Ponding on floating membranes - WIT Press · Ponding on floating membranes Greece National Technical UniversityofAthens, ... (Fig. lb). The case of the while in the second case the

Ponding on floating membranes

GreeceNational Technical University of Athens,Department of Civil Engineering,M. S. Nerantzaki &L J. T. Katsikadelis

Abstract

advantages of the pure BEM.illustrate the method and demonstrate its efficiency. The method has all theequilibrium state of the membrane. Example problems are presented, whichdeflection surface. Iterative schemes are developed which converge to theand the liquid reaction are not a priori known as they depend on the producednonlinearity, the ponding problem is itself nonlinear, because the ponding loadsources. The problem is strongly nonlinear. In addition to the geometricalthe problem to the solution of three uncoupled Poisson’s equations with fictitiousmembrane are solved using the Analog Equation Method (AEM), which reducesnonlinear equations in terms of the displacements governing the response of theconsidered, which result from nonlinear kinematic relations. The three coupledspace created by the deflection of the membrane. Large deflections aredisplacements, is subjected to the weight of a liquid (e.g. rain water) filling theinitially flat membrane, which may be prestressed by edge in-plane tractions orMembranes subjected to ponding loads and floating on a liquid are analyzed. The

1 Introduction

reached when all forces acting on it, namely the weight of the liquid filling theand floats partially on it. The equilibrium configuration of the membrane isaccumulated liquid until the membrane touches the liquid (e.g. oil) in the tank(e.g. rain water) fills the space. This space increases under the graduallyalong its boundary. Under its own weight, the membrane is deflected and a liquidwhich may be prestressed by edge in-plane tractions or displacements, is fixedshape serving as caps of oil storage tanks is studied. The initially flat membrane,In this paper the ponding problem on floating elastic membranes of arbitrary

© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved.Web: www.witpress.com Email [email protected] from: Boundary Elements XXIV, CA Brebbia, A Tadeu and V Popov (Editors).ISBN 1-85312-914-3

investigated.maximum allowable level of the liquid in the tank to avoid overflow is alsosmaller than the space created by the deflection (Fig. lb). The case of thewhile in the second case the ponding is due to a given volume of liquid, which isIn the first case, the liquid fills all the space created by the deflection (Fig. la),the tank, are in equilibrium. The ponding on the membrane may be full or partial.membrane, the reaction of the boundary support and the reaction of the liquid in

known, as they depend on the deflection of the membrane and vice versa.on the membrane and the reaction of the liquid in the tank are not a prioriinteraction, namely from the fact that the load distributions of the liquid pondingresponse of the membrane. The other nonlinearity results from the fluid-structurecoupled nonlinear partial differential equations governing the deformationdue to nonlinear strain-displacement relations and is reflected in the three

The problem is strongly nonlinear. Two types of nonlinearity arise. One is

t z t z

(a) Full ponding (b) Partial pondingFigure 1: Floating membranes under ponding loads.

solution technique.numerical examples illustrate the effectiveness and accuracy of the presentedof that developed for ponding on membranes [4] without liquid support. The[3], while the iterative scheme to determine the equilibrium state is modificationthe equilibrium configuration. The membrane problem is solved using the AEManalysis of membranes under a given load and the iteration procedure to reachthe problem requires the solution of two separate problems, that is the nonlinearaxisymmetric and, thus, one-dimensional [1,2]. The presented here approach toprestressed membranes, where the problem is highly simplified as it becomessolutions (FEM) to these problems which however, are restricted to circular non-not available in the literature. There are some approximate and numerical

Solutions for the general ponding problems of membranes as stated above are

2 The nonlinear membraneproblem

x, y plane bounded by the K+ 1 curves To,TI,...rK(see Fig. 2). The membranegeneous linearly elastic material occupying the two-dimensional domain C 2 mConsider a thin flexible initially flat elastic membrane consisting of homo-

acting along the boundary r = U:l,"rj . Large deflections are considered result-

is prestressed either by imposed displacement U, ,Ct or by external forces f n , 8


