Machine Learning Assisted Evaluations in 3D Graphic Statics

Hao Zheng, Vahid Moosavi, Masoud Akbarzadeh

##### Abstract

Manipulating the force diagram to control the magnitude of the internal forces in structural form finding is a unique property of geometry-based structural design methods such as 3D graphic statics. For instance, subdividing the force diagram and its polyhedrons using various rules results in a variety of structural forms for a given boundary condition with different load bearing capacities. A recent study showed that the load-bearing capacity of funicular forms could be significantly improved if the force diagram is subdivided for internal members and exterior forces. However, in each subdivision step, the complexity of the structure including the number of nodes and members drastically increases. Since the solution space of all possible forms resulted from various subdivision techniques for a given boundary condition is enormous, it is almost impossible to iterate through all possible forms, and calculate the structural performance for each, and find the best solution within the time limit. Recent advances in machine learning techniques offer promising data-driven approaches to learn the nonlinear and high-dimensional relations between forces and the structural performance of the generated forms.

As a part of ongoing research, in this paper, we show how by using different predictive machine learning techniques such as feeding forward neural networks and ensemble methods one can train a surrogate model to accelerate the structural performance assessment of various possible forms without the need to go through the slower process of geometric operations. After training, the surrogate model has the ability to evaluate an inputted data including the boundary conditions and subdivision rules, and output the predicted value of the load path and the buckling force within milliseconds. Then by comparing the outputs of all possible solutions, the form with smaller load path and larger local buckling force can be found in a much shorter time. Further, this will result in more advanced sampling methods, where the machine learning models assist the designers in choosing different design strategies.

##### BibTex

*Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE*, Barcelona, Spain, October 7-10, 2019.

```
@inproceedings{zheng2019machine,
title={Machine Learning Assisted Evaluations in 3D Graphic Statics},
author={Zheng, H. and Moosavi, V. and Akbarzadeh, M.},
booktitle={Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE},
year={2019},
month={October 7-10},
address={Barcelona, Spain}
}
```

From Polyhedral to Minimal Surface Funicular Spatial Structures

Mostafa Akbari, Mohammad Bolhassani, Masoud Akbarzadeh

##### Abstract

Exploring an unlimited variety of funicular spatial structural typologies is now feasible by using 3D (polyhedral) graphic statics. These structural forms have polyhedral configurations including vertices, edges, polygonal faces, and cells where the edges of the cells can be materialized as members of a structure with a cross section associated with the magnitude of the force (area of the reciprocal face) in the force diagram. The structural performance of such systems relies heavily on the buckling performance of the members. Besides, if the loading scenario changes, the mechanical behavior of the system will immediately change. Moreover, the fabrication process of such systems is quite challenging, due to its complex spatial configuration. Unlike polyhedral systems, minimal surface structures consist of a continuous surface where the internal forces are distributed on a surface with consistent mean curvature rather than the cross-section of the members. Recent studies show that the high surface-to-volume ratio in minimal surface geometries enhances cell proliferation and cell-to-cell interactions, maximizing both porosity and mechanical performance.

This paper proposes a novel approach to translate a polyhedral geometry of a funicular form designed by 3D graphic statics with n-manifold volumetric meshes to a minimal surface structure with two-manifold meshes for materialization purposes. Using such minimal surface structure for materialization in place of polyhedral geometry might improve the structural performance of the system and facilitate its fabrication process. Moreover, spatial shell structures can expand the boundaries of the thrust lines on their surface and thus improve the load-bearing capacity of the system under asymmetric loading scenarios. The proposed approach introduces a new typology of funicular spatial structures consisting of a minimal surface for given boundary conditions. Followed by introducing the computational procedure to generate such geometries, the paper concludes by evaluating the mechanical performance of a cellular specimen and a spatial continuous surface structure based on the same polyhedral geometry with the same volume of construction material.

##### BibTex

*Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE*, Barcelona, Spain, October 7-10, 2019.

```
@inproceedings{akbari2019polyhedral,
title={From Polyhedral to Minimal Surface Funicular Spatial Structures},
author={Akbari, M. and Bolhassani, M. and Akbarzadeh, M. },
booktitle={Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE},
year={2019},
month={October 7-10},
address={Barcelona, Spain}
}
```

Behavior of Modular Components in a Funicular Glass Bridge

Mohammad Bolhassani, Cory Byrnes, Joseph Robert Yost, Masoud Akbarzadeh, Jens Schneider, Andrei Nejur

##### Abstract

The recent evolution of two-dimensional graphic statics (2DGS) into the third dimension (3DGS) has broadened the opportunity for extensive exploration into the design of efficient funicular structural systems. Furthermore, the design and use of structural glass have expanded significantly in recent years. This research exploits the current potential to which 3DGS and computational form finding can be used to extend the boundaries of optimization in the design of glass structures. The ultimate research objective is the construction of a fully-transparent, high-performance pedestrian bridge composed entirely of glass plates, which are oriented in a double layer, funicular, compression-dominating configuration. The funicular form of the bridge, developed using 3DGS, maximizes structural performance and minimizes the use of materials and resources: making it both architecturally unique and structurally efficient.

