ISOFF
Concepts and prototypes for flexible moulds for production
of double curved elements
Peter EIGENRAAM*, Roel SCHIPPERa
*Faculty of Architecture, Dep. Architectural Engineering and Technology, Sec. Structural Mechanics, Delft University of Technology
Julianalaan 134, 2628 BL Delft, The Netherlands P.Eigenraam@tudelft.nl
a Faculty of Civil Engineering and Geosciences, Delft University of Technology
Abstract
Flexible moulds for production of double curved elements could offer a solution for challenges in realizing freeform architectural design which tends to be costly and time consuming. However developments are still ongoing and promising methods that result in highly accurate elements still remain to be proven. When surfaces are deformed into a specific shape inaccuracies occur easily due to bucking or wrinkling of the surfaces. In this paper a number of concepts are presented which can be used to prevent these effects. The concepts are based on a fundamental relationship between change of Gaussian curvature of surfaces and in-plain strain. Various prototypes that have been made using these concepts will be presented.
Keywords: flexible mould, Gaussian curvature, shear deformation, prototypes
1. Introduction
Within architectural design a large variety of complexly shaped buildings can be found. Since modelling software became available freeform shapes play an increasingly important role in contemporary architecture. The geometry of these structures poses big challenges to fabrication and construction. One way of facing these challenges is the development of flexible moulds which can be used for fabrication of double curved elements. At the moment the main application of these elements lies in cladding systems like in the ones shown Figure 1 and 2.
Developments in production methods of these elements already become visible. Figure 1 shows the Heydar Aliyev Cultural Centre in Baku (Azerbaijan). The elements for the façade have been created using CNC milled moulds. In general, this fabrication method results in high quality surfaces of the elements. However it also tends to be time consuming and very costly. The reuse of the milled moulds is limited to small series or even to zero. Figure 2 shows the Arnhem central station during construction. The first elements have been placed on the roof. The elements were already precast using a flexible mould system by the company mbX (http://www.mbx.nl/). The curvature of the elements is limited, though. This example illustrates that elements can be produced in large scale and can be offered within a competitive market.
Figure 1 Assembly of the facade elements of the Heydar Cultural Centre in Baku (Wintech [1]).
Figure 2 Assembly of the façade elements of the Arnhem Central Transfer Hall (UN Studio [2]).
Flexible moulds seem to have great potential to be used in construction of freeform structures. However, further development of such moulds should still demonstrate that they can fulfil requirements for production and accuracy. In general, the main benefit of flexible moulds would be the possibility to create uniquely shaped elements using the same, reconfigurable, mould. Flexible mould systems than could provide an alternative for CNC milled moulds and also be implemented in efficient large scale production (Schipper et al. [3]).
The focus of this paper is on geometric and mechanical aspects of the mould system which is the result of the Master’s thesis research by the first author (Eigenraam [4]) under supervision of the second author. Also, some of the more general aspects of this research will be discussed in a second paper in the parallel conference of the IASS. Although this research was performed for development of a flexible mould for the production of concrete elements the presented results are generally applicable for flexible moulds designed for other (e.g. hardening or thermos-setting) materials.
The general concept of making elements as done in this research is illustrated by Figure 3 and 4. Starting with an initially flat mould surface, the concrete is cast on top of a highly flexible interlayer and between edge profiles. After a short curing period the mould surface is deformed by lowering the entire surface until all supports reach their final position.
Figure 3 Cast concrete between edge profiles on an initially flat mould surface.
Figure 4 Deformed surface and element reached by lowering supports.
In general, when trying to deform surfaces like thin wooden plates or paper it can be observed that the surface buckles or wrinkles. This was also observed in a prototype by Janssen [5]. The phenomenon is unwanted for production of smooth double curved elements. Based on a fundamental relation between change of Gaussian curvature and strain of surfaces, concepts have been developed that, when put into practice, show promising results in bringing production of double curved elements one step closer.
2. Gaussian curvature and surface strain
Deformation of surfaces can be described in terms of strain in plane of the surface. Change of Gaussian curvature (Gκ) and its relation to strain is essential in gaining more insight into the limits of
deformation of mould surfaces without buckling. The Gaussian curvature (κG) is defined as the
product of the two principle curvatures (κ1 and κ2) of a surface and named after C.F. Gauss who first
described a measure of curvature. Deforming a surface, e.g. a flat surface (κG = 0) into a double
curved surface (κG ≠ 0) without strain is fundamentally impossible (Gauss [6]). Since material resists
to straining, the change of Gaussian curvature implies also in-plane stresses which above a certain level will cause buckling to occur. Therefore strain needs to be minimized in order to prevent buckling and the question arises: how can a maximum of change of Gaussian curvature be obtained using from as little in-plane strain as possible?
This relation between change of Gaussian curvature and in-plane strain is, both mathematically described and visually illustrated by Calladine [7, Chap.5 and 6]. In-plane strain has two components of axial strain and one of shear (εxx, εyy, γxy). Calladine showed that the relation between change of
Gaussian curvature and the in-plane strain is described with the following partial differential equation (1): 𝐺𝐺κ= −𝜕𝜕 2𝜀𝜀𝑥𝑥𝑥𝑥 𝜕𝜕𝑥𝑥2 + 𝜕𝜕2𝛾𝛾𝑥𝑥𝑥𝑥 𝜕𝜕𝑥𝑥2 − 𝜕𝜕2𝜀𝜀𝑥𝑥𝑥𝑥 𝜕𝜕𝑥𝑥2 (11)
From this equation it can be noted that there are different possibilities to obtain the same level of change of curvature. It is possible to obtain change of Gaussian curvature using only one of the three terms. Figure 5 and 6 show examples of how flat surfaces can be transformed into double curved surfaces. The rectangles in Figure 5 are only strained in axial direction as they are further away from the unstrained centre. Compared to an initially flat grid the squares become tapered. Figure 6 implements the same level of Gaussian curvature, however, now only shear deformation was applied. The latter is the main inspiration for the development of a flexible surface that can be used for flexible mould. It will be shown that using a special design the mould surface can be deformed similarly.
Figure 5 Double curved surface obtained from an initially flat surface by using axial strain only.
Figure 6 Double curved surface obtained by using shear deformation only.
This experiment can also be done by using this paper. The two shapes In Figure 7 can be cut from paper and the author hereby suggests to the reader to take up scissor and cut up this paper for scientific purposes.
3. Concepts for flexible moulds
A deformation of surfaces involves relatively large displacements in both parallel and perpendicular direction of the surface. Also, the mould surface requires a supporting structure in order to obtain a curtain shape. The mould surface as in this research is supported by rods which are adjustable in height. Challenges arise in pursuit of accurately obtaining the required shape. Based on the previous described relation between change of curvature and in-plane strain a number of concepts have been developed and implemented which will now be presented.
3.1. Composed surface
By the previous examples it was shown that double curvature can be obtained from an initially flat surface by using shear deformation only. Also, it was found that a continuous surface is not capable of deforming without buckling. It was found that a surface consisting of multiple slim components can allow in-plane deformation, especially shear deformation. By using strips or wires that can deform individually the double curvature can be obtained. This is illustrated in Figure 8 and 9.
Figure 8 Surface composed of two layers of strips in perpendicular direction.
Figure 9 A wire mesh interwoven to create a surface that allows shear deformation.
Individual components can be subjected to bending in two directions while no change in length is required. The components could be applied in two layers perpendicular to each other. At the point of intersecting the components can be connected such that relative in-plane rotation is still possible. For strips a single bolt is sufficient. A wire mesh can be interwoven and therefore no specific connection needs to be made. In the development of the flexible mould at Delft University of Technology the step towards separate strips was already made in the work of Janssen [5]. However, the question why it seemed to work still remained unanswered. Also, the strips were not attached to each other which introduced large inaccuracies.
The components, strips or wire mesh, would be loaded in and out of plane. Structurally it would be preferable that they have equal second moment of area perpendicular and in-plane of the surface. Strips are only a view a millimetres thick. Making strips of equal thickness and width would be unpractical to work with since there would be many. A compromise was made of strips with approximately 30 mm width and 2 millimetres thick. In order to make the strips more flexible in-plane the strips where given notches which reduced stiffness significantly. The width of the cross section was locally reduced to half. Therefore the second moment of area is eight times smaller. For the first prototype, the strips were given notches by using ordinary equipment as can be seen in Figure 10. Both the notched strips and the bolted connection are shown in Figure 11. Apart from the increase in flexibility, another advantage was that no surface area was lost to support the element on the surface.
Figure 10 In-plane stiffness of the strips was reduced by sawing notches on both sides.
Figure 11 Two layers of perpendicular strips with notches connected with a bolt at the intersection.
A steel wire mesh, like in Figure 9, could have much smaller distance between the wires and therefore creating a supporting surface. Recently, within a 3TU Federation project in which Delft and Eindhoven University of Technology cooperated (names of team members in acknowledgement), also a prototype was build using some of the concepts presented in this paper (3TU Lighthouse projects [8]). The wire mesh used is a readymade product directly applicable. Figure 12 shows the initial and Figure 13 the later prototype which both show a very smooth surface.
Figure 12 First wire mesh prototype. Figure 13 Later prototype with wire mesh surface [Source: Pronk A.].
3.2. Pendulum supports
In order to control the shape of the mould surface the supports are adjustable in height and connected to the surface in such a way that they can pull down and push upward. At the point of this connection the surface translates in both horizontal and vertical direction as can also be understood from Figure 5 and 6. This effect increases significantly further away from the centre and becomes clearly visible for highly curved shapes. Since the support will move sideways they would hinder deformation in case the supports only allow vertical displacements. To prevent jamming and simultaneously provide the possibility to push and pull the surface, a multi-hinged concept was needed for the supports. The developed 3D pendulum support fulfils these requirements. Initially the hinged connections were
made using universal joints. The joints were made using ordinary material available in the hardware store and additionally an old aluminium tent pole since the work was done on a small budget. Figure 14 shows the first prototypes of the pendulums and Figure 15 shows the first application in a full scale prototype.
By equipping all the supports with this pendulum supports the surface would become unstable. Therefore some supports require fixation in one direction in order to maintain a stable surface. All supports in line with the middle support have ordinary hinges that can only tilt towards the middle. Since there are two lines of supports perpendicular to each other the surface becomes stable. Moreover, no pendulum was used for the middle support. In Figure 15 three front pendulums can be seen. The leftmost one has a different connection for this purpose. All supports in line with the middle support have been given simple hinge connections. The pendulum supports showed to be able to follow the horizontal displacements well. For highly curved shapes the angle of the pendulums became so large that the height of the support needed to be compensated in order to obtain the required shape. To determine the amount of compensation of the support height a calculation method was developed, which is not further discussed in this paper. It is mentioned, though, because it illustrates the challenge of obtaining an accurate final shape of the surface.
Figure 14 Double curved surface obtained from an initially flat surface by using axial strain only.
Figure 15 Double curved surface obtained by using shear deformation only.
3.3. Additional weights for loading the mould surface
Not in all cases the self-weight of the material of the element is sufficient to bend the mould surface into its desired position, due to the elastic spring back of the mould surface. In these cases an additional pulling force is required as mentioned in previous section. To apply this force, weights were added to those supports that needed this. The weights were made of concrete and are cylinder shaped with a slot from the side as can be seen in Figure 16 and 17. They can be placed easily if and where required. In automated systems a force can be generated by a small motor. The advantage of manually adding weights is that it is easy and cheap. The operator of the mould visually checks all supports and can decide whether additional weight is needed. Most of the times only a few additional weights are needed. Extreme shapes tend to require more weights. Their application can be seen in Figure 24.
4. Prototypes and findings
Multiple prototypes have been built, all based on the presented concepts in order to explore the possibilities and limits of the mould surface. Prototyping became an important part of the methodology since studying deformation and stresses in the material became complex. Although the research started from a theoretical point, a practical approach proved very helpful. The prototypes provide a lot of insight in the making of elements and into the context of production. In that way it opened the eyes for many unexplored topics such as accuracy, tolerances of surface and edges, stability of the mould under asymmetrical loading, unsmooth surfaces and approximation of maximum reachable curvatures.
Figure 16 Formwork of the weights. Ordinary sewage pipe and a piece of wood were used.
Figure 17 Resulting weights with slots.
4.1 The first prototype using new concepts
The first prototype in which the concepts were applied can be seen in Figure 22 and 23. It had a surface of approximately 600 x 600 mm2 and centre-to-centre distance between supports of 15 centimetres in both directions. The prototype functioned well and demonstrated smooth surfaces for both concave and convex shapes with radii down to one meter. The shear deformation could be observed clearly as can be seen in Figure 18 and 19. Moreover the pendulum supports displaced without jamming and while the surface remained stable. Horizontal displacement can be seen in Figure 20 and 21.
A estimation of the accuracy of the surface was made using a 3D laser scanner which could obtain a point cloud of over 1.3 million points for one shape. The obtained points were compared to the intended shape. Two comparisons for the mentioned concave and convex shape show a maximum deviation of approximately 5 mm with mean distance, median distance and standard deviation of respectively (1.22-1.25, 1.04-1.05, 0.90-0.98) mm. Structurally it was expected that the accuracy would be less near the edges. The bending moment perpendicular and near the edge will be zero and
since bending moment and curvature are directly related the curvature will be zero too. In case only the inner part was taken into account (one field distance from the edge was neglected) the measured deviation was reduced to (0.86-0.96, 0.72-0.82, 0.67-0.72) mm. Efforts were made to determine the errors in the measurements by making a scan of a flat surface. Results were (0.73, 0.59, 0.59) mm. This shows the initial error is quite large compared to the results. More effort would be required to determine the accuracy more precisely. However the measurements do provide a first insight into the deviation which seems to be in terms of millimetres. Also the measurements show that the inner part of the mould is more accurate. From the latter, it can be concluded that using only the inner part of the mould would increase the accuracy of the final elements.
Figure 18 In-plane deformation of the strips can clearly be observed.
Figure 19 Initially straight strip deformed in-plane.
Figure 20 Horizontal displacement of pendulum supports.
Figure 21 Middle support remains vertical and others tilt towards the middle.
4.2 Second larger prototype
A second larger prototype was required to gain more insight into possible scale effects. As can be seen from Figure 6 the shear angle increases fast for areas further away from the middle. The question how far we could push the curvature was still unanswered. Therefore a second prototype was build which is shown in Figure 24. The mould surface was enlarged to 1080 millimetres and two rows of support were added. The centre-to-centre distance of the supports was enlarged to 180 mm. In order to meet specific requirements for connections the universal joints were 3D printed and can be seen in Figure 25.
In practice, the solution for stability using ordinary hinges for supports in line with the middle support was found to be sensitive. During one casting of an element of 2 centimetres thickness the mould collapsed due to overloading of the mould surface. The hinges were not strong enough and stability was not maintained. Since all but one support displaced downward because of tilting of the pendulums the full weight of the element was taken by the middle support which was deliberately not equipped with a pendulum. The support therefore punched entirely through the mould and element. From this experiment it became clear that stronger connections and/or additional stabilizing means were required.
Furthermore, tests with the larger prototype showed that the strain was indeed bigger and caused buckling for some shapes with a radius of 1.2 metre, although not for all. This suggested that orientation of the intended geometry on the mould is of importance. It could be that for each element specific orientation is beneficial. The observations showed a new boundary in terms of curvature. A significant improvement compared to solid plates as was observed by Janssen [5]. Not all shapes can be made with the current mould. Some still cause buckling of the surface because the required strain is simply too large. On the other hand the obtained radii of curvature where down to 1.2 meter which is a level of curvature rarely found in large freeform structures.
4.3 Third prototype with wire mesh
As mentioned a third prototype was made in a cooperation between Delft and Eindhoven University of Technology. Using the presented concepts, different solutions were implemented to realize them. One of the aims of the collaboration was to expand the applicability for materials other than concrete. Results of this research are also being published separately. The prototype was built more robust out of stainless steel which would be important for fabrication environments and also resistant to high temperatures. This is necessary in case the mould will be used for materials such as polycarbonate or glass. For the mould surface, a wire mesh was used and the pendulum supports were made from springs in tubes instead of universal joints as can be seen in Figure 27. The latter have the advantage of returning to their initial straight position and thereby reducing or possibly solving the stability issue. The shapes obtained from the wire mesh are smooth and even more smooth that when using strips. The limits of the curvature that can be reached are yet to be found. This prototype shows that the presented concepts can be work out in various ways. Using the wire mesh and spring connections, the obtained shapes are promising; these improvements bring feasible manufacturing technology for freeform architecture one step closer.
Figure 22 First prototypes (600 millimetres). Figure 23 Freeform shape obtained by the mould.
Figure 24 Second prototype (1080 millimetres). Figure 25 3D printed connections.
Figure 26 3TU project meeting. Figure 27 Third prototype using wire mesh.
5. Future work
Although the results look promising still quite a lot of aspects require more research, of which 5 will be discussed here; (1) In general the accuracy of the produced element requires attention. E.g. for elements to fit closely it is important to position the edges of the mould precisely so that after
deformation of the mould the edge is in the right position. First steps have been made to research this and find a practical way of obtaining accurate positioning using laser projection. (2) Optimization of the orientation of the elements within the mould could improve the accuracy of the final element. This could push the boundaries of producible shapes. (3) There are multiple aspects parameters that influence the stiffness of the mould surface. Various dimensions and materials can still be compared in order to find an appropriate solution. Possibly a different one for different projects will be required. (4) Since the mould once collapsed during one of the tests, stability of the mould should be improved. The third prototype was built more robust. Other methods can be explored too. (5) Application of double curved elements is often limited to façade or wall cladding. However, they have potential to be applied in self-supporting structures, like freeform shell structures. History shows limited, but impressive, examples of these structures but only few in recent years. The development of the flexible mould could bring this type of structure back to life.
Conclusions
Research on a flexible mould for production of double curved elements has been performed using both a theoretical and practical approach. From the present study the following can be concluded:
- The Gaussian curvature of a surface can be changed by using shear deformation only;
- Using a surface that consists of slim components, the Gaussian curvature of the mould surface can be changed in order to obtain a shape that can be used to produce double curved elements;
- 3D Pendulum supports provide means to apply pull and pushing force onto a mould surface while at the same time horizontal deformations can be followed;
- Weights are an easy and cheap method to apply additional force to a mould surface;
- The middle part of the mould surface is more accurate than the area of one support distance from the sides of the mould;
- The maximum curvature that can be obtained without buckling of the surface reduces as the mould size increases.
Acknowledgement
Since a large part of the content of this paper was found during research for the first authors Master’s thesis. I would like to express my appreciation to the members of the graduation committee Roel Schipper (second author), Andrew Borgart and Jeroen Coenders. Your assistance has been most helpful and therefore I would like to thank you.
In corporation with the company mbX useful insights were gained. The authors would like to thank Pieter Nap and Bjorn van Overveld for their cooperation and creating an environment in which the research was challenged to fulfil practical requirements.
Within the 3TU project there was a pleasant and productive corporation for which the authors like to thank the whole team. From the Delft University of Technology: Steffen Grünewald, Matteo Soru, Ivan Gavran and Mattias Michel. From the Eindhoven University of Technology: Arno Pronk, Hisham El Ghazi, Mitchell Janmaat, Tobi Lusing, Erwin van Rijbroek, Niek Schuijers, Martijn Verboord and Robin Versteeg. From SolidRocks: Dick Erinkveld. From the 3TU federation: Siebe Bakker.
References
[1] Wintech, Completion of testing for Heydar Aliyev Culture Centre, Baku, Azerbaijan, http://www.wintechtesting.com/2012/03/07/completion-of-testing-for-heydar-aliyev-culture-centre-baku-azerbaijan/, March 2012.
[2] UN Studio, Projects, http://www.unstudio.com/projects/arnhem-central-transfer-hall, May 2015. [3] Schipper R., Grünewald S., Eigenraam P., Raghunath P., Kok M., Production of Curved Precast
Concrete Elements for Shell Structures and Free-form Architecture using the Flexible Mould Method, Journal of New Building Materials & Construction World, 2015; 20; 100-112.
[4] Eigenraam P., Flexible mould for production of double-curved concrete elements, Master thesis, Delft University of Technology, 2013.
[5] Janssen B., Double curved precast load bearing concrete elements. Master’s thesis, Delft University of Technology, 2011.
[6] Gauss C.F., General investigations of curved surfaces. Royal Society of Göttingen, 1827. [7] Calladine C.R., Theory of shell structures, Cambridge University, 1989.
[8] 3TU Federation, Lighthouse project Kine-mould,
