Moody's Chart

.

A WATER RESOURCES TECHNICAL PUBLICATION

ENGINEERING MONOGRAPH NO. 27

Moments and Reactions for Rectangular Plates

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- taining computed data for use in the design of components of structures which can be idealized as rectangular plates or slabs. Typical examples are wall and footing panels of counterfort retaining walls. The tables provide the designer with a rapid and economical means of analyzing the structures at representative points. The data presented, as indicated in the accompanying figure on the frontispiece, were computed for fivl: sets of boundary conditions, nine ratios of lateral dimensions, and eleven loadings typical of those encountered in design.

As supplementary guides to the use and development of the data compiled in this monograph, two appendixes are included. The first appendix presents an example of application of the data to a typical structure. The second appendix explains the basic mathematical considerations and develops the application of the finite difference method to the solution of plate problems. A series of drawings in the appendixes presents basic relations which will aid in application of the method to other problems. Other drawings illustrate application of the method to one of the specific cases and lateral dimension ratios included in the 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

LOAD I

CASE 2 CASE 3 CASE 4

L-p-A

LOAD n

“NWORY LOAD OVER e/3 THE “EIBHT OF THE PLITE

LOAD H

BOUNDARY CONOITIONS

LOAD PII

LOAD IU

“NlFORY LOAD OVER 113 THE HEIOHT OF THE PLATE

f------- 0 L!kl I-

id- pd

LOAD Ip

“WlFORYLI “ARIINO LOAD OVER THE FVLL HEIGHT OF THE PLATE

LOAD Pm LOAD iI

“NlFORYL” “AWlNO LOAD UNIFORM YOYEW ALOW UNIFORM LINE LOAD OVER l/6 THE “ElB”f IHE soce y - b FOR ILOWO WE FREE EOBE 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.

CASE 5

PLATE FIXED *Lowe FOVR EOBES

f---- G i- d ;pd

LOAD Y

“NlFORYLl “ARIINO LOAD OVER e/3 TM ns,en* OF THE PLATE

-H p L-

LOAD I

“WIFORYL” “m”I*e LOAD D- o ALOWO y-b/e

INDEX OF BOUNDARY AND LOADING CONDITIONS

-FRONTISPIECE

Contents

Preface and Acknowledgments -----____-___________ --__

Frontispiece __------________________________________---------

Introduction ________ ________________.______ - _________________

Method of Analysis ______________________________ -- ______

Results ________________________________________------- -- ______ Effect of Poisson’s Ratio- ___________________________ - ________

Accuracy of Method of Analysis---------------------

Appendix I ______________________________ - _________________ An Application to a Design Problem-- - - _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _

Appendix I I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The Finite Difference Method- _ _ _ _ _ _ _ _ __ _ _ __ _ _ __ _ _ ___ _ _ _ _ _ - _-

Introduction____________________________---------------- General Mathematical Relations- _ _ _ _ _ -__ ____ _ _ ___ _ __ _ _-_ _ Application to Plate Fixed Along Three Edges and Free Along

the Fourth__________________________________---------

List of ,R e erences------------------------------------_____ f

LIST OF FIGURES Number

1. Plate fixed along three edges, moment and reaction coefficients, Load I, uniform load- ------------------ > ---------_--------------

2. Plate fixed along three edges, moment and reaction coefficients, Load II, 213 uniform load _____ __ __ _ _ __ _ _ __ ____ __ ____ _ ____ _ _ __ _ ___

3. Plate fixed along three edges, moment and reaction coefficients, Load III, l/3 uniform load--------- ______________________________

4. Plate fixed along three edges, moment and reaction coefficients, Load IV, uniformly varying load ____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _

Page . . . ill

iv

1

3

5

6

43

45

45

49

49 49 49

54

89

PW

7

8

9

10 V

vi

Number

CONTENTS

5. Plate fixed along three edges, moment and reaction coefficients, Load V, 213 uniformly varyingload _---__--_______ ------- ________

6. Plate fixed along three edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load -------- ____________________

7. Plate fixed along three edges, moment and reaction coefficients, Load VII, l/6 uniformly varyingload--_-----..------ ______________

8. Plate fixed along three edges, moment and reaction coefficients, Load VIII, moment at free edge------- _____ ---------- ____________

9. Plate fixed along three edges, moment and reaction coefficients, Load IX, lineload at free edge---------------------- _____________

10. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load I, uniform load---- - - - _ - _ _ _ _ _ _ _ _ _ _ _ _

11. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load II, 213 uniform load_ - _ _ _ _ _ _ _ _ _ _ _ _ _ _

12. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load III, l/3 uniform load- - - - __ _ _ _ _ _ _ _ _

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

14. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load V, 213 uniformly varying load--_ _ _ _ _ _

15. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VI, l/3 uniformly varying load_- _ _ _ _ _

16. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VII, l/6 uniformly varying load- _ _ _ _ _

17. Plate fixed along three edges-Hinged along one edge, moment and reaction coefficients, Load VIII, moment at hinged edge- - - - - - _ _

18. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load I, uniform load--- _ _ _ _ _ _ _ _

19. Plate fixed along one edge-Hinged along two opposite edges, moment and react,ion coefficients, Load II, 213 uniform load _ _ _ _ _ _ _

20. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load III, l/3 uniform load- _ - _ _

21. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IV, uniformly varying load.

22. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load V, 213 uniformly varying load----_______-----____________________------------------

23. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load_-__-------_________________________------------------

24. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VII, l/6 uniformly varying load- ---__-_-_-----__------~~~~~~~~--~~------------------

25. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load VIII, moment at free edge- -

26. Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coefficients, Load IX, line load at free edge-

27. Plate fixed along two adjacent edges, moment and reaction coefll- cients, Load I, uniform load--- __________- ------------------

me

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

CONTENTS Number

28. Plate fixed along two adjacent edges, moment and reaction coefficients, Load II, 213 uniform load------------- ______________

29. Plate fixed along two adjacent edges, moment and reaction coefficients, Load III, l/3 uniform load- _ - - - - - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

30. Plate fixed along two adjacent edges, moment and reaction coefficients, Load IV, uniformly varying load- - _- - _ - - - - _ _ _ _ _ _ _ _ __ _

31. Plate fixed along two adjacent edges, moment and reaction coefficients, Load V, 2/3 uniformly varying load- _ - - - - - - - _ _ _ _ _ _ _ _ _ _

32. Plate fixed along two adjacent edges, moment and reaction coefficients, Load VI, l/3 uniformly varying load- _ - - - - _ _ _ _ _ _ _ _ _ _ _ _

33. Plate fixed along two adjacent edges, moment and reaction coefficients, Load VII, l/6 uniformly varying load-- _ _ __ _ _ __ _ _ _ _ _ _ _

34. Plate fixed along four edges, moment and reaction coefficients, Load I,uniformload____-----------_____________________--------

35. Plate fixed along four edges, moment and reaction coefficients, Load X, uniformly varying load, p=O along y=b/2------- __________

36. Plate fixed along four edges, moment and reaction coefficients, Load XI, uniformly varying load, p=O along x=a/2---------------..

37. Counterfort wall, design example---~---------------~-~~~~~-~..~ 38. Grid point designation system and notation- _ - - - _ - _ - - _ _ _ _ _ _ _ _ _ _ 39. Load-deflection relations, Sheet I _______________ --__--_-------- 40. Load-deflection relations, Sheet II----------------------------- 41. Load-deflection relations, Sheet III---------------------------- 42. Load-deflection relations, Sheet IV __________ -___--_-------_---- 43. Load-deflection relations, vertical spacing: 3 at h; 1 at h/2, Sheet V- 44. Load-deflection relations, vertical spacing: 2 at h; 2 at h/2, Sheet VI- 45. Load-deflection relations, vertical spacing: 2 at h; 1 at h/2; 1 at h/4,

SheetVII___-_-----______________________----------------- 46. Load-deflection relations, vertical spacing: 1 at h; 3 at h/2, Sheet

VIII--__-----_-_----------------------------------------- 47. Load-deflection relations, vertical spacing: 1 at h; 1 at h/2; 2 at h/4,

SheetIX___-_________-_-______________________----------- 48. Load-deflection relations, vertical spacing: 1 each at h, h/2, h/4,

and h/8, Sheet X------------- _____________________________ 49. Load-deflection relations, vertical spacing: 4 at h/2, Sheet Xl _ _ _ _ _ 50. Load-deflection relations, vertical spacing: 1 at h/2; 3 at h/4, Sheet

