To Engineer Is Human Page 6
But if the classic sketch of Galileo’s cantilever is ornate, his analysis of the beam’s strength is spare and no-nonsense. He correctly observed, no doubt after actually breaking some cantilever beams, at least in his mind, much as our shipwrecked house builder might have broken some yardsticks, that as the weight is increased the beam cracks and breaks at the juncture with the wall. But since Galileo was apparently the first to attack this problem in a rational way, it is perhaps understandable that he made some mistaken assumptions about how the breaking force is actually distributed across the depth of the beam. (That would not be known correctly for another seventy-five years, when in 1713 a Frenchman named Parent would publish two memoirs on the bending of beams.) Nevertheless, even with his mistaken notions about the way the beam resists breaking, Galileo came to the correct conclusion that the beam’s strength is proportional to the square of the depth of the beam. This is consistent with our experience that it is much easier to break a piece of lumber by bending it through its small rather than its large dimension, and a one-by-ten does indeed appear to have about ten times as much resistance to bending through its ten- as across its one-inch depth. Thus Galileo’s result tells us what we already know about a yardstick held in our fist at one end only: the stick will resist being bent more one way than another. Trees, those organic vertical cantilevers that are subject to bending by the wind, conveniently have near-circular trunks that give equal resistance to the wind no matter what its direction. Skyscrapers, on the other hand, are generally not circular in plan, for structural considerations such as wind resistance are seldom as dominant as architectural or functional factors in determining the shape of a tall building.
Galileo’s analysis of the cantilever beam illustrates an extremely important point for understanding how structural accidents can occur: he arrived at what is basically the right qualitative answer to the question he posed himself about the strength of the beam, but his answer was not absolutely correct in a quantitative way. He got the right qualitative answer for the wrong quantitative reason. Thus Galileo could correctly have advised any builders how to orient their beams for the best results, but should he have been asked to predict the absolute minimum-sized beam required to support a certain weight so many feet out from a wall, the answer he calculated from his formula might have been too weak by a factor of three. We shall come back to this kind of error when we deal with the concept of factors of safety, but the important point here is that apparently right answers can be gotten through wrong reasoning. As a very simple illustration of this phenomenon, consider the theory that the product of a number with itself is the same as the sum of the number and itself. This theory works perfectly when the number is 2, for 2 × 2 = 4 gives the same result as the calculation 2 + 2 = 4.
If one were so inclined to believe this theory that one did not test the hypothesis beyond the single case of the number 2, one might believe one were correctly calculating the squares of all numbers by merely doubling them. Consider further that a number squared represents the amount of load applied to a beam. One might even get away with such an error as long as one had no need for great accuracy and had no need to square numbers other than those close to 2. However, if one day the square of 20 were needed, the erroneous method would give only 40 instead of 400, and the order-of-magnitude mistake might finally be uncovered because the beam would break when the theory predicted it should not have. Such a failure would indeed contribute more to uncovering error than all the successful “verifications” of the incorrect hypothesis.
It is easy now for sophomore engineering students taking their first course in strength of materials to see the error in Galileo’s analysis, but that is not to say that the error should have been obvious to him or his contemporaries. Hindsight is always 20/20, but most of us have at some time experienced myopia when we have had to stand back and criticize our own work. Who has not been amazed at the typographical error missed in countless proof-readings? Who has not been frustrated by the arithmetical error that stands through numerous attempts to balance a checkbook?
Engineers today, like Galileo three and a half centuries ago, are not superhuman. They make mistakes in their assumptions, in their calculations, in their conclusions. That they make mistakes is forgivable; that they catch them is imperative. Thus it is the essence of modern engineering not only to be able to check one’s own work, but also to have one’s work checked and to be able to check the work of others. In order for this to be done, the work must follow certain conventions, conform to certain standards, and be an understandable piece of technical communication. Since design is a leap of the imagination, it is engineering analysis that must be the lingua franca of the profession and the engineering-scientific method that must be the arbiter of different conclusions drawn from analysis. And there will be differences, for as problems come to involve more complex parts than cantilever or even simply supported beams, the interrelations of those various parts not only in reality but also in the abstract of analysis become less and less intuitive. It is not so easy to get a feel for a mammoth structure like a jumbo jet or a suspension bridge by flexing a paint store yardstick in one’s hands. And the hypothesis that a structure will fly safely through wind and rain can be worth millions of dollars and hundreds of lives.
5
SUCCESS IS FORESEEING FAILURE
All the successes of engineering as far back in history as the pyramids and as far into the future as the wildest conceptions of mile-high skyscrapers may be imagined to have begun with a wish to achieve something without failure, where “without failure” to the engineer means not only to stand without falling down but also to endure with what might be called “structural soundness.” Unsound structures—those that are eaten away by rapid corrosion, those that have repeated service breakdowns under ordinary conditions, those that suffer from fatigue cracking after not so many years of use—may be thought to have been failures as surely as if they had collapsed during construction. And no matter how ingenious or attractive his conception may appear in his imagination or on paper, if a designer overlooks just one way in which his structure may fail, all may be for naught.
The earliest engineering structures may have been designed by trial and error, and it may be argued that the Egyptian pyramids were built using that method. Although the pyramids’ construction process, with all its staggering numbers and sizes of blocks of stone and crews of workers, may never be known with certainty, it is not too hard to imagine why the shapes are what they are. The pyramid shape is an extremely stable one, perhaps suggested by the shape taken by a pile of sand at the bottom of an hourglass. But it is a timeless shape, one that resembles mountains, and one that in its monolithic appearance when viewed from afar looks as if it can resist being toppled over in even the fiercest sandstorm. And even though the pyramid is not a regular tetrahedron and thus not one of the Platonic solids, it is certainly an appealing if not a mystical shape. The Egyptian pyramids have more or less square bases and rise in square-cornered tiers, which is a much more natural way to assemble squarish blocks of stone than in the triangular pattern a tetrahedron would demand. Yet even with all of these more or less natural decisions made willy-nilly, the decision of exactly how to pile the stones and exactly how steep to make the sides of the pyramid rise is a crucial one that is not so naturally arrived at.
A pile of sand will assume a natural conical angle as the sand is dripped from the fist, but that angle can vary with the kind of sand and with the conditions under which the pile rises. If the pile rises on a patch of desert that is itself but a greater pile of sand, the added weight can lead to little avalanches, as beach sand acts when we try to pile it too high. The earliest pyramids are believed to have evolved from mastabas, which were rather low, rectangular tombs enclosed with sloping walls of brick. About 2700 B.C. Imhotep, the first builder known by name, apparently was charged with constructing a tomb for the Egyptian Pharaoh Zoser of the Third Dynasty, and Imhotep chose to elaborate on the mastabas of
the First Dynasty. He first faced a conventional mastaba design with stone, then piled stone upon its top in what might be called a stepped pyramid. Pharaoh Zoser’s pyramid thus grew in stages, with Imhotep perhaps gaining confidence to add more and more stone as each succeeding course did not fail.
Once Imhotep succeeded in raising a stepped pyramid, others could copy it with confidence against failure. But, apparently not satisfied with a pyramid with great steps, subsequent designers elaborated on Imhotep’s success and the stepped profile evolved into one with the steps filled in to give the familiar straight edge and flat sides now commonly associated with the pyramids of Egypt. The success of the Meidum pyramid, whose faces rose more steeply than those of its predecessors, gave still later designers something else to better. They attempted to do this in the pyramid at Dahshur, whose sides began to rise at the previously untried angle of 54°. However, something appears to have happened during construction to change the original plan, for approximately halfway up, the Dahshur pyramid suddenly changes not only in the kind of stone used but also in the slope of its sides. From 54° the walls drop to a 43° inclination—hence the structure’s descriptive name, the Bent Pyramid. One theory holds that the break in the Bent Pyramid resulted from a structural failure attributed to construction at the theretofore untried steeper angle, and that the designer’s original aspirations had to be lowered. What has been interpreted as great masses of debris about the base of the Bent Pyramid lends credence to this early example of structural design being pushed by hubris to the limits of failure. Subsequent pyramid builders, who built higher but dared not build steeper, seemed to be content with humbler successes.
Egyptian pyramid builders are not unique in their encounters with the limits of structures and their desire to do what had not been done before. Medieval cathedrals were certainly much more complex structurally than the pyramids, yet there is still considerable evidence that the cathedrals evolved through a process of experimentation and trial and error not unlike that of the Egyptian megaliths. Even the layman Henry Adams, in his avuncular Baedeker to the cathedrals of France, had to remark on how the architects of churches erected only forty or fifty miles apart around Paris in the late twelfth and early thirteenth centuries must have watched and been influenced “almost from day to day” by each other’s experiments. One builder’s structural and aesthetic successes and failures were challenges and lessons to the others.
In the year 1284 the cathedral at Beauvais suffered a major collapse, and this incident has been regarded as a turning point in the development of Gothic structures. Thenceforth architects are generally assumed to have been more conservative in their structural adventures, although thoughtful critics see a resurgence of structural innovation and aspiration in the fourteenth century. Robert Mark, who has used modern engineering models to analyze the forces acting on Gothic cathedrals, sees significant new achievements in height and slenderness exhibited in the nave of the Palma, Majorca, cathedral, for example, though he acknowledges that the accomplishments are by no means as major as those that culminated in Beauvais.
But pyramids and cathedrals may be said to belong to the pre-rational age of structural engineering, for there was apparently much more reliance on physical experiment and mid-construction correction than on any predetermined and inviolate set of plans for the final version of the structure. The flying buttresses of the cathedrals seem certainly to have been added and elaborated upon in response to undesirable cracking in the medieval masonry. Even the addition of apparently decorative pinnacles seems to have been in response to a functional need for more weight to keep further cracks from opening up under the great forces of the wind to which a massive cathedral was subjected as it rose exposed from among all the ground-hugging buildings of a medieval town.
The great structures of the nineteenth and twentieth centuries are the iron and concrete bridges and skyscrapers that have a slenderness and structural daring perhaps undreamed of by the builders in stone. The tenacity of steel has added a tensile dimension to structures that brings a release from the dominance of compression as a stabilizing force almost as if from the pull of gravity itself, but the memory of countless iron bridge failures during the nineteenth century keeps modern engineers from getting swell-headed over how fast they can build longer and taller structures even today. Indeed, the chronic ailments of iron bridges that evolved with the expansion of the railroads is to this day one of the most discussed and most chronicled chapters in the history of structural engineering.
Perhaps this is so because of the differing symbolic nature of bridges compared with monumental architecture such as pyramids and cathedrals. The latter two were erected as tributes to earthly and heavenly rulers, but bridges are principally functional structures. When they do assume symbolic proportions, it is almost always as an afterthought, as when the Brooklyn Bridge took its place in art and literature only after it had become firmly established as political link across the East River. If anything, the modern bridge is a tribute to man’s technological achievements, and no other structure shows off its structural bones or flexes its structural muscles the way a bridge does.
The nineteenth-century expansion of the railroad brought new challenges to man the builder. Pyramids were by and large great piles of stone, and the only spaces the stone was required to span were the long but narrow labyrinthine corridors leading to burial chambers no more ambitious in scale than a bedroom for today’s common man. Cathedrals, on the other hand, along with their great domed predecessors in Rome, strove not for height with mass and bulk but for height with delicacy and openness. Neither economics nor mass production was the first concern of the builders of the monumental architecture, but crass realities became principal ingredients in the evolution of bridges. That is not to say that there was less concern for an iron bridge to be successful than for a Gothic cathedral. Indeed, driven by economics, the railroad companies were perhaps even less inclined to risk structural failure than were the patrons of cathedrals, whose motivation lay beyond material profit or loss. Yet it might also be said that the railroad companies were still more daring, for they had to construct bridges under ever-new conditions, and merely copying what had stood the test of time was not always an option.
With the growth of the railroads came new demands on engineering structures. Railroad bridges had to withstand not only the sheer weight of the heavy locomotives and the rest of the rolling stock but also the dynamic action of the engine’s reciprocating parts and the constantly changing position of the train on the bridge. Railroads brought together the machines of the mechanical engineer and the stationary structures of the civil engineer, and the demands of each provided stimulus for the evolution of the other—but not without mishap. As the tentacles of the railroads reached further and further out to provide service, there were more and more incentives to have heavier trains traveling at greater speeds over more rugged terrain. Any hill that did not have to be climbed or any valley that did not have to be entered was time and energy saved, and that meant money. But soon the strength required for a railroad bridge built to accommodate an early generation of iron horses was exceeded by later, heavier generations, and collapses occurred. Each defective bridge resulted in demands for excess strength in the next similar bridge built, and thus the railroad bridge evolved through the compensatory process of trial and error. As Ralph Waldo Emerson observed in his contemporaneous essay on compensation, “Every excess causes a defect; every defect an excess.”
There was a technological awareness in the nineteenth century that recognized railroad trains and their infrastructure as a part of the changing environment. Thus William Wordsworth, the worshipper of nature, wrestled with the problem of a new culture intruding into the older, more pastoral one. The Industrial Revolution was changing the face of the English countryside, and in the poem, “On the Projected Kendal and Windermere Railway,” he exhorts the reader to share his disdain for what Wordsworth views as a disfigurement of the Lake District:
br /> Hear Ye that Whistle? As her long-linked Train
Swept onwards, did the vision cross your view?
Yes, ye were startled;—and in balance true,
Weighing the mischief with the promised gain,
Mountains, and Vales, and Floods, I call on you
To share the passion of a just disdain.
But disdain was tempered in another poem, entitled “Steamboats, Viaducts and Railways,” in which Wordsworth acknowledged technology to be but another manifestation of a greater nature:
In spite of all that beauty may disown
In your harsh features, Nature doth embrace
Her lawful offspring in Man’s art; and Time,
Pleased with your triumphs o’er his brother Space,
Accepts from your bold hands the proffered crown
Of hope, and smiles on you with cheer sublime.
Wordsworth’s honest ambivalence between improved and unimproved Nature was the poet’s response to the engineer’s introduction of amendments to a given nature. Indeed, Wordsworth, in a later version of the last line, replaced the somewhat distant “smiles on you” with the more positive “welcomes you.” His expression was as subject to improvement as the bridges of the nineteenth-century engineers. But the numerous bridge failures did not make it any easier for Wordsworth or any of his contemporaries to embrace the new technology. They were all too aware of the trying and erring going on not only in Britain but also in America.
In 1843 Nathaniel Hawthorne took The Pilgrim’s Progress, John Bunyan’s seventeenth-century religious allegory of the good man’s pilgrimage through life, as a model for a tale set in the midst of the Industrial Revolution. In “The Celestial Railroad” Hawthorne’s traveler is accompanied from the City of Destruction to the Celestial City by a Mr. Smooth-it-away, who points out how the new technology might improve the condition of man, which had escaped improvement for so long. But the traveler seems constantly distracted by the condition of the bridges over which he travels. One, though “of elegant construction,” he imagined was “too slight … to sustain any considerable weight.” Another “vibrated and heaved up and down in a very formidable manner; and, spite of Mr. Smooth-it-away’s testimony to the solidity of its foundation,” the traveler “should be loath to cross it in a crowded omnibus, especially if each passenger were encumbered with … heavy luggage.” Structural failure is also on the fictional character’s mind as the Hill Difficulty is approached: