f methods or computing the strength AND STIFFNESS OE PLYWOOD STRIPS IN PENDING Revised December 1946 L i ■ ■ ■ - • y * SEP 2 1948 r> ATLANTA BftANOH ATLANTA, G£ORQU U.S. DtKOt>i iu' This Report is One of a Scries Issued In Cooperation with the ARMY-NAVy-CIVIL CCMMITTCE on AIRCRAFT DESIGN CRITERIA Under the Supervision of the AERONAUTICAL BOARD Nc. 1304 UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison, Wisconsin In Cooperation with the University of Wisconsin Digitized by the Internet Archive in 2013 http://archive.org/details/metcomputiOOfore M ETHODS OF COMPUTING THE STRENGTH AN? STIFFNESS OF PLYVOCD STRIPS IF BENDING^ By ALAN D. FKEAS, Engineer INTRODUCTION This report gives methods of computing the strength and stiffness of plywood strips in bending. The methods have "been developed with a background of nearly U,000 static bending tests, and are believed to be reasonably accu- rate. Further analysis of the data may, however, result in changes in 3ome of the details. The methods are limited in application to plywood having the grain of adjacent plies at 90° and having the grain of the face plies either parallel or perpendicular to the span, fata are not yet available for other cases. STIFFNESS OF PL WOOD STRIPS IN BENDING The deflection of the center of a simply supported plywood strip under center loading will be given approximately by the equation w = */ d) kg E C I where w = deflection of the center of the strip P = load a = span E- = apparent modulus of elasticity in bending I = moment of inertia of the whole cr©3s section of the strip about its central line (i.e., I = bh-/l2) —This is one of a series of progress reports prepared by the Forest Products Laboratory relating to the use of wood in aircraft. Results here reported are preliminary and may be revised as additional data become available. Mimeo. No. 130U _1_ b - width of strip h = depth of strip T'he apparent modulus of elasticity in tending, E c , may "be f and i i - n 'S V T S c * L=JU ( 2 ) where S^ = modulus of elasticity of the i tn ply measured parallel to the span I. = monent of inertia of the cross section of the it* 1 ply about the n eutral axis of the cross section and the other terms are as defined above. For plywood constructed symmetrically, that is, with pairs of .dies of equal thicknesses at equal distances from the center, the neutral axis of the strip will be at the center of the plywood. This will usually be the case, at least for carefully made plywood; but occasions may arise when unsymraetrical plywood will be encountered. In such an event, the location of the neutral axis may be found by i = n \ " AiEidi a na _ . " \ A ' na i s n r A i E i ■ i - 1 where d n „ = distance from tov) of bean, to neutral axis &• = distance from top of beam to center of ith ply Aj_ = cross-sectional area of ith ply For plywood having all plies of equal thickness, for example, values of E c will be given by use of equation (2), as: No. 1304 (4) „ rm + 26 „ _ 1 + 26rT< _ 3-ply; E x = x ?n \ E 2 - - R -w- v _ SSr T + 99 _ 26 + 99r? ™ D— :3.ly; £j-> = — — Xjt Jj n = ili T 1 125 L 2 125 L 7-oly; I. = 95r ? + 244 E T E„ = " + 244rT E L - J 1 .343 L 343 L n - . ™ 244r T + 435 _ _ 244 + 485rfp - 9-ply; Si = E T e p - -^t 1 729 L ^ 729 L where E, '= E for r>lywood strips having the grain of the outer plies parallel to the span Eo = E c for plywood strips having the grain of the outer plies perpen- dicular to the span E 7 = modulus of elasticity of wood in the longitudinal direction (parallel to the grain) Tr-. ~ jir^/Er i. LI _j Em = modulus of elasticity of wood in the tangential direction Equations (4) above apply strictly only to plywood made of rotary- cut veneer, since the ratio rj has been used. Eor plywood made of quarter- sliced veneer, the ratio r-^ = 3~/Et, should be used, since the modulus of elasticity in ,he radial direction would be involved. While equation (l) for a simply supported beam loaded in the center has been used as an example, values of E c as found from equation (2) may be used in the appropriate equation for deflection corresponding to the method of loading, substituting E for E in the usual equations. If the effect of shear deformation is to be taken into account, equation (l) (and the corresponding equations for other loadings) would be modified by another term, as, for example, • ■ *H? (i + • $ (5) where e - a term involving the elastic constants of wood and the construction of the ■olywood No. 1304 -3- h -• depth of beam a = span of bean Published values of Er based on static bending tests made •'. the Forest Products Laboratory are about 10 percent below the true values or? St, for the material because of the effect of shear deformation. Where no correction for shear deformation is to be made, B c may be calculated with the Forest Products Laboratory published values of IV. £hus, values of S c so calculated will give values of deflection correct for cases when the effect of shear deformation amounts to about 10 percent. Usually these values of E c will be sufficiently exact. Where, however, a. correction for shear deformation is to be made, the published values of Ej should be increased 10 percent. Seme values for (e) are given in table 1. A discussion of their uses and limitations will be found in the discussion accompanying table 1. SEOTGER OP PLYWOOD SI -.IPS IN BENDING Fo r Plyw ood Hf ivi ns the Gra in of ' the Outer Plie: j J." ara llel to ■«-•, Ox le Span The strength of plywood strips of this type in bending will be given by M sKri 5 (°) l c where M = resisting moment of plywood strip K = constant S = stress in outer fiber