With the number of large capacity applications in core network increasing, the bandwidth requirement of optical connections in conventional Wavelength Division Multiplexing (WDM) networks keeps enhancing, so that the Orthogonal Frequency Division Multiplexing (OFDM) technology is adopted to provide higher spectrum efficiency and flexibility in the future elastic optical networks. Meanwhile, survivability in the conventional WDM optical networks has been widely studied as an important issue to ensure the service continuity. However, survivability in OFDM-based elastic optical networks is more challenging than that in conventional WDM optical networks because each fiber usually carries even more connections. Therefore, it is necessary to study the new lightpath protection algorithm in elastic optical networks. Since p-cycle protection scheme has short restoration time and simple protection switching procedure, in this paper, we study the static Survivable p-Cycle Routing and Spectrum Allocation (SC-RSA) problem with providing an Integer Linear Programming (ILP) formulation. Since RSA is a NP-hard problem, we propose a new heuristic algorithm called Elastic p-Cycle Protection (ECP) to tolerate the single-fiber link failure. For each demand, ECP scheme can compute highly-efficient p-cycles to provide protection for all of the on-cycle links and the straddling links. We also consider the load balancing and choose the proper working path for each demand. Simulation results show that the proposed ECP scheme achieves better performances than traditional single-line-rate survivable schemes.