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Int. J. Structural Engineering, Vol. 1, Nos. 3/4, 2010 217

Copyright © 2010 Inderscience Enterprises Ltd.

Some issues on the design and analysis of pneumatic structures

Ruy Marcelo de Oliveira Pauletti Department of Structural and Geotechnical Engineering, Polytechnic School of the University of São Paulo, P.O. Box 61548 – 05424-970 São Paulo, Brazil Fax: +55.11.3091.5181 E-mail: [email protected]

Abstract: Pneumatic structures are the tensile structures par excellence, since only with them it is possible to have all elements working in tension. This paper describes the evolution of pneumatic structures, as well as several issues related to its design, such as different pattering methods, the distinctions between high and low energy systems, and whether large strain material models, pressure variations or cable sliding are required to proper design. Selected civil engineering applications, designed by means of simple numerical models, are also presented.

Keywords: pneumatic structures; tension structures; membrane structures; historical evolution; design and analysis; sliding cables.

Reference to this paper should be made as follows: Pauletti, R.M.O. (2010) ‘Some issues on the design and analysis of pneumatic structures’, Int. J. Structural Engineering, Vol. 1, Nos. 3/4, pp.217–240.

Biographical notes: Ruy Marcelo de Oliveira Pauletti is a Civil Engineer by the Federal University of Rio Grande do Sul, Brazil (1983) and a Specialist on Thermonuclear Fusion Engineering by the University of Padua, Italy (1986). He received his DSc in Civil Engineering at USP in 1994. He is an Associate Professor at the Polytechnic School of USP since 2003, researching in the fields of membrane structures and non-linear analysis of structures, having conducted the structural analyses and design of several relevant membrane structures in Brazil, and worked as a Consultant for the national industry, mainly in the automotive, power generation and construction sectors.

1 Introduction

Pneumatic structures are the tension structures par excellence, since only with them it is possible to have all structural elements working exclusively in tension. There are three basic subtypes of pneumatic structures: an insufflated structure consists of a membrane enclosing a space under an inner pressure slightly larger than the atmospheric pressure; in an aspirated structure, inner sub-pressure is employed; an inflated structure uses pressurised balloons, functioning as beams, columns and arches as load-carrying elements. Besides these three basics types, sketched in Figure 1, sails and parachutes can be considered as a fourth subtype, the open pneumatics.

Although tension structures constitute perhaps the oldest and most spontaneous structural system, their modern configurations are quite recent, since relevant

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manifestations require sophisticated materials, building techniques and theories, such as the synthetic films, the high strength cables and the non-linear computations employed in their design. In particular, practical applications of pneumatic structures started at the beginnings of the 20th century, to become, a hundred years later, rather usual in areas as distinct as large span stadia roofs or everyday’s life furniture, like sofas, boats and toys. Even an inflated church was produced in 2003, in England!

After a brief outline on the historical evolution of pneumatic structures, this paper discusses some issues related to the behaviour, the design and the analysis of pneumatic structures, such as the distinctions between a priori and structural pattering, or between high and low energy pneumatic systems, and whether consideration of large-strains, sophisticated material models, internal pressure variations or cable sliding are required for design. The paper also presents some particular civil engineering applications, analysed mostly by means of simplified numerical models and common engineering sense.

Figure 1 Different types of pneumatic structures (a) insufflated (b) aspirated (c) inflated (see online version for colours)

(a) (b)

(c)

2 On the evolution of pneumatic structures

The last years have offered several notable manifestations of pneumatic structures to public appreciation. It seems appropriate to briefly connect these last achievements to the classical developments on the field. More comprehensive historical descriptions can be found on the books by Otto (1958, 1967, 1982), Dent (1970), Herzog (1977) and Topham (2002), as well as on the papers by Forster (1994) and Chi and Pauletti (2005).

The first experiments on pneumatic structures trace back to the development of hot air balloons. Although Brazilian priest Bartolomeu de Gusmão conducted a pioneering experiment as soon as 1709, in Lisbon, the development of balloons effectively started at the end of the 18th century. In a famous experience, in 1783, in France, the Montgolfier brothers built an 11 m diameter hot-air balloon, made by linen and paper. At the same year, Jacques Charles built the first hydrogen balloon. Also in France,

Some issues on the design and analysis of pneumatic structures 219

Alberto Santos-Dumont, another Brazilian inventor, pioneered the construction of dirigible balloons. Figure 2 clearly shows the neat pattering adopted to produce the liner of Santos-Dumont’s Dirigible Number 1 (1898).

