cheng wang paper

7/30/2019 Cheng Wang Paper

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Finite Element Method in Form-finding Process for Membrane StructuresaJin Wang,

bCheng Yan Chie

aMaster of Civil Engineering,[email protected]

bMaster of Civil Engineering, [email protected]

Part 1AbstractOur project concentrateson the most-widely-used methods for Form-finding analysis of membranestructure: finite Element Methods. Form-finding analysis, in non-linear finite element,of two types ofmembrane structure isconducted and verified with the aid ofANSYS. The influence of mesh density andself-balanced iterations on accuracy of form-finding isstudied.Part 2IntroductionMembrane structure has been rigorously developed in the late twentieth century. Its highpeaks and

sweeping curved lines, as well as allowing clear open space within the building, create desirable aesthetic

values to infrastructures. On top of that, it is a new -form of structure for sustainable development

(Shaffer, 1996; Lewis 1998). This construction method carries load mostly in tension, with limited

compression and bending. Some well-known papers (K.Ishii, 1995; D.S. Wakefield, 1999; M. Saitoh2001) in the industry demonstrated remarkable models.

As most tension membranes have to be reinforced by interior and perimeter cables. Therefore, some

research approach is using compressive rigid member as masts, assembled to the global structure to

provide adequate stability. Their focuses are on membrane element, cable elements and compressive

membranes which has been recently verified (Talvik, 2001; Li 2004). These are all based on the

fundamentals on non-linear continuum mechanics.

With an aim mainly focuses on the membrane analysis, several models were built from the process of pre-

stressing to the membranes which allowing themto withstand loads (Crisfield, 1991; Belytschko 2000;Holzapfel 2000). Such form-finding analysis is actually the process to pre-stress the membrane to find the

minimum surface area, which can satisfy the boundary conditions and aesthetic requirement. In the other

words, stress is equally distributed throughout the membrane.

This project is set up to investigate on the form-finding analysis of membrane structure by using finite

element methods. This analysis is based on the theory of non-linear finite element. With the help of

ANSYS computation, experimentsaboutmesh density and self-balanced iterations are designedin orderto find theirinfluence on theaccuracy of form-finding.Part 3Finite element implementationNonlinear Finite Element Method is used and our approach modified as the ones stated in Gil (2006) and

Valdes et al. (2009).Thegeometric nonlinearity is considered when establishingthe equilibrium equation.Therefore, all equilibrium equations are built upon the deformed configurations. Asthe strain is smallcomparedwith the deformation, we assume both membranes and cable arelinearelastic. In the firstprocess, form-finding, the membrane structures will undergo large displacement to achieve equilibrium.

Membrane structure, illustrated as space triangle, in the coordinationsystem xyz with displacement vector u and element nodal

displacement a.

Displacement vector { }=T

u u v w

Element nodal displacements { }1 1 1 2 2 2 3 3 3=Te

a u v w u v w u v w


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The element shape function matrix N when z is in terms of x and y:

1 2 3

1 2 3

1 2 3

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

N N N

N N N N

N N N

=

where

1 2 3 3 2 2 3 3 2( - )+( - ) +( - )N x y x y y y x x x y=

2 3 1 1 3 3 1 1 3( - )+( - ) +( - )N x y x y y y x x x y=

3 1 2 2 1 1 2 2 1( - )+( - ) +( - )N x y x y y y x x x y=

Displacement of arbitrary node: = eu Na

Since the relative position between element and the local coordinate could be arbitrary, the triangle

element could lie entirely in the xy plane, with its direction in x.

When1 1 2

=0, =0, =0x y y and the area of element is2 3

2A x y=,

theshape function can be furthersimplified:

1 2 3 3 3 2- +( - )N x y y x x x y= 2 3 3-N y x x y= 3 2N x y=

The equilibrium equation is-1

= +

j

T T T

v Av

E dv u pdv u qdA

Strain increment for an arbitrary node in the element: { }=T

x y z and = +

L NL

Linear and nonlinearstrainincrement is = eL L

B a andN

1 1=

2 2

e e

L NLB a AG a =

Nodal displacement in each time step { }1 1 1 2 2 2 3 3 3=Te

a u v w u v w u v w

Matrix connecting linear strain and displacement, LB ,

3 3

3 2 3 2

3 2 3 3 3 2

- 0 0 0 0 0 0 0

1= 0 - 0 0 - 0 0 0

2- - 0 - 0 0 0

L

y y

B x x x xA

x x y x y x

NLB is the matrix connecting nonlinear strain and displacement

0 0 0

= 0 0 0

u v w

x x x

u v wA

y y y

u v w u v w

y y y x x x

3 3

3 3

3 3

3 2 3 2

3 2 3 2

3 2 3 2

0 0 0 0 0 0 0

0 - 0 0 0 0 0 0

0 0 - 0 0 0 0 01

- 0 0 - 0 0 0 02

0 - 0 0 - 0 0 0

0 0 - 0 0 - 0 0

y y

y y

y yG

x x x xA

x x x x

x x x x

=

Using the equations above, element equibrium equations in local coordinate can be obtained as

