THE UNIVERSITY OF MJICHIGAN INDUSTRY PROGRAMl OF THE COLLEGE OF EN1GINEERING POSTBUCKLING BEHIAVIOR OF RECTANGULAR ELASTIC PLATES Chin Hao lihang A dissertation submitted in partial fulfillilent of the requirem-lents for the degree of Doctor of Philosophy in The University of Michigan Dept. of Engineering Mechanics 1961 October, 1961 IP- 3

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TABLE OF CONTENTS PREFACE,.. 0.......0 LIST OF TABLES................ LIST OF FIGURES.............. PART I. PLATES lfWITH FOUR EDGES SIMPLY SUPPORTED Page * ii iv V 1. INTRODUCTION................ 2. OUTLINE OF THE FIRST METHOD......... 3. SOLUTION FOR A SQUARE PLATE...... 4. AAPPLICATION OF FUNCTION SPACE METHOD TO TIE SQUARE PLATE............. 5. DISCUSSION AND CONCLUSION.......... PART II. PLATES WJITH TWO FREE EDGES 1. INTRODUCTION............... *. 2. FORMULATION AND GENERAL SOLUTIONS OF BOUNDARY LAYER EQUATIONS............. 3, AN EXAMPLE................. APPENDIX - SOLUTIONS OF THE DIFFEREPNTIAL EQUATIONS FOR THE MEMBRAJE DISPLACEMENTS BIB LI OGRALPHY.0*... *. * 1 5 14 78 81 83 91 93 105 -iii-.iIi

LIST OF TAB3LES Table Page I, Constants Contained in Deflection Function,,..,... O *.. 26 II. Constant Coefficients in Equations (3o32b) * *.............. *. 35 III. Constant Coefficieants in Equations (3,32c).... 36 IV. Constant Coefficients in Equations (3,33b) and (3o33c)......... 37 V. Constant Coefficients in Equations (3 35@c).. * * e e 0. *..~ ~. 38 VI. Constant Coefficients in Equations (3o35c) (continued)........, 39 VII. Constant Coefficients in Equations (3 35 d) ~ ~ 40 VIII. Constant Coefficients in Equations (3.35e), for m n.. 0. 41 IX. Constant Coefficients in Equations (3,35e), for m # n,,..,, 42 X, Integrated Constants in Equation (3,42) 51 XI. Amplitude Coefficients and Stability., 51

LIST OF FIGURES Figure I, An Assumed Membrane Stress Field. 2. Solutions of the Characteristic Equation (316b)...... 3. Buckling Modes in -Direction.. 4. Functions " and for m n '......... 5. Functions I i and; for m $ n 6. Normal Stress in -Direction Along the Loaded Edge......* * 7. Thrust vs. Shortening Curves.... 8, Diagram in Function Space for Primary Buckling............ 9, Stress and Enveloping Tangent Cones in Function Space........... 10. Result of the Buckling Problem of the Square Plate on a Function Space Diagram.......... 11. A Plate with Two Free Edges..., Page 16 * 25 * 25 * 43 4 0 57 0 59 63 * 77 83

PART I. PLATES WITH FOUR EDGES SIMPLY SUPPORTED lo INTRODUCTION The buckling behavior of plates with lateral edge support is quite different from that of a column. In the case of a column, when the applied thrust reaches the critical buckling load (Euler load), if there is no relative displacement of the two ends prescribed, the column will collapse~ Hence the Euler load is the upper limit of the buckling strength of a column. It is the upper limit since for an ordinary column to a greater of lesser degree there are physical imperfections which make its buckling strength lower than that of an Euler load. For plates with lateral edge support, however, after the thrust reaches its critical buckling load, the plate buckles into a primary buckling mode, Nevertheless due to the lateral support from the undeflectable edges, the buckling mode can stand, and its amplitude is uniquely determined by the amount of the thrust applied. For some cases, (for instance, a very thin plate of high strength material), if the thrust increases further secondary buckling occurs before it collapses, Hence, there is more buckling strength than the critical buckling load that can be developed from such a plate0 The need of utilizing this postbuckling strength is very urgent for engineers, particularly for those in -1 -

-2 -aeronautical engineering. This is because a plate is a primary element in aircraft structures which is highly likely to buckleo Because of the relatively large deflections (compared to the thickness of plate)9 the classic linear equations are no longer effective in governing the behavior of plates in the buckled state, A system of nonlinear equations governing the performance of them has been derived by Karman(l)* Due to the nonlinearity, the difficulty in solving them is tremendous especially when secondary buckling is includedo A number of solutions almost all of them approximate9 have been produced in recent decades(2) to (12), In most of them9 attention is directed to primary bucklingo Among them there are two outstanding workso One is given by Friedrichs and Stoker(8) representing a set of complete solutions (in the sense that they are applicable to the whole range of the loading parameter) for a circular plate with axially symmetric buckling mode, The other one is for rectangular plates done by Levy(4) a formally exact solution of Karman's pair of equations by Fourier series approach It might be noteworthy that due to the convergence of Fourier series9 the more terms in the series that are taken the closer the sum will approach the primary buckling "lumbers in parentheses refer to the bibliography at the end of the thesis

3 ^ ^ patterno But no solution in connection Cwith secondary buckling can be expected by this approach However secondary buckling has been observed by experiments (13) to (I!) C From an analytical point of view ' the instability of the axially symmetric mode in the case of circular plate has been pointed out by several. authors8 () (10) For rectangular plates9 in Marguerres (2) as well as Timoshenko s(3) work9 in addition to the primary buckling mode9 another coordinate function was included to accompliLsht the speculation about the secondary bucklingo The evidence of the change of buckling patterns in the deep post buckling domain has also been noticed by Koiter(6) Alexeev(1 ) and Stein(l2) through their analytic resultso Howvever in their studies the primary buckling (for n 19 n being the number of one f.h:alf waves in the lI.oaded dir etion) and secondary buckling (for n: 2) wre treated separately iHence no interactions between these two buckling patterns were -to be seen and no criterion for the stability for each of tVhese buckling configuyrations could be investigatedo In this part of this thesis an attempt is made to study the behavior of pl ates with four edges simply supported in the postbuekl:iLng domar.in Both primary and secondary buckling modes are includedo The investigation will include the modes th emse lves, the stabilities of each of the -tw o modes and of the combination. of them, and the transition from one mode to the othek:o

The problem is approached by two methods, One is similar to Marguerre s(2) The other is a function space method. Since the former which will be referred to as the first method is more familiar, it is given first in the next section. Then a complete solution for a square plate follows in the third section. In the fourth section, an extension of the function space method to problems including secondary buckling is established. A comparison of this method to Marguerre s is given. The application of this method to the square plate is also demonstrated thereo A comparison of the present result with other works and discussion of it conclude this study0,

2. OUTLINE OF THE FIRST METHOD Consider an elastic isotropic plate of a thickness h which is assumed to be small compared with the other dimensions of the plate, subjected to a prescribed edge thrust AT7 on portion B' of boundary B' and to a displacement A U4 on portion B", the remainder of B. (The index notation is used in the general discussion. The subscript letters, J etc, standing for 1 or 2 refer to the Cartesian coordinates on the middle plane of the plate). When the positive increasing parameter,A stays below a certain critical value, sayA, the plate remains in its plane. This is a plane problem in elasticity. The membrane stresses A ~t in the region of plate satisfy <yjVJ~ =,4,i, F, (2. 1) and on boundary ',-= p o (202) where 3 is the direction cosines of the outward normal of the boundary. The displacement field C^ is related to the strain field Ae&j by the equations ani oi b d it t ) (2,3) and on boundary it satisfies on B5' (214)

-6 -The stresses for homogeneous and isot.rpi ematerials are related to the corresponding strains by Hook's Law io0e -z= 7{etJ +4 7^ e ] (2,o) in which &' is the Kronecker delta; E, Young's modulus of elasticity; > 9 Poisson s ratio, As the parameter A increases and exceeds Ao, the equilibrium state described in this manner becomes unstable, Then the plate buckles into a configuration (k, x) whic.h is governed by:~.w._,y ~.~, (2~6) and appropriate boundary conditionso In the present problem the edges are simply supported, hence both the lateral deflection of the plate and the normal resisting moment-from the boundary vanish on boundary Bo In Eq: (2o6), D is the flexural rigidity of the plateo Due to this configuration, in general, the membrane stresses /, W the associated strains e, and the displacements U are different from those in the unbuckled state0 Let t, 9 eC and U4 be respectily their differences between the two stateso Then one would have in the buckled state stx=^tJ 4 1 (207) ^t/^^ J8)/ e~,= 5"+ et,/'- (2,8)

aU-= T +O i a, (2o9) These additional elements, on account of equations (2.1), (2.2) and (2,4) obey =ar. o= (2,10) At J on 07 ' (2. 11) and Ion Ar " (2.12) The same stress-strain relations given by Eq. (2.5) hold for them, i.e. t/ = 7is _t 7/ det T,= j. (2.13) But the strain-displacement relations now become /,.' ~j ' -7 65>-Z = i Cit y 4 f U h L <.y7, (2.14) The preceding formulation of. the problem which is ecuivalenti to that done by Karman was given in reference (9), The difference between the present method and Marguerre's is in the first steps. In the present method an arbitrary but reasonable membrane stress field is assumed first, such that T/= 'A^t - TA (2.15)

-8 - in0sv...ic..h 72. is a 'Istat'icall homoigene ous st-ress field, / 1 "r.......ild The term "statically homogen.eouts" means that r satisfy th~ie ijomog II<Y eneCo{s equ li-."i u quatios (2.T10) and he homo g.::'.-e efous:,.dar coni..tifis,2,s1 -1) Repa:.icting / (which.ma...y....e...o.w.n.. a..s actu..al. m eba.rane stress..ie l.,d rin con: tctra"s with the art.ificial t/ ) b eq uation (2 8) -may' 3e-...........;e;"... ' _,o & e a i ab~ro, i o. t 'i' W' "' 4 (........ ' '...... W. wif (o w4 wh is~ atua 1. 9l one) This W is supposed to satisfy t the a Dpp:Jr opriatus e bo u.ndar y co^Dn.dition. as en!tI- S.ione bef ore a..nd1, a ca te risti; eqiatin a isig A..b ndary c onditinss G n e t.hi s is a l.ne ar eigen5val e probl4e, Fo;r thie purnpose oo 2f stuvdyin g Ih he bui c1Lng beob,aiori o f pates C n the rdteep posbklig doa n pestabl suc a S pa A rt iar egenval. e can '' th be - t i. i. t,t bt p rin s a. zd t ]ive Iconficg.f r ati ol nful nctliot o f r-thes e e modes and An and A- be r co-eson ir a itude~ coefficmerts^ toen t.e o o ", t........-.b- of.4 on..e - s e se:. i",.i. by " " 'S a iy" 1tM7,/ /L -J / A-,2.16 ra d ly r foowe up to d eta ne:ri i eff ici en A<,, A A: Eqs~ (2i.2) and (2 1..4.) ee.d: n. be d ' sa"tisie ' s f4 i_. r held t.neee'i.d "not rb.e ^o o' ie etia i'.rt om g e 013 S -::

-9 -Now let 7,., E, and,/ be respectively the additional membrane displacement, strain and stress fields associated with the configuration W(xl, x2). All of them shall satisfy Eqs, (2.10) to (2,14). Hence one has - = if A + /7I '7 (2.17) and SC' = U/'''' - + W W(2.18) thus (2 19) Substituting this expression of ', into Eqs. (2,10) and rearranging the result, one gets (-~(~k/- J + i; /+ = -[f(/-S) +(.,Jw. j (2,20) These two simultaneous, second order, non-homogeneous partial differential equations combined with boundary conditions (2,11) and (2o12) characterize the additional membrane displacements / e With this set of equations solved, E.: and i' can be computed accordingly. Hence a new membrane stress field denoted by t may be obtained as ~ = 1; t o r;/ (2 2 -7-.,,</' -^ -^(2,21

