The growing need to understand the behaviour of un-reinforced masonry URM), subjected to repeated light man-made earthquakes caused by the extraction of gas in the north-eastern part of The Netherlands has resulted in intense research to determine the exact process of crack initiation and propagation. The historical masonry buildings and Dutch terraced houses in Groningen are prone to light damages which become severe upon repeated lateral earthquake loading. Although there are material models that describe the behavior of modern brick masonry, they do not accurately represent the mechanical properties of 19th century clay brick masonry. This led to a large-scale research into the mechanical behavior of un-reinforced masonry and an orthotropic continuum macro-model called the Engineering Masonry Model (EMM) was proposed. The existing tension constitutive model in EMM assumes a secant unloading-reloading branch which does not consider the strength degradation of URM under repeated loading. Since tension mode-I fracture results in cracking of URM, it is important to study the effects of repeated loading on the propagation of the crack and its effects on the capacity of the structure. This thesis presents a degradation model to represent the strength deterioration of URM observed during repeated loading. The constitutive model formulated in this thesis is based on hyperbolic functions along with a secant slope for the unloading-reloading branch. To justify the model assumptions, a single linear 4-node element is analysed with the new model and the effect of varying different components of the constitutive equations is established. The window bank spandrel sample modeled as a 4-point bending test is analysed using the new model for 10, 30 and 100 repetitions. It is shown that the hyperbolic model can predict accurately the stress reduction within each repetition displacement set and also represent the crack width widening and crack propagation accurately when compared to the experimental results. The new model is tested on a wall with a window opening sample and the results closely matched that of the experiment. Finally, recommendations are provided for further development of the hyperbolic model and calibration of the material properties.

Recycled aggregate concrete is emphasized more and more nowadays due to its importance towards the construction industry and general welfare of our planet. Being such a significant feature with regard to universal sustainable development, every aspect of its properties should be evaluated, examined and optimized so that there is a well-known, well-described and ready to use product. This project aimed at correlating some of the established parameters of conventional concrete such as water absorption of the coarse aggregate fraction and the resulting compressing strength, only applied in the field of recycled aggregate concrete. Along with these two properties, a prediction model was sought for which would be able to forecast the mechanical strength of concrete based on its coarse aggregate composition. A series of experiments were performed on samples fabricated for water absorption tests and compressive strength assessment. In total, 112 water absorption samples and over 250 concrete specimens were prepared and examined for the respective attribute. These samples used natural aggregate as the base of the coarse aggregate portion of the mix design, alongside a series of so-called contaminants which replaced the gravel in certain concentrations. These contaminants included bricks, ceramic tiles, glass, wood, gypsum, plastics, mineral fibers and recycled sand concrete and were entirely based upon the C&DW composition globally. Further investigation was done on the inclusion of air within concrete so that contaminated concrete results could be equated to this constant. All water absorption tests were performed according to the current standards and concrete specimens were crushed after 7 and 28 days of curing in order to obtain their strength in compression, while the predictive model was initiated via the Minitab statistical software. The results indicated some interesting and promising trends. From both sets of experiments, it was evident that the water absorption of the coarse aggregate fraction was not the main contributor towards the strength development of recycled concrete. It had a minor influence, especially compared to the type of contaminant present and how much volume it took. Furthermore, samples with identical water absorption fabricated concretes with different strengths. Overall, plastics and wood had the most negative effect in terms of compressive strength, while bricks, tiles and glass seemed to affect this aspect in neutral or even slightly positive manner. EPS foam in very limited amounts yielded a notable 30 to 35% strength drop, while on the other hand, brick replacing 20% of the NA improved the strength by approximately 7% after 7 days and 2% after 28 days. In the end, based on all experimental input, a predictive model was developed, optimized and validated in several steps so that it was able to predict the water absorption of coarse aggregate, compressive strength after 7 and 28 days and equivalent air content based on the composition of the coarse aggregates. The back-end of the model is provided within the report as a MATLAB code.

