Astm stp 91 a 1963

F ft O F E R T Y O F The university of Michigan Liabraries 01 A ft T E S 8C1ENTIA VB RIT AS A GUIDE FOR FATIGUE TESTING AND THE STATISTICAL ANALYSIS OF FATIGUE DATA Prepar'ed by COMMITTEE £-9 ON FATIGUE AMERICAN SOCIETY FOR TESTING AND MATERIALS 963 Reg U S Pat Off ASTM Special Technical No 91-A* (Second Edition} Price: $5.00; to Members: $4.00 Published by the AMERICAN SOCIETY FOR TESTING AND MATERIALS 1916 Race Street, Philadelphia 3, Pa ©BY AMERICAN SOCIETY FOR TESTING AND MATERIALS 1963 Library of Congress Catalog Card Number: 63-16331 Printed in Baltimore, Md February, 1964 F O REWO RD The First Edition of this Guide was the composite work of many people who contributed a great deal of time to the discussion and writing of the text under the guidance of Task Group Leader, F B Stulen A major portion of the statistical section was written by Miss Mary N Torrey George R Gohn not only contributed to the discussion and planning, but also edited and arranged for the printing of the advance copies of the text The coordination of contributions and discussions was done by H N Cummings Appreciable contributions to the statistical parts of the Guide were also made by D H Shaffer In addition to the above, R E Peterson, H F Dodge, D P Gaver, R Hooke, W T Lankford, R B Murphy, W C Schulte, P R Toolin, and M B Wilk contributed to the discussions at various conferences The original Task Group was organized under the leadership of J T Ransom, and a first rough draft was prepared in 1954 and revised in 1955 Other contributors to these drafts were E W Ellis, W T Lankford, F A McClintock, R E Peterson, E H Schuette, F B Stulen, and E J Ward In 1956, F B Stulen became Leader of the Task Group and the Guide was completed under his direction Upon the formation of Subcommittee VI on the Statistical Aspects of Fatigue, this subcommittee was asked to review the First Edition and to make any revisions necessary to bring the Guide up to date As a result of this study, extensive revisions have been made in various sections as printed in this Second Edition They include: (1) revisions in the definitions (Section II) and their separate publication as ASTM Tentative Definitions E 206,1 (2) an expansion of Section IV on the number of test specimens, (3) changes in Section V on tests of significance, and (4) the preparation of a new section, Appendix IV, on the use of the Weibull distribution function for fatigue Me This work was carried out by four Task Groups headed by S M Marco, H E Frankel, Miss M N Torrey, and C A Moyer, respectively Others who assisted in the preparation of the Second Edition were W N Findley, R A Heller, J H K Kao, H N Cummings, W S Hyler, B Ruley, and G R Gohn, Chairman of Subcommittee VI Definitions of Terms Relating to Fatigue g and the Statistical Analysis of Fatigue Data (E 206), 1962 Supplement to Book of ASTM Standards, Part iii NOTE.—The Society is not responsible, as a body, for the statements and opinions advanced in this publication CONTENTS I II III IV V PAGE Purposes of Fatigue Testing Definitions, Symbols, and Abbreviations Test Procedures Minimum Number of Test Specimens and Their Selection Analysis of Fatigue Data Appendices Miscellaneous Reference Tables Additional Technique for Distribution Shape Not Assumed Analysis of Correlation Between Two Variables The Weibull Distribution Function for Fatigue Life References Index 16 22 55 68 69 71 78 81 LIS T O F T A B L E S TABLE PAGE 1.—Allocation of Test Specimens for "Probit" Method of Test 2.—Minimum Number of Specimens Needed for Determining 95 Per Cent Confidence Intervals of Stated Width for a Population Mean, p 3.—Minimum Number of Specimens Needed for Determining 95 Per Cent Confidence Intervals of Stated Width for a Population Standard Deviation, a 4.—Minimum Number of Specimens Needed to Detect if the Standard "Deviation of a Population Is a Stated Percentage of a Fixed Value 5.—Minimum Number of Specimens Needed in Each Sample to Detect if a Standard Deviation of One Population Is a Stated Multiple of the Standard Deviation of Another Population 6.—Minimum Number of Specimens Needed to Detect a Stated Difference Between a Mean and a Fixed Value 7.—Minimum Number of Specimens Needed to Detect a Stated Difference Between the Means of Two Populations 8.—Median Percentage of Survivors for the Population 9.—Confidence Intervals for the Median 10.—Approximate Confidence Intervals for the Mean 11.—Confidence Intervals for Percentages 12.—Fatigue Test Data 13.—Fatigue Test Data 14.—Percentages Surviving 108 Cycles 15.—-"Probit" Test Data 16.—Computations for Fitting a Response Curve by Method of Least Squares 17.