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Asymptotic solutions of steadily spinning shallow shells of revolution under uniform pressure Lin, Yi Han 1983

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ASYMPTOTIC SOLUTIONS OF STEADILY SPINNING SHALLOW SHELLS OF REVOLUTION UNDER UNIFORM PRESSURE By Y I HANJLIN B . S c . , F u d a n U n i v e r s i t y , 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f M a t h e m a t i c s and I n s t i t u t e o f A p p l i e d M a t h e m a t i c s and S t a t i s t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BR IT ISH COLUMBIA A u g u s t 1983 © Y i Han L i n , 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my w r i t t e n p e r m i s s i o n . Department o f Mathematics The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date (Ont. £ , { 1 % I ) E - 6 ( 3 / 8 1 ) A b s t r a c t The p r o b l e m o f a s t e a d i l y s p i n n i n g s h a l l o w e l a s t i c s h e l l o f r e v o l u t i o n u n d e r a u n i f o r m p r e s s u r e d i s t r i b u t i o n i s i n v e s t i g a t e d by p e r t u r b a t i o n and a s y m p t o t i c m e t h o d s . A c c u r a t e n u m e r i c a l s o l u t i o n s a r e a l s o o b t a i n e d t o c o n f i r m the adequacy o f p e r t u r b a t i o n and a s y m p t o t i c s o l u t i o n s . The l i m i t i n g c a s e o f a f l a t p l a t e i s f i r s t s o l v e d f o r t h e e n t i r e r a n g e o f v a l u e s o f t h e two l o a d p a r a m e t e r s . The r e s u l t s p r o v i d e a d i f f e r e n t i n t e r p r e t a t i o n o f t h e e x i s t i n g n o n l i n e a r membrane s o l u t i o n f o r t h e same p r o b l e m . D e p e n d i n g on t h e p a r t i c u l a r g e o m e t r i c and l o a d c o n f i g u r a t i o n , t h e boundary v a l u e p r o b l e m f o r t h e s h e l l c a s e i s r e d u c e d e i t h e r t o the s o l u t i o n o f a sequence o f l i n e a r p r o b l e m s o r t o t h e s o l u t i o n o f a p r e v i o u s l y s o l v e d n o n l i n e a r p r o b l e m ( i n c l u d i n g one f o r a f l a t p l a t e ) m o d i f i e d by t h e s o l u t i o n s o f a sequence o f l i n e a r p r o b l e m s . The a n a l y s i s f o r t h e s h e l l p r o b l e m u n d e r a comb ined c e n t r i f u g a l and p r e s s u r e l o a d i n g shows a c o m p l e x i n t e r p l a y among t h e l o a d and g e o m e t r i c p a r a m e t e r s . I t a l s o e n a b l e s u s t o c o n s i d e r a number o f p r e v i o u s l y i n v e s t i g a t e d p r o b l e m s as s p e c i a l c a s e s o f o u r p r o b l e m and o f f e r s a u n i f i e d t r e a t m e n t o f t h e s e p r o b l e m s . i i TABLE OF CONTENTS P a g e ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF TABLES V L I S T OF FIGURES v i NOMENCLATURE v i i ACKNOWLEDGEMENT i x CHAPTER 1 INTRODUCTION 1 CHAPTER 2 FORMULATION 4 2 . 1 A x i s y m m e t r i c D e f o r m a t i o n s o f S h a l l o w S h e l l s o f R e v o l u t i o n . . 4 2 . 2 S t r e s s R e s u l t a n t s , S t r e s s C o u p l e s / D i s t r i b u t e d L o a d s and E q u i l i b r i u m 7 2 . 3 R e d u c t i o n t o Two S i m u l t a n e o u s E q u a t i o n s 10 2 . 4 B o u n d a r y V a l u e P r o b l e m s 12 CHAPTER 3 FLAT PLATES 15 3 . 1 D i m e n s i o n l e s s F o r m u l a t i o n 15 3 . 2 S m a l l t o M o d e r a t e L o a d M a g n i t u d e Range 17 3 . 3 The R e l a t i v e l y F a s t S p i n n i n g Range 22 3 . 4 The R e l a t i v e l y H i g h P r e s s u r e Range 25 3 . 5 D i r e c t a n d B e n d i n g S t r e s s e s 28 3 . 6 C o m p a r i s i o n w i t h F i n i t e D i f f e r e n c e S o l u t i o n 32 3 . 7 On N o n l i n e a r Membrane T h e o r y 39 CHAPTER 4 SHALLOW SHELLS 42 4 . 1 I n t r o d u c t i o n 42 4 . 2 The L i g h t t o M o d e r a t e L o a d M a g n i t u d e Range 44 4 . 3 C u r v e d P l a t e B e h a v i o r a t H i g h L o a d I n t e n s i t i e s 49 4 . 4 T h i n S h e l l B e h a v i o r a t H i g h L o a d I n t e n s i t i e s 52 4 . 5 Summary o f R e - s c a l i n g s and A s y m p t o t i c S o l u t i o n s 54 CHAPTER 5 CONCLUDING REMARKS 57 BIBLIOGRAPHY 59 APPENDIX A SOLUTIONS FOR VARIOUS CLASSES OF PLATE PROBLEMS 61 i i i P a g e A . 1 P e r t u r b a t i o n S o l u t i o n s f o r S m a l l P r e s s u r e and S l o w l y S p i n n i n g C a s e 61 A . 2 A s y m p t o t i c S o l u t i o n s f o r t h e F a s t S p i n n i n g C a s e 64 A . 3 A s y m p t o t i c S o l u t i o n f o r t h e H i g h P r e s s u r e Range 67 APPENDIX B F I N I T E DIFFERENCE SOLUTIONS 71 i v L I S T OF TABLES P age TABLE 1 NUMERICAL SOLUTIONS FOR REGION(1 .1 ) OF THE FLAT PLATE C A S E . . . . 76 TABLE 2 NUMERICAL SOLUTIONS FOR REGION(1 .2 ) OF THE FLAT PLATE CASE 77 TABLE 3 NUMERICAL SOLUTIONS FOR REGION(2) OF THE FLAT PLATE CASE 78 TABLE 4 DIMENSIONLESS BOUNDARY LAYER WIDTH NEAR THE INNER EDGE OF THE FLAT PLATE I N REGION (2) OF THE PARAMETER SPACE 79 TABLE 5 NUMERICAL SOLUTIONS FOR REGION(3) OF THE FLAT PLATE CASE 80 TABLE 6 SCALINGS I N PARAMETER SPACE REGIONS 81 v LIST OF FIGURES page FIG.l COMPONENTS OF AXISYMMETRIC DISPLACEMENT OF A SHELL OF REVOLUTION 83 FIG. 2 SLOPE AND CURVATURE CHANGES.... 83 FIG. 3 HORIZONTAL FORCE EQUILIBRIUM 84 FIG.4 VERTICAL FORCE EQUILIBRIUM 84 FIG. 5 MOMENT EQUILIBRIUM '. 85 FIG. 6 A NORMALLY LOADED,SPINNING SHALLOW SHELL 86 FIG. 7 k -k PARAMETER PLANE FOR FLAT PLATES 87 H V FIG. 8 k G " k v " k H PARAMETER SPACE FOR SHELLS 88 FIG. 9 REGION (I) OF THE PARAMETER SPACE FOR SHELLS 89 FIG. 10 REGION (II) OF THE PARAMETER SPACE FOR SHELLS 90 vi Nomenclature A * extensional f l e x i b i l i t y D • bending r i g i d i t y E » Young's modulus v - Poisson's ratio h = thickness of plate and shell V, H • axial and radial stress resultant i|» - rH M^ , Mg « stress couples N^, Ng • tangential and circumferential stress resultant u » steady angular velocity of the rotating shell P v - axial load intensity 2 p u = phu> r, radial load intensity n P 0 - (3 + v )pho» 2r 0 Q m transverse shear resultant Rg = circumferential curvature radius r, 9 - polar coordinates of a point on the midsurface of the shell r^ " radial coordinate of outer edge of the midsurface of the shell frustum r^ - radial coordinate of the inner edge of the shell frustum u, w • radial and axial displacement component of the midsurface of the shell x - r/ rQ» nondimensional radial coordinate x i " r i / r 0 z • axial coordinate of a point on the middle surface of the shell e r - radial component of midsurface strain e. B circumferential component of midsurface strain 9 v i i < r , K g = curvature changes of the midsurface of the shell £ • meridional slope of midsurface of the shell (= dz/dr) £Q • £ at outer edge p • density of the shell material 0 r * °8 " r a < ^ ^ a l a n (* circumferential stress component $ • meridional change of slope of the midsurface (= dw/dr) v i i i ACKNOWLEDGEMENT The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s was s u p e r v i s e d b y P r o f e s s o r F r e d e r i c Y.M.WAN whose a d v i c e i s g r e a t l y a p p r e c i a t e d . The a u t h o r w o u l d a l s o l i k e t o e x p r e s s h i s t h a n k s t o The U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r p r o v i d i n g h i m w i t h t h e needed f i n a n c i a l s u p p o r t i n c l u d i n g a Summer G r a d u a t e F e l l o w s h i p . i x I. Introduction This report investigates the elastostatic behavior of steadily rotating, shallow, elastic shells of revolution under a uniformly distributed pressure. The shells are in the form of a hollow frustum bounded by two parallel circular edges in planes normal to the axis of revolution. With different types of edge constraints, these structures, including the limiting case of a flat disc, have many applications in space exploration [1,2]. They will be analyzed in this report by way of a shallow, thin, elastic shell theory which takes into account the simplifications associated with the inherent axisymmetry of our class of problems [3,4,5]. To make the report self-contained, the development of this shell model will be sketched out in Chapter II. A two-point boundary value problem (BVP) for two coupled second order nonlinear ordinary differential equations (ODE) appropriate for the analysis of our particular class of shell problems will be formulated in section (4) of that chapter. Except for a narrow range of geometric and load parameter values, an appropriate dimensionless form of this BVP contains one or more small parameters. Perturbation and asymptotic methods [6] are therefore appropriate for the analysis of the BVP. Accurate numerical solutions have also been obtained to confirm the predictions of our asymptotic analyses. To be concrete, we consider shell frusta with an outer edge free of edge tractions and an inner edge clamped to a rigid hub. Shells with other edge conditions can also be similarly investigated. Within the framework of a shallow shell theory, there is no distinction between normal and vertical loadings. 2 The main d i f f i c u l t y i n an asymptotic ana lys i s of the BVP c e n t r a l to our s h e l l model l i e s i n a proper s c a l i n g of the unknown dependent var i ab le s to a r r i v e at a dimensionless form of the BVP. The proper s c a l i n g va r i e s with the range of load and geometric parameter va lues , and the dimensionless quant i t i e s which play the r o l e of small parameters a l so change accord ing ly . The correct sca l ings i n most cases are deduced from l i m i t i n g s i t u a t i o n s for which exact or adequate approximate so lu t ions are known (or can be obta ined) ; i n other cases, they are in fe r red by p h y s i c a l reasoning. Once the cor rec t s c a l i n g i s known, an approximate s o l u t i o n by per turba t ion and/or asymptotic methods [6,7,8] i s usua l ly s t ra ight forward . In cases where an asymptotic so lu t ion cannot be obtained i n terms of elementary or s p e c i a l funct ions , the a p p l i c a t i o n of per turbat ion or asymptotic methods s t i l l enables us to deduce some q u a l i t a t i v e features of the s o l u t i o n . The order of magnitude of the unknowns, the exis tence of a l ayer phenomenon, the width of the boundary or i n t e r n a l l a y e r , and other q u a l i t a t i v e s o l u t i o n features contr ibute s i g n i f i c a n t l y to an e f f i c i e n t and accurate numerical s o l u t i o n fo r the same problem [13,14,22,23,26] . Because of i t s ( r e l a t i v e ) mathematical s i m p l i c i t y , the Foppl-Hencky nonl inear membrane theory [9,10] has been widely used i n engineering analyses of shee t l ike e l a s t i c bodies . J . G . Simmonds [1] appl ied t h i s theory to a l i m i t i n g case of our problem, namely, a s t e a d i l y spinning d i s c subject to uniform pressure . The Foppl-Hencky theory seems p a r t i c u l a r l y appropriate for the problem as i t gives bounded transverse de f l ec t ions throughout the f l a t d i sc even when the inner hole s i ze shr inks to zero . (In c o n t r a s t , a l i n e a r membrane theory gives an unbounded transverse d e f l e c t i o n at the center for t h i s l i m i t i n g case.) For a s u f f i c i e n t l y h igh pressure load magnitude, the 3 theory also predicts compressive circumferential stresses occuring in some region(s) of the disc. As an ideal elastic membrane would wrinkle in the presence of compressive stresses, i t was concluded in [1] that the axisymmetric nonlinear membrane analysis should not be used for such a high pressure magnitude (relative to the magnitude of the radially directed centrifugal load distribution induced by the steady rotation). Our asymptotic analyses for this problem (in Chapter III) provide a different perspective for the Foppl-Hencky theory. In particular, i t w i l l be shown that, for the high pressure range, the nonlinear membrane solution is in fact the leading term outer asymptotic expansion of the exact solution, and i t alone provides an accurate approximate solution for the in-plane stretching action throughout the entire plate. This and other related results enable us to give a new interpretation of Simmonds' work [1], The insight on the nonlinear membrane theory gained from our asymptotic analyses i s one of the highlights of this report. For the general case, the undeformed shell shape introduces an additional degree of complexity. Aided by the results for some special cases obtained in [2,13,14,22,23], the shell behavior in different regions of the load-geometric parameter space w i l l be analyzed in Chapter IV, again by taking advantage of the presence of one or more parameters after proper sealings. In a l l cases, the nonlinear BVP for shells may be reduced either to a sequence of linear problems or to a nonlinear problem already solved herein or elsewhere, corrected by terms determined by linear problems. As such, the shallow shell problem may be considered completely solved. Accurate numerical solutions of the same BVP confirm the results of our asymptotic analyses. 4 I I . Formulation 1. Axisymmetric Deformations of Shallow S h e l l s of Revolut ion In c y l i n d r i c a l coordinates ( r,6 , z ) , the middle surface of a shallow s h e l l of r e v o l u t i o n may be described by z = z ( r ) . (II .1) A s h e l l i s shallow i f ( d z / d r ) 2 « 1 so that 1+ ( d z / d r ) 2 =1. For s h e l l s subject to axisymmetric external loads which induce only axisymmetric r a d i a l and a x i a l deformation, components of displacement, s t r a i n and s tress are independent of the angular coordinate 6. Let u(r) and w(r) be the r a d i a l and a x i a l displacement components, r e s p e c t i v e l y , of po ints on the middle surface of a s h e l l of r e v o l u t i o n . For an i n f i n i t e s i m a l l y smal l s t r a i n theory, the midsurface s t r a i n components i n r a d i a l and c i r c u m f e r e n t i a l d i r e c t i o n s , e and e . , can be expressed i n r 9 terms of u and w as fol lows [3-5]: e e = * (H.2) e r = u ' + z 'w' + y ( w ' ) 2 . ( I I .3) where a prime denotes d i f f e r e n t i a t i o n with respect to r . Eq . ( I I .2) and the f i r s t term of the righthand s ide of ( I I .3) are the same as the l i n e a r plane s t r a i n case [11]. The second and t h i r d terms of the r i g h t hand side of ( I I .3) are the cont r ibut ions to by w due to the curvature of the deformed s h e l l . They are easy to exp la in geometr ica l ly as shown i n F i g . 1. Arc AB of length ds with 2 2 2 ds = dr + ( z 'dr ) , 5 is moved to A'B' due to displacement components w and u. A'B' has length ds' with (ds') 2 - ( l + u') 2dr 2 + (w'+z') 2dr 2 = [ l + 2u' + (w' + z') 2]dr 2 where, consistent with the small strain approximation (II.3), we have neglected the quadratic term in u'. Hence, the change in length i s given to a good approximation by d s ' - d s s [ l + u ,+Y(w' + z , ) 2 - l - | z , 2 ] d r = [ u'+|(w') 2 + w'z'] dr and the relative change by ds' - ds ds' - ds ds' - ds , , , 1 , ,s2 ,TT T - — - g ~r~ £ u' + z'w' +-r (w') . (II.4) ds / « 0 0 dr 2 /dr 2 + z , 2 d r 2 The approximation of ds by dr in the denominator is consistent with the shallow shell restriction. That completes the derivation of (II.3). The midsurface curvature changes, K Q and K , are given by [3-5] o r <„ = - — , < = -w' . (11.5,6) 6 r ' r ' It i s not d i f f i c u l t to see why curvature change in the meridional direction is given by (II.6). The derivation of (II.5) i s helped by Fig. 2. As shown in that figure, we have , r 1 z' After deformation, z' becomes z'+w*. Hence, 1/RQ becomes A- ~ z' +w' _ z' + w' _ z' +w' = r + u " r ( l + e e ) = r (with 1 + £ Q = 1 for infinitesimal strains) so that o 1 _1_ z' + w ' Z ^ w^  < e = H " Re = r " r = r • The negative sign in (II.5) and (II.6) i s consistent with our definition of M„ and M and the relations between MQ, M and K Q , K . 9 r w r o r 7 2. Stress Resultants, Stress Couples, D i s t r i b u t e d Loads and E q u i l i b r i u m As i n the theory of f l a t p l a t e s , s t r e s s resultants N , N Q and r o s t r e s s couples Mf, Mg for shallow s h e l l s are given i n terms of stresses a^. o e as follows [4,5,12]: •r [N ,N_} = r 6 h/2 {a ,o } dS h/2 r 6 (II.7) fh/2 •n/z {M ,M } = l°r,°6h r 0 Lh/? r w -h/2 The s t r e s s - s t r a i n r e l a t i o n s are a l s o taken i n the same form as for f l a t plates [4,5,12]: e r - A(N r-vNg), Eg = A(Ng-vN r) (II.8) M r = D ( K r + V K Q ) , Mg = D ^ g + V K ^ . For a homogeneous material, we have i n terms of Young's modulus E, Poisson's r a t i o v, and the s h e l l thickness h, 1 Fh^ A - D = 2 • (II.9) ^ 12(1 -v ) The stress resultants and couples of the s h e l l must be i n e q u i l i b r i u m with external surface force i n t e n s i t y ( F i g . 3) f o r any portion of the midsurface. Force and moment equilibrium equations f o r the c l a s s of problems of i n t e r e s t here may be taken i n the form [3-5]: (rH)' - N e + r p R = 0 (11.10) (rV)' + r p v - 0 (11.11) (rM r)' - Mg - rQ - 0 (11.12) 8 where p„, p„ are radial and vertical surface load components. (11.10) is H V the equilibrium equation in radial direction. As shown in Fig. 3, a differential element with side length rd9 and ds is considered. The area of the element is rdOds and the radial component of applied load i s p rdOds. n The radial components of the force vectors are: Hrd6 + 4~ (HrdO)ds - Hrd6 « 4~ (Hrd6)ds ds ds and - NQdsd9. Hence, the equilibrium equation in radial direction i s 4~ (Hrde)ds - NQdsd9 + p„rdsd9 - 0. as o n For a shallow shell, i s replaced by -j^- , and the equation becomes ^ ( r H ) - N 0 + r p H = 0. (11.10) (11.11) i s the equilibrium equation in vertical direction. As shown in Fig. 4, the equation should be Vrd9 + ^-(Vrd9) ds + p„rdsd9 - Vrd6 = 0 ds V or ^ ( r V ) + r p v = 0. (11.11) (11.12) i s a moment equilibrium equation. As shown in Fig. 5, this equilibrium equation should be M rd9 + -£-(M rd9) ds - M rd9 - Qrd9ds - MQdsd9 - 0 r ds r r 6 9 or ^ ( r M r ) - Mg - rQ - 0. (11.12) Consistent with the shallow shell approximation, the radial and axial stress resultants, H and V, are related to the meridional stress resultant N and ' r the transverse shear resultant Q by the relations (see Fig. 6): Q - V - (z' +w')H N r - H + (z'+w')V » H. (11.13) 1 0 3. Reduction to Two Simultaneous Equations Equation (11.11) can be integrated to give r V ( r ) - ^ - p y(x)xdx (11.14) J r i where i s a constant of integration. Let * - rH • - w't £ - z\ (11.15) Then, (11.10) becomes Ng - (rH)' + r p H - * ' + r p H (11.16) while (11.13) and (II.8) become N r = H = | , Q . V - ( 5 + * ) | (11.17) and e r - A ( £ - v\i>' - vrp R) , eQ - A(i/»' + r p R - v ^ ) , M. --D(*' + ^ <fr), Mg = -D(v<t>' + ±t<), (11.18) respectively. With (11.17) and (11.18), Eq. (11.12) becomes D[<fr" +^ <fr' - \ $ ] - £ ( £ + • ) * --V. (11.19) r Substituting (II.2) into (II.3) to eliminate u, we get a compatibility equation e r =  i T E e y + ( z ' + I * H * (11.20) 11 We now use (II.8), (11.15), (11.16) to write (11.20) as A [ ¥ " + ^ - \ n +7<5 + I*)* =-A(rp H)« - ( l + v)Ap H. (11.21) r (11.19) and (11.21) are the two basic differential equations for the class of problems investigated in this thesis. 1 2 4. Boundary Value Problems We consider in this thesis a shell frustum subject to a uniform normal pressure distribution and to a meridionally varying radial load distribution induced by steadily rotating the shell about i t s axis of 2 revolution. In this case, we have p = constant and p *» phw r so that V n - A ( r p H ) ' - ( l + v)Ap H (3 + v)Aphu>2r =-Ap0x (11.22) and *0 1 f r ^0 2 2 _ v ( r ) = _ _ P _ + I P v X d x =__JL + _ Y. ( r2_ r2). ( I I . 2 3 ) where i s a constant of integration. We limit our discussion to the case of a shell with a traction free outer radial edge r • r Q so that V(r Q) » H ( r Q ) = M r^ ro^ ™ °* '* i i e condition V(r^) determines FQ so that -V(r) «= - J Z ^ l ' * 2 ) • (11.24) With (1.22) and (1.24), the governing differential equations (11.19) and (11.20) become 2 2 r — r DOT + - -^•) - ^ (5+*) * - - P v -^2r~ (11.25) r W + 1^,1 - -L^) + 1(5 + =-Ap Qx. (11.26) r The remaining traction free conditions at r » r^ may be expressed in terms of * and $ by way of (11.15) and (11.18) 1 3 r = r n : ^ = 0, - D ( ^ - + -<j>) = 0. (11.27,28) O r dr r The inner edge of the shell at r B ^ is clamped (to a ri g i d hub) so that r = r ^ (j) = u = w - 0. (11.29) The condition of no radial displacement u = 0 may be expressed in terms of i> by way of u • re.. Thus the f i r s t two conditions of (1.27) may be written as r = r ^ • - 0, A ( | j : + rp R - v £ ) = 0. (11.30,31) The two ODE (11.19) and (11.21), and the four boundary conditions, (11.27), (11.28), (11.30) and (11.31), form a two point boundary value problem (BVP) for the determination of $ and IJJ. A l l remaining quantities which describe the shell behavior can be obtained from (f> and ^  and/or their f i r s t derivative. The only exception is the axial displacement w which i s give n by w - f 4>(t) dt (11.32) J r i where we have made use of the third conditions in (1.29) for a clamped edge. The remaining chapters of this thesis are devoted to a characterization of the solution of the BVP for different load magnitudes p v and p Q. Edge conditions different from (11.27,28) and/or (11.30,31) can also be analyzed. We mention here only the special case of a shell closed at the apex of a dome shaped shell of revolution. For this case, the boundary conditions (11.30,31) are replaced by the regularity conditions: r - r,: • - • - 0. (11.33,34) The second c o n d i t i o n along with the boundedness of i|>' impl ies the symmetry c o n d i t i o n of no r a d i a l displacement at the apex. 15 I I I . F l a t Plates 1. Dimensionless Formulation Let r. x = — , x = , f ( x ) „ i . , g( x) = ± ( I I I . l ) r0 0 <f> * where $ and \J/ are scale factors to be s p e c i f i e d . With the help of these dimensionless q u a n t i t i e s , the two point BVP for <(> and ^ defined by (II.9), (11.21), (11.27), (11.28), (11.30) and (11.31) with £ = 0, become r Q X 0 r 0 x 0 where now a prime indicates d i f f e r e n t i a t i o n with respect to x . To allow for f i n i t e deformations, <J> and must be chosen so that the nonlinear terms are not of higher order. This requirement can be met by ( t e n t a t i v e l y ) s e t t i n g 2 r ' 2 r ' (IH.4) r0 0 r0 ° or — /DA — D • = — — , <!» = — • (III.5) 0 0 With (III.5), Eqs. (III.2) and (III.3) become 2 f + x f - "2 f - x f g " " k V ( I I I ' 6 ) X g" + I + ^ f 2 = - V (HI.7) x where 16 4 3 ro ro k v = - — p v , k H = — p (III.8) D/DA while the boundary conditions (11.27), (11.28), (11.30) and (11.31) assume the following forms: g(1) = 0, (III.9) f'(1) + vf(D = 0, (III.10) f(x ) = 0, (III.11) k g'(x i) - g ( X i ) + 3^ = 0. (III.12) i The dimensionless load parameters ky and k H w i l l play an important role i n the c h a r a c t e r i z a t i o n of the solu t i o n of our BVP. If k y < 1 and k^ < 1 the above dimensionless form of the BVP i s appropriate f o r a perturbation s o l u t i o n The s p e c i a l case with k << 1 and k << 1 w i l l be referred to as V H low pressure and slowly spinning case; a l i n e a r theory i s expected to be an adequate approximation f o r th i s case. Other p o s s i b i l i t i e s of scalings have been worked out for k y >> 1 and/or k H >> 1. B r i e f l y , the (ky»k H) plane i s 2 divided i n t o four regions: Region (1.1) with k < k < 1, Region (1.2) V H 2 2/3 k o < K „ < 1» Region (2) with k > 1 and k,/ < k , and Region (3) with k > 1 H V M V n V 2/3 and k H < k v . The so l u t i o n of the BVP exhibits d i f f e r e n t q u a l i t a t i v e features i n each of these regions. Accordingly, the method of so l u t i o n w i l l be d i f f e r e n t i n d i f f e r e n t regions and w i l l be discussed i n separate sections of t h i s chapter. 17 2. Small to Moderate Load Magnitude Range In this section we consider the moderate load magnitude range with k, < 1 and k < 1. For t h i s range of k and k values, a regular V n V H perturbation s o l u t i o n of the BVP i s appropriate. Without further r e s t r i c t i o n s on k and/or k , two parameter perturbation series i n k and H V H ky for f and g would be necessary and are obtained i n Appendix A. To have a a better understanding of the q u a l i t a t i v e behavior of the sol u t i o n , we 2 2 consider separately the two cases k A „ < 1 and k /k > 1 . A separate H V H V treatment of the two cases enables us to use a perturbation series i n one parameter only. 2 Region (1.1): k < k < 1. V n For the s p e c i a l case kyA H K < 1 (including ky = 0), we expect the s o l u t i o n to approach that of a rotating disk, [11]. Therefore, we set g(x) = k g(x) , f(x) = k f(x) (III.13) n V and write (III.6) and (III.7) as 1 1 kH i 2 f " + - f' - i - f - - 2 . f g = _ iz2L_ (III.14) x 2 x * 2x x ^ x 1 . 1 Mm .2 g" + - g ' - - 2 g + ^ f = " x ( I I I . I S ) X 2 where k < 1 and u = k /k u < 1. Correspondingly, the three boundary n m V H conditions (III.9)-(III.11) re t a i n t h e i r form with f and g replaced by f and g while (III.12) becomes 2 v x i g ' ( x i ) " xT^V + 3+^ = ° * (III.16) 18 2 For u < 1 (k < k ), we may seek a perturbation s e r i e s s o l u t i o n m V H in powers of u with k as a parameter m H oo {f,g} = I {f <x;k u),g ( x ; k u ) } u n . (III.17) _ rt n n n H m n=u The leading term solutions, g Q and f Q , are determined by two uncoupled l i n e a r BVP g o + x G6 " h go = _ x ( x i < x < 1 ) x 2 V 1 ) = G6<V -x7g0 (xi> + 3 T v = ° (III.18) k 2 f 3 + i f i - V o - r V o = - i r ( X . < X < D (III.19) f 0 ( x . ) = f'(1) + v f 0 ( 1 ) = 0 . We f i r s t solve (III.18) for g Q and then use the r e s u l t i n (III.19) to get a second order l i n e a r ODE for f Q . The BVP (111.18) i s i d e n t i c a l to the well known problem of a rotating disk free at the outer edge and clamped to a r i g i d hub at the inner edge*. If k << 1, the contribution of the f ng-. term H 0 0 i n (III.19) i s n e g l i g i b l e . Thus, to leading order, the bending and 2 str e t c h i n g actions of the plate are completely uncoupled f o r = k y / k H < < J 1 and k « 1. H *The s o l u t i o n of th i s problem i s given l a t e r i n (III.36) 19 For moderately smal l values of \i (and k not smal l compared m H to u n i t y ) , h igher order c o r r e c t i o n terms i n (III .17) w i l l have to be c a l c u l a t e d . I t i s not d i f f i c u l t to see that these higher order terms are a l so determined ( sequent ia l ly ) by l i n e a r boundary value problems. Higher order correc t ions w i l l be analyzed i n more d e t a i l by the two-parameter per turba t ion se r i e s i n Appendix A . I f k__ << (y <) 1, we may take instead of ( I I I . 17) a H m per turbat ion s o l u t i o n i n powers of k (keeping y as a parameter): H m {f,g} = I { F n (x ; y m ) , Gn(x;ym)}kJJ . ( III .20) n=0 The lead ing term so lu t ions FQ and GQ i n that case are a l so determined by two uncoupled l i n e a r BVP: F " + - F ' *0 x 0 1 - X 2x F Q ( x i ) = F £ ( l ) + v F 0 ( l ) = 0 (III .21) ( G" + — G' - -L G r t + ^ - P ? - - x J0 x 0 x 2 0 2 x 2 0 < x, G 0 ( 1 ) = G 0 < x i ) - x ; G 0 ( x i > + = 0 (III .22) We f i r s t solve (III .21) for F Q and then use the r e su l t i n (III .22) to get a second order l i n e a r ODE fo r G Q . The BVP (III .21) i s j u s t the c l a s s i c a l problem of axisymmetric bending by uniform pressure of a f l a t c i r c u l a r p l a t e . 20 clamped at the inner edge and free at the outer edge. I f , we have (in 2 2 addition to k„ « 1) p = k„/k u « 1 also, the contribution of the FN term ri m V H 0 i n (III.22) i s n e g l i g i b l e ; the bending and stretching plate action uncouple as we pointed out previously. For moderately small values of k (and u not small compared to H m un i t y ) , higher order c o r r e c t i o n terms i n (III.20) w i l l have to be c a l c u l a t e d . I t i s not d i f f i c u l t to see that they are also determined (sequentially) by l i n e a r boundary problems. Higher order c o r r e c t i o n terms are also d i s c u s s e d i n A p p e n d i x A i n t h e f ramework o f a t w o - p a r a m e t e r perturbation s o l u t i o n . 2 Region (1 .2): k < k„ < 1 . " H V 2 For the s p e c i a l k << k < 1, (including the l i m i t i n g case k = 0 H V H and ky * 0), we expect the s o l u t i o n to be approximately that for bending by uniform pressure [12]. Therefore we set f = k y f ( x ) , g = k yg(x) (III.23) and write the OOE (III.6) and (III.7) as k 2 2 1 1 V 1-x f" + x " f ' - I f --*- f * " -"25- ( I I I ' 2 4 ) x g- ^ g ' - : L 2 * + k*2--i» ( I I I ' 2 5 ) x m 2 - 1 2 where k, < 1 and u = k„/k„ < 1« Correspondingly, three boundary v m H v conditions (III.9)-(III.11) r e t a i n t h e i r form with f and g replaced by f and g while ( I I I . 12) becomes 2 x. g'(x. ) - — g(x. ) + — T ^ — r = 0. (III.26) l x. l v (3+v) i m 21 2 With ky < 1, we may seek a perturbation series s o l u t i o n i n powers 2 of ky. The leading term solutions, again denoted by and g^, are determined by two uncoupled l i n e a r BVP: f J + r f i - ^ f , - 1 " X 0 x 0 2 0 2x ' X f 0 ( x . ) = f i d ) + v f 0 d ) = ov ... + l g . _ 1-g = . i f 2 _ l x s 0 x y0 2 y0 2 x 0 u x m (III.27) (III.28) 2 x. g n O ) - g'(x. ) - — g_(x. ) + - , = 0. 0 30 i x. 0 i ji (3+v) l m We f i r s t solve (III.27) for f Q and then use the r e s u l t i n (III.28) to get a l i n e a r BVP for g Q . The BVP (Hl.27) i s j u s t the problem for the l i n e a r bending by uniform pressure of an annular plate clamped at the inner edge to a r i g i d hub and free of t r a c t i o n at the outer edge. Whether or not we have 2 -1 k << k (vi << 1), the bending and stretching action of the plate are H V m always weakly coupled as (III.28) gives a n o n t r i v i a l s o l u t i o n even for the l i m i t i n g case of a non-rotating plate with Ic^ = 0 (so that U ^ = 0). This i s i n contrast to the plate behavior i n region (1.1). m Note that higher order terms i n the perturbation s e r i e s are also determined by l i n e a r BVP and are discussed in Appendix A. The s p e c i a l case of = 0 has been solved i n [12] by a power s e r i e s s o l u t i o n i n x with no r e s t r i c t i o n on k^. 22 3. The R e l a t i v e l y Fast Spinning Range. 2 3 In region (2) where k > 1 and k„ < k , the plate i s expected to M V H experience s i g n i f i c a n t in-plane extension throughout. This suggests that we l e t kV g(x) = k u g ( x ) , f(x) = — f(x) (III.29) H KH and write (III.6) and (III.77) as T [ ' - ; " - ^ - J " - ! 3 ! «»•»> H X 3 1 1  v I 2 9 " + 7 9 ' " ~2 9 + l x " f X (III.31) x where 3 *V - — . (III.32) K H Three of the four boundary conditions (III.9)-(III.11) r e t a i n the same form with f and g replaced by f and g. The fourth (III.12) becomes i d e n t i c a l ( i n form) to (III.16). 3 With < 1, a perturbation s o l u t i o n f o r f and g i n powers of i s appropriate. The leading term solutions f Q and g Q s a t i s f y the ODE k l*0 ' x '0 2 *0 J x "0^0 2x H x J _ r f - + l f . _ l _ f 1 - l f g ; l r 0 x 0 2 0 J x 0 y0 (III.33) % + x- g0 - - T g 0 = " X (III.34) x and the same boundary conditions as the o r i g i n a l problem. The s i m p l i f i c a t i o n of (III.31) f o r the leading term approximation i s s i g n i f i c a n t as g r t may now be determined separately. With the boundary conditions, 23 2 x. g 0<1)-0, g « ( X i ) - ^ - g 0(x.) + ^ « 0, (111.35) we have 1 1 1 3 1+v (3*v)+(1-v)xf g 0(x) - A(x - -) • -a- x 3,, A - i W . \ / 1 \ 2 ' ( I I I ' 3 6 ) (1+v)+(1-v) X i Upon substituting (III.36) for g Q(x), (III.33) becomes a linear ODE for f Q 1-Ak k 2 f o + 7 £ o " ^  + V* + I> " IT x 2 ] f o - " *H ^ • ( I I I - 3 7 ) X The two boundary conditions for fQ are f Q ( x i ) - 0, fl(1) + vf Q(1) = 0. (III.38) The solution for this linear BVP can be obtained by asymptotic or numerical methods. For moderately small values of u , higher order terms in the 3 perturbation series i n may be necessary for a more accurate solution. These correction terms are also determined sequentially by linear boundary value problems similar to those for the leading term solution. In 3n particular, we have for the coefficients u^, g- + 1 g. - 1- g - - 1- y f f y n x y n 2 9 n 2x * zk n-1-: k=0 (III.39) g n(1) =0, g; ( X i) - £ - g ( X i ) =0 l and l ,1 k H k u n - 1 £; ^ ' i - h ' r V ' l ' n - r l V . . . x k=0 f (x ) - 0, f'(1) + vf (1) - 0. n l n n (III.40) 24 2 3 Thus, the restriction of fast spinning, < 1 (k^ < k^), reduces the nonlinear problem to a sequence of linear problems with the bending and stretching action of the plate weakly coupled through the forcing terms in the governing (linear) differential equations. For k » 1, an asymptotic solution of the linear BVP for fn is given in Appendix A. Except for a particular solution, the same asymptotic solution i s also applicable to f •. Even without an explicit solution, i t is clear from the form of the BVP (III.37)-(III.38) that the (dimensionless) -1/2 layer width near x = x. is 0(k ) while, because g n ( l ) = 0, the -1/3 (dimensionless) layer width near the turning point x = 1 i s 0(k ). H Furthermore, the layer strength is considerably weaker near the outer edge. 4. The R e l a t i v e l y High Pressure Range 3 2 In Region (3) where k > 1 and k < k . the plate i s expected to V n V experience s i g n i f i c a n t bending throughout. This suggests that we l e t g(x) - k j / 3 g ( x ) , f(x) = k y / 3 f ( x ) (III.42) and write (III.6) and (III.7) as 2 ^ [ f . + I f . . J 3 f ] . I f , . . l g - «„!.„„ k v x X g-+ ; g ' ' h * + k f* = ' $ (III-44) x H £ where u i s as given i n (III.32). The three boundary conditions ( I I I . 9 ) -(111.11) r e t a i n the same form with f and g replaced by f and g. The fourth (111.12) , becomes 2 g'(x.) -^-gU , ) •^T-JllvT " 0. (HI.45) With > 1, a regular perturbation s o l u t i o n i n powers of y~ 1 i s p o s s i b l e . The leading term f„ and q^ of such the perturbation series are determined by the d i f f e r e n t i a l equations and boundary conditions for f and but with terms m u l t i p l i e d by 1/u. replaced by zero. Evidently the BVP f o r fg and g n remains a nonlinear problem and an elementary so l u t i o n i s not p o s s i b l e . On the other hand, for extremely high pressure loading so that 2/3 ky >> 1, a s o l u t i o n by the method of matched asymptotic expansions i s appropriate for a l l £ 1. The leading term outer (asymptotic expansion) s o l u t i o n s , denoted by FQ(X) and G n(x), are determined by 0 x ° x^0 26 and the boundary conditions GQ(1) = 0 and (III.45) with g replaced by G Q . The two equations in (III.46) are just the governing differential equations for the Foppl-Hencky nonlinear membrane theory [9,10] (in dimensionless form) used in [1], Methods of solution for these equations have been discussed there as well as in [15-21]. Higher order correction terms of the outer solution may also be obtained by a regular perturbation series -2/3 solution in powers of . Evidently, the outer solution can only satisfy two of the four boundary conditions as the governing differential equations for the n-th order correction terms G and F constitute only a n n ' second order system. It is therefore necessary to obtain an inner (asymptotic expansion) solution valid in the neighborhood of each edge which satisfies a l l four boundary conditions on that edge. The actual asymptotic solution to the original problem is then obtained by matching the outer and inner solutions in some overlapping (intermediate) region of validity and forming a uniformly valid composite solution. In general, the outer solution GQ cannot be obtained in terms of elementary or special functions. The matched asymptotic expansion technique nevertheless provides us with many qualitative features (such as the order of magnitude of f and g in various regions of the solution domain, the width of the boundary layers adjacent to each edge, etc.) of the exact solution of the original BVP very useful in obtaining numerical solutions for that problem. For the present problem, we can do more. Even without an explicit solution for G ^ and F^, i t is possible to show (see Appendix A) that GQ i s in fact the dominant term in the exact solution for g of our problem throughout the plate. The nonlinear membrane solution thus provides an accurate approximate solution for the stretching actions of the plate 2/3 whenever i s l a rge compared to u n i t y . We w i l l re turn to t h i s point l a t e r i n Sect ion (7). 5. Direct and Bending Stresses In structural designs, i t i s often important to know whether the bending or stretching action of the plate is dominant. Therefore, the relative magnitude of the direct (membrane) and bending stresses, o and a u B in the plate for a particular loading i s of interest. The order of magnitude of o_ and o*0, given by D B N = e °D ~h~ 6M _ e_ °B u2 n generally varies from region to region in the (ky,^) plane. 