On the Design Process of Tensile Structures Rosemarie Wagnerl Fachhochschule Miinchen Fachbereich Architektur Karlstrasse 6,D-80333 Miinchen,Germany R.Wagner@fhm.edu Web page:http://www.lrz-muenchen.de/~architektur Summary.The influence of the development of computer programmes and auto- matic generation of cable nets and membrane structures will be shown in some er- amples.The main interest is laying on new evaluation methods of cable nets and membrane structure and the design process of membrane structures,integrating the material behaviour of coated fabric. Key words:Design process,cable nets,membrane structures,inflated structures 1 Introduction The design process of pretensioned structures such as cable nets and membrane structures is influenced by the development of computational methods.While the first methods of form finding had been physical modelling with fabric,wire nets or soap films,today several numerical methods of form finding are developed based on the force density method [1,2],the principle of minimal surfaces [3,4]using dynamic relaxation [5,6]or other approaches in fulfilling the three-dimensional equilibrium Further process has been carried out in the form finding with an anisotropic stress distribution [7].All methods have in common that no material laws are necessary finding an equilibrium of the three dimensional shape for given stress distributions, boundary conditions and supports.These shapes of equilibrium should ensure in the built structure a homogeneous distribution of the tension stresses.In reality the material behaviour,process of cutting patterns,manufacturing and pretensioning on site influencing the stress distribution,wrinkles and regions of over stress are obvious.can be seen and measured. The design process of pretensioned structures needed to be extended taking into account evaluation methods for shapes of equilibrium in relation the mate- rial behaviour and process of prestensioning.More realistic modelling of membrane structures is necessary including the strips in width,orientation of the fabric and seams for analysing the load charring behaviour.The process of reassembling flat- ten strips had already been proposed for a rotational symmetric hat type tent [8]. The process of form finding can be embedded in a design process including cutting pattern und structural behaviour under external loads.The load bearing behaviour 1 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,1-16. 2005 Springer.Printed in the Netherlands
On the Design Process of Tensile Structures Rosemarie Wagner1 Fachhochschule M¨unchen ¨ Fachbereich Architektur Karlstrasse 6, D-80333 Munchen, Germany ¨ R.Wagner@fhm.edu Web page: http://www.lrz-muenchen.de/∼architektur Summary. The influence of the development of computer programmes and automatic generation of cable nets and membrane structures will be shown in some examples. The main interest is laying on new evaluation methods of cable nets and membrane structure and the design process of membrane structures, integrating the material behaviour of coated fabric. Key words: Design process, cable nets, membrane structures, inflated structures 1 Introduction The design process of pretensioned structures such as cable nets and membrane structures is influenced by the development of computational methods. While the first methods of form finding had been physical modelling with fabric, wire nets or soap films, today several numerical methods of form finding are developed based on the force density method [1,2], the principle of minimal surfaces [3,4] using dynamic relaxation [5,6] or other approaches in fulfilling the three-dimensional equilibrium. Further process has been carried out in the form finding with an anisotropic stress distribution [7]. All methods have in common that no material laws are necessary finding an equilibrium of the three dimensional shape for given stress distributions, boundary conditions and supports. These shapes of equilibrium should ensure in the built structure a homogeneous distribution of the tension stresses. In reality the material behaviour, process of cutting patterns, manufacturing and pretensioning on site influencing the stress distribution, wrinkles and regions of over stress are obvious, can be seen and measured. The design process of pretensioned structures needed to be extended taking into account evaluation methods for shapes of equilibrium in relation the material behaviour and process of prestensioning. More realistic modelling of membrane structures is necessary including the strips in width, orientation of the fabric and seams for analysing the load charring behaviour. The process of reassembling flatten strips had already been proposed for a rotational symmetric hat type tent [8]. The process of form finding can be embedded in a design process including cutting pattern und structural behaviour under external loads. The load bearing behaviour 1 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 1–16. © 2005 Springer. Printed in the Netherlands
