Plate Formulas

PLATE, FORMULAS Other books by the author: HANDBOOK OF FORMULAS FOR STRESS AND

BEAM FORMULAS SHELL FoRMULAs

(in

WILLIAM GRIFFEL Mechanical Engineering Scientist Department ol the Army Picatinny Arsenal

preparation)

FREDERICK UNGAR PUBLISHING CO. NEW YORK

('opyright @ 1968 by lircdcrick Ungar Publishing (kr., Inc.

I'rintcd in the Unitad States ol Atncrica

Library of Congress Catalog Card No. 67-26127

PREF'ACE

This book presents a series of tables containing computed data for use in the design of comporlents of structures which can be idealized as llat, circular, rectangular, square, triangular and elliptical plates. A total of 139 tabulated cases with most common, and some not so common' loadings and supports*typical of those encountered in design-cover the subject of "Plate Formulas" quite thoroughly. In addition, the book contains a detailed treatment of large deflections of plates which many referonce books ignore completely, although such deflections are frequently met with in practice. This is the case where the deflections are of the order of magnitude of the thickness. A solution of statically indeterminate plates, encountered when there rurc more supports than necessary to maintain the stability of the plates, is trcated in detail. Removal of a redundant support would impair not only lhc structural integrity of the plates, but also that of affiliated components. Superposition is the usual procedure for solving statically indeterminate problems; however, the generalized equations of deflection, moments or skrpe must be known for the analysis. Such equations are presented for trniform load acting on a concentric circle of a thin, flat, circular plate. Also, a treatment is presented for cases of varying load distribution in which constant force, divided by the radial distance squared acts on a thin,

llat, circular plate. The general tone of the book reflects the author's approach towards thc solutions of stress problems, namely, simplicity and accuracy. At this cra of the race to the moon, an era of sophisticated structures the likes of wlrich were only in the imagination of designers a generation ago, it is inrpcrative that the present-day engineer take advantage of all the means tlrirt twcntieth-century technology has to ofier. In particular, reference is rnadc to thc electronic computer which was instrumental in putting this lxxrk toscther.

vL In order to present study of the subject n ^ltu"o-pr"tr"nrive ^ "" "r! engineering matter, an exhaustive review of literature was made (see bibliographical references at the end of this book). over 200 technical papers were reviewed with the purpose of presenting to the practicing engineer the most complete and useful data. once the reference material was gathered, there was the problem of presenting it in the most simplified and practical form. To reprint the equations in their original form was found to be impractical because, more often than not, the solutions to such equations are prohibitively time consuming. Furthermore, when one contemplates the unknowns in applied loads, in metal characteristics, and even in the dimensions of assembled structures, it becomes obvious that even the most rigorous calculation method may result in values of questionable accuracy.

The basic simplification used in the book was the assumption of a of 0.3, a value used for steel and aluminum. Then, by a technique of mathematical conversion, and with the help of a digital computer, the conventional formulas were brought into a greatly simplificd form, whereby dimensional ratios and loading patterns have been consoridated into one dimensionless "K factor" with the body of the formula retaining two principal dimensions, a material factor and a load factor. calculated K-factors, over applicable ranges for the variables, are presented in tabulated form for different cases of loading, support and types of plate. All tables and formulas presented here were published and copyrighted from time to time in: (1) "Journal of Applied Mechanics,', (2) "Argonne National Laboratory," (3) "product Design and value Engineering" (Canada), (4) ',Machine Design,,' (5) .,product Engineering,,, and (6) "water Resources Engineering Monograph." Thus, opportunity afforded for criticism has been of considerable advantase. To the publishers of the above journals and otiers who have gencrously permitted the use of material, the author wishes to express his Poisson's ratio

thanks.

Wrrrralr

Gnrnpnr,

CONTENTS

Chapter

1

FLAT PLATE DESIGN

L 2. 3. 4. 5. 6. 7.

J

Symbols and units Eftect of Poisson's ratio

J

Equations Edge conditions Assumptions Flat circular plates with concentric holes Moments and reactions for rectangular plates

5 6 7 7 8 10

Chapter 2

I]ENDING OF CIRCULAR PLATES UNDER SYMMETRICAL LOAD

8. 9.

10. 11.

65 68

68 68 72 76

Theoretical aspect considerations

Numerical example

Chapter 3 BENDING OF CIRCULAR PLATES UNDER LOAD ON A CONCENTRIC CIRCLE

14. 15. 16. L7. 18. 19.

65

Edge restraints System of units Assumptions

1,2. Design

13.

A VARIABLE

A

UNIFORM 127

Edge restraints System of units

727

Theoretical development

1,29

Generalized cases Design considerations

130 130

Statically indeterminate circular plates

r32

r28

Chaptar 4

I.ARCE DEF[,8(]'I'ION OF PLATES

20. Strgsscs ancl clcllcctions 21 . ('it'crrlirr solirl lllato with clartrpccl cclgo

139

139

t41 vii

viii

Contents

22. 23. 24. 25,

Circular solid plate with simply supported edge Elliptical plate with clamped edge Rectangular plate with uniform load and all edges simply supported Rectangular plate with uniform load and two edges

1,43

supported and two edges clamped

t49

t45 t48 ,I'N

BULATION OF' PLATES

Chapter 5

BENDING OF RECTANGULAR PLATES UNDER SIMULTANEOUS LATERAL AND END LOADS 26. Uniform lateral load w and tensile or compressive forces P acting on a pair of opposite edges (simply supported) 21

.

Uniform lateral load and uniform tension acting on all four edges (simply supported)

151

Itable

151 159

('rt,sr I

Inner Edge Supported. Uniform Moment Along Outer Edge

17

('rt.sc

Outer Edge Supported. Uniform Moment Along Inner Edge

T7

Inner Edge Supported. Uniform Load Along Outer Edge

t7

Inner Edge Fixed and Supported. Uniform Load Along Outer Edge

l7

Outer Edge Fixed and Supported. Uniform Load Along Inner Edge

T7

Outer Edge Fixed and Supported. Inner Edge Prevented From Rotating. Uniform Load Along Inner Edge

t7

Chapter 6

. 2.

SANDWICH PANELS WITH UNIFORM SURFACE PRESSURE

( tt,st'-J.

AND UNIAXIAL COMPRESSION

('tt,st'

28.

Uniform surface pressure and uniaxial compression

r63

THICK CIRCULAR PLATE WITH AN

29.

Circumferential stresses

L7L

('(r,\t,

3I.

System of units Mass Force and weight

32. 33. 34. Pressure 35. Acceleration of gravitY BIOGRAPHICAL REFE,RENCES SUBJECT INDEX

().

17t ('rt,:(

Appendix MASS VERSUS WEIGHT 30. Introduction

4.

(',r,rr,5.

Chapter 7

ECCENTRIC/CONCENTRIC HOLE

I

I,'OITMULAS FOR CIRCULAR PLATES WITH ( ONCENTRIC HOLES

175

t75 L75

r77

(

7.

rr,rr'8.

('tr.t(

().

r18 179

r79 181

195

r (

I

ttsr

10.

tr,s( I

tt,tr

l.

12.

Inner Edge Supported. Uniform Load Over Entire

Actual Surface Inner Edge Fixed and Supported. Uniform Load Over Entire Actual Surface

18

Inner Edge Supported. Outer Edge Prevented From Rotating. Uniform Load Over Entire Actual Surface

18

Inner Edge Fixed and Supported. Outer Edge Prevented From Rotating. Uniform Load Over Actual Surface

18

18

Outer Edge Supported. Uniform Load Over Entire

Actual Surface Outer Edge Fixed and Supported. Uniform Load Over Entire Actual Surface

( ,r.st 13. Outcr Edgc Supported. Inner Edgc Prcvcntcd From Rotating. Unilorm Load Ovcr Entire Actual Surfacc ('rt.st'14. Outcr Eclgc Fixcd and Supportccl. lnncr Eclgc Prcvcrrtcd Iirorr l{otatins. Unil'ornr Loacl C)vcr lintiro Actual Surfaco

18

18

t9 t9 ,T

Tabulation ol Plates

Tabulation ol Plates

Table 2

( 'tt.rc 32

SOLID CIRCULAR PLATE WITH UNIFORM LOAD Case

15.

Case

16.

r

20

Edges Fixed. Uniform Load Over Concentric Circular 20

Area of Radius r Case

Case

17. 18.

Outer Edge Supported and Fixed' Uniform Load on Concentric Circle of Plate

a

Outer Edge Simply Supported. Uniform Load on a Concentric Circle of Plate

20 2T

19.

20. Case 2 L

Case

('tr,te 34. ('tt,sc 35.

('rtse 36.

( 'rtsc 38,

Outer Edge Supported. Uniform Load Over Entire

Actual Surface

21

Outer Edge Supported. Uniform Load Along Inner Edge

21

Inner Edge Supported. Uniform Load Over Entire Actual Surface . Case22. Outer Edge Fixed and Supported. Uniform Load Over Entire Actual'Surface Case 23. Outer Edge Fixed and Supported. Uniform Load Along Inner Edge Case 24. Outer Edge Fixed and Supported, Inner Edge Fixed. Uniform Load Over Entire Actual Surface Case 25. Outer Edge Fixed and Supported. Inner Edge Fixed' Uniform Load Along Inner Edge Case 26. Inner Edge Fixed and Supporteb. Uniform Load Over Entire Actual Surface Case 27. Inner Edge Fixed and Supported. Uniform Load Along Outer Edge Cetse 28. Outer Edge Supported. Inner Edge Fixed. Uniform Load Over Actual Surface Cust:29. Both Edgcs Fixed. Balanccd Loading (Piston) Ctn'a 30. Inncr Ilclgc Strpportcd. Urrilbrm Load Along Outer Edge ('tt,tc -l I . Iltttcr litlgc Strpportctl. ()trtcr Iiclgo Prcvcrltotl llrotrt l{otlrl.irrs. tJtrilirrttr Ltllttl ()vcrr I irrtirc: Actttlrl Sttrlitcc

('tr,se 33.

t'tt|;a 37.

CIRCULAR PLATE WITH CONCENTRIC HOLE (CIRCULAR FLANGE) Case

Outer Edge Supported. Inner Edge Prevented From

Rotating. Uniform Load Over Entire Actual Surface

Edges Supported. Uniform Load Over Concentric Circular

Area of Radius

.

2t 22

( 'rtsc 39

( 'ttsc 40

.

('tt,tc 41.

22

(

22

( 'tt,tc

22

.

(

tr,tc

tt,st

42, 43. 44.

22 ('rt.s( 45,

23 23 23 23 ZJ

t

rrtr' 46.

t ttst'47. ('tt.st

'llJ.

xt

Outer Edge Fixed and Supported. Inner Edge Prevented From Rotating. Uniform Load Over Entire Actual Surface

24 a/1

Inner Edge Fixed and Supported. Outer Edge Prevented From Rotating. Uniform Load Over Actual Surface

24

Outer Edge Fixed and Supported. Inner Edge Prevented From Rotating. Uniform Load Along Inner Edge

24

Outer Edge Simply Supported. Inner Edge Fixed. Uniform Load on Inner Concentric Circle of Plate

25

Both Edges Supported. Uniform Load Over Entire Actual Surface

25

Both Edges Supported and Fixed. Uniform Load Over

Entire Actual Surface

25

Outer Edge Supported and Fixed. Inner Edge Fixed. Uniform Load on a Concentric Circle of Plate

26

Outer Edge Simply Supported. Inner Edge Free. Uniform Load on a Concentric Circle of Plate

26

Outer Edge Simply Supported. Inner Edge Fixed. Uniform Load on a Concentric Circle of Plate

26

Outer Edge Supported and Fixed. Inner Edge Free. Uniform Load on a Concentric Circle of Plate

26

Outer Edge Supported and Fixed. Inner Edge Free. Variable Load Over Entire Actual Surface

27

Outer Edge Supported anC Fixed. Solid Plate. Variable Load Over Plate Bounded by Circles of Inner Radius and Outer Plate Radius

27

Outer Edge Simply Supported. Solid Plate. Variable Load Over Plate Bounded by Circles of Inner Radius and Outer

Plate Radius

27

Outer Edge Supported and Fixed. Inner Edge Fixed. Variable Load Over Entire Actual Plate

28

Outer Edge Simply Supported. Inner Edge Free. Variable Load Ovcr Entire Actual Plate

28

Outcr Eclgc Simply Supportcd. Inner Edge Fixed. Variablo l,oacl Ovcr llutiro Actuitl Platc

28

Tabulation ol

Tabulation ol Plates

xu

(.etse

67. All Edges Fixed. Distributed Load Varying Along

CIRCULAR PLATE WITH END MOMENTS Case

49.

No Support. Uniform Edge Moment

50. Outer Edge Fixed. Uniform Moment Along Inner Edge Case 51. Inner Edge Fixed. Uniform Moment Along Outer Edge Edge Case 52. Outer Edge Supported' Uniform Moment Along Inner Cqse 53. Inner Edge Supported. Uniform Moment Along Outer Edge

Case

Table 3 RECTANGULAR, SQUARE, TRIANGULAR AND ELLIPTICAL PLATES surface Case 54. All Edges Supported. uniform Load over Entire

Case55'AllEdgesSupported.DistributedLoadVaryingLinearly Along Length Case 56. All Edges supported. Distributed Load varying Linearly Along Breadth

Case5T,AllEdgesFixed.UniformLoadoverEntireSurface Cese

58.

Long Edges Fixed. Short Edges Supported' Uniform Load Over Entire Surface

Case

59.

Short Edges Fixed. Long Edges Supported'

60.

One Long Edge Clamped' Other Three Edges Supported'

Length

EQUILATERAL TRIANGLE, SOLID

29

Case

29

68.

xiii Linearly 32

29

29

Plates

Edges Supported. Distributed Load Over Entire

Surface

33

Surface

33

(IIRCULAR SECTOR, SOLID (lase

29

69.

Edges Supported. Distributed Load Over Entire

I'ARALLELOPIPED (SKEW SLAB) ('use

70. All Edges Supported.

('use7l. 30

Distributed Load Over Entire Surface 33

Edges b Supported. Edges a Free. Distributed Over Entire Surface

30

I{IGHT ANGLE ISOSCELES TRIANGLE, SOLID ('use72. Edges Supported. Distributed Load Over Entire

30

ITI,LIPTICAL, SOLID

30

('use73. Edge Supported. Uniform Load Over Entire Surface ('use74. Edge Fixed. Uniform Load Over Entire Surface

30

Surface

('ttse75.

Case

61.

One Short Edge Clamped' Other Three Edges Supported' Uniform Load Over Entire Surface

3L

Case

62.

One Short Edge Free. Other Three Edges Supported' Uniform Load Over Entire Surface

31

t'tt.va77. Rectangular Solid Plate. All Edges Supported. Uniform Load w, Uniform Tension P lb Per Linear in. Applied

Case

63.

One Short Edge Free' Other Three Edges Supported' Distributed Load Varying Linearly Along Length

3T

Cuse

64.

One Long Edge Free' Other Tlhree Edges Supported'

Case

('usa

65.

30

Uniform Load Over Entire Surface

30

Uniform Load Over Entire Surface Onc Long Edgc Frec' Other Thrce Edges Supported' Distributcd l,oacl Varying l'irrcarly Along Lcngth

3L

('tt',;c66'AlllitlgcsStllrprlrtctl'l)istrilltrtctll,rlat|irrliornroIa 'l'l'ilrttgttlirr l'ristlt

('trsc

76.

to All

31,

31,

34

34 34

I{I'CTANGULAR PLATES UNDER COMBINED LOADS Rectangular Solid Plate. All Edges Supported. Uniform Load w, Uniform Tension P lb Per Linear in. Applied to Short Edges Rectangular Solid Plate, All Fdges Supported. Uniform Load w, Uniform Compression P lb Per Linear in. Applied to Short Edges

Uniform Load Over Entire Surface

33

Edges

35

35

35

S()tJARE. SOLID

('ttsc78.

Corners Held Down. Edges Supported Above and Below. Uniform Load Over Entire Surface

('tt,\'tt79. Corners Free to Rise.

Edges Supported Below Only.

Uniform Load Over Entire ('tt,vc

80. All

36

Surface

Edqcs Fixcd. Uniform Load Over Entire

36

Surface

36

Tabulation ol Plates

xtv

Tabulation of Plates

AI.L EDGES SUPPORTED, PARTIALLY LOADED RECTANGULAR PLATES Case 81. Uniform Load Over Central Rectangular Area Case 82. Uniform Load Along the Axis of Symmetry Parallel to the Dimension a (b, very small)

JI

CORNER AND EDGE FORCES FOR SIMPLY SUPPORTED RECTANGULAR PLATES Case 83. Uniformly Loaded and Simply Supported Rectangular Plate 31 Case

84.

Case

85.

Case

B6. 87.

a)

Case Case

Case

Case

Case

Cuse ('u,sc

88. 89. 90. 91. 92. 93.

b

94.

Simply Supported Rectangular Plate Under a Load in the Form of a Triangular Prism, q > b

l0l.

48 48

Plate Fixed Along Three Edges and Supported Along One 49

Along One

Edee. 1,/3 Uniformly Varying Load Edge.

('ase I05

49

l/6

Uniform Varying Load

50

1/3 Uniform Load

('tt,s'e

44 ('tt,sL,

53

109. Plate Fixed Along One Edge and Supported Along Two

Opposite Edges.

44

52

53

t'tne 108. Plate Fixed Along One Edge and Supported Along Two Opposite Edges. Uniformly Varying Load ('tt,s'c

Ilclgcs ancl Frcc Alonu Ono Edgc.

5Z

Plate Fixed Along One Edge and Supported Along Two Opposite Edges.

+J

45

.

51

Plate Fixed Along One Edge and Supported Along Two

('use 106. Plate Fixed Along One Edge and Supported Along Two Opposite Edges. 2/3 Uniform Load

ia

43

.

Opposite Edges. Uniform Load

( 'ttse I07

Plate Fixed Along Three Edges and Free Along One Edge.

Platc Fixed Along Three Edges and Free Along One Edge. 1/6 Uniformly Varying Load

Plate Fixed Along Three Edges and Supported Along One Edge. I/3 Uniform Load

('ase 104. Plate Fixed Along Three Edges and Supported Along One Edge. Moment at Supported Edge

42

Load

47

Edge.2/3 Uniformly Varying Load

4I

Plate Fixed Along Three Edges and Free Along the Fourth Edge. 2/3 Uniform Load

Plate Fixed Along Three Edges and Free Along One Edge. 2/3 Uniformly Varying Load

Plate Fixed Along Three Edges and Supported Along One Edge. 2/3 Uniform Load

('ase 103. Plate Fixed Along Three Edges and Supported Along One

42

Plate Fixed Along Three Edges and Free Along the Third Edge. Uniformly Varying Load

47

Edge. Uniformly Varying Load

40

Edge. Uniform Load

Plate Fixed Along Three Edges and Free Along the Third Edge. I/2 Uniform Load

Edge. Uniform Load

Case 102. Plate Fixed Along Three Edges and Supported

Plate Fixed Along Three Edges and Free Along the Fourth

9-5. l)latc lrixccl Akrng'l'hrco Mrltttsrtt irl lit'cc litluo

Case 99.

39

Simply Supported Rectangular Plate Under a Load in the Form of a Triangular Prism, a < b

1/3 Uniformly Varying

Case 98.

Case

BENDING MOMENTS AND REACTIONS FOR RE,CTANGULAR PLATES Case

Plate Fixed Along Three Edges and Supported Along One

Case 97.

38

b

Simply Supported Rectangular Plate Under Hydrostatic Pressure,

Case

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Flat Plate Design

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Flat Plate Design

16

Tabulation

,IABLE

ol

I7

Formulas

1.

FORMULAS FOR CIRCULAR PLATES WITH CONCENTRIC HOLES (See Figs. 1,2, 3)

II

l.

I I

supported. Unilirrm moment

I T

.ol tl

:lo -l x

ae.

Inner edge

a

:rlong outer ctlge.

w

(at inner edge)

0: y

:

100N1

IOD,

rt')"

;:i d ,1. Outer edge d

'5

supported. Unilirrm moment :rlong inner t'tlge.

'>tt

M1-

#

ff{

(at inner edge) Mt

^

l5srMy

lJS2Mo

-

0-

looN2

v-

roD,+F

w

\o

(atinner edge) M1- StW

bb

qe G'

l.

O?Crloq

+__

Inner edge

N

portcd. Uniform Ixrtl along outer

I I I

lgc.

cr

I

olI

I lo ol

r;i Lrr.

6

i.

H

Outer edge lrrctl and sup-

n

I'ollcd. Ijniform lo:rtl along inner , r|gc.

I .r; bb

+?

!: Fh#

orrtcr edge.

,1. Inner edge lrrccl and sup-

.-o

d:N3

Unilirlrn load along

:,upported.

W

(atinner edge) M,- FS;W

o--B'Nn#

!: Fsh# (atinner edge) M7- SsW Wa

- P'No Y: F"Dr# o

(, ()uter edge lrrctl and supt'()r'lo(1. Inner lrllqc prevented lr oln rotating, l lrrilirlm load .rkrng inner lr

|11c.

(at inner edge) M,

-

!: FBD.#

FS;W

t8

Flat Plate Design

Tabulation

13. Outer edge supported, inner edge prevented

(SeeFigs.4,5,6)

M1-

(atinner edge) 7. Inner

edge

supported. Uniform load over entire actual

a

!:Nz

wag

Et,

M,-

(atinner edge) 8. Inner

edge

9. Inner

urrdnJ-

e

from rotating. Uniform load over entire actual surface, 10. Inner edge fixed and supported, outer edge prevented

from rotating.

wd4

- p'Nr$ M1-

ilJbt{j4

pzSswd

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FuHlq-

pSgwaz

-dtr

F"D*

(at outer edge)

edge

supported, outer edge prevented

!:

weg 0-pN, -ntr

M,-

(at inner edge)

y:

Uniform load

B2Snwa2

w44

p'DLo-E*_

e--o

over actual surface,

11. Outer edge supported. Uniform load over entire actual

(atinner edge) M6- BSlwaz

IIIUII-

Y

-Drr#

0:

surface.

( at

N11

wa3

Etr

outer edge) M,

12. Outer edge fixed and supported. Uniform load over entire actual surface.

'e

-

83D,,

0

-

p')Nr,

:

waa

Ett

#;

from rotating. Uniform load

p2 S pw az

14. Outer edge fixed and supported, inner cdge prevented l'rom rotating. Uniform load over entire irctual surface.

I9

Formulas

I[[Jij[-

over entire actual surface.

wa4 0:D7 _ET

lIJJSlIij*

surface.

fixed and supported. Uniform load over entire actual surface.

BS7wa2

ol

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(atinner edge)

!

M,-

B2SBw&2

w# :9',Dre -Ett

0:

pNyr

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(atinner edge) M,- BzSywa2 wa4

!: F"DU _Etr 0:O

Tabulation

Flat Plate Design

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Chapter 2

BENDING OF CIRCULAR PLATES UNDE,R A VARIABLE SYMMETRICAL LOAD* 8. EDGE RESTRAINTS The basic equations for deflection, slope, and moment for a thin, flat, circular plate, under a symmetrical variable load, for a constant force divided by the square of the radial distance, have been developed in Ref. 36. Six cases have been derived. The first four cases cover the variable load acting over the entire actual plate: 1) (2) (3) (4 )

Outer edge supported and fixed, inner edge fixed. Outer edge simply supported, inner edge free. Outer edge simply supported, inner edge fixed. Outer edge supported and fixed, inner edge free. The final two cases are for a solid plate having the acting variable load bounded by circles of an inner radius and the outer support-load radius: (5) Outer edge supported and fixed. (6) Outer edge simply supported. The treatment for the six cases of varying load distribution in which constant force, divided by the radial distance squared acts on a thin, flat, circular plate is schematically delineated in Fig. 7. A computer program was developed for ascertaining deflections and moments. To simplify the determination of deflections, moments, and slopes when only one or two calculations are required, various dimensionless terms in the derived equations have been computed and presented in tabular form. The maximum deflection constants for these six cases have been graphically depicted. Bending-moment diagrams for these six cases have been obtained for a set of parameters. The maximum deflection and bending moment constants are presented in a table for rapid computations using prescribed conditions. (

'r Reprintcd

from Rcf.

36.

65

Circular Plates under a Variable Load

66

Notations

67

NOTATION British

Metric

Units

Units

Description

A a b d CI C2 Cs

Area

in.2

cm2

in.

r

Outer plate support radius

cm

Radius of plate

Radius of uniform load on inner concentric circle and/or inner plate radius Radius of uniform load on concentric

V

Shearing force per unit circumferential

in.

cm

W

in.

cm

l/in.

I/cm

in. in.

w ws

cm cm

z

circle Constants

of integration for outer por-

tion of plate, bounded by uniform load on concentric circle and outer plate sup-

1/cm cm cm

radius

D

Flexural rigidity of plate, symbolically

lbrin.

kgrcm

Modulus of elasticitv

lbr/in.2

kgr/cm2

Unifoim plate thickness

in.

cm

length

lbr-in./in.

kgrcm/cm

Radial bending moment per unit length Radial bendir g moment per unit length at outer plate support radius Radial bending moment per unit length at inner plate radius Tangential bending moment per unit

lbrin./in.

kgrcm,/cm

lbrin./in.

kgscm,/cm

lbrin./in.

kgrcm,/cm

lbrin./in.

kgrgnt/clrr

lbrin./in.

k91-cm,/cltt

lbpin./in.

kgr-crn,/cttt kgt kgt

Eh!/12(t

E /r k kd k,, k, kt

-

v2)

Metric

Units

Units

kg1/cm

plate

lbt

kgt

Deflection of plate

in.

cm

in.

cm

lbt/in.2 lbt/in.2 lbr/in.2

kg1/cm2 kg1/cm2 kg1/cm2 kg1/cm2

Uniform load on a concentric circle of Deflection of plate at uniform load on concentric circle Poisson's ratio

omar Maximum unit stress

cr crb c5 +

cm

lbt/in.

length

port radius

C4 Constants of integration for inner por- 1/in. Cb tion of plate, bounded by uniform load in. C6 on concentric circle and inner plate in.

British Description

Radial unit stress

Radial unit stress at inner plate radius Tangential unit stress Bending angle

lb1/in.2 rad

Deflection constant Maximum deflection constant Maximumbending-momentconstant Radial bending-moment constant Tangentialbending-momentconstant

M,,o* Maximum bending moment per unit

M, M,, Mrrt Mt

length

Mro Mtr, P 1l

Tangential bending moment per unit length at outer plate support radius Tangential bending moment per unit length at inner platc rirdius Constant forcc

lbt

Ilctlrrrtrllrrtt loltl

Ib1

Fig. 7. Variablc Symmetrical Load Distribution on Circular Platc; Schcmatic Diagram

rad

Il.

Circular Plates under a Variable Load

9.

Theoretical Aspect

69

SYSTEM OF UNITS

In this chapter, the unit force-mass system is used sincc it provides compromise between the absolute and gravitational systcms, and is automatically a self-containing reference systen. (See Appendix)

a

10.

ASSUMPTIONS

1. The plate under consideration is assumed to be perfectly elastic, isotropic (modulus of elasticity and Poisson's ratio are the same in all directions), and homogeneous. 2. The plate initially is flat and of uniform thickness. 3. Maximum deflection in comparison with thickness is small, say no more than half the thickness. 4. Deformation of the plate is symmetrical about the cylindrical axis. 5. During deformation, the straight lines in the plate initially parallel to the cylindrical axis remain straight but become inclined. 6. The middle surface of the plate is not strained by bending. 7. All forces, loads, and reactions are parallel to the cylindrical axis. 8. Shear effect on bending is negligible, thickness limited to no more than one-quarter of the least radial dimension.

l+

..J

"## ,t

-,;;:::il

+

11,. THEORETICAL ASPECT

l**,t

The ensuing theoretical compendium has been included with several thoughts in mind, viz., (l) it is an abbreviated version, (2) it relates all necessary formulas, and (3) it eliminates acquiring a reference if a quick

Fig. 8. Bending-Deflection Relationships for Element on Thin, Flat. Circular Plate

review is desired. The derived bending moments, slope, and deflection equations are the ones ascribed to Grashof and Poisson.

The pertinent unit-strain equations, according to Hooke's law for

plane stress and the geometric relations illustrated in Fie. g are

u,:+-r+:y# ,,:7-r+:y+

M,:

(1)

Solving for the radial and tangential unit stresses, we cbtain

o,:T?,,

(#.,+)

o,: -!! l-v2 -

(-9\ r

If it bc assumcd

+,

_$-\ ,1, )

I::,,,

Mt:

cr,y dA/unitlength

: o L# * " +f

a.y dA/unitLength

: o l+ *' #l

I::'-',,"

(2)

that unit strcsscs arc proporliorr:rl to tlrc clistuncc front tlrc nricldlc surl'ircc, tlrcrr, throtrgh trsc ol liigs. ll unrl 9 arrd ljc;. (2) tltcr rirrlill irncl tlrrrgcrrtiirl lrcrrtliltu rrrorncrrls per. rutit lt'rrgtlr ltrc

(3)

where

EI

^ ":-r-17:

Eh\

nn-11

Summation of thc moments about the ccnter tangential axis of the element slrown in Fig. 9 givos

Circular Plates under a Varioble Load

70

11. Theoretical Aspect

Soo 2

v*#4, 6t

u-+dlkr "r"* "l

v

dg.

ju,

1"

P ,.^ r - -T t'T

E, ) -l -:-[' ov

and for the unloaded region,

O

71

2nr dr o."'u' -

1 rz V

(7)

b

:0.

(8)

The general equations for the load-distribution region, as a consequence of substituting Eq. (7) into Eq. (6) and then integrating, are

$



: - + l*u" r)2 - tn r (k D]-r c{

-#-+:- #l(*#)' -,"#- (hb1z*+] .CrC, rt-r r-z

,Circular

2M,-

o

Plate

I

i

(, l| -"'-'" : ( u,+ dy.' dr)do -' )J-' dr ar \

M,r

d0

-2Mta,

r

dr dr dV dr \. +(v++ '\ ctr l'l(r*dr)4-dotvrt-ot z

!,'

_ (tnb)2+ +l *_ D 1)' z) / ^ "":, - t rz - C2lnr I Ca |(

The general equations for the unloaded region obtained by substi(8) into Eq. (6) and integrating, are

tuting Eq.

d!

*

2

ur,

w:-

where the trigonometric sine function has been assumed equal to the angle.

