MS14.2- Moving Loads

This work addresses the dynamical response of slender structures crossed by systems of equally spaced transverse forces which travel at constant velocity along the main length of the structure. The study of the dynamic behavior of elastic structures acted upon by moving loads is a topic of great importance in civil and industrial engineering and has been the subject of many investigations over the years. Different external action models of various complexity have also been proposed in the literature, but the model of the moving force, despite its simplicity, represents a good schematization when the inertia of the traveling mass is negligible with respect to the mass of the entire structure, which is the case considered in this study. Specifically, we investigate the dynamical response of a slender elastic structure schematized as an Euler-Bernoulli beam, which is crossed by a system of equally spaced forces travelling at constant velocity, with the distance between two adjacent forces equal to an integer submultiple of the beam length. This problem represents a natural generalization of a problem recently studied by Luongo and Piccardo (Continuum Mech. Thermodyn., 28, 603-616, 2016) and is solved analytically in the present work. Starting from the case of the single moving force, the effect of the aforementioned train of forces is analytically accounted for via a linear map, which transforms the system state (that is, system position and velocity in modal coordinates) at the time instant tk (time at which one of the equally spaced travelling forces, say Fn, leaves the beam from one of its ends) to the system state at the subsequent time instant tk+1 (time at which the preceding force Fn-1 of the train of forces leaves the beam from the same end). The reiteration of this linear map provides the discrete-time response of the beam. The analytical solution obtained in this work enables the study of the dynamic behavior of the system and, in particular, highlights the presence of critical values of the train velocity for which instability phenomena by response accretion may occur. Comparisons with solutions obtained by resorting to purely numerical methods confirm the accuracy of the proposed analytical solution.

Railway transportation has received increasing attention recently, especially in the context of climate change due to its capability of running fully on electricity, which can be generated from green sources. With this increasing demand on railway transport, the previously considered acceptable deterioration of the infrastructure is rapidly becoming a limiting factor in its sustainable development. One such situation is encountered at transition zones in railway tracks, which are locations with a significant variation of track properties (e.g., foundation stiffness) found near man-made structures such as bridges and tunnels. These zones require maintenance more often than the rest of the track as they are prone to pronounced differential settlements. In general, the measures to counteract these differential settlements were not successful, some due to poor design or deficient implementation, but the majority due to the lack of understanding of the governing mechanisms that cause the pronounced differential settlements at transition zones in railway tracks. To identify and investigate the underlying degradation mechanisms, next to experimental investigations, researchers and engineers used computational models ranging from simple phenomenological 1-D models to complex predictive 3-D finite element models (FEM). While the latter are important in situations when specific predictions are required, the interference and interaction of multiple phenomena makes it difficult to study specific mechanisms in detail, for which the former models are sometimes preferred. Nonetheless, also these phenomenological 1-D models have drawbacks, such as the local nature of the foundation as well as the neglect of wave propagation in the underlying soil medium. This study attempts to bridge the gap between simplified 1-D models and complex 3-D FEM ones by formulating a phenomenological model in which the soil is represented by a 2-D (plane strain) continuum layer. The railway track is modelled as a beam on a nonlinear and inhomogeneous spring-dashpot layer that rests on the soil continuum and is subject to a moving constant load. The settlement and variation of track properties is restricted to the spring-dashpot layer, as the soil continuum is assumed to be linear and homogeneous. This investigation aims to shed light on the influence of incorporating the foundation nonlocality and the situations in which this is necessary for accurate predictions of the settlement. To this end, the response of the aforementioned 2-D system is analysed in depth and is compared to an equivalent 1-D system that does not incorporate the foundation nonlocality. This study can help researchers and engineers in better understanding the limitations of simplified modelling approaches depending on the specific problem investigated.

Transition zones are critical in high-speed railways because of the concentrated occurrence of track and substructure degradations. Even if the rail geometry irregularity has been well controlled by intensive maintenances, amplification of dynamic responses can be commonly found at the various substructures interfaces and their vicinity. These phenomena could be explained by the abrupt change of the subgrade stiffness. Combining with a typical transition zone configuration of high-speed railways involving wedge-shaped backfills, this study is expected to explain the reason why components of transition zone easily deteriorate from the perspective of energy concentration. In addition, considering the background of constant increase of train speed, the elastic field state and the energy distribution when the load speed is close to the critical velocity will be focused on. In this study, the two-dimensional plane-stress model, in which two elastic layers with different physical properties are coupled by an inclined interface, is established. Both layers have a free surface and rigid bottom. The vehicle is simply regarded as a constant load moving along the free surface with a constant velocity. The elastic field of the model is composed of an eigenfield and a free field. The eigenfield describes the steady-state response part of the elastic layers subjected to the moving load. The free field, composed of guided waves that are excited when the load moves over the inclined interface, is the key to reveal transition radiation of such a physical model. The model is divided into a finite-difference region containing the inclined interface and continuous media regions on both sides. Based on hybrid method, the semi-analytical solution of guided wave modal coefficients and elastic field are solved in turn. The investigation of energy distribution in the space–frequency domain is realized by assessing the energy flux through enclosing surfaces. This study carries out a single factor analysis, in which a series of models with different model parameter values (e.g., inclination angle, stiffness ratio of two elastic layers and load speed) are compared. The results indicate that the power of transition radiation and the energy distribution around the interface change when the inclination angle changes. Stronger energy radiation is brought by larger stiffness ratio and such energy is in general accumulating in the softer layer. Analogies can be drawn between the abovementioned conclusions about energy and viewpoints obtained from existing research on transition zones of high-speed railways, which proves that the model is meaningful for revealing some physical phenomena in transition zones. When the load speed approaches the critical velocity, the magnitudes of the spectral density of transition radiation energy in the whole frequency range are significantly magnified. It provides explanation for the restriction of train speed increase