-10.Because of the undetermined amplitude coefficients of W, the magnitude of this improved stress field is also indefinite; otherwise it is uniquely determined. The principle of minimum potential energy will be employed for the determination of these coefficients - It can be shown tha ththe minimization of the potential energy of this system is equivalent to the minimization of its additional potential energy due to the lateral deflection, The latter is composed of three parts. They are the bending strain energy, Ubt work done by the edge thrust, We, and membrane strain energy, Um, ieeo V(li;)= Ub(JW + We I(1 )+ (M) (2.22) in which A, t&-z = fi. - /- 29 96j' %* / 'z i dj TWejAe = tJ (2023) The first two in the above expressions are well known. (19) The third one is derived from U,="l,;i y ^ d>. (2,24) By relationship (2o19) and in view of the symmetry of the stress strain matrix:

-11 - _..f t,,z 4 '-' "'f v',~;,:' "; a.'. (2.25) The first integral in the above equation vanishes. This can be seen if one applies Green's Theorem to it. Then all the resulting terms vanish based on the consideration of equations (2,10) to (2.12). An equilibrium state is characterized by the vanishing of the first variation of the potential energy V, that is, o = O. (2.26) In the present case, the function W is in the form given by (2.16). Hence the potential energy is a function of A1 and A2, and consequently the variation of V with respect to W shall be performed independently with respect to Al and A2. Thus condition (2.26) becomes Since (A1 and ~.A2 are arbitrary, two equations -- -o0 and -- O (2,26a) are available for the determination of Al and A2. The stability of the configurations so determined requires the second variation of the potential energy to be positive- definite(20) i e S(T?) > o

-129 For the present problem, this leads to 4 iA,)2l +^ wa (A,)(SA ( > o(4 (2 27) Considering the independence between gA1 and &A2, the necessary and sufficient conditions for the above inequality to be held are J' > and [d 12 ]2 V; (2027a) aT >, 0 an (2027b) The stability so characterized may be called relative stabilitye The counterpart, absolute stability, will be discussed in Section 4o The criterion for the change of buckling patterns of plates from one to the other has rarely been discussed, Koiteros, Alexeev's and Stein's conjectures (see po 3 ) were based on their computations that for certain loading parameter, the antisymmetric buckling mode resulted in a smaller thrust than that resulting from a symmetric mode. This is similar 41 f to what was proposed by Karman and Tsien(21) for buckling of circular shells. However, such kind of criterion for shells has been rejected and a criterion based on energy consideration has been establishedo(22) According to this, the buckling pattern will change from one configuration to

-13 -another when the potential energy associated with the latter is lower than that associated with the formero For the present problem, when the plate has undergone large buckling deformations, the configuration resembles that of a shallow shello It is therefore reasonable to apply the shell criterion to the buckled plate. Hence, the primary buckling mode may change to the secondary one when V1 V2 (2,28) where V1 is the potential energy associated with the primary mode and V2 is the one associated with the secondary mode,

3* SOLUTION FOR A SQUARE PLATE The approach to postbuckling problems outlined in the previous section can be applied to plates with any aspect ratio. In what follows, there is no particular aspect ratio specified up to the computation for eigenvalues. From there on, for simplicity, an aspect ratio equal to one will be assumed, Let the plate considered have a length a in the loaded direction, a width 2b, and a thickness h, It is simply supported in the lateral direction of the plate, A set of rectangular reference axes has its origin at one corner of the plate with xl in the loaded direction, X2 in the other. Prior to buckling, the middle plane of the plate coincides with the x1 x2 - plane. Let a relative displacement of two loaded edges be prescribed in proportion to a positive increasing value A such that = -\ where /= -and let it be free from shearing stress along these loaded edges, Let the edges, along x2 = 0 and 2b, be immovable in the x2 direction and let the displacement along these edges in the' xl- direction be linear in x1. Hence the boundary conditions for the membrane displacement field UW are -14 -

-15 - ZJ/'( ), X ) = O._U o( TV, 2D) -k ----- 1. (3.la).) Along the center line, the symmetric condition must be satisfied, namely U,~(Xi, =0 1 o (3,* b) G2 AN,, h)= 0 J All of these boundary conditions are satisfied by assuming l,l Z A = X- / ~' (3.Ic) yr Uaz (x,. ) o From this displacement field one easily gets its corresponding stress field a,. It is.n *' h / / k.. 4 - ~~~c~i ~~/ tl4cz (3.2) For a start of the present approach to the problem, a membrane stress field 7,' shall be selected. In order to study the behavior of the plate in a deep postbuckling

~16~ domain, it is reasonable to choose the stress distribution for ultimate strength of buckling of thin plates suggested by Karman, Sechler and Donnell(18) 0 It1 is. shown schematically in Figure 1, T T1T1 III T iT~h ] I. U, I I -A tItLf itTIt 'j == ~A + ' Figure 1l An Assumed Membrane Stress Field The width of stress strlp; e, show n:F.Figure 1 is known as the effective width in the literatureo, This is expressed then by the equations /l 77 =- 0 r-l= o 2 h T/ l (H l)TI' "- - '11 I c< x C, C < /2,6 o 1 X, A 6/ o < X/ C,/ 4~~~~~~ -(3 o3)!22 h o. X, c, = 0. R.

-17 -As before, positive stress represents tension and the negative, compression. The "statically homogeneous" additional membrane stress field 7/j so selected satisfies the equilibrium equation (2.10) as well as the boundary conditions (2.11). The satisfaction for the former one is quite obvious; while for the latter9 because no normal stresses are prescribed along the edges, it is satisfied automatically. Transforming the coordinate variables xl and x2 into dimensionless quantities ~ and t such that = c and and d replacing b- by 7y, the equilibrium equation (2.6) takes the form Plw^^tv, 2 i^^ lft, - ^ -l T CN V Wv- = ~ (3~4) for o y:- / where c2 comes from the transformation of the variables. For o <. CL- and c c c C one has;i m]^~t y X +2V>t- o (3.5) The bar over W' in the above equation is used to distinguish between the coordinate functions in the different regions. This notation will be employed in similar occasions below. The simply supported boundary conditions in the lateral direction of the plate may be stated as follows:

W(o,. ~)= W, (, o. J = J 7 Along the central line on account= (3) o Along the central line, on account of symmetry, one has WY- (g, 6J _ * W ^0 J-O. c Wrr ( c. (3 7a) Along 1 the deflection, slope, moment, and shear must be continuous from one region to the other, that is, ~n (S~ OI 01-1 W7,vr (;, /)-= w rr (f / I (3~8) To solve this system of equations for W, first let w =f(r) in r, W/tere r=,- c (3o9) then the first of the boundary condition (3o6) is satisfied automaticallyo Substituting Eqo (3o9) into Eqo (3o+) and denoting the derivative with respect to f by prime "', the partial differential equation (3o4) becomes / "'_ F"_j/ /ljf = o (301O)

-19 - in which e2 NA/ (3.la) The solution of the above equation with the satisfaction of the second of the boundary conditions (3.6) is in the following form:* f(r) = Afsu/nc fi + /hAl/7 j (3.11) whe re ~P -A l f -I3 / l /P (3.11a) and where A and B are undetermined coefficients. Similarly by letting e = if( ) in 1 / g (3.12) the condition (3.7) is satisfied. From Eq. (3.5), one has f -- /zr '"-/ = (3.13) The solution of this equation with the fulfillment of the boundary condition (3.7a) is found to be )=-A+ ( +H ] + )7 eH o' ~+fA C-)T / (3-143 +[6 +Nl(" _r). 't u} c i,r *It can be shown that f n, that is, < is real.

-20 -where G and H are again undetermined coefficients. The conditions of continuity (3.8) now become J=Jf f-fi f-f Q7d f /-f/ ^ or = / (3.15) The satisfaction of the above four equations leads to four linear homogeneous equati- ns in the undetermined coefficients. In order for this system to have a non-trivial solution it is necessary and sufficient that the determinant vanish. The result is a characteristic equation: Cos + 7 b / r (3.16) /-he o //th r *-9)i ( f fom / Jf — / 7n /iA 4a/~t ( /j - /J^t g c/S ji == 0 This characteristic equation gives two groups of eigenvalues. One group is from Cos o< = O (3.16a) The other is from //hcz V (-/7- / /- ^ 1/7 717 e-2 4- f + 7}P {/.. (3.l6b) ^~~~~~~~~~ 7~/G/-y ^ ^,[3^ - ^

-21 - _- f /i -/},[1 + / / (3.16b) -/- + i Ap l = - V From (3.16a), one has o(.2l^-) A^ ^ -d.. t X(3.17) Substituting < and ( given by Eqs. (3.11a) and (3.9) respectively into the above equation, one will find ~Pf - _ _- (3 1l8) It can be verified that when c is less than b, this mode leads to a higher eigenvalue than the mode to be investigated in the following'paragraphs. It is, therefore not considered further beyond noting that, after minimizing it with respect to n, it corresponds essentially to the Euler load (except for a correction associated with the particular type of boundary conditions assumed in the present problem.) Eq. (3,16b) is so complex that it is impossible to give an explicit solution. However, the eigenvalue P can be computed for a corresponding "effective width" c, if the dimensions of the plate, a and b, and the number of half waves, n, are specified, Once an eigenvalue f is determined, and by going through equations (3.15), a set of corresponding coefficients, B H and G can be calculated as follows:

-22 -22 ~ (/- ' f/t y f (/ s- + ' rr<^ g a6se &3n =e -----? --- —--------- ------ U-ere? v + ( -? e~ 2Ky/+e 54, (f^; f3" ^(3,19) in -which the su'bsecript r is attached to all. the values except the eigen-mvcallue. P in order to indicate their dependence on the parameter n. The only remaining amplitude coefficient1 Am- will be determined by energy consideration, It is well known that the primary buckling mode is a multiple of half square waves, if the aspect ratio of the plate is an integer, Hence, without significant loss of generality, one may assume the aspect ratio of the present plate to be one. i,e, - / T Then, by designating the paramaeter n toa be 1, 2- 3 and 4*, a group of eigenvalues f wrlth corresponding c r are computed from the charcteristic equation C(3.6b) for ~ = 0,3, ~1A.-tarii~ts, f1r.omr i ith hr.r A lImt ase for n approachi.ng i nf inity was studied by Koiter)x(6) wvo used the same ini+tial stress field, but found h.is eontclusions to be seriously at variance with ex~p er.eimeta a7d o.othe r cons aid 1ela.

-23 -Now one half of the total load P from Figure (1) and Eqs. (3*10a) is p2 p 2 F=NC /1C 27 c The one-half of the critical load prior to buckling for the plate can be found to be. =Al = z7 /9t9 x (3.20) Hence the ratio of the load P to PO is.. = //p f2 _ With the computed values of f and c a set of curves with C as ordinate and P as abscissa are depicted b Po as shown by Figure 2. These curves show that as the ratio b decreases PO (due to the increases of the load), the lowest value of P may change from one curve to the other. The intersecting point of the curves of n = 1 and 2, at which P - 35.3 P -26 or /-3- (3.21) gives the possibility that with this eigenvalue, either the first mode of buckling or the second one or the both *Put ito=- E-~, +2- o -— e wi 2 find the ei a - - into Eq. (2.6) one will find the eigenvalue AO /.Z

-24 -of them can exist. This particular point is selected for the further investigation. Then, in general, the configuration W/(', 6) can be expressed as W(K, r)-= A, ' tA2 w - (3.22) where Wn - () /rl 7 - /. 2 (3.22a) and f~rjz- ~5/W 7L 55 L B7S/rf'-?( 7 (3.22b) For the interior region yW(=, = — A, w'/ + Az w2 (3.23) where w J," - J"7 //I)., /" 7 O, (3.23a) and f: n — f e-{ 4 1 /1n on (C,/n +r 7- /(3.23b) in which 2C= (- //n ) - D.=( ^ 4 (3.24) f obtained from Eq. (3.14).

