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Equilibrium as the common ground: Introducing embodied perception into structural design with graphic statics

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The analogy between the human body and architectural structures dates all the way back to ancient times and has significantly shaped the design of buildings and structures. The article examines the body's historical influence on how structures are perceived and designed , demonstrating how the body shapes the "technical truth" dimension of structural design while oblivious to the importance of an "artistic truth" or perceptual dimension. This article aims to connect recent neuroscience findings and their implications for structural design through graphic statics and its design methods. Finally, this article proposes an equilibrium-based structural design approach for designing embodied structures based on graphic statics.
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RESEARCH ARTICLE
Equilibrium as the common ground:
Introducing embodied perception into
structural design with graphic statics
Shuaizhong Wang
a,
*, Toni Kotnik
b
, Joseph Schwartz
a
,
Ting Cao
c
a
Department of Architecture, ETH Zu
¨rich, Zu
¨rich, 8093, Switzerland
b
Department of Architecture, Aalto University, Espoo, 02150, Finland
c
Department of Architecture, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China
Received 7 November 2021; received in revised form 19 December 2021; accepted 3 January 2022
KEYWORDS
Structural design;
Body;
Equilibrium;
Perception;
Graphic statics;
Neuroscience
Abstract The analogy between the human body and architectural structures dates all the
way back to ancient times and has significantly shaped the design of buildings and structures.
The article examines the body’s historical influence on how structures are perceived and de-
signed, demonstrating how the body shapes the “technical truth” dimension of structural
design while oblivious to the importance of an “artistic truth” or perceptual dimension. This
article aims to connect recent neuroscience findings and their implications for structural
design through graphic statics and its design methods. Finally, this article proposes an
equilibrium-based structural design approach for designing embodied structures based on
graphic statics.
ª2022 Higher Education Press Limited Company. Publishing services by Elsevier B.V. on behalf
of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction: the body in structural design
Comparisons between buildings and the human body have a
long history and have become more prevalent in architec-
tural theory and practice since the Renaissance. For
example, Leon Battista Alberti describes the art of building
using the metaphor of bones, muscles, ligaments, and skin
(Alberti, 1988). The analogy between body and structure
was frequently mentioned in construction and architecture
research, particularly during the nineteenth century, as
building science developed. Among them, Thomas Tredgold
compared structure and construction to architecture’s
anatomy in his study of the science of the human body,
arguing that the concept of the body and its parts as well as
the mechanisms that govern its various behaviors, are
* Corresponding author.
E-mail address: shuaizhong.wang@arch.ethz.ch (S. Wang).
Peer review under responsibility of Southeast University.
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Please cite this article as: S. Wang, T. Kotnik, J. Schwartz et al., Equilibrium as the common ground: Introducing embodied perception
into structural design with graphic statics, Frontiers of Architectural Research, https://doi.org/10.1016/j.foar.2022.01.001
https://doi.org/10.1016/j.foar.2022.01.001
2095-2635/ª2022 Higher Education Press Limited Company. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Available online at www.sciencedirect.com
ScienceDirect
journal homepage: www.keaipublishing.com/foar
Frontiers of Architectural Research xxx (xxxx) xxx
strikingly similar to architecture as a whole in terms of its
constituent parts and material structure (Tredgold, 1820).
This fundamental and critical criterion is frequently over-
looked during the architect’s design process, leaving the
structure solely in the hands of the structural engineer-
dleaving the architectural structure as a purely intellec-
tual activity rather than derived from any particular bodily
experience, effectively suffocating the bodily meaning of
structure (Picon, 2005).
Fortunately, a group of architects and structural engi-
neers have constantly incorporated the human dimension
into the process of structural design and thought by drawing
inspiration from or imitating the body and experience. To
reintroduce the human dimension into structural design and
thought, this article reviewed various historical uses and
conceptions of the body in structural design and extended
them to include recent neuroscience findings regarding
embodiment and its relationship to structural perception.
Based on the critical role of equilibrium in the embodi-
ment, the following paper will review and analyze various
embodied structures from selected architectural examples
to propose an equilibrium-based structural design approach
for designing the embodied structures based on graphic
statics.
2. Anatomical perspective
As early as the 15th century, Leonardo da Vinci used the
diagram of the human spine in Codex Atlanticus to describe
the logic of the supporting structure of the dome. In the
nineteenth century, Viollet-le-Duc’s research on Gothic
churches expanded this anatomical analogy between body
and structure. Influenced by the comparative anatomy of
French paleontologist Georges Cuvier, Viollet-le-Duc used
anatomy as an analytical method to study the relationship
between structure, function, and form in the Gothic
church. Cuvier’s comparative anatomy began by examining
how the organs of an organism work together to form a
unified whole, both formally and functionally. Inspired by
Cuvier, Viollet-le-Duc used graphical representation
methods such as exploded diagrams and sections to reveal
the inextricable relationships between each structural
element in the Gothic church as well as the synergistic
relationships and mechanics that underpin their organiza-
tion (Viollet-le-Duc, 1990). Thus, Viollet-le-Duc argues that
the form of architecture is determined by structural prin-
ciples, such as organic and rational nature; once structural
and construction principles are established, the form or
“style” will naturally follow (Viollet-le-Duc, 1854e1868).
Similar to the exploded view used to dissect the human
body, Viollet-le-Duc used it to disassemble the structural
elements of Gothic churches, demonstrating clearly and
visually the static relationships between the parts and the
steps involved in how the ribs, for example, transferred the
stress from the vault to the flying buttress below (Fitchen,
1981)(Fig. 1). This anatomical representation of the
structural system demonstrates unequivocally that the
organic structural system of Gothic architecture is derived
primarily from force equilibrium, as a result of the mutual
counteraction and overall equilibrium of the forces acting
and reacting between the elements, thus “each stone with
a function such that no stone could be removed without
compromising the entire structure” (Viollet-le-Duc, 1990,
pp. 259e260). Furthermore, Viollet-le-Duc views this dia-
grammatic representation of equilibrium condition not only
as an interpretation or reproduction of current architec-
tural styles but also as a tool to rationally stimulate the
“active imagination” (Viollet-le-Duc, 1987). Compared with
the Beaux-Art system, which was dominated by passive
memory, this active anatomical graphic perspective is a
more rational and scientific method of conveying and even
manipulating knowledge. This abstract representation not
only assists the viewer in comprehending and absorbing the
structural principles underlying the optical forms but also
Fig. 1 Exploded drawing of the Gothic arch. Source: from
“Dictionnaire” (Viollet-le-Duc, 1854-1868) by Viollet-le-Duc,
"Construction," volume 4; public domain.