—Computation of Standard Deviation, s 18.—Method of Computing 95 Per Cent Confidence Limits for Per Cent Survival Values 19.—Method of Computing 95 Per Cent Confidence Limits for Fatigue Strength Values 20.—Computations for Significance Tests 21.—R R Moore Rotating Beam; Step Tests of 42 Specimens 22.—Analysis of Data in Table 21 23.—Prot Test Computations v 11 19 19 20 20 21 21 24 26 27 28 29 31 32 33 35 36 37 38 46 50 51 52 vi CONTENTS TABLE PAGE 24.—Prot Test Computations 25.—Minimum Per Cent of Population Exceeding Median of Low Ranking Points 26.—Unpaired Rank Test2 27.—Percentiles of the x Distribution 28.—Areas of the "Normal" Curve 29.—Values of t 30.—Percentiles of the x2 /d-f- Distribution 31.—Mo.25 and uo.yis for Runs Among Elements in Samples of Sizes Ni and Nz 32.—F Distribution 33.—k Factors for S-N Curves (Normal Distribution Assumed) 34.—Working Significance Levels for Quadrant Sum 35.—Ordinate Locations Corresponding to Per Cent Failed Values 36.—Mean-Rank Estimates of the Per Cent Population Failed Corresponding to Failure Order in Sample 37.—Typical Fatigue Test Data 38.—Typical Fatigue Test Data, Without Runouts 39.—Typical Fatigue Test Data, with Runouts 53 56 58 60 61 62 63 64 65 67 69 72 74 75 75 77 LIS T O F F I G U R E S FIGURE 1.—Probability-Stress-Cycle (P-S-N) Curve for Phosphor-Bronze Strip 2.—Response or Survival Tests 3.—Illustration of Staircase Method 4.—Representation of "Step" Testing of Single Specimen 5.—Graphical Illustration of Prot Data 6.—"Normal" or Gaussian Distribution Curve 7.—Response Curves for a Particular Type of Steel 8.—Per Cent of Specimens Having at Least the Indicated Fatigue Strength at 107 Cycles 9.—Prot Test: Stress as Linear Function of Stress Cycles 10.—Log-Log Plot of Prot Data 11.—Scatter Diagram 12.—Typical Weibull Distribution Curves 13.—Construction of Weibull Probability Paper from Log-Log Paper 14.—Estimation of Weibull Distribution Function Parameters for Data in Table 38 15.—Per Cent Failed at Weibull Mean 16.—Estimation of Weibull Distribution Function Parameters for Data in Table 39 RELATED ASTM PUBLICATIONS Abstracts of Articles on Fatigue (STP 9) Fatigue Manual (STP 91) (1949) ) Statistical Aspects of Fatigue (STP 121) h (STP 137) (1952) Fatigue, with Emphasis on Statistical Papers on Metals (STP 196) (1956) Fatigue of Aircraft Structures (STP 203) (1956) Large Fatigue Testing Machines and Their Results (STP 216) (1957) Basic Mechanisms of Fatigue (STP 237) (1958) Fatigue of Aircraft Structures (STP 274) (1959) Acoustical Fatigue (STP 284) (1960) Fatigue of Aircraft Structures (STP 338) (1963) 10 11 12 14 15 22 34 51 53 54 70 72 73 75 76 77 STP91 A-EB/Feb 964 GUIDE FOR FATIGUE TESTING AND STATISTICAL ANALYSIS About 15 years ago, ASTM Committee E-9 on Fatigue prepared a Manual on Fatigue Testing That Manual attempted to standardize the symbols and nomenclature used in fatigue testing, described the principal types of testing machines then in use, presented detailed instructions for the preparation of test specimens, outlined test procedures and techniques, and gave some suggestions for the presentation and interpretation of fatigue data Since the Manual was first prepared, a number of new techniques have been developed for evaluating the fatigue properties of materials Furthermore, the application of statistical methods to the analysis of the test results of samples offers a means for estimating the characteristics of the population from which the samples were taken To take cognizance of these developments, this guide has been prepared I PURPOSES OF FATIGUE TESTING The purposes of fatigue testing are (1) to estimate the relationship between stress- (load-, strain-, deflection-) amplitude and cycle life-to-failure for a given material or component, and (2) to compare the fatigue properties of two or more materials or components In order to specify the reliability of these estimates, they must be based on the results of testing a sample of fatigue specimens which have been drawn at random from a population of possible fatigue specimens and tested in accordance with acceptable testing procedures The principal acceptable procedures discussed in this guide are: A "Standard" tests (constant amplitude or classical Wohler method) Single test specimen at each stress level A group of test specimens at each stress level B Response tests (constant amplitude) "Probit" method Staircase method Modified staircase method C Increasing amplitude tests Step method Prot method The primary purposes of the statistical analysis of fatigue data are: (1) to estimate certain fatigue properties of material or a component (together