Strength of materials - Morley

This book has been written mainly for Engineering students, and covers the necessary ground for University and similar examinations in Strength of Materials; but it is hoped that it will also prove useful to many practical engineers, to whom a knowledge of the subject is necessary.

In some sections of the work well-established lines have been followed, but several special features may be mentioned. In Chap. II. the different theories of elastic strength are explained, and subsequently throughout the book the different formulae to which they lead in cases of compound stress are pointed out. Considerable use has been made of the method of finding beam deflections from the moment of the area of the bending-moment diagram, i.e. from the summation my attention was called to the very simple application of this method to the solution of problems on built-in and continuous beams developed in Chap. VII., by my friend Prof. J. H. Smith, D.Sc. Other subjects treated, which have hitherto received but scant attention in text-books, include the strength of rotating discs and cylinders, the bending of curved bars with applications to hooks, rings, and links, the strength of unstayed fiat plates, and the stresses and instability arising from certain speeds of running machinery. Most of the important research work bearing on Strength of Materials has been noticed, and numerous easily accessible references to original papers have been given. Most of the results involving even simple mathematical demonstrations have been worked out in detail; experience shows that careful readers lose much time through being unable to bridge easily the gaps frequently left in such work. Many fully worked-out numerical examples have been given, and the reader is advised to read all of these, and to work out for himself the examples given at the ends of the chapters, as being a great help to obtaining a sound and useful knowledge of the subject. Many readers will have the opportunity of seeing and using practical testing appliances, and this portion of the work has been treated somewhat briefly in the last three chapters, ample references to works on testing and original papers being furnished.

22. Elasticity. A material is said to be perfectly elastic if the whole of the strain produced by a stress disappears when the stress is removed. Within certain limits (Art. 5) many materials exhibit practically perfect elasticity.

Plasticity. A material may be said to be perfectly plastic when no strain disappears when it is relieved from stress.

In a plastic state, a solid shows the phenomenon of "flow" under unequal stresses in different directions, much in the same way as a liquid. This property of "flowing" is utilized in the "squirting" of lead pipe, the drawing of wire, the stamping of coins, forging, etc.

Ductility is that property of a material which allows of its being drawn out by tension to a smaller section, as for example when a wire is made by drawing out metal through a hole. During ductile extension, a material generally shows a certain degree of elasticity, together with a considerable amount of plasticity. Brittleness is lack of ductility.

When a material can be beaten or rolled into plates, it is said to be malleable; malleability is a very similar property to ductility.

23. Tensile Strain of Ductile Metals. If a ductile metal be subjected to a gradually increasing tension, it is found that the resulting strains, both longitudinal and lateral, increase at first proportionally to the stress. When the elastic limit is reached, the tensile strain begins to increase more quickly, and continues to grow at an increasing rate as the load is augmented. At a stress a little greater than the elastic limit some metals, notably soft irons and steels, show a marked breakdown, the elongation becoming many times greater than previously with little or no increase of stress. The stress at which this sudden stretch occurs is called the "yield point" of the material.

Fig. 25 is a "stress-strain" curve for a round steel bar 10 inches long and i inch diameter, of which the ordinates represent the stress intensities and the abscissae the corresponding strains. The limit of elasticity occurs about A, the line OA being straight. The point B marks the "yield point," AB being slightly curved. After the yield-point stress is reached, the ductile extensions take place, the strains increasing at an accelerating rate with greater stresses as indicated by the portion of the curve between C and D. Strains produced at loads above the yield point do not develop in the same way as those below the elastic limit. The greater part of the strain occurs very quickly, but this is followed without any further loading by a small additional extension which increases with time but at a diminishing rate. The phenomenon of the slow growth of a strain under a steady tensile stress has been called "creeping" by Prof. Ewing. The stress necessary to initiate yielding is probably considerably greater than that necessary to continue it, and when a ductile metal is able to relieve itself of stress, yielding (up to a strain much greater than that at the elastic limit) will continue with a very considerable reduction in the stress applied. Messrs. Cook and Robertson using a slender bar of mild steel in parallel with two stout bars, found a reduction of 23 per cent, of that necessary to start the yielding. On account of the part which takes time to develop, the total amount of strain produced by a given load and the shape of the stress-strain curve will be slightly modified by the rate of loading. At D, just before the greatest load is reached, the material is almost perfectly plastic, the tensile strain increasing greatly for very slight increase of load. It should be noted that in this diagram both stress intensity and strain are reckoned on the original dimensions of the material.

During the ductile elongation, the area of cross-section decreases in practically the same proportion that the length increases, or in other words, the volume of the material remains practically unchanged. The reduction in area of section is generally fairly uniform along the bar.

After the maximum load is reached, a sudden local stretching takes place, extending over a short length of the bar and forming a "waist." The local reduction in area is such that the load necessary to break the bar at the waist is considerably less than the maximum load on the bar before the local extension takes place. Nevertheless the breaking load divided by the reduced area of section shows that the "actual stress intensity" is greater than at any previous load. If the load be divided by the original area of cross-section, the result is the "nominal intensity of stress," which is less, in such a ductile material as soft steel, at the breaking load than at the maximum load sustained at the point D on Fig. 25. Fig. 31 shows the stress-strain curves for samples of other materials in tension; each curve refers to round specimens 1 inch diameter, and 8 inches long. The elastic portions of the curves are drawn separately, with the strain scale 250 times as great as that for the more plastic strains.

24. Elastic Limit and Yield Point. The elastic limit (Art. 5; in tension is the greatest stress after which no permanent elongation remains when all stress is removed. In nearly all metals, and particularly in soft and ductile metals, instruments of great precision (see Art. 174) will reveal slight permanent extensions resulting from very low stresses, and particularly in material which has never before been subjected to such tensile stress.

Commercial Elastic Limit. In commercial tests of metals exhibiting a yield point, the stress at which this marked breakdown occurs is often called the elastic limit; it is generally a little above the true elastic limit.

There are, then, three noticeable limits of stress.

(1) The elastic limit, as defined in Art. 5.
(2) The limit of proportionality of stress to strain.
(3) The stress at yield point the commercial elastic limit.

In wrought iron and steel the first two are practically the same, and the third is somewhat higher.

The suggestion has been made that failure of perfect elasticity just below the yield point is due to small portions of the material reaching the breaking-down point before the general mass of the material. This supposition is supported by the fact that ductile materials of very uniform character show the yield point more strikingly than inferior specimens of the same material.

When the necessary stress is applied, the yielding certainly does not take place simultaneously throughout the mass, but begins locally at one or more points (probably due to a slight concentration of stress), and spreads through the remaining material without further increase of the load. This spreading of the condition of breakdown may be watched in unmachined iron and steel; the strain in the material is too great to be taken up by the skin of oxide, which cracks and flies off in minute pieces as the yielding spreads. In highly finished drawn steel the oxide chips off so as to form interesting markings on the surface of the bar; two systems of parallel curves, equally and oppositely inclined to the axis of the bar, are formed. A similar phenomenon may be noticed on a polished metallic surface when the metal is strained beyond the elastic limit.