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International Journal of Space Structures
2016, Vol. 31(2-4) 177 –189
© The Author(s) 2016
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DOI: 10.1177/0266351116660799
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Introduction
In structural design, models are used to predict the physi-
cal behavior of structures. Since the rise of the digital cul-
ture during the late 20th century, computers have played an
increasingly essential role for these predictive behavior
models.1 However, most of the currently used computa-
tional models are not appropriate for the conceptual struc-
tural design phase.2 Either they require a large, clearly
defined set of yet unknown information (such as topology,
geometry, and material properties) or they are restricted to
explorations within specific structural typologies (e.g.
membrane structures). Whereas with the first approach,
the designer does not have active control over the struc-
tural behavior, with the latter strategy the designer loses
direct control over the formal articulation.3 Furthermore,
the described approaches have in common that they focus
very early on metric and quantifiable properties of the
structure and on the relationship between form and force
only. The observation by Billington4 that there can be “no
(quantifiable) optimum in structures, but only many rea-
sonable choices, allowing the designer the freedom to
express his own ideas” underpins this shortcoming (p.
142). Cecil Balmond5 with his informal/sequential under-
standing by way of illustration uses the words “rhythm,”
“fluctuations,” or “episodes” in space to describe what
structures can provide. In order to be useful in the concep-
tual design phase, an approach should consequently be
able to describe visually the influence of “lateral shifts,”
which are transformations in between topologically fixed
design spaces6 in order to ensure that several radically dif-
ferent solutions from different design spaces are known.
According to Marples,7 these fundamental variations are
indispensable to get a clear picture of “the real nature” of
the problem during the coevolutionary matching process
of problem and solution in the conceptual design phase.
Therefore, a theoretical structural design framework and a
digital tool that makes this explicitly available to the
designers would be of great benefit.
The research presented here tries to overcome the
described shortages and develop a theoretical framework,
named combinatorial equilibrium modeling (CEM), to
combine the control of qualitative structural behavior with
formal articulation in the conceptual design phase in a
novel and comprehensible way. This article presents an
innovative computer-aided modeling approach for equili-
brated spatial structures with any combination of compres-
sion and tension forces including the ability to interactively
modify the qualitative structural behavior by means of
topological operations and combinatory variation
Combinatorial equilibrium modeling
Patrick Ole Ohlbrock and Joseph Schwartz
Abstract
This research develops a theoretical framework (based on graphic statics and the graph theory) that allows for the
incorporation of the control of qualitative structural behavior in the conceptual design phase in a novel way. This article
presents an innovative computer-aided modeling approach for spatial pin-jointed networks in equilibrium with any
combination of compression and tension forces.
Keywords
combinatorial equilibrium modeling, graph theory, graphic statics, real-time structural design tools, structural design,
strut-and-tie models, topology
ETH Zurich, Switzerland
Corresponding author:
Patrick Ole Ohlbrock, Institute of Technology in Architecture, ETH
Zurich, Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland.
Email: ohlbrock@arch.ethz.ch
660799SPS0010.1177/0266351116660799International Journal of Space StructuresOle Ohlbrock and Schwartz
research-article2016
Article
178 International Journal of Space Structures 31(2-4)
(compression or tension) in order to make “lateral shifts”
in between design spaces possible. Design examples based
on this approach will show that it is at the same time pos-
sible to control structural and formal properties in an intui-
tive and active way. The structure of this article is as
follows: first, the basic concepts and the background of the
theoretical framework for the CEM are discussed; the mid-
dle part describes the main properties and technical pecu-
liarities of the CEM approach; finally, the article presents
a first prototypical implementation within the computer-
aided design (CAD) software package Rhinoceros and two
case studies in which the approach is tested for the design
of an arena roof and a fictive design scenario for a multi-
story tower.
Concepts
Theory of plasticity and graphic statics
Structural behavior is usually described in terms of forces.
