ELAT PLATES Of PLYWOOD UNDER UNIFORM OR CONCENTRATED LOADS March 1942 INFORMATION REVIEWED AND REAFFIRMED March 1956 L- U.S. DlkOSi iO" i THIS REPORT IS ONE Or A SCRIES ISSUED TO AID THE NATION'S WAR PROGRAM No. 1312 UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison 5, Wisconsin In Cooperation with the University of Wisconsin Digitized by the Internet Archive in 2013 http://archive.org/details/flatplaOOfore FLAT PLATES OF PLYWOOD UNDER UNIFORM OR CONCENTRATED LOADS" By H. W. MARCH Head Mathematician Table of Contents Paee Introduction 3 Assumptions made regarding properties and structure of wood. k Section 1. The elastic "behavior of wood 5 Plates with small deflections 6 Section 2. The differential equation for the deflection of a plywood plate 6 Section 3- Rectangular plate under uniformly distributed load. Edges simply supported 9 Form of the deflected surface 11 Central deflection of a plate expressed as a fraction of that of the corresponding infinite strip. Determination of W /h from a curve 1^ Tests Ik Section h. Rectangular plate under uniformly distributed load. Edges clamped : 15 Form of the deflected surface 16 Central deflection of a plate expressed as a fraction of that of the corresponding infinite strip 17 Tests 13 Section 5- Infinite strip. (Long narrow rectangular -elate.) Load concentrated at a point. Ede-es sijrroly supported 19 Section 6. Infinite strip. Uniform load applied over a small area- Edges simply supported 19 Section J. Rectangular plate. Load" concentrated at a point or apolied over a small area. Edges simply supported 20 Central deflection of a plate expressed as a fraction of that of the corresponding infinite strip 21 Plates with large deflections 22 Section g. Differential equations for the deflection of a plywood plate 22 * This mimeograph is one of a series of progress reoorts issued by the Forest Products Laboratory to aid the Nation's defense effort Results here reported are preliminary and may be revised as additional data become available. Mimeo. No. 1312 Section 9* Infinite strip (Long narrow plate). Uniformly distributed load. Edges simply supported ?3 Section 10. Infinite strip. (Long narrow plate). Uni- formly distributed load. Edges clamped 2b Section 11. Rectangular plate. Uniformly distributed load. Edges simply supported. Approximate method. . . 28 Section 12. Rectangular plate. Uniformly distributed load. Edges clamped. Approximate method 3" Appendix 1. Stress-strain relations in orthotropic material 3^ Appendix 2. The differential equation for the deflection of a plywood plate. Small deflections 35 Appendix 3- Rectangular plate under uniform load. Edp-es simoly supported. Small deflections k° Appendix U. Rectangular plate under uniform load. Edges clamped. Small deflections 50 Appendix 5- Infinite strip. (Long narrow rectangular plate). Load concentrated at a point. Edges simply supported 56 Appendix 6. Infinite strip. (Long narrow rectangular plate). Uniform load applied over a small area. Edges simply supported o2 Appendix 7* Rectangular plate. Load, concentrated at p point or applied over a small area. Edges simply supported Appendix 8. Differential equations for the deflection of a plywood plate. (Large deflections) 06 Appendix 9. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges simply supported 70 Appendix 10. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges clamped 82 Appendix 11. Rectangular plate. Uniformly distributed load. Edges simply supported. Approximate method 87 Appendix 12. Rectangular plate. Uniformly distributed load. Edges clamped. Approximate method Appendix 13. General notation 9^ ta 1 to 5°- 59 ires 1 to IS 1-7 -2- Introduction This report presents the results of a study made by the Forest Products Laboratory of the behavior of flat plates of plywood under uniform or concentrated loads. The information concerning the elastic properties of plywood developed in this study will, it is hoped, be useful in the treatment of further problems such as the buckling of flat and curved plywood panels. After a discussion of the elastic properties of wood the re- mainder of the report is divided into two main portions, one dealing with plates under such loads that the deflections are small, the other with plates under such loads that the deflections are large. For plates with small deflections, direct stresses ("membrane stresses") throughout the thickness of the plate due to the deformation of the middle surface, are negligible and bending stresses only need to be considered. These consist in tensile stresses on one side of the plate and compressive stresses on the other side. For a plate to be considered in the class of those with small deflections the deflection must usually be less than the thickness of the plate, in certain cases less than one-half of this thickness. For larger loads and consequent larger deflections direct stresses are developed to such an extent that they cannot be neglected. The loads are carried partly by such stresses and partly by bending stresses. In such cases the linear relationship between load and deflection, which holds for small loads, is no longer maintained. To the reader who does not wish to follow the mathematical analysis it should be pointed out that the majority of the results are presented in forms which do not presuppose for their application a know- ledge of their theoretical derivation. It is suggested that such a reader should first become familiar with the significance of the two mean Young' s moduli in bending (Section 2) , after having read Section 1 on the elastic behavior of wood. The curves of figures 10, IS, and 26 can then be used to determine the maximum deflections of plates with small deflections under uniformly distributed loads or concentrated loads. Correspondingly simple means of determining maximum bending stresses have not been worked out. However, for uniformly loaded plates whose lengths are greater than moderate multiples of their breadths, the stresses in the central nortions can be easily approximated by calculating the stresses in similarly loaded infinitely long strips to which essentially the simple beam formulas are applicable. The details of the procedure are explained in Section 3. For stresses in the vicinity of concentrated loads the methods of Sections 5 and 6 are available. The tables for bending moments have been calculated only for three-ply plates. Formulas requiring considerable computation are given by which the moments can be calculated for other types of plates. It appears that bending moments in the vicinity of the loads can in most cases be calculated with sufficient accuracy by considering the plates to be infinitely long. For plates with large deflections the approximate formulas of Sections 9 and 10 for long narrow plates can be used for uniformly 1312 distributed loads to find both the maximum deflection and stress in a plate whose length exceeds its breadth by a moderate amount. In Sections 11 and 12 approximate formulas are given for the maximum deflection of a plate of smaller length-breadth ratio with large deflection under a uniformly dis- tributed load. The tables presented for different types of plates enable one to estimate the ratio of length to breadth beyond which a plate may be considered as a long narrow plate for the purpose of calculating de- flections and stresses in its central portion. The formulas of these last two sections are to be considered as only moderately accurate approxima- tions, permitting an estimation of the relation between deflection and load. Assumptions Made Regarding Properties and Structure of Wood. In the analysis wood is taken to be an orthotropic material, i.e., a material having three mutually perpendicular planes of elastic symmetry. 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 report are not intended to apply directly to partially or completely impregnated plywood or compregna.ted wood, although it is to be expected that many of the results can be applied to such material. Under these assumptions the differential equations are set up for the determina- tion of the deflection of plywood plates for the cases of both small a.nd large deflections. The form of the differential equation for the deflection of an orthotropic plate, in the case of small deflections, is well known.- For plywood plates, which are made up of layers of orthotropic material, tne co-- efficients in the differential equation are given in terms of the elastic constants of the constituent wood in the author's paper,— "Bending of a Centrally Loaded Rectangular Strip of Plywood." The derivation of these coefficients under certain simplifying assumptions is £iven in the present report. Although actual plywood will seldom possess the structure assumed as ideal, nevertheless the procedure used in arriving at the coefficients brings to light the essential factors determining the stiffness an^ other elastic properties of plywood. Then, in a given situation a rational allowance for the effects of variation from the ideal structure can be mad.e. The principal results are given in the body of the renort wnile the mathematical analysis leading to them is placed in a series of appendixes whose numbers are the same as those of the corresponding sections of the text. r ~~ Gehring, P., Dissertation. Berlin, ISoC; Voigt, W. , Theor. Phys. I, p. 451, 1895; Huber, L". T. , Bauingenieur 4, pp. 3^4 and 392, 1923, 6, pp. 7 and 46, 1925, Zeits. fur ang. Hath. u. i.iech. 5, 22?, 192b; Iguchi, S. , Eine Lo'sung ftlr die Berechnung der biegsamen rechteckigen Platten. Berlin, Springer, 1933; Seydel, 3. , Zeitsch, f. Flugtechnik u. Ilotorluftschiffahrt, 24, No. 3, 1933. 2 "Physics 7, 32-41, 1936. ■4- 1312 In the numerical calculations, made to illustrate the applica- tion of the formulas, it is necessary to use a species of wood for which the appropriate elastic constants are known. For plywood with flat-grain plies, five of the twelve elastic constants of the wood of the species under consideration are needed, namely, two Young's moduli, two Poisson 1 s ratios, and one modulus of rigidity. The elastic constants of spruce have "been determined carefully. • It is for this reason that the illus- trative calculations have been made for plywood plates of spruce. In a number of instances the numerical results are expressed in a form involving a factor containing Young's modulus along the grain. It is to be expected that satisfactory values of the corresponding results can frequently be obtained for plywood made from wood of another species by replacing the Young' s modulus along the grain for spruce by the corre- sponding modulus for the second species. When this is done it must be realized that the assumption is made that the elastic moduli all change in the same ratio in passing from one species to the other and that the relevant Poisson' s ratios are the same for the two species. Experience will show that considerable variations from this assumed relationship can occur without greatly affecting certain of the results. However, the basic formulas take into account the elastic properties of the particular species under consideration. The behavior of plates made up of plies of wood of two different species can also be determined with the aid of these formulas. Section 1. The Elastic Behavior of Wood The visible structure of wood suggests that it may be considered to nave three mutually perpendicular planes of elastic symmetry, namely, the planes perpendicular to the longitudinal, radial, and tangential directions, respectively, as shown in figure J>. A substance having such properties of elastic symmetry is said to be orthotropic. If wood is orthotropic it will have (see appendix 1) three Young's moduli, Ex , Ep, and E m , the letters L, R, and T denoting the longitudinal, radial, and tangential directions, respectively; three shearing moduli ^-jjj- , PxRj and l^-pij, ; and six Poisson' s ratios, (Jim, C^L' G RT > a TR; a LR and tf^L ^here , for example, Cf-^n is the Poisson' s ratio associated with tension parallel to the direction L and contraction parallel to the direction I . Among these 12 constants there are three relations of the type (see (1.7) in appendix 1). E T cf - 2 a L T^ -T LT . Table 1, giving the valuesof these constants for several species Of wood as determined on the assumption that wood is orthotropic , is taken from a report by C. P. Jenkin "Report on Materials of Construction Used in Aircraft," Aeronautics Research Committee (London, 1920).! -See also E. Carrington, Phil. Mag. kl, 206, gl|g, 1921; 1+3, 871, 1922; . V+, 2gg, 1922; U5, 105, 1923- """ 1312 ~ 5 ~ The values given below of the elastic moduli of Douglas-fir at 10 percent moisture content, were obtained from a limited number of testsi Because of the small number of tests, these values are to be considered as tentative. Lb. /in. 2 3L = 1,960,000 Et = 113,200 Br = 155,800 M-LT = 123,800 ^LK " 110,600 ^pm - 7,100 The Poisson's ratios ffrn and C^t have not been determined. In the cal- culations made later in this report they are taken to be the game ^s for spruce. It is probable that the error thus introduced is not large. The assumption that wood may be treated as an orthotrocic material is reasonably well confirmed^- by the experimental evidence at present available. Plates with Small Deflections Section 2. The Differential Equation for the Deflection of a Plywood Plate. Small Deflections In deriving the differential equation for the deflection of a plywood plate the usual assumptions underlying the theory of thin plates are made. In addition, wood is taken to be an orthotropic substance and the following assumptions are made concerning the structure of the plywood: The material of the individual plies is accurately flat-grain, at is, the directions of the grain and of the annual rings are parallel to the faces of the -plies. The directions of the grain in adjacent nlies art perpendicular to each other and parallel or perpendicular to the respective edges of the plate. The analysis apnlies equally well to edg wood. It is only necessary to substitute R for T throughout in the subscripts of the elastic constants. Each ply is homogeneous. This implies tnat the variations of th£ elastic constants from springwood to summerwood are disregarded and aver* values of the constants are used. sik. 12, 369, 1931. 1312 Tec -0- The plate is symmetrical, both geometrically and as to arrange- ment and properties of the naterial, with respect to the plane z - 0, the axes of coordinates being chosen as in figure 1. If a plate is not of symmetrical construction- with respect to the plane z = approximately correct results should be obtained by using in place of the flexural rigidities D-j_ and D2 the flexural rigidities of strips of unit width parallel to the edges of the plate. The elastic constants of the wood are the same in all plies. This assumption can be omitted without materially complicating the dis- cussion, provided that the other assumptions are retained. Let h denote the thickness of the plate and p the load per unit area acting normal to the face z *= -h/2 in the direction of the positive axis of z in figure 1. The deflection w of points in the middle surface satisfies the differential equation2 dx H o-xby*- i> y + which in derived in appendix 2. The flexural rigidities D^ and Dg are proportional "go two "mean moduli- in bending" E-. and Ej as explained in appendix 2 where the plate is .^assumed to be of symmetrical construction with respect to the middle plane, z = o. Equation (2.12) may be expected to apply with small error to plates of unsymmetrical construction if E-j_ and E2 are determined from the flexural rigidities of strips of unit width parallel to the X and Y axes, respectively. Thus, for a strip of unit width parallel to the X-axis, E, is defined by the equation E, I = 2 (E ) . I. 1 v x'i 1 5 -An equation of this form for orthotropic material is well known. See, for example, the references to Ruber on page k. The dependence of the coefficients in this equation upon the elastic constants of wood and upon the structure of the plywood plate is discussed in appendix 2. -Price, A. T., Phil. Trans. A. 228, 1, 1928. The definition here given differs somewhat from that used by Price for the apparent Young's modulus in bending. The essential difference is in a term whose value is small. See his equations (13.82) and 'his discussion of plywood on pages 50 and 52. 1512 .7. where the summation is extended over all of the plies; (E. r ). is the Young's modulus of the i^h ply measured parallel to the X-axis; J. is the moment of inertia with respect to the neutral axis, of the cross section of the i u made by a plane perpendicular to the X-axis; and I = h^/12 is the moment of inertia of the entire cross section with respect to the central line z = 0. An approximate formula in which the error is very slight is obtained for EtI by taking the sum of the products (E X K 1* 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. The flexural rigidity E~I is calculated in a similar way. These definitions for E-i and Ep ma y also be applied in dealing with plates of symmetrical construction. In the case of such plates they are identical with the definitions of equations (2.18) and (2.19) of Appendix 2. In order to have a definite situation in mind it will be assumed from this point on, unless the contrary is explicitly stated, that the plates are of symmetrical construction and that all plies are of the same thickness .^ However, the application of the formulas obtained is not limited to plates of this assumed structure. Irregularities in the state of stress at the junction of two riies were neglected in deriving equation (2.12). This situation is dis- cussed, on pages lU, 15, 50. a n( i 51 of the paper by Trice, to which reference is made in footnote- 6. The effect of these irregularities in the state of stress would be to increase slightly the flexural rigidities of the plate above those calculated from the mean moduli E]_ and Eg. These effects could not be clearly detected in a long series of static bending tests of strips of plywood and are, therefore, considered to be so small that they may be neglected. With the elastic constants for spruce and for Douglas-fir as previously given the values for E]_ and Ep in tables 2 and 3 were found, the grain of the face plies being parallel to the X-axis. form where The differential equation (2.12) can be reduced to the simpler w ■ w w P ,- ^. r— rr + 2 x P P + r-zr = -w— (2.25) x = K /(D X D 2 ) 1/P (2 ; g6) by making the substitution t] = ey (2.27) -8- where s = C^/Dgj 1 /* = (E 1 /E 2 ) l/if (2.2?) The equation (2.25) ( or 2.12) is to be solved under appropriate boundary conditions to determine the behavior of a given plate under a given load. In deriving this differential equation it has been assumed tnat the deflections are so small that direct stresses are not developed to an appreciable extent. This implies maximum deflections of, roughly speaking, less than one-half of the thickness of the ulate. After equation (2.25) has been solved for the deflection as a function of the coordinates, the components of stress can be found with the aid of equations (2.5). Or the stresses can be expressed in terms of the bending and twisting moments which in turn are expressed in terms of the deflection in equation (2.23). Section 3« Rectangular Plate Under Uniformly distributed Load. Edges Simply Supported- ^-' - In this case the edges of the plate x = o , x = a, y= o, and y = b ; are simply supported and the load p per unit area is a constant, the same at all points of the plate. The deflection is found as the solution of the differential equation (2.25) subject to the conditions that on the edges x = o and x = a, the deflection w and the bending, moment in x vanish and that on the edges y - o and y = b the deflection w and the bending moment m v vanish. The plate is assumed to be held down at the corners. ~The exact solution (3. 15) mentioned briefly in the earlier part of this section and discussed in greater detail in appendix 3 is due to D. 3. Zilmer who, as a. graduate student at the University of Wisconsin on a fellowship supported by the Forest Products Laboratory, undertook the solution of this problem and that of the clamped plywood elate (see section U) at my suggestion. The calculations based on this exact solution were performed by the Computing Division of the Laboratory. The present' author is responsible for the approximate method and for the discussion of this section and that of appendix J,. Solutions of the problem for the simply supported plate of ortho- tropic material have also been given by Huber and by I^uchi in the rjapers to which reference was made in footnote 1. g -The reader who wishes a quick and easily applied method of finding the approximate deflection at the center of a plate should turn at once to page 13. -9- 1312 Since the differential equation is written with the variable t) instead of the variable y it is convenient to think of the plate as having been transformed into one having as edges the lines x = 0, x = a, 77 = 0, and 17 = J3 where /3 = eb (3.1) and to express the deflections and moments in terms of the variables x and .7 instead of in terms of the variables x and y. The deflection is found to be given by the equation (see appendix 3). n = 1, 3. . c •.Mere P = pa^/E^, k « 1 - y^cr^ (3.16) A n = n77/a (3.7) md T in defined by (3. 12) appendix 3. The bending moments m^. and my and the twisting moment n,,. f can be calculated from (3.15) with the aid. of (2.29) if appendix 2. The ratio of the deflection w at the center, to the thickness is readily calculated from (3. 15) a ^d found to be given by equation (3»1?) in appendix 3 in terms of an infinite series which converges so rapidly that it may usually be replaced by its first term only. Approximate formula for the deflection at the center . — As in many cases the satisfactory behavior of a plate will be determined by its deflection at the center, it is desirable to have a simple approximate formula for calculating this deflection. Such a formula can be obtained by making use of the fact that in a configuration of equilibrium the sum of the potential energy of deformation of the plate and that of the applied load is a minimum as compared with other configurations satisfying the same boundary conditions. In the application of this principle a plausible simple form is assumed for the deflected middle surface of the plate, this form containing the deflection w at the center as an undetermined para- meter. The potential energy of deformation of the plate in bending V^, and the potential energy of the load V^ are then calculated as a function of w . The total potential energy V of the system is the sum of V- D and Vjr ♦ On equating to zero the derivative of V with respect to w an equation is obtained connecting w with p, the applied load per unit area. A better approximation to the form of the deflected middle surface is obtained by assuming for it an expression containing several parameters 1312 -10- and determining all these parameters in such a way that the total potential energy is a minimum. In the fallowing analysis a second parameter 7 (or c ) wh»se significance will he explained is introduced in addition to the para- meter w , the defleotiT at the eenter. Let www sln_2L5 when f < p< "b-p w = w sin-iLSsi n ^ -a 2c when © < y < o, (See figs. 4 and 5.) An expression corresponding to the latter is assumed for the portion of the plate for which h-c, contains trigonometric functions of the variable 1 . This implies the possibility of a wave form alone any line X = constant and in particular along the central line x = %. "Because of the presence of this wave form the deflection at the center does not in- crease steadily to an asymptotic value with increasing ratio of length of plate to breadth. For a certain value of this ratio the central deflection has a maximum which is greater than the deflection at the center of a very long plate, under the same uniform load per unit area. The trigonometric terms are not present in the term Y n of the formula to which (3. 15) re ~ duces for the isotropic plate. Hence a wave form of the deflected surface along the line x = constant is not to be expected in this case. A wave form in the surface of an orthotropic plate under con- centrated load was noted by Huber. At my suggestion D. E. Zilmer— in- vestigated carefully the behavior of uniformly loaded plates of the type 3X. He found that terms after the first in equation (3. 15) could be neglected in studying an effect of the order of magnitude under con- sideration so that the wave form indicated by the first term of this equation could not possibly be obliterated by subsequent terms. He found that the central deflection considered as a function of k = -- , attained 3. a maximum value at k = 1.4-9 that was about 3 percent greater than the asymptotic value of this deflection for large k . This conclusion agrees with the results of table 5 for plate 3X. Table 7 for plate 5X shows that the maximum central deflection as a function of k = b/a occurs for k = 2 approximately and that this maximum deflection is 5-7 percent greater than the asymptotic value of the central deflection for larse k. In a number of tests with plates of commercial plywood, the surfaces were observed to take wave forms. The material of the plates w?s not sufficiently uniform to warrant comparison of the observed i: 'ave form with that predicted by the formula, since the effect predicted is so small that it would be easily masked by small variations in the material of the plates. In figures 6 to 9 are shown the deflection along the central line X = a/2 of a number of plates of commercial olywood under uniformly distributed load with differing ratios of length to breadth. In order to compare the shapes of these sections of the deformed surfaces, dial read- ings corresponding to the same central deflection are plotted and the distances from one end of the plates have been expressed as fractions of b, the length of the plate. The method of loading the plates and measuring the deflections is described in the latter part of this section. 9 -Thesis to be presented at the University of Wisconsin, 1312 - 12 ~ The curves for plates Nos. k (type 3X) , 6 (type 5X) , and S (type 5X) show clear indications of a change in shape of the deflected surface associated with the presence of a wave form. This effect does not appear clearly in the curves of figure 7 for plate No. 5 (type ^Y) . According to table 6 the maximum effect of wave form may be expected for larger values of k. = b/a in the case of plates of type 5Y than in the case of those of type 5X. Central Deflection of a Plate Expressed as a Fraction of That of the Corresponding Infinite Strip. Determination of W /h from a Curve . The deflection w at the center of a rectangular plate, simply sup-ported at its edges and under a uniformly distributed load, can be ex- pressed as a fraction of the deflection along the central line of an infinitely long plate similarly loaded. From this standpoint the deflection of a finite plate is regarded as that of an infinite strip multiplied by a corrective factor to take account of the effect of the ends of the finite plate. The deflection at the center of a uniformly loaded infinite strip simply supported at its edges is given by the formula v = 5 (1 - o LTgIL)paj. = 0.15^7 - P ±-.- (3-39) °~° 32 E lh 3 E lh 3 This is the formula for the central deflection of a uniformly loaded beam of unit width except for the factor in parentheses. This factor has been taken to be 0.99, the value which it has for spruce. If the deflection at the center of a finite rectangular plate is denoted by w we can write—- «o = Wo- <3 - 90) The factor 7 is found to depend almost entirely upon the value of p/a = (b/a)(E 1 /E P ) ' ', the ratio of the sides of the transformed plate and very little upon the type of plywood in the plate except insofar as this influences the value of the ratio 3 /a. The curve of figure 10, representing 7 as a function of p/a is a smooth average curve for points determined from the exact values of a in tables k - 9. These points are shown in the figure. More points than those obtained directly from the tables were secured by interchanging the 10 . A presentation of the results of an approximate analysis in essentially the form (3. 90) was made by Morris, C. B. , Hardwood 'Record, Way 1937'. Because of the approximations involved, the deflections calculated from his results are too small, a fact which he recognized would be the case. -13- 1312 J axes to which the plates in the tables were referred and utilizing the data of the tables to calculate the factor 7 for plates for which the ratios b/a of the actual dimensions were less than one. The factor 7 was also calculated from the results of the approx- imate formulas. This was done for plates 3X, J>Y , 5X, 5Y, and 7X with all plies of the same thickness and also for plates J>X, ^>X, and 7X with the face rlies one-half as thick as the remaining plies. All of these points are shown in figure 11. The curve in this figure is that of figure 10, namely, the average curve for points determined by the exact formulas. The curve is evidently sufficiently accurate for types of plywood similar to those under consideration in this report. The controlling elastic properties of the plate are manifested in the stiffness in the direction parallel to the X-axis which determines the deflection of the infinite strip, and in the ratio of the stiffnesses in the X and Y directions as it appears in the factor £ a (E n /E ) ' which is used to obtain P = e b of the transformed plate. Hence, to use the curve of figure 10 for a given plate it is only necessary to kno^ the t"o apparent Young's moduli in bending, E n and E of the plate. They can be determined from static bending tests or estimated from the structure of the plyood in the plate. The factor 7 corresponding to p/a = b/a (E^/ii^ ) can then be read from the curve. The central deflection vr Q is then given by W = ^o — ( 3' U °) where \.j is to be calculated by equation (3*39) • The curve of figure 10 can be constructed from the values in table 10. Tests— In table 11 are shown the results of a number of tests made with uniformly loaded plates of commercial plywood. A description of the method of making the tests will be given below. The factor 7 ^ r , was calculated as follows. The tests on a given plate yielded a mean value for the r^tio p/vr where p is the load per unit area and v Q is the deflection pt the center. The moduli E-, and E were determined by static bending tests on stripe cut from the plates after the tests on the plates themselves were completed. From the formula 5(1 -o LT a TL ) p / '°™~ '" 32 Elh 3 n tests described here and in sections U and 7 * r: cprri>. d out under the direct supervision of Alan D. Freas, Assistant Engineer. 1312 -1U- we obtain ( ) W o 3f l h u % ^ Obs ¥ noo 5(1 - a 7 mCTrnj )a 4 5(1 - ^LT^TL^ 8 - P 'O ^x^ The values of 7 thus found were compared with those of *Y-tv ie Q 1 , obtained from the curve for 7 as a function of the ratio p/a. As we have 'seen this curve represents a fairly good approximation to the theoretical values of 7 . In all instances the ratio of 7obs to ^theor is less than unity. The fact that the observed deflections are smaller than they would be expected to be can be attributed to a certain amount of restraint at the edges which could not be entirely eliminated. The variability of the results can be attributed partly to lack of uniformity of the plywood in a given plate and partly to varying degrees of constraint at the edges of the plate. It appears that the curve for 7 = h r /w cx=> ma y be used in predicting the deflection at the center of a plate if reasonable allowance is made for the effect of constraints at the edges and for variability of the material. In making the tests the plates were placed between two rectangu- lar frames made of heavy channels. The frames were 12 feet long and U feet wide. A cross section of the apparatus is shown in figure IP. For the case of simply supported edges the plate rested on circular rods 1/2 inch in diameter. The pressure was applied by inflating three rubber ba^s, approximately U feet square and 6 inches deep, with compressed air. Heavy planks bolted to the channels as shown in the figure formed the back of the chamber containing the bags. A run was made with the load on one side of the plate and then on the other side by moving the planks and bags. The plates were tested in the vertical position to eliminate the effect of gravity. The deflections were read on Ames' dials placed at various positions on the plates. The air pressure was measured ^ith a water manometer. Tests were first made on a 12 by U-foot plate. Then U feet were sawed off and the resulting g by U-foot plate tested. Finally this plate was sawed in two and tests were run on a U by U-foot plate. As the 12 by U-foot plates were made by joining up shorter lengths of plate by scarf joints there were frequently considerable variations in the elastic constants from one end of the plate to the other. In addition, there were present defects in manufacture and variations in direction of grain. Section U. Rectangular Plate Under Uniform Load. Edges Clamped In this case the deflection w and its normal derivatives vanish along the edges of the plate. The solution — of the differential equation (2.25) subject to these bound ar y conditions will be foun d in append ix U. vr~ ; This solution is due to D. S. Zilmer. (See footnote 7, p. 9.) The calcu- lations based on this exact solution were performed at the Forest Products Laboratory under the direction of the present .author who is responsible for the discussion to be found in section U and in appendix U. He is also responsible for the approximate methods. The clamped orthotropic plate was also treated by S. Iguchi by a somewhat different method in the paper to which reference rr a s made in footnote 1. See r»lso footnote 28. 1312 -15- Approximate formula for the deflection at the center .— ^.^As in tJac of the plate with simply supported edges it is possible from a con- sideration of the potential energy of the system to find a simple approxi- mate formula for the deflection at the center of a uniformly loaded plate with clamped edges. The procedure is the same as that employed in section 3. The plate is divided into three regions by the lines y - C and y = b-C uhere c = Tfi/2, T being a parameter to be determined. The assumed form of the deflection of the middle surface is given by the equations u = w„ sin 2 TtX 'o a when c < y < b-C (U.2?) . 2 -jT y .2 tix w = < , lU, and 15 are shown the deflection along the central line x = a/2 of a number of plates of commercial oly^ood. In order to compare the shapes of these sections of the deformed surfaces dial readings corresponding to the same central deflection are plotted. The distances from one end of the plates have been expressed as fractions of b the length of the plate. The curves show even more pronounced indications of a change in shape of the deflected surface associated with the presence of a. wave form than those for pla.tes with simply supported edges. In fact, the effect is so pronounced in the case of plate k that one hesitates to accept it as real. That this effect, which is so pronounced in the case of the k by 8-foot plate, is actually present in this plate at all stages of loading is shown in figures 16 and 17, which give the shape of the -olnte at successive intervals of loading. In order to compare readily the shapes of the curves the central deflections have been reduced to 0.100. This means that for small deflections experimental errors have, been multiplied by a large factor. The curves for this plate are published without further comment merely to show what actually happened in the case of this particu- lar plate. It may be remarked that for the larger deflections the relative heights of the maxima are reduced. This may presumably be attributed to the affect of membrane stresses. Central Deflection of a Piatt- Expressed as a Fraction of That of the Corresponding Infinite Strip . As in the case of the plate with simply supported edges it is possible to represent the deflection at the center of a plat-, witn clamped edges as a fraction 7 of the deflection at the center of an infinite strip with clamped edges. Thus > vr = yv (U.35) ' O <=x=> The factor 7 is agsln found to depend almost entirely uoon the atio B/a = (b/a) (E./E^)- 1 -/^ so that the results of the theory, so far as deflection at the center of the plate is concerned, can be represented with sufficient approximation by a curve in which 7 is plotted as a function of B/a. This curve, constructed from the exact values of a in tables 12 to 15, is shown in figure lg. More points than those obtained directly from the tables were secured by interchanging the axes to which the plates in the tables were referred and utilizing the data of the tables to calculate the factor 7 for other values of the ratio B/a. -17- 1312 For an infinite strip clamped at the edges, the deflection at a point on the central line is given by the formula 1 --T - 4 T I.T~TL pa "*» = 32 3 < 4 - 36 ) Eh In which the factor 1 -^jr^m-p ma « v ^ e t^ken to be 0.99. Except for this formula (4.36) is that for a uniformly leaded beam of unit width with fixed ends. e curve of figure 18 can be constructed from the values in table 21. In figure 19 the curve is that of figure 18 and the points are ■ imputed from the values of i given by the approximate formula, the ■ ^es of w jq being those given by the exact formula. Except for the " that the ao^roximate values of y do not show a maximum in the vicinity of 3/a = 2 the agreement is satisfactory. That the aporoximate values of t s v a maximum in the vicinity of /3/a = 2 is to be attributed c the incomplete representation of the deflected surface by the forms assumed in (4.22). Since the exact analysis clearly ooints to the existence of a maximum point on the curve, it may safely be assumed that the curve represents approximately the true situation for the seven-ply and nine-ply plates in addition to that for the three-ply and five-ply plates for which it was constructed. 'ests In table 21 are shown the results of a number of tests of uni- formly lradpd plates with clamped edges. The clamping at the edges of the plates was accomplished by removing the circular rods shown in figure 12 and clamping the plate between the channels of the two frames. The same plates »f commercial plywood were used in these tests as in the tests of tes with simply supported edges. The factor 7^-u.q was computed from the mean of the values of the quantity p/w for a given plate by the formula 3 v 32E, h w o 1 o ■ r * S W oco ( l ^LT^a 4 P o<- rre spending factor y , was taken from the curve of figure 16 ^ ' theor fcr the appropriate value r f ,-js.. As was to have been expected, owing the impoasil cf securing perfect clamping at the edges, the 13] 2 -18- observed factors 7 and consequently the observed central deflections are greater than those predicted by the curve, which is based on tne exact- analysis of ideal cases. On the average they are about kO percent greater. Hence in using the- results for plates with clamped edges con- siderable allowance must be made for the effect of imperfect clamping. The formula predicts the central deflection for the case of nerfect clamping, a situation that is rarely met in practice. It is realized in the case of a plate extending over a network of rectangular openings, all in the same plane. Ideal clamping will be found on the edges of interior rectangles of such a ne"twork. Otherwise elastic yielding reduces the clam-ping effect to a greater or less extent, depending upon the particular situation at the edges in a. given case. Section 5- Infinite Strip (Long, Narrow, Rectangular Plate). Load Concentrated at a Point. Edges Simply Supported Consider an infinite plywood strip with edges x = o and X = a along which it is simply supported and under a concentrated load F applied at the point x = U, y = on the X-axis as in figure 3.Q. As in the case of the isotropic striplit a solution of the differential equation (2.25) is obtained for the case in which a load of uniform intensity is distributed along a segment of the X-axis including the point (u, o) in its interior. By allowing the length of the segment to decrease while at the same time the intensity of the load increases in such a way that the total applied load is unchanged, we obtain the solution for the limiting case of a point load. This solution, expressed in terms of an infinite series, is given by (5*7) of appendix 5« From this expression for the deflection, the bending moments inl- and Ely can be calculated. It is found that it is possible to express the sums of the infinite series for these moments in closed forms in terms of two functions which are the real and imaginary components of a function of a complex variable. Replacing the series by closed forms reduces greatly the necessary calculations. (See appendix 5») It is clear that these moments should become infinite at the point of loading as they do. The values of the deflection and bending moments at certain points of an infinite strip of plywood of type }X having a concentrated load at a point on its central line are given in table 23 and shown in the curves of figure 20. Section 6. Infinite Strip (Long Narrow Plate.) Uniform Load Applied Over a Small Area. Edges Simply Supported A uniform load acts over a small rectangular portion, as shown in figure 21, of an infinitely long strip whose edges are simply supported. By integrating the effect of the loading of a narrow strip of the rec- t angle, con sidered as a loaded line segment, formulas are obtained in jtf — ~~~Nadai, A., Elastische Platten, pp. 7S-82 and S5~95 . -19- 1312 appendix 6 for calculating the resulting deflections and moments at any point of the infinite plate. From these formulas the moments can be cal- culated with the aid of equations (2.29). Table 2k and the curves of figure 22 give the deflections and moments at certain points clue to a load distributed uniformly over a small square "hose center is on the center line of a three-ply plate of spruce nlyood and whose sides are equal to one-tenth of the width of the plate. If tnble ?h and figure 22 are compared with table 23 and figure 20, the deflections in the case of a load applied over a small square area are seen to bo practically identical with those due to a similarly situated point load equal to the total load applied over the square. The bending moments are also practically the same except in the immediate vicinity of the loads. Section 7* Rectangular Plate. Load Concentrated at a Point or Applied Over a Small Area.. Edges Simply Supported The deflections and moments due to a concentrated load on -j rectangular plate with simply supported edges can be found by calculating the effects of a suitable distribution of -oositive and negative loads on an infinite plate, using the results of section 5 or section 6. It is only necessary to distribute the loads in such a \:rj that the deflections and bending moments vanish on the edges, y = o and y = b of the plate, The distribution is shown in figure 23- A positive load is denoted by a dot (.) and a negative load by a cross (x) . If the loads are numbered I, II, III, IV, V, etc., as shown in the figure, the deflection w at any point in the plate will be given by combining the deflections due to the separate loads, that is, T,T ~ T-J -f W + TsT + I II III In like manner expressions for the bending moments can be obtained. Calculations ^ere made for a square plate of type 3X having in one case a point load at its center and in another case a uniform load distributed over a central square whose sides !,r ere taken to be on^-tenth of those of the plate. The results are shown in tables 2^ and 26 and in the curves of figures ?k and 25- The choice of axes is to bp noted ^s slightly different from the customary choice for a finite plate in this report. The calculations for the case of a uniformly load.ed small central souare area have not been carefully checked. However, the behavior indicated by the results is in good agreement ™ith what was to be expected from the other cases considered in this section and in section 6. In the neighborhood of the load the deflections for the square plate are nearly the same as for the infinite plate while near the edges the effects of the loads V and II must be taken into account. In case of a five-ply square plate the effect of these loads would be noticed at greater distances from the edges of the plate. -20- 1312 Central Deflection of a Plate Expressed as a Fraction of Tha t of the Corresponding Infinite Strip The maximum deflection of a rectangular plate under n given con- centrated load P occurs at the center of the plate with the concentrated load placed at the center. For an infinite strip the central deflection due to a point load on the central -lino is given by (see (f .h) appendix 7) w = 1.051 isB- 6\e (M) *l h P* For a finite rectangular plate under a given central load concentrated at a point, the central deflection w Q can be expressed as a fraction 7 of the central deflection of a similarly loaded infinite strir>. Thus T-/ = -yw (7- 5) O ' For the purpose of determining this factor y } it is advantageous to replace the method explained earlier in this section by one used by Timoshenkoi-2. for isotropic plates. In this method the load is limited to a position on the central line. In appendix 7 the analysis is carried out for a plywood plate. The calculation of the factor 7 in (7«5) can then be readily performed. (See appendix 70 It is found that 7 can be represented T : : ith little error by a curve as a function of the ratio Si iA 3/a = (b/a)'(J0 • In table 27 are given the values of 7 for plates of several types of plywood. In figure 26 a smooth average curve for 7 is drawn from the points given in table 27« Further calculations indicate that this curve is satisfactory for other types of -nly^ood than those listed in table 28. This curve should be used only for plates of tne types con- sidered in this report. In particular, the directions of the grain of the wood in adjacent plies are mutually perpendicular and the constant k lies between 0.2 and 0.5 or not far outside of this interval. The results of tests on a number of plywood plates with concentrated central loads are shown in table 28. The same plates of commercial ply-'ood i1r ero used as in the tests described in sections 3 an| i ^« The plates -vre tested in a horizontal position. The edges were simply supported, resting on naif- inch rods as shown in figure 12. The loaded ares was in all cases a square U by U-inches. Only one side of each plate wa.s loaded. Tne numbers in the column headed 7 , were calculated from the observed ratios T r /p by the following formula which is obtained from (7.H): 3-i , 3 p-yt 1 __0 . 7 = VT o 00 (l.05l)6\ea 2 F 15 "TTimoshenko, S. , Bauingenieur , 3, 51, 1922, -21- 1312 The numbers of the column headed 7+.^,,^,,, '-ore taken from the curve of figure 26. The agreement bctreen the observed and theoretical values of 7 aprears to be satisfactory. The average value 0-921 of the ratio 7 ^o/%v, pn indicates some restraint at the edges but not so much as in the case of plates with uniformly distributed loads. This appears to be reasonable. Flates With Large Deflections Section g. Differential Equations for the Deflection of a Plywood Plate. Large Deflections When a plate with prescribed edge conditions is subjected to a succession of increasing loads it is known that at first the deflection at the center of the plate increases prooortionally to the load but that it does so only during the early stages of the loading. As the load is in- creased, the stresses remaining below the r>rooortional limit, it is found that the deflection increases less rapidly than ^ould be expected from the earlier linear relationship between deflection and load. When the de- flections are small the load is carried entirely by the bending stresses that are developed, that is, by compressive stresses on one side of the neutral plane and by tensile stresses on the other side. For moderately large values of the deflection, of the order of magnitude of the tnickness of the plate, appreciable tensile (or in certain cases compressive) stresses are developed throughout the thickness of the plate. They are associated with the extension (or compression) of the material accorrnanyin,?; the deformation of the plate from its originally plane form. These stresses may be conveniently referred to as direct stresses. If they are tensile stresses the term membrane stresses is a very descriotive designation for them. The load is thus carried partly by the stiffness of the plate an; 1 partly by direct stresses that are developed in the middle surface and in surfaces parallel to it. The determination of the deflection and the stress distribution of an isotropic plate with large deflections can be shown to depend uoon the solution of two simultaneous partial differential equations of the -1 c fourth order. — The maximum deflection is taken to be small in coirraarison with the length and breadth of the plates. It is easy to modify the steps taken in the derivation of these equations for an isotropic platei-L to obtain the corresponding equations for the plywood plate. This is done in appendix 8. The equations obtained are (g.ll) and (g.lU). We shall have occasion to use them in determining the deflections and stresses of uniformly loaded infinite strips IF von Karman, Th. , Enc. d. Math. Wiss. IVu, 3U9 , 1910. 17 Sqo Nadai, A- , Elastische Platten, pp. 2gU-287. -22- 1312 (practically, long narrow plates). The solution of those equations for finite rectangular plates has not been found. Considerations of energy- will be used later to determine the approximate deflections at the center of such plates. However, the results obtained from the consideration of infinite strips will perhaps be found to have a wide range of amplication. Even for isotropic nlates solutions have not been found for the equations corresponding to (8.11) and (8.lH) except in the cases of in- finitely long strips and circular plates. i£. Section 9» Infinite Strip (Long Narrow Plate). Large Deflections. Uniformly Distributed Load. Edges Simply Supported We shall now make use of the differential equations (8.11) and (8.lU) to find the deflections and stresses of a uniformly loaded infinite strip with simply supported edges when the loads are such that the de- flections are large and direct stresses are developed. It is assumed that there is no displacement of the edges associated with the direct stresses. This implies that the edges are restrained from moving in a direction perpendicular to the length of the plate. The solution obtained will be applicable in the central portion of a plate whose length is only moderately greater than its breadth. Approximate formulas are derived from a con- sideration of the exact formulas that are obtained for the infinite strip. These formulas are much simpler than the exact formulas and are sufficiently accurate for practical calculations. The approximate formulas here referred to are for the infinite strip. Approximate formulas will also be obtained in section 11 by the energy method for finite rectangular plates. The exact solutions of equations corresponding to (8.11) ant? (8.lU) were given for infinitely long plates of isotropic material with either simply supported or clamped edges by Stewart Way in a lithographed oreprint of a T?aper presented to the American Society of Mechanical Engineers in 1932. It has apparently not been published in any other form.=^- From a reference in Way's ^oaper it would appear that essentially the same solutions were given by I. Boobnoff in a book published in I91U for the use of naval architects of the Russian Navy and not readily available in American libraries. The corresponding solutions for infinite strips of plywood are given below. The edges of the strip are taken to be x = O and X = a. Under the assumed uniform loading, the deflection and the strain and stress components are independent of y . It can then be shown (see apnendix 9) that the mean direct stress component X, (the mean being taken over the thickness of the plate) is independent of x and is therefore a constant g for a given load p. This information corresponds to that which would be 19 See references to I. Boobnoff and S. Way in section 9. The substance of this paper is found on pp. k-lf of Timoshenko' s Theory of Plates and Shells, I9U0. -23- 1312 furnished by (8.11). The equation (8.1U) becomes Dj&i = P + gh^f (9.5) dx dx It is to be observed that g is a constant for a given load p but that it will have a different value when p is changed. The quantity g therefore enters the solution as a parameter whose value for a given load p must be determined in the course of solving the problem. A little consideration of the complications that arise in connection with the simple equation (9»5) will lead to an appreciation of the difficulties associated with the solution of the system of equations (8.11) and (8.lU) in the general case. The procedure to be followed in utilizing the solution of (9*5) subject to the conditions on the simply supported edges x = and x = a to find the maximum bending stress, the direct stress, and the relation between deflection and load, is discussed in appendix 9- However, in practical calculations it will not be necessary to follow this procedure, since it is possible to replace the exact formulas by quite accurate approximate formulas. With their aid the calculations involved in any given case are greatly simplified. These formulas whose derivation from the exact formulas is found in appendix 9 ^ re the following*: (a) Relation between load and deflection p = a!!o_ + B( X Io)3 (Q.28) h h where oa S L h A = id -i (9.29) B = £0.6 L a (b) Maximum bending stress in a face ply b = a *£ (£) 2 £° (9.3D 1312 ^ where a may be taken to be k.k. A method of obtaining a more accurate value of a from a curve is explained below. The latter method is easy to apply and is to be preferred. (c) Mean direct stress 2 v •572 3&(^) (9,25) In these formulas E-, denotes the mean modulus in bending, E„ j_ d the mean modulus in stretching, and E x the actual modulus in a face ply, all measured parallel to the X-axis. The mean modulus in stretching E is merely the arithmetic mean of the E's in the various plies measured in a direction parallel to the X-axis. Thus for three-ply plywood, Having all plies of the same thickness and the grain of the fac« nlies parallel uO the X-axis In like manner E-, -denotes the mean modulus in stretching in the direction parallel to the Y-axis. The calculated values of the ratios Eg/Ey^ and £, /Ej for various types of spruce and Douglas-fir plywood, using the values of -Ex and Erp given in table 1 and on page f> , are shown in tables 29 and 30. The grain of the face plies is taken to be parallel to the X-axis. If the grain of the face plies is parallel to the Y-axis, E r) /ET and E-, /E T as given in the tables are to be interchanged. In formulas (9-25) to (9-31) X may be taken to be 0-99- This is its value for spruce. For wood of other species this value may probably be used without appreciable error. It will be noted that the first term of (9t2g>) expresses the result obtained from the usual theory of thin plates when the deflections are assumed to be small. Of the three formulas (9-2g), (9-. 31), and (9-25), the second, namely, (9. 31), is the least accurate if a is taken to be k.h. Actually a ranges from U.g for small deflections to U.O for large deflections. Satisfactory values of a can be readily obtained from the curve of figure 27 where a, is plotted as a function of the quantity T] which is connected with the ratio W /h by the formula (see (9.24) appendix 9) -77S (^) ^1 o h 1312 -25- When a is determined in this way the only approximation in- volved is that in the equation last written. This error is never large. In this connection see equation (9- 24) of appendix 9 and the accomnanving discussion. The curve of figure 27 is plotted from the following data. ^0123^56739 10 a 4. SO 4.7S 4.64 4.50 4-3S 4.28 4.21 4.16 4.13 4.10 4.08 If the maximum "bending stress, s, is found "by the method just described, it may "be more convenient to calculate the mean membrane stress g from equation {$.1%), appendix 9. instead of from (9.25). The value of Tj neeaed in (9 -18) has been found in the calculation of s . Prom the mean direct stress and the maximum bending stress in a face ply as ^iven by (9. 18) and (9.20) or (9.25) and (9. 31), the corre- sponding stresses in another ply can "be found. The method of doing this is explained in appendix 9» Useful formulas applicable to long narrow plates of isotropic material can be obtained from formulas (9-28) o , (9-31). and (9*25) "by setting En = E a = E x = E and X. = 1 - d c ~ where denotes Poisson' s ratio for the material under consideration. Tables 43 to 47 contain a comparison of the results obtained by using the approximate and exact formulas for various types of plates. At the time that the calculations for these tables were made, the use of the curve for a had not been considered and a was taken to be 4.4. Section 10. Infinit e Strip (Long Narrow Plate). Uniformly S i s t ri b uted Load. Ldges Clamped. Large Defl ; • ;ns Exactly as in the case of an infinite strip with simply supported edges (see section 9 and appendix 9) exact formulas can be obtained for the case of an infinite strip with clamped edges and large deflections. From these formulas the deflection at the center, the mean direct stress, and 1 maximum bending stress can be calculated. These formulas are derived in appendix l n wh< re the tables necessary for their utilization are .riven. However, it is possible to replace the exact formulas by apixfoximate formulas which are much simpler to use and are sufficiently accurate. The derivation of the approximate formulas is given in appendix 10. In this section as in section 9 the edges of the plate are restrained from moving inward. The approximate formula connecting the load and the deflection is P = A V l° + B(ISL) (10.13) li h 1312 -26- where A. = _1£L _1 (10. 14) X E L 21 ^a (10>15) and 7^ -p _ pa . For the definitions of E_. , E •. X see section t - rr 1 a' 9 or the table of notations. Tor the maximum bending stress in a face ply we have the approximate formula «■ s = C !!° + D(!°.) (10.20) h h where 162 . 2 C = X ( n) (10.21) \ a P.9gE v E Q h 2 D = \ x =A (it) (10.22) \ E^ a The symbol E x denotes the Young' s modulus in a face ply and in tne direction parallel to the X-axis. The range of values of W_/h within which this formula mav be used is discussed in appendix 10. A better approximation to s is obtained from the formula S = «^(|) 2 ^ (10.16) W and the curve of figure 23 where the argument r\ is connected with ~ by the enuation n v 1/2 " c n -zrc (1\ -n (10.10) _ = 0.366 (jr-) T) 9. -27- 1312 I The curve of figure 28 can "be plotted from the following data: n,0123^567g9 10 a 16.00 16.S2 lg.03 20.31 23.20 2b. UU 29.92 33. 5S 37.30 lU.l«5 4^.00 If this method is used to find s it may be more convenient to calculate the direct stress g from (10-7) of apnendix 10 instead of using (10.11) below. The approximate formula for the mean direct stress is s = — ^ (|) ( r °-) (10.11) From the- direct stress and the maximum bending stress in a face ply as given by (10.11) and (10.20) the corresponding stress in any given ply can be calculated by the method explained in appendix 9* See equations (9.33) to (9.36). Tables 1+8 to 5? i- n appendix 10 show comparisons of the results of calculations made with the exa.ct and the approximate formulas. In the tables the maximum bending stress was calculated by formula (10.20) instead of by the more accura.te procedure based on equation (10. lb) and the curve of figure 22. These formulas are also applicable to long narrow isotropic plates. It is only necessary to use S in place of all letters J that have subscripts and take \ equal to 1 - & : - where o is Poisson 1 s ratio. Section 11. P.ect angular Plate. Uniformly Distributed Load. Edges Sinvoly Suppo rted. 20 Approximate Method — The solution of the differential equations (8.11) and (fS.lU) that describe the behavior of a flat plate when the deflections are large, has not been found for the rectangular plate because of the mathematical diffi- culties associated with the fact that these equations ?ro not linear. To obtain an approximate expression for the deflection at the center of the plates considerations of energy rre employed as was done in 20 The analysis of sections 11 and 12 and appendixes 11 and 12 was carried out by the author during a semester in which he was relieved from teaching duties under a grant to the University of Wisconsin fro,;) the Wisconsin Alumni Research Foundation. The numerical calculations were made by the Computing Division of the Forest Products Laborator . -28- 1U2 sections 3 and k in obtaining approximate formulas in the case of olates with small deflections. In the present case since the middle surface of the plate is in a state of strain it is necessary to assume suitable ex- pressions for the components u and v parallel to the X and Y axes, respectively, of the displacement of points in the middle surface of the plate, in addition to a suitable expression for the deflection. These expressions contain certain parameters which are to be chosen in such a way that the total' potential energy of the plate and applied load is a minimum. •s The potential energy of deformation is considered to be made vv of two parts, that of the state of strain associated with the bending stresses and that of the state of strain associated with the direct stresses This implies that both states of strain are considered to be so small that the potential energy of the sum of the two states of strain is aryoroximately equal to the sum of their respective potential energies. In the expressions assumed for the deflection and the disnlace- ment there are four parameters which are to be determined so that the total potential energy is a minimum. It is found that, the determination of one of these parameters, T(see apDendix 11), in this way involves calculations that are too complicated. Accordingly, T is taken to be the.* same as in the case of small deflections. This determines the general shape of the assumed deformed middle surface when the deflections are small. The sub- sequent argument, for the case of Large deflections, rests upon the assumption that the form of the middle surface does not change greatly as the deflections become larger,- all ordinates of the ( middle surface for a. small deflection being considered to be multiplied by a common factor of proportionality. v The parameter t having been chosen in this way, the remaining parameters can be found. After performing the calculations (see arpenrlix 11) the following formula is found connecting the quantity F = pa and the ratio w /h of the deflection at the center to the thickness: P=H^ + .Q(^£) 3 (il.23) The factor K is the reciprocal of the factor a in (3. 7 >7), the corre- sponding formula for a plate with small deflections. This is to be expected, since from the way in which it was derived (11.23) must agree with (3-37) when T ' c /h is small. A formula for calculating the factor Q, is found in -29- 1312 appendix 11. Tables 31 to 35 give the factors H and Q, for rl^tes of spruce rlywood of several types and for plates of isotropic material. It is to be recalled that the plies are assumed to be of equal thickn°ss in a ) j.s the reciprocal of ar u - (U. 35) • That this should be so follows at once from the way in which the formulas were obtained. The remarks made concerning the approximate . r . ■- . . I'll, 23) also apply to equation (12.3)- The curve of ly in is that if the ratio o/ a _ fh/al (E / r V^" is £:rPater than l«75i the central portion of th? plate can be treated as part of a Ions; narrow plate. The methods of section 10 can then be employed to deter- mine not only the deflections but also the bending and direct stresses in the central portion of the plate. For isotropic rectangular plates with large deflections an roximate treatment has recently been given by S. Way— using expressions different from (12.1) and (12.2) and containing a larger number of para- meters. He obtained curves connecting the load and the maximum deflection and also curves connecting the load and the maximum stress, for three rectangles, the ratios of whose sides are 1, 3/2, and 2, respectively. Because of the larger number of parameters employed his results undoubtedly resent a better approximation to the actual solution than those based on equations (12.1) and (12.2) although the amount of numerical calculation is much greater with the increased number of parameters. For isotropic plates the loads associated with a given deflection calculated by the two methods differ by less than 12 percent, usually by much less, for deflec- tions in which ^0_ lies between 0.5 and 2. h Nc attempt has been made to calculate the maximum stresses on 3 basis of equation-. (12. 1) and (12.2). It is probable that equation (12.3) yields values of f for which the error is of the order of magnitude of 10 percent. The labor involved in obtaining more accurate values of F for each type of plywood plate by the energy method is prohibitive. As noted above, the methods of section 10 can be exnected to yield satis- factory values of the deflection and stress in the central -nor t ion of lA plat<=s for which p/a = (b/r.) (E /E ) is greater than 1.75- Tables 3^ to U0 give the values of H and Q, for several type* of spruce plywood, the plies bein^: assumed to be of equal thickness. re is close agreement between the approximate formula (12. 3) for the case k = b/a = 00 and the approximate formula (10. 13) for the rite strip derived in appendix 10 as an approximation from the exact formulas. Th* formula (10. 13) was ■2! * _£ + 23 X ^0.) , 2_3 H ( 1 \ 1 ~h \ a (12.3) becomes, for k = <=* E t ^z (1.6) E, X * E, V E z z z where, with /\ denoting the determinant of the coefficients of (1.3) _L - BC-F 2 , _L - CA-G Z J. _ AB-H* ,, , E x ~ A E y - A ' E z ~ A <1,6) (Jxz- From (1.4) it follows that there are three shearing moduli 11 - |_ /£ ZX ~M and //- X y = N. Among these twelve constants there are three relations, namely, those expressed by (1.7) and two similar equations. -34- Appendix 2. The Differential Equation for the Deflection of a Plywood Plate. Small Deflections In addition to the assumptions explicitly stated in section 2, it is assumed as is usual in the theory of thin plates that the points of a straight line which is normal to the undeformed plane middle surface, 21 = 0» °f *ke plate, remain in a straight line which is normal to the middle surface after deformation has taken place. The deflections are assumed to be so small that direct stresses (see section 8) are not developed to an appreciable extent. Under these assumptions and with the choice of axes shown in figure 1, the components, (JL and V , parallel to the X- and Y-axes, re- spectively, of the displacement of a point whose coordinate with respect 23 to the middle plane is Z are expressed — "by the equations: where W denotes the deflection of a point in the middle surface. From these equations the strain components are found to he p = - 7 i!l, P = - 7 i!w p 3 -? 7 £^ (2.2) The stress component 2 z is taken to "be negligible in com- parison with Xx an< *- Ty ** follows from (1.5) that at a point in a given ply X x = -y- (e X x + 0-yxevy) ^ ^xxt u yx c yy< Ey_ , H'xy e xy V v = — S^ (e yy + y 2 Eyz: A ? i/. [d Z w z/A)dz , b 2 - / (E y zfA)dz , tyi d -ty t (2.8) c = Z / /^ z 2 dz Prom the relation (1.7) it follows that d 2 = b, The vertical shearing forces p x and Py are defined "by the following equations: P x = / Z x dz , p y = / Z y dz (2>9) They are represented "by vectors in figure 4, c. The conditions for the equilibrium of moments with respect to the X-axis, and Y-axis, respectively, lead to the equations: -37- dm x . ^m xy K * d* dy (2.10) _ ^m;y ^m xy p > a y 3x while the condition for the equilibrium of forces acting in the direc- tion of the Z-axis leads to the equation iEa + ^l + _ . a (2-11) dx <3y p u It is to be recalled from section 2 that h denotes the thickness of the plate and V the load per unit area on the face z =-h/2 From (2.7), (2.10), and (2.11) the following differential equation for the def lection, W, is found: n (? 4 W + pu d\±_ , D i^W _ (2.12) where D.'Q,, D 2 = b 2) K- -L(q 2 +b, + 2c) (2.13) If the plies in the plate are all flat grained, as assumed, the expressions for D, , D 2 ^d- K can be simplified. The sub- scripts L and"! being used to refer to the longitudinal and tangential directions in the wood, it is clear that for plies in which the grain of the wood is parallel to the X-axis E X =: E L , E y = E T) /* xy = / u Lr> (2.14) ^xy = ^t , ^x " <^l while for plies in which the grain of the wood is parallel to the Y-axis E* = E-T ) Ey = E L ) /^Xy = /^TL = /^LT > (2.15) Cf X y - D* = 12 \ ' Ul " \Z\ (2.16) where (2.17) IP r*6 Z 2 d2 =~T3 / ExZ l dz (2.18) -h/ 2 t/fc (2.19) EC 25 , and L2. may he called the "mean moduli in bending" under couples whose axes are perpendicular to theX £ an & YZplanes, respectively. As soon as the structure of the plywood is known these moduli are readily calculated in terms of the Young's moduli E L and Ey °^ *^ e w °°d. in question. For a plate whose construction is not symmetrical with re- spect to the middle plane, it is to be expected that E, and E^, as defined on page 7 , may be used with slight error in the formulas of this report, although these formulas were derived from an analysis that assumed a symmetrical construction of the plate. The quantities E, and E 2 determine the stiffness of the plate in the two principal directions. The term in the differential equation involving shear is independent of the situation as to symmetry so long as all the plies are flat grain (or all edge grain). Since the plies are all assumed to be flat grained and since, in accordance with (1.7) E l ^l - E T 6^ X y being identical with /4 LT for all plies. The expressions (2.7) for the "bending and twisting moments can nov; "be written in the forms: (2.23) where m x -- u, I <* x 2 T w, d y 2 / m y = -D« (d z \N \dy z + ?* + ^ c3x 2 / m xy - 6 dx ^7 -41- Appendix 3. — Rectangular Plate Under Uniform Load. Edges Simply Supported . Small Deflections The differential equation (2.25) v 4 3 w v o , 4 w i o w - P ax 4 *-* b% z d7) z drj* D, where p is a constant, is to be solved subject to the conditions stated below that hold on the edges X = 0, X = CL 7?-Q 7} -/3 where (See (2.28) ) = eb (3.1) The boundary conditions on the edges X - and X - OL are (See (2.29) ) ,2 w = 0, $$ + ate'fy -0 M The corresponding conditions on the edges 77 -Q and 7)~/3 (that is, y = D ) are w-o, e-^M^'O For the constants (J~ s , 01 and £ see equations (2.24) and (2.28). We choose first the following solution of equation (2.25): iU (x 4 -2ax 3 + a 3 x) w, =25d ( \x--tw* " *y (34) This solution satisfies the boundary conditions (3.2). It represents, in fact, the deflection of a uniformly loaded infinitely long strip of plywood having its edges, X=0 and X = CL , simply supported. It will be convenient to write this solution in the form w « = A fc_ -y~5 sinA n x , n =1,3,5 (3.5) -42- whe re and A = 4p/aD, (3.6) A n = n n/a (3.7) The satisfaction of the boundary conditions on the edges 7f ~ U and 7}~pJ will be secured by combining with W, a solution, W z , of the equation (2.25) with its right hand member set equal to zero. Let W z = ~Y_ X 5 Y " Sin *" X (3 8) where Y n is a function of 7J . In order that W 2 may satisfy (2.25) with its right hand member set equal to zero, Y n must satisfy the differential equation Y n "- z^A'y* + a;y„ = o On setting W n = Q "' it is readily found that whe re >n " A n /0 ; S n - A n (T (3.9) - - l\+tC -- /I- ft P ~\J Z > ° V Z (3.10) Then Y^ may be chosen as a suitable linear combination of the following functions: sinh Y n 7] s\r\8 n y , smh^ y cosS n y (3>n) coshr n ^ sin(5 n 7, cosh^,^ cos <5 n ^ -43- In obtaining these solutions it has been assumed that -ft is less than I . This appears to be true for all types of plywood. If fC^ I appropriate modifications in the functions (3.11) can be made. It is clear that W 2 satisfies the conditions (3.2). It remains to choose the coefficients of a linear combination of the foregoing solutions (3.11) in such a way that W t + NA/ 2 satisfies the conditions (3.3). It is found that when II is odd Y n = C n \k [sinh v n q sinSf, (fl-7J) t s\r\\\V n (/?-y)sin8 n y ] + n/|-# 2 [COSh/ h 7/ ZQ^>\(/3'7}) where + coshy n (fl-y) cos£ n 157]} (3.12) (3.13) \[F1F : (coshX/3 + cosS n /?) and that when fl is even Y„=0 Then the deflection is given by W = W, + Wi = H -4s 0-Y n )sinAnX written = 4b£ r-i-O-YjsinXnX, n-!,W.... tt 5 D, ^ n 5 ' (3.14) On recalling that D, = E ( h/I2.A equation (3.14) may be n = 1,3,5.... -44- where pa 4 p= E h 4 J ^ = '"^LT^L < 3 - 16 ) Using (3.15) and (2.29) expressions can readily be found for the deflections and moments at the center or at any other point of the plate. At the center of the plate the ratio of the deflection \A/ to the thickness h is found to "be given by the equation, w o _4SA E L r f# r / . v n /5> 8 n /5 + vT^cosh^ cos ^)] (3.17) n = I, 3, 5 . . . An approximate formula will now be obtained for the deflection at the center by assuming a plausible form for the deflected middle surface and determining certain parameters that appear in this assumed form, in such a way that the sum of the potential energy of deformation of the plate and that of the applied load shall be a minimum. (See discussion in section 3.) Let (See figures 4 and 5, section 3.) w - w sin-g— > when c ^ y ^ b~C (3.18) . 7rx . ?ry w = w sin -jj- sin -^ when < y < c A form corresponding to the latter will be assumed for the portion of the plate for which b~C < y< b but it need not be written down since the potential energy of the plate can be calculated as twice the potential energy of the portion of the plate for which 0< y Vy, and Xy their values as given by (2.3) in terms of the strain components e XX) e y v , an( i 6 X v • The integrand is then a quadratic function of these strain components. For the strain components we then substitute their values in terms of . ^ •> , ^y »and ? ^ as given by (2.2) and perform the in- d x a <2y z dxdy tegration with respect to I . We thus obtain -46- where *.=i *5*=- E L (3.22) Since the definition of W in (3.21) is different in the middle and end portions of the plate it is simpler to calculate separately the potential energy of deformation of these portions. The potential energy Vh m is obtained by integrating the same expression as in (3.21) over the central portion of the plate, Weeing there defined by the first of the expressions in (3.18). In like manner V! is found by taking twice the result of extending the integration over the portion of the plate between the lines y ~U and y = C , W being defined by the second of the expressions in (3.18). Using the abbreviation k = b/a we find from (3.18), (3.20) and (3.21) that (3.23) _ 4q 2 pw T + 7T (k-r)] (3.24) E L h W 7[ Ki _ E L h 3 w V v bm" 48 Aa 2 CK r)Hx (3.25) (3.26) The parameters Wq and T are to be determined from the requirement that V - V, +-v k . ♦ v, be bm (3.27) shall be a minimum as a function of these quantities. From (3.24), (3.25) and (3.26) the expression f or V can be written in the form: V = Lw 2 -Mpw -47- (3.28) where L=^fe^-,r+z*,K) (3.* 9 ) M = 4fi[r + £ (k-r)] 5 = Z t — ~PC -\ , \- \~ Sinhd m X + sinc m x sinhd m (X-a)]sin d m = ^wP , c m = >- cos 5m X? +coshx,/? rri l/U; f/D/cr)sinS n /9 + sinh7 n /9 (4.14) p fa) = cosc m a+ coshdmO, ( 4 . 15 ) ° m (/cf&)s\r\ c m a+ s'mhdmcx 26 — ^Hencky, H., Darmstadt Dissertation, 1913. ?7 —March, H. W., Trans. American Math. Soc. 27, 307-17, 1925, -51- From this point on in appendix 4, the numbers TT\ and n will be considered to be odd integers. For purposes of computation it is convenient to write equations (4.12) and (4.13) in the forms B \ — b m = ^T - 2ri_Q. m n F n (/3) 0 and y<0 separately. Consider the region y > 0. The deflection W satisfies the differential equation (2.25) with p= at all points of this region. The boundary conditions are: W = when X = , X = a , and when y = QO 5 (5.1) ■£-^ = v;hen X = , X = Q. (5.2) d W _ when v . _ n (5.3) A further condition on W is found from the distribution of vertical shear, p y , along the line y = The load on the segment (u~~ CX U. + Oi.) can be represented by the series ^Po f* sin A n u sinA n ol . _ _, > ~ sin A n x 7T ^y n n (5.4) where ~\ — _D_Z£ . We accordingly require that A "~ a lim p v = w~ T_ 7T s,n A n usinA n cxsinA n x (5 - 5) 29 — Nadai, A., Elastische Platten, pp. 78-82 and 85-95. Huber, M. T., Bauingenieur 6, 1925. -56- From (2.10) using the variable 7) ~€y instead of y we can find the expression for Py in terms of W . Entering this ex- pression in (5.5) we obtain: Mm r-D 2 e 3 4^-Ke -&$-) 7-> + \ C7^ 3 OX* 07} ) = — 7p- Y -^sinA n usin A n oisin A n x (5.6) n=l ]3y (5.5) and (5.6) it has been arranged that the dis- continuity in vertical shear along the X-axis is given by (5.4). Corresponding to (5.5) the limiting value of p y as y approaches zero from below is equal to the right-hand member of this equation with its sign changed. The solution of (2.25), with p = Oj which satisfies (5.1), (5.2), (5.3), and (5.6) is: oo w = 2~ A n e" n * (cos£ n 57+c sin S n 7) sin A n x (5.7) where p sinX n a sinA n ot _ p g sinA n usmA n ot An ~ npDtC 3 nA n 3 "tT/oQ" nA* (5,8) c = r n /S n = p/cr. The remaining symbols are defined in the table of notations and in section 3. We now allow OC to approach zero and p to increase in such a way that Z p (X remains constant and equal to P. The coefficient A^ in (5.7) becomes a - a sin An u (5 ' 9) M n - /A -\3 -57- where A = -^oTp (5 - 10) The "bending and twisting moments will he calculated hy (2.29). ' If we let 2 e "**oiri* *, cm A v (5.11) 1p - YL AnAj, e"" 7 sin6 n 7 sinA n x n^r X = > A n A n e cos(5 n 7 sin A n x (5.12) then and from (2.29), where m y = -&- (/3X+oLCp) (5 * 14) denotes the real part of the expression following it. How, setting e = z (5.20) -69- = - , ^ = <9 where vo and cJ denote the real and imaginary parts, respectively, of the expressions following them. It follows from (5.19) and (5.21) that: *Hr#- ^(c^) /z (5 - 22) v, - Aa + ,„-> BCC-H) , . V ~ -TfiC ™n CH + B Z (6 - 23) where H= cosh — cos-^ - cos- (x-a) (524) n „• u W : ^^7 (5.25) B= Sinh— r*- sin — - — a • a -60- 7COT? TTCT? 7t , '5 26) C = cosh-^ cos— ■—- - cos- (x + u) 5 ' 26) In the vicinity of the point of loading (U.,0) the follow- ing approximate expressions forX and ?