The objective of this article is to detect the location and severity of structural damage according to direct model updating of physical properties by a Moore– Penrose inverse problem. The proposed method is based on expanding the dynamic orthogonality conditions in a damaged structure for attaining the difference between physical properties of undamaged and damaged structures. Hence, a two-stage damage detection process consisting of localization and quantification of damage is established by using linear objective functions which are applied in the expanded orthogonality conditions. Thus, an error matrix which is characterized as damage index is determined to identify the location of damage. Subsequently, damage extent is quantified by applying a linear objective function in the extended eigenproblem of the damaged structure. Eventually, two numerical examples are utilized to validate the proposed damage detection approach. In these examples, the modal data are considered to be incomplete and the inverses of rectangular matrices are accomplished by the Moore–Penrose technique while no multipliers are used. Furthermore, in all damage investigations, the predicted damage is compared with the preset values of induced damage. It can be concluded that the damage localization approach proposed in this study can precisely identify the location of damage through updating process. Moreover, the obtained results confirm this technique as being appropriate to predict the severity of damage.

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