Generally, the calculations of the inner forces are based on
the Theory of Elasticity and carried out with numerical
methods such as finite element method (FEM). Using
FEM, the designer gets an overview of the structural
behavior but can hardly use it for a synthetic process, espe-
cially in the conceptual design phase.8 Furthermore, the
results are highly influenced by the designer’s assump-
tions on material properties, boundary conditions, the load
history, and the future behavior of the structure during the
use. The fact that this theory is not able to describe the
behavior of many construction materials in an appropriate
manner underpins the need for a change.9
The main characteristic of the Theory of Plasticity,
which represents an alternative to the Theory of Elasticity,
is that it concentrates on the ultimate rather than on the
serviceability state of design. The Theory of Plasticity is
therefore based on the assumption of a rigid-plastic mate-
rial behavior. While neglecting the material stiffness and
the deflections, the Lower Bound Theorem, as a part of
the Theory of Plasticity, places the focus on equilibrium
solutions. Through the application of the Lower Bound
Theorem, it is further possible to use graphic statics to
visualize and construct these equilibrium states of a
structure. Graphic statics directly illustrates the recipro-
cal relationship between form and force and thereby
makes it intuitive and understandable.10 This argumenta-
tion is also supported by the scientific community in this
research area, exemplarily by Caitlin Mueller:2 “Human
designers are highly visual and can process and evaluate
information much more quickly and fully when it is pre-
sented graphically” (p. 41). Furthermore, graphic statics
promotes an active involvement of the designer at the
interface between form and force. In comparison to
a black-box-like analysis tool, this involvement of
“crystallizing and refining, matures thinking about the
relationship” (p. 40).11 This observation is underpinned
by the physicist Victor Weisskopf,12 who told his students
at the MIT: “When you show me exclusively computer-
ized results, the computer understands the answer, but I
don’t think you understand the answer” (p. 81). It is
therefore the aim of this research to keep the designer
actively involved and the events under his control in the
computer-aided CEM approach.
Multiplicity of solutions and topological thinking
Another property of the Theory of Plasticity and its Lower
Bound Theorem is that it allows for a multiplicity of quali-
tatively differing structural behaviors. That is, for a given,
internally statically indeterminate structure (described
only with geometric boundaries) and considering the same
load scenario, infinite equilibrated options are possible, as
long as the forces fulfill equilibrium conditions, are lying
within the geometric boundaries and satisfy the yield con-
ditions, defined by the material (p. 9).9 As already briefly
described in the introduction, the work with a certain
amount of differing structural alternatives appears to be
the key for the early design phase and supports the applica-
tion of graphic statics as its theoretical backbone. A well-
known way of working with different alternatives is based
on structural typologies. A system-inherent problem of this
rather conventional approach is pointed out by Laurent
Ney et al.:13
A typology has a name, and the form and the relationship
between the elements is described. The advantage of this is
that it is easy to talk about structure, but the disadvantage is
that how the structure looks is predetermined … This approach
has a perverse effect: the vocabulary freezes the object, and
the objects thus frozen assume a sort of inviolable legitimacy.
In order to arrive at new forms and concepts we have to free
ourselves from such pre-defined typologies.
In the first instance, another aim of this work is to over-
come those pre-defined typologies by extracting and ana-
lyzing important equilibrium properties and their basic
relations, their topological structure in a meaningful way.
An analogy of this approach can be found in the guiding
principle of Le Ricolais work in which the topological
arrangement of the structure plays the key role,14
Furthermore, it is the goal of this research to show that
based on synthetic combinations and superpositions of
very basic equilibrium elements, new structures can
emerge which are not associated with pre-defined
typologies. This thinking has a reference to Herman
Hertzberger’s15 structuralistic manifesto, where he states
that each design, no matter at which time, no matter where
it is designed, is nothing else than an interpretation (a com-
binatory variation) of the archetype. This guiding principle
can be easily implemented with the use of graphic statics,
since it is not fixed to any type of structure but is able to
Ohlbrock and Schwartz 179
describe the geometrical relationship between form and
force for any combination of tension and compression
forces.
