.

A

WATER

RESOURCES

TECHNICAL

ENGINEERING MONOGRAPH

PUBLICATION

NO.

27

Moments and Reactions for Rectangular Plates UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU

OF

RECLAMATION

A

WATER

Engineering

RESOURCES TECHNICAL

Monograph

PUBLICATION

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, preserving 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. istration.

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 WASHINGTON : 1978

OFFICE

Preface presents a series of tables containing 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 s of counterfort retaining walls. The tables provide the designer with a rapid and economical means of analyzing the The data structures at representative points. 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. THIS

MONOGRAPH

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 computations by W. S. Young, J. R. Brizzolara, and D. Misterek. H. J. Kahm assisted in the computations and in checking the results obtained.

The figures were prepared by H. E. Willmann. Solutions of the simultaneous equations were performed using an electronic calculator under the direction of F. E. Swain.

CASE

I

CASE

2

CASE

3

CASE

4

CASE

5

PLATE FIXED FOVR EOBES

BOUNDARY

CONOITIONS

0f------IL!kl

L-p-A LOAD

LOAD

I

id- pd

n

LOAD

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

IU

LOAD

Ip

;pd

LOAD

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

“NlFORY LOAD OVER 113 THE HEIOHT OF THE PLATE

f---G i-d

H

LOAD

PII

LOAD

Pm

LOAD

UNIFORM YOYEW ALOW IHE soce y - b FOR OASES I, L. AND 5

“NlFORYL” “AWlNO LOAD OVER l/6 THE “ElB”f OF T”E PLATE

iI

UNIFORM LINE LOAD ILOWOWE FREEEOBE FOR OASES I AND 3

fP 7-P-q k-----a----+ LOAD

H

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

LOADING

CONDITIONS

NOTES The variaus cases ratios of o/b. Coses I, e, and 3:

Cose

4

Case 5 All results

INDEX

: :

are

are

analyzed

I/B,

3/a, I, ond 3/z. 314, and I. 3/4, 7/e, ond I. on a Poisson’s ratio of 0.2.

l/8, 310,

bored

OF BOUNDARY

for

1f4,

3/s,

I/Z,

l/4, I/S?,

3/0, s/8,

I/2,

AND

the indicated

LOADING

-FRONTISPIECE

CONDITIONS

Y

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

-H LOAD

*Lowe

LOAD

p LI

“WIFORYL” “m”I*e D- o ALOWOy-b/e

LOAD

Contents Preface

and Acknowledgments

-----____-___________--__

Frontispiece

__------________________________________---------

Introduction

________________________.______ - _________________

Method

of Analysis

Page .. . ill

iv 1

______________________________ -- ______

3

Results ________________________________________-------- ______

5

Effect of Poisson’s Ratio- ___________________________- ________

Accuracy Appendix

of Method of Analysis--------------------I ______________________________ - _________________

An Application to a Design Problem-- - - ___________- ___ ________

Appendix

6 43 45 45

I I ________________________________________________

49

The Finite Difference Method- _ _ _ _________________________- _Introduction____________________________---------------General Mathematical Relations- _ ____-__ ______________-_ _ Application to Plate Fixed Along Three Edges and Free Along the Fourth__________________________________---------

49 49 49

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

OF

54 89

FIGURES

Number

PW

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 _________________________________ _

7 8 9 10 V

CONTENTS

vi Number

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 edge27. Plate fixed along two adjacent edges, moment and reaction coefllcients, 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 V44. Load-deflection relations, vertical spacing: 2 at h; 2 at h/2, Sheet VI45. 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 XVII56. 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-__-_------_--_--________________________________

vii page

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

CONTENTS

Viii Number

PW

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-----------

76 77 78 79 80 81 82 83 84 85 86 87

LIST OF TABLES NUmb6T

2.

3. 4. 5. 6.

Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed along Four Edges _______--_-___-___----_-----_____--Comparison of Coefficients of Maximum Bending Moment at the Center of a Uniformly Loaded Rectangular Plate Fixed along FourEdges____________--________-____________________-____ M, for Heel Slab at s------- ________--- _____-_---- ______ M, for Heel Slab at s- ____-- ______--__- _________-- ______ M,forWallSlabats ____ -- _-____________________- _____ M, for Wall Slab at s- __ _______________________________

Pap

1. Effect of Poisson’s Ratio (p) on Coefficients of Maximum

6 43 47 47 48 48

Introduction CERTAIN COMPONENTS of many structures may be logically idealized as laterally loaded, rectangular plates or slabs having various conditions of edge . 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 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

FINITE difference method is based on t,he usual approximate theory for the bending of thin plates subjected to lateral loads.‘* The customary 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 The deflections are determined edge conditions. 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. THE

*Numbers page 89.

in superscript

refer to publications

in List of References

on

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 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 s. 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 representative 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

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

6

dimension ratios of 318, l/2, 518, 3/4, 718 and 1 were studied, subjected to Loads I, X, and XI. For this case, a and b denote the full lateral dimensions. All numerical results are based on a value of Poisson’s ratio of 0.2. The arrangement of the tables is such t,hat each coefficient, both for reaction and moment, appears in the tables at a point which corresponds geometrically to its location in the plate as shown in each accompanying sketch. Effect of Poisson’s

can be determined easily, since the deflections computed from finite difference theory are independent of Poisson’s ratio. Futhermore, the bending moments at, and normal to, the fixed edges are unaffected by this factor. It is reasonable then to conclude that insofar as the moments which are most important in design are concerned, the maximum effect for this case will occur at the center of the slab. 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 ratio of a/b for which Case 5 was computed. For a change in Poisson’s ratio from 0.2 to 0.3 it is noted that the maximum effect on the bending moment coefficient occurs at a/b= 1, where the change in the coefficient is less than 8 percent.

Ratio

A question which frequently arises is: What effect does Poisson’s ratio have on the bending moments in a plate? For the plate fixed along four sides, a clear understanding of this effect TABLE

l.-Effect

of Poisson’s

Ratio (p) on Coeficienk of Maximum Bending Moment Loaded Rectangular Plate Fixed Along Four Edges

-% “;I 0. 375 0. 5 0.625 0. 75 0. 875 1. 0

at the Center

Values of M./pa* 0

0.1

0.2

0.3

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

-0.0424 - 0.0407 -0.0367 -0.0311 -0.0251 -0. 0195

-0. 0424 -0.0411 -0.0376 -0. 0324 -0.0267 -0.0213

-0.0425 -0.0415 -0. 0384 -0.0337 -0. 0283 -0. 0230

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

-..

of a Uniformly

RESULTS

Moment-: Reaction

(Coefflctent)

(pb’)

: (Coefftclent)

(pb)

X

POSITIVE

FIGURE l.-Plate

.tixed along three edges, moment and reaction coeficients,

SIGN

Load I, uniform

CONVENTION

load.

8

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

IO

32 i-.0039 1

IO

IO

lo

0

IO

n

I

I

n.

.“a-<

0

31-.029sl

Moment Reaction

I

0

I

0

I

I

0

I

= (Goefficient)(pfl) = (Goefficient)(

pb)

POSITIVE

FIGURE 2.-P&e

0

$xed along three edges, moment and reaction

coeflcients,

SIGN

Load ZZ, 913 uniform

CONVENTION

load.

0

RESULTS

Y

MI Moment

= (Coefficieni)

(pb’)

Reaction

= (Coefficient)

(pb)

-Rx RV

@

POSITIVE

$xed along three edges, moment and reaction

I “._.-..-.-

~..

coeflcients,

Load III,

tX

’ 0 My I

W

FIGURE 3.-Plate

P v

SIQN

CONVENTION

l/S uniform

load.

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

10

I

I

t

I

I

\I

I 0

I+ IlId7

1.0 I+ 3177

,

..“T”..

I+ on99

lt.oe57r;

Moment

= (Goefficient)(pb*)

I?eOCflOn=

(Coefficient)(pb)

I/’ MI -+-.I 4 0 RV %iiJ,’ Mv I

WV

POSITIVE

FIGURE k--Plate

fixed along three edges, moment and reaction coeficients,

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

.______-

Load IV,

+X

SIGN

uniformly

CONVENTION

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

”

1 MI Moment

: (Coefficlent)(

pb2.)

Reaction

= (Coefflcient)(

pb)

-+’

IJP

Rx b

X

0 w W POSITIVE

FIGURE

B.-Plate

Jixed along three edges, moment and reaction coeficients,

”

SIGN

Load V, .9/S uniformly

M,

CONVENTION

varying

load.

Moment = (Coefficient)ipb’) Reaction

= (Goefficient)(

pb )

I

W. POSITIVE

FIQURE

B.-Plate

fixed along three edges, moment and reaction coeficients,

-.-..--.__-.

.._

Load VI,

SIGN

l/S uniformly

CONVENTION

varying

load.

RESULTS

0

*-0_-_ ---O---~ iGee _, , --t ill.ii -i-, FIGURE 7.-Plate

Moment Aeoction

f

(Coefficient)(pb*)

= (Coefficient)(

pb)

jixed along three edges, moment and reaction coejkients,

+X I

W POSITIVE

Load VII,

SISN

l/6 uniformly

CONVENTION

varying

load.

f-P Moment : (Coefficient)( M) Reaction : (Coefficient)($)

1 FIGURE 8.-Plate

jked

along three edges, moment and reaction

M. --JI p Al--l. I

. X

FOSITIVE SIQN CCNVENTION 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

I-.OOOOI-.o001

1+.0003

-~~~ 0.2

I-.OiSO

I-.0004~*.0004~+.0006~+.0004[+.0001

.“..,T -39111

n

I

n

$0036~

+:0056/+.007g/ +.001, +.0022

+:0010

I

n

I

;I+.0746

Moment Reaction

n

I

+.0666I+.O655

n

t.0024

I

+.0644]

= (GoeffIcient) = (Goefflcient)(F)

I

W POSITIVE

moment and reaction

coeflcients,

Load IX,

SIGN

n

CONVENTION

line load at free edge.

16

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

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

Moment = (Coefficient)(pb*) a

a Fxa UaE

IIriIll

lO.-Plate

Reoctiin

= (Coefficicnt)(

pb)

---- i - -X

jixed along three edges-Hinged

POSITIVE

along one edge, moment

and reaction

SION

coeficiente,

CONVENTION

Load I, uniform

load.

RESULTS ,

17

4 0.8

I .o

+.0017 +.0020 0 0

0

1 0.2

[ 0.4

1 0.6

[ 0.6

[ 1.0

0

IO

lo

IO

I”

In

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

LE Il.-Plate

jized along three edges-Hinged

pb )

along one edge, moment and reaction coejicients,

Load ZZ, d/S unifol vn load.

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment Aeoction

= (Coefficient)(pb’) = (Coefficient)(

pb)

X POSITIVE

FIQURE l2.-Plate fixed

along

three edges-Hinged

along one edge, moment and reaction coefficients,

SIQN

CONVENTION

Load III,

l/S unifor .rn load.

RESULTS

Moment = (Coefficirnt)(pb*) Reaction

= (Coefficicnt)(

0

FIGURE 13.-Plate

pb )

X

POSITIVE

fixed along three edges-Hinged

along one edge, moment and reaction coeficients, load.

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

SIGN

CONVENTION

Load IV, uniformly

varying

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment

=

(Coefficient)(pff)

Reaction

=

(Coefficient)(

pb ) +-X

POSITIVE

FIGURE 14.-Plate

fxed

along three edges-Hinged

along one edge, moment and reaction varying load.

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

SIQN

coefkients,

CONVENTION

Load V, d/3 uniformly

Moment = (Coefficient)(pb*) Reaction

= (Coefficient)(

pb)

0

FIGURE 15.-Plate

POSITIVE

fixed along three edges-Hinged

along one edge, moment varying load.

and reaction

coejkients,

SIGN

CONVENTION

Load VI,

l/3

uniformly

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment = (Goefficient)(pb? Reaction

= (Coefficient)(

pb)

POSITIVE

FIGURE 16.-Plate

jixed along three edges--Hinged

along one edge, moment and reaction

varying

load.

coejicients,

SIGN

CONVENTION

Load VII,

l/6 uniformly

RESULTS

Moment = (Coefficient)( Reaction

23

M)

= (Coefficient)($),

0

FIGURE 17.-Plate

POSITIVE

$xed along three edges-Hinged

along one edge, moment hinged edge.

and reaction

SIGN

coeficiente,

CONVENTION

Load VIII,

moment at

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Y ~+--*--+*

._-f

I I 0

Moment

= (Coefficlent)

(pb’)

Reaction

= (Coefficient)

(pb )

--_ L-x 0

FIQURE 18.-Plate

POSITIVE

fixed along one edge-Hinged

along two opposite edges, load.

and reaction

SIGN

coefkients,

CONVENTION

Load I, uniform

25

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

r1-.02’351 0

Moment

= (Coefficient)

(pb’)

Reoctlon

= (GoeffIcIent)

(pb)

~-.0140~-.0220~-.026lt-.0277kO262i

M. -+-I/ f&l’ R.^ . POSITIVE

FIG

E lg.-Plate

fixed

along one edge-Hinged

along two opposite uniform load.

edges, moment

and reaction

SIGN

CONVENTION

coeficients,

Load

II,

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Y

MI Moment

: (Coefficient)

(pb*)

Reaction

= (Coefficient)

(pb )

--

P v

’ h fb

tX

’ I My W@ POSITIVE FIQURE

20.-Plate

jkced along one edge-Hinged

along two opposite

uniform

load.

edges, moment

and reaction

SION

CONVENTION

coejkients,

Load III,

l/S

RESULTS

Moment

= (Coefflclent)

(pb’)

Reoctaon

= (Coeffxlent)

(pb)

IA Ma -+’ R.Rv d&iJI

X

MY

W. 0

FIGURE 21.-Plate

POSITIVE

jixed along one edge-Hinged

SIGN

along two opposite edges, moment and reaction coeflcients, varying load.

- .-._-

__--

CONVENTION

Load IV, unifol

28

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

--_ f Moment f?eactlon

= (caefflclent)(pb’) = (Coefflcient)(

pb) +X

I

W POSITIVE

FIGURE 22.-Plate

fixed

along one edge-Hinged

along two opposite edges, moment uniformly varying load.

and reaction

SIGN

CONVENTION

coeficients,

Load

V,

29

Moment = (Coefficlent)(pb*) Reaction

= (Coefficient)(

pb)

POSITIVE

FI :GuRE 23.-Plate

fixed

along

along two opposite edges, moment uniformly varying load.

and reaction

SIQN

CONVENTION

coejicients,

Load

VI,

11s

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

i Moment heOCtiOn

= (Coofficiant)(

pb’)

= (Coefflcient)(

pb)

W POSITIVE

FIGURE 24.-Plate

fized along one edge-Hinged

along

two opposite

uniformly

varying

edges, moment load.

and reaction

SION

CONVENTION

coeficients,

Load VII,

l/6

Moment Reaction

- (Coefflcient)(

M)

= (Coeffxwnt)(j-1

0

FIG URE 25.-Plate

POSITIVE

jixeu along one edge-Hinged

along two opposite edges, mom&t at free edge.

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

SIGN

and reaction coeficients,

CONVENTION

Load VIII,

momen tt

32

Moment

= (Coefftcient)(

Fb)

Reoctlon

: (Coefficient)(

F )

POSITIVE

FIGURE 26.-Plate

fixed

along one

along two opposite edges, load at free edge.

and reaction

SIQN

CONVENTION

coqjicients,

Load IX,

1ine

RESULTS

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

0

““151-.00241-.00301 ~00061+~00l71+

.ol76

0

I.0 0

+.6290+.

1977

+.7827

6.

0.6

I+.0060

-~3 \ 0

LO.2

1t.0002

0

1 +.OlZl

l .9739

t.0011

+.I576

t.3024

+.5696

+.0952

t.0296

-.0059

~0162

‘---

--

-

---

1

I-

.003Zl- .0023

~+.0377~+.OlZOl+ 1

0

1+.00291+

I+.0067

0066~+.0159~+

I+.0020

l-.0018

0

0

02361+,0304

\-.0401 Rx

~_

0

00321+.00471

I+.0741 d

~.0282~+.0144(+.0077~+

.OllO

__..

0

0

^

0

0

^.__

_---

- .0079

OOOII-

.0026

1

.00721+

.0061

1

0

+.0041

+.0125

+ .0221

+ .0326

-.0696

+.0333

+.2595

+.4574

+.7920

-.0696

0 1

-0

~~+.0165~+.006’~-.00051-.004z~-.UUb~(-.~0 0

1

+I.1266 ,

LY ; I

1

---.-

0

0

.---.+.

I-

.oesrl

I+ .00721+.0207)+

.03451+.04941

0

0

~1+.03621

I

I

I --Y04~~~/~~~/~~~~J FIGURE

0

27.-Plate

x

Moment

= (GoefficIent)

(pb*)

Reaction

: (Coefficient)

(pb)

W

p4 ;p&

fixed along two adjacent

POSITIVE

edges, moment and reaction

coeficients,

SIGN

CONVENTION

Load I, uniform

load.

34

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

0.2 1

-co

1

I.0

l-.0014

0.8

~+.00751+.OOll

0.4

0.6

+.00021*.00021+.0002l+

0.6

0001

1+.0001

~+.0006~+.0006~+.0004~+

+ 0021

+.0008

+.0001

0

t.0009

+.0013

0

+.0505

+.

-0.6. 0.4 ,Ib

t.2733 +.3352

+.0102 +.0213 + ~-~ t.0384

[~~+:I&]+

t.0026

OOl2(+

1050

+.0106 +.0149 -r

-+.0165

+.0065 +.009

.%$,@7 0

+.2030

l .3661 +.0026

L -

1 +.0011

t.0031

-.0069~+.0125~+.1333

--+-.-0003

+.034l

+.0542

+.0113

3114

+

5432

00021+.00021+

0

j 0.6 I

1 I. 0 1

I

I

0

0

0002/+.00031+.00031

OOOII-.00021

-,----A !

-.0003

-

t.0079

0010

0

1

0

0

( !+.0077~~.00241-

I.

c

0.

‘1

0

1

0

, I 0~~-.00~6~~.00~6~/~.0ll6

I

0

I

4 1

0 0

+.22851+.39631+.5629

+.0019

+.0524

+

0

1-.0022

+.0055

+

0053

+.00691+.0125

-.0250~+.0436[+.l939~+.3C7l

-I!

I

+ 0011

+.0055

r t.0063

l 2023

1061

+ 0020

0

~+.00~~j+.00071+.00031+.00001-

+.ooos

+.0031

0.d 1 ! 0.6 I

0

+.0040

+.0028:

0

I+

+.0001

y-.0822 Ra -.0462 t.0819

(

0002/+

0.2 I

0

0 +.0046j-.Ob61

I.0 0. e

0

+.0067

l .0046 -.0167

0

0

00021

~6~+.0035~+.0019~+.0008)+.0001 +.I209

1.0 I

+.0153

I

0

1+.46931+.6544 +.0026

-.0022

0

I

-.0042

-.00741-.0051

t-7

Moment

: (Coefficient)

(pb’)

Reoctlon

= (Coefficient)

(pb) 3-X I

W POSITIVE

FIQURE 28.-Plate

jbed along two adjacent

edges, moment and reaction

coeficients,

Sl(iN

Load II,

CONVENTION

$713 uniform

load.

RESULTS

I

-

I

0.2

f

~+.1774~+.0131~+.0020~-.00171-.00241-.00171

0

I+.Ofi261-.OOl71-.00471-.00641

R.

