This thesis aims to provide a better understanding on the relationship between the mechanical and geometrical properties of shell structures. In search hereof, an attempt is made to describe the thrust surface; a geometrical representation of the in-plane force trajectories through a structure. There exists a well-known relation between the parabolic shape of a (two-dimensional) catenary and the moment line of a simply-supported beam subjected to a distributed load, a relation which is similar to the three-dimensional case of shell structures. In this thesis, this relation is further exploited in the creation of various shell structures, using the moment hill of various simple plate cases. The moment hills of twistless plates as shell structures pose promising results with respect to shell-like behaviour. In the process of generating shell structures from moment hills of twistless plates, establishing the correct boundary conditions has proven to be essential in obtaining shell geometry with maximum shell-like behaviour. The Airy stress function is utilised to get further insight into the mechanical behaviour of shells. Exploiting the reciprocal relation between this Airy stress function and the diagram of forces allowed for both the design and analysis of shells. Taking the reciprocal figure of a discretised version of the Airy stress function results in a force polyhedron which by nature is in equilibrium. It was proven that the rules of graphic statics apply here, the angular defect between two planes then act as force vectors. By means of the force polyhedron a distinction can be made between tensile and compressive forces through any structure. Calculating the angular defect in an edge of the Airy stress polyhedron results in the force through its corresponding edge in its reciprocal figure. This thesis proposes multiple parametric tools with which the reciprocal figure of any Airy stress function can be created. These tools provide insight in the structural behaviour of a shell structure, and aid in the design of shells in the preliminary design stage.

‘Form-finding of branching structures supporting freeform architectural surfaces’ is the graduation thesis of the master track Building Technology at the University of Technology in Delft. This thesis is made by Alex kouwenhoven in the time-span of three-quarter of a year and tutored by Andrew Borgart and Ate Snijder.

With the increasing complexability of the built environment, the challenges for both the architect and the engineer are becoming bigger. Architects tend to make new shapes or free-forms and the engineers are challenged to make it buildable. A shape often used as a support of this free-form architecture is the tree-like column or branching column. A structure branching out and redirecting forces from a big roof surface to one single point.

However, the design of these complex columns is often done by the computer and optimizing software. There is a missing link in the field of structural mechanics and these types of columns, the knowledge of forces flowing through these complex three-dimensional structures.

In this thesis, a method of designing these complex structures is proposed. By an analytical approach, an efficient structural column can be made and multiple optimization strategies are proposed. Following these strategies, a design can be made. In the final chapters of the thesis, an example of a design is made using this strategy.

This thesis aims to exploit active bending as an approach to create complex curved geometries, which are structurally self-supporting systems. The simplicity of creating complex curved geometries from initially planar elements is the leading motivator for this research. The central research question is: ‘How can double curvature be exploited in a structural system composed of elastically deformed planar elements’’? This research answers questions like what double curvature is and how it can be measured and controlled. What is the relationship between an elastically deformed double-curved geometry and the structural characteristics of active bending structures? The literature review is composed of part A: research into double-curvature, and part B: the structural behaviour of active bending structure.

For the question to what extent planar elements can deform, the measure of Gaussian curvature has been used. The Gaussian measure provides the relationship between bending and torsional curvatures. Bending moments relate to these curvatures. However, the Gaussian curvature is a purely geometrical measure without incorporation of material properties. The problem is that deformation of planar elements results in double curved geometries rather than, according to the Gaussian measure, in single curved geometries.

The inclusion of material properties such as Poisson’s ratio and bending rigidity in the Gaussian measure allows for prediction of curvature and the related bending moments. The basic geometry of a rectangular plate has been used to do physical tests, followed by computational test and FEA analysis. It has been proved that the stiffness of an elastically bend plate increases for a large Poisson’s ratio. This makes the Poisson’s ratio an essential parameter for active bending plate structures.

Further stiffness can be achieved through torsion. Torsional displacement of a clamped planar plate leads to increased tension, which leads to increased stiffness. The combination of bending and twisting a planar plate leads to a structurally stiff arch, which is both a structural element and an architectural design component.

Suitable materials for active bending plates structures are composites. They offer a relatively high Poisson’s ratio and a high strength to flexibility ratio.

For a planar plate of epoxy glass fibre composite of fifteen by one meter, only 20 mm thickness is required to make it structurally sufficient.

The feasibility and application of an active bending arch plate depend on the building method. The building method aims to keep the bending process and building sequence simple and modular. The erection process of the arch will be performed through the motion of the ends of the plate in a single direction over a modular gliding rail system, which allows for a simple assembly process.

The double curvature, the structural performance and the aligned building method prove the feasibility of active bending plate structures. Active bending does not have to be considered just a formation process, but a structural system itself. The exploitation of active bending is found in the nuance of double curvature as a result of the material dependent Poisson’s ratio. The simplicity of deforming planar elements, which is feasible with the developed building method results in an active bending arch system which can be used as a cantilever, bridge support, or cladding system.

