fc^o. ! M 7«£ T .-**3 PREPRINT. — This preprint is subject to correction and modification and is not to be republished as a whole or in part pending its formal release by the American Society for Testing Materials through its Secretary-Treasurer. It is issued primarily to stimulate written discussion, which may be trans- mitted to the Secretary-Treasurer for presentation at the approaching Eight- eenth Annual Meeting, June 22 to 26, 1915. THE FAILURE OF MATERIALS UNDER REPEATED STRESS. SUMMARY. Nearly all our ideas of the action of materials under stress are derived from considerations of static action. The action of materials under repeated stress differs from static action in important respects. Under stresses slightly lower than the elastic limit as ordi- narily determined materials fail under oft-repeated stress, and the relation of stress at failure to number of repetitions may be expressed by an exponential formula, as shown by Basquin and others. A proposed explanation of this formula is given, based on the structural damage done to material by inelastic action during application and removal of load. For still lower stresses no positive evidence of a fixed endur- ance limit has been found for commercial materials, and various methods used for locating such a supposed limit give varying results from the same test data. A modification of the exponential formula is proposed for low stresses. Tentative values of constants for the proposed formulas are given and also some general discussion on action under repeated stress. THE FAILURE OF MATERIALS UNDER REPEATED STRESS. By H. F. Moore and F. B. Seely. Behavior of Materials. Materials under Static Stress and under Repeated Stress . — The behavior of materials under repeated stress shows important variations from the action under static stress. Nearly all the ideas of repeated stress have been developed from considera- tions of static loading. One very common idea is that for any given material there is a definite elastic limit below which the behavior of the material is perfectly elastic. Under static loads such a conception may be regarded as exact without involving serious error, though careful writers on the mechanics of mate- rials have for a long time recognized that no absolute elastic limit has ever been fixed for any material . 1 In structures under static load local stresses of considerable magnitude — frequently beyond the yield point of the material — exist without producing any appreciable effect on the stability or the deforma- tion of the structure as a whole, and such stresses are frequently neglected in structures subjected to static load. If, however, the load on a structural part or a machine member is repeated many times such local overstress may cause a crack to start which, spreading, eventually destroys the member; or inelastic action too small to be detected even by delicate static tests of material may by cumulative action cause serious damage under oft-repeated loading. An illustration of the difference between static and repeated loading is furnished by the action of wire ropes bent around sheave wheels. The fiber stress due to bending is high and frequently causes inelastic action, which, however, is confined to a small portion of each wire. This inelastic action is very difficult to detect by means of static tests, but as the wire is 1 Thurston, “Text-Book of Materials of Construction,” p. 348; Burr, “The Elasticity and Resistance of Materials of Engineering,” p. 200; Wawrziniok, “Handbuch des Materials- prufungwesen,” p. 9; Unwin, “The Testing of Materials of Construction,” p. 13. ( 2 ) Moore and Seely on Repeated Stress. 3 repeatedly bent around sheave wheels this high local stress starts cracks which eventually cause rupture of individual wires. In this case, as in many others, the conception of perfect elastic action, allowable for static loading, must be discarded for re- peated loads. Materials under Repeated High Stress . — For a range of fiber stress extending from the yield point of the material (for brittle materials the ultimate strength) down to a stress slightly lower than the elastic limit, as determined by laboratory tests of the usual precision, repeated stress will cause failure, and there seems to exist a fairly definite relation between fiber stress and the number of repetitions necessary to cause failure. This relation was pointed out by Basquin 1 before this Society. It may be expressed by the formula: S = KN q in which S = intensity of fiber stress in pounds per square inch, iV = the number of repetitions of stress to cause failure, and K and q are experimentally determined constants. A similar relation was noted later by Eden, Rose and Cunningham 2 and by Upton and Lewis . 3 Whether for still lower stresses such a law holds, or whether there is an “ endurance limit” below which failure will not occur under any number of repetitions of stress, however many, is a question which will be discussed later. Within the stress limits named above, if material is subjected to a cycle of stress involving application and removal of load, delicate measurements of deformation will show that the rela- tion between stress and deformation is represented not by a single straight line, but by two curved lines, one for application and one for removal of load. Even if such deviation cannot be detected after a single cycle of stress it has been shown by Bairstow 4 that the deviation may become appreciable after several thousand repetitions, and that the stress-deformation curve for a cycle of stress after a few repetitions becomes a closed lM The Exponential Law of Endurance Tests,” Proceedings, Am. Soc. Test. Mats., Vol. X, p. 625 (1910). 2 “The Endurance of Metals,” Proceedings, Inst. Mech. Engrs. (British), Parts 3 and 4, p. 389 (1911). * “The Fatigue Failure of Metals,” American Machinist, Oct. 17 and 24, 1912. 4 “The Elastic Limits of Iron and Steel under Cyclical Variations of Stress,” Philosophical Transactions, Royal Soc., A Vol. 210, p. 35 (1910). 4 Moore and Seely on Repeated Stress. curve with a general form like that shown in Fig. 1 ( a ). These curves of stress-cycles resemble the “ hysteresis” curves of mag- netic material; the area enclosed by the loop represents energy lost during a cycle of stress, and this loss of energy is spoken of as “ mechanical hysteresis.” If mechanical energy is dissipated during a cycle of stress it would seem that the lost energy must be transformed into heat, that there must be some form of internal friction in the material, that wear and structural damage take place, and that Fig. 1 . — (a) General Form of Stress- Deformation Curve for a Cycle of Stress. (6) Approximate Form of Curve. if the action is continued long enough the material will be ruptured. It would seem reasonable to consider the amount of structural damage during a cycle of stress to be proportional to the energy transformed into heat, or, in other words, to the area of the mechanical hysteresis loop. The shape of this loop remains similar to the shape developed under early cycles of stress. Referring to Figs. 1 ( a ) and ( b ), this area is seen to be very nearly equal to that of two triangles placed base to base, or to \(Si—S^)e. If S 2 is denoted by QS i ( Q is negative if the cycle involves reversal of stress) the structural damage done during an early cycle of stress (A') is given by the equation: A' =ASi(l—Q)e Moore and Seely on Repeated Stress. 5 in which A = some constant and e = the width of the hysteresis loop. An examination of the various tests which have come under the writers’ notice, especially those of Bairstow, 1 indicate that e is some function of the range of stress (l— Q)Si and may be taken as proportional to 5^(1 —Q) p in which p is some con- stant. The structural damage done during a cycle of stress is then given by the equation: A‘=A'Si il+p) (l (1) in which A' is some constant to be determined experimentally. The damage done by successive cycles is not the same. An examination of Bairstow’s results, of the results of tests by Sondericker 2 and of tests in the Laboratory of Applied Mechanics Fig. 2. — Relation between Damage per Cycle and Number of Repetitions of Stress. of the University of Illinois indicates that the damage A changes slightly with successive cycles and that the relation between damage per cycle, A, and the number of repetitions of stress, N , may be represented by the equation: A = A' N n in which A' = the damage done per cycle during the earlier cycles of stress, and n = a constant which has a very small numer- ical value. This equation is represented by a curve similar to that shown in Fig. 2. 1 “The Elastic Limits of Iron and Steels under Cyclical Variations of Stress,” Philosophical Transactions, Royal Soc., A, Vol. 210, p. 35 (1910). 2 “Repeated Stresses,” with a discussion by J. E. Howard, Technology Quarterly, March, 1899. 6 Moore and Seely on Repeated Stress. The total structural damage done by any number of repe- titions, N, may be denoted by T, and will be represented by the area under the curve up to the value of N or by the expres- sion J~A dN which is equal toJ~ A 'N n dN ( a is some small number — the first few cycles are apt to be irregular). Integrat- ing the above expression there is obtained : ^ A' 1 +n (iV r(1+w) — a (1+w) ) and as N is large as compared with a this may, without serious error, be written: T = A' 2V (1+n ) 1 +U but from equation (l) A' =A' Sj 1+P \l hence T = 0 (i+/o il / 5i (1+#) (l-C) (1+ ^ (1+ * ) 1 +n If the structural damage spreads regularly across the most- stressed cross-section of a member, and if the amount of struc- tural damage necessary to cause failure be denoted by W, then at failure under repeated stress: _r r __ A r S ^ 1 +p \ 1 - Q ) (1 +P) N {1 +n 1 1 +n W = T = Grouping the experimental constants W, A ', 1-j-p, and 1 +n, denoting W{lA-ri)/A f by B^ l+P \ and 1+n/l+p by q, solving for Si and dropping the subscript, there is obtained: (1 -Q)N* For any given set of experiments with a constant value of Q, the stress to cause failure under repeated stress might be expected to be proportional to some negative fractional power of the number of repetitions, which result is in harmony with the exponential law proposed by Basquin, as a result of his direct study of results of tests under repeated stress. Moore and Seely on Repeated Stress, 7 Number of Repetitions before Rupture 8 Moore and Seely on Repeated Stress. Equation (2) may be written: log 5 = log B — log (1 —Q)—q lo gN (2a) Equation (2) is represented by a straight line if plotted on logarithmic cross-section paper. As shown in Basquin’s paper, also by Eden, Rose and Cunningham, and by Upton and Lewis, the results of nearly all repeated stress tests, if plotted on logarithmic paper, do follow a straight line up to a value of N slightly greater than 1,000,000. In Figs. 3, 4 and 5 are given some test results compiled by the writers, including some not given in Basquin’s paper. These results also follow a straight line up to a value of N of about 1,000,000; the corresponding value of S is slightly below the elastic limit which would be determined by a static test of the usual laboratory accuracy. The writers wish to acknowledge the loan of collected test data by Professor Basquin, which was of material assistance. The analytical discussion of the cumulative damage done by repeated stress, which has been given, seems to yield results in accordance with the results of tests, and is submitted as an explanation of the failure of materials under repeated stress within the stress limits named; that is, for stresses ranging from the yield point (or ultimate strength for brittle materials) down to a stress slightly below the “ elastic limit” as usually deter- mined in static tests. Materials under Repeated Low Stress . — As builders of machines have to design parts to withstand many times one million repetitions of stress, a problem of greatest importance is the determination of the action of repeated stresses lower than those considered in the foregoing paragraph. Is cumula- tive damage done under these lower stresses, and will they finally cause failure? Or is there an “ endurance limit” below which no damage is done to the material, and below which the material will withstand an infinite number of repetitions? The latter view is the one which has been widely held, and the endurance limit has been regarded as coincident with the “true” elastic limit of a material. In favor of this view may be cited the fact that the number of repetitions of stress neces- sary to cause failure increases very rapidly as the fiber stress is lowered; that a number of tests have been made in which test Moore and Seely on Repeated Stress. 9 specimens withstood tens of millions of repetitions of stress without failure; and that at low stresses, even with delicate measuring apparatus, there can be detected no signs of struc- tural damage. Various methods have been used for determining Fig. 5. — University of Illinois Tests. the value of the endurance limit for a material; these different methods yield widely varying results, as will be shown in detail later. In favor of the view that damage is done to materials under 10 Moore and Seely on Repeated Stress. low stress and that there is a probability of their eventual failure under repeated low stress, the following considerations may be cited: 1. The occurrence of “slip lines” 1 in metal under repeated stress seems to be the result of cumulative damage within a crystal of metal. No sharply defined lower limit has been found, either for the appearance of these slip lines or for their tendency to spread and develop into cracks. 2. The gradual development of permanent set, under repeated stress so low that preliminary static tests had shown no measurable set, seems to the writers to be an indication of damage at low stresses. This development is shown especially by the tests of Bairs tow 2 for cycles of stress not involving com- plete reversal. Bairstow found that the set for any stress gradually increased, though for a single cycle of stress no mechanical hysteresis could be detected; and that after several thousand repetitions the set did not further increase during the test. Whether this set would have shown further increase with an increase in the number of repetitions, or whether increase would have been shown by more delicate instruments, is an undecided question. In the opinion of the writers the significant fact is the cumulative development of permanent set under repeated low stress. 3. The sudden sharp breakages which occur in repeated stress tests, even of ductile materials, would seem to indicate that structural damage may be done to material without any undue deformation of the member as a whole. The fact that no undue deformation can be detected is no sure sign that a mate- rial is free from danger of failure under repeated stress. 4. Data of tests involving more than a million repetitions of stress are very few, yet frequently machine parts must be designed to endure several hundred millions of repetitions. The repeated stress problems of the time of Wohler and Baus- chinger were mainly problems of railroad bridge members and other structures and machines which would be called on to with- 1 Ewing and Rosenhain, “Crystallin Structure of Metals,” Philosophical Transactions, Royal Soc., Vol. 193, p. 353 (1899); Ewing and Humfrey, “Effect of Strain on the Crystallin Structure of Lead,” Philosophical Transactions, Royal Soc., Vol. 200, p. 241 (1902). 2 "The Elastic Limits of Iron and Steel under Cyclical Variations of Stress,” Philosophical Transactions, Royal Soc., A Vol. 210, p. 35 (1910). Moore and Seely on Repeated Stress. 11 stand only a few million repetitions of stress. From the view- point of these earlier investigators experiments under a few million repetitions covered the ground; for machines of to-day reliance on the results of such experiments involves enormous extrapolation of test results. The data seem hardly sufficient for establishing an endurance limit for infinite repetition, or even for repetitions numbering hundreds of millions. More- over, the results of some tests, if taken alone, seem to indicate that some exponential law of endurance holds up to the limit of experimentation. These unusual tests are discussed later. 5. As instruments of increased delicacy are used in measur- ing deformation, evidences of mechanical hysteresis are found at lower and lower stresses in static tests. 1 In actual material these evidences have been found at stresses not much above ordinary working stresses. When the cumulative action of repeated stress is considered, the indefiniteness of the elastic limit becomes apparent. While the statically determined elastic limit has some significance for static loading, it appar- ently has no significance as a criterion of endurance strength. 6. If elastic vibrations are set up in metal test specimens such vibrations soon die out. 2 This dying out would seem to indicate loss of energy in heat, with accompanying internal friction, wear, and structural damage. The following quotation from the careful work of Bair- stow 3 is of interest: “For equal (reversed) stresses after the specimen had been fixed in position and before it had been loaded in either direction a reading was taken of the unstrained length. A similar reading was recorded after the tensile loads had been applied and removed, and a third reading after putting on and removing the compressive load. The three readings were alike and indicated complete elasticity within the accuracy of the measurement. The stress, =*= 31,000 lb. per sq. in., was then repeated automatically, and for some time the straight line continued to represent the cycle of extensions. As the number of repetitions became greater the ‘cyclical permanent set’ became measurable and gradually increased until after 19,000 reversals of stress it had 1 Moore, “The Physical Significance of the Elastic Limit,” Proceedings, Internat. Assn. Test. Mats., Article XXVIII (1912). 2 Boudouard, “Break-Down Tests of Materials,” Proceedings, Internat. Assn. Test. Mats., Article V3? (1912); Kelvin, “On the Elasticity and Viscosity of Metals," Proceedings, Royal Soc., May 18, 1865; also “Collected Mathematical and Physical Papers,” Vol. III. * “The Elastic Limits of Iron and Steel under Cyclical Variations of Stress,” Philosophical Transactions, Royal Soc., A Vol. 210, p. 35 (1910). 12 Moore and Seely on Repeated Stress. become 1 1 per cent of the original elastic extension. . . . Raising the stress to 33,000 lb. per sq. in. produced an immediate increase in the ‘cyclical per- manent set’ . . . Finally stresses of =*= 44,440 lb. per sq. in. were imposed and at 29,280 reversals . . . the width of the hysteresis loop was very great, but even for this case the lines (showing release of loads, both tension and compression) are parallel to the original elastic line. . . . “The behavior of this specimen illustrates the necessity for Bauschinger’s hypothesis relating to primitive elastic limits, as the extensometer was incap- able of showing the first deviation from elasticity. At a slightly lower range, probably =*= 28,600 lb. per sq. in., the specimen would have been really elastic, as.no number of reversals would have produced a hysteresis loop.” The writers feel that the conclusion given in the last para- graph is more sweeping than is justified by the data, and that the conclusion in such a case (which is typical) should rather have been, that a slightly lower stress applied to the specimen would have required many more repetitions in order to make it evident that damage was being done. After a study of the data of many series of repeated stress tests, the writers feel that no “ endurance limit” has been surely found below which any material may be relied on for indefinite endurance. It seems to the writers that more repetitions of stress would prob- ably have broken the test specimens, more delicate instruments would probably have shown evidences of inelastic action, rise of temperature, or other sign of eventual failure. To the writers it seems that a negative argument against the use of a definite endurance limit is furnished by the indefi- niteness of its determination. A common method of locating this limit is to plot from test a curve with stresses as ordinates (. N ) and number of repetitions to cause failure as abscissas. This method is shown in Fig. 6 (a) for a typical set of test results from Wohler. This curve becomes nearly horizontal at a few millions of repetitions and the horizontal line to which the curve is asymptotic is judged by the eye. The ordinate of this horizontal line is taken as the endurance limit, and in this case gave a value of 18,000 lb. per sq. in. Another method is to plot stresses as ordinates and values of 1 /N as abscissas. The endurance limit is taken as the ordinate of the intersection of this curve (extended) with the zero axis. This method is shown in Fig. 6 ( b ) and for the same test data as the first method gives a value for the endur- ance limit of 17,500 lb. per sq. in. Stress in Thousand Pounds per Square Inch Moore and Seely on Repeated Stress. 13 A third method is to plot stresses as ordinates and some root of 1/A as abscissas. The fourth root of 1/A has been sug- gested by C. E. Stromeyer. 1 Fig. 6 (c) shows the application 30 20 to . o F j&2_ ± 0 20 40 60 80 100 120 *0 Millions or Repetitions op St* ess — N (a) 30 20 10 0 30 20 10 O 30 20 /O 0 * £ * * j L 2 «/o 7 4*i o 7 , 6*io 7 8* /o' 7 i. 2 */o b 77 W Fig. 6. — Diagrams Showing Variation in Endurance Limit (. En ) with Method of Determination. of this method to the given test data, and the endurance limit is found to be 15,000 lb. per sq. in. Fig. 6 (d) has been plotted 1913. 1 "Memorandum of the Chief Engineer of the Manchester Steam Users Association” for 14 Moore and Seely on Repeated Stress. with values of the eighth root of 1 /N as abscissas, and the value of the endurance limit thus found is 7,000 lb. per sq. in. This series of tests involved one test at 19 million repetitions and one at 132 millions. If these various methods were applied to tests covering no more than one million repetitions of stress the results would show still greater variation. Formulas. Modification of the Exponential Equation for Repeated Stress. — The exponential equations for repeated stress, (2) and (2a), give for high values of N, stresses which are lower than those which material has withstood in several tests. This was shown by Basquin, by Upton and Lewis, and is illustrated in Figs. 3, 4 and 5. Some modification of these formulas would seem to be necessary for low stresses and high values of N. If a test specimen or machine member fails under a few repetitions of high stress the damage done by each cycle of stress is con- siderable; an appreciable proportion of the cross-sectional area is affected by this damage, and the spread of damage is regular and rapid. If a test specimen or machine member is sub- jected to cycles of low stress the damage done during each cycle is much less; a much smaller proportion of the area is affected, and the location of the little areas affected is dependent on the homogeniety of the material and the regularity of dis- tribution of stress. For example, a short cylinder under com- pression has a more uniform distribution of stress under a load sufficiently heavy to cause firm bearing between the cylinder and its bed, than it has under a load so light that the extremely small areas projecting from the end of the cylinder are not all mashed flat. Under low stress the rate of spread of damage becomes somewhat a matter of probability and chance. The affected areas may be so grouped that damage proceeds regularly, but probably they will be somewhat scattered; the damage will proceed with corresponding slowness and the endurance of the member will be increased. The writers suggest that the expo- nential equations, (2) and (2a), modified by the addition of a “probability factor,” be used for estimating the fiber stress which will cause failure under any given number of repetitions of stress, N, and range of stress, Q. They suggest as such a Moore and Seely on Repeated Stress. 15 factor 1 -\-kN m in which k and m are constants to be derived from experiment. The modified exponential formula then becomes 5 = B (1 +kN m ) (3) or log S = \og B— log (l —Q) —q log A+log (l +kN m ) (3a) These formulas seem rather formidable, but tabular values for Fig. 7. — Tests of Cold-Rolled Steel. Watertown. Arsenal. 1 -\-kN m , and graphical charts for the solution of the formulas are given later which facilitate their use. The variability in the action of metals under repeated stress is well illustrated by the results of a series of tests of cold-rolled steel made at the Watertown Arsenal. 1 The results of this series of tests is plotted logarithmically in Fig. 7. It will be seen that for the higher values of stress the range of values of N is less than for low values of stress. A straight line fairly Tests of Metals,” 1893, pp. 161-165, 170, 171, 176-178. 16 Moore and Seely on Repeated Stress. well represents the results for minimum values of N ; this straight line corresponds to an exponential law, equation (2), and for such minimum values it would seem that the damage caused by repeated stress proceeded regularly. For the other test speci- mens the damaged areas were more scattered, or for some other reason damage proceeded more slowly, and this effect became more marked at lower stresses. The objection may be raised to the formula proposed by the writers in that it involves failure of any material under any stress, however small, if that stress is repeated a sufficiently great number of times; and it may be stated that there is no proof that such failure will occur. It has not been conclusively proved that very low repeated stresses will ultimately break materials, but on the other hand, no endurance limit for indefi- nitely repeated stress has yet been positively established for any material. There is practically no direct experimental evi- dence concerning the behavior of material under more than 100 million repetitions of stress; there is very little experimental evidence for more than 10 million repetitions, yet some machine parts must be designed to withstand 10 billion repetitions of stress. The writers believe that a formula which assumes that the same destructive action which occurs under high stresses will continue to act with diminished intensity under low stresses has some indirect evidence in its favor, and is a safer guide for the designer than is a fixed endurance limit below which destruc- tive action is assumed to cease. It should be noted that even for values of N as great as 10 billion, the modified exponential formulas proposed by the writers give stresses which, though somewhat lower than the values usually given for the endurance limit, are yet of very considerable magnitude. This will be discussed under the head of “ Constants for the Formulas.” In addition to the above considerations, it should be noted that for very long endurance of machines or structures deteriora- tion due to other causes than repeated stress — wear, corrosion, change of current practice, and the like — becomes a criterion of endurance. Constants for the Formulas .— The writers have studied the available data of repeated stress tests and submit tentative Moore and Seely on Repeated Stress. 17 values for the constants in the formulas proposed, namely, equations (2), (2a), (3) and (3a). B has been determined for a number of materials from the data of repeated stress tests. It is the value of the ordinate for N = 1 (line extended backward) and Q = — 1 (stress com- pletely reversed). In Table I are given tentative values of B for some common metals. The value of Q (ratio of minimum stress to maximum stress) is generally known for any structural part or machine member. If the stress is wholly or partially reversed Q is negative. If Q approaches +1 in value, care should be taken that the safe static stress is not exceeded; the safe static stress is always effective as a criterion of safety. Table I. — Values of the Constant B in the Exponential Formulas for the Endurance of Materials under Repeated Stress. Material. B Log B Structural Steel and Soft Machinery Steel 110 000 5.0414 Cold-rolled Steel Shafting 275 000 5.4393 Steel, 0.45 per cent Carbon 175 000 5.2430 Wrought Iron 100 000 5.0000 Hard Steel 250 000 5.3979 Hard-Steel Wire 400 000 5.6021 The effect of range of stress has been studied in connection with the work of Bauschinger and Wohler. Equations (2) or (3) gives values of S for complete reversal (Q— — l) one-half as great as the values for repetition of stress from zero to a maxi- mum (Q = 0). This seems to agree fairly well with test results, but the effect of range of stress furnishes a most promising field for further experimental study. A value of q = J seems to fit fairly well the results of a wide range of tests of various-size test specimens of different metals tested on a variety of machines. Considerable variation from this value may be found, but the writers have been able to find no systematic variation; and as was pointed out by Basquin 1 1 “The Exponential Law of Endurance Tests,” Proceedings, Am. Soc. Test. Mats., Vol. X, p. 625 (1910). 18 Moore and Seely on Repeated Stress. a considerable variation in q makes a comparatively small variation in the value of S for a given value of N. Values of constants for the probability factor \+kN m must be very largely a matter of judgment. Referring again to the series of tests shown in Fig. 7 the minimum values of S follow an exponential relation (shown by the straight line). This relation corresponds to a value of 1 -\-kN m of unity for all values of N. As noted previously, for values of N above one million, such a result seems unusual though it is sometimes found. As our test data for long-time endurance tests are few, for structural parts and machine members, whose failure would endanger life, it would seem advisable to use a probability factor of unity for all values of N in which case the proposed formula for repeated stress becomes: •( 4 ) (4a) or log S = log B — log(l — Q) — f log N For cases in which failure would not endanger life, the writers propose as tentative values, 0.015 for k and f for m. These values were chosen after a study of the scanty data of long endurance tests, and represent values of S rather lower than those given in some of the long-time tests studied. The formula for repeated stress for cases in which failure would not endanger life then becomes: S = - — r (l+0.015iV») (1 -Q)Nl K ( 5 ) or log .S = logZ? — log (l — Q) — i logA+log(l +0.015 .... (5a) The determination of the value of 1+0.015 N* is rather cumbersome and in Table II are given values of this factor for various values of N. Charts for the graphical solution of equations (4) and (5) are given in Figs. 8 and 9. Fig. 8 is for the solution of equation (4), and Fig. 9 for the solution of equa- tion (5). The method of using either Fig. 8 or Fig. 9 is as follows: Enter the diagram at the bottom at the given value of Q for the problem, pass vertically to the desired value of N (this will in general lie between two curves, and its location must be Moore and Seely on Repeated Stress. 19 20 Moore and Seely on Repeated Stress. judged by interpolation). From this point pass horizontally to an intersection with the diagonal corresponding to the value of B for the material, then from this intersection vertically to the upper edge of the diagram, where the value of S may be read off. It should be remembered that 5 is the breaking stress. Factor of Safety for Repeated Stress . — In design involving static conditions the working stress must be less than the ultimate strength of the material. Reduction of stress by means of the so-called factor of safety is the only means through which a safe design is assured. It has already been pointed out that in no case must the law of fatigue be considered to Table II. — Values of the Probability Factor (1+0.015 N *) for Various Values of N. N (1+0.015 JVs) N (1+0.015 Nh) 100 000 1.063 100 000 000 1.150 500 000 1.077 500 000 000 1.183 1 000 000 1.084 1 000 000 000 1.200 5 000 000 1.103 5 000 000 000 1.244 10 000 000 1.112 10 000 000 000 1.267 50 000 000 1.138 50 000.000 000 1.326 Note. — For structures and machines whose failure would endanger life it is suggested that the value of the probability factor be taken as unity for all values of N. For other structures or machine parts the values as given by the above expression and here tabulated are suggested. hold for stresses greater than the yield point of the material, and that the requirements of static design must always be met. When the endurance strength of a machine member is less than the yield point of the material, a factor of safety must be applied to it in order to obtain a safe working repeated stress. Fatigue involves two factors, stress and the number of repeti- tions, and the factor of safety may be applied either to the stress or to the number of repetitions. Since the stress must also satisfy the requirements for proper static design, the “life” of the material or machine would seem to be properly insured by applying the factor of safety to the number of repetitions; that is, a design should be made with a stress corresponding to Moore and Seely on Repeated Stress. 21 failure for a number of repetitions the machine is to withstand multiplied by x, the factor of safety. This places emphasis upon endurance rather than on strength, but since the stress is low when determined by the condition of fatigue, this emphasis seems properly placed. While this method reduces the stress considerably less than would the application of the same factor of safety to the stress, it should be remembered that the function of the factor of safety is, in part, to guard against excessive stress due to non-homogeneous material, initial stresses, and local stresses due to fabrication, and that such stresses are reduced in effect after a few repetitions. Moreover, excessive stresses may be resisted for a considerable length of time with- out producing more than a small percentage of the total dam- age required for rupture. On the other hand, it should be remembered that repeated stresses tend to destroy the primitive ductility of the material. Material has failed in practice under extremely low repeated stresses, suggesting that defective material, local flaws, and the like may play a much greater part in determining the endurance of a material than they do in determining its static strength. Two examples of the use of the proposed formulas are given : 1. The eyebars and chords of a railroad bridge truss are to be made of structural steel, and should be designed to with- stand 2,000,000 repetitions of stress, varying from dead load to dead load plus live load. The d.ead load is about one quarter of the live load. What value of working stress should be used? The value of Q for this case is + 0.20, the value of B (taken from Table II) is 110,000, and the value of N may be taken as six times 2,000,000 or 12,000,000. As the failure of the bridge would endanger life equations (4) or (4a) are to be used. Substituting the above values in equations (4) or (4a), there is obtained for the proper value of 5 = 17,900, lb. per sq. in. The same result may be obtained from the use of Fig. 8. As the safe static stress would hardly be taken higher than 16,000 lb. per sq. in. it is seen that static conditions govern the design. 2. A line shaft is to be designed to withstand 500,000,000 complete reversals of bending stress and is to be made of a special steel for which B has been experimentally determined 22 Moore and Seely on Repeated Stress. as 300,000. What fiber stress should be allowed? The yield- point strength of the material is 90,000 lb. per sq. in. As the failure of a line shaft does not, in general, involve danger to life, use equations (5) or (5a) or Fig. 9. Q= — 1.0, B is given as 300,000: Take N as six times the designed endur- ance or as three billion. Substituting the above in equation (5a) or using Fig. 9 there is obtained for the working stress S = 12,100 lb. per sq. in. As this is far below the allowable static stress endurance conditions would govern in this case. Various Effects. Effect of Rapidity of Repetition of Stress. — A certain amount of time is required for any member of a machine or structure to assume the deformation corresponding to any given load, and if repetitions of load follow each other at intervals shorter than this time, the deformation in the member, the stress set up, and the number of repetitions it will withstand may be appreciably affected. A few recent British tests 1 of material under repeated stress seem to indicate that for small mem- bers there is no appreciable effect produced by varying the rapidity of repetition of stress below about 2000 repetitions per minute. Above that speed very little test data are available. Effect of Rest on Resistance to Repeated Stress. — If metal is stressed beyond the yield point so that plastic action is set up, its strength and its elastic action are improved under sub- sequent stress, if the material is allowed to rest. Recent experi- ments by British investigators 2 seem to indicate that, for steel and iron at least, the effect of rest on the resistance to repeated stress is negligible for unit stresses below the yield point of the material. Effect of Sudden Change of Outline of Member. — Every sharp corner in a piece subjected to repeated stress facilitates the formation of micro-flaws in the piece. From results of re- 1 Stanton and Bairstow, “On the Resistance of Iron and Steel to Reversals of Stress,” Proceedings, Inst. Civil Engrs. (British), Vol. 166, p. 78 (1906); also Engineering (London), Vol. LXXIX, p. 201 (1905). Turner, L. P., “The Strength of Steel in Compound Stress and Endurance under Repetitions of Stress, Engineering (London), July 28, August 11 and 25, Sept. 8, 1911. s Eden, Rose and Cunningham, “The Endurance of Metals,” Proceedings, Inst. Mech. Engrs. (British), Parts 3 and 4, p. 389 (1911). Moore and Seely on Repeated Stress. 23 peated stress tests made by Stanton and Bairstow, at the British National Physical Laboratory, on test specimens of different shape, the superiority of the test specimens in which sharp corners are avoided is obvious. The relative values for strength under repeated stress for the shapes tested seems to be about as follows: Rounded fillet 100 Standard screw thread 70 Sharp corner 50 Service Expected from Various Machine and Structural Parts . — We do not know with certainty whether any material can resist an infinite number of repetitions of any stress however small. The safest view for an engineer to take seems to be that under repeated stress materials of construction have a limited “life.” The exponential formula for repeated stress gives results in accordance with this view. If this view is held, the number of repetitions which any structural or machine member will have to withstand in normal service becomes of importance. The following list gives the numbers of repetitions of stress which may be expected to be applied to various machine and structural members. The list is intended to be suggestive rather than to serve as an exact guide. The members of a railway bridge carrying 100 trains per day for a period of 50 years would sustain about 1,826,000 repetitions of stress. The stress would vary from the dead- load stress to a live-load stress averaging somewhat below that caused by the passage of the heaviest locomotives. A railroad rail over which 250,000,000 tons of traffic passes would sustain something like 500,000 repetitions of locomotive wheel loads, the stress being slightly more severe than a repe- tition from zero to a maximum. The rail would have to stand, in addition to the locomotive wheel loads, something like 15,000,000 repetitions of stress caused by car wheel loads. The stresses set up by car wheel loads would be about half as great as the stresses set up by the locomotive wheel loads. A mine hoisting rope bent over three sheave wheels and operating a hoist 100 times a day, in a term of service of five years would sustain 550,000 repetitions of stress. If the sheave 24 Moore and Seely on Repeated Stress. wheels are so placed that they reverse the direction of the bending of the rope the range of stress would be nearly a com- plete reversal; if bending takes place in one direction only the range of stress is from nearly zero to a maximum. The piston rod and the connecting rod of a steam engine running at 300 r.p.m. for 10 hours per day, 300 days per year for 10 years, sustains 540,000,000 repetitions of stress, and the range of stress involves almost complete reversal. A band saw in hard service for two months sustains about 10,000,000 repetitions of stress varying from nearly zero to a maximum. A line shaft running at 250 r.p.m. for 10 hours a day, 300 days per year, sustains during a service of 20 years 900,000,000 repetitions of bending stress due to force transmitted by belts, gears, and driving chains. The stress is almost com- pletely reversed. It should be noted that for the line shaft the torsional stress is not repeated nearly so often as is the bending stress. The shaft of a steam turbine running at 3,000 r.p.m. for 24 hours per day, 365 days in a year during 10 years service sustains 15,768,000,000 reversals of bending stress caused by the weight of rotating parts and the tangential force of the inrushing steam. Bibliography. Andrews, “Microscopic Internal Flaws Inducing Fracture in Steel,” Engineering (London), July, 1896. Arnold, “Factors of Safety in Marine Engineering,” Transactions , Insti- tute of Naval Architects, Vol. L, p. 260 (1908). Bairstow, “The Elastic Limits of Iron and Steel under Cyclical Varia- tions of Stress,” Philosophical Transactions , Royal Society, A Vol. 210 , p. 35 (1910). Baker, “Some Notes on the Working Stress of Iron and Steel,” Trans- actions , American Society of Mechanical Engineers, Vol. 8 , p. 157 (1886). Basquin, “The Exponential Law of Endurance Tests,” Proceedings , American Society for Testing Materials, Vol. X, p. 625 (1910). Bauschinger, “ Mittheilungen des Mechanisch-Technischen Labora- toriums der kgl. techn. Hochschule,” Munchen, Heft 13; also, Dingier' s Journal , Bd. 224, and Civilingenieur for 1881. Boudouard, “Break-Down Tests of Materials,” Proceedings , International Association for Testing Materials, Article V 3 (1912). Moore and Seely on Repeated Stress. 25 Burr, “The Elasticity and Resistance of Materials of Engineering/' p. 200. Eden, Rose and Cunningham, “The Endurance of Metals," Proceedings , Institute of Mechanical Engineers (British), p. 839, parts 3 and 4 (1911). Ewing and Humfrey, “Effect of Strain on the Crystallin Structure of Lead,” Philosophical Transactions, Royal Society, Vol. 200, p. 241 (1902). Ewing and Rosenhain, “Crystallin Structure of Metals," Philosophical Transactions, Royal Society, Vol. 193, p. 353 (1899). Gardner, “Effects Caused by the Reversal of Stresses in Steel," Journal, Iron and Steel Institute, No. 1, p. 481 (1905). Gulliver, “Internal Friction in Loaded Materials," International Asso- ciation for Testing Materials, Fifth Congress, Art. VIII (1909). Hancock, “Recovery of Nickel and Carbon Steel from Overstrain,” Proceedings, Indiana Engineering Society (1907). Hopkinson, “Effects of Momentary Stresses in Metals," Philosophical Transactions, Royal Society (1905). Howard, “Notes on the Endurance of Steels under Repeated Alter- nations of Stress," Proceedings, American Society for Testing Materials, Vol. VII, p. 252 (1907); compiled from “Tests of Metals," Watertown Arsenal Reports (1898-1910). Howe, “Are the Effects of Simple Overstrain Monotropic?" Proceedings, American Society for Testing Materials, Vol. XIV, p. 7 (1914). Kelvin, “On the Elasticity and Viscosity of Metals," Proceedings, Royal Society, May 18, 1865; also “Collected Mathematical and Physical Papers,” Vol. III. Kommers, “Repeated Stress Tests of Steel," American Machinist, p. 551, April 1, 1915; also “Repeated Stress Testing," Proceedings, International Association for Testing Materials, Arts. V4a and V46 (1912). Moore, “The Physical Significance of the Elastic Limit," Proceedings, International Association for Testing Materials, Art. XXVIII (1912). Muir, “Recovery of Iron from Overstrain," Philosophical Transactions, Royal Society, Vol. 193, p. 1 (1900). Osmond, Fremont and Cartaud, “Les Modes de Deformation et de Rupture des Fers et des Aciers Doux,” Revue de Metallurgie, Part 1 (1904). Rasch, “Method for Determining Elastic and Critical Stresses in Mate- rials by means of Thermo-Electric Measurements,” Proceedings, International Association for Testing Materials, Art. VII (1909). Renolds and Smith, “On a Throw-Testing Machine for Reversals of Mean Stress," Philosophical Transactions, Royal Society, Series A, .Vol. 192 (1902). Roger, “Heat Treatment and Fatigue of Iron and Steel,” Journal , Iron and Steel Institute, No. 1, p. 484 (1905). Rosenhain, “The Plastic Yielding of Iron and Steel," Journal , Iron and Steel Institute, Vol. 1, p. 335 (1904). Smith, “On Stress Distribution during Tension Tests," Engineering (London), p. 593, Oct. 29, 1909, and p. 796, Dec. 10, 1909. 26 Moore and Seely on Repeated Stress. Sondericker, “Repeated Stresses," with a discussion by J. E. Howard, Technology Quarterly , March, 1899. Spangenberg, “ Ueber das Verhalten der Matalle bei wiederholten Anstrengungen,” Zeitschrift fiir Bauwesen, 1874, 1875. Stanton, “A New Fatigue Test for Iron and Steel," Journal , Iron and Steel Institute, No. 1, p. 54 (1908). Stanton and Bairstow, “On the Resistance of Iron and Steel to Reversals of Stress,” Proceedings , Institute of Civil Engineers (British), Vol. 166, p. 78 (1906); also Engineering (London), Vol. LXXIX, p. 201 (1905). Stromeyer, “Memorandum of the Chief Engineer of the Manchester Steam Users Association,” for 1913; “Elasticity and Endurance of Steam Pipes," Transactions, Institute of Naval Architects (1914). Thurston, “Text-Book of Materials of Construction,” p. 348. Turner, C. A. P., “The Thermo-Electric Determination of Stress," Transactions, American Society of Civil Engineers, Jan., 1902. Turner, L. P., “ The 'Strength of Steel in Compound Stress and Endur- ance under Repetitions of Stress,” Engineering (London), July 28, August 11 and 25, Sept. 8, 1911. Unwin, Presidential Address before Institute of Civil Engineers (British) for 1911, Part 1; also Engineering (London), Nov. 10, 1911, with editorial comment; “The Testing of Materials of Construction," p. 13. Upton and Lewis, “The Fatigue Failure of Metals,” American Machin- ist, Oct. 17 and 24, 1912. Van Ornum, “The Fatigue of Cement Products," Transactions, American Society of Civil Engineers, Vol. 51, p. 443 (1903), and Vol. 58, p. 294 (1907). Wawrziniok, “Handbuch des Materialprufungswesen," p. 9. Wholer, “Ueber die Festigkeitsversuche mit Eisen und Stahl.” A good account of these results is given in English in Engineering (London), Vol. II. A summary of Wohler’s work is given in “The Testing of Materials of Con- struction,” by Unwin. The original publication of Wohler’s results was in Zeitschrift fiir Bauwesen, Vol. X, XIII, XVI, and XX. Tests of Metals, 1893, pp. 161-165, 170, 171, 176-178. “Notes on Some Cases of Fatigue in Steel Material of Steamers," Transactions, Institute of Naval Architects, June 25, 1913; also Engineering (London), p. 891, June 2, 1913.