Moments and Reactions for Rectangular Plates

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A WATER RESOURCES TECHNICAL PUBLICATION ENGINEERING MONOGRAPH NO. 27

UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU OF RECLAMATION A WATER RESOURCES TECHNICAL PUBLICATION

Engineering Monograph NO. P7

Moments and Reactions for Rectangular Plates

By W. T. MOODY

Division of Design

Denver, Colorado

United States Department of the Interior

BUREAU OF RECLAMATION As the Nation’s principal conservation agency, the Department of the Interior has responsibility for most of our nationally owned public lands and natural resources. This includes fostering the wisest use of our land and water resources, protecting our fish and wildlife, preserv- ing the environmental and cultural values of our national parks and historical places, and providing for the enjoyment of life through outdoor recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Department also has a major responsibility for American Indian reservation communities and for people who live in Island Territories under U.S. Administration.

First Printing: October 1963 Revised: July 1963 Reprinted: April 1966 Reprinted: July 1970 Reprinted: June 1975 Reprinted: December 1976 Reprinted: January 1978 Reprinted: April 1980 Reprinted: March 1983 Reprinted: June 1986 Reprinted: August 1990

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON : 1978 Preface

THIS MONOGRAPH presents a series of tables con- As supplementary guides to the use and devel- taining computed data for use in the design of opment of the data compiled in this monograph, components of structures which can be idealized two appendixes are included. The first appendix as rectangular plates or slabs. Typical examples presents an example of application of the data to a typical structure. The second appendix explains are wall and footing panels of counterfort retaining the basic mathematical considerations and develops walls. The tables provide the designer with a the application of the finite difference method to rapid and economical means of analyzing the the solution of plate problems. A series of structures at representative points. The data drawings in the appendixes presents basic relations presented, as indicated in the accompanying which will aid in application of the method to figure on the frontispiece, were computed for fivl: other problems. Other drawings illustrate appli- sets of boundary conditions, nine ratios of lateral cation of the method to one of the specific cases dimensions, and eleven loadings typical of those and lateral dimension ratios included in the encountered in design. monograph.

Acknowledgments

The writer was assisted in the numerical The figures were prepared by H. E. Willmann. computations by W. S. Young, J. R. Brizzolara, Solutions of the simultaneous equations were and D. Misterek. H. J. Kahm assisted in the performed using an electronic calculator under the computations and in checking the results obtained. direction of F. E. Swain. CASE I CASE 2 CASE 3 CASE 4 CASE 5 PLATE FIXED *Lowe FOVR EOBES

BOUNDARY CONOITIONS 0f------f---- I-L!kl G L-p-A i- id- pd ;pdd LOAD I LOAD n LOAD IU LOAD Ip LOAD Y “NWORY LOAD OVER “NlFORY LOAD OVER “WlFORYLI “ARIINO LOAD “NlFORYLl “ARIINO LOAD e/3 THE “EIBHT 113 THE HEIOHT OVER THE FVLL HEIGHT OVER e/3 TM ns,en* OF THE PLITE OF THE PLATE OF THE PLATE OF THE PLATE

-H p L- LOAD H LOAD PII LOAD Pm LOAD iI LOAD I “NlFORYL” “AWlNO LOAD UNIFORM YOYEW ALOW UNIFORM LINE LOAD “WIFORYL” “m”I*e LOAD OVER l/6 THE “ElB”f IHE soce y - b FOR ILOWO WE FREE EOBE D- o ALOWO y-b/e OF T”E PLATE OASES I, L. AND 5 FOR OASES I AND 3

f- P 7--

P-q k-----a----+

LOAD H “WIFORYLI “ARIIYB LOAD p - 0 ALON0 x - a,*

LOADING CONDITIONS

NOTES The variaus cases are analyzed for the indicated ratios of o/b. Coses I, e, and 3: I/B, 1f4, 3/s, I/Z, 3/a, I, ond 3/z. Cose 4 : l/8, l/4, 3/0, I/2, 314, and I. Case 5 : 310, I/S?, s/8, 3/4, 7/e, ond I. All results are bored on a Poisson’s ratio of 0.2.

INDEX OF BOUNDARY AND LOADING CONDITIONS

-FRONTISPIECE Contents

Page . . . Preface and Acknowledgments -----____-______--__ ill

Frontispiece __------______------iv

Introduction ______.______- ______1

Method of Analysis ______-- ______3

Results ______------______5 Effect of Poisson’s Ratio- ______- ______6

Accuracy of Method of Analysis------43 Appendix I ______- ______45 An Application to a Design Problem-- - - ______- ______45

Appendix I I ______49 The Finite Difference Method- ______- _- 49 Introduction______------49 General Mathematical Relations- _ _ _ __ -______-_ _ 49 Application to Plate Fixed Along Three Edges and Free Along the Fourth______------54

List of ,Re f erences------_____ 89

LIST OF FIGURES Number PW 1. Plate fixed along three edges, moment and reaction coefficients, Load I, uniform load------>------_------7 2. Plate fixed along three edges, moment and reaction coefficients, Load II, 213 uniform load ______8 3. Plate fixed along three edges, moment and reaction coefficients, Load III, l/3 uniform load------______9 4. Plate fixed along three edges, moment and reaction coefficients, Load IV, uniformly varying load ______10

V vi CONTENTS Number me 5. Plate fixed along three edges, moment and reaction coefficients, Load V, 213 uniformly varyingload _---__--______------______11 6. Plate fixed along three edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load ------______12 7. Plate fixed along three edges, moment and reaction coefficients, Load VII, l/6 uniformly varyingload--_-----..------______13 8. Plate fixed along three edges, moment and reaction coefficients, Load VIII, moment at free edge------_____ ------______14 9. Plate fixed along three edges, moment and reaction coefficients, Load IX, lineload at free edge------______15 10. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load I, uniform load------_ - ______16 11. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load II, 213 uniform load_ - ______17 12. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load III, l/3 uniform load- - - - ______18 13. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load IV, uniformly varying load - ______19 14. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load V, 213 uniformly varying load--______20 15. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VI, l/3 uniformly varying load_- _ __ _ _ 21 16. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VII, l/6 uniformly varying load- __ _ _ _ 22 17. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VIII, moment at hinged edge------_ _ 23 18. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load I, uniform load--- ______24 19. Plate fixed along one edge-Hinged along two opposite edges, moment and react,ion coefficients, Load II, 213 uniform load ______25 20. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load III, l/3 uniform load- _- _ _ 26 21. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IV, uniformly varying load. 27 22. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load V, 213 uniformly varying load----______-----______------28 23. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load_-__------______------29 24. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VII, l/6 uniformly varying load- ---__-_-_-----__------~~~~~~~~--~~------30 25. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VIII, moment at free edge- - 31 26. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IX, line load at free edge- 32 27. Plate fixed along two adjacent edges, moment and reaction coefll- cients, Load I, uniform load--- ______------33 CONTENTS vii

Number page 28. Plate fixed along two adjacent edges, moment and reaction coefficients, Load II, 213 uniform load------______34 29. Plate fixed along two adjacent edges, moment and reaction coefficients, Load III, l/3 uniform load- _ ------______35 30. Plate fixed along two adjacent edges, moment and reaction coefficients, Load IV, uniformly varying load- - _- - _ - - - - ______36 31. Plate fixed along two adjacent edges, moment and reaction coefficients, Load V, 2/3 uniformly varying load- _------______37 32. Plate fixed along two adjacent edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load- _- - - - ______38 33. Plate fixed along two adjacent edges, moment and reaction coefficients, Load VII, l/6 uniformly varying load-- ______39 34. Plate fixed along four edges, moment and reaction coefficients, Load I,uniformload____------______------40 35. Plate fixed along four edges, moment and reaction coefficients, Load X, uniformly varying load, p=O along y=b/2------______41 36. Plate fixed along four edges, moment and reaction coefficients, Load XI, uniformly varying load, p=O along x=a/2------.. 42 37. Counterfort wall, design example---~------~-~~~~~-~..~ 46 38. Grid point designation system and notation- _- - - _- _ - - ______50 39. Load-deflection relations, Sheet I ______--__--_------56 40. Load-deflection relations, Sheet II------57 41. Load-deflection relations, Sheet III------58 42. Load-deflection relations, Sheet IV ______-___--_------_---- 59 43. Load-deflection relations, vertical spacing: 3 at h; 1 at h/2, Sheet V- 60 44. Load-deflection relations, vertical spacing: 2 at h; 2 at h/2, Sheet VI- 61 45. Load-deflection relations, vertical spacing: 2 at h; 1 at h/2; 1 at h/4, SheetVII___-_-----______------62 46. Load-deflection relations, vertical spacing: 1 at h; 3 at h/2, Sheet VIII--__-----_-_------63 47. Load-deflection relations, vertical spacing: 1 at h; 1 at h/2; 2 at h/4, SheetIX___-______-_-______------64 48. Load-deflection relations, vertical spacing: 1 each at h, h/2, h/4, and h/8, Sheet X------______6.5 49. Load-deflection relations, vertical spacing: 4 at h/2, Sheet Xl _ _ _ _ _ 66 50. Load-deflection relations, vertical spacing: 1 at h/2; 3 at h/4, Sheet XII------67 51. Load-deflection relations, vertical spacing: 1 at h/2 ; 1 at h/4; 2 at h/8, Sheet XIII------______-______68 52. Load-deflection relations, vertical spacing: 4 at h/4, Sheet XIV--- - 69 53. Load-deflection relations, vertical spacing: 1 at h/4; 3 at h/8, Sheet xv ____ ------70 54. Load-deflection relations, vertical spacing: 4 at h/8, Sheet XVI---- 71 55. Load-deflection relations, horizontal spacing: 4 at rh/2, Sheet XVII- 72 56. Load-deflection relations, horizontal spacing: 3 at rh/2; 1 at rh, SheetXVIII_------_------73 57. Load-deflection relations, horizontal spacing: 2 at rh/2; 2 at rh, SheetXIX ______--- ______-_-_-_- ______74 58. Load-deflection relations, horizontal spacing: 1 at rh/2 ; 3 at rh, Sheet xX-__-_------_--_--______75 Viii CONTENTS

Number PW 59. Load-deflection relations, horizontal spacing: 4 at rh, Sheet XXI- _ 76 60. Moment-deflectionrelations--- ______--_--___-- ______77 61. Moment-deflection relations, various point spacings- ______78 62. Shear-deflection relations, Sheet I--- _ _ - - - - __ - - - ______79 63. Shear-deflection relations, Sheet II------____- ______80 64. Shear-deflection relations, Sheet III- ______81 65. Load-deflection coefficients, r=1/4, p=O.2------____ -- ______82 66. Plate fixed along three edges-30 equations for determining unknown deflections. a/b=114 ______--______--- ____--- _____ 83 67. Plate fixed along three edges, deflection coefficients. a/b=114 Variousloadings______--_---______---___----______84 68. Plate fixed along three edges-20 equations for determining unknown deflections. a/b=114 ______---___---- _____ 85 69. Numerical values of typical moment and reaction arrays, r=1/4, p=o.2 __-____-___------______------86 70. Plate fixed along three edges, deflections-reactions-bending moments,Load I. a/b=1/4, p=O.2------____ ----- ______87

LIST OF TABLES NUmb6T Pap 1. Effect of Poisson’s Ratio (p) on Coefficients of Maximum Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed along Four Edges ______--_-___-___----_------_____--- 6 2. Comparison of Coefficients of Maximum Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed along FourEdges______--______-______-____ 43 3. M, for Heel Slab at Supports------______--- _____-_---- ______47 4. M, for Heel Slab at Supports- ____-- ______--__- ______-- ______47 5. M,forWallSlabatSupports ____ -- _-______- _____ 48 6. M, for Wall Slab at Supports- ______48 Introduction

CERTAIN COMPONENTSof many structures may be II of this monograph, makes possible the analysis logically idealized as laterally loaded, rectangular of rectangular plates for any of the usual types plates or slabs having various conditions of edge of edge conditions, and in addition it can readily support. This monograph presents tables of take into account virtually all types of loading. coefficients which can be used to determine An inherent disadvantage of the method lies in moments and reactions in such structures for the great amount of work required in solution of various loading conditions ,and for several ratios the large number of simultaneous equations to of lateral dimensions. which it gives rise. However, such equations can The finite difference method was used in the be readily systematized and solved by an electronic analysis of the structures and in the development calculator, thus largely offsetting this disadvan- of the tables. This method, described in Appendix tage.

