##### Algebraic 3D graphic statics: Reciprocal constructions

Márton Hablicsk, Masoud Akbarzadeh, Yi Guo

##### Abstract

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagram based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two.

##### BibTex

*Computer-Aided Design*, 2018.

```
@article{márton hablicsek 2018algebraic,
title={Algebraic 3D graphic statics: Reciprocal constructions},
author={Márton Hablicsek and Masoud Akbarzadeh and Yi Guo},
journal={Computer-Aided Design},
year={2018}
}
```

##### Developing Algebraic Constraints for Reciprocal Polyhedral Diagrams of 3D Graphic Statics

Masoud Akbarzadeh, Márton Hablicsk, Yi Guo

##### Abstract

3D graphic statics using reciprocal polyhedral diagrams (3DGS) is one of the recent developments in the field of geometry-based structural form finding and is a powerful method in generating spatial structural forms and their force diagram in three dimensions. However, constructing reciprocal polyhedral diagrams in 3D is quite challenging and the research lacks a rigorous mathematical definition formulating the geometrical and reciprocal relationship between the form and force diagrams in 3DGS. Having been used for the past 150 years, 2D graphic statics has recently been formulated algebraically that allows better topological understanding the relationship between the form and the force diagrams in 2D. Such algebraic formulation is crucial in developing interactive tools enabling designers and practitioner to exploit the potentials of working with the form and force diagrams by computationally drawing the reciprocal diagrams for each design iteration which was otherwise quite tedious and cumbersome. This paper provides initial formulation of the reciprocal relationships between polyhedral form and the force diagrams in 3DGS and lays a foundation for further research in algebraic implementation of 3DGS.

##### BibTex

*Proceedings of the IASS Symposium 2018, Creativity in Structural Design*, MIT, Boston, USA, July 16-20, 2018.

```
@inproceedings{akbarzadeh2018developing,
title={Developing Algebraic Constraints for Reciprocal Polyhedral Diagrams of 3D Graphic Statics},
author={Akbarzadeh, M. and Hablicsek, M. and Guo, Y.},
booktitle={Proceedings of the IASS Symposium 2018, Creativity in Structural Design},
year={2018},
month={July 16-20},
address={MIT, Boston, USA}
}
```

##### Graphic Statics: Constrained form finding for parallel system of forces using Corsican sum

Georgios-Spyridon Athanasopoulos, Masoud Akbarzadeh, Allan McRobie

##### Abstract

The field of graphic statics focuses on the development of geometric methods to facilitate and seek optimal solutions for structural design. Such methods include the construction of reciprocal form and force diagrams that can be used as a basis for form finding tools. Building on the work of Rankine, Akbarzadeh has made significant contribution on expanding analogies between 2D and 3D extraction techniques of reciprocal diagrams to explore three-dimensional equilibrium. Additionally, McRobie has further extended Rankine’s approach using geometric algebra with the notable introduction of the Corsican sum. Following the theme of the symposium, “Creativity in structural design”, we explore a method for novel form finding through 3D graphic statics with the innovative use of the Corsican sum. We construct a hypothetical system of parallel forces in 3D with given boundary conditions on a plane and we intend to explore form and its corresponding force diagram. The final 3D form derives from member directions resulting from the Corsican sum in such a way that global equilibrium is accomplished given that the focus is on axial only forces. We begin by investigating a triangulation method that distributes the applied forces to the supports and gives the 2D projection of the form. We continue with the shifting of the form pieces in 2D in order to construct the vertical force diagram. Boundary conditions and equilibrium requirements guide the construction of the final 3D reciprocal force diagram. This leads to angle values for each of the structure’s members. We compare the final axial forces with those resulting from alternative triangulations to proof the validity of our pattern logic.

