Stability of Elastic Grid Shells by Romain Mesnil Dipl^ome d'ing´enieur Ecole´ Nationale des Ponts et Chauss´ees, 2013 Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of the Requirements for the Degree of Master of Engineering at the Massachusetts Institute of Technology June 2013 © 2013 Romain Mesnil. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created Signature of Author: Department of Civil and Environmental Engineering May 10, 2013 Certified by: John A. Ochsendorf Professor of Building Technology and Civil and Environmental Engineering Thesis Supervisor Certified by: Cyril Douthe Researcher and Structural Engineer at IFSTTAR, Institut fran¸cais des sciences et technologies des transports, de l'am´enagement et des r´eseaux Thesis Supervisor Accepted by: Heidi Nepf Chair, Departmental Committee for Graduate Students Stability of Elastic Grid Shells by Romain Mesnil Submitted to the Department of Civil and Environmental Engineering in May 10, 2013 in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and Environmental Engineering Abstract The elastic grid shell is a solution that combines double curvature and ease of mounting. This structural system, based on the deformation of an initially flat grid without shear stiffness was invented more than fifty years ago. The apparition of new materials such as GFRP increased the potential of such structures whose properties depend on the deformation, or equivalently pre-stress of an initial structure. Elastic grid shells seem particularly promising as shelters, lightweight roofs, or kinetic structures. Although fundamental to the behavior of the strucure, the influence of the pre-stress on the stability of elastic grid shells has yet to be studied. Understanding this phenomenon could allow engineers to design more efficiently elastic grid shells. This thesis studies the influence of pre-stress on the stability of elastic grid shells. The research conducts a parametric study that focuses both a pre-buckled arch and initially flat circular elastic grid shells with different grid spacing and levels of pre-stress. Realistic values of the parameters are determined from existing projects. The buckling analysis as well as the form-finding of the different structures are performed using finite element analysis. The tools are validated with comparison of the shape and buckling capacity of a pre-buckled arch with existing experiments. The parametric studies lead to recommendations aiming to facilitate the design of elastic grid shells. Keywords Elastic grid shell, Low-Speed Dynamics, form-finding, linear buckling analysis Thesis Supervisor: John A. Ochsendorf Title: Professor of Building Technology Civil and Environmental Engineering Thesis Supervisor: Cyril Douthe Title: Researcher and Structural Engineer at IFSTTAR, Institut fran¸cais des sciences et technologies des transports, de l'am´enagement et des r´eseaux Stabilit´edes Grid Shells Elastiques´ par Romain Mesnil Pr´esent´eeau D´epartement de G´enie Civil et Environnemental le 10 Mai 2013 dans le cadre de l'obtention du Dipl^omede Master of Engineering in Civil and Environmental Engineering R´esum´e Les grid shells ´elastiques sont une solution pr´esentant le double avantage de la double courbure et de la facilit´ede montage. Cette typologie de structure, fond´eesur la d´eformation ´elastique d'une grille plane sans raideur en cisaillement, a ´et´einvent´ee il y a plus de cinquante ans. L'apparition de nouveaux mat´eriaux, tels les mat´eriaux composites `abase de fibre de verre ont accru le potentiel de ces structures dont les propri´et´es d´ependent de la d´eformation, ou de fa¸con ´equivalente de la pr´e-contrainte du mat´eriau. Les grid shells ´elastiques semblent particuli`erement prometteurs en tant que constructions temporaires, abris, ou couvertures l´eg`eres. Bien qu'essentielle au comportement de la structure, l'influence de la pr´e-contrainte sur la charge ultime de flambement des grid shells ´elastiques n'a pas encore ´et´e´etudi´ee en d´etail. La compr´ehension de ce ph´enom`ene pourrait permettre aux ing´enieurs de dimensionner plus efficacement les grid shells ´elastiques. Cette th`ese ´etudie l'influence de la pr´e-contrainte sur la stabilit´eau flambement des grid shells ´elastiques. Le travail de recherche consiste en une ´etude param´etrique d'une poutre flamb´ee utilis´eecomme une arche et de grid shells ´elastiques construits `apartir de grilles initialement circulaires avec des niveaux de pr´e-contrainte diff´erents. Des