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
a<
46
Case 100. Plate Fixed Along Three Edges and Supported Along One
Simply Supported Rectangular Plate Under Hydrostatic Pressure,
Plate Fixed Along Three Edges and Free Along One Edge. Line Load Along Free Edge
Case 96.
36
xv
2/3 Uniformly Varying
Load
54
I10. Plate Fixed Along One Edge and Supported Along Two Opposite Edges. 1/3 Uniformly Varying Load
54
l l I. Plate Fixed Along One Edge and Supported Along Two Opposite Edges. 1/6 Uniformly Varying Load
55
( 'tt,tr I 12. Platc Fixccl
45
Along One Edge and Supported Along Two Opposito Eclgcs. Momcnt at Frcc Edge
56
xvi Case
Tabulation ol Plates
Tabulation ol Plates
lI3.
Plate Fixed Along One Edge and Supported Along Two Opposite Edges. Line Load at Free Edge
l14. Plate Fixed Along Two Adjacent Edges. Uniform Load Case I 15. Plate Fixed Along Two Adjacent Edges. 2/3 Unlfotm Load case I16. Plate Fixed Along Two Adjacent Edges. 1/3 Uniform Load Case l17. Plate Fixed Along Two Adjacent Edges. Uniformly
Case
Varying Load
Case 118. Plate Fixed Along Two Adjacent Edges.
UniformlY VarYing
Load
57
CaseVIII.Outer Edge Simply Supported' Inner Edge Free' Uniform Load on a Concentric Circle of Plate
58
Case
58
59 59 60
1/3 Uniformly
IX.
CaseX. /'
xvu
92
Outer Edge Simply Supported' Inner Edge Fixed' Uniform Load on a Concentric Circle of Plate
94
Outer Edge Supported and Fixed' Inner Edge Free' Uniform Load on a Concentric Circle of Plate
96 98
Case
Xl.
Outer Edge Supported and Fixed' Inner Edge Fixed' Uniform Load on Inner Concentric Circle of Plate
Case
XII.
Outer Edge Simply Supported' Inner Edge Free' Uniform Load on Inner Concentric Circle of Plate
100
6I
CaseXlll.Outer Edge Simply Supported' Inner Edge Fixed' Uniform Load on Inner Concentric Circle of Plate Free' Case XIV. Outer Edge Supported ancl Fixed' Inner Edge of Plate Circle Uniform Load on Inner Concentric
Load
62
Case
XV.
case 122. Plate Fixed Along Four Edges. Uniformly varying
62
106
case 123. Plate Fixed Along Four Edges. Uniformly
Load varying Load
Outer Edge Supported and Fixed' Solid Plate' Uniform Load on a Concentric Circle of Plate
63
Case
XVI. Outer Edge Simply Supported' Solid Plate' Uniform Load on a Concentric Circle of Plate
108
Case
l19.
Case
120. Plate Fixed Along Two Adjacent Edges' 1/6 Uniformly Varying Load
Case
121. Plate Fixed Along Four Edges. Uniform
Plate Fixed Along Two Adjacent Edges.
VarYing Load
DESIGN FORMULAS FOR CIRCULAR Case
Case
Case
Cqse
Case
60
PLATES
78
l.
Outer Edge Supported and Fixed. Inner Edge Fixed' Variable Load Over Entire Actual Plate
78
lI.
Outer Edge Simply Supported. Inner Edge Free' Variable Load Over Entire Actual Plate
80
III.
Outer Edge Simply Supported. Inner Edge Fixed' Variable Load Over Entire Actual Plate
82
lV. V.
CaseVL
Outer Edge Supported and Fixed. Inner Edge Free'
Variable Load Over Entire Actual
Plate
Outer Edge Supported and Fixed. Solid Plate' Variable Load Over Plate Bounded by Circles of Inner Radius and Outer Plate Radius
84
86
Outer Edge Simply Supported. Solid Plate' Variable Load Over Plate Bounded by Circles of Inner Radius and Outer
Plate
Radius
Cu,st'VII. Outcr liclgc Supportctl arttl Irixctl. lrrrrcr Iitlgc lrixctl" [Jnilorlrr l,oircl tlr ir ('rtttccttlt'ic: ('it'c|": ol l)litlo
88
90
ro2 104
PLATE FORMULAS
Chapter
I
FLAT PLATE DESIGN (References 10, 117, 123, 124, 131, 136, 138, 142' 151, 156, 203)
1. SYMBOLS AND
UNITS
symbols and units is not used. The preferred method is to indicate, in each discussion and at the head of each table of formulas, the notation and units there employed. Furthermore, to facilitate comparison, it is sometimes considered advantageous to adopt a notation identical with that used in sources to which reference is made, even though such notation might difter from that used elsewhere in this book. The symbols listed below, however, have been used in all
In this book a master tabulation of
CASES:
E: G .I M: ,t : z-
modulus of elasticity (psi) modulus of rigidity (psi) moment of inertia of an area (in.a) bending moment (in.-lb.) unit stress, with subscript indicating kind or direction (psi) Poisson's ratio (nondimensional)
In all formulas, unless other units are specified, the unit of
distance areas are in
is the inch and the unit of force is the pound. Therefore, all square inches, all moments of inertia are inches fourth, all distributed loads are in pounds per linear inch or pounds per square inch, all moments trc in inch-pounds, and all stresses are in pounds per square inch.
2. E/Ject ol Poisson's Ratio
Flat Plate Design
a: a/b
B- a-b
Notation
G
shear modulus of elasticity
Vu
(psi)
W w
total applied load (lb. ) unit applied load (psi), also, p outer radius of plate (in.) inner radius of plate, (in. ) modulus of elasticity (psi) plate dimensions (in.)
a b E a, b K
L M r0 S
Y axis (lb./in.) vertical deflection (in.) slope of plate measured from horizontal (radians) length (in.) end moment (in.-lb./in.) radius of a circular area on which load is acting (in. )
loading-support factor for
unit stress at surface of plate (psi) .So stress in the direction of dimension a (psi)
stress, also corner force
Sr,
stress in the direction of
S"
dimension b (psi) unit stress at surface of plate
loading-support factor for deflection, also edge force fac-
tor
Kr
y 0
edge force distributed along
f
(dimensionless)
actor (dimensionless )
K2
loading-support factor for
R V*
slope (dimensionless) corner force (lb.) edge force distributed along X axis (lb./in.)
S, I z
in radial direction (psi) unit stress at surface of plate in tangential direction (psi)
Subscript Subscript Subscript Subscript
(Thus,
M,
R,,
R,
Kn Ku ot p x, !
For Cases 1-14:
S N I) 2
Loading-support factor for moment (dimensionless) Loacling-support factor for slope (dimensionless) l.rllclirrg-support factor f or dcllcction (climcnsionlcss) lrrtcrrsily ol llrcssrrrc, rrorrrirl lo llrc nirrrrc ol pllrtc (1tsi)
refers
to moment factor, Case 1, for tangential moment
at
2. EFFECT OF POISSON'S RATIO A question which frequently arises is: What effect does Poisson's ratio have on the bending moments in a plate?
The maximum stresses, and maximum deflections, are here conveniently expressed by simple formulas with numerical coefficients K that depend upon the ratio of plate dimensions and upon the chosen value of Poisson's ratio v - 0.3, a value used for steel'and aluminum. The Poisson's ratio of other materials are: 1. Brass 0.33
plate thickness (in. ) Poisson's ratio
Bending moment per unit length acting on planes perpendicular 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 and y axes respectively Factor for moment M, (dimensionless) Factor for moment M, (dimensionless) Factor for reaction R, (dimensionless) Factor for reaction R, (dimensionless) Rectangular coordinates in the plane of the plate
5116
inside edge. )
2. Copper
0.33
3. Cast IronO.27 4. Wrought Iron 0.28
Note: The tabulated values of K, K1 are for maximum deflection and maximum stress.
M*,
I refers to tangential moment r refers to radial moment a refers to maximum moment at outside edge b refers to maximum moment at inside edge
5.
Brass 0.33
6. Copper 0.36 7. Malleable Iron0.27
It was suggested that with the Poisson's ratio of 0.3 and by tolerating f3* percent discrepancy, the numerical coefficients tabulated here may be rrsed for any structural material. Table A shows a comparison of maximum bending stress coefficients at tlrc center of a uniformly loaded plate for several values of z and for various rirtios of a/b. For a change in Poisson's ratio from 0.2 to 0.3 it is noted tfrat the maximum eftect on the bending stress coefficient occurs at a/b : l, where the change in the coefficient is less than 8 percent.
to be satisfactory for design purpose. In some in order to indicate signilicant ligures for many conditions which would have no significance to three tk:cinral placcs. -Ihis should not be taken as an indication that the percentage * Such accuracy is considered
cuscs
K
coelficients have been computed to four decimal places
irccuracy is grcatcr than noted above.
6
Flat Plate Design
5. Assumptions
'l'nrrr-ri
A-ElJect of Poisson's Ratio (v) on Coefficients ol Maximum Bendirtg Stress at the Center of a Unilormly Loaded Rectangular Plate Fixed
Circular Plates with End Moments (Cases 49-53)
Along Four Edges
,-a*:
VALUES OF STRESS COEFFICIENT
s,,,o*:
ly
0
2.7 0.254 2.0 0.242 r.6 0.215 1.3 0.179 1.1 0.141 1.0 0.106
0.1
o.2
KwLa
0.3
/max
0.254 0.254 0.255 0.244 0.247 0.249 0.220 0.226 0.230 0.187 0.194 0.202 0.151 0.160 0.170 0.rr7 0.128 0.138
-
S-"*:
EtB
c _
umax
-
(.IRC-ULAR PLATES WITH CONCENTRIC HOLES (CASES 1-14) (Ste TabLe I\
f",,x
pplietl [,oad (Cases I S-48)
KWa2 __
__'
l,:t:l
.. *
,\,,,,,
K,W -., tL
!#
p
(Cases 82-87)*
ing cquations:
ll
Et:l
Corner and Edge Forces lor Simply Supported Rectangular plates
The presentation of equations for deflection, stress, and edge slope for plates of various forms, loaded and supported in various ways, follows the ostablished pattern used throughout this book, namely, an introduction, into the basic equation, of a dimensionless load-support factor K. The text book equations for the deformation of thin plates contain the poisson,s rertio and are rather long, and tedious. They are based on certain assumptions as to properties of materials, regularity of form, and boundary cond! tions that are only approximately true. Also, they are derived by mathematical procedures that often involve further approximation. In general, thcrefore, great precision in numerical work is not justified as the result it yields does not correspond to a real condition. Therefore, it is suggested that the solutions for one particular value of poisson's ratio, v 0.3, a valuc used for steel, aluminum, and magnesium be presented. A -considerablc change of the Poisson's ratio will only slightly change the stress and clcflcction of the plate. Thus, the tabulated values of K presented here could be used for any structural material of plate. All that is needed is to sclcct from the tables the loading condition of the plate and to find the numorical values of K, Kr, K2, which will be used with any of the follow-
"r
K2Ma
K,W
V-KwL
R-Kywab
A
-
Partially Loaded Rectangular Plates with atl Edges supported (case gI )
EQUATIONS
('irculur Platcs with
o
Rectangular, Square, Triangular and Elliptical plates (Cases 54-g0)
a/b
3,
!#
KrWu
-- I,:,J-
Moments and Reactions lor Rectangular plates with various Boundary Conditions (Cases B8-1 2 3 )
M"R"Ro:
Mu
K,(pbz) Ku(pb2)
o4pb) p@b)
4. EDGE CONDITIONS Quite often it is difficult to decide whether a plate should be calcul.tcd as freely supported or fixed, whether a load should be assumed unilirrmly or otherwise distributed. In any such case it is good practice to culculate the desired quantity on the basis of each of two assumptions lcrprcsenting limits between which the actual conditions must lie.
5.
ASSUMPTIONS
The equations of this section apply only if; (1) the plate is flat, of rrniform thickness, and of homogenous isotropic material; (2) the thickrrt'ss is not more than about one-quarter of the least transverse dimension, ;rrrcl thc maximum deflection is not more than about one-half the thickness: ( I ) all forces-loads and reactiels-.1s normal to the plane of the Pl:rtcsl and
(4) the plate is nowhere stressed
beyond the elastic limit.
'i' N. r'u: A rcctangular plate supported in some way along the edges and loaded l,rtr'r:rllv will rrsrrally inclucc not only reiictions, distributed along the bJundary of the (t'rlgc lirlccs) brrt also conccntratecl reactions at the coiners (corner forces). l'l;rlt I'rrr't'tlrc ('or'ncrs ol'lhc platc havc a tcnclcncy to rise up trnder the action of the 'rPIlit'rl l.rrtl rlrc c()rccnrlir(crl frlrcc /l rrrust hc lppliccl to prcvcnt.it.
6. l,lut ('itt.ttlur I'lrttt,t ryitlr ('rtttt't'tttt'it'llttltl
I'lrtt I'htlr I lttiritt
(r. lf l,n'l' ('llt('t ll.n lt l'l,n'l'lili Wl'l'll ('()N('liN'l'l{l(' llOl.nS* llr ctrst.ol'pl:rlt's willr corrcctttric
ltole:s, tlrc tlbtrlirtccl ccluations apply
only il':
K 'l'lrc total appliccl load W : 'l'aking E
For Cases 3,4, 5 and 6
For
For Cases
lI,
X
10u psi we obtain:
.)'n,u*
Exnuptp 2: ^the plate of Example I is partially fixed at the edges, so that when the uniform load of 3 psi is applied, the plate, instead of rcmaining horizontal at the edges, assumes a slope there of 0.25 deg. It is rcquired to determine the stress at the center and at the edge under these
(zu,g"ot_ \
-
r++) at'/
L2,13 and 1.4
sorurroN: The principle of superposition is used. The stresses at the cdge and at the center are first found on the assumption of true fixity. Then the uniform edge moment (case 49) necessary to cause an edge slope of 0.25 deg. is found. Finally the stresses produced by such an edge moment irre superposed on the stresses formerly found; the results represent the true stlesses. Iror fixed edges (Case 16)
,,-E#L( '- #-I#")
The shear stress becomes large when B approaches 1, in cases where there is load or support on the inner edge. The formulas should not be applied to the case of a solid plate. When p approaches 0 the beam formulas apply quite accurately and may be used above p - 0.2. The curves, Figs. 1, 2, and 3 in conjunction with tabulated equations, Table 1, cases 1to 6, give the moment, deflection and angular deflection (slope) of plates with edge loading only, while Figs. 4, 5, and 6 contain the same data for the uniformly distributed loading in form of pressure w. Bending Stress equals 6M/t2 using M from Figs. I and 4.
1: A circular steel plate, 0.2 in. thick and 20 in. in diameter is supported along the edge and loaded with a uniformly distributed load of 3 psi. It is required to determine the deflection and the maximum
At edge: For a/r1, - I, K, 0.239 -5,630 0.239 psi. X Q4Z/.04) -K, At center: For a/ro I, 0.159 andS:0.159 I Q42/.04) -3,740psi. and S
From Case 49, K2 - 8, K1 - 6 0.00436 radian
For 0 - 0.25 deg. 0.00436
stress.
SolurroN: This comes under Table 2 Case 15 from whete we obtaid
: I
* Absttlrctctl
-
: -ro*#*#'tr -
333
x
and
M therefore
Ex.q.N{prs
[or a/ r,,
-
13.08 in.-Ib., per in.
13.08 _ s_ 6 x0.04
.,960
1
psi
'l'he resultant stress at the edge: ^s
_
5,630
_
1,960
_
3,670 psi
+
1,960
-
5,700 psi
'l'hc resultant stress at the center: fr'orn l{cf. l3tl.
0.083 in.
conditions.
Cases 7 ,8,9 and L0
y,:84!L Ut
30
'O.212irncl K1 O.39fl 3 X 3.14 X 100 -- 942lb.
942v1OO _ort, : n.. Wa, nt, : v.zLz ldt,ory w 0.398 - /942\ 9,370 psi Jn,"* : K, (ffi,l t, -
- 2/3 (u /r) lirr sirrtply sLrpportcd edges t: l /2 (u b) for lixity of onc or both edges y :i t/2 t
Whcrc thc plate proportions are such that the thickness is greater than that Iixccl by the limits just given, the deflection may be calculated by taking into account the additional deflections due to shear, as determined by the following formulas:
-
t)
s
-
3,740
to-6M
10
7. Moments and Reactions for Rectangulur
Flat Plate Design
Exevrprr 3: A diaphragm having a clamped inner edge is loaded with a total loadW - 5.8 lb. uniformly distributed along the outer edge. These
are the edge and bending loading conditions of Table 1, Case 4. The dimensions and material constants are as follows: a b
r -
F-@-b)/a-O.ss E-30 1 106psi
L.l2 in. 0.5 in.
0.017 in.
From Fig. 1, Case 4, for p - 0.55 a value S+"r : 0.21 is found. M - FS;W - 0.55 X 0.21 X 5.8 - 0.86 in.lb./in.
Thus,
The maximum stress*
: : 6y t2
From Fig. 3, Case 4, for B
-
0.oI722
-
0.55, we obtain Da
wa2 u.IUor ^e ^ ^ '663 XU./ y:lr"r,t4-EF: r. o.t 7.
6=X,99f
17,800 psi
:
0.7. Thus,
v.vLtz.. 30X -i'9),l06X1{T*
0.0056 in.
MOMENTS AND REACTIONS FORRECTANGULAR PLATEST
Certain components of many structures may be logically idealized as laterally loaded, rectangular plates or slabs having various conditions of edge support. Table 3 presents 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 difierence method was used in the analysis of the structures and in the development of the tables. This method, described in Ref. 10, makes possible the analysis of rectangular plates for any of the usual types of edge conditions, and in addition it can readily take into account virtually all types of loading. An inherent disadvantage of the method lies in the great amount of work required in solution of the large number of simultaneous equations to which it gives rise. However, such equations can be readily systematized and solved by an electronic calculator, thus largely olTsetting this disadvantage. The finite difference method is based on the 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. Solution by finite differences provides a means of determining a set for cliscretc points of a plate subjected to given loading andclcllcctions ol
r"lltc nraxittrurr
stlcss
IAhslritclcrl llont llcl.
ol tltis cxrrtttplc is it lirdirrl strcss itt thc insiclc cdgc ir. 10.
plates
lI
edge conditions. The deflections are determined in such a manner that the deflection of any point, together with those of certain nearby points, satisfy
finite difference relations which correspond to the differential expressions of the usual plate theory. These expressions relate coordinates and deflections to load and edge conditions.
The coefficients in Table 3 are for the rectangular components of the maximum bending moment and for maximum reactions of the support for the following edge or boundary conditions. cesns 88-96: Plate fixed along three edges and free along the fourth edge.
cesss 97-104: Plate fixed along three edges and hinged along the
fourth edge.
casss 105-113: Plate fixed along one edge, free along the opposite edge, and hinged along the other two edges.
cesss 1.14-120: Plate fixed along two adjacent edges and free along the other two edges.
Cnsps
l2l-123: Plate fixed along four
edges.
The loads, selected because they are representative of conditions frequently encountered in structures. are: Uniform load over the full height of the plate. Uniform load over 2/3 the height of the plate. Uniform load over l/3 the height of the plate. Uniformly varying load over the full height of the plate. Uniformly varying load over 2/3 the height of the ptate. Uniformly varying load over I/3 theheight of the plate. Uniformly varying load over I / 6 ihe height of the plate. Uniform moment along the edge y - b of the plate. Uniform line load along the free edge of the plate. Uniformly varying load, p - 0 along y - b/2. Uniformly varying load, p - O along x - a/2. Plates with the following ratios of lateral dimensions, a, to height b,
were studied for Cases 88-120:
It will
I/8,I/4,3/g, I/2,3/4, I,3/2.
be noted that for cases 8g-113 which have symrnetry about a vertical axis, the dimension c denotes one-half of the plaie width, and for cases 114-723, unsymmetrical cases, a denotes the full width. For cases 127-123lateral dimension ratios of 3/9, l/2,5/9.3/4,7/g and,l were studied. For these cases, d and b denote the full lateral dimensions. All numerical results are based on a value of poisson's ratio of 0.2.
7. Moments and Reactions for Rectangular plates
Flat Plate Design
I2
I3
> o,t lr>bo
S
'--------lvt i -'f--r | I
rf
t
mb0
c+- i.-i
a rrl F rl
V) ,F6
i\
a
F+---iI
q
O
F
3
Q
x
rrl
I
r-----Q------!
z
E
F
U
x
f4
q--- -"-l I
U)
F
..F
ztrl
9 t! A
G -t-c o o 9^f
v
A. + -,:
5 F€
o\ E &o# (t) X! d d
> _r_
F g -3 E.
F*€o .V
':E
x
vl Eo
D.!c
j :.r Y\
cdo . .^i o-i-':
X F Y X* d v -6 o+ ^€'E -o .A --$ FV\\ i ,Ao\oF- -i ^,^ 4i ^6t- w.:
'-
_ar\\ e\io
I I
>ro!g
T_
*
__L
E
HEE
Fl
(tru6-
r 3 r Eg" 2 iq-EB' ?, o e a "E
iiEtE
--1
-b-'6
I I
d boo
I
FV
.09 o; E st /.-
_I
(g
....G o..Oo rr .^oc.tct tr d =*-*
x9 s o €.d
€oq
go '!'d HGO
ts
:i ts----- 0------!
i
\-..--_-----.---I|-
i
\'
--,@ooL vaaa vt
cd c€ 6=
g)'s
trotl o"A ,J=o
e 6,9
|JIIC)
L6l '<
-l
ro
lt
€oa)
q;--;
z
!i
-€"o -i rf L- : y'6s rJ O,q g
ZFOUU<
c>
U
5-9&
tro.F
F;*df # lTaaaii
z z
btl
'-
v)
F H-
-I
H
6
oo
O
x
I
q l'----- -----"-'l
__I
O
-6 > R: H
E iEE -,.} gh:Es de
I
)i :li -i6 .] ^€ - @.t 5'€\\^v 6\Ohg d .$ c-l 'ri ,-s\\ g\i* -oo€ *€\\; )u\*o
b rrN i il: h\"
I
-, i o
!,
O
oo€ o
+
t=b0 F-.voEo
T
z
Hg eE€ g
3 x ec*g€
> _J_
l*---
i
l;*,€:
P fi *i *9i ?,,EEiE;3
TU)
;
I :. ^c H--q+--H
o
; E'iE€&
z
Es $ n 600
| !! >oE !>b0 o'5 '- tq-T-T_rrl--r S ' ,E a l-\llll *-r-l_J .}"r H
It
+ C9
tl D
>s xQrr
^o^
oI o c)€ c.t trtr,o9
--:-!
al ! r, N il\
< a'\,lt
JE cl 4a
d
"
F- € o '! rnm €
Ntr T-
oE
r i*
O
,H >o.qo;; q
>o< h >-ox.9 -
d
oH=.qc oH
-jcf
.l r'ii^ o
:;'ds , o.d
h-=
---------------l*',
0r
-V
Lt
.E*-e
'E ^"io E Jf ;\q |J.= * o
Flat Circular Plates with Concentric Holes
Flat Plate Design
14
o
9 q
0:
o
qt ol
q I
I I
n'o
3. ao. nx o.= q6 o& o
q,G!o oo
qq ol
o
o
-ol
tlo q ol ct
-:
II
t:
':l o
x-
T5
3 bo
E
-olI
q
llN ol
q
cn
||.o
-rl
r.l
e6 o!
:
c.i
.$ tIi
o
o
o
|:qa
t|.to! ooo
ooo
{--s
o
o
E E
--
.E
a +
ii
o
-N
1 0l
\
I
e?
o
?
:1"
d
q
{.
-
r.l
X 6
o
(,)
X cl
*
-:
co
o
lJr
bb
tr.
q91t': oooo
q o
oo0 N'-
+-Q
t oo
"r:
bo
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
Y: F'DN#
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
/$ffi-
(atinner edge)
!
M,-
B2SBw&2
w# :9',Dre -Ett
0:
pNyr
#
(atinner edge) M,- BzSywa2 wa4
!: F"DU _Etr 0:O
Tabulation
Flat Plate Design
H L
!l ()
v
tr ()
tr)
N<
v?
c.l
(.)
K
co rr)
o\
t-.
R
...:
(.)
c.l
lr)
iltl
sA nH
H
()
o
d
A
:
ca)
N
ca
oo
N<
c.l
ra)
\ n
co
I
s @
.f,
N
rn
A lr)
,r?
N<
t)
tr)
n
fr
lr)
(-)
N v?
tr)
q
z
ra)
i
v?
F-
I
N
II tl I
rr) c.l
h
(\
oo
00
cl
ca
co
D
oo
R s
h
co
o
ca
oo
ol co
A
v
oo ra)
@
N
tr)
o
Y
c.l
A O
c.)
-f,
@
ol
A
cn
a
,,?
\
c.l
co
n
c.l
n
#
R
oo 00
00
(\
v? c.l
(\
q A
l6 tq
v? I
(\
N
co
t-
A
\
crl oo
lr) rn
co
ra)
ca
c.i
A A
co
A A
r.)
>'
la)
s (\ A
rrl ra
\
z
N<
Fl
trl
(..* l*
r--
J
-T? lrl Ir-'l
O
U Fl v) c'i
H
Fl I-<
e6: ,^y ? iib0ts -.=u. ;
Eb3E vr,:
o d
ocd 6oo.i eill o { 9E'EE '=s tEJ o-o ,^ o 6 ;irc Xno C ! !
L V=
tr 8,: : j tr. p
:;5h
.
A;
co
Q
q
o
T=r
k x i.! (!OU
U
jhl
t\\
a
Ug aa
E"E.g
qr
-o .Yoq
.o dE g d tr'd
o9?; .F'"I
r-- F d..::
-Zoa
bI)
F-
il1
s
U
z
.o/
U
\
\ \
,ti
r
+ c.)
oo oo
A
A
.;
t--
oo
(\
sv
oo
c.l
.j
Fl
EE v Xq: F.e L
U
"dFEE
Q
r{-
cn
t--
I'r
A
=q
sca \
N
c..l
c..l
v
tr-
o
A
-i
L) la)
sr-
lr)
A A
A
sN
lr)
v
-i
-i
oo o9
@
oo
ca
cl t-.
T T c.j
*
tr)
c.l
T T
|.-
tca $
t--
o I co
r- N
ca
oo
oo
c.l
C{
00
o
A
.j
o A A
\
00
N
N
-{.
|..
oo
ra)
A -i
sx* l
ts
\=
tltl
4
;9)
I
h
atr rd
oo =1
N H
EO
o.=
g6
Sc a; E 3.3 od=
A v9; O? O\< Ht
i9) o.5 OC)
C)
O
6
o'a0 io oo otr ,^o ,'= to d, o
\{.
oo
ra)
I
c.l
r.r)
00
ca
v
"l
*v
N
co
c{
C)
v v
s
rrl
o1l
N
ca t..i
.
tr
d
N
q
F
v
r
tFi
F B
()
1-{.
F I. N z iv i
tTl
E.
\r (hl
co
t.-
H
/;'\ l
=xv
^6
9rU LQts
c\|
O
frl
/i
co
Fl
'Tr
I
l* l*
il
c.l
F--
h
N v q
---T
S/+ S
rrl
o
H
v rn v tr) trr co A A
lr)
o'\ c.l
vlif
oo
v?
t.-
c..l
=+ r- ca v oo A A
Ca
tn
A
cn
t--
..j
F.l
Tr
F
l-r
v v
II tl
co ca;
t{
v
Ftl
ti
()
r.r)
cl
N
II ll
N<
l^
-i
s
se
ca
v
v
\
c.l
c{
v? ca
N<
rn a
lr)
A
|.r
I-r
F\r + A *.
00
t--
H
d
\<
oo
o -; n
\
ti()
c.)
