Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity

The properties of the first reflection of curves can be investigated by their connections. It is giving us the possibility to construct new reflecting curves and reflecting systems with the demand properties. The studied reflective properties of plane curves as sections of reflecting surfaces are preserved if these sections are generators of surfaces of revolution and rotative surfaces (the normal to the section at a given point coincides with the normal to the surface at the same point) and the source of the incident rays is in the same plane section. Caustics, orthotomics, and reflective curves with a source at an infinity are considered. For a given orthotomics, there is a one-parameter set of reflective curves with a source at an infinity. To select one reflecting curve from this set it is necessary to put straight line, perpendicular to the direction of the incident rays. The shaping of a reflecting curve and its caustics according to its given orthotomics, as well as the shaping of orthotomics according to a given reflective curve with a source at an infinity, is studied, and algorithms for these shaping are proposed. Equations are written for all studied curves. Caustics, orthotomics, and reflective curves with a source at infinity are a promising means of using the methods of geometric optics for modeling natural objects in ergo design.

Analytical Model to Predict Residual Flexural Capacity of Recycled Aggregate Concrete Beams with Corroded Longitudinal Rebars

This paper presents the results of research on the flexural behavior of recycled aggregate concrete (RAC) beams. The correlation between flexural behavior and the corrosion level of longitudinal rebar was analysed. Based on theoretical analysis and experiment results, the influence of corrosion on flexural cracking moment was analysed and a model to predict the residual flexural capacity of RAC beams with corroded longitudinal rebars was established. The experimental results show that the development degree of cover cracks deepens with the increase of the corrosion level, and the experimental data also demonstrate that the strain distribution of concrete in the midspan of beams conforms to the plane section assumption better when the corrosion level is little but no longer satisfies the plane section assumption when the corrosion level is high.

An approximation of geodesic circle passing through three points on an ellipsoid

In this paper we present an approximation method for a geodesic circle passing through three points on an oblate ellipsoid. Our method uses a prolate ellipsoid passing through the three points, and the new approximation curve is the intersection of the oblate and prolate ellipsoids, which can be obtained algebraically without iterations. The advantage of our approximation method is that it yields a significantly smaller approximation error. Compared to the plane section curve passing through the three points on the oblate ellipse, our method reduces the approximation error by at least 98\hspace{0.1667em}\% when the radii of geodesic circles are 100 km∼1000 km on the surface of the Earth. We illustrate the results using numerical examples.

Some curvature properties of paracontact metric manifolds

The purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying{\nabla_{\xi}h=0}and such that the sectional curvature of the plane section containing ξ equals a non-zero constantc. Also, we study paracontact metric manifolds satisfying the curvature condition{Q\cdot R=0}, whereQandRare the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed.

A Parameter Ascertainment Method for the Skeleton Curve of Corroded Reinforcement Concrete Members with Flexure[tnq_nbsp] and Compression

This paper presents an effective method of parameter ascertainment for the skeleton curve of corroded compression-bending members to establish its restoring force model. An assumption which considers the skeleton curve of damaged and undamaged members has similar shape is introduced into the fitting process of parameters. Meanwhile, two-dimensional plane section assumption is used to simplify the mathematical model and reduce the computational cost. Several sets of experimental data were compared with the prediction by the method developed in this paper, for its verification. The case studies show that the experimental trends can be captured very well.

The fixed-effects analysis of the relation between SDAS and carbides for the airfoil blade traces

The primary objective was to test a usefulness of the specific fixed-effect model for the analysis of quantitative relationships gathered from the image analysis of the material microstructures. The dataset was obtained from the investigation of turbine blades made from superalloy IN713C. The analysis based on the general linear model resulted in informative plots revealing mutual relationships between secondary dendrite arm spacing and the mean plane section area of carbides in the material. Directions for further research also were obtained.

Research on Shear Lag Effect of T-shaped Short-leg Shear Wall

Longitudinal displacement of cross section of T-shaped shortlegshear wall was simplified to three parts: shear lag warpingdisplacement, plane section bending displacement and axialdisplacement. Shear lag warping deformation was assumed ascubic parabola distribution along flange, and based on minimumpotential energy principle, differential equations were deduced;with boundary conditions, a calculation theory for shear lageffect was established. With two T-shaped short-leg shear wallmodels, vertical stresses of flanges were obtained by calculationtheory and finite element calculation respectively, and comparisonbetween theoretical analysis results and numerical calculationresults was made. At last, parameter analysis was carriedout, and the influence of shear force, shear span ratio andheight-thickness ratio on shear lag coefficient was obtained.Research shows that numerical calculation results are in goodagreement with theoretical analysis results, and each parameterhas different influence on shear lag coefficient.