Fundamental mode shape and its derivatives in structural damage localization

Various techniques for structural damage detection involving modal parameters have widely been used over the past few decades. This is because the modal parameters of a structure can easily be obtained from forced, free or ambient vibration measurements. In many of these techniques, mode shape curvature has been used for localization of damage. In this paper, a mathematical basis is provided to show the correlation between a structural damage and a change in the fundamental mode shape and its derivatives. This has been achieved by deriving the expression of a damaged mode shape utilizing a perturbation approach. For a cantilever shear beam, discretized into a large number of elements, this approach demonstrates that the change in the fundamental mode shape due to any damage is an excellent indicator of damage localization as it is found to be discontinuous at the location of damage. Further, the change in higher derivatives (i.e., slope and curvature) of the fundamental mode shape is shown to be sensitive enough in damage localization. A numerical study involving a shear building and a steel moment-resisting frame is conducted to show the effectiveness of the proposed approach in damage localization. It has been found from this study that spline fitting of mode shapes in case of a limited number of degrees-of-freedom, which is generally adopted for plotting mode shapes and their derivatives, may lead to false detection of damage.