MM2 Numerical Theory MEMBRANE STRUCTURES

8/16/2019 MM2 Numerical Theory MEMBRANE STRUCTURES

1/81

MM2:Numerical Theory

Lecture NotesMarch 2010

Prof. Dr.-Ing. K.-U. Bletzinger

Dipl.-Ing.(FH) Falko Dieringer M.Sc.Dipl.-Ing. Johannes Linhard


2/81


3/81

MM2: Numerical Theory Index

Index 1 Introduction to Form Finding ................................................................................1 - 1

1.1 Introduction...........................................................................................................1 - 21.2 Statical problem of form finding .......................................................................... 1 - 31.3 Methods for numerical form finding ....................................................................1 - 7

1.3.1 Reduction of equations.....................................................................................1 - 71.3.2 Force density method and updated reference strategy.....................................1 - 81.3.3 Dynamic relaxation ......................................................................................... 1 – 9

2 Differential Geometry of Surfaces and Continuum Mechanical Basics.............2 - 12.1 Differential geometry............................................................................................2 - 1

2.1.1 Description of a point in space.........................................................................2 - 22.1.2 Description of a spatially curved surface in space ...........................................2 - 3

2.2 Continuum mechanical basics .............................................................................. 2 - 62.2.1 Deformation gradient F....................................................................................2 - 72.2.2 Nonlinear strain measures ................................................................................2 - 82.2.3 Stress measures ............................................................................................. 2 – 10

3 Plane Stress State and Principal Stresses .............................................................3 - 13.1 Plane stress state ................................................................................................... 3 - 2

3.1.1 Reduced stress tensor .......................................................................................3 - 33.1.2 Reduced strain tensor .......................................................................................3 - 43.1.3 Hookean law for the plane stress state.............................................................3 - 5

3.2 Principal stresses................................................................................................... 3 - 63.2.1 Principal stress determination .......................................................................... 3 - 63.2.2 Mohr’s circle of stress......................................................................................3 - 7

4 Numerical cutting pattern generation ...................................................................4 - 14.1 Partitioning of the structure into strips ................................................................. 4 - 2

4.1.1 Definition of cutting lines ................................................................................4 - 24.1.2 Geodesic line calculation during form finding ................................................ 4 - 34.1.3 Geodesic line calculation on a found form ......................................................4 - 3

4.2 Flattening and compensation of the strips ............................................................4 - 44.2.1 Simple triangulization technique .....................................................................4 - 54.2.2 Optimization techniques ..................................................................................4 - 5

4.2.3 Mechanical approach .......................................................................................4 - 6

5 Finite Element Method............................................................................................ 5 - 15.1 Linear elastic plane stress: Principle of virtual work in matrix notation ..............5 - 25.2 Short introduction to the Finite Element Method ................................................. 5 - 4

5.2.1 Simple 3- and 4-node isoparametric displacement elements ........................... 5 - 45.2.2 Convergence behavior.....................................................................................5 - 6

5.3 Numerical integration ...........................................................................................5 - 95.3.1 One Dimensional Rules ...................................................................................5 - 95.3.2 Two Dimensional Rules.................................................................................5 - 11

5.4 The assembly process .........................................................................................5 - 14

5.5 Virtual work of a surface stress field, non-linear formulation............................ 5 - 19


4/81

MM2: Numerical Theory Index

5.6 Material Law.......................................................................................................5 - 205.7 Internal and external virtual work.......................................................................5 - 215.8 Soap film analogy to find minimal surfaces ....................................................... 5 - 215.9 Discretization of the governing equations ..........................................................5 - 225.10 Linearization .......................................................................................................5 - 24

5.10.1 Membrane element.........................................................................................5 - 245.10.2 Cable element.................................................................................................5 - 26

5.11 Application: 3-node membrane element in 2D...................................................5 - 285.12 Application: 3-node membrane element in 3D...................................................5 - 325.13 Solution of non-linear finite element equations..................................................5 - 34

5.13.1 Newton-Raphson method...............................................................................5 - 345.13.2 Time stepping, path following, numerical continuation ............................... 5 - 34


5/81

MM2: Numerical Theory Literature

Literature

[1] Otto F. and Rasch, B., 1995. Finding Form. Deutscher Werkbund Bayern, Edition A.Menges.

[2] Berger, H., 1996. Light structures - structures of light, Basel: Birkhäuser.

[3] Holzapfel G, 2000. Nonlinear Solid Mechanics. Chichester: Wiley

[4] Onate E. and Kröplin B. (Eds.), 2005. Textile Composites and Inflatable Structures.Wien, Springer.

[5] Forster B. and Mollaert M. (Eds.), 2004. European Design Guide for Tensile SurfaceStructures. Tensinet.

[6] Koch, K.-M. (ed.), 2004. Membrane Structures. München, Prestel.

[7] Carmo, M., 1976. Differential geometry of curves and surfaces. Englewood Cliffs,

Prentice-Hall.


6/81


7/81

1 Introduction to Form Finding


8/81

MM2: Numerical Theory Introduction to Form Finding

1.1 IntroductionThe main characteristic of membrane or cable net structures is their slenderness: Onedimension of a membrane structure is considerably smaller than the other ones. Thesestructures mainly extend to two dimensions and are therefore classified as surface structures

(in contrast to continua, where all the dimensions are of the same magnitude). For a cableeven two dimensions are smaller than the third. They are therefore consequently called linestructures.

This slenderness of these lightweight structures is also the reason for their characteristicmechanical behavior: Membranes and cables are only capable of withstanding tensile forces,while if they are subjected to compression forces, they lose their stability: If compressed, themembrane tries to withdraw itself from carrying the load by wrinkling, a cable will buckle.This can easily be illustrated with a simple experiment: If a piece of thread is pulled, one hasto pull the stronger the more the thread is stretched (for linearly elastic material the force andthe stretch are proportional to each other and linked over the Young’s modulus). In contrast tothat, almost no force at all has to be applied in order to compress it (Fig. 1.1).

load

displacement

F > 0(tension)

cable

F < 0(compression)

Fig. 1.1: Mechanical behavior of a cable under tension/compression

At the very least, a possible loss of stability through compression forces is only local and justaffects the visual appearance in a negative way. But it can also have worse effects: Because of

bending and frequent changes of curvature the outer layers of a wrinkled membrane aresubjected to higher strains than usual. Their mechanical properties may suffer, if they areexposed to this situation for a longer period of time. The worst case is a global system failure,if the whole structure loses its stability: This is especially dangerous for statically determinatestructures, where the failure of one element is enough to make the whole system collapse.

It is therefore obvious that this obliges the designer to take special care in determining the“right” form of the structure: A form has to be found, which is not only capable ofwithstanding the significant load cases under certain restrictions such as maximum alloweddeflection and stress, but also shows an even usage of the material: This leads to aminimization of necessary building material, which has the nice side effect that it leads to areduction of building costs as well.

For a membrane structure made of an isotropic material (like foils) the optimum shape isreached, when the stress distribution over the whole structure is uniform. For an orthotropicmaterial it may be advantageous to match the prestress with the different material propertiesfor each fiber direction.

1 - 2


9/81


This “form finding” can be done in various ways: Before the large availability ofcomputational calculation power, mechanical models - such as soap films or stretched tissues- were used to find a form, which is the equilibrium shape of a certain stress distribution in thestructure with external loads. Nowadays the numerical simulation has overcome theseapproaches, since the increasing complexity of the structures, although mechanical

simulations are still sometimes used for verification of the numerical calculations.

In the process of form finding the resulting stresses under a given load case (mainlymechanical prestress, but also external loads like pressure) are prescribed. Now the stillunknown final shape, on which these stresses act, has to be determined. This is doneconsidering large deformations, so that the static and mechanical formulations become fullynonlinear. In this process the numerical calculation needs to be stabilized as can be seen lateron. Until now no material parameters were needed: The presence of the prescribed stresses isassumed, no matter how they were generated. The material properties come into play, whenthe reference geometry is needed. For membrane structures this reference geometry are thecutting patterns. They have to be cut out in such a way, so that when they are stretched in the

boundary conditions, the resulting stress state is as close as possible to the desired one.

It can easily be seen that the form finding procedure is of reversed order compared to a“normal” static analysis: Here, the reference geometry (e.g. the dimensions of a concrete bar)and the external loads (self-weight, wind, snow, etc.), for which the static analysis shall becarried out, are given. With the help of the governing equations of the material, the kinematicsand the equilibrium it is possible to determine the deformed geometry. In a last step, thestresses inside the structure can be calculated with the usage of the material law.

Because of this reversed order form finding is often referred to as an “inverse process”.

external loads,reference geometry

deformedgeometry

stresses

stresses,external loads

referencegeometry

material, kinematicsequilibrium

material

deformedgeometry

kinematicsequilibrium

material

Form findi ng Patterning

Static analysis

Fig. 1.2: Comparison: Static analysis – Form finding

1.2 Static problem of form findingThe final shape of the form finding process is the equilibrium shape for a given load case,which is in general only mechanical prestress. The governing equations for the calculationand their characteristic features are presented exemplarily for a simple plane cable structure,

but they can be expanded for more complicated, three-dimensional cable net and membranestructures in the same manner.

1 - 3


10/81


In our example, a plane weightless cable with only mechanical prestress is given. Two rigidsupports, which are separated by the distance L, keep the cable fixed at the edges. Thediscretization of the cable is approximately done with two linear elements and oneintermediate nodal point K. The displacements of K in the horizontal (x) and vertical (z)direction are the only degrees of freedom in this system. It is quite obvious that the cable must

be straight to achieve equilibrium.

l 2 l 2 l 2 l 2L

K

x

z

x1 = x x 2 = L - x

l 1 l 2

Fig. 1.3: Plane prestressed cable in reference configuration

Formally the equilibrium condition states that the sum of horizontal as well as vertical forcesacting on node K must vanish. The results are two nonlinear equations for the two unknownvariables x and z:

z z

x2x1

S 1 S 2

( )0S

zxL

xLS

zx

xS

xS

x:0H 0

220222

2

21

1

1 =+−

−−+

=−=∑ll

(1.1)

( )0S

zxL

zS

zx

zS

zS

z:0V 0

22

2

022

2

22

11

=+−

++

=+=∑ll

(1.2)

S1 and S 2 stand for the respective cable forces, which are assumed to be constant for everyconfiguration and equal to the desired prestress force S 0. The cable forces can be calculatedfrom the cross section area A and the prestress σ0.

0021 ASSS σ=== (1.3)

It is easy to see that the second equation is only fulfilled, if the vertical displacement z = 0,which is in accordance with our first assumption. Now we have only the first equation left todetermine the horizontal displacement x. Using our result z = 0 we obtain:

( )( ) 0S11S

xL

xLS

x

x00

20

2

=−=−

−− (1.4)

1 - 4


11/81


This equation is always fulfilled, that means it is independent of x. Therefore we can’t use itto determine the horizontal position of the node K. Mechanically this can be interpreted thatevery position of the node K is allowed, as long as the two cable elements form a straight line.This means that there is not a unique solution for the form finding problem.

K…

K K

Fig. 1.4: Suitable solutions

If we expand our problem type now to three-dimensional surface structures (like membranesor cable nets), we observe the same phenomenon: We can’t determine a unique position of thenodes (nodes of the cables net or nodes of a FE discretization) on the equilibrium surface withonly the help of the equilibrium conditions: The nodes can “float” on the surface and still theydescribe the same geometry, only out-of-plane movements are not allowed. Out of theoriginally three equations only one equation remains to determine the three coordinates ofeach point. Therefore we need to modify the equation system, such that a unique solution can

be found.

3D view top view

Fig. 1.5: Suitable solutions for a hypar like shape (four point tent)But before that, a little rewriting of the governing equations needs to be done: Theequilibrium equations are identified as the derivatives of the virtual work of the whole systemw.r.t. to the various degrees of freedom.

0xWx

Ax

A:0H 022

2202

1

11 =∂

∂=σ−σ=∑l

ll

l (1.5)

0z

WzA

zA:0V 02

2

202

1

1 =∂∂=σ+σ=∑

ll

ll (1.6)

1 - 5


12/81


Both equations can now be combined to one equation for the virtual work δW:

zz

Az

Axx

Ax

Azz

Wx

xW

W 022

2021

1022

2202

1

11 δ⎟⎟ ⎠

⎞⎜⎜

⎝

⎛ σ+σ+δ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ σ−σ=δ

∂∂+δ

∂∂=δ

ll

ll

ll

ll (1.7)

The equation above states that in a state of equilibrium no work is done, if the systemundergoes virtual (infinitesimal small) displacements.

Furthermore all the terms of the virtual work are sorted in such a way, that the correspondingterms of each cable element are combined:

0d Ad AAA

zz

xx

Azz

xx

AWWW

21

02ea01ea02ea201ea1

022

22

2202

121

1121

=σδε+σδε=σδε+σδε=

=σ⎟⎟

⎠

⎞⎜⎜

⎝

⎛ δ+δ−+σ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ δ+δ=δ+δ=δ

∫∫ll

llll

lll

lll

(1.8)

In this representation all quantities refer only to the current configuration (and not to thereference or any other configuration): l i are the actual cable lengths, δεea i are the so calledvirtual “Euler-Almansi” strains and σ0 is the “Cauchy” stress. The Euler-Almansi strains andthe Cauchy stresses form an energetically conjugated stress-strain couple and are bothreferring to the actual configuration.

Since we have started our calculation from a fixed reference configuration, where the lengthsof the cables were L i, we can again rewrite the equilibrium equations, so that these initiallengths appear:

0xWL

L

xLA

L

L

xLA:0H 0

2

222

220

1

121

11 =∂

∂=⎟⎟

⎠

⎞⎜⎜

⎝

⎛ σ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −⎟⎟

⎠

⎞⎜⎜

⎝

⎛ σ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ =∑

ll (1.9)

0z

WL

L

zLA

L

L

zLA:0V 0

2

222

201

121

1 =∂∂=

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ σ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +⎟⎟

⎠

⎞⎜⎜

⎝

⎛ σ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ =∑

ll (1.10)

Now the virtual work equation is rewritten in the same manner:

0dLAdLALALA

LzLzx

LxLALz

Lzx

LxLAW

21 L2 pk 2gl

L1 pk 1gl2 pk 2gl21 pk 1gl1

02

222

22

220

1

121

21

11

=σδε+σδε=σδε+σδε=

=⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛ σ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ δ+δ−+⎟⎟ ⎠

⎞⎜⎜⎝ ⎛ σ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ δ+δ=δ

∫∫ll

(1.11)

Here all quantities (as well as the integrals) refer to the initial reference configuration: Theenergetically conjugated stress-strain couple in the reference configuration consists of thevirtual “Green-Lagrange” strains δεgl i and the “2 nd Piola-Kirchhoff“ stresses σ pk i .

1 - 6


13/81


Both the stress and strain measures in the reference as well as in the actual configuration arewell known in nonlinear continuum mechanics and serve as a foundation for the largedeformation analysis.

While the Cauchy stresses are real physical stresses, the 2 nd Piola-Kirchhoff stresses are

quantities, which are only used for calculation, but don’t have any direct physical meaning.They can be transformed into Cauchy stresses over the ratio of reference to actual length:

0i

ii pk

L σ=σl

If there is a difference between the cable lengths of the reference and actual configuration,there is also a difference between the Cauchy and 2 nd Piola-Kirchhoff stresses. For smalldeformations (1 st order theory) the ratio L i/li is negligible and therefore the difference betweenCauchy and 2 nd Piola-Kirchhoff stresses is often omitted. But as we are dealing with largedeformations in the form finding procedure, this difference has to be taken into account.

1.3 Methods for numerical form findingSince it is not possible to determine a unique solution for the position of each node purely outof the equilibrium conditions, several different strategies have been developed to overcomethis rank deficiency.

1.3.1 Reduction of equationsAs seen in the previous example the three equilibrium conditions provide us only one usefulequation for the calculation of the nodal positions regarding the out-of-plane movements, butnot for the tangential movement. Some strategies make use of this and imply the restrictionthat a node is allowed to move only perpendicular to the surface. By that the equation systemis diminished and can be solved, although it is still non-linear and needs therefore to be solvediteratively. After convergence, the equilibrium regarding the other two independent spatialdirections should be checked.

initial mesh overlappingedge cable - surface

tangential adjustmentof the surface mesh

Fig. 1.6: Tangential adjustment of the surface mesh

1 - 7


14/81


This simple method works fine with fixed boundaries, while it is not suitable for structureswith edge cables: In general cables, which form the boundary of the membrane, have to movealso tangentially to the surface in the form finding process in order to find an equilibrium

position. If on the other hand the membrane is only allowed to move perpendicular to thesurface, an overlapping between the edge cables and the surface mesh will occur, which

makes a tangential adjustment of the surface mesh necessary.