XII------------------------- -------------------------- --- 51. Load-deflection relations, vertical spacing: 1 at h/2 ; 1 at h/4; 2 at

h/8, Sheet XIII--------------- _______-_____________________ 52. Load-deflection relations, vertical spacing: 4 at h/4, Sheet XIV--- - 53. Load-deflection relations, vertical spacing: 1 at h/4; 3 at h/8, Sheet

xv ____ -------------------- ------------------------------ 54. Load-deflection relations, vertical spacing: 4 at h/8, Sheet XVI---- 55. Load-deflection relations, horizontal spacing: 4 at rh/2, Sheet XVII- 56. Load-deflection relations, horizontal spacing: 3 at rh/2; 1 at rh,

SheetXVIII_-------_------------------------------------- 57. Load-deflection relations, horizontal spacing: 2 at rh/2; 2 at rh,

SheetXIX _______________________ --- _________ -_-_-_- ______ 58. Load-deflection relations, horizontal spacing: 1 at rh/2 ; 3 at rh,

Sheet xX-__-_------_--_--________________________________

page

vii

34

35

36

37

38

39

40

41

42 46 50 56 57 58 59 60 61

62

63

64

6.5 66

67

68 69

70 71 72

73

74

75

Viii CONTENTS

Number

59. Load-deflection relations, horizontal spacing: 4 at rh, Sheet XXI- _ 60. Moment-deflectionrelations--- ______________ --_--___-- ________ 61. Moment-deflection relations, various point spacings- _ _ _ _ _ _ _ _ _ _ _ _ _ 62. Shear-deflection relations, Sheet I--- _ _ - - - - _ _ - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 63. Shear-deflection relations, Sheet II-------------- ____ - __________ 64. Shear-deflection relations, Sheet III- _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 65. Load-deflection coefficients, r=1/4, p=O.2------- ____ -- _________ 66. Plate fixed along three edges-30 equations for determining unknown

deflections. a/b=114 _______________--______ --- ____ --- _____ 67. Plate fixed along three edges, deflection coefficients. a/b=114

Variousloadings________--_---____________---___----_______ 68. Plate fixed along three edges-20 equations for determining unknown

deflections. a/b=114 _______________________ ---___---- _____ 69. Numerical values of typical moment and reaction arrays, r=1/4,

p=o.2 __-____-___-------______________________------------ 70. Plate fixed along three edges, deflections-reactions-bending

moments,Load I. a/b=1/4, p=O.2----------- ____ ----- _______

PW 76 77 78 79 80 81 82

83

84

85

86

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 COMPONENTS of many structures may be logically idealized as laterally loaded, rectangular plates or slabs having various conditions of edge support. This monograph presents tables of coefficients which can be used to determine moments and reactions in such structures for various loading conditions ,and for several ratios of lateral dimensions.

The finite difference method was used in the analysis of the structures and in the development of the tables. This method, described in Appendix

II of this monograph, makes possible the analysis of rectangular plates for any of the usual types of edge conditions, and in addition it can readily take into account virtually all types of loading. An inherent disadvantage of the method lies in the great amount of work required in solution of the large number of simultaneous equations to which it gives rise. However, such equations can be readily systematized and solved by an electronic calculator, thus largely offsetting this disadvantage.

Method of Analysis

THE FINITE difference method is based on t,he usual approximate theory for the bending of thin plates subjected to lateral loads.‘* The custom- ary assumptions are made, therefore, with regard to homogeneity, isotropy, conformance with Hooke’s law, and relative magnitudes of deflections, thickness, and lateral dimensions. (See Appendix II.)

Solution by finite differences provides a means of determining a set of deflections for discrete points of a plate subjected to given loading and edge conditions. The deflections are determined in such a manner that the deflection of any point, together with those of certain nearby points, satisfy finite difference relations which correspond to the differential expressions of the usual plate theory. These expressions relate coordinates and deflections to load and edge conditions.

*Numbers in superscript refer to publications in List of References on page 89.

In this study, for each load and ratio of lateral dimensions, deflections were determined at 30 or more grid points by solution of an equal number of simultaneous equations. A relatively closer spacing of points was used in some instances near fixed boundaries t’o attain the desired accuracy in this region of high curvature. For the a/b ratios l/4 and l/8, one and two additional sets, respectively, of five deflections were determmed in the vicinity of the x axis. Owing to the limitations on computer capacity, these deflections were computed by solutions of supplementary sets of 20 equations whose right-hand members were functions of certain of the initially computed deflections as well as of the loads. In each case, the solution of the equations was made through the use of an electronic calculator.

Computations of moments and reactions were made using desk calculators and the appropriate finite difference relations. The finite difference relations used are discussed in Appendix II.

FIGURES 1 through 36 present the results of these studies as tables of dimensionless coefficients for the rectangular components of bending moment and for reactions at the supports. The studies were carried out for the following edge, or boundary, conditions :

Case 1: Plate fixed along three edges and free along the fourth edge.

Case 2: Plate fixed along three edges and hinged along the fourth edge.

Case 3: Plate fixed along one edge, free along the opposite edge, and hinged along the other two edges.

Case 4: Plate fixed along two adjacent edges and free along the other two edges.

Case 5: Plate fixed along four edges. The loads, selected because they are represent-

ative of conditions frequently’ ‘encountered in structures, are :

Load I: Uniform load over the full height of the plate.

Load II: Uniform load over 2/3 the height of the plate.

Load III: Uniform load over l/3 the height of the plate.

Load IV: Uniformly varying load over the full height of the plate.

Load V: Uniformly varying load over 213 the height of the plate.

Load VI: Uniformly varying load over l/3 the height of the plate.

Load VII : Uniformly varying load over l/6 the height of the plate.

Load VIII: Uniform moment along the edge y=b of the plate for Cases 1, 2, and 3.

Load IX: Uniform line load along the free edge of the plate for Cases 1 and 3.

Load X: Uniformly varying load, p=O along y=b/2.

Load XI : Uniformly varying load, p = 0 along x=a/2.

Plates with the following ratios of lateral dimensions, a, to height b, were studied for the first four cases: l/8, l/4, 318, l/2, 314, 1, 312. The analysis was carried out for these cases using Loads I through IX and all dimension ratios, except that Load IX was omitted from Case 2 for obvious reasons, and Loads VIII and IX and the ratio a/b=312 were omitted from Case 4. It will be noted that for the first three cases, which have symmetry about a vertical axis, the dimension a denotes one-half of the plate width, and for the fourth, unsymmetrical case, a 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 bend-

ing moment coeflicients at the center of a uniformly

Effect of Poisson’s Ratio loaded plate for several values of p and for each 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. 375 0. 5 0.625 0. 75 0. 875 1. 0

- 0.0423 - 0.0403 -0.0358 -0.0298 -0. 0235 -0.0177

0.1

-0.0424 -0. 0424 -0.0425 - 0.0407 -0.0411 -0.0415 -0.0367 -0.0376 -0. 0384 -0.0311 -0. 0324 -0.0337 -0.0251 -0.0267 -0. 0283 -0. 0195 -0.0213 -0. 0230

0.2 0.3

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

RESULTS

Moment-: (Coefflctent) (pb’)

Reaction : (Coefftclent) (pb)

POSITIVE SIGN CONVENTION

FIGURE l.-Plate .tixed along three edges, moment and reaction coeficients, Load I, uniform load.

X


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)


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

RESULTS

Y

FIGURE 3.-Plate $xed along three

Moment = (Coefficieni) (pb’)

Reaction = (Coefficient) (pb)

MI v P

--

@

Rx

RV

’ 0 My

tX

W I

POSITIVE SIQN CONVENTION

edges, moment and reaction coeflcients, Load III, l/S uniform load.

I “._.- ..-.- ~..


t

I \ I , . . “T”. .

I I 0 I+ IlId7 I+ on99

I I 1.0 I+ 3177 lt.oe57r;

Moment = (Goefficient)(pb*)

I?eOCflOn = (Coefficient)(pb)

MI I/’ -+- -. I %iiJ

4 RV 0 , ’ Mv +X

WV I


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

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

Moment : (Coefficlent)( pb2.)

Reaction = (Coefflcient)( pb)

1 MI IJP -+-

’ Rx

w

b 0 X

” M,

W


FIGURE B.-Plate Jixed along three edges, moment and reaction coeficients, Load V, .9/S uniformly varying load.

Moment = (Coefficient)ipb’)

Reaction = (Goefficient)( pb )

W. I


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

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

RESULTS

*-- 0 _-_ i _, ---O---~ --- Gee , t

ill.ii -i-, 0

Moment f (Coefficient)(pb*)

Aeoction = (Coefficient)( pb) +X

W I

POSITIVE SISN CONVENTION

FIGURE 7.-Plate jixed along three edges, moment and reaction coejkients, Load VII, l/6 uniformly varying load.