( ) =e e e e eL NLK K a R F+ whereLinearand nonlinearelement stiffness matrix are given by

=e T

L L L

v

K B DB dv and = Ge TNLv

K G M dv

Element load vector in local coordinate e T T

v A

R N Pdv N qdA= +

Element nodal forces in local coordinate e T

L

v

F B dv=

As =

v

dv At (where A is area and t is thickness of the element) the above equation s can be further

illustrated as =e T

L LK AtB DB, G

e T

NLK AtG M= , e T T

v A

R N Pdv N qdA= +


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Self-weight and external load can be ignored in the form finding process, eR =0.

Element stress matrix, M

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

X XY

X XY

X XY

XY Y

XY Y

XY Y

M

=

is the membrane element stress vector. During form-finding, { }= 0T

x y .

The transform matrix between local and global coordinate Tis0 0

= 0 0

0 0

T

Where

cos ( , ) cos ( , ) cos ( , )

= cos ( , ) cos (Y, ) cos ( , )

cos ( , ) cos (Y, ) cos ( , )

X x Y x Z x

X y y Z y

X z z Z z

The element equilibrium equation in global coordinate is:

( ) =e e e e eL NLK K U R F+

Where e T eL NL

K T K T= ,e T e

NL NLK T K T= =

e T eU T a

e T eF T F= ,

{ }0 0 0 0 0 0 +3

Te AtR p p p= { }

1

3

T

X X Y Y Z Z X X Y Y Z Z X X Y Y Z ZA q A q A q A q A q A q A q A q A q

In the above formulae, p is the body force working on membrane element in the global coordinateX

A ,

YA ,

ZA are project of the membrane elements area on YOZXOZXOYplane, respectively and

Xq ,

Yq ,

Zq are the element force per unit area in the direction of X, Y and Z. During the form finding, the

self-weight and external load can be ignored, Xq , Yq , Zq

are zero.

Part 4 Mathematical Model assumptions

1. Only tensile force is applied to membrane and its associated cable. There is no compressi on load and

flexural resistance. And the membrane and cables are under pinned-pinned conditions.

2. The model remains certain rigidity under slack and wrinkle conditions.

3. Under the tensile stress working condition, the material nonlinear is ignored. The membrane is taken asorthotropic and cable is linear elastic.4. The model adopts triangle element for membrane and line element for cable.

5. The cables cross-sectional areas remain unchanged and the strain is small.

6. There is no relative displacementbetween the membrane and its cable throughout the analysis.Part 5Pre-stressed structural membranes analysis UsingANSYSThe analysis in ANSYSSoftware is based on the nonlinear finite element method. We starttheiterationwith a 2D plane configuration. By changing the coordinates of the controlled points gradually and

conducting equilibrium iterations, the model will reach the desireshape. One of the crucial assumptionsisthe small elasticity modulus. The virtual value is generally set as one-thousandth or one-ten thousandth of

the actual one so that themodel can then undergo free deformation. Meanwhile, theinternal force of thesystem remains unchanged.

In this way, with the uplift of supporting nodes in the system, the initial 2D plane can be modeledas thenon-linear structural membrane. The internal force in final equilibrium condition should equal to the


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default value.

Procedure of ANSYScomputationStep 1: Construct the membrane with 2D geometric front, side,top viewsStep 2: Set the virtual elasticity modulus, normally 3-4grades lower than the actual levelsStep3: Set the element type and its constants, andpre-stressto defaultvalues of the modelsStep4: Define the structural properties, model and conduct meshing, and set boundary condi tions of themodel.

Step 5: Uplift the nodes on the boundary, and obtain the pre-stressed condition by decreasing temperature.

Step 6: Select "FULL" mode and large deformation mode, and start self-equilibrium iteration.

Step 7: Reset the elasticity modulusof the membrane to the actual value, andtemperatureto default valuefrom the result before further loading analysis.

Part 6 NumericalResultsExample 1: Hyperbolic Paraboloid membrane structure in rhombus plane

The surface equation for this problem should be2 2

2 2= (- + )

x yz f

a b Let a=b=5m f=1mthe origin lies at the leftmost node of the rhombus

andthe vertical distance between the adjacent nodes is 2m

Then the surface equation becomes

2 2( 5)z=

25

y x

In ANSYS, membrane element shell41is adopted. Atthe beginning, the followings are set:Elastic modulus E is

3 23 10 KN m , thickness t is 1.2mm, thermal expansion coefficient is 15 and

reference temperature is0

0 C . During the self-balance iteration: Eis6 2

3 10 KN m , is 0.015.

The initial pre-stress condition could be simulated by lowering the temperature. Using the equation

T=tE

, we could obtain the change of temperature with regard to the corresponding pre-stress in the

membrane, which is -

1o

C in our problem.Set the modelmeshing as20 sections along the radius direction andset self-balanced iteration times as40.The meshing and final Mises stress diagrams are asbelow which thecurvature is at minimum.

Figure 3 Meshing Figure 4 Mises stress diagram

Table 3-1(In appendix A) shows the comparison between calculated and theoretical value of 15points along the positive x direction. According to the comparison, the error of our calculation is

acceptable, with a maximum error, 1.53%. This alsoprovesthe validity of our program in ANSYS.


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Example 2: Rotary Catenoid Membrane structure

Rotary Catenoid hasthe following analytic solution,

Let1

= =5a r m , =12h m ,2=27.78r m ,

Then the surface equation becomes 2z=-5(ln(r+ -25)-ln 5)+12r

Here membrane element shell41is adopted. At the beginning, the followings are set:

Elastic modulus E is3 2

3 10 KN m , thickness t is 1 mm, thermal expansion coefficient is 15 andreference temperature is

00 C . During the self-balance iteration: Eis

6 23 10 KN m , is 0.015.

The initial pre-stress condition could be simulated by lowering the temperature. Using the equation

T=tE

, we could obtain the change of temperature with regard to the corresponding pre -stress in the

membrane, which is -1o

C in our problem. Aside from that, in order to furtherexplore the influence ofmeshing density and self-balanced iteration, two experiments are designed as indicated below:Set the model meshing as 20 sections along the radius direction and self-balanced iteration times as

100,400 and 800 respectively;

Setself-balanced iteration times as 100 andmesh the model into10, 20and 30 sections along the radiusdirection respectively.

Table 1: Experiments A and B

# Meshing sections# Iteration times

A-1 20 100

A-2 20 400

A-3 20 800

B-1 10 100

B-2 20 100

B-3 30 100

The form-finding results of all the experiments are given in the Appendix B.

According to table B-1, B-2 and B-3, the maximum errors for iteration times 100, 400, and 800 are

2.95%, 0.85% and 0.8% accordingly. From the results we could see the trend that the larger the iterationis, the less the maximum error will be. Since the maximum error when iteration for 800 times isacceptablysmall, 0.8%, our form-finding result could beadopted as final result.According to figure B-4, B-5 and B-6, the denser the meshing is, the more uniform the stresses are

distributed and the smoother the surface will be. According to table B-4, B-5 and B-6, the maximumerrors for10,20 and 30 meshingsectionsare 4.66%, 2.95% and 2.64%,respectively.

2 2z=- [ ln ( + - )- ln ]+a r r a a h 1 2r r r


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Part 6Conclusions

From the comparison between the experimental result and the analytical solution, our maximum errorsinexample 1 and example 2 are 1.53% and 0.8%, respectively, which are acceptably small. Therefore,minimum surface with uniformly distributed stresseshas beenobtained, which means our methods areproved to becorrect.Meanwhile we come to the conclusions: the larger the iteration is, the less the maximum error willbe; thedenser the meshing is, the more uniform the stresses aredistributed, and the smoother the surface will be.

Reference:

1. D.S. Wakefield, Engineering analysis of tension structures: theory and practice, Eng. Struct. 21(1999) 680690.

2. M. Saitoh, A. Okada, Tension and membrane structures, J. IASS 42 (135136) (2001) 1520.3. H. Berger, Form and function of tensile structures for permanent buildings, Eng. Struct. 21 (1999)

669679.

4. K. Ishii, Membrane Structures in Japan, SPS Publishing Company, Tokio,1995.5. W.J. Lewis, Lightweight tension membranes: an overview, Civ. Eng. 126 (1998) 171181.6. R.E. Shaeffer, Tensioned Fabric Structures, American Society of Civil Engineers, New York, 1996.7. T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley,

New York, 2000.

8. M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, volume 1: Essentials,Wiley, New York, 1991.

9. G.A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Wiley, NewYork, 2000.

10.Jin-Jun Li, Chan Siu-Lai, An integrated analysis of membrane structures with flexible supportingframes, Finite Elements Anal. Des. 40 (2004) 529540.

11.Talvik, Finite element modeling of cable networks with flexible supports, Comput. Struct. 79 (2001)24432450.

12.Finite element analysis of prestressed structural membranes, Gil, Antonio J , Bonet, Javier ,FiniteElements in Analysis & Design, ISSN 0168-874X, 2006, Volume 42, Issue 8, pp. 683 - 697

13.Nonlinear finite element analysis of orthotropic and prestressed membrane structures, by Valds, J.G;Miquel, J; Oate, E,Finite Elements in Analysis & Design, ISSN 0168-874X, 2009, Volume 45,

Issue 6, pp. 395 - 405

14.Hoaward B. Wilson (2003) Advanced Mathematics and Mechanics Applications usingMatLab Chapman Hall/CRCSaeed

15.Moaveni (2008) Finite Element Analysis: Theory and Application with ANSYS ThirdEdition Pearson


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APPENDIX A

Table A-1 Comparison between calculated and theoretical values for Hyperbolic Paraboloid (Unit:

m)

# X Y Z Z(Theoretical) Error (%)

1 10.0000 0.0000 1.0000 1.0000 02 9.6477 -0.2212e-6 0.8635 0.8641 0.06

3 9.3210 0.9702e-6 0.7453 0.7468 0.20

4 8.9851 0.1604e-5 0.6329 0.6352 0.37

5 8.6492 0.1688e-5 0.5298 0.5327 0.55

6 8.3130 0.1785e-5 0.4359 0.4391 0.73

7 7.9771 0.1921e-5 0.3514 0.3545 0.90

8 7.6417 0.2092e-5 0.2762 0.2792 1.06

9 7.3073 0.2229e-5 0.2104 0.2130 1.21

10 6.9742 0.2262e-5 0.1538 0.1559 1.33

11 6.6424 0.2174e-5 0.1064 0.1079 1.4412 6.3120 0.1987e-5 0.0678 0.0689 1.51

13 5.9828 0.1742e-5 0.0380 0.0386 1.53

14 5.6546 0.1478e-5 0.0169 0.0171 1.43

15 5.3271 0.1224e-5 0.0043 0.0043 0.62


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Table B-3: Comparison between calculated and theoretical values for Rotary Catenoid inExperiment A-3(Unit: m)

Iteration: 800times# X Y Z Z(Theoretical) Error (%)

1 25.2493 0.8196e-5 0.4904 0.4872 0.66

2 21.4373 0.8992e-4 1.3354 1.3252 0.77

3 17.6018 0.1002e-3 2.3641 2.7428 0.80

4 15.0248 0.3344e-4 3.2029 3.1775 0.80

5 12.4370 0.2073e-4 4.2271 4.1936 0.80

6 9.8482 0.9875e-4 5.5473 5.5037 0.79

7 8.5715 0.2920e-4 6.3817 6.3322 0.78

8 7.3384 0.4870e-4 7.3913 7.3353 0.76

9 6.2173 0.20744e-4 8.6406 8.5782 0.73

10 5.3561 0.1339e-4 10.1913 10.1239 0.67

Figure B-1: Iteration100 times

Figure B-2 Iteration 400times


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Figure B-3: Iteration 800times

u For Experiment BSet self-balanced iteration times as 100 and mesh the model into10, 20 and 30 sections along the

radius direction respectively.

Table B-4Comparison between calculated and theoretical values forRotary Catenoidin ExperimentB-1(Unit: m)

Meshing section #: 10times# X Y Z Z(Theoretical) Error (%)

1 25.1219 0.1140e-3 0.5362 0.5130 4.53

2 22.4606 0.1045e-2 1.1359 1.0858 4.62

3 19.7877 0.5184e-3 1.8187 1.7379 4.65

4 17.1062 0.2690e-3 2.6108 2.4946 4.66

5 14.4160 0.1520e-4 3.5555 3.3974 4.65

6 11.7222 0.4310e-4 4.7280 4.5188 4.63

7 9.0522 0.1720e-3 6.2749 6.0007 4.57

8 6.5542 0.3206e-3 805093 8.1532 4.37

Table B-5Comparison between calculated and theoretical values forRotary Catenoidin ExperimentB-2(Unit: m)

Meshing section #: 20

# X Y Z Z(Theoretical) Error (%)

1 26.5227 0.3232e-4 0.2426 0.2365 2.57

2 20.2113 0.5092e-4 1.6759 1.6286 2.91

3 16.4005 0.7415e-4 2.7950 2.7154 2.93

4 13.8477 0.7764e-4 3.7188 3.6124 2.95

5 11.2940 0.4593e-4 4.8648 3.6124 2.95


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(c) Form-finding result (d)Form-finding result side view

Figure B-4 10 sections Meshing

(a) Meshing (b) Stress distribution diagram


Figure B-5 20sections Meshing


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(a) Meshing (b) Stress distribution diagram


Figure B-6 30sections Meshing

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