**Some counterstressed dual-cable structures (German), D Jawerth, Proceedings
of the IASS Colloquium on hanging roofs, continuous metallic shell roofs and
superficial lattice roofs, Paris, 1962.**

The suspension structure in the article, the Jawerth system, is made up of two suspension cables which are stressed against each other using a zigzag of struts. The upper cable forms a concave link polygon and the lower cable a convex one. The erection of various buildings with this system is presented.

**Steel cable creates novel structural space systems, L Zetlin, AISC
Engineering Journal, First Quarter, 1964.**

This paper presents means and design methods to achieve structural systems for suspension roofs which do not exhibit the phenomenon of aerodynamic instability. It does not deal with, nor attempt to determine, the behavior of a suspension roof during flutter. For practical purposes, if factors contributing to self-exciting vibrations are eliminated from a structural system to begin with, investigation of, and design for, flutter becomes irrelevant. The structural system presented in this paper consists of cables interconnected in such a manner that there is always flow of energy from a cable which tends to flutter to other cables which are at a lower energy level.

**Tension Structures: Jawerth System, M Majowiecki, Acier Stahl Steel, # 4, 1971.**

Some important examples in the natural development of structural design have been registered in recent years, including a new system of roof construction: the Jawerth System. This uses steel wire ropes of high strength as its main structural members, balancing the external loading by positive forces only, that is, by tensile stresses. The structural system - a tension or suspension type - does not, however, stand any risk of reaching a state of unstable equilibrium. The adoption of high-strength materials has the advantage of reducing weight and reducing overall costs. In spite of the inertia, which weighs against the adoption of any new constructional methods, these structures have, made good progress by reason of their outstanding essential merits, wider unencumbered spans and reduced dead weight. This kind of suspended roof has found its place in the provision of large internal spaces, such as are required for a great variety of purposes, public or industrial.

**The design and testing of a cable beam structure for prefabrication, H
Buchholdt, B McMillan, V Gill, Proceedings of the IABSE Ninth Congress, Amsterdam, 1972.**

This paper contains the description and testing of a cable beam structure which has been developed for prefabrication and low cost. The experimental results nave been compared with the results from two different non-linear theories indicating for which cases the two different theories are applicable.

** Stress strain relation for coated fabrics, R Testa, L Yu, Journal of engineering mechanics,
Vol 113, # 11, ASCE 1987.**

A constitutive relation is formulated for coated fabrics that exhibit both elastic and inelastic, but time-independent, responses. The nonlinear orthotropic response is represented by an elastic portion related to yarn stretch and coating shear, and an inelastic part related to crimp interchange. A method is proposed to compute the path-dependent inelastic strain and the elastic strain for any loading once material parameters are evaluated from uniaxial tests in the principal directions of elasticity.

**A Pade approximant to the inverse Langevin function, A Cohen, Rheologica
Acta, Vol 30, # 3, 1991.**

Application of the methodology of Pade approximants to a Taylor expansion of the inverse Langevin function led to an accurate analytical expression. The approximation, retaining a finite extendibility of the Langevin spring, enables a convenient analysis of experimental data and analytical manipulations of material models.

**Equivalence of a hanging cable net to an orthotropic membrane, G Yamada,
Y Kobayashi, S Nakae, Journal of Sound and Vibration, Vol 145, # 1, 1991.**

In this short note, the equivalence of hanging cable nets to orthotropic membranes is discussed. Cable nets are approximated by the lumped masses at intersections of the cables, and by substituting the kinetic and strain energies of the net system into Lagrange’s equation, the equations of motion of the system are derived. In this case, the intersections of the cables are assumed to be connected by the frictionless pin-joints and the sides of the cable nets to be supported without bending.

**On the feasibility of using large scale photogrammetry to accurately determine in-service strain distribution across three-dimensional textile roofs, L Grundig, E Moncrieff, H Schewe, Euromech 334, Textile composites and textile structures, Lyon, 1995.**

This paper presents the results obtained from a proof of concept simulation study into the feasibility of using a photogrammetric measurement system to accurately determine in-service strain distribution across full scale three dimensional textile roofs. The simulation has been carried out using synthetic images generated by photo-realistic raytracing software. Such a strategy allows for the incorporation of poor lighting and other potentially degrading effects. Particular attention has been addressed to the expected effects on measurement precision of image resolution; target shape, size, number and contrast; and operator access.

**Numerical simulation of the flow over sails in real sailing conditions, T
Charvet, F Hauville, S Huberson, Journal of Wind Engineering and Industrial
Aerodynamics, Vol 63, # 1-3, 1996.**

We present recent results concerning the numerical simulation of flow around ship sails. The main purpose of our work has been to derive a numerical model which includes real effects to the utmost. The basic numerical model is made of a lifting surface code for the sail and a vortex method for its wake. This description of the flow is supplemented by considering the fluid structure interaction problem, the sea surface influence on the flow, the effect of non-homogeneous incoming flow (vertical wind gradient and gusts), the ship hull motion, and sail interactions.

**Application of a non-convex model of fabric deformations to sail cut
analysis, O Maitre, S Huberson, J Cursi, Journal of Wind Engineering and
Industrial Aerodynamics, Vol 63, # 1-3, 1996.**

In this paper, we present a computer model based on an elastic string network representation for sail deformation. The equilibrium equation for this model is written in the form of a minimization problem. The latter is non-convex because of the unilateral-stress behavior of strings. The method of deconvexification has to be used in order to obtain an equivalent problem which is easier to solve. The resulting model is applied to sail cut design problems: bi- and tri-radial cut plans are compared, as well as variations of the elasticity modulus in warp and weft directions. The results are found to be very similar to what is usually observed on actual sail boats.

**Finite element analysis of nonlinear orthotropic hyperelastic membranes,
S Kyriacou, C Schwab, J Humphrey, Computational Mechanics, Vol 18, # 4,
1996.**

A finite element method is presented for geometrically and materially nonlinear orthotropic hyperelastic membranes. The constitutive relations are formulated in terms of the invariants of the 2D right Cauchy-Green strain tensor and the resulting system of nonlinear equations solved using a Newton-Raphson approach. Example problems are solved for isotropic and orthotropic membranes, and the effect of various parameters investigated.

**Geodesic and semi-geodesic line algorithms for cutting pattern generation of architectural textile structures, L Grundig, L Ekert, E Moncrieff, Proceedings of asia-pacific conference on shell and spatial structures, IASS, Beijing, China, 1996.**

The general field of stressed membrane surface structures is first introduced. The design processes typically involved in the design of such membranes, namely Form-finding, Statical Load Analysis and Cutting Pattern Generation, are next described. Specific consideration of Geodesic Line Generation and the requirements for a practical solution strategy are then given. A novel procedure for addressing the complex problem of generating Semi-Geodesic lines is next presented. All the line generation tools implemented in the widespread lightweight structure design system, Easy, including Geodesic and Semi-Geodesic are then described. Finally, these tools are demonstrated using a typical example surface. Due to the general importance of good quality seam line generation, the strategy described is applicable for all cutting pattern generation systems.

** A three-dimensional fe nonlinear analysis of membranes, B Wu, X Du, H Tan, Computers & Structures,
Vol 59, # 4, 1996.**

This paper presents a three-dimensional finite element method for computing stress and deformation of membrane structures. A conjugate pair of strain and stress tensors, the Green strain tensor and the Second Piola-Kirchhoff stress tensor are adopted to give the constitutive equation of Mooney material. By means of exploiting the plane stress condition of membrane, the hydrostatic pressure is directly expressed as functions of strains. Complex membrane structures, like those initially in box shape with abrupt changes in geometry, can also be analyzed by the proposed method.

** Formulation of a curved quadrilateral element for surface definition, P Gosling, W Lewis, Computers & Structures,
Vol 61, # 5, 1996.**

Based on a finite element discretisation, the numerical form-finding of geometrically non-linear surfaces spanning arbitrary boundaries and subjected to an initial prestress is presented. A 24 dof quadrilateral finite element is formulated to represent a general curved elastic (or inelastic) geometrically non-linear surface. The proposed isoparametric element is C0 continuous, of constant thickness, and assumes a plane-stress criterion. A rigorous derivation of the expressions describing the strains within a curved surface is offered, while the element equations are written with special consideration of the effects of large strains and large displacements. Assuming small incremental displacements, expressions are derived to explicitly include the adequate representation of rigid body rotations in the element geometric stiffness matrix.

**Form-finding of prestressed membranes using a curved quadrilateral finite
element for surface definition, P Gosling, W Lewis, Computers & Structures,
Vol 61, # 5, 1996.**

Form-finding of minimal surface membranes is investigated in this paper. A curved quadrilateral finite element is used to provide a numerical representation of a thin surface (structural membrane) established between fixed or flexible boundaries. Pre-stress is introduced to generate the form. Application of the matrix based finite element method to the vector based dynamic relaxation algorithm is presented. When analysing minimal surfaces, the assumption of large strains is shown to lead to a stress deviation at equilibrium. Various techniques are proposed to improve the numerical stability of the solution algorithm. The resulting final numerical model adequately represents both large displacements and large summative strains. Comparisons between numerical and experimental solutions to several minimal surfaces demonstrate the accuracy of the proposed formulation.

