Line Geometry for Lightweight Structures
Sommer School 2018
Summerschool LGLS Line Geometry for Lightweight Structures The international summer school “Line Geometry for Lightweight Structures” took place from 10th to 28th September in 2018 at the Technical University Dresden and was conducted by Prof. Dr.-Ing. Daniel Lordick. It was part of the research project “Thin-walled Concrete Structures with Line Geometry” in the DFG program SPP 1542 “Lightweight Construction with Concrete”. During this project the teams of Prof. Mike Schlaich (TU Berlin and Prof. Daniel Lordick (TU Dresden) developed tools and methods for the formfinding and dimensioning of components from ruled surfaces. The goal was to combine a mathematical approach with the requirements of engineering. The core program of LGLS focused on design projects, which were developed in small cross-disciplinary working groups during the three weeks of the summer school. The support program took place at different research institutes as well as some famous cultural institutions of Dresden. The main objective of LGLS were to design lightweight structures using ruled surfaces. Ruled surfaces play a major role in architecture and civil engineering like in the works of Vladimir Shukhov, Antoni Gaudí, Felix Candela, Santiago Calatrava and many others. LGLS is both a summer school to enhance your knowledge and abilities and a design space exploration of
ruled surfaces, especially in lightweight construction applications. To get a deeper insight into the different projects, all groups have written an explanation for their projects. It aims to give an insight into the different working methods and approaches and improves the understanding of the exhibits.
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∞ 44 – The String MODEL Dr. Iva Kodrnja – Mathematician – Croatia Dr. Helena Koncul – Mathematician – Croatia
This string model is a homage to the first material mathematical models used to study ruled surfaces in the 19th century. It is a representation of a 4D cube called a hypercube or a tesseract, a polytope with 4 edges going out of each vertex. Since this is a model of a 4D object in a 3D world, for a non-mathematical observer it is just a small cube in a bigger cube, but in 4D it should be 6 cubes creating a hypercube so that every side of an ordinary cube is again a cube. It is actually impossible to make an accurate model, without being in 4 dimensions, therefore we gave a new twist to the ordinary representation of the hypercube by replacing the inner cube with its dual body the octahedron, and the four connection between vertices is replaced with four ruled surfaces, hyperbolic paraboloids. The outer frame is a cube, a Platonic solid with 8 vertices and the inner object is an octahedron, a Platonic solid dual to the cube, with 8 edges. Both frames of the solids are made of an HDF panel cut by a laser cutter. Each ruled surface is containing one edge of the cube and one edge of the octahedron. These ruled surfaces ( the hyperbolic paraboloids are surfaces of 2nd order) are represented with wool strings which are one set of the lines (rulings ) entirely contained in the ruled surfaces. The octahedron is decorated with 2 spirals which pass through 3 vertices of the hypercube, which is made of one piece 4
of steel wire. Since everything points to number 4, this geometrical structure of the model is also a homage to number 4. From idea to creation When we arrived to the room where the LGLS summer school took place there were some string models of the TU Dresden archive of the mathematical models and the walls were filled with pictures of buildings and sculptures that contain ruled surfaces. Also, the lectures we heard were very inspiring, even though as mathematicians we knew a lot about ruled surfaces, but not much
about lightweight structures. When we got our assignment to make a project the idea was already in the air – we wanted to make our own string model, but we wanted it to be something more, not just to have a ruled surface in a box. It was an appealing idea, since we have some old models of ruled surfaces made by students at our workplace. Now we were at a place where all the tools for making such a model were there and we can make our own. The main idea was to make a hypercube. After modeling few versions of the hypercube in Rhinoceros with some ruled surfaces in it, we tried to make some small models. The first was a cube
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dimensions 10 x 10 x 10 cm with holes on the edges to create ruled surfaces made of strings, just to experiment how will it look like, which strings to choose (elastic, or non-elastic). The elastic string was not a good idea, it needed a lot of adjusting and every idea that something is movable did not work. Then we tried with a bigger 30 x 30 x 30 cm cube and octahedron that can fit in the cube to experiment will the ruled surface be enough to hold the octahedron in place. To make a big model we had to solve few problems: how big it would be, how to cut the frames, how wide should the edges be for the model to stand but not to lose the esthetics, how long string would we need, where to put the holes for the strings, and what will be the distance for two holes. The laser cutter has a limited cutting area and the dimensions of the HDF panels which we chose to make the model is 84 x 54 x 0.3 cm. So, to use the maximum of a panel the model can be the dimensions 80 x 80 x 80 cm, but for that we had to make 24 pieces to create a cube. For the cube to be stabile it was enough to have each piece with the width 3cm. Each edge of a cube was made of two pieces that can fit together as a puzzle and that can be attached to another edge, therefore they were labeled so that they can be glued properly. Only four pieces had holes in them for strings where the rulings of a ruled surface have to pass. The octahedron was a bit easier since it should be smaller, it is made of 5 pieces, but it defines how many holes should an edge have, in other words how many lines will the ruling of a hyperbolic paraboloid have. The holes should be wide enough to insert a string through it and not overlapping, therefore we put 20 holes on each edge of the frame where the surface will pass.