One reasonable simplification in this research is to
work with discrete resultant forces (strut-and-tie networks)
instead of their underlying stress fields. This makes the
handling, especially for the conceptual design phase, much
easier but is, nevertheless, accurate to represent the struc-
tural behavior (p. 4).16 Connecting the presented concepts
leads to the observation that the multiplicity based on the
Lower Bound Theorem and a visualized combinatorial
approach expressed through graphic statics gives potential
for the design of non-typological structures at an early
design stage.
State-of-the-art research
Graphic statics has its roots in the 19th century, when
Culmann formalized a method that describes the geometri-
cal relationship between the form of a structure and the
intensity of forces in it. Due to the reasons described
before, graphic statics is experiencing a revival in leading
practices like Skidmore Owing Merrill17 and design
schools around the world (p. 240).1
In the field of three-dimensional (3D) graphic statics,
Block18 developed in his doctoral thesis, thrust network
analysis (TNA), a force-density based method, in which
an algorithm finds equilibrium for compression-only
vaults, based on the designer’s choice of the structure’s
plan and its inner force distribution explicitly during the
form-finding process, using the planar projection of the
form and force diagram (p. 48). Rippmann et al.19 devel-
oped RhinoVault which is an interactive form-finding
plug-in for a commercial CAD software and based on
the theoretical backbone of TNA. The major advantage
of this application is that it is fully interactive. The
designer can actively control form or force quasi in real
time. However, the restriction of TNA and RhinoVault to
compression-only shells does not correspond to the
diversity of possible solutions across typologies in the
conceptual phase of structural design (p. 80).2 Jasienski20
stated that there
is a shortage of design tools for shaping equilibrium of three-
dimensional structures that are more complex than grid shells,
for instance structures that combine tension and compression
in a geometry that is out of plane and cannot be described by
3D curved surfaces. (p. 1)
Lachauer21 overcomes this typological shortage in his
dissertation, in which a novel and non-linear extension of
the Force Density Method is presented. His interactive
approach can generate equilibrium solutions close to the
geometry of a given pin-jointed, kinematic model and fur-
ther allows the definition of geometric constraints for
nodes and also numerical bounds on the force densities (p.
127).21 Although this tool-based approach is fostering new
and efficient structures, it has the problem that it requires
deep understanding of the equilibrium principles and
experience in structural design to define feasible condi-
tions. Otherwise, the algorithm cannot find an equilibrium
state (p. 128).21 Fivet and Zastavni22 and Jasienski20 are
working on constrained-based graphic statics in which the
designer can build and modify interactively geometric
constraints that control the shape and its static equilibrium
simultaneously and in an entirely graphical way. This
method allows for the creation of any planar strut-and-tie
network with operations that are geometrically channeled
so that it can remain in static equilibrium at each step of
the process.22,23 Although this work might overcome some
previously described limitations, it is not yet evident how
this approach can enrich/be embedded in an interactive
design process of 3D structures. Lee et al.24 recently pre-
sented a grammar-based generation of equilibrium struc-
tures through graphic statics. This method enables the
designer to evaluate automatically generated non-typolog-
ical, unexpected equilibrium solutions and to explore dif-
ferent design spaces. This method shows similarities to the
present research; however, the control of the structural
behavior is mainly incumbent upon a graphic computation
engine and not the designer.
The recent developments in this research area are prom-
ising. The present contribution is trying to provide a novel,
robust, and interactive method that can flexibly generate
combined tension and compression systems in space with-
out the necessity of deep knowledge in equilibrium
modeling.
CEM
Qualitative behavior and quantitative behavior
A key aspect of this research is the way structures and their
behaviors are represented. Conventional graphic statics
generally focuses on the geometrical relationship between
the form diagram F and the corresponding force diagram
F*. The form diagram F represents the geometry of the
structure with edges fi_j connecting the nodes position vec-
tors pi and pj; hence, it directly visualizes the edges length
λi_j and their direction vector ei_j. The force diagram F* rep-
resents the equilibrium of the structure by means of closed
force polygons. Per definition, to each edge fi_j in the form
diagram corresponds two inner forces and as such two par-
allel vectors fi_j
* in the force diagram; hence, the two dia-
grams share the directional unit vector ei_j. In the force
diagram, the magnitude µi_j of each force is represented by
the length of the vector fi_j
*.
On top of the conventional representation of the quantita-
tive geometrical behavior between the form diagram F and
the force diagram F*, a visual representation of the qualitative
180 International Journal of Space Structures 31(2-4)
behavior is proposed (Figure 1). This two-dimensional (2D)
topological diagram T, on one hand, visualizes topological
dependencies of members in a pin-jointed force network, as
it was in a similar way presented by Fivet and Zastavni23 and
Van Mele and Block.25 Additionally, the members ti_j in this
directed graph T are colored in order to represent and control
the combinatorial state of the inner forces (blue for compres-
sion, red for tension).Within the present research, the mem-
bers of a main trail, hence members on a shortest route from
an external applied load (external member qi) to a support
force (external member ri), are called force trail members.
Members that connect those force trails in sequence K and
are responsible for their redirection are called deviation
members. The differentiation of those two categories is for
regular and simple directed graphs (as they are shown at this
stage of the research) a feasible task. For irregular and com-
plex networks, which are not yet addressed, this task might
require assistance from a computational method. The topo-
logical diagram is metrically related to the form diagram
through the force trail member’s lengths λi_j and to the force
diagram through the deviation member’s force magnitude
µi_j.
The horizontally arranged members (continuous lines) in
the directed graph T represent the force trail members running
from the left (external applied loads) to right (external support
forces) on a shortest route, while the vertical members (dashed
lines) represent the deviation members in sequence K.
Sequential construction of form and force
diagrams for a given topological diagram
The following section describes the main scheme of the
CEM, mainly how the form diagram F and force diagram
F* can be sequentially constructed based on a given
Figure 1. Relationships between the three diagrams.
Ohlbrock and Schwartz 181
topological diagram T, knowing the edge lengths λ of the
trail members, the force magnitudes µ of the deviation
members and considering initial position vectors pi in the
form diagram F as well as the external force vectors qi
* in
the force diagram F*:
1. For a given sequence K with m trails, the deviation
connectivity matrix CK with dimensions m × m
can be developed, based on a given topological
diagram T and the combinatorial state of each
member. If a branch connects nodes i and j with a
compressive edge, then the element (i, j) equals −1,
and element (j, i) equals 1; if the same branch con-
nects nodes i and j with a tensile edge, then entry (i,
j) equals 1 and element (j, i) equals −1; all other
elements are 0
CK=
−
−−
L
MOM
L
c
c
iK jK
iK jK
,_,
,_,
(1)
with ciK jK,_,=1 if tension and ciK jK,_,=−1 if
compression.
2. Determine matrix of unit direction vectors EK
(with dimensions m × m) for those members that
are connected in the topological diagram
E
e
e
K
iK jK
iK jK
=
−
−
L
MOM
L
,_,
,_,
(2)
with e
pp
pp
iK jK
jK iK
jK iK
,_,
,,
,,
=
−
−
.