Figure 2: Domain S 2 occupied by the membrane.

given asthe deflection surface in the strain components. Thus, the strain components areing from nonlinear kinematic relations, which retain the squares of the slopes of

Ex = 24; +U,, +'W:; Ey = v,; +v,))+1W,;2 2

(IC)

(WJ)

Yxy =%;+v>: +U>))+v>, W,))

displacements due to the deflection.its plane; and U = U(X, y ) , v =v(x,y ) are the additional membranemembrane is subjected to the load g = g ( x , y ) acting m the direction normal todue to the prestress; W = w(x ,y) the transverse deflection produced when thewhere U' = u"(x,y ) , v' = v"(x,y) are the m-plane displacements components

equilibrium equations m terms of the displacementsUsing the procedure described in Katsikadelis et al.[31 we obtain the

= 0 >

GC)

Nxyx +Ny>y = 0 GwJ)

( X +N, h x x +N , h,+v; +N y> W y y +g = 0

in C2 ; N,",A$, NZy and N, ,N y,Nxy are the membrane forces given as

N," = C(u,y +m,; ) , Ny" = C(v,; vu,^ ) , NZy = Cti(u,; +v,: ) (3a,b,c)2


N y = . [ ( v , y + ~ W , ~ ) + v ( u , . + ~ - . : ) ]

Nxy= +v,. +VI,. W'J2 (4c)

principal directions to avoid wrinkling of the membrane, namelyshould be paid, so that the prestress results m tensile forces N f ,N ; in themodulus of elasticity and v the Poisson ratio; t is the thickness. Attentionwhere C = &/(l - v 2 ) is the stiffness of the membrane with E being the

Substituting eqns (4) into eqns (2 )yields

l - v

-V2v + + v , ~) , y = - w , ~( w , ~ ~+yW , = )-FW,. w,xy (6b)

*V2u +=(U,. +V,);),. = - W , . (W,= +~ w , ~ ~)-?W,. (6a)2 2

l-v2 2

+ j n i ~ + c [ ( v , , + ~ w , 2 , ) + v ( u , x + ~ W , : ) ] } W , ) y = - g

+ 2 N y +y " ( u , , +V,. + W X W,y > > W ,W,"+C[(u,,+~W,~)+.(v, ,+'W,:, l ,W:,2 2

(1-v)(6c)

2 2

; = c = + = 0boundary conditionsFor a fixed boundary it is, the displacements should satisfy the following

(7ahc)

3 The AEM for large deflections of membranes

here.However, for the completeness of this paper the method is concisely describeddeflection analysis of membranes is presented in Katsikadelis et al. [ 3 ] .Analog Equation Method (AEM). Detailed description of the AEM for largeThe boundary value problem described by eqns (6) and (7) is solved using the

Poisson's equations ( u1,u2,u3 stand for the functions U,v, W , respectively)According to the concept of the analog equation, eqns (6) are replaced by three

V2ui= b, (i = 1, 2, 3 ) (81

where b, = b,(x1,x 2 ) are fictitious sources.The fictitious sources are established using the BEM. For this purpose b, is

approximated as


B o u d u r - y E k w w ~ t sX X I V 223

b, =5di)fi (i=1,2,3)J (9)j= l

ii, and a particular one u p , U,+up . The particular solution is obtained asto be determined. We look for the solution as a sum of the homogeneous solution

where f, are approximating radial basis functions and a:1) are 3M coefficients

(10)J=l

where iii is a particular solution of

v 2 f i j =f, (11)

The homogenous solution is obtained from the boundary value problem

Writing the solution of eqn (12a) in integral form, we have

c& = - L ( u * , , ~ - i i , u , z)ds , i = 1,2,3

eqn (8) for points inside R ( c = 1) is written asP E Q u I- , respectively. On the base of eqns (10) and (13), the solution ofLaplace equation and c = 1, 11 2, 0 depending on whether P E R , P E r ,with U* = !nr / 277 , Y =j P -Q 1 , Q E r being the fundamental solution of the

(13)