Method of Analysis

THE FINITE difference method is based on t,he In this study, for each load and ratio of lateral usual approximate theory for the bending of thin dimensions, deflections were determined at 30 or plates subjected to lateral loads.‘* The custom- more grid points by solution of an equal number of ary assumptions are made, therefore, with regard simultaneous equations. A relatively closer spac- to homogeneity, isotropy, conformance with ing of points was used in some instances near Hooke’s law, and relative magnitudes of de- fixed boundaries t’o attain the desired accuracy in flections, thickness, and lateral dimensions. (See this region of high curvature. For the a/b ratios Appendix II.) l/4 and l/8, one and two additional sets, respec- Solution by finite differences provides a means tively, of five deflections were determmed in the of determining a set of deflections for discrete vicinity of the x axis. Owing to the limitations on points of a plate subjected to given loading and computer capacity, these deflections were com- edge conditions. The deflections are determined puted by solutions of supplementary sets of 20 in such a manner that the deflection of any point, equations whose right-hand members were func- together with those of certain nearby points, tions of certain of the initially computed deflections satisfy finite difference relations which correspond as well as of the loads. In each case, the solution to the differential expressions of the usual plate of the equations was made through the use of an electronic calculator. theory. These expressions relate coordinates and Computations of moments and reactions were deflections to load and edge conditions. made using desk calculators and the appropriate finite difference relations. The finite difference *Numbers in superscript refer to publications in List of References on page 89. relations used are discussed in Appendix II.

FIGURES 1 through 36 present the results of these Load V: Uniformly varying load over 213 studies as tables of dimensionless coefficients for the height of the plate. the rectangular components of bending moment Load VI: Uniformly varying load over l/3 and for reactions at the supports. The studies the height of the plate. were carried out for the following edge, or boun- Load VII : Uniformly varying load over l/6 dary, conditions : the height of the plate. Case 1: Plate fixed along three edges and Load VIII: Uniform moment along the free along the fourth edge. edge y=b of the plate for Cases 1, 2, and 3. Case 2: Plate fixed along three edges and Load IX: Uniform line load along the free hinged along the fourth edge. edge of the plate for Cases 1 and 3. Case 3: Plate fixed along one edge, free Load X: Uniformly varying load, p=O along the opposite edge, and hinged along along y=b/2. the other two edges. Load XI : Uniformly varying load, p = 0 Case 4: Plate fixed along two adjacent along x=a/2. edges and free along the other two edges. Plates with the following ratios of lateral Case 5: Plate fixed along four edges. dimensions, a, to height b, were studied for The loads, selected because they are represent- the first four cases: l/8, l/4, 318, l/2, 314, 1, 312. ative of conditions frequently’ ‘encountered in The analysis was carried out for these cases structures, are : using Loads I through IX and all dimension Load I: Uniform load over the full height ratios, except that Load IX was omitted from of the plate. Case 2 for obvious reasons, and Loads VIII and Load II: Uniform load over 2/3 the height IX and the ratio a/b=312 were omitted from of the plate. Case 4. It will be noted that for the first three Load III: Uniform load over l/3 the height cases, which have symmetry about a vertical axis, of the plate. the dimension a denotes one-half of the plate Load IV: Uniformly varying load over the width, and for the fourth, unsymmetrical case, a full height of the plate. denotes the full width. For Case 5, lateral 5 6 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

dimension ratios of 318, l/2, 518, 3/4, 718 and 1 can be determined easily, since the deflections were studied, subjected to Loads I, X, and XI. computed from finite difference theory are in- For this case, a and b denote the full lateral dependent of Poisson’s ratio. Futhermore, the dimensions. All numerical results are based on bending moments at, and normal to, the fixed a value of Poisson’s ratio of 0.2. edges are unaffected by this factor. It is reason- The arrangement of the tables is such t,hat able then to conclude that insofar as the moments each coefficient, both for reaction and moment, which are most important in design are concerned, appears in the tables at a point which corresponds the maximum effect for this case will occur at geometrically to its location in the plate as shown the center of the slab. in each accompanying sketch. Table 1 shows a comparison of maximum bending moment coeflicients at the center of a uniformly loaded plate for several values of p and for each Effect of Poisson’s Ratio ratio of a/b for which Case 5 was computed. A question which frequently arises is: What For a change in Poisson’s ratio from 0.2 to 0.3 effect does Poisson’s ratio have on the bending it is noted that the maximum effect on the bending moments in a plate? For the plate fixed along moment coefficient occurs at a/b= 1, where the four sides, a clear understanding of this effect change in the coefficient is less than 8 percent.

TABLE l.-Effect of Poisson’s Ratio (p) on Coeficienk of Maximum Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed Along Four Edges

Values of M./pa* -%“;I 0 0.1 0.2 0.3 0. 375 - 0.0423 -0.0424 -0. 0424 -0.0425 0. 5 - 0.0403 - 0.0407 -0.0411 -0.0415 0.625 -0.0358 -0.0367 -0.0376 -0. 0384 0. 75 -0.0298 -0.0311 -0. 0324 -0.0337 0. 875 -0. 0235 -0.0251 -0.0267 -0. 0283 1. 0 -0.0177 -0. 0195 -0.0213 -0. 0230

__.-- .-...-.-._-_ -.. RESULTS

Moment-: (Coefflctent) (pb’) Reaction : (Coefftclent) (pb) X

POSITIVE SIGN CONVENTION

FIGURE l.-Plate .tixed along three edges, moment and reaction coeficients, Load I, uniform load. 8 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

IO IO lo IO IO I

32 i-.0039 1 0 n I n.

.“a-< 31-.029sl 0 I 0 I 0 I 0 I 0 I 0

Moment = (Goefficient)(pfl) Reaction = (Goefficient)( pb)

POSITIVE SIGN CONVENTION

FIGURE 2.-P&e $xed along three edges, moment and reaction coeflcients, Load ZZ, 913 uniform load. RESULTS

P MI -- v

Moment = (Coefficieni) (pb’) Rx Reaction = (Coefficient) (pb) RV tX ’ 0 My @ W I

POSITIVE SIQN CONVENTION

FIGURE 3.-Plate $xed along three edges, moment and reaction coeflcients, Load III, l/S uniform load.

I “._.-..-.- ~.. 10 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I \I , ..“T”..

t I I 0 I+ IlId7 I+ on99

I I 1.0 I+ 3177 lt.oe57r;

MI -+-I/’ Moment = (Goefficient)(pb*) -.I 4 I?eOCflOn= (Coefficient)(pb) RV 0 , +X %iiJ ’ Mv WV I

POSITIVE SIGN CONVENTION

FIGURE k--Plate fixed along three edges, moment and reaction coeficients, Load IV, uniformly varying loud.

_.-.-.--.--.- .______- RESULTS

I I I 0 I- 0155 I+ 0025 I

. . t m\bI+, 001, I+ 1712 I+ 2595

t .3224 +. 3329 t.3356 I # ---- I --- I I Y ”

1 MI -+- IJP Moment : (Coefficlent)( pb2.) ’ Rx Reaction = (Coefflcient)( pb) b X 0 ” M, w W

POSITIVE SIGN CONVENTION

FIGURE B.-Plate Jixed along three edges, moment and reaction coeficients, Load V, .9/S uniformly varying load. Moment = (Coefficient)ipb’) Reaction = (Goefficient)( pb )

I W.

POSITIVE SIGN CONVENTION

FIQURE B.-Plate fixed along three edges, moment and reaction coeficients, Load VI, l/S uniformly varying load.

-.-..--.__-. .._ RESULTS

*--0_-_ ---O---~ iGee _,, ---t Moment f (Coefficient)(pb*) Aeoction = (Coefficient)( pb) +X

I -i-, W 0 ill.ii POSITIVE SISN CONVENTION FIGURE 7.-Plate jixed along three edges, moment and reaction coejkients, Load VII, l/6 uniformly varying load. f-P M. --J I p Moment : (Coefficient)( M) Reaction : (Coefficient)($) Al--l. I . X

1 FOSITIVE SIQN CCNVENTION

FIGURE 8.-Plate jked along three edges, moment and reaction coejkients, Load VIII, moment at free edge. RESULTS 15

+ 0185 t 0226 i.0261 +.0264 +.wg, ~,-.012’,+~~O036~~1-- +.0064 l-’ .0107 t.0147 +01j4 + 0164 -~~~ $0036~ +:0056/+.007g/ 0.2 I-.OiSO I-.0004~*.0004~+.0006~+.0004[+.0001 I-.OOOOI-.o001 1+.0003 +:0010 +.001, +.0022 t.0024

.“..,T -39111 n I n I n I n I n I n

;I+.0746 +.0666I+.O655 +.0644]

Moment = (GoeffIcient) Reaction = (Goefflcient)(F)

I W POSITIVE SIGN CONVENTION

moment and reaction coeflcients, Load IX, line load at free edge. 16 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

.__-F.,~-++I ---+I _--- hinged T Moment = (Coefficient)(pb*) a Reoctiin = (Coefficicnt)( pb)

---- i - -X a IIriIll POSITIVE SION CONVENTION

Fxa UaE lO.-Plate jixed along three edges-Hinged along one edge, moment and reaction coeficiente, Load I, uniform load. RESULTS 17

, 4 0.8 I .o 0 1 0.2 [ 0.4 1 0.6 [ 0.6 [ 1.0 +.0017 +.0020 0 0 0 IO lo IO I” In

Moment = (Coefficient)(pb’) Reaction = (Coefficient)( pb )

LE Il.-Plate jized along three edges-Hinged along one edge, moment and reaction coejicients, Load ZZ, d/S unifol vn load. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Coefficient)(pb’)

Aeoction = (Coefficient)( pb)

X POSITIVE SIQN CONVENTION

FIQURE l2.-Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load III, l/S unifor .rn load. RESULTS

Moment = (Coefficirnt)(pb*) Reaction = (Coefficicnt)( pb ) X

0 POSITIVE SIGN CONVENTION

FIGURE 13.-Plate fixed along three edges-Hinged along one edge, moment and reaction coeficients, Load IV, uniformly varying load.

-.--_._-_-.._-.-- --..--. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Coefficient)(pff)

Reaction = (Coefficient)( pb ) +-X

POSITIVE SIQN CONVENTION

FIGURE 14.-Plate fxed along three edges-Hinged along one edge, moment and reaction coefkients, Load V, d/3 uniformly varying load.

.._...- ..-.---..--.-- Moment = (Coefficient)(pb*) Reaction = (Coefficient)( pb)

0 POSITIVE SIGN CONVENTION

FIGURE 15.-Plate fixed along three edges-Hinged along one edge, moment and reaction coejkients, Load VI, l/3 uniformly varying load. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Goefficient)(pb? Reaction = (Coefficient)( pb)

POSITIVE SIGN CONVENTION

FIGURE 16.-Plate jixed along three edges--Hinged along one edge, moment and reaction coejicients, Load VII, l/6 uniformly varying load. RESULTS 23

Moment = (Coefficient)( M ) Reaction = (Coefficient)($),

0 POSITIVE SIGN CONVENTION

FIGURE 17.-Plate $xed along three edges-Hinged along one edge, moment and reaction coeficiente, Load VIII, moment at hinged edge. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Y ~+--*--+* ._--f I Moment = (Coefficlent) (pb’) I 0 Reaction = (Coefficient) (pb )

--_ L-x 0 POSITIVE SIGN CONVENTION

FIQURE 18.-Plate fixed along one edge-Hinged along two opposite edges, and reaction coefkients, Load I, uniform load. 25

gt-.00371-.00531-.00631-.0067i

r1-.02’351 0 ~-.0140~-.0220~-.026lt-.0277kO262i

M. -+-I/ Moment = (Coefficient) (pb’) ’ R. Reoctlon = (GoeffIcIent) (pb) f&l ^ .

POSITIVE SIGN CONVENTION

FIG E lg.-Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coeficients, Load II, uniform load. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

P MI -- v

Moment : (Coefficient) (pb*) ’ h Reaction = (Coefficient) (pb ) fb tX ’ I My W@

POSITIVE SION CONVENTION

FIQURE 20.-Plate jkced along one edge-Hinged along two opposite edges, moment and reaction coejkients, Load III, l/S uniform load. RESULTS

Ma -+-IA Moment = (Coefflclent) (pb’) ’ R. Reoctaon = (Coeffxlent) (pb) Rv X I d&iJ- MY W. 0 POSITIVE SIGN CONVENTION

FIGURE 21.-Plate jixed along one edge-Hinged along two opposite edges, moment and reaction coeflcients, Load IV, unifol varying load.

- .-._- __-- 28 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

--_ f

Moment = (caefflclent)(pb’)

f?eactlon = (Coefflcient)( pb) +X

I W

POSITIVE SIGN CONVENTION

FIGURE 22.-Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coeficients, Load V, uniformly varying load. 29

Moment = (Coefficlent)(pb*) Reaction = (Coefficient)( pb)

POSITIVE SIQN CONVENTION

FI :GuRE 23.-Plate fixed along along two opposite edges, moment and reaction coejicients, Load VI, 11s uniformly varying load. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Coofficiant)( pb’)

heOCtiOn = (Coefflcient)( pb)

W POSITIVE SION CONVENTION

FIGURE 24.-Plate fized along one edge-Hinged along two opposite edges, moment and reaction coeficients, Load VII, l/6 uniformly varying load. Moment - (Coefflcient)( M) Reaction = (Coeffxwnt)(j-1

0 POSITIVE SIGN CONVENTION

FIG URE 25.-Plate jixeu along one edge-Hinged along two opposite edges, mom&t and reaction coeficients, Load VIII, momen tt at free edge.

