Master thesis final submission

Compression-only based structures.
Discrete element assemblies, double curved surfaces, shell structures, mesh tessellation and digital production

This master thesis is a research based project which explores compression-only based structures, covering such subtitles as – discrete element assemblies, double curved surfaces, shell structures, mesh tessellation of symmetrical and asymmetrical geometry together with digital production used for testing structural behavior of masonry structures.
The background of the current research project was based on the works of Block Research Group. During this stage two different softwares were analyzed in comparison to each other (RhinoVault and Kangaroo2 for grasshopper) and the physical model of both structures was made. First experiments as compression-only based physical models were 3D printed from PLA and failed. Moving forward the further research methodology was developed – to start testing from simple structures – like arches and step by step by adding different variables explore the complex systems such as vaults, domes both single and double curved and finally – symmetry and asymmetry. This comprehensive analysis from the simplest element – arch to the most complicated sample – asymmetrical free form vault helped to deepen an understanding of behavior of such structures while testing the impact of thickness, curvature degree and curvature type, tessellation type and joints between the elements design.
Furthermore, project covers digital production and fast prototyping techniques – 3D modelling, parametric modelling and 3D printing together with all challenges which can occur during translation of digital model into the physical model.
Conclusions of the research project and implementation into the architectural field are proposed as analysis of adaptation to flat and uneven terrain, together with the analysis of different possible combinations of shell structures into the one whole system called mereology – in philosophy and mathematical logic, mereology is the study of parts and the wholes they form.

Discrete element assemblies, shell structures, curved surfaces, tessellation

Further development of the thesis project – testing 3 supports supported dome structure worked. Although the side elements tend to fall away and do not have any structural performance in the whole model.

Having a medial spine helps during the assembly process and once it is assembled all other elements can be easily attached to each other.

Documenting symmetrical 3 supports supported dome structure with hexagonal tessellation pattern

Some additional information according to the tessellation of symmetrical and asymmetrical shell structures could be found in the diagrams below.

Tessellation of a symmetrical shell
Tessellation of an asymmetrical shell

While working with symmetry gives an opportunity to automate the process in Grasshopper, working with an asymmetrical shapes requires some background knowledge, starting from medial spine to understanding of a force flow in the discrete element shell geometry to size of each voussoir and as for now is drawn manually. Furthermore, it is impossible to stick with a same number of vertexes for each detail, so the whole tessellation becomes an n-gon pattern.

Final model for testing asymmetrical discrete element shell on a flat terrain condition, size – 70x70x20cm, number of elements ~380 pieces.
Fragment of an asymmetrical model on a flat terrain condition
Layout of details

And finally first experiments with different thickness of a shell from DE.

Mesh offset by face normals using remaped amplitude
Scheme showing the different thickness of a shell having the thicker supports and thinner top elements

Curved surfaces, shell structures and discrete element assemblies

‘Flat pieces cost one dollar, single curvature pieces cost two dollars, double curvature pieces cost ten dollars. The good thing about the computer is that it allows you to keep a close control over the geometry and the budget.’ – F. Gehry

Single Curved Surface – only one of two curves is actually curved, making this shape developable (in mathematics, a developable surface is a smooth surcafe with zero Gaussian curvature, meaning it could be flattened on plane without distortion).

Double Curved Surface – non-developable surface. Saddle shell has compressive stresses along the convex curvature and tensile stresses along the concave curvature. In the 2nd dome example when both dimensions are curving in the same direction meaning that the dome is under compression everywhere.

Hyperbolic paraboloid is a double curved shape which can be created with straight lines, yet non-developable. Both hyperbolic and conical paraboloids has excellent tensile and compressive properties and can be built with straight members, that making the production relatively cheap and easy.

Source: The Function of Form by Farshid Moussavi

Vaults are structural system which are composed of curved ribs and surface elements that distribute loads along ribs and planes, and have greatest efficiency when resisting evenly distributed loads. Vaults vary in kind according to the arrangement of the columns and arches, and the degree of subdivision of the surfaces in between.

Can be divided into 6 primary subsystems defined by the degree of subdivision of the vault surfaces.

There are many great examples of vault architecture built in the mid-centuries, mostly used for sacral building. For the most of these examples building process took even several centuries. Most of them were arranged as a tessellation of the same symmetrical element, although even then some elements in the building system were trying to break the symmetry.