The frst and second derivatives for points inside R are obtained by directdifferentiation of eqn (14). Thus, we have

Ui,k =-&L,; G,, - i i , U , ; k ) d s +%%j,k ( k = 1,2) (154j = 1

Using the BEM with N constant boundary elements, discretizing eqn (13)and applying it to the N boundary nodal points yields

Cii, = H Z , - ( i = 1, 2, 3 )

from the integration of the kernels on the boundary elements. Eqns (14) and (15)c at the N boundary nodal points and H,G are N x N matrices originatingwith C being an N x N diagonal matrix including the values of the coefficient

(16)


224 B O L M ~ ~ I - J ~Elcmcnts X X I V

Total NBoundary nodes

Total MInterior nodes

l 0

and eqn (16)yields after eliminating U and U, by virtue of the boundary conditions (12b)are subsequently applied to Mpoints inside the domain (c= 1) (see Fig.3). This

Figure 3: Boundary discretization and domain nodal points.

ui = Da(') +Eii,

u ~ , ~ ,= D,,a(') +Ek,iii

ui,, = Dka(')+E,iil

substitute the derivatives of ui using eqns (1 S) and (19). This yieldsThe final step of AEM is to apply eqns (6) to the M points inside R and

coefficients.where D, E,....,E,, are known matrices and a(') is the vector of the unknown

and the corresponding vectors a(') and ac2) are computed fi-om eqns (2Oa,b).Eqns (20) are solved iteratively. A initial guess for vector ac3) is assumed

a(l) = Fl(ac3') ac2) = F2(ac3)), ac3)= F3(a('),a(2),a(3)) (20a,b,c)

subroutines use the Newton-Raphson method or the method of minimization ofexamples, ready-to-use subroutines of the IMSL were employed. Thesegood for the solution of coupled nonlinear algebraic equations. In ow numericalac3)computed. Eqn (2Oc) is nonlinear and can be solved using any techniqueSubsequently, these vectors are introduced in eqn (2Oc) and a new vector

displacements and their derivatives are computed fiom eqns (17)-(19). Finally,decreasing prestress. Once the vectors aci) ( i = 1,2,3 ) are known theconvergence is achieved for prestressed membranes, which becomes worse with

the function S(a,(3),af),...a:)) = laC3)- F 3 (ac3))12. In general a good

the stress resultants are computed fi-omeqns (4) and t ie boundary reactions tiomP I

T, =N,cosa+N,,sina, Ty =Nxycosa+Nysina (21a,b)

V = T,w,~+ T y ~ , y P l c )



U = U, v = V and G = 0 , where U,V are displacements producing the prestress.prestress are obtained by solving the membrane problem with g = 0 andwhere a = angle(x,n) . The displacements and the stress resultants due to the

4 The ponding problem on floating membranes

weights of the liquid 1 and 2; R, cR = ponding region of the liquid 1,A = area of the domain Q (cross-sectional area of the tank); y I ,y2= specifich = level of the liquid 1 before ponding; z = level of the liquid 1 after ponding;solved iteratively. In solving the problems the following symbols are introduced:We distinguish two problems corresponding to Fig. l a and Fig. lb. Both are

the membraneC l z c Q = region of reaction of liquid 2; g, = unifonn load due to the weight of

is obtained as follows.In this case the liquid 1 fills all the space created by the deflection. The solution

4.1 Ful l ponding ( F i g l a )

(a) If max d l ) h ,a new deflection surface W ( ' ) is determined under the load

determined under the load g = g, +y,V l A . Then:membrane (for reasons of simplicity uniformly), a deflection surface wil) is

Starting with any volume V of liquid arbitrarily distributed over the

g,= g, +y , d ) in R (22)

(b) If max d ' j h ,the region R ~ 'is determined and a deflection surface d 2 j

is established under the load

max W h no floating occurs.nodal displacements in the k-th iteration. Then it is set z = z k and W = W @ ) . If

criterion is max l w y ) - 1 5 E , , ( i = 1,2. . .,M) , where are the interiorThis procedure is repeated until convergence is achieved. The convergence

r

z - h outside the membrane. Thus, referring to Fig. la, we findsubmerged part of the membrane with that from the increase of the level byThe computation of zk can be performed by equating the volumes of the

The above procedure requires the knowledge of zk and QLk) in each step.