R
@ I
2T
d
v v v
N
Formulas
bo
d
\ I
d
.f,
o
o
L
v
co
rn
tr
H
C) d
ol
a6
S'u c; #60 d o.c H-
qEa E h;i{ E H< N,
O
dJ
Tabulation
Flat Plate Design
22
l^l^ tktH t()lo
IE
la la la tot()lo VIEIEIE
I it lv d
N
t€
Mlv
| d
lv lv tl tl oo c.t lo\
lv ttl lv lv r-- lo lN lN
-t-ttl
"l-l-1" ltl
!o Il$ loo O lc-l lO A t-{ t-i
rn
tn \o O A
.t
lo tt lc.l IA
l\o to\
lO IA
-t-tll
ro lrn lg $ lco lO\
O lci lO -i t,{ t-i
ao
tl
-t il| .+ lo\ et lo l\o o lc.r lo
c.l
-t-trrrl
A I,i
IA I
oo loo llr)
I$ IF
Q v
v?
lF l^
tl
^
lv lA
oot # lcq lA lco l- lX la )( Y IY IY
n q
I
-l"l-
N
v tlll
o
lv I il lv il lv
lo lc.) lN c.l lco lrn llr) co
rrr loo loo # lH lco c.t lO lta v^'| lF : t-i lv
ltl H lO lO'
lO lH lr.r t-i lv
loo
r# lFl\O lc.) ltr) l$ l.+ A lA lA l.i ttl * lo. l- lo. oo l\O ltn l\O
O lc'r l$ lc.l A l-i lA l.{
"l-l-lttl
$ lr- lO I'n c..t 100 lc.l l# O lO lca li .i IA I.i IA
ttl
l- l-= lt^* I-'t>. ^
l^r
lA
l+
Y lY l- l'i
o lo lo lo
l^ lr
lv tl lv
tl
tr* l.'t I H co lo\ lN O l* lC'{
a
r-sEi_:
i l\o lN cO lr li O l* lN A l.{ lA
-l"ltl
l= lla) c.l lt lo\ q l-1 l'1 l^ lcA
v^ lv tl
lv
o = O .i
lco
tll+
lO ln l* l-{ lilA
I iI
i
il
lv tl lv
tl + lt ) d lo lco li H lH l$ A I_: IA
ol lco O,
lco lO\ O lO\ Ica A IA IA
-t-ttl
ls l$ d l.i ld tl
l* tl
o l- l"- ? l$ l5 c.l IO l|r: O
-q l- l-
v^
A v
lr^ lv
lA lv
tl
lY
lq l^ lv
l-1 l-
-t-t-
I+ IN A l-i l-i .+l \o lc) lco o lc.t l\o O lc'l lt -i lA l/l I
O IN IO * lca l$
+
cr)
c.l
lo q l- le 5l* IA IA o lo lo ,i-t-t-
ra)
9llO li lo\
c.l lO\ t* lc.l ti -
tl co o l+ l#
lor
lo' c..l l16 l+
cj
l-j tl li
o, l^ c'l lct
l^
o l* In tl lo tl oo I cq lo\ @ lrn lco O lF- lc.l
-t"ttl
A l-i l^i tn c.I O A
lco lh lc.r lo
l* l*
t-i t,i -t"tFt ,t
cl
h O o A
v
l:
I I
lr-IN lc.r ti
l\o
I+ lo ti
-t-t-
*J
N
ts
t :t-
N
O lr) l$ l+ cq lr- ll]. tl
N
9=
it
l- l-
v?
s
F
rn
q lc'! le o lo lo tl
-l-t-
vtEtl lv lv
lol0)
lv tl lv tlllr) ld
lo
l.- nO lO \o lr- ltlrlO l{ ^o lli ltr- lco *I ltIA IA -i lA l-i -t-t- "l-ltl
vtEtl t.r tv lv lv I
cr l* lco
:d lE ls
l^ I^ lklk
l()lo
lv tl lv E
l^l^ l!lli
N
t: l: -l"l- -t-t- i"l-ltl
b-
\=
I
tl -tl-lo
d ld li
f-
,d-
l^ l^
lHll.r la tot() tot!2la lqls N4l.El3 VIEIE NclEll lv lv lv lY le lJ
tl
-
_\
\=
| *t
N
lv lv tl tl
co l\o l@ \o lc'r lo
Itr) lca .i- lA lA
"t-ttl
ol
Formulas
l^ lk
l^l^
l^ l^ l^ l^
vlE le
VIEIE
V IE IE I.E IE lv lv lv lv
l()
|,-
t-_
lrloi I
A IA
tl <'lco lo
v
lc..l
"t-ttl rn "o ls l* * o -{ v
tllc.l lO\
tv I I
o\ l* cn lt.) a l|.t A
IA I
i
tll^ lil^ lc.i
a r- l$
i;1515 li i lc.i l-; tl tl F- IO IO cn
\ o
I
*
co l\O
l* l* lA lA lv tl lv
-t-
cq l@ 9 tv
818ls l-- l+ lo\
$ lc..l O lc.) -t-
lco
tl
lv tl lv
A v
I
lc.l
li tv\
Y IIY lA l-
-t-t^
.n-
"i=f * =fi I=T
O lta) .: t-i F-l l^ ^ (3
loo
: t-
o lo
l()l6)lqlc)
lv I il lv d
l\o l\o lo9 lc'l
l* l-
tl
I
lnlo\ O lco lt oo
I
l!llilt
lol()
I
* I r.) \o lro O IF-
d l$
lLlk
lv
** lllo\ lo\
H l-t lN l-i v.{ lA lv lv
23
IA tv
IA tv
\o Irn lt
)
d l-; l-: tl
tv tttl tv
ltn lcr\ li lO loo lN lv lco l,i lA lA lr
-l-l-l-1" - l- lm lo lro K l3l8l€ * lfl I|.n lc-l lcq lC.I -i l.{ lA lA l-i en ls lO lO li
-l"l-l-l-
ilq lR ln ls ls lcl lcl lci l'1 ta lv
v^
€9
:o
a
2,
.*6 oo l9 Ed
u
r39
EC)
t^ lv
l- l^ l^ -o\* ll@ l* lro lO lf- lF- ll-- lc.l lO l* lO li lA l,i lA lA i l* l: O,i-t-t-t-ttl co l"o lco l+ ls la
In
F- lo l<) l-- | s l0o co l* lF-
i l* l-i
* lr |tn ll,n l$ o lco lc.l lco loo e lq l-1 lq lq l^ lA lA lA
"t-l-t-l^
--J-
S=. stl
I
s
Nll
E-+ i .--{
F
t
,' \
---l
'00 q:
dg !.1
5
gb0
o€ qo 5tr
\
3.9
ggET
"FE5
i#58
6bo
xtr
ocB o: .x! ,-Y gotr
: E'd -*
89" : BS'; xE co .x6 dE e Go P^9 €'Eg SE EO.Sg o'; o o.s x u.r r*i 'J gg Ea 96 tr bt;E E?!-q o. E A€E -o 5 6-=i:: " ; $sIg \o 5'= o @
,'
oo Xtr
..
i
orv
N
v
Jkh
S;E5H
E
-)
o.n ?=
4 oo
oX
6Ue H X.; ,: . d'= ;b F Jq€
oo
o
o0vE o
E€ 3.H . o i'i1 ogc 5 EUEo r.r€E € N.5,=,:
o d
,E* -.2 s?
-=
:a A d'X. ;.i ca
5
S)
6
s $H.*: d _..8 o.g
6i5
€Ee caE9
c)
69
o'" o0 lJ
E!
tA lv
lO lco rn lc.t l\o ca
a
U-
sE
r3o
€0) o.
ltIt
tA lv
lltl
tl
d
lv
* ll* lo lo loo lt-- lc-l lO
c{ \O * A
v
v
v
tN<
o\ I lo lo loo t- lcr) ltn lv lr'rlqr li H lo\ lo lco lT lco .j lc) lJ ld lo lc;
F'
q
il.dl
I il
bs
HH
t:
s#E E 385 H
-O^
!
i,Y
6
H
Flat Plate Design
1 ,'t
k()
H
"o
o
v
v
N<
@
rr)
co
cl
V?
A t
$
o i
rr)
t--
\r
C..l
o\
v
A
c.l
o
{co
@ c.i
.
i
A c.l
\<
.1
oo
co
lir
(r.l
v
-; F-
t--
O
i
A
ra)
6l
lr) co
ts
c.l
A
v?
v
A
N
q
.i. r)
t{
(r)
f-
lr)
c-l
co
ra)
^i
K
N
c.!
lltl
Ft--
co
c\|
.{-
co
N
c.l
co
h
A A
@
tr.if
-i
o
tat ca
A A A h v t-. ca (-r cl ra) v c.) A A
*
t-cO
ca
a
a
q
I
I
s
a
s
ts
F
ts
\
\
d
E(5
I
!
o- h I *U-Yi :2'= 9
19D
s,
-qI 3A.ET
bbE
q
l
i
\
!-o^bbo x !.=
eo:; LJ6
g
F v
E ss6
.EETg !=^
:'1
9^otr o L€ ifytride U.i
cd :o d a .;
A
O N !
^o"E -
g 6.E tr t
EcE " bE9*h
u x !-,t,.: q . dE i i 6 J u,1 q
9{6 9;
-i
ki J
!#
r'6
!
oo9Hf, bo:
E6ef .^: C
d.li
tr
! !* ! Y o; d - x .AE d ! T 5 O, \:J (.\ trt 2/ a
"g
o dd):bo^?"< x'o.= d
F .Er f'h e dE; e:.F
(tro:o
'6 ;Fo 9^ d vE
^'"qFd Xf-A g ilq: tr v H.9": , Fc;
d bo
T o,:'o a '/, (.ti,>rJ
v 6
EEE
f E'\ q/J
.if 00
oq
A ca
co loo ca tF
r N tr) -i
*l vl
r<
u
-l .j
r{]
cl *
R
ra)
I
|
11 oo
ra) II lco lca col cIA t-l c.i
c\
I
A
|
I
I
I
-\d
I
ora
22=5 -€ o.r1i botr Y
ED;
r 9'5 oxH @c> odO
q
I
'l
lh lh lN tA
Hq:
E u o
I
rr)l ool
Yci TE 9 9 6!
-'l-1
-.i
c.r @
|
\
o o
tl
vl
IN
*
d6
I
cOl
)
I
t--
\+
o! {d q
I
-t-Irr) ^l It\
s_
oo a,
I
A ,i lA
co
c.J
-.; -; col @
; t5 C.l It
A
to
lr)
oo
-l
rl -to |tn
oo
&
O\l t-'
lN l;l-1 l^ tv
ci ld ^' l-
a
=Y9HE
:
^
*'
.:
N
-l
lI
^A v^Y
N
;$) i
s
@ oq
-l
o llr)
Ir-
co
\ ':
tl
I
N
a* N
oo
trc..l
.+ co
I
t.-
tV?
:
H
AI
t-
v
ral .
N
o tlr) ool $ c.l co 6] l* tr) Y IY \ol AI
lr)
c\l
I
;l
I
co
oo
lr)
v
$
A .j
$l
xY I: tx
-
-i
co
ra)
F
\i'
lr)
A -i
N
I
vlv
F-
co
tr-
C.l
@
ca
ca
@ c.l
ta)
tltl
rn
o.l
oo
v
A i
N
-{-
cl
..;
co
A A
t--
\r
s
i tn
co
*v
ca
c.l
X
v?
c..l
A A c.i
00
q q
-i
q
\
cO
N
O
oo co
!r) tn lr)
N<
A A
c\|
i
=f,
\
\r ca
tr) F-
A
v
N
c.:l
lr)
-i A
i
ot()
r.)
vco ol
N
c'l
A A A
co
i
00
rn
t/)
@
O
<.
i
A
@
ca
A A c\l
A
oo
t--
O
co
tr)
i
t-i A
ra)
A
v
s tr-
o i
d
a-l oo
v?
Tr)
oi
cn
q
s
cl
o
F-
@
ra)
Ft--
cn
c.l
()
l/)
c..l
q
o
ll-l >
,^IA tiltr
Li
v
tr-
co
25
R
$
c.l
Formulas
N<
c.l
A
ol
d
@ @
ca
la)
co
R
eil
v v
cl
ral
FF-
()
T tr-
c.)
s#
k
H
co
co
co
Tabulation
tlll A
h
Tabulation ol Formulas
Flat Plate Design
26
27
0)l
-':x
AAA ANN
U= )a
qcd
-:iJSN 5\ls -O':i
(E to .Fdc) 3c=so 'o
O!J
-^ N N l.s .. ll ll f 9r F \ \ 9I6^^ Z il '.= * c.l
O
oo
q
9.
*e
c{ .t
v
c.l
rYl
(t) o
\,o
.5Eo
o
O#
U
v
*.
c.l
C o
s v
m
ca
c.t
s
|'r
co
v s $ \r F-.
c.l
c\
f
c.l
F-
s
a
co
r-ll $-ri
.{
-il
;'
'-kt
ii
ao
.E 6
c= a,:
g
E E* Y
$
v6,6+:! A !t: tr h
U
,
: O+
,!sLo
v
dit
.{
# t
!F
-o 'l^i e
€ 3*
c{
rf,
tA
@
O
v
@
N
co
F-'
c.l
co
(r)
N
.f,
(--
ta)
ca
@ c.I
r*
c.l c.l
rn lr)
:{-
c.l
n q
tv
v
oo
@
c..l
la)
00
co
lal
vN
c.l
c'l
A A
+ c! (\ t-- rn
c.l
l--
V
c-
cal
t-N
o
.j
sA
cl
oo c.)
..;
c.)
ol
co
+ N
oo
-i
A
o
N
|r)
$ co
N
vc-
.{-
cq
c.l C.l
c.l 00
oo
$ $ (--
tr) c.l
"-.il":.8 q. -. Atr'F C
00
*\i.
A
ci
t t.- c.) .t N A
n
@ t-r
+
ra)
A co
rn
t.-
cFl 6iiJ
:
cl
r/)
lr)
q
c.l
co co
U
V
,r?
c\l
a.
c.)
o\
c.I 00
la)
A
t.
r{-
co
ra)
v -i
c..l
c.l
\
F.
v
A<
ca
-;
tltl
la)
:
-i
n N
c.l
ra)
oj
\$
c.i
1\>
(\ vN<
co
.f, oq
C.l
ca
|,r
c{
F\
a\i
q
co
q
llrl
d
o,t
@
-q
-f,
s
c.l oq
C.l
A A
00
'rf,
o\
cal
vv il
ll
c.i
N
cc.l
A
@
^s& aca
v n t-
r.)
.!-\
d;
o\
.ir
ca
A n
o\
o'Lo
lal *{-
c.)
A
o 4 ca
ca
N
s
:
6.*
u
i6a
d
o
XO (tr
A
c? i
ra
n q N
N
N
c.!
o
oO
C.l
co
oo
00
-i
\ ol
O
|r)
cO
-i c\
@
s
lr)
o rc.) rt
lr)
oi
<J-
c.l
ca
00
c.l
F-
v ca 6i A
,r?
ol
n v A A A A -i A co
v
@ oo
ca
c"l
*
rn c..l
il
o\
:f, N
* lr)
ca;
A A o
c..l
oo
co
q co q A
co
N A
@ 00
r-
@
co
00
oo
-; A o o oo
ca
co tl
.ail
t-tr-
-.i
cl lr)
C.l
i
\r co
r.if
@
|'r
$
co
A
-i
-i
|
t- a.l @ o\ (\ cl co A o
c.)
co
cal
s
tl
Y^
€Eie'u A h: x.9 O
A
fTl (h
:J X ;€
v
$
.if
t- co A A
rn
v
\r
A
c.l
c.t
U)
co
oa *
E
rrl
o\
69.
o !c
Na
I
YA
..-
rn
v
F*
v?
.d.
c{
00
A A A A
frl
O
lt 4
oo
tr) |'r
r.-
v
+ sa- c..l c.l stn vrr @ oo ca) ca ca A A A A A -; A Ll
A
:::.9 F:.:=
"&
.O\O+
-R++ P (.) c=
9 d#c'5 - !.9 v: x X h,! L F:.r= .i 5EE b V
v v
.eE -,',' v
q
Hi;^
v v
I l;:
9
I;
nio oi
Q:d
.5
.
s,"9 I I Edsii
VXFNd .1,3 ci" a h-a
U cl
;9E o90
Cb 'o ia I . .;9 -3'1 EEE 'o.i !Eo.E"9E; Y Bg E ! iE
+;3i€
H fl
!'llo a.r d
a:q
i
; E
iaa
s
e€
E
!.5;
3;e9-b'I5
!.1 x d Ei:.Jy5 ",-
5
F:€i
Fd
H; F F;€ ^"t' dd>-E HI
Flat Plate Design
28
Tabulation
-.l-l-
ll^ I
N
F-
+ v q c.) v
\
\o
h ca
-i
co
q
v?
N
00
A
\ s $
tltl c.l
co
t--
q n co oo
ca)
T rtr) \'n -\ fl
N-\ R{
c.l
v') c\l
c'i
q
rlr)
c.l
o
A
& K tl
.if
c{
V;
q
llll
ll
>\
ct
A R .+
ca
ll
llll N
i
tr)
t{r
vv il ^-
ca
co
L,t
ca;
rn
F-
tr)
q
ca)
tr)
cl
co
c'l
-i
oo
$ A
to
"j
rlrlr
!
rlrlr
,l,l'
v lv I
v
lv
lN \o l* JF-
lv lv *{ lo l.lO l*
O I* IN oO loo lt
o l^ H lo lO\ loo
Il# \o II
tl
]||l l- l-
lr)
-o l-
alt
-_T
l--.
rt
Li_.]
o
i
l-
l''l^ tl
I.E lv I
it
t6 lv
- l'l-
lc';
$ l* ot\ l* lc.) cj l+ 1,.;
tn l* l\f h I lt- lO\ ,-i I od l.cj
tllo
tl l* -"9 lo o In le
n o o N
l'ltl
lro lio lo9 l--:
lo tllr- llo
lca |c\l lc.i
i li
tl tltlt tl
v
tl
t.E t.s lv lv it
H lO lF-
c.l ltr: l*
* l- ltl l-\o lo loo lA 'lrtl l*
oo
rn
lo lt/)
7,
A<
a ',/,
ri^
l.{
:)
o -.oo l
\JYL6d uD I d.'" -t -;,:) )
d) i'o o . oo :18 bO 6 .5 oE o=
9r, oo
I9!.9E
d o fi
E
'=
U E,b.F
E
U Kb F g
l-f
o d
t 'f
i c't'Aa
J c:: 1,,*r'
t)
N
t'_
V 1l
tl
h
il a,
8S a.9 otr
lQ li cl It l\o co
rn lr.l lo\
t-t tl
lco \o lO lco lc.l vd lc.i l<> c.t
l*l*
*co lo lo |tr) l\o
l"-e lo lc lq
o lo lo ln "q r- lq l\o l\o
lo l.+ 9 o ln l09
lq ^ lo 19 l"?
co c-t
l\O
.o
l.-
l-
lo - l(..l lco lc.)
i
16 loi
tl
rlrlr 't't'
ls tl
l*l*
\O
Itn lc.l tl rr- l\o lO co lco lo\ co
t-t-
l* l'
o.'
lS l+ l^ lo ll
o *
tltll tl
l*llo lo
l\o lO l* lod
l^ l^
.lo .o lo l.o lO
1..; l --;
lca tl lv-'
tltlt tl
=T
-?
-J
-J Ebo
7.e
orv
o9 dF
oP oF_: o tr;i
-:
aQ .. o
16,
9ao
o
6d q
oE; l-bo
=!o 6 v=
a
n r.=
ri
x:
OG
l"^o"
N
lv tl lv
-l*l* $ l* *l-l'
oi
d
!6 uc
l5 lv
.d 1,"; l.j
a oo b0 o
il
lv;
b0
9
lio '=
llr)
I
A<
qF
(tr .?, & rf9cd !
lc.r
l.E lv
kl
:f
E o! XA
v
d
lv tl lv
Irl
o
lol0)
'/,
o
c!
l^ l^ ll.ilH
lol() tdtd I
N
,.; lod lr--
l+
29
l^ l^ lkl!
l^ la lHlk lol() td t€
l.s lv
co
,a?
co
,;
.if
c.i
A j
v?
vv il ^t
09
09
.
rn
s
lr)
Formulas
llr to
I
!l-
ltN
oo
ol
hFJ.5
E€ ;D
x 6
Flat Plate Design
30
^l? blE
FI Yti E tv {)l
Lr
(.)
€td
5 tgi vl
ol vl
CJ
(J
vl
v tv I
,^I kl Olr
AI HI
()l YI
ol | vl
E ol tv
N.I I
!2
lr
H
E (JI tv
E (.) tv
q.)
vl
YI
E (.) tv
vl
ol
tzl
I
ul, (.)
c.i
U)
rrl
s
s
A A
F
F.l
|.-
00
la)
A
tl U
$ c.) A A lr)
co
A oo
s
F F.l F.l
rrl
*if
trf--
A
A
tr)
s
A R
z
c{
co
c..l
? Jp z
ca
A
c'1
|r) c.!
c.)
c.l
A N
-{-
.i
oo
s
c.l
f-
@
ca
c.l
ra)
co
A co
c.l
ra)
A
ra)
rr)
s
oo
c.l
@ c.l
|r) N
A
A co
N
A
-i A
tr)
c\
co
oo
\
o\
c..l
r.)
A
A
t-.
.
c.l
n
-i t--
N
.if
A -i
s.if
eo
N
o
-i. 00
r) A
;
c.l
A T \r
A A vi
N
co
A
|r)
r-
A
o
\J
c.l
T
a
k
&
.q qe
\)
r.l
p
rl
r:1
\
z
5> hE { JV "o"
qd
F
OO
U rrl $
\] X
^ hl
rl
,iP
L
o
d)
+
co
t.)
s co A A
-;
d
lr)
t*
9.F
t>
x> dil 90
474 .a: hx h!:
i'l
o
co C.l
lr)
N $ A
I \:rPI \:ll vll l.r
X
I
\ tLl
o
oo
oI)r @!
od
€': Od
\Fl:
.t Fl Ye
F€
od
htl-o
9.t
co
^
A cO
n
v
(.)
O
r-
co
t--
co
@
A oo
lr)
c! co
co
A
c't
s
A
e.l
ra)
H
.j
A
A
t-tr-
Ed
--i
qi
o
oo X,
ca
oo
-i
A
lr)
^""8
oO9
oi5
xo6 ;qO
rt Or,
A oo
s
A la)
A ral
.if
.
t--
A \i'
A
cr-
.
-i
A
@
o
cn
cl
v q
9tr
o
hO
Ol
oE
v
ol
F*
cO
-i
-i
A
q
,#g v:ox o.ii
mo o oAo x>
x-;
-Aao r*J
{ v
boi
A9:
v o(E
v N A A
co
ol
oo0
o
d;xv i
Yid
*
9tE r,: Y^.d
oJt
gi: oti:
^,-;,c oE,x;
o n€ .60F .i5 ;;)
d'O € ot 'q
*
3;
oq
L'
00
AA
o!
X.r:
A
oE
bod .o :o! :F9 e:-
ra)
r>l
oo E6 a .>
r)
tr-
.ai o:
ob0
N ra)
@
A A
Ptr E9
Po
:U"
co
a o
oo
q
|r)
eo
E
^"olv
oo
$
F-
r}
A A
co
la)
\q
rr)
ol
A o
A
F-
@
co
cl
oo0
@ co
=f,
A
ra.)
A
t
N
(--
c{
I
A
-i oo
I
I
c.l
ca)
A
d oi
.-i
tltl
-7 oe A U€
{
ra)
tr-
o A
A
-i
co
co
rn
ra)
t--
I
\n
00
A
oo
@
r
lv
\4
I
\
tr-
oo
oo
JO
bI)
l$ I
a
o
F> o(B
.;.E ij
.o"
o
o d o
o
v
tr-
^ \ @
.a
c.l
A
c.l
00
o
^
ots
o
\
U
J
<€
o
O
o
()
o
|
@
o
v
C)
a
o o
Fr
4 F
c..l
t-t
-i
rr'l
"a;
ca
A
c,)
| |
0)
rl6 il18 v vl tV
I
H
N1
v
\
A ra)
b0
olJ
.v
+ s
\n
A A
.i-
c\|
A A
rA A
oo
Fc.:l
A 1
ca
o\
rn
|1.)
3T
oo
oo
N
A
oo
@
00
-; A
-i
F
ra)
|.-
lr)
c.l
A
c.l
A
N
c.l
oo
Formulas IA
1]l
-l()I*
I
.d.
ol
Tabulation
6ets
6o;E ^^o YF
a
^N 9
R.o
". =: Ca^
d6
o: tr -l>F'i 4
.
i-i ya1 0
lltl l-.rt
rl
I
\': lol
l-r
\ti
l
-j
Tabulation of Formulas
D=a
32 C)
k
:.Y ErE
9€ Yq
9
E.^
il
et.; oi)h
da
h+l d
x,,,, €lttl
Exx
! ll otl Firi
'!
>'
"1
ci
dII
Ep
il
oio 6o kll otl tJr X
kll otl
Irr
I
v
:f,
al
tl
v
v
A
E.
rrk ilo
,.
trc qtr j
N
o\
x.i cdX d6
x'x
%X
ran
Y
q
$
v
I
$
'+
o
o
v I
\o
v
v a(
v
v
€
o
€
F-
N
{ {3
N
f-
v
r
f-
r
r s r
Y
|-.
vN r
Ea
vt--
q)
b0
s
-i
v
I
lo l*'!i
I I
o k
do gh x>
@d
I a*
oo
o: q =-o <.E H
AE
vA rH
hk
3: 3fi9 *s I ,X 6=
e
-.E
HO! .:a rHI-]
ts 5 b
r
fr'l
K
€
V
Frf,
vt--
r
i
t--
s
rs
A
F-i tsH do AZ
,C H.?
o& !t|{
t-:l
^(n iq) \n Htt :-
a9= ilr:t:.|.
?
ol.9
-X { E.: .J
'
l.H
...9 !
9r'
o
\]riti
<
F^
q?
jr
o Fl
O O
< :
r
E
V{J/ \\/N
\1// tl
BV/
*r.
L o
@ .o
.+
€
v
o
.a
d)€ EO
O
leb
dc) x>
G
.
Hrd
\
C) cg
"! L: lr.U b05 o o II'Oo F€ IiE$ >e :
€3;
A
u
.E IEY I6 lEo tE9 to.x
tl tl
h
C)
o !1 >
A F-
o
v
Elg
EO
vv
!+
$
v
o
r
a(
I
T
X
ol
v
v
v
v
v v
6t
ol
f
\o
scl
al
v
o\
v
6
q a.lctd
v l<
v
>,6
v
I
v
l--
l-€
0)
a)
O
v
€
tl
li
M
q
B
r
r+
\<
q
o
tr-
F
j
R
ci
s
-a
v
o bI)
6l
F-
v
\]
tl
tl
I
OI
{
{
dtl
33
'l
t-'
E dH ?
f;a
|
|
l-"'l -f-N NI 'J
I
PI -at
I
o
a 6
Flat Plate Design
34
N
q
v
o
v
el
A<
oq
a
v
r.-:
g0 6:l
x'i o
e. q4 ar
F1
'*l 3
cl
N
v ll!t
t=
v
oo
o
v
Y
a.l
q
N
Y c.l
s
r
r
!o !2>
b0
ax
ot)
eE
aa
oai tr-o
N6E
4A
rrl Fi
(t
(t)
rFl
F7Z
H;<
Y9A &=?
o
o
O
Jh YO r^
4
j
U H
F
N.l
il
(rl
v
i€; x rn
€
tiqJ
\
c.l
N
1
cr
it*
o 0)
I
I
09
a--
6l 6
.+
s€
ol
l<
€ a
6l
v
T N
r
v
\
s
c.l
v
J
YE
xx
obots Hd .;o
^, ,li
U
ci
v
k o
or
b0ts .
