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Proceedings of the International Association for Shell and Spatial Structures (IASS)
Symposium 2015, Amsterdam
Future Visions
17 - 20 August 2015, Amsterdam, The Netherlands
Prototypical Hypar: an operative form-making method
based on hyperbolic paraboloids
Ting CAO*, Joseph SCHWARTZa, Chi ZHANGb
*PhD Student, Institute of Technology in Architecture, ETH Zurich
Stefano-Franscini-Platz 5, HIL E 43.2
8093 Zurich, SWITZERLAND
caoting@arch.ethz.ch
a Professor, Institute of Technology in Architecture, ETH Zurich
b PhD Student, Department of Built Environment, TU Eindhoven
Abstract
This research aims to develop a form-making method based on hyperbolic paraboloids (hypars),
which creates forms possessing both architectural aesthetics and structural efficiency. Hypars can be
considered as prototypical examples for the correlation of architectural and structural design by means
of geometry. In this paper, a method based on graphic statics is introduced to analyze the structural
behavior of general hypars disposed in any arbitrary positions in space. This method illustrates the
interplay between structural behaviors and geometric forms of hypars, which is relatively easy to
operate and understand. In view of these analysis, the design strategy based on combinations of hypars
is proposed with an example.
Keywords: hyperbolic paraboloids, operative form-making method, interaction between architectural
and structural design, graphic statics
1. Introduction
Since the profession of engineering was formally separated from architecture in the 19th century, the
search for interactions between architectural design thinking and design of building structures has
never stopped (Saint [11]). Such interaction aims to design and build forms that not only meet
aesthetical and functional demands from the architectural perspective, but also be mechanically
efficient to fulfil structural and constructional requirements. Widely used methods such as finite
element method (FEM), which are mostly applied to analyze and refine well-developed design
proposals, concern the structural engineering so late in the design process that it can hardly affect the
formative stage of design (Kotnik and D’Acunto [6]) . More integrated approaches like form finding
methods involve structural considerations from the very beginning of design processes (Bechthold [1]),
but usually can only generate free-form surfaces, which are relatively more difficult to be fabricated
than descriptive geometries (Flory et al. [5]).
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
To overcome these limitations, some architect-engineers like Felix Candela, Pierluigi Nervi and
Eduaro Torroja, have attempted to use a descriptive geometrical type, the hyperbolic paraboloid
(hypar) as a new prototype for building forms (Pogacnik [10]). Due to hypars’ doubly curved and
doubly ruled shapes, they have advantages in different aspects: high structural stiffness, rich aesthetic
potentials and constructional convenience (Bechthold [1]). Therefore, a design method based on this
geometrical type has potentials to facilitate the interdisciplinary interaction between architecture and
engineering. In most built projects, hypars were used as basic modules combined to create efficient
structural forms as well as spatial enclosures for many types of architectures, including industrial
buildings, residential houses and public service buildings (Enrique [4]).
To develop the early exploration further into a handily operated form-making method, complex
technical questions should be solved in an easily understandable way. Currently, only a special group
of hypars, the ‘typical hypar’ (Bechthold [1]) , which has the geometrical rise parallel to the gravity
(fig.4), can be more easily interpreted by designers without strong engineering background, since its
mechanical behavior can be intuitively illustrated with diagrams of forces (Muttoni [8]). Regarding to
general hypars disposed in arbitrary positions in space, only engineers can calculate the stresses with
FEM tools or mathematics formulas (Candela [2]). Such limitation reduces the possibility for the
interplay between the geometry and the structural behavior due to the knowledge gaps between these
two disciplines.
To address the issues mentioned above, we propose an easy-to-use method based on graphic statics to
intuitively analyze the mechanical behavior of general hypars and their combinations. The structural
information is visualized as strut and tie models with only compression and tensile forces, and they
are developed further into a digital tool, which enables a real-time interdisciplinary interaction in the
conceptual stage.
In the following section, a brief overview about projects developed by previous designers is provided
with analysis on their methods and limitations. In section 3 and 4, the analysis of general hypars with
graphic statics is presented with a tool implementation. In section 5, the strategy to utilize the
generated structural information for combination of hypars is illustrated with an example. A
discussion of current limitations and future plan related to the proposed method concludes the paper.
2. Related research
The early explorations on the use of hypars for building forms are mostly done by architect-engineers.
Most of these efforts, however, mainly focus on the use of hypars in specific design projects. For
example, the roof of Saint Mary’s Church in San Francisco designed by Pier Luigi Nervi (fig.1), is
formed by a combination of eight inclined hypars, which have rises non-parallel to the gravity. To
understand the mechanical behavior of the roof and prove its stability, many resin models were tested
in different scales. Another civil engineer Lenord F. Robinson was also involved to test it with
mathematic modelling (Olmo and Chiorino [8]). All the efforts made in this project, however, only
studied several specific hypars, but didn’t summarize a method to analyze the behavior of general
hypars.
Felix Candela is an exception among these architect-engineers. By utilizing the hypar as a prototype,
he designed a serial of buildings with various shapes and different functions (fig.2). Based on his
pragmatic experiences, he developed a theory which covers theoretical considerations and formulas
for stresses investigation of such structures (Candela [2], [3]). As he mentioned, “in the case of
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
anticlastic shells like hyperbolic paraboloids, the intuitive analysis is not so easily visualized as in the
dome” (Candela [3]), all his analysis was based on complex mathematic calculations. Being a designer
with the ability of transdisciplinary transgression, he could receive benefits from these complex
calculations even in the formative stage of his designs, but this mathematic method constitutes a
threshold for other designers.
Figure 1 (left): Saint Mary’s Church in San Francisco designed by Pier Luigi Nervi
Figure 2 (middle): Church of la Medalla de la Virgen Milagrosa designed by Felix Candela
Figure 3 (right): Philips Pavilion in Brussels designed by Le Corbusier
The use of hypars in design by architects focuses more on architectural space than structural
efficiency. In the case of Philips Pavilion (fig.3) in Brussels for 1958 World's Fair designed by Le
Corbusier, mathematical procedures were once used to determine the characteristic form of the
structure, but this logic no longer proceeded, and the compositional arrangement was finally decided
on the basis of intuition (Petit [9]). This example suggests the limitation of the mathematic analysis.
Even with the explanation from engineers, it’s still difficult for architects to utilize the structural data
to artuculate their designs.
In conclusion, although forms developed from general hypars possess advantages both in structural
efficiency and architectural aesthetics, complex mathematic calculations of their mechanical behaviors
limit the study only to individual projects or designers with strong engineering background. An
analysis method based on graphic statics is introduced in this paper, which enables the design
philosophy based on hypars to be developed into a commonly understandable and easily operated
design method.
3. Analysis of general hypars with graphic statics
3.1. Surface definition
Structural and aesthetic characteristics of hypars are closely related with their special geometrical
properties: doubly ruled and doubly curved surfaces. To explain the method based on graphic statics
introduced in this section, the surface definition of a hypar is given at first.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Assuming four non-coplanar points A, B, C, D in space and connecting them in turn, we get four
segments of AB, CD, BC and AD, which are the four edges of the hypar. Connecting middle points of
diagonals AC and BD, the gotten segment r is the rise of the hypar (fig.4). Line h and i are middle
rulings of the hypar. Lines hm intersecting edge AD, BC and parallel to plan hOr, define the surface.
They are the first group of rulings. In the same way, another group of rulings im, which intersect edge
AB, CD, and parallel to plane iOr, can also define the same surface. These two groups of rulings
define a hypar as a doubly ruled surface, which also includes two groups of parabolas with upward
and downward curvatures. These parabolas are parallel to plan AOC and BOD respectively. Rises of
these parabolas are always parallel to the rise r of the hypar (fig.4).
