Theory of structures and strength of materials

The present work treats of that portion of Applied Mechanics which has to do with the Design of Structures.

Free reference has been made to the works of other authors, yet a considerable amount of new matter has been introduced, as, for example, the Articles on “Surface Loading” by Carus Wilson, “The Flexure of Columns” by Findlay, and “The Efficiency of Riveted Joints” by Nicolson; also my own Articles on “Maximum Shearing Forces and Bending Moments,” “The Flexure of Long Columns,” “The Theorem of Three Moments,” etc.

I am much indebted to Messrs. C. F. Findlay and W. B. Dawson for valuable information respecting the treatment of Cantilever Bridges, Arched Ribs, and the Live Loads on Bridges.

To Messrs. J. M. Wilson, P. A. Peterson, C. Macdonald, and others, many thanks are due for data respecting the Dead Weights of Bridges.

I am under deep obligation to my friend Prof. Chandler, who has kindly revised the proof-sheets, and who has made many important suggestions.

I have endeavored so to arrange the matter that the student may omit the advanced portions and obtain a complete elementary course in natural sequence.

At the end of each chapter, a number of Examples, selected for the most part from my own experience, are arranged with a view to illustrating the subject matter - an important feature, as it is admitted that the student who carefully works out examples obtains a mastery of the subject which is otherwise impossible.

The various Tables in the volume have been prepared from the most recent and reliable results.

A few years ago I published a work on “Applied Mechanics,” consisting mainly of a collection of notes intended for the use of my own students. The present volume may be considered as a second edition of that work, but the subject matter has been so much added to and rearranged as to make it almost a new book. I venture to hope that this volume may prove acceptable not only to students, but to the profession at large.

CHAPTER III - DEFINITIONS AND GENERAL PRINCIPLES

I. Definitions. The science relating to the strength of materials is partly theoretical, partly practical. Its primary object is to investigate the forces developed within a body, and to determine the most economical dimensions and form, consistent with stability, of that body. Certain hypotheses have to be made, but they are of such a nature as always to be in accord with the results of direct observation.

The materials in ordinary use for structural purposes may be termed, generally, solid bodies, i.e., bodies which offer an appreciable resistance to a change of form.

A body acted upon by external forces is said to be strained or deformed, and the straining or deformation induces stress amongst the particles of the body.

The state of strain is simple when the stress acts in one direction only, and the strain itself is measured by the ratio of the deformation to the original length.

The state of strain is compound when two (or more) stresses act simultaneously in different directions.

A strained body tends to assume its natural state when the straining forces are removed: this tendency is called its elasticity. A thorough knowledge of the laws of elasticity, i.e., of the laws which connect the external forces with the internal stresses, is absolutely necessary for the proper comprehension of the strength of materials. This property of elasticity is not possessed to the same degree by all bodies. It may be almost absolute, or almost zero, but in the majority of cases it has a mean value. Hence it naturally follows that solid bodies may be classified between two extreme, though ideal, states, viz., a perfectly elastic state and a perfectly soft state. Perfectly elastic bodies which have been strained resume their original forms exactly when the straining forces are removed. Perfectly soft bodies are wholly devoid of elasticity and offer no resistance to a change of form.

Bodies capable of undergoing an indefinitely large deformation under stress are said to be plastic.

Stresses and Strains. Every body may be subjected to five distinct kinds of stresses, viz.:

(a) A longitudinal pull, or tension.
(b) A longitudinal thrust, or compression.
(c) A shear, or tangential stress, which may be defined as a stress tending to make one surface slide over another with which it is in contact.
(d) A transverse stress.
(e) A twist or torsion.

Under any one of these stresses a body may suffer either an elastic deformation, of a temporary character, or a plastic deformation, of a permanent character.

Specific Weight; Coefficient of Elasticity; Limit of Elasticity; Breaking Stress.

Before the strength of a body can be fully known, certain physical constants, whose values depend upon the material, must be determined.

(a] Specific Weight. The specific weight is the weight of a unit of volume. The specific weights of most of the materials of construction have been carefully found and tabulated. If the specific weight of any new material is required, a convenient approximate method is to prepare from it a number of regular solids of determinate volume and weigh them in an ordinary pair of scales. The ratio of the total weight of these solids to their total volume is the specific weight. It must be remembered that the weight may vary considerably with time, etc.; thus a sample of greenheart weighed 69.75 lbs. per cubic foot when first cut out of the log, and only 57 lbs. per cubic foot at the end of six months. When the strength of a timber is being determined, it is important to note the amount of water present in the test-piece, since this appears to have a great influence upon the results.