~Y~+.00481+.16151+.2512~+.2874~t.3312~+.3489~ I.0

t

.0052

0. a

t.0356

0. 6

+ .0430

t.0104

t

I

.0059

+.0016

-.0006

-.OOl

t.0116

l .0053

t.0011

-.0008

t .Ol26

t.0041

- .0002

- .OOl6

I

0

0

0

0

0

0

0

-.OOlO

0

+.0023

t .OOl2

+ .0002

- .0007

- .0014

- .0020

- .0013

0

f.0025

t.0008

- .0012

-.0026

- .OOM

- .0046

- .0053

- .0069

- .0075

- .0060

- .0055

- .0043

- .0029

- .0020

II -

0.4

+.I052

+.0135

+.OOl3

-.0022

-.0025

-.0017

0

t .0027

- .002

0. 2

+ .I682

f.0094

-.0015

-.0023

-.0015

-.0005

0

t

- .0045

0

It.oos

+ .0033

+ .0057

+.0072

t .I3383

0

+ .2052

+ .2808

t

t .3372

.3064

t

.0019

I

0

.3473

Y

Moment

= (Coefficient)

(pb*)

Reaction

= (Coefficient)

(pb)

POSITIVE

FIGURE 29.-Plate

fixed along two adjacent

edges, moment and reaction coeficients,

Load III,

SIQN

CONVENTION

l/S uniform

load.

36

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I

m

+.0075

+.0007

+.0006

+.0004

+.oooz

+.0001

0

;.0252

+.0017

t.0011

+.0006

+.0003

+.OOOl

0

l .0056

-.0008

I

0.4

~+.2050j+.~246~+.0122

0.2

[+.I413

0 + .0003

.o

~+.0045~+.0002

~+.0162~+.0056~+.0010

)-.0005

l-.0012

1

0

(+.0049~+.0017~.

I-.0004)

0

I+.oo52/+

i? I, I

1

p

I

1

0.8 0.6

+.2303 +.2067

+.0459 +.0467

+.0195 +.0242

+,0064 +.0046

-.0002 -.0023

-.0036 -.0027

0

+.0097 +.0092

+.0040 +.oozo

-.0044 0004

-.0093

-

0.4

+.2363

t.0360

+.Ol2l

+.OOlO

-.0030

-.0029

0

+.0072

-.0003

-.0060

-.0096

-.Oll7

0 I,+,0032

+.oooa

+.0019

+.0050

+.ooe3

+.01oLl

+.0070

+.0110

+.014a

0

+.0147

+.0352

l .0546

+

t.0696

561+.44891+.5505i I

----1

+.0056

-.0059

-.0077

0

~~

0.2

I

I

~*.1193~+.01~0~+.0042~+.0005~+.0006(+.00l6 -.0194

+.0029

I

n v-.0194~+.1105~+.2399(+.32: R”

I I.0

-

0

0.6

+.I917

)+.23641+.0518

+.0662

+.0291

I+.0173

+.0046

l+.OOO4!-.00591-.0054

t.0103

t.0152

Moment Reoctton

0

30.-Plate

$xed along two adjacent

0740

I 0 I

’

0

I

l .0197

0

+.0232

= (Coefficient)

(pb*)

= (Coefftcient)

( pb)

c 0-A

FIG IURE

0129

+.0515

POSITIVE

edges, moment and reaction coeficients,

Load IV,

+.0759

SIGN

+.0987

+

1157

CONVENTION

uniformly

varying

load.

RESULTS

Mx

I 0.2

2 II c,

I

-

s II s

I

37 MY

0.6

0.4

O.SlI.0

0

lo2looln~lnnlln -.-

-.

.

-.v

.,.-

..”

1.0

-.OOOJ

+.oooo

+.oooo

+.oooo

+.oooo

+.oooo

0

0

0

0

0

0

0. e

+.0001

+.0001

+.0001

+.0001

+.0001

+.oooo

o

+.oooo

+.oooo

+.oooo

+.0001

+.0001

+.0001 ..+.OOOi

+.0001

+.0001

+.0001

+.0001 .-

-

-.OOOI

0.6

+ 0144

+.OOi I +.0007

t.0004

+.OOOZ

+.OOOI

o

+.oooe

+.0002

0.4

+.os&

+.0030

+.0019

;.&so

+.0004

+.0001

0

+

+.0004

0.2

+.0843

+.0044

+.0026

COO13

COO04

+

0

+ ones

0.6

I+ 00311+.00171+.00141+

0000

o

00111+.0007(+.0003)

I 1+.2646/+.35441

-.

0

+.0022

+ 0061

I.0

-.0169

+.0022

+.0027

+.0023

+.0016

+.0006

0

0

+.0147

l .0053

t.0040

+.0027

t.0015

+.0006

0

+.oooe

. +.0565

l .0099

+ 0059

+.0029

t

-.OOOO

0

+.0009 -~-.__-..+.0012

+.0009

0.6

+.ooll -t.0020

+.0005

0.4

+.1351

t.0148

+.0069

+.0020

-.0004

-

0

+.0030

+.0008

-.OOlZ

0.2

r.1353

T.0117

t.0040

+.0003

-.OOll

-.OOll

0

+.0023

+.OOOO

0

+.Ol25

0

+.oooe

+.0019

+.0031

+.0043

0

0

+.0038

0

0

0011

WY I?.

+.0125

+.0460

+.I236

+.I740

I.0

-.0220

+.0052

+.0053

t.0039

+.0023

t.0010

0

0.8

+.0298

+.0095

+ 0065

+.0036

+.0016

0

+.0753

+.0143

+.00!6

t.0030

t.0004

t.0005 ~~~~~~__ -.0005

0.4

+.I506

+.0176

+.0068

+.OOlO

-.0015

-.0017

0.2

+.1313

+.0119

+.0030

-.0005

-.0014

0

t.0050

0

+.0013

+.060

+.0050

+.0755

0.

6

WY R.

0

0

t.0103

0.0

0010

boo0

-7Mms

-

t.0146

t.0181

1+.ooo3~+.ooo3~+.ooo4~+.ooo57+.ooo6~

,

I I+.1661

0006

0 +.0001

0

0

0

+.0009

+.0010

-.OOOl

-.0006

-.0009

-.0029

-.0042

-iOil

-.0019

-.0033

-.0042

-.0049

+.0097

t.0156

+.0213

+ 0257

_.-..

+.25381+.3159 0

0

+.OOlO

+.0006

0

COO19 t.0014 ~~~ -~ -.~ +.0029 +.OOl4

-.OOOl

0

+.0036

+.0002

-.0026

-.OOlO

0

+.0024

-.0005

+.0046

+.0060

0

+.0063

+.I616

+.2132

+.2673

+.3382

~~-

0

I.0

-.0036

+.0130

+.0092

+.0045

+.OOlZ

-.0003

0

0

0

0.8

+.0540

+.0163

+.0090

+.0036

+.0005

-.0006

0

+.0033

+.0016

0.6

+.093

t.0191

+ 0079

+.0016

-.OOlZ

-.0016

0

+.0036

0.4

+.I561

CO193

t.0048

-.OOll

-.0027

-.0022

0

+.0039

0.2

+.I218

r~.ollo

l .ooll

-.OOlZ

-.OOlb

-.oooz

b

t.0022

0

-.oooo

0

+.0024

+.0649

+‘.0070

+:OOEB

0

0

-.OOOO

+.I274

+.2163

+.2612

t.3210

-.

0

0

+.0004

+.0003

-.0014

-.0025

-.0031

-.0052

-.0070

-.0062

-.0025

-.0036

-.0041

-.0044

+.0146

t.0226

+.0301

+.0358

0

0

0

+.oobs

0

-.0007

-.0016

-.0023

+.0009

-.0020

-.0043

-.0060

--.0072

-.0014

-.0055

-.OOSl

-.0097

-.01oa

-.0013

-.G23

-.OO:;

-.0006

+.0002

+.01&i

+.0246

+.0348

+.0438

+.0507

r.3569

1.0

+.0261

+.Ol96

+.OlO3

+.0027

II

0.8

+.0667

t.0207

+.0069

+0017

0.6

+.0943

+.0209

+.0064

1:0004

0.4

+.15;2

+.OI88

+:0026

-.0024

0\”

0.2

+.I166

-.OOlZ

-.0003

+.0008

0

+.ool9

-.0016

-.0008

+.0016

l .004l

+.0059

t.0016

+.0096 0

+.OOOO

0

+.0035

+.0066

+.ooee

+.0107

0

0

+.o174

To329

+.0440

+.0534

l .0603

+.OOlS

t.1676

r.2512

+.2879

+.3346

+.3567

VY Y b .__._

0 _____ y

Moment Reaction

= (Goefficient)(pb*) = (Coefficient)

(pb)

I

W. POSITIVE

FIQ URE 31.-Plate

fixed along two adjacent

edges, moment and reaction coefkients,

S ION

CONVENTION

Load V, S/S unijormly

varying

load.

38

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I

,,

ia

1

0.6

l .oolo

+.0007

*.0006

+.0005

+.0003

+.0001

0

+.0001

0.6

+.0039

+.0015

+.OOll

+.0006

t.0003

t.0001

0

l ,000

0.4

~+.0207(+.0032~+.0016~+.0006~-.00001-.0002[

0.2

+.0756

+.0051

+.0221

0

+.00011+.00021+.00021+.0002lt,0003

0

+.0013

-.0004

-.0006

-.0007

o

0

+.OOlO

-.0002

-.0013

-.0021

-.0026

-.0030

0.6 0.4

+.0244

+.0036

+.0016

t.0003

-.0003

-.0004

0.2

+.0747

+.0049

+.0006

-.0006

-.OOlO

-.0007

0

I+.0201

0.6

0.4

1

0

0

~+.0007~+.0015)*.0020~+.0025~

~+.0066~+.0024~+.0014(+.0006~+.0001

l-.0001

1

.~+.0256~+.0041~*.0010(-.0003~‘.0006/-.0005[

0

0.6

0.2

1t.02151

I+.0091

0

I

0

~+.0005~*.0003~+.00021+:00001-.00011-

0

0

)+.0012(+.0021~+.0027~*.0030~

I+.0031

~+.0014~+.0003~-.00021-.00031

0

l+.OOOfi

fl

0

d

~+.0037~+.0074~+.01021*.

l+.O(

~+.0006~+.0003~+.0001

I

I

II-.(

1

+.0111

+.0034~+.0011~-.00001-.0004)-.00031

+.0246

+.0039

+.0004

-.0006

-.0007

-.0004

0

*.0006

-.0004

-

1 + 0723

+.0034

-.0006

-.OOlO

-.0006

-.0003

0

+.0007

-.0019

-.0025

0

I

I

1

1+.00071+.00031-.00031-.00071-.OOlOi-.OOl2 0013

I J

-.0017

-.0019

-.002l

-.0023

-.oOl9

-.0017

Y A

-+- IA

M. Moment Reaction

:

(Coefficient)

(pd)

= (Coefficient)

(pb)

’ R, 5

-3-X

0 ’ MY

@ W. POSITIVE

ho ,URE 32.-Pkztejixed

along

edges, moment and

reaction

coeficients,

SIGN

CONVENTION

Load VI, i/3 uniformly

varying

locrrd.

