Finite element model (FEM) is a broadly used numerical tool in structural damage detection. In such applications, damage parameters in FEM are estimated by minimizing the differences between experimental modal analysis data and the corresponding FEM model prediction. Very limited works exist on analyzing the identifiability of the FEM used in such applications. In this paper, the identifiability of FEM-based structural damage detection is investigated for undamped elastic beams. We theoretically proved that damage severity at a given location in a uniform beam is identifiable by reformulating the FEM into a linear time invariant (LTI) system. A numerical algorithm is also proposed for checking the identifiability issue of multiple damage locations. Numerical case studies are provided to validate the effectiveness and usefulness of the proposed framework.