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SUMMAIAY OC IFCIMULAS II= FIlAT PILATEIS OUf PLYWCOII UNiEl UNIFUIM Ci CONC[NTIRAThD LOCADS PJ! 11 *. October 1941 2 4, 4 r441 Nj ~' TuliS r1OIT IS ONE Or A SEr1IES ISSUE) TO AlL Tll NATION'S WAIR IPIOGIRAM .No. 1300 L t LL i;.,,'i, UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison, Wisconsin In Cooperation with the University of Wisconsin oftEpST E f *SJ Foreword A basic study of plywood that is under way at the Forest Products Laboratory has included as one phase the mathematical anal ysis of the deflection of flat plates under uniformly distributed or concentrated loads. This theoretical analysis has progressed sufficiently to permit the publication of the formulas presented in this mirieofrar'h. Some of these formulas have been checked against test results, and the others are believed to afford reasonably accurate results. Other phases of the study of plywood relate to basic strength in compression, tension, bending, and shear; resistance to combined stress; criterion for buckling in flat and curved plates and shells and behavior after buckling; and methods of re inforcing. It is planned that as rapidly as significant results become available, they will be presented in this series of reports. Forest Products Laboratory ST'I.!'.Y OF FORMJLAS FCT. FLAT PLATES OF PLYWOOD tDER TYIFORT OR CONC::rRA:ED LOADSI 3y W. IARCH Special Consulting Mathematician, Forest Products Laboratory, Forest Service, U. S. Department of Agriculture Tr.e material herewith presented comprises a summary of the principal results of a more extensive report soon to be issue, by the Forest Products Laboratdry. Reference should be made to the extensive report for the derivation and discussion of the results contained in this summary. Rectanular olywood plates will be considered in which the directions of the grain of the wood in adjacent plies are mutually perpendicular, and perpendicular or parallel to the edg:e of the plate. The plies are acued to be either flat rain or eodge rain. The choice of axes in Thov.wn in figure 1. The effect of the glue other than that of securin, adherence of adjacent plies is assumed to be negligible. Consequrntl>, the formulas and methods of this summary are not intended to nn ly directly to partially or completely impregnated plywood or coimpregnatel wooL, although it is t be expected that many of the results of the extn sive report apply to such material. Notation a = width of plate b = length of plate h = thickness of plate wo = deflection at center of plate p = load per unit area P = concentrated load (section 5) P = pa/!F, (sections 6 and 7) W = deflection at center of an infinitely long plate of width a under a specified tnpe of load F,"' 1,/4 B = ) where EI Pnti E3, are Jefined in section 1 1. Stiffness in bending of strips of olywood Consider a strip of plywood with its edges either parallel or perpendicular to the gr?irn of the face plies and denote by x thr iir,/c tion parallel to the lenrIth of the strip. The stiffness of the strip is 'This mimeorr.,':. is one of a series of progress reports issued by t.e Forest Products Laboratory to aid the Nation's defense effort. Results here reported are preliminary and may be revised as additional data be come available. Mimeo. No. 1300 determined by a modulus E, defined by the equation Bj I = S^W i where the summation is extended over all of the lies numbere1, for exJle, as in figure 2; (F9)i is the Young's modulus of the ith liy mesured in a direction parallel to the ler.nth of the strio; Ii is the momnnt of inertia, with respect to the neutral axis, of the area of thb cross section of the ith rI m'de by a lane perprendicular to the length of the string; and I is the moment of inertia of the entire cross section of the strip wvith respect to Its central line, that is, I = h9/12 for a strip of unit width. An orproxi mate formula in which the error is very sliht is obtained for E I : tkin! h s'um of the products (E ).I. formed for only those plies in 4hich the ;rain is parallel to the length of the strip. Exception is to be made of a txreeply strip having the grain of the face plies perpendicular to the lergth of the strip. In the case of a rectangular plate with sides a and b, the modulus E1 would determine the stiffness of a strip cut from the plate with its edges parallel to the side a as in figure 1. The modulus E2 similarly defined, namely, B21 determines the stiffness of strips parallel to the side b. As in the case of E1 the calculation of E2 can be based on the parallel plies only, except in the case of a threeply strip having the grain of the face plies perpendicular to the length of the strip. 