Architecture within the last twenty years has developed in many ways. One such direction was reintroducing the physical world and physical phenomenon back to architectural language and design space. This is especially true as buildings have had to become more and more environmentally sustainable and developable with programs such as LEED, BREEAM, and Zero House popping up. While less pronounced, the same is true for other aspects of architecture too; such as pedestrian flow, constructability, and structural design. Each design in hi-tech architecture has become a compromise of these systems. While significant funding has gone into understand how to design for energy efficient buildings, architects typically shy away from structural systems leaving these problems for structural engineers to solve. However, great design comes to fruition when the two systems work hand in hand. This method of design, integrated design, is starting to appear more and more in the design space with gridshell structures becoming classic icons of the beauty of this marriage. Gridshell structures are becoming more widely known as architects such as Foster + Partners, Asymptote, and John McAslan & Partners have developed work that has become widely lauded and incredibly awe inspiring. These slender shapes still allow for large amounts of natural light as well as enable architects and designers to cover vast areas without the need of massive beams and a large amount of columns, instead finding rigidity in its own form. This research project, attempts to create a system for architects and designers to play with in order to develop an efficient and useful gridshell concept structure and does this by approaching the perspective from two points. On one side a computational system is built based on Finite Element Analysis (FEA) data and signal processing while on the other, structural concepts are explored in order to determine a geometric relationship between load, form, and principal stresses.

The main objective of this graduation is to find a new structural principle that is emulated by the design by nature. Observing, analysing and transforming the natural design principles and laws could derive a new application in the field of structural design. There are many concepts, approaches and directions to translate biology to architecture. The way to get from biological observations, to physical phenomena, describing these mathematically in the hope to come up with an integrative innovative structural principle for architecture is done through different levels of mimicry.

The observation phase led to a categorisation of biological processes into physical phenomena. It is a necessity to mention that firstly, the twelve phenomena are definitely not the only twelve and secondly, these twelve are not in a specific order or hierarchy. Some phenomena overlap others. To come up with a proper design direction based on natural examinations, a transformation methodology is proposed based on partial mimicry. Every level has its unique grade of abstraction. In the end, after having walked through all the separate levels of the methodology, the derived information will form the basis for the design direction. There are various ways to interpret the methodology. Looping, skipping, going back a few steps or even starting at a totally different level can therefore often be very useful, resulting in more in depth concepts.

The combination of the physical phenomena ‘minimal surface’ and ‘fractals’ formed the basis for my design concept. These phenomena were translated into architectural forms; the anticlastic ruled surface and the branching structure. An anticlastic ruled surface is a negatively double curved surface that can be described using straight lines. This is called a ‘hyperbolic paraboloid’ - in short ‘hypar’. The branching structure is based on the natural fractal of a tree. A split of the main ‘trunk’ is called an iteration. A branching structure defines itself by having one, two, three or even four or more iterations.

Using nature as inspiration combined with mathematics enables us to come up with structurally rational designs. These rational designs are because of its simplicity appropriate for smart solutions on detailing as well. Besides, the research on the structural performance of natural dendriforms contributed to the sustainable development in such a way that material usage was minimized. Ludwig Glaeser rightfully state in his book on the work of Frei Otto that by applying his minimal theories to support elements and space frames, Otto arrived at lighter structures by reducing the buckling lengths of their compression members. This reducement of buckling lengths is also applied to branching structures.

The design development was done one the branching connections, the assembly of the structural components and the water management of the structure. After having elaborated on these facets briefly, the next step was the building method. There are basically three stages when it comes to the building method. The prefabrication off-site, the transportation to the site, and the assembly on-site. For the stage of assembly a so called ‘method statement’ with building sequence is visualised. Finally, the detailling was done, incorporating the design functionalities and the desired design freedom.

There is still a lot to gain from our observations towards nature and phenomena that are all around us. With both of the selected phenomena separately several structures are built and have been proven to work. However the final product of this graduation project; ‘an anticlastic surfaced roof supported by a fractal-like branching structure’ has never been done before and therefore the main objective proposed in the beginning is achieved.

Throughout the long history of architecture, the effect of gravity is always present in any structures. Surprisingly in tensegrity composition, this primary law of nature seems to be absent. This is because the discontinuous set of struts in the continuous network of proportionally thin cables makes the structures look like floating in the air. However, there has not been much application of tensegrity principle in the construction field due to the lack of design methods and its complexity. By developing a large-span structure using tensegrity systems, a design method and an analysis technique are introduced along to define double-surface tensegrity systems. The design approach explores an innovative way to determine the structural topology and geometry of such the systems. The form-finding process and structural analysis are conducted to discover an appropriate way of analyzing tensegrity and ensuring that the systems are stable.