S. Wang, T. Kotnik, J. Schwartz et al.
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guiding the viewer in refining and reconstructing knowl-
edge, which is necessary for progressing toward
an analytical memory and even creating a new “style”
(Vinegar, 1995).
As a continuation of Viollet-le-Duc’s rational investiga-
tion of the body and structure, in 1866, under the influence
of the anatomist Georg Hermann von Meyer, Karl Culmann
described a vector-based graphical structural design
methodology on the basis of the equilibrium condition of
tension and compression curves to represent the inner
stress condition in load-bearing structures in his book, Die
graphische Statik (Graphic statics) (Maurer, 1998). Culmann
used graphic statics to analyze the stress patterns in curved
structures such as the Fairbairn crane and discovered that
they are remarkably similar to the internal structural pat-
terns of the proximal femur drawn by von Myer in 1867 (von
Meyer, 1867)(Fig. 2). The Berlin surgeon Julius Wolff
confirmed this similarity in 1870 after photographing the
internal structure of the sliced bone (Morava
´nszky, 2019).
These remarkable investigations exemplify Viollet-le-Duc’s
anatomical perspective on the rational relationship be-
tween form and structure. Thus, the way nature constructs
the skeleton and the human body is highly consistent with
the logic and purpose of structural design.
Culmann’s student, Maurice Koechlin, sketched the steel
structure of Eiffel Tower at Gustave Eiffel’s firm under the
influence of graphic statics. However, this result of utilizing
the fewest possible materials to achieve the most efficient
structural design was widely discussed at the time. On the
one hand, people were impressed by the new forms created
by the new materials but were scandalized by the fact that
the “fleshless” or “massless” structural skeleton did not
meet the project’s artistic and aesthetic requirements: “.
the human skeleton is surely the most perfect work of en-
gineering. But for my eye, when it is in search of beauty, it
is the blooming flesh that is decisive” (Lux, 1910, pp. 3e4).
Although the design of the Eiffel Tower exemplifies Viollet-
le-Duc’s rationality that form follows structure, these ar-
guments also demonstrate the omission of other systems
such as muscle and skin from the analogy between struc-
ture and the human body, resulting in a visual imbalance in
the skeletondthe experience and aesthetics of the body as
a whole cannot be seen separately. This idea indicates that
focusing exclusively on structural techniques precludes
structural design from an active comprehensive shaping of
architectural space.
3. Ontology and representation of the
structure
The debate over the Eiffel Tower’s “bone” and “flesh”
dates all the way back to Karl Bo
¨tticher’s discussion of the
art-form and core-from of structure. Even before Viollet-le-
Duc, the German theorist Karl Bo
¨tticher was concerned
with the representational aspect of Greek architecture and
the ontological aspect of Gothic architecture (Frampton,
1995). In his book, Die Tektonik der Hellenen, he coined
the German terms, Kernform (core-form) and Kunstform
(art-form), as analytical tools for interpreting the structural
design. Both of these terms “associate the separation be-
tween static structures from its artistic apparel (Mayer,
2004, p. 18).” The core-form refers to an architectural el-
ement’s material and static function, while the art-form is
designated as how this static function becomes apparent
and acquires meaning. Bo
¨tticher sees the relationship be-
tween Kernform and Kunstform in architectural structures
as a kind of corpus or Ko
¨rper bilden, arguing that archi-
tecture should focus on the appropriate interconnection of
structural elements to generate an expressive Kunstform
through the construction of Kernform (Schwarzer, 1993).
Similar to Bo
¨tticher, Gottfried Semper argued that
Viollet-le-Duc’s theory constrained artistic freedom and
imagination and expanded his famous Raiment theory,
which divides the structure into scaffolding and cladding
respectively; these two correspond to the ontology and
representation of the structure (Gottfried, 1989)(Oechslin,
2002). However, Semper’s view was later criticized by
people such as Hendrik Petrus Berlage, Otto Wagner, and
August Schmarsow, who argued that the two should not be
studied separately, and that the internal skeleton must be
considered alongside its ornamental artistic expression to
return to the inseparable “full-body” (Berlage, 1905, p. 24)
(Frampton, 1995, p. 89). The Viollet-le-Duc’s limitation is
that he considers the Kunstform of the structure as a nat-
ural consequence of the Kernform, starting only at the
level of construction and technology. While Semper’s
Raiment theory transitionally emphasizes the representa-
tional dimension of ornament, it obliterates its technical
part. Similar to how the bones and the skin of the body are
inextricably linked, the relationship between Kunstform
and Kernform should be complementary rather than
dichotomous. In this regard, Fritz Schumacher proposed the
“double truth concept,” in which “technical truth” serves
as the foundation for the realization of “artistic truth;”
additionally, “artistic truth” serves as an enhancement or
symbolic representation of “technical truth” (Schumacher,
1838, p. 228) (Morava
´nszky, 2019).
Viollet-le-Duc et al. have conducted many scientific
studies on the “technical truth” of the structure of the
Medieval Church through anatomical representation.
Fig. 2 Left is Culmann’s analysis of the patterns of internal
forces in a Fairbairn crane by graphic statics; Right is Julius
Wolff’s drawing of trabecular structure in the proximal femur.
Source: from “On Growth and Form” (Thompson, 1942)by
D’Arcy Thompson; public domain.
Frontiers of Architectural Research xxx (xxxx) xxx
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Similarly, since the 19th century, long-time scientific in-
terests and debates have existed regarding the “artistic
truth” of structural design.
1
One of the most representative
figures is Heinrich Wo
¨lfflin. In his study of “force” in Re-
naissance and Baroque architecture, Wo
¨lfflin focuses not so
much on the building statics but rather on how the body has
been used as a “metaphor of force” to empirically “feel”
the psychological tension and compression (Wo
¨lfflin, 1994)
(Forty, 2000). He reveals that the “force” of structure ex-
ists not only on a physical level but also on a psychological
level through Einfu
¨hlung (empathy) brought about by
embodiment. Empathy explains the capacity to understand
and “feel into” other things through the sympathetic pro-
jection of the human body. It primarily explains the unified
human perception of structural expression. The study of
empathy on how humans psychologically and biologically
perceive the expression of structures through the human
body has been addressed in numerous architectural designs
and research throughout history.