At the 20 century there were two architects – engineers working with curved surfaces – Felix Candela – concrete single surface shells and Eladio Dieste – mainly exploring the masonry structures.

In the works of E. Dieste we can find the same repetition of symmetrical elements, thus making the production easier, cheaper and faster.

In the F. Candela’s approach we can find radian symmetry and repetition of the same element.

Same as vaults, shells are composed of surfaces that distribute loads in plane and have the greatest efficiency when resisting evenly distributed loads.

Going forward to present:

Block Research Group (BRG) for the past 10 years are researching both shell and masonry structures of double curved surfaces. Due to the computational and digital fabrication tools being able to break the symmetry and work with free form structures.

In 2016 at Venice Biennale presenting the Armadillo vault project, which was built from flat stone pieces, without any glue/cement, meaning that structure was compression-only based.

There I want to discuss two different types of joints – the structural and the holding or orientational ones (such as mortise tenon joint). Structural joints – such as found in Frei Otto buildings or Japanese temples are the elements which basically holds all the structure and loads, meaning that without them it is destabilized and without equilibrium, meaning that they are extremely expensive and complex.

When the structure is in equillibrium itself – meaning it is stable by itself, the joint ensures the stability of it according to the external factors, such as wind, earthquake, explosions, hurricanes. For the long-term structures such joints was cement or grooves, sometimes both of them.

The primary approach for the design strategy in this project was based on BRG research methodology, where it consisted of two main stages – design and materialization. Design consisting of form finding, which can be influenced by external or internal constraints (such as – terrain, sunlight, the size and width with height or openings defined by the architectural program, function, vistas, flows (as movement).

And structural analysis with tessellation of a shell.

Production is mainly consisting of two main parts – fabrication and installation.

During the background research I was working with two different plugins – Kangaroo2 by D. Piker for grasshopper and RhinoVAULT by M. Rippmann for Rhino5, the detailed workflow comparison of both of them can be found in the image above.

As both experiments with the structures made with Kangaroo2 and RhinoVAULT failed I was able to formulate further research development, defining these main parameters:

  • Sliding moment or in other words – angle between two elements;
  • Tessellation pattern;
  • Thickness of elements;
  • Orientational joint design;
  • Planarity of elements (meaning that touching surface is reduced by element’s shift during the planarization);

And decided to change 1 thing at the time.

… And experiment failed again.

All that helped me to develop my own research methodology – making things clear – to start from simple things, such as arches, where I could test the impact of curvature degree and thickness.

And gladly, this time that worked.

Where I could make clear conclusion about the matter of thickness not so much of a curvature degree.

Moving forward to add two new variables – curvature type and tessellation pattern testing in it on more complex geometry – vaults.

The conclusions made from these series of experiments were:

  • Running hexagon tessellation has greater behavior of a self interlocking elements;
  • In convex curvature stripes tend to separate from each other by arches both in stacking quads and running hexagon bond tessellation;
  • While in concave – convex curvature by forces coming from the side stripes the problem disappears;
  • Higher degree of curvature or in other words angle between elements has an impact of greater behavior of a compression-only based structures;
  • Hexagon tessellation in concave curvature tends to higher distortion thus becoming more similar to a quad. If the concave curvature would be even higher – hexagon with tends to have two opposite corners going inside the 2D geometry (reminding a ribbon shape);

And to move even more forward – testing single and double curvature with variables such as tessellation pattern and thickness on even more complex geometry – domes.

  • In convex running hexagon tessellation the self interlocking behavior of a pattern feels even stronger, the structure is extremely stable, comparing to all other experimental models.
  • While convex-concave stripes tend to separate themselves into individual arches, although the structure is stable and distribute even loads very well, able to carry 20x of it’s self weight (doesn’t work for a point loads at all…)

As the production strategy in this project is equally or even more important than a design process I would go through key points:

  • Twisting and planar geometry – rapid and easy production is impossible having twisting geometry, therefore discrete elements needs to be planar and 3D printable or easy to cut;
  • As from the previous research it was clearly seen that points loads with the radius of 2mm and the height of 1mm are not enough for alignment between elements and due to the sliding moment in between them causing the collapse of the entire structure they were improved to linear alignment joints;
  • All elements are numbered, but this part could be improved by defining the neighboring elements’ numbers, thus making the assembly possible without any additional instruction;
  • As for the 3D printing strategy – challenges found there consist of tolerances, shrinkage and expansion of the material, so called elephant’s foot at the bottom of the element – all these can be solved by testing, trying and experimenting with your machine. As for the elephant’s foot – chamfering the bottom edges of the elements worked pretty well.