1.3.2 Force density method and updated reference strategyThis method is one of the earliest methods for numerical form finding. It has been originallydeveloped for the calculation of the roof of the Olympic stadium in Munich 1972. Although itwas only applicable for cable nets, it is still widely used for form finding of membranestructures: The particular membrane is then approximated by a cable net.

The basic idea behind the force density method is the assumption that the ratio of the force ina cable S i to its length l i – the so called force density q i = S i/l i - is constant throughout theform finding process. By that the system of equations is modified and all equations becomelinear and can be solved in one step.

If we recall the first example, the governing equations are now:

0xq xq 2211 =− (1.12)

0zq zq 21 =+ (1.13)

The force density is typically evaluated in the reference configuration for a given cable forceand length. Now for every other configuration the resulting stresses in the cable can bedetermined by the following equation:

i

i

i

i

i

iii LA

SA

q ll ==σ (1.14)

If the actual length of the cable l i differs from the reference length L i, the actual stress statediffers from the desired stress state as well. Therefore a new iteration step has to made inwhich the resulting configuration of the previous step is the reference configuration of theactual step.

Formally the assumption of constant force densities coincides with the assumption of constant2nd Piola-Kirchhoff stresses during the form finding process.

0xL

Ax

L

A2

2

2 pk 1

1

1 pk =σ

−σ

(1.15)

0zL

Az

L

A

2

2 pk

1

1 pk =σ

+σ

(1.16)

1 - 8


15/81


With that knowledge, the force density method can also be applied to membranes. Since it isnecessary to update the reference (for evaluating new force densities or 2 nd Piola-Kirchhoffstresses) in each iteration step, the generalized method is called updated reference strategy.

1.3.3 Dynamic relaxationAnother approach to overcome the indeterminacy of the tangential positions of the nodes isapplied in the dynamic relaxation method, which is physically based on the second Newtonianlaw of motion: The structure is modeled by nodes with concentrated masses on which notonly static forces, but also inertia and damping forces act. The nodes are linked by discretefinite elements representing membranes and cables. The equilibrium shape is seen as the rest

position of a fictive damped oscillation of the system.

For our example the static equations are expanded with inertia and damping terms (resultingfrom d’Alembert’s principle). In the time step t n they can be written as follows:

0xSxSvCam n22

2n1

1

1nx

nx =−++ ll

(1.17)

0zS

zS

vCam n

2

2n

1

1nz

nz =+++ ll

(1.18)

For the mass m and the damping coefficient C suitable assumptions are made. This is possible, since we don’t want to analyze the real dynamical behavior of the system. Theaccelerations a i and the velocities v i are approximated with a finite difference scheme for afixed time increment ∆t:

( )

( )n1n21nz

n1n21nx

zzt

1v

xxt

1v

−∆

=

−∆

=

++

++

(1.19)

( )

( )21ni21nini

21ni

21ni

ni

vv21

v

vvt

1a

−+

−+

+=

−∆

= (1.20)

After some manipulation of the equations, we get the following recursive formulas for thevelocities v x and v z:

4 4 4 4 34 4 4 4 21lll

nxR

2

2n

2

2

1

121nx

21nx L

Sx

SStCm2

t2v

tCm2tCm2

v ⎥⎦

⎤

⎢⎣

⎡−

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +

∆+∆−

∆+∆−= −+ (1.21)

1 - 9


16/81


4 4 34 4 21llnzR

n

2

2

1

121nz

21nz z

SStCm2

t2v

tCm2tCm2

v ⎥⎦

⎤

⎢⎣

⎡

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +

∆+∆−

∆+∆−= −+ (1.22)

Finally the position of the node in the next time step can be calculated by extrapolation:

mR

2t

vmitvtxx0x21

x21n

xn1n ∆−=∆+= ++ (1.23)

This scheme can be applied to bigger systems in a similar way: The simple recursive formulasstill persist and only relatively small equation systems have to be solved. The disadvantage ofthis approach is the dependency of the numerical stability to the choice of mass, dampingcoefficient and incremental time step: For example, the incremental time step has to besmaller than its critical value, which is the inverse of the eigenfrequency of the undampedsystem:

maxkritK m2tt π=∆


17/81

2 Differential Geometry of Surfacesand Continuum Mechanical Basics


18/81

MM2: Numerical Theory Differential Geometry and Continuum Mechanical Basics

2.1 Differential geometry

2.1.1 Description of a point in spaceIn order to uniquely describe the position of a point in space, we need to introduce a fixedreference frame: This is usually a global Cartesian coordinate system , which consists of three

base vectors ei (i = 1,2,3) of unit length which are orthogonal to each other: The scalar product of two different base vectors is 0, while it is 1 between the same base vectors.

(2.1)⎩⎨⎧

=≠

=⋅ ji, 1

ji, 0 ji ee

This type of basis is called orthonormal basis .

Another requirement is the orientation: The base vectors have to be orientated in such a waythat they form a right handed coordinate system: If the thumb of your right hand points in thedirection of the first base vector and your index finger in the direction of the second, thedirection of the third base vector is described by your middle finger. In a mathematical waythis can be written as:

213 eee ×= (2.2)

Now the position vector x of a point P (pointing from the origin of the coordinate system tothe point P) can be uniquely described through a linear combination of multiples of these basevectors:

(2.3)ii3

1ii

i3

32

21

1 xxxxx eeeeex ==++= ∑=

x

e 2e 1

e 3

x1e 1

x2e 2

x3e 3

P

Fig. 2.1: Position vector x of a point P

The three scalars x i are the global Cartesian coordinates of the respective point (it is importantto know that coordinates are only meaningful in a combination with the corresponding

basis!).

2 - 2


19/81


In the equation above Einstein’s summation convention was used: If an index is repeated(only once) in the same term, it is called a dummy index and a summation over the range ofthis index is implied (unless otherwise indicated). Note, that Latin dummy indices usually runfrom 1 to 3, while Greek run from 1 to 2.

2.1.2 Description of a spatially curved surface in spaceOne possibility to describe a spatially curved surface, which is still a two-dimensional entityalthough it lies in the three-dimensional space, is the so called parametrical description: Toevery point P on the surface two independent surface coordinates or surface parameters θ1 and θ2 are assigned. The position vector x, which points to the point P, is therefore a functionof these two surface parameters:

( )21 ,θθ= xx (2.4)

θ1θ2

x(θ1,θ2)

g 2 g 1

e 2e 1

e 3

P( θ1,θ2)

Fig. 2.2: Parametric description of a spatially curved surface

If one parameter is kept constant and only the other one is varied, all points P, which fulfillthis restriction, form a coordinate line of the varied coordinate. These lines are generallyspatially curved.

In order to describe the local behavior of the surface at each point, a local reference frame isneeded: The so called covariant base vectors g1 and g2 are defined by differentiation of x with

respect to θ1

and θ2

:

2211 ;

θ∂∂=

θ∂∂= xgxg (2.5)

The covariant base vectors are tangential to the corresponding coordinate lines, e.g. g1 istangential to the coordinate line, which is generated by varying θ1 and keeping θ2 constant.The tangential plane of the surface at point P is spanned up by g1 and g2. The surface normalis determined by g3 which is defined by the normalized cross product of g1 and g2:

1 ;321

21

3 =

×

×= ggg

ggg (2.6)

2 - 3


20/81


θ1

θ2g 2

g 1

g 3

ϕ

Fig. 2.3: Surface normal direction g3

The covariant base vectors, which span up the tangential plane, are in general not orthogonaland not of unit length. The θ1 and θ2 are consequently called curvilinear coordinates.

