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JOURNAL CF RESEARCH of the National Bureau of Standards— A. Physics and Chemistry Vol. 64A, No. 2, March-April 1960 A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars S. Spinner, T. W. Reichard, and W. E. Tefft (December 17, 1959) The relations from which Young's modulus may be computed from mechanical flexural and longitudinal resonance frequencies have been established by an empirical method using two sets of steel bars. Both sets contained rectangular and cylindrical specimens. For longitudinal vibration of cylindrical specimens, the agreement between the empirical curves and Bancroft's corresponding theoretical relation was within experimental error if Poisson's ratio for both sets is taken to be 0.292. For flexural vibrations, the agreement between the empirical curve and the corresponding theoretical relation developed by Pickett is also with- in experimental error for about the same value of Poisson's ratio for the rectangular speci- mens of both sets; but for cylindrical specimens, the empirical values are somewhat lower than those predicted by the theory. 1. Introduction In a, previous paper, [1] l an empirical relation was established from which the shear modulus could he calculated from the torsional resonance frequency using uniform steel bars of different rectangular cross sections. The empirical relation was compared with corresponding theoretical approximations. The pur- pose of the present paper is to establish similar rela- tions from which Young's modulus may be deter- mined from the flexural and longitudinal mechanical resonance frequencies for bars of round and rectangu- lar cross section. These empirical relations are also compared with corresponding theoretical equations when feasible. As in the previous work, advantage* is taken of the fact that these resonance frequencies can be deter- mined to an accuracy winch, when combined with comparable accuracy of dimensions, is sufficient to yield empirical results good to four significant figures. In fact, it is this increased accuracy to which modern experimental techniques have advanced dynamic elastic measurements that has made it possible to check in a more precise manner the theoretical results of such analysts as Rayleigh, Kelvin, Poisson, and Stokes [2]. As usually happens in such cases, this increased experimental accuracy has, in turn, led to a refine- ment and extension of the theory. Some equations had lain dormant for many years because, though presumably "complete" and "correct," they were nevertheless expressed in so general a form that numerical solutions for most real cases were too cumbersome to be of practical value. Such equa- tions have recently been solved for given boundary conditions. These solutions have often taken ad- 1 Figures in brackets indicate the literature references at the end of this paper. vantage of modern computing devices. A particular case in point is the set of Pochhammer-Chree equa- tions, relating the most general case of elastic waves in rods to their elastic moduli. These equations, almost forgotten for more than 50 years, were solved by Bancroft [3] for the case of longitudinal waves and by Hudson [4] for flexural waves. A recent article by Davies [5] presents a comprehensive re- view and bibliography of the advances in this field up to the present I ime. For Young's modulus, the problem of establishing an empirical relation is complicated by the fact that the cross sectional correction for both flexural and longitudinal vibrations depends upon Poisson's ratio as well as the dimensions. This is in contrast with shear for which the cross sectional correction is independent of Poisson's ratio. Consequently, the results to be presented here are more limited than those previously given for shear since these results apply mainly to those materials having Poisson's ratios approximately equal to those used here. Furthermore, when comparing the empirical with theoretical relations, any error in the value of Pois- son's ratio assigned to the specimens would result in a corresponding error in the comparison of the cor- rection factors. This error would increase as the ratio of cross section to length increased. 2. Experimental Procedures 2.1. General The basic experimental approach consists in de- termining the flexural and longitudinal resonance frequencies of specimens of known mass and dimen- sions, and assuming their uniformity of Young's modulus and density, to derive the empirical relation needed for the determination of Young's modulus from the mechanical resonance frequency. Data to 147 be presented later will show that the assumption of uniform density and Young's modulus is quite justified. 2.2. Specimens Two separate sets of steel specimens were used in this investigation. Each set of specimens was cut from its own parent piece to insure the greatest possible uniformity of intrinsic Young's modulus and density from specimen to specimen within each of the two sets. One source was a cylindrical bar of cold drawn steel about 1 in. in diameter, designated as SAE No. 1020. Originally 18 specimens, 1-5 through 1-22, ranging in length from about 3 to 12 in. were cut from the parent stock. Subsequently some of these rods were further shortened or machined to square cross section to extend the range of the experimental data. All of the specimens from this source are henceforth classified as set I. Exact dimensions and other related data for set I are given in table 1. Table 1. Data for specimens of set I & Speci- men Length I Density p /(long)b /(flex)b k/l ° v/vo empirical T empirical I-l 1-2 1-3 cm 5.050 5.118 5.354 5.639 g/cm* 7.849 7.850 7.849 7.850 cps 50, 253 49, 605 cps 31,310 30, 650 28, 624 26, 401 0. 