Wu YS Master Thesis

i

SHAPE ANALYSES FOR ISOTONIC SURFACE WITH COMBINATION OF

SOAP FILM ELEMENTS AND AXIAL LINES

by

Wu Yusheng

A thesis submitted in partial fulfillment of the requirements for the degree of Master of

Engineering in Civil Engineering, Saga University

Examination Committee: A. Prof. H. OBIYA (Chairperson)

Prof. K. IJIMA

Prof. ISHIBASHI

Previous Degree: Bachelor of Engineering in Civil Engineering

Zhe Jiang University of Science and Technology

Department of Civil Engineering and Architecture

Graduate School of Science and Engineering

Saga University

Japan

August, 2010

ii

ACKNOWLDEGEMENTS

This thesis has been carried out at the Division of Structural Engineering, Department of

Civil Engineering, Saga University, Saga, Japan from October 2008 to September 2010. I

have completed my research under the supervision of Professor Hiroyuki OBIYA.

First, I would like to express my appreciation to my advisor Hiroyuki OBIYA. Thanks

for offering me the opportunity to be part of his group. He is such a good advisor. He

always was there to listen to my problems and gave me good advices. He taught me how to

ask questions and express my ideas. He showed me different ways to approach a research

problem and the need to be persistent to achieve any goal. Under his encouragement and

constant guidance, I overcame some very tough time and completed my research during

two years staying in Japan.

Second, I also would like to thank the other two members in division of structural

engineering, Professor Katsushi IJIMA and Technician N.Kawasaki. Thanks for everything

you offered during my studying period. Without your supports, I would have not able to

finish my research.

Last but not least, I would like to thank my colleagues in Obiya Laboratory as well as

Ijima Laboratory for interesting discussions and being fun to be with.

Saga, Japan 2010

iii

SHAPE ANALYSES FOR ISOTONIC SURFACE WITH COMBINATION OF

SOAP FILM ELEMENTS AND AXIAL LINES

08547004 Wu Yusheng

1 Introduction

The shape analysis is to find the mechanically rational form for many kinds of structures,

for example, membrane structures, suspended cables, and so on. The isotonic surfaces can

be adopted for the primary shape of these structures. Purpose of this study is to discuss

about the discrete solutions obtained by soap film analyses. Under the assumption of

"triangle isotonic element", the determined solution should be obtained as "a polyhedral

equilibrium shape".

Physically soap film element has no stiffness to the tangent direction of its surface.

When 1D analysis only to normal direction is applied for shape analysis of soap film,

dispersion of the element area is a fatal problem. However, 2D axial lines (The axial line

elements are projected to tangent direction of its surface, and have 2 degree of freedom)

have an ability to optimize the node distribution even large increment of compulsory

displacement is adopted. Therefore, this study proposes a 3D analysis to obtain the

polyhedral solutions by parallel calculation of soap film analysis only to normal direction

and the 2D axial line analysis in tangent plane of the surface. By using this technique, node

distribution becomes more rational and large incremental of compulsory displacement can

be adopted when the simultaneous control is used as the incremental method. Moreover,

there is another usage of axial lines that can be converted to cable elements with real

stiffness after form-finding. This type of axial line elements which is named "3D axial

line" has degree of freedom to all of 3 dimensions. When 3D analysis with 3D axial lines is

applied for shape analysis of soap film, the polyhedral solutions can be expected to

determine composite structures of membranes and cables.

2 Analytical Method

2.1 3D Analysis

In this thesis, this study proposes a technique which is the parallel calculation of the

soap film analysis and the axial line analysis. This technique defines two independent and

parallel coordinate, as shown in Fig.1. One has only the one axis toward normal direction

of the surfaces and that is prepared for the isotonic surface analysis. The other has two axes

and perpendicular each other and both are to tangent direction of surfaces. These two

axes are prepared for 2D axial line analysis.

iv

2.2 2D axial line and 3D axial line

The Fig.2 shows an axial line element. The element forces of the axial line elements are

projected to and coordinate axes. Thus this type of axial line elements has only 2

degree of freedom, so the author calls it "2D axial line" in this thesis. According to the

effect of the 2D axial lines, the mesh distribution becomes more uniform, and the

polyhedral equilibrium solutions become more similar to the isotonic surfaces.

Moreover, there is another usage of the axial line elements that can be converted to cable

elements with real stiffness after form-finding. This type of axial line elements has degree

of freedom to all of 3 dimensions, and the author calls it "3D axial line". 3D axial lines can

restrict the deformation of soap film during the process of form-finding. Therefore, the

combination of the soap film elements and 3D axial lines can be expected to determine a

composite structure of membranes and cables.

3 Numerical Examples and Discussions

The geometry of initial shape is shown in Fig.3. The green lines denote 3D axial lines

and the 2D axial lines to unify the mesh distribution are

located on 3 sides of all triangular elements. The tensile of

the soap film elements is 3.0kN/m, and the coefficient of

line elements C=1. Simultaneous control is used as an

incremental method in this example. The increment of the

compulsory displacement is given as Dc=2m and Dc=0.1m

respectively for 1D soap film analysis (Fig.4 [a], [b]). In

contrast, for 3D analysis with 3D axial lines, the increment

of the compulsory displacement is given as 2m (Fig.4 [c]).

Because of the effect of 2D axial lines, node distribution becomes more rational and

i

Fig.3 Initial plane

N

N

Wi

Ui

Vi

i

Wj

Uj

Vj

j

i

i

i

i

R=8m

Fig.1 Coordinate system with normal direction

and tangent direction Fig.2 Axial line element

v

large increment of compulsory displacement can be adopted. Meanwhile, 3D axial line can

restrict the deformation of soap film during the process of form-finding. The determined

shape can be expected to composite structure of membranes and cables.

The mesh connectivity is shown in Fig.5, and the power of 3D axial lines n is 4. The

Fig.6 shows the equilibrium solutions compared with two kinds of mesh connectivity.

Fig.4 The equilibrium shapes

[a] 1st connectivity

Fig.5 Initial plane

[b] 2nd

connectivity


[a] 1st connectivity [b] 2

nd connectivity

n:Power of 3D axial lines, Dc: Increment of compulsory displacement

[a] 1D analysis, Dc =2m

[b] 1D analysis, Dc =0.1m

[c] 3D analysis with 3D axial

lines, Dc=2m, n=5

R=8m

vi

18m

3@6m

On the next example, the geometry of initial shape is shown in Fig.7, and the green lines

denote the 3D axial lines. Further, the 2D axial lines to unify the mesh distribution are

located on 3 sides of all triangular

elements. The tensile of the soap film

elements is 3.0kN/m, and the

coefficient of line elements C=1.

The Fig.8 shows the equilibrium

shapes in cases with 3D axial lines

and without 3D axial lines

respectively.

4 Conclusion

In this thesis, a technique for form finding of isotonic surfaces is shown, and it is to use

the combination of triangular soap film elements and the axial lines. The characteristics of

this technique and results from the numerical examples are concluded as follows:

1) Because of the effect of 2D axial lines, node distribution becomes more rational and

large increment of compulsory displacement can be adopted when simultaneous control

is used as the incremental method. Therefore, the polyhedral equilibrium solutions

become more similar to the isotonic surfaces.

2) 3D axial lines can restrict the deformation of soap film during the process of form

finding. Furthermore, these axial lines can be converted to cable elements with real

stiffness after form-finding. Therefore, the combination of the soap film elements and

3D axial lines can be expected to determine a composite structure of membranes and

cables. In addition, in some cases of boundary shapes, we have to notice that the mesh

connectivity has influence to determined shape.

