This is pretty simple, actually.
Whenever you take damage while
you are concentrating on a spell, you must make a
Constitution saving throw to maintain your concentration.
The DC equals 10 or half the damage you take,
whichever number is higher.
You make a Constitution saving throw “whenever you take damage”. So when you take 30 damage, you make a Constitution saving throw. Since you make this when you take the damage, you use the stats you have when you take the damage. So it will be a DC 15 Constitution saving throw which you make with a +5 bonus.
The same reasoning applies to the 60 damage case. So when you take 60 damage, you make a DC 30 Constitution saving throw using a +5 bonus. There is a caveat, though.
This suggests that the 30 carryover damage will count as being taken, so you’ll have to make another Constitution saving throw, this time at DC 15, using your own +0 modifier.