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1 -jif "-""IIMJ tlnl! HdllllUI M. M-&': I. "a ;iy^^-g ?ra^."i:^ ; *->^ ^"^fc^ivtv^. ratWMTMKM iiitrj^Mei'nTnntnir'-iiTii -r ^^-^.-|T)fgB. Finite Bifierence Equations lot the Analysis of Thin Rectangular Plates With Coinblnations of FitMVaM. ^ e>':E ^es '. : '.- (Hon :);. E rtohj M* V, ' : BRXi-li' . University of Texas, Defense Research L*afo., Austin (Same). E'aa 28- tables, diagr (Same): Atig 48/ 3?neUss*. : finite difference and differential e qu ti ns are developed for the analysis of thin rectangular plates with . one edge fixed and'-three- edges free , as a preliminary' study of the structural characteristics of thia swept ^ings and fins*, Emphasis is placed on the finite difference equations, and egressions are presented for the determination f deflections .. shears, moments, and reactions for a rectangular plate with various combinations of fixed and free edge conditions and transverse loadings The deflections and stresses for a square plate fixed along One edge and carrying load at the center of the other edge was evaluated.

2 These quantities are- compared with experimental results. Good agreement is obtained between theoretical and experimental results,". Copies of this -report obtainable from, 'C . Pia'Wo - -f&&messes Structures ft) Equations, Differential Theory and; Analysis (2); Wings i Swept. * Stress analysis DOCUMENTS DIVISION, t-2. RSGROFItM, Me. L-1 .. " "I- A- , "v* " ! *-i*( "SP'SMAT* -^*sxm s^K^p- **^';^as^,:f*~*~Ts ;f,"'. >%ft^!'fi^cW*wi!?|Eg LiE FENS E RESEARCH L,A BOBATGRY. THE OF ^TEWAS. Gepj HOC j S? OF-IOO:? finite Difference Equations-, for, the Analysts- of TMn Rectaaag lar/. Plates with Combinations of , '. 'by'. ,;">. ** M. S&rt n . SJ^rt'. 1. ' 31 . wszr. ^?^^^s^r##^^:-^^lf^^^^^i,,rnITl^. DEFENSE RESEARCH LABORATORY. THE UNIVERSITY OF TE^AS. : s .GF--10O5. DEL-l?^..A gUst 6>. igfrS.

3 Finite, Difference,.i&uations for,the,Analysis i1 .Thin ^ular ~; ' Plates with\ComlftnaTffona "ofFixed' .aM/FreCEdges- "". 1: .. - --'_;..;." *y . - - . ; ' " ". ~ - M. 'i Barton Abstract- As a preliminary study of the structural characteristics of thin swept wings and fins,; the equations for the analysis of thin rectangular plates with one edge fixed are. developed in differential and finite difference form. Emphasis is: placed oh the finite difference equations and expressions are given for the determination of deflections, shearsj moments, and reactions for a rectangular plate with various combinations of fixed and free - edge conditions and transverse loadings. A numerical evaluation is made of the deflections and stresses for a square plate fixed along one edge and carrying a load at the center of the other edge.

4 These quantities are compared with experimental results. Beasonahle correlations (within about 5 percent:) between theoretical and experimental results are obtained. Introduction The present work is the beginning of more comprehensive project to determine the elastic and dynamic properties of skew plates which are similar to the sveptback fins used on supersonic guided missiles. The information to be obtained ultimately is expected to consist of elastic characteristics, such as the deflections., stress distributions, leietic axis positions and so forth for thin plates of low ratio, with one edge clamped and three edges free.,, for various load conditions, as well as dynamic characteristics suchas modes of vibrations, natural frequencies^ and so forth. Analytical and experimental, procedures: are to he used.

5 Such information should he of considerable, value in faciliteting the structural design of wings and fins and in providing much, needed information on. the structural phase of aer - elastic problems. In order to "prove in" methods and techniques, the relatively simple problem of the rectangular plate is first studied. This report consists of .an account of the plate equations ^or the static structural analysis p 'I rectangular thin plates with one e je fixed and three edges free, in terms f g!S{tRjas<jf*j<pM|> j( ay; jwjWtMjwWJf'ay 9*'lfl^a ftl6fr*Vr fh"\i i tf ' i ^ 11 i ii '-^'^ tf ^ JT'r)i"il T "-f ~'V tfTIf-**- DEFENSE RESEARCH LABORATORY. THE UNIVERSITY OF TEX^S. M$B:fs -2~ .or--1005. August 6>-19l}8 EfiL~lT5. tjrx-i-M-3. the differential as well s finite difference equations t in addition.

