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- Permanent Link:
- https://ufdc.ufl.edu/UF00098426/00001
## Material Information- Title:
- Axisymmetric buckling of annular sandwich panels
- Creator:
- Amato, Amelio John, 1944-
- Place of Publication:
- Gainesville FL
- Publisher:
- [s.n.]
- Publication Date:
- 1970
- Copyright Date:
- 1970
- Language:
- English
- Physical Description:
- xi, 75 leaves. : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Approximation ( jstor )
Bending ( jstor ) Boundary conditions ( jstor ) Buckling ( jstor ) Coordinate systems ( jstor ) Equilibrium equations ( jstor ) Labor ( jstor ) Potential energy ( jstor ) Resultants ( jstor ) Sandwiches ( jstor ) Buckling (Mechanics) ( lcsh ) Elastic plates and shells ( lcsh ) Sandwich construction ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida, 1970.
- Bibliography:
- Bibliography: leaves 171-183.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030341164 ( AlephBibNum )
AEQ5192 ( NOTIS ) 16640373 ( OCLC )
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AXISYMMETRIC BUCKLING OF ANNULAR SANDWICH PANELS By AMELIO JOHN AMATO 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 1970 UNIVERSITY OF FLORIDA 3 1262 08552 2943 To my wife, Carol ACKNOWLEDGMENT S The author wishes to express his appreciation to the members of his supervisory committee: to Dr. I. K. Ebcioglu, chairman, who originally suggested the topic and, through his constant guidance and encouragement, made this work possible; to Drs. S. Y. Lu, V. H. Kurzweg, M. A. Eisenberg, and E. H. Hadlock for their helpful discussions with the author and many valuable suggestions. The author also wishes to express his thanks to the NDEA Title IV program for the financial support accorded to him. Finally, the author wishes to thank his wife, Carol, without whose patience, understanding and encouragement he would not have been able to complete this task. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . LIST OF TABLES . . . . . . . . . . . . . LIST OF FIGU ES . . . . . . . . . . . . KEY TO SYMBOLS . . . ABSTRACT . . . . CHAPTER I. INTI ODUCTION . . . . . . . . . . . II. BASIC CONCEPTS . . . . . . . . . . 1. Description of 2. Displacements Sandwich Panel and Notations 3. Strain-Di splacement Relations . . . . . . 4. Stress-Strain Relations . . . . . . . 5. Stress Resultants . . . . . . . . III. DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES . . Total Potential . . . . . . . . Theorem of 1Minimum Potential Energy . . . Equilibrium Equations and Boundary Conriitions Simplification of Equations . . . . . IV. EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS IN CYLINDRICAL COORDINATES . . . . . . . . Application of Covariant Derivatives . . Transformation of Equations . . . . Axial Stress Distribution . . . . Reduced Equilibrium Equations . . . Boundary Conditions . . . . . . Comparison with Other Theories . . . . . . . . . . iii vii . . . viii . . . . . . . . . . xi 6 S. . 8 TABLE OF CONTENTS (Continued) CHAPTER Page V. AXISYMIMETRIC DUCKLING OF ANNULAR SANDWICH PANELS . . 42 1. Uniform Axial Stress Distribution . . . ... 42 2. General Solution . . . . . . . . . 46 3. Successive Approximations . . . . . . 53 4. Numerical Results and Discussions . . . ... 60 5. Error Bound . . . . . . . . . . 66 6. Remarks on Convergence . . . . . . 68 VI. CONCLUSION . . . . . . . . ... . . 71 BIBLIOGRAPHY... . . . . . . . . . ... . 73 BIOGRAPHICAL SKETCH . . . . . . . . . . . 75 LIST OF TABLES Table Page 1. Comparison of Equations Governing Stability of Single- Layer Panels and Sandwich Panels . . . . .... . 40 2. Lowest Value of pi Satisfying Equation (86) . . ... 44 3. Approximate Values of (N /A) for 2 = 0 . . . . 67 LIST OF FIGURES Element of Sandwich Panel Annular Sandwich Panel Minimum Critical Values of Minimum Critical Values of Radius of Convergence (N /A) 0 (N /A) for N = N. 0 1 for N. = 0 1 Page . . 34 . . 63 . . 64 . . 68 Figure 1. 2. 3. 14. 5. KEY TO SYMBOLS V V 3 U11' w, x, y, z S z tc' tf t 'V Uc' U W 11 V q' N' P. C, 3, y, i, j, k eij' ep, eu3' e.33 1 .. 7, T 7 33 13 Ccp 0C3 33 G c Ef D A, N -a = Displacements = Displacement components = Cartesian coordinates = Coordinates, defined b3 equation (1) = Thicknesses of core and faces, respectively = Superscripts indicating the lower and upper face quantities, respectively (t 4 t )/t c f c - Total potential = Strain energies = Work performed by external forces and moments = Indices taking on values x or y = Indices taking on values x, y or z = Components of strain - Components of stress = Shear modulus of the core = Poisson's ratio of the face - Young's modulus of the face = Bending rigidity of the face =L(1-'-)/291 5 6 + 6 6. +[(2v )/(0-910f ~f [I y I,' '1 i P fP yfu = Kronccker daela = Pre-bucklliig nxial forces per unit length viii q 3 MB N (n) I nq Xq n,.,q,s,t,m,p Lm X Ys n n r, 9 2 u u Ni, N i o n, b i), E F, G, H, I = Lateral load per unit area = Externally applied moment per unit length = Transverse shear forces per unit length = Bending moments per unit length = Axial forces per unit length corresponding to small deformations during buckling = Bracketed index indicating physical compo- nent of a tensor = Chri stoffel symbol of the second kind = General coordinate variable = Indices taking on values 1, 2 and 3 -= Metric tensor = Arbitrary tensors = Polar coordinates = Laplacian operator = Pre-buckling lateral displacement of middle plane of faces = Lateral displacement of middle plane of faces for axisymmetric equations = Compressive axial forces per unit length applied to inner and outer edges, respectively = Inner and outer radii of annular panel, respectively = a/b = Defined by equation (65) dw dr = Defined by equation (72) A, B = Defined by equation (73) f -= r/b N, R, Q = Defined by equation (94) t = 11 1 Ak = Coefficients of power series P = Buckling coefficient for uniform axial stress distribution C1 = Arbitrary constant of integration Jl Y1 = Bessel functions of order one p = Radius of convergence Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AXISYIIETRIC BUCKLING OF ANNULAR SANDWICH PANELS By Amelio John Amato June, 1970 Chairman: Dr. Ibrahim K. Ebcioglu Major Department: Engineering Science and Mechanics The buckling of an annular sandwich panel is investigated using the theorem of minimum potential energy. Governing equations, derived in cartesian coordinates, are transformed into cylindrical coordinates by means of covariant differentiation. Considering the faces to be membranes and assuming an axisymmetric buckling mode, the equilibrium equations are uncoupled through the application of an improved technique. For a clamped outer edge and "slider" inner edge a series solution is applied to the general problem of radially varying in-plane stresses. Critical stresses are plotted versus ratio of inner and outer radii. The fifth approximation is shown to yield acceptable results. CHAPTER I INTRODUCTION A sandwich panel is defined as a three-layer panel, consisting of two thin outer layers of high-strength material between which a thick layer of low average strength and density is sandwiched. The two thin outer layers are called faces, and the intermediate layer is the core of the panel [1]. Among the main advantages of sandwich construction are: a high rigidity to weight ratio, good thermal and acoustical insula- tion, and ease of mass production. Some examples of core materials are bnl sa wood, cellulose acetate and synthetic rubber. HolAever, in more recent times thin foils in the form of hexagonal cells perpendic- ular to the faces have been employed. Depending upon the intended application, faces may be constructed of aluminum alloys, high-streng-th steel, etc. T'he practical importance of sandwich construction came into prominence with the advent of the aircraft and space industries. With the need for lighter, stronger and more stable structural com-- ponents, great emphasis was placed upon the design and analysis of workable sandwich pane-ls. Numbers in brackets refer to the Bibliography, The essential difference between the analysis of single-layer panels and that of sandwich panels is that the shear deformation associated with the core of a sandwich panel may not be neglected. Moreover, initially plane sections no longer remain plane during bond- ing, and the existing plate theories [2,3] require extensive modifi- cations. Numerous authors have contributed to the development of mathematical theories describing the behavior of rectangular sandwich panels. Two of the more noted of these are Hoff [4] and Eringen [5] who, early in the 1950's illustrated a concise and straightforward approach to the problem using the theorem of minimum potential energy. In 1960 Chang and Fbcioglu [6] introduced continuity conditions for displace- ments across the interface of two adjacent layers. This modification has been shown [7] to contribute appreciably to the accuracy of the derived equations, while introducing no additional mathematical compli- cations. In comparison, circular sandwich panels have been the subject of relatively little investigation. In 1949 Eric Reissner [8], neg- lecting the bending rigidity of the faces, solved the problem of a circular sandwich panel subjected to axisymmietric transverse loading. Later Zaid [9] included the effects of the bending rigidity of the face layers. Huang and Ebcioglu [10], using a technique similar to that employed by Zaid, recently investigated the axisymmetric buckling of a circular sandwich panel subjected to uniform axial compression. In their final results, the faces were treated as membranes. Prior to the advent of sandwich construction, the stability of circular and annular single-layer panels were extensively investigated [3]. Mcissner [11] analyzed the axisynmmetric buckling of a single- layer annular panel subjected to uniform compression along the outer boundary. In this work, the inner boundary was considered free and the outer boundary simply supported or clamped. Olsson [121 extended Meissner's analysis by considering the outer boundary clamped and the inner boundary "slider." Such a condition could be approximated by allowing a shaft or rigid cylinder to occupy the central hole (see Figure 2). Prompted by the foregoing sequence of investigations, the main objectives of the present analysis are: (i) to parallel the sequence of analysis present in the literature of single-layer panels by inves- tigating the axisymmetric buckling of annular sandwich panels; (ii) to achieve a more satisfactory formulation of the theory through the use of continuity conditions [G1 ; (iii) to modify the uncoupling procedure introduced by Zaid [91 and later employed by HIuang and Ebcioglu [10], thereby sign Ficantly reducing the complexity of the uncoupled equilib- rium equations; and (iv) to apply the boundary conditions employed by Olsson [12] to the present work. For the initial derivation, the set of basic assumptions employed in the present analysis are: (Al) The effect of transverse normal stress in each layer is negligible. (A2) The core undergoes shear deformation only. (A3) Displacements in each layer are linear functions of distance from the median plane of the layer. (A4) The median plane of the core remains neutral during small deflections of the panel. (A5) Each layer is homogeneous and isotropic. (A6) The core is attached to the faces securely. (A7) Hooke's law is valid throughout. (AS) Local buckling of the panel does not occur. These assumptions are similar to those used by Hoff [4] and Chang and Ebcioglu [13]. Equilibrium equations and boundary condiLions are derived, using the above assumptions and the theorem of minimum potential energy. These equations are then transformed into polar coordinates through the application of tensor analysis. In order to solve our problem, the resulting equations are then simplified further using the following additional assumpti ons: (Dl) Lcrnoulli-Navier hypothesis is valid for the faces. (132) Faces are considered to be membranes. (13) Arnular sandwich panel subjected to uniform axial compression buckles axisymmetrically. The first assumption is valid when the shear deformation of the face layers is negligible. Kim [7] has shown this to be true for most practical applications. The second assumption, which has been employed by many authors [8,10], is discussed at length by Plantema [1]. Finally, the assumption of axisymmetric buckling cannot be rigorously verified without solving the more difficult problem of unsymmotric buckling. However, such an assumption would seem reasonable since Olsson [12] has proved its validity for the analogous problem: of a single-layer panel with similar boundary conditions. 5 When comparing the present work with existing and subsequent theories, the words panel and plate may be considered interchangeable. Numerical computations are carried out by a desk calculator. CHAPTER II BASIC CONCEPTS 1. Description of Sandwich Panel and Notations An element of the sandwich panel to be considered in this wurk is shown in Figure 1. For convenience, three coordinate systems are defined having common x- and y-coordinates, and transverse coordinates related as follows: t + t c f z = z --- (1) t + t a c f z = z +- where t is the thickness of the core and t is the thickness of the face layers which are identical. 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 unprimed quantities are used for the core as well as for general relations. Subscripts c and f denote quantities related to the core and face, respectively. The total thickness of the panel is assumed to be very small in comparison with the lateral dimensions. Also, each layer is con- sidered to be isotropic and homogeneous, with the properties of the faces characterized by E and V and those of the core characterized by Ge. IFx Sz /7" t ' Figure 1. Element of Sandwich Panel Arbitrary transverse load q, axial forces N ,, and external moments are the only externally applied forces or moments which may contribute work to the system. For the initial derivation and formulations, indicial notation will be employed. Greek indices Q, $, y and p will take on values 1 or 2, while Latin indices, when employed, will take on values 1, 2 or 3. Any repealed index denotes summation, while any number of indices preceded by a comma indicates partial differentiation with respect to the coordinate variable represented by those indices. 2. Displacements In the following relations, which are valid for small displace- ments, it will be assumed that within each layer of the sandwich panel plane sections are preserved, although not necessarily perpendicular to the deflected middle surface. This will constitute an approximation since the presence of transverse shear suggests a non-linear variation of the longitudinal displacements through the thickness. Also, while the panel is geometrically symmetric with respect to the middle surface, the displacer.int relations will not, in general, reflect this symmetry, since unequal bending moments may be externally applied to each face layer. Therefore, with the aid of equation (1), our displacements can be written in the following form: V (x,y,z') = u (x,y) + z'tg(x,y) V (x,y,z') = w(x,y) V (x,y,z) = z (x,y) V 3(x,y,z) = w(x,y) (2) V-(x,y,z") = n (x,y) + z" (x,y) V\(x,y,z") = w(x,y) C1 ce Qa where (V ,V3) is the displacement of a generic point in the sandwich panel; u. represents the displacement in the xy-plane of a point lying in the median plane of either face; .,1 represents the angle that the normal to the median plane of each layer rotates when the planes are deflected; and i% is the transverse deflection which is assumed to be constant through the entire thickness. In order tc ensure continuity at the interface of any two adjacent layers, the following conditions must be imposed on equation (2): t t (3) t t a f I C u + T t Using (3) to eliminate and u11 from (2), our displacement relations become: v u u + z V z / V (2u tt) V (4) Vt t C3 (4) t V = '+ z' + ( ) V= w 3. Strain-Displacement Relations Linear strain-displacement relations are defined by [11]: 1 e -- (V 9+ V ) e (, + V5) 3 2 ,3 3,0 e33 \3,3 Using relations for (4) in conjunction with (5), the strain-displacement each layer become; e = u + z + u' z ''] ^3 = 2 Q p B, + / el 1rW + %I- e(3 33 Ze = t' + 2u -t ] o f0 e 3 = t[u + (tcf, C 33 0 t e -u ( ) + z t e" = o 33 4. Stress-Strain Rolations The generalize H]looke's law for a homogeneous isotropic body can be written in the following forn [15]: SE ij 1-V [Cij 1-2\ ij kk where the Latin indices i, j and 1: take en the values 1, 2 and 3. When the transverse normal stress, T33, is neglected, (7) may be rewritten as [15]: E v T = -[e + 6 eCp ] C3 = 3'r = 2G e --3 T= 3 (8 T33= 0 or, F Ep 1 2 Ac/,, e . (9 E 7 =- -- e o3 1+V 0'3 where, wi th, A (5 + 2 + ) C4y4 2 cp 1-i-w K' 0 Y A = A- = A A Or.Yk poyp. &piPY Yl-^ If the lateral, (9) reduces to: in-plane stresses, T , are also neglected, E a =- e = 2Go c'3 l+v C3 l Assumption (Al) implies that equations (9) may be used for the face layers, while assumption (A2) suggests the use of equation (10) for the core. 5. Stress Resultants In order to reduce the three-dimensional elasticity problem to a two-dimensional one, the following stress resultants are defined: t 2 t f 2 tf 2 t 2 Q = 'iT,3 dz t _f 2 t 2 dz' ; N S- T dz' t 7 2 t z dz' ; M T z dz 2 t f 2 2 t c 2 dz" % Q T a3' dz t 2 where M, N, and Q denote internal moments, in-plane stress resultants, and shear resultants, respectively. Expressions for stress resultants in terms of displacements arc obtained by substituting strain-displacement relations (6) into equations (9) and (10), yielding relations for the faces and core, respectively. hTien these