H,Y
P -is F- F/ rii ip
o I
q
\
Fl
J
2'".1 di-
o
ii:
s€ - :,sI
99
v
q
o
X
s
lltl
x
4X
I,^ IN N
-q
F
e
fill FS
c.l
n
v
c'l
v
v
l3 lco llll
H' lllv
r v
(J H
o d
X
ql
F-
al
v
a
\
cl
c..l
liI'
=lH
a.l
v
a k
a_
F..
v
cl
v
v
q
.f
d
$
6
v
c.i
v
.1 lN Fal.\
il
€ v?
l^
lr
t-j
E)
q
aX
cj
XO x
v
tl
o
I c,l
iltl
Fa
q v?
A
€ o
v
3l*
I
..;
.+
v
.: N4
"Q
v
v
{
v
at
X
vj
v
t-
q
nl
o
N
c-l
v
o
io y{9
'FX 9'€
q
n i
35
,!
dv
s
v
Formulas
$
d)
o
ol
aa.
N
o o
Tabulation
v n
9FAE
o L.i i:q
6
.= E'E a
@
9"
o4FX
:6 ih:
*63E;# 3- Eb 6 d tr dt
j
d g*E VIJ
E;"
t'*
F Air{:d
Q
*.: h,il -'b :.E C)ao
6o=EE
-;I o >5Q o -^ X E o*\.:
6dFPr
c g,E F'= a S*.E H HE,
OJ
()
o E'r
BFEF ?r6ir-9 F3.!
s
ri &u *&9H;E F qc c3"d o= "i 5.9.9 Fk
F-.i: tr ll o l--Ot;cd
ao.
\ X q)
H
c)
ts
Tabulation
Flat Plate Design
36
o 0)
o bI)
d
C)
oo
6i
6.9
xo
dg
69? ov
cio
69?
al N ll a
Y
6Le
'-x
XX
'C
c4X
%).
%>
al
c.l
4
a rrl
s
Y
z
F
tsl
F.l
a rri
o o
lst Y.= .o E^.
€F! :oo
o
9"o c :TH
fiEb d us F- vbl
c.l
t+
F-
c')
\ {
eN a a
H
n €
al
.1
tl
T
a
F
U E]
o
i.o
da-y H.9
.do o> XO a= 9^X
AJ
r:i/
r
v
F-
T
N
€ € A -i
j
€
rrl
N
r ol
rl
ll
v
F.l
Fl
al
€
.+ og
€ € N al
al
a
A
>'
r
^;
a
t--
F-
n
ll h4
F-.
=q
c,)
n
c?
A ol
{x
A
\ -i
tl
rrl rrl
il
F.l
*.
@
r I
$ A
:6
au J
|'{] ;'
lo
c.l
zo HO,
r, v
\
tl 4
-i
trl
v
€
rf,
r
A;ae
ntl
E:! E
;;uH
Y
J
6
€
€ -q
a $
a j
€ r
F-
€
s
J.(
j
$
:
€ n
Y
zs
\J .$
vil
-;
YO
'tr* I tr 6 k'63 +E€ 3EE
i
;5 rE: d :-i i.ll 'oPdF-> j ie hes 9r
i
&_drsi
o HO
z! y! l!o
zx t-Y di
@
r A
tr
s= 'JF
r
n
A
A
.tr
$
r
A
LI? c.i
-i
.+
"a{
tr
;
A
n
(n5
ta
.r
€ v
-i
€ €
o
'.1
N
\i A a
O
o
A
d
;
F
A
o
A
z
6l
$
vl
rl
rl [] ol
N
F "l
$
{cd
.f, ,-) lr
rrl
;)
ll
-i
s
c.l
oo oo
al v
€
G.l
oo
(n
F.1
{v;
ctl
lr) F-i
v?
-i
f-
ct
t-.
oo
cr)
vv il
v
6l
(.)
tnF rrl f.v
;;i
\; :{}
q
A
N
al A
A
-i
O
.: € q c{
rrl
F
rc.l
rtv
r:-
."i
A
ll
v
N
6
oo
s
o o
; \
v
d
-1 c-l
Fr
A
A @
v
F-
\
H H
oq
€ € .;
€
ti
-i
N
al
X
{
j
j
t*-
al
d
<.9 r
N
n
c.l
z
q
ti
oO!a -tr
\
€
-i
N<
tsE438 -f p aE* 6 o(h
€ @
q ve r
o
>o Oo
s
F-
d
p
o
d
(J
F.l
II
Y
o
j
o
\jj
r
ti
C)
F.l
9d
€
llII
F
.+
$
4
)a
0)
v
sv?
d .+ €
-j
OF
€ c.l
r- € oo
o
6'w
sv € r..)
37
\
tcl
s r A -i
of Formulas
v
N v
ol \i.
X d
X
qNi \ tIlt
H
\
v
A<
-i
A
Flat Plate Design
38
Tabulation
.l!2Sci
F.Fn<S a,o pda
(J
F F a6
a 'Y k
&v
c)
Hq2
.,/.
Fr^
:\tr3utj 6
!t Aoo::e E .o \ \ € € s .a ll ll ^!!H
ca
A
&3
co co
A
t---
cA
o
c.)
A
N<
\J5X =d z-Y
1l
969
&*/ q.: Ht^a
Frgf
a.b.E o:!
v
F.F: hoP.,r
rc at\ c,Fd )(t ;'i
C-.1
N
A
i
A A
*1 s l
>p I tu?F9
O v
@
-; i
A
s
v
oo
N
ca
A
: .b E 6
\
q
c\l
c.')
c?
s$
co
A tn
ci
d
cn
O N
i i
co
tr)
c?
ca
ca
N
c.l
co
tr-
c.i
ca
A A lr) ra)
V1
A
n
&,
./\
d
AE 26"
ca
c.l
N
c.l
A
s$
oo
c{
ol
al
A
i
ca
v?
c.) cA
c..l
c! co
c.l ca co
(\ -f,
c.)
c.t
\
ca
F-
A
-i
sco
co co
..Y
vNaF ltltss llll-i
"a
,r \\oullll v v q4 D b \n< tr-
v
\:
d
-X rlg Y5 Za FE
\
QE
N
oo
A
oc
6i
o
&o a)H :o Fb0 't
6n
c.i
;'i
.iil aN.
v
'; @
C.l
co c.)
s
vco
lr) c.)
@
c.)
A
s
Fr
ca
rr
c.)
A
r-
co
oo
c.t
A A (r)
ol
l/)
ra)
c..l
c'l
A
A A oo
.t
\
co
@ cA
ca
-i
A A
N A
c.l
c.)
cn
s
c.l
ca
ca
A rn cn
A A
\o cll
A
ra)
C.l
c.l
c.)
F-
|"n
c.l
c.l
h ol
c.l
i -
A
co
ra)
N
lr)
-i A A
o
'd $
c!
co
c.l c.l
O
.v
t-.
ra)
c.l
o A
co
tn
A
co cJ
@
cJ
oo
F-
oo
t--
cl
c.l @ c.l
A
O
c.)
+
o\ c.l A
c.l
-;
O
A o A
O
o
oq
\
ta)
+
a?
q
A
n
8
o\
oo rn cn A v r$ s Y $ co co c..l c.l H c] cl c-.| (\ N q c.l cl A A -i A A A o
rn
s I
q)
'!t
,) (t
o.?
M
lr)
A
bO
o
tll li9
39
KL
/,k ta)
Formulas
d
cn
A A A
s
()
\ \\ n
P!
A co
-o
H
*z rr'B
-; A
-;
|r)
A
q
c.)
|.-
A
c.l
h
A1
F F u)
li
oo
c.l
:
-y
c.l
c.l
Irc\tl
cib
-s
E
HO Oio
.j!
t g o-r I X !j€,9 <.l e FE.=!,:.:.i c,l-i: i
ll
AJ o*
a
vv J -r ll e s Hi*B =pE\ D D O EEE \ q
F
Hq =a
zE Dr
o
ol
H
Tabulation
Flat Plate Design
40
F
F
F
z
a J H 6 ()
o
a
rrl 1i: -:o
d
9>
o
u)r
a)
il
?: vo
ll
x
.;
o o0
lF- l\o |rn l$ lco li
l^ l^ 1919lv
lA
la lo\ loo lo\
lg lY ln lY lX lY lx lY lF lY lA lY ls lY l= lY | lA lv
l^ lv
l^ 19lv
lA
lA Iv
lA lv
IA lv
I I I
lA lv
I
X ra)
(\
v
c.l
c.)
c.l
*cl
\ {
s
td A -i
ca tr*
c.l
co
C.l
@
F-
oo
A -i
+ @
la)
lr)
A
d A
^4O €v €l&
O
Pi *O
()
()
il
(J 1
il
co
I
c.l
co
v
(\ q
-ic-l
c.l
(\ c.l
-i
oo
c.l
A
h .if c.l
ra)
c.l
6
d,< nJ
€q
a.4 1H (hl
q
a?
n
v?
\
oq
c.l
c.)
8
nx -:A F.- )/
@P
I le le lq le lq le lq lq le lq le lq lo lo lo lo lo lo lo lo lo lo lo lo
'd
,2 q
!H I!, FV 4Zq l'\4 :/\
_ lo lco l+ l\o loo lor l- lco l,n lro lF- loo I l- lc.l lc.l lc..l lN lc.l lcn lco lca lco lco lC.)
M
-q ri-
:z Lh
tr)
U)*
c.l
A A A
00
E
= -
.r
ll il.l 3t , \\nrr aa FrFi\v /J
ig
|
ad
^:A -v Zu
9q.o (l .S
^g zh
B q In l.i lE lV lB lY
v
a
U
()
dg
\M
X.
q
a k
,r4
\\kk
oo
',6
c|c49
sil -o {s lt > VV L \\rrl
I^
s:,E Y. P,1 6 AH FTi tsv HZ O< Ari
z"a
qH,s
co loo
&E FA
\\Hlr
Jr)
F.t Zr
r-l )Ar
41
qa
F
o
r',fr l-l
Formulas
rrl
trl
.l
ol
k
O
!)
O
X v?
v
\ X Cg
0 oO 6
r
@
ral
tr)
co
A A
c.)
o
co
r-
ci
oo
c-.1 I
\
F-
n
A
-i
N s
to
c.l ca
to
A
o
:i-
t-.
oo
c.l c.l
o\
oo
o
A
\
tr)
*oo
oo
ol
c.l
c\|
ca co
H
-i
-i
c?
cl
E
3
v
8
oq
6i
n
h
cJ
c-
v A
Tabulation of Forntulas
Flat Plate Design
i)
rif \o € OOO-Qn
bq .9 rl
a<
oc
l-{
>'
>,
*
>'
llil
.l
>'
n
-
aa
tlr
Z
vl
*
$
@
(-r
c.)
h
C\
ca
A
I
trl
ca
a
p cA
c.l
co
co
V)
p
c.l
c.l
cl
rr
l
s
i
$
@
n
C) (n
o tr-
sca co
(,
FN
&
l1
>'
{.Q
^^ ll ll ll | \ a-6"6 A>,>,a r _l') ddddV V a co-
il l||l l||t
ll
(a6,a.;)< i
a<
h k 6 xJ.:-
x F-
ca
.. l4 F
o
t-t--
':r co
o
co
T
a
o
ll ry o
,-.1
q
ta
o tl*
XXXX {iB*i; 41AA
s -m
oo
@
fr
c.i ail
la)
lr)
a-
@
F
o
tr-
a..l
i i
i
Fl
N
Z J
cf)
co
A
o co ca
bi
an
A
.9q 3i z qR o
z
x
-z
f''1
s
@
z
co c.)
$ v-)
I -Ha
a
ca
O
co o A -i
@
i
F
$
*
co
!i-
co
A A
t--
\o
ca
A
a.l co
c{
c.i
ta)
@
sco
a.l
t--
rr)
v-)
i o -
Z
R
*
o
s
:T
A
i
i
ca
aca
$
oo
co ca
cA
(--
o A A
F-,]
4.
co
i
a /,
v^)
co
rrl
t!
@ co
F
F-
s
oo 00
c.l
^ i
o t trco
.w co
A A
co
\r tr-
co
c.l
i
O
o
s
Fl
s $ co
00
c.l
A
to
i
co co
\r c.l
o .if co
c\ -i
I&
Fl
d
|ll L!
ca
i i
@
o
o r) \r ca s
O
cl
FT[fff[f[T[-[ftT]-fI.
oo
c.l
h
ca ca
A o
.
c.l
t *if #
A
o r,.)
ca co
o i
a
I
t|f 1|t ilt ?a-, co
c.l
O
.if
tr-
:ir
cn
oo
al a.) c\
+ * t
i i
i
oo
ca
cl
,I
v
A<
K
cta)
F
i -.
il
>
.:i
* * *,j
,
<s!* <4aA
5 z
f\
.+
ln O
z
t-
q s \r tr) h
co
o
N
F
z 9.]
co
v)
z
=i-
i o
\_
O
z rl
I
o
co ca
v
z i Fl
s oc.l v
oo ral
i
r/) c.j
U)
rn
F]
$ O
F v)
s
Or
4,
z o o O rrl
@ oO
c\
i o
o la)
rFt
Fi
co
F-
A
c\|
z @J F.1
H
o c.l -(r c.l
N t)
o tr-
ra)
cl
q
co
o
il runmffir _*.,_tU T? (J
\r
f-
o v il
t-epj
ra)
* o i o :-i la)
X
X
tr-.]
,j
c.)
FI
o
o
4,
N
A A A o
4,
,_l
lr)
F
(,
J
I-r trN
c..l
F
/, o
@
frl
C\t
@
o -f, $
cn co
o i
tr)
*
.if
i
A
F '-.1
o @
ca
o ca
a i
*.
UJ
I<
h
rrl
o s s oo A$
@ p.]
i t*
rr)
't o
s
-.i\\
v
daa
oo@o> {q'
tl
q
!J
X
F
lr..1
ca ca
El
IJ..]
co
A
a
h€
F ri
z
c..l
(J
rt
lF- c\ lrltr) loo N lN ttJca t^' i
i.:'e
ililil1||||1
rrl
li.l
tl] IJ.]
o
4,
v)
le)
l(9
coco
Lr..l
!r v v
la
tl
>r
I
I tl9 T
fTi
@
*if
tr.l
o co
z
5
rrl
F
'/,
lo
ll I ll il*a"s^^ \Nss
Z,-hA!!-_A.a.,
ll
F
tr..l
(,
x
'-.']
@
c.)
a
t-tl lz I l()
I I
!l
:c
z
a
><
r-l
o
i T i
x
O
z
tli t1 t!2 Ls
ll
v
lJJ
rTl
o
>'
! ri
I
I !
A
>'
tl
(/)
..-.
i
ll
A
s
lal
t-
il
C)
7,
U rrl |-|
cin*o -a o? oo"aoaa
*a r
z
O
-i
l-i
ri
u
z()
i
I
F
t!
F
>'
a,
N
- -
:Ii F
lr
E
z a c.)
E]
a)
c)
(,)
<
zp
-Fl
1)
\<
v
rI]
a
U
V
v
c.l
v
I
\Ol
I
1r)l
o ol
Tabulation oJ Formulas
Flat Plate Design
44
NN
"A "A "a olv\O
Fl
bq
z .9q
cl
c.] F.l
II tl
()
A
(J\
Q< z o (n
ll r<
z
rh
z
i
tltl
^
tltl
tltl
h
h
llit
tltl
C) C)
x
U)
i
tltl
tl tl
a
r<
F.l ,a?
t-
oo
i
-i
c.l
zir
@
s
co co
la)
co ca
A
@
t-'
z A A A
F-
Fl
ri
tl
t o-
cl
bi
z .9q
ddddv v x6 il il il rr ll il ll XXtrR
Fl @
^ ^-i-l N N ^^ ilil illl ilil ililss ^ ^' >' >' >\ >!5:: a. a.
./.
-
A66]riE > ( €', xxxx .. rrr66dcd
,
F Z
a
*
c)\
q.
R
d < rn
lltl
tltl
>'
\i
w
Fr
,Z
z
a
A
h
lltl
x ca
tltl
tl
tltl
tl
x |r)
*;€ \*,QS ^^
cl
x ol cn ca
+ ca
6
F @
@
@
I
rq
c.l
s
i
A
f-t.)
a.l
N
c.)
i
E]
z z
+
()
co
s.+
A
-i
rrl
z
o
@
rn co
-) tr)
c\| ca
o
Fl
z
aJ F]
v A
Fc.i
co ca
t--
:f,
z
(J
U
tll
A
z
za)
c.l
rq
d
F
=
c.l
z
|r)
rr) ca
N
o
Fh
A A
cl
at rrl
oo
ca
oo
lr)
o t-tr)
v
trl
i
A
z ={.
o vco
co
o oo
N
A
Ftr-
U
z ,l
$
t--
co co
ol
co
o
oo c\l
@
i
i+arr
lr)
A
trl
F
*.1
eil
oo
rr)
F-
cA
N
A
(\
C.l
tt)
co
(--
c.l
oo
ao
N
o o
a.l co
N
co oo
@
A A A A
x @
v?
oo
tr)
@
ll
ooSS,{€ .. x x n rrl66cdd
>i
\
j
z
$
\c.l N
I
rrl
^rl
z
^Fl
oo
^z
-i I I
oo cn
ca
|.-
al
.+ ca
N N
ca
N
c..l
F-
A I
rn co
xco \
..;
I
c.l
ot
sco
s
ra)
.; T
c?
oC)
t
I
H
a.l
tr-
rrl rrl
lr)
v
N
oo
co
s
r|.r
rJ
A
^
a]
lir
+ o
=r
N
\o N
09
("1
c.) co
rn
IJJ
X rr
@
!
.if
rtl
ra)
co
c.l
A A O
ra)
F
4
,.,1
A,
oo
la)
r-
c..l
n
r;
.T
rrl
v
V
ca
4
LJ
co ca
I
z
\T
oo
^/. 1l
a.l
a O
n
|'r A
A A A A
i F 4, J
o
rFl
rr')
tr..l
oo
t!]
F
J
\ * i N
n
(')
O
rr
lr)
trtr-
rl
F
.t, N<
II tt
tl
rl
rrl
^
\<
F 4 F
tl
ll ll il il il
F
4 q
v
n .if
tlll
x
&.
tlII
a
*
s
A
A
@
oe
rrl rrl
(, a
L!
l<
A
tlr
d;
v
co oo
!..1
&
lr.
oo
i o i
I
trJ
X @
ca
.t
u-.1
tr-
i
t!
oi
r! v) .(
oo
o A
A o
tr-
ca
rFt
co
X
.l
lr)
lTr
rrl
o
\r
'/,
tt.1
F
rr')
._______________ ______-i..{f.9
3
Lq+jd
c.l
F
rtl F -j
i
rFl
@
T
Fl
z
lr)
a rrl
v)
rrl
|.-
rr)
tl tl
h
rrl
J
tlr
tr)
lr)
N
't
trc\
E
rI] la)
H
tl
h
aa
11s,=tt s.\::E odvvx6>
z
a
rrl @
(i
a)
i ^ o
@
I I
tt.l
a
co
P
.9 ji
a
z N F-
Fl
.a
11
H
ol
ll ll ll ll no i-l d{c
tltl
x
o o o
D
{d';;
ilil
bi
A<
v
HI
xl Hl"
ll "S.S lrt trt a-a.
-t
xY'-& & dVM oco
.9
llllllllll
trl ()o Cd
llll
A
z
XXXX c6cd.d€
;
F.l
lltt
A
z H
rr.
Z
zp
O
rrl
,a?
a .if
co ra)
la)
A
A
p a
tri
z
n to
c.l
:ir o\
A
=FBEEI
co
co
ra)
.f,
['r
co
t
co co
@ ta)
(\
ra)
A
c.t
Fh
(\ lr)
A A
r-
Il.1
rrl
\ \
bd
ssae r rf ii
3E
x'
(r) *if
ca
A
N $ A
z
lr)
a U
,!
t-.{-
i i o
lr)
F-
#
s
s co o o i
.ir
|'-
oo ca
z
*lr)
Q
A
tr-
s
co
c! .(r
ca
tn
A -i -; rn
c! lr)
co
c.) *if
oo oo c..l
A
oo
co
ca
rn
(t
c..l
A
F-
H
scO
r! rrl
c.l
o
F
rrl
lr)
z
c.l
i
+
lr)
-+ FN
tr)
@ V
A
i
H
r
(4
A
(\
|.-t .if @ c{ co A A A A n c.l
ra)
ca
trl
'rt
za)
c.t
co
t']
r:
lr)
t-s oo A A
c..|
co
a rrl @
la!
vca
$ co
A A
F
--T
(, w
ra)
(\
ca
F-
co
v
ra)
@
co
c..l
o
lr)
A A
z
F.1
c\ rrl
u.]
X
rn
oo
*
c.l
$
tr)
co
cl
A A
o
F
X
4
r.] pr 1
oo
N lr)
co cO
$ lr) C-l
A
(\ c.l
v
i-
rrl
F
|t d
A A
-.-
l/)
ca co
c.l
h
cn
al
t--
lif
T-I
I
-Tv -' I
tr-.t
a
4
oi
s
v
l-
CA
(J
v v
- -'-'- --i
o
A
r ,'r
+a U
co
Er t-rl
U
co
n
us
a
p..l
ts
ca
.if
z
3
:<
*:.
tl
llllt
zal
F
F.l
tlil
\T
ra) a.]
rFJ
F.l
ti
>'
Fi
h "i
x
N
oo
co
lr)
F.l
X
tl
S
F z r!
s
V)
ii
tl
€i". IFgg
rl
F
6
tlil
tl
R
(\ co
z
(,
a
n
I
H a
F
d
tltl
II tl
F
zr-
H
a
h
47
H
-t
fr
lttl
tltl
J'
OR
erEEqq v:.
n<
()d
4ts5p5
i..]
n
Eq
rr
:lEt.
of Formulas
Tabulation
Flat Plate Design
46
a
i
-t* _Y__-
o
.:,
Flat Plate Design
T'abulation ct {- \o
cl
cl
a
illl
II II
>'
>'
II tl
>'
z
II tl
x
x
tltl
ilII
x
h
z
^:" a
tltl
()
tltl
C)
()
a
()
lltl
z
C)
i
()
j
rn
,a?
co
@
c.l co
h ca
A A
F-
ca
z o i
@
cn ra) c..!
@
tr)
F-
clR
bi
NQ.
Ee
dddgvxE. ET e;
il il il il il il ll A,;AS sd d i.--""xx rrt666S .
n v)
U)
II tl
39
od OR
6@
':i':-
v ca
sF-
ca
c.l
i o ^
r! co
rI.]
z
co
@
cn
H
-z z
oo 00
"+ co
e]
c! co
r.)
ca
c.l co
f--
c.l
A -; #
co
s
N
A
s
o\ \ir 00
A
ca
c.t
A A
FJ
z
co
a E] F D (/)
cj
r
cO
ral
a
O
o cl
o
@
N
a.l ca
z o
A
tr) .
D
cl
rrl
00
c-
c\l c.l co
F
=
(A
:ru..i?
3
a
cil
@
c.)
co
N c.)
oo
cA
oo
00
@
E] E]
El
A
lT'!
@
|'c{
c..l
i
A
c.l
i
on t/.)
A
,! .l
@
a
'if
t!
ca
tN
)
ri .tr
rn
s
ca
tr-
A
A
z @$
c.l
c!
l!
Fl
@
ca cn
v
A A
@
s A
U
t--
trr
(a)
co
cO
A -i @
N
r')
/
|'h lr)
@
t--
l|il|t._? li i*qi*i
F-
; i
c'l
A
L rl F.l
o
o
@
co
O
co
.if
i i i
v
co
t.l *v
o
O
r/) O
i
-r i
\
V
v
-tU
v v
-x-
-t-
i'r', -r'+ t7,
',& o
Y
-
.E
,",,
*t o
a
n
n
H
= o@ii11 .. xxxx ($666
rrr
aq
n< ,F+
N
ca
A A
I I
l!
* N ^ o ca
l/)
@
H
c.t
z z
cn
o]
F
z
rq
t-cn
oo
|r)
r!
z
cq
c.) d
co
@
o\
$
lr)
.{tr-
r}
co c..l
z
A
U
zrFt
i
A
cO
co
sco c-l
@
co
A
ol
(\
co
t-'
tr)
c.l
(h
lr)
c.l c\l
o
s
oo ra)
o oo
s
F]
F
A
*if
c.l
co (--
ca
lr)
00
n
F-
c.l
A
-i -i
U)
t F a
o - A
p (A
z
co
-
{ *
\n
rn
rrl
rn .$
A
i o
o .if
eo
c.l
ol
oo
o @ c.l
J
@
tr)
r!
ol
co
f-
t.-
ZA A: J
i
o A
nZ HF
o
o
r!. F'
c.l
oo
c.)
x/, |:<
i ^ v v
\r
oo
A
A -l*
O']> .1.
6iY
-z;) Er
oo
F-
00
@
v
|.-
A o
F
F
B
lr)
@
tlil
(.7
6
I'
k li.e.D
II t!
F
cn
o A A oo
o
ra)
F.1
o
Oa
r! U)
t--
X
X H
F
@
li
^u"
II tl
II ll
a
',)
t--
F
lltl
Fl
@
--T|TTf[r.'r
llII
z
s
c!
h
;
'/,
r r,)
@
x
U
u.]
>,
Y
ll\d6'6' i >, >'-i -s a..a: 1: n' tl ddvvx6, a: x l r I n l It dU 5X It
F.
@
@
lLl
cn
x a
'if
o U
a
A i
|.r)
N
>'
C)
()
llII
&
H
tl-l
z
@ c.l
t--
co
ll
llll
II
o A A -; A
I
L!
(]
O r11
@
z
trl
ri
z
z() @
v?
A
N
o
U)
co
oo a] r) tr) q A a A oO
0.)
F
F c.l
A
c)
z
N
cn
.9q
I
@
49
Formulas
bI)
lli!6;. >,x>,-p-=Na ll Il
F.l
ol
oo
lr)
lr)
co
i
t<-
--
A A A
x
A ._
i|--
A
q+*--;
c.l
cn
o{
o
U) v v <3 u*-
lr)
t--
$
-c-----+"1
l-I
00
I
A A
Y -T--
t,-x-
.F
c
-
v
o
Tabulation
FIat Plate Design
50
6l -i^aaa o ---N ta llil l' ll'a Ir il Rqro:Q ^^
Eq llll
.9q
>.
o\
()
o< z €
II
U)
B
() (h
c.)
o $ co
0) C)
@
l?
i @
@
i
@
to
A
zo
tlll
II tl
A >, ht<"- s.
>'
h d
II ll
II II
d h0 d;
ddv \4'Y{
x
ll il ll ll ll ll ll a 5ts s XXX A,a6E > q q xxxx -. rrrci6s6
oo
z
o co
A A
Oio
A A @
o
ol
cil c.)
o co
t!
o ca
c\l
rrl
ri
oo
i
O
rrl
H
r!
F
z z U
z
I
a rrl
ir r
o -+ t-
Fi
v< 'zA -r: \iH
?o
+
nZ
xx li< F'
-Z rl]iJ
lr) C.l
@
.f, c.l
i
#
o s A A
@
|'ca N
\o
di ,t \
F
rn
e \r
v?
r-.
t'-
o @
V
o
lr)
+
={
tr-
.-;
n (--
4
N
N
rr)
o
v)
oo
c.l
n
c.l
c\ U? .v
trl
oo ra)
tr-
;D 't u) Fr
<.
r.;
I
trr -f, @
.. **** E jji:
*-F
z
F
zrrl Z ()
\_
z
I
a
.:
rrl
E a H @
n I
o H
,^ ,r; I
ra)
co
o I
o
sr N
'i r. '.-z -t-
-
cn co
ca
A I
o oo
c-
T
Ir)
v F14
tr;
X
H-----I'
I
oo
o
@
Ir-
d;
h
t-
A
cO
tl N\\\
iAA
.ro
o]
n A r
I
v v
\\\
lr)
L!]
v v
5-
o@> > o< q'
I
c.l
I
'l& '!
o cn
v
I
Ofi rrl H
x
I
I
'zQ ,-..1
tl
t--
I
oo A
=o AC) arE
uao-
g
F
I
oo
q
irl
-._i-.]q7..-
@
R n
O
l||||ld
z
o co
F-
o ! o Ed ob0
I
tr)
I
L!
co
tr-
or
lJ..l
Yi i ii ;q+i.
lr)
o
r-l
tt.l
t-t F-
-H
(/)
o r-
c.l
c.l rr)
Il.,1
(a L!
co
oo
0.)
doVV Bk
II
x
I
$
z(J
h
v)
F".l
o A A
O 00
za)
()
I
=q
ri
z
c.l
A -i
a>
t= A/
o
-
51
" "€€\\
\
t-.-
co
z
oo
o
x
I
H
c\l
N
z
{'l t--
tlII
i
I
o
\o
N
tl
co co
A
ca co
x
h
t? tn
oo
A
tl
>'
Formulas
ll ^\\\\ >g__;<< ''N n b-v
tl
F
o A A A
fTt
Q< d
-,i- *
Ir-
tr-
II
n<
{
oq
o
()d
=tn
(t)
or
.9
oo
@
co
bo
E?
a.
\, d
bi
@.
@
ol
-x-
-l-
0--------> \ \ \\\
\ \ \ \ \ \\
I
|
v)
O
Fl
z
o\
rJ
.9 a< o\ OR
\<s
d n ri v?
tltl
ll
>'
d i illl
II tl
l<
a
>'
tr a
ll
(,) (,)
{n
lr)
t--
B
o]
a E,l
(--
oo
o
* c.)
f-
|.@
ca
c.l
i T a.l
F
A
A
N
co
$
6
c.l co co
o -; o
z
tr-
t--
O
co co
=+
s\o
$
co
s
i
i c\
co
ra)
s
N
.
o A
o
cn
l ll.1
z
.
t--
(..l
$
a.l
a
c1
X
t-co
$ c.l
s a1
Ir-
t--
co
' -
U)
o co co
t-
ra)
F-
o i
o
rA
s
N
o o o co @
@
oo
@
O
F-
o
I
ca
ll
z
II tl
tl tt
tl
H
a
o c.)
rn -i-
A
ta)
cl lr)
i
c.)
o IrF-
al
\o N
o @ c.l c.l
F-
co
t--
ln
f
co
@
sca
o d
rn
c.l
ca
@
O
A
i
o o i
O
3
ta)
|r)
$
oo
|,r
ca
a.l
n
@
F-
t--
v
co
c.)
co
i
O @
co
o
s
@
s A A
a)
U)
o @ .rf ta)
i
)
dc;ss$doVVXE
a.
ll il lt il il il il lt il il tl q D o@@@@@o>>(q i*'i x ,;i.. ddd6 \ k R t x k *,1.$
-F A
:s=s 4444
z
lr)
N
ca
t\
ca ca
ca ca
i i
ca
s
A
O
.t,
z ra)
z
o t:c.)
O
z
*
rt (t
s
F]
@
H
]Ev)
@
c'l
ra)
c! ca A
al
t-r
ta)
co la)
@
Irra)
t--
o ^ i
\-
@
t
o
r/)
al
o ca
@ c.l
i -;
oo
c'l
o
@
r) ca
Fd
A
o
U @
ca
lr) ca
I
t-(--
co
r.1
c.l
/1 Fl
A A
<
xz
o
c\
tr-
c-
\T
O A
tr-
A
& 11 j. ,r,
ittl"
o\
@
oo co
V
r^i i.-qi-! x
o ca
c!
\-
co
@
A a I
\<
{
l.-:----- q -------r1
o
xa'l i U,J
f-
c...1
Ir.1
v
\r |r)
I
tua
a- i!
@
o
s>
o r:r
i
f-
z
A h
I
oo
rrl
t a
I
o i
t-.1
r):
T
(?
'a lll
,.l
F-
co
i i
I
O
.1,
+ F-
@
@
1J
.t,
co
O
@
@ co
I
/, 1l
,-.1
trco
i
!-.1
,)
F
c..l
ra
o tr)
.t,
a
o
I
o o
co C\
i o
Ill
I
A<
z()
II tl
h
I
4\
,J
I I
v
\-
o @
c.l
o
I
ca
c.l
o o
z
u.l
(')
R
@
@
tr..l
ffirl
tl
(.)
(h
F O '/,
o!
-lilffi_i; i."o+.--
il
,l
o
A
a
o
o o
',4
v v
()
t/.)
Oi Oi
tu
o]
i o o o o
co
C)
a l?
tltl
>'
z
"a
o
tu rI.]
tltl
q)
ts
I
I
@
H a
rr)
A
OR
r o
U
o.l
.ir
O\i
c,
z
A A rn
ca
.9q
"s ^o c'!.+ \o -o A:^:-t^^^ NVNNv {, llilttililil[-$"s^^ ,r il rj il il lr rr \\iQ_Q h >, >, h >. >, h: -5.a.
r!
a
z(,
o
N
bn<
Lr-.]
F Z
co
53 .Q
(,
lr)
I
i i
tll
U
)
xx
@
@
I
z
(t
/,
oo
I
F]
tli
a
A
r) c-
c..l V^)
O
ri
ri
4 5a
o
z
v-)
c..l
rrl rrl
rl
disidvv':'E ll il ll ll ll|||l oooool I ( tr
i ..
I
I
F
,..1
\
I
I
!
Ilil llil ll[ ltl,,QN_o_o ll'a "a ^^ >. >\ >, A >,'i: a.
I
I
z
z
o
9!kkk
@
@ c.)
i o
A
O
F rt
^El
.J
v)
h
o
oci-oo"aCC
H
@
a.l
co
F
(n
z()
U)
R
()
tl tl
"o .a
I
$
p
rr.
zfl z
A n n c.l .j
rrl
>'
0.)
H
\r oo
co cn
F-
r.l
Tabulation of Formulas
Flat Plate Design
52
A<
-l
ra)
I
,t
rrl
a O
Tabulation
Flat Plate Design
54
.A 'S "A cl=19
ol
A
o
taq 0)
o>
3v z v V)o
lttl
>'
(.)
>'
z
()
c,)
a
lltt
II
A
>'
zc)
C)
R
illt
a
H
I
U)
@
ri
v?
rrl
@
c\ O
co
to
rn
o
v)
A
@ A tf) c.)
bi
hAh>,xYX\q
.3
sssssqv'xE
x
n
a
@
Ir)
.FJ
co
rrl
tr) tr-
tr-
oo
tr-
F
co
i
F
z
lr)
A
co rq
F
s i
oo
co
lr) tr-
-{-
o A i
p (n
N
trA
ca co
@
l.)
c.)
i
z
i
F
i
F-
@
@
\ir
fa
c-t c-l
i
z
ln tr)
N
s
'
A
rl*cf-l l,f
o
rrl
ca)
sl
z
ca
o
$ cl
i
a\
o
c)
rrl E]
Z
:H
nZ
s
:H
!.] q)
lr)
V1
H
o A
E' xH
fi< H> F4 i> A&
@
Ir-
oo
cr)
q
ar)
@
o.l
o
I
oo
o r) co
o |r)
c) I
;)
{6 (J ci
v< zd /1 :
@
A<
k
o.l
s i i a-
Fl
O c\ O
c..l
F
0 ,z o
r-
ral
c.i
@
F
6
co @
z
ca
\r
c\ h N
co
A
o
A
+ O
o] c.l
i i
^tt.l F il.
n ?
co
F
oo
zrrl
z O
v
z
c.)
@
(4
A
trl |--r
-A )oi
rt)
c.l
n
oo
ca
A
'/,
oo
$
co @
A A
4
i
A
tr)
c.l
co
xe ii<
YO
4,o
\r v
i
A
/,o O F]
J (1 ./
o
tr-
oo
co
c,
tn co
i
o.l ca
.+
-{-
N
tr.l
A
',Fr
l_-'1
co
.rs
i.
cf-l rrl
$
c.l
sca
O tr) F-
A A o A
(-)
nl)
V
@
------_--------_ i ---T*1,."
a
\
,)4
r'
tr)
i
\-
.
c.! oo
o A
;i>
6
v
A A A
',2
I
k
|.-
nZ 1,.
-r9 i
*z
co
'/,
a \J<
I
a4&
r! a i)
oo oo
co
r!
l6
o
oo
|]"]
\n d
N O
"
a
co
^z r!> o lrl
F-
0t
O
@
I
zd ^-i ?o
v
Lr..l
O tr-
0r
z(,
U
tl l
oo
;
I
a)
al
---------------- ii 3*
I
r!
F 0
F-.1
--qlf
x
i o o
IJ.,]
(J
E]
co
s V) tri o o o N
F
=:f
\o
A a A
@
A
N
z
@
v?
lr]
z
(n
ca co
co
A
-
$
\r) tr)
co
ri Z a) rl
I
@
N
;
tn (')
F z r!
@
c.)
@
@
.if
Fl
@
i
>' tl
v'xE ilil1|l SSqe )ixxx dddd
v
A
lJ..]
z
U) @
lt
55
I
i o A
A
1t
II
"1
F
a rrl
A
n<
Formulas
tg^^ \A,"a'o
ol
3ti q<
il il il 1|l ll l||l a $^\\vi' < \ \ (yei(?(r(9< x .. >i x xd.d6d nXlikX
I
I
ooo-ao5.Kll ll ll ll rr* ^€ ^^ 'r q+ios
ol
\J
v v
Flat Plate Design
s6
Tabulation
ll-s ll oo^^::
lltt
()
o
z
O
I
trl rrl l4 _!
a rp kRkRh z r:\66(ejra>: q qs
llll
tlll R
; i
H
d a
o |r)
o
C)
c.)
0)
"
@
fl z
rn
F-
r- cn N n {
A IY tl
N
c.l
i
rr)
co
q
N @ F-
$
F
i
@
F-
oo
$
oi
q
lr)
.'r H
Har =\J
XE]
a'l
q
R
co
@
v-)
trt--
i
o
@
ca
co
o
.r;
co
z
J
F
zrrl
co
v
H
ca
i
o v
ca
u..l
oo
Lt.l
co
'z
n
'q
rn
Z
s
O
tr-
\
z
(, a
-
r
.
|r)
U)
Fco
+ A \r
q
r) A fco
o c.l
a)
t!
rrl
'-.r
r)
AH H rrl
a
$
o co
u)
fF-
|r)
s oV) co
cO
h
:i'z
@ a?
N
s
f-
rr)
c'l
i i * v v
,u r----n
Ll
co
N
i
x k------- q-------->1 tl
cO
I
{i''1
co
I
oo
at
\r
o A
U lli
r\
co
I
'/,
'1
o
tr-
(,
lU I rt
\<
N
o
f-
I
t! (')
I
v
oo
o.l
ca
i (l
ca
F-
A
o
c.l
=zlI]
HE < l-)
ca
c! tn
.l\
qFq
*tr\ F-
I
ot
I
o
s<
>>>>
a
I
I
oo
$
Lr
o
cO
[! ,', Fii
a
cl
s
I
I
s
z
@
ra)
ca
rh
Fl
q oo -{
\ n
j
o
z
.. *:.tt
H
I
I
z
tr-
o@>>qq' z
ca
z
i
I
I
co
c)
co
z
O.
fr
oo
\n
il
n
tl
U
.f,
O
^i
z
cn ca
N
@
z
x
$.,33t.t.
o IXA< da1
il il ii ll ll
d
R
ta?
@
z
ct
il
h II
x
o
tr-
s s
a.l
U)
qd
*xxx 6dd6
@
$
p
>,
I
@
F.l
A
I
Ft
-ri
na
+
I I
a
.9 +i
illlllllllllllllll
a
57
il6 A il:: ^ils!^^
EF<
.\ ^€ ^ >';533 cj$osdVVuco'
>'
v)
o
I a rrl
ll
Formulas
rr09
>ll>'llll\\ll . rr - rr rr<<{!
H
ol
N
l!(/)
I
Q
of Formulas
Tabulation
59
Flat Plate Design
58
nq d.s NN ^^ llll >ril >'tlll "Q.A A. A."A S. YY A."A
&
bc< .9 !
llll
F<
II (.)
>'
6,)
(J\
o< z 0)
llll
C)
>.
C,)
z
z
ri Z
0-)
()
a
tltl
a)
C)
x a
U)
o
>'
II
tlll
a
R
llll
O
C)
R
iltl rlI
R
dd Niv'XE. il llll illlllll xxx r@o SS,qC )i)i*x
ct$\o€ ,-: -i -i -i *d ^ ^^cr'
bi
c.l
.9q
tltl
ilII
>'
h .i
3i o. 1
ol
A
d tlll
II tl
tlll
x
R
R
o C) () (h
;
o
@
A
s s q q
oo
oo
@
oo
Ft-'
N
(..l
..:
..:
cn
t-
co
o
A A
q) q)
x v)
,.i II tl
ll ll ll ll ll "A -o ^^ " \ 4...o a AhAhA-i"_A.a. I .:'v v OOooO
ilililililllilllll
XliltkXHb-_
o@@o@>>(q .. xddcg(B rrr 'i )i >i l==== F 4A1A z
ra)
c.l
II tl
z
'ii F
o
rr)
llll
A
z
z o
tl tl
st.-
z
ral
$
c.l @
(\ t.- o \o tn oo oo v la) cl v') A o
@
c.)
F.f, co
o
A
z F
z
F
zrq
.t c.)
Fl
o
@
f*
t--
o
rcr
oo
c{ @
A
o N
F.l
00
(\ .i
ol E
-i A
z
@
ca
cl
O
sF-
? I
sr-
o
@
c.l
a
lr)
at
F-
ca
ra)
00
r
i
c-
I
o
.v
co
c.l
s A
@
GI
A
rn
rt--
^
(\
@
ztJ]
oo
d
oo
ra)
ca
c.l
n
t-
O
00
o A
ca
la)
U
q
t.)
(,
00
z
ra)
(n
H (t')
sln
A
i
co
F,
*f,
c-
t--
F
0
c.l
v
co
ro
ca
A
F-
tA
@
$
d
ca
+ co
@
ca ca
srn
-i
-i
i
st
@
ca
c.l
(h
Fc.1
@ =q
: -ru-IITTIi; .*.-r{t
F
F I
.f,
v co A
rrl
rrl
F-
oo
oo |.r
-r
lrl
lf)
a
tif
z
A
zp
z
z
@
@
@
d
rn
co
.f,
oo
A
@
ct r
ro
a
c.l rr)
-i
@,ri
oo
*co
oo co
-i
A
ca
(\
co
*la)
co
ca
z o U
co
s
o'l
A
ra)
co
-;
ra)
cl
r 2 Q
rn
oo
n
a
F
F
F $
o ca to
-f, co
i o o co
c-
co
@
U
ri
t-
rr)
z r-l
(\
@
@
oo
co c.l
@
(\
-i -i
$
cl
c.l
$ r.) A
t
v
cO
N
A
,|
+t4 f tl
oo
c.l
-i
A
o
trN
c.l c.l
rrl
F F.l
tA
co
@
i
@
oo r
c.l
A
d
*q
@
ca
c.l
c.l
ra)
c.l
i
A
c.t
c.l
o -+
o
A A
lr)
F.l
A
N
\o
co
s i
? ql= Lqi; J -TrTrTnr-+ |l1il|il r!(t)
rI]
rl X H
o
z
X co
r'r
ra)
fra)
A
oo
c.l c.l
a F F.1
o
lrl
c-
x
t--
I'
c.l
I
t-
o
rrl
I
I I
vi
'n.
rI] (n
q
A
A
Fr
o oo
lal
v
V
(! a
v
v v
a
v v
>I
o
Flat Plate Design
60
.9q O\ o< z
tl
()
ll
l
(.)
a
tl C)
h
z()
II tt
U)
x
o
.A "a -A c.t d \o
q
cl
bq
Tabulation
|"<
tltl
tlll
A
i
:Aaiaa e--o N ilil ilil lls-a^^ rr ct A.s,o a. a. >,t
dddv v'xE il il ll ll ll ll
Itl lq
bi
XX>l
a b-
@@@> >
.:i F
.9
ll -
It
llll
od
o<
( €'
I
**[5
\in
iltl
>'
>'
tl
tl
tltl
(.)
z
II
rl (t
>t
Fl
co
A
N
G.l
lr)
N
z
@
A
I-r h
co
o A
i-]
ca
ri
oo
z
ao c..l
|'-
il::^^
AA-5-sd.d.
ddvvE6 illl l|l ilrl o@>s{€ RR
tlll
(.) C)
H
g *
.. xxnx
rrrCddd
F
z @
6t
Forrnulas
N O"O^^
cl
n<
ol
z @
$
Fl
tr-
al
z
z
A A
tr)
f-
co oo
r\l
A A
F
z
@ F]
.f,
co
F-
rn
c.)
-i
@
lr)
rl tr)
--+
t-
z
ca
c'l F-
|'-
O
A A O
A A
z
z
;)
J
oo
.+
m
z
rr.
(t
z
z
@
co
s
^F
soo
to C.l
zrrl
o A
z (-)
@
ol
oo
|'-
@
oo n ca n
@
A A
ci
00
co ca)
O
I
6 IJ] ri
V
n v t-s t-r-.
E
a
A A A
c.l
a rr)
cl
I
co
$
A A
rt
o
co
zrt
c.l
v@
i
A
rrl
o\
(n H
F
oo
oo
s
co
O
t-. lr)
co
TN
c\l
co
A
i o @
o \i
o cn
@
A
@
ol
z lal
(J
^-l
rr)
00
lr)
I *
F-
co
A
$ d
r.) ca)
i a
-{-
i
v
@
F-
z v
A
t--
@
A
z
(-o.l
A
o
i
q
F Fl
c..l
c.l
i
A
.f, oo
A
F.l
c.l
o co
N
@
c.i
a
-i
/l
r
oo
v
c.l
Ir.l
i
A
FI
F
o
___--____-_____ ti L_-..--1+,-:i
-;q?r* N oo
o |.- rc\ ca O
|.n
rrl q
oo
O
i
O N
rrl
(n
-f,
II
oi
v v
N N A
X co
lr)
oo co
o
B
rl
o c.)
oo
co
ca
rFl
X oo
O
q9-; IL/ '' F
ril
o
1: l.-, eo i+
zt!
F F
Fl
a
F
F @
v v
a
(,
v v
o
Tabulation of Formulas
Flat Plate Design
62
63
ddt
1r)
bq .9
n<
vl llll
x
tltt
x
tltl R
o\ rn
lr)
lltl
tl tltl
tlil
tl
O\
o< \
tl tl
q
lr)
oo
co
.i
o ltll
rr)
>'
la)
>'
tltl
tl
tl tl
tl
H
o A
x
Q\s
c^l
x
ll
h ra)
A
II
tl
s
bi -tr
vvx6. il il ll ll
.9
>>iq'
O\
o<
XHrr
H
al
tl
tl
b0
s
o0
al
*f, oo
00
co
A A n n
oo *{-
n<
A
lltl
tlll
h
h cl
h
N
tltl
tltl
tltl
tltl
oo
Fr
A
.if tl
tl
.i
z z
la)
A
A
x
co
A A
rn
rl
x
x
.f,
$ o\
ca
oo
F-
lr)
A
s n r- lr) n s co
co
00
F.l
lr)
rr'1
z' A A
rn
z
Ctl
H
(\
v)
$
00
*t o\ N
rn
A n
ca
z
r.J
\t
$
co C.l
F-
c.l
oo
I
t) E]
oo
rr1
tr-
d rn tr- h
o
p rh
N
F]
lal
oo
@
rn
co
co
s
oo
lr)
n
A
ol
v
$
oo ca
rn
$
-; A
oo
ta
co oo
\n rn
A
-a] U
E>qq' )ilili*
ddccd
z
$
rn
ca oo
co
o.l
A
s to
A
o
A
cl N
U)
00 oo
cl cI
r\
c.l rrl
A
(\ z co N
F-
c.l
A
c.t d
i-.1
-i A
-i
@
ol
@
q
ca
F-
la)
v
c.l
ro c.l
co
.i
hffi+il lllllr
rr rrrrrr
ra)
A
z I
t
a ri
oo
F-
rfi
o\
*
c.)
A
rrl
ri
cn
co
A
-T r! +(a
E]
\n c.)
z
+
oo
Fl
lr)
c!
r-
*
o\
o\
co
A
rrr--r
X
Fl
v)
tl
6
X
rrl
oo
F-
cl
rrl
F
Fl
A
rn lr)
tJi
\
z
il ll ll
E U)
Fl
sl
vvx6.
A
Q
N
ls
z
I
oo
*q\l
h
v?
ct
T
rrl
F Fl
|.. rn h to
oo
Fra) ra)
A o n c\l
ra)
oo
oo
to
6l
{ v
t! o U
v v
F*
ol
A
oi
€
oo
.. tD\ ttl
\
\/t \d\.
tFl
\HI \HI
B
NJ
M
o La-.I
OI
rrl (t)
()
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
$
<,r>
: - + 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
!t .d
atu (/l
q
ld
l^
6tH
d
al@
ll
.:x -d 9c
0Q" 0
N
dlF
-
dl k
<-i-i
*
j+
s-c
dlP :
lo !lNIord
N
+l
+
I
dtF
=-!.r
>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
<.-
:
tlt
*dl, g4tN
NIN hl
-lN I
rl t r l*
:It
k
dla
dl,
;E
+
+l+
I
'l,l+
crr ;F
NNNIN
tdl.o
Itl
o
lp
I t+
*l-
NIN pt k rrN
=.1= =! ++
;
!---v-
&
-lN
N lN !l c
N_
N
0
l+ -la
I
tl
o
I
I
ilN
dl,
:! !a
;i;;
'n
--=_
I
olN (t
NIN !l
; NIN tl
dl!
d lN kt d
N lN .ol x I
N
d
d
+l+
'P I l+
:8
'{
-1.
l3
>
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
-^l-_
.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
NIN !t
I
k
dlr NIN pl
N lN
rl
c
tl
Nt+!tdN
.l^'
Yl
*_
€ I
it6
*
g
*
.
Fl5
: o o
A
alo
ts
d
kld
, !l F, ,_ 'l' ' I+
*; Fl
d
+
_to
,Nlf --ti: .1.,
lo
lN
i
6 k
>tts rrN - o
>:
d
N IN^ dl 'i IN Id
'
rloF
>l
oln
FIS
d
NIN !l
....T-
-^
t N l-i-:dl
Fl6
n
kk
d
'-=r'
NIN
Nlt 6l-'
NIN .ol
,__€<_
dlk
,
lll
o
itBEE o*
r_9
>
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 >
+l+ ol_ -lN
vl vl
,l
d ts
rl
x
P
t-
co .:. 0
F
rE o
rlN
lN
?? ?+
'lN
-Jr
€t
hl
I
-T* NIP -
dl
rtd
N ld d !l + dl
I
_13.
d
'n
d +
-e->l > I l+ 5.]4 ilo
N lN !l it r
+NIN ,l
.
-
d
NIN
N lN ill I
dlN
d
:+ NIN hl
dlN
t0
-.-v!
IN ^, 1!
-'tl^' I d
Fl5 Fl$ rls >*
d
tr I rl+ rla
+
n{il
El
d
--=-
>l> rl+
: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
-T-
t
5
dts NIN pl
Ttr-l+
-.ol*"
";_lN_
NIN 61 6 '
ile
QO * : r^A
+
+ dl'u
d
.__s_
--l;Nl!
e
|
I
+
?
N
^l!
NINNIN Pl k
I
-TF
-lN + NIN !l
k
+
-la
kld
dl€ ca
kl€
tlt +l
;Ft ,l+
alN
---:-e+ .:il
nil5 FrE
d
NIN Pl
td
NIN^:1, kl d rrd
---L>l> I l+
)l> t ili l+
,--i 1ru. lru ---r--e -__g
Fl$ Fl$ 56 kc
-
ni*lN-
.-NINNIN-1
xa ", tg f F.! f;"| s Et 'fr| 5 E 5r 6t
(
--
d
_.:s N lN 'old+
dtE
+
++
ol
ot Ol
dtd
'
NIN
,
;"+
--{l'^ N I :
^^ l^--:1* dl k aa
d
>
-l-
dlilnl.! CE
d't P6
ol
NIN
>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$ls
tru '0 P €td itr : LDO
o
io
I
5
of Formulas
Tabulation
Circular Plates under a Variable Load
94
95
rl{ I
ilr --1NIN ,t d
!lnl
o
rl€
iliii
fl 3l
rlt
!.98:; uto
6l r i r 9l , >'€li tl fl EI
ol
I
:g EB
aat trtl
iii ..
rl r +l
3l
r
-
ili
rld I Nld ,t
:"
kl! e NIN !l
.1r
IE -;l;-
Ei;
niiEi
it
rl > +l ,
- dld'
+
-li +l
NIN !t
i
'.;t^F fi *l* fi, iljl ilj I
=1
Ht url
ol
+l'
kt !t
NIN !t d
-1.
d
"€ E
a
,l+ >l rl rr > +l ,l rlt+l tttt
,:t-,
I
I
=l$ n
-li
l
;
t 0 .A
Bl r €t€
NIN
NIN €l
-;l;.
:1""F; +l >lr -l | +l I
o
t
A
E
o
ot
E>
a
6
:
o
LJ
r
NIN !t
d
k
--_=>l >
+l
I
---tz
--StNI_--.--..1
,-tN,
T-i-------'] r-f-i---------'l lN lN lN lN rl 'l + +l rlr -l| +l
d
I
|
IN IN d | !l
!l ( +l -l _l >t > | +l '
-l
|
Fls n
!l{
'ls l
o
aa
0 o
6
; ilr €tE
a
NIN tt a rlr +l '
+
'
to
+
d
NIN €l
I
NIN !t 6
d
r | -l-
lala
!l$
o
o 0
tt
rl
r
!t
a NIN
-ld
alo
ll
"la ilf
'ls
NIN tt(
+
---=vl
3
>l,t
NIN pl
+
NIN kt d
r-_i--]--1 'ol^* l',1". -rl+|I :t? I I
?
o
6: oc
--i-
qd%iq
NIN
NIN I !t
NIN k !l
ts
+l
---5-
NIN !t a
,
,
o i o
NIN tt d ---5_lN
F-1"!t d
,t
NIN k !l
I l+
d
dlt
NIN
,t+
s
:t: -;
pt
,
.l€
-;
NIN 3lh
E
;
I
=;
, -lt, !t
-5ir
E
a
._a;
E
>lr +ll
+
NIN r !l
NIN I ,t
--; rd-
p
NIN
+l
e
d
ct€
d
5i:+l I al!
dtk
cl I
dl! I
;
NIN rt 6
+
-;-
a
NIN ,t d
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
Nld{lN r !l
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<' ..
^^l-----NIN r'l d r
;l;
-^l *|ru'4 I o
d+N
E'
^> 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!
.
