Computational Methods for Tension-Loaded Structures

Arch. Comput. Meth. Engng. Vol. 11, 2, 143-186 (2004) Archives of Computational Methods in Engineering State of the art reviews

Computational Methods for Tension-Loaded Structures Thouraya Nouri-Baranger Centre de M´ecanique Universit´e Claude Bernard Lyon 1, France E-mail: [email protected]

Summary This paper deals with tension loaded structures made of coated woven fabric, cables and rigid frames such as mats and hoops. It describes in details a general framework for modelling and numerical simulation of their mechanical behavior. Several methods, developped in these last decades, are presented and compared. The principal particularity of these structures is that they derive their stiffness and their stability from the surface geometry and tensile stress field coupling. This particularity is combined with nonlinearities which can be due to possible large deflections, material law behavior and local instabilities due to wrinkling effects. In addition, a great number of design parameters must be taken into account in order to optimize the mechanical behavior of the structure. Therefore, the design and the analysis of such structure are complex and involve extensive computational costs. The principal steps of the analysis process are: form-finding, structural response of the structure to loads, cutting pattern and optimization. In this work, after a short description of the methods developed in this field as well as a critical comparison, new approaches are proposed.

1INTRODUCTION

A growing number of architectural structures are today protected by fabrics that are tensioned over structures such as plywood, concrete and metal. The most significant membrane structures include the Haj Terminal at Jeddah (Saudia Arabia) whose roof extends over a 430000 m2 surface, the arenas of Nˆımes (France) and Zaragoza (Spain) and the Mil- lennium dome− in Greenwich (United Kingtom). Textile materials have enabled architects to design new shapes erstwhile impossible to achieve with standard materials. However, besides the aesthetic considerations, these techniques offer several advantages: the structures are lightweight, are able to cover large surfaces with no ground attachments, are deployable, convey natural light and can easily be adapted to existing structures. Several engineers and architects have been singled out for their work on these types of structures since the beginning of the twentieth century. These pioneers include Otto Frei [19] and Kasuo Ishii [31]. Owing to specification requirements and thermal and acoustic properties of textile materials, these structures are limited to public and standard industrial buildings with no special requirements. For example: sports & leisure facilities,train stations and airport terminals, exhibition halls, industrial facilities. Tensile fabric structures present a nonlinear mechanical behaviour under various external loads. The flexibility of the fabric and the tensile requirement lead to a coupling between the surface shape representing the roof and the prestress loads required to keep the fabric taut. The fabric is an orthotropic composite and the free-form surface is obtained from plane fabric cutting patterns. The design, engineering analysis and implementation of a tensile fabric structure calls for elaborate procedures requiring a certain number of specific tools. Several CAD tools have been developed since the 1980’s. The following is a list of some of these: Easy software [23], Archimedes software [63], Meftx software [17], Tensys software [81], Artex software [80] and MecTex [53]. All these tools follow the same global approach except Artex, which only deals with purely geometrical design aspects and does

c 2004 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: October 2003 144 Thouraya Nouri-Baranger not provide an engineering analysis. The design and analysis of the mechanical behaviour of such structures goes through four major steps:

1. Deﬁnition of an initial geometrical design conform to functional, architectural and aesthetic aspects (generally dictated by the architect).

2. Deﬁnition of a stable equilibrium state for the above-mentioned geometry and determination of an initial state of stress to keep the tensioned structure in equilibrium.

3. Modeling of the mechanical behaviour of the structure under external loads and analysis of the rupture criterion.

4. Deﬁnition and optimization of the fabric cutting patterns to determine the shape of the structure.

The fabric is a material where only tensile stress can be generated. It naturally does not have any elastic stiﬀness with bending nor with compression. It is therefore essential to stretch the fabric allowing it to respond to bending loads but also to impede direct response to compression stress through the appearance of wrinkles. Therefore the geometrical shape of a tensile structure and the stress state of the membrane are closely related and cannot be dealt with separately as there is coupling between the loads and the geometry. Therefore the execution order of these steps cannot be pre-established and successive iterations between certain geometrical shapes are required in order to reach favourable results. The result should comply with safety standards, feasibility studies and should take into account the membrane boundary conditions.

2THEFABRIC

The technical fabric used in architecture is a multi-layer composite comprising the woven fabric and the coating. The result is a flexible composite with a woven reinforcement and securing a macroscopically periodic structure. This fabric is made up of interwoven threads (high tenacity polyester or fiberglass) which follow two perpendicular directions, the warp and weft. The type of cloth, taffeta or satin used determines the geometry of the basic shape. The coating ensures impermeability as well as the strength and weldability of the fabric. The coatings most used are as follows: PVC (polyvinyl chloride), Teflon (polytetrafluorethylene) and Silicon. The macroscopic behaviour of the coated fabric presents a nonlinear geometry in view of the weaving and coating operations. In practice, fabric behaviour rules can only be perceived realistically after specific tests have been carried out on samples [49]. Parallel to these tests, theoretical models can also lead to the numerical determination of the behaviour rules [42]. This type of method assumes an exact appreciation of the basic shape geometry and awareness of the rheological properties of the fibers and coating. The following is a list of influences or parameters, which characterize the mechanical behaviour of the fabric:

the density, or number of threads per unit length according to each warp and weft • direction.

the contraction which describes the fact that the weaving is more or less slack and • therefore that a warp and weft yarn alteration step is conducted during the 1st loading stage. The contraction phase is assimilated when the fabric is ﬁrst tensioned and corresponds to a 0.5% to 5% deformation according to the weaving. Computational Methods for Tension-Loaded Structures 145

The hysteresis phenomena characterizes the invariability of the ﬁrst deformations. • This phenomena has been the subject of many papers including [9]. For clearer understanding, our models only include the values obtained by averaging the stress curve-biaxial tension study results. See [13], [63] and [9]

Once the fabric has been tensioned, we can consider that it has near linear characteristics. This is the model used by most authors carrying out research on the mechanical analysis of tensile fabric structures. The fabric is a very thin ﬂexible material, it does not resist to compression stress and only has bending strength if it is pre-taut. There- fore, to represent the behaviour of the fabric a two-dimensional elastic membrane model is used. The weaving does not generally present a weaving symmetry therefore we used a two-dimensional orthotropic model. The main orthotropic directions are the warp and weft. The elastic stiﬀness matrix is formulated in these orthotropic axes as :

E ν E w fw w 0 (1 νwf νfw) (1 νwtνtc) C = ⎡ −νwf Ef −Ef ⎤ (1) 0 ⎢ (1 νwf νfw) (1 νwf νfw) ⎥ ⎢ − − ⎥ ⎢ 00Gwf ⎥ ⎣ ⎦ where w and f refer to the warp and weft directions, Ew, Ef are Young ’s modulus, νwf is Poisson’s ratio and Gwf is the shear modulus. In any point of reference with an θ angle with regard to orthotropic directions, the elastic stiﬀness matrix is characterized as follows:

t C = TθCT( θ) (2) −

c2 s2 cs 2 2 − Tθ = s c cs (3) ⎡ 2cs 2cs c2 s2 ⎤ − − ⎣ ⎦ where c =cos(θ)ands =sin(θ), Tθ is the rotation matrix.

3 FORM FINDING OF AN INITIAL SURFACE IN EQUILIBRIUM STATE Unlike certain flexible structures such as sails which are kinematically undetermined, we studied configurations where it was essential to stretch the fabric in order to obtain significant bending stiffness while avoiding slack and the appearance of wrinkles (the kinematic condition is lifted following the geometric stiffness set by the prestress). The prestress state required depends on the limits with the rupture of the materials, the boundary conditions of the structure, the shape required and the external loads. A tensile structure presents a nonlinear mechanical behaviour in response to any kind of loads. This nonlinearity is geometric and mainly related to large displacements and prestress. The strain field is assumed to remain within acceptable limits for minor disturbances. Hence, the above assumptions must be taken into account to formulate the initial equilibrium state of the structure. This step is therefore essential and is referred to as theFormFindingprocedure. Several methods have been developed for this procedure and these can be divided into two groups:

1. Methods based on a purely geometric concept. 2. Methods based on a geometric and mechanical concept. 146 Thouraya Nouri-Baranger

Studies have shown that nature always operates in the most simple and efficient manner. Natural objects are often optimized for utility, efficiency and economic reasons. The first tensile structures were inspired from surfaces modeled on soapfilm wire structures. These surfaces are referred to as minimal surface area and are defined as follows:

A surface resting on a given form and whose surface area is smaller than any other surface in proximity with the same topology.

Minimal surfaces are defined as surfaces with zero mean curvature From a mathematical point of view, a minimal surface is defined as surface with zero mean curvature. Otto Frei[19] was highly inspired by these surfaces when he created his models before each construction. The minimal surface area problem was first referred to as the Plateau problem. Joseph Ferdinand Plateau (1801-1883) (see [22]), was the first to physically execute these surfaces with wire structures dipped into a soap and glycerine solution, but he did not have the mathematical skills to investigate the problem of existence of a minimal surface area. Weierstrass, Riemann and Schwarz [74] worked on this question and then presented mathematical proof of the existence of such surfaces. However, the problem was finally solved, in a much more general form, in 1930 by Douglas and Radó [16], [65]. Generally, if one considers a given boundary form Γ there can be one, several or no surfaces resting on this form. Therefore it is not possible to build a minimal surface area from any kind of structure, hence the necessity to introduce the notion of stability. This can be expressed by overall tension in all directions and on all points of the surface. The local equilibrium equation of a tensile fabric structure is characterized by the main stress values σ1 and σ2 and the main curvature radius R1 et R2.

