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Computing Rebar Layouts Aligned
with the Principal Stress Directions
1 Render of a rebar layout aligned
with the principal stress direc-
tions on a folded shell.
Rafael Pastrana
Princeton University
Zhao Ma
ETH Zurich
A Distance-Constrained Tracing Approach
1
ABSTRACT
Reducing the consumption of carbon-intensive materials such as rebar steel is crucial
to mitigate the environmental impact associated with architectural surfaces built with
reinforced concrete. While digital fabrication and modern structural analysis tools offer
opportunities to decrease rebar consumption, new computational approaches to create
material-minimizing rebar layouts are required to effectively harness such potential.
This paper presents a computational method to generate rebar layouts aligned with
the principal stress directions on architectural surfaces. This method combines a rein-
forced concrete design module based on current structural engineering codes, and a
distance-constrained algorithm with adaptive seeding that iteratively traces evenly spaced
rebars that follow a structurally optimal force ow. After its application to a at slab and a
folded shell, we demonstrate that the principal stress-aligned rebar layouts require up to
32% less steel than a single orthogonal rebar grid to resist an applied load.
Our work highlights the potential of integrating design computation and structural
engineering to advance research in the eld of digital reinforcement, and to foster environ-
mentally-aware design practices.
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INTRODUCTION
Reinforced Concrete
Ranging from at slabs to free-form shells, the breadth of
architectural surfaces built with reinforced concrete (RC)
highlights the relevance of RC as one of the most prom-
inent materials in the building industry (Figures 2.1 and
2.2). The inherent strength of RC, achieved by combining
the compressive properties of concrete with the tensile
capabilities of reinforcement steel, enables the design of
durable structures capable of spanning long distances and
supporting signicant external loads (Addis 2015).
However, RC is also one of the most carbon-intensive
materials in the construction sector, and its use aggravates
current environmental concerns. The production of cement,
a key component of concrete, is responsible for approxi-
mately 7% of the global greenhouse gas emissions (Andrew
2019). The extraction, processing, and transportation of
other raw materials involved in the production of concrete
inate its carbon footprint.
Fostered by advancements in computation, structural
optimization, and digital fabrication, the rising eld of digital
concrete opens up possibilities to design material-ef-
cient structures by depositing concrete only where it is
structurally necessary (Wangler et al. 2019). By employing
technologies such as additive manufacturing, innovative
designs have been materialized, including 3D-printed
columns (Lloret et al. 2015; Anton et al. 2020), trusses
(Lowke et al. 2021), slabs (Jipa et al. 2019; Oval et al. 2023),
and bridges (Bhooshan et al. 2022), showcasing the poten-
tial of digital concrete to foster sustainability.
Digital Reinforcement
Although minimizing concrete consumption is crucial,
diminishing the amount of rebar steel in RC structures is
also necessary. The production of steel for construction
contributes to nearly half of the greenhouse gas emis-
sions of RC (Wang et al. 2007). In architectural surfaces
built with RC, like at slabs, rebar accounts for up to 60%
of the embodied carbon (Miller et al. 2015). Therefore, the
growing progress of digital concrete should be comple-
mented with advancements in digital reinforcement
research (Asprone et al. 2018) to holistically curb the nega-
tive impact that RC surfaces have on the environment.
Promising efforts are underway in this direction. These
include the application of winded carbon ber reinforce-
ment in slabs (Oval et al. 2020) and rebar cages that
dispense formwork for doubly curved surfaces (Hack
and Lauer 2014; Mirjan et al. 2022). We note that the
latter example still adheres to arranging rebar in a single
2.2
3.2
3.1
2.1
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orthogonal grid, akin to conventional on-site practices,
instead of fully leveraging the capabilities of robotic arms
to deposit materials along arbitrary directions with high
precision (Figure 3.1). In contrast, recent studies (Ma et al.
2020; Gantner et al. 2022) have demonstrated the mate-
rialization of reinforcement layouts that follow the ow of
forces through digital fabrication techniques (Figure 3.2).
Principal Stress Directions
The principal stress directions (PSDs) encode an optimal
ow of forces in a structure. These directions represent
the local orthonormal basis of a state of plane stress where
axial stresses are maximized and shear stresses vanish
(Hibbeler 2017). Notable architectural works by Hecker and
Nervi (Hecker and Hecker 2009; Nervi et al. 2010) exem-
plify the benets of aligning load-bearing material with the
PSDs at the architectural scale, as the alignment reduces
the amount of material required to resist loads (Pellis and
Pottmann 2018; Ma et al. 2020).
The integration of PSDs creates a network of curves,
known as the principal stress lines, which can be used
as the geometry of a material-minimizing rebar layout.
Such integration can be done analytically (Michell 1904)
or numerically with modern structural analysis tools
(Michalatos and Kaijima 2014; Preisinger and Heimrath
2014). However, the majority of such analysis tools are
designed for visualizing stress ow, and not for generating
rebar layouts that comply with distance or fabrication
constraints, due to the presence of undesired geometric
discontinuities, bundles, and intersections. Tam and
Mueller (2015) proposed a rule-based algorithm to address
these geometric issues, but the efcacy of their approach,
like other current methods, depends on the manual selec-
tion of appropriate seed points (Figures 4.1 and 4.2).
