Proceedings of the International Association for Shell and Spatial Structures (IASS)
Symposium 2015, Amsterdam
17 - 20 August 2015, Amsterdam, The Netherlands
Form Exploration of Timber-based Folded Plate Domes
Andreas FALK*, Peter VON BUELOWa, Anahita KHODADADIa
* KTH Royal Inst. of Technology
KTH Building Materials, SE-100 44 Stockholm, Sweden
a, b University of Michigan, Ann Arbor, USA
This paper presents a study on timber-based plate-shell domes with a set base diameter and a variety
of topologies using different combinations of perforation ratios. Using a combination of geometry
generation and performance optimization, parameters of folds, depth of folds, height of dome and the
effect of perforations on structural efficiency, interior lighting and acoustics are explored. The
combination of a visual database with both structural and architectural oriented performance
parameters, gives the designer added insight in overall form determination. The overall geometry and
its tessellation are also discussed in terms of environmental performance.
Keywords: perforated, plate shells, cross-laminated timber, ParaGen, daylighting, acoustics
1.1. Development of timber-based construction
Cross-laminated timber (CLT) is a robust engineered wood product (EWP), which to date has been
proposed for and applied in a wide range of structures, so far mainly due to the renewability and
workability of the raw material and high element stiffness of the produced surface elements. In terms
of production economy it is so far available at comparably higher cost than light timber-frame systems
and corresponding prefabricated systems based on concrete elements. As environmental issues are
rising in priority on the political and societal agendas globally, however, reduced resource
consumption during production – compared to many non-bio-based materials and products – is getting
increased recognition as well. In the cradle-to-grave perspective the energy consumption during
extracting and refining of building materials have gained more interest lately and the tools for life
cycle assessment (LCA) have been developed to enable true comparison between different material
categories (Erlandsson et al. 2013 ). Through this harmonisation work considering e.g. the
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
agreement about system boundaries for evaluation, LCA tools at hand today which enable robust and
fair comparisons of different materials for the building sector and from environmental point of view
this will increase the competitiveness of bio-based materials and products in the construction sector, a
condition valid even for timber-based products such as CLT, which require relatively higher energy
consumption during production than light timber-frame systems.
1.2. Plate-based structural systems
The dimensional stability, rigidity and efficient force distribution in the CLT-based plate elements
leads to generally good structural performance of CLT-based structures. This study uses integrated
performance values based on structural behaviour, lighting characteristics and acoustic response, to
explore a range of geometric solutions. The study is conducted using ParaGen, a method, which uses a
GA to fill a database with well performing solutions based on a parametric model. ParaGen, allows for
open interaction between the designer/designers and the computer tool regarding the comparison of
multiple design factors, and the process of evaluating the generated results. Thereby performance
criteria and other design considerations can be easily linked to and taken into account in the process.
Studies of the effects of perforations on the structural properties of CLT elements have been carried
out on orthogonal plate structures and in experimental architectural projects. The effect of freely
placed openings for e.g. windows, doors, roof-lights etc., either already in the factory or on the
building site, can be seen in examples from the Wood Studio at Aalto University in Finland and in the
so called “Naked House” from 2006 by dRMM architects. The robustness of the CLT elements offers
many possibilities to vary both the architectural use of the plates and the level of prefabrication of the
structural parts – the workability of the timber material enables perforations to be made also on site.
Structural efficiency, material utilization and resource minimizing efforts, as well as lighting and
acoustic behaviour all have important impact on the design of a structure aiming for varied functions
and architectural use. Both the global geometry of the panel-based system as well as the local
perforation of individual CLT panels with glazed inserts impact a range of performances including
structural, lighting and acoustic, areas that are currently combined in an integrated performance
exploration of plate-based wooden domes. Acoustic behaviour of large open spaces is also a critical
design consideration especially in sport facilities. Sustained crowd noise in an acoustically active
space can reach levels, which are physically damaging to the hearing of the occupants. Both the
geometry and surface composition (glazing vs. CLT) of the structure play an important part in the
performance of the space. Also the design of daylighting through the use of the panel perforations
brings with it another set of performance considerations. In order to evaluate the space and structure
for all of these integrated performances, ParaGen stores the values in a SQL database that can be
explored both with sorting and query commands as well as parallel point and scatter graphs to find
Pareto optimal and other well performing solutions in all areas. Because the method is visually based
and displays images of the solutions for the designers, qualitative aspects of the design can also be
1.3. Folded plate-shells
In terms of structural performance, folds offer a beneficial alternative for plate-based steel structures
to obtain efficiency and folded plate-based vaults have been in focus in previous studies where
tessellations and depth of folds have been varied, and comparisons between curved plate shells with
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
supporting structures and facetted self-supported vault-structures have been made (Falk and von
Buelow 2011  and Falk and von Buelow 2009  respectively). The performance of facetted
typologies can be varied by changing the plate properties by perforating or altering the materials of the
plate-elements. Changes of the resulting performance can also be obtained by truncating the folds,
which opens the structure, offering insertion of elements with other properties, such as glass panels.
Figure 1 shows the basic plate-shell typology, which is in focus in this paper.
Figure 1. Basic plate-shell (left & middle); a plate-shell with fewer elements/larger openings (right).
2. Timber-based domes
Timber-based domes have been developed and studied in different contexts. Shell types offer many
different solutions both for structural systems and for modifying the interior environmental
performance and qualities. The traditional timber-based structural systems have been based on 1-D
elements of posts and beams and as a consequence dome-shaped structures have been built up as
reticulated systems. The stability issues and buckling phenomena of such systems are dependent on
the characteristics of linear elements, member density and nodal joints, as studied in e.g. Pan and
Girhammar 2003 , where the ring beam stiffness is in focus. The function as a tight roof is obtained
through addition of decking, which adds to the rigidity of the dome through bracing of the purlins. Pan
and Girhammar 2005 , discuss this in relation to the mesh density expressed as a relative timber
volume, defined as timber volume divided by the timber volume of a mesh density corresponding to n
= m = k = 10 (n is the number of sectors, m is the division of the length of the arc, k is the division of
the bottom ring length). In the performed study mesh densities between 0.27 and 0.63 are considered.
Increasing the mesh density, i.e. increasing the volume of used material is found to have marginal
effect on the structural performance. The function as support for cladding/decking has to be
considered but in case of the discussed densities, the effect is marginal.
The increase of used material volume can be evaluated in terms of added value and/or added
functions. In a resource efficiency perspective the addition of material should benefit the structure in
some way – preferably with more than one added function – and if it has little or no effect on the
structural performance it could still have effect on other aspects of the structure and its enclosed
volume. By going from a reticulated structure to a plate-based structure the material utilisation is
altered and the structural elements also serve as bracing/stabilisation of the geometry, and can be
potentially used for roofing function as well. The structure gains practical/utility qualities through this
change within the duality discussed by Wester, 1984 , where lattice and plate structures are defined
as interrelated anti-poles, by incorporating a surface (structurally interconnected surface elements)
utilised for force transfer. Cross-laminated timber as a product offers this function efficiently but
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
could as well be replaced by a typologically related product like LVL, laminated veneer lumber,
which is also available in the format of structural plate elements. This type was utilised in e.g. the
Metropol Parasol in Seville from 2011 by J. Mayer H. Architects in collaboration with Arup, even
though it is not primarily used for covering purposes in that very project.
2.1. Global geometry
Internationally, contemporary architectural trends result in experimental geometries and experiments
with visual material effects, which challenge both material and structural properties of the elements
and systems. The relation between architectural and structural design varies with the attitude towards
structural clarity and interest in structural and construction related aspects and if the Guggenheim
Museum in Bilbao by Frank Gehry Partners inaugurated in 1997 may be taken as example of one
extreme, where the structure is hidden and expected to anonymously support the undulating lines and
curves of the façade, Qatar National Convention Centre by Arata Isozaki and RHWL Architects
finished in 2011, may represent one anti-pole, where the structural concept is exaggerated to form the
main design feature of the project. Beijing National Aquatics Centre from 2008, more well known as
the Water Cube, designed by a consortium consisting of PTW Architects, Arup international
engineering group, CSCEC (China State Construction Engineering Corporation), and CCDI (China
Construction Design International) of Shanghai, is another example where the structural system was
developed as the main design characteristic of the object’s visual appearance.
The relation between structural form and the geometry of the building envelope is gaining importance
as architectural complexity increases at the same time as the necessity of resource efficiency becomes
more obvious, for environmental reasons. The potential synergies of the utilisation of structural
materials and the design of the spatial characteristics of the building skin appear in that light as both
interesting and developable from a combined structural/architectural/environmental point of view.
The material properties may provide to and co-act with the environment of the enclosed volume
(humidity and temperature buffering effects studied in e.g. Falk, Turrin and von Buelow, 2010  and
Falk, von Buelow and Kirkegaard, 2012 ) and combinations of different surface and material
properties in the interior have potential effect on the interior environmental performance. Surface
properties and geometry of the roof and wall structures have effect on acoustic and light properties of
the indoor environment through the way the surfaces reflect and/or absorb sound and light. Depending
on the arrangement of perforations and the local geometry, angles and surface finishing around them,
both sound and light performance can be tailored differently. The structural form in the case of the
CLT-based dome-shaped shells considered in this study, constitutes the form of the building envelope
and thereby both the external and the internal surface geometry. The geometry and element design of
the plate-shell is studied and optimized for proper structural function but also regarded in terms of
their additional environmental effects on the interior space. The plate shell presents a folded structure,
which combines plate and shear-plate action with increased structural efficiency obtained through the
depth of the folds. The arrangement of the folds may also be used as an optimisation parameter.
In structural perspective perforations can be made to save material and optimize the material
utilisation. In architectural and holistically functional perspective perforations are made to let light in,
to vary and improve the interior light conditions and ditto quality, to vary the expression of the
geometrical form, etcetera. In that way void and removal of material enables improved functional
performance as spatial values are potentially enhanced. For proper structural function the optimisation
of remaining mass gains increased importance and the studies of nature inspired skeletal analogies are
innumerable, e.g. Wester, 2004  and Phillips, 2012 . Phillips among others refers to Cullmann’s
graphical statics from 1866 and the discussion on trajectories by von Meyer 1867. The growth of
bones had been noted to develop a structurally optimized pattern in response to gravity and the
patterns of the resulting inner structure were in focus of their discussion. In his study of Cullmann’s
and von Meyer’s works, Phillips models and runs an optimisation sequence of the structural
composition of the thighbone close to the hip joint as a combination of shell and beam elements,
where the shell thickness and the distribution of the beams are varied. This discussion on material
saving is fully driven by the aim for structural efficiency, more or less as the I-beam is saving material
without making other use of the basic principle than decreasing the dimensions of the web.
Architecturally however, void is as important as mass. The structural materials offer the mass and the
design and location of their mass creates the preconditions and the characteristics of the void, i.e. the
The high in-plane rigidity of CLT plates offers efficient transfer of in-plane forces and this property is
efficiently utilised in folded plate-shells of CLT elements. Contemporary production technology
allows cutting-patterns for minimised waste of material and the dimensions of the CLT elements are
produced in more or less tailored build-ups, which can be easily chosen in response to the specific
load-conditions for each design situation. Thereby a wide range of forms is enabled. The timber-based
folded plate dome seen to the right in Figure 1 is based on truncation of the folds enabling openings in
the structural shell, which are filled with glass panels. The point of departure is the typology shown to
the left in Figure 1 and in the onward development the number of elements, depth of folds and size of
truncations/openings are varied.
2.3. Additional utility aspects
The effects of perforations and variations of materials on the indoor environment comprise apart from
light conditions and acoustic properties also air quality and humidity due to e.g. air streams and
materials’ varying capacity of interaction with the indoor air, as well as tactile experiences of the
users. Tools like ParaGen allows for consideration of a wide range of aspects in the design process
and different types of analyses along the process. It makes it feasible to easily include different factors
for environmental performance and by that enabling encompassing of a building’s compatibility with
not only structural and architecturally functional requirements but also environmental aspects such as
qualitative evaluation of geometry and building skin in the same procedure. Those additional aspects
are not yet added in this study.
3. Results of modelling
As mentioned in Section 2, a variety of topologies and geometries were explored using the ParaGen
method. In the runs made thus far, quantitative assessment has been based on geometric features and
structural performance. Further planned assessments include lighting and acoustic simulations.
ParaGen employs as variety of simulation software to collect performance data on solutions generated
using parametric modelling software. In this trial, Formian was used to generate the dome geometries
based on origami patterns. The structural analysis was performed using STAAD.Pro from Bentley
Systems. Also Rhinoceros by Robert McNeel and Associates was use to manipulate the geometry and
prepare renderings together with DIVA for lighting simulation and analysis. An analysis based on
Sabine’s equation was used for the acoustic simulation (Sabine, 1922 ).
3.1. Dynamic configuration processing of folded plate diamatic domes
“A dome is a structural system that consist of one or more layers of elements that are arched in all
directions” (Makowski, 1984 ). A dome might be a part of a sphere, ellipsoid, paraboloid or a
curved surface with different patterns of braced elements or plates. There are diverse types of domes
and 'diamatic domes' are frequently used in practice because of their specific characteristic. In some
types of domes 'element cluttering' around the crown is a significant issue and this not only causes
some construction problems but also aesthetic features, which normally are less preferable. In
contrast, diamatic domes are based on such patterns within which the population of elements look
almost consisting from the crown to the base. This advantage can be seen in Figure 2 in which a
lamella dome is compared with a diamatic one. Two notable domes of this type are the Superdome in
New Orleans and the Astrodome in Texas, see Figure 3, left and right.
In this paper, a diamatic type is opt for to avoid the problem of plate cluttering at the crown of the
folded plate dome. A diamatic dome consists of a number of 'sectors' and the pattern of each sector is
in a fashion that the side boundaries are along two meridians of the circumsphere of the dome and the
bottom boundary is along a parallel of the circumsphere. The number of elements along a boundary of
a sector is referred to as the 'frequency' (Nooshin and Disney, 2001 ).
Figure 2: Comparison of a lamella dome and a diamatic dome (Nooshin and Disney, 2001 ).
Figure 2: Sectors and frequency of elements in a diamatic dome (left). A typical form of a diamatic
folded plate dome that are to be explored (right).
In this paper, the configuration of a diamatic folded plate dome with a 20-meter span is processed
using concepts of Formex algebra and its associated programming software, Formian. Formex algebra
is a mathematical system that allows a designer to define the geometrical formulation of forms
through concepts that effect movement, propagation, deformation and curtailment ((Nooshin, Disney,
and Champion, 1997) ). Within this mathematical system the plates are defined as descrete
surfaces, which are arranged besides their edges.
The domes investigated in this paper have perforations for providing daylight to the interior. The
proportion of perforation area is defined in percentage. The geometry of the dome is exported in two
separate DXF files, one for the structural timber plates and the other for the non-structural glazed
surfaces. The folding pattern of the dome in this study is inspired by origami patterns (see Figure 4),
with one pattern forming the base topology of the dome. In Table 1, constant parameters, variables
and also their acceptable intervals, used in the parametric formulation of the dome are described.
Figure 4: Some origami-based forms, Yoshimura Pattern variations.
Table 1: Geometrical parameters of the folded plate diamatic dome.
Acceptable interval/ value
Span of the dome
Rise of the dome
Number of sectors
m should be integer. [2, 10]
Frequency of plates along the left
border of the each sector
n should be integer. [5, 20]
Depth of folds
[0.2, 2) m
Percentage of perforations within
(0-100%) is recommended.
3.2 Structural simulation
The dome is simulated using two materials: CLT panels based on softwood properties (Spruce-Pine-
Fir) and plate glass. STAAD.Pro uses 3 or 4 node planer plate elements with variable thickness. For
this analysis the CLT thickness was set to 9 cm and the glass thickness at 0.5 cm. The stiffness of the
glass panels was turned off during the analysis assuming that the glass would not transmit stiffness to
the system. In addition to self-weight, an imposed downward vertical load of 2.0 kPa was applied to
all plates. The performance values, which were taken from the analysis included the modal frequency;
the calculated weights of panels and their relative proportion by area (CLT to glass); the von Mises
stress values for plates (including a stress contour plot); deflection values; and total reaction loads.
Using the ParaGen interface, the results could be explored based on these performance parameters.
For example, Figure 5 shows a grouping of solutions sorted by the panel complexity. Below each
image is a short list of other parameters that describe the dome.
Figure 5: An array of solutions sorted by numbers of panels.
3.3. Simulation of lighting
The daylight analysis, see Figure 6, begins with setting a surface as the ground, and then importing the
DXF file of the dome into the space model. A point along the central axis with a height of 80 cm is
defined as a station point in calculating the daylighting factor and solar irradiation value of the
models. Furthermore, another surface at the same height of 80 cm is created to survey the light
distribution and illuminance quality. This surface has the same dimensions and location for all the
dome models. Next, appropriate materials are assigned to the structural parts and perforation surfaces.
DIVA daylighting simulation factors are set as follows: ambient bounces (ab) = 3, ambient divisions
(ad) = 1500, ambient super-samples (as) = 20, ambient resolution (ar) = 300, ambient accuracy (aa) =
0.1. The simulation then yields the solar irradiation, daylight factor and illuminance values. The
location is taken as Amsterdam, Netherlands (52.37° N, 4.90° E).
Figure 6: DIVA daylight analysis.
Table 2: Definitions and effects of different Daylight Factors.
DF and appearance, thermal performance
The room looks gloomy
Artificial lighting is required
The room looks lit, but
supplementary artificial lighting
Artificial lighting may be required for some times.
The room appears strongly lit
Daytime artificial lighting rarely needed, but glare and
solar gain may cause problems due to overheating in
summer and heat losses in winter.
Daylight Factor is a ratio that represents the amount of illumination available indoors relative to the
illumination present outdoors at the same time under overcast skies. The DF may influence our choice
of suitable design solutions more than other factors. Daylight Factor is typically calculated by dividing
the horizontal work plane illumination indoors by the horizontal illumination on the roof of the
building being tested and then multiplying by 100. Table 2 above, describes the quality of different
range of the DF. Comparing the value of daylight factor at a certain point within the solutions allows
us to make suitable design decisions accordingly.
This study considers dome-shaped structural systems comprised of a combination of flat CLT and
glazed plates. A sampling of solutions is generated based on a parametric model that ranges in
topology from a nearly smooth continuous surface to a reticulated shell composed of beams.
Regarding the typology with a curved plate-shell and a reticulated dome as anti-poles, it is obvious
that the material can be distributed – and utilised – in completely different ways. In the study the
solutions are each evaluated primarily for structural performance characteristics, but also acoustic and
daylighting quality are considered. The structural performance follows the range of the topology –
surface-active systems versus section active systems. Figure 7 shows this range and the corresponding
von Mises stress plots. In the analysis the stiffness of the glass panels have been set very low
(assuming they would be gasket mounted) which forces the CLT panels to provide the overall rigidity.
Figure 7. A comparison of the range of perforation and its effect on topology and structural behavior.
When perforating the plate-shell system (valid also in case of perforation of individual elements),
shear-plate action will remain and the system can be structurally optimized as a surface-active system.
This enables the designer to model the relation between opaque and transparent parts of the structure,
an option, which clearly differs it from the reticulated dome. In the design procedure the designer may
include several optimization aspects, which should be handled in relation to each other, such as
structural efficiency, material volume and utilisation, qualities and effects of lighting, acoustic
performance, actual view through the perforations establishing the experienced relation between
indoor and outdoor environment. Other aspects not explicitly considered in this study are e.g.
efficiency of production and deployability.
By varying the parameters and depth of folds and the height of dome the enclosed space is varied
markedly both in terms of enclosed volume and in terms of preconditions for sunlight penetrating
through the openings. Deeper folds will result in a structurally more efficient system and may enable
certain reduction of the plate-element thickness. Fewer facets, i.e. larger scale of the tessellation will
result in a relatively smaller number of openings with larger dimensions. An increased height of the
dome changes the angle of the shell surface towards the sun, and as the pitch of the dome surface
increases the lower parts of the dome change nature from mainly roof function to mainly wall
function. Sunlight will thereby penetrate the structure not only midday when the sun is high, but also
during morning and evening hours. This will increase the amount and intensity of daylight entering
the indoor volume.
As a final note: The perforated folded plate-shell offers an architecturally variable typology where the
enclosing structure also forms the geometry of the building envelope. Practical aspects such as water
drainage of the folds and additional layers protecting the timber elements from moisture and
weathering as well as sealing joints and insulating layers ensuring a comfortable indoor climate of
course have to be added. Furthermore, as a built structure with intended structural use, larger openings
for entering the structure are usually also needed. The entrance location may change the boundary
conditions for the dome. These aspects have not yet been considered in the current study.
So far, the results of the modelling are limited, since the simulations have been just recently started,
but the procedure of ParaGen is under steady development and show very useful results, not the least
when adding environmental performance and compatibility to the set of considered design aspects.
The potential to add different softwares and designer’s qualitative input to the optimisation process
provides a robust design process where the designer is in full control throughout.
The CLT-elements are not yet used in practice for the studied structural typology, but this type of
engineered wood product demonstrates both developable potential of CLT-based structures and a
potential to innovate and develop new bio-based structural elements and systems. The workability
enables fast processing of the material both off and on site and the environmental aspects of
prefabrication and CAD-CAM supported production with high level of resource efficiency and low
waste, and of materials interacting with the indoor climate, could be further utilised.
The variability of the building system for the studied dome type offers interesting possibilities to
develop the structural efficiency and material utilization in relation to their effect on the desired
environmental performance, providing a wide range of options with tailored indoor spatial properties.
These interrelations will be further modelled and analysed through the upcoming stages of the study.
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