The straining of a structure is generally largely due to its own weight.

The total load upon a structure includes all the external forces applied to it, and in practice is designated dead (permanent) or live (rolling), according as the forces are gradually applied and steady, or suddenly applied and accompanied with vibrations. For example, the weight of a bridge is a dead load, while a train passing over it is a live load; the weight of a roof, together with the weight of any snow which may have accumulated upon it, is a dead load ; wind causes at times excessive vibrations in the members of a structure, and although often treated as a dead load, should in reality be considered a live load.

The dead loads of many structures (as masonry walls, etc.) are so great that extra or accidental loads may be safely disregarded. In cold climates, great masses of snow and the penetrating effect of the frost necessitate very deep foundations, which proportionately increase the dead weight.

(b) Coefficient of Elasticity. Generally speaking, a knowledge of the external forces acting upon a structure, discloses the manner of their distribution amongst its various members, but the deformation of these members can only be estimated by means of the coefficient of elasticity, which expresses the relation between a stress and the corresponding strain.

In practice it is usually sufficient to assume that a material is elastic, homogeneous, and isotropic, and its deformation under stress may be found, if the coefficients of elasticity, of form, and of volume are known.

In a homogeneous solid there may be twenty-one distinct coefficients of elasticity, which are usually classified under the following heads:

(1) Direct, expressing the relation between longitudinal strains and normal stresses in the same direction.

(2) Transverse, expressing the relation between tangential stresses and strains in the same direction.

(3) Lateral, expressing the relation between longitudinal strains and normal stresses at right angles to the strains; i.e., a lateral resistance to deformation.

(4) Oblique, expressing other relations of stress and strain. If a body is isotropic, i.e., equally elastic in all directions, the twenty-one coefficients reduce to two, viz., the coefficients of direct elasticity and of lateral elasticity. Such bodies, how- ever, are almost wholly ideal. In a perfectly elastic body E would be the same both for tension and compression. In the ordinary materials of construction it is slightly less for compression than for tension; but if the stresses do not exceed a certain limit, the difference is so slight
that it may be disregarded.

The coefficients of direct elasticity for the different metals and timbers are sometimes obtained by subjecting bars of the material to forces of extension or compression, or by observing the deflections of beams loaded transversely. The coefficients for blocks of stone and masonry might also be found by transverse loading; they are of little, if any, practical use, as, on account of the inherent stiffness of masonry structures, their deformations, or settlings, are due rather to defective workmanship than to the natural play of elastic forces.

The torsional coefficient of elasticity, i.e., the coefficient of elastic resistance to torsion, has been shown by experiment to vary from two fifths to three eighths of the coefficient of direct elasticity.

(e) Limit of Elasticity. When the forces which strain a body fall below a certain limit, the body, on the removal of the forces, will resume its original form and dimensions without sensible change (disregarding any effects due to the development of heat) and may be treated as perfectly elastic. But if the forces exceed this limit, the body will receive a permanent deformation, or, as it is termed, a set.

Such a limit is called a limit of elasticity, and is the greatest stress that can be applied to a body without producing in it an appreciable and permanent deformation.

This is an unsatisfactory definition, as a body passes from the elastic to the non-elastic state by such imperceptible degrees that it is impossible to fix any exact line of demarcation between the two states. Fairbairn defines the limit more correctly, as the stress below which the deformation is approximately proportional to the load which produces it, and beyond which the deformation increases much more rapidly than the load. In fact, both the elastic and ultimate strengths of a material depend upon the nature of the stresses to which they are subjected and upon the frequency of their application.

Generally speaking, then, the limit of elasticity of a material subjected to repeated stresses, is a certain maximum stress below which the condition of the body remains unimpaired,

Bauschinger's experiments indicate that the application to a body of any stress, however small, produces a plastic or permanent deformation. This, perhaps, is sometimes due to a want of uniformity in the material, or to the bar being not quite straight initially. In any case, the deformations under loads which are less than the elastic limit, are so slight as to be of no practical account and may be safely disregarded.

The main object, then, of the theory of the strength of materials, is to determine whether the stresses developed in any particular member of a structure exceed the limit of elasticity. As soon as they do so, that member is permanently deformed, its strength is impaired, it becomes predisposed to rupture, and the safety of the whole structure is threatened. Still, it must be borne in mind that it is not absolutely true that a material is always weakened by being subjected to forces superior to this limit. In the manufacture of iron bars, for instance, each of the processes through which the metal passes changes its elasticity and increases its strength. Such a material is to be treated as being in a new state and as possessing new properties.