.4.358- %.3 1 T-~n= I.2 - - - - - 4 5 6 7 5.26 P/Pcr Figure 2. Solutions of the Characteristic Equation (3.16b) 2.0 1.6 - 12 f2 C1 f2 04 0 _ --- —---------------------------- ---- ----- 3.0 17 Figure 3. Buckling Modes in -Direction

26 -With the eigenvalue given by (3.21) the parameters ~^, ~ ~, ~ and the coefficients involved in Eqs. (3.22) and (3,23) are readily computed by their appropriate definitions from Eqs. (3,11a), (3.19) and (3.24). The results are given in Table I, with i = 0.3. TABLE 1 CONSTANTS CONTAINED IN DEFLECTION FUNCTIONS n `n o<n Bn Gn Hn n Cn _Dn 1 0.56301 1.34822 1.,5675.14656 -.63988 3,33867 0O425614 428 2 1.12602 1.7324 2.35311 0.043735 -.95092 14.9877.043648 -0.0072295 The configuration functions 1 (7) and (J in (-direction are shown in Figure 3, So far, one has found a particular deflection function W(~f) Jsatisfying the equilibrium equation (2,6) under the assumed membrane stress field 7,. and including both symmetrical (n = ) and antisymmetrical (n 2) modes. The next step is to calculate its associate membrane stress field I). To do this, one has to solve for the membrane displacement uz from the set of Eqs. (2.20), On substitution of Eq. (3,22) to the right side of Eq. (2.20), one has

-27 - (t-V~)) W< k -I- (/;P) Ij' e4r; = A^r.nji/-r^ p^^r^ n/ <-^ + I^ u~Jj~/ (325) in which m and n take the values 1 and 2. The summation convention of index notation applies to the repeated indices of m and n only when one is a subscript and the other is a superscript, For instance, 4 W/'"= W /AW But f/^/ means A/, or z1 z Such a convention will be used henceforth. Also note that the expression in the square bracket is symmetric with respect to m and n. Let the C7 in equation (2,20) be Z' = 4 /I /y" (3o26) Then the Eq. (2.20) becomes f'L (/ ) Wf f ( W (3 27) The dropping of the amplitude coefficients An, in the above equations is based on the independence of these

-2ocoefficients and the homrogeneity of the boutdary conditions. Hence in Eq. (3027) there are, due to the s metry of n and m, three sets of simultaneous, second order and nonhomogeneous partial differential equations for y7,^/ for the region of oQ ~ < f- and ^o:,. / Similarly there are another three sets of simultaneous equations for the interior region, They are (/ ) - J j (' /-t, 1rf w:,~) 9 -, --- r /?,.u - - f - * /< " yf=(e2^>-4/....j,/ Iv, (3.28) for o z <i._ and /< t * _ r In the above equations it was assu C.med tha it was asstred that i / / — // A. / i U^ I -- mAh U (3 29) and Eq, (3,23) was used. Sin..ce t.t-e t rmembrane disspla.memnts a're prescribed all around the plate (see Eqs, (3,1)), the boundary conditions on the additional mnembrane displacements according to Eq, (2,12) are homogeneous From Eqs, (3,1a) ~"i~~.::: ti,"c2c 6a ry D_ 7f fJ ur^^^c^^) —^^^" - ^'Y?r-) - ~(, o) 0 u( v B e/ r C Irom Eqs. (3,l.b) t6f"v s~ / K; 7,0 (3 30)

-29 -Along o 1, the displacements on each side must match, ioeo0 ^ ', ~ 9= / '.' U (~. -- 2 rf,. (3 31a) Due to the continuity of the additional membrane stress field?2j, the normal stress in -direction, h2 and shearing stress, <e 1 must equal their correspondents 2zz and ', respectively. In turn these conditions may be expressed on the consideration of the stress-strain, strain-displacement relationships and condition (3o31a) as zz(,l)=,2 (a,,( J=, U (a,)2 (3.31b) Hence the additional membrane displacement fields Uz" and <^ have been characterized by the sets of differential equations (3o27) and (3o28)9 boundary conditions (3o30), and conditions of continuity (3o31a) and (3o31b)o The mathematical computations in solving these systems of equations are quite involved, The details of doing so will be shown in the Appendix, The resulting solutions of c4'" are in the following forms: {rn =?^ + f( 7 )^ 7 14 = c Li (f5/n at Jl I (3,32a) (U 2 Hmr^)COS + J,n*(^ci/J l

-30 -where 1n4 = (M 17;7, = - r - /7. Note that when m = n, the second part in and COs ^,, = / Let, (J //* ( r), " r )( cd"/ vanishes and Jmn ( be considered as the complimentary parts of their corres ponding solutions and,, (Tr, _7Z,/, Zn / and J^ ( 7,) as their particular solutions, then they are in the following forms: L Io * and 'on j t it Ann Jm7ri = (b~tb, ~nr/cdS~Pnr f~d~ ~ L~qr ~nnrj-l~nA~m(3o32b) The last set of solutions is for m $ n, while ml m n Ja d = b, t b And 1 and Imn does not exist. And *Solutions of similar forms are collected together. The differences among them are only in constant coefficients. This equation should be strictly written.as mJi n)4j t |(amn) {(M)& 'a ll)4 (a^)| i,; hi 7 For simplicity, all subseripts m, n, and G (or H) are omitted. This understanding will be used for similar occasions without notification,

-31 - f C3 COs4 (f5 -/3n) 5 7L C, oS1 C> 1/"3 7 - /7 coscA A7r CSacr - C ce2S&<C4sP44/s ' u' >(3032c) ^J"K~- ~: A(ftnl kr t d 5z// (gem-P) 7f t 4 (4m +n f < 4 d5 cos6 hpf S/7 non -t dc/6 jsnin 4/7 + d7 5/gt7^ cOSdi 7 + d8f coY5s f sn/i^7 All the above solutions are valid for o c r ' /. For the interior region, / c 7 c, one has the solution of Ut,. as =77 4 | r(j 5 ^~ - J ( () I /m rx 17 ur/2m=-C^[-t/ }^^(r)cO~at 7 (nJ +7^A^ >z j t(3,33a) c to. ZIS -74-_T~~~~~~~~~~~

-32 - in which rimnfmnj (3.33b) for m = n; A,4 == ({ p,-f^^^ r)^ (^ ^^r^^f}P4 (3.33c) t P7 +/;fn j f9 gmf 7 for m = n. Both the complimentary and particular solutions are included in the above solutions. The numerical results for all the foregoing solutions are given in Table II to IV. With the obtained 46/" and 4"" combining with w" and M given by Eqs. (3.22) and (3.23), the additional membrane stress field Zj' is readily computed from Eq. (2.19); this leads to: =/. =, A. A f z,/A ( ' -/ <" (33) (3.3t) ~a(li^TfIlk} J

-33 - and fL ( pa (6 -k m Al / I ^/-T^^/^^~61 ^7^. v?,-., ^T 4 FVO" w (3.3+) Let in which yl =e/ JasC,, + S,;^Cos,, 4 7 =y?7c r) co45 5 ~ Q.(r) (COSe 4 for i 2 Q for i J, and where IV - k CO5 (c4- O(q m t cosk c('c 0)r tg(3C05/(r~e#I/ #t+ t~(0/; G^ J~ ~9 } (3.35a) (3.35b) (3,35c)' g *It should be noted that strictly writing K - (g/)'. s 1 to 12S So for gss in (3o35d) and S's; t s in (3.35e).

for -34 -i = j, l C'> J = 7A' (? -',",,) ' -7 42 5,>7 m'7, 3( ) 7 Co { s7 I ' 44," J )c 5z / 7 t (4 -'^ ^ '' 7 ' f '.6 5?js,<4, s/ }/7+ (3.3 5dk) e 9 /A <#< <r^47 7 7 t -^> t72 7 # -/ le:/i "t^^^tr)f~ for i # j; and v p-wn L r/ = (J/ + 4, / v5w5, f 29e^) '77 L^'J t (JJ$ -t 5X_ X,* +7 5 S::gi 2r 2J e- ~ i; 7 * (t/ ' tL 4^ + i t/ 72) e ^; * + f7 (c fm7 ^ -y e 7 (3* 3 5e) The computed numerical values of these constant coefficients are given in Tables V to VIII. The results of functions,46, Q and P 6 ", Q6' are shown by Figure s V and ' Figures 4 and %.

TABLE II,, CONSTANT COEFFICIENTS IN EQUATIONS (3._52 b) ___al a2 a3 a4 b< b2 b3 b4 b2fsq( ____} a1 a2 a3 bab b2 b3 b2 Gl1.05442 -.01398 -.02903.02520 -. G22. 04108 -.01194 -.02479.01245 G21.05932 -.01713 -.03557.02644 0 I21 - -- - -.29587. 43479.90301 -.29445 HIl -- -.02520 -.002085.01398 -- -- - -. IAJ 22- -.01245 -.01523.01194 - - - - -- H21 -- 02644 -.004417.01713 Jll -- --. — -- -- --- -- -.00567 J2 -- -- -- -- -.07288 J21- -- -- -.29445 -.31567 -.43479 -- _ _ _ _. I I I- J I,.. I I

TABLE III CONSTANT COEFFICIENT IN EQUATIONS (3.32 c) cos (.*,,-o ) cc'So (047 4)r c/(4,-41)r cA(4SM,|, )r s s/47'n 5/tS(gmf 5 7C5Odsin7 i Ca5O4,ZCA,4/ __ C1 C2 C3 C4 C5 C6 C7 C8 G1l -.02786 -.03519 -- 000756.001185 --.007875 -- G2 |.02009 -,07038 --.0001346.002825 --.009075 ~Gg1,02144 -.09631 -.003279.0006205 -.0006425.01350.004710,008465 I21.24968 -,01183.002451 -,000049431 00577.06793 -.01232.01566 2.,..ooo...3.... S (in no m-) X5/,( nra 'a)f'n C sA(-(4nS )a m 2f(fi4 f n) C n Cm/n4 t|t S/%i, 7 s/7Cc5so(3 7 c0dnm7 "sCl dl d2 d3 d d5 d6 d7 d8 'Hil - -.16853 - -.004204 -. 09918 -- -12353 HB2 -.21656 -.0005626 -.04339 -- -.06120 _ H2Pl -| 25769 -.18955 | 004329 -.001511 -.01250 -.07050 -.02570 -.06351 J1 | --.07986 --.002184,05141 --.05454 J22 --.09456 -,.0003007.02038 --.02288 J2l 1.2777.17628.005355.001592.01422 o06155.02213.05037.21 35 ON I

