antonio j. gil - structural analysis of prestressed saint venant–kirchho hyperelastic membranes...

8/6/2019 Antonio J. Gil - Structural analysis of prestressed Saint VenantKirchho hyperelastic membranes subjected to modera

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Structural analysis of prestressed Saint VenantKirchhoffhyperelastic membranes subjected to moderate strains

Antonio J. Gil *

Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea, Swansea, SA2 8PP, UK

Received 7 January 2005; accepted 1 February 2006Available online 17 April 2006

Abstract

This paper presents a complete numerical formulation for the nonlinear structural analysis of prestressed membranes with immediateapplications in Civil Engineering. The membranes will be considered to undergo large deformations but moderate strains. Therefore,Nonlinear Continuum Mechanics principles dealing with large deformations on prestressed bodies will be accounted for. The constitutivemodel adopted for the material will be a prestressed Saint VenantKirchhoff hyperelastic one. To carry out the computational resolutionof the structural problem, the Finite Element Method (FEM) will be implemented according to a Total Lagrangian Formulation (TLF),by means of the Direct Core Congruential Formulation (DCCF). Different numerical schemesfirst and second-order unconstrainedoptimization techniqueswill be presented to solve the resulting geometrically nonlinear problem, which involves the minimizationof the total potential energy system functional. These ones will be improved by a parametric line search algorithm according to a poly-nomial interpolation. Eventually, numerical examples will be introduced to verify the robustness of the aforementioned formulation. 2006 Elsevier Ltd. All rights reserved.

Keywords: Saint VenantKirchhoff hyperelastic material; Prestressed membranes; Total Lagrangian formulation; Finite Element Method; Optimizationtechniques

1. Introduction

The subject of this paper is the geometrically nonlinearanalysis of prestressed membrane structures with arbitrarygeometry undergoing moderate strains. An increasingapplication of this structural models is reaching extremelydifferent knowledge fields, which move from the well-known Architectural field to the recently discovered Bio-

mechanical field. In all these cases, it is feasible to findmembranes undergoing large deformations and subjectedto a previous state of prestressing. In this paper, we willfocus on those particular ones where strains can be mod-elled as moderate, despite having large deformations.

First, an initial arbitrary shape for the membrane isdefined by means of any shape finding technique. This ini-tial membrane structure does not undergo at this stage any

prestress whatsoever. Then, two different and successiveloading steps may be distinguished according to theireffects on the stabilization of the prestressed membrane.The first load step of prestress loading is developed to pro-vide the necessary in-surface rigidity to the membrane. Thesecond load step of in-service loading is comprised of awide group of loads: snow, wind or live loads amongothers.

The theory of hyperelastic membranes, as for example,propounded by [1,2] or [3] treats the problem from an ana-lytical viewpoint, arriving after complex algebraic manipu-lations to final formulae of difficult application. Somesimplicity, although not much, may be accomplished ifthe Von Karman compatibility equations are used (see[4,5]). Regardless of the important implications of thisapproach into the theoretical understanding of these struc-tures, a main disadvantage is that it flows into a nonlinearpartial differential equations of impossible analyticalsolution.

0045-7949/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2006.02.009

* Tel.: +44 0 1792 295902; fax: +44 0 1792 295903.E-mail address: [email protected]

www.elsevier.com/locate/compstruc

Computers and Structures 84 (2006) 10121028
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Because of this lack of numerical results, variationalapproaches ought to be taken into consideration as the bestones to provide feasible solutions from a practical stand-point. Regarding this approach, some authors have treatedthe problem of finite hyperelasticity set on rubber-likemembrane materials by means of the Finite Element

Method (FEM). By following this path, interesting papersare those due to [611], where initially unstressed mem-branes are analyzed when undergoing large strains. Forthese cases, the Updated Lagrangian Formulation (ULF)is considered to be the most suitable for the derivation ofthe tangent stiffness matrix.

The discussion to follow is divided into six parts. Section2 reviews the classical nonlinear strong form equations fora general structural problem starting from an initialunstressed configuration. Section 3 presents in detail thesame formulae but considering a prestressed configurationas the initial one. In these both sections, concepts such asGreenLagrange strain tensor, PiolaKirchhoff or Cauchy

stress tensors are introduced as basic tensorial tools tocarry out the forthcoming numerical approach. Analo-gously, Helmholtzs free energy functional is defined forlater calculations. The consideration of the Saint VenantKirchhoff hyperelastic material as the adopted model willbe of great effect to end up with a linear constitutive rela-tionship of easy implementation.

Section 4 entails a comprehensive explanation of theFinite Element semidiscretization of the previouslyobtained strong form. After the weak form is derived in astraightforward manner, the displa cement field is interpo-lated by means of shape functions based on a Lagrangian

mesh geometry. The resulting formulation will be the socalled Total Lagrangian Formulation (TLF). Afterwards,the exact linearization of the Total Lagrangian weak formof the momentum balance is carried out in detail. For thesake of further computing implementation reasons, theDirect Core Congruential Formulation (DCCF) isreviewed as the most appropriate one.

With the objective of giving a complete and robust for-mulation to analyze the whole structural problem, anothervariational approach in terms of the Total Potential Energyfunctional (TPE) is introduced in Section 5. Thus, the nextsection introduces the different numerical algorithms thatbased upon incremental-iterative schemes were used inthe present research. Eventually, based on a linear isopara-metric three-node triangular finite element, numericalexamples are provided in Sections 7 and 8. These cases willshow adequate performance as the required quadraticallyconvergence of the NewtonRaphson method is obtained.The conclusions are presented in the last section.