The construction of the new airport of Mexico City (NAICM) is challenging, as the subsoil at the site is very weak. The airport, which is under construction at the moment, is located on top of thick layers of Mexico Clay soils. These clayey layers consist of a mixture of both clay particles and pyroclastic volcanic materials. These unique soils are characterised by a high water content and compressibility. Upon loading, the soft soils will settle as a result of the consolidation process. This consolidation is defined as the dissipation of excess pore water as a result of an increase in stress. The rate of consolidation is governed by the hydraulic conductivity of the soil, which is in general very low in soft soils, and the drainage path. In order to speed up this time consuming process, vertical permeable elements can be installed in the soil. Upon installation of these elements, a reduction in drainage path is realised. This reduction in drainage path allows faster dissipation of excess pore water pressure, hence reducing the consolidation time. Due to the conditions at the site, the design of the runways is challenging. The differential- and residual settlements need to be within strict boundaries in order to keep the runways operational. Prior to the design of the runways, trial embankments have been constructed in order to measure the performance of different soil improvement techniques at this specific site. Based upon this field test results, prefabricated vertical drains (PVDs) were selected as the most efficient soil improvement technique. Besides the PVD trial, the sandpile trial also showed promising results. The PVDs and sandpiles both decrease the drainage path length inside the soft soil layers. The main difference is the relative high stiffness of the sandpiles, the stiffness of the PVDs is negligible. By installing stiff elements in the soft soil, the loads applied on the top of the soils are transferred to deeper and stiffer soils. Hence, the total settlements of the top soft soil layers are reduced. The sandpiles could therefore be feasible alternative to PVDs, applied at locations in which the total settlement needs to be limited. The investigation into the behaviour of sandpiles in Mexico Clay soils is performed in multiple stages. First, analytical models are used for gaining knowledge about the expected sandpile behaviour. In the second stage the FEM software Plaxis 2D is used for the numerical modelling of the sandpiles, using an axisymmetric model setup. The model is constructed in multiple steps, with increasing complexity. The first models simulate the material behaviour with the Mohr-Coulomb model, in the advanced models the Soft Soil Creep and Hardening Soil models are used. The final numerical model is verified by the measurement data obtained from the field trials. After fitting the numerical model to the field data, the verified model is used in the sandpile sensitivity analysis. In this analysis the sensitivity of the material properties and geometry of the sandpile on the performance are researched. The residual settlements or performance, which is defined as the difference in settlement after construction and over a period of 8 years, is affected by the length, radius and centre to centre (ctc) distance of the sandpile. Adjusting the pile stiffness has minimal effect on the performance of the sandpile. An optimum in performance is found by varying both the pile radius and ctc distance. A combination of a small pile diameter with a small ctc distance results in the best performance. When comparing the performance of both sandpile and PVD, the PVD is found to be more effective in terms of performance. The additional stiffness of the sandpile is not reducing the total settlements, on the contrary, the self-weight of the piles increases the total settlements by providing an additional load to the soft soil layers located underneath the pile tip. In conclusion the sandpiles have no additional benefit over the use of PVDs, therefore the application of sandpiles at the NAICM site is not a feasible alternative to the use of PVDs.