TABLE IV CONSTANT COEFFICIENTS IN EQUATIONS (3.33 b) AND (3.33 c) o-~3x ~ e-,,,q ' /o-2xe4,,7 e- M.~. t / I r /<rz I4' T I nn! '7 / 1! r 7 / $'m' I_ __ _ A PI P2 P3 P4 P5 P6 1 q2 3 5 q6 G21 2.7525 -0.5653 0.02941 0.43465 0.14399 0.36386 -1.0883 3.5912 -1.3250 -0.6672 1.0955 -0.306 121 -2.6975 0.47535 -0.02121 -1.2761 -0.9346 -0.26229 0.544 3.9017 -1.8341 1.0890 1.7637 -0.434 H21 -1.3250 0.3980 -0.02941 2.3006 -1.8313 0.36386 -2.376 1.0451 0.4416 -1.2189.8841 -0.102 J21 9750 -1.1025 0.12720 -2.9201 1.1985 -0.78835 -4.1034 -7.026 1.8341 1.8687 1o0404 -0.434 4.950'21.12 2 62 -54 21.54 /C - 3t e ' e- ~..y. / ///,,, ~,, 2, 2 / 7 2 / (JJ P1 P3 P4 P5 P7 P9 P10 G11 9.9050 -6.2010 -0.9888 0.10298 -0.01081 -0.52951.15001 -.13845.02204 -- G22 0.19483 -0.0620 0.004925 -0.32465 0.014816 1.4240.03400.03218 -.002562 111 -3.0070 3.7470 -0.9888 -0.53926 -1.3253 0.52951 -.35149.11187 -- H22 -0.12312 -0.07462 -0.004925 1.2446 -3.0187 1.4240 -.08118.01300 - - Jl 8.1635 -5.7485 1.0722 -0.36742 0.5243 -0.5742 -.27192 -.05857 -.02099.004454 J22 0.32235 -0.11034 0.009064 -1,8993 2.8239 -2.7428.66409 -.06941 -.017327.0004546

TABLE V CONSTANT COEFFICIENT 2N EQUATIONS (3o35 C) Xo5(|-} ko Cdm,-,,,) C c, /-, ) f c,^/~),/j4,3,2/,, s|,or,?,7/,7 s'f f,,co-ea,, codckkrk 7 kl k2 k3 k 4 k5 | k6 k7 i k8 pll 11 Qil 11 11 22 P^ 22 22 Q22 22 p21 11 Q21 11 p21 22 Q21 22.10058 -.036056.0007744.25181 - 21276.22708.25980 - 34260.045828 - 023195.09842 -.044823 - 14448 - 13957.004175 - oo004661.001254.0002718 o0009440 -.00005519 -.00001934 - 00000938.021139.020206.000938.025231.020777.004447.009525.013376 -.0004452 -.0007595 - 0018922.0017288 ooo0009676.007956.014310,034908.032269.002187 o010009 -.006201 o00620 -.014309 -.01013.006379 - 005570.0002156.005570.01960.034259.015954.0018147.01594 i I

TABLE VI COEFFICIENTS IN EQUATIONS CONSTANT (3.35 C) (Continued) | S >A f54/'e, j ^<mf j fc f CAkf f5ifTP/l - i_____f _ Kl_ __Kg l Kl_ o l Kll_ K ' K9 KO0 Kll K12 K K10 K1 K1 K1 pll 11 11 Ql1 11 22 11 Q22 22 Pll p22 11 21 Q22 22 P21 22.027963 -.005932.04776 -.01013.05141 -.010905 -.019859.019859 -.01960.01960 -.031257.051257. ~~~~~~~~~~~~~~ -.05206.012336.07380.03460 - 08457.022053.011017 -.011017.01882 -. o882.02025 -.02025 I I -.43497.09227.11604 - 11604.42016.061925 -.17135.17135

TABLE VII CONSTANT COEFFICIENTS IN EQUATIONS (3.35 d) _______ _______ __1_2_A 4 _____ _ 4 {7 48 1 g / _2 ( 3 |v5 1 1 I 7 8 Q 12 = - --... o006136 - 009750 Q12 - -.. o.00 o6359 -.o18375 p2 o.06716 -.0255o8 -.0013531)..0023742 -.011782 -o004850.0001660 -o10196.002289 o003697 i oo0000169.00550 4 -025054. o.007088 1 -007151 I ____I_ I000228 9 4 c. - i0 97070- -0 ____ o I /12 /9 1/10 l 112 -....... "I --- ' I u - -- --- / -_ -_ _ _ _-,_-.... _._ - Q12 -- ' a;-oo.032197.011017.016949 -.019859 Q22 o -- i-.05420.018815.028945 -.019600 12 21 o05412 -. 17135 26362 1 604 Q2 - - -l533.02025 o031156 -.031256 _ __.s ~~.05331 005, ~ g o

TABLE VIII CONSTANT COEFFICIENTS IN EQUATION (3535 e) for M=N _ _ __lR e.__ _ _.__.. " 7 ~/ _ _.. 72 S______ s2 1S 4 s S6 s S8 S4~~~~ " 7 p11 11 Q Q11 pil 21 22 -11 Q1 11 -22 11 -22 p22 -22 Q22 -22 12 13.03 1.757 -8.856 -1.752 1.501.3753.02961 -.055338 -4.022 2.770 0.5496.10497 -.15090 2.502 -2.177.18206 -.04021.04788 -.3753.3753.01508.00384.07370.11403.26095.8360 -.30167 -.29831 -.30812 -.32128 -.08068.12030 -2.1239 -2.1559 - o00582.80383.20096 -.20096 -.20096 4.3233 1.o809 -1.0809 -1.0809.025812.034737.25181 -.26564 -.10913.05916.00805.2598 -.12508 -.05072 -.06948.21826 -.06948.21826.06948.10146.10146.01615 -.01615 4 I -0o1615.12696 0o4404.00384 1.0752... -.... I I... I.. — - - I......

TABLE 3X CONSTAIqT COEFTIC'TonsTS IN EQUATIONS (3.35 e) FOR M4Nm / ______ /.94/ 2 / __ ___ / __ _ ~s S2 03 8_ 85 06 __ ___ t3 _t t5 t6 P11 -6.4195 -1.3846.,0752.64883 -.41331.93209.3.064 1.5798 -.3483 -.47250.14095 -.080 11i.7355 -.2372.01666 -.22135 -.96270.20713 -.20016.22442.17407.51442 -.30499.040 ~PJ| -1.0708.18729.008359 -.60163 -.36886 -.10356 8.8125 -.29729.3483.67396.18124.080 Q21 -.1310.3040 -.01666.90396 -.13418 -.20713 -3.0578 4.2461 -1.5665 -1.31442 1.29518 -.362 P 1 2.1515 -.51165.02508 -.69990 -.48363 -.31070 -5.8405 2.2763 -.3483.49268.02017.080 Q-2l 1.0055 -.27000.01666.76979.54844 -.20713 -.22946 -.37101.5222 -.86981.67339 -.120 7 57 28 )57;52 '57 >84 I to IN I

.8.6.4 E:-.2 c -.2 -.4 -.6 1.2 1.0.8.6.4 c C E'-.2 0% -.2 -.4 -.6 -.8 0 0.5 1.0 1.5 2.0 2.5 279 3.0 m l a) m=n= 0 0.5 1.0 b) m=n=2 Figure 4. Functions m Func io P P; 1.5 2.0 2.5 2.79 3.0 1) and Q M, for m = n

C E:-' c o C a 1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 I I 0 0.5 1.0 1.5 2.0 2.5 2.79 3.0 r) Figure 5. Functions p ar bJ antv and Qtj for m ~ n

In order to determine the amplitude coefficients Am and to examine the stability of the corresponding configuration9 the potential energy as defined by Eqso (2,22) and (2o23) needs to be calculated, For the present case, because of the simply supported boundary conditions in the perpendicular direction to the plane of plate, the bending energy term may be simplified to &T 2 = {J(,4 k))f[2, 4 (3036) where =_- =dz, 2 o After transforming it to da/ 't the integration is performed by taking from o to a - from o to 1 and then from 1 to b so that only one half of the area is covered. With the gZ given by Eqso (3~2) the work done term (see Eqo (2o23)) may also be reduced somewhat to the form ^^ j((Wm (441, 4wy(,-(tJ (3037) The membrane strain energy term, with ', given by EqSo (3o35a) may be expressed as 6=a. JR; 7) J ^ J/ "-g.2>JR (3.38)

-46 -NOw, let the integrals involved in Ub, and We be denoted by and 2.. =, z~ / ')f ( ),,) ( ('f: -.k z 2J=#Z2 2= 2 JV ~ 7Z = ^ij^)^2^ ^}^(<^J^ K-^(^h^;^;^^^^n /$c I /^ J 1 (3.39) J 1,1 1(3.40) j respectively, and those in Um be denoted by Z 3 3 J4^'/'J ~ a/RI ~i~.,,~ ~l 3=ar z /6^ '?//.;-c,', /. 7 and w/ H'~ 'I". > (3.41) (The computed numerical values of these constants are given in Table IX.) Then by summing the three energy terms, the potential energy assumes the form V=p (-3 /-,),4+(:'-2A)A, +2. T,,,,442) (3.42)

-7 -where A -- and A is given in Eq. (3.20). Note that the equality l /D / v 4z =a,,,/ e'-1 (3 43a) has been taken into consideration in the constant 2MI, It can be seen as follows, In view of the symmetry of the stress-strain matrix 2. 2 a @ 2S / J3 (3.43b) V ~ Replace the strain field,f on the both sides of the above equation, by its corresponding displacement field,, and W1 according to Eq. (2o19), then apply Green's Theorem to the integral involving i"' o This integral will vanish because on the boundary, Af ~ (for the present case; otherwise S0 = o), and in this region, the equations of equilibrium are satisfied by ". The remaining integral is the equality (3.43a). Now the two equations for the determination of Al and A2 based on Eqs (2026a) are readily to be obtained from Eqo (3,42). They are 0,~ /4g(J2J$2A +2M() 1 ( / j (3)I 4) -o1,. — ~

-48 -Since A1 and A2 are irdependent there are three possible cases. Namely, case (1) with A2 = 0, case (2) with A1 = 0, and case (3) neither of them vanish. Case (1): Let A2 = 0, from Eqs. (3.4+) (-, i / -J) > / /> (3. 5) The subscript on (-A) indicates the number of case. Case (2): Let Al = 0 in Eq$. (3.44) (h 24'(111J2) >/ / k-<2 (3.46) Case (3): Solving Eqs. (3.44) simultaneously for A1 and A2 (h) 2? v 2 -/412/ - /( - / <) f, // A,> 7o,/ - Jj f(3.47a) Af2 -LK, Note that M2 - LN o for the present case (see Table IX). And ( 22 / {,?/ (A/ -/ j / '-( 17- )2 (3.4'7b) /1&, - t2

Now the stabilities of these modes are going to be examinedo From Eq. (3o42) or Eqso (3,44)? =:[ 2 6 (' ) -1 2t ~J 21,1,] 9X =Z (( - A A'^^ -, 7-4 H F (3o48) and S^ -I "?4 1 )A2 ~ g/ds~k7S For case (1), with Eq. (3.4 5) A2 0, 'and A-) given by ^8L(i: a1L 2^ _ 4 (AS ) Z A ('. _I-L T v3, "1 I j, (3049) Hence, according to condition (2.27a) the primary buckling mode is table if and only if I^",,.,0 " >; I I: —. 4) O~~oell~~ ap~~~d /4> (3 0o) For case (2), with A1 = 0 and Eqo (3o46), given by

-50 - 2Y _ -/A/ ) /f A,2_ '- 20 AA, 4 j d'2V = Xa (h/222 jZ (3.51) 2/ =o Thus again, according to condition (2.27a), the secondary buckling mode is stable if and only if (- h3.> /, A / (3.52) No'i for case (3), with 4-J and J obtained in Eqs. (3.47a) and (3.47b)..,t A /'A (3.53) -A - -8 A/ A)s /A ) It follows that 2 c(l/,/.) (3)54) 1J 2 64 /1 T.iis fact shows that the co figuration of the combination of the two modes is uns.;able:n the whole postbuckling domain. The numerical resul: s of the amplitude coefficients A1 and A2 and the sta!.lity of each case are given in Tab.e X.

