such as have been obtained with the same class of material and under exactly the same circumstances; for only in this way can the effects of varied loading be ascertained independently of foreign influences. by Lippold. He starts from the following law: "To break a bar a certain amount of mechanical work is necessary, and this can be accumulated in the material at once, just as well as by repeated loads. These loads must, however, be applied instantaneously, or so rapidly The variable resistance to fracture, a, that vibrations arise." It is essential to is in Germany called, according to Launthis view, that only by means of vibra- hardt, "ultimate working strength tions can equal strain (i. e., proportion- (Arbeitsfestigkeit). The terms, too, of ate extension, or proportionate shorten- tensile strength, compressive strength, ing, as the case may be), and with it shearing strength, &c., are to be taken equal stress and equal work, be attained in a wider sense than hitherto, to mean by a load below that which produces t as by t itself, while less weight is attached to the repetitions of the straining action. Lippold arrives at formula for a, according to which the strength varies in the same manner as when based on Wöhler's law. It is doubtless possible that in time, through speculation and experiment, safe foundations for a correct theory of properties relating to strength will be reached. The author, however, is not satisfied to rest contented till then with the certainly false assumption of conerrors and stant strength. Smaller greater safety for structures can already be attained through the results of Wöhler's experiments. IV. ON THE STRENGTH OF MATERIALS. To start from Wöhler's law, regarding it as purely empirical: the ultimate working strength a for tension, compression, shearing, &c. Three special values of the ultimate working strength are of particular interest. I. The amount per unit of section of the resistance to fracture by a single application of a statical,.or at any rate very gradually applied load, which hitherto has been alone considered, may be called "statical breaking strength," t, (Tragfestigkeit). II. If, after every application of the load, the bar reverts to its original unloaded condition, and if all stresses are in the same sense,' * e. g., tension, compression, or shearing in one sense only, the greatest stress which can be sustained under the specified number of repetitions is called, according to Launhardt, the "primitive strength," u, (Ursprungsfestigkeit). III. Finally, for strength as regards alternating stresses of equal intensity in The words "often repeated" (II.) may opposite senses, i. e., for tension and be suppressed, and it may be left an compression, or for shearing in opposite open question what influences are con- senses, these being due to vibrations cerned in the fracture of the piece. If, taking place about the original position for the method hitherto used in determ- of unstrained equilibrium, the author ining dimensions, a new method is to be has introduced the term "vibrationsubstituted, the latter must be sufficient- strength," s, (Schwingungsfestigkeit). ly well founded, simple in its applica- Of course the statical breaking tion, and in conformity with experience. strength t and the primitive strength u At the same time it may be remembered apply to either tension, compression, or that it is not a question of ingenious shearing. If it be assumed, with Wöhlaws of unattainable accuracy, but of a ler, that repetitions of the straining useful guide for practical purposes. In action are essentially dangerous to the the first place it has to be determined material, u and s must be less the more by a formula how the ultimate working strength a varies with the difference between successive stresses. In developing or judging of this formula the results of experiment should find application, but if possible, only *"Direction" of a force, or a stress, is any line parallel to its line of action, and may be regarded indifferently as from left to right or right to left. "Sense distinguishes between these two alternatives. Thus, if a force is exerted from right to left it is exerted in one sense; if from left to right, in the opposite sense. The two opposite senses may, if required, be distinguished as positive sense and negative sense. This * Vide Organ für die Fortschritte des Eisenbahn- use of "sense" is universal among French writers on wesens, 1879. mechanics, and helps precision of statement. where the+sign must be chosen before and not less than u; since, as defined in the radical, because a must be positive V., its least value is u, and greatest t. The most complete results are those which Wöhler obtained in making bending experiments with unhardened Krupp spring steel. This material showed for a statical breaking strength of about 1,100 centners per square inch (Rhenish), a primitive strength of u=500 centners per square inch; whence from equation (9) the ultimate working strength in this case must be a=250+/62,500+600 a'. There follows for..... By Launhardt's equation.... By Wöhler's experiments... .a'= For unhardened spring steel from Mayr of Werben the experiments also give u=500, t=1,100; while for hardened Krupp spring steel, the lowest values are u=600, and t=1,200. ver, 1873. * Vide Zeitsch. des Arch. und Ing-Vereins zu Hanno+Wöhler "Uber die Festigkeitsversuche mit Eisen und Stahl;" Berlin Zeitschrift für Bauwesen, 1870, pp. and 14 of the reprint. As it is merely a question of comparison, the author retains Wöhler's original values and denominations. 