In this paper a new method for crack detection in structures based on first three mode frequencies and Modal strain energies using least square support vector machine has been proposed. Since the mode shape vectors are equivalent to nodal displacements of a vibrating structure, therefore in each element of the structure strain energy is stored. The strain energy of a structure due to mode shape vector are usually referred to as Modal strain energy (MSE) and can be considered as a valuable parameter for crack identification. Also, change of natural frequencies is effective, inexpensive, and fast tool for non-destructive testing. So, the proposed method uses the first three natural frequencies and Modal strain energies as the input parameters and crack states as output to train the least squares support vector machine model.

Civil structures inevitably undergo damage over time due to various reasons such as environmental changes, material aging, load variations, and insufficient maintenance. Monitoring these structures, especially aging ones, is crucial to detect damage early on and implement suitable retrofitting measures, ensuring their continued safe and reliable operation without unexpected failures. Consequently, there has been significant research in this field, focusing on damage detection in both simple and complex structures. Health monitoring of highway bridges is essential for achieving a reliable transportation system. The vibration-based damage detection method uses changes in the vibrational properties of structures to detect damages and ensure a healthy state. In this study, the absolute value of the Modal flexibility damage index and the Modal strain energy damage index simultaneously are utilized to prevent unsafe decisions. These absolute values of Modal strain energy and flexibility damage indexes are utilized as the bases for training deep neural networks (DNNs). These indexes are applied to provide safe decisions and reliable damage evaluation in steel girder of the highway bridges. The convolution neural network (CNN) is utilized for damage quantification estimation. The CNN is one of the deep learning models that can currently be applied in 2D dominant approaches, such as pattern recognition and speech recognition. In addition, these networks can utilize the 1D time domain and vibrational signal data via the convolutional layer. The initial stage of CNN model comprises combined convolutional and pooling layers that apply different filters to extract features. Following this, fully connected layers, similar to a hidden layer of a multilayer perceptron are incorporated. Ultimately, these layers are classified together with a softmax layer. The convolution layer acts as a filter that convolutes the input layer with a set of weights, adding bias and applying an activation function to the outcome. Gradient descent momentum methods (SGDM) can be employed to optimize the parameters in CNN network architecture. SGDM estimates the gradient with high velocity in any dimension. This method mitigates issues such as jittering and saddle points by utilizing high-velocity inconsistent gradient dimensions and the SGD gradients, respectively. Additionally, when the Current gradient approaches zero, the SGDM provides some momentum. The convolution neural network is trained to utilize damage indexes obtained from numerical simulation of the validated finite element model of the bridge. The damage indexes as the inputs for the neural network, which are achieved from different damage scenarios. Once network training and validation are completed, a well-trained neural network is used to detect, localize, and quantify the intensity of unknown damages. The proposed method overcomes previous damage detection problems such as false positive indications, the unreliability of a single damage index, and insufficient precision in determining the intensity.  The results revealed that the presented method, based on the dual updated damage indexes and CNN, practically and accurately identified unspecified single damages' location and severity in multi-span beams. The new training method of deep neural network systems overcomes some shortcomings in ANN. Moreever, this deep neural network training scheme can reduce the need for huge amounts of input data and enhance the accuracy of network training. The method is capable in predicting single damage scenarios in steel beam.

Change in Modal strain energy is one of the indicators used to detect damage in structures. However, in structures with high degrees of freedom, such as double-layer grids, this method requires a relatively large number of mode shapes which in practice is difficult to determine. Therefore, it is necessary to reduce the number of required mode shapes. In this study, a damage detection technique based on Modal strain energy and Dempster-Shafer evidence theory is presented for locating damage in double layer grids using only a few number of mode shapes. First, by calculating mode shapes of the grid in undamaged and damaged states, the Modal strain energy based index for each mode shape is determined. Then, the results obtained from separate mode shapes are combined using Dempster-Shafer theory to achieve better results. In order to investigate the effect of noise on damage detection, 3% random noise is added to mode shapes. To demonstrate the performance of the proposed method, different single and multiple damage cases with different damage intensities are considered. Numerical results show that using 5 mode shapes, the presented technique can detect up to 3 damaged elements with different damage intensities in different parts of the grid with good accuracy (probability of 92. 3%). Considering the fact that the classical Modal strain energy method fails to distinguish even 1 damaged element in the double layer grid, the result shows significant improvement.

This paper presents an extended cross Modal strain energy change method to estimate the severity of damage associated with limited Modal data in beam-like structures. This method takes in account the correlation between the analytical Modal data and the measured incomplete Modal data. A procedure was proposed and the analytical elemental stiffness of the damaged element after it is localized is included in quantification of the measured single damage extent. A three-dimensional numerical beam model with different damage cases is used to simulate the CMSE method application and to getting the bending displacements of the damaged element. An experimental Modal analysis (EMA) on a cantilever beam with and without crack was carried out to evaluate the effectiveness of the extended CMSE method. The severity magnitude of the damage was predicted within an acceptable error range through the using validation process. Results reveal that the proposed damage estimation method successfully evaluates single damage severity in beam like structure and can be useful in maintenance technology and structural health monitoring system.

In this study, an efficient method for determining the accurate location of damage to structures is introduced using an optimal sensor placement (OSP) and Modal strain energy-based index (MSEBI). The research is implemented in two main stages. In the first stage, a correlation function between the reconstructed mode shapes using the iterated improved reduced system (IRS) method and the complete mode shapes of a structure is defined and then the function is minimized via the binary differential evolution (BDE) algorithm to find the optimal sensor placement of the structure. In the second stage, the location of damage is determined using MSEBI based on the optimum place of sensors obtained already. In order to assess the efficiency of the proposed method, two standard examples, including a two-dimensional (2-D) frame structure with 45 elements and a 2-D truss with 47 elements, are examined. Numerical results, considering different conditions, demonstrate that the integration of OSP method and MSEBI can provide an efficient tool for accurate and rapid identification of the damage location. The parametric study shows that the proposed method has a low sensitivity to the number of modes and noise level, and it can properly identify damage by considering a few modes and the high level of noise.

As one of the most important components of an offshore oil and gas complex, Catwalk (access bridge) provides support for equipment and act as a passage for staff. Therefore, any damage in this structure may result in casualties as well as financial and environmental losses. Hence, identifying the location and severity of damage in these structures is of a great importance. As a common SHM method, Modal strain energy uses the changes in the dynamic properties of the structure for identifying the damage location and severity. Considering natural frequencies in the process of the damage localization is one of modifications that has been successfully applied to this method. In order to show the robustness of this method for identifying damages in real class offshore structures with a large number of elements, the improved Modal strain energy (IMSE) method is applied for damage localization and quantification in the access bridge of Foroozan platform in the Persian Gulf. The results showed that the IMSE damage index is more accurate than the original Stubbs index. Both the single and multiple damages were predicted with a good accuracy with this method. However, the method was more accurate in locating the damages in horizontal elements as well as the elements far from the supports of the structure.

Changes in Modal strain energy of elements before and after damage is a robust damage detection index. However, in structures like double layer grids which have large number of elements, this method has some problems. First, In large structures this method needs more mode shapes to detect damage through Modal strain energy method which in practice is difficult to determine. Second, this method introduce some healthy elements as damaged element. To overcome these problems, in this paper a two stage damage detection technique based on Modal strain energy method is presented for detecting damage in double layer grids. First, the Modal strain energy Based Index (MSEBI) for each mode shape is determined. Then a data fusion technique based on Bayesian theory is used to combine MSEBI values obtained from each mode shape to find damaged elements. Then Charged System Search (CSS) optimization method which is a powerful optimization method is employed to optimize an objective function based on natural frequency to determine damage severity of damaged elements. To demonstrate the performance of the proposed method, a large double layer grid with 1536 elements and different single and multiple damage cases is considered. Numerical results show that the proposed method can successfully find damaged elements and their severities using only few first numbers of mode shapes and frequencies.

Foroozan platform is one of the most important Iranian offshore facilities lies along the border of Iran and Saudi Arabia. As time passes by, the possibility of presence of some damage in the structural members of this platform increases. As a consequence of these damages, the operation of the platform may be disrupted and if the damage grows, especially in sensitive areas such as deck, joints or splash zone that are prone to failure, cause more damage in the future. This highlights the necessity of structural health monitoring of this platform. One of the most common structural health monitoring techniques is the Modal strain energy-based damage index which is known as Stubbs index. In recent years, modifications have been made to original version of this index, one of which is to consider the natural frequencies in determining the location of damage. In this paper, using the improved Modal strain energy method and considering the natural frequencies in determining the location of damage, the location and severity of damage in the Foroozan platform have been identified. One of the differences between this study and similar studies is the large number of elements of this real platform. The results showed that the improved method has a higher accuracy in locating the damage than the original method (Stubbs index). Also, single and multiple damages, with low and high intensity, were predicted with appropriate accuracy by this method.

