* by Lippold.* He starts from the fol- such as have been obtained with the lowing law : “To break a bar a certain same class of material and under exactly amount of mechanical work is necessary, the same circumstances; for only in this and this can be accumulated in the mate way can the effects of varied loading be rial at once, just as well as by repeated ascertained independently of foreign loads. These loads must, however, be influences. 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 the ultimate working strength a for tenby t itself, while less weight is attached sion, compression, shearing, &c. Three to the repetitions of the straining action. special values of the ultimate working Lippold arrives at formula for u, accord- strength are of particular interest. ing to which the strength varies in the I. The amount per unit of section of same manner as when based on Wöhler's the resistance to fracture by a single law. application of a statical,.. .or at any rate It is doubtless possible that in time, very gradually applied load, which hiththrough speculation and experiment, erto has been alone considered, may be safe foundations for a correct theory of called “statical breaking strength,” t, properties relating to strength will be (Tragfestigkeit). reached. The author, however, is not II. If, after every application of the satisfied to rest contented till then with load, the bar reverts to its original unthe certainly false assumption of con- loaded condition, and if all stresses are stant strength. Smaller errors and in the same sense, e. g., tension, comgreater safety for structures can already pression, or shearing in one sense only, be attained through the results of Wöh- the greatest stress which can be susler's experiments. taiped under the specified number of repetitions is called, according to LaunIV. ON THE STRENGTH OF MATERIALS. hardt, the “primitive strength,” u, (Ur sprungsfestigkeit). To start from Wöhler's law, regarding III. Finally, for strength as regards it as purely empirical : 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,” 8, (Schwingungsfestigkeit). ly well founded, simple in its applica Of 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. *"Direction" of a force, or a stress, is any line paral lel to its line of action, and may be regarded indifferIn developing or judging of this for- ently as from left to right or right to left. mula the results of experiment should if a force is exerted from right to left it is exerted in find application, but if possible, only one sense; if from left to right, in the opposite sense. The two opposite senses may, if required, be distin guished as positive sense and negative sense. * Vide Organ für die Fortschritte des Eisenbahn- use of “sense" is universal among French writers on mechanics, and helps precision of statement. course Sense This wesens, 1879. a u max B repetitions are intended, and hence If B denotes the total load on a bar minima for a bridge which is to last for of any section, alternating between ever. In giving numerical values in the maximum and minimum values, the ratio sequel the latter case is contemplated. of the less to the greater limit of stress is V. LAUNHARDT'S FORMULA. min B a' 0 (7) Suppose a bar whose sectional area is max R unity to experience stresses in the same so that sense which vary between the maximum a and the minimum a'. min B t-u (8) of stress is here d=a-a', The is Launhardt's formula.* It is and a=d+a'. (3) applicable when a member of a strucAccording to Wöhler's law a dimin- ture is always subjected solely to tenishes as d increases. From (3), and the sion, solely to compression, or solely to definitions of the statical breaking shearing stress in one sense, the values strength t and primitive strength u, the of t and u only, the statical breaking two limits of value for a are strength and the primitive strength for for a' =0, a=d =u, tension, compression, or shearing, as the " d=0, a=a'=t. case may be, having to be substituted. As according to Wöhler a should be a VI. REMARKS ON LAUNHARDT'S FORMULA. function of d, a=fd In order to ascertain whether Launwhere f is at present an unknown mul- hardt's choice of the coefficient f is aptiplier. It is known, however, that plicable also to intermediate cases, equafor d=0 since a=t, f= 0, tion (6) is to be solved for a, giving " d=u a=d, f=1. For intermediate values of a the varia +(t– u)a', (9) 2 4 tion of f must be determined by considering experimental results obtained un- the radical, because a must be positive where the+sign must be chosen before der circumstances, as nearly as possible, and not less than u ; since, as defined in identical. The coefficient chosen by Laur hardt V., its least value is u, and greatest t. The most complete results are those t-u f= (5) which Wöhler obtainedt in making t-- a corresponds to