2
However, since the early
twentieth century, the theory of empathy has been pri-
marily driven by technology-oriented formalism. Further
developing the “artistic truth” part of structural design was
restricted by a lack of scientific basis to explain or
demonstrate the principles underlying how humans read
and resonate with structural expressions (Mallgrave, 2013).
Therefore, it cannot provide detailed guidance for the
structural design.
4. Neuroscientific approach
Recent rapid advances in the field of Cognitive Neurosci-
ence enable a new way of scientifically conceptualizing the
metaphor and analogy between structural design and the
human body; it also expands our theoretical framework
through the notion of embodiment and embodied simula-
tion derived from theories such as empathy, thus marking
the beginning of a new chapter in the history of the body in
structural design (Mallgrave, 2013). With the goal of
investigating representational and artistic mechanisms of
perception, cognitive neuroscience provides an egocentric
perspective on human perception and understanding of
structures (Freedberg and Gallese, 2007). Similar to how
the development of biological sciences has inspired and
promoted structural design and thinking throughout history,
the findings from neuroscience can provide a rigorous
explanation of the human embodied perception of
structural expression, which complements the research on
the representational aspect of structural design.
As a milestone in neuroscience, the discovery of the
Mirror neuron in the mid-1990s established that the neural
circuits used to simulate other people’s actions are located
in the same areas of the brain as those used to perform our
own actions (Rizzolatti et al., 2006). Thus, traditional
psychology and cognitive science’s perception principles,
which consider perception as a computer-like processor of
visual signals in the brain, are flaweddthis passive and
disembodied dualistic view of human perception is one-
sided (Pe
´rez-Go
´mez, 2015). According to the mirror
neuron, the same neural structures involved in our own
bodily experiences contribute to conceptualizing what we
see and feel in the world (Ebisch et al., 2008). This
embodied perception process is based on the mechanism by
which humans initiate unconscious perception (System I)
prior to consciously analyzing it (Kahneman, 2011)d
embodied perception does not begin with a specific and
precise analysis. It is a precognitive or pre-reflective
instant perception occurring prior to conscious awareness
and is evocative of a previous similar bodily experience
(Gallese and Gattara, 2015, p. 162). Therefore, the mirror
perception system is “a direct form of ‘experiential un-
derstanding’ of others, achieved by modelling their be-
haviours as intentional experiences, based on the
equivalence between what the others do and feel and what
we do and feel” (Gallese, 2007). This idea implies that
human body is necessary for us to have an empathic rela-
tionship with the world (Rizzolatti et al., 2006). Thus, when
individuals observe a gesture that resembles a previous
bodily memory, they directly and unconsciously evoke the
previous bodily experience and mood associated with this
bodily gesture, thereby demonstrating their ability to read
into things. This phenomenon may account for the
perceptual similarity between the structural designer and
an untrained observerdtheir initial unconscious reaction to
the same structural expression will be very similar due to
their nearly identical bodies. Following the unconscious
impression, the structural engineer’s knowledge as well as
the knowledge of other individuals with varying educa-
tional/psychological backgrounds and cultural sensibilities,
will manifest in the conscious and analytical reading of the
structural expression. On the basis of the mirror neuron,
the notion of “embodied simulation” was proposed as an
extension to explain how humans not only “see” the built
environment but also feel and simulate emotions and ac-
tions from the world through the medium of the body and
experience (Gallese, 2007)(Thompson, 2007). This idea is
similar to Merleau-Ponty’s view that “the body is the
vehicle of being in the world (Merleau-Ponty, 1962).” The
findings of the embodied simulation were based on the
premise that perception and cognition are intrinsically
dependent on the organisms’ interaction with their envi-
ronment (Varela et al., 1991)(Thompson, 2007)(Jelic
et al., 2016). Which meanings, other than the mirrored
projection of a static bodily gesture, embodied perception
could simultaneously emerge from active dynamic action
and movement. The recent research indicates that our
embodied simulation is not limited to the social world.
Humans possess the “precognitive capacity to mirror the
tactile values of all objects or forms in our environments,
1
See for example M. Merleau-Ponty, Phenomenology of Percep-
tion, London: Routledge and Kegan Paul, 1962; H. Wo
¨lfflin,
“Prolegomena to a Psychology of Architecture,” in Empathy, Form,
and Space: Problems in German Aesthetics, 1873e1893, Santa
Monica, Getty Center Publication Programs, 1994; K. Frampton,
Studies in Tectonic Culture: The Poetics of Construction in Nine-
teenth and Twentieth Century Architecture, Cambridge, Massa-
chusetts: MIT Press, 1995 (2001).
2
See for example Schmarsow, August. Das Wesen Der Archi-
tektonischen Scho
¨pfung: Antrittsvorlesung Gehalten in Der Aula
Der K. Universita
¨t Leipzig Am 8. November 1893. Leipzig: Karl W.
Hiersemann, 1894; S. Holl, J. Pallasmaa and A. P. Go
´mez, Questions
of perception: phenomenology of architecture, SanFrancisco,CA:
William Stout Publishers, 2006.
S. Wang, T. Kotnik, J. Schwartz et al.
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both living and non-living” (Mallgrave, 2015). Vittorio Gal-
lese, a co-discoverer of mirror neurons, also proposes that
embodied simulation be used in place of the traditional
term “empathy” (Mallgrave, 2015).
The embodied simulation shows that our ability to read
force, balance, and aesthetics of structures is dependent
on our bodily memory. It is the sensation of muscular
contraction during the gesture that is memorized as well as
the position of the limbs in relation to one another and the
body as a whole. Thus, bodily gestures and movement
patterns can serve as a reference for designing structural
expressions. Whether it induces or inhibits the perception
of the built environment, one could argue that the funda-
mental goal of all design thinking is to establish an aspira-
tional dialogue between the human body and architectural
space, thereby enhancing the interconnectedness and
balance of matter and space (Jeli
c, 2015)(Drake, 2005).
The mechanism by which spatial order is established
through motion and embodied experience of spatial struc-
ture is linked to proprioception and bodily schema in
neuroscience.
Proprioception is the ability to grasp one’s own position
in space, including limb sensations, movement in space,
and sense of effort (Charlton, 1888)(Goodwin et al., 1972).
They helped develop the human balance system, along with
vestibular sense and kinesthesia. In addition, propriocep-
tion is the study of how the body’s parts unconsciously
coordinate with one another to maintain flexible balance
and gesture and how the body as a whole coordinates with
its surrounding space to maintain balance in space,
whether static or moving (Angelaki and Cullen, 2008). The
embodied sense of balance was defined by the sense of
force generated by the body in response to sensory signals
from muscles, joints, and skin receptors in response to
stretch and compression of body tissues (Colombo et al.,
2018). The notion of bodily schema
3
is a representation of
this constantly changing muscle configuration that uncon-
sciously controls our body’s shape and posture, our actions,
and our movement in space (Jelic et al., 2016). Notably,
interoceptive emotional inputs in the form of motivating
tendencies to act are included in body schemas on the
bases of balance (Gallagher and Bower, 2014). Accordingly,
it represents both physical and psychological equilibria.
These findings from cognitive neuroscience corroborate,
on the one hand, the scientific rationale behind Viollet-le-
Duc and Wo
¨lfflin’s hypothesis that the body serves as a
“metaphor of force” for reading and perceiving the logic of
forces behind structures; on the other hand, they also fill in
a gap in their empirical findings on people’s perception of
structures: the influence of dynamic interaction on the
body’s relative relationship to the built environment. Sup-
pose we interpret Viollet-le-Duc’s analogy between the
composition of structural elements and the composition of
body parts in medieval churches through diagrams such as
exploded drawings or sections through the lens of embodied
simulation and proprioception. This diagrammatic
representation aids the viewer in comprehending the rela-
tionship between forces and equilibrium because people
were “rendering” and simulating the observed structural
composition using their muscle memory of the body parts,
thereby revealing the sense of “force” behind it. Addi-
tionally, one can gain a better understanding of the purpose
of structural expression in Gothic churches as described by
Nikolaus Pevsner, which is “to enliven inert masses of ma-
sonry, to quicken spatial motion, to reduce a building to a
seeming system of innervated lines of action” (Pevsner,
1943, p. 90), that is, to stimulate bodily movement and
interaction with the structure of the building.
5. Equilibrium as the common ground
Considering that structural design encompasses both tech-
nical and artistic aspects, using the body as a design
reference or method should be analogous to both the
physical and psychological balance of the structure’s bones
and muscles. Otherwise, the structural significance of
architectural space cannot be fully expressed. Conse-
quently, balance is a critical reference point and tool in
architectural and structural design. For instance, balance is
a recurring theme in a number of Auguste Choisy’s works.
As with Viollet-le-Duc, Choisy’s anatomical interpretation
of architectural structure implies that balance is a struc-
tural strategy as well as an aesthetic principle. Accordingly,
Choisy frequently begins his designs by addressing the
aesthetic feature of balance before moving on to structural
form (Etlin, 2010).
Cognitive neuroscience demonstrates that what enables
us to comprehend the balance of “forces” is not only the
formal relativity of the medieval church’s structural ele-
ments as described by Wo
¨lfflin but also the dynamic balance
between our bodies and the structure. The latter’s refer-
ence system encompasses the relationship between one’s
limbs and the relationship between the person and the
structural elements that surround it. This aspect is the
critical factor in dynamic interaction with structures, as it
results in the perception of equilibrium as described in the
embodied simulation rather than simply the projection of
“force” by the body’s geometric relations. Therefore, the
influence of the structure on balance and bodily schema
which can alter one’s perception of and emotional response
to the building is twofold. First, structural forms imply a
state of balance in body posture. Second, structural form
implies the tendency and manner of interaction and move-
ment (position, orientation, scale) in relation to the body in
which it is placed. The former is analogous to implying the
muscular sensations of a body in fragile balance or out of
balance in a frame in the bodily schema series of motions
(Fig. 3). By contrast, the latter is intended to influence the
possibilities of interaction with the body through the
structure’s relationship to the body (Fig. 4). Both intend to
stimulate the arousal of different bodily schemas in people’s
experiences through the design of structural balance.
In particular, the dynamic interaction with structures is
not limited to the body’s action because the structure is
static. Moreover, predicting the person’s relative orienta-
tion, the field of view, and velocity in relation to the struc-
ture is difficult. Apart from direct stimulation of the
3
For other details about bodily schema, see Cuzzolaro, Massimo.
Body Schema and Body Image: History and Controversies. In: Cuz-
zolaro M., Fassino S. (eds) Body Image, Eating, and Weight. 2018,
Springer, Cham, pp. 1e24.
Frontiers of Architectural Research xxx (xxxx) xxx
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movement, the primary way structural design can convey
the perception of “force” rather than purely physical dis-
placementda concept coined by Rudolf von Laban as “im-
pulse or effort”dthrough the expression of dynamic bodily
action (Arnheim, 1974, p. 408).
4
In neuroscience, this
manifestation of force that results in a movement or the
proclivity to move is referred to the term enactive, and it is
the primary trigger for body movement and perception.
5
By
incorporating “Immobile Motion” between the structure and
the persondan invisible psychological effect of the
“tension” between us and the structural elementsdstatic
structures can still create “Directed Tension” within them
(Arnheim, 1974).
6
Recent research has even demonstrated
that people experience more kinetic sensations when con-
fronted with implied motion than when confronted with less
dynamic actions (Proverbio et al., 2009).
More importantly, the structure should transition from
introspective to more compatible structural design on the
basis of the body’s physical and psychological characteris-
tics. Similar to Viollet-le-Duc, the equilibrium system of
great Gothic architecture is honored by distinguishing be-
tween Roman structures’ “inertia” and Gothic structures’
“active” principle (Viollet-le-Duc, 1863, p. 270). By proac-
tively designing a pre-reflective structure, the structure
enables a more interactive spatial experience while car-
rying loads. This active design of structural expression is
comparable to what Arnheim refers to as the Acropolis’s
column perception: we perceive the column as standing
upright and bearing the weight of the roof not only because
we project ourselves onto it but also because the column’s
relative position, proportion, and shape enable and compel
Fig. 3 Bodily schema of carrying a load in (a) and its graphical representation (b), which is similar to the graphics statics’ force
flow in (c), and the tendency to lose balance when adding even a small horizontal force in (d), therefore expressing the fragile
bodily balance behind this schema.
