Scale

by Don Friedman on August 18, 2011

The world is not the same when examined at different sizes, a fact that has long been known. In order to see the asymmetrical nature of size, all one has to do is picture an elephant or an ant at the size of a dog and the adaptations to the true sizes of the creatures is clear. Galileo not only pondered this issue, he correctly identified the governing facts: the weight of an object is proportional to its volume while the strength of its support, a leg for example, was proportional to that support’s cross-sectional area. To paraphrase Galileo’s example, if you take a man and double his size by doubling every dimension of his body to create a 12 foot high giant, his weight will increase by a factor of 8 because the volume of his body will have increased by a factor of (2³ =) 8, while his legs (and leg bones) will only have increased in cross-sectional area by a factor of (2² = ) 4. The compressive stress in his femurs when he stands will be twice as great as that of an ordinary man. Create a bigger giant by multiplying the dimensions of an ordinary man by 4, and his weight will be 64 times as great, his femur area 16 times as great, and the stress in his femurs 4 times as great. Such a giant might suffer broken bones from the stress of everyday life – a good example of bad design.

This discussion of theoretical giants is a simplified structural analysis. The facts that it contains regarding the effect of scale are so thoroughly inculcated in engineers during their training that they can be considered in building design and analysis without conscious thought. The more interesting cases come when more extreme changes in scale take place: the difference between everyday buildings and very tall ones. Tall building design is governed by lateral forces – wind and seismic loads – to such an extent that gravity is a secondary concern. Why? Because the effect of a sideways push on a tall building increase by the square of the building height (assuming for the sake of simplicity that the building has no setbacks), while the resistance the building’s structure provides is (roughly) based on the building’s base width. Even though the resistance is based on the square of the base width, meaning that a giant would be roughly as stable when pushed sideways as ourselves, real buildings have difficulty with lateral loading because the base width is constrained by non-engineering forces. Tall buildings in a city are usually constrained by the width of individual blocks, which are as small as 200 feet wide in Manhattan. Theoretical designs for mile-high skyscrapers, from Frank Lloyd Wright’s Illinois tower proposal of 1956 to more recent and more realistic designs almost always exceed this width so that the giant can have the same stance as the ordinary man.

The most important effect of size is glossed over in the proceeding discussion: we have a limited palette of materials to work with. If the giant’s bones could made of steel, the stress in his femurs would not be a concern at the scale discussed. In the case of buildings, we already build using the strongest commonly available construction materials; barring a revolution in the economics of strong materials, tall building structure will be made of steel and concrete for the foreseeable future. While these materials have become stronger since their introduction through better production methods, and will probably continue to do so, the increases are incremental.

The use of strong materials such as steel for small-scale objects such as chairs has created a misleading impression of engineering’s limits. At a small scale, gravity and lateral load can be ignored or treated casually. Marcel Breuer’s famous cantilevered chair works as a chair, but would be a poor model for a building. The chair design is not rational in that cantilevers the main load (the weight of the person sitting) when for a small increase in structural weight legs could be provided at the rear for direct support. In short, rational structural design is not necessarily required at the scale of a chair, or to return to the analogy, an ant can have legs much more slender than a man.

Unfortunately, not every poor role model is as clear as the Breuer chair, and irrational design permeates small scale structures, from furniture to ornamental stairs to small houses. In most cases the irrational design has been chosen for aesthetic or usage reasons. These designs may or may not be successful in terms of their appearance and functionality; “irrational” refers solely to engineering criteria that are not necessarily important. For example, the cantilevered design of a pedestal table follows from the need to fit chairs and legs below.

The engineering difficulty comes if we try extend an irrational stair or chair design to the scale of a building. It is not impossible to do so, as Rem Koolhaas and the Office for Metropolitan Architecture demonstrated at the CCTV headquarters building, but it comes at the expends of very heavy and expensive structure.

And this issue will not be changing. We can build at the scale of microchips, but then a house built to the standards of a chip fabrication plant would cost hundreds of millions of dollars. Building with details at the natural scale of the human hand and hand-held tools will be less expensive than building larger or smaller as long as human beings are performing the construction. A futuristic scenario where small- (or large-) scale robots are constructed and then, in turn, construct buildings using details at a smaller (or larger) scale is feasible but not near. Every stage in industrialization since the early 19th century was hailed as the time when the human element would be greatly reduced in construction, but it has not yet happened. Our buildings, regardless of size, are generally built for people and by people, using the same materials that have been common for the past 100 years. Until some portion of this description changes, our buildings will be subject to the same constraints of size that are familiar now.

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