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SUJMMAiY OU IFCMULAS IfCI [FLAT PLAITS COf PLYWUOOU UNIDU UNIFOCM Cu CONCIENTIATI LOAUS Information IPc i cd and IPcaffirmcd January 1953 INFORiM.ATION REVIEWED AND REAFFIRMED JUNE 1959 DATE OF ORIGINAL REPORT FEBRUARY 13, 1942 ro E5  S  ~ No. 1300 UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison 5, Wisconsin In Cooperation with the University of Wisconsin Foreword A basic study of plywood that is under way at the Porcjt Products Lboh:.tory has included as one p.ase the mat iematical analysis of the deflection of flat plates under uniformly dis tributed or 'cnccritrated loads. This theoretical analysis has progressed sufficiently to permit the publication of the fornmlas presented in this mimeograph. 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 uc ling; and methods of re inforcing. It is planned tlat as rapidly as significant results become available, they will be presented in this series of reports. Fore t Products Lcboratory Digitized by the Internet Archive in 2013 http://archive.org/details/su m0Ofore STWMEY 0;W' 7OR.ITrLAS FOR FLAT :'L=T, OF FPLYWTOOD LDER UNIFORII OR C CUf?'..ThD LOADS By I. 1,. MARCH Special Consu.tling !,thcr.atician, 2 Forest Products Laborrt.ry, Forest Service U. S. Department of Agriculture The material herewith presented comprises a summary of the principal results of a more extensive report soon to be issued by the Forest Products Laboratory. Reference srr&c5, be made to the extensive report for the derivation and discussion of the results contained in this summary. Rcct.ngular plywood plates will be considered in which the direc tions of the grain of the wood in adjacent plies are mutually perpendicular, and per)endicular or parallel to the edges of the plate. The plies are assumed to be either flat grain or edge grain. The choice of axes is shown in figure 1. The effect of the glue other than that of securing adherence of adja,ent plies is assumed to be negligible. Consequently, the formulas and methods of this summary are not intended to apply directly to partially or completely impregnated plywood or compregnated wood, although it is to be expected that many of the results of the extensive report apply to such material. INotation a = width of plate b = length of plate h = thickness of pic .e ro = def.lection at center of plate p = load per unit area P = concentrated load (section 5) P = pa4/ELh4 (sections 6 and 7) W = deflection at center of an infinitely long plate c width a under a specified type of loi.d B = b where E1 and E2 are defined in 2 /section 1___ _ Ihis mimeograph is one of a series of progress reports issued by the Fo. zt Prof.ucts Laboratory to aid the !ation's defense effort. Results hcre re porLed are preliminary and may be revised as additional dat become available. 2Mainaincd at I adison, :s, in c. re:.ation with tt. University of .h'isconsin. Agricult re Ibdison Report liNo 1300 1. Stiffness in bending of strips of plywood Consider a strip of plywood with its edges either parallel or perpendicular to the grain of the face plies and denote by x the direc tion parallel to the length of the strip. The stiffness of the strip is determined by a nmodculus E, defined by the equation 1 (E i where the summation is extended over all of the plies numbered, for example, as in figure 2: (Ex)i is the Young's modulus of the ith ply measured in a direction parallel to the length of the strip; Ii is the moment of inertia, with res.:ect to the neutral axis, of the area of the cross section of the ith ply made by a plane perpendicular to the length of the strip; and I is the moment of inertia of the entire cross section of the strip with respect to its central line, that is, I = h3/12 for a strip of unit width. An approxi mate formula in which the error is very slight is obtained for ElI by taking the sum of the products (Ex)iIi formed for only those plies in which the grain is parallel to the length of the strip. Exception is to be made of a threeply strip having the grain of the face plies perpendicular to the length of the strip. In the case of a rectangular plate with sides a and b, the modulus E uld 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, rcmely, E2lY 1E3. determines the stiffness of strips parallel to the side b. As in the case of E' the calculation of E2 can be based on the parallel plies only, except in the case of a threeply strip having the grai. of the face plies perpendicular to the length of the strip. 2, Young's modulus of a strip of plywood in tension or compression As in section 1, consider a strip of plywood whose edges are pararlal to the Xaxis or to the side a of the rectangular plate of figure 1. Tlc mean modulus E. in tension or compression may be defined by the quLation  Eah = (Ex) ihi where (Ex)i has the same meaning as in section 1, hi is the thickness of the ith ply and h is the thicknesG of the strip.  Report No. 1500 In like mra:'ner for a s *ip parallel t ";'he side b Ebh = (Ey)h The moduli Ea and E az< rtdeIdf in ci3se3 "'..r, addition to Lending stressc_d ar." c.. 'e .ej T e mod'ili can be calcuAtI: with little error by considc: 22' oy those plies vh.ch are parallel to the length of the strip, 3. Rectangular pLate. Un:,.formy o.stributed .5. Edges simply support d,. c.a=.3 deflections The method presented blowi may be taken to apply if the loads are such "hat the deflections do rnot exceed the thickness of the plate. Appreciable direct stresses develop at deflections of the order of ag nituC'e of the thickness of the plate and the deflection will be less tlan that found by the method presented. For a plate whose length exceedS its breacdth 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 given plate of width a and length b, calculate first the central deflection W of a similarly loaded very long plate (infinite strip) of width a and of the same con struction. Now w X 0.99 Za (1) W 32 = o.l547 IL (1) El E1h3 Elh3 Except for the factor 0.99, which expresses the .....te effect (in wood practically negligible), this is the formula for the central deflection of a beam of unit width under a uniformly distributed load. Then the deflection at the center of .he given plate can be found approximately from the formula wo = f W( where f is a factor to be tV?. from the curve of fiche 3 corresponding to the argument 3 b (3) a a \E * Repo 'I No, 1500 The points shown in figure 5 were determined by an exact method, using the elastic constants of spruce, for various types of plywood. The curve is merely a smooth average curve determined by these points. A consideration of the extended analysis discloses the fact that the essen tial factors determining the central deflection of a plate under the con ditions of loading and support of this section are the two moduli E1 and E2 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 3.3 Hence it appears that this curve may be used for plywood of the type described at the beginning of this sum mary, independently of the species of wood used. The constants E1 and E2 must be known. They can be determined by calculation or by static bending tests of strips of matched material. The maximum shown in figure 5 in the vicinity of B/a = 2, which at first sight appears 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. Because of the approximations involved, the deflections calculated from his results are too small, a fact which he recognized would be the case. 4. Rectangular plate. Unifor.rjly distributed load. Edges clamped. Small deflections. In this case the central deflection of the corresponding infinite strip is given by w= 0.99 = 0.0309 (4) 32 Elhh 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 (5). The points shown near the curve in figure 4 are the exact values of f for various types of ply wood. The elastic constants of spruce were used, but the curve may be usd for wood of other species as pointed out in section 3. 3It was convenient to calculate the points shown in some of the figures for pl^ts in which the plies are all of the same thickness but the formulas and curves of this report are not restricted to such plates. 4 Hardwood Record, May 1957. Berort No. 13:0 The actual deflections will usually be considerably larger 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 length is moderately greater than its breadth. 5. Rectangular Tlate. Load concentrated at the center. Edges simply supported. Small deflections. The central deflection of the corresponding infinite strip is given by 1A 2 S1.051 x 6 x 0.99 jE)(6) Pa 0.8 Tr 2 E(6) 0.252 E) 1 a p (7) \E2/ Ej^ The central deflection of a finite )plate can be found from the formula wo = f W (8) where f is to be taken from the curve of figure 5. In formula (6) the number 0.8 is an approximate figure for a constant whose values may rangre from 0.76 to 0.86, for various types of ordinary plywood. A means of calculating this constant is to be found in the extended report. 