Figure 2 Santos-Dumont Dirigible Number 1 (1898) (see online version for colours)

Source: Santos-Dumont (1904)

In 1918, the English engineer F.W. Lanchester presented the first practical proposal for an insufflated dome, to be used as a campaign hospital. His patent was approved in 1918, but never actually constructed, due to the lack of adequate membrane materials or appeal to possible clients. However, only four years later, the Oasis Theatre, in Paris, sported a pneumatic hollow roof structure that was rolled into place when it rained (Shodek, 1980). Soon after, in 1929, in Japan, Kaneshige Nohmura invented the air-inflated membrane tent, with inflated tubes replacing rigid arches and masts (TKC, 1991).

During the 2nd World War, with the invention of nylon, pneumatics was used in military operations, as emergency shelters and decoys. After the end of the war, several radars in the US territory were protected from weather by large pneumatic domes developed by Walter Bird. The prototype for these radomes – as they have been called – was a 15 m diameter dome built in 1948 at the Cornel Aeronautical Lab. In 1956, Bird established Birdair Structures, the company which led, during the 1960s, the commercial application of pneumatics, as covers for warehouses, swimming pools, sport facilities and factories. Two remarkable structures of this period were the inflated lenticular membrane roof of Boston Arts Center Theater, 44 m in diameter, designed by Carl Koch and Paul Weidlinger, in 1959, and the Atoms for Peace Pavilion, a double-layer insufflated pavilion designed by Victor Lundy, which hosted an exhibition of the US Atomic Energy Commission, travelling through Central and South America, in 1960. A parallel development occurred in Japan, where the Nohmura Company, started as soon as in 1922. Interestingly, in 1947 the company changed its name to Taiyo Kogyo (‘Sun Industries’) and in 1989 it became the major shareholder of Birdair.

While W. Bird and K. Nohmura pioneered the commercial applications and the acquisition of empirical knowledge on pneumatics, Frei Otto was the first to undertake extensive academic investigations, especially about the form finding process. Through the IASS Pneumatic Colloquium (Stuttgart, 1967) and several publications and designs, Otto broadened the landscape, not only of pneumatics, but of tension structures in

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general. Pneumatic structures were also part of the repertoire of Buckminster Fuller, whose proposal for a pneumatic dome to cover New York (1962) is a famous example of Utopian pneumatic architecture. The exhibition Structures Gonfables, held in Paris, raised the interest of architects and designers from Europe, USA and Japan. One of the main works in exhibition, the Dyodon, showed J-P. Jungmann’s investigations on pneumatics’ shapes, which soon inspired many temporary and itinerant exhibitions.

At the Expo’70 in Osaka, three pneumatic structures especially relevant were displayed. The Fuji Pavilion, designed by Yutaka Murata and engineered by Mamoru Kawaguchi, was built by the Nohmura Company, and awed the public with its unusual form, composed by 16 inflated tubular arches. The Floating Theatre, produced by the same team, was composed by three inflated tubes highly pressurised, connected by a single layer membrane, with the inner space kept under a negative pressure, thus providing a rare case of aspirated pneumatic structure. The American Pavilion, designed by Davis Brody, David Geiger and Walter Bird, introduced a low aerodynamic profile dome with oval plan, which soon inspired several projects employing cable reinforced, insufflated membranes, for sport stadiums in the USA and Canada, in the 1970s and 1980s. These roofs drastically reduced the cost per seat, compared with conventional stadium, and have worked satisfactorily, except for some operational problems, due to accumulation of snow. It can be appointed as a paradox, that the main factor driving to construction of closed environments – harsh winter – is also the foulest enemy of the large pneumatic domes. Thus, in spite of more recent projects, like the Tokyo ‘Big-Egg’ Dome (1988, designed by Nikken, Sekkei and Takenaka, with David Geiger), having avoided problems with snow using larger internal pressures, smaller distance between cables and higher profiles than previous domes, the use of large insufflated domes is being superseded by other structural systems, like cable or tensegrity domes.

Nevertheless, inflated and insufflated domes, from small to medium size, are still frequently employed, when a temporary structure is required, of quick assembling and disassembling. Eloquent examples are given by the itinerant, colourful, organic pavilions produced by artists like Maurice Agis and Alan Parkinson. Companies such as Buildair, in Spain, routinely produce neat examples of temporary pavilions. Festo, in Germany, developed interesting adaptive pneumatic pavilions and roofs, able to adapt to variable wind loads or incidence of natural light, by means of pressure variations. Airlight, in Switzerland, developed the tensairity system, a combination of pressurised tubular fabric beams, externally reinforced by cables and slender metallic ribs.

Inflated lenses, such as the 1988 roof of Nîmes Roman Arena and the Expo’92 German Pavilion, in Seville, have lower operational costs than insufflated domes, although production costs can be higher. Their anchorages are usually lighter, since they do not tend to ascend due to internal pressure. Inflated lenses made of ETFE films are showing good performance as complementary elements to stiff structural systems. The trend started with two project by Nicholas Grimshaw: the National Space Center, inaugurated in 2000, in Leicester, presents wide transparent lenticular walls, and the Eden Project, located in Cornwall and completed in 2001, the world’s largest greenhouse, composed by geodesic domes covered with hexagonal lenses, has just 1% of the weight but up to 95% of the light transmittance of a comparable glass cover. These projects broadened the way to successful sport stadia: the Swiss architects Herzog and de Meuron’s Allianz Arena (Munich, 2005) and the 2008 Beijing’s ‘Water Cube’, designed by PTW Architects (Australia). This last one, comprising an irregular steel space frame, whose geometry was based on the Weaire-Phelan theoretical model for the structure of


soap foam, is the largest ETFE clad structure in the world, with over 100,000 m² of ETFE pillows, some as large as 9.14 metres across.

Along the years, the IASS – International Association for Shell and Spatial Structures meetings played an important role in the diffusion of knowledge on pneumatic structures. More recently, considerable thrust to computer methods is being provided by the International Conferences on Textile Composites and Inflatable Structures, held every two years in Barcelona and Stuttgart, which gather in-depth contributions to the non-linear structural analysis of pneumatic structures and special membrane materials.

3 On the behaviour and design of pneumatic structures

Except for a few particular cases, a pneumatics’ shape cannot be arbitrarily imposed, since membranes do not withstand bending. Thus, as is generally the case for tension structures, the design of a pneumatic structure involves the determination of an initial, viable configuration, encompassing the structure’s shape and the corresponding initial stress field. The viable configuration has to accommodate both architectonic (form and function) and structural requirements (resistance and stability). In brief, design of tension structures is necessarily integrated to analysis, in a process that, besides shape finding, encompasses also procedures for patterning and load analysis. Some references on the subject are Haber and Abel (1982a, 1982b), Knudson (1991), Moncrief and Topping (1993), Barnes (1994) and Pauletti (2008).

Before the widespread use of computational tools, the design of tension structures was performed either analytically or by means of physical models. Analytical solutions exist for stresses in spheres, cylinders, cones and toroids, under uniform and variable pressure loads. Formulas for these geometries are given by Otto (1958, 1967, 1982), Dent (1970), Bulson (1973), Herzog (1977) and Firt (1983), as well as in national standards and guides, such as the AFSI Air Structures Design and Standards Manual (1977). Barnes et al. (2004) refer to several national codes on tension and pneumatic structures, comparing their different design criteria. Many books on shell structures include a chapter on membrane theory. Differential equilibrium of membranes leads to the fundamental Laplace-Young equation, relating forces, acting transversally to the membrane surface, to the in-plane membrane stresses and normal curvatures. Although exact solutions exist only for regular shapes, approximate application of basic formulas may suffice for the quick dimensioning of a broad class of practical structures.

Functional physical models of pneumatic structures require the use of impermeable fabrics or films, able to sustain the pressure gradient that keeps a model in its inflated configuration. It is necessary to verify that the final shape of the model is close enough to the desired one, working at full pressure, without too many wrinkles or excessive stressing. If these requirements are not found, the designer modifies the initial cutting patterns, by educated guessing, in such a way that imperfections at the inflated configuration are minimised.

Computers render this trial-and-error process much easier, at the same time that circumvents problems with materials. Besides, once an appropriate shape is known, they allow the unstressed patterns to be found through a flattening procedure. Therefore, the level of guessing required from the designer is reduced. Moreover, the evaluation of the stresses acting in the membrane can be much more precise with computers than using physical models in reduced scale.

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Until the last decade, it was a prevailing notion that the process of design and analysis of tension structures was a special one, which required the use of special purpose programs (Tabarrok and Qin, 1997). Fortunately, nowadays, with knowledge accumulation and availability of efficient design tools, this process starts to be understood as a standard one, which may indeed be performed with the aid of many general purpose programs, able to reliably perform non-linear structural analyses (Pauletti and Brasil, 2003).

3.1 Alternative methods for static analysis

Several numerical methods have been used in the analysis of tension structures. Two alternative methods prevail. Newton’s method usually yields the best algorithm for the solution of non-linear static equilibrium of this type of structures, since the method presents quadratic convergence rate, in a sufficiently narrow vicinity of the solution. However, cables and membranes have no bending stiffness and thus work as continuous mechanisms, whose stability depends on the geometric stiffness of the system. In the absence of a proper tension field onto the whole structure, its tangent stiffness matrix may become non-positive defined, leading to divergence of the iterative process, either because the structure really goes slack, or simply due to the method’s numerical limitations.