**Dynamic analysis of tension structures, B Tabarrok, Z Qin, Computers &
Structures, Vol 62, # 3, 1997.**

The equations of motion for curved membranes are determined via Hamilton’s principle and subsequently solved by the finite element method. Initially linearized equations are determined by considering small amplitude oscillations about the position of static equilibrium: Later, more accurate equations are determined by taking account of geometrical nonlinearities in the displacement-strain relationships.

**The form-finding of structures possessing a constant surface stress, PhD
Thesis, T Lewis, Department of Engineering, University of Warwick, England,
1997.**

A method for the generation of structures of a constant surface stress is presented; the prime application being the form-finding of light-weight tension structures. However, rigid forms generated from the principle are also discussed. The numerical approach adopted for the solution to the geometrically non-linear problem of form-finding is that of the Dynamic Relaxation method, incorporating kinetic damping. The numerical solutions produced describe surfaces of a minimum surface area and a zero mean curvature.

** Formulation of constitutive equations for fabric membranes based on the concept of fabric lattice model,
S Kato, T Yoshino, H Minami, Proceedings of the fifth international conference on computational plasticity, Barcelona, Spain, 1997.**

The present paper discusses on a new formulation for continuum constitutive equations of fabric membranes based on a fabric lattice model. The equations consider the material nonlinearities of yarns and coatings and include crimp interchange between warp and weft yarns. The equations are formulated in an incremental form which can be directly applied to FEM. The validity of the constitutive equations is discussed by comparing with those obtained from testing.

**Analysis of membrane structures based on fabric lattice model considering viscous characteristics,
S Kato, H Minami, T Yoshino, T Namita, Proceedings of the IASS international symposium on shell & spatial structures, Singapore, 1997.**

The present paper aims at proposing an analysis method for stress-deformation of tensioned membrane structures based on fabric lattice model including the viscous characteristics. In this study the fabric lattice model previously proposed by the first author for the static case is extended to include the effects of the material viscosity into the time dependent constitutive equations.

**Large displacement analysis for ideally flexible sails, O Maitre, J
Cursi, S Huberson, European Journal of Mechanics - A/Solids, Vol 17, #
4, 1998.**

We consider the equilibrium of a sail under aerodynamic field of external forces. The sail is considered to be an ideally flexible structure, having the behavior of a network of stress unilateral strings: all the internal efforts are traction efforts. This model leads to a Non Convex Optimization Problem and a complete theory can be established, leading to relevant results of uniqueness for the field of stresses, even if configurations of equilibrium are not unique.

**Form finding and optimization of membranes and minimal surfaces, K Bletzinger, Lecture notes, Technical university of Denmark, Lyngby, 1998.**

Membrane structures are very attractive alternatives to span large distances. They are very light, elegant, and effective. The material is optimally used since the structures are subjected only to membrane tension stresses. The art of form finding means to find the optimal deflected and finally visual shape due to a given stress distribution acting on the deformed structure. The problem is very closely related to the determination of minimal surfaces.

**Experimental and analytical study on visco-elasto-plastic characteristics of ptfe-coated glass fiber fabric under cyclic loadings, S Kato, H Minami, S Segawa, T Yoshino, Proceedings of the lightweight structures in architecture, enginnering and construction IASS/IEAust/LSAA international congress, Sydney, Australia,
1998.**

The paper discusses the validity of the constitutive equations, previously proposed by the present authors, for the visco-elasto-plastic behaviors of fabric membranes by comparing the simulated behaviors with the experimental results recently performed also by the present authors. The constitutive equations are formulated based on Fabric Lattice Model where important material elements in the fabric material are replaced into intrinsic bar elements with time dependent behaviors as well as material non-linearities for which material constants are assumed to be as much compatible with measured results in experiments. The experiments are performed under two conditions; one is the relaxation test under a constant bi-axial strains with initial bi-axial tensions and the other one is the cyclic test under re-tension by five times after each relaxation in each re-tensioning process. The experimental results are compared with the simulated behaviors and the comparison shows a fair agreement.

**The surface stress density method as a form-finding tool for tensile membranes, B Maurin, R Motro, Engineering Structures,
Vol 20, # 8, 1998.**

Form-finding for membrane tension structures is a delicate operation, which must ensure both the absence of compressive areas and interactive control of the forms generated. Until now, methods have generally been based on large displacements and strain analysis that provide non-linear formulations; resolution and computation are, therefore, too complex and cumbersome. This paper describes a new method of form-finding which reflects a wish to provide architects with a simple, effective and reliable investigations suited to their needs. The surface stress density method uses surface triangular elements with an isotropic stress tensor and leads to an interactive procedure, which converges, on configurations that satisfy the laws of static equilibrium.

**A simple orthotropic, transversely isotropic hyperelastic constitutive
equation for large strain computations, J Bonet, A Burton, Computer methods in
applied mechanics and engineering, Vol 162, # 1, 1998.**

This paper presents a simple isotropic hyperelastic constitutive equation that can be used to model fiber oriented elastic materials in the fully nonlinear range. The hyperelastic strain energy function that defines this material is given in terms of three new material parameters and equations relating these material parameters to the Poisson ratio and the Young modules of the material along the fiber direction and on the orthogonal plane are derived. Expressions for the second Piola-Kirchhoff tensor, the Cauchy stress tensor and the Lagrangean and Eulerian elasticity tensors are also obtained. Static and dynamic applications of this material are used to illustrate its performance.

**A new approach to geometric nonlinearity of cable structures, A Kwan,
Computers & Structures, Vol 67, # 4, 1998.**

The basic structural principles surrounding nonlinear behavior of cable networks are explained through the example of a two-link structure. The nonlinear static response to load for this structure is then derived explicitly using the proposed simple approach. The proposed approach is then tested on three three-dimensional cable networks and the results compared with those obtained by three other techniques, namely geometric stiffness matrix, dynamic relaxation and general minimum energy.

**Multiparametered formfinding method: Application to tensegrity systems, N
Vassart, R Motro, International Journal of Space Structures, Vol 14, # 2, 1999.**

A method allowing a multiparametered formfinding for prestressed reticulated systems with tensile and compressive members is presented. Known methods, based on geometric analysis and dynamic (dynamic relaxation) considerations have been developed for these systems but they allow generally the evolution of only one parameter. The proposed numerical method exploits the force density method. Two sets of parameters can be identified: prestress coefficients of members and coordinates or redundant nodes.

**Computer shape finding of form structures, J Meek, X Xia, International
Journal of Space Structures, Vol 14, # 1, 1999.**

This paper presents a nonlinear finite element procedure for shape finding of form structures. Through the investigation of a simple numerical method of locating geodesic lines, a convenient shape finding approach is presented. The geodesic lines are generated to be used as seams for cutting pattern. To cut the patterns efficiently and economically, the maximum widths of strips should be approximately equal. Then the goal of layout optimization is achieved.

**Cutting pattern of fabric membranes with the stress composition method, B
Maurin, R Motro, International Journal of Space Structures, Vol 14, # 2, 1999.**

The designer must focus on the determination the plane fabric strips which once assembled together and placed into position in space with anchorage points define the required membrane configuration. This paper describes a new method devoted to the calculation of the strips with the aim of overcoming the drawbacks of the traditional processes. The used mechanical approaches allow to reduce distortions and errors by taking into account the whole geometrical characteristics of the strip, its stress distribution and the material constitutive laws. The theoretical formulation associates the operations of development and reduction and is based upon an iterative numerical procedure which converges to appropriate plane strips where distortions are minimized by least square methods. Several illustrative applications point out the efficiency of the method.

**Form finding analysis in consideration of cutting patterns of membrane
structures, K Ishii, International Journal of Space Structures, Vol 14, # 2,
1999.**

Methods for shape determination analysis and cutting pattern analysis which have been made so far are summarized, and their features/problems are discussed and a new method is shown. This method is a numerical shape finding method satisfying cutting patterns conditions (geodesic line conditions). The co-ordinates of the nodal points of triangular finite elements are controlled during shape determination analysis to satisfy a geodesic line laid inside the strip. By rearranging triangular finite elements from the obtained shape on the plane surface, cutting patterns can be obtained automatically. Some examples for membrane structures are shown and discussed in the paper.

**Form finding and analysis of tension structures by dynamic relaxation, M
Barnes, International Journal of Space Structures, Vol 14, # 2, 1999.**

The paper describes numerical procedures, based on the method of dynamic relaxation with kinetic damping, for the form finding, analysis and fabrication patterning of wide-span cable nets and grid shells, uniform or variably prestressed fabric membranes and battened membrane roofs. The historical development of the method is briefly reviewed and a full description is then given which accounts for cable or strut elements, membrane elements and spline beam elements. All of these elements are implemented in their natural stiffness form allowing for gross geometrical and material non-linearities, with automatic controls to ensure stability and convergence of the method.

**Geometrical nonlinear analysis of tensegrity systems, K Kebiche, M
Kazi-Aoual, R Motro, Engineering Structures, Vol 21, # 9, 1999.**

A calculation method for structures with large deformations and displacements is developed so as to determine the tangent stiffness matrix and the internal stress vector. The formulation is established for a bar element. Application of this method for tensegrity systems allowed the study of behavior for a simple self-stressed system, the four-strut tensegrity system, in case of traction, compression, flexion and torsion loading. A structure, generated by assembly of several four-strut tensegrity systems has been calculated. The behavior under uniformly distributed load can be related to the isolate cell behavior.

