# Form Follows Force: A theoretical framework for Structural Morphology, and Form-Finding research on shell structures

### Abstract

The springing up of freeform architecture and structures introduces many challenges to structural engineers. The main challenge is to generate structural forms with high structural efficiency subject to the architectural space constraints during the conceptual structural design process.

Structural Morphology is the study of the relation between form and force, which can be considered the guiding theory for this challenge. The relation between form and force is important for all types of structures during the entire structural design process. Thus, Structural Morphology has a wide range of related research subjects and multiple research approaches. Therefore, Structural Morphology has gained neither a clear definition nor a unified methodology.

In the present research, a theoretical framework for Structural Morphology has been proposed, that provides an effective solution to the challenge mentioned above. To enrich the proposed framework of Structural Morphology, systematic Form-Finding research on shell structures is conducted. Shell structures, the structural efficiency of which depends strongly on their 3D shape, have particular problems regarding the relationship between form and force. To obtain a structurally efficient shell, the form should follow the flow of forces, and a process of Form-Finding can achieve this. In this thesis, Form-Finding of shells indicates a process of generating the equilibrium structural forms of hanging, tent or pneumatic physical models.

In Chapters 2 and 3, a theoretical framework for Structural Morphology is established.

- Structural systems are divided into two categories based on their responses under the loads: ‘Force-Active’ and ‘Force-Passive’. A ‘Force-Active’ structural system can significantly and actively adjust its shape due to the loads, while a ‘Force-Passive’ system cannot. A generic conceptual model of the numerical analysis process of structural systems is presented, which is suitable to both categories of structural systems. This conceptual model includes three parts: (1) the initial system described by five categories of parameters: geometry, material distribution, material properties, boundary conditions and forces; (2) the setup of equations and calculation methods to handle the above parameters; and (3) the structural performance described by two categories of parameters: the structural form and its mechanical behaviour (Chapter 2).
- A conceptual model of Structural Morphology is proposed by adding further requirements of the structural form or the mechanical behaviour and an optimisation process into the above conceptual model of the numerical analysis process of structural systems. Then, a corresponding conceptual formula of Structural Morphology is concluded. Thus, a theoretical framework of Structural Morphology is established. Subsequently, its feasibility is validated by a comprehensive discussion of the two main aspects of Structural Morphology, including ‘Form-Finding’ and ‘Structural Optimisation’. In this research, Form- Finding relates to Force-Active structural systems, which means the generation of multiple equilibrium shapes subject to architectural space constraints. Structural Optimisation relates to Force-Passive structural systems, which indicates the adjustment of relevant parameters of the initial structural system with the aim of improving its mechanical behaviour. The methodology of both aspects is presented. Research achievements completed by the author’s research groups from Harbin Institute of Technology (HIT) and Delft University of Technology (TU Delft) are presented to validate the feasibility. These achievements cover the research on Form-Finding of cable-nets and membrane structures, and on the Structural Optimisation of shells and gridshells (Chapter 3).

In Chapters 4 to 7, the proposed theoretical framework for Structural Morphology is enriched by systemic Form-Finding research on shell structures.

- To study the form of shell structures, the curvature analysis of the surface is displayed. To study the mechanical behaviour of shell structures during the conceptual structural design process, an assessment strategy based on its linear static behaviour and buckling behaviour under two different load cases is proposed. To comprehensively study the linear static behaviour of a shell structure where bending moments may or may not be dominant in this shell, the membrane over the total stress ratios and strain-energy ratio are introduced (Chapter 4).
- The Vector Form Intrinsic Finite Element (VFIFE) method is a recently developed numerical analysis method. At the beginning of this research, few studies on the Form-Finding of shell structures using the VFIFE method were found in the literature. The VFIFE method is applied to generate equilibrium shapes of Force-Active structural systems and thus the structural geometries of shells. A MATLAB script and a plug-in in the Rhino-Grasshopper platform are developed (Chapter 5).
- Form-Control of Force-Active structural systems aims to generate form-found structural forms subject to the required architectural space constraints. Two Form-Control strategies are developed by combining two simple optimisation algorithms (the Newton-Raphson method and the inverse iteration method) with the VFIFE method. These strategies can help designers determine the structurally efficient forms more easily and more efficiently than some relatively complicated and time-consuming optimisation algorithms (Chapter 6).
- Based on the proposed theoretical framework of Structural Morphology, multiple structural forms of form-found shell structures are obtained by adjusting the five categories of parameters of the initial structural systems. This work can efficiently and effectively provide multiple structural forms with reasonable mechanical behaviour for designers from the perspective of structural engineers (Chapter 7).

