Transition zones in railway tracks are locations with a significant variation of track properties (i.e. foundation stiffness) encountered near structures such as bridges and tunnels. Due to strong amplification of the track’s response, transition zones are prone to rapid degradation. To investigate the degradation mechanisms in transition zones, researchers have developed a multitude of models, some of them being very complex. This study compares three solution methods, namely an integral-transform method, a time-domain method, and a hybrid method, with the goal of solving these systems efficiently. The methods are compared in terms of accuracy, computational efficiency, and feasibility of application to more complex systems. The model employed in this paper consists of an infinite, inhomogeneous, and piecewise-linear 1-D structure subjected to a moving constant load. Although the 1-D model is not particularly demanding computationally, it is used to make qualitative observations as to which method is most suitable for the 2-D and 3-D models, which could lead to significant gains. Results show that all three methods can reach similar accuracy levels, and in doing so, the time-domain method is most computationally efficient. The integral-transform method appears to be efficient in dealing with frequency-dependent parameters, while the time-domain and hybrid methods are efficient in dealing with a smooth nonlinearity. For multi-dimensional models, if nonlinearities and inhomogeneities are considered throughout the depth, the time-domain method is likely to be most efficient; however, if nonlinearities and inhomogeneities are limited to the surface layers, the integral-transform and hybrid methods have the potential to be more efficient than the time-domain one. Finally, although the 1-D model presented in this study is mainly used to assess the three methods, it can also be used for preliminary designs of transition zones in railway tracks.

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Transition radiation is emitted when a source moves along a straight line with constant velocity and acts on or near an inhomogeneous medium [1]. It occurs, for example, when trains cross areas with substantial variation of track properties (e.g., foundation stiffness) encountered near rigid structures such as bridges; these areas are called transition zones. Apart from potentially giving rise to vehicle instability, transition radiation has been addressed as one of the causes of track and foundation degradation due to the often strong amplification of the stress and strain fields. This leads to a high frequency of maintenance required for transition zones in areas with soft soils, which can be 3–8 times higher than for the regular parts of the railway track. Wave radiation is also generated due to the periodic variation of the support stiffness (i.e., periodically placed sleepers). The periodically supported beam interacting with a vehicle has been addressed in studies on vehicle instability and on the resonant behaviour of the system [2]. Moreover, it has been shown that the periodic supports play a role in the fatigue and corrugation of both wheel and rail [3]. However, most studies consider either the local variation of the foundation stiffness (i.e., transition zones) or the periodic variation of the support stiffness, and not the combination of the two. Furthermore, studies that consider both scales make use of complex supporting structures and vehicle models such that the influence of the discrete supports on the transition radiation is not clear. This work aims at studying the influence of accounting for the discrete supports on the transition radiation (inside transition zones) and on the plastic deformation that develops in the supporting structure. To this end, a 1-D model is formulated, consisting of an infinite Euler-Bernoulli beam discretely supported by nonlinear springs and dashpots whose characteristics locally vary in space, interacting with a moving loaded oscillator. The solution is obtained using a time-stepping method (i.e., Newmark-β) for the temporal dimension of the system while the finite element method is used to discretise the spatial dimension. The infinite extent of the system is ensured through a set of non-reflective boundary conditions obtained using the Floquet theory. The model presented here can be used for preliminary designs of transition zones in railway tracks. Given the stiffness dissimilarity, the optimum length of the transition zone and the train’s maximum velocity can be obtained such that the damage in the railway track is minimized.@en

