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Proceedings of the IASS Annual Symposium 2016
“Spatial Structures in the 21st Century”
26–30 September, 2016, Tokyo, Japan
K. Kawaguchi, M. Ohsaki, T. Takeuchi (eds.)
Copyright © 2016 by Eversmann, Schling, Ihde & Louter
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
Low-Cost Double Curvature: Geometrical and Structural
Potentials of Rectangular, Cold-Bent Glass Construction
Philipp EVERSMANN*, Eike SCHLING
a
, André IHDE
b
, Christian LOUTER
c
*ETH Zurich
Zwinglistr.43, 8004 Zurich
eversmann@arch.ethz.ch
a
TU München
b
TU München, Pfeifer Seil- und Hebetechnik GmbH Memmingen
c
TU Delft
Abstract
The realization of doubly curved façades often requires large investments in fabrication equipment and
produces additional waste through subtractive fabrication processes and non-reusable molds. In glass
construction, elastic bending techniques can be used for small curvatures. This paper continues
previous research of the authors on bending rectangular glass elements into irregularly curved panels.
First, we analyze the stresses occurring in cold bent glass during assembly, thus defining a particle-
spring model which is able to compute approximate stresses in real-time during the bending
procedure. In a second step, we compare the structural performance of the bent glass with that of flat
panels using FE-analysis. Finally, we illustrate the implementations on multi-panel façade layouts. We
analyze the dependencies between curvature, gap-tolerance and panelization. We present a method to
minimize gap-tolerances by optimizing the distribution of surface curvature. Our results highlight the
structural and geometrical potentials and possible applications for curved glass construction.
Keywords: glass, cold-bending, FE-analysis, particle-spring model, optimization
1. Introduction
In our previous research we have shown how rectangular glass panels can be deformed to approximate
double curvature (Eversmann et al. 2016). It would seem logical that a flat panel, after bending, would
remain a developable (singly-curved) surface. However, the process of bending glass onto curved
frames creates additional strain, which allows some deviation from single into double curvature. In the
chapter 2, we show how the bending process functions physically and how it can be simulated using
particle-spring modeling. We present a method which allows computing approximate stresses during
the simulation in real-time, enabling deformation up to a defined maximum design stress. We proceed
by calculating the maximum curvature for a range of surface typologies. In the chapter 3, we compare
the structural performance of various curvatures of the surface typologies to that of a flat glass panel.
We analyse the maximal deflection, as well as principal and shear stresses on the middle, top and
bottom surface of each panel. In chapter 4, we analyze the geometric implications for multi-panel
façade layouts. We show geometric possibilities of using a standard 1.0 x 2.0 m panel for doubly-
curved design surfaces. We present a simplified method to model large façade layouts, and analyze the
dependencies between the curvature of the target surface and the resulting gaps. An optimization
method is proposed to simultaneously maximize curvature and minimize gaps. Finally, we suggest
application possibilities for future studies.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
2
2. Bending Process and Computational Simulation
In this chapter we will explain the bending process, its computational simulation and deduce a list of
criteria, which can be used for subsequent investigations on geometry and structure.
2.1 Methods: Particle-Spring System
The particle-spring model from previous studies was rebuilt in the software Rhino/Grasshopper using
the Kangaroo 2 simulator. We implemented the model through custom Python programming in order
to calculate stresses resulting from the bending procedure in real time. We were therefore able to bend
the panels only as far as the maximum bending stress would allow. The goal of the model was not to
replace a proper FE-analysis but to allow the designer to get approximate results on the possible
curvatures and surface quality that can be achieved on a design target surface. In the particle-spring
system a glass panel is modeled as a series of nodes which are inscribed in a quadrilateral mesh. For
each node, or particle, all relationships with its neighbors are then described. For each quad, the
neighboring particles are connected by linear springs on which we assigned a constant axial stiffness
and initial length. We calculated the stiffness values S of the springs via the formula (1):
S’/²²
(1)
Additionally, we applied a bending stiffness between spring pairs via
K = EI where E is the Young’s modulus in N/mm2, and I is the second moment of area in mm4. (2)
Bending Procedure
In order to bend a flat glass panel to achieve double curvature, forces need to be applied
simultaneously on multiple points on each side of the panel. Physically this was achieved in the
previous study [1] by a laminated aluminum frame which was bent on a 5-axis, milled wooden frame
by simultaneously tightening a predefined set of screws.