2. First load step and preliminary results

Let us consider a material body B in an initial unde-formed configuration B0 with domain X0 frontier oX0,defined within the 3D-Euclidean space. The material coor-

dinates of a bodys particle at a time t = 0 will be described

according to X = (X1, X2, X3). As a result of the applicationof a displacement field, the current or spatial position ofthe bodys particle at a time t may be obtained as a func-tion of its material coordinates as1 xi = xi(XA, t). Therefore,the body will move to a new configuration Bt with domain

Xt and frontier oXt (see Fig. 1).For convenience here and in what follows the standard

summation for repeated indices is adopted, as well as theclassical indices convention to express spatial derivatives:

;k ooxk

. The balance of linear momentum at a local level,in the absence of inertial effects, may be expressed in Eule-rian description as follows:

rji;j qbi 0 in Xt; fi ti dC rjinj dC on oXt 1

The conservation of linear momentum is set up in the cur-rent configuration Bt of domain Xt, and the traction vectort may be deduced from the Cauchy stress tensor r and theunit normal n on the frontier oXt. Analogously, Eq. (1)

may be obtained in Lagrangian description

PAi;A q0bi 0 in X0; f0i t

0i dC0 PAinA dC0 on oX0

2

In this case, the conservation of linear momentum is for-mulated at a local level in the initial undeformed configura-tion B0 of domain X0, and the traction vector is impliedfrom the nominal stress tensor PAi and the unit normal non the frontier oX0. The upper and lower case indices standfor the initial and current configurations, respectively. Thismakes this stress tensor to be considered as a two-pointtensor (see [12]), as a difference of the Eulerian consider-

ation of the Cauchy stress tensor. It should be recalled thatthe nominal stress tensor is the transpose of the so calledfirst PiolaKirchhoff stress tensor (see [13]).

Another stress entity must be introduced for the sake ofconvenience: the second PiolaKirchhoff stress tensor SAB,which can be framed as a Lagrangian stress tensor. Themathematical relationships among the different enumer-ated stress tensors, that is, Cauchy, nominal and secondPiolaKirchhoff stress tensors, are shown right below by

1 Capital (A,B,. . .) as opposed to (i,j,. . .) indices have been used in order

to clearly distinguish the initial configuration B0 from any other.

Fig. 1. Motion of a body.

A.J. Gil / Computers and Structures 84 (2006) 10121028 1013


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means of the well-known deformation gradient tensorFiA = xi,A and its determinant or jacobian J:

Jrij xi;APAj xi;Axj;BSAB 3

where

xi;

A

oxi

oXA4

Once the balance of linear momentum equations havebeen established in the two descriptions par excellence, thisis, Eulerian and Lagrangianformulae (1) and (2), respec-tively, another required equation is the one which gives theconcept for the GreenLagrange strain tensor EAB and theright CauchyGreen strain tensor CAB

EAB 1

2CAB dAB; CAB xi;Axi;B 5

In the mathematical description of the material behav-iour, the response of the material is characterized by a con-

stitutive equation which gives the stress as a function of thedeformation history of the body. For the structures pre-sented in this paper, hyperelasticalso known Green elas-ticmaterials are considered for the following derivation.Furthermore, due to the fact that the rotating componentR of the deformation gradient tensor Faccording to thepolar decomposition theorem F = R U, see [12]is smallenough, the Saint VenantKirchhoff constitutive modelcan be concluded as the most appropriate for our purpose.Eventually, the formal mathematical formulae for thismodel may be summarized by using the Helmholtzs freeenergy functional (see [33]) or internal strain energy func-

tional wint as follows:

SAB owint

oEAB 2

owint

oCAB6

SAB CABCDECD 7

The equations gathered at (6) and (7) relate the secondPiolaKirchhoff stress tensor with the GreenLagrangestrain tensor by means of a fourth-order tensor, knownas the material elasticity tensor, which contains all the elas-tic moduli of the material. Therefore, the classical explicitexpression for the strain energy functional per unit of vol-ume would be

wint 1

2CABCDEABECD 8

3. Second load step: strong formulation

Let us consider now three possible configurations of thematerial body: in one hand, an initial undeformed state B0and, in the other hand, a primary state Bt and a secondarystate Bt , for the time instants t and t*, respectively. Amongthese two stages, a displacement field u = (u1, u2, u3) may bedefined in R3. From now on, quantities which proceed

from the movement from the primary to the secondary

state will be considered as incremental (see Fig. 2). It is thusfeasible to obtain the spatial coordinates for the time t* of aparticle as a function of its material coordinates in the ini-tial unstressed configuration B0 according to yi= yi(XA, t*).

As a consequence, the complete deformation path is

built up from the composition of two successive steps:the first one, from the initial configuration to the primarystate and a second one, which can be traced from the pri-mary state to a secondary state. By recalling the chain rule,relations among deformation gradient tensors can be dis-played as follows:

oyioXA

oyioxj

oxj

oXA9

Formula (9) can be rewritten in index notation asfollows:

yi;A yi;jxj;A 10

Analogously, the relation between jacobians is given as

J J0J 11

where J represents the jacobian at the primary state, J*

stands for the jacobian at the end of the secondary stateand J0 symbolizes the jacobian as a consequence of theincremental deformation. The scope of the next sectionswill be to set up the equations presented in the former sec-tion for the secondary state by adopting the primary one asthe reference configurationsubjected to a previous stressfield.

3.1. GreenLagrange strain tensor

A material particles position in the primary and second-ary states may be expressed in terms of the incremental dis-placement field as follows:

yi xi ui ) yi;A xi;A ui;A xi;A ui;jxj;A 12

The GreenLagrange strain tensor for the secondarystate is

EAB 1

2yi;Ayi;B dAB 13

From (12) and (13), the GreenLagrange strain tensor

can be rewritten as

Fig. 2. Motion of a prestressed body.

1014 A.J. Gil / Computers and Structures 84 (2006) 10121028


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EAB 1

2xi;Axi;B dAB

1

22xi;Axj;Beij ui;Aui;B 14

where

eij 1

2ui;j uj;i 15

The difference between the GreenLagrange strain ten-sor for the primary and secondary states can be obtained as

EAB EAB xi;Axj;Beij 1

2ui;Aui;B xi;Axj;BE

0ij 16

where

E0ij eij 1

2uk;iuk;j 17

where the incremental GreenLagrange strain tensor E0ijhas been introduced for the sake of convenience and it rep-resents a relative measure of the strain at the secondarystate by taking the primary one as an adequate reference.

It should be remarked that this later one is not an un-stressed state.

3.2. Nominal and second PiolaKirchhoff stress tensors

As it was introduced formerly, the Eulerian Cauchystress tensor may be transformed to a Lagrangian and atwo-point stress tensor by considering an initial and a cur-rent stressed configurations. This two new tensorial entitieswere referred to as the second PiolaKirchhoff and nomi-nal stress tensors respectively. By considering as initial con-figurations the initial undeformed configuration and the

primary state configuration, the following formulae canbe obtained respectively:

Jrij yi;APAj yi;Ayj;BS

AB 18

J0rij yi;kP0kj yi;kyj;lS

0kl 19

By considering relation (11), Eqs. (18) and (19) can bemodified to obtain

P0ij J1xi;Axk;Byj;kS

AB 20

S0ij J1xi;Axj;BS

AB 21

The expressions gathered at (20) and (21) summarize theexisting relationship between the nominal stress tensor P0ijand the second PiolaKirchhoff stress tensor S0ij expressedin the prestressed configuration Bt, with respect to the sec-ond PiolaKirchhoff stress tensor SAB represented at theinitial undeformed state B0.

3.3. Linear momentum balance law

The conservation of linear momentum of the materialbody in the secondary state may be depicted with respectto three possible descriptions: B0, Bt and Bt , accordingto a Lagrangian formulation for the first two ones or an

Eulerian formulation for the later one. Thus

rji;j qbi 0 inX

t; f

i t

i dC

rjinj dC

on oXt 22

PAi;A q0bi 0 inX0; fi t

0i dC0 P

AinAdC0 on oX0 23

P0ji;j qbi 0 inXt; fi ~t

i dCP

0jinj dC on oXt 24

where

rji;j orji

oyj25

PAi;A oPAioXA

26

P0ji;j oP0ji

oxj27

The formula (24) along with the boundary and continuityconditions, synthesizes the strong formulation of the struc-tural problem according to a Lagrangian description withrespect to a reference stressed configuration. This equation

will be used throughout the remainder of this paper.

3.4. Constitutive law

By accounting for the Saint VenantKirchhoff constitu-tive model adopted for the material behaviour, the expres-sion (6) can be reformulated by means of a Taylor seriesexpansion truncated after the first-order as follows:

SAB owint

oEAB

owint

oEAB

o2wint

oEABoECDECD ECD 28

The accuracy of this Taylor series depends directly onthe smallness of the step E

CDECD. For tension membrane

structures in Civil Engineering applications, as it was afore-mentioned, this is a valid assumption. Thus, from (16) and(28), the following expression can be written down:

SAB SAB CABCDxi;Cxj;DE0ij 29

By recalling (21) and (29)

S0ij J1xi;Axj;BSAB J

1xi;Axj;BCABCDxk;Cxl;DE0kl 30

The fourth-order tensor of elastic moduli can be referredto the prestressed configuration as follows:

Cijkl J1xi;Axj;BCABCDxk;Cxl;D 31

Eventually, Eq. (30) may be reformulated to give thefinal expression

S0ij rij CijklE0kl 32

This final formula is set up to show the constitutive law fora prestressed Saint VenantKirchhoff hyperelastic material.The second PiolaKirchhoff stress tensor is expressed interms of an easy linear relationship which depends on threetensorial entities: Cauchy stress tensor in the primary state,fourth-order tensor of elastic moduli and the incrementalGreenLagrange strain tensor of the secondary state re-

ferred to the primary one.