^ are readily obtained: „ Aa , ^ 2 [^ 4 + 2«Cx-u) 2 7 2 +Cx-a)' 1 ] )4 4X l0 9 9 .f. „ oc z*u\ (5 - 27) Za (J-cos — cl — ) It is to be observed that X becomes infinite at the point of loading and that the limiting value of l/j depends upon the path of approach to this point. The deflection at any point in the strip can be calculated from equation (5.7) with the coefficients given by (5.9) as the series is rapidly convergent. The bending moments m x and tTl-y can be cal- culated by the use of (5.13) and (5.14), the functions 9C and lb that are needed, being found from (5.22) and (5.23). For isotropic material p — I and (X ~ In this case the function /C as given by (5.23) is i dent ical # apart from a constant factor, with the function Q) which was obtained by lladai— in another way. The function lp reduces to zero. But the product C^ which occurs in the expressions (5.13) and (5.14) for the moments, a^iroaches the limit -y -x — as % approaches I. The expressions for the moments m x and ITly accordingly reduce to the known expressions for the isotropic case. 30 — Nadai, A., Elastische Platten, p. 95, -61- Appendix 6. — Infinite Strip (Long Harrow Bectangular Plate). Uniform Load Applied over a Saall Area. Edges Simply Supported A uniform load (^ per unit area acts over a rectangular area C Oi by CY whose center is at ( 11, O) as shown in figure 21. In appendix 5 the deflection associated with a line load Pq per unit length of the segment of the X-axis between X— UL"~OC and X = U+(X was found to be given by (5.7) for points in the upper half of the infinite strip. The loaded line segment will now be re- placed by a uniformly loaded strip of width dy. The corresponding deflection at points for which y (or 77) is positive is again given by (5.7) if the coefficients A n (see (5.8)) are modified by replacing Pq by ^.^y where C\ is the uniform load per unit area. It will be convenient to use the variable 7J instead of y and write The properly modified forms of (5.7) and (5.8) are then w -) B n d9?e (cos5 n 7/ + c sin y • *y • n x = h X' x , n y = hYy , n x> = hX^ (8 . 5) Since the deflection is assumed to be small in comparison with the length and breadth of the plate the conditions for equilib- rium of the forces fix, H^, and n x ^ or of the stress components X x , Vy, and X y —Price, A. T., Phil. Trans. A228, 1-62, 1928. Apparent Young's modulus for stretching, p. 41. The definition of this modulus as here given differs from that given by Price by a term whose value is small. See his equation (13.72) and his discussion of plywood on pages 50-52. -67- %iill be the same as if the plate were plane and in equilibrium under forces acting in its plane. These conditions are JX* + J»<*_ =0 dx dX' v dy aY *=o (8.6) dx dy Accordingly, there exists a stress function F such that v' - d F y' _ d z F w' __ d z Y d x dy (8.7) The elimination of UL and V from the system of equations (8.1) leads to the following relation connecting the components of strain: d exx . <3 eyy _ d e X y _ / d w \ _ d w d w ^y 2 - dx 2 dx<)y \dxdy/~ dx z c^y 2 - (8.8) This equation replaces one of the conditions of compatibility of the strain components. From (8.3) and (8.7) , _ E b a'F E L cr TL d z F e *> "Hay 2 ~TT~ dx 2 - E a d z V E^ T . a z F w~ H ax 1 H ay 2 e *y "" jll lt ~Jxdy (8.9) -68- H = E-g Eb " Ei- 0"tl A (8.10) The substitution of (8.9) in (8.8) leads to the following differential equation: Ea _^F_ + f_±_ _ 2E L cr T A c? 4 F E b d 4 F H Jdx^y H ax 4 \/z LT H ydx^y 2 H dy 4 2 .* (8.11) ax 2 ay ydxdy/ The condition for the equilibrium of an element of the plate under the vertical components of the forces acting on it is expressed "by the equation: ~fc + -W + "x-a^ + Zn *>lZd-y +n vlv + p = ° (8 ' 12) where p x and Py are defined "by (2.9). Now (See (2.1l) and (2.12) ) iP* + iR*_~_/n-^ + PK-^- I a * w "\ (8.13) ax dy ~ V U ' ax 4 <^ax 2 a y 2 + ay 4 y Using (8.5), (8.7) and (8.13), equation (8.12) becomes n a 4 w i ?w ,a 4 w , n <3 4 w D -^ +2K ?x^7 D ^y 4 (8 . 14 ) „ h ri!E-c3!w P d a F a z w , a 2 F a 2 w i - p + n L7y* ax 4 - ^dxdy ax ay + ax 2 ay 2 J Equations (8.1l) and (8.14) constitute a pair of simultaneous equations from whose solution under appropriate boundary conditions the deflection and stresses of a given plate under a given load are to be determined. -69- Appendix 9. — Infinite Strip (Long Narrow Plate). Large Deflections. Uniformly Distributed Load. Edges Simply Supported Consider an infinite strip of plywood of width Q. Let the uniformly distributed load per unit area be denoted by p. The edges X = and X — Q are taken to be simply supported and restrained from movement in a direction perpendicular to the length of the plate. Under the assumed uniform loading the deflection W will be independent of y. The component V of the displacement (the component parallel to the Y-axis) of points in the middle surface will vanish. Further, the component UL of the displacement (the component parallel to the X-axis) of such points will be independent of V. Consequently ^ = dy (9.1) It follows from (8.1) and (9.l) ; since V vanishes and UL and W are independent of y , that e yy = , xy = (9.2) Then from (8.3) the mean components of the direct stress system are: y y A x;= o XX e'. XX (9.3) From the equations of equilibrium (8.6), since Xy = 0. it follows that X*. is a function of y alone. But this function must reduce to a constant since, from the type of loading, it is clear that all components of stress and strain are independent of y. Hence x;= g (9.4) where is a constant. -70- It follows from the first of equations (9. 3) that e xx is constant and hence from the second of (9.3) that Yy is constant. Since the deflection W is independent of y the differential equation (8.14) becomes D, d 4 w dx« P + gh^yr (9.5) In writing this equation d * F = X' *y has been replaced by the constant 9 ^ n accordance with (9.4). Be- cause of the simplicity of the stress system all the information that could be obtained from the differential equation (8.11) is already contained in (9.4) combined with (9.3). It is to be observed that g is constant for a given load p but that it depends upon p. The quantity Q therefore enters the solution as a parameter. Equation (9.5) can be written d 4 w wz d*w _ _£_ 4 * ^ktt dx dx D. (9.6) where K 2 - gh/Q (9.7) The solution of (9.6) is D X 2 w = c, x + c z + A sinh kx + B cosh kx ~ ?k*D (9 ' 8) On determining the constants in (9.8) to satisfy the conditions on the simply supported edges, viz., w = 0. d a w _ dx 2 it is found that when X - and when X — GL 2 coshkO--f)-CoshJ^ , =TA?Q\-W ^ihl^ ^ +x(a-x) (9.9) -71- With the aid of this expression it is possible to obtain a relation connecting p and g or p and k, since q and k are connected by (9.7). From (9.3) Then '0 Further from (8.1) (9.10) e ' x * dx + 1 /dwV z^dx; Hence (dx/ dX (9.11) ! since, under the assumed conditions at the edges, the displacement LL vanishes when X = and when X = OL . On equating the right hand members of (9.10) and (9.11), calculating H w from (9.9) and performing the integration it is found that Aag _ E a 2 k'D, P)2 c +™u ka p- ka *t- ,J f4M1" The quantity Q is to be expressed in terms of K and D, with the aid of (9.7) and D, in terms of E, and h with the aid of (2.14). The resulting equation can then be solved for the quantity, H ITh* (9-13) -72- in terms of the quantity It is found that 4 P = 3A/3" (-fcf-fc^r^ (9.14) (9.15) where #(?)» £ p5tanh7)/y-5 +tanh 2 7 + ZjJ 3] Yt - (9.16) Prom (9.9) the deflection on the central line X - Q/2. is .given "by w= P 2k 2 D, "2 _k 2 l-cosh -^ cosh-^ + q2 4 Using the abbreviations (9.13) and (9.14) it follows readily that v% 3PAE coshff-1 E, cosh ^ (9.17) Prom (9.7) the following expression for CJ , the mean direct stress over the thickness of the plate, is obtained: 9 3A a* V (9.13) The bending stress (tension or compression) at a point in the plane whose coordinate with respect to the middle plane is Z is given by A * A e ** A 3x* (9.19) -73- where t_x denotes the value of L in a direction parallel to the X-axis in the plane in question. Then at a point on the surface of the plate, z - h/2. j v Exh p / cosh k(x- -§-) EX k 2 D,\ cosh ka This stress in a face ply is clearly a maximum along the central line, X ~ Q/2 , of the plate. Denoting this maximum bending stress in a face ply by S it follows that c- t_, a I' (9.20) where _, , I cosh 7- 1 e ^ = "^ cosh^ (9 ' 21) and P and 7) are defined by (9.13) and (9.14). Using (9.21) the relation (9.17) can be written Wo = 3PAEl_ h 8E, m where ?t0=Y [1-2 Ofy)] 0.23) The central deflection, the mean direct stress and the maximum bending stress in a face ply that are associated with a given load are expressed by equations (9.22), (9.18), and (9.20), respectively, in terms of the parameter 7). The value of this parameter corresponding to a given load can be determined from equation (9.15). Values of the function ft(77) appearing in this equation are given in table 42. The quantity vJAkql f 3 Lq. ) is to be calculated for the given load 4 e, VT/ -74- using the definition (9.13) of P. This is the value of the function 1£(7J) associated with the given load. The corresponding value of the parameter 7} is to "be found from table 42 or from a curve con- structed from this table. The parameter 7] having been determined, the values of the central deflection, the mean direct stress and the maximum bending stress in a face ply can be calculated with the aid of equations (9.22), (9.18), and (9.20), respectively. The values of the functions ty) and frfy) that are needed in these calculations are given in table 42. Approximate formulas. — It is possible to replace the exact formulas just obtained by very accurate approximate formulas connect- ing the load and the stresses with the deflection at the center. With the aid of these formulas the calculations involved in any given case are greatly simplified. From equations (9.15) and (9.22) we obtain = 7?F(7/)( where 34 With the aid of table 42 it is found that F(?f) is nearly constant — and that it may be replaced by F=- 0.36Qthe maximum error being less than 2 percent for the range of values of 7J in which we are interested. Hence the following linear relation holds approximately between W,/h s^d. 77 J ^=0.36<4)\ From (9.18) and (9.24) it follows that 9 " ^F^ ft)' &)' — This fact is shown by the following table of values: 7 2 4 6 8 10 F(?) 0.3661 0.3642 0.3612 0.3585 0.3568 0.3559 7 12 14 16 18 20 Hf) 0.3553 0.3549 0.3546 0.3544 0.3544 -75- For spruce A may be taken to be 0.99. For other specie* this value may probably be used with slight error. An approximate relation connecting p and Wo/h can be obtained from (9.17). For large values of 7] we have approximately Solving this equation for P we have, to the same degree of approxima- tion, 16E, w 8E, „i w H " 3AE L h + 3AE L ' h On substituting for 77 its expression in terms of Wg/h from (9.24) ' P.J6& v^, 8 Eg /vvo_x (9>26) H " ^XE L h + 3AF*E L ^ h where F = 3.6 Equation (9.26) is approximately correct for large values of 7J and hence by (9.24) for large values of the deflection. For small values of 7] we obtain from (9. 17), using the Maclaurin's series for COsh 7] and equation (9.24), the approximate relation n 32 E, w 2.602 _E^ /W\ 3 E L \h/ 5AE L h AF 2 E L Vh7 (9 . 27 ) On comparing (9.26) and (9.27) it is seen that the second terms agree to within about 2.5 percent while the first terms differ to a greater extent. Since the first term is important in comparison with the second when W /h is small the value of this term for small values of W Q /h (or of 7] ) as it is found in (9.27) is to be used in setting up an empirical formula, especially since this term becomes -76- of diminishing relative importance with increasing W f\\. Similar reasoning leads to the use of the second term as found in (9.26) corresponding to large values of W /h (or of 77 ) . Hence we write where P = A-^- + BP^) A= &4E]_ (9.29) AE L A t L It will "be noted that the first term of (9.28) expresses the result obtained from the usual theory of thin plates when the deflections are assumed to be small. An empirical formula for the maximum bending stress in the face plies can be set up with the aid of (9.20) and (9.22). These eauations lead to ft)' "Xl 1-^- where CL = 4®(7])/f(7}) (9. 3 ia) and E x denotes the value of E. in a face ply in a direction r>er- pendicular to the edge of the plate. It is found that QL ranges from the value 48 for small 77 (or W /h ) to 4.0 for large 7], By using the intermediate value (y, = 4.4, the bending stresses associated with small deflections will be underestimated while those associated with large deflections will be overestimated. The re- sulting percentage error in total stress, that is, direct stress plus maximum bending stress, will usually be small. Accordingly we write S-4<^©> (9.32) -77- A more nearly exact value of the factor QL which is taken to be 4.4 in formula (9.32) can "be obtained from the curve of figure 27 of section 9 with the aid of (9.24). The latter procedure is recommended. Equation (9.32) is an approximate expression for the maximum bending stress in a face ply. The stress in an adjacent ply is to be calculated by the formulas to be given below. The approximate formulas (9.25), (9.28), and (9.31) can be used for isotropic plates. When so used all letters E with subscripts are to be replaced by E a^cL A ^y \— It follows from (10. 10) and (10.7) that &J&) y a From (10.5) we obtain for small values of 7] } using the Maclaurin's series for tanh 1_ ' X-Wkfi-l*) (10.18) h 32E, V 10 / On comparing the approximate expression (10.12) with the exact ex- pression (10.5) it is found that a better approximation is secured for the range of values in which we are interested if the factor J - W z /\0 in (10.12) is replaced by | — Z?Vl0.4 « Solving the resulting equation for P we have approximately P = 32 E, w t 4E, mZ w AE L h ' l.3AE L ; h ^The behavior of the function Y\(w) is shown by the following table of values: T) 2 4 6 8 10 Hfy) 0.3698 0.3696 0.3682 0.3662 0.3638 0.3622 7 12 14 16 18 20 Hty) 0.3610 0.3599 0.3590 0.3586 0.3580 -84- On substituting for 7j its expression in terms of Wq/ h f rom (10.10) we obtain w. . „( ? ) P = A -it + Bhr ( 10 - 13 ) where A = ^ 2 ^' (10.14) _ 23Eq (10.15) B " AE L With the aid of (10.5) and (10.8) an approximate formula — for the maximum bending stress in the face plies can be set up. From these equations we obtain ■ - o-k ft)'* where OL= 4U(7JyM(7J) (10.17) For small values of 7} we find by expanding H(v) and M(#) in Maclaurin's series that ' OL = 16 (l+^-) (10-18) On comparing the values of OL given by (10.18) corresponding to the range of values of 7} in which we are interested, with those given by the exact formula (10.17), we are led to replace the factor 1/30 in (10.18) by 1/40 . —An alternative procedure is explained immediately after equation (10.22) -85- Then ol= 16(1 + 0.025 ? 2 ) (10.19) The expression for S "becomes (l6+0.4 7 ')(^)' E* /•._.., ,-Yh\ w On expressing ^7 in terms of W /h by means of (10.10) we obtain where r 16 E x /h V C - -^ ("J (10.21) = 2.98 E X E Q (W V (10 . 22) I Instead of using formula (10.20) in whose derivation con- siderable approximation is involved, the maximum bending stress in a face ply can be calculated directly from equation (10.16) with the aid of a curve giving 06 as a function of 77. This curve which is the graph of (10.17) is given in figure 28. In using this curve the value of 7J associated with a given deflection is to be found from equation (10. 10). The only approximation involved in the process is that contained in equation (10. 