2 In region (1.1) where < 1 and ky/k^ < 1, we have <fr - f f - ^ [k f(x)] r 0 v and correspondingly M . - D ( v ( v f . + I f ) 9 dr r r r„ v x As differentiation with respest to x does not change order of magnitude, we have 29 °B _ , 6/DA k V . n / v , — = 0 ( - r — j^-) - 0 ( — ) °D " KH kH Therefore, bending action dominates i f >> and stretching action dominates i f k^ « kfl. 2 In region (1.2) where k y < 1 and k^/ky S 1, we have ^ . ff . [ k f ( x ) ) ] r0 * = £g 8 3 _T" [kyg(x) ] 0 v and correspondingly D/DA M, e 2 V v f + i f ) , NQ - A k J [ g , + _ x _ ] r0 r0 m As there are no layer phenomena and p > 1, we have m °B 6Vh2 „,6/5A, . , 1 , so that a i s 0 ( a ) at most and is considerably smaller i f k << 1. D D V 2 3 In region (2) where k^ £ 1 and k^ < k^ , we have ro 'Si ro H and correspondingly, D/DA N , ,. . 1 « „ _ D , . , x 2 r2 k ^ * + x t ; ' 6 ^ ' V 8 3+v' l0 " 0 30 In the i n t e r i o r of the pl a t e , we have then o h 2 K 2 ' ^ 3/2 1/2 ' ' therefore, o i s at le a s t of the same magnitude as a ; i t i s considerably 2 3 larger i f e i t h e r k^ >> 1 or k y << k^. Near the edges however, f ^ e x h i b i t s a boundary layer phenomenon i f k^ >> 1 so that we have f ' = 0(^k^f) at most while g' remains 0(g). Therefore, we have near an edge r - " ^ - 0 ( 4 ^ ] ) . o . 4 » r ^ » * „ either o Q or may be dominant depending on the magnitude of the r a t i o k V / k H / 2 ' 3 2 F i n a l l y i n region (3) where ky > 1 and < k^, we have + -Tf - ^ I k i / 3 f ( x ) ] . • - * g - ^ " [ k j / 3 g ( x ) ] 0 0 and correspondingly 2 x r - ,1/3, . , , 1 . , M D .2/3, , ^ [e. r \ [ v f + x f ] ' e = ~2~ Sr [ g + jT,(3+v) * n r_ -C With > 1, we get °B _ /DA 1 x n n , " l / 3 , a ^ - 0 ( l T ^ 7 3 ) = 0 ( k V > so that o„ i s 0(o_.) at most and i s s i g n i f i c a n t l y smaller i n the plate o D interior i f ky >> 1. Evidently, in this range of parameter values, the plate resists the predominantly transverse loading by i t s membrane action. Near the edges, f^ and g^ both exhibit a boundary layer phenomenon i f ky >> 1. From the results of section (4), we have near the inner edge f «= 0 ( k v / 3 ) , g» = 0(1) so that ^ - o < f - oa,. Thus, for ky >> 1, the bending and direct stresses of the plate are of the same order of magnitude near x = x.. Similarly, o i s much smaller i B than o n near x • 1 as well as in the plate interior. 32 6. Comparison with Finite Difference Solution A FORTRAN computer program has been developed to calculate finite difference solutions of the dimensionless two-point BVP defined by (III.6), (III.7), (III.9)-(III.12). For a given set of input values of the parameters v = Poisson's ratio x^ = r^/r^, the inner-to-outer hole radius ratio ky = p^r^/D/DA, the dimensionless pressure load magnitude 3 2 4 kg = PQ ^ Q /D = (3 + v)phw r r t /D , the dimensionless radial load magnitude, the computer program obtains f(x) and g(x) to a specified accuracy and then calculates the corresponding distributions of stress resultants, stress couples and displacement components from the auxiliary formulas, (11.15)-(11.18), (11.32) and u=re Q. The actual methods used in the various steps o toward a finite difference solution are described in Appendix B. Numerical solutions for a number of check cases have been generated by the above computer code to validate the code; these include cases examined in [2] and [22]. For a typical case, the interval [x^,l] is usually divided into 320 subintervals. (With x^ = 0.1, the subinterval length is 0.0028125.) The local truncation error associated 2 -5 with finite difference scheme is 0(h ) < 10 . The computation typically begins with a coarse mesh using some limiting solution as an i n i t i a l guess. After convergence for the given mesh, the program proceeds to a finer mesh (usually obtained by doubling the mesh points) with (the linear interpolation of) the last solution as an i n i t i a l guess; the process continues until the 33 mesh contains 320 subintervals* At every step, solution is substituted back into the corresponding f i n i t e difference equations to compute the maximum (residual) error. The process terminates when the maximum-absolute residual error i s below the specified tolerance, which i s taken to be less than 10" ; the solution i s said to have converged for the given mesh. The above computer code has been used to generate numerical solutions for a large number of new cases to confirm the theoretical conclusions of sections (2)-(4). A small selection of the numerical solutions w i l l be presented below to bring out the qualitative features of the solutions in different regions of the k^k^-parameter plane. The selection usually consists of (i) f ( l ) = r ^ (r^)/JliR, ( i i ) mr(x^) = f'(x.) + v f ( x i ) / x i , ( i i i ) i ( x ± ) = r 0 i K r ± ) / D , (iv) n* = max [g' + l^x /(3+v)] The f i r s t two quantities give typical features of the out-of-plane bending actions of the plate while the last two give typical in-plane stretching plate actions. The two stress measures n Q m and mr(x^) give the maximum direct stress and the maximum bending stress, respectively; they are important in design studies. In a l l cases reported herein, v i s taken to be 0.3. A l l numerical results given are accurate at least to the number of digits displayed, with n m being read from computer output for discrete 0 values of x. 2 Region (1.1): ky < k^ < 1 From the results for k^ = 10°, 10 - 1, 10~2 and 10 _ 4 and for u m = ky2 / l ^ = 10°, 10 - 1, 10 - 2, IO - 3 and 0 given in Table (1), we make the following observations in support of the asymptotic results of section (2) for this region of the parameter plane: The mesh size would be halved again only i f the solutions for the last two meshes s t i l l d iffer by more than a prescribed tolerance. 34 (1) For a f i x e d value of u , numerica l so lu t ions fo r g (x . ) and m x nj 1 1 are e f f e c t i v e l y p r o p o r t i o n a l to k^. (2) For a f i x e d value of u , numerical so lu t ions for f ( l ) and m m (x ) are e f f e c t i v e l y p r o p o r t i o n a l to /kT! (= k //T) and r i n v m there fore to ky . (3) For a f i xed 1^, g U ^ / k g , n ^ / k , , , f ( l ) / k y and m.Up/ky do not change apprec iab ly as decreases and hard ly change at a l l for u << 1. m (4) As y + 0 , g ( x , ) , n 0 m , f ( l ) and m (x.) tend to the m l o r l corresponding values f o r the l i m i t i n g case of no pressure load (which i s the l ead ing term of a pe r turba t ion s o l u t i o n i n u ) ; i n p a r t i c u l a r f ( l ) and m (x.) tend to zero as y + 0 . m r I m Thus, the accurate numerical s o l u t i o n s conf irm the f ac t that a truncated per turba t ion s e r i e s s o l u t i o n i s expected to be adequate fo r u^ < 1. 2 Region (1 .2 ) : k^ < ky < 1 From the r e s u l t s for ky = 1 0 ° , 1 0 _ 1 and 10~ 2 and for y m = 1 0 ° , 1 0 1 , 2 3 A 6 10 , 10 , 10 and 10 given i n Table (2) , we make the fo l lowing observat ions i n support of the asymptotic r e s u l t s of s ec t ion (2) for Region (1.2) i n the parameter p lane : (1) For a f i x e d value of u , numerical so lu t ions for f ( l ) and m m (x^) are e f f e c t i v e l y p r o p o r t i o n a l to ky whi l e g(x^) and n^ 2 are e f f e c t i v e l y p r o p o r t i o n a l to ky . 3 5 (2) For a f i x e d value of ky , the four q u a n t i t i e s do not change apprec iab ly for l a rge values of u and a l l tend to a l i m i t m as y 0 0 corresponding to the case of no sp inning , i . e . m Si 0. (3) For y s u f f i c i e n t l y large (with the c r i t i c a l va lue of y m m depending on k y ) , compressive c i r c u m f e r e n t i a l s t resses appear at and adjacent to the outer edge of the p l a t e . For very l a rge k T T , the c r i t i c a l value of y co inc ides wi th the V m corresponding c r i t i c a l value f o r the nonl inear membrane s o l u t i o n observed i n [1] . I t appears that f o r smal l va lues of u m ^ » the (truncated) per turbat ion s o l u t i o n proposed i n s e c t i o n (2) f o r Region (1.2) gives an accurate approximate s o l u t i o n fo r the problem. Region (2) : ky 2 < , k R > 1 0 1 2 3 A From the numerical r e s u l t s for k = 10 , 10 , 10 , 10 , 10 and n 10 5 and f o r y £ = k^l3/k^ = 1 0 ° , 1 0 _ 1 , 1 0 - 2 , l ( f 3 , 1 0 - 4 and 0 given i n Table (3) , we make the f o l l o w i n g observations i n support of the asymptotic r e s u l t s of s e c t i o n (3) f o r t h i s reg ion i n the parameter p lane : (1) For a f i xed value of y^ < 1, numerical so lu t ions for g(x^) and n^ are e f f e c t i v e l y p r o p o r t i o n a l to k^ as p red i c t ed by the asymptotic s o l u t i o n . (2) For a f i xed value o f y ^ < 1, the numerical so lu t ions fo r f (1) 3/2 I— 2 are e f f e c t i v e l y p r o p o r t i o n a l to y ^ / k ^ = ky/k^ fo r k^ > 10 3/2 3/2 2 and to v [ ^ " * v f ° r \ < 10 . This i s due to the fact -2 that g(x) i s numerica l ly s m a l l , of order 10 near x ^ so that 2 we have f = 0(1^) for k^ = 0(10 ) or smal ler . 36 (3) For a f ixed value o f k^, g ^ ) , n™ , f C D / d y ^ ) and in^x^/Cky/k^) do not change appreciably with y^ as y^ decreases to zero. In the l i m i t as (i^ + 0, i t i s expected A A " * m that f and tend to zero and g(x^) and n Q tend to the r o t a t i n g d i s c s o l u t i o n and they do. (4) For >> 1, the (dimensionless) boundary l ayer width -1/2 (near x^) i s seen from Table (4) to be 0(1^ ) as pred ic ted by the asymptotic a n a l y s i s . For a measure of the l a y e r width , we use the dimensionless distance Ax * x - x^ wi th x be ing the l o c a t i o n where m decreases to one tenth of i t s r value at x^. Table (4) gives Ax for a range of values of -2 k R and for y^ = 1 and 10 taken from our numerical so lu t ions of the BVP for f and g. The l a y e r width near x = 1 has a l so -1/3 been v e r i f i e d to be 0(k ' ) . H (5) For kjj >> 1 and y^ << 1, the magnitudes of f and g are approximately those of the l i n e a r membrane s o l u t i o n , i . e . the leading term outer s o l u t i o n of our BVP; the l a y e r s o l u t i o n near x^ i s needed only to make f(x^) = 0. The magnitude of iii^ i n the l ayer adjacent to x^ i s seen from the numerical so lu t ions to be l a rger than the outer s o l u t i o n by 1/2 a f ac tor p r o p o r t i o n a l to k as pred ic ted by the asymptotic H a n a l y s i s . On the other hand, the strength of the l ayer s o l u t i o n adjacent to the outer edge of the p l a t e i s weak and has very l i t t l e e f fec t on f and g; i t s main funct ion i s to ensure m r ( l ) = 0. This i s again cons i s tent wi th the behavior of the asymptotic s o l u t i o n i n the neighborhood of the turn ing point obtained i n s e c t i o n (2) of Appendix A . 37 The above observations confirm that a truncated perturbation solution in p^ i s adequate in Region (2). In fact, the numerical solution for ky • = 10 "* agrees with the leading term perturbation solution to within 2%. Region (3); k 2 / 3 / ^ > 1, k y > 1 From the numerical results for ky = 10^,103,10^ and 10^ and 0 1 2 3 A for = 10 ,10 ,10 ,10 and 10 given in Table (5), we make the following observations in support of the asymptotic results of section (4) for this region of the parameter plane: * " m (1) For a fixed value of p^, g(x^) and n^ are effectively 2/3 proportional to ky while f ( l ) and mr(x^) are effectively 1/3 proportional to ky when ky i s sufficiently large compared to unity. (2) For a fixed value of ky, the same four quantities (and in fact the solution of the BVP i t s e l f ) do not change appreciably for p^ > 1. The larger i s the value of p^ , the sooner they reach their respective limiting values. (3) From a table similar to Table (4) for Region (2), the width of the boundary layer adjacent to x = x^ is seen to be -1/3 proportional to ky for sufficiently large ky. (We w i l l not display the table for Ax here.) The layer width near x = 1 is seen to be proportional to k^^. (4) For g, the strength of these layers are weak compared to the nonlinear membrane component of the solution. For ky = 10 * 2/3 the numerical solution for g = g/ky differs from the 38 corresponding nonlinear membrane solution by less than 0.2% for > 1. (5) For f, the strength of the layer near x^ i s comparable to the nonlinear membrane component of the solution. The strength of the layer near the outer edge i s considerably weaker than the nonlinear membrane component. In fact, outside the layer, near x • x^, the numerical solution for 1/3 f/ky <= f differs from the nonlinear membrane solution by less than 0.4% for k y = IO"* and £ 1, the layer region adjacent to the outer edge included. The above observations confirm that a truncated perturbation solution in U p * should be adequate in Region (3). 