Constructing hybrid gridshells using bending-active formwork

Abstract

Aiming to promote a more sustainable approach to material utilization in architecture, this article presents an alternative construction method for lightweight and efficient concrete structures. The strategic focus is placed on building wide-spanning roof structures like vaults, domes, and freeform shells with the lowest possible input of raw materials and energy. To achieve this objective, the article explores the novel idea of using bending-active structures, made from millimeter-thin carbon fiber strips, as lost formwork and structural reinforcement for the production of hybrid gridshells. After a general introduction to the state-of-the-art in concrete construction, the authors discuss opportunities and challenges related to flexible formwork designs and their possible impacts on the building industry. Relying on the example of a built prototype, the authors present a promising design methodology and analyze the gridshell’s geometrical and structural characteristics throughout different stages of the construction process. The article concludes by discussing the added value of this research and identifying the key aspects that need to be considered in the further development of this construction method.

Keywords

bending-active structures composites concrete shells construction flexible formwork gridshells lightweight structures

Introduction

Concrete and especially steel-reinforced concrete is by far the most popular construction material in the world and is valued for its low cost, widespread availability, and impressive durability. Its worldwide success is mainly due to its beneficial material composition that renders the possibility to build almost any kind of structure. While concrete offers high resistance to compression forces, its relatively low tensile strength is balanced out through the integration of steel rebar, which gives the material more strength under tension.

In the last decade, however, there has been a growing recognition of the downsides related to the use of concrete, such as high CO₂ emissions, large water consumption, and reduced pervious surfaces throughout our urban areas. In fact, the building and construction sector is responsible for nearly 40% of the global greenhouse gas emissions, 8% of which are due to the production and excessive use of concrete alone.^1,2 In light of these alarming numbers, it is clear that our excessive material consumption must not go on like this. First initiatives to improve concrete on a material level through low-cement mixtures or recycled construction waste are definitely steps in the right direction but do not go far enough. What is needed is a new approach to the design and construction of our built environment.

In buildings, concrete is employed in very different ways, not all of which are environmentally responsible nor economical. For instance, concrete is used in large volumes to make foundations strong and heavy. In columns and walls, the material needs to be heavily reinforced to carry high vertical loads and to brace buildings laterally. And finally, the concrete used in slabs, ceilings, and roofs enables structures to span horizontally. Especially the latter application is fascinating because it can exemplify a more sustainable and efficient use of this material.

For concrete to span over wide distances, however, architects and engineers had to exploit the material’s malleable property to create efficient yet often geometrically complex structures. It is well-known that some forms like folded plates, vaults, and funicular shells, especially when they feature high single- or double-curvature, can carry loads far more effectively because they are primarily in compression and require minimal additional reinforcement. Good examples for this material-efficient approach are the shell structures of Heinz Isler, Felix Candela, Eduardo Torroja, or more recently the Block Research Group.^3
–6

Designing lightweight concrete shells, however, is rather difficult and poses two major challenges. On one hand, the final geometry of the structure is often not known right from the beginning and requires the integration of physical or digital form-finding techniques. On the other hand, the actual construction of complex concrete shells can be very difficult. It usually relies on expensive, custom-made formwork that is produced with high expenditure of material and energy. Thus, investigating new ways to make the construction of concrete shells easier and more affordable would be a valuable contribution toward a more sustainable built environment.

Bending-active structures as formwork

The conventional way to construct concrete shells is by pouring or spraying the material in its liquid state onto a prefabricated and often modular formwork.⁷ This structure usually consists of two distinct elements: a mold and a scaffold. The task of the mold is to accurately reflect the shell geometry and to guarantee a desired surface finish, while the function of the auxiliary scaffold or falsework is to keep the mold in the correct position. Dependent on the size and complexity of the shell, the production of these elements can quickly become very cumbersome and often results in significant labor and construction costs.

