‘x— ‘w STATIC AW DYNAMIC MODULUS OF ELASTICIYY O“? HARDBOARD The“; {or flu Degree 9‘ M. S. MICHIGAN STATE UNIVERSE‘TY James Gibson Bair Jr. 1964 THESIS LIBRARY Michigan State University i_____qg ABSTRACT STATIC AND DYNAMIC MODULUS OF ELASTICITY OF HARDBOARD by James Gibson Bair Jr. The purpose of this study was to determine the relationship between static and dynamic modulus of elasticity of standard S-l-S hardboard and how these moduli vary with increasing moisture content. It was also decided to determine (1) if the effects of shear and rotatory inertia could be neglected in calculating the dynamic M. E. and (2) the behavior of the logarith- mic decrement with increasing moisture content. The dynamic and static modulus of elasticity and the logarithmic decrement were determined at each of five moisture content levels. The dynamic M. E. was determined from the fundamental frequency and the first five overtones for each moisture content level. It was concluded that the dynamic M. E., calcu- lated from any of the six modes of vibration is valid, that is, that the effects of shear and rotatory inertia are negligible and that the dynamic and static M. E. can be highly correlated. Due to creep, the static M. E. drops off at a more rapid rate than the dynamic M. E. James Gibson Bair Jr. with increasing moisture content. It was also observed that the logarithmic decrement increases with increasing moisture content as a result of greater plastic defor- mation in the hardboard at higher moisture contents. STATIC AND DYNAMIC MODULUS OF ELASTICITY OF HARDBOARD BY James Gibson Bair Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Forest Products 1964 ACKNOWLEDGMENTS The author wishes to express his sincere appreci- ation to Dr. Otto Suchsland, of the Department of Forest Products, for his guidance and counsel in the prepara- tion of this thesis. Appreciation is also extended to Dr. James H. Stapleton, of the Department of Statistics, for his guidance in the preparation of the statistical procedures used herein. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . LIST OF TABLES LIST OF FIGURES . . . LIST OF APPENDICES CHAPTER I. II. III. IV. VI. INTRODUCTION THEORY OF DYNAMIC TESTING. Transverse Vibration Boundary Conditions for a Free-Free Beam Logarithmic Decrement EXPERIMENTAL PROCEDURE . Design of Experiment Samples Conditioning Dynamic Modulus of Elasticity Logarithmic Decrement Static Modulus of Elasticity RESULTS Shear and Rotatory Inertia Effects Dynamic and Static Modulus of Elasticity Logarithmic Decrement DISCUSSION. Shear and Rotatory Inertia Effects Dynamic and Static Modulus of Elasticity Logarithmic Decrement Nomogram CONCLUSIONS . . APPENDICES . . . . . . . BIBLIOGRAPHY iii Page ii iv Vii 16 34 46 59 61 66 LIST OF TABLES TABLE Page 1. Summary of Design of Experiment . . . . . l7 2. Positions of Nodes . . . . . . . . . 25 3. Dynamic M.E. Uncorrected for Shear or Rotatory Inertia . . . . . . . . . 35 4. Results of Static and Dynamic Tests and Logarithmic Decrement at 5 M.C. Levels . . 40 5. Comparison of Dynamic and Static M.E. . . . 47 iv LIST OF FIGURES FIGURE Page 1. Sections of Bar During Transverse Vibra- tion 0 O O O I O O 0 O O O 0 O O 5 2. Plan Showing how Samples were cut From Commercial Hardboard Sheet. . . . . . . l9 3. Equipment for Dynamic Testing . . . . . 21 4. Block Diagrams for Dynamic and Static Tests . 23 5. Representative Photograph of the Decay of Free Vibrations of Cantilever Beam . . . . 27 6. Equipment for Static Testing . . . . . . 29 7. Representative Section of Chart Used with the Static Tests . . . . . . . . . . . 31 8. Dynamic M.E. from the Fundamental Frequency and first 5 Overtones, uncorrected for Shear and Rotatory Inertia . . . . . . . . . . 36 9. Regressions of Static and Dynamic M.E. on MOCO o o o o o o o o o o o o o o 38 10. Regression of Static M. E. on Dynamic M. E. with 95% Confidence Limits. . . . . . 41 ll. Logarithmic Decrement Plotted Against M.C. . 44 12. Dynamic M.E. Obtained from the Fundamental Frequency and First 4 Overtones, Uncorrected for Shear and Rotatory Inertia, for Spruce and Oak . . . . . . . . . . . . 48 13. Logarithmic Decrement for Spruce and Oak . 54 14. Nomogram with Example for Determining the Dynamic M. E. . . . . . . . . . 56 LIST OF APPENDICES APPENDIX Page 1. STATISTICAL PROCEDURES . . . . . . . . 62 Regression line and Correlation Coefficients 95% Confidence Limits 2.. NOMOGRAM. . . . . . . . . . . . . 