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CONCEPTUAL DESIGN The origins of the design of modern tensile roof structures are based on physical modelling techniques, particularly for the conceptual form-finding process [1]. These physical procedures have ranged, in order of accuracy, from minimum surface soap films, through stretch fabric nylon models and the use of specially formed hexagonal weave models, to the use of silkscreen fabric (with very low stretch) or uniform mesh wire models. The latter procedures were sufficiently accurate for fabrication patterning of simple structures provided that, for prestressed membranes, the material used in the real structure had adequate flexibility, or that boundary turnbuckles were incorporated in cable networks to adjust tension distributions on site.
31 FORM AND STRESS MODELLING OF TENSION STRUCTURES Michael Barnes SYNOPSIS This article gives a brief overview of the development of form-finding applications for modern tensile roof structures, particularly CAD methods with stress or fabrication control of form. Selected projects are chosen to illustrate the application to cable networks, woven steel mesh systems, prestressed mechanisms, coated fabric membranes and air-supported structures. The purpose is to review the form-modelling of direct force structures in which the form must physically be a reflection of the prestress distribution - to provide a background description, principally for architects who will work with engineers in the design process. CONCEPTUAL DESIGN The origins of the design of modern tensile roof structures are based on physical modelling techniques, particularly for the conceptual form-finding process [1]. These physical procedures have ranged, in order of accuracy, from minimum surface soap films, through stretch fabric nylon models and the use of specially formed hexagonal weave models, to the use of silk- screen fabric (with very low stretch) or uniform mesh wire models. The latter procedures were sufficiently accurate for fabrication patterning of simple structures provided that, for prestressed membranes, the material used in the real structure had adequate flexibility, or that boundary turnbuckles were incorporated in cable networks to adjust tension distributions on site. A Fig 1 Fig 2 The most sophisticated design to be carried through to the stage of fabrication patterning using such techniques, with precisely measured wire models, was probably the cable network for the German pavilion at EXPO 67 (Figs 1 & 2), designed by Frei Otto in collaboration with Gutbrod, Leonhardt & Andrea, and Linkwitz. Whilst other structures may have been technically more precise through the use of analytical descriptions for prestressed shapes, the EXPO 67 pavilion represented a truly free- form system; and it was perhaps this structure, more than any other, which captured the imagination of both the architectural profession and the gneral public and led to the popularity of tensile structures. During the design of the Munich Olympic Games stadium, it became apparent that even quite large-scale and accurate wire models of the network system could not be sufficiently precise for design purposes; and since this time, numerical methods for form-finding, load analysis and fabrication patterning of both prestressed membranes and cable networks have played an essential role in the engineering design process and in the development of conceptual models [2,3,4]. 32 Fig 3 COMPUTER-AIDED DESIGN The aim of CAD procedures might be stated as the replacement of physical modelling techniques by computer programs enabling realistic graphical display of form and stress levels and interactive control of these aspects together with the patterning and detail design features. An "expert systems" approach incorporated in these CAD procedures might further provide a guide to conceptual design, and coupled with CFD procedures may eventually lead to far better wind load definitions for complex shapes. In reality, simple stretch-fabric physical models (Fig 3) continue to be useful for the initial form studies of complex free-form systems. In addition to the tactile value of such models in conceptual design, they provide a means of communication between various members of the design team and the best learning process for new members of a team. They also yield an estimate of surface curvatures and hence stress distributions and, when stiffened by a reinforced resin coating, the basis for wind tunnel model tests. Models may, although now unusually, also provide a guide to suitable patterning arrangements and mesh generation topology for finite- element type analyses. For cable network systems, this entails the determination of preferred mesh orientation, generally with a uniform mesh (or grid of equal length links), while for prestressed membranes a model clarifies the choice of panels for fabric patterning - generally employing "geodesic" seam lines. These geodesies describe lines that follow a minimum distance over the surface - equivalent to the trajectories that would be followed by finite width tapes without shearing distortion. The geodesic seam lines can thus define the edges