12565 . 12398 . 11850 . 11251 0. 9851 .9857 2. 1234 2. 0997 1. 9998 1-4 1. 9205 1-5 1-6. 1-7 1-8 7.511 7. 584 7.760 10. 117 7.851 7. 851 7.851 7.847 34, 051 33, 752 32, 975 25, 370 16, 605 16, 341 15, 733 9,939 . 08447 . 08366 . 08176 . 06272 .9929 .9938 .9934 .9965 1. 5424 1. 5320 1. 5077 1. 3085 1-9 1-10 1-11 1-12 10. 224 14. 790 14.958 14. 968 7.846 7.851 7.853 7.854 25, 107 17, 389 17, 190 17, 185 9,755 4,965 4,859 4,854 . 06208 . 04289 . 04241 . 04238 .9966 .9985 .9982 .9986 1. 3024 1. 1475 1. 1447 1. 1438 1-13 1-14 1-15 1-16 15. 118 15. 235 19. 736 19.985 7.853 7.851 7.851 7.852 17, 014 16, 883 13, 037 12, 875 4,765 4,695 2,866 2,799 . 04196 .04164 . 03214 . 03174 .9985 .9986 .9989 .9989 1. 1410 1. 1395 1. 0852 1. 0829 1-17 1-18 1-19. 1-20 22. 951 25. 641 29. 987 30. 201 7.849 7.850 7.852 7.853 11,213 10, 039 8,587 8,525 2,139 1,725 1,271 1, 253 . 02766 . 02475 . 02116 . 02101 .9991 .9993 .9997 .9995 1. 0664 1. 0525 1. 0360 1. 0359 1-21. 1-22 30. 500 30. 554 7.852 7.850 8,442 8,427 1,229 1,225 . 02080 . 02076 .9996 .9995 1. 0353 1. 0350 I-12b .. 5.177 6.942 14. 968 19. 736 25. 641 7.849 7.849 7.850 7.848 7.847 26, 704 16,415 4,043 2,370 1,421 . 10020 . 07473 . 03464 . 02627 . 02023 1. 7649 I-12c _ 1. 4445 I-12a I-15a I-18a 17, 188 13, 039 10, 038 .9988 .9991 .9992 1. 1016 1. 0610 1. 0360 a All specimens except those followed by a letter are 2.5378 cm in diameter. Those followed by a letter are 1.796-cm squire. b Fundamental longitudinal and flexural resonance frequencies. c k = radius of gyration of the cross sectional area about an axis perpendicular to the plane of vibration for round specimens /c = H diam= 0.63445 cm. for square specimens k = edge/ Vl2= 0.51846 cm. for cylindrical specimens, B/X=2k/l where D=diameter of specimen and X= wavelength of longitudinal wave. The other source of specimens was a bar of 2-in. square stock of hot rolled and annealed tool steel designated by the trade name "Stentor." The original specimens from this source were the same 12 pieces (II— 1 through 11-12) of equal lengths but different rectangular cross sections that were used in the investigation for shear modulus [1]. As for set I, some of these specimens were further reduced in dimensions or machined to circular cross section. All specimens from the second source are classified as set II and data for these specimens are given in table 2. The dimensions of both sets of specimens are accurate to ±.001 cm. The density was calculated from the mass and the dimensions of the specimens. The average density for the specimens of set I was 7.851 g/cm 3 , and that of set II was, as previously given [1], 7.814 g/cm 3 . The standard deviation of this measurement was 0.002 g/cm 3 for both sets. This small variation in density is good evidence for the intrinsic uniformity of the specimens of each set. Although the density variations are within the error of the measurement, the mass and dimensions of each particular specimen were used in most calcula- tions rather than the average value of density. The density of some randomly selected specimens of both sets was also checked by weighing in air and while immersed in liquid and was found to agree with the above values within the error of their determination. Actually, for the specimens of set II, the density, p, by the immersion technique was found to be 7.816 g/cm 3 . Subsequent determination of p, calcu- lated from the mass and volume of two specimens machined to a higher degree of accuracy than the others (specimens II-^! and II-4r 2 ), agreed with the value obtained by immersion and is believed to be the most reliable value for the specimens of set II. 2.3. Method The mechanical longitudinal and flexural resonance frequencies of both sets of specimens were deter- mined by the dynamic method previously described [1]. Briefly, one of the mechanical resonance frequencies of the specimen is excited by an electro- magnetic driver. The increased amplitude of vibra- tion of the specimen at resonance is detected by a crystal pickup whose output, together with a signal of the same frequency, produces a Lissajou pattern on a cathode-ray oscilloscope. The different types of vibrations are obtained by appropriate placement of the driver and pickup with respect to the specimen. As with torsional vibrations the longitudinal and flexural resonance frequencies were excited and detected by more than one method. In the firkt method the specimens were supported on foam rubber in the vicinity of the nodal points and driven through air by a tweeter type driver. A crystal pickup placed lightly against the proper part of the specimen detected the vibrations. Both longitudinal and flexural vibrations could be ob- tained by this method. The second method could be used only to obtain flexural vibrations and was most appropriate for the lighter specimens. This method consisted in sus- pending the specimens from two cotton fibers, one fiber being attached to a phonograph record cutting head as the driver and the other fiber being attached to a crystal pickup. Unlike the case for torsion, it 148 Table 2. Data for specimens of set II Rectangular specimens Specimen * Length I Width Depth d Density p Flatwise Edgewise / (long) k/l*> /(flex) T fc/Jb /(Ilex) T II-l II-2 II-3 II-4 ^11-5 II-6 II-7 II-8 II-9 11-10 11-11 11-12 ll 2a cm 15.202 1 5. 