miki, kawaguchi - extended force density method for form finding of tension structures - 2010

8/22/2019 Miki, Kawaguchi - Extended Force Density Method for Form Finding of Tension Structures - 2010

1/13

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J.IASS

291

EXTENDED FORCE DENSITY METHOD FOR FORM-FINDING

OF TENSION STRUCTURES

Masaaki MIKI*1

and Kenichi KAWAGUCHI*2

*1Graduate Student, Department of Engineering, University of Tokyo, [email protected]*2Proffessor, Institute of Industrial Science, University of Tokyo, Dr. Eng., [email protected]

Bw502, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan

Editors Note: The first author of this paper is one of the four winners of the 2010 Hangai Prize, awarded for outstandingpapers that are submitted for presentation and publication at the annual IASS Symposium by younger members of theAssociation (under 30 years old). It is re-published here with permission of the editors of the proceedings of the IASS2010 Symposium: Spatial Structures Temporary and Permanent held in November 2010 in Shanghai, China.

ABSTRACT

The objective of this study is to propose an extension of the force density method (FDM)[1], a foregoingnumerical method for the form-finding of tension structures. FDM has a great advantage for the form-finding of

cable-nets. However, some difficulties arise when it is applied to the pre-stressed structures that consist of a

combination of both tension and compression members. Therefore, the FDM has scope for extension.

In this paper, we identify the existence of a variational principle in the FDM, although, the mathematical forms

used by original FDM are different from those related to the variational principle. Thus, a functional that can be

thought to be selected by FDM is clarified and it enables us to extend the FDM by considering various

functionals as a generalization of it. Such newly introduced functionals enable us to find the forms of complex

tension structures that consist of a combination of cables, membranes, and compression members, such as

tensegrities, and suspended membranes with bars, etc.

Keywords: Force density method, Form-finding, Tensegrity, Membrane, Cable-net, Variational principle,

Principle of virtual work

1. INTRODUCTION

Tension structures such as cable-nets, suspended

membranes, and tensegrities are stabilized by

introducing prestress. Hence, they require a special

process to ensure that they have a self-equilibrium

state, so-called prestress state. This process is called

form-finding because the existence of the prestress

state of a structure is highly dependent on its form.

For the form-finding, various numerical methodshave already been proposed by many researchers.

The force density method (FDM) [1] is one of such

method proposed to determine the form of

cable-nets. The purpose of this study is to propose

an extension of the FDM [1].

In Section 2, we describe the original FDM and its

great advantage. In addition, we identify the

limitations of the FDM when it is applied to

prestressed structures that consists of a combination

of both tension and compression members, e.g.

tensegrities. Therefore, the FDM has scope for

extension.

As an analytical expression for form-finding, two

types of mathematical expression have been mainly

adopted. One is a set of equilibrium equations,

whereas the other represents a stationary problem of

a functional based on the variational principle. In

general, the equilibrium equations and the

stationary problems of a functional are closelyrelated.

In Section 3, we show the existence of a variational

principle in the FDM, although, the mathematical

forms used by the original FDM are different from

those related to the variational principle. Therefore,

we can propose a functional which can be thought

to be selected by the FDM by considering the

variational principle. This functional enables us to

extend the FDM.


2/13

Vol. 51 (2010) No. 4 December n. 166

292

In Section4 and 5, we describe the extended FDM

and some of its applications in the form of

numerical examples, by considering various

functionals as generalizations of the selected

functional. The newly introduced functionals enable

us to find the forms of complex tension structures

that consist of a combination of cables, membranes,

and compression members, such as tensegrities, and

suspended membranes with bars.

2. FORCE DENSITY METHOD

2.1 Original Formulation

The FDM was first proposed by Schek and

Linkwitz in 1973. The main characteristics of the

FDM are divided into two parts. The first part

consists of the definition and use of a quantity

called force density. The force density qj is definedas

jjj Lnq / , (1)

where nj andLj denote the tension and length of the

j-th cable, respectively, as shown in Fig. 1(a). In the

FDM, each cable is assigned a force density as a

known parameter, whereas nj andLj are unknown

(to be determined). Therefore, some trials must be

carried out to obtain an appropriate configuration of

force densities.

The second part consists of the linear form of the

equilibrium equations. When the force densities are

assigned and the fixed nodes and their coordinates

are prescribed, the self-equilibrium condition for

cable-nets is represented by

,

,

,

ff

ff

ff

and

zDzD

yDyD

xDxD

(2)

where D is the equilibrium matrix andx, y, andz

are the column vectors containing the nodal

coordinates of each node. The terms with the

subscript f are related to the fixed nodes, whereas

those with no subscript are related to the free nodes.

The right-hand sides of Eq. (2) represent the

reaction forces from the fixed nodes. Hence, Eq. (2)

can be considered as an analogue of Hooks law,

i.e., fkx . Here, note that the linear form of Eq.

(2) is not approximated.

There are just three unknown variables, i.e., x, y,

andz, therefore, the nodal coordinates of the free

nodes can be obtained as follows,

.)(),(

,)(

1

1

1

ff

ff

ff

and

zDDzyDDy

xDDx

(3)

Once the nodal coordinates are obtained, the tension

in each cable is calculated using Eq. (1).

Eq. (3) represents the standard procedure for

solving a set of simultaneous linear equations;

hence, it enables the implementation of the FDM

very concise. Thus, the FDM has a great advantage

over other methods, which require non-linear

iterative computation.

Using the FDM, we can determine the form of

cable-nets by changing the coordinates of the fixed

nodes or the force densities of the cables, as shown

in Fig. 1(b).

n

L

q= n /L(a) Definition of Force Density

(b) Form-finding Analysis using FDM

Figure 1. Force Density Method

2.2 Limitation of FDM

In the application of the FDM to the form-finding

of self-equilibrium systems that consist of a

combination of both tension and compression

members, e.g., tensegrites, some difficulties arise

when negative force densities are assigned to the

compression members and positive force densities

are assigned to the tension members [2][3][4][5].


3/13


293

Let us consider the form-finding of a prestressed

structure, e.g., an X-Tensegrity, as shown in Fig. 2.

An X-Tensegrity is a planar prestressed structure

that consists of 4 cables (tension) and 2 struts

(compression). As in the case of general tensegrities,

the cables connect the struts and the struts do not

touch each other.

For such self-equilibrium systems that have no

fixed nodes, Eq. (2) reduced to a simpler form:

.,, 0zD0yD0xD

(4)

WhenD is a regular matrix, only a trivial solution is

obtained, i.e.,

.0zyx

(5)

On the other hand, when D is a non-regular matrix,

Eq. (4) has complementary solutions. Such

solutions are obtained by analyzing the null space

ofD. However, even if we analyze it, the FDM

would lose its conciseness as follows:

When the assigned force densities are in theproportion 1:1:1:1:-1:-1 for the 4 cables and

the 2 struts respectively, many solutions are

obtained. An example of D and the

corresponding solutions are shown below.

,

1

1

0

0

0

0

1

1

1

1

1

1

,

1

10

0

0

01

1

1

11

1

,

1

1

0

0

0

0

1

1

1

1

1

1

,

1111

1111

1111

1111

ihg

andfed

cba

z

y

x

D

(6)

where ai are arbitrary real numbers. This

implies, for example, that both Fig. 3(a) and

(b) satisfy Eq. (4). The first terms of the right-

hand-sides denote the position of the center

point, i.e., [adg], and the other terms denote

some symmetry that all solutions must have.

When the assigned force densities are not inthe proportion 1:1:1:1:-1:-1, the obtainedsolutions do not denote a form. An example of

D and the corresponding solutions are shown

below.

.

1

1

1

1

,

1

1

1

1

,

1

1

1

1

,

1322

3122

2231

2213

candba zyx

D

(7)

This implies that all nodes converge at one point,

i.e., [abc].

Figure 2. X-Tensegrity

(a) (b)

Figure 3. Self Equilibrium Forms

3 VARIATIONAL PRINCIPLE IN FDM

The functional that is thought to be selected by

FDM is given by

j

jjLw )()(2

xx , (8)

where wj and Lj denote an assigned weight

coefficient and a function for the length of the j-th


4/13

Vol. 51 (2010) No. 4 December n. 166

294

cable, respectively. The column vectorx contains

the x, y, andz coordinates of the free nodes. It is

generalized as an unknown variable vector by

Tnxx 1x . (9)

Note that the coordinates related to the fixed nodes

are eliminated from x beforehand and directly

substituted inLj, because they are prescribed.

The stationary condition of Eq. (8) is expressed as

0x

x

j

jjj LLw2)(

, (10)

where is the gradient operator defined as

nxf

xfff

1x. (11)

Eq. (11) denotes the direction of steepest increase in

n-dimensional space.

In this paper,Lj is given by

2212

21

2

21 zzyyxxL . (12)

In this case, L represents two normalized vectors

attached to both ends of the member, as shown inFig. 4(a). On the other hand, let us consider a linear

member resisting two nodal loads applied to both

its ends, as shown in Fig. 4(b), and let its axial force

be n. Comparing Fig. 4(a) and (b), a general

expression for a self-equilibrium state of prestressed

cable-nets is obtained as

j

jj Ln 0

. (13)

On the basis of Eq. (13), the principle of virtual

work for pre-stressed cable-nets is expressed as

j

jj Lnw 0 , (14)

where jL is defined as x jj LL , and x is

defined as an arbitrary column vector, Tnxx 1 .They are usually called the variation ofLj and the

virtual displacement, respectively.

Substituting Eq. (1) in Eq. (13), an alternative form

of Eq. (2) is obtained as

j

jjj LLq 0

. (15)

Here, because of the mathematical equivalence ofEq. (10) and Eq. (15), Eq. (8) is thought to be the

functional that is selected by FDM. In addition, it is

assumed that the assigned weight coefficients play

the same role as the force densities in the

form-finding process for cable-nets.

Although Eq. (2) and Eq. (15) appear rather

dissimilar, they are actually identical when each

function Lj is represented using Eq. (12). On the

other hand, when the unknown variables do not

represent Cartesian coordinates, (e.g., when they

represent polar coordinates), Eq. (15) remains valid,whereas Eq. (2) becomes invalid. Thus, Eq. (15) is

more general expression than Eq. (2).

On the basis of Eq. (15), the principle of virtual

work for the FDM is expressed as

.0 j

jjj LLqw (16)

Similarly, on the basis of Eq. (10), the principle of

virtual work is expressed as

j

jjj LLww 02 . (17)

Additionally, Eq. (17) is mathematically equivalent

to 0 x . Thus, we obtain a variational

principle as

0 , (18)

where is defined as x and usually

called the variation of .

In conclusion, it is important to note that in the

original paper [1], Eq. (8) is mentioned by the

following theorem:

THEOREM 1. Each equilibrium state of an

unloaded network structure with force densities qj is

identical with the net, whose sum of squared way

lengths weighted by qj is minimal.


5/13


295

(a) L (b) Equilibrium State

Figure 4. Linear Member

4. EXTENDED FORCE DENSITY METHOD

4.1 Generalization of Selected Functional

In the previous section, we obtained a functional

that is selected by the FDM. It is possible to extend

the FDM by generalizing this functional.

Let us reconsider the form-finding of the

X-Tensegrity. The same difficulties that is pointed

out in subsection 2.2 would also arise from the

stationary problem of Eq. (8), when negative weight

coefficients are assigned to the struts and positive

weight coefficients are assigned to the cables,

because of the mathematical equivalence of Eq. (2)

and Eq. (15).

In the case of cable-nets, the coordinates of the

fixed nodes are given as kinematic conditions,

whereas no kinematic conditions are given in thecase of the X-Tensegrities. Thus the length of each

strut of the X-Tensegrity is assigned as a kinematic

condition instead of a weight coefficient. In this

case, according to theLagranges multiplier method,

a modified functional is obtained as

k

kkk

j

jj LLLw ))(()(),(2

xxx , (19)

where the second sum is taken for every strut, and,

k and kL are Lagranges multiplier and the given

length of the k-th strut, respectively.

However, Eq. (19) does not completely eliminate

the aforementioned difficulties. If the assigned

weight coefficients of the cables are in the

proportion 1:1:1:1, and the given lengths of the

struts are in the proportion 1:1, both Fig. 3(a) and

(b) satisfy the stationary condition of Eq. (19). By

using the famous Pythagorean Theorem, i.e.,

c2=a2+b2, it can be easily verified that the sum of

squared lengths of the cables takes the same value

for both the shapes.

Here, another functional, e.g.,

k

kkk

j

jj LLLw ))(()(),(4

xxx , (20)

can be considered because the functionals such as

Eq.(8) and Eq. (19) do not represent any physicalquantity, such as energy. Thus, it is possible to use

higher powers ofLj.

Solving the stationary problem of Eq. (20), Fig. 3(a)

becomes the unique solution when the weight

coefficients of the cables and the lengths of the

struts are assigned in the proportion 1:1:1:1 and 1:1,

respectively. On the other hand, when the weight

coefficients and the lengths are assigned in the

proportion 1:8:1:8 and 1:1, respectively, Fig. 3(b)

becomes the unique solution.

Let us investigate the following generalized

functional:

k

kkk

j

jj LLL ))(())((),( xxx . (21)

The stationary condition of Eq. (21) with respect to

x is given by

0x

x

k

kk

j

j

j

jjLL

L

L

))((. (22)

Then, when Eq. (22) is satisfied, comparing Eq.

(13) and Eq. (22), the following non-trivial

configuration must satisfy Eq. (13):

r

m

mm

L

L

L

L

1

1

11 ,)()(

n . (23)

Eq. (23) represents one of the self-equilibrium

modes. Thus, any functional compatible with Eq.

(21) can be applied to such a problem. Henceforth

in this paper, we call j the element functional, andapply it to a stationary problem by selecting it.

Thus, we propose the following two policies:

Perform form-finding analysis by solving astationary problem of a freely selected

functional

When difficulties arise, test the otherfunctionals.

Let us consider the relation between Eq. (23) and


6/13

Vol. 51 (2010) No. 4 December n. 166

296

the FDM. If wjLj2

is selected as the element

functional, according to Eq. (23), we have the

following relation:

jjjjjj LnwLwn 2/2 . (24)

Therefore, it is verified that the assignment of the

force densities is virtually equivalent to the

assignment of the weight coefficients. On the other

hand, ifwjLj4is selected, we have

334/4 jjjjjj LnwLwn . (25)

Therefore, it is verified that defining and adopting a

new quantity wj=nj/4Lj3

is equivalent to selecting

wjLj4

as the element functional. We Call the new

quantities, such as wj=nj/4Lj3, the extended force

density.

Without the linear form, the key features of FDM

are reconsidered as follows:

The coordinates are assigned to each fixednode as kinematic conditions.

The force densities qj=nj/Lj are assigned toeach cable as known parameters.

On the other hand, when wjLj4

is selected as the

element functional, the key features of the extended

FDM are as follows:

The lengths kL are assigned to each strut askinematic conditions.

The extended force densities wj=nj/4Lj3 areassigned to each cable as known parameters.

Thus, it is verified that the extended FDM is quit

similar to the FDM. Moreover, the general form of

the functionals, i.e., Eq. (21), enables us to select

various non-linear computational methods when wecarry out the extended FDM.

As an alternative to Eq. (13), the following form

can be used to express theprinciple of virtual work

for not only the tensegrities but also general

pre-stressed structures that consist of a combination

of both cables and struts:

.0 k

kk

j

jj LLnw (26)

4.2 Additional Analyses

In this subsection, some additional numerical

analyses are reported for further comprehension of

the extended FDM.

Let us consider an analytical model that consists of220 cables connecting one another and 5 fixed

nodes, as shown in Fig. 5. The coordinates of the

fixed nodes are also provided in the figure.

Fig. 6 shows the results of a series of optimization

problems applied to the analytical model. These

problems are to minimize the sum of lengths with

powers ranging from 1 to 4.

On the other hand, Fig. 7 shows the results of the

same optimization problems applied to another

model, the Simplex Tensegrity, which is aprestressed structure that consists of 9 cables and 3

struts. In this case, the optimization was only

performed for the cables, whereas the lengths of the

struts were kept constant at 10.0.

A comparison between Fig. 6(ii) and Fig. 7(ii)

implies that different element functionals are

required for cable-nets and tensegrities.

Figure 5. Analytical Model

(i) min jL (ii) min2 jL

(iii) min3 jL (iv) min

4 jL

Figure 6. Optimization Results of Cable-nets


7/13


297

(i) min jL(ii) min

2

jL

(iii) min3 jL (iv) min

4 jL

Figure 7. Optimization Results of Simplex Tensegrity

5. NUMERICAL EXAMPLES

In this section, we report some numerical examples

as applications of the extended FDM.

In general, the unknown variables (to be

determined) in a stationary problem of a functional

represented by the form of Eq. (21) are nxx 1x and r1 . However, we just minimized

j as an objective function by keeping thelengths of the struts constant at kL . Hence, in this

case, only nxx 1x are the unknown variables.

General non-linear computations require

appropriate initial values for the unknown variables;

however, we roughly assign the random numbers to

the unknown variables (from -2.5 to 2.5).

Regardless of such a rough initial configuration, we

always obtained an expected solution.

5.1 Structures Consisting of Cables and Struts

Let us consider 20 struts assigned with sequential

nodal numbers at every end, as shown in Fig. 8(a).

For example, 1 and 2 are assigned to the 1st strut, 3

and 4 are assigned to the 2nd strut, and so on.

To determine a form of tensegrities with 9 different

connections, let N be an arbitrary number from 1 to

9. To make every end being connected to 4 other

ends by 4 cables, let the i-th node be connected to

the i+2N-th, i+2N+1-th, i+(40-2N)-th, and

i+(40-2N-1)-th nodes. If the calculated nodal

number is greater than 40, no connection is added.

Thus, we obtain 9 different connections for the

form-finding of tensegrities which consist of 20

struts and 80 cables. Fig. 8(b) shows 8 cables

connected to the 1st and 2nd nodes when N = 6.

Using the structure with obtained 9 connections, we

ran many analyses by solving the following

problem:

.stationary

))(()(),(4

k

kkk

j

jj LLLw xxx

(27)

The corresponding principle of virtual work is

expressed as

0.43

k

kk

j

jjj LLLww (28)

In this series of analyses, the weight coefficients of

half of the cables connecting the i-th node to the

i+2N+1-th and i+(40-2N-1)-th nodes are set to 1.0,

whereas those of the other half of the cables

connecting the i-th node to the i+2N-th and

i+(40-2N)-th nodes are set to w, a variable

parameter.

Fig.9 shows 9 results obtained when w was set to

2.0. Interestingly, these results are just a fraction of

the many possible results when w is kept constant at

2.0. Fig.9 shows the most frequently obtained

results. This fact implies that the functional is

multimodal.

(a) Indices of Nodes (b) 8 Cables Connected to 1stand 2nd Nodes (N=6)

Figure 8. Instruction for Configuration


8/13

Vol. 51 (2010) No. 4 December n. 166

298

Figure 9. Discovered Tensegrities

5.2 Structures Consisting of Cables, Membranes

and Struts

For the form-finding of the structures that consist of

cables, membranes, and struts, Eq. (21) is extended

as follows:

,stationary))((

))(('))((),(

k

kkk

j

jj

j

jj

LL

SL

x

xxx

(29)

where the first summation is taken for every linear

element; the second, for every triangular element;

and the third, for every strut. A function LjandSj

are defined, which gives the length of thej-th linear

element and the area of the j-th triangular element,

respectively. The forms of the cables are

represented by the linear elements, and the forms of

the membranes are represented by the triangular

elements.

The stationary condition of Eq. (29) with respect to

x is given by

.

))(('

))((

0

x

x

x

k

kk

j

j

j

jj

j

j

j

jj

L

SS

S

LL

L

(30)

Replacing the partial differential factors by

j

jj

j

j

jj

jS

S

L

Ln

)(',

)(

, (31)

the general form of the self-equilibrium state is

obtained as

0x

k

kkj

j

jj

j

j LSLn , (32)

and the principle of virtual work for general

self-equilibrium systems is expressed as

0 k

kk

j

jj

j

jj LSLnw . (33)

We report the form-finding analysis by using the

analytical model shown in Fig. 10 and by solvingthe following problem:

,stationary))((

)()(),(24

k

kkk

j

jj

j

jj

LL

SwLw

x

xxx

(34)

and theprinciple of virtual work is expressed as

.0243

k

kk

j

jjj

j

jjj LSSwLLww (35)

The model is based on a cuboctahedron, and it

consists of 24 cables, 6 membranes, and 6 struts.

Each cable is represented by 8 linear elements, and

each membrane is represented by 128 triangular

elements. Fig. 11 shows a standard result. By

varying the parameters wj and kL , various forms

are obtained, as shown in Fig. 12.

Figure 10.Analytical Model

Figure 11.Result


9/13


299

Figure 12.Variety of Forms

5.3 Structures Consisting of Cables, Membranes,

Struts and Fixed Nodes

In this subsection, we report the form-finding of a

suspended membrane structure based on the famous

Tanzbrunnen. It is located in Cologne (Kln),

Germany, and designed by F. Otto (1957). The

form-finding was carried out by solving the

following problem:

,stationary))((

)()(),(24

k

kkk

j

jj

j

jj

LL

SwLw

x

xxx

(36)

where, as in the previous subsection, jL and jS

do not denote the lengths of the cables or the

surface areas of the membranes, but the lengths of

the linear elements and the element areas of the

triangular elements, respectively.

As shown in Fig. 13, the form-finding was carried

out effectively and conveniently by changing the

weight coefficients and the lengths of the struts.

Note that the form has been improved very much

via process from Fig. 13(a) to (f)

(a) (b)

(c) (d)

(e) (f)

Figure 13.Form-finding of Suspended Membrane

6. CONCLUSION

We proposed the extended force density method

that enables us to carry out form-finding of

general pre-stressed structures that consist of a

combination of both tension and compression

members. We identified the existence of a

variational principle in the FDM, and we extended

the FDM by generalizing the functional that is

assumed to be selected by FDM. Moreover, we

found that various functionals can be selected for

the form-finding of tension structures. By using

the newly introduced functionals, various

self-equilibrium forms were obtained.


10/13

Vol. 51 (2010) No. 4 December n. 166

300

REFERENCES

[1] Schek, H. J., The force density method forform finding and computation of general

networks, Comput. Methods Appl. Mech.

Engrg., 3, pp. 115-134, 1974

[2] Connelly, R., and Back, A., Mathematics andtensegrity, American Scientist, 86,

pp.142151, 1998

[3] Tibert, A. G. and Pellegrino, S., Review ofform-finding methods for tensegrity

structures, Int. J. Space Struct., 18(4), pp.

209-223, 2003

[4] Zhang, JY., and Ohsaki, M., Adaptive forcedensity method for form-finding problem of

tensegrity structures,Int. J. Solids Struct., 43,pp. 5658-5673, 2006

[5] Vassart, N., Motro, R., Multiparameteredformfinding method: application to tensegrity

systems, Int. J. Space Struct., 14(2), pp.

147-154, 1999

[6] Maurin, B., Motro, R., The surface stressdensity method as a form-finding tool for

tensile membranes, Eng. Struct., 20(8),

pp.712-719, 1998

[7] Barnes, M. R., Form Finding and Analysis ofTension Structures by Dynamic Relaxation,

Int. J. Space Struct., 14, pp.89-104, 1999

[8] Goto, K., Noguchi, H., Form FindingAnalysis of Tensegrity Structure Based on

Variational Method, Proceedings of The

Forth China Japan - Korea Joint

Symposium on Optimization of Structural and

Mechanical Systems, pp. 455-460, 2006.

[9] Lagrange,J.L. ( author), Boissonnade, A. C.and Vagliente, V. N. (translator), Analytical

mechanics, Kluwer, 1997


11/13


301

APPENDIX (A) PHOTOGRAPHS

Handmade model based on the famous

Tanzbrunnen.

By changing the weight coefficients, better formwas obtained (corresponds to Fig. 13(f)).

This model corresponds to Fig. 8(d).

This model corresponds to Fig. 9(e).

This is another local minimum of the above model,

which implies that the functional is multimodal.

This model corresponds to Fig.11.


12/13

Vol. 51 (2010) No. 4 December n. 166

302

APPENDIX (B) VISUALIZATIONS

By changing the parameters, it is possible to break the usual symmetry of the optimized forms.

Discovered tensegrities consisting of 40 struts and 160 cables.


13/13


303

Form-finding ofTanzbrunnen.

Direct integration of three popular light-weight systems: tensegrity, cable-net and tension membrane.

miki, kawaguchi - extended force density method for form finding of tension structures - 2010

Documents