Post-Tensioned Modular Inflated Structures Romuald Tarczewskil Wroclaw University of Technology romuald.tarczewskiOpwr.wroc.pl Pneumatic structures are usually considered as inalterable,with predeter- mined features,e.g.shape etc.According to this approach,they cannot be easily rebuilt or modified.Application of techniques current in other sectors of construction industry leads to the structures that are much more flexible and adaptable. 1 Introduction The prestressing technique was applied to concrete and steel constructions in first decades of 20th century.It appeared after a period of development of these constructions,and its usage is obvious and common at present.Yet in reference to pneumatic constructions,the conception of prestressing ap- peared at the beginning of their development.The first known documentary evidence of the conception to use structural pneumatic elements comes from engineer Joachim A.Sumovski.He obtained,in 1893,an American patent on air-inflated structures [1].One of the drawings included in patent specification presents the structure that obtains its shape and loading capacity as a result of prestressing,Fig.1. Fig.1.An example of air-inflated structure invented by J.A.Sumovski 221 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,221-239. C 2005 Springer.Printed in the Netherlands
Post-Tensioned Modular Inflated Structures Romuald Tarczewski 1 1Wroclaw University of Technology romuald.tarczewski@pwr.wroc.pl Pneumatic structures are usually considered as inalterable, with predetermined features, e.g. shape etc. According to this approach, they cannot be easily rebuilt or modified. Application of techniques current in other sectors of construction industry leads to the structures that are much more flexible and adaptable. 1 Introduction The prestressing technique was applied to concrete and steel constructions in first decades of 20th century. It appeared after a period of development of these constructions, and its usage is obvious and common at present. Yet in reference to pneumatic constructions, the conception of prestressing appeared at the beginning of their development. The first known documentary evidence of the conception to use structural pneumatic elements comes from engineer Joachim A. Sumovski. He obtained, in 1893, an American patent on air-inflated structures [1]. One of the drawings included in patent specification presents the structure that obtains its shape and loading capacity as a result of prestressing, Fig. 1. Fig. 1. An example of air-inflated structure invented by J.A. Sumovski 221 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 221–239. © 2005 Springer. Printed in the Netherlands
222 Romuald Tarczewski However in air-supported structures,particularly of large span,strength- ening cables are nowadays often applied [2]-prestressing is not yet obvious in air-inflated structures. Another,well-known structural concept,which provides with many ad- vantages in terms of creating structures -is the use of repeatable,modular units,that can be assembled in a larger,complex structure.The spectacular example of application of repeatable air-inflated units for constructing a large structure-was the pavilion of Fuji Group,at Expo'70 exposition in Osaka 3. The advantages of modular structures are especially exposed,when units of relatively small dimensions are used.This allows constructing structures of various shapes,while limited quantity of different units are applied.The units can be used repeatedly many times.Shape of the structures created in this way is limited only by topological restrictions of division of considered surface.This applies in particular to the shell structures. Post-tensioning is a way of splicing small elements but also a way of shap- ing the structure.Coupling of these operations raises the efficiency of solution. Because of specific properties of pneumatic structures,suitable technical so- lutions are required,especially concerning supplying the elements with air, stabilization of their shape as well as method of connection. 2 Modular Air-Inflated Elements Applied to Shell Structures 2.1 Principles of Composition Depicted structures present a class of spatially curved surface girders.Their rigidity and loading ability are strongly related with the shape.Spatial cur- vature itself,is not a permanent,generic feature of the structure (as it is in concrete shells for example),but is achieved and maintained by means of post-tensioning,causing very large initial deformation.Thus,the final shape and properties of the structure are function of initial configuration and the course of post-tensioning process. Internal structure of modular air-inflated shells distinguishes them from other air-inflated structures and from other shell structures.They consist of the following basic elements: air cushions (main modular elements) tension cables cross-braces (optional) Air cushions and cables appear in shells of all types,while cross-braces only in the shells with increased structural height [4.After assembling,structure forms complete roofing and does not require any additional membranes to cover space.General view and components of an exemplary structure is shown on Fig.2