Fig.7 Initial plane

[a] Without "3D axial line" [b] With "3D axial line", n=6


vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

Title Page i

Acknowledgements ii

Abstract iii

Table of Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

1.1 Motivation and objective 1

1.2 Background 3

2 Tangent Stiffness Method 10

2.1 Outline 10

2.2 Tangent Stiffness Equation 11

2.3 Derivation of Tangent Geometrical Stiffness Matrix with Energy Principle

11

2.4 Iterative procedure of Tangent Stiffness Method 14

3 Shape Analysis 18

3.1 Element Potential Function 18

3.2 Triangular Soap Film Element 19

3.3 Axial Line Element 20

3.4 Nodal Coordinate Definition 21

3.5 Simultaneous Control 22

4 Numerical Examples and Discussions 26

4.1 Low-rise equilibrium shape by combination of soap film elements and axial

lines (Load control) 26

4.2 Equilibrium shape by combination of soap film elements and axial lines

(Simultaneous control) 33

5 Case Study by Using 3D Axial Lines 50

5.1 Case 1 50

5.2 Case 2 54

viii

5.3 Case 3 57

6 Minimal Surface 60

6.1 Introduction 60

6.2 Catenoid 61

6.3 Scheck surface 68

7 Conclusion 73

8 Reference 74

ix

List of Figures

Fig.1.1 Soap film constrained by a rectangular frame with a moveable side AB ................. 5

Fig.1.2 Surface tension forces acting on a tiny patch of surface ........................................... 6

Fig.1.3 One curvature of A soap bubble with two surfaces ................................................... 8

Fig.1.4 The process of inflating with air pressure increasing ................................................ 8

Fig.1.5 The process of inflating with air pressure decreasing ............................................... 9

Fig.1.6 Inner pressure Volume curve ................................................................................ 9

Fig.2.1 Energy vectors relationship in Tangent stiffness Method .................................... 12

Fig.2.2 Iterative procedure of Tangent Stiffness Method .................................................... 16

Fig.2.3 Flow chart of the program for the form finding ...................................................... 18

Fig.3.1 Triangular soap film element .................................................................................. 20

Fig.3.2 Normal direction at a node #i .................................................................................. 22

Fig.3.3 Coordinate system with normal direction and tangent direction............................. 23

Fig.3.4 The process of simultaneous control ....................................................................... 25

Fig.3.5 Elements connected to node #i .............................................................................. 27

Fig.4.1.1 Initial plane and connectivity ............................................................................... 28

Fig.4.1.2 ~ Fig.4.1.12 Equilibrium shapes and convergent process of unbalanced force

respectively .................................................................................................................. 29~34

Fig.4.2.1 Initial plane and connectivity ............................................................................. 35

Fig.4.2.2 Inner pressure-volume curve ................................................................................ 36

Fig.4.2.3 Equilibrium shapes (Left: 1D analysis; Right: 3D analysis) .............................. 36

Fig.4.2.4 Equilibrium shapes when the total compulsory displacement = 6 m ................... 39

Fig.4.2.5 Initial plane and mesh connectivity ...................................................................... 40

x

Fig.4.2.6 Equilibrium shapes when the total compulsory displacement= 5m ..................... 41

Fig.4.2.7 Straight line 1 and straight line 2 ....................................................................... 42

Fig.4.2.8 The curve of the straight line 1 ............................................................................. 43

Fig.4.2.9 The curve of the straight line 2 ............................................................................. 44

Fig.4.2.10 Straight line 1 and straight line 2 ....................................................................... 45

Fig.4.2.11 The value of n for straight line 1 ........................................................................ 45

Fig.4.2.12 Equilibrium shapes when the total compulsory displacement= 5m ................... 46

Fig.4.2.13 The curve of the straight line 1 ........................................................................... 47

Fig.4.2.14 The curve of the straight line 2 ........................................................................... 48

Fig.4.2.15 Initial plane and connectivity ............................................................................. 49

Fig.4.2.16 Inner pressure-volume curve ............................................................................ 50

Fig.4.2.17 Equilibrium shapes of two mesh connectivity ................................................... 51


Fig.5.1.2 Inner pressure-volume curve .............................................................................. 53

Fig.5.1.3 Equilibrium shapes (Left: 3D analysis without 3D axial lines; Right: 3D analysis

with3D axial lines) ............................................................................................................ 55


Fig.5.2.3 Equilibrium shapes ............................................................................................... 58


Fig.5.3.2 Inner pressure-volume curve .............................................................................. 60

Fig.5.3.3 Equilibrium shapes ............................................................................................. 61

Fig.6.1 The surface of catenoid ........................................................................................... 63

Fig.6.2 The relation between Z*and C ................................................................................. 64

Fig.6.3 The relation between Z and r ................................................................................... 64

xi

Fig.6.4 Primary shape and the connectivity ........................................................................ 65

Fig.6.5 Minimal surfaces of Catenoid ................................................................................. 66

Fig.6.6 Stable solution of Catenoid ................................................................................... 67

Fig.6.7 Unstable solution of Catenoid ............................................................................... 68

Fig.6.7 Element mesh of Scheck surface ........................................................................... 70

Fig.6.8 Minimal surface of Scheck surface ......................................................................... 71

Fig.6.9 Curve comparison of Scheck surface ...................................................................... 73

List of Tables

Table 1.1 Surface Area of each polyhedron ........................................................................... 4

Table 6.1 Comparison of c value in stable solution ............................................................. 67

Table 6.2 Comparison of c value in unstable solution ......................................................... 69

1

1. INTRODUCTION

The general research question of the study is related to the design problem of membrane

structures while the aim is to adopt the tangent stiffness method for generating minimal

surfaces, and form finding of the initial shape with given boundary shape by combining

axial line elements and triangular soap film elements.

The main concept is based on the simulation of a potential tensioned membrane such as

a soap film within a pregiven boundary. This study proposes a new technique to obtain the polyhedron solutions which can express smooth surfaces by parallel calculation of soap

film analysis only to normal direction and the "2D axial line" analysis in tangent plane of

the surface. By using this technique, node distribution becomes more rational and large

compulsory displacement can be adopted when the simultaneous control is used as the

incremental method. Further, when using the "3D axial lines" that have degree of freedom

to all of 3 dimensions, we can assume it to be converted to the real cable elements after

form-finding.

1.1 Motivation and Objective

Plateau's problem is to find the minimal surface with a boundary shape, and many

researchers have spent their effort to solve the shape of soap film for over 100 years.

On the other hand, for practical use of isotonic surfaces as an initial shape, such as for

suspended membrane roofs or pneumatic structures, computational form-finding methods

have been developed since the end of last century. Recently, we can determinate an

isotonic surface as precise solution by general type of the finite element method, in the

case of the application to membrane structures with low-rise and with simple in-plane

boundary shape.

However, application of membrane structures is not only for large span roofs. We can

expect the possibility to adopt the isotonic surfaces to the primary shape of more various

styles of structures, for example, the underwater structures such as the storage tank of pure

water floating on ocean, underground structures which is so called geo-membrane', and

space structures on some extraterrestrial planet.

If we apply the shape of the isotonic surface to these new uses for membrane structures,

the technical consideration for computation will be required as follows:

(1) Loading condition should be applied not only to constant inner pressure but also to the

location dependency load such as water or soil pressure.

(2) Sure convergence property should be guaranteed even if in cases of complicated

boundary shapes.

2

(3) Development of a rational mesh division technique to express smooth shape of curved

surface is expected.

The authors' former paper has concluded that the tangent stiffness method is very useful

for the form-finding of soap films [1]-[5]. The method uses the iterative process of the

tangent stiffness equation that derived from the equilibrium equation between the nodal

forces and the element edge forces. Furthermore, the authors mentioned that the

simultaneous control, which is an incremental technique, makes it possible to apply to the

location dependency load listed in above (1). For the problem of (2), the primary

equilibrium shape constituted by the axial line elements instead of the soap film elements

makes it easier to get the solutions, even if in case of the complicated boundary shape. The

axial line elements can produce the solutions very close to isotonic surfaces, and can let the

mesh distribution be uniform [2], [4].

