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Front Cover 1 Front Cover 2 Table of Contents Page 1 Page 2 Introduction Page 3 Page 4 Page 5 Plates with small deflections Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Plates with large deflections Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Appendix 1. Stress-strain relations in orthotropic material Page 33 Page 34 Appendix 2. The differential equation for the deflection of a plywood plate. Small deflections Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Appendix 3. Rectangular plate under uniform load. Edges simply supported. Small deflections Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Appendix 4. Rectangular plate under uniform load. Edges clamped. Small deflections Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Appendix 5. Infinite strip. (Long narrow rectangular plate). Load concentrated at a point. Edges simply supported Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Appendix 6. Infinite strip. (Long narrow rectangular plate). Uniform load applied over a small area. Edges simply supported Page 62 Page 63 Appendix 7. Rectangular plate. Load concentrated at a point or applied over a small area. Edges simply supported Page 64 Page 65 Appendix 8. Differential equations for the deflection of a plywood plate. (Large deflections) Page 66 Page 67 Page 68 Page 69 Appendix 9. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges simply supported Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Appendix 10. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges clamped Page 82 Page 83 Page 84 Page 85 Page 86 Appendix 11. Rectangular plate. Uniformly distributed load. Edges simply supported. Approximate method Page 87 Page 88 Page 89 Page 90 Page 91 Appendix 12. Rectangular plate. Uniformly distributed load. Edges clamped. Approximate method Page 92 Page 93 Appendix 13. General notation Page 94 Page 95 Page 96 Page 97 Page 98 Tables 1 to 52 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Figures 1 to 31 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 |
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F1I MAUlS (Af I)L-YW()41)1
UJNIAP IJNrIk)A?4 u4 CONCIENTIPATIH) 1IAU~S March 19P42 INFORMATION' REVIEWED AND REAFFIRMED March 1956 II U- rOTSERIct ~~TATOFA TIS 1PUPT~ IS CNIE if A S1E1115 ISSUED I TC AIDy 1111 NATION'S WAIP IMCC4IPAM No. 1312 UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY Madison 5,Wisconsin In Cooperation with the University 91 Wisconsin ~y) 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 H ead Mathemat idian Table of Contents Introduction .......... ........................................... 3 Assumptions made regarding properties and structure of wood. 4 Section 1, The elastic behavior of wood .................... 5 Plates with small deflections .............. ;...................... of Section 2. The differential equation for the deflection of a plywood plate ............................ 6 Section 3. Rectangular plate under uniformly distributed load. Edges simply su-o-orted ................. 9 Form of the deflected surface ............................... 11 Central deflection of a plate ex-oressed as a. fraction of that of the corresponding infinite strip. Determination of Wo/h from a curve .. .............................. I Tests ....................................................... 14 Section 4. Rectangular plate under uniformly distributed l o a d E d g e s c l a m p e d . . . . . . .. 1 5 Form of the 'deflected surface ............................... 16 Central deflection of a plate expressed as a fraction of that of the corresponding infinite strip .................. 17 Tests .. . . .l.. . . .. . . . . . . Section 5. Infinite strip. (Long narrow rectangular nolate.) Load concentrated at a point. Edg-es simply su-oported ..................................... 19 Section 6. Infinite strip. Uniform load applied over a small area. Edges simply supported ........... 19 Section 7. Rectangular plate. Load concentrated at a point or applied over a small area. Edges simply supported .............................. 20 Central deflection of a plate expressed as a fraction of that of the correspoonding infinite strip .................. 21 Plates with large deflections ..................................... 22 Section 9. Differential-equations for the deflection of a plywood plate ............................... 22 This mimeograph is one of a series of progress reports issued by the Forest Products Laboratory to aid the Nation' s defense effort Results here reported are preliminary and may be revised as additional Data become available. P,-)FC Mimeo. No. 1312 Pa-e Section 9. Infinite strip (Long narrow plate). Uniformly distributed load. Edges simply supported ..... Section 10. Infinite strip. (Long narrow plate). Uni- formly distributed load. Edges clamped ....... Section 11. Rectangular plate. Uniformly distributed load. Edges simply supported. Approximate method... Section 12. Rectangular plate. Uniformly distributed load. Edges clamped. Approximate method ............. dix 1. Stress-strain relations in ortbotropic material..... dix 2. The differential equation for the deflection of a plywood plate. Small deflections ................. dix 3. Rectangular plate under uniform load. Edges simply supported. Small deflections ..................... dix 4. Rectangular plate under uniform load. Edges clamped. Small deflections ....................... dix 5. Infinite strip. (Long narrow rectangular plate). Load concentrated at a point. Edges simply sup-or ted ......................................... dix 6. Infinite strit. (Long narrow rectangular plate). Uniform load applied over a small area. Edges simply sunoorted .................................. dix 7. Rectangular plate. Load concentrated at P voint or anelied over a small area. Edges simply supported ......................................... dix S. Differential equations for the deflection of a 'plywood plate. (Large deflections) ............... dix 9. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges simply supported ......... dix 10. Infinite strip. (Long narrow plate). Uniformly distributed load. Edges clamped .................. dix 11. Rectangular plate. Uniformly distributed load. Edges simply supported. Approximate method ....... dix 12. Rectangular plate. Uniformly distributed load. Edges clamped. Approximate method ............... dix 13. General notation ................................... Arren Apren ATren Apoen Apren Appen Apren ATren Arren Arren Arren Apr,en Atren 92 94 99 1 _7 F Lares 1 to 1 .................................................. -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 ply-wood 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 suchloads that the deflections are large. For plates with small deflections, direct stresses Otmeribrane 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 1"Tith 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 Young1s moduli in bending (Section 2), after having read Section 1 on the elastic behavior of wood. The curves of figures 10, 19, and, 26 can then be used to determine the maximum deflections of plates Tivith 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 notions 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 -3- 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 compregnated 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 and large deflections. The form of the differential equation for the deflection f an orthotropic plate, in the case of small deflections, is well known.- For plywood plates, which are made up of layers of orthotropic material, the co-* efficients in the differential equation are given in terms of the elastic constants of the constituent wood in the author's Daper,- "Bending of a Centrally Loaded Rectangular Strip of Plpvood." The derivation of these coefficients under certain simplifying assumptions is given in the present report. Although actual plywood will seldom possessthe structure assumed as ideal, nevertheless the procedure used in arriving at the coefficients brings to light the essential factors determining the stiffness and other elastic properties of plywood. Then, in a given situation a rational allo.-nce for the effects of variation from the ideal structure can be made. The principal results are given in the body of the report while 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. Gehring, F., Dissertation. Berlin, 1860; Voigt, W., Theor. Phys. I, p. 451, 1895; Huber, Y. T., Bauingenieur 4, pp. 354 and 392, 1923, 6, pp. 7 and 46, 1925, Zeits. far ang. .1ath. u. 1,lech. 5, 22?, 1926; Igchi, S., Eine Lfsung fGr dieBerechnung der bie.samen rechteckigen Platten. Berlin, Springer, 1933; Seydel, E., Zeitsch, f. Flugtechnik u. !Hotorluftschiffahrt, 24, No. 3, 1933. 2ysics 7, 32-4, 1936 -T1,ysics 7, 32-41, 1936. 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 Youngts moduli, two Poisson' s ratios, and one modulus ofrigidity. The elastic constants of spruce have been determined er-refr,11y;. 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 ex-pressed in a form involving a factor containing Young' s modulus along the grpin. 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. Ex-perience 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 7 articular 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 niave three mutually'perpendicular planes of elastic symmetry, namely, the planes perpendicular to the longitudinal, radial, and tangential directions, respectively, as shown in figure 3. A substance having such properties of elastic symmetry is said to be orthotropic. If wood is orthotropic it will have (see appendix 1) three Youngls moduli, E, ERI and Tthe letters L, R, and T denoting the longitudinal, radial, and tangential directions, respectively; three shearing moduli VLT, 11LR, and LRT ; and six Poisson' s ratios, CTLT' IGT L, 'IRT) (TR, OLR and CFAL where, for example, CIT is the Poisson's ratio associated with tension parallel to the direction L and contraction parallel to the direction T Among these 12 constants there are three . .elations of the type (see (1.7) in appendix 1). E yTJ=ET YLT Table 1, giving the valuesof these constants for several secies Of wood as determined on the assumption that wood is orthotrooic, is tnken from a report by C. F. Jenkin "Report on Materials of Construction Used in 'Aircraft," Aeronautics Research Committee (London, 1920) .3 See also H. Carrington, Phil. 1.,ag. 41, 206, s4s, 1921; 43, 871, 1922; 44, 299, 1922; 45, 105, 192 . 1312-5 The values given below of the elastic moduli of pouls-fir at 10 percent moisture content, were obtained from a limited number of tests. Because of the small number of tests, these values are to be consider as tentative. Lb./ n ,L= 1,960,000 D 113,200 R = 155,900 11LT = 123,800 LLR 110,600 CPT = 7,100 The Poisson s ratios CYL!2 and a TL have not been determined. In the cJI culltions made later in this report they are taken to be the spm, ai for srruce. It is probable that the error thus introduced is not lars e. The assumption that wood may be treated as an orthotrocic material is reasonably well confirmed! by the experimental evidence at ,resent av liable. 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 eflection of q 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 nly~ood: The material of the individual plies is accurately flat--rain, that is, the directions of the grain and of the annual rings Pre rprallel to tn7 faces of the plies. The directions of the grain in adjacent nlies - ro errendicular to each other and parallel or perpendicular to the r:-s.cive edges of the plate. The analysis applies equally well to edge train rlyood. It is only necessary to substitute for T tthrouhoa t in he subscripts of the elastic constants. Each ply is homogeneous. This implies tnat the vpritions of tLs elastic constants from springood to suimmrrood are disreprdeprJ an vrao vluGs of tne constants are used. A. T. Price, Phil. Trans. A. 22 1, 1929. 1. H3rio, Zeits. f. Tecbr. Pysik. 12, 353, 19k11 ~ 71-0- The plate is symmetrical, both geometrically and as to arrange- ment and properties of the material, 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 = 0 approximately correct results should be obtained by using in place of the flexural rigidities D1 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 equation D1 ~g- + 2K + Do= p, (2.12) which is derived in appendix 2.6 The flexural rigidities DI and D2 are proportional to two "mean moduli6 in bending" El and E2 as explained in appendix 2 where the plate isassumed 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 El and Eg are determined frou the flexural rigidities of strips of unit width parallel to the X and Y axest respectively. Thus, for -a strip of unit width parallel to the X-axis, E1 is defined by the equation Ell = (Ex)i Ii 2An equation of this form for orthotropic material is well known. See, for example, the references to Huber on page 4. 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. 