TECHNICAL FABRICS IN CONSTRUCTION OF LARGE SCALE ROOFS - NUMERICAL AND EXPERIMENTAL ASPECTS

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1 TECHNICAL FABRICS IN CONSTRUCTION OF LARGE SCALE ROOFS - NUMERICAL AND EXPERIMENTAL ASPECTS Paweł Kłosowski 1, Andrzej Ambroziak 1, Adam Zagubień 2 1 Gdansk University of Technology, ul. Narutowicza 11/1., PL Gdańsk, Poland, klosow@pg.gda.pl, ambrozan@pg.gda.pl 2 Koszali Universiry of Technology, ul. Śniadeckich 2, PL Koszalin, Poland, zagubien@wbiis.tu.koszalin.pl Received ; accepted Abstract. Designing and construction of textile hanging roofs is the challenging subject for engineers. This type of structure needs geometrically and very often also physically non-linear calculations. Due to different behaviour of two families of threads (warp and weft), which during deformation can change the angle between their directions, they require special constitutive modelling. Also the loading of hanging roofs is very special. The wind loading should be taken as the following forces, and also its distribution on the roof surface is not unique. Also the snow loading (for permanent structures) can not be usually taken according the national standard. The authors present their experiences with the finite element calculation of hanging roofs at the example of the Forest Opera open area theatre in Sopot (Poland) new designed roofing. Keywords: hanging roof, technical fabric, warp, weft, climatic loading, constitutive modelling, form finding 1. Introduction The hanging roofs made of a technical fabric become more and more popular nowadays. They can be used to cover large scale areas (Fig. 1), as well as roof s of small objects (Fig. 2). Fig 1b. Large scale textile roofs temporary tent for Hevelius bear fest in Gdansk (Poland) (right) Fig 1a. Large scale textile roofs roof of the Gottlieb-Daimler Stadium in Stuttgart (Germany) In this kind of roofs the fabric play two important roles. It gives protection against atmospheric phenomena s and it is also caring constructional element. Different sets of loading must be taken into account when a roof is permanent and different when it is a temporary structure. Also different types of textile fabric can be used mainly according to economic and architectural reasons but also due to their different physical properties. As the textile

2 membrane roofs can be subjected to tensile stresses only, the important problems accrues during the shape forming and the system of support design. produced by Mehler (Germany) (polyester threads, both sides PVC covered) has been applied (Fig 3a). Fig 3. Forest Opera in Sopot hanging roof (state from 1999) (a) and current state with additional rain protection over stage area (2006) (b) Fig 2. Small textile roofs temporary garage (up), permanent small roofing near the commercial center in Prague (Czech Republic) (down) In the paper several aspects of design of large scale textile roofs will be discussed on the basis of the Forest Opera in Sopot (Poland) open-air theatre roof. 2. History of the Forest Opera in Sopot open-air theatre roof The open-air theatre in the forest in Sopot (15 km from Gdansk) has been built in 1909 by Paul Walther Schaffer (German architect). Up to sixties of previous century it was used without roofing, what caused many problems especially during rain falls. As that time in the Forest Opera in Sopot started to be the place of well known song festivals, it was necessary to protect the stage, as well as audience, against rains. The first roof was constructed in 1964 on the basis of PVC covered cotton fabric, and due to bad properties of the fabric was several times damaged by wind and humidity. At the beginning of seventies the supporting system of the roof was redesigned and new more weather condition resistant fabric was used. The roof of this shape is operating until now, only in 1990 the new textile fabric VALMEX P50 Fig 4. Model of new Forest Opera roof The roof is working as the temporary structure and every autumn must be disassembled, as it is not designed for snow loading. These every year assembling disassembling procedures have caused some damages of the roof surface, therefore in 2006 it was necessary to construct additional protection over the stage (Fig. 3b). In 2005 the authorities of Sopot town decided to announce the competition for new design of the Forest Opera,