With regard to problem (3), for instance, when the deformation of the surface from a

primary shape to the determined shape is too large, dispersion of the element area also

grows so large. In such a case, we cannot keep 'the significance' as approximate solutions

for curved surfaces anymore. Now, we discuss about an isotonic surface as a discrete

solution. Under the assumption of "triangle isotonic element", the determined solution

should be obtained as "a polyhedral equilibrium shape". Moreover, originally, the isotonic

surface does not have any stiffness toward normal direction of the surface, and the position

of every node as discrete solution can be located anywhere on physical isotonic surface.

Namely, there is the discrete equilibrium solution innumerably on an isotonic surface.

Therefore, we should keep the proportionate mesh distribution all through the incremental

process of shape analysis.

Based on the above background, in this paper, the authors propose another technique

which is the parallel calculation of the soap film analysis and the axial line analysis. The

technique defines two independent and parallel coordinate. One has only one axis toward

normal direction of surfaces and that is prepared for the isotonic surface analysis by

triangular soap film elements. The other has two axes perpendicular each other and both

are to tangent direction of surfaces. The element forces of the axial line elements are

projected to these two coordinate axes. Thus this type of axial line elements has only 2

degree of freedom, so we call it "2D axial line", in this paper. According to the effect of the

2D axial lines, the mesh distribution becomes more uniform, and the polyhedral

equilibrium solutions become more similar to the isotonic surfaces.

In this paper, we also mention another usage of the axial line elements that can be

converted to cable elements with real stiffness after form-finding. This type of axial line

elements has degree of freedom to all of 3 dimensions, and we call it "3D axial line" to

distinguish these axial line elements from 2D axial line. Therefore, the combination of the

soap film elements and 3D axial lines can be expected to determine a composite structure

of membranes and cables.

3

1.2 Background

1.2.1 Soap bubble and soap film

Soap bubble, like a balloon, is a very thin soap film surrounding a volume of air. Now,

imagine a simple experiment. If dip a curved metal wire into a mixture of water and dish-

washing detergent, and then pull it out. After that you blow a slight air into the soap film, a

beautiful soap bubble forms. No matter what shape a bubble has initially, it will try to

become a sphere. The sphere is the shape that minimizes the surface area of the structure,

which makes it the shape that requires the least energy to achieve.

The Table 1.1 illustrates the surface area of each polyhedron in nature.

Table 1.1 Surface Area of each polyhedron

Soap films are the classical examples of minimal surfaces. A minimal surface is more

properly described as a soap film with zero mean curvature. A soap bubble, due to the

difference in outside and inside pressure, is a surface of constant mean curvature.

In addition, when a soap film is in a state of equilibrium, the surface tension is the same

on the surface. There would be an extremely valuable application in architecture. As we all

know, the knowledge of the form is a very important aspect of design of structures. Having

the least area property minimal surface is most used for light-roof constructions, and form-

finding models for membrane structure. Among the surfaces having the same boundary,

minimal surface is the surface of the least area. Therefore, its weight is less and the amount

of material is reduced on minimum.

Shape # of sides Volume Surface Area

Tetrahedron 4 1 cubic meter 7.21 square meter

Cube 6 1 cubic meter 6.0 square meter

Octahedron 8 1 cubic meter 5.72 square meter

Dodecahedron 12 1 cubic meter 7.21 square meter

Icosahedrons 20 1 cubic meter 7.21 square meter

Sphere infinite 1 cubic meter 7.21 square meter

4

1.2.2 Minimal surface

The Plateau's problem is to find the shape of the minimal surface constrained by a given

boundary. In 1873 a physicist named Joseph Plateau observed that soap film bounded by

wire appeared to form minimal surfaces. The problem named after him.

Minimal surfaces are defined as surfaces of the smallest area spanned by a given space

curve. The theory of minimal surfaces is a branch of mathematics that has been highly

developed, particularly recently.

In spite of the fact that it seems that soap film easily solves mathematical problem of

finding minimal surface for the boundary curve, many mathematicians devoted much of

their life to solve some basic problems as well as to give description of minimal surfaces in

mathematics for over 100 years.

The main fields of mathematics contributing to minimal surface theory are differential

geometry, theory of partial differential equations and calculus of variations.

In the recent time, as computer has made a great progress in science, this new

technology enabled researchers, especially in the field of differential geometry of

mathematicians, to study infinitely periodic minimal surface. This field has enjoyed a

lively renaissance with the help of modern visualization techniques.

Theoretical investigation of these surfaces is useful for application of this knowledge in

further investigation of forms in architecture.

5

1.2.3 Energy consideration

In order to obtain an expression for the energy of a soap film, let us consider a soap film

contained by a rectangular wire frame (Fig. 1.1), with one side of the frame AB, free to

move in the direction perpendicular to AB. The thickness of soap film is t. If this side is

initially at a distance x from the parallel fixed side and undergoes a further displacement

x, maintaining the temperature of the film constant, the work done against the film

tension force is

(1.1)

Where, is the film tension and is the increase in the volume of the soap film. From

Eq. (1.1), is seen to be the energy gained per unit increase in volume. Thus the total

energy necessary to increase the volume of a soap film from zero to V is given by F,

(1.2)

F is called the free energy at constant temperature. In the case of is constant and result

becomes

(1.3)

The free energy of the soap film is thus proportional to the area of the film under the

conditions of constant and t.

When a soap film is in stable equilibrium, any small change in its area A will produce a

corresponding change in its free energy F, if remains constant.

Fig.1.1 Soap film constrained by a rectangular frame with a moveable side AB

A

B

A

B

Y

X X

6

1.2.4 Equilibrium of soap film

A soap bubble consists of a thin soap film surrounding a volume of air. Inside the film is

air or gas at a greater pressure than the external atmospheric pressure. By examining the

equilibrium on a tiny patch surface of a soap bubble, we can relate the excess pressure

inside the bubble over the surrounding pressure P, and the film tension .

If no force acts normal to a tensioned surface, the surface must remain flat. But if the

pressure on one side of the surface differs from pressure on the other side, the pressure

difference times surface area results in a normal force. In order for the surface tension

forces to cancel the force due to pressure, the surface must be curved, as shown in Fig.1.2.

The weight of a tiny patch surface of fluid can be neglected compared with either of these

forces. When all the forces are balanced, the resulting equation is known as the Young

Laplace equation.

(1.4)

Where:

P is the pressure difference.

is surface tension.

Rx and Ry are radii of curvature in each of the axes that are parallel to the

surface.

Fig.1.2 Surface tension forces acting on a tiny patch of surface.

x and y indicate the amount of bend over the dimensions of the patch.

FT

FT

FT

FT

Rxx

Ryy

7

In the case of soap bubble, the principal radii of curvature, Rx and Ry, are equal to its

radius, that is Rx = Ry = r. So the excess pressure across each surface of the bubbles is,

from the Eq. (4)

(1.5)

For a sphere, the radius of curvature is simply its radius.

(1.6)

Where, is the curvature of the soap bubble.

8

1.2.5 Inner pressure-volume curve

When a soap film with a given boundary is inflated under the inner pressure, one

curvature of a soap bubble has two curved surfaces, as shown in Fig.1.3.

When the air pressure is applied on the soap film from inside, as the inner pressure

increases, the volume of soap film and curvature of curved surface also increase. When the

soap film is in the state of half sphere, the curvature of curved surface became a maximum

in Fig.1.4. Therefore, before the soap film is in the state of half sphere, the inner pressure

and volume of soap film is increasing. In this stage, we call it "expansion with inner

pressure increasing".

Initial state Half sphere

Curvature0 Maximum

Inner pressure0 Larger

Volume0 Larger

Fig.1.3 One curvature of a soap bubble with two surfaces

Fig.1.4 The process of inflating with air pressure increasing

9

However, when the inner pressure is over the state of the maximum point of its

path, volume is increasing while the curvature of curved surface and inner pressure is

decreasing, as shown in Fig.1.5. In this stage, we call it "expansion with inner pressure

decreasing".

