Two-dimensional Modeling Approach Research Articles

Promoting precision surface irrigation through hydrodynamic modelling and microtopographic survey

Precision irrigation aims to deliver water and nutrients to crops at exactly the right time, in the right place and in the right amount. While surface irrigation is often perceived as less precise, accurate water distribution, wise use of resources and high efficiency can still be achieved with careful land preparation, astute irrigation management and rigorous performance monitoring. In this study, we advocate the innovative concept of Precision Surface Irrigation, centered around three key design and operational principles that are: well-organized field geometry (and microtopography), precise control of hydraulic-hydrological variables and, finally, regular performance evaluation. These pillars are then integrated into a simulation environment capable of capturing the intricacies of surface irrigation dynamics. Conducted in northern Italy's Padana plain, the study contrasts one-dimensional (WinSRFR) and two-dimensional (IrriSurf2D) irrigation dynamic modeling. Data collection includes boundary geometries, inflow rates, intervention durations, and microtopography, facilitating spatial performance assessment from both the models. The results show that the versatility of the two-dimensional modelling approach was able to reproduce well the observed water depths and the phases of water advance and depletion both in time and space within the studied border irrigation strips, even in complex situations where the strips were hydraulically connected. The RMSE between observed and simulated maximum water depth and waterfront advance time was less than 2.1 cm and 1.9 min, respectively. The two-dimensional approach was also able to detect the cross variability of irrigation dynamics, and to provide a spatial assessment of irrigation performance at high resolution. In conclusion, while the one-dimensional hydrodynamic approach to describing the hydraulic behavior of surface irrigation and field-scale irrigation performance remains valid, the two-dimensional approach provides, in our case study and reasonably elsewhere, a valid simulation environment for spatially characterizing irrigation dynamics in the context of Precision Surface Irrigation.

Quantifying damage zones width in strike-slip faults: Insights from a two-dimensional finite-element modeling approach

We use a two-dimensional finite-element modeling approach to investigate the structural evolution of strike-slip faults under five different confining pressures and determine how their damage zone widths vary depending on the kinematic. Three representative fault segments are modeled, accounting for movement obliquity forming in pure strike-slip, oblique convergence and oblique divergence scenarios. The elastoplastic constitutive model (Drucker-Prager) was employed to couple an associated flow rule into the initial stress state of a geostatic stage. We applied two empirical methods for determining the width of the damage zone from progressive model response. The first method involves using the plastic strain distribution curve's inflection points, to mathematically fixate the width. The second method uses the standard deviation obtained from a trend Gaussian/Normal probability distribution, with over 98% fit to the plastic strain curve data, simplifying the inference of the strain propagation along the model. The simulation results indicate inverse proportional relationships between confining pressure and both porosity and plastic strain intensity, while showing a direct proportional relationship with the damage zone width. For fault zone width quantification, results show convergence between the methods, revealing widths of 2.46–2.65 m for oblique divergence (displacement of 0.20 m), 2.65–3.55 m for pure strike-slip (displacement of 0.20 m), and 4.36–4.46 m for oblique convergence zones (displacement of 0.70 m). The pure strike-slip and the oblique convergence damage zone width results align well with outcrop observations of faults with similar kinematics from the literature, underscoring the significance of numerical modeling as a valuable tool for measure and study the mechanics of characterizing the nucleation of fault zones and quantifying the width of the damage zone.

Joint flexural-density modeling of the Taltal, Copiapó, and Iquique hotspot ridges and the surrounding oceanic plate, offshore Chile

Abstract Based on gravity and bathymetric data and using a novel two-dimensional joint flexural-density modeling approach, this work studies the physical properties of the oceanic Nazca plate around the Taltal, Copiapó, and Iquique hotspot ridges offshore northern Chile. The area is located westward of the Chilean Trench where the Taltal and Copiapó Ridges collide with the continental margin. The results show that the variability in density structure at different scales is a key factor in explaining the observed gravity signal, playing an important role in the lithospheric flexure and hence the elastic properties of the Nazca plate in this setting. The results can be interpreted as evidence of spatial and temporal heterogeneities in the plate-weakening process at the hotspots, magmatic underplating, and crustal and upper mantle fracturing and/or hydration. These processes might be relevant for the ascent of magma pathways of later (secondary) volcanism and influence the mechanical segmentation of the oceanic plate. The latter is critical in explaining the active seismogenic contact between the oceanic Nazca and overriding South America plates.