**Finite element analysis of dynamic response of wrinkling membranes, S
Kang, S Im, Computer Methods in Applied Mechanics and Engineering, Vol 173, # 1-2, 1999.**

A new iterative scheme for finite element analysis of wrinkling membranes, originally devised for static analysis, is extended for analyzing dynamic response of wrinkling membranes. The scheme is found to be successfully implemented with an explicit total Lagrangian finite element code based upon the central difference method. The finite element implementation of the scheme is straightforward, and only minor modifications are needed for existing membrane finite element codes. The validity of the scheme is demonstrated via a numerical simulation of an inflating automotive airbag, made of orthotropic membranes, under impulse pressure loading.

**Form finding and analysis of tension structures by dynamic relaxation, M
Barnes, International Journal of Space Structures, Vol 14, # 2, 1999.**

The paper describes numerical procedures, based on the method of dynamic relaxation with kinetic damping, for the form finding, analysis and fabrication patterning of wide-span cable nets and grid shells, uniform or variably prestressed fabric membranes and battened membrane roofs. The historical development of the method is briefly reviewed and a full description is then given which accounts for cable or strut elements, membrane elements and spline beam elements. All of these elements are implemented in their natural stiffness form allowing for gross geometrical and material non-linearities, with automatic controls to ensure stablility and convergence of the method.

**Woven fabric composite material model with material nonlinearity for
nonlinear finite element simulation, A Tabiei, Y Jiang, International Journal of
Solids and Structures, Vol 36, # 18, 1999.**

The objective of the current investigation is to develop a simple, yet generalized, model which considers the two-dimensional extent of woven fabric, and to have an interface with nonlinear finite element codes. A micromechanical composite material model for woven fabric with nonlinear stress-strain relations is developed and implemented in ABAQUS for nonlinear finite element structural analysis. Within the model a representative volume cell is assumed. Using the iso-stress and iso-strain assumptions the constitutive equations are averaged along the thickness direction. The cell is then divided into many subcells and an averaging is performed again by assuming uniform stress distribution in each subcell to obtain the effective stress-strain relations of the subcell. The stresses and strains within the subcells are combined to yield the effective stresses and strains in the representative cell. Then this information is passed to the finite element code at each material point of the shell element.

**Formulation of constitutive equations for fabric membranes based on the
concept of fabric lattice model, S Kato, T Yoshino, H Minami, Engineering
Structures, Vol 21, # 8, 1999.**

The present paper proposes and discusses a new formulation of continuum constitutive equations for fabric membranes for architecture. The formulation is based on a fabric lattice model where the structure of the fabric membranes is replaced into an equivalent structure composed of truss bars representing yarns and coated materials. The equations consider the material nonlinearities of yarns and coatings and include crimp interchange between warps and wefts. Since the equations are formulated in an incremental form in terms of in-plane strains, the formulations can be directly applied to FEM analysis of the fabric membranes. The validity of the proposed constitutive equations is discussed by comparing the simulated results with those obtained by testing the materials under uni-axial, bi-axial and shear loadings.

**Generation of surfaces via equilibrium of forces, Y Zhang, B Tabarrok, Computers & Structures,
Vol 70, # 6, 1999.**

This paper deals with the generation of minimal surfaces subject to a prescribed volume constraint and surfaces with a given mean curvature. The generation of a surface with a desired curvature is related to finding the equilibrium state of the surface under a uniform state of tension and under a prescribed pressure. Assuming an initial configuration, the surface is discretized into nodes and elements, and its final equilibrating configuration is found in a stepwise manner. In each step, the nodal forces are calculated and the surface is allowed to undergo displacements proportional to the nodal forces. Thus, the equilibrium state of the surface is obtained iteratively. The generation of a minimal surface subject to a prescribed volume constraint is also considered and analyzed by a generalization of the method outlined. Numerical examples are given to illustrate the described approach.

**Tensioned fabric shape-finding, A Caner, R Hsu, Journal of Structural Engineering,
Vol 125, # 9, ASCE 1999.**

Tensioned glass fiber-reinforced fabric has been used in roofs and canopies for various permanent structures such as stadiums and airport terminals all around the world. At its final state of geometry, the fabric shall be all in tension in its natural stable shape. The natural shape of the fabric can be generated by refining a 2D computer model to determine the 3D state using a geometric nonlinear analysis program for the personal computer. This paper presents an actual design case using a general purpose structural analysis program to reduce the complexity involved in finding the true natural shape of the tensioned fabric roofs.

**Numerical analyses of cable roof structures, Licentiate Thesis, G Tibert,
Department of Structural Engineering, Royal Institute of Technology, Sweden,
1999.**

An extensive literature survey, concerned with both practical and theoretical aspects of cable roofs, is presented. The simple force density method is presented in detail and applied to a number of different types of cable roof structures. The method worked well for structures composed of only cables, but not for structures with compression members. Three analytical finite cable elements are presented. Two elements are mathematically exact and can accurately model both taut and slack cables using only one element per cable. A static analysis of the Scandinavium Arena in Gothenburg has been performed.

**Computer cutting pattern generation of membrane structures, X Xia, J Meek,
International Journal of Space Structures, Vol 15, # 2, 2000.**

This paper sets out a simple design procedure for generating the cutting patterns of a membrane surface which can then be used for a variety of purposes, including fabric of tension surface. It does this by pre-defining the requirements of the cutting pattern which is set out by the designer on an initially flat surface. A cable net is used for the approximating surface and the cables approximate the warp and weft directions of the fabric. The present work designs the procedure and gives examples of results of the shape finding of a surface with a number of different cutting patterns. An example of a compression membrane surface is also given.

**Geometric effects in an elastic tensegrity structure, I Oppenheim, W
Williams, Journal of Elasticity, Vol 59, # 1-3, 2000.**

Tensegrity structures are under-constrained, 3-dimensional, self-stressing structural systems. They demonstrate an infinitesimal flex and when loaded they display a nonlinear geometric stiffening. In earlier work many examples of the resulting force–displacement relationship have been demonstrated numerically, and some aspects of the force–displacement relationship have been derived analytically. In this article an energy formulation is presented for the case of a simple but representative tensegrity structure, yielding an exact solution for the force–displacement relationship.

**Numerical determination of mechanical elastic constants of textile composites, X Peng, J Cao, 15th Annual technical conference of the american society for composites, Texas, 2000.**

This paper presents a novel procedure for predicting the effective nonlinear elastic moduli of textile composites through a combined approach of the homogenization method and the finite element method. The homogenization method is used first to obtain the effective elastic moduli of the fiber yarn based on the properties of the constituent phases. A unit cell is then built to enclose the characteristic periodic pattern in the composites. Various numerical tests such as uni-axial tension and trellising test are performed by 3D finite element analysis on the unit cell. Characteristic behaviors of force versus displacement are obtained. Meanwhile, trial mechanical elastic constants are imposed on a four-node shell element with the same size as the unit cell to match the force-displacement curves. The effective nonlinear mechanical stiffness tensor is thus obtained numerically as functions of elemental strains. The procedure is exemplified on a plain weave glass composite and is validated by comparing with 30-degree bias trellising and bi-axial tensile test results.

**Nonlinear finite element for plain woven fabrics, O Kuwazuru, N Yoshikawa, 20th International congress of theoretical and applied mechanics,
Chicago, 2000.**

A new planar finite element is formulated to analyze the nonlinear behaviour of plain woven fabrics. The difficulty for the finite element formulation of the fabrics is on the handling of the nonlinearity caused by the discontinuity between threads. We introduce a strain-displacement relationship in lieu of the conventional one based on the continuum mechanics, since the discontinuity between threads violates the hypothesis of the continuum too much. Assuming that the friction among threads and the flexural rigidity of threads are negligibly small, we categorize the deformation of fabric into three types, that is, skewing, straightening and extension of threads. Each deformation is characterized by the new strain-displacement relationship with newly defined crimp parameter. The finite element is formulated on the principle of virtual work and solved by the Newton-Raphson method, since the obtained strain-displacement relationship gives rise to a kind of geometrical nonlinearity.

**New numerical model of composite fabric behaviour, J Billoet, A Cherouat, Advanced composites letters,
Vol 9, # 3, 2000.**

The present study concerns the modelling of the behaviour of pre-impregnated woven fabric during the forming process. The mechanical approach is based on a mesostructural model. It allows us to take into account the mechanical properties of fibres and resin and the various dominating mode of deformation of woven fabrics during the forming process. Shear and tensile tests of composite fabric specimens are proposed and compared with the experimental results in order to demonstrate the efficiency of our approach. Different numerical simulations and experiments of shaping process have been carried out in order to validate the proposed computational formulation. The various forming parameters examined have included the initial shape of fabric, fibre orientations and viscosity of resin.

**Developability conditions for prestress optimization of a curved surface, M Ohsaki, J Fujiwara, Architectural Information Systems Laboratory, Report
# 00-05, Kyoto university, 2000.**

Developability conditions are presented for prestressed curved surfaces to be reduced to plane sheets by relaxing the stresses. Those conditions are applied to an optimization problem for minimizing the deviation of stresses from the target values. Formulations based on the displacements defined by the local and global coordinates of finite elements are presented, and it is shown that the umbers of constraints by those formulations are same if triangular elements are used. Performances of those formulations are compared in the numerical examples, and the local formulation is shown to leads to more accurate estimation of stresses that are actually generated by connecting and stretching the plane sheets.

