Working Stress; Elastic Failure.— The mechanics of elastic bodies is of limited value in rational designing and serves only as a re fined rule of thumb until it is decided how large a stress or strain a material can safely resist, how large a stress causes ultimate rup ture and finally whether failure is produced by normal or shear stress or by some combination of them. Researches initiated by Willer in 1870 and continued by Bauschinger, Reynolds and Smith (Phil. Trans., Roy. Soc., 1902, p. 265), Stanton and Bairstow (Proc. Inst., Civil Engineering, 1906, Vol. CLXVI, p. 78), and others have shown that alternating stresses (tension to compression) well below the elastic limit may produce rupture when repeated often enough. A familiar illustration of this phe nomenon is that springs do not last forever, although they might never be stressed beyond the so-called limit. It used to be be lieved that these reversals caused the material to crystallize or become granular. This is now known to be incorrect; all inorganic bodies are crystalline under the microscope, and re versed stresses, or even fluctuating ones always of the same kind, weaken the material by caus ing grinding and abrasion on planes of cleav age.. Investigations solely with the aid of testing machines will throw probably little light on the subject. The most fruitful results are undoubtedly to be looked for in the compara tiVely new field of metallography. This is con cerned with the microscopic examination of internal structure. The first attempt to bring the microscope to the aid of the testing ma chine was made in 1864 by Sorby of Sheffield, whose work, however, remained unknown until it was independently taken up again by Mar tens in Germany and Osmond in France. The subject is too large to be discussed in a short article. Consult Mellor, 'Crystallization of Iron and Steel' (1905) ; Rosenhain, 'Introduc tion to Physical Metallurgy' (1917), and Humphrey, of Strain' (1919).
Both strength and stiffness need to be con sidered in designing. No structural member may yield or give more than a certain amount apart from the question of strength. For in stance, if a floor beam sags by an amount greater than about one-third per cent of its length a plastered ceiling under it will crack. Questions of rigidity or stiffness are decided by means of mathematical formulas derived either from the principles of mechanics or em pirically. They are based on Hooke's law and on the assumption that the elastic limit is not exceeded. The usual method of checking them is to compare calculated with experimentally found deformations; the calculated stresses are difficult to verify because we have no direct method of measuring stress within a body. However, use has been made of the well-known fact discovered by Brewster in 1815 that an isotropic transparent medium is made doubly refracting by stress and breaks plane polarized light into orthogonal components; likewise, cir cularly polarized waves become elliptically pol arized. Wilson, Carus (in Phil. Meg., Vol. XXXII, 1891, p. 481) ; Hiinigsberg (in Zeit schrift 5st, Mg. Verein, 1904, Nr. 11); Mesnager (in Annales des Ponts et Chaussees, Fasc. iv, 1913) and Coker (in Engineering, Vol. XCI, p. 1), examined the optical properties of various structural elements made of celluloid and glass and found a remarkably close agreement be tween theory and fact. Of the many books dealing with the formulas of the mechanics of materials the most complete are Morley, 'Strength of Materials' (1908 and later edi tions), and Bach, 'Elastizitat and Festigkeit' (5th ed., 1905).
The working stress is the greatest stress to which a member of a structure or a machine is subjected during use or operation. If it ex ceeds the elastic limit there may occur per manent deformations not allowed for in the design, change of properties of the material, and when the stresses alternate or vary, ulti mate failure. Indeed, the working or safe
stress must be taken considerably below the elastic limit to allow for deterioration due to wear, unavoidable imperfections in workman ship and manufacture, lack of uniformity of material, accidental overloading, but not for poor design. As it is difficult under commercial conditions to determine the rather ill defined elastic limit it is customary to state the safe stress as a certain fraction, 1/n, of the breaking stress; n is called the factor of safety and is de termined by experience. According to the ex periments of Wailer and Bauschinger on steady, fluctuating, and alternating loads, the safe stresses are in these cases roughly in the ratios 3:2:1, which are, therefore, taken as a guide in determining the factors. The building laws of many cities in the United States specify definite, safe stresses instead of factors of safety. For recent data consult Marks, 'Mechanical Engineers' Handbook' (1917) ; Merriman, 'American Civil Engineers' Pocket Book' (1916).
A material is simple tension or compression, or simple shear is considered to be overloaded when the corresponding safe stresses, as defined above, are exceeded. The immediate cause of failure in the case of compound stress— a number of stresses acting simultaneously— is still a matter of dispute. A material fails when the yield point is reached. Lame assumed that the largest principal stress produced yieldin; this theory is sometimes called after because he gave it considerable authority by using it in his 'Applied Mechanics' (1858), and it is generally followed by American and Eng lish engineers. In it the effect of the other principal stresses is disregarded, which implies that a block under compression on top and bottom is just as strong as it would be if it were prevented from expanding laterally. Pon celet and Saint Venant believed that excessive straits caused failure, i.e., stress is harmful only in so far as it produces molecular separation. From this point of view, the equivalent simple stress must not exceed the working stress. Coulomb, Tresca and G. H. Darwin proposed the theory that the difference between the prin cipal stresses, i.e., the greatest shear— eq. (7) above—is a measure of strength. Recent ex periments have given some confirmation to it, at least for ductile metals. Consult Guest, 'Strength of Ductile Materials under Combined Stress' (in Phil. Mag., 1900, Vol. L) ; Han cock, 'Effect of Combined Stresses) (ib., 1906, Vols. XI, XII) ; Scoble, 'Strength of Ductile Materials under Combined Stress' (ib., 1906, Vol. XII). The latest theory is that of Mohr (in Zeit. Verein d. Ing., 'Vol. XLIV, 1900, p. 1524, Vol. XLV, 1901, pp. 4, 740), the essential hypotheses of which are: (a) failure depends only on the normal stress on the surface of rupture; (b) the surface of rupture is normal to the planes of greatest and least principal stresses [for space stresses there are three principal planes], i.e., is parallel to the principal plane of intermediate stress and is in dependent of that stress.
In the absence of conclusive experiments it is necessary to design independently for greatest shear, greatest normal stress and greatest equivalent simple stress; this errs on the side of safety. Metallographical researches have shown such great differences in the molecular structure of materials that a complete theory of elastic strength will probably have to take structure into consideration instead of, as now, dealing only with the geometric arrangement of the stress vectors on the one hand and the stress-strain curve on the other.