: 2 JUL 1945

: JUL 2 1945 NON-LINEAR LARGE DEFLECTION BOUNDARY VALUE PROBLEMS OF RECTANGULAR PLATES Chi-Teh Wang B.S.M.E., National Chiao-Tung University, China 1940 M. Aero. E., Rensselaer Polytechnic Institute 1942 Sc.M. (Applied Math.), Brown University .i i 1943 Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF SCINCE From the Massachusetts Institute of Technology 1914 Signature of Author....................................... Department of Aeronautical Engineering, Oetober 9, 1944 Signature of Professor in Charge of Research lb00-AWWW"4'- Signature of Chairman of Department Committee on Graduate Students V . . .. r... LRA_ Cabridge, Massachusetts October 9, 1944 Professor George W,. Swett Secretary of the Faculty Massachusetts Institute of Technology Cambridge, Massachusetts Dear Sir: I hereby submit a thesis entitled, "Non-Linear Large Deflection Boundary Value Problems of Rectangular Plates", in partial fulfillment of the recquirement;for the degree of Doctor of Science in the Departmentof Aeronautical Engineering. Respectfully, Chi-Teh Wang 27 1 %· ii:; ,. ,· TABLE OF CONTENTS .·c; h .,i " Acknowledgements r·. i Abstract Nomenclature -i' i .i Chapter I. r Introduction Chapter II. ............... .. Review of Previous 'Wfork ........... . ........... . * * l . ·II-l-II-3 ·. II-3-II-4 - i 3. The Finite Difference Solutions ........... . z 4. The Fourier Series Solutions Chapter III. 1. ........ Bending of Thin Sheet Plates . . . . . II-4-II-7 .II-7-II-14 . . The Governing Differential Equations . II-1-II-4 1. Introduction ................ . * 2. The Energy Theory I--I-6 . ... .. . . . III-1-III-18 .III-1-III-5 .s 2. Differential Equation of the Deflection Surface . 3. Compatability Equation III-5-III-9 . .III-9-III-i 4 ............... 4. Summary of Equations for the Deformation of Thin Plat,es .III-15-III-16 5. Non-DimensionalForm of The Equations. . . . . Chapter IV. Formulation of the BoundaryConditions . . . . r . III-16-III-18 . IV-1-IV-8 ;i ·I 1. General Discussion of Boundary Conditions ...... . IV-1-IV-3 2. . IV-4-IV-8 nalytic;l Expressions of the Boundary Conditions . r,) Chapter V. The Finite Differences Equations of the Boundsay Value Problems ............... .V-1-V-L3 ......... 1. The Calcalus of Finite Differences 2. Tie Finite Differences Exression Ce.ses 9 Pr ·a 3. . of Two-Dimensional . . . . . . . .V-3-V-5 Conversion of the Governing Partial Differential u; Equations into Finite Differences Equations ..... ; i;· Ii r-i-, :- "·I· ;: :i .V-5-V-6 4. Finite Differences thpressions of the Boundary Conditions . . . . . . . . . . . . . . . V-7-V-9 5. The Boundary Value Problem in Terms of Finite Differences Exressions v v. ......... V-9-V-lS Chapter VI. The MJIethod of Successive Approximations . I. Outline of the 2. Crout's Method . . .. . 3 erivaton of the Crout'sMethod . 4* The lethod of Successive Sample Eoaations VI-3-Vi-8 . . ... rt ...l ns .p . VI-8-VI-11 VI-11-VI-ll . uccessive Aproximations: Calcul.tions 1. The Finite v-l-vI-3 . . lrethodfor Solving System of Linear hac-ter VII The Method.of VI-1-VI-13 . . . . . ifferences VII-l-VII-33 o±litions of the rrl.P1 .... action Teory 2, The Large Deflections Problem n=2 3. The Lar . nDeflecions Problem, n=3 . . . CY3t~ErVIII. The Rela xation . . . ethod . . w . . . VI]-1-VTI-5 VI-5-VTT-1 i. The iethod ,plined . . . . . . . . . , VIEl-Tr 2. The Small Deflection ?roe Thn Deflection L-rge . Rectanguar Pltes. ' Pr oblea of $ the Rect;angila. ........... VITI- Cha "l'RC t a1nd Co IX. Ter Concusiwon s esea.rc'l 4, TabLes a-.n! d Curve DiOgrail -hy . . . .. .s...... IX-4-IX-5 *. 3. Su-,ogestions for Furthser References TX-l-X-6 .. r.r*+......I .**..a...IX--IYX-4 of the results 1,Dincussion 2. ,clusn' VTTTII s . . . . . . . .. .. . *. . . i . . . . . . . . . X-7.-I.-16 ._ ._P . . .- -- ACKNOiiTEDG ENTS The author wishes to e-:presshis indebtedness to Professor J. S. Newell, who suggested the problem and.under whose supervision this thesis was written. The author wa.salso very fortunate to work this thesis under the frequent counsel of Professor PRichardvon Mises, of Harvard University. To both Professor Newell and Professor von Mises the author is grateful for their suggestions, enthusiastic encouragement and patient guidance throughout the investigation. Yhile working on this problem, the author has had the opportunity of discussing it with Professor R. V. Southwell, of Oxford University, England, with Professor H. W. of Harvard University, and mmons and Dr. A. Vazsonyi ith Professor Eric Reissner, of M.I.T. For all their inspiring discussions the author is extremely thankful. Thanks are also extended to Professor Shatswell Ober and Mr. R. W. Gras for lending references and providing the computing facilities. To Professor P. H. Smith, the author wishes to acknowledge his kind interest and encouragement; and to Professor I. S. Sokolnikoff, of the University of Wisconsin, Professor S. Timoshenko, of Stanford University, Professor W. Pr;ager,of Brown University, the author wishes to express his appreciation for the early training which they gave him in the theory of elasticity while a student in their course. a:t Bro.!rn University. ABSTRACT This thesis presents a theoretical analysis of an initially flat, rectangular plate with large deflections under either normal pressure or combined normal pressure and side thrust. As small deflections of a flat plate are governed by a single linear equation, large deflections introduce nonlinear terms into the conditions of equilibrium and are governed by two fourth-order second-degree partial differential equations. These so-called von Karman's equations are studied in this thesis by the finite differences approximation. The differences equations are solved by two methods, namely, the method of successive approximations and the relaxation method. Neither of these methods is new, but their applications to non-linear problems require new techniques. The problem of a uniformly loaded square plate with boundary conditions which approximate the riveted sheet-stringer panels is solved by the method of successive approximations. The theoretical center deflections show good agreement with the recent experimental results obtained at Calofornia Institute of Technology when the deflections are of the order of the plate thickness. This perhaps suggests the range in which these von Karman's equations are to be applied. Other problems-of thin plates with large deflections are discussed from te point of view of an aeronautical engineer. The boundary conditions which approximate the various cases are formulated and the methods for solving these problems are outlined. Since the method presented in this thesis is general, it may be applied to solve the bending and the combined bending and buckling problems with practically the results any boundary conditions, and may be obtained to any degree of accuracy reauired. Furthermore, the sme method may be applied to solve the membrane theory of the plate which applies whenthe deflection is very large in comparisonwith the thickness of the plate. I, NOMENCLATURE a, b = length and width of the plate, respectively h = thickness of the plate x, = coordinates of a point in the plate y, u, v = horizontal displacements in directions x, y of points in the middle surface; their nondimensional forms are ua/h 2, va/h2 , respectively = deflection of middle w surface out of its initial plane; its non-dimensional for is w/h = normal load on plate per unit area; its non- p dimensional form is pa4 /Eh4 = elastic constants, Youngts modulus and Poisson's ratio = Eh3 /12(1-), D V 2 + 2 cx 2 4 flexural rigidity of the plate 2 ay 2 C34 + Y4 4 sy )x2 = membrane stresses in middle surface; dimensional forms are and I Y xa./Eha, x' la2 /h 2 their nony'a 2 / , 2 ., respectively #1 = extreme-fiber bending and shearing stresses; their non-dimensional forms are y"a2/ 2 , and r'a 2 /Eh2 , x"a 2 /1h2 , respectively x' y, xy = membrane strains in middle surface; their non- dimensional formsare Wxy a2/h2, it · y? I x'a 2 /h 2, respectively. = extreme-fiber bending and shearing strains; their non-dimensional forms are Y F 5'a 2 /h 2 , and x a2/h2, 22 2/h2 y" a2h , and Y a/h2, respectively F = stress function; its non-dimensional form is F/E2 i, 2 ,... en = the difference of first, second, .... , and nrth order, respectively A x,Ly = the difference of first order in x and y directions, respectively : q .i .r t I i * .i _I CHAPTER I INTRODUCTION The classical theory of the bending of a thin elastic plate expresses the relation between-the transverse deflection of the middle surface of the plate and the lateral loading of intensity p by the equation (1) DVw where D I 3= = p is the flexural rigidity of the plate. It is known that the theory is restricted in application, for on the one hand its basic assumptions can be questioned unless the plate is thin, and on the other hand it neglects an effect which must be appreciable when w has values comparable with the thickness. This is the "membrane effect" of curvature, whereby tension or compression in the middle surface tends -to oppose or to reinforce p. The effect is negligible when w is very small, provided that no stresses act initially in the plane of the middle surface; but even so it operates when w is small becausestretchingthe middlesur- face is a necessary consequence of the transverse deflection. Whenthe deflection gets larger and larger, the "membrane effect" becomes more and more prominent until for very large w the "membraneeffect" is predominant (2), (5), (6) while the bending stiffness is comparatively negligible. As small transverse displacements of a flat elastic plate are governed by a single linear equation, large displacements entail stretching of the middle surface and consequent tensions which, interacting with the curvatures, introduce non-linear terms into the conditions of equilibrium and -so make those equations no longer independent. (3) The large-deflection theory of flat plates is due to A. Foppl and the second order terms were formulated by Th. on CrmAn in 1910. (4) 1-2 The amended (large-deflection) equations have been solved, however, only (5)-(21) in a few cases and then with considerable difficulty. Essentially there are three problems concerning flat plates with large deflections. 1. They ares The Bending Problems, where the flat plates are sub- Jected to lateral loading perpendicular to the plane of the plates but to no side thrust which is applied in the plane of the plates. 2. The Buckling Problems, where the plates are subjected to side thrust applied in the plane of the plates, but to no lateral loading. 3. The Combined Bending and Buckling Problems, where the plates are subjected to both lateral loading and side thrust. In the case of metal airplanes, where weight is of primary importance, the metal sheets used must be thin and the deflections of the plates are usually large in comparisonwith their thickness. To obtain the design formulae or charts for proportioning such plates the large deflection theory must be used. The bending problem is important in the design of seaplanes. Seaplanes are subjected to a severe impact during landing and take-off, especially on rough water. The impact must be withstood first bottom plating and then by a system of transverse and longitudinal bers, to which the bottom plating is attached, before it the body of the structure. by the is mem- carried into The bottom should be strong enough not to dish in or "washboard"permanently under these impact pressures. Such "washboarding"is undesirable, both because of the increased friction I-3 between the float bottom and the water and because of the increased aerodynamical drag in flight. The bottom plating of seaplanes is, as a rule, subdivided into a large numberof nearly rectangular areas by the transverse and longitudinal supporting ribs. Each of these areas will behave subtantially like a rectangular plate under normal pressure. Bending of rectangular flat plates may therefore be used to study the "washboarding"of seaplane bottoms, provided the boundary conditions at the edges can be formulated ust as in the seaplane. The buckling problem is important in determining the strength of sheet-stringerpanels in end-compression. The use of stiffened to carry compressive loads is increasingly popular airplane wings in sheet the box beams for and in other types of monocoque construction. Inasmuch as the sheets used as aircraft structural elements are generally quite thin, the buckling stresses of these sheet elements is therefore confronted with the probare necessarily low. The designer lem of using sheet metal in the buckled or wave state, end of determining the stress distribution and allowable stresses in such buckled plates. The combined bending and buckling problem has become a problem of importance with the increasing use of stressed-skin type wings and the pressurized fuselage construction for high altitude flying. During flying the wing is subjected to a pressure difference between the two sides which gives the lift. The normal pressure acts directly on the sheet covering and is then distributed to ribs and spars. At the same time the sheet panels are also subjected to a side thrust due to bending of the wing. In an airplane of pressurized fuselage construction an attempt is made to keep the pressure inside the cabin at a comfortable level for the passengers, no matter howhigh the ship may be flying. 1-4 Thus, there is a pressure differential across the fuselage skin with an internal pressure higher than that outside. The fuselage skin is usually subdivided into a numberof rectangular curved panels by longitudinal stringers and rings. These panels are subjected to the pressure difference and also side thrust resulting from bending of the fuselage. As (57) pointed out by Niles and Newell, the strength of curved sheet-stringer panels can be determined approximately from the flat sheet-stringer panels. The problem is then essentially that of determining the strength of flat plates under combinedlateral loading and side thrust. (21) Levyhas shownthat the effective width of a square plate with simply-supported edgesdecreases vrith the addition of lateral pressure, and the reduction is appreciable for pa4 /Eh4 > 2.25. Therefore, a panel is unsafe in strength if the design is based upon side thrust only and the study of combined loading is of great significance. (22)-(54) A great number of authors have studied the buckling problems and considerable experimental work has been carried out. As a result, design formulae are available and seem to be of good accuracy for practical usage. The bending problems, however, have been studied only (5)-(21) by a few persons, conclusion, ( (56),( 65) are too and test results few to reach any The combined bending and buckling problem has been studied, (21) so far as the author knows, only in one case and yet the results are incomplete. Among the solutions of the large deflection problems of rectangular plates under bending or combined bending and compression, Levy's solutions are the only ones of the theoretically exact nature. However,his solutions are limited to a few boundary conditions and the numerical results are tremendously laborious to carry out. I-5 The purpose of this thesis is to investigate whether it is possible to develop a simpler and yet sufficiently accurate method to solve the bending and the combined bending and buckling problems for engineering uses. It is quite fortunate that the author has been able to develop such a method by means of the finite differences approximation. Solving the partial differential equations by finite differences equations is nothing new. However,to solve the resulting differences equations is still a problem. In the case of linear differences equa(58) tions, solutions by successive approximation are always convergent the work is only tedious. and Besides, one may apply Southwell's Relaxation Methodwithout toomuchdifficulty. But,to solve the non-linear differences equations, the successive approximation methodcannotbe always applied because it does not always give a convergent solution. The Re- laxation Method, since it is nothing but intelligent guessing, can be ap(18) plied in a few cases and then with great difficulties. A study of the finite differences expressions of the large deflection theory reveals that although the successive approximation is actually divergent, the meanof certain quantities of two consecutive cycles is convergent. The system of non-linear difference equations can (62) then be solved by successive approximation using Crout's method to solve a system of linear simultaneous equations with rapid convergence. A square plate under uniform normal pressure with boundary conditions approximatingthe riveted sheet-stringer panel is studied by this method as an illustration. Non-dimensional deflections and stresses given under various normal pressures. (21) are The results are consistent with LBevy's approximate numerical solution for ideal simply-supported plates .I I-6 (17) and Wayts approximate solution for ideal clamped edges, and the center (65) deflections check closely with the test results by Head and Sechler for pa4 /Eh4 ratio up to 120. The deviation above pa4 /Eh4 = 120 is probably due to the approximate assumptions used in the derivation of the governing differential equations. While the method is general, it may be applied to solve the problems of rectangular plates of any length-width ratio with various boundary conditions under either normal pressure or combined normal pressure and side thrust. I 4 I IL CHAPTER II