Fig. 1: Physical Prototype and panel assembly from previous study
As shown in figure 2, the frame geometry and force applied by the screws are able to transmit bending
forces into the surface of the glass plate. In the simulation, bending is achieved through springs on the
panel edges which increase their pulling stiffness until the final geometry is achieved.
Fig. 2: Left: Particle-spring system. Right: Bending force vectors
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Stress Calculation
For the stress calculation we implemented a custom Python routine for a Kangaroo Goal object to
solve the normal stresses and shear stresses. The normal stresses
σx,σy
are calculated via the change
in length of the springs:
σx,σy
∗
γwhereγ
LnewlengthL′initallengthL
initallengthL
(3)
The shear stresses are calculated via the angle of distortion of the quads:
τ
Gγ
whereGistheshearmodulusin
γ
initalAnglenewAngleinradians
(4)
The principal stresses can now be calculated using Mohr’s Circle theory [2] via the formula:
σ1,σ2
2
2
2
2
(5)
For solving the stresses on the upper and lower surface of the glass plate, the particles are offset in the
normal direction of the mesh surface.
Calibration FEM
Various calibration tests were effectuated in order to achieve corresponding results between the
particle-spring model and the FE-analysis effectuated in Strand7. Figure 3 shows the corresponding
stress distributions of shear and principal stresses.
a1 b1
a2 b2
a3 b3
Fig. 3: Calibration of particle-spring model a) and FE-Analysis b), showing the shear stresses in the upper glass layer in the
1
st
row (a1,b1), the first principal stresses in the second row (a2,b2) and second principal stresses in the third row (a3,b3).
Results show a similar distribution of shear stresses, with values varying around 23%. Principal stresses show less detail in
the maximum stressed regions and are less precise.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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2.2 Results on Various Surface Types
For this study, we applied the particle-spring model on samples of surface types of synclastic and
anticlastic typology while trying to maintain a maximum design stress as described by Nicklisch et al.
[3] for heat-strengthened glass HSG at 29 N/mm
2
. Results were compared with FE-analysis using an
automated interchange of model parameters via custom-programmed Excel files. Figures 4-6 show the
resulting stress distributions for a 4 mm HSG panel. We also compared the difference to the target
surface and analyzed the resulting Gaussian curvature and minimum principal curvature radii of the
deformed glass panel.
Target: Hyperparboloid, ruled surface
Resulting Geometry:
Min. Principle Curvature (PC) radii : 11.8 m / -11.8 m
Z+
max 27.9/ 19.4 -1.5/ 0.6 27.3/ 10.3
min 1.5/ -0.1 -27.9/ -15.5 -8.5/ 1.7
0 max 14.6/ 18.9 -0.8/ 0.7 9.6/ 3.2
min 0.8/ -2.2 -14.6/ -8.0 -14.1/ -3.25
Z-
max 21.2/ 19.4 -2.2/ 0.6 -2.2/ -1.7
min 2.2/ -0.1 -21.3/ -15.6 -21.2/ -10.3
σ1 σ2 τ
xy
Difference to target surface in mm Gaussian curvature of target (left) and bent (right) panel
Fig.4: Particle-spring simulation/FE results for bending a rectangular 1.0 x 2.0 m HSG Panel of 4 mm thickness on a target
surface of a hyperbolic paraboloid. Results show that the peak stresses occur on the upper surface of the panel close to the
corner points. This coincides with the location of maximum Gaussian curvature. The difference to the target surface
maximizes in the center of the panel.