228j 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 x )i is the Young's modulus of the Ith ply measured parallel to the, X-axis; I1 is the moment' of iinertia with respect to the neutral axis, of the cross section of the Ih ply made by a plane perpendicular to the X-axis; and I = h3/12 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 Eli by taking the sum of the products (Ex)i Ii 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 E2I is calculated in a similar way. These definitions for.E1 and E 2 may also be applied in dealing with plates of symmetrical construction. In the case of such plates they are identical with the definitions of equations (2.1g) and (2.19) of .. Ap-pendix 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 lies were neglected in deriving equation (2.12). This situation is dis- cussed, on pages 14, 15, 50, and 51 of the paper by Price, 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 El and E2. 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 Dourla~s-fir as previously given the values for E,1 and E2 in tables 2 and 3 were found, the grain of the face plies being parallel Cto the X -axis. The differential equation (2.12). can be reduced to the simpler f orm .... 4 T IT + 2 4 W +V T P( 2 2 5 ) by m where :(D,/D 2) /4 = (El /E,)1/4 (.P 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 t 'at 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 th6 -plate. 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 (25). 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 Supported7' - In this case the edges of the plate x 0, x L= Y=0,an 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 mx vanish and that on the edges y = 0 and y =b the deflection w, and the bending moment m y 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. B. Zilmer who, as a graduate student at the University of Wisconsin on a fellowship sup-ported by the Forest Products Laboratory, undertook the solution of this problem and that of the clamped plywood plate (see section 4) at my suggestion. 'The calculations based on this exact solution were performedby 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 apendix 3. Solutions of the problem for the simply supported plate of ortho-I tropic material have also been given by Huber and by Iguchi in tie papers to which reference was made in footnote 1. - The reader who wishes a quick and easily applied method of finding the approximate deflection at the center of a plate should turn t once to page 13. . 9- 131-2 Since the differential equation is written with the variable 7 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 = O, x = a, 7 = 0, and = where S= eb (3.1) and to express the deflections and moments in terms of the variables x and q instead of in terms of the variables x and y. The deflection is found to be given by the equation (see appendix 3). 8 EL P (1 Yn) sin knx (3.15) h-a5 n5 n= 1, 3, 5 :;ere P = pa/ = 1 7 LTTL (3.16) An = n/a (3-7) anOd Yn in defined by (3.12) appendix 3. The bending moments mx and my and the twisting moment .., can be calculated from (3.15) with the aid of (2.29) of appendix 2. The ratio of the deflection wo at the center, to the thicknes; is readily calculated from (3.15) and found to be given by equation (0.17, 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 P minirmm as compared with other configurations satisfying the sasme 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 wo at the center as an undetermined para- Teter. The potential energy of deformation of the plate in bending Vb and the potential energy of the load V, are then calculated as a function of wo. The total potential energy V of the system is the sum of Vb and V1. On equating to zero the derivative of V with respect to wo an equation is obtained connecting wo 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 -10- andl determining all these parameters in such a way that the total potential energy is a minimum. In the following analysis a second parameter 7(or c ) whase significance will be explained is introduced in addition to the para- meter w., the defleotin at the center. Let w w stia when e < y< b-e W = W s i-Exsin_ S wo a 2c When e < y plate can be calculated as twice the potential energy of the half of the plate for which y which is onnected with c by the equation c = 'a. Thus the parameters T 2 and w ae to be determined in such a way that the total potential energy If the system is a minimum. The form of the assumed middle surface along the line x = a/2 is shown in figure 5, In the form assumed for the deflected middle surface, the curv- attire is dis intinuous along the lines y = c and y -= b-c. This dis- continuity could be removed by a slignt change in the form of the deformed middle surface in the vicinity of these lines with a consequent small change in the total potential energy. After making the necessary calculations (see apnendix 3) the follwing formula is found connecting wo /h and p. w,/h' =P (3.37) where P pa4/ELh4 (3.38) where a is given by (3.36) in appendix 3. Tables 4 9 were calculated using the values of the elastic constants of spruce. Values of the factor a as computed from.the exact and approximate formulas are also given in these tables. It appears that the approximate formulas are reasonably accurate. T he factor a is obtained from the exact formula (3.15) as (wo/h)/P. Form of the Deflected Surface Examination of the exact values mf the factor for the plyvcod plates discloses an interesting and unexpected phenomenon, The exact value 1312 -11- of cL and hence of the central deflection uwO for a given load increases to a maximum value as k = b/a increases to a particular value and then decreases asymnptotically to a limiting value as k is further increased. If the calculations for plate 3Y had been carried out for larger values of the ratio k = b/a the phenomenon would have been found in this case also, as rough calculations show. The possibility of such a behavior is revealed by a careful examination of formula (3.15) for the deflection. The term Yn, given in equation (3.12) of appendix 3, contains trigonometric functions of the variable 9 This implies the possibility of a wave form along any line X=constant and in particular along the central line X = at 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 Yn of the formula to which (0.15) 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. Zilmer9- in_ vestigated carefully the behavior of uniformly loaded plates of the typ-e 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 rave form indicated by the first term of this equation could not possibly be obliterated by subsequent term. He found that the central deflection considered as a function of k = attained a' a maximum value at k =1.49 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 large k.. In a number of tests with plates of commercial plywood the surfaces were observed to take wave forms. The material of the plates was not sufficiently uniform to warrant comparison of the observed wave 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 ae shown the deflection along the central line x = a/P of a nimnber of plates of commercial plywood under uniformily. 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- sisto be presented at the University of Wisconsin. 1312 -12- The curves for plates Nos. 4 (type 3X), 6 (type 5X), and g (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 5Y). 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 woih from a Curve. The deflection wrO at the center of a rectangular plate, sir ply supported 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 b-r a corrective factor to take account of the effect of the ends of the finit. plate. The deflection at the center of a uniformly loaded infinite strip simply supported at its edges is given by the formula 5 (1 Cr LTCTL)pa4 2LI WO4-= 32 Elh O1547Elh3 (3.39) 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 bee,-n taken to be 0.99, the value which it has for spruce. If the deflectionl1 t the center of a finite rectan,-,l,)r plate is denoted by WO0 we can write - We TW C*oo (3.90) The factor ry is found to depend almost entirely upon the value ot P/a = (b/a )(El1/E2l/, 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 O/a.. The curve of figure 10, representing "Y as a function of P/aq is a smooth average curve for points determined from the exact values of Cc in tables 4 -9. These points are shown in the fip:7,ra. Nere -points the-:n 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 Norris, C. B., Hardwood Record, !Jay 1937. Because of the approximations involved, the deflections calculated from his results are too small, a fact hich he recognized would be the case. 1312 -5 axes to which the plates in the tables were referred and utilizing the data of the tables to calculate the factor y 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 anorox- imate formulas. This was done for plates 3X, 3Y, 5X, 5Y, and 7X with all plies of the same thickness and also for plates 3X, 5X, and 7X with the face plies 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 plyood 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 E = (E1/E2)1A which is used to obtain B = Eb of the transformed plate. Hence, to use the curve of fi-ure 10 for a given plate it is only necessary to know the t-'o ano-nrent Young, s moduli in bending, E1 and E2 of the plate. They can be determined from static bending tests or estimated from the structure of the plywood in the late. The factor y corresponding to p/a = b/a(El/E2)1/4 can then be read from the curve. The central deflection iwO is then given by ,'o = oWo sa (3.40) ..here iTmo.nS to be calculated by equation (3.39). The curve of figure 10 can be constructed from the values in table 10. 11 Testsl - In table 11 are shown the results of n. number of tests made "ith uniformly loaded plates of commercial plywood. A description of the method of making the tests will be given below. The factor yobs was calculated as follows. The tests on a given plate yielded a mean value for the ratio p/rO There p is the load per unit area and wo is the deflection at the canter. The moduli E1 and E2 were determined by static bending tests on strips cut from the plates after the tests on the plates themselves were completed. From the formula 5 (1 -LT TL pa4 o0= 32 Elh3 1 -The tests described here and in sections 4 and 7 ere carried out under the direct supervision of Alan D. Freas, Assistant Engineer. we obtain w 3 2Elh3 wo obs woom- 5T 1- TO TL ) a P The values of y this found were compared with those of 7t!Por 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 of7 In all instances the ratio of Tobs to 7theor 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 1-,yood in a given plate and partly to varying degrees of constraint at the edges of the plate. It appears that the curve for y = wo/wo may be used in predicting the deflection at the center of a plate if reasonable allowTance 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 4 feet wide. A cross section of the apparatus is shown in figure 12. 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 bags, approximately 4 feet square and 6 inches deep, with compressed air. Heavy planks bolted to the channels as shovn 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 nlaced at various positions on the plates. The air pressure was measured ,.ith P ,Tater manometer. Tests were first made on a 12 by 4-foot plate. Then 4 feet were sawed off and the resulting 8 by 4-foot plate tested. Finally this plate was sawed in two and tests were run on a 4 by 4-foot plate. As the 12 by 4-foot plates 7ere made by joining up shorter lengths of late 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 4. 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 boundary conditions will be found in appendix 4. solution is due to D. E. Zilmer. (See footnote 7, P. 9.) The calcu- lations based on this .exact solution were performod at the Forest Products Laboratory under the direction of the present author vho is responsible for the discussion to be found in section 4 and in anoendix 4. He is also responsible for the approximate methods. The claLmmed orthotropic plate was also treated by S. Iguchi by a somewhat different method in the paper to which reference was made in footnote 1. See nlso footnote 28. 1312 -15- Approximate formula for the deflection at the center.1--As in th:. c-sF of the plate rith simply supported edges it is possible from n con- si eration of the potential energy of the system to find a simple aprroxi- mate formula for the deflection at the center of a uniformly loaded Plato 7ith 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 = lb-c where c = T a/2, T being a parameter to be. determined. The assumed form. of the deflection of the middle surface is given by the equations J = sin 2 x 0 a rhen c y < b-c (4. = sin ~ sin2 ,;hcn o < (c An expression corresponding to the latter is assumed for the region b-c following formula is found connecting wO/I and p. W /h = cXP (4. 33) v-!ere 4.4 P = pa /Eh (4. 