I

1 y/b

9

h%‘I

0

.

IO.2

IO.4

IO.6

IO.8

1 1.0

1.0 0. 8

-.oooo -.oooo

+.oooo +.oooo t.oooo + 0000 t.oooo +.oooo t.OOOO +.oooo +.oooo t.oooo

0. 6

-.OOOl

+.oooo

0. 4 0. 2 0

-.OOOI +.oooi +.OOOI +.oooo +.oooo +.oooa t.0137 l .0006 +.0004 +.oooz +.0001 t.oooo t.0050 0 +.0001 +.oooz +.0003 t.0004

I

+.oooo

+.oooo

+.oooo

1

0 0

+.oooo

0

0 0

0 +.oooo +.oooo +.oooo +.ooot

0

0

0

IO.2

IO.4

I

0 +..oooo +.oooo +.oooo +.0001

0 +.oooo +.oooo +.oooo +.0001

0.6

0 +.oooo +.oooo +.oooo +.oooo

IO.8

0 +.oooo +.oooo +.oooo +.oooo

1 I.0

0 +.oooo t.0000 +,oooo +.oooo

I

t.0004

t.0010

+.0016

+.ooee

+.0027

~Y~+,0050~t.0I69~+.0463~t.0669~+.09611(c.11651

I. 0

0.6 0.6

-.0007

+.oooo +.ooot

+.ooot

+.0001 +.oooo

0

0

0

0

0

0

0

+.0001

+.0001

+.0001

+.0001

+.0001

+.oooo

0

+.oooo

+.oooo

+.oooo

+,oooo

+.oooo

+.0001

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

Ill-.00021-.0002l-.000I1

nl-

I I+

nnnsl+

0001~-.ooo3~-.0002~-.oaa1

m-ml+

nnnll-

I

ooool-.ooool

0

1+.0003)+.00001-.OOOZ/-.00041-.00051-.00061

0

1+.00021-.00011-.00041-.00051-.0006l-.0007l

0

)+.00021-.00031-.00051-,00061-.0006(-.0006l

0

It.ooOl

017

lt.OOOll+.OO~OOl-.OOOOl-.OOaol-.oooll

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

Y

Moment Jeoction

q

(Goefficlenf)(pb?

: (Coefftclent)

(pb)

M, -+-IA ’ R” 0 RY I 45J-

X

MY

W

POSITIVE

FIGURE 33.-Plate

fixed along two adjacent

edges, moment

and reaction

coeficients,

Load VII,

SIGN

l/6

CONVENTION

unijownly

varying

load.

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

Moment Reaction

=

(Coefficient)(po*)

= (Coefficient)(po)

ieo-rl FIGURE

34.-Plate

$zed along four

POSITIVE

edgeqmcnnent

and reaction coeflcients,

SIQN

CONVENTION

Load I, uniform

load.

Moment = (Coefficient)(po*) Reoctmn = (Coefficaent)(

po)

X

POSITIVE

FIG

m 35.-Plate

fixed

along four

edges, moment

and reaction y= b/2.

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

coeficients,

Load

uniformly

SISN

CONVENTION

varying

load, p=O

al0 lng

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

42

Moment Reaction

=

(Coefflcient)(

=

(Coefficient)(po)

pa*)

POSITIVE

FIGURE

36.-Plate

fixed along four

edges, moment

and reaction x= al.%

coeficients,

Load

XI,

uniformly

SIGN

CONVENTION

varying

load, p=O

alox

Accuracy of Method of Analysis 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 progression 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 inconsistencies 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 THE

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

TABLE

Valuar of M./pa* from b/a

Timoshenko

1

Method of this Monograph 2

-

1. 1 1.2 1.3 1.4 1.6 1.7 1. 8 1. 9

-

0.0264 0.0299 0.0327 0.0349 0.0381 0.0392 0.0401 0.0407

- 0.0269 - 0.0301 - 0.0329 - 0.0352 - 0.0384 - 0.0395 - 0.0404 -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

44

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

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 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 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

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES DESIGN

DATA

Unit Weights Concrete Moist earth Saturated earth Water Surcharge Pressures Vertical Horizontal Equivalent Fluid Weights Moist earth Saturated earth

150 120 135 62.4

Ib/ft3 Ib/ft3 lb/+ Ib/ft3

360 Ib/fte 120 Ib/ft’ 40 Ib/ft3 75 Ib/ft’ Water surface elevatiorv’t --- -! -

FRONT

ELEVATION

END

COUNTERFORT DIMENSIONS

RETAINING

AND

TYPICAL

,

ELEVATION

WALL

DESIGN

LOADS

I

1L L pw -H WATER

pq-A

LOAD

COUNTERFORT IDEALIZED

WALL

i-- ps-4

i.6

SURCHARGE

LOAD

SLAB

DIMENSIONS

AND

-

EARTH

INTERIOR

COMPONENT



LOADS

Ffq+/iii??j w----pu ----H

NET LOAD ON HEEL SLAG

COMPONENT

COUNTERFORT IDEALIZE0

FIGURE

HEEL

SLAB

DIMENSIONS

37.-Counterfort

.- .--- ----

-

AND

~---p”----~ LOADS

INTERIOR

COMPONENT



LOADS

wall, design example.

-_-..

LOAD

PORE

PRESSURE

LOAD

APPENDIX I 3.-M.

TABLE T

Values of pb2+

0 0 0 0 0 0

0. 2 0.4 0.6 0.8 1.0

I-

Moment coefficients

-

1118.5

-Y b

-1032.3

PU

1.0 0. 8 0.6 0.4 0. 2 0 0 0 0 0 0

0.0852 0.0807 $0.0712 + 0.0545 + 0.0250 + +

x -ii 0 0 0 0 0 0

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

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

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

+ 95.30 $90.26 +79.64 + 60.96 $27.96 0 $2.13 +5.59 +8.95 $11.18 $11.97

0

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

TABLE

4.-M,

for

Heel Slab at

--

--

- 15.59 -22.30 -28.18 - 28.59 - 16.52 0 -1.45 -3.41 -5.16 -6.30 -6.71

+79.7

+68.0 +51.5 $32.4 $11.4 0 $0.7 +2. 2 +3.8 $4.9 f5.3

s Moments (foot-kips) Total moment (foot-kips)

-1032.3 M,

PV

M.

0

0

0

$0. 0043 $0. 0055 + 0.0055 +O. 0032 0 +o. 0068 $0. 0167 + 0.0252 $0. 0307 + 0.0325

+ 18. 01 + 15. 88 $12. 19 f5.59 0 + 10. 51 f28. 19 + 44.63 t-55. 81 f59. 73

-4.44 -5.68 -5.68 -3.30 0 -7.02 -17. 24 -26. 01 -31. 69 -33. 55

$13. 6 +10.2 +6. 5 +a. 3 0 $3. 5 +11.0 $18. 6 +24. 1 +26.2

Pu

--

M”

-

-

1118.5

--

-

-

Moment coefficients

--

Total moment (foot-kips)

-_ M"

0

$0.0019 + 0.0050 + 0.0080 +o. 0100 $0.0107

I-

Moments (foot-kips)

PP

--

-

Velues of pbb

for Heel Slab at s

-

-x a

47

-0

0

0.0161 +O. 0142 +o. 0109 +o. 0050 0 + 0.0094 +O. 0252 + 0.0399 $0. 0499 + 0.0534 +

-

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

48

TABLE

T

Moment

-i-985.5

-

-* s

Y

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

-_--

PW

b

--

Pa

+ 0.0000

+O. 0 +o. +o. +o. +o. +o.

0009 0032 0002 0005 0007 0009 0010

-

-

-985.5

x a 0 0 0 0 0 0 0. 0. 0. 0. 1.

?!

_-

2 4 6 8 0

-

PW

b

1. 0. 0. 0. 0. 0 0 0 0 0 0

--

_-0.0000 $0.0000

+o. 0002 +O. 0006 0 +o. 0011 +O. 0025 + 0.0036 + 0.0043 +O. 0046

-

Moment

6.-M,

M0ment.Y (foot-kips)

-_

-

- --

M”

P8

-+ 0.0004

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

-

0 +O. 0026 +O. 0027 $0.0026 +o. 0020 0 4-O. 0015 +o. 0041 -i-o. 0066 +o. 0082 $0.0088

0 +o. 0005 +o. 0011 +O. 0016 +O. 0016 0 + 0.0014 +O. 0036 +O. 0056 + 0.0069 +o. 0074

-

$0.56 +l. 67 $4.74 f9.47 + 10. 45 0 +O. 42 +O. 84 +1.53 -I- 1. 95 +2.09

+“o. 57 +1.33 +2. 10 +2. 67 +2. 86

-

--

f5. +9. +17.1 +25. +24.0 0 +O. +l. +3. +4. -k4.

0 1 7

8 8 2 0 2

-

T 1392.9

--

PW

_---

+2.29 +5.34 + 10. 29 + 15.05 + 15.05

--

for Wall Slab at s

coefficients

PS

M.

-_

-

-

1905.4

Total moment (foot-kips)

M.

_+o. 00 -0.00 -0.00 -0. 89 -3. 15 0 -0. 20 -0.49 -0.69 -0. 89 -0.99

+o. 0012 +o. 0034 + 0.0068 + 0.0075 0 +o. 0003 +O. 0006 +o. 0011 +o. 0014 +o 0015

-

157.7

-0

0 8 6 4 2

1399.9

_-

+0.0012 +o. 0028 + 0.0054 +o. 0079 + 0.0079 0 +o. 0003 + 0.0007 +o. 0011 +o. 0014 +o. 0015

TABLE

Values of pbz-1

-

_-

-

-

T

PO

-- -+o. 0133 +o. 0131 $0.0134 +o. 0133 +o. 0103 0 +o. 0003 +o. 0009 +o. 0013 +O. 0016 +o 0018

- 0.0000 +o. 0000 +o.

1905.4

-.

_-

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

we&icients -7

157.7

for Wall Slab at s

5.-M.

Moments (foot-kips)

---

0 +o. 0002 +o. 0007 +o. 0014 +o. 0015 0 + 0.0014 +O. 0036 +o. 0055 + 0.0068 +O. 0072

+:.

0 +o. 41 +o. 43 +o. 41 +O. 32 0 +O. 24 +O. 65 fl. 04 +1.29 +1.39

00

-0.00

-0. 20 -0.59 0 -1.08 -2. 46 -3.55 -4.24 -4.53

-

_-

_-

-

M,

_0 +o. 95 +2.10 +3.05 +3.05 0 +2.67 +6. 86 + 10. 67 +13.15 + 14. 10

-

Total moment (foot-kips)

M.

MQ

M”

PO

-- --

-

-

-0 +l. +3. +5. +4. 0 +3. +10.1 +15. +19.7 +21.0

0 +O. 28 +O. 98 +1.95 +2.09 0 +1.95 +5.01 f7.66 +9.47 + 10.03

-

-

6 5 2 9 8 8

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.’ throughe 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 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 The addismall as compared with its thickness. tional differential expressions relating the deflections to the boundary conditions, moments, and (See, for shears are perhaps equally well known. instance,‘.) They will therefore only be stated here, using the notation and sign convention shown in Figure 38. 49

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

50

(0)

INTERIOR

POINT

(bl GRID

POINT

DESIGNATION

I

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’

POSITIVE

SIGN

GRID

SYSTEM

0 (C)

SUB-DIVIDED

:

x

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 ed 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 Partial

differential

equation:

(1) Fixed edge conditions : w=o,

(2.01)

bW T&=0.

(2.02)

w=o,

(3.01)

$+p s?&J

(3.02)

Free edge conditions : (4.01)

(4.02)

Free corner conditions : (both directions) J

s

g+h4

(5.01)

=0 (both directions), d2W --

(5.02)

(5.03)

bndt-”

Bending moments : (6.01)

1

M Y=D Twisting

b2W~~d2~ by2 3x2 *

(6.02)

moments : (7)

Shears : V

x

=

-

V YE-D

D

(8.01) b3”+ b3W bY3 bx2by

In the above expressions

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).

Hinged edge conditions :

g=O

51

1

the partial

.

(8.02)

derivatives

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 advantageous 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 Certain of will be referred to as grid points. 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.

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

52

Based on the usual methods of finite differences,” the difference quotient relations required in this development can be written directly and are given below. All of the difference quotients are given with reference to the focal point, lettered Z. Aw 1 (9.01) bx=2rh (E--W), A2w 1 z=13h2 W--274+W), A3w 1 -s---2?h3

A4w -== Ax4

1

(EE--2E+2W-WW),

(EE-4E+6Z--4W+WW),

g=$

$=$

A3w

e=2h3

‘$=;

1

A%V -=-1

A3w

(9.03) (9.04)

(N-S),

(9.05)

(N--2Z+S),

(9.06)

(NN-2N+S-SS),

(9.07)

(NN-4N+6Z-4S+SS),

AxAy 4rh2

(9.02)

(NE-NW+SIW-SE),

(9.08)

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

(NE-2E+SE-2N

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

(9.12)

The approximate counterparts of the basic relations (1) through (8.02) may now be written. 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=&

This may be considered as an operator, and the portion within the brackets can be conveniently portrayed as an array of coefficients. This expression, multiplied by h4, is shown in array form at (a) of Figure 39. Each element of the array represents the coefficient of the deflection of one of the active grid points in a group similar to that shown at (a) of Figure 38. The location of the coefficients in the array is congruent to the physical locations of the points and the heavily outlined coefficient applies at the focal point-the point for which the relation is to be determined. Since the solution deals with discrete points, the distributed load intensity p in the right-hand 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 at any grid point is a function of the distribution of p on the four ading grid elements. If each of these elements is considered as an infinitely rigid plate ed at its four corners, then the 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~~~

(9.11) A% 7=&4 Ax Ay’

(10)

(11)

1

(9.10)

A3w

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

(9.09)

(NE-2N+NW-&+2S-SW), L\X2ay=2r2h3 -=k3

-4(l+r’)(E+W)--4r2(1+r2)(N+S)

[EE+WW+r4(NN+SS) +2r2(NE+SE+SW+NW)

in which PZNE represents the contribution from the grid element Z-N-NE-E and similarly for the other right-hand . 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 ading any focal point Z, that the magnitude of the statically equivalent average load is: Pz/rh2=(1/6)(p~+pE+Ps+p~S2P~), where pN represents the intensity etc. The approximate counterpart be written: v4y=m2*

PZ

(12)

of p at point N, of (1) may now

(13)

APPENDIX II Multiplying both sides of (13) by h4 and replacing V% by the deflections as given by (10) leads to :

53

In like manner n, and s: W.,----M,)h+

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

for elements with

centers at w,

Wrxnna-M,.,,)rh+V+rh2=0, (16.02)

+2r2(NE+SE+SW+NW) -40

+3(E+W)--4r2(1

Of,, ---M&h+

+r%N+S)

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

g*

(MxYne--Mxy,,)h+V,,,rh2=0, (16.03)

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

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

(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,,)

(17)

An approximation to each moment in 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) rh2

[NE-N+Z--El.

(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)+$

(Mxz -M&+

=Pe.

(NW-2N

(M,,B-M,.,Jrh +NE-2W+4Z-2E+SW-2S+SE) +w.,+vx,>

However,

if

the elements

are sufficiently

r;=o. small,

f (v.,+v.,) may be replaced with VXe so that ME---M.&+

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

OLne -M,.,,)rh+V.~rh2=0. (16.01)

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

54

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

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 edgesand 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 lefthand 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 procedure 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 is always (P&h*) (h’/D). Since these load are to be expressed as coefficients of ph’/D, it remains to evaluate the Pz/rhg in of p for each point and each loading. At Point 45 the right-hand 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 ading Point 45. For this reason, the more general method expressed by (11) must be employed. In particular for Load II, the elements 45-3536-46 and 4546-56-55 carry no load, and accordingly 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 righthand (load) 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 The equation spacing of the grid points involved. for Load I is

Substituting numerical values for various deflections, this becomes

-28w21+21Owzz+

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

-SW,,+?

low,,+

176~31-936~~~

~~,-364w~~+~

5057

+ 176w51- 936w&w~=4

w42

‘;=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. Comparison of old and new values shows that they approach closely for the points where y/b=O.4. 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. 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 :

the

e (h2) (g)

[--(32)(0.004944)-(16)(0.021325) +(128)(0.007860)-(32)(0.009833)] =(0.03125+0.192016)ph2=0.223266ph2.

3 ph4 D-~44.

Substituting for Point 44, its deflection as determined from the 30 equations gives, for the right-hand member (0.75-0.100572)

R3,,=0.03125ph2+

PsO and

R,,,=R3&h=0.893064ph, or in of b R,,,=O.l78613pb, 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 Mx23=(;)

@)

numerical

values

[(16)(0.015283)

t-(0.2)(0.029914)-(32.4)(0.043935) +(0.2)(0.046526)+(16)(0.073156)] =0.006818ph2=0.000273pb2. Upon completion of computation of the reactions, a partial check of the solution may be For obtained from equilibrium considerations. 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.

56

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES + r4 + 2r* +I

-4r*-

-4-4r*

4r4

+6+8r*+

+ 2r*

- 4r*-

+ 2r*

6r’

-4

- 4re

4r4

+ 2r*

INTERIOR

POINT

+I

=

p rh2O

=

P rhe

h’ .

+ r4 (0)

+ r4

I

(b)

/ / / /

POINT

I

ADJACENT

TO A FIXED

X-EDGE

+ 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.

h4 T’

APPENDIX II

(0)

POINT

ADJACENT

TO

A

57

MOMENT-FREE

CORNER

=

(b)

POINT

ADJACENT

TO A MOMENT-FREE

X-EDGE

=

(c)

POINT

ADJACENT

TO

A MOMENT-FREE

=

POINT

ADJACENT

TO

A MOMENT-FREE

X-EDGE

P rhe

h’ -6’

P -z-b’

h’

Y-EDGE

++

(cl)

-- P rh2

AND

A

FIXED

Y-EDGE

=

P rkQ

FIXED

X-EDQL

+r4

+ 20

-40

- 4r4

+(2-p)r* -2-212-p)

0

Y-EDQE

AN0

////////////////////////////////////////////////~ (a)

POINT

ADJACENT

TO A MOMENT-FREE

NOT&--For

general

FIGURE 40.-Load-deflection

notes see Figure relations,

A

39.

Sheet ZZ.