**Finite element analysis of air supported membrane structures, J Bonet, R
Wood, J Mahaney, P Heywood, Computer methods in applied mechanics and
engineering, Vol 190, # 5-7, 2000.**

This paper deals with the finite element analysis of closed membrane structures that contain an enclosed fluid such as air. The change in the fluid pressure resulting from the application of external forces is evaluated and taken into account in the formulation of the equilibrium equations. The membrane formulation presented avoids the need for local co-ordinate axes by using the isoparametric finite element plane as a material reference configuration. The membrane material is modelled using standard large strain hyperelastic constitutive equations. The volume of fluid is obtained by integrating over the membrane surface and Boyle's law is used to determine the changes in air pressure that result from changes in volume. Full linearization of the internal pressure forces and the resulting additional term in the tangent operator are derived.

**Finite element analysis of membrane wrinkling, K Lu, M Accorsi, J
Leonard, International Journal for Numerical Methods in Engineering, Vol
50, 2001.**

New results are presented for the finite element analysis of wrinkling in curved elastic membranes undergoing large deformation. Concise continuum level governing equations are derived in which singularities are eliminated. A simple and efficient algorithm with robust convergence properties is established to find the real strain and stress of the wrinkled membrane for Hookean materials. The continuum theory is implemented into a finite element code. Explicit formulas for the internal forces and the tangent stiffness matrix are derived. Numerical examples are presented that demonstrate the effectiveness of the new theory for predicting wrinkling in membranes undergoing large deformation.

**Finite element modelling of orthotropic material behaviour in pneumatic
membranes, S Reese, T Raible, P Wriggers, International Journal of Solids
and Structures, Vol 38, # 52, December 2001.**

In this paper, we develop a model to describe the hyperelastic material behaviour of pneumatic membranes reinforced with roven-woven fibres. A generalized stored energy function is developed via a series of loading tests on a representative sample of this composite material. The exponents in the effective law are chosen so as to fulfil basic restrictions, discussed in the body of the paper, as well as to match certain experimental values. Numerical examples demonstrate the application of the approach to inflated rubber matrix materials, as well as laminated shells.

**Form finding of membrane structures by the updated reference method with
minimum mesh distortion, J Bonet, J Mahaney, International Journal of Solids
and Structures, Vol 38, # 32-33, 2001.**

This paper presents a new technique for the solution of the well-known form finding problem in membrane structures. The technique proposed is based on the updated reference configuration method proposed by Bletzinger in which an area functional is minimized within the context of a finite element discretisation. In this paper an additional functional term is introduced with the aim of minimizing the mesh distortion during the form finding process. This new term provides in-plane stiffness which prevents the emergence of mechanisms without the need for ad hoc changes of the tangent matrix.

**Review of form-finding methods for tensegrity structures, G Tibert, S
Pellegrino, Accepted by International Journal of Space Structures, 2001.**

Seven form-finding methods for tensegrity structures are reviewed and classified. The three kinematical methods include an analytical approach, a non-linear optimisation, and a pseudo-dynamic iteration. The four statical methods include an analytical method, the formulation of linear equations of equilibrium in terms of force densities, an energy minimisation, and a search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. It is concluded that the kinematical methods are best suited to obtaining only conﬁguration details of structures that are already essentially known, the force density method is best suited to searching for new conﬁgurations, but affords no control over the lengths of the elements of the structure. The reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations.

**A bendable finite element for the analysis of flexible cable structures, P Gosling, E Korban, Finite elements in analysis and design,
Vol 38, 2001.**

An element formulation is developed for an analysis of cable structures where load reversal (tension to compression) may be significant. Strain energy associated with the deformation of an individual element is defined to include both membrane and bending components. Through the assumption of large strains and prestress, geometric non-linearity is introduced. Isoparametric absolute and relative interpolation schemes are combined to describe a C0 eight-degrees-of-freedom cable element applicable to three-dimensional space subsequent to transformation. Coupled with the Newton-Rapson scheme, solutions to numerical examples validate and demonstrate the capabilities of the proposed methodology.

**New concept of pseudo-continuum model for plain-weave fabrics, O Kuwazuru,
N Yoshikawa, Advancing affordable materials technology, pages 564-573, Proceedings of the 33rd SAMPE international technical conference, Seattle, 2001.**

A new concept of pseudo-continuum model is proposed to analyze the complicated nonlinear behaviour of plain-weave fabrics. Subjected to an extension in an arbitrary direction, plain-weave fabric undergoes three kinds of thread deformation, that is, skewing, straightening and extension. On the assumption of no friction and no flexural rigidity with respect to the threads, a pseudo-continuum model is constituted by a new strain-displacement relationship which translates the three kinds of deformation to two kinds of strain, that is, axial extension and transverse compression of warp and weft. The finite element is formulated by means of the principle of virtual work in the total Lagrangian form. The mechanical nonlinearity and anisotropy of the plain-weave fabrics caused by geometrical nonlinearity of threads are elucidated through numerical examples concerning the uniaxial tensile tests.

**FE analysis of large deformations of membranes with wrinkling, M
Stanuszek, Finite Elements in Analysis and Design, article in press, accepted 12
November 2001.**

The numerical analysis of large deformations of flexible membrane structures is considered in this paper. Taking advantage of the natural approach applied to the tension systems, complex membrane structures of arbitrary shape with wrinkling allowed are analyzed. In the study, the Finite Element Method with triangular membrane elements is used. The geometrical nonlinearity as a result of large displacements, shape dependent loads (internal pressure) and unilateral static boundary conditions add considerable complexity to the analysis of such objects. In the case of wrinkling a new concept of taking wrinkles into account based on the cable analogy is proposed.

**Cutting pattern design of membrane structures considering viscoelasticity
of material, J Fujiwara, M Ohsaki, K Uetani, Proceedings of the IASS Symposium
on Theory, design and realization of shell and spatial structures, Nagoya,
Japan, 2001.**

A method is presented for determination of cutting patterns of membrane structures considering viscoelasticity of material. A constitutive law is proposed to represent viscoelastic behavior of material in the range around the target stress level. By using the proposed constitutive law, relaxation behavior of a membrane structure is estimated without time-history analysis. The effectiveness of the proposed method is discussed in the example.

**Non-linear dynamic analysis of
cable-suspended structures subjected to wind actions, M Lazzari, A Saetta, R
Vitaliani, Computers & Structures, Vol 79, # 9, 2001.**

The numerical analysis of the response of wind-loaded flexible structures is presented. Initially the modeling and simulation of wind velocity are studied, by considering stationary, multivariate stochastic process, according to its prescribed cross-spectral density matrix. In the second part of the paper, geometrically non-linear structures subjected to wind loads are investigated, by means of a finite element approach. One test example is presented to show the reliability of the numerical procedure to solve geometrically non-linear problem in dynamic field. Finally, the study of a real structure characterized by an initial pre-tension layer subject to wind action is carried out.

**Deployable tensegrity structures for space applications, Doctoral Thesis,
G Tibert, Department of Mechanics, Royal Institute of Technology, Sweden, 2002.**

The analysis and design of deployable tensegrity masts, with three struts per stage, is described. A routine for the manufacturing of physical models is proposed and evaluated. Different schemes for deployment are investigated. A way to deploy the struts using self-deployable hinges is introduced and demonstrated by four- and eight-stage mast models. The requirements for a deployable reflector antenna used on small satellites are formulated. A concept, which uses a triangulated cable network to approximate the reflecting surface, is adopted. The kinematically determinate triangulated cable network is thoroughly analyzed.

**Form controlled for tensegrity formfinding Snelson and Emmerich examples,
R Motro, A Smaili, O Foucher, Lightweight Structures in Civil Engineering, IASS,
Poland, 2002.**

Tensegrity systems realize a coupling between the visible geometry and the invisible state of self-stress. They are complex to design and require form finding processes. Two classes can be identified: form controlled and force controlled processes. Two main contributions to form controlled design are developed. The work that has been achieved by Snelson begins with three main sculptures: one to another, one to the next and X-shape. David Georges Emmerich also developed form controlled studies based mainly on polyhedra. Several examples are described with considerations on their equilibrium: prisms, antiprisms, stella octangula, regular and semi regular are examined. Results obtained with models, and numerical models are produced.

**Linear dynamics of tensegrity structures, C Sultan, M Corless, R Skelton,
Engineering Structures, Vol 24, 2002.**

The linearized equations of motion for tensegrity structures around arbitrary equilibrium configurations are derived. For certain tensegrity structures which yield particular equilibrium configurations of practical interest, the linearized models of their dynamics around these configurations are presented. Evidence which indicates that these equilibria are stable is given and some stiffness and dynamic properties of these structures are investigated.

**Accuracy in sail simulation: Wrinkling and growing fast sails, P Heppel,
High Performance Yacht Design Conference, Auckland, 2002.**

This paper describes some techniques for improving accuracy in the computation of structural membranes. Of importance in membrane computations is the modeling of the wrinkling of the surface that occurs to relieve compression. The paper describes an improvement to current theory by consideration of optimum-shift in the stiffness matrix derivation. The paper also describes the way that wrinkling interacts with the kinematics of the element grid. Results are presented demonstrating the effects of varying grids, with and without the modeling of wrinkling.

