Recent Developments in the Analytical Design of Textile Membranes Lothar Griindigl,Dieter Strobel2 and Peter Singer2 1 Technical University Berlin Sekr.H20,Strasse des 17.Juni 135 10623 Berlin,Germany gruendig@inge3.bv.tu-berlin.de 2 technet GmbH Stuttgart Pestalozzistrasse 8,70563 Stuttgart,Germany Dieter.Stroebel@technet-gmbh.com http:/www:technet-gmbh.de Summary.The task of determining appropriate forms for stressed membrane sur- face structures is considered.Following a brief introduction to the field,the primitive form-finding techniques which were traditionally used for practical surface design are described.The general concepts common to all equilibrium modelling systems are pre- sented nert,and then a more detailed erposition of the Force Density Method follows. The ertension of the Force Density Method to geometrically non-linear elastic anal- ysis is described.A brief overview of the Easy lightweight structure design system is given with particular emphasis paid to the formfinding and statical analysis suite. Finally,some eramples are used to illustrate the feribility and power of Easy's formfinding tools. The task of generating planar cutting patterns for stressed membrane surface struc- tures is considered nert.Following a brief introduction to the general field of cutting pattern generation,the practical constraints which influence tertile surface structures are presented.Several approaches which have been used in the design of practical structures are outlined.These include the physical paper strip modelling technique, together with geodesic string relaration and flattening approaches.The combined fattening and planar sub-surface regeneration strategy used in the Easy design sys- tem is described in detail.Finally,eramples are given to illustrate the capabilities of Easys cutting pattern generation tools. 1 Introduction Contrary to the design of conventional structures a form finding procedure is needed with respect to textile membrane surfaces because of the direct relationship between the geometrical form and the force distribution.A membrane surface is always in the state of equilibrium of acting forces,and is not defined under unstressed conditions. In general there are two possibilities to perform the formfinding procedures:the 29 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,29-45. C 2005 Springer.Printed in the Netherlands
Recent Developments in the Analytical Design of Textile Membranes Lothar Gr¨undig1, Dieter Str¨obel 2 and Peter Singer2 1 Technical University Berlin Sekr. H20, Strasse des 17. Juni 135 10623 Berlin, Germany gruendig@inge3.bv.tu-berlin.de 2 technet GmbH Stuttgart Pestalozzistrasse 8, 70563 Stuttgart, Germany Dieter.Stroebel@technet-gmbh.com http:/www:technet-gmbh.de Summary. The task of determining appropriate forms for stressed membrane surface structures is considered. Following a brief introduction to the field, the primitive form-finding techniques which were traditionally used for practical surface design are described. The general concepts common to all equilibrium modelling systems are presented next, and then a more detailed exposition of the Force Density Method follows. The extension of the Force Density Method to geometrically non-linear elastic analysis is described. A brief overview of the Easy lightweight structure design system is given with particular emphasis paid to the formfinding and statical analysis suite. Finally, some examples are used to illustrate the flexibility and power of Easy’s formfinding tools. The task of generating planar cutting patterns for stressed membrane surface structures is considered next. Following a brief introduction to the general field of cutting pattern generation, the practical constraints which influence textile surface structures are presented. Several approaches which have been used in the design of practical structures are outlined. These include the physical paper strip modelling technique, together with geodesic string relaxation and flattening approaches. The combined flattening and planar sub-surface regeneration strategy used in the Easy design system is described in detail. Finally, examples are given to illustrate the capabilities of Easys cutting pattern generation tools. 1 Introduction Contrary to the design of conventional structures a form finding procedure is needed with respect to textile membrane surfaces because of the direct relationship between the geometrical form and the force distribution. A membrane surface is always in the state of equilibrium of acting forces, and is not defined under unstressed conditions. In general there are two possibilities to perform the formfinding procedures: the 29 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 29–45. © 2005 Springer. Printed in the Netherlands