2 Rosemarie Wagner can be evaluated by redundancy,flexibility or a stiffness value in relation to the curvature and the elastic strain of the materials. 2 State of the Art The design process of tension structures such as double curved cable nets or mem- brane structures such as tents,air support halls or airships can be divided into form finding,static analysis and cutting pattern.The result of the form finding is a shape of equilibrium for a certain stress distribution and boundary conditions.The shape of equilibrium ensures the geometry of the double curved surface which has only tension and avoids compression in the surface.From this geometry the structural behaviour is exanimated and the cutting pattern is made of.The flattening of the double curved surface is a geometrical process without considering the stress dis- tribution and the material behaviour.In the recent development of cutting pattern methods the stress distribution is taking into account 9,10.The analysis of the structural behaviour is carried out without the influence of the width of the strips, the seams,the orientation of the fabric and the process of pretensioning.The separa- tion of the structural behaviour and the cutting pattern leads in built structures to highly inhomogeneous stress distributions which can be seen in wrinkles and mea- sured in stresses which are two times higher than required.The difference in the stress distribution and geometry between the numerical found shape of equilibrium and the real structure causes in the non consistent design procedure see Fig.1. Structural Behaviour Shape ofequilibrium no influence of warp and weft orientation given stress distribution no influence of the seams no material behaviour no influence of width of the strips no influence of the shear deformation of the fabric no consideration of the process of pretensioning Cutting Pattern No material behaviour No stress distribution Fig.1.Common design process of membrane structures 3 Enhanced Design Process of Tension Structures An enhanced design concept will be based on 5 design steps defining the shape of equilibrium,generating the cutting pattern,reassembling and pretensioning the cut- ting pattern,the structural analysis of the reassembled structure and the evaluation of the structural behaviour.The material behaviour is considered in the last three steps:flattening the shape of equilibrium,reassembling and load bearing behaviour. The length and the width of the strips has an influence to the shear deformation of the coated fabric.The orthotropic behaviour of coated fabric influences the process of pretension and the stress distribution in the reassembled structure.The numerical process allows after evaluation modifications to reach better results in the reassemble structures considering stress distribution and deformations
2 Rosemarie Wagner can be evaluated by redundancy, flexibility or a stiffness value in relation to the curvature and the elastic strain of the materials. 2 State of the Art The design process of tension structures such as double curved cable nets or membrane structures such as tents, air support halls or airships can be divided into form finding, static analysis and cutting pattern. The result of the form finding is a shape of equilibrium for a certain stress distribution and boundary conditions. The shape of equilibrium ensures the geometry of the double curved surface which has only tension and avoids compression in the surface. From this geometry the structural behaviour is exanimated and the cutting pattern is made of. The flattening of the double curved surface is a geometrical process without considering the stress distribution and the material behaviour. In the recent development of cutting pattern methods the stress distribution is taking into account [9,10]. The analysis of the structural behaviour is carried out without the influence of the width of the strips, the seams, the orientation of the fabric and the process of pretensioning. The separation of the structural behaviour and the cutting pattern leads in built structures to highly inhomogeneous stress distributions which can be seen in wrinkles and measured in stresses which are two times higher than required. The difference in the stress distribution and geometry between the numerical found shape of equilibrium and the real structure causes in the non consistent design procedure see Fig. 1. Fig. 1. Common design process of membrane structures 3 Enhanced Design Process of Tension Structures An enhanced design concept will be based on 5 design steps defining the shape of equilibrium, generating the cutting pattern, reassembling and pretensioning the cutting pattern, the structural analysis of the