REVIEW OF PREVIOUS WORK 1. Introduction The large deflection theory of flat plates is due to A Froppl, and the difficulty of solving non-linear equations has been noted by '(4) The earliest Th. von Krman. work to deal with these differential (5),(6) tions is perhaps by H. Hencky equa- who devised an approximate method to solve the cases of circular and square plates when the deflection is very large and then the bending stiffness is negligible. Following the same (7) procedure, Kaiser solved the case of a simply-supportecdplate with zero edge compression under lateral loading. His theoretical results have checked closely with those from his experiment. In the case of circular plates with large deflections, because of the radial symmetry, the two governing partial differential equations which contain the linear biharmonic differential operator and quadratic terms in the second derivatives can be reduced to a pair of ordinary non-linear differential equations, each of the second order. For both the bending and buckling problems, "exact" solutions are available. (8) The bending problem has been solved approximately by Nadai (9) (10) and Timoshenko, and exactly by Way when the plate is under lateral pressure and edge moment. Way presented his solution in terms of power series for a rather large range of applied load. (11) The buckling problem has been solved by Federhofer and Friedrichs (12) (13) () and Stoker. Federhofer gave the solution for both simply- supported and clamped edges, which yields accurate results up to N about 1.25, where N is the ratio of the ressure applied at the edge to the II-2 lowest critical or Euler's pressure at which the buckling just begins. Friedrichs and Stoker gave a complete solution for the simply-supported circular plate for N up to infinity. three methods which three methods perturbation is To cover this range they employ are different, but which interlock. suitable for Each of the a particular range of values of N; namely, method for low N, power series method for intermediate N, andthe asymptotic solution for N approaching infinity. There is no solution, however, for the case of circular plates under combined lateral pressure and edge thrust. Theexact solution for a thin infinitely long rectangular (15) strip with clamped or simply-supported edges was obtained by Boobnoff (16) (17) (18) and the other cases were discussed by Prescott, Way, Green and Southwell, (19), (21) (20) Levy, Levy and Greenman. Prescott gives an approximate solution for the simply-supported plate with no edge displacement. While Prescott's approximation is rather rough, Way presented a better approximate solution by Ritz' Energy Method (7) Kaiser transformed for the clamped plates. the differential equations into finite differences equations and solved them by the cut and trial method. Green and Southwell extended the finite differences study into finer divisions and solved the differences equations by a technique based on relaxation method. (21) Levy gives a general solution for the simply-supported plates and numerical solutions are given for the square and rectangular plates with a width-span ratio of 3 to 1, under sonic combined lateral and side (19), loading conditions. (2o) Levy and Greenman also extended the solution for simply-supported edges to clamped edges. However, their conditions are limited to the case where the edge supports are assumed to clamp the plate rigidly against rotations and displacements normal to the edge, but L ' II-3 to permit displacements parallel to the edge. They presented a numerical solution for square and rectangular plates with a width-span ratio of 3 to 1, under lateral pressure. In short, the problem of rectangular plates with large deflections has been solved by three methods; namely, the energy method, the finite difference equations method, and by means of Fourier series. They are briefly outlined as follows: 2. The Enery Method (17) The method of attack used by Way is the Ritz Energy Method. Expressions are assumed for the three displacements in the form of algebraic polynomials satisfying the boundary conditions, then by minimizing the energy with respect to the coefficients, a system of simultaneous equations is obtained to solve for these constants. The energy expression for the plates with large deflection is vSelw)2 - q w + 6 V = //J + (wx2 + L + y2) + 2, ux 2 + (uyvy 2 ywy2 + VY 2 vy - ++ ux ) (Uy2 + 2 UyVx + Vx2 + 2UywxWy+ 2vxWxwy)] dxdy... (2-1) where u, and v, w are the dimensionless horizontal and vertical displacements q = pa4 /16Dh, and the subscripts indicate partial differentiation. For u, v, and w, to satisfy the boundary conditions for clamped edges, Way assumes (Figure II-I): u = (l-x 2 )(/ v = (1-x 2 ) (f 2 -y 2 )x(b 00 + 2 _ 2 )y(c 0 2 2 2 2 +CO 2 Y +c2 0 x +c22 x y ) w = (1-x) 2 ( /-y where f b/a 2 )2 2 2 +b 2 2 +bo2y2+bx 2y 2 2x (ao +aO2 y2 + a20 x2 ) ) ........... (2-2) A6 j + - 0 -5 | -- -. T-PE -~~~~~~~~~~~~~~~~~~~~ - + I - 6 X Z s. - / II-4 Minimizing V with respect to aij, bij and cij, eleven simultaneous equations are obtained, corresponding to the eleven constants. They are: 'V = O; =. 2±. a00 v = ; aboo a v= c cOo = O; dvV= =o(2-3) dv V =. - a02 v a = ; a v = ; a v =0 ab 0 2 ; (2-3) a20 V =; C 02 b20 a v =; 0c20 (2-4) ab22 a v =0 ( c22 (2-5) These equations are not linear in the constants. The first three, Eq. (2-3), will contain terms of the third degree in the a's. The equations (2-4) and (2-5) are linear in the b's and c's and quadratic in the a's. Way solved equations (2-4) and (2-5) for b's and c's in terms of a's and then substituted these expressions in Eqs. (2-3). There then are left three equations of third degree involving the a's alone. These were solved by Way by successive approximations. Way gives the numerical solutions for the cases 2 for / = 1, 1.5 and = .3 up to q = 210. Since he assumed the displacements by finite terms of algebraic polynomials, the solution is essentially an approximate one. By comparing with Boobnoff's exact solution for infinite plate, Way estimated that the error of his solution for 3. = 2 is about 10% on the conservative side. The Finite Difference Solutions 3. Kaiser writes the non-dimensionized KArman' s equations in the form of five expressions, as follows: II-5 2 ( 2 w )2 _ 2 w 2 w -K Ax ay vx y2 72F = S 2-- A x - PG 2 2F a2w xay xay 1 - 2F c32w- aday F 2 ax w_ (2-6) V 2M P -P V2w = M and then transforms them into finite differences equations. His procedure is to assume w's at all the points and then solve for S's. F's. M's and ws. If the calculated w's do not check with the assumed ones, he assumes a new set of ws and repeats the process. In i i doing so, the work is tedious and elaborate. In fact, if one does it j by successive approximation, the process is actually divergent as will i be pointed out later. Kaiser solved the simply-supported square plate with zero edge i compression under a uniform lateral pressure of pa4 /Eh4 = 118.72. His numerical solutions checked with his experimental results with good aci i curacy. (18) Southwell and Green solved four examples of the problem with a technique based on the former's "Relaxation Method". The fundamental requirements to work with the relaxation technique are a simple finite differences pattern of the variables and a simple expression of the boundary conditions.- In doing so, they expressed the differential equations in terms of the displacements u, v and w, which then gave simple boundary conditions. Instead of using "exact" relaxation patterns they worked with the patterns which are given by the linear terms of the dif- II-6 ferential eqLuations, the non-linear terms being combined w,iththe "residue" and making corrections from time to time. v+I 3 r 2wu+ 47 w=+ D h2 L 72x x2 / 3x ' ay2 av a'x + y y w ) ax ay a (au +fav+ 1-- 7+ x :,x 0Y yx y /x y w + aw2 w2+ aZy 1-_ w 2 + (aw)2 a)xy ax 2 _+a = ax - (Zu d5Y aFx + av)+ 1-2/ 5y 172W = 1-:: +* ay+M aw 2 + (J )2 + a Y O l+/L ay It is readily seen that in order to obtain a simple expression for boundary conditions, not only the number of the partial differential equations is increased from two to three, but the form of their terms is more complicated and the number of terms is increased. It is again tedious and very difficult to operate. Equations (2-7), being conditions of equilibrium, could have been derived by minimizing the total potential energy V, which is given by the expression L2 V = II + 12 + 13 h2 D . ..................(2-8) (2-7) !I-7 where I1 _ Jf (V 2 w) = 3 + 2 Jfe ( cO - cJfw I3 be-ing the lateral x dy, 2x+ .3 and 2 x + 2e e + Iz ee) dx dy, dx dy, loading. The relaxation technique consists of assuming a set of answers first and then changing them accordin boundary conditions. convergent. attern and to the relaxation In performing the operation they found the process To obtain a more rapid convergence, they multiplied the -ziven values of w by k, and substituted L2 V --i V D 4 2 = k I,!I++ k k them n enerzr ex-oressionto obtain + c·k· 2+ ~kI3 ......................... ( 2-9 ) whicn was then minimized with respect to k, namely, to put = 0, k to give 2k 2k1 ++ 43 4I2 From the third -0 - T3 O= order Eq. (2-10), 0(2-10) .............................. (2-10) k can be solved and would ive a set of w values which are closer to the true values. 4. The Fourier Series Levy(9)'(21) Solutions. and Levy and Greerman(20 obtained general solutions of the rectangular plates under combined bending and side thrust with large deflections by means of Fourier Series. solution is outlined as follows, (Fig. T-2. (a) Sismply-supportedrectangular plates. The approach of the ,I I· il, r c1, I 7 I ' a~ ~~a V h L1 a ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I C =I v > f C 1,1~ L i f IC 61 1 t i " -/,(= ~~~~~~~~~~~~~~~2- - w Jr~~~.. TT _s To satisfy ounaar}v conditions, the -wcan C .. , exopressed by Ibe the Fourier Series oo CO wf 4,- sin I i - m= ,,3 n=1,2,3 a sin n - b '.he nor al pressure may i-e expressed as a ourier Series = Pz To r-l,2,3 s-1,2, Prs 2 - 7ihlere ,Dx a as q a C p=0,1,2 q-o,,2 Biq;O p- ++ , (B i a] q-1 . Z - (2-1 (2-13) [.: (p-ri) (q-t) + "I4 -k2 (! -t)2] 4 + B5 + ~6 Ik, " (p + B7 + B8 -)( q-t) k=-1 tl and pO0 . oo B 2 = and wh.ere I42b b+q2 q if .... (;-i) pressure in p, are constants equal to the averaze .membrane the x- and y- direction if b ............. bn,qos 1 2 2 G sin s_ is found to be atisfy the compatability equation, = - P"xY and sin r t a 31 = 0 q-1 t k1=1 t1 oo B~3 =~Z q- lf= tl +iK2(q-t) 2 [t(+p)(qt) B 2 = 0 if q= q#C. if q = 0 or p = O. +p kt -t a(rn),\-t) O. 1 [(~L+p)kt (q-t) + (+p)2 ,t 3 j w+) kq-)t) -4 and ,. / 0. A 1 k=l ' 1 IL 5 p oo 0 and if r · . Y = 0. tt+q)(p-k) + 2t2 t-l IC= - f Ba = 0 =-1 '.Cor + k2(tS+c)2?J 2-k)(+ , [ t , 0 4L. - 0. q = :,+q C ,p-k, if p = 0 or q = 0. , 0O = B6 k=l if qs -2(t+q)9 q 0 and 00 t=l p O if p= B7 . 2- kt(k+p)(t+q) 8 0 and oO 9 p O.0. 0 or q - 0. - (r+p) 2 (t+q) 2 t=1 k=l . q -·t!tcl Z_ [ kt(t+q)(k+-) - k2t2JWk,(t+q) w(k+p),t k=7 4- t 0. ok - klt (k+-o)(-Uq) t=l B8 = O,if p = W(k+p), t W(t+¶) = 0. or 0o k=l kt - (k+p) 2t21 W(k+p),(t+q) t= k,t . (2-14) The equilibrium equation is satisfied if Pr, = D wr,3 (r 2 r. l 2 + s2 ,·r 2 T ) 2 s7 ' PX h sr I a 2 IT2 b2 + bTT4 a2 4ab 2 r ~ -k =! - S t-1 2 (s-t) k - (r-k)t b(r(s-t) k,t kt br-t.c),(s-t) 212 [t(+r) + - - + Z (t+s)b t k ( ) Z [( +kr)(+s)- tJ b t+s) w(k+r,) t k=O t=l o0o 1 21[tk + 2 00 (k+r - (k+r)(t+s)J] t 'T, t S) k=!l tO 21[(b-t.s s - - k=l r (k+r)t] b(k+r),(t+s) k,t t:O 2 +s) 2 (r-1k,t [tk + + =1 tl [(t+s,k + (r---)tj + Wk(t+S) 4+s Wkt t=l k=O S 00 15) ) ,t ..... - t( + Z Z [(s-t) k + tk+)b(k k=l t!1 i-len the lateral tion tie 2-15) represn.t eq1u;ati ns of¢<e ress-re a cub>l. family, is criven, P ,s infinite fil· the coeffic can be determined. of equations. ients b p, "juia- In each of g may be revp,,"ed teir ·I· values as iveny C. ,Catior (2-14 ). The resultin,? equations will involve the kno,2. normal ressi.re coefficients coefficients :· w directions, p , and the aera.e sirnsor i te po and p can be determined from o-olatesare eithier s'h:,ected to known edge compres- knovwn edge displacements. t.o 4lthenls.er rmerLbrane pressures in the -.- and the y- and p , respecti ely. the condLitions tlat Ss the cubes of the deflection The nber ' of ,l- nov. def].ection cefi'nin e . of these euations nts,F' .. i... is equal II-12 i__ Now, the procedure is hith the known values of Pr s to assume Wl,1 and solve the other coefficients by successive approximation. The work in doinr so, however, is tremendous and it is very easy to make mistakes. As in the case illustrated by Levy, for a relative simple case or square plates if six deflection coefficients are used, each equation contains sixty third-order terms. And for each normal pressure applied one has to solve these six sixty-term, third-order equations once by successive approximation. The method is too laborious to be of practical use. (b) Clamped Rectaneular Plates. Lev,-and Greenman solved the case of the clarmpedrectangular .i plate by assuming that the edges are clamped rigidly against rotations and displacements normal to the edge but are permitted to move freely ~?~,:¥~, -parallel to the edge. The reaquirededge moments mx and my are replaced by an auxiliary pressure distribution a tw,y) near the edges of the plate. The auxiliary pressure can be expressed in terms of a Fourier Series as follows. @-:4 . pa(x,) - a2My -- ~ 00': xress r sin a ..... 4u~2 ; ~ ~ ~ ft and by.and a a Fourier Series, where 's krr are coefExpress ~ ~by e S andSre, ficierts to be determined, i. i. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~L-...~ ........... '.. ~ ·-- ·------··-·· I1-13 ::: 2 4b o 2 .................. 1T3 s=1,3,5 * Inserting equation (2-17) --- n a, k sin b in equation (2-16) gives rl,5 r=l,,7,5 Ch .. xx = 1,3,5 1,3,5 .. (rks+skr)sin Sr. a sin _x.:-- .-.'S: 1:i .--- ,..;:' where 2'-" ,,. .,?. r=I,2,3 r=1,2',3 .. s E 1,2,3 r,s 1,2,3 .. Pr,s . ) 2 Pr,s T sin r a (rpk + Pkr) Prs sin sTy b .... (2-19) ( 2-2G3 I) by the . smpl -supported !. ~~(28~~~~ . Since the edge moments-.x and my hiave been replaced auxiliary pressure distribution Pa (x,y), t (2-18) -'' . st = po(xy) >;- r, - .... Combining with the normal pressure pz, Eq. (2-12), we have ,:( :-/.. --.. (2-17) . - (X =(4 ) :"~ ... 00 rectangular plate the general solution for the (Eqs. 2-11 to 2-25) can be applied to the clamped ones and the only thing one has to do is to determine .. .. the values of k at the edges ,_ . o and kr . This is obtained by the condition of zero slope the plate. '._ Setting the slope perpendicular to the edges .. _' x = 0, x = a, F., r =0 a.nd y = b, to zero ives d-'~~~~~~~~~~~~Jl = -dxx = , x = a '=l mT 3,5 a 375 o, , (S71 ,- . y _ b 1,3 ,5 nIT .. n=-,- , 5 Q h* W Wm,n sin bY w sin -( 2- 21) ru, n a Ia Equations (2-21) are equivalent to the family of equations I ,,3 0 . ..... w w 1+: 3 ° ! 0 ~-~~ i , 3 5,3 :~:~ :.-s~.:,1 Now the deflection amily '"and of equation kof 5 + (2-22) .. , ...... ,5 coefficients mn must be solved from the for the linear term in terms of the cubic terms the pressure coefficients Pr,s' stituted !::9..::'C' (2-15) 1,5 + 5 w3 w3,1 + 3 I-14 ~' w The -wn nthus obtained are nowsub- into Equation (2-22), and for the pressure coefficients Prs, are substituted their values by Equation (2-20). The resulting family equations contains linear terms of pkr and pks and the cubes of the deflection functions w. The rethod of obtaininf the recuired values of the deflection. coefSficients nn and the edge moment coefficients assumin: va.lues for 1ll a Wpks3 ,3 , ...... andthen solvin for Eh :--: nk . ..... pkr,Pks consists of h by successive approxi-Tations from the sitiultaneous equations. The procedure is even more laborious than that for simply- supported plates. problem of 15. :hi.;, of a square Twonumerical solutions are given, nTely, the bending plate and a rectangular plate with length-wi.dthratio r ~ i~~~~~~~~~~~~~~~~~ITI-1 CHAPTER III THE GOVLRNIING DIFIEIREINTIAL EQUATIONS 1. Bending of Thin Sheet Plates A cylinder is called a plate if its height is small compared with the linear dimensions of its cross-section. We shall designate the height of the cylinder by h and shall call it the thickness f the nlate. The mirldlr nlann. of the nlat vi:llbe taken to coincide with the xy-plane of the coordinate i i I i system, and as a plane of elastic symmetry. After bending, the points of the middle plane are displaced and lie on some surface wh:ichwe shall call the middle surface of the plate. The displacements of the points of the middle plane in the direction of the z-axis will be denoted by w and will be termed deflection of the points of the plate. We consider the case where the deflections are large in comparison with the thickess but are at the same time small enough to justify the following assumptions: (1) Sections of the plate cut by planes normal to the middle plane before deformation remain plane and normal to the deformed middle surface. (2) The normal stress r- perpendicular to the faces of the plate is negligible in comparison to the other normal stresses. To investigate the state of strain in a bent plate we suppose that the middle surface is actually deformed, with but 'I slight extension of any linear element, so that it becomes a surface differing only slightly from one or other of the surfaces which are applicable upon the unstrained middle surface. The first hypothesis is equivalent to saying that points originally on a normal to the undeformed middle surface remain approximately on a normal to this surface in its deformed state. - and That is, the shear strains 'v/Az+w/Mv are neplitible. of beams subjected to pure bending. xz This and yz oraU/az +w/3x is bhasd on t.hehavinr- It is the consequence of this hypothesis that the middle surface of the plate undergoes no ex- i i tension, and hence is a neutral surface. I The second hypothesis disregards in effect the contribution to the deflection of the compressive stress 6 z produced by the load p. Since, for z = O, u and v are zero, and by the first assumption, the displacements for any point in the plate are given by u = z(u/z) and v = zavA l( u 2 (=a z xz or za- ax _ aw v. ay therefore aw u=- z- /~ ~ ~ ~ v -=- Z ~ y a z). w =o x Again by the first assumption, III-3 These are the displacements at any point in the plate due to bending alone, the effect due to the extension of the middle surface being neglected. I 11 But the components of the strain tensor are related to i iI = the displacementsu and v by the formulas (x ii |(y= = 7'xy v/ay, Bu/Zx, 1/2( v/ax + Zu/Z y) so that by making use of these relations we can write ZX2 X X2 2 Y = S (3-1) ay2 2 w w by 11OOK'S - aw, since X = A(( 0Y we nave - u + X 6 y) + 2 A( Ex+ (3-2) Y)+2Vfy V y . 7- Substituting Eq. (1) into Eq. (2), we obtain x '- - z(Av 0Y = 22 w + 2 z(A 72W + 2 ~ 2w ) x2 ' aY2 ) (3-3) ~2w2 where A and V are elastic constantsand V ".'~ - is 2/ x2 + 2/ 2 III-4 rII i Consider an element of the plate (Fig. III-1) and take the of the forces about the x-axis. moment Neglecting third order terms, the equilibriwm conditions require that (a) f+h/2 7- +h/2 D -y dix h dy + d z dz xy a z -2 / x /----/ dy dz = 0 Similarly, take the moment of the forces about the y-axis and set it equal to zero. (b) -h/2 ao T- dxhdy+f Z - x I-h/2 2C dxdydz -/ +h/2 ar- / -h/2z a Y dxdydz = O By rearranging terms, we have 2 &y j+h/ h / z z YZ /-h/2 (c) hit - Substituting -dz + Y h/2 , -h/2 Eq (3-3) ito + ax (= iy [+ [lVW+ dy (h) [A a2 =_ h z ini Ecl.(c), and integrating, ""1zd h'l' = -h/2 ? r7, x fh/2 *· a+9 2v ) + 22 we have 3 +2 V-I 0. ) VY Vw 1' w I J v2w (3-4) 12 (A+ 2) -- yz * ( ,xz-- _ a~y a,x VW h2 12 The normal stresses distributed over the lateral (3-5) sides of the element cani be reduced to couples, the magnitudes of which ., .... I 1k'i lik.. - r dX 7 Z F/,9 .·J7- / III-5 per unit length evidently must be eual Mx 'and In this way, y. to the external moments we obtain h/2 ldy =I o- z dy dz /-h/2 0- '.Adx . 11/2 z dx dz Y Substituting expressions for x and 02 h2 and integrating, we have 2w (3-6) 2. Differential Euation of the Deflection Surface Consider the equilibrium of a small element cut from the plate by two pairs of planes parallel to the xz- and yz-coordinate planes. al forces (Fig. III-2). be Sx, Sy and note that Projecting Let the magnitudes of the laterSxy = Sy x these forces on the x- and y-axes and assuming that there are no body forces or tangential forces acting in those directions at the faces of the plate, we obtain the following equations of equilibrium: Sx ax + Saxy ay = 0 (3-7) Sxy +-=___ y :.' 0 x · c -x . l ,.1 -xd, a'--w aA aK I I i I I I I I- I I I I § I I I I I I I Yx I I I I '.5 Js-t 0x +a -VS,+ SpKt doYI I i J I I I T d' _-.X - y." -- ax x2 da III-6 In considering the projection of the forces on the z-axis, we must take into account the bending of the plate and the resulting small angles be-tweenthe forces Sx and S that act As a result of this bend- on the opposite sides of the element. ing the projection of the normal forces Sx on the z-axis gives (a) -SxdY (a a Sx X -+ d)(Sx Dw 2w + d) xdy+ aX- cx dxdy = Sx -' The last step was obtained by neglecting small quantities of higher than the second order. Similarly the projection of the normal forces Sy on the z-axis gives (b) a 2W Sy 7 dxdy+ *Sy bw yy dxdy Regarding the projection of the shearing forces Sxy on the z-axis, we observe that the slope of the deflection surface in aw/y the y-direction on the two opposite sides of the element is and ~w/y + ( 2 Hence the projection of the wxy)dx. shearing forces on the z-axis then can be written as d 2W Sxy a xay dxdy+ Yw wd dy An analogous expression can be obtained for the projection of the shearing forces Sy x = Sxy on the z-axis. The final expression for the projection of all the shearing forces on the z-axis then can be written as III-7 dxdy dx dy + ~ x Sxy aZXy An analogous expression can be obtained for the projection of the shearing forces Syx = Sxy on the z-axis. for expression The final the projection of all the shearing forces on the z-axis then can be written as x dy + x dy + 2 Sxya y ax dy Adding expressions (a), (b) and (c) to the load p dx dy acting on the element and equating the z-component of forces to zero, we obtain the equation of equilibrium: (d) W+ l2w + x 8 x ~w S Sx x2 + x ax where 7 xX and y2 +h + h-X 2 s + y + y W ay X ;-s+ p = are the average values over the height h. 7yz From eqs. (3-7), x we have ay =r--b III-8 Substituting into equation (d), we have w +S + 2 S S - +2S p ++s Z)+ + xz-x+ () h( Ce) n(-a~ / P -+S 2 x xyax y a 'ax ay 2 2-= Assume that the deflections are large in comparison with the thickness but are at the same time small enough to justify the application of simplified formulas for curvatures of a plate. Then by the condition resulted from pure bending, we have (A+ 2) 4 w h2 4 =-_1 E V w; 12 and eq. (e) becomes h3 -V12 (f) 4~2 E 1 _2 2 + 2 Sxy axw 2 + p + Sx x2 2 + Syy =0 2 where E is the Young's Modulus and /A is the Poisson's Ratio. Introducing a stress function F defined by -h= Sx ' 2F 2 y 2F ; h2 S =-h y= 2F S xy x - x y the resulting equilibrium equation is h3 12 B~ ~~ E 22V4w + p +h -2F 2F x 2w 2 h 2a W y 2F 'xy ~x2 =0 2w a3y x y O III-9 V + a2 or 4w =h/D(p/h a2 -X --2+2 ~ aa 2-2 - 2F xaa 2 ) (3-8) 3 where D = plate. Eh and is called the flexural rigidity of a 12 (1-/ ) 3. CompatabilityEquation. Considering the strain in the middle surface of the plate during bending, (Fig. III-3) 0 - a w -- tan x cos cC - 1 - 2 Take an element of length dx. - = 1 -1/2 w/x)2 After deformation, its length is dS ' dx +cos a ux a/ dx (1 + ." 1 - 1/2 ( w/ax)2 - dx l1 + u/ax + 1/2 (w/0x)2 ]. The x-component of the strain is therefore x =au/x + 1/2 (w/dx) 2 (a) r - /0 a I xn OIX .dK t II I; 7101,1I~~z I I I I¥S ii I t" I 4,cX 111-11 Similarly the y-component of the strain is 2 av 1Ay + 1/2 PwA/y) y (b) For the determination of shearing strain due to displacements w we take two infinitely small linear elements OAand OBin the x- and y-directions, figure. as shownin the following Because of the displacements w, these elements come to the positions 01 A the angle A1 OBland and 01 B1 . /2 The difference between is the shearing strain corresponding to the displacements w, (Fig. III-4). OlAl = [1 w/ x)+ ( +x)2 ( [1 + 1/2 (w/ x)2 1/2d x)2 + 1/2 (avx) The directional cosines of 0iA 1 dx/OAl V/X 0.x 1a1 - arwAx)2 - 1/2 a/ 1 - 1/2 dX _ d - 2 dx are )2 with x-axis with y-axis v/x awg/x with z-axis Similarly, the directional cosines of OlB1 are with x-axis By 1 - 1/2 ow/agy '. t/2 Ar - L wA y) 2 _ 1/2 (u/ay) with 0B1 z-axis cosA1B1 with y-axis A A, Y 3 Z F/' . af- X -ax F III- 13 cos o1 COS = I - 2uty cosC12 + cos/ cos 2 +au/·x +(aw/4x)(W/Ay) COs 2 c) The strain components are therefore I E = u1 x + 1/2 ( wx) i E =av/ay + 1/2 ( W/y) 2 Yxy = auy + vAx (3-9) +(a waX)( -a§Y) By differentiating these expressions, it can be shown ay 2 -3y + 2 2 ax2 ax 2y = ax ) X y , a X2 2 y2 (3-10) Hook's Law is x = l/E ( °- -/ (¢: =1/E( t Ox) F) t Y-xy =1/G (d) Xy Noting that o7 = /h = 1h ay2 y ' = S/h = 1/h a2F 2 y 7- k xy= Sx/h = /h v y2 x axay (e) r III-14 Therefore, 2 -a F <X= lhE (-ai2 y (f) x2 Y g a2 F E- E xy ,3y2 9;;i-y' Substituting these expressions in Eq. (3-10), we obtain 2 ()xa :E )a. ~F L a x2 y2 Y _ra 22 or V4F = Ed2 or v4=EL(kay) ':I 22 ) 2 a x2 a2w ya w (3-zz) III-15 Smay of Thin Plates. of Euatinsefor the Deformation The fundamental equations governing the deformation of thin plates developed in Sections partial differential equations are: the as follows: X4 + 2 -341 ; XI+ 2 and 3 may be summarized = E rL~ ~~~a2w \ +- axY2 2y4 , _ a4 ax2 2 4 - 2 a2 ] ) y2 a2 ax2 w ;26a 4w 2-x42 a a 2 aw 4 2 = /D + p/D 2F 2a 2 where the median-fiber stresses are x = a2 Ft _ Tty = DZF 2 a xxy andthe median-fiber strains are 'x = 1/E ,( ty = 1/E Y' =- - 2 - 2 ) 2 _/; xy III-16 The extreme-fiber bending and shearing stresses are: 0**" 2w a- Eh 2(1-) y (a Y2 7Xf " xy 2(1+) 5. a) Non-dimensional Form of the ECuations. Writing F' = F /I2E XI = x/a = w/h y' = y/a w pt = P a 0o' = E =6(a/h)2 ' Eh4 (a/h) 2 where a is the smaller side of the rectangular plate and h is the thickness, we have the following differential 4F l+ a X;_4 2 4t +2 ax-4 Dax2 4 F Cy)2 a,4ww axi'.,2 a y 4. 4 a% w -374 ( + 12 (1-,r/ w2 2 2w, 12 (1 - equations: , a2 ,w ,a x2 y/2 ) p2 1-)p ) 2 F' 2W' XL2 3y7-2 2F'/,y'2 2w'/x' 2 a2 w' ax(yt' (x'y' ?F' _rM III-17 2 Letting/ aluminum alloys, differential of and drop the primes, we have the partial equations in non-dimensional form as +2 bx 4w = 0.1, which is characteristic a 4'= ta)2W I xzy 4F x2ay2 2-- a + 2 = 10.8 p + 10.8 4 ~¢ 2 a F 2 2 a 2 ay ax ay a2F 2 a2 w C . 2 (3-12) a2F ;)2 .xy ] (3-13) The median-fiber stresses in non-dimensional forms are -r = x Ci2 a7 2 (3-14) y F2 b z', w x. xY and the median-fiber strains are Z2 F x a2F 32 2F a2F y x2 - 2 (1 +) L- I -0B (3-15) Cy axaF III-18 i The extreme-fiber bending and shearing stresses i I are I i r I 1 r-nx= I 0-fl 'px y P - (= 1 -- -·-- I 1 2 (1+1,e) --- ( a2 + ax2 ( 2(1-t.) L : d 21 2(1-,AA ) I -·- 2w ay 2 12w a xaY wB2 as2 2 w +C4 )x2 (3-16) V IV-1 CHAPTER IV FORIULATION 1. General Discussion OF BOUNDARY CONDITIONS of Boundar Conditions The governing differential equations as derived in Chapter III are two two-dimensional fourth order simultaneous partial differential equations. To obtain an unique solution, in the case of rectangular plates, there must be four given boundary conditions at each edge. Before proceeding to the actual case, two theoretical boundary conditions may be mentioned. (1) Simply-supported plates, where the edges can rotate freely about the supports and can move freely along the supports. (2) Clamped or built-in plates, where the edges are clamped rigidly against rotation about the supports and at the same time they are prevented from having any displacements along the supports. Actually, it is to be expected that neither of these conditions will be fulfilled exactly in a structure. Nowlet us proceed to the bending problem where the bottom plating of a seaplane is to be studied. The behavior of the sheet approximates that in an infinite sheet supported on a homogeneous elastic network with rectangular fields of the same rigidity as the supporting framework of the seaplane. L IV-2 The displacement in the plane of the sheet and the slope of the sheet relative to the plane of the network must be zero from symmetrywherever the sheet passes over the center line of each supporting beam. Each rectangular field will therefore behave as a rectangular plate clamped along its four edges on supports that are rigid enough in the plane of the sheet to prevent their displacement in that plane. At the same time these supports must have a rigidity normal to the plane of the sheet equal to that of the actual supports in the seaplane bottom. The rigidity of the supports will lie somewherebetween the unattainable extremes of zero rigidity and infinite rigidity. The extreme of infinite rigidity normal to the plane of the sheet is one that may be approximated in actual designs. It is pro- bable that the stress distribution in such a fixed edge plate will, in most cases, be less favorable than the stress distribution in the elastic-edge plate. tained from the theory The strength of plates ob- will therefore be on the safe side if plied in seaplane design. ap- Reference might be made in this con- nectionto a paperby Mesnager (63) in which it is shown that a rectangular plate with elastic edges of certain flexibility will be less highly stressed than a clamped-edge plate. It may also be seen clearly by comparing the extreme fiber-stress calculations by Levy (21) and Way (17) for simply-supported plates and clampedplates. However, the impact pressure on a flying boat bottom in actual cases is not even approximately uniform over a portion of IV-3 the sheet covering several rectangular fields. Usually one rectangular panel of the bottom plating would resist a higher impact pressure than the surrounding panels, and the sheet is supported on beams of torsional stiffness insufficient to develop large moments along the edges. The high bending stresses at the edges characteristic of rigidly clamped plates would then be absent. To approximate this condition the plate may be assumed to be simply supported so that it is free to rotate about the supports. At the same time the riveted joints prevent it from moving in the plane of the plate along and perpendicular to the supports. Ac- cording to the same considerations as in the case of rigidly clamped edges, the result would be on the safe side. This case has never been discussed before, and it seems to be of importance to study such a problem. For the combined bending and buckling problems the same considerations will be true. It is evident, however,that as soon as the side thrust is applied, there are displacements perpendicular to the supported edges in the plane of the plate. Gall (6 4) has found that a stiffener attached to a flat sheet car- rying a compressive load contributed approximately the same elastic support to the sheet as was required to give a simply-supported edge. In combined bending and compression problems, therefore, it seems also important to study the ideal simply-supported plates. The analytical expressions for all these boundary conditions are formulated in the following sections. ---- -rw IV-4 f 2. Analytical Expressions of the Boundary Conditions (1) Simply Supported Edge If the edgey = 0 of the plate is simply su'pported, the deflection w along this edge must be zero. At the same time this edge can rotate freely with res!:ect to the x-axis; i.e., there is no bending momentiy along this edge. This kind of support is rejpresen.tedin te following igure. The analytical expressions of the boundary conditions in this case are (w)ro ( = 0 1) (W) o w I Similarly, if the edge x O of the p;late is simply supported, the boulndaryconditions are kw) O) 2) ib W x+Z /1 a a y / O= IV-5 Now since w = 0 along y = 0, w/Bx and a2w,/dx2 must The boundary conditions can therefore be be zero also. ritten (W) O 0 (w)n0 = (4-la) 0 Similarly, for the edge x = O, we have (w)0 = (4-2a) 0 .( =0 x/x=O If the plate has ideal simply-supported edges, it must be free to move along the supported edges in the plane of the plate. Thisis equivalent to saying that the shearing stress alongthe edges in the plane of the plate is zero. Analytically, it is xyy0 or ( 2F ) = o (4-3) One more boundary condition is required to solve the plate problems uniquely, and this may be obtained by specifying either the normal stresses or the displacements along the edges. For a plate having zero edge compression, the normal stresses along the edges are zero. as (Ox') ~.or or 1 = o It is analytically expressed IV-6 (ya a) X =0 (4-4) y .,~2 = Xy ;0 - , o It is shown in Section3, Chapter III, that the strain in the median plane is +2 ) 2x x= 2 E v I = Y. BY Therefore, we have u ax av f(2xS 2 1 1 = ay (4-5) aW 2,i Y and the displacement of the edges in the x-direction is u , aw)1 LX2 (Td-X. (4-5a) while the displacement of the edges in the y-direction is j~ vv =J tICX tI. 1 (a w YF 2dy (4-5b) The addition of side thrust may be expressed in terms of the change in displacement Expressing Ex' of the edges. and E ' in terms of the stress function we have F, we have IV-7 ay2F _ ax 2 (4-c) a v- y (2) Clamped, or Built-in Edge. If the edge of a plate is clamped, the deflection along this edge is zero, and the tangent plane to the deflected middle surface along this edge coincides with the initial position of the middle plane of the plate. If the x-axis coincides with the clamped edge, the .boundary conditions are (w), =0 (2W/~~ Sa w \ ~~~~~~(4-6) =0 If the y-axis coincides with the clamped edge, the boundary conditions are (W)x = 0 ad . . ~(4-7) ~~~~~~~~~ 0 If the edge is clamped rigidly against any displacement along its support, the strain in the median fibers must be zero along that adge. Tlheboundary conditions are therefore (e ) = Y. =0 2F v2F or ) (/ (4-8) The one additional condition required is again furnished by specifying displacements of edges as in Eq.(4-5c). (3) Riveted Panel with Normal Pressure Higher than the Surrounding Ones. As discussed in Section 1, the boundary conditions which would approximate this case are: (a) =o = o = 0 2 ya -y )=O a F y .... 2F (- 1o -__~w 2 dx=O if y = 0 is one of the edges. The first two expressions are those of simply-supported edges, the third one gives the condition of zero strain along the supports, and the last one specifies that the displacement of the edge is zero. .., J . ::: V-1 CHAPTER THE FINITE DIFFERENCES V EQUATIONS OF THE BOUNDARY VALUE PROBLEMS 1. The Calculus of Finite Differences Brooks Taylor, in his book "The Method of Increments" (1715-1717), was the first to consider equations of finite differences. They are used more and more frequently in recent years as approximations to differential equations. In fact, the very large deflection problem has been studied by Henky ( ) '( 6 ) and Kaiser (7 ) by using finite differences approximations. Before we proceed to convert our partial differential equations into finite differences expressions, some fundamental concepts about the finite differences approximation may be mentioned. Let us assume that a function f(x) of the variable x is defined for equidistant values of x. If x is one of the values for which f(x) is defined, f(x) is also defined for the values of x + kax, is the interval between two successive values where &x of x, and k is an integer. y = f(x) for x + kx, We denote the value of the function for the sake of simplicity, by writing y with a subscript: f(x + kx)= 7 x+k (5-1) We now define the first difference or the difference of the first order Ayx of y at the point x as the increments of the value of y in going from x to x +x: ' V-2 (5-2) ax -xYX Yx It is seen that we have chosen arbitrarily the step in the direction of increasing x; we could also define 6Yx by the difference YS- Yx-ox' Continuing this process, we call the increment of the the difference of first difference in going from x to x +ax second order of y at x; i.e., 2 Yx+ ax - Yx = AYX (5-3) YX+2 x - 2yx+ x + Yx In general, we define the difference of order n by Znyx n-lyn-l x - a 1y x = A X (5-4) yx x equal to unity, then we write If we choose YX+n Ax = Yx+n Using this notation, the sequence of differences becomes YX y + - X Yx = Yx+2 - 2x+l a YX Yx = + 3 x+l - +3- y ( - 1) r=O * Y r(n x - r) Y+n-r (5-5) In many physical problems only differences of even order occur. In such cases it is more convenient to define the V-3 differences A 2yx 2 i.e., A 2 y x in another way. We shall write yX = Y-1 - 2yx + x+tl (5-6) is the increment of the first difference taken on the right and left sides of the point x. In general Q2m(yx) = 42( 4 2M2 x) (5-7) In this case a difference of order 2m represents a linear expres- x .... * sionin Yx-m'Yx-m+l' Sx+ l Yx+ 2. The Finite Differences Expression of Two-Dimensional Cases. In replacing the partial differentials by the finite differences expressions, we have to consider the differences corresponding to the changes of both the coordinates x and y. We shall use the following notations for the first differences at a point Amn with coordinates max and nay. The notation used in designating adjacent points is shown in the following figure. Fig. V-1 Y L 4mn . - w I A,7f,/, V--/ -Iz1, c ,r ,' PV .A/n",-l 0 ,''9. V-/ - V-4 ax Wm-l,n = wm,n - m-l,n' = Wm+n, n - Wm.n, Ax Wm,n Ay Wm,n-l m,n mnn-l' = Wm,n+l - Wm,n ay Wm,n Having the first differences, we can form the three kinds of second differences as follows: X K W h xx m,n ` = L a2 x m,n = wm+l,n- wm,n - (wm,n -- - wm-l,n) a `5 = m+l 2wm,n + m-l,n n y Wm,n YYy m,n = Wm,n+l - Wm,n - (wm,n - Wm,n-1) ·· :·; = Wm,n+l - 2wm,n + i ,n-1 I- y - AyWm,n Axy Wm,n = AWm+l,n -t; :., = wm+l,n+l- m+l,n = m+ln+l - m+l,n- Wm,n+l+ m,n - (m,n+l - wm,n) :Hl~ (5-8) And the three kinds of fourth differences which will be used later on are: xxxc w,n m = W,n I:I Wm+2,n - 4Wm+l,n + yyySy m,n y 6 Wm,n - 4wm-1.,n+ Wm-2, n wm,n Wm,n+2- 4Wmn+ + 6wm,n 4Wmn-l + m,n-2 I 'i, A i. i: wm n = iS2 wm,n = wm+l,n+1 - 2 +l,n + Wm+l,n-1 - 2 Wm,n+1 + 4 Wm,n 4. II : .1 ,.I .z. . .- Wm,n_ + m-l,n+l- 2 Wml,n + Wm-l,n-l (5-9) r'' V-5 2 3. Conversion of the Governing Partial Differential Equations Into Finite Differences Equations. Partial differentials may be approximated by finite differences as follows: LsW a 2w a2w .x%, _ dy2 4x2 X2 z aY 'x Ax = AY A3P2 a2w 4 a4W Axy axI 3w 4 _Y g_ &x 4 ' a ax2 y2 w s-lo) A 3c y2 By the above relations, our governing partial differ- entialequations (3-12), (3-13) may be replaced by the following difference equations: eA 2F y , 4,w x A + / z& 2 = ~Y2 - aw;y2 yw A + Ay4 Ay a.2 w X .~i..'.. Ay2 . . ax2 AX2 +10.8 ' y &j4 - \4F 2 x y AX2- 6y2 = 10. p A X2 F A y_2w A'x2 A2 4 F 4X w Axy A XYF-,AXxdy (5-11) I V-6 If we take Ax = (5-9), Eqs.(5-11) F - 8F Fm+2,n Ay = -f, and use the relations (5-8), may be written as +20F ln - mn F m-,n +F m-2,n + F 8Fm,n-1 + Fm,n-2 + 2Fm+l,n+l+ 2m+l,n- 1 + + -8F Fm,n+2 Fm,n+l 2Fm-l,n+l 2 F + 2Fm-l,n-1 m+l, (Wm+l,n+l.- - wm,n+1 n mn - +lm, 2w2W l,) +L, n 2m n Wm-,n M,n)2 ,n+l min +w (5-12) ,n- and m+2 n - 8m+l, + 20w n m - 8wm-l,n + Wm-2,n + m,n+2 - 8wm,n+ 1 -8wm,n1+ ,n2 +2m+l,n+l +2Wml,n+l 2 + + 2+2 Wm-l,n-l = l0.8(4l)4 p +10.8(Fm,+ - 2Fm,n + m-n) - 2m,n + Fm-,n) (Wml,n- 2m,n + Wm-ln) + (Fmtl,n (wm,n+l- 2 wm,n+ w,n-l) - 2(Fm+l,n+- Fml, n - Fm,n+l+ Fm,n) (5-13) W ln+l - wm+l,n - Wm,n+l+ wmn) Eqs. (5-12) and (5-13) may seem to be very complicated. In ac- tual3writing out of these equations the following finite differences patterns or so-called Relaxation Patterns are useful. i. J .j r m. n m,-EE r, 4 xe 4xy EE hto Or I; / -4- X .,x +. F'9. 7-1e Re/sx xy 1,a + ,xy t- ,y LI-V foon Asff eon T ~V-7 ,4.e Finite Differences Exoressions of the Boundary Conditions (1) Simply-Supported Edge. The boundary conditions for the simply-supported edge y.= 0 are: (w)=o = o (W_) = =0 dxdy and ( 2 F/,x2) = 0 for plates with zero edge compression, or 2 daF 1(W2 w dx u for plates with zero or known edge displacements. Let n = U denote the ecge points along y = u. Ine finite differences expressions of the boundary conditions are: Wm,o =o and ( m, o 2F)s [ = 2F rsF-r where m = o, k represents A (5-1 ) - M--O (x A)) 2 M~Im,i points along tle ed i. (5-16) (-16) two edges x = O and r, -1 V-8 x = a, and i is a point along any line y = constant along the plate. (2) Clamped Edge. The boundary conditions for the clamped edge y = 0 are: (w)_,, = O =0 47 w d d _ 4 F_ 1 W 2dx = u Following the samenotations, the finite differences expressions are wmo m,o =0 (Ay )m,o (Ay F -/ -=0 Ax 2F)m,o 1 k-l ( i=O (3) (5-17) =0 = Ui w)2]m,i Riveted Panel with Normal Pressure Higher Than The Surround- ing Ones. The boundary conditions which approximate this case are: //' %'):w :::. kl'W. -.. '. .......-. .. ~,,y2/ y--O = 0 V-9 =-° x 1-- 2 2 -2 -J if y = 0 is one of the edges. Expressed in terms of finite differences, they are: m, o = 0m,o0 = (i n)m, Y0 (5-18) k-1 - E [aioF m, i x2F -(x w)2 =0 5. The Boundary Value Problem in Terms of Finite Differences ExDressions. Now the boundary value problem, which approximates the riveted sheet-stringer panel subjected to uniform normal pressure higher than the surrounding ones, may be formulated in terms of finite differences. To start with a simpler case, the square flat plate will be discussed, because the condition of symmetry requires only one-eighth of the plate to be studied. The finite differences approximation of any differential equation requires that every point in the domain must satisfy the governing differential equations. If the points to be taken are infinite in number, the solution of the differences V-10 equations is the exact solution of the corresponding differential equations, and if the points to be taken are finite in number, the solution will be approximate, and the degree of approximation will increase as the number of points taken is reduced. Since the diagonals of a square plate are axes of symmetry, if the boundary conditions along the four sides are the same, wi,k = Wk,i' and Ei,k = Ek,i The conditions for zero +++ (i,)i,n + !+ +ee + edge displacements may be put into different forms, as follows. These conditions are: ('x)o,i + (x)ji 1 k-l 2 + 2 x)m,i k-l 1 i EM (W (Y )io+ + (f)k,i Wm,i i) )' +( + (Y)i k-l k k-l 12n=O (aw).n =2 - (Win+l w,) Adding these two equations together and noting that wi,k = Wk,i, and (f)i,k = (y)k,i' (i + by we know that (i+2')j, i, k-l + (x+ fi,k-l = n 1 +I y)n iin + + 2 (i,n+l - wi,n) To obtain a better approximation, the left side of Eq.(5-19) may be re-written to give ,I i A . (5-19) V - // r,, I-----~ ~-/ I I I I I I I I I I i -, I I I I I I I I I / -----X r'9- - V- 3 ------ ~~~~~~~~~~~~~ /I - +(x+,)i + -/_ V-11 A 1 -Cxx + y)io + (fx + + . *. +f )i,k-1+ 1(x f k-I 2 (Wi,n+ - w Now, (5-20) ) x y I2F 2F 2IF ~~+ dy2 -1-- dy2 d21~ dxa2 = (V2F) (1 -, Note that y)i,n here 72 a2/a X2 ( 72F)i, 2 (vF)i - (5-21) ) + a2/a . Eq.(5-20) then becomes + 2 + ~ik + 2(77 2( 2 F)i + *F 2 1F) i,k k-1 (W - w. )2 (5-22) =3(,n1l This simplification is not necessary, but it will be used in applying the Relaxation Method. (l)n=l Referring to Fig. V-3, points 1', 2' are fictitious points outside the plate in order to give a better approximation for the boundary conditions. ii. Using / 2 = 0.1 or L= 0.316,228 for aluminum alloy, the compatibility equation is 20Fo - 32F1 + 8F2 + 4F1 = K o where K = (w2 - 2w1 + wo) 2 - (2w1 - 2wo)2 . rium equation is L (5-23) Then the equilib- V-12 20 Wo - 32 w1 + 8 w2 + 4 wl' = p' + 10. (2 w ,-1 2 wo) - 2w z )(a2) 4 -wherep' = 12(1 /- -- 2F 2(5-24) + + w( {2(2F 1 - zFo) ) = .675p,sinceAJ= 1/2. The boundary conditions are: (a) wl = O w2 = 0 (b) w' - 2w1 + w = 0 (c) F0 - 2F,1 + F1 -/# (2F2 - 2F,) = O (d) (F 1 - 4F0 ) + (F + 2F2 + F 1 (w where S - w 0 )2 2/1-a = (w - - 4F 1 ) = S ) /.341886 problem determines the values Nowthe boundary vlue of w uniquely, and the values of F with an addition of an unknown constant. Since tis may define it by letting to the problem, we constant is irrelevant F2 = 0. Solving wll,wz', F1 ' from the boundary conditions, we have Wi = -W W2 = -W1 = 0 )F1 F' t= -F + 2(1 -/ Substitute these values into Eqs. (5-23), (5-24) and (d), and the resulting equations are 16ff 0 - 26.529824F, = -3wo2 (5-25) 16w0 = p' + 43.2wOF 0 -4F0 + 1.367544f1 = W .341886 Note: The nine or ten significant figures used in these equations result from the use of a computing machine hraving10 columns. In order to get satisfactory results in subsequentcom,.;utationsit is convenient to retain a number of figures beyond those normally considered justifiable in view of the precision of the basic data. r V-13 (2) n = 2 -4, points 3', Referring to Fig. titious points. The compatibility equations are: 20F0 - 32F + 8F2 4F3 = K0 + -SF0 + 25F1 - 16F2 - 8F3 where 0, 1, 4', 5' are again fic- 6F + F3 ' = K1 2F0 - 16F1 + 22F2 + 4F3 -16F 2F5 + 2F ~'t' 0o,K, K 2 are equal to [( xyw) 2 - ax2W y2w] at points conditions are: 20w 0 - 32w, + W 2 + 4w 3 = p + 21.6 (w1 - wo) - ao0 + /30 ' ) y'(wO - 2w1 + W2 )j -8w o + 25w. - 16w 2 - 8w3 + 6w4 +w 3 {e,1(2w2 - 2 1) + t 1'(w +2w0 - 16w, + 22W2 where = K2 2 respectively. The euilibrium {( a1 ( 5-26) = p + 10.8 + 3 ) - 2, 1'(w 4 - w + (5-27)' 2+ 4w3 - 16w4 + 2w5 + 2w4. = pt + wI)] 10.1 + 32t)(w4 - 2w2 + w1) - ~~21w 2 Ya'(w + wa) 5 - 2w4 4+W) 2 c', a', ?' are 2 x F, A I 2 F,, xg at the points indicated by the subscripts respectively. The conditions for zero edge displacements are: -2FO- 3F1 + 4F2 - 2F3 + 2F4 +F3 2 - F0 - 5F + 2F3 + F 5 + F where S1 - 416 $~ __ L ' .3~4886. = S2 f(3486 w- wO) + (w3 - w ) J ( - 1 )2 w) ( + ( ) w (5-28)= (5-28) V- /,3eL II I i I I I~~~~I ~ ~ ~ S -----T - ~~~~~I -1 IIo I I / I I I / u; /~~~ / // / /'0 .o I -/'9. L / / / / / _ _ / 1' I I I _ _ ___ __1Jo~~~~~~~ q *11- 1.4., I I I / 3 eO' V-I4 Thae boLdar7- conditions .re: (a) w = 0, w 5 = = 0, (b) w31 - 2w3 + w = v4 2-4 + w2 = 0 5 - 2w5 + W4 = 0 (c) F - 2 2F - F2 3 + F3' - (2E F' (F + F4 - 2F5 + 5 ' - - 2F3) + = O F3) O = 0 For the same reason as explained in the case of n = 1, let F 5 = 0. Solving of the boundary conditions equations gives (d) w3 ' = -, w4' = -W2 w5' =O (e) F 5 ' = -F 4 F4 ' = 2F4 + , (F3 - 2F4 ) - 2 F3 , =2F 3 + F(2F 4 - 2F3 ) - F1 Combining Eqs. (d) and (e) vith Eqs. (5-26), and (5-28), (5-27) we have 20F - 32F1 + 4' 3 F2 + = KO -8FO + 24F - 16F2 - 6.632456F3 + 6.632456F 4 = K1 2f 0 - 16F1 + 20F2 + 4.632456F 3 - 13.264912F 4 = K2 -2Fo - 4 F2 - .6324566 + F0 - 6F 2 + 2.316 2$F 3 3 (5-29) = S1 * .632456F 4 + 1.367544F S2 and [20 + 21.6( aO' + 3o' + 2 Yo)] 1v+ ( + P0' + 21.6Y')W Y 0 ')] w - 32 + 21.6(a 0 ' + 2 = p' (5-30) cont'd next page I V-15 -(8+ 0.,')wo +[2++ 21.6( a1, I+ + I - [16+ 21.6( + ,')] ( a2 + 0f2' + Y')] Here p' = 12(1 -/ = pt W2 2wo -t16 + 0.( + , )] 1 end of (5-30) ,2' + (32')] w +[ 20 + 21.6 5 w2 = P' p = .042175p, since 2)(al ) = 1/4. (3) n = 3 Referring to Fig. V-5 and noting that 6', 7', ', 9' are fictitious points for reasons explained in the case n = 1, we have the compatibility euations as follows: 20FO - 32F1 + 8F2 + 4 3 = KO -F 0O + 25F 1 - 16F2 - SF3 + 6F4 + F 6 = K1 2Fo - 16F 1 + 22F +4F3 - 16 4 + 2F5 F0 - 8F + 4F2 + 20F3 - 16F4 2F5 - + K2 7 F6 + 4F 7 + F6' = K3 3F1, - 8F - 8F F33 +23 + 23F4 - 8F5 + 8 6 - 8F7 + 3F8 + 7' = K4 2F 2 + 2F3 - 16F4 + 20F 5 + 4F7 - 16F8 + 2F9 + 2Fg' = K5 (Axy()2 where K, K1 , K 2 , K4 , K5 are equal to points 0, 1, 2, 3, 4, XZ 2 wy2W] at and 5 respectively. The ecquilibriumequations are: 20wo - 32w, + 8w2 (Il - wO)- Xo'(w - 2w + 4w3 = p - o' + 30') + w2 )I 2w1 + w3 ) - 2 Y1'(w p' + 10.8 w3 + 6w -8w0 + 25w - 16w - + pI'(w O 21.6 {( + 1 2al'(w 2 -4w) = w 2 - w3 + WN 4 )} 2w0 - 16w1 + 22w2 + 4w3 - 16w4 + 2w5 = p' + l0.8 {( a + P32 ')(w, - 2 2 + Wi 4) - 2 2 '(w 2 - 2w4 + w5 )} (5-31) V-/-' - _ _ _ _-- _,_ - _- _ _ j9' id' l 7' l l l ! ! r I 7 16 / I / 4 . . / . / -1 / . "Al 19' I I 1 / I I / 4 7 '17' ly '3 6 6' I I I / a I / / / / 9 / / r v-1 6 - w - 8w1 + 4w2 + 20w3 - 16v4 + 2w5 - 8wG 6 + 4w7 + w6 ' = rI + 10o . + 4- 2w3) + 3(I - 2w 3 + w6) - 2 3( '7 )} 3w1 8-'2 p, + 10.8 (W,- w 3 4 (3 wv - + + 23w - 3w5 + 2w6 3 - - 2w, + w5 ) + w7 ' +3 134'(W2 - 24 - 2 Y'47) + )} +5') +v) -+ 110. 8 '+ (142w5 -251 (5-2w +w 5)( 9)}(5where d', (3 and 20w + 477 - 16w 16 + 2w9 + 2w 16 3 - 2 are a' 2 x F, y F, F corresiondin- xy' 2 4 the subscripts respectively, and p' = 12(1 - / ) ( = .Q08,333,33p, since ) p= 1/6. The conditions for zero edge displacements + 0 - 42 + 3 3 3F4 + 2F5 + F6 - 2Ff7 + F+ +F1 + 2F2 - 2F 3 - 2F4 - 5 where S k-1I - 1-A T m= ( F 6 +3FFF 7 +F9 + 2 i (Wm+li w,i) The boulndary conditions are: (a) w6 = 0, (b) are: - 2F1 + 4F 2 - 5F3 + 4F4 - 2F 6 + 42F 7 + '6' - 2F w7 to 0, O w6 ' - 2w6 + w. = 0 w 7 ' - 2w 7 + W4 = 0 w 8 ' - 2w8 +. w 5 = 0 w' - 2w9 + w8 = O = 0, W9 = O 7 = S2 = S3 = S - 7 (c) 3 - 2ff6 + i1' - ( 2 ? -26) L4 - 2F7 + E7' / - (ig 5 - 2F8+ F9 - 2 ' + '- z6 - 2F7) = O0 + F7 6 '= w7 = 2F- = o Solving the boundary, conditions equations uat (d) = + ives -w'3 -4 wS' = -w5 WI9 =O 9 (e) F6 ' = -F3 + 1.367544F6 + .632456F7 71 = -"4 + 1.367544F7 + .31622[F6 + .31622SFg F8 ' = -F 5 + 1.367544F 8 + .316228F7 F9' = -F8 where we assume £9 = 0 for the same reason s explained in the case of n = 1. Combinining, we have: 20Fo - 32F 1 + 8F2 + 4L 3 = O - 8FO + 25E' - 16F2 - 8E3 + 6F4 + F6 = K1 2L0 - 16F' + 22F2 + 43 - 14 +F5 + '7 = K2 F0 - 8F1 + 4F2 + 18F3- 16F + 2F5 - 6.632456F6 + 4.632456F£7 = K3 3F1 - 8F2- 8F3 + 22f4 - $E5 + 2.316228F6- 6.632456F7 + 3.3162218F = K4 2F2 + 2F. - 16 + 1F -2F - 6F3 + 44 - 2F1 + 42 + .632456F ..,>~~ ~~~~~~~~2 ' F~ ~~~~~~ ' 3 - 13.264912F = K5 - .632456F6 + 2.632456F 7 = S1 - %-0 + 1.3162 ; + J3-4 25 1`4 6 - .3245617 + 1..16223 1Ki$= S 2 +2 = 1.367,54418 - 2_ - 2F 4 - 5 + + 3.316228F 7 + S3 (5-34) and [20 + 21.6( )1 7W o + 0 + [ /o'+ ol')] wo - [32 + 21.6( Z, 8 ++ 21.6 2 + 3 = - 3 + 1o..83t w + 25+ 21.6(,1 - [16 + 21.6( 1 + 't.)j w2 - L + + (6 + 21.6 'lt)w4 = ,2+ 2')w5 + Y1 )] wI + 2 Y 1 )] W3 w + 22 + 21.C6(2, + 2'+ 4 y,2)] w4 = p' Vi - [ + 10.3311 o ] + Y3')] w3 - [6 + 21.6(3 3w1 - (8 + 10.8 / 34') (a (2)] ')] w2 + 4173- [16 + 10.(, + (2 + 21.6 + p' 2w0 - [16+ 10o.( a2 + + lo.8( p' + + y3) 2 - + 19+ 21.6(a3, +/3 v"4+2w5 = p' ( + 10.8 + 34 + Yt)] Yw 4 - [8 + 10.(xt + 2j X')A 3 +22 + 21.6 w = I9 2w2 + 2w3 - 16 + 10.8( a5 + /35)] wr 4 + 1 + 21.6 ( 5i + 5 + Yt5)] w5 = P' (5-35) I CHAPTER VI THE METHOD OF SUCCESSIVE APPROXIiv1ATIONS 1. Outline of the Method. After we have expressed our boundary-value problems in terms of finite differences equations, the problem now is to solve the systems of non-linear simultaneous equations. We have now two sets of simultaneous equations. The first set consists of the compatability equations and the equations specifying the conditions of zero edge displacements. These equations contain linear terms of stress function F and the second-order terms of deflection ratio w . They are of the form OO F 0 + C0 1 F 1 + .... + cn Fn = K c1 0 F0 + 11 F1 + COn Fn = K1 .+ (6-1) and where K c 1 0 0F ( + cl xyw) 2 1 -(a F + x w)(A + cln . y w) at points 0, 1, etc. corresponding to the subscripts of K; Si and Co0, C0 1 , ... , c'1 0 , c'1 1 Fn = S -1 ... are given i m( w constants. The second set is that of the equilibrima equations, which contain the linear terms of the deflection ratio coefficients which contain linear terms of F. form w with They are of the VI-2 + booo (ao 0 + bl + b 'o) W + (a 01 /'o0 + b"oo 00 00 "0 ) 'o + b'0 1 S 0 + b"01 w1 +.... +(aOn + bnoL'o + b'On 'p + b"onr'O )w ....... where 0.......... ............. A A= = aX2 &t F, p' = Adz F, y etc. corresponding to the subscripts of ively, and a 0 0, a 1 , b"01 , .... = p- .... , b, b 1, ... ', (6-2) F at pointsO, 1, B b' ', b'01 y' respect- b"00, etc. are given constants. If we assume a set of values of w at each net point and have the values of K i and Si computed, Eq. (6-1) is a system of linear simultaneous equations in F and can therefore be solved exactly by Cront's method (6 2) for solving systems of After the values of F have been computed from linear equations. Eq. (6-1), o ', Then Eq. ', r ' can be found without any difficulty. (6-2) becomes another system of linear simultaneous equations and may be solved exactly by Cront's Method again. If the values of w found from Eq. (6-2) check with those assumed, we have the problem completely solved. In most cases, however, the values of w will not check with each other. Following the usual method of successive approximations, the computed u's will now replace the assumed ones and the cycle of computations will be repeated. If the value of w at the end of the cycle still does not check with the one assumed in the beginning of the cycle, another cycle will be performed. In our problem, however, if we should follow the ordinary method, the results would be oscillatory divergent. VI-3 Therefore, a special procedure must be devised if we want to make the process convergent. A close examination of these equations shows the values of F depend directly upon the values of Ks assumed and a plot of the assumed K's and those computed at the end of each cycle by the ordinary method of successive approximations reveals that these values of K oscillate about their true values which is someNow, if we use where near the mean of two consecutive cycles. the mean values of the K's from two consecutive cycles and repeat the cycle, the process is convergent, and the rapidity of convergence depends upon the accuracy of the assumed values of K. We will discuss the method of assuming values of K to get better approximation a little later. 