On the other hand, in the dynamic relaxation method, the static equilibrium problem is solved by means of a pseudo-dynamic analysis, with explicit time integration, and fictitious diagonal mass and damping matrices, arbitrarily chosen to control the stability of the time integration process. If constant external forces are suddenly applied, and the system is left to undergo damped vibrations, it eventually converges to the static solution. Although the dynamic relaxation method shows no advantage for small to medium sized, typical membrane problems, it may be economical for very large problems. It also provides an interesting alternative to solve complicated non-linear equilibrium problems, especially when derivatives of the force vector are not directly available (Pauletti et al., 2008, 2009).

En passant, we remark that the well known force density method (Linkwitz and Schek, 1971) can be understood as a special case of Newton’s method, when 2nd Piola-Kirkhoff stresses are imposed at a reference configuration, very conveniently yielding a linear system of equations (Bletzinger and Ramm, 1999; Pauletti and Pimenta, 2008).

3.2 A priori patterning

Since a closed envelope will always reach an equilibrium shape when pressurised, regardless the shape of the unstressed patterns, the design of a pneumatic is somewhat less stringent than the case of tents. Consider, for instance, the patterning of the (ideally spherical) soccer ball. In principle, any pressurised convex polyhedron gives an approximation to the sphere (with very different approximation errors, of course!). The traditional soccer ball is produced with a truncated-icosahedral patterning, as shown in Figure 3. In this case, an exclusively geometrical patterning is defined a priori, and the equilibrium configuration of the inflated envelope provides a fairly good approximation to the sphere.


Figure 3 Truncated-icosahedral patterning of the traditional soccer ball (see online version for colours)

This a priori pattering can be applied also to the design of an inflated dome, as shown in Figure 4. Patterns are conveniently defined as trapezoidal segments. The equilibrium shape of the model, under internal pressure, is determined through a standard non-linear analysis. Since the pressurised model shrinks in the longitudinal direction, while slightly increasing its width and reducing its height, knowledge of the deformed shape allows proper compensations to be made on the initial geometry and patterns, to achieve some desired final dimensions. It also allows the production of a flat, rigid base, according to the final geometry assume by the ends of each segment, thus avoiding wrinkles at these regions.

Figure 4 Initial, unstressed configuration and final configuration (stressed by internal pressure) of an inflated dome (see online version for colours)

The inflated shape of the dome shown in Figure 4 presents longitudinal and vertical segmentation lines. The vertical segmentation is a natural consequence of the diaphragms connecting the inner and outer liners, but the longitudinal segmentation may be minimised, starting the analysis with an initially smooth geometry, and proceeding with a standard shape finding procedure. Some difficulty arises from the fact that if the material is allowed to deform, overall dimensions of the dome may also vary too much. After a reasonable pressurised shape is achieved, a standard pattering procedure can be undertaken, dragging all the nodes of each pattern to a convenient plane. The resulting flat patterns will have curved, convex sides, and manufacturing will require some more care than the trapezoidal patterns of Figure 4.

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3.3 Deformations, high/low energy systems and stiffness

As a general rule, the production of double curvature membranes, starting from flat panels, requires a conscious pattering, to avoid excessive material straining, wrinkles or creases. As an example of the importance of pattering, Bulson (1973) considers the problem of deformation of an initially flat, circular membrane fixed at the border and pressurised to assume a rounded shape [Figure 5(a)]. Analytical solution for this problem – Hencky’s problem (1915) – were given by Fichter (1997) and Campbell (1956). Bouzidi et al. (2003) compared results of these studies with a numerical solution considering a hyperelastic material and large deformation elements.

Figure 5(b) depicts the profiles assumed by an initially flat membrane with radius R = 0.1425 m, a product Et = 311488 Nm, Poison ratio ν = 0.34 and inflation pressures of 100 kPa, 250 kPa and 400 kPa. Henky’s results, indicated by some ‘+’, superimpose quite well with Fichter’s, given by continuous lines. For moderate deformations, isotropic strains are approximately ε 2(δ / r)2 / 3. Thus, strains are about 3.5%, 7% and 10%, for the three pressures intensities considered. Clearly, even the smallest of these strains is over the admissible deformations of materials currently used in pneumatic structures of civil or architectonic relevance, and the spherical shape has to be previously approximated by an adequate pattering.

Figure 5(c) shows the deformed and undeformed configurations of a finite element model developed with the aid of the SATS finite element program, which implements Argyris’ natural triangular membrane element, and is capable to solve membrane structures both via Newton’s and dynamic relaxation methods (Pauletti et al., 2005, 2009). Even thou the SATS considers only linear elastic materials under small strains, results for 100 kPa internal pressure superimposes quite well to Fichter’s and Henky’s. For pressures of 200 kPa and 400 kPa, the SATS’ model gives somewhat higher displacements. Deformations for these pressures are of course outside the suitable range for the material model adopted by SATS.