Comparative 3D inversion of magnetotelluric phase tensors and impedances reveals electrically anisotropic base of Gawler Craton, South Australia

Isotropic three-dimensional (3-D) inversion has become a standard tool in the interpretation of magnetotelluric (MT) data. 3-D anisotropic inversion codes are under development, yet the number of unknowns increases by a factor of 6 rendering the problem extremely ill-posed. The presence of anisotropy is usually inferred from (i) spurious sequences of conductive and resistive bodies or (ii) comparison with two-dimensional anisotropic modelling approaches. Here, we investigate the 3-D structure of the Gawler Craton down to ∼250 km depth using 282 sites of the AusLAMP array located in the southern half of South Australia. Inversions of the MT impedance as phase tensors and real and imaginary parts result in diverging structures at depths > 70 km. We demonstrate that a unifying model that explains all data types similarly well is suggestive of an anisotropic resistivity structure at the base of the Gawler Craton lithosphere at depths of 120–210 km. Depth location and orientation of the anisotropy agree well with results from the analysis of seismic receiver functions. We suggest that electric anisotropy in the Gawler Craton is a result of lattice-preferred orientation of olivine crystals and metasomatic processes with macroscopic preferential orientation. Our results illustrate that inversion of phase tensor data is superior for the direct imaging of anisotropic resistivity contrasts in otherwise isotropic resistivity models; inversion models obtained with impedances may miss such structures. “Comparable” overall RMS misfits are often meaningless when comparing inversion results for various data types since sensitivities differ between data types. Reliable inversion results consistent with the entire data set can only be recovered if data fits are assessed systematically for all data representations. We also discuss the influence of error settings for phase tensors on the inversion. Our study also revealed that, if persistent across large areas, (i) parallel orientation of phase tensor major axes, (ii) constantly high phase tensor maximum phases or (iii) diverging directions of phase tensor major axes and induction arrows are suggestive of anisotropic structures and corresponding hypotheses should be evaluated.

Two dimensional Hydrodynamic Simulation in Labuan Bajo Waters, Indonesia.

Based on its strategic location, Labuan Bajo is a golden triangle between Komodo Island and Toraja Land as a national tourism sector. This makes information about the condition of Labuan Bajo waters very important. One of the oceanographic parameters that have an important role in determining the condition of a body of water is ocean currents. Ocean currents are the movement or displacement of water masses caused by waves, wind, water density, and pressure. In their movement, currents have a direction and speed that can form ocean current patterns. One method that can see patterns of sea surface currents is two-dimensional hydrodynamic modeling. This study aims to identify patterns of surface currents and tides that occur in Labuan Bajo waters. The method used in this study is quantitative descriptive with a two-dimensional hydrodynamic modeling approach along with nested-model techniques to provide the model results with a smaller resolution to get the phenomenon that occurs in those waters. The data used is bathymetry, wind, tide, and High-Frequency Radar (HF Radar) data as current validation. The results of surface current simulation in Labuan Bajo waters have the highest current of >1.12 m/s in the Western season and the lowest speed of 0.88 m/s in the Transition I season. The wind pressure gradient in the Western season affects the current speed in that season.

Validation and verification of three-dimensional finite element target scattering models and Green's functions

A configurable three-dimensional (3D) finite element framework is used for modeling acoustic scattering from various elastic targets on seabed and for the numerical determination of Green's functions used in far-field target scattering prediction. The 3D selection is chosen for possible inclusion of complex seafloor geometries, such as rough, rippled, cluttered, or variably layered seabeds, in which two-dimensional modeling approaches cannot capture the full acoustic interaction with the target and the seabed. Model results are verified and validated against analytic scattering theories and measured results from the 2013 Target and Reverberation Experiment (TREX) and the 2017 CLUTTEREX experiment.