σ σ 1 + 2 = p (4) R1 R2 where p represents the surface distribution of normal external actions on the surface. Two cases can be observed:

The surface has a double negative or reciprocal curvature. Equilibrium is possible for • a p = 0 value (the radii have opposite signs). This is the case for tensile structures. The surface has a double positive curvature (the radii have the same sign) and equi- • librium only exists if p is diﬀerent from zero. Therefore, there is permanent internal pressure p. This is the case for inﬂatable structures.

The first methods developed for the definition of an initial form were only based on the geometric considerations of minimal surface areas, these methods are known as surface splines. Since the 70’s, methods have been developed on the mechanical stability. The surface to be modeled undergoes quadrilateral or triangular meshing and represents a network of cables or membrane elements. Two mechanical aspects were studied. The first is based on form-finding through given stress or load fields. In this case, only the geometry can evolve towards a stable equilibrium state. The problem is then determined statically. Equilibrium no longer depends on the rheologic properties of the fabric. This approach has been widely adopted and uses two types of modeling:

Modeling with reinforced cable elements: this corresponds to the force density method. • The Easy [23], Tensys [81], Archimedes [63] and Artex [80] software programs all use this method. Computational Methods for Tension-Loaded Structures 147

Modeling by triangular and quadrilateral membrane finite elements. This method was • initially developed by Ishii [31] and [32], Haber [24] and [25], D’Uston [17], Tsubota [77] and Nouri-Baranger [58], [53] [57]. The second aspect is based on a nonlinear elastic large deflection analysis and considers strained elements. Large deflections and stress factors constitute a set of parameters, which evolve during the iterations. This aspect was implemented in 1974 by Argyris and al. [3] and [4]. However, although it is efficient, it is the most difficult for the designer to conduct.

3.1 Cable Modeling 3.1.1 Force density method This method was perfected by H. J. Scheck and K. Linkwitz [69] and [35]. The surface is represented by a network of reinforced cables. A di coeﬃcient, referred to as force density, is deﬁned for each element i:

Fi di = (5) Li where Fi is the tensible force in the cable element and Li represents the length. The equilibrium for each node is formulated as :

pi Fi = Fjext for j =1..n (6) i=1 where n being the number of nodes and pi the number of elements connected to node j. Using di coeﬃcients, the equilibrium equations are:

pi Lidi = Fjext for j =1..n (7) i=1 Both the definition of initial meshing and external loads are used to define the equilibrium state through a system of n nonlinear equations, whereby the force density coefficient di intervenes for each element. By isolating this coefficient considered by the user as a set parameter, the equilibrium equations then become linear. These can be resolved by one of the direct classic or iterative methods. The coordinates of each unbound node, and therefore the length and tension of each element can be determined. The equilibrium equation can be written as follows:

pi diΔXi = FjextX i=1 pi diΔYi = FjextY for j =1..n (8) i=1 pi diΔZi = FjextZ i=1 There is only one possible equilibrium state for each force density coeﬃcient. The advantages of this method reside in its simple implementation and highly reduced calculation times making it extremely user-friendly. For an identical force density throughout the network, we obtain a minimal surface area. This minimizes the sum length of the square network elements. We then obtain an equilibrium state and tension in the cables. The major drawbacks of this method are as follows: 148 Thouraya Nouri-Baranger

the force density coefficient does not have any physical bearing and only a skilled and • experienced designer can determine the surface form according to the choice of values credited to these coefficients; the force density coefficient is a ratio of the tension and length of the element and • therefore the form obtained is dependent on the meshing density and anisotropy; a network of cables can only represent a fabric if the meshing is very fine and the • mesh size tends to the design size representative of the warp and weft structure; it is impossible to determine the stress state within a fabric from the tension modeled • in the cable network, unless mathematical tricks are used even though these are not based on established theoretical calculations.

To conclude, this method can only be used to define a first initial equilibrium state for a choice of given force densities. However the stress fields associated with this equilibrium state remain unknown. In several cases, the surface equilibrium obtained is still far from that required (design-stress coupling), and this leads to a complex nonlinear calculation on analysis of the surface behaviour to external loads. 3.1.2 Dynamic relaxation method This method was developed at the beginning of the 70’s by M. R. Barnes [5] and [6] as form-finding method for tensile fabric structures. The basis of dynamic relaxation is to follow, step by step, by t time increment, the damping movement of the structure. The initial configuration is generated by a network of cables that represent any type of meshing. This network defines a group of nodes of which some are free and others, referred to as pilot nodes are moved in increments by the designer to their final position (resting on external frame). This method uses dynamic calculation with a fictitious damping of the structure in order to determine the static equilibrium position. Since the mass and viscous damping matrixes are hypothetical, these can be selected as diagonal in modal form. The movement equations are then decoupled and a system of linear equations appears. The easiest way to resolve this system is to use the finite difference method. The finite element method is too complex. On each t increment, residual loads induced to nodes, node speeds and new nodes co- ordinates are calculated to the equilibrium position of the system. The disadvantages of this method are listed as follows:

Complex iterative calculations which greatly reduce interactive possibilities. The t • increment should be sufficiently small to ensure numerical stability of the method. A great number of parameters (set by the designer) which control the equilibrium of • the structure. No physical representation of mass and damping parameters which stabilise and con- • trol movement of the structure. 3.1.3 Haug method This method was developed by E. Haug [29] and is based on a nonlinear static calculation. The basic data is the same as for the dynamic relaxation method: a network of cables including a series of free nodes and a series of pilot nodes that the designer can move in increments to their final position (resting on external frame). The rigidity of each cable is ES equal to l (S being the cable section area, E the Young module and l the element length). The designer introduces load variations to the cables for each displacement increment required. This results in the appearance of residual loads in the nodes. The equilibrium is Computational Methods for Tension-Loaded Structures 149 then calculated via a suitable iterative algorithm such as the Newton-Raphson method. This algorithm requires calculation of the elastic stiffness and geometric stiffness matrixes. Nonlinear problems are solved until all pilot nodes reach their final positions. The main advantage of this method rests in the fact that the designer has perfect control over the evolution of the surface form by successive movement of the frame. The parameter manip- ulated by the designer has physical significance. The main drawbacks of this method are as follows:

Calculations are complex and costly. • Successive movements generate substantial and unrealistic loads in the cable elements, • making it necessary to re-initialize all the element lengths each time the elastic distortion boundary is reached. Further iterative calculations are then necessary to determine the new loads required to stabilize external loads. The stress ﬁeld of the fabric is unrecoverable through the loads in the cable element. • Several alternative methods have been developed by university research teams and industrial groups such as Arcora and Alto [37]. The main drawbacks of these methods are as follows:

1. These are all inflexible form-finding tools, they do not provide any information on the stress field of the fabric. 2. There are often merging problems for certain boundary cases and it is subsequently impossible to find a form in an equilibrium state. 3. The results depend on the anisotropic meshing.

3.2 Membrane Models 3.2.1 Geometric stiffness method Haber [24] and Ishii [31] were among the first to develop a method using membrane element modeling to determine forms in equilibrium state. This method is based on the data of the area in the reference configuration and the stress tensor σo within the fabric in equilibrium. o From this data and an initial random form Gini we can search for an equilibrium position which neutralizes all residual loads. The Updated Lagrangian Formulation (ULF) is used to describe the equilibrium of the structure. Equilibrium equations are inferred from principles of virtual work:

p p 1 δw = σδε dG − (9) p 1 G − δwp represents the virtual work of external loads on increment p, σ is the second Piola- Kirchhoff stress tensor (PK2)andδε is the virtual Green-Lagrange strain tensor measured in the strain configuration (p 1). The Green-Lagrange strain tensor is nonlinear and is organized in linear and quadratic− terms: δε = δe + δη. The stress tensor σ is organized o by the elastic strain of the fabric and by an initially imposed part: σij = σij + Cijkl εkl. The Updated Lagrangian Formulation requires, on each p increment, the reconversion of the PK2 stress tensor resulting in a Cauchy stress tensor in p configuration for the next increment. The principle of virtual work is as follows:

p 1 o p 1 p o p 1 Cijkl εkl δεij dG − + σ δη dG − = δw σ δe dG − (10) p 1 p 1 − p 1 G − G − G − 150 Thouraya Nouri-Baranger