Figure 4: Hypars’ surface definition, doubly ruled (left), doubly curved (right) and the rise r of a hypar
3.2. Discrete strut and tie model of typical hypars
In this subsection, we show a typical hypar as the starting point to present the proposed method based
on graphic statics. In figure 5, the self-weight of a symmetrical hypar with the rise parallel to the
gravity is discrete as concentrated loads applied at four nodes on the surface. The smooth parabolas
intersecting at these nodes are discretized as polylines, which correspond to the tangents of parabolas
and represent the largest forces in parabolas. Such discreteness makes sure that polylines intersect at
least one node on each edge, but it causes each node on the surface to be divided into two sub-nodes,
which are in the same action line of the gravity. Each concentrated load is split in two and applied on
sub-nodes, so polylines with upward and downward curvature each take half of the gravity (fig.5b).
The forces in segments of a polyline can be calculated with force polygons (fig.6a, fig.6b). Following
three-dimensional graphic statics, force polygons for each node can be created to get supporting forces
on the edges (fig.6c, fig.6d). This model suggests no matter how other geometrical parameters change,
if rises are parallel to the gravity, hypars loaded with self-weights always work as shells with only
membrane forces.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Figure 5: Concentrated loads representing the hypar’s self-weight (a), hypar’s geometrical discrete
representation (b).
Figure 6: The distribution of loads on polylines with downward (a) and upward (b) curvatures, free-
body diagram of a typical hypar with self-weight (c) and its force polygons (d).
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
3.3. Discrete strut and tie model of general hypars
Based on the same rules in section 3.2, a discrete model is set up for a general hypar loaded with self-
weight. In this case, the rise of the hypar is not parallel to the gravity, so the concentrated loads
applied on each node have to be divided into three components: one component fr-n parallel to the rise,
the other two components fh-n and fi-n parallel to two rulings intersecting at this node (fig.7). Assuming
that only components fh-n and fi-n are applied on the hypar, they can be balanced with supporting forces
parallel to rulings (fig.8a). This is the first subsystem of the whole structural model. The second
subsystem is only loaded with components fr-m parallel to the rise (fig.8b), which is similar with case
study in section 3.2. But as the components fr-m applied on the second subsystem are not equally
distributed, we have to redistribute them to make sure that forces applied on each node of a polyline
are the same. The basic logic of the redistribution is to bring unequally distributed forces to the sub-
nodes located in the symmetric positions of polylines.
Figure 7: Each concentrated load can be split into three components: fh-n, fi-n along rulings and fr-n
parallel to rise.
In the second subsystem, to illustrate clearly the redistribution of loads at nodes of each polyline,
polylines AOC, EH and FG are explained as examples in detail. There are two components fr-3 and fr-2
applied on polyline AOC, EH and FG (fig. 8b). Following the set rule in section 3.2, we divide fr-3, fr-2
equally into two parts. Comparing half of fr-2 and half of fr-3, the latter is bigger. Then we add their
difference with the other half of fr-3, the sum is applied on sub-node 3’’ of polyline EH. The sub-node
2’’of polyline FG and sub-nodes 2’, 3’ of polyline AOC still take half of component fr-2 (fig.9a, fig.9b).
After the calculation of forces applied on each node of polylines, the force polygons of the second
subsystem can be developed. These polygons suggest the necessary supporting forces along edges and
rulings (fig.9c, fig.9d). The last step is to combine the first and the second subsystem all together, by
simply adding forces at each node (fig.10). At the end, we can get a complete strut and tie model of a
general hypar and its force polygons.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Figure 8: The first subsystem loaded with fh-n, fi-n and its supporting forces (a), the second subsystem
loaded with fr-n (b).
Figure 9: The redistribution of loads in the second subsystem (a, b), free-body diagram (c) and force-
polygons (d) of the second subsystem.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Figure 10: Adding the first and the second subsystems, the gotten free-body diagram (a) and force-
polygons (b) of a general hypar under self-weight and supported with two edges.