The scalar products g ij of the covariant base vectors – the components of the covariant metrictensor – reflect the metric of the surface, i.e. the length of the covariant base vectors and theangle between them:ϕ

( )2121212112

222222

211111

,cosgg

g

g

gggggg

ggg

ggg

ϕ=⋅===⋅=

=⋅=

(2.7)

The three covariant base vectors gi form the local covariant basis . A local contravariant basis – consisting of three contravariant base vectors gi – is defined by the rule

(2.8)⎩⎨

⎧

≠=

=δ=⋅ ji, 0 ji, 1i

j j

i

gg

where is the Kronecker delta: If the indices of the Kronecker delta are identical, its value is

1, if they are different, its value is 0.

i jδ

θ1

θ2g 2

g 1

g 1

g 2

ϕ

Fig. 2.4: Covariant and contravariant base vectors

This means that e.g. g1 is orthogonal to g2 (vice versa g2 is orthogonal to g1). The scalar products and are 1. The contravariant base vectors g1

1 gg ⋅ 22 gg ⋅ 1 and g2 span up the same

plane (the tangential plane) as the covariant base vectors g1 and g2.

2 - 4


21/81


Alternatively, the contravariant base vectors are defined as the derivative of the surfacecoordinates with respect to the geometry of the surface:

xg

xg

∂θ∂=

∂θ∂=

22

11 ; (2.9)

Since g1 and g2 are orthogonal to g3 it follows that and33 gg = 13 =g .

It can be proven that the contravariant metric tensor is the inverse of the covariant metrictensor and vice versa:

(2.10)( )1

2221

12111ij2221

1211ij

gg

ggg

gg

ggg

−−

⎥⎦

⎤⎢⎣

⎡==⎥⎦

⎤⎢⎣

⎡=

Co- and contravariant base vectors can be transformed into each by use of the metric tensors:

j (2.11)iji j

iji g ;g gggg ==

2.1.2.1 Differential line elementThe differential line element d s is linking the point P on the surface with the position vectorx(θ1,θ2) with another point in the vicinity of P with the position vector x(θ1+dθ1,θ2+dθ2). Itcan be calculated as follows:

22

11

22

11 dθdθdθθdθθd gg

xxs +=∂

∂+∂

∂= (2.12)

The differential length ds can be obtained by evaluating the length of the vector d s:

( ) ( )222221122111 dθgdθdθg2dθgddds ++=⋅= ss (2.13)

2.1.2.2 Differential area elementThe differential piece of area da is defined as the area of the vector parallelogram which isgiven by the vectors g1dθ1 and g2dθ2:

( )( ) ( ) 21ij21321

2121

22

11

22

11

dθdθgdetdθdθ

dθdθdθ,dθsindθdθda

=⋅×=

×=ϕ=

ggg

gggggg (2.14)

The total area of a given surface can be expressed in terms of the surface coordinates as

∫ ∫∫ ∫θ θθ θ

θθ×=θθ=1 21 2

2121

21 dddd ja gg (2.15)

where j is the determinant of the Jacobian matrix .

2 - 5


22/81


θ1θ2

x(θ1,θ2)

g 2dθ2 g 1dθ1

P( θ1,θ2)

x(θ1+d θ1,θ2+dθ2)

ds ϕ

Fig. 2.5: Differential line and area

In the following we will describe any vector or tensor as the weighted sum of the co- orcontravariant base vectors. E.g. the vector a or the second order tensor T may be written interms of either gi or gi.

(2.16) jiij jiiji

iii TT ;aa ggggTgga ⊗=⊗===

The normal letters represent the coefficient of a tensor, the bold faced letters thecorresponding basis. is the tensor product.⊗

If a tensor is written in matrix notation, it implies that the corresponding basis is a Cartesian basis. The entries of the matrix are then the tensor coefficients.

E.g., the second order unit tensor I is defined in terms of the co- or contravariant base vectorsas

(2.17) jiij jiij gg ggggI ⊗=⊗=

2.2 Continuum mechanical basicsIn continuum mechanics we want to observe how a body deforms during a certain process:

Starting from a known reference configuration , we trace each of the material points of the body throughout the deformation process until the current or actual configuration . By that weare able to calculate deformations, stretches, etc.

This material description is often referred to as Lagrangian description . The so called Eulerian description on the other hand doesn’t pay attention to a material point, but to a fixed point in space: We look at this specific point and study what happens there during thedeformation process (e.g. we observe the velocities of the particles passing that point). Thisapproach is widely used in fluid mechanics.

2 - 6


23/81


2.2.1 Deformation gradient FWe now look at a surface (membrane) which is parametrized with the two convective surfacecoordinates θ1 and θ2. The coordinate lines are quasi carved in the surface - the

parameterization doesn’t change during the deformation process. As a simplification weassume that the thickness of the membrane stays constant throughout the deformation.

We adopt the convention: All quantities indicated with a capital letter refer to the initial orreference configuration, quantities with a lower case letter to the current configuration:

x(θ1, θ2)

e 2 e 1

e 3 θ1

θ2

g 2

g 1

θ1

θ2

G 2 G 1

X(θ1, θ2)

u (θ1, θ2)

reference configuration

current configuration

deformation

P

P

Fig. 2.6: Deformation of a surface

A point P on the actual surface with position x(θ1,θ2) had originally the position X(θ1,θ2) onthe reference surface. The point is identically identified by its surface coordinates θ1, θ2 whichstay invariant during deformation. The displacement u of point P is defined as:

( ) ( ) ( )212121 ,,, θθ−θθ=θθ Xxu (2.18)The definition of the base vectors G i and G i is as before and stems from the differentiation ofX with respect to the surface coordinates.

The reference configuration is transformed into the actual configuration by the deformationgradient F . It can be calculated by deriving the actual geometry with respect to the referencegeometry.

iiTi

i ; gGFGgXx

F ⊗=⊗=∂∂= (2.19)

The inverse of the deformation gradient F transforms the actual into the reference geometry:

2 - 7


24/81


iiTi

i1 ; GgFgG

xX

F ⊗=⊗=∂∂= −− (2.20)

E.g., base vectors are transformed from one configuration to the other one as follows:

(2.21)

i ji j

i j

jiTi

i ji j

i j

jiTi

i j jii

j ji

1i

i j jii j jii

GGggGgFG

ggGGgGFg

GGggGgFG

ggGGgGFg

=δ=⋅⊗=⋅=

=δ=⋅⊗=⋅=

=δ=⋅⊗=⋅==δ=⋅⊗=⋅=

−

−

A differential piece of area transforms by

2121

21

2121

21

dddetdJddetdAdet

ddd jdda

θθ×=θθ==

θθ×=θθ=

GGFFF

gg (2.22)

21

21

J j

detdAda

GG

ggF

××

=== (2.23)

The surface area of the actual configuration can now be written with respect to the referenceconfiguration:

∫∫∫ ∫∫ ==θθ==θ θ AA

21

a

dAdetdAJ j

d jddaa1 2

F (2.24)

2.2.2 Nonlinear strain measuresThe stiffness of membrane structures is relatively low compared with conventional structures(e.g. structures made of concrete or steel). Thus large deformations can be observed when thestructure is loaded. In order to simulate this mechanical behavior of a membrane in a correctway, nonlinear strain measures have to be applied: Their main feature is that rigid bodymovements mustn’t lead to any strains.

One strain measure fulfilling all these conditions is the Green -Lagrange strain tensor E :

( ) ( ) jiijijT Gg21

21

GGIFFE ⊗−=−= (2.25)

It can be calculated with the deformation gradient or with the difference of the metriccoefficients. In order to get physical strains the usually contravariant basis has to betransformed into a Cartesian basis: The corresponding coefficients represent the strains.

ExampleWe look at a straight cable linking the points P 1 and P 2. The cable is parametrized with onecoordinate θ1 ranging from 0 (P 1) to 1 (P 2). The initial cable length is L, the deformed cable

length is l .

2 - 8


25/81


e 2 e 1

e 3

θ1

g 1

G 1

X1=X(θ1=0)

reference configuration

current configuration

deformation

X2=X(θ1=1)x1=x(θ1=0)

x2=x(θ1=1)

P 1

P 2

P 1

P 2

θ1

Fig. 2.7: Example: Strain calculation at a cable

The position vectors of the cable points in the reference and actual configuration are:

( ) ( ) 211112

11

11

1

1

xxx

XXX

θ+θ−=θθ+θ−=θ

(2.26)

Since a cable is only a one-dimensional entity, we have only one meaningful covariant basevector (the other two are perpendicular to the first, but their direction is not uniquely defined):

1211

1211

xxx

g

XXX

G

−=θ∂∂=

−=θ∂

∂= (2.27)

The covariant metric coefficients are:

221211

221211

g

LG

l=−=

=−=

xx

XX

It can be seen that the length of the covariant base vector is the length of the cable of thecorresponding configuration.

The contravariant metric coefficients can be calculated from the inverse of the covariantmetric tensor (which reduces to a scalar for this example):

2 - 9


26/81


211

211

1g

L

1G

l=

= (2.28)

A local Cartesian basis ie~ is introduced in the reference configuration, whose first base vectoris parallel to the cable. The contravariant base vectors can therefore be written as:

( ) 1121221111 ~

L1~L

L

1

L

1G eeXXGG ==−== (2.29)

Now we can evaluate the Green-Lagrange strain tensor:

( ) ( ) 11222

1122111111

~~

L

L21

L21

Gg21

eeGGGGE ⊗−=⊗−=⊗−= l

l (2.30)

The coefficients of the Green-Lagrange strain tensor, whose basis is Cartesian, represent the physical strains: In this example we have only one strain component (parallel to the cable

axis), which is constant over the cable length and has the value2

22

L

L21 −l

.

2.2.3 Stress measures

2.2.3.1 Cauchy stress tensor

In general when we talk about physical stresses, we mean the Cauchy stress tensor withoutactually knowing it: The Cauchy stress tensor “lives” in actual configuration: Its natural basisis the covariant basis in the actual configuration (and therefore components are contravariant):

(2.31) jiij ggσ ⊗σ=

Using the Cauchy stress tensor we can calculate the force d t acting on a differential piece ofarea da on the boundary of a body with the normal vector n as follows:

dad nσt ⋅= (2.32)

2 - 10


27/81


F

reference configuration current configuration

F -1

n

dA daN

dT=P N dA=d t

dt=

n da

dT’=S N dA= F -1dt

Fig. 2.8: Various stress measures

2.2.3.2 1 st Piola-Kirchhoff stress tensor P

It is often helpful to have a stress measure which is based not in the actual, but in thereference geometry. A first step is the 1st Piola-Kirchhoff stress tensor P . Using P you cancalculate the force d T acting on a differential piece of area dA with the normal vector N:

dAd NPT ⋅= (2.33)

This force d T has the same magnitude and direction as the force d t .

The 1 st Piola-Kirchhoff stress tensor can be obtained by a transformation of the Cauchy stresstensor:

(2.34) jiijT

Pdet GgFF σP ⊗=⋅= −

It can be seen that P lives with its “first leg” in the actual configuration, while it lives with its“second leg” in the reference configuration. It is therefore called a two point tensor . The 1 st Piola-Kirchhoff stress tensor is in general not symmetric, which can be a disadvantage.Because of that we introduce a symmetrical stress tensor in the reference configuration:

2.2.3.3 2 nd Piola-Kirchhoff stress tensor SThe 2nd Piola-Kirchhoff stress tensor S is generated by a pull-back operation of the Cauchystress tensor:

(2.35) jiijT1 SdetS GGFσFF ⊗=⋅⋅= −−

The 2 nd Piola-Kirchhoff stress tensor is in contrast to the 1 st Piola-Kirchhoff stress tensor asymmetrical tensor.

Using S a force d T’ acting on a differential piece of area dA with the normal vector N can becalculated:

(2.36)tFNST ddA'd 1 ⋅=⋅= −

2 - 11


28/81


Note, that the force d T’ in general hasn’t the same magnitude and direction as d t , althoughthey are linked by the deformation gradient.

2.2.3.4 Summary of the transformation rules

The transformation rules of the various stress measures are:

T11 ji

ij

T ji

ij

TT ji

ij

detS

detP

det1

det1

−−−

−

⋅⋅=⋅=⊗=

⋅=⋅=⊗=

⋅⋅=⋅=⊗σ=

FσFFPFGGS

SFFF σGgP

FSFF

FPF

ggσ

(2.37)

The coefficients transform as follows:

ijijij

ijijij

ijijij

detPS

SdetP

Sdet

1P

det1

σ===σ=

==σ

F

F

FF

(2.38)

2 - 12


29/81

3 Plane Stress State andPrincipal Stresses


30/81

MM2: Numerical Theory Plane Stress State and Principal Stresses

3.1 Plane stress stateThe mechanical model of a membrane is typically a two dimensional surface, nevertheless thereal membrane is of course a three dimensional body: The main reason for this reduction ofone dimension is to reduce the number of degrees of freedom and thus simplify the model.

Whenever a structural model with less dimensions than the real structural element is used, onehas to know the mechanical behavior in the dimension which is not included in the model!This information is then used to construct a two dimensional constitutive model of anoriginally three dimensional body.

The usual assumptions for a 2D membrane element are:

• Thickness is much smaller than the other two dimensions of the membrane

• Stresses are constant over the thickness (only normal forces as stress resultants, no bending moments)

• No stresses perpendicular to the midplane of the membrane (simulation of forcetransmission is not possible)

• The membrane can freely deform itself in thickness direction

All these assumptions can be summarized as plane stress state .

reference geometry deformed geometrymidplane

Fig. 3.1: Plane stress state of a differential membrane element

The opposite of the plane stress state is the plane strain state : In this case the strains perpendicular to the midplane are zero instead of the stresses. This model is e.g. applicable tostructures, which are much longer in one direction than in the other two and which arecontinuously loaded and supported perpendicular to this direction (e.g. a dam). It is usuallynot necessary to analyze the whole structure, but sufficient to cut out a two dimensionalsegment, as the strains along the longitudinal axis have to be zero.

3 - 2


31/81


plane stress: membrane plane strain: dam

1

3

2 132

Fig. 3.2: Comparison: plane stress - plane strain

3.1.1 Reduced stress tensor

We now look at a point P in the midplane of a membrane: At this particular point weintroduce a local Cartesian coordinate system in such a way that the third base vector e3 is perpendicular to the midplane ( e1 and e2 lie in the midplane).

Pe 2

e 3e 1

σ11

σ11

σ22

σ22

σ21

σ21

σ12

σ12

Fig. 3.3: Definition of local Cartesian coordinate system and positive stress components

The stress state at this point P is defined by the stress tensor σ .

(3.1)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

σσσσσσσσσ

=⊗σ=333231

232221

131211

jiij eeσ

The result of a post multiplication of the stress tensor σ with a unit vector n is a tractionvector t acting on a cutting plane with the normal vector n :

nσt ⋅= (3.2)

The traction vector t can be additively decomposed into a component which is perpendicularto the cutting plane (thus parallel to the normal vector n ) and a parallel component:

||ttt += ⊥ (3.3)

3 - 3


32/81


The length of is the normal stress σ, the length of the shear stress τ (note that stressesare scalar quantities).

⊥t ||t

Another possibility to evaluate the normal stress σ is to project the traction vector t on thenormal vector n .

( )nttt

nnσntt

⋅−==τ⋅⋅=⋅==σ ⊥

||

(3.4)

The tensor coefficients σij directly represent the physical stresses, if a Cartesian basis wasused for representation. The first index i indicates the direction of the stress, the second index

j the normal vector of the cutting plane on which the stress acts (note that this convention mayvary along different authors!). Two coinciding indices indicate normal stress, two differentindices shear stress.

As we assume to have a plane stress state, all stress components in the third direction vanish:

(3.5)0σσσσσ 3332233113 =====

We additionally know that the stress tensor has to be symmetric to guarantee equilibrium ofmomentum:

(3.6)2112 σ=σ

Thus only three significant stress components are left which can be written in a vector insteadof a matrix, which is widely known as Voigt notation .

(3.7)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

σσσ

=12

22

11

σ

3.1.2 Reduced strain tensorA similar procedure is done for the strain tensor:

The full strain tensor at the point P is:

(3.8)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

εεεεεεεεε

=⊗ε=333231

232221

131211

jiij eeε

No shear deformations perpendicular to the midplane occur:

03113 =ε=ε (3.9)

Again, the strain tensor has to be symmetric:

3 - 4


33/81


2112 ε=ε (3.10)

We now have four strain components left: The three components, which correspond with thecomponents of the reduced stress tensor, are again written in a vector:

(3.11)⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

εεε

=12

22

11

2

ε

Note the factor 2 in front of component ε12 describing the shear deformation. This is due tothe convention of the engineering community to work with the shear angle γ instead of thetensor components ε12!

122ε=γ (3.12)

The strain component ε33 describing the thickness change is a function of the components ofthe reduced strain tensor and the material. It is therefore not an independent parameter.

3.1.3 Hookean law for the plane stress stateThe stresses in a body are linked to the strains over a constitutive model. For the simplest caseof an isotropic, linear elastic material this relation can by written as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

εε

ε

⎥⎥⎥

⎥

⎦

⎤

⎢⎢⎢

⎢

⎣

⎡

ν−ν

ν

ν−=⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

σσ

σ=

12

22

11

212

22

11

22

100

01

01

1E

εCσ

(3.13)

C is the so called constitutive or elasticity matrix (in general it is a fourth-order elasticitytensor ). E represents the Young’s modulus, ν the Poisson’s ratio.

The inverse relationship is as follows:

( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

σσσ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ν+ν−

ν−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

εεε

= −

12

22

11

12

2211

1

1200

01

01

E1

2

σCε

(3.14)

The strain in the thickness direction can be determined as:

( ) ( 2211221133 1E ε+εν− )ν−=σ+σν−=ε (3.15)

3 - 5


34/81


3.2 Principal stressesIn the previous chapters we have seen that the magnitude of the normal and shear stressesdepends on the considered cutting plane: If the cutting plane is changed, the normal and shearstresses change, although we still have the same stress tensor σ .

There exists a cutting plane, on which no shear stresses act and the normal stresses become amaximum: The cutting plane is rotated until the traction vector t acting on this plane is

parallel to the normal vector n , which is necessary to fulfill the condition of vanishing shearstresses.

n

t = σ0n

n PP t

t ||t

cutting plane witharbitrary stress state

cutting plane withprincipal stress state

Fig. 3.4: Different stress states due to rotation of cutting plane

The traction vector t can thus be written as a multiple of the normal vector n (which has unitlength):

nt oσ= (3.16)

σ0 represents the so called principal stress , the direction of the normal vector the principal

stress direction .

3.2.1 Principal stress determinationWe now want to determine the principal stresses and their direction of a stress tensor which isgiven in matrix notation referring to an arbitrary local Cartesian basis:

(3.17)⎥⎦

⎤⎢⎣

⎡

σσσσ=

2221

1211σ

We have seen that the traction vector t has to be parallel to the normal vector n to fulfill thecondition of vanishing shear stresses. This can be written as:

( ) nInnσt ⋅σ=σ=⋅= 00 (3.18)

I is the identity tensor, which can be written in matrix notation as the identity matrix:

(3.19)⎥⎦

⎤⎢⎣

⎡=10

01I

3 - 6


35/81


The governing equation can be rearranged and leads to an eigenvalue problem for the principal stress σ0:

( ) 00 =⋅σ− nIσ (3.20)

The principal stresses σ0 are the eigenvalues, the principal stress directions the eigenvectorsof the stress tensor.

Since the vector n is defined to have unit length, the equation can only be fulfilled if thetensor ( is singular: A tensor is then singular if its determinant vanishes:)Iσ 0σ−

(3.21)

( )

( ) ( ) ( )( ) 0dettr

det

02

0

21222110

221120

02221

120

11

0

=+σ−σ=

=⎟ ⎠ ⎞

⎜⎝ ⎛ σ−σσ+σσ+σ−σ=

=⎥⎦

⎤⎢⎣

⎡

σ−σσσσ−σ=σ−

σσ

Iσ

The resulting equation is the characteristic polynomial , which is quadratic for a plane stressstate. The generally two independent solutions are the principal stresses of the stress tensor σ .The principal stress direction n for a particular principal stress can be obtained by solving thegoverning equations of the eigenvalue problem ( σ0 has now a fixed value and isn’t a variableanymore). Note that only the line of action of the eigenvectors is defined, but not their lengthand orientation.

It can be proven that the principal stress directions are orthogonal to each other (as alleigenvectors). So if the cutting plane is parallel to e.g. principal stress direction 1, the stressacting on that plane is principal stress 2 (and vice versa).

3.2.2 Mohr’s circle of stressIf the normal and shear stresses on two cutting planes which are perpendicular to each otherare known, it is possible to determine the principal stresses (or every possible stress state)graphically via Mohr’s circle of stress.

As an example the following situation is given:

σ1 = 2.0σ1

σ2 = 5.0

τ1 = 1.5

τ2 = 1.5

τ2

τ1

σ2

e 2

e 1P

Fig. 3.5: Example - Given stress state

3 - 7


36/81


We now denote the stress states of the two different cutting planes in a diagram: Thehorizontal axis of the diagram represents the normal stresses σ, the vertical axis the shearstresses τ. If the shear stress is turning anti clockwise it is defined as positive shear stress.

In the next step Mohr’s circle of stress is constructed: The center of the circle M is situated at

the intersection of the normal stress axis and the line connecting the two points whichdescribe the two stress states. The radius is the distance from the center to one of the two

points.

Every point on the circle is describing a possible stress state: The extremal normal stresses areobtained at the intersection of the circle with the normal stress axis (these are the principalstresses). The maximum shear stress is equal to the radius of the circle.

τ

σmin=1.38 σM σmax =5.62

(σ1=2.0, τ1 =1.5)

(σ2=5.0, τ2=1.5)

r

Fig. 3.6: Example - Mohr’s circle of stress

The results can be proved analytically. The characteristic polynomial of the eigenvalue problem is:

( )

075.70.7

10

01

0.55.1

5.10.2detdet

02

0

00

=+σ−σ=

=⎟⎟

⎠ ⎞

⎜⎜

⎝ ⎛

⎥⎦

⎤⎢⎣

⎡σ−⎥⎦

⎤⎢⎣

⎡=σ− Iσ

The two principal stresses are the solutions of this characteristic polynomial, which

correspond with the solutions obtained by Mohr’s circle of stress:

62.5 ,38.1

2

75.747.07.0

maxmin

2

1,20

=σ=σ

⋅−±=σ

The principal stress directions can also be determined over Mohr’s circle of stress, but the procedure is not as straight forward as the procedure for determining the principal stresses.Therefore only the analytical method is presented:

We now want to determine the direction of the first principal stress σmin . After plugging it inthe governing equation system we get:

3 - 8


37/81


⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡⋅⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡

0

0

n

n

62.35.1

5.162.0

n

n

10

0138.1

0.55.1

5.10.2

2

1

2

1

The two equations are linearly dependent (multiplying the first equation with 2.41 results the

second equation). Therefore we can choose any of these two equations. Since initially thelength of the eigenvectors is not defined we just set n 1 = 1 and use the first equation:

[ ] 0n5.162.0n

15.162.0 2

2

=+=⎥⎦

⎤⎢⎣

⎡⋅

By that we obtain the second component of the direction vector:

41.05.162.0

2 −=−=n

The vector is now normalized to have unit length:

( ) ⎥

⎦

⎤⎢⎣

⎡

−=⎥⎦

⎤⎢⎣

⎡

−−+=

38.0

93.0

41.0

1

41.01

122

minn

As we know that the eigenvectors are orthogonal to each other we can easily determine the principal stress direction for the maximum normal stress σmax as:

⎥

⎦

⎤⎢

⎣

⎡=93.0

38.0maxn

σ1 = 2.0σ1

σ2 = 5.0

τ1 = 1.5

τ2 = 1.5

τ2

τ1

σ2

e 2

e 1P

σmin = 1.38

σmin

σmax = 5.62

σmax

n min

n max

Fig. 3.7: Example – Principal stresses and principal stress directions

3 - 9


38/81

4 Numerical cutting patterngeneration


39/81

MM2: Numerical Theory Numerical cutting pattern generation

The result of the form finding process is generally a structure with a doubly curved surfacewith an inherent prestress state. The building material – e.g. fabrics, foils, etc. – on the otherhand comes in plane unstretched panels. This discrepancy has to be overcome in the

patterning process : Patches are cut out of the material according to certain cutting patternssuch that the resulting form and prestress state is as close as possible to the desired one when

they are attached together and moved into their boundary conditions.

The general procedure for cutting pattern generation is as follows:

4.1 Partitioning of the structure into stripsThe three dimensional structure is cut into strips along certain cutting lines . The governingfactors for the placement of these cutting lines are:

• material properties (especially the shear stiffness)

• available panel width and length

• curvature of the surface

• main load carrying paths

• aesthetical reasons

• etc.

4.1.1 Definition of cutting linesThe simplest way to define cutting lines is to intersect the surface with a plane. Thedisadvantage of this approach is the banana-like shape of the resulting cutting patterns, whichincreases the cut-off of membrane material.

Therefore it is proposed by several authors that cutting lines should follow geodesic paths onthe surface: A geodesic path or line on a doubly curved surface is the equivalent to a straightline on a plane. They are commonly defined as the line with the minimum distance betweentwo points on the surface (although this is not the exact mathematical definition): It has to be

paid attention to the fact that this includes also local minima. There may exist more than onegeodesic line with different lengths connecting two points P 1 and P 2 (e.g. on a cylinder):

While the shortest distance represents the global minimum, the other distances represent thelocal minima.

4- 2


40/81


geodesic line 1(global distance minimum)

geodesic line 2(local distance minimum)

P 1

P 2

Fig. 4.1: Several geodesic lines on a cylinder

4.1.2 Geodesic line calculation during form finding

It is possible to include the task of geodesic line calculation in the form finding process: Cableelements with high prestress are included in the FE mesh along an initial guess for the desiredgeodesic line. For each iteration step the resulting residual forces of these cable elements arecalculated for each node of the FE mesh and all components normal to the surface are set tozero. If the node is either the starting or end point of the geodesic line the whole residual forcevector is set to zero. As a result of this the cable elements representing the geodesic line affectonly the in plane positions of the nodes and not the general form finding process.

The advantage of this integrated procedure is that all finite elements are automatically pulledin the “right places”. That means the geodesic line doesn’t intersect any elements andtherefore no remeshing is necessary.

The disadvantage is that an initial guess for the geodesic line has to be made on a shape whichmay be far from the final one. This makes it often necessary to make two form finding runs:First without and the second time with geodesic line calculation

4.1.3 Geodesic line calculation on a found formIf a geodesic line is to be calculated on an already existing shape, one can make use of a

physical analogy: An elastic cable with high prestress is fixed on the starting point of thegeodesic line and pulled to the end point. By allowing it to slide over the surface it proceedsto a configuration with a minimum of potential energy. The final position of the cablerepresents the geodesic line. The problem here is that the cable nodes must have the

possibility to move freely on the surface described with the FE mesh. Therefore a contactformulation between the cable and the surface is necessary which makes it numerically morecomplicated.

Another possibility, which was originally developed in the computer graphics sector, is thegeodesic line calculation over triangulated surfaces: As indicated in the name it is onlyapplicable if the structure is meshed with linear triangles. The advantage of linear triangles istheir well defined geometry – which is essentially a plane – between the nodal points.

At first a discrete geodesic line consisting only of element edges of the FE mesh is created.This is usually done with a fast forward marching algorithm: For each node the minimum

4- 3


41/81


geodesic distance to the starting node is calculated. Then one starts from the end node andlooks at all neighboring nodes: The one with the minimum distance to the starting node isnow included in the geodesic line. This new node is now set to be the end node of thegeodesic and the procedure of finding the neighboring node with minimum distance andincluding it to the geodesic line is continued until the starting point is reached.

This discrete geodesic line is then optimized using an angular approach: The positions of thenodes of this geodesic are now not restricted anymore to coincide with the positions of the FEnodes. They are also allowed to be part of the edges linking two FE nodes. The nodes of thegeodesic line are moved along the edges of the FE mesh in such a way that the two angles

between the edge of the FE mesh containing the node of the geodesic line and the twoadjoining elements of the geodesic line are identical. Geometrically this can be explained asfollows: When the two finite elements containing the edge with the node are developed into a

plane the geodesic line should form a straight line.

surface

n

n

P start

P end P end

P start

α α

discrete geodesic on

triangulated surface optimized geodesic on

triangulated surface

P start

P end

α α

developmentinto a plane

Fig. 4.2: Geodesic line calculation on triangulated surfaces

Both of the presented methods for geodesic line calculation on a found form have in commonthat the resulting geodesic line is generally intersecting the FE mesh which requires at least aremeshing of the intersected elements. It has to be paid attention to the fact that the quality ofthe FE mesh may suffer if only the intersected elements are remeshed and not the wholemesh.

4.2 Flattening and compensation of the str ipsThe strips which are now obtained by cutting the three dimensional structure along thegenerated cutting lines are still doubly curved and prestressed. The following cutting patterngeneration generally consists of two parts: The doubly curved strip is first flattened into a

plane and then compensated (scaled down) to generate the prestress in the three dimensionalstructure. This flattening and compensation can be combined into a single process.

4- 4


42/81


The main difficulty here is the general non-developability of the membrane strips. A spatialsurface is only then developable into a plane without any distortions when the Gaussiancurvature K, which is the product of the two principal curvatures at each point, is zero atevery point of the structure. Since this is generally not the case several methods have beendeveloped to minimize the resulting distortions which can lead to deviations from the desired

prestress state.

4.2.1 Simple triangulization techniqueThe idea behind this technique is to make a generally non-developable surface developable:This is done by remeshing the membrane strips with linear triangles: All nodes except thenodes along the cutting lines of the longer edge are deleted. Then the patch is remeshed withtriangles according to the initial segmentation of the cutting lines (only one element along theshorter edge of the strip). Now it is possible to develop the geometry described by this meshinto a plane: The mesh is simply unfolded.

The disadvantage of this method is that it doesn’t take into account any material parametersand the inside geometry. Therefore it is only suitable for strips where the curvature of themembrane parallel to the shorter edge is relatively low.

The compensation is usually done with empirically determined scale factors.

simple triangulization spatial surface FE mesh planar development

Fig. 4.3: Simple triangulization technique

4.2.2 Optimization techniques

More sophisticated approaches use optimization techniques for the cutting pattern generation.Since a huge variety of different methods exists, only a few are presented in note form:

4.2.2.1 Minimizing the cable length differenceIn this approach the FE mesh is seen as a cable net. The edges of the elements represent thecables. If the cable net is then flattened, it is impossible to keep all cable lengths constant inthe case of an originally doubly curved surface. A least-squares minimization technique isapplied to minimize the sum of the squares of the length difference between the 3D and 2Dconfiguration.

The objective function, which is to be minimized, is:

4- 5


43/81


(4.1)( ) minimizellPedgesn

1i

2i,D2i,D3 →−= ∑

=

l 3D,i represents the length of the ith cable element in the three dimensional configuration, l 2D,i

the length in the flattened two dimensional configuration.Again it is not possible to take into account any material parameters.

4.2.2.2 Minimizing the stress differenceThe difference between the resulting and desired stress state is minimized in theseapproaches. The constraints are the equilibrium conditions. Here it is possible to include theeffects of anisotropic material. Since these methods need more mechanical and mathematical

background, this is beyond the scope of the course.

4.2.3 Mechanical approachA mechanical analogy is used here for cutting pattern generation: The doubly curvedmembrane strip is assumed to consist of an elastic material.

In a first step, the membrane is pressed into an arbitrary plane. It has just to be taken care thatthe flattened membrane mustn’t overlap itself. Due to this deformation elastic stresses arise inthe membrane which are not in equilibrium anymore. In a second step, the membrane isreleased in the plane while keeping the boundary conditions statically determinate. Thus themembrane can relax itself until it reaches a residual eigenstress state. The correspondingequilibrium shape is the cutting pattern.

It is possible to do simultaneous compensation: Then the prestress is included, which causes acontraction of the membrane.

The mechanical modeling has to be done under the hypothesis of large deformations.Therefore the resulting equations become non-linear and must be solved iteratively.

The advantage of this method is the possibility to take into account anisotropic material behavior and prestress states.