2. Grout's Method for Solving Systems of Linear Equations. To describe the method it is best to do so by an illustrative example. Let the given system of equations be specified by its given matrix, thus x1 x2 X3 X4 12.1719 27.3941 1.9827 7.3757 6.6355 8.1163 23.3385 9.8397 4.9474 6.1304 3.0706 13.5434 15.5973 7.5172 4.6921 3.0581 3.1510 6.9841 13.1984 2.5393 (6-3) the first equation being 12.1719 x 7.3757 x4 = 6.6355. + 27.3941 x2 + 1.9827 x3 + The solution requires the formation of one VI-4 matrix and a set of final results; thus we have an auxiliary matrix Xi x2 X3 X4 = 12.1719 2.2506 0.16289 0.60596 0.54515 8.1163 5.0720 1.6793 0.0057629 0.33632 3.0706 6.6327 3.9585 1.4193 0.19891 3.0581 -3.7316 12.7526 -6.7332 0.060806 (6-4) ard a final matrix x = 0.15942 x2 = 0.14687 3 = .011261 x 4 = 0.060806 (6-5) Theprocedure for obtaining the auxiliary matrix from the given matrix is contained in the following rules. (1) The various numbers, or elements, are determined in the following order: elements of first column, then elements of first row to right of first column; elements of second column below first row, then elements of second row to right of second column; elements of third column below second row, then elements of third row to right of third column; and so on until all elements are determined. (2) The first column is identical with the first column of the given matrix. Each element of the first row except the first is obtained by dividing the corresponding element of the given matrix by that first element. (3) A , Each element on or below the principal diagonal is equal . VI-5 to the corresponding element of the given matrix minus the sum of those products of elements in its row and corresponding elements in its column (in the auxiliary matrix) which involve only previous computed elements. Each element to the right of the principal diagonal is given (4) by a calculation which differs from rule (3) only in that there is a final division by its diagonal element (in the auxiliary matrix). It might be mentioned here that a matrix is a rectangular array of numbers, or elements. Those elements which have the same row and column index form the principal diagonal which slopes down to the right starting with the element in the upper left corner. The diagonal element of any element to the right of the principal diagonal is that element of this diagonal which lies in the same row as the given element. The diagonal element of any element below the principal diagonal is that element of this diagonal which lies in the same column as the given element. The procedure for obtaining the one-columned final matrix from the auxiliary matrix is containing in the following rules. (1) The elements are determined in the following order: last, next to last, second from last, third from last, etc. (2) The last element is equal to the corresponding element in the last column of the auxiliary matrix. (3) Each element is equal to the corresponding element of the last column of the auxiliary matrix minus the sum of those products of elements in its row in the auxiliary matrix and corresponding VI-6 elements in its column in the final matrix which involve only previously computed elements. Mathematically, if we denote the given matrix, the auxiliary and , matrix, and the final matrix by II Gij 1j tIFil J1, respectively, the method ing equations: - Aii = G. - is contained Aiij 1 , in the follow- _1 k=l Aik ki j -1 Aij = Gi k =1 ij - ij [ k=l Aik Akj AikAkj ] if i > j if i A j n and Fil = Ai, n + 1 - k=i+l Aik Fkl (6-6) where i, j denote the row and column, respectively, and any whose lower limit exceeds its upper vanishes. As examples we have the following typical calculations made in obtaining (6-), the letters R and C representing the words "row" and "column", respectively. A13, or R1C3: A2 2 ' or R2C2: A4 2, or R4C2: A2 5 , or R2CS: 0.16289 = 1.9327 12.1719 5.0720 = 23.3385 - 8.1163 x 2.2506 -3.7316 = 3.1510 - 3.0581 x 2.2506 0.33632 = (6.1304 - and a typical calculation .1163 x 0.54515)-. 5.0720 made in obtaining (6-5)is - 1.6793 x 0.11261 - .0057629 F21 or R2C1: 0.14687 = 0.33632 x .060806 _A L VI-7 Since a modern computing machine gives in one continuous operation a sum or difference of products with or without a final division, we see that each element of the auxiliary and final matrices is given by a single machine operation. When the number of equations gets larger and larger, the elements of the given and auxiliary matrices increase with the square of the number of equations. It would save much time if one would write a "check column" and make continuous check on calculations. The "check column" may be written at the right of the given matrix, each column being the sum of the elements of the corresponding row in the matrix. The column is now treated in exactly the same manner as the last column of the given matrix, the calculations being carried along with those for the other columns, and the result being the addition of corresponding "check columns" to the auxiliary matrix and the final matrix. The check columns thus obtained for (6-3), (6-4), and (6-5) are, respectively, 55.560 52.372 44.421 28.931 ' 4.5646 3.0214 2.6182 1.0608 and 1.1594 1.1469 1.1126 1.0608 (6-7) These columns provide checks at all stages of the computation, because (1)- In the auxiliary matrix any element in the check column is equal to one plus the sum of the other elements in its row which lie to the right of the principal diagonal. (2) In the final matrix any element in the check column is equal to one plus the sum of the other element in its row. vI-S I For example, noting (6-4), (6-5), and (6-7), two of the checks are 1 + 1.6793 + .0057629 + .33632 = 3.0214 1 + .11261 = 1.11261 These check columns are not necessary but they are strongly recommended from the author's experience in working with this method. The check column for the final matrix may be omitted, however, and the results may be checked by substituting them in one of the given equations. The satisfaction of the check guarantees the correctness of the solution. 3. Derivation of the Crout's Method. Define an augmented matrix of a set of equations to be one which contains the coefficients of the unknowns and a additional column composed of the right hand sides of the equations. and Now we want to determine the relations between A.. 1j Gij if the set of equations having the augmented matrix G11 G1 2 ..... G1 , n + 1 G21 G22 .... G2, n + 1 Gnl Gn2 Gn. n+l 1 . And that having the augmented matrix (6-8) VI-9 1 A12 1A 3 A14 Aln Al, o 1 A23 A24 ... An A 2, n+l O O 1 A34 .... A3n n+l n+l (6-9) o 0 0 0 .... 1 A, n+l Since the solution of the equations have the same solution. corresponding to an augmented matrix is not altered if the rows are multiplied by any numbers, or if to any row there is added a linear combination of the other rows, the two sets of equations would have the same solution if the augmented matrix (6-9) can be derived from the augmented matrix (6-8) by such combinations. We see that this is true if [i-th row of (6-9)] = l/Aiif [ i-th row of (6-8) - A l [ 1st row of (6-9)] - Ai2 [ 2nd row of (6-9) - .... - A i, i-l (i-l) th row of (6-9)] or more explicitly Aik 0= [Gij- ZE i -ik Ai j 1A j -Aii (6-10) j = 1 2,3, .... , i - 1 i-1 Aij= ij k=t i i. j = i+ 1,i + 2, .... , n + 1. Rearranging, we have VI-10 i-i k=l j-1 Giji ij i j Aik j kl (6-11) i-1 [Gij ij A k l A Ii k which gives the rule for determining the elements in the auxiliary matrix and is the exact form as Eq. (6-6). This proves that if the elements of an auxiliary augmented 1 Aijllare determined by the relations (6-11) matrix from a given augmented matrix I Gij II , the two sets of equations corresponding to these two matrices are exactly equivalent. Now, consider another augmented matrix (6-12). 1 0 0 ...... O F 0 1 0 ...... 0 0 00 F2 1 O~~ O 1 1 ...... o ...... e............ O 0 0 ...... 0 FF31 3 31 ~ (6-12) Fnl The set of equations corresponding to (6-12) are equivalent to those of (6-9) if [i-th row of - Aip i+l -Ai = (6-12)] i+2 [(i [i-th row of (6-9)] + l)-th row of (6-12)] r (i + 2)-th row of (6-12)] ....i, i - + n [(i + n)-th rowof (6-12)J or more explicitly, n Fil = Ai' n+l - k-i+l Aik Fkl ... (6-13) VI-l1 which is exactly the same rule for determining the final matrix as given in Eq. (6-6). This therefore proves that the equations corresponding to the final augmented matrix found before are exactly equivalent to equations corresponding to the given augmented matrix. the F1 2 , ... In other words, Fll Fn are the solutions of the given set of equations. We shall now see the check columns. If in a matrix the last column is the sum of the other columns, this fact evidently remains true if a row be multiplied by a constant, or if to any row be added a constant times another row. Since (6-9) (6-12) from (6-9) by just such can be obtained from (6-8), and operations, we see that if a check column is annexed to (6-8), the corresponding columns obtained in the auxiliary matrix and the final matrix are the sums of the columns of (6-9) and (6-12), respectively; and hence have the required properties. 4. The ethod of Successive Approximatins. Let us examine a simple case first. Consider the boundary-value problem with n = 1 under the normal pressure p = 100. Eq. (5-25) can be easily reduced to the form w or w o p1 = 3 + .424,507 w 16 + 37.690, -19 W O O 0 - 1.790,888 = 0 (6-14) This third order algebraic equation can be easily solved by the standard method, and the roots of this equation are VI-12 w = 1.098,254, and (-.549,127 + 1.152,878 i). (6-15) For our problem, we are only interested in the real root, because the imaginary roots do not have any physical meaning. Now let us try to solve Eq. (6-14) by the usual method of successive approximations: w assume = 1 w w2 0 If we assume . 2 00,000, wo2 = 1.440,000, e - = 70.27474 = .960,516 960, .922,591 wo 2 = .922,591 for the second cycle, and the value wo2 found from the second cycle as the assumed value for the of third cycle, and so on, we have the values of w found from 2 various cycles as follows: 1.767,416, respectively. .667,554, 2.689,324 etc. We see these values are oscillatorily divergent. A plot of these values against cycles shows that they oscillate about the true value, 1.206,161, and the true value is approximately the mean of the values from two consecutive cycles. If we now take as the assumed value of of this value and w 0 w 0 2 (Fig. VI-1) = 1/2 (1.440,000 + .922,591) = 1.181,296 for the second cycle, and the mean the value found from the second cycle as the assumed value for the third cycle, etc., we have the values of w 0 found from various cycles as follows: 2 cycles w 2 assumed 2 1.181,296 3 1.212,550 4 1.204,658 5 1.206,524 w 1.243,805 1.196,766 1.208,390 1.1.204,526 6 1.206,075 7 1.206,182 1.206,289 1.206,131 ' found 1 IMF VI-13 The process is convergent and w converges to the real root of Eq. (6-14). The value of w found at the end of the 7th cycle is 1.098,240, and is accurate to four figures at the end of 5th cycle, where it is found to 1.098,010. The results are plotted against cycles in Fig. (VI-2). It is to be noted that K n = 1. = - 3 wo2 in the case of The method of successive approximations would converge if we assume K It is found that to be the mean values of two consecutive cycles. this property is the same for n > 1. If we take the mean of K's or S's found from two consecutive cycles, the process is convergent; but is oscillatorily divergent if we follow the usual way of successive approximation. It may be pointed out here that for the special case n = 1, if we assume the mean of the values of w 0 from two consecutive w cycles, the process is also convergent, and if we assume 0 2 of second cycle to be equal to the sum of .6 times the assumed value for the first cycle and .4 times the value found from the first cycle, and so on, the convergence is much more rapid (Fig. VI-3). But it is not true for the cases with n > 1. The rapidity of the convergence depends upon the accuracy of the assumed values of K's and S's for the first trial. To have a good approximation, the following procedure is found to be valuable. (1) Always start with less divisions (smaller n) to more divisions and with smaller normal pressure to larger ones. Because when p is small, the values of w determined from the linear small deflection theory would give a good approximation for the VI/- /~. V/- / i ji I t I I t2A 0 /6 i I' .o is ( !t 4 o e r- 0 7C C ec/C I V/ ,-'/ <2 V/ - atl /4 /.2 0 ii .6 o 3i,4 ,V, ,ber of- CYc/es Es v/-/6 I /4 .6 .6 A7- / -'d .4 0 P· -Al'n e )Y4 C 5" /es VI-17 first trial, and the small deflection problem is very easy to solve. (2) For the case n = 1, we first start our computation with a small p and use the value of w. determined from linear small deflection theory at the corresponding p as a first trial. Then we can plot w against p after we have found the first o 0 w, because we know that the w 0 - p curve for the large deflection would become tangent to the at the origin. w 0 ~ p curve for the small deflection curve For a larger value of p, w can be now estimated by extrapolation. (3) at a small p. i i case For the cases n l, we again start our computations For n = 2, we can now assume w n = 1 as a first trial. w2, 0 as found from the w 3 are still difficult to assume. i We may use the values as found from the small deflection theory as a first trial. It is found, however, that we could have better results if we use the ratios (w2 /io) and (W3 from the and small deflection theory, and compute o) as found w2 the values of W 3 by multiplying these ratios by the assumed value of w. 3 0 When the deflections have been assumed at every net point, the values of K and S can be computed. may be used as a first trial. the true values of and (en o)'s are These are the values which By successive approximation, W's are then determined. The values of w now plotted against p to estimate the corresponding values at a larger p. The values estimated by extrapolation may be used as the trial values corresponding to that p. process repeats until the maximum p is reached. For n = 3, The wo from n = 2 is used as a first trial, the other procedures are the same. VI-18 A sample calculation is given in the next chapter. CHAPTER VII THE AJETHODOF SUCCESSIVE APPROXIMATIONS: SAMPLE CALCULATIONS 1. The Finite Differences Solutions of the Small Deflection Theory. Now let us study the small deflection theory of simplyThe differential equation is supported square plate first. 4W = p/D (7-1) and the boundary conditions are w = 0 along four edges axx = 0 along x = + a/2 (7-2) ;2W ay = 0 along y = a/2 where a is the length of the sides (Fig. 7-1). Non-dimensionizing Eqs. (7-1) and (7-2) by letting w' = w/h, p' = p a4 /E h4 , x' = x/a, y' = y/b, where w', p', x', y' are non-dimensional deflection, pressure and length, respectively, and dropping the prime, we have our boundary value problem as follows: V w = 12 (1 _<2) p w = 0 at x = + 1/2, y = + 1/2 a2wx va2 = at x = + 1/2, = 0 at y = + 1/2 (7-3) (7 a~~~~~~~~~~~/ yr ! I a. - cx -z I I , V// - / a VII-2 Following the notations as used in Chapter V, the finite differences equations for the roblem become + 5t4yw = p, b4x w + 22xyw (W)x= + 1/2,y = + 1/2 = 0 (7-4.) ( 2 x w) = + 1/2= (a2 W)y = + 1/2 =0 where p' = 12 (1 _2/A)( (1) n = 1 )4 p. (Fig. V-3) The finite-differences equation after combiningwith the boundary conditions is 16 w = p' (7-5) .w 0 = .0625 p' = .042,188 p for /A, (2) (7-6) = .1. n = 2 (Fig. V-4) After being combined with the boundary conditions, the finite differences equations are: 20 w0 - 32 w1 + 8 2 = p' -8 w0 + 24w 1 - 16 w2 = p' 0 - 16 +2 wO 1l + 20 22 (7-7) Pt Use Crout's Method to solve these equations, and denote the check collrmnby C.C. A i. The given matrix is VII-3L t C.C. 1 IV2 +20 -32 + 8 +1 -3 - 8 +24 -16 +1 +1 + 2 -16 +20 +1 +7 = ' The auxiliary matrix is +20 -1.600,000 + .400,000 +.050,000 -.150,000 - +11.200,000 -1.142,857 +.125,000 -.017, 57 -12.800,000 +4.571,430 +.54 6 , 75 +1. 