202 15.202 15.202 15.202 15.202 15.202 15.202 15.202 15.202 15. 202 15. 202 7.882 7.010 7.546 7.249 15.202 15.202 cm 3. 1496 3. 1433 3. 1433 3. 1433 3. 1433 3. 1433 3. 1433 3. 1433 3. 1433 3.1433 3. 1433 3. 1433 3. 1433 3. 1433 3. L433 3. 1433 2. 0574 0. 6426 cm 3.1496 2. 5405 1.9055 1.5875 1.4300 1.2708 1.1120 0. 9530 .7943 . 6363 .4773 .3172 2. 5405 2. 5405 1.9055 1.9055 1). f 13(13 . (13(13 g/cm 3 7.817 7.819 7. Ml 7.817 7.814 7.816 7. ML' 7.812 7.813 7. SI 5 7.811 7.814 7.814 7. SI 7 7.818 7.817 7.816 7. M 1 0. 05981 .04825 .03(110 .03015 .0271(1 . 02413 .02112 .01810 . 01508 .01208 .00906 . 00002 .09306 . 10463 . 07290 . 07589 .01208 .01220 cps 641J 5379 1 1 83 3538 3208 2867 2523 2172 1820. 6 1 013. 1 1101.1 733.0 16,941 20, 432 1 1. 983 16,052 1460.7 1.2915 1. 1936 1.1112 1.0775 1.0C.3;. 1.0514 1.0103 L.03O9 1.01 so 1.0118 l.OOOl 1.0028 1.(1(100 1.8297 1.4255 1.4584 1.01 17 0. 05981 .05900 . 05969 . 05969 . 05969 . 05969 . 05969 . 05969 . 05969 .05969 . 05969 . 05969 .11513 . 1 2945 . 1 2020 . 12510 . 03907 cps (1111 0399 6398 6400 6397 6399 6399 6394 6400 6400 0100 6396 19, 203 22, 934 20, tor. 21, 789 4475 1.2915 1.2910 1. 21)23 1.2909 1. 2926 1.2915 l. 2921 1. 2941 1. 2916 1. 2912 1. 2919 1.2927 1. 9849 L'. 2232 2. 0732 2. 1535 1. 1314 cps 17,046 17.053 17,0(12 17, 065 17,066 17,066 17,071 17, 067 17,075 17,070 17,070 17, 064 II-2b II-3a II-3b ll 10a II-10b 17,100 Cylindrical specimens Length I Diameter D Density p kll ° /(flex) T /(long) v,v Il-lr II-2ar II-3or II-4n II-4r 2 14.364 7.315 ti. 117 1 1.669 1 1. r.cs 3.1252 2. 4400 1.8124 1.2845 L.2843 7.816 7.812 7.812 7.816 7.816 0.05439 .08359 .07029 .02189 ,02189 (1303 17. 102 17,141 270S. 2708.3 1 238 1 1 . 5329 1.3882 1.0404 1.0407 IS, 057 35, 329 40, 145 17,714 17,715 0. 9978 . 9944 . 9957 . 9996 .9996 a Letter following specimen number indicates that the specimen has been redimensioned. Number denotes original specimen. A second letter indicates a second change in dimension. »> /c=radius of gyration of the cross sectional area about an axis perpendicular to the plane of vibration. For rectangular specimens in flatwise vibrations, k=d/ ^12; for edgewise vibration k=w -yfl2. o For cylindrical specimens, /c = D/4. Since, for the fundamental longitudinal resonance frequency, \=2l, kjl=d/2X. was not necessary for the points of suspension to be at opposite faces of the specimen. A third method, combining certain features of the first two, consisted in suspending the specimens from two cotton fibers as in the second method but driving them through air with a tweeter and detect- ing the vibrations with a crystal pickup as in the first method. This third met hod could be used to obtain both flexural and longitudinal vibrations and was satisfactory for heavy as well as light specimens. The highest resonance frequencies could be obtained most readily by this method. The accuracy of the resonance frequencies obtained by the last two methods was usually somewhat better than that obtained by the first method. However, by any of these methods, the accuracy of the resonance frequencies was usually better than 1 part in 4,000 [1]. The fundamental longitudinal and flexural reso- nance frequencies for the specimens of sets I and II are given respectively in tables 1 and 2. Inasmuch as the specimens of set II are rectangular in cross section, two separate flexural resonance frequencies occur about both longitudinal planes of symmetry (flatwise and edgewise). The fact that the edgewise flexural frequency is the same for specimens II-2 through II-l 2 is of considerable significance as will be explained. Table 3 gives frequencies of overtones of longitudinal resonance vibrations of four speci- mens of set I and one specimen of set II. It may be observed that longitudinal resonance frequencies of over 50,000 cps are recorded for both fundamentals and overtones. These remarkably Table 3. Overtones of logitudinal resonance vibrations of several cylindrical specimens I Fundamental (n=l) 1st overtone (n=2) 2d overtone (n=3) 3d overtone (n=4) 4th o ver tone [n = 5) [Fundamental (n=l) Qot r 1st overtone (n=2) Soecimen 1-21 V 2d overton e (n=3) specimen wi 3d overtone (n=4) Uth overtone (w=5) (Fundamental (n=l) 1st overtone (n=2) 2d overtone (w=3) 3d overtone (n=4) Set I fFundamental (n = l) Specimen 1-16 \lst overtone (n=2) Hot tt [Fundamental (n=l) Soecimen II-lr1 lst overtone (« = 2) bpewmen n ir [ 2 d overtone (n=3) Frequency Df\=r/l V/Vo* 8525 17027 25491 33898 42218 0. 04202 . 08405 . 12607 . 16809 . 21012 0. 9995 .9982 .9963 .9936 .9900 8442 16858 25238 33560 41800 . 04160 . 08321 . 12481 . 16642 . 20802 .9996 .9981 .9961 .9935 .8999 8587 17154 25683 34146 . 04232 . 08464 .12696 . 16928 .9997 .9985 .9966 .9938 12875 25692 . 06348 . 12696 .9989 .9967 18057 35823 52884 . 10878 . 21756 . 