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Finally, we colored the models, attached a steel wire in a shape of a spiral to the octahedron. Now the “stitching� of the ruled surface started, but first we temporarily fixed the octahedron with 8 strings to keep it in place. To make the rulings of 4 hyperbolic paraboloids we needed something over 44 meters of recycled wool.
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∞ 44 – The BAMBOO Model This star-shaped bamboo sculpture is the second representation of a hypercube, with a similar geometrical frame. The outer shape is an 8-star, stellated octahedron, named Stella Octangula by Johannes Kepler in the early 17th century, whose 8 vertices make a cube with invisible edges. It was made so, because a cube is not a stable figure if it is made only by bamboo fixed with rubber bands, while a tetrahedron can be made. The cube is thus constructed as a convex hull of a regular compound of two tetrahedrons, self-dual Platonic solids with 4 vertices. Geometrically, these two tetrahedrons are congruent and their intersection is an octahedron. Their combined Boolean difference consists of 8 smaller congruent tetrahedrons. In each of the small tetrahedrons two hyperbolic paraboloids are placed to contain among them all the edges of the tetrahedron. Small bamboo sticks present one system of rulings of each hyperbolic paraboloid and they are all parallel to the same vertical plane. From idea to creation The idea of this model came from the steel sculpture L’Etoile d’Ouchy made by the architect Angel Duarte in Lausanne, France, 1973. First, we started to experiment with little models, just to see what can be made with bamboo and rubbers. Since we realized that the tetrahedron is the best solid for this project we started to investigate what polyhedrons contain tetrahedrons, whereby the star-shaped solids were preferable because of its appealing look. Also, we played with what ruled surfaces would
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look interesting in a tetrahedron. Hence, we made a little tetrahedron with two hyperbolic paraboloids in it, just one set of rulings for each, passing through two skew edges of the tetrahedron. Finally, for the project we chose the stellated Octangula which we made of three sizes of bamboo sticks, 12 pieces of big (210cm), 8 pieces of medium (150cm) and over 200 pieces of smaller ones (80cm). From the big bamboo the whole stellated octahedron was made, the medium pieces were needed to create the edges inside the polyhedron so that the ruled surfaces could be placed in it. The small bamboo sticks represent the rulings of the 8 hyperbolic paraboloids placed in the stellated octahedron pointing out its star shape.
LSP – Line Space Portal Zlata Tošić – Architect – Serbia Nikolaos Xenos – Architect - Stuttgart Milan Varga – Architect - Belgrade Katrin Pohl – Civil Engineer – Leipzig 1. The theme “Line Geometry with Lightweight Structures” was the topic of the LGLS school in TU Dresden, Germany. The line geometry can seem rather simple and not usable in terms of contemporary Architecture. Nevertheless, there is one of most interesting type of surfaces in line geometry, that should not be neglected, called „Ruled Surfaces“. They enable Engineers to play with various double curved forms, but also stay in the area of lines, where lightweight structures are dominant, which is one of the properties all constructions want to obtain. The Architects: Candela, Calatrava, Torroja etc; used them in their designs which are shown to be one of the most efficient constructions that also have special poetic meaning in Architecture. All their results are based on the extensive positive properties of this shapes. One of the most important are: ruled surfaces are made by motion of the line, they can be double curved which extendes the good quality of their aesthetic as well as construction characteristics, their line geometry enables the production process much easier. In this project all of these advantages are taken into account and shown through our model. 9
2. The idea
3. Design process
In the past ruled surface geometry in Architecture was used mostly by cutting the basic double curved shapes such as: hyperbolic parabolic, cilindroid, conoid or hyperbolic of one sheet... These shapes were either cut or rotated and multiplied and as a result we would usually get symmetrical designed shapes for very large span constructions. To change this trend for a moment our idea for this project was to implement ruled surfaces in more contemporary design cases and carry out a futuristic solution. We wanted to show the power of ruled surfaces where we could make unusual free-form like shapes. It can be said that we wanted to look into future usage of line geometry, so what better solution then to make a pavilion with motive of intergalactic spaceships? They have very complex smooth geometry and can take us to the future!
Initial solution was a variety of designed spaceships that were very futuristic and without any constrains. With the help of plug-in called “Line Geometry“ that research group of professor Daniel Lordick from TU Dresden created, we were able to design freely in Grasshopper and explore the advantages and possibilities of line geometry. We pushed the limits of the “standard“ ruled surface designs and went from some first simpler to more complicated solutions for our pavilion. We have printed some of them in the 3D printer of the SLUB Marker space, which was the available working space for us during workshops
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4. Recourses The idea was to make big pavilion in order for visiors to walk through it and explore its geometry and shape during exhibition. Nevertheless, we had to think about our resources and make
most out of our design goals. The material that we arranged to use was 24 Styrofoam boxes 0.5 x 0.5 x 1.0 m. In the SLUB Marker space we were provided with 4 axis hot wire cutting machine which we are supposed to use for cutting the boxes to make and later connect the elements of our structure. The size of the boxes, quantity of material, limited possibilities of the machine and time were very much limiting the process of production. The hot wire cutter had limitations when it comes to: angle of cutting, size of the cutting unit and speed. Taking all this into account we decided that every styrofoam box was one unit element of the structure with limited angle of curvature. In that way, we to be able to make large cutting elements and assemble them on spot.
