Originally Posted by
Reynolds
But are there any aircraft where all the major structural parts are carbon fiber, like in bikes?
You mean like the Boeing 787 and, I believe, the A350??? The 787 fuselage is all-carbon. That's a fairly major structural part.
Similarly, the control surfaces as mentioned by the previous poster: a) experience considerable stress b) are critical to maintaining stable, controlled flight - failure of one would ruin your day.