Murcia J.

VIGLIALORO, GIUSEPPE

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Proceedings of the 12th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 2012July, 2-5, 2012.A not so common boundary problem related to the membraneequilibrium equationsGiuseppe Viglialoro1and Juan Murcia21Department of Mathematics, University of Cadiz2Instituto de Ciencias de la Construcción Eduardo Torroja, CSICemails: giuseppe.viglialoro@uca.es, murcia@ietcc.csic.esAbstractThis paper presents a mathematical problem related to the equilibrium analysis ofa membrane with rigid and cable boundary for the so called prestressing phase. Themembrane and its boundary are respectively identied with a regular surface and aset of regular curves. The equilibrium is directly expressed by means of a boundary(dierential) problem, in terms of the shape of the membrane and its stress tensor. Themembrane-cable interface equilibrium leads to take into account a singular condition;as a consequence, an unusual elliptic problem will appear.Key words: membrane, footbridge, equilibrium, rigid boundary, cable boundary, el-liptic problem.1 IntroductionThis work studies the bi-dimensional and continuous equilibrium of a membrane for theprestressing phase, when the prestressed membrane is ready to the serviceability phase.More exactly, a new membrane technology for footbridges has been developed in Spainand a membrane footbridge prototype has been built near Barcelona (see [2] for the maintechnological aspects and the Figure 1(a) for a real picture). The prestressing is introducedand obtained by means of the membrane's boundaries, that are one-dimensional elementsdened by spatial curves, that may or may not have curvature (straight elements).In the paper the membrane tensions are identied with a positive second order tensor andits shape with a surface with negative gaussian curvature. So, the equilibrium is directlydened by partial dierential equations, one of them involving the product between the©CMMSE ISBN:978-84-615-5392-1A not so common boundary problem related to the membrane equilibrium equationsvariables associated to the membrane and to its stress tensor. Hence, we will discuss thefollowing problem: once the stress tensor is given nd the shape of the membrane thatachieves the equilibrium.Along with the equilibrium equations the boundary conditions must be dened; theseconditions depend on the used boundary elements; herein both rigid and cable boundary willbe considered. Contrary to what happens with the rigid elements, the shapes of a cable andthe corresponding membrane are linked by a compatibility relation (see [3] and [4]), makingthat the form of the membrane (i.e. the unknown) has to verify a not ad hoc boundarycondition for a second order problem.2 Equilibrium equations of a membraneLet us identify the membrane with a surface S with a negative gaussian curvature (Figure1(c)); S is parameterized byS → ϕ(x, y) := (x, y, z(x, y)) ∀ (x, y) ∈ D,where z = z(x, y) is a regular function dened in a domain D ⊂ R2.If σ := Nαβ = Nαβ(x, y) (α, β = 1, 2) is the projected stress tensor of the membrane (infact, force per unit length), the membrane equilibrium equations in terms of Nαβ , neglectingits weight and considering no external load, are expressed by (see [5] for the details):Nxx,x +Nxy,y = 0 in D,Nxy,x +Nyy,y = 0 in D,Nxxz,xx + 2Nxyz,xy +Nyyz,yy = 0 in D.(1)The system (1) shows that if a positive tensor σ is xed the function z has to solve anelliptic equation; then, the problem we will consider consists of xing the stress tensor of themembrane and nding its shape.3 Boundary equilibrium equationsOnce the membrane equilibrium equations are given (system (1)), the problem has to becompleted by dening the corresponding boundary conditions on Γ = ∂D. We will put Γrthe subset of Γ corresponding to the rigid boundary and Γc the one corresponding to thecable boundary; of course Γ = Γr ∪ Γc (see Figure 1(b)).3.1 The rigid boundary: equilibrium equationsLet us consider the equilibrium on Γr; as you can check in [5], Γr can assume any shapeand moreover it is not necessary to impose any kind of restriction between its own shape©CMMSE ISBN:978-84-615-5392-1G. Viglialoro, J. Murcia(a) Picture of the footbridge in Spain. (b) Typical projected domain.d S O x y zMembrane stressesProjected membrane stressesd D (c) Membrane element and stress tensors.Figure 1: Characterization of a membrane footbridge.and the shape of the membrane (the unknown z). In this case, the corresponding boundarycondition is the usual Dirichel conditionz = g on Γr,being g the value of z on the same Γr, i.e. the 3−D shape of the rigid boundary of membrane.3.2 The cable boundary: equilibrium equationsLet us consider the equilibrium on Γc; the cable tensions are tangent to the cable and theybelong to the osculator plan of the curve C, that represents the 3−D shape of the cable. On©CMMSE ISBN:978-84-615-5392-1A not so common boundary problem related to the membrane equilibrium equationsthe other hand, the stress tensor of the membrane belong to the tangent plan of the surfaceS; of course, in order to reach the equilibrium these two plans have to coincide.As a consequence (you can nd the proof in [5]), the boundaries equilibrium returns thefollowing cable-membrane compatibility equation:z,xx + 2z,xyy′ + z,yyy′2 = 0 on Γc. (2)4 Denition and properties of the mathematical problem4.1 Mathematical problemWith reference to the equilibrium expressed by the system (1) and the above notation, letNαβ be a positive and symmetric second order tensor such that∑2β=1Nαβ,β = 0 (α = 1, 2),in a bounded domain D of the plan xOy.Find the surface z, dened in D, such thatz,xxNxx + 2z,xyNxy + z,yyNyy = div (σ · ∇z) = 0 in D,z = g on Γr,z,xx + 2z,xyy′ + z,yyy′2 = 0 on Γc.