Since the initial equilibrium of a structure is independent of the rheologic properties of the material, any value can be taken for Cijkl. To simplify the problem, take all Cijkl terms equal to zero. This invalidates the ﬁrst integral of the previous equation and we obtain:

o p 1 p o p 1 σ δη dG − = δw σ δe dG − (11) p 1 − p 1 G − G − If one does not take into account external loads, the virtual work wp equals zero and the equilibrium equation is simpliﬁed and becomes:

o p 1 o p 1 σ δη dG − = σ δe dG − (12) p 1 − p 1 G − G − By introducing a ﬁnite element approximation and taking the nodal displacement vector −→U equal to −→X , vector of the node co-ordinates for the structure in equilibrium, the previous equation− is written under the following matrix form:

t o p 1 t o p 1 BNL(−→U )−→σ BNL(−→U ) dG − X = BL−→σ dG − (13) p 1 − p 1 G − G − Kg−→X = −→F (14)

Kg is the updated lagrangian geometric stiffness matrix, BNL(−→U )andBL are the interpolation matrixes dependent on the form functions of the finite elements used. −→F represents the nodal load vector. The drawback of this formulation lies in the fact that Kg depends on the node co-ordinates therefore this system can only be resolved through a suitable iterative algorithm. If the user specifically indicates the values of all the matrix terms, the system becomes linear and can be easily solved. However, this is a highly sophisticated solution and is not easy for the designer to achieve. R. Haber and J. Abel tried to make up for the disadvantages of the previous methods. The process aims to be independent from the material and enables the designer to distribute a stress field in the membrane. However, the formulation itself renders the method highly complex. B. Maurin and R. Motro [40] and [41] have recently studied Haber’s method and have formulated the problem in a more simple form in order to obtain control of the prestress field in the fabric and avoid the appearance of a compression zone. This method, referred to as Surface Stress Density, is based on the definition of an isotropic prestress field: σo o o o −→σ = σ avec σ > 0 (15) ⎧ 0 ⎫ ⎨ ⎬ This stress field is not specifically specified but is determined via coefficient q referred ⎩ ⎭ to as surface stress density. σo q = where S is the element area (16) S From any initial meshing with clearly defined reference considerations, the initial equilibrium for this density is obtained by considering the nodal equilibrium equations. The resolution procedure is iterative. This method, combined with the force density method for boltrope cables provides satisfactory results for most classic surface areas whether minimal surface areas or not. However, this method diverges if the data cannot be used to create a stable form. It is also difficult to estimate the state of the actual stress from these densities when equilibrium has been achieved. Computational Methods for Tension-Loaded Structures 151

3.2.2 Stress ratio method A brief synopsis has been given above on the main research work carried out on tensile fabric structure forms in equilibrium, as achieved and presented over the past two decades. Using the extensive literature on the subject, it appears interesting to reformulate the problem to try and anticipate the actual stress state in the fabric, during the search for an initial o o equilibrium form. Given that the (G , −→σ ) couple is the departure point for a structure behaviour analysis under external loads, the outcome greatly depends on the error brought o about on this data. The −→σ field is used to introduce geometric stiffness in the structure in order to initialize calculations and also intervenes in the calculation of the displacement o and stress fields. However we have already mentioned that this −→σ field is closely linked to the Go geometric form and can therefore not be decoupled without interfering with the dimensioning process based on this data. The technical fabric is an orthotropic material. It is characterized by different Young modules according to two orthotropic warp and weft directions and different boundary limits on the stress admissible on these directions. This data represents the main characteristics measured and supplied by the fabric manufacturer. It is therefore extremely logical to use this data to define a stress field for the initial form required. An important point must be made: a certain number of studies have often been restricted to form-finding with an isotropic stress field. There are however forms in equilibrium that have been achieved with a non-isotropic stress field. A stress field isotropy is very difficult to achieve with materials such as fabric, which does not have any soapfilm properties. It is therefore obvious that is not necessary to impose this criteria. Ew and Ef , σw and σf represent the Young modules and the stresses according to the warp and weft o o directions. Initial mesh and a stress field −→σ are defined on any first geometry G : o σw o σo o o o −→σ = f with σw > 0, σf > 0etσwf = 0 (17) ⎧ σo ⎫ ⎨ wf ⎬ This ensures a state⎩ of stress⎭ within the fabric, and we define a coefficient r equal to the ratio of the warp and weft components: o σw r = o (18) σf Thestressfieldisthenformulatedas: r o o o −→σ = σf 1 = σf −→r (19) ⎧ 0 ⎫ ⎨ ⎬ This r coefficient will enable us to define⎩ initial⎭ non iso-stress equilibrium forms, which provides a much wider range of forms than that represented by minimal surface areas. These correspond to an r value equal to a unit. The r ratio is always positive and can be supported by extremes defined from the characteristics of the fabric used. max max If (σw , σf ) are the admissible boundary values of the traction stress in the fab- min min ric, and (σw , σf ) are the minimal values defined by the designer to ensure minimum geometric stiffness σmin σo σmax (20) w ≤ w ≤ w

σmin σo σmax (21) f ≤ f ≤ f 152 Thouraya Nouri-Baranger

rmin r rmax (22) ≤ max≤ min max σw min σw with r = min and r = max (23) σf σf We assume that the warp and weft directions remain perpendicular after straining. This assumption is even more feasible since shear aﬀects should remain low in order to avoid o wrinkling. From an initial arbitrary meshing Gini, secure or adjustable reference points, warp and weft directions and an r ratio, the principle of virtual work is drafted in Total Lagrangian Formulation as:

t o o t −→δU (BL + BNL(−→U ))−→σ dGini −→F = −→δU ϕ(−→U ) (24) Go − ini where −→F is the external loads vector. In equilibrium ϕ(−→U ) the residual load vector is invalidated, whereby:

o o o o BL−→σ dGi −→F = BNL(−→U )−→σ dGini (25) Go − − Go ini ini

−→R Kg −→U −→R is the residual load vector required for the stress state in the fabric and external loads, o Kg is the geometric stiffness matrix associated with the stress field −→σ and −→U is the nodal displacement vector. For an initial form-finding search, we generally take −→F = −→0.Inthis case, the following problem must be solved:

o o BL−→rdGini = BNL(−→U )−→rdGini (26) Go − Go ini ini −→Fr = Kgr−→U (27) The initial equilibrium now only depends on the r warp and weft stress ratio. The system of equation 27 is linear and can be resolved by using the preconditioned diagonal conjugate gradient algorithm with a penalization technique to take into account boundary conditions. A Total Lagrangian Formulation is used in this method, where the stress state o is defined by the second Piola-Kirchhoff (PK2) tensor in the initial configuration Gini, contrary to R. Haber who used an Updated Lagrangian Formulation therefore a PK2 stress tensor reconverted to Cauchy at each iteration. The Cauchy stress tensor in its current configuration Go is obtained by simple rotation of the PK2 field as defined in the o max min initial configuration Gini. By modifying the value of the r ratio between r et r , the designer can modify the Go form within the admissible stress value limits. Contrary to the force density method, the form obtained in this way is independent of the meshing anisotropy (see [58] and [57]). 3.2.3 Methodology The stress ratio method requires insight on the stress field according to the warp and weft o directions in the initial configuration Gini (not initially prestressed). This first geometry can be plane or not. The warp and weft directions are generally known when the fabric is cut. It is therefore logical to define the orientation of these directions on plane cuts. Two cases may arise: Computational Methods for Tension-Loaded Structures 153

Warp and weft directions

w w ww θ

f f

oo GGi i w θ

Planar Planar breadth breadth number number i i

Figure 1. Initial planar geometry established from overall planar breadths. The warp and weft directions are deﬁned in this geometry for each breadth

o 1. The initial form Gini is plane. In this case, the domain is cut into sub-domains and the warp direction is set according to one of the borders of each sub-domain, see Figures 1, 2 and 3. o o1 2. The initial Gini form is a spatial form. In this case, a first form in equilibrium G is determined for an r = 1 ratio. Therefore any orientation will concur with the warp and weft directions. Then, θ = 0 is taken at this step. From the Go1 geometry, spatial breadths are defined and flattened. Hence, the actual warp and weft directions are set to these cutting patterns. An update of the Go1 form is then carried out for a ratio r = 1 and the actual stress value σo to provide the final configuration in equilibrium f of Go1. To illustrate this method, the first example is a surface of revolution referred to as catenoid. It is a minimal surface area that can be obtained with soapfilm between two rings with a radius of a1 and a2. This surface can be obtained from the rotation, around the y axis, of the curve as shown in the following equation:

2 2 y = a1 ln x + x a ln(a2) (28) − 2 − The weft direction is deﬁned according to the circular direction y = Constant and the warp direction according to θ = Constant. From the plane meshing between two circles in the design (x, z), a displacement is requested of the small circle with a1 radius to its position on the equilibrium conﬁguration Go.Ther stress ratio is set to 1 and we then obtain a form in iso-stress equilibrium, which corresponds to the minimal surface area obtained analytically. Figures 4 and 5 display the form obtained for r = 1 and the comparison with the exact analytical solution. 154 Thouraya Nouri-Baranger

y Triangular finite element of membrane y Triangular finite element of membrane f (weft direction) w (warp direction) f (weft direction) w (warp direction) Fabric Fabric

Z Z x x

X X

Figure 2. Warp and weft directions of the fabric on a membrane element

Figure 6 features three different configurations obtained for three values of ratio r.To lift the surface or provide more curvature, alter the value of the r ratio. In this example, it is easy, according to the warp and weft directions, to envisage the direction of evolution of the structure according to that of r. To check the coherence between the stress field σ o and the geometric form Go,one o o−→ evaluates the response of G under the action of −→σ , via a nonlinear elastic calculation. We then note that the displacements generated are too small and negligible. The maximum displacement never overstepped 1%. In the case of cutting modifications, it is essential to redefine Go according to the new warp and weft directions. The following example is a minimal surface area referred to as Parabolic-hyperbolic surface. The initial geometry Go is a plane square. The warp and weft directions are parallel to the borders of the square. The final surface Go Figure 7, is obtained by fixing two opposite corners of the square and by moving the other two vertically with r = 1. The resulting shape represents a minimal surface area. After a nonlinear elastic calculation under the o action of −→σ , the distribution of the stress field is the same as that imposed initially with an error of 5%. Figure 8 shows the influence of the meshing anisotropy on the form in equilibrium obtained by the force density method. It is important to note that the meshing anisotropy has the opposite effect to that of the force density. Figure 9 shows that there is no meshing influence on the form obtained by membrane finite elements and the stress ratio method.

