"add strength," 1941, from college slang, from beef (n.) in slang sense of "muscle-power" (1851).
c.1300, from Old French buef "ox; beef; ox hide" (11c., Modern French boeuf), from Latin bovem (nominative bos, genitive bovis) "ox, cow," from PIE root *gwou- "cow, ox, bull" (see cow (n.)). Original plural was beeves.
"to complain," slang, 1888, American English, from noun meaning "complaint" (1880s). The noun meaning "argument" is recorded from 1930s. The origin and signification are unclear; perhaps it traces to the common late 19c. complaint of U.S. soldiers about the quantity or quality of beef rations.
To strengthen; reinforce •Beef up! is attested as an exhortation to use more strength by 1890: The Patriots beefed up their defense by adding an all-star lineman (WWII armed forces and industry)