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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.
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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].
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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
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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.
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(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
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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
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(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
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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
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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.
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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
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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.
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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.
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Form-finding ofTanzbrunnen.
Direct integration of three popular light-weight systems: tensegrity, cable-net and tension membrane.