Fig. 4 Different structural representations could influence different body interactions, thus arouse different embodied
perceptions.
4
Rudolf von Laban distinguishes between body displacement and
the visual expression obtained from body dynamic action through
the study of dance. Although he thinks that the body displacement
is defined simply by the attributes of physical vectors, the
expression of human motor behavior concerns the impulse that
gives rise to the movement.
5
Varela et al. argue that our perception arises from interactions
with the surrounding environment, accomplished in an active and
dynamic relationship. To study the generation of this embodied
view of mind, they stablished the enactive approach; the term
enactive depicts a concept that “a living being is an autonomous
agent that actively generates and maintains its own cognitive
domain through continuous reciprocal interactions of the brain,
body, and the world.” For other details, see F. J. Varela, E.
Thompson and E. Rosch, The Embodied Mind: Cognitive Science
and Human Experience, Cambridge: The MIT Press, 1991.
6
Arnheim’s account of “Immobile Motion” emphasizes visual
perception and is dismissive of bodily perception, whereas neuro-
science demonstrates that the body plays a significant role in his
description of people’s perception of “tension.”
S. Wang, T. Kotnik, J. Schwartz et al.
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us to do so. Conversely, a poorly designed structure will not
resonate with us (Arnheim, 1974). Then, the question is,
which specific operational tools should be used to design
structures that actively incorporate physicalepsychological
equilibrium?
As demonstrated by Viollet-le-Duc, the exploded diagram
graphically illustrates the forces acting on the structure by
emphasizing the organic connections between adjacent
parts. Furthermore, the exploded diagram can be used to
guide participatory cognitive behavior of forces within the
structure by re-enacting how the ribs transfer pressure from
the vault to the flying buttress in consciousness via psycho-
logical equilibrium (Fitchen, 1981, pp. 75e77). By contrast,
the exploded diagram is merely a disassembly and inter-
pretation of a pre-existing structural system. It is incapable
of serving as a medium for the design process, which is about
operation and transformation. It is comparable to finite
element analysis (FEM), which is widely used in structural
design today to analyze the structural system rather than to
facilitate continuous deformation or iterative design oper-
ations (Kotnik & DAcunto, 2013). In comparison, Maurice
Koechlin’s graphic statics approach to designing the static
system for the Eiffel Tower provides significant advantages
and the ability for diagrammatic manipulation.
Graphic statics is a vector-based construction of the
equilibrium condition.
7
It has already been formalized as a
methodology for designing building structures by Karl Cul-
mann (Maurer, 1998). It is a simplified, abstract, and
graphical representation of forces equilibrium that is
fundamentally different from mathematical structural
analysis and calculations. It is practically the resultants of
stress fields in structural materialsda spatial network
composed of compression and tension forces in equilibrium,
also known as force flows (Muttoni et al., 1997).
Several books introducing graphic statics or structural
design all explain equilibrium through the use of force flows
and human body motions
8
(Fig. 5). Predictably, the mirror
neuron enables people to easily comprehend the tension and
compression involved in bodily motions because we share
nearly the same equilibrium experience with our bodies and
develop empathy for these forces in response to a specific
bodily gesture (Vignemont, 2010). “Architectural design is
the specialisation of a balanced bodily tension so that, while
moving, the body maintains this equilibrium (Ionescu,
2016).” As pointed out in mirror neuron, our conceptuali-
zation of things begins in the unconscious, through the
evocation of pre-existing experiences. In addition, we are all
born experts in equilibrium; our bodies’ experiences have
prepared and accumulated an infinite variety of muscle
memory in the bodily schema and associated emotions. This
equilibrium experience, which is stored in each body, can be
used as an already prepared vocabulary in the design of
abstract equilibrium systems by graphic statics.
As previously stated, proprioception is the sensation of
tension and compression forces applied to our muscles,
joints, and skin. This definition is consistent with the graphic
nature of force flows based on compression and tension in
graphic statics, allowing for a visual and straightforward
interpretation of bodily experiences as equilibrium of ab-
stract compression and tension forces (Fig. 6). Thus, the
equilibrium diagramda graphic representation of force
flowsdcan integrate human embodied perception and
structural design. Graphic statics is not only the logic for
achieving physical structural equilibrium but also the prin-
ciple that guides the design of architectural geometries and
geometric compositions with the potential to emotionally
project bodily experience.
For example, the force flow in Fig. 3(c) can be used to
describe the bodily equilibrium condition and muscle
stresses in Fig. 3(a) and the feeling of possible momentum in
Fig. 3(d) when subjected to horizontal forces. Using the body
schema as a guide, graphic statics can use force flow distri-
bution as core-form to design structures. By composing and
deforming graphic statics, we can directly guide the con-
ceptual design of core-form, guiding the conceptualization
of the holistic conceptual structure model. Moreover, our
body experience has muscle memory and an emotional
memory associated with these gestures. Therefore, when we
use the experience of body balance to guide the design of
structural equilibrium, we include both the physical and
psychological aspects of balance, which contain people’s
emotions. For example, the entangled relationship between
bone and flesh in the Eiffel Tower can be reconciled by
balancing physical force flow with psychological perception.
Intuitive thinking informed by bodily experience is crit-
ical during the conceptual design phase, when architects
collaborate to quickly outline the composition and thinking
behind the building’s overall structural and spatial concept.
Given that the conceptual structural design phase excludes
detailed structural analysis and verification, considering its
overall structural logic is more critical. Moreover, this
embodied structural design approach allows for incorpo-
rating human body experience into the process of designing
equilibrium, adding a dimension of perception and experi-
ence to the structural design and allowing the structure to
define the expression of space better. Many projects by
Santiago Calatrava are the illustration of this structural
thinking. He was inspired by analogies and metaphors for
bodily gestures, incorporating them into the structural
design process to achieve expressive structural tension.