Half sphere Determined shape

CurvatureMaximum Smaller

Inner pressureMaximum Smaller

VolumeLarge Larger

By the means of simultaneous control, we can get the determined shape which are over

the maximum point and have high-rise. The Fig.1.6 shows inner pressure volume curve

analyzed by simultaneous control.

Fig.1.5 The process of inflating with air pressure decreasing

Volume (m3)

Expansion with Ip increasing

Inner p

ressure (k

pa)

Fig.1.6 Inner pressure Volume curve

Expansion with Ip decreasing

10

2. TANGENT STIFFNESS METHOD

2.1 Outline

This chapter presents the derivation of Tangent Stiffness Equation and Tangent

Geometrical Stiffness Matrix; also the chapter descibes iterative process for unbalanced

force equation solution and geometrical nonlinear structure analysis technique for Tangent

Stiffness Method.

The Tangent Stiffness Equation can be easily derived by differencial calculus of

Equilibrium Equation that derived from the equilibrium equation between the nodal

forces and the element edge forces. Element edge force that works on element end is

completely prescribed by Element Force Equation. In this equation, element stiffness is

independent from the element displacement; it is caused by the nonlinearity of the Tangent

Geomtrical stiffness. Also the Element Force Equation precisely represents element

behavior, avoiding approximation even for complex cases.

Furthermore, by applying Principle of Stationary Potential Energy, it is possible to

express a symmetric matrix for Tangent Geometric Stiffness. Using this Tangent

Geometric Stiffness matrix makes possible to express strict element behavior that

prescribed in the Element Force Equation.

Tangent Stiffness Method avoids the derivation of complicated nonlinar element

stiffness equation with nonlinear material properties, which is quite difficult to formulate.

Only strcit Compatibility Equation and Equilibrium Equation are used for iterative

process to converge the Unbalanced Force.

Itervative solution in Tangent Stiffness Method is mathematically equal to Newton-

Raphson Method technique. Comparing the Tangent Stiffness Method to Newton-

Raphson Method applied in Finite Element Method(FEM), it shows overwhelmingly

high efficiency of Tangent Stiffness Method in convergence performace.

11

2.2 Tangent Stiffness Equation

Let the vector of the element edge forces independent of each other be indicated by S,

and let matrix of equilibrium which relates S to the general coordinate system by J. Then

the nodal forces D expressed in the general coordinate follow the equation:

(2.1)

The tangent stiffness equation is expressed as the deferential calculus of Eq. (2.1),

(2.2)

In which, K0 is the tangent element stiffness, and it is caused by the relation between the

increments of axial forces and the nodal displacements. Namely, K0 provides the elements

own behavior inside of the local coordinate moving with the rigid body displacements.

KG is the tangent geometrical stiffness caused by the relation between the changing of

the equilibrium conditions and nodal displacements. The cosine vectors and the length of

elements construct the equilibrium equation. Changing of cosine vector means the rotation

of the element. Therefore, KG provides the rigid body rotation and displacements. is

nodal displacement vector in general coordinate.

2.3 Derivation of Tangent Geometrical Stiffness Matrix with energy principles

Regarding to Eq. (2.1) and Eq. (2.2), Tangent Geometrical Stiffness KG can be expressed

as

(2.3)

According to expansion of Principle of Stationary Potential Energy, and element

force vector obtained from primary balance condition leads into expression of Tangent

Geometrical Stiffness Matrix.

12

V#cV

V#

V

#U

U

U

Stiffness Equation Equilibrium Equation

Nodal

Displacement

Element Force

Element Force Equation Compatibility Equation

Element Deformation

Fig.2.1 Energy vectors relationship in Tangent stiffness Method

Nodal Force (External Force)

13

The Fig.2.1 shows Nodal displacement vector, Element Deformation vector, Element

Force vector and Nodal Force vector, expressed as scalar quantities versus energy. The first

quadrant represents Tangent Stiffness Equation, the second quadrant Equilibrium

Equation, the third quadrant Element Force Equation, and the last quadrant

Compatibility Equation. In addition, the inner rectangle shows primary balance condition,

and the outer rectangle presents balance condition after nodal force increment when

external force has been applied. Here, if Stain Energy is U and External Energy V is, the

Total Potential Energy is

(2.4)

After deformation, Total Potential Energy can be expressed as follow:

(2.5)

If balance condition before and after deformation is assumed to be constant, according to

the Principle of Stationary Potential Energy,

(2.6), (2.7)

In Fig.2.1 shows that is not being influenced by the increment of nodal displacement

. Therefore,

(2.8)

The differential for increment of nodal displacement for V#, , U# and becomes

(2.9), (2.10), (2.11)

(2.12)

Therefore, Eq. (2.7) can be rewritten as follows.

(2.13)

Thus, it is clear that Eq. (2.13) and Eq. (2.1) are identical.

(2.14)

14

According to Eq. (2.2) and Eq. (2.3), the Tangent Geometrical Stiffness is

(2.15)

If a finite increment of element deformation vector expands as a quadratic function

, the work quantity that performed by element force S and increment .

Tangent Geometrical Stiffness KG can easily be expressed by second differential of ,

according to geometrical quantity and dynamic quantity in primary balance condition.

Therefore, the Tangent Geometrical Stiffness Matrix from Eq. (2.15), expressed in

particular form is

(2.16)

2.4 Iterative procedure of Tangent Stiffness Method

The procedure of Tangent Stiffness Method can be implemented in the way describe

below. The iterative technique proceeds until convergence of unbalanced force to a strict

balance position by using loading steps. The common procedure steps are:

Primary displacement:

Primary load:

Load increment:

Displacement for step of iteration (r):

Element deformation vector:

Element force vector:

Element force Nodal Force Conversion Matrix:

Tangent Stiffness Matrix:

The Fig.2.2 shows a convergence concept diagram for Tangent Stiffness Method. The

calculation goes clockwise. The first quadrant represents relation of displacement and load.

In Tangent Stiffness Method, nonlinear stiffness equation is not involved in calculation

process and it is marked as a dotted line on the graph. The fourth quadrant represents the

condition of compatibility, which expresses relation between nodal displacements in

15

global coordinate system and element deformation vector S from element deformation

vector. The second quadrant represents Equilibrium Equation, which is obtained from

Element Force vector and the coordinate transformation for the current displacement. This

is necessary to calculate in order to seek Nodal Force vector.

In Fig.2.2, iterative process in Tangent Stiffness Method begins form given state of

balance, which starts from the point O, and the following steps explain the iterative

solution process until convergence.

O

A

B

C

D

EF

G

H

I

J

KL

M Z

dS

s

2s

1s

1d

2d

1D

2D

2S 1S

1d

2d

1D

2D

1D

2D

0D

D

O

A

B

C

D

EF

G

H

I

J

KL

M Z

dS

s

2s

1s

1d

2d

1D

2D

2S 1S

1d

2d

1D

2D

1D

2D

0D

D

Fig.2.2 Iterative procedure of Tangent Stiffness Method

Stiffness Equation Equilibrium Equation

Element Force Equation Compatibility Equation

16

is obtained by solving Tangent Stiffness Method at point O, for given value of load increment . (O A B)

Calculate by adding to displacement in primary state.

From displacement obtain strict compatibility equation that is used to solve element deformation vector . (B C)

From the Element Force Equation , the state of equilibrium, F is provided by calculation for using the Equilibrium Equation. (C D E F)

Unbalanced force is calculated by cyclic calculations, from the difference of load condition and load vector which is in balance state in displacement position .

In this stage, condition F is considered as the primary state, and it finds solution by Tangent Stiffness Method which corresponds to . (F G H)

New displacement for nodes is calculated using the previous

Calculation for the next state of balance L is performed in the similar way.

(H I J K)

The rest of the steps is repeated continuously until the state of balance in first quadrant get s closer to Z, and the convergent solution is obtained for this calculation procedure.

Summary, iterative process in Tangent Stiffness Method can be expressed as

(2.17)

Thus, from the comparison between iterative process of Finite Element Method, Eq.

