《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）12 Fabric Membranes Cutting Pattern

Fabric Membranes Cutting Pattern Bernard Maurinl and Rene Motro2 1 Laboratoire de Mecanique et Genie Civil maurinolmgc.univ-montp2.fr 2 Laboratoire de Mecanique et Genie Civil motroolmgc.univ-montp2.fr 1 Introduction Tensile fabric membrane design implies successive stages,each of one related to particular problems requiring well adapted approaches and appropriated results. The first step of the analysis deals with the form-finding process that corre- sponds with the coupling in lightweight structures between the form(geome- try)and the forces (initial tension).The objective is to determine the shape of the membrane associated to its prestress distribution.A good control on the tension in the fabric must be obtained in order to have suitable stresses, for instance that ensure the absence of compressive zones. The following stage focuses on the realization of the tensile membrane calcu- lated during the form-finding.More precisely,the objective is to determine the starting configuration(set of plane strips)which,once assembled on the site according to specified anchoring conditions,will lead closely to the required surface,that is to say to the theoretical one (target strip)calculated during the shape-finding procedure with its associated characteristics of form and prestress.The erection process indeed generated deformations on each strip that will define in the end a mechanically equilibrated geometry coupled with a prestress distribution.The purpose is to minimize the differences between the target state and the therefore obtained state.It corresponds with the cut- ting pattern stage. In case of low deviances,the prejudice will be essentially aesthetic such as disgraceful folds (Fig.1 left)but,in case of higher differences,the integrity of the whole structure could be affected since large membrane zones may be less or not tensioned,this leading to major risk of failure(wind fluttering, horizontal areas with stagnant rain water...,Fig.1 right).The cutting pat- tern necessitates the specification of the surface cutting lines also called strip edges.This procedure has to take into consideration several parameters like- wise technology,geometry,mechanics and aesthetics. Each strip being so identified,the designer must next calculate the associated 195 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,195-212. C 2005 Springer.Printed in the Netherlands

Fabric Membranes Cutting Pattern Bernard Maurin1 and Ren´e Motro2 1 Laboratoire de M´ecanique et G´enie Civil maurin@lmgc.univ-montp2.fr 2 Laboratoire de M´ecanique et G´enie Civil motro@lmgc.univ-montp2.fr 1 Introduction Tensile fabric membrane design implies successive stages, each of one related to particular problems requiring well adapted approaches and appropriated results. The first step of the analysis deals with the form-finding process that corre￾sponds with the coupling in lightweight structures between the form (geome￾try) and the forces (initial tension). The objective is to determine the shape of the membrane associated to its prestress distribution. A good control on the tension in the fabric must be obtained in order to have suitable stresses, for instance that ensure the absence of compressive zones. The following stage focuses on the realization of the tensile membrane calcu￾lated during the form-finding. More precisely, the objective is to determine the starting configuration (set of plane strips) which, once assembled on the site according to specified anchoring conditions, will lead closely to the required surface, that is to say to the theoretical one (target strip) calculated during the shape-finding procedure with its associated characteristics of form and prestress. The erection process indeed generated deformations on each strip that will define in the end a mechanically equilibrated geometry coupled with a prestress distribution. The purpose is to minimize the differences between the target state and the therefore obtained state. It corresponds with the cut￾ting pattern stage. In case of low deviances, the prejudice will be essentially aesthetic such as disgraceful folds (Fig. 1 left) but, in case of higher differences, the integrity of the whole structure could be affected since large membrane zones may be less or not tensioned, this leading to major risk of failure (wind fluttering, horizontal areas with stagnant rain water..., Fig. 1 right). The cutting pat￾tern necessitates the specification of the surface cutting lines also called strip edges. This procedure has to take into consideration several parameters like￾wise technology, geometry, mechanics and aesthetics. Each strip being so identified, the designer must next calculate the associated 195 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 195–212. © 2005 Springer. Printed in the Netherlands

196 Bernard Maurin and Rene Motro plane fabric cutting patterns.Most of the used methods split the process into two different stages.In the first one,the 3D strip is flattened onto a projection plane;in the second,the pretension of the membrane is considered so as to reduce its dimensions. Fig.1.Folds at strip edges;compressive zones 2 Strip Edges Determination This process results in determining the balance between various,and some- times opposite,requirements. 2.1 Technological Issues The design must firstly take into account the maximum strip widths in connec- tion with the products available from fabric manufacturers.Generally,mem- branes are supplied in the form of 2m width rolls [5].After cutting,the strips are assembled by thermo-welding (fusing of the material between high fre- quency electrodes and pasted by applying a pressure);the resulting membrane is next transported to the erection site

196 Bernard Maurin and Ren´e Motro plane fabric cutting patterns. Most of the used methods split the process into two different stages. In the first one, the 3D strip is flattened onto a projection plane; in the second, the pretension of the membrane is considered so as to reduce its dimensions. Fig. 1. Folds at strip edges; compressive zones 2 Strip Edges Determination This process results in determining the balance between various, and some￾times opposite, requirements. 2.1 Technological Issues The design must firstly take into account the maximum strip widths in connec￾tion with the products available from fabric manufacturers. Generally, mem￾branes are supplied in the form of 2m width rolls [5]. After cutting, the strips are assembled by thermo-welding (fusing of the material between high fre￾quency electrodes and pasted by applying a pressure); the resulting membrane is next transported to the erection site

Fabric Membranes Cutting Pattern 197 2.2 Geometrical Issues Some designers consider as necessary to have strip edges along geodesic curves ([2]and [14).It allows indeed,in the particular case of surfaces that are devel- opable onto a plane (on the mathematical meaning),to generate straight lines in accordance with an economical objective of minimal material wastes.This approach may however be relativized since,in the case of double curvature ge- ometries,the surfaces are not developable:we know that such operation leads to unavoidable distortions.It is then judicious to use low dimension strips on a surface zone with high total curvatures.A numerical method devoted to the calculation of membrane curvatures is presented in appendix.Nevertheless, this consideration has to be balanced with a resulting increase of the cutting operations and welding lengths and therefore of the total cost. 2.3 Mechanical Issues The production of the fabric does not end in a perfect isotropy between the warp and weft directions (higher strength and stiffness for the warp)even if improvements in production processes aim to reduce this difference.The low shear strength of the fabric has also to be taken into account.Thus,an ideal configuration will be related to the positioning of the strip edges,that corre- spond after cutting approximately with the warp direction,along the direction of the main forces,that is to say the maximum principal stresses.In that case, the fabric weft is thus turned on to the minimum stresses directions with re- sulting shear forces close to zero.All of these theoretical considerations have however to be balanced with practical aspects:inexistence of exact solution, knowledge of stresses due to the initial stresses in the fabric and to climatic effects.If the action of wind is paramount (pressure normal to the surface), then the directions of maximum stresses correspond with the directions of the membrane maximum curvatures.For snow (vertical action)the answer is much more complex but some basic cases such as the radial positioning of the strip edges at the top of anchoring masts(Fig.2).In addition to these requirements dealing with the surface,others considerations are related to the membrane edges.Since the initial pretension is applied by progressively tensioning edge cables,it is necessary that strip edges be orthogonal to these cables. However,so as to point out the problems associated to particular situations, we quote the case of the design of Mina Valley tents in Mecca build for pilgrims [10].The project,realised in two stages in 1997/98,is composed of 40000 tents with a rectangular frame(from 4x4m to 8x12m)with a vertical mast at middle. The membranes build during the first stage are based upon the basic radial positioning of strip edges,but the difficulties in tensioning the fabric with the mast have lead to prohibited folds on the surface.The designers of the second team have then decided to set the strip edges parallel to the anchoring sides

Fabric Membranes Cutting Pattern 197 2.2 Geometrical Issues Some designers consider as necessary to have strip edges along geodesic curves ([2] and [14]). It allows indeed, in the particular case of surfaces that are devel￾opable onto a plane (on the mathematical meaning), to generate straight lines in accordance with an economical objective of minimal material wastes. This approach may however be relativized since, in the case of double curvature ge￾ometries, the surfaces are not developable: we know that such operation leads to unavoidable distortions. It is then judicious to use low dimension strips on a surface zone with high total curvatures. A numerical method devoted to the calculation of membrane curvatures is presented in appendix. Nevertheless, this consideration has to be balanced with a resulting increase of the cutting operations and welding lengths and therefore of the total cost. 2.3 Mechanical Issues The production of the fabric does not end in a perfect isotropy between the warp and weft directions (higher strength and stiffness for the warp) even if improvements in production processes aim to reduce this difference. The low shear strength of the fabric has also to be taken into account. Thus, an ideal configuration will be related to the positioning of the strip edges, that corre￾spond after cutting approximately with the warp direction, along the direction of the main forces, that is to say the maximum principal stresses. In that case, the fabric weft is thus turned on to the minimum stresses directions with re￾sulting shear forces close to zero. All of these theoretical considerations have however to be balanced with practical aspects: inexistence of exact solution, knowledge of stresses due to the initial stresses in the fabric and to climatic effects. If the action of wind is paramount (pressure normal to the surface), then the directions of maximum stresses correspond with the directions of the membrane maximum curvatures. For snow (vertical action) the answer is much more complex but some basic cases such as the radial positioning of the strip edges at the top of anchoring masts (Fig. 2). In addition to these requirements dealing with the surface, others considerations are related to the membrane edges. Since the initial pretension is applied by progressively tensioning edge cables, it is necessary that strip edges be orthogonal to these cables. However, so as to point out the problems associated to particular situations, we quote the case of the design of Mina Valley tents in Mecca build for pilgrims [10]. The project, realised in two stages in 1997/98, is composed of 40000 tents with a rectangular frame (from 4x4m to 8x12m) with a vertical mast at middle. The membranes build during the first stage are based upon the basic radial positioning of strip edges, but the difficulties in tensioning the fabric with the mast have lead to prohibited folds on the surface. The designers of the second team have then decided to set the strip edges parallel to the anchoring sides

198 Bernard Maurin and Rene Motro Fig.2.Radial strip edges Fig.3.Strips used for the Mina project stage 2 It was allowed by the absence of snow and has resulted in the vanishing of the folds (Fig.3). We emphasize herewith on the fact that small structures design may gen- erate more difficulties than wider membranes design since the dimension of the fabric rolls appears as important with reference to the dimensions of the structure. 2.4 Aesthetical Issues The approach could however be modified when architects play a role.Their creativity may for instance leads to the making of geometric drawings by using fabric samples of different colours.Moreover,since the visual perception of the surface is dependant on the strip edges positioning,mainly at night,this architectural feature could lead to specific patterning strategies

198 Bernard Maurin and Ren´e Motro Fig. 2. Radial strip edges Fig. 3. Strips used for the Mina project stage 2 It was allowed by the absence of snow and has resulted in the vanishing of the folds (Fig. 3). We emphasize herewith on the fact that small structures design may gen￾erate more difficulties than wider membranes design since the dimension of the fabric rolls appears as important with reference to the dimensions of the structure. 2.4 Aesthetical Issues The approach could however be modified when architects play a role. Their creativity may for instance leads to the making of geometric drawings by using fabric samples of different colours. Moreover, since the visual perception of the surface is dependant on the strip edges positioning, mainly at night, this architectural feature could lead to specific patterning strategies

Fabric Membranes Cutting Pattern 199 3 Cutting Shapes Determination 3.1 Background Before presenting several used methods,we aim to point out some significant principles. Since the objective is to have a good adequacy between the target state and the real state,it is thus necessary for each strip to evaluate the result in the light of the morphological parameters of forms and forces: -If the geometry of the strip put into place is close to those theoretically de- termined during the form-finding stage,we will say that it exists a geometrical equivalence between the two strips.However,one point has to be respected: an edge belonging to two strips must have the same length on the plane cut- ting shapes so as to allow their future assembly. -Similarly,if the prestress field generated in the strip is close to the required one,the sthenical equivalence is ensured.We may observe that it implies the perfect knowledge of the selfstress state determined during the form-finding process. Nevertheless,these two principles only represent a virtual reality since it is illusory to expect a complete equivalence but very particular cases.A pattern- ing method without taking into account all the geometric and sthenic data will however not offer the possibility to have an optimal solution to the prob- lem.The same comment is also relevant if these considerations are not seen as indissociable and so envisaged as two separate steps(flattening and then reduction).We remark that,as far as we can know,most of the used methods are based upon such splitting. Let's now have a look on the existing flattening processes. The first technique is the simple triangulation method (Fig.4).The 3D strip (a)determined by form-finding is mapped with a series of triangles between the longest edges,leading to the geometry (b).The obtained triangles are next successively flattened onto a plane by keeping identical the lengths of their sides (c).Since this method is quick and easy to apply,it is at the core of numerous CAD tools. (a) (b Fig.4.Simple triangulation method

