NOTE. The printer having altered the numbering of the Articles from page 137 to 184 of this work without the knowledge of the author, and without a corresponding alteration in the numbering of the Articles to which reference is made in these pages, the following is the reading of the Articles of reference :

For the Articles 111, 114, 116, 122, 124, 125, referred to in pp. 137-184, read 109, 112, 114, 120, 122, 123.

PART I.

THE RESISTANCE OF MATERIALS,

CHAPTER I.

INTRODUCTORY REMARKS AND FIRST PRINCIPLES.

INTRODUCTORY REMARKS.

THE Resistance of Materials is a branch of science which, in its nature, is partly physical and partly mechanical. It falls within the province of physics, inasmuch as it involves the different properties of the molecular structure of bodies; in its mechanical character, it comprises the laws of equilibrium and motion. Its object is made evident by its name. Bodies, even such as are tenacious in a very high degree, are capable of being broken or divided into small portions or molecules, when such bodies are acted upon by forces of sufficient intensity. This, it needs scarcely be remarked, is a primary law in physics. Wood, stone, earths, iron, steel, &c., and hence all artificial constructions, are capable of being broken, crushed, extended or compressed, by mechanical agency.

The laws of the Resistance of Materials have been based on facts, ascertained and established by accurate experiments. Theory and practice are thus constrained to proceed hand in hand as it were together. Such experiments, directed or personally carried out at various periods by men whose name is a guarantee of the soundness of their inquiries, have led in general to results, in which practice has verified a very close approximation to the truth. These results have been registered or tabulated. Thus

B

the strength or tenacity of bodies up to a certain limit is known, and therefore we can ascertain that when the estimated limit is passed, rupture or fracture will almost necessarily ensue.

Hence we see that this branch of science, viz. the "Resistance of Materials," has for its peculiar object, the determining, à priori, from various given circumstances, under what stress or strain the fracture of a material will take place. The stability of constructions is, in general, fully determinable by the application of its principles. But the whole scope of the subject may be comprised in the following propositions:

I. All the external forces which act on a body being known, it is required to determine the intensity of the interior forces which are brought into play at every point of the body, and upon which depends either stability or rupture.

II. All the external forces being known, it is required to determine the alteration of the form and of the dimensions of the body to which such forces are applied.

The strict theoretical method of investigating the general problem thus enunciated, by means of the equations of internal equilibrium, is of extreme difficulty, as may be seen on reference to M. Lame's Traité de l'Elasticité. But the cases which occur in practice, and to which this treatise will be chiefly confined, can generally be solved with sufficient accuracy by approximate methods which are comparatively simple. Certain laws are assumed, which, if not exactly true, are known to give approximations to the truth sufficiently close in practice. For instance, it is assumed amongst other things (Hooke's Law, “ut tensio sic vis"), that if a certain weight suspended at the lower extremity of an elastic vertical prism produces an elongation 7, a double weight would produce under the same circumstances a double elongation 2 7, and so on. Now this law is sufficiently correct for all practical purposes in material constructions, when the tensions or strainings do not pass the limits compatible with stability. But it is found that if the straining load approaches closely to the conditions under which fracture would ensue, this proportion between the forces which strain bodies and their effects, ceases to hold; and, consequently, it is to be expected

that the formulæ of solution then give results contrary to those of experience.

In the applications of this science, it is important to distinguish between the breaking load of a material and that load which would produce the greatest amount of strain without destroying the tenacity of such material. The former has been called the absolute strength or ultimate strength of the material, the latter, its proof strength. To each of these there is a corresponding coefficient or constant, called respectively the coefficient of rupture, and the coefficient of proof strength. These coefficients are determined by experiment. In order to provide for unforeseen contingencies, there is also a coefficient of safety which is less than that of proof strength.

The earlier writers on the strength of materials assumed that when the intensity of a straining force was such as to produce a set in the material on which it acted, that is to say, a change of form of the material from which it did not recover on the removal of the straining force, its tenacity was destroyed. But Hodgkinson has proved by experiment, that a set is produced in many cases by a force which is perfectly consistent with safety.

In some cases a problem, which in ordinary statics is indeterminate, assumes a determinate form in the Resistance of Materials. On this part of the subject the following interesting remarks of Bresse may be cited.

Suppose we have found (Bresse's Mécanique Appliquée), a solution for the case in which we know all the exterior forces acting on a body whose resistance we are desirous of verifying. There will be cases in which this solution will be insufficient. To give a very simple example of such-let us suppose a piece of wood resting horizontally on two fixed supports, and which is loaded between the supports. In this case all the external forces are not given-the reactions of the fixed supports having to be determined by calculation. But the calculation here needed is free from all difficulty; the elementary theory of the statics of solid bodies is sufficient for accomplishing it. But exceptions to this general facility present themselves, and a notable exception occurs, for instance, when instead of two supports, there are three in a straight line. The statics of solid

bodies cannot then furnish any more than two equations between the three unknown reactions, and the problem viewed in this way is indeterminate.

We conceive à priori, that there ought to be an indefinite number of systems of three reactions exerted by the supports which can balance the given load. For let us imagine that the intermediate support is removed. Then the reactions of the two others are known by elementary statics, and the reaction of the middle support is zero. The equilibrium will still be preserved if we again put the suppressed support in contact with the load, but without pressing much against it. That being done, we shall be able to increase gradually the pressing of this support against the load, and consequently its reaction, without disturbing the equilibrium. This is a proposition, the truth of which results clearly from acquired experience. Now this pressure exerted at the middle of the load is arbitrary in certain limits, and on the other hand it cannot vary without involving a corresponding variation in the reactions of the extreme supports. We shall then have as many systems of values as we please for the unknown reactions, by satisfying always the conditions of equilibrium of elementary statics-so that we cannot thus know which is the system that really exists. Yet these systems can be distinguished from each other by a particular circumstance that of the height or versed sine of arc (more or less in magnitude) which is produced in the material towards the support of the middle-which height of arc in fact diminishes in proportion as the reaction increases. If we indicate amongst the data of the problem, that the three supports are at the same level, the value of the intermediate reaction will be that of a force which if applied at the same point would correspond to the height of arc produced by the removal of the support.

It is possible then to remove the indeterminateness just mentioned-but for that we must take into account the distortions produced on the material by the load.

The "Strength of Materials," has not yet been so clearly or so completely elucidated as to render further research useless or unnecessary. The reverse indeed of this would appear to be the