n... A STUDY OF S£MPLY SUPPORTED SQUARE PLATES EY THE MORE ME’FHOS AND BY FINITE MFFERENCES Thai: for flu Dogma of M. S. MiG-{83AM STATE UNIVERSITY B. have Raiu 1958 . “"f‘o -. This is to certify that the thesis entitled A STUDY OF SIMPLY SUPPORTED SQUALLE PLATES BY THE MUIRE METHOD AND BY FINITE DIFFELtBNC-LIS presented by B. BASAVA RAJU has been accepted towards fulfillment of the requirements for Master degree in Agglied h‘ec hanics $124.3.“ 613ng Major professor Date New. ZS; I455 B. BASAVA HAJU. ABSTRACT A study of simply supported plates was made by the Moire Method and by the use of finite differences. The three principal purposes of the study were: I (a) To determine the distribution of moments in a simply supported, laterally loaded square plate, (b) To determine the effect of square and rectangular cutouts in a simply supported plate on the distribution of moments, (c) To study the accuracy of the Moire method as applied to simply supported plates. In the Moire method, the slopes are determined for any point and in any desired direction by first photographing the reflection of a fine grid by the unloaded model and then, on the same negative, photographing the reflection by the loaded plate. This results in characteristic interference fringes called Moire Fringes . From these fringe photos, the curvatures were then found by plotting the curves showing the variation of slope along given lines and measuring the slopes of these curves; the deflections were determined by numerical integration of the area under the slope curves. The simple support was provided by two knife edges which press gently against the plate model. The constant for plate model was found by comparing with the constant for a theoreti- cally solved circular clamped plate. A uniformly distributed B. BASAVA RAJU load was used throughout this investigation. Three types of cutouts were used at two different positions in the plate model. The analysis of the plate by finite differences was made using two different grid spacings and extrapolating the values between the two. The moments and deflections found by the Moire method and by finite differences were plotted on the same reference axes so as to provide a comparison of the two methods. The Moire method for a simply supported and uniformly loaded souare plate gives a solution which is 5 to l? percent smaller than the solution by finite differences, and the solution given by M. Levy. The possible sources_and the approximate percentage amount of errors are: (a) Membrane action of the plate due to large deflections, up to 5%, (b) Calibration, (c) Measurement of load, up to 2%, (d) Reduction of the data. A comparison of the effect of various cutouts on the maximum moments and maximum deflections was made. The effect of cutout on a simply supported souare plate extends a dis- tance from the cutout which is little greater than the size of the cutout. ' B. BASAVA RAJU. Because of high speed computers, finite differences offers a powerful method for plate analysis. For unusual structural forms like the irregular cutouts, formulation of the boundary conditions becomes formidable. Then access could be had to the Moire method. Thus, the Moire method and finite differences could be used to check the accuracy of one on another and also they could be used as alternative methods if one method fails for a particular problem. A STUDY or SIMPLY SUPPORTED SQUARE PLATES av THE MOIRE METHOD AND sv FINiTE DIFFERENCES By B. BASAVA RAJU A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the redulrements for the degree of MASTER OF SCIENCE Department of Applied Mechanics l958 ACKNOWLEDGEMENTS The writer is indebted to Dr. William A. Bradley for his guidance and help throughout this investigation. Sincere appreciation is expressed to Dr. Charles 0. Harris for his encouragement. The writer is also grateful to the Engineering Experiment Station and to the Department of Applied Mechanics for help in various ways. Acknowledgements are also due to Mr. V. Swarny for his help in making some of the drawings. TABLE OF CONTENTS CHAPTER PAGE ACKNOWLEDGEMENTS LIST OF ILLUSTRATIONS NOTATION I . II III IV VI VII INTRODUCTION................................... l PRINCIPLES..................................... A DIAGONAL PHOTOGRAPHS........................... ll THE EQUIPMENT.................................. l4 Simple Support Model Calibration REDUCTION OF DATA.............................. 26 FINITE DIFFERENCE ANALYSIS..................... )0 Introduction Boundary Conditions Plate Analogy Summary of finite difference analysis TEST. RESULTS.................................. 42 Plate AB -- No cutout Plate CDl -- SQuare central cutout Plate C02 -- Rectangular central cutout Plate CD} -- Square central cutout Plate EF -- Square cutout at X=0-Sa., Yao-zsat Summary of Test Results TABLE or CONTENTS CONTINUED CHAPTER PAGE VIII SUMMARY AND CONCLUSIONS.......................... 64 BIBLIOGRAPHY........................................... 66 APPENDIXOOOOOOOOOOOOOOOOOOOOOO...OOOOOOOOOOOOOOOOOOOOO. 67 iv FIGURE i 2 l2 l3 I4 l5 l6 l7 l8 Moments and LIST OF ILLUSTRATIONS forces acting on slab element...... Schematic diagrams for Moire Principle......... Rotation of plate to give photograph which is light throughoutOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Moire fringes for rectangular plate with two edges clamped and two edges free, loaded at the center with a concentrated load.......... Notation for diagonal moments.................. Diagonal fringe patterns for rectangular plate with two edges clamped and two edges free, loaded at the center with a concentrated load Genera. V'ew Of test frame.................0... Side view of test frame........................ View Of teSt frameOOOOOOCOOOOOOOOOOOOOOOOOOOOOO Simple SUDport SketCheeeeeeeeeeeeeoeeoeeeeooeee Components for the support of uniformly loaded simply supported BQuare plate................ A'r ce"80.0COOOOOOOOOOOOOOOOOOOO0.0......IO... Set-up for measuring model thickness........... Set-up for calibration of the material......... cutout arrangementGOOOOOOOOOOOOOOOOOOOOOOOOO... Plots of fringe values for determination of curvature in figure Notation of Plan of the and twist. Refer to fringe patterns 2000OOOOOOOOOOOOOOOOOOOOOOOOOOOOO. points in the Difference Method.... SIaDOOOOOOOOOOOOOOOOOOOO.0.0.0.0... V PAGE ii l2 :2 l5 l6 l7 '9 20 20 20 23 24 28 30 3| LIST OF ILLUSTRATIONS CONTINUED FIGURE PAGE I8A Boundary conditions along a free edge........... 33 I9 Forces acting on elements on analogous structure at a point midway between the corners of Cut- outOOOOOOOOOOOOOOOO00......OOOOOOOOOOOOOOOI... 37 2O F'n'te Difference gr‘dOOOOOO..OOOOOOOOOO'OOOOO... 40 2| Fringe patterns for simply supported square plate A8, with uniform ioad................... 44 22 Moments for plate Model AB...................... 46 23 Deflections in plates........................... 47 24 Fringe patterns for series CD................... 48 25 Moments for plate CDl........................... 52 26 Moments for plate CD2........................... 53 27 Moments for piste C03............................SA 28 Fringe patterns for plate EF.................... 55 29 Moments for plate EF............................ 57 30 Comparison of maximum moments and deflections in simply supported square plate with different cutoutaOO0.0GOO...OOOOOOOOOOOOOOIOOOOOO0.0000. 62 vi Dc Mx, My Mxy NOTATION Side dimension of square plate Distance from plate model to camera lens Distance from horizontal axis of Moire set-up to a given point on the model Spacing of lines on the grid = Et3/ (l-‘Ffi, which is the plate constant Plate constant for the calibration plate Modulus of elasticity Bending moments per unit of width acting on sections perpendicular to the x and y axes, respectively, with positive directions as shown in figure l Twisting moment per unit width in the x and y directions Distributed load per unit area, with positive load downward Radius of calibration plate Horizontal distance measured along the camera axis, from plate model to the Moire Screen Slab thickness or model thickness Thickness of calibration plate Deflection of slab, positive downward Horizontal coordinate axes. In all fringe photographs, origin is taken at upper left corner, with x positive to the right and y positive downward Vertical distance between the point. on the screen reflected to the film by a point of the unloaded model and the point reflected by the same point of the loaded model Spacing of grid used in the difference procedure vii Poisson's ratio Slope of plate model, or change in slope at a point, in radians viii CHAPTER I INTRODUCTION In the classical analysis of simply supported rectan- gular plates the first solution was due to Navier, who used for this purpose the double trigonometric series. Levy gave a solution of the simply supported plate by using a simple series. The energy principle offers another method of analyti— cal solution to simply supported plates, which again assumes a double trigonometric series for deflection. But in practical designs, (as for example the cut outs), the solution of the differential equation by assuming a double trigonometric series becomes impossible because of the boundary condition. A numerical method like the finite difference approx- imation to the plate eduation, offers a powerful tool in sol- ving relatively complicated problems that are not solved by either LEVy's method, Navier's method, or energy principle. This of course leads to the solution of a large set of linear equations that could be solved by high speed computers. As the plates form a structural element of utmost im- portance, it is understandable that many efforts have been made to use powerful experimental methods, like photoelasti- city. An independent attack on the problem was made in re- cent years by employing the "Photo Reflective Principles". There are two methods having this principle. The first of these is applied commercially by Fresan Corporation and has been successfully used in the floor slab design of many large buildings of unusual shape. The second method is due to F. K. Ligtenberg, who developed an experimental method whereby--- with simple apparatus and a reasonable amount of work--it is possible to determine the moment distribution in plates with sufficient accuracy. The Fresan method yields better accuracy. But, in the Presan method, curvatures are determined directly, while the Moiré method determines the slopes, from which de- flections are found by integration and curvatures by differ- entiation. The purpose of this study is summarized as follows: (a) To find the moment distribution in a simply supported, laterally loaded square plate. (b) To find the effect of square and rectangular cutouts in a simply supported square plate on the distribution of moment. (c) To study the accuracy of the Moiré method as applied to simply supported plates by comparing the experimental results of (a) and (b) with finite difference analysis. The principles of the Moiré methodcne reviewed here 1’or the completeness of the presentation, although they are presented elsewhere.'v 2 .The edulpment used in this investigation is the same as described in reference i. Only the simple support has been newly introduced, a description of which is included in this presentation. 'See "The Determination of moments and deflections in plates by the Mot1§ Method and by finite differences with application to the square clamped plate with souare cutouts" --a doctoral thesis by Dr. William A. Bradley, University of Michigan, l956. 2See F. K. Ligtenberg, "The Math? Method-~a new experimental method for Determination of moments", Proceed- ing§_of the Society for Experimental Stress Analysis, Vol. XII, No. 2, l9§§, pp. 82-987Pand C. G. J. Vreedenburgh and H. van Wijngaarden, "New Progress in our Knowledge about Moment Distribution in Flat Slabs by means of the Moiré Method", Proceedings of the Societygfor Experimental Stress Analysis, VoT. XII, No. 2, i955, pp. 99-iifi. CHAPTER II ELEQLEL F. S. The equation characterizing small deflections of an isotropic plate under lateral loads has the form B4w+234w +31% :3 3x1; 5x23y2 Byu D l where 0. Et3 75-77-,2) . The moments and shears are given by BMX Mxy-i- .QML‘J (IX 3X Qx+§9a§dx ax Figure l. Moments and forces acting on a slab element. Directions are positive as shown. a MR:-D(%E¥1+S§$) , htr=-VD (ééet‘TI gag) ’ Mx¥=-D (i— 9);}.‘37 , s—Ct‘a it), eta—‘3: as. From the nature of the above eduations it istseen that, ; if curvature of the neutral surface at all pointsfdliplhte‘ is known by some method, then the moments and shears are easily determined. Also, curvature can be looked upon as "slope of the slope curve". Thus giggle found by finding the slope of 35—}:- .Curve with respect to x and gig“ is found by find- ing either slope of gal curve with respect to y or 96.1.: with respect to x . The deflections are determined by numerical integration using the slope curve. Principle: (Figure 2) Let D = Change in slope from the unloaded condition to the loaded condition q = The distance from the axis, of the point on the sur- face of the screen reflected to the negative by a given point P on the unloaded model. SCREEN ‘ MODEL T L 0) MODE; 4/07 é OMIév I l [’4f-__f‘ __ f . U.‘ 2 _ _ 5:! _ S b (y Mama A 0/1050 #6. 2 soc/5M4 776‘ 0/4 GPA/Ms FOR M0095 PIP/”(Wolf r : The distance to the screen point reflected by the same point P with the model loaded. q-r=A= 26¢+28¢ Cz/DZ l + 23 fl b ’2 28¢ (l + c?) 32 (Since d is small, c is small with respect to b) 2&2 is small) The same relation holds if the initial slope of the model is not zero. P = £> 28 (l + c§7b2) If the point P on the slab were not too great a distance from the axis, the term 5; would be small and could be neglected for a flat screen. This error can be theoretically eliminated and practically reduced to a negligible amount by the use of a curved screen, the derivations of which are shown in Appendix A in reference I. Measurement of ZS: The measurement of Axis accomplished by the use of: (a) The Grid, consisting of alternate white and black lines _placed on the screen; these lines are of equal widths d/2, so that the spacing from the center of one line to the next one of the same colour is g. (b) The Camera, which is used to record two exposures on the same negative; first, the reflection of the grid by the 8 unloaded model, and second, the reflection of the grid by the loaded model. a) Unloaded Model b) Loaded Model - entire plate rotated through angle 3b Figure 3. Rotation of plate to give photOgraph which is light throughout. In the illustration, the first exposure is made with point P reflecting a black line, and if developed, would merely have recorded a reflection of the grid. Now, the model is rotated after the first exposure through an angle <¢ , figure 2, such that a point P now reflects a white line, and a second exposure is made on the same negative; the result will be a negative which has been sensitized throughout, and a light print will result. In this CaSPAA = l/2 d, and since ¢ : éi , then ¢ : ifi . Thus, since distance S and g are known, we can find the slope of the model, 525 On the other hand if the model had been rotated further after the first exposure so that a dark line was reflected to the negative by point P, the film would not be sensitized; the whole print would simply appear as the grid itself, with alternate light and dark lines. In this case, the slope of the model i356: d . , as Whene er the slo was i d 2 d d V pe (55” (asl’ 3 (as? . . . n(g_) the original grid would appear with alternate 28 black and white lines. When the slope was L(g_), (g_), 2 2S 2 28 the print will be entirely light. 0 o o QH-I d ’ T (is) But in any actual model the slopes will vary from point to point, and there will result on the prints a series of light regions and other regions in which the original grid appears. Interpretation of Moiré fringes: Figure (4) The model in these photos was a rectangular plate with 2 edges clamped and the two vertical edges free with a concentrated load at the center. Since the top and bottom edges are fixed, the slopes along these edges is known to be zero. Figure (4b). Moving downward, the successive . 'aw \ dark fringes show all points for which:§=§>= g_, gg, etc., 28 23 The light fringes will show points for which is by = <5“- 5%,)» g(%§). and so on. For convenience, the magnitude of these. fringes can be designated by order numbers. Thus the order of the fringes in figure (4b) is, fr0m top to bottom, l/2, l l/2, 2 l/2, 3 l/2, A 1/2, 5 l/2, 6 l/2, 6 l/2, 5 l/2, 4 1/2, 3 l/2, 2 l/2, l l/2, l/2, -1/2, -l l/2, 10 . -2 1/2, -3 1/3, -4 1/2, -5 1/2, -6 1/2, -6 1/2, -5 1/2, -4 1/2, -3 1/2, -2 1/2. -l l/2. -|/2. Similarly 3.?! can be found from photo (46.). X /Oq .QVQV QNKVQKQMURQU T \\k\\~.\ QNKxfiU moxo‘k k‘ QNQTQV smownKux MMWQM. ng Qxaxux QMQQ‘RNU “NO va QQK \ka\\x .Nkvfifix %‘ VQWRRKNMW‘ me.‘ WNOQRXK N%\Q\.\ .‘ .W\h\ not 23 CHAPTER III DIAGONAL PHOTOGRAPH The purpose of diagonal photographs is to provide a possible check on results as well as to give a picture of dia- gonal curvatures. The diagonal relationship of slope, curvature and moment with respect to x, y slopes, curvature and moments are fly : Dx'Sin O + Dy'CosO Ax = px'Coso - ¢y8in9 em = B x'SlnG +Bg'y CosG By y B)’ B x :Bg'x CosG ~3Q'y Sine x ‘ox B x 62w + 32w = 32w + 62w 3x2 By? an: at: Mn : Mx + My + Mx - My 00820 + MxySin2G 2 2 Mnt : - Mx-M Sin29 + MxyCos29 Y l2 Figure 5. Notation for diagonal moments. Principal moments: 3;} = -—.——M + M . Jester + tan 29' = W /3 .QQQV QNK‘WKRNVxVfib V §\\\V Nkkxxuob NSR KR QNQNQQV WNnKU. MNWQM‘ ORR Qx‘V QNK‘VVU MWWQHN ank \\k\\_\ Mk «\VK *TV\VW\<‘k.UmVnw\ Uthx “auxwkkmxbx Nw>\\nk\ VVQQW V.\Q .m ..%\.U\ CHAPTER IV THE EQUIPMENT The test set up consists of: l. The grid with the reduired lighting. 2. The camera. 3. The simple support. 4. Arrangement for application and measurement of load. 5. The model. 6. Provision for calibration of model material. A detailed description of the above are given in reference I, except the simple support and the model. Figures (7-l4) illustrating the above are included for the completeness of the presentation. The Simple Suppprt: If the edge Y:O is simply supported, the deflection ! along this edge must be zero. At the same time, this edge should rotate freely with respect to the .X - axis, ie, there are no bending moments along this edge. In this investi- gation such an edge condition is created as shown in figures (lO),(il). Two knife edges press very gently against the plate model so that there are no deflections and at the same time allowing the edge of the plate to rotate freely. The two half inch thick plates with nine inches souare opening 14 ,. F76. 7. GEN£PAL VIEW OF 7557 ”AME l5 F/6.3. 5/06 V/EW OF 7557' F/i’A/Hé' l6 F76. 9. VIEW OF 7557 FRAME l7 are held together by means of l6 bolts and nuts, which are tightened by hand only. Since the plate model to be tested is supported in a vertical position, it is prevented from falling by suspending the plate to the outer plate frame by means of fine string which passes through the holes made in plate model and in the plate frame. The Model: There are two logical choices for a plate model-- i. The Metallic plates 2. The Non-metallic plates The merits and demerits of both can be enumerated as follows: (a) The calibration of metallic plates is usually direct and simple, since the engineering constants, namely, the Poisson's ratio and the Young's modulus of elasticity can be found easily by using strain gauges and the Universal Testing machines. (b) The temperature and the humidity does not affect the calibration of the metallic plates. (C) The thickness of metallic plates would be fairly uniform. The disadvantages of the metallic plates are: (a) They are difficult to polish to a fine enough mirror surface. SIMPL Y SUPPORTED EDGE ___.__V___T + ' + + + . + i + 1' + t_.----._.. ] l + -l- : l + + fi‘ 1 + g 24'; 'i‘ + + + __ 2 _,_J FLA TE M0051. 1‘76. /0 /9 MODEL [741475 20 WWM‘LQ\\\R NNQQx.‘ wéxkxchtmax “wok RS - km“. m.\ .%\i\ m o RC Let: .N\ .mxk wok apex SSSGW QNkmeQmKQW >Vn\\<\h. QQQVQQ \. QQRVQK§<§ HQ k WAQKQQVW Nxxk nva.‘ Wk \<.N>\0W\\<= O-Sa, y: 0°25 figure 28. First, enlargements of the pictures were made. Sizes of l0" x lo" and 8" x 8" were tried. It was found that unduhy large sizes gave no advantage,and hence E" x 8" size was finally used. The pictures were printed for maximum contrast. Next, the photos were lined with grid spacing of O-l25a where g is the side length of the square model. Each half fringe was then followed, and each point where it inter- sected a grid line was marked with a very fine pin-hole. The curves of 95335 versus X, 26).; versus ‘i , Pal; versus Y , 2%: versus X were next drawn. Each half fringe is located along each grid line, with the location determined in terms of a. For example, for the curve %J$! versus Y at X : O-25a, the locations are as given in table below. m Fringe y interms ofa Fringe y intermsofa 3 1/2 ' o-12sa -|/2' 0-523a 2 '/2 O-239a -l l/2 0.648 I I/2 0-327a -2 1/2 0.773 1/2 O-4l2a -3 1/2 0-922 26 27 For 1 fringe,¢: .2— : 0“0 '= 0.002 in/in. A similar 2* 2(25) . tabulation of these locations is made for all lines required for drawing the four sets of curves listed above. The figure (l6) shows the curves of g¥versus y and EE% versus x. Determination of Curvatures: The curvature is found as the slope of the slope 2 curve. gggy is found by measuring the slope of curveta) figure l6 and Sfiyi‘s found by measuring the slope of curve (b) To expedite the measuring of slope, a small instrument which is comparable to anangle measuring protractor was used. Instead of angles, this instrument was calibrated with tangents of angles, so that the slope of any curve at a point is read off directly. In finding the curvature at the free edges in the direction perpendicular to the edge, the boundary condition is made use of. Since the moment in this direction is zero, curvature normal to edge = - (‘9) (Curvature tangent to the edge). Moment Determination: The expression for moment is given by hdx::_ +§l:“:) a? as _ t 5 9a 3.1! A. _._Dc(_{.c) [3.3 + 32!]0L 2. 2. Nix..- is.” 231’ 95.31%. 1 REV D° t.) [a ax..+ 3er GTE?» .-.i.-... . .r . w !'- 1 i .- gain/v.1 €13”qu 29 The right hand side of the last equation is a dimensionless value in which DC : Flexuspal rigidity for the calibration plate, t 2 Thickness of model at point considered, to : Thickness of calibration plate, v : Poisson's ratio, q : Load intensity in Pfiv a : Side dimension of model. Deflections: There are three possible methods of finding the de- flections: (a) By direct counting of the squares (b) By applying formula's like Simpson rule (c) By mechanical devices such as the planimeter. In' this. investigation, the numerical integration of the curves is done by the first and third method. CHAPTER VI FINITE DIFFERENCE ANALYSIS The analytical check for the Moiré method re6u|ts was made by finite difference analysis. The plate equation, for small deflections of a thin plate, Biiw 31+“, CNN 0+ 374+ 2. a ”Taxt'T “574 = ’5 . can be approximated by replacing the partial derivatives by finite differences. This reduces the fourth order and first degree differential plate eQuation to a system of linear algebraic eQuatlon whose variables will be the deflections of nodal points. At each nodal point, we can write down the plate eqoation and hence this reduces to a solution of a set of linear algebraic equations which gives the deflections at each pivotal point.knowing the deflections, the moments, shears can be found from the various differential relations. 0 X NIN o—XXo 1 Nw N NE I'Y WW w 0 E EE xx: 7\‘f a m _ SW s as $5 Y Figure l7. Notation of points in the difference method. \— 3O 3i Consider a plan of the slab on which is imposed a square grid. It is desired to write the plate equation in the difference form at 0. (fl)& Vila—W0 ax )s x"'% aw (SW) = (%!'i) Xwg - I167) Koo-1} a WI;__W__9 — Wo—WH bx? )\ g I WE‘ZWo+fl_W 70. BAW = LO: ( 3“” ____ WEE' 4W5 +6Wo-A-WI +WWV bxfi' as} ‘57@) ‘)3 Similarly in = Wgs—A-Wg-,j;§Wo-4;-WN+WNN 8‘“- )3- a4’yj ._. WNE +W$I +Nsw +Www -?_WB-'2Ws ~2WwL-1WN 4- 4-319 31331‘ 14. Substituting these in the plate eduation, we have We: +Wss +Www +Wxx J1-'?.(Wiis +Wss +Wsw + Www) -8(W1+W5+WN +Ww) +Zow. = EEK. The expression for moments takes the following form: Mx:-Q2 [(We-2Wo+ww)+i(w$-2Wo+wn)] ‘A Myz-Q [(Ws-2W0+Wn)+SLWe-2W0+Ww)] x2 Mxy = - l-9 ( Wse + an - Wne - Wsw) 4A The Boundary Conditions: 0 Q- A x b C B Y Figure l8. Plan of the slab. 32 If gfi,is a simple support, the analytical expressions of the boundary conditions in this case are (W)~ivo =0 | 8w ETw -O (757 bx?)'°"0 2 Replacing the partial derivatives in the second equation by their finite difference approximations, we get- (Ws-2W0+Wn)+ §(We-2W0+WW)=O. From the first equation. WO 2 we = ww O at y:O. Hence, for a point ouside the boundary, the fictitious dis- placement is determined ln terms of the diaplacement of an interior point. So the plate equation for a point next to the boundary can be written down making use of the relation 3. If an edge of a plate, say X =a (figure l8), is entirely free, it is natural to assume that along this edge there are no bending and twisting moments and also no vertical shearing forces, ie (MXIx'ct = 0. (MXY)x-a= 0, (OX).....= O The three boundary conditions are too many because only two boundary Conditions are sufficient for the complete determination of deflections w satisfying a4 B4“ ‘b‘w _ 31; ”axll‘T Z yea—TIT ave " o- Mxy 'i-BM"Y CIT / / / BY ft __--y -———--—3_Y_-2. dY/ J, Mxy (at) (b ) 1 p Z Figure l8A. Boundary conditions along a free edge. The bending of a plate will not be changed if the horizontal forces giving the twisting couple Mxy.dy acting on an element of the length dy of the edge x :a are replaced by two vertical forces of the magnitude Mxy.dy, dy apart as shown in figure l8A. Such a replacement does not change the magnitude of twisting moments and produces only the local changes in the stress distribution at the edge of the plate, leaving the stress condition of the rest of the plate unchanged. Considering the two adjacent elements of the edge, we find that the distribution of twisting momentswuy is statically edui- valent to a distribution of shearing forces of the intensity o'x = - (3W 1,... Hence the joint requirement regarding twisting moment Mxy and shearing force 0)! along the free edge x=a becomes Vx : (Qx - 2%;§_) = 0 Substituting for Ox and Mxy their expressions = _ ’a 8w + 31w Q" 0' 3-; (are 7572le Mxy = -D (l- 9) 'azw 3x29}! 34 we get 3 3 (ON 4- (2-9) a" z 0 5x3 Bx.‘ay L... a The first condition reduces to (4%.» 9. 31112)” .=0 b B Equations (a) and (b) represent the two necessary boundary conditions along the free edge x=a of the plate. The pattern approximating the plate equation for an ordinary interior point of a plate is shown below. Nw N NE +2. '8 +2. +2 -8 +2 S S ”’ 5E For one of the interior points, the plate eQUation pattern is developed here. Other cases are shown in reference 1. N N ww N we w w o E E W E SW SE L_____._..1 AtO, My : O ”I“ ‘ 3 "it-'0 + W8) + 9 (Tie - 2 to + WW) = 0 $5 = -‘i(he - 2 ho + Ww) + 2 W0 - Wn l, :- 1(We+WW)+2wO (l +1)-Wn. 23w -9 EL. 8Y5 I)?“ ) :3an =0 (“88 - 2 W5 4» 2 Wn - Anni + (2 - 9) [Vise + ‘A'isw 2 )t3 Wne - hnw - 2 W5 + 2 WIT]: O 2? §C§ ”viss : (6 - 2i)1is + (29- 6) Vin + Wnn + (2-2) (Wne + pnw - fisw - Ase) : (- 511 + 292) (we + 111w) + (49 -l2)11\n + (6-2T)2(l +9)W0+Wnn + (2--9) (woe «ewnw - wa - Woe) 36 Substituting for w$s and We in plate eduation we have Wee + 2 Wnn + Www + (4 -‘9) Wne + 9.Wse + 1 Wsw + (4- 9) an + (2‘92 + 2‘9 -8) (We + Ww) (4‘9-l2) Wn + (l6- 89-41)?) W0 = g)“ For ‘9: l/3, the pattern of plate equation would be N N 1 Nil N us. «1.0. ‘& +11 6 6 S w 9 E E CNN" w —‘ Nl" II N d 1.2. 11 ~32; s +fi§ 9 w t "—57 “E +46 I'Ss The "Plate Ahaiogy"' offers another method by which the pattern for plate eduation could be derived. In this method the plate is replaced by an analogous structure which is made up of a series of rigid bars and blocks joined by springs. The moment Springs transfer the moments Mx , My and the shears, Qx and Qy between the bars and the joint blocks; the torsional springs, connected at the mid-points of the bars, transmit only torsional moment, Mxy . The loads are applied only at the joints. To illustrate this method, the above plate eduation pattern is derived below. lDetermination of Moments by Moir? Method and by Finite Differences--Dr. William A. Bradley. 37 9 w Mow-wk f yin-u} L -‘ E M»%(L , (I 1mg I %3NOI.% ”we 7 We " 0 we / Q‘s. Figure l9. Forces acting on elements of analogous structure at a point midway between the corners of cutout. Considering bar ow and taking moments about w. (g ° Qow) = Mow - Mwo - Mow-n 2 2 = 9__ [(Wd- 2W0 + W) + §(Ws - 2W0 + Wn)] 279 +0 [( Wo-ZWW+an)+§(Wsw-2Ww+an)] E73 ' + 0 (hi) ( W0 +an - WW - Wn) 73. But My : 051(Ws - 2 We + Vin) : - ‘NWe - 2 W0 + WW) ow = D [- (l - $2) (‘r'le —2Wo + WW) + (W0 - 2 Ww + an) + 9( WSW - 2 Ww + WnW) + 2 (l - 9) (W0 + an - Ww - Wn)] : o , W0 ( 5 - 2i- 2%?) + Wn (2i -2) 37x? ‘ +ite (- i +$2) + Ww ( -5+92) +an (2-9) + Www + w) WSW. ] 38 Similarly Qoe. 7. = 0 [W0 (-.5 + 2) + 292) + m (2-29) _ 2 '2'"??- + We (5-91) + Ww (l- 9a) + Wne (-2 +9) + Wse(-9) - Wee] Qon.7\= 9171 [lilo (6-4y) + We (-2 + 29) + w (-2 + 2i) + Wn (-8) + 2Wne + "2an.] Summing the vertical forces on the joint at O, as in figure ('9)and substituting the values for Q'sas found above gives the following plate equation. (l6 - 8%- hi?) W0 + (-8 + 29 + 2%?) We + (49 42) Wn + (-8 + 29 + 292) m + (An 9) Wne + (4- 9) an + i Wse +§Wsw + 2 Wnn + Wee + Www : -%7\+ For‘9zl/3, the pattern of plate eduation would be as shown before. Analysis of plate with and without cutout Difference Solution: The plate series AB, CD, and EF were solved by finite difference method to provide a check on the experimental results. In the case of AB and CD, (see figure 20) due to symmetry only l/8 of the plate was considered, except in the case of the rectangular cutout. The solution was carried out for two grid spacings-—7\= a/8 and a/l6. The eduations were formulated using the plate eQuation patterns. For EF series 39 plate, one half plate was considered. For plate AB, (without cutout), 36 equations resulted for a grid spacing of a/l6, and lO equations for a grid spacing of a/S. These eQuations were all solved by "MISTIC DIGITAL COMPUTER". Program L2 was used. To solve 39 equations, on this program, we need only four minutes. Because of the limitation in memory space, the Mlstic can handle only 39 eQuations at a time. Also, since these equations for plates lead fa symmetric matrix, the solution of greater number of eQUations should be possible by reprogramming. All these eqUations were solved to ll decimal places. The values of the deflection g were substituted back into the original equations to determine the residues. The residues were found to be almost negligible. Richardson's2 extrapolation formula: 2 2 A = Af ”f - Ac "C (hf? - ncz) ( nf2 - nc? ) was used to extrapolate between the coarse grid and fine grid. Summary of finite difference analysis: The results of finite difference analysis shows that by using fairly fine grid with)»: a/8, we could get good values which would compare closely to a finer grid with7\= a/l6. A still finer grid, besides exceeding the limit on the number 2Numerical Methods in Engineering-~Salvadori and Baron p. 77. Prentice-Hall, New York, l952. F/N/ TE O/FFE/Pf/Véf 69/0 lake -3?! I Ll; l . 2. PLATE M0052 AB. i I Z 3 4. 5' 6 7 8 9 lo // lz ,. ,. J 4 PLATE MODEL 60.2. __.h \ A—e PLATE MODEL 630 3. LIL; . // (a sis- air/K _ L4 ’9_ l2; 4T1: V. 40 PLA TE M0062 CD]. / 4‘! 3 ¢— 5’ 6 7 3 ' 9 lo // /2 [3 Id 41’ /6 f I? l5 lg to I 2/ Z: Z) 2 f 25' Z 6 8 7 % P44 7’! MODEL 5F 4| of linear eduations that Mistic can handle, does not modify the values too much. Instead, an extrapolation between a rough grid like a/8, and fine grid like a/l6, improves the values considerably. The finite difference analysis reduces the advantage of Moire Method in the case of simple, rectangular cutouts. But the formulation of boundaries for finite differences, in the case of irregular cutouts presents a problem which is not met in the Moire Method. Also, the analysis given for the points near an interior corner is no longer valid. To get a better approximation a very fine grid could be imposed near the interior corner. CHAPTER VII TEST RESULTS Plate AB -- No Cutout. In figure 2i are shown the fringe patterns for uniformly loaded, square simple supported plate. Moments’and deflections have been found by both Moire Method and finite differences. Moments found as above are compared in the figure 22. Dashed lines show moments determined by finite differences, solid lines show moments determined by Moire method. Deflections determined by Finite Differences are entered in Table I. The deflections determined by finite differences as well as Moife Method are compared in figure 23a. Plate Series CD -- Cutouts. The fringe patterns for this series of plates are shown in figure 24. The moment curves are compared in figure 25. through figure 27. The deflections for this series are entered in Table II through Tablelv. The deflections for a particular line for each individual cutout are compared in figure 23b -through figure 23d. Plate EF. The fringe patterns for this plate ocre shown in figure 28. Moments are compared in figure 29, while deflections are 42 43 compared in figure 23s. The deflections by finite differences are shown in Table V. The figures in parenthesis represent the difference solutions, while the figures not in the paren- thesis represent Molré solutions. ¢¢ .Q‘Q u \SVQ\\>\§ \\.\\\~\ Wu‘ Mk V w\ %\\Q%h. QNKRQRKQW l uhxhxxm, .YQK in. >xkuak K. ‘Qx .mvm\<\m\.u\ .\N.W\h\ DEFLECTIONS 83: : Point DEFLECT TAELE I IONS IN PLATE AB .ua \0 (numuibvu h) —~ O\OCI)\JO\U1 bun) - DJMFDMIU— \NNJ—O I '\)i\)i\) 03me 29 UUUUWUW O\Ui cum -0 Grid Spacing a/l6 Grid Spacing a/S Extrapolated Value 8. 8. .78I06 .0737} .09689 .86686 .4094! .76506 .24205 .9CI96 .l25l2 .74252 .8347i .A3569 .5992l .33l60 .92662 .7I7C3 .7l989 .96720 .5ll58 .38034 .86729 .3354} .80798 .32828 .222A2 .62l85 .8292} .8734l .74776 .0829} .09896 .88769 39. 40. Ono —\_NU1UIO\—\AJ>O\\)\) 352l5 20896 l8363 60492 .268296 29.373l66 27.334877 2l.3445h9 37. .067690 704365 40.547588 \JJ U1 40 .233053 .ll5895 .9246A8 .589964 .380340 .3356l4 .762225 lO9383 .624030 W 4e M Mxy _ - /E& M (MOMENT VAZd/fs f0 55 Mad 77px 7 , _, , V, V X ‘ 2 —* [/55' . ,g/Ls” 5V /0 ¢ g0. j I I '33?) 52%) ggs/rsnws T /“ I ,u ; g;:sfi”‘ ‘ I s “Q M Q S \ / . . . i . I /72;} ' ES. A“? {3.4 _ < N {‘1 _ __- — ”a” K!" :2; _ feast) {269/ race) I I I c_-- — ' /” ’ I {/22) . (W454;- ‘ , - r -- - . ' I l r ‘ II/ A I I I as J "'I , / f -~ \ *- " my _ .. l~ l I '\ \\ '/ \ON . — 6 /7}. , I 57) . I IRE // Q ‘I' IFS: "/ 257 3&2- -- " 43:; (7%) (774/ - ' ' "27y .( N \"311/ I“ 1.. ,-_-__. _. \, /5 7 M/ ‘ _ -- » {2374/ IO t _ _ ._ I I ~- * If 075) 4- __ if--. I ' . U , I“ I I ' I I , , 7Q / __.___ s _ i - .«7—1 3i ~ ' ‘il l ‘5 ii .. —~— ~- - 5/7 ‘ fi/f’ (we) [95) l y 19 _ a, \ /95' . ‘57)- “ (ad/j ____ _ _ ,. 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I iI \J I‘5\J / a l i .. «e . i i “a r” I I 2 I N I i / g§ Q§ §$ ! , . Q‘ I II,/ .. w i i .2 «#6 .4 l 1 ‘ l I r\ ’I\ ,\ I ’\ U , no to . g8 &:‘ Q” ha I ;\Q) . N\, “3% “\5 I‘”/ i ,\ -\ . \‘3 \ \\g R) \ E8 \ QR « git) \x. \ \\J \ .‘\ \\4 iNQ) .4; i I X x\i l 14 g K i ‘T ~— _____--_ ___.fi __ MAX/MUM [2.942 £67/0A/ E254 E A7 CE/VfEAD (7F £17700; ‘ £065: #104560?” ./a 0 fl = 50.00 213... m 0 4 —¢ 90 I 29.9 /6.5' (35) (42) (75-2) /7¢ 2/? 7 /5/ {/37} , IK/fv {/67) 228 (223) wé) ‘26 7 {2/ ¢/ _+/33-9 z 25* (£25) /75’ £03 /66 05/) 097/ 6/67) mg /a 7 /6/ (3 a) . (M2) {/52} F/é. 2 6 - MOME/VTS 1 /K5 (do) "/69 05¢) M w 53/” (3”) 2:77 (287/ "43/ (z 3/) #3 2/30 [ /o 9 {/30 1:30 (83/) ”237 (237) p... .315 f3”) FOP PLATE M0054 (“0.2. w i498 [av/"i (25/) 4 . {109/ , {/05‘) I i . 225’ /7/ nf/ / , ' I (86;) +733} - , IK/ofl) I .203 I47 /?f~/’/" f2 :0) 42:42) , {/M/ I I I ‘ {at 7 7 I 4/44)- - - w , i . 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D d ~¢ f ' [33. 3/) 2-3... 0 7 6 p 56’ .hxnfl kaxvn‘ W‘th .menKmvkk‘K NWxSkK .NN .\\.\ 4+ 4444444444444444444 4 I" I III IIWII III I III IIIIIIIIIII IIIIIIIIIII IIIIII IIIIIIIIII III ' III III III II IIIIIII IIIII III I II I II II \I H I4 I. . .4 4 _ M I“ II 44444 :44-4 4-4 4 4 4 44444 444 444 44444444 4444 4:4 4 4444444444 4 4 4 444444 4 II II I|||I II III III III I'll Ill-Ill M I III III III I III III III Ill llll Illlu || I II III M ‘|| 'IIIIIIIIIIIlIIl l||||| III ||||Il ||||||II . H“ I II II I l l l I I II I I) III I! I, W l. 1:],4' IIIIIIIIIII “:44 ;::; III II III III III I V II III III II I I I I I I I III I "W I III"! I l I! III I II, “a II I II I W M “IIII III I III .. I I III I l‘l ‘ I a m I. I II I m III 4 I II II II‘III IIII I \I III \II II‘I‘II \I \ ‘\I‘ II“ III. \ I‘ I‘IuIIu II\‘ In“: II. IIIIIIIIIIIIIIII IIIIIIII III II IIIIIII IIIIIIIIIIIII I * III I WNW IIIIIIIIH.IIIIIIIIIM 1|! #1] WIIIIIIIIIIIIIIIIIWIIIW IHI . ””ImWHII I. 41”” 4—;221" I 56 TABLE v DEFLECTIONS IN PLATE EF <11ini . Point DEFLECTIONS a. Grid Spacing a/8 20.105017 18.709138 13.469297 7.216595 34.000531 24.056982 12.837644 46.923704 41.437834 30.113289 16.175111 44.326580 40.710126 31.030140 16.949549 39.183487 36.316835 28.103013 15.491216 29.872320 27.766783 21.622612 11.990916 16.394859 15.262790 11.935829 27 6.658674 M M m_———-——-— OWOUNOUW bwm-Oxomwmm #9110— MM M..- MMMM gunk»: 7 E M MW M [MOMEA/r VALUES 0 5 9 . 1 MUéT/PAAFD 5y /0 9 ¢ 2 1 _ . _ 1 /J _-1,_--3_.-_ ' T "” 7 1 297 1.255- 1.543 {M 3’7 1 1’ 1 fl/ ”“3---” ,2] W ,7 I 1 1 1 ‘ [30 8) (26’?) ”fig/"77f” 1 1 1" / // ’ 1 , 7’“ ##fl,- - _. _‘..“1: 7 1 ' ['1 l/ / 1 1”, a. J'— : 1’: --_— ~ ~ 1 1 1 ’\ / ’\ 11 '\ 1 (333/ '1 . 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El? / “ / 1 1 __ __. 1 \Q \0 ,/ /.;._ 1 1 fl fiifiw 1 211-3 ; \J 1/32 339 - 7333 67¢) (46) . 152/0 / 3018-9 ‘ “ f H x. 1 1 1/ / / K 1 . 3 3 6/ 1 ~ ; 1 $13 / / 1 1 1:6 , 9 2 _ 1 - *1--~-—--~ —‘ 235— , -m‘wmfi _ _- *- '—_~ _ / / 7 / 1 1 1 Li?) «SK-3J3) -.. _ £5657 _ -_ {3/7) 05-) ad 41¢ ) O”) 1 1 / / 1/ 1 1 1 ‘; 3/- “ , 1 1 ' 1 2.3 1 3 . 1 1 1 2-31-“ ,1 i 1 7/ ’ J 1 29 1 93 _, Mir-WW ' w 3/ 1 M Ma v ”33”“- W 1 1 4,4 3 "’3'”- ' 1 ' 1 ' : 1 “ 3 1 3 43/ 1 1 1 § _ 1 1 -—————— Q Q / 1955 1233- 1 33?‘"”"”"1309 1 my; if]; ”1:7? - 2653”")? 1 1 7 771 ——-v——— / 1 1 {/13} 15:79) 1499/ ‘35) £— —-- 1 1 ,____ 1 1 , #_”;_;=W ‘ 1 1 1 1 1:73 . /, / -/ 1’ ,3 g 1 L /1 n 1 / z 25‘ a 2 3 M; 1/51: _, ‘ ———L \\' ? Me? ’5 5' "‘— 23: (z 55) (23v 1 "’7' 5") ‘ 0/14) 175") (z 2.5) K ) 72:3:fi_ 324 ” "'1 1 1 1 - 1 1 1 1 1 J ! 1’ z 7 1 1 _ - w 26 3/3/9157 / ,A 1 1%. (53¢) 2515'), - .. 0 g J::“::"—':—:” 1 MA X/MUM 0522' EC f/O/V 1 ._ #4257 4 F . (A ' 1 FORD p; ,4 7' E Mo. DEA E . a_ :51{_ .2151 H ___ 4’ 47' X ___, . 501.1 .4713. 2 9 MOMEA/7’5 1 v 3 .375 «g L 1 14/ . 4/. 20 23.. 1 1 y 9 0 -¢ 3647-001L'4é “0 fl 58 InteTQLetation of Test Results The Moire method for a simply supported and uniformly loaded square plate gives a solution which is 5 to l2 percent emailer than the solution by finite differences, and the solution given by M. Levy.3 The possible sources of error could be listed as follows: (i) Membrane action of the plate due to large deflections (2) Calibration (3) Measurement of load (4) Reduction of data Membrane Action: To get an approximate solution for a simply supported rectangular plate a simple method consisting of a combination of the known solutions given by the theory of small deflections and the membrane theory can be used. We assume that the load q can be resolved into two parts Q. and 02 in such a manner that the part q' is balanced by the bending and shearing stresses calculated by the theory of small deflections, the part q2 being balanced by the membrane stresses. The de- flection at the center for a square plate with sides a by the theory of small deflections is 4 wo = 0.00406. “ya 0 4 x q, 0134 ' = 0.0443 qIa Eh} Eh q‘ = W6 . Eh3 I 0.5333 a” 3Theory of plates and shells--S.‘Tlmoshenko, pp. 349-350. 59 Considering the plate as a membrane and using formula (204) in Timoshenko's plates and shells, WO :1: 0.802. (8/2) 3 3.2: J2Eh q = |6W03.§h 2 . a 2 q = q‘ + Q2 3 = Wo.Eh3 (22.5 + 3|.0 31.122) a4 “2 Now, D = Eh3 120-12) Eh3=0 .12 . (l-iz) Dc (1)3 . l2 . (l-iz) tc §2.2 x l.473 x l2 x 8 9 = i025 4 Substituting l, 2, and 4 in 3, and taking 0 = 2.5 inches of water, We. have, I735 Wo3 + 22.5 W0 - 0.599 = 0, solving for w0 W0 = 0.0253 inches q‘ 2.38 inches of water 02 : 0.l2 inches of water Hence, it is concluded that membrane stresses balance about 5% of the load and only the remaining is balanced by bending 60 and shearing stresses. By applying the corrected value of load, all the Moire method solutions are increased by 5;. For a better approximation the 'Theory of Large deflections' as discussed in Appendix could be used. Calibration: The calibration plate used in this investigation had a fixed edge condition, while the plate model was simply supported. if the calibration plate were simply supported, the maximum. deflection at the center iss w max = ES +92 034 +1 c Wmax : 4.08 . 34 EEOC The deflection in this case is four times as great as that if ‘9: 0.3 for the plate with clamped edges. This would introduce the membrane action of the plate, which would take a part of the load. Eut, plate model also is acting as a membrane. Hence, by applying the same load to both calibration plate and plate model, the effect of membrane action could be partially can- celied out in the plate since the value of Dc in this case would be increased proportionately. .5 Theory of Plates and Shells-~S. Timoshenko. p. 62. 6i Load and Thickness Measurement: The errors in load measurement may be due to small leak- ages in air circuit. Reduction of g££2i This type of error is introduced while drawing the slope curves, and in measuring the slope. If the fringes are small in number, very few points will be available for drawing the curves. About 2% of error could be accounted due to this cause. A comparison of the effect of the various cutouts is made in figure 30. For sauare cutouts with side dimensions eQUal to or less than i/h of the plate side length, the greatest“ variation in maximum moment from the maximum moment in the uniformly loaded and simply suppdrted square plate is about i} percent. For larger cutouts the variation in bending moment is greater. The maximum deflection, in the case of plate with cutout up to l/h span, varies from the simply supported plate to an extent of 20 percent. As the cutout size increases, the maxi- mum deflection tends to become smaller than the maximum deflection of simply supported plate, because of the reduction of the load on the plate. Hence, it is concluded that the effect of cutout on a simply supported plate extends a distance from the cutout which is little greater than the size of the cutout. 5/015 fl/NE/VS/flll/ 0F CO/éflf ——" COMPAF/fio/V 0F MAW/MOM MéME/VTS A'A/D (5?. I k l in: i § S a”. i”m \3M 1 § .. i a‘ l “é l ix 6 § ‘V_w‘i\ ., / ;;j’//"'"’—_“‘ x “‘ ~\ ,.. § 5 : ".:/:K//V -’ \~\.‘\\"-:\’f/‘M’9X~ 5/” 3‘ i l \ “4 ¢ “ MAX. 7/3/15 7/445 \_ § '- ”0/”fo \\:§. s / -., WI§2M¢+=w-A mm- 2...; u, 3 2 § 1 ‘1) | § / l . § 254. 500. 4 ¢ 9,6212 MAX. DEch—‘Cf/a/V ._.,. F/é- Jfi D ._\ \ \7 2 2\ 7/ A\ \ Aid/Y. pfFC FL” 7/JN\‘--- _ ._ ‘ .... n... __.___ -I‘ ’ __ “T‘ l l , l ‘.\K -‘ ‘XH l , i . i \\_ ___e-___-_____._-l __ W __ ‘504. 5/05 D/ME/l/S/fl/i/ 0F Cflfédf DEFZECT/fl/i/S //i/ S/MPLV JflP/DO/F'TED SQUAFE Ply/7:. W/f/7’ D/Ffifff/Vf CVfflUfS. .Ill\|..|.l| {I‘ll-I il. ‘1: I 63 Also from the figure 30 it is seen that the deviation of twisting moment is less than the deviation of bending moment. The size of the cutout influences the maximum bending moment in two ways. Up to the cutout size of l/A side span of the plate, the bending moment is increased, afterwards, as the cut- out size increases, the maximum bending moment decreases in value in comparison to the maximum moment in the plate with no cutout. But, the maximum twisting moment increases steadily with the increase of size of the cutout. The manner of varia- tion of maximum deflection compares closely to the variation of maximum bending moment. CHAPTER VIII SUMMARY AND CONCLUSION The moments and deflections in the case of uniformly loaded, simply supported sqaare plate are found by the Moire Method. The variation of deflections and moments due to cut- outs are also studied by this method. Photographs for different loads and diagonal photographs are used where necessary. As a check for experimental method, finite difference analysis is presented here. In solving the linear algebraic eduations for finite difference, MISTIC Computer was used. Conclusions: From an examination of the fringe photographs for plates with cutouts, an idea of the moment distribution can be gained. Also as the fringes becomes closer, the curvature in that re- gion is great and hence the moment. The values of maximum moments and maximum deflections given by Moiré's Method differ from finite difference analysis by about l2 percent. Of this nearly to the extent of 5 percent is accounted due to the membrane action of the plate. The re- maining 7 percent is due to the errors in calibration, reducing the data and possible leakage in air circuit if any. At the cutout corner, the exact value of moments are not determined in this study either by Moiri's Methdd or by finite difference. 64 ' 65 The effect of cutout in a uniformly loaded, simply supported square plate extends to a distance from the cutout slightly greater than the size of the cutout. The size of the cutout has little effect on maximum moment and deflection unless the cutout size exceeds a quarter of the plate span. Because of the high speed computer, finite differences offers a powerful method for plate analysis. From the results it is concluded, that the use of a very fine grid offers little advantage. Instead, by extrapolating between a coarse grid, and a medium fine grid, good results could be obtained. For unusual structural forms like irregular cutouts, formulation of boundary conditions becomes formidable. Then access could be had to the Moire Method. Thus, the MoirE Method and finite differences could be used to check the accuracy of one on an- other and also they could be used as alternative methods if one method fails for a particular problem. BIBLIOGRAFHN Bradley, w. A. "The Determination of Moments and Deflections in Plates by the Moire Method and by Finite Differences With Application to the Square Clamped Plate with Square Cutouts"--A Doctoral Thesis Submitted to the University of Michigan,‘T256. _ Ligtenberg, F. K. "The Moire Method--a New Experimental Method For Determination of Moments", Proceedings of the Society for Experimental Stress Analysis. vol. XII, NO- 2. F955, 92- 83-98. w._ _ Salvadori, M. G. and Baron, M. L. Numerical Methods in Engineerigg. New York, Prentice-Half, Inc., T952. Sokolnikoff, I. S.--A lecture notes on Mathematical Theory of Elasticity, TQET, BFEwn UnivEFsity. it Timoshenko, S. P. Theoty of Plates and Shells. New york, McGraw-Hili Book Company,‘i950. Vreedenburgh, C. G. J. and van Wijngaarden, H. "New Progress in our Knowledge about Moment Distribution in Flat Slabs by Means of the Moiré Method", Proceedings of the Society for Experimental Stress AnaTyst. Vol. XII, No. 2, F955, pp. 99-TFE. .2- "Photo Reflective Stress Analysis", a pamphlet publiShed by Presan Corporation. Los Angeles, ‘953° 66 APPENDIX THEORY OF LARGE DEFLECTIONS In the theory of large deflections* the plate equation takes the following form: balm: 2M 32w a—x, + algae-waif; 363,-» Nx—z + Ny—1 a + a Nx‘l 3x37) , where Mx, Ny, ny now depend on the strain of the middle plane of the plate due to bending. Assuming that there are no body forces in the xy hiane TicvmaI and that the load is to the plate, the eQUations of equili- brium of an element in the xy plane are ‘aNx + any = O l ?T? 23y ’ BNx + 3N = O 2 75-1.. 75% The third equation is provided by compatibility. ate. a‘ey air“ 3%: a 31w 33w W +W’Bx31 ”(R—3‘7)“ 320531?- Replacing strain components by the equivalent expressions, Ex = . (Nx -9 My) , l TIE €=|.N-3)N Y =l.N yr: (y X), xym xy The third eQuation in terms of Nx, Ny, nyis obtained. *Theory of plates and shells--S. Timoshenko, pp. 299-30i. 68 69 The solution of these equations is greatly simplified by the introduction of a stress function. Nx = h. 32F 3y? Ny = h. 32F (b) 3x2 ny : -h. 32F 'Dx.8y where F is a function of x and y. From (a) and (b), the compatibility gives 2 2' 3v: 3%! B‘F if. .2 .331.) _. .__ . .1 Bx4+ a sxtax + 3% EUme a“ 3‘ The second equation necessary to determine F and w is obtained from the plate equation 1w'a 33 m Bw_h,fl-_+le=3wfi§_w 38": w Bx~+asfi~a~it+ 314 D in +aviaxl+axt bi?— Esta 3‘16be These two equations,together with the boundary conditions determine the two functions F and w. Having the stress funchbn F, the membrane stresses could be found. From w, the bending and shearing stresses could be found by using the equation for snmll deflections. Thus the investigation of large deflection of plate reduces to the solution of the two non-linear differential equations. 70 The solution of these equations in general is not known. Appro- ximate solutions like finite differences might be used for further study of the combined membrane action and bending of thin plates. E‘E‘I" C23 ~32“! 2...: g, .1” {/3 -. TI}. r'vm I ear! nan-c. TF3 s: "3 .I {0‘ ' s—Il o 5%...“ who 0 M1,, W‘ ‘s.- ‘W‘hk _- _. MICHIGAN STATE UNIVERSITY LIBRARIES I o 1 5 3 757 15 I II 31293