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3.5. Internal strain energy balance

Another important feature which needs to be obtained isthe incremental strain energy accumulated into the struc-ture along the deformation path from the primary to thesecondary states. By performing again a Taylor series

expansion truncated after the second-order, the internalstrain energy functional per unit of undeformed volumemay be developed as

wint wint X ! 33

where

X owint

oEABEAB EAB 34

! 1

2

o2wint

oEABoECDEAB EABE

CD ECD 35

The terms X and ! can be expanded as

X SABxi;Axj;BE0ij JrijE0ij 36

! 1

2CABCDxi;Axj;BE

0ijxk;Cxl;DE

0kl

1

2JCijklE

0ijE

0kl 37

By substituting (36) and (37) back into (33), the incre-mental internal energy per unit volume is obtained as

wint wint J rijE0ij

1

2CijklE

0ijE

0kl

! Jw0int 38

where w0int represents the incremental energy per unit vol-ume measured in the prestressed configuration. By inte-grating over the initial undeformed volume X0 and by

applying the mass conservation principle from this volumeX0 to the prestressed one Xt, the total incremental energy isgiven as

DWint

ZX0

wint wint dX0

ZX0

Jw0int dX0

ZXt

w0int dV 39

Therefore, the internal strain energy functional per unitof volume of the primary state takes the final form

w0int rijE0ij

1

2CijklE

0ijE

0kl 40

4. Finite element semidiscretization

4.1. From the strong to the weak formulation

The aforementioned primary and secondary states canbe understood as an initial prestressed state Bpret and afinal in service loading state B due to the consideration oflive and dead load. Henceforth, the coordinates of any par-ticle will be renamed as Xpret for the initial prestressed stateand x for the final in service loading state. These ones arerelated by means of the incremental displacement field u

as follows:

x Xpret u xi Xpreti ui 41

According to this new nomenclature, the strong formula-tion of the problem in a Lagrangian description with respectto the prestressed configuration is summarized in Fig. 3.

Tensor P0ji has been replaced with Pji, tensor S0ij with Sij

and tensor E0ij with Eij. Thus, the weak form may be devel-

oped in a Total Lagrangian format (TLF) by means of theso called Principle of Virtual Work. Neglecting inertiaforces, this gives:

dWintdui; ui dWextdui; ui 42

Both terms of the above equation can be expanded as

dWint

ZXpret

dFijPji dV

ZXpret

dEijSij dV 43

dWext

ZXpret

duibidV

ZCpret

duiti dC 44

where the work conjugacy property of the tensors S and P

with E and FT

, respectively, has been employed for theequalities at (43); b is the body force vector and t are thesurface tractions.

4.2. Semidiscretization of the weak form

The weak form equations obtained formerly may becombined with a finite element discretization of the dis-placement field in terms of the nodal values and shapefunctions NI as

ui uIiN

I; i 1; 2; 3; I 1 . . .N nodes 45

Fig. 3. Strong formulation for a Lagrangian description.



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This enables the nodal equivalent internal and externalvector forces, fint and fext, respectively, to be obtained ina straightforward manner for a given node Iin tensor nota-tion as

fIinti ZX

pret

PjioNI

oX

pret

j

dV ZX

pret

FikSkjoNI

oX

pret

j

dV 46

fIexti

ZXpret

biNI dV

ZCpret

tiNI dC 47

Or in a more compact matrix notation as

fIint

ZXpret

PTrNI dV

ZXpret

FSrNI dV 48

fIext

ZXpret

bNI dV

ZCpret

tNI dC 49

Assembling these forces for all the nodes of the Lagrangianmesh gives the global equilibrium equations

fint fext ) fres fint fext 0 50

where fint is the global vector of internal forces, fext is theglobal vector of external forces and fres is the global vectorof residual forces. This last vector represents clearly the outof balance forces as a result of the strong nonlinearity con-tained into the structural problem.

4.3. Linearization of the global equilibrium equations

The set of equations depicted at (50) presents a geomet-rically nonlinear feature, so an iterative solution scheme

will be required. Among all the available methods, the sec-ond-order NewtonRaphson one accomplishes the bestconvergence properties. The total tangent stiffness matrixrequired by the later one is formed by linearizing the globalequilibrium equations (50) in the direction of the incremen-tal displacement u.

By carrying out the linearization of the global vector ofinternal forces, it turns out to be

dfIint dfmatI

int dfgeoI

int KmatIJ Kgeo

IJ

duJ 51

where KmatIJ

and KgeoIJ

stand for the elemental material orconstitutive stiffness matrix and the elemental geometrical

or initial stress stiffness matrix, respectively. Each one ofthese matrices can be expanded and represented in tensornotation as follows:

KmatIJ

ij

ZXpret

FikoNI

oXnCnklmFjl

oNJ

oXmdV 52

KgeoIJ

ij dij

ZXpret

oNI

oXmSmk

oNJ

oXkdV 53

By assuming that the body forces b and external surfacetractions t not associated to pressure forces remain con-stant and by taking into account that the pressure compo-nent is dependent upon the geometry due to changing

orientation and surface area of the structure, the lineariza-

tion of the global vector of external forces is given throughthe following derivation:

fIext

ZC

tNI dC p

ZC

nNI dC 54

where p is the pressure scalar acting on the considered

material body. By applying the Nanson rule for the unitnormal n see for instance [12] or [14] for details, it canbe deduced:

fIext p

ZCpret

JFTnpretNI dC 55

Particularizing for an isoparametric three-node linearfinite element, both in matrix and in tensor notation

fIext pCpret

3

ox

onpret1

ox

onpret2

fIexti pCpret

3ilm

oxl

onpret1

oxm

onpret2

56

where npret1 and npret2 are the local plane coordinates, ilm is

the so called alternating third-order tensor and standsfor the classical cross product between 3-D vectors.