9
7
The exploration of equilibrium in graphic statics is based on the
simultaneous use of location plan (form diagram) and force plan
(force diagram). While the location plan depicts the geometrical
equilibrium condition, the force plan depicts the magnitude of the
individual forces. The application and operation of graphic statics
discussed in this paper focus on the graphical representation of the
structures in relation to embodiment under the use of form dia-
grams in the conceptual design phase. Therefore, the force dia-
gram, which is more critical for structural analysis and calculation,
receives less attention in this paper.
8
See for example recent textbooks on structural design such as
Allen, E. & Zalewski, W.: Form and Forces: Designing Efficient,
Expressive Structures, Wiley, 2009, Muttoni, A.: The Art of Struc-
tures: Introduction to the Functioning of Structures in Architec-
ture, EPFL Press, 2011.s.
9
See for example from many of his drawing on the analogue
between body and structure in C. Politakis, Architectural Colossi
and the Human Body: Buildings and Metaphors, New York: Rout-
ledge, 2018; Calatrava, Santiago, and de A. C. Carrillo. Santiago
Calatrava: Drawing, Building, Reflecting, 2018.
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7
More importantly, the force flows in graphic statics serve
as more than a simplified method of modelling structure. It
also has the potential for operability. Graphic statics not
only can visualize and diagrammatize the abstract equilib-
rium relationships of structures during the design process
but also that “the reduction to the simple concept of
vectorial equilibrium and the transformation of calculation
into simple geometric operations reduces the amount of
necessary expertise and opens up structural design to
empirical and intuitive understanding, thereby allowing for
increased referentiality” (Kotnik & DAcunto, 2013). The
constructive and generative nature of the vector-based
operations of graphic statics allows the designer to
continuously deform and adjust to different constraints on
the bases of the force flow, thereby achieving the desired
intersection between the structure’s rational logic and the
artistic concept of architecture. It enables architects and
designers to take a more operational and communicative
approach to design.
6. Design of the embodied structure
To describe the relationship between stable form and its
force flow within bone structures, Viollet-le-Duc uses
Fig. 5 Human bodies are used to describe different forces and equilibrium in graphic statics. From “The Art of Structures:
Introduction to the Functioning of Structures in Architecture” (Muttoni, 2011) by Aurelio Muttoni.
S. Wang, T. Kotnik, J. Schwartz et al.
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exploded diagrams, whereas Culmann uses graphic statics.
However, they are all limited on the research of core-form
of structure in terms of stability. In addition, this paper
attempts to incorporate the art-form into the structural
design process through embodied perceptual principles to
design a pre-reflective structure that the human body can
perceive and experience in the future.
To advance the neuropsychological embodied under-
standing of structures to a more operational level for
design, deconstructing bodily gestures and movements in
relation to structural design principles becomes critical.
Neuroscientific research enables us to reduce the artistic
dimension of structures’ relatively abstract dimension to an
expression of the degree of embodiment that can be
created. That is, one gains an understanding of the me-
chanics of an architectural structure’s “bodily” form by first
grasping the mechanics of transmission in one’s own body
and the structure’s relative relationship to our body. The
critical part here is the ability to map and motivate the
body structure to the architectural structure. Thus,
depending on the architectural intent, we can employ a
variety of degrees and types of structural embodiment to
dialogue with the body.
6.1. Design of force flow and material distribution
The manipulation of graphic statics can directly influence
the metaphor and guidance for the bodily schema. The
distortion of the force flow based on basic geometrical
operations allows for abundant structural and spatial
variations. Additionally, taking embodiment as the basic
concept, bodily schema can serve as the vocabulary, while
graphic statics serves as the grammar. This function allows
us to frame, bridge, cantilever, and materialize the
structures according to the design concepts. Thus, we can
construct a twofold embodied structure: the design of
force flow and the design of material distribution.Onthe
one hand, directing the potential representations of the
structure on an abstract level is possible by directly
operating and deforming the structural topological re-
lations of the force flow. On the other hand, by designing
approximate material allocations between form and
forces to inform and guide the materialization of final
structural forms to approach embodiment principles,
constructing an art-from based on the structural core-
from is possible.
Fig. 6 Vitruvian Man cropped and erased measurements with the representation of graphic statics.
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For the design of the force flow, graphic statics facilitates
a design-oriented understanding of the inner forces within a
building structure that is an active engagement with the
pattern of distribution of forces within space (Muttoni,
2011). By collapsing or splitting the forces (Fig. 7) or trans-
forming the forces between tension and compression
(Fig. 8a1ec1), graphic statics can bring the force flow
approach or influence the compositions of the bodily equi-
librium gestures and movements, thus resonating with the
corresponding psychological balance and its emotions
(Fig. 8a2ec2). Through the series transformation of the
force diagram in Figures (a1) to (c1), the force flows of the
graphic statics demonstrate their ability to correlate with
the various bodily gestures in Figures (a2) to (c2), which can
elicit different bodily feelings and emotions. Additionally,
through the re-composition of the force flow units (a1), (b1),
and (c1), we can construct a more complex global equilib-
rium. For instance, in the case of (d), they can be combined
into Fig. 6 to create a global bodily equilibrium for the
Vitruvian Man. This possibility opens up architectural form
as a structural design topic by using the topological flexi-
bility of force flow to guide the material allocation in space
and tectonic expression.
Additionally, the freedom inherent in graphic statics
design extends to the materialization of this pattern as
structural elements. In general, graphic statics is an
approach to structural design that is material-independent
similar to Culmann’s graphic statics analysis of bone
morphology and historical discussion of bones and skin in
architecture. In each case, the shape of the structural
element is interpreted as the minimal envelope possible for
the force pattern to be carried. By incorporating the yield
stress or ultimate stress of a material as a parameter, such
an interpretation of the force-form relation can result in
the shaping of a structural element, which is a design de-
cision. The material only needs to act as a medium for the
transmission of forces through space. Consequently, no
strict correlation exists between the inner force flow and
the realized form.
Additionally, with the constructed force flow as an
inscribed distribution pattern, the materialized envelope
can be interpreted more freely and receive its shape in
relation to other design criteria (Fig. 9). Thus, the
constructed pattern of forces can be viewed as a diagram
for the form of a structural element; it is not the form of
the building structure but rather serves to inform it.
Therefore, this correlation allows the force flow and the
enveloping material to obtain a second layer of design
freedom that can be integrated with the embodiment into
the structural materialization process.