32634 .9978 .9898 .9741 a For set I specimens, »o=5152 m/sec; for set II specimens, tfo=5199 m/sec. 149 high values of resonance frequency which can be excited and detected by a method sometimes de- scribed as "sonic," are explained by the nature of the specimens and also by the fact that the response of both driver and pickup, though reduced, persists at frequencies considerably above their rated upper frequency limit. This reduced response is amplified and detected as a recognizable pattern on the scope. Since the upper frequency response obtained is more than 2}i times higher than the nominal upper limit of the sonic range and will probably go higher as experimental techniques improve, it is felt that the term, "mechanical resonance methods," would be more appropriate than "sonic methods" to de- scribe the experimental procedure used. 3. Calculations, Results, and Discussion 3.1. Longitudinal Resonance Frequencies for Cylin- drical Specimens of Sets I and II The following relations for this type of vibration are recalled. First, a rod vibrating in this manner satisfies the condition that l=rik/2 (1) where 1= length of specimen, n= order of resonance frequency. For the fundamental, n=l, for the first overtone n=2, etc., \=wavelength of the vibration in the specimen. This leads to the well-known relation between the velocity of the longitudinal wave, v, and the longitudinal resonance frequency, Jri) v=^- (2) n The subscript after the / indicates the order of the resonance frequency. For an infinitely thin specimen of length /, v becomes the "rod velocity," v . Rayleigh's familiar approximation, given below, shows the amount by which v, in a specimen of finite circular cross section is reduced from % -4 1+ (T)T (3) where /x = Poisson's ratio and r=radius of the rod. The relation showing the effect of cross section is often expressed in terms of v/v as a function of D/\, D being the diameter of cross section. This convention will be followed here. From eq (1) it is seen that D/\=nr/l. Empirical values of v/v can be conveniently cal- culated after the value of v has been determined. We consider first the specimens of set I. Substitu- tion of the appropriate values for the longest speci- men, 1-22, (having the lowest r/l) in eq (3) shows that ^0.9996?;o. Specimens 1-19, 1-20, and 1-21 are sufficiently long to give an average for the ratio, v/v , from eq (3) equal to that for specimen 1-22. At the low values of r/l associated with these specimens, the choice of a proper value of \i is no problem, since any reasonable variation about the selected value of 0.3 (say from 0.26 to 0.32) will have only a negligible effect on the result. Also, at such low values of r/l any difference between Rayleigh's and corresponding equations, such as Bancroft's, will also be negligible. Therefore, the average value of #0=5152 m/sec, obtained by sub- stituting the values for these four longest specimens in eq (2) and dividing by 0.9996, is taken to be the rod velocity for the specimens of set I. The empirical values of v/v for the remainder of the specimens of set I are then found from the following equation v/v c = 2lf n , 5152ft (4) These v/v values are given numerically in table 1 and graphically, as a function of D/\ in figure 1 . The empirical values of v/v for the specimens of set II are found in the same manner. The reference specimens in this case, having the lowest value of r/l, were 11-4?^ and II-4r 2 . These are recalled to be the specimens that were more accurately machined. The purpose of this was to obtain a more reliable base value. For these specimens, v/v = 0.9996, w =5199 m/sec, and empirical values for the other specimens are found from the equation v/v = . 2lfn "5199ft (4a) V^o Figure 1. Effect of the ratio of diameter (D) to wavelength (X) on the ratio of the velocity of the longitudinal wave (v) to the rod velocity (vq) for two sets of cylindrical steel specimens. Theoretical curves are included for comparison. 150 Numerical values of v/v Q for these specimens are given in table 2 and are plotted along with those of set I in figure 1 . For both sets of specimens, v/vq for higher values of D/X was obtained both by vibrating the shortest specimens at their fundamental resonance frequency and also (since D/X=n r/f) by vibrating some speci- mens at higher overtones. The data for these over- tones are found in table 3. It is observed from figure 1 that not only do the empirical points fall on the same curve, within experi- mental error, whet her determined from the funda- mental or overtones of either set of specimens, hut also the points for both sets of specimens also fall on this same curve. Since v and the density, p, are known, Young's modulus, E, can be determined for each set of speci- mens from the equation, E- --v Q - p (5) for set I, E= 2084X10° dynes/cm 2 =2084 kilobars, and for set II, £'=2113 kilobars. Bancroft's [3] numerical solution of the Poch- hammer-Chree equation for longitudinal waves has already been mentioned. His values for ^=0.25 and jLt=0.30 are plotted, along with values based on Rayleigh's equation for /z=0.25, in figure 1. Ban- croft's solution is seen to reduce v/v by a greater amount than Rayleigh's for a given value of p. Since Bancroft's solution is considered more exact than Rayleigh's, comparison of the