Rearrangement of terms and neglect of higher-order derivates of Eq. (4)

t M,-M, :_v d, -T

dM,

(s)

The equilibrium equation in terms of the bending angle and radius is now ascertained bytaking the derivative of the first expression in Eq. (3), substituting this expression and the expressions of Eq. (3) into Eq. (5); thus,

dr+,1.

d6__+:df

#+ , i-

,,

-

dr

I d ('+)J .l

L;i

V :--5-

(6)

Referring to Fig. 7 the shearing fotce per unit tangential length at any radius within the load distribution region, b 4 r 4 a, is established ar

- c+r

-#-+:+,++

(4)

yields

(9)

$ ,'-CsInr*Ca

(10)

If Eq. (3) is used and the derivative taken of the second expressions of Eq. (9) and Eq. (10), the bending:rnorreot equations for the loaded region become

: - *,' +,tl(t, #)'*(#)," * - (n b)2 - +(+t+)l + s* G * v) - + G -v) M,: - *,' +,>l(, +)'ffi*) ," t- enb)2 M,

+

+(-+=+)]

+

s*0

*v) + s*0

-v)

(11)

Considerations lt'*u*: k6Pa2/Efui

12. Design

Circular Plates under a Variable Load

72

and the bending-moment equations for the inner region become

tv)-!;-6-,)l I rr\ cM,-Dlttrtv)++(1-z)l

M,:D-LZ [9rt

(12)

-1

L

The six cases presentbd here in tabular form, as cases

I thru vI

were

derived by using the appropriate equations that fulfill the continuity conditions and/or boundary conditions. The equations used in obtaining the integration constants were the last two expressions of Eqs. (9) and (10), plus the first expression of Eqs. (11) and (12). The continuity conditions and/or boundary conditions for each case are shown in the upper right corner of the tabulations. As an example, consider Case III' The boundarv conditions are

w-O Mr:0 dw

#:0

whenr:c

r: when r: when

(13)

b

are

where "maximum" signifies magnitude only, or maximum absolute value. Figure 10 depicts the maximum deflection constant for the six derived cases for ratios of the outer plate suppmt and load-distribution radius. to the inner plate radius and/or load-distribution radius, from one through four. The determination of these deflection constants is based on a poisson,s ratio of 0.3. Numerical values of the,deflection constant, calculated for several values of the ratio a/b and v - 0.3 are tabulated in Table 5. Since the bending moment must be an absolute maximum in determining the maximum stress, location and magnitude of the bending moment are a prerequisite. Because of the complexity of the moment equations, and because Poisson's ratio depends upon the material and related

-+(fi+)] ++(1*z) -+Q-v) o-- #l-(hb)2+il++b++

be verified by the customary mathematical procedures.

vr could Theoreticalry,

/

^ Pa2 l/, o _1)'_(tnb)2++l-+a2-C2tnq*Cs L) + ':-E l\''-T t o - - *,' * rl(^ #)' * (+#)," t - enb)J

(9)

(15) The maximum bendine moment in all six cases can be expressed by the form M^u*- k^P (16)

parameters, only the absolute maximum bending moment of case

a

Hence, the three equations to be solved for the constants

where the second and third expressions of Eq.

73

Fig. 10. MaximumDeflection Constant Ver-

sus Ratio

(14)

and the first expression

of Eq. ( 11) were used. To facilitate the moment, slope, and deflection qomputations, variog! terms in the derived formulas have been computed and are related in Table 4.

1.2. DESIGN CONSIDERATIONS Normally, the maximum deflection and the maximum bending moment are the major design criteria. For these six cases, the maximum deflectiol can be represented by a formula of the type

of

Outer

Plate Support

and

Load Distribution Ra-

dius to Inner Plate and/or Load Distribution Radius for v

:0.3

ilI 12. Design Considerations

Circular Plates under aVariable Load

'7/

TABLE

4

Computation Terms

l

{,

,z D

al

/,

,z 2

1.0

L

00000

j.l l. z

0,a2645 0,69444

1.3 1.4

o.591? z

1.5

0,44444 0,39063 0,34602

r.6 1.7 1.8 2.

O

z.l 2.3

z.? 2.8 3.0

3.4

3.8 4.0

0.00000

2.00000

0.17355

r.92645

0.30556 0.40828 0.48980

0,51020

1.59r7

0.25000

0.?5000

0.2261 6

o.7

0.20661 0.18904 0.17361

0.79339 0.81096 0. a2639

7

t.44444 l,39063 t.34602

l.

30864

t.21?O\

r. L

324

14793

l. l3?r?

L IZ7 55 r. I 1891

o. a7 245

0.88109

0.llllt

0.88889

1.1Ittr

0.10406 0.09766 0.09183

0. 89594

r.10406

z.449ZA

1.04t61

0.80000

l.

0.64t03

1.64I03

0.52910 0.44643 0,38314

1.52910

80000

r.44643 1.38314

-1.00000 -0)90469

0,03324

-0.81?58

0.06883 0,11321

-0,731 64

0.40547 0.47000 0,53053 0.54779 0.64185

0. r6441 0.22090 0.2815? 0.34550 0,41 197

-0,59453 -0.53000 -o.46937

0.693I5

-0.30585 -0.25806

69374

-0. t6?09 -0, 12453

o.74t94 0.?aa46

r.233t0 t.2r008

0.8329r

0"

L

r9048

r.1736r

r. 1 5898 r. I4620

0. 12500 0. I16I4 0. 10823

-0.4rzzr -0. 358 I 5

-o.

zll54

0.08948

0.12811 0,12650 0. r2600 o. t2342 0.12179

0. t49zz

0.08664

0.

09470

I.22378

0.08163 0.0??16 0.07305 0,06925 0.065?5

r.

08889

0.922a4 o.92695 0.930?5 o.93425

r.

l6

0,0836I

I.08361

1.5694r r.640?8

1.07305

r.

t.07 440 r.0?03?

r.33500

1.7aZZ3

1.065?5

0.0?880 0,0?440 0.0?03?

1.2527 6 l. 28093 1, 30833 1. 36098

t.85ZZ7

0. z5z7 6 0.28093 0.30833 0.33500 0.36098

1.06250

o.06661

r. 0666?

r.3a629

L

0.38629

k,o

k

l.?r173

92180

o.oI1z7

t220? r1773 t1359 l 0964 I 058?

0, 09245

Case IV

se III

k,o

m

r.06290

o. t0zz6 0.09884 0.09557

r.

0,08889

m

0. I 3433

0.06389 0.0?892 0.09507 0. ll2z3 0, 13031

0.09470

k,q

0. 91018

0.95136 0.99035

0.19392 o.22318

19392

Ca

0.12482

0.02662 0.03?60 0.05008

28007

0.13t33

m

t20ll

_z

t.

oz7 46

1. 09686 r. 12949

r.16095

l

19131

t.

22068

l.24913 r.27

].

67

4

30356

u2 -7- l,l."tl"tZ

0.50000 o,453A6

0. o0000

0.04324

0.549r?

0.4r436

o. o7 555

0.59668

0.38023 0.35049

0.099?5 0. ll?93

o.68696

o.32438

0.13153

0.30128 0.280?6

0.14r60

o,2624i 0.24592

0.14898 0. t5424 0.15784

0,23105 0.21?58 o.20533 0.19415 o. ta39z o. t7453 0.16589 0. r.5?91 0. t5053 0. 14369 0. t3?33 0. l3l4l 0. I 2s88 0.

tzol

z

0.50000

o.64259 o.7 2985 o.77 tza 0.81139 0.85020

Zz

a -Z-

o.29594 o.36250 0.43055 o.4997 4

0.56981

o.9z4zo o.95952 o.99379 r. o2706

0.6406t

1.05939

o.927 41

o.15992 0. r585t

l. 09082 l. I z140 r.ls1r5

o.999sZ

0.15684 0.15499 0. r5z9a 0.15087 0. 1486? o. t4642 0. t44l 3 0. 14 182

t,35502 r.3797?

0.lrl36

4039C 1.42? 45

r.45046

0.10310 0.09933 0.09578

0.13950 0. I37r9 0.13489 0. l326I 0.13035

t.4? 293

0.09242

0. I28lZ

t.18015 r.20839 L.23594

0.?rI90 o.7 8357

o.45545

1, 0715

t4339 l. 2l5l I

r.24659

t. z6zao

1.35781 t.4za7 4

r.28904 1.3r464 r.3396?

1.63947

1.36412

l.

38803

r.41t43 t.43432

r.49935 r.56957 r. ?0891 t.7 77 97 1.84661

r.91483

t.4567 5

l.9826l

l

z. 04993

1,47 A7

Case V

o

m

ld

.094

0.025

-0.072

0. 041

- 0.064

0. 309

0.076

0,565

A )\L

0.287

0. 148

- 0. 185

0.159

- 0. r59

0.827

0, zIz

2.326

0. 857

0. 605

0.510

0

.344

- 0. ztl'l

0.304

- 0.248

| .346

0.360

0.31I

2.912

1, 140

0. 993

0.7 36

0.538

- 0. l7,l

o.453

- 0.327

r .827

0.507

0.360

o.4ZZ

3. 390

I .4t

I .40t

0.956

0.?31

- 0. 44ll

o

.596

- 0.397

z.263

0. 649

TABLE

0.4

o.5l(,

t . ',

0.785

Maximum Deflection untl Momcnt Constants Where v

0,27

2.0

0.057

- 0. 132

r qql

)q

0,141

n ?l

tr

3.0

0

.246

7tt

11{\

3

3

l.(it"

I. /')ll

l.

I {,()

0. ,)0'/

1

r.

0

0. 693

F"tl

o.00000 o.0523z o. r08?g 0. r6858 o. 23t14

0. 04I

- 0, 049

t_ al

0. l60r5 0.16143 0, r6189 0. t6t?l 0. r5102

0. r r589

0.107I0

I al -za z t"tl

o,8877 7

l,32965

l.

z

al -z-b Illntl

0.00000 0, I ?408 0.30893 o.41760 0.50814

o. t2844 0. l3tl4 0. 13306

0. I63r5

L

1.08163

uase rl

t6az4 r6290 I5745

t.20694

1.0865t

0.93?50

0.86644

0.13410 0. 13320 0. t3zl3 0,13090 0. I2955

0.9r83?

0.06250

o. tzotz

t7 329

0. 0. 0. 0, 0.

0.9\349

880

o. o. 0. o.

0.0091 z

0.0865I

r. 06925

0.09416 0.06659 0.04475 0.02792

0. l8l4z 0. 17780

0. 0985 I 0. 13140

r.497 64

0?

0.6s360 o.7 t4z4 o,1 6921 0.81965

0. 12660

0.I0tll

0?7

o.o9743 0. 10664 0. l14t2

o.08629

0.004I9

1

r.09183

r,

0.07 301

0,1836r

0,0647

I.

r,10Il1

0, 18021 0.18360

1.13361

-0, 04449

-0.00675

0.90817

09?66

0.35347 0.28090 0.22031 o. t6992 0, tz8z7

o. 04073

o.o5776

I.060lz

r.oz962

r.15315

0. r2661 0. r5524 o. t7 t67

0. 13533 0. 13522 0. r 3480

r.0647I

l. o986I I. I3I40

0.66850 0.54411 0.44021

0.00000 0. 007 50 0.02308

0. o296z

0.9t629 0.9555r

r.

0.00000 0. o7 877

0. I466 I 0. t4t35 o. t3624

-0.083?r

o.99325

00000

t.

0.00701 0.00198 0.00005 0.00088

0.83959 0.91300 0.98555

I

a

0.8I846

0.15t99

0.76645

.2\ Dl

2

{* *)

0.0r551

0.87 541

t. I 2500 I. tl614 I oaz3

-0.66353

0.90234

o

.o

0.00000 0,00908

t.29326 t.26042

0.009

,l

0.00000 0.09531 o. ta23z 0.26236 0.33647

0,48046 0.5504? 0.62167

o.19048 0. I?36r 0.15898 0. t4620 0.13495

1.16000

|.

