Modern bicycles are most commonly made of either steel, aluminum, carbon fiber or alloy. The material used to build the frame defines how it must be built. For example, steel is very strong but also very heavy, the thickness and radius of the tubing being used do not have to be very large for the frame to be able to support great amounts weight. While aluminum is not as strong as steel but much lighter, aluminum tubing must be thicker than steel tubing. Even though an aluminum frame must be thicker than steel it will still result in a lighter frame than a steel frame. Frames are not made to be perfectly stiff. they have to be able to withstand a both static and dynamic loads.
http://www.exploratorium.edu/cycling/frames2.html
If a frame was perfectly stiff it the rider would have to absorb every bump that they encounter. Bicycles made for roads are made to be able to flex but they are not built with any type of shock system. Mountain bikes are most commonly built with either front suspension or full suspension, depending on how much the rider wants the bike to absorb. While suspension is good for riding over rocks and bumps it also mean that it more power from pedaling will be lost in the suspension. in the past 10 years bike builders have made suspension systems that the rider can lock while going up hills or in areas where they need minimal suspension.
https://29erbigwheeledballyhoo.wordpress.com/tag/full-suspension-29er/