# An Effective Tangent Stiffness of Train-Track-Bridge Systems Based on Artificial Neural Network
## 1 Introduction  
Dynamic response analysis of a train-track-bridge (TTB) system is a challenging task for researchers and engineers, partially due to the strong nonlinearity and complex contact conditions of the wheel-rail interaction (WRI). In recent years, many researchers have proposed different WRI models based on various methods. [PERSON] et al. [1; 2; 3; 4] developed a novel explicit two-step method and a new family of predictor-corrector integration algorithms used in the solving of dynamic problems, based on which they developed a computer program named TTISIM to predict the vertical and lateral dynamic responses of the vehicle-track coupled system. [PERSON] et al. [5] proposed a wheel-rail contact formulation for analyzing the nonlinear train-structure interaction that takes into account wheel and rail geometry and used a WRI element to model the behavior of the contact interface based on [PERSON]'s theory and [PERSON]'s laws. [PERSON] [6] presented a general wheel-rail contact detection formulation and calculated the contact forces using [PERSON] linear theory, a heuristic nonlinear model, and the Polach formulation. [PERSON] et al. [7] developed a 3D WRI element by wrapping the WRI into an element where the contact force is obtained by using [PERSON] linear theory, nonlinear Hertz theory, and the Polach formulation. The WRI element has been demonstrated to be efficient and accurate, able to simulate complicated TTB systems, and is therefore used in the dynamic responses analysis of the TTB systems in this paper. When [PERSON]'s method is used to solve implicit nonlinear finite element equations, e.g., the TTB system, the consistent tangent stiffness (CTS) is essential to guarantee the quadratic convergence rate [8; 9; 10; 11; 12]. For the WRI element mentioned above, the computation of the CTS requires the differentiation of the internal forces with respect to the nodaldisplacement, following the complicated calculation process of the internal forces. Although the CTS is accurate and efficient and is able to guarantee the quadratic rate of convergence of [PERSON]'s method [13], its derivation and software implementation require significant efforts. Therefore, it is necessary to find an alternative method for computing the tangent stiffness that can reduce the efforts of derivation and software implementation while maintaining high computational efficiency. Recently, artificial neural network (ANN) has become increasingly popular in solving a wide range of engineering problems. The ANN can directly obtain a potentially good tangent stiffness by a trained relationship between input nodal displacement/velocity and output tangent stiffness. There are relatively less formula derivation and programming work. The ANN model has been used to solve complex nonlinear problems. [PERSON] et al. [14] presented a rate-dependent ANN constitutive formulation to analyze the time-dependent behaviors of concrete. [PERSON] et al. [15] proposed a novel ANN-based model for fast predicting the backbone curves of nonlinear flexure- and shear-critical columns. [PERSON] [16] proposed an ANN-based method for the identification of a discrete-time nonlinear hysteretic system under strong earthquakes. [PERSON] et al. and [PERSON] et al. [17] proposed an ANN-based health monitoring model for predicting the displacement responses of gravity dams. The ANN has been applied to various fields as an accuracy and efficiency method; however, it has not yet been applied to solving the tangent stiffness of complex nonlinear TTB systems. In this paper, a BP neural-network-based tangent stiffness (BPTS) method is presented for the WRI element proposed by [PERSON] and co-workers [8]. The input of the BP neural network is the displacements/velocities of the wheel node and rail virtual node, while the output is the tangent stiffness matrix of these nodes. The element tangent stiffness is obtained straightforwardly based on the output tangent stiffness.