Teaching the design of cable-membrane structures

Transcription

1 Teaching the design of cable-membrane structures P. Iványi Pollack Mihály Faculty of Engineering, Department of Information Technology, Pécs, Hungary Index terms: special structures, design methodology, practical design. I. INTRODUCTION One of the main characteristics of cable-membrane structures that they have no stiffness against loading perpendicular to the line of the cable or the surface of the membrane. [1] This behaviour is illustrated in Figure 1. The figure shows the displacement of a beam and a cable structure when they have to withstand a perpendicular load. Figure 1b shows that the cable exhibits large displacements. The large displacements result in a significant change of the geometry and therefore lead to a geometrically non-linear design and analysis procedure. (a) Beam Fig. 1. Displacements of beam and cable structures. (b) Cable The stiffness of a cable-membrane structure can be increased in two ways: using special geometry (for example increasing the sagging height) or using prestressing. Furthermore the structural behaviour of cable-membrane structures can be characterized by the following equation: T 1 + T 2 = F (1) R 1 R 2 where T 1 and T 2 are the internal forces in the membrane in the direction of the principal curvature, R 1 and R 2 are the radii of the principal curvature and F is the external load. In the case of cable-membrane structures only tension forces can develop in the structure which means that: T 1 > 0 and T 2 > 0 (2) and if there is no external force the following formula can be obtained from Equation 1: T 1 = T 2 R 1 R 2 (3) Formula 3 suggests some very important information about the shape of the cable-membrane structure, as to satisfy this formula the only possibility is if the sign of the radii of the principal curvatures are different. Such shapes are called anticlastic and an anticlastic surface is shown in Figure 2a. On the other hand when there is an external, supportive force, like air pressure Equation 1 can be satisfied in different ways. Figure 2b shows a clastic surface, which is the general form for air-supported structures. II. DESIGN PROCESS OF CABLE-MEMBRANE STRUCTURES It was shown that Equation 1 is very important, as it dictates certain aspects of the geometry of cablemembrane structures. This leads to the problem that the initial, equilibrium shape of cable-membrane structures must be designed by a form-finding process. [1], [2] The form-finding process determines the equilibrium shape of the structure for a given geometry, prestress and material selection. Different parameters will result in different shapes. This type of form-finding, where even the initial shape of the structure is 1 peteri@morpheus.pte.hu

2 (a) Anticlastic surface (b) Clastic surface Fig. 2. Classes of surfaces for cable-membrane structures. not known in advance, is not common for other civil engineering structures, for example for bridges, halls, building, frames, bars, etc. Another problem is the manufacturing of cable-membrane structures. Figure 2 shows that double curved surfaces are commonly used for cable-membrane structures, however these structures are manufactured from industrial textiles which are produced in rolls. From the roll of material only plane surfaces can be obtained, therefore the double curved surfaces must be flattened. The usual procedure for flattening is that strips of materials are determined on the curved surface (as it can be seen in Figure 3a), then the strips can be easily flattened. The flattened pieces make up a cutting pattern. An example cutting pattern is shown in Figure 3b. The definition of cutting pattern is that it is a production plan, which is generated from an equilibrium state of the structure to determine the stress-free side lengths of the membrane pieces in plane. The generation of this cutting pattern is also unique for cable-membrane structures. (a) Generation of a strip of material (b) Generated cutting pattern Fig. 3. Cutting pattern generation for cable-membrane structures. The following list summarizes the steps of the design process for cable-membrane structures which can be compared to the design process of other civil engineering structures: 1) Shape definition 2) Creation of engineering model 3) Form-finding 4) Cutting pattern generation 5) Analysis under loading 6) Detailing and construction studies III. TEACHING THE DESIGN OF CABLE-MEMBRANE STRUCTURE The above described special properties of cable-membrane structures require a different approach in their teaching. One of the main emphasis in the teaching of cable-membrane structures is that the students must

3 assimilate the basic rules and techniques. To make clear for the students how these formulas work and what kind of behaviour the students can expect from a cable-membrane structure a design program has been introduced into the course. By using the design program the students can try out different geometries, configurations, prestressed states. Figure 4 shows the window of the design program. Fig. 4. Design program for cable-membrane structures. A. Design program for cable-membrane structures There are several numerical methods to design cable-membrane structures: grid method [3], Stuttgart direct method [4], force density method [5], surface density method [6], dynamic relaxation [1], finite element method [7]. Out of these methods the combination of force and surface density methods is used in the program. The reasons for this selection that they are very simple, they model the structure correctly considering geometric non-linearity and the combination of them uses an iterative process. The iterative process is very important from the pedagogical point of view, as the students can follow the development of the surface. In this way the students see what kind of changes occur in the shape. Furthermore if during the process no equilibrium can be achieved it is possible to change the parameters of the design and it is possible to continue the process without going back to the initial shape. Figure 5 shows the initial and the final shape for a structure. From the figure it is clear that there is no way to determine how the final shape has evolved. The evolution of the surface is important from another point of view as the numerical problems can indicate where structural problems may occur. (a) Initial shape (b) Equilibrium shape Fig. 5. Form-finding of cable-membrane structures. B. Examples of cable-membrane structures In the teaching of cable-membrane structure it is also important to show already built structures, case studies. Through these examples the students gain a better understanding what has been built and what is possible to build. Furthermore the detailing of cable-membrane structures is also unique for almost every

4 structure. However in engineering practice it is an important technique to copy and adapt previous designs and therefore through the examples previous solutions for complicated joints between cable-cable, cablemembrane, cable-anchor, membrane-anchor and so on can be studied. C. Student projects During the course the students have to design a cable-membrane structure. To ensure that everybody will create a unique design the students receive different shapes of areas which should be covered by the cable-membrane structure. The shapes of areas include: rectangle, triangle, circle, rhomboid, 5 and 6 sided polygons and so on. There is no further specification, for example where the masts (fix) points should be, what should be their height and so on. It is interesting to observe, that throughout the years, students who have attended the classes and experimented with the program could create very nice structures and have a better understanding of their behaviour. On the other hand every year there is a group of students who cannot perform the design task satisfactorily and their performance in the exams are also lacking. Figure 6 shows the results of some student projects. IV. SUMMARY AND CONCLUSION This paper has presented some unique properties of cable-membrane structures and it also discussed why these properties require some special design procedures and why it is mandatory to teach this subject through examples and experimentation. REFERENCES [1] B. H. V. Topping and P. Iványi: Computer Aided Design of Cable-Membrane Structures, Saxe-Coburg Publications, Stirling, [2] Michael Barnes and Michael Dickson (Editors): Widespan roof structures, Thomas Telford, London, [3] J. Szabo and L. Kollar: The structural design of cable-suspended roofs, Halsted Pr, [4] K. Linkwitz: Formfinding by the Direct approach and pertinent strategies for the conceptual design of prestressed and hanging structures, International Journal of Space Structures, vol. 14, pp , [5] H. J. Scheck: The force density method for form finding and computation of general networks, Computer Methods in Applied Mechanics and Engineering, vol. 3, pp , [6] B. Maurin and R. Motro: The surface stress density method as a form-finding tool for tensile membranes, Engineering Structures, vol. 20, pp , [7] T. Shimada and Y. Tada: Development of a curved surface using a finite element method, In 1st International conference on computer aided optimum design of structures, pp , 1989.

5 (a) (b) (c) (d) (e) Fig. 6. Student projects.