Based on the analysis mentioned above, a hypar under self-weight and supported with two edges,
generally works as a combination of a shell and a wall. With variations of geometry and position in
space, the main behavior of a hypar is changed between a shell and a wall. Like the typical hypar
studied in section 3.2, components fh-n and fi-n, which can cause a wall-like behavior in a hypar are
zero in this special case, so it works only as a shell. In another example, when the rise is perpendicular
with the gravity, components fh-n and fi-n, are much higher than fr-n, and the hypar gets much higher
internal forces along rulings than parabolas, so the load bearing behavior of a wall is governing.
3.4. Simplified model of general hypars’ supporting forces
In the combination level, supporting forces of a hypar are related more with the design than internal
forces, because they will affect the geometrical relation of adjacent hypars directly in a combination.
In the discrete model in figure 10, continuous supporting forces parallel to one group of rulings are
discrete as three resultants. They should be simplified further into one resultant in order to reduce the
complexity in the combing process. This section introduces how to achieve this simplification.
The same hypar in section 3.3 is utilized as an example. Because only supporting forces are studied
here, strut and tie model can be reduced to the simplest case: self-weight of the whole hypar is
represented by only one concentrated load applied on point O. This concentrated load can be split into
three components, which are respectively parallel to the rise r and middle ruling h, i. The components
along rulings can be directly balanced with supporting forces parallel to ruling h and i, the other
component parallel to rise r can also be balanced with the supporting forces along the four edges. At
the end, all the supporting forces in this model are combined into six resultants: four along the edges,
and two along the middle rulings (fig.11).
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
To verify such simplification, this paragraph proves that supporting forces in the discrete model with
one concentrated load (fig.11) is statically equivalent to the one with four concentrated loads (fig.10).
Figure 12 shows the difference between supporting forces of these two models in figure 10 and 11.
According to Extended Graphic Statics (Schrems and Kotnik [12]), the resultant of these differential
forces in figure 12 can be proved as zero.
Figure 11 (left): Simplified model for a general hypar’s supporting force (a), force polygons of
supporting forces (b). Figure 12 (right): The resultant of differential supporting forces between models
in figure 10 and figure 11 can be proved as zero.
According to this simplest discrete model, a hypar under self-weight generally needs to be supported
continuously by at most two edges, which provide supporting forces in the same directions of edges
and rulings. Only in some special cases, when the rise is parallel to the gravity, it only needs
supporting forces along edges. When the gravity is parallel to the plan defined by two middle rulings,
only supporting forces along rulings are necessary.
4. Implementation of a digital tool
The method described in section 3 is implemented as a Grasshopper function with python scripts in
Rhino 5.0. By defining four non-coplanar points in space, this tool can construct the discrete model
and force-polygons of a hypar under self-weight and supported with two edges. It represents both
internal forces and supporting forces of a hypar and visualizes the interplay between geometry and
forces flow of hypars. With help of this tool, designers can easily find out behaviors of each individual
hypar and understand how adjacent hypars structurally interact with each other in the combination.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Figure 13. Grasshopper function for hypars’ internal forces and six resultants of supporting forces.
5. The strategy for combinations of hypars
This section suggests the possible strategy for combing hypars together by referring to the structural
data obtained with the method in section 3.
In most built projects, forms combined from hypars are always stiffened by beams along the edges.
This is a simple and effective method to resist forces accumulating there. In fact, it is possible to
reduce the accumulation of forces and avoid these stiffening beams by properly combining hypars
together.
According to the analysis in section 3.4, resultants of a hypar’s supporting forces are always along
edges and middle rulings. When two hypars are connected, there are supporting forces along three
directions intersecting at one node of the shared edge. To achieve the equilibrium at the edge, these
supporting forces in three directions should be coplanar. This means the shared edge and middle
rulings of adjacent hypars should intersect at the same node, and should also be coplanar (fig.14).
