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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.

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).

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.

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.

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).

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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.

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

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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.

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Acknowledgement

We would like to thank Pierluigi D’Acunto from ETH Zurich for his valuable suggestions both on the

research and the paper.

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