Intersection Of Plane Research Articles

Manufacturing a curved profile with fine grains and high strength by differential velocity sideways extrusion

Abstract For structural and aerodynamic reasons, curved profiles are widely used in the transport industry for manufacturing lightweight structures. In the present work, a curved AA1050 bar with fine grains and high strength was manufactured by a novel forming technique, differential velocity sideways extrusion (DVSE). The evolution of grain structure and micro-texture during DVSE and the mechanical properties of the formed bar were studied, and the grain refinement mechanism was revealed. Due to the severe strains arising in the process, (greater than that for conventional one pass equal channel angular extrusion), significant grain refinement in the curved bar (grain size ∼3 μm) was achieved from the original billet (grain size ∼357 μm) in one extrusion operation. Coarse band-like structures containing subgrains with low angle boundaries in the shearing zone gradually transformed into fine shear band-like structures containing equiaxed (sub)grains with a mixture of low and high angle boundaries. The fine shear band-like structures inclined approximately along the shear intersection planes. Severe plastic deformation induced a high dislocation density that initiated subgrain walls with low angle boundaries, which gradually transformed into grain boundaries with high misorientation, indicating that refinement of AA1050 grains in the DVSE process is due mainly to continuous dynamic recrystallization (cDRX). Due to the appearance of greater effective strain on the inner bend of the extruded bar, the grain refinement degree and high angle boundary fraction of the material on the inner bend are slightly greater than those of the material on the outside. DVSE resulted in a weak C-type shear-texture component which can be determined by a proper rotation of the negative simple shear texture. Compared with the billet, significant increase of the hardness, yield strength and ultimate tensile strength by 134.8%, 354.0% and 116.8% respectively was achieved in the formed curved bar, although the elongation to fracture was decreased by 60.0%.

Компетентностный подход в преподавании темы «Касательная плоскость и нормаль»

Qualified presentation of the topic "Tangent Plane and Surface Normal" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.

Pose Estimation for Straight Wing Aircraft Based on Consistent Line Clustering and Planes Intersection.

Aircraft pose estimation is a necessary technology in aerospace applications, and accurate pose parameters are the foundation for many aerospace tasks. In this paper, we propose a novel pose estimation method for straight wing aircraft without relying on 3D models or other datasets, and two widely separated cameras are used to acquire the pose information. Because of the large baseline and long-distance imaging, feature point matching is difficult and inaccurate in this configuration. In our method, line features are extracted to describe the structure of straight wing aircraft in images, and pose estimation is performed based on the common geometry constraints of straight wing aircraft. The spatial and length consistency of the line features is used to exclude irrelevant line segments belonging to the background or other parts of the aircraft, and density-based parallel line clustering is utilized to extract the aircraft’s main structure. After identifying the orientation of the fuselage and wings in images, planes intersection is used to estimate the 3D localization and attitude of the aircraft. Experimental results show that our method estimates the aircraft pose accurately and robustly.

Non-Roundabout design of cancel the intersection signal light on horizontal plane

Traffic congestion is a world problem and an important factor restricting urban development. In order to solve the problem of urban traffic congestion, this paper takes the traffic flow theory and the intersection channel design theory as the research foundation, and conducts in-depth research on the causes of congestion at the intersection and the corresponding solutions, and proposes to cancel the traffic lights at the intersection without any stagnation. This paper proposes a new intersection design scheme, which is like the veins of the flower veins to channel the design intersection, cancel the signal light, and the vehicle can pass through the intersection without stagnation. It proposes a new solution to solve the traffic congestion problem. This new design allows the traffic flow to be spatially separated on the horizontal plane, and due to the cancellation of the signal lights, there is no signal waiting at the intersection, and the vehicle can travel without stopping at the intersection. At the same time, this paper also establishes a plane intersection service capability evaluation system based on simulation and quantitative calculation, which provides an evaluation index and proof basis for the non-stagnation driving channel design of the non-roundabout intersection.

Near-threshold excitation of the λ132.2 nm resonance line of the Tl<sup>+</sup> ion by electron impact

Background : Among all processes of collision of electrons with atomic particles the study of the interaction between electrons and ions is extremely important and necessary for a better understanding of the basic collisional processes in plasma. In the processes of excitation, recombination, and ionization of ions by electron impact autoionization states play an important role. Their radiation decay makes an additional contribution to the photo-recombination process and leads to dielectronic recombination of an ion. Such contribution is most significant in the case of heavy many-electron ions such as the Tl + ion. Excitation of the Tl + ion by an electron impact at electron-ion collisions is practically not investigated up to date. Methods: Experiments were performed using a spectroscopic method in conditions of electron and ion beams intersecting at an angle of 90°. The ribbon electron beam (cross section 1×8 mm 2 , energy E e = 6–14 eV, current I e = (2–10)×10 –5 A) was formed by a low-energy three-electrode electron gun. The ion beam (cross section 2.5×2.5 mm 2 , energy E i = 800 eV, current I i = (5–7)×10 –7 A) was obtained using a low-voltage arc discharge source. The radiation due to decay of the ion states excited during the collisions was detected perpendicular to the plane of the beams intersection. Spectral separation of the radiation in a vacuum ultraviolet region was carried out by 70° vacuum monochromator based on the Seya-Namioka scheme. Results: Thorough measurements of the near-threshold region of the energy dependence of the effective cross section of the electron excitation of the resonance l 132.2 nm line (6 s 6 p 1 P o 1 → 6 s 2 1 S 0 ) of the Tl + ion in the 6–14 eV energy range revealed resonant features both after and before the excitation threshold of the line. The widths of the before-threshold peculiarities is defined by the dispersion of the excited electrons energy. This is indicative of their resonant nature, i.e., they are due to the formation and decay of the atoionization states (AIS) of the Tl atom. Most likely, they are related to the formation and decay of the AIS of the 6 s 6 pnp configuration ( n ≥ 7) to the excited levels of the Tl atom. The nature of the after-threshold structure indicates that it is also the result of resonance processes and is due to the formation and decay of the AIS, most likely of the 5 d 9 6 s 2 6 p 2 configuration, in the electronic channel. The absolute values of the excitation cross sections of the dielectronic satellites were determined by comparison of their intensities with the intensity of the resonance l 132.2 nm line which effective excitation cross section was obtained by normalizing the experimental data on the theoretical calculation by the tight-binding method of two states of the excitation cross section of the 6 s 6 p 1 P o 1 level of the Tl + ion at the 100 eV energy. Conclusions: The energy dependence of the effective excitation cross section of the dielectronic satellites of the Tl + ion resonance line showed that the absolute values of their excitation cross sections are ~ 10 –16 cm 2 and are of the same order of magnitude with that for the resonance line. It is shown that the main excitation mechanism of the resonance line satellites is dielectronic recombination which effectiveness essentially depends on the ratio of the probabilities of the AIS radiation and electron decay. In the case of the relativistic Tl + ion a strong configuration mixing of levels leads to significant increase of the AIS radiation decay probability.

КОНСТРУКТИВНІ АСПЕКТИ ПРОЕКТУВАННЯ ТА ЕКСПЛУАТАЦІЇ ОСНАЩЕННЯ ДЛЯ ГВИНТОВОЇ ЕКСТРУЗІЇ

The present paper introduces a twist extrusion (TE) process and special equipment capable of processing of round and prismatic bars with uniform deformation and reports analytical, and numerical modeling of the process. It is shown that the ability to treat round bars can be achieved by design of special off-axis TE dies in which the axis of the twist surface is displaced from the central axis of the bar being processed. Analytical model developed in the present study explains why using off-axis TE dies leads to uniform deformation and how this deformation uniformity depends on the die geometry. The main conclusions made upon analytical modeling are confirmed with complement finite element simulations. The simulations also show that the main deformation mode in off-axis TE is simple shear at the intersection planes between the twist and the straight channels of the die. Features of design and operation of dies for metal working with pressure from the point of view of design features and working conditions of equipment for severe plastic deformation of blanks by TE are considered. Based on the analysis of required modes of deformation of titanium and iron-nickel alloys, the require-ments for the construction of matrices, materials for their manufacture, lubricating-technological environments, as well as the working conditions of the twist matrices. Recommendations are proposed for the development of a rational design of technological equipment for deformation of blanks by twist extrusion in a batch production. On their basis, technical solutions and constructive measures to improve the design of the tooling for the process of twist extrusion of high-strength alloys for aircraft engineering are proposed. Examples of developed various structural schemes for the implementation of twist matrices are shown. It is shown that a matrix with grooves in a screw channel evolves into a promising construction of the matrix for the realization of screw extrusion of blanks of different sections. This allows to reduce friction between the blank, the counter-blanks and the matrix. The developed equipment for severe plastic deformation by TE can be used for technological processes of processing different aviation alloys and technological process of high-effectiveness titanium materials producing from powders.

