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It clarifies and expands geometric representation of force equilibrium in polyhedral frames, redefining topological relationships from Rankine’s reciprocity between frame form and force diagrams. The work also provides a computational framework to construct reciprocal diagrams from convex polyhedral cells.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/3d-graphical-statics-using-reciprocal-polyhedral-diagrams-doctoral-thesis/665469/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/3d-graphical-statics-using-reciprocal-polyhedral-diagrams-doctoral-thesis/665469.png","ImageObject",300,407,{"name":92,"@type":93},"Angel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main goal of this research thesis?","Question",{"text":112,"@type":113},"To establish methods of 3D graphic statics based on the concept of 3D reciprocal diagrams, extending the historical 2D graphic statics methods.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the thesis use Rankine’s concept?",{"text":117,"@type":113},"It applies Rankine’s reciprocity between the form of a polyhedral frame and its force diagram, and redefines the topological relationships for 3D graphic statics.",{"name":119,"@type":110,"acceptedAnswer":120},"What methods are introduced for constructing equilibrium and forms?",{"text":121,"@type":113},"It provides step-by-step geometric procedures to construct spatial funicular forms under given boundary conditions and loads, and describes procedures using force polyhedrons to show equilibrium of internal and external forces.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665469,1793304372,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Research Collection  \nDoctoral Thesis  \n3D Graphical Statics Using Reciprocal Polyhedral Diagrams  \nAuthor(s):  \nAkbarzadeh, Masoud  \nPublication Date:  \n2016  \nPermanent Link:  \n[https://doi.org/10.3929/ethz-a-010867338](https://doi.org/10.3929/ethz-a-010867338)  \n[Rights / License:](Rights / License:)  \nIn Copyright-Non-Commercial Use Permitted  \nThis page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.  \nETH Library  \nDiss. ETH No. 23582  \n3D Graphical Statics  \nUsing Reciprocal Polyhedral Diagrams  \nA thesis submitted to attain the degree of  \nDoctor of Sciences of ETH Zurich  \n(Dr. Sc. ETH Zurich)  \npresented by  \nMasoud Akbarzadeh  \nB.Sc. in Civil and Environmental Engineering Zanjan University, 2004  \nM.Sc. in Earthquake Engineering and Dynamics of Structures Iran University of Science and Technology, 2007  \nM.Arch.  \nMassachusetts Institute of Technology, 2011  \nS.M.Arch.S. in Design and Computation Massachusetts Institute of Technology, 2012  \nborn on 20.09.1981 citizen of Iran  \nsupervised by  \nProf. Dr. Philippe Block  \n(ETH Zurich)  \nco-supervised by  \nWilliam F. Baker  \n(SOM Chicago)  \n2016  \nii  \nAbstract  \nMore than 150 years after Culmann (1864) established the methods of 2D graphic statics at ETH Zurich, this research aims to establish the methods of 3D graphic statics based on the historical concept of 3D reciprocal diagrams. It clarifies and develops the concept of geometric representation of the equilibrium of forces in polyhedral frames based on the proposition by Rankine (1864) in 1864 . It uses Rankine's proposition on the reciprocity between the form of a polyhedral frame and its force diagram and redefinesthe topological relationships to be used as the basis for the 3D graphic statics methods. It also provides a computational framework to construct 3D reciprocal diagrams from convex polyhedral cells.  \nUsing 3D structural reciprocity, this thesis provides methods to find global equilibrium for systems of forces in 3D and establishes step–by–step geometric procedures to construct spatial funicular forms that are geometrically constrained to given boundary conditions and applied loads. Moreover, it describes the procedures to show the equilibrium of internal and external forces in the members of general polyhedral frames using force polyhedrons.  \nIn addition to the 3D graphic statics methods, this research introduces valuable design and optimization techniques for form finding of complex spatial structural systems by aggregating force polyhedrons and subdividing the global equilibrium in the force diagram. These methods are valuable in deriving complex compression–only structural solutions with different topological properties for given boundary conditions. Lastly, this research provides additional examples to show the extensive design potential of these methods to generate non–conventional structural systems with a combination of compressive and tensile forces in their members.  \niv  \nAbstrakt  \n150 Jahre nachdem Culmann die Methoden der zweidimensionalen graphischen Statik an der ETH Zrich bestimmte, begrndet diese Forschung die Methodender dreidimensionalen graphischen Statik basierend auf dem historischen Konzept der dreidimensional-reziproken Diagramme. Sie erklrt und entwickeltdas Konzept der geometrischen Darstellung des Gleichgewichts der Krfte in polyedrischen Rahmen, welche 1864 von Rankine eingefhrt wurden. Es nutzt Rankins Aussage ber die Gegenseitigkeit zwischen der Form eines polyedrischen Rahmens und dessen Krfteplan und definiert ihre topologischen Beziehungen neu, um sie als Grundlage fr die Methoden der dreidimensionalen graphischen Statik zu verwenden. Sie stellt auch rechnergesttzte Grundlagen bereit, umdreidimensional-reziproke Diagramme von konvexen polyedrischen Zellen zukonstruieren.  \nMit Hilfe der dreidimensionalen reziproken Diagramme stellt diese Arbeit Methoden vor, um ein globales Gleichgewi","cbCaipsydIr4v0Ww","https://ap.wps.com/l/cbCaipsydIr4v0Ww","pdf",31996724,201,"English","# Abstract\n# Abstrakt\n# Acknowledgements","[{\"question\":\"What is the main goal of this research thesis?\",\"answer\":\"To establish methods of 3D graphic statics based on the concept of 3D reciprocal diagrams, extending the historical 2D graphic statics methods.\"},{\"question\":\"How does the thesis use Rankine’s concept?\",\"answer\":\"It applies Rankine’s reciprocity between the form of a polyhedral frame and its force diagram, and redefines the topological relationships for 3D graphic statics.\"},{\"question\":\"What methods are introduced for constructing equilibrium and forms?\",\"answer\":\"It provides step-by-step geometric procedures to construct spatial funicular forms under given boundary conditions and loads, and describes procedures using force polyhedrons to show equilibrium of internal and external forces.\"}]","3D Graphical Statics Using Reciprocal Polyhedral Diagrams - Doctoral Thesis | PDF",507]