KW.i 
 
 summary or 
 
 fORMLLAS POR LLAT PLATES Of 
 
 PLYWOOD UNDER UNIFORM OR 
 CCNCENTRATEP LCAPS 
 
 Infcrmaticn Reviewed and Reaffirmed 
 
 January 1953 
 
 INFORMATION REVIEWED AND 
 
 REAFFIRMED JUNE 1959 
 DATE OF ORIGINAL REPORT 
 FEBRUARY 13, 1942 
 
 
 Nc. 13CC 
 
 U.S. DbKUanu^n 
 
 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 Forest 
 Products Laboratory has included as one phase the mathematical 
 analysis of the deflection of flat plates under uniformly dis- 
 tributed or concentrated loads. This theoretical analysis has 
 progressed sufficiently to permit the publication of the formulas 
 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 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 Laborator}"- 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/sumOOfore 
 
SUMMARY OF FORMULAS FOR FLAT PLATES OF PLYWOOD 
 UNDER UNIFORM OR CONCENTRATED LOADSi 
 
 By 
 
 H. W. MARCH 
 
 Special Consulting Mathematician, 
 Forest Products Laboratory, 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 should be made to the extensive 
 report for the derivation and discussion of the results contained in 
 this summary. 
 
 Rectangular plywood plates will be considered in which the direc- 
 tions of the grain of the wood in adjacent plies are mutually perpendicular, 
 and perpendicular 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 
 adjacent 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. 
 
 Notation 
 
 a = width of plate 
 b = length of plate 
 h = thickness of plate 
 
 v o = deflection at center of plate 
 
 p = load per unit area 
 
 P = concentrated load (section 5) 
 
 p = -par /Ejjnr (sections 6 and 7) 
 
 W = deflection at center of an infinitely long plate 
 of width a -under a specified type of load 
 
 iA 
 
 where Ej_ and E 2 are defined in 
 section 1 
 
 ±This mimeograph is one of a series of progress reports issued by the Forest 
 Products Laboratory to aid the Nation's defense effort. Results here re- 
 ported are preliminary and may be revised as additional data become available, 
 
 o 
 
 -Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 
 
 Report No„ 1300 Agriculture -Madison 
 
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 modulus E^ defined by the equation 
 
 E 
 
 i 1 a ywA 
 
 where the summation is extended over all of the plies numbered, for example, 
 as in figure 2: (E x ) i is the Young's modulus of the i** 1 ply measured in a 
 
 direction parallel to the length of the strip; 1^ is the moment of inertia, 
 
 with respect to the neutral axis, of the area of~~the cross section of the i*" 
 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 - \P /12 for a strip of unit width. An approxi- 
 mate formula in which the error is very slight is obtained for E^I by taking 
 
 the sum of the products (Ey)^!^ formed for only those plies in which the 
 grain is parallel to the length of the strip. Exception is to be made of a 
 three-ply 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^_ 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, 
 
 v-5jva 
 
 determines the stiffness of strips parallel to the side b. 
 
 As in the case of E-j_ the calculation of Eg can be based on the 
 
 parallel plies only, except in the case of a three-ply strip having the 
 grain 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 
 parallel to the X-axis or to the side a of the rectangular plate of figure 1. 
 
 The mean modulus E in tension or . compress ion may be defined by 
 the equation — 
 
 E a h = 
 
 <Vi h i 
 
 where (E x )^ has the same meaning as in section 1, hj is the thickness of 
 the i tn ply and h is the thickness of the strip. 
 
 Report No. 1300 -2- 
 
In like manner for a strip parallel to the side b 
 
 V'JiVft 
 
 The moduli E a and Ev are needed in cases where the deflections 
 
 of the plates under consideration are so large that direct stresses in 
 addition to bending stresses are developed. These moduli can be calculated 
 vith little error by considering only those plies which are parallel to the 
 length of the strip. 
 
