Curvature Of Curve Research Articles

Weak curvatures of irregular curves in high-dimensional Euclidean spaces

Weak curvatures of irregular curves in high-dimensional Euclidean spaces

Influence of redundant hole on the degradation and failure of round-hole tubes under cyclic bending

ABSTRACT This article introduces the effect of redundant holes (RDHs) on the performance of 6061-T6 aluminum alloy round-hole tubes (Al-RHTs) subjected to cyclic bending. In this study, Al-RHTs with a fixed hole diameter of 2, 4, 6, 8 or 10 mm were drilled to acquire an RDH of the same size on the same cross section but at different positions: 45°, 90°, 135° and 180°. The moment–curvature relationship demonstrated a steady loop from the first cycle. The RDH at any position showed a slight influence on the relationship. However, the ovalization–curvature curve displayed ratcheting, increasing, asymmetrical, and bowtie-like trends as the number of cycles increased. The RDH position shows substantial influence on the relationship. Although the Al-RHTs were tested with five different hole diameters and different RDH positions, the curvature-number of cycles needed to initiate failure relationships in a log–log scale displayed parallel straight lines for each hole diameter. Finally, an empirical formulation proposed by Lee, Weng, and Pan in 2019 was modified and improved to simulate the above relationships. The simulation result was compared with the experimental results, and it was found that the theoretical analysis could reasonably reproduce the experimental results.

Safety Effects of Horizontal Curve Design and Lane and Shoulder Width on Single Motorcycle Accidents in Norway

Factors related to the road infrastructure contribute to the occurrence of motorcycle accidents. This study investigates how design parameters of the existing rural two-lane road network in Norway influence the occurrence of single motorcycle accidents. The design elements considered in this study are horizontal curvature (curve type, degree of curvature, and adjacent curve requirements) and lane and shoulder widths. A matched case-control study design was applied to investigate the safety effects of these elements. Cases were defined as segments experiencing at least one single motorcycle accident during the study period from 2013 to 2017, while controls were defined as segments not experiencing an accident in the same period. In order to identify the segments, a GIS analysis was performed on data collected from the National Road Database (NVDB). In case-control studies, matching allows us to control for confounding variables. AADT and speed limit were used as matching variables in this study. A matching ratio of 4 : 1 (i.e., four controls per case) was used, resulting in 752 controls being matched to 188 cases. The results indicate horizontal alignment to have a more significant effect on single motorcycle accidents compared to lane and shoulder widths. Segments with several adjacent reverse curves, with high curvature (R < 200 m), have high odds for an accident. Further, if the requirements for adjacent curves are not fulfilled (i.e., considerable variation in adjacent curve radii), the odds increase even more. While the results are not statistically significant, trends seen indicate that wider lanes were associated with increased odds for an accident, while wider shoulders were associated with decreased odds. In comparison with a similar study considering passenger vehicles, the results of this study also indicate that horizontal alignment has a greater effect on single motorcycle accidents than on passenger vehicle accidents.

Experimental and numerical investigation on the flexural behavior of concrete-filled elliptical steel tube (CFET)

Concrete-filled elliptical steel tube (CFET) is a novel type of composite member, which has great potential to be used as piers and columns in bridges and airport terminals because of its low flow resistance coefficient and reasonable distribution of major-minor axis. Pure bending loading conditions are not frequently occurred in practical engineering, but the flexural researches are important since it's the basis of mechanical property researches under complex loading. However, little attention to the flexural behavior of CFETs has been paid at present. Hence, the main objective of this paper presents a combined experimental and numerical investigation on the flexural behavior of CFETs. Flexural tests were carried out to investigate the behavior of CFETs under bending. The moment vs. curvature curves during the whole loading process was analyzed in detail, and all the specimens behaved in a very ductile manner. Besides, the composite action between the steel tube and core concrete was discussed and clarified based on experimental results. Furthermore, FE (Finite element) model with precise material constitutive models was established to investigate the flexural mechanical of CFETs comprehensively, in terms of the full curves analysis on the moment vs. curvature curves, development of the stress and strain distribution in the steel tube and core concrete, and composite action. It should be noted that there existed bond slip in the interface between the core concrete and steel tube during the loading process. Then, the parametric studies with various factors were performed. Finally, design formulas were proposed to predict the ultimate moment and flexural stiffness of CFETs. The predicted results showed satisfactory agreement with the experimental and FE results. Moreover, the difference among the experimental, FE and predicted results using the related design codes were illustrated. The results of this study are expected to provide a reliable reference for design and application in concrete-filled elliptical steel tube (CFST) structures.

