Abstract
Framed structures are deeply studied in civil engineering since they provide a numerical model for the analysis of the static and dynamic response of multi-storey buildings. In order to evaluate the vibrational properties of these structures, an eigen-problem, which involves the stiffness and mass matrices of the frame, must be solved. Both matrices can be assembled by means of standard methods, which take into account the numbers of degrees of freedom of the frame. The occurrence of concentrated damage in some vulnerable sections modifies the degrees of freedom and therefore both the stiffness and mass matrices. Very often, the critical sections are located in the joints between the structural elements of the frame where the bending moment reaches its maximum value. Assuming that the joints are rigid in the undamaged configuration of the frame, it is possible to take into account their loss of stiffness due to the presence of eventual damage by means of hinges with rotational springs of variable rigidity. In this paper, an original algorithm that allows us to evaluate the stiffness and mass matrices and therefore the natural frequencies of vibration of undamaged and damaged planar frames with an arbitrary number of beams and columns is presented. The proposed algorithm for the stiffness and mass matrices determination requires a few input data which can be provided in a text file and therefore allows us to speed up the procedure with respect to the application of an FEM approach which requires the construction of single models for each considered frame. The results obtained by means of the proposed algorithm have been validated through a comparison with those provided by an FEM model implemented in SAP2000. The natural frequencies obtained by means of the proposed approach are used for the solution of two different inverse problems, which concern the identification of, respectively, the mechanical characteristics of the constitutive material and the location and intensity of the damage. Both the proposed identification procedures deal with optimization algorithms that are based on opportune fitness functions. Applications to frames of different size confirm the validity of the presented identification algorithms. Furthermore, an iterative procedure, able to reduce the required computational burden related to the identification of the location and intensity of damage, is presented and applied in a parametric study concerning frames with increasing size.
Highlights
E key point in all the procedures concerning damaged structures consists in the definition of an accurate and reliable numerical model of the structure
A great number of studies are based on the comparison of the static or dynamic response of models of damaged structures with respect to the undamaged configuration [1,2,3,4,5,6,7,8,9] and propose damage identification techniques. e latter often require the measurement of different kind of data on the existing structure; these, for example, may concern the variation of dynamic characteristics, such as natural frequencies [10,11,12,13,14,15,16,17,18,19,20], mode shapes [21, 22], modal curvature [23, 24], and time-frequency features [25], or static quantities, such as displacements or strains induced by applied
Damage identification procedures are often based on the solution of an inverse problem that compares numerical response data, evaluated on a model of the structure, with the corresponding experimentally measured ones [29,30,31]
Summary
Due to the aging of a very high number of existing structures in the urbanized areas of the various continents, it has been necessary over the years to verify their conditions of integrity. e repeated stresses to which the structures are subjected and the action of atmospheric agents can cause damage of varying intensity, which reduces the bearing capacity of structures and inevitably leads to structural failure. e object of study, usually called “structural health monitoring,” has attracted a lot of interest from researchers who have published numerous scientific papers. E aim of the present study is to give a contribution to the evaluation of reliable numerical models of undamaged and damaged frames with arbitrary numbers of beams and columns. E stiffness and mass matrices of undamaged and damaged frames (in which one hinge with rotational spring is located in an arbitrary critical section) have been evaluated by means of the proposed algorithms and used to determine the natural frequencies of vibration of undamaged and damaged frames. For frames of large size having a great number of critical sections, an automatic procedure, which allows us to iteratively converge to the exact damage position without evaluating all the fitness functions, and reducing the computational burden, is illustrated
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