-5 - TABLE X CONSTANTS IN EQUATION (3.42) INTEGRATED J1. J2 K1 K2 L M '.7453 3,0650. o6244 1,0293 1,8680 3.342 2o034 3.464. _... I i,::::i,,,,, i _::::...... -..::::::::::: ___: TABLE XI AMPLITUDE COEFFICIENTS AND STABILITIES Case (A)2 Min\ ()2 Min. Stability ~?. 2 ' 0 1671 A 1,98 0 Stable if.1995 A' 1,19 2 0 v.1486 A 2,98 Unstable if.4423, t' 6.00 Stable if ___.._.____.._______!__'> 6.00 3.1359 A - 6,00, 01746 A - Unstable.8152.4104! ___ i________________________+_ It is seen from Table Xlthat the symmetric configuration starts at A' 1,19 and it is stable from there on, The antisymmetric configuration could exist after A - 2,98, but it is stable only after A exceeds 6,00o Note that an actual symmetric buckling mode starts at A = 1o The difference between this value and the present

one (1.,19) is due the approximation involved in the configtzation function, The primary buckling mode may change to the secondary one when the potential energies associated with each mode are equal, (See Eq0 (2,28)), On substitution of Eqs, (3,45`) and (34 6) into Eq. (3,42) separately, one obtains I/ - -— 12 and ' '325) respectively, Equating these two equations and solving for Ai results in ~Ah~ =- ~-::.-...3,,, 55) i v J/ _t /Z -. P w ~f~ K.~',' /', Ae Using the constants given in Table:X, one finrds _._J ' —,/ ' /-:,<'4.d (3 55a) The first vailue in the. ab( e results is in a - range in which izdo<Fes- not exlts, THene, at 7/ ll H14-2 t he transition may oc aur it thl the deter mmined 'a teral b"ckllng a.mpol; ir;ude coefficients (-)j an (' — te t additional' mebrane stress field 2< as given by Eqjs (3e35) i ns unique.ly d etermined, In t.urn one may f.ind 'the.s::new membrane stress f.iel.d a' s as defined by Eq. (2.21) witht the e< given by Eqs. (3o2). For instance in case e(il) usLing A, = -~ -

C53 - o,, - + 3 (/A, 2(3. 5,,6 A. - (2 / J2 By changing (EgTj <'/ to ( 22 2 in the above equations the stress field of case (2) may be obtained, The distributions of the normal stress along the edges of = 0 are depicted in Figure 6 for both cases, A relation between the total thrust P and the parameter A' can easily be found by integrating the normal stress?, along 5 = O. The resulting equation for the ease (1), is P- =C0.553c+ct 5~-,?7.or 9'>v/9 (3055a)* '/0 for the case (2), P o =.3075Af2. o6/3 /or /\2, 9& (3 55b) Po where Pi and P2 are the one-half thrust in their respective cases, while Po is the prior buckling load equal tob e Equations (3,55) are shown graphically in Figure 7o *The conventional coefficient of A/ is e05; the discrepancy is due again to the choice of the deflection functionz

-54 - 10 8 6 Z 4 o c 2 0 0 z 2 4I 4 C - T O _- 1 _ I h n-^T n 3on oS 1 ~ Mn 0 V.V I. V 1. a) A2=O 2 il..V i.v 2.79 "' 12 10 8 s 6 Wa< A i4 - 0) 4. o 0 Z2 _ 4 0 F 0.5 1.0 1.5 r7 b) A =0 'igure 6. Normal Stress in Along the Loaded Edge 3.0 -Direction

9 8 7 6 p Pcr 5 -_ UNSTABLE BASED ON THE CRITERION STABLE OF " RELATIVE STABILITY l | / |.^ / THE BUCKLING MODE OF Ai=O, A2#O IS __ _ ~1' / (3{ _ _0 "RELATIVELY STABLE" IF X> 6.00 ^^,,S S I I A TRANSITION MAY OCCUR AT X'= 11.42 ___Vr____ ____ ___|_(~) AT WHERE THE POTENTIAL ENERGIES ARE EQUAL.THIS CAUSES A DROP AT THE THRUST. L__ ___________ 4 3 2 0 O I I 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 X' Figure 7. Thrust vs. Shortening Curves

4. APPLICATION OF FUNCTION SPACE METHOD TO THE SQUARE PLATE The function space method was initially suggested by Prager and Synge as an approximate method for problems in elasticity. The application of the concept of this method to buckling problems of plates and the basic principle involved was first given in a previous paper(9) on which the present study leans heavily. However, the method suggested there was primary for problems without including secondary buckling. An extension of this method in this direction is made in the present study. The illustration is given to the square plate discussed in Section 3. A principle of minimum strain energy of the additional membrane stress field was established in reference (9). The inequality, in present notation, iz4 )-^ j ) l2vWe i J S).o ( L 1 ) is used to characterize a surface which divides the space into two parts. Any stress field Jj satisfying (4.1) is on one side and is stable. Those of the other side are unstable. In the inequality (4.1), W ( jJ>c) is any nontrivial configuration which has piecewise continuous second X See REFERENCES > in reference (9) -56 -

derivatives and satisfies the geometric boundary conditions relating W on Bo Since the actual w also satisfies these conditions therefore, U(/j; J > ~, The equality holds when the membrane stress field is also an actual one, i.eoe 0 (6., )=o o This can be shown by multiplying Eq. (2.6) by w and integrating by parts, Hence, (;, '.-, h,) > o ioeo h T.,/. -v c7 0. f (4.2) which, in view of Eqo (2o14), can be written as i ( t. - /i - ~/ ) ^/ d > ^. (4, 3) Applying Green s Theorem to the second integral in (4,3), the result vanishes because Y- - on B" (T,,' ')=o v J V ( on B and (7,-.)-= o in R. Y In view of the symmetry of the stress-strain matrix 2 4j; T; '-( - ''- in which E* is the strain field associated with 7y Then the remaining integral in (4~3) may be written as /R ^^^^Z e

This is the principle of minimmun additional membrane strain energy State in words tthe strain n erln y associated with the additional membrane stress is as small as possible subject to the requirement of static admissibility"' Some basic aspects about the function space are these, A vector in function space represents a stress field. The inner product between two vectors is defined by ^Z. tk-/tz Ei7 /f7/2gasa - i"" z'; c" >A t(4,6a) from which the length of a vector is given by while the angle 0 between two vectors can be found from (4 6a)e Applying these concepts to the inequality (4,e4) one now may write it as 7-7-^' -- 73t ":T ( t (4, 7 ) This indicates that the swrfaces S defined by (4.1) is concave to the opposite side of the or.igi.n 0 as shovn.n Figure 8. In this figure P is a generic point associated with Tv and p is at a point such that vector op indicates the actual additional membrane stress field. It also has been proved that the vector ~-, repre.senting the additional membrane stress field -, wihich is the statically homogeneous part of a stress field

-9 - Figure 8. Diagram in Function Space for Primary Buckling associated with a strain field derived from ' / -it' is normal to the surface at the corresponding generic point as shown in Figure 8. In fact, the stress fields 7i and ~ are defined in the formulation for the first method by Eqs. (2,15) and (2.21) respectively, The stress field 2L, due to the undetermined amplitude coefficient in W (in the present discussion, W includes primary buckling only), is also indefinite; otherwise it is uniquely determined. The undetermined coefficient can be determined in the following wray. Let (Figure 8) o0 = T' PZ = To/ (4*8) Taking inner product of the above vector with 2A and considering the fact that

-60 - one has r- _Z- T (4.9) This equation enables one to compute the coefficient. Note that referring to the inequality (4.7), the length of vector 7' represents an upper bound to the actual one, the length of z. The length of vector o~ is the lower bound to it, since this length does not exceed the length of t' due to the convexity of the surface S. Hence, the length of Z' so determined is the lower bound of the actual one. For different buckling modes which are characterized by different eigenvalues, there are different surfaces in function space. Each one represents one mode. Usually they are independent' and separate from one to the other. However, in the case of the problem discussed in Section 3, there are two surfaces, say S1 and S2 for the buckling modes of n = 1 and n = 2 respectively, passing through the point P which is located by the vector T' given by Eq. (3.3). This is because of the particularly selected eigenvalue (see (3.21)). The additional membrane stress vector,?/, is in the form of

ZT t _ E^ 7"#2//iz?^T 7 ]2 (4+lO) where t-t are the vectors representing the stress fields 0of 7^ given by Eqs, (3o35). It is normal to the surface S1 when A2 0. It is normal to the surface S2 when A1 O 0O When neither Al nor A2 vanish, and by varying these two coefficients, the locus of the vector 7 ' is an elliptic cone. This can be shown in the following way, Write Z-'= ~c~/V"'#2Y4/U +, ~22 (4,ll) 22 in which //V and /2 are normalized stress vectors of Z"/ and 72 2 respectively, If one takes inner products of the vector ' given by Eqo (.410) with A2 = 0 and of the one given by Eq. (4oll) with 4 s o to themselves separately, then comparing the results and since /A/"=- /, one obtains And > (4 12) 2 2 E / / 2__ L / 2 is obtained by a similar way. It can be shown that by substituting the stress field M"" given by Eqso (3,35) into Eq. (4.6a) fintodEq.

f4? A ioe s (413). This means for the present problem, //'" is orthogonal 7 't '.:2'.,. / ( il4a) /ai v7 1$ One also has. and ' -, t. 4 The anrgle -, i-nlude. 'by /" a.nd Ab.e o'r7 by the equivalences " J" and 722 can be computed by CCr 529/- 722 --— ~.~ ~_i (4 ) Referring to Eqs, (3,41) and Table 'X Let a J and k 'be three mit vectors along x, y, and i, the three Cartes ian oi:. na'.te axes:respecti'vely as shown in Figure 9, Then one has

-63 -/ =- CO -/- S/a/7 /7 22= C-5 & 5'- 5/n6' ~~~ 6 k~ (4.16) 55 Cone Fc ve/o/;v q /I:,-, 00.77 Figure 9. Stress and Enveloping Tangent Cones in Function Space Hence, Eq. (4.11) can be converted to where = (d- Y2+)c/s (4.16a)

-64 - and 7'=2 =2&( The projection of the stress vectors zg and 7r on the plane 2^= / 17 CO 9 where can be found by eliminating < and /v in 7 and * The result is / y )2 + / J2-/ (4.17) Imt5,/ '4/ 1 GThis is an ellipse with axes c 7 =; 9= m It is a circle if s = sin 9 Let this cone be called stress cone. A tangent plane d/ which is normal to Z' is given by the equation 57. Ad =o (4.18) The totality of all vectors /' as it has been shown, lie on a cone. Hence, the envelope of the tangent planes also forms a cone. The equations of this enveloping cone for which c( and i are varying parameters, may be derived from

-6 5 - aor( ' \ )c o d/31 U. hi7 = 0 which, on account of Eq 0 (4^11) become an Ud / + U* /2 = and, J (4o19) _ - /- -2 Z2 -V -,; a, 1,V 1.2 4 Ig a, /A/ respectively, Eqs, (4 19) have non-trivial solutions if and only if (/47'/) (6/ ~/ - (U& '//) 2 = o (4-, 20) which is the equation for the enveloping tangent cone as shown in Fig, 9, From Eqs. (4.19) / _ 6 /'// / _ 0 C6/^ * j7/,; ~/ d~ aL (4.21) Substituting it into (4o11) T'= [(/ / 2 /- 2(.//^'9//'2 + (/7. 27 where p is a proportionality constant such that (4+ 22) U. N 22= 0 611 IY2 -- 0//-/Ol o,,c/ e, /_d f (4,23) Now in order to determine the coefficients 0< and 43 in a similar way as shown by Eq. (4,8) put