3. Consider the matrix µK (with dimensions m × m)
containing the force magnitude values µi, K_ j, K of
the deviation members as a parameter (absolute val-
ues) that can be defined and controlled by the user
µµ
K=
−
−
L
MOM
L
µ
µ
iK jK
iK jK
,_,
,_,
(3)
4. Develop the Hadamard product HK (with dimen-
sions m × m). The Hadamard product is defined as a
matrix where each elements entry (i, j) is the prod-
uct of elements entry (i, j) of the original matrices
(with the same dimensions)
HCE
KK
KK
=⋅⋅
µµ
(4)
5. Obtain column vector DK
* (with dimensions m × 1)
containing the resultant deviation vectors diK,
* for
every node i by adding all entries j of Hk in row i
Dd H
K
*
iK
*
k
==
=
∑
,
j
m
1
(5)
6. Impose equilibrium to obtain the column vector
fK+1
* of unknown forces fiK,
*
+1 (with dimensions
m × 1) of the consecutive force trail members
ffDq
KKKK+++
+=
10
**** (6)
ff
Dq
KK
KK
+=−
−−
1
**
**
(7)
7. Determine the column vector pk+1 of the unknown
position vectors piK,+1 considering the combinato-
rial state of each force and vector
λλ
K+1, containing
the edge lengths
λ
iK,+1 of the consecutive force
trail members. If the consecutive trail member is a
tensile edge, then
λ
iK,+1 is larger than 0. If it is a
compressive edge, it is smaller than 0
pp f
f
KKK
K
K
++
+
+
=+
11
1
1
λλ
*
*
* (8)
Following the same procedure, all the unknown forces
f* and position vectors p can be determined in a sequential
way until the support nodes are reached.
Variations
To illustrate this procedure and possible variations, a
generic topological diagram T with four trails, each discre-
tized in a tripartite sequence of nodes, eight external mem-
bers, four deviation members, eight force trail members,
and the combinatorial state (compression–tension) is con-
sidered here (Figure 2).
Based on this set-up, it is shown for the first sequence
(K = 0), how to find the unknown position vectors (p1,1,
p2,1, p3,1, and p4,1), depended on the set of the correspond-
ing parameters (gray shaded boxes). The connectivity
information from the topological diagram plus the position
vector pi,0 allows to determine the required unit vectors.
The multiplication of these unit vectors with their corre-
sponding force magnitudes µ0 leads to the deviation forces
(dashed lines) that are added to the external force vector
qi,
*
0 in the force diagram F
0
*. Closing the force polygons
determines the subsequent trail forces f
ii
,_,
*
01
in this force
diagram. The multiplication of the unit vectors of these
forces with their corresponding edges lengths λ1 leads then
to the position vector pi,1 of the subsequent nodes when
this product is finally added to the preceding position vec-
tor pi,0 (Figure 3).
In the next sequence (K = 1), these determined position
vectors (p1,1, p2,1, p3,1, and p4,1) serve as an input parameter
pi,1. In this sequence, only two nodes (P1,1 and P4,1) are
182 International Journal of Space Structures 31(2-4)
connected with the tensile deviation member t1,1_4,1. The
external force column vector qi,
*
1 equals zero. Since the
nodes P2,1 and P3,1 are not connected to a deviation
member, the directional unit vector of the trail members
t2,1_2,2 and t3,1_3,2 remains invariant regarding their preced-
ing trail members t2,0_2,1 and t3,0_3,1, respectively. The other
two trail forces f11 12,_,
* and f41 42,_ ,
* are the result of clos-
ing the preceding trail forces and the tensile deviation
force in the force diagram F
1
*. The position vector pi,2 of
the following nodes is determined as in the sequence
before (Figure 4).
One key aspect of the CEM is to facilitate a controlled
variation of the topology during the early design stage. In
order to illustrate this possibility, sequence (K = 1) is
repeated, specifically, with the difference that an addi-
tional compressive deviation member t2,1_3,1 is added in the
topological diagram T between the nodes P2,1 and P3,1. The
comparison of the form diagrams F2 in Figures 4 and 5
illustrates the consequences of this additional deviation
member. The position vectors p2,2 and p3,2 are now pushed
outward by the additional compressive force. The other
two unknown position vectors p1,2 and p4,2 in this case are
not affected by the topological variation and remain invari-
ant compared to the previous state.
