Rational and applied mechanics

This book is written primarily for students entering upon their second collegiate year. Their knowledge of mechanics is limited to what was gained during their study of physics and from practice in laboratory and shop; hence no apology is needed for making matters plain and easy in the first chapters of a work which later on leads up to the Theory of Structures, Internal Stress, Motion under Complicated Conditions, Forms of Energy, and higher Graphics.

It is assumed that at every stage of progress the student can by reasonable effort read the book understandingly by himself (always with pencil and paper at hand), so as to be ready to grasp the conditions of concrete or ideal problems brought up in class, and follow statements made in the language of mathematics and mechanics. In other words, he must here learn to translate the laws of the world of matter, motion, and force into mathematical terms.

There is here no attempt to embrace the special work of any one branch of Engineering or Architecture; but the aim has been to make an opening into every field and to make it possible for the student in any technical branch to intelligently read and use carefully prepared professional Papers and Manuals, and to solve new problems as they arise. Some readers may think that I have ventured into too many fields, but not far enough in some. I shall not have wholly failed, if the student says as he finishes a Chapter, - “I wish the Author had gone further in this interesting subject.“ The Author has been Teacher, and his effort now, as always, has been to guide, and help his young companions to climb alone; and when he thinks he has helped them sufficiently, he has turned to another group.

Accordingly, Chapter XI is devoted to Elementary Graphical Statics, and incidentally, some of the laws which obtain in Framed Structures are presented; while the subject of Redundant Members in Frames is deferred to Chapter XXIII, and even there, the treatment is purely elementary, inasmuch as a thoro discussion of Indeterminate Problems belongs in an advanced professional course. In the same way Hydraulics and Thermodynamics are introduced in Chapters XXVI and XXVII.

It was a most important step in the development of Applied Mechanics, when Internal Stress and the theory of Elasticity were introduced into the study of solids which had previously been regarded as rigid. The Theory of Structures is impossible without the Moduli of Elasticity and the Distribution of Stress. The behavior of a loaded beam with a fixed end is quite beyond the scope of many books on Mechanics: Hence these matters are introduced as strictly appropriate and even essential.

The Author hastens to acknowledge his great obligation to Professor Rankine, the eminent Scotch engineer and author, the perfect master of Mechanical Analysis. His profound insight into the conditions of a problem, and his ready command of the methods of Mathematics made difficult things seem so easy to him, that he failed to see the pits into which his students fell. His reasoning was almost sub-conscious, and he was given to writing down the second equation first, and then deriving the first from the second. Hence it was always a great achievement, when a student could truthfully say that he could “read Rankine.”

Newton explained his superiority to other men as due to his ability to hold his attention steadily upon the details of a problem without a waver for a longer time than other men. Rankine seemed to see as by a flash of lightning.

In the book which follows, the reader will find no lightning flashes; but it is hoped that he will find some evidence of a steady light, and a series of logical steps which a strict attention to business will enable him to surmount. In the first few chapters the steps are short and easy, and the exposition full. In the later chapters the steps are longer and higher, as the climber is supposed to have greater reach, and stronger mathematical legs.

Thruout the book the aim has been to be rational, to make every step reasonable, and every demonstration intelligible. The secret of a lucid analysis is, like that of untangling a snarl of yarn, viz: to get hold of the right end of the thread.

It is often necessary to repeat what has been said before, and to call special attention to a new application of a principle which may have been forgotten. The Theory of Couples will be found as service- able as the Parallelogram of Forces.

No attempt is made in this book to deal with commercial matters. Prices like fashions change, and markets rise and fall, but the fundamental laws of mechanics are eternal; and once they are fairly mastered, the engineer should never be at loss under new conditions. He must at times partly make, and wholly utilize, a new environment.

The Author has not cared to multiply problems. To go beyond clear illustrations of general principles and useful methods of analysis, is to waste time and opportunity. To ring endless changes upon bodies falling in a vacuum, and projectiles flying thru empty space, is to kill time and “keep students busy.” No problems are more interesting and useful than those one finds in the shops, yards and mills of his own neighborhood, if he has eyes to see and lives near a busy com- munity. A personally conducted tour of a few hours, should fill a note book with problems.