Target: Sphere
Radius: 8.6m
Resulting Geometry:
Min. Principle Curvature (PC) radii : 16.1m/ 24.5m
Z+
max 28.4 /29.0 -1.5/8.5 27.0 /15.7
min 1.6 / -1.3 -28.3/-20.3 -24.9/-15.4
0 max 25.2/ 22.1 -1.4/ 0.9 20.7/15.8
min 1.4/-6.6 -25.4/-26.4 -24.1/-16.1
Z- max 24.0/ 29.0 -1.4/ 8.5 22.3/0.8
min 1.4/-1.3 -24.3/ -20.3 -22.9/-0.8
σ1 σ2 τ
xy
Difference to target surface in mm Gaussian curvature of target (left) and bent (right) panel
Fig 5: Particle-spring simulation/FE results for bending a rectangular 1.0 x 2.0 m HSG panel of 4 mm thickness on a target
surface of a sphere. Results show that the peak stresses occur on the upper surface of the panel close to the corner points. The
Gaussian curvature analysis maximizes in the middle and minimizes on the edges. The difference to the target surface
maximizes on two clearly separated fields in the middle of the panel.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Target: Conoid
Resulting Geometry:
Min. Principle Curvature (PC) radii : 13.1 m/ 64.7 m
Z+
max 20.1/ 28.1 -1.0/ 6.5 19.0/ 13.9
min 1.1/ -9.4 -20.0 / -20.2 -18.7/ -13.8
0 max 11.9/ 15.3 -0.6/ 0.8 11.0/ 3.1
min 0.5/ -2.8 -12.0/ -4.7 -11.4/ -3.1
Z- max 23.7 / 23.7 -1.3 /8.7 22.6/ 11.9
min 1.2 / -7.4 -23.8/ -21.2 -20.9/ -11.9
σ1 σ2 τ
xy
Difference to target surface in mm Gaussian curvature of target (left) and bent (right) panel
Fig. 6: Particle-spring simulation/FE results for bending a rectangular 1.0 x 2.0 m HSG panel of 4 mm thickness on a target
surface of a conoid. Results show that the peak stresses occur on the upper and lower surface of the panel close to the two
corners of the flat edges. Shear stresses are also considerably high in these regions. This coincides with the area of maximum
Gaussian curvature. The difference to the target surface maximizes in the center of the panel.
3. Structural Analysis
In this chapter we analyze the structural performance of bent glass based on the previous construction
criteria and geometric investigation. We look at the behavior of a single bent-glass panel in
comparison to a flat panel.
3.1 Modelling
For the verification and calibration of the previously described particle-spring model, a method was
developed to import the geometry, meshes and support movements into Strand7 via Excel. Before the
analysis, the imported movements were applied to a flat glass panel as defined support displacements.
For the investigation of the structural behavior of already bent panels, additional surface loads were
applied using the stage manager of Strand7. These loads represented climatic loads, such as wind and
snow, and were defined to +1kN/m² (representing pressure) and -1kN/m² (suction). The used non-
linear-solver with Kirchoff element-theory (material and geometric nonlinearities) is a direct sparse
solver with very accurate termination criteria (deflection criteria displacement zero 1x10
-8
m) and
maximum defined 1000 iterations. All calculations converged under 50 iteration steps. The results are
shown without extrapolation to the nodes to adapt to the discrepancy between the real linear supports
and the node-oriented simulation.
def (m) z- (MPa) middle (MPa) z+ (MPa)
Surface
type Load σ1 σ2 τ σ1 σ2 τ σ1 σ2 τ
Flat
wind + max 0. 010 10.03 -1.64 10.35 4.35 0.97 1.41 18.84 10.73 10.08
min -0.002 -5.32 -17.86 -10.35 -0.44 -6.29 -1.41 -1.47 -9.41 -10.08
wind - max 0.002 18.84 10.73 10.08 4.35 0.97 1.41 10.03 -1.64 10.35
min -0.010 -1.47 -9.41 -10.08 -0.44 -6.29 -1.41 -5.32 -17.86 -10.35
Ruled 1
bending max -0.222 6.23 0.09 -1.59 3.79 0.00 0.80 6.51 0.10 3.98
min -0.245 -0.09 -4.99 -4.10 -0.50 -2.12 -0.81 -0.08 -4.77 1.63
wind + max 0. 011 15.92 -0.70 7.01 4.56 0.64 1.31 21.58 9.41 15.81
min -0.002 -5.57 -20.99 -13.38 -1.80 -4.18 -1.43 -1.45 -14.94 -6.15
wind - max 0.002 21.76 9.48 6.47 4.89 0.62 1.45 16.59 -0.04 13.57
min -0.011 -1.57 -15.23 -16.18 -1.85 -4.00 -1.41 -5.64 -21.16 -7.22
Ruled 3
bending max -0.140 15.09 0.15 -2.03 11.40 0.02 2.40 14.89 0.13 8.31
min -0.190 -0.09 -11.88 -8.31 -1.60 -6.06 -2.45 -0.11 -11.86 2.12
wind + max 0. 014 24.09 1.40 5.57 11.72 0.42 3.65 26. 54 9.34 21.85
min -0.003 -7.54 -26.41 -16.07 -3.37 -5.75 -2.21 -1.02 -22.09 -5.36
wind - max 0.003 26.22 9.22 5.02 11.15 0.42 2.15 23.31 0.51 15.97
min -0.013 -0.77 -21.46 -21.54 -3.32 -5.48 -3.38 -7.35 -26.33 -5.35
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Saddle 1
bending max -0.098 10.18 3.03 3.69 3.10 3.10 0.94 11.76 11.76 2.77
min -0.125 -1.01 -6.85 -3.77 -1.14 -1.14 -0.95 -1.81 -1.81 -2.75
wind + max 0.020 24.77 2.51 9.07 15.43 0.00 3.83 36.20 36.20 13.97
min -0.004 -14.88 -38.29 -9. 02 -2.31 -8.93 -3.84 6.02 6.02 - 14.06
wind - max 0.001 15.02 8.56 9.10 4.77 12.73 1.77 11.83 11.83 8.81
min -0.004 -1.49 -10.81 8.93 -0.62 -8.01 -1.76 -6.50 -6.50 -8.88
Saddle 3
bending max -0.076 30.33 7.77 9.15 9.01 0.05 2.72 23.46 2.20 7.51
min -0.131 -4.50 -11.89 -8.77 -2.57 -4.75 -2.71 -5.45 -22.43 -7.39
wind + max 0.005 23 12.04 8.89 14.38 0.81 3.43 22.98 0.28 8.96
min 0.001 -7,63 -16.57 -8.75 -4.48 - 5.21 -3.43 -5.25 -27.58 -8.91
wind - max 0.001 23.48 8.35 11.48 14.38 2.17 2.46 21.79 2.94 9.87
min -0.002 -2.80 -13.20 -11.35 -4.48 -6.55 -2.47 -6.16 -23.81 - 10.11
Conoid 3
bending max -0.091 4.88 0.17 4.18 1.26 0.04 0.36 7.46 0.80 3.54
min -0.119 -0.53 -6.07 -4.20 -0.38 -0.53 -0.35 -0.19 -4.65 -3.52
wind + max 0.007 10.41 -2.63 10. 97 5.51 1.10 1.76 13.91 12.33 10.77
min -0.002 -6.31 -16.96 -11.00 -0.52 -8.99 -1.76 -1.05 -9.81 -10.75
wind - max 0.003 26.12 8.35 17.96 8.05 0.24 2. 50 22.19 1.94 13.07
min -0.014 -1.07 -19.44 -17.98 -2.34 -4.04 -2.51 -6.30 -26.10 13.08
Sphere 1
(r = 12m)
bending max -0.129 15.89 0.75 11.31 0.99 0.55 0. 48 18.60 6.39 11.21
min -0.172 -5.24 -17.71 -11.36 -0.42 -0.99 -0.48 -0.93 -14.95 - 11.14
wind + max 0.002 13.00 0.89 7,32 4.45 1.53 1.90 15.95 10. 15 9.72
min -0.001 -7.29 -19.94 -8,62 0.00 -7.49 -1.90 -1.04 -12.80 -9.66
wind - max 0.001 13.00 0.60 15.52 10.27 0.06 1.73 23.21 4.47 12.53
min -0.003 -7.29 -21.89 -15.58 -1.40 -5.37 -1.73 -0.83 -16.95 - 12.45
Fig. 7: Excerpts from results of FE-analysis: For each surface typology, three curvature variations for design stress of 29 Mpa
with additional wind loads were calculated. The table shows the resulting deflection in m, principal stresses and shear
stresses in Mpa on the top surface (z+), middle and bottom surface (z-) of each glass panel. The plates are clamped only
vertically. Results show that there are some significantly different values depending on the geometry of the curved panel in
comparison to the flat panel. The best performing panel is the ruled (1) surface with a utilization of 74% compared to 65% of
the flat panel under wind load. The worst performing is the saddle (1) geometry, with a utilization of 132%.