empirical points will be made with Bancroft. Graphical inter- polation between Bancroft's values for jii=0.25 and M=0.30 at D/\=0.25 shows the empirical curve to agree with Bancroft for the case where /x=0.292. That is, if the n of the specimens of sets I and II is 0.292, then agreement of the empirical with Ban- croft's solution would be within the error of the measurement. It would be desirable then to obtain an independ- ent value of ju as a further check. The method thai appeared most feasible for this was to determine the shear modulus, G, from the torsional resonance frequency and then, since E is already known, to compute /x from the well-known relation between E and G for isotropic materials, E i (6) For the specimens of set II, G is already known from the previous investigation [1] to be 822.1 kilobars. For the specimens of set I, however, it was not pos- sible (at first) to detect the torsional resonance fre- quency of the round bars by any of the variations of the method previously described. 2 To circumvent 2 Subsequently, by the use of an improved driver and suspension of the speci- mens from strings as already described, the torsional resonance frequencies were obtained for cylindrical bars of sets 1 and II. The average values of 0, calculated from these resonances, were in agreement with those given in the text. this difficulty, three of the specimens were machined to square cross section. This was in fact the original reason for squaring some of the round specimens of sei I. (These squared bars incidentally provided additional specimens for which longitudinal and flex- ural resonance frequencies could be determined. It can be seen from table 1 that for specimens of this size, (he effect of cross section in reducing the rod velocity is of the same order of magnitude as for circular cross section.) For square specimens the torsional resonance frequency, and hence G, can be obtained in the manner described in the previous paper [1]. Two of the longer and one relatively short speci- men were selected. A square rather than rectangular cross section was chosen because the shape factor for square cross section is believed to be more accurate [1] and would therefore lead to a more accurate value of G. The value of G obtained for specimens I-12a, I-15a, and I-18a, of set I, were 820.5, 821.6, 820.5 respectively, with an average of 820.9 kilobars. Substituting the known values of E and G for both sets of specimens in eq (6) yields the following values for fi: For set I, jli=0.2G9 and for set II, M=0.285. The physical constants obtained for sets I and II are now summarized in table 4. It appears far more likely that the value of M=0.292 is closer to the true value for both sets of specimens than the values obtained from eq (6). This belief is supported by the following evidence. Table 4. Physical constants of two different sets of steel specimens m/seo.- g/cmA_ kilobars. - kilobars.. Set I Set II Po, "rod velocity'' />. densitj E, Young's modulus. O, shear modulus n, Poisson's ratio 5152 7.851 2084 820.9 0. 269 * 5199 7.816 2113 822.1 0.285* I derived from E and values and eq (6). From Bancroft, ^ = 0.292 for both sets. The value of jjl for steel usually found in the literature is around 0.29. Markham [6], for instance, measured E and G for 10 different types of steel by an ultrasonic method and, from these elastic moduli, calculated /a. His values for /x varied between 0.286 and 0.292 with an average of 0.289. Analysis of Markham's data shows that the variation in values of /x given for the different types of steel can be accounted for completely by precision in measure- ments of E and G, given by Markham, rather than by any differences in the values of E and G themselves. Therefore, the average value of /x= 0.289 for all the steels may be taken as characteristic of each of them. Thus it appears that though E and G may be differ- ent for different types of steel, these elastic moduli vary concomitantly so that p. remains constant. If jit — 0.292 is correct for the specimens of sets I and II, then a possible explanation for the lower 535595—60- 151 values of ju obtained from eq (6) lies in the fact that a preferred crystal orientation develops in the steel during the process of manufacture. Consequently, the assumption of macroscopic isotropy resulting from a completely random crystal orientation is not entirely fulfilled, and eq (6) which is based on this assumption, is not entirely valid for these specimens. Frankland and Whittemore [7] have demonstrated that the average E for specimens of " black" sheet steel cut perpendicular to the direction of rolling is significantly different from the average E of speci- mens cut parallel to the direction of rolling. In this connection, it is noteworthy that for the speci- mens of set I, in which the process of repeated cold working of the parent stock results in a more pro- nounced crystal orientation, the value of /x departs by a greater amount from the "correct" value, than for the specimens of set II, where the process of annealing of the parent stock largely restores the random crystal orientation. Indeed, the value of /* in the specimens of set II from eq (6) is in good agreement with the values found in the literature and with that based on Bancroft in this investigation. Also, it