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 corresponds with the coupling in lightweight structures between the form (geometry) 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 calculated 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 cutting 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 pattern necessitates the specification of the surface cutting lines also called strip edges. This procedure has to take into consideration several parameters likewise 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 sometimes opposite, requirements. 2.1 Technological Issues The design must firstly take into account the maximum strip widths in connection with the products available from fabric manufacturers. Generally, membranes 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 frequency 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 developable 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 geometries, 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 correspond 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 resulting 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 generate 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 determined 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 cutting 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 patterning method without taking into account all the geometric and sthenic data will however not offer the possibility to have an optimal solution to the problem. 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