34) and aO is given by formula (4.32) of appendix 4. Values of the factor a as computed from the exact and approximate formulas are given in tables 12-20. The exact factor a is obtained from the exact formula (4.11) together '7ith (4.4) and (4.6) qs ( ,To/1)/ )/P It ap-pears that the approximitoe formulas arc repsonpbly accurate. The tables -ere cnlculktcd using the elastic constants of spruce. The number k denotes the rntio b/: of the sides. Form of the Deflected Surface Ex-aminntion of the exact vw-lues of thu frctor r, for .ltus of tre 3X and 5X sho-rs thpt here, just !s in thoe case of plywood plrnates 7,ith simply supported edges, the deflection for 7 given lond. increases to a mnxi:,.-m value as k = b/at increases to a particular v-lue and then do- Creses to an~ symptotic value as k is further increased. This effect 13 'An rrproximnate method, for finding thc- deflection at the: center, -hich requires no kno,:-ledre of the mathemnticnl analysis, rill be found on rae 17. -16-~ is somewhat more pronounced than in the clFse of pla tes -:ith simply supported edges. As before, this effect is to be associated ith the fact that the deflected surface assumes a wave form. The possibility of the existence of such a wave form is shown by the presence of,.trigonometric functions in the expression for the deflection. .. In figures 13, 14, and 15 are shown the deflection along the central line X = q,/2 of a number of plates of commercial -nly-rood. 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 chfang,,e 1in shape of the deflected surface associated with the presence of a wave form than those for plates with simply supported edges. In fact, the effect is so pronounced in the case of plate 4 that one hesitates to accept it as real. That this effect, which is so pronounced in the case of the 4 by g-foot plate, is actually present in this plate at all states of loading Is shown in figures 16 and 17, which give the shape of the pnlqte at successive intervals of loading. In order to comTpare r,.ivthe sha-pes of the curves the central deflections have been reduced to 0.100. This means that for small deflections experimental errors have been multi-olied 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 paqrticu- lar plate.. It may be remarked that for the larger deflections the i-olative heights of the maxima are reduced. This may presumably be attributed to the affect of membrane stresses. Central[ Deflection of a Plate 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 plati with clamlped edges as a fraction 'y of the deflection at the center of an infinite strip with clamped edges. Thus VT ry~r(4.35) 0 -- cTo Th fator yisa nfudto deedamostotirl h ratio P/ct = (b/a) (EI/2 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 y 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 18. 17ore points than those obtained directly from the tables vWere secured by interchanging the aLxes to which the plates in the tables vrere referred. and utilizing the data of the tables to calculate the factor yf for other values of the ratio P/a. -17- F r an infinite strip clamped at the edges, the deflection at a point nn the central line is given by the formula 1 -LTTL pa4 - 3 3 (4.36) Elh E In which the factor 1 -Tr-MT may be taken to be 0.99. Except for this factor formula (4.36) isl ht for a runiformly loaded beam of unit width with fixed ends. The curve of figure 18 can be constructed from the values in tale 21. In figure 19 the curve is that of figure 18 and the points are those Cmuted from the values of t given by the approximate formula, the -7 -s of w X be.ng those given by the exact formula. Except for the tfa-t tht the ap roximate values of y do not show a maximum in the vicinity 1f 3/a = 2 the agreement is satisfactory. That the approximate values of 1 i not shrw a maximum in the vicinity of f/a = 2 is to be attributed Sthe incomplete representation of the deflected surface by the forms ssumed in (4.22). Since the exact analysis clearly points to the existence f 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 irn addition to that for the three-ply and five-ply plates for which it was constructed. Tests In table 21 are shown the results of a number of tests of uni- formly loaded 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 of commercial plywood were used in these tests as in the tests of plates s with simply supported edges. The factor Yobs was computed from the mean of the values of the Istit. p/w, for a given plate by the formula 3 w 32E1 wh o 1 oc Yobs -wo (1i -LT TL)a4 p T crrresp-nding factor ytheor was taken from the curve of figure 18 fr t e aR; ropr)e value -f ;/a. As was to have been expected, owing Sthe i:Dessibi .~tr cf securing perfect clamping at the edges, the 31 ; ] observed factors y and consequently the observed central deflections are greater than those predicted by the curve, which is based on the exact analysis of ideal cases. On the average they are about 40 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 oDenings, all in the same plane. Ideal clamping will be found on the edges of interior rectangles of such a network. Otherwise elastic yielding reduces the clamping effect to a greater or less extent, depending upon the particular situation it 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 P applied at the point x = u, y = o on the X-axis as in figure 30. As in the case of the isotropic strip14 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 analied load is unchanged, we obtain the solution for the limiting case of a ooint 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 mx and m 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 ty pe 3X having a concentrated load at a point on its central line are given in table 23 and shown in the durves of figure 20. Section 6. Infinite Strip (Long Narrow,, Plate.) Uniform Load Alied Over a Small Area. Edges Simply Supported A uniform load acts over a small rectangular portion, as show.,n 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- tangle, considered as a loaded line segment, formulas are obtained in 1 4 a a ~Nadai, A., Elastische Platten, pp. 78-82 and 85-95. -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 24 and the curves of figure 22 give the deflections and moments at certain points due to a load distributed uniformly over a small square hose center is on the center line of a three-ply plate of spruce -oly,ood and whose sides are equal to one-tenth of the width of the plate. If tqble 24 and figu-re 22 are compared with table 23 and figure 20, the deflections in thoencse of a load applied over a, small square area, are seen to boe practically idlontical with those due to a similarly situated point load equal to the total loaJ aplied over the square. The bending moments are also -orpctically 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 i. rectangular plate with simply supported edges cain be found by calculating the effects of a suitable distribution of positive and negative loads on an infinite plate, using the results of section 5 or section 6. Tt is only necessary to distribute the loads in such a ,ay that the deflections and bending moments vanish on the edges, y = 0 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 lenas are numbered I, II, III, IV, V, etc., as shown in the fizz are, the deflection 'TP at any point in the plate rill be given by combining the deflections due to the separate loads, that is, 'TT=TTI +I I +-Till+ In like manner expressions for the bending moments can be obtained. Calculations were 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 17ere tnken to be one-tenthi of those of the plate. The results are shown in tables 29 and. 26 and in the curves of figures 24 and 25 The choice of axes is to be noted. ns slightly different from the customary choice for a finite plat n hsreot The calculations for the case of a uniformly loadedI small central square area have not been carefully checked. However, the behevior indicated by the results is in good agreement -rith 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 rlate- are nearly the same as for the infinite plate ,hile 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 Frction of That of the Corresponding Infinite Strin The maximum deflection of a rectangular plate under .a 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 eflection due to a point load on the central ;line is given by (see (7.4) aoondix 7) W 1.051 P 6kE (7.4) 0- h3 3 For a finite rectangular plate under a given central lond concentrated at a point, the central deflection W o can be expressed as F fraction 7 of the central deflection of a similarly loaded infinite strip. Thus "Po = co (7.5) For the purpose of determining this factor y, it is advant-geious to replace the method explained earlier in this section by one used by Timoshenko5 for isotropic plates. In this method the load is limited to a position on the central line. In appendix 7 the analysis is cnrried out for a plywood plate. The calculation of the factor y in (7.5) can than be readily performed. (See appendix 7.) It is found that y can be represented with little error by a curve as a function of the r-tio Z-11 1/4 =1 B/a = (b/a)( =) 2 In table 27 are given the values of y for plates of several types of plywood. In figure 26 a smooth average curve for y is drRlawn from the points given in table 27. Further calculations indicate that this curve is satisfactory for other types of nlywood than those listed in table 28. This curve should be used only for plates of tnc tynrs con- sidered in this report. In particular, the directions of the grcin of th_ wood in adjacent plies are mutually perpendicular and the constant x lies between 0.2 and 0.5 or not far outside of this interval. The results of tests on a number of plywood plates 7ith concentrated central loqds are shown in table 28. The same plates of commercial ply ood m1ore used as in the tests described in sections 3 and 4. The plates eure tested in a horizontal position. The edges were simply supported, resting on half- inch rods as shown in figure 12. The loaded arsP "as in all cases a square 4 by 4-inches. Only one side of each plate reas loaded. The numbers in the column headed Yobs were calculated from the observed ratios re/P by the following formula wThich is obtained from (7.4): '3 3 w3 Eh w Wo = 1 o P oc- (1.051)6ksa2 P 15 Timoshenko, S. Bauingenieur, 3, 51, 1922. 1312 The numbers of the column headed ytheor -ere taken from the curve of figure 26. The agreement between the observed and theoreticnl values of y rap7ets to be satisfactory. The average value 0.921 of the ratio y ob theor indicates some restraint at the edges but not so much as in the case o plates with uniformly distributed loads. This appears to be reasonable. Plates With Large Deflections SectioD 8. 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 proportionplly 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 nronortional limit, it is found that the deflection increases less rapidly than would 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 thickness 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 acompanying the deformation of the plato 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 descriptive designation for them. The load is thus carried partly by the stiffness of the plate and. 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 unon the solution of two simultaneous partial differential equations of the 16 fourth order.- The maximum deflection is taken to be small in comparison with the length and breadth of the plates. It is easy to modify the steps taken in the derivation of these Iequations for an isotropic platell to obtain the correstondinr equations for the plywood plate. This is done in appendix 8. The equations obtained are (8.1l) and (8.14). We shall have occasion to use them in determining the deflections and stresses of uniformly loaded infinite strips 16 von Karman, Th., Enc. d. Math. Wiss. IV4, 349, 1910. 17 Seo Nadai, A., Elastische Platten, rn. 284-297. -22- 1312 (practically, long narrow plates). The solution of those equat"o~ for finite rectangular plates has not been found. Considerations of e-nrgy 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 application. Even for isotropic i lates solutions have not been found, for the equations corresponding to (9.11) and (g.14) except in the cpses of in- finitely long strips and circular plates.l- Section 9. Infinite Strip (Long Narrow Plate)." Large Deflections. Uniformly Distributed Load. Edges Simply Supported We shall now make use of the differential equations (g.11) and (8.14) to find the deflections and stresses of a uniformly loaded infinite strip with simply supported edges when the loads axe such that the de- flections are large and direct stresses ate 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 wrill 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.