5.76190

0.33333 0.29326 o.26042 0.23310 0.21008

25000 22676

l. 20661 t. I8904 l. l?36r

0,84000 o.85201 0.86283

0, r6000 0, r4193 0. I371? 0. tz1 55 0. I l89r

Z

l.5l0z0

0.60931 0.65398 0.69136 0.1 zz99

0. 30864 0.217 0r

6 4.1 6190 z.21213 t.449ZA t. 04t67

Case I at

75

0.rrtirt

().731

- 0.459

z. 658

5

:

0.3

q :' ,i

76

t

Circular Plates under a Variable Load

13. Numerical

rubber. six

cases, Figs. 11 through 16 show the radial and tangential bending moments

divided by the force constant, where a/b - 1.5 through 4.0 in intervals of 0.5, and v - 0.3. Table 5 lists the maximum bending moments computed. These maximum moments are located at the outer plate radius, or the inner plate and/ot load distribution radius, for values of a/b equal to 1.5 through 4.0 in increments of 0.5, where again Poisson's ratio is 0.3.

Referring to the tabulated equations of Case vI, transposing the rrr.ximum deflection equation, and substituting the flexure rigi$ity expressirn into the transposed equation, the following equation is asclrtained for tlrc uniform plate thickness:

':[u#4t(+# *#),.#

From these moment diagrams, numerical computations, and specified conditions, the following general statements can be made concerning the maximum bending moment and its location: C.rsB I. From Fig. 11 either M,o or M,6 is the maximum. Equating the absolute M,o and M,6 eqttations and solving, one finds ln a/b - 1 or a/b - e - 2.71828. . . . Therefore, M,u is the maximum when a/b I " and M,t is the maximum when a/b ) e. C.tsn II. M6 yields the maximum bending moment for all ratios of a./b

- (+++)('- #)ll"' 7

--Ltv-

1.5 through 4.0 for the conditions imposed. (See Fig. 12.) Cesn III. M,6 is the maximum bending moment for this case with

q/b

-

t

.

_

tlrc lirrrrr I'/r: :rcliny, on rr srrrllt'c lrourrtlt'rl lry cilt'lcs ol irrr inncr rrclitrs :uttl lltt' ottlcr r'rl11t' sttppotl. llt't':tttst' ol lltt' t'ortsltut liort ol llrc orrtel clttl

-

+

cm)2(1

40.85 cm/21.50 cm

-

{0.63426 cma [(2.50376

-

:

-

0.33r)

(18)

-

1.900

(1e)

1.75

1

88

(0.t zzos

{0.32810}trr

-

+

O.2jjTt)0.64185

117, r+

0.757 cm (0.298 in.)

(20)

l lro maximum moment becomes M

has

1-Lu

Eq. (17), computed terms, and Table 4 the required

Hence, using

lrickness is

h

1.3. NUMERICAL EXAMPLE

is h:rll' thc platc thickrrcss. 'l'lrc vlrrilblc load

rtv

To use Table 4, the ratio of the inner load radius to the outer radius is

Ca,sB V. M,o is the maximum calculated bending moment throughout the range covered. Transition a/b rutio is about 6.55, v 0.3. (See

pcrmissiblc dcllcction

:0.50376

ffi-':0.63426cm1

(See Fig. 14.)

Determine the optimum, uniform, plate thickness, the maximum bcnding moment, and the maximum bending strcss tlf a symmetrical, variably loadcd, flat, solid, circular, coppcr plato whcrc thc maximum

tt

--21v Ltv - 1.75188 \ Pa2(l v2) _ 3(150 kg1) (40.85 D-

- 0.3. (See Fig. 13.) Case IV. flere the maximum bending moment must be established according to specifications. With v - 0.3, M,,,is the maximumwhen a/b is 3.56 or less, and Ms, is the maximum when a/b is greater than 3.56. v

Calculations for plotting moment diagrams were performed with the CDC 3600 computer using Argonne National Laboratory program 1837 /PAD 143. For any given combination of values for z and the ratio a/b, this program computes and tabulates the deflection constants and the radial and tangential moments per force in all six cases where r/b ranges from 1 to the selected a/b in increments of 0.1. (Ref. 36)

(r7 )

lirom the maximum bending-moment equation and the rearranged equation, tlrc following terms containing poisson's ratio are first computed:

-

Fig. 15.)

7Z

t'tlgc support, the plate is considered to be simply supported. Given plate ;r'd load specifications are: outer plate and load radius, a - 40.g5 cm ( 16.083 in.); innerload radius, b 21.50 (g.465 cm in.); and load consrant, P 150 kg1 (330.7 lb1). The following mechanical properties apply lirr the specified copper: modulus of elasticity, E - 10.55 X 10i kg1/cmz ( l-5.0 X 106 lbilin.z); and Poisson's ratio, y :0.33.

Poisson's ratio can have a value from zeto to 0.5; e.g., Poisson's ratio is approximately zero for cork, and nearly 0.5 for materials like paraffin and

To obtain a better insight into the bending moments for these

Example

I

:

(

50) (0.33250) t0.41 197

+ 0.50376 (0.641 85 ) - (1/2)(0.s0376) (0.72299)l _ 2j.5gks_cm/cm (zr)

^o*

1

lsirs this obtainccl m:rximtrrn m()rncnt, the maximum unit stress is

(r,,,,,* ' !\tt' i llil'1.()

,

6(21.59 kgrcm/cnt)

k111,/t.lrr:r(.1,

(0.7.57 crrr)r

I lO lb,/ilr.:r

)

(22)

78

Tabulation of Formulas

Circular Plates under a Variable Load

tt;

rl E; €l F B 5l

rl

o o

r il

-;E-a

l> dl+

l> >l+

o'

+

+

''

klp

hlp

'!

;i;l+

-r-l ru l! tl^'

E

?l r ll.s ltg

l-

>l > I l+

3l

r

SH do

NINC !l -s k tl > I l+ dri

adT

$gj;t .91 o05 d

o

o

!!i.

I N

rtlp I

NIN ,t

IN

-.1-i '.1*f I d .1., +\ *'-t

H

[[

o

0hh

E>

o

sl

I

&l+ -

I

ilil

dlp t r

N

t

d

lr0 +: d+

tN

N

n

it

kl,

dle

a

c

g.*r

NINNIN hl d \---r

I

*dlg

dte

k

g

I

klF

Al.r, il

n

o a o o

ttk dtd

pl

*e

la

I

.ir a

it k3

dlt g

N

hl,

.n

NIN

Alr'

-

+

'.1-f 'l'l

-{-

--=-/

ld

N

Nt!

rd

N-lN. Ht o tN

I

g;;

dtp

I

NIN pt k

rnl,a

,!i]

N

;Ti

.lP tl.'

dti

dlp q r5 Nl

NIN 3l d N

lN

I

dtF

I

h

;i; ,l+

6

it l.o

El 6 5s fil s a 6l

'olru' pl

j

i

+X

I

'iE

3

dl k

dlF

l'o N

T! c

F

d.9 qda lto

ld

dl

d

36 dr

\

l^

Fh

h

a

ro

l! F

+

0N

;E

;? 6E

N

rla

; N

79

OE

3rFo

d

80

hl Sl

dlp

c

_:

-"-i..IN

-3

Nl4 dl

rilr

kk B> El ol

I

lN td

NIN Pl

k

.-.i.\ N l.o dl r IN ld

dt

;

'l+

d

NINNIN .ol h

6.EE

HI

-=-++ Hl,o rl \-----\-

pl

-

FI

-=:NlNiNl! kl d ,

I

I

l+

l+

N

h

+

-t-

r

; ;l;

_ =-l rto-

+ I} I

-l+ lr

dr^

:dNlNl r++

=--

-3kl.o

\--\z-

NIN 3t

fi; ..-

d

IN

Nt^ dl Nlo ld

N

+ I

N

htp

c Al{

p.

Al$

Nkl rN

\---_-v*

N-

lT

o

:

o

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atu (/l

q

ld

l^

6tH

d

al@

ll

.:x -d 9c

0Q" 0

N

dlF

-

dl k

l> +l+

o lP Pl |

tc

t]5 NIN

=*Pld rl hlp i-

;

+

Al$

lt.

+

I'

-l+ '-

rti

lN^ Nlsidl.o dl |

>l)

ft 'l'I

tN

dl IN td

g-

NIN Pl k

'dl.o

JN Nl!Nl! dl IN ' td

i

dl.o cts

ld

;

dtp g

>

dl* ,-r N5

I

lN

d l.o

dlp EF

's$i

3|

I

't+

NN

5b o. qoo

II

>t

^.ll^^1

>l>

T

l+ *t*

-t-

lo XF ot -q 6o

it l I t+

't' I

>l> I l+

6:

+l

I

d

Er''._

>l>

+ hLo

hlp

flge'i

NIN !l

+

Ndt!

+ I

cr o66 .91 h fl L

-.lp-

NIN Pl r

Ell

,F!rod

81

--t' ;

-;i;

k

-g >rt El olooo ol

Formulas

I l+ *td

dntA [[x h k El ot I c

Et

ol

Tabulation

Circular Plates under a Variable Load

iltl

-

-g-

>l>

al

.ll \--__-

. dl.le

o;l-

,

Circular Plates under a Variable Load

82

3l

rl

rr

-1,

dt!

;

dl.o

!l

d

htrutrutru .ol hl Pl +l+ tlt tlt +l l +l

.i

ritd ll

' |I

dt/ '

+

c

'n

X

I

rll

+ r----;--------.', NINININ pr rl Fr d + tlr l tlt

E

+l' l*l

:NIN pt

NIN pl

'

+

ktp

dtp

+

tlt

.nlp

-=-

I

;

N

+l

| +l

I, I li

rl+ - I rlt | +1, llilal

H

E> o

,l

+

d

N

N

ol

lN

ol

| >l > | +ll -l-

dlp I

N

N

l.o

h l"o

tlt I +l | ' | '-'\p. t.

I

+ N

dtp

kLo

;G

\-.-*-yJ

NIN tst l

d

+

N l^

ale

Als

dlE ll

9 ots

l'.l'o ;*.

N

I

&l$

*o

+ h

=+

dlp

| >t>

I

rutrulNlN pl hl Pl I

;

hlp

--\,-

I

+ ili

h

*t,

-t-

'

+

NI N

dtp

*t-l+

+

dLo

N

dl HI

; t-.

*]:.F.f-.,i=-

6'T;

+

Flp

+

dl

+ N

d

+ -lN

l>

'l*l-

;i, +l

lp

d

+

-lN

+

-a-

l-

t

ldrdlHlil

d

l>

-l*la

Er *$i gil Fl 6s5 el

d l.o

N

6tn

dl.a

N

+

N

-l'l-

g

>l> I l+

*t*

r----------------

3l

/.-+.--L ' l>

I

dlF

tlt I t+

''

?E Ft E#ri

83

I

dtp

>* its El '

tf; H,

Formulas

l> I ^l+

dl+

sl;;; 3j:

ol

Tabulation

6

:

NIN

\---pt

d

N.ln

fi16 ll X

d

pd

E

o(')Q

Tabulation

Circular Plates under a Yariable Load

84

ol

Formulas

85

r---l'--1

l>

dl+

t:;

:

*l ;;; , it,$

lt Hl+ l-

l-

B Nd-

s

>"

ll

:

t!1,

t5

.!

I

d's

tF E5

r....,....T-1

--=dlN

NN !t k

I

>l>

+l

,

tlt

'n

klo

n -l-

d

>l

;

+l ',

r-i----*-l IN IN lrt c -l'ir tlil

l{, u

H

o

-

l+

NIN Ft d

+

+ d

I

I t+

-tI

'!

I

d

.l

+

;ii

I l*r^ l!l'd

|

N

I

ktp

;T;

*\.-

NIN hl

I

AIT

-vJ

I tl

I

=-

+

+

I

I N

I

N

z

I

t> dt+ lp

'!

I

l+

Nl^ dtH

al@

il

d

NIN pt d \------.-v+ N.l^ 6tH g.l@ n

o (,

k

:

d o

ll r €t{

d

I

N

hlP

l+l t-

AIN

NIN pl

I

l> I -l+ ld

l- tl> I I

ftlN

+

l> dl+ l-

a

3l

NIN Pl t{ N

+

N

+

l.o

NIN kl

' II alr*l '

a

lF

NIN 4t

dlp

N

I

N

I

I lf

klP

I

-lN

rl+

NIN 3l

I

I3

X

d

t

N

-l-

r0

.!

>l>

dl3

klp

A

r!

q

rNIN

N

!!i

-li

I >r> lv

dlp

+

'-$; El el 595

dl!

_F.-

+l * lilj :.lj , ll,^rl-. .

+

H3i El

NIN 3l

€ .!

N

'€

-----1lNldri ololrl'i .ot tst _

N

,E .do dH

r;

X

I

d!

dt!

;JEC aa

al,

+

dlp

N-

dlp g/\

l>

dl+I

rl+

I

,l

-.;

I

.+

--r;-

dl€

:

r-

ri

il €E{ 3l

l-

+

rl

x o

u 0

{,

'

,

Tabulation of Formulas

Circular Plates under a Variable Load

86

3!.

; l; !

ig ,;i

&

k

f;s f;E f;6 " :l 'il €l ' ;n tu

El El El ht

31 rql

o

n

E

€

" &: ! flt .ql f 5 i ol I

i

.El

il'Fl .!l

r ll':

,;i NIN kl

.

>'

"jE

NINNIN .ol k

E

>l> l+ dfi'

6l

ol

--*-

e

t-a NIN ,ot

$i3 t

,

og ;,r

ai

r? iti?

-5d

d l.o

c -

NtN+1+---

-lN

"l ' r.-Enir"5-l;

+

dlp

+l ';-]i.' .ol ojifilld+ *

g g ;E

rr f d it:' ?l

N

dlp

i ; !i:;.!

BI Ol

q+ NIN pl >l

ad

-_l..-1 * .tJ

d

dlpPld*€o^NtN | d

-lN

Al$ l

o

+

' dld

L --+

il o d

H

oo 6.h .o

3l

;;+-tr-

dlp hlO

dl8

Al$

'll

o o. a

€t'd 'lF

lal+ l-

) ;i'^

l' >l+ l*

dt.o

;

dr.o

{F,

Blp

3

.+ ut.o

N

>l> t.

E :

>l> t.

+

E 'X 3 %l*.

| dl

rN N

-lN

.i" E

.E ;

'lo €-

.l

'

F!,o -!=-

^ NtNCN.S pl d ,,

-: =-

+ dl.o

,iB n

rro,,"rtilrl

^

-E

N ;i;

. I

N :

F

-'FNC 6lJ hl

+-g--9hr^-Nl_ g

iltl k 5

J N lN pl d

ilN

:-:rLa.iNlN

L,-J

> r

dlh

,b' .,

dln

";-"-oN

dt9

;

ala

G'

rlo .+E-F

d

-tN

HrN to

-l: -Trr

NIN !l

>l'

.ol .,

d

_e_-,

h

!l

;

N- lN pldNlNNlN

E

d d

87

dlH 3 &r$ -oo F

d

,'

:d

:o

nE

tsi qA

h

ar{ a

i:

ar$

N- ln

al6

,

>

a.r@

.a

oE 2>F.Ald6, oA

d

88

Tabulation

Circular Plates under a Variable Load

ddo"F tr ril h*Hgo c ie

ii B ' 6ro0.il"E:

Eliln"3l:ll f;r ?rs .ql 3l E a-l ;l ilel r >" llr !l El

ol

4l

' r

--NIN !l

g

NIN Dl

fi

El

3l ol

;ii

il'E >'

NIN rl

+

l.o

'E

;

-

NIN Ol

+

t

i:ai

;

NIN !l dt

fi

rl+ -l-

>l > I l+

I i{;

t fi lil

al+

;

il

alc d -

I

tir

I l+ ild -:d

tl I

'l*

l

+Nlo dtl.o 4 NI

+ *dlg

dilp .L

il x

A.lil I

d,

q o g o

to B

o

I

il

r!

x

rt,lli E IA oo

:6

I

t

>l>

+l+ rol-

;

6 il

tl

o o

tr I

:l:

I n

k

'dlg

l.o

h

o 6

!l ! I I t+

-t-

;

rl!

| l+

a

or( o.i

3lr-

k o

al

I c

o

hr!

!t

d

t!

I'o

!

r0

+ N

rlp I

Alr,

Al{

- dlp I

l!

.L

d::a

N

p

5Ndl.o

Ila

NIN !td +

N

;

>t ++l

nil[ o a o

rd

I

ad

Al{

'tr

dt3

c

il



l+

I

I l+

k

sl

I

dt

-lN Nlo !t

;

l

H

>t

+

I

iE ? i . iE e, $

d

+l+ rol-

d

t:

ia

89

I

it

d

Formulas

NIN pl

3'i b 1

E

ol

AIT I

il

I

N

rl! A

NIN hl

'"le g. lo I

t

d

Tabulation

Circular Plates under a Variable Load

90

of Formulas

91

oF dE

ab ok

ld9

ttc 000

ttt

dtr --5-

aia --=_-

Eh

ooo

--_,..olk

dlk

EO o: ;o oo Ad &o oq

NIN d !l N

.=6

E NIN ,olkNlN >t > I atrl+

dt€

nln NIN

-t

tlk dt€

tlk €t€

,

r ll& >"

"-6't?-

NIN €l

NINd .of d

9u d'i

NIN dl

t6 bd ad ooo ,x.i!

$sJ

tr-t.,t k c

>t

,t

d

dld

qEI -o

F3r

n6k XE ao .I

o

:N€lNd ,

NIN ,t

t::NIN

-l!dl ' 'd

F

=ll{

dtE jalN

NIN €l

dlE

olNodlIN 'td *l >lNF llil !a

d

I

-.ll+ lN ld

+

iF

>lF -t$

.o

rlQ

;

Fls I

n

k+E-

:>>H"

o

6

6td 'lh

d

'Et

6l

d

: NIN tl

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.l^,

l-o - '-l'. I

--.N l^

(l?r! !=l@ .YXX *r!;

=lT 'l+ rlr

ld

NIN rl d I ----1_-

-lo \----vd

NIN 6l d rt

Vl

---:*lo I

.-la

k

NIN €l

k

I

NIN hr

d

NIN !l

n

+ dlN

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I

k

dlr NIN pl

N lN

rl

c

tl

Nt+!tdN

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Yl

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€ I

it6

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g

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.

Fl5

: o o

A

alo

ts

d

kld

, !l F, ,_ 'l' ' I+

*; Fl

d

+

_to

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lo

lN

i

6 k

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>:

d

N IN^ dl 'i IN Id

'

rloF

>l

oln

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d

NIN !l

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t N l-i-:dl

Fl6

n

kk

d

'-=r'

NIN

Nlt 6l-'

NIN .ol

,__€

elo -=--, ":,ji

d

I

c

NIN ts !l r-NlN

d

klt

NIN pt

NIN kt d

I

I

tN

€l

+

llr

NIN

'

Qr

n

-lo

NIN

dts

dl

+

IN

d

dl€

,l

d

NIN pld N

dlN

-lN

ilN

N lN

'

to lt.

_to

6

rNlN _

d NIN !l

.{

l rld

_:,

ru14 -lN ld

--:--

ol!)l "lN ld

kl

ts

.5

IN

d

NIN &l

NIN

+ dt€

----5-

Nht === dt d

dlN

..-.i-

3l

*l-' ld

+

>l> I l+ *li

dlN

€ il tslN Yl

NIN !l

ts

-t-

NIN €t

El

trjl

NIN

>

;

{ ri

+ 6lk I +t

IN

+

_t;

X

g iE nl Ft s o 6 31 ';

H

NI'

€l h

'd; ca nq €X

E

_:_

=s-+

a;

NIN !l

'd

dl k 5l I >l> I itdl+

kl€ g

5 dt!

d

NIN PI N

cl€

l

d

,rts

d lrJ

o

o

t

Tabulation

Circular Plates under a Variable Load

92

ol

Formulas

93

dt!

e--*dt€

Jia

-e-

oF ;.s Eb dd3 niln ooo

6l

cl ol

ol ol

t}'

fil

:lol

EI

ol

{ltl rEE

tl EI ol ol

ol

€l

+l

Ad ko OF

.E$

---

N lN !l d 'lN NIN kt d

ril€>

dl

FH 6O

5d o.

dt€

dl€

E 86 .6TE $ s.,e;

'!ld

C

dl d

|

*

€l

-*

rl> 'l+ ilH

t?

.a

u

|

d

>l> ild'l+

-fN

-lN

lp -,-Rr

'--;;\ l!

^.1 ,t.l

\Il

=(

Hl

I

VI

o d

*,l^, -,l^,

-- 'lFlr

t: c

NIN dt

"

*-+-

+ N-l l^-oI

t_i;a pt k

'"lN

rl>

+l

3

ol

lN to

rlO EI - r+E t.:

r

,

llE 'lk

=C+ IN - ol| ! - lN \-j__-:: i NIA dlts ,

d o

.

6lk

'oltu

r+ hl€ rNlN

NIN !l -

d

d

dlh i -t -

lN^

dl€

:

"olI

dl,

N lN t{l k

N tN

+

rl> f l+

-

.n

>t >

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vl vl

,l

d ts

rl

x

P

t-

co .:. 0

F

rE o

rlN

lN

?? ?+

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€t

hl

I

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dl

rtd

N ld d !l + dl

I

_13.

d

'n

d +

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N lN !l it r

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.

-

d

NIN

N lN ill I

dlN

d

:+ NIN hl

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t0

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Fl5 Fl$ rls >*

d

tr I rl+ rla

+

n{il

El

d

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:q:-t.: -ali '-ali

-_l__e 0

=l@ ild

dlN '+---i7-

+

I irdl+ + NIN ,{t k _NIN N lN

olo I

=-=--,

lN | d

6ll d

.3.iT\

N lN hl d

{Ji:--

---_:_ NIN.dlr! kl r!

'olt.

t{lk j-NIN +l

k

d

d

N lN pl d

r-.=t + N^1N.. + -r'NlNNrN I 6l k NIN

-lN

Bl

Fl

>r I +l r dld

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)l> t ili l+

,--i 1ru. lru ---r--e -__g

Fl$ Fl$ 56 kc

-

ni*lN-

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xa ", tg f F.! f;"| s Et 'fr| 5 E 5r 6t

(

--

d

_.:s N lN 'old+

dtE

+

++

ol

ot Ol

dtd

'

NIN

,

;"+

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^^ l^--:1* dl k aa

d

>

-l-

dlilnl.! CE

d't P6

ol t

;o

Lrjl

--.61

'I

x

hl kl

--=-,

-:t:

co io Oo

ol

ol

dlk

NIN !l d -i:-:-

rla tl5

--F: N l{

l^' -J--9 -N16 d

' i$ls

tru '0 P €td itr : LDO

o

io

I

5

of Formulas

Tabulation

Circular Plates under a Variable Load

94

95

rl{ I

ilr --1NIN ,t d

!lnl

o

rl€

iliii

fl 3l

rlt

!.98:; uto

6l r i r 9l , >'€li tl fl EI

ol

I

:g EB

aat trtl

iii ..

rl r +l

3l

r

-

ili

rld I Nld ,t

:"

kl! e NIN !l

.1r

IE -;l;-

Ei;

niiEi

it

rl > +l ,

- dld'

+

-li +l

NIN !t
i

'.;t^F fi *l* fi, iljl ilj I

=1

Ht url

ol t > | +l '

-l

|

Fls n

!l{

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o

aa

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6

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a

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+

'

to

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d

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d

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lala

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o 0

tt

rl

r

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a NIN

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alo

ll

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'ls

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+

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3

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r-_i--]--1 'ol^* l',1". -rl+|I :t? I I

?

o

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qd%iq

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ts

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,

,

o i o

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F-1"!t d

,t

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I l+

d

dlt

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s

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a

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p

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e

d

ct€

d

5i:+l I al!

dtk

cl I

dl! I

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+

-;-

a

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NIN !t d

:ir ;t^n

la

I -15 ir dr o

Circular Plates under a Variable Load

96

Tabulation

ol

Formulas

97

al€ a

.l€

:g a!

caroL x nil kkkuio

€

!;

dld !t i

gl

3l

3l

" i. ---

il

;l'

rrr

=" El 5l

'

ll+

dld

,t

NIN kt

6

d

'ri5'l+

>"

NIN

+

+

NIN tt

a

rl d rl€

dld

!

;

E

!t r^d

F-

>lr +l+

!5

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a

TI:

3t EE

NIN ( !l !l

.9

!s B.

N l^r' ,t d

i tc

NIN ,l

h

i I !i il i

a

--_=- T; ';rd-

" tldNIN !t

TNIN -----l---NIN dl d

d

+

-ld

I

-{ l!l ol

3l

I

klNlN tl >l !l + P -l'l r': r'-'-l ;i

E. o-

*

n3! IE: ;>

!t5

d

i

al h

,r\!4 !l *l+

'--.--.-_'r^r^' lNlN' rllld ,l.t olr ' -t,l >l>*

,ldl-,r >r

F

lN

>l rr I 'l+l -tr'l+ Htit

I

I-t" -1.l'l*

I

'l{

'lS .

o

}lh

N-

s .9r o

ln

5i:,l+ -ta

| !l d dl | >t> | 'l+ dt! Fl6

;lE

x

6

uk

.g .5 q

r

6E rI a

6

!l

r

>l

;

NIN ,t I

+

!l

d

NIN Et

NIN

6t

!.1

-

'l{ >-

tt I

l+

' l+

;

k

i

NIN !l

k

NIN !t

d

r+ {lN rt

E',lo1,r.o' /\pt

-l/

/t \ ./ | >|i

d

rlfl. I| rrd\l\

rlA

3l$ E

n

ts

rl: lN

f-;-------:J' lN hr ' f l..lJ -l "-l ' rt'l !r+ d 'l*"$\. +/ rl'l rir 'r'l.rl ',' rl >r> ,l+ l.1l* rL__\J ,'\dt_, l,l+ d t-'tr-rI r lF'.

:'

NIN ,t a

+

--

-t* \

d

) -ta

d

I

lN

*l$

d

+

dri l+

I

I

\.il

?ok

a

-lN llr+.-=-.' ';F dtd ' ,l+ N lN E 4 rl/ !l r N t4 L_t n, Fr{ ifN'. , rlN

d

t

,t

+ d

rl

NIN dt

dlr

.r1.,

r

i

r-T-------1 IN IN

I

il

ili -l:

r--ii-.];--]

n

..1..

----:---

j

+

-lN

>l

| ,l*

;

!l rN k

N

Nld !t

-lN

N\NlN

|

N

NIN rt

I

.i.

=;

+l+

NIN

)l>

NINNIN !l d

>l:

!l

l--ZTr------1

,.---

*l| 'rr | I+

;

.--=-

! I d

-l-

ilN

'y-;.---

H\

dl k

Eg +

E-F,t d

Nld

+l aE

---

-t-

I l+ rtd

NIN !t d

kt€

+l | -td

>l >

I

t

+

NIN kt d

-;til

r

!l

I

'r l^.

---

----=:---

fiir -.!:

$

a

NIN Dl h

;rr

1TT

d l'{ 3ta Nld !t a

olr

hl€

+

,l d

€l!

NIN kl a +

,-----idtd

k

!l

r+ kl€ NIN

!lk

NIN ,t c

-1G-

/---.:':-' dld

+

NIN !l

NIN ,t d

d

N

all I

+

.lt

Bl

!I

ts

a

4

-;

e

!A

:: ili !

>l

.lt al

* e i

5 S8 Bt: !t tAAatsa

alts

.!l

o dtr tlk '' 8. a

1

*

d

r---i-----ti-tN td

lry'd-

-l-

:ir

ll

| -l-

.-

lo

s ils dr

Tabulation

Circular Plates under a Variable Load

98

ol

Formulas

99

nil kk oo @l

tt

r----:-l dlk

6l

.91 ooo

T

6l

ol

>l

rl

rr

B

=l

tt

d t,.o

ll

-€NIN .Ol

Eli llr

tl

I

3l

olr NIN ht d

-;E-

-'E EE

fi3 e6 Eo tE E..E Rt ad€

I

k

NIN Pl k

-T; rl+

,_-* >l>

*t-

6

rl+

kl€ rd

^,--l | t '-lN ld:

lN rulF | ol IN rdNl'o +

r---.---t '!l,o

dl^r

?IN _la

+ -16 +.+

.---!-, !l{ o o

tl

Nl.i

|

il+ ,,

rl* tl

L

(t t{

lN

)

:

€

[ x,,

d" F

>tts -l kl

>--/

' d

'FNIN il

,

Fls

-lN

-

IN

dl

N- lN !l d

d+E-t NlruFl H rr.! ri l'o .E-lN N lru .ol d

r-.4-.-l ----1I

| --t

Olnr.

- dlp d

NIN !l d + .dlk

r-----1 dLo d 4d

dlk

----';--o

ol

---.-

kl! NIN !l

nl

NIN kt d

*-dti

$s"J; dr t f B! 'E.l I E *; fil 6 I 5-e s '6 ;tFl

HI

NIN Pl + kl!

.'"11ENIN !l

.h

H

rd

N

I =' < IICll d:id

--':1.-

---.^, -.1| "o -

^,-dll.o'

'

l-. --'-;

lN,6

N l^

.rlo l* '|s

{l? Flcb

'ol

o I

----*tru

*.

lQ

Ft6

.dX

> :iiF€l€ fl ' (,)o

'-

Circular Plates under a Varioble Load

100

Tabulation

ddp

ol

t0I

Formulizs

r:::= dlp d

qr I 9 I

dlr.

6I EEE tl ooo 8l

Ul hl

---iN^llrr,

tt tt lr kh

Fl

r> €l sl

>l> +l I

el

N

--,ilp-

'i

-.! 6!l 9:

dtp

E

NIN !l

F

69

>t

+l

dlp

ooo bPbod.

T

NIN "ot d

kl€

lN

NIN

ptr

i*.

I

I ! 5t fit dl

tlP a

al

T

NIN Pl .!

'd

+

dll. F

lN t! Nlr dl lN ld

-

3 H

oll 0H

3l

E> o

:

dlF

olh + NIN

pl

dlp n

I

---* )lr tl+

G t! t"l til ' l.l l

+

+

rlt

rl{ tl

d

t

dlk

'-rll.o ^, lN ' Ird r

>l>

rl+ -le

, -lN ,

l-.

NIN Pl

>rr I

l+ dtd

tp

+l I

lN ' ld

N T

l-.

I

NIN

NIN kl (,

l>

ilN

-l+td

?

+

-lP ,nl

FIS

t

ilN

rlQ

rls

N ,a

I I

(,

Blr dtd

T .9

IA

I

>l )

>l> +l+ ola

lN ld

s

Pl

I

----i?n-

g

Fl,s ; (!c

IN

'*l

(t

I

f

nX

""lT >l)

+

>il

s€

tdl tr

A

+

lN

,

+

Nl. l" dl

l.o

lN. tG

-o [t.

E

rN --'--1

N^lt-o

c

+

Eg

.d

I

dlp

CL

FE6 '6'rt

ru|ru

> ,

+

hlP

5.H

li

hl

NIN Fl h

*h

El kl

+

}|lF

EI

H.5

E5 8t b 5

lN ld

o U c,

o

$lE ilI

-l+ olr dlN

--ro

glS r,

rn

102

ol

Tabulation

Circular Plates under a Variable Load

Formulas

103

dlp

_:_ dlk NIN

.ol

d

q-r +l I

-;

ala

ddp lnil

kl,

;;;

r .El .:l ooa ' orl4a

NIN .ot NIN tst

'

EI

-r

-;l;-

5l

4l

.i

.t

>l r

+l

I

dt!

;T; +l

-o

Fl kt t s 33 o

;

lt* +

Nl+ --:-' L >t

+l

'^l^.

',l*,

!

--'--iNIN tst

3

j

-^

t>

lN

I

Fl$

$r ok

E:

---='Ti

pl

d

-l

li -T" lN

:l -l 'l> | +l '

,l-l!-

-----:Nhr

l> >l+

-l+ r---

lt-

NIN Pl k

r

d

;:=:."rdtrt

Fl$

:

NIN pl d

il

\..--i.,J

+

lo lo l 'ol 'n _l >t+ > I +l , |I HIH

>r >

rlQ l

Blk 6td

I

I

---=/

Fl$

n

>l)

+l NIN 3l

| +lr I elr

x

d

+

I oqGt O

N.IA #lE

d I

NIN tst 6

I

-l| I

FIS

+l

IN IN d l!l

l-^l^-' +l ,l'o +l*. fl1" rl >r> -l'l iti | :l: l

I

rl:

,--r=-,

rtN U

r

-t-

>

NIN !t

N lN !l d

6

{l

+

rl+t> *l:tl

ri

--;-

NIN !l

--i.r

;

>lr +ll

Tl.. -

tn

.-F-

t!lk

=dtp

NIN !l

I

.d

I

c-fikl ri

llr -t-

K

NIN !l

;Ti +l

dlr

*l^ 3l k

dlh

TT; $cs. i E F$

(t

6tE

NIN ,I

d

I

d

c

NIN pl

dl3

I

NIN !l

dlr

E;'!

3t

T'i;-

-

dt!

[5 Br i3

HI

1,"

NIN 5l

NIN !t

B'$

H

G_

dt4

fl

3t

d

+

3l trrtr B >"'€l€ dl

Ht

,n

il-l^

'

l.--r^ lp! .

I

-l

tt t | +l I

lr Nto *15 tl

x

d

,c

I

104

Circular Plates under a Variable Load

Tabulation of Formulas

il

105

l!

dI

t

ts

NIN ,t c dl! E

NIN ,td kle

Et i f

:*l .

{

o

3l tr n '$l

,

dtN

o

,l

+

l

Er;

dlts

>.

al

4l

dl,

V p

-rlo-

o

z o

NIN

I o

NIN !l

Et-4 d ot o 9l k .=t e iit : ql ol HI

A

d Xo

:t

l

k

+

rl!

x

rl> +l I

r-_-:dl3

dlr

NIN !t d

Nlo !t

NIN !t

NIN !t

'3q D?

+

o

NIN !l

d

I

l>

>

dlN

;

I lt

a

d k

T;-l;>l !l d rl

| 'l+ t'-l-l *lF

H

3r ok

5:

il I

+

tt r | ' l+ l:ls,

.

Fl,s

'l ,:, -l l+ r

liti

Fl$

l d

d

l

r-r-i--------'l

>l rl Yl 'l |

'

,l

IN

+

>t>

,l+ -l-

Fl,5 fl[ !d

r

o

i

>

NIN

rdl-,

,l

d I

NIN hl

d

dlN .

l^;l;.r | ,t d

6

+

,l+ .1t1-

' l+

ts

d

tt

al r

-'i-r -l+

dld

-IN

-r

NIN ,l d

I

l t"--:*lNpltN d I

rl: +t+ ol-

>

NIN !l

3t

>ll>+

lN lN .l | ,l

+

-l I >l >

>l

dl!

.

l

uJl (rrl

NIN !l

kl!

t

+

dt!

:-

ts

+

o

'

d

tl

NIN €l d

tlt I t+

NIN

A

5iT +l

d

-al!-

I N

NIN ptd

kl! J

o

a

NIN !l

t

r

!l

€

E-Ekt d

-

6 o

o u

dl,

e

I N

o x

d

NIN ht d

o

tr

dlN

t--T--------'l

l^^fn

tl ttt | l+ tlildr '

dl

| >l

rlili

F

rlo 'lFI

6

tlh dtd

8r o

sts

-

'lNlN'pl I l+

I 'l+

> r

*tE l X

o

t

d

Tabulation ol Formulas

Circular Plates under a Variable Load

106

oF ;'i 6.3

ddo xiln

HA

fr i

Eri

107

ot EO

$l

o.9E rr io jod oo cl9 9EE E.E

ll

3l 31tr $l 'rHEu ;l r il€ >'

gl

3

tt

(t

.ql

a)

t E

'd (,

Tfi EE

x

rd

do

o

oF g6 !rt aa

----5-

f F ic $t 3l

)'+

I

3nE

nl s kl

tl

NIN dt d

AO

-lN

dlr

NIN !'l

E

H ;ii.o

3

?

+

'6

g

I

-

NlNl |rl

N l,rr tl rl I

'irt

VI ot

o>

-.dl

ilN dld H

ol$

-9oil

;a:46€l! E =* *En

lrr

N-ln

_

gli g-16xF .9rr!.: 1.tE;

U}'Q

H

6

d

Nld dl

-

d

oh og

*tE rr

r

'l+ -lr_

'ii

lh E

d

?

' dl k NIN trt d +

_

dl

6

h

Nlru rdl td

F 3 ;; E

.fi t!

-

t

NIN tt d

r'l+ r;

NIN tl

+

NIN

.!lk

d dt G \__s_ __e__ i

>l$ '

ci

p

"

E=. g

d

.'---5NIN tl

dlk r-t

dlN

klQ

rl$ "

Ft$

Nl^

I

".

,,

d

I

k

-lN I

e'l$j*[

*.

NIN ht

I

L---J

d

g

^r'stlN d

v *a-

dlN i-ilN

|

N l^

rt

-' '-;;-

d,

, ilN,

E>

E:

+

rl

dld

'dl d -_5| d lJ -|ru

I

I

,E"

al 3l

---:-NlNiNlN

NIN

-,.-=.,NIN

d+

rhrt

+{t

rd

N lru 'dl'nH

il

ul

_d_

I

rls

H

:-;:'rl-6 rril

TtA

#tE .q

#.E

tl

'

i!

Circular Plates under a Variable Load

r08

Tabulation

ol

Formulas

109

f;,8 a8 eA vt

.6el EI'

El dk kc

k

of

t!

El .ti.t r ' o, El oo 3l >lnxil

El u E

3l bl

E

{!

ht x {l;l t>El€ 3l ' llt

3l

6

€l * &

E

>"

.gl al

.-|-;.

H

o

d

TT E6

+

N lN' rilc

d9

33 it .A' F8 '5? sgl 6t3{.8

I

>l

ilj

dlti

)

E

IJ *lN

NIN ril6

t*l^,r -TG-

Fl$

-;;-

l

d

E

N l^l 6ta

il

.{:F

€> Vl k

Vl OA

E

I

H

-Ctrl r tl+

n

=-

a

Ftr

3l

I

oox

i1t! 59sFo 6.E4fu -oe H€ oz d

tl

$l{ n)

+

I

5iF

ili +l+ t-------r

ft-.i-- o-r--

d

vl

,i o

-y -#13 f F#lS " n,, r i N-tO

0ci

ag .o

lN

€l

ok

I

>

I l+

€l k \3_ >l> >t> rl+ rl+

dti

dti

N lN dl d

rlru

-lN.

--r-

l+ a

>lr

+

+l+

old

NIN dl

k I

NIN kl

d

-----=I

dlN

-f^

>l+ =l+ t,

E>

o

-riB

I

dd

on

NIN dld

d!

I

)l

t(

I

NIN €l d

>tl +fd

o

f;

€t'*t

+ N

NIN dl

F

NIN dlrtNlN r6lH NlNl

I

>l>

-=-

.klo

a ,,::

>l> rl+ N lN -Iddld

rdlNE . lJ

;

oll t{r Oil

H

-

t ,!l;-

NdlNd

NIN kt d +

h

t

F

E

tH

It:Fia

itl

-,ffi,

+

+

tl3l e ; git

E

dlr

..

bt F

Fls

aaSirld

I ,'

!!

6

^.lQ

I Fl6 Etr

*r 0

Radial and Tangential Moments

Circular Plates under a Variable Load

:tlnd utnn]utJ

und unntun

$I

lor

Circular Plates

Jo 1

t

€

_l

-l "l

6ll

I

I

il (l tl qu) :T ti

I

€€ €€

Fig. 11. Radial and Tangential Momcnts par Forcc Constant Diagrant for Circular Platc Having Fixcd Supportcd Outcr liclgo ancl liixed lnncr [!clg,t' (Caso l, u '0.3)

ztt"/'

( (( Fi

.i

I

Ai

+++

liig. 12. Radial and Tangcntial Momcnts per Force Constant Diagram for Circulerr Platc Having Sirnply Supportcd Outcr Edgc and Frcc Inner Edge (Caso ll, z :: : 0.3)

It2

Circular Plates under a Variable Load

Radial and Tangential Moments

lor

Circular Plates

113 o

ilnd

rnn3ut3 Jo 1

trrta unncutc lo

I

qqqu!qu! =66NN;

€€€€€€

Fig. 13. Radial and Tangential Moments per Force Constant Diagram fot Circular Plate Having Simply Supported Outer Edge and Fixed Inner Edgc

(CaseIII,v-Q,3)

liig. 14. Radial and Tangential Moments per Force Constant Diagram for (lircular Plate Having Fixed Supported Outer Edge and Free Inner Edge

(CaseIV,v-0.3)

Circular Plates under a Variable Load

114

||tl

-lrtl

Radial and Tangential Moments

lor

tI5

Circular Plates

L

tr! 5t

o

Fi

,l iltndunncuts$lf

\-t I I

|||IIr,, I|| ||lflrlTr[rr '\l

Itna unnlutc :o

I

-

6.1

9uo lw

S=i

I

'w

ai di ri :' €€.e e

s

TI

I

_l *l

I

-l

I I I

t

t I

-llltllrll I I

I

I

_r_\ l rttllrrrt\rri

Fig. 15. Radial and Tangcntial Momcnts pcr Forcc Constant Diagram lir Solicl Circular Platc [{aving [,-ixccl Suppurtccl Outor liclgc (Caso V, z ==, 0.1] )

lrig. 16. Radial and Tangential Moments per Force Constant Diagram for Solid Circular Platc Having Simply Supported Outer Edge (Caso VI. z - 0.3 )

Radial and Tangential Moments

Circular Plates under a Variable Load

].'FQ Fr) rort |('@9 |ort rt|oo .i i oU -': $i o rt ||[{trtl