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I'm curious about how exactly the Martial Arts class feature modifies damage done with magic weapons that add extra damage. Here's the relevant bullet point of the class feature (which is on page 78 of the PHB, and in the basic rules).

• You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.

The kind of scenario I'm curious about is a 10th level monk (Martial Arts die: 1d8) wielding a Frost Brand. Here's the relevant part of the rules for the weapon (from page 171 of the DMG, and in the basic rules):

Frost Brand

Weapon (any sword), very rare (requires attunement)

When you hit with an attack using this magic sword, the target takes an extra 1d6 cold damage. [...]

A shortsword version of a Frost Brand is a valid monk weapon, and it does 1d6 piercing damage as well as the additional 1d6 cold damage described in the quote. If our monk was attacking with it and wanted to use Martial Arts, which dice would be replaced? Just the base die for the weapon, or both dice? If both dice would be replaced, what damage type would the 1d8 damage be?

Does Martial Arts replace non-dice based damage modifiers? Would a Shortsword +1 do 1d8+1 damage, or just 1d8? Would Martial Arts replace the extra +7 damage you'd get when a Mace of Smiting gets a crit? Obviously for a lot of weapons, switching out the damage may be a bad idea, but its unclear if you can just swap the base damage die and still keep the additional magical damage.

(This topic came up in discussion of a question I asked about a homebrew tweak to a monk subclass. I realized I don't actually know how the rules-as-written work, which makes it a bit trickier to homebrew!)

The Martial Arts feature replaces the sword's piercing damage, not the bonus cold damage

Martial Arts states:

You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.

The general rules for Damage Rolls states:

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Notice how none of the weapons on the weapon table feature any sort of bonus. They only list a number and type of dice.

This is because any additional bonus to the damage is just that: a bonus. The normal damage of a weapon means the standardized base damage. Any value added to this normal base damage is a bonus and Martial Arts does not replace bonuses. You would never take a weapon that deals 1d6+Dex and instead roll 1d8+1d8 (replacing both the weapon's die and the bonus with the Martial Arts die value).

In this same way, the +1d6 cold damage is a (variable) bonus that is supplemental to the sword's normal base damage. Just as you can't replace a static bonus with a Martial Arts die, you can't replace the elemental bonus damage either.

As a result, a monk with a d8 Martial Arts die, while wielding a Frost Brand Shortsword, would deal 1d8 + dex + 1d6 damage. The first two would be piercing, per the sword's damage type, and the d6 would be cold, per Frost Brand's bonus damage.