Sensitivity Based Methods

Nowadays the sensitivity based methods are the most popular since they overcome the limitations of the direct methods. In these methods, the model updating problems are posed as optimization problems. They set the errors between analytical and experimental data as an objective function, and try to minimize the objective function by making changes to the pre-selected set of physical parameters of FE model. The optimum solution is obtained using sensitivity-based optimization methods. Because of the nonlinear relation between the vibration data and the physical parameters, an iterative optimization process is performed. This approach is able to update the relevant physical parameters and to locate erroneous regions of the model. Link [13] gives a clear overview of the sensitivity-based updating methods. These methods can be further classified according to the data used in optimization process as modal domain methods [7, 14-18] and frequency domain methods [19,20].

In frequency domain methods, one can use the input error, the output error or the error in frequency response data [13, 21]. Since the number of measured DOFs is generally much smaller than the number of analytical DOFs, it is necessary for most residual types to expand the measured vector to full model size or to condense the model order down to the number of measured DOFs. Furthermore, an implicit weighting is performed which depends on the proximity of the chosen frequency points to resonance. Therefore, the weighting is less controllable. In civil engineering, however, the approach using modal data is most often applied since the frequency response functions of heavy civil structures are not available over a wide frequency domain. The FE model updating can also be performed with a neural networks algorithm, as reported in Atalla and Inman [22].

A major problem in model updating is the relatively low information content of the measured data. Rade and Lallement [23] and Nalitolela et al. [24] increase the information content of the data, by testing the structure in different configurations so that the areas of model uncertainty are stressed in different ways. The alternative is to reduce the number of updating parameters, which is done in Teuguels et al. [25] through the use of damage functions. Fritzen and Bohle [26] proposed a parameter reduction technique for damage identification problems based on the correlation between the change in the dynamic stiffness matrix and the residual vectors.

Parameterization is the key issue in FE model updating. It is important that the chosen parameters should be able to clarify the ambiguity of the model, and in that case it is necessary for the model output to be sensitive to the parameters. Element stiffness parameters, such as the element’s Young’s modulus, are most often used as updating parameter as in Friswell and Mottershead [4] and Link [13]. Mottershead et al. [27] used the geometric parameters, such as offsets in beam elements, for the updating of mechanical joints and boundary conditions. Gladwell and Ahmadian [28] and Ahmadian et al. [29]

demonstrated how an element stiffness matrix can be adjusted by modifications to its eigenvalues and eigenvectors, and Mottershead et al. [30] used both, the geometric as well as the element modal parameters, in a generic element method to update mechanical joints.

FE model updating is used for the parameter estimation and damage assessments of structures. The works that have been reported in the literature in this aspect are summarized below.

1.4.2.1 Parameter Identification of Structures

The sensitivity-based FE model updating technique can be used as a parameter identification technique and belongs to the class of inverse problems. Inverse problems

typically involve the estimation of some quantities based on indirect measurements of these quantities [31]. In parameter identification process, the inverse operation is performed in which the model parameters are determined by fitting the model to the measured output values. Zhang et al. [32] identified various structural parameters, including connections and boundary conditions, of a cable-stayed bridge in Hong Kong by minimizing the discrepancies in eigenfrequencies and in literature [33], they reported similar work on a scaled suspension bridge model. Brownjohn et al. [34] quantified the effectiveness of upgrading works on a short-span highway bridge in Singapore through subsequent model updating, i.e., before and after the refurbishing and strengthening of the bridge. Ventura et al. [35] updated the Heritage Court building structure in Vancouver, Canada, by adjusting the stiffness and mass properties. Brownjohn and Xia [36]

investigated the application of the model updating technology to the dynamic assessment of a cable-stayed bridge in Singapore, by adjusting the Young’s modulus of the concrete and the structural geometry. Gentile and Cabrera [37] performed a similar study on a curved cable-stayed bridge at Malpensa airport in Milan.

1.4.2.2 Damage Detection of Structures

As FE model updating procedures are used to identify unknown physical properties and to build a representative FE model applicable to structural dynamics, they can also be used to detect and identify damage on structures. In 1996, Doebling et al. [38] made a detailed review of the vibration based damage identification literature. It gave a brief overview of global nondestructive methods based on the fact that structural damage usually causes a decrease in the structural stiffness, which produces changes in the vibration data of the structure. Fritzen et al. [6] examined the problem of detecting the location and extent of structural damage from measured vibration test data using FE model updating. It is noted that the mathematical model used in the model updating is usually ill posed and the special attention is required for an accurate solution. Wang et al. [39] implemented FE model updating to establish the baseline modal values (modal frequencies and mode shapes) for a long-span bridge. They suggested that model updating might be used in automated on-line monitoring on bridges. FE model updating method was successfully applied to the damage assessment of structures using frequency and mode shape residual with the introduction of damage functions [25, 40].