3.2.4 Non iso-stress form ﬁnding It is obvious that the architectural forms used are not often minimal surface areas. If we also take into account the orthotropic behaviour of the fabric, it is unnecessary to pursue this criterion in form ﬁnding. The following example represents a roof with a surface comparable to a Chinese coolie hat. This roof has a conical shape deviated outwards, two sides and a corner of fabric are embedded. The other two sides are cabled and free, see Figure 10. o The initial form Gini is a square with a hole in the middle representing the hoop. We Computational Methods for Tension-Loaded Structures 155

Z y Z y Warp Warpdirection w Weft direction w Weftdirection f direction f θ x x

Three dimensional element Three dimensional element

Weft f f Weft directiondirection FlattenedFlattened element element y y Y Y

wWarpWarp w directiondirection

θ α

XX BreadthBreadth edge edge x x

Figure 3. Definition of the warp and weft directions compared to the local reference of the membrane finite element. The angle between the warp and weft directions of the fabric is assumed to be always equal to 90 in the planar and three-dimensional configurations ◦

12000 8000 z 4000

0 10000 x20000 30000 20000 30000 0 500010000 y

Figure 4. Catenoid imposed a displacement of the hoop and one of the corners to their ﬁnal positions, a stress o 6 state such as σf =24MPa with r =2.5andT =10 N for cable tension. The warp directions are parallel to the sides of the square. Figures 11 and 12 represent the actual distribution of the stress ﬁeld in Go following a nonlinear elastic calculation. The values of these stresses vary by 10% when compared to the values set initially. We observed a stress concentration around± the hoop and on the edges, which is normal. The second example reveals a swimming pool roof. This roof includes two rows of three hoops inclined 156 Thouraya Nouri-Baranger

Stress Ration Method solution 16000 Analytical solution 14000 12000 10000 y 8000 6000 4000 2000

0 5000 10000 15000x 20000 25000 30000 35000

Figure 5. The analytical solution compared to that obtained by the stress ratio method for r =1

r = 0.25 12000 r = 0.5 r = 1 10000

8000 y 6000

4000

2000

0 5000 10000 15000 20000 25000 30000 x

Figure 6. Proﬁle of the Catenoid obtained with three values of r

20 15 10 20 5 15 10 5

Figure 7. Surface parabolic-hyperbolic surface

at a 10◦ angle with respect to the horizontal direction. There are twenty anchorage points and± twenty boltrope cables. This roof measures 37 7 7 m. Figure 13 represents the surface in equilibrium obtained by the force density method× × for densities equal to the unit overall except on the edge cables where they are worth 1000. Figure 14 represents the o surface in equilibrium obtained by the stress ratio method with r =1,σt =5MPa and Computational Methods for Tension-Loaded Structures 157

0 5 10 y 15 20 25 0

10 x 15

Figure 8. Inﬂuence of the meshing on the force density method. The blue meshing represents a5 5 discretisation. The red meshing represents a 4 5 discretisation × ×

0 5 10 y 15 20 25 0

10 x 15

Figure 9. Inﬂuence of the meshing on the stress ratio method. The blue meshing represents a5 5 discretisation. The red meshing represents a 4 5 discretisation × ×

10000 8000 z6000 4000 2000 0 0 0 5000 5000 10000 10000 15000 15000 y 20000 20000 x 25000

Figure 10. Example of a non iso-stress surface

tension in the edge cables of T =104 N. The latter presents a more flexible appearance with more curvature than the first. This example shows that a representation of the fabric behaviour through a network of bars can be used to obtain a shape in equilibrium, which often appears extremely rigid with minimized curvatures. The aspect of this form can be altered by adding more curvatures through the force densities (from configuration 13 to configuration 14). This equilibrium is however closely related to the meshing density. It is therefore practically impossible to extract the stress field related to this form through the density and meshing information. The stress ratio method offers basic advantages without requiring additional resources or creating dilemmas for the definition of data. The forms in equilibrium obtained through this method present perfect coupling between the geometry and the stress state imposed. 158 Thouraya Nouri-Baranger

Figure 11. Stress according to the warp direction

Figure 12. Stress according to the weft direction

3.3 Form-Finding under Loading The stress ratio method in its initial form, as represented by equation 25 takes into account external loads. The system of equations to be solved is represented as follows:

f o o Kg−→U = −→R with −→R = −→F ext BL−→σ dGini (29) − Go ini −→R represents the residual loads vector relative to the structure and −→F ext the external load vector. A surface with no cables and a constant r ratio can be formulated as: 1 K U = F Bt rdGo (30) gr−→ f −→ext L−→ ini σ − Go o ini Computational Methods for Tension-Loaded Structures 159

z12000

35000 25000 0 15000 5000 x 10000 5000 0 y

Figure 13. Swimming pool roof. Surface in equilibrium obtained via the force density method

z12000

35000 25000 0 15000 5000 x 10000 5000 0 y

Figure 14. Swimming pool roof. Surface obtained via the stress ratio method

1.5 z 1 0.5 0 –6 0 –4 2 –2 4 y 0x 6 2 8 4 10 6

Figure 15. Initial equilibrium for r =1

In this case it is possible to conclude the stress state required and the r ratio to balance the system. If the structure requires cables then it will be more complex to solve system 30 since there are many unknown factors such as: the stress field, the r ratio and the cable tensions. This problem has not yet been solved and still requires more research. Figure 15 160 Thouraya Nouri-Baranger represents a roof for an industrial building. This structure is saddle-shaped and rests on two 45◦ arches tilted outwards and two horizontal segments representing the wall boundaries. The warp direction is parallel to the y axis and the form in equilibrium obtained for the o following ratio: r =1andr = 2 and for any value of σf are shown in Figure 16. Figures 17 and 18 represent the two configurations for r, with and without loads. The stress level o σf and loads are the same in both cases. This procedure is obviously only a first approach to the evaluation of the theoretical stress state coupled with the geometric form required. Additional developments are necessary to broaden the field of application.

1.5

z 1

0.5

0 0246810y

Figure 16. Equilibrium in blue for r =1andingreenforr =2

1.5

z 1

0.5

0 0246810y

Figure 17. Equilibrium for r = 1, in blue without load and in red with load

1.5

z 1

0.5

0 0246810y

Figure 18. Equilibrium for r = 2, in green without load and in red with load Computational Methods for Tension-Loaded Structures 161

4 MECHANICAL BEHAVIOUR OF A TENSILE FABRIC STRUCTURE Tensile fabric structures are submitted to static and dynamic loads. Having obtained the o couple (Go, −→σ ), the analysis of the response of this structure to loading is one of the main stages of dimensioning and checking. The first procedure developed in this sense was based on fabric modelling via a network of cables (see the papers written by Argyris [3] and [4], Gründig [23], Nishino [50] and Frei [19]). The disadvantage of this procedure rests in its inability to reproduce the actual behaviour of the fabric: stress state, wrinkling and warp and weft direction orientations according to the pattern cutting. The second procedure uses modeling by triangular or quadrangular membrane finite elements. The bibliography presents several papers using this model in various applications: Pauli [63], Ishii [31] and [32] and Mutin [46]. A tensile structure generally consists of a membrane tensioned by cables or directly fixed to extremely rigid structural elements (walls, beams, posts or braced frames). The latter are often regarded as external structures. The mechanical behaviour of the fabric and cables present a nonlinear nature which can be classified into two categories: 1. nonlinear nature due to large displacements (rotation and translation). 2. nonlinear nature due to the fact that neither the fabric nor the cables can be subjected to compression loads. Therefore, they do not present any compression stiffness. However,thestrainfieldinthefabricandcables is considered small. The material associated with the fabric is considered to have orthotropic linear elastic properties, see equation 2. The same assumption is made for the cables. Finite membrane elements and cable elements have been implemented for the development of a tool able to analyze the mechanical behaviour of a tensile structure under different static external loads. The tool developed is first used to verify perfect adequacy between the initial stress state and the geometry (Go, o −→σ ). An analysis is then carried out on the stress, displacement and strain fields as well as the reactions generated by the external loads, in order to provide final validation of the o o o o (G , −→σ ) couple. Otherwise, modifications on (G , −→σ ) will be necessary to ensure an adequate and reliable response of the structure to the external loads envisaged. 4.1 Equilibrium and Nonlinearity A Total Lagrangian Formulation was adopted to implement the problem. This formulation was chosen for its minimal calculation costs. The Total Lagrangian Formulation is characterized by the definition of strain and stress fields relative to the reference configuration of the structure at time t = 0. The use of a fixed reference configuration has the advantage of a constant integration domain. At each Total Lagrangian Formulation iteration, the displacement, strain and stress fields are actualized although their definition remains the same. This formulation uses the Green-Lagrange tensor for strains and the second Piola- Kirchhoff tensor, also referred to as PK2, for stresses. These tensors are defined in the initial configuration Go, see Bathe [7]. However, at the end of the iterative process, the designer will need to be aware of the stress distribution in the final G configuration. This stress tensor referred to as Cauchy’s strain tensor can be obtained by simple rotation of the PK2tensorfromGo to G, when there are small strains, see Crisfield [12].

4.2 The Total Lagrangian Formulation Where Go and G are respectively the initial and ﬁnal conﬁguration. At each increment of the calculation process, the equilibrium equations are concluded from the principles of virtual work:

δwint = δwext (31) 162 Thouraya Nouri-Baranger

y y v v 3 3 w 3 w 3

u u 3 3 3 b 3

cc vv v v 11 w 2 2 w w 11 w 2 2

uu u u Z Z 11 2 2 x x 11 a 2 2 YY a

Figure 19. Membrane element

Whereby fj represents the component j of speciﬁc volumic load density and pj the compo- o nent j of surface density load. S represents the surface where pj is applied. The principles of virtual work are then formulated as:

o o o o Cijkl εkl + σij δεij dG = fjδujdG + pjδuj dS (32) o o o G G S The above equation is nonlinear, its solution is carried out via an iterative algorithm within an incremental loading process. The physical movement of Go to G is generated by a series of small perturbations, see Bathe [7].

4.3 Finite Element Formulation 4.3.1 Membrane element A triangular ﬁnite element was used for the fabric modeling, see Figure 19. The surface meshing is topologically similar to that used for the initial Go form-ﬁnding.

In this case, the Green-Lagrange strain tensor is written as:

∂u 1 ∂u 2 ∂v 2 ∂w 2 εxx = ∂x + 2 ∂x + ∂x + ∂x 2 2 2 ⎧ ε = ∂v + 1 ∂u + ∂v + ∂w ⎫ (33) ⎪ yy ∂y 2 ∂y ∂y ∂y ⎪ ⎪ ⎪ ⎨⎪ ∂v ∂u ∂u ∂u ∂v∂v ∂w ∂w ⎬⎪ γxy = ∂x + ∂y + ∂x ∂y + ∂x ∂y + ∂x ∂y ⎪ ⎪ ⎪ ⎪ The displacement vector⎩⎪ at node i is written: ⎭⎪

ui −→u i = vi (34) ⎧ w ⎫ ⎨ i ⎬ ⎩ ⎭ Computational Methods for Tension-Loaded Structures 163

The displacement ﬁeld interpolation on element k is formulated:

u 1 −→ t u = N1 u 1 + N2 u 2 + N3 u 3 = N1 N2 N3 u 2 = −→N −→U k (35) −→ −→ −→ −→ ⎧ −→ ⎫ u 3 ⎨ −→ ⎬ with ⎩ ⎭

N1 = a1 + b1x + c1y t −→N = N2 = a2 + b2x + c2y (36) ⎧ N = a + b x + c y ⎫ ⎨ 3 3 3 3 ⎬ The strain ﬁeld can be written⎩ under the following matrix⎭ form:

1 ε = B + B ( U ) U (37) L 2 NL −→k −→k where BL is a (3 9) constant matrix, BNL is a (3 9) matrix dependent on the −→U k ﬁeld. × × By disregarding the volume loads fj, for an element k we can write:

1 o Ksk = BL + BNL(−→U k) C BL + BNL(−→U k) dSk (38) T o 2 k

t o −→F ext k = −→N −→pdSTk (39) − So Tk After all the elements have been assembled, the secant formula is written:

Ks(−→U )−→U = −→F ext (40) The incremental formula, also referred to as tangent is written as follows:

KT (−→U )Δ−→U = −→ΔF ext (41) with

o t o o KNL = BL + BNL(−→U k) C BL + BNL(−→U k) dG + G SGdGdG (42) Go Go

t o o o −→ΔF = −→N −→pdS BL + BNL(−→U k) −→σ dG (43) So − Go

1 o σ = C BL + BNL(−→U k) + σ (44) −→ 2 −→ where S represents a matrix containing the stress vector components and C represents the elastic stiﬀness of the fabric, as shown in equation 2. 164 Thouraya Nouri-Baranger

y y v v v1 v 2 1 w 1 2 w 2 w 1 w 2

u EE S S u u Z Z 1 2 2 x x 11 2 2 Y Y LL

X X

Figure 20. Cable element

4.3.2 Cable element The formulation of the cable ﬁnite element follows the same procedure as that of the triangle, see Figure 20. The only diﬀerence lies in the fact that the stress and strain tensors are written as εxx and σxx. The form functions are linear and are written as:

x N1 =1 L −x (45) N2 = L

∂u 1 ∂u 2 ∂v 2 ∂w 2 ε = + + + (46) xx ∂x 2 ∂x ∂x ∂x σxx = Eεxx (47)

4.4 Local Instability As mentioned further, the cables and fabric do not show stiffness under compression or bending. This characteristic generates instability revealed as pleats in the fabric. These local instabilities can, in some cases cause damage to the structure. It is therefore essential for the designer to know where local instabilities appear. These instabilities should then be dealt with on each iteration, or the process will experience deviant solutions. Behaviour laws are rarely examined for textile materials in compression. Some research has however been carried out in order to represent as best as possible the areas in compression. Mans- field proposed a model based on a maximum strain energy theorem on a membrane in compression [38] [39]. Additional studies have been presented by Miller [45] in 1985, by Norihiro [52] in 1986 and by Contri [10] in 1988. Norihiro and Contri presented modeling of the surfaces in compression and used the technique of finite elements with variable stiffness and a stress state modified in the compression areas. A new modeling was recently proposed by Roddeman [67] and [68] in 1987 and by Dong [15] in 1992. Their analysis takes into account a physical measurement of the surface in compression by modifying the strain tensor in the compression areas. 4.4.1 Management of wrinkles regarding a membrane element All the models referred to are not able to provide a geometrical description of the surface undulation. The appearance of a compression area on the surface of the membrane is considered as a physical nonlinearity combined with a significant displacement finite element. Computational Methods for Tension-Loaded Structures 165

Wrinkled surface Wrinkled surface

Fictive plane surface Fictive plane surface

e2 Z Z e e1 1 Y Y

X X

Figure 21. Wrinkled surface

The numerical model presented in this paper and used by D’Uston [17] is based on two hypotheses: 1. The membrane is not able to endure major negative stress. The appearance of a 2 2 major negative stress σp produces a pleat perpendicular to σp. 2. An average ﬁctitious surface replaces the distorted surface by eliminating the pleat, as shown on the Figure 21. The stress state in an element is determined by:

o −→σ = C−→ε + −→σ in local coordinate system (48)

1 1 σp p σ = σ2 and = 2 (49) −→p ⎧ p ⎫ −→p ⎧ p ⎫ ⎨ 0 ⎬ ⎨ 0 ⎬ with σp the major stress vector⎩ in the⎭ principal directions⎩ ( e ⎭1, e 2) as shown on Figure −→ 1 2 −→ −→ 21, where σp σp and −→ p major strain vector in the principal directions (−→e 1, −→e 2). Two cases may occur≤ requiring a local modiﬁcation:

2 1 1. Uniaxial tensile state: σp 0andσp 0. In this case, Equation 48 is reduced to that ≤ ≥ 1 1 of uniaxial behaviour according to the first major direction: σp = A p + B,whereA o 2 and B are expressed according to C and −→σ p, based on the equation σp =0.The stiffness matrix of the element is then determined by using the new elastic stiffness matrix Cuni and stress vector −−→σuni.

A 00 Cuni = 0 Aη 0 in the principal directions and with η 1 (50) ⎡ 00Aη ⎤ ⎣ ⎦ 166 Thouraya Nouri-Baranger

1 1 σuni = σp σ = 0 in the principal directions (51) −−→uni ⎧ ⎫ ⎨ 0 ⎬ ⎩ ⎭ 1 1 σuni = σp σ = 0 in the principal directions (52) −−→uni ⎧ ⎫ ⎨ 0 ⎬ ⎩ ⎭ 2. Biaxial compression state: σ2 0andσ1 0. In this case, the compression area p ≤ p ≤ is inactive. The stress vector is invalidated: −→σ = −→0 , and the new elastic stiﬀness matrix in the primary reference point is given as:

Aη 00 Cbiax = 0 Aη 0 in the principal directions and with η 1 (53) ⎡ 00Aη ⎤ ⎣ ⎦ 4.4.2 Compression state management for a cable element The same procedure was adopted for the cable elements. The stress state is checked each time the calculation process is reiterated. If the σxx stress appears negative or equal to zero, the stiﬀness matrix of the element in question will no longer be included in the calculation of the global stiﬀness of the system and the stress will be reset to zero.

