MODULUS OF ELASTICITY Of WOODDETERMINED 13Y DYNAMIC METHODS

April 1954

No.N 19/7

UNITED _STATES (DEPARTMENT OF AGRICULTURE(FOREST SERVICE

OREST —PRODUCTS LABORATORYMadison 5, Wisconsin

In Cooperation with the University of Wisconsin

MODULUS OF ELASTICITY OF WOOD DETERMINED BY DYNAMIC METH=

By

E. R. BELL, PhysicistE. C. PECK, Technologist

andN. T. KRUEGER, Physical Science Aid

Forest Products Laboratory,1 Forest ServiceU. S. Department of Agriculture

Suranar,

Tests at the Forest Products Laboratory prior to World War II showedthat the modulus of elasticity of wood could be measured by use ofsound waves. This report presents data from more recent studies madeto compare results obtained by sonic (dynamic) methods with those fromstatic tests. Samples were tested in compression parallel to grainand in static bending according to American Society for Testing MaterialsStandard D143-50; they were also given both longitudinal and transversesonic tests.

The effect of shear deformation in both the static and sonic transversetests reduced observed values of modulus of elasticity by about 10 per-cent. After a correction was made for this effect the modulus ofelasticity as determined by sonic methods was about 11 percent higherthan that obtained by static methods. The reason for the higher valuesis not wholly clear, but they may be caused by the difference in theamounts of deformation of the wood brought about by the two methods;the deformations with sonic methods are minute compared to those withstatic methods.

Introduction

The determination of modulus of elasticity by dynamic methods is notnew. The elastic properties of gases were determined many years agoby vibration methods that measured the velocity of sound in air and ingases. There are many methods of obtaining dynamic moduli. Theseinclude slow flexural and torsional vibrations, usually below the

1–Maintained at Madison, Wis., in cooperation with the University of

Wisconsin.

Report No. 1977 -1- Agriculture-Madison

frequency of sound, obtained by mounting the specimens as flexural ortorsional pendulums; the vibrating-reed type of test using small speci-mens mounted for electrical excitation; longitudinal and transversevibrations of bars and beams in resonance induced by electrical means;and the measurement of the velocity of sound transmission through thespecimen using short pulses of ultrasonic waves.

Ide„?.. working at Harvard University's Cruft Laboratory, determined thedynamic modulus for several solids, This work was done particularly inconnection with the travel of sound through terrestrial materials ingeological studies. The agreement between his values and those found inhandbooks and other accepted tables is very close.

In 1943 licBurney3 showed fairly close agreement between the modulus ofelasticity determined by sonic methods (1,616,000 pounds per square inch)and that determined by the standard static compressive test parallel tograin (1,679,000 pounds per square inch) from tests on a series of Sitkaspruce specimens. The specimens had an average specific gravity of 0.382and 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 pointsin such a test is difficult because of the uncertainty of the proportionof the deflection that results from shear.

4Hearmon- in 1948 reported studies using both the transverse and longitu-dinal methods on wood. His studies included slow flexural and torsionalvibration as well as electrically induced flexural and longitudinal vibra-tions in the sonic range. This work indicates dynamic values from 10 to 12percent above static values. He considers the fact that the dynamic testsare adiabatic and the static tests are isothermal. However, over a widerange of frequencies Hearmon reports that the theoretical difference be-tween the isothermal and the adiabatic conditions amounts to less than 0.1percent for the longitudinal modulus of elasticity. The difference is lessthan 1 percent for moduli of elasticity perpendicular to grain and is zerofor 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 bythe Sonic Method. U. S. Forest Products Laboratory. Advance ProgressReport on Forest 2roducts Laboratory Wood, Plywood, and Glue Researchand Development Program Prepared for the Ninth Conference of the ANCExecutive Technical Subcommittee on aequirements for 4bod Aircraft andWood 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 BritainDepartment of Science and Industrial Research, Forest Products ResearchLaboratory Special Report 7. London. 1948

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In 1948 Kuenzi2 used low-frequency vibration methods for determiningelastic moduli of thin sheets of veneer, plywood, and aluminum, usingthe torsion pendulum and the vibrating cantilever. He concluded thatthe modulus of rigidity may be determined fairly accurately by means ofthe torsion pendulum test, provided the length to width ratio is greaterthan 10 and the specimen is thin, He also concluded that the modulus ofelasticity of plywood beams can be determined by using vibrating canti-levers without end weights.

In 1949 Lesliel in reporting on the pulse vibration technique usingultrasonic waves for the testing of concrete, mentioned preliminaryresults on tests of sound and rotted wood,

7 8 9Horio and others,–'–'– (1951) reported studies made with forced vibrationof reeds as a method of determining viscoelastic properties of severalhigh polymers and of pulp and paper in the audio-frequency range*

10McSwain and Kitazaw&-- (1951) report results of tests on 25 specimensof assorted hardwoods and softwoods with average values 8 percent higherfor dynamic than for static bending moduli on 2- x 2- x 30-inch testbeams, Their results show also that the longitudinal resonant modulusexceeds the transverse bending modulus by an average of 16 percent, thetransverse modulus being uncorrected for deflection due to shear*

5–KUenzi / E. W. Low Frequency Vibration Methods for Determining Elastic

Proj. 1,246-1A,72, Unpublished S, Forest Products LaboratoryReport, April 29, 1948.

-Leslie on Pulse Techniques Applied to Dynamic Testing, American Societyfor Testing Me.terials Special Tech, Pub. No, 101, June 1949.

7Horio/ PL / Onogi, S., Nakayama/ C., and K. Yamamoto, Viscoelastic

Properties of Several High Polymers, Journal of Applied Physics22(7):966-970, July 1951.

8Norio, M., and Onogi, S. Dynamic Measurements of Physical Properties

of Pulp and Paper by Audiofrequency Sound. Journal of AppliedPhysics 22(7):971-977, July 1951.

2Horio, 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. TecoProj. No, B-33, Aug. 1951.

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In most cases reported and in the results presented in this paper,Youngs moduli determined by dynamic methods exceed those determinedstatically by approximately "8 to 15 percent. No completely satisfactoryexplanation of this difference has been made, but it has been found toexist over a very wide range of frequencies. Ide has suggested thatthe "dynamic modulus is obtained for minute alternating stresses, farbelow 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 oflongitudinal and transverse moduli of elasticity-by resonant vibrationin the audio-frequency range on 40 specimens of second-growth Douglas-firthat had been prepared for standard strength tests, and on a few speci-mens of other species. The specimens were caused to vibrate eithertransversely or longitudinally, and the resonant or fundamental frequen-cies were determined. These data were obtained in connection with otherwork and are submitted as further evidence of the possible usefulness ofthis type of test procedure. They indicate the accuracy that can beexpected when compared to the conventional static test methods,

Equipment

The initial Work on transverse vibration was done with apparatusarranged as shown in figure 1. The signals produced by the audio-frequency oscillator were delivered to the . audio7frequency amplifier,where they were amplified,and delivered to the recording_ at amaximum level of 15 Watts, The recording head moved in a lateral direc-tion and imparted to the specimen a lateral Vibration. The recordinghead was placed near one end of the specimen, which was mounted on twopieces of sponge rubber placed approximately at the ouarter points ofthe specimen. A variable-reluctance cartridge commonly used for thepickup in a record player was used as a vibration detector. The outputof the cartridge was fed to a one-tube amplifier, and then to anoscilloscope and a vacuum-tube voltmeter in parallel. The amplitude ofoscillation of the specimen, as indicated by the variable-reluctance-cartridge pickUp, was observed on both the oscilloscope and the vacuum-tubevoltMeter, As the audio-frequency oscillator was varied through aconsiderable range of frequencies, the amplitude observed on the oscil-loscope , and,voltmeter reached its maximum when the oscillator frequencybecame equal to the natural frequency of vibration of the specimen.This is called its fundamental frequency of vibration. Most specimensshowed a pronounced fundamental-frequency response. There were some,however, in which it was rather difficult to select the true fundamentalfrequency.

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Apparatus used in the study of longitudinal wave motion of wood speci-mens differed from the transverse set-up only in the means of excitingthe vibrations. Figure 2 shows how the output of the audio amplifierwas fed through a condenser to the aluminum plate that acted as thehigh-voltage electrode of the exciting condenser, the other side of whichwas held at ground potential. A direct-current voltage was also fed tothe aluminum plate through a 2-megohm resistor. In this manner boththe variable audio frequency and the direct-current voltage were appliedto the exciting condenser. The wood specimens were stood on end on thehigh voltage electrode. The exciting-condenser arrangement was taken

efrom drawings in Ids articlee-11- Me forces established between thetwo plates of the condenser when the voltage was applied were sufficientto set the specimen into vibration in a longitudinal direction. Thepickup was placed against the side of the specimen near its top. Theoutput of the pickup was fed to a preamplifier, and the amplified wavewas observed on the oscilloscope screen and its amplitude on the vacuum-tube voltmeter. The dielectric used was glass cloth impregnated withresin.

Determination of Modulus of Elasticity by Transverse Vibration

The value of modulus of elasticity of a wood specimen when vibratingtransversely was calculated by use of the following equation:1g

E 0.002443 W f2 (1)TR

ab

where

rTH = modulus of elasticity, when vibrating transversely,

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.

either in a tangential or radial direction

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The effects of shear deformations and the rotation of the beam are notincluded in the formula,

Table 1 gives results on 40 specimens representative of approximately250 specimens of second-growth Douglas-fir tested by transverse vibrationin the tangential direction at a moisture content of 12 percent, andtheir 'Young's moduli determined by formula (1), These same specimenswere also statically tested as centrally loaded beams on a 28-inchspan, 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 aLamb's roller extensometer of 6-inch gage length to obtain a value of themodtlus. The moduli of elasticity resulting from the three types oftest -- the transverse vibration test, the static bending test, and thestatic compression test (computed by means of formula (1) for thevibration' test and by the usual engineering formulas for the statictests) -- are given in table 1. Neither the data from the static bendingtests nor that from the vibration tests were corrected for shear deforma-tions.

Values of modulus of elasticity at resonant frequencies were determinedalso for 6 specimens of various species including southern; yella y pine,Sitka spruce, and oak, all at a moisture content of 9 percent. Themodulus of elasticity was obtained also by statically testing these samespecimens as centrally loaded beams and using the usual engineeringformula. The span used in each case was 2 inches shorter than the lengthof the specimen. Table 2 gives the test data and the ratios between thedynamic and static moduli. Neither the dynamic values nor the staticvalues were corrected for shears

Determination of Modulus ofElasticity by Longitudinal Vibration

When the wood specimen was vibrating longitudinally, modulus of elasticitywas calculated by using the following equation;13

EL = 0.010352 w (2)

13lbid

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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

in an elastic medium, V =f P Velocity is equal to frequency times

wave length; the wave firth, in the case of longitudinal vibration, isequal to twice the length the specimen. Another equation was set upthat included Poisson's ratio. It was found that a correction forPoisson'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. Adifferential equation was also set up by Herman U. March of the ForestProducts Laboratory to include a frictional force within the wood speci-men. The frictional force was assumed to be proportional to the velocityof motion of a particle within the specimen. It was found that thecorrection for internal friction was negligible,

Thirteen of the specimens of second-growth Douglas-fir that were used toobtain the data given in table 1 were vibrated in the longitudinaldirection, and their resonant frequency for longitudinal vibration wasdetermined, The modulus of elasticity, calculated by the use of formula(2), was determined for each specimen by means of the resonant frequencyin longitudinal vibration and by two static tests for compressionparallel to the grain. One of these static tests was made on the samespecimens that were vibrated longitudinally. These specimens were 30

inches long and the strains were measured by means of a Lamb's rollerextensometer having a '6-inch gage length mounted at the center of thespecimen. The other static test was made on end-matched 8-inch specimensas previously described. The moisture content of the specimens was 12percent at the time of both static . and dynamic tests. The results aregiven in table 3.

Discussion

In a beam bent transversely, deflections result from shear deformationswithin 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 inproducing deflection. The value of modulus of elasticity as calculatedfrom static bending tests is, therefore, not , the true modulus.

The proportion of the deflection of a. beam that is due to shear isdependent upon the relation between modulus of rigidity and modulus ofelasticity and upon the ratio of span to beam depth, For a span-depthratio of 14, such as was used in the static bending tests reported here,the correction to values of modulus of elasticity for shear daformationsare about 10 to 11 percent. The average difference of moduli ofelasticity from static tests in compression and bending of 11.5 percentas shown in table 1 is in general conformity to this relationship. Aswas indicated earlier, the formula for computing modulus of elasticityfrom dynamic transverse tests neglects the effects of shear deformations.The computed sonic modulus values for transverse specimens, like thosefor static bending, are accordingly in error by about 10 percent,

A number of references quoted it the introduction report values of dynamicmodulus somewhat-higher than the static modulus, the difference usuallybeing on the order of 10 percent. This is in agreement with the averagedifference of 10 percent shown in table 1,

The average:*auep 'Of:dynamicmodulUs from bending tests and of staticmodulus from.compression tests as-sitomi in the tables are approximatelythe same, numerically, This ' aPparentsimilarity of -values should notbe taken to indicate that =duli determined by the two methods are thesate, Shear deformations have not been - taken into account in computingthe tabulated values of dynamic modulus. True values of modulus, computedby taking into account shear deformations in the transverse vibration testswill be about 10 percent higher than those tabulated, and will no longerbe similar to the static Modulus from the static compression test,

The trend quoted above for Douglas-fir is confirmed by limited testsof three additional species (table 2). The data indicate a smalldifference between ratios of dynamic to static modulus depending uponwhether the deflection was in the radial or the tangential direction.It is doubtful that this difference is significant, It is based on only6 tests and results from the difference, in the 2 directions, of thestatic data; the dynamic data are essentially the same in the 2 directions.

The Sitka spruce specimen tested shows dynamic moduli less than staticmoduli. The other data, however, show the dynamic moduli to averageabout 10 percent higher than the static moduli.

Vibration in the longitudinal direction , gave results (table 3) similarto those resulting from Vibration in the transverse direction. Based onthe static tests on the 30-inch specimen, the dynamic moduli averageabout 9 percent higher than the static moduli, One specimen showed thereverse of the general trend, If its values be excluded, the dynamicmodulus is about 10 percent higher than the static modulus.

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Table 4 permits direct comparison of specimens that have been testedboth transversely and longitudinally, The similarity of individualratios of dynamic to static moduli in both bending and compression,with a few exceptions, is evident. On the average, the two ratios arevery nearly the same,

Method of Inducing_Vibrations

The excitation of vibrations by means of the vibrating electrode of thecondenser offers greater possibilities than does exciting by a recorderhead because of the larger energy output of the condenser electrode.Although, in the experiments reported here ; the electrode was used onlyto excite longitudinal vibrations, it can be used also to excite trans-verse vibrations. The large energy output obtainable from a condenserexciter suggests the possibility of testing relatively large timbers.Although the data presented were determined on specimens not over 2 by 2by 30 inches in size, pieces up to 6 by 6 inches by 6 feet have beentested in other experiments. The limiting dimensions of timbers thatmight 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 3respectively. Both figures indicate a general trend with specificgravity. In both, however, two points deviate considerably from thetrend. In each figure, the two points represent specimens 41-N6-aand 41-W6-a. Thus, both the transverse and the longitudinal vibrationtest methods have picked out the two specimens in which the modulusvalues were especially low for their specific gravity. Thus, forapplications where modulus of elasticity is important, both methodsoffer an opportunity, when used in connection with specific gravitydeterminations, of excluding low-line pieces.

Conclusions

Based on a limited number of static and dynamic tests, the followingconclusions may be drawn:

(1) Young's modulus for wood may be determined from tests in whichthe specimens are caused to vibrate, either longitudinally or transversely.The data required in the evaluation are the resonant frequency of vibrationand the weight and dimensions of the specimen.

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(2) The value of Young's modulus determined from vibration testswill be about 10 percent higher than that determined from a comparablestatic test.

(3)The value of Young's modulus determined fronLallongitudinalvibration test will be somewhat higher than the apparent value determinedfrom a transverse vibration test, but when the effect of shear deformationin the transverse test is taken into account, the values are of the sameorder.

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