with measures of the reliability) from a given set of fatigue data, obtained by testing a sample of fatigue specimens in accordance with one of the previous test procedures, and (2) to provide objective procedures for comparing two or more sets of fatigue data to determine whether or not the data come from similar populations Statistical theory also provides information on (a) Fatigue Manual, ASTM STP 91, Am Soc Testing Mats., 1949 The term "standard" test, as used here, does not imply an ASTM standard Copyright© 964 by ASTM International www.astm.org FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA the most efficient use of a limited number of test specimens and (b) the number of test specimens required to give a specified degree of confidence in the test results Even with some basic training, it is difficult to locate the techniques particularly useful in fatigue testing in the statistical literature The purpose of this guide is to describe some statistical treatments that are suitable for the analysis of fatigue data obtained in any one of the foregoing test methods and to present these statistical treatments in a form useful to the test engineer Definitions of certain statistical terms are included, but only enough of the basic concepts of statistics are included to make the methods understandable; theory is left to the references Test procedures are discussed hi Section III while techniques for analyzing the data obtained in these tests are given in Section V and the Appendices II DEFINITIONS, SYMBOLS, AND ABBREVIATIONS Relating to Fatigue Tests and Test Methods: To encourage uniformity of terminology, the terms dealing primarily with fatigue testing and test methods are also published in ASTM Definitions E 6.3 The symbols used are, hi general, those recommended in the American Standard Letter Symbols for Mechanics of Solid Bodies Fatigue (Note 1).—The process of progressive localized permanent structural change occurring in a material subjected to conditions which produce fluctuating stresses and strains at some point or points and which may culminate in cracks or complete fracture after a sufficient number of fluctuations (Note 2) NOTE 1.—The term fatigue in the materials testing field, has—in at least one case glass technology—been used for static tests of considerable duration, a type of test generally designated as stress-rupture NOTE 2.—Fluctuations may occur both in stress and with time (frequency), as in the case of "random vibration." Fatigue Life, N —The number of cycles of stress or strain of a specified char- acter that a given specimen sustains before failure of a specified nature occurs Definitions to 19, inclusive, apply to those cases where the conditions imposed upon a specimen result or are assumed to result in uniaxial principal stresses or strains which fluctuate in magnitude Multiaxial stress, sequential loading, and random loading require more rigorous definitions which are, at present, beyond the scope of this section Nominal Stress, S.—The stress at a t calculated on the net cross-section by simple elastic theory, without taking into account the effect on the stress produced by geometric discontinuities such as holes, grooves,'fillets, etc Stress Cycle.-^-The smallest segment of the stress-time function which is repeated periodically Definitions of Terms Relating to Methods of Mechanical Testing, 1962 Supplement to 1961 Book of ASTM Standards (E 6), Part ASA No Z10 3—1948, Am Standards Assn., 1948 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA 72 FIG 12.—Typical Weibull Distribution Curves, from Kao (35) TABLE 35.—ORE)INATE LOCATIO NS CORRESPON DING TO PER CENT FAIL!]D VALUES F(N) X 100 10 12 14 15 16 18 20 22 24 25 26 28 30 32 34 35 36 38 40 42 44 45 46 48 50 log 1 - PUt) 0.0088 0.0177 0223 0269 0362 0.0458 0.0555 0.0655 0.0706 0.0757 0.0862 0.0969 0.1079 0.1192 0.1249 0.1308 1427 1549 0.1675 0.1805 0.1871 0.1938 0.2076 0.2218 0.2366 0.2518 0.2596 0.2676 0.2840 0.3010 NOTE.—All logs are to the base 10 F(N) X 100 52 54 55 56 58 60 62 63 64 65 66 68 70 72 74 75 76 78 80 82 84 85 86 88 90 92 94 95 96 98 Iog 1 - F(N) 3188 3372 3468 3565 3768 3979 4202 4341 4437 4559 0.4685 0.4949 0.5229 0.5528 5850 6021 6198 6576 0.6990 0.7447 0.7959 0.8239 0.8539 9208 000 1.097 1.222 1.301 1.398 1.699 failure prior to N0 life Later it will be n how to test for N0 values greater than zero, but if it is reasonable to assume N0 = 0, the frequency distribution function is simplified Since the data are usually obtained in an ordered manner in fatigue testing, it is easy to fit a cumulative distribution function to fatigue life The cumulative function for the fraction of population failed prior to life N is WEIBULL DISTRIBUTION FUNCTION FOR FATIGUE LIFE 73 This function can be transformed into the straight-line relationship which allows a simple graphical method for fitting the Weibull distribution to the data and the subsequent graphical estimation of the parameters (b, N0, and Na) in the formula 95 FIG 13.