appears from the fact that the empirical points of v/v for sets I and II lie on the same line, that the value of n for both sets of specimens is the same. This does not prove that the value of \x based on Bancroft is "correct" but it does make it improbable that sets I and II should have different values as the results based on eq (6) would indicate, for it would be a most unlikely coincidence that any error resulting from interpolation from Bancroft should lead to the same value of /jl, if the /x of both sets of specimens were actually different. Further- more, the agreement in ju for both sets of specimens is in accordance with Markham's data. The alternative possibility to explain the discrep- ancy in ju, is that the values based on eq (6) are correct, and that Bancroft's correction for cross section and consequently, the value of jit based on it are incorrect. Inasmuch as this alternative involves the (at least partial) rejection of Bancroft's theoretical equation as well as the value of ju for steel found in the literature, both widely accepted, its correctness appears most unlikely. 3.2. Longitudinal Resonance, Rectangular Specimens The longitudinal resonance frequencies of the rectangular specimens of both sets are listed in tables 1 and 2 but will not be considered here. It is planned to discuss these in a subsequent paper. It will merely be noted here that specimen 11-10 b, having a small nearly square cross section, had a considerably higher resonance frequency (17,100 cps) than the other specimens of the same set. Substituting the resonance frequency and length for this specimen in eq (2) yields a value of #=5199 m/sec in agreement with v for this set. This value would be expected for a round specimen of the same k/l. 3.3. Flexural Vibrations, Sets I and II Flexural vibrations are probably of more practical importance than longitudinal as a means of de- termining Young's modulus because flexural vibra- tions can usually be excited more easily than longi- tudinal. This is especially true for thin specimens. Thus, for these thin specimens where any error in E due to an error in the correction for cross section would be minimized, the longitudinal resonance frequency is relatively difficult to obtain, whereas the flexural resonance frequency becomes experi- mentally easier to excite. Hence a reliable relation- ship from which E may be determined from the flexural resonance frequency becomes important. Hudson's [4] numerical solution of the Poch- hammer-Chree equations for flexural waves has already been mentioned. Unfortunately, no com- parison can be made between Hudson's results and the empirical ones, because no simple or clear-cut relation has been found to exist between the length of a traveling flexural wave in a very long bar and the length of bars vibrating in flexural resonance. Consequently, one relies on a direct relation between Young's modulus and the flexural resonance fre- quency. Goens [8] has solved Timoshenko's [9] equation relating Young's modulus to the flexural resonance frequency for bars of different cross section. Pickett [10] has further simplified Goen's solution. Goen's solution can be expressed in the following form: (7) where /, in this case, is the flexural resonance fre- quency; k is the radius of gyration of the cross sec- tional area about an axis perpendicular to the plane of vibration. For a rectangular cross section k = t/->l\2, t being the dimension in the direction of vibration. (The depth and width interchange as t depending on whether the vibration is flatwise or edgewise.) For a circular cross section, k=r/2; m is a constant which has higher values for higher overtones, for the fundamental m=4.730; J 7 is a correction factor which varies with k/l and ju. Pickett used subscripts for m and T since both factors vary with the order of vibrations. Since only the fundamental flexural resonance frequency is considered here, the subscripts are dropped. For cylindrical bars eq (7) becomes £=1.2619 r^l'pT, and for rectangular bars eq (7) becomes (7a) #=0.9464 \ l ~f 'm (7b) 152 Pickett gives algebraic equations relating T to kjl for p=0, %, and }{. In addition lie gives numerical solutions of these equations for particular values of k/lam\ graphs based on these solutions. The graphs for n $and % the two values which span the range of interest for steel, arc reproduced in figure 2 from Pickett's numerical values. 7 7 approaches 1 as kjl approaches for all values of /u. 2 I -ROUND '-SQUARE -ROUND '-RECT. Figure 2. Empirical and theoretical curves showing the effect of k/1 on the correction factor, T, for flexural vibrations. kfl is the ratio of the radius of gyration of the cross sectional area about an axis perpendicular to the plane of vibration to the length of the specimen. Sets L and II represent two separate sets of steel bars. According to eq 7b, for a given value of E, the flexural resonance frequencies of rectangular speci- mens are independent of the dimension perpendicular to the plane of vibration. Pickett shows in the appendix of his paper, which deals with the problem more rigorously, that ia the extreme cases of an infinitely thin bar or an infinitely wide slab, this dimension (perpendicular to the plane of vibration) does slightly affect the flexural resonance