30 Lothar Gruindig,Dieter Strobel and Peter Singer physical formfinding procedure and the analytical one.The physical modelling of lightweight structures is characterized by stretchening a soft rubber type material between the chosen boundary positions in order to generate a physically feasible geometry.It has limitations with respect to an accurate description due the small scale of the model.The computational model allows for a proper description by discretizing the surface by a large number of points:a scale problem does not exist any more.Therefore the computational modelling of lightweight structures becomes more and more important;without this technology advanced lightweight structures cannot be built. 2 Analytical Formfinding The analytical formfinding theories are based on Finite Element Methods in general: the surfaces are divided into a number of small finite elements like link elements or triangular elements for example.In such a way all possible geometries can be calculated.There are two theories established in practice:The linear Force Density Approach which uses links as finite elements and the nonlinear Dynamic Relaration Method based on finite triangles. The Force Density Method The Force Density Method was first published in [1]and extended in [2-3,9].It is a mathematical approach for solving the equations of equilibrium for any type of cable network,without requiring any initial coordinates of the structure.This is achieved through the exploitation of a mathematical trick.The essential ideas are as follows. Pin-jointed network structures assume the state of equilibrium when internal forces s and external forces p are balanced. In the case of node i in Fig.1 sa cos(a,)sb cos(b,)sccos(c,)sd cos(d,)p sa cos(a,y)+s cos(b,y)+se cos(c,y)+sa cos(d,y)=Py sa cos(a,z)+s cos(b,z)+se cos(c,z)+sd cos(d,z)=p= where sa,s,se and sd are the bar forces and f.i.cos(a,r)is the normalised projection length of the cable a on the r-axis.These normalised projection lengths can also be expressed in the form (zm-x)/a.Substituting the above cos values with these coordinate difference expressions results in 会m-)+若g-)+总-)+普国-)=pm m-n)+若-)+兰s-+学n-W=P四 Q 2(m-4+若名-)+告(-动+学a-2)=p In these equations,the lengths a,6,c and d are nonlinear functions of the coor- dinates.In addition,the forces may be dependent on the mesh widths or on areas of partial surfaces if the network is a representation of a membrane.If we now apply
30 Lothar Grundig, Dieter Str¨ ¨ obel and Peter Singer physical formfinding procedure and the analytical one. The physical modelling of lightweight structures is characterized by stretchening a soft rubber type material between the chosen boundary positions in order to generate a physically feasible geometry. It has limitations with respect to an accurate description due the small scale of the model. The computational model allows for a proper description by discretizing the surface by a large number of points: a scale problem does not exist any more. Therefore the computational modelling of lightweight structures becomes more and more important; without this technology advanced lightweight structures cannot be built. 2 Analytical Formfinding The analytical formfinding theories are based on Finite Element Methods in general: the surfaces are divided into a number of small finite elements like link elements or triangular elements for example. In such a way all possible geometries can be calculated. There are two theories established in practice: The linear Force Density Approach which uses links as finite elements and the nonlinear Dynamic Relaxation Method based on finite triangles. The Force Density Method The Force Density Method was first published in [1] and extended in [2-3, 9]. It is a mathematical approach for solving the equations of equilibrium for any type of cable network, without requiring any initial coordinates of the structure. This is achieved through the exploitation of a mathematical trick. The essential ideas are as follows. Pin-jointed network structures assume the state of equilibrium when internal forces s and external forces p are balanced. In the case of node i in Fig. 1 sa cos(a, x) + sb cos(b, x) + sc cos(c, x) + sd cos(d, x) = px sa cos(a, y) + sb cos(b, y) + sc cos(c, y) + sd cos(d, y) = py sa cos(a, z) + sb cos(b, z) + sc cos(c, z) + sd cos(d, z) = pz where sa, sb, sc and sd are the bar forces and f.i. cos(a, x) is the normalised projection length of the cable a on the x-axis. These normalised projection lengths can also be expressed in the form (xm − xi)/a. Substituting the above cos values with these coordinate difference expressions results in sa a (xm − xi) + sb b (xj − xi ) + sc c (xk − xi) + sd d (xl − xi ) = px sa a (ym − yi) + sb b (yj − yi) + sc c (yk − yi) + sd d (yl − yi) = py sa a (zm − zi) + sb b (zj − zi) + sc c (zk − zi) + sd d (zl − zi) = pz In these equations, the lengths a, b, c and d are nonlinear functions of the coordinates. In addition, the forces may be dependent on the mesh widths or on areas of partial surfaces if the network is a representation of a membrane. If we now apply
