Skip to main content

Full text of "A comparison of experimental and theoretical relations between Young's modulus and the flexural and longitudinal resonance frequencies of uniform bars"

See other formats

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 

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 


be presented later will show that the assumption of 
uniform density and Young's modulus is quite 

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 & 






k/l ° 














50, 253 
49, 605 

30, 650 
28, 624 
26, 401 

0. 12565 
. 12398 
. 11850 
. 11251 

0. 9851 

2. 1234 

2. 0997 
1. 9998 


1. 9205 






7. 584 
10. 117 


7. 851 


34, 051 
33, 752 
32, 975 
25, 370 

16, 605 
16, 341 
15, 733 

. 08447 
. 08366 
. 08176 
. 06272 


1. 5424 
1. 5320 
1. 5077 
1. 3085 





10. 224 
14. 790 
14. 968 


25, 107 
17, 389 
17, 190 
17, 185 



. 06208 
. 04289 
. 04241 
. 04238 


1. 3024 
1. 1475 
1. 1447 
1. 1438 





15. 118 
15. 235 
19. 736 


17, 014 
16, 883 
13, 037 
12, 875 



. 04196 
. 03214 
. 03174 


1. 1410 
1. 1395 
1. 0852 
1. 0829 





22. 951 
25. 641 

29. 987 

30. 201 



10, 039 


1, 253 

. 02766 
. 02475 
. 02116 
. 02101 


1. 0664 
1. 0525 
1. 0360 
1. 0359 



30. 500 
30. 554 





. 02080 
. 02076 


1. 0353 
1. 0350 

I-12b .. 

14. 968 
19. 736 
25. 641 


26, 704 





. 10020 
. 07473 
. 03464 
. 02627 
. 02023 

1. 7649 

I-12c _ 

1. 4445 




17, 188 
13, 039 
10, 038 


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 


Table 2. 

Data for 

specimens of set II 

Rectangular specimens 

Specimen * 








/ (long) 










ll 2a 


1 5. 202 




15. 202 
15. 202 





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. L433 
3. 1433 

2. 0574 
0. 6426 


2. 5405 


0. 9530 

. 6363 

2. 5405 
2. 5405 

1). f 13(13 
. (13(13 

g/cm 3 

7. Ml 


7. ML' 

7. SI 5 

7. SI 7 


7. M 1 

0. 05981 


. 02413 

. 01508 
. 00002 


. 10463 
. 07290 
. 07589 



1 1 83 


1820. 6 
1 013. 1 

20, 432 
1 1. 983 


1. 1936 




1.01 so 




1.01 17 

0. 05981 
. 05969 
. 05969 

. 05969 
. 05969 
. 05969 
. 05969 

. 05969 

. 05969 
. 05969 

. 1 2945 
. 1 2020 
. 12510 

. 03907 







19, 203 
22, 934 

20, tor. 

21, 789 



1. 21)23 

1. 2926 
l. 2921 
1. 2941 

1. 2916 
1. 2912 
1. 2919 

1. 9849 
L'. 2232 

2. 0732 
2. 1535 

1. 1314 


17, 065 

17, 067 

17, 064 




ll 10a 



Cylindrical specimens 




kll ° 





II-4r 2 


ti. 117 
1 1.669 

1 1. r.cs 

2. 4400 





17. 102 


1 238 1 
1 . 5329 

IS, 057 
35, 329 
40, 145 


0. 9978 
. 9944 
. 9957 
. 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) 





0. 04202 
. 08405 
. 12607 
. 16809 
. 21012 

0. 9995 



. 04160 
. 08321 
. 12481 
. 16642 
. 20802 



. 04232 
. 08464 
. 16928 



. 06348 
. 12696 




. 10878 
. 21756 
. 32634 


a For set I specimens, »o=5152 m/sec; for set II specimens, tfo=5199 m/sec. 


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 



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, 


v=^- (2) 


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 


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 , 



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 




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. 


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, 


--v Q - p 


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 

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 


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, 

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 



kilobars. - 


Set I 

Set II 

Po, "rod velocity'' 

/>. densitj 

E, Young's modulus. 

O, shear modulus 

n, Poisson's ratio 



0. 269 * 



I derived from E and values and eq (6). From Bancroft, ^ = 0.292 for both 


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 



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 

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- 

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: 


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 


#=0.9464 \ l ~f 




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. 



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 


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 ' 


^ 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 

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. 


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) 

[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 

[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).