The strength of materials - Andrews

The advance in the application of scientific methods to architectural and engineering problems has made increasing demands upon the theoretical knowledge required by architects and engineers; it is the aim of the present book to present in as simple a method as is consistent with accuracy, the principles which underlie the design of machines and structures from the standpoint of their strength.

The subjects commonly called respectively the Strength of Materials and the Theory of Structures have much in common; much of the subject matter contained in the author's books upon the latter subject has, therefore, been incorporated, the same general method involving the use and application of graphical methods in preference to purely mathematical methods having been adopted in the other branches of the subject. An attempt has been made to present more clearly than is general the various theories as to the cause of failure in materials and the effect of these theories upon design.

Although the author hopes that the book will be especially useful for students reading for the Assoc. M. Inst. C. E., and University degree examinations in Engineering, he has attempted to present the subject in sufficiently practical form for it to be of greater assistance in practical design than is the case with an ordinary class book; with this in view many diagrams and tables have been incorporated for enabling the formulae to be applied with a minimum of time and trouble.

A large number of numerical examples are worked out and further exercises are given; the student is recommended to work for himself all such examples and to pay particular attention to the assumptions which are made in deriving the various formulae. Nearly all engineering formulae are only approximately correct; in the present branch of the subject this is chiefly because there is no material known which conforms exactly to the simple laws of elasticity upon which the subject is based. We cannot condemn too strongly the blind application to a particular practical problem of formulae which were never intended to be so applied; the unfortunate distrust which practical engineers so often have to "theory" is to some extent brought about by the fact that the theories that they see employed are often inapplicable. It is essential for us to acknowledge the limits of theoretical methods and not to attempt to express our results to a greater degree of accuracy than the nature of the problem will allow.

Repetition or Variation of Stresses. In the design of machines and structures, we very often have to deal with cases in which the stresses vary in amount from one time to another; such cases occur in nearly every machine part subjected to rotary and reciprocatory movements and in structures which have to resist wind-pressures and rolling loads. In recent years, a large amount of investigation has been carried out on the strength of materials which are subjected to alternating stresses. The stress required to cause rupture in a material which is gradually increasingly stressed is called the static breaking stress, and is the stress obtained in the ordinary testing machines.

Fairbairn discovered in connection with some tests on wrought-iron girders, that a girder can be ruptured by repeatedly applying a load equal to about one-half of the static breaking load.

The first exhaustive investigation on the subject was conducted by Wohler on behalf of the Prussian Ministry of Commerce, and was published in 1870. Wohler's experiments extended over a period of twelve years, and had results which at the time were very startling, and the importance of which has only in comparatively recent years been appreciated by engineers.

The general result of these and subsequent experiments is to show that the stress necessary to rupture a material when such stress is repeated a very large number of times is considerably less than the static stress.

In Wohler's experiments, which were carried out in tension, bending and torsion, some of the variations were from zero to a maximum in tension or compression and some were for a complete reversal of stress.

In one form of Wohler's apparatus for testing by reversal of stress in bending, the specimen was in the form of a projecting beam or cantilever A (Fig. 41) clamped at the end of a shaft E, mounted between bearings B. The shaft was rotated by means of a belt surrounding a pulley C, and the specimen was of circular section and loaded at the end by a spring D, and in the rotation the compression and tension sides changed places gradually, thus giving a gradual reversal of stress. To balance the forces on the machine, a specimen was mounted at each end of the shaft.

In another form of apparatus a beam was mounted upon knife edges to one of which a spring was connected by levers. The load was applied in the centre by a spring rod which was lifted periodically by a crank upon a rotating shaft, thus gradually applying the load and taking it off again.

Full accounts of the experiments will be found in Unwin's Testing of the Materials of Construction.

The Fatigue of Metals. The phenomena described above are often referred to as the "fatigue of metals"; the suggestion being that the stress causes a change in the molecular structure of the metal and that the metal gets fatigued after a time and so breaks down under a smaller load. The bulk of the evidence, however, appears to be against that view and in favour of the theory that ultimate failure will occur only if the elastic limit is exceeded and thus the effects of overstrain become accumulative.

