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THERMAL BENDING OF SANDWICH PANELS UNDER UNIAXIAL LOADING By EDWARD LINDE BERNSTEIN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1964 ACKNOWLEDGMENTS The author wishes to express his appreciation to the members of his supervisory committee: to Dr. I. K. Ebcioglu, chairman, for the advice and encouragement which he tendered throughout the course of this investigation; to Dr. W. A. Nash, whose many kindnesses have been of great assistance; and to Drs. A. Jahanshahi, J. Siekmann, and R. W. Blake for their helpful discussions with the author. The author wishes words were adequate to express his appreciation to the only member of his special committee his wife, Eileen. She has served as secretary, editor, ad visor, and most importantly, inspiration to success. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ...................................... ii LIST OF TABLES .. ... .......................................... v LIST OF FIGURES .................... .. ............... vi LIST OF SYMBOLS ...................................... vii ABSTRACT ............ .... ........................... xii Chapter I. INTRODUCTION ................................. 1 Historical Background ....................... 1 Definition of the Problem ................... 5 II. DERIVATION OF THE EQUATIONS ................... 9 The Strain Energy Expression ............... 9 The Equilibrium Equations ................... 18 The Boundary Conditions ..................... 24 III. SOLUTION OF THE EQUATIONS ..................... 32 Transformation into an Uncoupled System ..... 32 Solution of the Membrane Force Equations .... 39 Solution of the Second Set of Equations ..... 41 Convergence of Series Solutions ............ 51 Development and Solution of the Ordinary Differential Equations .................... 55 Solution for a Simplified Case .............. 68 IV. COMPARISON WITH EXPERIMENTAL RESULTS .......... 85 iii TABLE OF CONTENTSContinued Chapter Page V. COMPUTATION AND DISCUSSION OF THE SOLUTION .... 92 Computation of the Solution for Various Parameters of the Panel ...................... 92 Discussion of the Results ...................... 96 VI. SUMMARY ....................................... 108 APPENDIX ............................................. 109 LIST OF REFERENCES ................................... ill BIOGRAPHICAL SKETCH .................................. 115 LIST OF TABLES Table Page 1. Experimental Data ............................. 87 2. Theoretical Results .......................... 90 3. Comparison of Buckling Load Values ............ 104 4. Experimental Temperature Gradient ............ 110 LIST OF FIGURES Figure Page 1. Dimensions of the rectangular sandwich panel .. 6 2. Variation of displacement u through the thickness of the sandwich panel ............. 34 3. Resolution of membrane forces into a force and couple system at the "neutral" surface ...... 34 4. Comparison of theoretical and experimental values of the deflection at the center of the plate ................................ 89 5. Deflection at the center of a plate under a temperature gradient R = R. sin 1T sin ITI as a function of edge loading ............... 97 6. Deflection at the center of a plate under a constant temperature gradient as a function of edge loading .................... 98 7. Deflection at the center of a plate under a temperature gradient R = R, sin 7r sin 7rq as a function of edge loading .............. 99 8. Deflection at the center of a plate under a constant temperature gradient as a function of edge loading ................... 100 9. Comparison of the deflection at the center of a plate under a temperature gradient R = Ro sin Tr sin 7T7 as a function of edge loading ......................... ......... 105 10. Dimensions and data points of the sandwich panel ....................................... 109 LIST OF SYMBOLS a  at b  bij Bf Cjii Ci  F ej CIj , D  D  Dji e., E  fo, Fi F  go, vii Length of panel in xdirectlon Elements of the determinant 6j Length of panel in ydirection Quantity defined in equations (3.34) B* Be B1j Fourier coefficients defined in equa tions (3.27) Quantity defined in equations (3.34) Function defined in equations (3.22) Cj Fourier coefficients defined in equations (3.27) Differential operator d/dg Plate modulus Determinant formed from 6j (see p. 63) e,y,ey Linear components of strain Young's modulus f/ Auxiliary functions S Function defined in equations (3.23) Fqj Fourier coefficients defined in equations (3.27) ax/ y g Auxiliary functions Gx, Gy Orthotropic shear moduli of core h, h" Distances to "neutral" surface ho, h, Auxiliary functions Hij Fourier coefficient Hx, Hy, Hy Couples formed from resolution of membrane forces H7 Thermal couple formed from resolution of thermal membrane forces J Function defined in equations (3.20) ke Dimensionless edge loading parameter oNy/Pe Ki Fourier coefficient m E' t/E"t" M,, May, My Bending moments in faces MT Thermal bending moment in faces M,* My, My*' Bending moments applied on edges of faces Ng, Nxy, Ny Additional membrane forces relative to state of initial stress N7 Thermal membrane force Nx,, Nxy, Ny Resultants of additional membrane forces NT Resultant of thermal membrane forces oNx, oNsy, oNy Initial membrane forces oNx, oN, y* Ny4 Applied initial edge loads .N<, oNy oNy Resultants of initial membrane forces viii 7Nx, Ney, 7NT Total membrane forces, defined in equations (2.18) Buckling load of a simply supported, infinitely long sandwich strip with a rigid core Sum of buckling loads of each face of a simply supported, infinitely long sandwich strip with a rigid core Distributed transverse load acting on faces Shear force resultants in core Vertical shear forces applied on edges of faces Dimensionless parameter Dimensionless temperature gradient parameter de fined in equation (3.13) (t + t')  (i + t") 2 q  Qx re  R  S'  S  Si  Si. t', t , tx I Pf  Function defined in equations (3.21) Sy, Sy Components of initial stress t Face thicknesses Core thickness 1 t' t" 1 + w(t + t + t t 2 2 tay, ty Components of stress relative to initial state of stress Change in temperature from a constant initial temperature T, Coefficients of the temperature distribution function Oy QY*_ TI, Xc, Yc XP, YU u, v  u, v  u VU U, u, U0, Uo/ U/ , Displacements of middle surface of upper face relative to "neutral" surface, defined in equa tions (3.1) Displacements of "neutral" surface, defined in equations (3.1) x and ycomponents of displacement in core ', v', s" x and y components of displacement in the faces Functions defined in equations (3.19) SVI, Va Coefficients of displacement distribution functions in the core Virtual work of edge moments Mx,* Mxy, My Virtual work of edge forces oN* oNy* oNy* Virtual work of transverse loading q Virtual work of edge shear forces Q,* Qy* zcomponent of displacement Dimensionless deflection parameter Function defined in equations (3.19) Strain energy of core Strain energy of faces Z Solutions to ordinary differential equations; see equations (3.25) , Z, Complementary solutions , Zp Particular solutions VM  v  v,  va  w  V? W, Wr  X, Y CK  oe  Alj Cx 6xy C  A;  Coefficient of linear thermal expansion Dimensionless parameter 2t/b Aspect ratio of panel b/a Determinant y Complete components of strain z s y/b Quantity in cubic equation Coefficient of exponent in complementary solution; see equations (3.33) Xj / Poisson's ratio x/a 1+m p Dimensionless parameter 2 (x, Wy, wx Components of rotation Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMAL BENDING OF SANDWICH PANELS UNDER UNIAXIAL LOADING By Edward Linde Bernstein April, 1964 Chairman: Dr. I. K. Ebcioglu Major Department: Engineering Science and Mechanics The deflections of a rectangular sandwich panel sub ject to a thermal gradient, transverse load, and combined edge loading were calculated. Two opposite edges of the panel contained rigid inserts and were simply supported. The other two edges were simply supported, clamped, or free. Small deflection theory was assumed, as well as an ortho tropic weak core. The effects of bending in the faces were also included. The five coupled, linear partial differen tial equations governing the deflections were transformed into two sets of equations, one