The Logic of Machines and Structures

by Paul Sandori

Overview

Based on common, everyday phenomena, the principles governing the balance of forces on machines and structures are extremely straightforward. Their expression in mathematical form, however, obscures their clarity. This volume exposes the principles of statics in their original simplicity, presenting them as an exercise in logic. The modern analytical method of reasoning is carefully preserved to assist students in their grasp of the thinking that underlies mathematical methods of analysis.
Suitable for architecture and engineering students as well as other readers with minimal background in mathematics, this unique treatment also restores enjoyment to the study of statics. Author Paul Sandori develops the subject using crucial highlights and discoveries in the field's historical evolution, noting the brilliant early insights and intuitions that contributed to the modern science. The text is complemented by illustrations of source materials from Galileo, Newton, and others that document the discipline's evolution.

Product Details

ISBN-13:	9780486816159
Publisher:	Dover Publications
Publication date:	09/21/2016
Series:	Dover Books on Engineering
Sold by:	Barnes & Noble
Format:	eBook
File size:	20 MB
Note:	This product may take a few minutes to download.

About the Author

A practicing engineer with a specialty in building science, and a Fellow of the Royal Architectural Institute of Canada, Paul Sandori served for 29 years on the faculty of the University of Toronto. He is now Senior Consultant with construction consultants Revay and Associates Ltd.

Read an Excerpt

The Logic of Machines and Structures

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By Paul Sandori

Dover Publications, Inc.

Copyright © 1982 Paul Sandori
All rights reserved.
ISBN: 978-0-486-81615-9

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CHAPTER 1

Machines and Structures

Things are simpler than they seem

The Intuitive Approach 1 — The Art of Weighing 4 — Forces and Equilibrium 10 — Example 13

THE INTUITIVE APPROACH

The man changing the wheel on his wagon (Fig. 1.1) is faced with a problem: before he can remove the wheel; he has to raise the heavy vehicle; but muscular strength is not sufficient for the job. He solves the problem by means of a familiar device — a lever. A more sophisticated version of the lever is standard equipment on most cars today and; in even more complex forms; it is part of various machines used to exert enormous forces and pressures in industry.

The modest lever is a so-called simple machine and does the same job as its complex derivatives: it multiplies the effort of the human arm to achieve something that; unaided; the arm could not possibly do. The advance of our civilization started with devices like that.

Once propped up the lever becomes a structure. It is no longer acting to raise the load. It is fixed; and it holds the load up against the pull of gravity. We should now call it a beam spanning its supports on the ground and on the prop. Lever or beam it is a simple enough device; familiar and easy to understand in its working. Let us now look at something less familiar.

The bridge over the Firth of Forth in Scotland (Fig. 12), completed in 1890; seemed an amazing feat of engineering at the time of its construction. Its twin spans of over 520 m each not only were the longest in the world but also had the strength and rigidity to support tremendous train and wind loads. Its huge size dwarfed anything built before: nearly 60 thousand tonnes of steel went into the bridge; held together by six and a half million rivets!

The bridge has been likened to the pyramids of Egypt; because of its monumental size and its significance as a landmark. People have been flocking to admire it ever since it was built. On the other hand; because of its unusual profile; it has also been likened to a dinosaur and a well-known contemporary aesthete; William Morris, called it "the supremest specimen of all ugliness."

Leaving aside aesthetics; let us try to answer a much more down-to-earth question: how does this bridge support the loads on it? What does what in the structure? What are the engineering, reasons that dictated the unusual shape? At first glance; the answers to these questions may seem very hard for a nonspecialist to find; but this is not so. Figure 1.3, a photograph taken from a contemporary publication; contains an extensive structural analysis of the bridge based entirely on intuitive knowledge.

As can be seen from the diagram on the wall behind the live model; only one of the twin spans is shown. The man sitting on the stick in the middle of the model represents the load on the span; a train. The stick is the central girder between two piers of the bridge; called the suspended girder. The supporting structure projects outward from the piers on either side and is represented by the arms of the men sitting on the chairs and by the sticks held in their hands at one end while the other end is butted against the chair. Such a structure projecting over the support is called a cantilever.

The load from the suspended girder is transmitted to the cantilevers. It tends to make the pier overturn but, on the opposite side of the support, it is balanced by a counterweight. In the model, the counterweight is a stack of bricks; in the bridge, it is in the masonry towers at either end of the bridge.

What happens inside the structure is equally easy to understand intuitively. The arms of the men and the sticks work together in resisting the load. The load pulls on the arms and pushes on the sticks. The arms are in tension; the sticks are in compression. In a structure, the compression elements — the sticks — are called struts and the components in tension — the arms — are called ties. Another part of the structure that is evidently in compression are the bodies of the two men. They have to make an effort to hold up their shoulders just as if the load had been placed directly on them.