-66 - By Eq. (4,ll):' = 2/" f " /z Z6' 22 (4.24) where y: Is known, given by Eqs, (3.3) and 4 / is on the enveloping tangent cone, Taking inner products of the vector given by (4.24) with //,,/"2 and /722 consecutively and considering the relations given in (4,13), (4,14b) and (423) 9 one obtains respectively 2 /3P2C05 2 _( 2 = ~2 wh2;ere/ =: > (4 25) 'cC0052<9 fi-3d-^2 where /:M = T /VbA (4, 25a) are known quantities, The three equations given by (4.25) serve to determine the three unknown parameters '(, f and / Solving for p from the second equation and substituting it into the other two equations, a pair of equations for o' and / are obtained as

-67 - > (4.26) To show explicitly that this way of determining vc and /6 is identical to Marguerre's method as used in Section 3, let the first two terms included in the potential energy / in Eq. (2.22) and defined by Eqs. (2.23) be converted into such form that X f Y^ e "" ^ / = J//^A,j '6 + 6 < -/ (/' - t_/Z-2 ' (427) = --- V* k. 7C8 The vanishing of the first integral can be shown by multiplying the equilibrium equation (2.6) by V, by integrating it over the region, and by applying Green's Theorem and boundary conditions to the integration. It vanishes because the equilibrium equation has been satisfied by it (=4z7 Jj) and W/ as shown by Eqs. (3.4) and (3.5). It can be shown by a similar way as was done for Eqs. (2.24) and (2.25) ijmh ' 4 b = / h 77 (4 28) The second equality in the above equation is based on (4.6a). The membrane strain energy term defined in Eqs. (2,23) can also be expressed in the vectorial form as

-68 - // h (429 U"n Z7 4 - / a14 2 - rT (4..29) Thus the potential energy obtained by combining Eqs. (4.28) and (4.29), written in present notation is v=.Z/2-(7 72. z f (4.30) Using the?-' given by Eqs. (4.11) and applying the relationships of inner products among the components (see Eqs. (4.13), (4.14a) and (4.14b)) and those between TY and ' (see Eqs. (4.25a)), to Eq. (4.30), one finds Marguerre' s method leads to Marguerre's method leads to a2C$-& -) —~ I ( $c0, 127-o~ ^-^ ^^^y^-^^^-^-/-^= (4.32) This set of equations is identical to those given by Eqs. (4.26). For the present problem it can be shown that,2 T= '//=2 -- Then the Eqs. (4.32) are reduced to be

o<'e g(259 CCNS2 - &>/J= 1 0 (4,32a) 2(2 S C )- - J These two equations differ from those given by Eqs. (3o44) only by a constanto As it has been discussed there9 that due to the independence of o( and,, there are three caseso Namely case (1) 13 O0 case (2) o' = 0, and case (3) o 09 0, Further, it also has been seen in Section 3 that the configuration in the last case is unstableO Hence hereafter the discussion will be confined to the first two cases only, The fact that r/~2 _,//" /V = / -. /v22 = o as mentioned before implies that the origin is on a plane8 normal to ///2 and containing 7- /" and /22 as shown in Fig, 10, Consequently, the solution of case (1) with - 0O is easily to be figured out from Figure 109 simply by setting o = 7=' / (4o33) It can also be seen from Eqs, (4o32a) that for C 0 but of =

.6 Y7"I " 7A ff - ' — 7/ )T. 1 H.x~ive (, z34 ICllt(1i0e:. _ /. Let r < ';be t '-,-.) den. oi;e0a. at..d t.he. potential ener.,gy when,,Jt ~, 0 hn..... applying:!e,:,u<0a~5ltt o th4 reul^3h4) to Eqo ( 31 ) =t 0 %::o,,,....s, 0he', ) i.g AIC d I ' // ".. increases ns tead of moving the coneS -',, on<e Xmay' consi der!he origin 0 be: l-ng oo:e,When / > i:f,o L*.a. sen f.o.i Table, ' e Xt s, t a.a/ /, By t'::,se &JYa.re proe:::ces one will.i:..d ithle l',1.... th 1 uti. on or 0 case ( 2) wi h:ss t., 0 o as / o /k, 7 A a............... s=^1 ncla sesFnsfead o h.o., m '. and: /: ase! ' / C,... 4 A s tIhon p. t..:: moe 4 l3 *e:'': of-... the cone s th s me 0 aod cc.> are o opps-i. e sldaes of thce h en&tra! l linee)

- /o//- /= k / Hence It occurs when A 1142 as known from Eq0 (3e 55a) There s another value gven there gthat when A 2 17 ' V1 is also equa to V2 However, this point is in a region where oz2 i.s in a negative- direction as shown in Fig. 10 o Hence is imaginary o This agrees with what was concluded thereo With these particular values of A' in mind and using a certain scale9 one is able to draw a straight line as the path, of the moving origin as shown in Figo 10 Let the variation of -the stress reactor ' (see Eqo (41ol)) 'be ~A^7 &.7Yt a. 2k (~24/22 (4 37) and that of the potential energy V (see Eqo (4.o30)) be zV= z(7 vte c')- ~71 Then one gets 4 P/- (r^^) ^ ^ ^ ^^t) (4o38) If only the terms with the second order variation of the amplitude coefficients o< and 3 are maintained using the tz and t' given by Eqs0 (4oll) and (4o37) respectively and also the relationships given by EqSo (4.13)

9 (4,l14) ) (4,14b) and (4 25'a) one assumres, (>. 2. ~,~:T -. /..,:C',J 2-f 2 -~:-', S-:;(a})('A );7, -32,..2,_...:....... (' 439) g; X A'- /2 be the coefficient of K(z1 g& f,( (2' and.- /-.i< respectively, then-the sufficient conditions for V to 2 / or l (44),, These e the criteria that shall be sed to test the stability of the conf guiratio is concerneds Note thor at thee bae ent om those give by Eqse (3I-8) which xpressed in present notation, derived fromn Eq, (4!31) are * ~,';yj}*' 2._,;S 2 4, K,2.~:;~ ~

"73 - /=?2 (2 $ s2-,'C,2z -'62 (4.42) It can readily be shown that the stability criterion according to Eq. (4,41) is more severe than the one based on Eqs. (4o42); in other words? stability according to the former implies also stability according to the latter, although the inverse is not true. Moreover, if Eqo (4.41) is used as a stability criterion it is easy to demonstrate that an absolute minimum of potential energy is attained, that is, that any other configuration, not necessarily in the neighborhood of the one being considered, exhibits a higher level of potential energy even if terms of order higher than the second are included, The term "absolute stability" is therefore applied to Eqo (4E41) although the latter represents only sufficiencys but not necessity conditions. Now for the present problem, F12 o Case (3) is unstable according to the criterion based on (4~42) or its equivalent (3,48). Thus it will be unstable by (4o41)e However the absolute stability of cases (1) and (2) needs to be examinedo For case (1) with a3 = 0 and o( F,/ seen from Eqe (434), one has V=co o V =, osZ - z (4.43) V o o J

< 74 -A comparison of th -e prot-nt-tia. eerry, g vt b,." E. (3n'1:42) with t gi en by qS i( 3, l )ie r el. i on 1' i p s between, a'nd 4, ',- aniven br y r Eqso (4 iol2) and.:.- OJ Ea:$dl of t-he c ria(.i),.. fiO nds t: fns the follow*s.... 1 i4ng I elatiosips a/ o 'g th- ostants 4/I, CC'~, 52 -2 -~', Taking the -a:.r. i a ales of `he a 'sta nt s fom... r c....- nTab le X, the r es'u.:lJt 'is (//,2^^34^X1 A^ /^A'^^^: (4 )LIf T in th n"L i 5 /-'' *.. f T.....fe5 Thi.s means the co iur to of Ease (. 'ir.a1)o o f c cases (.with 0: starts a~i.t 7- 1 )19 o.ne! can co -al,, ude th.at tbs,nf...gra tin xists amd 'is absouteo y sta'ble only. in te g tof OL hs o"S c a(se (2) \a 0<, 0 and,. / (t ee Eq i. ( 36 )f t) h -' coe ff iciet i rn t s d emfi ned "by, Es" ( 0 1) h s Inu a s 1Sm wi m'f e

-75 -= \COG C 29 -5 > - ( - o) -( /V -X2 I4.46) - -/. 05 4 -. 2oo00 0 Vqf-O V f = o Thus the antisymmetrical mode given by case (2) is unstable in the whole postbuckling domain according to vhe present criterion of absolute stability. To interpret the foregoing results geometrically, it would be convenient to give a name for every point as shown in Fig, 10 on the. path line. The line starts from point a, where A = 1.19. When it is between a and b, oi, is positive; while o2 measured in the opposite direction, is negative. Since,32=// /, the negative o32 means 36 is imaginary. Hence there is no possibility of having the antisymmetrical mode of buckling exist in this region. After it passes the tangent surface or the tangent line in Fig. 10, marked by o(= o, o/, and Of2 both are positive, that is both o< and $ exist. However, in the region between b and d, o/, is always greater than 0o2. In view of Eqs. (4.35) and (4.36), this fact means in this region, V1 < V2. From the point of view that a configuration with lower potential energy is more stable,

-76 -hence the symmetrical mode of buckling is maintained up to the point d. But this mode itself is absolutely stable only up to the point c. Right on this point, all the secondary variations given by (4.43) vanish. Then this is a neutral state. From this diagram when A reaches 6.45, the point f/ coincides with P and which may be considered as a singularity. Hence before the moving origin hits the extension line of /V, the configuration with = 0 is stable. This interpretation may also be applied to the case (2) with ( = 0. If the A - path line had gone through the extended stress cone, then beyond the extension line of /2/, in that region /oj// </o2z/ hence V2 < V1, the antisymmetrical mode would be stable if it stayed out of the cone. However, for the present case, the A -path will never pass through the other side of the cone. Therefore it is unstable as it is indicated by Eqs. (4.46).

0- a=O EXTENDED STRESS N A' I / j,STRESS CONE d (11.42) l __ \_ 28= 36~54' ORIGIN\ I b(2.98) \ N22 a V: Q Figure 10. Result of the Buckling Problem of the Square Plate on a Function Space Diagram

5. DISCUSSION AND CONCLUSION A difficulty arises when one intends to make a comparison of the present work with others. In most of the other works, the edges, paralleled to the thrust were allowed to move in their normal membrane direction. However, in this study the edges are immovable in this direction. Only the case with zero displacement along these edges in that direction was given by Marguerre.(2) But his solution for this case included the symmetric primary buckling mode only. No such case was investigated in his "refined solution". However, since the results are given in dimensionless quantities, it is still possible to make a rough comparison. For instance, the bifurcation point of two modes on load (P) versus shortening (A) curves according to Stein's work occurs at - about 2, A about 3. But Alexeev claimed for P6 same type of plate, it would occur for A >/o (K = 40 in his notation. No -~ value was given). The present result seen from Fig. 7 it is at - = 3.97, A= 6.22. The experiments done by Yamaki(l6) with four different boundary conditions in the normal to the plane of the plate direction show that up to = 3 no change of buckling patterns were observed. This of course only tells that patterns were observed. This, of course, only tells that -78 -

-79 -the present result is reasonable and the experimental results favor it. The change of buckling patterns, in view of the present result would occur when V1 = V2 at A = 11.42. This value is much greater than that at the bifurcation point. Hence, if there is a change of buckling patterns, the external thrust will have a drop (shown in Fig. 7). This phenomenon has been observed in experimental works. () It has been shown in the last section that the two methods used in solving the buckling problem of the square plate lead to the same results in the determination of the amplitude coefficients, The conclusions about the stability of the two modes, based on different criteria agree with each other only for A < 6.00. From there on different conclusions are reached. From the point of view of the absolute stability, the symmetrical mode is stable only for A < 6.45. Because of the singularity of the point P and the instability of the two modes for AX 6.45, no behavior of the buckling of the plate can be told in this region, A further study along this line of approach is desirable. From the point of view of the relative stability, a transition from the symmetric mode to the unsymmetric one may occur at a relatively large value of \- (= 11.42). This value is far from that an usual experiment can reach. To corroborate the analytic prediction, a special (say,

.80 -for a plate of very thin and of high strength material) test program is highly desirable, The solution obtained in this study is an approximate one, However, it shows some behavior of the buckled plate that has not been shown by other analytical works, Moreover, while quantitative corroboration is still lacking, the results show adequate qualitative agreement with available experimental data.