In Chapters 8 and 9, the specific influence of curved supports on the structural forms and the mechanical behaviour of these shells is studied. Intuitively and qualitatively, designers may be able to select the correct shapes for the supports of shells. However, there was a need to quantify the consequences of designing particular shell supports. In this work, form-found shells with slightly different support shapes are analysed numerically and experimentally.

- Four hexagonal form-found shells generated from hanging models with different support shapes but with the same target point are generated. The following four support shapes are considered: straight supports, outwardcurved supports, inward-curved supports and strongly inward-curved supports. From the numerical comparison, slight changes of the support shapes have a relatively small influence on the equilibrium structural forms but have a considerable influence on the mechanical behaviour of these form-found shells. It is concluded that we can improve structural efficiency by slightly curving the supports during the Form-Finding process, which would not significantly change the architect’s design scheme (Chapter 8).
- In the experimental research, three scaled plastic shell models (with straight supports, outward-curved supports, and inward-curved supports) are tested, and the shadow Moiré method is used in the observation of the deformation of the shells. Form these tests, the influence of the support shapes on form-found shells is studied visually by these obtained Moiré patterns, which represent the buckling modes of these shell models influenced by the curvature distribution near the supports as well as thickness distribution (Chapter 9).

There are still issues that need to be solved in future research. For instance, the theoretical framework for Structural Morphology needs to be enriched with Structural Optimisation work, more complicated design constraints need to be considered in the Form-Finding process of shell structures (for example, the stress level or distribution in the shell), and more influence factors of the form-found shells need to be researched (for example, the number or length of the supports, and edge beams).

### References

Adriaenssens S., Gramazio F., Kohler M., Menges A., Pauly M., editors. (2016). Advances in Architectural Geom- etry 2016. VDF Hochschulverlag AG an der ETH Zürich, Switzerland.

Paulo J. S. Cruz, editor. (2016). Structures and architecture beyond their limits : proceedings of the third Interna- tional Conference on Structures and Architecture (ICSA2016). Taylor & Francis Group, London.

Thomsen M.R., Tamke M., Gengnagel C., Faircloth B., Scheurer F., editors. (2015). Modelling Behaviour: Design Modelling Symposium 2015. Springer International Publishing, Switzerland.

Vambersky J.N.J.A., Schipper H.R., editors. (2010). Precast 2010: Assembling Freeform Buildings in Precast Concrete. Delft University of Technology.

Mungan I., Abel J.F., editors. (2009). Fifty years of progress for shell and spatial structures. Multi-Science Pub- lishing Co. Ltd, Essex.

http://www.designboom.com/architecture/arata-isozaki-himalayas-Centre-zendai-08-11-2015/

http://www.archdaily.com/448774/heydar-aliyev-Centre-zaha-hadid-architects

https://en.wikiarquitectura.com/building/heydar-aliyev-cultural-Centre/

https://www.iconeye.com/architecture/features/item/12384-harbin-opera-house

http://www.archdaily.com/778933/harbin-opera-house-mad-architects

http://www.gebouwvanhetjaar.nl/entry/station-arnhem-centraal/

Horikx M.P. (2017). A Methodical Approach on Conceptual Structural Design. PhD thesis, Delft University of Technology.

Shen S., Wu Y. (2014). Structural morphology and modern space structures. Journal of Building Structures, 35(4): 1-10. ( in Chinese)

Coenders J.L., editor. (2008).Reader of the course ‘CIE5251 - Structural Design - Special Structures’. Delft: Delft University of Technology.

Motro R., editor. (2009). An Anthology of Structural Morphology. World Scientific, London.

Motro R., editor. (2009). Structural Morphology and Configuration Processing of Space Structures. Multi-Sci- ence Publishing Co. Ltd, Brentwood.

https://en.wikipedia.org/wiki/Battle_of_Luding_Bridge

https://iam.tugraz.at/workshop14s/2014/05/04/tanzbrunnen-tent-in-cologne-by-bodo-rasch/

https://www.stylepark.com/en/news/fabric-makes-home

Adriaenssens S., Block P., Veenendaal D., Williams C., editors. (2014). Shell structures for architecture: Form-Finding and optimisation. London: Routledge Taylor and Francis.