of outermost longitudinal ply c = distance from neutral axis to outer fibor of outermost longitudinal ply r i = e i/ :b i Por Plywood Ha v ing the Grain of the Outer Plies Perpendicular to tho Span The strength of plywood strips of this type in bondin- will given by Ho. 1504 -4~ M = Kr ffi ^ ( 7 ) where r ffi = E n /E] E m = E c for a strip of plywood having the outer plie3 at right angles to the span, hut considering that the outer ply on the tension side does not act. The position of the neutral axis for this case may be found by use of equation (3) except that there will be (n - l) plies, the outer ply on the ten- sion »ide being omitted from consideration in calculating E^. (The value of I as before is based on the full cross section about its center line, including all plies. ) PLYWOOD MADE OF MORS THAIT QUE SPECIES The formulas previously presented in this report have been checked by tests of specimens made with all plies of the same species. A form of plywood commonly met in aircraft work is that made of two or more species. In the absence of test data to serve as a check, and pending the development of more accurate methods, the following procedure is suggested for calculating the bend- ing strength and stiffness of plywood of mixed species; In computing deflections, equation (2) for E c may still be used, except that in this case the values of E^ for the parallel plies will not be the same for all plies, but will be those corresponding to the species. Similarly, the values of E^for the perpendicular plies will be those for the species under consideration. In computing strengths, equations (6) and (7) may still be used. The values of r will be determined from E c as indicated above. The value of S, which for plywood made with all plies of the same species would be the same f»r either face grain direction, may now be different for the two directions, but in any case will correspond to the outermost ply whose grain direction is parallel to the span. VALUES OF THE RATIOS r T = E T /E L and r R = S^/Ex Values cf Em and Ep from which to compute the ratios r m and r R are available for a few species. The data are, however, incomplete in that only a few species have been tested and that little is known about the variation of these ratios with variations in -Jpe^if io gravity or with changes in moisture content. No. 130U -5- Such data as are available are presented in "Elastic Properties of Wood," Forest Products Laboratory Report No. 152S and supplements thereto, Ag more data become available, additional supplements will be issued. Data from these publications should be used when applicable. For cases in which no data are available, the use of values of r™ = 0.05 and r^ = 0.10 is suggested. As pointed out later, even approximate values of these ratios will generally give results only slightly in error, so that lack of exact values ic not a serious deterrent to the use of the proposed methods. ERRORS INTRODUCED SY NEGLECTING TRANSVERSE PLIES The transverse plies may, in general, be neglected with relatively small error except in the case of 3-ply plywood having the grain of the outer plies perpendicular to the span. For plywood having the grain o^ the outer plies parallel to the span, such neglect will generally result in small errors. For plywood having the grain of the outer plies perpendicular to the span, however, the error is somewhat larger, and for 3-ply plywood of this type the error may be consider- able. Figures 1 to U show the error that may be expected from neglect of the transverse plies for 3> 5> 7» and 9~Ply plywood of two types for various values of r. While certain approximations were used in calculating the values for these figures, they will be useful as a guide in determining whether or not neglect of the transverse plies will be a serious matter. It will be recog- nized that interpolation in the curves of figures 1 to h is not possible. The connection of the plotted points by straight lines has been done only to emphasize the trend, as the number of plies is varied, of the ratios concerned. The foregoing discussion also permits the conclusion that, for most cases, reasonably accurate results may be obtained even with assumed values of r. Therefore, reasonable results may be obtained for species for which no values of r™ and r-n are available by using the approximate values previously given as a guide in estimating values for other species. VALUES FOR (e) FOR USE IN EQUATION (5) The factor (e) for use in correcting deflections for the deformations due to shear has been evaluated for 3» 5» 7. and 9~pl y Douglas-fir and Sitka spruce plywood having all plies the same thickness for the case of a simply No. 130U _6- supported "beam under center load. Corresponding values for other cases have not "been evaluated. Values of ( e) for a uniformly loaded, simply supported beam may be expected to be slightly smaller than for a simply supported beam with center load, hut no exact values are available. It should he noted, however, that reasonable corrections for shear deformations may be made even with values of (e) considerably in error. Assume, for example, that the proper shear correction was (l + 0.10) and that the value of (e) used was 20 percent low. The computed correction would then be (l + 0.08), which is only about 2 percent in error, so that the corrected deflection would he only about 2 percent in error. From this it appears that the values of (e) given in table 1 may be used as approximations of the correct values for other types of plywood and plywood of other species, and may also be used to estimate