**Numerical and experimental aeroelastic analysis of sails, D Coiro, F
Nicolosi, F Scherillo, U Maisto, High Performance Yacht Design Conference,
Auckland, 2002.**

A computer code has been developed to perform the viscous aerodynamic analysis of a multi-sail system including mast effect. The code is based on 3D vortex lattice method coupled to 2D boundary layer solution along streamlines. Three nodes iso-parametric triangular elements have been chosen for the finite element method. To validate the structural computer code an experimental test has been set up and a simple machinery has been built allowing shape measurements of a rectangular membrane under constant pressure load.

**Cable nets for bat habitat preservation, J Kretzmann, New Mexico
Abandoned Mine Land Bureau, Santa Fe, New Mexico, 2002.**

Cable-supported structures have been used in architectural and engineering practice for long spans, such as suspension bridges, and to cover large areas with a minimum of support columns, such as sports arenas and aviaries. Similarly, in bat habitat preservation in underground mines and caves, use of cable nets is particularly well adapted to large span, usually vertical, openings. Nets also provide a solution at smaller vertical openings where equipment access is constrained because of steep slopes or other barriers. A critical design and construction requirement for cable nets is the necessity for solid anchorage, generally into competent rock around the opening.

**A finite-element model for the analysis of wrinkled membrane structures, R
Ziegler, W Wagner, K Bletzinger, Universitat Karlsruhe, Institut fur Baustatik,
2002.**

The problem of wrinkling in membrane structures has been a field of research since the publication of the tension field theory for plane structures. Significant progress in wrinkling analysis of arbitrarily shaped membranes has been made with the development of numerical methods. In the paper we present the enhancements of a standard finite element membrane formulation which allow to depict the wrinkles within the plane of the structure. A mathematical-numerical method is derived, which describes a valid stress state by minimizing the differences in the stress density function while observing the wrinkling conditions. A consistent linearization of the proposed algorithms ensures quadratic convergence behavior.

**A new technique for optimum cutting pattern generation of membrane
structures, J Kim, J Lee, Engineering Structures, Vol 24, # 6, 2002.**

In general, the cutting pattern for membrane structures is determined by dividing the complicated curved 3-D surface into several 2-D plane strips by using the geodesic line method or flattening technique. In this paper, a new analytical method for determining an optimum cutting pattern considering material properties is presented. The optimization method proposed can diminish the deviations occurring from numerical errors as well as from material properties.

**Finite element analysis of thin membrane wrinkling, MSc Thesis, J Mansson,
J Soderqvist, Department of Mechanics, Royal Institute of Technology, Sweden,
2003.**

The wrinkling of thin membranes has been analyzed in many stages and there are a number of theoretical approaches that have been developed over the years. These theories consider membrane elements and are therefore only valuable to identify wrinkled regions and not the actual configuration of these wrinkles. In the finite element method, the general approach assumes that the wrinkles are associated with bifurcation buckling. This approach involves a linear eigenvalue analysis with a highly localized type of instability and a post buckling analysis with hand-tuned numerical stabilization. The finite element package ABAQUS is presented and the choice of relevant features when modeling thin membranes is described. The analysis is divided into three main steps: the initial conditions, the eigenvalue buckling analysis, and the post buckling analysis.

**Formfinder - concept for a software-tool to assist architects in the
preliminary design of form-active structures, PhD Thesis, R Roithmayr, Vienna
University of Technology, Austria, 2003.**

The text describes the concept for the realisation of a software-tool entitled Formfinder. The tool is intended to assist architects in the preliminary design of form-active systems. Form-active structure systems are structure systems of flexible, non-rigid matter, in which the redirection of forces is effected through particular form design and characteristic form stabilization. The designer outlines the desired design just as he or she would use a pen and a sheet of paper. The software visualizes the behaviour of a form-active system and guides the designer to a possible next step. A model-based recognition algorithm analyses the sketch and compares analogies with information stored in a data base.

**Equilibrium conditions of a tensegrity structure, D Williamson, R Skelton,
J Han, International Journal of Solids and Structures, Vol 40, # 23, 2003.**

This paper characterizes the necessary and suffcient conditions for tensegrity equilibria. Static models of tensegrity structures are reduced to linear algebra problems, after first characterizing the problem in a vector space where direction cosines are not needed. This is possible by describing the components of all member vectors. While our approach enlarges (by a factor of 3) the vector space required to describe the problem, the advantage of enlarging the vector space makes the mathematical structure of the problem amenable to linear algebra treatment. Using the linear algebraic techniques, many variables are eliminated from the final existence equations.

**The making of a tensegrity tower, H Klimke, S Stephan, MERO GmbH, 2003.**

A tensegrity tower was conceived for the fair in Rostock (Germany). The modules of the tower consist of three compression members of about 10 m length and nine cables, six horizontal cables and three diagonal cables. The key problem of tensegrity structures with respect to the production is the big movement of each module due to prestressing of the cables. All deflections have to be anticipated in the design of the components to eventually meet the desired geometry of the tower.

**Finite element analysis of membrane structures, R Taylor, University of
California at Berkeley, 2003.**

This report summarizes the formulation for a large displacement formulation of a membrance composed of three-node triangular elements. A formulation in terms of the deformation gradient is first constructed in terms of nodal variables. In particular, the use of the right Cauchy-Green deformation tensor is shown to lead to a particulary simple representation in terms of nodal quantities. This may then be used to construct general models for use in static and transient analyses.

**Approximate solution for a nearly flat square membrane subject to a
uniform force per unit area, V Arcaro, A Palisoc, Proceedings of the IASS
international symposium on shell & spatial structures, Taiwan, 2003.**

This text presents an approximate solution for a nearly flat square membrane subject to a uniform transverse force per unit area. The total potential energy of the membrane is minimized with respect to the parameters of an assumed function for its displacements. This text corrects an error in the expression presented in the book Theory of Plates and Shells by Timoshenko and Woinowsky-Krieger (1959) and also in the book Tension Structures by Leonard (1988), extending the analytical solution for any value of the Poisson ratio. Comparing the analytic solution with the solution from ANSYS shows reasonable agreement.

**Stability and mechanism order of isotropic prestressed surfaces, B Maurin,
R Motro, International Journal of Solids and Structures, Vol 41, # 9-10,
2004.**

The surface stress density method has been proposed as an effective form-finding tool for the design of fabric membranes. It enables tensile shapes to be determined by considering the isotropic prestress tensors in the membrane. The first objective of this paper is to demonstrate that the forms calculated in accordance with this mechanical property are stable. The second is to calculate their mechanism order. The approach is based on an energy criterion, pointed out by writing out the potential strain energy of the system and by using Lejeune-Dirichlet's theorem.

**A class of orthotropic and transversely isotropic hyperelastic
constitutive models based on a polyconvex strain energy function, M Itskov, N
Aksel, International Journal of Solids and Structures, Vol 41, # 14,
2004.**

In the present paper we propose a set of orthotropic and transversely isotropic strain energy functions that (a) are polyconvex, (b) are proved to be coercive and (c) satisfy a priori the condition of the stress-free natural state. These conditions ensure the existence of the global minimizer of the total elastic energy and for this reason are very important in the context of a boundary value problem. The proposed hyperelastic model is represented by a power series with an arbitrary number of terms and corresponding material constants.

**Deformation analysis of inflated cylindrical membrane of composite with
rubber matrix reinforced by cords, B Marvalova, T Nam, XXI International
Congress of Theoretical and Applied Mechanics, 2004.**

We present the orthotropic hyperelastic material model for numerical simulation of the loading of the cylindrical membrane. The coefficients of strain energy function of the hyperelastic orthotropic material are fitted to the experimental results by the nonlinear least squares method. The components of the deformation gradient are determined from measured displacements of the grid points drawn on the cylindrical surface of the spring. The stress tensor is calculated from the membrane theory.

**Optimization of class-2 tensegrity towers, M Masic, R Skelton, SPIE 11th
Annual International Symposium on Smart Structures and Materials, 2004.**

This paper concerns the optimal mass-to-stiffness ratio design of class-2 tensegrity towers. For different loading scenarios, the procedure seeks the topology and geometry of the structure that yields an optimal design satisfying common constraints. The domain of feasible tensegrity geometries is defined by imposing tensegrity equilibrium conditions on both unloaded and loaded structure. Remaining constraints include strength constraints for all elements of the structure and buckling constraints for bars. The symmetry of the design is imposed by restricting the domain of geometric variables and element parameters. The static response of the structure is computed by using a nonlinear large displacement model. The problem is cast in the form of a nonlinear program. The infuence of material parameters on the optimal shape of the structure is investigated.

**An integrated analysis of membrane structures with flexible supporting
frames, J Li, S Chan, Finite Elements in Analysis and Design, Vol 40, #
5-6, 2004.**

Conventional analysis and design for tensioned membrane structures are separated by two assemblages, fixing the support positions and determining the equilibrium shape of the cable-membrane at first and checking the adequacy of the steel structure against support reactions. Under this methodology, the interaction between the cable-membrane and the steel structure is neglected. An integrated nonlinear finite element (FE) analysis, including cable element, membrane element and beam element in the FE library, is proposed in this paper for analysis of tensioned membranes supported by steel structures. The interaction between the support structure and the cable-membrane is examined through numerical study of a saddle shade pavilion structure.

