MS14.4 - Moving Loads

Moving load problems are ubiquitous in engineering, where an engineering structure is subjected to a moving point mass source, for instance, a vehicle moving on a bridge, a train moving on railway tracks, and a pantograph moving on a catenary, among others. The moving point source induces forces on the structure, which subsequently causes the deformation and displacement of structures. One physical quantity of interest for engineers is the deflection of the structure, determining which benefits the design and maintenance of the structure. In practice, numerical studies are carried out to compute the deflection profiles. However, in the case of railways, where the tracks are hundreds of kilometres long and the deflection of the tracks is in the scale of micro and nanometers, obtaining the complete deflection profiles of the railway tracks and catenaries is challenging. Mathematically, moving loads on beam like structures can be modelled as Partial Differential Equations (PDEs), including a Dirac delta function modelling the moving point mass source. These PDEs could be simulated to obtain complete deflection profiles at all space time locations. In recent years, considerable interest has shifted towards employing deep learning based methods to simulate and model differential equations. Traditionally deep learning based methods require a substantial amount of data; however, collecting the data for deflection profiles in case of moving loads is experimentally expensive. One possible way to mitigate this challenge is to leverage the physical equations directly in the neural network's loss function. Recently, a deep neural network based architecture termed as Physics Informed Neural Networks (PINNs) has been proposed to solve physical equations that use the PDEs in the loss function along with initial and boundary conditions and does not need any experimental data. Although PINN has shown significant progress in solving systems of physical equations, including non linear engineering systems, there are also several limitations of PINNs, including its inability to tackle shock type behaviour. The Dirac delta function in moving load PDE models an instantaneous load change at one location, which poses a challenge in the PINNs training process. We aim to tackle this challenge and accurately simulate the moving load problem through vanilla PINN architecture. In this work, we approximate the Dirac delta function as a Gaussian distribution. This results in a smooth approximation of the Dirac delta function, mitigating the challenge for the vanilla PINN architecture. In this paper, numerical experiments will be presented, highlighting the successful implication of the proposed methodology for moving load problems having applications in railway engineering. The major outcome of this research will be 1.) To the author's best knowledge this is the first work which simulates the moving load problems through Physics informed machine learning. 2.) Several experiments will be presented for the train track and catenary pantograph interactions, establishing the proposed methodology's validity. 3.) From the perspective of machine learning, this work would propose an approach to tackle the Dirac delta function efficiently.

The issue of the environmental vibration caused by subway trains has become a hot topic in both academic and engineering communities. An accurate train-track-tunnel-soil interaction model plays a decisive role in the prediction of subway train-induced environmental vibration, which is conducive to engineering design and construction. To this end, a comprehensive train-track-tunnel-soil coupled dynamics model is proposed in this study by combining the train-track sub-model and the tunnel-soil sub-model, in which the lining and grouting zone of the shield tunnel are regarded as cylindrical composite shells, and the soil is simulated as a viscoelastic artificial boundary. In addition, based on Kirchohhff plate vibration theory and Mindlin plate vibration theory, the motion of track slab is formulated to provide a more accurate simulation. Then, the accuracy and reliability of the train-track-tunnel-soil system dynamic model are validated by comparing the results calculated by the model with the data obtained by the field test under the subway train operation. Finally, the train-track-tunnel-soil coupled dynamics model is used to analyze the influence of different track structures, including monolithic track and three kinds of floating slab tracks, on the vibration characteristics of tunnel-soil system induced by subway trains. Results show that the floating slab tracks have great vibration isolation performance compared with the monolithic track, but have an amplification effect for the vibration around 10 Hz. The dynamic stress fluctuation of the soil under the floating slab track is slighter than that under the monolithic track, which indicates that the usage of the floating slab track can alleviate the stress state of the soil under the cyclic dynamic loads of the moving trains.

The dynamic analysis of continuous beams crossed by moving loads is a topic of great interest in the field of structural engineering. Indeed, the transversal deflection and stresses due to moving loads could be considerably higher than those observed under static loads. Usually, the damping of beams is assumed to be of the viscous type. This leads to a frequency dependence of the viscous mechanism, while experimental tests indicate that the damping forces are nearly independent of frequency [1]. Recently, it has been widely recognized that the most realistic model to describe the dissipation of the energy of materials is the viscoelastic one involving fractional derivatives [2]. The present study addresses the dynamic analysis of fractional viscoelastic Euler-Bernoulli beams crossed by moving loads. The motion of such beams is governed by a fractional differential equation. Specifically, since the structural systems can be considered quiescent before the motions appear, the fractional constitutive law involves Caputo’s derivative [3]. The main purpose of this paper is to propose a step-by-step procedure for the numerical integration of the fractional differential equation. To this aim, first, the Grünwald–Letnikov approximation of the fractional Caputo’s derivative is adopted [4]. Then, the so-called improved pseudo-force approach, introduced by Muscolino [5] for evaluating the response of linear structural systems governed by classical ordinary differential equations, is extended to fractional differential equations. Numerical results show that the proposed procedure reduces the computational burden and increases the accuracy with respect to numerical integration procedures proposed in the literature. References [1] Clough R.W., Penzien J. 1993. Dynamics of Structures. MacGraw-Hill, New York. [2] Di Paola M., Pirrotta A., Valenza A. 2011. Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mechanics of materials 43: 799-806. [3] Podlubny I. 1999. On Solving Fractional Differential Equations by Mathematics. Science and Engineering, vol. 198. Academic Press. [4] R. Scherer, S.L. Kalla, Y.Tang, J. Huang. 2011. The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications 62: 902–917. [5] Muscolino G. 1996. Dynamically modified linear structures: deterministic and stochastic response. Journal Engineering Mechanics (ASCE) 122: 1044-1051.