Starting from a rigorous mechanical formulation, a numerical procedure is developed for the form finding of minimal surfaces and pre-stressed membrane structures. The method is based on an iteratively adapted regularisation of the stiffness matrix which gives the method its name: the updated reference strategy. The method can be applied to any finite element discretization of cable and membrane structures subjected to pre-stress as well as lateral pressure or other external loading. It is very robust and reliable as is shown by many illustrative examples. Further analysis states that the well known force density method is the special case of applying the updated reference strategy to cable nets.
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References
1.
OttoF.RaschB., Finding Form, Deutscher, Werkbund Bayern, Edition A. Menges, 1995.
2.
SchockH.-J., Segel, Folien und Membrane, Birkhäuser, Basel, 1997.
3.
BergerH., Light structures – structures of light, Birkhäuser, Basel, 1996.
4.
DierkesU.HildebrandtS.KüsterA.WohlrabO., Minimal surfaces I and II. Grundlehren der mathematischen Wissenschaften 295–296, Springer, Berlin, Heidelberg, 1992.
5.
SuzukiT.HangaiY., Shape analysis of differently stressed surfaces by the finite element method, in: SrivastavaN. K.SherbourneA. N.RoodaJ. eds., Innovative Large Span Structures (vol. 2): Proc. IASS-CSCS International Congress, Toronto, Canada, 1991, 400–411.
6.
OttoF., Zugbeanspruchte Konstruktionen, Vols. I and II, Ullstein Verlag, Franfurt, Berlin, 1962, 1966.
7.
BarnesM., Form and stress engineering of tension structures, Structural Engineering Review, 6, 1994, 175–202.
8.
LewisW. J.LewisT. S., Application of Formian and dynamic relaxation to the form finding of minimal surfaces, Journal of the IASS, 37(3), 1996, 165–186.
9.
LinkwitzK.SchekH.-J., Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen, Ingenieur-Archiv, 40, 1971, 145–158.
10.
SchekH.-J., The force density method for form finding and computations of general networks, Computer Methods in Applied Mechanics and Engineering, 3, 1974, 115–134.
11.
LinkwitzK., Least squares methods in non-linear form finding and analysis of pre-stressed and hanging nets, in: Non-Linear Analysis and Design for Shell and Spatial Structures: Proc. SEIKEN-IASS Symposium, Tokio, 1993, 601–610.
12.
SingerP., Die Berechnung von Minimalflächen, Seifenblasen, Membrane und Pneus aus geodätischer Sicht, PhD Thesis, University of Stuttgart, 1995.
13.
MaurinB.MotroR., The surface stress density method as a form-finding tool for tensile membranes, Engineering Structures, 20(8), 1998, 712–719.
14.
HaugE.PowellG. H., Finite element analysis of nonlinear membrane structures, Report UCSESM 72–7, University of California at Berkeley, 1972.
15.
HaugE.PowellG. H., Finite element analysis of nonlinear membrane structures, in: Tension and space structures (vol.2): Proc. 1971 IASS Pacific Symposium, Tokio and Kyoto, 1972, 165–175.
16.
HaberR.AbelJ., Initial equilibrium solution methods for cable reinforced membranes, Computer Methods in Applied Mechanics and Engineering, 30, 1982, 263–284.
17.
BletzingerK.-U., Form finding of membrane structures and minimal surfaces, in: RozvanyG.OhlhoffN. eds., Proc. 1st World Congress of Structural and Multidisciplinary Optimization, Goslar, Germany, 1995, 563–568.
18.
BletzingerK.-U., Form finding of tensile structures by the updated reference strategy, in: ChiltonJ.LewisW. eds., Structural Morphology – Towards the New Millennium, Proc. IASS International Colloquium, August 15–17, 1997, University of Nottingham, UK, 1997.
19.
BletzingerK.-U., Shape optimization by homotopy methods with special application to membrane structures, in: Proc. 6th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, Washington, USA, 1996, 122–129.
20.
AllgowerE. L.GeorgeK., Numerical Continuation Methods, Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg, New York, 1990.
LeonardJ. W., Tension Structures, McGraw-Hill, New York, 1988.
23.
RammE., The Riks/Wempner approach – An extension of the displacement control method in non-linear analysis, in: HintonE.OwenD. R. J.TaylorC. eds. recent advances in non-linear computational mechanis, Pineridge Press, UK, 1982, 63–86.
24.
ReitingerR.BletzingerK.-U.RammE., Shape optimization of buckling sensitive structures, Computing Systems in Engineering, 5, 65–75, 1993.
25.
WagnerW., Zur Behandlung von Stabilitätsproblemen der Elastostatik mit der Methode der finiten Elemente, Habilitationsschrift, Bericht-Nr. F 91/1, Institut für Baumechanik und Numerische Mechanik, Universität Hannover, 1991.
26.
WriggersP.SimoJ. C., A general procedure for the direct computation of turning and bifurcation points, International Journal Numerical Methods in Engineering, 30, 1990, 155–176.
27.
SchweizerhofK., Nichtlineare Berechnung von Tragwerken unter verformungsabhängiger Belastung mit finiten Elementen, PhD. Thesis, University of Stuttgart, 1982.
28.
WahlC., Numerische Formfindung von vorgespannten Seil- und Flächentragwerken, Diploma Thesis, Institut für Baustatik, University of Stuttgart, 1995.
29.
KrapfA., Formfindung und Statik vorgespannter Membrantragwerke, Diploma Thesis, Institut für Baustatik, University of Stuttgart, 1996.
30.
SeebacherF., Formfindung, Zuschnitt und Statik vorgespannter, textiler Membrankonstruktionen, Diploma Thesis, Institut für Baustatik, University of Karlsruhe, 1998.