frequency. However, for the specimens used in this investiga- tion, this correction would be insignificant. This means that if the specimens of set IT are really uniform with respect to E as well as p, then the edgewise flexural resonance frequency of speci- mens II-2 through 11-12 should all be equal, since the only variable for these specimens is the dimension perpendicular to the plane of vibration. The degree of agreement in (his frequency is a critical indication of the intrinsic uniformity of the specimens. The variation in frequency is insignificantly small, as shown in table 3. Therefore, the specimens must be uniform with respect to E as well as pt The importance of this result can hardly be overempha- sized, since the uniformity of the specimens with respect to E and p is the foundation of the entire empirical approach. For the specimens of set I no such conclusive check on the uniformity of E is possible, so that the uni- formity of p must serve as indirect evidence of the uniformity of E. However, the evidence just pre- sented for the specimens of set II makes it more likely that the same situation holds for the specimens of set I. The empirical values of T are obtained by substi- tuting the base value of E for each set of specimens, given in table 4, and the other appropriate param- eters for each specimen, all of which arc known from tables 1 and 2, in eq 7a or 7b. It is interesting to compare the values of E which result from a determination based on eq 7a and 7b, using T obtained directly from Pickett, with the base values of E used above. For this purpose, only those specimens of each set are used which have t he lowest values of Jc/l because for these, as was the case for longitudinal vibrations, it can be seen from the theoretical curves in figure 2 that an error in the choice of p would cause only a negligible error in T. Values of E for these specimens of low kjl are given below: Set I, average of specimens 1-19, 1-20, 1-21, and 1-22—2085 kilobars; Set II, average of specimens II 4r, and II-4r 2 — 2113 kilobars; Set II, average of specimens II 11 and T I— 1 2 — 2115 kilobars. These values are seen to be in excellent agreement with those based on longitudinal vibrations and given in table 4. The values based on longitudinal vibrations are used in establishing the empirical values of T because the equations on which they are based are established by long usage. The empirical values of T are given numerically in tables 1 and 2 and are plotted as a function of kjl in figure 2. The average value of T obtained from the edgewise flexural resonance frequency for specimens II— 2 through 11-12 of set II provide a single point which is designated in the figure. Figure 3 shows the same data in an expanded form. The values for T from Pickett, for p=0.29 given in figure 3, were obtained by a quadratic interpola- tion from Pickett's equations for p=Q, 1/6, and 1/3. 3 3 The actual equation used for this interpolation was: T= 1+79.02 (1+0.0752 m+0.8109m 2 ) (W 5 \l) 1+76.06 (1+0.14081 m+1.536 At 2 ) WO 2 ' 153 ^ I .OK T PICKETT (fj. -- 29) T PICKETT {fj. -- 1/6 ) .0 2 .04 .0 6 Figure 3. Expanded method of showing data in figure 2 illus- trating (a), separation of empirical correction factor for round and rectangular specimens, and (6), departure of all empirical points from theoretical {solid) curve for n = 0.%9. Square symbols represent rectangular specimens; round symbols represent cylindrical specimens; hollow symbols, set I; solid symbols, set II; symbols with crosses, special group (footnote 4). The computations involved in obtaining T for a given value of jjl from this equation are obviously more cumbersome than from the corresponding one given in the ASTM Book of Standards, pt. 3, p. 1355, 1955 (C215-55T). However, the equation given in the ASTM is inadequate necessitating the use of the equation given here. Inspection of figures 2 and 3 shows the empirical points to fall on two distinctly separate curves. The points determining these curves are not grouped on the basis of which set of specimens they are comprised but rather on the basis of whether the specimens are cylindrical or rectangular. All of the points forming the upper curve are derived from the rectangular specimens of sets I and II, while all of the points forming the lower curve are derived from the cylindrical specimens of both sets. Inasmuch as the empirical curves are developed without any assumption for the value of p, these data support the conclusion drawn previously from longitudinal vibrations; namely, that the speci- mens of both sets have the same value of p. However, the separation of the empirical points into two curves, one for cylindrical and one for rectangular specimens, is unexpected; for, according to Pickett, the value of T at any given k/l should depend only upon /x and not upon whether the bars are circular or rectangular in cross section. 