PART IC. PLATES TITH TI.O ' ' FRE E DGES I INTRODUCI TON!It has been seen in the.last -:)a rt that there is a consi.de"rablec strength in the postbuckl% ng domain of t he plate Wit1. all edsges simpliy supp orted (see ig 7)Q and th e intens.it y of the normal stress in the loading d ire tion I as -the tendency of shifting tcoward the two edges paral.lel. to the th!.aust (see Figo 6)F I.n the imS t almost &al thire buckling strengthl is developed along those edgeso Tbhe.re- is even tension produced in th le interior portiono On tlIe cont.ra:ry for plates 7w.ith tw;o free edges w'hich are parall.el to the thrust".tl.e bucklin.ig stren"lgtTh is limited to th1e critical. bu.ckling strength- Euler load However9 f the shortening, instead of force 9 is prescribed9 the d.eflection surf1 ace is 'primarily a cy"lind.r ical type wi.lth a generlator perpendicular to thel free edrges, Bu t it will sbe seens l.ater tohatL the boni da ry conditions aloong the free edges can nrot be fully satisfied bNy thIe 0 cyl nd-rical. deflecticon s urf aceo A "boun)lldary layer'" al on.g each of 'these edges is developed so that the rema.ini ng bo'unda.ry condition is sat isied.. Ctonsequently a nmembranest str ess field will be developed and a -tensile n.orrmai str:ess may be prod.uced along thie trims of the free edges Nevertheless, the total buckl1insg stre ngth of the pl ate remai. ns '-)ou-tnded.o ~$1,

-82 -In this part, a general discussion on the postbuckling behavior of plates with two free edges and two loaded edges arbitrarily supported is given. An example of the two loaded edges simply supported follows.

2o FORMULATION AND GENERAL SOLUTIONS OF BOUNDARY LAYER EQUATIONS Consider an isotropic and elastic plate of thickness h with a set of reference axes x1 and x2 as shown in Fig, 11o It has two free edges at x2 - 0 and 2b, It is arbitrarily supported at x1 0 and a, However the supports are statically stable. Each of the supported edges is attached to a rigid bar, But there is no shearing stress produced between the plate and the rigid bar, A relative shortening of the bars is prescribed, Before buckling, the middle plane of the plate coincides with xlX2-plane.!i' I Figure 11.... Plate.ith Two Free Ed-. Figure 1lo A Plate w.ith Two Free Edges The pair of basic' equations governing the behavior of large deflection of thin plates given by Karman(l) are equations of equilibrium i:n the lateral direction of the plate

74W- gZ /^ / /21<r z 2 k (2ol) and equation of compatibility V,= -- -( - /22,,) (2.2) in which / ( x, X 2), D and E are defined in Part I. /(x,,x2) is the Airyts stress function such that 522 ^ 4y%; ^^ ^ / — 2^ C (2,3) Along the free edges both moment and shearing force in the lateral direction of the plate vanish. Expressing them in terms of j,r one has (''o22 ' //1g == x2 on x 0.: 2, (2,4) /<^ -t (^'^ /L -=O 0 6<2 2 / Also there is nfeither normal nor shearling membrane stress. Hence, /// - 0,0r x -.O,zb (2. F) On the other two edges, the boundary conditions on r depend upon the conditions of supports. However each end supplies two. Totally there are four boundary conditions on (F availablee From statics one requires 216 hA '^2Q2/ -P J =. ~ (2,6) t=

-85 -and with no shearing stress /2z =- o n X -=, o (2.6a) First, one may assume that the deflection surface is a cylindric type of function of X,. If we let J= 4t W j1) (2 7) / A/ 2. F: Z - 2 (2.8) where k is the amplitude coefficient, then Eq. (2.1) becomes 4/// D — // (2.9) and its solution is W ( J)== i,y /Jc 0-i-, -f z i5C^5 U &'%16 *(2.10) The three arbitrary constants B, C, E, and the eigenvalue N may be determined by the four conditions at = 0 and a. The Eq. (2.2) is satisfied identically. However, the reduced boundary condition from conditions (7.4) which is i //=0 or x=- o (2*11) cannot be satisfied by functions (2,10). Applying a technique similar to that used in connection with boundary

-86 -layer phenomena in fluid mechanics(23) to the present problem let (2.12) Substituting these two equations into Eqs. (2.1) and (2.2) and on account of Eq. (2.9) they result in (2.13) - V 4 = -/ /. t / The boundary conditions on u and a are q22 +^,/ — / l on —.=i, 2X (2.14) from Eq. (2.5), and from conditions (2.6) ~// -— ~ / --- to C? 6^ =- G (2.15) The system of equations defined by Eqs. (2.13) to (2.15) is no better than the original set if one intends to find an exact solution of this system. However, a set of approximate equations —so-called boundary layer equations — can be derived by considering the additional functions u and 'i to be confined in a boundary layer along the free edges. This is accomplished by introducing a new variable

-87 - so that x2 -coordinate has been "blown up" by the parametric variable k. The set of Eqs. (2.13) now become,,, ', ' 77,. =x,, +,,:'.... t-X<-4t<,,+ I -tig/47r f kLt 'i7 4- r-~ ^ (2.16) The equations are divided by k2 and k is permitted to approach to infinity, a set of boundary layer equations for u and < are obtained. They are @2t7 - # 2/ = ~/ 0 J(2.17) in which -( (,x^) is known from Eq. (2.10). By a similar procedure, the boundary conditions (2.14) and (2.15) are reduced to the following forms: o ^ r=o (2.18) 64<r ~0 and, - =r (2.19)

-88 -respectively. The functions u, S and their derivatives shall vanish as - o Now let.ft r=,4 e ~ (2 20) where A, B and F are arbitrary functions of Put them into Eqs. (2.17), which become A/~D - B =,, o The necessary condition for A and B to be nontrivial is that the determinant formed by the coefficients of the above simultaneous equations must vanish. This results in where,g4 =2 v2 Hence in which (2.21) Thus /^g = e ~(4s Cc^-, 4 5/,,7')2 (2. 22) - e g 'c, ' /o: f D, 3/^ ^ )

- ' <(c2 ^s 2t'r< 2 s/o Since bboth -the fun.ct:i.ons shall vansh at r =a Through Eqs (2, 17) one will find ~. <:,9 3Y (2c-,23) -=-Z; / C1 2 - 2 Y ) 4 'boundary conditions (2,18) gives ( 1424) p The first of A, — ^^/ Y j~82( - V B ( 2 25a. and the second of the conditions yields A./4, - (2 25t) With the coeffiients given by the fo regoing three sets of equations the ftorm C: <: " —.tat-2,f=2tc 'si,i p t Intera'tg d f these equations twie with respect Cto r result.s i

"' -T r '4 '<c;"f C 1 $_ ~ '"q74-~ ~ ~ ~ ~ ~ ~ ~ ~~~~ ' OS The i.ntegratio; fi c.t.e'ons. fl and. f2.u-tt vanmi sh D as ause Isthe f untcton U wil vani ash when - " The a tisfactio'n oIf bo-ndarry ondi:.t ons (2 919) yie ds - )6 / _ " gec1~8;;"~~ — e f- 4i ~~~~t~~~~~"-' (20,28....*-* t-t, - " t; J": These linear functions. however contribute nothing to stressesu both of thlem may 'be neglected The final forms of the solutions (2.27) are 244, 4,i4~~'.~ - ~ -4 - 42 -- t4.,' >fC44~ ' 7T77~~~~~~~ C-'. ~<,~, 4-..; C 4444, '"G.? Y * 2 A7 9 r IIRL --- i i B ~-~~,C, I I I II c- f-.orC (\2,29) <>eff.ect+44.,, e:"f'k ct ~,..L' i' At SL ^ _ ^/ 4" an ld from Eqs ( ( 2,) ad (23 3 0 2i2 ha -..,' 4 - - c,.' *- >:,. c y,,, )f c.... r5/...7. (2 30) [44 "-'"<4 — I ~,

3 f0. EXA'MP LE Take a plate whi ch is simply supported along the two loaded edges as an example, The boundary conditions along these edges are that the lateral' deflection and moment vanisho They are satisfied by setting the constants B, C and E in-Eqe (2e 0) equal to zero and N = 2 o Hence wf (}= 5' 1'(3 I) SubstAtStution of the last equation into Eqo (2221.) gives <1z-i// (7// / _1 977//O (3e2) Then one has the soluitions for the additional functions u and 5 inmmediately by putting expressions (3.l) and (3o2) into Eqs, (2,29), They are Utd= 7W <eJ$7ays^- /7^7g (3o3) <f=_v]/e ~Y't/aoj^, 0 (30o4) These solutions are similar to what have been given by Ashweli' (2) Fung and Wittri ck 25) in 'the study of large deflection of the same plate by lateral bending^ =9-1 -

^92 -The solution given by Eq. (3,3) shows the additional deflection due to the anticlastic curvature i1n ' the Classic theory which now is confined along the two free edges with a maximum at 7=o, Since p =/2-/l- and considering O - 0..3 one has the maximt:n U(,2 x'J- ' '- h^ (3.5 The additional normal stress by Eq, (2o31) and after simplificationr is,;= — M k ^/ (;Z)2SZre'Xffg f —S/Kq^rl (3 6) which means that a 'thin boumdary layer of indefinitely increasing tensile stress is formed along the free edges, Although this result can be visualized with some effort, it is nevertheless considered startling in view of the overall compressive stress in the platee

APPEND IX SOLUTIONS OF THE DIFFERENTIAL EQUATIONS FOR THE MEMBRANE DISPLACEiENTS The pair of differential equations (3,27) governing the displacement components U 6 are expanded as for i = 1, 2 J/// _ (/-& 4'22 4 / l,/ = - (Aei ) for i = 29 2az ~ /~* <*" /-_;, f4n z - /7 2 6/ a # -"- (/i'i = -_- n where _:-: ce/' n [, (,c-^,. %y " +0,] On account of the functions ~T4 given by Eqso (3o Z = [^ (A! fd Y4Jc^J, ^, in which ot ()=ttf; /_A^/ 2rg / (A 2) 22) (A.3) (Ao3a) and A/k M&)- r r f^ i /_ 7 ) 4I- / i/ / 7 1 9' (-A- /3 (A o'3b)

However, MM and Nnm are of the same focrms as that of Min and Ntn; respectively by changing th.e su.Kbscripts m and n accordingly. This kind of notation will be applied to similar occasion s without further remarks+. And ~} 3 -^+ '-C W 7C 4z-t /f (A.4a) Mr J;}- Z - + z 7 (A.4b) Notice that when m n, 0. s the second funictiornal term in i'n i vanishes and that in J" is the function of { only, It might be noteworthy that the components of the membrane displacement field, introduced by Eqs, (3.26) and (3.29), characteri>zed by Eqs. (3,27) and (3,28) and boundary conditions (3,30) and (3,31) are symmetrical about m and n, Now for simplicity, but without loss of generality, let t"&- /e { H..:r2....X ~*. m, ' ' -^. 3f (A ^ 5