reassembled structure and the evaluation of the structural behaviour. The material behaviour is considered in the last three steps: flattening the shape of equilibrium, reassembling and load bearing behaviour. The length and the width of the strips has an influence to the shear deformation of the coated fabric. The orthotropic behaviour of coated fabric influences the process of pretension and the stress distribution in the reassembled structure. The numerical process allows after evaluation modifications to reach better results in the reassemble structures considering stress distribution and deformations. Shape of equilibrium í given stress distribution í no material behaviour Structural Behaviour í no influence of warp and weft orientation í no influence of the seams í no influence of width of the strips í no influence of the shear deformation of the fabric í no consideration of the process of pretensioning Cutting Pattern í No material behaviour í No stress distribution
On the Design Process of Tensile Structures 3 Shape of Generation of Reassembling Structural be- Evaluation equilibrium cutting pattern and preten- haviour sion Fig.2.Enhanced design process of membrane structures [11] 3.1 Shape of Equilibrium The development of Computer Aided Geometric Design (CAGD)marked the start of changes in geometry endorsing new and free forms.This generation of double curved 3-dimensional surfaces is restricted by few limitations.Theoretically there are an unlimited number of forms to be numerical generated and represented.How- ever,the manufacture and realization of such double curved surfaces are subject to numerous boundary conditions and restrictions.Using cables and membranes for the load transfer only tension forces can be carried,the cables and membranes can not withstand bending moments and compression forces in a global point of view. The structures have to be pretensioned activating the geometric stiffness or to be able carrying compression forces by reducing the pretension.The shape of equilib- rium defines a pretensioned geometry of a doubled curved surface for a cable net or a membrane structure.The relation between the tension stress,geometry and equilibrium allows three possibilities to introduce the tension into the membranes and influences the shape of equilibrium,see Fig.3. Pretension against rigi Pretension as res t o e ia Pretension as result internal o n aries tion or es ul 一u2 (soil-.fluid-or gas-)pressure p R2 S traig t ire tions osite r e ire tions Same direction of the curvature S1 is in e en ent o S2 兴-是=0 +器=p Fig.3.Relation between tension forces and curvature The pretension against rigid boundaries enables plain tension structures,ten- sion structures with single curvatures and double curved tension structures if the direction of the cables or yarns is along the evolution line of a hyper parabola.The tension forces are independent from each other in this case
On the Design Process of Tensile Structures 3 Fig. 2. Enhanced design process of membrane structures [11] 3.1 Shape of Equilibrium The development of Computer Aided Geometric Design (CAGD) marked the start of changes in geometry endorsing new and free forms. This generation of double curved 3-dimensional surfaces is restricted by few limitations. Theoretically there are an unlimited number of forms to be numerical generated and represented. However, the manufacture and realization of such double curved surfaces are subject to numerous boundary conditions and restrictions. Using cables and membranes for the load transfer only tension forces can be carried, the cables and membranes can not withstand bending moments and compression forces in a global point of view. The structures have to be pretensioned activating the geometric stiffness or to be able carrying compression forces by reducing the pretension. The shape of equilibrium defines a pretensioned geometry of a doubled curved surface for a cable net or a membrane structure. The relation between the tension stress, geometry and equilibrium allows three possibilities to introduce the tension into the membranes Pretension against rigi o n aries traig t ire tions S1 is in e en ent o S2 Pretension as res t o e ia tion or es u1 −u2 osite r e ire tions S1 R1 − S2 R2 = 0 Pretension as result internal (soil-, fluid- or gas-) pressure p Same direction of the curvature S1 R1 + S2 R2 = p Fig. 3. Relation between tension forces and curvature The pretension against rigid boundaries enables plain tension structures, tension structures with single curvatures and double curved tension structures if the direction of the cables or yarns is along the evolution line of a hyper parabola. The tension forces are independent from each other in this case. Shape of equilibrium Generation of cutting pattern Reassembling and pretension Structural behaviour Evaluation and influences the shape of equilibrium, see Fig. 3. S1 S1 S1 S1 S2 S2 S2 S2 S1 S1 S1 S1 S2 S2 S2 S2 u1 = – u2 u2 R2 R1 S1 S1 S1 S1 S2 S2 S2 p S2 1 R2 R1 p2