2. Young's modulus of a strip of plyn'ood in tension or compression As in section 1, consider a strip of lywood 'hose ,des are parallel to the Xaxis or to the side a of the rectangular plate of figure 1. The mean modulus V in tension or compression may be defined by the equation 1 = (Ex)ihi E~h = _ where (2,)i has the same naning as in section 1, hi is the thickness of the ith ply and h is the thickness of the strip. In like manner for a strip parallel to the side b " Ebh = (Ey)ihi 2 The moduli Ea nrid Eb are needed in cases where the deflections of the plates under consideration are so large that direct stresses in addition to hendinm stresses are developed. These moduli can be calculated '"ith little error by considering only those plies which are parallel to the length of the strip. 3. Rectangular plite. UTniformly distributed load. Ed.es simply suprort. i. S7;i! d'1 1 1o., The method presented below may be t<:n to apply if the lo',d3 are such that the deflections do not exceed the thickness of the late. Appreciable direct stresses develop at deflections of tne order of mar nitude of the thickness of the plate and the deflection will be less than that found by the method presented. For a plate hose len,th exceeds its breadth by a moderate amount the method of section 6 can be applied in this case. It is assumed throughout that the corners of the plate are held down. To find the deflection wo at the center of a riven plate of width a and length b, calculate first the central deflection 7 of a sim ilarly loaded very long late (infinite strip) of width a and of thr. same construction. Now W=5x 0.99 pa =0.1l547 () 3 E2 Elh3 Except for the factor 0.99, whichh expresses the plat, effect (in woo1 practically negligible), this is the formula for the central deflection of a b, r,, of unit with inlrr a uniformly distribut e loa. Then the deflection at the center of the given plate can be found approximately from the formula o = f 7 (2) where f is a factor to be taken from tihe curve of fi.,ure 3 corresponiin7 to the argument B b l El l1/4 a a'. ^ / 3 The points shown in figure 3 were determined by an exact ,nthod, using, the elastic constants of spruce, for various types of plywood. Th. curve is merely a smooth average curve determined by those points. A 1300 consideration of the extended analysis discloses the fact that the essen tial factors detrrminin.g the central deflection of a plate under the con ditions of loading and support of this section are the two moduli E1 and ET that enter into the determination of W and B. Variations in other elastic constants will account for variations of the order of magnitude shown by the points in figure 3.2 Hence it appears that this curve may be used for plyTrood of the type described at the beginning of this sum mary, independently of the species of wood used. Th constants E1 and E, must be known. Tr'y can be determined by calculation or by static bend ing tests of strips of matched material. The maximum shown in figure 3 in the vicinity of B/a = 2, 7hich at first sight arpears to be impossible, is found in the exact analysis. It is associated with a wave form of slight amplitude that is assumed by the deflected surface of a plywood plate. A presentation of the results of an approximate analysis in essentially the form (2) was made .by C. B. Norris.3 Because of the approximations involved, the deflections calculated from his results are too small, a fact vhich he recognized would be the case. 4. Rectangular plate. Uniformly distributed load Edges claimed. Small deflections In this case the central deflection of the corresrondir: infinite strip is given by 4 4 0.99 pa = 0.0309 pa (4) 32 Elh3 Elh3 The deflection of a finite plate can be found from the formula wo f W (5) in which f is to be taken from the curve of figure 4 where it is shown as a function of B/a, B being related to b by (3). Th points shown near the curve in figure 4 are the exact values of f for various types of plywood. The elastic constants of spruce were used, but the curve may be used for wood of other species as point'cd out in section 3. The actual deflections will usually be considerably lar. r than those calculated by (5) because perfect clamping of the edges is rarely realized in practice. If the edges are restrained from moving inward, direct stresses will develop at moderate deflections. In this case the methods of section 7 are available for a plate whose lerqth is moderately greater than its breadth. 2It was convenient to calculate the points shown in some of the fiurcs *for plates in which the plies are all of the same thickness but the formulas and curves of this report are not restricted to such plates. 3ardwood Tecord, May 1937. 130o 5. Rectangular plate. Load concentrated at the center. Edges simply supported. Small deflections The central deflection of the corresponding infinite strip is given by 1/4 w = 1.051 x 6 x 0.99_E \ Pa (1) 0 .S Tr3 \^^ El 0 .2521 _ E, \ ) 1E1h3 The central deflection of a finite plate can be found from the formula w f 7 (M) o where f is to be taken from the curve of fi.'ure 5. In formula (6) the number '0.8 is an approximate figure for a constant Those values .a" range from 0.76 to O.g6, for various types of ordinary ol.vyood. A measns of calculating this constant is to be found in the extended retort. 