— 95 -in which Gmnr HEm n Imn and Jm are yet arbitrary furnctons of Due to the nonsymmetry of s, n one shall ha ve However e, — 6 a/ still hold and hence the computatioins will be carried out for i' 9 I an d L/ only; So for, Sbstuituting fun"ctions (Ao2) and (A.5 int.o the first of Eqse (A, ) and equa<ting the functional coefficients of 5/1 f7 and s, '1- 5^ separately, one gets two independent equations. Repeating this process with the second of Eqs, (A.i1) one obtains two more equations, After rearrangeme nt they are (A"6) and.- (AA7) 2f'* ^2, - ~.J.(/ —2f V/#/J \;/, ('ji ~~4) 22Ij I(/t~~-1/j,', Z7Ii These two pairs of equations characterize the functions of G mnr, Hm iA an d Jmni Note in the case of T " C 0O when m n, the second of Eqs. (A 7) becoomes J -, / Z. (A,8)

-96 - while the first of them does not exist. Repeating the procedures to U,0, let ow +- { 6[ (72J 9A 4, - ] - r, o-, r -- T"()::C az 6r-cs-Yrr,I~O~J (A.9) from Eqs. (3.28), one will have -i? < i + 4/-^ ^ - i / =-z-/ i' nr" | and ' 2 J</ ).z y - J 17 (/ ^ i ^ -where- - - {/S where (A.10) (A. 11) -n -f- I - /f '. -N/ K.. i) -i f t- 2 / / b- 4I/ < I / U, %~ I / (A.12) /r,( (2j =y g zf 7# / /n Ivan~~~~7 VIj (A.13)

In the tase of,0 the 'air of1 equai.aons (A. 11).s reduced to...... /....(A14) Because;all the condi.tions along O0 and ae C satisfied identi.ca.lly by the assumed forms of U/.? and.f}/'"t (see Eqs, (A5') and (A.9) respectively), the boundary oonditions (3,30) beeocime = Ot1%= ( and the conditions of continuity (3,31) now are, on / (A.l6) To solve Eqs0 (A,6) for G(r) and H( ) for instance, one may solve for their complementary solutions first" To do this, taking derivative with respect to tr t he homogeneous part of the first of Eqsc (A,6) one has inC- ~ t/iC/I Solving for (J from the homogeneo"us part of ith sec:ond of the equations and substituting it ani its third. de riva tiv

_98. <.'. Sinto the above equation, the result, after simplification, yields ~0w7-2 C ^^ ^ ^ =~>o7C (A 1l6) The solution of this equation is easily found as 'Zn 6 rn2 rfs4 / a (A.17) where 4,,,,, and are arbitrary constants, The super "t" on lHmn indicates it is the complementary solution of Hmn. On substitution of the so-found bA[ and hence the obtainable (/,4)" into the homogeneous part of the second of EqSe (A 6) and after integrating the resulted equation with respect to 7 once, one will get the complementary solution of Gmn as 0( {A Y 4 r~ ~(A,18) where \m /(A l8a) 5^^A = n(A2- f4 AA), 5*4 nr - The integration onstant in the above solution has been omitted, This is based on the consizderation of the

satisfaction of these solutions to the fJirst of Eqse (A.6) Proceeding to find out the particular sol.tions of Eqs. (A.6), one computes the functions Mnm and Kun first from Eqs 3 (Ao3a) and (A.4a) awith the functions fm given by Eq0 (3e22b). The results are in the following forms tL ( y + )C ^^ - (cZ k^g4 ^ ^ 7 7 V -e// r,:2 J,2I (A 19a) 7/t= 2 d>m {x2 (/J-1 4;; raj C7- = i2og on2 47 a, f= 2; gX+ 6 r ^m PN ^ -J^//^ ^,~n~l~MO~ j

-.100 -+ g + r /,v7?/4) {ffc -sF and (Ao 20) where —, z -jr24n t (; 7 z(A. 20a) ^/7P2 ^ < ^-i^'^^^~~~~~~a~~ /^=.'^^ ^^^ Pr4~aL~?~~/r~7;i ^w^ ^ ' ^~~~~~\~/s

Then let the particular sol-uticons of GQ and Hmnr:respectively 'be i, C^,sg 's~~^ - f:"es(;c'>A7 and. 7{p Op, " A Lii. ~j/ c. /h y s, (A 2 w'hee <C., " and,,' a:.'r: cons:;ant coefficients~, They a: 'be...iied sb 3t.,-i.g the b t.wo a ssumed func: tljon;s jf4'... aa:cI.:to h.e two e:: t ons (A, 6K...w.:th r- f' g~,v.. -q,, (A,.......

-102 -and (A.20) and equating the constant coefficients of each corresponding terms on each side of the equations and then solving for them from the resulted algebraic equations. For the interior part, following a similar approach, first one will get an equation parallel to Eq. (A.16) for the homogeneous solution of h/, from Eqs. (A.10). That is /H/ -24, -AL V' f = O (A.23) of which the solution is (A, 24) where j, / p. and /t are arbitrary constants. The homogeneous solution of its conjugate function obtained from the homogeneous part of equations (A.10) is (A.25) in which / -(:- ),,2 =?,Ia /~ (td *,2AFn) S = " (A.2 5a) // ^ - ^ (fw = p/, t;i

-103 -The particular solutions of c and,, can be obtained by the same way as that done for A and /^, Compute /An and ~-t first; assume two functions for $4 and w/ rhich are the same in forms as that of /,/, (or K,, ); then determine their coefficients through the differential equations. The results are: =( k^ + Ken f K A" f^ Je 7 (A 26) for m Z n; while m = n l(^ 3 r^re (A.26a) - Pf -/, S^, -# 4 9 /?/P are also in the above forms but with different coefficients. The eight arbitrary constants involved in the complementary solutions of Gmn, Hmn, Gmn and Hmn are determined by the eight conditions; four from the boundary conditions

-104 -(A.15) and the other four from the conditions of continuity (A.16). Following the same procedures as that did for,,, t//,y and,, //t, one can find the solutions for Z^n ~, J and i!,, J. * Except when m = n, -2, and Z, vanish and (A.27) which can be easily 'integrated. The final solutions for,_, and J, are in the same forms as that of,, and H,, respectively but change 4, to ^,. They have been given in Section 3. (See Eqs. 3.32b) and (3.32c)). Those for I and J~ are in the same form as that of G,. (See Eqs. (3.33b) and (3.33c)). The complete solutions of,, /4,, -~ Jot and 4C, /4, -a, jM of which the coefficients are computed after the constants given by Table I, are tabulated in Tables II to IV.

IB ILI OGRAPHY 1, von Kr.a:an, Th. "cs tls:7ei tsprobl:.e im Il;'as,, cinenbau, Enx: lo aedlie er.:.athema tischen Wis sens chaften, Vn7, 5, Pt., (1 ':1) Art. 27,. 349, 2. MIargue". e -.l:, '"arn:a WJJidth 'o th'e Plate i-n Com1r::r-ss 3 1n" I ACA TI.,. 1937) 3. Ti noshieo - S, TTe or " f.S'ilastic S tabilit lcGraw Hill pLe,: f::, C i1e c":3 n.3 -. 3 f -, 4.0 Levy, Sa ei —l. 7BneTl, 7t'.:.:1f R:-ectang:ilr Plates S;ith Large De l.ct i' -. t, rACA Ri.;"'", (i0 )i ^5'.C i'[ " "3. Cox, H. i:,'1.:c^; S.;.1 T '.~i. Plate i n:: Co -:l'ession", a AerI i S. —ir Coi:: 1yw^ t! ( 14, (1933) 6. Koiter, ',. l D,: nlee(dr:;.': '.rate bia. roote oversJ 'i 4i n - i -n -f. r c -*ji.: o;;. -n; 7 ~ *' or::o s v.? i llende inkle -..i.- (T:- P. Eff ective Width of F -. 1.:: ':s for 'V':.r'';s LongCitludl i -,nal 3dge Condition-. atc L:::ads Far 3i,' e Bluc klin: Lc:: ). National 2l3. r....;vaaut L.o.::O. o.riUj:: Ai 'ister:ia'^, Repot I- i —Zz> (: 3 3) 7. Yama:moto I.,.;':. Ko'ndo K, '"': kling and Fanilutre of Thin Rect a,<lar 1: ato.s Co: e,:,ssion ' Aeiro Res. Inst _. o l...... V... _.^... 1 o12 ( Vo l. 10, No. 1, A1'.-.'~ l r 3. 8. Friedrichs. Le o andl StoCker J J. "Buckling of the Circular:'l.ie:t::-yond t!e Ciritical Thrust',, AppI. IMech. Trans. i-31'E Vol. '4 1, l42) A. 7. See also A uthor S C`ur, sa e ol:._ A192. 9e Masur? E. F. "On t1Ie Analysis of Bu c!kled Plates" Proc. Third U. S. Nat, Coons. A.1. Mliech., June 1958 AS"iE 19H7 pp. 411-4177. 10. Yanowitch, I,, "Non-linear Buclling of Circular Elastic Plates" Crli omimlunication on ',r e and Aplied Ma' th Vol. IX (956), 7 p 6l.-672. 11. Alexeev S. A. "The Postcritical Behavior of Flexible Elastic Pl.:-tes" ApJlield n'athematics and Miechanise (U.S.S,R.) Vol XX, No, 6( 19 p7' p. -3-679. Translated by Tro J. H. Taylor. -105 -

-lo6 - 12, Stein, Manuel. "Behavior of Buckled Rectangular Plates", Proc. ASCE Journal of the Eng Meech. Div., April 1970, Part I, pp. 59-76. 13, Schuraan, L. and Back, G. "Strength of Rectangular Flat Plates under Edge Compression", NACA TR 356, (1936). 14. Ramberg, W., McPherson, Ao E., and Levy, S, "Experimental Study of Deformation and of Effective Width in Axially Loaded Sheet Stringer Panels", NACA TN 6849 (1939)0 15. Ojalvo, Mi and Hull, F. I "Effective Width of Thin Rectangular Plates", Proc. ASCE, Vol. 84, EM3 Mech. Divo P. 1718, July 1955. 0 -16. Yamaki, N. "Experiments on the Postbuckling Behavior of Square Plates Loaded in Edge Comparison", Journal of A~~!o Ilech. June 1961, pp. 238-244. 17. Lahde, K. and Wagner, H. "Experimental Studies of the Effective Width of Buckled Sheets", NACA TM 84, (1936). 18. von Karmnan, Th., Sechler, Ernest E. and Donnell, L. H. "The Strength of Thin Plates in Compression", ASME Trans., AMP14P=, Vol 54, No, 2, (1932), pp. 37. 19. Timoshenko, S. "Theory of Elastic Stability", 1st Edition (1936), IMcGraw Hill Book Co., ppo 302-314. 20, Hill, R. "On Uniqueness and Stability in the Theory of Finite Elastic Strain", J. 1Mech. Phy Sol, Vol, 5 pp, 229-241, 21. von Karman, Th., and Tsien, Hsue-Sheno "The Buckling of Spherical Shells by External Pressure', Jour. Aero. Sco., Volo 7, No0 2, Dec. 1939, pp. 43-50. 22. Tsien, Ho S. "A Theory for the Buckling of Thin Shells", Jour0. ero Sci. Volo 9, No, 10, Aug. 19429 ppo 373-384. 23. Schlichting, H, "2Boundar Laer Theory", MGraw-Hill, (1955). 24, Ashwell, D. Go "A Characteristic Type of Instability in the Large Deflections of Elastic Plates. I. Curved Rectangular Plates Bent about One Axis. II. Flat Square Plates Bent about All Edges", Proc.. Soc. A. 214 1952. 25. Fung, Y. C. and Wittrick, W. H. "A Boundary Layer Phenomena in the Large Deflection of Thin Plates", Qat. J. Mech. & AI. Iath. 8, Part 2, pp. 191-120, June 1955.

unmvcnoi i T ur MLbnmiaAm 3 9015 02829 976911111111 3 9015 02829 9769