Maximally writhed real algebraic links

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots and links in $$\mathbb {RP}^3$$ which is not a topological isotopy invariant. In this paper we study real algebraic links of degree d with the maximal value of this invariant. We show that these links admit entirely topological description. In particular, these links are characterized by the property that any of their planar diagram has at least $$(d-1)(d-2)/2-g-1$$ crossing points where g is the genus of the complexification. Also we show that these links are characterized by the property that any generic plane intersects them in at least $$d-2$$ real points. In addition we give a complete topological classification of these links.

Right intersectional transection plane based on portal inflow in left trisectionectomy

Left trisectionectomy (LT) extending to the segment I with bile duct resection for perihilar cholangiocarcinoma (PHC) is a technically demanding procedure with high morbidity. Liver transection during LT is generally conducted to expose the right hepatic vein (RHV) on the remnant side. In clinical practice, we have often encountered a discrepancy between the theoretical RHV-oriented plane and the actual right intersectional plane. To enable anatomical LT safely, the three-dimensional right intersectional transection plane based on portal inflow was investigated using multidetector-row computed tomography, and it was compared to the theoretical RHV-oriented plane in 100 patients with hepatobiliary disease. The posterior portion of RHV just below the diaphragm was supplied by the dorsal portal branches of segment VIII in 85 cases of 100 (85.0%). The median volume of this portion was 82mL (25-169mL). On the other hand, the anterior region of the peripheral RHV was supplied by a few small ventral portal branches of segment VI in 24 of 90 cases (26.7%). The median volume of this portion was 53mL (20-104mL). In ten cases with a large inferior RHV, the RHV trunk was relatively short and did not reach the caudal part of the liver. The portal inflow-oriented right intersectional plane does not coincide with the RHV-oriented plane in most cases. The cranial part of the actual transection plane becomes hollow, whereas the caudal part is protruded in relation to the RHV. Hepatobiliary surgeons should recognize this complicated transection plane to avoid postoperative complications when performing LT for PHC.

Contribution of Torque Terms to Capillary Equilibrium Along Triple Junctions in Polycrystalline Copper

In the present study, the capillary equilibrium along the triple junctions of grain boundaries (GB) was analyzed. The values of GB energy torque terms were estimated for an 18-μm-thick copper foil with {110} texture from the dihedral angles formed between the intersections of the GB planes and the sample surface and dihedral angles between the GB planes joined at a triple junction. The average value of the torque term was found to be greater than 15 pct of the GB energy.

Line–planes deflectometry

The key issue in phase measurement deflectometry is the solution of incident light which is used to determine the surface normal. To restore the 3D mirror surface information, line–planes deflectometry is proposed to determine the surface normal from the reflected light and the incident light planes containing incident light. To this end, a point light source is used to replace the continuous surface light source of traditional deflectometry. Accordingly, the incident light is calculated by the intersection of incident light planes which are determined by two or more projected lights of a point light source. Finally, the precise mirror surface is restored through the radial basis function interpolation from the gradient which is transformed from the normal information. To verify the proposed method, a line–planes deflectometry system is developed with two projectors and one camera. The system can be applied for two measurement modes concerning different measurement requirements: the ‘line–plane model’ and the ‘line–planes model’. The ‘line–plane model’ can be inferred with a measurement error of less than 0.25 mm to measure objects with low accuracy and non-connectivity. In contrast, the ‘line–planes model’ can be used to measure objects with high accuracy and good area connectivity, and the measurement error is about 1 µm.

An Effective Data-Driven Method for 3-D Building Roof Reconstruction and Robust Change Detection

Three-dimensional (3-D) reconstruction of building roofs can be an essential prerequisite for 3-D building change detection, which is important for detection of informal buildings or extensions and for update of 3-D map database. However, automatic 3-D roof reconstruction from the remote sensing data is still in its development stage for a number of reasons. For instance, there are difficulties in determining the neighbourhood relationships among the planes on a complex building roof, locating the step edges from point cloud data often requires additional information or may impose constraints, and missing roof planes attract human interaction and often produces high reconstruction errors. This research introduces a new 3-D roof reconstruction technique that constructs an adjacency matrix to define the topological relationships among the roof planes. It identifies any missing planes through an analysis using the 3-D plane intersection lines between adjacent planes. Then, it generates new planes to fill gaps of missing planes. Finally, it obtains complete building models through insertion of approximate wall planes and building floor. The reported research in this paper then uses the generated building models to detect 3-D changes in buildings. Plane connections between neighbouring planes are first defined to establish relationships between neighbouring planes. Then, each building in the reference and test model sets is represented using a graph data structure. Finally, the height intensity images, and if required the graph representations, of the reference and test models are directly compared to find and categorise 3-D changes into five groups: new, unchanged, demolished, modified and partially-modified planes. Experimental results on two Australian datasets show high object- and pixel-based accuracy in terms of completeness, correctness, and quality for both 3-D roof reconstruction and change detection techniques. The proposed change detection technique is robust to various changes including addition of a new veranda to or removal of an existing veranda from a building and increase of the height of a building.

Unveiling the role of super-jogs and dislocation induced atomic-shuffling on controlling plasticity in magnesium

Magnesium (Mg) alloys are promising metals for many lightweight structural applications. However, the fundamental deformation mechanism in Mg that lead to poor ductility, formability and anomalous thermal-hardening response remain elusive. Here, atomistic simulations were utilized to unveil the mechanisms responsible for these undesirable properties. We show that pyramidal 〈c+a〉 screw dislocations glide in hexagonal-close packed crystals by a fundamentally new mechanism that involves atomic shuffling and kink pair formation on the trailing partial. Due to local fluctuations in the stresses in the dislocation core, stable super-jogs subsequently form due to inhomogeneities in the shuffling process along the dislocation length as well as due to kink pair collisions. The screw dislocation then moves while dragging these super-jogs, sometimes leaving debris behind (e.g. vacancies or interstitials) as well as long faulted loops on the basal plane that are aligned along the basal plane intersection with the pyramidal-II plane. These dragged out super-jogs dissociate athermally on the basal plane and exert a strong drag-effect on dislocation glide, resulting in dramatic work-hardening at small strains between 0 and 600K, which correlates with the experimentally observed low-ductility during c-axis compression of Mg. Furthermore, this new mechanism also accounts for the experimentally reported anomalous thermal-hardening as a result of the increase in the number of super-jogs per dislocation unit-length due to thermal activation.

Structural origin of high catalytic activity for preferential CO oxidation over CuO/CeO2 nanocatalysts with different shapes

Understanding how the shape of a nanocatalyst alters its catalytic performance is essential for the design of active catalysts at the nanoscale. This report describes novel findings of an investigation of CuO/CeO2 nanocatalysts with different shapes for preferential CO oxidation (CO-PROX), aiming at unraveling the structural origin of the high catalytic activity. CuO/CeO2 catalysts with a series of shapes, including octahedron, rod, cube, sphere and spindle morphologies, were synthesized by a combination of hydrothermal and impregnation methods. By probing the structure, elementary valence and reductivity of the catalysts using an array of techniques, an intriguing structure-activity synergy is revealed, pinning the origin of the highest catalytic activity to the intersection of the nanocrystal planes. The CuO/CeO2 catalysts with spheres and spindles featuring exposed {111} and {002} crystal planes of CeO2 were found to exhibit the lowest reaction temperature in terms of 50% CO conversion and a wider temperature window for the complete CO conversion in CO-PROX. The catalytic synergy reflects a combination of high surface area, reduced state of copper sites, high oxygen vacancies, as well as stronger interaction between CuO and CeO2. This synergy was further substantiated by analysis of the results based on DFT calculation, showing that the oxygen vacancies were the highest at the intersection of {111} and {002} facets, but the lowest on CeO2 {111} surface. The finding of {111} and {002} facets with an increased oxygen vacancy and a stronger metal-support interaction for improving the catalytic performance has important implications for designing facet-tunable nanocatalysts for efficient PROX.