6. Infinite strip. (Long, narrow, rectangular plate). Uniformly distributed load. Edges simply supported Th: formulas of this and the next section are applicable ,rhen the maximum deflection is small in comparison "ith the v'idth of the strip, although possibly equal to several ties the thi.'kness of the strip. In addition to the conditions stat.' in tht heifin., the ces are assumed to be restrained from moving in,, ar.. UTsin7 the notation 340 (9) 1300  the following approximate relation connects the load and the central deflection W. P H )+ K 3 (10) 'where H = 6.46 El/EL (11) K = 20.8 Ea/EL (12) 'here Z_, denotes Young's modulus in the direction of the rain of the wood. If the plywood is made of 77ood of more than one species, equation (10, can be multiplied through by EL. There results a relation in 'hich only the moduli 31 and 'Ea enter. The mean direct stress is given by the formula g = 2. 0 1(]h 2 a, 1)2 (13) This is the mean direct stress averaged over the thickness of the plate. The direct stress in tny nly can be calculated by observing that the stress in any ply is proportional to the 3 of that ply in a direction parallel to the Xaxis. This follows from the fact tnat the strain associated with the direct stress is constant across the thickness of the plate. The maximum bending stress in a face ely can be calculated by the approximate formula s'= 1.01 a( EB i ( U) % a) h (l'i) where a is to be taken from the curve of figure 6. In this figure the argui..mnt Th is connected with the deflection by the formula = 2.77,9 E_ l/2 The bending stress at any point in any' other rlv can bo cl culted by noting that the associated s'trair. varies linearly V:itn the distance from the neutral plane and that the corresronling stress is the prolict of this strain and Ex at the roint under consideration. The forruls just given can bo used for a plate whose length exceeds its breadth by a moderate amount and for sr.ll as well as large de flections. An inspection of figure 3 indicates that these formulas 11 e 1/4 can bhe used with small error if b 1 .roater than a akv) 6 1300 1.75. It is to be expected that the stresses calculated in this way will be satisfactory approximations to the stresses in the central T'or tion of such a plate. 7. Infinite strip. (Long, narrow plate). Uniformly distributed load. Edges cl amLp e d The edges are further assumed to be restrained from moving inward. Using the notation Spa Lh the following approximate equation holds P H +J K 3() where H = 32.3 EL (17) K 32a (17) E K 2 23. 2 ( Equation (1i) may be multiplied through by EL and thus be made arnlii cable if the plywood is made of wood of two or more different srcies. (See discussion following (10)]. The mean direct stress is obtained from the approximate formula g= 2.51 a (a (19) and the maximum bending stress in a face ply from the aprroximate formula S 1.o1 a E (^ ( (1Y0) 7 1300 where a is to be taken from the curve of figure 7. In this curve, the argument T is connected with the deflection by the formula h 2732 ) 1/2 (21) An inspection of figure 4 justifies the conclusion that the formulas just written can be used to calculate approximately the Cantral deflection and the stresses in the central portion of a plate for which B b 1 1/74 a a (2) isgreater than 1.5. l3c,, a 0 X FIG. I SIDE AND AXES DESIGNATIONS FOR A RECTANGULAR PLATE 4 FIG. z SECTION OF A,4 PLYWOOD STRIP SHOWING NUIMBERING OF PLIES 0.9 0.8 0.7 i % 0.6 0.5 LEGEND: EXACT METHOD ALL PLIES SAME 7THICKtVESS 0.4 03X 03Y tA5X A5Y 0.3 0.2 O.Z   o./ ! 0.1 0 I 2 3 4 5 6 5/a, 4 FU/NCT/ON OF THE DIMENSIONS OF THE PLA7 FIG. 3 FAqCTOR f [FORMULA (2) ] ,4S 4 FUNCTION OF B/a, WHERE UNIFORM LOAD. EDGES SIMPLY SUPPORTED. 2 UL 1 ,*. F 7 8 'E B S=b (E,/E,) t U.b LQ< 0.5 LEGEND: EXACT METHOD ,4ALL PLIES SUY7IE THICKNESS 0.4 03 X 93Y A5y 0.3 0.2  0.I 0 0 I Z 3 4 5 6/a, A FUNCTION OF THE DIMENSIONS OF TIHE PLATE FIG. 4 FACTOR f [FORMULA (5) ]JS FUNCTION OF 5/a, WHERE B =b (E,/E2). UNIFORM LOAD. EDGES CLAMPED. Z & 397o3 7 t I T a .4> 0.41 i . Io  LEGEND: A} ALL PLIES SAME THICKNESS. *FACE PLIES ONEH,4LF 4S THICK AS REMAINING PLIES. 0" v I IJ IJ 0 I Z 3 4 5 B/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE FIG. 5 FACTOR f [FORMULA (6) ] AS FUNCTION OF 8/a, WHERE B=b(E,/E). CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED. Z ': ". 17o4 F 0.8 0.7 S0.6 K ^ f ? Al I I LL 0 2 4 6 8 10 VALUE OF FIG. 6 THE COEFFIC/ENVT a IN THE FORMULA (/4) ,S A FUNCTION OF '7 50 4 30 S/0 0 0 2 4 6 8 I0 VALUE OF y7 FIG. 7 THE COEFFICIENT a IN THE FORMiULA (20) AS A FUNCTION OF 77 z 5)7o5 F UNIVERSITY OF FLORIDA 3 1262 08866 6374 