Unique to this structure is a modular approach to construction, where the bridge is built using a collection of inter-connected polyhedral hollow glass units (HGU). The proposed modular approach simplifies construction, improves quality control and allows for removal and replacement of an individual HGU in the event of damage. The research will initially involve the study of a single HGU. The behavior of an individual HGU will be related to the boundary and edge geometry of the glass plates, the connection detail between glass plates, and the collective interaction between glass and connection. The initial study will involve fabrication methodology, finite element modeling, and experimental testing. Following the single HGU study, will be an investigation of a reduced scale bridge for the purpose of understanding the inter-HGU connection behavior. Finally, a full-scale bridge will be built, modeled and tested. This paper will present the overall research methodology with an emphasis on construction, analysis, and experimental testing of an individual HGU.

##### BibTex

*Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE*, Barcelona, Spain, October 7-10, 2019.

```
@inproceedings{bolhassani2019behavior,
title={Behavior of Modular Components in a Funicular Glass Bridge},
author={Bolhassani, M. and Byrnes, C. and Yost, J. R. and Akbarzadeh, M. and Schneider, J. and Nejur, A.},
booktitle={Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE},
year={2019},
month={October 7-10},
address={Barcelona, Spain}
}
```

Algebraic Formulation for Controlling the Face Areas of 3DGraphic Statics

Masoud Akbarzadeh, Marton Hablicsek

##### Abstract

Geometric construction of the reciprocal polyhedrons as the form and force diagrams are at the heart of 3D/polyhedral graphic statics. Moreover, the face areas of the force diagram are associated with the magnitude of the applied loads and internal forces in the form diagram. Therefore, controlling the face areas in the design process is quite crucial in structural form finding and optimization. In a previous paper, authors presented an algebraic formulation for the construction of reciprocal polyhedral diagrams by computing the edge lengths of closed polygons around each edge of the primal diagram. These polygons thus represent the faces of the dual and reciprocal diagram.

Although this formulation is a novel approach in constructing the reciprocal diagrams, it does not address how the areas of the faces of the resulting polyhedron could be controlled for form finding and optimization purposes. In this paper, we will expand on the algebraic relationship between the area of a face and its edges in the context of reciprocal polyhedral diagrams. In addition, the paper explains the solution space of the equation system comprising of the equilibrium equations and the equations describing the areas of the faces specified by the user.

The approach of this research allows controlling the areas of convex as well as self-intersecting (complex) faces with positive and negative regions per face. With the approach of this paper, a designer can even assign zero areas for certain faces of the force diagram in the form finding process and remove their corresponding force component/member in the form diagram.

##### BibTex

*Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE*, Barcelona, Spain, October 7-10, 2019.

```
@inproceedings{akbarzadeh2019algebraic,
title={Algebraic Formulation for Controlling the Face Areas of 3DGraphic Statics},
author={Akbarzadeh, M. and Hablicsek, M.},
booktitle={Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE},
year={2019},
month={October 7-10},
address={Barcelona, Spain}
}
```

Graphic statics in a continuum field and its application for strut-and-tie modeling

Salma Mozaffari, Masoud Akbarzadeh, Thomas Vogel

##### Abstract

The geometry-based method of graphic statics has long been used to generate truss-like solutions, with the advantage of assuring the equilibrium conditions and providing force magnitudes. Although this method has been successful in the creative and efficient design of discrete or compression/tension-only non-discrete structures, there is a lack of investigation on its capabilities for providing reasonable solutions in continuum fields, where both tension and compression could be carried. This research tends to examine graphic statics for the design of reinforced concrete elements, where a truss model is normally developed to provide an understanding of the load transfer mechanism and an insight into the reinforcement distribution. The computational generation of truss models, also known as strut-and-tie models, has been extensively relied on optimization methods such as layout optimization. While these optimization algorithms are helpful for the initiation of a load path inside a continuum accounting for the design domain and boundary conditions, the inadequacy of control over or creativity in the outcomes is recognizable. In addition, the optimized load path output sometimes requires simplifications or refinements in order to be useful or practical for instance, for the purpose of concrete design.

