After the quasi two-dimensional simulation of the hanging roof, a fully three- dimensional case will be discussed: a four-point tent structure under wind loading.
The aim of this simulation is the qualitative assessment of the occurring effects and their magnitude.
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Fig. 12.Implicit coupling procedures using Aitken’s method for stabilization: Iterations and results.
The inflow profile is logarithmic according to the requirements of German build- ing code DIN 1055-2. The four-point tent structure resembles a saddle surface of a membrane with a uniform prestress of 2.5 kN/m stabilized by four cables at the edges prestressed with 50 kN. The two masts are steel tubes with a diameter of 88.9 mm and a thickness of 6 mm. The bracing consists of two sets of two guy cables with a diameter of 13.8 mm and prestressed with a force of 41 kN. The membrane ma- terial is a polyester fabric with PVC coating of type I and a thickness of 1 mm, the edge cables are spiral strands with a diameter of 16 mm. The dimensions are given in Figure 14.
The setup of the analysis follows the flow chart given in Figure 1, including the form finding procedure described in Section 2.1 to acquire the proper initial geo- metry. In the fluid domain only the membrane itself is modeled, the influence of cables and masts on the flow field are neglected. For the form finding and the struc- tural analysis, masts and cables are fully taken into account. The interface is treated as a two-sided infinite thin surface in both the structure and flow analysis.
The simulation setup is totally 3D (Figure 15) using a tetrahedral mesh for the CFX simulation with refinement by prism layers nearby the membrane and the bot- tom. A SST-turbulence model with appropriate wall functions is applied. In the struc- 156
Fig. 13.Deformation of membrane roof and velocity vectors.
Fig. 14.Geometry and dimension of the four-point tent structure.
tural analysis 3-node membrane elements are used for a nonlinear analysis. The fluid is air at 25◦C.
As a first approach, a steady-state analysis of the tent was performed. The max- imum wind speed is 30 m/s. The most susceptible configuration with respect to the deformations as well as the pressure load on the membrane is reached when the higher tip of the structure heads towards the wind. As Figure 16 shows, an upwards deformation occurs in the front part while the rear part deforms downwards.
The dynamic behavior of the coupled system was analyzed in a transient FSI simulation. With regard to the low fluid density and the large deformations of the structure, the computation was carried out in a fully implicit manner. The time in- tegration used on the fluid side was a second order backward euler scheme, on the structural side the generalized-αmethod was applied.
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Fig. 15.Visualization of flow and pressure at the membrane.
Fig. 16.Deformation for wind flow in they-direction.
The fluid simulation was started from the result of a steady-state solution of flow around the undeformed membrane at 20 m/s. Therefore, the undeformed membrane first needs to find its equilibrium for this inflow velocity. During this initial stage, the inflow velocity was kept constant until the structure stopped oscillating att =4.0 s.
In the following the maximum wind speed was varied between 10 m/s and 30 m/s in a gust-like behavior. Figure 17 shows the maximum inflow velocity and maximum displacement of the structure: the structural deformation follows the variation of the wind speed without any delay due to its prestress and its little mass. This supports 158
Fig. 17.Max. infow velocity and max. displacement over simulation time.
Fig. 18.Response of the structure at different time steps.
earlier results which showed very little influence of the mass of the membrane on the inertia of the system. Figure 18 presents the state of deformation at different time steps.
These results provide a first assessment to the structure’s behavior under wind loading, as well in magnitude of the deformation as in the occurring frequencies.
Thereby, a possibility to quantify the effects, which are expected due to the geometry and the problem setting, is given.
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4 Conclusion
In this paper a modular software environment using methods of fluid-structure inter- action for the simulation of wind effects on light-weight structures like tents, shells, or membrane roofs was presented.
To take into account all demands arising from the physical problem, a partitioned approach using highly developed codes for the single field computations, namely CARAT and CFX-5, is applied. In the case of membrane structures the coupling surface consists of an infinitely thin structure which is represented by two interface meshes on both wetted sides, respectively.
It is demonstrated that the special load carrying behavior of membranes requires additional considerations concerning prestressing and form finding. Strongly and weakly coupled iterations schemes are possible. It is pointed out that in the case of flexible, nearly massless structures interacting with incompressible fluids iterative coupling is necessary. For the stabilization of strongly coupled iteration schemes an under-relaxation technique is applied using adaptive relaxation factors determined by the Aitken method.
The application of the proposed software environment and the frame algorithm is shown in two examples. The first example concentrates on a 2D transient numerical experiment: a hanging roof structure subjected to wind flow. In the second example the stationary and transient analysis of a wind loaded tent structure is performed after the geometrical definition of the coupling interface was done by means of the updated reference procedure.
Considering the requirements towards the occurring turbulence and the boundary conditions, it is obvious that further improvements are necessary in order to precisely approach the effects of wind loads. However, they provide an idea about the occur- ring effects and their magnitude, showing the possibilities that the application of methods of FSI opens in wind engineering.
References
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2. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-αmethod.Journal of Applied Mechanics 60:371–375.
3. Farhat C (2004) CFD-based nonlinear computational aeroelasticity. In: Stein E, de Borst R and Hughes TJR (eds),Encyclopedia of Computational Mechanics. John Wiley & Sons, Chichester, pp. 459–480.
4. Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical sys- tems.Computer Methods in Applied Mechanics and Engineering190:3247–3270.
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6. Menter FR (1994) Two-equation eddy-viscosity turbulence models for engineering ap- plications.AIAA Journal32(8):1598–1605.
7. Kuntz M, Menter FR (2004) Numerical flow simulation with moving grids. In:TAB Con- ference, Bremen.
8. Mok DP, Wall WA (2001) Partitioned analysis schemes for the transient interaction of in- compressible flows and nonlinear flexible structures. In: Wall WA, Bletzinger KU and Schweizerhof K (eds),Trends in Computational Structural Mechanics, CIMNE, Bar- celona, pp. 689–698.
9. Wüchner R, Bletzinger KU (2005) Stress-adapted numerical form finding of pre-stressed surfaces by the updated reference strategy.International Journal for Numerical Methods in Engineering64:143–166.
Inflatable Tubes. New Analysis Methods and Recent Constructions
Eugenio Oủate1, Fernando G. Flores1,2and Javier Marcipar3
1International Centre for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC), Edificio C1, Gran Capitán s/n, 08034 Barcelona, Spain;
E-mail: onate@cimne.upc.es
2National University of Córdoba, Casilla de Correo 916, 5000 Córdoba, Argentina;
E-mail: fflores@efn.unc.edu.ar
3BuildAir Ingeniería y Arquitectura SA, Muntaner, 335, 08021 Barcelona, Spain;
E-mail: marcipar@buildair.es
Abstract. This paper shows applications of a recently developed thin shell element adequate for the analysis of membrane and inflatable structures. The element is a three node triangle with only translational degrees of freedom that uses the configuration of the three adjacent elements to evaluate the strains in terms of the nodal displacements only. This allows us to compute (constant) bending strains and (linear) membrane strains using a total Lagrangian formulation. Several examples, including inflation and deflation of membranes and some prac- tical applications to the analysis, design and construction of membrane structures formed by low pressure inflatable tubes are presented.
Key words: shell elements, rotation free shell triangle, membrane structures, inflatable struc- tures, low pressure inflatable tubes.
1 Introduction
Inflatable structures have unique features. Because of their foldability and air- or helium pneumatic stabilization they cannot be compared to any classical structural concepts.
Inflatable structures have become increasingly popular in recent years for a wide range of applications in architecture, civil engineering, aeronautic and airspace situ- ations.
The use of inflatable structures can be found in temporary and/or foldable struc- tures to cover large spaces or to support other elements, in permanent roofs or shelters
©2008Springer. Printed in the Netherlands.
E. Oủate and B. Krửplin (eds.), Textile Composites and Inflatable Structures II,163–196.
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with a high degree of transparency, in mobile buildings as temporary housing in civil logistic missions (e.g. environmental disasters and rescue situations), in the construc- tion of tunnels and dams, in antennas for both ground and aerospace applications, as well as in extremely light airship structures among other uses [1–11].
Some efforts have been made in the past years to develop inflated structures formed by assembly of high pressure tubes. The obvious disadvantages of these structures are the design of the joints and their big vulnerability to air losses. In general, high pressure inflated structures are difficult to maintain and repair and have high costs.
Inflatable structures formed by an assembly of self-supported low pressure tubu- lar membrane elements are ideal to cover large space areas. They also adapt easily to any design shape and have minimal maintenance requirements, other than keeping a constant low internal pressure accounting for the air losses through the material pores and the seams.
The simulation of the inflation of membrane structures is normally performed with membrane finite elements, i.e. no bending stiffness included. The formulation of such elements is simple as they only requireC0continuity [12], in contrast with elements based on thin shell theory whereC1continuity implies important obstacles [13] in the development of conforming elements. Triangular elements are naturally preferred as they can easily adapt to arbitrary geometries and due to the robustness of the associated mesh generators.
Membrane structures components have some, although small, bending stiffness that in most cases is disregarded. However in many applications it is convenient to include bending energy in the model due to the important regularization effect it in- troduces. Shell elements are typically more complex and expensive due the increase in degrees of freedom (rotations) and integration points (through the thickness). In the last few years shell elements without rotation degrees of freedom have been de- veloped [14–22], which make shell elements more efficient for both implicit and explicit integration schemes.
When only the final configuration of the membrane is of interest implicit schemes are normally used, including special algorithms due to the lack of stiffness of the membrane when no tensile stresses are yet present. When the inflation/deflation pro- cess is of interest, the explicit integration of the momentum equations is largely pre- ferred. Modeling of complex deformation with constant strain shell triangles, such as those occuring in the inflation-deflation process of inflatable membranes account- ing for frictional contact conditions typically require fine discretizations. These type of simulations can be time consuming due to the time increment limitations. In this paper a rotation-free triangular shell element with similar convergence properties to the linear strain triangle, but without its drawbacks, is used.
The outline of this article is as follows. The next two sections summarize the rotation-free shell triangle used. Section 4 summarizes the procedure for aer- oelastic analysis. Section 5 presents examples of application to the analysis of inflat- 164
able membranes. The paper concludes with practical examples inflatable structures formed by low pressure inflatable tubes designed and analyzed using the technology described in the paper. Finally, Section 6 presents some conclusions.
2 Formulation of the Rotation Free Shell Triangle