Analogously as above, this vector can be linearizedalong the direction of the displacement vector u, hence thisresults in

KpIJ

ij pCpretilm

3Fm2dlj

oNJ

onpret1

Fl1dmjoNJ

onpret2

57

5. Direct Core Congruential Formulation

From the computational viewpoint, a very elegant pro-cedure termed Direct Core Congruential Formulation(DCCF) may be applied to perform the implementationstage of the formulation developed above. This methodol-ogy, hardly used in the existing literature, presents as pio-neer studies the ones due to [15,16]. The main ideasbehind this formulation can be discovered in the notablepaper due to [17]. A more recent paper about the topic is[18].

The scope of the DCCF is to establish the set of globalequilibrium equations whose unknowns are the compo-nents of the displacement gradient tensor G which is given

as

Gij oui

oXpretj

58

The aim is to establish a set of core equations as indepen-dent as possible with respect to subsequent finite elementdiscretization criteria. According to [17,18], the term coreemphasizes this issue. These core equations are basicallythe set of equations to be satisfied (equilibrium, compatibil-ity and constitutive law), expressed in a continuum manner(at a particle level). Complete independence is obtained ifthe relationship between the displacement gradient tensor

and the nodal displacements of a finite element model is



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linear. In this paper, this linear relationship will always bethe case as the finite elements used for all the numericalexamples will be isoparametric and linear. At the sametime, this approach helps to distinguish the differentsources of nonlinearity that arise in the analysis of pre-stressed structures.

Afterwards, every single component of the displacementgradient tensor may be easily expressed in terms of thenodal displacements of the Lagrangian mesh (accordingto a prescribed finite element model). Naturally, it is at thisstage that properties concerning geometry and discretiza-tion are brought to light. The consideration of only trans-lational degrees of freedom for the nodes of the Lagrangianmesh makes the DCCF particularly simple and easy toimplement. This methodology, as stated in [17,18], is notrestricted to isoparametric linear finite elements, howeverit makes it simpler. In this paper, as the main objective isto show the performance of the technique rather than itscomputational implementation, isoparametric linear finite

elements will be preferred. Fig. 4 shows a summary of thisformulation.

By employing a classical vectorization of the displace-ment gradient tensor (see for instance [12,14]) into a col-umn vector g and by expressing this one in terms of thenodal displacements, it turns out to be

g Bu 59

where B is the matrix of the gradient of the shape functionsand u is the column vector that gathers the nodal displace-ments for the N nodes of a single finite element

uT u11 u12 u

13 . . . u

N1 u

N2 u

N3 60

Analogously, the 2 2 submatrix of in plane componentsof the GreenLagrange strain tensor E may be vectorizedinto a column vector e by following the kinematic Voigtrule (see [12]) as

eT E11 E22 2E12 61

Every component of the new vector e may be expressed interms of the vector g as

ei hTi g

1

2gTHig 62

where hi and Hi are a vector of order ng and a symmetric

matrix of order ng ng, where ng represents the dimensionof the vector g. Both of them are constituted of numericalvalues comprising (1,0). Eventually, the in-plane compo-nents of the second PiolaKirchhoff stress tensor may betransformed into a vector by means of the kinetic Voigtrule as

sT S11 S22 S12 63

By substituting Eqs. (62) and (63) back into (43) the vectorof global internal forces may be rewritten in an easier wayas

fint ZVpret

BT/int dV /int sihi siHig 64

where summation is implied for repeated indices accordingto Einsteins notation. By proceeding in the same manner,the contributions to the total tangent stiffness matrix Kmat

and Kgeo may be obtained as

Kmat

ZVpret

BTMmatB dV Kgeo

ZVpret

BTMgeoB dV 65

Mmat hi HigCijhTj g

THj Mgeo siHi 66

where the fourth-order tensor of elastic moduli has beentransformed into a 3 3 matrix by applying the Voigt rulevectorization procedure to Eq. (82) to come out with:

s rpret Ce 67

As it can be observed, the second of the equations in (64)and equations in (66) represent the so called core equa-tions. On the other hand, the first of the equations in(64) and equations in (65) represent the transformationfrom the core space to the physical space (see Fig. 4).Eqs. (64)(67) constitute the fundamentals of a Finite Ele-ment code program. This core formulation provides a veryflexible framework to model any nonlinear structural prob-lem under a Total Lagrangian Formulation. As a conse-quence, the numerical examples presented later on in this

paper, have been performed under this numerical basis.

6. Energy principles

The mathematical formulation of the structural problemhas been collected in the formula (50), which summarizesthe global equilibrium of the membrane by means of a sys-tem of nonlinear equations imposed on the nodes of theLagrangian mesh. The solution of this system of equationsby means of a NewtonRaphson scheme may flow into abadly convergence algorithm. The NewtonRaphson algo-rithm satisfies local convergence. However, it lacks globalconvergence. This means that for a sufficiently close initialapproximation to the final solution, the algorithm con-verges. Otherwise, namely for highly nonlinear problems,the algorithm might not converge appropriately. Reference[29] explains in detail this issue. For the sake of this reasonand given the implicit characteristics of the structural mem-brane to be analyzed, an alternative approach based uponEnergy principles may be taken into considerationsee[19] for a further explanation of this technique.

Let us consider once again the material body B in an ini-tial prestressed configuration Bpret-primary state, whichafter the application of a displacement field, it maps intoa configuration B-secondary state. Therefore, it is feasible

to obtain the incremental relationship among the initialFig. 4. DCCF scheme.

