The phrase "took damage" is correct and can be used in written English.
This phrase usually means that something has been harmed or injured. For example: "The boat took damage after the storm."
"There were high winds last week so a lot of the plants took damage and we have been spending the last week repairing the damage.
Some years back, after their home took damage in heavy weather, the Trainas learned their losses were not bad enough to be covered under flood insurance.
Also, the upper parts of the logs, which will be used, took damage.
Their frustration was increased when the protagonist took damage from enemy attacks as he or she executes the unintentional command to change facing.
The waist-high-barriers-everywhere cover system, so common now, felt uniquely Gears in 2006, as did the spread of red that stained the screen as Marcus took damage.
If one of your structures takes damage, you will hear a beep and your HUD will highlight the damaged structure.
They act as a kind of second life for you - you can take damage then eject out.
If product wood takes damage, the wood reduces its value.
Facebook still stands to take damage from the trumped up release of these "leaked" guides.
These fighters have more strength and health and protect you from taking damage.
However, these robots may take damage when trees fall down against them.