,t
rrr +l+
NIN ,t d
' I+ -;-
NIN
;
t+
{l+
-;-
d
dlk
-Trrl+
I
-Tr
Nl+
Nld !t
dl I
+
NIN d !l
d
-'l;-
E
0
o
>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 -': $i o rt ||[{trtl
<€< <€ a! ! !! ltt o ddct Nrt\r dd Nn c lil[flll
-r
N-otr [ trx il 'oNtrta 4t -ol! 4r Elt tt Etr tt
\\\
\\\
qc
n
6
-\
I
IT t/
q
!r rt trn -cta
117
Plates
/
oF l{) orot{) o@ r) o,olo qq il qr'?r?
t].
for Circular
(\l
t: !T
tr
ti
a
oo
q N
7",', 3rv']d uvrncurc
Jo
l
3tv']d uv''lnculc
(,oct
q a oln,lW? '^!
o()
o
o |r! o
q
ci I
o Frf' o o @ r{)
,4t
9! trx
].)@ tr
aq N[tr
4t tt 4l lto
t tt It t !tt
o qc i$ qq r)ot a l.'N tr xn a lt .cr
o C'O
f,
trtr tl C'o
o
#ro# "
r)O o o @ r,o rt@ q eqt(, aq NN 3 n
\i
Jo
ll
4l 4t !t !t
xtr ntr rl ,a! 4l 4l !t tt !tt
o qq qq d $ rt Nlf ro Al tr xn 'lt lta o c' oo
I
Ix
.ct
.Cl.CI
oo
at 4! C'
o I
Fig. 17. Radial and Tangential Moments per Uniform Load on Concentrlo Circle for Circular Plate Having Fixed Supported Outer Edge and Fixod Inner Edge (Case
VI[, v *
0.3),
lrig. 18. Radial and Tangential Moments per Uniform Load on Concentric Circle for Circular Plate Having Simply Supported Outer Edge and Free Inner Edge (Case VIII, v - 0.3)
a
|D f,
qqc
Ittn
T'N-
,o ao5 ! gt tt
qq
qqq
tr
aara !€a \\\ \\\
il-' qqc 1:a '-1'- qqq r
rt N $TON .CI
Plates
n or,F l() of.r@
3 Fc tt E
for Circular
Radial and Tangential Moments
Circular Plates under a Variable Load
Nrlrf Nrr9
xtr[t tt .cl 4t tt C' oC'E
!sj. 3lv1d uvrncurJ
Jo
1
p8
oo ,h o ci ,ll UO;.J{ In -
(o
: tr
4t
!,
rt
t.) rg
3
tri 3 f, f, 4t 4l
n
tt tt
c aol 14,
tr
ts
rO
ll
a a
o o
r|
ot qq ro (\l tr
tl
C'
o
tl a
Fig. 19. Radial and Tangential Moments per Uniform Load on Concentrlo Circle for Circular Plate Having Simply Supported Outer Edge and Fixod Inner Edge (Case IX, v - 0.3)
o Frt (oro ofr) (o o qa q qri'l on N tl l! It
o
U'
lL rl| uo l!'yll
o o
N I
o (' o oI
:
r)N : -trtrtrtr o 4l.Ct .o 4r o lttt aE t
o qq cq c I r)N tr) (\l otrtrtr
rll
It .Ct .o 4t oo oC'
Ol
Fig. 20. Radial and Tangential Moments per Uniform Load on Concentric Circle for Circular Plate Having Fixed Supported Outer Edge and Free Inner Edge (Case X, v - 0.3)
., itr
""1
Circular Phtes under a Variable Load
Radial and Tangential Moments
;l.f r? lT;
u
.ct
\\\ ooo \\\ ooo lrlt4t
3rv-rd uv'rncurJ Jo 1
tr
tt o
o AI
+80 rl{
:
qqc,4q'q q'orl NNnotrotrtr It .ct 4t 3 4l lt
\\\ ooo
.tr
3lvld uv'lncutc Jo 1
#3"+
=
Circular Plates
o()o u?q rtt
-r rriri NN o -13ttu tt lt 4ta ' \\\ cr0 0
o|oo |oo|o alrrt
lor
qqqqqq . r trtrrNN-
\\\ C,C'c'
=-llo[trtrtrtr
<<<<<{ ooooC'o
q t Fig. 21. Radial and Tangential Moments per Uniform Load on Inner Con. centric Circle for Circular Plate Having Fixed Supported Outer Edgo ald Fixed Inncr Edge (Case XI, z - 0.3)
l:ig.22. Radial and rangential Moments per uniform Load on Inner concontric circle for circular Plate Having simply supported outer Edge and Free Inner Edge (Case XII, z 0.3 )
-
I
I
#o ('in'ulur I'lules utultr u Variuhfu Load
Radial and Tangential Moments
for Circular
plates
,a?
AI i 4t
o
3rv']d 8v']nc8t3 lo
()o auo
;
In
31v']d uvlncuts Jo tr
a;
3 #ro#
'rl[
q"?q'qqq
9rrrtNAl.t .t ^: \\\\\\
='l':ooctct00 l.l
sf n
='1. 4
|r)o o rt r.t ot Aj xx
tr
tr
tr
.ct.o ll tt ..t o oo o o o
Fig.23- Radial and rangential Momonts pcr Uniform l-oacr on Inncr concentric Circle for Circular Platc [-laving Sinrply Sultportc:rl Outcr Edgc ancl ljixccl lnrrcr lidgc (('lsc Xllt, z . 0.3)
trig.24. Radial ancl Tangcntial Monrcnts pcr Uniform Load on Inncr Con* ccntric Circlo lirr Circular l)latc Having liixccl Supportccl Outcr Eclgc and lirr:cr lrrncr lllgcr ((';11;1y XlV. z 0.3 )
Circular Plates under a Variable Load
Radial and Tangential Moments
lor
Circular Plates
I 25 o
1r o('o
r'r; r; tr|tr oD4l 000
q N t !
OroO|oO|')
o
a lr tt.4r ct \\\\\\ ooooC'C'
t,.ir; nicinxtrnxn
t
-cr I
4t
:
31v'ld UV'lnCUtC
JO
i\ I
o|,)o()o(, rririaiAij trtrtrontr
qu? ro ro (\l
atrtt.o.o.c| \\\\\\ oooooo
il
trtr
a 4t
4t
o oo
Fig.25. Radial and Tangential Moments per Uniform Load on Concentric Circle for Solid Circular Plate Having Fixed Supported Outer Edge
(CaseXV,v*0.3)
l('
n
It
o
Fig.26. Radial and Tangential Moments per Uniform Load on Concentric Circle for Solid Circular Plate Having Simply Supported Outer Edge
(CaseXVl,z-0.3)
I
Chapter 3
BENDING OF CIRCULAR PLATES UNDER A UNIFORM LOAD ON A CONCENTRIC CIRCLE* 14. EDGE RESTRAINTS Here are presented basic equations of deflection, slope, and moments for a thin, flat, circular plate subjected to a uniform load on a concentric circle. Ten cases are presented and tabulated as cases VII through XVI. The loading considered here is schematically shown in Fig. 27. If there are more supports than necessary to maintain stability of the plates, the solution of statically indeterminate plates is inevitable. Removal of redundant support would impair not only the structural integrity of the plates, but also that of affiliated components. Superposition is the usual procedure for solving statically indeterminate problems; however frequently the generalized equations of deflection, moments, or slope must be known for the analysis. This chapter presents four generalized cases for a uniform load acting on a concentric circle of a thin, flat, circular plate (schematically depicted by Fig. 27) for solving statically indeterminate plates. The four generulued cases are:
(1) outer edge supported and fixed, inner edge fixed; (2) outer edge simply supported; inner edge free;
)
outer edge simply supported, inner edge fixed; outer edge supported and fixed, inner edge free. From these four generalized cases, six simplified cases are derived. The first four simplified cases have the uniform load along the inner plate radius, and boundary conditions complying to the four generalized cases. The last two cases are for a solid plate where the outer edge conditions are fixed-supported and simply supported. (3
(4)
* Reprinted from [37].
t27
Circular Plates under a Uniform Load
128
16, Theoretical Development
129
W
16. THEORETICAL DEVELOPMENT
--------l
i
The fundamental equations for thin, flat, circular plates subjected to symmetrical loads are as follows: (1) The radial and tangential bending moments per unit length are
(!L +,i-\ r/ \4r M,-D(* *,#) M,-
and
(2) the equilibrium
4* r dr2 *' +
D
(23)
equation is
+lJ- dr ''t'J : - ( +dr -er! - dr Ir +t,+)l
D
rz+t
- -dw/dr andD: Eh]/12(l -v') From Fig. 2l the shearing force per unit tangential length at any
where4
radius for the outer portion is
,:#
(2s)
and for the inner portion is
Fig. 27. Uniform Load Acting on a Concentric Circle of Thin, Flat, Circular Plate; Schematic Diagram
V:0
a
Substituting these shearing forces into the equilibrium equation, integrating thrice, and taking the derivation of the 4 expression, one obtains, for the outer portion of the plaie, d 1 r 4 a,
dw
- dr:+:A
for resolving deflections and in the solution for statically indeterminate circular plates. When only one or two computations arc computer program was developed
moments of these ten cases and
for
assisting
required, various dimensionless terms in the developed equations have been computed and tabulated to simplify deflection, qtoment, and slope calculation. (See Table 4.)
15. SYSTEM OF UNITS Here, as in the previous chaptcr tho unit forcc-lnass systcm is r-rsccl sincc it proviclcs a contpromisc bctwcclr tho trbsolrrto ancl gravitational systcnrs uucl Appcntlix.
)
is atrtontrtically lr
scll-corrtirirrt:tl lcl'crcrrt:c syslonr. (Scc
(26)
W
/,
1
\ , Cr
4rD,'l//?r-Z)+ir-t
da W /, t\ , Cr C2 V;:4rD (,/'?/+ , )* 2 - ," r(lnr - l) - 1-r!-C2lnrlCs .:nlp W.C, dr
C1
-a-
-;-r_||zr CN
4
(27 )
Cs
!9--c^ -cr drZ12 l1/::
r
zrzd
and for the inner portion of the plate, b
dw
C2
,
(28 ) C;,ln
r{
C1y
130
18. Desipn Considerations
Circular Plates under a Uniform Load
Upon substitution of the appropriate portions of Eqs. (27) and (28) into Eq. (23), the moment equations become, for the outer portion,
i v) #lo * v)tn, + +(r - v)fI + Dl+g Lz C, ,, .l -n{t-vt1 M,: -#l,r * z) +1tL I v)tn, - +{r z -,)-l) + rl+(r Lz (2e) +?(t-,)] M,: -
and for the inner portion,
M.-
Dl*"
- #o-,) ] Mt- Dl*rr *v) + +(t -,) ] *v)
(30)
17. GENERALIZED CASES The four generalized cases presented in tabular form (Cases VII through X) were derived by applying the appropriate expressions that fulfill the continuity and boundary conditions. The integration constants were dctermined from the first and third expressions of Eqs. (27) and (28), and lrom the expression of Eqs. (29) and (30). The upper right corner ol cach tabulated case shows the continuity conditions and/or boundary conditions.
18. DESIGN CONSIDERATIONS Normally, either the maximum bending moment or the maximum deflection establishes the design criteria for thin, flat, circular plates. Thc maximum deflection can be expressed by a formula of the type
l?-o*:
k6lla2/Ehg
Thc maximum bending moment can be represented by the expression
M^"*:
ftn,ll
whcrc "nraximum" clcnotcs magnituclc clnly, ilr maximum absolrrte valuc. As was pointccl otlt in thc prcvious chaptcr thc absolutc maxinrurrr bcrrtlirrg Ill()lllcllt. tlctcrtrtiltcrs thc nrirxirrrrrrrr rrrrit strcsscs; its llcation ltltl tttitl',ltittttlc ttttlsl bc kttttwtt. llccrrLrsc ol llrt't'onr;llt'xily 9l'llro rnontLlllt
131
equations for the four generalized cases and the dependency of Poisson's ratio upon the material and other parameters, only the absolute maximum bending moment of Case VIII could be established by studying the derived equations for any combinations of a/b and d/b or a/d ratios and Poisson's ratio. The maximum bending moments for Cases XI, XII, XIII, and XVI, the four simplified cases having any Poisson's ratio and a/b or a/d ratio, were resolved by examining the equations. Theoretically, Poisson's ratio has a value ftom zeto to one-half; e.g., for materials like paraffin and rubber, the ratios are almost one-half; for cork, the ratio is approximately zero. Poisson's ratio was taken to be 0.3, as is normally done, and the maximum bending moment and its location were determined for simplified
XIV and XV. Table 4 contains various terms in the derived moment, slope, and deflection formulas to facilitate computations. To eliminate numerous laborious hand computations, a computer program was developed. This Argonne National Laboratory program 211,7 /PADI46 computes and tabulates the deflection constants, the radial moment per load, and the tangential moment per load at predetermined r/b ot r/dratios, and for any given combination of values for z and ratios a/b and d/b or a/d.For Cases VII through XIV, the tabulated values of r/b runge from one to the selected value of a/b, in increments of 0.1. Also, Cases VII through X include r/b - d/b. For Cases XV and XYl, r/d varies from zero to the selected value of a/d, also in increments of 0.1 To acquire a better insight into the bending moments of these ten cases, especially the four generalized cases, numerical data for the radial and tangential moments divided by the load, plus the deflection constant, were ascertained with the aforementioned program. The following data were used: (1) Poisson's ratio equals 0.3 for all cases; (2) a/b for Cases VII through XIV, and a/d for Cases XV and XVI, range from 1.5 through 4.0, in intervals of 0.5; and (3) the uniform load on a concentric circle for the four generalized cases is equally positioned at two locations between the inner and outer plate radii, i.e., the third distance of d - b + (a - b) /Z or d/b - (a/b + 2)/3, and d : b + (2/3)(a - b) or d/b (2a/b * r) /3. Numerical values of the maximum deflection constant and maximum bending moment per load are tabulated in Table 6 using the stipulations of the preceding paragraph. Figures I7,18,19 and 20 depict the bending moments per load for the generalized cases for a/b ratios of 2.0, 3.0, and 4.0. The bending-moments-per-load diagrams for the six simplified cases are illustrated by Figs. 2l through26. The following statcments czrn bc madc for the maximum bending monrcnt ancl its location" with tho alorcmcntionccl prcscribed critcria: Cases
132
19. Statically Indeterminate Circular Plates
Circular Plates under a Uniform Load Cese
VII.
M,o is the maximum bending moment for all ratios studied.
6\o r€ hFoh 6i4 hd
(Fig. 17) Cnsn
VIItr.
M76 is the maximum bending moment
(Fig. 18) C^l.se
IX.
for all ratios studied.
a
M,6 is the maximum bending moment for all ratios studied.
Clsr X. M,, is the maximum (Fig.20)
CesB XI. M,6 is the maximum bending moment for all ratios studied, predicted from equation study. (Fig. 21)
as
predicted from equation study.
cess cass
as
XII.
!
bending moment for all ratios studied.
as
X H
(FiS.22)
according
U t
XIV. The maximum bending moment must be established to specificatians. M,,, is the maximum bending moment from
and lhen Mu, is the maximum bending moment from 3.0 through 4.0. The transition a/b ratio is approximately 2.6. (Fig.
*\a
F
Casn XV. The maximum bending moment must be determined according to specifications. M,o, is the maximum bending moment from 1.5 through 3.0, and then the constant moments of the inner portion of the plate (M, : M7 : M,a - M7) predominate from 3.5 through 4.0. The transition a/b ratio is approximately 3.13. (Fig. 25) Cass XVI. Moments of the inni:r portion of the plate were maximum, constant, and equivalent for all ratios as predicted from equation study. (Fig.26)
U
!
.1
o.l F- ('\ F- * \io\ \o o N
-i-i
\j. r|N :i-al oo i 4i
o\H f-r F-o oo ::
ro\ O\c) Od ro -l -i
olF\Oo rh io ^:.i
r{-r €€ alh *o ,-i -{
€.+ rol o\O *o .-l -'
6h ONi OC : -i
r$ Na r6 c)o ::
rS hr N\O +C i ^:
rO\ OO t--ct\ ;-C -:
dO\ rO O(> NF ;. -,'
O.O \On 6Cl N,-i,i
enor *o oo -:+ o\o hr *o io -i::;;;,{-i,i-{
m@ -6 €€ O6 \of- €€ *o *o
rr O\\.o mcQ *o
hh t--O\ o\€ *o -ii
*h O\\O o\O o: -i;::
h\o h\O rDC{ $ ct -i.-i
<j'o\ f-o $ a] -ia:
€
c)al hN o* <- al
o* <"*:i' f-6 .i'N -:-i
II tl
A
\O-i-n \i-€ ^l\t fqNh -nr €-+ oo rN -o 11.N h!a)..1hff
L
Vn\Ot'-6O\ X
\O rd-
tttl
a
r)
v\o@**o cir@oh\o s'6*vh\o
dF-O\N\O* oh:i-NNO
€Eqqiq ttl
rOoHOa.l
S*oNO\FN60r-O\* OO***51 -i ,-i -i ,- -l i
U
o\O'itcr\F-o inF-O@N rci\oo(\h
q-:-1qq.l X o\a-OhO\€ $r6O\ho N€hOno\ OO*Nc]o.l -i -i -i -i ,.{ -i
U
-N
llttllllll
s
1.9. STATICALLY INDETERMINATE CIRCULAR PLATES
\oh €v .i-'-.Oh OC -O
W)
6
(J
The derived equations will now be applied to solving statically indeterminate, thin, flat, circular plates subjected to various loadings.
a]€ Oa oO
6s No -i-i
*h -€ r,-
*c.l 6lO. €-
F-o\.+{ot\f\o\oiiF\: O\+€*o\O c.lNo]ooo -i -{ -{ ,-l ,-'-i
X
\c)*oF-\o€ €al*o:o *rd60N vl \?\\\\
U
\oh oo t\o o\o o\o o6 661 c)c) €h cl* hc.l \Oc{ O€ €ol O* ao Oo Oo
o
I
The structural integrity of the copper diaphragm shown in Fig. 2[i is to be determined. The following criteria apply: outer plate and loacl
^l +l ;
\oho*n@ \Or:1-FF-n
-1-
4.00 in.; inner plate and load radius, b - Z.OO in.; plate thick_ ness, h 0.125 in.; constant force, P 324 lb1; maximum permissiblc unit stress, orurr., r 6,000 lbr/in.r; modulus of clasticity, E : 15.0 y 1g,; lb1/in.2; arnd Poisson's ratio, u 0.3.
-
(J
\)
q)
-o
^16
tt
:
a.l
t-m 6r €6 .:cl.19o€ *
oo \Oo c)O F-o nq 9.1 Hat:C\1
ot6\O 0?9 *ci
oo OO qC No
3I3::: -i -i -i ,-{ --' -i
X O E
t['c
-
Sinco tlro rc,cltrirccl clcl]cction lillrrtullrs rrrc krrown, tlrc nrcthocl
$\O n€
Oc.l
q
24)
radius, a
mO \o\o O\al .l.i
X
a/b - 1.5 through 2.5,
Numerical Exarnple
rV o{$ $r \O- oN €O.t .{-i,:-i
e
XIII.
M,7,is the maximum bending moment for all ratios studied, predicted from equation study. (Fig. 23)
CesB
N€ -n ol 6 ao -O 6-;;,-i-: 01
Mt6 is the maximum bending moment for all ratios studied,
*oholO\i' @€O\f-6lO\ a.loO\o\OF\ clo6sv\+ ai-i-ii;-i
Q
O
(Fie. 1e)
v@clri-6lh l-N@OhO\ hO\*mln\O
AO{ €O\ rr c)o os. h\o t--o Hat \OO \Ofi \OO
hO
lrllrltrlltl
X
133
-s
{ q t-<
--
qa.nq cl
cl
.o
m
:t
s
tr
$00hoom €[email protected] O a.:l \+ € F- O\ -i ,- -i -i -i -i rnOt4OraO
*'c.i ci.., c,
=r
Circular Plates under a Unilorm Load
134
19. Statically Indeterminate Circular Plates
2M*-
13s
(34)
-17.0in.-lbrlin.
and
f b2 /' a \2 2M,o:I{, q--6zla(.'"-T) -h+flD! +( + :o' R l- 1 b2 , af -TL- 2 +-F=V,n-Tl 2Mno- 9.8 in.lbtlin. From these computations, the maximum radius; thus,
cfy6: =
Fig. 28. Statically Indeterminate Diaphragm Subjected to a Symmetrical Variable Load radius. The deflection at the redundant support being zeto, to write
Jw
-
0
-
w(variableload)
(35)
unit
6Y!!- * 6!;LJ;?) ___W_: - = fo_:Iff
-
stress occurs
-F6,s30
rbr/in.2
it is possiblo
- w(redundantload)
(31)
(32)
Numerical Example 2 Determine the uniform plate thickness, the radial and tangential bending moment diagrams, the maximum unit bending stress of the symmetrically loaded, statically-indeterminate, circular aluminum plate illustrated in Fig.29. Specifications are: outer plate and load radius, a - 61.2 cm; redundant support radius, d 40.8 cm; inner plate and load radius, -
b-
24.0 cm; maximum permissible deflection at inner plate radius,
('or by substitution:
R-324++++ ' 0.02327 -- llrtbt
(33 )
The maximum bending morhent for the redundant load occurs at tho inner radius, while the maximum bending moment for the variable load occurs at the outer radius for a/b - 2.0. Therefore, moments at the inncr radius and outer radius must be computed. By superposition, the moment expressions and numerical values at the inner radius and outer radiut become, respectively,
R f
-Tl-
-1
T
bz
r/' a\2 *-rll ',oll l\''z) e!
a1 +, V,=F'0-T)
(36)
The imposed unit stress criterion has not been met; hence, the design must be revamped before the analysis is continued.
If the maximum deflection formulas of Case I and Case XI are substituted in Eq. (31), the result is
2M,6:4{t+:o' + ( u-
at the inner
Fig. 29. Statically fndeterminate Circular Plate Subjected to Symmetrical Variable Load and Uniform Inner Edge Load
"@f
Circular Plates under a Uniform Load
136 wmax
:
load, W
19. Statically Indeterminate Circular Plates
137
0.08 cm; variable load constant, P - 147 kg1; inner circular edge 1,430 kgt; modulus of elasticity, E - 70.3 X 10n kgt/cm2; and
-
0.3. Poisson's talio,v To determine the redundant reaction at radius d, the method of super-
-
position is used again; i.e., the deflection at this support is zero. MathematicallY, 2wu
0
-
-
w(variable load)
-
k6(vatiable load)
-fta(redund
f
w(inner load)
ffi +
antloail
-
k.(innernail
I
ffi (37)
ffi
d to be determined from and IX:
where the k,1's are deflection constants at appropriate equations of Cases
€o
w(redundant load)
III, XI[,
the
w^o*(variable load)
-lrr-u*
-
f
ffi
+k*o*(inner rcail
o
(41)
Eh\
- 0.63442; ft,,o*(innerload) load) - 0.1126
cascs
0.1.5914
the proper values are inserted in Eq. (41), and the equation is
'
"'--.1-"*,^ri
',
)
10'63442(141)
0. lll2(t(21'l(trll I
''"
4
Fig. 30. Bending Moment
2M, -
M,(variable load)
|
f
M,(inner load)
M,(redundant load) k,(vaiable load) P { k,(inner load) W k,(redundant load) R M/variable load) f M1(inner load) M6(redundant load) kt(variable load) P f ftt(inner load) W k3edundant load) R
-
(42)
(44)
From Fig. 30, the maximum unit stress at the inner radius becomes
solvcrtl
(ry.,:
(6rrt::
lrl
ol
Figure 30 depicts the superpositioning of the moments.
I'or platc thickness, the result is
(
6l
l
>M,
ffi
{$
k*o*(variable load)
If
EI
The radial and tangential moments are ascertained by superposition:
whcre the ft*u*'s refer to the maximum deflection constants of the used. From the program results, or by hand computation,
ft-u*(redundant
o = =
.l
il('l
w*u*(inner load)
load)
'}\:v994.9*|.1.".-, -k*u*(redundant
Diagram, Numerical Example 2
(redundant load)
k."*(variable load)
2 LI
LI
The required plate thickness can now be determined; i.e.,
-
kl JI cl
o = z
- 0.45998; k6(inner load) : O'lL1'26 (38) ka(redundantload) - 0.08287 where a/b - 2.55, a/d.- 1..5, and d/b - 1'70. Hence, Eq. (37) becomes 0 - 0.4se98 (r47) + 0.1,1126(1430) - 0.08287R (39) (40) R - 2736kg1 from which,
r!l
J'
(9
ka(vaiable load)
Iw-u*
I
E
0'l'sel4(1430)
|1.0():l7lrli' 1.030c1r (43)
t
62U'o
-+564kgt/cm, (1.030) h2 :- *- !!9!2
(45)
Chapter 4
LARGE DEFLECTION OF PLATES 20.
STRESSES
AND DEFLECTIONS
In the case of bending of beams, a large-deflection theory is needed only when the deflection curve is of the order of magnitude of ong, that is, when the deflections are of the order of magaitude of the length of the beam. Such deflections are almost never experienced in practice and one is thus prone to think of large-deflection theories as of academic interest only. But the situation is entirely different with flat or curved plates which are bent in such a way that stretching or compressing of the plate must accompany the flexure, as is the case when a developable surface is bent to a nondevelopable one. The simple theory of small deflection becomes inaccurate when the defleetions are of the order of magnitude of the thickness. Such deflections are frequently met with in practice. Plate that undergoes large deflections supports a lateral loading partly by its bending resistance and partly by membrane action. Such plate is stiffer than indicated by ordinary theory. Stresses for a given load are less and stresses for a deflection are generally greater than the ordinary theory indicates. The followins cases will be summatized here:
Circular solid plate with clamped edge Circular solid plate with simply supported edge
Elliptical solid plate with clamped edge Rectangular plate with all edges simply supported Rectangular plate with two edges simply supported and two edges clamped 139
W 21. Circular Solid Plate with Clamped Edge
Large Defl'ections ol Plates
140
141
NOTATION
21,, CIRCULAR SOLID PLATE WITH CLAMPED EDGE+
: hw E zy D: a:
so that rotation and radial displacement are prevented at the edge. The plate has radius r and thickness ft and is loaded by a uniformly distributed load w per unit area that causes a maximum deflection which is large relative to the thickness of the plate. It is assumed that the plate is so tightly clamped that no radial displacement at the edge is possible. The radial membrane stress at the edge is due to the tensile forces which must be applied radially to prevent edge dis-
r
b
:
The plate under consideration is a circular plate whose edge is clamped
plate radius, in. plate thickness, in. intensity of uniform load, psi Young's modulus, Psi Poisson's ratio vertical deflection of Plate, in.
v'), flexural rigidity of plate length of semiminor axis of elliptical plate, also, half-width of
Ehy/12(l
rectangular Plate, in. length of semimajor axis of elliptical plate, also, half-length of rectangular Plate, in' P44F 4o.
placement. As for the deflection, it is not a linear function of the load. Values of y / h for various values of wra / Eha are shown in Fig. 3 3. Also, values of the stress multiplied by the quantity 12/Elxz as ordinates and values of maximum deflection y divided by the thickness /z as abscissas are shown in Fig. 34. It wiil be noted that the dimensionless ordihates and abscissas in Figs. 33 and 34 make it possible to use the curves for plates of many dimensions, provided that other conditions are the same.
It will
141
T
be seen from Fig. 33, that variations in Poisson's Ratio have very little effect on the behavior of plate. ExeNlprs: A numerical example will illustrate the use of these curves. Consider a plate of thickness, h :0.02 in., radius r - 2 in., and load 0.3. w : 3 psi. Let E - 30 X 106 psi and let a From Fig.33 for wra/Eha - 10 we-obtain y/h - 1.055 so that
y 143
I
D
-
0.0211 in.
145
--tI
S
148
S
s F
lfig. 31.
F
t49
Fig. 32. Circular Plate with Clamped Edge and Uniform Lateral Load. * Abstractcd l'rom llof. 203.