##### BibTex

*Proceedings of the IASS Symposium 2018, Creativity in Structural Design*, MIT, Boston, USA, July 16-20, 2018.

```
@inproceedings{ athanasopoulos2018graphic,
title={Graphic Statics: Constrained form finding for parallel system of forces using Corsican sum},
author={ Athanasopoulos, G. S. and Akbarzadeh, M. and McRobie, A.},
booktitle={Proceedings of the IASS Symposium 2018, Creativity in Structural Design},
year={2018},
month={July 16-20},
address={MIT, Boston, USA}
}
```

##### Constrained Manipulation of Polyhedral Systems

Andrei Nejur, Masoud Akbarzadeh

##### Abstract

This paper presents a method for the manipulation of groups of polyhedral cells that allows geometric transformation while preserving the planarity constraints of the cells and maintaining the equilibrium direction of the edges for the reciprocity of the form and force diagrams. The paper expands on previously investigated single-cell manipulations and considers the effects of these transformations in adjacent cells and the whole system. All the transformations discussed in this paper maintain the initial topology of the input system. The result of this research can be applied to both form and force diagrams to investigate various geometric transformations resulting in convex or complex (self-intersecting) polyhedra as a group. The product of this research allows intuitive user interaction in working with form and force diagrams in the early stages of geometric structural design in 3D.

##### BibTex

*Proceedings of the IASS Symposium 2018, Creativity in Structural Design*, MIT, Boston, USA, July 16-20, 2018.

```
@inproceedings{nejur2018constrained,
title={Constrained Manipulation of Polyhedral Systems},
author={Nejur, A. and Akbarzadeh, M.},
booktitle={Proceedings of the IASS Symposium 2018, Creativity in Structural Design},
year={2018},
month={July 16-20},
address={MIT, Boston, USA}
}
```

##### Structural Behavior of a Cast-in-Place Funicular Polyhedral Concrete: Applied 3D Graphic Statics

Mohammad Bolhassani, Ali Tabatabai Ghomi, Andrei Nejur, Masoud Akbarzadeh

##### Abstract

Although geometry-based structural design methods like 3D Graphic Statics (3DGS) allow for exploring a variety of spatial funicular geometry and their force equilibria. However, the material properties are not involved in the geometric form finding and there is no experimental data on the actual mechanical behavior of such systems. This paper will explore the structural performance of a funicular polyhedral geometry using experimental testing. The geometry of the physical prototype for the presented study is designed using 3DGS method. The specimen is constructed as a cast-in-place concrete structure, and the geometry of the sample is comparable to the standard concrete cylindrical test. High-performance, self-consolidating concrete is used for casting. Experimental results validated the 3DGS force distribution in the structure and showed that the magnitude of internal force in the members of the sample can be accurately predicted by 3DGS as long as the ultimate strength of the specimen is known.

##### BibTex

*Proceedings of the IASS Symposium 2018, Creativity in Structural Design*, MIT, Boston, USA, July 16-20, 2018.

```
@inproceedings{bolhassan2018structural,
title={Structural Behavior of a Cast-in-Place Funicular Polyhedral Concrete: Applied 3D Graphic Statics},
author={Bolhassan, M. and Tabatabaei Ghomi, A. and Nejur, A. and Akbarzadeh, M.},
booktitle={Proceedings of the IASS Symposium 2018, Creativity in Structural Design},
year={2018},
month={July 16-20},
address={MIT, Boston, USA}
}
```

##### Effect of Subdivision of Force Diagrams on the Local Buckling, Load-Path and Material Use of Founded Forms

Ali Tabatabai Ghomi, Mohammad Bolhassani, Andrei Nejur, Masoud Akbarzadeh

##### Abstract

This paper investigates the relationship between the topology of a structure, load-path values and material efficiency for given boundary conditions in structural form finding using 3D Graphic Statics (3DGS) methods. Subdividing the force polyhedron is a technique in graphic statics that allows generating topologically-different structural forms for a given boundary conditions. This method is used to deal with buckling problems in long members by substituting them with multiple members with shorter lengths. However, subdivision methods result in more members and nodes in the structure that adds to the construction costs and material use. This paper investigates the effect of subdivision techniques on the change in the load-path values and local buckling load of various developed funicular polyhedral systems and the volume of the construction material. Multiple subdivision algorithms are developed to generate series of bar-node compression-only spatial structural systems for a given boundary condition, and relevant algorithms are designed to calculate the volume, load path and maximum local buckling force. The results of 41 different specimens show that by applying subdivision on global force diagram, generally the maximum local buckling force would increase as well as load path and volume. However, the slope of increase in local buckling force is higher. Furthermore, subdividing the applied forces as well as internal forces causes a better local buckling force than the subdivision of interior geometry.