The other main aspect that allows to open up the design
space is the combinatorial variation. In order to visualize this
variation, the compressive (blue) member t2,1_3,1 is changed
into a tensile member (red). Compared to the previous state,
the nodal position p2,2 and p3,2 are now pulled into the inside
of the equilibrium network in the form diagram F2. Again,
the other unknown position vectors p1,2 and p4,2 are invariant
compared to the previous states. The thin dashed lines (blue)
in the form diagram F
1
* and their corresponding edges in the
form diagram F2 show the influence of a variation in µ1. For
Figure 2. Topological diagram T with four trails and four
deviation members.
Figure 3. Topological diagram T, form diagram F0, force diagram F0
*, and form diagram F1 for K = 0.
Ohlbrock and Schwartz 183
one extreme, in case µ2,1_3,1 equals 0, the trail member is not
redirected. For the other extreme, in case µ2,1_3,1 equals infin-
ity, the trail member is parallel to the responsible deviation
member t2,1_3,1. A variation of λ2,1_2,2 which is part of λ2 is also
visualized in the form diagram F2. It enables to move the
position of node P2,2 in the form diagram along the line of
action of the preceding trail member t2,1_2,2 (Figure 6).
Following the sequential definition which was intro-
duced in this chapter, equilibrium is guaranteed at every
moment of the process. Due to the low computational
effort, the consequence of any variation can be observed
in real time. Furthermore, all the presented variations
can be individually combined and executed in any
sequence.
Figure 4. Topological diagram T, form diagram F1, force diagram F
1
*, and form diagram F2 for K = 1.
Figure 5. Topological diagram T, form diagram F1, force diagram F
1
*, and form diagram F2 for K = 1 and a topological variation.
184 International Journal of Space Structures 31(2-4)
Designing with CEM
Implementation of the interactive tool
To test the approach of CEM in 3D, an interactive CAD
prototype, named “DEVIATE” has been developed within
the CAD software Rhinoceros, which is a commonly used
tool in 3D design. The required code was written in the
scripting environment RhinoPython, which can be accessed
via Grasshopper, the associative modeling plug-in for
Rhinoceros. The developed prototype allows two slightly
different modes of using it. Within the first relative one, the
user has not only to specify the topological diagram T and
the other necessary inputs but also to model the geometry
of a boundary volume V, in which the spatial force network
is then deployed. Hence, in this relative mode, the edges
lengths are not absolute values but range from 0 to 1.
Whereas 0 signifies that the distance from the last position
vector to the subsequent is zero, 1 implies that the next
position vector is located at the intersection of the line of
action of the preceding edge and the boundary volume. The
required intersection calculations are automatically carried
out by the prototype. In the other, the absolute mode, only
the position vectors of the starting nodes as well as the load
vectors have to be specified on top of the defined topologi-
cal diagram T and the required metric λ and µ values.
Relative mode—arena roof design
This exemplary design process starts with the modeling of
a simple elliptical prism, representing the geometrical
design boundaries of an arena roof and the definition of a
governing load case. Based on a variation of the topologi-
cal diagram, several radically different spatial force net-
works can be actively created and explored with ease.
Figure 7 shows six different topological diagrams T and
possible corresponding form diagrams (a)-(f) that are
designed with this prototype for constant λ and ν in each
network sequence. In a second step, the structural concept
of (e) is selected and its force flow resolution is increased
to get a better impression on how the structure geometri-
cally could look like (Figure 8).
Absolute mode—tower design
In this design example, neither a boundary design space
nor the location of the supports is defined. Hence, the
parameters λi_j are regarded as absolute values. The fictive
task is to design the primary structure for a tower with ca
40 m height, based only on combinatorial variations of the
simple and regular topological diagram given in Figure 9.
A circle with a radius of 10 m, which describes the approx-
imate outline of the tower, is discretized in eight pieces to
allocate the initial position vectors of the form diagram.
Here, self-weight is taken into account at each node (visu-
alized with green diagonal crosses at the affected nodes in
the topological diagram). The nodal load is calculated with
the length of the connected elements multiplied by a spe-
cific weight factor controlled by the user.
Figure 10 shows an excerpt of the matrix of possibili-
ties that the CEM approach is enabling for the described
Figure 6. Topological diagram T, form diagram F1, force diagram F
1
*, and form diagram F2 for K = 1 and a combinatorial variation.
Ohlbrock and Schwartz 185
scenario. Based on the regular, given topological diagram
with five sequences in Figure 9, it is possible to generate
form and force diagrams for (232 (32 trail members that can
be compression or tension) *332 (32 deviation members
that can be zero, compression, or tension)) 7.95e + 24 dif-
ferent qualitative structural behaviors (vertical axis).
Within each of them, there are infinite different metric
equilibrium states conceivable (horizontal axis). The first
qualitative option (a) corresponds to a very simple column
like system. None of the deviation members is active in the
topological diagram. The intensities of the compression
forces on the force trails are increasing at each node (due
to the self-weight) but do not change their directions. The
intensities of the forces are qualitatively visualized with
the line thickness in the form diagram. To the right of this,
a metric variation (a2) is visualized. If the starting outline
is changed from a circle into another shape (e.g. ellipse),
the position vectors of the input nodes changes. Hence, the
form diagram automatically adapts to this change in the
geometric input. In the second created option (b), the qual-
itative structural behavior is transformed by activating the
deviation member in between neighbor trails in all the four
sequences of the topological diagram. On the right, the
form diagram for reduced force magnitudes of the devia-
tion members is visualized. If these values are all zero, this
option corresponds to the first option (a1). The third design
Figure 7. Six alternative form diagrams that are based on a variation of the topological diagram (a) - (f).
186 International Journal of Space Structures 31(2-4)
option (c) shows the result of closed and active compres-
sion rings in all sequences. The result is a rotational sym-
metric compression-only dome. To the right (c2), every
second edge length λ is set to zero. This leads to a triangu-
lated force network which still follows exactly the same
rules as described. In order to reach the intended height of
the tower, the edge length λ in the last sequence is increased.
Another well-known structural concept (d) for towers con-
sists of a hanging structure on the outside that on top is
connected to the core, which brings the collected forces
down to the foundations. In order to achieve the height of
the tower, the edge lengths λ in the last compression
sequence are increased to 45 m. On the right of this fourth
option, the force magnitude values in the first sequence are
set to zero. The last option (e) in Figure 10 can be created
by alternating sequences of tension and compression rings.
On the right side, specific metric values are changed indi-
vidually (e2), in order to illustrate that the CEM approach
has the potential to allow also irregular shapes in equilib-
rium to emerge. As discussed earlier, one motivation for
this research and the CEM approach is to overcome typol-
ogies and their implied formal limitations. The presented
case study, although consisting in a very regular topologi-
cal diagram and a conventional combinatorial state force
pattern, signifies that more complex but also intriguing
structural forms in equilibrium are hidden in the depth of
this CEM-based approach.
Discussion and future work
On top of the well-known form and force diagram, this
research introduces a planar topological diagram, a sys-
tematic graph theory–based representation of the qualita-
tive structural behavior. Due to the sequential character of
the CEM, it is now possible to control the spatial form and
the force diagram for any combination of tension and com-
pression forces with simple geometrical operations only.
The developed prototype shows very robust characteris-
tics. Even a designer without equilibrium knowledge is
able to generate a huge variety of different spatial networks
easily. The first example shows that it is possible to create
and explore several radically different, conventional but
also unexpected non-typological force networks for a
given boundary geometry in real time. The second case
study illustrates that a CEM-based design process can be
synthetic, chronology free, and that form and force can be
actively controlled simultaneously from the conceptual
stage on. As described in section “Concepts,” it is a major
aim to actively involve the designer in the digital creation
of his artifacts. With the present framework, a designer, by
being actively involved, can easily learn how forces can be
disposed in space, be deviated, and how they can be intui-
tively manipulated. The developed tool is conceptually not
used to narrow or predetermine anything. Ideally, it serves
as an explorative sketch board with a programmed guard-
rail that ensures equilibrium at every moment. Eventually,
it offers like “architectural sketches do, pictures of possi-
bility”26 (p. 40) and by this “allowing the designer the free-
dom to express his own ideas” (p. 142),4 but always in
equilibrium.