(2.17) and calculation performed by Tangent Stiffness Method, it is necessary to

formulate or apply any approximation concept. Although it is almost impossible to express

strict solution using nonlinear stiffness equation in reality, (actually, it is quite impossible

to be expressed with any analytic technique, and furthermore, it is not necessary to express

any of it in this method). The results from the iterative process which was explained

previously, is presented by the dotted line in Fig.3.2 that shows nonlinear stiffness equation

which is solved strictly while passing through the curved line in a random step.

In addition, Fig.2.3 is a flow chart for a geometrical nonlinear structure analysis

program performed by Tangent Stiffness Method.

17

INPUT PRIMARY VALUES START

INPUT COMPULSORY DISPALCEMENTS

CAL LOAD INTENSITY

CAL UNBALANCED FORECES

REACTION FORCES AT CONTROL POINT

CONVERGE?

COMPOSE TANGENT SRIFFNESS MATRIX

SOLVER SIMULTANEOUS EQUATIONS

RENEW NODAL POSITIONS

YES

NO

Fig.2.3 Flow chart of the program for the form finding

18

3. SHAPE ANALYSIS

3.1 Element potential function

In general type of finite element analysis, the strain is measured with respect to an

element (local) coordinate and relationship between displacement and strain is formulated

as a nonlinear equation. In contrast, in the tangent stiffness method, the relationship

between the vector of the element edge forces and nodal forces is formulated as linear

geometric stiffness. Therefore, when we need to calculate the finite strain in all elements, it

is not necessary to solve any complicated nonlinear equation. Instead, we just calculate

nodal displacements after deformation, and the elements behaviour in any conditions can

be tracked.

In order to regulate the element behaviour in element (local) coordinate, we define the

element measure potential which is expressed as the function of measurement such as

element length or element area. Defining element measure potential is equal to assuming

the "virtual" elemental stiffness. And it has no relationship with material's stiffness.

Let element measure potential is P, and let the vector of elements' measurements whose

component is independent of each other is s

(3.1)

Then we can get the element edge force S.

19

3.2 Triangular soap film element

Let the area of triangle is A, and unit membrane tensile is t (=constant, where is

tensile stress of the soap film element, t is the thickness of element). The element measure

potential is expressed as the function of element area,

tAP (3.2)

We can get the element edge force by differential calculus of element measure potential,

)3,2,1(2

1

itr

l

PN i

i

i (3.3)

Where, ri is the distance from the orthocentre of triangle (point 0) to each node.

As the element edge force can be set to the direction along the side of triangle, and

therefore, we can use the tangent geometrical stiffness matrix as same form as one of

triangular truss block. So, the form of KG becomes as follow:

G2 G3 G3 G2

G G3 G1 G3 G1

G2 G1 G1 G2

k + k -k -kK -k k + k -k

-k -k k + k (3.4)

( ), ( 1,2,3)i

i

Ni

l TGik e - (3.5)

In which, are the components of cosine vector alongside direction, and e is 33 unit

matrix.

Fig.3.1 Triangular soap film element

)3,2,1()( il

N Tii

i

i

Gi ek

20

Considering that the value of element stiffness K0 is much less than the value of tangent

geometrical stiffness KG [1], we ignore K0, and substitute K0 = 0 to the Eq. (2). From the

above equations, we get the tangent stiffness equation,

1 1S

2 G 2

3 3

U uU K uU u

(3.6)

3.3 Axial line element

The line element is connected with nodal point 1 and nodal point 2. Supposing that the

element measure potential is proportional to the power of length of line element, the

element measure potential can be expressed as:

nClP (3.7)

The axial line element force can be obtained by differential calculus of element measure

potential:

1 nnClN (3.8)

Where, C is the coefficient of axial lines.

Let are the components of cosine vector alongside direction, and we can rewrite the Eq.

(1) as:

N

1

2

U -U

(3.9)

Substituting the Eq. (11) to the above Eq. (2), and make it matrix.

L1 1T

2 2

U uK

U u (3.10)

2( 2) ( 2)( 2) ( 2)

n n nnCln n

T TL

T TT

e e K

e e (3.11)

For the Eq. (10), in the case of n=1, the element forces become constant, and for the Eq.

(11), the tangent geometrical stiffness of line elements becomes the same form as truss

element's. Therefore, form-finding with fixed axial forces can be realized.

21

In addition, in the case of n=2, axial force is proportional to the length of line element,

and Eq. (12) is linear. However, in the case of n>2, iterative steps are required because of

nonlinearity. The magnitude of n become larger, the length of all line elements on the

solution surface tend to be more uniform [2]

3.4 Nodal coordinate definition

3.4.1 Surface normal definition

As shown in Fig.3.2, the node i was connected with m (in this case, m=6) pieces of

elements (j=1~m). The area of element #j can be written as aj. When we consider that node

i shares 1/3 of each triangular area where inner pressure affects, we can express area vector

of each element as,

3

ij

ij

aA (3.12)

And normal directional cosine vector of surface at node i can be written as follows:

1 1

m m

i

j j

ij ijA A (3.13)

Therefore, the tangent stiffness equation Eq. (8) can be rewritten as

1 11 1

2 1 1 2

3 31 1

T

T

T

Z zZ zZ z

S

GK (3.14)

Thus, the resultant force on each node can be projected to normal direction.

Fig.3.2 Normal direction at a node #i

22

3.4.2 Tangent plane direction definition

In this paper, the line elements which can make the node distribution more uniform are

located along the 3-sides of triangular elements. We propose the independent coordinate

which has two axes perpendicular each other and both are to tangent direction of surfaces.

By using this coordinate, the axial forces of line elements can be projected to tangent plane

directions.

As shown in Fig.3.3, the normal direction of node i is i , and two tangent directions are

i and i , which are perpendicular with normal direction. The tangent stiffness equation Eq.

(12) can be rewritten as follows:

11 1

1 1 1 1 1

2 22 2 2

2 22

T

T T T

T T T

T

X xY yX xY y

S

GK (3.15)

The Eq. (16) and Eq. (17) are perfectly independent tangent stiffness equations. We

parallelly solve the Eq. (16) and Eq. (17) to get the incremental displacement of every node.

After that, we renew the nodal displacements, and calculate the unbalance forces in new

state. The new iterative calculation is continued until the unbalanced forces converge.

3.5 Simultaneous Control

3.5.1 Outline

When a soap film with a boundary is inflated under the inner pressure, the normal load

control is used to find shapes of curved surfaces. As the volume increases, the inner

pressure and curvature of curved surface also increase. However, when the inner pressure

is in the state of the maximum point of its p- path, we cannot get the solutions in

exceeding the maximum point of its path by using this method. Therefore, we need to

figure out another way to input the increments.

Fig.3.3 Coordinate system with normal direction and tangent direction

23

Simultaneous control is an incremental method, and it is effective to find shapes which

are over the maximum point and have high-rise. The technique controls both of

compulsory displacements on a control point and the inner pressure derived from the shape

in each iteration step at the same time, when it is applied to shape analyses under inner

pressure. When the unbalanced forced on every node as well as the average of converted

inner pressure converge, we can get the equilibrium shape with large volume.

3.5.2 The process of Simultaneous Control

Here, the Fig.3.4 is an example that simultaneous control is applied to isotonic surface

analysis. The primary shape in Fig.3.4 [a] is a low-rise equilibrium shape which is the

perfect balanced solution at all the nodes. However, our current purpose is to obtain the

equilibrium shape with very large volume and high-rise in Fig.3.4 [c]. This can be

considered as a post-buckling problem in wide sense, because this large volume solution is

on over the maximum point of its equilibrium path.

To begin with, we give the compulsory displacement of Dc to only this control point,

and the Fig.3.4 [b] is the shape of the first step of iteration. Next, we need to calculate the

intensity of inner pressure of P1, which is depending on this current shape. After this, shape

is changed continually, and also load intensity is modified in each iterative calculation.