Evaluation of the potential stranding risk for aquatic organisms according to long-term morphological changes and grain size in alpine rivers impacted by hydropeaking

Hydropeaking is one of the major hydropower-related disturbances of natural processes in river systems. The artificial flow fluctuations that are caused by the on-demand production of electricity are known for their severe impacts on aquatic ecosystems. These particularly affect those species and life stages that are not able to adjust their habitat selection to rapid up- and downramping rates. To date, the stranding risk has both experimentally and numerically mainly been investigated with variable hydropeaking graphs over stable river bathymetries. There is a lack of knowledge on how single, discrete peaking events vary concerning their impact on the stranding risk when the river morphology changes in the long-term perspective. The present study precisely addresses this knowledge gap by investigating morphological changes on the reach scale over a period of 20 years and the related variability of the lateral ramping velocity as a proxy for stranding risk. Two alpine gravel bed rivers impacted by hydropeaking over decades were tested by applying a one-dimensional and two-dimensional unsteady modelling approach. Both the Bregenzerach River and the Inn River exhibit alternating gravel bars on the reach scale. The results of the morphological development, however, showed different developments in the period 1995–2015. The Bregenzerach River displayed continuous aggradation (uplift of river bed) over the various selected submonitoring periods. In contrast, the Inn River showed continuous incision (erosion of river bed). The stranding risk exhibited high variability on a single cross-sectional basis. However, on the reach scale, no significant changes in stranding risk were calculated for either river reach. In addition, the impacts of river incision on the substrate composition were investigated. Here, in line with preceding studies, the results show that the coarsening of substrate increases the stranding risk and that especially the d90 (90 % finer of the grain size distribution) must be considered. The present study reveals that the quantified stranding risk of aquatic organisms is a function of the general morphological (bar) characteristics of the impacted river and both the morphological and grain size development have an impact on the potential stranding risk of aquatic organisms and should be considered in the revision of licences in the management of multi-stressed river systems.

Code Benchmark of the HTTF Pressurized Conduction Cooldown Test Using SAM

The High Temperature Test Facility (HTTF) at Oregon State University is an integral system test facility to simulate postulated reactor transients of prismatic high-temperature gas-cooled reactors(HTGRs). A series of test campaigns was launched, providing abundant test data that could be used to benchmark reactor system analysis codes like the System Analysis Module (SAM). In this study, a SAM model of the facility is developed based on the two-dimensional (2D) ring model approach. All components including the ceramic matrix, graphite heaters, helium coolant channels, core barrel, upcomer, pressure vessel, and reactor cavity cooling system are modeled as concentric cylindrical rings. The model is used to simulate one of the benchmark problems—Pressurized Conduction Cooldown (PCC)—within the scope of the Organisation for Economic Co-operation and Development Nuclear Energy Agency International HTTF Benchmark. The simulations consist of two parts. In the first part, operating and boundary conditions as well as thermophysical properties of materials are specified for the benchmark problem. Results from the first part will be used in code-to-code comparison. In the second part, the SAM model is used to simulate Test PG-27, which is the first PCC test carried out in the HTTF, with only two of the ten heater banks activated. The results in the second part are used for code-to-data comparison. Because the helium coolant flow rate is not measured in this facility, it is estimated using the input power and inlet/outlet coolant temperatures. Additionally, radial heat flow in the ceramic blocks is complicated by hundreds of cylindrical coolant channels and heater rods embedded in them. As such, it is necessary to deduce an effective thermal conductivity for the ceramic to analyze the core thermal behavior. SAM predictions of the helium coolant and ceramic temperatures are compared with test data measured in three equivalent sectors. Overall, the SAM results agree reasonably well with test data within the variation of data among the three sectors, which demonstrates SAM’s capability in capturing transient effects in HTGR using the simplified 2D ring model.

Does the model reflect the system? When two-dimensional biomechanics is not 'good enough'.

Models are mathematical representations of systems, processes or phenomena. In biomechanics, finite-element modelling (FEM) can be a powerful tool, allowing biologists to test form-function relationships in silico, replacing or extending results of in vivo experimentation. Although modelling simplifications and assumptions are necessary, as a minimum modelling requirement the results of the simplified model must reflect the biomechanics of the modelled system. In cases where the three-dimensional mechanics of a structure are important determinants of its performance, simplified two-dimensional modelling approaches are likely to produce inaccurate results. The vertebrate mandible is one among many three-dimensional anatomical structures routinely modelled using two-dimensional FE analysis. We thus compare the stress regimes of our published three-dimensional model of the chimpanzee mandible with a published two-dimensional model of the chimpanzee mandible and identify several fundamental differences. We then present a series of two-dimensional and three-dimensional FE modelling experiments that demonstrate how three key modelling parameters, (i) dimensionality, (ii) symmetric geometry, and (iii) constraints, affect deformation and strain regimes of the models. Our results confirm that, in the case of the primate mandible (at least), two-dimensional FEM fails to meet this minimum modelling requirement and should not be used to draw functional, ecological or evolutionary conclusions.