**Numerical solution of hyperelastic
membranes by energy minimization, R Bouzidi, A Van, Computers & Structures,
Vol 82, # 23-26, 2004.**

A numerical approach is presented for solving problems of finitely deformed membrane structures made of compressible hyperelastic material and subjected to external pressure loadings. Instead of following the usual finite element procedure that requires computing the material tangent stiffness and the geometric stiffness, here we solve the membrane structures by directly minimizing the total potential energy, which proves to be an attractive alternative for inflatable structures.

**Finite element formulation for modeling sliding cable elements, B Zhou, M
Accorsi, J Leonard, Computers & Structures, Vol 82, # 2-3, 2004.**

Sliding cable elements are developed to solve the general problem of constraining a string of cable elements to continuously pass through a prescribed moving node. These elements can be used for a wide variety of applications and in the current work are used to model various features in parachute systems. The principle of virtual work and total Lagrange formulation are used to derive the element internal force vector, tangent stiffness matrix, and time-dependent mass matrix and body forces. The element equations are implemented in a geometrically nonlinear, transient implicit finite element program.

**Development of a wrinkling algorithm for orthotropic membrane materials, T
Raible, K Tegeler, S Lohnert, P Wriggers, Computer Methods in Applied Mechanics
and Engineering, Vol 194, # 21-24, 2005.**

A commonly known problem while investigating membrane structures is the well wrinkling phenomena. In this work, we present and compare robust algorithms to predict wrinkled regions within complex membrane structures. Special focus is set on the application to isotropic and orthotropic membrane material formulations. As reference the solution of a 3-dshear test calculation is used. Special focus is set on the local and global characteristics of wrinkling.

**Description and comparison of existing methods for static membrane
structure formfinding, MSc Thesis, D Cooper, University of Stuttgart, Germany,
2005.**

Different mechanically oriented methods and algorithms developed in the past to determine the form of membrane structures will be presented to give a state of the art. Two methods will be picked out and they will be discussed in more detail and illustrated with numerical examples. The results will then be analyzed to compare these methods.

**David Georges Emmerich Professor of morphology, A Chassagnoux,
International Journal of Space Structures, Vol 21, # 1, 2006.**

David Georges Emmerich taught morphology at the Ecole des Beaux Arts, and later at the Paris La Villette School of Architecture from 1965 to 1990. An architect and engineer by training, convinced of the modern movement’s inability to provide mankind with the architectural space needed, his research led to constructive systems using cheap, industrialized components, with wide scope for self help housing as well as a broad range of architectural structures. His extensive study of regular partitioning in space, natural shapes, the resistance of shapes and combinatorial analysis led him to developing stereometric systems and, more specifically, to the invention of self tensioning or tensegrity structures.

**Form-finding of nonregular tensegrity systems, L Zhang, B Maurin, R Motro,
ASCE Journal of Structural Engineering, Vol 132, # 9, 2006.**

The potential applications of tensegrity structures are not only increasing in civil engineering but also in fields like biomechanics. The key step in designing tensegrity, the form-finding problem, has been investigated by many researchers but until now they have tended to focus on methods for regular shapes. Since there is an increasing need for design tools devoted to more various and complex systems, the objective of this paper is to present the form-finding of nonregular tensegrity structures with a numerical approach. It is based on the dynamic relaxation method with kinetic damping, and new tensegrity configurations in more intricate and creative forms can be obtained this way. During the form-finding process, either the force or length of some elements can be fixed by an appropriate choice of related stiffnesses. The application of the process is illustrated by several numerical examples.

**A direct approach to design of geometry and forces of tensegrity systems,
J Zhang, M Ohsaki, Y Kanno, International Journal of Solids and Structures, Vol
43, # 7-8, 2006.**

In the process of designing a tensegrity system, some constraints are usually introduced for geometry and/or forces to ensure uniqueness of the solution, because the tensegrity systems are underdetermined in most cases. In this paper, a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively, under the equilibrium conditions and the given constraints, to satisfy the distinctly different requirements of architects and structural engineers. The proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved, and no equation of kinematics or material property is needed. Some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations.

**Numerical form-finding of tensegrity structures, G Estrada, H Bungartz, C.
Mohrdieck, International Journal of Solids and Structures, 43, 2006.**

A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favorably with all the leading techniques in the field.

**Finite element analysis of prestressed structural membranes, A Gil, J
Bonet, Finite Elements in Analysis and Design, 42, 2006.**

A very powerful approach by means of the Nonlinear Continuum Mechanics theory is introduced for the analysis of prestressed membrane structures. Membranes and cables in taut, wrinkled or slack state are considered adequately in the numerical procedure. A finite element approximation along with a Newton-Raphson numerical scheme provide a very elegant and accurate way to solve the structural problem. To reveal the flexibility and robustness of the procedure, a complete assemblage of fabric textile, reinforcing cables and rigid members will be analyzed from its initial design stage to its final loaded configurations.

**Eight-node quadrilateral double-curved surface element for membrane
analysis, D Hegyi, I Sajtos, G Geiszter, K Hincz, Computers & Structures,
Vol
84, # 31-32, 2006.**

The dynamic relaxation method is applied to membrane analysis using an eight-node quadrilateral element. The element uses second order shape functions to approximate the geometry of the structure. The element is based on the element of Gosling and Lewis. They used a finite element approach. In this paper exact tensorial calculation is used to determine the exact deformation between the deformation-free state and the actual state.

**RANSE investigations of downwind sails and integration into sailing yacht
design processes, K Graf, H Renzsch, 2nd High Performance Yacht Design
Conference, Auckland, 2006.**

A fluid structure interaction method has been developed to calculate the viscous turbulent flow around flexible trimmable downwind sails. The structural part of the method is based on a Finite Element presentation of the sail modelled as a membrane. Minimization of the total potential energy function using a quasi Newton type method is carried out to calculate the displacement of the sail under aerodynamic loads. The flow around the deformed sail is calculated solving the Reynolds Averaged Navier Stokes Equation using a Finite Volume approach.

**Wrinkled membranes I: experiments, Y Wong, S Pellegrino, Journal of
Mechanics of Materials and Structures, Vol 1, # 1, 2006.**

This paper presents a detailed experimental study of the evolution and shape of reversible corrugations, or wrinkles, in initially flat, linear-elastic and isotropic thin foils subject to in-plane loads. Two sets of experiments were carried out, on a rectangular membrane under simple shear and on a square membrane subjected to two pairs of equal and opposite diagonal forces at the corners.

**Wrinkled membranes II: analytical models, Y Wong, S Pellegrino, Journal of
Mechanics of Materials and Structures, Vol 1, # 1, 2006.**

We present a general analytical model for determining the location and pattern of wrinkles in thin membranes and for making preliminary estimates of their wavelength and amplitude. A rectangular membrane under simple shear and a square membrane subject to corner loads are analysed.

**Wrinkled membranes III: numerical simulations, Y Wong, S Pellegrino,
Journal of Mechanics of Materials and Structures, Vol 1, # 1, 2006.**

This is the third and final part of a study of wrinkles in thin membrane structures. High-fidelity, geometrically nonlinear finite element models of membrane structures, based on thin-shell elements, are used to simulate the onset and growth of wrinkles. The simulations are carried out with the ABAQUS finite element package.

**Application of the Murnaghan model in analysis of non-linear elastic
material properties of pvc-coated fabric, A Ambroziak, Task Quarterly, Vol
10, # 3, 2007.**

The aim of the present paper is to propose a method of laboratory tests necessary for identification of non-linear elastic properties of the PVC-coated Panama fabric often used for hanging roofs. Two methods of describing the fabric’s non-linear behavior are investigated: piece-wise linear relations between stress and strain are assumed and the Murnaghan model of solid behavior is applied. The material parameters are specified on the basis of uniaxial constant strain rate tensile tests in the warp and weft directions. Techniques based on the least squares methods are applied in the identification process.

**Aerodynamic analysis of sails with OpenFOAM, M Ledri, M Poian, R Russo,
ESTECO,
Second OpenFOAM Workshop, Zagreb, June 2007.**

To model a sail boat different problems have to be addressed: 1) Free surface flows. 2) Aerodynamic study of hull appendices. 3) Aerodynamic analysis of sails. The current technology already provides empirical relations to model all the hydrodynamic and aerodynamic forces on the boat, once some geometrical parameters are given (Velocity Prediction Programs). So far the design of sails has been mainly based upon experience or at most upon direct experimental measurements. CFD analysis is not straightforward either: 1) Massive flow separation and intrinsic unsteadiness, above all for downwind sails. 2) Wind loaded sail shape (flying shape) is not easily predictable. The aim of this work was to set up applications to address this kind of problems.

**Improved procedures for static and dynamic analyses of wrinkled membranes,
A Shaw, D Roy, Journal of Applied Mechanics, Vol 74, 2007.**

An analysis of large deformations of flexible membrane structures within the tension field theory is considered. A finite element procedure is proposed to study the wrinkling behavior of a membrane element. The state of stress in the element is determined through a modified deformation gradient corresponding to a fictive non wrinkled surface. The model uses a continuously modified deformation gradient to capture the location orientation of wrinkles more precisely. It is argued that the fictive non wrinkled surface may be looked upon as an everywhere taut surface in the limit as the minor (tensile) principal stresses over the wrinkled portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere differentiable tensors.

