In this paper, we study a scheduling problem involving a set of tasks and a continuously-divisible renewable resource of limited capacity shared by the tasks. The resource requirement is a continuous function of time that must be determined. We perform an analysis of the structural properties of the problem for realistic concave piecewise linear functions. We adapt the energetic reasoning satisfiability test as well as the hybrid branch-and-bound algorithm developed for the linear case to the case of concave piecewise linear functions.