# Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating
## 1 Introduction  
With the worldwide dissemination of high-speed railways, studies have been conducted in order to maintain high-speed railway structures, such as roadbeds, tracks, overhead systems, and bridges, by using numerical model updating [1; 2; 3; 4]. Since the damage to high-speed railway bridges caused by accidents can be more severe compared to other structures, numerical models that can accurately express the response characteristics of actual structures are very important for performing system identification, damage assessment, and structural health monitoring [5; 6; 7]. In general, the initial numerical model based on structural design documents is different from the results measured from the high-speed railway bridge [8; 9]. The difference is particularly more significant for high-speed railway bridges in operation due to construction errors and deterioration resulting from aging and structural damage [10; 11; 12]. Therefore, it is necessary to confirm that the deterioration of high-speed railway bridges is within the allowable range and no structural problems exist. Numerical model updating using the results measured from high-speed railway bridges in operation is very important in securing structural safety and performing efficient maintenance [13; 14; 15]. Reliable numerical model updating can reduce maintenance costs during the service period by recognizing and responding to the deterioration of high-speed railway bridges at an early stage. It can also be used as basic data to perform proper maintenance on significantly deteriorated high-speed railway bridges. The main methods used to perform numerical model updating are the direct method and the iterative method. The direct method has been developed in a variety of ways since the development of the Lagrange multiplier method [16; 17; 18; 19], matrix mixing method [20; 21; 22; 23], and error matrix method [24; 25; 26]. The direct method optimizes an objective function with constraints composed of the modal data measured from a structure and the system matrix of a numerical model. System matrix error is prepared as a mathematical function so that it can be corrected with a single calculation. Since the direct method attempts to correct the change in stiffness and mass directly from the mathematical function, accurate model updating is difficult if the structure has a complex degree of freedom. In order to address these issues, iterative methods were developed. Iterative methods can be divided into the sensitivity method that relies on the sensitivity of natural frequency and mode shape to changes in variables to be updated and the sensitivity method that uses the frequency response function of a structure, with the penalty function being mainly used [27; 28; 29; 30; 31]. Sensitivity methods can be divided into directly updating the element matrix of the member and updating physical variables. The iterative method that uses the sensitivity matrix is effective in identifying variables, such as materials, geometrical properties, and boundary conditions, that directly affect the dynamic properties of a structure. However, the sensitivity matrix must be prepared for all variables to be updated. The iterative method based on this sensitivity method always requires the differential functions of matrices on dynamic properties, such as natural frequency and mode shape, for each parameter, and its applicability for a complex structure is significantly low. It is also difficult to apply a commercial finite element analysis program due to the complexity of the inverse analysis algorithm. A method of updating the stiffness of the numerical model using displacement response was suggested in order to improve such applicability. However, it was applied to general railway bridges without interaction between the bridge and the train [32]. Most methods that use the iterative method have only been applied to very simple structures, and reduction and expansion of the system matrix is required in order to match the measurement degree of freedom.