As the construction of sub-aerial and submarine geotechnical structures increase in amount, rate and size, so do their associated risks. Often, with use constitutive models, finite element analyses (FEAs) are performed in order to identify and mitigate these risks. NorSand, which is a consti- tutive model based on critical state soil mechanics for particulate materials (e.g., sand), is one of the first models to integrate the state parameter ψ into its constitutive framework to model dense and loose sands material with the same parameter set. Importantly, it is able to identify the liquefaction potential as it can simulate softening behaviour due to pore pressure increase of loose soils in undrained conditions. Since NorSand has recently been implement into PLAXIS, a geotechnical analysis software capable of performing FEAs, it must be verified, validated and applied with the software, which is done in this report. First, stress-path and parametric analyses were conducted at single stress points. The stress- path analyses show how the state variables evolve for different triaxial conditions. Systematically changing the input parameters to extremes found in literature helped determine their influence on the evolution of stresses and strains. The resulting figures can be used to help future calibrations to experimental lab test data. The PLAXIS implemented NorSand (PLAXIS NorSand) was verified by comparing it with an implementation written in Visual Basic for Applications (VBA NorSand) by the authors of the model Jefferies and Been. Verification in this context means determining if PLAXIS NorSand is able to produce outputs as intended by the authors. Various testing conditions, both triaxial and direct simple shear, showed overlap and agreement between the outputs of both implementations, verifying PLAXIS NorSand. Then, the model was compared to, albeit not in a traditional sense, an ’analytical solution’, which is the relationship between the mobilized friction ratio Mi and state parameter ψ in its simplest form. The mobilized friction ratio and stress ratio at peak strength of PLAXIS NorSand and the ’analytical solution’ were compared. The values between both showed less than 3% difference, further verifying PLAXIS NorSand. The constitutive model was then validated - i.e., established that PLAXIS NorSand is able to approximate soil behaviour as intended. First, by using the soil parameter set that had been derived from lab tests of Erksak sand as a baseline, the input parameters were varied until PLAXIS NorSand was calibrated to individual triaxial tests as best as possible. Then, triaxial tests of Erksak, Nerlerk and Ticino sand were approximated with PLAXIS NorSand without changing the soil parameters determined from lab test data. PLAXIS NorSand is able to follow lab test data decently well with one parameter set. And if one decides to take the time and calibrate individual lab tests, and deviate from soil parameters determined from a set of lab tests, they can be matched even better. Additionally, it showed a consistent need for activation of the softening flag (S = 1) in order to appropriately model loose soils in undrained conditions. Furthermore, NorSand exhibits indefinite hardening in dense soils during undrained loading, which can be avoided by employing a ’cavitation cut-off’. The last part of this report tested PLAXIS NorSand by applying it in FEAs of a simplified submerged landslide, which was subjected to 20 centimeters of displacement at the crest through a rigid slab in undrained conditions. First, the difference in slope behaviour due to change in soil density within NorSand was determined: dense soil resulted in the slope to be able to bear the full 20 centimeter displacement, whereas increasing the void ratio (i.e., increasing the positive value for the state parameter) gave the effect of even quicker slope collapse and a lower bearing capacity. In other words, when using NorSand, the looser soil the further the failure surface moves up and the quicker the structure fails to maintain equilibrium. Lastly, NorSand was compared to Modified Cam-Clay and Mohr-Coulomb to highlight the differences in their ability to model static liquefaction, while being triggered by unrealistic loading conditions. Even though none of the FEAs showed actual liquefaction, since it is accompanied with the fluidization and loss of structure, they still gave in indication of the liquefaction potential. NorSand, contrary to the other constitutive models, showed the expected high sensitivity to forced displacement resulting in clear shear bands resembling Prandtl-type failure mechanism and early onset soil body collapse.

In concrete structures, shear failure is one of the failure mechanisms that should be estimated carefully due to its brittle nature. As the technology advances, the capacity of shear in concrete structures can be calculated with the analytical formulation in codes and standards and simulated with NLFEA. The shear capacity is affected by the size which is a phenomenon observed from experiments where the change in the structure size does not have a linear relation to the structural strength. This can be referred to as the experimental size effect. When estimating the shear capacity with NLFEA, this effect must be included in order to avoid over-estimation of capacity. However, another size effect can be found from the result of NLFEA. The change in structural size has an influence on the global response of concrete structures with different numerical configurations in a simulation with NLFEA. A numerical configuration