4.5 Numerical Analysis The resolution of system 40 is carried out via the Newton-Raphson algorithm. At each iteration, a system of linear equations 41 are solved via the Conjugate Gradient algorithm. The Conjugate Gradient algorithm was chosen as the stiﬀness matrix is not altered during the iterative process. This enables rigorous and optimal management of the memory. Only non-zero values are then stored. This storage is independent of the numbering. Boundary conditions such as rigid ﬁxing, slide rails and imposed displacements are taken into account through the penalization technique. The conjugate gradient algorithm is preconditioned to improve the convergence relative to penalization.

5 SENSITIVITY ANALYSIS AND OPTIMIZATION

In this section, optimization problems relative to the mechanical behaviour of tensile fabric structures are studied. When a tensile fabric structure is designed, the designer first defines an initial shape in equilibrium Go from the architectural and mechanical considerations for o o o o a pre-stressed state −→σ coupled to G . This first shape in equilibrium (G , −→σ )) is then submitted to externals loads. If undesirable effects are generated subsequently damaging the aesthetics and safety of the structure, the designer should then carry out the necessary o o modifications on the couple (G , −→σ ) to avoid this effects. This procedure leads to a series of successive tests and generates more important calculation costs given that the model is nonlinear and therefore results are unpredictable. The undesirable effects to be avoided are as follows:

Appearance of over-stressed zones, damaging the fabric. • Computational Methods for Tension-Loaded Structures 167

Appearance of under-stressed zones, causing wrinkling and pockets and therefore • generating local instabilities. Generation of large displacements. • These consequences can cause permanent damage to the structure. The designer should therefore carry out modifications, classified into two categories: 1. Basic modifications, causing an overall review of the structure architecture. These modifications are very difficult to automate. 2. The topology of the structure remains the same. Modifications are only made to the anchorage and support positions, the initial pre-stressed field of the fabric and the cable tension. The second category of modifications can easily be automated. Here we face the following opposite problem: o What is the consequence of an anchorage position modification, of field −→σ and tension T in the cables on the parameter to be improved. What values should be attributed to these parameters to obtain optimal behaviour? This problem should be examined as an improvement process and calls for notions of sensitivity analysis and optimization. In existing papers, few studies relate to the optimization of flexible structures. This was noticed by Ramm in 1997 [66] when he revealed the po- tential of these techniques. However, there has been some research on the optimization of cable-network structures. These studies have examined the optimization of mechanical behaviour according to sections and pre-stress loads, see the papers drafted by Nishino [50] and Swaddiwudhipong [75]. We first focused on the implementation of a CAD tool based on a sensitivity analysis, then on the development of an optimization tool with the same design parameters in order to obtain an optimal configuration corresponding to our objectives. 5.1 Sensitivity Analysis The purpose of the sensitivity analysis is to determine the effect, on the stress and displacement field, of the design parameter variations without having to carry out an in-depth analysis of the problem. In this study, the design parameters are: the anchorage positions, o the stress field intensity −→σ and the T tension in the cables, see Figure 22. For these cases, a sensitivity analysis can be used to rapidly discover the influence of a parameter variation on the stress, strain and displacement field resulting from the analysis of the structure under external loads. This tool can also be used to determine the influence rate of defects caused on implementation of the structure on-site. For example incorrect positioning of an anchorage point or cable over-loading. According to the influence rate of these errors, particular attention can be brought on implementation, to the setting of the most significant parameters. The theoretical grounds of the sensitivity analysis as well as a detailed description on the different analytical, semi-analytical or numerical methods are described in detail in the papers written by Haftka [26] or Kirsh [33]. The sensitivity analysis of field r in relation to a parameter di is determined by calculating the derivative of r in relation to di.Ifr is a field of size n and di a scalar parameter, r represents the modified field corresponding to di +Δdi, the first order expansion is formulated as:

∂r1 ∂r ∂di r = r + si(r, di)Δdi with si = = : (54) ∂di ⎧ ∂rn ⎫ ⎨ ∂di ⎬ ⎩ ⎭ 168 Thouraya Nouri-Baranger

Hoop Displacements ?

Material Properties ?

Anchorages Displacements ?

Cable tension ?

Figure 22. Design variables

where si is the r field sensitivity vector in relation to parameter di. In practice, several di parameters can be modified at once. If −→d represents the p design parameter vector, the r sensitivity in relation to the −→d variation is as follows: ∂r1 .. ∂r1 ∂d1 ∂dp r = r + s(r, −→d )Δdi avec s(r, −→d )= :: (55) ⎡ ⎤ ∂rn .. ∂rn ∂d1 ∂dp ⎣ ⎦ The response of a structure is first evaluated by the calculation of the −→U displacement field, solution of system 40. We then calculate s(r, −→d ) by differentiating the equilibrium equation 40

∂−→U ∂Fext ∂Ks Ks = −→U (56) ∂di ∂di − ∂di

The resolution of the previous equation provides the sensitivity of −→U in relation to di from which it is possible to infer the strain and stress field sensitivities. The derivative of the stiffness matrix and the second member vector can be carried out analytically or numerically. The method is essentially selected per type of design variable and calculation speed. For easy implementation and integration within the existing code, we adopted a numerical differentiation based on the first sequence diagram of finite differences:

∂−→U −→U (di +Δdi) −→U (di) = − (57) ∂di Δdi Due to the nonlinearity of problem 40, the finite difference method is often costly in calculation terms if the same −→Uo(di) departure point is used to calculate fields −→U (di +Δdi)and Computational Methods for Tension-Loaded Structures 169

−→U (di) thus minimising the calculation error on the derivative. However, if the −→U (di +Δdi) departure point is −→U (di), the convergence is extremely fast but the error on the derivative is often unacceptable. We therefore adopted an alternate method as presented by Haftka in [27] and referred to as modified finite difference method. The algorithm proposed by Haftka is as follows: if −→U (di) procured from −→Uo(di)istheexactsolutiontothe non-disturbed problem Ks(−→U )−→U = −→F ext problem, we search for −→U (di +Δdi)=−→U Δ as solution to the following disturbed problem:

Ks(−→U (di +Δdi))−→U(di +Δdi) = Ks(−→U (di))−→U (di) +

KsΔ−→U Δ Ks−→U −→F ext(di +Δdi) −→F ext(di) (58) − −→F ext Δ −→F ext − −

KsΔ−→U Δ = Ks−→U + −→F ext Δ −→F ext (59) − − The previous system is nonlinear. Its solution is achieved via a Newton Raphson algorithm in which at each k iteration, the following tangent linear system is solved:

k 1 k k 1 k 1 KT (−→U Δ− )−→U Δ = −→F ext Δ −→F ext + Ks(−→U )−→U KsΔ(−→U Δ− )−→U Δ− (60) − − − This algorithm converges in 2 to 3 iterations with sufficient precision on the derivatives. Further details can be obtained from [27]. The calculation of the derivatives from the displacement field according to the design variables is used to arrive at the strain and stress derivatives in relation to these same variables. The small cost in sensitivity analysis calculation times enables us to envisage an interactive CAD tool based on this method. This tool would enable the designer to rapidly determine, at little expense, what kind of modifications could be brought to the structure to improve its behaviour. This qualitative scenario then leads us to consider a quantitative scenario, which would enable us to define within a reasonable interval, the values to be given to the design variables chosen to be modified. This last objective can be reached by solving the following global optimization problem:

Deﬁne the values of the variables selected. These should minimize the target functions used to improve the behaviour of the structure.

5.2 Optimization The objective here is to define the values to be given to the design variables selected by the designer following a sensitivity analysis. These values are solutions to the global optimization problem as mentioned above. We should now define what exactly are the objective functions and constraints to which the design variables are submitted. There are two target functions, the displacement field and the stress field. These functions express the effects detrimental to the structure and are defined as follows:

1. In order to avoid the appearance of pockets (mainly due to the horizontal surfaces) or too much fabric movement, the ﬁrst objective is to rigidify certain zones of the 170 Thouraya Nouri-Baranger

fabric. This may lead to an operation in view of minimizing the node displacement of the membrane.