7 Wöhler obtained from bending experi- d increases, and is a function of d, so ments with iron for axles made by the that as in V., Phoenix Company t=550, u = 300; whence follows a=150+/12,500+300 a'. For example, if a'=240, equation (9) gives for the ultimate working strength q=fd. (10A) for a=u, f=1, " a=s, f=4. Intermediate cases should depend on the results of experiment which, however, at present do not exist. In the meantime the two conditions just mentioned are fulfilled by the coefficient a=440; which agrees exactly with the therefore the following conditions must value given by his tension experiments. be fulfilled: Unfortunately only a few among Wöhler's experiments give t, u, s, and intermediate values of a. Considering, however, that absolutely exact laws for constructive materials will certainly never result from experiments, that even in brands of iron acknowledged to be good, differences in the statical breaking strength t of as much as 40 per cent. occur, and that it is merely a question of finding a substitute for the still more hence, from (10a), rough and incorrect assumption of a constant a, even the preceding might suffice for practical purposes until more facts are accumulated. a= f= u-s U-s 2u-s-a and therefore d= U-8 2u-s-a u-s a' (11) ·(a+a'), u(1— u—3 a) . . . (12) и Launhardt's formula (4) is the expression of Wöhler's law. By the latter the limits of value for f are also determIf, now, for a bar of any section, max ined. In the first instance, the choice B is the numerical value of the absoof the interpolation formula for f is ar- lutely greatest load, max B' the numeribitrary. Nevertheless, this choice ap- cal value of the greatest load in the oppears to be well justified by the only ex-posite sense, distinguished from the periments of Wöhler's suited for com- other as negative, then the ratio of the parison. Neither results from other less to the greater limit of stress is sources, nor experience, nor practical feeling are against it. Even for more exact determinations than those under consideration such an approximation whence from (12) ought to be considered satisfactory. To the author's mind, it would appear sufficient if the deviations of the real values of a from those given by the formula did not exceed the deviations from one another of real values of a in good and commonly used materials. u=3,510, s=2,050; senses. The results which are obtained by formula (17) do not at any rate contradict practical feeling or assumptions hitherto customary (vide X). IX. ADMISSIBLE STRESS FOR IRON. In determining the numerical values of the admissible stress per unit of area, Wöhler's experimental results must be applied. At the same time every illusion as to the universal applicability of the latter is to be put aside; just as formerly a new series of experiments on the statical breaking strength t was accepted without their results being henceforth exclusively used. If, for instance, in such cases a statical breaking strength of t=3,870 kilogrammes per square centimeter resulted, 3,500 perhaps, was taken as a low value; so, if not still more cautiously, similar judgment must be exercised in the choice of a. Beyond this, the help of factors of safety was always for shearing stress in opposite senses resorted to, and this must still be so in with the same material u=2,780, 8=1,610 centners per square inch (Rhenish). It is a decided defect that the choice of the coefficient for the intermediate stages cannot for the present be checked by ex periment. This is of course true of all empirical formula, which may be con structed on Wöhler's data for stresses in opposite senses. When, however, it is admitted that s and u are not equal, or in other words, that the same ultimate working stress cannot apply in cases of alternation between tension and compression that applies to intermittent tension alone, it becomes evident that an interpolation formula must be adopted. The author considered that it ought to be built up by reasoning similar to Launhardt's, and hence in Germany the formula, a=u (1 + 1+ = u( 1 + a=u the future. In the following determinations particular reference is made to iron bridges. For simple compression, in accordance with the usual practice, the same limiting stresses as for tension are adopted. Alternating stresses, either exclusively tensile, or exclusively compressive.-In what follows the numerical intensities of stress are expressed in kilogrammes per square centimeter. With iron for axles of the "Phoenix" Company, bending experiments of Wöhler's gave t=4,020, ; and by (16) u=2,190; whence t-u и t-u 5 = 5 u (1+ Φ )=2,190 (1+0). a=u 6 If the formula is to afford the desired security for any tensile or compressive stress, a start must always be made from the most unfavorable case of this stress. For the same iron, with the direct appli -") for positive . (16) cation of tensile stress, u=2,190 also: Þ) for negative Þ. (17) are always applied side by side. In both (16) and (17) t, u, and s are numeri- very approximately; and suppose the cal values without sign [+or-], while ultimate working strength be further the ratio of the least to the greatest reduced by rounding off the value of u limit of stress is positive or negative on the side of safety, to according as the extreme alternate stresses are in the same or different 1 a=2,100(1+0). 2 Alternation between tensile and compressive stresses. -For the Phoenix iron above-mentioned Wöhler found u=2,190, U-8 7 s=1,170; hence in this case U 15' and by (17), 7 a=u(1+"=30) =2,190(1+ P). 15 Rounding off these values on the safe side by diminishing 2,190 to 2,100 and increasing to, since is negative, there results t=3,150, u=2,100, 8=1,050. Smaller values of strength for bridges than these figures indicate are certainly not necessary, and have not been hitherto customary. Notwithstanding this, as has already been remarked in I., these values can have no universal application, and should they be thought too high for whence, with the factor of safety the the material and the purpose, the followadmissible stress per square centimeter ing may, perhaps, be suitable: v=640, becomes m=n; whence a=2,100 (1+P), It has been shown in the above that F= max B 640 (1 + 2/2) (24) Different values also may be assigned to m and n; for steel, for instance, different values cannot be avoided.* fraction Exceptiont has been taken to the choice of 6-700 might have been conWith equal justice the sidered dubious, and 687 preferred. It may suffice to answer that, in these objections, general and special facts have been confused. When for booms of girders p is the the stress per square centimeter may be dead load, q the total load per lineal generally expressed, meter, and = min B p max B q then b=v (1+m these latter numerical values for wrought iron being according to IX. When the numerical values assumed for b, the admissible stresses for a static load, for alternations between tension and an unloaded condition, and for alternations between tension and compression of equal amount, would stand in *Vide Weyrauch "Strength and Determination," &c., pp. 60-68. + Vide" Engineering," vol. xxix., p. 263. |