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

f-P

1

Moment : (Coefficient)( M)

Reaction : (Coefficient)($) Al--l . M. J I p --

. I X

FOSITIVE SIQN CCNVENTION

RESULTS

0.2

-~~~ $0036~ +:0056/+.007g/

I-.OiSO I-.0004~*.0004~+.0006~+.0004[+.0001 I-.OOOOI-.o001 1+.0003 +:0010 +.001, +.0022 t.0024

+ 0185 t 0226

~,-.012’,+~~O036~~1-- -’

i.0261 +.0264 +.wg,

+.0064 l .0107 t.0147 +01j4 + 0164

.“..,T -39111 n I n I n I n I n I n

Moment = (GoeffIcient)

Reaction = (Goefflcient)(F)

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

15

W I


moment and reaction coeflcients, Load IX, line load at free edge.

16

a

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

.__- ~-++ ---+I F., I _--- hinged T

IIriIll a

i ---- - -X

Moment = (Coefficient)(pb*)

Reoctiin = (Coefficicnt)( pb)

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.


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.

Moment = (Coefficient)(pb’)

Aeoction = (Coefficient)( pb)

RESULTS

0

Moment = (Coefficirnt)(pb*)

Reaction = (Coefficicnt)( pb ) X


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

-.--_._- _-.._-.-- --..--.


Moment = (Coefficient)(pff)

Reaction = (Coefficient)( pb ) +-X


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

.._.. .- ..-.---..- -.--

0

Moment = (Coefficient)(pb*)

Reaction = (Coefficient)( pb)


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


Moment = (Goefficient)(pb?



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

RESULTS 23

0

Moment = (Coefficient)( M )

Reaction = (Coefficient)($),


FIGURE 17.-Plate $xed along three edges-Hinged along one edge, moment and reaction coeficiente, Load VIII, moment hinged edge.

at


Y

~+--*--+* ._- --_

0

-f

I I

0

L-x

FIQURE 18.-Plate fixed along one edge-Hinged along two

Moment = (Coefficlent) (pb’)

Reaction = (Coefficient) (pb )


opposite load.

edges, and reaction coefkients, Load I, uniform

25

FIG

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

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

Moment = (Coefficient) (pb’)

Reoctlon = (GoeffIcIent) (pb) f&l M. I/ -+-

’ R. ̂ .


E lg.-Plate fixed along one edge-Hinged along two opposite edges, moment and reaction coeficients, Load II, uniform load.


Moment : (Coefficient) (pb*)

Reaction = (Coefficient) (pb )

Y

MI v P

--

@

’ h

fb

’ I My W

tX


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

l/S

RESULTS

0

Moment = (Coefflclent) (pb’)

Reoctaon = (Coeffxlent) (pb)

Ma IA -+-

d&iJ- ’ R. Rv X

I MY

W.


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

- .-._- __--


--_ f

Moment = (caefflclent)(pb’)

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

W I


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

29

FI :GuRE 23.-Plate fixed along

Moment = (Coefficlent)(pb*)


along two opposite edges, uniformly varying load.

moment and reaction coejicients, Load VI, 11s



Moment = (Coofficiant)( pb’)

heOCtiOn = (Coefflcient)( pb)

i

W


FIGURE 24.-Plate fized along one edge-Hinged along two opposite edges, moment and reaction coeficients, Load VII, l/6

uniformly varying load.

0

FIG

Moment - (Coefflcient)( M)

Reaction = (Coeffxwnt)(j-1


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

tt

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

32

FIGURE 26.-Plate fixed along one

Moment = (Coefftcient)( Fb)

Reoctlon : (Coefficient)( F )

along two opposite edges, load at free edge.

and reaction coqjicients, Load IX, 1 ine


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 -

Rx \-.0401 t.0011 +.I576 t.3024 +.5696 l .9739

I.0 +.6290+. 1977 +.0952 t.0296 -.0059 ~0162 0 0 0 0 0

0 6.

0 0 __.. +.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 ,

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

I

- - - . - 0 .--- .+. I Moment = (GoefficIent) (pb*)

Reaction : (Coefficient) (pb)

I 04~~~/~~~/~~~~J

--Y- x p4 W

;p& POSITIVE SIGN CONVENTION

FIGURE 27.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load I, uniform load.


1 ! 0.6 j 0.6 1 I. 0 0.2 0.4 0.6 0.6 1.0 0 0.2 0.d I 1 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-.0822 Ra - 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

,Ib

0.4 +.3352 t.0384 +.0165 t.0063 - 0003 -.0022

[~~+:I&]+ .%$,@7

L .- 1 ! 0 !+.0077~~.00241- 0~~-.00~6~~.00~6~/~.0ll6 1

-.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

Moment = (Coefficient) (pb*)


Y


FIGURE 29.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load III, l/S uniform load.


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

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

I l .0056 -.0008 .o I

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

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

i? 0.6 0.8 +.2067 +.2303 +.0467 +.0459 +.0242 +.0195 +.0046 +,0064 -.0002 -.0023 -.0027 -.0036 0 +.0097 +.0040

-

0004 0 +.0092 +.oozo -.0044 -.0093 - 0129

I, 0.4 +.2363 t.0360 +.Ol2l +.OOlO -.0030 -.0029 0 +.0072 -.0003 -.0060 -.0096 -.Oll7

1 p 1 0.2 ~*.1193~+.01~0~+.0042~+.0005~+.0006(+.00l6 I 0 I, I I ~~ I I

+,0032 +.oooa +.0019 +.0050 +.ooe3 +.01oLl

-.0194 0 +.0029 +.0070 +.0110 +.014a 0 0 +.0147 +.0352 l .0546 + 0740 t.0696

I R” n v-.0194~+.1105~+.2399(+.32: 561+.44891+.5505i I ----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

c 0-A

Moment = (Coefficient) (pb*)

Reoctton = (Coefftcient) ( pb)


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

RESULTS 37

I Mx MY

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

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

2 0. e +.0001 +.0001 +.0001 +.0001 +.0001 +.oooo o - +.oooo +.oooo +.oooo +.0001 +.0001 +.0001

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

s II s

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 -- -~-.__-..- _.-..

0.6 . +.0565 l .0099 + 0059 +.0029 t 0010 -.OOOO 0 t.0020 +.0012 +.0005 -.OOOl -.0006 -.0009

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

0 +.Ol25 0 +.oooe +.0019 +.0031 +.0043 0 0 +.0038 +.0097 t.0156 +.0213 + 0257

WY +.0125 +.0460 +.I236 +.I740 +.25381+.3159 I?.

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

WY +.0050 +.0755 +.I616 +.2132 +.2673 +.3382 R.

- II

\” 0

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

0.8 +.0667 t.0207 +.0069 +0017

0.6 +.0943 +.0209 +.0064 1:0004

0.4 +.15;2 +.OI88 +:0026 -.0024

0.2 +.I166 +.0096 +.OOOO -.OOlZ -.0003 +.0008 0 +.ool9 -.0016 -.0008 +.0016 l .004l +.0059 -

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*)


W. I

POSITIVE S ION CONVENTION

edges, moment and reaction coefkients, Load V, S/S unijormly varying load. FIQ URE 31.-Plate fixed along two adjacent

38

ho ,URE 32.-Pkztejixed along


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~+.0001 I-.( I I 1 I 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

Moment : (Coefficient) (pd)


edges, moment and reaction coeficients, Load VI, i/3 uniformly varying locr

Y

A

M. IA -+-

@

’ R, 5

0

’ MY W.

-3-X


rd.

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

9 0. 8 -.oooo +.oooo t.OOOO +.oooo +.oooo t.oooo 0 +.oooo +..oooo +.oooo +.oooo +.oooo +.oooo 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

Moment q (Goefficlenf)(pb?

Jeoction : (Coefftclent) (pb)

Y

M, IA -+-

45J-

’ R” RY 0 I X

MY

W


FIGURE 33.-Plate fixed along two adjacent edges, moment and reaction coeficients, Load VII, l/6 unijownly varying load.


ieo-rl POSITIVE SIQN CONVENTION

FIGURE 34.-Plate $zed along four edgeqmcnnent and reaction coeflcients, Load I, uniform load.

Moment = (Coefficient)(po*)

Reaction = (Coefficient)(po)

FIG m 35.-Plate fixed along four edges, moment

Moment = (Coefficient)(po*)

Reoctmn = (Coefficaent)( po)

and reaction coeficients, y= b/2.