Specimens cut out of pieces that have been fractured by repetition of stress do not exhibit any weakening that the fatigue idea suggests.

The subject is still full of difficulties from the point of view of a satisfactory explanation of the results. For instance, the effect of overstrain is to cause the material to become brittle, and yet the more brittle kinds of steel (the high carbon steels) show less effect than the mild steel.

One explanation, called Foster's Theory, is that the mechanical hysteresis (p. 59) causes a very small permanent strain at each repetition and that the effect of these permanent strains is cumulative so that ultimately the permanent strain becomes sufficient to cause failure.

The appearance of the fracture in these experiments is always different from that for ordinary static tensile tests; that for mild steel being more like a hard steel. This is probably due to the effect' of overstrain upon the properties of the metal.

WORKING STRESSES

The Conflict between Theory and Practice. An engineer has been tersely described by a somewhat characteristic American as "a man who can do for one dollar what a fool can do for two." Although from an aesthetic standpoint this seems to be a somewhat too mundane description of the engineer's vocation, we must not forget that the most scientific construction is the one which best fulfils the conditions for the least cost.

There is in reality no conflict between theory and practice in designing; each has its own place, and each is dependent on the other. The theory will tell us what is the best design as far as the economical arrangement of material goes. The best-designed structure is one which would be about to collapse at all sections at the same time; or, in other words, the various parts are so designed that the stresses in them are equal. This is all that the theory sets out to do. Practice, on the other hand, determines whether the theoretical design is in reality the cheapest in the end. Questions of workmanship, cost of erection and upkeep have to be considered, and it is only by balancing these with the theory that the really scientific design is obtained.

In dealing with the theoretical side of design we must never forget that, if we are to be guided by theory at all, we should see that we use the best theory. The disdain for theory that ultra-practical men often possess is largely due to the fact that their theoretical knowledge is not sufficiently comprehensive; they have not realised the conditions which have to be fulfilled before a certain theory is applicable, and so they probably use some formula for a case for which it was never intended.

Another point to be remembered is that practical rules for use in design are not necessarily sound because the machines or structures resulting therefrom satisfactorily fulfil their function. Such rules may make the design much heavier, and therefore much more costly, than necessary. Our aim in the theoretical investigations should be to eliminate as many uncertainties as possible, and not to be merely content with erecting something which will stand.

Working Stresses and Factor of Safety. The question of the working stresses to adopt in practice is of the utmost importance, and if our design is to be of any real value we must have clear ideas as to such working stresses.

In dealing with working stresses we often speak of the factor of safety. This may be defined as the factor by which the working stresses may be multiplied to give stresses which will result in failure. This phrase is one which is often used glibly without any real meaning; and it has been suggested that in many cases it would be better called the factor of ignorance. If we design a structure with a factor of safety of four, say, we certainly do not as a rule mean that the structure could bear four times the load without failure. This is because there are certain contingencies that we do not allow for in our design. Our aim should be, however, to make our calculations so that the factor of safety has as exact a meaning as possible. This can be done only by choosing our working stresses skilfully and by making allowance for as many points as possible. For steel-work it is common to adopt as a working stress in tension one-quarter of the breaking stress in tension and to say therefore that the factor of safety is 4. Many designers forget, however, to make the due allowance for live -or variable loads., The basing of the factor of safety on the breaking stress is also open to a very serious objection, viz. that the elastic limit of the material is the point which really determines the safety of the structure. If the stresses are above the elastic limit, failure is almost certain to ensue, especially in the case of compression members or struts. It would, therefore, be better to base the working stresses on the elastic limit or the yield point, since in pure tension the two points are close together, and the yield point is much easier to measure and specify for a definite minimum value of such limit in the steel. The point commonly urged against this method of procedure, viz. that the elastic limit is a much more variable quantity than the breaking stress seems to us to be one in favour of its adoption. It is certain that stresses beyond the elastic limit are very dangerous, and if this quantity is a variable one we ought to know it for the material that we are using, and base our working stresses on it accordingly. We would suggest that the dead-load or static working stress should be taken as one-half of the natural elastic limit.