Hyperbolic Paraboloid Research Articles

Mean pressure coefficient distributions over hyperbolic paraboloid roof and canopy structures with different shape parameters in a uniform flow with very small turbulence

The hyperbolic paraboloid with two high and low corners is the most basic anticlastic double curved shape of tensile surface structures. However, this shape is currently not considered in the existing wind load standards and only very few investigations are documented in literature. Therefore, this study aims to map the mean aerodynamics over hyperbolic paraboloid roofs and extremely thin canopies to help engineers understand the general aerodynamics of these structures. In this study, scaled wind tunnel testing is performed on rigid models of a roof with enclosing walls and a canopy without enclosing walls to draft mean Cp-distributions and standard deviations for various wind orientations. These mean Cp-distributions are validated with the limited studies in literature and used to calibrate and benchmark a numerical wind tunnel, using CFD RANS simulations, that allows to further reduce the thickness of the canopy compared to the wind tunnel models. The numerical wind tunnel is used to study the mean wind load distributions for hyperbolic paraboloid roofs and canopies with different surface curvature for the most important wind orientations. The results in this paper show good agreement between wind tunnel experiments and CFD simulations, with similar variations as established by different studies of the Silsoe cube. First mean Cp-distributions obtained with scaled wind tunnel testing are presented for hyperbolic paraboloid canopies of only 5 mm thick, with similar mean net Cp and only slightly larger suction over the upper and lower face compared to the numerical results for a zero-thickness canopy. The numerical study of different surface curvatures indicates different geometrical patterns in the mean Cp-distributions when the flow starts to separate for highly curved hypars. In general, suction and pressure increase for higher surface curvature, with similar geometrical patterns in the mean Cp-distributions, as long as the flow remains attached. If the flow separates for highly curved hypars, the geometrical pattern differs and the mean Cp drops due to the different behavior of the flow. Finally, the presented mean Cp-distributions give insight in the aerodynamics of hypar roofs and canopies with different surface curvature and could form a basis for future analysis towards peak loads for the design of these structures.

Una alternativa para la producción de superficies de doble curvatura en láminas de fibrocemento a través de mecanizado de tres ejes

El presente trabajo describe el proceso y resultados del mecanizado de un material compuesto de matriz de cemento reforzado con Sílice y fibras naturales, comercialmente conocido como fibrocemento. El mecanizado se realizó en una lámina comercial de 10 mm de espesor, a partir de una superficie de doble curvatura en un centro de mecanizado ROLAND - MDX 40A, por medio de un proceso de desbaste y posterior acabado en 3 ejes, con el uso de herramientas de corte comerciales de carburo de tungsteno. Para la obtención de la geometría, se utilizó Rhinoceros 5 con el complemento Grasshopper para la parametrización del modelo y RhinoCAM para la generación de los códigos G que permitieron ejecutar los procesos de mecanizado. La geometría parametrizada en la lámina partió de un módulo de un paraboloide Hiperbólico, el cual tomaba sus dimensiones, según las medidas de la placa en su largo, ancho y profundidad y el número de veces que el usuario deseaba replicar en un patrón rectangular este módulo. El proceso de mecanizado se realizó sin refrigeración y con herramientas de corte estándar, por lo que la velocidad de avance, profundidad de corte y avance de la herramienta se ajustaron a parámetros mínimos que permitieron conservar la vida útil de la herramienta y el cuidado del equipo durante los procesos de mecanizado y así obtener un acabado superficial aceptable, esto tuvo como consecuencia, un tiempo lento los procesos, los cuales podrían ser optimizados utilizando sistemas de refrigeración o herramientas especializadas en el mecanizado de materiales cerámicos. La técnica de manufactura empleada en este material y desarrollada en este trabajo, puede ser aplicada en la fabricación de piezas con volúmenes bajos de producción o elementos especializados, en sectores tales como la Ingeniería y manufactura, el sector Eléctrico, la industria de la construcción y arquitectura, el arte y diseño y la fabricación de Mobiliario.

Determination of stresses in thin-walled steel fiber reinforced concrete roofs in the form of hyperbolic paraboloid

The authors have determined the absolute values of deformations of steel fiber reinforced concrete (SFRC) in a thin-walled shell in the form of a hyperbolic paraboloid that on the basis of which the experimental stresses arising in the shell are determined. The article presents the results of theoretical studies of stresses in thin-walled shells of steel fiber reinforced concrete in the form of hyperbolic paraboloid (hypar), which were determined on the basis of the developed calculation method based on the equations of the moment theory of shallow shells. To solve the basic equations, the authors generalized the method of double series of Fourier, which takes into account the curvature of torsion of the hypars. The solution for the non-fixed hypars with rectangular shape was found through the double series of Fourier with additional application of the iteration method. The coefficients in the series are determined analytically using recurrence formulas. An approach to the study of the stress-strain state of rectangular hypars with a fixed contour is proposed.

Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)

This paper discusses two approaches for deriving the equations of motion for a ball that rolls without slipping on the surface of a rotating hyperbolic paraboloid. We analyze two possible methods for defining the surface on which the ball rolls, and show the relationship between the two methods. We describe how the stability of the ball's rotation at the saddle point depends on the radius of the ball, in the case where the stability analysis is made in dimensionless parameters.

Analysis of Natural Light Distribution in the Building

Global warming affect the condition of the sky and also affect the temperature on the earth surface. Changes in sky conditions affect the distribution of light into the building. Sunlight is a natural light source in the morning, afternoon and evening. Utilization of natural lighting into buildings, can reduce energy use as an artificial lighting source. Phinisi Tower Building has a unique form of hyperbolic paraboloid, so it can be a guide in designing buildings. This building needs to be analyzed, how the distribution of light into the building, so it can become a guide in buildings design. The purpose of this study is how the influence of position, floor height and window openings on the building envelope to the distribution of natural light. Quantitative research method is to analyze the level of illumination inside and outside the building in the form of tables and graphs. The result of research is illuminance level of daylight depends on the sky condition and it have affect the light distribution into the building. The increased light distribution at the height of the floor. As well as of window opening on the building envelope (top lighting and side lighting) have effect on the level of illuminance in the building. The distribution of natural light will increase during the day, if the sky condition does not change in one day.

Invariant and smooth limit of discrete geometry folded from bistable origami leading to multistable metasurfaces

Origami offers an avenue to program three-dimensional shapes via scale-independent and non-destructive fabrication. While such programming has focused on the geometry of a tessellation in a single transient state, here we provide a complete description of folding smooth saddle shapes from concentrically pleated squares. When the offset between square creases of the pattern is uniform, it is known as the pleated hyperbolic paraboloid (hypar) origami. Despite its popularity, much remains unknown about the mechanism that produces such aesthetic shapes. We show that the mathematical limit of the elegant shape folded from concentrically pleated squares, with either uniform or non-uniform (e.g. functionally graded, random) offsets, is invariantly a hyperbolic paraboloid. Using our theoretical model, which connects geometry to mechanics, we prove that a folded hypar origami exhibits bistability between two symmetric configurations. Further, we tessellate the hypar origami and harness its bistability to encode multi-stable metasurfaces with programmable non-Euclidean geometries.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

We present a restriction theorem for the Fourier transform to a 2-dimensional conical surface of finite type, obtaining a sharp result, which improves previous work by Barcelo.

Determinación geométrica de los paraboloides hiperbólicos de las bóvedas que configuran el pórtico de acceso a la Cripta de la Colonia Güell de Antoni Gaudí

La Cripta de la Colonia Güell (1898-1914) diseñada por el arquitecto Antoni Gaudí, es un bien de interés cultural y patrimonio histórico de España. Según algunos historiadores y arquitectos, las primeras bóvedas de la historia de la arquitectura con forma de paraboloide hiperbólico se encuentran en el techo del pórtico de acceso a la cripta. El presente estudio pretende comprobar la anterior afirmación con un método geométrico objetivo. Con este proceso determinamos: 1) Cuáles son los 19 paraboloides hiperbólicos que mejor se ajustan a las superficies generadas por los 19 fragmentos de bóvedas que conforman el pórtico; 2) Ofrecemos una medida objetiva de tal ajuste y, 3) Presentamos una clasificación de tales paraboloides mediante sus parámetros geométricos.

Mode frequency analysis of antisymmetric angle-ply laminated composite stiffened hypar shell with cutout

Abstract This article deals with finite element method for the analysis of antisymmetric angle-ply laminated composite hypar shells (hyperbolic paraboloid bounded by straight edges) that applies an eight-noded isoparametric shell element and a three-noded beam element to study the mode-frequency analysis of stiffened shell with cutout. Two-, 4-, and 10-layered antisymmetric angle-ply laminations with different lamination angles are considered. Among these, 10-layer antisymmetric angle-ply shells are considered for elaborate study. The shells have different boundary conditions along its four edges. The formulation is based on the first-order shear deformation theory. The reduced method of eigen value solution is chosen for the undamped free vibration analysis. The first five modes of natural frequency are presented. The numerical studies are conducted to determine the effects of width-to-thickness ratio (b/h), degree of orthotropy (E 11/E 22), and fiber orientation angle (θ) on the nondimensional natural frequency. The results reveal that free vibration behavior mainly depends on the number of boundary constraints rather than other parametric variations such as change in fiber orientation angle and increase in degree of orthotropy and width-to-thickness ratio.

Hyperbolic Paraboloid for a Ruled Preform of an X-Fitting: Calculation of Parameters

A design and manufacturing process for composite fittings (a part of a truss structure of aircraft) is discussed. Intersecting tubular rods are connected via ruled surfaces such as a hyperbolic paraboloid. A mathematical model of X-fitting surface is developed, and numerical simulation of the fitting is performed to optimize its geometry. Due to the thing that interface surface is ruled, reasonable composite application and, as a result, high weight efficiency of the structure may be achieved.

HYPAR ROBOTIC FABRIC FORMWORK FOR CREATING HYPERBOLIC PARABOLOID PANELS

HYPAR ROBOTIC FABRIC FORMWORK FOR CREATING HYPERBOLIC PARABOLOID PANELS

Influence of ambient conditions on wind speed measurement: Impact on the annual energy production assessment

The Annual Energy Production (AEP) estimations are crucial to analyze the potential of wind energy projects. To calculate the AEP of a wind farm, it is necessary to accurately measure the wind speed, because small errors in these measures lead to significant deviations in the wind turbine power curve. In-field wind speed is usually measured by means of cup anemometers, which are calibrated within wind tunnels. In-field ambient conditions differ from those at the laboratory, which increases the uncertainty of the wind speed measures performed at the location of the turbine. The present work is focused on analyzing the effect of the following ambient parameters on the cup anemometer behavior: temperature, humidity and atmospheric pressure. In order to reach this target, experimental tests in a wind tunnel were performed, which allows minimizing the effect of the rest of influence parameters: turbulence, average flow inclination angle and flow direction. With this work it is determined how flow air viscous forces affect the cup anemometers, changing its rotation frequency. This explanation concludes that a variation on air temperature, humidity and/or pressure modifies moist air kinematic viscosity, which leads to change the friction between air and cups and, consequently, the cup anemometers rotational frequency. In most cases, the kinematic viscosity is inversely proportional to the air density and therefore, higher in-field densities, compared to those at the laboratory where the anemometer was calibrated, lead to underestimate the wind speed, and vice versa. The fact that this effect has been quantified during the calibration process is quite important, since it allows removing the influence of the environmental parameters studied; so that by modifying the calibration methodology, the accuracy of cup anemometers would be optimized. In order to clarify how the moist air kinematic viscosity influences the calibration curve, the calibration measurements of a real cup anemometer are mapped into a new dimensionless abacus, with the Tip Speed Ratio (TSR) and the Reynold’s number as coordinates. The key idea is that the rotation frequency of the cup anemometer is related to both wind speed and moist air kinematic viscosity. This relation is mathematically described by the equation of a hyperbolic paraboloid surface up to a value of wind speed of 15.3 m/s.

Analysis of an experimental tension structure

This article builds on the article written last year. The paper is the comparison between the results from the first numerical pre-analysis of the construction of an adaptive hyperbolic paraboloid, the results of which were used as input data for the design of the experimental device in the laboratory at the Faculty of Civil Engineering of the Technical University in Košice with the newest results gained from the numerical model created according to documentation from the manufacturer of this device. The structural numerical model was modeled in regard to geometry, materials and cross-sections characteristics, weight distribution, existing eccentricity and it was created in Dlubal software based on finite element method.

Complex failure analysis of a cable-roofed stadium structure based on diagnostics and tests

Results of comprehensive failure-related diagnostics, non-destructive and destructive tests, geodetic surveying, numerical modelling, time-dependent nonlinear static analysis and reliability assessment of a stadium structure with a suspension cable roof in the form of hyperbolic paraboloid cable net after fifty years of operation are presented in the paper. The main goal is the determination of the current degree of damage and resistance of the structure. Results of the metallographic analysis confirmed that the upper layers of the steel wires of carrying and stabilizing cables which were in contact with condensed moisture in the areas of the roof located above the ice surface are damaged by atmospheric corrosion. Current resistances of the diminished cable members obtained by tension tests and their creep constitutive equations are characterized. The geodetic surveying of the selected nodes of the cable net confirmed the significant change of the initial geometry of the cable structure subjected to corrosion and creep over a period of fifty years. Results obtained by time-dependent nonlinear finite element analysis considering rheological phenomena (shrinkage, creep and degradation of mechanical properties) affecting concrete arches and reliability assessment of the cable net with current properties showed that the reliability conditions are not fulfilled, preventing the cables from safely performing its function and consequently requiring the replacement of the cable net with a new cable structure exhibiting the required properties.

