Advanced Timber Structures

Table of contents Introduction 1

6

Folded plate structures

13

2

Advanced architectural geometry

49

3

Active bending

91

4

Form-finding and mechanical investigations of active bended systems

125

5

Customized construction

189

The Research Laboratory IBOIS at the EPFL Lausanne

236

Picture credits

237

5

2. 2

T1

Iterative geometric design for architecture

T2

Ivo Stotz, Gilles Gouaty, and Yves Weinand

T3 T4

Fig. 1

This interdisciplinary research project is presented by a group of architects, mathematicians, and computer scientists who research new methods for the efficient realization of complex architectural forms.

The present work investigates methods of iterative geometric design inspired by the work of Michael Fielding Barnsley. Several iteratively constructed geometric figures will be discussed in order to introduce the notion of transformation-driven geometric design. The design method studied allows interaction with the design, forming affine transformations and generating discrete geometries.

Furthermore, the handling of specific constraints is discussed. Geometrical and topological constraints aim to facilitate the production of architectural free-form objects. A surface method based on vector sums is studied, allowing the design of free-form surfaces that are entirely composed of planar quadrilateral elements. The combination of the proposed surface method and transformation driven iterative design provides new form-finding possibilities while fulfilling a number of material and construction constraints. Finally, the findings are tested on a series of applications. The studied test scenarios aim to evaluate the advantages of discrete geometric design in terms of efficient integrated production of free-form architecture.

Keywords

architecture, applied discrete geometry, IFS, timber construction

58

1

Introduction

In order to present the geometric design method studied here, the mathematical background must first be clarified. Before explaining the principles of transformation-driven geometric design, a series of historic examples will be examined. This will introduce the reader to the methods of iterative geometric design. The relationship between the mathematical method of geometric surface design and the physically constructed building will be shown by examples in the second part of this presentation.

2 2.1

Mathematical background

Monster curves The Cantor set (Fig. 1), also called Cantor dust, is named after the German mathematician Georg Cantor. It describes a set of points that lie on a straight line. At the end of the 19th century, this figure attracted the attention of mathematicians because of its apparently contradictory properties. Cantor himself described it as a perfect set, which is not dense anywhere.1 Further properties, such as self-similarity, compactness, and discontinuity, were studied years later.

The geometric construction of the Cantor set can be explained as follows: Take a straight line segment, divide it into three parts of equal length and remove its middle third; divide again each of the resulting line segments and keep removing their middle thirds. If you repeat this for each of the new line segments, you will end up with the Cantor set.

The von Koch curve is one of the best-known fractal objects and among the first found. In 1904, the Swedish mathematician Helge von Koch described it for the first time. 2 The curve is constructed step by step.

Beginning from a straight line, a meandering curve with strange properties is created: – It does not possess a tangent, which means that it cannot be differentiated.

– The length of any of its sections is always infinite.

The geometric construction of the von Koch curve is iterative, where each of the construction steps consists of four affine geometric transformations. The primitive is a section of a straight line, which is scaled, rotated, and displaced by each of the transformations {T1…T4}. Four duplicates are generated per construction step and each of these will, in turn, produce four more duplicates in the following construction step (Fig. 2).

2.2

Iterative geometric figures The peculiar properties of the aforementioned objects led mathematicians to name them “monster curves.”

In 1981, Barnsley defined a formalism based on Hutchinson’s operator 3 that was able to describe objects such as these in a deterministic way. 4 His IFS-method (see section 2.3) consists of a set of contracting functions that are

Fig. 2

Fig. 1

Cantor Set

Fig. 2

Von Koch Curve

Example of a free-form object designed using the presaged surface method

Fig. 3

applied iteratively. In our case, a function is an affine geometric transformation. Iterative means that the construction is done step by step. The input of a construction step is the result of the previous step.

What is really new in Barnsley’s work is that the resulting geometric figures are not defined by the primitive used, but rather by its transformations. As shown in fig. 4, the construction of a Sierpinski triangle may use a fish as a primitive. Analogous to this, the von Koch curve might be constructed on the basis of the letter “A.” The end result remains exactly the same.

