brings me here is a question about the Rage class feature, and more precisely about the bonus damages.
Basic rules state that (emphasis mine) :

While raging, you gain the following benefits if you aren’t wearing heavy armor:

• You have advantage on Strength checks and Strength saving throws.


• 
  When you make a melee weapon attack using Strength, you gain a bonus to the damage roll that increases as you gain levels as a barbarian, as shown in the Rage Damage column of the Barbarian table.


• You have resistance to bludgeoning, piercing, and slashing damage.

From what I understand, when taking the Attack action using, for example a Greataxe, if the attack hit, the damage will be :

1d12 + Strength Mod + Rage bonus damage

Also, the bonus damage is linked to the melee weapon attack, not to the Attack action. Which lead me to not being sure how does this bonus damage apply when the barbarian is wielding two weapons.

Also from what I understand, if the barbarian takes the Attack action while wielding say, two handaxe, and both attacks land, each attack damage will be :

1d6 + Strength Mod + Rage bonus damage

Thus, that would lead to double the rage bonus damage, which seems pretty strong to me and makes two weapon fighting more valuable than just have one two-handed weapon, as two weapon fighting also guarantee at least 2 damage, increasing a bit the average damage for the same max damage.

Does it work as I understood it? If yes, won't it make two-handed weapons useless for a barbarian?

Two-handed weapons are not useless: bonus action and TWF

The two weapon fighting that a barbarian can use is not the same as the Fighter/Ranger Class Feature "two weapon fighting." Only that class feature allows the Fighter/Ranger to add the Ability Score Modifier to the second weapon's damage.

Each successful attack (using Strength) gets the Rage damage bonus

Assume a barbarian with a Strength of 16. (+3 damage modifier)

Two-Weapon Fighting (Chapter 9)

When you take the Attack action and attack with a light melee weapon
that you’re holding in one hand, you can use a bonus action to attack
with a different light melee weapon that you’re holding in the other
hand. You don’t add your ability modifier to the damage of the bonus
attack, unless that modifier is negative.

The Barbarian has to use a bonus action to trigger the rage.
Rage (PHB, Chapter 3)

In battle, you fight with primal ferocity. On your turn, you can enter
a rage as a bonus action. snip
When you make a melee weapon attack using Strength, you gain a bonus to the damage roll that increases as you gain levels as a barbarian, as shown in the Rage Damage column of the Barbarian table.

Let's see how a three round battle will go.

1. Round 1, trigger rage and one 2H weapon attack, or, one hand axe attack plus rage.


2. Round 2. One attack with 2H weapon, two attacks with two weapons, STR bonus damage on only one


3. Round 3, One attack with 2H weapon, two attacks with two weapons, STR bonus damage on only one

Damage comparison

Two-handed sword versus two hand axes: damage if we presume that all attacks hit

1. Round 1: 2d6 + rage + STR (2d6+5) versus 1d6 + rage Str mod. (1d6+5)
   
   Advantage 2H weapon.




2. Round 2: 2d6 + rage + Str (2d6+5) versus 1d6 + 1d6 + rage + rage +
   str mod (only on first blow) (2d6 +7)
   
   Total after two rounds is: WH 4d6 + 6 + 4 (24) vs 3d6 + 6 + 6 (22.5) Slight lead to 2H weapon



3. 
   

   Round 3: Same as round two, but now TWF takes a slight lead, since:

   
   
   

   All told, we see 6d6 + 15 from the 2H weapon. (Average 36)
   
   All told, we see 5d6 + 19 from the TWF choice. (Average 36.5)

   
   

Note: you gotta hit it to damage it.

An advantage accrues to the TWF approach if the fight lasts that long due to having more chances to hit. There are two chances to hit on the second and third rounds, whereas with the 2H weapon, there is only one chance to hit on each turn. You'll have a total of 5 attack rolls, rather than 3 attack rolls, if the dwarf goes with the two hand axes.

For what it's worth...

If the Barbarian takes the Dual Wielder Feat at level 4, they can use 1d8 damage dealing warhammers, battle axes, or long swords. The longer the battle goes on, the more two weapon fighting approach looks to be more advantageous. But, after level 5 where the Barbarian gets two attacks, this changes things enough that you may see the 2H weapon being a good or better choice.

• 
  

  We have a number of Q&As that compare TWF with 2H-weapons that go
  into far more detail about that. Since you are dealing with a
  beginner that's not germane for this question. If she gets to level 4 and considers that feat, I suggest you drop a question here to ask about the pros and cons of the Dual Wielder Feat for a Barbarian versus fighting with two-handed weapons.

  
  
  

  The same three round fight with the dual wielder feat, at level 4, using battle axes (1d8) as one handed weapons, yields this result after three rounds

  
  
  

  All told, we see 6d6 + 15 from the 2H weapon. (Average 36)
  
  All told, we see 5d8 + 19 from the TWF choice. (Average 41.5)
  Round 1 is still "advantage 2H weapon" and round 2 is roughly a tie.

  
  
  

  This looks to be a little more attractive, except during the first round, and the 5 versus 3 attack rolls means that there will be fewer misses. Each miss by the 2H weapon is a bigger loss of damage to the target.

  
  
  

  The 2H weapon gets another spike in attractiveness at level 5, so early rounds tend to favor the two handed weapon.

  
  
  

  The Two Handed Weapon Barbarian gets two attacks. TWF gets three on rounds 2 and subsequent, using the bonus action. Damage summary:

  
  
  

  Round 1: 4d6+2Str+2Rage 2H vs 2d8+ 2Str + 2 Rage for TWF
  Round 2: 4d6+2STR+2Rage vs 3d8+ 2Str + 3 Rage
  (All subsequent rounds are per Round )

  
  
  

  The fight has to go on for a bit longer than two rounds for the TWF to begin to show advantages.

  
  
  

  If the Dwarf does not take a feat, but instead boosts Strength to 18 at level 4, all attacks hit more often, all attacks get another damage point boost from strength, and the 2H weapon approach begins to look more attractive.

  
  

The rage bonus is added to all your melee weapon attacks that use Strength

However, you do have your damage calculations wrong, as I'm not sure where you are getting a "Mastery bonus" from.

The attack with a greataxe (or any single weapon) would be:

Weapon damage die + Strength modifier

See Damage rules where it says:

When attacking with a weapon, you add your ability modifier--the same modifier used for the attack roll--to the damage.

Nothing else is added by virtue of the basic rules.

If you are raging then you add your rage bonus, so the full damage would be:

Weapon damage die + Strength modifier + Rage bonus.

Two weapon fighting

See the rules here. To quote:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

The emphasis is mine. So if you qualify for two weapon fighting (using light melee weapons), then your bonus attack would just be:

Weapon damage die + rage bonus (no strength mod)

Note - There is a 'fighting style' - an option for fighters and some other classes - that allow you to add your modifier to the damage of your bonus attack. But this is not accessible to a (single-classed) barbarian.