MODULUS OF ELASTICITY Of WOOD
DETERMINED 13Y DYNAMIC METHODS
April 1954
No.N 19/7
UNITED _STATES (DEPARTMENT OF AGRICULTURE
(FOREST SERVICE
OREST —PRODUCTS LABORATORY
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
MODULUS OF ELASTICITY OF WOOD DETERMINED BY DYNAMIC METH=
By
E. R. BELL, Physicist
E. C. PECK, Technologist
and
N. T. KRUEGER, Physical Science Aid
Forest Products Laboratory,1 Forest Service
U. S. Department of Agriculture
Suranar,
Tests at the Forest Products Laboratory prior to World War II showed
that the modulus of elasticity of wood could be measured by use of
sound waves. This report presents data from more recent studies made
to compare results obtained by sonic (dynamic) methods with those from
static tests. Samples were tested in compression parallel to grain
and in static bending according to American Society for Testing Materials
Standard D143-50; they were also given both longitudinal and transverse
sonic tests.
The effect of shear deformation in both the static and sonic transverse
tests reduced observed values of modulus of elasticity by about 10 percent. After a correction was made for this effect the modulus of
elasticity as determined by sonic methods was about 11 percent higher
than that obtained by static methods. The reason for the higher values
is not wholly clear, but they may be caused by the difference in the
amounts of deformation of the wood brought about by the two methods;
the deformations with sonic methods are minute compared to those with
static methods.
Introduction
The determination of modulus of elasticity by dynamic methods is not
new. The elastic properties of gases were determined many years ago
by vibration methods that measured the velocity of sound in air and in
gases. There are many methods of obtaining dynamic moduli. These
include slow flexural and torsional vibrations, usually below the
1
–Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
Report No. 1977
Agriculture-Madison
-1-
frequency of sound, obtained by mounting the specimens as flexural or
torsional pendulums; the vibrating-reed type of test using small specimens mounted for electrical excitation; longitudinal and transverse
vibrations of bars and beams in resonance induced by electrical means;
and the measurement of the velocity of sound transmission through the
specimen using short pulses of ultrasonic waves.
Ide„?.. working at Harvard University's Cruft Laboratory, determined the
dynamic modulus for several solids, This work was done particularly in
connection with the travel of sound through terrestrial materials in
geological studies. The agreement between his values and those found in
handbooks and other accepted tables is very close.
In 1943 licBurney3 showed fairly close agreement between the modulus of
elasticity determined by sonic methods (1,616,000 pounds per square inch)
and that determined by the standard static compressive test parallel to
grain (1,679,000 pounds per square inch) from tests on a series of Sitka
spruce specimens. The specimens had an average specific gravity of 0.382
and were tested at an average moisture content of 12.5 percent.
In the sonic tests, the specimen was placed in a horizontal position and
"supported at its two nodal points, 0.227 of the length from each end,
by means of knife edges." The accurate location of the true nodal points
in such a test is difficult because of the uncertainty of the proportion
of the deflection that results from shear.
4
Hearmon- in 1948 reported studies using both the transverse and longitudinal methods on wood. His studies included slow flexural and torsional
vibration as well as electrically induced flexural and longitudinal vibrations in the sonic range. This work indicates dynamic values from 10 to 12
percent above static values. He considers the fact that the dynamic tests
are adiabatic and the static tests are isothermal. However, over a wide
range of frequencies Hearmon reports that the theoretical difference between the isothermal and the adiabatic conditions amounts to less than 0.1
percent for the longitudinal modulus of elasticity. The difference is less
than 1 percent for moduli of elasticity perpendicular to grain and is zero
for
modulus
of
rigidity.
2
-Ide, John M. Some Dynamic t:sthods for Determination of Young's Modulus.
Review of Scientific Instruments 6(10):296-298. Oct. 1935.
2McBurney, R. S. Evaluation of Modulus of Elasticity of Sitka Spruce by
the Sonic Method. U. S. Forest Products Laboratory. Advance Progress
Report on Forest 2roducts Laboratory Wood, Plywood, and Glue Research
and Development Program Prepared for the Ninth Conference of the ANC
Executive Technical Subcommittee on aequirements for 4bod Aircraft and
Wood Structures in Aircraft. Madison, Wisconsin June 14- 18, 1943.
Madison, U. S. Forest Service, FPL, June 1, 1943. pp 32-34.
Ilearmon, R. F, S. Elasticity of Wood and Plywood. Great Britain
Department of Science and Industrial Research, Forest Products Research
Laboratory Special Report 7. London. 1948
Report No. 1977
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In 1948 Kuenzi2 used low-frequency vibration methods for determining
elastic moduli of thin sheets of veneer, plywood, and aluminum, using
the torsion pendulum and the vibrating cantilever. He concluded that
the modulus of rigidity may be determined fairly accurately by means of
the torsion pendulum test, provided the length to width ratio is greater
than 10 and the specimen is thin, He also concluded that the modulus of
elasticity of plywood beams can be determined by using vibrating cantilevers without end weights.
