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## Transformable structures by the movement of rigid panels or structural segments

As mentioned in the introduction, first designs for retractable covering of sport stadiums stem from the crane technology. F. Otto classified these convertible roofs by a movement matrix (Fig. 2.4). different directions. The panels can overlap while retracting or move independently. The first retractable dome structure is said to be the circularly sliding retractable roof of the Pittsburgh Civic Arena (Fig. 2.5) opened in 1961 and closed in 2010. The 127 m-span roof consists of eight 300 ton sections, six of which are able to rotate by five motors per panel. All panels are fixed on the top to a gigantic, 80 m tall steel truss cantilever. The roof could be opened in about two minutes [Ishii, 2000]. [Helvenstone, 1959] The structural form of the civic arena is initially optimal as bending moments are minimal due to geometry. Unfortunately for retractability this optimal shape had to be sliced in parts, thus the cost was the huge cantilever that supports the panels, and the bigger structural height. A similar geometry was achieved by a more recent construction that did not apply an external structure to hold the panels. The Fukuoka stadium in Japan (Fig. 2.6) opened in 1993 frameworks, with remarkable bending moments. Though careful shape correction was performed for the geometry of individual parts (Fig. 2.7a) to avoid singularities in reaction forces at the inclination lines [Ishii, 2000], the structural height is still gigantic. Each panel is four meters thick, and the total roof weighs 12 000 tons. The sliding rotation of the two panels is enabled by 24 bogie wheel assemblies (Fig. 2.7b-c). It takes approximately 20 minutes to open the roof.

### Transformable structures by pantographic systems

A large number of structures that can be opened and closed are based on the well- known concept of the lazy tong system. The minimum component of this system is the so- called scissor-like element (henceforth SLE). The SLE consists of two bars connected to each other with a revolute joint. By the parallel connection of SLEs the simplest 2D deployable structure, the lazy tong is constructed. Connecting at least three of SLEs through complete pin joints a ring is formed, providing a secondary unit of this frame structure (Figs 2.17a-d). By the further connection of secondary units almost all kind of 3D-shapes can be formed folding into bundle (Fig. 2.17e-h). Adding tension components like wire or membrane to its developed form, it becomes a 3D-truss and gets effective strength, thus towers, bridges, domes and space structures can be rapidly constructed [Atake, 1995].

#### Deployable structures folding into a bundle

Using scissor-like deployable structures for architecture was pioneered by the Spanish engineer, E. P. Piñero. He presented a foldable theatre (Figs. 2.18-9) in 1961 [Piñero, 1961], and elaborated several other deployable designs. The biggest drawbacks of his designs were the relatively heavy and big joints due to eccentric connections (Fig. 2.19) and necessary temporary support as the structure was stiffened by intermediate bars or tension elements that were added after the structure was deployed into the desired configuration [Gantes, 2010]. Despite of all the disadvantages of his structures, Piñero inspired several researchers. This was the case with Professor F. Escrig, who designed the 30 m×60 m deployable roof for a swimming pool in Seville [Escrig, 1996] (Fig. 2.20).

**Retractable pantograph structures**

The application of structures that can fold into bundle when continuous transformability needed could be difficult to get. The American engineer, C. Hoberman made a considerable advance in the design of retractable roof structures by the discovery of the simple angulated element [Hoberman, 1990, 1991]. By the refraction of the two straight rods of a single SLE the angulated element is formed (Fig. 2.25). This element is able to open and close while maintaining the end nodes on radial lines that subtend a constant angle [Pellegrino, 2001]. Considering a classic SLE (Fig. 3.9a) ― consisting of two identical straight rods, hinged together with a cylindrical joint at ― the relationship between , the angle subtended by the lines and defined by the endpoints of the rods (, and , C) and , the angle between the two rods (deployment angle).

**Other deployable lattice designs and mixed systems**

For architectural and special applications several deployable lattice systems were invented using ideas differing from the already mentioned pantographic or tensegrity systems and some use a mixture of the mentioned types. The categorization of these lattice systems is quite cumbersome as each invented system is very unique based on ingenious inventions [Pellegrino, 2001]. Herein only a few examples will be presented without scoping an exhaustive list.

For instance, the coilable mast system invented by Mauch [Webb and Mauch, 1969] is derived from the idea that any elastic rod can be pushed to a helical shape [Love, 1944] (cited by [Pellegrino, 2001]). His lattice column is deployed through compressing the longitudinal elastic bars (called longerons) into a helical deformed shape. In the deployed configuration the stiffness is reached by bracing bars (battens) perpendicular to the longerons, and diagonal prestressed cables (Fig. 2.39a).

**Table of contents :**

Preface and acknowledgement

Declaration

Abstract

Absztrakt

Contents

**1 Introduction **

**2 Architectural background: Transformable Structures in Civil Engineering and Architecture**

2.1 The history of architectural transformable structures

2.2 Transformable structures by the movement of rigid panels or structural segments

2.3 Transformable lattice structures

2.3.1 Transformable structures by pantographic systems

2.3.2 Deployable structures folding into a bundle

2.3.3 Retractable pantograph structures

2.3.4 Pantadome erection

2.4 Tensegrity structures

2.5 Other deployable lattice designs and mixed systems

2.6 Soft, membrane structures

2.7 Foldable membrane structures

2.8 Pneumatic structures

2.9 Pneumatic systems for the erection of spatial structures

2.9.1 Formwork for thin concrete shell structures

2.9.2 Erection of segmented concrete or ice domes

2.10 Summary of architectural background

**3 Analysis of a simplified planar model **

3.1 Aims and scope

3.2 Analytical investigation

3.2.1 Mechanical analysis of the basic segment

Kinematical equations

Equilibrium equation

Constitutive equation

Equilibrium path

Calculation of critical state – nonlinear instability problem

3.2.2 Mechanical analysis of multi-storey, ‘alternately stiffened’ structures

Tracing equilibrium paths: uniform and bifurcated packing

Snap-back behavior of bifurcated packing

Packing path with ‘post-packed phenomenon’

Stability analysis

Taking self-weight into account

3.2.3 Mechanical analysis of masts without intermediate stiffening

Tracing equilibrium paths

Stability analysis

3.3 Numerical analysis

3.3.1 Methodology

3.3.2 Software description

3.3.3 Simulation of the packing of the basic unit

3.3.4 Numerical analysis of ‘alternately stiffened’ multi-storey masts

Unrestricted simulation

Restriction of ‘post-packed phenomenon’

3.3.5 Numerical analysis of multi-storey masts without intermediate stiffening

Unrestricted simulation

Restriction of ‘post-packed phenomenon’

Investigation of packing pattern and the critical force

3.4 Summary of the investigation of simplified planar models

**4 Analysis of packing antiprismatic deployable lattice structures **

4.1 General characteristics

4.2 Mechanical characteristics – analytical investigation

4.2.1 Analysis of the basic unit

Kinematical equations

Equilibrium equation

Constitutive equation

Equilibrium path

Calculation of critical state

4.2.2 Analysis of ‘alternately stiffened’ multi-storey structure

4.2.3 Analysis of a ‘non-stiffened’ k-storey structure

4.3 Mechanical characteristics – numerical analysis

4.3.1 Numerical analysis of the basic unit, parameter analysis

Basic assumptions, numerical model

Parameter analysis

4.3.2 Numerical analysis of ‘alternately stiffened’ multi-storey structure

4.3.3 Numerical analysis of a ‘non-stiffened’ multi-storey structure

4.3.4 Dynamic analysis of antiprismatic structures

4.4 Summary of investigating antiprismatic deployable structures

**5 Realization― experiments by physical models, ideas for control and applications**

5.1 Physical models

5.2 Ideas for control and application

6 Summary, further research perspectives

6.1 Summary

1. Parametric analysis of basic antiprismatic and planar segment, proposal for

approximations of main mechanical parameters for preliminary design

2. Analysis of the complex alternately stiffened planar and antiprismatic structure

3. Non-smooth packing of the alternately stiffened structure

4. Analysis of non-stiffened antiprismatic structures

6.2 Further research perspectives

**Appendix A:** Static and kinematic determinacy of antiprismatic structures

**Appendix B:** Analysis of the snapping-through of a shallow truss

Quasi static analytic and finite element formulation for tracing equilibrium path and for finding critical points

Numerical examples

**Appendix C:** Analysis of the basic unit of a snap-through type pantographic deployable

structure

**Appendix D:** Energetical approach for the calculation of planar and antiprismativc selfdeployable

structures

D1. Calculation of the planar structure with energetic approach

D1.1 Analysis of basic segment

D1.2 Analysis of ‘alternately stiffened’ multi-storey structures

Equations

Methodology

D1.3 Analysis of ‘non-stiffened’ multi-storey structures

Equations

Methodology

D2. Calculation of the antiprismatic mast with energetic approach

D2.1 Analysis of basic segment

D2.2 Analysis of ‘alternately stiffened’ multi-storey structures

D2.3 Analysis of ‘non-stiffened’ multi-storey structures

Annex E: Choosing constitutive laws in small and large displacement domain

List of Figures

List of Tables

**List of References **