Axial compressive behavior of GFRP tube confined seawater coral aggregate concrete reinforced with epoxy-coated bars

An experimental study on the mechanical behavior of glass fiber reinforced plastic (GFRP) tube confined seawater coral aggregate concrete reinforced with epoxy-coated steel bars (GSCACESB) under axial compression were systematically performed. Three different parameters were considered: concrete type (seawater coral aggregate concrete and ordinary concrete), GFRP tube thickness (4 mm, 5 mm and 6 mm) and reinforcement ratio (longitudinal and circumferential reinforcement ratios). Typical influences of concrete, GFRP tube and reinforcement on the bearing capacity, deformation and load-deformation curve of GSCACESB were investigated. Test results showed that the failure pattern of GSCACESB was similar to that of GFRP tube confined ordinary concrete (GCOC). However, the bearing capacity of GSCACESB was significantly decreased compared to GCOC due to the less confinement between seawater coral aggregate concrete and outer tube. The peak deformation and bearing capacity of GSCACESB increased with the increases of tube thickness, and longitudinal and circumferential reinforcement ratios. The load-deformation curve of GSCACESB was divided into elastic-plastic and hardening stages. The more the reinforcement ratio, the less the curve curvature. An analytical expression considering the influences of SCAC was also proposed, which could predict the structural response of GSCACESB. Finally, the numerical simulation analysis was undertaken to investigate the GSCACESB mechanical behaviors.

Study on the nonlinear performance degradation of reinforced concrete beam under chloride ion corrosion

The chloride-induced corrosion deteriorates the performance of reinforced concrete (RC) structures. This paper proposes a novel method for the nonlinear analysis of RC structures under chloride ion corrosion based on the cellular automata (CA) method and the fiber model of FEM. The CA model is established to simulate the chloride diffusion in cross-section of RC continuous T-beam, then the time-varying chloride ions concentration in cross-section are evolved. The damage indices corresponding to the corrosion of steel bar and deterioration of concrete have been defined and their variations with time are discussed. Following that, the CA model is coupled with the fiber model of FEM to compile an matlab program to analyze the performance of corroded RC continuous T-beam. Thus the deflection-time curve, the load–deflection curve, the strain distribution of cross-section, the bending moment–curvature curve and the time evolution of bending moment for the T-beam are obtained to discuss the time-variant nonlinear performance degradation process of corroded RC beam. It can be concluded that the proposed CA model can effectively and accurately simulate the diffusion process of chloride ions in concrete. Besides, the damage model introduced in this paper can quantifies the relationship between the damage caused by chloride ions and corrosion time. Furthermore, the matlab program established in the present study can be effectively applied to the nonlinear analysis of corroded RC beams. Considering the co-action of geometric and material non-linearity and corrosion, the calculation results can more accurately describe the actual stress process of the RC structure. Finally, the proposed method can be used as a effective theoretical tool for the life assessment of RC structures in corrosive environment.

Research on Carrying Capacity of T-Shaped Stiffened Concrete-Filled Steel Tubular Columns Subjected to Eccentrically Compressive Load

In this paper, in order to further study the mechanical behavior of T-shaped stiffened concrete-filled steel tubular (CFST) columns, numerical programs of eccentrically compressive T-shaped stiffened CFST columns were developed to calculate moment M-ϕ curvature curves of cross section and column. The calculated curves with the numerical programs agree well with the experimental results. A parametric analysis was carried out to calculate M-N correlation curves of cross section and column to investigate influence of concrete strength fck, steel yielding strength fy, steel tube thickness t, slenderness ratio λ, and loading angle θ. On the basis of parametric analysis, simplified resistance models of T-shaped CFST section and column were proposed and verified by the numerical analysis results. The simplified resistance models are reliable to predict the mechanical behavior for engineering application.

Nurbs modeling for design waterline based on the variation of sectional area curve

Hull form design from parent ships transforms the ship's parameters based on the variation of theoretical sectional area curve of the Lackenby method. The correction and modification of the theoretical sectional area curve is essentially the change of ship displacement, hull form coefficients, and the longitudinal center of buoyancy from the parent ships. In the preliminary design stage, the hull form design approach from parent ships minimizes the risks compared to the new design while still retaining hydrostatic and hydrodynamics' advantages. However, the Lackenby method of ship hull form variation uses a linear or quadratic function to shift the sectional area curves, regardless of the ship's hull form faring, especially the curvature's discontinuity the bow, stern, and midship. Therefore, the computer graphic algorithm based on the B-spline function is studied and applied; simultaneously, the mathematical model for the designed waterline is built in the form of a continuous curve instead of the B-spline segments. In this study, the mathematical model for the coastal container ship's design water line is constructed, ensuring continuity and fairing throughout the continuous B-spline curve. The geometry continuity evaluation results are expressed through the parameter curve's curvature and resistance component calculations' performance by computational analysis.