10) in which the error is small. After the maximum bending stress in a face ply has been calculated by one of the processes just described, the corresponding stress in other plies can be calculated with the aid of (9.36) or (9.37). Formulas (9.33) and (9.35) can be used to calculate the direct stress in a given ply from the mean direct stress Cj . In all of these formulas the plies are taken to be of equal thickness. Tables 48 to 52, inclusive, contain a comparison of results calculated by the exact and approximate methods. The dimensions of the plates and the notation are given in appendix 9 in connection with the description of tables 43 to 47. The maximum bending stress was calculated by using (10.20). It would have been better to use (10.16), taking the value of OC from the curve of figure 28 as explained above. -86- Appendix 11. — Rectangular Plate (Large Deflections). Uniformly Distributed Load. Edges Simply Supported. Approximate Method An approximate formula for the deflection W Q at the center 37 of the plate will "be obtained by assuming the following expressions— for the deflection W and the components U and V parallel to the X-and T-axes of the displacement of points in the middle surface of the plate. It is assumed that the edges of the plate are restrained from moving inward. When < y < C (See figure 4), 7TX • 7TV w= w sin^-p- sin-jjH- u - c, sin— — sin-^- v = csin- 2 ^ sin^ When c < y < b - c 7T x w= w sm a (n.2) v= Expressions corresponding to (11. l) are assumed for the region b — C < V < b Further let c- ra/2 (11.3) —Expressions corresponding to (11. l) were used by A. and L. Fbppl Drang and Zwang I p. 227. For plates which are not square it seems best to choose the forms (11.2) for the central portions, -87- The state of strain in the plate is made up of two parts, one associated with the bending stresses and given by (2.2), the other associated with the direct stresses and given by (8.1). The corre- sponding potential energies of deformation will be denoted by \ffo and \^j . Each of these is calculated as the sum of two parts arising from the end and middle portions of the plate, respectively. These parts will be denoted by the additional subscripts G and ITl respectively. Thus v b = v be +v bm (11.4) V d = Vde +V dm To these potential energies we have to add the change in the potential energy of the load, due to the deflection. This will be denoted by\£ The parameters W , C, , C 2 , and T (or C ) occurring in (11. l) and (11.2) are to be determined in such a way that the total potential energy, \J = \C + \/^j +\£ of * ne system will be a minimum. The expressions for \£ and "V^ were calculated in appendix 3 (see (3.24), (3.25), and (3.26) ). The strain energy per unit area associated with the state of strain (8.1) is equal to ~~5~ l-X-X ^xx ** *y 6yy + Xy6xy| = — b_TF p /£ +?F c ' w ° 2 + °* c *i (n,7) A.- *' 256 ^;(9*ar 1 +^ + 2 the quantities C, and C z can be expressed in terms of W . When these values of C, and C 2 are entered in the equation, d V —Q t ttle following relation between W^ and p is obtained: 3 v, . „ W p -"-^ + l: l- (>..,» -90- where n = P Q4 • (11.20) H E L h 4 7T 6 -^ Hz 8 H=™ -*= r - + -$r + «,(2k-r) < n - 21 > 92A r + -f (k-r) B 2 F-BCG +C*[ 4 DF - G * Q = *-L -Al 4 DF - G * (11 .22) w 2A ~ - * A = (A,+A 2 )a 2 , B =(B, + B 2 )a, C=C,q, D = D,+D,, F = F t| G=G, (11.23) When K = OOj that is, when the plate is an infinitely long strip with edges parallel to the Y-axis it is readily found that (11.21) and (11.22) become: H = 7r 5 #,/48 A (n.24) Q = lt 5 H Q /\ i d\ (11.25) -91- Appendix 12. — Rectangular Plate (Large Deflections). Uniformly- Distributed Load. Edges Clamped. Approximate Method Using the assumed forms (12.1) and (12.2) of section 12 for the deflection and the displacement we obtain by the procedure of section 11 and appendix 11 the following approximate formula where P = H £*Q© : M I2A 3# t 8 r- r + *> (8 k - 5 r) 2k -r (12.3) (12.4) [ A B 2 F-BCG+C 2 D 1 ^|_ A 4DF-G 2 J where T is found from (4.30) appendix 4, K = fcy Q and n- 8 A = 7t 05*r a r + I05^i + 2 4096 -t-^f-Va (k-r) r i) (12.5) (12.6) B - C = D = F = -7T ^f ( l6 ^ -4cr TL +2A^) (12.8) 7T* (4K a T + £i) + 8K Z K a (k-T) (12 - 9) ,.(- 4 « b A v r ) (12.10) -92- G= -^(o- TL + A^) a2-ii) The numbers it^ t K^ etc., are defined in appendix 11. When k = b/Q-Ooit is readily found that (12.4) and (12.5) become: H = 7r 4 ^,/3A (12.12) Q = 7T 4 *w4A (i2.i3) -93- Appendix 13. — General Notation Choice of Axe 8 and Designation of Type of Plate In figure 1 with the choice of axes shown, the load is con- sidered to "be applied to the upper surface producing a deflection in the direction of the positive Z-axis. In figure 2 only the XY-plane is shown. The conventions as to signs of tending moments will be explained in appendix 2 in connection with figure 29. In using these conventions in connection with figure 2, the Z-axis is to "be thought of as drawn outward from the paper and the load as applied on the under side. The abbreviations 3X, 5X, ..., denote three-ply, five- ply, ... plates with the grain of face plies parallel to the X-axis. The abbreviations 3Y, 5Y, ..., denote three-ply, five-ply, ... plates with the grain of face plies parallel to the Y-axis. Symbols GL, b — lengths of sides of rectangular plate as shown in figure 2. Q. ^ b [} , Dz — Coefficients of flexural rigidity defined by the equations E,h 3 _ E 2 h* D, = TTT » D, = I2A " 2 ~ \2X e xx 6 X y ... — Components of strain. (Love, A. E. H., The Mathematical Theory of Elasticity* Art. 8.) Ex, E-y, E 2. — Young* 8 moduli at a given point asso- ciated with tension or compression parallel to the X-y Y-, or Z-axes, respectively. E | — Mean Young's modulus in bending under a couple whose axis is perpendicular to the XZ-plane. See equation (2.18) of appendix 2 and page 7. E. 2 — M 88121 Young's modulus in bending under a couple whose axis is perpendicular to the YZ-plane. See equation (2.19) of appendix 2 and page 8. -94- Ccl — Mean Young's modulus in stretching in a direction parallel to the X-axis. See equation (8.4) of appendix 8. . E-b — Mean Young's modulus in stretching in a direction parallel to the Y-axis. See equation (8.4) of appendix 8. 9 — Mean direct stress .X* in sections 9 and 10. h — Thickness of plate. [^ =. -pL — Ratio of length to breadth of rectangular plate — In sections 9 and 10. + ZjUL ») 2A — a functional symbol in section 10. L, r\ t — Subscripts denoting directions parallel to the longitudinal, radial, and tangential directions, respectively, in v/ood; that is, L denotes the direction parallel to the grain, R the direction at right angles to the annual rings considered to be plane, and T the direction parallel to the annual rings and perpendicular to the grain of the wood. See figure 3. fTI % — Bending moment per unit length of a vertical section of the plate perpendicular to the X-axis, ITly — Bending moment per unit length of a vertical section of the plate perpendicular to the Y-axis, fflxy — Twisting moment per unit length of a vertical section of the plate perpendicular to either the X- or the Y-axis . P — Load per unit area. qq4 — In case of uniformly distributed load. P = p r-a (In sections 3, 4, 9, 10, 11, and 12 and t-L-H in corresponding appendixes.) -95- P — Load in case of a load concentrated at a point or total load in case of load distributed over a small area. (In sections 5, 6, 7 and in the corresponding appendixes.) R — Subscript denoting radial direction in wood. S — Maximum bending stress in face plies. — Subscript denoting tangential direction in wood. UL — Displacement parallel to the X-axis of a point in the plate . V — Displacement parallel to the Y-axis of a point in the plate. W — Deflection of a point in the middle surface of the plate; that is, the displacement of this point parallel to the Z-axis. W Q — Deflection at the center of a plate. W Qoo — Deflection on the center line of an infinitely- long plate. J\. x ) Xy — Components of stress. (Love, A. S. H., The Mathematical Theory of Elasticity, Arts. 41, 47.) (3 = eb r n = **p S n = A n or S = Zcr rL + AXv ka 7} —

s^ 5! CN t- . oo sJ O 05 00 o en o •* 5 ex ex CO © CM 00 ex en o • ec ^ o> * rH * i— 1 « s CO ec ex ex „• en « T* O ex . «* «• «o "^S ~* en © O „• •* ex "« f-f b .H ec .? to jia 5 *•» £ ° « oo H -*i i CO « co b « OO J- ec © • .« t- J H ec b • © ■*♦< io _ {- en ij en H ex >4 ^ H ec j ex H ex j cc »J ec (H rH J lO Ch ex -1 »o H ■*« ■J ec H ec b to to o to IO to o to CO to o to CO b o 3 £ as ec to <=> to « a; ex b o . ex bj ec j oo b "5 b © 3 en a! io a o ca a CD p. a o 03 w 6* o o ex o 8 00 H is. __C« O •H« lO o •o o o •o CO o OO o M 11 a. o CX o ex © 00 o o ex ex 8 »o ex 08 8 o o CO o o en co o o o e en ex ex <=> o T 4 ^ 8 "ST en o o o ex oo io ec o O ^H o ex o ^H 2 O ex 8 CO o ex 00 ec ex 8 o IO CM o lO O o • ec t» • •* en -*» •««< ^H CO t- o o i-i o ex o lO en 8 g co o -1> © o 00 8 8 ec co s o 00 • en (J> • CO IO • 00 ec • ex 3 -H IO t^ «-H •o CO ex ec t- w < •* a . a o -s ts* -a # g o Xx ii *■-< do w n , o r^ 1 cZ> — r^J ~7*3 "~r X —7-— 1 ~rp "~r a . a a « a . d a — -t> •« -a o •« -w -3 -ii O "5 < 4a .»■« w ,m^ . qd Ct cc y ^pB . qB r^ oo CO g^ ex 2 o» d » ex gh h t!H • >S »4 Sp«^ »ec J§^dg- •r 1 " .spa. CO 'o .0 <1^ ^•4-1 CSS "* ei «o CD <0 CC CD O Q S *»" d St- hi ® 3 ^ CO CD hH CO p 3 hH in £0 O- H-» •P CD O CD 3P Jh P P hH CO a a a 00 .2152 131? Table 9.— Isotropic pistes.- (I = . ?5 ) Uniform . load. S&ges simply sv prior ted 'k T . oc a ( approximate) : \ exact) 1.0 1.00 . 04b8 : O.OU56 l*-5 1.50 .039? : .0868 2.0 1.83 .1146 : .1139 3-0 • 2.09 .132U .1376 • 4.0 ■2.19 .1382 . 1442 00 .1470 .1465 Table 10. — The factor 7 of fornul.n (3-^3) p/a :"0.50 0.750 " 1.000 . I.250 1.500 1 . 7^0 • 2.000 7 : .06 .245 • . 490 •765 .897 .968 1 . 010 &/a : 2.25 2.500 : ' 2.750 1 3-000 : 3 • 500 4.000 : U.500 7 ■ .1.03 1.038 ■ 1.040 : I.036 : ■ 1.025 1 . 010 1.0C3 1312 CO r0 o U o o S CD o ,£ ^ O Ql| erf 1 u ^ 1 o SB ►» 8D PJ C «s| q (D H P o 03 Q. O VO CM o H O a Q) r) 0) 01 ■ : o CO CO .h r— r — vo co r*- o VO LOT — -3- coj* CO OJ LO co co c— OJ CO CO VO CO OJ CO h-CT\ LPNVO -T\CT\ CO CO J- OJ CO o o CO c~ st 1 c^ r— i o CO 1 CO O .. .. .. .. •« •• •• " OJ o OJ LO O CO COvO LO coj" iO O CO OJ oco lo O coco O O r— ro^t rH o om-o O co o o c— ro O O OJ o CO CO o O I ■ CO 1 O I • H rH r-i r-i rH rH H r-i (H rH rH 1 .. .. • •• ■• • • • • .. cO--f- r»-vO O ON^ . r-H r"OCO C— IO rH CO rH OJ COVO CO co co io LO J" CO CJV3- co co r— oj CTN O LT\ ro COCT\ CO OJ OJ St LO O 1 • O 1 • CO 1 O • • .. ... •• •• •• •• O CO VD ^o LOVO VD lovo r— • O J" CO COCO OJ CO LOOJ LOvO co VO CT\ CO'CTN r — vo crs rH LO 1 CO 1 rOrH OJ r-i J" OJ rH OJ rH st OJ OJ CO CO CO 1 .. .. .. .. •• O CO oj^t OJ OJ st O CO inocr\ rH OJ st lo loo nzf LO o co co KO rH 1 oj rocn OJ CO co r— - r-i c— VO CO o St ro r-i OJ 1 O 1 • OJ o rH OJ OJ • . • ■ .. .% .. .. .. r-\ r— OJ J" OJ OJ OvD O VO OJ VO St CO rH rH rH LO co r— o co rooj COCO rH o o o LPvO CO o o rH CO oj co CO 1 — o OJ OJ o O CO ro O O I • LO 1 CO 1 OJ 1 • o rH OJ OJ I . . .. .. .. .. .. .. o o co o rH LO OJ OJ o-3- o rH VO OJ vjD O CO rH OJ ^t rH rH O O VO CO LO OJ -3" co-d/ lo o o o r-K (T\ OJ vo oj r^- OJ \<~\0\ o o oj r— CO f — O CO CO VO o o LO rH o r-i CO OJ O i • O 1 C— 1 rH 1 o r-i OJ OJ .. .. .. .. ■-.. .. .. •• •• •• •• O CO CO -O rH rH QiflJ- rH rH OJ OJ COOJ k> r— oj oj vo rr\ r-i ^ st o o rH CO LO I — o o CTN CO 1 CO I CO o rH r— I rH ,-\ rH r-i H oj o co r— \T\st -=t- co j- ^f COCO -4"-rt ro CO VO OJ' TN O CO ro J- oj o J- cr> OJ r'A ro UP\ LT\VD r-i OJ VO OJ LT\VO r-i r — co o OJ LO OJ CO CO i CO 1 CO 1 rH rH rH rH rH r-i r-i r-i OJ O r— co COCO co co co st st st ■ CT\ O CO CO O CO --t m-rt- cn r— vo o o o vo vo vo o o rH r-i VO vD CO CO st 1 — LO CO CO rH | O I • O 1 • o rH I .. .. .. .. •• •• •• •• LOLO LO lOLO o LOLO LfMTMO LTMT» LP» LO LO LO I OJ OJ st _-r OJ OJ OJ St St St OJ OJ OJ OJ OJ OJ OJ OJ S?Jt OJ St OJ J- OJ St OJ 1 • st • •• .. .. •' ■• •• * • * LOLO LOLOLO O LO LO LT\LT\ LT\ ir\ lT\ \T\ LO LO LO 1 O OJ CTSJ-J- CO o OJ rH CO O OJ COCO st rH CO O OJ r-\ CO o r-i CO CO ' r-i co CO r-i co CO r-i CO 1 rH 1 '.< 1 ro CO >-• >H >H LOLO LO io '.o lT> p-* pH t >H lt> .rN m rH LTV LO 1 xi ! J r— i G uk 1 VO l 1 CO I 1 o rH 1 r-i rH 1 I cO OJ | CD rH | ^ Table 12. — Plates of type 3Y. Uniform load. Edges clamped k : T : a a ! (approximate) : (exact) 1.00 : (See table for 3X) 1-33 2.00 » : 0.0943 : 2.00 : 0.2753 ■ : .2698 3.00 : 2.46 : -3SS0 4. 00 : 2.53 : .4055 ! ..<-•*-». 5.00 2.55 : .U115 ! . . -^.- . . O-f-O • • ••••• s .4235 : .4321 Table 13 .— f: .ates of type 3X. Un: Lform load. Edges clamped k T a a (approximate) (exact) o.so 0.50 0.01722 0.01682 •75 66 ■ .02821 .02984 1.00 69 .03007 .03341 1.25 70 .03058 .0^33? 1.50 : 70 .03086 .03279 2.00 70 .03113 3-00 71 .0313^ . . .- . -.- . . oxr.i •••"•• .03163 : .03209 Table 14. --Plates of type 5Y. Uniform load 1312 Edges clammed k 0.5 • 1.0 2.0 3.0 : T (See tal 1.74 1.79 ' 1.31 : : a. . (apDroximate) Die for ^X} 0.1210 .1262 .1274 .1292 : a : (exact) .11840 4.0 o<> .1^080 Table 15. — Flptes of type 5X. Uniform lonri Edges clamped a (approximate ) (exact) 1.00 1.25 1.50 1-75 2.00 2.50 ^.00 0-93 •96 •97 .93 .98 • 98 • 98 0.03488 0.03539 .0^647 .03850 .03709 .04093 .03737 .04119 .037^4 •03973 .03763 .03*21 .03782 • . • 1 r . • .03816 .03872 Table 16. — Plates of type 7Y. Uniform lor>d Sdge s clamped k a (approximate) 1.00 : 1.00 0.0^706 1.25 1.25 .06262 1.50 1.50 .08396 1.75 1.58 .09058 2.00 1.60 .09276 3-00 1.63 •09533 • 4.00 . 1.64 .09597 5.00 : 1.64 .09625 \ .09704 Table 17.— PI? ?tps of type 7X. Uniform load. Sdges clampec 1 (atroroximate) 1312 1.00 : 1.0C 0.03706 1.25 I.05 . 04008 1.50 1.07 .04098 2.00 1.08 .04151 3-00 1.C8 . : .04186 4.00 1.08 .04198 0*3 . 04225 Table lg. — Plates of type 9Y. Uniform load. Edges clamped (approximate) 1.00 1.00 0.03706 1.25 . 1.25 .06020 1.50 1.U7 .076^0 2.00 1.53 .02190 3.00 1.55 .0S36?. 4.00 1.56 .OSUlg- 00 .0^50-3 Table 19.—: Plates of typ - 9X. Uniform load. Edges clamp 3d k •' : t i (a a prroximate) 1 . oc •" : 1.00 0.03 706 1.25. . 1.10 . Qk2kl 1.50: 1.12 .okiUS 2.00 1..13 .0UU19 3.00 l.lU .0Ui+5g 00 .0k^S2 H12 Table 20. — Isotropic plates ( cr = . 25 ) . 1 o ad . Edges clamped Uniform 1.00 1.00 O.OlUUU • O.OIU06 1.25 1.1U .01941 .02032 1.50 1.23 .02248 .02463 2.00 1.32 .02532 .02864 0-0 *• • ■ .02887 .02930 Table 21. — The factor 7 of formula (^-35) P/a 0.500 O.75O 1.000 1.250 7 .065 ' .285 •575 - -835 P/a 1.500 1.750 2.000 : 2.250 7 i960 - 1.015 1.642 1.051 (3 /a : 2.'500 : 2-75Q - 3.000 3 • 500 7 : 1 i 044 ! ' 1.026 : 1.015 1.002 1312 r £ -p ^ en £i O *-■■ <3D_ cti S > <3 pq nj P- o •H J^ w v0 1 CJ o ■ MXH vo i H O M X » ft -u CI CD r-j O OJ CM rH f<~\ rH CO OJ rp, G> On O vx> . CT.60 60 r-pco O CMtAl ro O r-i r-i r-i LP, LP. OJ O 1 — rH VD 60 _d- IPX VD LP O 1 ro LP. 1 O VD 1 Jf rH r-i .—I rH rH rH rH rH r-l rH rH rH rH rH r-i r-i 1 r-i .. .. .. .. • ■• •• - •• •• 1 • rH "P O O o o o o o o O r-P O OJ LP, O OJ LP. O O 60 o o o J" 0 O O r^» O LP, O rH o o rH O rH O O cr. o o LP 1 O 1 O I r-i r-i rH rH r-i r-i r-i rH rH rH rH rH rH 1 .. .. .. .. .. .. •• •• •• •• 1 ^t 60 r-i fP r-i 60 OJ CO CT\ O O LPi On>X3 r — rPrH r-i OJ KA.H/ O rH rH rH cr. co vx> rH 1 >XD LP 1- X LP, CTi 1 LP 1 U3 1 r-i i-H rH rH rH rH rH rH rH rH r-i rH rH rH 1 .. .. .. .. .. .. •• •• •• •• O 60 VX) 0-3" 60 epeo OJ r-PLPOJ LP>vX) CO vx> cr. r^\CT, r — en VD on rH LP 1 r^n I t*~\ r-i r-i _"t OJ rH OJ rH J" OJ . OJ r^» r^n r*n i ■■ •• •■ •• .. .. - •• •• •• LPVX5 CO hP -4" 60 CO rH V£> CT.OJ O rpr-p rH OJ LP rH OJ r— <^\ PiH C\J o J- J- 60 LP,^- LP. rH cr, rH o p O r^n i o rH rH rH rH OJ OJ OJ OJ r^» rx LP 1 •• •• •• •• .. .. •• •• •• • • 1 r-i O CT\ r-H r— LP. r-pr — rH LPt LPOn 60 On O 60 ^t 0> t"Pr *^X> IP. CO r— r-p ,4- an Onr^rH O on oj -two O -J- LP. rH OJ ^t cr> rH oj lp. r^, vx> -dr U3 CO rH O rH o CO O I CO 1 CO 1 o rH rH rH rH OJ OJ OJ OJ r^i LP. LP 1 .. .. - •• .. .. •• •• •• •• O 60 H> CO r-i r-i CO =t to o rH OJ oj r*-\ oj vx> vx> r~P OJ VD 0> rH o o rH LP. r — O o T\ 1 on i CO 1 O rH rH rH rH rH rH .. .. .. .. .. .. •• •• ■• •• OJ LP 60 1 — LP. .7t to J- r-pan ,4 n-\ C0VO OJ O^Q CO KVt OJ cm N^ro LP. LPiUD rH OJ -X3 OJ LP.VX) rH P. 1 co o OJ LP, OJ CO 60 1 on i r-t rH rH rH rH rH rH rH .. .. .. .. .. .. •• ■• -• .. OJ O r~- co O^ CO =t .