Load

X

POSITIVE SISN CONVENTION

uniformly varying load, p=O al0 lng

_--_---._-_-


Moment = (Coefflcient)( pa*)

Reaction = (Coefficient)(po)


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 approximate. A factor directly affecting its accuracy is the closeness of spacing, hence the number, of grid points. In obtaining the solutions presented in this monograph, a maximum number of points was used, consistent with the objectives of the study and the capacity of the available electronic calculator.

A few instances may be found where there appear to be irregularities in the orderly progres- sion of the coefficients as the ratio a/b changes. Such instances are most likely to occur in the 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- sistencies are undesirable from an academic standpoint, they are not of sufficient magnitude to affect materially the usefulness of the results.

As a general check on the finite difference method, problems for which “exact” solutions are known have been computed. The results indicate that for spacings comparable to those used in this study, errors in the maximum moments may be of the order of five percent. Such accuracy is

considered to be satisfactory for design purposes. Percentage errors for small numerical values of the coefficients may, of course, be somewhat higher.

For Case 5 a comparison is given in Table 2

TABLE 2.-Comparison of Coeficients of Maximzcm Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed Along Four Edges

Valuar of M./pa* from

b/a Timoshenko 1 Method of this

Monograph 2 -

1. 1 - 0.0264 - 0.0269 1.2 - 0.0299 - 0.0301 1.3 - 0.0327 - 0.0329 1.4 - 0.0349 - 0.0352 1.6 - 0.0381 - 0.0384 1.7 - 0.0392 - 0.0395 1. 8 - 0.0401 - 0.0404 1. 9 - 0.0407 -0.0410

1 These values taken directly from page 223, Reference 1, with due regard for difference in sign conventions.

2 These values interpolated from the column for p=O.3 of the preceding table.

43


between values found on page 228 of Reference 1 and directly equivalent values obtained by the method of this monograph. In this particular case, the relative differences are, for the most part, less than one percent.

Comparisons have also been made with other existing results 2 for full uniformly varying load and certain ratios of a/b. These indicated very

good agreement. All coefficients have been computed to four

decimal places for consistency and to indicate significant figures for many conditions which would have no significance to three decimal places. This should not be taken as an indication that the percentage accuracy is greater than no ted above.

Appendix I

An Application to a Design Problem

THIS appendix illustrates use of the tabulated coefficients by an application to a typical design problem. Figure 37 shows essential dimensions and typical loads acting on an interior panel of a counterfort retaining wall. Both wall and heel slabs approximate the condition of a plate fixed along three edges and free along the fourth. The variations in thickness of the wall slab and the relatively great thickness of the heel slab compared with its lesser lateral dimension are both, perhaps to some degree, in violation of basic assumptions. Ignoring these, however, is done with the conviction that results obtained in this manner are more nearly correct than what might be determined by other available methods.

Center line dimensions have been used for both slabs. The net loads, as determined from equi-

librium conditions, have been broken into components similar to certain of the typical Loads I through XI. These are illustrated together with a table of their numerical values in Figure 37.

It will be noted that for the wall slab, r=a/b= 0.2. This requires interpolation on r for the various loads and in the case of pB, interpolation both on r and the load. For the heel slab, r=a/b=1/2, and since both component loads act over the full area, no interpolation is required.

For illustrative purposes, moments have been computed along the assumed lines of support for both the wall and heel slabs. Where interpolation was required to obtain the moment coefficients, second degree interpolation was used. The moment coe5cients and actual computed moments are given in Tables 3 through 6.

45


FRONT ELEVATION

DESIGN DATA

Unit Weights Concrete 150 Ib/ft3 Moist earth Saturated earth

120 Ib/ft3 135 lb/+

Water 62.4 Ib/ft3 Surcharge Pressures

Vertical 360 Ib/fte Horizontal

Equivalent Fluid Weights 120 Ib/ft’

Moist earth 40 Ib/ft3 Saturated earth 75 Ib/ft’

Water surface elevatiorv’t

--- -- ! -

, END ELEVATION

COUNTERFORT RETAINING WALL DIMENSIONS AND TYPICAL DESIGN LOADS

L pw -H pq-A i.6

WATER LOAD SURCHARGE LOAD EARTH LOAD

COUNTERFORT WALL SLAB - INTERIOR PANEL IDEALIZED DIMENSIONS AND COMPONENT LOADS

Ffq+/iii??j NET LOAD ON w----pu ----H ~---p”----~

HEEL SLAG COMPONENT LOADS

COUNTERFORT HEEL SLAB - INTERIOR PANEL IDEALIZE0 DIMENSIONS AND COMPONENT LOADS

FIGURE 37.-Counterfort wall, design example.

I

1L i-- ps-4

PORE PRESSURE LOAD

.- .--- ---- -_-..

APPENDIX I

TABLE 3.-M. for Heel Slab at Supports

47

T - Moment coefficients I- Values of pb2+ Moments (foot-kips) -

--

-

Total moment (foot-kips)

1118.5 --

x a

-Y b PU

0 1.0 + 0.0852 0 0. 8 + 0.0807 0 0.6 $0.0712 0 0.4 + 0.0545 0 0. 2 + 0.0250 0 0 0 0. 2 0 $0.0019 0.4 0 + 0.0050 0.6 0 + 0.0080 0.8 0 +o. 0100 1.0 0 $0.0107

-1032.3

PP

q-o.0151 $0.0216 + 0.0273 + 0.0277 $0.0160

0 +o. 0014 + 0.0033 + 0.0050 $0.0061 + 0.0065

-

-_

-

M" M” --

+ 95.30 - 15.59 $90.26 -22.30 +79.64 -28.18 + 60.96 - 28.59 $27.96 - 16.52

0 0 $2.13 -1.45 +5.59 -3.41 +8.95 -5.16

$11.18 -6.30 $11.97 -6.71

-

--

+79.7 +68.0 +51.5 $32.4 $11.4

0 $0.7 +2. 2 +3.8 $4.9 f5.3

TABLE 4.-M, for Heel Slab at Supports - I

- Moment coefficients

Velues of pbb Moments (foot-kips) -

--

--

-

--

--

-

Total moment (foot-kips)

-1032.3

PV

0 $0. 0043 $0. 0055 + 0.0055 +O. 0032

0 +o. 0068 $0. 0167 + 0.0252 $0. 0307 + 0.0325

1118.5

x -ii

s b

Pu

0 1. 0 0 0 0. 8 + 0.0161 0 0. 6 +O. 0142 0 0. 4 +o. 0109 0 0. 2 +o. 0050 0 0 0 0. 2 0 + 0.0094 0. 4 0 +O. 0252 0, 6 0 + 0.0399 0. 8 0 $0. 0499 1. 0 0 + 0.0534

M.

0 + 18. 01 + 15. 88 $12. 19

f5.59 0

+ 10. 51 f28. 19 + 44.63 t-55. 81 f59. 73

M,

0 0 -4.44 $13. 6 -5.68 +10.2 -5.68 +6. 5 -3.30 +a. 3

0 0 -7.02 $3. 5

-17. 24 +11.0 -26. 01 $18. 6 -31. 69 +24. 1 -33. 55 +26.2


TABLE 5.-M. for Wall Slab at Supports

T T -

--

-

Moment we&icients M0ment.Y (foot-kips) -i- -7 Total

moment (foot-kips)

--

f5. 0 +9. 1

+17.1 +25. 7 +24.0

0 +O. 8 +l. 8 +3. 2 +4. 0 -k4. 2

-

_-

_-

-

-985.5

PW

- 0.0000 +o. 0000

+ 0.0000

+o. 0009 +O. 0032

0 +o. 0002 +o. 0005 +o. 0007 +o. 0009 +o. 0010

157.7 --

Pa

+o. 0133 +o. 0131 $0.0134 +o. 0133 +o. 0103

0 +o. 0003 +o. 0009 +o. 0013 +O. 0016 +o 0018

1905.4

PO

--

+0.0012 +o. 0028 + 0.0054 +o. 0079 + 0.0079

0 +o. 0003 + 0.0007 +o. 0011 +o. 0014 +o. 0015

1399.9

P8

+ 0.0004 +o. 0012 +o. 0034 + 0.0068 + 0.0075

0 +o. 0003 +O. 0006 +o. 0011 +o. 0014 +o 0015

-- * s

0 0 0 0 0 0 0. 2 0. 4 0. 6 0. 8 1. 0

-

--

-

-_

_-

-

-.

--

-

-_

--

-

-

_-

-

-

--

-

-- M.

