Green’s Dictionary of Slang

short of a sheet phr.

also short of a bit
[the sheets of bark used to roof early dwellings ]

used of someone who is not very intelligent; also as short of a sheet of bark; thus short sheet n., a simpleton.

[UK]Farmer & Henley Sl. and Its Analogues.
[UK]J. Curtis There Ain’t No Justice 86: Maybe she was short of a bit. It was on the cards that Arthur could not serve her good. He looked a weakling sort of a bastard.
[Aus]Baker Popular Dict. Aus. Sl.
[US]Maurer & Baker ‘“Aus.” Rhyming Argot’ in AS XIX:3.
[Aus]N. Pulliam I Travelled a Lonely Land (1957) 238/2: short sheet – simpleton.
[UK]J. Franklyn Dict. of Rhy. Sl.
[NZ]McGill Reed Dict. of N.Z. Sl. 187: short of a sheet of bark Mentally deficient. ANZ.

In phrases