Fig. 3

Advanced architectural geometry

59

Weinand / Advanced Timber Structures 978-3-0356-0561-7 December 2016 www.birkhauser.com

Fig. 5

Fig. 6

Fig. 7

Fig. 8

7.1

7.3

7.4

Intermediate-scale prototype of three-directional timberfabric, configuration study one, photographed near completion

Fig. 5

7

Fig. 6

Connection of modules in longitudinal direction

Fig. 7

Connection of modules in lateral direction

Fig. 8

Connection of modules in radial direction

Three-directional timberfabric

The previous section showed how multiple modules and their basic components can be combined into oneand two-directional timberfabric assemblies. For twodirectional assemblies, two modes of aligning laterally adjacent modules have been identified with regard to potential additional connector elements in a lateral direction. In three-directional timberfabrics, these principles are taken to the next level by superposing and connecting two layers of two-directional timberfabric of identical or different construction. To make this possible, the exterior layer needs to be based on modules with a larger radius or apothem. Several other factors that need to be taken into account are discussed below.

112

Offset between layers In all variants except one, the superposed layers circumscribe a cylindrical segment and have a common central axis (that runs in the x-direction) while the exterior layer is perpendicular to the interior layer. The offset between the two layers is achieved by increasing the size of the modules (the elements’ width is neglected here) that are designated to be on the exterior layer.

Maximum offset In principle, there is no upper limit for the offset distance between two layers. One could even argue that a greater distance is desirable as it results in a higher structure overall. On the other hand, the combined height of the two layers is only effective if they work as an entity, which in turn depends on the connections between them. Among other considerations, these connections have to be moment-resisting, in order to prevent relative displacement of the layers in the span direction in the case of horizontal loads, as such a displacement lowers the overall rigidity of the structural system. Greater distance leads to greater leverage and hence also increases the demands on the connections’ resistance. Furthermore, increasing the distance also has a visual impact: it decreases the visual coherence of the two layers, which to a certain degree can be understood as analogous to the structural logic.

7.2

Minimum offset Technically, the value of the offset can be freely chosen. However, several aspects have to be considered in the process. If the offset is too small, the units cannot be rotated about each other, as this would cause a collision between the low-lying extremities of the upper module and the prominent center of the lower module. As a selfimposed rule of design, the offset has to be large enough to allow for two modules to be superposed while being mutually rotated against each other by half a module.

Configuration principle of vertically adjacent modules The character of the layers and their alignment relative to one another is defined with regard to their constituent modules. Five basic principles of alignment have been established and form the basis for the timberfabric configurations subsequently developed. The alignment principles are conceived in order to provide for connections between the low- and high-strength zones of the modules.

8

Timberfabric demonstrators

As part of the research, various configuration studies of three-directional timberfabric have been carried out, two of which are discussed below.

In this configuration, the layers are mutually rotated around O and shifted in the x-direction. The interior layer is built up of a series of identical arch-shaped

113

Active bending

4. 6

Braided structures: applying textile principles at an architectural scale Marielle Savoyat

Design

IBOIS—Laboratory for Timber Constructions/ EPFL, Swiss Federal Institute of Technology, Lausanne, Switzerland Prof. Yves Weinand and Dr. Markus Hudert (researcher)

Research and completion

2007–2013

Fig. 1

184

The research undertaken between 2007 and 2013 by Markus Hudert within IBOIS, the Laboratory for Timber Constructions at the Swiss Federal Institute of Techno­ logy, analyzes the use of textile techniques at an archi­ tectural scale. It soon became apparent that principles of knitting, braiding, and weaving offer great potential for varying structural possibilities when applied to the scale of architecture.

The common denominator between all of these textile techniques is one basic element: that of a thread interlaced with another thread. This starting principle can be transferred onto two interlaced planks of wood.

To put this concept into practice, a first prototype called a textile module was created, which demonstrated how the application of a textile technique, when combined with the properties of wooden material, could lead to a particularly efficient freestanding structure.

Axonometry of a braided arch

Fig. 2

Underview of the model

Form-finding and mechanical investigations of active bended systems

185

Weinand / Advanced Timber Structures 978-3-0356-0561-7 December 2016 www.birkhauser.com