—Construction of Weibull Probability Paper from Log-Log Paper Construction of Probability Paper Although Weibull probability paper can be purchased from a source such as Cornell University, Ithaca, N Y., Columbia University, New York, N Y., or Technical and Engineering Aids for Management, 104 Belrose Ave,, Lowell, Mass., it can be constructed rather simply from square log-log paper, that is, log-log paper in which the cycles are the same size in both directions The paper is prepared by the marking off on the vertical logarithmic scale of the probability percentages F(N) corresponding to the values of given in Table 35 For example, in Fig 13, the ordinate of the 90 per cent failure value is 1.000 on the vertical logarithmic scale Similarly, the ordinate for the 20 TABLE 36.— MEAN -RANI C ESTI MATE 3° OF THE P ER CE NT PO PULArDION i^AILEI) COR]RESPO NDINCJ TO F AILUEJE ORIDER IJf SAM!PLE Samp' e Size, n 10 12 11 13 14 15 Order No., q 16 18 17 19 20 No No No No No No 50.00 33.33 25.00 20.00 66.67 50.00 40.00 75.00 60.00 80.00 No No No No 10 No No 12 No No No 15 No No 17 No 18 No 19 No 20 Mean-rank estimates = 100 16.67 33.33 50.00 66.67 83.33 14.29 28.57 42.86 57.14 71.43 85.72 12.50 25.00 37.50 50.00 62.50 75.00 87.50 11.11 22.22 33.33 44.44 55.56 66.67 77.78 88.89 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 9.09 18.18 27.27 36.36 45.45 54.55 63.64 72.73 81.82 90.91 8.33 16.67 25.00 33.33 41.67 50.00 58.33 66.67 75.00 83.33 91.67 7.69 15.38 23.08 30.77 38.46 46.15 53.85 61.54 69 23 76.92 84.62 92.31 7.14 14.29 21.43 28.57 35.71 42.86 50.00 57.14 64.29 71.43 78.57 85.71 92.86 6.67 13.33 20.00 26.67 33.33 40.00 46.67 53.33 60.00 66.67 73.33 80.00 86.67 93.33 6.25 12.50 18.75 25.00 31.25 37.50 43.75 50.00 56.25 62.50 68.75 75.00 81.25 87.50 93.75 5.88 11 76 17 65 23 53 29.41 35 29 41.18 47.06 52.94 58 82 64.71 70.59 76.47 82.35 88.24 94.12 5.56 11 11 16 67 22 22 27.78 33 33 38.89 44.44 50 00 55 56 61.11 66.67 72.22 77.78 83.33 88.89 94.44 5.26 10 53 15 79 21 05 26.32 31 58 36.84 42.11 47.37 52 63 57.89 63.16 68.42 73.68 78.95 84.21 89.47 94.74 00 10 00 15 00 20 00 25 00 30 00 35 00 40 00 45 00 50 00 55 00 60 00 65.00 70 00 75 00 80.00 85.00 90.00 95.00 76 52 14 29 19 05 23 81 28 57 33 33 38 10 42 86 47 62 52 38 57 14 61 90 66 67 71 43 76 19 80 95 85.71 90.48 95.24 WEIBULL DISTRIBUTION FUNCTION FOR FATIGUE LIFE 75 per cent failure line is 0.0969 on the logarithmic scale On such paper, the tangent of the angle is an estimate of the Weibull "slope," b, for the population line The angle may be measured with a protractor, or the slope of the line may be computed Plotting Positions on Probability Paper: The fatigue data for any one sample are first ordered from shortest to longest life, each specimen being given an order number, q, from through n The horizontal plotting position is its individual life value All runouts are assumed to have longer lives than the last ordered specimen that failed, but such data are treated separately below under "Estimates of the Distribution Function Parameters." The vertical plotting position of the per cent failed (Fig 13) is the estimate of the per cent of the population failed, F(N), based upon the specimen order number Mean-rank estimates of the percentages of the population failed at successive TABLE 37.—TYPICAL FATIGUE TEST DATA Order, q No No No No No No No No Number of Revolutions to Failure Lot 1.1 X 106 2.3 4.0 6.5 8.6 Lot X 106 0 20 23 5 11 13 TABL,E 38.—TYPI CAL FATIGUE 1PEST DATA, WITHOUT RUNG UTS Plot of N Versus 1?(N) Nonlinear Order, q Specimen Number of Revolutions to Failure No No 4.0 X 105 No No No No No No No No No No No No No No 5.0 6.0 7.3 8.0 9.0 10.6 13.0 order numbers are given in Table 36 for sample sizes ranging from through 20 Mean rank, q/(n + 1), is an unbiased estimate of F(N); such estimates are recommended by Gumbel (36) and Weibull (37) Blom(38) suggests modified mean-rank estimates For the data given in Table 37 for the sample taken from lot 1, the abscissa for the first specimen is plotted at its life value of N = 1.1 X 106 revolutions and the ordinate at F(N) X 100 = 16.67, the plotting position for the first of a sample of five based upon mean ranks given in Table 36 Estimates of the Distribution Function Parameters: An estimate of the population cumulative distribution