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Design modelling has benefited from computation but in most projects to date there is still a strong division between computational design and simulation leading up to construction and the completed building that is cut off from the computational design modelling. The Design Modelling Symposium Berlin 2013 would like to challenge the participants to reflect on the possibility of computational systems that bridge design phase and occupancy of buildings. This rethinking of the designed artifact beyond its physical has had profound effects on other industries already. How does it affect architecture and engineering? At the scale of engineering and building systems new perspectives may open up by engaging built form as a continuous prototype, which can track and respond during use and serve as a real world implementation of its design model. This has been tried many times from intelligent façades to smart homes and networked grids but much of it was only technology driven and not approached from a more holistic design perspective.

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Design Modelling Symposium Berlin, 28/09 - 02/10/2013

Advisory Committee

Christoph Gengnagel, UdK Berlin

Axel Kilian, Princeton University

Julien Nembrini, UdK Berlin

Norbert Palz, UdK Berlin

Fabian Scheurer, designtoproduction, Zurich

Ioannis Zonitsas, Visual-Dream, Berlin

Organising Committee

Prof. Dr.-Ing. Christoph Gengnagel, UdK Berlin

Prof. Norbert Palz, UdK Berlin

Dagmar Rumpenhorst-Zonitsas, Daglicious Coordination

Scientific Committee

Sigrid Adriaenssens, Princeton University

Marc Alexa, TU Berlin

Jussi Ängeslevä, UdK Berlin

Olivier Bavarel, UR Navier, Université Paris-Est

Philippe Block, ETH Zürich

Alexander Bobenko, TU Berlin

Peter von Bülow, University of Michigan

Neil Burford, University of Dundee

Jeroen Coenders, TU Delft

Christian Derix, Aedas London

Günther Filz, Universität Innsbruck

Al Fisher, University of Bath

Christoph Gengnagel, UdK Berlin

Michael Hensel, Oslo School of Architecture and Design

Johann Habakuk Israel, Fraunhofer IPK Berlin

Axel Kilian, Princeton University

Toni Kotnik, ETH Zürich

Oliver Tessmann, KTH, Stockholm

Stefan Peters, TU Graz

Julian Lienhard, Universität Stuttgart

Julien Nembrini, UdK Berlin

Norbert Palz, UdK Berlin

Fabian Scheurer, designtoproduction Zurich

Volker Schmid, TU Berlin

Paul Shepherd, University of Bath

Martin Tamke, CITA Copenhagen

Florian Förster, Buro Happold Berlin

Roland Wüchner, TU München

Tobias Wallisser, ABK Stuttgart, LAVA Berlin

Main sponsor

Co-Sponsors

Contents

Foreword

Structural Design

Physical and Numerical Prototyping for Integrated Bending and Form-Active Textile Hybrid Structures

From Shape to Shell: A Design Tool to Materialise FreeForm Shapes Using Gridshell Structures

Designing Regular and Irregular Elastic Gridshells by Six DOF Dynamic Relaxation

Shaping Structural Systems

Bridging the Gap

Funicular Funnel Shells

From Structural Purity to Site Specificity New Canopies for the Entrance Gates of the Messe Frankfurt

Architectural Design

DesignScript: A Learning Environment for Design Computation

Frequencies of Wood – Designing in Abstract Domains

Enhancing Free-Form Architecture with Conical Panels

Embodied Prototypes: The Interaction of Material System and Environment

Combined Self-Organising Systems for Spatial Net Structures

A Framework for Flexible Search and Optimisation in Parametric Design

Operative Diagramatology: Structural Folding for Architectural Design

The Materiability Research

Ascending Curve - Digital Realisation of Shanghai Tower

MIKADOweb – Innovative Light-Weight Structure

Fuzzy Modelling with Self-Organising Maps

ALIVE - Designing with Aggregate Behaviour in Self-Aware Systems

Integrated Design Methods for the Simulation of Fibre-Based Structures

Complex Geometry

Behavioural Prototyping: An Approach to Agent-Based Computational Design Driven by Fabrication Characteristics and Material Constraints

From Generic to Specific - Prototyping a Computational Growth Model

Linear Folded V-shaped Stripes

Free-Form Shading and Lighting Systems from Planar Quads

Climate Design

Optimisation of the Building Skin Geometry to Maximise Solar Energy Collection

Analysing The Performance-Based Computational Design Process: A Data Study

Climate-Specific Mass-Customisation of Low-Technology Architecture as Part of a High-Technology Process

Digital Fabrication

Examples for Tool Integration in Design Concepts and Production Methods of Load Carrying Structures

Unlocking Robotic Design

Autonomous Tectonics - A Research into Emergent Robotic Construction Methods

Material Products: How Data is Successfully Transformed into Real-World Objects

The Design Carport – Prototyping Matter

Sketch-Based Pipeline for Mass Customisation

Prototyping

Prototyping Robotic Production: Development of Elastically-Bent Wood-Plate Morphologies with Curved Finger-Joint Seams

Design and Manufacturing of Self-Supporting Folded Structures Using Incremental Sheet Forming

Architectural “Making” Modes in Relation to Prototype Notions The Stripe Pavilion: Progression from a Bespoke to a Parametric-Algorithmic Mode

Blended Prototyping Design for Mobile Applications

Design Workflows for Digitally Calibrated Heterogenous Building Elements

Sustainability-Open: Why Every Building Will Be Sustainable in the Future

Prototyping Helixator

Validation Framework for Urban Mobility Product-Service Systems by Smart Hybrid Prototyping

Porsche Pavilion - Designing the World‘s Largest Seamless Monocoque Shell

The Generator 2.0

A Process where Performance Drives the Physical Design

Gradient Grid - A Spring Mesh with Different Zones of Flexibility

Serial and Persistent Prototyping Addressing Architectural Acoustics

Author Index

List of Contributors

Foreword

Prototypical Models of Design

At this Symposium, we look forward to discussing the relationship between prototypes and models of design. The term prototype stands for an implemented design step rather than a trial run for mass production, as an extension to the thought and computational constructs that make up the model of design. The term models of design stands for the idea and all underlying abstractions and assumptions that define the design process.

The relation between model of design and prototype gains importance as our understanding and relating of material systems to their simulated abstract models improves and computation increasingly becomes embodied in physical constructs replacing complex mechanical assemblies with computational feedback and control.

In architecture, the mechanical complexity has usually been lower than in other engineering fields; but obviously much of architecture’s complexity lies in its cultural context and the human occupation due to its scale and the social density of the built environment. Buildings need to evolve due to their potential long lifespan and are essentially evolving prototypes of the initial design intent reflected in the design model. Bridging the gap between design abstraction during the design development and the operation of the built structure is an ongoing challenge. Inherent to the use of digital tools for design is a tension between using simulation and computational processes to develop robust physical constructs that work as physical assemblies but independent of their computational simulations, or whether to move the computational processes into the built form and further sophisticate the feedback and control cycles and adaptability of physical constructs. In other words, computational processes may be found at many levels whether implicitly as computationally crafted material behaviour and/or explicitly in the computational capabilities of construction elements.