Contributions
Digital fabrication technologies and structural analysis
tools enable the construction and mechanical assessment
of bespoke rebar congurations. However, it is necessary
to develop computational methods that support the design
of material-minimizing rebar layouts to effectively diminish
rebar consumption in the eld and the associated emis-
sions produced by RC surfaces.
In this paper, we introduce a computational method that
automates the generation of rebar layouts aligned with the
PSDs (Figure 1). This method supports the exploration of
the geometry of such layouts during the conceptual design
stage for different loading and support conditions, as well
as for distinct rebar diameters and tracing parameters.
Our method combines an RC design module based on
current structural engineering codes with a distance-con-
strained tracing algorithm to generate material-minimizing
layouts. This algorithm iteratively traces rebar following
the optimal ow of forces on a manifold mesh, while it auto-
matically adjusts the placement of seed points to achieve an
evenly spaced distribution of rebars graded according to
the tensile stresses acting on an RC surface.
DESIGN METHOD
Figure 5 provides an overview of the three parts that
compose our method. Besides the numerical parameters,
the input is a mesh representing the surface to be designed,
and the output is the geometry of a rebar layout, generated
as one set of polylines per PSD. We implement this method
using COMPAS, a framework for computational research
in architecture and structures (Van Mele et al. 2017), and
Karamba3D (Preisinger and Heimrath 2014).
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
4.2
3 Robotically-fabricated reinforcement on RC surfaces:
3.1 Mesh mould prefabrication (Mirjan et al. 2022)
3.2 Reinforcement bers deposition (Gantner et al. 2022)
2 Architectural surfaces built with RC:
2.1 A at slab in a multistory building
2.2 The curved roof over the Stuttgart 21 rail station
4 The placement of seed points affects the quality of the PSDs integration:
4.1 Manual seeding
4.2 Adaptive seeding
4.1
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Structural Analysis
We conduct a linear elastic structural analysis to obtain
the internal forces in an RC surface, given specic load and
support conditions.
The analysis takes as input an orientable, manifold mesh
with sufcient density to accurately capture the mechan-
ical behavior of the structure. We then apply the material
properties, cross-sections, support conditions, and loads
to the mesh. We also dene two orthogonal vectors u and
v per mesh face that set the reference axes along which
the analysis results are expressed. In our analyses, all
the structural information is output at the centroids of the
mesh faces.
Figure 6 illustrates the conguration of the calculated
internal forces, which includes the magnitude of the
three axial forces ( nuu , nvv , nuv ) and the three bending
moments ( muu , mvv , muv ). We also estimate the pair of
PSDs at three locations along the thickness of the struc-
ture: the top ( σ1
t , σ2
t ), the middle ( σ1
c , σ2
c ), and the
bottom ( σ1
b , σ2
b ) layers. We select the PSDs with the
highest tensile force among the three layers at each mesh
face and use these to trace a rebar layout. We denote these
direction vectors as σ1 and σ2 in the following. The locations
where we extract the internal forces and the PSDs are in
line with the sandwich model we use for RC design.
Reinforced Concrete Design
We implement a modied version of the three-layer sand-
wich model (Lourenço and Figueiras 1995; Blaauwendraad
2010) to estimate the area of rebar steel and the spacing
between rebars that the RC surface requires to withstand
tensile stresses.
Mesh
Load and support
conditions
Rebar diameters
k, si, dl, ul
Tracing parameters
Structural
analysis
RC
design
Layout
tracing
Rebar
layout
Principal stress directions, σ1, σ2
Rebar spacings
Internal forces
dsu, dsv
nu, mv
Inputs
5
The mathematical formulation of the sandwich model
is compact and accounts for the combined action of
membrane forces and bending moments. Moreover, it is
included in current engineering codes (ECS 2005; IFSC
2008), making it compatible with practical design appli-
cations. In the sandwich model, every mesh face on the
surface is split into a top, a middle core, and a bottom layer.
The top and bottom layers are separated by a distance h
measured between the midpoints of each layer (Figure 7).
The sandwich model outputs the cross-section area or rein-
forcement per width unit, asu and asv , at the top and bottom
layers of the mesh faces. The model inputs are the surface
thickness t, the steel yield stress fy and the rebar diameter
ϕu and ϕv along axes u and v, respectively. We assume
adequate bonding between concrete and rebar, and that
the thickness of the RC surface is sufcient to resist shear
forces without rebar stirrups.
Conventional applications of the sandwich model result in
two rebar layouts, one hosted in the top layer and another
in the bottom layer. We generate a single rebar layout
instead because the out-of-plane bending moments muu,
mvv and muv exert tensile stresses only on one side of the
surface, either at the top or at the bottom. As we target the
design of material-efcient structures, it would be redun-
dant to add two layers of rebar if only one sufces. Hence,
we shift the geometry of the layout to the top or to the
bottom layer based on the acting tensile stresses (Figure 8).
