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

in Science andEngineering,CMMSE 2012

July, 2-5, 2012.

A not so common boundary problem related to the membrane

equilibrium equations

Giuseppe Viglialoro

1

and Juan Murcia

2

1

Department of Mathematics, University of Cadiz

2

Instituto deCienciasde la Construcción Eduardo Torroja, CSIC

emails: giuseppe.viglialoro@uca.es,murcia@ietcc.csic.es

Abstract

This paperpresentsamathematicalproblem relatedto theequilibrium analysisof

amembrane with rigid and cable boundaryfor the socalled prestressing phase. The

membrane and its boundary are respectively identied with a regular surface and a

set of regular curves. The equilibrium is directly expressed by means of a boundary

(dierential)problem,intermsoftheshapeofthemembraneanditsstresstensor. The

membrane-cableinterfaceequilibrium leadsto take into account asingular condition;

asaconsequence,anunusualellipticproblemwillappear.

Key words: membrane, footbridge, equilibrium, rigid boundary, cable boundary,

el-lipticproblem.

1 Introduction

This work studies the bi-dimensional and continuous equilibrium of a membrane for the

prestressing phase,whenthe prestressed membraneisready to theserviceabilityphase.

Moreexactly,anewmembrane technology for footbridges hasbeen developed inSpain

and a membrane footbridge prototype has been built near Barcelona (see [2] for the main

technological aspectsand theFigure1(a)for areal picture). The prestressingisintroduced

and obtained by means of the membrane's boundaries, that are one-dimensional elements

dened byspatial curves,that mayor maynot havecurvature(straight elements).

Inthepaperthemembranetensionsareidentiedwithapositivesecondordertensorand

its shape with a surface with negative gaussian curvature. So, the equilibrium is directly

variables associated to the membrane and to its stress tensor. Hence, we will discuss the

following problem: once the stress tensor is given nd the shape of the membrane that

achieves the equilibrium.

Along with the equilibrium equations theboundary conditions must be dened; these

conditionsdependontheusedboundaryelements;hereinbothrigidandcableboundarywill

beconsidered. Contrarytowhathappenswiththerigid elements,theshapesofacableand

thecorrespondingmembranearelinked byacompatibilityrelation(see[3]and [4 ]),making

that the form of the membrane (i.e. the unknown) has to verify a not ad hoc boundary

condition for asecond orderproblem.

2 Equilibrium equations of a membrane

Let us identify the membrane witha surface

S

witha negative gaussiancurvature (Figure 1(c));

S

is parameterizedby

S → ϕ(x, y) := (x, y, z(x, y)) ∀ (x, y) ∈ D,

where

z

= z(x, y)

isa regularfunction dened ina domain

D ⊂ R

2

.

If

σ

:= N

αβ

= N

αβ

(x, y)

(

α, β

= 1, 2

)istheprojectedstresstensorofthemembrane(in fact,forceperunitlength),themembraneequilibriumequationsintermsof

N

αβ

,neglecting its weight andconsidering no external load,are expressedby(see [5]for thedetails):











N

xx,x

+ N

xy,y

= 0

in

D,

N

xy,x

+ N

yy,y

= 0

in

D,

N

xx

z

,xx

+ 2N

xy

z

,xy

+ N

yy

z

,yy

= 0

in

D.

(1)

The system (1 ) shows that if a positive tensor

σ

is xed the function

z

has to solve an ellipticequation;then,the problemwewillconsiderconsistsofxingthestresstensorof the

membrane and nding its shape.

3 Boundary equilibrium equations

Once the membrane equilibrium equations are given (system (1)), the problem has to be

completed by dening the corresponding boundary conditions on

Γ = ∂D

. We will put

Γ

r

the subset of

Γ

corresponding to the rigid boundary and

Γ

c

the one corresponding to the

cable boundary;ofcourse

Γ = Γ

r

_∪

Γ

c

(see Figure1(b) ).

3.1 The rigid boundary: equilibriumequations

Let us consider the equilibrium on

Γ

r

; as you can check in [5],

Γ

r

can assume any shape

(a)PictureofthefootbridgeinSpain. (b)Typicalprojecteddomain.

d S

O

x

y

z

Membrane stresses

Projected membrane stresses

d D

(c)Membraneelementandstresstensors.

Figure1: Characterizationof a membrane footbridge.

andtheshapeofthe membrane(theunknown

z

). Inthis case, thecorresponding boundary conditionisthe usualDirichelcondition

z

= g

on

Γ

r

,

being

g

thevalueof

z

onthesame

Γ

r

,i.e. the

3−

Dshapeoftherigidboundaryofmembrane.

3.2 The cable boundary: equilibrium equations

Letus considerthe equilibrium on

Γ

c

;thecable tensionsare tangent to thecableand they

theotherhand,the stresstensor ofthemembranebelongto thetangent plan of thesurface

S

;of course,inorder to reach the equilibriumthese twoplans haveto coincide.

Asaconsequence (youcanndtheproofin[5]),theboundariesequilibriumreturnsthe

following cable-membrane compatibilityequation:

z

,xx

+ 2z

,xy

y

0

+ z

,yy

y

02

= 0

on

Γ

c

.

(2)

4 Denition and properties of the mathematical problem

4.1 Mathematical problem

With reference to the equilibrium expressed by the system (1) and the above notation, let

N

αβ

be a positive and symmetric second order tensorsuch that

P

2

β=1

N

αβ,β

= 0 (α = 1, 2)

, in a bounded domain

D

of the plan

xOy

.

Find the surface

z

,dened in

D

, such that











z

,xx

N

xx

+ 2z

,xy

N

xy

+ z

,yy

N

yy

= div (σ · ∇z) = 0

in

D,

z

= g

on

Γ

r

,

z

,xx

+ 2z

,xy

y

0

+ z

,yy

y

02

= 0

on

Γ

c

.