546,875 8 +2 and the final matrix is +1.031,250 + .750,000 + .546,S75 The solutions of Eq. (7-7) are therefore W0 = +1.031,250 p' = +.043,50 p (7-8) W1 = + .750,000 p' = +.031,641 p w2 = + .546,875 p' 2 where/ 2 is taken to be equal to 0.1. w (3) = +.023,071 p n = 3 For = .3, = +.032,989 p. 0 (7-9) (Fig. V-5) After being combined with the boundary conditions, the finite differences equations are: 20 w 0 - 32 w1 + 8 w 2 + 4 w3 = P' -8 w 0 + 25 wl - 1 6 w2 +2 W0 - 1 6 w w3 + 6 w = pt + 22 w 2 + 4 w 3 - 16 w4 + 2 w 5 = p' + wO - 8 w 1 + 4 w + 19 w3 - 16 w + 2 w5 = p +3 W1 8 2 - 8 + 22 w4 +2 2 + 2 w3 - 16 w4 + 1 8 w5 = p 5 = p' (7-10) VII-4 ) o 4 C ct;O It \ 0 c\- L(ZC - t -':t E- 1 Q c, o c \ to 0O 0cN0 , 4 tt + + I + + + o0 lt'4t N t\% tw "D N-ot \. o V 0 H + H I n + + r+ 0 t- to 0 a H ri s mo r * 4 it*\ 4 -V to -) c ,,0 0 0 O 0 0 4 o ++ + + ++ + + + ++ + ci in 0 0 N + N + rC to0 I + 4,U) -c 4+ d) rd a0 Fr4, .) rl + I U] I + I O rC - H I HO I in to ~ ot 0", 0, 0 0 n to to 60 N+n-t I I O 00 0c 0 -t 0 -o + I + + + I + 0 -H ricq N U) to0 + N + + to I I I O O 0to 4' 00 cN- O H 4O C) 0% 0 rtl 9 + I I + 0000 -000 r03 X N I \0 I + I M + H to C + o (H H I 0 0 to cxH 0 0 + I + + w t + I I 0000 * L N I 0 0 0 00 0 0 0 0 to I + + Nc H + + 4. N to 0000 00 00o 0 + 0 I- 4 N + 0 O r-i I + .H I H I + 0 m -- ON r + '- U) to + H 4, rd VI-5 +5.246,672 +A 597. _ (q +4.031,250 +2.735,207 +2.402,367 +1.439,164 The solutions of Eq. (7-10) are: w = 5.246,672 pt = .043,722p W 1 = 4.597,633 p' 2 = .038,314 p = 4.031,250 p' = .033,594 p (7-11) W 3 = 2.735,207 p' = .022,793 p w4 = 2.402,367 p' = .020,020 p w5 = 1.439,164 p' = .011,993 p if /2 is assumed to be 0.1. If /e is assumed to be .3, we have w0 = .044,208 p Timoshenko gives the exact value of (7-12) w for simply-supported square plate as (Ref. 1, p. 133): w0 = .0443 p. Therefore the solution by finite differences with n = 3 has an error of 0.23%. It is seen to be sufficiently accurate for engineering purposes. The agreement of the finite differences approximation with the more exact results of Timoshenko is sufficiently close to encourage one to apply the finite differences approximation to the problems with large deflections. 2. The Large-Deflections Problem, n = 2. After combining with the boundary conditions, the two sets of finite differences equations are: A i VIi-6 20 F0 - 32 F 1 + F2 + 4 F3 = K0 + 24 F1 - 16 F2 - 6.632,456 F3 + 6.632,456 F4 = K1 -8 F +2 FO - 16 F 1 + 20 F2 + 4.632,456 F3 - 13.264,912 F4 = K 2 -2 F - 4 F 1 + 4 F2 - .632,456F3 + 2.632,456F + F = S1 - 6 F 2 + 2.316,228 F3 + 1.367,544 F4 = S 2 (7-13) and [20 + 21.6 (%+ ' + 'o 0) W0 - [32 + 21.6 ('0 + f 'o + 2 y 0 )]1 + (8 + 21.6 Y0) -(8 + 10.8;'l ) wo + = P' [24 + 21.6 (, - [16+ 21.6( 2 w2 1+ - [16+10.3 (0( 2 +p' 2 1)] )J + + +,1) 2 =P l + [ 20 + 21.6 (0(12+fl2 + 12 )JW =' (7-4) It is to be noted here that the left-hand side terms of Eq. (7-13) do not change if we vary our assumed values of K and S. Eq. (7-13) can be therefore solved uniquely in terms of K's and S's. The given, auxiliary and final matrices are obtained by Crout's Method as in Tables 7-1, 7-2 and 7-3 respectively. Tdore significant figures than required are used to ensure good results. The solutions of Eq. (7-13) are as follows: F0 = -.048,703 K0 - .265,696 K 1 -.225,111 K2 -.304,114 S1 -.309,525 S 2 F 1 = -.111,203 KO -.307,363 K1 -.235,527 K 2 -.262,447 S1 F2 -.238,692 S2 -.103,085 K0 -.311,962 K1 -.221,052 K 2 -.162,880 S1 -.317,642 S 2 F3 = -189,937 K0 F4 = - .506,498 K1 - .316,561 K2 -.253,249 S1 -.126,624 S2 .094,968 Ko - .316,561 K 1 -.269,077 K 2 - .063,312 S1- .221,593 S2 (7-15) rVII-7 0 X4 H "!? CN Hi 4 N m co O o 0 it O C) -(N N t - ON -4 0% 0 t . 9 VI\ N\ -t (N \0 · 9~ C0\~ ic 'I~ I * + + 00 00 00 0 00 0 00 o \Q I 0 too 0 O (N N (N \D - \ .- n 4 Cr N c4 D\ \O 'lo N H CD I 4- o 0O 0 0 0 0 IE I 0OH0 O 0 O 0 O O 0 0 0O 0 09+ 0I + 0 0+- 0I O 0 0O 00 00 O + OI OI O 000 0 0 0 00 0 000 0O 00 0C0 00 I + + CI2~ 0 .4 L I o0 0 Hr + mr, 0o\ C O 0 0 CD 0 + 0 0 I + vii-8 I r OQO b- to M ri\\U ) to 0 t/ 0 c't to t O, N CO c C12ON r N( + I 'n I C1 r O c\%% O O o uD -o < O N--- ONto r-t N rC r C'O '~ OC 0 + I I o N - ci 1-% to Cd 0 -tN>'Nt o +N o to -4 I -t Co I rE r-4 0 N H to H o -t r- N 1> *, -4c wH .H C· I t I 0 tO 0 .) C4 r4 .O \.Q N-H C 4ri 9l + I + I I + cQ1 00O OO00 00000 I o -H LtoO", ++ 00000 +I+ + ,+ C1-n I I o COHtO' r-I O", O .O O H NO O -l\ HO - n OO + + r- o CTl 'O C,, tO CV ,O 00000 00000 4% *N(TND c"J 0', L' 0,4D H CH 4 C% · OOr-O +++ · · H ® I I VII-9 t) 0 4t"0 I H tO r0 CN to ', \C , rt IH ' < -It r 0 0O -t c. 0 X -. rV . o o 90 I t.. H .F 0l . . .' 0 -- N H , H n 0 (N c oC-H cV \ r-A E-Q -0 C S C.t' ' fwO r\ Ho (NI , 0" (N o40 0C r4 0 r H \) tO- O0 ' 0 \ to 4 -O cro I I I -0' 0t ' I I VII-10 Let us now give Let p a numerical example of the computation. l00 p' = .042,1875p From the wa/W p,w 0 4.218,750 P, 2wO p curves (Fig. 9-1, 7-2, 7-3 respectively), it is estimated that w = 1.135 wl/w = .7535 w /wO = .5775 We have, therefore, for the first trial, w0 = 1.135 w1 = .855 ,222 w 2 = .655,463 Write these values at the right corners below the corresponding net points. Use the finite differences patterns as described in Chapter V (Fig. 5-2), find , , and S at the net points (Fig. 7-4). %= Y, (wn+ w n) and then K As an example, we have 0 = -2 (1.135,000- .855,222)= - .559,556 = 1.135,000 + .655,463 - 2 x .355,222 = + .080,019 K0 = (.030,019)2- (-.559,556) 2 = - .306,700 Similarly, we find K1 = - .189,997 K2 = + .221,966 S1 = + 2.368,276 S2 =+ 1.373,368 From (7-15), we obtain the values of F's as follows: xxN I i : ; i j i i I i r : I I ! , i i i . i ; I . i i i ) i i 1 I i , : 1 i I I (1 N N· NI ! I i i ; t l i ; .i i ; : i : i i ic ~1~ I i i 11 i ; ; i : i i ! I i ; li'i 4. IZ Pi 4.) N Rn MS 11) I I N\ N N~ f I I I I I i . . !. i . I I .~~~~~~ i I i : I ., i ! i . . i j i - I j i ' N: I J--- )~~-: u VII-13 F0 = - 1.129,866 F1 = - .977,802 F2 = - .780,162 F3 = - .689,444 F4 = - .424,723 These values are substituted in any one of Eq. (7-13) to ensure correct computation and then are recorded at the net Similarly, we have the values of o ', points as on Fig. 7-4. P', ¥' recorded below the corresponding values of F. Now Eq. (7-14) can be written and the given matrix is W 0 w1 +34.122,771 -47.107,213 + 8.984,442 +4.218,750 + -12.269,024 +36.930,948 -20.392,900 +4.218,750 + 8.487,774 + 2.000,000 -19.408,458 +28.313,451 +4.218,750 +15.123,743 W2 w = 2 C.C. p, .218,750 The check column can be written by the following relation: C.C. -4 + P' +10.8 ' + 6 + 21.6 ('2 + p' 2) + P The sum of the elements in a row should be equal to the value of the element of the same row in the check column. This procedure provides with a check for the correct substitution made in the given matrix. The auxiliary and final matrices can now be computed and are as follows: +34.122,771 - 1.380,5213 + .263,2975 +.123,6344 + .006,4107 -12.269,024 +19.993,2990 - .858,4124 +.286,8773 + .428,4649 + 2.000,000 -16.647,4154 +13.496,5082 +.648,11,1 +1.648,1118 VII-14 +1.117,078 + .843,225 + .648,112 The first approximation gives therefore w0 = +1.117,078 w1 = + .843,225 w2 = + .648,112 A similar computation as outlined above gives K0 = - .293,781 K1 = - .184,115 K2 = + .214,841 S 1 = +2.299,072 S2 = +1.339,974 As a second trial, assume K0 = 1/2 (-.306,700- .293,781) = -. 300,241 K1 = 1/2 (-.189,997 - .184,115) = -. 187,056 K2 = 1/2 (+.221,966 + .214,841) = +.218,404 S1 = 1/2 (+ 2.368,276 + 2.299,072) = +2.333,673 S2 = 1/2 (+1.373,368 + 1.339,974) = +1.356,671 The results of the second trial are shown in Fig. Following the same procedure, the results trials 7-5. of the third and fourth are shown in Fig. 7-6, and 7-7 respectively. The corresponding assumed and computed values of the fourth trial are assumed computed K -.300,446 -.300,006 K1 -.187,407 -.187,472 K2 S1 S, +.218,738 +2.337,941 +1.360,090 +.218,786 +2.338,531 +1.360,631 VII-15 The first three figures check with each other, and the results corrected to the third figure after the decimal points are wO = 1.1269 w1 = .8502 w2 = .6528 and this is our required result. 3. The Large Defectins roblem, n =3. When n is taken to be greater than 2, the same procedure of computation as that in the case of n = 2 is still valid. As an example, let us consider the case of n = 3. When the square plate is subjected to a uniform pressure of p = 100. After combining with the boundary conditions, we obtain sets of differences equations (5-34) and the two in Chapter V. Eq. (5-35) as given (5-34) can be solved in terms of Kts and S's, and the solutions are given in Table 7-4. From the w 0 '' p, w4/o&p, - w1 Aw p, w 2/ w 0 -vp, w5/w ON-pcurves (Fig. 9-1 , Figs. 7-8 to 7-12), we obtain the following values by extrapolation: O = 1.1247 = .8891 = .7932 w3 /wO= .5516 w o = .5037 w i = .3497 wlW 0 w2A For a first W3 /o 0 -p, trial, we assume: T i /// - // ii i ii I F,';. V//- 4 a a Asr-revm c &: r)= o />rsf Approx. Ccz,f ed: S= . 29 7o ~se= A .339 ~7# - 7 o, ' / 799 oc =- -. SJ,7oy (=f' P+. Y--t. o,8ad4 Y_ +. -. z, 7. /S,46/ Ale=# 2 /9 6-/ / - cossued /C of/e: - compefed (4ZI-4= '7 95776 (.4 wf,- -&1077 /35,oto 03'-O,304/Z r- .04iO/,9 jf - ,,/m dA6 , /. fE -. 3O6,700 a=1oo/ - -.9,7¢, o8 -. o<.= -. sl,, v'= . o07 o/ = -. a4e4 , 9,-7s, /,= - a9, 997 /.//7o7d / =&-. ,4, 7°6' Cx -.- .. ,?. - <o= -. 93. 7d/ = -.39,e2 Y= - / . //, X/-- d , /- 66g, - 0 T V11 - /7 - - a a = S, = .9e X p=/ /0 avz~dd Se co/·l>/ Approx. -. 770,4 /6 ",- - -f/,9 -IF/ a d=4 = _V "9o7 y· . 63.07 "(d.6 d16 K = -. z/ f 5 7-9/ o . o =,A=-. Sfp9e Y' -t. Of ,6/ Y= o79,-OS-// 19f A - :2?7.,)s A<=---3,o6 - -. 6 o6./, O -. 367. {a·& /- +.dJ. 332 Y'-. odo7/ A=r -.-./?7.9 A:= -9,34 Y= - .1,7. 4,ee <- - Id, a0v V/i'- /d V//- 6 I 0 a o--= t .j, o = /00 Sa = *+/. , 3*3 n/=p 7-rci' Appro .. -.4o,74 -.772.z6 Y-$' = . Y'= +-.o ogp , _.'P. 4 SS . A·--+.2-/8eA.e t. x =,e . .- 0Oay3* 7d,s e - .?3z.eo- o=, #,=- O ,<=P5 . so. /d . o4A, ~gp o<_ - . ,*7-0, a7 / ~=.39 ,,y. /<= .- /d. ds o 0 v/' - F'/9. V/ - 7 . l 0 a S, -=v.3d, 6/ ls - O, 63 / Fo urft -. 77, 7.3? '"P'= *-.,(..34 Y'=+ *.od, 067 * 4eoS, C9106 oea =- fv3 *- - u! ~ 'p /K . 79,. 2 -. -,oo o 7, - lwq owt 0 P r= +·o5r ede6 o -.r6, 786 ot -. //d,797 /-/.? /9 bY = Y/- -t. -f-f- Aprox. 0 ? .6-foa-2f7 ft = - . J 'j ? 16-f.6 6 ?3 ,79%3/ rC - ,p " *" yO' +..o g,/ Y=-/97 4c9 K/' -+. -./BrZ 4;dP 9 VII-20 w0 = 1.124,700 w = W2= w = W = w4 = 5 .999,971 .892,112 .620,385 .566,511 .393,308 Again write these values at the right corners below the corresponding net points. With the values of C, p , , W r W cotnm- puted, we have K 0 = -.061,945 K== -. 052,063 K2 = -.024,186 K = -.023,043 K3 = +.001,252 K4 = +.106,245 S = +1.592,696 S2 = +1.282,838 " = +.548,700 By Table 7-4, the values of F's are found to be F0 F1 F2 F2 = -1.095,495 = -1.028,996 =.950,911 = - .868,159 F3 - .762,520 F4 = - .505,761 F 5 = - 675,850 F6 = .546,620 F 7 = - .239,729 Write the values of F's at the left corners below the corresponding net points, and compute the values of oc', Substituting the values of Eq. (5-35) and noting that p' = ', .008,333,33 p ', 4', and '. and ' into x p = 0.833,333, we have the given matrix of the equations as in Table 7-5 ad the auxiliary matrix as in Table 7-6, and the soluticns of Eq. (5-35) given by the final matrix are N i ii : i i : i I : ; I : b 4 N i 9C i : i r N 1 £ i 5i t i i i. z I ; IT . i i I ! 1 . *1 "4 NI I,) t t C" 0\ I i I ; : i t I d1 A vs N1) t1) y n) N N : i i . : 'Ž1" i ' 1 t I i j i ; i ; i . i l ; I .I I . I i I i I; I I i . i 1 I I . : N N i i I : i KOx i t i t i . i i . I I I ; ; I i i i i I i i I I I i I : I t i : ;I I i i i ! i i . i . I . : i i I I : . ; i i i : - I I I p %' >4 i i I, > \ 'V tz '4) N 1 NZ Cu N' i !i i i :I I i ½ . r i . i I i I . i I I : 'Nr K N' ½ i i I ) IC 'C) 1' I:· :: i : i i I I : . . 'Li 1 D O L; L I I Oi I I i P i o o a I 1) - " 'a "I 1.) il 14, viI-Z6 r -4 \. 1 c ' 4. U0 tt, ®N N · "\ 0"0 " N c· C 0 N · - · 1>N * . . . * I I I I I L'" >C .<*- ->-N -4 c\) 4"~ -4c0 4 N HtO N H 0 N '-C t)0to4'.ot -4~ft * . 6 . I "..1'"-t - '...1'-Lr\ N to N "-.t -- -rl -r- H u ", Ht tc\ o '. u N 0 \o o c -,t C 0 "0r'.4: 0o 0 , -4 06% Oe'\0 4SHto c c, o a . . 6% c, ,N 0o \ 06% 6% r6 fO H .-'cH 0 H cr\H N O'xH r- ,- I I I I I I I I I CmO 0oc H -ix0toC (O' CO 0 N Vt rH (OtoN 0 C O 0' ' \ iG 0 NL r to tr0 0 CS .- 0 , H (TN c* *- o t\ - oo 0 N H H N Cn N r -i -. . . . . toH ONN O I I I I I II II I ii II II II rX4 F% F4 r* c 4- PC4 . u' P4.0 F44 I -rO \C n, \ n t O 0 0 a' H 1--i\t' --. 0 "L) toCO 0 (v4 rO- L 0\ * m I rHI L. 1- H N H 0 0 I ( 'N;O0 . -4 iS 1>-H 1>H C" . .-', -;Cr -4- 'r\ O 0(1 N. C - (N' i II II fk4 PZ to to O cC o VI-? VI _ w It =1.123,384 w 1 = .993,956 w2 .891,465 w 3= .620, 342 w 4 .565,591 v 5 = .390,999 ight be pointed out here that the check column of the given matrix may be obtained by a direct substitution using the following relations: C.C. pt -1 + pt -2 + p' 2 x 10.8 / 4 + p' 1 x 10.8 '5 + pt 6 + 21.6 ('6 + " 6 ) + p' This procedure would give a way of nowing the correct substitution in the given matrix since the sum of the elements in any row should be equal to the element of the same row in the check column. Find K,0 values of ws. K K2 , K3,K ,4 S1 , S2, S3 from the computed K 5, Use the mean values of the K's and S's assumed and those computed as the trial cycle and so on. for the second At the end of the third trial, we have the assumed and computed values as follows: A values first N (r 0S1 t' crA'\ N NoH ' ON NO N ON Cf Cr~ Cr\ ON tfo > > * , O O O CN 0' -- O--tCO + O m +N 0 + v 0+ rH H N H Mo N N tC\ 0 \N H O -4', O0 O 0 cN 0 0* H o N \C 0 N(D 0N 0N 0 0 N (-\ CT 0 0 I + (0 O 0 0 00 Q)@+ '4\ ; t'\ ON 0 00 Co i C. N0 + + 0 0 o 0', + O H ul to \ COs " riX 0C\ C\ , Cr| + C<A\ * NO + Iv + - + C\ * oI ' C'\ CC + i H I+I + O 0 + + 0 r-I q E-.. o N oN ,Ot'. -0 : C ON \4} to 0 0 \N 0 H . -4t 00 N 0CD CT, OO 0O O . to + to t - I + + 0N C c 0 04 O -4 (', 0 0O O I+ 00 to o' C o 0 I + I i + ((N O 00 N'. O,, + NO+ O NO 0 00 O On N O O (N O N H + I + + 0 0 + 0 < VJ-r-n 0 Or. t- ,,© IC0 (" u~ tO 0 -'t (rO2 cffi °+ G + O C' ' (e r-I to" ,\rC C rl 0 + C ,- , rl · ~' r- tnJ l (S\ r-ti -It o cj 0 + + 0', * + d~ '. '0 +I 0)+I to 0 I- to, r -t tH r\H to, I r-I to 4 H b "4 - ] 0 *.t H 0 m) O t.) 0 0 0 '" +(3 1 ,-G o CA- 0', + r 0 '.tO t ~ O to tO + oU)C Ov~~~ O' r I ON +t CD + -0 0 o (D (., ~ -t +- C o0 90 - 0 0 '00+ + '~~~ciXCd0 ( to O > (N O 'O H D '.. 0 00 4J\ C 0 H ( oo '. r-t O4 Cq + - i+ + O to (3" H O H - O O O I + I I+ O 0 O + + Hs + I 00 O VII-30 .023, Assumed Comouted K0 K1 K2 3 K K3 K4 - .061,763 - .051,947 - .024,660 + .001,614 + .106,177 S ++1.592,106 - .61,695 - .051,894 - .024,799 - .023,477 + .001,697 + .106,204 +1.592,078 S2 +1.2831,878 +1.281,814 S3 + .546,560 + .546,173 These values check with each other to the fourth figures after the decimal points. accurate to The deflections at the various net points, the fourth figure after the decimal points, are wO = 1.1240 wI = .9995 W2 = .8920 w3 = .6207 w4= .5660 w5 = .3915 The results of various trials are shown in Fig. 7-13 to 7-15. -AL F7', V/ - . 3 - . a A /. re,4. 36 SS =i 5 ,o 700 ,.J-", S. /00 n=3 /. z.~o, o80 0 For'sf AvcroX. ...ss'oc d A - %s4J~ 0pf =)S~ i-n e C- 4 ,rv- l -- - 3. 39a 72) -. .o..76l .33,3 0 '= - . .o2o7y ot=/=-. S0/ · ,= 026303 o t.69, //4 0l //,jooc=~e- 5qsq 9/ =~* )/'_+.od6 -. -. Al 7742 Y=A-. 4/d6+. e a =-. o, /6 7464.f0 * 021,=f 02 / -+ /20 Y,-!-+",,, 4 - 9. -- Y-+.o/..f6 Y=-+.0/,70 A.=--.o6 y /= . . y= +.o/6.7 -- O. o,of d , 379,f 6 · -~~~~~~~ -ol.ooj9 +. 9-99,971 *66. = *.o9. r, .o-rY (. o../ .0 d -,.6/Jo 1'= --. /. 7/6 Y r. /.3 0 -b _ ,,-7/0 Y=+ .1/73. .7 -c- -. ,/d6 -. / = -. -7 ,e 9 . A05, 7 0 .f 654 dg t .620Jf a = - f0, 7. 0?/, -,O-,7z I Y +: t.Opj"SgP X - +. vo6, C /=- -.// , 797, J( . /, 7-r .. -1W .3 o,=,,,0'Z. 7o 2 r=t-3o 999 Ks . A = +. /74;4; .-- +-./s · ++..,9,2,9 vo. 99sq oc =-. , o *3A. 6/ /07.s. -te2 of=+. o65As - '^1, 729 Y= -. ov, 679 _--J 0: t. 6420392 0<-. 24/, 7d / =-. /_ 3 =- ,so. o 4PU-,'d97 72 0.j' V/ -& Fi9. V// - / a SI' / 76 / /. f - 0 /00 - 2, 6 S, = S3 * 6S2ie sS o/rw S Ap ra . -.^sof894 .39/. *Od X =/-.dV, / Y'== -0od,46 o0 = -. 'd X- .- P/r LT f 0 6/0 /(==* lode0d o / oZ ,0d4 T _9 7/0 otB +. + /, 2/S x =/=,- 4.e,/2 t= + -.or44 A = -. dA e - .7, 7.2 , %fr6/ oe - . "IP4j~ 4..6 /74,dJ of'--+. a-eO ,''=.oo Y= -. -..S4. 7 7.4, t K -FOCo/ 72+ Y'_t./0Z@d3 . 2,Jo -A 07 4, / ..' 1-04"'Pz &JC~~~d 04= _=-. 18 + QJP +, o Y= -. o/K.~ Y= - 014',Pf f,/o .97s970-4 -/, 0-, 7?3 .·7P9 77 --. o%,ejz7 Y- . or,.r4q = -.-. y _ . a,-, ao0 X = -.oa@/ Y17 Y -. oe . all oiUX7 *6eo,667 151" oc-.-.9, Y = +.od'. 747 A = .oej,9 -. d67*0* 0 a o I T- VI/-.3 i ii I i I II a ,/. S -, 078 3= /-= /00 /. .d/,6/4 · 46. /73 Th Apo o. , .fsol 7/7 39-R 91 P9. ot=0 +.~h+ 09 -. -6 144 6 C /(z=~y~ -.o4.z.7p / C. ./7./Oe A= - -/,-,4, fd/.ey=,= /.t0o-Z4d16 -/.,, * dC' . u79do ,= -.#.d3 // *~7. 7) -/,. .,979 s44 o,$ -.a67-7a /I 4'D 0- Y= 0d,9_7/ O-. A =-. og, 6- y Y-t. Ou.gdO6 Y== . .jo- _ x = _ .Xo/, d9 ,74o YC+. ,r 7 o97 d-' 6eo,7*6 d = -. o9,94d * Y'-=* - - Ki" = +.. -. 4,od38O 1-'=. Y--+. r. 700 A~=-oU..#7 --d7,,/7 0 ITI -1 CAPTER. III THE 1. LIPTO PETHOD The I.Method Explained. Wren a more accurate result is needed, the plate rru.st be divided into a set of finer nets. The numrLberof simultaneous equations increases as the number of nets is increased, and the labor involved in thleirsolution increases rpidly. In avoidinF the solution of simultaneous equations, an insreniousmethod has been developed by Southwell(9)(G)(6) which is named by its inventor as the "Releation In the compu ations by zlvxta.u>u PLetho!'. Lethod,as explainedby Southwell, "we replace exact'differential equations by a finite-difference approximation, then seek values at the nodal points of a regulr 1lthouP h, this method has been discussed in a book for l (61 ) net. and in a number of papers by Southwell and his collaborators, it has never been explained, at least so it seems to the author, to a degree to make it easily under- stood. Essentially, the idea behind the treatbmentby the relaxation method is just the same as that by the Cross's method of moment distribution in the case of bending of continuous beams. It seers, therefore, most easily.to explain the relaxation method by a comparison with the momentdistribution method, since t+;helatter is well-accepted and is familiar to most structural engineers. Let us examine the redundant beam as shown in Fi., WTII-1. ) V/I- X 6O 0 / 6 U o/6. /on. /k. // 4. 4 i~ +~ rT F If o( 30I z ~ w i/ /1 &OI.,be' r 77 .III- 60,. V/// +oO90 * ? oao - 600 1-1 001, / 9000 -+1o-00 J 00 V/I - 2 VII - 3 Th.e -.r'cd,r. of *..