64 vi CHAPTER I INTRODUCTION Hardboard is a wood based product, manufactured in sheet form from wood fibers. The properties of hard- board are functions of the raw material and process variables. Commercial standards have been adopted which specify maximum and minimum values for these properties, regardless of the raw material or process used. In the present commercial standards the modulus of rupture is used as an indicator of quality, whereas the modulus of elasticity* is not required to be deter- mined. This is due to two reasons (1) the modulus of rupture is determined by a simple test, whereas the determination ofNLEL is more difficult and relatively time consuming and (2) hardboard is not considered an engine- ering material. However, it is because of (1) that the bending strength of hardboard is specified in terms of the modulus of rupture. One important disadvantage of the modulus of rupture is that the material under consideration must be stressed * Hereafter the modulus of elasticity will be designated by M.E. or E. beyond the elastic limit to obtain this property, hence the result is not as accurate an indicator of the elastic properties as is the M.E. Defects or discontinuities in the material will have more effect on the modulus of rupture than they will on the M.E. Since the M.E. is a better indicator of the elastic properties of a material, it would be highly de- sirable to obtain this property by a rapid, simple test. Determination of the M.E. by vibrational methods could be such a test, and it is for this reason that this study was initiated. This vibrational method of testing or dynamic test- ing, as it is often called, has two important advantages. (1) The method can be easily carried out and the M.E. calculated easily under certain conditions and (2) this method of testing is non-destructive and this allows the same samples to be tested under different conditions, for example varying moisture content or temperature, and elim— inates "between groups" variation. The most important disadvantage is that the effects of shear and rotatory inertia may cause the calculated M.E. to be low. It is quite difficult to compensate for these effects if they are appreciable. These effects are due to too large a depth/length ratio and if this ratio is such that the effects of shear and rotatory inertia are small enough to be neglected, then the M.E. can be obtained easily. In general, the dynamic M.E. does not equal the static M.E. because in the short time periods involved in the dynamic test, time dependent properties such as creep and relaxation do not affect the M.E. as they do in the conventional static test. Because of this and the fact that the dynamic test does not measure M.E. directly, but rather the frequency of the vibrating beam, statistical and mathematical relationships are necessary to correlate the dynamic M.E. to the static M.E. Another property of interest is the logarithmic decrement which indicates the internal friction of the material. The greater the internal friction, the greater is the plastic deformation. Specifically, the purpose of this study was to (1) determine the effect of shear and rotatory inertia with increasing depth/length ratio and increasing moisture content (2) to establish the relationship between dynamic M.E. and static M.E. as a function of the moisture content and (3) to determine the logarithmic decrement as a func- tion of the moisture content. CHAPTER II THEORY OF DYNAMIC TESTING Transverse Vibration The classical differential equation describing the transverse bending of a uniform beam is developed in the following manner (see references 1, 2, 3, 4): In Figure (1A) the element is compressed by a length re, from the original length Sx as a result of the bending of the beam. Therefore (1) E: 176 ’ 3x wheregz strain in the element and since where F = force necessary to compress the element '3' = stress A = cross sectional area of the element and 6‘: EE (3) the force on the element may be written re 31' , The moment of this force about the center line is M=EA§ir (5) u Fig. l.