6.2. Body as the method for structural design
By using bodily schema as a vocabulary and graphic statics
as grammar, we can re-read the embodied expressions and
interconnections present in various structures and develop
them into a structural design method.
The perception of the body is implied or emphasized in
many famous architectural structures. For example, the
vertical force flow in the caryatids (Fig. 10) of the Temple
of Erechtheion (406 BC) was sculpted as multiple female
bodies rather than bare columns. Considering that its
gesture could evoke people’s bodily experience of sup-
porting a heavy load overhead with an upright body (Similar
to Fig. 8 b2), it structurally corresponds to a column’s
balanced load-bearing behavior in compression. As a result
of the analogy between supporting bodily experience and
structural geometry, the embodied perception of the
structure’s heaviness is stimulated, expressing a sense of
stability, harmony, and the social metaphor of
responsibility.
The structure of the entrance staircase in Studio di
Architettura Livio Vacchini (1985) (Fig. 11) suggests a
gesture resembling legs apart (similar to Fig. 8 a2). The
massive solid structure employs the leg gesture as its vo-
cabulary, assisting the structure in resisting the horizontal
lateral thrust of the structure as a whole while also psy-
chologically implying stability. Therefore, it alleviates the
sense of instability generated by the thin walls on the
ground floor’s two sides.
Marcel Breuer’s cantilevered roof structure at the
entrance to the Whitney Museum (1966) (Fig. 12) can be
interpreted as a vocabulary of straightened arms (similar to
Fig. 8 c2), expressing the tension created by maintaining
the arms horizontal for an extended period. The arms then
become fatigued, creating a sense of tension and empha-
sizing the entrance’s position through its unusual expres-
sion. Additionally, the complex geometry of the entrance
stimulates a variety of embodied perceptions from various
vantage points. For example, the bottom support resembles
a tiny leg supporting a massive body, reinforcing the
structure’s expression.
In Villa Ale
´m (2014) by Valerio Olgiati (Fig. 13), the wall
that encloses the courtyard of the building induces a
seemingly unbalanced fold through the deformation of the
force flow. This folding can easily provoke the past expe-
rience of bending or leaning forward of the body (Fig. 14),
which through its unstable form provokes a sense of
imbalance. The obliqueness-oriented instability and
oppression expressed in this structure responded to the
site’s procession-like quality that Olgiati has consistently
been instilled in the project (Woodman, 2015), and
the leaning bodily vocabulary becomes the medium to
express it.
Fig. 7 By collapsing and splitting forces, the addition of an
equilibrium point to a linear force flow creates a new force
flow pattern.
S. Wang, T. Kotnik, J. Schwartz et al.
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Similarly, the way graphic statics constructs the rela-
tionship between structure and body not only can stimulate
embodiment as in the previous cases but can also obscure
the structure’s stimulation of embodied perception in
accordance with the design intent.
In the Forsterstrasse apartment (Fig. 15) designed by
Christian Kerez and Joseph Schwartz, the building achieved
a discontinuity between the upper and lower floors of the
structure via force flow distortion, thus breaking up the
repetition and regularity in space and form (Kerez, 2009).
The interruption in the flow of forces aroused by the
misalignment of the wall, both on the fac¸ade and inside,
implies the metaphor of an incomplete or dismembered
bodydin contrast to the balanced bodily experience be-
tween the limbsdevoking a sense of abnormality and cu-
riosity and stimulating a rewarding mechanism of
exploration and movement (Mallgrave, 2013). This skepti-
cism stems from the perception-absence of holistic equi-
librium relations: individuals are unaware that the slab is
also suspended from a wall above (Fig. 16). Furthermore,
the Forsterstrasse apartment is also characterized by the
materialized form of the walls. Rather than directly
responding to the flow of forces similar to a truss (Fig. 16),
the structure is enclosed by similar-looking walls (similar to
Fig. 8 Through the series transformation of the force diagram in Figures (a1) to (c1), the force flows of the graphic statics
demonstrate their ability to correlate with the various bodily gestures in Figures (a2) to (c2). Additionally, through the re-
composition of the force flow units (a1), (b1), and (c1), we can construct a more complex global equilibrium.
Fig. 9 Different materialized geometry based on the same force flow represents very different design concepts and the degree of
embodiment they can offer.
Fig. 10 Temple of Erechtheion, Athens. Photo: Sharon Mol-
lerus. Source: flickr Creative Commons.
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Fig. 9 d). This materialization reduces the embodiment of
individual structural elements but reinforces the relation-
ship between walls, allowing for various possible in-
teractions with the structures as one moves through them.
Similar to Forsterstrasse, Junya Ishigami’s Kanagawa
Institute of Technology (KAIT) Workshop (2008) (Fig. 17)
blurs the embodied understanding of structure by splitting
the force flow (Fig. 18). Not only do the ultra-thin columns
themselves reduce the sense of embodiment, but Ishigami
has materialized all the vertical elements, whether in
compression or tension, as an identical white rectangular-
shaped component. The presence of the columns in the
building is thus extremely diminished (in contrast to the
caryatids in the Temple of Erechtheion), emphasizing their
relationship. The random-like structural expression and
differentiated column spacing can imply various body
movements. People can elicit different bodily emotions and
meanings by making subconscious embodied judgments
about the local arrangement of the columns based on tree
arrangement, evoking the memory of people walking freely
Fig. 11 Entrance of Studio di architettura Livio Vacchini.
Source: Public domain.
Fig. 12 Entrance of Whitney Museum. Source: Petr Kra-
tochvı
´l/Fulbright-Masaryk Grant, online at: https://
www.archiweb.cz/en/b/whitney-museum-of-american-art
Fig. 13 Villa Ale
´m. The view from the Garden. Source:
Archive Olgiati (Villa Alem / Valerio Olgiati).
Fig. 14 (a) The tilted human body; (b) The expressed equi-
librium regarding its geometry; (c) The actual equilibrium of
the structure.
Fig. 15 Fac¸ade of Forsterstrasse apartment. Source: Hisao
Suzuki, online from https://www.subtilitas.site/post/
156949653449/christian-kerez-apartment-building-on.
S. Wang, T. Kotnik, J. Schwartz et al.
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and comfortably through a tree-filled forest in the sunlight
as Junya Ishigami imagined (Ishigami, 2008).