of shear-free panels for fabrication purposes. In most cases now, only an imagined physical model is necessary for the definition of an initial computer model. From an initial definition of topology, CAD procedures for membranes control the seam geometry to provide precise cutting patterns with optimum use of material, the overall structural form being governed simultaneously by specified stress distributions in the surface for a weightless prestress state. The neglect of self-weight in form-finding is important since it allows the shape to be determined purely on the basis of stress ratios in the principal weave directions (the warp and weft of the fabric), with subsequent scaling of stress levels to satisfy load state design conditions (including self weight). For fabrication patterning, the "compensations" that must be applied in the factory cutting of initially stress-free fabric panels are also governed by the choice of design prestress levels and the warp/weft stress ratios, with allowances made for in- service load distributions and relaxation of the prestress. For cable networks the CAD process will start from an initial coarse definition of topology with equal length links in each net direction (apart from end links intersecting with curved boundaries). Tension ratios in each net direction are also specified to cuntrol the surface curvatures - as with membranes. However, the process is then usually one of gradual refinement - doubling and then qudarupling the mesh density - until a net is found with no untensioned links and sufficient accuracy for fabrication patterning; "sufficient" meaning that patterms in a large surface net can be obtained to the nearest millimetre - steel cables are far less forgiving than coated fabrics. During all of this process the topology must be changing, with links disappearing from boundaries in some regions and new links appearing in others. It can be seen from the above description of the interrelations between form, patterning, design loads and stress distributions that the use of CAD procedures is an essential feature in both the conceptual and developed design stages for tension structures; and that, together with changes in support geometry and topology, the specification of required stresses, or forces in discrete components, is of prime importance in the control of form. Fig 4 33 Fig 5 Fig 6 CABLE NET AND UNIFORM MESH STRUCTURES The initial form investigations for cable networks and analysis checks for various load states may often be based on an equivalent membrane model with appropriate properties and with one of the cable traverse directions approximately parallel to the warp (or geodesic direction) lines in each surface region. A coarse grid cable-net model may subsequently be numerically assembled over the membrane surface by choosing one of the warp lines as a control traverse, from which the network can be set out link by link until intercepting with the boundaries. Networks numerically constructed in this way may be of three types: (a) uniform grid nets (the most commonly used in practice); (b) geodesic nets - since all warp lines in the membrane surface are geodesies, one set of cable traverses will be parallel to these lines throughout the surface and the other set can be assembled so that angles of incidence onto the first set are equal to their angles of departure; in such geodesic nets all cables have constant tension along their lengths in the prestress state. (c) hybrid nets in which one set of cables are geodesies over the surface with equal link lengths and the other set have constant tensions throughout their lengths. Only the third type of net can theoretically exactly fit the membrane surface - provided that the equivalent membrane stresses are uniform and equal in all directions (as a soap fdm model). However, for all three types, the procedure yields a good starting topology for the net analysis and its form adjustments. The procedure is closely analogous to the initial generation of nets from fabric models used for concept studies. Figure 4 shows a fabric study model (for Gatlinberg community centre) which was subsequently fibreglassed for wind tunnel testing (Fig 5). The physical modelling in this case progressed in parallel with the numerical modelling which is shown in figures 6-9. In this coarse CAD model the cable traverse spacings represent four grids of the real uniform link net and the colours represent tension levels: light blue or yellow for the target or low but acceptable tensions, dark blue for high tensions and red for slack (or buckled) links. At an early stage in form-finding from an initially flat net the tensions are rather random with slack or highly stressed links concentrated in the boundary or mast support areas (Fig 6). The first true equilibrium state is shown in figure 7. In arriving at this state, as much as possible in the CAD process may be automatic - for example, in short traverses with shallow curvature the average tension can be checked and end link lengths adjusted automatically to meet target values. But for longer traverses with more complex curvatures, increased stressing at their ends in an attempt to eliminate large slack areas (see buckled band in fig 7) will merely over-stress other areas of the net, particularly around mast supports. In such cases, a substantial shearing of the net and/or adjustment of mast configurations is necessary in order to force the slack bands to follow greater distances over the surface - the adjusted prestress state, with no slack or over-stressed links, is shown in figure 8. After evaluating network tensions under wind and snow loadings further adjustments of the form may evidently be necessary; for example, under snow loading, the tensions around the hoop and radial support cable ring of the central mast were too high, and the effective bearing diameter of this ring was therefore increased (fig 9). Fig 7 34 Fig 10 Fig 9 The form-finding and analysis of large or finely spaced networks may be carried out using comparatively coarse grid models. For fabrication patterning, corrections to the grid lengths of the numerical idealization must subsequently be made to allow for local curvatures. This may be achieved by fitting splines thriough the grid traverses and shorteneing the slack grid lengths by the difference between surface arc and chord lengths. The final stage of patterning involves precise boundary and mast support zone analyses using a fine grid spacing, with geometry and tension distributions interpolated from the overall analysis to set the local zone analyses. A structure which perhaps more vividly illustrates the concept of net angle (or shearing) distortions to fit a surface shape is Munich Aviary (Fig 10). The continuous surface is formed from butt welded constant width rolls of crimped stainless steel fine mesh (Fig 11). The wires are held in place mechanically by the crimping, but allow in-plane shearing angles of up to 30°. The feasibility of constructing such a system without wrinkling (and hence, in this case, parting) of the mesh depends on the following factors: (a) the number and height of masts and tie-down systems; (b) the number of convex indentations and concave curves forming the plan boundary geometry (in relation to the number of masts); (c) the freedom given, assuming a fixed plan, for adjustments in the heights of the mesh attachment along the boundary curve; and (d) the precise arrangement of the trellis supporting plates around each mast or tie-down (Figs 12 & 13). Fig 11 Fig 12 Fig 13 Fig 14 35 4* Fig 16 A simple analogy is of laying an open weave cotton net over a sphere: at the top the net will be orthogonal, but as the net progresses down the sphere it must shear to hold to the surface. At a particular depth parts of it must depart outwards from the surface otherwise it will wrinkle. In contrast to the structures reviewed above, the concept of overlapping shingle plates for the Radolfzell concert sail dictated that as many plates as possible were standardized within a tolerance governed by the size of the clamp bolt holes (Figs 14 & 15). The 0.55m cable grid was thus arranged to be nearly orthogonal throughout the surface. To achieve this the surface was split into regions by means of ridge cables which also limited deflections and provided security for the main mast. A similar but more free-form structure was proposed for the Munich Zoo large cat enclosure (Fig 16). The orthogonal net orientations for these shingle structures are clearly dictated by the drainage requirements. Since the sides of plates are parallel with the net cables the ideal orientation is at 45° to the steepest gradient direction, though with a practical tolerance of ±20°. An alternative which could provide a stiffer structure, with the net traverses more closely following lines of principal curvature, would be to orientate the net at 45° to the plate edges, although perhaps at the expense of aesthetic appearance. Since the shingles are physically attached to the net only at opposite corners this also suggests the use of a hybrid geodesic type net (type C referred to earlier). A third alternative of a true principal curvature net is probably impractical from the point of view of standardization of the cladding panels. 36 Fig 21 The same glazed shingle system as in the Radolfzell concert sails was used for the undulating wall enclosure of the German pavilion at EXPO 92 in Seville (Figs 17 & 18). Again, a major objective was to employ the greatest number of standard panel units. In this case, however, there were to be no regional splitter cables; the entire surface was continuous. Instead of splitting the surface into regions, two geodesic bands (AA and BB in figure 19a) were employed to induce more equal tensions in the longer set of traverse cables (inclined at about 30° to the horizontal). To disguise these "tucks" in the otherwise uniform network the necessary shortening of the traverses was spread over three consecutive links in each geodesic band - figure 19b shows a region of the final network viewed normal to the plane 123. Whilst the shingle panels adjacent to the band AA are non-standard, the great majority of the surface panels are orthogonal. The main roof of the pavilion was a pneumatic lens membrane structure 90m x 50m, with an elliptic internal boundary truss, which was intended to give the impression of floating over the main assembly / functions area (Figures 20 and 21). The structure was supported by a single main mast which, because of its inclination, allowed the system to be stabilized only by slender cables (varying from 22 - 42mm 0) attached to the perimeter of the pneumatic lens (Fig 22). At the base of the mast the reactions in the self-weight state are components vertical and horizontal (parallel with the main axis of the ellipse); the vertical component is resisted by purely vertical cables around the perimeter, with greatest tensions in cables at the end nearest the mast in order to counter the overturning weight. The horizontal component is resisted by just two cables (shown thicker in the image), taken from either side of the lens structure to strong points at the top of the exhibition building. To accomodate variations in live load, particularly the lateral components due to wind, two further cables are taken from these strong points to a single point at the apex of the lens furthest from the mast; this arrangement of cables also resists overall torsion of the structure [5]. In effect the entire system is "docked", rather analogously to a ship which is moored to a dockside by fore, aft, and shear lines. A similar principle, though with more complex structuring, was used in the design of the Guthrie pavilion in Singapore (Fig 23). Fig 22 Fig 23 Fig 25 Fig 24 PRESTRESSED MEMBRANE AND AIR SUPPORTED STRUCTURES As with cable systems, certain controls in CAD form- finding of membranes, such as controls on stress distributions, can be interactive. A very simple example is shown in figures 24 and 25, in which necking contraction of a conic is occurring because of insufficient warp stress (in the radial direction). The system can be restored to a desired form (Fig 26) by altering the single parameter of warp line tension to induce increased warp stress in the necked areas. Other adjustments, for example of panel and warp orientations, will entail some resetting of topology - yet this can be established by specifying only a few principal control lines in each surface, with automatic interpolation between them. The colour display of behaviour (Fig 25), form and patterning (Fig 30), and stress distributions (Figs 35 and 36), are clearly useful aids to guide the design process. The display and simple storage of stress contours is important from an engineering point of view for the comparison of different load states (and adjustments to form or material properties). The same graphics routines for stress plots are also used for contours of height (to examine possible ponding in shallow areas), or any other variable such as slope, discontinuity of slope and wind incidence - the latter to aid load definitions. The two main entrances at EXPO 92 both employed widespan cable and membrane shade structures [5]. The Fig 26 Fig 27 "Diadema", shown in figure 27, employed a porous fabric partly in order to alleviate the high wind loads on the structure which had a maximum height of about 55m and span of 77m. The main surface structure was a wide grid cable net of equal spacing in one direction and equal traverse tensions in the other, with the lightly stressed fabric acting solely as shade covering. The "Oleada" entrance structure, shown in figures 28 - 31, employed a more highly stressed heavy-grade fabric, together with cable reinforcing in the surface whose principal function was to stabilize the central compression booms. Fig 28 Fig 29 The design of the Oleada was intended as a continuation and reflection of La Barqueta bridge which joined the EXPO site to Old Seville. An initial functional aim of the structure was to constrain and then open out the view of the EXPO site. The sculptural form that emerged was enhanced (Fig 30) by adjusting surface stress ratios and the positioning and heights of the boundary mast points so that the surface reflected the motion of a bird in flight - "Oleada". The membranes span from perimeter guyed masts to two main central arches of 60 and 70m span, one inflected downwards and the other upwards. The first downward arch is suspended and pre-compressed by cables from 65m high V-form masts, with the compression force in the arch balanced by ties to the ground at its free ends. The second, upward arch is pre- compressed by light ties to ground level with the main free end thrust in the arch sustained by cables to the top of the V masts (Figs 29 & 31). In order to decouple the interactions between these two systems the masts were additionally guyed by independent cables to ground anchorages. Fig 30 39 The original concept for the design was that the two main arches should be tensegrity type systems - in the sense that each was to consist of eight pin-jointed slender compression booms stabilized by spiral cable bracing for torsional and general stability (Fig 29), assisted by the mast stay cables or ground ties for stability in the vertical plane and by transverse cables in the membrane surfaces to enhance lateral stability. In fact, because of the complexity and high cost of connection details for the tensegrity system, and the difficulty of construction in terms