This paper presents a stress analysis of a membrane modelling the 32-panel soccer ball. The most popular soccer ball type and three variations are considered. The discretized mesh of the stress-free polyhedron-shaped membrane is subjected to internal pressure and the sphericity and the stress distribution of the models are compared

Hincz, Krisztián

Lengyel, András

English

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530VI International Conference on Textile Composites and Inflatable StructuresSTRUCTURAL MEMBRANES 2013K.-U.Bletzinger, B. Kröplin and E. Oñate (Eds)STRESS ANALYSIS OF INFLATED POLYHEDRA FOR THE32-PANEL SOCCER BALLANDRÁS LENGYEL∗ AND KRISZTIÁN HINCZ†∗Department of Structural MechanicsBudapest University of Technology and EconomicsMűegyetem rkp. 3, H-1521 Budapest, Hungarye-mail: lengyel@eik.bme.hu, web page: https://www.me.bme.hu/en† Department of Structural MechanicsBudapest University of Technology and EconomicsMűegyetem rkp. 3, H-1521 Budapest, Hungarye-mail: hincz@eik.bme.hu, web page: https://www.me.bme.hu/enKey words: Soccer ball, polyhedron, inflation of membrane, dynamic relaxationAbstract. This paper presents a stress analysis of a membrane modelling the 32-panelsoccer ball. The most popular soccer ball type and three variations are considered. Thediscretized mesh of the stress-free polyhedron-shaped membrane is subjected to internalpressure and the sphericity and the stress distribution of the models are compared.1 INTRODUCTIONIn spite of that soccer has been one of the most influential sports games for decades,scientific analysis on the geometry of the soccer ball has received significantly less atten-tion. Since its origin in the 19th century the history of modern soccer witnessed a vastvariety of balls shapes. They mostly consisted of flat panels of leather sewn together andinflated by means of a bladder. The 32-panel icosahedral ball type has become the mostwidely used model since the 1970’s and is still one of the models sold in the largest num-bers worldwide. Though materials used for fabrication turned from leather to synthetics,its composition remained an assembly of twelve pentagonal and twenty hexagonal panelsrepresenting one of the Archimedean solids, the truncated icosahedron. It served as theofficial ball for several world championships.A recent approach in soccer ball design introduced curved panels thermally bonded inorder to create a more precise approximation of the sphere (e.g. Teamgeist or Jubilani),however, the 32-panel ball has remained the most popular ball.Early attempts to create a round ball of flat pieces based on heuristic constructionslacking probably any scientific foundations. As the game gained increasing importance1Stress Analysis of Inflated Polyhedra for the 32-panel Soccer Ball531András Lengyel and Krisztián Hinczball shapes based on regular and semiregular polyhedra were created requiring geometricalbackground for the analysis. There have been a number of approaches for the definitonof the roundness of spatial solids from different fields of science. Among them the mostwidely used term considered in mathematics is the so-called isoperimetric quotient (IQ),introduced by Polya [1] in its present form:IQ = 36πV 2A3(1)where V and A denote the volume and the surface area of the body, respectively. Thescaling factor 36π is introduced so that IQ yields 1 for the unit sphere. A central problemin mathematics of spatial bodies is the isoperimetric problem, which states: among a setof bodies of given surface area, which one encloses the largest volume (largest IQ), i.e.which one approximates the sphere the most closely?In a previous study the authors of this paper addressed the issue of roundness of the32-panel icosahedral soccer ball [2], using a set of geometrical properties to quantify theroundness of a spatial solid introduced previously by the first author [3]. This paperconsiders four different types of 32-panel soccer ball geometry, all commercially available:the truncated icosahedron, which is an Archimedean solid, the Geo 0.84 model by Nike,the equal panel area ball by Puma, and the HyperballR© by Prof. P. Huybers [4]. Ouraim is to simulate the inflation of a polyhedron-shaped membrane representing the soccerball. Discretized geometric models of all types are created and subjected to internalpressure. The key parameters of the analysis are the density of the finite mesh, theapplied pressure, and the material stiffness of the membrane. The roundness of theinflated shapes are evaluated and compared. The next three sections outline the modelfor the computations, the most important results, and the major conclusions.2 MODELLING2.1 Geometrical modelThe initial (undeformed) shape of the ball is defined by the truncated icosahedronor its variations. The first model is the Archimedean truncation of the icosahedronrepresenting the most popular ball configuration. The Archimedean truncation yieldsregular pentagons and hexagons, i.e. all edges are of the same length. The second modelis Geo 0.84 by Nike where the ratio of edges between hexagons and edges of the pentagonis approximately 0.839. This configuration is obtained by a nonregular truncation of theicosahedron resulting in larger pentagons. It is claimed that such truncation results inthe optimal stress distribution in the inflated structure. The third model (Puma) is atruncation of the icosahedron in a way that it yields pentagons and hexagons of equalsurface area. The fourth model is not purely a truncation of the icosahedron but a moresophisticated form even though it is made of 32 panels. The geometry of the Hyperballis derived from the isodistant truncation of the icosahedron (i.e. when all panels are at2532András Lengyel and Krisztián HinczFigure 1: Triangular mesh models for four ball types: (left to right) the truncated icosahedron(Archimedean), Nike Geo 0.84, Puma equal face area ball, Hyperball.the same distance from the centre of the inscribed sphere). This polyhedron is modifiedby further truncation of the common edges of hexagons forming rectangular faces. Bothtruncations are defined in an isodistant way meant to produce the roundest shape. Thekey feature of this model is the reduction of the large number of faces by dividing therectangles to triangles and trapezoids, which then can be joined with the neighbouringpentagons and hexagons. (A detailed description is given in [4].) Sketches of the fourmodels are given in Figure 1.In order to have a realistic approximation of the sphere, all polyhedron faces are dividedinto a number of triangles. The triangulation has been performed in a way to preservethe icosahedral symmetry of the surface. All pentagons, hexagons, and rectangles arefirst divided into five, six, and four equal triangles, respectively by drawing radial lines tothe vertices from the centroids of the faces, which are then further divided into smallertriangles by placing points along the edges at equal distances. In the case of the fourthmodel, the ratio of the sides of some of the triangles were inconveniently large, thereforean alternative triangulation has been applied in parts of the domain, also preservingsymmetry.2.2 Structural modelThe surface of the polyhedron is regarded as a thin membrane of isotropic materialcharacterized by its in-plane stiffness and Poisson’s ratio. The triangular elements maydevelop stresses and strains in the plane determined by the position of their vertices andfree rotational displacements may occur at the edges. Internal pressure inside the closedmembrane is represented by a distributed load of equal intensity on all elements.We apply the dinamic relaxation method (DRM) [5] to determine the equilibriumshape of the structure. It is an iterative technique widely used for tensile structures.Fictitious lumped masses are assigned to the vertices of the mesh. In each time step ofthe iteration, the position, the velocity, and the acceleration of each vertex is computed,and the equations of motions are numerically solved. The final deformed shape is obtainedwhen the vertices are balanced and have zero velocity.3533András Lengyel and Krisztián HinczIsoperimetric quotient IQ0,99750,99800,99850,99900,99951,00004 5 6 7 8 9 10 11 12 13 14 15 16NT1T2T3T4 Figure 2: Isoperimeric quotient IQ for the four ball types (1: the truncated icosahedron (Archimedean),2: Nike Geo 0.84, 3: Puma equal face area ball, 4: Hyperball) against mesh density parameter N .3 RESULTSAll initial (uninflated) models are scaled to have volume equal to that of a 100 mmradius sphere. Once DRM yields the final equilibrium positions of all nodes in the mesh,geometrical and mechanical evaluation of the deformed shape is performed. The isoperi-metric quotient IQ is computed via the surface area A and volume V . Values of IQ showconvergence with respect to the density of the mesh. The density is characterized by thenumber N of the sections along the edges of any face of the polyhedra. (E.g. N = 3 isapplied for the generation of shapes in Figure 1.) It is found in a series of test compu-tations that IQ does not vary significantly for N > 12 (see Figure 2) therefore densityN = 12 has been applied for further analysis.The best isoperimetric quotient has been obtained for the standard model and for Geo0.84, followed by the equal face area model, and the Hyperball.The stiffness of the membrane and the applied pressure have been set to obtain realisticinflated shapes. FIFA Inspected quality for soccer balls requires balls to have circumfer-ence 68 to 70 cm among various specifications. We set normal stiffness S = 40 kN/m,Poisson’s ratio ν = 0.2, and internal pressure p = 1.0 atm for the analysis.Stresses for all four models have been computed, see Figures 3 to 6. (In this paper theterm ‘stresses’ refer to specific forces in sections of the membrane in units of force/length ascustomary for tensile structures and not in units of force/area as in continuum mechanics.)4534András Lengyel and Krisztián HinczFigure 3: Distribution of maximum (left) and minimum (right) principal stresses in the membrane formodel 1. Blue and red colours denote high and low stress levels, respectively.Figure 4: Distribution of maximum (left) and minimum (right) principal stresses in the membrane formodel 2. Blue and red colours denote high and low stress levels, respectively.Figure 3 shows the distribution of principal stresses for model 1. The maximum principalstresses (left) are large in the middle of the hexagonal panels and slightly smaller inthe pentagonal faces. Strips of large stresses connect the centres of neighbouring facesindicating the directions of characteristic stretching in the membrane. Local low stressregions are formed around the vertices of the panels. Minimum principal stresses (right)range over a wider interval. Peaks are measured in the middle of the hexagonal panelswhereas peaks in the pentagonal faces are slightly smaller. Stresses along the edges areapproximately uniform with medium intensity, and the smallest values are again obtainedaround the vertices. It indicates that these regions are moderately affected by inflation.5535András Lengyel and Krisztián HinczFigure 5: Distribution of maximum (left) and minimum (right) principal stresses in the membrane formodel 3. Blue and red colours denote high and low stress levels, respectively.Stresses of models 2 (see Figure 4) and 3 (see Figure 5) are characterisically similar tothose of model 1. The pentagonal faces remain regular but the symmetry of the hexagonalfaces reduces to D3. With larger pentagons, the ratio of the long and the short edges ofhexagons increases. Distribution of stresses corresponds to the symmetry of the shapeof the structure. It is found that peaks of the maximum principal stresses occur in thestrips perpendicular to long edges while stresses at the centres of panels and in strips nearshort edges decrease. Distribution of minimum principal stresses varies moderately. Asthe size of the pentagonal faces increase relative to that of the hexagonal ones, the peaksin the middle of all faces increase or decrease with the surface area, accordingly, comparemodels 1 to 3.The layout of model 4 (see Figure 6) is characteristically different from the other threeregarding the number of faces and the ratio of edges. Three sides of each hexagons aresignificantly smaller than the rest resulting in a significantly different stress distribution.Regions around the vertices are joined in pairs, and the hexagonal patterns become prac-tically triangular. Largest values of the maximum principal stress occur in the hexagonaland the rectangular faces forming a contiguous region of nearly uniform stresses sur-rounding isolated pentagonal parts of smaller stress. It indicates that the loadbearing isprimarily provided by the frame of hexagons and rectangles. (In the actual fabricationrectangles are divided and joined with the neighbouring faces.) The minimum principalstresses are more evenly distributed than in the previous cases. Smallest values againoccur around the vertices.4 CONCLUSIONSMaximum and minimum stresses as well as isoperimetric quotients for all inflated mod-els are summarized in Table 1. Ranking of the models with respect to IQ in decreasing6536András Lengyel and Krisztián HinczFigure 6: Distribution of maximum (left) and minimum (right) principal stresses in the membrane formodel 4. Blue and red colours denote high and low stress levels, respectively.order is Nike Geo 0.84, the standard truncated icosahedron, the Puma equal face areaball, and the Hyperball. It is found that the largest σ1 stresses are practically in inversecorrespondence with roundness: σ1,max decreases with IQ increasing except that Hyper-ball has rank 3. It suggests that lower peak stresses are expected with better roundness.Also, with increasing IQ, the smallest σ1 values in the membrane increase closing the gapbetween maximum and minimum, i.e. more evenly distributed stress field is expected.Principal stresses σ2 vary more, though it can also be concluded that among the trun-cated icosahedral models the gap between the maximum and minimum decreases with IQincreasing. It is interesting to note that Hyperball performs significantly better in thisrespect, which can be attributed to the double truncation to smoothen the peaks of thepolyhedron. Overall, Geo 0.84 and the standard ball produce the best roundness valuesand are the most efficient in reducing the largest stresses and their deviation. The specialgeometric configuration of the Hyperball proves capable of involving vertex regions intothe loadbearing.ACKNOWLEDGEMENTSThe research reported here was partially supported by OTKA grant no. K81146 awardedby the Hungarian Scientific Research Fund. This paper was also supported by the JánosBolyai Research Scholarship of the Hungarian Academy of Sciences.REFERENCES[1] Pólya, G. Mathematics and Plausible Reasoning Vol. I. Induction and Analogy inMathematics, Princeton University Press, 1954.[2] Lengyel, A. and Hincz K. Optimization of the roundness of the soccer ball. In:7537András Lengyel and Krisztián HinczTable 1: Comparison of four ball models with respect to the largest and the smallest values of principalstresses σ1 and σ2, and the isoperimetric quotient IQ.Principal stresses (kN/m)Model σ1,max σ1,min σ2,max σ2,min IQ1 Standard TI 6.350 4.769 6.082 3.310 0.9997107262 Nike Geo 0.84 6.279 4.819 6.032 3.274 0.9997255903 Puma equal face area 6.399 4.666 6.107 3.109 0.9996396434 Hyperball 6.366 4.419 5.943 3.627 0.999519980Domingo, A, Lázaro, C, (eds) Evolution and Trends in Design, Analysis and Con-struction of Shell and Spatial Structures: Proceedings of the IASS Symposium 2009,Valencia. Valencia, Spain, 2009.09.28-2009.10.02. pp. 1100-1110.[3] Geiger, A. and Lengyel, A. On the roundness of polyhedra for soccer ball design. IntJ Space Struct 23:(3) pp. 133-142. (2008)[4] Huybers, P. A new kind of football with improved roundness, International PatentApplication PCT/NL2008/002421, dated 17 March 2008.[5] Day, A.S. An introduction to dynamic relaxation. The Engineer (1965) 219: 218-221.8

Stress analysis of inflated polyhedra for the 32-panel soccer ball

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Stress analysis of inflated polyhedra for the 32-panel soccer ball

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