Fig. 4.4: Cutting pattern generation using mechanical approach

4- 6


44/81


45/81

MM2: Numerical Theory Finite Element Method

5.1 Linear elastic plane stress: Principle of virtual work in matrix notationWe start from the principle of virtual work for a 3-dimensional body:

(5.1)( ) ∫∫∫ΓΩΩ

Γδ−Ωδ−Ωδεσ=δ+δ−=δ− dugdu bdwww iiiiijijextint

where σij and εij are the components of stress and strain tensors, b i and g i the componentsof body and surface loads, respectively.

x, u

y, vz, w

g

b

x, u

y, vz, w

g bt

A

E

3D continuum plane stress/strain continuumFig. 5.1: Reduction of general 3D continuum to plane stress/strain

The special case of a thin 2-dimensional plane stress/strain structure is considered bysplitting the integrals of 5.1 into an integral over the thickness and into one over the mid-

surface A or along the edges E:0dtdEugdtAdu bdtdAw

t Eii

t Aii

t Aijij =δ−δ−δεσ=δ− ∫ ∫∫ ∫∫ ∫

As the stresses are constant through the thickness the integration over t can be done inadvance (pre-integration) which transfers stresses into stress resultants and external loads toarea q or line loads p, respectively. Using a matrix notation the principle of virtual work nowdisplays as:

0dEdAdAwE

T

A

T

A

T =δ−δ−δ=δ− ∫∫∫ upuqn

where

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

δ

δ=δ

⎪

⎪

⎭

⎪⎪

⎬

⎫

⎪

⎪

⎩

⎪⎪

⎨

⎧

δγ

δε

δε

=δ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪

⎪

⎭

⎪⎪

⎬

⎫

⎪

⎪

⎩

⎪⎪

⎨

⎧

γ

ε

ε

=

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

τ

σ

σ

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

τ

σ

σ

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

= ∫

v

u;;

v

u;

.constt;gt

gt

p

p;

bt

bt

q

q;dt

t

t

t

n

n

n

xy

y

x

xy

y

x

y

x

y

x

y

x

y

x

t

xy

y

x

xy

y

x

xy

y

x

uu

pqn

5 - 2


46/81


py

px

nynyx

nxy

nx

x

y

qx

qy

Fig. 5.2: In-plane loading and internal forces of 2D membrane

The constitutive equations are also written in matrix notation, introducing the constitutivematrix C :

Cn t=

plane stress plane strain

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ν−ν

ν

ν−=

2

100

01

01E

21C ( )( )

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ν−ν−ν

νν−

ν−ν+=

221

00

01

01

211E

C

At any point the strain is related to the displacement u by

⎭⎬⎫

⎩⎨⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

==ε vu

xy

y0

0xuL (5.2)

The matrix L is a differential operator matrix.

Putting all together:

uL

uLCCn

δ=δ==

tt

and, finally:

0dEdAdAtwE

T

A

T

A

TT =δ−δ−δ=δ− ∫∫∫ upuquLCLu (5.3)

5 - 3


47/81


This equation represents equilibrium of plane stress / plane strain states in terms of theunknown displacement field u T = (u(x,y), v(x,y) ). It is the basis for finite element proceduresto determine an approximate solution of u . The virtual work equation is also called the “weakform of equilibrium”.

5.2 Short introduction to the Finite Element Method

5.2.1 Simple 3- and 4-node isoparametric displacement elementsThe principal idea of the finite element method is to reduce a continuous problem to a

problem of a finite number of discrete parameters. The solution of the discrete problem givesan approximation of the continuous one. Here, we assume that the displacement field can bedescribed by number of discrete displacement values which are defined at the finite elementnodes. Several nodes together form a finite element (e.g. 3 node triangle, 4 node rectangle,Fig. 5.3) . Inside the element, i.e. between the nodes, the displacement field is approximated

by a linear combination of shape functions, each of them related to one node of the element.

3-node triangle 4-node rectangle

x

y

1

2

3

x

1 2

3

m

4 a

b

Fig. 5.3: Two simple plane stress/strain finite elements

The shape functions can be defined with respect to the x,y-coordinate system as:

3-node triangle:

( ) ([ ])

( ) ( )[ ]

( ) ( )[ ]yxxxyyyxyx21

)y,x( N

yxxxyyyxyx21

)y,x( N

yxxxyyyxyx21

)y,x( N

122112213

311331132

233223321

−+−+−∆

=

−+−+−∆

=

−+−+−∆

=

where ∆ is the element area: ( ) ( ) ( )[ ]213132321 yyxyyxyyx2/1 −+−+−=∆ 4-node rectangle:

( )( )( )(( )(( )( η−ξ+=

η−ξ−=η+ξ−= )))

η+ξ+=

114/1)y,x( N

114/1)y,x( N

114/1)y,x( N

114/1)y,x( N

4

3

2

1

5 - 4


48/81


with

( ) ( )

( ) ( )

4121

41m21m

mm

yy bxxa

yy21

yxx21

x

yy b2

xxa2

−=−=

+=+=

−=η−=ξ

For the triangle the displacement field is defined as:

( )( )

( )

( ) ( ) ( )

( ) ( ) ( ) ⎪⎭⎪⎬

⎫

⎪⎩

⎪⎨

⎧

++

++=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

332211

332211

vy,x Nvy,x Nvy,x N

uy,x Nuy,x Nuy,x N

y,xv

y,xuy,xu

or by separation of shape functions N i and nodal displacements u i, v i using a matrix for-mulation:

( ) ( ) vNu y,x

v

u

v

u

v

u

N0 N0 N0

0 N0 N0 Ny,x

3

3

2

2

1

1

321

321=

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡= (5.4)

Inserting 5.30 into the strain-displacement relation (5.2) yields

( ) ( ) ( )

( ) ( )

( ) vB

v

vN

vNLuL

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−−

−−−

=

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

==

∆

∂∂

∂∂

∂∂

∂∂

y,x

yyxxyyxxyyxx

xx0xx0xx0

0yy0yy0yy

y,x0

0

y,x

y,xy,xy,x

211213313223

123123

211332

21

xy

y

x

The differential operator matrix B is the discrete equivalent of L , now relating discretenodal displacement with an approximation of the strain field . The procedure is equivalentfor the 4-node rectangle or any other displacement finite element.

5 - 5


49/81


All expressions are inserted into the virtual work equation (5.3):

( )

( ) ( ) ( ) ( ) vf vNpNq

vk vvBCBvvBCBv

upuquLCLu

δ=δ⎥⎦

⎤⎢⎣

⎡+=δ

δ=δ=δ=δ−

=δ−δ−δ=δ+δ−=δ−

∫∫∫∫

∫∫∫

T

E

T

A

Text

T

A

TT

A

TTint

E

T

A

T

A

TTextint

dEy,xy,xdAy,xy,xw

dAtdAtw

0dEdAdAtwww

And, finally, the element stiffness matrix k and the equivalent nodal force vector f are de-fined as:

∫∫

∫

+=

=

E

T

A

T

A

T

dEdA

dAt

pNqNf

BCBk

which contribute to the system stiffness matrix and force vector.

5.2.2 Convergence behaviorThe quality of a finite element analysis is shown for the example of a cantilever beam discre-tized by 4-node elements:

P = 1

h = 1

t = 1

E = 4 ⋅106; ν = 0wl = 10

33

exact 10EIP

w −== l

60WP

max ==σ l

Fig. 5.4: Cantilever beam with concentrated load

5 - 6


50/81


Element discretization:

160 elements, 205 nodes, 400 degrees of freedom (dof)

3

2


1


Fig. 5.5: Several different discretizations

Convergence behavior:

0

0,20,4

0,60,8

1

0 200 400

do f

w / w

e x a c t

Fig. 5.6: Displacement of cantilever trip

01020304050

60

0 200 400

dof

/

e x a c t

Fig. 5.7: Convergence of surface stresses

5 - 7


51/81


2

20 30 40 50 60

4

6

8

10

σsurface

x

exact solution

10

1

2 3

Fig. 5.8: Distribution of surface stress

5 - 8


52/81


5.3 Numerical integrationIntegration is essential to determine stiffness matrices and load vectors, e.g. ( 5.48) and ( 5.49) .It is only possible, however, to evaluate the integration analytically for some few geometrictypes of finite elements, as e.g. linear triangular, rectangular and rhombic, plane geometries.

The reason is that the nominators of the contravariant base vectors are functions of the naturalcoordinates ξ and η. As a consequence, the integrand is a non-uniform rational functionwhich cannot be integrated analytically in a closed form.

For the most of elements, in particular isoparametric elements , the standard practice has beento use numerical Gaussian integration rules because such rules use a minimal number of sam-

ple points to achieve a desired level of accuracy. This property is important for efficient ele-ment calculations because we shall see that at each sample point we must evaluate a matrix.The fact that the location of the sample points in the Gauss rules is usually given by non-rational numbers is of no concern in digital computation.

5.3.1 One Dimensional RulesThee standard Gauss integration rules in one dimension are:

(5.5)( ) ( )∑∫=−

ξ≈ξξ p

1iii

1

1

f wdf

Here p ≥ 1 is the number of Gaus

Date post:	06-Jul-2018
Category:	Documents
Upload:	mile
View:	234 times
Download:	0 times

MM2 Numerical Theory MEMBRANE STRUCTURES

Documents