3.2 Assessment of the Results
As already described in chapter 2.2 and shown in figure 7, the general distribution of stress shows very
similar results. At few points the maximum stresses can be higher than in the particle spring-model.
This is due to the fact that the particle-spring model operates at a much smaller resolution, meaning
that peak stresses might occur on points which are not evaluated. The calculated stress and deflection
results for the bent elements show significantly different values compared with the flat panel. The
closest geometry is the ruled (1) surface with a utilization of maximum glass stress of 74% referring to
a 65% at the flat panel under wind load. The maximum deviation could be found at the saddle (1)
geometry. The maximum stress of 36 Mpa (132% utilization) is much higher than for the flat glass
panel. As expected, the deviations between flat vs. bent are not dominant. All bent elements show a
slightly lower deflection because of increased overall stiffness of the bent panels. This effect is due to
the small double curvature of the panels.The calculation results already show maximum stresses close
to or locally exceeding the allowable stresses of 29 MPa. As previously mentioned, the bent panels are
not supported as a shell should be, meaning that the panel would need to be clamped for horizontal
movement as well. Shell-like support for bent glass panels should reduce stresses from additional
loads. This must be analyzed and compared with the general rule to support glass panels statically
determined to avoid internal stresses due to temperature loads and loads coming from deformations of
the superstructure, such as a grid shell or a cable net.
4. Geometric Effects and Surface Design
We have shown a method to bend rectangular glass panels into double curvature. This method allows
for fast and low-cost assembly on site. What kind of façade surface can be covered with this method?
The following chapter discusses the geometric implications of larger panel layouts, focusing on the
use of only one standard panel format to cover doubly-curved design surfaces.
4.1 Façade Panelization with a Single Format
The research field of Architectural Geometry has extensively addressed the problem of paneling
freeform surfaces and reducing the number of unique panels. However, there are very few examples in
which only one panel format is used.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Research on seamless tessellation with flat panels has provided some strategies to avoid the variation
of panel dimensions. Using only one format however, can only be accomplished by either restricting
the overall geometry to be developable (singly-curved) or tolerating kinks between panels (Huard,
2014). Alain Lobel has classified such meshes, built exclusively from equilateral triangles
(http://www.equilatere.net). In a design process however, the faceted appearance is often not intended.
Triangular panels are less material-efficient for both fabrication and construction. Furthermore, the
process of discretization into straight elements inevitably shifts complexity to the nodes.
A so-called “semi-discrete” approach suggests the subdivision of a surface using a continuous curve
network and curved panels. When looking at built examples of smooth, doubly-curved façade
coverings, the following strategies can be distinguished: shape restriction, individual tailoring of
panels, positive and negative (overlap) gap tolerance and smoothness tolerance (kink between panels).
In the following chapter we will only use gap-tolerance to cover a freeform shape with rectangular
glass panels. Section 4.2 will describe a computational method to model large panelizations. Section
4.3 shows the analysis of the geometric dependencies between the curvature of the design surface, the
resulting gap between panels and the extent of the panel layout. Section 4.4 illustrates effects related
to the paneling grid and the alignment of glass-edges. In the final section 4.5 our method is
implemented, to design freeform façades. An optimization method is proposed to simultaneously
maximize curvature and minimize gaps.