The Arctic presents a great opportunity for two major industries. First, since the region is expected to contain a significant amount of hydrocarbon reserves, it is very attractive for the oil and gas industry. Second, the receding extent of sea ice is making the region more accessible for shipping and, therefore, an opportunity is emerging for the shipping industry. In order to exploit both economic opportunities in a safe and sustainable manner, a thorough understanding of the interaction between ice and floating structures is needed. The most common method for studying ice-floater interaction (IFI) is via numerical modeling, which the fluid is a major component of. As fluid-ice interaction is challenging to model, a wide range of simplified and sophisticated models are employed to meet the challenge. A literature study was performed on the usage of fluid models employed in IFI and it was found that they can be divided into four categories: hydrostatic models, models based on potential flow, models based on Reynolds-averaged Navier–Stokes or a similarly advanced method, and effective fluid models. The hydrostatic models are by far the most prevalent despite only accounting for buoyancy. Most IFI models that account for hydrodynamics make use of potential theory. These models account for fluid flow and surface waves, which together alter the dynamic behavior of floating ice, resulting in hydroelastic effects. The surface-wave-based coupling between ice and floater has not been studied before and there are still open questions regarding the effects of hydroelasticity on the bending failure of ice. The advanced fluid models are a recent trend in IFI and, consequently, most of those are still under development. These models are very promising and may be the future of IFI modeling. Finally, the effective models avoid the practical issues associated with hydrodynamic models in terms of development and calculation time by capturing hydrodynamics in an effective manner, employing, for instance, added mass and damping coefficients. While several studies investigated the efficacy of these models, currently no satisfactory effective fluid model exists. The main goal of this thesis is to further the understanding of how hydrodynamics affects the interaction between ice and a sloping structure and to assess whether it is possible to create an effective model that can replicate the observed effects. The full scope encompasses three smaller studies. First, the surface-wave-based coupling between an elastic ice sheet and nearby floater structure is investigated. This interaction has not been studied before and the solution method that is developed for this problem is also used in the subsequent two studies. Second, a thorough study of the effects of hydrodynamics on the interaction between a sloping structure and level ice is accomplished. This study resulted in the identification of the parameter range wherein hydrostatic models are valid, which is essential given that they constitute the majority of all models. In addition, this study improved the understanding of the effects of hydrodynamics by means of investigating the importance of various components such as the rotational inertia of the ice, axial compression, and the nonlinear hydrodynamic pressure. Furthermore, the relation was analyzed between the temporal development of the contact force and the velocity dependence of the breaking length. Lastly, based on the findings of the second study, an attempt was made to develop an effective fluid model for ice-slope interaction. The efficacy of this model was studied in this thesis for a range of parameters. The main findings of the three studies are summarized next. In the first part of this thesis, the interaction is investigated between an ice floe and a floater through surface waves. This problem is considered first as the Green's functions that are derived for this problem are required for the subsequent studies on ice-slope interaction. The floater is modeled in-plane as a thin rigid body that floats on the surface of a fluid layer of finite depth. On one side of the floater, an ice floe is present which is modeled as a semi-infinite Kirchhoff-Love plate. The floater is excited by external loads and the resulting motions generate waves. Those waves hitting the ice edge are partly transmitted into the ice floe and partly reflected back towards the floater. The reflected waves exert pressure on the floater, altering its response. The resulting motions of the floater were analyzed, revealing several interesting facts. The study showed that below a certain onset frequency, the waves are almost fully transmitted into the ice floe and, consequently, the response of the floater is unaffected by the presence of the ice. The susceptibility of a floater to the waves reflected by a nearby ice floe can thus be estimated by checking how much of its open water response occurs above or below the onset frequency. The onset frequency is sensitive to changes of the ice thickness and insensitive to changes of the Young's modulus and water depth. Above the onset frequency, the waves reflected by the ice have a pronounced effect on the response of the floater. In certain frequency ranges, quasi-standing waves occur within the gap between ice floe and floater. Within these frequency ranges, the response of the floater is significantly altered. Depending on the phasing between the reflected waves and the floater's motions, resonance or anti-resonance can occur which can greatly amplify or reduce the floater's motions when compared to the case when no ice is present. Even when there is no gap between ice and floater, the amplitude of the floater can still be amplified and its natural frequency somewhat increased. The second study of this thesis focuses on the effect of hydrodynamics on the bending failure of an elastic ice floe due to the interaction with a downward-sloping floater, i.e. the effects of hydrodynamics on ice-slope interaction (ISI). A novel, semi-analytical in-plane ISI model is proposed that is based on potential theory in conjunction with the nonlinear Bernoulli equation to describe the fluid pressure. The ice is modeled as a semi-infinite Kirchhoff-Love plate. The predictions of the hydrodynamic model are compared with those of a hydrostatic ISI model, thereby obtaining a quantitative measure of the effect of hydrodynamics on ISI. The comparison revealed several interesting facts. First, the importance of several components of the model was investigated to determine which ones are essential for ISI. It was found that the contribution of the rotational inertia of the ice, axial compression and the nonlinear hydrodynamic pressure is insignificant. Being able to ignore the last two components greatly simplifies the modeling of ISI as it removes all sources of spatial nonlinearity. The terms that were found to be essential for ISI, listed in the order of importance, are: bending of the ice floe, linear hydrodynamic pressure, hydrostatic pressure and the inertia of the ice floe. The contribution of the fluid's inertia is on average four to ten times bigger than that of the inertia of the ice. The study also demonstrated that the effect of wave radiation on ISI is minimal. Second, the relation between the temporal development of the contact force and the velocity-dependence of the breaking length was studied. The study showed that the breaking length has two regimes which are separated by a transition velocity. When the ice velocity is below the transition velocity, the ice fails during the initial impact. Alternatively, when the ice velocity is above the transition velocity, the ice floe survives the impact and fails with a breaking length that is close to the static breaking length. The transition velocity of the hydrodynamic model is much lower than the transition velocity of the hydrostatic model, 0.0725 m/s compared to 0.275 m/s. This major difference in transition velocity is the primary reason for the limited applicability of the hydrostatic model. The results show that the hydrostatic model should not be used when the ice velocity is higher than 0.6 times the transition velocity of the hydrodynamic model as its predictions will deviate significantly, with errors ranging from 30% to 100%. This upper bound corresponds to values between 0.02 m/s and 0.1 m/s for the parameters considered. Lastly, this study underlined the stochastic nature of the breaking length of the ice floe. When the floe fails, a relatively large segment of the floe is, in fact, close to failure. A defect in the ice can locally amplify the stresses, causing the ice to fail at the defect rather than at the location predicted by a homogeneous model. This can cause the breaking length to vary by 10% to 30%. The last part of this thesis builds on the knowledge gained in part two by attempting to create a simple effective fluid model (EFM) that captures the effects of hydrodynamics on ISI as observed in part two. Based on the observations, an EFM is proposed that uses frequency-independent added mass and damping coefficients. This EFM was added to the hydrostatic model, thereby obtaining an ISI model that contains all four essential components of the ISI model. The resulting effective ISI model is very simple and, consequently, its implementation is trivial compared to a true hydrodynamic model such as the one proposed in part two. Its simplicity should help to improve the adoption of hydrodynamics in ISI. The performance of the effective ISI model is assessed. Investigated are the velocity-dependent breaking length, the maximum contact force that occurred during the interaction, and the contact force as a function of time. The predictions of the effective model are far more accurate than those of the hydrostatic model. The coefficients of the EFM were found to be relatively insensitive to changes in the parameters, allowing the effective model to be used for a fairly broad range of parameters. @en