Two-Dimensional Modelling Approach for Electrokinetic Water Transport in Unsaturated Kaolinite

Although electrokinetic treatments for soil remediation and ground improvement have gradually undergone significant experimental development, one of the most important physical phenomena has received very little attention in recent years. Electroosmotic flow, especially in unsaturated conditions, has not been simulated in domains of more than one dimension. In the present work, a modification of the M4EKR code was used to study water movement in partially saturated soils under electrokinetic treatments. Two different configurations were studied: electrodes placed in electrolyte wells and electrodes directly inserted into the soil, and the treatment was started under unsaturated conditions for both. Due to the unrestricted availability of fluid in the first case, soil saturation is inevitable, but in the second case, only a spatial redistribution of water content can be observed. A detailed analysis of a variety of configurations involving several electrolyte wells showed that the number of electrodes, the distance between them and, above all, the ratio between the two magnitudes determines the efficiency of soil saturation and the energy consumed in the process.

An approach for 2D modelling of laterally loaded piles

Despite a considerable progress in the analysis and design of monopiles, many methods are based on complex mathematical structures with doubtful or hard assumptions to verify. Therefore, there is still a need for simple and yet accurate methods for the analysis of monopiles under drained and undrained lateral cyclic loading conditions. In this work, a simple yet efficient two-dimensional modelling approach for the analysis of monopiles is proposed. To account for out-of-plane frictional forces, counter-forces derived from virtual frictional forces generated at the out-of-plane pile interface are applied along the pile length together with the scaled pile stiffness. The predictive capabilities of the proposed approach were validated by back-calculating two different experimental sets. The first consists of a small-scale field monopile test on a coarse-grained soil subjected to lateral cyclic loading under drained conditions. The second is a centrifuge test involving a fine-grained soil subjected to lateral cyclic loading under practically undrained conditions. Simulation results with the proposed approach suggest an accurate prediction of pile displacements and bending moments under both drained and undrained lateral cyclic conditions. The method is, however, unable to reproduce pore water pressures generated behind the pile in low permeability materials.

Tunnel-soil-structure interaction mechanisms in a metallic arch bridge

Tunnel excavation causes ground movements that can affect the serviceability of surface structures, including bridges. These often feature compliant joints between structural components, such as bearings and expansion joints, which reduce stresses under operational loads. However, joints affect the stiffness of the structure and may impact soil-structure interaction mechanisms in unexpected ways. To develop an improved understanding of these mechanisms, a two-dimensional parametric modelling approach is formulated in this study, which enables rapid generation of model geometry to investigate the influence of tunnelling on arch bridges. Elastic Euler beam elements are used to model the arch, deck and spandrel elements while concentrated zero-length springs are used to idealise bridge joints. The soil is modelled using non-linear Winkler springs, distributed at the embedded sections and base of the piers. Tunnelling-induced greenfield displacements are applied to the soil elements to simulate soil-structure interaction. First, an application of the numerical model to simulate the response of a steel arch railway bridge affected by Thames Tideway tunnels is presented. This case study indicates good agreement between the proposed models and field monitoring data and highlights the key role of bridge joints on tunnel-soil-structure interaction mechanisms. To explore these aspects further, numerical simulations are carried out for superstructure arrangements with and without bridge joints, various soil conditions and tunnel eccentricities. The influence of these aspects on bridge structural integrity is quantified using the force and strain distribution in the structural elements, and is summarised with the utilisation ratio. Due consideration is also given to bridge serviceability by investigating salient displacement measures, such as roller bearing travel and deck elevation changes.

Guidance in the calibration of two-dimensional models of historical floods in urban areas: A case study

The statistical characterization of extreme floods largely benefits from the reconstruction of historical floods, making use of various documentary sources and of hydrodynamic models. The two-dimensional (2D) modeling approach provides a higher accuracy and detail with respect to one-dimensional (1D) modelling. On the other hand, 2D models require a large amount of topographic and hydraulic information for calibration, having several parameters to be determined, which makes the calibration prone to underdetermination. To address this issue, we provide here practical guidance for the effective calibration of 2D models for the reconstruction of historical floods in urban areas with a limited amount of information available. Specifically, we show how to constrain model parameters by distinguishing among “static”, “dynamic” and “global” data and their role in the calibration procedure. The latter is exemplified by a relevant application to the reconstruction of the great 1870 flood of Rome, Italy.

