An iterative algorithm to optimize the stress
distribution of an equilibrium form by
using the force density method
Dr. Fevzi DANSIK, Dr. Meltem SAHIN
Assistant Professor at Mimar Sinan Fine Art University and
Partners of Fabric Art Membrane Structures
Abstract
The force density method has been developed to solve the non-linear system of
equilibrium equations for a network of elements, which have no flexural rigidity and
are pin connected. This solution is achieved by defining a ratio between the force
and the length of each element, which gives the name for the method. Consequently,
the non-linear system of equations becomes a linear one for the given set of force
densities. Hence, it is possible to obtain a different equilibrium form for each
different set of force densities for the same network. Furthermore, different element
forces and stresses are obtained for each different set of force densities.
The introduced algorithm also works networks with compression element such as
tensegrity systems. Furthermore, it is also possible to obtain minimum surfaces by
defining proper boundary conditions.
Keywords: Form Finding, Architectural Fabric, Geometric Non-Linear Analysis
1
Introduction
The force density method has been developed to solve the non-linear system of
equilibrium equations for a network of elements, which have no flexural rigidity and
are pin connected [1]. This solution is achieved by defining a ratio between the force
and the length of each element, which gives the name of the method as ‘Force
Density’. The non-linear system of equations becomes a linear one for the given set
of force densities. Consequently, the equilibrium form of the network is achieved by
solving the linear system of equation for the given force densities. Hence, it is
1
possible to obtain a different equilibrium form for each different set of element force
densities. To find the suitable or desired equilibrium form among the endless
possibilities depends on many different engineering and architectural requirements
and/or requests. Meanwhile, to find the set of force densities that corresponding with
the equilibrium form which fulfils these constraints is not an easy task to tackle. In
this paper, an iterative algorithm is introduced to find the corresponding set of
element force densities which minimizes the variation between the minimum and
maximum stresses of a pin-connected network [2].
2
Force Density Method
The main idea of the force density method is that the equilibrium form of
configuration can be found by solving a linear system of equations if the ratio of the
element force to its length is held a constant. The method deals with a pin-jointed
configuration. This requires that all the nodes in the configuration can rotate and
displace freely without introducing any bending moment into the system. Also, the
configuration may have only external loads at its joints.
Consider the bar element which has only axial stiffness in Figure 1 and let it be a
member of a pin-jointed configuration. The components of the internal force of the
bar element at node j with respect to the global coordinate system can be obtained
by using the directional cosines of the vector passing through node j to node i as
given in Equation 1. Neither the directional cosines nor the element force are known
in Equation 1. Furthermore, the length of the element is a non-linear function of the
node coordinates. Consequently, the system of equation is a non-linear one. Hence,
Shcek [1] has suggested to use the ratio of the bar force to its length as a selected
constant which is referred to as the force density of an element and denoted as q, in
Equation 2. When this force density is substituted in Equation 1, they will become a
linear set of equations for the selected force density as in Equation 3. The
components of the internal force of the bar elements at node i can be found in the
same way or from the simple equilibrium considerations as in Equation 4. These
equations for the bar element may be represented by a single matrix format as in
Equation 5 which is also written in a symbolic notation in Equation 6, where Pe is
referred to as the ‘end force vector’ of element e, Qe is referred to as the ‘force
density matrix’ of element e and Ce is referred to as the ‘end coordinate vector’ of
element e [2]. After writing all the element force density matrices, the primary force
density matrix of the configuration can be obtained by assembling the element force
density matrices in the same manner as the stiffness matrix of the finite element
method. The linear system of equilibrium equations is achieved by equalising the
internal forces to the external forces and implementing the boundary conditions. The
solution of this system of equations gives the coordinates of the nodes, and hence
the equilibrium state of the considered configuration for the selected element force
densities.
2
pe
j
e
le
pe
z
i
y
x
Figure 1: a bar element with only axial rigidity in a pin jointed configuration
Equation 1:
,
,
Equation 2:
Equation 3:
,
Equation 4:
Equation 5:
,
,
=
Equation 6:
3
Iterative Algorithm
The algorithm starts with a set of force densities, which refers to a group of elements
with different mechanical properties of the considered network. In order to
elaborate, consider the cable net configuration in Figure 2 and let the cross sectional
3
area of the boundary cables is 5 times more than that of the inside cables. The steps
of the algorithm are described as follows:
Step 1: To begin with, let the force densities of the element of the initial form in
Figure 2 be equal to the cross sectional area of the cables in order to impose
the relative stiffness difference between the cables.
Step 2: Obtain the equilibrium form by solving the linear system of equations for
the defined force densities in Step 1.
Step 3: Calculate the element lengths, forces and stresses.
Step 4: Calculate the average element stress by simply dividing the overall sum of
the element stresses with the total number of the elements.
Step 5: Find the minimum and maximum stresses.
Step 6: Calculate ‘the stress ratio’ by dividing the minimum stress with the
maximum stress. If the stress ratio is 1, which is the limit, there will be no
stress difference over the surface and hence the surface can be considered
as ‘a minimum surface’. Although it is possible to reach the limit with the
proper boundary conditions, the aim of the algorithm is not to seek
minimum surface but to minimize the stress variation over an equilibrium
form. Hence, the relative difference between two consecutive steps is
considered as the termination criteria of the algorithm.
Step 7: Terminate the iteration procedure if the relative difference of the stress
ratios between two consecutive steps is equal or less then the defined
convergence limit.
Step 8: If Step 7 is not satisfied, terminate the iteration algorithm if the allowed
number of step is reached. This is necessary if the process is not
converging.
Step 9: Calculate the element force density for the next iteration cycle by using the
average force, which is simply obtained by multiplying the average stress
with the area of the considered element, and its length in the equilibrium for
obtained in Step 2.
Step 10: Go to Step 2, but this time use the force densities calculated at Step 9 and
continue until the conditions in Step 7 or 8 are satisfied.
At each repetition of the above process, the stress ratio changes. This can be
observed from Figure 4 where the stress ratio is plotted for each repetition of the
above process. In the present case, after 16 iteration steps, the stress ratio of the
configuration is found to be 0.99999. This value is very close to 1 which is the
‘limit’ value for the stress ratio of a configuration. The closer the stress ratio is to the
limit, the more regular the stress distribution of the configuration.
It can be seen from Figure 4 that the change in the stress ratios for any two
consecutive iterations becomes smaller as the iteration process progresses. For
example, the stress ratios for the first and second iteration steps are respectively
0.587 and 0.860. The difference between the stress ratios for these two iteration
steps is 0.273. At the seventh iteration step, the difference is less than 0.001. The
changes become smaller in a nonlinear fashion as the iteration process progresses.
4
Finally, the iteration process is terminated at the sixteenth step when the change in
the stress ratios for two consecutive steps is equal to 0.00001. The resulting
equilibrium form at this iteration step is given in Figure 3, along with the
equilibrium form obtained at Step 2. In the figure, solid lines indicate the
equilibrium form obtained by using the iteration algorithm, and dashed lines indicate
the equilibrium form at Step 2.
y, U2
z, U3
y, U2
z, U3
3
3
13
1
1
13
8
2
5
5
2
8
4
6
12
9
6
12
9
10
7
4
10
7
Mode 2
Mode 1
11
11
x, U1
Figure 2: Initial configuration
x, U1
Figure 3: mode 1: equilibrium form at
Step 2, mode 2: equilibrium form after
the completion of the iteration algorithm
1.00
Stress Ratio
0.90
0.80
0.70
0.60
0.50
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Number of Iterations
Figure 4: The changes of the stress ratios for the present case
The changes in the maximum, minimum and average stresses for each iteration step
are presented graphically in Figure 5. As can be seen from the figure, the maximum
and minimum stresses converge rapidly to the average stress as the iteration process
progresses. At the end of the iteration process, all the element stresses are almost
identical. Despite the fact that the stress ratio has changed significantly and has
5
become almost the same, the equilibrium form has not changed significantly as can
be seen from Figure 3.
2.25
Maximum Stress
Average Stress
Minimum Stress
Stress
2.00
1.75
1.50
1.25
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Number of Iteration
Figure 5: The changes of the maximum, average and minimum stresses of the
present case
The algorithm for the above process is shown as a flowchart in Figure 6. In the
flowchart;
 q denotes the force density of an element,
 A denotes the cross-sectional area of an element,
 R denotes the stress ratio of the configuration,
 L denotes the length of an element,
  denotes the limit on the difference between the stress ratios for two
consecutive iteration steps, referred to as the ‘convergence tolerance’,
 P denotes the internal force of an element,
 S denotes the stress for an element,
 Smax, Smin and Save denote the maximum, minimum and average stresses in the
configuration, respectively,
 M denotes the maximum number of iteration steps allowed,
 superscript i denotes the number of iteration step and
 subscript e denotes the number of an element
6
Initial element force densities
q0e=Ae
Initialise the stress ratio
R0=0
Apply the force density method
Calculate the element lengths:
Lie
Calculate the element forces:
Pie = q(i-1)e * Lie
Calculate the element stresses:
Sie = Pie / Ae
Calculate the average
element stress:
Siave
Calculate the element force densities
for the next iteration step:
qie = (Siave * Ae) / Lie
Find the maximum and minimum
element stresses:
Simax and Simin
No
Calculate the stress ratio
Ri = (Simax/S imin)
|Ri - R(i-1)| <= 
No
Yes
The results
i>M
Yes
WARNING:
The maximum number of
iteration is reached.
Figure 6: The flow chart of the algorithm
7
4
Implementation of the algorithm for tensegrity systems
All the elements of the configuration in Figure 3 are defined as cables and hence
they can only take tensile forces. However, the force density method works for
elements with only axial stiffness regardless the direction of the force. If an element
has to be in compression, its force density is simply defined as a negative value. If a
configuration consists of only compression elements, the algorithm will work with
the same principles as described in Section 3. If a configuration consists of tension
and compression elements such as tensegrity systems, the force density method will
work without any problem other than singularity problems that may occur due to
zero diagonal elements of the primary force density matrix for the system. However,
the iteration algorithm cannot be used for the following reasons:
 Firstly, since there are positive and negative stresses in a tensegrity form, the
average stress of the configuration may become zero, very small, a positive value
or a negative one. This causes divergence for the iteration process. More
importantly, it is not logical to seek the same stress values for two different kinds
of elements.
 Secondly, the iteration process is terminated when the difference between the
stress ratios for two consecutive iteration steps is less than or equal to the defined
convergence tolerance. However, as discussed above, the stress variation of a
tensegrity form cannot be defined by using one overall stress ratio. Instead, two
stress ratios are necessary to define the stress variation of a tensegrity system,
one for the tension elements and one for the compression elements.
Consequently, the termination criterion of the iteration process must be satisfied
for the both tension and compression elements.
With the above observations in mind, the iteration process is developed in the
following manner:
 Instead of calculating one average stress for a tensegrity system, two average
stresses are calculated, one for the tension elements, positive stresses, and one
for the compression elements, negative stresses.
 The element force densities of the next step are calculated by using the positive
average stress for the tension elements and using the negative average stress for
the compression elements. The maximum and minimum stresses are found out
for the tension elements and the compression elements, separately.
 Then, the stress ratios for the tension elements and the compression elements are
calculated.
 This process is carried out until the termination criterion is satisfied for both the
tension and the compression elements.
To elaborate, consider the configuration shown in Figure 7. This configuration is a
double layer tensegrity system where the top and the bottom layers consist of
tension elements and the vertical web elements are compression elements. The
compression elements are indicated with double lines in Figure 7. Let the axial
stiffness of the compression element be 5 times more that of tension elements.
Hence, let the element force densities of the initial configuration be 1 for the tension
8
elements and -5 for the compression elements in order to impose the relative
stiffness difference between the elements. Then, the same iteration steps can be
carried out with the above modifications. The resulting equilibrium form for the
configuration of Figure 7 is shown in Figure 8.
Figure 7: The initial configuration of the tensegrity system
Figure 8: The equilibrium form of the configuration of Figure 7 obtained by using
the iterative algorithm
The changes of the stress ratio are given graphically in Figure 9. It can be noted
from Figure 9 that the stress ratio for the compression elements converges to the
limit ratio relatively quicker than that of the tension elements. Also, it is seen that
the compression elements satisfy the termination criterion at the 6th step of the
iteration process. However, the iteration process is kept going until the tension
elements also satisfy the termination criterion at the 20th step.
9
1.00
0.98
Tension Elements
Compression Elements
0.96
0.94
Stress Ratios
0.92
0.90
0.88
0.86
0.84
0.82
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Number of Iteration Step
Figure 9: The progress of the iteration process carried out to reach the equilibrium
form in Figure 8.
5
Conclusions
The introduced algorithm minimises the difference between the minimum and
maximum stresses drastically and rapidly. It is also possible to reach to the limit
value, which is 1, as in minimum surfaces as long as the boundary conditions are
properly defined. The extensive research has been carried out in reference [2] for the
introduced iterative algorithm. Furthermore, another iterative method [2] has been
developed to minimize the variation of element lengths in the same manner
introduced in this paper.
The other advantage of the algorithm is that the force densities are not directly
defined as constant values, but instead they are derived by the iteration algorithm
according to their mechanical properties with respect to their relative correlations.
The algorithm works for the network with different element properties as in
tensegrity systems. This can be extended to the network with more element types
such cables, struts and membrane elements. Different types of element force density
matrices such as triangles and quadrilateral can also be developed as briefly
described in Section 2.
References
[1] SCHEK H J, The force density method for form finding and computation
of general network, Computer Method in Applied Mechanics and
Engineering, 4, 1974, pp 115-134
[2]
DANSIK, F, Force Density Method and Configuration Processing,
Ph D Thesis, 1999, University of Surrey, Guildford, U.K.
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