6. Infinite str..p. (Long, narrow, rectangular plate). Uniformly distributed load. Edges simply supported. The formulas of this and the next section are applicable when the maximum deflection is small in comparison with the width of the strip, although possibly equal to several times the thickness of the strip. In addition to the conditions stated in the heading, the edges are assumed to be restrained from moving inward. Using the notation p I2 (9) ELh4 the following approximate relation connects the load and the central deflection W. P = H ( W h (10) b( h Report No. 1300 where H = 6.46 El/EL (11) K = .8 E (12) where 1 denotes Young's mcdulus in the direction of the grain of the wood., T the plywood is made of wood of mcre tanr. one species, equation (10) can be multiplied through by EL. There results a relation in which only the aod;AlJ E1 and Ea enter., t..;.< mean direct stress is given by the formula h )2 W 2 g = 2.60 E ( ~ ( ) (13) This is the mean direct stress averaged over the thickness of the plate. The direct stress in any ply can be calculated by observing that the str=;s in any ply is proportional to the E of that ply in a direction parallel to the Xaxis. This follows from the fact that the strain associated with the direct stress is constant across the thickness of the plate, Th' maximum bErlur stress in a face ; ly an be calculated by the approximate formula s = 1.01 a E, a (14) x a E where a is to be taken from the curve of figure 6. In this figure the argument n is connected with the deflUtion hw the formula i =2.778 f/2 (15) The bending stress at any point in any other ply can be calculated by ,ioting tbut the associacd strain varies linearly with the distance from the "Iutral plane ar. that the corresponding stress is the Tprod1ct of this strain ;rd Ex at the point under 'cr Ideration. The formulas 'ust given can be used for a plate whose length exceeds its breadth by a modercte amcunt and for small as well as large de'ILe. tions. An inspection of figure 5 indicates that these formulas can be used with small error if N1/4 B b ' a b E2 is greater than 1.75. It is to be expected that the \ ,/ Report 1. 15CO 6 stresses clIculated in this way will be atisfactory approximations to the stresses in the central oron of E..ch. a plate. 7. Infinite strip. ( ' Uniformly distributed load. Lde.5 i, ed. The edges are further assued to be restrained from moving inF ward. Using the notation P pal EL4 the following approximate equation holds P = (h) +K () (16) where H = 32.3 E (17) Fa K 23.2 E (18) E'L Equation (16) may be multiplied thrc_gh by EL and thus be made applicable if the plywood is made of wood c: two or more different sLecies. LSee discussion following (10) . The mean direct stress is obtained from the approximate formula g : 2.51 E ( 2 ) 2 (19) and the naximum beading stress in a face ply from the approximate formula S 1.01 a E ( a ) ( ) (20) x a h Report No. 1300 where a is to be taken from the curve of figure 7. In this curve, the argument 'n is connected with the deflection by the formula 1 2.732 (y w I7EJ ) (21) An inspection of figure 4 justifies the conclusion that the formulas just written can be used to calculate approximately the central deflection and the stresses in the central portion of a plate for which ( E1 /4 \E2 / is greater than 1.5. RcTort No. 1300 B b a a 0 SIDE AND b a X x FIG. I ,AXES DESIGNATIONS FOR A RECTANGULAR PLATE FIG. 2 SECTION OF A PLYWOOD STRIP SHOWING NUMBERING OF PLIES Z l 397oi F 0.9 0.8 0.7 S 0.6  k 0.5 0.4 0.3 02  O.I 0 0 FIG. 3 F,4CTOR f[FORMULA (2) ] AS A FUNCTION OF B/5a, WHERE B=b(E,1/E%. UNIFORM LOAD. EDGES SIMPLY SUPPORTED. Z U 39762 F / 2 3 4 5 6 7 5/a, 4 FUNCTION OF THE DIMENSIONS OF THE PL,4TE S U.b q  0.5   ~LEGEND: EXACT METHOD ALL PLIES SftYME THICKNESS 0.4 03X 03Y A5X A5Y 0.3 0.2  o./ 0 O I   0 / 2 3 4 5 8/a, 4 FUNCTION OF THE DIMENSIONS OF THE PLATE FIG. 4 FACTOR f [FORMULA (5) ] 4S 4 FUNCTION OF B/a, WHERE B = b (E,/E2). UNIFORM LOAD. EDGES CLAMPED. Z h 39753 F ~~1 ft U.Y " 0.8 0.7 0.6  s0.5 0.4 0.3 LEGEND: 0 }ALL PLIES SAME THICKNESS. 0 0FACE PLIES ONEH4ALF AS THICK / AS REMAINING PLIES. 0.1 O I I  0 0 I Z 3 4 5 b/a, f FUNCTION OF THE DIMENSIONS OF THE PLATE FIG. 5 FACTOR f/[FORMULA (8) ] AS A FUNCTION OF 8/a, WHERE B=b(E,/E2). CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED. Z ) 47o4 F n f*A U  0  4.6 4.2 _ 4.0 0 2 4 6 8 10 VALUE OF ;7 F16. 6 THE COEFFICIENT d IN THE FORMULA (14) 4S A FUNCTION OF y 50 4 0  30 L. IZO 10 0 0 4 6 8 /0 VALUE OF FIG. 7 THE COEFFICIENT a IN THE FORMULA (20) AS A FUNCTION OF 7 Z 1 39765 F UIJIVERSITY OF FlI OROIDA 3 1262 08866 5913 