4 Rosemarie Wagner The tension forces in surfaces with negative Gaussian curvature result of the deviation forces at the nodes and this leads to a relation between tension forces and curvature.Fulfilling the equilibrium at each node the tension forces are related to the radius of curvature in the both directions.Equal forces in both directions require the same curvature of the cables.The ratio of tension forces and radius of curvature is constant by meaning the higher the forces the lower the curvature to ensure equilibrium. Stabilising the membranes with internal pressure leads to surfaces with positive Gaussian curvature and a dependency between the internal pressure,the tension forces and the curvature.The tension forces are directly related to the internal pressure and the lower the curvature the higher the forces. Cable nets with square meshes are cinematic systems.the thin membrane with- out bending stiffness is statically determined.In both cases the double curved surface is a result of the three dimensional equilibrium at each node for given tension forces in a cable net or at each point for a given stress distribution in a membrane.The equilibrium is fulfilled without taking into account the material behaviour and is influenced by the boundary conditions such as high points,boundary cables or rigid boundaries. For cable nets the numerical solution is based on the constant ratio of cable force and length in the first step.The ratio of cable force and length is described as force density [12]and the shape of equilibrium is calculated from a plane net with a square gird by moving the nodes in the third direction,forced by the fixed points and boundaries which don't lie in the same plane as the cable net.Depending on the change in length from the cable links in the plane into the three dimensional surface the cable forces changes,the longer the cables the higher the forces.The result is a doubled curved surface with a steady change in the forces along each cable related to the change of curvature of the surface.This method can also be exceed to cable nets which are statically indeterminate such as nets with triangle meshes because of the constant ratio of force/length.Both forces and length of the links are free parameter searching for the three dimensional equilibrium. ===omt Plane net with square meshes Shape of equilibrium and constant forces in each link change in forces Fig.4.Shape of equilibrium fulfilling vertical equilibrium Fulfilling the equilibrium in the tangential plane at each knot allows adjusting the link length and leads to constant forces in each cable.The cables are oriented along geodesic lines onto the surface and the angles are not constant at the nodes between crossing cables. In membranes the equilibrium has to be fulfilled at each point of the surface. Plane state of stress assumed shapes of equilibrium are also the result of a given stress
4 Rosemarie Wagner The tension forces in surfaces with negative Gaussian curvature result of the deviation forces at the nodes and this leads to a relation between tension forces and curvature. Fulfilling the equilibrium at each node the tension forces are related to the radius of curvature in the both directions. Equal forces in both directions require the same curvature of the cables. The ratio of tension forces and radius of curvature is constant by meaning the higher the forces the lower the curvature to ensure equilibrium. Stabilising the membranes with internal pressure leads to surfaces with positive Gaussian curvature and a dependency between the internal pressure, the tension forces and the curvature. The tension forces are directly related to the internal pressure and the lower the curvature the higher the forces. Cable nets with square meshes are cinematic systems, the thin membrane without bending stiffness is statically determined. In both cases the double curved surface is a result of the three dimensional equilibrium at each node for given tension forces in a cable net or at each point for a given stress distribution in a membrane. The equilibrium is fulfilled without taking into account the material behaviour and is influenced by the boundary conditions such as high points, boundary cables or rigid boundaries. For cable nets the numerical solution is based on the constant ratio of cable force and length in the first step. The ratio of cable force and length is described as force density [12] and the shape of equilibrium is calculated from a plane net with a square gird by moving the nodes in the third direction, forced by the fixed points and boundaries which don’t lie in the same plane as the cable net. Depending on the change in length from the cable links in the plane into the three dimensional surface the