Force is an action between two bodies. It is not a tendency, it is a real thing as experience readily shows. A pushes or pulls B, and at the same time B pushes or pulls A; there are two bodies and one action or force. There can be no force unless there be at least two bodies. The bodies may be at rest or they may move. There may be an action, a push or pressure, between two bricks in a wall which does not move. There may be a pull or tension between two cars which are in motion.

 In Applied Mechanics the general surface of the earth is assumed to be at rest. In Celestial Mechanics full account is taken of the motions of the earth and of other heavenly bodies.

All forces are more or less distributed, either over surfaces or thru volumes. The pressure between one's foot and the floor, or between steam and a piston, is distributed over the surface of contact; in one case unevenly, in the other uniformly; in one case without motion, in the other with motion. The earth, in a most mysterious way, pulls upon every particle of matter: — an apple hanging on a tree, a brick in a wall, and an iron ball flying thru the air; and of necessity the apple, the brick, and the ball pull the earth. This is called the force of gravitation. Of all the forces with which mechanics deals, gravitation is the most common, and the most inexplicable. We know, however, that nothing can escape it, and that it is not sensibly affected by such distances as are generally considered in this book, or by intervening objects.

 It is highly important that the student regards the force of gravitation as a pull, distributed thruout the whole volume of a body. Moreover the action (or pull) of gravitation upon a body must not be confused with the action between the body and the platform or foundation upon which the body may rest. When a body like a block of stone rests upon the ground, there are two actions between the stone and the earth, viz: a pull down and a push up, which exactly balance, one action being distributed thru volumes, the other being distributed over the surface of contact.

The plural word "volumes" is used advisedly. One volume is that of the stone which is made up of an infinite number of particles; the other volume is that of the earth which also consists of an infinite number of particles; and there is an action between every particle in the stone and every particle in the earth. What we call the weight of a body is the resultant of all the separate pulls, and its direction is towards and away from the earth's center. The stone pulls the earth just as much as the earth pulls the stone. See "Attraction"
in a later Chapter.

This law of Newton does not explain the stability of the block of stone which rests on the ground in a former illustration. In that case there were two actions between the earth and the stone, a pull and a push, that is, an attraction and a pressure; and the pull and push balanced.

It is true in the case considered, that if there were no pull, there would be no push, because we have assumed that the stone was at rest, and hence the two actions must balance. If there was but one action, the stone would not be at rest; this must be made clear.


Stress. Thus far we have thought of A and B as two separate bodies, but it is evident that they may be separate parts of the same body like adjacent strata in the earth, adjacent leaves in a book, adjacent links in a chain, adjacent particles in a steel rod or a concrete post. When A and B are adjacent parts of a continuous body, their mutual action, no matter what its character may be, is called Stress. If it be a pull it is called tensile stress^ and the continuous body, rope, wire, rod or bar is said to be in tension. If the adjacent layers press against or upon each other in a post, strut, block or wall, the post, strut, block or wall is in compression, and the stress is called compressive stress.

The link of a chain illustrates fairly well adjacent parts of a continuous body. Suppose in Fig. (1) that the stone is suspended to the derrick boom by means of a chain. The chain is in tension and every link is acted upon by two forces (independently of the earth's pull or attraction upon the material in the link) the downward pull of the link below it, and the upward pull of the link above it. Furthermore, it is evident that the upward pull of the topmost link must equal in magnitude the downward pull of the lowest link, plus the pull, or attraction, of the earth upon all the material in the inter- mediate links.

Magnitude of forces. Units. We are immediately conscious that forces or actions have magnitude; one pull or attraction is greater or less than another; one push, pressure or repulsion is greater or less than another. In order to express magnitude with precision we must have a well known unit of force with which all other forces may be compared numerically. A certain pull or push shall be called one and like other units in common use, it shall have a name. The unit of time in mechanics the world over is a second (or a multiple of a second); the unit of length or distance most commonly used in the Anglo-Saxon countries is a foot (or a multiple thereof), though the meter, a French unit, is in common use in physical text-books and laboratories. The unit of value in North America is a dollar. In all these cases we know by observation, experience and frequent use just what these units are, and what a given number of seconds, feet, meters or dollars means.