+2.29 +5.34

+ 10. 29 + 15.05 + 15.05

+“o. 57 +1.33 +2. 10 +2. 67 +2. 86

-

-_

-

Y b

1. 0 0. 8 0. 6 0. 4 0. 2 0 0 0 0 0 0

M”

+o. 00 -0.00 -0.00 -0. 89 -3. 15

0 -0. 20 -0.49 -0.69 -0. 89 -0.99

M.

$2. 10 s2.07 f2.11 +2. 10 $1. 62

0 +o. 05 i-0. 14 4-o. 21 -l-O. 25 +O. 28

M.

$0.56 +l. 67 $4.74 f9.47

+ 10. 45 0

+O. 42 +O. 84 +1.53 -I- 1. 95 +2.09

TABLE 6.-M, for Wall Slab at Supports

T Moment coefficients Moments (foot-kips)

Values of pbz-1 Total

moment (foot-kips)

-

--

--

-985.5 157.7

-

_-

1905.4 1392.9

?! b

PW PS PW

--

-- PO

_- -- --

1. 0 0 0 0 0 0. 8 -0.0000 +O. 0026 +o. 0005 +o. 0002 0. 6 $0.0000 +O. 0027 +o. 0011 +o. 0007 0. 4 +o. 0002 $0.0026 +O. 0016 +o. 0014 0. 2 +O. 0006 +o. 0020 +O. 0016 +o. 0015 0 0 0 0 0 0 +o. 0011 4-O. 0015 + 0.0014 + 0.0014 0 +O. 0025 +o. 0041 +O. 0036 +O. 0036 0 + 0.0036 -i-o. 0066 +O. 0056 +o. 0055 0 + 0.0043 +o. 0082 + 0.0069 + 0.0068 0 +O. 0046 $0.0088 +o. 0074 +O. 0072

- -

-

_-

-

--

--

-

-

_-

-

-

_-

-

-

_-

-

x a

0 0 0 0 0 0 0. 2 0. 4 0. 6 0. 8 1. 0

M”

+:. 00

-0.00 -0. 20 -0.59

0 -1.08 -2. 46 -3.55 -4.24 -4.53

MQ

0 +o. 41 +o. 43 +o. 41 +O. 32

0 +O. 24 +O. 65 fl. 04 +1.29 +1.39

M.

0 +o. 95 +2.10 +3.05 +3.05

0 +2.67 +6. 86

+ 10. 67 +13.15 + 14. 10

M,

--

0 0 +O. 28 +l. 6 +O. 98 +3. 5 +1.95 +5. 2 +2.09 +4. 9

0 0 +1.95 +3. 8 +5.01 +10.1 f7.66 +15. 8 +9.47 +19.7

+ 10.03 +21.0

-

Appendix II

The Finite Difference Method

Introduction

The bending of thin elastic plates or slabs subjected to loads normal to their surfaces has been studied by many investigators.’ through e A large number of specific problems have been solved by exact or approximate means, and these results are available. (See, for instance,3.) Exact and certain approximate methods are frequently difficult to apply except to structures where some symmetry exists and where a simple loading is used. The finite difference method, however, is readily adaptable to rectangular plates having any of the usual edge conditions and subjected to any loading.

In Denmark, as early as 1918, N. J. Nielsen applied the finite difference method to the solution of plate problems. In his book 4 he has analyzed t,he problem in considerable detail and has given numerical solutions for a number of cases. H. Marcus published an excellent book 5 in Germany in 1924 on this subject in which he included numerous examples. In the United States, Wise, Holl, and Barton u.7 a have contributed to the literature of finite difference solutions for

rectangular plates, and Jensen e has extended the method to provide a useful tool in the analysis of skew slabs.

General Mathematical Relations

The partial differential equation, frequently called Lagrange’s equation, which relates the rectangular coordinates, the load, the deflections, and the physical and elastic constants of a laterally loaded plate, is well known. Its application to the solution of problems of bending of plates or slabs is justified if the following conditions are met: (a) the plate or slab is composed of material which may be assumed to be homogeneous, isotropic, and elastic; (b) the plate is of /a uniform thickness which is small as compared with its lateral dimensions; (c) the deflections of the loaded plate are small as compared with its thickness. The additional differential expressions relating the deflections to the boundary conditions, moments, and shears are perhaps equally well known. (See, for instance,‘.) They will therefore only be stated here, using the notation and sign convention shown in Figure 38.

49


(0) INTERIOR POINT

P Q, b

h r Y

Z,N,E,...NE:ki

n,e,s ,... SW,“W

“3 t Ml, MY

by * Myx VI, VY

‘?I, Ry

P R E I

P D

v’

(bl SUB-DIVIDED GRID

GRID POINT DESIGNATION SYSTEM

0 I : x

(C) POSITIVE SIGN CONVENTION

Intensity of pressure, normal to the plane of the plate. Lateral dimensions of the plate. Loterol dimension in the y direction of the grid elements of the plate. Ratio of lateral dimensions of the grid elements. Deflection of the middle surface of the plate, normal to the XOY plane. Rectangular coordinates in the plone of the plate. Designotion of active grid points. Also used to represent the value

of the deflection of the plate ot the point so lettered. Designation of additional points on sub-divided grid. Subscripts used to indicate directions normal ond tangential to on edge. Bending moment per unit length acting on planes perpendicular to the x and y axes respectively. Twisting moment Ser unit length in planes perpendicular to the x ond y axes respectively. Shearing force per unit length acting normal to the plane of the plate, in planes normal to

the x and y axes respectively. Shearing reactions per unit length acting normal to the plane of the plate, in planes normal to

the x ond y axes respectively. Concentrated load acting at o grid point: positive in the some direction OS D. Concentrated reaction acting at 0 supported grid point; positive direction opposite to thot of p Young’s modulus for the material of the plate. Moment of inertia per unit length of o section of the plate. Poisson’s ratio for the material of the plate. Flexural rigidity per unit length of the plate; 0 = EI/(I-$1.

Difference quotient operator: V’w = +- + 2 & + $.

NOTATION

FIGURE 38.-Grid point designation system and notation.

APPENDIX II 51

Partial differential equation:

Fixed edge conditions :

w=o,

bW T&=0.

Hinged edge conditions :

w=o,

$+p s?&J

Free edge conditions :

Free corner conditions :

g=O (both directions) J

g+h4 s =0 (both directions),

d2W -- bndt-”

Bending moments :

M =D Y b2W~~d2~ by2 3x2 * 1

Twisting moments :

(1)

(2.01)

(2.02)

(3.01)

(3.02)

(4.01)

(4.02)

(5.01)

(5.02)

(5.03)

(6.01)

(6.02)

(7)

Shears : V =

x - D (8.01)

V E-D Y b3”+ b3W bY3 bx2by . 1 (8.02)

In the above expressions the partial derivatives

with respect to n indicate rates of change in a direction normal to the edge, and those with respect to t indicate rates of change tangential to the edge.

A solution to any specific problem consists of determining a deflection surface which satisfies the basic equation (l), and the appropriate sets of boundary conditions (2.01) through (5.03). The moments and shears required for design purposes may then be computed from (6.01) through (8.02).

In general, it is difficult to obtain an analytical expression for a deflection surface which satisfies all of these conditions. If, however, an approximate solution is acceptable, it is always possible in analyzing a rectangular plate to determine a set of deflections for a finite number of discrete points such that approximate relations corresponding to (1) through (5.03) are satisfied. From these deflections it is possible to compute moments, reactions, and shears at the selected points, using relations similar to (6.01) through (8.02).

The approximate relations referred to above are obtained by replacing the partial derivatives by corresponding finite difference quotients. Such relations are simplest if the discrete points determined by values of the independent variables are equally spaced with respect to both variables. However, in this application it will be advan- tageous for the relations to be developed on the more general basis of having the equal spacing in one coordinate direction bear a given ratio to the spacing in the perpendicular direction.

Figure 38(a) represents a portion of the interior of a plate subdivided by grid lines into rectangular grid elements. The grid lines are spaced h units apart in the y direction and rh units apart in the x direction. The int,ersections of the grid lines will be referred to as grid points. Certain of these, lettered for identification, will be spoken of as active points, and the central point of the active group will be called the focal point. For simplicity in writing the equations, the identifying letters for each active point %ill also be used to represent the value of the deflection, w, of the middle surface of the plate at that point. The double letters refer in every case to the deflection at the individual point so lettered; they do not indicate products of deflections at points designated by only one letter.


