A crack or a localized damage in a structure provokes a discontinuity in the rotation. Consequently, mode-shapes are nonsmooth at the damage position and the first derivative (strictly related to the rotation) presents a jump discontinuity. Based on this simple concept, a new approach has been developed in order to predict the location of the mode-shape derivative discontinuities, and therefore the location of damage, without the need to directly differentiate. This approach applies a Gaussian process regression to the mode-shape data, using a covariance function which allows for a nonsmooth point; this point, which indicates the crack position, can be determined by a maximum likelihood algorithm. Using a finite-element model of a cracked beam, the performance of this methodology has been analyzed for the case of single crack and multiple cracks, for increasing amounts of noise. The effects of several parameters (damage position, damage severity, number of measurement points and of considered mode-shapes) on the approach accuracy have also been investigated. Finally, the method has been verified using experimental data coming from a vibrating steel beam with a cut. Results are encouraging and indicate that further developments of the technique for nondestructive testing of beam-like structures would be highly worthwhile.