Models to cope with large deformations of pneumatic structures are given by Noor and Tanner (1985), Noor and Peters (1996), Bonet et al. (2000), Reese et al. (2001), Bouzidi and Le van (2004), Weinberg and Neff (2008) and Mosler and Cirak (2009), among others. Hyperelastic, large-strains models may indeed be necessary to analyse high-energy system, such as highly pressurised balloons, when the pneumatic envelopes undergo large displacements and large strains, during pressurisation. These systems can be rendered quite stiff, but their pressure envelopes are prone to collapse explosively, like a party balloon punctured by a needle. Therefore, architectonic application of high-energy pneumatic systems demands a lot of caution.

On the other hand, an insufflated pneumatic is usually a low-energy system, undergoing large displacements but small strains during pressurisation. A puncture on the pressure envelope does not lead to tear propagation, under internal pressure only, and pressure losses can be compensated by increasing the power of the pumping system.

By its turn, an inflated structure, like an inflatable boat, is usually a moderate-energy system, undergoing large displacements during pressurisation, whilst strains and stresses are kept small to moderate. Tears may propagate, but not as explosively as in high-energy systems. Most probably, a puncture will lead to the slow deflation of the pressure envelope.


Figure 5 (a) Axisymmetric pressurised membrane; (b) comparison between FE and theoretical results and (c) finite element model in SATS (see online version for colours)

(a)

(b)

(c)

Source: (a) and (b) adapted from Bouzidi et al. (2003)

Under external loads, typical inflated envelopes are usually stiffer than insufflated ones, but both still develop small strains. Thus, very schematically, we propose that inflated structures are more suited to applications where a given shape is to be fairly preserved, whilst insufflated structures are adequate to applications where shape variations are not a major concern.

Anyway, except in the less frequent case of high-energy systems, pneumatic structures are intrinsically flexible and therefore they only provide a reasonable structural solution when sharp geometric precision is not at stake. Thus for many practical pneumatics applications, a limit load analysis, based on simple analytical or numerical models, may suffice for design. Furthermore, most pneumatic structures for civil and architectonic purposes can be conveniently represented by simple, small-strains material models.

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Also, it is a well established notion that tension structures are endowed with both material and geometric stiffness. The first corresponds to a reluctance of the structure to alter its stress field, by material straining. The second corresponds to a reluctance of the structure to modify its geometry, by angular variations. Internal pressure also offers a geometric contribution, associated to geometric distortions, plus a volumetric stiffness, which corresponds to a reluctance of the pressure envelope to alter its volume. The higher the internal pressure, the more relevant this term can be. Bonet et al. (2000), Bellmann (2005) and Hassler and Schweizerhof (2005, 2008) present some alternative approaches to cope with volume variations. But there again, the hypothesis of an adiabatic process, of (practically) constant internal pressure and volume, may suffice for a broad class of practical, low-energy systems, as those treated in the sequel.

4 The liner of a biodigester tank

An interesting application is given by the envelope of a large, 40 m × 85 m rectangular biodigester tank. With a prescribed sag of 6 m, the membrane has a curvature radius about R = 41.5 m. An external nylon net is used to limit the deformations of a rubber liner, as shown in Figure 6(a). Figures 6(b) and 6(c) show the initial geometry (given by the intersection of three cylindrical segments) and the finite element mesh used to study the response of this structure to wind. The internal pressure is p0 = 250 N/m2. A basic wind velocity v0 = 30 m/s was extracted from the wind chart provided by Brazilian standard NBR-6123/88 – wind loads on structures, depicted in Figure 7(a). Location of the structure is roughly indicated by a circular dot. Considering a ten years recurrence period, there results a dynamic wind pressure q = 250 N/m2, which has to be multiplied by the external pressure coefficients for the most adverse load condition, given by a lateral wind, as shown in Figure 7(b).

Figure 6 (a) A biodigester tank (with similiar pneumatic liner, but different geometry); (b/c) initial geometry and mesh (see online version for colours)

(a)

(b) (c)


Figure 7 (a) Basic wind velocity chart in Brazil (in m/s); (b) assumed pressure coefficients for a lateral wind (positive: compression; negative: suction) (see online version for colours)

(a) (b)

Although some simple analytical estimates are possible for this problem, based on a cylindrical approximation, we herein present a numerical analysis developed with the aid of the Ansys finite element program. An initial, uniform stress field σ0 = p0R / t = 9.0 MPa is specified. The pneumatic envelope is constituted of EPDM rubber, with very low elasticity modulus Er = 12 MPa, breaking strength, σu,r = 10 MPa and thickness t = 1.15 mm, reinforced by a nylon net, whose strands have an equivalent elastic modulus EN = 2 GPa and breaking strength, σu,N = 80 MPa. The rubber membrane and the nylon net are assimilated to an equivalent material, with an elastic modulus given by Eeq = (ArEr + ANEN) / (Ar + AN), where Ar and AN are the two phases’ cross-sectional areas. For this problem, we have assumed Eeq = 0.7 GPa.

Figure 8(a) shows the field of vertical coordinates for the liner, under internal pressure loads. The prescribed sag is roughly achieved. Figure 8(b) shows the 1st principal stresses field (σ1), under internal pressure. There are some stress concentrations at the corners of the model, due to the consideration of an isotropic equivalent material. However, the stiffness of the equivalent material is due mostly to the reinforcement nylon net, which actually offers no resistance to distortion.

Figure 8(c) shows that consideration of an equivalent orthotropic material, with a low shear modulus (Geq,xy = 0.02 GPa), removes the stress concentrations at the corners, whilst stresses outside these regions are practically unaltered. However, a low shear modulus also introduces some numerical instability, leading to a slightly non-symmetric σ1 stress field. Thus, we rather continue the analysis with an isotropic material, simply disregarding stress concentrations at the corners, as seen in Figure 10(d).

Figures 8(e) and 8(f) show the 1st and 2nd principal stress fields resulting from the combined action of internal pressure and lateral wind loads, disregarding stress concentrations at the corners. Also shown are the relative distributions of anchor loads, whose intensities are required for the dimensioning of the concrete anchorages. The maximum σ1 stress in the equivalent membrane, along the transversal direction, is about 12 MPa, which means a superficial stress about S1 = σ1t 14 kN/m. The maximum σ2 stress, acting along the longitudinal direction, is 9.5 MPa, that is, S2 11 kN/m. Experimental tests with the safety net determined an equivalent superficial breaking stresses Srup = 40 kN/m. Considering that the material presents a loss of strength about 30% during the operational life of the equipment, there results a safety factor γ = 2.0, which may be acceptable for an agro-industrial environment, of low frequency access.

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Figure 8 Biodigester liner: (a) vertical coordinates field; (b) σ1 field, under internal pressure, for an equivalent isotropic material; (c) idem, for an orthotropic material; (d) idem, for an isotropic material, disregarding stresses at the corners; (e) σ1 field under internal plus wind pressures, disregarding stresses at the corners; (f) idem, for σ2 (see online version for colours)

(a) (b)

(c) (d)

(e) (f)

4 An insufflated dome reinforced by sliding cables

Figure 9 shows a perspective view of a large cable-reinforced, insufflated dome designed to cover the site of a new nuclear power plant, during the process of ground preparation. Its design life-time is very short, only six months. The structure has a rectangular plant, 110 m × 86 m, 30 metres high, with rounded corners, and is anchored to a perimeter concrete wall. One of the smaller sides presents a semi-circular inclusion, which breaks the symmetry of the system. The membrane will be constituted by a PVC-coated, polyester fabric, and will be reinforced by seven cables laid over the membrane, aligned


with the longitudinal direction and allowed to slide in the tangential direction. Cables numbering is given in Figure 9(b).

Figure 9 Perspective view of a large pneumatic envelope (110 m × 86 m) reinforced by sliding cables (see online version for colours)

Figure 10 Assumed external wind pressure coefficients for (a) ‘LC1’ – transversal wind and (b) ‘LC2’ – longitudinal wind; (c) numbering of the reinforcement cables

(a) (b)

(c)

An internal pressure load of p0 = 100 N/m2 was initially specified. Self-weigh is disregarded. Considering the site of the structure (roughly indicated in Figure 7(a) by a square dot), the NBR-6123/88 wind code provided a basic wind velocity v0 = 32.5 m/s. Due to the short lifetime, a reduced design wind velocity vk = 20 m/s was adopted, and thus, q = 245 N/m2. Wind tunnel tests were deemed too costly, and the response of the

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structure for wind loads was studied considering two limit conditions, given by the cylindrical and spherical domes, for which pressure coefficients are known. In this paper, we consider only the spherical dome limit, for winds acting along the longitudinal and transversal directions (respectively, load-cases ‘LC1’ and ‘LC2’), with pressure coefficients given in Figure 10.