-f CT.O M CTNO CT> ,-t iX\ rf o o o vO VD U) o o rH rH 60 r — LP. CP\ on rH 1 O I • O l O rH .. .. .. .. .. .. •• •• •• • • LP LTi LP LP LP. LP. LOi ir\LPi lp, LP. LP, LP, LP LP, ■-Px 1 OJ OJ OJ OJ OJ OJ OJ .--t J- .--1- OJ OJ OJ J* -=f J* OJ OJ OJ OJ OJ 4 OJ -3- .. .. .. .. .. .. •• •• •• .. LP LP. LPi LP. LP. IPS LP. LP. LP. lP\ in lp, IP !X^ LP LP 1 O OJ CT^.rt O OJ CO O 0J r-pon^t rH 60 O OJ r^cTN J- rH 60 O rH CO rH CO rH CO rh >h ;-i |X^ ip. ic\ LP. LP, LP. X LP, X! ! J r— i a 1 1 cti OJ I cd £h O nb P nj Q o d rH M Ti ;> CD rH -p Q a K t* • H +3 1/3 £ CD 03 O CD a tf o Td o F3 ^ — « ■d ir o • o r— ^ * — * *3 r— 1 CO CO -=f ■=+■ r"A O o O o o O r— CO o VD C\J J- rH OJ CVJ O o o rH fH rH rH VJ3 o O r-~ .=!- H r*> 3- o o o C\J O r— ^D x> LT\ r« CT\ o T\ cr> o .TV H r^> OJ •A r<~\ H LTl UD -o JO CO • • «. o •• •• •• .T> rH CM .H CM r — LP\ r^- 1^ — r> LT» LT\ *0D O r. r^\ LO in IT\ m LO> CM CM CM CO "O .0 rH rH H ^.o VD vO o O o -Hi X! M « i-\ LT» C^ p< Pi •H C! •H t rH cd rM o •H 03 CO" «H rH | I O PI o CO CD •H 0) o fa h o o S 0) o O H cd o] o w Q cm o rq XrH 1 H O rd .-3 X} CD 'Tj Q) Ph Eh CO (T\ r-i r-i o nn o o VO O r — co r— co o o :.r»LP> r»lTv O CO X X r-oro J- vx> r~- i — BO CJM^ r— vd co to cno> r— J- co cm r-i ur\^o -=t" O co o cr> r— r— CO o I.O r-i o LO CTA O CT\ o o o no uo O co O cor— . O O O O LO o o m J Oin O o o o o o o o O co co o o o o o o o o O OU3 o o o o u VO CT\ CT\ CO rH CO o> CO rH CO to CT\ i — r--,-H- rH rH rH ^t C\J HO l^ co h m c\j co i — co cr« a> o wu3 C\J I — r — CO O ^f LO r-i o in o> CTn o cr\ o O CO LOvO UD OJ - w CO LO CM * VD CTN CM O O O O rH in O VO CO J" CM rH CM J" CM O O O O O O O r-H VD CM VX) J" O rH J- O mC— O LT\ m cm ro LOi rH O LT\H; CO WON co co in p- — r-— r — » ro ro c~\ O ^X> m CM coco co cocM nai h VD U3 CTs r — [ — (-i r— r— co vx) ro CO O r-i CM CTs r-~ r-i r-i r~i CM CM rococo cok>k> ro ro ro ro Ki ^DVJD i-I ,— I rH rH r-i r-i r-i rH rH rH rH COCO CO CO CO COCOCO CO CO CO CO CO ro r-i CO r^ r-i r-i CM in jHt CO,zf to i — jht c "i o LT\J3- ^f -=t rO CO -JD CM cr> o co roj- CM rH rH rH O -=f OA rH LO LOvO 1 \ 60 o CM m CM CO r-i r-i CM O • G> CT\ CO r— CO CTv CXVO r^\^> ,3- J- _h- HHH O^OM (T\0 cn o o o '.O VX) U3 o r-i tr> lpiu~\ LO LT\ir> mirtin ur» LP» LO LP. CM C\: CM CM Cvl LOIT4LO CM CM CM J" _t J" LTMrMTN CM CM CM CM J- J- J" jH- LP> LT» LT\ LT> CM LT\ CM LP> CM LPv CO O CM k-< >n $* l CO O CM !x} :xj X! LTVLTMO I WOW CO r— • >-* >-« rH I CO CO X I O CO rH I I CO ro CO CO CM o in LP> o O CO CO O CO O CO 4- CM m CM VD UD CO CM O CO O CT\ T^, CO a>co O VD-Ztf — r-o r— CM CT\ o CO O CTN r-i r-i rHrHCM ,t H/ 4 1 J-jH/J- ^f r-i I CT\ CO CO CTN Ki CO LO rH .rfr ^t a> o r— h- CTN o LT\ CM LTN CO X I CM Table 29. — The ratios E„/Et and St./E-t for spruce plywood. Plies of equal thickness Flies y\ Number 3 : 0.679 : 0.357 5 : .6lU : ,k22 7 : -5*7 = .UUQ 9 : .572 : .HSH Table 30. — The ratios Z /S T and E v /2- for Douglas-fir pi; rwood. Flier 1 of squal thickness Fli es i V E L ' — '- / J - , T Numb er 3 o.Cs6 0.372 5 .623 .toH 7 • 596 .4-61 9 •581 .k-(S Table 31.— Fl? )te of tyve 3Y V : T , T- n 1.0 : (See table 32.) 1.5 : 1.5 . 1.739C > : ^.S62 2.0 2.C .900^ : U.729 3-0 3-C . p '277 : 4.279 U.o ■ 3-5 .U706 : 4.720 5.0 : 3-5 1 .U685 : 5.312 0-0 .... . . . : .1*637 : ■ "?7 1312 Table 32.— Plate of type 3X T H Q 1.0 0.953 6.327 2.0 .967 ! 6.236 3.0 .969 . 6.226 CXD 6.208 13.30 13.11 13.11 13.12 Table 33.— Plate of type 5Y u 1.0 (See table 340 1.5 1.50 : 2.408 6.U65 2.0 2.00 1.679 S.434 3-0 2.45 1.478 5.962 4.0 : 2.43 I.492 6.684 00 1.520 g.154 Table 34.— -Plate of tyi -0 5X k T H Q 1.0 1.00 6.165 : 12. 3"* 1.5 1.34 4.571 9.36 2.0 1.32 S.044 : 10.27 3-0 1.31 : 5.09S IO.96 4.0 1.31 5.103 : 11.20 <=*«o 5-145 11.86 1312 Table 35* — Isotropic plate j -- : T « * "^ 1.0 : 1.00 21.360 : 29. 2R 1.5 : I.50 ll.lUo : 16.77 2.0 : 1.83 8.727 : 15. 43 3-C : 2.09 7.552 : 16.61 k.o : 2.19 7.237 : 17.5b 5.0 : 2.2k 6.986 : lg.lU CKS 6.800 20 . UO Table 36.— -PI ate of type 3Y k : t H 1.0 (See table 37 •) ?..o : 2.00 3.632 : 6.02U 3.0 2.1+6 2-577 : 6.678 k.o 2.53 2.1+66 : 7.U09 5-0 2.55 2.U3C 7-777 2.361 S.7$?l Table 37.— -Pi ate of tyj se 7 ,X k = t : ^ 1 Q 1.0 ': O.oS : 33.26 : 27.86 2.0 : .70 : 32.12 : 20.30 3-0 : •71 : 31.91 : 18. 91 k.O : .71 : 31.82 : 13.29 00 31.62 : Id. 70 1312 Table 3g.— Plate of type 5Y X. T Q 1.0 2.0 3-0 4.0 (See table 39.) 1 74 : 1 79 : 1. 81 : 8.265 7-927 7-848 7.7^0 g.156 9.1^7 9.574 10.^80 Table 39.— Plate of type 5X k T H Q 1.0 2.0 3.0 4.o 0.93 •93 •98 •99 28.67 17.74 26. 6U 15.74 26.44 15.48 26.37 15.37 26.21 15.10 Tablfl '40. — Isotropic plate- ( 0.2S) k T H 0, 1.0 1-5 2.0 3-0 4.0 : 1.00 69.27 : 41.47 : 1.23 44.48 26.98 : 1.32 39.49 26.65 : 1.38 36.89 26.06 : 1.40 36.09 25-97 34.63 25. 9g 7able 41. — The coefficients— a and b in equation (4.l) te= Type : b/a : a i ; a 3 ; a 5 b : : 1 : b 3 : 1? : 5 3X : 0.50 0.15989 0.000417 . -0.000045 : 0.13755 : 0.000299 : 0.000046 5X .75 .13960 .000260 - .000070 .44736 ! .002650 . .000040 3X 1.00 . .12907 .000220 - '.000084 .88629 : .009890 .000550 3X 1.25 : .12780 . 000340 - .000040 1.43460 : .023900 .001600 3X 1.50 . .12831 .000467 - .000002 2.09460 : .045550 .003720 3X 2.00 .12996 .000900 .000100 3.75890 : .109020 .011990 5X 1.00 .13792 .000724 .000046 .38688 : .002525 .000181 5X 1.50 .12321 .000824 ! .000083 1.01714 : .012707 .001010 5X 1.75 . 12291 .000900 .000110 1.41460 : .022350 .001850 5X ! 2.50 .12468 .001289 .000219 2.94680 : .076670 .007460 ~~In the calculations the elastic constants cf spruce were used. The values as writ .en indicate an accuracy to a greater number of decimal places than one is warranted in assuning from the nature cf the calculations. 1312 Table 42. --Values of x(t)), ( rj ) , \j/(r]) ? k( r\ ) , m(t)) and n(t}) x(t) e(n) 0.4167 K(n) M(n) 0.O8333 »(t]) 0.3333 .0 .1 •17 .2 .4 •5 .6 • 1 .8 •9 X .0 1 .1 1 .2 1 •3 1 1 .4 1 P" 1 . b _1_ . 7 1 1 .8 ]_ •9 2 p\ 2 .2 2 .4 2 •5 2 .6 2 .3 7 j .0 3 ■ D 4 4 5 PC 5 5 6. 6. 5 7. f • 5 8. Q 8. "5 9- 9. 5 10. 11. 12. 13. in. 15. 16. 18. 20. 1.297 1.676 2.093 2.554 3.067 3.639 4.277 4.990 5.734 6.666 7.646 8.728 9.921 11.23 12.67 14.24 15-95 19-82 24.35 29.58 35.53 42.41 63.48 91.00 125.89 I69.O7 283 • 9'4 442.96 653.45 922.76 1,258.2 2,157.0 3,^08.7 5,071.7 7,204.9 9,867.4 0.5 4913 4687 4346 3942 3519 3109 2730 2391 2093 1835 1614 1424 .1261 .1121 .1001 .O767 .0602 .0483 .O39U6 . O3279 .02764 .02360 .02037 .01776 .OI56I .01383 .01235 .01108 . 00999 .00826 .00694 .00592 .00510 .00444 .00391 .00309 .00250 4100 3912 3633 3306 2962 >626 2316 2038 1794 1R82 1399 1242 .1106 .0990 .0889 .O69I .0550 .0446 .03684 .03089 . 02624 .02255 .01958 .01715 .01513 .01346 .01204 .01083 .00980 .00813 .00685 .OO5S5 . 00505 . 00440 .00388 .OO307 . 00249 0.112 1.008 1.809 3.610 4.547 5.516 6.516 7.553 8.633 9.760 10.941 12.179 13.481 14.851 16.296 17.819 19.427 21.124 22.914 24.805 28.9C4 33.460 38.511 44.102 50.264 68.434 91.016 118. 57 151.66 236.6^ 350.22 496.71 680.35 905. 4 1,176.0 1,496.5 1,871.1 2,304.1 2,799-6 3,361.9 4,704.1 6,364.4 .04410 •03773 .03236 .02791 . 02421 .02114 .OI857 .01641 .01459 .01304 .01172 .01058 .00960 .00874 .00800 .OO676 .oo p .79 .00501 .00437 .00385 .00342 .00274 .00225 .0830 .0820 : .3324 : .3298 .0804 : .3256 .0783 : .3199 •075S : .3130 .0729 .0697 '.'o664' : .2921 .O63O : .0^96 : .2687 .0514 : . 2454 ■2239 2046 '1877 1729 1600 ,1488 1389 .1225 .1093 .0983 .0900 .0826 ,0764 ,0710 0663 ,0622 ,0586 0525 0475 •T3T?" Table U3. — Plate of type Tf . Uniform load. Edges simply supported Exact and approximate methods q i r : : : h : a a 5a ; 1 : 8 2 S 2a ; g 2 ; & 2a Lb . per Lb. ner : Lb . per : Lb. ner : Lb. per : Lb . -oer sq. ft. sq. ft. : sq. in. : sq. in. : sq. in. : so. in. 1 : 0.4638 1.001 : 1.0010 : 83-7 : 81.8 : 67-4 : 66.6 6 : .8992 6.089 : 1.0150 : 151:7 : 158.5 : 2U9.6 : PRO. 2 20 : 1.3410 : 19-400 : •9700 : 219.7 : 2^6. 4 : 57?. : 556. 4 50 : 1.8270 : U8. 290 : .9658 : 296.7 : 322.1 : 1073-0 : 1032.8 200 : 2.9180 : 19U.600 : •9730 : 474.1 : 51U.5 : 2707.O : 2634.7 Ta.ble 44. — Plate of type 3%. Uniform load. Edges simply supported. Exact and approximate methods q ; w : h qa : q : s l la ; e i ! e la Lb . per Lb . per : Lb. per : Lb . per : Lb. per ■ Lb. sq. ft. sq. ft. : sq. in. . sq. in. : sq. in. ! so. in. 4 : O.U360 : 4.066: 1.0160 : 249.2 230.6 : 57.1 ! 58.8 20 : .9797 : 20. 270 : l.OlUo : 541. g . 518.2 ?94.7 : 297.0 65 I.360O : 66.210: 1.0190 : 830.6 . 825-2 746.1 : 753-1 175 2.2240 : 176. 900: 1.0110 : llUO.O : 1176.0 : 1527.0 : 1530.0 500 : 3.1850 : U97.7OO: •995U : 1593-0 : lbg^.O : 3160.0 : 3140.0 1312 Table U5. — Plate of type 5Y. Uniform load. Edges simply supporter! Exact and approximate methods w q : _£ : : h ! q a 2§L •• q ' S 2 J S 2a ; g 2 ! 6 2a Lb. per : : Lb . per : ; Lb. per : Lb. per : Lb. ner : Lb . per sq. ft. : ': BCt. ft. 5 sq. in. : sq. m. : sq. in. : sq. m. 8 : O.36U8 : 7.92g : 0.9910 : 3UU.9 : 321.5 : 118. 7 : 11U.U 50 : .8281 : 50.380 : 1.0080 : 7^7-8 : 729.9 : 59I.I : 589. k 250 : I.U770 : 2U6.200 : ,.98U8 : 1261.0 :' 1302.0 : 1919.0 : 1875.0 750 : 2.1350 :. 715.OOO : •9533 : 1770.0 : 1882.0 :' 1*108 .0 ': 3919.0 Table 1+6. — Plate of type 5%. Uniform load. Edges simply sun-port erl Exact and approximat e methods q : : h : q : a : . q a ! : q : s i ; la ; Si : gla Lb. per ; : Lb. per : : Lb. per : Lb. per : Lb. ppr : Lb. per sq. ft. : • : sq. ft. • • sq. in. : sq. in. : sq. in. : ; sq. in. 25 : O.U231 : 25. 1+5 : 1.0180 : 67I.I : 621.7 : IU8.9 . : • 153.9 150 : 1.0190 : 151.60 : 1.0110 : 1557.0 : IU97.3 : 890.5 : .892.6 500 : I.62U0 : 509 . kO : 1.0190 : 238U.O : 2386.0 : 22U8.0 : 2268.0 1250 : 2.2370 : 12U8.00 : -998U : 3171.0 : 3287.0 1+3^8.0 U301.O 1312 Table U7« — Steel plate. Uniform load. Edges simply supported. Exact and approximate methods W 12. • q. h a a g a Lb . per Lb . per sq. ft. : : sq. ft. " 100 : 0.5009 : 102.0 ' 1.020 500 : 1.02S0 : 512.0 : 1.024 1000 : I.33U0 : 1,015.0 : 1.015 2000 : 1.7200 : 2,048.0 : 1.024 Lb. per : Lb . -per : Lb. -per : Lb. per sq. in. : sq. in. : sq. in. : sq. in. 4,748.0: 4,43^.0: 1,261.0: 1,299.0 9,301.0: 9,101.0: 5,410.0: ^,473-0 11,887-0: 11,811.0: 9,157-0: 9,216.0 14,865.0: 15,228.0: 15,133.0: 15,3?1.0 Table 48. — Plate of type JI . Uniform load. Clamped edges. Exact and approximate methods q : h ^ ^a : q S 2 ! S 2a : So ; C \ g 2a Lb . per Lb. per Lb. per : Lb . per : Lb. per : Lb . per sq. ft. sq. ft. sq. in. : sq. in. : sq . in. : sq. in. 2 : 0.4614 1-973 : O.9865 366.7 : 35^-0 : 63.O : 63.5 6 : .7780 5.967 : .9945 796.4 : 777-3 : 178.4 : 180.6 10 : .9562 : 9-890 : .9890 : 1122.0 : 1130.9 . 273.5 272.8 15 : 1.1240 : 15-020 . 1,0010 : 1481.0 : 1561.? 375-6 377-0 25 : 1.3550 : 24.820 •9928 : 2093-0 : 2342.4 556.2 547.8 1312 Table U9. — Plate of type J>X. Uniform load. Clamped ed^es, Exact and approximate methods q • h q a ! ^a ': : q : S l ; S la ; gi ! & la Lb. per sq. ft. j Lb. per sq. ft. Lb. per sq. in. : Lb. per : sq. in. : Lb. per : sq. in. ; : Lb. per : sq. in. 15 : 0.U209 ilk 95 : 0.9967 : 836.6 : 228. 4 : 52.2 : : : 52.9 40 : .3809 : 39-94 : .9985 : 1S97-0 : 1866. 7 : 227.3 231.6 100 : 1.4bl0 : 98.85 : .9885 : 373^-0 : 3596.1 : 63I.3 : 637.I 250 : 2.2170 : 231.30 . 1.0050 : 7034.0 : 7010.0 . 1435-0 I467.O Table 50. — Plate of type 5Y. Uniform load. Clamped edges. Exact and approximate methods q : : : h : q : a ; ^a q ; s 2 : ': S 2a e 2 : r : 2a Lb . per : : Lb. per : :Lb. per : Lb. per ■ ■ Lb . per : Lb. per sq. ft. • '• sq. ft. : : sq. in. : sq. m. • sq . in. " : sq. in. 30 : 0.4022 : 29.89 : 0.9963 : I377.O : 1358.4 . : 131.6 : 134.1 50 : .5670 : U9.26 : .9852 : 2069.0 : 2011.^ : ; 266.6 : 266.4 150 : 1.0230 : 147.30 : • 9820 : 4592.0 : : 4419.8 : 871.7 867.4 300 : : 1.3920 : ■ 298.10 : •9937 : 7403.0 :' 7336.0 : ' 1^82.0 : 1606.0 1312 Table 51. — Flate of type 5X. Uniform load. Clemmed edges. Exact and approximate me thods q : h : q a 5a ! q S l . la . g i ; g la Lb. per : : Lb. per ; Lb. per : Lb. per : Lb. per : Lb . per sq. ft.' ' : sq. ft. 1 sq. in. : sq. in. : sq. in. : sq. in. 100 : 0.1*329 : 99-55 : 0.9955 : 2399-0 : 2375.1 = 153-6 : 155.1* 2C0 : .7366 : 199. ko . •9970 : H290.0 : 1+241.4 : UU0.9 : UU9.8 350 : I.0UU0 : 3^8.20 ■ .99^9 : 6621.0 : 6UU8.6 : 886.3 : 903.5 ■ 600 : 1.3750 : 585-00 : •9750 : 9736.0 : 9335-0 : 1596.0 : I567.O Table 52. — Steel plate. Uniform load. Clamped edges, Sxact and approximate methods q : h : q a : ^a : s 1 S r, > a ; p' : g a Lb. per Lb. per Lb. per : Lb. -oer Lb. -cer Lb. per sq. ft. sq. ft. sq. in. : sq. in. sq. in. so. in. 100 : 0.172^ : 96.8 O.968O ' 5,597-9 ! 5, 1 +2l*.0 11*6.9 11*1*. g 3 co : .1+606 : ' 291.3 I • .9710 ' 15.U62.0 ll*, 963.0- 1,015.0 1,031.0 500 : .666U : U97.O : .991*0 : 23,569.0 : 23,228.0 : 2,185.0 : 2,218.0 7 r 'C .860H : 7H5.O : -9933 : 32,320.0 : 31,518.0 : 3,616.0 : 3,697-0 1000 ! 1.0060 : 982.0 : •9820 : 39,936.0 : 38,1+90.0 : 5,072.0 : 5,055-0 1312 FIG. 1 CHOICE OF AXES FOR PLYWOOD PLATE FIG. Z MIDDLE PLANE OF PLYWOOD PLATE 127 FIG. 3 THE THREE PRINCIPAL DIRECTIONS IN WOOD -~i r -b Lj = h-C FIG. 4 SUBDIVISIONS OF A PLATE IN THE APPROXIMATE METHOD z u )«o3 F FIG. 5 CROSS SECTION OF ASSUMED FORM OF THE DEFLECTED SURFACE OF A PLATE IN THE APPROXIMATE METHOD 1 r /f w •/' ! f/i ii ii 1 ' i? . / o ft! I I «J •* 1 Ci u ' CJ ^ «o \ -^ . % il i 1 1 > 1 * I - 5 \ ( .X n, 'Ay 1 \\V7I 19 \ °3 ^ 'A 9 ■$* vv 3 '^ % w ? A \ 1 V \ i c S * // // // i I / / _L 1 1 1 I 1 «5 •* 5 ■a <5 ^1 % 1 5 : 1 ^ 1 * <* y ^ ^ 1 •* ,\ - M " 7 V ■5 A 5 \\ \\ \\ \\ 1 1 3 § ^ £?5 5 ^ (Hjm)noii03iJ3a (H3Nl)N0U331J3a /1 / 1 ? 1 / 1 / 1 / t 1 1 1 p T 1 / 1 1 1 1 1 1 p ' T / kl 5 1 1 I / / / / 1 kj 1 / 1 / 1 1 / \l [j <* Ml .5 1 A \ \ f 1 1 1 1 1 \ \ \ V \ O \ \ \ 9 \ \ \ \ <=> kj vo £ * vj C] X % *-> ^ ._ ^ s * ? /r //: if ' f 1 /6 1 / ' ii j, 1 1 1 1 1 ,1 1 ii * i U| t. 1 5 a. 1 *J * \ 9 ««; v v -^ * \ . x V V A 9 ^ ^ A % x * A\V 5 3 ; 3 5 3 I 1 « ?a kj *! kj ■-) li kj ^1 kj (HONl) NO II V 31330 (H3Nl)fJ0llD3133a /./ 1.0 0.9 0.8 0.1 ^ 0.6 5 0.5 0.4 0.3 02 0.1 / • — — — — o — • LEGEND: EXACT METHOD ALL PLIES SAME THICKNESS o-3X • -3Y A-5X A -5Y / 2 3 4 5 6 7 /3/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE FIG. 10 FACTOR T [FORMULA (3.40)] AS f\ FUNCTION OF 0/a, WHERE j3 =6 (E,/E z )*. UNIFORM LOAD. EDGES SIMPLY SUPPORTED. /./ 1.0 0.3 0.8 0.1 ^ 0.6 ^ 0.5 0.4 0.3 02 0.1 -e ^< ■ ^>^i^ ^^e- d d a o D A | • o ■e- o S- - U — \J • ► LEGEND \ Ai o-3X* • -3Y *- 5X A-5Y □ - 7X -©■-jx] .a. ^ y ^PROXIMATE METHOD > ALL PLIES SAME THICKNESS \ [FACE PLIES ONE -HALF AS &-7X; 1 HICK AS 1 UMAININ G PLIES J I Z 3 4 5 6 7 /3/a,A FUNCTION OF THE DIMENSIONS OF THE PLATE 8 FIG. It FACTOR r [FORMULA (3.40)] AS A FUNCTION OF 0/a, WHERE /S=b(E,/E z f. UNI FORM LOAD. EDGES SIMPLY SUPPORTED Z il 3^966 ? COMPRESSED AIR \ RUBBER BAG 3 PLANK r~4xZl CHANNEL > SUPPORT FOR DIAL GAGES FOR CLAMPED EDGES, STEEL RODS ARE REMOVED AND THE PLYWOOD IS CLAMPED AS SHOWN ABOVE 6x3 CHANNELS- -g BOLTS SPACED 4 CENTER TO CENTER FIG. IZ CROSS-SECTIONAL VIEW OF APPARATUS FOR UNIFORMLY LOADING A PLYWOOD PLATE, EDGES SIMPLY SUPPORTED Z k 39695 r FIG. 13 DEFLECTION ALONG CENTER LINE OF PANEL N 0.4 -TYPE 3X EDGES CLAMPED 0.100 k FIG. 14 DEFLECTION ALONG CENTEP LINE OF PANEL N0.6 -TYPE SX EDGES CLAMPED F/G /S DEFLECT/ON 4LONG CENTEX i/A/£ 0^ o./ * X 0.2 > OJ ■> 1 II II »« X ii ii •• ^\ ^3^ ^i } a N >--l fc) N & n i • X • X • " I Si 5C 5* V) 5$ 5 I-. K. Y w/(Pa*/D,) QOS0.04 0.03 0.0N =^ X 0= X ^ . ( - "O K ^ r- ^ <=> ^ ~J Q. F- 2?=X X = X 3" i + ii i 3 kj ^; ^ l^r ^-j ki Q ^C ^ k ^> Ci C^ K ^ < ^ wj £ 2 x <0 <0 Uj ^ K F^ ^ u. ^ ^s v: c>