that corresponds to the data plotted in Fig 13 can be fqund quickly by drawing a line by eye through the failed points More refined techniques for calculating this line can be found by referring to Gumbel (36), Lieblein , or Kao (35) It is possible to calculate this line by the method of least squares, as illustrated in Section V A4 of this guide For example: and 76 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA Comparisons using these methods as against the graphic method sho.w,,however, that the latter is usually adequate for small samples An estimate of the characteristic life, Na , is obtained from Fig 13 by reading off the life value corresponding to the intersection of the fitted line and a horizontal line corresponding to F(N) X 100 = 63.2 per cent An estimate of the median life is obtained by reading off the life value corresponding to the intersection of the straight line of Fig 13 and a horizontal line corresponding to F(N) X 100 = 50 per cent FIG 14.—Estimation of Weibull Distribution Function Parameters for Data in Table 38 In Fig 13, the minimum life, N0 , is assumed to equal zero, since the plot of the fatigue data is approximately linear The plotted data from Table 38 result in a line which curves downward (Fig 14(a)); thus the existence of a finite minimum life value greater than would be suspected To find an estimate of minimum life, N0 : (1) note the life value which the curve approaches asymptotically, (2) obtain the quantity N — N0 for each point by subtracting the N0 value from each individual specimen life, and (3) plot this life difference on Weibull paper versus the same per cent failed values as before Thus, by trial and error, the best estimate of N0 will be found so that the data shown in Fig 14(o) will, when transformed, plot as a straight line, as shown in Fig 14(6) The slope parameter, b, is equal to the tangent of the angle shown in Fig 13 Another estimate of b can be made by computing the tangent of from the logarithms of the ordinates and abscissas of two widely separate points, NI and Nz, on the fitted line Thus estimate of b b; The skewness of the Weibull distribution varies with the shape parameter, and the Weibull mean, in general, may occur at various per-cent-failed values; WEIBULL DISTRIBUTION FUNCTION FOR FATIGUE LITE 77 TABLE 39.—TYPICAL FATIGUE TEST DATA, WITH RUNOUTS Order, No.'l No No No No No No No q Specimen Number of Revolutions to Failure No 1.30 X 106 1.60 1.75 2.10 2.35 2.70 runout runout No No No No No No No FIG 15.—Per Cent Failed at Weibull Mean FIG 16 —^Estimation of Weibull Distribution Function Parameters for Data in Table 39 78 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA that is, the mean does not coincide with the median Using the estimated Weibull slope, b, it is possible to read from Fig 15 an estimate of the per cent failed at the Weibull mean and then refer back to the estimated population line on Weibull probability paper, as in Fig 13, to read off the estimated mean life from the1 curve Gumbel (36) and Kao (35) give methods for calculating the Weibull mean when the characteristic life Na and the slope b are known For data containing run-out specimens (Table 39), the n' broken specimens (6 in the example, Fig 16), out of a total of n specimens tested, are plotted on probability paper at the mean-rank plotting positions, corresponding to a sample size n (8 in the example, Fig 16(a)) The line drawn through these points will approach a horizontal asymptote, F/racture , which is equal to the ratio of the first plotting positions corresponding to sample sizes n and n', respectively (Fig 16(a)) The parameters of this distribution may be obtained graphically by plotting only the n' broken specimens at mean-rank plotting positions, corresponding to a sample size n' versus N — N0 , where N0 is again the estimate of the vertical asymptote approached by the curve The slope of the resulting straight line (Fig16(6)), tan = b, can be obtained as described in this Section Na , at the probability level of 63.2 per cent, is taken directly from the plotted line The estimated equation of the probability function for the complete sample of size n will then become where F/ = rfrac ture The curve of Fig 16(a) may now be replotted by using, as ordinates, Fracture times the ordinates of the straight line and, as abscissas, N0 plus the abscissas of the straight line Note that Na is, in this case, no longer the estimate of the? characteristic life parameter of the complete distribution, F(N) The value of N at the 63.2 per cent probability of failure level may be obtained from the plot in Fig 16(o) Weibull mean: where T — the