Eventually, when both of the unbalanced force of all nodes and the reaction force of the

control point converged almost to zero, and the load intensity also should converge to a

constant value, we can find the perfect equilibrium shape in Fig.3.4 [c].

Fig.3.4 The process of simultaneous control

[a] Primary shape [b] First step of iteration [c] High-rise equilibrium shape

24

3.5.3 Converted inner pressure

In case of form finding of soap film, which is inflated by inner pressure, we calculate the

intensity of load in every iterative step as the average of the resultant of element edge

forces concentrated at every node. Namely, we set the inner pressure to be balanced for

every node.

As shown in Fig.3.5, the node #i was connected with m (in this case m=6) pieces of

elements (j=1~m). Let the vector of element edge forces of element #J connected to node

#i be indicated by Nij, and let matrix of equilibrium which relates Nij to the general

coordinate system by Jij. Then the nodal forces Uij expressed in the general coordinate

follow the equation,

(3.16)

Therefore, resultant of element edge force at node #i

(3.17)

The soap film has stiffness only towards normal direction of the surface and normal

directional cosine vector of surface at node #i can be written as follows:

(3.18)

Therefore, the resultant of element edge force can be projected to normal direction as,

(3.19)

The inner pressure P is a distribution load which is applied on the soap film from inside.

The direction of P is always normal to the changing surface during all stages of

deformation. So, balance equation to normal direction can be expressed as,

(3.20)

25

Therefore the converted inner pressure PCi (i =1, 2n) can be written as,

(3.21)

Let n be the number of all the nodes, the average of the converted inner pressure can be

obtained as,

(3.22)

We use this PAV as the inner pressure of each iteration step, so eventually the load term

becomes follower force. But we never need special modification for this follower force.

And it is enough to use the simple symmetric tangent stiffness matrix as same form as

when we use normal load increment.

i

j=1

j=2

j=m

j=m1

j= j=3

j=

i

j=1

j=2

j=m

j=m1

j= j=3

j=

Fig.3.5 Elements connected to node #i

26

4. NUMERICAL EXAMPLES AND DISCUSSIONS

4.1 Low-rise equilibrium shape by combination of soap film elements and 3D axial

lines (Loading control)

Fig.4.1.1 shows the primary shape and the connectivity. One side of initial shape is

divided into 18. The green lines denote the 3D axial lines to be converted into cables after

form finding. Further, the 2D axial lines to unify the mesh distribution are located on 3

sides of all triangular soap film elements. The tensile of the soap film element is 3.0kN/m,

and the coefficient of line element C=1. The magnitude of n is set 1, 2, 4, 6, and 10

respectively in 3D analysis with 3D axial lines.

Fig.4.1.2 ~ Fig.4.1.12 show the equilibrium shapes and convergent process of

unbalanced force respectively.

Fig.4.1.1 Initial plane and connectivity

18m

3@6m

27

3-D analysis (without 3D axial lines)

Tensile of the soap film element : 3.0kN/m

Acceptable minimum unbalanced force: 10-5

kN

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Volume=346.3 m3, Ip=1.0 kpa

Fig.4.1.3 Convergent process of unbalanced forces

Iterative steps

Unbalan

ced fo

rces

Normal direction

Tangent plane direction

Fig.4.1.2 Equilibrium shapes when the volume close to 347m3

28

3-D analysis (with 3D axial lines, n=1)



kN

Volume=347.3m3, Ip=1.077 kpa, n=1


n: Power of 3D axial lines; Ip: Inner pressure

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22 25 28 31 34 37


Iterative steps

Unbalan

ced fo

rces

Normal direction


29




kN

Volume=347.4m3, Ip=1.19 kpa, n=2



1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22 25


Iterative steps

Unbalan

ced fo

rces

Normal direction


30




kN

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22

Volume=347.0m3, Ip=1.425 kpa, n=4




Iterative steps

Unbalan

ced fo

rces

Normal direction


31




kN




1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22 25


Iterative steps

Unbalan

ced fo

rces

Normal direction


32




kN

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1 4 7 10 13 16 19 22 25 28 31 34





Iterative steps

Unbalan

ced fo

rces

Normal direction


33

4.2 Equilibrium shape by combination of soap film elements and 3D axial lines

(Simultaneous Control)

4.2.1 Effect of the 3D axial lines

As mentioned above, this thesis also mentioned another usage of the axial line elements

that can be converted to cable elements with real stiffness after form-finding. This type of

axial line elements has degree of freedom to all of 3 dimensions, and we call it "3D axial

line". In this chapter, we will make three discussions. The first one is the comparison of the

proposal with 1D soap film analysis without 3D axial lines, and the sencond is the

influence of power n in the 3D axial lines. At last is influence of mesh connectivity in

determined shape.

As shown in Fig.4.2.1, the radius direction of initial shape is divided into 8, and the

circumferential direction is divided into 8. The tensile of the soap film element is 3.0kN/m,

and the coefficient of line element C=1. For 3D analysis, the green lines denote the 3D

axial lines to be converted into cables after form finding. Further, the 2D axial lines to

unify the mesh distribution are located on 3 sides of all triangular soap film elements.

Simultaneous control is an incremental technique adopted in this example. The control

point is set in the center of the initial shape, and compulsory displacement is given along

vertical direction. Compulsory displacement of each step is given as Dc=0.1m for the 1D

soap film analysis. In contrast, for the 3D analysis, the compulsory displacement of each

step is given as Dc=1m.


R=8m

Control point

34

The Fig.4.2.2 denotes inner pressure volume curve. The tendencies of curves are so

similar. However, "3D axial line" can restrict the deformation of soap film during the

process of form-finding. Therefore, larger air pressure is required to get the equal volume.

The Fig.4.2.3 shows the equilibrium shapes in cases with 3D axial lines and without 3D

axial lines respectively.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5000 10000 15000 20000 25000

Inner p

ressure (k

pa)

1 DOF, Dc=0.1m

3 DOF, Dc=1m, n=2


Fig.4.2.2: Inner pressure-volume curve

Volume (m)

35

R=3.0m, Ip=0.36kpa, V=453.6m

R=3.0m, Ip=0.39kpa, V=467.1m

R=6.0m, Ip=0.61kpa, V=1029.6m

R=9.0m, Ip=0.66kpa, V=1782.8m

R=6.0m, Ip=0.55kpa, V=990.2m

R=9.0m, Ip=0.61kpa, V=1688.7m

36

R=12.0m, Ip=0.64kpa, V=2821.1m

R=15.0m, Ip=0.60kpa, V=4241.4m

R=18.0m, Ip=0.55kpa, V=6143.4m

R=12.0m, Ip=0.60kpa, V=2615.3m

R=15.0m, Ip=0.57kpa, V=3830.3m

R=18.0m, Ip=0.52kpa, V=5388.4m

Fig.4.2.3 Equilibrium shapes (Left: 1D analysis; Right: 3D analysis)

R: Rise of the control point, Ip: Inner pressure, V: Volume

37

In Fig.4.2.4, the increment of the compulsory displacement is given as Dc=2m and

Dc=0.1m respectively for 1D soap film analysis (Fig.4.2.4 [a], [b]). In contrast, for the 3D

analysis with 3D axial line elements, the increment of the compulsory displacement of

each step is given as Dc=2m (Fig.4.2.4 [c]).

When the 1D analysis is applied, too large increment makes the area of triangular

elements around the control point large. Further, because of no stiffness to the tangent

direction, the uneven distribution of the size of triangles would be kept all through the

process of the analysis. If the magnitude of increment becomes larger than 2m, it causes

divergence. Therefore, the compulsory displacement in each iterative step should be less

than 0.1m in case of the 1D analysis.

On the other hand, the 3D analysis brings the stable calculation process, even though the

increment is larger than 2m. Compare with 1D soap film analysis, 3-Danalysis can obtain

better equilibrium solution because 2D axial line elements optimize the node distribution.

Meanwhile, 3D axial line can restrict the deformation of soap film during the process of

form-finding. Therefore, the determined shape can be expected to composite structure of

membrane and cables.