In 1949 Lesliel in reporting on the pulse vibration technique using
ultrasonic waves for the testing of concrete, mentioned preliminary
results on tests of sound and rotted wood,
7 8 9 (1951) reported studies made with forced vibration
Horio and others,–'–'–
of reeds as a method of determining viscoelastic properties of several
high polymers and of pulp and paper in the audio-frequency range*
10
McSwain and Kitazaw&-- (1951) report results of tests on 25 specimens
of assorted hardwoods and softwoods with average values 8 percent higher
for dynamic than for static bending moduli on 2- x 2- x 30-inch test
beams, Their results show also that the longitudinal resonant modulus
exceeds the transverse bending modulus by an average of 16 percent, the
transverse modulus being uncorrected for deflection due to shear*
5
–KUenzi / E. W. Low Frequency Vibration Methods for Determining Elastic
S, Forest Products Laboratory
Proj. 1,246-1A,72, Unpublished
Report, April 29, 1948.
-Leslie on Pulse Techniques Applied to Dynamic Testing, American Society
for Testing Me.terials Special Tech, Pub. No, 101, June 1949.
7
Horio/ PL / Onogi, S., Nakayama / C., and K. Yamamoto, Viscoelastic
Properties of Several High Polymers, Journal of Applied Physics
22(7):966-970, July 1951.
8
Norio, M., and Onogi, S. Dynamic Measurements of Physical Properties
of Pulp and Paper by Audiofrequency Sound. Journal of Applied
Physics 22(7):971-977, July 1951.
2
Horio, PL 1 and Onogi, S. Forced Vibration of Reed as a Pbthod of
Determining Viscoelasticity. Journal of Applied Physics 22(7):
977-981, July 1951.
10
--McSyain, George, and Kitazawa, George. Application of Ultrasonic
and Sonic Vibrations for Improvement and Testing of Wood. Teco
Proj. No, B-33, Aug. 1951.
Report No, 1977
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In most cases reported and in the results presented in this paper,
Youngs moduli determined by dynamic methods exceed those determined
statically by approximately "8 to 15 percent. No completely satisfactory
explanation of this difference has been made, but it has been found to
exist over a very wide range of frequencies. Ide has suggested that
the "dynamic modulus is obtained for minute alternating stresses, far
below the elastic limit, which do not give rise to complex ;creep'
effects or elastic hysteresis,"
Ss...ape of Study.
The experiments discussed in this report cover the determination of
longitudinal and transverse moduli of elasticity-by resonant vibration
in the audio-frequency range on 40 specimens of second-growth Douglas-fir
that had been prepared for standard strength tests, and on a few specimens of other species. The specimens were caused to vibrate either
transversely or longitudinally, and the resonant or fundamental frequencies were determined. These data were obtained in connection with other
work and are submitted as further evidence of the possible usefulness of
this type of test procedure. They indicate the accuracy that can be
expected when compared to the conventional static test methods,
Equipment
The initial Work on transverse vibration was done with apparatus
arranged as shown in figure 1. The signals produced by the audiofrequency oscillator were delivered to the . audio7 frequency amplifier,
at a
where they were amplified,and delivered to the recording_
maximum level of 15 Watts, The recording head moved in a lateral direction and imparted to the specimen a lateral Vibration. The recording
head was placed near one end of the specimen, which was mounted on two
pieces of sponge rubber placed approximately at the ouarter points of
the specimen. A variable-reluctance cartridge commonly used for the
pickup in a record player was used as a vibration detector. The output
of the cartridge was fed to a one-tube amplifier, and then to an
oscilloscope and a vacuum-tube voltmeter in parallel. The amplitude of
oscillation of the specimen, as indicated by the variable-reluctancecartridge pickUp, was observed on both the oscilloscope and the vacuumtubevoltMeter, As the audio-frequency oscillator was varied through a
considerable range of frequencies, the amplitude observed on the oscilloscope , and,voltmeter reached its maximum when the oscillator frequency
became equal to the natural frequency of vibration of the specimen.
This is called its fundamental frequency of vibration. Most specimens
showed a pronounced fundamental-frequency response. There were some,
however, in which it was rather difficult to select the true fundamental
frequency.
Report No. 1977
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Apparatus used in the study of longitudinal wave motion of wood specimens differed from the transverse set-up only in the means of exciting
the vibrations. Figure 2 shows how the output of the audio amplifier
was fed through a condenser to the aluminum plate that acted as the
high-voltage electrode of the exciting condenser, the other side of which
was held at ground potential. A direct-current voltage was also fed to
the aluminum plate through a 2-megohm resistor. In this manner both
the variable audio frequency and the direct-current voltage were applied
to the exciting condenser. The wood specimens were stood on end on the
high voltage electrode. The exciting-condenser arrangement was taken
11
e articlee- - Me forces established between the
from drawings in Ids
two plates of the condenser when the voltage was applied were sufficient
to set the specimen into vibration in a longitudinal direction. The
pickup was placed against the side of the specimen near its top. The
output of the pickup was fed to a preamplifier, and the amplified wave
was observed on the oscilloscope screen and its amplitude on the vacuumtube voltmeter. The dielectric used was glass cloth impregnated with
resin.
Determination of Modulus of Elasticity by Transverse Vibration
The value of modulus of elasticity of a wood specimen when vibrating
transversely was calculated by use of the following equation:1g
E
TR
f2
0.002443 W
(1)
ab
where
rTH =
modulus of elasticity, when vibrating transversely,
either in a tangential or radial direction
W = weight of specimen in pounds
R
length in inches
a
width in inches
b
= depth in inches (in the plane of vibration)
f = frequency in cycles per second
11
--Ide, Review of Scientific Instruments.
12
--Den Hartog, Jacob Peter, Mechanical Vibrations. Ed. 3. 478 pp.,
illus. New York, 1947.
Report No. 1977
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The effects of shear deformations and the rotation of the beam are not
included in the formula,
Table 1 gives results on 40 specimens representative of approximately
250 specimens of second-growth Douglas-fir tested by transverse vibration
in the tangential direction at a moisture content of 12 percent, and
their 'Young's moduli determined by formula (1), These same specimens
were also statically tested as centrally loaded beams on a 28-inch
span, Further, when the specimens were originally cut from the timbers,
additional specimens 2 by 2 by 8 inches were cut and end-matched to them.
These additional specimens were tested in end compression by using a
Lamb's roller extensometer of 6-inch gage length to obtain a value of the
modtlus. The moduli of elasticity resulting from the three types of
test -- the transverse vibration test, the static bending test, and the
static compression test (computed by means of formula (1) for the
vibration' test and by the usual engineering formulas for the static
tests) -- are given in table 1. Neither the data from the static bending
tests nor that from the vibration tests were corrected for shear deformations.
Values of modulus of elasticity at resonant frequencies were determined
also for 6 specimens of various species including southern; yella y pine,
Sitka spruce, and oak, all at a moisture content of 9 percent. The
modulus of elasticity was obtained also by statically testing these same
specimens as centrally loaded beams and using the usual engineering
formula. The span used in each case was 2 inches shorter than the length
of the specimen. Table 2 gives the test data and the ratios between the
dynamic and static moduli. Neither the dynamic values nor the static
values were corrected for shears
Determination of Modulus of
Elasticity by Longitudinal Vibration
When the wood specimen was vibrating longitudinally, modulus of elasticity
was calculated by using the following equation;13
EL = 0.010352 w
13
lbid
Report No. 1977
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(2)
where
E = modulus of elasticity when vibrating longitudinally
W = weight of specimen in pounds
/1?
=
length in inches
A = area in cross section in square inches
f = frequency in cycles per second
Equation 2 comes directly from the expression for the velocity of sound
P
Velocity is equal to frequency times
=f
wave length; the wave firth, in the case of longitudinal vibration, is
eq ual to twice the length
the specimen. Another equation was set up
in an elastic medium, V
that included Poisson's ratio. It was found that a correction for
Poisson's ratio would increase the results, as determined by equation
(2), by about 1 percent. This correction factor was not used,,however,
because correct values for Poisson's ratio are difficult to obtain. A
differential equation was also set up by Herman U. March of the Forest
Products Laboratory to include a frictional force within the wood specimen. The frictional force was assumed to be proportional to the velocity
of motion of a particle within the specimen. It was found that the
correction for internal friction was negligible,
Thirteen of the specimens of second-growth Douglas-fir that were used to
obtain the data given in table 1 were vibrated in the longitudinal
direction, and their resonant frequency for longitudinal vibration was
determined, The modulus of elasticity, calculated by the use of formula
(2), was determined for each specimen by means of the resonant frequency
in longitudinal vibration and by two static tests for compression
parallel to the grain. One of these static tests was made on the same
specimens that were vibrated longitudinally. These specimens were 30
inches long and the strains were measured by means of a Lamb's roller
extensometer having a '6-inch gage length mounted at the center of the
specimen. The other static test was made on end-matched 8-inch specimens
as previously described. The moisture content of the specimens was 12
percent at the time of both static . and dynamic tests. The results are
given in table 3.
Discussion
In a beam bent transversely, deflections result from shear deformations
within the beam as well as from deformations due to longitudinal stress.
The ordinary engineering formulas relating deflection and modulus of
Report No, 1977
-7-
elasticity do not take into account the effects of shear strains in
producing deflection. The value of modulus of elasticity as calculated
from static bending tests is, therefore, not , the true modulus.
The proportion of the deflection of a. beam that is due to shear is
dependent upon the relation between modulus of rigidity and modulus of
elasticity and upon the ratio of span to beam depth, For a span-depth
ratio of 14, such as was used in the static bending tests reported here,
the correction to values of modulus of elasticity for shear daformations
are about 10 to 11 percent. The average difference of moduli of
elasticity from static tests in compression and bending of 11.5 percent
as shown in table 1 is in general conformity to this relationship. As
was indicated earlier, the formula for computing modulus of elasticity
from dynamic transverse tests neglects the effects of shear deformations.
The computed sonic modulus values for transverse specimens, like those
for static bending, are accordingly in error by about 10 percent,
A number of references quoted it the introduction report values of dynamic
modulus somewhat-higher than the static modulus, the difference usually
being on the order of 10 percent. This is in agreement with the average
difference of 10 percent shown in table 1,
The average:*auep 'Of:dynamicmodulUs from bending tests and of static
modulus from.comp ression tests as-sitomi in the tables are approximately
the same, numerically, This ' aPparentsimilarity of - values should not
be taken to indicate that =duli determined by the two methods are the
sate, Shear deformations have not been - taken into account in computing
the tabulated values of dynamic modulus. True values of modulus, computed
by taking into account shear deformations in the transverse vibration tests
will be about 10 percent higher than those tabulated, and will no longer
be similar to the static Modulus from the static compression test,
The trend quoted above for Douglas-fir is confirmed by limited tests
of three additional species (table 2). The data indicate a small
difference between ratios of dynamic to static modulus depending upon
whether the deflection was in the radial or the tangential direction.
It is doubtful that this difference is significant, It is based on only
6 tests and results from the difference, in the 2 directions, of the
static data; the dynamic data are essentially the same in the 2 directions.
The Sitka spruce specimen tested shows dynamic moduli less than static
moduli. The other data, however, show the dynamic moduli to average
about 10 percent higher than the static moduli.
Vibration in the longitudinal direction , gave results (table 3) similar
to those resulting from Vibration in the transverse direction. Based on
the static tests on the 30-inch specimen, the dynamic moduli average
about 9 percent higher than the static moduli, One specimen showed the
reverse of the general trend, If its values be excluded, the dynamic
modulus is about 10 percent higher than the static modulus.
Report No. 1977
-8-
Table 4 permits direct comparison of specimens that have been tested
both transversely and longitudinally, The similarity of individual
ratios of dynamic to static moduli in both bending and compression,
with a few exceptions, is evident. On the average, the two ratios are
very nearly the same,
Method of Inducing_Vibrations
The excitation of vibrations by means of the vibrating electrode of the
condenser offers greater possibilities than does exciting by a recorder
head because of the larger energy output of the condenser electrode.
Although, in the experiments reported here ; the electrode was used only
to excite longitudinal vibrations, it can be used also to excite transverse vibrations. The large energy output obtainable from a condenser
exciter suggests the possibility of testing relatively large timbers.
Although the data presented were determined on specimens not over 2 by 2
by 30 inches in size, pieces up to 6 by 6 inches by 6 feet have been
tested in other experiments. The limiting dimensions of timbers that
might be tested by this method have not been determined.
Application of Dynamic Testing
Figures 3 and 4 present, in graphical form, data from tables 1 and 3
respectively. Both figures indicate a general trend with specific
gravity. In both, however, two points deviate considerably from the
trend. In each figure, the two points represent specimens 41-N6-a
and 41-W6-a. Thus, both the transverse and the longitudinal vibration
test methods have picked out the two specimens in which the modulus
values were especially low for their specific gravity. Thus, for
applications where modulus of elasticity is important, both methods
offer an opportunity, when used in connection with specific gravity
determinations, of excluding low-line pieces.
Conclusions
Based on a limited number of static and dynamic tests, the following
conclusions may be drawn:
(1) Young's modulus for wood may be determined from tests in which
the specimens are caused to vibrate, either longitudinally or transversely.
The data required in the evaluation are the resonant frequency of vibration
and the weight and dimensions of the specimen.
Report No. 1977
-9-
(2) The value of Young's modulus determined from vibration tests
will be about 10 percent higher than that determined from a comparable
static test.
(3)The value of Young's modulus determined fronLallongitudinal
vibration test will be somewhat higher than the apparent value determined
from a transverse vibration test, but when the effect of shear deformation
in the transverse test is taken into account, the values are of the same
order.
Report No. 1977
-10-
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