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and final coordinates (see Eq. (41)). The consideration ofthe loading process undertaken over the material body B,from the configuration Bpret to the one B, enables the def-inition of three classical mathematical functionals knownas

External potential energy functional

Pext

ZXpret

ZBBpret

qpretbi dui dX

ZCpret

ZBBpret

ti dui dC

68

Internal potential energy functional

U

ZXpret

ZBBpret

Sij dEij dX 69

Total potential energy functional

P U Pext 70

The former functionals gather the external, internal andtotal potential energy accumulated in the structural mem-brane along the loading path. The process is consideredto be adiabatic and kinematically slow, so the generationof thermal and kinematic energies can be neglected.

The system of global equilibrium nonlinear equationsrepresented in the Eq. (50) was obtained through the semi-discretization procedure starting from the weak form of thestructural problem or, equivalently, by means of the well-known Principle of Virtual Work (PVW). From a mathe-matical perspective, this means that both internal andexternal nodal forces are derived from potential function-als, known as U and Pext, respectively. This mathemati-cal property makes the mechanical system to be namedas conservative. This principle is nothing more than thevariational formulation of the total potential energy, alsoknown as the Minimum Potential Energy Principle.

dP dU dPext dWint dWext 0 () dWint dWext

71

Eq. (71) represents the minimization of the total potentialenergy functional with respect to the displacement field.This formula may be understood as a classical uncon-strained optimization problem for an objective function

which results to be the total potential energy P. As a con-sequence, any appropriate technique may be used to carryout this mathematical optimization.

This alternative approach of understanding the struc-tural problem by means of minimization techniques canbe found in some References, for instance, [2025]. Fig. 5

summarizes the different steps to reach the equilibriumsolution: the total potential energy functional (zero-ordermethods), its gradient or unbalanced force vector (first-order methods), or even its hessian or total tangent stiffnessmatrix (second-order methods). A comprehensive explana-tion of these procedures is developed in [26,27].

7. Numerical techniques

So far we have considered the derivation and calculationof the equilibrium equations of a particular finite elementsystem. In this section, a brief description of the numericalmethods employed to solve the resulting geometrically non-

linear problem is presented. Any of the developed numericalalgorithms may be understood as an incremental-iterativetechnique, whereby the spatial description of the membraneat the increment n and at the iteration k+ 1 is obtainedstarting from the known configuration at the same step nand at the former iteration k. An admissible direction rep-resented by dkn along with a step coefficient a

kn included with

the purpose of increasing the convergence of the eventualalgorithm, are related as follows:

xk1n xkn a

knd

kn 72

Different incremental-iterative schemes are presented right

below for three different families of numerical methods:steepest descent method, conjugate gradient method andNewtonRaphson method. The first two of the former onesare first-order methods, whilst the last one can be classifiedas a second-order one. This classification is establishedaccording to the ideas shown in the previous section.

Steepest descent method

dkni xkn

ofxkn

oxkni73

NewtonRaphson method

dkni xkn

o2fxkn

oxknioxknj

1ofxkn

oxknj74

Conjugate gradient method

d0ni x0n

ofx0n

ox0ni75

dkni xkn

ofxkn

oxkni bk1n d

k1ni

76

bk1nFR

gk

T

n gkn

gk1T

n gk1

n

bk1nPR

gkn g

k1n

T gkn

gk1T

n gk1

n

77

Fig. 5. Variational formulation of the structural problem.



9/17

The coefficient bk1n that appears into the conjugate gra-dient method derivation, may be formulated according tothe FletcherReeves approachbk1nFR or the PolakRibi-ere onebk1nPR (see [29]).

The parameter ak is obtained by a line search techniquethat allows to raise enormously the convergence of the final

algorithm. Its implementation according to a backtracingstrategy within the interval (0,1) must be introduced for

the sake of convenience. In [5,14,28], different appropriatetechniques are presented.

Although the quadratically mathematical convergenceof the NewtonRaphson method is much higher than theone gathered in the remainder of the above presented pro-cedures, however the former one does not satisfy the global

convergence theorem for nonlinear numerical descentschemes (see [29] for further details). This important reason

100 50 0 50 100

100

50

0

50

100

OX axis (in)

OYaxis(in)

1 2 3

4 5 6

7 8 9

10 11 12 13 14

15

16

17

18 19 20 21 22

23

24

25

100 50 0 50 100

100

50

0

50

100

OX axis (in)

OYaxis

(in)

1

2

3

4 5

6

7

8

9

10

11

12 13

14

15

16

17

18

19

20 21

22

23

24

25

26 27

28

29

30

31

32

Fig. 6. Numerical example 1: Discretization.

1 2 3 4 5 64

3

2

1

0

1

2x 10

4

Iterations

TotalPotentialEnergy

1 2 3 4 5 610

12

1010

108

106

104

102

100

102

Iterations

InfiniteNormon

ResidualForces

Fig. 7. Numerical example 1: Convergence curves.



10/17

has to be taken into consideration specially when treatingwith highly nonlinear structural problems. The implemen-tation of these algorithms altogether allows flexibility atthe time of choosing the right method to approach theequilibrium solution.

8. Numerical example 1

This example is considered in [30]. It is a squared planemembrane initially prestressed. The edges of the mentionedmembrane are completely fixed. The side length is 240 in.and the thickness is measured as 0.004167 in. Mechanicalproperties for the material have the following values:30,000 Ksi for the Young modulus and 0.3 for the Poissonratio. The prestressing effect is considered to be 80 Ksi iso-tropically distributed. These magnitudes can be convertedto standard SI units by accounting for: 1 in. = 2.54 cm,1 Ksi = 6897 KPa and 1 Kip = 4448.4 N.

The in-service load is composed of a point transverseload applied right in the middle of the membranes domain.The load which takes a value of 10 Kip is considered down-wards. According to Fig. 6, the Lagrangian mesh is com-prised of 32 isoparametric three-node linear elements and25 nodes.

To accomplish the final solution, the employed methodwas the NewtonRaphson one according to one load incre-ment. Two convergence curves are gathered in Fig. 7. The

Fig. 8. Numerical example 4: Displacements OX and OZ.

Table 1Numerical example 1: Displacements (in.)

Node Levy and Spillers Present work

u v w u v w

1 0.015 0.015 1.431 0.014 0.014 1.4232 0.000 0.017 2.605 0.000 0.017 2.600

5 0.000 0.000 6.642 0.000 0.000 6.626

Fig. 9. Numerical example 4: Cauchy stresses rI and rII.



11/17

first one shows the evolution of the total potential energyalong the iterations path, whilst the second one representsthe infinite norm on residual forces vector with respect tothe number of iterations as well. The later one shows per-fectly the required quadratic convergence of the NewtonRaphson algorithm.

Fig. 8 shows the displacements field along the cartesianaxis OX and OZ, respectively. Both representations revealin a clear manner the axisymmetry of the considered mem-brane. Table 1 details the displacement values for three dif-

ferent nodes of the mesh. The accuracy of the numericalexample can be observed by checking the results with theones obtained in [30].

Analogously, principal Cauchy stresses rIand rIIcan beviewed in Fig. 9. Table 2 presents the numerical values forthree different elements of the membrane as well as its com-parison with the ones of [30]. Perfect agreement can bededuced.

9. Numerical example 2

This example represents a more realistic prestressedmembrane. It is shade pavilion composed of a fabric textile

reinforced by means of cables in the interior and in theperimeter of the prestressed membrane. The necessaryanchorage has been achieved by means of pinned masts.An isometric view of the structure is displayed in Fig. 10and a plan view is shown in Fig. 11. As it can be observed,the membrane presents symmetry with respect to the OYaxis, so hereafter only half of the model is to be studiedwith suitable boundary conditions.

Three sorts of finite elements will be considered for thefollowing numerical simulation. The fabric textile will be

modelled by an isoparametric three-noded finite elementaccording to the formulation described in previous sec-tions. The cables will be represented by an isoparametrictwo-noded finite element according to the same formula-tion described in previous sections. Further details of thisimplementation can be found in References [37,39].Finally, the pinned masts will be represented by a classicalisoparametric finite element. To prevent compression stres-ses being generated in any of the membrane or cable ele-ments, the wrinkling algorithm proposed in [38] wasimplemented.

To define an initial arbitrary shape, the Force DensityMethod was used (see [32]). For simplicity, this technique

Fig. 10. E2: Initial configuration, isometric view.

Table 2Numerical example 1: Cauchy stresses (psi)

Element Levy and Spillers Present work

rxx ryy rxy rxx ryy rxy

1 97377.6 85212.4 2801.5 97300.1 85163.9 2796.73 83510.2 96859.1 8657.1 83501.5 96830.3 8630.7

11 144691.0 97830.7 15615.6 144470.8 97849.2 15582.4



12/17

was preferred over some other available techniques,namely, dynamic relaxation (see [34]) or updated referencestrategy (see [35]). For both internal and perimeter cables,the considered force density factor was ten times higherthan the one for the interior domain. The kinematic bound-ary conditions for nodes along the membranes perimeterare displayed in Table 3. Fig. 12 shows a plan view ofthe initial configuration, where the node numbering ofthe selected Lagrangian mesh can be observed. Fig. 13shows the isometric view of the structure after form findinganalysis has taken place, where cable and membrane ele-ments can be easily distinguished.

As a result of this shape finding analysis, an initial shapeis achieved under a controllable prestress loading. More-over, this control is set in terms of the relative values forthe force density coefficient among the different compo-nents of the membrane. In other words, this shape is not

dependent on the absolute values of the prestress but onits relative ones, so this permits the reduction of the abso-lute prestress as much as desired. Therefore, an initialshape with a negligible prestress loading is obtained.

9.1. First load step: Prestressing

Once the initial equilibrium shape is obtained, an appro-

priate and realistic prestressing loading is applied to the

Fig. 11. E2: Initial configuration, plan view.