Contour Primitives of Interest Extraction Method for Microscopic Images and Its Application on Pose Measurement

This paper proposes a suite of methods to realize high precision pose measurement in 3-D Cartesian space based on a multicamera microscopic vision system. Since it is inefficient to develop a specific image algorithm for each kind of object and the imaging condition might be unsatisfactory, we propose a method of contour primitives of interest extraction, which allows flexible reconfiguration for novel object image and owns robustness under different imaging conditions. The object is detected in grayscale image based on a template of contour primitives. Edges are extracted according to derivatives along the normal vectors of these contour primitives. The positions and directional derivatives of these edges are used for feature extraction and autofocus, respectively. The point features and line features extracted from multiview images are utilized to measure 3-D vectors and orientations, respectively, based on image Jacobian matrices. Cameras’ linear motions are considered in the imaging model, so that the measurement range is expanded beyond the limitation of microscopes’ shallow depths of field. The affine epipolar constraint and focused planes intersection constraint between cameras are applied to improve the real time performances of image feature extraction and multicamera autofocus, respectively. A series of experiments are conducted to verify the effectiveness of the proposed methods. The root mean square errors of pose measurement are evaluated as 3 ${ {\mu } }\text{m}$ in position and 0.05° in orientation, while the measurement range is about 5000 $ {\mu }\text{m}$ in position and 20° in orientation.

Slip transmission of high angle grain boundaries in body-centered cubic metals: Micropillar compression of pure Ta single and bi-crystals

Here we seek to probe and understand the mechanical behavior of grain boundaries in pure Ta as a model bcc material using micropillar compression experiments. Three high angle grain boundaries are chosen with varying crystal orientations. Multiple bi-crystal pillars are prepared containing a single, nearly vertical grain boundary in the approximate center of the pillar and compared against their single crystalline pillar counterparts. The main phenomenon of interest was slip transmission or strain transfer in the bi-crystals which was considered to occur when slip traces aligned across the grain boundary. This occurred in two of the three bi-crystals. These observations were compared against two slip transmission factors, m'=cos(ψ)cos(κ) and LRB=cos(θ)cos(κ), where ψ,θ,andκ are the angles between the slip vectors, slip plane normals, and the intersection of the slip planes with the grain boundary from the slip systems on either side of the grain boundary. Additionally, transmission was compared against the stress-strain response and overall deformation (i.e., sheared boundary) of the bi-crystal. The transmission factors exhibited a consistent behavior with slip transmission occurring for high transmission factors and not occurring for low transmission factors. For example, slip transmission occurred for m'≥ 0.85 and did not occur for m'≤ 0.46. The engineering stress-strain response and overall deformation behavior did not show correlations with the presence or absence of slip transmission. High angle boundaries in bcc metals are shown to represent a diverse set of responses in bi-crystalline micropillar compression experiments.

The Search for the Centre of Resistance of a Tooth Using a Finite Element Model and the Continuum Mechanics

Abstract In orthodontics, the CR is defined as the ideal point for the positioning of the bracket. This paper presents numerical analyses of a premolar tooth, which aim at searching the centre of resistance (CR) by applying several external torques. We built the geometry of the tooth by making a tomographic renderisation with the software MIMICS. The numerical simulation is performed in the software ANSYS WORKBENCH (ANSYS® Academic Research). The domain of the tooth is discretized using three-dimensional solid elements. The search for the position of CR is made through a heuristic analysis by applying external torques to the model and, afterwards, by seeking the intersection of the neutral planes (where there is no translation). The numerical results showed that the CR is not a point, but rather a region that strongly depends on the relative stiffness between the tooth and the Periodontium (PDL).