222 Romuald Tarczewski However in air-supported structures, particularly of large span, strengthening cables are nowadays often applied [2] – prestressing is not yet obvious in air-inflated structures. Another, well-known structural concept, which provides with many advantages in terms of creating structures – is the use of repeatable, modular units, that can be assembled in a larger, complex structure. The spectacular example of application of repeatable air-inflated units for constructing a large structure – was the pavilion of Fuji Group, at Expo ’70 exposition in Osaka [3]. The advantages of modular structures are especially exposed, when units of relatively small dimensions are used. This allows constructing structures of various shapes, while limited quantity of different units are applied. The units can be used repeatedly many times. Shape of the structures created in this way is limited only by topological restrictions of division of considered surface. This applies in particular to the shell structures. Post-tensioning is a way of splicing small elements but also a way of shaping the structure. Coupling of these operations raises the efficiency of solution. Because of specific properties of pneumatic structures, suitable technical solutions are required, especially concerning supplying the elements with air, stabilization of their shape as well as method of connection. 2 Modular Air-Inflated Elements Applied to Shell Structures 2.1 Principles of Composition Depicted structures present a class of spatially curved surface girders. Their rigidity and loading ability are strongly related with the shape. Spatial curvature itself, is not a permanent, generic feature of the structure (as it is in concrete shells for example), but is achieved and maintained by means of post-tensioning, causing very large initial deformation. Thus, the final shape and properties of the structure are function of initial configuration and the course of post-tensioning process. Internal structure of modular air-inflated shells distinguishes them from other air-inflated structures and from other shell structures. They consist of the following basic elements: – air cushions (main modular elements) – tension cables – cross-braces (optional) Air cushions and cables appear in shells of all types, while cross-braces only in the shells with increased structural height [4]. After assembling, structure forms complete roofing and does not require any additional membranes to cover space. General view and components of an exemplary structure is shown on Fig. 2
Post-Tensioned Modular Inflated Structures 223 air-inflated cushions tension cables cross-braces (optional) Fig.2.Composition of modular air-inflated shell 2.2 Structural Components Air cushions Pneumatic modular elements have a form of cushions shaped in order to fit geometrical constrains of the prospective surface.Basic shapes are:rectangle, square,rhomb,hexagon and triangle.Following conditions must be fulfilled in order to enable effective application of modular system: shape of the cushions must correspond with final shape of the structure;it is obvious that the usage of as few varying shapes as possible is profitable elements must be small enough in relation to the final structure,to form a smooth,easily deformable surface(at least ten times smaller);additional criteria can be also applied,such as easiness of in-site manipulation e.g. one workers should be able to carry the cushion the connections between the elements have to assure their suitable inte- gration as well as continuity of transmission of internal forces the cushions should be suitably equipped with guides allowing the usage of post-tensioning cables Other components Bar members (i.e.cross-braces)are optional-they appear only in some types of shells.These additional bars are used in order to increase the structural height.They can be made of any lightweight material capable of carrying com- pression,like steel,aluminum,wood or composite.Naturally,closed sections (tubular)are better than the others,for this purpose.Connection of bars and cushions must prevent damages (e.g.by means of strengthened pockets)
Post-Tensioned Modular Inflated Structures 223 Fig. 2. Composition of modular air-inflated shell 2.2 Structural Components Air cushions Pneumatic modular elements have a form of cushions shaped in order to fit geometrical constrains of the prospective surface. Basic shapes are: rectangle, square, rhomb, hexagon and triangle. Following conditions must be fulfilled in order to enable effective application of modular system: – shape of the cushions must correspond with final shape of the structure; it is obvious that the usage of as few varying shapes as possible is profitable – elements must be small enough in relation to the final structure, to form a smooth, easily deformable surface (at least ten times smaller); additional criteria can be also applied, such as easiness of in-site manipulation e.g. one workers should be able to carry the cushion – the connections between the elements have to assure their suitable integration as well as continuity of transmission of internal forces – the cushions should be suitably equipped with guides allowing the usage of post-tensioning cables Other components Bar members (i.e. cross-braces) are optional – they appear only in some types of shells. These additional bars are used in order to increase the structural height. They can be made of any lightweight material capable of carrying compression, like steel, aluminum, wood or composite. Naturally, closed sections (tubular) are better than the others, for this purpose. Connection of bars and cushions must prevent damages (e.g. by means of strengthened pockets)