-. h’

h’ D

58

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I

+ r4

I

(0)CONANT-FREE

X-EDGE

+ I + 4(1

(b)

(C)

POINT

ON

A

POINT

ON

MOMENT-FREE

A

MOMENT-FREE

X-EDGE

(P-R) - r h*

=

ADJACENT

Y-EDGE

TO

A

MOMENT-FREE

= t(2

- fi)rp

h’ 0’

Y-EDGE

(P-R)

h4

rh*

-D--’

-2(i-u)r~2(1-u~)r4 ++(i-gz)r4

(d)

POINT

(13)

ON

POINT

A

MOMENT-FREE

ON

A

MOMENT-FREE

Y-EDGE

ADJACENT

X-EDGE

TO

ADJACENT

MOMENT-FREE

A

TO

A

X-EDGE

FIXED

Y-EDGE

FIXED

X-EDGE

++(I-p*)r4

(f)

POINT

ON

A

MOMENT-FREE

NOT&-For

Y-EDGE

ADJACENT

TO

general notes see Figure

FIGURE 41.-Load-dejlection

relations,

A

39.

Sheet III.

APPENDIX II

(0)

POINT

ON

A

MOMENT-FREE

59

CORNER

(b)

POINT

ON

A

FIXED

*+ k (d)

POINT

ON

A

CORNER

FIXED

Y-EDGE

+2r2

L r4 (0)

POINT

ON

ADJACENT

A

FIXED

TO A

X-EDGE

FIXED

*

-4-4r*

9

CORNER

+I

*

(t)

POINT ON ADJACENT

A TO

FIXED A FIXED

-l.,;,,,fi/,z ~

Y-EDGE CORNER

=

(P-RI r h2

h’ T-’

!!I, D (0)

POINT TO

ON A

A FIXED MOMENT-FREE

X-EDGE

ADJACENT Y-EDGE (h)

POINT

ON A

TO

(i

1

POINT

ON

A

FIXED

X-

ht0htE~T-FREE

Y-CORNER

Nom.-For

( j)

POINT

ON

A

general notes see Figure 39.

FIQURE 42.-Load-deflection

relations,

Sheet IV.

A FIXED MOMENT-FREE

FIXED

Y-

Y-EDGE

ADJACENT X-EDGE

MOMENT-FREE

X-CORNER

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

60

+r’

I

i P h

+

2 rp

-

4 rz

-

I

4 r4

+ 2re

I -P

= +2r*

1

+--rh-s+<

-4r*-

6r4

____ rh--.+

(0)

+2r*

____ rh ____ ++-rh--+l

INTERIOR

I

Th

+

POINT

r4

I

I

=

7

,+

(b)

POINT

I

+2

r,,

-+,+.-

rt

-4P

-

rh --__

ADJACENT

6r4

+

TO

+

----

A

(2

P X0’

-pL)r*

rh -----

,j

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

b---rh---+j+

(c)

POINT Nom-For

FIGURE 43.-Load-de$ection

ON

_____

A

-PI

r’

rh----A

MOMENT-FREE

Y-EDGE

general notes see Figure 39. l’elations,

P 37

vertical

spacing:

S

at h; 1 at h/B, Sheet V.

h’

h’ T’

APPENDIX II

61

+ r4 I

s +128

-&+Zr*

= I-&

1+$+4r’

p--rh---*

_____

rh

(a)

----*

----

rh

INTERIOR

____

*---rh--q

POINT

+ r4

I k-rh--*--

(b)

POINT

+8r4 I

rh ---+

ADJACENT

__-_-

TO

A

rh ----

+/

MOMENT-FREE

Y-EDGE

+t(t-pe)r4

+*

- & +(2-tc)r*

+*-2(l-p)rc

-3(1-p*)

r*

I 7

1 +4(1 km-(c)

POINT

rh---e ON

NOTE.-FOI FIGURE 44.-Load-dejlection

A

-P)r’

____ rh _____ 4 MOMENT-FREE

Y-EDGE

general notes see Figure 39. relations,

vertical

spacing:

d

at h; d at h/d; Sheet VI.

- P rh'

-.h' 0

62

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

5 +128 +105 IJ =

128

I I 7

+&+4+

-64

-&.

r’++

--__

-64

I

+ 64 r4 3

I ,+t--

7

+&+4rr

t3rt-40r4

I

r,, ---+

(a)

L&+.

I

__---

+-++-~‘,--~

INTERIOR

POINT

%

=

i $‘h 7 +=+4r=

-&

f

-z

35

f ‘h k

- are-40r4

+64

/-+- rh--*

(b)

ADJACENT

TO

+&+2(2-p)r*

____ rh _- ____ 4

A

MOMENT-FREE

+

+

(I -pe)

Y-EDGE

r4

= -is-

+

$

+ 2(2-p)r*

____

(c)

rh---+-------rh

POINT

ON

NOTE.--For

FIGURE

45.-Load-deflection

relations,

IP-R) h’ . rh2 0

-&-4(1-p)rL-20(i-p1)P

+

p

rh2

r4

3

____ rh ____ +f+--

POINT

PA!.

+

(I-pz)r*

_______

A general

4

MOMENT-FREE notes

see Figure

vertical spacing:

Y-EDGE 39.

2 at h; 1 at h/2; 1 at h/4, Sheet VIZ.

D

APPENDIX II

63

7h

P h’ = --. rh* D +4rP

-are-

32r4

I (a)

+4re

+6r4 I

INTERIOR

I

x

POINT

+$

r4

I

% i,h *

=

+‘h +4rp

+ +‘h P

-Ore-

32r’

+2(2

--. I' rh*

-p)r*

+0r4

+-rh--4---rh

(b)

POINT

----

ADJACENT

TO

+ 2(2

I 7

-PC)

A

MOMENT-FREE

r*

++-+ 0(1 -p)r* + 661,-pcljr4

I (c)

POINT

rh ----*

ON

NOTE.-For FIGURE 46.-Load-deflection

=

(P-R) --. rh'

h4 0

-~-p)r*-i6(1-pe)r’

+2(2-p)r*

k---

Y-EDGE

-4wF12(1-pV

-4(2-pL)r’

-I

+t

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

_____

A

rh

+4(1

- ___-

-p*)r’

4

MOMENT-FREE

Y-EDGE

general notes see Figure

relations,

vertical spacing:

39.

1 at h; 3 at h/B, Sheet VIII.

h4 D

64

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES + f r’

-I

- Lg -,2p

105 +256

r’

+&+gre

-Tk

b--rh--e

-g-

5

--___

rh

(a)

INTERIOR

----+

I F

+ &-+

I +i56

+105 256 0r r

=

?.

-35

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

POINT

- 6r’-

I 4oP

I

G-l6r*-192r4

I

I

+

I b--rh--e----

rh

POINT

c

----_

ADJACENT

64t’

-&

+2(2-pL)r

-$

e

-_____

TO

-p)r*

+2(2

A

I

+ & t 4(2-p)+

rh’

0

+&t4(2-p)r

rh------d

MOMENT-FREE

+$(I

-2)

Y-EDGE

r* II

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

II

=

-6s

Lx..

I

I

5 +256 I

- &

- 9(1 -p)r~9ql-p*)

+32(1

P-R) rh*

h’ . D

r*

-/4r’

II km-(c)

rh ---*

POINT

__---___ ON

NOTE.--For FIGURE 47.-Load-deflection

relations,

A

0

--A

= +&+Br

+A!..

rh

105 +isd

-6

(b)

______

105 =-,2rg

_

++r . 25

-&+4rg

.

16rL - 192r4

I +zz

496

++$+24r’+-j-r

rh------A

MOMENT-FREE

Y-EDGE

general notes see Figure vertical spacing:

39.

1 at h; 1 at h/2; 2 at h/4, Sheet IX.

APPENDIX II + $

65

r4

--.P rh*

= -7

+&-

+8r’

-$+

-16r’

126

7 +3-i-++rP

-320r’

h' D

7 -128 A

+ 51e 3

+-rf+-+

r4

_____ rh ____ 4---rh---~--rh--~

(a)

INTERIOR

++

POINT

r4

- *

+&

+ 69

-$$

-16r*-320r’

+w

be-

(b)

rh --+

POINT

_____

+4(2-p)r*

r*

rh ----+

ADJACENT

+&

h' D

--.P rh*

=

______

TO A

rh ____-

4

MOMENT-FREE

+ f

(I -p*)

Y-EDGE

r*

f

$ * fh x f;, f

5 +e56

-&

+ 2(2-p)r*

+A

-4(1-p)r*-20(1-p*)r’ =.

I

P-R)

rh'

r)

-T&i

+*

+4(2-p)r

*

h’

D

-&-8(1-p)rc-i60(i-$)r4

+?(I

-p*)r4

II b--fh---e (c)

POINT

______rh _______4 ON

Nom-For FIQURE 48.-Load-dejlection

relations,

A MOMENT-FREE

Y-EDGE

general notes see Figure 39. 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,, -----

(0)

+-s-v

INTERIOR

I

POINT

I

+gr*

32r’

+2(2-p)r*

+4rc

- 8rg-

32r4

+2(2-p)

_____

______

__-_

rh

ADJACENT

POINT

ON

FIGURE 49.-Load-dejfection

rh

_____

r*

++

TO A MOMENT-FREE

____ *-

NOTE.-For

T

+2(2-p)

.f+----rh

r,,+..,

POINT

-8r’-

I

(Cl

+--

+4rz

b--rh---+/.+

(b)

rh -----

rc

Y-EDGE

I -4(1-p)r1-16(1-p)r411

__---

rh -----

4

A MOMENT-FREE

Y-EDGE

general notes see Figure 39. relations,

vertical spacing:

4 at h/8, Sheet XI.

APPENDIX II

67

+ 64 r. 3 + 9r*

-l6r*-

+6rP

- 16r*

192r4

-

+6r2

256r’

I

+64r4 J b-rh--+----rh

____

(0)

+----rh

----+--rh--d

INTERIOR

I

l

POINT

+y

r’ I

=

p

--.P rh*

I ~-rh-~---

(b)

POINT

+64r*

I

rh---++g-----rh

ADJACENT

TO

____ 4

A

MOMENT-FREE

+y

$h 4

I

+4(2

I-++

-+

(I-p*)+

-6(l-p)r’-

-p)”

96(1-p’)

r,, -+

POINT

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

_______

ON

NOTE.-FOT FIGURE 50.-Load-deflection

r4

-8(2-p)P

+ 32(l

(c)

II (I

+4(2-p)r’

+-c-m

Y-EDGE

A

r,, - _____

-PL)r4

II

+,

MOMENT-FREE

Y-EDGE

general notes see Figure 39.

relations,

vertical spacing:

1 at h/B; 3 at h/4, Sheet XII.

h’ D

68

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

I r)

+IoI

-j+$-24r’

512

-

&

+&

+z+46rt+yr4

+ 16r’

-j$-24rt

-.&-32~1536r4

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

____ h

(a)

+&

+s

+ 16r’

=

f’

h’

rh’

D

- &

_____ rh _____ *---rh--~

INTERIOR

+ 84 3

POINT

r4

p At. -3-D

-!r4 +&+16r*

-&

- &

-32r’-

1536r4

+ &

+ 6(2-p)

_____

4

r*

+512r4

k--rh-e

(b)

-____

POINT

rh

ADJACENT

____

+j+-----

TO A

I 5

-&+4(2-p)rE

+sle

t$

rh

MOMENT-FREE

++(I

-p*)r*

Y-EDGE

II

-6(l-~)+l6OfJ+)+

= -I

+&+6(2-p)r*

256

-&

- 16(1-/~)+766(1

+256(1

(P-R) h’ --. rh’ D

+,r’

-PC)+

II b---rh

(cl

____ +

POINT

ON

NOTE.--For FIQIJRE 51.-Load-deflection

relations,

_______ rh _______ 4

A MOMENT-FREE

Y-EDGE

general notes see Figure

39.

vertical spacing:

1 at h/d; 1 at h/4; 8 at h/8, Sheet XIII.

.

APPENDIX II

$ x

- 16r*-

69

256r4

f'h

=

x f'h +6r*

f flh

-16rz-

9

266r.

P -;i;p

+ 6re

+64r4

bt---rh.--+

_____rh ____e

(a)

-I f:h * t:” * flh *

_____rh ____+-rh--d

INTERIOR

POINT

+64r4

I 7Ic+T +Ort

-l6r’-

256r’

=

+6re

--16+

-256r4

+4(2-p)

P h' Ti;fO'

rt

+@h k

+64r4

f+--rh---*

(b)

POINT

____ rh _____ *

ADJACENT

TO A MOMENT-FREE

,+--r,,/+

(c)

POINT

ON

NOTE.-For FIGURE 52.-Load-deflection

______ rh _____ -f

Y-EDGE

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

A MOMENT-FREE

Y-EDGE

general notes see Figure 39. relations,

vertical spacing:

,$ at h/4, Sheet XIV.

h' T-'

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

70

I

+16r’

-32r’

b--rh--+

-_____

rh ----

(a)

(

*-----rh

- 32r’

+,6,.F1

____

INTERIOR

+ l6r*

f fh

- 1536r’

4--,-h--d

POINT

-

1536r4

+6(2

-p)rc

= m$+.

-+q++&

+I++

+ l6r*

-32r*

- 2046r’

+6(2-p)r*

+ 512r4

(b)

POINT

ADJACENT

TO

I + 6(2

k--rh

(c)

POINT

______

ON

NOTE.-For FIGURE

53.-Load-deflection

MOMENT-FREE

+?(I

-pE)r4

-160 +)r’-

-p)r*

--+

A

A

rh _____

relations,

II

766(1-/L*)

r’

-4

MOMENT-FREE

general

Y-EDGE

Y-EDGE

notes see Figure

vertical spacing:

39.

1 at h/4; S at h/8, Sheet XV.

71

APPENDIX II +512r4

P rhL +16r’

- 32re-

2046P

+l6r*

+ 512 r’

kc--

rh ---*-----rh

_____ *

(a)

-____ rh _____ G---rh

INTERIOR

--+

POINT

I

+512 r4 + 16r’

- 32r*-

2046P

+16r’

- 32 r’-

2046

r4

+6(2

-pL)r*

+s12r4

k--rh---*

(b)

POINT

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

ADJACENT

TO

A MOMENT-FREE

+256(1-

+6(2

-pLr*

+6(2-p)+

+c---rh--+

(c)

POINT

ON

NOTE.-For FIGURE 54.-Load-dejlection

Y-EDGE

PL) r’

--16(1 -~)r’-lO24(1-~~

-16(1-~)+1024(1-)

____ -rh------4

A MOMENT-FREE general notes see Figure relations,

vertical

spacing:

Y-EDGE 39.

4 at h/8, Sheet XVI.

h’ 0’

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

b-irh

+--+rh---h--s

rh---h-irh-F(

+

I

4r’

I

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

=

-16r*-

I

--.(P-R) rh*

h' D

tzr' +4r4

I

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

+ + +4rz

-6rz-

r' 2r4

+4r'

=

(P-ax, rh'

NOTE.-FOT FIGURE

55.-Load-deflection

general notes see Figure

relations,

horizontal

spacing:

39. .$ at rhl%, Sheet XVII.

D

APPENDIX II

73

=

(P-R)

‘.

rh’

- 16P*-

+ 6P

16f’

0

+er’

+4r4

I 7 + 6r’

- 16f’-

16f4

=

P rhTD

2.

P

P 7

h’ -= D

+ 6P

+4r’ +-+rh

-+--$

rh---+--+rh

I

---+-

+ 4 r4

rh

-4

I

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

I

Th

+$r

l

I

f i” + 4r’

-

$.,, r2

8r’

-

3r’

rh

+4r’

+ $r*

1

+ 4r*

1 -

,+,,++$

50.-Load-dejlection

-

rh ---wf+--3rh

NOTE.--For FIGURE

6~’

relations,

2r4

1

+4r*

--+-

rh -4

general notes see Figure

horizontal

spacing:

1

39.

3 at rh/d; 1 at th, Sheet XVIII.

74

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES +6f'

=

+ 8rP

-

12r*

-

24r'

(P-A) 7 rh

h' --

D

+ 4r*

+6r4 1 b-irh-rt<---$rh--+

____

rh---+--rh-+

+2r'

= +

6fP

-

i2r2-

24r4

P --. rh'

h'

P --. rh'

h4

0

+ 4rg

+6r'

b-$rh-+---irh

--+f+----rh---*--rh--,../

+

L

Jr’ 4

=

+6 r' +6rp

-

12r*

-

16r.

D

+4r*

+6r'

h---+rh---Zt,

b-$rh

____

rh---+--rh--+

+Jr’4 +4rz

-

6r*

-

3r' =

P 2

rh

h4 -0

+2r*

b-irh-+---krh

--+----,-h

---*--rh-+

+ Jr’ 4

I

I

=

-

P rh'

-6r*

+4r*

+zr*

Nom.-For FIGURE

57.-Load-deflection

relations,

Jr4

+2r2

4

general notes see Figure horizontal

spacing:

39.

2 at rh/d;

2 at rh, Sheet XIX.

h' -. D

APPENDIX II

75

=

+6r'

I

b-krh*

+

=r'

___- rh ---e

D

rh ---e--,-h-+

1

=

k-irh-4

h4

rh'

I

_-__ ,-h---e---

I3

(P-RI --.

---_

,-h ---*-,-h

P I--. rh

h'

P

ha

2-* rh

D

0

-4

Th f

h

khrh

4--rh

--e---rh---*-rhd

+r'

;f” i;” $,, r2

I 7 +2

r*

-4r*

+

bhrh

6r'

+

2rP

8r' 3

-4

b--rh--+=-/+---rh--+-j+-rh

I

+r4

I

=

-

+2r*

4r"

-

4r4

--. P rh*

+ 2rp

+r*

birh

+---rh

Nom.-For FIGURE

58.-Load-desection

rh

--e---rh--W/N-

general notes see Figure

relations,

horizontal

spacing:

4

39.

1 at rh/8; 3

at

rh, Sheet XX.

h' D

76

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES +6r4 +4rt

-

-

6r*

32r'

+

4rt (P-R) --. rh'

=

+4r'

- 6P

-

+

32r'

+

h' D

4r'

6r'

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

+

Lr' 3

P --. rh*

= +4r'

-6r'-

32r' +

h' D

+4r'

6r*

+r4

I 7

P

=

+P

h4

77-D' +4r'

-

er'-

24r4

+4r*

+6r'

b-rhh---rh

--+---rh--4--rh-c(

+r4

=

+2r1

-

4r'-

6r'

P

h*

rh'

D

--.

+2r'

++rb

b-r h-4---

rh --4--+

rh ---h-

rh --f

r*

=

+2re

-4r'

-

4r'

+2r*

+r* +-rh-+---

rh

NOTE.-For FIGURE 59.-Load

---+---

rh--+-

rh-4

general notes see Figure

deJEection relations,

horizontal

spa&a:

3% .4 at rh. Sheet XXI.

--.

P

h'

rh*

D

APPENDIX II

I

77

I

I

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

J

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

b)

(a)

l+rh+t+rh+ (4

INTERIOR

POINT

Mx = $31 My = 0

-4

t+-rh-+-rh k4

(e)

*

tJI

+2rx b=?pry!t

-t

+e(l-p)r*

4

*

My=++

*

H Mx

=

M,

pm,

=

0

(i)

(h)

EDGE

AND

CORNER

POINTS

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

I+-rh-+I+-rh-+I

(i)

(k)

INTERIOR

AND

EDGE

Mr

POINTS

= 0

(4

EDGE

(ml

- NONUNIFORM

SPACING

MI

AND

CORNER

POINTS

- FRACTIONAL

VERTICAL

NOTES 4

=

M xv =

M,

=

MYI =

0 0

at ot

NOTE.--For

either a fixed or moment-free any point on o fixed edge.

general notes see hgure

FIQURE 6O.-Momentdejlection

-

0 (t)

1s)

corner.

39.

relations.