 5 • Rectangular plate. Unifo rml y Distributed l oad , 
 
 Edges simply supported, ginc.ll deflections , 
 
 The method presented below may be taken to apply if the loads are 
 such that the deflections do not exceed the thickness of the plate. 
 Appreciable direct, stresses develop at deflections of the order of mag- 
 nitude of the thickness of the plate and the deflection will be less than 
 that f ound by the method presented. For a plate whose length 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 w Q 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 = 1x0,99 h!L = 0.151*7 WL (1) 
 
 Except for the factor 0.99, which expresses the plate 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 the given plate can be 
 found approximately from the formula 
 
 w Q = f W (2) 
 
 where f is a factor to be taken from the curve of figure 3 corresponding 
 to the argument 
 
 a 
 
 g _ b ( El V» (5) 
 
 Report No„ 1300 -3- 
 
The points shown in figure 3 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 E^ and 
 
 Eg 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.*^ 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 E^ and E^ 
 
 must be known. They can be determined by calculation or by static bending 
 tests of strips of matched material. 
 
 The maximum shown in figure 3 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. 
 
 U. Rectangular plate. Uniformly distributed load . 
 Edges clamped. Small deflections . 
 
 In this case the central deflection of the corresponding infinite 
 strip is given by 
 
 w . 2^22 jsj . 0.030O jgj ( i0 
 
 The deflection of a finite plate can be found from the formula 
 
 w Q = f W (5) 
 
 in which f is to be taken from the curve of figure U where it is shown as 
 a function of B/a , B being related to b by (3). The points shown near 
 the curve in figure h are the exact values of f for various types of ply- 
 wood. The elastic constants of spruce were used, but the curve may be 
 used for wood of other species as pointed out in section 3. 
 
 lit was convenient to calculate the points shown in some of the figures 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. 
 
 k 
 
 -Hardwood Record, May 1937. 
 
 Report No. 13C0 -U- 
 
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 plate. Load concentrated at the center . 
 
 Edges simply supported. Small deflections . 
 
 The central deflection of the corresponding infinite strip is 
 given by 
 
 W = 1-051 x 6^x 0.99 ( Ei ) _p£ (6) 
 
 E x h 5 
 
 (7) 
 The central deflection of a finite plate can be found from the 
 
 formula 
 
 w Q = 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 range 
 from O.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 strip. (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 = JsJL <9) 
 
 the following approximate relation connects the load and the central 
 deflection W. 
 
 P = H ( jf)f K ( \ >' (10) 
 
 Report No. 1J00 -5- 
 
where 
 
 H = 6.k6 \/\ (11) 
 
 K = 20.8 E^ (12) 
 
 where E^ denotes Young's modulus in the direction of the grain of the 
 
 wood. If the plywood is made of wood of more than one species, equation 
 (10) can be multiplied through by E^. There results a relation in which 
 
 only the moduli E]_ and E & enter. 
 
 The mean direct stress is given by the formula 
 
 s . 2.6o E a ; g f ( I , s (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 
 stress in any ply is proportional to the E of that ply in a direction 
 parallel to the X-axis. This follows from the fact that the strain 
 associated with the direct stress is constant across the thickness of 
 the plate. 
 
 The maximum bending stress in a face ply can be calculated by 
 the approximate formula 
 
 s = 1.01 a E _ (£) £ (Ik) 
 
 x a h 
 
 where a is to be taken from the curve of figure 6, In this figure the 
 argument ^ is connected with the deflection by the formula 
 
 n - 2.778 (jfcY A g (15) 
 
 The bending stress at any point in any other ply can be calculated 
 by noting that the associated strain varies linearly with the distance 
 from the neutral plane and that the corresponding stress is the product of 
 this strain and E at the point under consideration. The formulas just 
 
 given can be used for a plate whose length exceeds its breadth by a 
 moderate amount and for small as well as large deflections . An inspection 
 of figure 3 indicates that these formulas can be used with small error if 
 
 B „ fay* 
 
 ~ ■ ~ V E 2 / * s greater than 1.75. It is to be expected that the 
 Report No. 1300 -6- 
 
stresses calculated in this vay will be satisfactory approximations to 
 the stresses in the central portion of such a plate. 
 
 7 . Infinite strip. (Long narrow plate ) . 
 
 Uniforml y dis tributed load. Edges c lapped .■ 
 
 The edges are further assumed to be restrained from moving 
 inward. 
 
 Using the notation 
 
 h 
 
 D _ 
 
 pa 
 
 h hk 
 
 the following approximate equation holds 
 
 where 
 
 P = H (g) + K (£) 5 (16) 
 
 E l 
 H = 52.3 - (17) 
 
 Ea 
 
 K = 23.2 -2 (18) 
 
 Equation (16) may be multiplied through by El and thus be made applicable 
 if the plywood is made of wood of two or more different species. (_See 
 discussion following (10) ~j . 
 
 The mean direct stress is obtained from the approximate formula 
 
 h 2 u 2 , . 
 
 g = 2.51 E a (|) {\ ) (19) 
 
 and the maximum bending stress in a face ply from the approximate 
 formula 
 
 S = 1.01 a E (- ) 2 ( ^ ) (20) 
 
 x a h 
 
 Report No. 1300 
 
where a is to be taken from the curve of figure 7- In this curve, the 
 argument r\ is connected with the deflection by the formula 
 
 i ■ s -"2 hr s (si) 
 
 An inspection of figure k 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 
 
 B b (s X f U 
 
 a = a I e" i is 6 rea-ter than 1.5. 
 
 \ 2 / 
 
 Report No. 1300 -8- 
 
FIG. 1 
 SIDE AND AXES DESIGNATIONS 
 FOR A RECTANGULAR PLATE 
 
 FIG. Z 
 SECTION OF A PLYWOOD STRIP 
 SHOWING NUMBERING OF PLIES 
 
 2 M 397ol F 
 
/./ 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.1 
 
 X 0.6 
 *A5 
 
 0.4 
 
 0.3 
 
 02 
 
 0.1 
 
 
 
 
 
 
 A /-N 
 
 
 
 
 / 
 
 • 
 
 
 
 
 
 o — 
 
 
 /• 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 
 
 
 
 
 
 
 
 
 LEGEND: 
 EXACT METHOD 
 ALL PLIES SAME THICKNESS 
 o-3X 
 • -3Y 
 &-5X 
 A -5Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 13 4 5 6 1 
 
 B/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE 
 
 8 
 
 FIG. 3 
 FACTOR f [FORMULA {z) ] AS A FUNCTION OF B/a, WHERE B*b(E,/£ 2 )* 
 UN/FORM LOAD. EDGES SIMPLY SUPPORTED. 
 
 z 1/ 39762 F 
 
/./ 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.1 
 
 S 0.6 
 "< 0.5 
 
 0.4 
 
 0.3 
 
 o.z 
 
 0.1 
 
 
 4 
 
 
 
 
 
 TA 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 LEGEND: 
 EXACT METHOD 
 ALL PLIES SAME THICKNESS 
 o- 3X 
 • -3Y 
 A-5X 
 A- 5Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12 3 4 5 
 
 B/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE 
 
 FIG. 4 
 
 FACTOR / [FORMULA (5) ] AS A FUNCTION OF B/a, 
 WHERE 8^b{E,/E z )*. UNIFORM LOAD. EDGES CLAMPED. 
 
 z u 397o3 F 
 
/./ 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.1 
 
 S 0.6 
 
 5 
 
 ^ 0.5 
 
 0.4 
 
 0.3 
 
 0.2 
 
 0.1 
 
 
 ft 
 
 
 
 
 
 
 
 \ 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1} ALL PI 
 
 • -FACE 
 AS 
 
 LEG 
 
 JES SAME 
 
 PLIES ONl 
 REMAIN/I 
 
 END: 
 
 TMCKNE 
 
 C .-HALF AS 
 iG PLIES. 
 
 ss. 
 
 THICK 
 
 
 
 
 
 
 
 I Z 3 4 5 
 
 B/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE 
 
 FIG. 5 
 
 FACTOR f [FORMULA Id)] AS A FUNCTION OF B/a, WHERE 
 B*b(E,/E 2 )*. CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED. 
 
 Z 11 39764 F 
 
4 6 
 
 VALUE OF rj 
 
 FIG. 6 
 THE COEFFICIENT a IN THE FORMULA (l4) 
 AS A FUNCTION OF 7] 
 
 4 6 
 
 VALUE OF rj 
 
 10 
 
 FIG. 7 
 THE COEFFICIENT a IN THE FORMULA (lO) 
 
 AS A FUNCTION OF 7] 
 Z U 397t>5 F 
 
UNIVERSITY OF Fl ORIDA 
 
 3 1262 08866 5913