A@ -2 ) = V ( 2 ) (24)


226 B o u ~ i w - ~ ~E l m c n t s X X I V

linear functionhorizontal plane at level z . Hence z can be established as the zero of the non

where V ( z )= (w-z)dR is a volume cut of the deflection surface by a

F ( z ) = A@ -2 )-V(z) (25)

geometrical problem, which is solved using the procedure developed in [4].surface w(x,y) by a plane parallel to the xy plane at a level z constitutes aThe determination of the region R z and the volume V(z) cut from a given

4.2 Partial ponding (Fig. lb)

as follows:by the deflection of the membrane under the load V*yl. The solution is obtainedIn this case, the volume V* of the liquid 1 is given and is less than that created

by a plane parallel to the xy plane, that isg = g , +yiV*/ A . Then a volume cut of this surface equal to V* is determined

The deflection surface W(’) of the membrane is determined under the load

(a) If max d l ) h , a new deflection surface W(‘) is determinedunder the load

where C$’ is the first approximation to the ponding region. Then:

volume cut equal to V‘.

and a new ponding region Q:’ c !2 is established by determining again a

under the loadas in the case of full ponding and the deflection surface W(’) is established

(b) If max d l ) > h the region 0:) of the reaction of the liquid 2 is determined

0:)c$!)=,/” +YI(W(l’- S > in

-y2 ( W ( ’ ) - z ) in (28)

g oin ~ -0 : )- R ( ’ )

z

volume cut equal to V’.

and a new ponding region R:’ cSz is established by determining again a

each iteration step the condition for partial ponding is checked, that is

m a x l ~ ~ ~ ’ ) - w ~ ’ - l ) l ~ & , ”( i=1,2 ..., M ) . If maxw(k)<h no floating occurs. InThis procedure is repeated until convergence is achieved, when

I



p ’ ’ & 2 ) v*

5 Numerical Examples

(29)

subroutine DNEQNF of IMSL.with a tolerance E,,, = 0.0001 . The solution of eqn (2Oc) was obtained using thePC. In both examples the ponding problems converged in less than 10 iterationsresults were obtained using the MS Fortran PowerStation 4.0 on a Pentium 111constant. Computationally, the method was found very efficient. The numericalf = ( r z+ c ~ ) ” ~have been employed; c is an appropriately chosen arbitraryanother of arbitrary shape. Radial basis functions of multiquadric type, i.e.program was written and two membranes were analyzed, one of rectangular andOn the basis of the procedure described in previous section a FORTRAN

5.1 Rectangular membrane

as shown in Fig. 4. The employed data are: a = 5.0m , b = 4.0m ; t = 3mm ;membrane. The membrane was prestressed by imposed boundary displacementsThe fidl and partial ponding problems have been studied for a rectangular

maxw,fuii= 0.441m and Vfua= 3 . 9 5 0 ~ ~ ~. The partial ponding formembrane was analyzed for full ponding with h = 0.32~1.It was foundE=l lOMPa: v=O.3 ; y1 =15kNlm3; y2=10kNlm3; i l=G=2cm.Firs t , the

V* = 0.6x Vfull= 2 . 3 7 d resulted m max wparriol= 0.393m . The obtained

hmin= 0.16m . In Fig. 6 and 7 the computed profiles of the deflection and theminimum allowable level to prevent overflow of liquid 2 was foundponding and reaction areas for partial ponding are depicted in Fig. 5. The

oxo/ ( o , ~ ) ~ U ~ ~versus the partial ponding volume V/ V& are shown i n Fig. 9.

the variation of the central deflection ratio W , ! ( w ~ ) ~ U ~ ~and stress ratiofull ponding volume V on the level h of the liquid 2 is shown in Fig. 8. Finallystress resultant N , along the x axis are presented. Moreover, the dependence of

y- ~ ~~5..~-i-y.-f-J;,a*

-.

