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: