Vibration-based structural health monitoring of civil structures relies on the repeated identification of dynamic structural characteristics of the structure from output-only vibration data. Natural frequencies and displacement mode shapes are the most commonly employed dynamic characteristics, yet their sensitivity to local damage of moderate severity is rather low with respect to their sensitivity to other factors such as temperature, necessitating data normalization. Strain mode shapes offer a higher sensitivity to local damage, but their accurate identification in a dense grid is challenging given the very small dynamic strain levels that are encountered under ambient excitation. In this paper, a method is presented for tackling this challenge. It consists of three stages. Firstly, fiber optic Bragg grating strain sensors are attached to the structure and interrogated with a tunable laser performing a wavelength sweep. In this way, the measured strain amplitudes have the required accuracy but synchronization errors are introduced between the different Bragg sensors. Secondly, a modal analysis is performed on the dynamic strain data using an accurate parametric system identification technique. This is followed by a synchronization step which compensates for the delays introduced by the wavelength sweep. Finally, the synchronized strain mode shapes are employed as damage-sensitive features, either directly or via a newly proposed quantity, the top-to-bottom strain ratio. The method is validated by progressive damage testing of a complex, prestressed concrete "roof" beam, reinforced with steel fibers. It is observed that the proposed method can identify both the presence and the location of the damage in a relatively early stage.