In other engineering disciplines, one can see a fascinating trend where complex and large scale mechanical assemblies for mechanical control are replaced by simpler mechanics empowered by computational controls such as for instance in the case of the development of helicopters to quadcopters or windmills to autonomously flying power kites.

Architecture and engineering structures are obviously different from aerospace constructs in terms of development costs and impact on the physical environment, but similar effects may be achievable in enabling existing infrastructure and structure to operate beyond their initial design intent and capabilities. Already actuated structures responding computationally to live loads thus simpler or lighter than conventional ones are being developed and constructed. Even the average eco-building corresponds to the definition of a robot with complex control algorithms linking sensors to actuators. Imagining coordination and collaboration on a building-to-building scale as well as at the scale of cities, think of smart grids, is not inconceivable.

However fascinating, such developments implicitly entail further vulnerability to system failure. Structures losing their control capabilities may collapse; automatically-shaded Passivehaus buildings overheat and become non-liveable. Directly embedding complex computational processes in the architecture calls for a careful balance between system performance and robustness.

Actually, long-going efforts in autonomous robotics suggest achieving robustness through embedding non-digital computational capabilities in physical constructs by exploiting system dynamics and non-linearities. Control only then provides the additional performance delta that makes the system reach the prescribed efficiency. Models, meaning our abstract understanding and invention of such processes play a crucial role in the development of new ideas and increasingly so as we rely more and more on their implementations in digital form.

We hope this collection of papers presents a range of insights at the cutting edge of the fields in addressing these questions and thank all participants for their contributions.

C. Gengnagel, University of the Arts, Berlin

A. Kilian, Princeton University, Princeton

J. Nembrini, University of the Arts, Berlin

Physical and Numerical Prototyping for Integrated Bending and Form-Active Textile Hybrid Structures

Sean Ahlquist, Julian Lienhardt, Jan Knippers and Achim Menges

1 Introduction

This paper describes research for the development and implementation of a functionally and structurally intricate textile hybrid architecture, entitled M1, built in Monthoiron, France as part of the La Tour de l’Architecte complex. The term textile hybrid stands for the mutual exchange of structural action between bending- and form-active systems based on textile material behaviour. The implementation of such a structural logic is critical to this particular project as its presence is minimally impactful to the site, which houses a historically protected, and decrepit stone tower from the fifteenth century’s, as shown in Fig.1. The design by Leonardo da Vinci employed an innovative buttressing system to structure the tower without a significant foundation. The buttresses have since been scavenged from the site, though the M1 structure seeks a minimal footprint to protect areas where traces of the original buttressing structure still exist.

To explore the complexities for minimal site imposition, lightweight material deployment and spatial differentiation, a set of multi-scalar and multi-modal prototyping procedures are developed. In both physical and numerical simulation, data towards eventual full-scale implementation is cumulatively compiled and calibrated, interleaving aspects of topology, material specification, force distribution and geometry. This paper defines prototyping as the interplay between modes of design in physical form-finding, approximated simulation through spring-based methods, and finite element analysis to form, articulate and materialise the textile hybrid structure. A particular feature in the exchange between and within these modes of design is the consideration of geometric input as a critical variable in the form-finding of bending-active behaviour.

Sean Ahlquist

University of Michigan, Taubman College of Architecture and Urban Planning, Ann Arbor, USA

Julian Lienhardt, Jan Knippers

University of Stuttgart, Institute for Building Structures and Structural Design, Stuttgart, Germany

Achim Menges

University of Stuttgart, Institute for Computational Design, Stuttgart, Germany

Fig. 1 Stone Tower and M1 Textile Hybrid at La Tour de l’Architecte, Monthoiron, France (Photos and drawings provided by Christian Armbruster, 2011; Ahlquist and Lienhard, 2012)

2 Multi-Hierarchical Textile Hybrid

The M1 textile hybrid project is formed via a multi-hierarchical arrangement of glass-fibre reinforced polymer (GFRP) rods of varying cross-sectional dimensions, which are structurally integrated with Polyester PVC membranes and polyamide-based textiles. The primary structure, in Fig. 2a, is formed of a series of interleaved loops emerging from only three foundations at the boundary. The meta-scale bending-active structure morphs between gridshell-like moments and free-spans stabilized by the tensile membranes. A secondary system (Fig. 2b) provides additional support through a series of interconnected cells embedded within the longest spanning region of the structure. Working to disintegrate the homogeneous nature of the textile membrane, the cells are differentiated in their form and orientation. The levels of hierarchy coalesce to form a clear span of up to eight meters with a total structure weighing only 60kg, while simultaneously generating variation in all scales of the spatial architecture.

Such articulation in behaviour and geometry is arrived at through an intricate exchange between various modes of form-finding. While the form-finding of tensile membrane structures considers stress harmoniously as an input variable, the form-finding of bending-active structures commonly results in varying stress distributions based on a comparatively large number of geometric and mechanical input variables. Therefore, the process of form-finding in the development of bending-active and, furthermore, textile hybrid structures eschews the consideration of structural optimisation. Aligning all input variables to form a functioning equilibrium, which satisfies both mechanical behaviour and contextual constraints, becomes the challenge within the form-finding processes and overall design framework. Due to this unique combination of freedom and complexity, it is shown through this research that a single computational technique alone does not offer the necessary flexibility and insight for developing textile hybrid structures. Rather, the combination and integration of multiple modes and techniques of design into a structured framework is shown to be necessary for the exploration and rationalization of complex textile hybrid structures.

Fig. 2 Multi-hierarchical textile hybrid system (Ahlquist and Lienhard, 2012)

These modes of design, in prototyping through form-finding, include physical models, spring-based computational studies and finite element analysis. Via physical experiments, specifications of topology and approximations of geometry are derived. Through spring-based modelling, also referred to as mass-spring methods or particle systems, variation is generated in the interactions between bending resistance and tensile forces (Ahlquist et al. 2013). In finite element analysis, fixed topological arrangements are inputs for exploration of specific mechanical relationships, force equilibria and further structural investigations (Lienhard et al. 2012). Each avenue serves to advance and articulate design aspects of the textile hybrid while also establishing the degree of fidelity towards the overall design framework.

3 Prototyping Framework for Textile Hybrid Systems

For designing a system formed of structural action, it can be decomposed into parameters of topology, structural forces, and materiality. Fig. 3 unravels these groups of parameters, as they would be addressed within a spring-based modelling and simulation environment. Topology specifies the count, type and associations of all elements within the system. Force describes the primary internal stresses, which the system will undergo, in this case tensile, compressive and bending actions. Materiality defines input parameters relevant to a material’s structural performance, while also translating values for computational or scaled behaviour into specific material definitions for fabrication and assembly. By distinguishing these parameters, particular relationships can be explored and exploited in their influence to material behaviour, as it forms force-active spatial architectures (Ahlquist and Menges 2011). This research describes the relationship between these aspects of material behaviour and relevant modes of design in physical form-finding, spring-based numerical methods, and simulation using finite element analysis.