consists of several numerical parameters. This effect is referred to as the numerical size effect. Therefore, it is necessary to incorporate the experimental size effect and reduce the influence of the numerical size effect in a numerical model in order to increase the accuracy and precision of the simulation with NLFEA. The aim of this study is to investigate how the numerical size effect influences the NLFEA results, which will help understand how to improve the accuracy of the simulation. The investigation is done by studying numerical models with a variation on several numerical parameters, such as load increment, error tolerance, the maximum number of iterations and mesh size. The study is focused on identifying the influence of these numerical parameters for cases with different structural sizes and comparable shear slenderness. This is done by observing the global behaviour of the structure and comparing NLFEA results to their respective experimental results. Any difference in the simulated global response is analysed by observing the difference between the experimental and the simulated crack pattern and the convergence state of the respective step. The study is done in three stages. The study is initiated by investigating how the flexural shear failure mode can be obtained with NLFEA. Two specimens with comparable shear slenderness (a/d), 3.70 and 3.91 for beam A123A1 and H123A respectively, are used as a reference for the simulation model. It was found that bad convergence was achieved at the later stage of the analysis and by not accepting non-converged steps, the results became dependent on the convergence state of the analysis where different peak loads were achieved in different numerical configurations for the same case study. The study continues by identifying the correlation of numerical parameters with the results of NLFEA by creating simulation models with different iterative incremental solutions, load increment, error tolerance, and the maximum number of iterations. All load steps are accepted in this stage, regardless of their convergence state, and the rotating crack model is used in these numerical models. The NLFEA results are post-processed by setting up an upper limit in the error tolerance in order to minimize the variation of the peak load and attain a higher consistency of the results. At the last stage, another study is performed to identify the correlation between mesh sizes with the results of NLFEA by simulating several models with different mesh sizes. All load steps are also accepted in this stage, regardless of their convergence state, and the rotating crack model is used in these numerical models The NLFEA results are also post-processed with the same criteria attained in the previous stage. It is concluded that the results of NLFEA are dependent on the following numerical parameters, the load increment, error tolerance, and the maximum number of iterations, due to different initiations and propagation of dowel crack in the non-converged steps, which influences the convergence condition in the later stage of the NLFEA. The formation and rapid propagation of the dowel crack results in the excessive change in the principal strain direction and magnitude, which induces high relative energy variation and out-of-balance force. These numerical parameters indirectly contribute to the error found in each step in the analysis. An additional study is done by replacing the crack model with a fixed crack model with a damage-based shear retention model, but it was found that this did not solve the problem due to the excessive change in the shear retention factor on the elements that lead to premature failure of the beam. The observation related to the numerical size effect reveals that the mesh size controls the propagation rate of the dowel crack. The use of a smaller mesh size will reduce the propagation rate of the dowel crack after its initiation due to a smaller increment of the crack width between each load step. As a result, the increment of the crack width becomes smaller as the mesh size decrease. This becomes more pronounced due to the effect of excessive change of the principal stress-strain direction and magnitude. Full Newton-Raphson method has been proven to give the smallest variation of the peak load ratio (Pnorm), ranging from 0.887-1.140 for specimen A123A1 and 1.055-1.246 for specimen H123A, compared to the other iterative-incremental method with the use of load increment ratio (load increment/experimental peak load) within 0.03-0.006 in combination with evaluation of acceptable analysis step with error tolerance, where the predefined error tolerance must be set to Gerr = 0.0001 and Ferr = 0.01, and non-converged steps can still be accepted if the relative energy variation and out-of-balance force of the corresponding step are below 0.03 and 0.58 respectively. An additional criterion related to the shear displacement on the major flexural shear crack (Δ) should be added to the evaluation of the acceptable analysis step in order to prevent an excessive crack opening on the major flexural shear crack at the peak load. The analysis step can still be considered valid if the respective step satisfies the criterion of the error tolerance and the vertical crack opening on the major flexural crack on the respective step is below the ultimate tensile crack width (CWt,u), which can be calculated from the Guidelines for Nonlinear Finite Element Analysis of Concrete Structures (Hendriks & Roosen, 2020). A mesh size ratio which larger than h/15 is required to have the correct simulated failure.

Asphalt concrete is the most used material in road construction. Analysing the behaviour and properties of this material is vital for modern day society. Currently, this is mostly done in labs using actual asphalt mixtures. However, this is a very expensive and time consuming process. An upcoming alternative approach is the use of computational models. The finite (FEM) and discrete element methods (DEM) have been used in the past, but both of these have shown significant disadvantages regarding the modelling of discrete particle movement and shape, respectively. An upcoming alternative is the use of physics engines, such as Bullet Physics, to model porous media. This method can have substantial benefits in terms of costs and research time. Also, phenomena can be visualised that cannot easily be observed in experiments. This research focusses on the utility of Bullet Physics for modelling hot asphalt compaction. Therefore, a complex contact model is implemented which can describe the contact forces of the bituminous mixture. The superpave gyratory compaction process is digitally modelled. The model has been programmed in PyBullet, an open source physics engine, programmable in Python. A parametric study has been performed, which reveals the significance of certain properties, which cannot easily be investigated during laboratory compaction. Bullet Physics was not initially designed for scientific research in the field of structural engineering. Therefore, alterations are needed to make the software usable. The implementation of a complex contact model is challenging. Although the Burgers’ contact forces can be correctly described, it proves difficult to implement custom contact forces directly in PyBullet. Two attempts have been made. In the case of a custom integration scheme, the computation time proved too long to be applicable for large scale simulations. In case of a direct implementation in Bullet Physics with the application of external forces, instability occurs. The only correct way to implement a custom contact model is by altering the source code. During the simulations with a simpler contact model, substantial improvements of digital simulations over actual experiments are presented. The consistency proves far better than the prescribed minimum. The influence of inertia and friction can be assessed. It turns out that inertia, as well as the friction of the mould, can be disregarded. Another phenomenon that can be clearly illustrated is the revolving of aggregates inside the mould. Further analysis has shown that the average contact area depends on degradation, but in a typical asphalt mixture does not depend on the size of the individual elements, and could rather be treated as a constant. Further analysis shows that Bullet Physics is incredibly efficient, thus yielding the possibility of performing large scale simulations within reasonable time.

With the continuous increase in transportation volume, bridges face the challenge of carrying higher traffic loads. As a result, bridges undergo fatigue due to repeated and increasing loads, leading to progressive deterioration of their structural reliability. While existing codes already account for several variables affecting the concrete's ability to withstand compression fatigue, some critical factors are still not considered. Furthermore, the modelling aspects of Finite Element Analysis (FEA) are often simplified or disregarded in practical applications, leading to an inaccurate estimation of fatigue life for concrete. As a result, a considerable number of bridges may be rejected for fatigue despite the possibility that they are not susceptible to this issue. To address these issues, this thesis presents a parametric study that explores and quantifies the influence of various parameters on the fatigue life of inverted T-girder bridges, focusing on modelling aspects and material degradation. For this reason, a special case study was specifically designed to fail in fatigue within its lifespan. The fatigue analysis process employed both analytical and numerical methods, with the fatigue failure of the beams determined based on the bending failure at the beam's midspan criterion. A basic case was established as a basis for analysis, and the effect of each parameter was evaluated by incorporating it into the basic case and calculating the fatigue life of the bridge. The analysis results demonstrate that stress distribution affecting parameters are significantly important in determining the fatigue performance of a bridge. The analysis identified that accounting for the gradual development of prestress losses, rather than instantaneous losses, and utilising area loads for vehicle modelling are critical aspects that substantially prolong the bridge's fatigue life. Furthermore, performing a historical lane configuration analysis is crucial for accurately assessing fatigue, as it significantly impacts the fatigue life of the bridge and identifies its critical components. The inclusion of material degradation caused by cyclic loading in the fatigue analysis is highly advantageous. It can even result in the bridge no longer being susceptible to fatigue. It is worth noting that accounting for the cracking of the slab in the lateral has a major negative effect on the fatigue life of the bridge. However, it is necessary to include it in the analysis to avoid an inaccurate and overly optimistic fatigue assessment. The study also provides specific equations to calculate the effect of time-dependent traffic volume on fatigue life. Lastly, the research investigated other parameters, such as the composite action of the structural components, which had a minor effect on the fatigue life of the bridge. It is important to note that the influence percentage of most parameters cannot be generalised to all bridges, as they are contingent on specific factors unique to each case. Nevertheless, this research provides insights into the magnitude and contribution of the investigated parameters' effect on the fatigue life of concrete bridges, outlining those that must be necessarily included in the fatigue assessment.