Falsework is made using Bowerbird plugin for grasshopper making it fast and easy both for the design and production.

Supports are modeled manually, but also evolving during the whole research – tolerances, joining separate parts of support structure, reducing mass for faster production, design etc.

Assembly strategy – working with smaller structures helper to very easily and intuitively understand the principle of going by stripes/arches and from medial spine while assembling the structure.

Force flow (black) and perpendicular stress lines (blue) on the shell structure.

To sum everything up, my objectives during this master thesis project is to develop a deeper understanding in the structural behavior of free form structures based on the scientific research method – learning by observation – meaning that infinitely strong digital clay model in digital environment should be tested as a physical model.

Be able to work with complex systems, investigate the possibility of adaptivity of shell structures to different terrain conditions, therefore the shift between the bespoke design and the universal design model.

From flat to 3D

In the beginning of the project I decided to work with folding and bending processes, from a flat sheet of a material making a 3D shape/volume/space. Current method has main advantage of no material waste and is usually used for facade elements such as panelling. At the same time there are several projects where folded sheets could be used as and serve for the structural purposes, as an example the bridge made by CITA (stressed skins or the one related to this).

For the first task I have developed the library of the “rules”, going from a straight line folding to curve, arc, spline folding etc. Having 1,2,3,4 numbers of creases on the sheet of the material.


In the scheme bellow I was looking into the relation between the radius of the curvature and the concave/convex curvatures on a folded sheet. The boundary of the sheet was chosen as simple as it could be, due to the fact of the primarily research. In the scheme bellow the current sheets are folded from 0 to 80 degrees (step every 20) and the radius of the curvature varies from 10-30 (every 10).


After this simple tests I started to combine more complex patterns having both mountain and valley folding creases on the same tile. Tested the outcome and volumetric/spatial qualities of the spaces it could produce (45 and 90 degree fold) at the same time looking into the material deflection depending on the pattern. Test were done on 3 independent and one combined “tile”.


From the results above depending from the pattern I get different results in deflections ratio and also at the 3D outcome from a flat sheet. It is more obvious exploring the sections of current patterns:


For the simulation of the folding I was using Kangaroo2 by Daniel Piker where I took a folding origami example file and looked into how the definition works. According to the rules I have introduced in the beggining I have developed the parametric ability to change the curvature of the line, the number of creases on the sheet and also the checking the deflection of each triangle in a folded mesh.


According to deflection one could think about removing the areas where material had been deflected or even research the ability to change material properties (for example by adding some perforation/dashed line opening  perpendicular to the folding line to achieve the elasticity in the material – but thus are only the thoughts and should be tested in the future research).


The project could be developed in both ways: modelling the 3D shape and unrolling it to flat sheets ot the one I have chosen – taking the flat sheet and discovering what spatial qualities it produces.

The scale of the project varies from furniture/panel, landscape architecture elements, concrete casting slabs or pavilion like buildings.  At the same time as these sheets have the possibility to be flattened on the plane and are developable, ruled surfaces there is a possibility to make straight lines (for example wooden beams) making these “curved” volumes.



Origami algorithm

MIT’s ultimate origami algorithm is unlocking the potential of 3D printers


Demonstration of IST Austria‚Äôs CurveUP technology, turning a 2D shape into a 3D model. Image via: Guseinov, Miguel and Bickel. Picture from

Aggregation and complexity

Testing density control by number of branches and different thickness of field for aggregation.

2D tile and 3D tile joints.

Test with 100 10 x 10 angles

3D space is defined by x,y,z variables. If one could reconsider the space by point cloud where every point could correspond to the landscape/city.


Relation between the part and the final volume.

Test with 100 10 x 10 anglesTest with 100 10 x 10 angles

Physical model of 2D tile:


Physical model of 3D tile and 3D printed joints:


3D print: