ABSTRACT
Funicular forms The term funicular, as it is used here, means “tension-only” or “compression-only” for a given loading. This is typically considered as the shape taken by a hanging chain for a given set of loads. Because flexible materials such as chains, ropes, and textiles offer no resistance in bending or compression, they can only form funicular structures that most efficiently follow the flow of tension forces. The most efficient way to resist a force is through axial tension, and the second most efficient mode of structural resistance is axial compression. Bending, the method of resistance used in frame structures, is relatively inefficient, at least in terms of the amount of material required for the structure. The relative virtues of different structural systems and geometries, and the choice of one system over another, involve multiple factors, but in terms of pure material consumption the answer is clear – funicular forms are extraordinarily efficient. A key symmetry in nature is the mirrored, inverse, relationship of tension and compression: if a flexible hanging chain (a funicular tension structure) has each of its links welded together, and is then flipped upside-down (inverted), it will stand as a funicular compression arch. The same strategy can be deployed in three dimensions using a hanging sheet of fabric loaded with a thin layer of wet concrete that is allowed to harden (see Chapter 11). Once inverted, the hard concrete acts as a funicular compression vault supporting its self-weight. This makes an architectural structure whose spatial surface is the shape of its own structural resistance to gravity. The earliest example of this kind of form-finding for an ideal arch in compression can be traced to the English scientist Robert Hooke (1635-1703) who, in 1676, first articulated this symmetry: “Ut pendet continuum flexile, sic stabit contiguum rigidum inversum” (“As hangs the flexible line, so but inverted will stand the rigid arch”) (Heyman 1997). The form of the ideal arch depends on the applied loading. For a chain of constant weight per unit length, the shape of a hanging chain acting under self-weight is a catenary (Figure 3.1). But if the load is uniformly distributed horizontally (as in a suspension bridge), the ideal arch would take the form of a parabola, which is slightly different. The chain or cable assumes different geometries according to the loading. Thus, even a simple two-dimensional arch has infinite possible forms that act in pure compression, depending on the distribution of weight and the rise of the arch. A simple experiment with string and small weights can be used to explore the families of funicular forms that are possible. The flexible string will immediately solve the structural form problem for any loading pattern by adjusting its shape accordingly. To continue the analogy with Hooke’s hanging chain, a three-dimensional web of hanging chains, technically called a cable net, can describe a variety of dome shapes. This is essentially how Robert Hooke envisioned the primary masonry dome of St Paul’s Cathedral in London in his collaboration with architect Christopher Wren (1632-1723) – a cubico-parabolical conoid form which is the ideal form of a compressive dome with zero hoop
Figure 3.1 Hooke’s hanging chain and the inverted rigid catenary arch, as depicted by Poleni (1748)
forces (Heyman 1998). However, many more forms are possible for shells. Three-dimensional funicular systems are considerably more complex than two-dimensional arches because of the multiple load paths that are possible through a three-dimensional surface. Unlike a two-dimensional arch, a three-dimensional shell can carry a wide range of different loadings, through membrane behavior, without introducing bending. When flat fabric sheets are used to form threedimensional funicular structures, the geometric possibilities become even more interesting. The warp and weft threads of a woven fabric form a kind of a cable net. By the shearing action of the warp and weft threads, a flat woven sheet can, to a certain extent, produce smooth double curvatures without buckling (see Figure 4.2, p. 53). However, deeper curvatures will start to produce buckles in the sheet. Concentrated tension forces in the sheet will tend to produce pull-buckles along the principal lines of force, as seen in Figure 3.2. (For more on pull-buckles see Chapter 4, pp. 60-3.) These buckled shapes have the potential to create structurally useful corrugations (see Chapter 11: Hanging sheet moulds, pp. 220-3). Such flatsheet behaviors introduce a new vocabulary for thin-shell structures, waiting to be explored by architects, engineers, and builders. Engineer Heinz Isler (1926-2009) derived such forms from hanging physical models to provide stiffening corrugations near the edges of his compressive shells in concrete (Chilton 2000). Structural designers can take inspiration from any number of sources, but Robert Hooke’s powerful axiom provides a clear path forward. The tensile capacity of the formwork membrane and the compressive capacity of concrete perfectly complement each other, while the very act of casting provides the geometric inversion that can “flip” a convex tension-net geometry into a concave compression shell geometry. By minimizing bending forces, designers can build more efficiently and can make better use of limited resources. By understanding and exploring the infinite possibilities for even highly constrained design problems, designers can continue to discover new structural forms for centuries to come.