5. Optimization process After reviewing the resources that we had aveliable for the production, it was neccesary to optimize our initial designs. During the form-finding process our top priority was on identifying the geometry that enables us to produce our design in large scale. We demonstrated the power of ruled surface geometry in design, but due to production limitations we decided to make a Line Space Portal to take you into the future where you can see and explore a wide range of all possibilities. Intriguing, right? Every portal has an entrance and the exit, so we decided that will be our main focus in design. We came up with 4 ruled surface strips with waves in interior detail and shape which will mimic the energetic transition from „now“ to „tomorrow“ and at the same time show the possibilities of line geometry.
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Shape, size and number of the elements were optimized to both fit into the available Styrofoam blocks and inside the machine’s workspace. On the up left of the picture it can be seen how the elements are perametric checked if they can be placed in the styrofoam boxes. The three shames demonstrate the yellow (unfitting elements) and blue (fitting elements). The minimum number of elements was 8 per strip. The change in the curvature of elements was tested in order to see the difference in results and possibilities of more efficient structure. Nevertheless, the analysis showed that there is no bigger difference in number of elements or size of the structure. Even though we decided to go for the more interesting shape, that solution had more complications during the production. 6. Production process For the production we developed and used algorithms in Grasshopper to segment the geometry into elements that are possible to produce with the 4-axis hot-wire cutter. This first phase was digital and, and it represented the placement of the elements into unit boxes, then positioning the styrofoam boxes on to the hot wire cutter table. Moreover, we had to make sure that all the cutting lines of the ruled surface fit into the frame of the machine in order to be able to cut it. This process was done before every cut. The second phase was cutting process. Each block had to be cut at least twice: once for top, bottom, front and back surface, and another time for side surfaces. To make the process possible and to position the styrofoam boxes it was necessary to number the verticies of the boxes and to take precise coordinates for the machine table. This made the cutting process time-consuming 12
and the curvature of the elements positioning of the Styrofoam blocks inside the machine workspace extremely challenging. 7. Assembling the model Considering the other’s teams projects and other students at University that had to use the machine the total hours of cutting time that were at our disposal in three days was in total 15 hours. For that period of time we were able to cut 11 pieces which is 22 cuts, average 40 minutes for digitalizing, positioning and cutting per cut. Assembling of the model was done partly in the SLUB Markerspace in four sections for transport and whole structure on the site of exhibition place. For the connections we used a combination of glue with small wooden sticks which showed to be the most efficient. 8. Exhibition At the end we can say that we are very satisfied and proud of our results. We were able to present our work to other teams and people from Univerisity during the exhibition. The summer school was very productive and inspiring on many levels because of special lectures, excursions, workshops and team work with people in different fields of study. We would like to thank the professor Daniel Lordick and his research team for making this experience possible. The process of the production was challenging and at the end the result is a growing project of a space portal that continues as we are passing through it.
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Choollapse Bar Maja Ilić – architect – Bosnia and Herzegovina Birgit Schuh – Artist – Germany Biljana Jović – Landscape Architect – Serbia Shinnosuke Tsubaki – Architect – Japan This heterogeneous team had a precise task – to build a BAR, but (luckily) was not given a clear explanation what the bar is… 1. The task – possibilities and limitations While other teams were formed by combining the wishes and aspirations of the summer school participants, with no topic limitations, this team was formed with a specific task - to make a stable structure in the space that could be a furniture for serving and disposing drinks in a closed public space. We called it - the BAR. Aldo we had all the tools and machines in the Maker space at our disposal, the project still had some limitations regarding the available material: bamboo sticks, Styrofoam, rubbers, concrete. At first, this does not seem like a problem if you want to make a small model, but if your task is a structure that must be stable, big and functional, then the struggle between practicality and desire is inevitable.