3.1 Physical Form-Finding and Computational Means

While physical form-finding provides agile means for studying relationships of materiality and structural action within a single model, there is a limitation for any such study to predict behaviour beyond its own specific arrangement and scale. With a homogeneous material description, bending-active behaviour is generally scale-able as long as the topological input is repeated (Levien 2009). To establish a vehicle for design search, an individual study must serve as a prototypical case, projecting a design space, which implies a new vocabulary for form, performance and generative means (Coyne 1990). When integrating textile behaviour into a bending-active system, the extensibility of any one prototypical constructional model becomes further limited as the structural and spatial performance of the textile shifts greatly between scales.

While the physical prototype projects a narrow set of parametric rules and material descriptions, it can be a resource in defining fundamental logics of topology, proportion and behaviour, for further computational exploration. In this research, computational explorations occur through two venues: modelling and simulation of relative material descriptions with spring-based numerical methods, and finite element analysis defining precise mechanical (material and force) relationships. Spring-based methods calculate force based upon linear elastic stress-strain relationships (Hooke’s Law of Elasticity); using a numerical integration method such as Euler or Runge-Kutta to approximate the equilibrium of multiple interconnected springs (Kilian and Oschendorf 2005). Such methods are deployed to primarily explore varied relationships between topology and force. Both conditions are easily manipulable during the process of spring-based form-finding, enabling immediacy for feedback and ability to extract how minute manipulations affect the overall system behaviour. A fundamental layer of this research is the continued development of a modelling environment, programmed in Processing (Java) with a particle-spring library, allowing for complex topologies and force descriptions to be initially generated then actively re-modelled through an interface.

Fig. 3 Decomposition of material behaviour for spring-based modelling and simulation (Ahlquist 2013)

Finite element methods (FEM), on the other hand, contribute to forming complex equilibrium structures in defining the complete mechanical behaviour of the system. The given necessity for simulation of large elastic deformations in order to form-find bending-active structures poses no problem to modern nonlinear finite element analysis (Fertis 2006). However, software using FEM does not serve well as an expansive design environment, specifically for textile hybrid systems due to the inability to manipulate geometry and behaviour during the form-finding process. This necessitates the input data, for the pre-processing of the simulation, to be based upon the unrolled geometry of either physical form-finding or a computational environment such as the spring-based methods described above. Though the advantage, as well as necessity, of FEM in the development of textile hybrids, lies in the possibility of a complete mechanical description of the system. Provided that form-finding solvers are included in the software, the possibility of freely combining shell, beam, cable, coupling and spring elements, enables FEM to simulate the exact physical properties of the system in an uninterrupted mechanical description. Such means allows the FEM environment to accomplish, in a single model, the complete scope of form-finding, analysis of performance under external loads, and finally, the unrolling and patterning for fabrication.

4 Cellular Structure: Exploring Topological and Geometric Variation

Where bending action is triggered in a material system, certain geometric values become necessary inputs to the form-finding process. In simple terms, the length of the bending-active elements must be stated prior to the initiation of the form-finding process. In physical form-finding, geometry is inextricable from topology. The components in their count, type, and associations, carry with them their material properties. This introduces a helpful constraint in managing the complexity of searching for states of form- and bending-active equilibria. In developing the cell strategy for the M1, the physical studies define a proportional geometric logic for the bending-active aspect of the system. The exact geometry of the multi-cell array is only realized when arranged within the interleaved macro-structure. As both, the region within the meta-structure and the proportional rules of the individual cell are three-dimensionally complex, the spring-based modelling environment is well suited to explore the variation of geometric inputs arranging the meso-scale cellular textile hybrid system.

4.1 Modelling and Active Manipulation of Material Behaviour

Within the range of linear elastic material behaviour underlying the spring-based methods, a single spring element may compute tension or compression, and, in a combined arrangement, also bending action. Bending stiffness is simulated by adding positional constraint to the nodes (particles) that form a linear element. Three commonly known methods for simulating this behaviour are crossover, vector position and vector normal (Provot 1995; Volino 2006; Adrianessens 2001). In modelling behaviour with springs, there is a unique consideration where certain springs define only a particular aspect of material behaviour such as shear or bending stiffness, while others simulate the totality of behaviour and display the resultant material form such as a surface geometry or linear bending element.

In defining the tensile surface of a textile hybrid system, a mesh of springs both simulates the tensile condition in warp, weft and shear behaviour, as well as defines the material surface. In simulating bending stiffness, a linear array of springs implies the material condition of an elastic element, but the springs simulating constraint at the nodes do not have any geometric representation, as shown in Fig. 4. The flexibility in which a spring may drastically shift behaviour, between tension and compression, along with how relationships of geometry and behaviour can be more gradually tuned has been implemented as the foundation of the modelling environment programmed in Processing (Java). The key capacity in this particular mode of design is how the characterisations of behaviour can be manipulated, in topology and force description, during the effort of form-finding allowing freedom to define behaviour of different material make-up and composition.

Fig. 4 Comparison of spring topology between simulating a surface and a linear element with bending stiffness (Ahlquist 2013)

4.2 Transferring Relational Logics from Physical Form-Finding to Computational Exploration

In the M1, the array of interconnected cells serve as secondary support to the overall structural system, while, more critically, providing a means for differentiating the spatial conditions underneath the primary membrane surface. The geometries of the bending rods are calibrated to act as stiffening struts spanning between the upper and lower level of the meta-scale bending-active network. Within the cellular structure, a series of tensioned textiles further stiffen the system and serve as the media for diffusing light. The fundamental relationships between the boundary condition for the cells, the structure of an individual cell and its relation to its neighbour are most readily represented in physical form-finding studies, as shown in Fig 5. Yet, due to the complexity of those combined conditions specifying the geometry, which successfully resolves all of those parameters and constraints, is more readily accomplished in the spring-based environment where active manipulation of local and global behaviours is possible.

The spring-based modelling environment in Processing exposes variables related to the simulation of bending stiffness. Using the vector position method, the ratio of stiffness in the springs defining the linear beam elements to the degrees of constraint in the nodes can be varied to express differing amounts overall stiffness and curvature, thus implying different material properties. The lengths of the linear beam elements are exposed locally and globally enabling for the acute management of bending-active behaviour when multiple elements interconnect. These two capacities allow initially simple topological and geometric arrangements to be formed into the complex relationships defined by the physical cell models and made suitable to the context of the interleaved bending-active structure, as shown in Fig. 6.

Fig. 5 Rules for bending-active cell structure (Ahlquist, 2012)

Fig. 6 Form-finding sequence for cells in spring-based modelling and simulation environment, programmed in Processing (Java) by Sean Ahlquist (Ahlquist, 2012)

5 Interleaving Structure: Developing Force Equilibria

The interleaving macro structure of the M1 exhibits how multiple modelling and simulation techniques can be used at various scales to develop an intricate structural system. The development of a bending-active system goes hand in hand with its form-finding which in contrast to membrane structures includes the consideration of a large number of geometric and material input variables. The instant feedback of mechanical behaviour possible with the construction of a physical model is indispensable in finding ways for shortcutting forces in an intricate equilibrium system. Holding an elastically bent element in your hands directly shows the spring back tendency of the system and thereby supplies direct feedback for the position and orientation of necessary constraints. When interlocking multiple elements in a physical simulation, the moment of overlap is malleable and easily adjustable. Therefore, complex but harmoniously stressed equilibrium systems may be readily found through methods in physical form-finding.

5.1 Resolving Geometry through Multiple Modes of Simulation

Such freedoms afforded in physical form-finding are not readily available in computational analysis. While the spring-based vector position method allows for the simulation of elastic bending on already curved elements, the input geometry for finite element analysis is required to be straight or planar in order for shape and residual bending stresses to be simulated accurately. The form-finding sequence shown in Fig. 7 shows the transformation of individual straight elements into a network of interconnected leaves. The resultant bending-active geometry is compared to the scaled physical model, which provides the initial topological input as shown in Fig. 8. The geometric difference measured in relative length, was found to be smaller than 3%. In the case of both the meta-scale interleaved structure and meso-scale cellular structure, the precedent for the computational explorations and analysis was established through a physically feasible system.

Fig. 7 Sequence of form-finding for bending-active structure of the M1 using FEM software Sofistik (Lienhard, 2012)

Fig. 8 Comparison between physical form-finding model and computational model in Sofistik (Photo by Ahlquist, 2013; Sofistik model by Lienhard, 2012)

5.2 Designing the Complete Mechanical Behaviour

For the M1, the importance of generating the complete mechanical behaviour was exhibited in defining the final geometry of the entire system. In physical form-finding and spring-based modelling the results are approximations due respectively to their scalar nature and the relative calculations of material behaviour. In this case, the behaviour of the forces in the tensile surfaces resolves the geometry for critical cantilever conditions. Several iterations are explored to define the geometry of the free-spanning edge beam condition, whose position is only realized in the exact equilibrium of bending stiffness in the boundary rod and tensile stress in the upper and lower membrane surfaces as shown in Fig. 9. While this is only a single feature within the textile hybrid system, it can be explored efficiently as the topology generation and form-finding process is automated as a programmed routine within the FE software Sofistik.

Fig. 9 Form-finding of the brow condition for M1, with various membrane pre-stress ratios (Lienhard, 2012)

Element length is a critical consideration not only for the effort of form generation but also for the construction of an architecture that relies upon continuous and integrated structural behaviour. In typical building structures the joining of elements is solved at crossing nodes or points where the momentum curve passes through zero. Though in bending active structures the beam elements pass through the nodes with continuous curvature as defined by bending stress. Adjoining elements at these moments is unfavourable. Rather, the locations of low bending curvature are targeted as the moments for adjoining elements. For the M1 this defined the location of crossing nodes and total length of elements, as shown in Fig. 10, in order to assure positioning the joints at the locations of smallest bending stress and, at the same time, maximizing individual element lengths.

Fig. 10 Topology map of GFRP rods for M1 (Ahlquist and Lienhard, 2012)

6 Conclusion

This research establishes the coordinated means by which aspects of material behaviour can be explored in forming complex textile hybrid structures. The critical consideration is in the priority of prototyping constructional and behavioural logics through physical form-finding. In the two cases between the meta- and meso-scale textile hybrid systems though there is a difference in the application of the physical prototype to further study. As applied to computational exploration through spring-based methods the prototype is referential to a series of topological, geometric and material descriptions. On the other hand, in furthering the design through FEM the initial physical prototype defines literal parameters of topology and geometry. The behaviour is then more accurately reformed by engaging real material values, internal pre-stresses and external forces.

Because of the complexities inherent in engaging material behaviour as a design agent, the architectures formed are often based upon repeating modules whose differentiation is shaped by a singular relation of material make-up to structural behaviour. With the development of the M1 a design framework is proposed, which allows for the development of a structurally continuous system that is based upon the alignment of multiple differentiated agents in material, force and geometric constraints.

Fig. 11 Textile Hybrid M1 at La Tour de l’Architecte in Monthoiron, France, 2012 (Ahlquist and Lienhard, 2012)

Acknowledgements

The research on bending-active structures was developed through a collaboration between the Institute for Computational Design (ICD) and the Institute for Building Structures and Structural Design (ITKE) at the University of Stuttgart. The research from the ITKE is supported within the funding directive BIONA by the German Federal Ministry of Education and Research. The student team for the M1 Project was Markus Bernhard, David Cappo, Celeste Clayton, Oliver Kaertkemeyer, Hannah Kramer, Andreas Schoenbrunner. Funding of the M1 Project was provided by DVA Stiftung, The Serge Ferrari Group, Esmery Caron Structures, and Studiengeld zurück University of Stuttgart.

References

Adrianessens, S.M.L.; Barnes, M.R., 2001: Tensegrity Spline Beam and Grid Shell Structures. Engineering Structures, 23 (2), pp. 29-36.

Ahlquist, S.; Menges, A., 2011: Behavior-Based Computational Design Methodologies – Integrative Processes for Force Defined Material Structures. In: Taron, J.; Parlac, V.; Kolarevic, B.; Johnson, J. (eds.): Proceedings of the 31st Annual Conference of the Association for Computer Aided Design in Architecture (ACADIA), Banff (Canada), pp. 82-89.

Ahlquist, S.; Lienhard, J.; Knippers, J.; Menges, A., 2013: Exploring Material Reciprocities for Textile-Hybrid Systems as Spatial Structures. In: Stacey, M. (ed.): Prototyping Architecture: The Conference Paper, London, February 2013, London, Building Centre Trust, pp. 187-210.

Coyne, R.D.; Rosenman, M.A.; Radford, A.D.; Balachandran, M.; Gero, J.S., 1990: Knowledge-Based Design Systems. Reading, Addison-Wesley Publishing Company.

Levien, R. L., 2009: From Spiral to Spline. Dissertation, University of California, Berkeley.

Kilian, A.; Ochsendorf, J., 2005: Particle-Spring Systems for Structural Form Finding. Journal of the International Association for Shell and Spatial Structures, 46 (148), pp. 77-84.

Fertis, D. G., 2006: Nonlinear Structural Engineering: With Unique Theories and Methods to Solve Effectively Complex Nonlinear Problems. Berlin Heidelberg, Springer.

Lienhard, J., Ahlquist, S., Knippers, J., and Menges, A., 2012: Extending the Functional and Formal Vocabulary of Tensile Membrane Structures through the Interaction with Bending-Active Elements. In: [RE]THINKING Lightweight Structures, Proceedings of Tensinet Symposium, Istanbul, May 2013. (Accepted, awaiting publication).

Provot, X., 1995: Deformation Constraints in a Mass-Spring Model to Describe Rigid Cloth Behavior. In: Graphics Interface 95, Quebec, pp. 147-154.

Volino, P.; Magnenat-Thalmann, N., 2006: Simple Linear Bending Stiffness in Particle Systems. In: Cani, M.P. and O’Brien, J. (eds.): Proceedings for Eurographics/ACM Siggraph Symposium on Computer Animation, Vienna. Aire-la-Ville: Eurographics Association, pp. 101-105.