Following this principle, the transition from one hypar to the adjacent one is always smooth, so there
is no folding line between them. Another advantage is that two adjacent hypars always need
supporting forces along the shared edge in the opposite direction, therefore they can be partly or
totally banlanced with each other. In this case, it effectively reduces the amount of forces
accumulating at the shared edge. In the direction of rulings, the accumulating forces can be finally
taken by the supports or cancelled by forces accumulating along other rulings. For example, in the
combination of five hypars (fig.14), the supporting force along middle ruling ia of hypar A can be
combined with a component of the supporting force along the shared edge, then change the direction
into ruling io of hypar O. In the same way, supporting forces along rulings hb, ic and hd can change into
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
the direction of ho or io. By refining the geometrical shape and position of each hypar, these forces
could be balanced at point O. Following the combining logic mentioned above, figure 15 shows a
design proposal for a canopy combined smoothly from 14 hypars.
Figure 14: Smooth combination of hypars, supporting forces along shared edges of adjacent hypars
can be balanced by each other, and the accumulation of forces happens along rulings. Figure 15: the
design proposal for a canopy based on the smooth combination of hypars.
6. Conclusion and future work
The final objective of this research is to develop an operative architectural and structural design
method based on combinations of hypars. Currently the research mainly focuses on individual hypars.
The analysis of their internal forces and supporting forces is developed as an easily understandable
graphical method, and implemented as a digital tool. The research presented in this paper builts the
foundation of the author’s PhD project. It prepares the geometrical data and information of force-
flows for the next step development.
In the future work, we are very interested in investigating the combination of hypars both from
architectural and structural perspectives. Based on the combination strategy proposed in section 5, an
optimization algorithm will be developed in the future to help designers improve design proposals
based on smooth combinations of hypars. The structural and geometrical data of individual hypars in a
smooth combination will be utilized as the starting point in a loop operation, which proceeds until the
global equilibrium of the whole combination is achieved.
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Acknowledgement
We would like to thank Pierluigi D’Acunto from ETH Zurich for his valuable suggestions both on the
research and the paper.
References
[1] Bechthold M, Innovative surface structures-Technologies and applications, Taylor&Francis,
2008
[2] Candela F., General formulas for membrane stresses in hyperbolic paraboloidical shells. Journal
of the American concrete institute, 1960, 353-371
[3] Candela F., Structural application of hyperbolic paraboloidical shells. Journal of the American
concrete institute, 1955, 397-415
[4] Enrique X. de Anda Alanis, Felix Candela, 1910-1997 the mastering of boundaries, TASCHEN
GmbH, 2008
[5] Flory S., Nagai Y., Isvoranu F. and Pottmann H., Ruled Free Forms. Advances in Architectural
Geometry 2012, L. Hesselgren et al. (eds.), Springer-Verlag/Wien, 2013, 57-66
[6] Kotnik T. and D’Acunto P., Operative Diagramatology: Structural Folding for Architectural
Design, Proceedings of Design Modelling Symposium 2013, 2013 193-203
[7] Muttoni A., The Art of Structures, Routledge, 2011
[8] Olmo C. and Chiorino C., Pier Luig Nervi: Architecture as Challenge, Silvana Editoriale, 2010
[9] Petit J., La Poeme Electronique de Le Corbusie, Le Collection Force Vivre, Editions du Minuit.
[10] Pogacnik M., Technology as Means of Expression in the Nineteenth Century--Architects and
Engineers in Dialogue. The Engineer and the Architect, Birkhauser GmbH, 2012
[11] Saint A., A study in Sibling Rivalry. Architect and Engineer, Yale University Press, 2007
[12] Schrems M. and Kotnik T., On the Extension of Graphic Statics into the Third
Dimension, Proceedings of the Second International Conference on Structures and Architecture,
Guimaraes, 2013, 1736-1742