Modeling the fractal geometry of Arctic melt ponds using the level sets of random surfaces

During the late spring, most of the Arctic Ocean is covered by sea icewith a layer of snow on top. As the snow and sea ice begin to melt, water collects on the surface to form melt ponds. As melting progresses, sparse, disconnected ponds coalesce to form complex, self-similar structures which are connected over large length scales. The boundaries of the ponds undergo a transition in fractal dimension from 1 to about 2 around a critical length scale of 100 square meters, as found previously from area–perimeter data. Melt pond geometry depends strongly on sea ice and snow topography. Here we construct a rather simple model of melt pond boundaries as the intersection of a horizontal plane, representing the water level, with a random surface representing the topography. We show that an autoregressive class of anisotropic random Fourier surfaces provides topographies that yield the observed fractal dimension transition, with the ponds evolving and growing as the plane rises. The results are compared with a partial differential equation model of melt pond evolution that includes much of the physics of the system. Properties of the shift in fractal dimension, such as its amplitude, phase and rate, are shown to depend on the surface anisotropy and autocorrelation length scales in the models. Melting-driven differences between the two models are highlighted.

External Validation and Optimization of the French Association of Hepatopancreatobiliary Surgery and Transplantation's Score to Predict Severe Postoperative Biliary Leakage after Open or Laparoscopic Liver Resection

Biliary leakage is a major contributor to morbidity after hepatectomy. A score to predict severe posthepatectomy biliary leakage (PHBL) was recently developed by the French Association of Hepatopancreatobiliary Surgery and Transplantation (ACHBT). The aim of the study was to validate and optimize the score on an external cohort. The ACHBT score uses 5 factors (blood loss, remnant ischemia, anatomic resection of segment 8, transection along right aspect of the left intersectional plane and associating liver partition and portal vein ligation for staged hepatectomy) to predict severe PHBL. The score was tested on an external cohort of patients undergoing hepatectomy without hepaticojejunostomy between 1994 and 2016 at a single center. Association between the score, pre- and intraoperative variables, and severe PHBL was assessed in an attempt to improve the score. Among 778 procedures performed (including 679 [87.3%] laparoscopic and 260 [34.3%] major hepatectomies), 31 (4.0%) were complicated with severe PHBL. The ACHBT score showed good discrimination (AUROC [area under the receiver operating characteristic curve] 0.747, 95% CI 0.652 to 0.841), calibration and accuracy (diagnostic odds ratio for a score ≥1: 6.217 [95% CI 2.642 to 14.627], for a score ≥2: 6.059 [95% CI 2.858 to 12.846],and for a score ≥3: 9.589 [95% CI 2.868 to 32.066]). On multivariable analysis, the ACHBT score was the only predictor of severe PHBL. A model combining the ACHBT score and conversion to open surgery was significantly more discriminating than the ACHBT score alone (AUROC 0.790 [95% CI 0.711 to 0.870], Delong's test p= 0.002). The ACHBT score represents an externally validated tool to predict severe PHBL. Inclusion ofconversion to open surgery as an additional factor to the score allowed it to improve its performance to predict severe PHBL after laparoscopic hepatectomy.

Laboratory evaluation of dynamic behavior of steel-strip mechanically stabilized earth walls

To investigate the influence of the peak acceleration, the loading duration, and the strip length on the dynamic behavior of steel-strip reinforced soil walls (SSWs), in terms of the dynamic reinforcement load distribution and the dynamic lateral earth pressure behind the surface, a series of 1-g shaking table tests was performed on five reduced-scale reinforced soil wall models. It was observed that the maximum axial force of the strips (Tmax) is mobilized at the intersection of the failure plane with strips in all rows. It was also discovered that, in the upper half of the walls, the Tmax values decrease with a decreasing strip length, while this trend is reversed in the lower half of the walls. Additionally, a proper convergence was found between the Tmax/H·γs·SV·SH and L/H′ ratio at different levels of acceleration and duration, so that Tmax/H·γs·SV·SH can be defined as a function of the L/H′ ratio and the seismic parameters for different rows of strips. On the other hand, it was observed that the values of earth pressure predicted by conventional methods under static and seismic conditions are too conservative and these methods predict the position of the resultant lateral force higher than the actual point.

On the ellipsoid and plane intersection equation

On the ellipsoid and plane intersection equation