Researchers around the world are therefore seeking new means of building shell structures that require less formwork or are easier to deploy. In recent years, notable improvements have been made through the development of textile formwork.⁸ In contrast to rigid molds, textile formwork uses high-strength fabric or steel cable nets and takes advantage of the concrete’s fluidity to create curved geometries. Good examples are the shape-optimized building elements by West and Araya or the thin shells of Veenendaal and Block.^9
–12 While this approach renders many practical advantages, the design space of textile formwork is limited to saddle-shaped, anticlastic geometries, and cannot easily be applied to dome-shaped, synclastic shells.

In response to this problem, the authors developed the idea of using bending-active structures as lost formwork and structural reinforcement for the construction of lightweight concrete gridshells. The core concept of bending-active structures is to use large elastic deformations of initially straight or planar elements for the construction of complex curved geometries and load-bearing systems.¹³ Recent work by the authors and their collaborators has successfully demonstrated the value of bending-active structures in various applications ranging from architectural pavilions, kinetic structures, and compliant mechanisms.^14
–18 Especially when manufacturing curved geometries, bending-active structures can offer a cost-efficient alternative.

One of the remaining challenges for this construction technique, however, is to predict the equilibrium shape of multiple bent components and to feed this information back into the fabrication process. Novel simulation tools such as the particle-spring models in Kangaroo Physics or the finite-element methods (FEMs) in SOFiSTiK have simplified this task significantly. With the help of these tools, it is possible to ascertain the precise geometry of bent objects and to analyze the forces and stresses within the materials. Another, even more fundamental, dilemma of bending-active structures relates to the conflicting goal of building strong and rigid forms from soft and flexible parts. This problem is exacerbated even more when considering bending-active structures as formwork which needs to carry significantly higher loads than its self-weight. It is precisely this research question that the authors seek to address in this article.

The idea of an elastically bent formwork is not completely new and was used before in the context of minimal falsework for masonry.¹⁹ More recently, a research group led by Cuvilliers et al.²⁰ presented an interesting shell that combined a textile- and bending-active formwork. By using a bent gridshell as formwork, the team was able to build a dome-like shell that would otherwise not have been possible with a pure textile approach. The auxiliary gridshell was made from flexible fiberglass rods and gave the concrete shell not only its funicular shape but also played an important structural role by remaining mechanically connected to the shell after the concrete has cured. The concrete in turn had the dual function of providing not only a rigid cover but also braced the flexible gridshell against in-plane shear movements. In this setup, the concrete helped to rigidify the structure, distribute the stresses, and reduce the danger of local buckling. With this prototype, the research team successfully proved the feasibility of using bending-active structures as formwork for the construction of thin concrete shells. Where their research may have left unintentional gaps is in analyzing the geometrical and structural characteristics of the shell during and after the construction process. Thus, the authors of this article would like to investigate these aspects further by designing and building their own experimental structure.

Design development and structural analysis

In the following sections, the authors will present the development of a prototypical hybrid gridshell that was built using a lost bending-active formwork. The paragraphs below will discuss the structure’s material composition, geometrical design, and changing structural behavior as it gradually transitions from a simple formwork to a hybrid gridshell in which the formwork and the applied concrete bond together. Special focus in this study was placed on comparing digital simulations with physical observations, and testing the actual performance of the hybrid gridshell after completion.

Material composition and geometrical design

The design of the hybrid gridshell is based on the opportunities and limitations of two main construction materials, as shown in Figure 1 and further defined in Table 1. For the elastic formwork, the team used strips of pultruded carbon fiber–reinforced polymer (CFRP) composites. This material came in two rolls of 10 and 5 cm width, and the prototype required a total length of 41 m from each roll. With a thickness of only 1.2 mm, the strips were very light and flexible. Prior physical tests on the CFRP strips resulted in a minimal bending radius of 40 cm and a maximum twisting angle of 30° for a 5 cm wide and 100 cm long specimen. For the concrete layer, a fast-setting, high-strength, and low-shrinkage premix mortar was used. The mortar had 28 days mean compressive strength of 48.3 N/mm² according to ASTM Procedure C109, which was tested on 50 mm cube specimens. The mortar set in 15–35 min and developed a mean compressive strength of 17.2 N/mm², 60 min after final set. In the case of this project, the concrete’s tensile capacity was additionally strengthened with an embedded grid of glass fiber–reinforced polymer (GFRP) composites.

Figure 1.

The main materials in this project were (a) strips of fiber-reinforced polymer composites (CFRP) and (b) concrete strips made from a premix fast-setting, high-strength mortar. (Figure 1(a) with the permission of Shanghai Horse Construction).

Table 1.

Material properties of the hardened mortar and CFRP.

Material	Density	Young’s Modulus	Compressive strength	Tensile strength	Ultimate tensile strain
Units	(kN/m³)	(N/mm²)	(N/mm²)	–	(%)
Rapid set mortar	23	35,286	48.3	4.0 (N/mm²)	–
CFRP	16	165,000	–	3100 (N/mm²)	1.69
GFRP grid	27.5	75,000	–	55.4 (kN/m)	2.4

CFRP: carbon fiber–reinforced polymer composites; GFRP: glass fiber–reinforced polymer composites.

Aware of the aforementioned material limitations, the design of the case study was driven by multiple objectives. First, the shape of the hybrid gridshell had to feature a challenging double-curved geometry with areas of high synclastic and anticlastic curvature. The reason was twofold: on one hand, high surface curvature results in forms with better structural capacity and, on the other hand, a system that can be applied to these two geometrical conditions is very likely to be applicable to other freeform geometries. The second design driver was the necessity to build the formwork from straight strips of CFRP and to respect the material’s limited bending and twisting capacity. And finally, the third objective in the design process was related to the overall dimensioning, which had to allow for a sequential application of the concrete and thereby enabling the analysis of the structure’s deformation as the load increases and the concrete begins to cure. Based on these considerations, the team designed a simple gridshell with a span of 2.40 m × 2.40 m in the modeling software Rhinoceros. The starting surface can be seen in Figure 2(a) and was created by sweeping three cross-sections along three arch-like rails. These arches are based on elastica curves as defined by the classic Euler–Bernoulli model for pure bending and can be seen as a post-buckling shape of a beam pinned between two supports.^21,22 It was possible to create different height-to-span ratios by parametrically adjusting the lengths of these arches. Figure 2(b) shows the control logic of the strip layout. The individual strips followed a grid of geodesic curves, which connect opposing edge points on the target surface. Informed by the locally changing tangent plane of the surface, these curves were then extruded to 15 longitudinal and 17 transversal strips, as shown in Figure 3. Modeling these strips with geodesic curves was chosen as a method because it resulted in developable surfaces that can be unrolled into straight strips of varying lengths.

Figure 2.

Generation of (a) target surface and (b) developable strips.

Figure 3.

Curvature analysis of the formwork’s bent and twisted CFRP strips.

The curvature analysis in Figure 3 revealed the required deformation in each strip that is necessary to match the target surface. Evaluating the Gaussian curvature helped to locate the areas in which the strips show the highest torsion, here indicated in dark blue. While a closer look at the mean curvature provided helpful information about how much uniaxial bending is needed to align the strips with the target surface. Both aspects were monitored closely when adjusting the geometry of the gridshell by modifying the height-to-span ratio of the arches and creating a final shape that complied with the material constraints of the CFRP strips.

Structural analysis of the formwork

To prepare the gridshell for structural analysis, two more elements were added to the geometric model, as can be seen in Figure 4. On one hand, a base frame that braced the supports and, on the other hand, a layer of 15 mm thick concrete strips that covered the gridshell in longitudinal direction.

Figure 4.

Model of the gridshell with base frame and strips of CFRP and concrete.

For the following structural analysis, the authors used the commercial FEM software SOFiSTiK. The structural model was generated by turning the center lines of each CFRP strip into beam elements with a cross-section of 100 mm × 1.2 mm. Before going into more detail, it should be noted that the imposed stresses in the carbon fiber strips that are due to the process of elastic bending were neglected in the following calculations. This is due to the fact that these stresses only utilize 3% of the mean tensile strength of the CFRP. To calculate the stresses due to bending, the authors used the following formula

\begin{array}{l} σ_{M} = \frac{E \cdot h}{2 \cdot r} = \frac{165, 000 N / {mm}^{2} \cdot 1.2 mm}{2 \cdot 1097 mm} \\ = 90.25 N / {mm}^{2} \end{array}

(1)

The first step in the analysis studied the structural behavior of one CFRP strip before being coupled to its neighbors and loaded with concrete. Figure 5, shows the individual Eigenmode of the arch-like strip in the center of the gridshell, which has a natural frequency of 5.28 Hz. Since the natural frequency only depends on a structure’s mass and stiffness,²³ it is a good measure to determine the global stiffness of a structural system. Structures with the same mass but higher stiffness will result in higher natural frequencies, while structures with same stiffness but higher mass will have lower natural frequencies.

Figure 5.

The first natural frequency of the CFRP strip at the center is 5.28 Hz (the thickness of the strip is amplified for better visibility).

The second step in the analysis investigated the formwork’s global structural behavior and studied its performance after all strips were connected with each other. The CFRP structure consists of 15 longitudinal strips with a cross-section of 100 mm × 1.2 mm and 17 transversal strips with a cross-section of 50 mm × 1.2 mm. As shown in Figure 6, the first natural frequency of the fully assembled CFRP formwork is 8.15 Hz, which signals a significant increase in stiffness in comparison to the single strip.

Figure 6.

First natural frequency of the assembled CFRP formwork is 8.15 Hz.

The third step in the analysis looked at the gradually changing structural performance of the formwork during the construction process. Here, the CFRP formwork was sequentially loaded with wet concrete strips with a cross-section of 100 mm × 15 mm and a dead load of 0.0375 kN/m. The full loading of the longitudinal arches resulted in stresses of 18.8 N/mm². While the maximal vertical displacement at midspan of the outermost longitudinal strip is 1.4 mm, the direct neighbor only shows a vertical displacement of 0.35 mm. It can therefore be concluded that the loading has only little effect on the geometry of the formwork, and it will remain close to the initially form-found shape. An ultimate load iteration of the CFRP formwork produced an ultimate uniformly distributed line load on the arches of 0.05 kN/m. However, this value does not account for possible deviations that may occur during the successive placement of concrete strips. The full-scale prototype was built by placing the first concrete strip on the center arch and then adding pairs of concrete strips to the left and right until the entire formwork was covered. The time needed to mix the concrete and shape the strips made this a rather slow process, taking the team 1 h to place just two strips. Since the rapid setting mortar achieved a compressive strength of 17.2 N/mm² after 60 min and created a strong bond with the CFRP formwork, the structure’s stiffness gradually increased during the construction process and the load-bearing capacity of the formwork grew with every new pair of concrete strips that were added to the system. An Eigenmode analysis was conducted that digitally recreated the sequential loading process to further investigate the structure’s change in rigidity. Figure 7 illustrates the gradual increase in stiffness represented by the first natural frequency due to an alternating stepwise layering and hardening of the concrete strips, beginning with the longitudinal strip in the center.

Figure 7.

Normalized first natural frequency and normalized ultimate limit load over increasing numbers of concrete strips.

The graph shows that after placing the first concrete strip, the global stiffness of the formwork has more than doubled. From this moment on, the first natural frequency is slowly increasing with every additional concrete strip that is placed until it makes a sudden jump to 29.2 Hz when Strip 13 of 15 is added to the system. This measurement can be explained by the fact that the first natural frequency is mostly governed by the behavior of the two outermost longitudinal strips, which for a long time remained unaffected by the placement of concrete at the center of the gridshell. With the hardening of the 13th concrete strip, however, there is only one carbon fiber strip left on both sides of the formwork. This affects the location of highest impact on the structure’s first natural frequency. Aside from the stiffness, also the normalized ultimate load is increasing slowly after having almost doubled when the first concrete strip had fully cured. Only when the very last concrete strip is placed, the ultimate limit load goes up to its final value of 2.06 kN/m. This can be explained with the fact that due to the low stiffness of the transversal CFRP strips, only directly neighboring concrete strips are affecting each other. Until the final strip has been covered, the strips without concrete will fail at a very low additional loading level. In conclusion, it can be said that the loading sequence with concrete strips and the timing of their hardening has a huge impact on the formwork’s load-bearing capacity during the construction process and will need to be balanced carefully in future explorations.

Structural analysis of the hybrid gridshell

The goal of this section is to examine the structural performance of the hybrid gridshell after completion when the concrete strips are fully cured and have bonded with the lost CFRP formwork. To simplify this study, the 15 longitudinal strips were modeled as pure concrete strips with a cross-section of 100 mm × 15 mm, while the 17 transversal strips were modeled as CFRP elements with a cross-section of 50 mm × 1.2 mm. With this setup, the entire hybrid gridshell has a natural frequency of 26.8 Hz. As shown in Figure 8, the first Eigenmode of the finished gridshell exhibits a much more local behavior at the outermost longitudinal strip. As stated before, this is due to the low stiffness of the transversal CFRP strips and the bigger spacing in the strip layout, which both result in a weaker connection of outermost longitudinal strips with the rest of the structure. The uniformly distributed ultimate limit of a line load on the concrete strips is calculated to 2.06 kN/m resulting in a total vertical load on the gridshell of 81 kN.

Figure 8.

First natural frequency of the finished hybrid gridshell 26.8 Hz.

In the next step, the deformations under a vertical and horizontal unit load vector of 0.1 kN was examined independently for two directions, as shown in Figure 9. For this study, the authors compared the CFRP formwork with the finished hybrid gridshell, which again was modeled with concrete strips in longitudinal direction and CFRP strips in transversal direction. The loads are first applied to the CFRP formwork and then to the hybrid shell. For the CFRP formwork, the vertical load is resulting in maximum vertical deflections of 66 mm at midspan of the outermost strip, and 0.41 mm at the same point in case of the hybrid gridshell. This means a reduction in deformation of 99%. The horizontal load is resulting in the maximum horizontal displacement of 8.38 mm in the lower third of the last arch in case of the CFRP formwork and 0.275 mm at the upper third of the last strip in case of the hybrid gridshell. This means a reduction in the horizontal displacement of 97%.

Figure 9.

Deflections under a point load of 0.1 kN in the vertical and horizontal direction at midspan of each arch.

The analytical ultimate moment M_r,u regarding the weak bending axis lies around 0.0385 kN/m, which governs the ultimate limit load. The governing factor is the tensile strength of the GFRP grid reinforcement that was inserted into the concrete strips during their fabrication. This reinforcement has an ultimate strength of 55.4 kN/m. Doubling the amount of reinforcement or increasing the concrete layer thickness would improve the structure’s ultimate load-bearing capacity. Alternatively, covering the transversal strips with concrete would also increase the load-bearing behavior of the hybrid gridshell. Numerical calculations showed an increased natural frequency of 39.33 Hz. The ultimate limit load would increase to 3.36 kN/m loading on the longitudinal strips. This effect results from a stiffer connection between the longitudinal and transversal strips. A hybrid gridshell with concrete applied in both directions shows a global buckling form that is comparable to previously discussed Eigenmode analysis of the CFRP formwork. Finally, one could also compare these results with a continuous concrete shell with 15 mm thickness, which would most likely have an even better load-bearing behavior but was left out of this study due to its much higher weight.

In the future, the authors would like to extend this investigation by modeling the hybrid gridshell with a compound concrete/carbon cross-section. This level of accuracy would be needed in order to describe the synergy effects created between the two materials and to answer the question how well the CFRP formwork performs as external structural reinforcement of the concrete strips. The authors would expect that the high tensile strength of the CFRP formwork will significantly increase the load-bearing and buckling behavior of the hybrid gridshell and make additional internal reinforcement or increased material thicknesses unnecessary.

Proof of concept

To test the feasibility of this concept, the authors built the proposed hybrid gridshell in full scale. The gridshell was constructed in 2 days in the College of Environmental Design at the University of California, Berkeley. The structure had a span of 2.40 m × 2.40 m and a height of 76 cm. The lost formwork was assembled by riveting together 32 CFRP strips with a total length of 82 m and a width of 10 and 5 cm, as shown in Figure 10(a). The fully completed hybrid gridshell can be seen in Figure 10(b) and featured 15 concrete strips with a cross-section of 100 mm × 15 mm that were placed on top of the lost formwork in a sequential laying and hardening process that took about 8 h. To provide a stronger bond between the carbon and the concrete strips, nails were glued to the formwork in 10 cm spacing, as can be seen in Figure 11(b). This enabled a more efficient transmission of shear forces between the two material layers.

Figure 10.

Full-scale CFRP formwork (a) and finished hybrid gridshell (b).

Figure 11.

CFRP formwork (a) is sequentially loaded with concrete strips (b), which results in the finished hybrid gridshell (c).

To get an actual feeling for its performance, the team put the finished hybrid gridshell to the test with a rudimental loading experiment, allowing students to step onto the structure. Simultaneously conducted digital calculations defined the ultimate uniformly distributed line load on the shell to be 2.06 kN/m. Our team consisted of eight people, and we assumed that each person would add a vertical load of around 0.7 kN to the system. One person after the other stepped onto the hybrid gridshell until the first three were standing on the center strip. Assuming their weight being evenly distributed over the strip’s length of 2.45 m resulted in a uniform load of 0.86 kN/m at this moment, which is 41% of the ultimate limit loading. Loading just a single strip in the digital simulation with this value showed almost no effect in the neighboring strips due to the soft transversal connection.

This observation was also confirmed in the practical test. However, after loading more longitudinal strips by allowing all eight students to gradually step onto the structure, the hybrid gridshell began to fail. More precisely, the collapse was initiated by a horizontal sliding of the friction support at the center strip. With the change in geometry, the bending moments increased and the failing of the center strip led to a cascading collapse of the entire structure, as seen in Figure12. Due to the use of internal GFRP reinforcement, the failure of the concrete strips was brittle. However, after removing the broken concrete pieces from the carbon strips, the CFRP formwork recovered and elastically sprung back to its initial geometry without fracture in any of the carbon strips. This shows that the flexible formwork has an exceptionally high resilience.

Figure 12.

Ultimate loading test reveals the hybrid shell’s resilient behavior.

Conclusion and outlook

The described approach introduces a new way of building hybrid gridshells using a lost formwork made from elastically bent carbon fiber strips, as shown in Figure 13. The archived shape is close to optimal shell geometry and the load transfer is dominated by in-plane forces. Adding a concrete layer to the CFRP formwork improves the load-bearing capacity and stiffness of the structure drastically. After the hardening of the concrete, additional live and dead loads such as glass claddings, wind, and snow loads could be accommodated. Choosing the right sequence in the layering process is important to fine-tune the changing load-bearing capacity of the CFRP formwork and to take the stiffness variations into account that occur during the construction process of the hybrid gridshell. The presented initial research on digital simulation and physical construction is very promising and renders the possibility to be scaled to larger spans. For the future, the authors see great potential in combining this construction method with the technology of three-dimensional (3D) concrete printing and thereby achieving the goal of building affordable, lightweight, and sustainable roof structures in an efficient and economical way.

Figure 13.

Finished prototype of the carbon/concrete hybrid gridshell.

Footnotes

Acknowledgements

The authors would like to thank the Department of Architecture at UC Berkeley’s College of Environmental Design for accommodating this study. Special thanks are also due to our industry partners Kreysler & Associates and SOFiSTiK for donating material and software licenses. Finally, this study would not have been possible without the great work of our graduate student researchers Yasaman Yavaribajestani, Konstantinos Moustakas, and the entire team at the Flexible Structures Lab.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Simon Schleicher

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