Quantitative and objective condition assessment for infrastructure protection has been a subject of strong research within the engineering community. To achieve this aim, methodologies of the routine inspections with fixed intervals or the continuous monitoring,

which provide constant information on safety, reliability or remaining lifetime of the structure, have been under development in recent years. Inspection of structural components for damage is vital to take decisions about their repair or retirement. Visual inspection is tedious and often does not yield a quantifiable result [41]. For some components a visual inspection is virtually impossible. Methods which are based on pure signal processing have only a limited capability for the early detection of damage and often do not allow unique conclusions to be drawn on the sources of the damage [6]. The importance and difficulty of the damage detection problem has caused a great deal of research on the quantitative methods of damage detection based upon physical testing.

Among those physical tests, the use of the modal tests has emerged as an effective tool to use in damage detection. The possibility of using measured vibration data to detect changes in structural systems due to damage has gained increasing attention [42, 43].

The methods are predominantly based on the change in eigenfrequencies, as in the paper of Hanselka et al. [44], Williams and Messina [45]. In an earlier work by Cawley and Adams [46], it was shown that the ratio of frequency changes in different modes was only a function of damage location and not the magnitude of damage. Salawu [47] reviewed the different methods of structural damage detection through changes in natural frequencies.

He emphasized the simplicity and low cost of this approach, but at the same time pointed out the factors that could limit successful application of vibration monitoring to damage detection and structural assessment since the changes in natural frequencies cannot provide the spatial information about structural damage. In order to localize the damage, mode shapes are used which provide spatial information about structural damage. Analysis of changes in mode shapes due to damage represents another subgroup of modal-based methods. Natke and Cempel [48] used changes in eigenfrequencies and mode shapes to detect damage in a cable-stayed steel bridge. Based on changes in frequencies and mode shapes of vibration, Ren and Roeck [49,50] proposed a damage identification technique for predicting damage location and severity. However, a large number of measurement locations are required to accurately characterize the mode shape vectors and to provide a sufficient resolution to find the damage location.

As an alternative for obtaining spatial information, Pandey et al. [51] introduced the use of mode shape curvatures and Maeck and De Roeck [52] extended this approach by using mode shape curvatures in a direct stiffness calculation technique which they applied to the damage identification in a prestressed concrete bridge. Ho and Ewins [53] states that the derivatives of mode shapes are more sensitive to damage, but the differentiation process enhances the experimental errors inherent in mode shapes, yielding a large statistical

uncertainty.

Changes in strain energy were used as an indicator to represent damage in many works.

Modal strain energy has been studied previously by Lim and Kashangaki [54] and Doebling et al. [55] in the identification of structural behavior and location of structural damage. Kim et al. [56] evaluated damage detection and localization algorithms based on changes in eigenfrequencies, mode shapes and modal strain energy. Stubbs and Kim [57] directly used the modal strain energy as a damage indicator. Shi and Law [58] and Ren and Roeck [59]

studied the change of the elemental modal strain energy before and after the occurrence of damage in the structure, and they verified that this parameter would be a very efficient indicator in structural damage localization.

Another class of damage identification methods uses the dynamically measured modal flexibility matrix. Catbas and Aktan [60] and Bernal [61] proposed the use of the flexibility matrix as damage indicator. Aktan et al. [62] proposed the use of the measured flexibility as a condition index to indicate the relative integrity of a bridge. Two bridges were tested and the measured flexibility was compared to the static deflections induced by a set of truck-load tests. Pandey and Biswas [63] presented a damage detection and location method based on changes in the measured modal flexibility of the structure. This method is applied to several numerical examples and to an actual spliced beam where the damage is linear in nature. Results of the numerical and experimental examples showed that estimates of the damage condition and the location of the damage could be obtained from just the first two measured modes of the structure. It is demonstrated that the modal flexibility is more sensitive to damage than the natural frequency or mode shape. Similarly, in the study of Zhao and DeWolf [64], the sensitivity study is carried out to compare the use of natural frequencies, mode shapes and modal flexibilities for damage detection and concluded that modal flexibilities are more likely to indicate damage than either natural frequencies or mode shapes. Reisch and Park [65] proposed a method of structural health monitoring based on relative changes in localized flexibility properties and applied for the damage detection of elevated highway bridge column. Topole [66] developed an algorithm to calculate the contribution of the flexibility of the structural members to the sensitivity of the modal parameters to change on the flexibilities of the members and applied to detect the damage of simulated structure with truss member.

The literatures for other specific issues of model updating are discussed in most relevant places throughout the thesis.