From Shape to Shell: A Design Tool to Materialise FreeForm Shapes Using Gridshell Structures

Lionel du Peloux, Olivier Baverel, Jean-François Caron and Frederic Tayeb

Abstract This paper introduces and explains the design process of a gridshell in composite materials built in Paris in 2011 for the festival Soliday. A brief introduction presents the structural concept and the erection methodology employed. It explains why composite materials are relevant for such applications. Following this practical case, the whole process from 3D shape to real-shell is then detailed. Firstly, the shape is rationalized and optimized to smooth local curvature concentrations. Secondly, a specific computing tool is used to mesh the surface according to the compass method. This tool allows designers to look for optimal mesh orientations regarding the elements curvature. Finally, a full structural analysis is performed to find the relaxed shape of the grid and check its stability, strength and stiffness under loads. The authors conclude on the overall relevance of such structures.

Olivier Baverel

UR Navier, Ecole des Ponts ParisTech, Champs-sur-Marne, France

ENSAG, Grenoble, France

Jean-François Caron, Frederic Tayeb

UR Navier, Ecole des Ponts ParisTech, Champs-sur-Marne, France

Lionel du Peloux

UR Navier, Ecole des Ponts ParisTech, Champs-sur-Marne, France

T/E/S/S, Paris, France

1 Introduction

The emergence of gridshell structures – intensively studied by the German architect Frei Otto – is a major step in the development of complex shapes in AEC (Architecture, Engineering and Construction). Since the 1970s this structural concept has led to emblematic realizations (Mannheim [Happold and Lidell 1975], Downland [Harris et al 2003], Savill, Hanovre [Ban 2006]). They have shown that beyond their architectural potential, gridshells are well suitable for complex shape materialization because of their intrinsic geometric rationality.

However, the very few number of gridshells constructed up to now attests that they are quite tricky to design compared to standard buildings. Architects and engineers would face both demanding conceptual knowledge in 3D geometry, form-finding techniques, non-linear behaviour, large-scale deformations, permanent bending stresses, etc. and real lack of tools dedicated to their design.

This paper presents a computing tool based on Rhinoceros & Grasshopper that aims at meshing NURBS surfaces with the compass method. This tool also includes a one-way interface for GSA (a structural analysis software from Oasys) to perform the structural analysis of the resulting grid. Thereby, this tool introduces shape-driven design of gridshells. Following a case study – the construction of the first composite gridshell to host people – a methodology to design these shape-driven structures is proposed. Finally, future prospects to their development are discussed.

1.1 Gridshell: Concept, Erection Process, Materials

Concept

A gridshell is a structure, which behaves like a shell but is made of a grid. Thus, the material is not spread continuously as shells, but it is organized in a discrete grid pattern. Like shells, gridshells derive their stiffness from their double curvature shape. These structures can cross large spans with very few materials. They offer a rich and voluble lexicon to express blob-shapes.

Erection Process

Usually, the grid morphology is not trivial and leads to design numerous costly and complex joints. To overcome this issue, an original and innovative erection process was developed that takes advantage of the flexibility inherent to slender elements.

A regular planar grid made of long continuous linear members is built on the ground (Fig. 1). The elements are pinned together so the grid has no in-plane shear stiffness. Thus, the grid can accommodate large-scale deformations during erection (Fig. 2). Then the grid is bent elastically to its final shape (Fig. 3). Finally, the grid is frozen in the desired shape with a third layer of bracing members (Fig. 4). The grid becomes a shell and the structure’s stiffness is multiplied by about 15.

Fig. 1 Regular grid on the ground

Fig. 2 Grid erection

Fig. 3 Erected grid

Fig. 4 Grid triangulation

Material Flexibility for Structural Rigidity

Composite materials like glass fibre reinforced polymer (GFRP) could favourably replace wood in this case where both resistance and bending ability of the material is sought. Thus, the structure’s stiffness derives from its geometric curvature and not from the material’s intrinsic rigidity. Moreover, using synthetic materials free us from the painful problematic of wood joining and wood durability (Douthe, Caron and Baverel 2010).

High Tech & Low Cost

Though gridshells require high-tech design techniques, they seem to be a low-cost way to materialise non-standard morphologies (200€/m2), because of their geometric rationality. The project complexity is shifted upstream.

1.2 From-Finding Versus Grid-Finding

One can identify two different ways of designing gridshells: those with a given outline and those with a given shape. The first approach considers the final shape a consequence of a form-finding process, driven by the supports of a grid which is thought to be an input data. The second approach consists of deriving a grid from a given shape. When erected on its supports, the grid should give back the intended morphology.

Form-Finding

A physical or numerical grid-model is handled until a structural shape is found, in compliance with the architectural intents. This way, Frei Otto designed the Multihalle of Mannheim using hanging funicular nets and photogrammetry (Otto and Hennicke 1974), see Figs. 5-6. Nowadays, this form-finding stage would probably be done by computer, relying on numerical methods such as dynamic relaxation or force density.

Fig. 5 Hanging net

Fig. 6 Resulting structure

An alternative method has been proposed by the Navier laboratory to achieve this form-finding stage by computer. Based on a dynamic relaxation algorithm which considers the elements bending stiffness, it leads to new shapes where free outlines express the grid natural stiffness (Douth, Baverel and Caron 2006 & 2007).

Grid-Finding

The compass method is used to develop the initial shape in a quadrangular mesh. Rebuilt on a plane, the mesh leads to a regular grid suitable to materialise the studied shape by a gridshell. An alternative method, taking into account the grid’s mechanical properties, was also proposed by the Navier Laboratory (Bouhay, Baverel and Caron 2009). This method uses explicit dynamic algorithms to pin an initially flat grid on a given shape, with a system of fictive forces.

2 Case Study

2.1 The Project

In June 2011, six students from École des Ponts ParisTech (French engineering school) supported by the Navier Laboratory gave birth to a structure for the association Solidarité Sida: a tent unlike any other, reminding blob architecture with its curved and rounded shape (Fig. 7).

Fig. 7 Photo of the final structure (http://vimeo.com/31341461)

Description

This structure of 300m² was the first gridshell in composite materials (GFRP) to receive an audience. It had to get a certificate of approval that involved administrative requirements and EUROCODE justifications, a first for such a structure. Beyond the technical performance, this large scale project designed to house up to 500 people at a time has shown the economic relevance of this concept. It became reality thanks to key partnerships including T/E/S/S and Viry.

Solidays Festival

Each year in June, Solidarité Sida organizes the Solidays Festival. It is a music festival that attracted around 160,000 people during four days and raised about 1.7 million Euros in 2011. It is also a forum. Its purpose is to raise awareness about AIDS and raise funds for medical research and outreach initiatives. The gridshell structure was designed to house the forum during the festival.

2.2 Design Process

The opposite diagram summarizes the design process from the initial architectural intent (a 2D sketch) to the final gridshell. Each step is then detailed.

2.3 3D Modelling

From the initial architectural intent we have modelled the shape as a wireframe (Fig. 9) including the outline and some sections. With those curves we can control the shape in-plane and drive its volumetry. The surface is then derived from the wireframe using a NURBS interpolation (Fig. 10). All along the design process adjustments were carried out on this surface until it reached the architectural and structural requirements. In this process, the initial surface was mostly deformed or sculpted by handling its control points.

Fig. 8 Step by step design process including different levels of structural control, based on curvatures and stresses checks (1,2,3)

Fig. 9 Wireframe geometry

Fig. 10 NURBS patch

2.4 Shape Optimisation

For now, we built a space sketch of the project. However, this NURBS surface has no reason to lead directly to a structural shape. Thus, the structural elements have to be checked to make sure they will support the stress field induced by grid shaping.

Stress & Curvature

Stresses in elements are mainly due to grid bending, that is to say geometric curvature imposes the grid stress state (Eq. 1). Thus, principal curvatures of the surface describing the shape are good indicators to evaluate if the structural members have the required mechanical properties. This preliminary control can be completed by an analysis of the curvature of the mesh elements. Finally, nothing but a true structural analysis considering members mechanical properties will allow us to find the exact relaxed shape and the stress field in the structure.

(1)

Where σ represents the total stress (compressive plus bending) induced by the shaping. E, v and I are respectively the longitudinal young modulus, the radius and the bending inertia of the profile. R is the radius of curvature of the profile.

Surface Optimisation

Before any attempt to mesh the shape, it is recommended to optimize the sketch shape regarding its minimal principal curvatures (Eq. 2). Using the curvature-analysis built-in function in Rhino it is easy to identify and smooth areas that are initially too curved (Fig. 11).

(2)

Different sketches are compared according to this criterion to smooth areas where curvature is excessive.

Sketch n°1

Sketch n°2

Sketch n°3

2.5 Shape Meshing

Following a decade of research on this topic at the Navier Laboratory, a specific tool has been developed on Rhino & Grasshopper for the design of such shape-driven gridshells (Fig. 12). This tool gathers several components that process basic operations (meshing with the compass method, grid-processing, structural analysis) required for the generation of a suitable grid for the materialization of a 3D shape by a gridshell structure.

Fig. 12 Grasshopper canvas (compass method, grid processing, structural model generation)

Compass Method

This process propagates a two-way mesh of constant pitch on any NURBS surface.

Fig. 13 Compass method principle

Two crossing guide-curves are drawn on the surface to mesh. These curves mark the boundary of four quarters. Each half guide-curve is then subdivided with a compass of constant distance w (the pitch). Finally, from two consecutive half guide-curves, quadrants are meshed with the same compass distance (Fig. 13).

Meshed Surface

The compass method does not allow meshing the entire meshing domain. Only a smaller part could be meshed and its area varies according the chosen set of guide-curves (Figs. 14-15). Thus, it is not possible to rely exclusively on the shape to be realized with the lattice. An extended surface - chosen carefully - has to be considered as the meshing domain.

Fig. 14 Two different meshes are obtained from two distinct sets of guide-curves. The meshed area never takes on the whole surface. Convergence phenomena could be observed (right picture).

Overall Process

To overcome this difficulty we propose a methodology, which relies both on the creation of a meshing domain (domainSrf) from the targeted surface to materialise (gsSrf) and on the identification of a suitable set of guide-curves.

Step One

We consider the gridshell surface (gsSrf) a part of a larger surface (domainSrf). Trimmed by a clipping plane or surface (cuttingSrf), this domain surface should give back the intended shape to build (Figs. 15-16).

Fig. 15 gsSrf

Fig. 16 domainSrf and cuttingSrf

Step Two

A set of guide-curves is chosen (Fig. 17) and the mesh is propagated on the domain surface according to the compass method (Fig. 18). The guide-curves have to be chosen so that the whole gridshell surface (gsSrf) is meshed. Several trials can be necessary to get a suitable mesh.

Fig. 17 Guide-curves set

Fig. 18 Resulting mesh on domainSrf

Step Three

The mesh is trimmed by the clipping surface (Fig. 19). The resulting mesh lays on the whole initial intended surface to mesh. The gridshell support-outline is given by the intersection of the clipping plane and the domain surface (Fig. 20).

Fig. 19 Trimmed mesh

Fig. 20 Final mesh and support outline

Mesh Optimisation

Since the form has been known and the procedure to mesh the surface is known now, an optimisation of the mesh can be performed (Fig. 21). The aim is to find a mesh that can mesh the entire surface and that creates acceptable stresses in the beams. To this end, the curvature of the elements is checked using the following equation (Eq. 3):

(3)

Different sets of guide-curves are chosen and the resulting meshes are compared according to this criterion:

Mesh n°1

Mesh n°2

Mesh n°3

Fig. 21 Mesh testing

Grid Processing

The generated mesh can be shaped in a matrix. This allows both its transformation in a planar grid of regular pitch (Fig. 22) and automatic definition of triangulation elements for bracing.

Fig. 22 Developed surface and derived grid.

Generation of Analysis Model

A procedure gathers and processes all the geometric information. It creates an import file for the automatic generation of an analysis model in GSA, a third-party software dedicated to structural analysis. Additional components assist the designer in the definition of complex load cases, such as non-uniform wind and snow loads, directly in Rhino & Grasshopper.

2.6 Structural Analysis

Once the structural model is built by our tools, it can be loaded in the structural analysis software to perform:

Computation of the permanent flexural stress and the relaxed shape using a dynamic relaxation algorithm (Fig. 23)

Loading analysis according to the Eurocode (self-weight, snow, wind ...)

Fig. 23 Final compass mesh and corresponding relaxed mesh (stress diagram).

3 Conclusion

This paper has presented the different steps for the design of a gridshell in composites materials built for the Solidays Festival in 2011 in Paris. The first step was the optimisation of the shape in order to avoid concentrations of curvature locally. The second step showed a tool to automatically mesh a surface using the compass method. With this tool the optimum orientation of the mesh is studied. The last step showed the details of the structural analysis of the gridshell. This construction demonstrated the technical feasibility and also the economical feasibility of the gridshell in composites materials.

Acknowledgements

The authors would like to thank the students E. Roux, E. Blache, J.-R. Nguyen, A. Grandi, G. Frambourt, T. Perarnaun, their university supervisor R. Mège and the association Solidarité Sida for their initiative and confidence. Special thanks also to T/E/S/S and Viry for their technical and financial support, which permitted this project to become real. Thanks also to all our partners who provided significant material assistance: Serge Ferrari, Top Glass & Solutions Composites, Owens Corning Reinforcement, DSM Resins, ENSG, Esmery Caron, Axmann, Chastagner and ¨Paris Voile.

References

Douthe, C; Baverel, O.; Caron, J.-F., 2007: Gridshell in Composite Materials: Towards Wide-Span Shelters. Journal of the I.A.S.S, Volume 48 Issue 155, pp. 175-180.

Douthe, C.; Baverel, O., Caron, J.-F., 2006: Form-Finding of a Gridshell in Composite Materials. Journal of the International Association for Shell and Spatial Structures, Volume 47, Issue 150, pp. 53-62.

Douthe, C.; Caron, J.-F.; Baverel, O., 2010: Gridshell Structures in Glass-Fibre Reinforced Polymers. Construction and Building Materials, Volume 24, Issue 9, pp. 1580-1589.

Bouhaya, L.; Baverel, O.; Caron, J.-F., 2009: Mapping Two-Way Continuous Elastic Grid on an Imposed Surface: Application to Gridshells. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposius, Valencia.