--A. Section of bar deformed during transverse vibration to show compression of an element at distance r from the center line. (8* is the original length of the element. B. Section of bar showing moment M and force F acting on one end and the resultant force F+ df and Opposing moment M + dM. acting on the other end. The length of the elemental section is dx. and the total moment of all elements about the center line is ' Mggfrei. <6) The radius of gyration is defined by k2 = % r2A, where k radius of gyration S = total cross sec- tional area of the bar. For a rectangular cross section where t thickness of beam Therefore 2 M = EeSk (8) For small deflections 2 dx dx (9) 0]." e — ‘QEX a§'_ dx2 (10) By substituting (10) into (8) the following is obtained: 2 = - 2 d M F‘s“ 7.13%? (11) The bending moment is a function of x, when x is measured along the bar. The difference in the bending moments at the two ends of an element of the bar is equal to the shearing force times the length of the element (see Fig. 1B) or dM = Fdx (12) Therefore F = dM (l3) dx Taking the derivitive of (11) with respect to x and substituting in (13) we have: _ _ 2 div F - ESk dx3 (1a) Since F is also a function of x there is a resultant vertical force dF = dE 1 dx 8, ( 5) acting on the element. This force is equal to the elements acceleration times its mass. ab: = 86 deg. (16) dx 6* F *dt where P = density t = time. Taking the derivitive of (14) with respect to x, we have dF = -E3k2.aix Hi dxu (17) By substituting (17) into (16) we obtain 4 2 fled—ah a. =.sa._w 2 dt (18) which may be written Ek2 d41 + d2y = o (19) 9 ex” at? which is the classical differential equation desired. In order to solve equation (19) the following solution is assumed: ‘y = a cos nt, where y = deflection at time ‘- t and distance X from the origin a 2 maximum amplitude of vibration for any given x :3 ll angular velocity t = time Then 2 Q_1 = -n2 a cos nt (21) dt2 and by substituting (20) into (21) we obtain dzy = -n2y . (22) dt2 Since y in equation (20) is a function of x and t, (a) is assumed to be a function of x, as cos nt is clearly a function of t. Let d (a cos nt) = duy dx dx4 (23) Then by substituting (22) into (19) we obtain d4 = n2 dx Ex? y (2”) Define the quantity m m = Pn2 EK2 (25) so that Equation (26) in operator form is (Du- m4)y = 0 (27) with the characteristic equation DL'L = m4 (28) and the roots are D1 = m D2 = -m (29) D3 = im -im 10 Then y = clemx + c2e'mx + c3eimx + cue'imx(30) y = cl(cosh mx + sinh mx) + c2(cosh mx - sinh mx) + C3(cos mx + i sin mx) + 04(cos mx - i sin mx) (31) Expanding and collecting terms this reduces to y = A cosh mx + B sinh mx + C cos mx + D sin mx, (32) where A = c1 + 02 B = c1 - c2 C = C3 + on D = 103 ’ 10” This is the deflection of the bar as a function of x only and is equal to (a) of equation (20). Considering y as a function of x and t both y = [A cosh mx + B sinh mx + C cos mx + D sin mx] cos nt (33) Boundary Conditions for a free—free beam Let x = O at 1/2 1 d2 = O at x 1/2 1 (no curvature at ends ) g_% = O at x dx The solution (33) includes both symmetrical and assymmet- n I+ 1/2 1 (no shear forces at ends) u |+ rical vibrations of the bar. If only the symmetrical vi- brations are considered (33) becomes y = [A cosh mx + C cos mx] cos nt (34) 11 From the assumed boundary conditions four equations are obtained to give a value of m which satisfies tan 1/2 ml = tanh 1/2 ml (35) The roots of (35) are obtained graphically and x =.gl = (s-l/4)U'tfi (36) where s = 1,2,3. . . (modes of vibration) I3==small quantity only appreciable at s = 1. Then, from (25) m=2_x=_2_i_s_1&lfl AVE; (37) l Ek and finally the frequency of vibration is fr.=.n__ 11(48-112 .E-JE_ 21! 8 l2 p (38) A similar method may be employed for the asymmetric fre- quencies and these are found to be proportional to (43 + l). Let A - 1 2n 1/2 2 +1 11 2 ( s 8 ) = [ ( gfl ) 1 (39) where p = s and define mp = 1/2 (2p + 1)fl (40) so that (38) becomes k 2 E fr=fiie ; (“1) 12 As mentioned on the previous page, these equations do not take into account the effects of the shear force or rotatory inertia which act on the bar. As the depth/length ratio of the beam increases, these effects become more pronounced. This ratio increases with higher overtones causing the observed fre- quencies to be too low which in turn cause the calcu- lated M.E. to be too low. The differential equation which takes into account these two factors is (4) Ek2 d4 + d2 d4 k2 a” = o, 76% (17%: -k(1 +35) 31%th +£—'Se a??? (42) where G shear modulus s = shear deflection coefficient to allow for the fact that the shear stress is not uniform over the cross section. Although there have been several solutions to this equation the following develops the relationship into a rather simple transcendental equation which, though it is approx- imate, gives the effect of shear and rotatory inertia (13). The theoretical frequency fg is defined as fg = fr 71" (113) where fr is from equation (41) T = the expression below T = l + K-E2[m2F2(m) + 6mF (m)] + k23E [m2F2(m)- 2mF(m)] l 12G' - 4nepsk2fg2 (44) G 13 Equation (42) may be simplified by the following expressions: m2F2(m) + 6mF(m) Flm (45) m2F2(m) - 2mF(m) F2111 (46) Equation (44) then becomes T = 1 + k2 F