the best of the results, Kruppspring steel. bending experiments with unhardened This material and fulfills the imposed conditions. showed for a statical breaking strength Hence by (4) of about 1,100 centners per square inch t-U t-u d= -la--a'); (Rhenish), a primitive strength of u=500 tra t-a centners per square inch ; whence from and therefore, reducing, equation (9) the ultimate working t-u a' strength in this case must be a=U1+ (6) a=250+ 162,500+600 a'. a= a=u(1 . For unhardened spring steel from * Vide Zeitsch. des Arch. und Ing-Vereins zu HannoMayr of Werben the experiments also ver, 1873. give u=500, t=1,100; while for hard- und Staht;" Berlin Zeitschrift für Bauwesen, 1870, pp: + Wühler “Uber die Festigkeitsversuche mit Eisen ened Krupp spring steel, the lowest 7 and 14 of the reprint. As it is merely a question of comparison, the author retains Wöhler's original values are u=600, and t=1,200. values and denominations. U-8 a= U-8 d= -a a 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., Phænix Company t=550, u= 300; I=fd (10A) whence follows Since, by the definitions of the primitive a=150+12,500+ 300 a'. strength u, and the vibration strength 8 in IV., For example, if a'=240, equation (9) for a'=0, a=u=d, gives for the ultimate working strength " a'=s, a=s=jd, a=440; which agrees exactly with the therefore the following conditions must value given by his tension experiments. be fulfilled : Unfortunately only few among for a=u,f=1, Wöhler's experiments give t, u, s, and intermediate values of a. Considering, a=8, fr. , however, that absolutely exact laws for Intermediate cases should depend on constructive materials will certainly the results of experiment which, hownever result from experiments, that even ever, at present do not exist in the in brands of iron acknowledged to be meantime the two conditions just mengood, differences in the statical breaking tioned are fulfilled by the coefficient strength t of as much as 40 per cent. f= (11) occur, and that it is merely a question of 2u-s-a finding a substitute for the still more hence, from (10A), rough and incorrect assumption of a constant a, even the preceding might (a +a'), suffice for practical purposes until more 2u-8 2u-s-a facts are accumulated. and therefore Launhardt's formula (4) is the expres a=ul sion of Wöhler's law. By the latter the (12) limits of value for f are also determ If, 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 max B' a' feeling are against it. Even for more 0: (13) exact determinations than those under max B consideration such an approximation whence from (12) ought to be considered satisfactory. To max B' the author's mind, it would appear suffi a=u1 cient if the deviations of the real values of a from those given by the formula This is Weyrauch's formula. It is did not exceed the deviations from one always to be applied in those cases another of real values of a in good and where a member of a structure is subcommonly used materials. ject to stresses alternating between ten sion and compression, or between shearVII. WEYRAUCH'S FORMULA. ing actions in opposite senses. By u and a are to be understood, respectively, Suppose a bar of unit sectional area the primitive strength and ultimate subjected to alternating straining ac- working strength for the numerically tions in opposite senses. If then a be greater of the two alternate stresses if the numerical value of the greater, a' they differ in amount. that of the lesser limit of stress, the VIII. REMARKS ON WEYRAUCH'S FORMULA. difference of stress is d=a+a', and therefore If equation (12) be solved for a, a=d-a (10) + -(u—s)a'. (15) According to Wöhler, a diminishes as 2 4 VOL. XXIV. No. 4-23. U-8 ť) ... (14) u max B u a= senses. In this formulua, from (12), The results which are obtained (u-8)a'=au-a'. by formula (17) do not at any rate conThe maximum of this expression is tradict practical feeling or assumptions found by differentiation with respect to hitherto customary (vide X). a, thus: IX. ADMISSIBLE STRESS FOR IRON. In determining the numerical values da of the admissible stress per unit of area, to occur when a= =$u, and the maximum Wöhler's experimental results must be apvalue of plied. At the same time every illusion as u? (u-$)a'=au-a'= to the universal applicability of the latter 4. is to be put aside; just as formerly a new Hence the expression under the radical series of experiments on the statical in equation (15) can never become imagi- breaking strength t was accepted withnary, and this refutes an objection in out their results being henceforth exclu“Engineering” (vol. xxix., pp. 304, 341). sively used. If, for instance, in such Wöhler found for alternations of ten- cases a statical breaking strength of sion and compression for “Phænix" t=3,870 kilogrammes per square centiiron meter resulted, 3,500 perhaps, was taken as a low value; so, if not still more cauu=2,190, s=1,170; tiously, similar judgment must be exerfor Krupp cast steel for axles cised in the choice of a. Beyond this, u=3,510, s=2,050; 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 the future. In the following determinau=2,780, s=1,610 tions particular reference is made to iron centners per square inch (Rhenish). It bridges. For simple compression, in accordance with the usual practice, the is a decided defect that the choice of the coefficient s'for the intermediate stages same limiting stresses as for tension are adopted. cannot for the present be checked by experiment. This is of course true of all tensile, or exclusively compressive. In Alternating stresses, either exclusively empirical formule, which may be con- what follows the numerical intensities of structed on Wöhler's data for stresses stress are expressed in kilogrammes per in opposite senses. When, however, it is admitted that s and u are not equal, of the "Phoenix square centimeter. With iron for axles Company, bending or in other words, that the same ultimate working stress cannot apply in cases of experiments of Wöhler's gave t=4,020, t-u 5 alternation between tension and com- u=2,190; whence 7; and by (16) pression that applies to intermittent tension alone, it becomes evident that an t-u interpolation formula must be adopted. a=u(1+ 6 The author considered that it ought to If the formula is to afford the desired be built up by reasoning similar to security for any tensile or compressive Launhardt's, and hence in Germany the stress, a start must always be made from formulae, the most unfavorable case of this stress. =u(1 Φ For the same iron, with the direct appli) for positive 0 . (16) cation of tensile stress, u=2,190 also: but t=3,290 only. Therefore take 2,190 =2 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 0 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 1 the extreme alternate according as a=2,100(1+0). stresses are in the same or different 5 * = и t-U au U-S a=U + u + ) (24) If } is chosen as a suitable factor of For iron, in particular, as explained in safety, the admissible stress per square IX., there may be assumed for tension centimeter then becomes and compression v=700, m=n=}, so that very simply, b=700(1+ · (18) 2 6-=700( 1+ Alternation between tensile and com 2 pressive stresses. – For the Phoenix iron max B (23) above-mentioned Wöhler found u=2,190, F= 7 1+ s=1,170; hence in this case 2 15' and by (17), By this threefold safety is attained if 7 t=3,150, u=2,100, 8=1,050. a=u(1+ © ) =2,190(1+ P). 15 Smaller values of strength for bridges than these figures indicate are certainly Rounding off these values on the safe not necessary, and have not been hithside by diminishing 2,190 to 2,100 and erto customary. Notwithstanding this, increasing it to }, since 0 is negative, as has already been remarked in I., these there results values can have no universal application, a=2,100 (1+0), 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 b=700 (19) 6=640(1+ 2 Shearing Stresses. - In most practical max B Fcases a and b for shearing stresses may be made of what they would be with 640(1+ 2 the same ratio of stress for tension and compression.* Therefore, gener. Different values also may be assigned to ally, m and n; for steel, for instance, differ ent values cannot be avoided.* b=560( 1+ (20) 2 Exception has been taken to the fraction 4. where 0 is positive or negative, accord- choice of 7=700 might have been con With equal justice the ing as the alternating limits of stress sidered dubious, and 687 preferred. are in the same or in opposite senses. may suffice to answer that, in these ob jections, general and special facts have X. REMARKS ON THE CHOICE OF b. been confused. It has been shown in the above that When for booms of girders p is the the stress per square centimeter may be dead load, 9 the total load per lineal min B generally expressed, P meter, and Ø= then for positive 0 by b=v (1+mo) · (21) max B 2 negative 0 " b=v (1+no). (22) b=v(1+m =7001+ (25) where v, m, n, are constants depending 291 on the nature of the material and sense these latter numerical values for wrought of stress, Ø the ratio of the less to the iron being according to IX, greater limiting stress. When the ad When the numerical values assumed missible stress per unit of area has been for 6, the admissible stresses for a static fixed, the necessary net sectional area is load, for alternations between tension found by and an unloaded condition, and for alter nations between tension and compresF= 6 sion of equal amount, would stand in Vide "Strength and Determination of the dimen * Vide Weyrauch “Strength and Determination," sions of Structures of Iron and Steel,” by Dr. P.J.J. &c., pp. 60-68. Weyrauch, pp. 81-88 ante. + Vide“ Engineering," vol. xxix., p. 263. v max B |