Application of two-stage evaluation and optimization update methods for the structural damage detection of a portal beam structure
Introduction
The development of detection methods to grasp the early stages of structural damage is an important topic of concern being explored in several structural engineering fields such as machinery, aerospace engineering, civil engineering, construction, and vibration control. The primary objective of damage detection is to identify structural damage at an early stage, so that action can be taken before repair is required or significant damage occurs. Structural damage detection can be divided into three levels: the first is the determination of damage occurrence; the second is the determination of the geometric location of the damage; and the third is the quantification of damage severity. When comparing the measured data-corrected structural damage model with the original model, the structural damage location and severity can be confirmed. The purpose of establishing and solving the optimization function is to reduce the difference between the measurement response and the numerical prediction.
Structural damage detection requires the comparison of finite element models (FEMs) and modal testing. Contemporary large-scale structures include mechanical equipment, vehicles, mega buildings, multi-span bridges, and offshore oil platforms. Owing to the complexity of materials, geometrical characteristics, and boundary conditions, it is difficult for engineers to build a finite element analysis (FEA) model with the same results as those yielded in the case of damaged structures. Inaccuracies in FEMs primarily manifest in three ways. (1) Model structural error arises because of some uncertain factors that affect the model control equations and is usually related to the selected mathematical model. (2) Model variable error arises as a result of environmental changes and production processes, such as steel welding methods, material unevenness, construction, and other procedure errors. (3) Model order error arises from the FEA discretization of the continuous model. Model structural error can be addressed by establishing a reasonable mathematical model. Model order error can be adjusted by improving the discrete density of the elements. Therefore, FEM updating is the best approach to reduce the error between the FEM and the actual structure caused by errors in the model variables. The use of modal testing measurement results to update the FEM becomes an effective structural damage detection method. In this regard, the most common modal parameters used to detect damage are the natural frequency and mode shape of structures. Physical model variables, such as the Young’s modulus, density, and thickness, which present physically meaningful information, are used to represent the structure material or geometrical properties. For the modal-based damage detection technique, certain physical model variables should be updated to ensure that the experimental and analytical modal parameters are in agreement. The process of updating physical model variables is the process of identifying the severity and location of structural damage through the FEM. The updated physical model variables can be interpreted to evaluate damage severity and location.
In recent decades, FEMs have been updated on the basis of sensitivity using direct and iterative methods, for example, the updated model has been extensively used for damage detection [1]. Direct methods compute a direct solution for the elements of global stiffness and mass matrices [2], and these matrices, which comprise the analytical model, are directly updated. However, the updated matrices are not directly related to the stiffness and mass values of the individual members of the analytical model. Furthermore, the updated matrices do not maintain structural connectivity, and the corrections are not always physically meaningful with regard to changes made to the initial analytical model.
Iterative methods utilize modal parameters sensitivity for analytical model updating [3]. They update the physical model variables by minimizing the discrepancy between experimental and analytical vibration results. Iterative updating methods also use modal parameters sensitivity to determine their change, and they are favorable for identifying parameters that directly affect the structure’s dynamic characteristics. These methods update the physical model variables and locate damaged regions within the model.
The model updating procedure is posed as an optimization problem. Owing to the nonlinear relationship between modal parameters and the physical model variables, an iterative optimization process is performed, whereby the modal parameters are approximated using a linear function.
The objective function is constructed using residuals between the measurement results and the analytical predictions. The three expressions used in the objective function are frequency, mode shape, and modal flexibility residuals. Several modal-based damage detection methods attempt to detect natural frequency changes in structures. However, the natural frequency change in different modes is a function of damage location only and not severity [4], and it cannot provide spatial information on structural damage [5]. Another type of modal-based method involves analyzing damage mode shape changes. Variations in the mode shape curvature are sensitive to damage. As the mode shape curvature corresponds to the strain energy at that location, the strain energy is used as an indicator to evaluate damage [6]. The natural frequencies, as well as mode shapes, can be applied as modal parameters in the updating procedure. However, the measured mode shapes are less accurate than the natural frequencies [7]. Therefore, the natural frequency residual is considered in this study as the objective function.
In this study, a modal-based damage identification method is proposed and applied to a portal beam structure. The damage location identification procedure involves two stages. In the first stage, the residual force vector method is used to detect possible damaged elements in the structure. In the second stage, the sensitivity-based method is used to confirm that the elements are indeed damaged. Element damage severity can be detected by identifying variations in the physical model variables.
The residual force vector plays different roles in the damage identification approach. (1) Previous studies have used the residual force vector method to determine the damage location and quantify the damage severity in the FEM [8], [9], [10], [11], [12]. (2) A two-stage method based on the residual force vector and response sensitivity methods is proposed [13]. First, the role of the residual force vector is only to determine the location of “to-update elements.” Then, a response sensitivity method is employed to identify damage severity. (3) In order to improve the accuracy of identifying the location of the “to-update elements,” this study further proposes a new two-stage evaluation approach combining the residual force vector and eigensensitivity methods. The residual force vector is only used in the first stage to filter the possible “to-update elements,” after which the eigensensitivity method is used in the second stage to select the “to-update elements.” The residual force vector can reduce the number of physical model variables in the eigensensitivity calculation. When the eigensensitivity matrix dimension is reduced, the calculation efficiency increases. Compared with other studies, the novelty of combining the residual force vector and eigensensitivity methods lies in the accuracy and efficiency improvement of the damage location identification. In addition, case studies have been discussed wherein each model node has only one degree of freedom (DOF), and each model element has only one physical model variable that can be updated [14]. This study further considers cases wherein each model node has multiple DOFs, and each model element has multiple physical model variables that can be updated.
Furthermore, the physical model variables of the damaged elements are then used to update the FEM. The model updating optimization algorithm is solved using the sequential quadratic programming (SQP) method [15], [16]. Elemental damage severity is indicated by the final updated physical model variables of the analytical model. The effectiveness of the proposed method is verified via comparison of the reference and updated physical model variables.
Section snippets
Residual force vector method
The correct structure stiffness and mass matrices are assumed as [K] and [M], respectively; the correct mode shapes and eigenvalues are and , respectively. Then, the characteristic equation is as follows:
If [K] and [M] are not completely correct (referred to as and , respectively), Eq. (1) cannot be satisfied. The residual force vector, introduced to indicate the degree of error, is defined as
and are obtained through the modal test results; and are
Case study
The portal beam structure used as the case study is depicted in Fig. 2. The structures on the left and right sides are divided into eight beam elements each, and the upper structure is divided into 10 beam elements. The 26 beam elements, positions 1–26, are shown in Fig. 2. The bottom of the structure is assumed to be fixed to the ground.
The structure is composed of steel, with a Young's modulus . The structure elements cross-sectional area A and second moment of inertia I are
Conclusions
The structural damage detection with two-stage evaluation and optimization updating methods developed in this study can be applied for structural damage monitoring and for an early warning system. A modal-based damage identification method that can be applied to portal beam structures was proposed herein. On the basis of changes in the structural modal parameters (natural frequency and mode shape) of geometric or material property variables, with errors in certain positions, damages in the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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