Distribution of DO (Dissolved Oxygen) and BOD (Biological Oxygen Demand) in the Waters of Karimunjawa National Park using Two-Dimensional Model Approach

Massive mangrove conversion into intensive pond farming has become environmental problem in Kemujan Island and Karimunjawa Island which affect the water quality. This research aimed to examine the dispersion of DO and BOD related to the current pattern in the seas west of Kemujan Island and Karimunjawa Island by using two-dimensional modelling simulations. Quantitative and descriptive methods were used to provide interpretation and analysis of the modelling simulation results. Modelling simulations were conducted in September 2019. The validation results show that the model and the field measurements has a very good. The results show that the current characteristics are dominated by tidal current, which moves westward with an average magnitude of 0.078 m/s. Furthermore, divergence, convergence and turbulence are also identified. Based on the simulation results, the prediction of DO and BOD concentrations fluctuate in Lagoon Mrican. During spring tide, the DO concentration changes from 7.95 - 8.1 mg/L into 8.55 - >9.45 mg/L and during neap tide, it changes from 8.55 - 8.7 mg/L into 9.15 - >9.45 mg/L. On the other hand, the BOD concentration increases from 0 - 0.08 mg/L to 0.88 - 0.96 mg/L during spring tide and neap tide.

Machine Learning Driven Prediction of Residual Stresses for the Shot Peening Process Using a Finite Element Based Grey-Box Model Approach

The shot peening process is a common procedure to enhance fatigue strength on load-bearing components in the metal processing environment. The determination of optimal process parameters is often carried out by costly practical experiments. An efficient method to predict the resulting residual stress profile using different parameters is finite element analysis. However, it is not possible to include all influencing factors of the materials’ physical behavior and the process conditions in a reasonable simulation. Therefore, data-driven models in combination with experimental data tend to generate a significant advantage for the accuracy of the resulting process model. For this reason, this paper describes the development of a grey-box model, using a two-dimensional geometry finite element modeling approach. Based on this model, a Python framework was developed, which is capable of predicting residual stresses for common shot peening scenarios. This white-box-based model serves as an initial state for the machine learning technique introduced in this work. The resulting algorithm is able to add input data from practical residual stress experiments by adapting the initial model, resulting in a steady increase of accuracy. To demonstrate the practical usage, a corresponding Graphical User Interface capable of recommending shot peening parameters based on user-required residual stresses was developed.

A new toolbox to distinguish the sources of spatial memory error.

Studying the sources of errors in memory recall has proven invaluable for understanding the mechanisms of working memory (WM). While one-dimensional memory features (e.g., color, orientation) can be analyzed using existing mixture modeling toolboxes to separate the influence of imprecision, guessing, and misbinding (the tendency to confuse features that belong to different memoranda), such toolboxes are not currently available for two-dimensional spatial WM tasks.Here we present a method to isolate sources of spatial error in tasks where participants have to report the spatial location of an item in memory, using two-dimensional mixture models. The method recovers simulated parameters well and is robust to the influence of response distributions and biases, as well as number of nontargets and trials.To demonstrate the model, we fit data from a complex spatial WM task and show the recovered parameters correspond well with previous spatial WM findings and with recovered parameters on a one-dimensional analogue of this task, suggesting convergent validity for this two-dimensional modeling approach. Because the extra dimension allows greater separation of memoranda and responses, spatial tasks turn out to be much better for separating misbinding from imprecision and guessing than one-dimensional tasks.Code for these models is freely available in the MemToolbox2D package and is integrated to work with the commonly used MATLAB package MemToolbox.