Based on the usual methods of finite differ- -4(l+r’)(E+W)--4r2(1+r2)(N+S) ences,” the difference quotient relations required in this development can be written directly and

+2(3+4r2+31d)Z]. (10)

are given below. All of the difference quotients are given with reference to the focal point, lettered

This may be considered as an operator, and the

Z. portion within the brackets can be conveniently

Aw 1 portrayed as an array of coefficients. This expres-

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

-- (EE--2E+2W-WW), (9.03) coefficients in the array is congruent to the physical

s---2?h3 locations of the points and the heavily outlined

A4w 1 coefficient applies at the focal point-the point

-== (EE-4E+6Z--4W+WW), (9.04) for which the relation is to be determined. Ax4 Since the solution deals with discrete points,

g=$ (N-S), the distributed load intensity p in the right-hand

(9.05) member of (1) is replaced by an average intensity P/rh” at each of the interior grid points. Here P

$=$ (N--2Z+S), (9.06) represents a concentrated load whose magnitude at any grid point is a function of the distribution of p on the four adjoining grid elements. If each

A3w 1 e=2h3 (NN-2N+S-SS), (9.07)

of these elements is considered as an infinitely rigid plate supported at its four corners, then the

‘$=; (NN-4N+6Z-4S+SS), (9.08)

A%V -=-1 (NE-NW+SIW-SE), (9.09) AxAy 4rh2

A3w 1 L\X2ay=2r2h3

- (NE-2N+NW-&+2S-SW),

(9.10) A3w

-=k3 (NE-2E+SE-NW-b-2W-SW),

(9.11) A%

7=&4 (NE-2E+SE-2N Ax Ay’

+4Z---2S+NW--2W+SW). (9.12)

force Pz, at the focal point, is equal in magnitude and opposite in direction to the sum of the reactions at all corners common to Z. This can be expressed mathematically as :

p,=p,,,+p,s,+p,,,+p~~~ (11)

in which PZNE represents the contribution from the grid element Z-N-NE-E and similarly for the other right-hand members. Thus it is seen that the concentrated loads Pz are the static equivalent of p.

It can be shown, if p varies linearly-a usual condition for structures-and if this variation is constant over the four grid elements adjoining

The approximate counterparts of the basic any focal point Z, that the magnitude of the

relations (1) through (8.02) may now be written. statically equivalent average load is:

For instance if V4w is used to represent the difference quotient equivalent to the left-hand member of equation (l), and the partial derivatives are replaced by their corresponding difference quotients, (9.04)) (9.08), and (9.12)) there results:

v’w=& [EE+WW+r4(NN+SS)

+2r2(NE+SE+SW+NW)

Pz/rh2=(1/6)(p~+pE+Ps+p~S2P~), (12)

where pN represents the intensity of p at point N, etc.

The approximate counterpart of (1) may now be written:

PZ v4y=m2* (13)

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:

$ [EE+WW+r4(NN+SS)

+2r2(NE+SE+SW+NW)

-40 +3(E+W)--4r2(1 +r%N+S)

+2(3+4ra+3r4)Zl=~z g* (14)

This is the general load-deflection relation for an interior point. It is written at (a) of Figure 39 in the convenient array form previously described. This general form of the equations has been used for the special cases which include the boundary conditions and, in fact, for all of the relations connecting the deflections with load, moments, reactions, and shears. These load-deflection equations establish a linear relation between the load at the focal point and the unknown deflections of the plate at that and the other active grid points. It is these linear equations which are to be solved simultaneously to determine the approximate deflections of the plate at the grid points.

Equation (14) may be derived directly by a second method which considers equilibrium of certain elements of the plate. Referring to the subdivided grid of Figure 38(b), consider the rectangular element ne-se-sw-nw with center at Z. Equilibrium of forces normal to the plate requires that

(V.,-V.,)h+(V,,-V,,)rh+Pz=O. (15)

For the similar element with center at e, equilibrium of moments about the center line ne-se requires that

(Mxz -M&+ (M,,B-M,.,Jrh

+w.,+vx,> r;=o.

However, if the elements are sufficiently small,

f (v.,+v.,)

may be replaced with VXe so that

ME---M.&+ OLne -M,.,,)rh+V.~rh2=0.

(16.01)

W.,----M,)h+ Wrxnna -M,.,,)rh+V+rh2=0,

(16.02)

Of,, ---M&h+ (MxYne --Mxy,,)h+V,,,rh2=0,

(16.03)

(M,,--M,s)rh+(M,,,e-M,,,)h+V,,rh2=0.

(16.04)

If equations (15) and (16.01) through (16.04) are combined to eliminate the shears, noting at the same time that MIY=MYX, there results

; CM,, --2M,,+M,,)+2(M,,,e--M,,,,+M,,,

--MxYBJ +r(M,,--2M,,+M,,) =Pe. (17)

An approximation to each moment in terms of deflections is obtained if the partial differentials of the definitions (6.01); (6.02), and (7) are replaced by their proper difference quotients corresponding to (9.02), (9.06), and (9.09). For instance,

M.,=-& [E-2Z+W+Lcr*(N-2Z+S)] (18)

and

M ‘une W-P) =--- [NE-N+Z--El. rh2 (19)

Substituting these and corresponding relations for the other moments into (l7), and multiplying both sides by h2/rD gives

f (WW-4W+GZ--4E+EE)+$ (NW-2N

+NE-2W+4Z-2E+SW-2S+SE)

+(NN--4N+6Z--4S+SS)=s

which, with some rearrangement, is the same as (14). This second method is easily adapted to de-

riving expressions involving nonuniform spacings, moment-free boundaries, etc. It was applied to obtain all of the load-deflection arrays shown in Figures 39 through 59, which were required in the solution of the problems covered by this monograph.


Where boundary conditions involve a reaction, the load P may be replaced by the net load, (P-R), which is the difference between load and reaction. Note that R represents a concentrated force whose positive direction is opposite to that of p, R. and R,, on the other hand, represent intensities of shearing reactions whose positive directions conform to V, and V,.

Relations connecting the deflections with moments and with shears are given in Figures 60 through 64. It should be noted that shears computed by finite difference methods are inherently less accurate than moments. This is because the shears are functions of odd numbered difference quotients which are determined by a grid spacing double the value found in the even numbered quotients which define the moments.

ApplicaEion to Plate Fixed Along Three &?ges and Free Along llie Fourth

As an example of the use of this general method, its application to the problem of a plate fixed along three edges and free along the fourth is given below. The a/b ratio of l/4 has been used to illustrate use of the 20 supplementary equations. Loads I, II, and IV only are included.

The plate is divided into grid elements and the grid points numbered systematically for identification. Layout of Plate, Figure 66, shows the method used in this case. Because of symmetry of the plate and loading about the line x=a, points which are symmetrical about this line will have equal deflections and are, therefore, numbered alike. This reduces considerably the number of unknown deflections to be determined.

With r=l/4 and p=O.2, the left-hand side of each of the loaddeflection relations yields an array of numerical coefficients corresponding to the type of point it represents. These values have been computed for typical points and they are shown in Figure 65. They are used in writing the left- hand members of the simultaneous equations. Solution of these equations determines the deflections.

One equation must be written for each grid point having an unknown deflection. The equation corresponding to any point is formed as follows :

a. Select the array of load-deflection coefficients having edge conditions and

spacings which correspond to those of the given point.

b. Orient the focal point of this array at the given point.

c. Multiply the unknown which represents the deflection of each active grid point by the corresponding coe5cient.

d. Equate the sum of these products to the load term for the given point.

For example, for Point 45 the array at (b) of Figure 65 must be used in order that the free edges correspond properly. Then, following the pro- cedure outlined above, the left-hand member of the equation for Point 45 is

+256wpI,+32wg,- 1088wa~+28.&Jw,,+w,, -68~,,+(1669+256)w,h--59.6~~

+32wM- 1088w,+28.8wM.

Noting that RZ=O along the free edge it is seen that in this case the general expression for the right-hand terms is always (P&h*) (h’/D). Since these load terms are to be expressed as coefficients of ph’/D, it remains to evaluate the Pz/rhg in terms of p for each point and each loading. At Point 45 the right-hand members for Loads I and IV may be obtained by direct application of (12). However, a discontinuity occurs in the magnitude of Load II within the grid elements adjoining Point 45. For this reason, the more general method expressed by (11) must be employed.

In particular for Load II, the elements 45-35- 36-46 and 4546-56-55 carry no load, and accord- ingly they make no contribution to P,. The elements 45-44-34-35 and 45-55-5444 each carry an equal portion of the uniform load. Under the assumptions leading to (11) it is found, by statics, that the contribution of each of these elements to P,, is ph*/144. Hence, P,,=ph*/72 and P&h*=p/18.

The complete set of 30 equations and the right- hand (load) terms are shown as two matrices in Figure 66. Simultaneous solution of the equations establishes a set of deflections for each of the 30 grid points, corresponding to each load. These results are tabulated in the upper portion of Figure 67.

The 20 supplementary equations used to determine the deflections of the row of points at y=ih

are set up in a similar manner. Equations are

APPENDIX II 55

written for each point of the 3-, 2-, l-, and 7-rows (see Figure 68). However, in writing equations for the 3- and 2-rows use is made of the previously computed deflections for the 4- and 5-rows. In addition, the solution of the 20 equations gives new and improved values of deflections for the 3-, 2-, and l-rows. For Point 42, for example, the array (f) of Figure 65 is used to conform with the spacing of the grid points involved. The equation for Load I is

Substituting numerical values for PsO and the various deflections, this becomes

R3,,=0.03125ph2+ e (h2) (g)

[--(32)(0.004944)-(16)(0.021325)

+(128)(0.007860)-(32)(0.009833)]

=(0.03125+0.192016)ph2=0.223266ph2.

-28w21+21Owzz+ low,,+ 176~31-936~~~

5057 -SW,,+? ~~,-364w~~+~ w42

This represents a concentrated force acting at Point 30. Assuming that it is uniformly distributed over a distance rh, it can be expressed ‘as an average shearing reaction per unit length

+ 176w51- 3 ph4

936w&w~=4 D-~44. R,,,=R3&h=0.893064ph,

Substituting for Point 44, its deflection as determined from the 30 equations gives, for the right-hand member

or in terms of b

R,,,=O.l78613pb,

(0.75-0.100572) ‘;=0.649428 ‘;.

The complete set of 20 equations for Loads I, II, and IV is given in Figure 68. Solution of these 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 approach closely for the points where y/b=O.4.

which is in the units used in Figures 1 through 33. Similarly, for example, the bending moment

M, at Point 23 is computed using array (g) of Figure 69. Thus

Again inserting numerical values

Having determined the deflections, reactions and moments may be computed by operating upon the deflections with the appropriate relations, typical samples of which are given in Figure 69. These numerical arrays were obtained similarly to those for the load-deflection relations, by inserting numerical values for r and p in the proper general expressions of the referenced figures.

Mx23=(;) @) [(16)(0.015283)

t-(0.2)(0.029914)-(32.4)(0.043935)

+(0.2)(0.046526)+(16)(0.073156)]

=0.006818ph2=0.000273pb2.

To illustrate the method of computation of reactions and moments, an example of each (Load I, a/b=l/4) is given below. At Point 30, for instance, using array (f) of Figure 69, the reaction is :

Upon completion of computation of the reactions, a partial check of the solution may be obtained from equilibrium considerations. For Load I, a/b=1/4, the total load on one-half of the plate is p(5h)(5h/4)=6.25 ph2. The 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.


+ r4

+I

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

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

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

+ r4

(0) INTERIOR POINT

I + r4 I

(b) POINT ADJACENT TO A FIXED X-EDGE

= P h4 rhe T’

/ + r4

/ / + /

-4rt- 4r4 + 2 r*

=

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

(C) POINT ADJACENT TO A FIXED Y-EDGE

+ 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

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

= P h’ -- rh2 D

P h’ rhe -6’

P h’ -z-b’

=

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

++ =

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

+r4

+ 20 -40 - 4r4 +(2-p)r*

-2-212-p) 0 = P h’ -.

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.


I + r4 I

(0) CONANT-FREE X-EDGE

+ I + 4(1 (P-R) h’ = - 0’ r h*

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

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

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

= (P-R) h4

rh* -D--’

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

(0) POINT ON A FIXED X-EDGE

ADJACENT TO A FIXED CORNER

-l.,;,,,fi/,z ~ (0) POINT ON A FIXED X-EDGE ADJACENT

TO A MOMENT-FREE Y-EDGE

!!I, D

* + +2r2

L r4 k * -4-4r* +I

9 *

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

= (P-RI

r h2

h’ T-’

(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.


i P h

T h

I +r’ I + 2 rp - 4 rz - 4 r4 + 2re

I -P

P h’ = 37 T’

I 7

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

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

(0) INTERIOR POINT

I + r4 I

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

P h’ = X0’

,+ 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

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

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

s +128 -&+Zr*

I-& 1+$+4r’

+ r4 I

= P h' - -. rh' 0

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

(a) INTERIOR POINT

+ r4

I +8r4 I

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


+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.


%

i $‘h

f f ‘h k

7 I I -64 +&+4+ -&. I

t3rt-40r4 I

+&+4rr

I + 64 r4

3 I

5 +128

IJ +105

128 = L&+.

7 -64

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

(a) INTERIOR POINT

-& 7 35 +=+4r= -z - are-40r4 +&+2(2-p)r*

+64 r4 3

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


= PA!. rh2 D

+ + (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

x % i,h * +‘h

+ +‘h P

+4rP -are- 32r4 +4re

I +6r4 I

P h’ = --. rh* D

(a) INTERIOR POINT

I +$ r4 I

+4rp -Ore- 32r’ +2(2 -p)r*

+0r4

= I' h4 --. rh* D

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


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

I

7 +t -I -4(2-pL)r’ ++-+ 0(1 -p)r*

+ 661,-pcljr4

+2(2-p)r* -~-p)r*-i6(1-pe)r’

I +4(1 -p*)r’

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

(P-R) h4 = --. rh' 0


NOTE.-For general notes see Figure 39.

FIGURE 46.-Load-deflection relations, vertical spacing: 1 at h; 3 at h/B, Sheet VIII.

64

I 105 +256

- Lg -,2p 496 ++$+24r’+-j-r

. - _ 105 =-,2rg +105 = +A!.. 256 rh 0 r’

-Tk +&+gre -g- 16rL - 192r4 + &-+ 0r r --A

b--rh--e --___ rh ----+ ______ rh----++--rh--4

(a) INTERIOR POINT

I ++r . I 5

+zz

105 +isd

-6

-&+4rg I

25 +i56 - 6r’- 4oP

I -& +2(2-pL)r

I F

= Lx.. rh’ 0


+ f r’

+&+Br ?. -35

I G-l6r*-192r4 +&t4(2-p)r

I

I + 64t’ I

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


c I +$(I -2) r*

II

5 +256 I -$ +2(2 -p)r* I

+A-4(1-p)r*-20(1-p*)+ II

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

+32(1 -/4r’

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


= P-R) h’ . rh* D


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’ -320r’

7 +3-i-++rP

7 126 -128

A

+ 51e r4 3

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

(a) INTERIOR POINT

++ r4

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

+w r*

be- rh --+ _____ rh ----+ ______ rh ____- 4

P h' = --. rh* D


f

$ * fh x f;, f

+ f (I -p*) r*

5 +e56 -& + 2(2-p)r* +A -4(1-p)r*-20(1-p*)r’

I =. P-R) h’ . r) rh' D

-T&i +* +4(2-p)r * -&-8(1-p)rc-i60(i-$)r4

+?(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.


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 _____ ++


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


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

$h 4

APPENDIX II

+ 64 r. 3

+ 9r* -l6r*- 192r4

+6rP - 16r* - 256r’

+64r4

+6r2 I

J

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

(0) INTERIOR POINT

I l +y r’ I

p

I +64r* I

P h’ = --. rh* D

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


I +4(2 -p)”

I- ++ -+ -8(2-p)P

+4(2-p)r’

+y (I-p*)+ II

-6(l-p)r’- 96(1-p’) r4 (I

-e(l-p)re-126(1-pc)r4

+ 32(l -PL)r4

67

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


NOTE.-FOT general notes see Figure 39.

FIGURE 50.-Load-deflection relations, vertical spacing: 1 at h/B; 3 at h/4, Sheet XII.


I r)

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

rh’ D

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

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

(a) INTERIOR POINT

+ 84 r4 3

-!- r4

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

+512r4

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


I ++(I -p*)r* II

p At. -3-D

5 +sle -&+4(2-p)rE t$ -6(l-~)+l6OfJ+)+

= (P-R) h’ --. rh’ D

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

+256(1 -PC)+

II b---rh ____ + _______ rh _______ 4

(cl POINT ON A MOMENT-FREE Y-EDGE


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

f'h

x f'h

f flh

9

-I f:h * t:” * flh *

+@h

k

- 16r*- 256r4

=

+6r* -16rz- 266r. + 6re

+64r4

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

(a) INTERIOR POINT

I I 7 +T c

P h' -;i;p T-'

+64r4

+Ort -l6r’- 256r’

P h' = Ti;fO'

+6re --16+ -256r4 +4(2-p) rt

+64r4

f+--rh---* ____ rh _____ * ______ rh _____ -f


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



FIGURE 52.-Load-deflection relations, vertical spacing: ,$ at h/4, Sheet XIV.