[c] 3-D analysis with "3D axial line"

Dc=2m, V= 990 m, n=5 [a] 1-D analysis

Dc=2m, V = 1070 m

[b] 1-D analysis

Dc =0.1m, V= 903 m

Fig.4.2.4: Equilibrium shapes when the total compulsory displacement = 6 m

n:Power of 3D axial lines, Dc: Increment of compulsory displacement, V:Volume

38

4.2.2 The power of n in 3D axial lines

As previously mentioned, the 2D axial line elements located on 3 sides of all triangular

soap film elements can unify the mesh distribution, and 3D axial line elements to be

converted into cables after form finding which can restrict the deformation of soap film

during the process of form finding. Here, we will discuss the influence of power n in the

3D axial line elements.

Case 1 Assuming that the power of n in the 3D axial line element force equation is kept the

same in all of 3D axial line elements.

In this situation, every 3D axial line element has the same stiffness. Here, the green lines

denote the 3D axial lines to be converted into cables after form finding. Further, the 2D

axial lines to unify the mesh distribution are located on 3 sides of all triangular soap film

elements, as shown in Fig.4.2.5.

In addition, the tensile of the soap film element is 3.0kN/m. Simultaneous control is

used as an incremental technique in this example. Compulsory displacement of each step is

given as Dc=1m along the vertical direction and the total compulsory displacement is 5m.

Here, we will discuss the influence of soap film by 3D axial lines. The power of n and

coefficient of 3D axial lines C are discussed in this example. The equilibrium shapes are

shown in Fig.4.2.6.

Fig.4.2.5 Initial plane and mesh connectivity

39

[a] Without 3D axial line

V = 800.6m, Dc=1m

[b] With 3D axial line, n=1

V= 809.6m, Dc=1m, C=1

[c] With 3D axial line, n=4

V= 893.9m, Dc=1m, C=1

[d] With 3D axial line, n=1

V= 912.5m, Dc=1m, C=10

[e] With 3D axial line, n=6

V= 1155.5m, Dc=1m, C=1

Fig.4.2.6 Equilibrium shapes when the total compulsory displacement= 5m

n: Power of 3D axial lines, C: Coefficient of 3D axial lines

V: Volume, Dc: Increment of compulsory displacement

[f] With 3D axial line, n=1

V= 1010.5m, Dc=1m, C=30

40

In the case of n=1 and C=1, axial forces of 3D axial line elements become unit force.

The equilibrium shape is almost the same as in the condition of without 3D axial lines

(Fig.4.2.6 [a]). However, as n or C become large (Fig.4.2.6[c] ~ [f]), the restriction of soap

film by 3D axial lines become more strong. Therefore, the combination of 3D axial lines

and soap film elements can be expected to determine a composite structure of membrane

and cables.

In the Fig.4.2.7, the green lines denote the 3D axial lines, and straight line 1 is composed

of 3D axial line elements. Here, we pick up straight line 1 and straight line 2 to discuss the

influence by the power of n, and n is set from 1 to 5.

The Fig.4.2.8 and Fig.4.2.9 denote the curves on the straight line 1 and straight line 2

respectively compare with different value of power of n in the 3D axial lines. Here,

(4.2.1)

In which, x, y is the coordinate of nodes on straight line 1 and straight line 2.

Straight line 1

Straight line 2

Fig.4.2.7: Straight line 1 and straight line 2

41

In the Fig.4.2.8, as the magnitude of n increases, the deformation of underneath curve is

restricted away from the green curve which is in the condition of without 3D axial line

elements. However, the deformation of upper curve is away from the green curve to

contrary direction. This is because the nodes are perfect fixed in the boundary of initial

shape, and power of n in the 3D axial lines is kept the same, which causes the restriction of

lower part is larger than uppers.

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9

n = 0

n = 5

n = 4

n = 3

n = 2

n = 1

R

Z

Fig.4.2.8 The curve of the straight line 1

Here, "n = 0" means without

3D axial line elements

Underneath

Upper

42

Next, the deformation of curve of straight line 2 is discussed.

In the Fig.4.2.9, as the magnitude of n increases, the curvature of curve becomes larger.

3D axial lines are only located on 3 straight lines. The soap film is divided into 3 parts.

When the soap film with a given boundary is inflated by air pressure from inside, some

parts of surface without 3D axial lines are expanded much larger.

0

1

2

3

4

5

6

0 2 4 6 8 10 12

n = 5

n = 4

n = 3

n = 2

n = 0

R

Z




43

Case 2 Assuming that the power of n is a linear function of 3D axial line element number

m on a straight line and incremental coefficient k.

In the Fig.4.2.10, the green lines denote the 3D axial lines, and straight line 1 is

composed of 3D axial line elements. Therefore, for straight line 1, the magnitude of n can

be obtained according to the Fig.4.2.11. Where, k is set from 0.1 to 0.5, and m = 8 in this

example.

The Fig.4.2.12 shows the equilibrium shapes with different magnitude of k when the

total compulsory displacement is 5m.

Straight line 1

Straight line 2

1 2 3 4 5 6 7 8 9

k = 0.1

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

n = 1.7 n = 1.6 n = 1.5 n = 1.4 n = 1.3 n = 1.2 n = 1.1 n = 1

k = 0.2 n = 2.4 n = 2.2 n = 2.0 n = 1.8 n = 1.6 n = 1.4 n = 1.2 n = 1

k = 0.3 n = 3.1 n = 2.8 n = 2.5 n = 2.2 n = 1.9 n = 1.6 n = 1.3 n = 1

k = 0.4 n = 3.8 n = 3.4 n = 3.0 n = 2.6 n = 2.2 n = 1.8 n = 1.4 n = 1

k = 0.5 n = 4.5 n = 4.0 n = 3.5 n = 3.0 n = 2.5 n = 2.0 n = 1.5 n = 1

Node 3D axial line element

Fig.4.2.11 The value of n for straight line 1

Fig.4.2.10 Straight line 1 and straight line 2

44

[a] Without 3D axial line

V = 800.6m, Dc=1m

[b] With 3D axial line

V= 822.2m, k=0.1, Dc=1m

[c] With 3D axial line

V= 839.6m, k=0.2, Dc=1m

[d] With 3D axial line

V= 861.7m, k=0.3, Dc=1m

[e] With 3D axial line

V= 888.3m, k=0.4, Dc=1m

[f] With 3D axial line

V= 918.5m, k=0.5, Dc=1m

Fig.4.2.12: Equilibrium shapes when the total compulsory displacement= 5m

n: Power of 3D axial lines, Dc: Increment of compulsory displacement, V: Volume

45

As previously mentioned, if power of n in the 3D axial line element force equation is

kept the same, the restriction of lower part is larger than uppers. Thus, the deformation of

underneath curve is different with the deformation of upper curve.

However, in this example, we adjust the power of n in the 3D axial line element force

equation to make the restriction of lower part is equal to the restriction of upper part. The

deformation of curve is shown in Fig.4.2.13. As the magnitude of n increases, the curvature

of curve becomes larger.

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9

n = 0

k = 0.1

k = 0.2

k = 0.3

k = 0.4

k = 0.5




Underneath

Upper

R

Z

46

Next, the deformation of curve of straight line 2 is discussed.

In the Fig.4.2.14, as the magnitude of n increases, the curvature of curve becomes larger.

The tendency of deformation of curve is similar to the situation which assuming that power

of n is kept the same in all of 3D axial line elements.

0

1

2

3

4

5

6

0 2 4 6 8 10 12

n = 0

k = 0.1

k = 0.2

k = 0.3

k = 0.4

k = 0.5




R

Z

47

4.2.3 In case that different shape is obtained depending on mesh connectivity

The Fig.4.2.15 shows the same initial shape is meshed by two kinds of different

connectivity. The radius direction of initial shape is divided into 8, and the circumferential

direction is also divided into 8. The control point is set in the centre, and compulsory

displacement is given along vertical direction. The green lines denote 3D axial line and its

power is n=4. Also in this example, the 2D axial lines to unify the mesh distribution are

located on 3 sides of all triangular soap film elements. The tensile of the soap film element

is 3.0kN/m. Fig.4.2.16 illustrates the inner pressure-volume curve.

Fig.4.2.17 shows equilibrium solutions compared with two kinds of mesh connectivity.

For 1st mesh connectivity, the surface is inflated with rotation and it looks "twisted". In

contrast, when the 2nd

mesh connectivity is adopted, the surface is inflated without rotation.

This is why the shape of the polyhedral solution is depending on the mesh connectivity.

Therefore, we have to be careful when we prepare the mesh connectivity in some cases like

this example.

[a] 1st mesh connectivity


[b] 2nd

mesh connectivity

Control point

R=8m

Control point

R=8m

48

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 7000 8000

1st mesh connectivity, Dc=0.1m, n=4

2nd

mesh connectivity, Dc=0.1m, n=4

Inner p

ressure (k

pa)


Fig.4.2.16 Inner pressure-volume curve

Volume (m)

49

R=6.0m, Ip=0.75kpa, V=1139.5m R= 6.0m, Ip= 0.68kpa, V=1062.5m

R=12.0m, Ip=0.82kpa, V= 4496.3m R= 12.0m, Ip=0.80kpa, V= 3804.2m

Fig.4.2.17: Equilibrium shapes of two mesh connectivity

R= 9.0m, Ip=0.86kpa, V= 2233.8m R= 9.0m, Ip= 0.80kpa, V= 1967.8m

[a] 1st mesh connectivity [b] 2

nd mesh connectivity


50

5 Equilibrium shape by combination of soap film elements and 3D axial

lines Case study (1~3)

5.1 Case 1





and the coefficient of 3D axial line C=1 in this example.

Compulsory displacement of each step is given as Dc=1m along vertical direction for

the 3D analysis with 3D axial lines and 3D analysis without 3D axial lines respectively.

The Fig.5.1.2 denotes inner pressure volume curve. Obviously, the tendency of curves is

quite different. "3D axial line" can restrict the deformation of soap film during the process

of form-finding. Therefore, larger air pressure is required to get the equal volume.

Control point

8m


51

The Fig.5.1.3 shows the equilibrium shapes in cases with 3D axial lines and without 3D

axial lines respectively.

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1000 1500 2000 2500 3000 3500 4000

Fig.5.1.2: Inner pressure-volume curve

Volume (m)

Inner p

ressure (k

pa)

3 DOF without 3D axial lines, Dc=1m

3 DOF with 3D axial lines, Dc=1m, n=4


52

R=5.0m, Ip=0.53kpa, V=784.3m R=5.0m, Ip=0.77kpa, V=873.6m



53

R=8.0m, Ip=0.60kpa, V=1466.3m

R=9.0m, Ip=0.61kpa, V=1749.6m

R=10.0m, Ip=0.60kpa, V=2065.7m

R=8.0m, Ip=1.0kpa, V=1820.7m

R=9.0m, Ip=1.05kpa, V=2323.4m

R=10.0m, Ip=1.08kpa, V=3010.3m

Fig.5.1.3 Equilibrium shapes

(Left: 3D analysis without 3D axial lines; Right: 3D analysis with3D axial lines)


54

5.2 Case 2

As shown in Fig.5.2.1, the radius direction of initial shape is divided into 7, and the

circumferential direction is divided into 7. The tensile of the soap film element is 3.0kN/m,

and the coefficient of line element C=1. The green lines denote the 3D axial lines to be

converted into cables after form finding. Further, the 2D axial lines to unify the mesh

distribution are located on 3 sides of all triangular soap film elements.


point is set in the center of the initial shape, and compulsory displacement is given as Dc =

0.1m along vertical direction.

The Fig.5.2.2 shows Inner pressure volume curve.

The Fig.5.2.3 shows Equilibrium shapes in case with 3D axial lines.


Control point

55

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1000 2000 3000 4000 5000 6000


Volume (m)


Inner p

ressure (k

pa)

3 DOF, Dc=1m, n=4

56





57

5.3 Case 3





and the coefficient of 3D axial line C=1 in this example.


point is set in the center of the initial shape, and compulsory displacement is given as Dc =

0.1m along vertical direction.

The Fig.5.3.2 shows Inner pressure volume curve.

The Fig.5.3.3 shows Equilibrium shapes in case with 3D axial lines.

Fig.5.3.1: Initial plane and connectivity

Control point

W =

40 m

L = 40 m

58

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 5000 10000 15000 20000 25000 30000 35000

3 DOF, Dc=0.1m, n=4


Volume (m)


Inner p

ressure (k

pa)

59

R=8.0m, Ip=0.14kpa, V=16382m R=10.0m, Ip=0.15kpa, V=21718m

R=12.0m, Ip=0.16kpa, V=27316m R=14.0m, Ip=0.17kpa, V=31099m



60

6. MINIMAL SURFACES

6.1 Introduction

Minimal surfaces are defined as surfaces of minimal area which are enclosed by a given

fixed boundary. Since centuries, they are one of the oldest toys of mathematicians. There

exists an enormous amount of knowledge of how to describe and to generate them. From

the mechanical point of view minimal surfaces are determined by an isotonic stress field

that can be experimentally simulated by the soap film analogy.

Minimal surfaces have gradually been translated from the field of mathematics into the

architectural design research due to their remarkable geometric properties. As we all know,

the knowledge of the form is a very important aspect of design of structures. Having the

least area property minimal surface is mostly used for light roof constructions, and form-

finding models for membrane structure. Among the surfaces having the same boundary,

minimal surface is the surface of the least area. Therefore, its weight is less and the amount

of material is reduced on minimum.

The minimal surface form-finding analysis of membrane structures is a nonlinear

problem. Its strong nonlinearity makes its computation to be a great challenge, because

calculation usually needs a lot of iterations and sure convergence of the calculation process

should be guaranteed. In this chapter, the author is to adopt the tangent stiffness method for

generating minimal surface. The method uses the iterative process of the tangent stiffness

equation that derived from the equilibrium equation between the nodal forces and the

element edge forces. This method can be used either for an approximate form of the

membrane surface in the primary design stage or for an initial solution for further

computation of real membrane analysis.

Two classical minimal surfaces (Catenoid and Scheck surface) will be preferred in

follow examples to compare with analytical solutions which are produced by this method.

The results show that this method is simple, efficient and reliable with highly satisfactory

accuracy.

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6.2 Catenoid

The Fig.6.1 shows a soap film between two circular rings. The rings are pulled apart

until the distance between the rings is about 0.6627 times the diameter, and the soap film

pops. I use the term soap film to denote a surface that does not enclose a volume, the way a

soap bubble does. A soap film forms a minimal surface, which means that any small

deformation of the surface would have more area. The mean curvature of a surface is the

average of the two principal curvatures. Every minimal surface has zero mean curvature.

The converse, however, is not true. A surface of zero mean curvature corresponds to a

critical point (a local minimum or a saddle point). In other words, some surfaces with zero

mean curvature are not minimal.

The Catenoid is a minimal surface of rotational symmetry which can be expressed by

formulation as follows:

(6.1)

Where, c > 0 is a constant and cosh(x) = (ex + e

-x)/2 is the hyperbolic cosine.

Thus, every surface of revolution with r = c cosh (z/c) has zero mean curvature. Assume

that the rings have radius r =1, and here z are replaced at z = z*. Then z* and c are

related by

(6.2)

While we cannot solve for c as a function of z*, we can solve for z* as a function of c:

(6.3)

Fig.6.1 The surface of catenoid

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The Fig.6.2 plots

. Note that for z* < 0.6627, there are two values of c.

The blue curve shows c corresponding to catenoids that are not minimal surfaces, even

though they have zero mean curvature. These soap films are unstable and will pop. The red

Wu YS Master Thesis

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