Integrated Parametric Shaping of Curvilinear Steel Bar Structures of Canopy Roofs

Shaping building objects is conditioned by many interrelated factors, both architectural and structural. Modern tools for shaping structures working in the environment of Rhinoceros 3D such as Grasshopper and Karamba 3D enable algorithmic-aided shaping structures, while allowing the free flow of information between the geometric model and structural model. The aim of the research is to use these tools to test the curvilinear steel bar roofs’ structures shaped based on Catalan surfaces as well as to select the most efficient structure. Three types of roof structures were analyzed: cylindroid shape, conoid shape, and hyperbolic paraboloid shape. In order to find the most preferred structural form, evolutionary structural optimization was carried out, which allowed, among others, to determine optimal discretization of the base surface, as well as optimal positions of supporting columns. As the optimization criterion, the minimum mass of the structure was assumed. The most effective structure turned out to be a structure based on hyperbolic paraboloid supported by multi-branch columns. The use of a roof with the above structure is beneficial not only because of the low weight of the structure compared to the analyzed structures, but also due to the possibility of using flat panels on the roof.

Prototyping of thin shell wind tunnel models to facilitate experimental wind load analysis on curved canopy structures

The topologies of membrane and shell structures are not covered by existing wind load Standards and wind tunnel testing should be used to obtain representative wind loads for these structures. However, accurate scale-models of these organically shaped and often open thin structures are complex, time-consuming and expensive to build. To stimulate experimental research on wind load distributions over these structures, this paper illustrates a prototyping methodology for double curved thin shell wind tunnel models with integrated pressure sensors. The production process is illustrated for a hyperbolic paraboloid roof structure. The obtained wind load distributions are validated with literature for a flat roof and canopy that is made according to the same methodology and for two hyperbolic paraboloid roofs. Results indicate that, compared to conventional wind tunnel models, these thin shell wind tunnel models yield more realistic wind pressure distributions over very thin canopy structures. Finally, Cp-distributions are shown for the hyperbolic paraboloid canopy with the high corner under attack. The production of glass-fibre reinforced composites in a CNC-milled mould is convenient and accurate and facilitates the production of wind tunnel models to be used for wind load measurements on organically shaped thin canopy structures.

Characteristics of vibrating composite stiffened hypars with cut-out at different modes

Vibration characteristics of laminated composite stiffened hypar (hyperbolic paraboloid shell bounded by straight edges) with cut-out are analysed in terms of natural frequency and mode shapes. A finite element code is developed for the purpose by combining an eight noded curved shell element with a three noded curved beam element for stiffener. Finite element formulation is based on first order shear deformation theory and includes the effect of cross curvature. The isoparametric shell element used in the present model consists of eight nodes with five degrees of freedom per node while beam element has four degrees of freedom per node. The code is validated by solving benchmark problems available in the literature and comparing the results. The generalised Eigen value solution is chosen for the un-damped free vibration analysis. New results are presented for first five modes of natural frequency by varying boundary conditions, ply orientation and curvature of the shell. The results furnished here may be readily used by practicing engineers dealing with stiffened composite hypars with cut-outs.

Effects of porosity and thermomechanical loading on free vibration and nonlinear dynamic response of functionally graded sandwich shells with double curvature

Resume The fundamental frequencies and nonlinear dynamic responses of functionally graded sandwich shells with double curvature under the influence of thermomechanical loadings and porosities are investigated in this study. Two material models are considered. The continuity requirement of material properties throughout layers are fulfilled by newly introducing refined effects of two porosity types regarding the average of constituent properties weighted by the porosity volume fraction. The first-order shear deformation theory taking the out-of-plane shear deformation into account is employed to obtain the Lagrange equation of motions. The number of primary variables reduces from five to three after introducing the Airy stress function. The system of dynamic governing equations is obtained by utilizing the Bubnov–Galerkin procedure. The natural frequencies are analytically computed by solving eigenvalue problems, and the fundamental frequencies are acquired by further assumptions about the inertial force caused by the shell rotation variables. The nonlinear dynamic responses of the functionally graded spherical, cylindrical, and hyperbolic paraboloid shells under the influence of different geometry configurations, loading conditions, and porosity types and degrees are obtained by applying the fourth-order Runge–Kutta method. The numerical results are presented and verified with available studies in the literature. Although porosities are usually considered material defects weakening the structure performance, this study has proved clearly that porosities stiffen the shell structures to some extent.

Geometry and Construction Optimization: An example using Felix Candela’s Church of St. Joseph the Craftsman in Mexico

This paper presents a contemporary architectural working method that encompasses digitalization and parameterization of existing buildings and optimization of new buildings designed with ruled surfaces. The method uses parametric modeling and computational structural analysis in order to simplify contemporary building processes. As an example of the application of these techniques, in this paper they are applied to Felix Candela’s Church of St. Joseph the Craftsman, a design which features hyperbolic paraboloids that are considered difficult to design, calculate and build. The optimization method introduced in this paper seeks to explore different possibilities for designing and modifying buildings designed using non-standard geometry allowing them to be built out of simplified elements but also keep construction and visual properties of their shape. This method is also useful for students and young engineers to expand their skills in structural analysis, parametric modeling and optimization methods with contemporary tools.

Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 3

The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.