cable forces changes, the longer the cables the higher the forces. The result is a doubled curved surface with a steady change in the forces along each cable related to the change of curvature of the surface. This method can also be exceed to cable nets which are statically indeterminate such as nets with triangle meshes because of the constant ratio of force/length. Both forces and length of the links are free parameter searching for the three dimensional equilibrium. S l = Hx lx = V lz = const. Plane net with square meshes and constant forces in each link Shape of equilibrium change in forces Fig. 4. Shape of equilibrium fulfilling vertical equilibrium Fulfilling the equilibrium in the tangential plane at each knot allows adjusting the link length and leads to constant forces in each cable. The cables are oriented along geodesic lines onto the surface and the angles are not constant at the nodes between crossing cables. In membranes the equilibrium has to be fulfilled at each point of the surface. Plane state of stress assumed shapes of equilibrium are also the result of a given stress S1 l1 lx 1 Hx,1
On the Design Process of Tensile Structures 5 Plane net with square meshes Shape of equilibrium and constant forces in each link constant forces at each link Fig.5.Shape of equilibrium fulfilling vertical and tangential equilibrium distribution.In general and in covariant description [13]the equilibrium normal to the surface is written as: =0,with baa as tensor of curvature Related to main axis o11b11+g12b12+o21b21+o22b22=0 The orientation of the coordinate system in direction of the principle stresses (o12=o21=0)or principle curvature (612 =621 =0) 011b11+o22b22=0 Tension stresses in both principle directions o11 >0 und o22 >0 requires a negative Gaussian curvature,with buand bthe equilibrium normal 1 to the surface results in o122 =0 R1 R2 The equilibrium in the tangential plane of the point can be written in covariant description as 8=0 Assuming the stress is constant at a certain point leads to o=agas with gas as metric tensor Substituted (cg8)1e=0→01B98+c9g2=0 ÷0 and results in 01398=0 The metric tensor has a certain value at each point in a double curved surface; this means the deviation of the stress has to be zero.This requires a constant stress distribution also to neighboured points and describes the hydrostatic state of stress. Therefore has to be o11 =022=constant and with R2-B1 =0 and R1 =R2 R1·R2
On the Design Process of Tensile Structures 5 Fig. 5. Shape of equilibrium fulfilling vertical and tangential equilibrium distribution. In general and in covariant description [13] the equilibrium normal to the surface is written as: σαβbαβ = 0 , with bαβ as tensor of curvature Related to main axis σ11 b11 + σ12 b12 + σ21b21 + σ22 b22 = 0 The orientation of the coordinate system in direction of the principle stresses (σ12 = σ21 = 0) or principle curvature (b12 = b21 = 0) σ11b11 + σ22b22 = 0 Tension stresses in both principle directions σ11 ≥ 0 und σ22 ≥ 0 requires a negative Gaussian curvature, with b11 = 1 R1 and b22 = − 1 R2 the equilibrium normal to the surface results in σ11 R1 − σ22 R2 = 0 The equilibrium in the tangential plane of the point can be written in covariant description as σαβ |β = 0 Assuming the stress is constant at a certain point leads to σαβ = σgαβ with gαβ as metric tensor Substituted (σgαβ)|β = 0 ⇒ σ|βgαβ + σgαβ |β ⇒0 = 0 and results in σ|βgαβ = 0 The metric tensor has a certain value at each point in a double curved surface; this means the deviation of the stress has to be zero. This requires a constant stress distribution also to neighboured points and describes the hydrostatic state of stress. Therefore has to be σ11 = σ22 = constant and with σ11 R1 − σ22 R2 = 0 ⇒ R2 − R1 R1 · R2 = 0 and R1 = R2 Plane net with square meshes and constant forces in each link Shape of equilibrium constant forces at each link
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
8 Rosemarie Wagner Unknown are Strain of the fabric ,[% 11 Stress of the fabric u Fig.10.Model describing the behaviour of a woven fabric 3.3 Reassembling and Pretensioning The tension forces can only be introduced into cable nets or membranes by elastic strain of the cables and coated fabric.The numerical process of reassembly requires the description of the material behaviour in which both the change of the geometry and the elastic strain is considered.The change in geometry is for cable nets mostly the in plane shear deformation reaching the double curved surface.The change of geometry in woven fabric is related to the elongation of the yarns.The simple model is useful enough describing the behaviour of a woven fabric,developed in 1978 [18], refined and tested in 1987 [19]and finally numerical transferred in 2003 [20]. Neglecting the influence of the coating the behaviour of a woven fabric can be described by the Geometry of the fabric such as thickness and distance of the yarns(warp A1,LI and inclination mi =A1/L1,fill A2,L2 and inclination m2 A2/L2) Stress-strain-behaviour of each yarn (warp F1,e1,fill F2,E2) The change in the thickness of the fabric (y)and The equilibrium of the deviation forces at each knot The ratio of unstrained to strained length is described by: 41=1+U1and2=1+U22 With the ratio of undeformed and deformed inclination of k1 =A1/p1 and k2 A2/u2 is the elastic strain of the yarns e1-1 十亚+1=0anda-a十 +1=0 V1+m阔 √1+m话 The constrain of the distance between the yarns at the knots is k141A1+k242A2-A1-A2-YF1 k1m1 =0 V1+km函 Equilibrium of the yarn F2- k2m2 kimi 1+m元-√1+k好m =0