4 Actu- ally, Pickett recognizes that an assumption is involved in the equality of Tfor specimens of circular and square cross section. Since the two empirical curves do diverge, especi- ally at higher values of k/l, it is relevant to inquhe which empirical curve is in better agreement with Pickett's theoretical relation. An estimate of the probable values of p for the two curves may be made on the basis of their relative positions from Pickett's curves for p=}i and p=%. Such an estimate leads to a value of p for the upper curve of around 0.26 to 0.30 while for the lower curve p appears to be around 0.17 to 0.19. Inasmuch as the p. value so estimated for the upper curve is in agreement with the literature as well as earlier parts of this investigation, while the similar estimate for the lower curve leads to an absurdly low value of p for steel, one concludes that the empirical curve for rectangular specimens is in better agreement with Pickett than the empirical curve for cylindrical specimens. It also appears that Pickett's curve for p = }i would give reasonably good values of T for cylindrical specimens having an actual value of ju~0.29. Inspection of figure 3 also shows that the empirical curve for rectangular specimens departs from the theoretical curve for p=% by an increasing amount as k/l increases. For the cylindrical specimens the curve appears to level off to a value slightly above Pickett's curve for p=%. 4. Summary 1. Empirical relations have been developed from which Young's modulus may be determined from the longitudinal and flexural resonance frequencies. Two sets of steel bars were used as specimens. Both sets were composed of cylindrical and rectangular specimens. These empirical relations have been compared with corresponding theoretical ones. The accuracy of the empirical determinations are such that numerical comparisons with the theory to four significant figures are justified. 2. For longitudinal vibrations, the empirically determined curve for the cylindrical specimens, agrees with the corresponding theoretical one (based on Bancroft's numerical solution of the Pochham- mer-Chree equations for this particular boundary condition) if a value of Poisson's ratio of 0.292 is assumed for both sets of specimens. This value is in agreement with that found in the literature for steel. 3. For flexural vibrations two separate empirical curves develop. One curve is formed by the rectan- gular specimens of both sets and a second curve is 4 This unexpected result was further tested using an entirely different group of six specimens all cut from the same soft steel. Special care was taken to have the specimens homogeneous and isotropic. The six specimens were divided into three pairs, each pair having the same length and the same value of k. One specimen of each pair was circular and the other square in cross section. If Pickett's assumption of the equality of the correction factor T for square and cylindrical bars of the same k/l is correct, then each of the above pairs should have the same flexural resonance frequency. However, the frequencies of each pair were found to differ from each other by an amount in agreement with the empirical results already obtained for sets I and II. Points representing these specimens are included in figure 3. 154 formed by the cylindrical specimens of both sets. The curve formed by the rectangular specimens is in fair agreement with the corresponding theoretical relation (based on Timoshenko, Goens, and Pickett) if a value of Poisson's ratio about 0.292 is again as- sumed. However, the empirical curve formed, by the cylindrical specimens would agree with the theoreti- cal otic only if a Poisson's ratio of about % is assumed for them. Since this value is obviously too low for steel, based on the literature and the present study, it is concluded that the experimental results agree with the theory for rectangular specimens but that Pickett's- equations give too high a value for the correction factor for cylindrical specimens. The authors express their appreciation for the invaluable help of J. B. Wachtman, both for in- formation of a general background nature and also for clarifying many particular problems which arose during the course of the investigation. Washington, D.C. (Paper (>4A2-37) 5. References |1| S. Spinner and R. C. Valore, Jr., Comparison of theo- retical and empirical relations between the shear mod- ulus and torsional resonance frequencies for bars of rectangular cross section, J. Research NBS 60, p. 459 (1958) RP2861. [2] H. Kolsky, Stress waves in solids, Preface (Oxford at the Clarendon Press, Oxford, England, 1953). [3] I). Bancroft, The velocity of longitudinal waves in cylin- drical bars, Phys. Rev". 59, p. 588 (1941). [1] G. E. Hudson, Dispersion of elast ic waves in solid circular cylinders, Phys. Rev. 63, p. 46 (1943). [oj R. M. Davies, Stress waves in solids, Surveys in Me- dia nies edited by G. K. Batchelor and 11. M. Davies, pp. 64-137. (Cambridge University Press, 1956.) [()] M. F. Markham, Measurement of elastic constants by the ultrasonic pulse method, Brit. J. Appl. Phys. Suppl. No. 6, pp. 556 to 563 (1957). [7] J. M. Frankland and H. L. Whittemore, Tests of cellular sheet steel flooring, J. Research NBS 9, 136 (1932) RP463. [8] E. Goens, Uber die Bestimmung des Elastizitatsmoduls von Staben mit Hilde von Biegung Schwingungen, Ann. der Phys., B. Folge, Band 11, pp. 649 to 678 (1931). [9] S. P. Timoshenko, On the transverse vibrations of bars of uniform cross section, Phil. Mag. Ser. 6, 43, pp. 125 to 131 (Jan. 1922). [10] G. Pickett, Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders, Proc. ASTM 45, pp. 846 to 865 (1945). 155