This paper investigates the capability of the reciprocal relationships of graphic statics for either refinement of the optimization results or providing other load path scenarios. Building upon a previously developed computational procedure, combining layout optimization and algebraic graph statics, first, a strut-and-tie model and its reciprocal force diagram are initiated for a given design domain and boundary conditions. Useful rules for transformations of the form and force diagrams are then established to simplify or rationalize the preliminary optimized form. Through design examples for reinforced concrete, it is explored whether graphic statics can further generate appropriate or innovative designs for reinforcement layouts, where their practicality and usefulness will also be evaluated.

##### BibTex

*Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE*, Barcelona, Spain, October 7-10, 2019.

```
@inproceedings{mozaffari2019graphic,
title={Graphic statics in a continuum field and its application for strut-and-tie modeling},
author={Mozaffari, S. and Akbarzadeh, M. and Vogel, T.},
booktitle={Proceedings of IASS Symposium 2019 and Structural Membranes 2019, FORM and FORCE},
year={2019},
month={October 7-10},
address={Barcelona, Spain}
}
```

Generation of strut-and-tie models and stress fields for structural concrete

Salma Mozaffari, Masoud Akbarzadeh, Thomas Vogel

##### Abstract

Strut-and-tie models are principally the discretized stress field patterns, which simplify the dimensioning and detailing of reinforced concrete members. The computational generation of stress field patterns currently relies on finite element analysis or optimization methods. This research addresses the limitations of optimization algorithms in producing reasonable strut-and-tie configurations. The computational procedure of this paper utilizes layout optimization and graphic statics cooperatively to create strut-and-tie models and stress fields for two-dimensional cases. The presented examples demonstrate the capabilities of the mentioned methods to produce some desired results for two-dimensional scenarios and suggest a similar approach for solving the strut-and-tie problems for three-dimensional cases.

##### BibTex

*Structures Congress 2019*, Orlando, Florida, April 24-27, 2019.

```
@inproceedings{mozaffari2019generation,
title={Generation of strut-and-tie models and stress fields for structural concrete},
author={Mozaffari, S. and Akbarzadeh, M. and Vogel,T.},
booktitle={Structures Congress 2019},
year={2019},
month={April 24-27},
address={Orlando, Florida}
}
```

##### The Design of an Ultra-Transparent Funicular Glass Structure

Masoud Akbarzadeh, Mohammad Bolhassani, Andrei Nejur, Joseph Robert Yost, Cory Byrnes, Jens Schneider, Ulrich Knaack, Chris Borg Costanzi

##### Abstract

This project presents novel research in structural design and analysis of an ultra-transparent pedestrian bridge made exclusively of glass sheets in a double layer, funicular, compression–only configuration. The funicular form of the bridge maximizes its structural performance and minimizes the use of materials and resources. The structural form of the project has been developed using 3D graphic statics (3DGS) that is a geometry-based structural design method allowing the extensive exploration of funicular structural solutions in three dimensions. Using the 3DGS method results in structural forms that are polyhedral geometries with planar faces. Therefore, not only does 3DGS find the efficient structural forms, but its planarity constraint facilitates the construction using flat sheet materials. The current structure of the bridge consists of three-dimensional polyhedral cells as hollow glass blocks with planar glass faces held together in compression by using a transparent silicon-based substance. The total span of the bridge is 10 m (32.81 ft) with a one-meter deck for pedestrian traffic. The asymmetric geometry of the bridge will significantly improve the behavior of the bridge under asymmetric and lateral loading conditions.

##### BibTex

*Structures Congress 2019*, Orlando, Florida, April 24-27, 2019.

```
@inproceedings{akbarzadeh2019design,
title={The Design of an Ultra-Transparent Funicular Glass Structure},
author={Akbarzadeh, M. and Bolhassani, M. and Nejur, A. and Yost, J. R. and Byrnes, C. and Schneider, J. and Knaack, U. and Borg Costanzi, C.},
booktitle={Structures Congress 2019},
year={2019},
month={April 24-27},
address={Orlando, Florida}
}
```

##### 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{hablicsek2018algebraic,
title={Algebraic 3D graphic statics: Reciprocal constructions},
author={Hablicsek, M. and Akbarzadeh, M. and Guo, Y.},
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{bolhassani2018structural,
title={Structural Behavior of a Cast-in-Place Funicular Polyhedral Concrete: Applied 3D Graphic Statics},
author={Bolhassani, 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}
}
```