I googled a little bit. The original literal meaning was "A 1.7(?)-meters-tall man". It underscores the matureness of a man. It began to be used to refer to husband in Chinese in the 10th centry and it gets the "study" signification in Japanese in the 16th centry.
丈夫 = robust; strong; solid; durable (etc...) Robust and sturdy are kind of synonyms. You use sturdy only with things/material (I think) and robust (like 丈夫) with (things and persons) . So robust/durable are the better translation and should be both correct.