Engel, H. (1997) Structure Systems, 3rd edition. Gerd Hatje Publishers: Ostfildern, Germany.

Dong S. (2009). The development history, innovation, classification and practical application of spatial struc- tures. Spatial Structures. 15(3): 22-43.

https://structurae.net/structures/deitingen-service-station

https://hiveminer.com/Tags/gifu%2Ckitagata/Recent

Shen S., Wu Y. (2014). Structural morphology and modern space structures. Journal of Building Structures. 35z(4): 1-10. ( in Chinese)

Veenendaal D., Block P. (2012). An overview and comparison of structural form finding methods for general networks, International Journal of Solids and Structures. 49(26): 3741-3753.

Veenendaal D. (2017). Design and form finding of flexibly formed shell structures: including a comparison of form finding methods. PhD thesis, ETH Zurich, Zurich.

Yao J., Lu Z. (2011). Vector Form Intrinsic Finite Element-overview, recent progress and future developments. 2011 International Conference on Multimedia Technology, Hangzhou, China.

Block P., Lachauer L. (2014). Three-dimensional Funicular Analysis of Masonry Vaults. Mechanics Research Communications, 56: 53-60.

Pottmann H., Eigensatz M., Vaxman A., Wallner J. (2015). Architectural geometry. Computers & Graphics. Graph. 47: 145-164.

Kilian A. (2007). The Steering of Form. Journal of the International Association for Shell and Spatial Structures. 48 (4): 17-21.

Li Q. (2013). Numerical simulation and applications of the inverse hanging method. MSc thesis, Harbin Insti- tute of Technology. (in Chinese)

Wu Y., Li Q., Shen S. (2014). Computational morphogenesis method for space structures based on principle of inverse hanging experiment. Journal of Building Structures, 35(4): 41-48. (in Chinese)

Li Q., Wu Y., Shen S. (2014). Computational generation of freeform shells based on the inverse hanging experi- ment. Proceedings of the IASS-SLTE 2014 Symposium, Brasilia, Brazil.

Ramm E., Wall W.A. (2004). Shell structures - a sensitive interrelation between physics and numeric. Interna- tional Journal for Numerical Methods in Engineering, 60: 381-427.

Tomás A., Martí P. (2010). Shape and size optimisation of concrete shells. Engineering Structures, 32: 1650- 1658.

Structural Morphology Group (IASS Working Group NO.15). (2007). Newsletter NO14.

Structural Morphology Group (IASS Working Group NO.15). (2008). Newsletter NO15.

Motro R., editor. (2009). An Anthology of Structural Morphology. World Scientific, London.

Motro R., editor. (2009). Structural Morphology and Configuration Processing of Space Structures. Multi Science Publishing Co Ltd.

Coenders J.L., editor. (2008). Reader of the course ‘CIE5251 - Structural Design - Special Structures’. Delft: Delft University of Technology.

Mungan I., Abel J.F., editors. (2009). Fifty years of progress for shell and spatial structures. Multi-Science Pub- lishing Co., Ltd, Essex.

[Stach E. (2010). Structural Morphology and Self-organization. Design and Nature V: Comparing Design in Nature with Science and Engineering, in: C. A. Brebbia, A. Carpi (Eds.), WIT Pres.

Wester T. (2009). The first 13 years of Structural Morphology Group - a personal view. An Anthology of Structural Morphology, Ed. Motro R., World Scientific Publishing Company, Chapter 1: 1-14.

Motro R. (2009). Structural Morphology. Fifty years of progress for shell and spatial structures. Ed. Mungan I., Abel J.F. Multi-Science Publishing Co. Ltd., Chapter 12: 451-458.

Wester T. (1994). The nature of structural morphology and some interdisplinary examples. Spatial, Lattice and Tension Structures, IASS-ASCE, American Society of Civil Engineering, New York, 1000-1009.

Ramm E., Bletzinger K.-U. (1993). Structural optimisation, International Association for Shell and Spatial struc- tures (IASS) newsletter 112, 6.

Motro R. (2002). Teaching of space structures with initial stresses, International Journal of Space Structures, 17(2&3):107-116.