224 Romuald Tarczewski Post-tensioning cables are always placed at the internal side of curved shell.In case of anticlastic shells,cables are placed in two layers -at the opposite sides of the shell.Direction of cables in each layer corresponds with the curvature of the shell,Fig.6.The cables can be placed directly below the cushions or below cross-braces.In both cases the cable should be able to slide freely through the nodes.Both,fiber ropes (made of natural or man-made fibers)and wire ropes can be used as post-tensioning cables.It is significant that the cushion was protected from damages caused by ropes,e.g.by means of protective jackets. 2.3 Transmission of Forces After completion,modular elements must transmit internal forces induced in the structure.It is possible due to the compression of cushions'sides(touching each other)and tension of their external cover and post-tensioning cables, Fig.3.Thus,the way of assembling the cushions must ensure a full contact of side surfaces and continuity of external cover. These basic principles allow setting a geometrical configuration of the shells of various size and shape,designed for various purposes. tension of external cover compression of side surfaces air-inflated cushion post-tensioning tension of cable bottom cable cross-braces Fig.3.Transmission of forces in modular air-inflated shells
224 Romuald Tarczewski Post-tensioning cables are always placed at the internal side of curved shell. In case of anticlastic shells, cables are placed in two layers – at the opposite sides of the shell. Direction of cables in each layer corresponds with the curvature of the shell, Fig. 6. The cables can be placed directly below the cushions or below cross-braces. In both cases the cable should be able to slide freely through the nodes. Both, fiber ropes (made of natural or man-made fibers) and wire ropes can be used as post-tensioning cables. It is significant that the cushion was protected from damages caused by ropes, e.g. by means of protective jackets. 2.3 Transmission of Forces After completion, modular elements must transmit internal forces induced in the structure. It is possible due to the compression of cushions’ sides (touching each other) and tension of their external cover and post-tensioning cables, Fig. 3. Thus, the way of assembling the cushions must ensure a full contact of side surfaces and continuity of external cover. These basic principles allow setting a geometrical configuration of the shells of various size and shape, designed for various purposes. Fig. 3. Transmission of forces in modular air-inflated shells
Post-Tensioned Modular Inflated Structures 225 3 Application of Post-Tensioning and Self-Erection The structure is stabilized by means of post-tensioning.This process induces internal forces that are shown on Fig.3.Distribution of these forces is invari- able during exploitation,though their values can change.Alteration of forces' direction effects in destruction of the structure -in consequence of opening of the gaps between cushions or "compression"of their external cover,Fig.4. tension of external cover and compression of side surfaces disappears gap between chusions starts to open Fig.4.Failure mode of the shell There are two ways of realizing the post-tensioning procedure: structure is post-tensioned and erected simultaneously (self-erection) post-tensioning is applied to previously erected structure 3.1 Self-Erection Procedure In that case,the flat structure is assembled at ground level as a near mecha- nism.It is stabilized and finally shaped by means of self-erection.This process unifies operations of post-tensioning,erection (i.e.construction)and spatial curving of the structure.The essence of the process is the introducing into the structure forces that cause its large deformation [5].In practice,the process starts simply with a shortening of the bottom cables.The cables are attached to the fixed supporting points while going through all the other joints to the opposite,mobile,supporting points. As a result-supporting points are brought closer to each other.Thus the deformation is introduced to the structure and it starts to erect.The process is continued till required position is obtained.Then the cables are fixed in the mobile supporting points.Fig.5 presents successive stages of self-erection process. Air-inflated shell can be post-tensioned either in one direction or in two directions.Unidirectional post-tensioning is applied in order to get structures with zero Gausian curvature (cylindrical),while bidirectional-when struc- tures with negative Gausian curvature (anticlastic)are required,e.g.hypar surfaces,Fig.6. Bidirectional post-tensioning is performed successively.At the beginning, cables of the first direction are tensioned to 50-60 of assumed value.Then