Recent Developments in the Analytical Design of Textile Membranes 31 Fig.1.Part of a cable network the trick of fixing the force density ratio sa/a =qa for every link,linear equations result. These read qa(Im-Ii)+9(Ij-Ti)+qe(Ik-Ii)+qd(xI-Ti)=Pr qa(ym-）+q96(y5-）+qc(yk-)+9a(-)=Py 9a(2m-24)+9b(2-2)+9e(k-2）+9(4-)=p= The force density values q have to be choosen in advance depending on the desired prestress.The procedure results in practical networks which are reflecting the architectural shapes and beeing harmonically stressed.The system of equations assembled is extremely sparse and can be efficiently solved using the Method of Conjugate Gradients as described in [3]. 3 Analytical Formfinding with Technet's Easy Software The 3 main steps of the Analytical Formfinding of Textile Membrane with the tech- net's EASY Software are described as follows: 1.Definition of all design parameters,of all boundary conditions as:the coordi- nates of the fixed points,the warp-and weft direction,the mesh-size and mesh- mode (rectangular or radial meshes),the prestress in warp-and weft direction, the boundary cable specifications(sag or force can be chosen). 2.The linear Analytical Formfinding with Force Densities is performed:the results are:the surface in equilibrium of forces,described by all coordinates of points on the surface,the stress in warp-and weft direction,the boundary cable-forces, the reaction forces of the fixed points.The stresses in warp and weft-direction and the boundary forces may differ in a small range with respect to the desired one from Step 1
Recent Developments in the Analytical Design of Textile Membranes 31 Fig. 1. Part of a cable network the trick of fixing the force density ratio sa/a = qa for every link, linear equations result. These read qa(xm − xi) + qb(xj − xi ) + qc(xk − xi) + qd(xl − xi ) = px qa(ym − yi) + qb(yj − yi) + qc(yk − yi) + qd(yl − yi) = py qa(zm − zi) + qb(zj − zi) + qc(zk − zi) + qd(zl − zi) = pz The force density values q have to be choosen in advance depending on the desired prestress. The procedure results in practical networks which are reflecting the architectural shapes and beeing harmonically stressed. The system of equations assembled is extremely sparse and can be efficiently solved using the Method of Conjugate Gradients as described in [3]. 3 Analytical Formfinding with Technet’s Easy Software The 3 main steps of the Analytical Formfinding of Textile Membrane with the technet’s EASY Software are described as follows: 1. Definition of all design parameters, of all boundary conditions as: the coordinates of the fixed points, the warp -and weft direction, the mesh-size and meshmode (rectangular or radial meshes), the prestress in warp- and weft direction, the boundary cable specifications (sag or force can be chosen). 2. The linear Analytical Formfinding with Force Densities is performed: the results are: the surface in equilibrium of forces, described by all coordinates of points on the surface, the stress in warp- and weft direction, the boundary cable-forces, the reaction forces of the fixed points. The stresses in warp and weft-direction and the boundary forces may differ in a small range with respect to the desired one from Step 1
32 Lothar Griindig,Dieter Strobel and Peter Singer 3.Evaluation and visualization tools in order to judge the result of the formfinding. The stresses and forces can be visualized,layer reactions can be shown,contour- lines can be calculated and visualized.cut-lines through the structure can be made. 4 Force Density Statical Analysis The Force Density Method can be extended efficiently to perform the elastic anal- ysis of geometrically non-linear structures.The theoretical background is described in detail in 3 where it was also compared to the Method of Finite Elements.It was shown that the Finite Element Method's formulae can be derived directly from the Force Density Method's approach.In addition,the Force Density Method may be seen in a more general way.According to [3 it has been proven to be numeri- cally more stable for the calculation of structures subject to large deflections,where sub-areas often become slack.The nonlinear force density method shows powerful damping characteristics. Prior to any statical analysis,the form-found structure has to be materialized. Applying Hooke's law the bar force sa is given by: Sa EAa-40 where A is the area of influence fore bar a,E is the modulus of elasticity,and ao is the unstressed length of bar a.Substituting sa by ga according to qa sa/a results in EAa a0= qaa+EA Because of a being a function of the coordinates of the bar end points,the materialized unstressed length is a function of the force density ga,the stressed length a and the element stiffness EA. In order to perform a statical structural analysis subject to external load,the unstressed lengths have to be kept fixed.This can be achieved mathematically by enforcing the equations of materialization together with the equations of equilibrium. This system of equations is no longer linear.The unknown variables of the enlarged system of equations are now the coordinates z,y,z and the force density values g. Eliminating q from the equations of equilibrium,by applying the formula above to each bar element,leads to a formulation of equations which are identically to those resulting from the Finite Element Method.Directly solving the enlarged system has been shown to be highly numerically stable.as initial coordinates for all nodes are available,and positive values or zero values for g can be enforced through the application of powerful damping techniques. The usual relationship between stress and strain for the orthotropic membrane material is given by: e1111 e2222 The warp-direction u and the weft-direction v are independent from each other; this means:the stress in warp-direction ouu f.i.is only caused by the modulus of elasticity e1111 and the strain euu in this direction.Because of this independency cable net theories can be used also for Textile membranes
32 Lothar Grundig, Dieter Str¨ ¨ obel and Peter Singer 3. Evaluation and visualization tools in order to judge the result of the formfinding. The stresses and forces can be visualized, layer reactions can be shown, contourlines can be calculated and visualized, cut-lines through the structure can be made. 4 Force Density Statical Analysis The Force Density Method can be extended efficiently to perform the elastic analysis of geometrically non-linear structures. The theoretical background is described in detail in [3] where it was also compared to the Method of Finite Elements. It was shown that the Finite Element Method’s formulae can be derived directly from the Force Density Method’s approach. In addition, the Force Density Method may be seen in a more general way. According to [3] it has been proven to be numerically more stable for the calculation of structures subject to large deflections, where sub-areas often become slack. The nonlinear force density method shows powerful damping characteristics. Prior to any statical analysis, the form-found structure has to be materialized. Applying Hooke’s law the bar force sa is given by: sa = EAa − a0 a0 where A is the area of influence fore bar a, E is the modulus of elasticity, and a0 is the unstressed length of bar a. Substituting sa by qa according to qa = sa/a results in a0 = EAa qaa + EA Because of a being a function of the coordinates of the bar end points, the materialized unstressed length is a function of the force density qa, the stressed length a and the element stiffness EA. In order to perform a statical structural analysis subject to external load, the unstressed lengths have to be kept fixed. This can be achieved mathematically by enforcing the equations of materialization together with the equations of equilibrium. This system of equations is no longer linear. The unknown variables of the enlarged system of equations are now the coordinates x, y, z and the force density values q. Eliminating q from the equations of equilibrium, by applying the formula above to each bar element, leads to a formulation of equations which are identically to those resulting from the Finite Element Method. Directly solving the enlarged system has been shown to be highly numerically stable, as initial coordinates for all nodes are available, and positive values or zero values for q can be enforced through the application of powerful damping techniques. The usual relationship between stress and strain for the orthotropic membrane material is given by: σuu σvv = e1111 0 0 e2222 εuu εvv The warp-direction u and the weft-direction v are independent from each other; this means: the stress in warp-direction σuu f.i. is only caused by the modulus of elasticity e1111 and the strain εuu in this direction. Because of this independency cable net theories can be used also for Textile membranes