The tension forces for rebar design in the top and bottom
layers nt
su , nt
sv , nb
su and nb
sv only include the contribution
of the bending moments (Equations 1 and 2). The effect of
the in-plane axial forces are accounted for in Equation 3:
5 Overview of the method to compute rebar layouts aligned with the PSDs.
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As negative values are expected from calculating the tensile
forces on the layers, denoting no need for reinforcement
(Blaauwendraad 2010), the terms nc
su , and nc
sv are added
only to the maximum values of nt
su , nt
sv , nb
su , and nb
sv. We
discard eccentricity effects. The calculated tensile forces
are then condensed into nsuu and nsvv which represent the
total tensile forces along the u and v reference directions:
In Equation 5, we compute the cross-sectional area of
rebar per unit width in the sandwich layers asu and asv
shown in Figure 9 based on the total tensile forces:
nt
su =muu
h+muv
h
nt
sv =mvv
h+muv
h
6
We then calculate the separation distances per reference
direction, dsu and dsv , with the input diameters ϕu and ϕv :
The distances obtained with Equation 6 are stored in
vectors dsu and dsv with pointers to their corresponding
location on the mesh to simplify their query during tracing.
These distances modulate the separation between the
rebars generated by the distance-constrained rebar
tracing algorithm we develop in the following subsection.
By changing the diameters ϕu and ϕv we can tune the
geometry of a rebar layout. Additionally, our method allows
the denition of an individual diameter per mesh face to
enable ner-grained control of the layout density. We illus-
trate the effect of choosing different global rebar diameters
with two examples in the structural applications section.
Rebar Layout Generation
We develop an algorithm to trace evenly-spaced rebars
along the PSDs, building on the work of Jobard and Lefer
(1997), Mebarki et al. (2005), and Tam and Mueller (2015).
This algorithm iteratively integrates polylines on the mesh
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
nvv
σ
t
muv
mvv
nuv
nuv
nuu
muv
muu
c
b
uv
σ
σ
7
T
op
C
ore
Bottom h
t
uv
nsu
t
nsv
t
nsv
c
nsv
b
nsu
b
nsu
c
6
Top
Bottom
asu
a
su
8
u
v
asu
asv
asu
asv
dsu
dsv
9
6 Internal forces in an RC surface.
7 Sandwich model for RC design.
8 Rebar area shift between sandwich layers.
9 Rebar area distribution per sandwich layer.
nb
su =−
muu
h
−
muv
hnb
sv =−
mvv
h
−
muv
h
(2)
n
c
su =nuu +nuv
2
nc
sv =nvv +nuv
2
nc
su =nuu +nuv
2n
c
sv =nvv +nuv
2
a
su =nsuu
fy
asv =nsvv
fy
dsu =πφ
2
u
4asu
dsv =πφ
2
v
4asv
n
suu = max(nt
su,n
b
su)+nc
su
nsvv = max(nt
sv,n
b
sv)+nc
sv
nsuu = max(nt
su,n
b
su)+nc
su
nsvv = max(nt
sv,n
b
sv)+nc
sv
(4)
nt
su =muu
h+muv
h
nt
sv =mvv
h+muv
h
(1)
(3)
(5)
(6)
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d
s
u
l
representation of the input surface via bi-directional
projective tracing. One polyline corresponds to the center-
line of one rebar. To avoid rebar cluttering, and to minimize
rebar consumption by placing material only where needed,
the algorithm maintains the distance between rebars
computed with Equation 6, while it traces rebar polylines.
The algorithm traces one polyline at a time until a priority
queue is empty, or until the generation of more rebars
is unfeasible due to distance, seeding or geometric
constraints. We run the algorithm twice to generate a rebar
layout, once per set of PSDs, σ1 and σ2. The four operations
involved in the generation of a rebar layout are:
• Automating the selection of good initial seeds.
• Constructing a priority queue to ensure adequate densi-
cation of the rebar layout.
• Calculating the rebar tracing direction.
• Controlling the separation between rebars.
Once the rebar layout tracing is complete, we lter and
translate the control points of each polyline by a distance
h/2 or -h/2 along the unit normals of the mesh, so that the
geometry of the rebars lies in the middle of the layer where
the principal tensile stress occurs. We rebuild the polylines
after shifting their control points.
Seeding Automation: Every rebar polyline starts from a
seed point. The location of a seed point can be arbitrary, but
the choice may lead to inadequate layouts. To procure an
evenly distributed layout, we derive all the possible seed
p
1
p
3
p
2
p
3
p
2
p
1
d
l
σ
i
σ
ii
σ
iii
points at a given distance from an existing polyline before
moving on to another (Jobard and Lefer 1997). Figure 10.1
depicts this seeding rule. Every polyline is sampled equi-
distantly by length ul and each of the resultant points is
offset on the mesh by a rebar separation distance dsu or dsv
(hereafter ds for notation simplicity) to the right and to the
left of the polyline.