39 7. On Nonlinear Membrane Theory Because of i t s mathematical simplicity, the Foppl-Hencky nonlinear membrane theory has been widely used i n engineering analysis of sheetlike elastic bodies. Of interest here i s the use of this theory in the analysis of a steadily spinning disk under a uniform normal pressure distribution [1]. With recent advances i n both nonlinear theories of plates and shells and in methods for asymptotic solutions of dif f e r e n t i a l equations, we now see that the Foppl-Hencky theory may be considered as the outer asymptotic expansion of the exact solution of the von Karman plate theory (see section 4 ) . Physically, the Foppl-Hencky theory i s an approximation of the von Karman theory neglecting i n the latter the effect of the small bending stiffness. In this section, we examine the range of applicability of the Foppl-Hencky theory taking advantage of the asymptotic results obtained i n sections (3) and (4) of this chapter. It is evident from our asymptotic results that even within the framework of a moderately small rotation theory, a nonlinear membrane theory 3 2 i s appropriate only in the relatively high pressure range, i.e., k H < ky, or in the notation of [1] ~ 2 3 k = 16 ky/hjj > 16. It is found in (1] that compressive circumferential stresses appear for k = k = 0 .3154 i n the limiting case of a vanishing small hole (x. •*• 0) and wp l for smaller k i f x i > 0 . Now, compressive stresses induce wrinkling in the elastic membrane and are therefore unacceptable for various design purposes. However, these same compressive stresses pose no design problems when the 40 membrane theory which generates them constitutes only the outer asymptotic expansion of the exact solution for the von Karman plate theory. The sheet does have a f i n i t e bending stiffness, however small, which would induce bending stresses whenever necessary to inhibit the occurrence of wrinkles. In particular, significant bending stresses develop in a region adjacent to a clamped edge of the annular plate where the outer solution ceases to be valid. A different type of asymptotic solution, which incorporates the significant effect of the small bending stiffness, i s appropriate in the edge zone. Thus, i n the framework of a matched asymptotic expansion solution, the results in [1] obtained from Foppl-Hencky nonlinear membrane theory continues to be useful whether or not the resulting membrane stresses are compressive. ~ 2 3 For k = 16k„/k0 < 16 and k > 1*, we know from section (3) that a V H n "quasilinear" plate theory resulting from a regular perturbation solution in 2 3 powers of ( f ty/h H) for the von Karman equations i s appropriate for our problem. As we found in section (3), this theory generates membrane stresses by a (quasi) linear membrane theory (III.34)-(III.35). The bending action of the plate is then determined by a linear bending theory (III.37)-(III.38) which incorporates the effect of the membrane solution just obtained. It has been argued [1] that a nonlinear membrane theory should be used as i t leads to f i n i t e transverse deflection throughout the sheet while the (quasi) linear membrane theory gives an unbounded transverse deflection at the center of an annular plate with an infinitesimally small hole. With the *It i s clear that for k R < 1 (and therefore k y < 1 also), a suitable linearized theory suffices as we are in region (1.1). 41 asymptotic r e s u l t s of s e c t i o n (3), we now see that a (quasi) l i n e a r membrane theory i s i n f a c t appropriate f o r the determination of the membrane stresses i n t h i s range of parameter values. However, i t should be supplemented by a (quasi) l i n e a r bending theory f o r the bending p l a t e ac t i o n , with $ „ $ ( k y / k ^ ) f Q determined by (III.37)-(III.38), r e s u l t i n g i n a bounded transverse d e f l e c t i o n everywhere. On the other hand, the use of a nonlinear membrane theory without a matching inner s o l u t i o n also may not give an adequate approximation of act u a l plate bending behavior, even though i t gives a f i n i t e transverse d e f l e c t i o n and an accurate approximation f o r the s t r e t c h i n g action throughout the sheet. IV. Shallow She l l s 1. In t roduct ion When 5 ± 0 so that the s h e l l i s not a f l a t p l a t e , we may again use the dimensionless q u a n t i t i e s introduced i n ( I I I . l ) a long wi th r ^ l i l i i, 5 ° r ° / o n »» 2 \ C 0 r 0 s(x) = , k_ = — — ; = /12 ( l -v ) —r— , — — ' J - ' - \ " / v, » (IV.1) ? 0 G /DA H where £ Q i s chosen so that max|s (x) | = 1, and t e n t a t i v e l y w r i t e (11.25) and (11.26) as 2 L [ f ] - i [ k G s ( x ) + f ] g = - k y ^ g " L[g] + i [ k G s ( x ) + y f ] f = - l ^ x (IV.2) with ky amd k^ given by ( I I I .8 ) and L[ ]=[}"+ h ] ' - \[ ] . (IV.3) x In terms o f f and g, the re levant boundary condi t ions for the s h e l l case have the same form as those for the p l a te case given by ( I I I . 9 ) - ( I I I . 1 2 ) . With k = 0, (IV.2) and (IV. 3) reduce to the corresponding p l a t e equations G ( I I I .6) and ( I I I . 7 ) . The appearance of the k s(x) term i n (IV.2) and G (IV.3) introduces an unusual degree of complexity which makes the ana ly s i s of the s o l u t i o n of our BVP considerably more invo lved . -I 4 3 Consider f o r example the extreme case when the s h e l l i s subject only to normal pressure load ing . We know from the r e s u l t s i n [13,14,22,23] that "po la r d impl ing ' may occur for some range of k va lues . That i s , G depending on the r e l a t i v e magnitude of k^, ky and k^, the s o l u t i o n of our BVP at a given l o c a t i o n may be e i t h e r f 2 0 or f « -2k s(x) and that an G i n t e r n a l l ayer phenomenon may ex i s t at a l o c a t i o n determined by the load magnitude. The occurence of po lar dimpling depends on the i n t e r p l a y among kg, ky and k^ i n a complicated way which i s not at a l l obvious from the analyses for the p la te case. To deduce the correct s h e l l behavior corresponding to d i f f e r e n t combinations of load and geometric parameter va lues , the (k^.ky.kg) parameter space w i l l be d iv ided i n t o three p r i n c i p a l regions ( F i g . 8 ) . In the next few s e c t i o n s , we w i l l work out the appropriate r e - s c a l i n g s of the BVP ( I V . 2 ) , ( I V . 3 ) , ( I I I .9 ) - ( I I I .12 ) i n the var ious subregions of these p r i n c i p a l regions and d i scuss the behavior of the s h e l l implied by these r e - s c a l i n g s . 44 2. The Light to Moderate Load Magnitude Range We consider in this section the load magnitude range 0 S < 1 > and 0 S < 1, designated as light to moderate load magnitude range in Chapter III. In the load-geometric parameter space, this range spans the unit square column with 0 S k < 0 0 and will be designated as Region I of G the (k^, kR, kg) space. We subdivide this region into three subregions: Region (1.1) with kg < ky, Region (1.2) ky < kg < 1 and Region (1.3) with k > 1 (see Figure 9 ) . G Region (1.1); kg < k y (and ky < 1, kg < 1) For kg/ky<< 1, we expect the shell to behave very much as a flat plate and the scaling for Region (1.1) or Region (1.2) of the plate problem applies. To be consistent with the flat plate results, we subdivide Region 2 (1.1) for shells into Region (I.1.1) in which ky < kg and Region (1.1.2) 2 * in which ky > kg. 2 For Region (I.1.1) with ky < kg, we use the scaling (III.13) and write (IV.2) as \ 1-x 2 L[f] - -f (uGs + f)g = - ^  (IV.4) Ug] + ^  (P Gs + \i )f =-x (IV.5) 2 where u_ = k_/k„ and y = k,, /k,, is as previously defined for plate and G G V m V H u < 1 in Region (I.1.1). The boundary conditions for f and g are exactly m It is possible to seek a perturbation solution in kg in Region (1.1) with a leading term identical to the kg = 0 case already treated in section (2) of Chapter III. However, for kg near (but s t i l l less than) unity, a different scaling from the one used in Regions (1.1) and (1.2) for flat plates will lead to a more efficient solution. The new scalings in Region (1.2) for shells also provide a gradual transition into Region (1.3). 45 as in the plate case. With u < 1, a (regular) perturbation solution in (powers of) u_ is appropriate. The leading term solution i s identical to the f l a t plate solution in Region (1.1) given in section (2) of Chapter III; i t requires no further comments. Higher order correction terms are determined by a sequence of linear inhomogeneous BVP which differ from each other only in the forcing term. 2 For Region (1.1.2) where k^ < ky , we use the scaling for plate for the same load parameter range given in (III.23) (namely, f = kyf and 2 -g = Ky g). Write the ODE for f and g as k v i 2 L [ f ] " x ( wG S + f ) g = " ( I V ' 7 ) L[g] + i ( u G s + | f ) f = - ^ x (IV.8) m 2 where = ky /k^ i s now greater than one. The boundary conditions for f and g in this region are identical in form to those for the f l a t plate in Region (1.2), namely, (III.9)-(III.ll) and (III.26). Again a perturbation solution in y (< 1) is appropriate. The leading term solution i s identical G to the f l a t plate solution in Region (1.2) and requires no further comments. Higher order correction terms in the perturbation series are agian determined by a sequence of linear BVP. Region (1.2): 0 £ k y < kQ < 1 (k R < 1) With k G/ky H y G > 1, the scalings used in Region (1.1) are no longer appropriate. As expected, the correct scaling in the new parameter range depends on the relative magnitude of ky and k^. Again, we separate two subregions: Region (1.2.1) where k G k H > ky and Region (1.2.2) where ^ < V 46 In Region (1.2.1), the radial load is relatively large compared to the pressure load; the shell response to the combined loading should have a significant component of rotating disc type behavior, especially when kgky >> ky. Therefore, we set g = k^g, f - k ^ f . (IV.9) Note that for k^ - ky, we have g «= O(k^) and f = O(kgk^) also in Region ( i . l . l ) . Thus, there i s a gradual transition in the magnitude of the solution from the interior of Region ( I . l . l ) to i t s border adjacent to Region (1.2.1). Our choice of scaling (IV.9) for Region (1.2.1) i s consistent with this transition. With (IV.9), the governing ODE become L[f] - ^ ( s + V ) g = - ^ - ^ L[g] + (s+lkjjf) f = - x . (IV.10) The boundary conditions become f(x ±) - 0, f ' ( l ) + v f ( l ) = 0, (IV. 11,12) 2 x. g(l) • 0, g ' ^ ) - ~ g ( x ± ) + 3^7- 0, (IV.13,14) With k,2 < k 2 k 2 < k < 1 and k 2 < k„/k„k_, a perturbation V b n n V V n U 2 solution in ky is appropriate for the new BVP. The leading term solution {fg.gg} i s determined by L[g QJ - - x (IV.15) Q 1 - x 2 L[f,J - — grtf„ " -gn - r-7— \ . (IV.16) 1 0 J x B0 fl x 60 k_ku 2x 47 Thus, to a f i r s t approximation, the problem i s uncoupled i n t o a r o t a t i n g d i s c poroblem and a problem of l i n e a r bending of a "p re - s t re s sed" f l a t p l a te by a (nonuniform) d i s t r i b u t i o n of pressure l o a d . In Region (1 .2 .2 ) , we have 2: k^k^ and the pressure load i s mainly re spons ib le for the s h e l l behav ior . There fore , we set G and w r i t e the governing ODE as L [ f ] - j ( s + / f ) g - ~ (IV. 18) G 2 L[g] A ( s + ^ f ) f = - ^ x - ( I V ' 1 9 ) G V Three of the boundary condi t ions for t h i s range are a l so given by (IV.11)-( IV .13) ; the fourth becomes kgk^ x 2 g ' ( x . ) - ^ g ( x . ) - , ^ 3 ^ = 0 . (IV.20) The form of the new BVP suggests that a per turba t ion s o l u t i o n i n 2 k would reduce the nonl inear problem to a sequence of l i n e a r boundary G value problems. The lead ing term s o l u t i o n again cons i s t s of a r o t a t i n g d i s c s o l u t i o n and the s o l u t i o n of a problem i n l i n e a r bending of a " p r e - s t r e s s e d " f l a t p l a te by a (nonuniform) d i s t r i b u t i o n of pressure l o a d . 2 On the other hand, i f k^ >> k^ /kg , then i t would be more e f f i c i e n t to work w i t h a per turba t ion s o l u t i o n i n k^/k^,. The lead ing term s o l u t i o n i n 48 that case would be a linear bending of a shallow shell of revolution by the combined radial and axial loads which appear on the right side of (IV.18) a and (IV.19). Region (1.3): kQ >_ 1 (ky.k^ < 1) Let * max [kyjky] and ff.g) = k ^ f . g ) . (IV.21) Then the ODE (IV.2) may then be written as ^ L [ f ] - i ( s + ^ f ) g = - T T T - 1 ^ (IV-22) G X kG L G j ^ L t g ] - ^ ( s + T ^ o f - -inH- <IV-23> kG X 1 kG Y G The boundary conditions now consist of (IV.11)-(IV.13) and k x 2 g , ( x i > -xT g< xi> +k7 3 T V = ° - < I V - 2 « > With k^/kg < 1 and ik^/k^, ^/I^) ^  1, the new BVP may be linearized by a perturbation solution in k^/k^ giving us as the leading term solution the conventional linear bending problem for a shallow shell of revolution under the combined radial and axial loads. 