Recent Developments in the Analytical Design of Textile Membranes 33 In [4 the Force Density Method has been applied very favorably to triangular surface elements.This triangle elements allow the statical analysis taking into con- sideration a more precise material behavior in case of Textile membranes.Actually the both material directions u and v are depending from each other;a strain u leads not only to a stress in u-direction but also to a stress ov in v-direction caused by the modulus of elasticity e1122.The fact that shear-stress depends on a shear- stiffness e1212 seems not to be important for membranes because of its smallness e1111 e1122 0 e2211 e2222 0 0 e1212 Using these constitutive equations Finite Element Methods should be applied. We are using in this case the finite triangle elements. 5 Further Extensions of the Force Density Approach The force density approach can be favorably exploited for further applications. According to [3]the following system of equations of equilibrium is valid: C'QCx=p C is the matrix describing the topology of the system,Q is the diagonal matrix storing the force density values,x contains the coordinates of the nodes of the figure of equilibrium and p the external forces acting on the structure.For linear formfinding C,Q and p are given,x is the result of the above equation. In some applications it might be of interest to know,how close a given geomet- rical surface will represent a figure of equilibrium.In this case C,x and p are given and q is searched for.As there might be no exact solution to the task described above the best approximating solution is achieved allowing for minimal corrections to the external forces.The system now reads: C'Uq=p+v U now represents a diagonal matrix of coordinate differences (C x),q the force density values,and v the residuals of the systems to be minimal. Solving the system of equations,applying the method of least squares,results in best approximating force density values for any given surface under external loads or subject to internal prestress,if some force density values are chosen as fixed in the structure. As shown in [3 the system can be extended even further,by choosing q and x as observables and enforcing the equations of equilibrium,according to the method of least squares condition equations.In this case an architectural design can be best approximated computationally,enforcing the necessary conditions.This extension proves to be a powerful optimization strategy. 6 Statical Analysis with Technet's Easy Software The statical Analysis of lightweight structures under external loads can be performed after three introducing steps:
Recent Developments in the Analytical Design of Textile Membranes 33 In [4] the Force Density Method has been applied very favorably to triangular surface elements. This triangle elements allow the statical analysis taking into consideration a more precise material behavior in case of Textile membranes. Actually the both material directions u and v are depending from each other; a strain εuu leads not only to a stress in u-direction but also to a stress σvv in v-direction caused by the modulus of elasticity e1122. The fact that shear-stress depends on a shearstiffness e1212 seems not to be important for membranes because of its smallness σuu σvv σuv = e1111 e1122 0 e2211 e2222 0 0 0 e1212 εuu εvv εuv Using these constitutive equations Finite Element Methods should be applied. We are using in this case the finite triangle elements. 5 Further Extensions of the Force Density Approach The force density approach can be favorably exploited for further applications. According to [3] the following system of equations of equilibrium is valid: Ct QCx = p C is the matrix describing the topology of the system, Q is the diagonal matrix storing the force density values, x contains the coordinates of the nodes of the figure of equilibrium and p the external forces acting on the structure. For linear formfinding C, Q and p are given, x is the result of the above equation. In some applications it might be of interest to know, how close a given geometrical surface will represent a figure of equilibrium. In this case C, x and p are given and q is searched for. As there might be no exact solution to the task described above the best approximating solution is achieved allowing for minimal corrections to the external forces. The system now reads: Ct Uq = p + v U now represents a diagonal matrix of coordinate differences (C x), q the force density values, and v the residuals of the systems to be minimal. Solving the system of equations, applying the method of least squares, results in best approximating force density values for any given surface under external loads or subject to internal prestress, if some force density values are chosen as fixed in the structure. As shown in [3] the system can be extended even further, by choosing q and x as observables and enforcing the equations of equilibrium, according to the method of least squares condition equations. In this case an architectural design can be best approximated computationally, enforcing the necessary conditions. This extension proves to be a powerful optimization strategy. 6 Statical Analysis with Technet’s Easy Software The statical Analysis of lightweight structures under external loads can be performed after three introducing steps:
34 Lothar Gruindig,Dieter Strobel and Peter Singer 1.To define stiffness values to all finite membrane and cable elements. 2.To calculate the unstressed link lengths by using the assigned stiffness valueses and the prestress of the membrane and the forces in the cables of the formfinding result. 3.To check if the result of the statical analysis with the loads of the formfinding procedure is identical with the formfinding result. After these three steps,the statical analysis without beam elements under ex- ternal loads can be achieved very easily. 1.To calculate the external load vectors as for example snow,wind or normal loads. 2.To perform the nonlinear statical analysis:the approximate values,which are needed in this nonlinear process,are given by the formfinding result. 3.Evaluation-and visualization tools in order to judge the result of the stati- cal analysis.The stresses and forces can be visualized and compared with the maximum possible values.Stresses,forces and layer reactions can be shown, contour-lines can be calculated and visualized,cut-lines through the structure can be made,deflection of the nodes can be calculated. If beam elements are included,the statical analysis under external loads has to be done as follows:all data for the beam-elements as cross-section areas,moments of inertia,local coordinate systems,joints,etc.have to be defined firstly.In order to set all these values in a convenient way the user is supported by a Beam-Editor in EasyBeam. Fig.2.The Easy beam editor
34 Lothar Grundig, Dieter Str¨ ¨ obel and Peter Singer 1. To define stiffness values to all finite membrane and cable elements. 2. To calculate the unstressed link lengths by using the assigned stiffness valueses and the prestress of the membrane and the forces in the cables of the formfinding result. 3. To check if the result of the statical analysis with the loads of the formfinding procedure is identical with the formfinding result. After these three steps, the statical analysis without beam elements under external loads can be achieved very easily. 1. To calculate the external load vectors as for example snow, wind or normal loads. 2. To perform the nonlinear statical analysis: the approximate values, which are needed in this nonlinear process, are given by the formfinding result. 3. Evaluation- and visualization tools in order to judge the result of the statical analysis. The stresses and forces can be visualized and compared with the maximum possible values. Stresses, forces and layer reactions can be shown, contour-lines can be calculated and visualized, cut-lines through the structure can be made, deflection of the nodes can be calculated. If beam elements are included, the statical analysis under external loads has to be done as follows: all data for the beam-elements as cross-section areas, moments of inertia, local coordinate systems, joints, etc. have to be defined firstly. In order to set all these values in a convenient way the user is supported by a Beam-Editor in EasyBeam. Fig. 2. The Easy beam editor
Recent Developments in the Analytical Design of Textile Membranes 35 Then-see above-the steps 1-3 follow.The Beam Editor is also used for checking the results as internal forces and moments,layer-reactions,flexibility-ellipsoids,etc. 7 The Complete Easy Lightweight Structure Design System The Easy system is composed of a number of program suites.These are represented schematically in Fig.3. EasyForm Formfinding of lightweight structures EasySan Nonlinear Statical Load Analysis (without Beam elements) EasyCut Cutting pattern generation EasyBeam Nonlinear hybride Membrane structures including Beam elements Easy Vol Formfinding and Load Analysis of pneumatic constructions 密国四 EasyForm EasySan EasyCut EasyBeam EasyVol Fig.3.The Easy program suites Easy Form comprises the programs used for data generation together with force density form-finding.When the EasySan programs are additionally installed stat- ical structural analysis of non-linear structures becomes possible.The EasyCut