gamma function; and for Weibull variance: STP91 A-EB/Feb 964 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA 79 REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) D J Finney, Probit Analysis, Cambridge University Press, 1952 E S Pearson and H O Hartley, Biometrika Tables for Statisticians, Cambridge Uni versity Press, 1954 R A Fisher and F Yates, Statistical Tables for Biological, Agricultural and Medical Research, Fourth Edition, Oliver & Boyd, London, 1953 T S Dolan, "Certain Mechanical Strength Properties of Aluminum Alloys 25S-T and X76S-T," NACA TN914, October, 1943 H T Corten, Todor Dirnoff, T J Dolan, and Masaki Sugi, "An Appraisal of the Prot Method of Fatigue Testing, Part II," Technical Report No 35 on the Behavior of Metals under Repeated Stress, ONR Contract N6-ori 071(04), University of Illinois June, 1953 E Prot, "Fatigue Testing Under Progressive Loading, A New Technique for Testing Materials," translated by Edward J Ward, Captain, USAF, WADC TR 52-148, September, 1952 E J Ward and D C Schwartz, "Investigation of Prot Accelerated Fatigue Test," WADC TR 52-234, November, 1952 A P Boresi and T J Dolan, "An Appraisal of the Prot Method of Fatigue Testing," Technical Report No 34 on the Behavior of Metals under Repeated Stress, ONR Contract N6-ori-71, T.O IV, University of Illinois, January, 1953 W J Dixon and F J Massey, Jr., Introduction to Statistical Analysis, McGraw Hill Book Co., 1957 E L Crow, F A Davis, and M W Maxfield, Statistical Manual, Dover Publications, Inc., New York, N.Y., 1960 The Design and Analysis of Industrial Experiments, edited by O L Da vies, Hafner Publishing Co., New York, N.Y., 1956 D B Owen, Handbook of Statistical Tables, Addison Wesley Publishing Co., Inc., Reading, Mass., 1962 J A Greenwood and M M Sandomire, "Sample Size Required for Estimating the Standard Deviation as a Per Cent of Its True Value," Journal, Am Statistical Assn., Vol 45, 1950, p 258 C P Ferris, F E Grubbs and C L Weaver, "Operating Characteristics for the Common Statistical Tests of Significance," Annals of Mathematical Statistics, Vol 17, 1946, p 178 A M Freudenthal and E J Gumbel, "Minimum Life in Fatigue," Journal, Am Statistical Assn., September, 1954 R B Murphy, "Non-Parametric Tolerance Limits," Annals of Mathematical Statistics Vol XIX, 1948, pp 581-589 E H Schuette, "A Simplified Procedure for Obtaining Design-Level Fatigue Curves," Proceedings, Am Soc for Testing Mats., Vol 54, 1954 E H Schuette, "The Prediction of Exceedances in Limit-Value Testing," Statistical Methods in Materials Research, proceedings for a short course conducted by the Pennsylvania State University, June, 1956 I£ R $air, "Table of Confidence Intervals for the Median in Samples from Any Continuous Population," Sankhya, Vol 4, 1940, pp 551-558 W J Youden, "Systematic Error in Physical Constants," Physics Today, Vol 14, September, 1961, p 32 "Tables of the Binomial Probability " Applied Mathematics Series 6, Nat Bureau Standards, U S Government Printing Office, Washington, D C., 1949 W H Kruskal and W A Wallis, "Use of Ranks in One Criterion Variance Analysis," Journal, Am Statistical Assn., Vol 47, 1952, pp 583-621 P G Hoel, Introduction to Mathematical Statistics, Second Edition, John Wiley & Sons, Inc., New York, N Y., 1954 A Hald, Statistical Theory with Engineering Applications, John Wiley & Sons, Inc New York, 1952, pp 550-551 N C Severe and E G Olds, "A Comparison of Tests on the Mean of a Logarithico- Copyright© 964 by ASTM International www.astm.org 80 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA Normal Distribution with Known Variance," Annals of Mathematical Statistics, Vol 27, No 3, September, 1956, p 670 (26) G W Snedecor, Statistical Methods, Fifth Edition, The Iowa State College Press, 1946 (27) E H Schuette, "The Significance of Test Results from Small Groups of Specimens," Proceedings, Am Soc Testing Mats., Vol 57, 1957 (28) F Wilcoxon, Some Rapid Approximate Statistical Procedures, American Cyanamid Co., New York, N Y., 1949 (29) C Eisenhart and F Swed, "Tables for Testing Randomness of Grouping in a Sequence of Alternatives," Annals of Mathematical Statistics, Vol 14, 1943, p 66 (30) P S Olmstead and J W Tukey, "A Corner Test for Association," Annals of Mathematical Statistics, Vol 18, 1947, pp 495-513 (31) W Weibull, "A Statistical Distribution Function of Wide Applicability," Transactions, Am Soc Mechanical Engrs.; and Journal of Applied Mechanics Vol 73, September, 1951, pp 293-297 (32) W Weibull, Fatigue Testing and the Analysis of Results, Pergamon Press, New York N Y., 1961 (33) R A Fisher and L H C Tippett, "Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample," Proceedings, Cambridge