I +16r’ -32r’ - 1536r’ ( +,6,.F1

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

(a) INTERIOR POINT

f fh

+ l6r* - 32r’ - 1536r4 +6(2 -p)rc

+I++ -+q++& = m$+.

+ l6r* -32r* - 2046r’ +6(2-p)r*

+ 512r4


I +?(I -pE)r4 II

+ 6(2 -p)r* -160 +)r’- 766(1-/L*) r’

k--rh --+ ______ rh _____ -4



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

APPENDIX II 71

+512r4

+16r’ - 32re- 2046P +l6r*

+ 512 r’

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

(a) INTERIOR POINT

P h’ rhL 0’

I

+512 r4

+ 16r’ - 32r*- 2046P

+16r’ - 32 r’- 2046 r4 +6(2 -pL)r*

+s12r4

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


+256(1- PL) r’

+6(2 -pLr* --16(1 -~)r’-lO24(1-~~

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

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



FIGURE 54.-Load-dejlection relations, vertical spacing: 4 at h/8, Sheet XVI.


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

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

APPENDIX II 73

Th

f

i”

$.,, r2

= (P-R) ‘. rh’ 0

+ 6P - 16P*- 16f’ +er’

+4r4

I 7

+ 6r’ - 16f’- 16f4 + 6P

+4r’

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

I + 4 r4 I

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

I +$r l I

+ 4r’ - 8r’ - 3r’ +4r’

+ $r*

1 + 4r* 1 - 6~’ - 2r4 1 +4r* 1

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

= P 2. rhTD

P h’ P 7 -= rh D


FIGURE 50.-Load-dejlection relations, horizontal spacing: 3 at rh/d; 1 at th, Sheet XVIII.


+6f'

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

+6r4 1

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

+2r'

+ 6fP - i2r2- 24r4 + 4rg

+6r'

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

+ Jr’ 4

L +6 r'

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

+6r'

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

+Jr’ 4

+4rz - 6r* - 3r'

+2r*

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

I + Jr’ 4 I

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

+zr* 4


= (P-A) h' 7 --

rh D

P h' = --.

rh' 0

P h4 = --.

rh' D

P h4 = 2 --

rh 0

P h' = - -.

rh' D

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

APPENDIX II 75

Th

f h

;f” i;” $,, r2

I +6r'

I

b-krh* _-__ ,-h---e--- rh ---e--,-h-+

I3 1 + =r'

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

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

P = h' I--.

rh 0

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

+r'

I P ha 7 2-*

rh D

+2 r* -4r* - 6r' + 2rP

+ 8r' 3

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

I +r4 I

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

+r*

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

P h' = --. rh* D


FIGURE 58.-Load-desection relations, horizontal spacing: 1 at rh/8; 3 at rh, Sheet XX.


+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

7 +P

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

+6r'

P h4 = 77-D'

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+-rh--+a---h-+-j

(a)

l+rh+t+rh+

(4

INTERIOR POINT

Mx = $31

My = 0 t+-rh-+-rh -4

k4 (e)

* +2rx

-t H b=?pry!t *

4

Mx = pm,

(h)

EDGE AND CORNER POINTS

I J

I-c-rh-+k--rh-+I

b)

+e(l-p)r* tJI My=++ * I

M, = 0

(i)

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.


Mx P ah*

L - 0

,--rh-+--rh -v,

MyP He

b-+-rh +-rh-w,


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

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

+r2

POINT ADJACENT TO A MOMENT-FREE EOQE

(e)

POINT ON A MOMENT-FREE EDGE

(4

D I v, =Tm

(f)

-r2(l-fi)

+2re(i-Cc) E 0

-2+(1-u)

+r2( I-u)

(9) (h)

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


FIGURE 62.-Shear-deflection Telations, Sheet I.


v,=P ’ h’ 2

(b)

POINT ADJACENT TO A FIXED EWE

(cl (4

POINT ON A FIXED EDGE

(4

/ / / * / , + -2

VY = -$-&r

IfI

POINT ON A FIXED EOQE ADJACENT TO A FIXED CORNER

(e) (h)

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


FIGURE 63.~Shear-dejlection relations, Sheet ZZ.

APPENDIX II 81

ph vx =- r3

b-rh-+-rhe

k- r h -+- rh+

I

b- rh +-rh+

-6(2 +r2) +4

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.


FIGURE &I.-Shear-de$ection relations, Sheet ZZZ.


+ 32 -66 +32

+256 - I066 + 1670 -1086 +256

+32 -66 +32

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

i [

+26.6 -59.6 +26.6

+ 256 -1066 + 1669 - I066 + 256

f +32 -66 +32

f +I

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

+ 256 -1066 +50(1 3

+32 -70

-;!;6 +256 ] f’[

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

(C) INTERIOR POINT

VERTICAL SPACING: 3 AT h; I AT +h

(d) POINT ON A FREE X-EDGE

f + 64 - 152 + 64

t I26 - 640 +G 3 - 640 +I26

+64 -I60 +64

+a

l tl

+ IO -6 -10 -6 + IO

t 210 -936 +y -936 +210

-26 + 176 - 336 + 176 -26

+64 3

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

(0) INTERIOR POINT

VERTICAL SPACING: IATh; SAT+h

(f) INTERIOR POINT

VERTICAL SPACING: 2 AT h; I AT fh; I AT fh

FIGURE 65.-Load-deflection coeficients, r=M, p=O.Z.

--x

i...


I I DEFLE(

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

1 1.0 1 +.000426

Deflection = (Coefficient)(ph’/D)

beep-t;

LOAO I

cc-p -4

LOAD Iz

k-p -ti

LOAD m

Y

IJP -+-

l@J


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.

i5 %

, ” I : i

.Nxxm

31 ay”w

-*“sIY l o

XIYMN


(0) POINT ON A FIXED Y- FREE X-CORNER [FIQURE 39 II,]

(b) POINT ON A FIXED I-EOQE ADJACENT TO A MOMENT-FREE X-EWE

[FIGURE 39 (II,]

(C) POINT ON A FIXED Y-EDBE VERTICAL SP.oI*G: h ANO + h

[FIGURE 42 iol]

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

(d) POINT ON A FIXED Y-EWE

“ERTlCAL SPACING: + II ANO + h

[FIGURE 44 to)]

(0) POINT ON A FIXED CORNER (f) POINT ON A FIXED X-EWE [FIGURE 49 (Ol] [FIGURE 49 IO,]

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

(j) INTERIOR POINT [FIGURE se (b)]

l++h -+fh tl

(k) POINT ON A FIXED EDQE (m) INTERIOR PQINT

[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

0 0 0 0 0

t.010522 I

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 __-_. , 5 + .020636 + .009346 + .000622 1 -.005565 1 -.009301 1 -.010539

+ 020917 7 +.009607 1 t .000693 1 -.005724 1 -.009592 1 -.010863 I

1 t -000553 1 -.005621 1 - .009305 1 - .010531 1 4 + .020516 + .009253

3 + .019562 + .006526 1 t .000273 1 - .005438 I -.006766 I - .009690 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 SE?1 0 UhllJ - I I 2 3 4 5 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

+ .006326 t .01006 I + .012566 + .013460 0 1 0 + .002349

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, McGraw-Hill, New York, 1940.

2. Anonymous, “Rectangular Concrete Tanks,” Concrete lnjormation Bulletin No. ST63, Portland Cement Association, 1947.

3. Westergaard, H. M., and Slater, W. A., “Moments and Stresses in Slabs,” Proceed- ings, American Concrete Institute, Vol. XVII, page 415,192l.

4. Nielsen, N. J., Bestemmebe aj Spaendinger i PZuder, JpLrgenson, Copenhagen, 1920.

5. Marcus, H., Die Th-eorie elaatischer Qkwebe, 2nd Edition, Julius Springer, Berlin, 1932.

6. Wise, J. A., “The Calculation of Flat Plates by the Elastic Web Method,” Proceedings,

American Concrete Institute, Vol. XXIV, page 408, 1928.

7. HoII, D. L., “Analysis of Plate Examples by Difference Methods and the Superposition Principle,” Jvurnd of Applied Me&nice, Vol. 58, page A-81, 1936.

8. Barton, M. V., Finite LX@rence Equutions for the Analysis of Thin Rectangular Plabs, University of Texas, 1948.

9. Jensen, V. P., “Analyses of Skew Slabs,” Bulletin S&s No. $%?, University of IIhnois, Engineering Experiment Station, 1941.

10. Scarborough, J. B., NumerieaZ Mahmuhal Andy&, John Hopkins Press, Baltimore, 1950.

89

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