Recent Developments in the Analytical Design of Textile Membranes 35 Then -see above- the steps 1-3 follow. The Beam Editor is also used for checking the results as internal forces and moments, layer-reactions, flexibility-ellipsoids, etc. 7 The Complete Easy Lightweight Structure Design System The Easy system is composed of a number of program suites. These are represented schematically in Fig. 3. EasyForm Formfinding of lightweight structures EasySan Nonlinear Statical Load Analysis (without Beam elements) EasyCut Cutting pattern generation EasyBeam Nonlinear hybride Membrane structures including Beam elements EasyVol Formfinding and Load Analysis of pneumatic constructions Fig. 3. The Easy program suites EasyForm comprises the programs used for data generation together with force density form-finding. When the EasySan programs are additionally installed statical structural analysis of non-linear structures becomes possible. The EasyCut EasyForm EasySan EasyCut EasyBeam EasyVol
36 Lothar Griindig,Dieter Strobel and Peter Singer programs enable the generation of high quality planar cutting patterns from Easy- Form output. In most situations the incorporation of geometrically non-linear bending ele- ments to lightweight structure models is not economically appropriate.Rather,it is more convenient to treat the beam supports as fully fixed points.The result- ing reaction forces on these points are then exported to conventional rigid frame design packages as applied loads.Such a decoupled analysis is appropriate if the resulting deflections are low.However for a sensitive structure decoupling may not be adequate,due the strong interaction of forces causing geometry changes of the membrane surface and the beam elements.In this case the Easy Beam add-on module permits the incorporation of geometrically non-linear frame elements 5]. Easy Form and EasySan together can deal with all standard pneumatic struc- tural configurations which have defined internal pressure prestress.In situations with closed volumes,such as high pressure air beams,this assumption is not valid. It becomes necessary to use more sophisticated algorithms which constrain the cell volumes to prescribed values,and vary the internal pressure accordingly. Fig.4.Formfinding and statical analysis under inner pressure and buoyancy 8 Cutting Pattern Generation of Textile Structures The theory,being used to project a 2D surface in 3 dimensional space to a 2D surface in a plane is very old;it is part of the mathematical field named map projection theory.For example the Mercator Projection dates back to the 17th century. The surfaces,which are used in practical membrane structure design are in gen- eral not developable without distortions.In addition there does not exist a geomet- rical shape without prestress.The map projection theories-used for the flattening of textile membranes-minimize the distortions with respect to lengths,angles and areas respectively
36 Lothar Grundig, Dieter Str¨ ¨ obel and Peter Singer programs enable the generation of high quality planar cutting patterns from EasyForm output. In most situations the incorporation of geometrically non-linear bending elements to lightweight structure models is not economically appropriate. Rather, it is more convenient to treat the beam supports as fully fixed points. The resulting reaction forces on these points are then exported to conventional rigid frame design packages as applied loads. Such a decoupled analysis is appropriate if the resulting deflections are low. However for a sensitive structure decoupling may not be adequate, due the strong interaction of forces causing geometry changes of the membrane surface and the beam elements. In this case the EasyBeam add-on module permits the incorporation of geometrically non-linear frame elements [5]. EasyForm and EasySan together can deal with all standard pneumatic structural configurations which have defined internal pressure prestress. In situations with closed volumes, such as high pressure air beams, this assumption is not valid. It becomes necessary to use more sophisticated algorithms which constrain the cell volumes to prescribed values, and vary the internal pressure accordingly. Fig. 4. Formfinding and statical analysis under inner pressure and buoyancy 8 Cutting Pattern Generation of Textile Structures The theory, being used to project a 2D surface in 3 dimensional space to a 2D surface in a plane is very old; it is part of the mathematical field named map projection theory. For example the Mercator Projection dates back to the 17th century. The surfaces, which are used in practical membrane structure design are in general not developable without distortions. In addition there does not exist a geometrical shape without prestress. The map projection theories -used for the flattening of textile membranes- minimize the distortions with respect to lengths, angles and areas respectively
