Detecting the position of supports within an elastic structure has many applications, particularly when these supports are not fixed. Previous studies have presented methods for the detection of translational support locations in elastic structures based on the minimization of the difference between the measured and computed natural frequencies. However these discrete supports were constrained to be at nodes of the finite element model of the elastic structure. This required a fine mesh and the numerical computation of the eigenvalue derivative with respect to the support location. This paper has the same purpose, namely to identify the support locations, but the position of the supports is now a continuous parameter. When the support is located within an element the shape functions are used to produce the global stiffness matrix. The advantage is that the support location now appears explicitly in the formulation of the problem, and hence the analytical computation of the eigenvalue derivative is possible. Furthermore, the mesh may be much more coarse, requiring fewer degrees of freedom to detect the support locations, with reduced computational effort. The effect of identifying both the support stiffness and location is also discussed. The proposed approach has been illustrated with simulated and experimental examples used in the earlier studies.