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2. The team For team it was necessary to adapt to the utilitarian challenges of the project. On the one hand, this may seem like a constrain, but on the other, this could also be seen as a - guidance. And we choose to look at it that way, especially because the team was heterogeneous and consisted of different professional and cultural profiles: an architect, a landscaped architect, an artist and a Japanese guy (also an architect), and indeed, some direction was needed. This interdisciplinary and multicultural team started with an idea of origami structure inspired by presence of Japanese member. At the same time, while struggling to connect and mix available materials, we had to think of the main topic of summer school – line geometry. So, we started with paper models which looked quite nice, especially when Japanese architect made them. Next attempt was to make real scale model similar to origami structure with small bamboo sticks which seemed to be not as stable and beautiful as we expected. We soon realized that the planar origami concept is not the most useful for the material we had at our disposal, and it hardly fits in the idea of line geometry. So, we switched to line elements of bamboo sticks and started to experiment with the stability of ruled surfaces. At the same time, we started to build virtual model and real fullsize model made of big bamboo sticks with strong connectors which made our structure stable enough and satisfying beautiful as well. Three times connecting and reconnecting big bamboo sticks made our fingers hurt but at the end we came out with quite 14
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nice infinite structure which could be modified depending on space characteristics. 3. The Idea – what is a bar? The problem between feeling and working free as well as playing with material on the one hand, and following instructions from “outside” to create a bar on the other hand led us to think about the definition of what a bar is in general – and what could be in our case. Another topic we had to discuss was if we develop our work in relation to the specific space in which the bar would stand. This is very important question for both the artist and the architects, and one would assume that these opinions would coincide, but it was not the case. From this perspective, it seems that this moment of learning about each other was the greatest benefit of the workshop. The artistic approach is more personal and related to the moment and the sense of THAT space in THAT moment, while the architectural is more modular, unified, adaptable and functional. We wanted this structure to be personalized for the space it was intended for, but at the same time to be "parameterized" and thus adaptive to any other space. So, we left the idea of the bar as a common barmen-customer concept, and contemplated a continual line-like lightweight structure that had one or more points of solid elements with a horizontal plane to put the drinks or glasses on. The main idea was to build a structure that does not emulate the bar, that floats through the space and emphasizes it, and at some point, it “catches it” and tames it. 16
The line structure consisting of several HP geometries provided us the perfect tool for this, pointing the “saddle” at the end of this continuity as a horizontal plate that would serve the purpose of the bar. This form, with its geometry, is adaptable and it can adjust to every space. It is a spatial continuum with certain “points of rest”. The line structure was made up of bamboo sticks that we have shortened as needed and fixed with the rubber bands. This bond was not stiff, it was elastic, and so the basic construction had to be connected with a minimum of three sticks in a single node, and tightened with the parallel secondary sticks linked to the outer contour of the structure forming a curved continual surface. The saddle, or the drink storage element, was made of Styrofoam using a styrocutter from Makerspace. The saddle had to be horizontal from the upper side, and from the bottom side it had to be placed on the central part of the last HP. The curvature of the saddle and the bamboo structure had to be the same. This has led to the formation of parallel work processes where each of us contributed with the part of him/herself that was necessary for the project. Landscape architect and artist worked on experimenting and assembling the physical model, while architects worked on the digital model and fabrication of the saddle.
This project was a perfect polygon for testing the workflow
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4. The parallel processes This project was a perfect polygon for testing the workflow between synchronizing a physical and digital model during the process. The structure had to be constantly tested for stability and position in the exhibition place, and at the same time it had to be well-defined in its dimensions because of the fabrication and styrocutting. It went through several phases – from a paper model, parametric digital model, through small scale wooden stick model, middle size bamboo stick model to the full-scale model. The position of the nodes in the structures had to be as accurate as possible, as they determined the curvature of HP's saddles. The physical model needed to be digitized to support the fabrication process of styrocutting, and the digital model had to be parameterized to be able to follow the constant changes that were made on the physical model. So, this was a constant process back and forward. When the final layout of the model was defined, a digital model for styrocutting was produced. The Styrofoam was first glued in the layers to obtain the required thickness and then the required shape was cut from the square. 1. The final touch Since the styrocutter was pretty busy at the time, it slowed down the process but on the other hand gave us some free time to think about the name for our artwork, to make the logo of our team and make few accessories inspired by the name of our product.
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The final structure was very light, but not so easy to transport because of its size. But the good thing was that the nodes were very elastic and adaptable. At the end, this project represents the constant battle between the artistic approach of liberating space and architectural approach of controlling space at the same time. It was the perfect mixture of personal and professional differences, with enough space for everyone to contribute, and finally to LINE UP with the infinite geometry of “the bar�.
Construction CINÉTIQUE Marie Bartz – Architect – Germany Julien Antoine Schmidtke – Architect – Stuttgart 1. The starting idea Oriented on the title of the Summerschool “LGLS” (Line Geometry for Lightweight Structures) at the beginning of the threeweek workshop, all participants started in an initial practice to generate line geometries based on own sources of inspiration. The results of this first exercise were still graphic patterns of sometimes more or a less complex nature and in the end, it was up to the observer these first approaches to the topic simply to look at as two-dimensional images or to consider them as projections of three-dimensional ruled surfaces into the two-dimensional space. The knowledge gain was simple but fundamental for the further course of the project. Line geometries arise everywhere and constantly by simply moving a line through a n-dimensional space. The line, in turn, is just a construct of a pair of two points in space and their belonging to each other. And with this simple insight and an Interest in this dynamic it was clear that our work as part of the Summer School should consider the two motives of movement and interaction.
The idea of creating a kinetic object in line geometry was born. 2. Role models and inspirations The initial Inspirations were, for example, two moving airplanes in the sky which in their choreographic flight create an abstracted line geometry and also the image of the wings of a bird. Which was interesting in several ways because the wings can be generated as a line geometry on the one hand, furthermore they are truly mobile objects and as part of the bird’s skeleton they represent a prime example of lightweight construction in nature. Both topics of the summer school, line geometry and lightweight construction and even the motive of the movement found themselves in this model again. And to maintain the theme of lightweight construction as an elementary component of the project the preferred material was pretty quick chosen. It was bamboo as another prime example of nature for the lightweight construction of a line-like structure. 3. Analog model studies Far from computer-aided design, we began to build physical models mostly by hand. Through this simple and analogue working method, it was possible in a very short time to get a feeling for how a line geometry is generated which on the one hand is able to make a movement at all and on the other hand one that is furthermore elegant and aesthetic at the same time. At the beginning we worked with small wooden sticks and elastic as well as inextensible strings and also fabrics. 19
Parallel to this process, the first digital and parametric models were created, which was still able to represent relatively complex structures at an early stage of the design phase. Exactly by this working method a sensitive understanding was developed of how complex the final kinetic artwork should become. Because in the context of the final exhibition it should work as an eye-catcher which introduces the visitor to the topic of line geometry in a playful way through the possibility of interaction and by this awakened curiosity. So, we started to pick from the repertoire of the first model studies these ones which were feasible both from the later usability by the visitor of the exhibition and also from the production within the scope of the Summerschool. There were more models produced which approached the final Mobile structure on their scale. And in the end, it came to the decision to produce a kinetic sculpture of at least 5-meter span out of bamboo sticks which makes it possible to create associations with the image of the bird’s wings. 4. Scaling problems Due to the circumstances of the later exhibition space, with its five-story airspace above the foyer, the decision was made to hang the kinetic construction as a mobile on a steel cable suspended centrally above the exhibition area. As with any mobile construction, the self-weight of all design-related components had to be taken into account, because of course the final figure in the attached state depends on it. 20
While with the still manageable analog models, we were able to quickly move the suspension points of the cords and with thus the balance, that was no longer possible with the 5 m exhibition object. First, because for the construction on site only a few hours were provided. Secondly, because the final structure, in order to achieve a good spatial effect of the overall object and the two axismirrored ruled surfaces, had to be constructed from many more individual bamboo rods than the previous smaller models. Also, there was no way to suspend the final mobile in advance to see how it behaves and changed its geometry when you pull on the foreseen rope, which is attached to the central axis. That’s why the entire construction had to be prefabricated so that it only had to be hung at the exhibition site. And above all, it should be clear before the first hanging which form the structure takes under its own weight and how the line geometry will behave under the action of the exhibition visitors. 5. Digital model The result to answer all our questions was a digital, parametric model of the entire construction in which certain basic parameters, such as the number of bamboo poles, the distance of the suspension points from each other and the model itself, the rod lengths and the length of the ropes for hanging, were freely adjustable.
Parallel to this process, the first digital and parametric models
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However, in order to be able to examine this model for its behavior in the real world, the parametric model also had to be combined with a live physics engine for interactive simulation. In this context, the bars were assigned a dead load and introduced certain force vectors simulating the gravity, acting on the bars. Using this model, it was possible to determine in advance a fairly accurate picture as well as behavior of the kinetic construction and to determine the final design. 6. Realization Thanks to the digital model, all production parameters could be precisely determined in advance and the structure of the construction of a total of 45 bamboo poles with a total of 84 movable connections could begin systematically. However, the last challenge was precisely these mobile connections. Because the entire construction should be realized as a lightweight construction of bamboo poles and in the digital model, only
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the weight of the rods, but not the joints was taken into account, these joints should weigh as little as possible or better nothing to have no influence on the shape and the movement behavior of the structure after suspension. Screwable elements made of metal or other heavy materials were therefore out of the question and that’s why a fabric-based connectivity was found for the construction. Interlocking loops made out of straps were attached to the bamboo poles with cable ties to create articulated connections without relevant own weight which also, unlike conventional mechanical connections, allowed a free 360-degree movement in between the bars. Even in the digital model, the construction had been simulated with such free movement. At the end, the movable structure was suspended at the planned anchor points and responded to the pull on the provided string by the observer with an elegant movement that reminds of the wing beat of a bird, especially because of the mirror-symmetrical arrangement of the two ruled surfaces.
Due to the interaction with the construction it is possible for the visitors to create a variety of ruled surfaces by own behavior and experience the principle of line geometry directly on their own.
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CUBE RULES Dr.-Ing. Maria Patricia Garibaldi – Civil Engineer – Dresden M.Sc. Iurii Vakaliuk – Civil Engineer – Dresden Dr.-Ing. Sebastian Wilhelm – Civil Engineer – Dresden CUBE, as a part of C3 Project (Carbon Concrete Composite), is assumed to be a building, constructed entirely of carbon concrete. Construction shall be completed by 2020. The planning started at the end of 2017. The novel carbon concrete construction will portray to the public numerous the advantages and potentials of carbon concrete as well as recent results achieved in the C3 project, such as: • • • • •
High structural efficiency. Lightweight, filigree and free-form elements. High durability as a result of high-quality concrete surfaces. Possible integration of multifunctionality. General technology sustainability and environmentally friendly features.
The main purpose of the building is to show innovative construction methods developed within the scope of the C3 project, as well as to provide with a testing facility site where required components of multiple future research projects can be implemented and tested under realistic conditions. Projects that in turn will exhibit other important aspects of the proposed TRC (textile reinforced concrete) technical solutions, such as the development of energy efficient building envelope to reduce the energy demand 24
of the future constructed projects. Additionally, the main goals of the CUBE project include: •
• • •
Presentation, execution and testing of the entire production chain – from planning approval, construction, monitoring to operation. Demonstration of the suitability of the carbon concrete material for the mass production and everyday application. Proof and assessment of carbon concrete as a building material for the construction industry. Combining results from the individual C3 projects.
The provided spaces and areas will be used by scientists and researchers. Therefore, the CUBE building will be adapted with laboratories to introduce and illustrate the latest findings in the field of innovative construction, such as functionally integrated prototypes for energy generation, integrated heating and lighting components, embedded photovoltaics as well as building envelopes with efficient shading system. Both indoor and outdoor spaces assumed to be used for exhibitions and numerous other event formats. Since the shape of the CUBE building load-bearing shell (twist element) was originally proposed as a ruled surface, it was decided to include it to the list of feasible topics within the scope of Line Geometry for Lightweight Structures (LGLS) summer school. Considering the special CUBE building features, and the agenda of the project at the time, the main tasks for the LGLS Summer School were the development of a CUBE building model with a scale of 1:50. Such a task was important to demonstrate the current state of the building design process, to verify different workflows for the production of complicated formworks for
concrete casting, and in general to obtain experience in the development of the ruled surfaces. Of special interest was the understanding of possible critical geometrical aspects and features regarding the modelling of load-bearing parts which were manufactured with high precision. The very first step was of course modelling of the load bearing shell (twist element) using Rhino 3D + Grasshopper 3D software solutions. The shell was modelled according to the latest updates that were received form the multiple CUBE project involved partners. Thus, the model was of great interest for all CUBE project participant, as it provided a 3D visualization regarding the structural scheme of the future building. The shell modelling procedure was relatively time-consuming, due to the features of the shell. It was decided to develop the twist element as a single shell with a thickness of 5 cm and with an embodied sequence of transversal ribs that follow the rulings of the surface. The so-called
edge beams were also integrated to provide a proper force flow at the edges of the twist element. The selected scheme had a huge amount of variable edges and trimmed curves that should be modelled carefully to avoid inaccuracies, collisions and improper geometry solutions, from an engineering point of view. According to the CUBE project, the building’s load-bearing shell should be covered with a special outer façade shell, but in order to provide visibility of the inner structures, it was decided to make the model just with the inner, load-bearing shell. After the development of the main part of the Grasshopper 3D twist element model script, a special script part was added to transform the real scale digital model to the required scale with all consequent model simplifications. Such an additional script provided appropriate and exact data for multiple production tools like a laser cutter, hot wire cutter, 3D printer, etc. To support the twist element, even in the small-scale model, a numerical model was 25
coupled with special parts that describe columns geometry, laboratories walls (so-called box) and slabs geometry. Supporting elements were modelled under consideration of the small-scale constraints such as material and manufacturing limitations. Within the scope of the modelling stage, a range of geometry analysis procedures and discussions were executed. Thus, for example, different methods for creating loft surfaces were compared as well as possible errors that come after application of the offset procedure to the ruled surface. To estimate such errors numerically, a special brief tool was scripted within the Grasshopper 3D environment. As a result, such programming tools are applied gradually to compare and evaluate different proposed solutions of the load-bearing shell and present a report for the CUBE project partners, after the summer school. Additionally, as part of the provided lectures, several topics of interest were discussed including various schemes and basic mathematical principles in the developing of the ruled surfaces and calculation or modelling related aspects. Calculating, modelling and developing of the Grasshopper 3D tool to generate the socalled net surfaces from ruled surfaces was also part of the summer school. This allows cutting correctly the textile reinforcement mats that fit the required shape of the model. The next stage was the manufacturing procedure of the CUBE small-scale model. For this stage, it was decided to use a combination of different possible production techniques. Thus, 3D printing production, using plastic material, was applied to fabricate the complicated load-bearing shell with multiple ribs. Laser cutting technology was a perfect fit for the production of the flat elements, such as columns and walls. And finally, hot wire cutter was used 26
to prepare a special Styrofoam formwork for the shaping of the glass fiber grid element. To summarize the experience from the manufacturing stage, it can be stated that 3D printing was perfectly suitable for the production of tiny and complicated objects such as thin, load-bearing small scale shell with multiple faces oriented with different directions. However, it is important to consider the required time, and the accumulated error during the production of the model, especially for large elements. Therefore, the element selected for 3D printing should be further analyzed to determine its optimized position to avoid printing defects and reduce material consumption. Supporting elements, such as columns and walls, were modelled as a layered structure, to be cut from the sheets of hardboard. In general, laser cutting procedure can be considered as one of the fastest and relatively simple methods for rapid prototyping. For this method, it is still important to provide vector data correctly and adjust laser power properly to burn the material in one step. It was decided that one part of the load-bearing shell was going to be manufactured using the hot wire cutter to prepare Styrofoam formwork to shape the glass fiber textile impregnated with epoxy matrix. But, like any other production technique, hot wire cutter has limitations. For instance, in order to cut shape that
has a significant angular amplitude in the wire inclination, it was required to subdivide the shape into fragments that fulfil the geometry of the model given the constraints of the hot wire cutter movement. Therefore, it was necessary to split the shape of the twist element into 4 parts. After the cutting, fragments were glued together using Styrofoam glue and prepared for the casting stage. Then, the glass fiber was placed in position over the negative shape Styrofoam blocks and coated with epoxy. Additionally, it is important to mention that a special adjustment of the hot wire cutting machine was required, as recommended by the mentors, to achieve high precision and proper, fast cutting. Afterwards, the last step was assembling of all separately manufactured parts and the subsequent painting of the assembled model. In general, it may be concluded that the model making within the scope of the LGLS summer school as well as research procedures regarding ruled surfaces properties was extremely useful and helpful for the proper modelling and evaluation of challenging projects such as the CUBE. It is envisioned that the knowledge gain can be applied to future projects as well. Thanks to the well-organised lectures, workshops and efforts of all mentors, the knowledge that was obtained during the summer school will help greatly and will improve the modelling and manufacturing of the ruled surfaces as parts of next projects related to the field of civil engineering projects that demand the realization of complicated shapes.
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Texperiments Anna Wetzel – Fashion Designer – Germany Marina Mudri – Architect – Serbia „Texperiments“ is an exploration of connecting the real world with textile simulation to find lightweight surfaces. The core of the „Texperiments“ is the rebound force of fabric in combination with a reinforcement. The basic idea was to prestress fabric and apply reinforcement, then release the fabric from stress and create 3Dshapes as a result. The contrasting properties of both materials make this possible. Additionally, we wanted to see how close a computer simulation can get to these physical results and we were curious if the computer simulation could help predicting the outcome of the experiments in the future.
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Part 1 (the physical experiments) The decision which fabric was going to be used was dependant on the outcome that should be created. We decided to choose spandex fabric that has 3 key properties: lightness, extreme elasticity and high resilience. Combined with reinforcement from veneer (ductile), cardboard (slightly ductile) and MDF (not ductile), the experiments show a variety of shapes through prestressing the fabric. Making the tactile experiments consisted of multiple steps. First, we had an exchange of ideas, which type of shapes/surfaces we would like to create. The desire was to try and achieve ruled surfaces. The ideas where to place the reinforcement came partly from the tensegrity pavilion, partly from own experience having worked with this method already. We sketched where to put the
The exact process of the physical experiments was as following
reinforcement on the prestressed textile and made first try outs with thin nylon fabric and cardboard stripes. We also experimented with attaching strings to the cardboard stripes after we had created an interesting surface, to see if we could alter it even more by pulling the strings. We didn't follow the idea of strings in the end. We decided to give the spectator the task to pull the fabric samples with their own hands. After we created the small tryouts we went into a bigger scale and made vector patterns on the computer from our hand sketches. The CAD patterns served to make laser cut stencils from cardboard, cutting the reinforcement and using one of them for further computer simulation. This will be explained in part two of the project.
A 40 x 40 cm wooden plate was laser cut with toothed edges, so the spandex fabric could easily be spanned onto it. One side of the plate needed to be covered with a layer of tape, to avoid glue sticking to it later in the process. The spandex was cut into 40 x 40 cm squares and spanned in different intensities and directions on the board (2,5% to 25% to 35%, horizontally, vertically or both). Afterwards the cardboard stencil was taped onto the fabric and the reinforcement was placed and glued into the gaps. Everything was let to dry under a light amount of pressure. Later, the stencil was removed, and the reinforcement checked to be adhered correctly. Then the stress of the fabric could be released by cautiously pulling the reinforcement and the fabric off the plate. Then the shape was finished. There was no postprocessing. We noticed that sometimes the surfaces we were trying to create manifested, some surfaces were unlike we hoped they would be. Nonetheless this was the power of the physical experiment: we came across unexpected new results. Finally, we chose our favourite experiments for the computer simulation. We wanted to recreate the physical model on the screen to see how close it can get to reality and if we could predict the outcome of the physical experiments in the future. Part 2 (the virtual representation) The computational visualization of the project consists of two opposing problems, i.e. the motion of the figure must be contained in both, the moving and the fixed, rigid parts of the object. It is worth pointing out that the same logical approach was used for all represented types of models. Accordingly, we can divide the 29
workflow into six segments, that then comprise the geometry part, the lists operation, the engine setup and refining, and the visual representation of the former. The figure below displays the whole definition for the model 1 (the model with parallel reinforcement). First comes the geometry, that is, the drawing of the model, and in this case, it was imported into Grasshopper from Rhino. Afterwards, the lists command creates constraints for the cloth, and constraints for the rigid bodies, which, for the real model are sticks glued to the textile. Constraints for the cloth are rigid bodies, and constraints for rigid bodies are springs, which simulate the force that generates motion. Here we can notice the difference between the physical and the simulated models. When comparing the final appearance of an object, if pull being a negative force and push being a positive force, physical models are affected by positive force and simulated models by negative force. Required after setting the geometry and two types of constraints, or rather before setting constraints, are Flexhopper engine functions. When having a working setup, the fifth segment in the workflow would be fine-tuning of the parameters and the whole definition. Last comes the visual output that can be adjusted to preferences. Hereafter, mostly dealing with lists will be interpreted, as it frames the core difficulties in the definition. If we were to design a simpler homogeneous system, managing lists could maybe be surpassed. When having a rigid system and a free system to act as one, precisely the question of their binding complicates the whole structure. After the geometry is imported, the cloth and rigids matching area needs to be defined, which indeed indicates isolating a list of points located on the cloth, only that match the bounds of rigid elements. This list firstly can be used to name its items as 30
Flexhopper particles, for the reason that those elements are meant to move by force of spring particles and this way there is a uniform distribution of holders. Considering particles for points now (because whatever their term, they are essentially referrals for the elements), they are not simultaneously perceivable to Flexhopper as rigid elements. To address them as such, rigid elements, the original list of isolated points can be used if shifted for its length number and number of spring points list, for a purpose to establish a list of items that are constraints for the cloth, which are really rigid bodies. Additionally, this list is merged with a list of points that represent the points of a height of the elements.