(9.ll) snd (g,.14) were given for infinitely long plates of isotropic material with either simply supported or clamped edges by Stewart Way in a, lithographed -creprint of a -aper presented to the American Society of Mechanical Engineers in 1932.,. It has apparently not been published in any other form.l-9- From j a reference in WayIs -paper it would appear that essentially the same solutions were given by I. Boobnoff in a book published in 1914 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 = 0 and x = a. Under the assu 'med uniform loading, the deflection and the strain a hd stress components are independent of y .It can then be shown (see apendix 9) that the mean direct stress component Xx1 (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 correspoonds to that which would be -See references to I. Boobnoff and S. Way in section 9. ig -The substance of this paper is found on pp. 4-17 of Timoshenko's ..oTheory of Plates and Shells, 194o. -23- 1312 . furnished by (8.11). The equation (8.14) becomes dw 27' D = p + ghX (9.5) ldx dx2 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 (9.11) and (8.14) in the general case. The procedure to be followed in utilizing the solution of (9*5) subject to the conditions on the simply sup-orted edges x = o 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 are the following: (a) Relation between load and deflection P A'O + B(_O)3 (9.29) h h where oa4 P E h4 EL A z_ =R_ (9.29) k L p 20.6 a 7T (9.30) (b) :aximum bending stress in a face ply q 7: nf "X (h- (9.31) -,24- where a may be taken to be 4.4. A method of obtaining a more accurate value of ca from a curve is explained below. The latter method is easy to apply and is to be preferred. (c) Mean direct stress E= (h 2 w 2 In these formulas El denotes the mean modulus in bending, -a the mean modulus in stretching, and EX the actual modulus in a face pll, all measured parallel to the X-axis. The mean modulus in stretching 'a is merely the arithmetic mean of the Els in the various plies measured in a direction parallel to the X-axis. Thus for three-ply plywood, n~avino, all plies of the same thickness and the grain of the fac, lies parallel to the X-axis 2nL + T a 3 In like manner b enotes the mean modulus in stretching in the direction parallel to the Y-axis. The calculated values of the ratios Ea/L and lb/EL for various types of spruce and Douglas-fir plywood, using the values of EL and ET given in table 1 and on page 6 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, ER L and ab/ L 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 nrob,bly be used without a-ppreciable error. It will be noted that the first term of (9.29) expresses the result obtained from the usual theory of thin plates when the deflections are assumed to be small. Of the three formulas (9.28), (9.31), and (9.25), the second, namely, (9.31), is the least accurate if a is taken to be 4.4. Actua-;lly cc ranges from 4.8 for small deflections to 4.0 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 9 which is connected with the ratio w o/h by the formula (see (9.24) appendix 9) = 2.778 ( -25- When q 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 accomoanying discussion. The curve of figure 27 is plotted from the following data. 0 1 2 3 4 5 6 7 8 9 10 a 4.80 4.75 4.64 4.50 4.3g 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 (9.18), appendix 9, instead of from (9.25). The value of p needed in (9.18) has been found in the calculation of s . From the mean direct stress and the maximum bending stress in a face fly as given 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 aptndix 9. Useful formulas applicable to long narrow plates of isotronic material can be obtained from formulas (9.28 (9.31), and (9.25) by setting E = E = Ex = E and X = 1 c where F 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. Infinite Strip (Long Narrow Plate). Uniformly Distributed Load. lidges Clamped. Large Defleo .<" Exercly 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 the maximum bendin stress can be calculated. These formn las are derived in appendix 1' nhre the tables necessary for their utilization are Civen. However, it is oiT sicle to replace the exact formulas by -r07roximate formulas which are much sinrler to use and are sufficiently accurate. The derivation of the aprroximate formulas is given in annen ix 10. In this section as in section 9 the edges of the plate are restrained from moving inwrd. The arrroximate formula connecting the load and the deflection is 12-'r 3 O + B(1.13) 1312 -26- where x=32 E1 (lo-14) X EL B -.3 E a (1015) and SpPa4 For the definitions of El, E, ) see section 9 or the table of notations. For the maximum bending stress in a face ply we have the approximate formula S C 'To(+_D (10. 2(-,- where 16F, (h ) 2(10.21) S a .98Ex Ea (h(022 The symbol EX denotes the Young' s modulus in a face -ply and in tne direction parallel to the X-axis. The range of values of TTo0/h within which this formula may be used is discussed in appendix 10. A better approximation to s is obtained from the formula (h) 'Fo-(10. 16) and the curve of firare 28 where the argument is connected ith 0-- 'by the equation h 0- O366 ) 7] 1.0 h La -27- The curve of figure 28 can be plotted from the following data,: p 0 1 2 3 4 5 6 7 9 9 10 a 16.00 16.52 18.03 20.31 23.20 26.44 29.92 33.58 37.30 41.15 45.00 If this method is used to find s it may be more convenient to calculate the direct stress E from (10.7) of apnendix 10 instead of us-ing (10.11) below. The approximate formula for the mean direct stress is =a (h) (FR)( From the direct stress and the maximum bending stress in n face tly as given by (10.11) and (10.20) the corresponding stress in Iny given ply can be calculated by the method explained in )ppendix 9. See equations (9.33) to (9.36). Tables 4S to 52 in appendix 10 show comparisors of the results of calculations made with the exact and the approximate formulas. In the tables the maximum bending stress was calculated by formula (10.20) instead of by the more accurate procedure based on equation (10.16) and the curve of figure 28. These formulas are also applicable to long narrow isotronic plates. It is only necessary to use E inoplace of all letters 2 that have subscripts and take X equal to 1 -C where 0 is Poisson's ratio. Section 11. Rectangular Plate. Uniformly Distributed Load. Edges Simply Supported. Approximate Method-- The solution of the differential equations (9.11) and (8.14) 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 are 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 from the Wisconsin Alumni Research Foundation. The numerical calculations were made by the Computing Division of the Forest Products Laboratory. -28- 1312 sections 4i oban n.- rroxirte forrafls in th,-'> of'rL- S oit> il~l dcf I ct ions. In tl- -present c~ase, !incf, the, TA-111 f o l'3t in' a stitc of ctririn it is ne.ces,,n;ry to -,'su -itnbl -;smons for --io comronen' u nnd v rpra,11(71 to thie X nn Y '' re s-re c tiv- 1-., of the disppcement of -Points in thc mildl( 1urf of -r, Int~ e~ dizior to a. suit,,ble expression for the dp'flection. Tlh,- rcso2contam ccrt.air pnr.9ie~ters which ,re to b:- ci- 0-en in sK-' w' Cof t-.ro a r 't, t I- t o f th e s t, t e o f s tr i n ass o c i. t-, d,% "i t'- t' ir s-~ ~ e ta ,and t of' the stnte. of strain :ssoci,)tf- ith thr, -ire c tv II7'i-~ sc t, t 0th s tte s o f s t rain -. re c o rsi,' r t o b~ 'o i 'he ro'PnLi-l cir-fof "he s- m of th( tro st-qtcs o str-in is' Yfi i r eK' 0o + of th-Fir ri' etctiv,-- -rotenti,- l ~~~i T~ 1t e7rr c's-ions -tssiimed for the cl,-flection an 1 th r , m--lrnt fo ir p'i-l-ret( r-, 1-hich nre to be dterminedr so tl-,,-t th ew ' -c- ftei eJ'w rer_,y is mnimim. It i-. fo7-mcl tutin,the d- termin,-ation of n- ,f se Tar,?meat, rs, q, (se,- apn cndix 11), in thlis, "Iny involve ca,-,i ,hat a-i -',cco com-plic'2tp-. Accordin.-'ly, is ta3k -n to beh aw K cas f 1de f 1v'l cif ot iorns. This dnermin s te P-enr,ral Fhelr ' S T,- d f orme d m i dr1ib7 sur fa-ce whe n t 1 dt-)f 1,,ct ions ar~ s~Tp. 7 - 11 t axr- --zl t, for thei c ,se of la&~defltetions, r -t c o' ssamtc v' -1 form of the middle siirface doi s not, cha'r- sre'' u "he Je jfl 1 -ct io ns becorneler" r,11l orclin tes of t'he r,-,iddl(- sur fr-c for : sm l~l lc f etion 1 emrs- congiderud to b multi-oli ,-, b1 -I col' mfn fa o" prarm,,t~r 'r hpvinf- baeen clho.;,,n in t~lis !"csY, th-~ t~ac rs can b(, found. Af ter -rcrf ormin, tne c,1cu (c-' t i U 'ollov':inge forval., is found connectin,- tho quiiAtitv ! nd raDtio r of the d ,flectiorn at thu, center to th, thic ca cn'7S: + -I-5 ~-Cac' or .i -q the reci-procal of the factor (, ir ~ )~ln cr-- srondtin, for77ila for a -rlate r,,ith small ( Cl et ions. Thi s i t o bc c t I ari e from tl ic -av 1n wh i t -w :s de r ive J (11. mus t a, r ~i 7 ri~~i S M.. ,1'r1. A formula for caic -atin ti- fa-ctor ~i m i 72 -9 arr endix 11. Tables 31 to 35 give the factors H and Q for nltes of spruce lywood of several tynes and for plates of isotronic material. It is to be recalled that the plies are assumed to be of equal thickness in a civen rate and that the elastic constants of the wood are those Priven in table 1. The letter 1. denotes the ratio b/aof the sides of the rect-ngle, the side b being parallel to the Y-axis. It is to be borne in mind that formula (11.23) is an approxmate one and that the errors involved in using it may be considerable. However, it is believed that the formula will give a reasonable estimate of the deflection associated with a given load for a given plate. If the length of a Dlywood plate is sufficiently greater than the breadth so that the ratio =I1/4 n/a = (b/a)/(E 2) 1/4 is greater than 2, the curve of figure 11 indicates that the plate Tnm'y be considered to be a long narrow plate. Then either the annroximate or ,x.ed methods explained in section 9 can be used. The an-roximate method ex lined in tl-hat section will be found to be sufficiently accurate. For such values of the ratio /,. the approximate method explained in section 9 can also be used to find the maximum bending stress and the direct stress, The ratio /a is to be distinguished from the ratio = b/a of tables 31 to 35. It is interesting to compare the approximate formula for the infinite strip as found for the limiting case k =,o of the exact theory. For the infinite string from (11.24) and (11.25) (see arndix 11), . w 3 2 ao 3(_) (11.26) h 1 h, t h while from (9-28) 6.4 "'o 20.6 o 3 P = -- x i + x "- 1K~~1 h X a h Section 12. Rectangalar Plate. Uniformly SDistributed Load. Edges Cl01amped. arge Deflections. Apnroximate Method The method of section 11 will be applied to find an approximate expression for the deflection at the center of uniformly loaded nlates with clmped edges. The following forms are assumed for the deflection and the com- ponnts of the display mesnt: -30- 1312 When o < y < c w w= sin2 18c sin2 -a o0c a u 0 gin ,y in r (12.1) 1 do a v = C2 sin LI-Z sin --_x 2 c a when c < y < b-c, V = w sin 7 X o a u = c1 sin I4x (12.2) a V = O Deflections and displacements corresponding to those given by (12.1) are assumed for the region b-c < y 4 b. The forms (12.2) were chosen to represent approximately the sitt,tion in an infinite strip, w being chosen to satisfy the conditions at thu edges and u in such a way that the mean direct stress X' is constant. The forms (12.1) were then x chosen to satisfy the conditions at the edges of the plate and to pass over continuously into the forms (12.2) along the line y = c. The following formula (see appendix 12) is found connecting the load and the deflection at the center: P = w __ + Q ( ) (12.3) h h where P Q a4 L Lh and H and Q are defined by (12.4) and (12.5) in appendix 12. -31- .he factvw 1 ir (12.)) is ,he reciproal of the fof. ca ..t (4.35) That this should be so follows at once from the way in which the formulas were obtained. The remarks made concerning the annroximate e 4 ,'mc, 71.23) also arl-y to equation (12.3). The curve of :i- .re 1) indicates that if the ratio -/a = (b/a) (E/E is greater than 1.75, the central portion of the plate can be treated as part of a long 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 isotrovic rectangular plates with large deflections an aprroximate treatment has recently been given by S. Way21 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 i, 3/2, and 2, respectively. ease of the larger number of parameters employed his results undoubtedly reTressnt a better approximation to the actual solution than those based on nations (12.1) and (12.2) although the amount of numerical calculation is mac. greater with the increased number of parameters. For isotropic plates the loads associated with a given deflection calculated by the two mehoda differ by less than 12 percent, usually by much less, for deflec- tions in which "o lies between 0.5 and 2. h No attempt has been made to calculate the maximum stresses on the basis of equations(12.1) and (12.2). It is probable that equation (12.L) yields values of P for which the error is of the order of magnitude of 10 percent. The labor involved in obtaining more accurate values of P for each type of plywood plate by the energy method is prohibitive. As noted above, the methods of section 10 can be expected to yield satis- factory values of the deflection and stress in the central portion of (-r 1/4 plates for which R/a = (b/a) ( /E 2 )1/4 is greater than 1.75. Tables 36 to 40 give the values of H and Q for several type- of spr).c:e Tlywood, the plies being assumed to be of equal thickness. There is close agreement between the approximate formula (1 .3) 'or the ease k = b/a =c> form' s. Tl formula (10.13) was TN .T 3 o + 2L3 X (_11) A i h A a h chie .formula (12.3) becomes, for k = oe, 2 x+"o + ( o) X 1_ T a n1 -- ,:, ZProc. [}th Internat. Cong. ArT-,l. Upch., Cambridge, !,'_,ss., 193. I 51i -7'2- Appendix l.