The search for a transparent, reversible, and reusable consolidation system for monuments led to the ambition to test possible cast glass interlocking brick designs using numerical calculations. The focus of this research is hence the design, simulation and evaluation of a possible cast glass interlocking geometry using Finite Element Analysis (FEA). For this purpose, design criteria are formulated from literature; considering glass, polyurethane (used as interlayer) and interlocking systems. As interlocking systems are determined by their boundary conditions, a case study of a monument is chosen to provide additional design criteria. The goal of the case study is to provide a reversible and reusable restoration and consolidation alternative for the current invalid restorations. The design criteria obtained then are combined into an initial geometry, whose parameters are varied to test their sensitivity to its shear capacity, using FEA. Christensen’s failure criterion is used to locate prone areas in the geometry, and to evaluate the theoretical moment of failure. This output value combines the three principal stresses into a failure envelope, hence can generate contour plots to envision peak-stress-prone areas. This is important especially for glass structures, as they are prone to peak tensile stresses. From the results design diagrams are created and applied on a conceptual cast glass interlocking consolidation design for the monument chosen as case study: The Lichtenberg Castle ruin. The initial design is moreover prototyped to check its interlocking capabilities, residual stresses and deviations introduced by shrinkage. Being able to evaluate possible geometries using FEA can decrease costs and time when searching for a new interlocking geometry. Prone areas are easily highlighted using the Christensen’s failure criterion output. Hence peak stress sensitive or invalid geometries can be discarded before reaching the prototyping stage, which is time consuming and costly. The creation of a methodology to predict this behaviour is hence valuable for further research on other cast glass geometries and can moreover be applied in any other field when analysing solid complex geometries. Another goal is to find a cast glass brick design which not only can consolidate the monument of the case study, but is moreover applicable in other projects or configurations. The brick then is not a one-solution design, but can be reused in other projects. The geometry hence is varied using Grasshopper plug-in for Rhinoceros. By exporting the geometry using a STEP-file, a solid can be loaded into DIANA FEA, where it can be analysed using their newly implemented output value of the Christensen’s failure criterion. The geometry of the monument is gained through a 3D laser scan, resulting in a point cloud. The point cloud is adapted using Autodesk Recap, then further processed in Rhinoceros. The Christensen’s failure criterion output is a proper and fast way to evaluate possible cast glass brick designs. Any compressive stresses on the interlocking brick geometry are beneficial for its shear capacity, as is an increase in interlocking amplitude or brick height. Increasing the amplitude however affects the allowable tolerance negatively, which is also the case for a decrease in brick height. Decreasing the brick height hence results in both negative effects. The conceptual design for consolidation of the Lichtenberg Castle tower can replace the current interventions with equal or higher capacity, even for all conservative assumptions and simplifications. The design can still be altered less conservative after more experimental results and simulations come available. The methodology applied can now be further developed and performed on other complex geometry designs. The presented multifunctional cast glass interlocking brick design, and its variations can be further investigated and applied in other projects.

This research project aims to implement an elasto-plastic Neo-Hookean Alpha material with a Desai flow surface with hardening and softening in Abaqus via the Continuum Mechanics framework. This is done by first defining the elasto-plastic potato diagram and deriving the corresponding stress tensor definition and dissipation inequality. The elastic deformation gradient is needed in the definition of the stress tensor, which is found by using the Newton-Raphson procedure for the maximization of the dissipation inequality subjected to the plastic flow surface. The Neo-Hookean Alpha elastic material, the Desai plastic flow surface and the definition of the hardening and softening curve are given and are substituted in the stress tensor definition and the corresponding Newton-Raphson procedure, which finishes the derivation of the theory needed to implement this model. This model is implemented in both Abaqus and Python, where the latter is used for rapid prototyping of the model. The methodology of verifying both implementations is presented. The results of the verification of the implementations are given and show that the behaviour of both implementations is as expected and that both implementations correspond to each other; although there are a few small issues. In the end, these issues are discussed, and future recommendations are given to solve them.