S
Large Deflections ol Plates
142
DEFLECTION OF CIRCULAR PLATE IVITTI THE EDGE CLATPED 1.2
22. Circular Solid Plate with Simply Supported Edge From Fig. 34 the stresses arel
1. Membrane
stress at edge:
0.56
Sr2/Eh2
S -
0.8
.9
143
1,680 psi
2. Bending stress at edge:
E o.l
_
srz/Ehz
f.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
3. Membrane stress at center:
Fig. 33.
STREsSES
AT EDGE AND CENTER
17,550 psi
-
S 9.0
Lood wr'lEhr
5.85
S -
sP/Eh2
(I'
1.07
3,210 psi
4. Bending stress at center:
0,3)
5rz/Eh2
_
2.57
S:
7,71"0 psi
22. CIRCULAR SOLID PLATE WITH SIMPLY
SUPPORTED
EDGE* Critical values of stresses and deflection for a simply supported, uniformly loaded plate are presented graphically in Figs. 36 and 37. Here, too, q
-+ I I
0.6 0.8 t.0 1.2 t.4 Dcflrcllon
y,/h
1.6
Fig. 35. Circular Plate with
Simply Supported Edge and Uniform Lateral
Load
Fig" 34. * Abstracted from Ref.
142.
"@
144
23. Elliptical Plate witk Clantped
Large Deflections ol Plates
Edge
145
it possible to use the curves for curves also give an indication of the These plates of many dimensions. is range of the use of the linear theory which usually quoted in references in cases where deflections are less than one-half the thickness of the plate. dimensionless coordinates are used making
aEn/2/il6$,SE't L//YE4R
3r
fi/foRf
23. ELLIPTICAL PLATE WITH
CLAMPED EDGE*
Ellipticat plates of uniform thickness are considered with ctramped boundary and subject to uniformly distributed loads. The treatment of the problem is extended to the limiting cases of the circular plate and infinite strip. Numerical solutions presented here apply to elliptical plates represented by the following ratios of minor to major axes: MEUa,(a/r/E
FoE
.D///E/YS/o/Y/.
ftt
*a,zt
DEfl fcr/au Yl
Fig. 36.
q/b
- 1 (circular plate of radius a) a/b - 2/3 a/b L/2 a/b :0 (infinite striP of widlh2a)
t/f,Ett a,tq orrt
The respective shapes of the four cases are shown in Fig. 38.
Fig. 38. Representative Shapes of Plates Solved
Numericallv
trig.37. * Abslnrclctl frrrrn
l{cl'. 124.
23. Elliptical Plate with Clamped Edge
Large Deflections of Plates
t46
detailed mathematical treatment of the large deflection problem was solved (Ref. 124) by the energy method and the results are summarized in Figs. 39 to 43. Also, exact solutions for two extreme cases, namely the circular and the infinite strip, are shown by dash-dot curves' Deflections obtained by the energy-method solution are close to the exact values, the deviation not exceeding one per cent. (See Fig. 39.) The elementary (small deflection) solution takes into account only that part of energy imparted to the plate which is due to bending, whereas, in the solution of large-deflection, the effect of shearing forces on the complementary energy of the plate is included. The deviation between the small-deflection and large-deflection is indicated in Figs. 39 to 43.In Fig. 39,lhe elementary (small-deflection) solution represents the initial tangents to the curves. The error involved in the elementary solution becomes significant when the center deflection of the plate exceeds 40 per cent of the
A
plate thickness. The two most important graphs are Figs. 39 and 43. Fig. 39 shows the dimensionless center deflection as a function of dimensionless load and Fig. 43 shows the dimensionless center deflection as a function of the dimensionless total stress at edge of plate. Also, Fig. 43 shows that the edge stress versus center-deflection relation in its dimensionless form remains indcpendent of the shape of the plate for the large-deflection solution as wcll as for the elementary theory. This is an important finding considering that the stress at edge represents the maximum stress over the plate and woulcl thus govern design considerations. If, therefore, the dimensionless centcr deflection can be obtained from Fig. 39 by interpolation for a given load on an elliptical plate of arbitrary dimensions, the governing stress cittl
s N
.\ \
N
'\
\ 5 \
Assume a clamped elliptical plate of thickness
of minor to major axes; a/b a
-
0.3.
- z/3;load w -
h
-
3 psi; E
7.4 or y
Function of Load
\'
t,6
\u
N, '2
(s N
*n
N'. x
4 6 DtMENtlo^llE
,tl2
I lo t to4D u7rA"
Fig. 40. Total Stress at Center
as
Function of Load
N
-
N,N
- 0.028 in. From Fig. 43, for y/h =- 1.4, wc obtain Sd/liF, l0 or by substittrtion, S 30,(X)0 psi.
as
i:
\
Solution:
y/h :
f ss Lo4 2 o ok,t "
:"$ 5^ 2o
Sra
From Fig. 39 we obtain:
/rrEN 5 /otYl
Fig. 39. Center Deflection
3t
lO
2+a6tOtz./1/6 D/
N
:
/
l.ff6E.ZEF/f,Ar/d/Y
Find the center deflection and the maximum stress.
The dimensionless load: waa/Eha
E/4.7 Jol,a//o/
s
O.O2
in.; the ra(io 30 \ 100 psi;
rM4//. OEF/.EC//O
N
be obtained immediately from the single curve shown in Fig. 43' (All curves are computed fot v - 0.3.) Exuvrpr-B:
147
la \
\
$e '\oso \\
ail N
B
_ __
_6Mttt DEFIECT/oN
_._._tI4c7 ,tatur/o/v
zEFl.ECr/oN _L4RGE y'cO.J
2166tOt? D//t4E^ts/o^/tE's
lrig.
4l.
toAD *a7ef
'lirt;rl Strcss irl litlgc
lrs
liunctiou
o1'
l,oad
25. Rectangular Plate with Uniform
Large Deflections ol Plates
148
TWO EDGES SIMPLY SUPPORTED AND TWO EDGES CLAMPEDX
$, - - -.tt',LL
N
pEflEcr/oil
A plot of the deflection at the center of plate is given in Fig. 45 for
a/b
$,, N
h'e \
!::iTi
N
"'a\ s' SE N
- l, 1.5, oo. The plot may be used for any value of z.
$\
a\
N
\n \ N.
N t g o.75 t;o Dl €/vt/ o N I E t t "2tf/ Ec r /o/v h ^t
o-?5 o.5
l<
Fig. 42. Interrelation of Total Stress at Center and Center Deflection
2,4
2.(
\
1,6
N
t.G
\ N
o,t
--snALL
o.1
N
l\ to
s
ilrc
\ \ $''
S
of y
60 80 100 t20 DlM8il5/o/u/tss 1eetr> * o\Ot,
Fig. 44. Central Deflection for Plates with
L48qE DEFtEcrloN
, ,!',o'?
0EFLECT/O/V
/qu lAtuEs o
DEFLECT|OAI
- - -sn4u
t,
A11
4 Edges
Simply Supported
-
i\8
4 b
,2
.\ .v
a
\r s 3, O P
149
25. RECTANGULAR PLATE WITH UNIFORM LOAD AND
s il
b
Load
,6
L,9 o.25 o.5 o.ts r.o t.25 r'5.. !/h D/
^4
EN S /ONLE1,
OErt
ec
r rc N
Fig. 43. Interrelation of Total Stress at Edge and Center Deflection
__<S/44U
4lL
24, RECTANGULAR PLATE WITH UNIFORM LOAD ANI) ALL EDGES SIMPLY SUPPORTED"F A plot of the deflection at the ccntcr of platc is givcrr in lii8. ,1.1 lilt a/b - 1, 1.5, 2, a.The plot m1y bc usccl filr any valuc gl'I)tlissgtt"s t';tlt' Ahstntclt'tl lrorrr l{t'1. I I l.
pEfltcl/Oil
/4'(6f,28f/tc//a4/
D/N/E,v,i/O/y/. E
ta/?tUEJAr
y
t5 totO wa/on
Fig. 45. Center Deflection for Plates with Two Edges Simply Supported
':'Ah:llirctctl lrtrrn l{el.
1.1 |
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 (srMPLY SUPPORTED)
b P per unrt
lcngth
F---
a Fig.46.
"{
The formulas presented here apply only if the madmUm de-flgclion does not exceed a fraction of the thickness of the plate: Arso, the poisson,s ratio
as-sumgd. 9=!*!,1 y_as
/max:
where:
"#
Sn(center)
:
p.
#
(Longitudinal Bending Stress)
Jr(center)
:
p,
#
(Transverse Bending Stress)
y.=
deflection (in.)
'r Abstracted
from Ref. 156.
tsl
W
26. Uniform Lateral and Tensile Loads
Bending of Rectangular Plates
152
Tensile end load
- lateralload, psi a, b - plate dimensions (in.) a, B*, Fo: constants dependent upona/b andP/Pn r2EtB ^ 3(l-v2)b2 ^"w
P
Etz:
Pa- 3(l (v ."
modulus of elasticity (psi)
Poisson's ratio
v2)b2
-
1.18
-
a
Values of a, 8,, Bn, are plotted against values of P/Pn for various values of a/b in Figs. 47, 48, 49, for tensile load and Figs. 53, 54, 55, for compressive load. In addition, for convenience, for tensile load only, a, Bn, Fo are plotted against values of a/b for various values of. P/Pa in Figs. 50, 51, and 52, respectively. While using these data it should be noted that although the values of stresses given here are for the center of the plate, they may be assumccl maximum for practical values of a/b and P/Pa such that in the case of compression, a/b 4 2 for all values of P/Pn. If the values of S, and S, are determined from the plots, the resultant stresses in the x and y directions are, respectively
4-
Maximum deflection
1lmar
:
Direct stress
EtB
10,000/0.5
0.342
to a frantc,
load.
Resultant stresses
Tensile End Load:
/ t2
-
0'387 in'
8830 psi
-
Compressive End Load. Data as in tensile case but with a longitudinal compressive load of 2000lb per in.
2000 8500
-
-
0.236
From Figs. 53, 54, and 55 it is found that
a
0.136, B*
-
-
0.330, pa
-
0.930
As before
wba Imar: o-ET : Longitudinal bending stress S,
Direct stress
-
0.136X9y404: (05;, lou
30
-
X
B,wb2/t2
4000 psi 4000 -f 19.000
2OOO/0.5
X
-
o.E:,tn. u.6J)
19,000 psi
- 23,OOO (compressive), 15,000 (tensile), psi Transverse bending stressS, - Bowbz/t2 - 39,700 psi Direct stress is zero .'. Resultant
t) psi
p *wb2
-
19,700 psi (tensile and compressive)
-
.'. Resultant stresses
Exnna pLps:
-
0.063x9x404 30 x lo. x (0.5)5-
-
Direct stress is zero.
P/Pa
example, it is obvious that bending moments will set up at thcse attachments. This is the case of a plate with built-in edges in which citse the yielding of the attachment will have to be considered; this is bcyorrtl the scope of this book. A particular case of zero end force in the plane of tlto plate and clamped edges is shown in Table 3, Case 57' To clarify the use of the graphs, two practical examples havs bcctt worked out and reported in Ref. L56: I) plate with a tensile end load, nttrl
w
:
value of
- 20,000 psi .'. Resultant stresses - 20,000 -f 8830 - 28,830 and 11,170 psi (bothpsitensile) Transvsrse bending stress S, 19,700 Bowb2/t2
for
Lcngth of platc: a - 80 in. Wiclth of plato , /r ,,. 40 iu. 'l'hic:kncss ol' plirtc I 0.5 irt. ltytlroslii(ic: [,lrrilirlrrr l)rcssttrc
0.153, Bu
_
awba
Longitudinal bending stress S,
Sn
a problem is encountered in which the plate is attached
2) with a compressive end
:
0.063, p,
-
sa
If
it is found, for this particular
Hence
0.3
P/t
8500 lb per in.
o.3)
P/Pa
(in.)
-
10,000 lb per in.
-
From Figs. 47,48, and 49 P/Pa and a/b - 2,that
end load (tension or compression), lb. per in.
plate thickness
P
-
12EtB
153
stresses
-
39,100 psi (tensile or compressive)
In many practical problems the lateral loading is not uniformly distributcd bccauso thc top cclgcs ol' thc platc arc for cxample nearer the
154
Bending
of Rectangular
26. Unilorm Lateral and Tensile Loads
Plates
water surface. Therefore it is necessary to be able to analyze the in which the intensity of loading varies uniformly from zero along the edge to a maximum w along the bottom edge. It will be obvious that if the end loads are unaltered, the values the deflections and stresses at the center of the plate will be one half those obtained for the case of uniformly distributed lateral loading. deflections and stresses will vary but little from the maximum values. A further example will serve to illustrate how the solution to such
t
$r qdi
+e
155
o q c.)
t-i od
$
bb
problem may be obtained. Tensile End Load. Data as in previous tensile case but with a hydro' static pressure varying from 9 psi along one edge to 11 psi along
opposite edge. From Figs. 47, 48, and 49 cu
0.063,
pn:0.153,
as before,
Bu
-
that
0.342
- 1 psi 1 I=4f , 0.043 in. deflection /r"-: aax- w V: =9'961X 106 Ets 30 X X (0.S;r -- "" "
Therefore, for Maximum
-
it is found,
a
uniformly distributed loading ol
1'
/2
(
1
1-9
)
Hence the total values of the deflection and stresses are:
y
^s, ss -
0.387
+
0.043
-
0.430 in.
+ L/9 (8,830) - 9,811 psi 19,700 + r/9 (19,700) - 21,889 psi
8,830
and the resultant stresses follow as before'
C)
a
F tr-
$
ry 156
Bending
ol Rectangular
26. Unilorm Lateral and Tensile Loads
Plates
{t-
157
.o o
() a
-
-o
(.)
14
roO
c{ tn
rr,
bb
bb
ol'o
Cg
o ,q ()
Ll
s hb H
+
\ ,\
\\ \\ \
\
\
\ \
\
rt
'JH.,
bo
\
*\
tNS$3$so ct ci d
\
>..l o
qici
rn :a
\
*j
qiu $ ilx ri i!
a
\
agqr
V^t
C)
t:
\
?^l clo-'
\
t!
\
ol'
N N
\
\
N
o'
ci
158
Bending
of Rectangular
Plates
27. Uniform Lateral and Unilorm Tension Loads
159
o
T_ Fig.
Fig.
53.
(ComPressive load)
55.
(Compressive load)
27, UNIFORM LATERAL LOAD AND UNIFORM TENSION ACTING ON ALL FOUR EDGES (SIMPLY SUPPORTED)*
Fig. 56.
?oJ
Deflection )mar Bending moments per unit length: Mn
b
Irig.
54.
:
Mr:
(CotrrPrcssivc load) t'Abstrlctccl frtnr ltcf.
1-5 1.
awba/EtB Bnwb2
Brwb2
W
160
Bending
ol Rectangular
Plates
zrEt|
,- 'L3(l-v2)b2
27. Unilorm Lateral and Unilorm Tension Loads
r2
^ u,: f(wD
where the constants a, p, and Bo are functions of P/Pa and a/b. The maximum values of a, Fn and B, as well as those occurring at the plate center, areplotted against a/b for various values of P/Pn in Figs. 57, 58, and 59 respectively; the value of Poisson's ratio assumed was 0.3. The maximum values of B, occur along the X'axis and generally
move further from the center with increasing a/b ratios. Also, there is a tendency for increasing values of P/Pn to move the location of the maximum value of Mo away from the center. Maxirnum values of Mo occut at the center of the plate for all values of P/Pn and a/b investigated with the exception of a/b - I-2 with P/PB - 1-5. The deviations of the maximum values of Mrfrom the center occur on the line x - a/2. Decreasing the value of P/Pe and increasing a/b generally decreases the distance that the maximum value of Mo occurs from the center. For the tange of. P/Pp and a/b investigated, the maximum deviation of M,1^u.1 from the center of the plate was about 0.45 a, while that of M,y6a*1 was about 0.30 b. If the values of M'1^u*1 and Ms@a,1 are determined for any specific values of P/Pn and a/b from Figs. 58 and 59 the maximum normal stresses in the X andY directions are, respectively
Pr t-r and
P
-l--:t
lrig. 57.
Fig.58.
6Mn(^u*) t2 6Ma(max)
'n,
t2
Fig. 59.
:frwb'
I6I
Chapter 6
SANDWICH PANELS WITH LINIFORM SURFACE PRESSURE,
AND UNIAXIAL COMPRESSION*
(Stresses and deflections)
28. UNIFORM SURFACE
PRESSURE AND UNIAXIAL
COMPRESSION Sandwich panels must often be designed to withstand uniform surface pressures and/or temperature differences between the faces in combination with uniaxial compression, Fig. 60.
Charts presented here permit rapid evaluation of maximum
stresses
and deflections caused by combined surface pressure and uniaxial compression. Since linear theories are used, results for temperature and pressure can be directly added by superposition.
Fig. 60. Simply Supported Sandwich Panel under Uniaxial Compression. Surface Pressure Not Indicated.
>--
*Reprinted fronr Ref. 41. 163 L
28. Unilorm Surlace Pressure and Uniaxial Compression
Sandwich Panels
t64
t6s
Notation
* D, Do D, :
a, b
: G" h -M: N, : ,E
Panel dimensions, in. Panel stiffness parameter Panel shear stifiness, in. per lb Panel bending stiffness,
lb-in. Modulus of elasticitY, Psi Core shear modulus, Psi Core thickness, in. Bending moment, lb-in. Axial compressive load, lb per in.
P: I/': w qn
: :
Normal surface pressure,
PSi
Plate thickness, in. Solid (isotropic) plate thickness, in. Panel deflection. in. Axial-load parameter
N,/DO : -Panel aspect ratio - a/b z : Poisson's ratio o : Stress, psi .tr.
tl I
Tt -T It l:
\ 0.5
\
-E
4 l
o1l --l
(r
I
I
I
I
/
/
v'/ /l-/
).05
'l -0.i 0
8 Ponel
l,'ig.
I
ryn ,fu
{----t .- --4 .- -:--1 = - -+ ...= --J
7I
I
"=r//,
\ \
/
I
,
Maxinrunr l)e:llcction ol' Sanclwich pancl
t0
Sheor Sliffness,
/,
28. Unilorm Surface Pressure and Uniaxial Compression
Sandwich Panels
l' l
6 5
4 3
2
I
0.5
5
5
4
4
l tl
?
z
\, b.
:-
I
05
0 o 6
6
I
lt
4
I
I
I
tl T
"t,'liti
3
2
2
I=
v77.,
l.
tt T, /
-0.05,/
/-
1 1
--L/ 0.5
0.5
o
I
I
5
2346810
4
Ponel Sheor litiffncss, /.),
Mlrxirrrrrttt Motttt'ttl, fiy',, ;rlrottl .v Axis t;l Slrrrtlwit'lt l'itrtel
20 Ponel Sheor Stiffness,
Irig. (r3. Mlrxirrrunr Morrrcrrl, M,,,
ltb<>ut
30
40
/p
x Axis of Sanclwich Pancl
7
168
28. Unilorm Surface Pressure and Uniaxial Compression
Sandwich Panels
rectanguThe design culves, F-igs. 61 and62 are for a simply supported I and lar sandwich panel consisting of two isotropic faces of equal thickness plane stress a core of thiikness h. The skins are assumed to deform in with a modulus of elasticity E, while the core possesses a uniform transstiffness (represented by a core shear modulus' G") but no
verse shear
in-plane stiffness. The following equations
Dn: D_ ua -
No _Ds
(46)
+ t)G" Et(h+t)2 (h
2(l
-
(48)
u2)
the panel is
(4e) Individual curves in Figs. 6I to 62 are based on the aspect ratio of the platc
I: -5 b
(s0)
Thefiguresshowthemostimportantparameterrangesforaspect ratio ),, of 0.67, 1.0, and 1.5. values for other aspect ratios may be found pancl from the figures by interpolation. Deflection) w, at the center of the Maximum 61' Fig' parameters, is given as= a functiott of th" pertinent from UeriOlng moments, Mn, Mo at the center of the panel are determined and 63. Stresses can be computed from these moment values.
Exeupr-n 1: A simply-supported rectangular sandwich panel is subjected to surface pr"rrtt., P : 5 psi and is compressed by an axial load deflcc+OO lb per in. Determine the resulting maximum stresses and ru,
-
tions.
a:= 16in';b - 24in; I - 0'020 in'; ft in.;E- 10.0X 106psi; G":2.A X lOnpsi,andv:O'3' Panel properties are:
Front Equation 47 Du
:
(0'38
+
0.02)2.0 X 10'
-
8000lb per in'
Iirom Equation 48
r)"
( r1l,()
and from Equation 50
x llllLt(?).lfil -l 0-02)'j0.'.\2 2(
|
)
r7,6(X) rb-in.
0'3tl
16
2a
-
0.67
Therefore,
,., '''
(47 )
described by the single Parameter
6i
r, _ (242) (80oo) u,:fiffi-26.s3
X:
For purposes of graphical representation, total stifiness of
Figs.
From Equation 49
aPPIY:
T":
169
:- *: Do
j=o=q 8000
_
o.o5
From Fig. 61, w*naD,/Pazb2 ore, wm in. - 0.56. Theref - 0.242 From Figs. 62 and 63, M;rz/Pab O.80, thus, M, 154.66 and M6r2/Pab 0.50, thlus Mo 96.30.
-
-
Therefore,
* ,=Mn , -r N"l I -^-f -"-l-(h+t)'2lt : ,*rr6.Us + 200):+: 0.020 -
'-9332 and29,332 psi
-
-+-
cru:!#rr_r+# 12,038 psi
Maximum stress along the x axis is psi on the lower face and -9332 29,332 psi on the upper face. Stress along the y axis is on the -12,038 lower face and 12,038 on the upper face. (compressive stress is positive.)
Exavrprr 2: Consider an isotropic plate subject to normal pressure only. using the data of the previous example, except with a total solid-plate thickness t' - O.3O in., determine maximum deflection. In defining an equivalent sandwich the dimensions are t - t' /2 and, (h + t) t'/\/3, (h thus, t 0.150 in. and t) O.l732in. + - 2.473 From Equation 48, Dr X.101 lb-in. For an isotropie plate, the effective shear modulus is infinite (shear deformation effects neglected), but Do - 100 can be used. From Equation 47, G" - 2.447 X 10u psi and from Equation 49, Dq - 4.237 X 104 lb per in. Then from Fig. 61, with ), - 0.67, w^traD,/Pa2b2 - 0.354. Thus w0.108 in. By comparison, for an isotropic plate, Table 3, Case 54,for a/b - 1,5, by intcrpolation, givcs w,, 0.084 Pa4/Ett 0.102 in. -
-
'W
Chapter 7
THICK CIRCULAR PLATE WITH
AN ECCENTRIC/CONCENTRIC HOLE (Reference 6)
29. CIRCUMFERENTIAL
STRESSES
A problem often encountered in engineering is the situation where a thick circular plate with an eccentric hole is stressed by an inside or outside radial pressure. rn order to determine the circumferential stresses for a similar plate with a concentric hole we use the standard Lame's equations for fixed cylinders. By transformation of Lame's equations it is possible to solve problems of circular plates with an eccentric hole as shown in Tables 7 and 8. These tabulated equations for circumferential stresses then also apply to thick cylinders. Notation R2 R1 se
sr
d
-
circumferential stress along the contour of the outside circle of radius R2, psi.
circumferential = R1, psi.
:P1 : P2 0
Outside radius of plate, ins. Inside radius of plate, ins.
stress along the contour of the inside circle of radius
Location of hole from center of plate, ins. Angle, as indicated in Tables 7 and 8, degrees. Inside pressure, psi. Outside pressure, psi.
I7I
29. Circumlerential
Thick Circular Plate
172
Stresses
^i:
t ^-l j^l +
rti b
|
F
.E
E % n
;t
tq
F
$ E E 6 $
I
qs E
"i-
El cLl
rl3l*s
al tl -l t3
.:^l G
'.Sl
ddl
I
sl; ot t :lc\l d rl, ol a3'
rt \ vt,A
Fit
€
nll
'1-
5t tr
Iti
t
-
I
+ \J
^l -l
tl"sl ll"S
13l N
|
\Jl dl
c'l
Hl** Nl^|\ +l r
I
a al oiR
FII vl
F(
asl
-T-
.t3
x
6.1
\l
I
I
ni
vl
RllNi'r nil -r
-T-
N
t\ lv
I
c\
ll
tl
lt
tl
d
u)
u2
q
ll
,i
u) ri
i
X
€l*
€l*
as-l nr
+ I tl
ddlq
Fr
I
.ts
l1
.*t
J
d'dl I
qt
\
I
tl
tf
v)
d.
*
dl^ tl
ivl sdill
trl E
qa qq FL
c
l
a
at)
I"r
-r
? 'fl^ ! ,.8
= €l*s \
^yl
au?
+
^=l Nl '-l^-vt, IN}T t\l rv .lD
+ e. *l +---LS ^*{ tl sDp
;. :v)4 '-4. * F-:rs
g-\
6l
l|tl qq
I
I
I
=
A5lon ',-
=
NdtdN
Rl""r
N
.b
I
.:-l
-r
|
r
ll\Nl
*l-l
Fl
lv 6l
I I
I
| N I -.1
tEl r il s El -l* ' |
-L
l€
1^ | ai FF 's| -rl"F sl ;3 :s :l^ Al qQ Rl ' E'l 'rs Fh :iS .se c\l'Itl 3r trl rltr-
asl s,
t\ Fil
6t
ss
rlrl'
te tv
6l
^
^^l
s.
-L
*E A€
-T-
:
a5'lv
I
N
I
^*
DI vl
I
+
rtv oih "r^l EI Fi
lv
-{-l
c\
I
e-l
Nl
h
G
!l r lAAl+ II\ 'lv
c'l + al
^l
l;-
173
I I
: 'rs I
,
d
|
, I
.<^l -Blnd E rl 'l=
iglt$ ld-
I
I
I
= +
+
€l
+l
nd
I xltg
NNIdN tul N^l
1:
* ht\
tr
II llll
6l
dl
I
orlas
o. 4q *x
tl
s "iJ
{
TABLE 7 PLATE WITH H9LE STRESSED BY A CONSTANT INSIDE PRESSURE Pl
TABLE 8 PLATE WITH HOLE STRESSED BY A CONSTANT OUTSIDE PRESSURE
P2
Appendix
MASS VERSUS WEIGHT* 30. INTRODUCTION Until space flight came into prominence, variations in gravity at difierent locations were considered negligible. Mass, an independent quantity, and weight, a lunction ol gravity, otten were used interchangeably. In this present age ol extreme accuracies and lantastic requirements, such neglect no longer is permissible. To kelp clear up the current confusion, here is a standard system ol units lor mass, weight, force, pressure, and acceleration. Whereas the acceleration of gravity varies up to 0.55% over the earth, thus changing the weight of a body accordingly, the mass of that body remains the same. In most fields of engineering the difference in gravity is negligible and, as a natural result, it has been common practice to use terms and quantities of mass and weight interchangeably. Unfortunately,
this practice, now a force of habit, has crept into the missile and rocket propulsion fields too. Masses are being catled weights, and sometimes (to make matters worse! ) mass data, which already had been obtained correctly from measurements, are converted back into weight units in an efiort to correct for a different acceleration of gravity. With the widespread confusion and inconsistency apparent in the use of terms and units of mass, weight, force, pressure, and acceleration, there is a definite need for the adoption of a standard procedure for their use.
31. SYSTEMS OF UNITS The absolute and the gravitational systems use L m/sec2 or 7 ft/secz of acceleration; the units of mass and force are such that 1 force
as the.unit 'r'
Reprinted from Ref, 98. 175
176
M
u,sls
v
t
r,s
t
r,r
W t ig
It
I
32. Mass
unit r:,: I nrass unit \ | accoleration unit. Therefore, the weight of a body is numcrically about 9.8 or 32.2 times its mass, depending on the units of mcersurcment. Therc are some mixed systems of units in use, employing thc mass unit in kilograms (kg) or pounds mass (lbm) from the absolute system and the force unit in kiloponds (kp) or pounds force (lbf) from the gravitational system. In these systems the numerical values of mass and weight are equal at standard acceleration of gravity (g"). This is very desirable; however, the current mixed systems still use '!. m/sec2 or I ft/secz as the unit of acceleration, which results in various confusing or incorrect forms of Newton's equation. When kilograms and kiloponds are used as units of mass and weight, the proper unit of gravity is calculated by:
*- 1kp lkg These (Table
9.80665 kg m,/sec2
lkg
or analogous English units are I).
-9.80665m/sec2- I
used
in this
g"
standard procedure
The above units are derived from the basic units of mass, length, and time. The kilogram is the international standard of mass as represented by the platinum-iridium cylinder at Sevres, France. The kilopond, defined aq the weight of 1 kg mass at the standard acceleration of gravity, is the new international unit of force. One kilopond is equivalent to 980,665 dynes. The unit of acceleration is one standard acceleration ol gravity, accepted
by international agreement as gs - 9.80655 m/sec2.It prevails at about 45" latitude, sea level. The following syrnbols are used for various accelerations of gravity: gs
: go,. : 91 gc:
g",o
standard acceleration of gravity in metric or English units standard acceleration of gravity in metric units standard acceleration of gravity in English units local acceleration of gravity
acceleration of gravity at the location where the instrument was calibrated with calibration masses.
Using these standard metric and English units, Newton's equation can be written as:
F(kp):m(kg)Xa(8,-) OI
F(lbf)-m(lbm)Xa(g"") A
givcn body has the sante numcrical valuc of nrass ancl wcight
lt
a loclttitltt lrlrving stlrnclurcl accclcrttitln ol' gravity, wlron thcsc rrnits arc usccl.
Sccirrg lltitt ltcc:clcrrrtiorr is boinu cxprcssctl in "g" lirrrn r;uitc l'rcclur:nlly lrllcirrly, llrc sl;utrlirrtl ulrits will lrr: very prtrcticirl irr ltrost cuscs.
32.
177
MASS
Mass is a quantity of rnatter or a measure of the inertia of a body. The unit of mass is the kilogram or pound mass where 1 kg - 2.204622lbm. Mass standards are available at the National Bureau of Standards (NBS) in washington, D.c. Secondary standards, or so-called "dead weights," which have been calibrated by comparison with a primary standard, are available from numerous sources. At different locations, the dead weights have the same mass, but difierenl weights; therefore, the term "dead weight" itself is misleading. More correctly, the standards could be called "dead masses." In the missile field such factors as fuel weight, tiftoff weight, etc., vary with location. They are actually masses and must be treated as such. Although NBS uses the word 'oweight" for the piece of metal representing a standard of mass, it is being used only as the name of a body used in the process of weighing. Force of habit will uphold the word ,,weight,' in that sense for some time, especially when the masses are determined bv weighing.
Most weighing systems are calibrated by caribration masses and the results are in units of mass, provided the calibration is made at the same
location. No correction, therefore, need be applied to these results. If load cells or proving rings have been calibrated at another location having gravity g", the mass measurements made with such instruments, at the local gravity gr, rnust be corrected by the factor g"/91. For example, instruments calibrated at NBS are compared with standards of masses at an acceleration of 9.8008 m/sec2. At Redstone Arsenal, the acceleration of gravity is 9.7964 m/sec2 and, as a result, more mass is required ro produce the same weight-force obtained for a certain mass in washington, D.C. when the mass of liquid oxygen (lox) or fuel filling is determined with differential pressure gages that have been calibrated at the same location by liquid columns or pressure balances, using calibration masses, the result is in units of mass. It is important that these data must not be multiplied by gt/g, to obtain correct pressure units. The Ap values for filling control or density measurements are not in units of pressure, but they are measures of mass. The units must be called psi(m), inches of Hg(m.;, etc. to differentiate them from standard pressure units. If the mass of lox, fuel, or other liquids is determined by weighing, a
correction must be made for the buoyancy of air. The buoyancy effect x cquals the mass (m,) times the specific gravity of air (yo:0.0012) divided by the specific gravity of the body (7,). The weighing systems are calibrated with masses having a specific gravity betwcon 6.8 and 8.4. The br"royancy effect is about:
on a body
35. Acceleration of Gravity
Mass versus Weight
178
0.0012
-
X fii,:0.00015 X Ino: O.Ul'vo
The results from the weighing systems have a buoyancy error of
/
o.oor2
0.0012
\
(j;__T)xtoj%.
This is about 0.11,Vo for water, O.l4% for hydrocarbon fuels, and 0.lO% for lox. These average figures for buoyancy corrections are accurate enough for the difierent densities of the respective liquids. Liquid hydrogen, however, has a larger variation in density, and its buoyancy effect may vary from 1,.5% to 4%l The buoyancy, as a result, must be calculated for each particular specific gravrty of liquid hydrogen. Similar requirements pertain to the buoyancy correction when weighing complete missiles; the empty "Jupiter" missile has a specific gravity of about 4.0, and its buoyancy efiect is only 0.015Vo. When the missile is fueled and the pressurized compartments are closed, however, its specific gravity is about 0.8 and its buoyancy effect is O.1.3Vo. This correction must be applied when the liftofi mass is determined by weighing. In turn, the buoyant force of 158 pounds must be added to the thrust at liftoff, and decreasing buoyant forces must be added with decreasing air densities during flight.
33. FORCE AND WEIGHT Force is the cause of deformations or the change in motion of a body; the weight of a body is the force with which the earth pulls that body toward its center. The unit of weight and force is 1 kilopond or 1" pound force. This is the weight of a unit mass under the standard acceleration
34.
179
PRESSURE
Pressure is a force per unit area. Its basic unit of measurement is 1 kp/cmz (atmosphere) or 1 lbflin' (psi). All the basic standards for calibration of pressure gages, including dead-weight testers, dead-weight pressure balances, and manometers with liquid columns, use certain quantities of mass to provide pressure, or force per unit area. In view of this, the calibration values again must be currected by gt/g" or A,g%, as described previously. Calibration curves must present corrected values and details indicating how they were obtained. NBS applies this correction when it calibrates pressure gages, yet it does not correct to true force values when calibrating load cells and proving rings.
35. ACCELERATION OF GRAVITY Table II lists the accelerations of gravity for difterent "missile locations" and the deviations of these values from the international standard. As noted previously, the international standard acceleration is a value adopted by agreement. The other values are measured or calculated. This table can be used, for example, in ascertaining the per cent correction necessary when employing an NBS calibrated load cell at the Redstone Arsenal. The mass measurements obtained with the NBS calibrated By per cent device must be corrected by g"/gt: 9.8008/9.7964 - 1.0004. Agt deviations given in the table, the correction is Ag"
-
-(-O'.lOVo):*0.04Vo.
-0.06%
of gravity.
At other accelerations of gravity 91, a mass m has a weight of m' X gr/g*. When load cells, proving rings, and other force-measuring
devices are being calibrated, therefore, the calibration masses must be multiplied by gt/g, or corrected by A'gVo . At present, the NBS calibrations of load cells and proving rings give the number of pounds mass and the output caused by the weights of these masses at local acceleration of gravity. Because 91 at Washington, D'C. is 9.8008 m/sed, the NBS data must be correcfed by the factor 9.8008/ (Table II), when NBS calibrated instru9.80665 - 0.994 or -O.O6Vo ments are used for force measurements or as standards for calibration of othcr forcc-mcasuring clcviccs. Al'tor this corroctiort, thc instrtrments mcaslrc lru( [ttr<'<' ltl lttty krcltliott.
Mass Force Acceleration
English Units
Metric Units
Property
Kilogram Kilopond
kg kp
Standard Gravity (9.80665 m/sec2)
8sm
Pound Mass Pound Force Standard Gravity (32.1,7398 ftlsec2)
I. Metric and English units on which the standard procedure is based.
TABLE
lbm
lbf 8se
Mass versus Weight
180
Location
l.
International Standard about 45o latitude, sea level
2. Chrysler Plant, Detroit 3. Bureau of Standards, Washington 4. Redstone Arsenal 5. Rocketdyne, Santa Susana 6. Patrick Air Force Base 7. Equator 8. Pole
m/secz
ft/sec2
9.80665 9.8031 9.8008 9.7964
32.17398 32.1623
9.7949 9.7924
32.127
9.7803
32.0815
9.8322
32.2578
AgVo
-0.M
32.t548
-0.06 -0.10
32.1404
-0.11
32.1266
-0.14 -o.27
+0.26
BIBLIOGRAPHICAL REFERENCES
II. The adopted international standard and the calculated accelerations of gravity are listed for various locations. The third column indicates per cent deviation from the standard' TABLE
1. Griffel, W. Handbook ol Formulas lor
Property
Metric Units
English Units
Force
Kilopond (kp)
Pound Force
Mass
kp
Acceleration
m/sec2
TABLE
III.
m-1
lbf
seg2
Units mostly used
ft-1
Stress and Strain. New Frederick Ungar Publishing Co., 1966. 2. Boutier, R. H. "Bending Moments in Circular Plates," Machine Design
(lbf)
3.
secz
ft/secz
(Aprrl 29, 1965). Isakower, R. L "Circular Plate with an Eccentrically Applied Spot-Load," PIdd. Engn. (December 20, 1.965).
Grffin, D. S. "Stresses and Deflections of Thick, Curved Plates" (Trans.
in the gravitational or
L Engng. Industr. (August t965),303-308. A. "A Direct-Stress Analysis of Orthotropic
ASME),
technical system for aeroballistic and other calculations.
Coult,
Cantilever Plates"
(Trans. ASME), Paper No. 64-WA/AMP-14 (1965). 6.
lbf kp
ke
I
m-1
o.L0t97t6
9.80665
I
2.204622 21.61996
0.453592
0.0462536 1.488162
32.t7398
14.593 88
I
0.0685219
=
7.
0.67t970 0.03 108 102 1
TABLE IV. Factors for converting the units of mass shown in TABLES I and III. The units of force are the same in both tables, and the units of acceleration of gravity gs-
sec2
(slug)
lbm
ses2
ft-1
9.80665 m/sec2 or gsu = 32,17398 ft/secz.
8.
Griffel, W. "Stresses in a Thick Circular Plate with an Eccentric Hole When Subjected to Inside or Outside Radial Pressure," Prod. Design Value Engn. (March-April 1965), 29-30. Nevel, D. E. "A Semi-Infinite Plate on Elastic Foundation," Research Report 136. Hanover, N. H.: U. S. Army Cold Region Research and Engineering Lab., March 1965. Panc, V. "Axisymmetrical Stress Conditions in a Shear-Sensitive Circular Plate" (In German), Apl. Mat. Ceskoslov. Akad. Ved., X, No. 5 (1965), 399-422.
9.
D. "Nonlinearity Criteria for Plates" (in Russian) , Prik. Mekh.,I, No. 8 (1965), L7022. 9a. Cheung, Y. K., and O. C. Zienkiewicz. "Plates and Tanks on Elastic Foundations: An Application of Finite Element Method," Internat. I. Solids Stuct., I, No. 4 (November t965), 451-461. 10; Moddy, W. T. "Moments and Reactions for Rectangular Plates," Engineering Monograph 27. Washington: Bureau of Reclamation of the United States Department of the Interior, 1965. 11. Powell, G. I{. "Cornparison of Simplified Theories for Folded plates" (Proc. Amer. Soc. Civil Engrs.), J. Struct. Div., XCI, ST 5 (December Livshits, Ya.
1965), 1-31 (Part
I).
I8t
Bibliographical Relerences
182
12.
B
of Clamped Apl. Mech., IY
Balachandra, M., and S. Gopalecharyulu. "Large Deflections
Plate by Modified Fourier Series" (Trans. ASME),
I.
(December 1,965), 943 (brief notes).
13.
Redwood, R. G. "Discontinuities in Bent Plates," Aeronaut. Quart., No. 4 (November 1965). 388-398.
XYl,
r4. Saakyna, S. M. "Flexure of a Rectangular Thick Plate with Clamped Edges" (in Russian), Aikakan SSR Gitutyunneri Akad, Zekuitsner, Dakladi Akal. Nauk SSSR, XL, No. 3 (1965), 137-L43. 15. Barta, J. "On the Deflection Analysis of a Membrane" (in German), Acta Tech. Acad. Sci. Hungaricae, L (Budapest, 1965),5-9 (Part I of 1.6.
11
4 parts). Reiss, M., and M. Yitzhak. "Analysis of Short Folded Plates" (Proc. Amer. Soc Civil Engrs.), J, Struct. Div., XCI, ST 5 (October 1965), 233-254 (Part I). Gravina. P. B. I. "Contribution to the Calculation of Circular Plates with Symmetric Ribs" (in Italian), G. Geni
Petyt, M. "The Application of Finite-Element Techniques to Plate and Shell Problems," Isvr. Report No. 120, Inst. Sound and Vibration Res., University of Southampton (February 1965). 19. Latsinnik, I. F., and Ya. M. Grigorenko. "Bending of a Circular Plate with Linearly Varying Thickness under Antisymmetric Loading" (in Russian), Prikh. Mekh., I, No. 7 (1965), 67-76. 20. Maslov, N. M. "Bending of a Circular Orthotropic Plate of Varying Thickness" (in Russian), Prikh. Mekh., II (1965), 67-73.
22.
zJ.
Kuenzi, E. W. "Minimum Weight Structural Sandwich,"
)2.
Service Res. Note FPL-086 (January 1965). Sayer, F. P. "A Problem of a Thin Circular Plate Transversely Loaded and Rigidly Held on Its Boundary," Proc. Camb. phit. LXI, No. 1 ^Soc.,
(January 1965), 289-293. JJ. Hulbert, L. E., and F. W. Niedenfuhr. "Accurate Calculation of Stress Distributions in Multiholed Plates" (Trans. ASME), Iournal ol Engineering lor Industry, LXXXVII, No. 3 (August 1965), 33L-336. 34. Wittrick, W. H. "Axisymmetrical Bending of a Highly Stretched Annular Plate," Quart. J. Mech. Appl. Math., XVIII, No. 1 (February 1965), 1L-24. 35.
Bernstein, R. L. "Thermal Bending of Sandwich panels under Uniaxial Loading." Unpublished Ph. D. thesis, University of Florida, 1964.
J. C. Bending of Circular Plates under a yariable Symmetrical Load, ANL 6882. Argonne, Ill.: Argonne National Laboratory, April
36. Heap,
1964. Itreap, J. C. Bending ol Circular Plates under a (Jniform Load on a Concentric Circle, .LNL 6905. Argonne, Ill.: Argonne National Labora-
tory, April 38.
1964.
Ross, D, S. "Assessing Stress Concentration Factors," Engineering Materials and Designs (June 1964), 394-398.
Kao, J. S. "Bending of Circular Sandwich Plates" (Proc. Amer. Soc. Civil l. Engng. Mech. Div. (August 1965), 165-176 (Part I).
40. Gellatly, R. A., and R. H. Gallagher. "Thermal Stress Determination Techniques for Supersonic Transport Aircraft Structu' :s. part II: Design
of the Elastic Behavior of Engng. Industr. (August 1965),
Tsai, S. W. "Experimental Determination 3
L
15-3 I 8.
25. Goldberg, M. A., M. Newman, and M. J. Forray. "Stresses in a Uniformly
27.
U. S. Forest
39. Griffel, W. "Plates with Holes," product Engineering (September 16, 1963; November 11, 1963; March 16, 1,964),98-104, LO4-113, L8-24.
Orthotropic Plates" (Trans. ASME),
26.
183
Bernstein, E. L. "On Large Deflection Theories of Plates" (Trans. ASME), J. Appl. Mech. (September 1965), 695-697 (brief notes). Engrs.),
.,^
nccs
Jt.
21. Wempner, G. A., and J. R. Baylor. "General Theory
r57-t77.
Re I e re
28. Platus, D. L., and S. Uchiyama. "Very Large Deflection Behavior of Corrugated Strips," AIAA J. (August 1965), L549-1552 (tech. notes). 29. Sakata, M. "Creep in Disks Rotating under Axial Loads and Having Asymetrical Profiles," BuL ISME (May 1965), 14L-149. 30. Bruce, J. "Cylindrical Bending of Long, Flat, Rectangular plates under Linearly Distributed Pressure," l. Roy. Aeronaut. ^Soc., LXIX. No. 649 (January t965), 59-63 (tech. notes).
18.
of Sandwich Plates with Dissimilar Facings," Internat. l. Solids Struct., II (May 1965),
ibliogra p hical
Loaded Two-Layer Circular Plate" (Proc. Amer. Soc. Civil Engnrs.), J. Engng. Mech. Div. (June 1965), 13-31 (Part I). Junker, A. T. "Bending of Thick Stecl Plates" ('frans. ASME), J..Basic Engng. (June 1965), 290-292. I-ing, C. B. "On thc S{rcsscs in:r Scnri-lnfinitc Slrip Strbjcctctl to a (lonccntralc(l Loltcl" ('l'runs. /SM I,:), J. l1tpl. Mtt lr. \.lturc l9(r5), 456-4-5fl (
blicl
rrolcs ).
Data for Sandwich Plates and Cylinders under Appliecl Loads and Thermal Gradients.",4 !D-TDR, LXIII (1964), 7g3. 4t. Gallagher, R. H., and R. A. Gellatly. "sandwich panels," Machine Design (December 17, 1964). 42, Kravitz, S. "Radial Stress and Deflection of Flat plates,', Design News (November 25, 1964). +). Basuli, S. "Note on the Large Deflection of a Circular plate of Variable Thickness under Lateral Loads" (in English), Bul. Inst. politechn. IASI, Serie Nova 10 (XIV) (1964), 63066 (Part I of 2 parts).
V. V. "Bcntling of Mr,rltilaycred Plates" (in Russian), Iryestiya Akadentii Nar.rft J.SJ/I Mckhanika i Musltiru*tr., No. I (1964), 6l-66.
44. Bolotin,
BibliograPhical
184 45. Dundrova,
Theory
of
Bibliographical Ref erences
Ref erences
V., V. Kovarik, and P. Slapek' "Application of New Bending Sandwich Plates" (in Czech'), Stavebaicky Casopis' XII'
No. 10 (1964),622-640. "simply Supported Rectangular Plates of Varying Flexural "{-\ 46. Desayi, P. $ nigdity Subjected to iateral Loads and Forces in the Middle Plane; aiproximate Solutions of Some Cases," l' Sci' Engng' Res'' VIII' No' J"1Y'"1.2^-6l), 219-226. 2 (India, -::irnp.ou"d Theory of Elastic Plates" (in German) ' Ingen'47. Panc, V. Arch., XXXIII, No' 6 (July 1964), 219-226' 43.Mendito,G...TheCircularPlateattheYieldPoint',(inltalian),Rendi. (1964)' conti Semanario Matematico e Fisico di Milano, XCVII, No' 1
61. Johanshahi, A., and J. Dundrus. "Concentrated Forces on Compressed Plates and Related Problems for Moving Loads" (Trans. ASME), L Appl. Mech., XXK, No. 1 (March 1964), 83-87. 62. Rajappa,
N. R. "On the Bending of
Rectangular Orthothropic Plates
with Small Initial Curvature," "/. Royal Aeronaut. Soc., XVIII, No. 640 (April 1964), 280 (tech. notes). 63. Wilson, P. E., and A. P. Boresi. "Large Deflection of a Clamped Circular Plate Including Effects of Transverse Shear" (Trans. ASME), J. Appl. Mech., XXXI, No. 3 (September 1964), 540-541 (brief notes). 64. Rajappa, N. R. "On the Deflection of Rectangular Orthotropic Plates under Lateral Loads,"
L Aeronaut.,Soc., VII, No. 643 (July 1964),
483-484.
335-356.
4g.Sengupota,P.R...NoteontheTorsionofaLargeThickCompositePlate under Shearing Forces Applied on Two opposite Faces," Bul. Calcutta Math. Soc. (March 1964)' 33-40. 50. Contri, L. "Exact Solution of a Problem of Rectangular Elastic Plates of Variable Thickness," t. Math. Pftys' (March 1964), 72-74'
51.sih,G.C.,andS'R.Rice...TheBendingofPlatesofDissimilarMaterials withCracks,'(Trans.ASME),J.Appt.Mech.(September1964),477.482. 52.Dhalinwal,R.S'..BendingofaRectilinearRegularPolygonalPlatewith (1964)' a Circular Hole by Concentrated Edge Couples," Appl' Sci' Res'
Peach, R. W. "Buckling in Rectangular Plates," Machine Design (December 19,1963). 66. Niedenfuhr, F. W., and A. W. Leisse. "Bending of a Square Plate with Two Adjacent Edges Free and Others Clamped or Simply Supported," ArAA 1.,I (1963), IL6-120. 67. Ross, D. D. "Stresses in Plates with Polygonal Holes," l. Mech. Engng. 65.
No.3 (1963). Sinha, S. N. "Large Deflections of Plates on Elastic Foundations" (Proc. Amer. Soc. Civil Engrs.), l. Engng. Mech. Div., LXXXIX, No. 1 (FebruSc., V,
68.
ary 1963).
43-45. Folde
53. Goble, G. C. "Analysis of Civil Engrs'), J. Siruct. Div., XC, ST
I. Appl. Mech', XXXI, No' 4 (December 1964) '
N. "Torsion of a Large, Thick, Nonhomogeneous Plate," Indian l. Theor. Phys., XI (June 1963), 59-73. 70. Wilson, B. "Bending of Rectangular Plates with Adjacent Free Edge," Trend in Engng. (April 1963), 16-19. 7r. Bufler, H. "Torsion of Nonhomogeneous Thick Plates" (in German),
Manig, J. "Discontinuity Solutions to Plate and Beam Problems," Internat' J. Mech. Sci., VI, No' 6 (Decembet 1964), 455-466'
ZAMM, XLIII, No. 9 (April1963),389-401. Reiss, E. L. "A Uniqueness Theorem for the Nonlinear Axisymmetric Bending of Circular Plates," AIAA 1., I, No. 11 (November 1963),
1 (February t964)'
93-1'24
I).
(Part
,'on Bending of Thin Elastic Plates in the Presence of couplc 54. Hoffman, o. Stresses" (Trans. ASME), 7
55.
t85
06-7
07 ( brief
69. Chaterjee, R.
notes ) .
56.ChicurelR.,andE'W.Suppiger'...TheReflectionMethodinElasticity and Bending of Plates," ZAMP, XV, No' 6 (November 1964)' 629-6311' 57. Westbrook, D. R. "The Torsional Rigidity of Tubes of constant Nortnal Thickness," Proc. Camb. Phil. Soc., LX, No' 4 (October 1964)' 1'9231926.
58. Tuncel, O. "Circular Ring Plates under Partial Arc Loading" (l'rart't' ASME), I. Appl. Mech., XXXI, No' 2 (June 1'964), 340-341 (bricl notes).
L. R. "Bending Analysis for clampecl Rcctlngtrlar Plalcs (Proc.Amer.Soc.CivilEngrs.),.l.Ilrtgttg'Mrclt'l)iv'(Jttncl96zl)'
59.
Herrman,
60.
7l-g$ (Part I). Wasil, ll. A., arrtl I). (1. Mcrcltltnl. "l)l:rlc-l)cllcctiorr Mcltsttrctttcttt lrY l)lr
72.
2650-2652 (tech. notes, comments). t), 1A
Hanuska, A. "The Bending of a Wedge-Shaped Plate" (in English), Arch. Mech. J/os., XV, No. 2 (1963),209-224. Shul, M. E., and L. W. Hu. "Load Carrying Capacities of Simply Supported Rectangular Plates" (Trans. ASME), L Appl. Mech., XXX, No.
4 (December 1.963) 61.7-621. t). Mitchell, T. P., and W. E. Warren. "The Stresses in a Thin Circular Disk Rotating about an Eccentric Circular Rigid Insert," Internat. J. Mech. Sci., V, No. 2 (March-April 1963), 137-148. 76. Okubo, H., and M. Seika. "Stresses in Heterogeneous Plates" (in English), Ingen.-Arch., XXXU, No. 4 (1963), 246-254. 77. Askentian, O. K., and I. I. Vorovich. "The State of Stress in a Thin Platc," .l . Appl. Murh. ML'clt., XXVII, No. 6 (1963), 162l-1643.
t86 78.
B ib
Stern,
liogr
aP hic
M. "Upper and Lower
al
Bibliographical Relerences
Re I er enc e s
Bounds
for the
Average Deflection of
Elastic Plates" (Trans. ASME), I. Appl. Mech., XXX (September 1963) (brief notes). '19. Shioya, S. "The Effect of an Infinite Row of Semicircular Notches on the Transverse Flexure of a Semi-infinite Plate" (in English)' Ingen'Arch., XXXII, No. 2 (February 1'963) 143-1'51. 80. Hellan, K. "Application of a Numerical Procedure to the Analysis of Thin Rectangular Plates of Variable Thickness," Acta Poly-tech. Scandi' navica Ci. No. 16 ( 1963). ,'Comment on the Use of superposition for Plates under 81. Anderson, w. J. Combined Loading," AIAA J.,I, No. 6 (June 1963), 1459-1460 (tech' notes, comments).
82. Tsai, S. W., and G. S. Springer. "The Determination of Modulii of Anisotropic Plates" (Trans. ASME), Appl. Mech. (September 1963) 467-468 (brief notes). 83. Dunturs, J.. and T. M. Lee. "Flexure by a Concentrated Force of the Infinite Plate on a Circular Support" (Trans. ASME), I. Appl. Mech'
(June 1963), 225-23L.
84. Ang, D. D., E. S. Folias, and M. L' Williams. "The Bending Stress in a Cracked Plate on an Elastic Foundation" (Trans. ASME), L Appl' Mech' (June 1963), 245-251.
g5. Vol'Mir, A. E. "survey of Investigations on the Theory of Flexible Plates and Shells Covering the Period from 1941 to 1957," NASA Tech'
g6. g7.
Trans. F-L80 (October 1963). Huffington, N. J., Jr., and R. N. Schumacher. "Twisting of orthogonally Stiffened Plates," Martin Research Report 44 (August 1963). yamaki, N. .'Bending of stiffened Plates under uniform Normal Pres-
sure," Report 137, Inst. High Speed Mech., Tohuku University, 1963,
tt5-134.
gg.
.,Bending of circular sandwich Panel under Uniform TemHuang, J. C.
perature Gradient and Edge Compression." Unpublished MS thesis, University of Florida, 1963. 89. Kendall, P. J., et at. "A Method for Predicting Thermal Response in Sandwich Panels," ASD-TDR, LXIII (1963)' 306. 90. Griffel, W. "Flat.Plate Design," Product Engineering (June 11' 1962)' g1. Durcelli, A. J., and c. A. sciamarella. "Elastoplastic Stress and Strain Distribution in a Finite Plate with a circular Hole Subjected to unidimcnsional Load," Trans. ASME,
LXII (L962),
1'52.
92. llbcioglu, I. K. "Thcrrnoclastic Equations for a
Sanclwich Pancl trntlcr'
Arbitrary Tcnrpcraturc DistribUtion, Transvcrse Load, ancl l'iclgc ('ont('ttttgr- A1t1tl. Matlt. (ASMIt) (1962),537' lrrcssion," I'roc.4tlt Il . S. Nul. 91. ('5rr, I l.-N. "lligh lrlcqrrcncy lrxlcrrsiontl Vibruliorts ol' Srtrttlwich l)latcs, .l . Attttr,tl. ,\rt<. tlttt., XXXIV 1l()(rJ), ll'11{-ll()O.
IU7
W. "Rectangular and Circular Plates," Design News (July 7\
94. Griffel, 1.962).
95. Griffel,
re62).
W. "Slope for Circular Plates," Design News (November 28,
96. Griffel, 1962).
W.