Numerical analyses were performed with the aid of the SATS finite element code. Figure 11(a) shows the initial mesh adopted for shape finding. Displacements at the borders were restrained, and the membrane was initially subjected to internal pressure load only (load-case ‘LC0’). A low elastic modulus was initially adopted, and the mesh was deformed under internal pressure, until an acceptable shape, shown in Figure 11(b), was achieved. Then material properties were subsequently changed to realistic values, and the resulting mesh used for wind load analysis. An initial diameter of 1 inch was adopted for all cables. Initial normal loads, listed in Table 1, were roughly estimated according to the distance between cables and the sag of each of them. Due to the distance between cables, there resulted also an initial superficial stress S0 = 1.0 kN/m, i.e., an initial stress σ0 = 1.0 MPa, for a nominal thickness t = 1 mm.

Figure 11 (a) Initial mesh modelled in SATS; (b) equilibrium geometry under internal pressure (see online version for colours)

(a) (b)

Table 1 Initial normal loads for the longitudinal cables [kN]

Cable 1 2 3 4 5 6 7

N0 100 60 50 40 50 80 100

Simplified analyses sometimes consider that reinforcement cables are adherent to the membrane, even if this is a condition hardly found in practice, where cables are usually transversally restrained, but free to slip longitudinally. The possibility of slippage of cables over a membrane has been noticed by several researchers (Li and Srivastava, 1974; Mitsugi, 1992; Pauletti and Pimenta, 1995; Matsumura et al., 1997; Bletzinger and Ramm, 1999; Ishii, 1999; Song, 2003; Noguchi and Kawashima, 2004; Zhou et al., 2004). In the present paper, we compare results based on both adherent and non-adherent, frictionless sliding cables, using the ‘sliding-cable super-element’ proposed by Pauletti and Martins (2009a, 2009b) and Pauletti et al. (2009). It is reasonable to suppose that the actual system behaviour is an intermediate situation between full-adherence and ideal, frictionless sliding.

Figure 12 shows, in greyscale, the displacement norms and the 1st and 2nd principal stress fields (σ1 and σ2 fields) for the case of internal pressure loading, with adherent


reinforcement cables (‘LC0A’). The analysis considers the mesh obtained during the shape finding process, shown in Figure 11(b). Since shape-finding was developed considering adherent cables, the maximum displacement norm, about 12 cm, is very small, if compared to the dome’s dimensions. Except from some stress concentrations at the corners and discontinuities (maximum σ1 about 4.1 MPa), the principal stress fields are fairly smooth, with a maximum σ1 stress about 2 MPa on the region of the top of the dome. Due to the adherence between cables and membrane, tractions are variable along the cables, as shown in Table 2, load case ‘LC0A’.

Figure 12 Results for ‘LC0A’ – internal pressure, adherent cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)

(c) (d)

Figure 13 shows the same quantities for the case of internal pressure and sliding cables (‘LC0S’). Since the initial mesh was determined considering adherent cables, maximum nodal displacements up to 40 cm are observed, tangentially to the longest cable, (cable number 6). These displacements are still fairly small, if compared to the structure global dimensions, thus initial mesh was not updated. Also principal stress intensities are rather preserved. Table 3 presents a comparison of normal loads on reinforcement cables, considering both adherent and frictionless sliding conditions. Average normal loads given by the adherent case are fairly comparable to the uniform cables loads given by the sliding case, with some deviation in the case of cable 6, for which larger tangential displacements were also determined, with sliding conditions.

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Table 2 Normal loads on the adherent cables [kN]

Load case Cable Naverage Nmax Nmin

1 107 117 103

2 58 65 41

3 47 50 43

4 40 47 20

5 44 53 27

6 80 120 47

LC0A internal pressure

7 101 122 73

1 198 224 187

2 137 148 108

3 139 157 117

4 130 180 62

5 154 203 117

6 138 214 89

LC1A transversal wind, adherent cables

7 46 92 30

1 258 290 236

2 137 180 107

3 97 145 43

4 81 136 0

5 100 153 12

6 205 307 131

LC2A longitudinal wind, adherent cables

7 227 310 120

Table 3 Comparison between normal loads on adherent and sliding cables [kN]

Internal pressure Wind loads

Trasversal wind Longitudinal wind Cable N0 LC0A (average)

LC0S (uniform) LC1A

(average) LC1S

(uniform) LC2A (average)

LC2S (uniform)

1 100 107 107 198 195 258 250

2 60 58 57 137 133 137 128

3 50 47 47 139 135 97 80

4 40 40 38 130 124 81 45

5 50 44 42 154 149 100 61

6 80 80 70 138 123 205 184

7 100 101 100 46 45 227 217


Figure 13 Results for LC0S – internal pressure, sliding cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)

(c) (d)

Figure 14 presents results for the transversal wind, with adherent cables (‘LC1A’). Maximum displacements about 2.4 m are observed at the weather side. A maximum σ1 stress about 14 MPa is observed at the border corners, again at weather side. In Figure 14(d), the darker region at the weather side indicates that a large portion of the membrane is in a wrinkled state (σ2 = 0) –SATS program adopts Akita’s model for wrinkling (Akita et al., 2007). As pointed out by Ligarò and Barsotti (2008), eventual slack regions may appear in a pneumatic if, and only if, the sum of internal plus external pressure is zero. This is marginally the case of the portion of the weather side region where the external pressure coefficient is +0.4, as shown in Figure 10, for which pext + p0 = 2 N/m2 0.