Looking more closely at the bridge itself (Fig. 1.4), we can easily differentiate between ties and struts. The ties are transparent, light, delicate-looking lattices; the struts huge tubes. Experience provides a the reason for that too. A long, slender piece of steel or wood can take a large pull in tension but far less in compression. It simply bends out under a comparatively small push. This type of structural behavior is called buckling. A tube is a particularly efficient structural shape when it comes to resisting buckling.

The stick on which the man in the middle is sitting is neither a tie nor a strut. Straight when unloaded, it bends under the load; it works as a beam just as the propped lever did in the first example. Bending is far more damaging to a structural member than either tension or compression. If we wanted to break a stick, we would not for a second entertain the idea of breaking it by pulling or pushing at the ends to produce tension or compression. The easiest way to break it is by bending it. Consequently, in the bridge we notice that the suspended girder is kept comparatively short and, moreover, it is made up of struts and ties, just like the rest of the bridge. Bending is, somehow, resolved into tension and compression. Such a structure is called a truss.

In the following chapters, we develop a method of analyzing structures and machines in terms of a small number of basic principles. This method — the result of centuries of thought and experience — reinforces intuitive and practical understanding of how machines and structures work while at the same time making us independent of it.

THE ART OF WEIGHING

The method of analyzing machines and structures that we are looking for originated many centuries ago in attempts to understand devices like the lever which, as if by magic, multiplied the effort of the person using them. (Structures did not attract much attention: there was nothing mysterious about the way a beam supports its load and, if one broke, it was simply replaced by a bigger one.) For the person seeking a logical explanation of machines and structures rather than a detailed engineering analysis, it is profitable to go back to the origins and start where the early investigators started: with simple, basic problems involving familiar devices. Only those texts that particularly suit the purpose of this book are discussed here.

In 1586 a Dutch engineer and mathematician, Simon Stevin (Fig. 1.5), published The Elements of the Art of Weighing, a book that dealt mainly with the balancing of loads. There are several reasons for choosing Stevin's work an excellent teacher for he had a genius for deriving his explanations from axioms (self-evident facts) without recourse to any previous knowledge of mathematics. Moreover, in his discussions he instinctively chose the path that, in an organized and developed form, constitutes the modern method of analysis.

Stevin's work on the art or science of weighing was paralleled by the writings of many scientists before and after him. We shall borrow from the work of just one of them: the greatest Italian scientist of the period, Galileo Galilei (Fig. 1.6). After his trial and condemnation by the Roman Inquisition in 1633 for his heretical views regarding the motion of the planets, Galileo was forced to retire. He was forbidden to express his opinions "in writing or speech or in any other way, on the movement of the earth and the immobility of the sun." Instead, he managed in 1638 to have a book published called Two New Sciences. One of the new sciences dealt with bodies in accelerated motion; the other explored the resistance to fracture of beams. For the first time structures received a scientific treatment.

At first glance, water, air and other fluids appear to be substances that have little to do with machines and structures, either in their application or in their behavior. Appearances are misleading on both counts. Stevin wrote a book on buoyancy and on the effects of the pressure of water, coming to some startling conclusions. He laid down the fundamentals of the subject but his work did not have much influence until the thread was picked up again by the French mathematician and philosopher Blaise Pascal, who in 1663 published his Treatise on the Equilibrium of Liquids and the Weight of the Mass of the Air. Pascal did not really go much further than Stevin regarding water pressure; what he did though was to produce many ingenious proofs and experiments covering the subject more systematically and clearly. In one respect he made a significant advance: he made use of his understanding of the behavior of liquids under pressure to invent a new machine — the hydraulic machine.

The pressure of the air was not touched upon at all by Stevin. The pioneering work in this respect was done by Galileo's pupil Torricelli and further developed by Pascal (as the title of his book indicates). Pascal showed that a large group of phenomena thought to be caused by an abhorrence of the vacuum, attributed to Nature ever since Aristotle, can be explained in terms of pressure of the air, closely resembling the effects of the pressure of water. At approximately the same time the mayor of Magdeburg in Germany, Otto von Guericke (Fig. 1.7), made many practical experiments involving air pressure, some with really spectacular results. These stimulated other scientists and eventually led to the invention of the steam engine that revolutionized industiy a century later.

Finally, we shall take Stevin's work on the workings of another laborsaving device, the pulley, as the starting point of a different approach to problems involving machines and structures. The results obtained in this way will, of course, be the same answers as before but seen from a new point of view and providing new insights. Stevin himself is often credited with developing this alternative. Actually, he did his best to suppress it as erroneous.

The objective of this infusion of historical examples, as noted at the start, is an intuitively and rationally understandable logic of structures and machines. However, there is another reason. The originality and sheer brilliance of the reasoning contained in the work of people like Stevin, Galileo, and Pascal is in itself enjoyable and worth study as part of our heritage in the fields of science and philosophy.

FORCES AND EQUILIBRIUM

Let us look at how Stevin applied his Art of Weighing to a problem — a treadmill crane that enjoyed widespread popularity in sixteenth-century Europe just as it did in Antiquity (Fig. 1.8). One or several men would make the large wheel of the crane turn by forever climbing up inside; this would wind a rope around the axle EF and cause the load H, hanging on the rope to be lifted. How much load can a single man lift in this way?