6 Numerical evaluation is presented for the deflections of a square rectangular cantilever plate with, a concentrated load applied at the center of the out- febkrd edge." Analytical results are compared with, test values. It is expected that later reports will deal with the ore complete solutions of rectangular plateB with a Y riety of loading and Will discuss the relative merits of methods of analysis such as the relaxation procedure and the solution of simultaneous equations, along with the effect of network size on the accuracy, of solutions of the. finite difference equations and the effect of Eoisson's ratio. Experimental results will also "be indicated. "A. similar procedure will "be followed In presenting the. results of skew plate analyses. Many investigators have solved a great variety f plate problems.

7 However, exact solution, by which is meant an analytical solution to the governing differential equations, have "been largely confined to rectangular or circular plates with the boundary ed^ss fixed, simply supported, or a combination of fixed, free and simply supported. An example of this type of solution is that of the "rectangular plate with two opposite edges simply supported, the third edge free and the fourth edge built in or Bimply supported". given by Timoshenko^' on page 215 of "Theory of Plates and Shells" The problems of the skew plate or the rectangular plate with more complicated the great Complexity of obtaining an "exact" solution or the tremendous labor involved in obtaining approximate numerical solutions by means of finite difference equations or other methods. Some work?

8 However, has been done on . these problems by means of finite difference equations. For example, Jensen*2'. solves a number of skew plate, problems for a. variety of lateral loads With two opposite edges simply supported and the other two edges free or elastically supported. HallOi has solved fche problem of cantilever rectangular plate for which the fixed edge is four times the cantilever length and which carries a concentrated load at the center of the outboard edge. differential ^Equations The equations of equilibrium of an element of thin plate in terms of rectangular coordinates save well known (see reference 1) and will only be cited here. The directions -corresponding to positive quantities are shown in Figure 1. 5. *5^"<JT*T? *TT . DEFENSE .RESEARCH LABORATORY. THE UNIVERSITY OF TEXAS.

9 *3- CF-10 5. August 6, 19)48 SfeL-175. If K-l-AA-3. >_. -'1. I/oad: dar yc 2w A V. Moments: M =. 2C. li;. = M = D{l-u} iLi (I). Shears: * 3y3 "S^y2. Qy Edge forces R. r*&3w i>%*. Concentrated H = 2Mw = 2D(l-|i]L 4 . corner force: - rv.<r7 -"i'.^v. ' *TT* _ * ;'"jf-1 > A4 -;: r """' -^B*^^S*?^?--Sr_i,.'% S. DEFENSE RESEARCH LABORATORY. THE NIVE-RSi-TY OF TEXAS. i-ii- CF-i G5 , August 69l$k8 BBL-lfl v ere- p =: intensity of distriMtsd load: fp .i). D :=. Eh? 12(1-^). E. modulus of elasticity (i Polsson's ratio & 0 plate thickness (in). v t= deflection of plat in Z direction, (in). Moments, shears, axti reaction are for a unit length of edge . The equations are limited "by the following assumptions. 1. h thickness of the plate is small in comparison with its other dimensions. 2. Deformations are snail so that the curvature is apprcximted "by the second derivative of the deflection with respect to coordinates la las plane of the plate.

10 3. Stretching of the middle surface of the plat is not considered. laBt assuHptioa has as a coroHary| the fact that the deflection is eipail coshered vith the thickness of the plat for certain 'boun&ary conditions. If this assumption is not made, then the so-called large deflection theory must "foe used which takes into account the effect of the msabrans forces in which case the relation "between load and deflection is not linear. Fortunately,, our experiments show that for the "boundary conditions used corresponding to the cantilever plat* * the load and deflections are essentially linear up to a deflection of at least 20 times the plate thickness. Hence, the use of the linear theory se@ss justified in this particular profit . Finite Difference Equations iwnnwi 11 iTiiimniiiiinnwi-niiwiiirniiniwiiBiiinii n iw For the square network shown in Figure 2, the approximate mlues of the derivatives in terms of finite differences of the values of the deflections at : 2TJ~''7 "~"C!