valeurs r´ealistes des param`etres sont d´etermin´ees `apartir de projets pr´ec´edents. L'analyse de flambement lin´earis´eeainsi que la recherche de forme sont effectu´ees `al'aide d'un logiciel d'´el´ements finis. Les outils utilis´es sont valid´esgr^ace `ala comparaison avec de pr´ec´edents r´esultats th´eoriques et exp´erimentaux. Les ´etudes param´etriques aboutissent `ades recommendations visant `a faciliter le dimensionnement de grid shells ´elastiques. Mots-cl´e: Grid shell ´elastique, Low-Speed Dynamics, recherche de forme, analyse de flambe- ment lin´earis´ee Co-directeur de th`ese: Cyril Douthe Titre: Chercheur et ing´enieur `al'IFSTTAR Co-directeur de th`ese: John A. Ochsendorf Titre: Professeur de G´enie Civil et Architecture Acknowledgments I want to express my sincere gratitude my advisors Cyril Douthe and John Ochsendorf. Thank you Cyril for having accepted to supervise this work from France. Despite the distance, your advice and insight were decisive for this work. Thank you John for being a passionate engi- neer and teacher and showing me how important enthusiasm is in a research work. I thank also professor Tomasz Wierzbicki for giving me the opportunity to be his TA, and for having been so kind to me throughout my last semester at MIT. I would also like to ac- knowledge Professor Jerome Connor for his great knowledge and open mind, our discussions have always been very pleasant and instructive for me. I thank also Professor Bathe who helped me to understand some subtilities of ADINA in the very beginning of this work and Reza Sharifi Sedeh, who kindly explained to me how to generate the input files. I also want to thank my sources of funding: la Fondation des Ponts and the Department of Mechanical Engineering at MIT. I am grateful to the trust that have been put in me. For that reason, I also sincerely thank Alain Pecker, Patrick de Buhan and Lutz Sch¨one. Beside being great engineers and a source of inspiration for me, they supported me in my application for MIT. Finally, I thank my family and friends, who supported me throughout this incredible experi- ence. And a very special thank you to Marion for helping me improving this document, but mostly for sharing her life with me. Table of Contents List of Figures 13 List of Figures 15 1 Introduction 17 1.1 Motivation . 17 1.1.1 Force and form . 17 1.1.2 Double curved structures in history . 18 1.1.3 What is a grid shell? . 21 1.1.4 Why lightweight structures? . 23 1.2 Potential of pre-stressed structures . 24 1.2.1 Ease of use . 24 1.2.2 Temporary structures . 25 1.2.3 Kinetic structures . 26 1.3 Problem statement . 27 1.3.1 Unresolved questions and design of elastic grid shells . 27 1.3.2 Thesis goal . 27 1.3.3 Approach . 29 1.4 Outline of Chapters . 29 2 Literature review 30 2.1 Introduction . 30 2.2 Design of elastic grid shells . 30 2.2.1 Overview . 30 2.2.2 Simple and double layers . 32 2.3 Mechanics of grid shells . 33 2.3.1 Form-finding . 33 2.3.2 Free-form . 35 2.3.3 Buckling of grid shells . 35 2.4 Mechanics of pre-stressed structures . 38 2.4.1 Overview . 38 2.4.2 Stability of 2D structures . 38 2.5 Summary . 38 9 TABLE OF CONTENTS 3 Methodology 39 3.1 Procedure for a parametric study . 39 3.1.1 Procedure for the finite-element analysis . 39 3.1.2 Choice of the parameters . 40 3.2 2D validation of computational method: example of a pre-buckled arch . 40 3.2.1 Form-finding . 40 3.2.2 Consistency of the Finite Element Model . 43 3.2.3 Buckling analysis . 46 3.3 3D validation of computational method . 49 3.4 Summary . 50 4 Application to a 2D Arch 51 4.1 Description of the problem and choice of the parameters . 51 4.2 Comparison of the pre-stressed and unstressed state . 53 4.2.1 Shape of the buckling modes . 53 4.2.2 Influence of the prestress on the buckling capacity . 53 4.3 Discussion . 56 4.3.1 Buckling capacity of pre-stressed arch . 56 4.3.2 Influence of the initial axial load . 57 4.4 Summary . 58 5 Application to pseudo-funicular grid shells 60 5.1 Expectations . 60 5.2 Background of the parametric study . 61 5.2.1 Introduction . 61 5.2.2 Form..................................... 62 5.2.3 Stabilization . 64 5.3 Parametric studies . 69 5.4 Results . 71 5.4.1 Influence of the pre-stress on the mode shapes . 71 5.4.2 Influence of the pre-stress on the buckling load . 74 5.5 Discussion . 77 5.6 Summary . 80 6 Conclusion 81 6.1 Summary of contribution . 81 6.2 Future work . 82 References 83 A Previous elastic grid shells 87 A.1 List of previous elastic grid shells .