Bagneris M., Motro R., Maurin B., and Pauli N. (2008). Structural morphology issues in conceptual design of double curved systems. International Journal of Space Structures, 23(2): 79-87.

Motro R. (2009). An approach to structural morphology. An Anthology of Structural Morphology, Ed. Motro R. World Scientific Publishing Company, Chapter 2:15-32.

Shen S., Wu Y. (2014). Structural morphology and modern space structures. Journal of Building Structures. 35(4): 1-10. ( in Chinese)

http://www.iass-structures.org/index.cfm/page/TechAct/WG15.htm

Borgart A. (2010). New challenges for the structural morphology group. Journal of the International Association for Shell and Spatial Structures. 51(3): 183-189.

Bletzinger K.-U. (2011). Form-Finding and morphogenesis. Fifty years of progress for shell and spatial struc- tures, Ed. Mungan I., Abel J.F., Multi-Science Publishing Co. Ltd., Chapter 12: 459-474.

Ohmori H. (2011). Computational morphogenesis - its current state and possibility for the future. International Journal of Space Structures. 26(3): 269-276.

Schek H.J. (1974). The force density method for form finding and computations of general networks. Computer Methods in Applied Mechanics and Engineering. 3:115-134.

Day A.S. (1965). An introduction to dynamic relaxation. The Engineer. 29(2): 218-221.

http://en.wikipedia.org/wiki/Finite_element_method

Adriaenssens S., Block P., Veenendaal D. and Williams C., editors. (2014). Shell Structures for Architecture: Form Finding and Optimisation. London: Routledge Taylor and Francis.

Li Q., Su Y., Wu Y., Borgart A., Rots J.G. (2017). Form-Finding of shell structures generated from physical models, International Journal of Space Structures. 32(1): 11-33.

Veenendaal D., Block P. (2012). An overview and comparison of structural form finding methods for general networks, International Journal of Solids and Structures. 49(26): 3741-3753.

http://en.wikipedia.org/wiki/Finite_element_method

Schek H.J. (1974). The force density method for form finding and computations of general networks. Computer Methods in Applied Mechanics and Engineering. 3: 115-134.

Day A.S. (1965). An introduction to dynamic relaxation. The Engineer, 29(2): 218-221.

Billington D.P. (2011). Heinz Isler: From Delft to Princeton and Beyond. Journal of the International Association for Shell and Spatial Structures (J. IASS). 52(3): 135-141.

https://www.pinterest.com/pin/317011261243639594/

Ramm E. (2011). Heinz Isler Shells – The Priority of Form. Journal of the International Association for Shell and Spatial Structures (J. IASS). 52(3): 143-154.

Li Q. (2013). Numerical simulation and applications of the inverse hanging method. MSc thesis, Harbin Insti- tute of Technology. (in Chinese)

Wu Y., Li Q. (2012). Inverted hanging method and its application on structural morphogenesis. 14th Academic Conference of Space Structures in China. Fuzhou, China, 2012. (in Chinese)

Li Q., Wu Y., Shen S. (2014). Computational generation of freeform shells based on the inverse hanging experi- ment. Proceedings of the IASS-SLTE 2014 Symposium, Brasilia, Brazil.

Wu Y., Li Q., Shen S. (2014). Computational morphogenesis method for space structures based on principle of inverse hanging experiment. Journal of Building Structures, 35(4): 41-48. (in Chinese)

Qian H. (2007). Theoretical and Experimental Research on Supporting Structure of FAST Reflector. PhD thesis, Harbin Institute of Technology. (in Chinese)

https://en.wikipedia.org/wiki/Five_hundred_meter_Aperture_Spherical_Telescope

Wu Y., Xiang Y., Shen S. (2001). Structural wind tunnel test of the cable-membrane structure of Weihai stadium. Building Structure. 6: 66-68. (in Chinese)

http://www.whnews.cn/news/2008-08/28/content_1372944_2.htm

Cao Z., Wu Y., Qian H., Shen S. (2011). Design and analysis of giant grid suspen-dome of Dalian Centre Gymna- sium. Steel Construction, 26(1): 37-42. (in Chinese)

Borgart A. (2010). An approximate calculation method for air inflated cushion structures for design purposes. International Journal of Space Structures, 25(2): 83-91.

Wu Y., Liu X., Li Q., Chen B., Luo P., Pronk A.D.C., Mergny E. (2017) . Form-Finding and construction of ice com- posite shell structures. Proceedings of the IASS Annual Symposium 2017, Hamburg, Germany.

Christensen P. W., Klarbring A. (2009). An Introduction to Structural Optimisation. Springer.

Masching H. (2016). Parameter Free Optimisation of Shape Adaptive Shell Structures. PhD thesis, Technischen Universität München.

Bletzinger K.-U. (2011). Form-Finding and morphogenesis. Fifty years of progress for shell and spatial struc- tures, Ed. Mungan I., Abel J.F. Multi-Science Publishing Co. Ltd., Chapter 12: 459-474.

Ohmori H. (2011). Computational morphogenesis - its current state and possibility for the future. International Journal of Space Structures, 26(3): 269-276.

Weise T. (2009). Global Optimisation Algorithms - Theory and Application. Self-Published, second edition. Online available at http://www.it-weise.de/.

Li X., Wu Y., Cui C. (2011). NURBS-GM method for computational morphogenesis of free form structures. China Civil Engineering Journal, 44(10): 60-66. (in Chinese)

Cui C., Wang Y., Jiang B. and Cui G. (2013). Study on the structural morphogenesis technique for single-layer reticulated shells of free-curved surface. China Civil Engineering Journal, 46(4): 57-63. (in Chinese)

Chang C., Borgart A., Chen A. and Hendriks M.A.N. (2014). Direct gradient projection method with transforma- tion of variables technique for structural topology optimisation. Structural and Multidisciplinary Optimisation, 49: 107-119.

Wu Y., Li Q., Hu Q. and Borgart A. (2017). Size and Topology Optimisation for Trusses with Discrete Design Variables by Improved Firefly Algorithm. Mathematical Problems in Engineering.

Hu Q. (2015). Topology optimisation analysis for framed structures based on the improved firefly algorithm. MSc thesis, Harbin Institute of Technology. (In Chinese).

Cui C., Jiang B. (2014). A morphogenesis method for shape optimisation of framed structures subject to spatial constraints. Engineering Structures, 77: 109-118.

Wu Y., Xia Y. and Li Q. (2015). Structural morphogenesis of freeform shells by adjusting the shape and thick- ness. Proceedings of the IASS Symposium, Amsterdam, the Netherlands.

Xia Y. (2015). Computational morphogenesis of freeform structures based on hybrid optimisation method. MSc thesis, Harbin Institute of Technology. (In Chinese).

Calladine C.R. (1983). Theory of Shell Structures. Cambridge University Press, New York.

Hoogenboom P.C.J. (2017). Notes on Shell Structures, reader of the course ‘ CIE4143 Shell Analysis, Theory and Application’. Delft: Delft University of Technology.

Lisle R.J., Robinson J.M. (1995). The Mohr circle for curvature and its application to fold description. Journal of Structural Geology, 17(5): 739-750.

Blaauwendraad J., Hoefakker J.H. (2014). Structural Shell Analysis: Understanding and Application. Springer, Dordrecht.

Borgart A., Eigenraam P. (2012). Scanning in 3D and analysing the models of Heinz Isler, the preliminary results. Proceedings IASS-APCS, Seoul, Korea.

Calladine C.R. (1977). The static–geometric analogy in the equations of thin shell structures. Mathematical Proceedings of the Cambridge Philosophical Society, 82: 335-351.

Borgart A. (2017). Lecture notes of the course ‘CIE5251-09 Structural Design, Special Structures’. Delft: Delft University of Technology.

Schuddeboom P. (2014). New Ways of Representing Finite Element Results of Shell Structures. MSc thesis, Delft University of Technology.

Chilton J. (2000). The Engineer’s contribution to contemporary architecture: Heinz Isler. Thomas Telford Pub- lishing, London.

Ramm E. (2011). Heinz Isler Shells - The Priority of Form. Journal of the International Association for Shell and Spatial Structures, 52 (3): 143-154.

Nerdinger W., With I.C., Meissner I., Möller E., Grdanjski M., Architecture Museum of Technical University Munich, editors. (2010). Frei Otto complete works: lightweight construction, natural Design. China Architecture & Building Press, Beijing.

https://structurae.net/structures/deitingen-service-station

https://www.pinterest.com/pin/457045062154696404/

Nicoletti M. (1999). Sergio Musmeci, Organicità di forme e forze nello spazio. Testo & Immagine, Turin.

https://www.pinterest.com/pin/317011261243639594/

https://www.pinterest.com/pin/481111172670145006/

Kromoser B., Patrick Huber P. (2016). Pneumatic Formwork Systems in Structural Engineering. Advances in Materials Science and Engineering.

Bini D. (2014). Building With Air. Biblioteque McLean, London.

http://www.khs82.com/dbpage.php?pg=khsphotos

Kokawa T. (2012). Building Techniques for Ice Shell as Temporary Structure. Proceedings of IASS-APCS 2012, Seoul, Korea.

Rippmann M. (2016). Funicular Shell Design: Geometric approaches to form finding and fabrication of discrete funicular structures. PhD thesis, ETH Zurich.

Day A.S. (1965). An introduction to dynamic relaxation. The Engineer, 29(2): 218-221.

Linkwitz, K., Schek, H.J. (1971). Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruk- tionen. Ingenieur Archiv, 40: 145-158.

Schek H.J. (1974). The force density method for Form-Finding and computations of general networks. Computer Methods in Applied Mechanics and Engineering, 3: 115-134.

Haug E., Powell G.H. (1972). Finite element analysis of nonlinear membrane structures. IASS Pacific Symposium Part II: on Tension Structures and Space Frames, Tokyo and Kyoto, Japan. p. 93-102.

Bletzinger K.-U., Ramm E. (1993). Form finding of shells by structural optimisation. Engineering with Comput- ers, 9: 27-35.

Block P., Ochsendorf J. (2007). Thrust network analysis: a new methodology for three dimensional equilibrium. Journal of the International Association for Shell and Spatial Structures, 48 (3): 167 - 173.

Vizotto I. (2010). Computational generation of freeform shells in architectural design and civil engineering. Automation in Construction, 19: 1087 - 1105.

Veenendaal D., Block P. (2012). An overview and comparison of structural form finding methods for general networks. International Journal of Solids and Structures, 49(26): 3741-3753.

Adriaenssens S., Block P., Veenendaal D., Williams C., editors. (2014). Shell structures for architecture: Form-Finding and optimisation. Routledge Taylor and Francis, London.

Ting E.C., Shih C., Wang Y.K. (2004). Fundamentals of a vector form intrinsic finite element: Part I. Basic proce- dure and a plane frame element. Journal of Mechanics, 20(2): 113-122.

Ting E.C., Shih C., Wang Y.K. (2004). Fundamentals of a vector form intrinsic finite element; Part II. Plane solid elements. Journal of Mechanics, 20(2); 123-132.

Shih C., Wang Y.K., Ting E.C. (2004). Fundamentals of a vector form intrinsic finite element: Part III. Convected material frame and examples. Journal of Mechanics, 20(2): 133-143.

Wu T.Y., Ting E. C. (2008). Large deflection analysis of 3D membrane structures by a 4-particle quadrilateral intrinsic element. Thin-Walled Structures, 46: 261-275.

Lien K.H., Chiou Y.J., Wang R.Z., Hsiao P.A. (2010). Vector Form Intrinsic Finite Element analysis of nonlinear behaviour of steel structures exposed to fire. Engineering Structures, 32: 80-92.

Wang R.Z., Tsai K.C., Lin B.Z. (2011). Extremely large displacement dynamic analysis of elastic-plastic plane frames. Earthquake Engineering and Structural Dynamics, 40:1515–1533.

Wang Z. (2013). Theory and application of thin shell element based on the Vector Form Intrinsic Finite Element method. Ph.D. Thesis, Zhejiang University, Hangzhou, China.

Wu T.Y. (2013). Dynamic nonlinear analysis of shell structures using a Vector form Intrinsic Finite Element. Engineering Structures, 56: 2028-2040.

Zhao Y., Wang Z., Peng T. (2015). Membrane element based on Vector Form Intrinsic Finite Element and its application in wrinkling analysis of membrane structures. Journal of Building Structures, 36(1): 127-135. ( in Chinese)

Brew J.S., Lewis W.J. (2007). Free hanging membrane model for shell structures. International Journal for Nu- merical Methods in Engineering, 71:1513–1533.

Block P., Lachauer L. (2011). Closest-Fit, Compression-Only Solutions for Free Form Shells. Proceedings of the IABSE-IASS Symposium 2011,London, UK.

Block P., Lachauer L. (2014). Three-dimensional Funicular Analysis of Masonry Vaults. Mechanics Research Communications, 56: 53-60.

Van Mele T., Panozzo D., Sorkine-Hornung O., Block P. (2014). Best-fit thrust network analysis: rationalization of freeform meshes. In Shell Structures for Architecture: Form Finding and Optimisation, 157–168.

Li Q. (2013). Numerical simulation and applications of the inverse hanging method. MSc thesis, Harbin Insti- tute of Technology. (in Chinese)

Wu Y., Li Q., Shen S. (2014). Computational morphogenesis method for space structures based on principle of inverse hanging experiment. Journal of Building Structures, 35(4): 41-48. (in Chinese)

Li Q., Wu Y., Shen S. (2014). Computational generation of freeform shells based on the inverse hanging experi- ment. Proceedings of the IASS-SLTE 2014 Symposium, Brasilia, Brazil.

Bletzinger K.-U., Ramm E. (1993). Form finding of shells by structural optimisation. Engineering with Comput- ers, 9: 27-35.

Ramm E. (2004). Shape finding of concrete shell roofs. Journal of the International Association for Shell and Spatial Structures, 45(1): 29-39.

Ramm E., Wall W.A. (2004). Shell structures - a sensitive interrelation between physics and numeric. Interna- tional Journal for Numerical Methods in Engineering, 60: 381-427.

Ohmori, H., Kimuraa, T., Maeneb, A. (2009). Computational Morphogenesis of Free Form Shells. Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium, 28 September - 2 October 2009, Universidad Politecnica de Valencia, Spain.

Tomás A., Martí P. (2010). Shape and size optimisation of concrete shells. Engineering Structures, 32: 1650- 1658.

Adriaenssens S., Block P., Veenendaal D., Williams C., editors. (2014). Shell structures for architecture: Form-Finding and optimisation. Routledge Taylor and Francis, London.

Marino E., Salvatori L., Orlando M., Borri C. (2016). Two shape parametrizations for structural optimisation of triangular shells. Computers & Structures. 166: 1-10.

Li Q., Borgart A., Wu Y. (2016). How to understand ‘Structural Morphology’?. Journal of the International Associ- ation for Shell and Spatial Structures, 57(2): 145-158.

Li Q., Su Y., Wu Y., Borgart A., Rots J.G. (2017). Form-Finding of shell structures generated from physical models. International Journal of Space Structures. 32(1): 11-33.

Kilian A. (2007). The Steering of Form. Journal of the International Association for Shell and Spatial Structures, 48(4): 17-21.

Li Q., Wu Y., Shen S. (2014). Computational generation of freeform shells based on the inverse hanging experi- ment. Proceedings of the IASS-SLTE 2014 Symposium, Brasilia, Brazil, 2014.

Heathcote E., Ledgard J. (2016). ’A’A’ Perspectives - The Droneport Project. LafargeHolcim Foundation.

Block P., Rippmann M., Van Mele T., Escobedo D. (2017). The Armadillo Vault: Balancing computation and traditional craft. FABRICATE 2017, Menges A., Sheil B., Glynn R. and Skavara M. (editors): 286-293, UCL Press London.

Veenendaal D., Bakker J., Block P. (2017). Structural design of the flexibly formed, mesh-reinforced concrete sandwich shell roof of NEST Hilo. Journal of the International Association of Shell and Spatial Structures, 58(1): 23-38.

Theocaris P.S. (1969). Moiré fringes in strain analysis. Elmsford: Pergamonpress.

Sciammarella C.A. (1983). The Moiré method-a review. Experimental Mechanics, 23(4): 446-449.

Kafri O., Glatt I. (1990). The Physics of Moiré Metrology. John Wiley & Sons, New York.

**A+BE | Architecture and the Built Environment**, [S.l.], n. 2, p. 1-278, feb. 2018. ISSN 2214-7233. Available at: <https://journals.open.tudelft.nl/abe/article/view/1993>. Date accessed: 12 aug. 2020. doi: https://doi.org/10.7480/abe.2018.2.1993.

This work is licensed under a Creative Commons Attribution 4.0 International License.