"Beam-Supported Plates," Machine Design (February
l,
97. Niedenfuhr, F. w.,
and A. w. Leisse. "study of the cantilevered Square Plate Subjected to Uniform Loading," L Aerospace Scj., XXIX (1962),
162-169. 97a' Ershov, v.
v. "compressive Buckling of sandwich plates of symmetric Construction," Izv. Vsch. tlcheb. Zavod., Aviats Tekh. No. 1" (1962), 1.20-t24.
98.
Schuler,
A. E. "Let's euit confusing Mass with weightn" ISA Iournal
(October 1960).
99.
Savin, G. N. ,stress concentration around floles. New
york:
persamon
Press, 196L.
100. Nevel, D. E. "The Narrow Free Infinite wedge on Elastic Foundation,', Research Report 79. Hanover, N. H.: U. S. Army Cold Region Research and Engineering Lab., July 1961. 101. Rao, B. N. s. "Elastic Equilibrium of a plate with a Reinforced ovaloid Hole," Proc. 7th Congr. Theor. Appl. Mech. (Kharagpur, Indian Soc.) (Bombay, December L96l). 102. conway, H.D. "Axially symmetrical plates with Linear varying Thickness," Trans. ASME, L (7961), A-6. 103. Aleksandrov, A. Ya. "Local Stability of Sandwich plates with Honeycomb Cores," Trans. Conf. Theory of plates and Shells, Lvov (Kiev: AN Ukr SSR Press, 196l), 347. 104. Mead, D' J., and A. J. Pretlove. "on the vibrations of cylindrically curved Elastic Sandwich plates," Report 1g6, university of Southampton, Dept. Aero. Astron. (196L). 105. chang, c.-c. "Transient and periodic Response of a Loaded Sandwich Panel,"
L
Aero. Sci.,
106. Essenburg,
F. "on
XXVIII (1.961), 382-396. surface constraints
LXr (1961), 105. 107. Nash, w. A., and J. R. Modeer. "certain ASME,
in
plate problems.', Trans.
Approximate Analysis of Non-
Linear Behavior of Plates and Shallow shells," proc. symposium Theory Thin Elastic shells, Internat. (Jnion Theor. Appl. Mech., Ig5g. Amsterdam: North Holland Pub. Co., Arnsterdam pub. Co., 1960, 331_354. 108. Yu, Y.-Y. "Flexual vibrations of Elastic Sandwich plates," J. Aero.-sci..
xxvll (1960), 272-282. 109. Yu, Y.-Y. "Simplified vibration Analysis of Elastic sandwich plates," XXVII (1960),894-900. Yrr, Y.-Y. "Flexural vibrations of Rectangular sandwich plates,', Tech. J. Aero. Scj.
ll0.
Notc
tJ.
I3r'ooklyn: Polytcchn. Inst., l960.
188
Bibliographical References
"An Approximate Method for Obtaining the Frequencies, Deflections, and Stresses in Sandwich and Cross-Stiffened Rectangular Plates," Technical Report l, I. G. Eng. Assoc. (1960). 112. Saelman, B. "Axially Loaded Columns and Plates," Design News, XV, 1.1.1. Greenspan, J. E.
No. 14. 113. Timoshenko, S., and J. N. Goodier. Theory ol Elastic Stability. New York: McGraw-Hill Book Co., 1960. 114. Shapiro, G. S. "Design of a Plate Conceived as an Infinite Band Resting upon an Elastic Foundation," Comptes Rendus (Doklady) de L'Academie des Sciences de L'URSS, XXXVII, Nos. 7-8, 202-204. 115. Ohashi, Y., and S. Murakami. "On the Elasto-Plastic Bending of a Clamped Circular Plate under a Partial Circular Uniform Load," Bull'
ISME, VII, No. 27, 491-498. 116. Galletly, G. D. "Optimum Design of Thin Circular Plates on an Elastic Foundation," Proc. Inst. Mech. Engn., CLXXIII, No. 27 (London, 1959). 117. Timoshenko, S., and S. Woinowsky-Krieger. Theory ol Plates and Shells.
New York: McGraw-Hill Book Co., 1959. 118. Yu, Y.-Y. "A New Theory of Elastic Sandwich Plates, One-Dimensional
/. Appl. Mech., XXVI (September 1959), 4L5-42t. 119. Yu, Y.-Y. "A New Theory of Sandwich Plates, General Case," AFOSR Case,"
TN, 5901163. Brooklyn: Polytech. Inst., 1959. L20. Mindlin, R. D. "Flexural Vibrations of Elastic Sandwich Plates," ONF ?R, No. 35. New York: Columbia University Press, 1959. 120a. Reissner, E. "Finite Twisting and Bending of Thin Rectangular Elastic Plates," L Appl. Mech., XXIV, No. 3 (1959), 39t-396. 121. Howell, G. H. "Factors of Safety," Machine Design (July 12, 1956), 76-8r.
L22. Horvay, G. "Stress Analysis of Perforated Plates," Report 56-RL-1611. Schenectady, N. Y.: General Electric Research Lab., 1956. t23. Georgian, J. C. "Uniformly Loaded Circular Plates with a Central Holo and Both Edges Supported" (Trans. ASME), J. Appl. Mech., XXIV, No. 2 (June L957),306-310. 124. Weil, N. A., and W. M. Newmark. "Large Deflections of Elliptical Plates" (Trans. ASME), J. Appl. Mech., XXIII, No. 1 (March 1956), 2l-26.
125. Williams, M. "Large Deflection Analysis for a Plate Strip Subjectccl ltr Normal Pressure and Heating" (Trans. ASME), J. Appl. Mech., XXll, No. 4 (Decernber 1955), 458-464. lt(r. Yrrssuff, S. "Thcory of Wrinkling in Sanclwich Construction," .1. llo1," Acro. Sot'., l-lX ( I95-5), 30-36. 127. Illryllrorrrlhwuitc, ll. M. "'l hc I)cllcctiorr ol l)lrrlcs in thc lilaslic-l)lltc ftirngc," I'rtx'. )rtrl Il. S. Nu.l. (;ttrt7i. Ap1tl. Nlttlt., ASMI'.. Ncw Yot'k: I t)-5 5, 5I I --5 l(r.
B ib
liograp hical Re I cr ences
189
128. cooper, R. M., and G. A. Shifrin. "An Experiment on circular plates in the Plastic Range," Proc. 2nd U. S. Nat. Cong. Appl. Mech., ASME. New
York:
1955, 527-534.
129. Haytl'tornthwaite, R. M., and E. T. Onat. '.The Load Carrying Capacity of Initially Flat Circular Steel Plates under Reverse Loading,,' I. Aero. Sci., XXII (1955), 857-859. 130. Lee, E. H. "Stress Analysis in Viscoelastic Bodies," euart. Appl, Math.,
xrrr
(1955),183-190.
131. Berger, H. M. "A New Approach to the Analysis of Large Deflection of Plates" (Trans. ASME), J. Appl. Mech., XIX, No. 4 (December 1955),465-s72.
132. Black, P.H. Machine Design. New york: McGraw-Hill Book Co., 1955. 132a. wang, A. J., and H. G. Hopkins. "on the plastic Deformation of Built-in Circular Plates under Impulsive Load," L Mech. phys. Solids, lII. London: Pergamon Press, Ltd., 1954. 133. Huang, M. K., and H. D. Conway. "Bending of a Uniformly Loaded Rectangular Plate with Two Adjacent Edges Simply Supported or Free," Trans. ASME, LII (t954), 14. 134. Radkowski, P. P. "Stresses on Plate containing a Ring of circurar Holes and a Central Circular Hole," Proc.2ncl U. S. Nat. Cong. Appl. Mech. (t954), 272-282. 135. Goodier, J. N., and C.-S. Hau. "Nonsinusoidal Buckling Modes of Sandwich Plates," J. Aero Scj., XXI (1954), 525-532. 136. Roark, R. J. Formulas lor Stress and Strain. New york: McGraw-Hill
Book Co., 1954. 136a. Hopkins, H. G., and w. Prager. "The Load carrying capacities of circular Plates," l. Mech. Phys. Solids, II. London: pergamon press, Ltd., 1953.
137. Young, D. "Deflections and Moments for the Rectangular plates with Hydrostatic Loading" (ASME Design Data and Methods), Appl. Mech.
(1953), 29.
138. Trumpler, w. E., Jr. "Design Data for Flat circular plates with central Holes" (ASME Design Data and Methods), Appl. Mech. (1953). 139. Fuchs, S. J. "Plates with Boundary Conditions of Elastic Supports," Proc. ASCE, 139a. Peterjohn,
LXXIX (June 1953).
R. E.
,S/ress
Concentration Design Factors, New
york:
John
Wilgy & Sons, Inc., L953. 140. Wang, C.-T. "Principle and Application of Complementary Energy Method for Thin Homogeneous Sandwich plates and shells with Finite Deflections," NACA TN 2620 (L952).
141. Goodier, J. N., and I. M. Neou. "The Evaluation of rheoretical critical Compression in Sandwich Plates," l. Aero. Sci., XVIII (1951), 649-656, 664. Sec conrnrcnts on this articlc in J. Aero. Sci., XIX (1952),281.
190
lliltlitryrupltit'ttl ltaft:rences
Bib liographical Re le r enc e s
142. Stippcs, M., and A. H. Hausrath. "Large Deflections of Circular Plates'n ('I'rans. ASMIi), .1. Appl. Mech., XlX, No. 3 (1952), 287-292. 143. Conway, H. D., and M. K. Huang. "The Bending of Uniformly Loaded Sectional Platcs with Clamped Edges" (Trans. ASME), J. Appl. Mech', XIX, No. I (March 1952). 144. Seeley, F. B,, and J. O. Smith. Advanced Mechanics ol Materials. New York: John Wiley & Sons, Inc., 1952. 145. Gerard, G. "Linear Bending Theory of Isotropic Sandwich Plates by Order-of-Magnitude Analysis," J. Appl. Mech., XIX (1952), 13-15. 146. Hemp, W. S. "On a Theory of Sandwich Construction," Report 2672,
157' Nash, w' A' "Efiect of a concentric Reinforcing Ring on stiffness and Strength of a Circular plate,,, Trans. ASME, XLVU isqg), l_tS. 158. Lely, S. "Large Deflections Theory for Rectangular plates,,, proc. Sym. Appl. Mech.,I (1949). 159. weinstein, A., and D. H. Rick, "on the Bending of a clamped plate,,,
Aeronaut. Research Council, 1952 (Report 15, Cranfield, Coll. Aeronaut.
t948).
147. Eringen, A. C. "Bending and Buckling of Rectangular Sandwich Plates," Proc. lst U. S. Nat. Cong. Appl. Mech., ASME (1952), 381. l,47a.Yallance, A., and V. L. Doughtie. Design ol Machine Members, New York: McGraw-Hill Book Co., 1951. 148. Grigor'ev, A. S. "Investigation of the Work of a Circular Membranc Deformed Well Beyond the Elastic Limit" (in Russian), Akad, Nauk USSR, Inzerernyi Sbornik, IX (1951), 99-112. 149. Teichmann, F. K., and C.-T. Wang. "Finite Deflections of Curved Sanclwich Plates and Sandwich Cylinders," S.M.F. Publ. Fund. Inst. Aero. Sci,, FF4 (1e51)
150. Stein, M., and J. Mayers. "A Small Deflection Theory for Curved Sanclwich Plates," NACA TN 2017 (1950) (NACA Rept. No. 1008 lL95Il). 151. Morse, R. F., and H. D. Conway. "The Rectangular Plate Subjected to Hydrostatic Tension and Uniformly Distributed Lateral Load" (Trans. ASME), I. Appl. Mech., XVIII, No. 2 (June 1951), 209-21.6. 151a. Bricksen, W. S., and H. W. March. "Compressive Buckling of Sandwich Panels Having Facings of Unequal Thickness," Report 1538-8, U.S. Forest Prod. Lab., 1950.
152. Hoff, W. I. "Bending and Buckling of Rectangular Sandwich Platcs," NACA TN 2225 (1950). 153. Perry, C. L. "The Bending of Thin Elliptical Plates," Proc. Sym. Appl, Math., III (1950), 131-138. 154. Wyrnan, M. "Deflections of an Infinite Plate," Canadian Journal ttl Research, V.A.,
){XVIII (1950).
155. Coan, J. N. "Large Deflection Theory for Plates with Small Initial Curvlturc Loaded in Edge Compression" (Trans. ASME), l. Appl. Mc<'|t., XVIII, No. 2 (1951), 143-151. 156. Conway, H.D."Bending of Rcctangr,rlar Platcs Subjoctccl to a Unil'ortrrly Distributccl [,atcral l-oacl ancl to 'Ionsilc or C)ornprcssivc Forccs in lltt l)lanc of thc l)latc" ('l'rans. ASMIi), l. A1tpl. Mc<"1t., XVI, No.3 (Soptcltttrct l949), 301-109.
I9I
Quart. Appl. Math., il (1949), 262. 160. Leggett, D. M. A., and H. G. Hopkins. ..sandwich panels and cylinders under compressive End Loads," Report 2262, Aeran Research council (L949) [Report SME 3203, Roy. Aircr. Est. (1942)]. 161. Reissner, E. "small Bending and stretching of sandwich Type shells,,, NACA I-N 1832 (1949). L62. Gleyzel, A. "plastic Deformation of a circular Diaphragm under pressute" (Trans. ASME), l. Appl. Mech., XV, No. f lS"pt"mber 1948) 288-296.
163' Libove, c., and S' A. Batsdorf. "A Generar Smail-deflection Theory for Flat Sandwich plates,,, NACA Rept. No. Sgg (194g). t64' Levy, s., A. E. Mcpherson, and E. c. smith. "Reinforcement of a small Circular Hole in a plane Sheet under Tension,, (Trans. ASME), J. Appl. Mech. XY , No. 1 (June 1948) , 1 60_ I 68. 165' wang, chi-teh. "Nonlinear Large Deflection Boundary varue probrems of Rectangular plates,,' NACA Tech. Note t42S (lg4g). 166. Ling, c. B. "on the stresses in a plate containing Two circular Holes,,, J. Appl.phys., XIX (1948),77-82. 167' cox, H. L., and E. pribram. "The Elements of the Buckling of curved Plates,"
L Roy. Aeronaut.
Soc. (1,94g).
168. Reissner, E. "Finite Deflections and Sandwich plates,,, lour. Aero. Sci., XV, No. 7 (L948). 169. wang, chi-teh. "Bending of Rectangurar prates with Large Deflections,,, NACA TN 1462 (1948). 170' Libove, c., and S' A' Bardofi. "A General small-deflection Theory for Flat Sandwich plates," NACA Report No. 899 (194g). 171. wang, chi-teh. "Bending of Rectangurar prates with Large Deflections,,, NACA Tech. Note 846 (April 1948). 172' conway, H' D. "The Bending of symmetrically Loaded circular prates of Variable Thickness', (Trans. ASME), t. A.ppl. Mech., XV, No. 1 (March 1948), I-6. 173. odley, E. G. "Deflections and Moments of a Rectangular prate clamped .on all Edges and under Hydrostatic pressure,,, .f . Appl. Mech., XIV, No. 4 (Dq-cember 1947), A-289. 174. Stiles, W. B. "Bending of Clamped plates', (Trans. ASME), t. Appl. Mech., XIV, No. I (1,947), A_55. 175. Reissner, E. "On Bending Elastic plates,', euart. Appl. Math. (L947), 55-68.
B
192
ibliographical
R
e
176. Chien, w. z. "Large Deflection of a circular clamped Plate under Uniform Pressure," Chinese I. Phys., VII (1947)' IO2-II3' 177. Sneddon, I. N. "Elastic Stresses Produced on a Thick Plate by the Application of Pressure to its Free Surfaces," Proc. cambridge Phil. soc.,
xvII
(1946).
l7g. Goodier, J. N. "Cylindrical Buckling of Sandwich Plates" (Trans. ASME), I. Appl. Mech., XUI (1946), 253-260I7g. cox, H. L. "sandwich construction and core Materials. III: Instability
of sandwich Struts and Beams," Report 9226, Aerc. Res. Council (1945). 1g0. Hoff, N. J., and S. E. Mautner. "The Buckling of Sandwich-type Panels," I. Aero. Sci., XII (1945), 285-297.
1g1. Beskin, L. "strengthening of circular Holes in Plates under Edge Loads" (Trans. ASME),I. AppI. Mech., XII, No' 1 (March 1945), A-140' 182. chien, wei-zang. "The Intrinsic Theory of Thin Plates and Shells," Quart. I. Appl. Math., I, Nos. L,2, 4 (1,944). 1g3. carrier, G. F. "The Bending of the clamped sectorial Plate," Trans. ASME, No. 66 (t944), A-134. 1g4. Hopkins, H. G., and S. Pearson. "The Behavior of Flat Sandwich Panels under Uniform Transverse Loading," Report SME 3277, Roy' Aircr' Est. (t944). 1g5. Stevenson, A. C. "Some Boundary value Problems of Two-Dimensiontl Elasticity," Phil. Mag., XXXIV (1943),7666. 186. Van der Neut, A. "Die Stabilitat Geschichteter Platten," Report 5286, National-Luchtvaart Lab. (L943). 187. Levy, S., and S. Greenman. "Bending with Large Deflection of a clampctl Rectangular ?late with Length-Width Ratio of L.5 under Normal Prosstre," NACA Tech. Note 853, (1,942). 1g8. Levy, S. "square Plate with clamped Edges under Normal Pressttro Producing Large Deflections," NACA Tech. Note 847 (1'942)' 1g9. Ramber, w., A. E. McPherson, and c. S. Levy. "Normal Pressure Tcsts of Rectangular Plates," NACA Rep. 748 (1942)' 1g0. Stocker, J., and K. Friedrichs. "Buckling of the circular Plate Beyond the: Critical Thrust" (Trans. ASME), t- Appl- Mech', L){IV (1942)' A-1 ' 1.91. Levy, S. "Bending of Rectangular Plates with Large Deflection," NAC/I Rep. No. 737 (1942). lg2. Levy, s. "square Plate with clamped Edges under Normal Pressttrc." NACA Rep. No. 740 (1942)' 1g3. Young, D. "Benrling Moments in the walls of Rectangular Tarrks," Proc. ASCE, XXVII (1941)' 1683. D. D. et al. "Flat Sandwich Pancls tttrtlcr ('orrtprcssivc lrrrrl Wifliams, 194. Loacls," Rcport AD 3174, Roy. Aircr. lrst. (1914)' 195. l,abrow, S. "'l'hc Strcsscs irr, trrrtl tlrc I)cllcc(iorr ol ('it'ctllal' Iilrrl l)lrrlcs, witlr ir ('cnlnrl llolc, rrttdct' Norlttlll liorccs," I'rtx. ltt,sl. Mcclt. I'.tt1i.,
('xl,v ( I().ll), ll5-12.5.
Bibliographical Ilc I arert<
I er ence s
cs
193
196. Gough, G. S,, et al. "The Stabilization ol a Thin Sheet by a Continuous Supporting Medium," l. Roy. Aero. Sot:., XLIV (1940), 12-43. 197. Young, D. "Analysis of Clamped Rectangular Plates" (Trans. ASME), J. Appl. Mech., VII, No. 4 (1940), A-139. 198. Pickett, G. "Solution of Rectangular Clamped Plate and Lateral Load by Generalized Energy Method," Trans. ASME (1940). 1.99. Evans, T. H. "Tables of Moments and Deflections for Rectangular Plate Fixed at all Edges and Carrying a Uniformly Distributed Load" (Trans. ASME), l. Appl. Mech., VI, No. 1 (March L939). 200. Young, D. "Clamped Rectangular Plates with a Central Concentrated Load" (Trans. ASME), I. Appl. Mech., VI, No. 3 (1939), A-1I4. 201. Marguerre, K. "The Apparent Width of the Plate in Compression," NACA TM No.833 (July t937). 202. Sturm, R. G., and R. I. Moore. "The B€havior of Rectangular Plates under Concentrated Load" (Trans. ASME), J. Appl. Mech., IV, No. 2 (1,937), A-75.
203. Way, S. "Bending of Circular Plates with Large Deflection," Trans. ASME, LXVI, No. 8 (August 1934), 627. 2O4. Wahl, A. M. "Strength of Semicircular Plates and Rings under Uniform External Pressure," Trans. ASME. LIV, No. 23 (1932). 205. Sibert, H. W. "Moderately Thick Circular Plates with Plane Faces," Trans. Amer. Math. Soc., XXXIII (1931). 206. Garabedian, C. A. "Circular Plates of Constant or Variable Thickness," Trans. Amer. Math. Soc., XXXIII (1931). 207. Inglis, C. E. "Stresses in a Plate due to the Presence of Cracks and Sharp Corners," Trans. Instn. Naval Archir. (London, 1931).
208. Wahl, A. M. "Stresses and Deflections in Flat Circular Plates with Central Holes," Trans. ASME, LII, No. 1, (1930),29. 209. Wahl, A. M., and G. Lobo. "Stresses and Deflections in Flat Circular Plates with Central Holes," Trans. ASME, LII, No. 3 (L930),29. 210. Nadai, A. Elastische Platen. Berlin: 1925. 211. Prescolt, I. Applied Elasticity. London: Longmans, Green & Co., 1924. 212. Wojtaszak, l. A. "Stress and Deflection of Rectangular Plates" (ASME Design Data and Methods), Appl. Mech. (1913).
SUBJECT INDEX
A
C
Absolute system of units, 175 Acceleration of gravity, 179
Angular deflection,
Circular Plates, see a/so Flat plates assumptions for thin, 7 under uniform moment along inner
see Slope
coefficients Assumptions, 7 Axial compression for sandwich plates, 163
B Bending of circular plate under simultaneous lateral and end loads.151 under uniform load on a concentric crrcle,127 under variable symmetrical load, 65 Bending moment diagram for circular plates
with fixed-supported outer
edge
and fixed inner edge, rtt, lt6, L20 with fixed-supported outer edge and free inner edge,
lLl, 1,L3, L19,723 with fixed-supported outer edge, 1.14,124 with simply supported outer edge and free inner edge, 1.tL,
rtz, ll7, l2l
with simply supported outer edge and fixed inner edge, 118, l22 with srmply supported outcr cdgc, 115, 125 Boundary conclitions, 79
edge, (table),21 under variable load over entire actual plate, 65 with outer edge fixed and supported, inner edge prevented from
rotating, (tables),21 with outer edge fixed and supported, (table),21 with outer edge supported, (table), 2L
with outer edge supported, inner edge prevented from rotating,
(table),17 bending of, under uniform load, 127
bending of, under variable load, 65 bending moment, diagrams of, see Bending moment diagram Combined, axial and lateral load, 15 surface pressure and uniaxial compression, 163
D Deflection of circular plates, (tables), 17, 2L
of circular sector, (table),
21
due to shear. 8 of clliptical platcs, (table), 2l largc, scr,r l,argc delloction,
ol' plrtllclcpipcd, (tublc), 2l 195
Subject Index
196
Deflection of circular plates (cont.)
of rectangular plates,
see
Rectangular plates
of square plates, (table),21 of triangular plates, (table),21 Direct stresses combined with bending, 151
E Edge forces in rectangular plates, see Rectangular plates
Elliptical plate, (table),
17
,\'rth
(table),17 circular sector, (table), 21 circular thick plate with eccentric hole, 171 circumferential stresses in thick circular plate, 171 corner forces in rectangular plates, (tabLe),21 edges forces, (table), 2l elliptical plates, (table), 21 large deflection, see Large deflection parallelepiped plate, (table), 2L rectangular plates, (table), 21 shearing force, 8
Finite differences method, 10 Flat plates circular, (tables), 17, 21 circular sector, (table), 21 corner forces in, (table), 21 deflection coefficiehts in, see Deflection edge forces, (table), 2I effect of Poisson's ratio on stresses rn,
thick circular plate, 17l triangular plate, (table),
large deflections in, 139 moments and reactions for
I Indeterrninate system, I 32
K Kilopond, definition of, 176
triangular, (table),
21
under simultaneous transverse and axial loading, 151 Forces corner, see Flat plates edgc, see Flat plates
Fornrr,rlasfor
ilt rcctangttlitr plirtcs, (tablo), 2l
bcrrcling nr()nrctlt
Panel, shear stiffncss, 163 bending stifl'ncss, I 63 Parallelcpipcd platcs, (table), 2L Plates, scc lilat plates
boundary conditions for, 79 circular hole in, 8 circumferential stresses for thick, deflection due to shear of, 8 deflection of, (tables), 17, 21 edge conditions of, 7 effect of Poisson's ratio on strcsscs and deflections in, 5 large deflection of, 139 sandwich, 163 slope of, (Iabl,es), 17,21
solid, 21 statically indeterminate, 132
in, (tables), 17, 21 symbols and units for, 3 171
under combined axial and lateral Ioad,151
under combined surface pressure and uniaxial compression, scc Sandwich panels
L
square, (table),21 stress, deflection and slope formulas
for, (tables), 17,21 statically indeterminate, 132 thick, with concentric or eccentric hole, l7l
P
thick,
2L
parallelepiped, (table), 2l rectangular, see Rectangular plates slope coefficients for, (tables), L7,21
(tables),17-29
Moment diagrams for circular 1;latcs, see Bending Moment diagranrs
stresses
rectangular, 10 moment coefficients for, (tables),
I7,
21
Gravitational system of units, L75
21
Moment coefficients for flat plltcs,
r7t
Grashof's equation, 68
)
elliptical, (table),
M
circular plate with concentric hole,
square plate, (table), 21
F
jct
Lame's equations, 171 Large deflection of circular solid plate with clamPed edge, 141 circular solid plate with simply supported edge, 143 elliptical plate with clampccl edgc, 145 rectangr-rlar platc with all cclgcs
sinrply sul.rportcd, I 4tl
rcclangular plittc with two ctlgcs sirrrply strpportctl lrtttl lwo orlgcs
cllrrrpcd,
141)
under uniform edge moment, 29 under uniform load, (tables) ,17,21 under uniform load on a concentric circles of plate, (table), 17 under unilorm load on inner concentric circle of plate,127 Poisson's equation, 68
t lrt,l,
lt)i
r
ftcr l.urlirrllrl plate, also see Flat plirlcr lrr.rr,lrrrl, ol, under simultaneotts Lrlr.r;rl lrrrd end loads, 151 IrcrrrIirr,', nrontent
R Reactions for rectangular plates, 7
see Bentlirrg
c()r n(.l l()r'(.cs; in, (table), 2l ctlgc lirr ct.s in, (table), 2l
S
Sirrrrlwit lr prlrcls, stresses and
(lcllecli()lls litr, 163 ('()r'(' {llL'.ll' rnrlrlttlus for, 163
Slrt'rrl cllccl orr bcnding, 146 Siurrrlllrrcous latcr.al and end loaclings olr rcctlnglrlur plates, 151
tlcllcclion tluc to, 151 lorrgilrrtlinll strcsses due to,
l5l
llltnsvcrsc bcrrding stresses due to, t-51
Slopc cocflicicnts l'or circular plates. ( lrrlrlcs
l,
17,
2l
Solid circr"rlar platcs, (table), 2l Square flat plate, see also Flat plates,
(table),21 with all edges fixed, (table),21 with corners free to rise, (table), 21 with corners held down, (table),21
Symbols and units, 3 Statically indeterminate analysis, 1 32 Statically indeterminate plates, 1 32 Strcss coefficients, for circular plates for circular sector, (table), 2l for elliptical plates, (table), 21 for parallelepiped, (table), 21 for rectangular plates, (table), 2I for square plates, (table), 2l for triangular plates, (table), 2l System of units, 175
Poisson's ratio
effect on stresses in plates, 5 limiting values of, 5
fot,
ilt(tnt(.nls
T Thick circular plate, with concentric hole, L7l
with eccentric hole, 17 | Thin plates, see Flat plates Triangular plates, (table), 2l
