model updating method

Damage in structural elements causes obvious changes in their physical properties such as stiffness and damping. These changes affect the stiffness and damping matrices of whole building so its mode shapes change. Therefore, mode shapes of existing building are widely used in damage detection methods. Since vibration test can only provide translational components of mode shapes, previous methods mostly worked with this type of data. This paper focused on considering the importance of using some rotational/translational components of the mode shapes to detect damages in structural frames. In order to analyze the frames and update them, an automatic iterative model updating program is developed in MATLAB software that works with OpenSees for conducting finite element analysis. The iterative program evaluates a set of objective functions in each step and tries to optimize them by means of nonlinear least square method. Objective functions are defined based on the combination of two criteria of these four items: comparison between frequencies and/or mode shapes of two situations, the modal assurance criteria (MAC), and the modal flexibility matrices. In each step of the analysis, based on optimization results, a new frame will be modeled in OpenSees software that its elements stiffness is changed according to new sets of data, then finite element analysis will be done and new modal data will be extracted and optimization process will be repeated by new data. To verify the effectiveness of the developed program, two three-dimensional steel structures are modeled and evaluated, one of them is a five-story moment resisting frame and the other one is a three-story brace frame. It has been considered that these frames suffered damages which are defined by three different scenarios for each of them. Damage scenarios consist of minor, severe and both minor and severe damages. Actually, in this study, damages are defined by reduction in elements’ stiffness. In fact, damage is a percentage of reduction of stiffness in damaged element in comparison with its healthy condition. Mode shape components and natural frequencies of damaged structures are the only needed input data for the program. To investigate the influence of rotational components in model updating, frames have been analyzed with three types of data in each scenario, all translational or rotational components, and all components of mode shapes. Extensive analyses show that among employed objective function, the one which compares mode shapes is the most successful one in damage detection, also modal flexibility can be effective when it works by only rotational components of mode shapes. The findings indicated that the translational components of mode shapes are not capable of detecting damages accurately. Results of model updating by use of only translational components of mode shapes indicate that not only the damages’ location and their intensities could not be predicted, but also several false damages are reported in undamaged elements. It can be concluded that using rotational data leads to more precise results in determining both damages’ locations and their intensities. Besides, the number of false damage detection has been decreased by use of rotational components. It means, when the rotational components are employed, the methods report no damage in healthy elements or the amount of detected damage is very small that can be ignored. Real data extracted from existing building are always polluted by noises due to human or machine faults or sometimes errors in numerical methods lead to inexact input data. Since the data employed in this study are exact numerical data, to consider the effects of these errors, analytical modal data has been polluted by some noises. These noises are generated by use of random function in MATLAB software. Surprisingly, the results show that even with noisy data, the proposed method can detect damages precisely.

In many vibration-based methods, stiffness matrix variations play an important role in determining the location and extent of damage. In this research, through a bilinear crack model, stiffness matrix components are directly linked to the cracking parameters with the aid of seven coefficients. An innovative method for identifying the number of breathing cracks is developed, in a manner that the information of the initial structure is not needed. To model structures in MATLAB software, a nonlinear dynamic analysis program with the ability to model bilinear cracks is developed. Breathing crack modeling has been done using the softness method and the concept of curvature of the member. In order to determine the open or closed state of the crack, at each moment of vibration, an indicator was applied based on the moment curvature of each element. Due to the crack opening and closure, frequency varies continuously over time. If there is only one crack in the structure, two frequency values for the first mode of vibration are obtained. It is clear that one is related to when the crack is open and the other corresponds to when the crack is closed. By increasing the number of cracks, the number of frequency bands increases nonlinearly. This feature applied to determine the number of damaged points, so that the vibration frequencies were determined and arranged throughout the vibration time. By plotting the ordered frequencies, the points of the same frequency were plotted at a certain level and formed the frequency steps. A significant relationship was found between the number of frequency steps and cracked points. By applying different crack topography in buildings of one, two and five stories, a relationship between frequency steps and number of cracks was formulated. In each incident scenario, the effect of different crack intensity and distribution is investigated through three different cases. The first case is, 20 different crack layouts with a depth of 0.1 of the cross sectional area and random distribution, the second case is, 20 different crack layouts with a depth of 0.3 of the cross sectional area and random distribution, and the last case is, 20 different layouts in which the location and depth of the crack as well as the crack propagation are considered random. The average and standard deviation of frequency steps for each 20 tries are obtained. This study shows that, the variation of frequency steps in terms of the cracks number is similar in different structures. The uniform change of the depth of all cracks is ineffective in increasing the number of frequency steps. However, diversifying cracking due to the combination of asymmetric distribution, increases the number of frequency steps. When about half of the members are cracked, the maximum standard deviation of frequency steps occurs. This dispersion reduces the identification accuracy up to two crack numbers in the studied structures. The curve of variation of frequency steps in terms of the number of cracks is divided into two parts. The first part of the curve is as long as the number of cracks is less than half of the number of members. At this region, the exponential relationship between the two parameters is established. The second part of the curve is when more than half of the number of members are cracked. In this case, the frequency steps show little changes with the change in the number of cracks. In this region, the relationship between the two parameters is different for each structure, but it has a trend of quadratic order. Some of the factors such as axial force which can change the stiffness of the frames at any moment of vibration can affect the number of frequency steps. Investigating the effect of these factors can be the subject of future research.