Post-Tensioned Modular Inflated Structures 225 3 Application of Post-Tensioning and Self-Erection The structure is stabilized by means of post-tensioning. This process induces internal forces that are shown on Fig. 3. Distribution of these forces is invariable during exploitation, though their values can change. Alteration of forces’ direction effects in destruction of the structure – in consequence of opening of the gaps between cushions or “compression”of their external cover, Fig. 4. Fig. 4. Failure mode of the shell There are two ways of realizing the post-tensioning procedure: – structure is post-tensioned and erected simultaneously (self-erection) – post-tensioning is applied to previously erected structure 3.1 Self-Erection Procedure In that case, the flat structure is assembled at ground level as a near mechanism. It is stabilized and finally shaped by means of self-erection. This process unifies operations of post-tensioning, erection (i.e. construction) and spatial curving of the structure. The essence of the process is the introducing into the structure forces that cause its large deformation [5]. In practice, the process starts simply with a shortening of the bottom cables. The cables are attached to the fixed supporting points while going through all the other joints to the opposite, mobile, supporting points. As a result – supporting points are brought closer to each other. Thus the deformation is introduced to the structure and it starts to erect. The process is continued till required position is obtained. Then the cables are fixed in the mobile supporting points. Fig. 5 presents successive stages of self-erection process. Air-inflated shell can be post-tensioned either in one direction or in two directions. Unidirectional post-tensioning is applied in order to get structures with zero Gausian curvature (cylindrical), while bidirectional - when structures with negative Gausian curvature (anticlastic) are required, e.g. hypar surfaces, Fig. 6. Bidirectional post-tensioning is performed successively. At the beginning, cables of the first direction are tensioned to 50–60 % of assumed value. Then
226 Romuald Tarczewski final position intermediate position initial position tension of the bottom cables Fig.5.Scheme of self-erection procedure air-inflated cushions cables in bottom layer tension of the upper cables cables in upper layer Fig.6.Bidirectional post-tensioning of air-inflated shell cables of the second direction are tensioned to the same value.Finally,the cables are alternatively rectified,to reach requested values. 3.2 Post-Tensioning Applied to Completed Structure Another way of assembling is to put cushions in position one by one-as in an igloo and then tension the cables to stiffen the structure.In that case,shell must be shaped as a self-stable before post-tensioning is performed.Certainly, scaffoldings can be used,however,this overthrows the whole system. Structures with positive Gausian curvature (synclastic),surfaces of rev- olution in particular,can be achieved in that way.Fig.7 presents such a structure,shaped as a hemisphere,made of hexagonal elements
226 Romuald Tarczewski Fig. 5. Scheme of self-erection procedure Fig. 6. Bidirectional post-tensioning of air-inflated shell cables of the second direction are tensioned to the same value. Finally, the cables are alternatively rectified, to reach requested values. 3.2 Post-Tensioning Applied to Completed Structure Another way of assembling is to put cushions in position one by one – as in an igloo and then tension the cables to stiffen the structure. In that case, shell must be shaped as a self-stable before post-tensioning is performed. Certainly, scaffoldings can be used, however, this overthrows the whole system. Structures with positive Gausian curvature (synclastic), surfaces of revolution in particular, can be achieved in that way. Fig. 7 presents such a structure, shaped as a hemisphere, made of hexagonal elements
Post-Tensioned Modular Inflated Structures 227 air-inflated meridian paralel cushions cables cables Fig.7.Construction of synclastic air-inflated modular shell 4 Adjustment of Rigidity and Hardening Systems 4.1 Structures with Variable Rigidity Curvature of air-inflated shells formed in self-erection process can be con- trolled by means of changing their rigidity along the span [6].It is an ef fective way of shaping this kind of structures,which allows fulfilment of the requirements.Initial stiffness of inflated shell is defined mostly by its struc- tural height.The change of height causes the change of stiffness.This can be achieved in two ways: either by use of additional rods,i.e.cross-braces,moving tension cables down from the cushions or by change of cushions'thickness Figure 8 presents those two methods. cushions with variable thickness cross-braces with variable length Fig.8.Methods of rigidity alteration in modular shells Two compared shells,of the same initial length and subjected to the same upthrust,i.e.nearing of supports,but with various,variable rigidities,demon- strate distinctly different final geometry.If the rigidity of the shell is increased in the central part-the curvature is smaller in the center than in peripheres (structure is more flat).On the other hand-the curvature is smaller on sides, when rigidity is decreased in the central part of the shell(structure is more