Bond of epoxy-coated steel bars to seawater sea sand recycled concrete

The bonded behaviour between epoxy-coated steel bars (ECSB) and seawater sea sand recycled concrete (SSSRC) were systematically analysed using pull-out tests. Five different parameters were considered: including types of concrete (SSSRC, recycled concrete and ordinary concrete) and steel bars (ECSB and ordinary steel bars (OSB)), the ratio of bond length to bar diameter (3, 5 and 8), and concrete strength (C20 and C25) and thickness of concrete cover (42 mm and 67 mm). The failure pattern, bond strength and bond-slip curve of specimen were analysed. The results indicate that the failure mode of specimen was pull-out failure, and the concrete type changed the failure surface of specimen. The specimen with OC exhibited high bond strength compared to specimen with SSSRC. Increasing concrete cover thickness increased the bond strength of seawater sea sand recycled concrete-ECSB (SSSRC-ECSB), while its bond strength decreased with the increase in the bond length. It can be found that the bond-slip curve of SSSRC-ECSB was similar to that of ordinary concrete-OSB (OC-OSB). SSSRC and ECSB obviously reduced the curvature of bond-slip curve, and the decline of curve changed with the types of concrete and steel bars used. Finally, a general expression of bond-slip curve was derived, which could provide a basis for the practical application of SSSRC.

Experimental and numerical studies of concrete-encased concrete-filled steel tube stub columns under uniaxial and biaxial eccentric compression

Experimental and numerical studies were conducted to investigate the structural performance and resistances of concrete-encased concrete-filled steel tube (CECFST) stub columns under uniaxial and biaxial eccentric compression. A test programme, consisting of 1 axially-loaded stub column, 3 uniaxial-eccentrically-loaded stub columns and 11 biaxial-eccentrically-loaded stub columns, was firstly conducted. The effects from uniaxial and biaxial loading eccentricities and end moment ratios on the structural performance of CECFST stub columns were studied. The key test results, including the failure load and mode of each specimen, crushing and spalling of the concrete cover, load–deflection and moment–curvature curves, were presented. It was found that the loading eccentricity and end moment ratio affect the ductility and failure mode of the specimen. The mid-height moment capacity of each specimen was maintained by the inner CFST after the outer concrete encasement was crushed. The test programme was followed by a numerical modelling programme, where finite element models were initially developed and validated against the experimental results, including the full load–deflection curves and end moment–rotation histories, and then adopted for conducting parametric studies to generate additional numerical data. The obtained test and numerical data were then used to evaluate the applicability of the codified design approaches in the American specification (Method 1 and 2), European code, and AIJ standard for composite columns under eccentric compression. The results of the evaluations revealed that the European code and American specification (Method 2) overestimate the cross-section resistances of CECFST stub columns under uniaxial eccentric compression, while the AIJ standard and American specification (Method 1) provide conservative resistance predictions. For CECFST stub columns subjected to biaxial eccentric compression, both the European code and American specification (Method 2) underestimate the cross-section resistances. Finally, new design methods were proposed for CECFST columns under uniaxial and biaxial eccentric compression, indicating a higher level of design accuracy over the design codes.

Solitons of the midpoint mapping and affine curvature

For a polygon x=(x_j)_{jin mathbb {Z}} in mathbb {R}^n we consider the midpoints polygon (M(x))_j=left( x_j+x_{j+1}right) /2. We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on mathbb {R}^n. These smooth curves are also characterized as solutions of the differential equation dot{c}(t)=Bc (t)+d for a matrix B and a vector d. For n=2 these curves are curves of constant generalized-affine curvature k_{ga}=k_{ga}(B) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

Gauss—Bonnet Theorems in the Lorentzian Heisenberg Group and the Lorentzian Group of Rigid Motions of the Minkowski Plane

The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are C2-smooth in the Lorentzian Heisenberg group away from characteristic points and signed geodesic curvature for curves which are C2-smooth on surfaces are studied. Using a similar method, we also studied the corresponding contents on Lorentzian group of rigid motions of the Minkowski plane.