-.tairn e the redundant -striril.ion bs .... ior is to assle , S asslltion, s zer fouznd vw'.tbouit we see tlat orinci i fPC ,;. .l The resuil. is not satisfied te , C' D, can be :ere TT-T are sa-isfied, and the "'he co-ndition of euiliri are uiialaned nonents iu and C. at method no-wWofers ;-a-rocedure to halanoe tihese -~°~~~~~~~~~~~~~~4 essenrlta11r satisfy -is show n i A an since there moments bo.a relaxation un.talanced ~ moment at each of the four suprort s the end-moments' atj A, ons -t;e boun.darr conditi TrIe moment distribution -- the -tat sope of' continui!.: is also satis-.ed Le horrever moments b the is well k-nov4m.The first step in the moent distrimwethod 1 ysi .a sui.nort the same. based eldrmowme.tsat , fboundaTrconditions on cosi-'stent 4 ,CS, ard r and '-he cn o itio defornaios are assused to ontinui ;r. The the relaxation -, unbalanced oments at "3and C are then distribted based on consistent defomations, Te difference lies in tat t-e relaxaioi .ethod offers more freedom in assu.ino te end imoments and tlere- .. ;more raiJd, fore could mak the ecorverren.ce of the oerations -other hland, however, t, becomes difficult; to asume these values. applies only to redundant As the method of momrentdistributions structui;res, the anlication of the re.axr.ation an} -j a-,7rlication to te mariial differential stud of enineering On the retbhod oes much frther, eati.,-nsor has brou.:r.htthe sciences into a new era, because the boundaryr condi- tioins are now no lonorer difficult ,s an exarlle, to bel described 1evmay illuzstrate and to be satisfied. the procedure by studying the Fi we I small deflection theory of thin plates. '~~}t~''l' ! . wL, where w Letting w and p are the nondimensional deflection and pressure, the equilibrium equation in terms of the finite difference is .. 2 w + y4 w w + 2 Axy 12 (1-A.2 )(A 4 . . (-i) .1 To solve the problem, the domain to be investioated is drawn and the net points chosen, Values of w are assumed to satisfy the boundary conditions and are then written in adjacent to each point.of the net. From these values of w, the residuals Q at points (m,n) : ) -2 + -_ 1 4 m+l,nW-l,n m,n W m,n + 1 (n.+!, n+l m+l nW+n+.+ i_ Wrva..l, n-+l + wim+, n + 2 +m, l m-2, n +Wmn - -z - 12(1-t 2)(A). are computed and recorded. (8-2) Q, thus cmuted, The can 'betought of as an unbalanced force which must be removed from the systerq.Now,instead of setting up a specific iteration process, it is merely observed that if the deflection l''~llllfixed, t one point (m,n) is altered, all others remaining the residuals will chane the "relaxation pattern". distribution according to the pattern, Fig. V-2, Each change of w at any point effects a re- of the residuals, Q, amonu- the net point, and such changes of w are desired as will move all the unbalanced forces to the boundary. These 'operation instructions" may appear to be very vague. Indeed, they are vague. >-";".l ~'ll~t~:'~- A but it is also the source Their vagueness is a disadvantage of the method of the great power of the method because it S , I I·-. ";·. ; ;· VIII-~' Bri: permits the computorto alter the procedure to attain more rapid ap- ;, ... I.·· ;:··;I- proach · to the final answer (of no residuals) without violating the recuirements of a fixed routine. There is only one way to appreciate fully the meaning of these remarks and that is to do a problem. le ,·1 · ;I" 2. The Small Deflection Problems of Rectangular Plates. se ·I --. q The non-dimensional equilibrium equation of a thin plate in bendinp-with sall .ar: 7-:·- deflections is A p and the boundar, conditions for a simply-supported plate are ,·. O alongx = +-, 2 w .· *-·-· a W = 0 along ·; "· .(8-3) ax rg,· x - 2 + - 2 Z2w = O alongy = + 1 ay 2a Sincew = 0 alon x along those edges. q.r. y_ + -( 2 b 2w + , Similarly, is ................ ................ w and ; (8-4) must also be zero 7lb). These + zeroalongy - properties enable one to write the boundary conditions as (. ··· w = O along x = + -I = ( + -;;j i-' = + (8-6 - f (b)...... a 'ri 72W = cr'. + -2 ax2 :1··1: .1,..-. w a Y2 O along x + -2 y ·- v - The differential equation can be written as ·. -'·:' ;· 2 ,2 ( Letting ··: w) = :l, we have our boundary-valued f' -F= p. roblems as follows: p i = 0 along the four edges ............................ (86) ·,- . I" ;;' :·L; '·· · :;i'·c: :" t. iri Oi ;-· :". "I, -.- ·--- -- ·- 1·1-:----;- -;i-~;;~,~;atr~-~~a~ · --- ·-- ·-- ~~,~~ · · ·?6~~- liiil .Y..'.F~~~ l~.r~PII·*~711 i- · Cr VIII -6; and V2 = ........... (8-7) 0 along the four edges ........... W The problems may now be solved in two steps, i. e., first Eq. (8-6) and then ER. (8-7). This transformation greatly reiduces the labor required in applying the relaxation method, because the rrelaxation pattern of the harmonic " "t: -biharmonic or Laplacian simpler th,an that of the type. Let us first the plate solve the boundary-valued problem of Eq. (8-6) when is a square one. process with n 4. be assumed. example, 1let us study the VII) as found f'rom the calcula- the values of w at all the , net points can By Eq. (8-7), }IJm,nwin+l, The values As an illustrative From the previous results tions with n - 3 (Chapter j I type is much 0of -i,n" n + Wm-l,n + Wm,n+ +.,n WM,n ...*9**.. 1- (8-8) are then recorded at the right of the corresponding net point, and the residuals +1 = ::[ + ~ Qm,n !*m+l,n +-ln -1,,n m-, n- - 4 4 .1rn I, = 4 w1 Fig. VIII-3 - and VIII-4. 4Wo = 4 x .0406 As an examplei, )(a)4 (8-9) are computed and are recorded at the left of these net pc)ints. sults are shown in 12(1-/ The re- we have - 4 x .0437 = -. 0124 'g4 = W2 + ws W3 + .=0377 + - + 7 - 4w 0316 + .0231 + .0163 - 4 x .0295 - .0093 :. · /. .' r(_. i__ C _11_ _ICr·__PII*UI___i3Ci( VIii -7 Qo 41 ! - 41 = 002,617 x (-.0117) - x (0.0124) -.002,617 + .000,163 + Q4 = + 1'i+ }7 - 4 .002,617 - -.0106 - .0093 - .0078 - .0054 - x (-. 3 - .002,617 = + .001,463. ;Tlere 2 )(j) , since 12 (1 -/(j .002,617 reatest I. " occurs in the vcInitv coumter-balanced 1 Thlelar-est the 0.1 and A of de-i ation of the assLunedvalues from the correct solutition, so chanfes are first made at this point. .1nrexaination s'horwsthat te ,Treataest residual occurs at oin 2, o Fi. . VIII-4 Sinrce =- ,,4 o40],'2 -a - a clanoe o 112 ; ou1d chance -..aheiaatt.t cally ,2 b-y an amount equa1l to ',here A denotes the aamount of chane Addin al! the other values of :J tom.emain chan'ed, of loment tiLes (-AZ2) 4 -42 equal to -. 000,637. four -. 0004 to IT2 , while assnining AQ 2 = + .0016 and is now If w-euse a nomencla.ure similar to t;hat ir the method distribution, this rocess can e ca1.led "balancing the unbalanced Q". A-sambol (bl) is put at the side of the value to indi cate this is the fi rst 'ba7ancin'. Now,we see tat l Q( = M and + 212 + = 2 + }'3 and4 13 + 1 3 + + - .002,Ei7 - 4 4 - .002,617. A o.) nge of M 2 with all the other M' fixed 'wol liange n , r Q by the relations as folicws: A , A =h T'Towiby "relaxing" the nets, :2 AQ = -. 0004 A4 and . .-0 (-.0004) Q, = + .001,26Z - .0008 = + .000,463 'Q4 = + .001,463 - .0004 = + .001,063 These operations may be called "carr-in,'-over" The whole process consists onerations b- similar calculations. of 20 and be denoted b, (l) balancinmt" and "carryin-over" - The detail onerations of te compnuta- tions are sovn- in Fig. TITII-4. After thd values of M's coDmputed,the residuals + QP~n ,n +1 ,:9n = "mNy+l, m-l,,n + m,n+ are comnuted and e naay determine of The detail alcul.ations. + 7f mn-l w r-, 7-ii ' the values of w by a similar oerations series are computations are shovmin The whole rocess consists of eleven "balancino" and "car- :~i~, rTTJT3 rrinT-over" oerations. The cter w o = .043790 def lecti on ratio thus ob;zainedis ? l for ,/ = 0.31i6,228. For/Z= ro = .043790 = .0443 x .3, 9 -wehave p p wh-ic!'checklsexactly w'lth tne exact" analytical solution. For thin plates with clam.pededges, the boundar- conditions are w = 0 ZX=0 along x ' ---2 +2 -_~ The relaxation pattern of the bihamnonic type must be used in this case. Althoutwhthe pattern is more complicated, the process is essentially the same. After the essential idea of the relaxation method is graphed, other roblems may be solved by rather obvious steps. It may 'rbe noted that no question of convergence can occur in the general relaxation process since no specific instructions are given. the residuals et worse the intellirent changes in the opposite direction. If, after some stepss, coomrutor oes back and makes These remarks, hrever, oversimplify the problem somewhat because of two facts; the computor may become confused as to whether or not the residuals are reallyrbetter, and secondly there i.salways a cq-estionof whether or not a solution with zero residuals exists. e O I, \\ o "3 b 4 9 X i O ei f '" I I Yb (u LI C·" o D O ·S Ir , j 9 w*ri PI i 1 i. i j ( I 1 . ' ' , ' i i - i, h i ;. , ., . ,[ : ' I ri \o 9 9 i ! I i i ; I I i j o"_ 1 1 I I 1 : o ' t ii V·9 z %% ii S V '' ,1 \j:4 i ~o-' Y t +' c +;e hcU OhP9 9$2% ft+ c h9>a6ti ++ t a ' Pf9 v s h __ ,, c hh 9?3?L, o !-- ! z Qb v C · P C 1 i I + .q Z ~ %1 ; b- ~,=1, I -- , %i.lao·o '.: 'trS 't t ef;l C!I s +eX , or u` ?, o; \Pd c; v o:l i14: %i" oi i : a at i h T1 T s Il , V,- tw - I . I, O j ~ 9, [~~~ _ 1 _ _ _ _ _. . _~~~~~~~~ i il a \1, I N. ; N *N 0` . UQ'i I N t ez .b· N; *i~: >x *N ; 1 o II R i 11 i i Qr i i I :b ' i : : e ' aXNll I I I ·r\ 1: 1 1) 4z0 i ' ji l v ji it$Y2 i 8 La -o'N t ,cteE 'e 4i : . ' iN . . . _ ___N i Z5h I i i 4. ,;;,~~~~~~~~~~, . ......... e"e e ee ., ~ ~ ~,, ~ 'ee vv ma,99 + hh .\. a h'V>s. fi 9 _P$ a e~% 1 X ~~~Cc + e ++: 5........... I + %a ci: i i 3 h, v· A 3. ": The Large Deflection P roblems of the i;Z The solution of ··I of the rectangular plates ectangular Plates. : -1 the general case of the large deflection by the relaxation problems method has been studied by I t St ; tPr Green and Southwell(l18 ) anLd t the method was outlined in Chapter II. worked with the three complicated They !ii equilibrium equations in terms of n., :e$ i.;- v, and w. The author has found, however, that it would be equally simple ,, simple r euations to use the two much a. F, and te x-; '·.;i in terns of the stress function, deflection, w. .· overning di fferential equations are Our :Fu V.4 F = K = 10.8 p ./1 ~Z ~ w where k and. k' = =( 2) s: :: +10.8 k' ................ ..... (s-10 ( 2 a,2 zxay 22 F 2 a2 _ _X- laxation pattern becomes r-ather may use the relaxation 2xaJ aa2w 2 These equations are non-linear, · i and therefore comulicated. the "exact" re- Instead of using it, we pat ;tern of the biharmonic tpe if the non-linear terms are taken care of by making corrections from time to time. The boundary w.:r forF conLditions But it is always possible those of interior oints. are usually the source of trouble. to solve the boundary values of Fs in terms of Here, the boundary values of F's are no longer constants but they vary fr'ointime to time as the interior values D change. . i: OW _- _ = TI II -11 The domain of the problem to be solved is first drawn, and the net points chosen and then/set of solutions of w and F guessed. i K then can be computed and the residuals ;71 . ;~i 4 Ax4F+ - A2 .-F.+ A 4 2 Ay2 are computed at each point. - k By the relaxation process carried out exactly as in the linear problems, the residuals Q are reduced somewhat. Before bothering to eliminate the Q completely, use the values of F thus obtained and the values of w assumed to compute k. On another sheet, values of "-Q'=x4 w + 2xy 2 W + y4 w - 10.8 are computed at each point. are reduced somewhat. - 10.8k' By the relaxation process, the residual Q' New values of k are computed and hence corrected values of the residuals attached to each net point. This process of re- I duction and correction is continued until sufficient accuracy has been attained. In the case of square plates with given edge displacements, ] the biharmonic type compatability equation can be transformed into two equations of the harmonic type. They are V2Ti= V2F = T ................... (8-11) j The condition of given edge displacements ,~'~":'~ -g :~., 1 1TToi T Tl~i+ 2] i'-'l·i * +TMT1"i +1 +2 mIi=sSi -,i 2 m (8-12) ViII -12 gives the boundary values of Ti, andte condition'io zsro searing stress along the edges (when t1he or zero strain plate is free to move along the edges) alonz the edge,s (when displacement along the edges) the plate is prevented from any gives the boundary values of F. If the edges of the plates are free to rotate about their supports, the biharmonic type equilibriun equation can be transformed into two equations of the haimonic type. .:2 Ei = 2 w They are 0 .8 k' 10.8 p + ....... .. and the boundary conditions MI = w .... ......... ... 2) .e. Lre 0 along the e3dges i ,dges.......... 0 along the e ...... .......... (8-13) The problem nowris transformed into four equations of the PoissonT s type. The relaxati -on pattern is much simpler at the sacri- fice of working with more eq uations.The relaxation method, however, iis ; j process of applying the just the same as that of the general case as described before. ii,, .3,j '~ z~~, ~ ?__~;¢.~-r~'?_~¢~! ~--''~.,L~ .. r- .%_:. %,~~ d';:,~¢~ ~ ~~.--~ ~ -ei&fi';-"·r?---·--a··-;---9·p: --- ·.- 3srp- '~~ ~ ,~'~-zl~%, ' ? ¢~,5~ .~:;~ ~ ~:¢". ? -. : I-1 IX CHAPTE RESULTS AND CONCLUSIONS 1. Discussion of the Results The bending problem of a square plate under uniform normal pressure, with the edges prevented from displacements along the supports but free to rotate about them, is studied by the The differences equations are finite differences approximation. solved by the method of successive approximation as described in Chapter VI. The computation starts with n = 1 to n = 3, in which case the plate is divided into 36 square nets with 25 inner points. normal pressure calculated is pa4/Eh4 The maximm = 250. After the values of w and F have been determined (Chapter VII), the stresses can be found by the following relations: 2F ~ ~ 22 o_ aF I4F/( ) 2= I/(A)2 A F(dP) I2 X 2 i 1----(A n2 w +/ LA 2w) "=2(l- 1 2 2) "1 2~~41 2(1-v 2 ( i (° +</ )2 iI I I C I~~~~~~~~~~~t . where r = ~12 ( 1 X X}LOC 2(1-_41 s2) (ot1-)2 r ) 1D -' and o-" are the membranestressand the extreme fiber bending stress respectively. The total stresses, , are the sum of the membrane and bending stresses at the section, and are maximum at the extreme fiber of the plate. x y X 5+ They are x At the center of the square plate, o t =/ , and W=c , and therefore we have '= x i = o /(a)D y Ory"= 0 = '/(t- )4 '-. =r 1__ . The deflections at various points determined in the cases n = 1, n = 2, and n = 3 are tabulated in Table IX-1 and Table IX-2. The center deflections are plotted against the normal pressure ratio in Fig. IX-1. The membrane stresses in the center of the plate and at the centers of the edges are tabulated in Table IX-3 and are plotted in Fig. IX-3. The bending and total stresses are tabulated in Table IX-4 and are plotted in Fig. IX-4. A study of the results shows that as n increases, the center deflection and membrane stresses tend to decrease, whereas the bending stresses and the membrane stresses along the sides tend to increase. The total extreme fiber total stresses at the center of the plate, however, still tend to decrease as n increases. The maximum error in center deflections is 0.47% for n=2 in comparison with n = 3 and the maximum error in center membrane stresses is 0.44%, both on the conservative side. Both occurred T IX-3 at pa ,/Eh4 = 250. 2% at pa4/Eh4 The error in the center bending stresses is 12.5 and is 0.83% at pa4 /Eh4 = 250, both on the = The error in the center extreme fiber stresses unsafe side. is 1.6% at pa4 /Eh4 = 12.5 and 0.17% at pa4 /Eh4 = 250, both on the safe side. The error in the membrane stresses at the center of I ' and the sides is 12% for both 8.9% for both ox and 6y' oy' at pa 4 /Eh4 = 12.5 and at pa4 /Eh4 = 250, all on the unsafe side. Judging from the above results, the error for n = 3 in comparison with n = 4 would be still smaller. Since in our case only the center deflections and stresses are to be investigated and the errors are sufficiently small for engineering purposes, the case n = 3 is considered to be satisfactory for the final results. The center deflections obtained by Way(17) Levy(19)(21) and Head and Sechler the present results. are plotted in Fig. IX-2 to compare with The center membrane, bending, and total stresses are plotted in Fig. IX-5 to compare with the results by Levy) It is seen from these results that the center deflections are in good agreement with C.I.T. test results up to pa4 /Eh4 = 120. The theoretical results seem to be too low at higher pressures. are It is interesting to note that thetestresults really for clamped edge plates. The clamping effect seems to be only local, and at the center of the plate the plate behaves just as tlough it were simply-supported, i.e. the plate is free to rotate about its edges. - .1 L IX-L From the point of view of the engineer designing the plate, the total stresses at the center of the edges are still much larger in the case of clamped edges than all the other cases, hence a design based on those stresses would give a structure on the conservative side. The center deflections, however, would give an idea of how much is the "washboarding" of a boat bottom while a seaplane is taxiing or landing. 