Among these cases, although the architects may be
unaware, we can conclude that they are all attempting to
actively introduce bodily schema and its corresponding
emotions into structural design by distorting the common
structural force flow or designing the material distribu-
tion, reinforcing or blurring the stimulation of the
embodiment of the structure according to the architec-
tural intention.
Ifthedesignofforceflowreectstheimpactof
embodied perception on the structural relational system,
the design of material distribution is more concerned with
the physical representation of the structure. Using
graphic statics as a guide, we can strike a balance be-
tween the physical and psychological aspects of the
structure: between core-form, which requires static effi-
ciency and minimal material distribution, and the art-
form, which emphasizes perceptual aspects of space.
However, the design of force flow and its materialization
are not mutually exclusive; they always appear concur-
rently or in a circular fashion during the structural design
process to address distinct spatial and architectural
requirements.
With the advancement of digital technologies, the
design of graphic statics has been expanded further: by
defining a basic force-flow topological model, an infinite
number of possible variations can be generated to interact
with the design concept
10
. While the interaction of humans
with their physical environment is abstract, force flow can
be used to direct body movements while maintaining
structural stability. Graphic statics can further integrate
human perception principles into structural design pro-
cesses by simulating human perception and behavior pat-
terns using technologies such as virtual reality or agent-
based simulation.
11
Consequently, the relationship be-
tween human perception and structural design is continu-
ously optimized.
Fig. 16 Graphic statics depicts the simplified equilibrium
condition and corresponding force flow in the Forterstrasse
apartment. The walls are both supported and suspended from
the slabs and walls below. By redirecting the forces within the
slabs, the walls can move freely in the horizontal plane while
maintaining equilibrium.
Fig. 17 Kanagawa Institute of Technology (KAIT) Workshop,
Japan, Junya Ishigami. Source: Junya Ishigami þassociates.
10
See for example the recent research 3D Graphic Statics and its
application on Machine learning: Saldana Ochoa K, Ohlbrock PO,
D’Acunto P, Moosavi V. Beyond typologies, beyond optimization:
Exploring novel structural forms at the interface of human and
machine intelligence. International Journal of Architectural
Computing. July 2020.
11
See for example the review of scientific methods for measuring
and experiencing architectural spaces: S
ule Tas
lı Pektas
. A scien-
tometric analysis and review of spatial cognition studies within the
framework of neuroscience and architecture, Architectural Science
Review, 2021, 64:4, 374e382.
Frontiers of Architectural Research xxx (xxxx) xxx
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13
7. Conclusion: the embodied structure
Architecture, as a discipline, needs to be more open to
the more comprehensive challenges posed by contempo-
rary society in all of its aspects. The architect and
structural engineer must also act as a composer that or-
chestrates building structures into the synchronization of
technology, function, emotion, and aesthetics through the
senses.
If Viollet-le-Duc’s anatomical perspective on building
structures inspired by biology allowed for a significant
shift in the physical equilibrium of designing architectural
structures, neuroscience discoveries would allow for a
“paradigmatic shift” (Eberhard, 2009)or“sensorialrevo-
lution” (Jeli
c, 2015) in the perceptual equilibrium through
the artistic dimension of structures. Taking body and
embodied principles as the method, the application of
Cognitive Neuroscience findings to architecture and
structural design via graphic statics could result in a new
and clearer scientific perspective on structural operation,
thus achieving an embodied structural design thinking
that connects structure, space, and the body. This idea is
very similar to Julius Wolff’s description in 1870 of Cul-
mann’s ability to see Meyer’s specimens as “an extraor-
dinary piece of luck for science” (Wolff, 1978, p. 111). A
series of neuroscientific discoveries, most notably the
study of proprioception, have enabled us to reconceptu-
alize and rethink the relationship between bones and skin
or the ontological and representational dimensions of
structural design practice. Unlike the majority of current
structural design methods, which focus exclusively on
force and form, the embodied understanding of structural
design encompasses both the initial and subsequent pha-
ses of structural design, transcendences structural art
from an engineering perspective based on the dry “tech-
nical truth” of Efficiency,Economy,andElegance (P.
Billington, 1983), into an “artistic truth” that can be
incorporated further into human perceptions and behav-
iors by determining the degree to which the structure’s
design and its materialization are relevant to
embodiment.
In practice, embodied structural thinking enables the
architect and structural engineer to communicate more
efficiently and seamlessly by intuitively understanding
one another’s intentions. In terms of education,
embodied structure research may also contribute to
future improvements in structural design education.
Incorporating body-related diagrams enables students to
grasp complex structural principles more quickly and
clearly by evoking motor sensory memory through bodily
experiences. In terms of health, investigating embodied
structures enables us to understand better how psycho-
logical needs are expressed structurally. This process may
benefit children and individuals with disabilities by
assisting them in maintaining their physical and mental
health. In the digital realm, investigating the embodi-
ment of structures can aid in the development of the
increasingly popular virtual reality interface and the
comprehension of spatial perception, specifically the
simulation of gravity and equilibrium associated with
structures. Additionally, the embodied structure can
potentially reintroduce a new definition of
Fig. 18 In KAIT, the simplified equilibrium condition is represented via graphic statics. (a): walls take up the equilibrium con-
dition under horizontal force, and the composition of walls defines the human circulation; (b): when the walls in (a) are subdivided
into a series of dense columns, the equilibrium condition in KAIT is approached. When a horizontal force is applied, the columns’
pre-stressing and density may resolve the issue similarly to how walls do. In this case, the arrangement of the columns could subtly
and softly imply the availability of human circulation and possible functions.
S. Wang, T. Kotnik, J. Schwartz et al.
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14
“sustainability” into structural design: the capacity to
provide a range of distinct and varied experiences rather
than fixed functions or spaces. Finally, it may provide a
new perspective on the reuse of space.
12
Embodied structural thinking can enable a building
structure to surpass its load-bearing capabilities and
transform the structural design from a collection of
incomprehensible numbers into a medium that connects
the materiality of architectural representations with the
abstraction of culture and aesthetics, thereby bringing
structures to life.
Declaration of competing interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgement
This study was funded by the China Scholarship Council
(Grant No. 202008170012).
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Chapter
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