Three-dimensional Graphical Statics using Polyhedral Reciprocal Diagrams

Akbarzadeh M.
doi: 10.3929/ethz-a-010867338

More 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 in 1864 . It uses Rankine’s proposition on the reciprocity between the form of a polyhedral frame and its force diagram and redefines the 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.

Using 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.

In 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.


    author      = "Akbarzadeh, M.",
    title       = "Three-dimensional Graphical Statics using Polyhedral Reciprocal Diagrams",
    institution = "ETH Zurich, Department of Architecture",
    year        = "2016",
    address     = "Zurich",
    month       = "October",
    doi         = "10.3929/ethz-a-010867338",
    note        = "",

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