Philosophical Soc., Vol 24, Part 2, 1928, p 180 Reprinted in Fisher's Contributions to Mathe matical Statistics, John Wiley and Sons, Inc., New York, N Y., 1950 (34) A M Freudenthal and E J Gumbel, "Physical and Statistical Aspects of Fatigue," Advances in Applied Mechanics, Vol 4, 1956, pp 117-158 (35) J H K Kao, "A Summary of Techniques on Reliability Studies of Components Using Weibull Distribution," Proceedings, Sixth Symposium on Reliability and Quality Control, Cornell University, Ithaca, N Y., January, 1960 (36) E J Gumbel, "Statistical Theory of Extreme Values and Some Practical Applications," Applied Mathematics Series 33, Nat Bureau Standards, Feb 12, 1954 (37) W Weibull, "A Statistical Representation of Fatigue Failures in Solids," Acta Polytechnica, Mechanical Engineering Series, Vol 1, No 9, 1949 (38) G Blom, Statistical Estimates and Transformed Beta Variables, John Wiley and Sons, Inc., New York, N Y., 1958 (39) J Lieblein, "A New Method of Analyzing Extreme-Value Data," Technical Note 3053, National Advisory Committee for Aeronautics, Washington, D C., January, 1954 SUPPLEMENTARY READING FOR APPENDIX IV (40) J H K Kao, "The Design and Analysis of Life-Testing Experiments," Transac' tions, 1958 Middle Atlantic Conference, Am Soc Quality Control; and Reliability Training, Inst Radio Engrs., 2nd Edition, Chapter II, March, 1960 (41) E J Gumbel, Statistics of Extremes, Columbia University Press, New York, N Y., 1958 (42) E J Gumbel, "Probability Tables for the Analysis of Extreme-Value Data," Applied Mathematics Series 22, Nat Bureau Standards, July 6, 1953 (43) W Weibull, "New Methods for Computing Parameters of Complete or Truncated Distributions," FFA Report 58, Aeronautical Research Inst of Sweden, February, 1955 STP91 A-EB/Feb 964 INDEX Abbreviations, p Analysis of correlation between two variables, Appendix III (p 69) Allocation of test specimens—probit method, Table (p 11) Analysis of fatigue data, p 22 Areas of the Normal curve, Table 28 (p 61) Arithmetic mean, p Average, sample, p X2/d.f distribution, percentiles of, Table 30 (p 63) X22 distribution, percentiles of, Table 27 (p 60) X ,P-3 Choice of distribution shape, p 40 Computations for fitting a response curve by method of least squares, p 34, Table 16 (p 34) Computation of significance tests, Table 20 (p 46) Computation of standard deviation of values about fitted line, Table 17 (p 36) Confidence coefficient, p Confidence interval, pp 5, 26, 27, 28, 42, Table (p 26), Table 10 (p 27), Table 11 (p 28) Confidence level, p Confidence limits (see confidence interval) Constant amplitude tests, pp 1, 9-13 Constant life fatigue diagram, p 14 Construction of Weibull probability paper from log-log paper, Fig 13 (p 73) Correlation between two variables, Appendix III (p 69) Cycle ratio, p Definitions, p Definitions relating to fatigue tests and test methods, p Definitions relating to statistical analysis, p Definitions relating to statistical analysis of fatigue data, p Difference among k means, p 47 Difference between two means, pp 21, 45 Difference between two standard deviations, pp 20, 44 Distribution, p 4, Table 27 (p 60), Table 30 (p 63), Table 32 (p 65), Fig (p 22) Distribution curves, Fig 12 (p 72), Fig (p 22) Distribution shape, choice of, pp 22, 40 Estimate, p , interval (see confidence interval) , point, p 81 Estimates of parameters, single stress level, pp 18, 40, 76 Estimates, mean rank—Weibull distribution function, Table 36 (p 74) Estimation, p 4, Fig 14 (p 76) Estimation, Weibull distribution function parameters, Fig 14 (p 76) Fatigue, p Fatigue data (see analysis of), p 22 Fatigue life, pp 2, 6, 27, 39 Fatigue life for a stated value of per cent survival, p 28 Fatigue life for p per cent survival, p Fatigue limit, p Fatigue limit for p per cent survival, p Fatigue notch factor, p Fatigue notch sensitivity, p Fatigue strength, p 6, Fig (p 51) Fatigue strength for p per cent survival at N cycles, p Fatigue test data, Table 12 (p 30), Table 13 (p 31), Table 37 (p 75), Table 38 (p 75), Table 39 (p 77) Fatigue tests, p F-distribution, Table 32 (p 65) F-ratio test, pp 45, 47 Frequency distribution, p Gaussian distribution curve, Fig (p 22) Group, p Increasing amplitude tests, pp 1, 13, Fig (p 14), Fig (p 15) Interval, pp 5, Interval estimate, p ^-factors for S-N curves, Table 33 (p 67) Least squares, method for fitting a response curve, Table 16 (p 34) Level, confidence, p Level, significance, p Level, tolerance, p Limits, confidence, p Limits, fatigue strength at N cycles, p 38 Limits, method of computing, Table 19 (p 38) Limits, tolerance, p Maximum stress, p Mean, confidence interval for, p 19 Mean, definition, p Mean, confidence limits for, Table 10 (p 27) Mean fatigue life, p 27 Mean, sample, p 21 Copyright© 964 by ASTM International www.astm.org 82 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA Mean rank estimates: per cent of population failed corresponding to failure order in sample, Table 36 (p 74) Means, confidence interval for, p 42, Table (p-1 9) Means, differences between two, p 45 Means, differences among k, p 47 Mean stress, p Mean, Weibull, Table 36 (p 74), Fig 15 (p 77) Median, confidence limits for, Table (p 26) Median fatigue life, pp 6, 26 Median fatigue strength at N cycles, p Median percentage of survivors for the population, Table (p 24) Median, sample, p 43 Medians, differences of group, p 29 Method of least squares, pp 34, 35, Table 16 (p 34) Minimum per cent of population exceeding median of low ranking points, Table 25 (p 56) Minimum stress, p Modified staircase test method, pp 1, 13, 48, 49 Moore rotating beam step test, Table 21 (p 50), Table 22 (p 51), Fig (p 51) Mo.025 and MO.QTS for runs among elements in samples of sizes NI and N%, Table 31 (p 64) Nominal stress, p Normal curve, areas of, Table 28 (p 61) Normal distribution curve, Fig (p 22) Normal distribution of fatigue hie, p 39 Normal distribution, ^-factors for S-N curves, Table 33 (p 67) Number of test specimens, minimum, pp 16-21, Table (p 19), Table (p 19), Table (p 20), Table (p 20), Table (p 21), Table (p 21) Parameter, pp 4, 18, 26, 40, Fig 14 (p 76), Fig 16 (p 77) Per cent of specimens having at8 least the indicated fatigue strength at 10 cycles, Fig (p 51) Per cent survival for a stated value of fatigue life, p 27 Per cent survival values at N cycles, confidence limits for, p 37 Percentiles of the x2 distribution, Table 27 (p 60) Percentiles of the x2/d.f distribution, Table 30 (p 63) Point estim_ate, pp 5, 26-29 Population, p Probability-stress-cycle curve, Fig (p 10) Probability paper, pp 33, 73 Probit test—allocation of test specimens, Table (p- 11) Probit test data, Table 15 (p 34) Probit test method, p 10 t test, pp 1, 15, 52, Table 23 (p 52), Fig (p 53) "Quadrant sum" correlation test, Appendix III (p- 69) "Quadrant sum," working significance level, Table 34 (p 69) Range of stress, p Rank test, pp 25, 30, 31, Table 25 (p 56), Table 26 (p 58) References, p 79 Response curves, pp 7, 18, 33, 34, 36, 38, Fig (p 35), Table 16 (p 34) Response or survival tests, pp 1, 10, 12, 13, Fig (p 11) Rotating beam tests, R R Moore, Table 21 (p 50) Run test, Appendix II (p 68) Runs along elements in samples of sizes NI and N2, Table 31 (p 64) Sample, p Sample average, p Sample means, p 21 Sample median, p Sample standard deviation, p Sample percentage, p Sample variance, p Scatter diagram, Fig 11 (p 70) Selection of test specimens, pp 16, 17, 19 Significance level, p 6, Table 34 (p 69) Significance level, for "quadrant sum," Table 34 (p 69) Significant, p S-N curve for 50 per cent survival, p S-N curve for p per cent survival, p S-N curves, pp 6, 7, 17, 19, 23, 25, 40, 41, Table (p 24), Table 25 (p 56), Table 33 (p 67) S-N diagrams, p Staircase test method, pp 1, 48, Fig (p 12) Standard deviation, pp 19, 44-46, Table (p 19) Standard deviation, confidence interval for, p 42 Standard tests, pp 1, 9, 23, 25 Statistic, p Steady component of stress, p Step test method, pp 1, 13, 50, Fig (p 14) Step tests, R R Moore rotating beam specimens, Table 21 (p 50), Table 22 (p 51) Stress, p Stress amplitude, p Stress concentration factor, p Stress cycle, p Stress cycles endured, p Stress ratio, p Survival tests, Fig (p 11) Symbols, p Test of significance, pp 6, 18, 20, 29, 30, 31, 45, 46, 47, Table 20 (p 46) Test procedures, pp 8, 9, 10, 12, 13, 15, Fig (p 12), Fig (p 14T, Fig (p 15) Test specimens, minimum number, p 16 Test specimens, selection of, p 16 INDEX Test-statistic, p Theoretical stress concentration factor, p Tolerance interval, pp 5, 41, 42 Tolerance level, p Tolerance limits, p 2-test, p 45 /-values, Table 29 (p 62) 83 Universe, p Unpaired rank test, p 30, Table 26 (p 58) Values of t, Table 29 (p 62) Variable component of stress, p Weibull distribution, p 71 Weibull mean, Fig 15 (p 77), Footnote p 78 Wohler test method, p This page intentionally left blank T HIS PUBLICATION is one of many issued by the American Society for Testing and Materials in connection with its work of promoting knowledge of the properties of materials and developing standard specifications and tests for materials Much of the data result from the voluntary contributions of many of the country's leading technical authorities from 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