Post-Tensioned Modular Inflated Structures 227 Fig. 7. Construction of synclastic air-inflated modular shell 4 Adjustment of Rigidity and Hardening Systems 4.1 Structures with Variable Rigidity Curvature of air-inflated shells formed in self-erection process can be controlled by means of changing their rigidity along the span [6]. It is an effective way of shaping this kind of structures, which allows fulfilment of the requirements. Initial stiffness of inflated shell is defined mostly by its structural height. The change of height causes the change of stiffness. This can be achieved in two ways: – either by use of additional rods, i.e. cross-braces, moving tension cables down from the cushions – or by change of cushions’ thickness Figure 8 presents those two methods. Fig. 8. Methods of rigidity alteration in modular shells Two compared shells, of the same initial length and subjected to the same upthrust, i.e. nearing of supports, but with various, variable rigidities, demonstrate distinctly different final geometry. If the rigidity of the shell is increased in the central part – the curvature is smaller in the center than in peripheres (structure is more flat). On the other hand – the curvature is smaller on sides, when rigidity is decreased in the central part of the shell (structure is more
228 Romuald Tarczewski scarped).Fig.9 presents a comparison of shapes of cylindrical shells with variable rigidity.More sophisticated shapes can be achieved when rigidity is changing according to a parametric function. rigidity decreased in the center 442 rigidity increased in the center Fig.9.Comparison of shells with variable rigidity 4.2 Adaptable-Hardening Structure Variation of rigidity allows constructing a structure that actively adapts to external loadings.If the internal lever arm increases together with increasing external loadings-deformations increment slowly.Load-deformation relation in this case is an exponentially growing function.If the internal lever in the structure can be self adjusted in order to find position of equilibrium,then the structure reveals a "hardening"characteristic.In numerous situations this specific type of structures is more advantageous than any other,with a linear response.An example of hardening system is described in [7].Its application in pneumatic shells [8 is shown on Fig.10. air inflated cush●ns basic spring units Fig.10.Example of air-inflated hardening structure
228 Romuald Tarczewski scarped). Fig. 9 presents a comparison of shapes of cylindrical shells with variable rigidity. More sophisticated shapes can be achieved when rigidity is changing according to a parametric function. Fig. 9. Comparison of shells with variable rigidity 4.2 Adaptable – Hardening Structure Variation of rigidity allows constructing a structure that actively adapts to external loadings. If the internal lever arm increases together with increasing external loadings – deformations increment slowly. Load–deformation relation in this case is an exponentially growing function. If the internal lever in the structure can be self adjusted in order to find position of equilibrium, then the structure reveals a “hardening”characteristic. In numerous situations this specific type of structures is more advantageous than any other, with a linear response. An example of hardening system is described in [7]. Its application in pneumatic shells [8] is shown on Fig. 10. Fig. 10. Example of air-inflated hardening structure
Post-Tensioned Modular Inflated Structures 229 5 Technological Concepts 5.1 Air-Inflated Cushions The cushions are the basic components of the shell.The air supplying system described below maintains internal pressure.Cushion is usually a flat element -its thickness is smaller than dimensions in plane.Two main surfaces (upper and bottom)and several side surfaces can be distinguished in the cushion.It is generally made of soft textile or foil,suitable for pneumatic structures The cushions are equipped with elements that allow attaching a tension rope.These are the flexible hoops enabling the rope to slide easily.The rope is dragged through the hoops during assembling.The hoops are placed at the cushion's corners,on one or on both main surfaces (e.g.for hypar shells). If the cross-braces are to be placed in the structure,strengthened pockets are made in the cushions.Strengthening prevents damage of soft fabric caused by bar edges. Cushions with flaccid main surfaces must be equipped with internal ele- ments ensuring flat shape after inflation.Fabric diaphragms(with openings) or set of threads can be applied in this case.These elements are not neces- sary if both main surfaces are made of rigid panels.Fig.11 presents a general scheme of the cushion entirely made of fabric. For the structures with variable rigidity,cushions with variable thickness can be prepared.Main surfaces of such a cushion are not parallel and side surfaces have various width,as shown on Fig.8. touch fastener strips bonded on upper face fabric welt-10 cm width at non-welt edges upper surface (bottom face covered with touch fastener strip) internal air pressure conduit junction module of air-supplying system hoops- post-tensioning side surface cables connectors approx.100+150cm bottom surface shape stabilizers (diaphragms) Fig.11.Internal structure of air-inflated cushion