which the fabric characteristics and position of forces equally partake for the change. In the simulation are shown two types of models, of which the second type demonstrates three different versions of ruled surfaces, as the topic for the project is the exploration of lightweight ruled surfaces, and as this particular one examines the communication between real and virtual.
Now, going back to the spring constraint definition. From the list of top points, it is selected branch of items that frame the specific rigid element. These are intertwined with the spring particles list, which force strength is controlled by length and stiffness of the springs. The pull is simulated by a contraction in the spring’s lengths. The equivalent between the real-world application of forces and the ones that have to be simulated does not exactly compare, as it is discussed in the introduction, regardless, we can get a similar output. However, this difference conditions the conception of which mimics which, i.e. the virtual models are not as easy replicated after real ones, only vice versa. Regarding the parameters fine-tuning, even if the cloth alone resembles, by features, the real-world fabric, the difference here is that by changing, for instance, only stretchiness degree, we get the exact same motion and shape, especially because the stretchiness and tension are not internally affiliated, in opposition to the real-world models for 31
torrojometry Juan Pablo Letelier – Civil Engineer – Berlin Jack Lehrecke – Civil Engineer – Berlin Max Dombrowski – Civil Engineer – Berlin Jan Philip Schulze-Ardey – Civil Engineer – Aachen Doubly curved concrete shells can exhibit an extremely efficient load carrying behavior due to their ability to redistribute spatial loads through their geometry. Because of this great sensitivity between geometry and performance, form finding and form optimization is integral to the realization of shell structures. When designed properly, shells can span large distances with minimal cross-sectional areas, greatly reducing overall material costs. However, this structural efficiency is often achieved through the
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application of a complex and doubly curved geometry, the construction of which can be extremely uneconomical in comparison to more conventional architectural forms. Ruled surface concrete shells offer a possible solution to this problem, allowing for the realization of complex curvatures using primarily linear formwork elements, greatly reducing construction costs. This potential of ruled surfaces for architectural application was most notably demonstrated in the mid-20th-Century. Pioneers such as Torroja, Candela and others utilized these linearly generated geometries to realize a number of elegant and structurally efficient forms. The concept behind the project Torrojometry was twofold and branched off into two separate designs. First, an existing structure, Hipódromo de la Zarzuela, was chosen as the base geometry for investigation. The large reinforced concrete structure, designed by Eduardo Torroja and completed in 1941, utilizes a
trimmed section of a one-sheeted hyperboloid and acts as an overhang for the stadium seating of a racetrack in Madrid. The slenderness of the resulting form in comparison to its span is truly impressive and a testament to the ingenuity of Torroja at the time. Our goal in the first stage of this project was to parameterize the geometry used for this structure and scale it to a model which we could recreate with a formwork created using the hotwire cutter provided by the LGLS. The scaled model would then be outfitted with textile reinforcement allowing for the construction of an ultrathin shell structure. The second aspect of the project was to explore variations of the hyperboloid geometry used for the Torroja model. Doubly curved shells derive much of their strength and ability to efficiently redirect incident loads through the degree of their curvature. The flatter a shell’s geometry is, the more it relies on bending as a load carrying mechanism and the more it ultimately becomes vulnerable to large deformations. To this end, we wanted to optimize the geometry of a trimmed one-sheeted hyperboloid to achieve a maximum curvature with respect to a given set of boundary conditions. We were limited by the size of the Styrodur block used for the formwork and the maximum angle that the hotwire cutter was capable of. Because the one-sheeted hyperbola is a doubly-curved surface, it has two principal radii of curvature at any given point. The Gaussian curvature is then defined as the product of these two curvatures and this was chosen as the value to maximize during the optimization. To carry out this optimization, a multi-objective evolutionary algorithm was applied to the given geometry with the constraints mentioned above. This resulted in a range of forms possessing an identical maximum Gaussian curvature, achieved 33
through variable degrees of curvature along the principal radii. Ultimately, a form equally curved along both axes was chosen as it highlighted the double curvature of the geometry.
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© TU Dresden – Institute for Geometry – Prof. Dr.-Ing. Daniel Lordick
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