The following excerpt appears in the book SpaceCraft, published by RIBA Press: http://www.ribabookshops.com/site/viewtitle.asp?sid=&pid=8540&HID=

All architecture implies structure, but structure does not imply architecture. This is the first dilemma that faces the designer. In many minds, the simple act of creating a structure means creating architecture. In the case of buildings, however, there is a qualitative difference between something earning the label architecture and something that is simply ‘building’. How the structural component of architecture is integrated into the overall concept is an important ingredient. It is one that the Modern Movement has increasingly put to the forefront. The argument that elegant structure is, by definition, architecture is somewhat suspect, however, as is the inverse – that compelling architecture necessarily has to resolve structure elegantly.

The advent of computing has renewed interest in ‘free-form’ design, and the ability to describe mathematically complex forms with complex curvature has resulted in a degree of experimentation that is only natural with the advent of this new technology. The current state of affairs is nevertheless a half-way house. Digital manufacturing has in no way kept pace with the formal possibilities given by the ease of access to mathematic libraries by modern CAD systems. As a result, many of the most interesting architectural techniques of today mediate between the ideal complex geometry and its manufactured manifestation in the real, full-sized world.

Following on from that is the expression of the manufacturing and assembly processes. The nuts and bolts of the assembly system do literally become architectural form, just as the extrusion process of material manufacture dictates the aesthetics of the total product. But an obsession with detail does not in itself make great architecture. The composition of the parts and the spatial manipulation lies at the heart of true architecture. So how do computational strategies help create a fusion between form, structure and an expression of material and process?

It is essential to approach the issues with techniques that actually enhance the product. One of the more spectacular failures is the notion of ‘automated design’ where some kind of process (random or inspired by, for instance, Darwinian selection) actually creates architectural forms. The architect is reduced to some kind of form selector rather than a designer driven by a self-given goal. Optimisation is a technique that promises more. In essence, the technique is one where a system delivers the ‘most efficient solution’. The devil, as usual, lies in the detail. What is our evaluation criterion? If we have multiple criteria, how is one traded for the other? What are the inputs and variables that can be tweaked? The issue lies not so much in the computation, which nowadays is very efficient, even in massive multi-dimensional spaces, but in the definition of the problem.

The classic problem that illustrates optimisation is that of the travelling salesman – how to find the shortest path connection between a set of destinations across a two-dimensional plane. This problem turns out to be ‘NP-hard’ (where NP stands for Nondeterministic Polynomial time), which means that solution time explodes exponentially with the set of points. However, the great feat of optimisation theory is to verify that there are efficient ways of solving the problem. To understand how this works, visualise a closed body in three-dimensional space. All the possible solutions lie inside the volume; however (and this is the key), all solutions that could be minimal will lie not just on the surface, not just on the edges, but absolutely on the vertices. Optimisation travels along the edges (which connect the vertices) and compares the vertices’ efficiency and then travels to the next more efficient vertex. This can lead to finding just a local optimum; surprisingly, in the case of the travelling salesman problem, this is exceedingly rare and the best solutions can be found reasonably quickly.

Most engineering problems are far more complex and need more variables to be evaluated against each other. These variables are connected by rules (just as Euclidian three-dimensional space is dictated by the rules of Pythagoras’s theorem, and the transformation from three to two dimensions is governed by the rules of projective geometry). For mathematical convenience these variables are called ‘dimensions’ or ‘degrees of freedom’, and they are generally visualised in abstract form through equations and matrices. Multi-dimensional spaces tend to frighten the uninitiated. It is, however, important to understand the general concepts and mechanics because so much of modern mathematics and computing depends on it.

We all understand (we think) ordinary Euclidian three-dimensional space. Abstractly, the importance of the three dimensions is the set of rules that binds them together; for instance, the fact that (thanks to Pythagoras) we can calculate the shortest distance between two points. Embedded within three-dimensional space exist lower-dimensional objects: one-dimensional objects such as lines with lengths, two-dimensional surfaces with additional properties such as area and Gaussian curvature, and three-dimensional solids with properties such as volumes and hence mass.

Once we are properly in three-dimensional space, volumes (hence mass) and cross-section areas have utterly profound consequences. The strength of a material depends directly on its cross-sectional area, and its mass depends on its volume. If you double the (linear) size of an object its mass will increase as the cube of the transform, but its strength only as the square (given the same geometry, but there are subtleties as we shall see).

Large objects are effectively less strong and this phenomenon not only dictates how we approach structure as an architectural tool, but it determines the structure of the universe – from galaxies to quarks. In architecture the consequence is that large buildings are very different from small ones. The effort of supporting large spans or cantilevering tall objects rises as the cube of the size. We cannot expect similar solutions to work at different scales.

The solutions will all obey the simple structural concepts first appreciated during the nineteenth century. Beams don’t get stronger by being thicker, they get stronger by being deeper. The distribution of material within the structural system is critical. We learnt all this in structure 101 and it is worth revisiting these fundamental rules for horizontal structural systems – the rules apply to every horizontal structure, be it a short beam or a large- span arch. The critical dimension is the structural depth, which translates into the ‘second moment of area’.

The second moment of area states that the vertical distance within a horizontal structural system is proportional to the square of the vertical distance, that is the vertical separation between the top structural member (normally in compression) and the bottom structural member (normally in tension).This simple fact dictates the configuration of all horizontal structural systems. The most common example where this is made explicit is the I beam. On the face of it, the arch seems to behave differently. However, as a structural system it contains two components: the arch itself and the forces tying it together at ground level (explicitly as a tie or implicitly by a foundation). In an arched enclosure we use the space within the structural system, where the rise of the arch is the vertical distance of the second moment of area.

Materials behave differently under different loads. Tension as a system is very important because, unlike compression systems, there is no buckling and thus tension-based systems on a per-weight basis are more efficient. Tension-based systems have very different configurations, the principal issue being that tension has to be maintained under all load conditions. Tent-like structures therefore use curvature as a means to maintain form and are saddle-shaped so that the opposite curvatures can both be in tension and thus maintain their shape.

Curvature is one of the most critical issues of built form after the fundamental configuration. It is striking that as a species we choose to want the most difficult system of all on the horizontal: the flat plane. Flat floors have serious problems – span and deflections are the two limiting factors in any multi-storey design.

Vertical walls, however, exploit gravity in a positive manner – we use gravity as a post-compression strategy. In walls, gravity does all the structural work for us.

Case study

The Pinnacle(formerly the ‘Bishopsgate Tower’), London

The Pinnacle is one of a number of tall buildings planned for the City of London (figure 1). The 100,000 square metre building has been configured with a visually fragmented profile to break down what could otherwise become a massive imposition on the London skyline; furthermore, a series of strategically important views, including views of St Paul’s Cathedral, have had to be respected. The design, therefore, has grown from a consideration of a number of key factors, including planning constraints, aesthetic considerations and commercial demands (figure 2).

The tower was designed primarily with Bentley’s Generative Components parametric modelling software (then in Alpha release stage, now a released product), while our ‘Bishopsgate-System’ was implemented in the C+ programming language as an in-process server to the parametric platform. The interactive system is composed of a series of object-oriented assemblies specialised on specific aspects of the design space. The drivers for the selection of this development configuration were dictated by the need for high-performance, interactivity and flexibility for accommodating the scale and complexity of the project.

The primary design focus for the tower was to produce geometry that could be relatively easy to manufacture and communicate. The fundamental decision was taken in summer 2004 to build the tower’s geometry entirely from lines and arcs. Subsequent design development has substantiated this approach. It has, among other things, yielded straight columns, together with beams that are either straight or simply bent to a radius (figure 3)

The geometric approach is based on a number of simple constraints, while including flexibility in the design process. This need for flexibility means that the focus in the design process moves away from designing the object towards designing the system that designs the object. The tower is therefore built on a sequence of parametric dependency models, always responding to the demands of the process.

The original form-finding search (early 2004) was essentially manual, when a wide variety of options from very simple geometries (cylinders, prisms, etc.) were evaluated (figure 4). From this original work it became clear that a roughly triangular floor plate provided the only reasonable solution for this particular site, capable of accommodating the floor space required to make a viable project. The problem with a triangular floor plate is that its townscape impact when viewed edge-on is essentially benign; however, when viewed face-on the visual impact is large.

The design team approached this problem by creating an envelope which divided the volume into two visual components. This pair of components was then tamed in townscape terms by creating a helix which cuts into the wrapped surface (figure 5). The helical cut starts at the height set by the adjacent Tower 42 and 30 St Mary Axe (at approximately 180 metres). The cut then traverses the surface to the pinnacle of the building (at around 288 metres). One advantage of the form is that its asymmetry allows clear orientation in relation to the city. The overall form is thus designed to achieve maximum slenderness for the considerable bulk of the building.

The geometric problem was one of finding a coherent geometric schema allowing for a tapering building where each face slopes differently to be built from simple geometry capable of simple construction. One of the team proposed a design consisting of flat tapering planes joined by sheared cones. This schema was applied to the whole building and became the foundation of the overall geometric system. Such a system has many degrees of freedom and we had to make choices as to how the dependency mechanism should be structured.

The first move was to create a polygon defining the springing points of the tapered planes (the control net). Each corner has a chamfered tangential arc. The first question that arose was how to control the circular fillet. There was no intrinsic advantage in controlling the arc radius explicitly, particularly since the radii at each corner would vary as a function of the sheared cone. A linear set-out was chosen, where the arc radii were controlled indirectly by giving a length to each flat face which established the tangent position.

The issue then was one of direction and the origin of setting-out. The obvious origin point was at the edge of the envelope self-intersection/wrapping. That point itself is located arbitrarily in space in relation to the complex site boundary. The first model set out the building anti-clockwise. This method inevitably resulted in the wrap intersection point not aligning with a putative division system.

At this point of the design it was assumed that the façade module would taper in alignment with the building. Hence, each linear façade segment was made of a multiple of a standard bay (which was varied between 1500 and 1800 grids and a number of variations between). It was obvious that this geometric schema would need somewhere to ‘take up the slack’. In September 2004, the decision was taken to reverse the setting-out, starting from the wrap intersection point (which would therefore coincide with a notional mullion and, more importantly, column one of the structural system). This created a variable dimension at the wrap intersection, an architecturally satisfying solution. The centres of the sheared cones all lie on a single vector. This vector (and the intersection with each floor plane defining the actual arc) can be computed by constructing the vector on the plane and its intersection with the next plane. Hence, only a single vector on a plane drives the taper of the cone. In a linear setting-out system, the only additional variables that need to be supplied are the inclination and taper of each plane. These two parameters indirectly control the radii of the sheared cones. While the design assumption was that the mullion system would taper in sympathy with the tapered planes, it was important that the taper was the same on each plane.

The initial design concept of tapering modules was abandoned in late 2004 in favour of a linear setting-out on a regular 1.5 metre module (figure 6). This new ‘snake skin’ of overlapping panels was configured with the help of a specialised optimisation algorithm (figures 7, 8 and 9).

The structural column setting-out is generated from the design skin, but the columns have combinatorial issues of their own. The arcs at corners had chord lengths added, rounded to the nearest possible module. Combined with an offset from the design surface, these provided location points for the columns. In addition to regular spacing, the decision was taken to position the columns symmetrically with respect to the principal arc corners. Even within this constrained set there were many possible combinations, most of which were evaluated. In addition, there were constraints imposed by foundation conditions and access roads. These resulted in a number of columns being born off by v frames. The final number of columns was 21, although solutions ranging from 19 to 23 columns were investigated. The huge benefit of the original design geometry is that every column is straight. However, no column is vertical – their inclination is governed by the design surface.

The shape of the helical cut was the next principal design element. In order to control the shape of the cut, a ‘normalised law curve frame’ was built (figure 10). Rather than representing the true unwrapped surface on the diagram, all near-vertical mullion lines and inflection points to arcs are represented as vertical lines. Equally, the heights are represented as true z heights ignoring the effect of the tilted planes. The law curve is controlled by a set of points such that the planes are cut by a straight line and the sheared cones have a smooth transform. This schema gave rise to a large set of variations driven by client and regulatory body input, and was explored on top of varying base forms. Each form was evaluated from a large number of critical street-level views; it was essential that the helical form could be appreciated from all important views.

As this description of The Pinnacle’s design demonstrates, the process of architectural design is one that sets the stage for the structural analysis. Analysis is simple if the design–analysis feedback loop is simple – given this shape and these constraints, which structural solution is ‘best’?

But design isn’t like that. Structural strategies impact on the design and how the designers think of the design. Some visible design aspects are vital; others can be finessed in various ways. The interesting fact is that with evolving design technologies, the loop can be far more close-coupled; structural analysis really can adapt form in a visible manner and in a fashion that enhances architectural concepts.

Space Craft is available from RIBA Bookshops and is priced at £35.00.  To order tel +44 (0)20 7256 7222 or visit www.ribabookshops.com