SPACING

I

78

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES MyP He

Mx P ah*

b-+-rh

L

,--rh-+--rh

-

+-rh-w,

0

-v,

Nom.-For

general notes see Figure

FIGURE 61.-Moment-dejlection

relations,

various

39. point spacings.

APPENDIX II

INTERIOR

POINT

vy=$&0 eF -(l-IL)

+I

+2

I I -u) t

-(I-P)

r2

-2(l

t

0 r2)

+I

+r2

(4 POINT

ADJACENT

TO

A

MOMENT-FREE

EOQE

E -r2(l-fi)

+2re(i-Cc)

D v,

I

=Tm

0

-2+(1-u) +r2(

(e)

(f) POINT

ON

A

MOMENT-FREE

EDGE

(9) POINT

ON

A

(h) MOMENT-FREE

NOTE.-FOT FIGURE

EDGE

ADJACENT

TO

A

general notes see Figure

62.-Shear-deflection

MOMENT-FREE

39.

Telations, Sheet I.

CORNER

I-u)

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

80

v,=P

h’ 2

’

(b) POINT

ADJACENT

TO A FIXED

EWE

(cl

(4 POINT

ON

A

FIXED

EDGE

/ / / / , VY =

+

-$-&r

(4

IfI

POINT

ON

A

FIXED

EOQE

ADJACENT

TO

A FIXED

CORNER

(e) POINT

*

(h) 0N.A

MOMENT-FREE

NOTE.--For

EDQE

ADJACENT

TO A FIXED

general notes see Figure

FIGURE 63.~Shear-dejlection

relations,

39. Sheet ZZ.

EDQE

-2

APPENDIX II

vx =-

81

ph r3

b-rh-+-rhe

b- rh +-rh+

-6(2

k-

r h -+-

+r2)

rh+ I

Note:

These orroys points on magnitude

opply Only where the load at opposite sides of the centerline but opposite in direction.

Nom.-For FIGURE

general notes see Figure

&I.-Shear-de$ection

relations,

39.

Sheet ZZZ.

corresponding is equol in

+4

82

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

+ 32

-66

i

- I066

+256

+ 1670

+32

+26.6

-59.6

+26.6

-1066

+ 1669

-

+32 -1086

-66

[ + 256

+32

f

+32

-66

f

(a)

+ 256

INTERIOR

+32

-66

-1066

+50(1

+ 256

POINT

ADJACENT

+122.66 -;!;6

+256

]

+32

+I

(b)

POINT

3

I066

+256

-517.

12

TO

A

+76X46

FREE

X-EDGE

-517.12

+122.66

f’[

-70

+32

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

(C) VERTICAL

INTERIOR

(d)

POINT

SPACING:

3

AT

I

h;

AT

POINT

ON

A

FREE

X-EDGE

+h

l

tl

f

t

I26

+ 64

-

- 640

+G

+64

152

+ 64 -

3

-I60

640

+ IO

-6

210

-936

+y

+

-

t

+I26

-26

+64

176

(0) VERTICAL

INTERIOR SPACING:

336 +64

+a

p-v

SAT+h

FIGURE

f ,,++&

(f)

POINT IATh;

VERTICAL

65.-Load-deflection

coeficients,

SPACING:

r=M,

-6

-10

2

p=O.Z.

AT

-936

+210

+ 176

-26

3

+ ,,++--

INTERIOR

+ IO

$ ,,-+--

f

h-4

POINT h;

I

AT

fh;

I

AT

fh

--x

i...

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

84

I

I

I

DEFLE(

IO.1

I

1 1.0

1 +.000426

+.005659*I

+.ol!

Deflection

=

(Coefficient)(ph’/D)

Y -+-IJP

l@J beep-t; LOAO

cc-p -4 I

LOAD

Deflection

Starred when from FIGURE

67.-P&e

values computed the corresponding the 20 equations.

=

Iz

k-p LOAD

-ti

POSITIVE

SIQN

CONVENTION

m

(Coefficient)(ph’/D)

NOTE from 30 improved

equations are discorded value is oljtoined

$xed along three edges, deflection coefiients.

a/b=%.

Various

loadings.

i5 %

”, I

: i

.Nxxm31

ay”w-*“sIY

l o XIYMN

MOMENTS AND REACTIONS FOR RECTANGULAR PLATES

86

(0)

POINT ON A FIXED FREE X-CORNER [FIQURE 39 II,]

Y-

(b) POINT ADJACENT

ON A FIXED TO A MOMENT-FREE [FIGURE 39 (II,]

I-EOQE X-EWE

(C)

POINT ON A FIXED VERTICAL SP.oI*G: h [FIGURE 42 iol]

tcfh-++hc( (d)

POINT

“ERTlCAL

ON

I+ A

FIXED

SPACING: [FIGURE

+ 44

II

Y-EWE ANO

(0)

+

POINT

ON A [FIGURE

h

FIXED 49 (Ol]

CORNER

(f)

POINT

Y-EDBE ANO

+

h

h -+-fh+

ON A FIXED [FIGURE 49 IO,]

X-EWE

to)]

REACTION-DEFLECTION r

=

COEFFICIENTS

l/4

p

=

0.2

T h

i l+h

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

k+h+++hd (0)

INTERIOR [FIGURE

(h)

POINT se

POINT

ON

A

[FIGURE

to)]

BENDING

FREE !N! Ml]

MOMENT-DEFLECTION r

=

(i)

INTERIOR

POINT

“E”TIOAL

scAaIw0: [ .=IGURE

h A”0 se (I,]

EDQE

COEFFICIENTS

l/4

jl

=

(j)

INTERIOR [FIGURE

POINT se (b)]

ON

A

[FIGURE

FIXED se

(m)

EDQE

(PI]

“ERTlCAL

-+fh

MOMENT-DEFLECTION r

=

COEFFICIENTS p

I/4

-

PQINT

SFAOIW:

h se

(MY)

0.2

NOTES To

find

the

net

compute the the deflection Figure these FIGURE

W.-Numerical

numbers numeric01

reaction products of the in

or

the

brackets orroys

values

bendinq

moment

of the coefficients correspondinq points refer were

to

qenerol

at

b

tl

INTERIOR [FIGURE

BENDING

+

(M,)

l++h POINT

h

0.2

+fh+-+h+i (k)

f

ony

focal

of the oppropiote ond multiply their expressions

point, orroy sum

from

by by (O/h’). which

computed.

of typical moment and reaction

arrays,

r=x’,

p=O.B.

AGO

km)]

87

APPENDIX II POINT Yfl

DEFLECTIONS 0

6

I

0

+.017022

-

I

1

1 t.049660

1 + .063466

1 t.

+.I16792

107935

1

0

+ .016122

t .046640

+ .076499

+. 101377

+ .I09650

4

1

0

+ .016030

t.046526

t .077914

+. 100572

+.I06761

1 +.043935

1 t .073156

1

+.015263

2

0

+ .010730

+ .029914

+ .046903

I

0

+ .004699

t.013261

+ .02 1325

7

0

+ .001835

+ .004944

+ .007660

0

0

0

04 03 02

0

1

t.125

+I.131256

+I.256256

+.I25

+I.131056

+ I .‘256056

.-_-.

+ .a32224

+. 190464

t.09375

t .736474

+ SO46675

+.I78392

+ .225267

+.03125

- .000992

+ .030256

01 07 00

[ t .Ol5625

IO

1 t .03125

I

I

-.056720

1 - .043095

1 + .029514

-.001712

I

I

+.I92016

+ .223266

+.I78613

+ .03125

+ .240512

+ .271762

t.217410

+ .03125

t .256320

+ .207570

+ .230056

+ .I41346

6 5 4 3 2 0

,

MOMENT

-

+.009607

1

I

* lncludrs

+6.249145

2

I 7

+ .023630

+ .03125

BENDING

only

of bo.

MJpb*

3

4

5

t .000693

1

-.005724

1

-.009592

1 -.010863

+ .000622

1

-.005565

1

-.009301

1

-.010539 - .010531

+ .020636

+ .009346

+ .020516

+ .009253

1

t -000553

1

-.005621

1

- .009305

1

+ .019562 + .013734

+ .006526 t .005335

1

t .000273

1

- .005438

I

-.006766

I 1

0

+ .000470

POINT NO.

- .000330

-.003930

-.005917

+ .001266

+ .002012

+ .0025 I7

BENDING

-T SE?1UhllJ 6 1

0

I

I

0

I

004127

I

MOMENT

-

.009690

I

2

3

4

5

0

0

0

0

0

+.001901

+ .00022

I

- .000949

-.001639

- .001666

+ .004104

+ .OOl625

t .00002

3

-.001264

- .002076

- .002344

+ .003912

t .001559

- .000364

-.001636

- .002734

- .003036

+ .002747

+ .000703

-.001051

- .002366

- .003177

- .003449

+ .002349

+ .006326

t .01006

+ .012566

+ .013460

70.-Plate

fixed along three edges, dejlections-reactions-bending

I

My/pbe

t

0

1 ,

t .002694

4 3 2

1

I

- .006549

!I

0

I

+. I I3077

+.I10096

POINT NO.

0

t.029536

+ .03125

I c”

+ __-_. 020917

t.010522

0

1

20 30 40 50

FIGURE

I

5

1310

I

w/(ph4/D)

I

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,” Proceedings, 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,

7.

8. 9.

10.

American Concrete Institute, Vol. XXIV, page 408, 1928. 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. Barton, M. V., Finite LX@rence Equutions for the Analysis of Thin Rectangular Plabs, University of Texas, 1948. Jensen, V. P., “Analyses of Skew Slabs,” Bulletin S&s No. $%?, University of IIhnois, Engineering Experiment Station, 1941. Scarborough, J. B., NumerieaZ Mahmuhal Andy&, John Hopkins Press, Baltimore, 1950.

89 *U.S.

GOVERNMENT

PRINTING

0mcE:

19~5-8341-386

.

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of the Bureau

of Reclamation

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of the interior is of the Nation’s

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