relations are then substituted into equa- tions (11), and the indicated integration performed, the following expressions for stress resultants and moments are obtained: M' I D A 4 = D A , f N' D A u tE t t fE f A F- t f (12) N", = A -u' + ( (' ) (12) 2 1-v2 2 L ,, 2 yv, YJ tE ft Q [ + w, ] & 2(1+L ) f , tEr Q [(1 + 'w a 2(l+vv) c ,a Q = 2G [u' + i(tw tu A)]1 c c a 2 c ,y f a where the bending rigidity of the faces, Df, is defined: 3 Et 12(1-v ) CHAPTER III DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES 1. Total Potential The total potential consists of the strain energy stored in the panel during a small deformation and the total work performed on the panel by the external forces and moments during that deformation. In general, the strain energy stored in the body is defined [14]: U J j 7 e. dv (13) 2 1J 1J In particular, as a result of assumptions (Al) and (A2), the strain energies associated with the faces and core are: U 1' (IT' e', + 2T3 e' ) dv' f 2 ? Do aS Cd3 c3 u =-' (T 0c + 23 e 3) dv" f 2 a c 3 o3 where the integration is performed over the volumes of the individ- unl layers. Also, since the only externally applied forces or moments which are allowed to do work are the transverse loading function, q, the axial forces N F, and the external moments M the work performed on the panel is defined by: W =IV qS w da a W = 3N ,(VI + VI + w w ) da' (15) N 4 --a$ CY, y ,aY ,p f = f (it N(' i VN + ) )n (i where the integration is performed over the areas and boundaries of the individual layers. In the second and third integral expressions of (15), which are analogous to those employed by Timoshenko [3] and Eringen [5], it is assumed that the total axial load is evenly distributed between the upj.er and lower faces. Therefore, the total potential, which is defined as r = U' + U + U, w W' W (16) f c f q N N (16) may be rewritten, using (14) and (15), in the following form: 1 Jjff (T e', + 2r' e' ) dv 2 fp 03 C 3 cY3 v + Jf (2T 3 e) dv + LJ (T&B e"# + 2T e"3) dv" v V qw da + :J f E (V' + Vt +w, w ,) da' a a + f N (V" + v" + w w ) da" 4 J-op ,8 g$, ,a ,z a f (M3 + / 3 ) ng ds (17) 2. Theorem of Minimum Potential Energy The theorem of minimum potential energy states that of all displacements satisfying the given boundary conditions, those which satisfy the equilibrium equations make the total potential energy an absolute minimum [14]. Therefore, equilibrium equations and boundary conditions for the sandwich panel are given by the variational equate ion: 6 = 0 (18) However, before applying this extremum principle to equation (17), we note that: 6 (T e) = 6 1 A ey e 2E 1I / 16 = 2r 6 ' oa ci and similarly, 6 (T e') = 2T3 6e' (20) S(3 e3 = 23 a3 (21) 6 (T3 e) 2T- 6e (21) 03 ea3 U3 a Q3 6 (r" e",) = 2T" 6e" (22) a3 a3 a3 a3 6 (T" e" ) = 2T" 6e", (23) Therefore, using equations (19-23) in conjunction with displacement relations (4) and strain-di splacement relations (6), the variation o. equation (17) yields: ' p = T L,(511 + z'6' ) + ai (64' + ), dv' Y L : Q, 'Y, a3 acy , S 2 6ll ( t- i r2T a'3 L 2 + 6a t 6 a J V c c + 61a, + z""r + t f + ,F6t" + 6w dv" q 6w da + 6 N 2w 6w + t (6' 6 ) da a a [M 65' + 6 n ds = 0 (24) In equation (241) the symmietry of the stress tensors and strain tensors has been employed. Also, the displacements associated with N are those occurring in the median planes of the face layers. Using relations (11) to reduce the volume integrals ill equation (24), and applying the twordimensional divergence theorer, of Gauss to the resulting area integrals, the final integral equation is obtained. As an example of the above procedure, consider the fi-'st teri-i in the integrand of equation (24): t /2 T a 6u1, dv'- T 6, dz' da a -1 /2 = f Nj 6u' da a = ff[(u' bu') N' 6u'] da = N' 6u' n ds -J N' 6u' da (25) s a After applying a similar procedure to the remaining terrs, and collecting coefficients of similar virtual displacements, a complete description of the sandwich panel in the equilibrium state is obtained in integral forn: i = f- q Q N w Q Q<, 6w + N'33 + 4N' 6u, + L - c c t t f/ f Ij U 2 + 1' f Qu + N wq + "6 + [N5 N/J 61, + tf N + m, + tf N - + N + M" 6 n ds 0 (26) ,T -0a L 4 -oS 2 of f g -a pj S^ Q 3. Equilibrium Equations and Boundary Conditions If .we apply the fundamental lemma of the calculus of varia- tions [15] to equation (26), we obtain the following equilibrium equations in terms of stress resultants and moments: N,, w - (Q + 0' + Q") = q 20 ,NI + N" 0 C t t -- M +0 N" = 0 t af aS,, a 2 Q@,P SN" + Q" = 0 (27) 2 at,P ap, a and boundary conditions, S + Q + Q + N w w ;n ds = 0 5 [ N' N" 5Su' n1 ds = 0 f N i + M' + N 6' n, ds = 0 ri o + M - N *16 nc s = 0 (28) If we substitute the stress resultant-displacement relations (12) into equations (27) and (28), and collect similar terms, we obtain the following equilibrium equations in terms of displacements: 4G tE c 3 ff [u + (t w t )] + A [- u + t ( t a 2 c ,a 2(-v 2) Oypv y, p f YP )] 0 -2t t fE fcu- + ( t ) + 2 [f + w tc ae 2 c ,a f a 2(1+V ) C t2E + A u D A- [4D 3" ] = 0 2(1-v yi y4,P f Y ,p y ,P -it2: tE Aaf-- A + u + [ + w D A [ 3 2 uy y (31v-) L ] f 0 (28) + 4 P] =0 (29) and boundary conditions, 2G [u' + (t w)] + w+ + f [2w c 2 c ,a ,p 2(l+v ) ,a + '+ 5] w n ds = 0 tf A [4u tf(f ]46u na ds = 0 s 2(1-v ) 2 ykl y, y -Y y a f tf j I 4 -t3 s {- 4 --^ tE t f 2 A (4 ) t u 2(1-v) Syp L2 yd y-2 -f dJ + D A 4 -M i n ds = 0 t- A 1 (E' ) --t 2 2(1-v 2 L 2 y, Y, f -Y, + D A 4" B3 84 ds = 0 O yk y1p a~ c Boundary conditions (30) state that either the quantities contained within the large brackets must vanish along the boundary, or the variation of the displacement must vanish along the boundary. Compared with equations (27) and (28), equations (29) and (30) constitute a formulation through which the buckling problem can be more readily solved. This becomes immediately evident by consider- ing the classical concept of instability which is associated with displacement fields rather than stress fields. 4. Simplification of Equations In order to simplify equations (29) and (30), we exclude the possibility of externally applied edge moments by specifying: II I 0 (31) which reduces the number of dependent variables from seven to five since (31) suggests: =t; ((32) Also, if we assume that plane sections remain perpendicular to the deflected middle surface of each respective face assumptionn Bl), we have, S= w (33) which further reduces the number of dependent variables to three. A final assumption (132) which simplifies the problem by reducing the order of the differential equations is [1]: t >> tf (34) which justifies neglecting the bending rigidity of the faces, D , compared with the bending rigidity of the panel as a whole. Consequently: when equations (29) and (30) are simplified through the application of (31-34), we obtain the following equi- librium equations: 2 c c t pa ,oan -aS ,cx C 2t E 2G [-u + w ] A u = 0 (35) c t ,a 2 oQyp y, P f and boundary conditions, f2t t G\ + t + N w } 6w n ds = 0 L c c L 2 ,cd -c-y ,+ 0! s C r {--1- A u 1 6u n ds = 0 (36) sI 2 , f Equations (35) and (36) can be rev.ritten in extended form as: 2 r2 - t G --u + w t- G 1F- u f w N c c t x,x ,xx c c Lt y,y ,yy -K w v. -N w =q -xx xx -xy xy -yy ,yy c Lt x ,x 2 f x, xx y, xy f + ( l-t) (u + u )v] = 0 f x,xx x,yy F2 tE 2G u v ) ( ) c t 1-v2 y,yy x,xy + (1-v ) (u + u y) f y,xx yyy)J 0 (37) and, SY{2t tG + w +N v, +N w 6 n d = 0 cc c- 2 ,x -xx ,x -xy ,yJ x - 2tG Y w +N w +N w Sw dx = 0 J2 c Lc Lt 2 ,y -yy ,y -xy ,x y { I uf + V u 1-v2 L x,x f yu j J x xdy y tE f )-V [u + u ] u n dy = O f f [u + u ] 6u n dx = 0 (38) 1+v x,y y,x x y t + t wherc ---- f C and primus and double primes have been omitted without confusion as a result of equation (31). Equations (37), which are a special case of equations (29), are identical with those derived by Chang and Ebcioglu [13] if thermo- elastic effects are neglected. It should be noted that when simplifi cation (33) is introduced into equations (29) and (30), equations (35) and (36) are not immc- diately produced. The last two of equations (29) must be returned to the area integral of equation (2G) where, as a rsu lt of (33), they become coefficients of -6w Consequently, a transformation proce- durec similar to that illustrated in equation (25) yields additional terms which contribute to clquatioets (35) pad (3G). CHAPTER IV EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS IN CYLINDRICAL COORDINATES I. Application of Covariant Derivatives Let us consider a covariant derivative of an arbitrary covariant vector, Xn, [16]: aX X = -". ( I X (39) n"q X q nq- 2 In (39) y1 is a general coordinate variable, { ] is the nq Christoffel symbol of the second kind, and all indices take on values 1, 2, or 3. Furthermore, when a vertical line precedes any number of indices, it indicates covariant differentiation with respect to the coordinate variables represented by those indices. In considering the second covariant derivative of an arbitrary vector we must recall that in order to sum two indices we must have one covariant and one contravariant index. Therefore, introducing the metric tensor, g (39) implies: X g X = g -1[- (' J IX (40) n q q nq And since the covariant derivative of an arbitrary second order mixed tensor, Y is [1 6: n 6ya ,,s n + s Mm m s we have, from (40) and (41): X gpt ps s gps { sq n 8x x [ sq fi s my q n mq q Im x ( [ nq gs g 1Xq J } (42) In cylindrical coordinates, the Christoffel symbols and metric tensor take the following form [16]: f 2 2 1 1 x 1 12 21 1 22 all other nI = O 11 33 11 33 = = g = 1 g, = (X1 2 ; 22 (112 (13) Equations (42) and (43) imply: X = X (44) Xn pt ntp (4) Therefore, with the aid of (-13) and (44), (39) and (42) may be expanded to yield: Xli 1 X2 1 212 2 -ix XI BX ax1 |112 2 X SX -1 1 2 S- X2 1 X 1 2 aX X ax3 x x1 311 ay 2 a "_ 21 1 11 1 2 2X l a(x ) 2X? 2 x a111 (; ) 2 2 21 222 22 aX ) 2J12 1a 2ax BX EX c 2 2 1 ax2 1 ax 1 1 ax 2 2X (X12 ax WI 2X 2 1 2 2 ax9 x ax -. 2X a + 2 ax 8X 1 ax2 x1 ax ax1 2X2 ax a x ax ax3 -1 2 ax2 2x -3 X3J11 = a(x1 2 3 1 3 X322 2 2 X (46) a(X ) ax In order to make each equation of (45) and (46) dimensionally homogeneous, we replace the right-hand members by their physical com- ponents, X (, through the relationship [16]: (n) = X (No summation) (47) (n) n where a bracketed indc:x indicates physical component. Furthermore, to ensure dimensional compatibility among equa- tions, the following relations are used to introduce physical compo- nents to the left-hand members of equations (45) and (46): 11)J ufg" r g X p1 x( = i /-T g Xn (Nc. sumiwation) (48) The left-hand members of the resulting equations may be interpreted as cartesian components. Therefore, after replacing 1 2 X by r, X by e, X(1) by ur, X(2) by Ug, and X(3) by w in the right- hand mc-mbors, and X() by u, X(2) by u and X(3) by w in the left- hand members, transforn.iation equations relating csrtesian and polar coordinates take the iollow'ing form: U =U Xy ux x = r, r 1 1 U I = U + -Ur y y r 6,9 r r 1 1 u = -- u - 11 xj y r r,9 I UyJx ,r Ix ,r x xx 1r, 1rr 1 2 1 1 u --u + -u --u "xyy 2 i,SS 2 U,5+r r,r 2 r 1 1 1 1 S = u - u + Sxy 2r r, r 2 r, 0 r 0,r 2 0 r r u ik =u S-1- 9,69 + 2 + -I r 1 2 1 1 + e 2 96 2 ur, + r Cr r 2 Ur I r y -xy r ,r 2 r rr 2 r 1 1 In equation (49), r and 6 are polar coordinates defined in the usual manner, while ur, u. and w represent displacements in the direc- tions associated with this new coordinate system. 2. Transformation of Equations If we interpret the derivatives appearing in equations (37) and (38) as covariant derivatives and thereby transform them into polar coordinates using relations (40), we arrive at the following equilibrium equations: - t 2G { L (r u ) t r- u t 2 c L r r r r 9, J 1 1 1 1 N 2N (- N (- N + -w = q -rr ,rr -r9 r ,rO 2 ,a -9W 2 ,9 r ,r r r 2G 1 2j t W 1 2 1u + f u CLt r 1-2 ur 2 r 2( 2 6,6 f -+ (u- u -.u )= 0 2r r,r6 0 rr r 20 2 tfEf r 2 1 cL2 u + r tE (1-v )(V u- 2 u) 1 1 (3-v - + (liv )(-u +i- u ) +- u 0 (50) r r and boundary conditions, S2t 2 + rr wr + + N,9 6w nr d6 0 c O02tf E 1 1 -r2 Lur,r r ,0 r r2Jr r ri--- -[ u - u + uJ } 6uL n dO = 0 (51) 1+f r r,e r e 9,J r r 2 2 1 2 1 where V + + ar2 r ar 2 a82 Since only complete annular or circular regions are to be considered, the use of continuous cyclical functions of 6 eliminates the need to specify boundary conditions along a radial boundary. In the present work, which investigates the buckling of annular sandwich panels, the transverse loading function, q, is not considered. Such a restriction does not limit the application of the obtained results, since, for snall deflections, the transverse loading function does not influence the buckling load [3]. Finally, as a result of uniform compression along the inner and (or) outer boundaries, an axisymmetric buckling mode is assumed (B3) to result from the lowest critical axial load. Such an assump- tion has been shown by Olsson [12] to be valid for single-layer annular panels. The governing equations thus become: tt G { _I2 (ru) + V2 1 d (rN ') = 0 (52) c t L dr r dr -rr dr 2trE d211 1 du 2 2G u + t + = 0 (53) c Lt drj 2 2 r dr 2_j c 1-v dr r 2 1 d d where V ( - r ) r dr dr and, 24 G- C + t -' + N 6w n d = 0 (54) r c 2 d + r -rr dr r -c S"er + v xi 8u a d 0 (55) In equations (52-55) the subscript r has been omitted from the displacement u, without confusion. Also, N has been eliminated by considering Ihcepre-buchling equilibrium of the face layers: dN N N -yr -rr --95 + 0 (5G) dr r( 3. Axi.al Stress Distribution For two-dincnsional axisyneiotric stress distribution in pol ar coordinates, the governing equations are [17]: 2- - du 1 du u S+, O (57) dr2 r dr 2 dr 2'1 t Ir 2 +dr f r/ J f 2t E N f f + v (59) 1-v2 r dr where u is the pre-buckling lateral displacement. The general solution of equation (57) is: u = C r I C /'r (60) where C. and C6 are arbitrary constants of integration. Substituting (6) into (55) yields: 2tF E (1- )F N C(lv1+) C-- (61) For the case of an annular panel subjected to uniform com- pression along the inner and outer boundaries (see Figure 2), boundary conditions are: N (a) = N. Krr 1 (62) N (b) = N -rr o where a and b are inner and outer radii, respectively, and N. and N are inner and outer compressive forces per unit length, respectively. Imposing boundary conditions (62) on equation (Gl), we obtain the following axial stress distribution: N = -- (63) -rr 2 N --- F (61) -zGe 2 Figure 2. Annular Sandwich Panel where, b 2(N -N ) N PN 1 1 1 5) and, aB (66) b Since N and N are considered to be much larger than -rr -06 the forces produced by bending during buckling, the axial stress distribution remains essentially unchanged during buckling. Also, if a = = 0, or N. = N it is easily verified that equations (63) and (64) reduce to: N = N = N (67) 4. Reduced Equilibrium Equations From Chapter IV, Section 2, the equilibrium equations (reproduced here for convenience) are: C2 71 d 2 1 d (5w) -t G -- I- (ru) + L vw ( r N 0 (52) c tc t Lr r r dr -rr dr) 2t E 2 2G r2 + d 2- -[du 1 du o (53) c Lt dr_ 2 2 r dr 2 c ]-v I r Zaid [9] and Hiuang and Ebcioglu [10] uncouple similar equa- tions by operating on the second with the linear operator 1 cl L(y) -r dr ) and eliminating the first large common bracket in both equations. The resulting equation is then directly integrated to yield: 1 f 1 dw C l r+ C2 r + t tE r -rr dr 2 2 2 rj fc f (68) where C', C', and C' are arbitrary constants of integration. 1 2' 3 Equation (68) agrees with Huang and Ebcioglu's results if N is defined by equation (67) instead of (63). It should be noted, however, that the uncoupling procedure described above yields five constants of integration, while only four boundary conditions are available. This is a direct result of the uncoupling procedure which initially increases the order of equation (53). As a consequence of the above-mentioned complications and other considerations which will be discussed later, a different technique is employed in the present work to uncouple equations (52) and (53). Multiplying equation (52) by r and integrating directly without the aid of equation (53), we obtain: t N C c -rr 1 1 u + + (69) S 2G 2tG r c c where, dw Y, J- (70, and C is an arbitrary constant of integration. Comparing equation (69) with equation (68), the advantage of simplicity becomes immediately evident. Also, it will be made clear in the following sections that the present procedure facilitates the application of boundary conditions, and suggests analogies between the classical single-layer plate theory and the present analysis of sandwich panels. Substituting equation (69) into equation (53),with the aid of (63) and (70), we obtain, after some simplification: 3 2 2 d 2 d 4 12 BCr (r ,2G)r r 4 (r -3G) r L + (Hr Ir 4 3G)= (71) 2 dr F dr where, D BE F+BD EF G= ; ; I =- ; F = A (72) 2F- F F 2 t 2G G (1- 2) A-c c c f A = c B (73) 2 tftEc (7f and D and E are defined by equation (65). Equations (69) and (71) represent the uncoupIl d equilibrium equations. Since (71) is a second order differential equation, two constants of intcgvration are generated. Together with C, and the constant introduced through the integration of (70), we have four arbitrary constants of integration to be determined by boundary con- ditions at the inner and outer edges of the annular panel. 5. Boundary Conditions From Chapter IV, Section 2, the boundary conditions (reproduced here for convenience) are: u dwl dw S2t G + + N 6w n dB = 0 (54) c +t 2 dr1 -rr dr r C S e C11+. 6u n. d = (55) Integral (54) requires the specification of either the trans- verse deflection w, or the resultant shear stress along the inner and outer edges. Similarly, integral (55) requires the specification of either the relative lateral movement of one face with respect to the other, or the moment produced by tensile stresses on one face and compressive stresses on the other, along the inner and outer edges. Since we are neglecting the bending rigidity of the faces, the large bracket in integral (55) represents the total edge moment, while u becomes analogous to the slope, y, used in the formulation of boundary conditions for the classical theory of single-layer circular plates [3]. Specifically, in the present work, integrals (54) and (55) are satisfied through the following choice of boundary conditions at the inner and outer edges: tt N At r = a: u C F c -rr ] cp = 0 u = 0 (74) c At r = b: w = O; u = 0 (75) Conditions (75) are analogous to boundary conditions termed clamped or built-in in the classical theory, while conditions (74), termed "slider" in the present work, have previously been employed for stability analysis of sandwich columns [1] and single-layer annular panels [12]. Physically, such a restriction could be approx- imated by allowing a shaft or rigid cylinder to occupy the central hole (see Figure 2). As the inner radius, a, shrinks to zero, conditions (74) become identical to boundary conditions present at the center of a circular sandwich panel, without central hole, constrained along the outer edge only. This limiting process provides a check for our final results since the stability problem associated with such a panel yields a relatively simple solution. 6. Comparison with Other Theories Comparison of the present work with existing theories can be facilitated through the use of Table 1. Referring to this table, an analogy between single-layer theory and sandwich theory is observed. Meissner's [11] equation, which can be solved by means of bessel functions, is reduced to a homogeneous differential equation (1-=0) if the shear resultant is made to vanish at any arbitrary radius. This becomes immediately evident if we compare the boundary condition which specifies zero shear with the second form of the equilibrium equation. Consequently, since the shear at the center of a circular panel must vanish because of symmetry, Timioshoeko's [3] bessel equa- tion of ordei one is a ho:mogoncous differential equation. CO u 4- -9 u C,) + C-l I ) 41 '4> <-4' a- N + + 411 Cl + 'Cfltf,- H C 0 II 0+0 Hr Cl -H N^ . N' CM') I -^ c! U tiC 4 ,I H S 0, 0 CO 44' Sa 41 9 ,- 01)- O 1-1-S- t-44 0 H 'C c'-; 0 a 0 9- 40 O 0 U N + SL-W t- + 0 41 44 0 a 41 I N9 n + -r4 T-f -.4 O- *r-t H 000t 00CC 0 4 H ( ) 0 M~ o 004441 0C o o 01 1 "i-s O 9- r-. Cl C 01 -- 0 0 o O rH -.+- - M 0 0o 0 0 0 9 N 44 O 0 N I I .-1 Cl o\ 'i- *Q In + 9 19 + -9-It 0 0 A 9 N 1aI C oM 4 0 H *-H' 1C) ^-~T o a 0I M . .II w 4-' 414 a i Cl 0 1+ <4 ''.4-' k I3 - 441H 41 + 41 1-3 + ,-1 44 0 At 01 o spi HO 41 *H 1-' 04 ' *-< 0 ',41 *r1 PIc C; ^ . *^ V IS 0 ~ I _~__ ____1___1 I ___II_ ____I_____ Analogously, one of two equilibrium equations describing an annular region for a sandwich panel reduces to a homogeneous equation (C1 = 0) if the shear resultant is specified to vanish at some arbitrary radius. Such a simplification also provides a direct correspondence between u and y (see equation (69)). In the case of a circular sandwich panel, a homogeneous equation again results as a consequence of the symmetry involved, and one of the two equilibrium equations yields bessel functions of order one, as is the case in the single layer theory. The second equilibrium equation for a circular sandwich panel, which is attributed to Huanig and Ebcioglu, is deduced from equation (69) rather than the original form in which it appeared (see equation (68)). Consequently, without employing the present uncoupling technique, the above stated analogies would not be evident. Since the present work deals with an annular sandwich panel, it becomes obvious that bessel functions cannot be employed unless N = N. (see Equations (65) and (67)). Solutions for this special case Sare obtained in Chapter V and the more general case are obtained in Chapter V. CHAPTER V AXISYMMETRIC BUCKLING OF ANNULAR SANDWICH PANELS 1. Uniform Axial Stress Distribution If the pressure along the inner edge of the annular sandxtich panel is equal to the pressure along the outer edge, or: N. = N (76) 1 o then, from (65), (67) and (72): D = G = 0; I = 1 ; E = N o N RN (77) N = -N F A ; H 0 -rr o 2 A-(N /2) 0 which reduces equations (71) and (69) to: 2 B 0C r Sdr LA- 2T r A-(N /2)(7 dtr o o c c Applying the first of boundary conditions (74) to equa- tion (79) imnredliately reduces (78) and (79) to: 2 r BN r - r2 d d o j 0 0 (80) r r 2 dr A-(N /2) ( dr rit N - u = Z-r[- p (81) 2tG c Since (80) is Bessel's equation of order one, a general solution of equations (80) and (81) takes the following form [18]: 1 1 r N 2 BN 2 A1 A-o/2) r + B2 1 A-(N /2) r (82) O 0 1 1 ft N BN 2 BN 2 u L2c 7- {A 1 A-(N /2) r+A2Y1 L (rO ) r (83) 0 ?0 c where J, and Y1 are bessel functions of order one of the first and second kind, respectively, and A, and A are arbitrary constants of integrati on, If we now impose the remainder of boundary conditions (74) and (75) on equations (82) and (83), we have: [o Al[(c) + A2 Y1(i()16 0 2tG c (84) SA (L + A Y (p)] 2- 2G 2 1 0 c where 1 B(N )c 2 a u [(N ) /2 b b (85) o cr These equations can be satisfied by taking A = A = 0. Then the deflection at each point of the panel is zero and we obtain the trivial, undeflected form of equilibrium of the panel. The buckling form of equilibrium of the panel becomes possible only if equations (84) yield values for Al and A2 different from zero, which requires that the determinant of the coefficients of these constants vanish. Therefore, after multiplying by (4/t t) and considering the first of equations (73), our critical condition becomes: [0) 2] rJ1()Y ) J1 (V1)Y(1) = 0 (86) Equation (86) closely resembles the critical condition obtained by Olsson [121 for a single-layer panel constrained in a similar manner. For a given value of p, the smallest corresponding value of [L for which the second large bracket in equation (86) vanishes is given in Table 2 below [12]. TABLE 2 LOWEST VALUE OF pV SATISFYING EQUATION (86) $ 0.0000 0.0256 0.0526 0.0909 0.1000 4 3.832 3.840 3.860 3.924 3.942 p 0.1111 0.1250 0.1433 0.1667 0.2000 p 3.966 4.000 4.045 4.116 4.235 $ 0.2500 0.3333 0.3956 0.5000 0.5461 p 4.445 4.905 5.355 6.394 7.016 @ 0.6285 0.6897 0.7634 0.8333 1.000 1 8.523 10.175 13.312 18. 73 w And, from the first of (85): N (2 Acr b2 2 /2 It can be shown that (N /A) given by equation (87) is a monotonic increasing function of p. Therefore, the lowest value of p results in the lowest value of (N /A) Also, since b B is always positive, o cr equation (87) yields values of (N /A) which are greater than or 0 cr equal to two; and we can therefore conclude that the lowest root of equation (86) is always given by equation (87) in conjunction with Table 2. (See Figure 3.) The first of boundary conditions (75) has not been used to obtain the above results since the buckling load is independent of a transverse translation of the panel as a whole. In this respect our analysis parallels the classical single-layer theory [3]. For a = 0, equation (87) becomes: 2 ) = (3.832) 2 (88) cr b B+(3.832) /2 Equation (88) agrees with Huang and Ebcioglu's [10] results if the present notation is used. The validity of this limiting process was discussed in Section 5 of Chapter IV. As G. approaches a very large value, the first term in the denominator of equation (87) becomes dominant, and we obtain: 2, (N 1 (S9) o cr 2 b where the "effective bending rigidity," D', is defined: D' = (90) 2(1-v ) f Equation (89) in conjunction with Table 2 agrees with the results obtained by Olsson [12] for a single-layer annular panel subjected to uniform inner and outer axial pressures of equal intensity. For a = = 0, equation (89) becomes: ) (3.832)2D o cr 2 (91) b Equation (91) is identical to the buckling load obtained by Timoshenl:o [3] for a circular single-layer panel clamped along the outer edge. This limiting process is intuitively expected, since, for G = m, the only structural function of the core is to control the distance between the face layers. A similar relationship exists between the web and flanges of an I-beam. 2. General Solution We now return to a general solution of equations (69) and (71). Introducing the dimensionless variable, 7], through the trans- formation: = r/b equation (71) becomes: 2 b2BC E where, 2 N = ; R = b N ; Q = IN (94) Sir.:ila;ily, equation (69) becomes: ft N C c F -rrl 1 1 u I Ij + (95) 2 2iLG 2btG c c Since we seek a series solution of equation (93), and choose to expand our series about the point T[= 1, the following additional transformation is introduced: S= 1 (96) The reasons for seeking a solution about the point ,1 1 (= = 0) will be discussed in Sections 5 and 6 of this chapter. With the aid of equation (96), and after some simplification, equation (93) becomes: 2 [(N+1) + 2(2N+1)5 + (6N+)1)2 + 4N 3 + N(4 d- w+ [(N-3) dr + 30(-1) + 3N72 + N 3] -+ [(R--Qi3) + (4R 2Q) 2 3 b1 C1(1+) + (6R-Q) + 41HE 4 R4 ]T = -- (97) To obtain a complementary solution of equation (97) the following infinite series is employed [18]: cP = Z A k (98) k=0 where the coefficients A, are functions of the elastic and geometric properties of the panel and the critical buckling load. The radius of convergence of series (98) can be shown (see Section 6, Chapter V) to be of sufficient magnitude for our particular problem. Substituting (98) into a homogeneous form of equation (97), and collecting coefficients of common powers of , we have: [2(N+1)A2 + (N-3)Al + (R-Q43)A ] + [6(N+1)A + 2(5N-1)A + (R-Q+3N)A + (4R-2Q)A ]o + [12(N+1)A + (27N+3)A3 1 o 4 3 4 (R-Q3]fN-l)A + (4R-2Q+3N)A + (6R-Q)A ] 2 2 1 o + [20(N+1)A5 1 4(13N+3)A + (45N+R-Q)A3 + (14N+41R-2Q)A + (N+6R-Q)A + 4RA ] 3 2 1 o 4 . . . . = 0 (99) In order that this series vanish for all values of in some region surrounding E = 0, it is necessary and sufficient that the coefficients of each power of vanish [18]. This produces the following relations in which some coefficients have been eliminated through the accumulative introduction of previously computed coefficients: A (R-Q+3) A 3)A (100) 2 2(N+1) 2(N+1) 1 A [(S-)(R-Q+3) (4R-2Q)] A r(5N-)(N-3) ( R-Q+3N) A (101) 2 fL -^ -^2 6(N+6)( 31)l A_ (6R-Q) (R-Q+3)(R-Q+18iN-1) (27N+3)(4R-2Q) 4 12(N+1) 24(N+1)2 72(N+1)2 (27N-i3)(5N-1)(R--Q+3)1 A (4R-2Q+3N) (N-3)(R-Q+18N-1) 72(N+1)3 L 12(N+1) 24(N+1)2 (273i3) (R-Q+3N) (27K+3) (5N-1) (N-3) A (102) 72(N+1)2 72(N+1) 1 [ (13N+3)(GR-Q) (13N+3) (R-Q+3) (-Q+18SN-1) 5 GO(N+1) 120(N+1)3 (13N+3) (27N+3) (4R-2Q) (13N+3) (27N+3)(5-1) (R-Q+3) 3 + --- -- 4 360(\+1) 360(N+1) (45N+R-Q)(5N-1) (R-Q+3) (453iX+R-Q) (4R-2Q) (R-Q+3)(14N+4R-2Q) 120(N+1)' 120(N+1)2 40(N+1) 4R (13N+3) (411-2Q3) (13+3) (-3) (R-Q+18N-1) 20N+ o 60(N+1)2 120(N+1) (13N+3) (27Nt3) (R-Q4 3N) (13N+3) (27N+3)(5N-1)(N-3) 360(;+1)3 360(N+1) (45N+R-Q)(5N-1)(N-3) (45N+R-Q)(R-Q+3N) 120(N+1) 3 120(N+1)2 (N-3)(14N+4R-2Q) (N+61R-Q) A (103) 40(N+)2 20(N+1)J 1 40(3+1) . . . . . . . . * * A- where A and A remain arbitrary. o 1 Because of the complexity of the computations, and the immediate requirements, no recursion formula is sought in the present analysis. With the aid of (100-103), our complementary solution becomes: ^ Ai -( (R-Q+3) 2 + [(5N-1)(R-Q+3) (4R-2Q) 3 c 2(N+ (1) G(N+1) 6(N+1) [ (6R-Q) (R-Q+3) (R-Q+18N-1) (27N+3) (4R-2Q) + 12(+1) 2 + 2 2(N+1) 24(N+1)2 72(N+1) (27N+3) (5-1) (R-Q+-3) 4 (13N+3) (6R-Q) 72(N+1) 60(N+1) (13N+3)(R-Q+3)(R-Q+18N-1) (133+3)(27N+3)(4R-2Q) 320(N+1)3 360(N+l)3 (13N+3) (27N43) (5N-1) (R-Q+3) (45N+R-Q) (5X-1) (R-Q+3) 360(N+1)4 120(N+1)3 (45X+R-Q)(4R-2Q) (R-Q+3) (14N+34R-2Q) 4R 5 120(N+1)2 40(N)+1)2 2 0 N+ + 4A 1{ (-3) 2 + (5N-1)(N-3) (R-Q+3N)3) 3 1 2(N+1) +L -.2 -6(N+ 1) ] F (4R-2Q-F3N) (N-3) (R-Q-18N-1) (27N+3) (Rf-Q+3N) L- 12(N+1) + ^ 2 .2 24(N+1) 72(N+1) (27N+3) (5N-1) (N-3)1 4 r~(13.N+3) (4R-2Q+3N) 72(N+1) 60(N+1) (13N+3)(N-3)(R-Q+18N-1) (13N+3)(27N+3)(Rfl-Q+3N) 120(N+1)3 360(N+1)3 (13N+3) (27+3) (5N-1) (N-3) (45N+R-Q)(5N-1)(N-3) 360(N+1)4 120(N+1)3 (45N+R-Q)(R-Q+3N) (N-3)(14N+4R-2Q) (N+6R-Q)f 5 120(N-l) 40(N+1)1) J + . . . . .} (104) Following a procedure similar to that employed in the previous section, equations (95) and (104), together with a particular solution of equation (97), are constrained according to boundary conditions (74) and (75). Imposing the first of boundary conditions (74) on equation (95) eliminates the need to find a particular solution of equation (97), since we have CI = 0. The subscript c can therefore be eliminated from equation (98) and our solution becomes a complete solution. From equations (92) and (96), we note that at r = b, 1] = 1 and 0 = 0. Therefore, from equation (95), recalling that C = 0 and N (b) = N (see equation (62)), the second of boundary condi- tions (75) becomes: [() 2 cp (:=0) = 0 (105) And since (N /A) A 2 in general, condition (105) imposed on equa- tion (104) yields A = 0 Similarly, the second of boundary conditions (74), evaluated at r = a ( =3-l), and expressed through equation (95) becomes: -i) 2 p (=0-1)= 0 (106) After substituting equation (104) into (106) with A = 0, and noting that in general (N./A) 4 2, we arrive at the general buckling criteria for an annular sandwich panel: (N-3) (1) 2 F(5N-1)(N-3) (R-Q+3N)] ( 3 S2(N+1) G(N+l)2 6(N+l) F- (4R-2Q+3N) (N-3)(R-Q+1SN-1) (27N+3)(R-Q+3N) 12 (NT+1) 24(N+1)2 72(N+l) 2 24(N+1)2 72(N+1)2 (27N+3)(5N-1)(N-3) (1)4 (13N+3) (4R-2Q+3N) 72(N+1) 1- 60 (N+1) (13N+3) (N-3) (R-Q+1-1) (13N+3)(27N+3) (R-Q+3N) 120(N+1)3 360(N+1)3 (13N+3)(27N+3)(5N-1)(N-3) (45N+R-Q)(5N-1)(N-3) 360(N+1)4 120(N+1)3 (45N+R-Q)(R-Q+3N) (N-a3)(14N+4R-2Q) (N+6R-Q) 0( -1 5 420(+1) (1 120(N+1)2 40(N+1)2 20(N+) + . . . . = 0 (107) As explained in the previous section, the undeflected form of a panel in equilibrium, i.e., When Al = 0, is of little interest. Therefore, approximate critical buckling loads of the panel under discussion may be computed by considering a finite number of terms in equation (307). The first of boundary conditions (75), as stated earlier, is not used in obtaining the above results, since the buckling load is independent of a transverse translation of the panel as a whole, 3. Successive Approximations From equations (94), (72) and (65), we have: N 2A(1-2 (N 0 N (108) 2 (N -N.) 2 (N -N.) 0 1 0 1 2b2B (No-2 N) B 0 i (109) 2 (N -N.) 0 1 2A(1-2 (No N 2 Q + 2b B (110) 2 (x -N.) 2 (N -N.) 01 O 1 We designate the first approximation to be that which considers only the first term of the infinite series (107), and consecutively add a term for each succeeding approximation. Therefore, with the aid of (108-110), and after some simplification, we have: First Approximation The first approximation yields no results since the loading functions are not present in the first term of series (107). Second Approximation S) ) 2[(N ) (N.) c r 2 +2 (p1) I 2A(1-- ) o cr 1c -31 0 (111) ( N[(N ) -(N.) i] 2[(N ) -(N.) ] L o cr 1 c o cr 1 Cr Third Approximation 6r 2A(1-02) L2 [(N ) -(N.) ] L 0 cr i cr [(N ) -2 (N.) o cr 1 cr 2 2[(N ) -(N.) ] S10OA(1-p ) ocr i cr [(N ) 2 (N.) I o cr icrY Socr i cr [( cr -2Ni cr +1 3 2A(1-2) $2[(N )[ (-() N ) cr-(N.) ] o cr i cr L o cr i cr I 2 [(N ) -_ (N.) cr I 2A(1-5 ) ocr -3cr-i) 2 o [(N) -(N.) ] 2[(N ) -(N.)]c 5[(N ) -52(N.) 1 2 ocr 2 cr l 2A(1-S) 2 S2[(N ) -(N.) c -2[(N ) -(N.) ] o cr i JL o cr i cr - 3] (1)2 2 [(N) -( 2(N.) S2A(1- ) o cr i cr 2[(N ) -(N.) ] (2[(N) c-(N.) ] S4A(1-2) -21(N ) -(N.) c] - 2b2B (-1)2 = 0 Fourth Approximat ion 2[(N ) -5 2(N.) 2 [(N ) -(N.) 0 cr 1 cri 52( oc^(icrl^ 2b2B[(Nocr- 2(N )cr 2 [(N ) -(N.) 1 O Cr I C (112) 2) 2A(1-5 ) 72 2A _;2 72Ls N 0) -(N.) ] ocr i cri [(N ) - (N.) ]- Scr- i cr 2 + 1 a cr a cr - - 36 F2A S2[(N ) -(N.) I cO Ca 1Ci [(N cr- 2(N ) 2 [(N ) -(N.) 1 O ~ CT1CT -3 2A(1-$2 ) - 3Ik cr-( L2[(N ) -(N.)] -cO Ca 1Cr [(N ) -5 (N.) ] o cr i cr 52[(N ) -(N.) ] o cr a cr -+1 (B-1) + (121 10A(1-3 2 ) LP [(N cr-(N) 5[(N ) 2 (N .) ] o (cr 1 crl [ 2A(1-82) S 2 [(N ) -(N.) I r [(N ) -2(N.) o cr 1 cr+ 2b 2B[(N ) 2( )] + 2 2[(N ) -(N.) ocr 1 cr 2[(N ) (N.) I o cr 1 cr [1 2 2A(1- -$ ) o cr(N 1 cr [3( -(N.-),, [(No cr- 2(Nicr 2 L(No~cr (Ni)cr] r 2 4A(1-B ) ) L2[(N ) (Ni) cr - 2b B (3-1) + 6 2 r 2 2A(1-3 ) 2[(cN ) -(N.) ] o cr i cr [(N ) -$2 (N.) 1 o cr i cr 3 3 [(N ) -(N.) ) ocr icr 12 I 2A(i-32) 2[(N ) 2(N.) I 2A(1-g5 32[(N,) ,-(N.)"I ocr-P i crI S[(N ) -c(N.) c o cr i cr Sb2 [(No) cr- 2(N) ' cr cr - 4b2j + [ 2A( _2) o cr-32 cr 82 [(N ) -(N.) o cr i cr [ 34A(1-22 ) 2 [(N ) -(N.)"I Oc C1 1Cr 1 2A(1-2 ) 112[(N ) -(N.) ] [5 o cr i cr 17[(NO cr 2 (N )cr S2[(N ) -(NV.) ] O crT 1C [(N ) -2 (N.) I 0 cr I cr-3 32[(N ) -(N.) r o cr 1 cr 32 [(No )cr-(N)cr 2 27(1-02 [(N) -32 (Ni ] - 2b B +1 + 2A (1-32) 2 c (N) cr-(N) c1 2 [(N ) -- (N) ] O C cr O Cr 1 cr 54A(1-5 2 ) S[(NO cr-(N cr] 27[(No cr- 2(Ncr 5V[(N )c-(N.)c 0 cr dcr' 4A(1- 2) 2 cr(N c 2 [(N ) ,-2(N .) ] 2( cr-(N )cr 27[(No c 2 (N. )cr 2[(No cr-(Ni cr 82[(N ) -(N.) ] ___ 2A(l-g2) o cr 1 cr Fifth Approximation 2b 2B(N o 2 (N.) ,] o cr 1 cra 3][1OA10(1-2) [(N cr-(N )cr [(No cr- 2(N)cr 2 [(N ) cr(Ni c o cr 'c 2b2B 54A(1-B 2 S 2[(N ) (N.) ] 5[(No cr 2( i cr S[(N ) -(N.) I ocr icr 1 - 3 (-1)3 = 0 (113) 360 2A(1-_2) 62[(N ) -(N.)cr- o cr a ci [(N) cr- (N icr p [(N ) -(N.)cr o cr 1 cr - 180 2 A(1-2) 2) - 0 [ c i cr [(N cr-B2 cr 0 cr cr 2A(1-B2) [(No cr(N i)cr [(N) -F2(N.)I o cr 1 cr1 2- N cr cr cr 2 cr S22A(1-B2) L [(No )cr- (N )cr (S-1) + 60 2 2(-n ) 2[(No cr- (Ni)c 2 +1 10A(.-2 ) 2[(NO cr-(N )cr [(N )cr- 2(N )cr [(No ) r-(N )c,] 5[(No) cr,-(N)i cr 2 [(No )cr-(N )c o-cri^cr - 60 2 2A(1-9-F2 . S[(No cr-(N i)cr [(No cr (N cr 3 4A(1-2 0 -1 cr B2[(N ) -(N.) c]r LB[(N ) -(N.) ] 0 ia1c o cr i cr 2[(N )cr- 2(Ni)c 2[(N ) -(N.) Scr i cr + 1 ' 2b2B[(N ) -2 (N.) ] ii ocr icr 2 2 +- 2-2b B (8-1) - 82[(N ) -(N.) J ocr i cr 2A(1-2 ) 0 cr i cr (2N( 2 ) (N.) o cr r cr r 2[(N ) (N.) I o cr 1 cr 2 o o ccr (N i cr S[(oN ) -(N.) o cr i crl 2A(1-$2 ) 2[(No) cr-(N )cr [( cr cr 2 [(No c) -(N )r 4b2B + 15 2c(12 S(N ) cr-(N.) 3 2A(1-2) ) [(N ) -(N.)cr [(No cr- 2()cr 2(NO cr- i cr 2 17[(: ) 3 (1-l o cl 2[(N ) --(N) c2[ (N) -2(N.) ] 2b2B[(N ) cr- (N cr cl ( r cr+ 2 cr -(N.) ] [2(N ) -(N.) 2r i cr o cr i cr [(N )cr (N c 82[(N ) -(N.) o cr i cr S541A(1-3 ) ' 2[(N ) -(x cr.) L o cr 1 cr 27[(N ) -c- (N.) 0ocr icr) 2 + cr- c 2 [(No )r-(N.)cr S4A(1-32) _ S (No cr i cr 2[(N ) (N.) ] 2 0 r i cr 2[(N ) cr-(N .) r 2b2 B[(N )cr- 2(Ni )cr 2[(No cr -(Ni)cr r 5 2A(1-S2) 2 [(N ) -(N.) ] Scr i cr [(No) c- (Ni)cr I F 2 [(-- c.- ( + 0 Cc 1 Cr 54A(1-82) F 2[(N ) -(N.) ] O cr i cr 27[(N ) 2 (N) 2 5[(N ( o cr 1(Ni cr 3 OA(1-2) 5(o ccr cr 82[(N) cr-(N cr [( ) -(N. )c i [(N ) -el cr(N. ( c c cr i cr cr icr - 2b 2B IJ 2A(1-2 ) S2[(N ) -(N.) 0cr 1 cr [(N ) -$ (N.) o cr i cr 2 [(N ) -(N.) ] ocr i cr 2A(1- 2) [(N )cr-(N)cr O Cr i Cr 13[(N ) -(N.) ] o cr 1 cr 8b [(N ) 2-()2 0 cr 1 cr Sb B[(N ) -a (N.) B2[(N ) -(N.) 1 o cr 1 Cr [(No cr-82(N )cr+ [(N ) -c (N.) a cr 1 Cd 2 + 5 [(N ) ,-(N.i ) 2A(1-82 ) [(N )cr -(Ni o cr icr '1 26A(1-32) JL2[(N ) -(N.)cr -(N -$ 2(N )crI 0cr r icir 2[(N ) (N.) 2] F2A(1-2 - 4bB 3 2 L ((N ) -(N.) I SO r 2T Cr [(N ) -5 (N.) J o cr i cr, a cr 1 Cr . 1 26A(1-5 2) L2[(N ) -(N.) ] cr 1 cr 13[(N ) ,- 2(N.)] 2 [(N ) -(N.) o Cr 1 er S o cr 2i cr 17 [(N ) -(N.) ] o cr i cr 182[(N ) --(N.) l] oc icr - [No cr- ci 34A(1-52 (N ) cr-(N.) IJ 2[(N ) -(N.) r a cr i cr a cr i cr 2b2 B[(N ) c- 2(N.) ] o cr i cr - 2b2B 1 2A(1-2) 0 cr 1 cr z3[(No)cr 2(Ni )c 2 O cr 1 C + 3 4IA(1-2 ) i 2L(N ) -(N.) o cr 1 cr [(No cr- 2(ai c + ^1 2 L(N,),,-(N i) cr ] 2[(N ) (N.) i] aocr i cr 2 [(N ) -(N.) cr cr 52[(No) -(N.)i c 2GA(1-52) 27[(N ) 2(N .) o cr a cr 2b2 I[(N ) -P2 (N.) ] o cr i cr 2[(N ) -(N.) I Oc iT1Cr + 3 ($-1)3 -i 2b2B 26A(1-82) 13[N) cr-8(N) cr 3 2( L c -(N. )" 2[(N) c (N cr ) S54A(1-2) 2 2 [(N ) -(N.) ] ocr icr 5[(N ) -(N. o cr 1 cr 02[(N ) -(N.) O Cr 1 Cr 44[(N ) -3(N.)] o cr 1 cr2 5[(N ) -2(N.) o cr 1 cr 44[ (N) -32 ), (N ) -2(N.) ] ocr (Ni cr ------------+ 3 [(N ) -(N.) I o cr 1 cr r 2 2A(1-4 ) 2[(N 0)cr-(N el I [(No cr-P2 (N i c 2[(N -(N.) ] a cr i cr [(N ) (N.) 0 Cr 2 1 cr 2A(1-) 2 ) $2 [(N) -(N) 0 cr i cr o cr2(N cr 2 ocr 1 cr 1OA(1- 2 L8 2 [(Ncr-(N cW (N 2 (N.) I Scr I crl S[(N ) -(N.) I o cr i cr Sr 1 CT 1 8SA(1-2 ) 2b B 21AI- ocr icr [(N ) -2(N.) ] o cr 1 cr 2 [(N ) -(N.) ] o cr 1 cr 1 - 31i j 88A(-2 ) I2 [(N) -(N.) ] o cr Icri 22 2b21 -B-r -_$2 [(N- ) ,-(N.) ]r o cr 1c 2[(N ) -2(N.) I 2b2B[(N ) -82 (N.) ] o cr -c c -icr 2 2b2 2[(N ) -(N ) cr] 2 [(N ) -(N.) II 2 -cr ) cr o cr crJ 2A(1-2) (No) cr- 2(N r1 2 2A(1-,2 2[(No ) ^c.) ] [(N ) -(N.) ] 2 [(N ) r-(N.) cr ocr icr ocr 1cr [ 2A(1-P 2 ) S [(N ) -( .) ]cl 1- o cr i cr [(N ) -$ 2(N.) ]r 24 12[(N ) 2(Ni .)cr ocr 1 c 24A(1-c) o cr a cr -o3 $2[(N ) -(N.) ] L2[(N 0)-(N .] )2[(No) -(N.)"] 8b 2B[(N ) 2(N.) c] r 2 2 cr cr 2402 2A(1-0 + o 4b Ba is8 2 [(No 0cr-(N )c 2[(No cr-(N )cr [(W ) -(N.) ] 12b2B[(N ) -2(N.) ]-N) cr 1 cr o cr i cr 2b2 (o-1)4 = 0 2 [(N ) cr-(N.) ] 2[(N)-( N ) .) o cr cr o cr a cr (114) Computation of the sixth approximation is unnecessary since it can be shown that the fifth approximation yields acceptable results (see Section 5, Chapter V). Equations (111-114), each being self-contained, represent the approximate critical buckling criteria for an annular sandwich panel constrained by boundary conditions (74) and (75) and subjected to uniform radial compressive loads, N. and N along the inner and outer edges, respectively. 4. Numerical Results and Discussions Once the inner compressive load, N., is prescribed to be some multiple of the outer compressive load, No, the approximate critical conditions given in the previous section are completely defined by three dimensionless parameters: (N /A) b B, and p. Thus we obtain o cr algebraic polynomials in (N /A) which increase in degree as the order of approximation increases (ranging from first degree in the second approximation to fourth degree in the fifth approximation). For obvious reasons we consider only the lowest positive value of (N /A) o cr satisfying each polynomial. Three possible loading conditions are analyzed in the present work: (1) N = N., (2) N = 0, and (3) N. = 0. However, it should be 0 1 0 1 noted that the techniques employed in these examples are applicable for any ratio of N to N.. O 1 N = N. o 2 If the inner and outer axial compressive loads are equal, then the exact solution is given by equation (87) in conjunction with Table 2. Thus, the approximation techniques employed in the previous section are unnecessary. For this case, Figure 3 shows the existing relation between (N /A) and 3 for various values of b 2B. o cr N = 0 o If the inner edge alone is subjected to axial compression, then, since equations (111-114) yield no positive values of (N /A) , it can be concluded that buckling never occurs. This would seem reasonable, since, from equations (63-65), such a reduction results in a relatively large tensile N compared with a relatively small -96 compressive N Analogously, for a rectangular single-layer panel subjected to compression along opposite edges and tension along adja- cent edges, Timoshenko [3] has shown that a large tensile load will prevent a significantly smaller compressive load from causing insta- bility. However, we must keep in mind that, while N and N can -xxe -yy be varied independently in a rectangular panel, such is not the case for a circular or annular panel, since the following equilibrium condition must be maintained: dN N N -rr -rr -68 (56) -dr + r = 0 (56) N. = 0 1 If only the outer edge is subjected to axial compression, the second approximation, equation (111), can be solved explicitly for (N /A) o cr (N 2(1+)(3-) (115) cr 3+23 + 33 Since b B does not enter into relation (115), it is obvious that further approximations must be considered. Due to the complexity of the calculations involved, a graphical solution is employed for the succeeding approximations. In Figure 4 approximations two through five are compared by plotting (N /A) o cr versus $ for various values of b B. All quantities appearing in Figures 3 and 4 are dimensionless. Buckling loads may be obtained in the appropriate dimensions by using the first of relations (73). Figure 3. Minimum Critical Values of (N /A) for N o o (1 /. p= aA Figure 4. Minimum Critical Values of (N /A) for N. = 0 0 1 The following conclusions may be drawn from Figures 3 and 4: 1. All critical values of (N /A) are less than or equal to two. 0 (Figures 3 and 4) 2 2. As the value of b B approaches infinity, the results approach those obtained for a single-layer panel in equation (89). (Figure 3) 3. For p = 0, the present theory coincides with Huang and Ebcioglu's results (equation (88)) for a circular sandwich panel. (Figures 3 and 4) 4. The second approximation in Figure 4 is the exact solution for b B = 0. 5. An annular sandwich panel subjected to axial compression along the outer boundary becomes stronger if an equal compressive load is also applied along the inner edge. (Figures 3 and 4) 6. In Figure 4, the third approximations yield more accurate results than the fourth approximations. This peculiarity and the error bound associated with the fifth approximations will be discussed in the next section. 7. A dual response is apparent in Figure 4. As the hole increases in relative size, the panel may become weaker or stronger depend- ing on the value of b B and the range of P being considered. Such a behavior is possible because both the shear and the bending stiffness of a sandwich panel enter into the analysis. An annular single-layer panel, which can be described by only two dimension- less parameters, exhibits no such dual response. 2 - 8. If buckling loads for values of b B or ratios of N to N., not O 1 considered in Figures 3 and 4, are required, equation (114) may be used directly. However, when considering only values of p greater than one-half, the third approximation, equation (112), yields acceptable results. (Figure 4) The above conclusions are valid only for the special case in which boundary conditions (74) and (75) are applied. 5. Error Bound Since we chose to expand the solution of equation (93) about the point = 1, the speed of convergence of the series solution obtained, equation (104), depends on the proximity of the entire annular region to that point. Clearly, as the hole increases in size (5 approaches one), the solution converges more rapidly. This fact is evident from Figure 4, and also from equation (107). Indeed, it can be concluded that the speed of convergence is the slowest when $ = 0. From equations (111-114) the approximate critical buckling parameter, (N /A) can be solved explicitly for the degenerate cas' of 0 = 0. The results of this simplification are found in Table 3. However, for this special case, the exact solution (reproduced here for convenience) is available from Section 1 of this chapter: 2 o 14.684 (88) cr b B + 7.342 TABLE 3 APPROXILATE VALUES OF (N /A) o cr FOR P = 0 Order ef (N /A) o cr Approximation First (No results) Second, equation (111) 2 12 Third, equation (112) 2 12 bB + 6 10 Fourth, equation (113) 2 b B + 5 Fifth, equation (114) 2 15 b B + 7.5 If we compare equation (88) with Table 3, it can be concluded that the fifth approximation, for = 0, is within 2.2 per cent of the exact solution, even for large values of b B. And, since the series solution converges more rapidly for other values of 5, equation (114) yields results that lie within 2.2 per cent of the exact solution for all values oC b2 B and . Following the same reasoning outlined above, it can be concluded that the third approximation yields more accurate results than the fourth approximation. This peculiarity can be easily verified by expanding a series solution of equation (78) about the point r = b ( = 0). If this is done, the resulting approximations, as expected, coincide exactly with those listed in Table 3. 6. Remarks on Convergence The series solution of an ordinary differential equation possesses a radius of convergence at least as great as the distance from the point of expansion to the nearest singularity [19]. Equation (93) possesses two regular singularities, one at = 0, and another at (NQl +1) = 0; and its solution was expanded about the point I = 1. Therefore, it must be demonstrated that these singularities do not inhibit the validity of our solution throughout the entire annular region of the panel. Referring to Figure 5, it becomes obvious that the singularity at T = 0 does not restrict the required radius of convergence, regard- less of the value of 3. It is therefore necessary only to show that the singularity occurring at (NJ 2+1) = 0, lies outside the annular region and its reflection illustrated in Figure 5, for all values of 3. refleded i ncae ot&JfZLdh o 1 2-p 2 Figure 5. Radius of Convergence With the aid of equation (108), (NT 2+1) = 0 becomes: 2 S[2A(1-B) (N -2 N.)] +1 0 (116) S2(N -N.) 1 O i Therefore, the position of the singularity associated with equation (116) depends on the value of the critical buckling load which becomes known only after the solution is obtained. For the particular case in which N. = 0, (11G) is satisfied if (N) 2(l- 2P (317) A /cr 2 2 However, from equation (115) and Figure 4, it is apparent that, for = 0, ( ) 2(1+g)(3-) (118) 3 + 2 + 332 for all values of h 2'. In order that conditions (117) and (118) be satisfied simul- 2 taneously, Tl must satisfy the following inequality: 2 2(3-) (119) [2(3-3) 2(1-3)(3+2+33 )] 2 2 ' Equation (119) constrains fl to be greater than (2-s) for all values of ranging from 0 to 1. It can therefore be concluded that the singularity associated with equation (117) lies outside the annular region and its reflection illustrated in Figure 5. Similarly, it can also be sho'n that, for No = 0, the singularity associated with equation (116) lies outside this critical region. Thus, the radius of convergence, p, of the series solution of equation (93), expanded about the point T1 = 1, is: P A 1 (120) which, as illustrated in Figure 5, is large enough to encompass the entire annular region of the panel. Care must be taken, however, when imposing boundary conditions other than (74) and (75) on the solution of equation (93). Critical loads resulting from boundary conditions or ratios of N to N. not O 1 considered in the present work may satisfy equation (116) within the needed radius of convergence. It would then become necessary to either expand the solution of equation (93) about some other point, or employ the techniques associated with analytic continuation. Tne difficulties encountered in the latter approach would be enormous. CHAPTER VI CONCLUSION The present work investigates the buckling of annular sandwich panels. Equilibrium equations and boundary conditions satisfying continuity requirements were derived in cartesian coordinates, using the theorem of minimum potential energy. These equations were then transformed into polar coordinates through the application of tensor analysis. Axisymmetric buckling being assumed, and the bending rigidity of the faces being neglected, the equilibrium equations were uncoupled by using a modified technique. The governing equations were then compared with existing theories for single-layer annular panels [11,121 and circular sandwich panels [10]. For the general problem of an annular sandwich panel subjected to unequal inner and outer compressive loads, and constrained by boundary conditions similar to those employed by Olsson [12], a power series solution was obtained. This series was shown to possess a radius of convergence of sufficient magnitude. Successive approxima- tions Aore then computed, and a graphical solution was employed for various ratios of outer to inner compressive loads. Results from the fifth approximation, which were shown to be within 2.2 per cent of the exact solution, were compared with those obtained from earlier theories [10,11,12]. 72 The present work represents the first attempt to analyze the stability of annular sandwich panels. Further extensions of the present theory may be carried out by including the effects of the bending rigidity of the faces or considering boundary conditions other than those employed here. Furthermore, continued efforts should be directed toward obtaining a solution to the unsymnetric buckling problem. In this way, the assumption of axisymmetric buckling could be justified, and the problems associated with angular dependent loading functions could be analyzed. BIBLIOGRAPHY I. F. J. Plantema, "Sandwich Construction," Wiley and Sons, New York, 1966. 2. S. Timoshenko and S. Woinowsky-Kricger, "Theory of Plates and Shells," 2nd ed. McGraw-Hill, New York, 1959. 3. S. Timoshenko and J. Gero, "Theory of Elastic Stability," 2nd ed., McGraw-Hill, New York, 1961. I. N. J. Hoff, "Bending and Buckling of Rectangular Sandwich Plates," NACA TN 2225, 1950. 5. A. C. Eringen, "Bending and Buckling of Rectangular Sandwich Plates," Proc. First U.S. Natl. Cong. Appl. AMech. 1951, pp. 381- 390. 6. C. C, Chang and I. K. Ebcioglu, "Elastic Instability of Rectangu- lar Sandwich Panel of Orthotropic Core with Different Face Thick- nesses and Materials," Transactions of the American Society of Mechanical Engineers, J. App. Mech., Vol. 27, No. 3, September 1960, pp. 474-480. 7. S. J. Kim, "Symmetric and Antisymmetric Buckling of Sandwich Panels," Doctoral Dissertation, Dept. Eng. Sci. and Mech., University of Florida, 1969. 8. E. Reissner, "Small Bending and Stretching of Sandwich-Type Shells," NACA TN 1832, 1949, 9. M. Zaid, "Symmetrical Bending of Circular Sandwich Plates," Pro\c. of the 2nd U.S. Natl. Cong. Appl. Mech., 1951, pp. 413--422. 10. J. C. Ihuang and I. K. Ebcioglu, "Circular Sandwich Plate under Radial Comprcssion and Thermal Gradient," AIAA Journal, Vol. 3, No. 6, June 1965, pp. 1146-1148. 11. E. Meissner, "Uber das Knicken krcisringfodrmiger Schihben," Schweiz. Bauztg., Vol. 101, 1933, pp. 87-89. 12. G. GOsson, "Knickung der Krei sringplatte vcn quadratisch veranderlicher Stcifigkcit," Ingr.-Arch., Vol. 9, 1938, pp. 205- 214. 73 74 13. C. C. Chang and I. K. Ebcioglu, "Thermoclastic Dehavior of a Simply Supported Sandwich Panel Under Large Temperature Gradient and Edge Compiession," J. Aero. Sci., Vol. 28, No. 6, June 1961, pp. 480-492. 14. I. S. Sokolnikoff, "Mathematical Theory of Elasticity," 2nd ed., McGraw-Hill, New York, 1956. 15. Y. C. Fung, "Foundations of Solid Mechanics," Prentice-Iall, Englewood Cliffs, New Jersey, 1965. 16. A. J. McComnnell, "Applications of Tensor Analysis," Dover Publications, Inc., New York, 1957. 17. S. Timoshenko, "Strength of Materials," Part II, D. van Nostrand, Princeton, New Jersey, 1956. 18. F. B. Hildebrand, "Advanced Calculus for Applications," Prentice- Hall, Englewood Cliffs, New Jersey, 1962. 19. L. R. Ford, "Differential Equations," 2nd ed. McGraw-Hill, New York, 1955. BIOGRAPHICAL SKETCH Amelio John Amato was born in Newark, New Jersey, on January 20, 1914. He was graduated from Seton Hall Preparatory School in June, 1962. In June, 1966, he received the degree of Bachelor of Science in Mechanical Engineering from Newark College of Engineering (New Jersey). In September of the same year he entered the Department of Engineering Science and Mechanics at the University of Florida as a National Defense and Education Act, Title IV Fellow. Here, in August, 1967, he received the degree of Master of Science in Engineering prior to pursuing the degree of Doctor of Philosophy. This dissertation was prepared undrc the direction of the chairman of the candidate's supervis-oi / croniuitt ec aniijd har' becin approved by all members of that committee. It a.'.s stul..mii tdcd to h11e Dean of the College of Engineering aind to the Crnduate Council, rind was approved as partial fulfillment of the rCequireinieits for the degree of Doctor of Philosophy. June, 1970 Pean, Coll gcz of En gicieering DC-Mn, Gr1.Ddu1itC Sc-1ool Supervisory Committee: /W-
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PAGE 1 AXISYMMETRIC BUCKLING OF ANNULAR SANDWICH PANELS By AMELIO JOHN AMATO A DISSERTATION PRESENTED TO THE GRADUATE COUNOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSETY OF FLORIDA 1970 PAGE 2 UNIVERSITY OF FLORIDA 3 1262 08552 2943 PAGE 3 To my wife, Carol PAGE 4 ACKNO\\LEDG?,IENT S The author wishes to express his appreciation to the members of his supervisory comnittee: to Dr. I. K, Ebcioglu, chairman, who originally suggested the topic and, through his constant guidance and encouragement, made this v.orl; possible; to Di-s. S. Y. Lii, V. H. Kurzv/or M, A. Eisenberg, and E. H. Hadlock for theii' lielpful discussions witii tlie autlior and many valuable suggestions. The author also wislies to express his thanks to the NDEA Title IV pi'ograiu for the financial support accorded to him. Finally, the autlior wishes to thnnl; his wife, Carol, without whose patience, understanding and encouragement he would not have been able to complete this task. PAGE 5 TABLE OF CONTENTS Page ACKNO\\'LEDGMENTS iii LIST OF TABLES vi LIST OF FIGURES vii KEY TO SYMBOLS viii ABSTKACT xi CHAPTER I. INTRODUCTION 1 II. BASIC CONCEPTS 6 1. Description of Sandwicli Panel and Notations . . , . 6 2. Displacements 8 3. Sti-ain-Di splacement Relations ...Â» 9 4. Stress-Strain Relations 10 5. Stress Resultants 12 III. DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES ... 14 1. Total Potential 14 2. Theorem of Minimum Potential Energy 16 3. Equilibrium Equations and Boundary Conditions ... 19 4. Simplification of Equations 22 IV. EQUILIBRIUM EQUATIONS AND BOUNT)ARY CONDITIONS IN CYLINDRICAL COORDINATES 25 1. Application of Covariant Derivatives 25 2. Transformation of Equations 30 3. Axial Stress Distribution 32 4. RedV'Ced Equilibrium Equations 35 5. Bounciaiy Conditions 38 6. Comparison with OthejTheories 39 PAGE 6 TABLE OF CONTENTS (Continued) CHAPTER Page V. AXISY.'.niETRIC BUCKLING OF AInT\TJLAR SANDWICH PANELS .... 42 1. Uniform Axj al Stress Distributioji 42 2. General Solution 46 3. Successive Approximations 53 4. Niuiierical Results and Discussions 60 5. Error Bound 66 6. Remarks on Convergence 68 VI, CONCLUSION 71 BIBLIOGR-APilY 73 BIOGRAPlJlCrVL SKETCH 75 PAGE 7 LIST OF TABLES Table Page 1. Comparison of Equations Governing Stability of SingleLayer Panels and Sandwich Panels ..... 40 2. Lowest Value of [j. Satisfying Equation (86) 44 3. Approximate Values of (N /A) for g = 67 o cr PAGE 8 LIST OF FIGURES Figui-e Page 1. Element of Sandwich Panel 7 2. Annular Sandwich Panel 34 3. Minimum Critical Values of (N /A) for N = N .... 63 o o 1 4. Minimum Critical Values of (N /A) for N = .... 64 o 1 5. Radius of Convergence 68 PAGE 9 KEY TO SYMBOLS Y V = Displacements a' 3 "a' \ >>, y, ^^ = Displaceir.ent components = Cartesian coordinates ^'^ 2," ~ Coordinates, defined bj equation (1) i f = Thicknesses of core and faces, respectively ^c' 'f ' " = Superscripts indicating the lov/er and upper face quantities, respectivelj' t = (t + t )/t c f c y Total potential \\ \j = Sti-ain ener[;ies c' i ^\' V,' V/ = W'ork performed by external forces and moment: q ' N ' M c^ p;j Y, P= Indices taking on values x or y i j^ k = Indices taking on values x, y or z e e ^. e , e, , = Components of strain ij .orp' a.-i -iA -f T ^ T T Components of stress ij (up gS 33 G = Shear m.odulus of t]ic core c vi = Poisson' s ratio of the face f E Young' s modulus of the face p = Bending rigidity of the face A . 5 = Kronecker delta = [(l-v)/2];5 6. +6 0. +[(2vJ/(l-vJ]5 6 YM^ N ^ Pre-huckliiig axial forces per unit length PAGE 10 q Lateral load per unit area Mo = Externally applied moment per \ini t length -dp Q Transverse sheaiforces per unit length a l\ . = Bending moments per unit length N r> Axial forces per unit length corresponding ^ to small deformations during buckling (n) = Bracketed index indicating physical component of a tensor I f ] = Chri stof f el symbol of the second kind 'nq-' X = General coordinate variable n, X,q , s , t ,m,p = Indices taking on values 1, 2 and 3 Xm ,, . Â• . e.g. Metric tensor ^ ' ^Xm X , Y = Arbitrary tensors n n r, G = Polaicooi-dinates 2 V = Laplacian operator u = Pre-buckling lateral displacement of m.iddle plane of faces u = Latei'al displacement of middle plane of faces for axisymmetric equations N. , N = Compressive axial forces per miit length applied to inner and outer edges, respectively a, b = Inner and oviter radii of annular panel, respectively P = a/b D, E = Defined by equation (65) dw ^ "" dr F, G, H, I = Defined by equation (72) PAGE 11 A, B = Defined by equation (73) Tl = r/b N, R, Q = Defined by equation (94) Â§ Tl 1 A = Coefficients of power series p, = Buckling coefficient for uniform axial stress distribution C= Arbitrary constant of integration J , Y = Bessel functions of order one D = Radius of convergence PAGE 12 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AXI SYMMETRIC BUCKLING OF ANNULAR SANDWICH PANELS By Amelio John Amato June, 1970 Chairman: Dr. Ibra]iim K. Ebcioglu Major Department: Engineering Science and Mechanics The buckling of an annular sandwich panel is investigated using the theorem of minimum potential energy. Governing equations, dei'ived in cartesian coordinates, are transformed into cylindrical coordinates by means of covariant differentiation. Considering t]ie faces to be membranes and assuming an axisyiiimetric buckling m.odc, the equilibrium equations are uncoupled through the application of an improved technique. For a clamped outer edge and "slidei-" inner edge a sej-ies solution is applied to tlie general problem of ?'adial]j' varying in-plane stresses. Critical stresses ai'e plotted versus ratio of inner and oxiter radii. The fifth approximation is shovm to yield acceptable results. PAGE 13 CHAPTER I INTRODUCTION A sandwich panel is defined as a three-layei' panel, consisting of tv.-o thin ontcilayers of high-strength material between wliich a thick layer of low average strength and density is sandwiched. The two thin outer layers are called faces, and the interniedi ate laj-er is the coi'e of the panel [1], Among tlie main advantages of sandwich construction are: a high rigidity to weight ratio, good thermal and accoustical insulation, and case of mass production. Some examples of core materials are balsa wood, cellulose acetate and synthetic rubber. However, in more recent times tliin foils in the foi-m. of hexagonal cells perpendicular to the faces liave been employed. Depending upon the intended application, faces may be construcied of aluminum alloys, high-strength steel, etc. The practical importance of sandwich construction came into prominence with the advent of the airci'ait and space industries. With the need for lighter, stronger and more stable structural components, great emphasis was placed upon the design and a.nalysis of workable sandwich panels. Numbers in ^rackets refer to the Bibliography, 1 PAGE 14 The essential difference between the analysis of single-layer panels and that of sandwich panels is tliat the shear deformation associated with the cere of a sandwich panel may not be neglected. Moreover, initially plane sections no longer remain plane during bending, and the existing plate theories [2,3] require extensive modifications. Numerous authors have contributed to the development of matliematical theories describing the behavior of rectangular sandwich panels. Two of the more noted of these are Hoff [4] and Eringen [5] wlio, early in the 1950' s illustrated a concise and straightforward approach to tlio problem using the theorem of minimum potential energy. In 1960 Qiang and Fbcioglu [6] inti-oduced continuity conditions for displacements across the interface of two adjacent layers. This modification has been shov.ri [7] to contribute appreciably to the accuracy of the derived equations, wliile introducing no additional mathematical complications. In comparison, cii'cular sandwi cli panels liave been tlie subject of relatively little investigation. In 1949 Eric Reissner [8], neglecting the bending rigidity of tlie faces, solved tlie pj'oblemi of a cii'culai' sandwich panel subjected to axisyimnetric transverse loading. Later Zaid [9] included tlie effects of the bending rigidity of the face layers. Huang and Ebcioglu [10], using a teclmique similar to that employed by Zaid, recently investigated the axisynmietric buckling of a circular sandwich paiiol subjected to uniform axial compression. In t}ieir final results, the faces were treated as momljranes. PAGE 15 Prioilo the advent of sanchvich construction, the stability of circular and annular single-layer panels were extensively investigated [3]. Jleissner [11] analyzed tlie axisyiimietric buckling of a singlelayer annular panel subjected to uniform compj-ession along the outei' boinidary. In this vork, the inner boundary was considered free and the outer boundary simiDly svippoi'ted or clamped. Olsson [12] extended Meissncr's analysis by considering tlic outer boundary clamped and the iniier boundarj' "slider." Sucli a condition could be approximated by allowing a shaft or rigid cj-linder to occupj' tlie central hole (see Figure 2) . Prompted by tlie foi-egoing sequence of investigations, the main objectives of tlie present analysis are: (i) to parallel the sequence of analysis present in the literature of single-layer panels by investigating the axisynunoti'ic buckling of annular sandwich panels; (ii) to acliicve a more satisf actorj' foi'mulation of tlie theory througli the use of continuity conditions [6]; (iii) to modify the uncoupling procedure introduced by Zaid [9] and later employed by Huang and Ebcioglu [10], tlicreby signi f icaiitly reducing the complexity of the uncoupled equilibrium equations; and (iv) to apply the boundary conditions employed by Olsson [12] lo the present work. For the initial derivation, the set of basic assumptions employed in the picsent anal3'sis are: (Al) The effect of transverse normal stress in each layciis negligible. (A2) Tlie core undergoes; shear deformation only. (A3) Displacements in each layer ai-e linear functions of distance from Ihe median plane of the layer. PAGE 16 (A4) The median plane of the core remains neutral during small deflections of the panel. (A5) Each layer is homogeneous and isotropic. (A6) The core is attached to the faces securely. (A7) Hooke' s law is valid throughout. (AS) Local buckling of the panel does not occur. These assumptions are similar to those iised by Hoff [4] and Chang and Ebcioglu [13]. Equilibi'ium equations and boundary ccndi Lions arc derived, using tlie above assumptions and the th.corem of minimum potential energy. These equations are then transformed into polar coordinates thi-ough the application of tensor analysis. In order to solve our problem, the resulting equations are then simplified further using the following add i 1 i on al as su)iii:>t i ons : (Dl) Bernou] li-Navier lij-potliesis is valid foi' the faces. (B2) Faces are considei-cd to be membranes, (D3) Annular sand\^Â•ich panel subjected to uniform axial compression buckles axisj-mmetrically. The first assumption is valid wlien the sliear deformation of the face layers is negligible. Kim [7] lias sho\ni this to be true for most practical applications. The second assumption, which has been employed by many authors [8,10], is discussed at length by Plantema [1]. Finally the assumi)tion of axisymmetric bvickling cannot be rigorously verified without solving the more difficult problem of unsyraraetric buckling. However, such an assumption v.ould seem I'easonable since Olssoji [12] h.as proved its validity for the analogous probleji. of a siogle-layer panel witli siifiilr.r boundary coi_di tions. PAGE 17 When comparing the present work with existing and subsequent theories, the words panel and plate may be considered interchangeable. Numerical computations are carried out bj' a desk calculater. PAGE 18 CHAPTER II BASIC CONCEPTS ^ Â• r)c scrlption o f Sandwich Panel End Notations An element of tlie sandwich panel to be considered in this work is sIiouTi in Figure 1. For convenience, three coordinate systems are defined having common xand y-coordinates , and transverse coordinates related as follows: t + t. (1) t + t^ // _ _c f Z Z 4 2 wlierc t is the tliickncss of the core and tÂ„ is tlie thickness of the c f face layers whicli are identical. 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 vmprimed quantities are used for the core as ivcll as for general relations. Subscripts c and f denote quantities related to the core and face, respectively. The total tliickness of the panel is assumed to be very small in comparison with the lateral dimensions. Also, each layer is considered to be isotropic and h'')mogencous, witl: the pi'operties of the faces characterized by E and V , and those of the core characterized by G^. PAGE 19 3 z 3 i' X -X -Fic-ure 1. Element of Sandwich Panel PAGE 20 Arbitrary transverse load q, axial forces N ^, and external Â— O'P moments H g are the only externally applied forces or moments which may contribute work to tlie system. For the initial dei-ivation and formulations, indicial notation will be employed. Greek indices 0! , p, y and |j, will take on values 1 or 2, while Latin indices, wlien employed, will take on values 1, 2 or 3. Any repented index denotes summation, wliile any number of indices preceded by a comma indicates partial differentiation witli respect to the coordinate variable represented by those indices. 2. Displacements In the following relations, w]iich are valid for small displacements, it will be assumed that within each layer of the sandwich panel plane sections are preserved, although not necessarily perpendicular to the deflected middle surface. This will constitute an approximation since the presence of transverse shear suggests a non-linear variation of the longitudinal displacements through the thickness. Also, while the panel is geometrically syimiietric with respect to the middle surface, the displacement relations will not, in general, reflect this syiimietry, since unequal bending moments may be externally applied to each face layej'. llieref ore, wi tli the aid of equation (1), our displacements can be v.ritten in the following form: V^(x,y,z') = u^(x,y) 4 z'.;^'^(x,y) V^(x,y,z') = w(x,y) VQ,(x,y,z) r. zv^,^(x,y) y^(.^,y,y) = w(x,y) (2) V^(x,y,z") = u^(x,y) + z/^.^(x,y) V^(x,y,z") :. w(x,y) PAGE 21 where (V ,V ) is the displacement of a generic point in the sandwich panel; u represents the displacement in the xy-plane of a point lying in the median plane of eitlier face; i}, represents the angle that the normal to the median plane of each layer rotates v.hen the planes are deflected; and w is the transverse deflection which is assumed to be constant througli the entire thickness. In Older tf ensure continuity at the interface of any two adjacent layers, the following conditions must be imposed on equation (2) t. t / _ _f I / _ __c I (3) "0^ "^ 2 ^'a ~ " 2 ^u Using (3) to eliminate ! and u from (2), our displacement relations become: \ \' ^ u, + z V (4) V^ -U^ + z A + (A 4 ) a ct 'Or 2 ^cy *Â» V3. w 3. S train-Pi s]>l acement Relations Linear strain-displacement relations are defined by [11]: Â®C^3 " 2-^cy,? ^ ^'g,Â»^ ^C.3 = l^^'c.,3 ^ ^.e.) <^> e -V 33 3.3 PAGE 22 10 Using (4) in conjunction with (5), the stiÂ—iin-displacement relations for each layer becomej e' = 33 u^ 4 Â— (yÂ„ Vr^ ) + z w ^ ] <3 = Ik ^ '\a^ e 33 4. Stress-Strain Relations (6) Tiie generalized Hooke' s law for a liomogeneous isotropic body can be v.:-itten in tlie following foi^n [15]: T. . --. -^[e. . + -^ 6. . e, , ] (7) where the Latin indices i, j and k take on the values 1, 2 and 3. PAGE 23 11 \Vlien the transverse normal stress, T , is neglected, (7) may be rewritten as [15]: Â©Â•3 1-V o;^ 1-v Â£<-3 YY T =: 33 or, ^ A ^p i-v^ ^f^Y^^ Y^^ (9) whei'e, A Â„ = 2:::!:i(6 6Â„ + 6 6, + ;^ 5 ,. 6 ) wi til , A, =AÂ„ =A_ =A Â„ C.cy[i ^oy[i a?i[Ly y[uyp If the lateral, in-plane stresses, T Â„, are also neglected, (9) reduces to: T =-^e^=2Ge^ (10) 03 1+v 0-3 Q'3 Assumption (Al) implies that equations (9) may be used for the face layers, while assumption (A2) suggests the use of equation (10) for the core. PAGE 24 12 5. Stress Resultants In order to reduce the tliree-dimcnsional elasticity problem to a two-dimensional one, the following stress resultants are defined: ^3 ^^ *f 2 ^f 2 PAGE 25 ^W = Â°f ^Â«3Y^l ^.^x m'' = D A Â„ ii" 13 N _ = A_ C^3 ^_^2 * C'^YMY'^^ Op f f ^ f 1 = -^ A ^ i -u' + -~ (t' f ' ) (12) t E // f f // ^a 2(l-iV',) "^^(7 .o' Q r. 2G [u' + -(t w t. ly')] O' c a 2 c ,Â» f ^cy whore the bending rigidity of the faces, D , is defined: ^f ^ f f 12(l-v^) PAGE 26 chapt]:r III DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES 1 . Total Potential The total potential consists of the strain energy stored in tlie panel dui'ing a small def oinnation and the total work pei'foi'med on the panel by the external forces and moments dui-ing that deformation. In general, the strain energy stored in the body is defined [14]: U r_irrr t e dv (13) 2 J^c> ij ij In pa^'ticular, as a result of assumptions (Al) and (A2) , the strain enerq,ies associated witli the faces and core are: < I rjr C^'. ^'r + 2t' e'j dv U = ifff (2T Â„ e Â„) dv (14) ""l-Ull '\^ V-^'^^3<3^ ^^" where the integration is performed over the volumes of the individual layers. 14 PAGE 27 15 Also, since the only externally applied forces or moments which are allowed to do work are the transverse loading function, q, the axial forces N ^, and the external moments M ^, the work performed on the panel is defined bj': W = rp qw da a 1 r-r X. K = T n N ,(V' o + V' + w w ) da' a K^ " 7 rr N n(v" + V" + w w ) da" (15) W = r (m'Â„ (' + M-'' (/') n_ds s ' where the integration is perfoi'med over the ai-eas and boundaries of the individual layei's. In the second and third integral expi'cssions of (15) , w]iich are analogous to tliose employed by Timoshenko [3] and Eringen [5] , it is assumed that tlie total axial load is evenly distributed between tlie uppei' and lower faces. Therefore, the total potential, whicli is defined as 1r = V' + V + U" W W' \/' W (16) f c f q N N M may be rewritten, using (14) and (15), in the following form: PAGE 28 16 y = ifff (t' e'Â„ + 2t' e' ) dv' V + Ini (2T . e ^) dv + i nj (t\ e% + 2t'' e\) dv" ^ V (m' i'' + M^' i;,'") n ds (17) -a3 cy -a3 V 3 2. Theorem of Minimum Potential Energy The theorem of minimum potential energj' states that of all displacements satisfying the given boundary conditions, those whicli satisfy t]ie equilibrium equations make the total potential energy an absolute minimui;i [14], Tliercfore, equilibrium equations and boundai-y conditions for tlie sandwich panel are given by the variational equation: 6 y = (18) However, before applying this extremum principle to equation (17) , we note that: ' PAGE 29 17 and similarly, 6 (t' e'^) = 2t' 6e',^ (20) 6 (T Â„ e ^) = 2T _ 6e Â„ (21) Q'3 cy3 , Q'3 Q'3 6 (T% e" J = 2T% 6e% (22) q3 qo q'3 q'3 6 (t" e". ) .= 2t'^ 6e'' (23) op QP Qp Q^ Therefoi'e, using equations (19-23) in conj\n:ic.tio3i with displacement relations (4) and strain-displacement i-elations (6), the variation of equation (17) yields: 6y = IT ^(ou r + 7. 6v ,) + T , (6(( + 6\v ) dv V .u;.| c'j L t. 2 .Qt Q' J V c c + Ht Â„ 6u + z 6i^ + (6^, _ 6-^ ) Â•Â•'<';' I. GP L <^,P CV.P 2 ^Q',3 ^Q,P J Â• v' + t'' 6 a' + 6w idv" Qf3 L Â» jOj -Â• ff q 6\v da + ff N . r2u6\v ^ + ~ (6i|i' Â„ 6a" ^) da r m' 6iV' + m" 6il''' n-, ds = (24) ^ L-G'P ^Q-cvS ^o-J p In equation (24) the syiimietrj' of the stress tensors and strain tensors has been employed. Also, the displaceme:its PAGE 30 associated with N Â„ arc those occuriing in the median planes of the face laj-ei's. Using relations (11) to reduce the volume integrals In equation (24), and applying the two-dimensional divergence theorer,; of Gauss to the resulting area integrals, the final integral equation is obtained. As an example of the above procedure, consider the first Xai-ia in the integrand of equation (24): t^/2 da TJ/t' 6u' , dv' = rr J t' 6u' , dz' / a -t^/2 ' V f = ff n' 6u' da a . n[(^\ 6u') , n' . 6u'"]da = f n' 6u' n ds rr n' Â„ 6u' da (25) s a After applying a similar procedure to the remaining terms, and collecting coefficients of similar virtual displacements, a complete description of tlie sandwich panel in the equilibrium state is obtained in integral form: 6r rr { Tq Q N ^ w _ q' q" "1 6w ''^ < L. Of, a -ap ,cyp a, a o:, PAGE 31 19 + Q' -^ n'' ^1 5ik' + r~ n" ^ Â„ ^f' Â„ + q" I 6f'| da a 2 0-3, f J ^a L2 cy^,3 oP.P oj ^d + r i Q + N Â„ w . + q' + q" 6w ^ I \_ a -op ,p a cyj L cc?> o.SJ 3 \_6 -a^ q3 2 q-3 -la-gj ^ P + ~ N ^ -^ n'' + m'' k' I 6-^''} n ds ^ (26) 3. Equilibrium Equations and Boundary Conditions If v.e apply tlie fundamental lomma of the calculus of variations [15] to equation (26), we obtain the following equilibrium equations in terms of stress resultants and moments: N Â„ w 5 (Q -I + Q ) = q 2Q T^-<3.3 "^''aP.B^ Â° t ^ t t ^ Q' Q-3 , 3 Q' 2 QÂ§ , 3 -^ n\ Â„ m'' .+ q" = (27) 2 Qi::> , 3 QP , P PAGE 32 20 r-t t -, If v;e substitute the stress resultant-displacement relations (12) into equations (27) and (2S) , and collect similar terms, we obtain the following equilibrium equations in terms of displacements: 1 S^f 2G [u' + (^ \^' tj' )] N Â„ w a -Â— i [2w c a,cr' 2 c ,(ya fa, a -a3 ,op 2(l4V ) ,aa + (^Â•' +'i" )] = q "a, a 'a, a 4G ^ t E -Â— ^ [u' + (t w t^il')] + Â— A ^ [4u' ^ + t (*' ^ t^ a 2 c ,a i^a'' 2(l-v^) ""'^"^'^ V.^^^ f \,P^l f' )] = -2t,G , t^E^ ^C^/ 1,, j,'sn ff ri' T Â— [u + (t w t i; )] + [(( + w ] t a 2 c ,Q' IV 2(l4V^) ^a ,a C 1 + ^ A u' D. A . [4^ Â„ 3il'" f. ] = 2(l-v ) ^'^"^^ '^'^'^ '^'''^^ "i'^VY.PM' -t^E t E Â— \ A Â„ u' Â„ + Â— iiÂ—[(/' + w ] D^ A ,, [3*' . 2(1-V^) ^">'^'' '^'^^' 3'^^+^-^f> ^ '^ ^ ^^v^^ *7.P^i + 4f' ] = (29) PAGE 33 21 and boundary conditions, f f f ^2G [u' + i (t w t^^')] + N p w . + ^,: ^ , [2w Leo2 c ,a f^QÂ— ^ -Â• "/-... \ -QfP ,P 2(l + v^) ,Â« + a' + ilr"] r 6w n ds = / \ ^A [4u' t (^' f )]ku' nÂ„ ds = iÂ— ^ Â„ + Â— AÂ„ Â— (i) -\k )-tu + D^ A -. \|f' M -,^ 6 PAGE 34 22 4. Simplification of Equations In ordeito simplify equations (29) and (30), we exclude the possibility of externally applied edge moments by specifying: m'Â„ = m" = (31) -ap Â—dp which reduces the number of dependent variables from seven to five since (31) suggests: il' = f (32) a a Also, if we assume that plane sections remain poi-pendicular to the deflected middle sui-facc of each respective face (asssumption Bl) , we have, (r = w (33) a .a whi c]i further reduces the number of dependent variables to three. A final assinnption (B2) which simplifies the problem by reducing the order of tlic differential equations is [1]: t Â» t. (34) c f whic]: justifies neglecting the bending rigidity of the faces, D , compared witli tlie bending rigidity of the panel as a whole. Conseqviently . when equations (29) and (30) arc simplified throxigh tjie application of (31-34), we obtain the following fc..;uilib.tium equations: PAGE 35 23 2 -t t G (-Â— u + Â£ V.' ) N Q w = q c c t cyjCv .Q'Q' -ap ,ap c 2t E 2G [-^ u + Â£ w ] ~ A ^ u ^ = (35) and boundary conditions, Â•{2t t G -^ + V,+ N o w r. I 'Sv.n ds = "J L c c Lt 2 jqJ -op ,PJ ex s c , r2t E . r 1 Â— Â— ^ A . u ^ 6u nÂ„ ds ^ (36) s I .^ Equations (35) rn;d (36) can be rewritten in extended form as: i: t G c c r 2 ^ ~\ ^ r 2 f '] Â— u + i w t t G -Â— 11 -I t w Lt X,:; ,xxj c c LtÂ„ y,y ,yyj N v.2X w N w = q -XX ,xx -xy ,xy -^-y ,yy T' 2 ^ "I ^f^f 2G I Tu + t w [d+V )(u + u ) c It X ,x I ,2 f x.xx y.xy "c ' 1-V ' ^ ' + (1-V^)(u + u )] f x,xx x,yy ; r ~ u -1 f -.v 1 Â—-4 [(l-v^)(u -;u ) c Lt^, y ..yj ,_^^2 i y,yy x,xy + (1-V ) (U +1.1 f y,xx y,yy)J ^0 (o/) PAGE 36 24 and, + N w r6'An dv = X -xy ,yj f -^ 2Â£ t G ~ 4 W + N V/ Â•J L c c Lt^ 2 ,xj -XX , X , pU <^ Â— j .^ f J2f 1 G -^ 4 w + N w + N w f 6w n Â•J L c c Lt^ 2 ,yj -yy ,y -xy ,xJ ^ l~T-L^,x-^^f "y,yj|^\"x^^=Â° f i llT^ f"x,y ^ "y,xV ^^ \ ^>' = Â° ^x 2t^E_^ Â•^ ^-1 ,2 y,y f x,x J y y f irÂ—-Â— [u +11 ][ 6u n dx :^ ^ LlHV^ x,y y,x J X y dx = y (38) 1 <; C f and primes and double primes have been oiiutted without confusion as a 2"e5ui t of equation (31). Equations (37), which ai-e a special case of equations (29), are identical with those derived by CTiang and Ebcioglu [13] if thermoelastic effects are neglected. It should be noted tliat when simplification (33) is introduced into equations (29) and (30), equations (,'3r->) and (36) are riot immediately pi'od^iced. The lasL two of equations (29) niust be i-eturncd to the area integral of equation (26) where, as a result of (33), tJicy become coefficients of -5w . Conseouently , a transformation procedirre similar to that i J lu;.ii-at en in eqi.a'.iou (25) yields additional terms v.hich contribute to equatiois (35) and (3G). PAGE 37 CHAPTER IV EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS IN CYLINDRICAL COORDINATES 1. Application of Covai-iant Derivatives Let us consider a covariant derivative of an arbitrarj' covari ant vector, X , [16]: n Sx X I = Â— i' { ^} X, (39) n|q ^ q 'nq-' i . In (39) y is a general coordinate variable, f 1 is the nq-" Chrjstoffel symbol of tlie second kind, and all indices take on values 1, 2, or 3. Eui'tliei'niore, when a vertical line precedes any nnniber of indices, it indicates covariant differentiation with I'espect to the cooi'diiiate variables represented by those indices. In considering the second covariant dei'ivative of an arbitrary vector we must recall tliat in order to sum two indices we must have one covariant and one contravariant index. Therefore, inti'oducing the metiic tensor, g , (39) implies: X ,^= g-^^^X , = g'^f-^{ } Xl (40) n| *= njq \_ q. "-nq^ ij And since tlie covariant derivative of an arbitrary second order mixed tensoi', Y , is [1 61: 25 PAGE 38 26 ox we have, fi'om (40) and (41): 'Sx" Bx 8X Sx Bx n|pt %s n|t Ss \^t L^ ^^q r m , r sq m ^Q r ^^ -i ^~1 1 , .^^ In cylinrlri cal coordinates, the Christoffel symbols and iTietric tensoi' take tlie following form [16]: ^2-^ ^21^ 1 ' '22-' ^ all other [ '( = 11 33 ^11 ~ ^33 " ^ " ^~ 1 *> 99 1 9 529 = PAGE 39 27 SX 11 ax ' OX ^1 1 2 " ~2' ~ "T ^2 ' dx X ^^'2 1 ^2|l 71 ~ "T\ ' Sx X 3 2 Sx, Sx' 3 1 (45) and, S"x, 1 11 ^, 1,2 S(x ) _^i a!!2 1^ ^. '1J22 2,2 1,2^^ . 1 ~ -^1 ' s(x ) X ax Sx 1 12 2 11 ., 12 .1^2 1 -, 1 ' d(x ) (X ) X Sx '"''^ , 1 ^\ 1 ^\ ^^ ^2l92 = 2~â€¢ ' ^^ ~2 + >^ ~T " 2X ' S(X ) 3x^ SX^ ^ a^x, PAGE 40 28 PAGE 41 29 u = u y e 1 1 y y r 6,9 r r 1 1^ xjy " r r,e r W; = W y r ,e X XX 1, rr 'xjyy ^ 7 ^.ee ~2 "e,e "^ ? ^,r " ~2 "r ^Ixy = I "r,re " "2 \,e " ^ ^G.r "^ \ "o yjxx G,rr V|yy " 3^" ''e,6e "^ "I ^,e ' F "e.r " "T "e V|xy i "e,r9 " -2 "0,6 ^ ^ ^^ , r " "^ ^ .1 . _ 2 yy 1 ^ , i ,, '2 ^'',00 ' r '\r (49) PAGE 42 30 In equation (-19), r and 9 are polar coordinates defined in the usual manTier, while u , vi and \v represent displacements in tlie direcr 9 tions associated with tliis new coordinate system. 2. Transf ormat j on of Eqviations If we interpret the derivatives appearing; in equations (37) and (3S) as covariant derivatives and thereby transform them into polar coordinates using relations (49), we arrive at the following equilibrium equations: -t to {AriA(ru).lu^ 1.?. V^w} c c Ll 1 r or r r 6,9 J J c ^r ^rr " ^^-G ^^ ".rO " ^2 ^e^ " ^96 ^^ ".99 ^ I \r^ ^ ^ r r ^^^f^V r 2 1 /''f"^ J Â— U 4 t w Â— 7 u Â— Tu cLt r ,1'J -, 2 L 12 c 1 -V . r r"("9,r9 -^^,99^]= Â° r 2 t 1 'f ""f r L.l^ t3 r ,9j i..,,2 L f 1 ^''~"'f' 1 u .) + -;iU ., -r r,rG ^ r,o_J (1+v^) -I Â— t E I If I ? 1 2GJ ~ uÂ„ 4 Â£., J _ -i-i I (l-v^)(v^ Ug -'^Uq) 1 1 ^^'-~^f^ + (l-i-v^)(-Ug p^^ + u.. ..J + u__ ., I -(50) r ' and boumlaj-y conditions, PAGE 43 31 + N ^ Â— w ^h 6w n dG ---^ , r Â— r9 r y, p-u A -, f -^2tt G -^ + w + N w >> l celt 2 ,rj Â— rr , : "c 6^2t E p 1 ~1^ r -j Â— j u + V (u -h u ) \\6u n de Â•-. Â•J L 2 L 1' , 1' f 1 , 9 r r JJ r r f r {r^ [u -u + u I } 6u n dG -(51) U+v Lr r,G r G 6|rJ J r 2 2 2 S IB 13 where V = --+ --+ -rÂ— 5-,2 r or 2 -,^2 6v r dG Since onlj' complete annular or circular regions are to be considered, the use of continuous cyclical functions of G eliminates the need to specify boundary conditions along' a radial boundary. In the present work, which investip;atcs the buckling of annular sandwich panels, tlie transverse loading function, q, is not considei-ed. Such a restriction does not limit the application of the obtained i-esiil'ts, since, for snail deflections, the ti-ansverse loading fiuiction does not influence the buckling load [3], Finally; as a result of uniform compression along cho inner and (oi-) oi'.ter boiindaries, an axisynunetric buckling mode is assvimed (B3) to result from the lowest critical axial load. Such an assumption has been sIiov.ti by Olsson [12] to be valid for single-layer annular panels. The governing equations thus become: PAGE 44 32 _ U G {A [I -f (ru)] f t V\} ^ A (, N ^) = (52) c c Lt I 1' <^'i' J J r dr Â— rr dr'^ c r ?' i, cl'-'H ^^i'^I Ttl^u 1 du u~l Â„ ,^Â„^ c L ^ cii' I , -^ L , 2 r dr 2 j "c -Â• 1-v dr r -' r.2 1 cl , cK wnere V = r(r -Â— ) r dr diand , e -^ 21;t Gj -Â—+---+ X Â—-^ 6w n dQ = (54) <-' L c c Lt 2 di'_J Â— rr drJ r ^ illii + V ^6u n dG .(55) Ldr 1 rJ r In equations (52-55) the subscript r has been omitted from the displacement u, witliout confusion. Also, N has been eliminated by considering Uie pre-buckling equilibrium of the face layers: dN N N.^ Â— rr Â—rr --9t ^ /Â•cr-\ + = (56) dr r 3. A^riEil^ S tress Distr ibution For two-dimensional axisymmotric stress distj'ibution in polar coordinates, th.e fjoveriiing equations are [17]: d!" ., i. ^ _ A ^ (57) , 2 r dr 2 dr r l^ ^ + Â„ _ (58) -^Â•^' 1 Vci7 + ^f 'f PAGE 45 33 2t E f f /u du\ ^99 = --7t ' '' d-rV ^-^^) where u is the pre-bi!ckling lateral displacement. The general solution of equation (57) is: u .= C_r + C /r (60) i) 6 where C and C are arbitrary constants of integration. Substituting O D (60) into (5S) yields: N^.^. --^ I C (l.v^) -C -^ (61.) l-v_^ r -^ For the case of an annular panel subjected to uniform compression along tlie inner and outer boundaries (see Figure 2), boundai-y conditions are: N (a) = N. Â—yy 1 (62) N (b) =:. N where a and b arc inner and outer radii, respectively, and N. and N are inner nr.d outer compressive forces per unit length, respectively. Imposing boundary conditions (62) on equation (61), we obtain the follO'VJng axial stress distribut j 01: N^r = "2 ^ <62) N^Q . -^ F (64) PAGE 46 34 PAGE 47 35 wliere , and. D . ^V; E -. -^ -i (65) e (66) b Since K _ and X are considered to be much larger than the forces produced bj' bonding during buckling, the axial stress distribution remains essentially unchanged during buckling. Also, if a = S r; , or N = N , it is easily verified that ' ' i o Â•^. equations (63) and (64) reduce to: N = N.. = N (67) Â— rr Â— 9G o 4. Reduced Equili'orium Equations From CHiaptcr IV, Section 2, the equilibrium equations (reproduced liere for convenience) are: _ U G fl [1 f (ru)] . i vU i f (r N ^ . (52) c c Lt I r dr J r dr \ Â— rr dr/ c c Lt dr -.^1,2 r dr 2 J c -' 1-v. "-dr r (53) Zaid [9] and Huang and Ebcioglu [10] unco\iple siinilar equations by ope:'ating on tlie second witla tlie linear opei-ator L(9) = -~ (r,o) T r dr ' PAGE 48 36 and eliminating the fii'st large coiiunon bracket in both equations. The resulting equation is then directly integrated to j-leld: 1-v^ r' r' r' t.t tE f c 1 Â— Ir N -rÂ— drdr + -Tri'lnr-Â— +Â— -r+ Â— r ^ J -rr dr 2 \ 2/2 r_ (68) where C, , C^ , and CÂ„ are arbitrary constants of integration. 1 ' 2 ' 3 Equation (G8) agrees with Huang and Ebcioglu' s results if Is is defined by equation (67) instead of (63). It sliould be noted, Â— rr hov.-ever, tliat tJie uncoupling procedure described above yields five constants of integration, whjle only four boundarj' conditions are available. This is a dii'ect result of the uncoupling procedure wliicli initially increases tiic order of equation (53). As a consequence of the above-mentioned complications and other considerations '.vhich will be discussed later, a different technique is employed in the present work to uncouple equations (52) and (53), Multiplying equation (52) by r and integrating directly without the aid of eq\iation (53) , we obtain: r~t t N -, C^ . c Â— rr I 11 .,,^. I^ 2tG -^ 2?G ' where ^ dw , .^ dr and C is an a^bi tvary constant of iiitegration. PAGE 49 37 Comparing equation (69) with equation (68), the advantage of simplicity becomes inmiediately evident. Also, it will be made clear in the following sections that the present procedure facilitates the application of boundary conditions, and suggests analogies between the classical single-laj^er plate theory and the present analysis of sandwich panels. Substituting equation (69) into equation (53), with the aid of (63) and (70) , we obtain, after some simplification: 3 2 BC T (,^0)r2 ^ , (,2_3c) r ^ , (jlr^ Ir^ . 3G)cp . -i(71) dr where,. c.JI, -f . -^S!. r.A-l t Â£^G G (l-v^) c c c f f c f and D and E are defined by equation (65). Equations (60) and (71) represent the uncoupled equilibrium equations. Since (71) is a second order differential equation, two constants of integration are generated. Together with C and the constant introduced through the integration of (70) , we have four arbitrary constants of integration to be determined bj^ boundary conditions at the iuiier and outer edges of the annular panel. PAGE 50 38 5. Bounclai-y Conditio ns From Cliapter IV, Section 2, the boundary conditions (reproduced here for convenience) are: /{2U G [f + I ^"j + N ^} 5w n de . (54) ^ l c c Lt 2 dr J Â— rr drJ r c -' ! {B + ^f ?} '^ \ ^'^ = <^5) Integral (51) requires the specification of either the transverse deflection w, or tlie resultant shear stress along the inner and outer edges. Similarly, integral (55) requires the specification of either llie relative lateral movement of one face witli respect to the other, or the moment produced by tensile stresses on one face and compressive stresses on t))o other, along the inner and outer edges. Since we are neglecting the bending rigidity of the faces, the large bracket in integral (55) represents the total edge moment, wliile u becomes analogous to the slope, cp, used in the formulation of boundary conditions for tlie classical theory of single-laj^er circular plates [3], Specifically, in the present work, integrals (54) and (55) are satisfied thi'oug]i t'.io following choice of boundarj' conditions at the inner and outei' edges: ptt N , -, At r = a: u + j -Â— + -^^ qi = ; u = (74) Â•^ 2tG -^ At r = b: w ;.: ; u = (75) PAGE 51 39 Conditions (75) are analogous to boundary conditions termed clamped or built-in in the classical thcoi'y, while conditions (74), termed "slidei'" in tlie present v/ork, have previouslj' been emploj^ed for stabilitjanalysis of sandwich columns [1] and singleÂ— layer annular panels [12]. Physically, such a restriction could be approximated by allov/ing a shaft or rigid cylinder to occupy the central hole (see Figure 2). As tlie inner radius, a, shrinks to zero, conditions (74) become identical to boundary conditions present at the center of a circular san.dv.ich panel, witliout central hole, constrained alo)"Â»g the outer edge only. This limiting process provides a check for our final results since the stability problem associated with such a panel yields a relatively simple solution. 6. Comparison witli Other Theories Coir.parison of the present work with existing theories can be facilitated through the use of Table 1. Referring to this table, an analogy between single-layer theory and sandwich theory is observed. Meissnei-' s [3 1] equation, wliicl: can be solved by means of bessel functions, is leduced to a homogeneous differential equation (1: = 0) if the sliear I'esultant is made to vanish at any arbitrary radius. This becomes immediately evident if we compare the boundary condition which specifies zero shear with the second form of the equilibrium equation. Consequent Ij", since the shear at the center of a circular panel must varnish because of symmetry, Timoshenko' s [3] bessel equation of oi'dei one is a horaogeneous differential ccuation. PAGE 52 40 PAGE 53 41 Analogously, one of two equilibrium equations describing an annular region for a sandwich panel reduces to a homogeneous equation (C = 0) if tlie shear resultant is specified to vanisli at some arbitrary radius. Such a simplification also provides a direct correspondence between u and cp (see equation (69)). In the case of a circular sandwicli panel, a homogeneous equation again results as a consequence of the symmetry involved, and one of the two equilibrium equations yields bessel functions of order one, as is the case in the single layer theoiy. The second equilibrium equation for a circvilar sandwich panel, which is attributed to Huang and Ebciog]n, is deduced from equation (69) ratlier than the original form in which it appeared (see equation (68)). Consequently, without employing the present uncoupling technique, the above stated analogies would not be evident. Since the present work deals with an annular sandwich panel, it becomes obvious tliat bessel functions cannot be employed unless N =N (see Equations (6r;) and (67)). Solutions for this special case o i and the more general case are olitained in Chapter V. PAGE 54 aiAPTER V AXI SYMMETRIC BUCKLING OF AXNTJLAli SAJvTiWICH PANELS 1. Uniform Axial Sti'ess Distribution If the pressure along tlie inner edge of the amuilar sandwich panel is equal to the pressure along the outer edge, or: N. == N (76) 1 o then, from (65), (67) and (72): D:rG=0; 1 = 1; E=N o N BN ^rr Â— ^^o ' ^^ ^ -F ' " = Ar(NT2y o wh.ich reduces equations (71) and (69) to: (77) o ^2. , rBN on B<^i^ 2 d cp av o 2 I 1 ^ 2 dr La-(N /2) J^ A-(N /2) dr o o pft N -, C c o 11 " =^ " "^ 7~ ^' ' ~~ r (78) (79) Applying the first of boundary conditions (7-1) to equation (79) immediately reduces (78) and (79) to: 2 rHN r^ 2 d cr, dp To , , ., . (80) Â•^ ,1l-iiN 1-I Ti -^ ^d7 ^ La::Tn-;^" 'J ^' ^ ' dr o 42 PAGE 55 43 ^H N -j u . -I-/-Â— Â°cp (81) 2tG c Since (80) is Bessel's equation of ordei' one, a general solution of equations (SO) and (81) takes the follov/ing form [IS]: 1 1_ rRN ,2 -. rBN 2 -, ^-wliirT^) d ^ Vi [(a^tIt^) ^J '''^ 1 1 p{;t ^ -] r r/ BN .2 -, r/ BN x2 -I -, Â• [-^ Â— ] k\ [(sttttIt) ] * \h [(^woTTIt) '] } Â«-^' 2tG where J and Y are bessel functions of oixler one of the first and second kind, respectively, and A and A are arbitrary constants of integrat i on , If we )iov." impose the remainder of boinadai-y conditions (74) and (75) on equations (82) and (83), we have: -tt PAGE 56 44 These equations can be satisfied by taking A = A =0. Then the deflection at each point of the panel is zero and we obtain the trivial, undeflectcd form of equilibrium of the panel. The buckling form of equilibrium of the panel becomes possible only if equations (84) yield values for A and A different from zero, which requires that the determinant of tlie coefficients of these constants vanish. Therefore, after multiplying bj' (4/t t) and considering the first of equations (73), our critical condition becomes: [(x) 2] [j^(,OY^(^P) J^ (^P)Y^(^)J= (86) Equal ion (86) closely resembles tlie critical condition obtained by Olsson [12] for a single-layer panel constrained in a similar mannei' For a given value of 3, the smallest corresponding value of [1 for wliicli the second large bracket in equation (86) vanishes is given in Table 2 below [12]. TABLE 2 LOV/EST V.ALUE OF [i. SATISFYING EQUATION (86) li PAGE 57 45 And, from the first of (85): N . 2 o\ u. (f A/cr ^2^ 2.Â„ b B+p, /2 (87) It can be shown that (N /A) given b\' equation (87) is a monotonic o cr increasing function of ji. Therefore, the lowest value of p. results 2 in the lowest value of (N /'A) . Also, since b B is always positive, o cr equation (87) yields values of (X /A) which are greater than or o cr equal to two; and we can therefore conclude that tlie lowest root of equation (86) is always given by equation (87) in conjunction with Table 2. (See Figure 3.) The first of boundary conditions (75) has not been used to obtain tlic above I'esults since the buckling load is independent of a transvei-se translation of the panel as a whole. In this respect our analysis parallels the classical single-layer theory [3], Foi' a ^ p = 0, equation (87) becomes: cr b B+(3.832) /2 Equation (88) agrees with Huang and Ebcioglu' s [lO] results if the present notation is used. The validity of this lim.iting process was discussed in Section 5 of Chapter IV. As G approaches a vei'y large value, tlie first term in the denominator of equation (87) becomes dominant, and we obtain: b PAGE 58 46 where the "effective bending rigidity," d' , is defined: Â£2 2 Â„ t t t^E^ T.' off D = 5Â— (90) 2(l-v_^) Equation (89) in conjimction with Table 2 agrees with the results obtai)icd by Olsson [12] for a single-layer annular panel subjected to uniform inner and outer axial pressures of equal intensity For a = p = 0, equation (89) becomes: ,Â„ X (3,832)^d' (\^r = 2 (91) b Equation (91) is identical to the buckling load obtained by Timoshonlco [3] for a circular single-layer panel clamped along the outer edge. This limiting process is intuitively exi^ectcd, since, for G = 0Â°, the only structural function of the core is to control the distance between the face layers. A similar relationship exists between the web and f ] aiigcs of an I-beam. 2. Genera ] Solution We nov/ return to a general solution of equations (69) and (71). Inl reducing the djmensionless variable, T], through the transformation: Tl = r/b (92) PAGE 59 47 equation (71) becomes: 2 ^ d\ 2 d-o 4 2 ^^^^1^^ T1^N11"+1) ?-| + TlCNll'^-S) 2^ + (RT]^_Qn^3)cp = ~^(93) where, 2 N = -; R = b HX ; Q = IN (94) SiKilaxly, equation (69) becomes: -^t N -I C C Â—IT I 1 ptt N -I C ^ Since ue seek a scries solution of equation (93), and choose to expand our series about the point T| =; 1 , the following additional transf ormatioii is intrcxluced: Â§ = 'q 1 (9G) The reasons for seeking a solution about the point T| 1 (Â§ = 0) will be discussed in Sections 5 and 6 of tliis chaptei'. V/i th tlie aid of equation (96), and after some simplification, equation (93) becomes: [(N^l) + 2(2X+1)^ + (CN+l)?;^ + 4NÂ§^ + Ng^] ^ + [ (N-3) + 3(K-1)Â§ + 3NÂ§^ + NÂ§^]$ + [(R-Qf3) + (4R 2Q)Â§ 2 3 4 bV (1+5) + (6R--Q)Â§ + 4R5 + RÂ§ ]cp zr ^ (97) PAGE 60 48 To obtain a complement ai-y solution of equation (97) the following infinite series is employed [IS]: CO 9^ = ^ A E,^ (98) "^ k=0 '^ where the coefficients A arc functions of the elastic and geometric properties of the panel and the critical buckling load. The radius of convergence of series (98) can be shown (see Section 6, Qiapter v) to be of sufficient magnitude for our particular problem. Substituting (98) into a homogeneous form of equation (97), and collecting coefficients of common powers of ^j ^'e have: {[2(N+1)A2 + (N-3)A^ + (R-qH3)A^] + [G(Nil)A^ + 2i5:;-l)A^ + (R-Q+3X)A, + (4R-2Q)A ]Â§ + [12(N+1)A^ + (27Xf3)A^ 1 o 4 3 4 (R-Q+}8X-1)AÂ„ + (4R-2Q+3N)A + (6R-Q)A ]t^ 2 1 o + [20(X+1)A^ 4 4(13N+3)A^ + (45XfR-Q)A^ 5 4 3 + (14X+4R-2Q)AÂ„ + (X+6R-Q)A + 4RA ]Â§ 2 1 o } (99) In order that this series vanish for all values of Â§ in some region surrounding Â§ = 0, it is necessary and sufficient that the coefficients of each power of Â§ vanish [18]. This pi'oduces the following relations in which some coefficients have been eliminated tlxrough tJie accumulative introduction of pieviovisly computed coefficients: PAGE 61 49 (R-Q+3) . ( N-3) r( 5N-l) (R-Q+3) (4R-2Q)' l r(5N-l)(N-3) (R-Q+SN)" ! , ,^ _^ , A fIHzSl(R-Q+3) (R-Q+lSy-l) (27N+3)(4R-2Q) ^^L-12(N+1) ^ 24(N+1)^ ' 72(N+1)^ ( 27N-; 3) (5N-1) (R-Q+3)~ j ^ ["_ (4R-2Q+3N) (N-3) (R-Q+lSN-1) 72(N+1)^ J o^L" 12(N+1) ' 24(N+1)2 ^ (27N^3)(R-Q+3X) _ (27N+3) (5N-1) (N-3)" 1 ^ ^^^^^ 72(X+1)^ 72(N+1)^ -* """ r(13N+3)(6RÂ•60(N+1)^ I "(13N+3)(6R-Q) (13N+3) (R-Q+3) (R-Q+18N-1) ^^L2 120(N+1)^ _ (13X+3) (27.\+3) (4R-2Q) (13X+3) (27X+3) (5N-1 ) (R-Q+3) 3 "^ 4 360 (X+1) 360 (N+1) (45X+R-Q) (5X-1) (R-Q+3 ) (45X+R-Q) (4R-2Q) (R-Q+3) (14X+4R-2Q) 120(N+1)"^ 120(X+1)^ 40 (N+1) 4R ~[ ^ r (13X+3)(4R-2Qt-3X) _ (13X+3) (N-3) (R-Q+18N-1) -20(N+1)J^^L 60(,.,,)2 i20(N+l)' (13N+3) (27N+3) (R-Q4 3N) (13y+3) (27N+3) (5N-1) (N-3) 3 ^ 4 360 (N+1) 360 (N+1) (45X+R-Q) (5N-1) (N-3) (45N+R-Q) (R-Q+3N) 120(N+1)^ 120(N+1)^ 14N+4R-2Q) (N+6R-Q)" ] . ,^_Â„, ~~;2 20 (N+1) J \ ^^"Â•^'' (N-3) ( + Â— 40 (N+1)' \ = v.'here A and A, remain arbitrary. o 1 PAGE 62 50 Because of the complexity of the computations, ancl the immediate requirements, no recursion formula is sought in the present analysis. With the aid of (100-103), our complementary solution becomes: rn (F^ A /i (^--Q^^^) f2 ^ r (5N-l)(R-Q4-3) (4R-2Q)~ ] ^3 f(6^^-Q> (R-Q +3)(R-Qfl8N-l) (27N+3) (4R-2Q) Â•'L" 12(N+1) "^ o......^2 "^ 2 24(N+1) 72(iN+l) (27N +3) (5N-1) (R-Q+3)~ l ^4 ^ r(1 3N+3)(6 72(N+1) -Â• ^ 60(N+1) (6 R-Q) 2 (13N+3) (R-Q+3) (R-Q+18N-1) _ (13X+3) (27N+3) (4R-2Q) 120(N+1)^ 360 (N+l)*^ (13Ni-3)(27N-t3 ) (5N-1) (R-Q+3) (45N+R-Q) (5N-1) (R-Q+3) 4 3 360 (N+1) 120(N+1) ( 45y+R-Q) (4R-2Q ) (R-Q4 3) (14X+4R-2Q) 4R ~| 5 " 120(N+1)^ ' 40(N+1)2 -20(N+1)_J^^ r ^X-3) _2 r(5N-l)(N-3) (R-Q+3N)~j p3 r (4R-2g:-3N) (y-3) (R-Q^,18N-1) (27N+3) (R-Q+3y) -" 12(N+1) ^ ^^^^.^^^2 ^^^^^^^2 (27N+3)(5N-1) (N-3)~l _4 r(13Ni-3) (4R-2Q+3N) I I + I 72(N+1) "60(X+1) (13N+3) (N-3) (R-Q+18N-1) (13N+3) (27N+3) (R-Q+3N) 3 3 120(N+1) 360 (N+1) (13N+3) (27.\+3) (5N-1) (N-3) (45N+R-Q) (5N-1) (N-3) + 4 3 360 (N+1) 120(yVl) (45N+R-Q) (R-Q+3N) (N -3) ( 1 4N+4R-2Q) (N+6R-Q)"] ^ 120(^.1)2 ' 40(N+1)2 "20(N;i)J (104) PAGE 63 51 Following a procedure similar to that eiaployed in the previous section, equations (95) and (104), together with, a particular solution of equation (97) , are constrained according to boundary conditions (74) and (75). Imposing tlie first of boundary conditions (74) on equation (95) eliminates tlie need to find a particular solution of equation (97), since ve have C = 0. The subscript c can therefore be eliminated from equation (98) and our solution becomes a complete solution. From equations (92) and (96), we note that at i' = b, T] = 1 and Â£ = 0. Therefore, from equation (95) , recalling that C = and N (b) = N (see equation (62)), the second of boundaj-y condirr o tions (75) becomes: [(f) ^] cp (Â§=0) -(105) And since (N /A) ?;^ 2 in general, condition (105) imposed on equation (104) yields A = o Similarly, the second of boundary conditions (74), evaluated at r i^ a (^=3-1), and expressed through equation (95) becomes: [(Â— ) 2lcp (Â§ = P-1) = (106) After substituting equation (104) into (106) with A = 0, o and noting that in general (N./A) ?^ 2, we arrive at the genei-al buckling criteria for an annular sandwicli panel: PAGE 64 52 . /^^ ^^ (^^-3) .. ^.2 r(5N-l)(N-3) (R-Q+3N)" | .^ ^, r (4R-2Q+3N) (N-3) (R-Q+18N-1) ( 27N+3) (R-Q+3N) ' L" 12(X-.l) ^ 24(K.1)2 ' " 72(N-M)^ (27N+3) (5X-l)(N-3)~]^Â„ _4 r(13N+3) (4R-2CH3N) 3 72(N+1) -^ Â•60(Nhl)' (13N+3) (N-3) (R-Q4I8X-I) (13N+3) (27N+3) (R-Q+3N) 120 (N+1) 360 (N+1)^ (13N+3) (27N+3) (5N-1) (N-3) _ (45N+R-Q) (5N-1) (N-3) *" 4 3 " 360 (N+1) 120(N+1) (45N+R-Q) (R-CVh3N) (N-3) (14N+4R-2Q) (N+6R-Q) |.., ^.5 120 (N+1 )'^ 40(N4l)^ ^^' N+6R-Q)~ | , 0(N+1) S'" 1 . I = (107) As explained in the previous section, the imdeflected form of a panel in equilibrium, i.e., Wien A 0, is of little interest. Therefore, ajjproximate critical buckling loads of the panel under aiscussion may be computed by considering a finite numbei' of terms in equation (107). The first of boundary conditions (75), as stated earlier, is not used in obtaining the above results, since the buckling load is independent of a transverse translation of the panel as a wliole. PAGE 65 3. Successive Approximations From equations (94) , (72) and (65) , we have: 53 N rr 2A(l-32) ^V^ ^i^ 5^(N -N.) P^(N -N.) o 1 o 1 (108) R = 2^\ (V^^\) ,2 (N -N.) ) o 1 (109) r(N -X.) 3 (N -N.) o 1 o 1 (110) We designate the first approximation to be that whicli considers only the first term of the infinite series (107), and consecutively add a term foi' each succeeding approximation. Therefore, v.'itli the aid of (108-110) , and after som.e simplification, we have: First Approximation Tiie first approximation yields no results since the loadiiig functions are not present in the fii'st term of series (107). Second Approximation 4A(l-3") 2[(N ) -3 (N.) ] o cr 1 cr + 2 3 [(N ) -(K.) ] r[(N ) -(N.) ] o cr 1 cr o cr i cr (p-1) 2A(l-3^) [(N ) -3"(N.) ] o cr 1 cr 3 le-'rdX ) -(N.) ] 3'^[(N ) -(X.) ] 1o cr 1 cr o cr i cr -^ r-O (111) PAGE 66 54 Third Approximation 2A(l-& ) [(N ) -fl^CN.) ] ""^ U^[(N ) -(N.) ] 3^[(N ) -(N.) ] io cr 1 cr o cr i cr 2A(l-3 ) & [(N ) -(N.) ] o cr 1 cr [(N ) -g^(N.) ] o cr 1 cr P^[(N ) -(N.) ] o cr 1 cr + 1 2A(l-3^) [(N ) -P (N.) ] o cr 1 cr P^[(N ) -(N.) ] 3^[(N ) -(N.) ] o cr 1 cr o cr i cr (p-1) 10A(l-p ) 5[(N ) -r(N.) ] o cr 1 ci' P"[(X ) -(N.) ] 3"[(N ) -(N.) ] o cr 1 cr o cr i cr 2A(l-g ) P [(N ) -(N.) ] o cr 1 cr [(N ) -3^(X.) ] , -^L^Â£ i--^ 3 I (P-l)2 3 [(N ) -(N.) ] o cr 1 cr OA/n r2^ [(N ) -3 (N.) ] 2A(l-p ) o cr 1 cr P^[(N ) -(N.) ] B^[(N ) -(N.) ] o cr 1 cr o cr i cr + 1 4A(l-g J P^KX ) -(N.) ] 5^1 (N ) -(N.) ] o cr 1 cio cr i cr 2[(N ) -P^(N.) ] 2b^B[(N ) -?^(N.) ] o cr 1 cr o cr i cr + [(N ) -(X.) ] o cr 1 cr 2 2 2b B (5-1) = (112) Fourth Approximation 72 2A(1-S ) [(N ) P (N.) ] o cr 1 cr ?^t(N ) -(N.) ] 0^[(N )^ -(N.) 1 o cr 1 cr o cr 1 cr 3 + 1 36 o A f T _o ^ ^<^o> cr-g'<^i^ r^ J P^[(N ) -(X.) ] P^[(X ) -(X.) ] o cr 1 cr o cr i cr 2A(1-P ) P [(X ) -(X.) ] o cr 1 cr [(N ) -P^(X.) ] ""^ o cr 1 cr H. 1 P [(N ) -(X.) ] o cr 1 cr (8-1) + (12 10A(l-3 ) LP ^^\Kr-^\^c.^ PAGE 67 55 5[(N ) B (N.) ] o cr 1 cr (1-B^ [(N ) -S (N.) ] o cr 1 cr o cr 1 cr 2A(1-R ) o cr i cr [(K ) -(N.) ] P^[(N ) "(N.) ] B^[(N ) -(N.) ] ^ o cr 1 cr-' J L'-' \ Q-'cr i cr o cr i cr 2A(l-3 ) [(N ) -S (N.) ] o cr 1 cr B^[(N )^,.-(N.)^ ] P^[(N ) -(N.) ] io ci' 1 cr o cr i cr + 1 ]2 2A(i-S ) B [(N ) -(N.) ] o cr 1 cr [(N ) -P (N.) ] o cr 1 cr P^[(N ) -(X.) J o cr 1 cr -I 2 + 1 4A(1-S ) 2[(N ) -3 (N.) ] o cr 1 cr 3 [(N ) (N.) ] 3 [(N ) -(N.) ] Io cr 1 cr o cr i cr 2b^B[(N ) -3^(X.) ] o cr 1 cr Â„, 2 + 2b B P l(X ) -(X.) ] o cr 1 cr (S-1)" + 2A(l-p^ 3 [(N ) -(X.) ] o cr 1 cr [(X ) -3 (X.) ] o cr 1 cr P^[(X ) -(X.) ] o cr 1 cr 2 r+ 1 oAM =2, [(X ) -r (X.) ] 2A(l-3 ) o cr 1 cr '[(X ) -(X.) ] 3 [(X ) -(X.) ] o cr 1 cr o cr i cr e, 2^, [(X ) -3^(X.) ] 8b B o cr 3 cr ~Z2 Tex"! ^~) r p o cr 1 cr 4b^B + 3 2A(l-3. ) o cr 1 cr [(X ) -p^(X.) ] o cr 1 cr 3^[(X ) -(X.) ] o cr 1 ci+ 1 2A(l-3 ) [(X ) -3 (X.) ] o cr 1 cr^ 3 [(X ) -(X.) ] 3 [(X ) -(X.) ] o cr 1 cr o cr i cr 34A(l-[-. ) o cr 1 cr O'er ^ -i'cr^ 2b2B ^ ^ V cr"^ ^\>cr^ 3^[(X ) -(X.) ] 3^[(X ) -(N.) ] 3^ f^^\^cr"^"'i^cr^ >o cr 1 cr o cr i cr 2b^B 1 2A(l-3^) f(\^r-P ^\Kr^ J 3^[(x ) -(X.) ] 3^[(x ) -(X.) ] -^ "o cr 1 cr o cr i cj+ 1 54A(l-3 ) 27 [(X ) -3 (X.) ] o cr '^ 1 cr + 3 4A(l-3 ) ^ ^^Vcr-(^"i>cr3 P't(N^)^^,-(X.)^^J -L^'^ (X^)^^,-(X. )^^.] PAGE 68 56 2[(N ) -S^CN.) ] 2b^B[ (N ) -3^(N.) ] o cr 1 cr o cr i cr [(N ) -(N.) ] o cr 1 cr B [(N ) -(N.) ] o cr 1 cr 2b B 54A(l-3 ) 3^[(N ) (N.) ] o cr 1 cr 27 [(N ) -?j (N.) ] o cr 1 cr + 3 10A(l-5 ) 5[(N ) -3^(N.) ] o cr 1 cr 1 [(N ) -(N.) ] B [(N ) -(N.) ] P [(N ) -(N.) ] o cr 1 cr J -'^ o cr i cr o cr i cr 2A(1-B ) [(N ) -P (N.) ] o cr 1 cr B^[(N ) -(N.) ] B^[(N ) -(X.) ] o cr 1 cr o cr i cr 3 )(?-l) = (113) Fifth Approximation 360 2A(l-p") [(N ) -B (N.) ] o cr 1 cr + 1 B [(N ) -(X.) ] p^[(N ) -(N.) ] o cr 1 ci' o cr i ci' 180 r 2 2A(1-B ) [(N ) -3 (N.) ] o cr 1 cr 3 4 1 '[(N ) -(N.) ] 3"[(N ) --(N.) ] o cr 1 cr o cr i cr 2A(l-3^) 8 [(i\ ) -(N.) ] o cr 1 cr [(N ) -? (N.) ] o cr 1 cr B^[(N ) -(N.) ] o cr 1 cr (R-l) -t (60 2A(l-p'') [(N ) -(N.) ] o cr 1 cr [(N ) -B'-CN.) ] o cr 1 cr -1 2 + 1 2 -^ I [(N ) -(N.) ] J o cr 1 cr -J 10A(]-p ) 5[(N ) -p'^CN.) ] 1 o cr 1 cr I P^[(N ) -(In.) ] P^[(N ) -(N.) ] o cr 1 cr o cr i cr 2A(l-3 ) [(N ) -p (N.) ] o cr 1 cr P^[(N ) -(N.) ] p^[(N ) -(N.) ] J Io cr 1 cr o cr i cr -J 60 2A(l-p ) .P f PAGE 69 57 2b^B[(N ) ,,-3^(N.) ] o cr 1 cr Â„, 2 + 2b B B [(N ) -(N.) ] o cr 1 cr (p-1) + 2A(l-p ) '[(N ) -(N.) ] o cr 1 cr [(N ) -p (N.) ] o cr 1 cr P^[(N ) -(N.) ] o cr 1 cr + 1 2A(l-3 ) [(N ) -3 (N.) ] o cr 1 c r P [(X ) -(N.) ] p^[(N ) -(N.) ] >o cr 1 cr ^ Â• ^ o cr i cr"^ 8b^B[(N ) -g^(lN.) ] o cr 1 cr ^2 + 2 ^^ ^ P [(N ) -(N.) ] o cr 1 cr + 15 2A(l-3 ) P [(N ) -(N.) ] o cr 1 cr [(N ) 3^(N.) ] o cr 1 cr B^[(N ) -(N.) ] o cr 1 cv + 1 2A(l-?5 ) [(N ) -(N.) ] o cr 1 cr [(N ) -3 (N.) ] o cr 1 cr 3^[(N ) -(N.) ] o cr 1 cr 3I r 34A(i-3^) 17[(N )^^-3^(N.) ] 2b^B[(N ) 3^(N.) ] o cr 1 cr o cr 1 cr '^^^'^c?c~.r^'^.">,J 3^[(N ) -(N.) ] 3^[(N ) -(N.) ] o cr 1 cr o cr 1 cr o cr i cr 2b B + 5 2A(l-3 ) [(N ) -3 (N.) ] o cr 1 cr ' B"[(N ) -(N.) ] p'^LCN ) -(N.) ] o cr 1 cr o cy 1 cr 54A(1-S ) 27 [(X ) -3 (N.) ] o cr 1 cr ^^f^^'o>r..-(^'> ^ B^[(N ) -(N.) ] o cr 1. cr o cr 1 cr + 3 4A(l-p ) 3 [(N ) -(N.) ] o cr 1 cr 2[(N ) o cr MN.) 3 2b B[(N ) -B (N.) ] 1 cr o cr ^ i^cr-" 2 4 2b B 3 [(N ) -(.\.) ] o cr 1 cr 3 [(N ) -(N.) ] o cr 1 cr 5 2A(1-S ) [(N ) -3 (N.) ] ^ o cr 1 cr , ! Â— + 1 LP'[(^o),,-(N.)^^.] b2[(N^)^^,-(N.)^^,] ':Lp'[(N,),,-(N,)^^] 54A(l-3 ) 27[(N ) -?^(N.) ] If ,,2 o cr __JL_lÂ£Â£._ 3 10A(l-3 ) 5[(N ) -3 (N.) ] o cr 1 cr ^ ^^^>cr-(^'i)cr^ -P t^Vcr-^^^^r^ ^'"^^ ^ V cr-^^^^cr^ 1 PAGE 70 58 2A(l-3 ) [(N ) -3 (N.) ] o cr 1 cr 3^[(N ) -(N.) ] S^[(N ) -(N.) ] o cr 1 cr o cr i or (P-1)" 2A(1-S ) [(N ) -e (N.) ] o cr 1 cr P^[(N ) -(N.) ] 3^[(N ) -(N.) ] >o cr 1 cr o cr i cr + 1 26A(l-3 ) [(N ) -(N.) ] o cr 1 cr + 3 ' ' '[(N ) -(N.) ] o cr 1 cr [(N ) -3 (N.) ] o cr 1 cr ^^^'^^r.K^-'^^-^.J ^^L(N ) -(N.) ] o cr 1 cr o cr i cr 8b^B[(N ) -p^(N.) ] o cr 1 cr ., 2^ , + 5 4b B I 3 3 [(N ) -(N.) J o cr 1 ci2A(l-3 ) [(N ) -(N.) ] o cr 1 cr [(N ) -3 (N.) ] o cr 1 cr 3^[(X ) -(X.) ] o cr 1 cr + 1 26A(1-S ) 3 [(N ) -(N.) ] o cr 1 cr 13 [(N ) -3 (N.) ] o cr 1 cr 3^L(N ) -(N.) ] o cr 1 cv + 3 2A(]-3") [(N ) -3 (N.) ] o ci' 1 cr B^[(N ) -(N.) ] P^L(N ) -(X.) ] o cr 1 cr o cr i cr 34A(l-3^) 3 [(X ) -(X.) ] o cr 1 cr 17 [(X ) -p^(X.) ] 2b^B[(X ) -3^(X.) ] o cr 1 cr o cr i cr Â„ 2 ~:27t: Â— rÂ— " ~2 2b B 1 [(X ) "(X.) ] o ci1 cr 3 [(X ) -(X.) ] o cr 1 cr 2A(1-3 ) [(X ) -(X.) ] o cr 1 cr [(X ) -3 (X.) ] o cr 1 cr 3^[(X ) -(X.) ] o cr 1 cr + 1 26A(l-3 ) 3 [(X ) -(N.) ] o cr 1 cr 13[(N ) .-3 (X.) ] L(X ) -(X.) ] o cr 1 cr .al r 2 54A(l-3 ) 3^[(x ) -(X.) ] o cr 1 cr 27 [(X ) -3 (X.) ] o ci" 1 cr '[(X ) -(X.) ] o cr 1 cr + 3 4A(l-32) 3^L(X ) -(X.) ] 3^[(X ) -(X.) ] o cr 1 cr o cr i cr 2[(X ) -B^CX.) ] 2b2DL(X ) -32(x.) ] o cr 1 cr o cr i cr 3 L(X ) -(X.) ] o cr 1 cr PAGE 71 59 2 2b B 26A(l-p ) 13[(N ) -3 (N.) ] o cr 1 cr P^[(N ) -(N.) ] 3^[(N ) -(N.) ] 'o cr 1 cr o cr i cr + 3 r 2 54A(l-p ) P^[(N ) -(N.) ] o cr 1 cr 27 [(N ) -3 (N.) ] o cr 1 cr P^[(N ) -(N.) ] o cr 1 cr + 3 10A(l-3 ) B [(N ) -(N.) ] o cr 1 cr 5[(N ) -?^(N.) ] o cr 1 cr P^[(N ) -(N.) ] o cr 1 cr 1 2A(l-p ) [(N ) -3 (N.) ] o cr 3 cr ; I p^[(N ) -(N.) ] 6^[(N ) -(N.) ] -Io cr 1 cr o cr i cr = ,.3 2A(1-P ) P [(N ) -(N.) ] o cr 1 cr [(N ) -P (N.) ] o cr 1 cr P^[(N ) -(N.) ] o cr 1 cr h 1 88A(l-0 ) 3^[(N ) -(N.) ] o cr 1 cr 44[(N ) -S (N.) ] 2b B[(N ) -3 (N.) ] ^ ] o cr 1 cr o cr i cr Â„, 2 i Â— + Â— 2b B '[(N ) -(N.) ] o cr 1 ci'[(N ) -(N.) ] o cr 1 cr 10A(1-) 3 [(N ) -CN.) ] o cr a cr 5[(N ) -3^(N.) ] o cr 1 cr 3^[(x ) -(X.) ] o cr 1 cr 2A(l-3 ) [(N ) -3 "(X.) ] o cr 1 cr ?^[(N ) -(N.) ] 3^[(N ) -(N.) ] J 1o cr 1 cr o cr i cr -" + 3 2A(l-3. ) [(N )^-3 (N.)^ 1 o cr 1 cr 2 3^[(N ) -(N.) ] p^[(N ) -(N.) ] >o cr 1 cr o cr i cr + 1 88A(l-3 ) P"[(N ) -(N.) ] o cr 1 cr 44[(X ) -3^(N.) ] 2b^B[(N ) -p^(N.) ] o cr 1 cr o cr i cr 2b^B 3 [(N ) -(N.) ] o cr 1 cr 3 [(N ) -(X.) ] o cr 1 cr 4A(l-3 ) 3 '[(N ) -(N.) ] o cr 1 cr 2[(N ) -3^(N.) ] 2b^B[(N ) -B^(N.) ] o cr 1 cr o cr i cr Â— -Â— , + _ '[(X ) -(N.) ] o cr X cr 2b^B [(X ) -(X.) ] o cr 1 cr + 9 2A(l-3^) [(N ) -3^(X.) ] o cr 1 cr 3 [(X ) -(X.) ] 3 [(X ) -(X.) ] o cr 3 cr ^ o cr i cr 2A(l-3 ) P. [(X ) -(N.) ] Â•o cr 1 cr PAGE 72 60 t(N ) -3 (N.)^,J o cr 1 cr 3 r 24A(l-3 ) 12[(N ) -P (N.) ] o cr 1 cr e^[(N ) -(N.) ] I P^t(N ) ,,-(N.)_] p^[(N ) -(N.) ] o cr 1 cr 1o cr i cr o cr i cr 8b^B[(N ) -S^(N.) ] _ o cr 1 cr ,2. + p 4b B P [(N ) -(N.) ] o cr 1 cr 18 2A(l-3 ) 3 [(N ) -(N.) ] o cr 1 cr [(N ) -3"(N.) ] o cr 1 cr -I1 ! P [(N ) -(N.) ] o cr 1 cr ~^^12b^B[(N ) -3^(N.) ] ^^Â£ L-Â£Â£_ _ sb^B 3 [(N ) -(N.) ] o cr 1 cr ^(3-1)^=0 (11^) Computation of tlie sixth approximation is unnecessary since it can be shov.ii that the fifth apj^roximation yields acceptable results (see Section 5, Cliapter V). Equations (111-111), each being self-contained, represent the approximate critical buckling criteria for an annular sandwich panel constrained by bomidary conditions (74) and (75) and subjected to uniform radial compressive loads, N. and N , along the inner and outer edges, i-especti vely. 4, Numea'ical Results and Discussions Once tlie inner compressive load, N. , is pi-escribed to be some multiple of the outer compressive load, N , tlie approximate critical conditions given in the previous section are comiiletely defined bj^ 2 three dimcnsio)ilcss parameters: (N /A) , b B, and p. Thus we obtain o cr algebraic polynomials in (N /A) which increase in degree as the order o cr of apj)roximatiorx increases (ranging from first degree in the second approximation to fourth degree in the fifth approximation). For PAGE 73 61 obvious reasons we consider only the lowest positive value of (N /A) o cr satisfying eacl^i polynomial. Three possible loading conditions are analyzed in the present work: (1) N = N. , (2) N = 0, and (3) N = 0. However, it sliould be o 1 o 1 noted that the techniques employed in these examples are applicable foi' any ratio of X to N. . ox N = N. If the inner and outer axial compressive loads are equal , then the exact solution is given by equation (87) in conjunction with Table 2, Thus, tlie approximation techniques employed in the previous section ai's unnecessai-y. For this case. Figure 3 shows the existing 2 relation betweeji (X ,''A) and 3 for vai'ious values of b B, o cr If the inner edge alone is subjected to axial compression, then, since equations (111-114) yield no positive values of (N /A) , o cr it can be concluded that buckling never occurs. This wo\ild seem reasonable, since, from eq^iations (63-65), such a reduction I'esults in a relativelj' large tensile N compared v.ith a relatively small Â— y B compressive N_ . Analogously, for a rectangular single-layer panel subjected to compression along opposite edges and tension along adjacent edges, Timoshenko [3] has shown that a large tensile load will prevent a significantly smaller compressive load from causing instability. Ho'vevev, '.\'e must keep in luind that, while N anrl X can ^ ' ' ' -XX -yy be varied j.ndependeni ly in a rectangular panel, such is not the case PAGE 74 62 for a circular or annular panel, since the following equilibrium condition must be maintained: dN N NÂ„^ _i2Z + :2Z =90 ^ (56) dr r N. = 1 If onlj' the outer edge is subjected to axial compression, the second approximation, equation (111), can be solved exiplicitly for (N /A) : o cr (t) ^"^Â°'"-" (115) cr 3+23 + 35 2 Since b B does not enter into relation (115), it is obvioiis that furtlier approximations must be considered. Due to the complexity of the calculations involved, a graphical solution is employed for the succeeding approximations. In Figure 4 approximations two througli five are compared bj' plotting (N /A) 2 versus 3 foivarious values of b B. All quantities appearing in Figures 3 and 4 are dimensionlcss. Buckling loads may be obtained in the appropriate dimensions by using tlie first of relations (73). PAGE 75 63 Lo Figure 3. Minimum Critical Values of (N /A) for N = N o o i PAGE 76 64 ^'approx. Ci'iy r '^T^f+Vpprox;. 6^e>^ /^^^ T r p--Vh -T" .8 ,^ /.o Fijxure 4. Minimum Critical Values of (N /A) for N. = PAGE 77 65 The following conclusions may be drawn from Figures 3 and 4: 1. All critical values of (N /A) are less than or equal to two. o (Figures 3 and 4) 2 2. As the value of b B approaches infinity, the results approach those obtained for a single-layer panel in equation (89). (Figure 3) 3. For 6 0, the present theory coincides with Huang and Ebcioglu's results (equation (88)) for a circular sandwich panel. (Figures 3 and 4) 4. The second approximation in Figure 4 is the exact solution for b^B = 0. 5. An annular sandwich panel subjected to axial compression along the outer boundary becomes stronger if an equal compressive load is also applied along the inner edge. (Figures 3 and 4) 6. In Figure 4, the third approximations yield more accurate results than the fourth approximations. This peculiarity and the error bound associated with the fift]i approximations will be discussed in tlie next section. 7. A dual response is apparent in Figure 4. As the hole increases in I'olative size, the panel may become weaker or stronger depend2 ing on tlie value of b B and the range of P being considered. Such a beliavior is possible because both tlie shear a)id the bending stiffness of a sandwich panel enter into the analysis. An annular single -layer panel, which can be described by only two dimensionless parameters, exliibits no sucli dual response. PAGE 78 66 2 8. If buckling loads for values of b B or ratios of N to N , not o 1 considei'G'd in Figvires 3 and 4, are required, equation (] 14) may be used directly. However, when considering only values of p gi'eater than one-half, the third approximation, equation (112), yields acceptable results. (Figure 4) The above conclusions are valid only for tlie special case in wliich boundary conditions (74) and (75) are applied. 5. Error Refund Since v/c chose to expand tlie solution of equation (93) about the point T| n. 1 , the speed of convergence of the series solvition obtained, equation (104), depends on the proximity of the entire annular region to that point. Clearly, as the hole increases in size (p approaches one), the solution converges more rapidly. This fact is evident from Figure 4, and also from equation (107). Indeed, it can be concluded that the speed of convergence is the slowest v.lien 3^0. From equations (111-114) the approximate critical buckling parameter, (N /A) _, can be solved exi^licitly for tlie degenerate casf; of 3=0. The results of this simplification are found in Table 3. Howe^'er, for this special case, the exact solution (reproduced here for convenience) is available from Section 1 of tliis chapter: \\ 1 ^-684 ^ A J " 2 cr b B + 7.342 PAGE 79 TABLE 3 APPnoXir.iATE VALUES OF (N /A) o cr FOR B = 67 Order cf Approximation (N /A) o cr First PAGE 80 68 6 , Remarks o n Convergence The series solution of an ordinary differential equation possesses a radius of convergence at least as great as the distance from the point of expansion to the nearest singularity [19]. Equation (93) possesses two regular singularities, one at 2 T| = 0, and another at (nT] +1) = 0; and its solution was expanded about the point T] = 1. Therefore, it must be demonstrated that these singularities do not inhibit the validity of our solution throughout the entire annul airegion of the panel. Referring to Figure 5, it becomes obvious that the singularity at T| = does not restrict tlie required radius of convergence, regardless of the value of p. It is therefore necessary only to show that 2 the singularity occurring at (>n] +1) = 0, lies outside the annular region and its reflection illustrated in Figure 5, for all values of p. -P ^ ^e-fleciecl I maae <}fcii\n.u let r ^ region -Prom h-^1 ! T ^ H P 1 2-/3 2 ' Figure 5. Radius of Convergence 2 With the aid of equation (108), (N1] +1) = becomes: 2 ^ [2A(1-S^) (N -P^N.)] +1=0 (IIG) o 1 PAGE 81 69 Therefore, the position of the singularity asfociated with equation (116) depends on the value of the critical buckling load which becomes known only after the solution is obtained. For tlie particular case in which N. =0, (116) is satisfied if UJcr ^2_p2 However, from equation (115) and Figure 4, it is apparent that, for N. = 0, -' ^ 2(l + R)(3-Â«) s "^"""^ ^ '' (118) A /cr 2 3 + 23 + 3p for all values of b B, In oi-der tliat conditions (117) and (118) be satisfied simultaneously, Ti must satisfy the following inequality: ^2 ^ , 2(3^,^)3! (,,,^ [2(3-0) 2(1-3) (3+2P+33 )] 2 2 Â• Equation (119) constrains T] to be greater than (2-3) for all values of 3 ranging from to 1. It can therefore be concluded that the singularity associated with equation (117) lies outside the annular region and its reflection illustrated jn Figure 5. Similarly, it can also be shov.ii that, for N =0, the singularity associated with equation (116) lies outside this ci^itical region. PAGE 82 70 Thus, the radius of convergence, p, of the series solution of equation (93), expanded about the point T] = 1, is: p ^ 1 P (120) which, as illustrated in Figure 5, is large enough to encompass the entire annular region of the panel. Care must be taken, however, when imposing boundary conditions other than (74) and (75) on the solution of equation (93). Critical loads resulting from boundary conditions or ratios of N to N not o i considered in tlie present work majsatisfy equation (116) within tlie needed radius of convergence. It would t}icn become necessary to eithei' expand the solution of equation (93) about some other point, or employ the techniques associated with analj'tic continuatiori. Tne difficulties encountered in tlie latter appi'oacli would be enormous. PAGE 83 CHAPTER VI COxXCLUSION The px-esent work investigates the buckling of annular sandwich panels. Equilibrium equations and boundary conditions satisfying continuity requirements were derived in cartesian coordinates, using the theorem of minimum potential energy. These equations were then transformed into polar coordinates througli the application of tensor analj'sis. Axisymmetric buckling being assumed, and the bending rigidity of the faces being neglected, the equilibrium equations were uncoupled by using a n.odified technique. The governing equations were then compared witli existing theories for single-layer annular panels [11,12] and circula]' sandwich panels [10]. For the general problem of an annular sandwich panel subjected to unequal inner and outer compressive loads, and constrained by boundary conditions sim.ilar to those emploj^ed by Olsson [12], a power series solution was obtained. This series was sliov.ii to possess a radixis of con\'ergence of sufficient magnitude. Successive appi'oximations were then computed, and a graphical solution was employed for various i-atios of outer to inner compressive loads. Results from the f if til approximation, which were shown to be within 2.2 per cent of the exact solution, were compared with those obtained from earlier theories [10,11,1?]. 71 PAGE 84 72 The present work represents the first attempt to analyze the stability of annular sandwich panels. Further extensions of the present theory raay be carried out bj' including the effects of the bending rigidity of the faces or considering boundarj' conditions other than tliose employed here. Furthermore, continued efforts should be directed toward obtaining a solution to the unsymmetric buckling pi'oblom. In tliis way, tlie assumption of axisynunetric buckling could be justified, and the problems associated with angular dependent loading functions could be analj-zed. PAGE 85 BIBLIOGRAPHY 1. F. J. Plant ema, "Sandwich Construction," Wiley and Sons, New York, 1966. 2. S. Timoshenko and S. Woinowsky-Krieger , "Theory of Plates and Shells," 2nd ed. , McGraw-Hill, New York, 1959. 3. S. Timoshenko and J. Gere, "Theory of Elastic Stability," 2nd ed. , McGraw-Hill, New York, 1961. 4. N. J, Hoff, "Bending and Buckling of Rectangular Sandwich Plates," NACA TN 2225, 1950. 5. A. C. Eringen, "Bending and Buckling of Rectangular Sandwich Plates," Proc, First U.S. Natl. Cong. Appl . Mech. , 1951, pp. 381390. 6. C. C. Chang and I. K. Ebcioglu, "Elastic Instability of Rectangular Sandwich Panel of Orthotropic Core with Different Face Thicknesses and Materials," Transactions of the American Society of Mechanical Engineers, J. App. Mech., Vol. 27, No. 3, September 1960, pp. 474-480. 7. S. J. Kim, "Symmetric and Antisynmietric Buckling of Sandwicii Panels," Doctoral Dissertation, Dept. Eng. Sci. and Mech., University of Florida, 1969. 8. E. Reissner, "Small Bending and Stretching of Sandwicla-Type Shells," NACA TN 1832, 1949, 9. M. Zaid, "Symmetrical Bending of Circular Sandwich Plates," Proc, of the 2nd U.S. Natl. Cong. Appl. Mech., 1951, pp. 413-'422, 10. J. C. Huang and I. K. Ebcioglu, "Circular Sandwich Plate under Radial Compression and Thermal Gi'adient," AIAA Journal, Vol. 3, No. 6, June 1965, pp. 1146-1148. 11. E. Meissner, "Uber das Knicken kreisringf ormiger Scheiben," Schweiz. Bauztg. , Vol. 101, 1933, pp. 87-89. 12. G. Olsson, "Knicknng der Kreisringplatte vcn quadratisch veranderlicher Steif igkeit ," Ingr.-Arch., Vol. 9, 1938, pp. 205214, 73 PAGE 86 74 13. C. C. Chang and I. K. Ebcioglu, "Thermoelastic Behavior of a Simply Supported Sandwich Panel Under Large Temperature Gradient and Edge Compression," J. Aero. Sci., Vol. 28, No. 6, June 1961, pp. 480-492. 14. I. S. Sokolnikoff, "Mathematical Theory of Elasticity," 2nd ed. , McGrav,--}Iill , New York, 1956. 15. Y. C. Fung, "Foundations of Solid Mechanics," Prentice-IIall , Englewood Cliffs, New Jersey, 1965. 16. A. J, McConnell, "Applications of Tensor Analysis," Dover Publications, Inc., New York, 1957. 17. S. Tim.oshenko, "Strength of Materials," Part II, D. van Nostrand, Princeton, New Jersej', 1956. 18. F. B. Hildebranri, "Advanced Calculus for Applications," PrenticeHall, Englewood Cliffs, New Jersey, 1962. 19. L. R. Ford, "Differential Equations," 2nd ed. , McGraw-Hill, New York, 1955. PAGE 87 BIOGRAPHICAL SKETCH Amelio John ^mato was born in Newark, New Jersey, on January 20, 1944. He was graduated from Seton Hall Preparatory School in June, 1962. In June, 1966, he received the degree of Bachelor of Science in Mechanical Engineering from Newark College of Engineering (New Jersey). In September of the same year he entered tlie Department of Engineering Science and Mechanics at the Universitj' of Florida as a National Defense and Education Act, Title IV Fellow. Here, in August, 1967, he received tlie degree of Master of Science in Engineer jng prior to pursuing the degree of Doctor of Philosophy. 75 PAGE 88 This dissertation was prepared under the direction of the chairman of the candidate' s supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and v/as approved as partial fulfillment of the requirements for the degree of Doctor of Philosopliy. June, 1970 Dean, College of Engineering Supervisox-y Committee: Dean, Graduate School >l7r^^'V A ^"^^/^.yk?^^ Aa. ^A\Cvvx-Â»-w>e \f tJLlMSd.^ |