of required tolerances, alternative segmental arches using thin walled large diameter tubes were eventually employed. In spite of their sizes (650 and 810mm diameter) they still had a slender appearance (Fig 31). Fig 32 In a subsequent study [6] the use of fabricated segments was investigated. Each segment consists of a central compression boom and three tie rods to either end braced apart in the centre of the segment by a triangular yoke. The segments are then prestressed into a pin-jointed tensegrity arch system using only three continuous longitudinal chord cables attached to the apices of the triangular yokes (Fig 32). The advantage of this system is that only the chord cables (or in fact, for a circular arc, only one of the chords) need prestressing adjustments, and the problems of tolerances on site should be alleviated. However, although the resulting arch is stiff in bending it has no torsional stiffness - since, in contrast to the Oleada concept model, there are no spiral bracing cables. In fact, this is quite acceptable and even beneficial provided the membranes (or cables within the membrane surface) act compositely with the arch structure: Figure 33 illustrates the stability of the system under extreme transverse wind load which induce greatest torsional load. The yokes twist along the arch and allow the stresses in the membrane regions on either side to equilibrate eachother. A more recent project taken to full engineering design by IPL, though not built, is a very large air-supported structure for covering industrial waste, principally employed in order to minimize the cost of contaminated water treatment. This low-rise structure has a kidney shape (Fig 34) with main and minor axis spans of 500m x 280m and was to be fabricated in very heavy-grade PVC polyester in standardized 20x20m panels with mechanical joints. The high strength of the fabric eliminates the need for a reinforcing cable grid, which otherwise would have to sustain the main spanning tensions because of the disparity between membrane and cable stiffnesses (particularly with longer term effects). In addition to its strength the membrane has very high visco-elastic damping which is of benefit in terms of dynamic behaviour. However, this also entails substantial creep, particularly in the weft direction of the material. A major aspect of the design and associated material testing was therefore to account for both first time loading elastic stiffnesses and long-term creep effects in order to ensure a reasonable degree of load sharing between warp and weft fibres under various load states - figures 35 and 36 show contour plots of warp and weft stress for one wind loading direction. W Fig 33 Fig 34 Fig 36 Another important consideration in the design of very wide span air-supported structures is the question of dynamic stability. High-rise structures with greater curvatures may have potentially lower membrane stresses but are likely to be subject to non-uniform, shape-deforming distributions of wind pressures and suctions. In contrast, low-rise systems are likely to have much more uniform wind suctions over the entire surface, but higher tensions will be induced because of the shallow curvature. Although these higher tensions incur penalties in terms of foundation loads they significantly improve large scale dynamic stability. A greater problem for these large scale air-supported structures (and other shallow membrane systems) is snow loading and the potential occurrence of ponding and consequent failure. This has been a difficulty with several of the largest built structures of this type and is perhaps the reason for the reduced use of air-supported systems in recent years. An alternative view is that control systems and the means of modelling wind and snow loadings and their interaction with structural systems have now substantially improved. These aspects are considered in the final two papers of this section on form and structure. ACKNOWLEDGEMENTS Consulting Engineers for the Gatlinberg Centre project and for Munich Aviary were Buro Happold. The Architects for the former were Sprankle Lynd and Sprague and for the latter was Jorg Gribl. Frei Otto was Consultant Architect for both projects. Consulting Engineers and Architects for all other projects illustrated were IPL under the direction of the late Harald Muehlberger. REFERENCES 1. Frei Otto: Tensile Structures, MIT Press , 1971 2. Haug, E, Powell, G H: Finite element analysis of non-linear membrane structures, IASS Symp. on Tension Structures and Space Frames, Tokyo, 1971 3. Argyris, J H, Angelopoulos, T, Birchat, B: A general method for the shape finding of lightweight structures, Int. Conf. on Tension Strucrtures, London, 1974 4. Barnes, M R: Applications of dynamic relaxation to the design and analysis of cable, membrane and pneumatic structures. Second Int. Conf. on Space Structures, Guildford, 1975 5. Barnes, M R, Renner, W, Kiefer, M: Case studies in the design of widespan EXPO structures, Proc. Conceptual Design of Structures, Stuttgart, 1996 6. Adriaenssens, S, Barnes, M R, Mollaert, M: Deployable torsion free tensegrity spines, Proc. Engineering a New Architecture, Aarhus, 1998