49 3. Curved Plate Behavior at High Load Intensities Having completed the analysis of the light to moderate load magnitude range characterized by ky < 1 and kg < 1, we now turn to the high load intensity range ky > 1 or k > 1. As in the f l a t plate case, we 3 2 expect the shell to respond in a qualitatively different way for k„ < k„ H V 3 2 and for kg > ky . We consider in this section the load magnitude ranges, 3 2 3 2 3 2 (ky > k G , ky > kg } and {kg > kg , kg > ky }. In the load-geometric parameter space {ky, kg, kg} with the kg axis pointing upward (Figure 10), the high load magnitude range spans the positive octant of the space outside the unit square column (anchored at the origin) and below the two inter-1/3 1/2 3 2 secting surfaces kg = ky and kg » kg . The two subregions kg < k y 3 2 and k > k are designated as Region (II.1) and. Region (II.2), respectively. H V Region (II. 1): k 3 < ky and k 3 < ky For kg << 1, we expect the shell behavior is platelike; therefore we use the scaling (III.42) (with f = k y' f and g - k y g) for the plate case and write the two governing ODE as ^ L f f ] 4 ( 7 i 7 3 s + f ) 8 " -Hr  (IV-25) L [ 8 ] + ^ ( _ T 7 T s + T f > f = ~ TT (IV.26) x k y / J 1 ul 3 2 3 where u£ - ky /k^ is as defined in (III.32). The boundary conditions take the form (IV.11)-(IV.13) and (III.45). It i s clear from (IV.25) and (IV.26) 1/3 that the same scaling may be used for a l l k_ < k , and a perturbation O V 1/3 solution in k /k i s appropriate. The leading term solution provides an c» v 1/3 accurate approximation of the exact solution i f kg << ky . This leading 50 term solution i s just the solution for the fl a t plate problem already analyzed in section (4) of Chapter III. 2/3 1/2 Region (II.2); k„ £ k „ / J and k_ < k u' H V G H The proper scaling In this region depends on whether we have kg < ky/ky or k„ > ky/k^. We designate these two subregions as Region (II.2.1) and Region (II.2.2), respectively. For the subregion (II.2.1) with k k < k , we expect the shell G H V to behave nearly as a f l a t plate, especially i f k << 1. Therefore, the scaling (III.29) (with f = (ky/k^f and g = k^g) for a relatively fast spinning plate should apply for k << 1. With this scaling, the governing ODE may then be written as i ^ u f ] - x - < i ^ 8 + f > 8 - -Lir-  (IV-27) k k L[g] + ( - ^ s + | f ) f = -x (IV.28) while the boundary conditions are as given by (IV.11)-(IV.14). Both 3" k k /k and u» are smaller than unity. We may therefore consider a G H V -c perturbation in k^k^/ky. The leading term solution in this range of the parameter values has already been worked out in section (3) of Chapter III. For the complementary subregion (II.2.2) with k^kH > ky, the pressure load is too small to prevent the flattening effect of the radial load. For the shell to be nearly flattened out, we have $ + £ = $(k s+f) = 0. G Therefore, we take in .this Region (II.2.2) £ - k G f , g = kyg (IV.29) 51 and write the governing ODE as ^ L I f ] - i ( . + f ) | . - ^ (IV.30) k2 L[g] + - ~ £ ( s + | f ) f - -x. (IV.31) The boundary conditions now take the form (IV.11)-(IV.14). 2 2 With kg /kg < 1, we seek a perturbation s o l u t i o n i n kg /kg to reduce the nonlinear problem to a sequence of l i n e a r problems. The leading term f o r g i s the so l u t i o n of a r o t a t i n g d i s c problem while the leading term s o l u t i o n f o r f i s the so l u t i o n for the l i n e a r bending of a "pre-stressed" f l a t plate by a nonuniform a x i a l load d i s t r i b u t i o n . When kg >> 1 and kgkg >> k^, we have to a f i r s t approximation, s + f = 0 except f o r layer phenomena. The s h e l l i s more or les s flattened out whenever the r a d i a l load i s s u f f i c i e n t l y high, i . e . when the s h e l l i s spinned at a s u f f i c i e n t l y high constant angular v e l o c i t y . 52 4. Thin S h e l l Behavior at High Load I n t e n s i t i e s The only range of the load and geometric parameter values not analyzed up to t h i s point i s the region i n the parameter space outside the u n i t square column, 0 £ < 1 and 0 < ky < 1, and above the two 1/3 1/2 3 2 i n t e r s e c t i n g surface kg = ky and kg = kg , i . e . , kg > ky and kg > kg. The p r o j e c t i o n of the i n t e r s e c t i o n curve on the kg,ky-plane i s the base 3 2 curve kg = ky. We designate t h i s remaining region as Region I I I and 3 2 consider separately the two subregions ( I I I . l ) where kg < ky and (III.2) 3 2 where kg > ky , Region ( I I I . l ) ; k R 3 < ky 2 (and k 3 > ky > 1) In t h i s region, we take f - kgf, g = k v 2 / 3 g (IV.32) and w r i t e the ODE (IV.2) as ,1/3 2 ^ l I f ] - 5 < , + f>*--(5] ^ (IV-3> (IV.34) Correspondingly, the boundary conditions take the form (IV.11)-(IV.13) and 1/3 g'(x.) - ^ - g(x.) + \'-^J\ - ^ - = 0 (IV.35) 2 3 2/3 with kg/kg < (ky/kg ) < 1, we seek a perturbation s o l u t i o n i n 2 kg/kg . The leading term s o l u t i o n i s i d e n t i c a l to the case of no spinning which has already been analyzed i n [13,14,22,23,26]. 53 Region ( I I I . 2 ) : k^ > k ^ (and k 2 > ^ > 1) The s h e l l i s l i k e l y to be f l a t t ened out i n t h i s r eg ion . Therefore , we take f = k G f , g = k G 2 g (IV. 36) and w r i t e the two governing ODE as \ - 2 -VMf]-±(s + f ) g = - ^ ^ (IV.37) k G k G L[g] + - ( s+ \t) f - Ax. (IV.38) k G Correspondingly , the boundary condi t ions take the form (IV.11)-( IV.13) and * , ( x i > - x T 8 ( x i ) + 7 f ( I V- 3 9 ) 1 k G With ky/kg < (kjj/kg) ' <, 1 (which fol lows from k y < k^ ) , a per turbat ion 3 s o l u t i o n i n ky/k^ i s appropr ia te . The lead ing term s o l u t i o n i s i d e n t i c a l to the s o l u t i o n for no pressure load already analyzed i n [2] . 54 5. Summary of Re-scalings and Asymptotic Solutions To summarize the results of this chapter, we write the re-scaling of I and g in the form f - c f f , g - c gg (IV.AO) with the choice of c. and c depending on the value of the parameters k , k g H V and kg. The re-scaled form of the two differential equations may be written as C v L [ f ] - ^ ( c V s s + c f g f ) g = - c ^ ^ - (IV.41) C H L [ 8 ] +^ ( cHs S + Kf f ) f - CHk X (IV.42) Of the four boundary conditions, the form of three of them remains invariant upon re-scaling; the fourth may be written as 8 , ( x i ) "x7 8 ( xi ) + CHB 3+T (IV.43) The eleven multiplicative factors [c £.c ]. c c c cVs fg Vk CHs C f f CHk l CHB ] completely describe the re-scaled BVP. The re-scaling of the BVP in the various regions of the parameter space w i l l now be summarized in the Table (6) with the help of these multiplicative factors. 55 The asymptotic solutions in different regions of the parameter space are summarized below: Region (1.1); A perturbation solution in u < 1 reduces the problem to G solving the corresponding plate problems (done in Chapter III) and a sequence of linear problems which give corrections to the plate solutions. Region (1.2): A perturbation solution in ky < 1 reduces the problem (in each subregion) to a sequence of linear problems. For the subregion (1.2.1), a perturbation solution in k^ < 1 also leads to the same result. Region (1.3); Small load amplitudes naturally lead to a perturbation solution in k^/kg < 1 (where k L = max [k^.ky] < 1) resulting in a linear shell problem (of the singular perturbation type i f k >> 1) G which has been extensively studied in the literatures [24,25]. 1/3 Region (II. 1); A perturbation solution in k /k < 1 reduces the G V problem to solving the corresponding f l a t plate problem and determining corrections to the plate solution from a sequence of linear problems. Region (II.2.1); A perturbation solution in k^k^/ky < 1 reduces the problem to solving the corresponding plate problem and determining corrections to the f l a t plate solution from a sequence of linear 3 problems. Alternatively, a perturbation solution in < 1 has as i t s leading term the well known rotating disc solution and the solution for the linear bending by a modified axial load of a fl a t plate pre-stressed by a nonuniform distribution of in-plane (radial) stress resultant induced by the stretching action of the 56 s h e l l . Correct ions i n the second per turbat ion s o l u t i o n are a l so determined by l i n e a r problems. 2 -1 Region ( I I . 2 . 2 ) : A per turbat ion so lu t ion i n k^ k^ g ives r e s u l t s 3 s i m i l a r to the per turbat ion s o l u t i o n i n for Region ( I I . 2 . 1 ) . Region ( I I I . l ) ; A per turbat ion s o l u t i o n i n the smal ler r a d i a l load 2 amplitude fac tor k /k < 1 has as i t s l ead ing term the s o l u t i o n for the same problem but without the r a d i a l l o a d . Th i s d i f f i c u l t nonl inear BVP has already been analyzed i n [13,14,22,23,26] . Correct ions to the lead ing term s o l u t i o n are again determined by a sequence of l i n e a r problems. Region ( I I I . 2 ) : A per turba t ion s o l u t i o n i n the smaller amplitude 3 fac tor ky/kg < 1 has as i t s l ead ing term the s o l u t i o n of the same problem for a s h e l l wi th no pressure load ing . Th i s d i f f i c u l t * nonl inear BVP has a lready been analyzed i n [2 ,5 ,22] . Correc t ions to the lead ing term s o l u t i o n are again determined by a sequence of l i n e a r problems. *For the case ^ / k ^ « 1 and k y / k 3 << 1, the BVP can be l i n e a r i z e d . 57 V. Concluding Remarks In the preceding pages, the problem of a s t e a d i l y sp inning shallow e l a s t i c s h e l l of r e v o l u t i o n under a uniform pressure d i s t r i b u t i o n has been analyzed and completely solved by per turbat ion and asymptotic methods. The r e s u l t s have been confirmed by accurate numerical s o l u t i o n s . For the l i m i t i n g case of a f l a t p l a t e , our asymptotic ana lys i s shows that the s o l u t i o n under the combined r a d i a l and a x i a l loading may be one of the fo l lowing four q u a l i t a t i v e l y d i f f e r e n t types depending on the p a r t i c u l a r load combination. For " l i ght - to-modera te " load magnitude, the p la te may e i ther behave e f f e c t i v e l y as a d i sc pre-s tressed by spinning with (secondary) out of plane bending by the uniform pressure , or as a t ransverse ly bent p late by the uniform pressure with the deformed slope producing a d d i t i o n a l (secondary) i n -plane extension beyond the e f fect of the steady r o t a t i o n . At high r o t a t i n g speed, the p la te again behaves e f f e c t i v e l y as a d i sk pre-s tressed by the c e n t r i f u g a l force and bent out of plane by the uniform pressure, with the (secondary) bending a c t i o n confined to a narrow layer adjacent to the p la te edges. At high pressure l o a d i n g , the p la te behaves e f f e c t i v e l y as a membrane i n f i n i t e deformation except for some l i n e a r bending a c t i o n i n a boundary l ayer adjacent to the p la te edges. For t h i s case, i t has been shown i n Chapter I I I , that bending s tresses are n e g l i g i b l e i n the p la te i n t e r i o r and are at most of the same order of magnitude as the d i r e c t (membrane) s tresses near an edge. For design purposes, a s o l u t i o n by the Foppl-Hencky type (or the Reissner type 127]) nonl inear membrane theory should provide a good approximation to the d i r e c t s tresses and therefore a lso a good estimate of the bending s t re s s l e v e l . Near the borders between load-parameter regions where one of these bas ic modes of deformations dominates, the expected mode would be modif ied considerably by the competing neighboring mode(s). 58 For shallow shells, our asymptotic analysis shows that the structure may exhibit one of three typical mode of deformation, depending on the relative magnitude of the load and geometric parameters. Loosely speaking, the shell is predominantly in a linear bending mode for sufficiently light load magnitude; "very shallow" shells (defined more precisely in chapter IV) are nearly plate-like while "very thin" shells at high load magnitude are usually in a nonlinear membrane or inextensional bending mode. For the first two modes, either bending or stretching action may be dominant in the shell interior, analogous to the flat plate case. The fact that a shell under the combined radial and axial loading may exhibit one of several modes of behavior is not unexpected. However, our analysis makes precise the ranges of the load and geometrical parameter values for which each of these modes would occur; it also shows how the solution procedure may simplify in each case leading to a complete solution of the problem for the entire parameter space. It is important to note that the same analysis, possibly with some straightforward modifications, applies to shells subject to more general loading and to nonshallow shells. We have focussed on a specific class of problems in order to give a unified treatment for these problems important in applications and to put known results for special cases (such as those obtained in references 11,2,12-23,26]) in a proper context. Our contributions, however, go well beyond the results for this class of problems; our analysis actually constitutes a general approach to the solution of (small strain) axisymmetric finite deformation problems for shells of revolution. 59 Bibliography 1. J.G. Simmonds, "The Finite Deflection of a Normally Loaded, Spinning, Elastic Membrane", J. Aerospace Sci., Vol. 29, 1962, 1180-1189. 2. E. Reissner and F.Y.M. Wan, "Rotating Shallow Elastic Shells of Revolution", J. Soc. Indust. Appl. Math. Vol. 13, 1965. 3. E. Reissner, "Symmetric Bending of Shallow Shells of Revolution", Journal of Mathematics and Mechanics, Vol. 7, 1958, 121-140. 4. E. Reissner, "The Edge Effect i n Symmetric Bending of Shallow Shells of Revolution", Communication on Pure and Applied Mathematics, Vol. 12, 1959, 385-398. 5. Wan, F.Y.M., "Mathematicsl Models and Their Formulation", i n Handbook of  Applied Mathematics (2nd Edition), Chapter 19, Ed. C.E. Pearson, Van Nostrand Reinhold Co., New York, 1983, 1044-1138. 6. J. Kevorkian S J.D. Cole, Perturbation Methods i n Applied Mathematics, Springer Verlag, New York-Heidleberg-Berlin, 1981. 7. J.L. Nowinski and I.A. Ismail: "Application of a Multi-Parameter Perturbation Method to Elastostatics, Dev. Theor. Appl. Mech., Vol. 2, Pergamon Press, 1965, 35-45. 8. Chuen-Yuan CHIA: Nonlinear Analysis of Plates, McGraw-Hill, Inc. 1980. 9. A. Foppl, Vorlesungen ubertechnische Mechanik, Vol. I l l , Teubner, Leipzig, 1907. 10. H. Hencky "Ueber den Spanneringszustand i n kreisrunden Platten", Z Math. Phys. Vol. 63, 1915, 311-317. 11. S. Timoshenko & J.N. Goodier, Theory of E l a s t i c i t y , (2nd Ed.), McGraw-H i l l , 1959. 12. S. Timoshenko & S. Woinowsky-Krieger, Theory of Plates and Shells, (2nd Ed.). McGraw-Hill, 1959. 13. F.Y.M. Wan, "The Dimpling of Spherical Caps", Mech. Today Vol. 5, (The Eric Reissner Anniversary Volume), 1980, 495-508. 14. F.Y.M. Wan, "Polar Dimpling of Complete Spherical Shells", Theory of  Shells, Proc. of the Third IUTAM Shell Symp. ( T b i l i s i USSR; August, 1978), Ed. W.T. Koiter & G.K. Mikhailov, North-Holland, 1979, 191-207. 15. H.J. Weinitschke, "Existenz-und Eindeutigkeitssatze fur die Gleichungen der kreisformagen Membrane", Meth. u. Verf. d. Math. Physik, Vol. 3, 1970, 117-139. 60 16. H . J . Weinitschke, "On Axisymmetric Deformations of Nonl inear E l a s t i c Membranes", Mechanics Today, V o l . 5. (The E r i c Reissner Volume) Pergamon Press , 1980. 523-542. 17. , "Some Mathematical Problems i n the Nonl inear Theory of E l a s t i c Membranes, Plates and S h e l l s " , Trends i n A p p l i c a t i o n s of Pure  Mathematics to Mechanics" (Lecce Shymp.), G. F ichera ( ed . ) , Pitman (London), 1976, 409-424. 18. R.W. Dickey, "The Plane C i r c u l a t E l a s t i c Surface under Normal Pre s sure " , Arch , Rat . Mech. A n a l . V o l . 26, 1967, 219-236. 19. A . J . C a l l e g a r i and E . L . Re i s s , "Nonl inear Boundary Value Problems f o r the C i r c u l a r Membrane", A r c h . Rat . Mech. A n a l . 31, 1968, 390-400. 20. F . Y . M . Wan and H . J . Weini t schke , " A Boundary Layer So lu t ion for Some Nonl inear E l a s t i c Membrane", I.A.M.S . Tech. Rep. U . B . C , Vancouver, Canada, 1983 (To appear) . 21. W. Nachbar, " F i n i t e Deformations of Membranes and S h e l l s under L o c a l i z e d Loadings" , Proc.2nd IUTAM Symp.on Thin S h e l l s , Ed . F . Niordson, Spr inger , 1968, 22. F . Y . M . Wan, "Shallow Caps with a L o c a l i z e d Pressure D i s t r i b u t i o n Centered at the Apex", Proc . EUROMECH C o l l o q . (No. 135) on F l e x i b l e She l l s Ed . by E . Axelrad and F . Emmerling, Spr inger -Ver l ag , 1984, to appear. (An expanded ver s ion of the same a r t i c l e i s publ i shed as IAMS Tech. Report 79-44, revi sed i n August, 1982). 23. D . F . Parker and F . Y . M . Wan, " F i n i t e po lar d impl ing of shallow caps under sub-buckl ing pressure l o a d i n g " , SIAM J . of A p p l . M a t h . , V o l . 43, 1983, to appear. (Also publ i shed as IAMS Tech. Rep. No. 80-10 ( r e v . ) , U . B . C . , Vancouver, June, 1891.) 24. H. Kraus , Thin E l a s t i c S h e l l s , John Wiley & Sons, I n c . , New York , 1967. 25. P. Se ide , Small E l a s t i c Deformations of Thin S h e l l s , Noordhoff I n t e r n a t i o n a l P u b l i s h i n g , Leyden, 1975. 26. F . Y . M . Wan and U. Ascher , " H o r i z o n t a l and F l a t Po int s i n Shallow S h e l l D i m p l i n g " , Proc . of BAIL I . Conf. (Dub l in ) , Ed . J . M i l l e r , 1980, 415-419. 27. E . Rei s sner , "On F i n i t e D e f l e c t i o n of C i r c u l a r P l a t e s " , Proc . Symp. A p p l . Math. V o l . 1 (AMS, Providence, R . I . ) , 1949, 213-219. 61 APPENDIX A. Solutions f o r Various Classes of Pl a t e Problems. 1. Perturbation Solutions f o r Small Pressure and Slowly Spinning Case. The equations and boundary conditions f o r t h i s case are given by (III.6)-(III.7), (III.9)-(III.12). With k„ « 1 and k„ « 1, a n V perturbation s o l u t i o n of t h i s BVP is.a p p r o p r i a t e . We take the two parameter perturbation s e r i e s solutions i n the form f(x) = k y f ^ x ) + k H f 2 ( x ) + k y f 3 ( x ) + k v k f l f 4 ( x ) + k ^ t x ) + ••• (A.D g(x) = k ^ C x ) + ^ ( x ) + k j g 3 ( x ) + VH94(x) + k H g 5 ( x ) + ( A , 2 ) These kinds of perturbation s e r i e s are widely used i n e l a s t o s t a t i c s [6-8]. For our problem, we expect leading terms to be proportional to load parameters ky, k^, f(x) to be an odd function of ky and g an even fun c t i o n of ky. Hence (A.1) and (A.2) may be s i m p l i f i e d to f(x) = k y f 1 ( x ) + k^k^f (x)+ ••• (A.3) 2 2 g(x) = k g ^ x ) + k H g 2 ( x ) + K v9" 3( x> + (A.4) where subscripts of functions have been changed. S u b s t i t u t i o n of (A.3), (A.4) i n t o Eq. (III-6)-(III.7) y i e l d s f" + - f! 1 x 1 f " + — f' 2 x 2 2 f2 1-x* 2x x f1 g1 (A.5) (A.6) q" + — q' y1 x y1 q" + — q' 9 2 x 92 q" + — q' 9 3 x 9 3 T G ! 2 g2 x 1 ^ g 3 = -x = 0 2x 1 (A.7) (A.8) (A.9) 62 S u b s t i t u t i o n of (A.3) and (A.4) i n t o B.C. (III.9)-(III.12) y i e l d s w = 0 f 2(x.)=0 f ] ( D + v f ^ D = 0 f l ( D + v f 2 ( 1 ) = 0 2 g i , ( x i > -i7 W + i b = 0 g 2 ( x i ) "x^W = ° ( A* 1 0 ' a _ j ) i g'(x.) -^Vx,) =o i g ^ D = 0 g 2 ( i ) - 0 g 3 ( D - o e t c . We solve (A.5) with (A.10 a,c) to get f , then solve (A.7) with (A.10 e,h) to g ^ e t c . For s i m p l i c i t y , we give here only the solutions f o r the case x. = 0 : l 1 (-1-v 3 . . •> f i = T? MTv x + x " 4 x £ n x J 91 = 8 x(1~x ) g 2 = o g„ = f r1 f l - v ^ 5 (1 - v ) 275 •. r1 f1~V V ^ 8 M + v J 6 (1 + v) 144 J _ L8 i-1+v J 3 512 3(1-v) 7 2(2+v) „ „ 2 , 3 r i . 5 + —7- + — ; ~ £nx + 2£n x Ix + I — £nx - — 1 _ V - ] x 5 - ^  x 7} (A.11 a-d) 12(1+ V ) J " 48 2 For the k << k << 1 case, we have H V f(x) ~ k y f 1 ( x ) 2 g(x) ~ k y g 3 ( x ) 63 or • (x) (A.12) 0 * ~ T" Kv g 3 ( x ) * ( A , 1 3 ) 0 2 For the ky << k H « 1 case, we get instead fix) ~ kyf^x) or g(x) ~ k g (x) n 1 • ~ V 2 ( X ) ( A ' 1 4 ) 0 ¥ ~ k g (x). (A.15) r Q H 1 64 2. Asymptotic Solutions for the Fast Spinning Case From the view point of an asymptotic analysis, the BVP for k » 1 (see (III.30) and (III.3D) i s a singular perturbation problem. H 2/3 When k H » k^' (in addition to 1^ » 1 ) , then the leading term perturbation 3 2 3 solution in y £ = k y A „ « 1 for g is (see (III.34)-(III.35) gn = Afx - -1 + U- - x 3) (A.16) *0 ^ xJ 8 vx } where (3+v)+(1-v)x 4 . 1+v L A = — — — r n • (A.17) 8 ( 3 + v ) d + v) + (1-v)x 2 with A • 1 /8 ' as x ± -»• 0. The leading term solution for f i s then determined by f o + \ f o - \-T V x ) + -Tl'o " " kH H s T  ( A ' 1 8 ) X f 0 ( X i ) = 0, fi(1) + vf 0(1) = 0 (A.19) The leading term outer (asymptotic expansion) solution of (A.18) i s evidently 90 x +1-8A i t does not satisfy either of the boundary conditions in (A.19) and requires boundary layer corrections. For the boundary layer correction adjacent to the inner edge, we write the leading term solution which is uniformly valid away from x = 1 as f Q ~f£ 0 )(x) + f Q ( y ) , y = ^ ( x - x ± ) (A.21) 65 Then fQ i s determined by the BVP JQ - a 2 f 0 - 0 (A.22) fo(0>+ • °- y > - ° » 2 where a = g (x ± ) / x . The s o l u t i o n of (A.22) i s 1-x2 -ai/lTCx-x ) S i m i l a r l y , we w r i t e the s o l u t i o n which i s uniformly v a l i d away from x = x^ as fQ ~ f ^ 0 ) ( x ) + f Q ( t ) , t = k j / 3 ( x - l ) . (A.24) With gQ(x) = B ( l -x ) + 0 ( [ l - x ] 2 ) , B = | ( 1 - 4 A ) (A.25) ~ l / 3 ~ ~ near x = 1, we f ind that f Q ( t ) = f ( t ) with f determined by the BVP + B t f = 0, (A.26) f ' ( 0 ) » C = 8 A ~ 4 „ , f ( t ) •* 0 as t + - « . ( 1 - 4 A ) Z The s o l u t i o n of (A.26) i s f ( t ) = CT ( j X f ^ ^ d k j j B d - x ) 3 ] 1 7 3 ) (A.27) where A^(z) i s the A i r y f u n c t i o n . 66 As f Q i s e x p o n e n t i a l l y s m a l l f o r x > x^ ^ and f ^ i s e x p o n e n t i a l l y s m a l l f o r x < 1, the u n i f o r m l y v a l i d s o l u t i o n f o r f g i n the whole i n t e r v a l 0 £ x £ 1 i s g i v e n by f Q = f < 0 ) ( x ) + f Q ( y ) + k j / 3 f ( t ) 1-x2 L - X i - ^ ( x " x i ) e 2 g Q ( x ) 2 g 0 ( x ± ) + k J ^ C r c l x ^ ^ A ^ t k ^ d - x ) 3 ] 1 7 3 ) (A.28) 67 3. Asymptotic Solution for the High Pressure Range Suppose we write the solution of our BVP in region (3) as ,A ~ C:) • C) where GQ(X) and Frt(x) are the nonlinear membrane solutions for f and g, respectively, as determined by (III.46) and the boundary conditions 2 G 0 ( l ) = 0, G j ( x ± ) - ^ - G 0 ( X l ) + ^ - j = 0. (A.30) Upon substituting (A.26) into the ODE (111.43,44) and the four relevant boundary conditions and observing (III.46) and (A.30), we get e 2 [ | . . + I f . _ _ l _ i ] - i £ i - i G 0 ( x ) £ - i F 0 ( x ) i - e 2 [ P S + ^ - ^ F 0 ] (A.31) g" + x g' _ ^ i + ^ f 2 + x F 0 ( x ) ? = ° ( A ' 3 2 ) x with i ( l ) = 0, f'(l) + vf(l) +Fjd) +vF Q(l) = 0 (A.33-36) g ' ( x i } ~x^ g ( x i } = °» f(x i)+F Q(x i) = 0' -1/3 where E = ky . The new problem for f and g is again a singular 1/3 perturbation problem for >> 1. We will not be concerned with the 68 outer s o l u t i o n of t h i s new problem as i t i s of h igher order i n e compared to GQ and F Q . For an inner s o l u t i o n near x ^ we note that a d i s t i n g u i s h e d l i m i t of the problem (A. 31-36) as E 0 cons i s t s o f _ X — X« i / o f*f 0(y), i ^ g 0 ( y ) . y 1 V ( x - x ± ) (A. 37) with f Q and i Q be ing the s o l u t i o n of the BVP f 0 ' = 0, I 0 ' + 2 x 7 ? 0 + x T F 0 ( x i ) f 0 " ° . ( A ' 3 8 ) f 0 ( 0 ) + F 0 ( X i ) - 0 , I 0 ' ( 0 ) = 0 (A.39) i 0(y), f0(y) + 0 as y (A.40) 2 where ( ) ' = d( ) /dy and a = G Q (x (known to be p o s i t i v e from the nonl inear membrane s o l u t i o n ) , and where the l i m i t of y -»• 0 0 i s understood to mean E 0, n(e) 0 T I ( E ) / E •*• 0 0 ( for some s u i t a b l e n(e)) and (the intermediate v a r i a b l e ) z = ey/n(e) = 0 (1 ) . The s o l u t i o n of t h i s problem i s - o v -a(x-x.) /E /» / i \ V y ) =-Fo ( xi ) e =-Vxi)e ( A' 4 } 5 ( y ) = 3B + c + J _ - t t(x-x ±)/ E _ l ^ ( x - x ^ / e , (A.42) V y ; 4a £ L 0 2 1 6 8 J a 2 where 6 = FQ (X^)/X^ and where CQ i s a constant to be determined i n conjunct ion wi th the next order term i n the asymptotic expansion (and may be set equal to zero for a l ead ing term s o l u t i o n ) . For an inner solution near x • 1, we note that a distinguished limit of the BVP in this case consists of f * / e f 0 ( t ) = k ^ f ^ t ) , I <v E 3 / 2 g Q ( t ) - k v " 1 / 2 i 0 ( t ) (A.43) x- 1 _ , 1/6 /I t = ^ -=^  - V " (x-1) with f^ and g^ determined by the BVP f o " - F o ( 1 ) 8 o = 0 ' V + F o ( 1 ) f o = 0 ( A ' 4 4 ) 80(0) = 0, f 0 ( 0 ) + F 0 ( 1 > + v F o ( 1 ) " 0 ( A ' 4 5 ) g 0 , f Q -»• 0 as t +- - (A.46) where ( )* = d( )/dt and where the limit t -»•-» i s understood to mean /F-+- 0 ? ( e ) •+ 0, C ( e ) / /e -»• » (for some suitable £ ( e ) ) and (the intermediate variable) s = /et / c(e) = 0(1). The solution of this problem i s ~f0 = ~ ~- [F 0' (D+vF^D] e Y t / / F c o s ( Yt//t). (A.47) 80 = T " [ F 0' (D+vF^l)] e ^ t / / 2 s i n (Yt//2) . <A.48) 2 where y = F ^ ( l ) . With {fp, gg) and {f^, g^ } exponentially small away from x=x^ and x = l , respectively, we may write the leading term composite solution as - a ( X - X J ) / E f * F Q (x ) - F 0 ( X i ) e 2 g * G n (x) + E i l l IF ' (1) + v F n ( l ) ] e ' r ( * ' 1 ) / , ^ c o B ( Y . U - 1 ] / ^ ) (A.49) i i * 3 + JL ( e - ( x - x i ) / e _ I.e-2«(*-Xi)/e]j O w L4a E ' a 2 / — 3 + [F 0 ' (1) + v F 0 ( l ) ] e Y ( x _ 1 ) / / r E s i n ( Y [ x - 1 ] / ^ ) (A.50) For k y " ^ >> 1, the nonl inear membrane s o l u t i o n FQ.