This paper provides background and overview of the methods and utilization of computing in the design and construction of tensile membrane structures. In addition the author outlines the general methodology employed in the use of automated processes in the design, fabrication and construction of these structures.
No other class of architectural structural systems is as dependent upon the use of digital computers as are tensile membrane structures. The shape or form and prestress of tensile structures are determined using true "computer aided design". Typical simple structural systems defy classical analysis. Structural behavior is simulated under load using computer modelling techniques. The procedures for prestressing the system are determined in similair analysis. Finally, the drawings or templates used to cut and fabricate the fabric mmembrane surface are typically computer generated.
The modern use of tensile membrane structures as a means of permanently covering large spaces has been wholly dependent upon the use of digital computing. Many of the developments in membrane structure technology have occured in the last twenty years precisely because of the accessibility of relatively powerful digital computers. Significant pioneering work of Frei Otto was accomplished using physical models(1), which while they well illustrate the desired form of a membrane are not conductive to the precise communication and/ or documentation of the membrane's structural characteristics in a manner necessary for the construction of large and/ or complex systems.
Tensile membrane structures have unique difficulties that have made them resistant to classical methods of design and analysis. Generally, they are non linear in behavior. Typically structures exhibit both geometric non-linearity due to large deflections in addition to material non-linearity. The nature of tensile membrane is such that much of their stiffness is acheived by the virtue of the initial prestress in the membrane and its supporting components. This prestress is an internal stress condition usually prescribed by the designer to achieve the desired performance of the structure and must be induced into the system in its construction.
The general methodology pursued in the design and construction of a tensile membrane structure is illustrated in the flow chart shown in Figure 1. Processes which are typically automated are highlighted. As with most design methodologies the process is iterative, such that anticipation of the results in the conception of a structure will reduce the general effort involved in the design and engineering of the system. While there are a number of algorithms presently used with success for each of the computer processes, the general methodology illustrated is appropriate for a wide variety of prestressed tensile systems.
In the simple case of air-supported structures the membrane prestress is acheived by synclastic shaped membrane with a differential air pressure. The simplest form of air supported structure for which the prestress can be easily determined is a spherical dome. Assuming that the unit weight of the membrane is small with respect to the internal operating pressure , the membrane stress at a given pressure is proportional to the radius of curvature of the sphere. While the analysis of such a structure under real wind loads is non-trivial, both membrane patterning and determination of the prestress are easily accomplished without the aid of computing. Consequently, it is not surprising that the first widely used air-supported membrane structures were the spherical air domes built by Birdair Inc.
The consideration of low profile non-spherical air-supported membrane structures required better analytical tools. Both the geometry or "shape" of the cablenet and the analysis of the air-supported roof of the US Pavillion at Expo '70 were accomplished on a digital computer by David Geiger Associates with assistance from Dr. Michael McCormick. This is beleived to be the first use of the digital computer in form finding and analysis of a built membrane structure. In this case as with all of the early air-supported engineered by Geiger, the fabric surfaces were patterned by "hand" as the surface geometry of the membrane was simple enough that this could be accomplished satisfactorily. The first computer patterned fabric membrane for a low profile cable restrained air-structure was used in the Minneapolis Metrodome roof, patterned by Birdair.
Prestressed anticlastic tensile structures present a more difficult problem. A wide variety of complex forms can be determined from physical models. As demonstrated by Frei Otto, minimal surfaces can be created using soap films . However, none of these techniques can precisely communicate to the fabricator the prestress and surface geometry information required to fabricate and stress the membrane shape. This became the pressing issue as desirable materials suitable for permanent structures, such as TEFLON (TM) coated fiberglass became available. Fabrics such as TEFLON (TM) coated fiberglass have desirable attributes such as their non-combustibility, however, they are significantly stiffer than other materials commonly used in tensile structures and require greater precision in pattern making or patterning. The development of algorithms for defining the surface form or shape of a general class of prestessed networks was the key to the general exploitation of tensile membranes in structures of significant scale.