Otto, F.; Hennicke, J.; Matsushita, K., 1974. Gitterschalen Gridshells, Institut für Leichte Flächentragwerke. IL 10, p. 340.

Happold, E.; Lidell, W.I., 1975: Timber Lattice Roof for the Mannheim Bundesgartenschau. The Structural Engineer, Volume 53 Issue 3, pp. 99-135.

Harris, R.; Rohmer, J.; Kelly, O.; Johnson, S., 2003. Design and Construction of the Downland Gridshell. Building Research & Information, Volume 31, Issue 6, 2003, pp. 427-454.

Ban, S., 2006: The Japanese Pavilion. In: McQuais, M. (ed): Shigeru Ban, Phaedon Press, pp. 8-11.

Designing Regular and Irregular Elastic Gridshells by Six DOF Dynamic Relaxation

Jian-Min Li and Jan Knippers

1 Introduction

An elastic gridshell defined in this paper is a single-layer or double-layer shell structure that consists of initially straight and continuous members, has equal/regular or various/irregular grid lengths and allows scissoring movements in the joints during the erection process (Fig.1). Designing a grid shell of this type needs to fulfil many geometrical and mechanical constraints. The grid pattern needs to be as close to the target surface as possible. Meanwhile, important from material aspects, an ideal grid pattern should exhibit the lowest strain energy compared with other patterns.

Most of form-finding methods for elastic grid shells are based on simplified structural models, which consist of only three degrees of freedom (DOF) per node [1][2][3] such that bending and torsion effects cannot be accurately considered. We use dynamic relaxation method (DR) with six DOF per node and Bernoulli beam elements, which have 12 DOF each, to solve the optimised grid pattern, which fulfils the geometric constraint and exhibits the lowest strain energy [4]. One advantage of our scheme is that it can handle both the form-finding and loading analyses. The obtained form can be directly transformed to a bearing structure. There is no need to transfer the form-finding result to another finite-element solver.

To address the feature of bending-active elastic gridshells [5], we demonstrate a simple method to create pre-stress in grid shells such that we can start the form-finding or the loading analysis from a strained geometry. With this method, we are no longer restricted from beginning in an unstrained state and, thus, the tedious and tricky pre-stressing procedure for bending-active structures is prevented.

The projection method and constraint force method are integrated into our scheme. The projection method is applied to find grid patterns that exactly fulfil the geometric constraints, while the constraint force method allows the obtained forms to deviate from the target surfaces. The fitness of the obtained forms to the target surfaces is controlled by the magnitude of constraint forces. Smaller constraint forces, which result in less fitness, generate grid patterns that exhibit lower strain energy.

The profile stiffness can be used as an active factor in the form-finding process. The real stiffness (a profile stiffness of a practical profile that can be used in the bearing structure) is useful to explore the structural behaviour of elastic grids under geometric constraints. Besides, the real stiffness facilitates the form-finding of a grid pattern in accordance with the pre-determined grid lengths. Meanwhile, the fictitious stiffness (a profile stiffness that is designed on purpose and is only used in the form-finding stage) enables the grids to have a larger range of strained lengths, which facilitates the reduction of bending stresses and can be utilised to generate smoothly curved grid patterns with various grid lengths.

Jian-Min Li, Jan Knippers

Institute of Building Structures and Structural Design (ITKE), University of Stuttgart, Germany

Fig. 1 Structural model of a single-layer grid shell with equal grid lengths

The rest of the article is structured as follows: In Sec. 2, we describe the structural modelling of elastic beams. In Sec. 3, we illustrate how to apply geometric constraints. In Sec. 4, four examples are given. In the final section, our conclusions are provided.

2. Beam Modelling

Our beam model is visualised as two separated nodes/particles with an interval as the beam length and each node is assigned a mass term and an inertia term. The nodes will shift and rotate freely if no force or torque is exerted. Once the internal forces are applied, the nodal movements become coupled and the individual nodes will behave as an integral structure.

2.1. Dynamic Relaxation with Six DOF per Node

DR is an explicit time integration method [6], which is applied in our scheme to simulate the above-mentioned system. With the information of the state of the previous time step and the forces currently exerted on the system, we can calculate the state of the subsequent time step.

Translational motion is described by nodal positions and translational velocities, while rotational motion is described by nodal orientations and angular velocities. The rates of change of translational velocities and rotational velocities are proportional to the residuals of translational forces and torques exerted on nodes. The update formulations for translation motion and rotational motion are listed in Tab. 1. The derivation and the application of these formulations are illustrated in our previous work.

Table 1: Translation and rotation formulations of DR

Translational part

Rotational part

With proper damping, the kinetic energy is reduced to zero and a static state can be derived. We use two different damping methods: For the translational degrees of freedom we use kinetic damping, setting every translational velocity to zero once the translational kinetic energy of the system begins to decrease. For the rotational degrees of freedom we use velocity damping, multiplying every angular velocity by a factor less than one at each time step [7].

2.2. Beam Mechanism

To calculate the internal forces in beam elements, we need to define additional orientations of the beam-ends. In this paper, we consider only rigid joints. That means each beam end will rotate simultaneously with the corresponding node and maintain their relative difference in orientations. The Euler-Bernoulli beam element, which has 12 DOF each, is used to calculate internal forces in beams. The axial force, bending moments and torque are determined by the corresponding positions and orientations of the two beam-ends of the beam element. The formulations for calculating internal forces are listed as follows:

Where θ is the included angle that is defined by the orientation of the beam end and the axial direction of the beam. Its geometric definition is shown in Fig. 2.

2.3. Pre-stress of Bending and Torsion

For a rigid joint its corresponding orientations of the beam-ends, orientations of the beam-ends connected to that joint, maintain their relative difference in the transient phase. For a straight joint, the two corresponding orientations of the beam-ends coincide with each other throughout the process. Therefore, by assigning orientations as in Fig. 3a, we can assign the pre-stress of an initially straight joint. The assignment of an initially angled joint is shown in Fig. 3b.

Fig. 2 (a) Definitions of included angles, θa,z and θb,z ; (b) Definitions of included angles, θa,y and θb,y ; and (c) Definitions of the beam axial direction, .

2.3. Pre-stress of Bending and Torsion

For a rigid joint its corresponding orientations of the beam-ends, orientations of the beam-ends connected to that joint, maintain their relative difference in the transient phase. For a straight joint, the two corresponding orientations of the beam-ends coincide with each other throughout the process. Therefore, by assigning orientations as in Fig. 3a, we can assign the pre-stress of an initially straight joint. The assignment of an initially angled joint is shown in Fig. 3b.

Fig. 3 (a) In the area enclosed by the dashed line, the orientations of the consecutive beam ends are equivalent. This builds up the pre-stress of moments of an initially straight joint; (b) In the area enclosed by the dashed line, the orientations of the two consecutive beam ends are the same as its own beam orientation. Hence no pre-stress of moments and torsion is built.

2.4 Weighted Stiffness

In our scheme, the profile stiffness can be used as an active factor in the form-finding process. The real stiffness - a profile stiffness of a practical profile that can be used in the bearing structure - is useful to explore the structural behaviour of elastic grids under geometric constraints. Besides, if the grid lengths are pre-determined, the real stiffness is helpful to solve a grid pattern in accordance with these pre-determined grid lengths.