**Evaluation of membrane structure designs using boundary web cables for
uniform tensioning, H Sakamoto, K Parka, Y Miyazaki, Acta Astronautica, 60,
2007.**

The present paper begins with a brief review of existing designs of membranes surrounded by catenary cables and shear compliant borders. The paper then introduces weblike cables that surround the membrane, and analyzes the consequent reduction in mass and volume. In addition, a series of quasi-static finite element analyses demonstrates the attenuation of wrinkles by the web cables when support points are perturbed. The paper concludes that the proposed design preserves a biaxially prestressed membrane even in disturbed conditions, with a minimal suspension-cable mass and volume.

**Advanced material model for coated fabrics used in tensioned fabric
structures, J Pargana, D Lloyd-Smith, B Izzuddin, Engineering Structures,
Vol 29, # 7,
2007.**

An accurate and reliable material model for plain weave coated fabrics used in the construction of Tensioned Fabric Structures is developed in this paper. The paper begins with an account of the material response of these fabrics, furnishing an understanding of the key elements that a successful material model should encompass. The paper also contains a review of existing models, which has highlighted the need for an accurate, reliable and fully calibrated material model. The proposed material model consists of a series of nonlinear elastic elements, frictional elements and rigid links to model the yarns, and an isotropic plate to model the coating.

**Modelling and optimization of sails, M Spalatelu-Lazar, F Lene, N Turbe, Computers &
Structures, article in press, 2007.**

The aim of this paper is to improve the quality and the performances of sails by using non-linear modeling, numerical experimentation and optimization methods. The sail is represented by a triangular structure made of composite materials (unidirectional or woven composite) well modeled by an orthotropic membrane behavior. Under the wind pressure, the sail is submitted to large displacements and small strains. Initial pre-tension load which ensures that the surface is reasonably free of wrinkles is required. The numerical solution is carried out by means of a modified Newton-Raphson method. The mathematical problem of optimization relates to the displacement in the transverse direction of the sail. The parameter of design is the fiber orientation. The optimization method uses the Nelder-Mead algorithm, efficient to solve non-linear problems.

**Form finding analysis of tensegrity membrane structures based on
variational method, M Shigematsu, M Tanaka, H Noguchi, Proceedings of the
IASS-IACM 2008, Cornell University, 2008.**

A tensegrity membrane structure is presented. Firstly, through a static nonlinear finite element analysis, structural response is investigated to ensure that this structure can be established. Then, several tensegrity membrane structures are demonstrated to show that this could be more rational and universal than tensegrity or membrane structures themselves and might be referred to the next generation of space structures.

**A New Tensegrity Module - Torus, X Yuan, Z Peng, S Dong, B Zhao, Advances
in Structural Engineering, Vol 11, # 3, 2008.**

Research on cylindrical and spherical tensegrity modules are extensively carried out. However research on other tensegrity modules is little reported. This paper presents an exploratory study on a new kind of tensegrity module – torus. The topology of the torus tensegrity is firstly introduced. Then the initial form-finding of the torus tensegrity is discussed. The static and dynamic analysis of the torus tensegrity shows that prestressing has a stiffening effect on infinitesimal mechanism modes if the geometry is properly arranged. A new cable dome with a torus tensegrity employed as its ring beam is finally proposed. The behavior of the new dome is also examined.

**Comparison between experimental tests and numerical simulations carried
out on a tensegrity minigrid, J Dube, N Angellier, B Crosnier, Engineering
Structures, Vol 30, # 7, 2008.**

Tensegrity systems are structures in equilibrium under an initial self-stress state. A continuous dialogue between numerical simulations and experimental tests made it possible to validate previous models. Static and vibratory measurements clearly show that the bending moment of the elements influences the behavior of the structure.

**FLEXSAIL - A fluid structure interaction program for the investigation of
spinnakers, H Renzsch, O Muller, K Graf, Innovation in High Performance Sailing
Yachts, Lorient, France, May 2008.**

A numerical method has been developed based on a RANSE solver for flow analysis and a Finite Element Method for the structural investigation of the sail. A twisted flow wind tunnel is used to validate the numerical method. The numerical method consists of a fluid structure interaction method, that is, forces from flow simulation are used to predict the deformed shape of the sail, while in turn the deformed shape is used to predict the flow forces. The structural part of the method is based on a Finite Element presentation of the sail modeled as a membrane. The flow around the deformed sail is calculated solving the Reynolds Averaged Navier Stokes Equation using a Finite Volume approach. The paper describes results from the numerical method as well as validation investigations for a spinnaker of a typical cruiser racer yacht.

**Whole skin locomotion inspired by amoeboid motility mechanisms, D W Hong,
M Ingram, D Lahr, Transactions of the ASME, Vol 1, February 2009.**

A locomotion mechanism for mobile robots inspired by how single celled organisms use cytoplasmic streaming to generate pseudopods for locomotion is presented. It works by way of an elongated toroid which turns itself inside out in a single continuous motion, effectively generating the overall motion of the cytoplasmic streaming ectoplasmic tube in amoebae. Descriptions of an early prototype and the preliminary experimental and finite element analysis results demonstrating the feasibility of the whole skin locomotion strategy are also presented.

**A FEM-MATLAB code for fluid-structure interaction coupling with
application to sail aerodynamics of yachts, D Trimarchi, C Rizzo, 13th Congress
of the International Maritime Association of the Mediterranean, Istanbul,
Turkey, 2009.**

A MATLAB code has been developed for deformation analysis of sails. Sails are modeled as isotropic homogeneous membranes reinforced with cables. The problem, fully non-linear, is resolved by assembling the global stiffness matrix of a mesh of membrane and cable elements in the MATLAB environment to get an n-equations n-unknowns system. Validation has been performed by comparing numerical results with analytical solutions of geometrically simple cases and with experimental data. A fluid structure interaction analysis of a main sail has been carried. The result is in accordance with the physics of the phenomena and engineering judgment.

**Finite element analysis for geometrical shape minimization, V Arcaro, K
Klinka, Journal of the International Association for Shell and Spatial
Structures, Vol 50, # 2, 2009.**

This paper presents a mathematical model for minimizing path length, surface area and volume using a line element, a triangle element and a tetrahedron element respectively. The triangle element together with the line element can be used to minimize a surface area with free boundaries through the unifying concept of minimizing shape volume. A quasi-Newton method is used, which avoids the evaluation of the Hessian matrix as required in a Newton method. The source and executable computer codes of the algorithm are available for download from the website of one of the authors.

**Finite element based form-finding algorithm for tensegrity structures, M
Pagitz, J Tur, International Journal of Solids and Structures, Vol 46, # 17,
2009.**

This paper presents a novel form-finding algorithm for tensegrity structures that is based on the finite element method. The required data for the form-finding is the topology of the structure, undeformed bar lengths, total cable length, prestress of cables and stiffness of bars. The form-finding is done by modifying the single cable lengths such that the total cable length is preserved and the potential energy of the system is minimized. Two- and three-dimensional examples are presented that demonstrate the excellent performance of the proposed algorithm.

**On the implementation of a wrinkling, hyperelastic membrane model for skin
and other materials, S Evans, Computer Methods in Biomechanics and Biomedical
Engineering, Vol 12, # 3, 2009.**

A number of researchers have studied the mechanical properties of skin and developed constitutive models to describe its behaviour. Typically, many of these studies have concentrated on the uniaxial tensile behaviour of the skin, on the grounds that it will wrinkle under in plane compression and have minimal stiffness. However, although there is a substantial body of literature on wrinkling models, the practical implementation of such a model of skin in a finite element setting has not been widely addressed. This paper presents computational details of a wrinkling, hyperelastic membrane model and aspects of its implementation and areas requiring further research are discussed.

**A finite element for form-finding and static analysis of tensegrity
structures, D Gasparini, K Klinka, V Arcaro, Journal of Mechanics of Materials
and Structures, Vol 6, # 9-10, 2011.**

This text describes a mathematical model for both form-finding and static analysis of tensegrity structures. A special line element that shows constant stress for any displacement of its nodes is used to define a prestressed equilibrium configuration. The form-finding and static analysis are formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy and the displacements of the nodal points are the unknowns. Analytical solutions for tensegrity prisms are presented and compared with the numerical results of the proposed approach. A quasi-Newton method is used, which avoids the evaluation of the tangent stiffness matrix. The source and executable computer codes of the algorithm are available for download from the website of one of the authors.

**Nonlinear dynamic response and design of cable nets, PhD Thesis, I
Vassilopoulou, School of Civil Engineering, National Technical University of
Athens, Greece, 2011.**

Firstly a simple cable net is studied, consisting of two crossing cables and the equation of motion is derived. A simplified single-degree-of-freedom cable net is assumed. Proceeding to multi-degree-of-freedom systems, a saddle-form cable net with circular plan view is assumed. The cable net boundary is considered either as rigid, with cable ends modeled as pinned, or as flexible, modeling the deformable edge ring. Modeling the ring is proved to influence significantly the symmetric vibration mode of the net, due to the ring’s in-plane mode, which induces a symmetric oscillation to the net. On the other hand, the anti-symmetric modes of the net remain unaltered irrespectively of whether the cable supports are considered as fixed or as flexible. Next, the influence of the spatial load distribution on the response of a cable net subjected to harmonic loads is investigated.

