the cables of a suspension bridge as well as the elements of many man-made structures are designed to behave elastically. whey it is absolutely necessary that such elements function elastically?

When loaded past their elastic limit, and the loads are repetitive, there is frictional heating and weakening of the element (hysteresis) and a permanent deformation that reduces the load-bearing ability, shifting the load to other elements until a failure occurs.

If there is only a steady load, there are situations in which loading beyond the elastic limit is permissible, but it is not good engineering practice. It is called "yielding" or plastic flow.