Future research will discuss and try to improve the opera-
bility of the CEM approach. Within this section, the role of
Figure 8. Topological diagram T, top view and perspective of
form diagram F of concept e) with a higher resolution of force
elements.
Figure 9. Regular topological diagram T with 32 trail members
and 32 deviation members.
Ohlbrock and Schwartz 187
Figure 10. Exemplary matrix of possible form diagrams in equilibrium that can be created out of the topological diagram in Figure 9.
188 International Journal of Space Structures 31(2-4)
the force diagram and its representation will be further
explored. The operative strength of the classical 2D reciproc-
ity is often lost in a fully 3D approach. This research will
develop a new proposal aiming to keep the interplay between
those two diagrams as operative and legible as possible.
Furthermore, there will be an intensive examination with the
question of multiplicity of solutions, how the difference
between design proposals could be evaluated in order to pro-
vide a complementary view in the early stage of a design pro-
cess. At this stage of the research, only regular topological
diagrams were developed in the CEM. Obviously, it is the
ambition of the authors also to handle more complex and
irregular topologies and by this also discuss the static and the
kinematic determinacy (including the instability for other
load cases) of the generated networks. It is further planned to
ask students to work with the prototype in order to understand
with their feedback and their results, how designers could
really use and benefit from this novel and unconventional
approach. Additionally, these results will hopefully also allow
a detailed discussion on the advantages and the challenges of
the CEM’s sequential and non-linear character.
Comments in a practical context
As a product of the CEM approach, the designer possibly
generates a spatial force network which is not a result of a
pure deductive analytical process, but rather the result of
an inductive, creative, and interactive process based on
graphic statics and equilibrium only. The presented
approach corresponds quite well to the structural design
philosophy of many successful practitioners such as Frei
Otto, Heinz Isler, and Sergio Musmeci27 who in the 1970s
stated that in his design “the form and not the stresses are
the unknown.” Nowadays, a tendency toward material-
independent equilibrium forms as a departure is detectable
among contemporary structural designers.26 It is obvious
that these force networks can only give a first but strong
idea of the structures global behavior.
To get a clearer picture of the behavior of the structure,
some further decisions have to be taken. For example, if the
forces are discretized as a visible truss-like structure or if
some forces constitute a more continuous surface structure.
Furthermore, the materialization and construction detailing
obviously play an important role in this context. It is clear
that further analysis for all load cases should be based on
these material-related decisions to ensure the structure’s ser-
viceability and prevention against buckling of the compres-
sive members. More to the point, this materialization is the
key for the architectural articulation of space but also crucial
for the constructive and tectonic logic of the structure.
Conclusion
The presented CEM approach, implemented within a CAD
environment, allows for the intuitive material-independent
exploration and even an active design of spatial force net-
works for arbitrary load scenarios and geometrical layouts.
In such a way, the global structural behavior can be con-
trolled by the designer and furthermore be aligned with the
architectural concept quickly, interactively, and based on
geometric operations only. By allowing lateral shifts
between different qualitative structural behaviors, the pre-
sented approach frees the designer from early determina-
tions and typologies and gives a new perspective on the
discussion on the interaction between structural behavior,
architectural design, and the use of computational models
in the conceptual design phase.
Acknowledgements
We owe thanks to Pierluigi D’Acunto for his suggestions and criti-
cal comments during the development of this paper. Furthermore,
we thank Juliana Felkner for careful and patient editing.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
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