3 including new concept of the roof. From many designs the butterfly-shaped, also textile roof has been chosen (Fig. 4). 3. General description of Forest Opera roof The structure of the roof consists of two over 100 m large and 27 m high steel circular arches. To these arches and 8 posts, two surfaces of textile fabric of the roof are connected. The area between two arches is going to be covered be glass panels. As one of the most important requirements has been that new roof is going to be the permanent structure, the snow loading has become the most important load case. After initial calculations (especially due to large snow loadings) the final shape of the Forest Opera roof had to be modified (Fig. 5). To protect the roof against large displacements which can cause snow bags the system of internal snow ropes has been added (Fig. 6). 4. Form finding Many important publications devoted to this subject can be found in the literature (e.g. [1-4]). There are several methods of such initial configuration finding. In the design of the Forest Opera roof two of them were applied. In the first one, the initial values of the stress components for each segment of the textile fabric and for each rope were selected, and each structure element has been divided into required number of elements as it is described in Fig. 5. However, this method (which was successful in this case) has same drawbacks. The main of them is lack of convergence of the iteration process after application of initial loadings. Therefore also the method of application of the initial stresses to the flat surface of the roof was also checked (Fig. 7). In such configuration the finite element algorithm had no problem to find the balanced configuration. In the next step the nodes of the roof supported by the arch and posts were moved to their proper positions in several increments (Fig. 8). Finally, in so obtained configuration exploitation loadings were applied. The first very important stage of a textile roof design is finding of the initial shape of the roof, which is in good correlation with the shape requirements and which guarantees proper distribution of stresses. Additionally, as most of the structures are calculated using the finite element method, the initial parameters of the structure must ensure stability of the nonlinear iterations process. The problem of so called form finding is one of the most important in the hanging roofs design. Fig 6. Snow and s system Fig 7. Flat initial configuration of the single wing of the Forest Opera in Sopot Fig 5. Main dimensions and shape of new designed the Forest Opera roof

4 1,8e+5 1,6e+5 1,4e+5 1,2e+5 polyester threads - warp polyester threads - weft glass threads - warp glass threads -weft Stress [N/m] 1,0e+5 8,0e+4 6,0e+4 4,0e+4 2,0e+4 Fig 8. Configuration of the roof after prescribed displacement of the supporting nodes 0, Strain [%] 5. Constitutive modeling of the textile fabric. Modern large scale hanging roof are made almost always made form from two types of fabric: polyester threads covered by PVC or of glass fibers covered by PTFE (Teflon). In the numeric calculations the fabric can be modeled by several models. Their description can be found in [5]. In the calculations of the Forest Opera the dense net model of the fabric was used. Full description of this model can be found in [6, 7]. In this model two threads families (warp and weft) are treated independently with possibility of change of angle between them during deformation process. As the textile fabric of the hanging roof works as the membrane shell the strains can be calculated from the equations ε x ε ε12 = = ε 2 2 y = Tε xy ε 2 cos α sin α sinαcosα γ xy where ε1, ε 2 are strain components in the warp and weft respectively, and α is the actual angle between thread families. Forces in each family of threads can be calculated using the uniaxial stress state constitutive equation and required physical model (1) σ σ1 F1( ε1) 0 ε1 12 = = 12 σ = 2 0 F2( ε2) ε F ε (2) 2 where F 1 and F 2 are proper constitutive functions in uniaxial approach. Finally stresses in the fabric (plane stress state) can be recalculated from 2 σ 1 cos α x 2 σ1 T σxy = σy = 0 sin α = ( T) σ 12 (3) σ 2 τ 0 sinαcosα xy Fig 9. Strain-stress graphs for polyester (9032 Architectural Fabric) and glass threads (Sheerfill) fabrics The actual value of the angle between threads can be obtained from the formula σ α = arctg x (4) τ To apply equation (2) it is necessary to establish experimentally functions F 1 and F 2 Here two types of experiments can be carried out: uniaxial tensile tests (according to national standards e.g. EN ISO 1421:1998) or biaxial tests with different proportions of the warp and weft forces. Typical results of uniaxial constant strain rate tests (according to EN ISO 1421:1998) in the warp and weft directions are presented in Fig. 9 for two kinds of fabric. stress [N/m] ,00 0,01 0,02 0,03 0,04 0,05 0,06 xy strain [.] uniaxial test biaxial test 1:1 biaxial test 1:2 biaxial test 2:1 Fig 10. Comparison of uniaxial and biaxial test for 9032 fabric warp It is easy to notify the differences in fabric behavior with respect to direction of the test (warp and weft direction) and with respect of threads material. Glass fibers have better mechanical properties but on the other hand they are more than 2,5 times more expensive (about 100 /m 2 for polyester/pcv fabric, and 260 /m 2 for glass/ptfe fabric). In Fig comparison between results of uniaxial and biaxial tests is presented. During the biaxial test it is difficult to keep constant ratio