Priority Queue: It is desirable to enhance the structural
integrity of a rebar layout by populating rst the mesh
regions requiring more reinforcement. We utilize a priority
queue to this end. Every time a new seed point is created,
we insert the point in the queue with a priority equal to the
area of rebar asu or asv needed at its current location on
the mesh. The queue then schedules the release of new
seed points, releasing points rst in areas of the structure
with the highest rebar demand. The priority queue stores
and releases seed points until it is empty. Then, the tracing
of rebar polylines stops.
Polyline Tracing: A polyline representing one rebar is
traced by placing points on the mesh over a nite number
of steps. At every step, the polyline grows by translating
the seed point on the mesh by an extension distance dl , in
the direction of the average of the k PSDs closest to the
point using rst-order Euler integration (Figure 10.2). Small
values of dl are ideal to generate rebars that closely follow
the PSDs (Halpern et al. 2013). After translation, the point
is projected back to the mesh. The rebar tracing continues
iteratively until the polyline hits the boundaries of the mesh,
or until any of the distance control thresholds are met.
Distance Control: To ensure an evenly distributed layout, we
aim to have no pair of rebars closer to each other than any
of the proximity thresholds p1, p2 and p3 . These thresholds
are computed based on the rebar separation distances ds :
p1=dss1p2=dss2p3=dss
3
(7)
10 Rebar layout generation steps:
10.1 Seed sampling on rebar polyline
10.2 Direction interpolation, k = 3
10.3 Distance control with proximity thresholds
10.1 10.2 10.3
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Proximity factor s1 denes the allowable distance of one
rebar to the others at the rst tracing iteration. Values that
range between 0.8 and 1 are adequate. Factor s2 controls
the separation among rebars at any other stage of the
tracing algorithm. Values under 0.5 lead to predominantly
long rebars (Jobard and Lefer 1997). In zones where
circumferential stresses exist, rebars tend to indenitely
turn around themselves. Therefore, the proximity factor s3
is introduced to check the proximity of a polyline to itself.
Values between 1 and 3 are suitable.
At every iteration, we calculate these proximity thresholds
based on the distance between the moving seed point of
a traced rebar and the control points of all traced rebar
polylines and itself (Figure 10.3). If any of the thresholds
is exceeded, then the tracing of a polyline stops and the
tracing process moves on to the next seed in the queue.
STRUCTURAL APPLICATIONS
We showcase the effectiveness of our method by gener-
ating rebar layouts for two RC surfaces: a rectangular slab
plate and a folded cantilevering shell.
The rst structure reproduces a design from Lourenço
and Figueiras (1993). The slab is xed along its perimeter
with pinned supports, and subjected to an out-of-plane,
uniformly distributed load of 15 kN/m2. The slab has a
constant thickness of t = 15 cm, and is shown in Figure 11.1.
The folded shell in Figure 11.2 is a 3D structure subjected
to an uplifting uniform load of 2 kN/m2. The shell is
restrained with pinned supports at the base, and has a
constant thickness of t = 17 cm. For both structures, the
yield strength of reinforcement steel is fy = 500 MPa. We set
the distance between top and bottom layers to h = 0.9t.
In these experiments, we generate a rebar layout aligned
with the PSDs for each structure, and benchmark it in
terms of the rebar demand against a single orthogonal grid,
a typical layout in construction practice. We set the refer-
ence axes u and v as the Cartesian vectors x and y for the
baseline orthogonal layouts, respectively; and as σ1, σ2 for
the layouts that follow the PSDs.
Figures 12 and 13 show the rebar distributions in the slab
calculated via Equation 5. The slab requires most rebar
at the bottom layer as it experiences positive bending
moments that exert tensile stresses on the soft. There,
rebar smears from 0 to 5.1 cm2/m on directions x and
σ1; and from 0 to 4.1 cm2/m along y and σ2 . In the bottom
layer, rebar has a similar cross-like spread along x and σ1
(Figures 12.1 and 13.1), but it differs between y and σ2: in
the latter case, rebar radiates elliptically from the midspan
and vanishes towards the four corners of the slab (Figures
12.3 and 13.3). In the top layer, we highlight that the align-
ment with the optimal force ow suppresses the need for
rebar along σ1 (Figure 13.2). In contrast, the orthogonal
layout requires up to 3.6 cm2/m of rebar along x in the top
layer (Figure 12.2).
The distribution of rebar along y and σ2 is comparable
for the cantilevering shell, requiring a maximum of 16.7
cm2/m at the base of the structure for both alignment
types (Figures 14.2 and 14.4). Along x and σ1, however, the
spread of rebar demand varies from one alignment to the
other, particularly where the surface folds and transitions
from the vertical to a horizontal cantilever (Figures 14.1
6 m
15 kN/m2
5 m
y
x
y
z
x
2 kN/m2
6.5 m
4.5 m
1.2 m
0.4 m
11.211.1
11 RC surfaces: geometry and applied loads:
11.1 Flat slab
11.2 Folded shell
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
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HABITS OF THE ANTHROPOCENE
a
su 3.6
0
5.1
0a
su
and 14.3). The maximum rebar demand along x and σ1 is
4.1 cm2/m, but the area with high reinforcement demand
occupies a larger portion of the structure in the baseline
alignment, thus indicating the need for more rebar along x
than along σ1 .
Figure 15 summarizes the total area of rebar steel calcu-
lated with Equation 5 required to carry tensile stresses
for both RC surfaces, detailing the contribution that every
layer has to the total steel demand. The reported areas are
normalized, with respect to the orthogonal layout areas,
to show changes in rebar demand relative to the baseline
conguration. Aligning rebars with the PSDs reduces the
rebar steel required to resist tensile stresses. The layouts
that follow the PSDs require only 68% and 90% of the total
area of steel per unit width as
total needed by the reference
layout: 32% and 10% less than the baseline, respectively.
Aligning rebars with the PSDs saves more steel in one
direction than in the other. In the slab, 24% of the steel
savings relative to the baseline stem from aligning the
rebar at the bottom layer with σ2. In contrast, there is only
a 1% difference between x and σ1 in the same layer. In the
folded shell, there is an 8% difference in rebar area demand
between the σ1 and x alignments in favor of the former.
Moreover, all the rebars are hosted in the bottom layer of
the shell (i.e., rebar is not needed in the top layer) due to the
bending moment caused by the applied uplifting load.
Besides the difference in the magnitude of the applied loads,
a plausible reason why rebar reductions are more prom-
inent in the slab than in the shell is that the former is a at
structure where out-of-plane moments are high and must
a
sv 4.1
0
a
sv
3.7
0
12.1 12.4
12.2 12.3
13.1 13.4
13.2 13.3
12 Rebar distribution per slab layer with the baseline alignment (x, y):
12.1 Direction x, bottom layer
12.2 Direction x, top layer
12.3 Direction y, bottom layer
12.4 Direction y, top layer
13 Rebar distribution per slab layer with the PSDs alignment (σ1, σ2):
13.1 Direction σ1, bottom layer
13.2 Direction σ1, top layer
13.3 Direction σ2, bottom layer
13.4 Direction σ2, top layer
ACADIA 2023
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rebar values ϕu and ϕv to the directions corresponding
to the reference axes u and v, respectively. We run the
tracing algorithm with dl = 2 cm; ul = 10 cm; k = 3; s1 =
0.9; s2 = 0.5; and s3 = 2. For the shell, we assign rebars of
constant diameter ϕu = ϕv = 0.8 cm on both directions and
keep the same tracing parameters as for the slab, except
for s2 , which we lower from 0.5 to 0.1.
In both structures and both alignment types, the layouts
computed with our method adaptively spaced out, guided
by the distances we calculate with Equations 5 and 6.
Our method traces and densies rebar to maintain the
features of the required steel distribution to bear tensile
stress, while depositing rebar only where it is structurally
required (Figures 16, 17, and 18). Using larger diameters
results in fewer rebars per layout, as the spacing between
4.1
0a
su
y
x
16.7
0
a
sv 4.1
0
a
su
16.7
0a
sv
14.1 14.2 14.3 14.4
14 Rebar distribution in the folded shell:
14.1 Direction x
14.2 Direction y
14.3 Direction σ1
14.4 Direction σ2
be resisted predominantly by rebar steel; whereas, in the
latter, the curved folded shape activates the in-plane axial
strength of the shell, thus decreasing the reliance on rebar.
This suggests that a PSDs-aligned layout can yield higher
material savings on at surfaces than on folded structures
whose main load-bearing mechanism is membrane action.
Next, we generate rebar layouts for the baseline single-
grid case and the PSDs alignment case on both structures.
These layouts correspond to the rebar area distributions
presented in Figures 12, 13, and 14. In the baseline case,
we only present the layout in the bottom layer of the slab.
For the slab layouts, we select two different groups of
diameters to study the effect that this parameter has on
the spacing between rebars: a) 1 cm and 0.8 cm, and b) 0.8
cm and 0.6 cm. Each group of diameters assigns constant
Structure Rebar orientation ab
su at
su ab
sv at
sv a
total
s
Flat slab Baseline (x,y)0.46 0.07 0.40 0.07 1.00
Stress-aligned (σ1,σ2)0.45 0.00 0.16 0.07 0.68
Folded shell Baseline (x,y)0.31 0.00 0.69 0.00 1.00
Stress-aligned (σ1,σ2)0.23 0.00 0.67 0.00 0.90
15
15 Normalized reinforcement areas for
the baseline (x, y) and the PSDs (σ1, σ2).
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
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HABITS OF THE ANTHROPOCENE
expands quadratically if the input rebar diameter increases
(Equation 6). Conversely, using smaller diameters results in
layouts with more rebars in order to provide the required
tensile capacity to the RC structures.
CONCLUSION
This paper presented a method to design rebar layouts
aligned with the PSDs on RC architectural surfaces
subject to a load case. By integrating a sandwich model
for RC design, a priority queue-based seed generator and
a distance-constrained tracing algorithm, our approach
produced rebar layouts that effectively grade reinforce-
ment steel based on the tensile stresses in an RC surface.
This method thus overcomes some of the limitations of
other current approaches and can support the design
exploration of layouts in the conceptual design stage.
Through two illustrative examples, we demonstrated the
efcacy of aligning rebars with the PSDs. Our method
resulted in a reduction of rebar area of 10% and 32%,
compared to the single orthogonal grid layout employed in
traditional construction. Most of the savings in rebar steel
stemmed from aligning structural material with one of the
two PSDs. This reduction in rebar consumption not only
enhances material efciency but also has the potential to
decrease the embodied carbon footprint of RC structures.
The proposed distance-constrained rebar tracing approach
is a greedy algorithm that operates based on local
heuristic rules. Consequently, one limitation of our method
is that it does not guarantee a globally optimal solution.
Nevertheless, our aim is not to nd a global solution,
which might be computationally expensive or intractable
for general structures. Instead, the goal of our method
is to facilitate the exploration of different locally optimal
rebar layouts. The convergence rate to a local optimum
is, however, contingent on the selection of an appropriate
input mesh resolution and parameter values, as well as on
the presence of singularities among the PSDs.
y
x
σ2
σ1
y
xσ1
σ2
16.1 16.2
17.1 17.2
18.1 18.2
16 Rebar layouts for the slab with the baseline alignment (x, y):
16.1 Diameters ϕu= 1 cm, ϕv= 0.8 cm
16.2 Diameters ϕu= 0.8 cm, ϕv= 0.6 cm
17 Rebar layouts for the slab with the PSDs alignment (σ1, σ2):
17.1 Diameters ϕu= 1 cm, ϕv= 0.8 cm
17.2 Diameters ϕu= 0.8 cm, ϕv= 0.6 cm
18 Rebar layouts for the folded shell, ϕu= ϕv= 0.8 cm:
18.1 Baseline alignment (x, y)
18.2 PSDs alignment (σ1, σ2)
ACADIA 2023
12
Looking ahead, we plan to incorporate higher-order
integration methods to make the convergence rate of our
tracing algorithm more robust to variations in the inputs.
Another crucial aspect to consider is the inclusion of fabri-
cation constraints during the tracing of the rebar polylines
to generate layouts that are not only structurally informed
and evenly distributed, but also increasingly suitable for
construction using both analog and digital fabrication tools.
While our focus has been on placing rebar steel along
an ideal ow of forces, our framework can be extended
to trace efcient reinforcement distributions for other
lament types, such as carbon bers or natural bers, on
different architectural surfaces.
In conclusion, our research advances the development
of new digital methods to design rebar layouts, opening
up opportunities in digital reinforcement research for
improved structural efciency, reduced material consump-
tion, and enhanced sustainability.
ACKNOWLEDGEMENTS
The basis of this work was developed as part of the master's
thesis of Rafael Pastrana at ETH Zurich, which was supported by a
CONACYT-Alianza FIIDEM scholarship. The authors thank Andrew
Liew, Clemens Preisinger, Matthias Rippmann, Alex N. Walzer, and
Mark Tam for their insightful conversations.
REFERENCES
Addis, William. 2015. Building: 3000 Years of Design Engineering
and Construction. London: Phaidon Press, Inc.
Andrew, Robbie M. 2019. “Global CO2 Emissions from Cement
Production, 1928–2018.” Earth System Science Data 11 (4):
1675–1710. https://doi.org/10.5194/essd-11-1675-2019.
Anton, Ana, Patrick Bedarf, Angela Yoo, Benjamin Dillenburger,
Lex Reiter, Timothy Wangler, and Robert J. Flatt. 2020. “Concrete
Choreography: Prefabrication of 3D-Printed Columns.” https://doi.
org/10.3929/ETHZ-B-000408884.
Asprone, Domenico, Costantino Menna, Freek P. Bos, Theo A.M.
Salet, Jaime Mata-Falcón, and Walter Kaufmann. 2018. “Rethinking
Reinforcement for Digital Fabrication with Concrete.” Cement
and Concrete Research 112: 111–21. https://doi.org/10.1016/j.
cemconres.2018.05.020.
Bhooshan, Shajay, Vishu Bhooshan, Alessandro Dell’Endice, Jianfei
Chu, Philip Singer, Johannes Megens, Tom Van Mele, and Philippe
Block. 2022. “The Striatus Bridge: Computational Design and
Robotic Fabrication of an Unreinforced, 3D-Concrete-Printed,
Masonry Arch Bridge.” Architecture, Structures and Construction
2 (4): 521–43. https://doi.org/10.1007/s44150-022-00051-y.
Blaauwendraad, Johan. 2010. Plates and FEM:
Surprises and Pitfalls. Vol. 171. Solid Mechanics and Its
Applications. Dordrecht: Springer Netherlands. https://doi.
org/10.1007/978-90-481-3596-7.
European Committee for Standardization (ECS). 2005. “Eurocode 2.