(3)Due to the second boundary condition, the previous is not a typical elliptic problem withusual Dirichlet or Dirichlet-Neumann boundary conditions (see [1]). By means of Hopf'sLemma, one can prove the uniqueness of the solution of the system (3); on the other handwe have to underline that the corresponding existence problem is still an open question.In the next section, we will propose a numerical method to solve the system (3).4.2 Solving the problem by a numerical methodIn [5] is shown that the problem (3) is equivalent to the following one, that is more direct.With the same notations of the previous problem (system (3)), nd z and h, such thatdiv (σ · ∇z) = 0 in D,z = g on Γr,z = h on Γc,z,y =h′′y′′on Γc.(4)Let us consider the following system−div (σ · ∇z) = 0 in D,z = g on Γr,z = h on Γc,(5)©CMMSE ISBN:978-84-615-5392-1G. Viglialoro, J. Murciaand the equationh′′ = z,yy′′on Γc. (6)By means of the Finite Element Method (see [6]), let us x a mesh for D; if nt is the totalnumber of nodes of D, nr those of Γrand nc those of Γc, putting z '∑ntj=1 zjNj andh '∑ncj=1 hjNj and replacing them into the system (5), we obtainz =Hh+Gg, (7)being z = (z1, . . . , zt), g = (g1, . . . , gr) and h = (h1, . . . , hc) the nodal values vectors of z inD, g on Γr and h on Γc, and H ∈ Mnt×nc(R) and G ∈ Mnt×nr(R).With reference to the equation (6), let us dene the nc−dimensional residual vectorR, whose components are(R)i = Ri(h) :=∫Γc(z,yy′′− h′′)Ni dΓc, i = 1, . . . , nc,and let us make it zero:Ri(h) =∫Γc(z,yy′′− h′′)Ni dΓc = 0, i = 1, . . . , nc. (8)Putting z '∑ntj=1 zjNj and h '∑ncj=1 hjNj , replacing them into the relation (8) andintegrating by parts, we obtainnt∑j=1zj∫Γcy′′Nj,yNi dΓc +nc∑j=1hj∫ΓcN ′jN′i dΓc = 0, i = 1, . . . , nc.DeningMij :=∫Γcy′′Nj,yNi dΓcand Wij =∫ΓcN ′jN′i dΓc,we concludeMz +Wh = 0, (9)with M ∈ Mnc,nt(R) and W ∈ Mnc(R).Comparing the systems (9) and (7) we have the solutions h and z given byh = −(MH +W )−1MGg and z =Hh+Gg,that are the nodal vector corresponding to the shape of the surface z on Γc (i.e., the shapeof the cable) and, the nodal vector corresponding to the shape of the surface z all over thedomain D (i.e., the shape of the total membrane).©CMMSE ISBN:978-84-615-5392-1A not so common boundary problem related to the membrane equilibrium equations5 Conclusion and future workIn this paper we have analyzed the membrane equilibrium equations, for the prestressingphase. The mathematical formulation has been written down and the corresponding problemleads one to consider an elliptic problem with a singular boundary condition for the unknown(the shape of the membrane). More exactly, it is possible to prove the uniqueness of thesolution but it is not known if the problem has or not always a solution; moreover thenumerical method herein proposed is generally not stable (see [4]).Finally, the authors are working on another resolution strategy for the same problembased on a x point procedure, being the nal goal the proof that the corresponding bound-ary equilibrium problem is well posed, in terms of existence, uniqueness and continuousdependence on data.AcknowledgementsThis paper has been partially supported by the Project MTM2010-16401 of the Ministeriode Ciencia e Innovación/FEDER and the Consejería de Economía, Innovación y Cienciaof the Junta de Andalucía.References[1] H. Brezis, Analyse functionnelle. Théorie et applications, Masson Editeur, Paris, 1983.[2] J. Murcia, Structural membrane technology for footbridges (in Spanish), Informes dela Construcción, 59-507 (2007) 2131.[3] G. Viglialoro, J. Murcia, F. Martínez, Equilibrium problems in membrane struc-tures with rigid boundaries (in Spanish), Informes de la Construcción, 61-516 (2009)5766.[4] G. Viglialoro, J. Murcia, Equilibrium problems in membrane structures with rigidand cable boundaries (in Spanish), Informes de la Construcción, 63-524 (2011) 4957.[5] G. Viglialoro, Análisis matemático del equilibrio en estructuras de membrana conbordes rígidos y cables. Pasarelas: forma y pretensado (in Spanish), PhD thesis, UPC.,(2006).[6] O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method, Butterworth Heine-mann (2000).©CMMSE ISBN:978-84-615-5392-1

A not so common boundary problem related to the membrane equilibrium equations

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