_.-.-.-__.-.- ._-__.. 32

Moment = (Coefftcient)( Fb) Reoctlon : (Coefficient)( F )

POSITIVE SIQN CONVENTION

FIGURE 26.-Plate fixed along one along two opposite edges, and reaction coqjicients, Load IX, 1ine load at free edge. RESULTS

~.0160~+.0061~+.0029~+.0002) 0

““151-.00241-.00301 ~00061+~00l71+ 00321+.00471 0 0 1+.00291+ 0066~+.0159~+ 02361+,0304

.ol76 I+.0060 1t.0002 0 1 +.OlZl I+.0067 I+.0020 l-.0018

0 - \-.0401Rx t.0011 +.I576 t.3024 +.5696 l .9739 I.0 +.6290+. 1977 +.0952 0 0 0 t.0296 -.0059 ~0162 __.. 0 0 0 0 ^ ^.__ _--- ~_ 0 6. +.7827 ‘------I- .003Zl- .OllO 1 0.6 1 - .0023 - .0079 ~+.0377~+.OlZOl+ OOOII- .0026 1 -0 ~~+.0165~+.006’~-.00051-.004z~-.UUb~(-.~0

1 -~3 LO.2 I+.0741 ~.0282~+.0144(+.0077~+ .00721+ .0061 1 0 1 \ 0 d -.0696 0 +.0041 +.0125 + .0221 + .0326 -.0696 +.0333 +.2595 +.4574 +.7920 +I.1266 ,

LI Y 1 0 I- .oesrl 0 I+ .00721+.0207)+ .03451+.04941 0 0 ~1+.03621 ;---.- 0 .---.+. I

Moment = (GoefficIent) (pb*) Reaction : (Coefficient) (pb)

I W --Y- x p4 04~~~/~~~/~~~~J ;p& POSITIVE SIGN CONVENTION

FIGURE 27.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load I, uniform load. 34 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

0.2 0.4 0.6 0.6 1.0 0 0.2 0.d 1 ! 0.6 j 0.6 1 I 1 I. 0 I.0 l-.0014 +.00021*.00021+.0002l+ 0001 1+.0001 I 0 0 I 0 I 0 I 0 I 0 I 0 1 -co 1 0.8 ~+.00751+.OOll ~+.0006~+.0006~+.0004~+ 00021 0 I+ 0002/+ 00021+.00021+ 0002/+.00031+.00031 ~6~+.0035~+.0019~+.0008)+.0001 ( 0 ~+.00~~j+.00071+.00031+.00001- OOOII-.00021

+.I209 +.0067 +.0040 + 0021 +.0008 +.0001 0 l .0046 0 +.0001 +.ooos t.0009 +.0013 0 +.0046j-.Ob61 +.0505 +. 1061 l 2023 + 3114 -.0167 +.0028: +.0031 t.0026 + 0020 + 0011

y-.0822Ra - OOl2(+ 1050 +.2030 l .3661 + 5432

I.0 -.0462 +.0102 +.0106 +.0065 +.0055 +.0026 0 0 0 I. 0. ‘1 0 1 0 I 0 0. e t.0819 +.0213 +.0149 +.009 -0.6. t.2733 + ~-~ -r -- - r --+-- -,----A - ( - c , I I 4 0.4 +.3352 t.0384 +.0165 t.0063 L - .-0003 1 -.0022 ! 0 !+.0077~~.00241- 0~~-.00~6~~.00~6~/~.0ll6 1 ,Ib [~~+:I&]+ .%$,@7 -.0003 - 0010 1 0 0 1 +.0011 t.0031 +.0055 t.0079 0 -.0069~+.0125~+.1333 +.22851+.39631+.5629

+.0019 + 0053 +.00691+.0125 0 -.0250~+.0436[+.l939~+.3C7l 1+.46931+.6544 -I!+.0524 +.034l +.0153 +.0026 -.0022 0

I I +.0542 +.0113 -.0042 -.00741-.0051 t-7

Moment : (Coefficient) (pb’) Reoctlon = (Coefficient) (pb) 3-X

W I

POSITIVE Sl(iN CONVENTION

FIQURE 28.-Plate jbed along two adjacent edges, moment and reaction coeficients, Load II, $713 uniform load. RESULTS

I - I 0.2 ~+.1774~+.0131~+.0020~-.00171-.00241-.00171 0 I+.Ofi261-.OOl71-.00471-.00641

f ~Y~+.00481+.16151+.2512~+.2874~t.3312~+.3489~R. I I.0 t .0052 t.0104 t .0059 +.0016 -.0006 -.OOl I 0 0 0 0 0 0 0 0. a t.0356 t.0116 l .0053 t.0011 -.0008 -.OOlO 0 +.0023 t .OOl2 + .0002 - .0007 - .0014 - .0020 - 0. 6 + .0430 t .Ol26 t.0041 - .0002 - .OOl6 - .0013 0 f.0025 t.0008 - .0012 -.0026 - .OOM - .0046 II - 0.4 +.I052 +.0135 +.OOl3 -.0022 -.0025 -.0017 0 t .0027 - .002 I - .0053 - .0069 - .0075 - .0060 0. 2 + .I682 f.0094 -.0015 -.0023 -.0015 -.0005 0 t .0019 - .0045 - .0055 - .0043 - .0029 - .0020 0 It.oos + .0033 + .0057 +.0072 t .I3383 0 0 + .2052 + .2808 t .3064 t .3372 t .3473 Y

Moment = (Coefficient) (pb*)

Reaction = (Coefficient) (pb)

POSITIVE SIQN CONVENTION

FIGURE 29.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load III, l/S uniform load. 36 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+.0075 +.0007 +.0006 +.0004 +.oooz +.0001 0 0

;.0252 +.0017 t.0011 +.0006 +.0003 +.OOOl 0 + .0003

l .0056 -.0008 .o I I

0.4 ~+.2050j+.~246~+.0122 ~+.0045~+.0002 l-.0012 1 0 (+.0049~+.0017~.

m 0.2 [+.I413 ~+.0162~+.0056~+.0010 )-.0005 I-.0004) 0 I+.oo52/+

i? 0.60.8 +.2067+.2303 +.0467+.0459 +.0242+.0195 +.0046+,0064 -.0002-.0023 -.0027-.0036 0 +.0097+.0092 +.0040+.oozo -.00440004 -.0093 - 0129 I, 0.4 +.2363 t.0360 +.Ol2l +.OOlO -.0030 -.0029 0 +.0072 -.0003 -.0060 -.0096 -.Oll7 I I ~~ I I 1 p 1 0.2 ~*.1193~+.01~0~+.0042~+.0005~+.0006(+.00l6 I 0 I,+,0032 +.oooa +.0019 +.0050 +.ooe3 +.01oLl -.0194 0 +.0029 +.0070 +.0110 +.014a 0 0 +.0147 +.0352 l .0546 + 0740 t.0696 n I R”v-.0194~+.1105~+.2399(+.32: 561+.44891+.5505iI ----1 I.0 +.I917 +.0662 +.0291 +.0056 -.0059 -.0077 0 I 0 I ’ 0

- 0.6 )+.23641+.0518 I+.0173 l+.OOO4!-.00591-.0054 I

+.0046 t.0103 t.0152 l .0197 0 +.0232 +.0515 +.0759 +.0987 + 1157

Moment = (Coefficient) (pb*)

Reoctton = (Coefftcient) ( pb)

c 0-A POSITIVE SIGN CONVENTION

FIG IURE 30.-Plate $xed along two adjacent edges, moment and reaction coeficients, Load IV, uniformly varying load. RESULTS 37

I Mx MY 0.2 0.4 0.6 O.SlI.0 0 lo2looln~lnnlln-.- -. . -.v .,.- ..”

1.0 -.OOOJ +.oooo +.oooo +.oooo +.oooo +.oooo 0 0 0 0 0 0 0

0. e +.0001 +.0001 +.0001 +.0001 +.0001 +.oooo o +.oooo- +.oooo +.oooo +.0001 +.0001 +.0001 2 0.6 + 0144 +.OOi I +.0007 t.0004 +.OOOZ +.OOOI o +.oooe +.0002 +.0001 +.0001 +.0001 +.0001 -. ..- II 0.4 +.os& +.0030 +.0019 ;.&so +.0004 +.0001 0 + 0006 +.0004 +.OOOi +.0001 - boo0 -.OOOI .- c, 0.2 +.0843 +.0044 +.0026 COO13 COO04 + 0000 0 + ones -7Mms -

I - I 0.6 I+ 00311+.00171+.00141+ 00111+.0007(+.0003) o 1+.ooo3~+.ooo3~+.ooo4~+.ooo57+.ooo6~

I I , 0 +.0022 + 0061 t.0103 t.0146 t.0181 I+.1661 1+.2646/+.35441

I.0 -.0169 +.0022 +.0027 +.0023 +.0016 +.0006 0 0 0 0 0 0 0 0.0 +.0147 l .0053 t.0040 +.0027 t.0015 +.0006 0 +.ooll +.0009 +.0009 +.oooe +.0009 +.0010 -- -~-.__-..- _.-.. s 0.6 . +.0565 l .0099 + 0059 +.0029 t 0010 -.OOOO 0 t.0020 +.0012 +.0005 -.OOOl -.0006 -.0009 II 0.4 +.1351 t.0148 +.0069 +.0020 -.0004 - 0011 0 +.0030 +.0008 -.OOlZ -.0029 -.0042 -iOil 0.2 r.1353 T.0117 t.0040 +.0003 -.OOll -.OOll 0 +.0023 +.OOOO -.0019 -.0033 -.0042 -.0049 s 0 +.Ol25 0 +.oooe +.0019 +.0031 +.0043 0 0 +.0038 +.0097 t.0156 +.0213 + 0257 WYI?. +.0125 +.0460 +.I236 +.I740 +.25381+.3159 I.0 -.0220 +.0052 +.0053 t.0039 +.0023 t.0010 0 0 0 0 0 0 0 0.8 +.0298 +.0095 + 0065 +.0036 +.0016 t.0005 0 COO19 t.0014 +.OOlO +.0006 +.0004 +.0003 ~~- ~~~~~~__ ~~~ -~ -.~ 0. 6 +.0753 +.0143 +.00!6 t.0030 t.0004 -.0005 0 +.0029 +.OOl4 -.OOOl -.0014 -.0025 -.0031 0.4 +.I506 +.0176 +.0068 +.OOlO -.0015 -.0017 0 +.0036 +.0002 -.0026 -.0052 -.0070 -.0062 0.2 +.1313 +.0119 +.0030 -.0005 -.0014 -.OOlO 0 +.0024 -.0005 -.0025 -.0036 -.0041 -.0044 0 t.0050 0 +.0013 +.060 +.0046 +.0060 0 0 +.0063 +.0146 t.0226 +.0301 +.0358 WYR. +.0050 +.0755 +.I616 +.2132 +.2673 +.3382 I.0 -.0036 +.0130 +.0092 +.0045 +.OOlZ -.0003 0 0 0 0 0 0 0 0.8 +.0540 +.0163 +.0090 +.0036 +.0005 -.0006 0 +.0033 +.0016 +.oobs -.0007 -.0016 -.0023 0.6 +.093 t.0191 + 0079 +.0016 -.OOlZ -.0016 0 +.0036 +.0009 -.0020 -.0043 -.0060 --.0072 0.4 +.I561 CO193 t.0048 -.OOll -.0027 -.0022 0 +.0039 -.0014 -.0055 -.OOSl -.0097 -.01oa

0.2 +.I218 r~.ollo l .ooll -.OOlZ -.OOlb -.oooz -. b t.0022 -.0013 -.G23 -.OO:; -.0006 +.0002 0 -.oooo 0 +.0024 +.0649 +‘.0070 +:OOEB 0 0 +.01&i +.0246 +.0348 +.0438 +.0507

-.OOOO +.I274 +.2163 +.2612 t.3210 r.3569

1.0 +.0261 +.Ol96 +.OlO3 +.0027 +.0667 t.0207 +.0069 +0017 - 0.8 0.6 +.0943 +.0209 +.0064 1:0004 II 0.4 +.15;2 +.OI88 +:0026 -.0024 \” 0.2 +.I166 +.0096 +.OOOO -.OOlZ -.0003 +.0008 0 +.ool9 -.0016 -.0008 +.0016 l .004l +.0059 0 - 0 t.0016 0 +.0035 +.0066 +.ooee +.0107 0 0 +.o174 To329 +.0440 +.0534 l .0603

VY +.OOlS t.1676 r.2512 +.2879 +.3346 +.3567

Y b .__._ 0 _____ y

Moment = (Goefficient)(pb*) Reaction = (Coefficient) (pb)

W. I

POSITIVE S ION CONVENTION

FIQ URE 31.-Plate fixed along two adjacent edges, moment and reaction coefkients, Load V, S/S unijormly varying load. 38 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

0.6 l .oolo +.0007 *.0006 +.0005 +.0003 +.0001 0 +.0001 +.00011+.00021+.00021+.0002lt,0003

0.6 +.0039 +.0015 +.OOll +.0006 t.0003 t.0001 0 l ,000 I ,, 1 0.4 ~+.0207(+.0032~+.0016~+.0006~-.00001-.0002[ 0 ia 0.2 +.0756 +.0051 +.0013 -.0004 -.0006 -.0007 o +.OOlO -.0002 -.0013 -.0021 -.0026 -.0030 +.0221 0

0.6 0.4 +.0244 +.0036 +.0016 t.0003 -.0003 -.0004 0

0.2 +.0747 +.0049 +.0006 -.0006 -.OOlO -.0007 0

0 I+.0201 1 0 ~+.0007~+.0015)*.0020~+.0025~ 0 I 0 ~+.0037~+.0074~+.01021*.

0.6 ~+.0066~+.0024~+.0014(+.0006~+.0001 l-.0001 1 0 ~+.0005~*.0003~+.00021+:00001-.00011-

0.4 .~+.0256~+.0041~*.0010(-.0003~‘.0006/-.0005[ 0 l+.OOOfi

0 1t.02151 0 )+.0012(+.0021~+.0027~*.0030~ fl 0 l+.O(

0.6 I+.0091 I+.0031 ~+.0014~+.0003~-.00021-.00031 d ~+.0006~+.0003~+.0001I I 1 II-.( I I 1 +.0111 +.0034~+.0011~-.00001-.0004)-.00031 0 1+.00071+.00031-.00031-.00071-.OOlOi-.OOl2 I J +.0246 +.0039 +.0004 -.0006 -.0007 -.0004 0 *.0006 -.0004 - 0013 -.0017 -.0019 -.002l 0.2 1 + 0723 +.0034 -.0006 -.OOlO -.0006 -.0003 0 +.0007 -.0019 -.0025 -.0023 -.oOl9 -.0017

Y A

M. -+- IA Moment : (Coefficient) (pd) ’ R, Reaction = (Coefficient) (pb) 5 0 -3-X ’ MY W.@

POSITIVE SIGN CONVENTION

ho ,URE 32.-Pkztejixed along edges, moment and reaction coeficients, Load VI, i/3 uniformly varying locrrd. I 1 y/b h%‘I . 0 IO.2 IO.4 IO.6 IO.8 1 1.0 1 0 IO.2 IO.4 I 0.6 IO.8 1 I.0 I 1.0 -.oooo +.oooo +.oooo t.oooo + 0000 t.oooo 0 0 0 0 0 0 0 0. 8 -.oooo +.oooo t.OOOO +.oooo +.oooo t.oooo 0 +.oooo +..oooo +.oooo +.oooo +.oooo +.oooo 9 0. 6 -.OOOl +.oooo +.oooo +.oooo +.oooo +.oooo 0 +.oooo +.oooo +.oooo +.oooo +.oooo t.0000 0. 4 -.OOOI +.oooi +.OOOI +.oooo +.oooo +.oooa 0 +.oooo +.oooo +.oooo +.oooo +.oooo +,oooo 0. 2 t.0137 l .0006 +.0004 +.oooz +.0001 t.oooo 0 +.ooot +.0001 +.0001 +.oooo +.oooo +.oooo 0 t.0050 0 +.0001 +.oooz +.0003 t.0004 0 0 t.0004 t.0010 +.0016 +.ooee +.0027 I ~Y~+,0050~t.0I69~+.0463~t.0669~+.09611(c.11651

I. 0 -.0007 +.oooo +.ooot +.ooot +.0001 +.oooo 0 0 0 0 0 0 0 0.6 +.0001 +.0001 +.0001 +.0001 +.0001 +.oooo 0 +.oooo +.oooo +.oooo +,oooo +.oooo +.0001 0.6 t.0004 '.0003 +.0002 +.0001 +.0001 t.oooo 0 +.0001 +.oooi t.0001 +.a001 +.OOOI t.0001

1+.0212l+.00~i~+.00031-.0001 I- 00021-.00011 0 1+.0003)+.00001-.OOOZ/-.00041-.00051-.00061

Ill-.00021-.0002l-.000I1 0 1+.00021-.00011-.00041-.00051-.0006l-.0007l

nl- 0001~-.ooo3~-.0002~-.oaa1 I 0 )+.00021-.00031-.00051-,00061-.0006(-.0006l

I I+ nnnsl+ m-ml+ nnnll- ooool-.ooool 0 It.ooOl lt.OOOll+.OO~OOl-.OOOOl-.OOaol-.oooll

017 1+.ooo21-.00041-.00061-.00051-.00051-.0005l

Y M, -+-IA Moment q (Goefficlenf)(pb? ’ R” Jeoction : (Coefftclent) (pb) 0 RYI X 45J- MY W

POSITIVE SIGN CONVENTION

FIGURE 33.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load VII, l/6 unijownly varying load. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Coefficient)(po*) Reaction = (Coefficient)(po)

ieo-rl POSITIVE SIQN CONVENTION

FIGURE 34.-Plate $zed along four edgeqmcnnent and reaction coeflcients, Load I, uniform load. Moment = (Coefficient)(po*) Reoctmn = (Coefficaent)( po) X

POSITIVE SISN CONVENTION

uniformly varying load, p=O al0 FIG m 35.-Plate fixed along four edges, moment and reaction coeficients, Load lng y= b/2.

_--_---._-_- 42 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Coefflcient)( pa*) Reaction = (Coefficient)(po)

POSITIVE SIGN CONVENTION

FIGURE 36.-Plate fixed along four edges, moment and reaction coeficients, Load XI, uniformly varying load, p=O alox x= al.% Accuracy of Method of Analysis

THE FINITE difference method is inherently considered to be satisfactory for design purposes. approximate. A factor directly affecting its Percentage errors for small numerical values of accuracy is the closeness of spacing, hence the the coefficients may, of course, be somewhat number, of grid points. In obtaining the solutions higher. presented in this monograph, a maximum number For Case 5 a comparison is given in Table 2 of points was used, consistent with the objectives

of the study and the capacity of the available TABLE 2.-Comparison of Coeficients of Maximzcm Bending electronic calculator. Moment at the Center of a Uniformly Loaded Rectangular A few instances may be found where there Plate Fixed Along Four Edges appear to be irregularities in the orderly progres- sion of the coefficients as the ratio a/b changes. Valuar of M./pa* from b/a Such instances are most likely to occur in the Timoshenko 1 Method of this Monograph 2 low values of the ratio where, to gain accuracy, - the number of points used in the analysis was increased as a/b decreased. Although these incon- 1. 1 - 0.0264 - 0.0269 sistencies are undesirable from an academic 1.2 - 0.0299 - 0.0301 1.3 - 0.0327 - 0.0329 standpoint, they are not of sufficient magnitude 1.4 - 0.0349 - 0.0352 to affect materially the usefulness of the results. 1.6 - 0.0381 - 0.0384 As a general check on the finite difference 1.7 - 0.0392 - 0.0395 method, problems for which “exact” solutions are 1. 8 - 0.0401 - 0.0404 - 0.0407 -0.0410 known have been computed. The results indicate 1. 9 that for spacings comparable to those used in this 1 These values taken directly from page 223, Reference 1, with due regard study, errors in the maximum moments may be for difference in sign conventions. 2 These values interpolated from the column for p=O.3 of the preceding of the order of five percent. Such accuracy is table. 43 44 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES between values found on page 228 of Reference 1 good agreement. and directly equivalent values obtained by the All coefficients have been computed to four method of this monograph. In this particular decimal places for consistency and to indicate case, the relative differences are, for the most part, significant figures for many conditions which less than one percent. would have no significance to three decimal Comparisons have also been made with other places. This should not be taken as an indication existing results 2 for full uniformly varying load that the percentage accuracy is greater than and certain ratios of a/b. These indicated very no ted above. Appendix I

An Application to a Design Problem

THIS appendix illustrates use of the tabulated librium conditions, have been broken into com- coefficients by an application to a typical design ponents similar to certain of the typical Loads I problem. Figure 37 shows essential dimensions through XI. These are illustrated together with and typical loads acting on an interior panel of a a table of their numerical values in Figure 37. counterfort retaining wall. Both wall and heel It will be noted that for the wall slab, r=a/b= slabs approximate the condition of a plate fixed 0.2. This requires interpolation on r for the along three edges and free along the fourth. The various loads and in the case of pB, interpolation variations in thickness of the wall slab and the both on r and the load. For the heel slab, relatively great thickness of the heel slab com- r=a/b=1/2, and since both component loads act pared with its lesser lateral dimension are both, over the full area, no interpolation is required. perhaps to some degree, in violation of basic For illustrative purposes, moments have been assumptions. Ignoring these, however, is done computed along the assumed lines of support for with the conviction that results obtained in this both the wall and heel slabs. Where interpola- manner are more nearly correct than what might tion was required to obtain the moment coeffi- be determined by other available methods. cients, second degree interpolation was used. The Center line dimensions have been used for both moment coe5cients and actual computed moments slabs. The net loads, as determined from equi- are given in Tables 3 through 6.

45 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

DESIGN DATA Unit Weights Concrete 150 Ib/ft3 Moist earth 120 Ib/ft3 Saturated earth 135 lb/+ Water 62.4 Ib/ft3 Surcharge Pressures Vertical 360 Ib/fte Horizontal 120 Ib/ft’ Equivalent Fluid Weights Moist earth 40 Ib/ft3 Saturated earth 75 Ib/ft’

Water surface elevatiorv’t --- -- ! -

, FRONT ELEVATION END ELEVATION

COUNTERFORT RETAINING WALL DIMENSIONS AND TYPICAL DESIGN LOADS

1L L pw -H pq-A i.6 i-- ps-4

WATER LOAD SURCHARGE LOAD EARTH LOAD PORE PRESSURE LOAD

COUNTERFORT WALL SLAB - INTERIOR PANEL IDEALIZED DIMENSIONS AND COMPONENT LOADS

Ffq+/iii??j

~---p”----~ NET LOAD ON w----pu----H HEEL SLAG COMPONENT LOADS

COUNTERFORT HEEL SLAB - INTERIOR PANEL IDEALIZE0 DIMENSIONS AND COMPONENT LOADS

FIGURE 37.-Counterfort wall, design example.

.- .------_-.. APPENDIX I 47

TABLE 3.-M. for Heel Slab at Supports

T - Moment coefficients Values of pb2+ - Moments (foot-kips) I- 1118.5 -1032.3 Total moment -- -_ (foot-kips) x -Y a b PU PP M" M” ------

0 1.0 + 0.0852 q-o.0151 + 95.30 - 15.59 +79.7 0 0. 8 + 0.0807 $0.0216 $90.26 -22.30 +68.0 0 0.6 $0.0712 + 0.0273 +79.64 -28.18 +51.5 0 0.4 + 0.0545 + 0.0277 + 60.96 - 28.59 $32.4 0 0. 2 + 0.0250 $0.0160 $27.96 - 16.52 $11.4 0 0 0 0 0 0 0 0. 2 0 $0.0019 +o. 0014 $2.13 -1.45 $0.7 0.4 0 + 0.0050 + 0.0033 +5.59 -3.41 +2. 2 0.6 0 + 0.0080 + 0.0050 +8.95 -5.16 +3.8 0.8 0 +o. 0100 $0.0061 $11.18 -6.30 $4.9 1.0 0 $0.0107 + 0.0065 $11.97 -6.71 f5.3 - - - -

TABLE 4.-M, for Heel Slab at Supports - - Moment coefficients Velues of pbb I- - Moments (foot-kips) 1118.5 -1032.3 Total moment -- -- (foot-kips) x s Pu PV M. M, -ii b -- --

0 1. 0 0 0 0 0 0 0 0. 8 + 0.0161 $0. 0043 + 18. 01 -4.44 $13. 6 0 0. 6 +O. 0142 $0. 0055 + 15. 88 -5.68 +10.2 0 0. 4 +o. 0109 + 0.0055 $12. 19 -5.68 +6. 5 0 0. 2 +o. 0050 +O. 0032 f5.59 -3.30 +a. 3 0 0 0 0 0 0 0 0. 2 0 + 0.0094 +o. 0068 + 10. 51 -7.02 $3. 5 0. 4 0 +O. 0252 $0. 0167 f28. 19 -17. 24 +11.0 0, 6 0 + 0.0399 + 0.0252 + 44.63 -26. 01 $18. 6 0. 8 0 $0. 0499 $0. 0307 t-55. 81 -31. 69 +24. 1 1. 0 0 + 0.0534 + 0.0325 f59. 73 -33. 55 +26.2 - 48 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

TABLE 5.-M. for Wall Slab at Supports T - Moment we&icients - T M0ment.Y -i- -7 (foot-kips) Total -985.5 157.7 1905.4 1399.9 moment -- - -_ -- -. _- -_ - - -- - (foot-kips) * Y PW Pa PO P8 M” M. M. M. s b -- _- -- -- _- -- _- -- -_ -- --

0 1. 0 - 0.0000 +o. 0133 +0.0012 + 0.0004 +o. 00 $2. 10 +2.29 $0.56 f5. 0 0 0. 8 +o. 0000 +o. 0131 +o. 0028 +o. 0012 -0.00 s2.07 +5.34 +l. 67 +9. 1 0 0. 6 + 0.0000 $0.0134 + 0.0054 +o. 0034 -0.00 f2.11 + 10. 29 $4.74 +17.1 0 0. 4 +o. 0009 +o. 0133 +o. 0079 + 0.0068 -0. 89 +2. 10 + 15.05 f9.47 +25. 7 0 0. 2 +O. 0032 +o. 0103 + 0.0079 + 0.0075 -3. 15 $1. 62 + 15.05 + 10. 45 +24.0 0 0 0 0 0 0 0 0 0 0 0. 2 0 +o. 0002 +o. 0003 +o. 0003 +o. 0003 -0. 20 +o. 05 +“o. 57 +O. 42 +O. 8 0. 4 0 +o. 0005 +o. 0009 + 0.0007 +O. 0006 -0.49 i-0. 14 +1.33 +O. 84 +l. 8 0. 6 0 +o. 0007 +o. 0013 +o. 0011 +o. 0011 -0.69 4-o. 21 +2. 10 +1.53 +3. 2 0. 8 0 +o. 0009 +O. 0016 +o. 0014 +o. 0014 -0. 89 -l-O. 25 +2. 67 -I- 1. 95 +4. 0 1. 0 0 +o. 0010 +o 0018 +o. 0015 +o 0015 -0.99 +O. 28 +2. 86 +2.09 -k4. 2 ------

TABLE 6.-M, for Wall Slab at Supports

Moment coefficients T Values of - - Moments pbz-1 (foot-kips) Total -985.5 157.7 1905.4 1392.9 moment ------(foot-kips) x ?! PW PS PW PO M” MQ M. M, a b _- _- -- _------_- _- _- --

0 1. 0 0 0 0 0 0 0 0 0 0 0. 8 -0.0000 +O. 0026 +o. 0005 +o. 0002 +:. 00 +o. 41 +o. 95 +O. 28 +l. 6 0 0. 6 $0.0000 +O. 0027 +o. 0011 +o. 0007 -0.00 +o. 43 +2.10 +O. 98 +3. 5 0 0. 4 +o. 0002 $0.0026 +O. 0016 +o. 0014 -0. 20 +o. 41 +3.05 +1.95 +5. 2 0 0. 2 +O. 0006 +o. 0020 +O. 0016 +o. 0015 -0.59 +O. 32 +3.05 +2.09 +4. 9 0 0 0 0 0 0 0 0 0 0 0 0. 2 0 +o. 0011 4-O. 0015 + 0.0014 + 0.0014 -1.08 +O. 24 +2.67 +1.95 +3. 8 0. 4 0 +O. 0025 +o. 0041 +O. 0036 +O. 0036 -2. 46 +O. 65 +6. 86 +5.01 +10.1 0. 6 0 + 0.0036 -i-o. 0066 +O. 0056 +o. 0055 -3.55 fl. 04 + 10. 67 f7.66 +15. 8 0. 8 0 + 0.0043 +o. 0082 + 0.0069 + 0.0068 -4.24 +1.29 +13.15 +9.47 +19.7 1. 0 0 +O. 0046 $0.0088 +o. 0074 +O. 0072 -4.53 +1.39 + 14. 10 + 10.03 +21.0 ------Appendix II

The Finite Difference Method

Introduction rectangular plates, and Jensen e has extended the method to provide a useful tool in the analysis of The bending of thin elastic plates or slabs sub- skew slabs. jected to loads normal to their surfaces has been studied by many investigators.’ throughe A large General Mathematical Relations number of specific problems have been solved by exact or approximate means, and these results are The partial differential equation, frequently available. (See, for instance,3.) Exact and cer- called Lagrange’s equation, which relates the tain approximate methods are frequently difficult rectangular coordinates, the load, the deflections, to apply except to structures where some sym- and the physical and elastic constants of a laterally metry exists and where a simple loading is used. loaded plate, is well known. Its application to The finite difference method, however, is readily the solution of problems of bending of plates or adaptable to rectangular plates having any of the slabs is justified if the following conditions are met: usual edge conditions and subjected to any loading. (a) the plate or slab is composed of material which In Denmark, as early as 1918, N. J. Nielsen may be assumed to be homogeneous, isotropic, and applied the finite difference method to the solution elastic; (b) the plate is of /a uniform thickness of plate problems. In his book 4 he has analyzed which is small as compared with its lateral dimensions; (c) the deflections of the loaded plate are t,he problem in considerable detail and has given small as compared with its thickness. The addi- numerical solutions for a number of cases. tional differential expressions relating the deflec- H. Marcus published an excellent book 5 in tions to the boundary conditions, moments, and Germany in 1924 on this subject in which he in- shears are perhaps equally well known. (See, for cluded numerous examples. In the United States, instance,‘.) They will therefore only be stated Wise, Holl, and Barton u.7 a have contributed to here, using the notation and sign convention shown the literature of finite difference solutions for in Figure 38. 49 50 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

(0) INTERIOR POINT (bl SUB-DIVIDED GRID

GRID POINT DESIGNATION SYSTEM

I 0 : x (C) POSITIVE SIGN CONVENTION

P Intensity of pressure, normal to the plane of the plate. Q,b Lateral dimensions of the plate. h Loterol dimension in the y direction of the grid elements of the plate. r Ratio of lateral dimensions of the grid elements. Y Deflection of the middle surface of the plate, normal to the XOY plane. Rectangular coordinates in the plone of the plate. Z,N,E,...NE:ki Designotion of active grid points. Also used to represent the value of the deflection of the plate ot the point so lettered. n,e,s ,... SW,“W Designation of additional points on sub-divided grid. “3 t Subscripts used to indicate directions normal ond tangential to on edge. Ml, MY Bending moment per unit length acting on planes perpendicular to the x and y axes respectively. by * Myx Twisting moment Ser unit length in planes perpendicular to the x ond y axes respectively. VI, VY Shearing force per unit length acting normal to the plane of the plate, in planes normal to the x and y axes respectively. ‘?I, Ry Shearing reactions per unit length acting normal to the plane of the plate, in planes normal to the x ond y axes respectively. P Concentrated load acting at o grid point: positive in the some direction OS D. R Concentrated reaction acting at 0 supported grid point; positive direction opposite to thot of p E Young’s modulus for the material of the plate. I Moment of inertia per unit length of o section of the plate. P Poisson’s ratio for the material of the plate. D Flexural rigidity per unit length of the plate; 0 = EI/(I-$1. v’ Difference quotient operator: V’w = +- + 2 & + $.

NOTATION

FIGURE 38.-Grid point designation system and notation. APPENDIX II 51

Partial differential equation: with respect to n indicate rates of change in a direction normal to the edge, and those with (1) respect to t indicate rates of change tangential to the edge. A solution to any specific problem consists of Fixed edge conditions : determining a deflection surface which satisfies the w=o, (2.01) basic equation (l), and the appropriate sets of boundary conditions (2.01) through (5.03). The bW moments and shears required for design purposes T&=0. (2.02) may then be computed from (6.01) through (8.02). Hinged edge conditions : In general, it is difficult to obtain an analytical expression for a deflection surface which satisfies w=o, (3.01) all of these conditions. If, however, an approximate solution is acceptable, it is always possible $+p s?&J (3.02) in analyzing a rectangular plate to determine a set of deflections for a finite number of discrete Free edge conditions : points such that approximate relations corresponding to (1) through (5.03) are satisfied. From (4.01) these deflections it is possible to compute moments, reactions, and shears at the selected points, using relations similar to (6.01) through (8.02). (4.02) The approximate relations referred to above are obtained by replacing the partial derivatives Free corner conditions : by corresponding finite difference quotients. Such relations are simplest if the discrete points deter- g=O (both directions) J (5.01) mined by values of the independent variables are equally spaced with respect to both variables. However, in this application it will be advan- g+h4 s =0 (both directions), (5.02) tageous for the relations to be developed on the more general basis of having the equal spacing in d2W one coordinate direction bear a given ratio to the -- (5.03) bndt-” spacing in the perpendicular direction. Figure 38(a) represents a portion of the interior Bending moments : of a plate subdivided by grid lines into rectangular grid elements. The grid lines are spaced h units (6.01) apart in the y direction and rh units apart in the x direction. The int,ersections of the grid lines M =D b2W~~d2~ (6.02) will be referred to as grid points. Certain of Y by2 3x2 * 1 these, lettered for identification, will be spoken of Twisting moments : as active points, and the central point of the active group will be called the focal point. For (7) simplicity in writing the equations, the identifying Shears : letters for each active point %ill also be used to represent the value of the deflection, w, of the V = D (8.01) x - middle surface of the plate at that point. The double letters refer in every case to the deflection V YE-D b3”+ b3W (8.02) at the individual point so lettered; they do not bY3 bx2by . 1 indicate products of deflections at points desig- In the above expressions the partial derivatives nated by only one letter. 52 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Based on the usual methods of finite differ- -4(l+r’)(E+W)--4r2(1+r2)(N+S) ences,” the difference quotient relations required +2(3+4r2+31d)Z]. (10) in this development can be written directly and are given below. All of the difference quotients This may be considered as an operator, and the are given with reference to the focal point, lettered portion within the brackets can be conveniently Z. portrayed as an array of coefficients. This expres- Aw 1 bx=2rh (E--W), (9.01) sion, multiplied by h4, is shown in array form at (a) of Figure 39. Each element of the array A2w 1 represents the coefficient of the deflection of one z=13h2 W--274+W), (9.02) of the active grid points in a group similar to that shown at (a) of Figure 38. The location of the A3w 1 coefficients in the array is congruent to the physical -- (EE--2E+2W-WW), (9.03) s---2?h3 locations of the points and the heavily outlined coefficient applies at the focal point-the point A4w 1 -== (EE-4E+6Z--4W+WW), (9.04) for which the relation is to be determined. Ax4 Since the solution deals with discrete points, the distributed load intensity p in the right-hand g=$ (N-S), (9.05) member of (1) is replaced by an average intensity P/rh” at each of the interior grid points. Here P represents a concentrated load whose magnitude $=$ (N--2Z+S), (9.06) at any grid point is a function of the distribution of p on the four adjoining grid elements. If each A3w 1 of these elements is considered as an infinitely e=2h3 (NN-2N+S-SS), (9.07) rigid plate supported at its four corners, then the force Pz, at the focal point, is equal in magnitude ‘$=; (NN-4N+6Z-4S+SS), (9.08) and opposite in direction to the sum of the reactions at all corners common to Z. This can be A%V expressed mathematically as : -=-1 (NE-NW+SIW-SE), (9.09) AxAy 4rh2 p,=p,,,+p,s,+p,,,+p~~~ (11) A3w 1 - (NE-2N+NW-&+2S-SW), L\X2ay=2r2h3 in which PZNE represents the contribution from (9.10) the grid element Z-N-NE-E and similarly for the A3w other right-hand members. Thus it is seen that -=k3 (NE-2E+SE-NW-b-2W-SW), the concentrated loads Pz are the static equivalent (9.11) of p. A% It can be shown, if p varies linearly-a usual 7=&4 (NE-2E+SE-2N Ax Ay’ condition for structures-and if this variation is +4Z---2S+NW--2W+SW). (9.12) constant over the four grid elements adjoining any focal point Z, that the magnitude of the The approximate counterparts of the basic statically equivalent average load is: relations (1) through (8.02) may now be written. For instance if V4w is used to represent the differ- Pz/rh2=(1/6)(p~+pE+Ps+p~S2P~), (12) ence quotient equivalent to the left-hand member of equation (l), and the partial derivatives are where pN represents the intensity of p at point N, replaced by their corresponding difference quo- etc. tients, (9.04)) (9.08), and (9.12)) there results: The approximate counterpart of (1) may now be written: v’w=& [EE+WW+r4(NN+SS) PZ v4y=m2* (13) +2r2(NE+SE+SW+NW) APPENDIX II 53

Multiplying both sides of (13) by h4 and replacing In like manner for elements with centers at w, V% by the deflections as given by (10) leads to : n, and s:

W.,----M,)h+ Wrxnna -M,.,,)rh+V+rh2=0, $ [EE+WW+r4(NN+SS) (16.02) +2r2(NE+SE+SW+NW) -40 +3(E+W)--4r2(1 +r%N+S) Of,, ---M&h+ (MxYne--Mxy,,)h+V,,,rh2=0, (16.03) +2(3+4ra+3r4)Zl=~z g* (14) (M,,--M,s)rh+(M,,,e-M,,,)h+V,,rh2=0. This is the general load-deflection relation for an (16.04) interior point. It is written at (a) of Figure 39 in the convenient array form previously described. If equations (15) and (16.01) through (16.04) are This general form of the equations has been used combined to eliminate the shears, noting at the for the special cases which include the boundary same time that MIY=MYX, there results conditions and, in fact, for all of the relations connecting the deflections with load, moments, ; CM,, --2M,,+M,,)+2(M,,,e--M,,,,+M,,, reactions, and shears. These load-deflection equations establish a linear relation between the load --MxYBJ +r(M,,--2M,,+M,,) =Pe. (17) at the focal point and the unknown deflections of the plate at that and the other active grid points. An approximation to each moment in terms of It is these linear equations which are to be solved deflections is obtained if the partial differentials simultaneously to determine the approximate of the definitions (6.01); (6.02), and (7) are re- deflections of the plate at the grid points. placed by their proper difference quotients corre- Equation (14) may be derived directly by a sponding to (9.02), (9.06), and (9.09). For in- second method which considers equilibrium of stance, certain elements of the plate. Referring to the subdivided grid of Figure 38(b), consider the M.,=-& [E-2Z+W+Lcr*(N-2Z+S)] (18) rectangular element ne-se-sw-nw with center at Z. Equilibrium of forces normal to the plate and requires that M ‘une=--- W-P) [NE-N+Z--El. rh2 (19) (V.,-V.,)h+(V,,-V,,)rh+Pz=O. (15) Substituting these and corresponding relations for For the similar element with center at e, equilib- the other moments into (l7), and multiplying rium of moments about the center line ne-se re- both sides by h2/rD gives quires that f (WW-4W+GZ--4E+EE)+$ (NW-2N (Mxz -M&+ (M,,B-M,.,Jrh +NE-2W+4Z-2E+SW-2S+SE) +w.,+vx,> r;=o. +(NN--4N+6Z--4S+SS)=s However, if the elements are sufficiently small, which, with some rearrangement, is the same as (14). This second method is easily adapted to de- f (v.,+v.,) riving expressions involving nonuniform spacings, moment-free boundaries, etc. It was applied to may be replaced with VXe so that obtain all of the load-deflection arrays shown in Figures 39 through 59, which were required in the ME---M.&+ OLne -M,.,,)rh+V.~rh2=0. solution of the problems covered by this mono- (16.01) graph. 54 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Where boundary conditions involve a reaction, spacings which correspond to those of the load P may be replaced by the net load, the given point. (P-R), which is the difference between load and b. Orient the focal point of this array at the reaction. Note that R represents a concentrated given point. force whose positive direction is opposite to that c. Multiply the unknown which represents of p, R. and R,, on the other hand, represent the deflection of each active grid point intensities of shearing reactions whose positive by the corresponding coe5cient. directions conform to V, and V,. d. Equate the sum of these products to the Relations connecting the deflections with mo- load term for the given point. ments and with shears are given in Figures 60 For example, for Point 45 the array at (b) of through 64. It should be noted that shears com- Figure 65 must be used in order that the free edges puted by finite difference methods are inherently correspond properly. Then, following the pro- less accurate than moments. This is because the cedure outlined above, the left-hand member of shears are functions of odd numbered difference the equation for Point 45 is quotients which are determined by a grid spacing double the value found in the even numbered +256wpI,+32wg,- 1088wa~+28.&Jw,,+w,, quotients which define the moments. -68~,,+(1669+256)w,h--59.6~~ +32wM- 1088w,+28.8wM. ApplicaEion to Plate Fixed Along Three &?ges and Free Along llie Fourth Noting that RZ=O along the free edge it is seen that in this case the general expression for the As an example of the use of this general method, right-hand terms is always (P&h*) (h’/D). Since its application to the problem of a plate fixed along these load terms are to be expressed as coefficients three edges and free along the fourth is given below. of ph’/D, it remains to evaluate the Pz/rhg in The a/b ratio of l/4 has been used to illustrate use terms of p for each point and each loading. At of the 20 supplementary equations. Loads I, II, Point 45 the right-hand members for Loads I and and IV only are included. IV may be obtained by direct application of (12). The plate is divided into grid elements and the However, a discontinuity occurs in the magnitude grid points numbered systematically for identifica- of Load II within the grid elements adjoining tion. Layout of Plate, Figure 66, shows the Point 45. For this reason, the more general method used in this case. Because of symmetry method expressed by (11) must be employed. of the plate and loading about the line x=a, points In particular for Load II, the elements 45-35- which are symmetrical about this line will have 36-46 and 4546-56-55 carry no load, and accord- equal deflections and are, therefore, numbered ingly they make no contribution to P,. The alike. This reduces considerably the number of elements 45-44-34-35 and 45-55-5444 each unknown deflections to be determined. carry an equal portion of the uniform load. With r=l/4 and p=O.2, the left-hand side of Under the assumptions leading to (11) it is found, each of the loaddeflection relations yields an array by statics, that the contribution of each of these of numerical coefficients corresponding to the type elements to P,, is ph*/144. Hence, P,,=ph*/72 of point it represents. These values have been and P&h*=p/18. computed for typical points and they are shown The complete set of 30 equations and the right- in Figure 65. They are used in writing the left- hand (load) terms are shown as two matrices in hand members of the simultaneous equations. Figure 66. Simultaneous solution of the equa- Solution of these equations determines the de- tions establishes a set of deflections for each of the flections. 30 grid points, corresponding to each load. These One equation must be written for each grid results are tabulated in the upper portion of point having an unknown deflection. The equa- Figure 67. tion corresponding to any point is formed as The 20 supplementary equations used to deter- follows : mine the deflections of the row of points at y=ih a. Select the array of load-deflection coefficients having edge conditions and are set up in a similar manner. Equations are APPENDIX II 55 written for each point of the 3-, 2-, l-, and 7-rows Substituting numerical values for PsO and the (see Figure 68). However, in writing equations various deflections, this becomes for the 3- and 2-rows use is made of the previously computed deflections for the 4- and 5-rows. In R3,,=0.03125ph2+ e addition, the solution of the 20 equations gives (h2) (g) new and improved values of deflections for the [--(32)(0.004944)-(16)(0.021325) 3-, 2-, and l-rows. For Point 42, for example, the array (f) of Figure 65 is used to conform with the +(128)(0.007860)-(32)(0.009833)] spacing of the grid points involved. The equation =(0.03125+0.192016)ph2=0.223266ph2. for Load I is

-28w21+21Owzz+ low,,+ 176~31-936~~~ This represents a concentrated force acting at Point 30. Assuming that it is uniformly distrib- 5057 -SW,,+? ~~,-364w~~+~ w42 uted over a distance rh, it can be expressed ‘as an average shearing reaction per unit length 3 ph4 + 176w51- 936w&w~=4 D-~44. R,,,=R3&h=0.893064ph,

Substituting for Point 44, its deflection as deter- or in terms of b mined from the 30 equations gives, for the right-hand member R,,,=O.l78613pb,

(0.75-0.100572) ‘;=0.649428 ‘;. which is in the units used in Figures 1 through 33. Similarly, for example, the bending moment The complete set of 20 equations for Loads I, II, M, at Point 23 is computed using array (g) of and IV is given in Figure 68. Solution of these Figure 69. Thus gives the deflections shown on the lower portion of Figure 67. Where improved values of the deflection were obtained, the former ones have been discarded as indicated in the figure. Com- parison of old and new values shows that they Again inserting numerical values approach closely for the points where y/b=O.4. Having determined the deflections, reactions Mx23=(;) @) [(16)(0.015283) and moments may be computed by operating upon the deflections with the appropriate relations, t-(0.2)(0.029914)-(32.4)(0.043935) typical samples of which are given in Figure 69. These numerical arrays were obtained similarly to +(0.2)(0.046526)+(16)(0.073156)] those for the load-deflection relations, by inserting =0.006818ph2=0.000273pb2. numerical values for r and p in the proper general expressions of the referenced figures. Upon completion of computation of the reac- To illustrate the method of computation of tions, a partial check of the solution may be reactions and moments, an example of each obtained from equilibrium considerations. For (Load I, a/b=l/4) is given below. At Point 30, Load I, a/b=1/4, the total load on one-half for instance, using array (f) of Figure 69, the of the plate is p(5h)(5h/4)=6.25 ph2. The reaction is : summation of the R/ph2 column of Figure 70 should agree with this, and it is seen to be in error by something less than 0.015 percent. 56 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+ r4

+ 2r* -4r*- 4r4 + 2r*

+I -4-4r* +6+8r*+ 6r’ -4 - 4re +I = p h’ . rh2O + 2r* - 4r*- 4r4 + 2r*

+ r4

(0) INTERIOR POINT

I + r4 I

= P h4 rhe T’

(b) POINT ADJACENT TO A FIXED X-EDGE

/ + r4 / / + -4rt- 4r4 + 2 r* / =

/ + - 4+ 4r4 +2rg / / / + r4

+ r4

(d) POINT ADJACENT TO A FIXED CORNER

NOTES Except where otherwise indicated horizontol spacing of grid points is rh units ond vertical spacing h units. An osterisk (*I indicates thot no coefficient is required because the fixed-edge deflection ot thot point is zero. An edge porollel to the X-Axis is designoted OS on X-Edge. An edge porollel to the Y-Axis is designated OS o Y-Edge. A fixed edge is indicated thus: T7T/777TTTT A moment-free edge is indicated thus: Any factor preceding on array of coefficients is o multiplier of each element of the orroy.

FIGURE 39.-Load-dejlection relations, Sheet I. APPENDIX II 57

(0) POINT ADJACENT TO A MOMENT-FREE CORNER

= -- P h’ rh2 D

(b) POINT ADJACENT TO A MOMENT-FREE X-EDGE

= P h’ rhe -6’

++ = P h’ -z-b’

(cl) POINT ADJACENT TO A MOMENT-FREE X-EDGE AND A FIXED Y-EDGE

+r4

+ 20 -40 - 4r4 +(2-p)r* P h’ -2-212-p) 0 = -. rkQ

////////////////////////////////////////////////~

(a) POINT ADJACENT TO A MOMENT-FREE Y-EDQE AN0 A FIXED X-EDQL

NOT&--For general notes see Figure 39.

FIGURE 40.-Load-deflection relations, Sheet ZZ. 58 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+ r4 I I (0)CONANT-FREE X-EDGE

+ I + 4(1 (P-R) h’ = - r h* 0’

(b) POINT ON A MOMENT-FREE Y-EDGE

(P-R) h4 = rh* -D--’

t(2 - fi)rp -2(i-u)r~2(1-u~)r4

++(i-gz)r4

(d) POINT ON A MOMENT-FREE Y-EDGE ADJACENT TO A MOMENT-FREE X-EDGE

(13) POINT ON A MOMENT-FREE X-EDGE ADJACENT TO A FIXED Y-EDGE

++(I-p*)r4

(f) POINT ON A MOMENT-FREE Y-EDGE ADJACENT TO A FIXED X-EDGE

NOT&-For general notes see Figure 39.

FIGURE 41.-Load-dejlection relations, Sheet III. APPENDIX II 59

(0) POINT ON A MOMENT-FREE CORNER (b) POINT ON A FIXED CORNER

(d) POINT ON A FIXED Y-EDGE

*+2r2 L * -4-4r* +I r4 +

(0) POINT ON A FIXED X-EDGE 9 * ADJACENT TO A FIXED CORNER

(t) POINT ON A FIXED Y-EDGE kADJACENT TO A FIXED CORNER

= (P-RI h’ -l.,;,,,fi/,z ~ r h2 T-’ !!I, D

(0) POINT ON A FIXED X-EDGE ADJACENT TO A MOMENT-FREE Y-EDGE (h) POINT ON A FIXED Y-EDGE ADJACENT TO A MOMENT-FREE X-EDGE

(i 1 POINT ON A FIXED X- ht0htE~T-FREE Y-CORNER ( j) POINT ON A FIXED Y- MOMENT-FREE X-CORNER

Nom.-For general notes see Figure 39.

FIQURE 42.-Load-deflection relations, Sheet IV. 60 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES I +r’ I i + 2 rp - 4 rz - 4 r4 + 2re I = P h’ P -P 37 T’ h

+2r* 1 -4r*- 6r4 +2r*

+--rh-s+< ____ rh--.+ ____ rh ____ ++-rh--+l

(0) INTERIOR POINT

T I + r4 I h

I P h’ 7 = X0’

I +2 rt -4P - 6r4 + (2 -pL)r*

,+ r,, -+,+.- rh --__ + ---- rh ----- ,j

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

I ++(I -p) r*

+ (2-p)? -2(1-p)r*-2(1-p*)r

+ (2-P)@ -2(l -/L)+3(l-Z)r’

++(I -PI r’

b---rh---+j+ _____ rh----A

Nom-For general notes see Figure 39.

FIGURE 43.-Load-de$ection l’elations, vertical spacing: S at h; 1 at h/B, Sheet V. APPENDIX II 61

+ r4 I s +128 -&+Zr*

= - P -.h' rh' 0 I-& 1+$+4r’

p--rh---* _____ rh ----* ---- rh ____ *---rh--q

(a) INTERIOR POINT

+ r4

+8r4 I I

k-rh--*-- rh ---+ __-_- rh ---- +/

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+t(t-pe)r4

+ * - & +(2-tc)r* +*-2(l-p)rc -3(1-p*) r* I 7

1 +4(1 -P)r’

km-- rh---e ____ rh _____ 4

NOTE.-FOI general notes see Figure 39.

FIGURE 44.-Load-dejlection relations, vertical spacing: d at h; d at h/d; Sheet VI. 62 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES +1285

+105128 = L&+.

7 +&+4+ -&. 7 -64 t3rt-40r4 +&+4rr -64 I I I I IJ + 64 r4 3 I I

,+t-- r’++ --__ r,, ---+ __--- +-++-~‘,--~

(a) INTERIOR POINT

% = PA!. i rh2 D $‘h 7 35 f -& +=+4r= -z - are-40r4 +&+2(2-p)r* f ‘h k +64 r4 3

/-+- rh--* ____ rh ____ +f+-- ____ rh _- ____ 4

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+ + (I -pe) r4

= IP-R) h’ . rh2 0

-is- + $ + 2(2-p)r* -&-4(1-p)rL-20(i-p1)P

+ + (I-pz)r*

p ____ rh---+------rh ______4

NOTE.--For general notes see Figure 39.

FIGURE 45.-Load-deflection relations, vertical spacing: 2 at h; 1 at h/2; 1 at h/4, Sheet VIZ. APPENDIX II 63

7- h

= --.P h’ rh* D

+4rP -are- 32r4 +4re

+6r4 I I

(a) INTERIOR POINT

+$ r4 x I I % i,h = --. I' h4 * rh* D +‘h + +4rp -Ore- 32r’ +2(2 -p)r* +‘h

P +0r4

+-rh--4---rh ---- *-----rh-----+/

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+ 2(2 -PC) r* -4wF12(1-pV

I ++-+ 0(1 -p)r* = --.(P-R) h4 -I -4(2-pL)r’ + 661,-pcljr4 7 +t rh' 0

+2(2-p)r* -~-p)r*-i6(1-pe)r’

I +4(1 -p*)r’

k--- rh ----* _____ rh - ___- 4

NOTE.-For general notes see Figure 39.

FIGURE 46.-Load-deflection relations, vertical spacing: 1 at h; 3 at h/B, Sheet VIII. 64 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+ f r’

105 496 . _ 105 - I - Lg -,2p ++$+24r’+-j-r =-,2rg +105 = +A!.. +256 r’ 256 rh 0 +&+gre -g- -Tk 16rL - 192r4 + &-+ 0r r --A

b--rh--e --___ rh ----+ ______rh----++--rh--4

(a) INTERIOR POINT

. I ++r I

5 25 -&+4rg - 6r’- 4oP -& +2(2-pL)r +zz I +i56 I I 105 = Lx.. F +isd rh’ 0

?. -35 -6 +&+Br G-l6r*-192r4 +&t4(2-p)r I I

64t’ I + I

b--rh--e---- rh ----_ e -_____ rh------d

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+$(I -2) r* I II c

5 -$ +2(2 -p)r* +A-4(1-p)r*-20(1-p*)+ +256 I I II = P-R) h’ . rh* D

-6s + & t 4(2-p)+ - & - 9(1 -p)r~9ql-p*) r*

+32(1 -/4r’

II km-- rh ---* __---___ rh------A

NOTE.--For general notes see Figure 39.

FIGURE 47.-Load-deflection relations, vertical spacing: 1 at h; 1 at h/2; 2 at h/4, Sheet IX. APPENDIX II 65

+ $ r4

= --. P h' rh* D -7 +&- +8r’ -$+ -16r’ 7 7 126 -320r’ +3-i-++rP -128 A + 51e r4 3

+-rf+-+ _____ rh ____ 4---rh---~--rh--~

(a) INTERIOR POINT

++ r4

= --.P h' rh* D

- * +& + 69 -$$ -16r*-320r’ +& +4(2-p)r*

+w r*

be- rh --+ _____ rh ----+ ______rh ____- 4

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+ f (I -p*) r* f 5 -& + 2(2-p)r* +A -4(1-p)r*-20(1-p*)r’ $ +e56 I =. P-R) h’ . fh* r) rh' D f;,x -T&i +* +4(2-p)r * -&-8(1-p)rc-i60(i-$)r4 f +?(I -p*)r4 II b--fh---e ______rh ______4

Nom-For general notes see Figure 39.

FIQURE 48.-Load-dejlection relations, vertical spacing: 1 each at h, h/d, h/4, and h/8, Sheet X. 66 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

b---r,, -+,+- --__ r,, ----- +-s-v rh ----- +-- r,,+..,

(0) INTERIOR POINT

I +gr* I

+4rz -8r’- 32r’ +2(2-p)r*

+4rc - 8rg- 32r4 +2(2-p) r*

b--rh---+/.+ __-_ rh _____ T ______rh _____ ++

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

I +2(2-p) rc I -4(1-p)r1-16(1-p)r411

.f+----rh ____ *- __--- rh ----- 4

(Cl POINT ON A MOMENT-FREE Y-EDGE

NOTE.-For general notes see Figure 39.

FIGURE 49.-Load-dejfection relations, vertical spacing: 4 at h/8, Sheet XI. APPENDIX II 67

+ 64 r. 3

+ 9r* -l6r*- 192r4

+6rP - 16r* - 256r’ +6r2 I

+64r4 J

b-rh--+----rh ____ +----rh ----+--rh--d

(0) INTERIOR POINT

l +y r’ I I

p = --.P h’ rh* D

+64r* I I

~-rh-~--- rh---++g-----rh ____ 4

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+y (I-p*)+ II

+4(2 -p)” -6(l-p)r’- 96(1-p’) r4 (I $h I 4 I-++ -+ -8(2-p)P +4(2-p)r’ -e(l-p)re-126(1-pc)r4

+ 32(l -PL)r4

II +-c-m r,, -+ ______r,, - _____ +,

NOTE.-FOT general notes see Figure 39.

FIGURE 50.-Load-deflection relations, vertical spacing: 1 at h/B; 3 at h/4, Sheet XII. 68 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I +IoI -j+$-24r’ +z+46rt+yr4 512 -j$-24rt +s = f’ h’ . r) rh’ D

- & +& + 16r’ -.&-32~1536r4 +& + 16r’ - &

b--rh--+-----rh ____ h _____ rh _____ *---rh--~

(a) INTERIOR POINT

+ 84 r4 3

-!- p At. r4 -3-D

+&+16r* - & -32r’- 1536r4 + & -& + 6(2-p) r*

+512r4

k--rh-e -____ rh ____ +j+----- rh _____ 4

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

++(I -p*)r* I II 5 +sle -&+4(2-p)rE t$ -6(l-~)+l6OfJ+)+ = --.(P-R) h’ rh’ D

-I -& - 16(1-/~)+766(1 +,r’ 256 +&+6(2-p)r*

+256(1 -PC)+

II b---rh ____ + ______rh ______4

(cl POINT ON A MOMENT-FREE Y-EDGE

NOTE.--For general notes see Figure 39.

FIQIJRE 51.-Load-deflection relations, vertical spacing: 1 at h/d; 1 at h/4; 8 at h/8, Sheet XIII. APPENDIX II 69 $ x - 16r*- 256r4 f'h = P h' x -;i;p T-' f'h

f +6r* -16rz- 266r. + 6re flh 9 +64r4

bt---rh.--+ _____ rh ____ e _____ rh ____+-rh--d

(a) INTERIOR POINT

-I +64r4 f:h * +Ort -l6r’- 256r’ t:” P h' * I I = Ti;fO' flh 7c +T * +6re --16+ -256r4 +4(2-p) rt +@h k +64r4

f+--rh---* ____ rh _____ * ______rh _____ -f

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

,+--r,,/+ _____- r,, ----- +,

NOTE.-For general notes see Figure 39.

FIGURE 52.-Load-deflection relations, vertical spacing: ,$ at h/4, Sheet XIV. 70 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I +16r’ -32r’ - 1536r’ ( +,6,.F1

b--rh--+ -_____ rh ---- *-----rh ____ 4--,-h--d

(a) INTERIOR POINT

+ l6r* - 32r’ - 1536r4 +6(2 -p)rc f fh +I++ -+q++& = m$+.

+ l6r* -32r* - 2046r’ +6(2-p)r*

+ 512r4

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+?(I -pE)r4 I II

+ 6(2 -p)r* -160 +)r’- 766(1-/L*) r’

k--rh --+ ______rh _____ -4

NOTE.-For general notes see Figure 39.

FIGURE 53.-Load-deflection relations, vertical spacing: 1 at h/4; S at h/8, Sheet XV. APPENDIX II 71

+512r4

P h’ rhL 0’

+16r’ - 32re- 2046P +l6r*

+ 512 r’

kc-- rh ---*-----rh _____ * -____ rh _____ G---rh --+

(a) INTERIOR POINT

I +512 r4

+ 16r’ - 32r*- 2046P

+16r’ - 32 r’- 2046 r4 +6(2 -pL)r*

+s12r4

k--rh---* ____ rh _____ pj+ ______rh ------~

(b) POINT ADJACENT TO A MOMENT-FREE Y-EDGE

+256(1- PL) r’

+6(2 -pLr* --16(1 -~)r’-lO24(1-~~

+6(2-p)+ -16(1-~)+1024(1-cp)

+c---rh--+ ____ -rh------4

NOTE.-For general notes see Figure 39.

FIGURE 54.-Load-dejlection relations, vertical spacing: 4 at h/8, Sheet XVI. MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

b-irh +--+rh---h--s rh---h-irh-F(

I + 4r’ I

birh+--+rh--+--+rh--+-$rh-4

= --.(P-R) h' rh* D

-16r*- tzr'

I +4r4 I

bkrh-*--irh--h--$rh--+-irh-4

+ + r'

+4rz -6rz- 2r4 +4r' = (P-ax, rh' D

NOTE.-FOT general notes see Figure 39.

FIGURE 55.-Load-deflection relations, horizontal spacing: .$ at rhl%, Sheet XVII. APPENDIX II 73

= (P-R) ‘. rh’ 0 + 6P - 16P*- 16f’ +er’

+4r4

I = P 2. 7 rhTD + 6r’ - 16f’- 16f4 + 6P

+4r’

+-+rh -+--$ rh---+--+rh ---+- rh -4

+ 4 r4 I I

b-$rhh--frh--h--irh--*-rh-c(

+$r l I Th I f P h’ P i” 7 -= rh D + 4r’ - 8r’ - 3r’ +4r’ $.,, r2 + $r*

1 + 4r* 1 - 6~’ - 2r4 1 +4r* 1

,+,,++$ rh ---wf+--3rh --+- rh -4

NOTE.--For general notes see Figure 39.

FIGURE 50.-Load-dejlection relations, horizontal spacing: 3 at rh/d; 1 at th, Sheet XVIII. 74 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+6f'

= (P-A) h' 7 -- rh D

+ 8rP - 12r* - 24r' + 4r*

+6r4 1

b-irh-rt<---$rh--+ ____ rh---+--rh-+

+2r'

P h' = --. rh' 0 + 6fP - i2r2- 24r4 + 4rg

+6r'

b-$rh-+---irh --+f+----rh---*--rh--,../

+ Jr’ 4

P h4 L +6 = --. r' rh' D

+6rp - 12r* - 16r. +4r*

+6r'

b-$rh h---+rh---Zt, ____ rh---+--rh--+

+Jr’4

+4rz - 6r* - 3r'

P h4 = 2 -- rh 0

+2r*

b-irh-+---krh --+----,-h ---*--rh-+

+ Jr’ I 4 I

P h' = - -. rh' D

+4r* -6r* - Jr4 +2r2

+zr* 4

Nom.-For general notes see Figure 39.