**Tensegrity applied to modelling the motion of viruses, S M Cretu, G C
Brinzan, Acta Mechanica Sinica, Vol 27, # 1, 2011.**

A considerable number of viruses’ structures have been discovered and more are expected to be identified. Different viruses’ symmetries can be observed at the nano scale level. The mechanical models of some viruses realized by scientists are described in this paper, none of which has taken into consideration the internal deformation of subsystems. The authors’ models for some viruses’ elements are introduced, with rigid and flexible links, which reproduce the movements of viruses including internal deformations of the subunits.

**Application of tensegrity to tensile-textile constructions: formfinding
and structural analysis, D Pena, I Llorens, R Sastre, D Crespo, J Martinez,
Journal of the International Association for Shell and Spatial Structures, Vol
52, # 2, 2011.**

This paper applies tensegrity to create an architectural structure such as those that could be used for sports arenas or other buildings requiring large, open spaces. This proposal generates an external tensegrity ring with a central dome, free of any interior support, by formfinding a diamond-shaped membrane with discontinuous struts in a double layer structure that finds its equilibrium through the pretension of the membrane. The tendons that are used in traditional tensegrity structures are replaced by membranes.

**Structural morphology of tensegrity systems, R Motro, Meccanica, Vol 46, #
1, 2011.**

The coupling between form and forces, their structural morphology, is a key point for tensegrity systems. In the first part of this paper we describe the design process of the simplest tensegrity system which was achieved by Kenneth Snelson. Some other simple cells are presented and tensypolyhedra are defined as tensegrity systems which meet polyhedra geometry in a stable equilibrium state. A numerical model giving access to more complex systems, in terms of number of components and geometrical properties, is then evoked. The third part is devoted to linear assemblies of annular cells which can be folded. Some experimental models of the tensegrity ring which is the basic component of this hollow rope have been realized and are examined.

**An Integrated Framework for Finite-Element Modeling of Mitral Valve
Biomechanics from Medical Images: Application to MitralClip Intervention
Planning, T Mansi, I Voigt, B Georgescu, X Zheng, E A Mengue, M Hackl, R I
Ionasec, T Noack, J Seeburger, D Comaniciu, Siemens Corporate Research, Image
Analytics and Informatics, Princeton, 2012.**

Finite-element models (FEM) of MV physiology have been proposed to study the biomechanical impact of MV repair, but their translation into the clinics remains challenging. As a step towards this goal, we present in this manuscript an integrated framework for finite-element modeling of the MV closure based on patient-specific anatomies and boundary conditions. Starting from temporal medical images, we estimate a comprehensive model of the MV apparatus dynamics, including papillary tips, using a machine-learning approach. A detailed model of the open MV at end-diastole is then computed, which is finally closed according to a FEM of MV biomechanics. Mitral annulus and papillary tips motions are constrained from the image data for increased accuracy. A sensitivity analysis of our system shows that chordae rest length and boundary conditions have a significant influence upon the simulation results. We quantitatively test the generalization of our framework on 25 consecutive patients.

**An overview and comparison of structural form finding methods for general
networks, D Veenendaal, P Block, International Journal of Solids and Structures,
Vol 49, # 26, 2012.**

This paper discusses and compares existing form finding methods for discrete networks. Well-known methods such as the force density method, dynamic relaxation, updated reference strategy and others are discussed by mathematically structuring and presenting them in the same way, using the same notation and combining terminology. Based on this, a single computational framework using a sparse branch node data structure is presented. It is shown how each method approaches the initial equilibrium problem, defines and linearizes the equilibrium equations applied to linear elements, and uses particular solving strategies. This framework marginalizes any differences related to operating platforms, programming language and style, offering a better baseline for independent comparison of performance and results. As a consequence, it is possible to more clearly relate, distinguish and compare existing methods, allow for hybrid methods and identify new avenues for research.

**Analysis of downwind sail structures using non-linear shell finite
elements - wrinkle development and fluid interaction effects, PhD Thesis, D
Trimarchi, University of Southampton, United Kingdom, 2012.**

The turbulent flow is here analyzed with a Reynolds Averaged Navier Stokes method implemented in the finite volume solver OpenFOAM. Shell finite elements of the Mixed Interpolation Tensorial Components (MITC) family are used for simulating the fabric. The use of these sophisticated Finite Elements allows for capturing the greater detail of the structural behavior and the generation of the wrinkles. Comparisons are presented between the results obtained with the shells and the membrane finite elements, traditionally adopted for the structural analysis of fabrics. The performances of the method are demonstrated with simplified validation test cases and applications are shown for realistic 3D devices.

**Form finding of tensegrity structures using finite elements and
mathematical programming, K Klinka, V Arcaro, D Gasparini, Journal of Mechanics
of Materials and Structures, Vol 7, # 10, 2012.**

We show that the minimization of total potential energy is the general principle behind the well-known rule of maximizing some lengths of a truss mechanism to define a tensegrity. Moreover, the latter rule is a special case, due to the usual high values of the modulus of elasticity. An innovative mathematical model is presented for finding the form of tensegrity structures, based on the finite element method and on mathematical programming. A special line element that shows constant stress for any displacement of its nodes is used to define a prestressed equilibrium configuration. Form finding is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy and the displacements of the nodal points are the unknowns. A connection is made with the geometric shape minimization problem, defined by a constrained nonlinear programming problem. A quasi-Newton method is used, which avoids the evaluation of the tangent stiffness matrix.

**Wrinkle development analysis in thin sail-like structures using MITC shell
finite elements, D Trimarchi, M Vidrascu, D Taunton, S R Turnock, D Chapelle,
Finite Elements in Analysis and Design, Vol 64, 2013.**

We propose a method of modeling sail type structures which captures the wrinkling behavior of such structures. The method is validated through experimental and analytical test cases, particularly in terms of wrinkling prediction. An enhanced wrinkling index is proposed as a valuable measure characterizing the global wrinkling development on the deformed structure. The method is based on a pseudo-dynamic finite element procedure involving non-linear MITC shell elements. The major advantage compared to membrane models generally used for this type of analysis is that no ad hoc wrinkling model is required to control the stability of the structure. We demonstrate our approach to analyze the behavior of various structures with spherical and cylindrical shapes, characteristic of downwind sails over a rather wide range of shape and constitutive parameters.

**Finite elements for geometrical minimal shape, V Arcaro, K Klinka, D
Gasparini, FORMA, Vol 28, # 1, 2013.**

This text describes a novel mathematical model that unifies all geometrical minimal shape problems by defining geometrical finite elements. Three types of elements are defined: line, triangle and tetrahedron. By associating a volume for each element type, the elements can be used together in the discretization of a geometrical shape. For each element type, its corresponding isovolumetric element is also defined. The geometrical minimal shape problem is formulated as an equality constrained minimization problem. The importance of this approach is that apparently distinct problems can be treated by a unified framework. The augmented Lagrangian method is used to solve the associated unconstrained minimization problem. A quasi-Newton method is used, which avoids the evaluation of the Hessian matrix. The source and executable computer codes of the algorithm are available for download from the website of one of the authors.

**Minimal surface computation using a finite element method on an embedded
surface, M Cenanovic, P Hansbo, M G Larson, arXiv, 2014.**

We suggest a finite element method for computing minimal surfaces based on computing a discrete Laplace{Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is done on the piecewise planar isosurface using the shape functions from the background three dimensional mesh used to represent the distance function. A recently suggested stabilization scheme is a crucial component in the method.

**Stiffness matrix based form-finding method of tensegrity structures, L
Zhang, Y Li, Y Cao, X Feng, Engineering Structures, Volume 58, january, 2014.**

The stiffness matrix and the total potential energy of the structure are utilized to direct the rapid convergence of the structural configuration to the self-equilibrated and stable state. In the programmed procedure, we employ the stochastic selecting algorithm to exclude rigid-body motions, the restricted step algorithm to guarantee the positive definiteness of the structural stiffness matrix, and the line search algorithm to minimize the total potential energy. A number of representative examples are given to demonstrate its accuracy and efficacy for both regular and irregular tensegrity structures of large scale.

**Modeling of tensegrity-membrane systems, S Yang, C Sultan, International
Journal of Solids and Structures, Vol 82, 2016.**

Tensegrity-membrane systems are a class of flexible multibody systems, which are composed of bars, tendons, and membranes. Due to advantages of lightweightness, multifunctionality and adaptability via control design, and the capacity of deployment inherited from tensegrity systems, tensegrity-membrane systems can be utilized in space applications such as solar sails and radar antennas. To study system’s prestressing conditions, bars are treated as rigid bodies, and an energy-based method is used to determine the equations for static configurations of general tensegrity-membrane systems. For symmetric tensegrity-membrane systems with multiple stages, the equations for static configurations can be significantly simplified. Explicit analytic solutions for the equilibrium conditions of one-stage symmetric systems are derived. The system dynamics is studied based on the nonlinear finite element method, and the total Lagrangian formulation is implemented.