Design of Concrete Structures - Part 2: Concrete Bridges- Design
and Detailing Rules.” European Standard. Brussels: BSI British
Standards. https://doi.org/10.3403/30096437.
Gantner, Stefan, Tom-Niklas Rothe, Christian Hühne, and Norman
Hack. 2022. “Reinforcement Strategies for Additive Manufacturing
in Construction Based on Dynamic Fibre Winding: Concepts and
Initial Case Studies.” Open Conference Proceedings 1: 45–59.
https://doi.org/10.52825/ocp.v1i.78.
Hack, Norman, and Willi Viktor Lauer. 2014. “Mesh-Mould:
Robotically Fabricated Spatial Meshes as Reinforced Concrete
Formwork.” Architectural Design 84 (3): 44–53. https://doi.
org/10.1002/ad.1753.
Halpern, Allison B., David P. Billington, and Sigrid Adriaenssens.
2013. “The Ribbed Floor Slab Systems of Pier Luigi Nervi.” Journal
of the International Association for Shell and Spatial Structures 54
(2–3): 127–36.
Hecker, Hans-Dieter, and Sigrid Hecker. 2009. Entwürfe Und
Bauten. Freiburg im Breisgau: Herder.
Hibbeler, Russell C. 2017. Mechanics of Materials. Tenth edition.
Boston: Pearson.
International Federation for Structural Concrete (IFSC). 2008.
Practitioners’ Guide to Finite Element Modelling of Reinforced
Concrete Structures: State-of-Art Report. 1. Bulletin / CEB-Fib
State-of-Art Report 45. Lausanne: International Federation for
Structural Concrete (b).
Jipa, Andrei, Cristián Calvo Barentin, Gearóid Lydon, Matthias
Rippmann, Georgia Chousou, Matteo Lomaglio, Arno Schlüter,
Philippe Block, and Benjamin Dillenburger. 2019. “3D-Printed
Formwork for Integrated Funicular Concrete Slabs.” In
Proceedings of IASS Annual Symposia, 2019:1–8. Barcelona, Spain.
https://www.ingentaconnect.com/content/iass/
piass/2019/00002019/00000006/art00006
Jobard, Bruno, and Wilfrid Lefer. 1997. “Creating Evenly-
Spaced Streamlines of Arbitrary Density.” In Visualization in
Scientic Computing ’97, edited by Wilfrid Lefer and Michel
Grave, 43–55. Eurographics. Vienna: Springer. https://doi.
org/10.1007/978-3-7091-6876-9_5.
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
TOPIC (ACADIA team will ll in) 13
HABITS OF THE ANTHROPOCENE
Lloret, Ena, Amir R. Shahab, Mettler Linus, Robert J. Flatt, Fabio
Gramazio, Matthias Kohler, and Silke Langenberg. 2015. “Complex
Concrete Structures.” Computer-Aided Design 60: 40–49. https://
doi.org/10.1016/j.cad.2014.02.011.
Lourenço, Paulo B., and Joaquim A. Figueiras. 1993. “Automatic
Design of Reinforcement in Concrete Plates and Shells.”
Engineering Computations 10 (6): 519–41. https://doi.
org/10.1108/eb023923.
Lourenço, Paulo B., and Joaquim A. Figueiras. 1995. “Solution for
the Design of Reinforced Concrete Plates and Shells.” Journal of
Structural Engineering 121 (5): 815–23. https://doi.org/10.1061/
(ASCE)0733-9445(1995)121:5(815).
Lowke, Dirk, Aileen Vandenberg, Alexandre Pierre, Amaury
Thomas, Harald Kloft, and Norman Hack. 2021. “Injection 3D
Concrete Printing in a Carrier Liquid - Underlying Physics and
Applications to Lightweight Space Frame Structures.” Cement and
Concrete Composites 124: 104169. https://doi.org/10.1016/j.
cemconcomp.2021.104169.
Ma, Zhao, Alexander Walzer, Christian Schumacher, Romana
Rust, Fabio Gramazio, Matthias Kohler, and Moritz Bächer.
2020. “Designing Robotically-Constructed Metal Frame
Structures.” Computer Graphics Forum 39 (2): 411–22. https://doi.
org/10.1111/cgf.13940.
Mebarki, Abdelkrim, Pierre Alliez, and Olivier Devillers. 2005.
“Farthest Point Seeding for Efcient Placement of Streamlines.” In
IEEE Visualization 2005 - (VIS’05), 61–61. Minneapolis, MN, USA:
IEEE. https://doi.org/10.1109/VIS.2005.39.
Michalatos, Panagiotis, and Sawako Kaijima. 2014. “Eigenshells:
Structural Patterns on Modal Forms.” In Shell Structures for
Architecture: Form Finding and Optimization, 1st ed. London:
Routledge. https://doi.org/10.4324/9781315849270.
Michell, Anthony George Maldon. 1904. “LVIII. The Limits of
Economy of Material in Frame-Structures.” The London, Edinburgh,
and Dublin Philosophical Magazine and Journal of Science 8 (47):
589–97. https://doi.org/10.1080/14786440409463229.
