Oxford English Dictionary-
   To go with a rush or dash; to drive. An American 'word', first appearing in 1819 in print.
It developed a rather broad set of meanings. Mark Twain spoke of "pedestrians staving by us in a rapid gait..."

It has nothing to do with 'stave' in the sense of bashing, or stave as stave in (stove in).

Now how many of you have used another 'stave', as in he tipped me a stave? (He sang me a song or verse).

And do you know what a stave of a chair is?

I called it an "American word," in the sense it is used in "Ground-hog." which is incorrect, but it is an American use of the word.

Stave is defined under different headings Stave in the OED because a variety of meanings belong to those five letters.