##### BibTex

*Proceedings of the IASS Symposium 2018, Creativity in Structural Design*, MIT, Boston, USA, July 16-20, 2018.

```
@inproceedings{tabatabaei ghomi2018effect,
title={Effect of Subdivision of Force Diagrams on the Local Buckling, Load-Path and Material Use of Founded Forms},
author={Tabatabaei Ghomi, A. and Bolhassan, M., and Nejur, A. and Akbarzadeh, M.},
booktitle={Proceedings of the IASS Symposium 2018, Creativity in Structural Design},
year={2018},
month={July 16-20},
address={MIT, Boston, USA}
}
```

Hedracrete; Prefab, Funicular, Spatial Concrete

Masoud Akbarzadeh, Mehrad Mahnia, Ramtin Taherian, Amir Hossein Tabrizi

##### Abstract

Hedracrete is a unique research project aiming to address three important topics in the field of digital design and fabrication. These subjects include efficient structural form finding in three dimensions, fabrication of complex spatial systems, and the innovative use of conventional materials.

##### BibTex

*DISCIPLINES & DISRUPTION: Projects Catalog of the 37th Annual Conference of the Association for Computer Aided Design in Architecture (ACADIA)*, 2017.

```
@inproceedings{masoud2017hedracrete;,
title={Hedracrete; Prefab, Funicular, Spatial Concrete},
author={Masoud, Akbarzadeh and Mehrad, Mahnia and Ramtin, Taherian and Amir Hossein, Tabrizi},
booktitle={DISCIPLINES \& DISRUPTION: Projects Catalog of the 37th Annual Conference of the Association for Computer Aided Design in Architecture (ACADIA)},
year={2017},
organization={Association for Computer Aided Design in Architecture (ACADIA)}
}
```

##### Addressing buckling of compression members using subdivision of force diagrams

Timo Harboe Nielsen, Masoud Akbarzadeh, Per Goltermann

##### Abstract

This paper demonstrates how buckling issues can be addressed early in a design process using graphic statics and force diagram modification. It explores how the insertion of a single node in a closed polygon of the force diagrams leads to the insertion of a polygon in the form diagram, and how this method can be used to effectively reduce the length of members and increase the buckling capacity of structures significantly.

The research highlights the degrees of freedom of the resulting form diagram, and respectively identifies important subdivision parameters that can be modified to establish a determinate reciprocal relationship between a subdivided force diagram and the corresponding form with no buckling in the compression members. To evaluate the performance of the resulting form, a buckling-adjusted load-path formula is used. It is closely investigated how the mentioned subdivision parameters can be chosen to effectively reduce the required amount of materials in the structure.

##### BibTex

*Proceedings of the IASS Annual Symposium 2017, Interfaces: architecture . engineering . science*, 2017.

```
@inproceedings{timo2017addressing,
title={Addressing buckling of compression members using subdivision of force diagrams},
author={Timo, Harboe Nielsen and Masoud, Akbarzadeh and Per, Goltermann},
booktitle={Proceedings of the IASS Annual Symposium 2017, Interfaces: architecture . engineering . science},
year={2017},
organization={IASS}
}
```

##### Prefab, Concrete Polyhedral Frame: Materializing 3D Graphic Statics

Masoud Akbarzadeh, Mehrad Mahnia, Ramtin Taherian, Amir Hossein Tabrizi

##### Abstract

This research describes the form finding and structural analysis of a prefabricated, concrete polyhedral structure designed by the use 3D graphic statics based on reciprocal polyhedral diagrams (3DGS). The form is a self-supporting, funicular polyhedral geometry with both compression and tension members Fiber reinforced, self-compacting, lightweight concrete is used to construct the members and the joints. The structure can be considered as the first built prototype designed based on the principles of the equilibrium of polyhedral frames and the methods of 3D graphical statics as the recent development of this principle.