Fabric Membranes Cutting Pattern 199 3 Cutting Shapes Determination 3.1 Background Before presenting several used methods, we aim to point out some significant principles. Since the objective is to have a good adequacy between the target state and the real state, it is thus necessary for each strip to evaluate the result in the light of the morphological parameters of forms and forces: - If the geometry of the strip put into place is close to those theoretically de￾termined during the form-finding stage, we will say that it exists a geometrical equivalence between the two strips. However, one point has to be respected: an edge belonging to two strips must have the same length on the plane cut￾ting shapes so as to allow their future assembly. - Similarly, if the prestress field generated in the strip is close to the required one, the sthenical equivalence is ensured. We may observe that it implies the perfect knowledge of the selfstress state determined during the form-finding process. Nevertheless, these two principles only represent a virtual reality since it is illusory to expect a complete equivalence but very particular cases. A pattern￾ing method without taking into account all the geometric and sthenic data will however not offer the possibility to have an optimal solution to the prob￾lem. The same comment is also relevant if these considerations are not seen as indissociable and so envisaged as two separate steps (flattening and then reduction). We remark that, as far as we can know, most of the used methods are based upon such splitting. Let’s now have a look on the existing flattening processes. The first technique is the simple triangulation method (Fig. 4). The 3D strip (a) determined by form-finding is mapped with a series of triangles between the longest edges, leading to the geometry (b). The obtained triangles are next successively flattened onto a plane by keeping identical the lengths of their sides (c). Since this method is quick and easy to apply, it is at the core of numerous CAD tools. Fig. 4. Simple triangulation method

200 Bernard Maurin and Rene Motro We may however note that it does not take into consideration a lot of points: those located inside the strip and some on top and bottom edges.Its use therefore requires caution so as to avoid important errors. Several authors have thus proposed improved processes.L.Gruindig has put forward a method which takes into account all the points belonging to the strip edges 7;this objective being also one of H.Tsubota's aims [13].In both cases,the geometry of edges is calculated by using minimization error processes.We may regret that data of internal points are still avoided.The method proposed by T.Shimada ([12 and [1])offers some improvements on that purpose.It consists in determining a plane domain composed of trian- gular surface elements which,once transformed into the 3D strip,leads to a minimal strain energy.The material characteristics are used in the mechanical formulation.However,parameters related to the prestress of the membrane are not considered. In each situation,the development of the strip must be followed by an opera- tion so as to take into account the initial stresses.Three main strategies may be pointed up: -The strip is not modified.Stresses in the fabric are generated by the displace- ments of anchoring zones (mast erection,shortening of edge cable lengths...). -If the strip is triangulated,every element is reduced along two directions in relation with the selfstresses determined by shape-finding.In the case of uni- form stresses within the strip(minimal area surface for instance)the solution in not difficult.On the contrary,specific methods are to be used to determine an accurate result. -The most commonly used technique results in considering a reduction scale factor for the developed shape.The designer applies it generally along the longer direction of the strip (the warp direction,with a factor from 2 to 3%) and the transverse reduction along the weft is obtained during the strips as- sembling (bilayer of the welded zone close to 2cm width). The experience and a good knowledge of the material,mainly obtained by mechanical testing (stiffness,creep...[6),play nevertheless a major role in these methods and the designer must proceed carefully. If every cutting pattern process unavoidably leads to errors,we may remark that the greatest part of them are directly related to the splitting of the technique into two separate operations of flattening (for the geometry)and reduction (for the prestress).Hence,it appears that a better solution obvi- ously relies in an integrated approach which takes into account at the same time the considerations of form,forces and material.The target state has to be determined in a comprehensive process without splitting of these pa- rameters.Several research teams have thus proposed innovative patterning methods based upon such integrated approach.We may quote the works of J.Kim [8]and the method developed at the Mechanics and Civil Engineering Laboratory at Montpellier University and called stress composition method [91

200 Bernard Maurin and Ren´e Motro We may however note that it does not take into consideration a lot of points: those located inside the strip and some on top and bottom edges. Its use therefore requires caution so as to avoid important errors. Several authors have thus proposed improved processes. L. Gr¨undig has put ¨ forward a method which takes into account all the points belonging to the strip edges [7]; this objective being also one of H. Tsubota’s aims [13]. In both cases, the geometry of edges is calculated by using minimization error processes. We may regret that data of internal points are still avoided. The method proposed by T. Shimada ([12] and [1]) offers some improvements on that purpose. It consists in determining a plane domain composed of trian￾gular surface elements which, once transformed into the 3D strip, leads to a minimal strain energy. The material characteristics are used in the mechanical formulation. However, parameters related to the prestress of the membrane are not considered. In each situation, the development of the strip must be followed by an opera￾tion so as to take into account the initial stresses. Three main strategies may be pointed up: - The strip is not modified. Stresses in the fabric are generated by the displace￾ments of anchoring zones (mast erection, shortening of edge cable lengths...). - If the strip is triangulated, every element is reduced along two directions in relation with the selfstresses determined by shape-finding. In the case of uni￾form stresses within the strip (minimal area surface for instance) the solution in not difficult. On the contrary, specific methods are to be used to determine an accurate result. - The most commonly used technique results in considering a reduction scale factor for the developed shape. The designer applies it generally along the longer direction of the strip (the warp direction, with a factor from 2 to 3%) and the transverse reduction along the weft is obtained during the strips as￾sembling (bilayer of the welded zone close to 2cm width). The experience and a good knowledge of the material, mainly obtained by mechanical testing (stiffness, creep... [6]), play nevertheless a major role in these methods and the designer must proceed carefully. If every cutting pattern process unavoidably leads to errors, we may remark that the greatest part of them are directly related to the splitting of the technique into two separate operations of flattening (for the geometry) and reduction (for the prestress). Hence, it appears that a better solution obvi￾ously relies in an integrated approach which takes into account at the same time the considerations of form, forces and material. The target state has to be determined in a comprehensive process without splitting of these pa￾rameters. Several research teams have thus proposed innovative patterning methods based upon such integrated approach. We may quote the works of J. Kim [8] and the method developed at the Mechanics and Civil Engineering Laboratory at Montpellier University and called stress composition method [9]

Fabric Membranes Cutting Pattern 201 3.2 Stress Composition Method We consider a 3D target strip calculated by form-finding process and with every elementary prestress tensor known fo=foy orL.The method relies in the determination of the plane cutting shape o such as its exact transformation into (that is to say by considering the geometrical equivalence as respected)generates these stresses (i.e.the sthenical equiva- lence as an objective).We will in the end go back on the relevance of this starting assumption. On that purpose,a starting domain is defined (determined by the orthog- onal projection of the target strip on the development plane)and is deformed into n hence generating the prestress fotocL.If these values are different from thenis modified into,so as the resulting stresses balance the observed deviation(Fig.5).The calculation of the tensor fotocL is achieved in accordance with the hypothesis of large displacements.The mechanical characteristics of the fabric are taken into account during every transformation. 3D strip Development plane 2 2222222220 SDH Fig.5.Used configurations The modification of n*into 0 is achieved according to the small displace- ments and small strains hypothesis(SDH),by acting on the boundary condi- tions of the frontier nodes ofin order to have close to the deviation {ooc}L-{a话c}L. The associated displacement {dmod)(Fig.6)may be determined with refer- ence to the matrix relationship: [A].{dmod)o=foiped). (1)

Fabric Membranes Cutting Pattern 201 3.2 Stress Composition Method We consider a 3D target strip ΩL calculated by form-finding process and with every elementary prestress tensor known {σff loc}L = {σx σy σxy}L. The method relies in the determination of the plane cutting shape Ω0 such as its exact transformation into ΩL (that is to say by considering the geometrical equivalence as respected) generates these stresses (i.e. the sthenical equiva￾lence as an objective). We will in the end go back on the relevance of this starting assumption. On that purpose, a starting domain Ω∗ is defined (determined by the orthog￾onal projection of the target strip on the development plane) and is deformed into ΩL hence generating the prestress {σloc}L. If these values are different from {σff loc}L then Ω∗ is modified into Ω0, so as the resulting stresses {σmod loc }∗ balance the observed deviation (Fig. 5). The calculation of the tensor {σloc}L is achieved in accordance with the hypothesis of large displacements. The mechanical characteristics of the fabric are taken into account during every transformation. Fig. 5. Used configurations The modification of Ω∗ into Ω0 is achieved according to the small displace￾ments and small strains hypothesis (SDH), by acting on the boundary condi￾tions of the frontier nodes of Ω∗ in order to have {σmod loc }∗ close to the deviation {σloc}L − {σff loc}L. The associated displacement {dmod f }0 ∗ (Fig. 6) may be determined with refer￾ence to the matrix relationship: [Aσ]∗ {dmod f }0 ∗ = {σmod loc }∗ (1)

202 Bernard Maurin and Rene Motro 2 Fig.6.Transformation of into 0 The equation is solved by using a least square method that provides a first approximate solution of o called 0(1).We consider next a second"starting" domain *(2)=0(1);it allows therefore to calculate a second approximation 0(2)of 0.This iterative process constitutes the background of the stress composition method (Fig.7). orthogonal projection Q 20 convergence 22) 202) 0 Ω° Fig.7.Stress composition method It converges in p iterations into the domain 0(P)characterized by a stress deviation)according to: Aoo(P)oel =ofe)-(le)+ore() (2) The sthenic convergence criterion is set by the designer accordingly to the required maximm value of△opl.The vectorial norm‖‖corresponds to the euclidian norm. 3.3 Application The searched 3D strip belongs to a minimal area surface characterised by an isotropic and uniform prestress {oeL={oooo 0L with oo=250 daN/m. The dimensions of the strip in plane projection are 10m x 2m with an elevation of 1m in the top vertex (Fig.8).The mechanical features of the material are those of a manufactured fabric:warp direction Young modulus equal to 24900 daN/m and 23000 daN/m for the weft direction with Poisson coefficients 0,097 and0,090. The stress composition method converges in seven iterations with a stress distortion=30%with reference to the target state

202 Bernard Maurin and Ren´e Motro Fig. 6. Transformation of Ω∗ into Ω0 The equation is solved by using a least square method that provides a first approximate solution of Ω0 called Ω0(1) . We consider next a second ”starting” domain Ω∗(2) = Ω0(1); it allows therefore to calculate a second approximation Ω0(2) of Ω0. This iterative process constitutes the background of the stress composition method (Fig. 7). Fig. 7. Stress composition method It converges in p iterations into the domain Ω0(p) characterized by a stress deviation ∆σL(p) 0 according to: ∆σL(p) 0 {σff loc}L = {σff loc}L − {σloc}L + {σmod(p) loc }∗ (2) The sthenic convergence criterion is set by the designer accordingly to the required maximum value of ∆σL(p) 0 . The vectorial norm corresponds to the euclidian norm. 3.3 Application The searched 3D strip belongs to a minimal area surface characterised by an isotropic and uniform prestress {σff loc}L = {σ0 σ0 0}L with σ0 = 250 daN/m. The dimensions of the strip in plane projection are 10m x 2m with an elevation of 1m in the top vertex (Fig. 8). The mechanical features of the material are those of a manufactured fabric: warp direction Young modulus equal to 24900 daN/m and 23000 daN/m for the weft direction with Poisson coefficients 0,097 and 0,090. The stress composition method converges in seven iterations with a stress distortion ∆σL(7) 0 = 3, 70% with reference to the target state.