Miller, Dane, Jeung-Hwan Doh, and Mitchell Mulvey. 2015.
“Concrete Slab Comparison and Embodied Energy Optimisation for
Alternate Design and Construction Techniques.” Construction and
Building Materials 80 (April): 329–38. https://doi.org/10.1016/j.
conbuildmat.2015.01.071.
Mirjan, Ammar, Jaime Mata-Falcón, Carsten Rieger, Janin
Herkrath, Walter Kaufmann, Fabio Gramazio, and Matthias Kohler.
2022. “Mesh Mould Prefabrication.” In Third RILEM International
Conference on Concrete and Digital Fabrication, edited by Richard
Buswell, Ana Blanco, Sergio Cavalaro, and Peter Kinnell, 31–36.
https://doi.org/10.1007/978-3-031-06116-5_5.
Nervi, Pier Luigi, Carlo Maria Olmo, Cristiana Chiorino, Christophe
Pourtois, Marcelle Rabinowicz, and Elisabetta Margiotta Nervi.
2010. Pier Luigi Nervi: Architecture as Challenge. Cinisello
Balsamo, Milano: Silvana.
Oval, Robin, Eduardo Costa, Diana Thomas-Mcewen, Saverio
Spadea, John Orr, and Paul Shepherd. 2020. “Automated
Framework for the Optimisation of Spatial Layouts for Concrete
Structures Reinforced with Robotic Filament Winding.” In .
Kitakyushu, Japan. https://doi.org/10.22260/ISARC2020/0214.
Oval, Robin, Mishael Nuh, Eduardo Costa, Omar Abo Madyan,
John Orr, and Paul Shepherd. 2023. “A Prototype Low-Carbon
Segmented Concrete Shell Building Floor System.” Structures 49
(March): 124–38. https://doi.org/10.1016/j.istruc.2023.01.063.
Pellis, Davide, and Helmut Pottmann. 2018. “Aligning Principal
Stress and Curvature Directions.” In Advances in Architectural
Geometry, edited by Lars Hesselgren, Axel Kilian, Samar Malek,
Karl-Gunnar Olsson, Olga Sorkine-Hornung, and Chris Williams,
34–53. Klein Publishing Ltd.
Preisinger, Clemens, and Moritz Heimrath. 2014. “Karamba—A
Toolkit for Parametric Structural Design.” Structural Engineering
International 24 (2): 217–21. https://doi.org/10.2749/101686
614X13830790993483.
Tam, Kam-Ming Mark, and Caitlin T. Mueller. 2015. “Stress Line
Generation for Structurally Performative Architectural Design.”
In Proceedings of the 35th Annual Conference of the Association
for Computer Aided Design in Architecture (ACADIA), 95-109.
Cincinnati: USA. https://doi.org/10.52842/conf.acadia.2015.095
Van Mele, Tom, Gonzalo Casas, Romana Rust, and Mathias
Bernhard. 2017. “COMPAS: A Framework for Computational
Research in Architecture and Structures.” https://doi.
org/10.5281/zenodo.2594510.
Wang, Tao, Daniel B. Müller, and T. E. Graedel. 2007. “Forging the
Anthropogenic Iron Cycle.” Environmental Science & Technology 41
(14): 5120–29. https://doi.org/10.1021/es062761t.
Wangler, Timothy, Nicolas Roussel, Freek P. Bos, Theo A. M. Salet,
and Robert J. Flatt. 2019. “Digital Concrete: A Review.” Cement
and Concrete Research 123: 105780. https://doi.org/10.1016/j.
cemconres.2019.105780.
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IMAGE CREDITS
Figure 1: © Matteo Lomaglio
Figure 2.1: © Shutterstock
Figure 2.2: © Werner Sobek
Figure 3.1: © Gramazio Kohler Research
Figure 3.2: © Gantner, S., T.-N. Rothe, C. Hühne, and N. Hack
Figures 4.1 and 4.2: © Jobard, B. and Lefer, W.
All other drawings and images by the authors.
Rafael Pastrana is a Ph.D. candidate at the Princeton University
School of Architecture. He develops computational methods and
tools to solve complex structural design problems. Rafael holds a
master's degree in architecture and digital fabrication from ETH
Zurich. His professional experience, at the intersection of struc-
tural and software engineering, includes working at Bollinger +
Grohmann, the Block Research Group, and Robert McNeel and
Associates.
Zhao Ma is a multidisciplinary researcher and educator with a
background in design, engineering, and computer science. He is a
lecturer and senior researcher at ETH Zurich, where he teaches
computational methods for design education, and conducts
research on computing applications for living systems, geometry
processing, and robotic fabrication. Zhao received two master's
degrees from MIT in 2017, and a doctoral degree from ETH Zurich
in 2021, conducting cross-disciplinary research across the elds
of architecture and computer graphics. His doctoral work, Stylized
Robotic Sculpting, developed a novel design and fabrication system
for carving clay sculptures utilizing robots.
Computing Rebar Layouts Aligned with the PSDs Pastrana and Ma