(X) alone i s an adequate approximation for f (x) away from the edge. However, i t i s c l e a r that FQ(X) i s not dominant near the inner edge and the edge zone s o l u t i o n IQCY) must a l so be i n c l u d e d . Furthermore, we have f^' (t) = 0(fg) near x = l , so that the outer s o l u t i o n alone does not g ive an accurate approximation of the bending s tresses of the p l a t e near the outer edge. In c o n t r a s t , the nonl inear membrane s o l u t i o n G „ ( x ) i s the dominant part of g(x) throughout the e n t i r e p l a t e , the edge zones i n c l u d e d , fo r k y 1 ^ >> 1. Furthermore, we have dg/dx = 0 ( / e )G 0 ' (x) at most for x.^  < x It fo l lows that the s t r e t c h i n g a c t i o n of the e n t i r e p l a t e i n reg ion (3) of the k , k -plane i s accurate ly descr ibed by the Foppl-Hencky nonl inear H V membrane s o l u t i o n alone whenever k ^ " ^ >> 1. 71 APPENDIX B. FINITE DIFFERENCE SOLUTIONS A FORTRAN program has been developed to solve the systems of nonlinear ordinary differential equations numerically. The equations are taken i n the form - - 2 f" + — f' - •—r- f - — s g - — f g = -D-^j—^ (B.1) x 2 x ° x ° 2x x g" + x" g ' " ~T 9 + lSf+ 2x~f2 = " 5 x (B*2) X where B,»»», H are constants depending on the load and geometric parameters of a specific problem. These two equations are supplemented by four boundary conditions: - 2 v x = x, : f = 0, g' + Ax - - g = 0 (B.3,4) x = 1: g = 0, f + vf = 0 (B.5,6) where A = r o p o (3+v)iJ> Upon using central f i n i t e difference method, the ODE's are replaced by system of nonlinear algebraic equations with the discretization errors 2 0(h ), where h is the length of step: f 1 = 0 -g 3 + 4 g 2 - [ 3 ( 2 h ) ] g l + 2hAx 2 = 0 i x. i 1 ± 1-x 2 + 2x ± 1 Dh2 = 0 ( l = 2,--,n-l) 2 _ 2 x. i i 1 + x . H h 2 = 0 ( i = 2 , " ' , n - l ) (3 + 2 v h ) £ n - 4 £ n . 1 + f n . 2 - 0 2 To get 0 ( h ) accuracy, second order i n t e r p o l a t i o n s fo r boundary po in t s have been used. For convenience, we denote (B.7) as F i ( f 1 , g 1 , f 2 , g 2 , " - , f n , g n ) =0 ( i = l , 2 , . - . , 2 n ) (B.S) where F . are nonl inear funct ions of f , , g , , * , , , f , g . The nonl inear i l i n n system i s solved by Newton's method. The s o l u t i o n procedure i s as f o l l o w s : 1. Take an i n i t i a l guess of the s o l u t i o n ( f ? , g?, • • • , f^ ,g^) . For our 1 1 n n problem the l ead ing terms of outer so lu t ions are used whenever a v a i l a b l e 2. C a l c u l a t e ( re s idua l ) e r ror s of the s o l u t i o n , i . e . , subs t i tu te the s o l u t i o n i n t o the equat ions , compute the r e s i d u a l s ^ O ' ^ O * * * * , e 2 n 0 and then form , 2 , 2 . , 2 .1/2 e 0 = ( e 1 0 + E 20 + + E 2 n 0 ) ' I f i s smal l enough (depending on the accuracy requirements) , computation i s f i n i s h e d . Because of the f i n i t e d i f f e r e n c e , accuracy 2 requirements should not be h igher than 0(h ) . 3. If the tolerance is not met i n step (2), find a direction and magnitude such that new solution yields a smaller error < e^. 4. Using the new solution as a starting point, search for the new "optimal" direction and magnitude for corrections to get a solution which makes the error smaller then etc. The process repeats u n t i l the accuracy requirement is sa t i s f i e d . reference: 1. Because we minimize the residual errors of solutions by substituting back to the equations, the procedure guarantees that the f i n i t e difference equations are satisfied up to some required accuracy. With the errors 2 between ODE's and Finite difference equations being of 0(h ), the total accuracy is under control. 2. The optimal direction in Step 4 of the procedure is known to be the one given by Newton's method i f the i n i t i a l guess i s good enough. Hence, we f i r s t calculate Newton direction in our program. The principle i s as follows. Form the linear Taylor approximations of (B.8): (B.9) Some features of the computer program are recorded here for Ag. + • ••+ i Af + n Ag„ + n • • • = 0. (i=1,2 , • • • n) (B.10) where 74 3 F i 9 F i ,.0 0 .0 0, lqm ( f l ' 8 , w I f f°,g°, ,f°,g° are close to the s o l u t i o n , we have reason to neglect 1 , n n higher order of A f ^ ••• A g n i n (B.10); then (B.10) can be wr i t t e n as [DF(y k)]{Ay k} = -{F(y k>} ( B . l l ) where we denote / A , r * k + l £ k k+l k .k+l .k k+l k, {Ayfc} = {f± - fv 8]_ - g ^ . . . , f n - f n , g n - gn> {F(y k)} M F ^ . F ^ y ^ , . . . , F ^ ) } . [DF(y. )] = 3 V V W ay, 3y, 3 F 2(y k) F 2(y k) 3 V V  3 y 2 n 3 F 2(y k) 3y. 3y. 3F (v ) 3F (v ) 2n yk' 2n Yk' 3y 2n 3 F 2 n ( V 3y, 3y, 3y 2n ( B . 1 2 ) In our problem [DF(y k)] is a band matrix, both the length of the lower band and the upper band are 4 (excluding the diagonal). A UBC library routine 75 DGBAND is used to solve Eq. ( B . l l ) . The band of coefficients matrix are stored column by column with some zero elements forming a one dimension array. Upon using the band, behavior, calculations are much simplified. The main feature i s Gaussian elimination with partial pivoting and back substitution. 3. As mentioned in (2), our program f i r s t calculates the Newton direction to reach y k + 1 from y^. If e < e^, the calculation terminates at the kth 2 step. If e f c + 1 > e^, we calculate the negative gradient direction of which guarantees that would descend most rapidly in the local region at the price of slow convergence speed. £k * F?(V + F2 ( yk> + — + F2n ( yk> - i V ( e K } = - F i ( \ ) ' V F i ( y k ) - F 2 ( y k ) , 7 F 2 ( V " - y y k ) - V F 2 ( y k ) = -[DF(y k)] T{F(y k)} (B.13) where [DFfy^)], (Fty^)} have already been obtained in (2). 4. One dimension search is used in both Newton direction and fast descent direction to get a smaller error. 5. The coarse subintervals are automatically equally divided to produce new finer subintervals for the improvement of accuracy. The new i n i t i a l guess is linear interpolation of results of coarse subintervals. Table ( 1 ) : Numerical S o l u t i o n s f o r Region (1.1) of the F l a t P l a t e Case (v = 0.3) \ 2 10° i o " 1 i o " 2 10" 3 0 g ( x ± ) 1.933-2* 1.902-2 1.899-2 1.899-2 1.898-2 n * m 1.1 -1 1.1 -1 1.1 -1 1.1 -1 1.105-1 10° f ( D 8.567-2 2.709-2 8.568-3 2.709-3 0 m r(x.) 9.568-1 3.026-2 9.569-2 3.029-2 0 g ( x ± ) 1.933-3 1.902-3 1.899-3 1.899-3 1.898-3 _ i * m n e 1.1 -2 1.1 -2 1.1 -2 1.1 -2 1.105-2 10 1 f ( D 2.742-3 8.671-3 2.742-3 8.671-4 0 m r(x.) 3.045-1 9.628-2 3.045-2 9.628-3 0 g(x.) 1.934-4 1.902-4 1.899-4 1.899-4 1.898-4 i o " 2 * m n e 1.1 -3 1.1 -3 1.1 -3 1.1 -3 1.105-3 f ( D 8.681-3 2.745-3 8.681-4 2.745-4 0 m r(x.) 9.634-2 3.047-2 9.634-3 3.047-3 0 g(x.) 1.934-6 1.902-6 1.899-6 1.899-6 1.898-6 i o " 4 - m n e 1.1 -5 1.1 -5 1.1 -5 1.1 -5 1.105-3 f ( D 8.682-4 2.746-4 8.682-5 2.746-5 0 m (x.) r l 9.635-3 3.047-3 9.635-4 3.047-4 0 The n o t a t i o n x± y means (x) * 10 Table (2): Numerical So lut ions for Region (1.2) of the F l a t P l a t e Case  (v • 0.3) 10° i o 1 i o 2 i o 3 10* g<*i> 1.933-2* 2.247-3 5.393-4 3.685-4 3.5 -4 10° /* m n9 1.1 -1 1.2 -2 2.7 -3 -1.56-3 — 1.8 -3 -2.03-3 — -1.7-3 2.1-3 f ( D 8.567-2 8.669-2 8.680-2 8.681-2 8.682-2 i r ( x 4 ) 9.568-1 9.627-1 9.633-1 9.634-1 9.635-1 g(x ±) 1.934-4 2.248-5 5.394-6 3.686-6 3.5 -6 i o " 1 ^ in n e 1.1 -3 1.2 -4 2.7 -5 -1.56-5 1.8 -5 -2.03-5 1.7-5 -2.1-5 mm -J i d ) 8.681-3 8.682-3 8.682-3 8.682-3 8.682-3 m r ( x . ) 9.634-2 9.635-2 9.635-2 9.635-2 9.635-2 g(x.) 1.934-6 2.248-7 5.394-8 3.686-8 3.5 -8 i o " 2 -> m n e 1.1 -5 1.2 -6 f" 2.7 -7 -1.65-7 1.8 -7 -2.03-7 " l.7-7" -2.1-7 f ( D 8.682-4 8.682-4 8.682-4 8.682-4 8.682-4 m r ( X i ) 9.635-3 9.635-3 9.635-3 9.635-3 9.635-3 * The n o t a t i o n x ± y means (x)x 1 0 ± y Table (3) : Numerical So lut ions fo r Region (2) of the F l a t P l a t e Case (v - .^o/ The n o t a t i o n x ± y means (x) * 10 79 Table (4): Dimensionless Boundary Layer Width Ax Near the Inner Edge  of the Flat Plate in Region (2) of the Parameter Space with v - 0.3. V H i o 1 i o 2 i o 3 10* i o 5 i o 6 10° 0.37 0.14 0.0475 0.0160 0.006 0.0016 i o " 2 0.37 0.24 0.125 0.044 0.017 0.0060 Table (5 ) : Numerical So lut ions fo r Region (3) of the F l a t P l a te Case (v -\ *v2/3  kv V" Si . 1 0 ° io1 io2 1 0 3 1 0 4 00 g(x ± ) 1.933-2 2.247-3 5.393-4 3.685-4 3.512-4 3.5 -4 10° 1 ~ m ne f ( D 1.1 -1 8.567 1.2 -2 8.669-2 " 2 .7 -3*1 -1.56-3J 8.680-2 [ 1.8 -3] |_-2.03-3J 8.681-2 L-2.07-3J 8.681-2 [-2'.l-3] 8.681-2 m r ( X i ) 9.579-1 9.627-1 9.633-1 9.634-1 9.634-1 9.635-1 g(x.) 1.190-1 4.243-2 3.480-2 3.404-2 3.397-2 3.4 -2 io1 * m ne f ( D 6.3 -1 8.062-1 2.1 -1 8.492-1 ' 1.7 -f] .-1.99-lJ 8.537-1 f 1.6 -l] I 2.01-lJ 8.542-1 f 1.6 -l l |_-2.01-lJ 8.542-1 8.542-1 m r ( X . ) 9.262+0 9.512+0 9.538+0 9.540+0 9.541+0 9.541+0 g(x.) 1.630+0 1.447+0 1.430+0 1.428+0 1.428+0 1.4 +0 io2 in ne * 7.8 +0*1 -5.26+OJ [" 6.8 +0"1 L-7.50+0J r 6.7 +oi [-7.74+0J [" 6.7 +0"| L-7.76+0J [" 6.7 +0" L-7.77+0 6.7+0" _-7.8+0^ f ( D 4.674+0 5.165+0 5.216+0 5.222+0 5.222+0 5.222+0 m r ( X i ) 6.895+1 7.155+1 7.182+1 7.185+1 7.185+1 7.185+1 g(x.) « m n 9 1.387+1 1.342+1 1.338+1 1.337+1 1.337+1 1.3 +1 io3 " 6.1 +f] _-4.16+lJ " 5 . 8 + 1 -5.50+1 L. f" 5 .8+1 -5.65+1 5.8 +1"] .-5.66+lJ r 5 . 8 +1 L-5.66+1 ] f 5.8+1 L-5.7+1  f ( D 1.210+1 1.353+1 1.368+1 1.369+1 1.370+1 1.370+1 m ^ x ^ 3.493+2 3.551+2 3.556+2 3.557+2 3.557+2 3.557+2 •g(x±) A m ne 2.154+3 2.120+3 2.120+3 2.120+3 io6 [ 7.2 +3j 1-4.67+3J f 7.0 +3 l 1-6.36+3 J f 7.0 +31 [-6.38+3J [ 7.0 +3] I -6.4 +3, f ( D i n ^ x ^ 1.283+2 3.362+4 1.516+2 3.388+4 1.519+2 3.388+4 1.519+2 3.388+4 io9 g(x ± ) ^ ID ne f ( D m (x ± ) 2.254+5 f 6.9 +5" |_-4.69+5_ 1.294+3 3.297+6 2.222+5 [ 6 . 8 +5 ^6.39+5 1.548+3 3.321+6 2.222+5 ("6.8 +5 [-6.41+5 1.551+3 3.321+6 2.2 +5 [ 6.8+5 L-6.4+5J 1.551+3 3.321+6 Table (6): Sca l ings i n Parameter Space Regions (I): k v < 1 k H < 1 Region ( I .D : (1.2): KG< 1y G > 1 ( I . l . l ) : . 2 m < 1 (1.1.2): m (1.2.1) W 1 (1.2.2): VG < 1 (1.3) KG> 1 (k E max[kv,.kH]) M u l t i p l i c a t i v e Factors [ k v , k R ] , VG V o Si m t k v ' k v ^' 2 2 kV\ V 1 -1 m -1 [ k v , y G _ 1 ] , , -1 -1 KLKG k7KG . . -1 \ . -1 KLKG k L KG [1] K1 > m [1] [VG] continued ... Table (6): Scalings i n Parameter Space Regions (continued) Region M u l t i p l i c a t i v e Factors ( I I ) : V*v 1 / 3 ( I I .1): H A " u k v > 1 (yG < l ) " k v " 2 / 3 K G k v " 1 / 3 1 i kGV1/3 1 ly£-11 ( I I . 2 ) : < 1 ( I I . 2 . 1 ) : V G < 1 V 1 V G 1 1 3 3 , i V G U £ Y £ _ [ i l ( I I . 2 . 2 ) : V G > 1 TKG»V» ^H"1 1 1 (VG ) _ 1 1 K G 2 K H " 1 KG 2 f cH" 1 1 _ , HI ( I I I ) : NT < *C ( I I I . l ) : 4 > i . k v ' > i , , -2/3 , x k ^ k ' 1 1 1 K V • • G *v 2 /V 1 1 V G 2 ( I I I . 2 ) : y/ < 1 K H > 1 [ k G , k G 2 ] , k G " 2 1 1 k v k G " 3 i 1 1 Vc"2 , [ k ^ - 2 ] 00 83 84 Hrdfi+dOird*) ds (a) (b) F i g . 3 H o r i z o n t a l f o r c e e q u i l i b r i u m F i g . 4 V e r t i c a l f o r c e e q u i l i b r i u m 85 KfldedB Med6 \\^*% ds ««dd Fig.5 Moment eq u i l i b r i u m 86 ( a ) l u b - s h e l l c o n f i g u r a t i o n V ( b j A s e g m e n t o f s h e l l F i g . 6 A n o r m a l l y l o a d e d , s p i n n i n g s h a l l o w s h e l l F i g . 7 * H - k v p a r a m e t e r p l a n e o f f l a t p l a t e s 00 F i g . 8 k - k - k p a r a m e t e r s p a c e o f s h e l l s 0 0 CJ V H R e g i o n ( I . l . l ) K e g i o n - ( I . l ) oo F i g . 9 R e g i o n ( I ) o f t h e p a r a m e t e r s p a c e f o r s h e l l s Fig.10 Region(II) of the parameter space f o r s h e l l s 

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