In the following section, the basic structural principle is illustrated by referring to traditional and contemporary construction techniques. Subsequently, graphic statics and its extension to surface structures, Thrust Network Analysis, are discussed. In the last section, the form-finding approach and its implementation as a design tool are tested through a digital design exploration, and finally verified using a 3D printed, structural scale model.
2 Structural Principle
The basic structural concept discussed in this paper is based on the use of compression-only vaults in combination with continuous tension ties. This combination is of course known from masonry domes where continues tension rings are often inserted to resist the tensile hoop force towards the supports. Felix Candela used a similar system of combining tension and compression forces for his umbrella shells in the 1950s.
Fig. 4 illustrates the use of partial compression arches balanced by tension ties, showing (a) a barrel vault, and (b) a modern concrete arch bridge, both during the erection stage. Both structures are in static equilibrium thanks to temporarily attached cables which counteract the horizontal thrust until the barrel vault, respectively the arch, is closed. In these two examples the tension tie is used as a restraint during construction. However, this temporary state could also represent the structural system of a finalized structure. Hence, these examples serve as a sketch to illustrate the half-arch tension equilibrium system.
Fig. 4 (a) Medieval vault construction using tensioned ropes (Fitchen 1961); (b) construction of the Froschgrundseebrücke, near Coburg, Germany.
2.1 2D Form-Finding Using Graphic Statics
The examples in Fig. 4 show how funicular structures such as half arches can stand in equilibrium by introducing tension elements. This structural principle to create large cantilevering structures is illustrated in Fig. 5 using simple graphic statics. Graphic statics is based on two diagrams: a form diagram, representing the geometry of the pin-jointed structure, and a force diagram, also referred to as (Maxwell-) Cremona diagram, representing the equilibrium of the internal and external forces of the structure (Cremona 1890). The power of graphic statics is based on its inherent bidirectional capabilities; one can either use the form diagram to construct the force diagram, or apply the inverse process and construct parts of the form diagram from an intended force diagram, i.e. either form or force constraints can drive the design exploration (Kilian 2006). Geometrically, the relation between the form and force diagram is called reciprocal (Maxwell 1864). Graphic statics also allows to mutually use compression and tension elements which can be easily identified based on their individual direction in the form and force diagram.
Fig. 5 shows the form and force diagram of a funicular arch in compression (a) and the same structure cut in half, with one half flipped horizontally (b). The equilibrium of each node in the form diagram is represented by a closed force vector polygon in the force diagram. Note that the upper ends of both arch segments (b) are in static equilibrium considering the appropriate reaction forces. These reaction forces can either be guaranteed by horizontal supports such as buttresses on both sides or by using a tension tie connecting both ends (Fig 5b).
Fig. 5 (a) Form and force diagram of a closed funicular arch in compression; (b) two, tied half arches in static equilibrium (tension forces marked in blue)
2.2 3D Form-Finding Using Extended Thrust Network Analysis
Imagine now that we take several of the tied half arches in Fig. 5b and arrange them in a radial configuration as shown in the equilibrium network G (Fig. 6). Instead of having a tension tie for each pair of aligned arches, this same spatial layout of arches can be balanced by a continuous, polygonal tension ring (Fig. 7). The horizontal equilibrium, and hence the plan geometry, of this funicular tension ring is defined by the (equal) thrusts of the half-arches, and can be represented by the reciprocal force diagram Γ* in Fig. 7.
Such reciprocal diagrams of form and forces are used in Thrust Network Analysis (TNA) (Block & Ochsendorf 2007; Block 2009) to explain and control the horizontal equilibrium of compression-only vaults. To guarantee compression, Block (2009) explains that corresponding directed edges ei and ei* in both diagrams need to have the same directions, i.e. not only be parallel, but furthermore have the same orientation, resulting in positive force densities q, which are the ratios of the corresponding edge lengths of the force diagram Γ and the form diagram Γ*. Tension elements have negative force densities, and their corresponding edges in form and force diagram thus have opposite directions. For two reciprocal diagrams, the sign of axial forces in the elements can thus directly be obtained by using the normalised dot product of the corresponding edges, -1 for tension, and +1 for compression (Eq. 1).
Fig. 6 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration based on two half arches as shown in Fig. 5b (tension elements marked in blue; for illustration purposes overlapping edges are shown offset to each other)
Fig. 7 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration with a continuous tension tie based on the configuration shown in Fig. 6 (tension elements marked in blue)
Fig. 8 shows the same configuration in plan as in Fig. 7, but with inverted support conditions. While the inner node is free to move vertically, the nodes along the outer boundary are fixed in the vertical direction. Note that this results in a simple dome structure with a continuous tension tie resisting the tensile hoop force at the supports.
Fig. 8 Equilibrium network G, form diagram Γ and force diagram Γ* of a radial configuration with a continuous tension tie resisting the tensile hoop force at the supports (tension elements marked in blue)
3 Results
In this section, the form-finding approach and its implementation as a design tool is tested through a digital design exploration, and is finally verified using a 3D-printed structural scale model.
3.1 Design Tool Implementation
To explore the design space of funicular funnel shells, the TNA method was implemented as an interactive, digital tool. It takes advantage of the inherent, bidirectional interdependency of form and forces represented in visual diagrams, which are essential for a user-driven and controlled exploration in the structural form-finding process. Thus, the implementation and design of the form-finding tool focused on design through exploration, underlining the visual and playful nature of the approach, mainly targeting the early structural design phases. The software was developed for in-house research but also released as simple design tool for compression-only structures, so without the option to include the continuous tension ties, under the name RhinoVAULT (Rippmann et al. 2012) as a free plug-in for the CAD software Rhinoceros (McNeel 2013).
3.2 Design Exploration
Figs. 9-11 show the design exploration of various funicular funnel shells generated with the tool. The variations of equilibrium networks in Fig. 9 are realized by simply changing the definition of free or fixed support nodes, the latter marked with blue dots. The only other modification of the diagrams is the uniform scaling of the edges of the force diagrams. The force diagrams are drawn to the same scale so that the overall magnitude of the horizontal thrusts in each structure can be visually compared. Based on these simple modifications during the design exploration process, the resulting funicular