##### BibTex

*Proceedings of the IASS Annual Symposium 2017, Interfaces: architecture . engineering . science*, 2017.

```
@inproceedings{masoud2017prefab,,
title={Prefab, Concrete Polyhedral Frame: Materializing 3D Graphic Statics},
author={Masoud, Akbarzadeh and Mehrad, Mahnia and Ramtin, Taherian and Amir Hossein, Tabrizi},
booktitle={Proceedings of the IASS Annual Symposium 2017, Interfaces: architecture . engineering . science},
year={2017},
organization={IASS}
}
```

##### On Structural Behavior of the First Funicular Polyhedral Frame Designed by 3D Graphic Statics

Mohammad Bolhassani, Masoud Akbarzadeh, Mehrad Mahnia, Ramtin Taherian

##### Abstract

##### BibTex

*Structures*, 2017.

```
@article{mohammad2017structural,
title={On Structural Behavior of the First Funicular Polyhedral Frame Designed by 3D Graphic Statics},
author={Mohammad, Bolhassani and Masoud, Akbarzadeh and Mehrad, Mahnia and Ramtin, Taherian},
journal={Structures},
year={2017}
}
```

##### Three-dimensional Graphic Statics: Initial explorations with polyhedral form and force diagrams

Masoud Akbarzadeh, Tom Van Mele, Philippe Block

##### Abstract

##### BibTex

*International Journal of Space Structures*, 10(1), 2016. #

```
@article{masoud2016three,
title={Three-dimensional graphic statics: Initial explorations with polyhedral form and force diagrams},
author={Masoud, Akbarzadeh and Tom, Van Mele and Philippe, Block},
journal={International Journal of Space Structures},
year={2016},
number={1}
}
```

##### 3D Graphical Statics Using Reciprocal Polyhedral Diagrams

Masoud Akbarzadeh

##### Abstract

150 years after Culmann (1864) established the methods of 2D graphical statics at ETH Zurich, this research aims to establish the methods of 3D graphical statics based on the historical concept of3D 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 redefines their topological relationships to be used as the basis for the methods of 3D graphical statics. 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 the 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 methods of 3D graphical statics, 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.

##### BibTex

*3D Graphical Statics Using Reciprocal Polyhedral Diagrams*. PhD thesis, ETH Zruich, Stefano Franscini Platz 5, Zurich, CH, 8093, July, 2016.

```
@phdthesis{masoud20163d,
title={3D Graphical Statics Using Reciprocal Polyhedral Diagrams},
author={Masoud, Akbarzadeh},
school={ETH Zruich},
year={2016},
month={July},
address={Stefano Franscini Platz 5, Zurich, CH, 8093}
}
```

##### On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

Masoud Akbarzadeh, Tom Van Mele, Philippe Block

##### Abstract

##### BibTex

*Computer-Aided Design*(63): 118-128, 2015.

```
@article{masoud2015equilibrium,
title={On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams},
author={Masoud, Akbarzadeh and Tom, Van Mele and Philippe, Block},
journal={Computer-Aided Design},
year={2015},
number={63},
pages={118-128}
}
```

##### 3D Graphic Statics: Geometric Construction of Global Equilibrium

Masoud Akbarzadeh, Tom Van Mele, Philippe Block

##### Abstract

Based on Rankine’s proposition for the equilibrium of polyhedral frames in 1864, this research provides a clear illustration of the theory of reciprocity between form and force diagrams in three dimensions. It explores the geometric relationship between three-dimensional networks to determine the equilibrium of general spatial frames. It extends graphic statics to a fully three-dimensional method to design and analyze spatial frames such as tree structures, curved frames, cellular structures, etc., under nonparallel, external loads or self-stressed, compression- or tension-only conditions.

##### BibTex

*Future Visions*, 2015.

```
@inproceedings{masoud20153d,
title={3D Graphic Statics: Geometric Construction of Global Equilibrium},
author={Masoud, Akbarzadeh and Tom, Van Mele and Philippe, Block},
booktitle={Future Visions},
year={2015},
organization={Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015}
}
```

##### Three-dimensional Compression Form Finding through Subdivision

Masoud Akbarzadeh, Tom Van Mele, Philippe Block

##### Abstract

##### BibTex

*Future Visions*, 2015.

```
@inproceedings{masoud2015three,
title={Three-dimensional Compression Form Finding through Subdivision},
author={Masoud, Akbarzadeh and Tom, Van Mele and Philippe, Block},
booktitle={Future Visions},
year={2015},
organization={Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015}
}
```