Post-Tensioned Modular Inflated Structures 229 5 Technological Concepts 5.1 Air-Inflated Cushions The cushions are the basic components of the shell. The air supplying system described below maintains internal pressure. Cushion is usually a flat element – its thickness is smaller than dimensions in plane. Two main surfaces (upper and bottom) and several side surfaces can be distinguished in the cushion. It is generally made of soft textile or foil, suitable for pneumatic structures. The cushions are equipped with elements that allow attaching a tension rope. These are the flexible hoops enabling the rope to slide easily. The rope is dragged through the hoops during assembling. The hoops are placed at the cushion’s corners, on one or on both main surfaces (e.g. for hypar shells). If the cross-braces are to be placed in the structure, strengthened pockets are made in the cushions. Strengthening prevents damage of soft fabric caused by bar edges. Cushions with flaccid main surfaces must be equipped with internal elements ensuring flat shape after inflation. Fabric diaphragms (with openings) or set of threads can be applied in this case. These elements are not necessary if both main surfaces are made of rigid panels. Fig. 11 presents a general scheme of the cushion entirely made of fabric. For the structures with variable rigidity, cushions with variable thickness can be prepared. Main surfaces of such a cushion are not parallel and side surfaces have various width, as shown on Fig. 8. Fig. 11. Internal structure of air-inflated cushion
230 Romuald Tarczewski rigid panel made of plywood. metal or plastic intemal air pressure conduit junction module of air-supplying system hinge hoops- elements post-tensioning cables connectors bottom surface side surface shape stabilizers (threads) Fig.12.Internal structure of semi-rigid air-inflated cushion For some applications,semi-rigid cushions are necessary.In this case,one or both main surfaces can be made of rigid panels(metal,plywood or plastic), Fig.12.Side surfaces are always made of textile,to assure a proper contact between cushions after assembling. 5.2 Connections of Cushions There are two points of connection: connection of main surfaces connection of side surfaces Side surfaces are connected by means of direct clamping,as it is shown on Fig.3.In order to ensure a proper connection,contact of surfaces can- not be restricted in any way (e.g.by protruding rigid panels,cross-braces or connectors of air supplying system). Connection of main surfaces should assure continuity of transmission of tensile forces in the shell,in all directions.Additionally,if the shell is used as a cover protecting against weather conditions,these connections should assure the tightness of the shell. If the main surfaces of the cushion are soft,a convenient type of connec- tion is "touch fastener"(e.g.Velcro).Welts with a bottom face covered with fastener strips are placed along some of cushion's edges.The edges at the reverse sides of the cushion are also covered with touch fastener strips.The continuity and tightness of the shell can be easily obtained during assembling, Fig.13. Another feasible connection is a "zipper"-a wire strand or flexible bar dragged through the small,tight-fitting eyes placed along edges,Fig.14.Both
230 Romuald Tarczewski For some applications, semi-rigid cushions are necessary. In this case, one or both main surfaces can be made of rigid panels (metal, plywood or plastic), Fig. 12. Side surfaces are always made of textile, to assure a proper contact between cushions after assembling. 5.2 Connections of Cushions There are two points of connection: – connection of main surfaces – connection of side surfaces Side surfaces are connected by means of direct clamping, as it is shown on Fig. 3. In order to ensure a proper connection, contact of surfaces cannot be restricted in any way (e.g. by protruding rigid panels, cross-braces or connectors of air supplying system). Connection of main surfaces should assure continuity of transmission of tensile forces in the shell, in all directions. Additionally, if the shell is used as a cover protecting against weather conditions, these connections should assure the tightness of the shell. If the main surfaces of the cushion are soft, a convenient type of connection is “touch fastener” (e.g. Velcro). Welts with a bottom face covered with fastener strips are placed along some of cushion’s edges. The edges at the reverse sides of the cushion are also covered with touch fastener strips. The continuity and tightness of the shell can be easily obtained during assembling, Fig. 13. Another feasible connection is a “zipper”– a wire strand or flexible bar dragged through the small, tight-fitting eyes placed along edges, Fig. 14. Both Fig. 12. Internal structure of semi-rigid air-inflated cushion