Experimental and numerical investigation on the behavior of concrete-filled rectangular steel tubes under bending

Pure bending loading conditions are not frequently occurred in practical engineering, but the flexural researches are important since it's the basis of mechanical property researches under complex loading. Hence, the objective of this paper is to investigate the flexural behavior of concrete-filled rectangular steel tube (CFRT) through combined experimental and numerical studies. Flexural tests were conducted to investigate the mechanical performance of CFRT under bending. The load vs. deflection curves during the loading process was analyzed in detail. All the specimens behaved in a very ductile manner. Besides, based on the experimental result, the composite action between the steel tube and core concrete was studies and examined. Furthermore, the feasibility and accuracy of the numerical method was verified by comparing the computed results with experimental observations. The full curves analysis on the moment vs. curvature curves was further conducted, where the development of the stress and strain redistribution in the steel tube and core concrete was clarified comprehensively. It should be noted that there existed bond slip between the core concrete and steel tube during the loading process. And then, an extensive parametric study, including the steel strength, concrete strength, steel ratio and aspect ratio, was performed. Finally, design formula to calculate the ultimate moment and flexural stiffness of CFRTs were presented. The predicted results showed satisfactory agreement with the experimental and FE results. Additionally, the difference between the experimental/FE and predicted results using the related design codes were illustrated.

Polar tangential angles and free elasticae

In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.

Accurate higher order derivatives of curvature and torsion of geodesic curves on freeform surfaces

Accurate higher order derivatives of curvature and torsion of geodesic curves on freeform surfaces

Optimization of Alchemical Pathways Using Extended Thermodynamic Integration.

Thermodynamic integration (TI) is a commonly used method to determine free-energy differences. One of its disadvantages is that many intermediate λ-states need to be sampled in order to be able to integrate accurately over ⟨∂H/∂λ⟩. Here, we use the recently introduced extended TI to study alternative parameterizations of H(λ) and its influence on the smoothness of the ⟨∂H/∂λ⟩ curves as well as the efficiency of the simulations. We find that the extended TI approach can be used to select curves of low curvature. An optimal parameterization is suggested for the calculation of hydration free energies. For calculations of relative binding free energies, we show that optimized parameterizations of the Hamiltonian in the unbound state also effectively lower the curvature in the bound state of the ligand.

Experimental and numerical behavior of eccentrically loaded square concrete-filled steel tubular long columns made of high-strength steel and concrete

The high-strength materials are increasingly used in the concrete-filled steel tubular (CFST) column to improve its mechanical behaviors. This study aims to evaluate the fundamental behavior of high-strength concrete-filled high-strength square steel tubular (HCFHSST) long columns under eccentric compression. A series of tests were conducted on eccentrically loaded HCFHSST long columns to investigate the mechanical characteristics and failure mechanisms. Numerical investigations were systematically performed to explore the load-carrying performance of the column. Meanwhile, parametric studies were implemented to evaluate influences of several key parameters on the strength increase factor (SIF), P–M, and P/P0–M/M0 curves of the HCFHSST members, respectively. Furthermore, the equations were proposed to predict the peak-loads of HCFHSST long columns under eccentric loading. The experimental and simulated peak-loads were compared with the results of various design codes and recommended equations to assess the feasibilities of those calculation systems. The results demonstrated that the bending moment–curvature curves of the experimental specimens could be categorized into three types considering the second-order effect. Additionally, it was numerically observed that using high-strength steel can enhance the yield load proportion of the column, and the concrete withstands most of the applied load when utilizing high-strength concrete as infill. Based on the peak-load calculations of codes EC4, AISC 360, and GB 50936, it was found that code GB 50936 presented a better result on average while was unsafe for some cases with high e/B ratios. The equations proposed in this study can provide an improved prediction and hence are suggested.

A simple differential geometry for complex networks

AbstractWe introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

Buckling performance of high-precision variable stiffness composites laminate based on automatic placement technology

The buckling resistance of the variable stiffness composite laminates laid by the automatic placement can be effectively improved by tailoring the in-plane stiffness. During the placement process, laying the trajectory path planning is one of the key technologies to achieve variable stiffness design. So, the fiber trajectories composed of angle linear variation, constant-curvature curve and quadratic Bezier curve were taken as the research object, and the parametric analysis of its compression buckling performance was carried out. Finally, the influence of tape width on the surface profile accuracy and buckling resistance mechanical properties was studied using a finite element model. The results show that under the compression condition, the secondary Bezier curve has the best buckling resistance, and the equal curvature curve is least affected by the curvature constraint. The width of the tape is negatively correlated with the area of the overlap area and the buckling resistance, when the tow width of the fiber placement head is constant. Using the largest tape size can minimize the overlapping area and improve the profile accuracy of the composite structure. At the same time, it is guaranteed that the buckling performance of the structure is improved by 37.3%.

A Lagrangian approach to extremal curves on Stiefel manifolds

<p style='text-indent:20px;'>A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.