8 Rosemarie Wagner Fig. 10. Model describing the behaviour of a woven fabric 3.3 Reassembling and Pretensioning The tension forces can only be introduced into cable nets or membranes by elastic strain of the cables and coated fabric. The numerical process of reassembly requires the description of the material behaviour in which both the change of the geometry and the elastic strain is considered. The change in geometry is for cable nets mostly the in plane shear deformation reaching the double curved surface. The change of geometry in woven fabric is related to the elongation of the yarns. The simple model is useful enough describing the behaviour of a woven fabric, developed in 1978 [18], refined and tested in 1987 [19] and finally numerical transferred in 2003 [20]. Neglecting the influence of the coating the behaviour of a woven fabric can be described by the - Geometry of the fabric such as thickness and distance of the yarns (warp A1, L1 and inclination m1 = A1/L1, fill A2,L2 and inclination m2 = A2/L2) - Stress-strain-behaviour of each yarn (warp F1, ε1, fill F2, ε2 ) - The change in the thickness of the fabric (γ) and - The equilibrium of the deviation forces at each knot The ratio of unstrained to strained length is described by: µ1 = 1 + U11 and µ2 = 1 + U22 With the ratio of undeformed and deformed inclination of k1 = A1/µ1 and k2 = A2/µ2 is the elastic strain of the yarns ε1 − µ1 1 + k2 1m2 1 1 + m2 1 + 1 = 0 and ε2 − µ2 1 + k2 2m2 2 1 + m2 2 + 1 = 0 The constrain of the distance between the yarns at the knots is k1µ1A1 + k2µ2A2 − A1 − A2 − γF1 k1m1 1 + k2 1m2 1 = 0 Equilibrium of the yarn F2 k2m2 1 + k2 2m2 2 − F1 k1m1 1 + k2 1m2 1 = 0 Unknown are Strain of the fabric U U 11 22 [%] Stress of the fabric V11 22 V V V U U L L A A F F H H J
On the Design Process of Tensile Structures 9 E1111 E222 E1122 C Fig.11.Young's Moduli and Poisson ratio as function of the fabric strain,PVC coated fabric [19] This set of 4 equations serves a non linear system of equations for the four unknown values e1,e2,kI,kz.After solving the equations the stresses of the fabric can be defined directly by 711 and √1+km 022= 1+m The calculated strains and stresses enable to define the stiffness E1111,E2222 and E1122.The elastic stiffness are non linear and closely related to the strain ratio in warp and fill direction.Even the Poisson ratio E1122 is non linear and depending to the strain ratio of the yarns. The numerical process of reassembling enables taking into account the behaviour of the fabric,the influence of the seams and the distribution of the tension stress through the whole surface.The plane strips have to be remeshed,sewed together and pretensioned by moving the sewed structure into defined boundaries,moving support points into their position after reassembling or putting internal pressure onto the system.The stress distribution and geometry of the sewed and pretensioned structure is different from the assumed stress distribution of the shape of equilibrium. The differences are depending on the curvature of the surface,the orientation of the strips in relation to the main curvature,the torsion of the strips,the distortion of the load transfer along the seams,the stiffness of the seams,the assumed compensation of the flatten strips,the width of the strips,the of the surface,the shear deformation of yarns and in the shown example of the load transfer between the boundary cables and fabric,see Fig.12. In the shown example the stress distributions varies in a single strip and changes from strip to strip.Relatively low tension stress in the middle strip can been see as result of less compensation.The influence of the stiffness of the seams can be shown in the difference between deformation in vertical direction comparing the geometry of the shape of equilibrium and reassembled and pretensioned structure.For the shown example the difference is app.20%of the span.The antimetric deformation is caused by the inhomogeneous stress distribution in the cross section along the high points.The tension stress perpendicular are unsteady,low stress leads to high vertical deformations and high stress kept the fabric down which can clearly seen in the up and down of the differences