2 2 2 Min F1,F1 = uj + vj + wj with j =1..Nnode (61) j 2. The second objective relates to the stress state of the fabric. We need to outline the minimum and maximum values of the stress calculated in the membrane elements in order to avoid the appearance of wrinkles (compressible zones or maximum shear in strain) and tearing (zones with too much tension). We then observe the primary major σ1i and minor σ2i principal stress values for each i element. Therefore, the

objective is deﬁned by the optimization constraint g1i and g2i for each element i,such as:

max max g1i : σ1i σ σ1i σ 0fori =1...Nelements (62) ≤ 1 ⇐⇒ − 1 ≤

min max g2i : σ2i σ σ σ2i 0fori =1...Nelements (63) ≥ 2 ⇐⇒ 2 − ≥ The design variables are as follows:

1. The pre-stress values in the cables: Ti with i =1...Ncables, 2. The displacement values imposed to the hoops, support points or walls in predeﬁned directions: with i =1...Nsupports where Ncables and Nsupports respectively represent the number of cables and supports deﬁned as design variables. These design variables are submitted to the following constraints:

1. The tension Ti in the cables are such that:

adm 0 Ti T , for i =1...Ncablesˆ (64) ≤ ≤ i adm Where Ti represents the possible maximum tension. 2. Support displacements can only be carried out within a time as deﬁned in the speci- ﬁcations sheet. For the displacement of each support i:

max 0 di d , for i =1...Nsupports (65) ≤ ≤ i

max where di represents the maximum displacement amplitude of support i. The problem observed here is that of nonlinear optimization under constraints since the various objectives characterized by functions F1 and g1i and g2i as defined above are nonlinear. The constraints g1i and g2i are not strict limitations but should be considered here in terms of objectives. This leads us to select a sequential unconstrained optimization method. This optimization procedure based on an iterative algorithm generates a succession of often costly calculations in account of the important number of finite element models used. It is therefore essential to use an efficient algorithm in order to reduce calculation costs. The Augmented Lagrangian Multiplier Approach has proved efficient in dealing with this problem. This method is coupled with a unconstrained minimization algorithm. The Computational Methods for Tension-Loaded Structures 171

AugmentedAugmented Lagrange Multiplier Lagrange MethodMultiplier Method Minimize objectiveMinimize function objective F function F1 subject to the constraints1 g and g subject to the constraints g and g 21 21

Create pseudo-objectiveCreate pseudo-objective function A function A which include F , g and g which include F2 , g1 and g1 2 1 1

Flectcher ReevesFlectcher Method Reeves Method Minimize pseudo-objectiveMinimize pseudo-objective function A as functionan A as an unconstrainedunconstrained function function

Determine theDetermine descent direction the descent direction

PolynomialPolynomial approximations approximations Determine theDetermine descent step the descent step

Fletcher-Reeves converged? Fletcher-Reeves converged?

Yes Yes

No Lagrange converged? Yes Exit No Lagrange converged? Yes Exit

No No

Figure 23. Optimization module algorithm latter is based on a unidimensional research technique used to define an optimal increment. A general algorithm of the optimization module is shown on the Figure 23. The Augmented Lagrangian Method as well as the Steepest Descent Method are presented in detail with specific applications in several papers, among which we can quote [79] and [26]. The Augmented Lagrangian Method belongs to the category of unconstrained sequential minimization methods. This method consists in building a pseudo-objective A(X) function referred to as Augmented Lagrangian and covering the objective function F (X) and the constraints gj(X). The Augmented Lagrangian is created by introducing variables λj referred to as Lagrange Multipliers and a penalization coefficient rp. A(X)isthen minimized by using a unconstraint minimization algorithm. The Augmented Lagrangian 172 Thouraya Nouri-Baranger

Method algorithm consists of the following steps as long as the convergence criteria has not been respected: 1. Calculation of the Lagrange multipliers and the penalization coeﬃcient. 2. Construction of the Augmented Lagrangian. 3. Minimization of Augmented Lagrangian. 4. Convergence test.

5.3 Example

The structure examined here consists of two rows of three hoops deviated at a 10◦ angle with respect to a horizontal direction, as shown on the Figure 14. There are also± 20 fabric anchorage points and 20 boltrope cables. The fabric consists of 40 distinct parts themselves made up of diﬀerent fabric panels. The ﬁnite element meshing is made up of triangular membrane elements at 9 D.D.L. to model the fabric and linear cable elements at 6 D.D.L. for the boltrope cables. We can enumerate 1559 nodal points, 2784 membrane elements and 112 cable elements. The hoops and fabric tips connected to the structure are considered rigid. For the boltropes, we assume that there is no slippage between the cable and the fabric. We will now study the results for a single load case, as follows: prestress combined with an equivalent pressure according to a horizontal direction of 500 N/m2 and a vertical overload of 300 N/m2, see Figure 26. The fabric properties are as follows:

Membrane (isotropic material) : Ec = Et = 300 N/mm ∗ and ν =0.3 • Cable : E = 110000 N/mm2 S = 139 mm2 •

Figure 24. Distribution of major principal stress σ1

An initial prestress of 5 N/mm, in warp and weft, is applied to the fabric. Three prestress values are deﬁned for the cables according to their length and bending radius: 10000 N, 18000 N and 20000 N. The stress ratio method is used to deﬁne a form in

∗The stress and module values are already multiplied by the fabric thickness. Computational Methods for Tension-Loaded Structures 173

o o initial equilibrium (G , −→σ ). Figures 24 and 25 show the major stress distribution σ1 and σ1 in the initial conﬁguration. Their values vary in an interval of [3.13, 7.83] N/mm, which is normal since this form does not a minimum surface area. Under the action of this load, the maximum value of the stress can reach 21 N/mm in areas around the hoops. We also notice the appearance of wrinkles in these areas characterized by minor invalid primary stress. Figure 27 shows the distribution of the vertical displacement component generated by the loads. The areas in red represent the maximum values reaching up to 385 mm. Figures 29 and 30 respectively represent the sensitivity of component σ2 and − the displacement component Uz foranincreaseincabletension as shown in Figure 28. The areas in blue represent the negative derivatives while the areas in red represent the positive derivatives. These sensitivities seem to indicate a decline in displacement which is what we were seeking to achieve. However, we also note that this generates an increase in the stress σ2. In this conﬁguration, we observe a very localized sensitivity of the structure response in relation to the cable tension variation.

Figure 25. Distribution of minor principal stress σ2

Fz=300 N/m² Fx=500 N/m²

4 x 10

1.4 1.2 1 3.5 3 2 2.5 4 1.5 2 x 10 1.5 4 1 x 10 1 0.5 0.5 0 0

Figure 26. Load occurrence examined 174 Thouraya Nouri-Baranger

Figure 27. Vertical displacement generated by the load

4 x 10

1.4 1.2 1 3.5 3 2 2.5 4 1.5 2 x 10 1.5 4 1 x 10 1 0.5 0.5 Cable tension 0 0

Figure 28. Cable tension variation

Figure 29. σ2 sensitivity to the cable tension variation Computational Methods for Tension-Loaded Structures 175

Figure 30. Vertical displacement sensitivity to the cable tension variation

Figure 31. Fabric displacement sensitivity as regards the increase of all cable tensions

Figures 31, 32 and 33 respectively represent the fabric displacement sensitivity as regards a tension increase of 5000 N in all the cables, the outward displacement of the fabric tips by 10 cm and the upward displacement of the hoops by 10 cm. In this example the stress state in the membrane elements is as follows:

Primary maximum stress: σ1 =7.83 N/mm. The average is σ1 =5.48 N/mm • Primary minimum stress: σ2 =3.29 N/mm. The average is σ2 =4.69N/mm. • We can therefore note a critical level of stress, which could lead to premature fatigue of the material. Therefore, while guaranteeing minimum rigidity, we wish to reduce the prestress level of the fabric. For the beneﬁt of this study, the adjusting devices can only be used to modify the prestress level in the cables. The optimization algorithm developed provides optimum conﬁguration. The design variables which correspond to the prestress values in 176 Thouraya Nouri-Baranger

Figure 32. Fabric displacement sensitivity as regards displacement of all fabric tips

Figure 33. Fabric displacement sensitivity as regards upward displacement of the hoops

the cables are shown on the Figure 34. Our objective, as given above is translated by integrating limitations on the stress values, that is :

3 N/mm < σ < 5 N/mm (66)

We must emphasize that these limitations should not be taken in a strict sense but should be considered as objectives to be met.

The convergence of the optimization algorithm was obtained after the fourth iteration. The results are given in Table 1. If we look at the design variables, we can observe a global decline in the prestress cable values. This decline represents 14% for boltropes B and can reach up to 45% for boltropes D. As regards the stress values for the membrane elements, we can observe an average reduction of 10%, which corresponds to our objective. This Computational Methods for Tension-Loaded Structures 177

E E E D D B B

C C

A A

C C

B B D E D E E

Figure 34. Design variables for the optimization problem reduction may appear relatively small, however it appears diﬃcult to achieve any better given that we only include the prestress of the cables and we give a minimum stress value as a limitation in order to guarantee a certain fabric stiﬀness.