--Stress-strain Relations in an Orthotropic Material The energy f deformation of an orthotropic material can be written in the form:-- 2 W = A ex + Beyy + Ce (. + 2Fey ezz. + 2 Gezz exx + 2 Hexx eyy +Le, + Mezx +Nez yz ezx xy , the coordinate planes being parallel to the planes of elastic symmetry. It is not necessary to discuss the significance of the coefficients A, C ... in equation (1.1). It is sufficient to remark that they are numbers that characterize the elastic behavior of the material. The usual elastic moduli are introduced in equations (1.5) and their relations to the numbers A,oC etc., are shown in equations (1.4), (1.6), and (1.7). The relations between the components of stress and strain are obtained from the equations = dw- W (1.2) x dexX dexy Then X, = A exx t Hey, + Gezz Yy = Hexx + Bevy + Fezz (1.3) Zz = Gexx + Feyy + Cezz 2-2See for example: Love, A. E. H., The Mathematical Theory of Elasticity, Art. 110; St. Venant in his annotated translation of Clebsch, Theorie de 1'Elasticiteo des Corps Solides, pp. 76-80; Price, A. T., Phil. Trans. 228A, 1-62, 1928. Love's notation for the components of stress and strain will be used throughout. -33- Y,= LeY , ZX= MezX Xy (1.4) The solution of the equations (1.3) for the strain components in terms of the stress components can be written in the form I er xx E x E x x y yy E x E Y z E z I (1.5) -xZ X L- Y e = de dterXi-a*Y + Z zz E x E y E x where, with A denoting the determinant of the coefficients of (1.3) I -CA-G2 Ey 6 y I BC-F2 Ex L x I (1.6) Srx CH -FG (1.7) XYY = E A There are two further equations that can be written down by cyclic permutation of the letters in (1.7). We observe accordingly that there are three Youne's moduli, Ex, Ey, and Ez ; six Poisson's ratios OXy Oyx, o-y 6zy, onzx dx-. From (1.4) it follows that there are three shearing moduli /z = L, /4zxM, and /xy N. Among these twelve constants there are three relations, namely, those expressed by (1.7) and two similar equations. -34- Nexy I i _AB-H. E A 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, -Z =0, of the 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, L 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: dw CdW u = -z 3 x v = -Z I y (2.1) where \Adenotes the deflection of a point in the middle surface. From these equations the strain components are found to be - z WW 2 (2.2) eyx = -z a Yz ey=- z y ex -z z Exx~~ CIp > By dx ag-Z ~ y The stress component Zzis taken to be negligible in com- parison with Xx and y. It follows from (1.5) that at a point in a given ply S. X+ 6yx yy E ly X (ey, + oxy e..) (2.3) X y = #xy e xy where (2.4) MSee, for example, Nadai, A., Elastische Platten, (Berlin 1925) p. 19. -35- 2--Tor the corresponding treatment of the isotropic plate, see Nadai, A., Elastische Platten, p. 20. Hence, using (2.2) dz W Sy + Vy Id2 Y\d + OX- 2 W 7w) Elyz Yy T~ (2.5) _ )2 w Xy=- 2 xy z ax ay The differential equation for the deflection VV is readily 24 obtained4 from the conditions for the equilibrium of an element of the plate (see figure 29). The bending moments are denoted byx1 and my, the twisting moment by xy, and the vertical shearing forces by Px and py, all measured per unit length of the edge of the elements along which they act. equations:The moments mx, my and mxyare defined by the following equations : mx = .M mnxy=hl MY =Y Vyzdz h"d Xx zdz (2.8) Xy zd The moments acting on a small rectangular element of the plate are represented by vectors in figure 29. From (2.6), using (2.5) -36- 2)2W mx = a, x my b, ax 2 mxy -C G x by d'W b? 2 Wy - ay - b 3yn dyw (2.7) where a, = (E, z/A) dz , h/? b, (E exy zY) dz c = / xy Z2 dz a Edhx b, = E, yzyA) dz (2.) From the relation (1.7) it follows that a. = b. The vertical shearing forces px and Py are defined by the following equations: , h/z py = Zy dz .h/2 Px = Zx dz , -h/z (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- PX a X p y + MXy + arnx'y (2.10) while the condition for the equilibrium of forces acting in the direc- tion of the Z-axis leads to the equation 3px + dpy + = 0 C x & y It is to be recalled from section 2 that h denotes the thickness of the plate and p 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 deflection, 1/, is found: D, ) 4 w + 2 K 4 1 + D a w (2- wo X4 eX )y a y4 whe re ) ) K = (o, + b, +2c D, = as, D2 = bz, (2.13) If the plies in the plate are all flat grained, as assumed, the expressions for D,, D and K can be simplified. The sub- scriptsL andT 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 Ex = E-L , Ey = ET, A4Xy = /AL T, (2.14) (5rxy dyx X =OTL while for plies in which the grain of the wood is parallel to the Y-axis Ex = ET L. Ey=EL ,Xy = # TL = 4 LT ) (2.15) a-x Or ( T -38- The coefficients D, and Oz of (2.12) are readily expressed in terms of certain mean moduli in bending E and E? Thus we write D, E, h 3 E? h 3 =1 Z2 N z-I (2.16) whe re ;k = I-Cr LT OTL (2.17) E, = hE, zz h/z dz I Z---- Ex z z (2.18) E2 h--- Ey z dz (2.19) The quantities E, and Ez may be called the "mean modul! g5 in bendinge' under couples whose axes are perpendicular to thex Z and YZplanes, respectively. As soon as the structure of the plywood is known these moduli are readily calculated in terms of the Youngs moduliEL and ET of the wood in question. For a plate whose construction is not symmetrical with re- spect to the middle plane, it is to be expected that FI 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 Ez 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) ELOTL = FT OrLT (2.20) 4-Srite, A. T., Phil. Trans. A 228, 1, 1928. It is pointed out in a footnote in section 2 that the definition of these moduli there used differs somewhat from that given by Price. -$9- it follows that the factors Ex yx /A EY 6 A and in the expressions for Qz and b, in (2.8) are the same for all plies and are equal to Eb L/A Hence h3 , =L O TL atz = b, = .a (2.21) and K =(E, o, 1h' 3, + Z, (2.22) pUxy being identical with #LT for all plies. The expressions (2.7) for the bending and twisting moments can now be written in the forms: 2 W mX D, a _~Llh3 #LTh mxy 6 + a-w 62 V ~aw dx by EL EL E . (2.23) where EL L E, (2.24) The differential equation (2.12) can be reduced to the simpler form adw +2 + -, -D, c x 4 + 2 e ;xa 2 + 7 4 D, (2.25) -40- where K ( ID (2.26) by making the substitution (2.27) whe re "-(DJ'D7)' The change of variables from X and Y to X and 7 corre- sponds to a simple extension or contraction of the plate parallel to the Y-axis in the ratio A plate of dimensions OL and b is, thus transformed into one of dimensions CI and C: b = b (E,/E?). The latter plate will be referred to as the transformed plate. In the new variables the expressions (2.23) for the bending and twisting moments become: =-n D, w my d a : I". LT Eh 3 rnXY 6 + Wc -qx2 N +w (2.29) -41- =(E,/E2) Y (2.28) Appendix 3.--Rectangular Plate Under Uniform Lq&d. Edges Simply SUported. Small Deflections The differential equation (2.25) S4 4 4 dw w d w p x+ 2 a xa ia+ 14 = D= where p is a constant, is to be solved subject to the conditions stated below that hold on the edges X == 0 0 7 1= where (See (2.28)) The boundary conditions on the edges x = 0 and X = CL are (See (2.29) ) w =x O + 67 '' = (.2) The corresponding conditions on the edges 7=0 and ?=8 (that is, y= b ) are w--O0, as if+oiC 0) z + Oz Tj (3..3) For the constants 0( 0 and 6 see equations (2.24) and (2.28). We choose first the following solution of equation (2.25): P ((3-4) w = x*, -, x3+a x 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 T, 1 sin An X n = 1, 3,5..%.. (3.5) -42- where A = 4 p/a D, . = n 71r/a (3.6) and (3.7) The satisfaction of the boundary conditions on the edges =0 and y=1 will be secured by combining with \, a solution, W2, of the equation (2.25) with its right hand member set equal to zero. Let co A nZI n Y, sin ,, x (3.8) where Yn is a function of 7. In order that WZ may satisfy (2.25) with its right hand member set equal to zero, Yn must satisfy the differential equation - 4 y,+ n Yf = 0 On setting Y= em"V it is readily found that m. = +'Y t 6S,, whe re Yn = X P P =+ ViE 1n n 0" (3.9) (3.10) Then Yn may be chosen as a suitable linear combination of the following functions: sinh % -/ sin Bay sinh -Y 7 cos 8n / cosh, 9/ sin &ny, cosh Yn 9 cos 8~ 7 (3.11) -43- In obtaining these solutions it has been assumed that is lees than 1. This appears to be true for all typee of plywood. If f 5 j appropriate modifications in the functions (3.11) can be made. It is clear that VVW 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, + W. satisfies the conditions (3.3). It is found that when n is odd Y. = C, 4 [sinh Yv. sine (F- 7) + sinh YV, (-) sin8, 7 J + /I-g' [cosh c cos ,(-) (3.1) where + coshv, (.-) cos &n ] } Cn = 1-46'z (cosh( +) cos 3, ) and that when n is even Y,=0 Then the deflection is given by W = V, "+ wz = A ( -Y,) sin ,Anx 4 pa 5 (1-Yn)sinhn x, n= 1,3,5.... 71s D, n (8.14) On recalling that D,= E,hlZ A equation (3.14) ma b written w 48 A EL I (3.1) h ~r 5 E, ") sn'ns n =1,3,5.... -44- whe re p= PCL 4 E~h' 4 A -eT d.;L16 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 W to the thickness h is found to be given by the equation, h 70' E, 11- n 2Cn V sinhTn/ n n + /- Z osh co y )](3.17) 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.) Lot (See figures 4 and 5, section 3.) w =w. sin ) when c Y 4 b-c (. w :Wosin 7txsin T." when O< < 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 Letf -45- The deflection \v^ is expressed as a function of the two parameters \WO and 7' which are to be determined in such a way that the total potential energy of the system is a minimum. Let Ve = change in the potential energy of the load due to the deflection. Vbe = potential energy of deformation of the portions of the plate at the ends. bm = potential energy of deformation of the middle portion of the plate. vb = Vbe + b Now /ba 0 0 It is known that the potential energy of deformation is given by Vb kx xex+Yy eyy + Xy exy)dz dxdy We substitute for Xx, and X their values as given by (2.3) in terms of the strain components exx0 eyy, and exy. The integrand is then a quadratic function of these strain components. For the strain components we then substitute their values in terms of dVV dV and ox V as given by (2.2) and perform the in- tegration with respect to Z. We thus obtain 3 w b' /a ', 2" dI Vb = 24 A/\,-a 1 d . +2 Z +2 4 Wv dxdy (3.81) -46- where e, E._ %6 V=AL (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 Vbmn is obtained by integrating the same expression as in (3.21) over the central portion of the plate, Wbeing there defined by the first of the expressions in (3.18). In like manner Vbe is found by taking twice the result of extending the integration over the portion of the plate between the lines Y = 0 and Y = C W being defined by the second of the expressions in Using the abbreviation k = b/A (3.23) we find from (3.18), (,3.20) and (,3.21) that S4 c'w [ +, 7'--(k 7) EL W 1' r" +--- + 2 +4 A V96 X all r or"" (3.25) Vbrn El h' 3WO C- ) Vb : 48 A a 2 ( -r (3.26) The parameters W0 and 7 are to be determined from the requirement that v = V Vbe + vbrn (3.27) shall be a minimum as a function of these quantities. From (3.24), (3.25) and (3.26) the expression for V can be written in the form: V = L wo z-Mpwo (3.28) -47- =L aEF h' -K 2 (k -70 r-, where L= 6 3 ,96 X ct 2 1(,CL C-F-I - ie, r + Z- %e, k) 6 + --- 7 (3.29) (3.30) (3.31) + 4)v 3 = o a The conditions 4) WO f3 V F3 (3.**) (3.33) =2L we - Mp =0 _ pW d dL J-r a lead to the equation 7 -1.-846 kr4 -3' $el + 2.752a kr mir (3.34) -5 ' Act 7* + 8.2 56 6k = 0 parameter 7 having been found the center is found at once by SM Wo 2L as a root of (3.34) the solving (3.32) for W The deflection at then (3.35) vberwe a = 4( 7r (3.36) + k V + 2 7s4 7' _48- MLL M = + (k 7 2LdM -Md = 0 dr d7 for the determination of 7 After some reduction this equation becomes Instead of (3.35) we can write --O- p (3.37) where p a4 (3.38) For a plate of isotropic material we can use (3.34), (3.35) and (3.37) if we note that in this case %6 =-6 A='-- and & -2 In the equation for A 6' denotes Poisson's ratio. -49- Aomendix 4.--Sectanaular Plate Ud.er Uniform Load. NdSes Claped. Sall Deflections The conditions satisfied by the deflection at the edges are: w =0, henx =, x =a, p =0, y/-f = b > (4.1) )w= 0, when X =0 C X =4 (4.2) -7 = O, when :=0 (4.3) The conditions at the edges 7/= 0 and 7if of the transformed plate correspond to those at the edges y= 0 and y = b of the given plate. The differential equation (2.25) Is to be solved subject to the conditions (4.1), (4.2) and (4.3). Choose as a particular integral of (2.25) satisfying the conditions (4.1) VVw, = A x (X -L) 7/(1 ) (4.4) where A = p/ 8D, e (4.5) A solution Wz of the homogeneous equation, obtained by settin p=0-O in (2.25) is to be found such that W = W +We satisfies all of the conditions (4.1), (4.2) and (4.3). Choose W as follows: w. = A[s ( ) ( -9)sinh rn (4.6) + sin n ?I sinh rn (7/ -) ]smn Xnx + Kb. [sincm (x- a) sinhd, x + sin cm x sinh dm (x-oa sinom a j -50- where the summations are extended over positive odd integral values of M and Y respectively and In= n7 r/a Cr = rn x/d (4.7) r n = Anp 6,n = Ano, dm = rmp Cm = 0m cr (4.8) H, (/f) = sin E41 + 6n sinh / (4.10) Km(a) = dmsin cm a + cm sinhdm a. The function Wz satisfies the conditions (4.1). It remains to deter- mine the coefficients (n and bm in such a way that the combination w = w, + w (4.11) satisfies the remaining conditions (4.3) and (4.3). The procedure is much the same as that for the corresponding problem in the case of the isotropic plate.26'27 It is found that CLn and bm vanish when their subscripts are even and that these letters with odd subscripts satisfy the equations bm a. n a 2 I + m/I n (4.12) m \"(On +am C L o- dm Gm (a) n (c +T + d2)-4 2- brm (4.13) where F (d)= cos8. S +cosh (4.14) n (/9/) sin an + sinh y74 G a) = cos cm a + coshdm a (4.15) m a (o/Cr) sin cmc+ sinh4ma 6Hienc H., Darmstadt Dissertation, 1913. &7March, H. W., Trans. American Math. Soc. 27, 307-17, 1925. -51- From this point on in appendix 4, the numbers M 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 be = Q mn Fn (D a,, (4.16 (I n r R nGm(a) bm (417 n3 n M = 1,3,5..... n= 1,3,5.....) where, writing b/a =k anda s=6k = ba = ,/C'. (4.1a) B= 8s' a/r C = 8s / 3r (4.19) Qm 7r m 4+ 2-rcm nsa +n4S4 (4.20) Rn 8 8s p nmn* rn SI 2 n n (4.21) nm~ r M4 +.7 2 -mmns' +n 4S4 and F( d) and Gn(a) are defined by (4.14) and (4.15). Because of the rapid decrease in value of CLn and bm with increasing n and M, the first few (1n's and bm Is can be found by solving-28 the finite system of equations obtained by replacing the unknowns with higher indices by sero in the first few equations. 28In D. E. Zilmer's thesis to be presented at the University of Wisconsin, he discusses rigorously the solution of the infinite system of equations for both the orthotropic and the isotropic plate. This was done for the isotropic plate to remedy a defect in the convergence proof of the present author's paper (Trans. Am. Math. Soc. 27, 307-317, 1925). This defect was due to the omission of the factor 72 in the last term on page 312 of that paper. The method of the convergence proof is not changed essentially. S. Iguchi (See footnotes 1 and 12) was also led to an infinite system of equations. He did not establish the con- vergence of the process used in solving this system. -52- The values of (1L, bf and b3 apart from the common factor Q 3 are given in table 41 for plates of various types of ply- wood and for various ratios of the sides b and CL. An approximate formula will now be obtained for the deflec- tion at the center of a uniformly loaded plate with clamped edges. As in the case of a plate with simply supported edges a plausible form, depending upon certain parameters, will be assumed for the de- flected middle surface. The parameters will then be determined in such a way that the sum of the potential energy of deformation of the plate and the change in the potential energy of the applied load shall be a miniimim. The plate is divided into three portions by the lines y a n nd Y = b-o The following forms are assumed for the deflection in each of these portions: w = W'oSln2 ar O< y < b-c w =wS .inZ 7['7X si Y < -< (4,22) and a form corresponding to the latter for the region, b- c < Y < b Let C 7"k/ 2 (4.23) Then the deflection W is expressed as a function of the two parameters WO and which are to be determined in such a way that the total potential energy of the system is a minizmim. Let the symbols Vt ,Vbe and Vbm have the same meaning as in section 3. Using (3.20), (3.21) and (4.22) it is readily found that V% = P V a c (2 k-) (4.24) = 96 A oL? 7 + Vbm- E''h3 l? (K- _r) (4.26) -53- 2 k r k- + + 8k-5 3 +t + 8,,-, where %,, Mg and 3 are defined by (3.22) and (3.31). Then the total potential energy of the system V = V + Vbe 'Vbm can be written in the form: V= Lwoz MPWo The conditions (4.27) dV w = 2 Lwo -Mp = 0 V o dL dM 7" -wo d7 PWo d" (4.28) (4.9) = 0 lead to the equation 2LdM -MdL =0 d- dr for the determination of 7 After some reduction this equation becomes 5- 1.2k r 0. 6 '" + 0.46k 2 -3 -, + 3.6 k = 0 /Y'l (4.30) The parameter 7 having been found as a root of (4.30), the deflection at the center is found at once by solving (4.28) for W Then pO4 Wo = ot E.h 3 (4.31) where 12 x ot= 4 (4.32) -54- Instead of (4.31) we can write WO = L P (4.33) h o whe re P E. 4 (4.34) -55- Apendix 5.--Infinite Strip (Long Narrow Rectanular Plate). Load Concentrated at a Point. Edges Simply Supported The edges of the strip, X = 0 and X = L figure 30, are simply supported. A concentrated load P is applied at the point X = LL, y = 0 on the X-axis. 29 At first the load9 will be considered to be uniformly dis- tributed along the segment, LL-- O < X < LL4O,of the X-axis, the intensity of the load per unit length being P0. The total load is then P= 2poOL. Later C will be made to approach zero. In this case pO will be taken to increase in such a way that the product 2po O= p remains constant. Because of the discontinuity in the loading, it is con- venient to determine the deflection and moments in the regions y > 0 and y< 0 separately. Consider the region Y > O. The deflection W satisfies the differential equation (2.25) with p= 0 at all points of this region. The boundary conditions are: W = 0 when X =0 X = 0, and when y = O 3 (e.I) x = 0 when X = 0 X = C; (5.2) when (5.3) =y 0 y= O. A further condition on W is found from the distribution of vertical shear, P along the line y = 0 The load on the segment (ut- Oc, L + O) can be represented by the series 4p, sin Xn u sinkn oL sin kn k 7 n in x (5.4) where ,. n We accordingly require that lim p_ 2P ~I sin Anu sinAn osin,x (5.5) y*+0 n=l 2Nadai, A., Elastische Platten, pp. 78-82 and 85-95. Huber, M. T., Bauingenieur 6, 1925. -56- From (2.10) using the variable 7 =Ey instead ofy we can find the expression for Py in terms of VV. Entering this ex- pression in (5.5) we obtain: * 3vv ) 3w Jim (-D2E 3 -Ke 3a , +o ) 0 =p Ip- sin A,,usin A oy sin nx (5.6) 7r n By (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 py 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 = 0, which satisfies (5.1), (5.2), (5.3), and (5.6) is: 0o w = An e" V (cos&.n/ +c sin 8/)sin .An x (5.7) where An Po sinX, u sinAhAo M sinAnu sin co (5.8) An D, Es3 n 7k r/oD, n A3 c = 'Yn/n = P/O-. The remaining symbols are defined in the table of notations and in section 3. We now allow OL to approach zero and Po to increase in such a way that 2 Po OL remains constant and equal to P. The coefficient An in (5.7) becomes sin An u (5.9) An=: A X3 nr whe re Th bending and twisting momesnts will be calculated by If we let (5.10) (2.29). 00 n-- 00 AAn e lsin 8,, sin An An A?' e' n cos(5,, sin An X (5.11) (5.12) then 2 Vy = -x + CPp and from (2.29), (El Ez) Y- E? ( O-x Cx c Y/) (,OX+ cc C PC ) (5.13) (5.14) whe re m EL OTL -(El Ez (5.15) -58- (5 .16) E~ +(E L) Y Zxpressions for X and Y/will be obtained in closed form. Let 0= (5.17) flko e sin At, x sin A, u (5.18) An-/=(Yn-(5n?/= nP i-7 From (5.18) we obtain i 0p nRi e n where Ir= Ir= (X +U Cl a ( X- U) and 66 denotes the real part of the expression following it. Now, setting -89- where (5.19) (5.20) X + i 0 a 4yn-iSM)'/ Xn e sin An X 00 A 1; n . AQ 2?r - n(4 +1r) '=0 Z" = -0t log (1- z) =- I log[ze 2 "'(cosh cos C)] Treating the second term in (5.20) in the same way o cosh" cos F' 9 coshg cost 0 4 r In the notation of (5.19) 0 AcL (5.21) Recalling (5.17), X and b can be found from the relations x= 6% , 3 = C where 9 and S the expressions that: denote the real and imaginary parts, respectively, of following them. It follows from (5.19) and (5.21) A ct - 47 (5.22) (5.23) where H= cosh cos Cos- 7 a a Q (x -u.) (5.24) (5.25) (11.21) S= sinh X0 Q sin -60- log + 1j - B(C-H) ton CH + 8 F C cs h C, 0 1 / CI In L2 VI. flty (C t 0 t ingC :proi Ie exrF ,' 47r 1uc ~1- x 2 oto 0 of loadinc rAt approachi to i v t ( t u t x i ~t /( ~ C I ii. te at theW noiit kI~ ~ d- -( Lo' tle n.-At ol from ution o~ r~ 1.t e is rc-mid17~ conver_- ~t T_. alb~t ~ ~ ~ d nb '1 culated by t i use thit .4) e fimct ins~ d t a~re needed, being o c rol- I ) nd (b "'73 For isotro:- : ,-rinl /0) 1 and 0- 0 the fIanct ion X :, ,-~ b: ( ile rt ir!, nr t way. Tne functim nZ 2e(Cs t Buaht tnc n Le occurs in t e cxre ssI 1 (.i~ 1 or tloc the limit -Y '~s /- oe Tile e )mressr. c oy mxI2 andC My ,Io th kncv,7 -o: 7, ca s e oo-l tn 'E w"Ii c " n~~ t r )"c aes I or t )e Appendix 6.--Infinite Strip (Long Narrow Bectangular Plate), Uniform Load Aplied over a Saall Area. Edges ,Simply Spported A uniform load q per unit area acts over a rectangular area 2 0cu by whose center is at (L, O) as shown in figure 21. In appendix 5 the deflection associated with a line load PO per unit length of the segment of the X-axis between X = U-O and X := LL+ C 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 7) is positive is again given by (5.7) if the coefficients An (see (5.8)) are modified by replacing Ph by Cdy where q is the uniform load per unit area. It will be convenient to use the variable 7 instead of y and write Po = qdy = q d y (6.1) The properly modified forms of (5.7) and (5.8) are then w = Bnd? e (cosSa?;+csinrj)sinx/ (6.2) where Bn sin?,LusinXov( np (6.) In writing (6.3) ve have used the relation D264- D. The coefficient C in (6.2), has the same meaning as in (5-7). An the transformation 7=y, is being used, the dimensions of the loaded area on the transformed plate are 2OLand 2 7 whe re " (6.4) The boundaries of the loaded area are the lines X = L- CO, x = U + ,OI = -', and 71=" To find the deflection due to the loaded rectangular area consider a horizontal strip of width dv at y=v within the -6Z- boundaries of the loaded area on the transformed plate and calculate the deflection at the point (X 71) due to the load on this strip. If ?/> V this deflection is given by (6.2) with 7/ replaced by 71- V an&d F by dv. If 7 < V/ these replacements are to be made in the equationi corresponding to (6.2) for 97 < 0 . For points for which 7/> r" (namely, Y > r" ) the deflection is obtained by integrating between the limits -* and *" the expression for the deflection due to the strip of width d V/ at 7/ =V. The result is w ni~ X{ ' W n sinXx e si n[6, (T'-9/ e h=1+e- 'I) sin[ 6, ("r+ 7/)+ 01 (6.5) where cosO=N;, sinO=11/-oe (6.6) For the region < 7/ in (6.5) is to be replaced by -/throughout. If -7" < 71 < Ir, the deflection is found as the sum of two integrals, one between the limits -7r and 7/) the other between the limits 7/ and 7". The result is W= noBnsinxn 2sini -e? 7+? sin[8, (7-+ ;7) + 0 + e S 'f i n [6n (r7-- 0]}1 (6.7) The moments can be calculated from (6.5) an4 (6.7) with the aid of (2.29). -63- Ed-wee Simn_ S marte 31Timoshenko, S., Bauingenieur 3, 51, 1922. Anpend 7.--Rectangular Plate. or AIplied Over a Small Area, Load Concentrated at a Point Edges Simply SupForted The method of a suitable distribution of positive and negative loads as described in section 7 can be applied for both types of loads. However, for the case of a point load acting at a point on the central line y = 0 (see figure 31) the method used by Timoshenko31 for the corresponding type of loading of an isotropic plate leads more directly to the result. With the choice of axes shown in figure 31, a concentrated point load is applied at the point (0, 0). Take the following solution of the equation (2.25) with p -= 0, for the upper half of the plate: n= Vt si n, x (7.1) where Y = a. sinh Yn 7/sin $n + b sinh Yvoy cos6, 7/ +c,coshvo y sin8, / + dncosh -y, /cosS 7 (7.2) he conditions on the edges X = 0 and X = C are satisfied by (7.1). The coefficients in (7.2) are to be determined so that SW-O on y= 0, that the condition for discontinuity in vertical shear along y= 0 is satisfied (see section 5 and appendix 5) and that \=0 and alw =0 on 4, 4_ b It is found that P sin Sa +psinh ,/ sinAn c a,-2 Qp p cos 6, # + cosh r n1 X3 sin Anc 3 Ps bn - 2ap P Cn= P bn (Y p sin 8, ( n cr sinh 1np cos 8n + coShni3 sin ;n c A~ P 2 p d. -2aD,po-" -64- With these values of the coefficients the deflection at the center X=c2, 7= 0 due to a load at this point is found to be given by the formula Pa 6 s I sinh r -p/ sinSn! (7.3) Wo- E, h p o n3 coshY,,p+cosS,6 n ='1,3,5 . In obtaining (7.3) the distance C in the expressions for the coefficients Qcn, bn, etc., has been set equal to a/2. The symbols which occur in (7.3) are defined in the table of notations. For an infinite strip, b-- Ocit is found from (7.3) that the deflection o o at the center due to a point load at the center of the strip is given by the equation: - Pan, 6A& l+- + + .. Wo- E, h pK3 53 7 ) Poe2 6 As - 1.051 > 7 E,h* (ts (7.4) a result that is in agreement with one which could be obtained from (5.7), (5.8), (5.9), and (5.10) of appendix 5. The factor 7Y in the equation wo = w o (7.5) is to be calculated from the formula II sinh 8/ /) sin S. (7.6) 1.051 n 3 cosh + cos8,0 n = I,3,5,7..... -65- Appendix 8.--The Differential Equations for the Deflection of a Plywood Plate. Large Deflections When the deflections of a plate become so large that direct stresses are developed in addition to the usual bending stresses the state of strain in the plate is a superposition of two states of strain, one associated with the bending stresses and a second asso- ciated with the direct stresses. The components of the former, given by equations (2.2) vary linearly across the thickness of the plate. The components of the latter are constant across the thickness of the plate. They are given by the equations-2 e -) +I aw e y y = +V , du + av +daw Ow xY- dy ax ax ay where U, V, and W denote the components of the displacement of a point in the middle surface of the plate. The corresponding direct stresses at a point in any one of the plies are given by (see (2.3) ) xE (e,, + 4, e ,) -Y A e Y(+e.Y ) E Y Yy y x (8.2) XY =-x y e'x y where Ex, Ey and xy denote the values of these constants in the ply under consideration and I; = I- (T xy ry The mean stress components X1 ; and are obtained by averaging the stress components Y; and X)y over the thickness of the plate. 32Nadai, A., Elastische Platten, pp. 270, 284-287; Prescott, J., Applied Elasticity, pp. 435-438. -66- It follows readily that, for plywood witi flat--gra nd pies x-, (Eo e' + E,cT,e' XY jx (F Eae 'xx + EL (TT L e8x V = (Eb ey + Ecrk e' ) y = /LT 8~ /Xy where E = Ex dz h /( Eb = ,Eydz A = -, ; The quantities Ea and Eb may be called the "mean moduli:LL in stretching" in the X and Y directions, respectively. Denote tne mean forces, resulting from the me,-n stress coi-o ponents, per unit length of edge of an element of the plate such as that shown in figure 29, by nx ) n. and n., Then nx = h Xx, n, hY nxy h (8.5) Since the deflection is assumed to be small in covrerison with the length and breadth of the plate the conditions for equilib- rium of the forces nx, n, and nxy or of the stress com:ponents X > y ond X, 33Price, A. T., Phil. Trans. A228, 1-62, 1928. Apparent Youn. :'s modulus for stretching, p. 41. The definition of this modulus s 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- will be the same as if the plate were plane and in equilibrium under forces acting in its plane. These conditions are -J dX, 8 x dx + (3XY dX dy (8.6) Accordingly, there exists a stress function F such that 2 = F > Y3 -, _aF Y a > - 2F X, --, a (8.7) The elimination of LL and V from the system of equations (8.1) leads to the following relation connecting the components of strain: 8 aW - a dyy 2 / 2 / e a ey ax ex y 3 z W a X agew (3yI a) Y (8.8) This equation replaces one of the conditions of compatibility of the strain components. From (8.3) and (8.7) e/ 4 a'F ex, -E day ,, EaF el Eo, H aF eYY H -- a, EL (YIL H ELOL H deF & xZ e I F eXy = -LL Tax2y (8.9) -68- 2TIa H Eck Eb EL Crrt ) (8.10) The substitution of (8.9) in (8.8) leads to the following differential equation: Eo 4 F I 2ELr & F Eb d4 F _ + + H d x9- Oxa + H y4 _ C 2 j'a (8.11) 8xd yx ay 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: x p, daw da g' _ + n + n 0 (8.12) 3x dy x a x'-X- X py y(8.12) where pg and Py are defined by (2.9). Now (See (2.11) and (2.12) ) dP, + -(D, 2Kw X4W 4W (8.13) ax dy + dax x y + )4 Using (8.5), (8.7) and (8.13), equation (8.12) becomes 4w dOw daw D, 8x4 + 2K 8x ay + Da C(.s Lj X4 ax,41 ) y (8.14) d'F d'F 'w da F a pF +w l = p + h L 2xay ax oy a a] Situations (8.11) 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 Stria (Lone Narrow Plate). Large Deflections. Uniformly Distributed Load. Ed4es, Simply Suwported Consider an infinite strip of plywood of width (I. Let the uniformly distributed load per unit area be denoted by p. The edges X = 0 and X -= OL 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 WV 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 ttL of the displacement (the component parallel to the X-axis) of such points will be independent of Y. Consequently dy It follows from (8.1) and (9.1) since V vanishes and UL and W are independent of Y that I~ = 0 e -0 (9.2) Then from (8.3) the mean components of the direct stress system are: Xx- A e' x Y 0 From the equations of equilibrium (8.6), since X, = 0, it follows that X;( is a function ofy alone. But this function mast reduce to a constant since, froin the type of loading, it is clear that all components of stress and strain are independent of Y. Hence 9 (9.4) where 9 is a constant. -70- It follows from. the first of equptllon ;99 n t P_' is constant and '.-ence front, t.,Ie second of ( P.3) thaot Y, is con-st Since thie deflection W is independent ol Y t e diffrential enuatiorn (8.14) becomes tDi x4 In writing t.i- ec uation~ ZF _ 2__- x has been replaced byt.- const~rnt (j in accorni,-_-e wit' (9.AO. Be- cause o-f the sim-olicity of tule stress sz.steii all thei~rrtiL.t could. be obtained. from tnie differert.;,4l equation, (8.11) is n contained in (9.4) combined with (9.3). It is to be observed ~ is constant for a given load P but that it de-nends unon P. fThe quantity 9 therefore enters the solution as P a r! i,,tor. Equation (9.5) can be written d 4W 2 d2w -p d X4 k dx 2 D ( 9 .6 ) whe re k = 9hD The solution of (9.6) is w =c, x + c, + A sinh kx + B cosh kx k On determining the constants in (09.8) to satisfy the conditions the simply supported edges, viz., W = 0) dx 1 (9.7 ) (9-8) on when X =O and when X ;- C it is found that p [2 cosh~x--a-)-oshkci p ___ 7__ __ _ 2 Z2Dl [k2 Cosh-)cs 2 -71- p + 9h dZW d x 2- + x (a-x)] With the aid of this expression it is possible to obtain a relation connecting p and 9 or p and k since g and k are connected by (9.7). From (9.3) S= A A exx Ea xa Then G.a e / dx =A( g n xx E (9.1o) 0 Further from (8.1) / du+ I d e == - ex( dx 2 dx) Hence exx dx dx (9.11) 0 0 since, under the assumed conditions at the edges, the displacement Vanishes when X = 0 and when X = (1 . On equating the right hand members of (9.10) and (9.11), calculating w from (9.9) and performing the integration it is caelculating dXaaaaaa found that Aag p'kaa E 2 k [D,2 tanh ka- Ea 2 k'l 2-.2 ka a ka 2. a 3] (9.12) + 2 tanh + 2. 2 The quantity ] 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, p4 P r -ELh,(9.13) -72- [(5tanh Y)/ 5 + ta~nhey + 2,73] Y in terms of the quantity ka It is found that P = 4 E., L 3 A V/3 LEa /EL ' (9.14) (9.15) whe re (9.16) From (9.9) the deflection on the central line X = G/?- is given by 1-cosh k. c+ cosh Using the abbreviations (9.13) and (9.14) it follows readily that (I- cosh ;-1 cosh?7 ) v 3PAEL h 8 E, (9.17) Prom (9.7) the following expression for C, the mean direct stress over the thickness of the plate, is obtained: E h 7 =3 A a? / (9.18) 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 Ex = exx = E dw (9.19) -73- p _2 W 2D 2 k* Ik where Ex denotes the value of E 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- 2 2 k'Dcosh '" This stress in a face ply is clearly a maximum along the central line, X = C/tC j, of the plate. Denoting this maximnum bending stress in a face ply by S it follows that where Icosh Using (9.21) the relation (9.17) can be written h 8E, ,,y whe re [1 ()](.3 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 V(?/) appearing in this equation are given in table 42. The quniy ,A E )Y is to be calculated for the given load 4 El k4 ,1 -74- using the definition (9.13) of P. This is the value of the function 1(7) associated with the given load. The corresponding value of the parameter 7 is to be found from table 42 or from a curve con- strutedfro thi tale.Theparameter ?/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 9 (7) and k?) t-hat are needed in these calculations are given in table 42. Approximate formalas.--!t 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 where With the aid of table 42 it is found that F(7) is nearly constant 3 and that it may be replaced by F-= 0.360 the maximum error being less than 2 percent for the range of values of 7/ in which we are interested. Hence the following linear relation holds approximately between Vj/h and 7/ *-: WOh= 0. 360'; (9.24) From (9.18) and (9.24) it follows that Z., 5 E a (9.25) 94his fact is shown byj the following table of values: S0 2 4 6 8 10 F(?/) 0.3661 0.3642 0.3612 0.3585 0.3568 0.3559 S12 14 16 18 20 F(7) 0.3553 0.3549 0.3546 0.3544 0.3544 -75- For spruce /X may be taken to be 0.99. For other species this value may probably be used with slight error. An approximate relation connecting p and VWo/h can be obtained from (9.17). For large values of ?I we have approximately Solving this equation for P we have, to the same degree of approxima- tion, S16 E, \No On substituting for from (9.24) 16r E, vvo P= ,3,EL h + + E ?Iz Wo its expression in terms of Wo/h (9.26) where F"= 3.6 0 Equation (9.26) is approximately correct for large values of 71 and hence by (9.24) for large values of the deflection. For small values of*71 we obtain from (9.17),using the Maclaurin's series for Co5h 7/ and equation (9.24), the approximate relation = 32 E, w. 2.60 Z EL 3 P=5 X, E, -h + r ,F2 EL (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 o/h is small the value of this term for small values of wo/h (or-/of 17 ) as it is f ound in (9.27) is to be need in setting up an empirical formula, especially since this term becomes -76- 8 Eo. wo_.3 3 A, F"-E,.(h ' of di::inishing relative importance with increasing / Similar reasoning leads to the use of the second term as found in (9.26) corresponding to large values of V'V h (or of 7/ ). Hence we write P A 0+ B (9.28) where A -6.4 E, (9.29) AE, B = 206 Eo.a (.so) E, It will be noted that the first term of (9.28) expresses the result obtained from the usual theor- of thin plates when the deflections are assumed to be small. An empirical formula for the maxiuTIm bending- stress in the face plies can be set up with the aid of (9.20) and (9.22). These equations lead to S = CL (9.1) E~ Wh A\ CL h where 4 eii, (9. Dla) and Ex denotes the value of E in a face ply in a direction ner- pendicular to the edge of the plate. It is found that OC ranges from the value 4.8 for small 7/ (or wo/h ) to 4.0 for large By using the intermediate value OL = .4, the bending stresses associated with small deflections will be .nderestim-ated while those associated with large deflections will be overestimated. The re- sulting percentage error in total stress, that is, direct stress plus maxi:-mun bending stress, will usually be szmll. Accordingly we write s = 4.4 2 (9.32) -77- A more nearly exact value of the factor OL which is taken to be 4.4 in f ormula (9.32) can be obtained f rom the curve of f igure 27 of section 9 with the aid of (9.24). The latter procedure is re comme nded. 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) and A by ]-G where a-" is the Poisson' a rat io. Formulas will now be given for calculating the direct stress in each ply in a given plate from the mean direct stress as found by (9.18) or (9.25). It is to be recalled that it has been assumed that the plies are of equal thickness. Three-Pl~y Plate Let 91 = direct stress in a face ply 9z = direct stress in a center ply S= mean direct stress Ex = the value of E in a face ply in the direction parallel to the X-axis Ey = the value of E in a face ply in the direc- tion parallel to the Y-axis r = FEy/ E) The ratio r will be E-T/E, for plates of type 3X and E,/ET for plates of type SY. Since el is the same for all plies and all other strain components are zero, it follows readily from (8.2), (9.2), and (9.3) that 9, -78- Now 2g=+gz 3 = Hence g,= 2 +r 9a= 2+r 9 (e.33) (9.33a) Five-Ply Plate Let 9, = direct stress in the face and center plies = direct stress in the intermediate o the values of E in a face ply and L in plates of type 5X and r = E L/ 91 3g, + 2g2 5 92 EX and Ey refer t Hence, r = ET/E of type 5Y. Then plies r = Ey/Ex. Er in plates Hence 5 9:J= 3 +2r 5 r gp3 + 2r (9.34) (9.s5) Formulas will now be obtained for finding the maximum bend- ing stress in a given ply in a given plate from the maximum bending stress in a face ply as obtained from (9.20) or (9.31). -79- Three-Ply Plate Let S = maximum bending stress in a face ply Sg = maximn bending stress in the center ply. Exand Ey refer to the values of E in the face plies as before. In accordance with (9.19) EX d'w dx __h d'w In the last equation Ey is the value of E in a direction parallel to the Y-axis in a face ply and therefore the value of E in a direction parallel to the X-axis in the center ply. It follows that s (9.36) Pive-Ply Plate Let S maximum bending stress in a face ply S? = maximum bending stress in an adjacent ply. Exand Ey refer to the values of E in the face plies. In accordance with (9.19) E,,h daw -2 dx2 3E h dpw s 10 dx a -80- Then S 3 5 Ex Tables 43 to 47, inclusive, contain a comparison of the results obtained by using the approximate formulas for load, maximum bending stress and direct stress with the results of calculations based on the exact formulas. In these tables q denotes the uniform load in pounds per square foot, 9 a direct stress, and S a maximum, bending stress. In calculating S the factor Cy, in equation (9.31) was taken to be 4.4. More accurate values of would have been obtained by taking Cv, from the curve of figure 27. The letters WO and h denote, respectively, the central deflection and the thickness of the plate. The subscript CL denotes a result calculated by an approx- imate formula. If the subscript CL does not k-ppear, the result was obtained by the exact method. The numerical subscripts 1 and 2 refer to the plies in which the stress is calculated, the subscript 1 re- ferring to a face ply and 2 to a ply just under a face ply. The stresses are calculated in the ply for which the total stress, direct stress, plus maximum bending stress is greatest. This does not necessarily imply that failure occurs in this ply instead of in other plies. In the calculations for the plywood plates the elastic constants of spruce were used. For the steel plate E = 3 x 107 pounds per square inch and 0- = 0.3. The width of all plates is '48 inches. The thickness of the three-ply plates and of the steel plate is 3/8 inch, that of the five-ply plates is 5/8 inch. _81- Appendix 10.