Abstract The passage of personnel and the placement of facilities on the access stairs of offshore complexes have made it very important to identify damage to these components. The Modal strain energy method is one of the non-destructive and practical methods that uses changes in the dynamic properties of the structure to identify the location and determine the severity of damage in the structure. In recent years, modifications have been made to the initial version of this method, one of which is to consider the natural frequencies in locating the damage. In this paper, using the improved Modal strain energy method and considering natural frequencies, a new relationship is presented to more accurately identify the location of damage in the structure, and three different damage indices in the offshore platform access bridge structure are studied and compared. The results show that the average error to accurately identify the location of damage in the average Stubbs index, the improved method and the novel method are 3.55, 2.82, and 2.21 percent, respectively, so the novel method can more accurately identify the location of damage in the structure. Also, comparing the results of different cases shows that the average damage location error decreases with increasing damage severity. The accuracy of identifying the location of the damage also increases when moving away from the supports.  Keywords: Offshore platform, Access bridge, Damage detection, Improved Modal strain energy, Stubbs index. INTRODUCTION One of the most important components of marine platforms is the access bridges between them. In addition to being used for the passage of personnel, these bridges are also the place for the passage of facilities. Due to the location in the corrosive environment of the sea and also the possibility of various damages to these members that cause huge financial and life losses, it is very important to check and control the health of these structures. To monitor the health of structures, various methods are used. One of the methods used in this regard is visual inspection, which leads to obtaining important information about the health of the structure. In addition to its special advantages, this method requires a lot of time and money. Also, due to the unavailability of some sections of the structure, it is not possible to detect the damage in them using visual inspection, and it is not possible to detect the internal damage of the structure and its origin using visual inspection. For this reason, in recent years, a lot of attention has been paid to non-destructive damage identification methods to increase safety and ensure the existing condition of the structure. One of these methods is to use the vibration characteristics of the structure to evaluate the damage at the structure level, which is used as a complementary solution next to visual inspections. In all vibration-based damage detection methods, the structure's Modal characteristics (natural frequency, mode shape, and Modal damping) are a function of its physical characteristics. Therefore, by using the change in the static or dynamic response of the structures, the change in their physical characteristics and as a result the structural damage can be identified in the initial stages, the maintenance costs of the structure can be reduced and the failure of the structure can be prevented. The studies conducted in the past show the high accuracy and ability of the Modal strain energy method in detecting damage in marine structures. Considering the long operational life of the country's offshore platforms and the existence of possible damages in these platforms, as well as the importance of access bridges between offshore platforms, in this article a comparison between the accuracy of different damage detection methods, including the Stubbs index method, the improved strain energy method and the new recommended method has been done by the authors to use this method in the identification of real damages in marine platforms. In the Stubbs index method, as the primary method based on Modal strain energy, only the shape of the structure's vibration modes has been used to identify damage. In the improved method, by using the natural frequencies of the structure, a more accurate estimate has been provided to identify the damage in the structures. This research tries to provide better results, especially for multiple damages in structures with a large number of members, by improving the improved method. METHOD AND MATERIAL When an elastic object is subjected to a force, tension is created in it, the object changes its shape and the state of its various points changes compared to the initial state. Changing the point of effect of forces applied to the body causes some work to be done when they are applied. The aforementioned work, which is accompanied by the change of shape of the object in the state of tension, causes the storage of some energy in the form of elastic energy in the object, which is called strain energy. Modal strain energy is a situation where no force is applied to the structure and the structure is in a state of free vibration, and the Modal strain energy of each member can be obtained by dynamic analysis and solving the presented relationships. Damage in a structure usually causes a decrease in the stiffness of the structure and does not affect the mass matrix of the structure. In this part, using the Modal strain energy method, the damage in the structure has been identified. This damage is defined by reducing the stiffness (reduction of the elastic modulus) of the member and the finite element model of the structure is written in MATLAB software. So, for example, the presence of 20% damage in member number 15 causes the difficulty level of that member to reach 80% of the initial state. Validation of the results was done by comparing the assumed amount of damage and the amount of damage obtained from the written program. The structure has 100 members and is modeled as a bending frame. The 4 connection points of the structure to the marine platform are modeled as girders. The rows and columns related to the supports have been removed from the mass and stiffness matrices of the structure. The elastic modulus (E) is equal to 210 GPa and the density of steel is equal to 7850 kg/m3. DISCUSSION AND CONCLUSION With the passing of the service life of marine structures, the need to monitor their health is felt to identify the place of damage under the structures. The communication bridge between marine platforms is of great importance considering that it is the place where personnel and pipelines pass. However, it has been less studied in previous studies. In this research, using the Modal strain energy method, which is one of the most suitable methods for non-destructive damage identification in structures, single and multiple damages were identified in the communication bridge in a marine platform, and the results of the three Stubbs index methods, the strain energy method The improved Modal and the new method presented by the authors were compared with each other. It was also tried to include different modes of damage; members whose failure causes more damage. The results obtained from this article are: Considering that the average error for the accuracy of identifying the location of damage in the Stubbs index, the improved method and the new method are 3.55, 2.82 and 2.21% respectively, so the new method can identify the location of damage in the structure with more accuracy. Comparison of the results of different modes shows that the average damage location error decreases with increasing damage severity. In the case of minor and small injuries that are in the initial stages of formation and growth, as well as large and highly developed injuries, the new method can identify the location of the injury with a high ability. The accuracy of damage location using the Modal strain energy method is higher in horizontal members than in restrained members. The accuracy of this method is higher for members that are on a horizontal plane than for members that connect the upper and lower members. The accuracy of identifying the damage location increases by moving away from the supports. The new method identifies the location of the damage in the structure with higher accuracy and it is recommended to use this method instead of Stubbs index and improved Modal strain energy methods. ACKNOWLEDGEMENT The authors would like to thank the Iranian Offshore Oil Company (IOOC) for providing the drawings of the access bridge of the Foroozan Oil complex.