of which is the wellknown equation of plane stress. The remaining set was simplified to neglect all edge loading except a uniaxial force and was xii solved by the use of Levytype infinite series, which reduce the equations to ordinary differential equations with con stant coefficients. The convergence of these series was investigated. A standard solution of these equations was obtained, and the resulting expressions for the displace ments were written in closed form. Values obtained from the solutions were compared with experimental values reported in the literature, and agree ment was found to be good, although shortcomings of the ex perimental values are mentioned. The variation of the deflections and the buckling loads for panels with varying elastic properties, dimen sions, and edge loading were calculated on the University of Florida IBM 709 computer, and these data are presented in graphs. The values of the buckling loads were found to be somewhat less than those reported in a less exact analy sis. It was noted that, in general, a reversal of deflec tions took place and that buckling modes were characterized by a large number of halfwaves. xiii CHAPTER I INTRODUCTION Historical Background Legend has it that the term "sandwich" arose in 1792 when the Fourth Earl of Sandwich devised a convenient way to have his lunch without interfering with his daily card game. Since that time the term sandwich has come to mean more than merely a way to eat. The term applies to any object con structed like a sandwich, that is, composed of three layers, the middle one being of a different material than the outer layers. Sandwich construction is a type of composition that has recently begun to achieve wide application. The con struction is characterized by two thin outer layers known as the faces and a thicker inner layer known as the core. The faces are made of a stiff dense material and resist bending and membrane stresses, while the core is made of a light ma terial offering resistance mainly to transverse shear and compression. The original, and still most widespread, application of sandwich construction is in air and space vehicles; its high strengthtoweight ratio is well suited for this pur pose. One of the earliest applications was for the skin of the World War II DeHaviland Mosquito bomber, in which birch 1 wood faces and a solid balsa core were used. At the pres ent time the faces are frequently made of steel or aluminum with a core made of either an expanded cellular material or a corrugated metallic sheet. Sandwich construction is also found in curtain wall 2 panels for office buildings; with a foam plastic core it is used in refrigerator construction; it is also suitable as a hull material for boats when a balsa core is used. Because of this wide use of sandwich construction, especially in panel form, the elastic analysis of a sandwich plate is a timely and important problem. The nature of the core necessitates the consideration of several effects not present in the usual analyses of thin plates. The strength of corrugated cores will obviously depend on the direction of the corrugations. Cores made of expanded materials are constructed of thin strips of material which run in one di rection. The properties of either type of core are thus orthotropic. Because of the relative thickness of the core, the effect of shearing deflection is important. Since cores contain air spaces, and thus act as thermal insulators, sandwich construction used in an air vehicle will be subject to a temperature difference between the two faces. Thermal stresses arise from this temperature gradient. The earliest study of a sandwich plate was performed 3 by Gough, Elam, and DeBruyne; they considered the problem of a thin sheet supported by a continuous elastic foundation in order to study "wrinkling," or instability of the faces characterized by a large number of buckling waves. Later, 45 Williams, Leggett, and Hopkins and Leggett and Hopkins5 examined the Euler type of buckling of sandwich plates as a whole; they accounted for shearing deformations in the core by using the "tilting method," as termed by March,6 in which a line originally normal to the middle surface tilts during deformation through an arbitrary angle. Use of an approxi mate strain energy method determined the buckling load. March and others extended this method to include the effects of orthotropic cores and orthotropic faces of differing thicknesses and elastic properties.7'8'9 Libove and Batdorf developed a method of analysis in which the sandwich plate is treated as an orthotropic plate with the effect of shear ing deformation included.10 Reissner developed differential equations for the finite deflection of sandwich plates.11 Hoff used the assumption of a weak core which resists only vertical shear and the principle of virtual displace ments to derive exact equations for the bending of a sand wich plate.12 He was able to solve the problem for a simply supported plate by assuming that rigid inserts were set into the core edges. In this way the boundary conditions were altered to permit a solution. In 1951 Eringen used the principle of virtual work to derive extremely general equations which account for the ef 13 fects of compression and flexure of the core. In 1952 Gerard, by an order of magnitude analysis, examined the limits of validity of the simplifications commonly assumed 14 in sandwich plate analysis.14 Goodier and Hsu considered 15 nonsinusoidal buckling modes. In 1959 Chang and Ebcioglu extended Hoff's theory to include an orthotropic core and dissimilar faces and found the buckling load for a simply supported plate.16 Recently, Yu, in a series of papers, has included the effects of rotatory inertia and shear in the faces.17.18,19,20 Thermal stresses in sandwich plates were treated by 21 Bijlaard, who used a homogeneous plate approximation.21 Chang and Ebcioglu extended their approach to include the effect of a constant temperature gradient,22 and Ebcioglu effect of a constant temperature gradient, 2and Ebcioglu later solved the problem for a temperature gradient which varies arbitrarily across the area of the plate.23 Utiliz ing Hoff's approximation for the boundary conditions, he presented a solution for the bending of a plate simply sup ported on all edges. Ebcioglu et al. also give experimental results for the deflection due to a thermal gradient.24 Definition of the Problem It is the aim of the present analysis (1) to extend Ebcioglu's work in order to develop equations which include the effects of bending of the faces, a linear temperature distribution through the thickness, and various boundary conditions, (2) to solve the equations for the exact bound ary conditions rather than making Hoff's approximations, and (3) to present graphs illustrating the effects of various parameters of the plate on the deflection and buckling load. A Cartesian coordinate system is defined for a rec tangular sandwich plate as shown in Figure 1. The middle plane of the core lies in the xyplane, with two adjacent edges lying along the x and yaxes; the zaxis points down ward. In the notational system employed here a single prime denotes quantities related to the lower face, a double prime those related to the upper face, and a superposed bar those related to the core. 11. 1 4MJ r4 Q) 0 Ca 'f 0 P4 a) 4 u fo 4J $4 0 0 r0 I a) rz The plate has lateral dimensions a and b along the x and ydirections respectively, and the thicknesses of the upper face, core, and lower face are t", t, and t' respec tively. The faces may differ in Young's modulus E, but have the same Poisson ratio /I. The core is assumed to be homo geneous and orthotropic. The temperature change T of each face is defined as the change from some constant initial reference temperature and is arbitrarily distributed in the x and ydirections; the variation in the zdirection through the thickness of each face is given by the following equations: T' = TI/ + T/ (1.1) T" = T//V + T2" where T', TI," Tz', and T"/' are functions of x and y only; =z s (1.2) where s = s' = (1 + t') for the lower face 2 s = s"= l( + t") for the upper face. 2 Arbitrary transverse pressures q' and q" act normal to the outer surface of each face, while on the edges of each face there act applied axial forces oNO oNxy*, and oN", applied moments Mx* Mxy* and My! and applied vertical shear forces Q,* and Qy* where each of these quantities is defined per unit length of the boundary. The subscripts follow the usual convention for stress notation. These forces are uni formly distributed in the zdirection on each face and do not act on the core, but on any edge where external loading is applied a rigid insert is set into the core for its pro tection. A more complete description of the insert is given in the section on boundary conditions. The Kirchoff hypothesis is assumed to hold for the deflections of the faces. The temperature is considered to be in steadystate equilibrium and the effect of deformation on temperature is neglected. Account is taken of the fact that elastic properties vary with temperature by permitting each face to have a different value of E. The "weak core" assumption is made, that is, the core is considered to be so weak in its resistance to bending and axial stresses that only the antiplane stresses T. Ty, and Tz are present. The core is considered incompressible in the zdirection. It is assumed that neither local buckling of the faces nor bond failure between the faces and the core occurs. CHAPTER II DERIVATION OF THE EQUATIONS The Strain Energy Expression A macroscopic approach has been chosen for the deri vation of the differential equations governing the five in dependent components of displacement. It is felt that this approach, based on the theorem of virtual work, more clearly shows the assumptions involved in formulating a mathematical model which adheres to physical reality yet allows a solu tion; this approach also permits a more straightforward derivation of the boundary conditions than would the alter nate approach based on the microscopic equilibrium of an element of the plate. The theorem of virtual work states that if an elastic body is in equilibrium, the variation of the strain energy equals the virtual work of external forces acting on the body. The expression for strain energy for each layer of the sandwich plate is written and the variation taken by applying the variational calculus. 10 The derivation of the strain energy expressions for the faces proceeds from the consideration that they are ini tially in a state of equilibrium under the action of large membrane stresses. From this initial state the faces under go additional deformations. The approach used will be based on that of Timoshenko25 and Herrmann.26 The initial components of stress are denoted by Sx, Sy, and Siy. They are assumed not to vary through the thick ness of each face, and to be of an order of magnitude higher than the stresses produced during the additional deformation. The following initial stress resultants are defined: oN, = tS, ,Ny = tSy (2.1) ONxy = tSxy Since the plate is initially in equilibrium, the fol lowing equilibrium equations may be written for each face: c) ,Nx 6 oN, y + = 0 ax 6 y (2.2) C +oNxy l)oNy + = 0 b x 6y The boundary conditions for the initial state are oNy*= oNy (2.3a) ONxy = oNxy on y = 0, y = b, and oN%* = ONK (2.3b) oNxy* = oNXY on x = 0, x = a, where oNx*, oNY*, and oN are the initially applied loads on the edges of each face. The additional components of strain are assumed small compared to unity; it is further assumed that the elongation and shear terms are of the same order and the rotation terms are of a smaller order. However, since products of initial forces and rotation terms will occur, these latter terms will be retained. Then the components of strain relative to the initial state are x e + (W2 + W (2.4a) If Y ey + 4(CJ 2 + .zz) (2.4b) 6z ez + (y2 + ) (2.4c) z =ez + y2 12 Y = ey 1 x y (2.4d) 6yz = eyz a.y 4Jz (2.4e) 2 ,zx  ezx Wgz Wx (2.4f) 2 where e., ey, e, exy, eyz, and ex are the linear strain components and wx, wy, and w. are the rotation components. Based on the Kirchoff hypothesis, the deflections in the faces are given by u = u __ ox (2.5) v = v  ay where u and v represent the displacement components due to the stretching of the middle surface in each face from the initial equilibrium state, and 9 is given by equation (1.2). The rotations coz are considered to be negligible for plate deformations. Then the use of equations (2.5) in equations (2.4) produces the straindisplacement relations: = au + 1(_2w), (2.6a) SC)x 2 c)x x(2.6a) Vy 1( W)Z (2.6b) ) 13 1u +v + w w 2t w( 2x x ~y +25 ) (2.6c) 2 ay 6x ax by x(wy C ( + l() (2.6d) 2 Sx 2 b x = 0 (2o6e) S= 0 (2.6f) The additional components of stress tx, t)y, and ty are assumed to be related to the additional strains by the DuhamelNeuman equations. The error introduced by this assumption is of the order of the ratio of the initial stress to the elastic modulus of the face.26 Furthermore, the transverse normal stress Ojz will be neglected in ac cordance with classical plate theory. Therefore, the fol lowing equations for plane stress27 will apply: tX = E (x + 4EY EOT 1 (E ET ty = E (CY + 1x) EoT 1 / 1  1 = E Ex S1 +4 Y where o is the coefficient of linear thermal expansion. The following stress resultants are defined in the faces. The membrane forces per unit length are Nx= tid = J /. + /1v + (6w) 6 ^ Et [1 x 1 /Z X Ny = tyd/ = [ Et[b J V1 /" C) y + p b b x (t/I r tEt )u J 2(1 +,/)[(y + w2 2 (y + )2 2 ay + "V+ aw 6w] Sx ax 5yJ where the thermal force per unit length is =1 E x S EocT2t Td =/ 1 / The bending moments per unit length are th7 ttd D 2 M, = jtd = ID Lx2 Iel M = txD[ +W] 2WM ti/I xy2 My = j tyd = D(l +/x) , /V .  N7 + ,5,1  Nr (2.7) (2.8) + ,. by  where the thermal moment Mr is MTr = D(I +/ )T, and the plate modulus is D Et3 12(1 (1 ) The strain energy WF in the faces is given by the expression27 Wr =1 it 6 + ty~y + 2txy Ery 2(xT(t, + ty)]dV V + j (Sx, + Syty + 2Sxyxy)dV (2.9) v and the integration is extended over the volume of the body in the initial state. The stress resultants (2.1), (2.7), and (2.8), to gether with the strains (2.7) and the temperature distribu tion (1.1), are substituted into expression (2.9). After integration through the thickness, there results W = D (V w)2 2(1 /x) w ( w)2 2 1 6x+ Ty2 yxdy + 2(1 + /1)xT, V2wdxdy + S{(oNx + )[bu + l(w)2] ( +2 c) x 2+ 2x (.N), + INY) av+ 1(2kw 2 lapy 2 by (,Ny + !:Ny [av+ u bw BW 2 a ox by +ox ~yj OTa (N, + Ny)}dxdy (2.10) The first integral in (2.10) represents the strain energy of bending, while the second represents strain energy of the stretching of the middle surface. Since a "weak" core that is inextensible in the z direction is assumed, only the shear stresses Tzx and Tzy exist. If the rotation terms wz are neglected, the ortho tropic stressstraindisplacement relations are f = 2 x= G ( + ) )z )x (2.11) = = Gy( +w) where Gx and Gy are the orthotropic shear moduli. It follows from equations (2.11) and from the assump tion that iz = 0 that the displacements in the core are given by the relations U = U, z + U, V = V, z + V, (2.12) where the functions U (x,y), U,(x,y), V, (x,y), and V,(x,y) are found from the conditions of continuity of displacements at the interfaces between the core and the faces. In this way U,  U . S= u" + U + + Ut + U/ U2 2 + 2 V/ V V V + Vp +Vr V2 = ++ 2 The core. The t" + t" 4w 2t bx t" t" w 4 ax t + t" w 2t by t t" Iw 4 ay following stress resultants are defined in the shear forces per unit length are  F u" A f !Fix dz = tGx( + t i) th bx Q,= Tdz = tG(V :' v" + ) ;12 3' v,,I (2.13) where A t t") t = (t + + ) (2.14) t 2 2 The strain energy W, of the core is of the form WC = (Tx^ The expressions (2.12) are substituted into the orthotropic stressstraindisplacement relations (2.11) and then into the strain energy term (2.15). After integration through the core thickness, there results WC = A I 6 aw We = J G[x(Ut + Tx + Gy (V2 + )y dxdy (2.16) Do " The Equilibrium Equations According to the principle of virtual work, SW/,' + SW," + SWo SVt SVN SV" SV' SV" SVY' SV"' 0 (2.17) where the virtual work due to the applied loadings is as follows: (1) from the transverse pressure JVq = (q" "0 "0  q')Swdxdy (2) from the edge forces on each face rbra SVN = (,N1*Su + Nxy, Sv)dy + (0Ny*Sv + 0Nxy*Su)idx (3) from the edge moments on each face *b r a j ax y v = M," 8(y)dy My"s J'o yo 3M"y* 10~ ks + f sy Swdy + 2 Swdx 2M, 0 0 (4) from the shear forces on each face SVO = QJo,*Swdy + QY'Swdy ^o 0o In forming the variations use is made of the formulas rxa d y x a j F dxdy = JF(xy) dy S~O x J ob Jo 3C 2W Iy dxdy = *a y = F(xy) dx 0 yo and of integration by parts. Equation (2.17) becomes f[(D' + D")V7w + V(Mi + Mr) (rN/' + N') aw *' ,T / W (W +T "" 62w q/ W 1 2( N + TNtY ),xy (TNy '+ TNy)" + q' q my ~ ~~~ axy()Ty1+V  ([ (u au" +) + aILW 'by + L x dx + x y oy fw ix., +_ y y + bxxy a (/ + a ) Su' ax ay t ax J  + Ty + G ( + 5 uw a INI' (.v' + t X')V ) +x C) t C) Sj c rTN _V/ _" ^ C) w It Ix y t y J by t ~ yl6vId + x ( + M,+)8( ) + (x + 2 r + ,N + fo [ lCa + t~x $ )u+ (  + (7N. oNXy*)Sv]dy + tt 8w by2 'Jl dy N )wy rN .,Y y I0 "0 (My + My)S (w) + my aM2 Y ( :'y+ 2 "'+rNyT + TN' I_.w 7y dy Cx y 'f x + J Y N, *)+ + ty a Qy')6w + (rNy y)u + (rNy oNy')8v dx + 2(M.y M.y') Sw = 0 (2.18) 0 o In expression (2.18) the total membrane forces are defined by 7N. = oN. + N, rNy = oNy + Ny (2.19) 7rN> = oNjy + Ny The integrals extend over the body in the initial state, and virtual displacements are taken as relative to that state. Since the virtual displacements Su', Su", Sv', Sv", and Sw are arbitrary within the plate, their coefficients in equation (2.18) must vanish. In this manner the follow ing equations of equilibrium result. In the first of these equations the terms (N' + N ,")*w (N,'+ Ny)T and (Nxy' + Ny,) are neglected. The terms in parentheses are membrane stresses due to bending of the plate after initial stress and are negligible. (D' + D") Vw (oNx' + Nc") a 2 2+ N ). W ,, 2,W 2 o(Mi + N[b~ awzM + (" a ul au" ^Pw) (,N + N+ )M + M") t G ( + tt . + y I V ax Sx. ax + y( y + tt y)1 + q' q"= 0 (2.20a) by C)y t y J aNx' 3 U' U" A 'my + NW' U + t = 0 (2.20b) +x ay =x 3N," N,," bN b./ / 0' Nx" +_ u" _w + t ) 0 (2.20c) ax xy ( x c)N y' +N T(v' v" ( 0w) X T )y G t oy D N. t ~" VI a +NX+ N/ (v" V" + t T) = 0 (2.20e) axy These five equations in five unknowns reduce to those de 23 rived by Ebcioglu23 for the case of constant temperature variation through the plate thickness and for small deflections. Equations (2.20) can be seen to contain the equations of homogeneous thin plate theory. If the core is assumed to be rigid, i.e., if Gz = Gy, then it follows from equa tions (2.11) that xz. = yz = 0 and there is no contribution from the core to the total strain energy. The coefficients of G, and Gy in equations (2.20) disappear and, except for the thermal terms, the resulting equations are the classical ones of SaintVenant, describing two plates constrained to have a common vertical deflection and subjected to lateral and edge loads. These plates correspond in their dimensions and properties to the faces of the sandwich panel. SaintVenant's equations also result if the thickness t of the core is allowed to go to zero, since in this case the integral (2.15) then degenerates to zero. In either case the effect of the core is only to constrain the two faces to have a common vertical deflection. The core has no stressresisting properties of its own, so the fact that it retains its thickness for the first case is of no importance. It is noted that a formal application of these limit ing processes to equations (2.20) is meaningless and pro duces incorrect results, since considerations preceding the derivation of this equation are involved. The Boundary Conditions The boundary conditions arise from consideration of the line integrals occurring in the variational equation (2.18).o These line integrals may be made to vanish by set ting either the variations or their coefficients equal to zero. The boundary conditions are linearized in the same manner as equation (2.20a). After use of equations (2.3), they are found to be: (1) on the boundaries y = 0 and y = b either u' =0 (2.21a) or Nxy = 0 (2.21b) either u" = 0 (2.22a) or N /y =0 (2.22b) either v' =0 (2.23a) or N = 0 (2.23b) either v' = 0 (2.24a) N = 0 either w= 0 (M + My) + 2 y(M.' + M."') + (.N + Ny)aw 3y y yx o y + ^ay = Q;+'+ Q+ )(M.;' + M.;) either 3 = 0 c)y Mi + M" = M*+ MY/f (2) on the boundaries x = 0 and x = a either u'. =0 or Nx'= 0 either UH = 0 or N= 0 either V/ = 0 (2.24b) (2.25a) (2.25b) (2.26a) (2.26b) (2.27a) (2.27b) (2.28a) (2.28b) (2.29a) Nx = 0 (2.29b) either V I = 0 N" =0 (2.30a) (2.30b) either w= 0 (2.31a) b(M/+, M+') + 2 (M, + M,;') + (,N,'+ N+ ) + tQ, = Q,*+ Q'I"+ (M4' + M,') either _w= 0 )x or Mx, + MI,= M:,+ M.*" (3) for the corners (0,0), (a,0), (0,b), and (a,b) either w= 0 (2.31b) (2.32a) (2.32b) (2.33a) I Iu a Mv C = Mxy 1(2.33b) 0 o 0 o0 The boundary conditions for a rectangular sandwich panel simply supported along the edges perpendicular to the yaxis and variously supported along the other two edges, containing rigid inserts set into the edges of the core that are perpendicular to the yaxis, may be selected from the preceding sets of boundary conditions. A simply supported edge may be supported either on knifeedges or by pins of negligible crosssectional dimen sions, about which the panel may freely rotate. If the latter case is considered for a sandwich panel, the pins may be imagined as running through both faces, parallel and arbitrarily close to the edges. They restrain the boundary from displacement perpendicular to itself, but permit paral lel motion. The restraint in the perpendicular direction will cause a normal stress at the boundary. If, on the other hand, the knifeedge type of simple support is envisioned, the edge of each face is considered to be in contact with a wedgeshaped support along the line forming the vertex of the wedge. In this case the edge is free to translate in directions both parallel and perpen dicular to itself. Finally, the effect of a thin rigid insert is con sidered. The insert is envisioned as being set into the core so that it is flush with the edges of the upper and lower faces and welded to these faces at its top and bottom edges. Its effect is to prevent the edges from having dis placements in the direction parallel to themselves. If the insert is perfectly insulated, so that thermal effects cause no expansion of it, boundary conditions (2.21a) and (2.22a) or (2.27a) and (2.28a) will apply. If, however, thermal expansion of the insert is allowed, the boundary conditions must allow for some displacement of the edges parallel to themselves. In general, vertical displacements of the edges would also occur, but it is assumed that the supports pre vent such displacements. The extreme case occurs when the insert offers no resistance to the free thermal expansion of the edges. Then boundary conditions (2.29a) and (2.30a) are replaced by v' = oc Tzdy (2.34a) v" = T"dy (2.34b) J 2 and conditions (2.21a) and (2.22a) are replaced by u' = O'j T,'dx (2.35a) %f/ u" = O" Tj"dx (2.35b) 0/2 In general, the displacement along the edge will be some where between zero and these values, depending on the co efficient of thermal expansion of the material of the insert. The vertical deflections at the center of the plate were calculated for both zero deflection and the extreme values in equations (2.35); only a slight difference resulted. The problem to be examined here is a plate containing a rigid insert along the two edges parallel to the xaxis, which are simply supported on knifeedges. The two edges parallel to the yaxis may be simply supported by knife edges, clamped, or free. Furthermore, the only external edge loading applied to the face edges is the initial load oNy! which is assumed to be constant along the edge. No transverse loading acts on the lower face (q' = 0). The boundary conditions to be used are as follows. (1) On the boundaries y = 0 and y = b u' = Tdx (2.36a) u" = T.'dx (2.36b) Ja/2 NY = 0 N/'= 0 w= 0 (2.36c) (2.36d) (2.36e) (2.36f) My'+ M/'"= 0 where the righthand sides of equations (2.35) should be set equal to zero if the inserts do not expand when heated. (2) On the edges x = 0 and x = a, for the case of simply supported edges, N,' = 0 N/y = 0 (2.37a) (2.37b) (2.37c) (2.37d) (2.37e) (2.37f) N,Y" = 0 w= 0 M/ + M."= 0 For the case of clamped edges U, = 0 (2.38a) (2.38b) U 1 = 0 Ny' = 0 (2.38c) Nxy"= 0 (2.38d) w = 0 (2.38e) )w = 0 (2.38f) ax For the case of free edges N,'= 0 (2.39a) Nx"= 0 (2.39b) Nxy = 0 (2.39c) Ny = 0 (2.39d) (ONx" + ON)Tx + x(Mx' + Mx') + 2 (Mx/ + Mx') + 6, = 0 (2.39e) M y+ My"= 0 (2.39f) CHAPTER III SOLUTION OF THE EQUATIONS Transformation into an Uncoupled System The solution of equations (2.20) is difficult because they are coupled in the four dependent variables u', u", v', and v". By a transformation of variables two sets of equa tions more amenable to solution may be developed. One set resulting from this transformation is composed of two equa tions in two variables. These equations are then the equa tions of plane elasticity including the effects of a body force. The second set of equations may be solved by a separation of variables. To effect this transformation, new displacement com ponents are defined as follows: u= m (u' u") l+m v = m (v" v") l+m (3.1) u= 1(U" + mu') l+m 1 + m =  (v"+ mv') l+m where E't' E" t" It is possible to give a physical interpretation of the quantities in (3.1). The variation of u and the varia tion of v through the plate thickness are given by straight lines with discontinuities of the slope. Figure 2 illus trates this variation for u. If, however, it is imagined that these variations are given by lines of constant slope, then for pure bending a neutral surface exists, which is defined by the quantities h" = 1 + m where h and h" represent the distance from the neutral h i A where h/ and h"' represent the distance from the neutral surface to the middle planes of the lower and upper faces respectively. Then u and v represent the displacements of the neutral surface, that is, since the neutral surface was defined for pure bending, u and ; represent the displace ments due purely to membrane effects. Furthermore, u and v represent the displacements of the middle surface of the upper face relative to the neutral surface of the plate. 7 t t' Figure 2. Variation of displacement thickness of the sandwich u through the panel  N." N % H N,  ; ^ J H ______ N.' Figure 3. Resolution of membrane forces into a force and couple system at the "neutral" surface F U" +t a u 2 ax u, + w 2 Sx t' w U 2 x h" h'/2 t'/2 4 Substitution of the transformations (3.1) into the equations for membrane force resultants (2.7) yields the corresponding transformations for these quantities: ONx = oNx'+ oNx" Ny = oNy' + oN/' I / If oNly = oNy + oN^y N, = Nx'+ N," Ny = Ny + Ny" Nxy = Nxy' + N~y" NT = N/'+ Nr" A H = t (N.' mN') l+ m Hy = (Ny' mN"') A Hy = tt (N,/ mN'y) Hy=1+ m Y y 1 + m 7' where t has been defined in equation (2.14). The quantities NK and Hx may be interpreted as the statically equivalent resolution of the membrane forces Nx' and N",. which act in the middle planes of the faces, into a force and couple system acting at the neutral surface (see Fig. 3). The interpretation of the other quantities follows in a similar manner. Substitution of the transformations (3.1) into the equations of equilibrium (2.20) yields (D' + D")V'w RN 2Ny ,N 2M t[Gx(1+ mm u + W w) + G +( m + )w) [ m Tx o )x m Ty t y2 (3.2a) 2m )t 2 + (1 (1 +1 2m(1 1.4 [2 ox2 6y 2 axdyj x+w 1 ) Hr 5,(+m u + t 2 mtx tt x E/t/ ) +2 c +(1 c)) 2m(l /"A) 2 +y2 ( x2 1 3 + y v1 + m) = 1 H mt y tt y bx^ My = aI~*y a~ ax +^ay= 0 (3.2b) (3.2c) (3.3a) (3.3b) Similarly, the boundary conditions become as follows. (1) On the boundaries y = 0 and y = b Ny = 0 Nxy= 0 U = m (oc'T' o"T/)dx 1 + m 2 HW= 0 w = 0 My/+ My"= 0 (2) On the boundaries x = 0 and x = a, for the case of sim ply supported edges, NR = 0 N"y = 0 Hx = 0 HXy = 0 (3.6a) (3.6b) (3.7a) (3.7b) (3.7c) w= 0 (3.4a) (3.4b) (3.5a) (3.5b) (3.5c) (3.5d) For the case For the case !_(M/ Qx M'+ Hmy"= 0 of clamped edges u = 0 Nuy = 0 u = 0 Hry = 0 w= 0 aw 0  = 0 ax of free edges Nx = 0 N~y = 0 Hx = 0 Hxy = 0 My+ My"= 0 + Mx') + 2 y(M"/ + M"') +t0 OX 0 (O~' + ax" (3.7d) (3.8a) (3.8b) (3.9a) (3.9b) (3.9c) (3.9d) (3.10a) (3.10b) (3.10c) (3.10d) Solution of the Membrane Force Equations The membrane forces in the initial state of stress are governed by equations (1.2) and (1.3). In general, a complete solution for the initial state of stress requires the further specification of the compatibility condition. However, since the forces oNx, oNy, and oNAy have been as sumed to be an order of magnitude higher than the linear additional stresses, the straindisplacement relations will include nonlinear terms. As a result, the compatibility equation becomes nonlinear and, in general, a solution of the state of stress is not possible. If the initial edge forces oN*, oNy*, and oNxy* are constant, then the initial membrane forces are constant. Since oN..= oNxy*= 0, there results oNx = oNxy = 0 (3.11) oNy = oNv and therefore oNX = oNxy = 0 (3.12) :Ny = yO In the case when the edges of the plate x = 0 and x = a are clamped, the second set of boundary conditions (1.3) is replaced by the conditions ou= 0 oNxy = 0 The solution for the membrane forces requires the specifi cation of the compatibility condition. If it is assumed that the plate is clamped after the initial stress is ap plied, then equations (3.12) will also apply to this case. The membrane forces Nx, Ny, and Nty arising from the additional stresses satisfy equations (3.3) and boundary conditions (3.4) and (3.6). These equations are identical to those just discussed for the initial stresses. However, since these forces are within the realm of linear elasticity, the compatibility equation may be expressed as27 V,(N + y) + (1 /i)VNT = 0 28 Now the classical methods for plane stress problems28 can be used for the solution of the state of stress. If Nx'= NY*= Ny* = 0 and Nr is a linear function of x and y, where N7 = 1(E'o'T2t' + E O'Tt") 1 i/.' then the equilibrium equations, compatibility equation, and boundary conditions are satisfied by = NY = Nxy = 0 and thus u=v=0 Solution of the Second Set of Equations Once the membrane forces have been found as indicated in the preceding section, the next step is to solve the set of three partial differential equations (3.2), which are subject to the boundary conditions (3.5) and (3.7), (3.9), or (3.10). First the system is transformed into dimensionless units. The scheme used by Ebcioglu23 is followed. Dimen sionless constants, variables, and reference quantities are defined as follows: a b 42 Pe= 7 E (h') (1 + m) b2 1 /AL Pf (D' + D") kie = Pe Pe t tGy b 1+ m% 2 = 2t b The temperature gradient is nondimensionalized by the intro duction of the quantity l+ m (3.13) The quantity Pe has been defined as a reference quan tity with dimensions of force per unit length; it represents the buckling load of a simply supported, infinitely long sandwich strip with a rigid core, calculated under the assumption that the faces act as membranes. PF represents the sum of the buckling loads of each face of the strip. Substitution of these quantities into the appropriate equations and boundary conditions yields the transformed equilibrium equations: rI = a u C) a 2 v 2p ^i Trr1 < A[2gA + (1 + (3d +AA ) 2v m'oc/3 w, 2mbre 6R 2 ~ 1 9 7 ) 31a r.2c2 /3 7(1 1A)a v + /'3(1 + u2v =___/ a)? r R (3.14b) 2/i TT'2 >i m~ r w w 2] u ;gzm a=w 21T bG 4a + 23 )w + 43w +m m I (r ek,, 1) 2W qmb mbVM1 /P VM= = (3.14c) 4 z 2GY 2GY and the transformed boundary conditions, written in terms of displacement components: (1) On the boundaries = 0 and 7 = 1 44 U = mb R(9)dt + 1( +<) JI bv + 13,ua u = mb R ;? at w = 0 Ir az MT (3.15a) (3.15b) (3.15c) (3.15d) (2) On the boundaries 4 = 0 and 1 = 1, for the case of sim ply supported edges, S+ = MbR u+ 1  a7q a w=0 Af WT i6 s= "MT For the case of clamped edges u = 0 au+ /A =0 377 at (3.16a) (3.16b) (3.16c) (3.16d) (3.17a) (3.17b) w = 0 (3.17c) w = 0 (3.17d) 51 For the case of free edges au + = mbR (3.18a) + /3 0 (3.18b) P f 3 w + z, w (3.18c) P# 3 [1 w 3 + W 1 + m rr [f3i 3 + (2 mA ()tal tGM u +b(wat.4) = A d + (t9tG = b d Mr (3.18d) It is seen that the system of equations (3.14) is linear and nonhomogeneous and is accompanied by nonhomo geneous boundary conditions. By the substitution of auxil iary functions29 the nonhomogeneous boundary conditions on the sides 7 = 0 and )7 = 1 can be reduced to homogeneous ones. Then it is possible to select trigonometric functions of that satisfy these homogeneous boundary conditions. These functions may be employed in the Levy method25 of separation of variables so that the partial differential equations are reduced to ordinary ones. It is necessary to expand the nonhomogeneous parts of the boundary conditions and equations in a series of these trigonometric functions; furthermore, the final solution is expressed in such series, and the convergence of these series is investigated in order to validate their differentiation. In order to reduce the boundary conditions (3.15) to homogeneous form, the following substitutions are made: u = u.( ,) + [f, (,) fO(g)] + fO(V) (3.19a) v 0 ~ag 1 df, (U) v = V. ( 7.) + 7 (t) 1, f ] (1 o [go(g) 4/ df() (3.19b) w = w.O,(,) + ?(, 1)[()7 + l)h,( ) 6 L + (2 )?)h(07) (3.19c) where the auxiliary functions denoted by f, g and h repre sent the following nonhomogeneous boundary terms: o mb R(, 0)dt fo () =/3(l + u ) f W mb R(, l)ds r f1( 3(1 + 1)J/ , g.() = mbR(t,0) 2 g, (1) = 1 mbR(9,l) 2 ho() = M(,(0O) h,() = MT(t,1) It may be readily verified that the desired reduction is accomplished by means of substitutions (3.19). The equa tions governing the quantities u., vo, and wo are then es tablished by the substitution of relations (3.19) into equa tions (3.14) through (3.18). The following equations result: re 2 zu 3 2u_ ) _v [ 2/ 2 + (1 /U)a7U2 + ( + u)  mgcx4 wo 2i3mbre )R c + Jf (,77) (3.20a) 2/o t r 2 4^  r) 2 v o 0v )zu, 2v S ; 1 + 1 art + (1 +] 2Vo t'4m awo 2mbre R+ R = + J) + (,) (3.20b) m [ o w +wo] U0 27T'bw Pf AvbS L, O + 2/,, 21#,, + 'w,j o3 u w3tgm =Zw0 avo amt2(rek,6 1) 32wo 2qmb 2 v 2MT + J3 (9,2) (3.20c) 2GF 2e cae y The boundary conditions on = 0 and = 1 become, for the case of simply supported edges, 13au, + A,)  mbR + S, ( (3.21a) auO + 13 av S M ( 7, ) (3.21b) w, = S3 (9,7) (3.21c) 13= M2 + 2.W ) (3.21d) For the case of clamped edges u0 = C, (, 7?) (3.22a) Wo= S (j) Cwo Fo = cs (c,?) For the case of free edges Uo _v au0 av 1132 + 1. mbR + S, (,?7) uo dVo P 2+ /1 0+ .3s) 7 ) = s + F7 ) (3.23a) (3.23b) (3.23c) P,/3 [ 2CJ+wo 3w + m(.aw. 72b [3C3 + (2 /) 1 m+ u bzotG,) .)t C3722 m b C) /3 dM7  2 + F, (M,2) b dt (3.23d) In the above boundary conditions the functions JL (4,), Sj (,??), Ci (9,), and Fj (t,), where i = 1, 2, 3, 4, repre sent the additional terms arising from the substitutions (3.19). The boundary conditions on ,7 = 0 and 7 = 1 are (3.24a) (3.22b) (3.22c) (3.22d) uO = 0 V" + 131A a u = Wa = 0 a2 62W W0 ) = (3.24b) (3.24c) (3.24d) It is seen that the boundary conditions (3.24) along the sides 7 = 0 and '7 = 1 are satisfied by terms of the form uoj = Xj (9) sin J7r?1 voj = Yj (E) cos j7T7 woj = Zj () sin J7Trj Thus the following solutions are assumed: U0 = 211 A=I V= I kI 00 WO = A1 Xj (t) sin Jir7 Yj (9) cos j7rr7 Zj (t) sin j1rr (3.25a) (3.25b) (3.25c) However, before attempting an infinite series approach, certain mathematical questions must be examined. Convergence of Series Solutions Once solutions of the form of (3.25) have been pro posed, it is desirable to develop some assurance that the series can satisfy the required equations, in other words, that termbyterm differentiation of the series will produce a series converging to a proper derivative. Such considerations must not be thought a mere mathe matical superfluity. Quantities of physical interest occur ring in the deformation of a plate involve second and third derivatives of the displacements; if the displacements are represented as infinite series, it may happen that termby term differentiation does not provide series that converge to their physical counterparts. For example, the deflection y of a vibrating beam of length 7T, simply supported, with an applied sinusoidal bending moment at one end is represented by the following equation:30 a Y + a2 )Y = 0 at2 a'x with the boundary conditions y(0,t) = y(7,t) = 0 a2y(ot) = 0 a 2Y(Tt) = b sin 4t )x If a solution is attempted by the method of finite sine transforms, the result will be the infinite series y = 2 a2 bn (1) (sin wt W sin an2t) sin nx 17 a2 n' W an which, upon termbyterm differentiation, will be found to satisfy neither the differential equation nor the nonhomo geneous boundary condition; nor will the internal bending moments calculated in this way converge to their true values. In order to be assured that termbyterm differentia tion of a convergent series, particularly a Fourier series, will lead to another convergent series, two theorems of per tinence may be noted. The first is as follows.31 Theorem 1. Let there be given a series fn(x) nmo whose terms are differentiable in the interval J = [a,b] and which converges at least at one point 00 of J. If the series 2 f,'(x) deduced from it by A0 termbyterm differentiation converges uniformly in J, then so does the given series. Furthermore, if 2: f,(x) = F(x) and I f.'(x) = p(x), then n.0 A0 F'(x) = o(x) It is therefore necessary first to perform formally the termbyterm differentiation and then to investigate the uniform convergence of the result in order to determine the validity of the operation. The second theorem relates specifically to the term byterm differentiation of Fourier series.32 Theorem 2. If f(x) is a continuous function of period 2i with an absolutely integrable derivative, then the Fourier series of f'(x) can be obtained from the Fourier series of f(x) by termbyterm differentiation. A corollary of this theorem applicable to the inter 32 val (0,1) is given as follows.32 Theorem 2(a). If f(x) is continuous and absolutely integrable on (0,1), then the Fourier cosine series I a, cos nnx of f(x) on this interval may be dif ferentiated term by term, while the Fourier sine series 72 bn sin nnrx can be differentiated term by term if f(0) = f(l) = 0. It is seen that the Fourier cosine series is necessarily always continuous everywhere; if the conditions in the corol lary are met, the sine series is also continuous everywhere. Theorem 2 applies to functions of one variable only. However, it may be extended to series of the form of equa tions (3.25) if the variable is treated as a parameter. It is now possible to proceed with the justification of the assumed solutions. From Theorem 2(a) and boundary condition (3.24a) the series for u0 may be twice differen tiated with respect to Condition (3.24a) implies that 6u  = 0 along ) = 0 and 7? = 1, and thus condition (3.24b) may be replaced by a 0 (3.26) The cosine series (3.25b) for vo can be differentiated once with respect to ), and condition (3.26) implies the permis sibility of a second differentiation. Finally, conditions (3.24c) and (3.24d), together with Theorem 2(a), imply that the series for w, may be differentiated four times with re spect to n. It is concluded that it is permissible to perform all partial differentiations with respect to q required to satisfy the differential equations and boundary conditions. The validity of taking derivatives with respect to , according to Theorem 1, can be established only after the actual solution is obtained and the differentiations in question performed. Some further remarks will be made when this stage in the analysis is reached. Development and Solution of the Ordinary Differential Equations The next step in the solution involves the develop ment of ordinary differential equations in the variable E. These relations arise from equating coefficients of trigo nometric terms after substituting the assumed solutions (3.25) into equations (3.20). In this process the nonhomo geneous terms of equations (3.20) are expanded in Fourier sine or cosine series on the interval (0,1) according to the following formulas. (1) The Fourier cosine expansion of a function f(x) defined on 0 5 x !! 1 is f(x) = + aj cos jinx J' where aj = 2 f(x) cos jflxdx (2) The Fourier sine expansion of a function f(x) defined on 0 x 1 is f(x) = iI bj sin jnx where 2ff(x) bj = 2 f sin J7Txdx 0 In this way the following relations may be written: 00 S2 KR (K ) sin j7r? ._=/ @0 () Kj () cos JIT;7 + 1 K,0 (6) arj 2 q = 2 K3j (4) sin j7Tr 00 V2MT = T2 KjJ (W) sin jrrj j=1 Furthermore, the remaining terms on the righthand sides of equations (3.20) are given by H,1 (9) sin JT?? J, (MO ) = j= J2(.i) = L J.,! H2 (j ) cos j7T? + H2 (9) H3j () sin j7T? In a similar manner the nonhomogeneous righthand sides of the boundary conditions (3.21) through (3.23) on S= 0 and = 1 are mbR(4,7) + S, (4,?) = S. = B ( 4=I S3 (91_) = Bj (9) sin JIT? *1=' Bej () sin jTTj 4 Mr(,) + s4(,1) = C, (,) = 1 Coj (k I B3j (9) sin jTri ) sin jrn? C.9 (4) sin jvTr (3.27a) (3.27b) (3.27c) (3.27d) c#(9,1) = (3.27e) JO T,' (3.27f) ) COS j'Wq + 1 B,. (t) cc M7(,) + F, (,) = 2 Fej (9) sin ju b dM( ) + F ((,t) = 7 Fy (4) sin jrr' ji (3.27g) (3.27h) As a result of applying the procedure described above, the following systems of ordinary differential equations re sult (D denotes the operator ): r D2 [r, (1 + + 2g Xj r/( DY 7r 11 7r mgt c/3 2/3nibre 2, DZj = 17 Kz + Hij 2/o 772 (3.28a) re3(l + A) j DX, + r32( D' (2rj + 2) Y+ Ir 1 7 1 t2 XmjT 2mbrc Z = KIj + H2j 21o J r" a (3.28b) fPlimb> Pm/32j2 + 32f Zgm] DZ p3DXf + pjYj + 2 b m [V + b o 4 ] Pf mj lT72 + [ 2bG, + (nm ,(l rek,e )jZ2TZ] 4 zE mb mb = + K KGj + Hjj 2Gy2GGy KK 3  (3.28c) r1,,(l /)D' 1 Yo = mbr K2o + H20 (3.28d) [2n' 7 7T 2 where j = 1, 2, .... The boundary conditions (3.21) on = 0 and = 1 become, for the case of simply supported edges, 3d ,uJY (k) = 2Boj () (3.29a) dd jnx, () + / d = Ba (=) (3.29b) Zj () = Bej (t) (3.29c) d z(U) = Bg ( ) (3.29d) dYo ( ) 1 dY = B' ( (3.29e) where j = 1, 2,.... The boundary conditions for the clamped and free edges similarly follow. The complementary and particular parts of the solu tions to equations (3.28) are denoted by the subscripts c and p respectively as follows: Xj = Xcj + Xpj (3.30a) Y = Ycj + Ypj (3.30b) Zj = Zc] + Zpj (3.30c) Yo = YOc + Yop (3.30d) Since equations (3.28) are linear with constant co efficients, solutions of the following form are assumed: X' = Aje j (3.31a) Ycj = Bi exk (3.31b) Zj = C.e j (3.31c) Yoc = B0e o (3.31d) Substitution of the above relations into the homo geneous form of equations (3.28) reveals that Aj satisfies the following quartic equation: 2 2P e 2Pf (ree2 ) biA ei g(i + bir ) rekt j eJ + j2ke + j2 re( 1) {[brz ( 2)  + j2k,e (1 0 (3.32) 1A IJ where e (= ) j 7T~ Further, A0 = + re(l /4) If there are no repeated roots of equation (3.32), the complementary solutions are given by 9 x = 2 Aj e" (3.33a) Xt'? t YCj= 21 Bjie e (3.33b) Zc' = 2 Cj1 e Aj (3.33c) Yoc = B, eXo + B,, eX (3.33d) The constants Bji and Cji are related to the constants Aji by the relations BjiL = bj, Aj, CjL = cji Aji where b irr r,/32 [2 1 +,1)] AI2 _7[r. (1)j + 2] 4A^ reg/32(l/ 1 Aj, 2 ,[2g(rej + 1) j2re(l+)] J (3.34) 2 1o 2r e 13' r .r ( 1 + /,A) j /3 j i b 2/o___ [2re i3aAjc" J___2g_ b__ CJ m,3 2' oyAj j 772 7( J) re2j biI The constants Aji are determined by substituting the complete solutions (3.30) into boundary conditions (3.29); eight simultaneous algebraic equations result, and their solution may be written in the form Aji = (3.35) 6j where Aj is an eightbyeight determinant A1 = Ia,I whose elements are given as follows for the case of simply supported edges: a/L = (/3Aj /,ujTbiL )ejL aLl = (/3Aj /#j;7bL )ejl a2L = (/3AA /AjTbjL ) a,, = (/3AjL ,.jffbjL ) ajL = (JTT +/3A;L bL )eJL, a.3,, = (Jrr + /3AjL bJL ) ebi aL = j rr + /3AjL bL a,^ = j + /AjL bjL a.L = cjL eAJL a,L, = cjL e'AL a6 = cJL a&,,,? = cjI. a = AL 2c eAjL a,,L, = AjL cj. ex a, = Aj 2cjL a1,, = Aj CjL where L = 1, 3, 5, 7. The Dj, are the determinants formed by replacing the ith column of Aj by the column vector 2B./ (1) 1g a 1 + /aj7Ypj (1) dX,. (0) 2B0j (0) / d ( + 1jfYpi (0) dYp, (i) B,4 (1) jxpj (1) 1 dYj(1) d4 dYo; (0) Bcj (0) jl7Xpj (0) /3 d( Be] (1) Zpj (1) B'1 (0) Zp, (0) B3j (1) ____ 1( B (0) d2Zpj (0) B31 (0)  Also, dY0P (0)] + dY0, (1) e Bo (0) B,(1)d /3 dE Boy = YZ ]d /3Ao(e A eO ) e[B(0) dYo (0)] B () + dY () B e, CO (0) d % B '%' ) d' /AoA(e" e ) For the case when the edges k = 0 and = 1 are clamped, the elements of Aj are given by aL = eJL a, ,, e AA aZ. i a2,L 1 a., = (j r + /3AjL bjL )eAJ aL/ aL = Jr +/3 A L bJL a%,, a., = CJ e, a,,,,, a. cLl. a=,Z,, alL = AjL CJL e JL a,,,, aL = Aji. CiL. a,,, where L = 1, 3, 5, 7. The Djj are vector (1) (0) (1) Bj (0) Bej Bej Cqj (1) (0) (1) C9i (0) = eL = 1 = (Jir + 3 AjL bjL )eA = jPT + /3 AjL bjL = cj eAis = CJL = CjL = AjL c e'JL = AJL CjL formed from the column  xpj (1)  Xpj (0) dYp1 (1)  jTTXpj (1) /31 dt  jItXpj (0) 1 dYpj (0) dt  Z ./ (1)  zpi (0) dZpi (1) d t dZpi (0) d4 For the there results case when the edges t = 0 and 4 = 1 are free, a,, = (/3 j, mjfbj, ) e AjL a2L = 3AjL /uJlbjL aJL = (jTT +/iAJL bij )eAjL aL = j1T + /3AjLbjL as, = (/32AjclCjL AJ7'i2CJ.L )eA'. a6, = /3AjL CJL Uj 7C a7L = {f!1 [32AjJ (2 )JZa7Aj] CJL /3 t j.A. e J b ItGcAjeJ aPL = /3 xJI3 (2 1)j2.2JL CJL arL 7 &b IAL j 1 zt )  t tG )c" Aj. = (/3AJL /4JTbjL )e'" = (3JL J7bj L ) = (j77 + /?Xj bJL )eJL = J1T + /3A.JL bJ, = (3'2AJLCj, ,.j2l72cjc )eAL = (/3"ALcJLj /,J1a rCi. ) { [32 ( ^J72 IPf ) [13 2 L p)jzIkjLJ CiL 17 zbr I 1 + min m t mn b (12t CjLjl e b I 1 + m  m tGX 1+ mi m a, L .l a.,, Z. ! a2.7 alL., a /2 3 (P'f [ 1] _l4 71A+ m tG. a L,L. = A (2 /7)j1 1 I JL m 2(t tG)cjLAjL b where L = 1, 3, 5, 7. The Djz are formed from the column vector dX, (1) 2Baj (1) /3 + ,hdjfrYe1 (1) 2Boj (1) /3 dXpi (0) f+ /.ijlrYpj (0) dt ddXpj (i) Be, (1) jrX1 (1) 13 dY ,j (1) d Bej (0) jTXpj (0) /3 dY (0) d 7r Fej (1) 1 dZj (1) + 1 J) iZpj (1) P" d +Z P F, (0) dzf.1 + /jI77Z,,j (0) 3 Ad' ,,i (1)1^ \1''J F,1 (1) P / [ # /(1) (2 j.ITL dZ(1)1 7r2b I dd Z j d dZ (l +  tGX p (1) + dZ tG (1) m b d Fyj (0) 1 3 [ djZe(0) (2 /4)jZ" dZ(0) 772b [ dt3 d7 + t m GX PJ (0) + ( ,)dzZ j(0) m b d Therefore, from equations (3.19), (3.25), (3.30), (3.31), and (3.35), the solutions for the displacement components u, v, and w are u(9.") = f, () + (1 I)f () + 21 { DBi e Xj + X'.' (k) sin JlTr F_ I J f :1 )~ (3.36a) [gj ( ,) / I df, () 1 (1 g [ M df0 () 2 d t 1 2 d I + LLi bjiD eAf + YP, ()cos jTr j e B,, + + B01e + Bi, eA'1 + yoP m w(.?) = (3.36b) 77 1 ( l) (? + l)h, () + (2 )ho( 6Pf + m j cji Ai eA + Z^ (e) sin jlrY[ 7z \ Y z, (3.36c) These equations completely determine the displacement components. Now that the solutions have been completely specified, further remarks can be made about the validity of forming partial derivatives of these quantities with respect to . Unfortunately, because of the extreme complexity of the terms in each infinite series, it is not possible to actually examine the uniform convergence of the derivatives. All that can be said is that actual evaluation of the series for w at the center of the plate indicated that the terms converged. Uniform convergence was not shown. Solution for a Simplified Case In the preceding part of this analysis the solution of a very general case of loading on a sandwich plate has been presented. The remainder of the analysis will utilize certain restrictions which aid in the tractability of the problem, while still representing a broad category of results. The following assumptions are now made: 1. Bending of the faces is neglected (Pf = 0); 2. The temperature is constant through the thickness of each face; 3. The temperature distribution and the edge loading are symmetric with respect to perpendicular bi sectors of the edges of the plate; 4. The transverse loading q" is constant. As a result of these restrictions, the following re lations hold: =r f'(4) = fm) b R(9,l)dt (3.37a) f' (= + ) g, () = g, (S) = mbR h, () = h0() = 0 (3.37b) (3.37c) (3.37d) MT = 0 The particular solutions can be determined from equa tions (3.28) after the following relations are noted: r,/3mb 2 _R T,,) = ( /. + 1 +/a)( ,1) + 2g 1)mb l JR( Rl)d 1 13(1 + U) T'(%,) = re/3~ /4 ( lmbl (1 1u) (2 1) mb(1 2 R_ / [ + (2 1)Inb (1 T(,) = /onmb(l + g 1 )R(+ ,) and after the following series expansions are noted: 00 1 si 4 sin j 1 /7 ,... (3.38a) (3.38b) (3.38c) (3.39a) 00 1 4 ~ j cos jr?7 (3.39b) i. ,j... 1? 2 17'~____ bR(t',) 2 [' )R(4,)) sin j7T d sin j7rl (3.39c) _R(_,__ 2 aR( ,',) (3.d9d) R(t) 0 =2 j R(, cos jTT dn cos JT (3.39d) i'',3 0 where the series in the last two equations contain only terms with an oddnumbered index because of the symmetry of R(t,7). Thus from equations (3.38) and (3.39) there result I K,j = 2 L R sin jrTT d7 (3.40a) fo 6 R K j = 2 cos jITJ? d?? (3.40b) J0? Jo Kjj = q0 (3.40c) K 1J = 0 (3.40d) K'j= 0 (3.40d) (3.40e) = 0 2 i +/f) 1)d ] j34f 4 r3 ___' R( i) H, =^ re lmbl ... H r. 2 + 1 + + mb(l + ((3 H2 = mb(l + g 1 R(4 1) + (3.40h) nPL!L g ( 1+/^~ )R(,l H80 = 0 (3.40i) where j = 1 3, 5, .... Then the particular solutions Xpj, Yj, and Zpy are found from equations (3.28) after P. has been set equal to zero and expressions (3.40) have been substituted for the quantities on the righthand sides of these equations. Furthermore, it is found that for any symmetrical tempera ture distribution Y0p = 0 The particular solutions for two specific temperature distributions were found. For a constant distribution R = RO these solutions are Xpj = Ajt + B Ywj = Cy Zp = Di (3.41) where A = Bamb R S r/3(il +/ )j [re(l /a)J3 + 2g] R Bj = 1 Aj 2 =. y41 ^ ^'3 J1 and = 2(rj + 1) l= 2(rej2 + 1) 1j 2 j7 2yo = [ /d )R  j 1=2J2 mb(1 +/ J 8r, mbgRo rr'[r e(l )j2 + 2g] .j = pjrr gy = r(mZ(l rek,e )7'2j2 S= Tj\ 2G +[ 1 +/" J 8pmb g2R ~ 7j(l +P)[re(l /L)j' + 2E] Here j = 1, 3, 5, .... For a double sinusoidal distribution of the form R = R, sin nt sin rrj, the solutions for j = 1 are Xpj = A1 cos 1T E Ypj = Ej sin Tr + F; (3.42) Zpj = Hj sin t + Ij where Aj = 9Sj pli p. pi Aj = Gi P5 j p. H1 = ,L P. IJt_ Here 2mbre S= R, Pj is the determinant V .j Vi x /3j where i = 2r,/ 3 + r e(l 14) + 2 X, = r,13(l +,) rj = log1 S= re/3'2(1 /u) + 2re + 2 ^~2 mgt of/ = IT 2,o .j= 4 (g , rekve + 1) and Pii is the determinant formed by replacing the ith column of Pi by the column vector 1 0 Further, F = A j I 3j 5 = A j 7. j /M.i where A = 4 mbqo Trji 2Gy For j = 3, 5, 7, ... Aj = Ej = Hj = 0 The coefficients in (3.27) are determined from equations (3.37) as Boj (0) = B, (1) = {mb[+ (1 a)]R(l,1)J Jo' + 2mb R(l,r) sin jTu d7 (3.' 4 cR( I1,1) ( (3. B, (1) = B,j (0) ;T7 mb 1) (3., Bej (1) = Bej (0) = 0 (3., B5i (1) = B,, (0) =0 (3. 43a) 43b) 43c) 43d) Bo (1) = B.o (0) = 0 (3.43e) C.j (1) = Coj (0) = 0 (3.43f) C,1 (1) = Cj (0) = 0 (3.43g) Fej, (1) = F,1 (0) = 0 (3.43h) m) tG b^ 1! 1 F3j (1) = Fj (0) 4 (1 M R(tul) dJ (3.43 ) ^ (l)7^l3(l + 14 JO .~ where j = 1, 3, 5, .... The assumption of a temperature distribution sym metric in the 9direction makes possible a simplification of the complementary solutions (3.33). The function E1(M) is defined as Ej () = ej(4' Y) + e)(j and is seen to be even with respect to the line = 1/2, which bisects the plate; the function j0, () = e kj( Y' ) e (' is seen to be odd with respect to F = 1/2. Each pair of terms of the form Aj eAj4 + Aj,., e ki that occurs in the complementary solutions (3.33) may be written in terms of the newly defined functions as 1l(Aj, e' Ai + A,,,, e Aj )Ej + l(Ai e'i Aj,., e'1/ AJ )0 Under the present assumptions, the horizontal dis placement component u will be odd with respect to 9 = 1/2, while the horizontal displacement component v and the verti cal displacement component w will be even. The expression (3.36a) for u in terms of the functions E, and Oj becomes U mb R(, l)d + 2 1{21 (DJ)2A e/AXJ 1 3(1 + ,a) 2 f A1 1e J/j= f, 3 + D,/ e'Jk )Ej, + D.;, , e , Dj, e ) iJ sin j17 AJ Since u must be odd and the functions Ejk and Oj, are lin early independent, the coefficients of Ejk must equal zero; there result Dz = e Dj, Dj, = e'2 Dj3 Dj6 = eAJs Djf Therefore the solutions (3.36) become u mb R(t,l)dt + 1+ s1 +s) J; + XpjI sin JT? J e j, [ e e* 1 Dji y.j.~J .. .I (3.44a) v = " J[ mb(l ) R(t,l) 2 1 +1( 1 D (3.44b) bj j ex + ej'j + Ypj cos J77( ^j j j 7,... * / '' w = DAj [eA1 + ei ] + Zpj sin j771 (3.44c) i., 1~... . Here the Ajj are the roots of the cubic equation (re, ej ) (gj + rek,, j'8O Zj'ke ) + j're(g 1) ej j'kie (1 ) = where e = ( 3) j Further, Aj is the threebythree determinant (3.45) Az = aKL (3.46) where K,L = 1, 2, 3; the elements aK. are, for the case when the edges = 0 and F = 1 are simply supported, a,,L = (e + 1) (/3AjL uj7TbjL ) a2L = (ezjL 1) (jrr + 13AjL bjL ) a3L = (exJL + l)CJL where L = 1, 2, 3. The Dyj are the determinants formed by replacing the ith column of Aj by the column vector dXp, (1) 2B.j (1) /3 d + /4jlTYJ (1) Bej (1) jTTXP (1) A dYpj (1) (3.47) d z>,, (1) where the coefficients B.1 and Bej are given by expressions (3.43a) and (3.43b). For the case when the edges = 0 and E = 1 are clamped, the elements of Ay are a. = ej 1 a2, = (e j 1) (jTr + /3AiL bL ) aJL = (e JL + 1) cJL and the Dji are formed from the column vector xpj (1) dYp1 (1) Bej (1) jnXe, (1) 1 a( zpj (1) For the case when the edges = 0 and = 1 are free, the elements of elj are a,L = (e*JL + 1) (/3A, /,jnbJL ) a2,L = (e JL 1) (jiT + /3AjL bjL ) a., = (eAJL l)[1+ m 13m Aj [ m b and the DjL are formed from the column vector dX,., (1) 2B.j (1) / d + /ujrrY/p (1) dk dYe, (1) B,1 (1) jnXj (1)  db T. ) J "^_ ni ^ dZp; (1) 1 + m 13 t Ed.) d F~j (1) M tG., Xp bi dt't, The coefficients F9j are given by expression (3.43i). The quantities b., and c,; are given by expressions (3.34) An observation can be made on the roots of the char acteristic equation (3.45). If the equation is expanded into a cubic equation of standard form A3 + aA2 + bA+ c = 0 (3.48) where A= A=j it is seen from the definitions and physical meaning of the coefficients that a< 0 b > 0 b>0 c < 0 for ke < J 1 + rej2 c > 0 for k,e > 1 + rj From Descartes' rule of signs it follows that the number of positive real roots of the cubic equation will be either one or three if ke < 1 I r or zero or two if ke > + er. 1 + r, j 1 + rej In general, then, as ke which represents the compressive edge loading, increases past this value, complex values of X are to be expected. A further observation is that when oNy, = 0, i.e., no compressive edge loading is present, a double root for Aj occurs; in this case the solutions (3.44) approach the fol lowing limits as oNyy 0: mb( R(,I)d + 2[ej" (1 eFj)e Z 1,3.. + 2 A [eAjA e""("] + XPi sin jTv (3.49a) mb (1 )R(_,_) v = (2n l) l )R(+1) 2 + 21 [ bj,[ ei + (1 )e + ej, [e' + e J + b [e + e "] + Ybj cos j+rr (3.49b) (* 1 w = r c1,[Sex'' + (1 )e'' ] ./,J... + gj, [e . 4 + e "1 (1]} Dj3 j A + c. [e'" + e""11) A + Z Pj sin jTr (3.49c) 4,1 +L ci I? ZJ Here the Ay are the roots of the equation and are easily found to be = ~j 17 7=7 /2g + rej2 ( /4) Aj = re,(l F/) and Aj represents the determinant AL = IaI whose elements are given by a,3 = /31 + (1 + Aj,)ej'] /,j7[e., + (by, + ey,)e'] a,. = jnei' + /3 {(Aj, ej, + b,/) + [(bj, + ei,)A\, + b,,] eJ'} a3. = gj, + (cj, + g,)eA, = (1 + = (e AJL = (eZ; e j ) (/3AjL /j7rbjL )  1) (ji +/3bJAJL ) + )cJL where L = 1, 2. The Dji are the determinants formed by re placing the ith column of Aj by the column vector (3.47). The bji and cji are given by equations (3.34); r. rjz(3 ) (1 ( ) + 2g ej n j rej2 (1 + P) (1 g) 2g g4o= morJz reJ (3 ,1) + g reJ(1 +1,) + 2 while Xpj, Ypj, and Zpj are the particular solutions of equations (3.28). CHAPTER IV COMPARISON WITH EXPERIMENTAL RESULTS Although there are many reports of experiments on rectangular sandwich panels without thermal effects, for ex 33 34 ample, those of Boiler33 and Kommers and Norris,34 the only experiment involving thermal deformation known to the author 24 is the one performed by Ebcioglu et al.24 In this experiment a rectangular sandwich panel was simply supported on all four edges and a temperature gradi ent was applied by simultaneously heating and cooling oppo site faces of the panel. No compressive load was applied and the deflection was measured at the midpoint of the panel. The properties of the panel were given as follows: t' = t" = 0.010 in. t = 0.395 in. /= 0.3 E" = E" = 30x106 psi (4.1) G = 39,825 psi Gy = 66,400 psi a = b = 6 in. and thus kle = 0 P, = 1,853 lb/in t= 1.0253 r, = 0.067214 0' = ac" = 7.34xl06 in/in OF (4.2) q = 0 g= 0.6 m= 1 ,4 1 From Ebcioglu et al. and the original data sheets of the experiment, it was found that for the four trials made in the experiment the distribution of the temperature gradi ent R, defined by equation (3.13), could be approximated by multiples of the function 4 R = (1.25 sin 77? + 1.373 sin rTy sin rrt)10 (4.3) A measure of the error caused by representing the tempera ture distribution by equation (4.3) was calculated from the formula E o aIa e(4 Error =  (4.4) A Tmax F_ i where the ai are the experimentally recorded values of the temperature gradient, the c. are the values of the gradient calculated at the same points from formula (4.3), and ATmax is the maximum difference among the experimentally recorded values of the temperature gradient. The following table gives the temperature gradient distribution, experimentally recorded values of the deflec tion at the center of the plate, and measure of the error as given by equation (4.4) for each trial. TABLE 1 EXPERIMENTAL DATA Temperature Measured Trial Gradient Error Deflection (in.) 1 R +0.005 0.013 0.056 2 1.1 R +0.023 0.0135 0.089 3 1.2 R +0.043 0.0195 0.079 4 1.8 R +0.065 0.021 0.080 