Fabric Membranes Cutting Pattern 203 Fig.8.Strip determined with the stress composition method The following graph represents the values of the principal stresses o1 and o2 for every 40 triangular elements used for the modelling;the ideal solution is drawn by the horizontal line o1=o1=oo=250 daN/m (target state).We observe a regular stress distribution within the surface and the solution appears as quite satisfactory. ol 回o2 20 6789101111101415:167w10202121日42502723月11)45%7第540 Fig.9.Principal stresses obtained with the integrated method If the same strip is now calculated with the simple triangulation method followed by a reduction,the determined shape may be characterised by a stress deviance△oo ()=26,62%(evaluated after remeshing the plane domain with 40 triangular elements by adding internal nodes without altering the geometry of edges). The values of the resulting principal stresses are presented in the following graph. We point out that,when all the geometry of the domain is not taken into consideration likewise during the mapping with simple triangulation,increas- ing deviances occur.Such result is however not so surprising.A more detailed analysis of the graph allows noticing a higher distortion between the principal stresses for the elements located in the top zone of the surface (right part of the graph).A possible explanation relies in the fact that these elements are

Fabric Membranes Cutting Pattern 203 Fig. 8. Strip determined with the stress composition method The following graph represents the values of the principal stresses σ1 and σ2 for every 40 triangular elements used for the modelling; the ideal solution is drawn by the horizontal line σ1 = σ1 = σ0 = 250 daN/m (target state). We observe a regular stress distribution within the surface and the solution appears as quite satisfactory. Fig. 9. Principal stresses obtained with the integrated method If the same strip is now calculated with the simple triangulation method followed by a reduction, the determined shape may be characterised by a stress deviance ∆σL(7) 0 = 26, 62% (evaluated after remeshing the plane domain with 40 triangular elements by adding internal nodes without altering the geometry of edges). The values of the resulting principal stresses are presented in the following graph. We point out that, when all the geometry of the domain is not taken into consideration likewise during the mapping with simple triangulation, increas￾ing deviances occur. Such result is however not so surprising. A more detailed analysis of the graph allows noticing a higher distortion between the principal stresses for the elements located in the top zone of the surface (right part of the graph). A possible explanation relies in the fact that these elements are

204 Bernard Maurin and Rene Motro Fig.10.Strip determined with simple triangulation method 00 290 口o2 210 21 67190121141511714920222刀425242刀路290引3034第617翼39 Fig.11.Principal stresses obtained with simple trangulation method subjected to the highest strains,which may generate shear forces and thus differences between the principal stresses. 4 Modelling of the Strip Prestressing All the cutting pattern methods previously presented are based upon an im- plicit hypothesis:the plane strip will be exactly transformed into the target 3D strip,that is to say the surface determined by form-finding will be,in the end,exactly obtained.Some comments may be pointed up.Firstly,we notice that such postulate is a strong one with reference to possible conse- quences.However it must also be kept in mind the difficulty to avoid such assumption in order to be able to determine the cutting shapes.Nevertheless, a rigorous process has to check both the geometrical and sthenical deviances between the target membrane and those obtained after prestressing the as- sembled membrane in the site.Cutting pattern and prestressing stages may be thus regarded as a "back and return"operation. The modelling of the prestressing operation could be envisaged according to two levels of complexity: -A global approach in which the designer studies the deployment of the whole

204 Bernard Maurin and Ren´e Motro Fig. 10. Strip determined with simple triangulation method Fig. 11. Principal stresses obtained with simple triangulation method subjected to the highest strains, which may generate shear forces and thus differences between the principal stresses. 4 Modelling of the Strip Prestressing All the cutting pattern methods previously presented are based upon an im￾plicit hypothesis: the plane strip will be exactly transformed into the target 3D strip, that is to say the surface determined by form-finding will be, in the end, exactly obtained. Some comments may be pointed up. Firstly, we notice that such postulate is a strong one with reference to possible conse￾quences. However it must also be kept in mind the difficulty to avoid such assumption in order to be able to determine the cutting shapes. Nevertheless, a rigorous process has to check both the geometrical and sthenical deviances between the target membrane and those obtained after prestressing the as￾sembled membrane in the site. Cutting pattern and prestressing stages may be thus regarded as a “back and return” operation. The modelling of the prestressing operation could be envisaged according to two levels of complexity: - A global approach in which the designer studies the deployment of the whole