Recent Developments in the Analytical Design of Textile Membranes 37 60W 36 309sD 6048 Mercator Projection Fig.5.Mercator projection Qo 03 0 以0 Ro Co o T Fig.6.Triangles non deformed (3D)and deformed(2D) The theory to be applied optimizes the total distortion energy by means of the adjustment theory. The surface,which has to be flattened is described using finite triangles.The distortion between the non deformed and deformed situation can be calculated and has to be minimized for all triangles. The paper strip method is exactly described in [10].Practical examples are described in [6-8]. Fig.7 illustrates the paper strip method.A paper is pressed on the surface of the physical model in such a way,that the seam line and the border of the paper are touching each other as good as(in general the paper strip will touch the surface only in one common line,with an increasing distance from this line the difference between paper strip and surface becomes larger.)In the next step a needle is used to perforate the paper strip in a certain number of equidistant points so that the neighboring seam or the boundary line is reached on the shortest way.In doing so the direction of the needle has to be perpendicular to the surface.The connection of the needle holes by straight lines on the flat paper strip leads to the patterns
Recent Developments in the Analytical Design of Textile Membranes 37 Fig. 5. Mercator projection Fig. 6. Triangles non deformed (3D) and deformed (2D) The theory to be applied optimizes the total distortion energy by means of the adjustment theory. The surface, which has to be flattened is described using finite triangles. The distortion between the non deformed and deformed situation can be calculated and has to be minimized for all triangles. The paper strip method is exactly described in [10]. Practical examples are described in [6-8]. Fig. 7 illustrates the paper strip method. A paper is pressed on the surface of the physical model in such a way, that the seam line and the border of the paper are touching each other as good as (in general the paper strip will touch the surface only in one common line, with an increasing distance from this line the difference between paper strip and surface becomes larger.) In the next step a needle is used to perforate the paper strip in a certain number of equidistant points so that the neighboring seam or the boundary line is reached on the shortest way. In doing so the direction of the needle has to be perpendicular to the surface. The connection of the needle holes by straight lines on the flat paper strip leads to the patterns
38 Lothar Gruindig,Dieter Strobel and Peter Singer 可☒ 四阳四压亚出团图可。正田P区 具均世11时 Fig.7.Paper strip method 9 Cutting Pattern Generation with Technet's Easy Software The Cutting pattern generation can be performed in the following steps: 1.Geodesic lines are created as seam lines. 2.Cutting procedures are used to cut the surface into different sub-surfaces ac- cording to these geodesic lines. 3.Ways of flattening are achieved:map projection,paper strip method. 4.Spline algorithms are applied to create equidistant points on the planar circum- ference. 5.Boundary adjustment is performed in order to produce identical seam lengths. 6.Compensation values are defined to compensate the strips. 7.Job-drawings are produced. 10 Flexibility Ellipsoids for the Evaluation of Mechanical Structures The geodetic network adjustment determines the geometrical position of points con- nected by link observables.In general the adjustment theory was invented by C.F. GAUSS by mimizing the residuals of the observations.The process is well known as Least-Squares-Method
38 Lothar Grundig, Dieter Str¨ ¨ obel and Peter Singer Fig. 7. Paper strip method 9 Cutting Pattern Generation with Technet’s Easy Software The Cutting pattern generation can be performed in the following steps: 1. Geodesic lines are created as seam lines. 2. Cutting procedures are used to cut the surface into different sub-surfaces according to these geodesic lines. 3. Ways of flattening are achieved: map projection, paper strip method. 4. Spline algorithms are applied to create equidistant points on the planar circumference. 5. Boundary adjustment is performed in order to produce identical seam lengths. 6. Compensation values are defined to compensate the strips. 7. Job-drawings are produced. 10 Flexibility Ellipsoids for the Evaluation of Mechanical Structures The geodetic network adjustment determines the geometrical position of points connected by link observables. In general the adjustment theory was invented by C.F. GAUSS by mimizing the residuals of the observations. The process is well known as Least-Squares-Method