4.2 Setting up a 3D-Modelling Process
The following investigation is based on the assembly of HSG-glass panels (4 mm) with dimensions of
1.0 x 2.0 m. We previously determined a minimal radius of curvature for the target surface to be 8.6
m. Our test setup is based on the assumption that the edge of a bent glass panel will follow a geodesic
line on the target surface. Geodesic lines are defined as the shortest path between two points on a
surface. They have no geodesic curvature. This property is achieved by rolling a straight strip onto a
surface, as is the case when bending the edges of the rectangular glass panels. Furthermore, we assume
that panels will not undergo drastic changes in edge and diagonal length. This provides us with the
information to iteratively measure and adjust the geodesic length of any edge. The digital model is
controlled by a particle spring system. Four points are projected onto a surface and connected with six
geodesic curves; four curves create a quad, and the other two create the diagonals. We then reposition
the points on the surface until all curves fulfill the expected length requirements (Fig. 9). The resulting
quad will approximate the outline of the bent glass panel with sufficient accuracy.
Fig. 9: Modelling process: A quadrilateral point grid is projected onto the target surface. The projected grid is then adjusted
to fulfill the expected geodesic-length requirements. Additionally, a seam is simulated by individually modelling each corner
of every panel. The resulting four points at each intersection can move independently.
Using this method, we can manipulate any quadrilateral point grid on a surface by adjusting the
geodesic distance of one point to its eight neighbors to the expected values of 2.0 x 1.0 m for the edge
dimensions, as well as 2.236 m for their diagonal dimensions. This process will yield a homogenous,
seamless grid of equidistant panels for any developable (singly-curved) surface, such as flat planes,
cylinders or cones. In case of a freeform surface however, the algorithm does not converge fully. It is
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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impossible for all panels to remain rectangular and congruent, and at the same time seamlessly cover a
doubly-curved surface. To counter this, we introduced a flexible seam between the panels. This seam
will adjust its width to ensure that the panel-grid can be transformed to the defined dimensions. We do
this by creating four points at every intersection in a façade grid; each point represents the corner of
one of the adjacent panels. These four points can move apart to allow the edges to fulfill our defined
relationship of geodesic lengths.
4.3 Curvature, Layout and Seams
For our analysis this computational setup is applied on a sphere and a hyperbolic paraboloid – two
surfaces with homogenous positive and negative Gaussian curvature. Any geometric effect on
panelization will be displayed most extremely on these two shapes. We first run the panel-fitting
algorithm and then measure the difference in width between smallest and largest seam. We call this
value the seam variance ().
Fig. 10: Two target surfaces: A sphere (top row) and a hyperbolic paraboloid (Hypar) (bottom row). Each panel has a
standard format of 1.0 x 2.0 m. The seam width increases with the greater layout size (left column) and smaller
curvature radii (right column). Synclastic curvature will result in a convex seam; anticlastic curvature causes a concave
seam. Seams are more pronounced along the long side of the rectangular panels.
Curvature and layout influence the progression and size of seams throughout the grid (Fig. 10). On
synclastic surfaces, the seams follow a convex, banana-like shape. Anticlastic surfaces produce a
concave, hour-glass-shaped seam. A smaller curvature radius of the target surface will create a larger
seam in the panelization. The layout size also has an increasing effect on the seam’s dimensions.
Rectangular panels cause wider seams along their long edge.
4.4 Uni-Directional Seams, “Stepping”- and “Bulging” Effect
The orientation of seams can be shifted to only one direction. This property can be useful when
designing with a linear substructure. This is achieved by enforcing collinear edges in the opposite
direction. It however can trigger a “stepping” effect transversal to the main seam. Fig. 11 shows two
panelizations of the same target surface with seams oriented in opposite directions.
As mentioned in section 3.2 the edges of each panel lie on a geodesic curve along the surface. The
panel grid however may deviate from a geodesic curve as shown in Fig. 12. Because of this deviation,
succeeding glass-edges do not follow a continuous curve. When aiming our view along the
panelization grid, we notice the panel edges “bulging” from node to node, creating a cloud-like or
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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serrated edge. In our example, this effect is most pronounced along the boundary (up to 2 cm) and
vanishes towards the center.