In this paper, the efficacy of an effective fluid model (EFM) is studied for replicating the effects that hydrodynamics has on the interaction between level ice, modeled as a semi-infinite Kirchhoff-Love plate, and a downward sloping structure, modeled as a rigid and immobile body. The proposed EFM is based on a distributed frequency-independent added mass and damping coefficient, as well as a damper located at the point of contact with the structure. The optimal value of the three coefficients of the EFM is obtained by minimizing the error of the predicted breaking length and maximum contact force over a range of ice velocities when compared to a true hydrodynamics ISI model that is based on incompressible potential flow. The resulting effective hydrodynamic ISI model has greatly improved performance compared to an ISI model that only accounts for hydrostatics, even when the parameters of the system are changed. Moreover, it is much easier to implement and has significantly faster calculation times than the true hydrodynamic ISI model.@en

In this paper, the bending failure of level ice caused by the interaction with a downward sloping structure is studied in 2D. The focus is on the effect of hydrodynamics on the interaction. This study is done by comparing the predictions of a model that includes both hydrostatics and hydrodynamics with one that only includes hydrostatics.
For both models, the ice is modeled as a semi-infinite Kirchhoff-Love plate that is assumed to float on an infinitely wide fluid layer of finite depth. The fluid pressure exerted on the ice is governed by the nonlinear Bernoulli equation. The ice moves towards the structure, impacts with its downward sloping hull, slides down the structure and ultimately fails in downward bending. Validation of this model shows good agreement with experimental data.
It is shown that the nonlinear term in the Bernoulli equation has a negligible effect on the interaction and can be ignored. The effect of hydrodynamics can thus be attributed to the linear part of the hydrodynamic pressure.
The effect of the rotational inertia of the ice and axial compression is negligible as well. At low velocities, ice fails in a quasi-static manner, while at higher velocities, the failure takes place shortly after the contact initiation. The
transition between these two regimes is marked by a transition velocity that is significantly lower for the hydrodynamic model than for the hydrostatic one. Because of this, it is not desirable to use the hydrostatic model for velocities above the transition velocity.@en

In this paper the effect of a nearby, semi-infinite, level ice sheet on the frequency domain response of a thin, floating, rigid body is studied using a 2D model. The ice is modeled using a dynamic Euler-Bernoulli beam and the finite depth water layer is described with the Laplace equation and the linearized Bernoulli equation. Eigenfunction matching is used to resolve the interface between the ice covered and open water regions. The body is excited by external loads, generating waves. The waves are partially reflected by the ice edge and these reflected waves influence the body's response. It is this influence that this paper focuses on. Below a certain onset frequency the amplitude of the reflected waves is insignificant and consequently the body remains unaffected by the ice. This frequency is only sensitive to the ice thickness with thinner ice resulting in a higher onset frequency. Above the onset frequency the reflected waves cause quasi-standing waves between body and ice. For frequencies at which half the wavelength of the surface wave in the water is approximately an integer multiple of the gap length, the amplitude of the standing waves is greatly amplified. This can result in (anti-)resonance depending on the phasing between the reflected waves and the body's motion.

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The bending failure of level ice in 2D is mainly characterized by its breaking length and interaction forces. In this paper a study is carried out to assert if a 2D ice-structure interaction model based on an incompressible, inviscid and irrotational fluid together with the linearized Bernoulli equation for ice can reproduce the dependence of the breaking length on the interaction velocity as observed in experiments. To this end a 2D model is composed in which the ice, modelled as a semi-infinite Euler-Bernoulli beam, is pushed into a rigid, immovable, downward-sloping structure with a fixed horizontal velocity. The beam rests on the finite depth, infinitely wide fluid layer. To assert whether this linear description of the fluid suffices, the velocity dependence of the breaking length predicted by this model is compared with the experimentally validated 2D model of (Valanto 1992). The comparison shows that linear hydrodynamics gives significant error in certain velocity ranges. Recommendations are given as to a fluid model that would result in an improved prediction of the velocity dependence of the breaking length.@en