The are a number of form finding algorithms in use. Geiger Engineers employs software based upon the force density method(2). This matrix method solves directly for the geometry of a general network of prestressed tensile components. Iterative tecniques allow the designer to prescribe desired prestress conditions for cable and membrane elements. Birdair Inc. successfully employs their matrix analysis algorithm for form finding. Basically elements are given a very low mechanical stiffness and a prescribed prestress. Equilibrium geometry is determined in an iterative analysis of the structure. Another method of form finding in common use is the method of dynamic relaxation with kinetic damping(3). This method is employed in the form finding used by FTL Associates.
The ability to generate shapes on a digital computer within a prescribed boundary with a prescribed prestress quickly lead to to computer patterning of shapes. The problem is to determine the pattern for flat strips of fabric which when seamed together will approximate the shape's surface. As the shape's geometry is determined for a prestressed condition, patterns must be compensated for strain in the fabric. Compensated strip patterns are used for cutting.
While the physical models are still used to study membrane forms, the geometry and stress conditions of the membrane surface are now almost exclusively determined by designers utililizing computing techniques. Data from form finding, typically comprised of connectivity, nodal geometry, and the element represents a complete model description of the membrane structure, and the element properties. Consequently, shape results can be utilized directly for analysis. Often, the the additional elements, such as struts and or beams are added to a shape model to create an anlysis model of a complete structural system.
General analysis of membrane structures requires geometric non-linear techinques. Typical matrix methods employ an iterative procedure using the Newton-Paphson method or a variant, often with a damped solution strategy. Many tensile structural systems are strain hardening. A variety of common tensile structural systems are initially strain softening and begin to exhibit to strain hardening behavior once sufficient load is applied. Consequently, non-linear solution stategies that anticipate strain hardening have been employed with success and can speed convergence in a wide variety of commonly encountered problems. There are significant exceptions, such as a class of "tensegrity" type structures that become strain softening as load is increased. The dynamic relaxation method is also used with success for the general analysis of geometrically non-linear problems.
Most architectural/structural fabric materials exhibit non-linear behavior, as a consequence of being woven composites. Almost all architectural/structural fabrics in use today are coated composites. However, material non-linearity is rarely modeled. Mechanical behavior of textiles is primarily dependent upon the properties of both the yarn and the weave. Coating properties also have an effect upon the composite's mechanical behavior, albeit at a lesser extent than the properties of the base cloth. Fabric is commonly modeled utilizing LST and CST finite element methods or a network of string elements. Both of these modelling approaches have been widely used with success while each has attendant limitations that the analyst must consider. Membrane elements that better simulate the non-linear behavior of woven composites have been developed(5). While the fabric material non-linerity is typically no modelled, it will likely prove to be useful when the mechanics of fabric failures are better understood and utilized quantitatively in a limit state design approach.
The ability to create, analyze, design and fabricate complex membrane forms has in turn created difficult construction problems. Prestress is as much a property of these structures as element properties and/ or geometry. A prestressed state for a structural system can be created without direct regard for the manner in which the prestress is developed in the structure. In a wide variety of structures, this is in fact preferred. Consequently, with redundant structures techniques to establish the sequence of stressing is necessary to assure that the structure will in fact realize the prestressed state desired. Moreover, in many complex tensile systems analysis of the stressing sequence is necessary to assure that various components of the system are not over stressed during stressing. A technique developed by Geiger Associates of analytical disassembly of a prestressed structural system in reverse order of stressing has been utilized by the author as well others at Gieger Engineers and Birdair Inc. with great success. The erection and stressing of some structural systems such as Geiger's Cabledome, it's variants, and other complex prestressed structural systems can be determined in this manner. Generally, the accurate construction of these structural systems would not be possible prior to the development of appropriate software and suitable techniques for the determination of stressing sequences. This was the key in the realization of many significant membrane structures including the Haj Terminal at Jeddah, Lyndsay Park Athletic Centre, Calgary Alberta, the Ontario Pavillion Roof at Expo '86 Vancouver B.C. and all the Cabledomes including the Florida Suncoast Dome and the Georgia Dome Roof system.
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