m + k2sE F, m - 4n293k2fg2 "I? l( ) 126 2( ) G (47) Combining (41) and (43) the following is obtained: f = k2muE g 4112139 T (48) Substituting (48) into (47) _ k2 sE _muk2 - (49) T _ 1 +.T2(Flm + 5_'F2 m[l FéIm) lZT ]) The last term in (49) is an approximation, but its numer- ical value is small in comparison with 1 and T is very close to 1. Hence a good approximation of T can be ob- tained by setting the right hand side of the equation (47) equal to l. The T obtained can then be inserted in- to the right hand side of (47) to improve the first estimate. This method may be repeated to obtain an even better estimate of T. Logarithmic Decrement This is defined as the logarithm of the ratio of the amplitudes of two consecutive cycles in the free vi- brating system of a cantilever beam or as the logarithm of the ratio of the amplitudes of two cycles, n oscil- lations appart and divided by n or 14 1mg X ézlnflzl 1'] X2 1’) where 6 = logarithmic decre- ment x1: amplitude of first cycle x2: amplitude of second cycle x = amplitude of nth cycle The logarithmic decrement is a measure of the loss of energy in a vibrating beam. With each cycle of the vi- bration a certain amount of the energy of the beam is lost as heat energy due to internal friction. Consider a purely elastic cantilevered bar which has been given a certain amount of energy to set it in motion. As the bar passes through the rest position, all the energy is of the kinetic form, but is continually converted to po- tential energy until it is all of this type and the bar will stop in a position of maximum deflection. It will immediately begin to move in the opposite direction and convert all the potential energy to kinetic energy by the time it passes through the rest position again. The bar will go to a position of maximum deflection in the other direction, then reverse and again pass through the rest position. This procedure will continue indefinitely. However, if the material is not purely elastic, but exhibits some plastic deformation also, part of the energy 15 is lost as heat energy in causing the plastic deformation, hence there is less energy after each cycle and the amplitude of the free vibration falls off with time or is "damped." CHAPTER III EXPERIMENTAL PROCEDURE Design of Experiment The experiment was designed so that one group* of specimens would be tested at each of the five moisture content levels for the dynamic M.E., then logarithmic decrement, and finally for the static M.E. This helped minimize variation between the two test methods, for the values determined here were used in the comparison of the two methods. The values obtained here were also used to determine the effect of moisture content on these properties. Another group was tested only by the dynamic method at each moisture content level to minimize variation from one level to the next. The results of the dynamic M.E. of both groups at each moisture content level were combined to determine the effect of shear and rotatory inertia at the overtones. Table 1 summarizes the design. Samples The samples used in this experiment were cut from one 4' x 8' standard commercial screenback hardboard as *Each group represents ten samples 16 17 TABLE 1 Summary of Design of Experiment Dynamic and Static M.E. Dynamic Dynamic Moisture Content Logarithmic M.E. M.E. from Level Nominal TEActual Decrement Only Overtones l O .68 Group 1 Group 13 Groups 1 & 13 2 4 3.52 Group 2 Group 13 Groups 2 & l3 3 8 7.34 Group 3 Group 13 Groups 3 & 13 4 12 11.80 Group 4 Group 13 Groups 4 & l3 5 20 20.44 Group 5 Group 13 Groups 5 & l3 indicated in Fig. 2. The samples were grouped at random into fifteen groups. Groups 1 to 5 and 13 were used as indicated in Table l, the remaining groups are intended for future experiments. Conditioning For moisture content level 1 all samples were placed over phosphorous pentoxide, P205, in two large covered aquariums. Each of the aquariums contained a fan for air 18 circulation and a light bulb to maintain the temperature somewhat above room temperature to speed attainment of equilibrium. The temperature in the tanks was approximate- ly 8OOF. For moisture content levels 2 to 5, the samples were placed in a controlled relative humidity cabinet which was properly adjusted for each moisture content level as the testing proceded. In the conditioning cabinet the temperature was maintained at approximately lOOOF. Dynamic Modulus of Elasticity After reaching equilibrium, a small piece of steel shim stock .5" x .5” .002” was bonded to each end of the sample on the smooth side, using Duco Household