2. Conclusions We can draw the following conclusions from this analysis: (1) The large deflection problems of rectangular plates can be solved apiproximately by the present method with any boundary conditions and to any degree of accuracy required. it is still difficult, ' Though the present method is nevertheless simpler than the previously used methods to give the same degree of accuracy. (2) For the case considered, the case n = 3 gives results of good accuracy and the results are consistent with the existing theories. (3) The clamping effect to be only local. of a clamped thin plate seems At the center, the plate behaves more like a plate with simply-supported edges, i.e. the fin plate is approximately free to rotate about its edges. (4) The test results shiow that at pa4 /Eh4 > 175, all the existing solutions of the differential equations give unsafe results for center deflection for a square plate. This suggests IX-5 that the differential equations may use some assumptions which neglect some quantities that are large when the deflections are large. (5) The present results of the center deflections and membrane stresses give good agreement with the test results when pa4/Eh4 < 120. 3. Suggestions for Further Research. Since the method presented here is general, it may be applied to solve rectangular plates of any length-width ratio with The case solved in this thesis is various boundary conditions. only one of many in which aeronautical engineers are interested. The other problems are as follows: The solution by the (1) Clamped rectangular plates. present method may serve as a good check to the approximate results obtained by Way. (2) Simply-supported rectangular plates. cases have been solved by Levy. Only two The present method may be used to solve all the other cases. (3) Rectangular plates with edges prevented from displacements along the supports, but free to rotate about them. The method gives good agreement in center deflections and membraneclamped plate in the case of a square plate. Other cases would probably yield similar agreement. (4) Combined bending and buckling problems can also be solved by this method and they are of interest to aeronautical engineers for reasons discussed in Chapter I . IX-6 (5) The membrane problems can also be solved by this method. There are two cases which ire to be solved. the rectangular One is plates with edges prevented from any displacements along the supports, and the other is rectangular plates with edges which are free to move along the supports. IX-7 TABL I-1 CENTZR DEFLECTIONS pa 4 wo/h n =·1 O I . n = 2 0 0 n= 3 0 12.5 .3888 .4062 .4055 25 .5844 .6092 .6083 50 .8184 .8474 .8460 75 .9757 1.0052 1.0031 100 1.0980 1.1269 1.1240 150 1.2888 1.3145 1.3104 200 1.4376 1.4616 1.4557 250 1.5623 1.5844 1.5770 ,, .. . ii 7 TABLE IX-2 DEFLECTIONS AT VARIOUS POINTS (n = 2) A_ pa4/'Eh4 · , 0 w/h I wo/h . .L 0 W2/h I- 0 0 12.5 .4062 .2980 .2198 25 .6092 .4508 .3363 50 .8474 .6332 .4791 75 1.0052 .7555 . 5766 100 1.1269 .8502 .6528 150 1.3145 .9966 .7713 200 1.4616 1.1116 .8648 250 __ 1.5844 ----- .... - 1.2076 -- I --------- .9431 I _ _-- T IX-9 TABLE IX-3 DEFLECTIONS AT VARIOUS 3) (n pa 4 /Eh4 0 wl/h w3 /h w5 /h w/h 0 0 0 0 0 O w2/h 12.5 .4055 .3564 .3136 .2139 .1890 .1159 25 .6083 .5365 .4738 .3249 .2892 .1822 50 .8460 .7494 .6650 .4592 .4131 .2711 1.0031 .8905 .7930 .5500 .4986 .3370 100 1.1240 .9995 .8920 .6207 .5660 .3915 150 1.3104 1.1677 1.0450 .7305 .6717 .4804 200 1.4557 1.2988 1.1641 .8164 .7551 .5531 250 i.5770 1.4081 1.2634 .8880 .8249 .6149 75 L wC7h POINTS IX-10 TABLE IX-4 IMEMBRANESTPESSES 2 'a oa2 ,Cr 'a a7 - Crpa4 Eh2 - 2 Eh4 n= 2 n=3 0 12.5 0 0 .6103 Cla Cry 2 Eh2Eh2 - n=2 0 .6089 22 n=3 0 n=2 n =3 0 0 .3338 .3795 1.055 1.200 .7612 .8574 2 .407 2.711 25 1.384 1.377 50 2.695 2.683 1.484 1.661' 4.693 5.254 75 3.806 3.792 2.096 2.341 6.628 7.401 100 -4.802 4.785 2.643 2.943 8.357 9.305 150 6.566 6.542 3.613 4.001 11.43 12.65 200 8.136 8.103 4.473 4.929 14.15 15.59 250 9.575 9.533 5.264 5.778 16.64 18.27 Note: Subscript e denotes the Subtscript center of the plate. denotes the center of the sides, x = + a/2. IX-'! TABLE IX-5 EXTEME FIBER BENDING AND TOTAL STRESSES i j AT CENTE OF THE PLATE I TotalStresses Bending Stresses j I I I i i i pa Oa a1a2 Eh4 E2 a2 Eh2 Eh2 n =3 n=2 n=2 n=3 0 0' 12.5 2.530 2.582 3.140 3.191 25 3.708 3.781 5.092 5.158 50 5.010 5.087 7.705 7.770 75 5.845 5.928 9.651 9.720 100 6.475 6.554 11.277 11.339 150 7.439 7.513 14.005 14.055 200 8.191 8.261 16.327 16.364 8.817 8.891 18.392 0 250 . . '0 ,, . 18.4!24 . .. i i ! ! I I I i I I j . i I qj i t ; i ;I : I ; i i i I i i I - i II : Il : I i I ii i I _. I I I i i - i u Ii i / I- ! i i . I i . i 11 I i i V i i. I i i I i i : I i i I i i I i I t : I N) i ; i f I : i i I i t I ; i i i i i I I I ; . : o I I i i i . o I i j w ICI 1: : 0 0 I1 V S Vo N0 A: N I . . i i i i I . I I : ! I tI i ! I I I iI . I ; V I~~~~~~~~~. Nr ?1S (I ; i I ; · C9 : i i 0' I (. x ; N IU N! Q\L 4 , Q) -. O I,, \ \ . j 5(44 QJZ i ;: :e~~( * b .V '4 IK N0k k ? '4 I) 'O I. i I, X i I . . ! . i I I ; I C * ! V I X I i CO : I i . .' 1) -- . 7 (i .V IV. i ; I i /, , I' i 4( ily : * ~ 14 I i f i I: : i i ! I I I I i I iI :L 1 : I i I 2< 1) N0 \1 N 10 'a. "4 V "4~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Iq 1 . i I i -11 i i I G I i i i j i - i i : I i I V, Kb i i N i i i . I i Ii It i I . i V i' i I t .i I i I i i i i . r i I 'S i V/ I i 0 i I : I i i: i I i i ; I i i i I : i I i I I I ! i N i i i i : I i i i i i 1 i ; Iq :: 0 ! .V V N- 10 (14 I l l il \i k \D o R-1 REFEREI CES -(1) See, for example, Timoshenlco,S p 88, Eq. 101, I cGraw-Hill (2) Ibid, p 344, Eq. 202. (3) Foppl, A.: Theory of Plates and Shells, Book Co., 1940. Vorlesungen n. techmische Miechanik, Bd. 5, 24. Leipzig, 1907. - (4) v. arman, Th.: Festigkeits probleme in 349, .achinenbau, p. Vol. IV of Encyk. der I.atn.'iss, 1910. (5) .lenky, iH.,Uber den Spannungoznetand in kreisrunden Platten mit versclhwindenderBiegungs-steifigkeit, Zeitschrift fur Matlhematik und Pysik, (6) Vol. 63, 1915, p. 311. ]-ernky, H., Die Berenchnung dunner rechteckiger Platten mit verschwindender Biegungs-steifikeiet, Zeitschrift fur angewandte (7) athimatatik und M,echnik, Kaiser, R., ol. 1, 1921, pp. 81, 423. ec'nmesischeund exper.nentelle Ermittlung der Durchbiegungen und Spannungen von quadratiscien Platten bei freier Auflagerung an den Randern, Gleichnlassigverteilter .. Last unter grossen Ausbiegungen, Z. F. a.' ,. 1J., d. 1, eaft 2, April (8) 1936, pp. 73-98. N.adai, A., Die Slastisclhe latten, p. 288, Springer, Berlin, 1925. (9) Timoshenko, S., Vibration Problems in Engineering, p. 317. D. Van Nostrand Co., New York, 1928. s- :,· · :-·-i . - R-2 (10) ;Wday,S., Bending of Circular Plates with Lar7geDeflection, A.S.Mi.E.. ·; -r· i-; Trans. A.P.i. - 56 - 12, Vol 56, No. , Aug. 1934, pp. 627-636. -- (11) L- Federhofer, K., Tragfahigkeit der uber die Beulgringe Belastetan Kreisplatte, Forschung auf dem Gebiet des Inginieurwesens, Vol. 11, 1940, p. 97. (12) Friedrichs, K.O., and Stoker, J.J., The lon-linearBoundary Value Problem of the Buckled Plate, Proc. of the ational -·-; - Academy of Sciences, Vol. 25, o. 10, pp. 535-540, 1939. :t·I,-.·- (13) 'c:3· j'·: · :ri Friedrisc'hs, .D0.,and Stoker, J.J., The Non-linear Value Problem of the Buckled Plate, Am. Journal of oundary athl., B ":" Vol. LXIII, No. 4, Oct. 1941. :.::i i-'i. (14) i Friedrischs, .0. and Stoker, J.., Buckling of the Circular c; Plate Beyond the Critical Thrust. j;· I I 21 I;Lil (15) -a 1914, Vol. II, See also Rof. 1. i i Boobnoff, J., Theory of Structures of Ships, St. Petersburg, , .4 4i (16) Prescott, J., Applied Elasticity, London, 1924, p. 474. (17) Wiay,,S., Uniformly Loaded, Clamped, Rectangular Plates with i' t Large Deflections. Proc. Fifth Int. Cong. Aopl. Mech., John 'W;iley & Sons, Inc., 1939, pp. 123-28. Ir ·t;t" (18) Green, J.R., and Southrell, R.V., Relaxation UlethodsApplied -7·":.:·i. to Engineering Problems, VIII A. I Transverse Displacements of Thin Elastic Plates, Phil., Trans. oy So. of London, of I, pp. 137-176, e Problems Relating to Large . ,- I - 1 I I-._ ·.;- - -----i.. Series C, Miath. and hy. Sc., No. 6, Vol. Feb. 1944. .- . . I II ...--I -. (19) Levy, S., Square Plate Pressure with Clamped Edges under Normal roducing Large Deflections, N.A.C.A.T.N. N!o. 847, 1942. (20) Levy, S. and Greeraan, S., Bending with Lare Deflection of a Clamped]Fectangular Plate with Lenfth-idth Levy, S., Bending of Rectangular Plates with Large Deflec- tions, (22) of Pressure, i,.1.C .A., T.1., No. 853, 1942. 1.5 uder ioral (21) ratio .A..A., T.N . o. 846, 1942. Schnadel, G., Uber niluckun von latten. -yerf t eederei iafen, 9. 1928, pp. 500-502. (23) cmadel G, Die Ubersclreitung Platten. Verh d. dritten (Stockholm), (24 ) Seztara, (25) Latsutada, L.iI., Kong. f. Tech. .ech. ;On the Buckling under Edge Thirusts of a Clearped at 1Four Edges. o. 3), Aero Donnell, Inter. 1931. pp, 73-81. III, RIectan.rular Plate T, der rni.ckgrinze bei durmen es. Inst., Sechler, Tokyo Imnerial E.E., and von arman, Th., Sechler, C.I.T. E.E.., and Dornell, of Thin Plates in Compression, jan. (27) 30, 1932, . ni-v., A ril, 131. Tech. ?'ub. 16, 192. L.H., The Trars., trength A.-.. -54-5, pp. 53-57. Sezawa, Katsutada, Das Ausknicken von Allseitig befestigten tmIn-edrucktenrect;eci.en Platten. , (28) 69, (Vol . on arman, Th., Survey of Problems of Thin 7falled Structures, (26) .e-ort Z.f.a..i2. '., 12, 132, 27- ,22 . Cox, 1H.L., Te .6uckling of Thin Plates in Compression, . No. 1554, Brit,. A..C., 1933. I:. R-4 (29) von Karman, Th., Survey of Problems of Thin Wl'alledStructures, 1.. Trans., Part III, Analysis of some Typical Structures, i.S.I, AER - 55-19 c, Oct., Dec., 1933, pp. 155-158. (30) Sechler, E., Compression. (31) The Ultimate Strength of Thin Flat Sheets in C.I.T. Tech. Pub. 27, 1933. Buckling Load for a Rectangular Plate with Taylor, G.I., Te Four Clamred Edzes. (32) d. 13, 1933, pp. 147-152. Z.f.a...., On the Elastic Stability of a Rectangular Leggett, D.M:A.A., plate when Sbjected: to a Variable Edge Thrust. (33) Sattler, 1935. 31, pp. 368-381, bridge Phil. Soc. E., Beitrag Proc. Cam1- dunnerPlatten. Mitt zur Knicktheorie 3, 1935, pp. 257-279. Forsch Anst. GTE-T-Konzein (34) Schmieden, C., Das Ausknicken eines Plattens treifens unter Schub-und Druck raften. Z.f.a.M.M., Bd. 15, Heft 5, Oct. 1935, pp. 278-285. (35) Trefftz, E., Die Bestimmung der Knicklast gedruckter, rechteckiger Platten, Z.f.a.d, pp. (36) d.. 15, Heft 6, Dec. 1935, 339-344. Yamamoto, M., and Kondo, K., Buckling and Failure of Thin Aero. Res. Inst., Tokyo Rectangular Plates in Compression. Imp. Univ. Report Nlo. 119, (Vol. (3) Kaufmann,'W., Uber unelastisches ten. 10, No. 1) April, 1935. nicken rechteckiger Plat- Ingenieur-Archiv 7. 1936, pp, 156-165. ~.~ ~~~ ---~~-~,..?...? · R-5 (38) iles,Aaron ., Stability dge. Supported at the of A.S ectanpular Plates :lastically .E., Trans. Vol. 3, o 2, June, 1936, pp. A47-.52. (39) Iguchi, S., Allrgeeine recteckige (40) 'latte-n. Inr:enieur-4rch'.iv Timoshenko, S., Theory of Elastic Book Co., Inc., (41) Losung der Knickungsaufgabe fur Barbre, R., (42) IcGraw-iill Stability. 1938. Stabilitat leichmassig Gedruckter Rechteck- platten mit Langs-oder uerstreifen. 1937, p, 7, l36, pp. 207-215. Incenieur-Archiv 8, 117-150. urchlard, ;., Schraasteife Beulspannunsfen der gqadratisch.-en unter Plat;te Druck bzw. Schub. In-enieur-Archiv it 8, 1937, pp. 332-348. (43) Grzedzielski, A., and Billewicz, tole flainbee. (44) :[ar-ruerre, 'Y., Sur ia rigidite Sprawozdanie I..T.L., arl., Die ittragende . 1, ten, Luftfahrtforschun, Blo. 1, 21, 137. reite der de la pp.5-22. edruckten Plat2C, 1937, pp. 20arc Lfr. 3, 121-128. (45) }largeurre, K., and Trefftz, lansbelasteten last Z_.f.a.-.[., E., Uber die Tragfahigkeit eines Plattentstreifens inachUberschreiten der eul- d. 17, eft 2, Aril .937, pp. 85-100. 40) Schman, L., and ack, G., Strength of Rectangular Flat Plates under Edge Compression. T.2,. o, 356, T.A.C.A., 1930. :r R-6 Iu (47) Schwartz, E.H., An Investigation of the Compressive Strength Properties of Stainless Steel Sheet-stringer Combinations, Part II. (48) Air Corps Tech., Rep. 4096, 1936. Mlatulaitis,J., and Schwartz, E.H., An Investigation of the Compressive Strength Properties of Stainless Steel SheetStringer Combinations. Part IV, Air Corps Tech. Rep. 4271, 1936. (49) Lahde, R., and Wagner, H. Experimental Studies of the Effective Width of Buckled Sheets. T.M. No. 814, N.A.C.A., 1936. (50) Newell, J.S. and arrington, J.H., Progress Report on Miethods of Analysis Applicable to _vlonocoque Aircraft Structures. Air Corps Rech. Rep. 4313, 1937. (51) Sechler, E.E., Stress Compression. Distribution J. Aero. Sci., in Stiffened Panels under Vol 4, No. 8, June 1937, pp. 320- 333 . (52) 'Gough, .J., and Cox, H.L. Son.eTests on the Stability 'oftThin Strip under Spearing Forces in the Plane of the Strip. Proc. Roy. Soc., (London), Ser. A., Vol. 137, 1932, pp. 145-157. (53) Lundquist, E.E., Generalized A.nalysisof Experimental Observations in Problems of Elastic Stability. T.N. No. 658, .A.C.A. 1938. (54) Rambrg, w., /IcPherson, .E., and Levy, S., Experimental Study of Deformation and of EffectiveWidth in -Aially Loaded Sheet-stringerPanels, TN., No. 684, N.A.C.A., - :'' ~ .,·7 1939. ' .. ....... ...... ."'~' '[ -':"'~---~'-- '-',-',,~-~ ~' ..... R-7 (55) iMcPherson, A.E., Ramberg, W'., and Levy, S., Normal-Pressure i Tests of Ci-rcularPlates with Clamped Edges, T.N. No. 848, I N.A.C.A., 1942. i i (56) 1 i Ramberg, AF.,MicPherson,A.E., and Levy, S., Normal Pressure E Tests of Ractangular Plates, T.N. No. 849, N.A.C.A., 1942. j i ·; (57) Niles, A.S., and 1Newell,J.S., Airplane Structures. John W'"iley& Sons, Inc., New York, Third Edition, 1943, Vol. I. p. 329. (58) von ises, R., and Geirenger, iH.,Praktische Verfahren der Gleicnungsauflosung. Z.f.a..M., Vol. 9, 1929, pp. 58-76, 152-164. (59) Cristopherson, D.G., and Southwell, R.V., Relaxation .ethods Applied to Engineering Problems. III. Independent Variables. Proble!ns Invoing Two Proc. Roy. Soc., Series A, Vol. 168, V: 1938, pp. 317-350. Southwvell,R.V., New Pathways in Aeronautical Theory. 5th Wfright Brothers Lecture. J. Aero. Soc., Vol. 9, No. 3, Jan. 1942, pp. 77-89. 6 Southwell, .V., Relaxation Methods in Engineering Science. Oxford Univ. Press, 1940, Chapters VII and VIII. (62) Crout, P.D., A Short iethod for Evaluating Determinants and Solving Systerrms of Linear Equations with Real or Complex Coefficients, Trans., A.I.E.E., Vol, 60, 1941. .i .,? .! (63) Miesnager,A., Calcul elemantaire ri-oureux des plaques rectangulaires. Inst. Assoc. for Bridge and Structural Eng., 1932, pp. 329-336. ,.--..7 `--T--;rl.-n*rTCIB914*·Si*i R-8 (34) Gall, Hi.d. CompnressiveS trenglth of ALuminumri Alloy, Thesis, C i IHead, R.-r., stiffened 1944. i;· i- · t 2: 1930. See also ef. (57) p, 327. (65) -I .I.T., tiffened Sheets of and Sechler, Flat Plates, E.E., M . ormial Pressure T.N. To. 943, Tests on Un- .A.C.A., September, BIOGRAPHY The author, Chi-Teh Wang, was born on May 6, 1918, in Hangchow, China. His father, then a senator from Kiangsu Prov- ince, after resigning from a position as a college professor, wished his son to be a statesman and gave him the name Chi-Teh, literally meaning 'developing morality since he had painfully witnessed the political situation in China at that time to be turbulent and corrupt to a severe degree. At the age of five the author was put into study under a private tutor in Chinese classics and mathematics. However, the author has, in a way, disappointed his father by showing that he always tends to be more interested in natural sciences than in politics and economics. He chose his career as an engineer and entered National Chiao-Tung University in the Department of Mechanical Engineering in 1936, where he graduated in 1940 with a degree of Bachelor of Science in Mechanical EngineerLng with an option in Aeronautical Engineering. While he was a sophomore, the Sino-Japanese war broke out and he was entirely cut off from his family. of Upon the recommendations everal professors, he was awarded a University Fellowship from 1937 to 1940 which enabled him to complete his undergraduate study. He was one of the five students.in the graduating classes who were elected to be members of the Phi Tau Phi Society, the only national honorary society for higher education in China. Affter graduation, he joined the staff of the same Due university to be an assistant in aeronautical engineering. to the departure of a professor in aeronautical engineering, he was promoted to be an instructor teaching aerodynamics and propeller design during the academic year of 1940 to 1941. Upon the University, of the faculty of Chiao-Tung recommendation from Rensselaer he was awardeda fellowship technic Institute in 1941, which enabled him to cometo the United States and to do some graduate degree of' Aaster Poly- work. He received of Aeronautical Engineering in his ay, 1942, and was elected to be a member of the Sigma Xi Society, the national honorary society for scientific research, and he was fortunate to be offered fellowships and scholarships fronmsix differenough ent universities. He studied at BrownUniversity in the AdvancedInstitute during the summers of 1942 and 1943 on fellowships of ivMechanics from Tsing--Hwa University, degree of Master of Science in Applied award from the Institute University durin and a fellowshipfromTsing-Hua the year 1942 to 1943. was awarded a Saltonstall Fellowship worked as a part-time assistant athematics in October, 1943. in September, 1942, on a scho- He came to this institute larship a ieceived and Brown UniversitTy, and From 1943 to 1944 he eh ti and at the sam time to Professor J. S. Newell. He has one paPer to be published in the October issue of Journal of Aeronautical Sciences, 1944, entitled "Contracting Cones Giving UniformThroat S-eeds" which he wrote jointly Pr 0ofessor R. H. Smith of 1.I.T. with

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