On the Design Process of Tensile Structures 9 Fig. 11. Young’s Moduli and Poisson ratio as function of the fabric strain, PVC coated fabric [19] This set of 4 equations serves a non linear system of equations for the four unknown values ε1, ε2, k1, k2. After solving the equations the stresses of the fabric can be defined directly by σ11 = 1 L2 F1 1 + k2 1m2 1 and σ22 = 1 L1 F2 1 + k2 2m2 2 The calculated strains and stresses enable to define the stiffness E1111, E2222 and E1122. The elastic stiffness are non linear and closely related to the strain ratio in warp and fill direction. Even the Poisson ratio E1122 is non linear and depending to the strain ratio of the yarns. The numerical process of reassembling enables taking into account the behaviour of the fabric, the influence of the seams and the distribution of the tension stress through the whole surface. The plane strips have to be remeshed, sewed together and pretensioned by moving the sewed structure into defined boundaries, moving support points into their position after reassembling or putting internal pressure onto the system. The stress distribution and geometry of the sewed and pretensioned structure is different from the assumed stress distribution of the shape of equilibrium. The differences are depending on the curvature of the surface, the orientation of the strips in relation to the main curvature, the torsion of the strips, the distortion of the load transfer along the seams, the stiffness of the seams, the assumed compensation of the flatten strips, the width of the strips, the of the surface, the shear deformation of yarns and in the shown example of the load transfer between the boundary cables and fabric, see Fig. 12. In the shown example the stress distributions varies in a single strip and changes from strip to strip. Relatively low tension stress in the middle strip can been see as result of less compensation. The influence of the stiffness of the seams can be shown in the difference between deformation in vertical direction comparing the geometry of the shape of equilibrium and reassembled and pretensioned structure. For the shown example the difference is app. 20% of the span. The antimetric deformation is caused by the inhomogeneous stress distribution in the cross section along the high points. The tension stress perpendicular are unsteady, low stress leads to high vertical deformations and high stress kept the fabric down which can clearly seen in the up and down of the differences. U22 E2222 U22 E1122 U U22 U11 E1111
10 Rosemarie Wagner Shape of equilibrium Cutting pattern Pretension of the plane strips Stress distribution Stress distribution Fig.12.Influence of the cutting pattern to the stress distribution of a membrane [21] Low High point point seams Cross section low points △zmm=20%of the span High point Low point Isometric view,scaling factor 1 100 Cross section high points Fig.13.Difference in z-direction between the reassemble shape and the shape of equilibrium 21] 3.4 Load Bearing Behaviour The load bearing behaviour is in general depended on: The flexibility of the structures including masts,bending elements such as arches, stay cables,anchorages and foundations The height of the pretension related to external loads The orientation of the cables or yarns related to the main curvature of the surface The curvature of the surface and The stress-strain-behaviour of the material The stability of the cable net or membrane structures is depending on the pre- tension.In structures with negative Gaussian curvature the pretension is only to reduce the deformation.The slag of the spanning direction causes a change in the system but no instability
10 Rosemarie Wagner Fig. 12. Influence of the cutting pattern to the stress distribution of a membrane [21] Fig. 13. Difference in z–direction between the reassemble shape and the shape of equilibrium [21] 3.4 Load Bearing Behaviour The load bearing behaviour is in general depended on: - The flexibility of the structures including masts, bending elements such as arches, stay cables, anchorages and foundations - The height of the pretension related to external loads - The orientation of the cables or yarns related to the main curvature of the surface - The curvature of the surface and - The stress – strain – behaviour of the material The stability of the cable net or membrane structures is depending on the pretension. In structures with negative Gaussian curvature the pretension is only to reduce the deformation. The slag of the spanning direction causes a change in the system but no instability. Shape of equilibrium Cutting pattern Pretension of the plane strips Stress distribution Stress distribution Cross section low points Isometric view, scaling factor 1 : 100 Cross section high points High point Low point High poin Low point t seams 'zmax = 20 % of the span