Design Variables: Principle Tension (N) Stresses (N/mm) max min moy moy A B C D E σ1 σ2 σ1 σ2 Initial values 18 103 20 103 10 103 18 103 10 103 7.83 3.29 5.62 4.69 Optimized Values 15186 15531 8602 9863 6564 7.28 2.81 4.72 4.04

Table 1. Initial and optimized tension values

6 CUTTING PATTERN OPTIMIZATION The last step before arriving at an industrially achievable form, is the cutting process and the flattening of the free-form surface determined via the process described above. This surface is obtained by cutting plane fabric panels. The panels are then sewed or glued together. The cut panels should both minimize the fabric cuttings and take into account the warp and weft directions so the fabric may have optimum resistance capacity. Once cut, assembled and the membrane in place, the geometry of the structure and the internal loads should be as close as possible to those calculated and no wrinkles or pockets should appear. Incorrect initial panel cutting can lead to or generate the appearance of a wrinkled surface and over-stress. This step therefore requires adequate definition of the surface cutting pattern and layout. The cutting pattern research process combines three major steps: the lines defining the different widths are generated on the free-form surface; • the widths defined are then developed into a pattern; • geometric corrections, referred to as compensations, are made to the cutting pattern • since the fabric widths are not tensioned during the cutting and assembly process. In 1986, Hangleiter carried out research in this field, see [28] when he applied geodesic surface patterning to a spatial cut patterning development process. In 1989, Tsubota [78], used the same method but made some modifications to take into account the strain generated on implementation. In 1989, Shimada [70] presented a method based on minimization of the strain energy calculated from the disparities between the three-dimensional form and the 178 Thouraya Nouri-Baranger planar form requested. This method applied elastic formulation by finite element, within an iterative resolution estimation. In 1990, in [21] the authors compared three cutting pattern methods. Two were based on a purely geometric technique to develop the triangles forming the width meshing. The third method used the research carried out by Shimada, based on a representation of the surface in equilibrium via square NURBS surface cut into fabric widths according to one of the main directions of each square. These three methods produce cutting patterns that do not take into account the strain generated when the fabric is tensioned. After assembly, the fabric widths generate a free-form surface where the prestress field equals zero, therefore impossible to achieve. According to previous research, tension can only be achieved following a reduction in width size. Phelan and Haber introduced the optimization concept, see [64], since the fabric is not tensioned when the cutting process is being carried out. Their method resides in the use of non-prestressed cutting patterns as the initial configuration for an equilibrium analysis and form optimization. They thereby attenuate the design problem by combining the form, equilibrium and cutting problems into one. A nonlinear equation is then to be resolved. However, to find the solution this method requires data on the following two initial geometries: the three-dimensional form and the cutting patterns. The two meshings should be topologically equivalent and care- fully selected. Within the framework of this paper and following the research carried out by Shimada and Phelan, a cutting pattern tool has been developed. It can optimize the stress field generated in the structure after assembly, by finding the adequate cutting pattern shapes. This concept is based on the data of a geometric form in equilibrium coupled to a o o sufficient stress state: (G , −→σ ), see Figure 35.

500

400

300

200

100

−100 1000

500 1000 500 0 0 −500 −500 −1000 −1000

Figure 35. Tensile fabric structure in a Chinese coolie hat shape

The geometry Go is cut into widths based on a representation of the surface via square NURBS segments, see [80]. The geometry of this example resembles that of a symmetrical Chinese coolie hat of which only one side appears on the Figure 36. The cutting pattern geometry is referred to as G2D. This geometry can fit any shape (one rectangle must be used for each width). The meshing for both geometries should be topologically equivalent. o The G form can theoretically be obtained by applying a −→U 2D >3D displacement field to the G2D planar cuttings. This represents a simulation of the assembly− independent of the mode used. This displacement field is the difference between the co-ordinates of Go and G2D. This geometric transformation adds to a distortion of the G2D elements which Computational Methods for Tension-Loaded Structures 179

2000

1500

1000 3D G 2D−>3D U 500

2D 0 G 3000

2000

1000

−1000 2000 2500 1000 1500 0 500

Figure 36. Building Go structure from the plane cutting

G2D 3D GG3D

ΔF

Figure 37. Distortion applied to an element

o are then superposed to the G elements. −→σ represents the stress ﬁeld generated on each element following this distortion. The goal is to minimize the −→Δσ variation between the o −→σ ﬁeld that we wish to create on assembly and that created in reality −→σ . Regarding each element of G2D.The−→Δσ variation creates a residual load vector −→ΔF which distorts the element in order to superpose it on its equivalent in the Go geometry, see Figure 37. Therefore, the problem set can be designated as strain energy minimization as follows:

1 t t Min J(−→U d)= −→U Kd−→U d −→U −→ΔF (67) 2 d − d

J(−→U d) represents the strain energy generated by −→ΔF , −→U d is the displacement ﬁeld re- o quired to superpose G and Gˆ 2D . Kd is the ensuing stiﬀness matrix. The minimization { } of J(−→U d) requires us to solve a system of linear equations 68. Rigid displacements are eliminated by adding boundary conditions to each width while allowing them to distort freely

Kd−→U d = −→ΔF (68) 180 Thouraya Nouri-Baranger

Figure 38 represents the cutting patterns before and after optimization. Contrary to that already developed in [21], the cutting patterns obtained depend on the materials used and generate the stress state as imposed in the spatial conﬁguration. Figure 39 represents the variations of the stress vector components, obtained after assembly of the Go shape, compared to the stress state imposed σo. Adequate correlation can be observed between both stress states in this example which corresponds to a nondevelopable surface. In developable surfaces J(Ud) is invalidated and we retrieve the stress state imposed.

1200

1000

800

600 Initial cutting pattern

400

Optimized cutting pattern 200

0 0 200 400 600 800 1000 1200

Figure 38. Initial and optimal cutting patterns

−1 MPa

σ 6 xx

3 σ yy 2

σ 0 xy

−1 0 5 10 15 20 25

Figure 39. Stress variation obtained through optimized cutting Computational Methods for Tension-Loaded Structures 181

7CONCLUSION

Cad tools and the analysis of a tensile fabric structure were perfected and validated on several models. We carried out research on four different modules: a module to find an initial form in equilibrium, a module to analyse the response of the structure to external loads, a sensitivity analysis and optimization module to analyse the behaviour of the structure according to various design variables, and a fabric cutting pattern optimization module. We do not feign to have solved the problem set by optimization of the design-stress factor in fabric structures. We have however developed several decision aid tools for easy comprehension when faced with this type of problem. The initial equilibrium form-finding module developed provides more flexibility at this stage and is also erstwhile more reliable. Workable geometric forms are therefore extended to non iso-stress forms, without adding extra complexity to the data definition. This tool can be used to impose a prestressed non-isotropic field: this stress field is determined by the designer from the technical characteristics of the fabric and within the orthotropic model defined by the warp and weft cutting directions. The forms in equilibrium thus found are in perfect bearing with this stress field. This is easily verified by a nonlinear analysis of the elastic behaviour of the geometric form discovered due to the sole action of the pre-stress field. When the warp and weft stress ratio and boundary values are introduced, we can then modify the geometric form by adding more or less curvature while still remaining valid. In this way, we ensure a more reliable perfect coupling between the geometric form and the stress. This method is used to identify the stress distribution throughout the overall surface. This information is essential to the next step, which is to analyze the behaviour of the structure following external stress (snow, wind, etc.). The method extends to loaded and inflatable structures and is used to solve ensuing counter problems. However, additional research ought to be carried out on this subject and should prove interesting. The behaviour analysis module of a tensile fabric structure under different static external loads is based on the finite element method. It uses surface modeling of the fabric and integrates membrane elements. To formulate the equilibrium problem, geometric nonlinearities due to large displacements and prestress issues are taken into account. The model implemented for the membrane element is orthotropic elastic and we therefore have to take into account the alignment of the different widths on assembly of the fabric. Instabilities generating wrinkles in the fabric or slackening of cables are managed by integrating material nonlinearity in the formulation of the elements. The sensitivity analysis and optimization module can be divided into two sub-modules: a sensitivity analysis module and an optimization module. In order for the designer to have a clear comprehension of the behaviour of the structure being studied, the sensitivity analysis provides a rapid view of the influence of the various design parameters. On synthesis of these results, the designer is guided through the modifications to be carried out on the structure design. When a designer uses this tool, he no longer has to carry out different analysis and the design approach is improved through a significant reduction in calculation times. The second tool relates to the automatic optimization of the setting parameters. When the structure is being implemented, various adjusting devices are included to help calibrate the fabric. These affect the position of the hoops and fabric tips as well as the tension in the cables. These various adjustment devices have a genuine influence on the rigidity of the structure and therefore on its capacity to sustain various external loads. In order to automatically determine the optimal configuration of the adjustment parameters, an optimization algorithm was developed and its accuracy was verified. A fabric cutting pattern optimization module was developed to take into account the elasticity of the fabric. Cutting pattern modifications are currently being made with no prior analysis and largely depend on the designer’s common sense. The method developed 182 Thouraya Nouri-Baranger is therefore used to amend the cutting patterns based on the stress field required for a particular structure. This tool was validated in theory on structures with developable and non-developable surfaces. The modules presented cannot be disjoined from one another. To arrive at an optimal and workable solution, research on a tensile fabric structure requires the use of these modules in a sequence as defined by the designer. This sequence depends on conceptual and technical constraints linked to the execution of the structure as well as the autonomy of the designer to modify the model. Research on the action of the actual stress on such structures is one of the viewpoints to be considered for this topic. For the requirements of this paper, the external load considered was represented by evenly distributed continuous stress. However, these structures experience atmospheric conditions such as wind, snow, rain and temperature variations. These effects are not constant in time and are distributed according to spatial variables. A study on the dynamic response of these structures as well as the sensitivity of their behaviour according to temperature variations and uncertainties regarding data values are logical perspectives to the research carried out on this subject.

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