It is interesting to point out that the (ideal) materials of a tension structure fail due solely to tension, never to compression. So, static wrinkling or slackening of this particular structure do not jeopardise its material safety. Perhaps a larger internal pressure is advisable, to avoid flutter. Full-redundancy is predicted for the pressurising system, so the internal pressure can be actively controlled. Anyway, this discussion will be avoided herein. We remark, however, that in the case of pressurised elements, such as beams, columns or arches, the onset of wrinkling is considered a condition of structural failure (Davids et al., 2007; Le van and Wielgosz, 2005a, 2005b).

234 R.M.O. Pauletti

Previous experiences with the design of membrane structures suggest that they usually respond to wind through a few basic mechanisms, by which external loads are transferred through the membrane to border cables and anchorages. In the current case, an ‘effective clasp’ working at about 10 MPa is formed, identified by a two-headed arrow connecting the vertices at the weather side, superimposed to the clear grey region. Borrowing, from limit analysis, the notion that any load transferring system that can be devised provides an upper limit for the ultimate load of a structure, we propose that identification of effective clasp mechanisms allows a simplified analysis of membrane structures. However, so far these mechanisms have been only identified a posteri, and since fabrics are not ductile material, the true ultimate load can be far lower than this upper limit.

Figure 14 Results for LC1A – transversal wind, adherent cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)

(c) (d)

In the case of transversal wind, with frictionless sliding cables (‘LC1S’, shown in Figure 15), a somewhat larger maximum displacement is observed, about 2.5 m, at the same region as the adherent case. Although the same stress pattern is also observed, maximum σ1 stress raises up to 18 MPa. Nevertheless, a similar effective clasp is observed, again with stresses around 10 MPa. Besides, there exists a similar wrinkled region at the weather side. It can be seen in Table 3 that also the uniform normal loads for LC1S compare pretty well to the average normal loads for the adherent cables, given by LC1A. In both conditions, the outer cable at the weather side (cable 7) reduces the load considerably, whilst the outer cable at leeside (cable 1) almost double its value.


In the case of longitudinal wind, with adherent cables (‘LC2A’, shown in Figure 16), a maximum displacement about 2.2 m is observed at the weather side. Maximum σ1 stress reaches 15.4 MPa at the weather side corners, whilst the effective clasp, connecting the circular inclusion to the farther weather side corner, presents σ1 stresses about 11 MPa. Large wrinkled regions are observed both at weather side and at leeside (the darker regions of Figure 16d).

Finally, in the case of longitudinal wind, with frictionless sliding cables (‘LC2S’, shown in Figure 17), the maximum membrane displacement rises significantly, up to 3.15 m. Nevertheless, the maximum σ1 stress is reduced to 11.1 MPa, and this value spreads out along the effective clasp, which becomes wider, connecting both the larger border sides. A wrinkled region similar to the adherent case is observed at the weather side, but little wrinkling is observed at leeside. It can also be seen in Table 3 that the longitudinal wind more than doubles the normal loads at the outer cables. However, in the adherent case (LC2A) this increase is observed also in the central, number 4 cable, whilst in the frictionless sliding case (LC2S) the uniform normal load acting on this cable is fairly unaltered, if compared to the case of internal pressure loads only.

All summed up, it is seen that at least for this particular structure, cable sliding does not alter significantly the response of the structure to winds transversal to the cables, but much larger displacements are observed when the wind acts parallel to them, whilst stress concentrations are smoothed out, and the typical ‘effective clasp’ spreads into a wider region.

Figure 15 Results for LC1S – transversal wind, sliding cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)

(c) (d)

236 R.M.O. Pauletti

Figure 16 Results for LC2A – longitudinal wind, adherent cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)

(c) (d)

Figure 17 Results for LC2S – longitudinal wind, sliding cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (see online version for colours)

(a) (b)


Figure 17 Results for LC2S – longitudinal wind, sliding cables: (a) field of displacement norms; (b) idem, isometric view; (c) σ1 stress field; (d) σ2 stress field (continued) (see online version for colours)

(c) (d)

Acknowledgements

The author gratefully acknowledges the support by Nohmura Foundation for Membrane Structure’s Technology, Japan.

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