Cement. In the determination of equation (41) the beam was considered to be free-free or unsupported. Since supports are necessary in order to test the beam they must be placed in such a position so as to cause a minimum of damp- ing. The nodes, or positions of zero amplitude, are chosen as the points of support, for at these points only, will the damping due to the supports be a minimum. The distances from the end of the sample to the nodes are listed in Table 2. In all cases the supports were placed at the nodes nearest the ends of the specimen. The supports were fine taut wires. The test set up is shown in Fig. 3 and in Fig. 4 a block diagram is given. Following the diagram, the test 19 Fig. 2.--Plan of how samples were cut from 4' x 8' sheet of hardboard. Each sample was 1" x 20”. 2O 21 Fig. 3.--Equipment used for dynamic testing, showing sample in place on wire supports, positioned for ob- taining the fundamental frequency. 22 II/ / - .- 5—5 .03 (or: L .1... Ir... D -f .1 23 Fig. 4.--Block diagrams of instrumentation for static and dynamic tests. 21L Emu... o_._.<._.m ¢m_m_4n_2< IJJNU Odo..— cuEZQIC I ho>.._ — Hmwh o:2o 5:3“.qu oco E — 05 r _ «32:8 4 , w , m Juzzuzuaou¢m . — .55... # Ila zmzauaml mica 5&3 3.2.514 «0:388 ‘ - , ‘ union. .3 , Egon. , - ‘ «momooum Juzzfia o»: 25 TABLE 2 Positions of Nodes Distance of nodes from one end in Mode of Vibration terms of 1 (length of beam) Fundamental frequency .2242 .7758 lst overtone .1321 .5000 .3679 2nd overtone .0944 .3558 .6442 .4056 3rd overtone .0734 .2770 .5000 .7230 .9266 operation is as follows: The desired frequency is gener- ated by the oscillator, amplified and fed into the drive coil, which is directly under the steel shim stock on the sample causing the sample to vibrate at the given frequency. The period of the frequency is monitored by the frequency counter and may be visually observed on channel A of the oscilloscope. The pickup or receiving coil is directly under the shim stock at the other end of the beam. This picks up the oscillations of the beam and sends the signal to channel B of the oscilloscope. At resonant frequency and at each overtone the vibrations reach a maximum which is easily determined by visual means on the oscilloscope. The period was recorded at each important node of vi- bration. 26 Logarithmic decrement This property was obtained from the free vibration of the sample clamped as a cantilever beam. The sanple was set into oscillation by a light tap. The oscilla- tions were picked up by the receiving coil and traced on the oscilloscope where they were photographed with a polaroid camera mounted on the oscilloscope. (Not shown in Fig. 3) The right side of the dynamic test set up in Fig. 4 was used to obtain this property. Fig. 5 is an example of the trace obtained from which the logarith- mic decrement was obtained. Static Modulus of Elasticity The static bending tests were carried out accord— ing to A.S.T.M. standard. The span used was 11 0“", the same distance as between the nodes for the fundamental frequency. This was done so that any defects in the sample would have the same effect on either the dynamic or static test. The test set up is shown in Fig. 6 and in Fig. 4 the block diagram is given. The Linear Variable Differential Transformer (LVDT) measures the deflection which is recorded on channel I of the recorder and the load cell measures the load applied, which is recorded on channel 2 of the recorder. From the two traces obtained on the dual channel recorder (see Fig. 7 for example of chart) a load versus deflection curve was plotted to obtain the slope for the formula: 27 Fig. 5.--A representative photograph of the decay of free vibrations of the cantilevered beam used to ob- tain the logarithmic decrement. 29 Fig. 6.--Equipment used for static testing, with sample in place in the compression cage of the testing machine. 30 31 Fig. 7.--A representative section of the chart ob- tained from the dual channel recorder used for the static tests. The trace starting at the left repre- sents the deflection and the trace starting at the right the represents the load. 32 “M lHVHD 'V'S'fl 'SVXBL 'NOLSOOH 'OELVUOdUODNI SLNEWflHENI SVXBL ' 33 where P1 load at any point within prOportion- al limit deflection at Pl Span width of sample thickness of sample CHAPTER IV RESULTS Shear and Rotatory Inertia Effects Table 3 tabulates the dynamic M.E. from the funda- mental frequency and first five overtones for each moisture content level. These values are uncorrected for shear and rotatory inertia, and are shown graphically in Fig. 8. Each of the