--Infinite Striao (Long Narrow Plate). Large Deflec- tions. Uniformly Distributed Load. Edges Clamped In this case the constants of euation (9.8) are to be deter- mined to satisfy the conditions W = O -L- 0 when x = 0 and when X = CL. It is again assumed that the edges of the plate are restrained from moving inward. sinh k_ It is found that cosh Ka - coshk (X- px (a-x) pct 2 kzD, 2ksD, (10.1) The relation connecting P and g is found as in the case of a strip with simply supported edges by equating the two expressions in (9.10) and (9.11) for the integral of e1xx over the width of the strip. Using (10.1) the following relation corresponding to (9.12) is obtained E p Z 2k 7O,2 (10.2) -3 Using (9.7) and (2.14) this equation can be solved P as defined by (9.13), in terms of the quantity It is found that h =3rEL whe re for the quantity = k/2 . (10.4) (10.4) K(7)= (12-9 coth9-3 zcoth' y + 5p From (10.1) the deflection on the central line, X = a/2 is found to be given by 3A PEL O 3 PE, h ( t o,5) -82- 2?ka+ _- coth ka 3 2 where From (9.7) it is found as in the case of the plate with simply supported edges that the mean direct stress is expressed in terms of 7/ by the equation E, h?- g 3= _7ai 7 (10.7) The bending stress on the face Z = -h/2 is calculated by the use of (9.19) and found to be 3 PEL Ex h?-F I I cosh k(X-)_- This stress is a maximumn at the edges X = 0 and X = CL. De- noting this maximum bending stress in a face ply by S we find thtt 3PELEX h2 N(08 ?_E NCOO.. where N )'= cot h / I//7Z (10.9) The calculations of the central deflection, mean direct stress and maximum= bending stress in a face ply are to be carried out essentially as in the case of a plate with simply supported edges. The value of the parameter *7 associated with a given load is to be found-.from (10.3). 'Values of the functions K(7/),) M (7), an d N are given in table 42. AD'oroximate For~las.--.As in the case of a plate with simply supported edges it is possible to replace the exact formulas just obtained for a uniformly loaded plate with clamped edges by very accurate approximate formulas connecting the load and the stresses with the deflection at the center. -83- k5The behavior of the function H )is shown by the following tbeo values: From equations (10.3) and (10.5) ,%(E, Yk h =c) where H(/ K K(7 /) M~(7 The function H(79) is approximately constant35. It can be replaced by the number 0.366 with an error of less than 2.5 percent for the range of values of 7/ in which we are interested. Hence the following linear relation between wo//h and 17 holds approximately: 0o.10) 0o.11) 0o.2) It follows from (10.10) and (10.7) that From (10.5) we obtain for small values of *77, using the Maclaurin's series for i'anh _ - h= 3REt (- 0 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 --?/Z/[ in (10. 12) is replaced by I -7Y0.4 Solving the resulting equation for P we have approximately p = 3 2 E, ",% 4 E, z , A EL. h + 1-3A,. L "h- 0 0.3698 12 0.3610 2 0.3696 14 0.3599 4 0.3682 16 0.3590 6 0.3662 18 0.3586 8 0.3638 20 0.3580 10 0.3622 11 HM 71 H (ij) -84- On substituting' for 7/ its expression in terms of W~h from (10. 10) we obtain P A Avh + B (10.1) where A= 32 E, (10.14) B 23 Ea (10.15) With the aid of (10.5) and (10.8) an approximate formula7- for the mximmm bending stress in the face plies can be set up. From these equations we obtain where ot. = 4 N (q/M q (10.17) For small values of 7/ we find by expanding N( and M(7/) in Maclaurin's series that,/, aL = 16 (1 + (10.18) On comparing the values of CX given by (10.18) corresponding to the range of values of 71 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 V4o. R6An alternative procedure is explained immediately after equation (10.22) -85- Than c = 1rb(1I+ 0.025 / 2) (10.19) The expression for $ becomes S= X (16 +0.4 (-a) h On expressing ?I in terms of Wo/hby means of (10.10) we obtain where D "2.98 ExEa (h7)z 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 M as a function of ?I. This curve which is the graph of (10.17) is given in figure 28. In using this curve the value of 7 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 a. 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 OL from the curve of figure 28 as explained above. -86- AppendiA ll.--Rectangular Plate (Large Deflections). Uniformly Distributed Load. Edges Simpl-y Supported. Approximate Method An approximate formula for the deflection W0 at the center of the plate will be obtained by assuming the following expressions37 for the deflection\VV and the components L and V parallel to the X-and Y-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 G'< Y < C (See figure 4), S7fX *7r w= wo sin a sinic 27x ,i- 113 (11-1) u = C, Sin Sin - CL 2 c v = c sin 7r sin ry Q C C When 7( X w= wo sin7- a (11.2) U= c, sin a v= 0 Expressions corresponding to (11.1) are assumed for the region b-c < Y< b Further let c = a/ (11.3) 3EBpressions corresponding to (11.1) were used by A. and L. Fppl - 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 Vb and "V'd 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 e and M respectively. Thus -b ="Ve + V.4n 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 byxt The parameters W~o, q C2. and r (or C ) occurring in (11.1) and (11.2) are to be determined in such a way that the total potential energy, V =-b + d +V ) of the system will be a minimum . The expressions for Vt and Vb 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 h [y(., e/ I ey -+ -1'e eX x + +YY X yx h [ F'. el 7 + 2 EL,, e' ey' + Eby +Y'zeXY where XX Y and Xlare the mean direct stress components, the average being taken over the thickness of the plate. In the re- duction to the second form the relations (8.3) and (8.4) have been used. Using (8.1) the strain components are calculated for the assumed displacement and deflection as given ty (11.1) and (11.2) -as- and substituted in the expression (11.5) for the strain energy per unit area, which is then integrated over the appropriate portions of the plate with the following results: hEL Vde =2 [A,we + B,Clo t C, c ,w 2 (11. 6) )+U ,- + rCe t G, c, czJ hEV[ c D? V,= A [Aw, + Bc,w,oz + Dec.- (11.7) who re (r 9Yb )~ a+ 2 a. + 4 A -gs a +l = 256 a2 r A, B, C, D, F, (11.8) (11.9) (11.10) (11.11) (11.12) =3a 3a 7 3a f2~cb 7'' + AV OTL = Fr a 7 = 7fz- + (7r + r 4 )7 = 9 (AV + OT GI (11.13) -89- 3 A = 32cs (k ) (11.14) Bz 3i2 d B ( -9) (11.17,) D, = 27rz lk--) (11.1) and S/EEE,, Ez k = b L c (11.17) ; .E !EL ,Rb: =Eb/ L ;/aT ,k -/# (i. and :, z, 6 and A have the same meanings as in (3.29). The parameters W, C, and 7 are to be so determined that V = Vt + V. + V, + .+% (11.V +) shall be a minimum. The determination of 7 by this method proves to be too difficult. Instead the value of 7 will be taken to be the same as that given by (3.34) for a plate with small deflections. ThiR pro- cedure is discussed in section 11. Using this value of 7", v be- comes a function of WO, C, and C. Since V is to be a minimum as a function of AWo, C,, and Ct its first partial derivatives with respect to these quantities must vanish. From the equations (r =0 and =0- the C) d C 2. quantities C, and C, can be expressed in terms of VV,. When these values of C, and Cz are entered in the equation, c =O,) the following relation between V and p is obtained: P= H + (19) h Q ho (11.19) -90- 7 + g (k 7) F-- BCG +C aD 4 DF G ? rr + k-7 7iT where (11.20) + 5 T + c,(2k-r) (11.21) H = S192 A 9r2 Q 2A (11.22) A = (A + A,) a', C= C,a, D= 0, + D B = (8, + B,) a, )Z, F = F, G = G, (11.23) When k = 00, 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= K si,/48 A Q =7 Ks o 6 N (11.24) (11.25) -91- pa P- 4 E, h Apendix 12.-Rectangular Plate (Large Deflections). Uniformly Distributed Load. Edges Clamped. Approximate Method (8 k 5 r) T Using the assumed forms (12.1) and (12.2) of section the deflection and the displacement we obtain by the procedure section 11 and appendix 11 the following approximate formula P = H-h + 3 12 for of (12.3) whe re C +a cr + e, 2k -7 H= 12A (12,.4) 8 [ BF BCG + C D] S-) A 4DF G2 where 7 is found from (4.30) appendix 4, k = b/ and A= 40 057T + 105 + 25 4096 (1 7 (12.5) (12.6) B -r 2, 4 , B 15 (6 ,7 2 rv 7 3R, (k-r) (12.7) (12,8) C=T S= 15 6 2Av gbC Ir z - 4 cT + D= 4 a 7 F= 7r '4b + 4 + 87 ro (k 7) S4 ) + Ar~u' 4 (12.9) (12.10) -92- S 12 (oT + (12.11) The numbers ;&t, 'b etc., are defined in appendix 11. When k = b/a= oo it is readily found that (12.4) and (12.5) become: H = r,4,/3A (12.12) Q = 4 /4 (12.13) -93- Avondix 13,--General lotatio Choice of Axes 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 bending 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 (1,O b -- lengths of sides of rectangular plate as shown in figure 2. a1 g- b 0D2 -- Coefficients of flexural rigiditydeid by the equations El h3 Eh3 D, = 12 D* = 12 X, exx e. Cy- omponents of strain. (Love, A. E. H., The Mathematical Theory of Elasticity# Art. 8.) Ex) Eyf, E z Young's moduli at a given point asso- ciated with tension or compression parallel to the X-, Y-, or Z-axes, respectively. E Mean Young' e modulus in bending under a couple whose axis is perpendicular to the XZ-plans. See equation (2.18) of appendix 2 and page 7. Z Mean 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- .CL Mean Young's modulus in stretching in a direction parallel to the X-axis. See equation (8.4) of appendix 8.. EFb --Mean Young's modulus in stretching in a direction parallel to the Y-axis. See equation (8.4) of appendix 8. g -Mean direct stress X1 in sections 9 and 10. h --Thickness of plate. K= b -- Ratio of length to breadth of rectangular plate. -- In sections 9 and 10. K = E + 2#L h'F K 7) -- a functional symbol in section 10. L) RT -Subscripots denoting directions parallel to the longitudinal, radial, and tangential directions, respectively, in wood; 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-]- the direction parallel to the annual rings and perpendicular to the grain of the wood. See figure 3. M X MY -Bending moment per unit length of a vertical section of the plate perpendicular to the X-axis. --Beniding moment per unit length of a vertical section of the late perpendicular to the Y-axis. rY 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. -- In case of uniformly distributed load. (In sections 3, 4, 9, 10, 11, and 12 and in corresponding appendixes.) -95- h\D, =pOL4 p = EI-hA C-Ir .a --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 Ma kximumn bending stress in face plies. T --Subscript denoting tangential direction in wood. LL --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. W0 Deflection at the center of a plate. VWooO Deflection on the center line of an infinitely long plate. XXx y -- Components of stress. (Love, A. E. H., The M athematical Theory of Elasticity, Arts. 41, 47.) k a =-- in sections 9 and 10. -96- K t() -- A functional symbol in section 9. E, E2 a-EL Eb Cb= EL ;k= 1-LT aCTL for flat-grain plies /A CL > n an integer /xy -- Modulus of rigidity associated with the shearing strain exy in a plane parallel to the XY- plane. There are two other moduli of rigidity, / yz and /Z ayX _ILT " -- Poisson's ratio in isotropic material. Not related to O" defined in preceding line. -97- C y -- Poison's ratio associated with a tensile stress parallel to the X-axis and contraction parallel to the T-axis. There are five other Poisson's ratios similarly defined: 0-y X,)CX Cxz X x C7.yz j a nd 0,-, _ E cr ELO-TL E, EL r = Ez for flat-grain plies. for flat-grain plies. In all sections except section 6 this symbol is defined by (3.18). In section 6 it is defined by (6.7). -98- Z 40186 F |