(3)

Dueto the secondboundary condition,the previousis nota typicalelliptic problemwith

usual Dirichlet or Dirichlet-Neumann boundary conditions (see [1]). By means of Hopf's

Lemma, one can prove theuniquenessof thesolution of thesystem (3); onthe otherhand

wehave to underlinethatthe correspondingexistenceproblemis still anopenquestion.

Inthe nextsection, we will proposeanumerical methodto solve thesystem(3 ).

4.2 Solving the problem by a numerical method

In [5]is shownthattheproblem(3 ) isequivalent to thefollowing one,that ismore direct.

With the samenotations of the previous problem (system (3) ),nd

z

and

h

, suchthat



























div (σ · ∇z) = 0

in

D,

z

= g

on

Γ

r

,

z

= h

on

Γ

c

,

z

,y

=

h

00

y

00

on

Γ

c

.

(4)

Letus considerthe following system











−

_{div (σ · ∇z) = 0}

in

D,

z

= g

on

Γ

r

,

z

= h

on

Γ

c

,

(5)

and theequation

h

00

= z

,y

y

00

on

Γ

c

.

(6)

By means ofthe FiniteElement Method (see [6]), letus x amesh for

D

; if

n

t

isthe total number of nodes of

D

,

n

r

those of

Γ

r

and

n

c

those of

Γ

c

, putting

z '

P

n

t

j=1

z

j

N

j

and

h '

P

n

c

j=1

h

j

N

j

and replacingthem into the system(5) , we obtain

z

= Hh + Gg,

(7)

being

z

= (z

1

, . . . , z

t

)

,

g

= (g

1

, . . . , g

r

)

and

h

= (h

1

, . . . , h

c

)

thenodalvalues vectorsof

z

in

D

,

g

on

Γ

r

and

h

on

Γ

c

,and

H ∈ M

n

t

×

n

c

(R)

and

G ∈ M

n

t

×

n

r

(R).

Withreference to the equation (6), let us dene the

n

c

−

dimensional residual vector

R

,whose components are

(R)

i

= R

i

(h) :=

Z

Γ

c

(z

,y

y

00

−

h

00

)N

i

d

Γ

c

,

i

= 1, . . . , n

c

,

and letus makeit zero:

R

i

(h) =

Z

Γ

c

(z

,y

y

00

−

h

00

)N

i

d

Γ

c

= 0, i = 1, . . . , n

c

.

(8) Putting

z '

P

n

t

j=1

z

j

N

j

and

h '

P

n

c

j=1

h

j

N

j

, replacing them into the relation (8) and integratingbyparts,weobtain

n

t

X

j=1

z

j

Z

Γ

c

y

00

N

j,y

N

i

d

Γ

c

+

n

c

X

j=1

h

j

Z

Γ

c

N

0

j

N

0

i

d

Γ

c

= 0,

i

= 1, . . . , n

c

.

Dening

M

ij

:=

Z

Γ

c

y

00

N

j,y

N

i

d

Γ

c

and

W

ij

=

Z

Γ

c

N

0

j

N

0

i

d

Γ

c

,

we conclude

M z

+ W h = 0,

(9) with

M ∈ M

n

c

,n

t

(R)

and

W ∈ M

n

c

(R)

.

Comparing the systems(9 )and (7)we have thesolutions

h

and

z

given by

h

= −(M H + W )

−1

M Gg

and

z

= Hh + Gg,

thatarethenodalvector corresponding to the shape ofthesurface

z

on

Γ

c

(i.e.,theshape

of thecable) and,thenodalvector corresponding to theshape of thesurface

z

all over the domain

D

(i.e.,the shape ofthe totalmembrane).

5 Conclusion and future work

In this paper we have analyzed the membrane equilibrium equations, for the prestressing

phase. Themathematicalformulationhasbeenwrittendownandthecorrespondingproblem

leadsonetoconsideranellipticproblemwithasingularboundaryconditionfortheunknown

(the shape of the membrane). More exactly, it is possible to prove the uniqueness of the

solution but it is not known if the problem has or not always a solution; moreover the

numerical methodherein proposedis generallynot stable(see[4]).

Finally, the authors are working on another resolution strategy for the same problem

basedonaxpoint procedure, beingthenalgoaltheproofthatthecorresponding

bound-ary equilibrium problem is well posed, in terms of existence, uniqueness and continuous

dependence ondata.

Acknowledgements

This paper hasbeen partially supported by the Project MTM2010-16401 oftheMinisterio

de Ciencia e Innovación/FEDER and the Consejería de Economía, Innovación y Ciencia

of theJunta deAndalucía.

References

[1] H.Brezis,Analysefunctionnelle.Théorieetapplications,MassonEditeur,Paris,1983.

[2] J. Murcia, Structural membrane technology for footbridges (in Spanish), Informesde

laConstrucción, 59-507 (2007) 2131.

[3] G.Viglialoro,J. Murcia,F. Martínez, Equilibrium problems inmembrane

struc-tures with rigid boundaries (in Spanish), Informes de la Construcción, 61-516 (2009)

5766.

[4] G.Viglialoro, J. Murcia, Equilibrium problems in membranestructures withrigid

andcable boundaries(inSpanish), Informesdela Construcción, 63-524 (2011)4957.

[5] G. Viglialoro, Análisis matemático del equilibrio en estructuras de membrana con

bordes rígidosycables. Pasarelas: formaypretensado(inSpanish),PhD thesis, UPC.,

(2006).

[6] O.C.Zienkiewicz, R.L. Taylor, TheFiniteElementMethod,Butterworth