moisture content levels is represented with a straight, horizontal line through the overall mean for each level. In all cases the dynamic M.E. from the funda— mental frequency is lower than the overall mean for each level. Each point represents the average of 20 tests. Dynamic and Static Modulus of Elasticity The results of the comparison are tabulated in Table 4 and presented graphically in Fig. 9. This figure presents the regressions of dynamic and static M.E. on moisture content. Each point represents the average of 10 tests, and at each moisture content level the points on both regression lines were obtained from the same samples. The correlation coefficients for the dynamic and static tests are .986 and .981, respectively. The statistical procedure 34 35 oooqdjm oooqwxm oooqamm oooamdm oooqamm oooqmmm $3.0m m oooqmm: oooqwm: OOOqHOm oooqmom oooawom oooammd O®.HH : OOO.:O© oooqwoo ooonon oooqwow oooqwow oooqmmm :m.w m OOquNQ Goonaww 000nmm© oooqmmw ooonmww OOOqMNw mm.m m oooqbaw oooqaaw oooamfib oooqdab OOOamHN oooqaaw mm. H humam zpmsom Chase Ucooom pmgflm accosvmpm poopcoo Hm>oq weeppm>o Hmpcmewoczm magpmfioz mHuLmCH hLOPMpom ocm pmmzm LOM UmpomghooCD .m.z OHEwczm m mqm¢e Fig. 8.--Dynamic M.E. obtained from the fundamental frequency and first 5 overtones, uncorrected for shear and rotatory inertia for all levels of moisture content on hardboard. [The samples are 1” wide x 20" long and vary in thickness from .215 to .240 inches approximately, depending on the moisture con- tent.] 37 ZO_._. no macs. m m e m m _ re. 3.8 i a . 1L2. 8.: L i I. crj o o a H ILL. 3. N. i W .x. New _ Fl Lr L... e ~—- 45 \T & \\ \\ \ 2. <2. 2. 1N3W3833C| OIWHLIHVQO'I -A ‘2 CHAPTER V DISCUSSION Shear and Rotatory Inertia Effects When the dynamic M.E. was calculated without cor- recting for shear or rotatory inertia (apparent M.E.) and plotted against the mode of vibration, straight horizontal lines were obtained, with the average of 20 dynamic tests falling close to this line (see Fig. 8). This demonstrates that the effect of shear and rotatory inertia is negligible. It can be seen that even at the 5th overtone the depth/length ratio, which is directly related to shear and rotatory inertia, is such that these effects are still negligible. Using spruce and oak specimens 8 x 10 x 200 mm Kollmann (5) found a considerable drop in M.E. obtained from overtones as a result of shear and rotatory inertia (see Table 5 and Fig. 12). Here the depth/length ratio is much higher than that of the hard- board samples used in this experiment. The slight increase in the average M.E. obtained from the overtones as compared to that obtained from the fundamental frequency is attributed to small defects or irregularities in the density alOng the sample. For 46 47 m m.m mo.H m.m so.H Im om m.m mo.H m.m so.H 0mm Dam s.oH oo:.mHH 0.: ooo.HsH mso\sm 0mm sates pow cmpooppoo .m.z esteem a :.oH oom.oHH 0.: 00:.Hms msox x mm empomssoo -ss .m.z ospspm s.HH ooo.mmH w.m oom.msa mso\sm omm sates Low Uopomhhoo .m.z oaswcmm N.HH oownmfla ©.m oowqama mEo\Qm Qm Umpomhhoo no: .m.z oHEmczm 3.0 mmm. m.m Hem. Eo\o pswfimz oamfiomqm & mwmum>< R mwmgo>< COHmcoEHQ coHpmHLm> mo coapmfipw> do ssmsoseamoo psmsoseamoo Xmo oosgmm .m.z oaptpm 6cm oflemsaa mo comHLmQEoo m mqm<8 48 Fig. 12.-—Dynamic M.E. obtained from the fundamental frequency and the first 4 overtones, uncorrected for shear and rotatory inertia for Spruce and Oak specimens 8 x 10 x 200mm, as reported by Kollmann (5). 49 ZO_._. LO woos. c m N f /./ »~._. __—_ 1P" MUD: / O Q N ,.m mama/d» '3'IN alwvmo S 9 m. 5O example, a defect at the center would have considerable effect on the M.E. determined at the fundamental frequency but almost no effect on the frequency of the first overtone where a node occurs at the center. The overall result is that as the frequency increases the effect of defects is less significant. In comparing the results on spruce and oak obtained by Kollmann (Fig. 12) this effect can explain the difference in the two curves between the first and second nodes of vibration. Spruce is quite homogeneous and there is little lowering of the M.E. at the fundamental due to defects or at the first overtone there is not much compensation for defects, hence the curve is quite steep. In oak, however, the wood is not so homogeneous and the presence of defects and irregularities lowers the dynamic M.E. at the fundamental frequency and at the first over- tone, the drop in M.E. due to shear or rotatory inertia is partially compensated for and the curve has little slope up to this point. Beyond the first overtone the shear and rotatory inertia effects dominate. Dynamic and Static Modulus of Elasticity The difference in the regression lines for dyanmicand static M.E. on moisture content is attributed to creep. Harris (6) defines creep as "a continuing deformation with time, at a constant stress” and relaxation as "the reduction in stress, over a period of time, at constant strain.” In 51 the tests for M.E. both stress and strain are increasing with time so that a combination of the two results. Moslemi (7) has shown that for hardboard, creep is a func- tion of time and level of stress. Both of these factors are important here, (1) the time of the static test is many times greater than the time for the dynamic test, the order of magnitude being approximately 9,000 to l, (2) the level of stress is increasing during the static test, whereas the stresses involved in the dynamic test are very small, due to the small deflections. Hence, from what actually amounts to a loss of load during the static test, the static M.E. is lower than the dynamic M.E. The difference in the slopes of the two regression lines is also attributed to creep, since creep increases with moisture content (7). This accounts for the large relative change from dry to wet conditions, for the creep in the static tests has much more effect than in the dynamic test, where the time is so small that time dependent prop- erties such as creep and relaxation cannot have much effect. Therefore, the static M.E. falls off at a more. rapid rate than the dynamic M.E. The negative slopes in Fig. 9 are due to the weakening. of the fibers by swelling. [As described above, the regression of static M.E. on moisture content is more 52 negative than the regression of dynamic M.E. on moisture content as a result of increased creep at the higher moisture contents.] The slope of the regression line in Fig. 10 is greater than 1 since the difference between the two M.E.‘s is less at the higher values of M.E. (lower moisture content) as can be seen from the equation ED + ES + 101, 250 + 508(x) The line lies completely below a 450 line through the points ED = ES since the dynamic M.E. is consistently higher than the static M.E. The 95% confidence limits are curved outward from the overall mean of all M.E. values. The reason for this is that in estimating the regression line of the form Y = a + b (X-i) where a = Y b slope if the estimate of b is in error, the slope will change and the greater the distance from the overall mean, the greater is the change of the line, hence the error is greater at the extremes and the confidence limits curve outward. Logarithmic decrement The dashed line in Fig. 11 is included because it has some resemblence to the curves obtained by Kollman 53 (5) and Pentoney (8). Kollmanns curves are reporduced in Fig. 13. These are for solid wood, and show a definite minimum at 8%, which indicates that at this moisture content plastic deformation is at a minimum. Moslemi (7) obtained similar results for hardboard. The results obtained in this study indicate that plastic deformation increases with moisture content in a fairly linear fashion, and thiS is supported by the fact that the static M.E. was linear with moisture content. If plastic deformation is greatest at high and low M.C. and at a minimum at 8% M.C. the expected difference in dynamic and static M.E. would be greater at high and low M.C. but there was no evidence of this obtained. Nomogram The nomogram in Fig. 14 is a graphical solution to the equation for dynamic M.E., without corrections for shear or rotatory inertia. See Appendix 2 for some details of the construction. The nomogram has an accuracy of i_2000 psi as constructed and is quite simple to use, as can be seen from following the example in Fig. 14. However, the chart could be simplified with some loss in accuracy, by assuming that the dimensions of all samples are within a certain tolerance. For example, if 54 Fig. l3.--Logarithmic decrement for Oak and Spruce as reported by Killmann (5). 55 .040 4 1 E C K U A R 0 P S 2 3 3 2 M m m 0 O O. O O 0. hzuzmmowo 0.2:.Em4004 IOO 60 4O MOISTURE CONTENT °/o 20 56 Fig. l4.--The nomogram prepared here is a graphical solution to the equation E fr213W 5.4x 10-6 = hb Where E = dynamic M.E. fr = fundamental frequency 1 = length of sample, inches h = thickness of sample, inches b = width of sample inches W = weight of sample grams The equation may be reduced to log form and written as the following set of equations 310g 1 - 3log h = A A - log b = B B + 2log fr = C E C + log W = log (5.4 x 10 _5) Example: Let 1 = 20” h = .25” b = 1” fr = 50 cps and W = 75 gms. E is determined by 1. Construct a line through 1 = 20" (l) and h = .25" (2) to obtain a point (3) on reference line A. 2. Construct a line through the point on A (3) and through b = l" (4) to obtain a point (5) on reference line B. 3. Construct a line through the point on B (5) and through fr = 50 cps (6) to obtain a point (7) on reference line C. 4. Construct a line through the point on C (7) and through w = 75 gms (8) to obtain point (9) the desired dynamic M.E. 57 mm Oc mm z 08 oh m. nmm E. m. EN 8. 8 con. oh oN ON. 0: 0: no. 6) mm. Om mm Om 58 the length is assumed to be 20.00 i .05" and the width 1 i.-Ol" the chart could be reduced to four graduated lines and one reference line. The result will then be approximately within the range of i 6000 psi. CHAPTER VI CONCLUSIONS From the results obtained it may be concluded that the dynamic test for modulus of elasticity is a valid measure of this property and that the M.E. can be ob- tained without applying a correction for shear or rotatory inertia if the depth/length ratio is small enough. If the overtones are more easily obtained, the dynamic M.E. may be calculated using these values. The dynamic M.E. can be correlated with the static M.E. to obtain the more conventional values for indica- tions of the quality. Since the dynamic test is nondestructive, it proves very useful in research, where the same specimens may be tested under different conditions, but would also be very useful in quality control work. The dynamic test combined with a simplified nomogram or §y§n_with the same type nomogram as prepared here could give the dynamic M.E. in a very short time, which would be highly desirable in quality control work. It may also be concluded that creep is of major importance in testing hardboard. Creep caused the static 59 60 M.E. to fall below the dynamic M.E., with the difference increasing with increasing moisture content. Plastic deformation, which increases with increasing moisture content, causes the logarithmic decrement to increase with the moisture content. AP PEN DI CBS 61 APPENDIX 1 STATISTICAL PROCEDURES Regression Line and Correlation Coefficients.--Let Y be the dependent variable and X the independent variable. From the data obtain the following statistics: 2X1 2 2x1 2Y1 SY? l i XiYi From these statistics the following are obtained: (2’- Xi) (5 Y1) Sxy =§X1Y1- n n - 1 3X -2x2 - (ix 2 n n - l 2 s -21}? - Elfi— Y n n - l The regression equation of Y on X is Where a = Y=§§r 63 The correlation coefficient is r = Sxy SXSy 95% Confidence Limits.--The interval (see (9)) at any point on the regression line is Y‘t <fl<‘.I x = X - APENDIX 2 NOMOGRAM The nomogram of Fig. 14 was constructed using the method of Giet (10). The equation for the dynamic M.E. is 4 E = erP-%2 (11.44 x 10'7), '2 Where (3 = S F4 mo” :3‘ i: H [0 fr = frequency W = weight of sample 1 = lenghth of sample b = width of sample h = thickness of sample The equation may be written E 4 = fr2l3W 5.4 x 10‘9 h3b and in log form is log(E/5.4 x 10-6) = 210g fr + 310g 1 + log w - 310g h - log b 64 65 This may be further simplified as a system of four equations 310g 1 - 310g h = A A-logb = B B + 210g fr = C c + log w - log (E/5.4 x 10-6) = 0 From these four equations the length and graduations of the reference lines were established as well as the distance between the lines. BIBLIOGRAPHY 66 lO. BIBLIOGRAPHY Hearman, R. F. S. 1958. "The Influence of Shear and Rotatory Inertia on the Free Flexurel Vibration of Wooden Beams'1 British Journal of Applied Physics, Vol 9, p. 381. Wood, A. B. 1930. A Textbook of Sound. New York: The Maxmillan Co. Goens, E. 1931. Ann. Phys. (Leipzig)p. 11. Timoshenko, S. 1921. Phil. Mag. 41, p. 744, p. 125; Collected Papers. 1953. New York: McGraw-Hill Book Co. Inc., p. 288; 1955, Vibration Problems in Engineering. New York: D. Van Nostrand, Inc. Kollmann, F. and Krech, H. 1960 ”Dynamiche Messung der elastischen Holzeigenschaften und der Dampfung," Holz Als RoH-Und Werkstoff. Harris, C. 0. 1959. Introduction to Stress Analysis New York: The Macmillan Co. Moslemi, A. A. 1964. "Some Aspects of the Viscoelastic Behavior of Hardboard." Unpublished Ph.D. disser- tation, Michigan State University. Pentoney, R. E. 1955. "Effect of Moisture Content and Grain Angle on the Internal Friction of Wood," Composite Wood Vol 2, No. 6, p. 131. Snedecor, G. W. 1957. Statistical Methods Ames, Iowa State College Press. Giet, A. 1956. Abacs or Nomograms. Translated by J. W. Head and H.D. Phippen. London Iliffe and Sons Ltd. 67 mIIIIIIIIIIILIIII'ILIIIIIIIIIIIIIIEIIIIIIIIIIIIVs