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Stress analysis, strain analysis, and shearing of soils
Published in Rodrigo Salgado, The Engineering of Foundations, Slopes and Retaining Structures, 2022
Mohr’s circles have interesting geometric properties. One very useful property that every circle has is that the central angle of the circle corresponding to a certain arc is twice as large as an inscribed angle corresponding to the same arc (Figure 4.5). Applying this property to the Mohr circle shown in Figure 4.3, the angle made by two straight lines drawn from any point of the circle to point S(σθ, τθ) representing the stresses on the plane of interest and to point S1(σ11, −σ13) is equal to θ. In particular, there is one and only one point P on the circle with the property that the line joining P to point S(σθ, τθ) is parallel to the plane on which σθ and τθ act. The point P with this property is known as the pole of the Mohr circle. Based on the preceding discussion, the pole can be defined as the point such that, if we draw a line through the pole parallel to the plane where stresses σθ and τθ act, this line intersects the Mohr circle at a point whose coordinates are σθ, τθ.
Cardiac Image Processing
Published in Wen-Jei Yang, Handbook of Flow Visualization, 2018
Shigeru Eiho, Michiyoshi Kuwahara
Figure 13 shows a schematic diagram of the reconstruction of a cross-sectional shape on a (horizontal) plane perpendicular to the axis of rotation. Three boundary curves shown in this figure correspond to the 3 different angles of projection at the same cardiac phase. By assuming that the X-ray radiation is done in parallel beam, one can obtain the left ventricular shape on the horizontal plane by the following steps: Assume an inscribed point on each side of the polygon (i.e., at the center of each side).Draw a smooth curve connecting these points. The spline curve of the third order is used.
Generalized formulation for resonance frequency of even polygonal microstrip antennas
Published in Electromagnetics, 2022
Deepankar Shri Gyan, K. P. Ray
For a polygon inscribed in a circle, as the number of sides increases to infinite, the polygon becomes a circle. This indicates that the nature of radiation of a polygonal MSA is going to be similar to that of a circular MSA. Figure 1 depicts the vertex angles for a square MSA (red dotted), octagonal MSA (green dotted), effective extent of magnetic wall (R(ns)), and the excitation along the longest diagonal. It is indicated in Figure 1 that an even polygonal patch behaves as a circular patch when excited along the longest diagonal (and this claim has been substantiated in the further analysis). It should be noted that if this condition is not observed, then the nature of polygon will get closer to rectangular MSA instead of circular MSA. When an even polygon is excited along its longest diagonal, resonance at the fundamental mode happens along the diameter of the circumcircle. The excited waves in such polygons produce cylindrical harmonics because of diametrical symmetry in structure. The cylindrical harmonics are governed by the variations of Bessel functions of first kind (Kline 1948). In case of odd polygons, this diametrical symmetry is absent, and hence, their harmonics are not similar to circular MSA.