If minimising the bending strain energy is a more critical issue than maintaining grid lengths, the fictitious stiffness - a profile stiffness that is designed on purpose and is only used in the form-finding stage - is used, which is defined by multiplying the bending and torsional stiffness by a weighted factor lager than one and keeping the axial stiffness unchanged. The fictitious stiffness enables the grids to have a larger range of strained lengths, which is helpful to reduce curvatures of the bending elements. Besides, this character can be utilised to generate smoothly curved grid patterns with various grid lengths.

3 Geometric Constraint

In contrast with the implicit integration method the explicit method does not require a global stiffness matrix. The motion update is treated locally for each node. The nodal motion is only affected by the physical quantities around the node. As a result, DR is an ideal method to deal with local perturbations that are caused by physical contacts or geometric constraints, because the local instability will not cause the entire system to crash. The geometric constraint discussed in this paper is a curve or surface defined by the non-uniform rational basis spline (NURBS). If a structural node is constrained, its movement will be restricted within the constraint domain. The static state derived in this manner is a state that exhibits the lowest strain energy while satisfying all geometric constraints, which is crucial for combining material aspects with geometric design requirements.

3.1 Projection Method

By counting only tangential nodal residuals, a smooth tangential movement on the constraint surface/curve is achieved. The slight deviation caused by the tangential movement is adjusted by projecting the constrained node to the constraint at the end of each time step.

3.2 Constraint Force Method

We can also manipulate the form by applying constraint forces. A nodal constraint force may be defined by a vector which points toward the closest point on the geometric constraint and is proportional to the distance between the node and the point.

The magnitude of the constraint force is used to control the fitness of the obtained form to the target geometry. Smaller constraint forces result in a form that exhibits less strain energy but with the price of a larger deviation from the target geometry. An agreeable result may be attained by tuning the magnitude of the constraint forces.

4 Examples

Four examples are presented in this section. The first example shows a beam that is relaxed from a strained geometry. The second example demonstrates the form-finding process of a single-layer grid shell. The third example and the fourth example show the form-finding processes of double-layer grid structures.

4.1 Relaxed Beam

This example is used to verify the pre-stress assignment as proposed in Sec. 2. If the pre-stress is correct, the strained curved beam will resume a straight line as in Fig. 4. The diameter of the barrel and the length of the barrel are 20m and 40m respectively. The initial geometry of the bended beam is defined as a helix on the barrel. The beam is composed of 36 elements with a square profile that has a side length of 5cm. Each element has a length of 141cm. The elasticity and shear modulus are defined by E=107KN/m2 and G=E/2, respectively.

Once the beam is released, the motion is triggered by the residuals and the system is eventually damped to a static state. The computation terminates once the residual of each node is less than 10-4KN. The variation in strain energy throughout the process is shown in Fig. 5.

4.2. Weald and Downland Gridshell

The Downland Gridshell, which is charcaterised for its unique triple-bulb geometry, is our first form-finding example. The task is to find a single-layer grid pattern that complies with the material aspects and satisfies the geometric constraints.

Fig. 4 (a) Pre-stressed beam on a barrel surface; (b) Beam relaxed from a strained geometry

Fig. 5 Strain energy versus time step

The grid, which is composed of 102 initially straight planks, has a uniform square profile and a width of 5cm (Fig.6). Each beam element has an unstrained length of 1m. The connections between the crossing rods are revolute joints, which enable free rotation along the local z-direction. The elasticity and shear modulus of the material are defined as E=107KN/m2 and G=E/2, respectively.

Two geometrical constraints are given. First, the grid nodes on the two longer edges must stay on the curved boundaries. Second, the remaining grid nodes must remain on the triple-bulb surface.

Fig. 6 (a) Initial grid and geometric constraints; (b) Transient state; (c) Equilibrium state under constraint; (d) Bearing structure after constraints are removed and bracing and bearing conditions are added; the enlarged part shows the nodal orientations.

4.3. Effect of the Weighted Stiffness (Irregular Gridshell)

In this example, we apply the weighted stiffness in the form-finding stage to the above grid structure to see its effect on the grid patterns: A factor of 10E4 is applied to the bending and torsional stiffness to enable the beam element to have a larger range of the strained length. This fictitious stiffness setting is only used in the form-finding stage. Once the form is found, the material stiffness is recovered and the strained lengths of the found form are taken as the new unstrained grid lengths.

The comparison of the strain energy between the two forms, which are found with the real stiffness setting and the fictitious stiffness setting, is showed in Tab. 2, and the comparison of their geometries is showed in Fig. 7. The in-plane bending strain energy of the irregular grid shell is largely reduced, which is only 25% of the in-plane bending strain energy of the regular gridshell.

Table 2: Comparisons of the grid length and the strain energy between the regular grid and the irregular grid

max. length[m]

min. length[m]

strain energy

[kNm]

axial strain

energy

[kNm]

torsional

strain

energy [kNm]

out-of-plane

bending strain

energy [kNm]

in-plane be

ding strain

energy [kNm]

regular grid

irregular grid

1.0003

1.0602

0.9996

0.9308

9.5E+01

6.7E+01

5.2E-01

3.2E-03

2.5E+01

2.6E+01

3.6E+01

3.3E+01

3.2E+01

8.2E+00

4.4 2D Hybgrid

Hybgrid, which was proposed by Truco and Felipe [8], is an innovative double-layer structural type that is composed of uniform flexible chord members and can generate various geometries by controlling the strut lengths between the chords (Fig. 8). Currently, the only available form-finding for Hybgrid is based on physical models. Thus, the testing of our method constitutes a significant benchmark.

The grid is composed of three chords: the upper chord, the middle chord and the lower chord. Every chord is composed of 40 beam elements; each contains an unstressed length of 12.88cm. The connections between the chords consist of revolute joints. Each chord has a uniform rectangular profile that exhibits a width of 80mm and a thickness of 4mm. The elasticity and shear modulus are defined as E=107KN/m2 and G=E/2, respectively.

Fig.7 Comparison of the grid patterns of the irregular grid and the regular grid, which are presented by the orange colour and the purple colour respectively. Both grid patterns are smooth, but the irregular grid has less in-plane bending curvatures.

Fig. 8 (a) Initial geometry; (b) Transient state; (c) Transient state; (d) Equilibrium state under constraint; (e) Bearing structure after boundaries are removed and struts and supporting conditions are added.

Fig.9. (a) Initial grid; (b) Transient state; (c) Transient state; (d) Equilibrium state under constraint; (e) Bearing structure after constraints are removed and struts and supporting conditions are added; (f) The enlarged part shows the nodal orientations.

The upper chord nodes that will be subsequently connected with struts after form-finding are constrained by the upper curve, only movable on the curve. Similarly, the lower chord nodes that will be subsequently connected with struts are constrained by the lower curve. The two constraint curves, which contain an interval of 0.375m between the curves, are equivalent. The constraint curve is defined by two arcs with a curvature of 4.4m.

4.5 3D Hybgrid