Impact of entry–exit loss on the measurement of flow resistivity of porous materials

This study experimentally investigates the impact of low pore density and thickness on the flow resistivity, hence permeability, of periodic open porous materials. An experimental setup is developed to measure the flow resistivity of porous samples adhering to the guidelines provided by the International Standards Organization 9053 and American Society for Testing and Materials C522 standards. The flow resistivity of three different porous microlattices is considered. Two of the three lattices are obtained from a monospherical void subtraction method, while the third is the octet-truss lattice. The monospherical void geometries are body-centered-cubic and A15 lattices. The pore density, measured in terms of pores-per-inch (PPI), was varied from 1 PPI to 12 PPI for all three microlattices considered. At low PPIs, the pore sizes are on the order of macroscale thickness. Hence, the volume-averaging or homogenized equivalent fluid model may not be valid. The thickness variation for a fixed PPI is also considered over limits where traditional volume-averaging methods break down. It is shown that at extremely low PPIs and thicknesses, entrance and exit losses become dominant, and the flow resistivity measured as per standards ceases to be a constant with PPI or thickness. Furthermore, existing two-dimensional modeling approaches for losses due to flow in pipes are adopted to predict the entry and exit pressure loss coefficients for the considered microlattices with appropriate modifications.

Modeling and simulation of hydrothermal oxidation of cutting oil in supercritical water

AbstractSupercritical water oxidation may be used as waste treatment technique that consists of oxidizing organic and inorganic matter using water at supercritical conditions. It was developed as an alternative technique in order to limit the risks of secondary pollution. The purpose of this study is the development of simulation tools in stationary state in supercritical water oxidation (SCWO) tubular reactor which has been numerically investigated basing on using a two-dimensional modeling approach applying the k-ε turbulence model, and on the international association for the properties of water and steam formulation (IAPWS-IF97). A Multiphysics simulation using Comsol Multiphysics 5.2 software for the supercritical water oxidation of “cutting oil Biocut 35” as a pollutant is reported. First, the heat transfer coefficient was estimated and the obtained temperature and species concentration profiles were compared to experimental data where a quite good agreement was obtained as confirmed by the acceptable coefficient of determination value. Finally, once the used model was validated, numerical simulations were performed to describe the behavior of the reactor investigating the effects of operating conditions such as temperature, pollutant feed concentration and reactor inlet flow rate, on dependent variables like outlet temperature, chemical oxygen demand removal (COD) and the highest temperature reached inside the reactor. This last parameter was considered in the study to take into account the operation safety. The significance of the studied factors and their interactions were quantified and analyzed by means of the full factorial design of experiment (DOE). The results showed that the effect of initial temperature was the most important for the three responses, followed by the feed concentration then to a lesser extent the flow rate. The effect of initial temperature was positive for outlet temperature, maximal temperature and cutting oil removal. The interaction temperature-feed concentration was the only significant one for the COD removal and the maximal temperature. The results defined an operation safety zone of the SCWO reactor based on the superposition of the three studied dependent variables and imposed constraints on the inside reactor temperature and the pollutant outlet concentration.

Capturing the hardness of coating systems across the scales

A two-dimensional multi-scale modelling approach that concurrently couples discrete dislocation plasticity and crystal plasticity finite element has been applied to study the hardness variation of coating systems across different scales, covering nano- to micro-indentation. The difference in indentation size sensitivity between film and substrate gives rise to three regimes of hardness, one typically dictated by the intrinsic coating indentation size effect, which is regulated by dislocations activity, and the other two linked to the continuum response of the coating and the substrate. We propose a new hardness formula that incorporates physics-based indentation size effects of thin films into established continuum hardness transition formulae. This formula is shown to substantially improve the hardness prediction of coating systems, particularly when relative indentation depth is at the nanometre scale.

Effects of soil arching on behavior of pile-supported railway embankment: 2D FEM approach

Abstract The construction of railway embankment on soft soil is a challenging task for practitioners. In the past, several approaches have been employed in practice to overcome this problem. Pile foundation is considered one of the reliable solutions for soft ground. In a pile-supported railway embankment, soil arching plays a vital role in the efficient load transfer to the piles. In this study, the soil arching phenomenon in a granular embankment subjected to train induced loading is demonstrated based on the two-dimensional (2D) finite element modeling approach. A 2D plane strain idealization is used to convert a real three-dimensional (3D) case into 2D. The effects of the piled railway embankment properties on soil arching are investigated. It is found that pile modulus, embankment modulus and friction angle affect the arching mechanism significantly. Pile spacing (s) variation affects soil arching (i.e., the arching zone can be increased by reducing s). In addition, the height and shape of arching are studied. A comparison with the available analytical design methods for embankments is presented which indicates inconsistencies in the previously published results.