X

~ .. cI ,

-\

U VFigure 4: Prestressing d;i mp&ed boundary displacements.


3.5

3-

2.5-

2-

1 . 5 7

1:

0.5-

Figure 5 : Ponding and reaction area in a rectangular membrane for V* = 0.6Vj7Ln

c)+--~-~ p,/

1 l----r--'~

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5+

0

0.02

0.04m3

0.06

0.08

0.1-0.4 -0.2 0 0.2 0.4

xlaFigure 6

4400 i ' I ' ~ ' ~ ' i

2800 410 20 30 40 50

h k m )Figure S

1

0.8

0.6

0.4

0.2

0

-0.4 -0.2 0 0.2 0.4

Figure 7xia

l I lI I l I l , l 1

l l ll 1 l

l I I1 l l ~

l l ll l l

l ' l ' l ' l '0 0.2 0.4 0.6 0.8 1

VNhAFigure 9



studied. Its contour is defmed by the closed curve (Cassini's curve)The full and partial ponding of a membrane with arbitrary shape have been

5.2 Membrane of arbitraryshape

V* = 0.6V, ,/[, Vfu,,= 19.224, are shown m Fig. 10.is the central deflection. The computed ponding and reaction areas forboundaries of the ponding and reaction areas cut the x axis, respectively, and wo

Numerical results are presented in Table 1; x, and x, are the points, where thet = 3 m m , E = l l O M P a , v = O . 3 , y1 =10kN/m3; y2=101cN/m3; h=0.50.

normal to it, while in the tangential direction it is U, = 0 . The employed data are:due to imposed displacements U, = 2 cm along the boundary and in the directionr = d 9 cos28 +JSl cos' 28 +175 , 0 5 BI2 n . The prestress of the membrane is

under partial and full ponding loads.Table 1. Deflections and stresses in a floating membrane with arbitrary shape

x, X R W O %ax

V",// (m> (m> (cm> (mm> (MPa) (MPa) (Mf'a)

V* 0,L+= cyL+"0,I x , y 4 1

4.573 2.914 65.8 36.4I 2.217 1.596

1 5.000 3.511 70.6 50.5 3.778 4.897 2.400

3.267

V* = 0 . 6 ~VfuN.Figure 10: Ponding and reaction area in a membrane for arbitrary shape for

6 Conclusions

deflection problem of the elastic membranes was solved using a boundary-onlynonlinear fluid structure interaction problem, has been solved. The largeThe problem of ponding on floating membranes of arbitrary shape, a highly


230 B o u ~ i w - ~ ~E l m c n t s X X I V

deformed shape of the membrane is negligible.order smaller than the transverse deflection W . Thus, their effect on theelastic in-plane displacements U and v were found very small, more than oneregions were determined using simple iterative fast convergent procedures. Thereaction loads of the supporting liquid, as well as the ponding and reactionponding loads, filling the space created by the deflection of the membrane, thesimplifies considerably the numerical modeling of the problem. The unknownmethod based on the concept of the analog equation, a solution procedure that

7 References

StructuralMembranes, Computers and Structures, 21, pp.443-451.[2] Epstein, H.I. and. Stmad, T.J., 1985. Liquid-Filled, Liquid-Supported Circular

computers, Computers and Structures, 11, pp.349-353.[l] Epstein, H.1, 1980. Floating Roof Analysis and Design Using Micro-

Computational Mechanics, 27(6), pp.513-523.method for large deflection analysis of membranes.A boundary-only solution.

[3] Katsikadelis, J.T., Nerantzaki, M.S. &L Tsiatas, G C . The analog equation

print).Membranes. An Analog Equation Solution, Computational Mechanics, (in

[4] Katsikadelis, J.T. &L Nerantzaki, M.S. The Ponding Problem on Elastic


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Ponding on floating membranes - WIT Press · Ponding on floating membranes Greece National Technical UniversityofAthens, ... (Fig. lb). The case of the while in the second case the