6 Rosemarie Wagner The tension stress in the surface is isotropic and homogeneous by meaning the stresses are at each point and in each direction constant and this is named as hy- drostatic state of stress.The stress can be set as a constant value and reduces the description of shapes of equilibrium to the geometrical problem searching for the minimal surface by given boundary conditions.Physical models of minimal surfaces are soap films,in earlier times one of the few methods describing double curved surfaces which are at each point under tension. Soap film [14] Numerical solution of the minimal surface [15] Fig.6.Minimal surfaces as soap film and the numerical solution 3.2 Cutting Pattern The shapes of equilibrium are characterized by no material behaviour or by the material behaviour of soap films without shear resistance.The real shape of the ten- sioned structures is influenced by the material behaviour and the difference between the shape of equilibrium and the materialized,pretensioned shape resulting in the non existing shear stiffness of a cable net,the orthotropic behaviour of coated fabric or the relatively high shear stiffness of foils.Known from the globe is the fact that double curved surfaces cannot be flattened without distortion.Furthermore the fab- ric is manufactured in width up to max.5 m and this requires the assembling of the whole cover with patches or strips of a certain length and width.The common way of generating the cutting pattern from the shape of equilibrium is described in four steps.The shape of equilibrium is cut into strips mostly using geodesic lines for the cutting lines.The whole structure is then divided into double curved strips.These strips are flattened with different methods such a paper strip method or minimiz- ing the strain energy while flattening the strips.The compensation as final step is necessary to introduce the tension forces by elongation of the fabric.All strips have to be decreased in width and length in relation to the stress and strain behaviour of the fabric in the built structure. Differences in geometry and stresses between the shape of equilibrium and the built structure are caused by the orientation of the fabric,the shear deformation of the fabric,the stiffness of the seams und the process of pretension.Reducing the mistakes in the cutting pattern which can be seen in wrinkles and can be measured in local stress peaks is possible by taking into account the jamming condition of the coated fabric.The load carrying compounds in a fabric are the yarns which are protected by the coating.In a woven fabric warp and fill will kept in place if the tension stress acts in direction of the yarns.Shear forces lead to a rotation of warp and fill against each other up to an angle when the yarns touch each other.The
6 Rosemarie Wagner The tension stress in the surface is isotropic and homogeneous by meaning the stresses are at each point and in each direction constant and this is named as hydrostatic state of stress. The stress can be set as a constant value and reduces the description of shapes of equilibrium to the geometrical problem searching for the minimal surface by given boundary conditions. Physical models of minimal surfaces are soap films, in earlier times one of the few methods describing double curved surfaces which are at each point under tension. Soap film [14] Numerical solution of the minimal surface [15] Fig. 6. Minimal surfaces as soap film and the numerical solution 3.2 Cutting Pattern The shapes of equilibrium are characterized by no material behaviour or by the material behaviour of soap films without shear resistance. The real shape of the tensioned structures is influenced by the material behaviour and the difference between the shape of equilibrium and the materialized, pretensioned shape resulting in the non existing shear stiffness of a cable net, the orthotropic behaviour of coated fabric or the relatively high shear stiffness of foils. Known from the globe is the fact that double curved surfaces cannot be flattened without distortion. Furthermore the fabric is manufactured in width up to max. 5 m and this requires the assembling of the whole cover with patches or strips of a certain length and width. The common way of generating the cutting pattern from the shape of equilibrium is described in four steps. The shape of equilibrium is cut into strips mostly using geodesic lines for the cutting lines. The whole structure is then divided into double curved strips. These strips are flattened with different methods such a paper strip method or minimizing the strain energy while flattening the strips. The compensation as final step is necessary to introduce the tension forces by elongation of the fabric. All strips have to be decreased in width and length in relation to the stress and strain behaviour of the fabric in the built structure. Differences in geometry and stresses between the shape of equilibrium and the built structure are caused by the orientation of the fabric, the shear deformation of the fabric, the stiffness of the seams und the process of pretension. Reducing the mistakes in the cutting pattern which can be seen in wrinkles and can be measured in local stress peaks is possible by taking into account the jamming condition of the coated fabric. The load carrying compounds in a fabric are the yarns which are protected by the coating. In a woven fabric warp and fill will kept in place if the tension stress acts in direction of the yarns. Shear forces lead to a rotation of warp and fill against each other up to an angle when the yarns touch each other. The
On the Design Process of Tensile Structures Dividing the surface by Separating the strips Flattening of the strips compensation geodesic lines along the geodesic lines Fig.7.Generation of cutting pattern [16 KIIIIININIIIN fI EITI I1I1I1101I1E1110181 DININIOINIIIOIIIIIOI HINENIN TEI IIE INI I I 0OD0O000011000110000 Fig.8.Shear deformation of woven fabric [17] maximum shear deformation is depended by the thickness of the yarns,the distance of the yarns and the flexibility of the coating.If the rotation of the yarns is larger than the required distortion to flatten the doubled curved strips the flattening is only a process of strainless deformation. If the process is invert and still definite needs further examination because the manufacturing of membrane structures is from the flat and assembled strips into the double curved and pretensioned structure.Already known is the shear deformation which is used to build double curved surfaces with cable nets.The cable net can be put onto the doubled curved surface just by changing the angles between the cables; the distance between the nodes is kept as constant.The rotation of the two layers of cables against each other is related to the curvatures of the surface. Plane net with square meshes Plane and double curved net Double curved net Fig.9.Shear deformation from the plane into the double curved net
On the Design Process of Tensile Structures 7 Fig. 7. Generation of cutting pattern [16] Fig. 8. Shear deformation of woven fabric [17] maximum shear deformation is depended by the thickness of the yarns, the distance of the yarns and the flexibility of the coating. If the rotation of the yarns is larger than the required distortion to flatten the doubled curved strips the flattening is only a process of strainless deformation. If the process is invert and still definite needs further examination because the manufacturing of membrane structures is from the flat and assembled strips into the double curved and pretensioned structure. Already known is the shear deformation which is used to build double curved surfaces with cable nets. The cable net can be put onto the doubled curved surface just by changing the angles between the cables; the distance between the nodes is kept as constant. The rotation of the two layers of cables against each other is related to the curvatures of the surface. Fig. 9. Shear deformation from the plane into the double curved net Dividing the surface by geodesic lines Separating the strips along the geodesic lines Flattening of the strips compensation Plane net with square meshes Plane and double curved net Double curved net