Fig. 11: Freeform surface with uni-directional seams. This can trigger a “stepping” effect transversal to the wider seam.
Fig. 12: Spherical and hyperbolic layout with minimal curvature radius 8.6 m. Shown are the resulting pattern of our panel-
fitting routine (black) and geodesics connecting the corner vertices (red). When deviating from a geodesic curve, the glass
edges do not follow a continuous curve. They “bulge” individually from node to node creating a tolerance of up to 2 cm.
4.5 Freeform Design Using Rectangular Glass Construction
To design an appropriate surface for the use with rectangular, bent glass, we have to avoid surpassing
the critical curvature radii. This can easily be done by simultaneously analyzing and manipulating a
design surface until no radius violation is detected.
In the following section, we aim to distribute positive and negative curvature evenly across a freeform
surface. This will minimize the size of the seams over a large panelization. The setup is based on an
optimization loop using a genetic algorithm. A large population of surfaces is generated via control
points by randomly varying the Z-value of a point-grid. The following three indicators are then
measured at regular intervals across the surface.
Fitness: F = K
SUM
* (1+K
PEN
) * L
RANGE
(6)
where K
SUM
is the sum of all curvature radii measured on the surface grid, as absolute values, K
PEN
is
the sum of penalties given for any curvature radius below a minimum value (8.6 m) and L
RANGE
is the
difference between the longest and the shortest curve along each direction of the surface grid,
indicating a homogenous distribution of surface area.
Fig. 13: Optimized freeform surface: Maximizing the overall curvature without violating the minimal radius criteria and
minimizing the differences of surface length measured along the panel grid. This results in a maximally curved surface, with
low variation of seams: Layout: 11 x 22 panels, R
MIN
= 8.61 m, = 4.9 cm
The algorithm evaluates each surface for its fitness F, favoring surfaces with the lowest F-value.
Eventually a set of solutions with minimal value F is found. These surfaces have the properties of high
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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curvature within the bounds of the minimal curvature radii, and a balanced distribution of surface area.
They are optimal for the use of curved rectangular panels, as they cause a minimum of seam
variations.
5. Conclusion
We have shown that the bending process can be efficiently simulated using a particle-spring model.
This modeling technique allows us to bend the panels into as much curvature as allowed within the
maximum defined design stresses. This is achieved by calculating and analyzing stresses in real-time.
Therefore, any target surface, defined by the architectural designer can be approximated into panels of
buildable curvature. We calibrated the modeling technique with the FE-analysis software Strand7 and
calculated the maximum Gaussian curvature for surfaces of doubly-ruled, sphere and conoid typology
for a given panel of 1.0 x 2.0 m of 4 mm HSG glass. This allowed us to define a minimal bending
radius for the subsequent investigation of geometric potentials on multi-panel façade layouts. In our
geometric analysis we set up a computational method, which allowed us to quickly simulate large
panelization layouts, using only rectangular panels. We analyzed the dependencies of curvature, layout
size and seams, and illustrated different effects on the resulting panelization grid. We implemented our
modelling method to design and optimize freeform surfaces.
Future investigations might integrate the calculation of external loads in the particle-spring model
allowing for multi-criteria optimization. Another research area is the ongoing development of the
physical façade detailing of doubly-curved, cold-bent glass construction. The combined behavior of
the substructure in connection (stiffening effect) with the bent glass panels will be subject of future
research. Future research may investigate the potential of orthogonal nodes in the curvilinear
substructure. Further cost efficiency could arise from a careful negotiation of tolerable seam
dimensions and tailoring of panels.
Acknowledgements
The physical prototypes were constructed during a Master’s elective course at EPF Lausanne held in
collaboration between Philipp Eversmann, Paul Ehret, Christian Louter, Manuel Santarsiero and the
students Andrea Baraggia, Dominik Baumann, Tomas Odelbo, Agnes Ulrika Charlotta Orstadius,
Francesca Rabbiosi, Itai Vander, Robbert Verheij and David Viladomiu Ceballos.
References
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