In the Netherlands, there is a large and widely used railway system that includes numerous railway bridges. This master thesis investigates the problems caused by these railway bridges. Since World War II, continu- ous welded rails have been used extensively throughout the Netherlands. These rails offer several advantages on embankments, such as reduced maintenance due to less vibration, increased passenger comfort, and reduced noise. However, continuous welded rails can cause problems when they run continuously over a bridge, as is the case in the Dutch railway system. Bridges are not continuous structures and have joints between the decks or between the deck and the transition structure. These joints allow the bridge deck to expand and contract due to temperature changes, move due to longitudinal forces from train braking and acceleration, and deflect due to the vertical load of the train, causing rotations at the ends of the bridge decks. This relative displacement between the bridge deck and the continuously welded rails creates addi- tional stresses in the rails. These additional stresses must not exceed the maximum allowable stress of the rail material, as this could cause the rails to buckle or fracture. Fixed points are often used in practice to limit the movement of the bridge deck and keep additional rail stresses within acceptable limits. This investiga- tion, however, focuses on understanding the interaction between the rails and the structure without the use of fixed points. The study assumes a railway bridge in the Netherlands with a ballast bed and uses concrete precast bridge decks. To gain initial knowledge of the problem and the various railway components and bridges, a literature review was conducted, and two existing railway bridge projects were analysed. It quickly became evident that there are two different bridge types relevant to the study of additional rail stresses: those without embankment influence and those with embankment influence. The study then identified parameters that could influence the magnitude of additional rail stresses to use them as variables in further investigations. Two longitudinal force models were created in SCIA Engineer: one without embankment influence (Model 1) and one with embankment influence (Model 2). These models are spring models where the stiffness of connections and elements is schematised as springs. The models were validated with hand calculations and used to obtain results for the influence of various parameters such as bridge deck span length, elastomeric bearings, bridge pier length, and foundation stiffness. The results of the two models were compared to understand the influence of the embankment. Additionally, the spring elongations of the non-linear springs between the bridge deck and the rails, representing the ballast bed, were examined to determine if they were in the linear or non-linear part of the spring characteristic to see if the springs slipped enforcing stress redis- tribution. Finally, the models assessed the individual contributions of three load cases (thermal, longitudinal traffic load, and vertical traffic load) to the combined additional rail stress. The results lead to the following conclusions: • When a structure is not influenced by the embankment, the magnitude of additional rail stresses depends on the stiffness of the substructure. The stiffness of the weakest component, in this case, the elastomeric bearings, has a significant influence. • When a structure is influenced by the embankment, the magnitude of additional rail stresses mainly depends on the dominant stiffness of the embankment, with the stiffness of the substructure having little to no influence. • Structures with embankment influence experience lower additional rail stresses due to the additional stiffness provided by the embankment. Problems with exceeding maximum permissible rail stresses occur mainly in relatively long railway bridge structures without embankment influence. • The vertical load has the largest contribution to the combined additional rail stresses for both structure types. • The thermal load has a larger influence on structures with embankment influence because these structures are more constrained by the embankment and thus more vulnerable to thermal deformations. • Linear summation of individual stress contributions from different load cases generally results in higher or equal stresses compared to non-linear combinations, making linear summation a conservative approach. The stress difference between linear summation and non-linear combination is usually only a few megapascals. If the springs between the bridge deck and the rails slip into the non-linear branch, this difference increases slightly but remains small. Based on these conclusions, the following recommendations can be made to prevent additional rail stresses from exceeding maximum permissible stresses without the use of fixed points and to streamline the process of longitudinal force analysis for engineers: • To prevent exceeding the maximum permissible stress in structures without embankment influence, consider using larger elastomeric bearings or less slender bridge decks that are less susceptible to deflection, thereby reducing additional rail stress, especially from vertical loads. • Since linear summation of individual contributions to additional rail stresses results in conservative and faster calculations, it is recommended to use linear calculations in the initial phase. Then, perform a final design review with non-linear calculations to ensure accuracy. This approach will significantly speed up the design process, as models will not need to run for hours or days each time.