Table 3E2: boundary conditions (m)

Node x y z

13 6.0 4.5 12.0130 6.0 0.0 0.0247 6.0 4.5 12.0

OY axis 0.0

0 1 2 3 4 5 6

4

3

2

1

0

1

2

3

4

247

130

13

234

26

143

117

246

12

129

221

39

156

104

233

25

142

116

245

11

208

52

220

38

169

91

155

103

182

78

195

65

128

232

24

141

115

207

51

168

90

194

64

181

77

219

37

154

102

244

10

206

50

127

167

89

231

23

193

63

180

76

140

114

218

36

153

101

205

49

192

62

166

88

243

9

179

75

230

22

126

139

113

217

35

152

100

204

48

191

61

165

87

178

74

242

8

229

21

125

216

34

138

112

Initial shape

OX axis (m)

151

99

203

47

190

60

177

73

164

86

241

7

228

20

124

215

33

137

111

150

98

202

46

189

59

176

72

163

85

240

6

227

19

214

32

123

136

110

201

45

149

97

188

58

175

71

162

84

239

5

226

18

213

31

122

135

109

200

44

148

96

187

57

174

70

161

83

238

4

225

17

212

30

121

134

108

199

43

147

95

186

56

173

69

160

82

237

3

224

16

211

29

120

133

107

198

42

185

55

146

94

172

68

159

81

236

2

223

15

210

28

119

197

41

132

106

184

54

145

93

171

67

158

80

235222209196

183

170

157

144

131

118

105

92

79

66

53

4027141

OYaxis(m)

Fig. 12. E2: Initial configuration, plan view.



13/17

structure. Firstly, masts are added to the structural model.This fact allows the tensioned membrane, the reinforcingcables and the masts to behave in an interactive mannerrather than analyzing the compressive members separate

from the membrane-cable assemblage. This methodologyis strongly encouraged in [36].Fig. 14 represents all the masts and reinforced cables

added to the analysis at this stage. Spatial coordinatesfor the masts extreme nodes are displayed in Table 4, seealso Fig. 14.

The reinforcing cables are taken with EA = 1.2e4 kN,where Estands for the Youngs modulus and A symbolizesthe cross sectional area. The masts pinned at their respec-tive foundations are considered to have EA = 2.0e5 kN.The textile fabric is assumed to behave isotropically withEt = 5.0e2 kN/m and m = 0.3, where in this case t denotesthe membrane thickness.

The prestress process is carried out by means of animposed displacement on nodes plotted in Fig. 14. Table5 summarizes the applied displacements u, v and w alongthe corresponding space directions OX, OY and OZ,respectively.

9.2. Second load step: In-service loading

Once the structure is prestressed and stabilized, an in-service snow loading was considered. The snow loadapplied on the structural membrane will consist of1.0 kN/m2 distributed across a central region of the mem-

brane which extends up to 6.0 m2

. This whole surface is

accounted for according to its projection on a plan view.Figs. 1517 show the displacements contour diagrams forthe different loading conditions.

Again, the interior and perimeter cables act as stabiliz-

ing members for the overall performance of the membrane

0

2

4

6

4

2

02

4

0

2

4

6

8

10

12

OXax

is(m)

Initial shape

OYaxis(m)

OZaxis(m)

Fig. 13. E2: Initial configuration, isometric view.0 2 4 6 8 10

10

8

6

4

2

0

2

4

6

8

10

252

251

253

250

130

247

13

OX axis (m)

249

248

OYaxis(m)

Fig. 14. E1: Reinforced cables and masts configuration.

Table 4E2: spatial coordinates (m)

Node x y z

248 5.000 3.500 0.0000249 5.000 3.500 0.0000250 7.042 10.41 0.0000251 11.91 5.542 0.0000252 11.91 5.542 0.0000253 7.042 10.41 0.0000130 5.0 0.0 5.0

Table 5E2: boundary conditions (cm)

Node u v w

250 5.0 5.0 5.0251 5.0 5.0 5.0252 5.0 5.0 5.0253 5.0 5.0 5.0130 5.0 0.0 5.0248 0.0 0.0 0.0249 0.0 0.0 0.0OY axis 0.0



14/17

and therefore, they reduce the displacements that resultfrom the different applied loads. From the strain point of

view, a maximum value less than 1.0e 2 was achieved.

This fact agrees with the moderate strain requirement to justify the application of the Saint VenantKirchhoff

hyperelastic model.

Fig. 15. E2: OX displacements. (a) Prestress load and (b) snow load.

Fig. 16. E2: OY displacements. (a) Prestress load and (b) snow load.



15/17

Figs. 18 and 19 display the contour diagrams for theprincipal Cauchy stresses rI and rII, respectively. As it

can be noticed, at the prestress stage the whole membraneis under pure tension.

Fig. 17. E2: OZ displacements. (a) Prestress load and (b) snow load.

Fig. 18. E2: Cauchy stress rI. (a) Prestress load and (b) snow load.



16/17

10. Concluding remarks

This paper has presented a complete numerical frame-work for the analysis of prestressed Saint VenantKirch-

hoff hyperelastic membranes. The structural problem hasbeen split into two successive loading steps, in such away that a prestressed configuration is adopted to be asthe initial one. With respect to the former one, a TotalLagrangian weak form is detailed. Afterwards, a finite ele-ment linearization technique is employed over the equiva-lent internal nodal forces to derive the total tangentstiffness matrix. A straightforward computational imple-mentation is allowed by means of the Direct Core Congru-ential Formulation.

An alternative general approach in terms of the totalpotential energy principle, brings then into light a set of dif-ferent numerical algorithms based upon unconstrained

optimization techniques. Eventually, the robustness of thewhole formulation has been demonstrated by some numer-ical examples.

Acknowledgement

Special gratitude is addressed towards Professor J.Bonet, Head of the Civil and Computational EngineeringCentre at Swansea University, UK, for many helpful com-ments and discussions.

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Fig. 19. E2: Cauchy stress rII. (a) Prestress load and (b) snow load.



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