FIGURE 57.-Load-deflection relations, horizontal spacing: 2 at rh/d; 2 at rh, Sheet XIX. APPENDIX II 75

= --.(P-RI h4 rh' D

+6r' I I b-krh* _-__ ,-h---e--- rh ---e--,-h-+

+ =r' I3 1

P = h' I--. rh 0

k-irh-4 ___- rh ---e ---_ ,-h ---*-,-h -4

f h

khrh 4--rh --e---rh---*-rhd

+r' ;f” I P ha 2-* i;” 7 rh D $,, +2 r* -4r* - 6r' + 2rP r2 + 8r' 3

bhrh b--rh--+=-/+---rh--+-j+-rh -4

+r4 I I

h' = --. P rh* D

+2r* - 4r" - 4r4 + 2rp

+r*

birh +---rh --e---rh--W/N- rh 4

Nom.-For general notes see Figure 39.

FIGURE 58.-Load-desection relations, horizontal spacing: 1 at rh/8; 3 at rh, Sheet XX. 76 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+6r4

+4rt - 6r* - 32r' + 4rt

(P-R) h' = --. rh' D +4r' - 6P - 32r' + 4r'

+ 6r'

j-e-rh-+--rh--+---rh---+--rh-4

+ Lr' 3

P = --. h' rh* D +4r' -6r'- 32r' +4r'

+ 6r*

+r4

I P h4 = +P 7 77-D'

+4r' - er'- 24r4 +4r*

+6r'

b-rhh---rh --+---rh--4--rh-c(

+r4

P h* = --. rh' D

+2r1 - 4r'- 6r' +2r'

++rb b-r h -4--- rh --4--- rh ---h- rh --f

+ r*

P h' = --. rh* D

+2re -4r' - 4r' +2r*

+r*

+-rh-+--- rh ---+--- rh--+- rh-4

NOTE.-For general notes see Figure 3%

FIGURE 59.-Load deJEection relations, horizontal spa&a: .4 at rh. Sheet XXI. APPENDIX II 77

I I I J

I+-rh--+a---h-+-j I-c-rh-+k--rh-+I (a) b)

l+rh+t+rh+ (4 INTERIOR POINT

Mx = $31 My = 0 t+-rh-+-rh -4

k4 (e)

* +2rx +e(l-p)r*

4 I b=?pry!t * My=++ * -t H Mx = pm, M, = 0 tJI (h) (i) EDGE AND CORNER POINTS

I+-rh--+--rh--4 I+-rh-+I+-rh-+I (i) (k) (ml INTERIOR AND EDGE POINTS - NONUNIFORM SPACING

Mr = 0 MI - 0 (4 1s) (t) EDGE AND CORNER POINTS - FRACTIONAL VERTICAL SPACING

NOTES 4 = M, = 0 at either a fixed or moment-free corner. M xv = MYI = 0 ot any point on o fixed edge.

NOTE.--For general notes see hgure 39.

FIQURE 6O.-Momentdejlection relations. 78 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Mx P MyP ah* He

b-+-rh +-rh-w,

L - 0 ,--rh-+--rh -v,

Nom.-For general notes see Figure 39.

FIGURE 61.-Moment-dejlection relations, various point spacings. APPENDIX II

INTERIOR POINT

-(l-IL) +2 I I -u) -(I-P)

vy=$&0 t r2 0

+I -2(l t r2) +I eF +r2 (4

POINT ADJACENT TO A MOMENT-FREE EOQE

-r2(l-fi)

+2re(i-Cc)

D I v, =Tm 0

-2+(1-u)

+r2( I-u) (e) (f)

POINT ON A MOMENT-FREE EDGE E

(9) (h)

POINT ON A MOMENT-FREE EDGE ADJACENT TO A MOMENT-FREE CORNER

NOTE.-FOT general notes see Figure 39.

FIGURE 62.-Shear-deflection Telations, Sheet I. 80 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

v,=P ’ h’ 2

(b)

POINT ADJACENT TO A FIXED EWE

(cl (4

POINT ON A FIXED EDGE

/ / / * / , + -2

VY = -$-&r

(4 IfI

POINT ON A FIXED EOQE ADJACENT TO A FIXED CORNER

(e) (h)

POINT 0N.A MOMENT-FREE EDQE ADJACENT TO A FIXED EDQE

NOTE.--For general notes see Figure 39.

FIGURE 63.~Shear-dejlection relations, Sheet ZZ. APPENDIX II 81

vx =- ph r3

b-rh-+-rhe b- rh +-rh+

-6(2 +r2) +4

k- r h -+- rh+

Note: These orroys opply Only where the load at corresponding points on opposite sides of the centerline is equol in magnitude but opposite in direction.

Nom.-For general notes see Figure 39.

FIGURE &I.-Shear-de$ection relations, Sheet ZZZ. 82 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+26.6 -59.6 +26.6 + 32 -66 +32

i [ + 256 -1066 + 1669 - I066 + 256 +256 - I066 + 1670 -1086 +256 +32 -66 f +32 +32 -66 +32 f +I

(a) INTERIOR POINT (b) POINT ADJACENT TO A FREE X-EDGE

+32 -66 +122.66 -517. 12 +76X46 -517.12 +122.66

+ 256 -1066 +50(1 -;!;6 +256 ] f’[ 3

+32 -70

tb-fh--*--fh--*--th--~--~h--~

l tl f + 64 - 152 + 64 + IO -6 -10 -6 + IO

t 210 -936 +y -936 +210 t I26 - 640 +G 3 - 640 +I26

+64 -I60 +64 -26 + 176 - 336 + 176 -26

+64 +a 3

p-v f ,,++& + ,,++-- $ ,,-+-- f h-4

(0) INTERIOR POINT (f) INTERIOR POINT VERTICAL SPACING: IATh; SAT+h VERTICAL SPACING: 2 AT h; I AT fh; I AT fh

FIGURE 65.-Load-deflection coeficients, r=M, p=O.Z. --x i... 84 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I I DEFLE(

IO.1 I +.005659*I +.ol! I 1 1.0 1 +.000426

Deflection = (Coefficient)(ph’/D)

-+-IJP

l@J

beep-t; cc-p -4 k-p -ti POSITIVE SIQN CONVENTION LOAO I LOAD Iz LOAD m

Deflection = (Coefficient)(ph’/D)

NOTE Starred values computed from 30 equations are discorded when the corresponding improved value is oljtoined from the 20 equations.

FIGURE 67.-P&e $xed along three edges, deflection coefiients. a/b=%. Various loadings. ”, .Nxxm31 ay”w-*“sIY l o XIYMN

: i5 i % I 86 MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

(0) POINT ON A FIXED Y- (b) POINT ON A FIXED I-EOQE (C) POINT ON A FIXED Y-EDBE FREE X-CORNER ADJACENT TO A MOMENT-FREE X-EWE VERTICAL SP.oI*G: h ANO + h [FIQURE 39 II,] [FIGURE 39 (II,] [FIGURE 42 iol]

tcfh-++hc( I+ h -+-fh+

(d) POINT ON A FIXED Y-EWE (0) POINT ON A FIXED CORNER (f) POINT ON A FIXED X-EWE “ERTlCAL SPACING: + II ANO + h [FIGURE 49 (Ol] [FIGURE 49 IO,] [FIGURE 44 to)]

REACTION-DEFLECTION COEFFICIENTS r = l/4 p = 0.2

T h

i l+h -++hd

k+h+++hd CC+h-Ct(-+h+

(0) INTERIOR POINT (h) POINT ON A FREE EDQE (i) INTERIOR POINT [FIGURE se to)] [FIGURE !N! Ml] “E”TIOAL scAaIw0: h A”0 f h [ .=IGURE se (I,]

BENDING MOMENT-DEFLECTION COEFFICIENTS (M,) r = l/4 jl = 0.2

+fh+-+h+i l++h -+fh tl

(j) INTERIOR POINT (k) POINT ON A FIXED EDQE (m) INTERIOR PQINT [FIGURE se (b)] [FIGURE se (PI] “ERTlCAL SFAOIW: h AGO + b [FIGURE se km)]

BENDING MOMENT-DEFLECTION COEFFICIENTS (MY) r = I/4 p - 0.2

NOTES To find the net reaction or the bendinq moment at ony focal point, compute the products of the coefficients of the oppropiote orroy by the deflection of the correspondinq points ond multiply their sum by (O/h’). Figure numbers in brackets refer to qenerol expressions from which these numeric01 orroys were computed.

FIGURE W.-Numerical values of typical moment and reaction arrays, r=x’, p=O.B. APPENDIX II 87

POINT Yfl DEFLECTIONS - w/(ph4/D)

0 I I 1 I 6 0 +.017022 1 t.049660 1 + .063466 1 t. 107935 +.I16792 5 1 0 + .016122 t .046640 + .076499 +. 101377 + .I09650 4 1 0 + .016030 t.046526 t .077914 +. 100572 +.I06761 1310 1 +.015263 1 +.043935 1 t .073156 2 0 + .010730 + .029914 + .046903 I 0 + .004699 t.013261 + .02 1325 7 0 + .001835 + .004944 + .007660 t.010522 I 0 0 0 0 0

04 1 t.125 +I.131256 +I.256256 03 1 +.I25 +I.131056 + I .‘256056 .-_-. 02 t.09375 t .736474 + .a32224 +. 190464 01 + SO46675 +.I78392 + .225267 07 +.03125 - .000992 + .030256 00 [ t .Ol5625 -.056720 1 - .043095 1 + .029514 IO 1 t .03125 I -.001712 I t.029536 I + .023630 I I 20 + .03125 +.I10096 + .I41346 +. I I3077 30 + .03125 +.I92016 + .223266 +.I78613 40 + .03125 + .240512 + .271762 t.217410 50 + .03125 t .256320 + .207570 + .230056 I c” +6.249145 * lncludrs only of bo.

POINT NO. BENDING MOMENT - MJpb* 0 I 2 3 4 5 6 + __-_.020917 7 , +.009607 1 t .000693 1 -.005724 1 -.009592 1 -.010863 I 5 + .020636 + .009346 + .000622 1 -.005565 1 -.009301 1 -.010539 4 + .020516 + .009253 1 t -000553 1 -.005621 1 - .009305 1 - .010531 1 1 t .000273 1 .005438 -.006766 .009690 3 + .019562 + .006526 - I I - ,I 2 + .013734 t .005335 - .000330 -.003930 -.005917 1 - .006549 0 0 + .000470 + .001266 + .002012 + .0025 I7 t .002694

POINT NO. BENDING MOMENT - My/pbe -T 2 3 4 5 UhllJSE?1 - 0 I I 6 1 0 I 0 0 0 0 0 I !I t 004127 I +.001901 + .00022 I - .000949 -.001639 - .001666 4 + .004104 + .OOl625 t .00002 3 -.001264 - .002076 - .002344 3 + .003912 t .001559 - .000364 -.001636 - .002734 - .003036 2 + .002747 + .000703 -.001051 - .002366 - .003177 - .003449 0 1 0 + .002349 + .006326 t .01006 I + .012566 + .013460

FIGURE 70.-Plate fixed along three edges, dejlections-reactions-bending moments, Load I. a/b= xi, p=O.6.

List of References

1. Timoshenko, S., l%eory of Plates and SheuS, American Concrete Institute, Vol. XXIV, McGraw-Hill, New York, 1940. page 408, 1928. 2. Anonymous, “Rectangular Concrete Tanks,” 7. HoII, D. L., “Analysis of Plate Examples by Concrete lnjormation Bulletin No. ST63, Difference Methods and the Superposition Portland Cement Association, 1947. Principle,” Jvurnd of Applied Me&nice, 3. Westergaard, H. M., and Slater, W. A., Vol. 58, page A-81, 1936. “Moments and Stresses in Slabs,” Proceed- 8. Barton, M. V., Finite LX@rence Equutions for the Analysis of Thin Rectangular Plabs, ings, American Concrete Institute, Vol. University of Texas, 1948. XVII, page 415,192l. 9. Jensen, V. P., “Analyses of Skew Slabs,” 4. Nielsen, N. J., Bestemmebe aj Spaendinger i Bulletin S&s No. $%?, University of PZuder, JpLrgenson,Copenhagen, 1920. IIhnois, Engineering Experiment Station, 5. Marcus, H., Die Th-eorie elaatischer Qkwebe, 1941. 2nd Edition, Julius Springer, Berlin, 1932. 10. Scarborough, J. B., NumerieaZ Mahmuhal 6. Wise, J. A., “The Calculation of Flat Plates by Andy&, John Hopkins Press, Baltimore, the Elastic Web Method,” Proceedings, 1950.

*U.S. GOVERNMENT PRINTING 0mcE: 19~5-8341-386

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