5 between forces in the warp and weft. Also registration of tests results is also more complex. Therefore even if they are made in the experienced laboratory, their results are much more difficult for interpretation. Additionally, in a real structure in each point forces ratio is different. This is the main reason that in numerical calculations of hanging roofs mainly the uniaxial tests results are used decided that the new roof will be the permanent structure. The main advantage of such a roof is easier and cheaper its exploitation. The main drawback of the new structure is the necessity of taking the snow loading into calculations. According to national standards the value of the snow loading depends mainly from the slop of the roof surface. Additionally, the kind of coating plays here the important role. The PFTE coating almost guarantees, that for properly designed roof, snow will push down itself from the roof surface. For PCV coating such assumption is not always true, especially when the roof is relatively flat. stress [N/m] ,00 0,02 0,04 0,06 0,08 0,10 strain [.] uniaxial test biaxial test 1:1 biaxial test 1:2 biaxial test 2:1 Fig 11. Comparison of uniaxial and biaxial test for 9032 fabric - weft 6. Climatic loadings Textile hanging roofs are light structures, very sensitive on wind and, if they are permanent, on snow loading. The Forest Opera roof is relatively flat, therefore wind suction forces should be carefully examined. Due to complicated shape of the roof it is very difficult to determinate forces distribution on the roof surface. Usually the test in a aerodynamic tunnel are necessary. Unfortunately, if the structure is opened like the designed structure the tests are difficult and expensive. The method proposed by authors in [8] gives good results as the value of distributed loading depends from actual slope of normal of each finite element (Fig. 12). The actual value of the loading can be calculated from the relation pk = qk Ce C β (5) where : q k is the basic value of the wind loading taken from the national standard for structure location, C e exposition factor, also taken from the national standard, C reduction factor taken from Fig. 12, β coefficient taking into account the blast of wind. Calculated in such a way wind loading has the following character. The snow loading plays important role in permanent structures. The current roof he Forest Opera must be every year in autumn disassembled and again assembled in spring next year. This operations and storage of the roof in the folded form cause many damages of the textile fabric. Therefore the investor C reduction coefficient inclination of element normal vector [ ] Fig 12. Relation of the wind loading and actual inclination of a finite element normal C Fig 13. Relation of the snow loading and actual inclination of a finite element normal In such situation the value of snow loading can be calculated from expression S = Q C (6) k k where Q k is the characteristic value of snow loading according to the national standard, C coefficient dependent from inclination of the element normal (see Fig. 13). 7. Results of calculations of the Forest Opera roof In the analysis of the calculation results for textile roofs not only maximum stresses and displacements should be considered. This type of structure can be easy destroyed by wind or rain falls when the surface of the roof is not properly tight. Also too big vertical displacements (in both directions!) can be very dangerous. Too big displacements in down direction can produce sacks which can be filled with rain water, too big up displacements cause change of curvature of the roof and leads to flatter.

6 Obtained initial configuration (under dead loading and tensile forces in the fabric and ropes) of the Forest Opera roof is presented in Fig. 14. In this configuration almost unique state of stresses (8 kn/m in warp and 6 kn/m in weft direction has been obtained) Fig 15. Vertical displacements of the roof wind suction case Fig 14. Vertical displacements of the roof in initial configuration Fig 16. Stresses in warp (left) and weft (right) - wind suction case, 1e+06= 1kN/m

7 2. Tabarrok, B., and Qin, Z. A finite element procedure for form finding of tension structures. Transactions of the Canadian Society for Mechanical Engineering Vol. 16, No. 3/4 (1992), pp Fig 17. Vertical displacements of the roof snow +25% of wind pressure case 8. Conclusions Calculation of textile roof is still challenging subject for engineers and designers. They have very interesting architectural form, are easy to construct, light and can cover large areas without internal supports. On the other hand they require non-linear numeric calculations and very often special programs or procedures to model properly their shape and properties. The authors presented main theoretical problems on this type of structures on the basis of their experience in designing of the Forest Opera in Sopot roof. We hope that the paper will promote textile hanging roofs also in new European Community Countries. Acknowledgment Most calculations presented in this paper have been made on computers of TASK Center in Gdańsk (Poland) References 1. Haber, R. B., and Abel, J. F. Initial equilibrium solution methods for cable reinforced membranes: Part I formulations. Computer Methods in Applied Mechanics and Engineering Vol. 30 (1982), pp Fig 18. Stresses in warp (left) and weft (right) - snow +25% of wind pressure case, 1e+06= 1kN/m 3. T. Nouri-Baranger: Form finding, analysis and computer aided design of tension structures. Computational structures technology. 2002, pp: Gründig L. and Moncrieff E.: Form finding of textile structures. Studiedag Textielstrukturen, 25/05/1993, Vrije Universiteit Brussel. 5. A. Ambroziak: Geometrically non-linear analysis of membranes applied for hanging roof structures with respect to different types of constitutive laws. Gdansk University of Technology, PhD thesis Kłosowski P., Zagubień A., Woznica K.: Investigation on Rheological Proprties of Technical Fabric Panama. Archive of Applied Mechanics Vol. 73, 2004, pp Ambroziak A., Kłosowski P.: Constitutive models for technical woven fabric state of art and future trends. Workshop on Advanced Mechanics of Urban Structures, September , Gdańsk, pp Ambroziak A. Kłosowski P. Nowicki M.: On problem of membrane structures designing. Inżynieria i Budownictwo 1/2005, (in Polish). ).