In the theoretical part of his Art of Weighing Stevin deduces from a very basic axiom what was then known as the law of the lever. The load H on the short arm of the lever and the effort E applied at the end of the long arm (Fig. 1.9) stand in a certain ratio to each other which depends on the relative lengths of the lever arms AG and GF. These ratios are given by the law of the lever (which we examine in Chapter 2). Stevin's analysis of the machine consists of showing that it functions as a simple bent lever. As the man climbs toward point A, he pulls down the long lever arm AG. The short arm of the lever is represented by the radius GN of the axle on which is exerted the pull of the load. Anybody who has mastered the art of weighing and knows the law of the lever can now figure out easily the answer to the problem of the crane.

The art of weighing does not exist any more. It has been replaced by the science of mechanics. The concept of weight has lost its special significance. Weight is the pull of gravity and is in no way different from the pull of a rope or the push exerted by a man. In mechanics, any push or pull, however caused, is called force. The weight of the load at one end of the lever is a force, just as the push of the man is a force at the other end.

Consider another illustration from Stevin's book, reproduced in Fig. 1.10. Stevin employs a variety of devices to show the pulls exerted on a prism. A string passing over a pulley with a load at the far end exerts an oblique pull. For an upward pull the string is attached to a balance with a load hanging at its opposite end — then the upward pull of the string is equal to the weight of the load. Whenever required, a helpful hand reaches down from the sky to hold any loose ends.

The modern equivalent of this illustration, seen in Fig. 1.11, is less vivid but much simpler. All the props, including the pyramid support, have been eliminated and replaced by arrows indicating forces following very specific rules which we shall see in a later chapter. This is one of the problems with modern mechanics: it is simple almost to the point of being dull.

The principles of mechanics are no longer concerned with the law of the lever or some similar device. Mechanics deals only with bodies, and all bodies, regardless of their shape, size, or function, behave in the same way under the action of forces. A lever is a body which, in practice, happens to be a simple machine; in mechanics, this does not entitle it to any special treatment. The old law is simple enough but we reduce it and all the other similar laws and rules to a small number of even more basic principles which concern forces and bodies. Armed with these principles we proceed in pretty much the same way as Stevin did in his treatment of the crane. We discover these same simple principles at work in all the complex problems that we may encounter.

The forces acting on a body are either balanced or not. If they are not balanced, the body will be subjected to accelerated motion. The branch of mechanics that deals with accelerated motion is called dynamics. It was developed by Isaac Newton (following some pioneering work by Galileo) in his famous book Mathematical Principles of Natural Philosophy, published in 1687. The branch of mechanics that investigates forces in a state of balance; or equilibrium, is called statics. This is the science that replaces the ancient art of weighing and is the subject of this book.

The world of statics is inhabited only by rigid bodies and forces. Everything in it can be measured in terms of two kinds of units: the units of force; newtons, and the units of length; meters. These are the units of the metric International System of Units (abbreviated SI); adopted in 1960 by a world conference on weights and measures. For uniformity we use SI units even in examples that go back to the time long before the metric system was invented. The simplicity of statics makes it ideally suited for a mathematical treatment and; in fact; mechanics was the first science that was put in a mathematical framework with all the other sciences later trying to approach this ideal.

Over the centuries; the mathematical side of statics has become extremely well adapted to the job and streamlined to perfection. In fact; many machines and structures are now possible not only because of the advances in technology but also because they can be analyzed mathematically. The CN Tower in Toronto (Fig. 1.12), the tallest freestanding structure in the world; is a good example: its design was performed to a large extent with the help of computers. However; this is a separate development. We can understand the structure of buildings such as the CN Tower perfectly well without a mathematical analysis. Statics offers a quick and reliable method of understanding the effect of forces on bodies or; in practical terms; a logic of machines and structures.

EXAMPLE

We can easily understand the structural action of the Firth of Forth bridge once we see the live model of Fig. 1.3. In fact, we can feel the work required of the various structural components of the bridge in our muscles and bones. This is not surprising. After all; the job that structures perform on a large scale — that of carrying and balancing loads — is on a smaller scale an everyday experience for humans and animals.

(Continues...)

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Excerpted from The Logic of Machines and Structures by Paul Sandori. Copyright © 1982 Paul Sandori. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Contents

1. MACHINES AND STRUCTURES Things are simpler than they seem, 1,
2. THE LEVER A paradox of nature explained, 16,
3. THE INCLINED PLANE "A mystery is not a mystery", 42,
4. THE BEAM And a general method of attack, 69,
5. WATER AND AIR More paradoxes and a horror that is not, 99,
6. A DIFFERENT APPROACH And some new insights, 127,
7. SUMMARY AND PROBLEMS Or statics in a nutshell, 154,
BIBLIOGRAPHY, 175,
INDEX, 177,