approximately the shear corrections for such cases. It is expected that these values, for the types of plywood specified in table 1, will give corrections somewhat too large if used for the case of a uniformly loaded, simply supported beam. Calculations based on the values in table 1 indicate that the correc- tion for shear deformation will be less than 10 percent in practically all cases if a ratio of span to beam depth greater than 48 to 1 is maintained for plywood strips having the grain of the outer plies parallel to the span, and greater than 2U to 1 for plywood strips having the grain of the outer plies perpendi- cular to the span. Table 1. — Values of (e) for plywood having all plies the same thickness and us ed as a simply supported beam with center load of pi Les (< Fo. Sitka spruce : Dougl< a.s-f if Outer plies parallel to I span Outer plies perpendicular to span Outer plies parallel to ! span : Outer plies : perpendicular : to span 3 18U.g : 8.6 : lUl.7 i 9.9 5 : 160.8 U6.3 118. U ! 36.5 7 1 1U7.I : 63.5 I 108. 5 : U8.8 9 ! 138.8 73.3 ! 102.5 55.7 No. 130U -7- VALUES 0? K FOR USE IN EQUATIONS (6) AND (7) The values of K suggested for use in equations (6) and (7) are: Stresses at or "below proportional limit — Outer plies parallel to span, K = 0.85 Outer plies perpendicular to span, K = 0.90 Stresses at or near modulus of rupture — Outer plies parallel to span, K = O.85 Outer plies perpendicular to span, 3-ply, K = 1.10 Others, K = 0.95 If the approximate method, neglecting the effect of the transverse plies, is used, the values of K for the outer perpendicular plywood will have to be increased somewhat, since this approximation results in values of moment lower than the true values. The difference will, of course, depend upon the construction of the plywood, hut an approximation of the correct factor may he obtained by dividing the appropriate factor above by the appropriate ratio given in figure 2 or U. For outer-parallel strips the approximate method gives results, in general, close enough to those of the exact method so that no modification of the factor is necessary. STRESS VALUES TO BE USED IN CALCULATIONS The stress values or modulus of elasticity values to be used in the calculations will depend upon the use for which the plywood is intended. U. S, Department of Agriculture Technical Bulletin U79, "Strength and Related Properties of Woods Grown in the United States," gives average strength and modulus of elasticity values for the more common species as found from tests on small clear specimens. U. S. Department of Agriculture Miscellaneous Publication I85, "Guide to the Grading of Structural Timbers and the Determination of Working Stresses," gives basic stress values for determination of working stresses for the more common structural species. ANC Bulletin 18, "Design of Wood Aircraft Structures," gives values for a number of species for use in aircraft design. The appropriate design values given in the sources above are applica- ble to calculations on plywood; that is, the appropriate values of stress or modulus of elasticity will be substituted for S or E L in the equations given in this report. No, I30U -8- SUMMARY OF NOTATION a - span of bean A- = cross-Sectional area of i th ply ■ b = width of "beam c = distance from neutral axis to outer fiber of outermost longitudinal ply cL = distance from top of beam to neutral axis dj = distance from top of beam to center of i ply e = a term dependent upon the elastic constants of wood and the construction of the plywood E = apparent modulus of elasticity in bending E-i = E for plywood strips having the grain of the outer plies parallel to the span Ep = E for plywood strips having the grain of the outer plies perpendicular to the span E-r = modulus of elasticity of wood in the longitudinal direction (parallel to the grain) Em = modulus of elasticity of wood in the tangential direction E-n = modulus of elasticity of wood in the radial direction Ej_ = modulus of elasticity of the i ply measured parallel to the span E m = E for a strip of plywood having the grain of the outer plies at right angles to the span, but considering that the outer ply on the tension side does not exist h = depth of beam I = moment of inertia of the whole cross section of the strip about its cent ral line (i.e., I = bh-^/12) 1^ = moment of inertia of the cross section of the i ply about the n eutral axis of the cro?;s se ct ion K = constant M = resisting moment of plywood strip No. 130U _9_ n = number of plies P = load r r. = V s l r i " h'\ r = E /E T m m' L S = stress in outer fiber of outermost longitudinal ply w = deflection of the center of the strip LI TERATURE C ITED (1) ARMY-NAVY- CIVIL COMMITTEE ON AIRCRAFT TESIGN CRITERIA 19^U. resign of wood aircraft structures. ANC Bulletin IS, 2l+7 pp., illus. (2) DOYLE, D. V., DROW, J. T. , and McBURNEY, R. S. 19^5 • Elastic properties of wood, Forest products Laboratory Report No. 1528 and supplements. (3) MARCH, H. W. 1936. Bending of a Centrally Loaded Rectangular Strip of Plywood. Physics 7(1), 32J+I. (U) MARKVARTT, L. J. and WILSON, T. R. C. 1935» Strength and related properties of woods grown in the United States. U. S. Dept. Agr. Tech. Bui. ^79, 99 pp., illus. (5) WILSON, T. R. C. 193^* Guide to the grading of structural timbers and the determination of working stresses, U. S. Dept. Agr. Misc. Pub. 185. 26 pp., illus. No. 130U -10- 2.10 2.00 190 1.80 1.70 ki £ 1-60 kj k ^ 150 o f^ 5 1.40 1.30 120 1.10 \ \ \ LEGEND \ \ (all plies same thickness) \ \ o o OUTER PLIES PERPENDICULAR \ \ o— «o OUTER PLIES PARALLEL , \\ \ \\ \\\ \\\, y-r T 0Rr R =O.O6