• TABLE OF CONTENTS
HIDE
 Front Cover
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Introduction
 The exact solution for plane...
 The exact solution for plane...
 The plane stress solution applied...
 Appendix
 Bibliography
 Biographical sketch
 Back Cover














Title: Effect of a rigid elliptic disk on the stress distribution in an orthotropic plate ..
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Permanent Link: http://ufdc.ufl.edu/UF00098043/00001
 Material Information
Title: Effect of a rigid elliptic disk on the stress distribution in an orthotropic plate ..
Physical Description: 83 leaves : ; 28 cm.
Language: English
Creator: Owens, Alvin Jewel, 1918-
Publication Date: 1950
Copyright Date: 1950
 Subjects
Subject: Elasticity   ( lcsh )
Strains and stresses   ( lcsh )
Wood -- Testing   ( lcsh )
Mathematics thesis Ph. D
Dissertations, Academic -- Mathematics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Dissertation (Ph. D.) - University of Florida, 1950.
Bibliography: Bibliography, leaf 82.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098043
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000551361
oclc - 13323005
notis - ACX5836

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Table of Contents
    Front Cover
        Page i
        Page i-a
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    The exact solution for plane stress
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
    The exact solution for plane strain
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
    The plane stress solution applied to a sitka spruce plate
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
    Appendix
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
    Bibliography
        Page 82
    Biographical sketch
        Page 83
        Page 84
    Back Cover
        Page 85
        Page 86
Full Text






EFFECT OF A RIGID ELLIPTIC DISK

ON THE STRESS DISTRIBUTION IN

AN ORTHOTROPIC PLATE










yE
ALVIN JEWEL OWENS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIl.. OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL. FUI.FIl.MENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHII.OSOPHY










UNIVERSITY OF FLORIDA


June. 19;0




















ACJC]OWLEDN;;;E:TS


The 2thn- c1:1 1e o to c::p:'cc S ;-et .l r to

Dr. C. ?. 3::,t> .r:- zu--crti:. t ,: :. l--. rcd f:?- :-

:uc:-.:.c tl.' lu l-I'ut t.e :-C 'rC -z: f t".c ',o, ::, rd t3 the

: e .f t:e. i c) t- t.t 7 :'o t.>"- ss =S tance in edc'li r.
theta pape:.














TABLE OF COUTFI!TS


Chca ter


I. I TRODUCTIO: . . . . . .

Ii. THE EXACT SOILTIO!: FOR PLAIT
STRESS . . . . . . .

The athiieatcal For lulatior of
thle iounrdar' Co;.d tions . .

Thr Deriv.lation of tih Diff-rential
E-u-atlon for an lOrtho'rorir
Plate in a State of Pia'.n Sti.,-ss


The Jenerlal Jolution of t!he
L"'fere.-tial Equat'on . . . . 12

The Stresses Ojtailned from- the
Stress nctio . . . . . .

Th, F:.-t-alicc' 3cur.c'ar-' C' ..,: t O"
C; ..c : . .l . . . . . .].

The Drr:v-atiro of th_ D's lace--its
u and . . . . . . .. .. 2.

A Lai 2 deter iln ": ?rocm flh. Ir.teri,:.
.,o .nridar- Conc di tions . . . . . 2

,. Ir:tro-ic St:-;s.r Furcltion s t' .
L'-itin- Cas of t.: Or'hatr:' ,c
StPr-ss Functio . . . . . .

II1. THE FACT SOLUTION: FOR PLAN~ I STAl~ .. '

Thc Dc:L'ivat.Lon of Lhe Diffrertr. trial
Equat.iol. for aln rt'hotro.)ic
Plate in a Stlt- Df Plane Stra-r- . 2

A' andi D' Detcrnir.ed from the
Inrteror Boundar3 Coditions . . 44


i1-i


C


Pa ,?









TABLE OF COINTENTS--Continued


Chapter P.

The Isotropic Stress Function
as the Limiting Case of the
Orthotropic Stress Function . . .

IV. ThE PLAME STRESS SOLUTION APPLIED
TO A SITrA SPRUCE PLATE . . . 52



BIBLIOGrRAPHY . . . . . . .












LIST OF TA.LLES


Table Pa;e

I. ELASTIC rMODULI OF SIThA SPRUCE . . .

II. CO:STAI'TS EVALUATED FOR SITKA SPRUCE. . 61

III. THE COISTAITS A1 ALD p1 EVALUATED
FOR SITKA SPRUCE . . . . . . 62

IV. VALUES OF THE NORMAL STRESS COMPOrl'rETS
AT POI;!TS ALOI;G THE x:-AXIS EXTERIOR
TO A RIGID CIRCULAR DISK OF RADI.1US
a WITH CENTER AT THE ORI'Il! FOR A
PLAIIP-SAWri PLATE OF SITKA SPRUCE . .

V. VALES OF THE NORMAL STRESS COrMPOIF7;TS
AT POINTS ALCIUG TH}E. --AXIS EXTERIOR
TO A RI3ID CIRCULAR DISK OF RADIUS a
WITH{ CENTER AT THE ORIGII: FOP A
PIAIL!-SAWTJ PLATE OF SITKA SPPLCE . .

VI. VALUES OF THE SifEAR STRESS COMPCI:EINT
X,. AT POINTS ALO;!G THE BOUiTDARY OF
A RIGID CIRCULAR DISK OF RADIUS a
WITH CETITFR AT TilE ORI0Ii: FOR A
PLAI!J-SAWIV PLATE OF SITiA SPRUCE . .

VII. VALUES OF THE SHEAR STRESS COrMPOl:ENT
X,, AT POINTS ALOIG THE LINE x = 2a
EXTERIOR TO A RIGID CIRCULAR DISK
OF RADIUS a WITH CE'ITER AT THE
ORIGIN FOR A PLAIN:-SAI..; PLATE OF
SITKA SPRUCE . . . . . .









LIST OF TABLES--Contrinu n-i


Table Pa

VIII. VALUES OF THE IIORI.AL STRESS COrPO:.EI:TS
AT POINTS ALOI:G TIF x-AXIS .EX(FRICO
TO A RIGID ELLIPTIC DIShJ (b : --)
I!ITH CEN.:TR AT Ti ORIGIi. FOR A
PLATII-SAW, PLATE OF S SIiA SPR;UC . .. .

IX. VALUES OF THE IIORMAL STRESS COr:POFET.ITS
AT POIIJTS ALOT;G T E :--AX IS W; (~ TF IOR
TO A RIGID ELLIPTIC DISK (.- fa)
WITH CEIITER AT THE ORIGINT FOR A
PLAlI:-SA!'7 PLATE OF SITA SPFR'.' . . .

X. VALUES OF '.HE SHEAR 3TFr-S COr'?OTT:
X AT POII:TS ALOT;G TIfE 30D'DARY OF
A RIJID ELLIPTIC DISi- (,1. -)
/IT1i CE:"iTE AT V'IlE OI> : O- 70- A
PLAIT--SAW%: PLA'TF OF 1 SIT'A PF"CF.....

XI. VALUES OF THEi IOR'.'AL STRESS 0O'-
POCT.:NTS AT POI.'TS ALO:'i; T; x-A IS
FXTEFRIOR TO A R3IED ELLIPTI.' DISK
(a -: ':,) :I'1,H CE"F .L A' P,. Of 1i]
FOR A PIAI: -SA.T. PLA'TF 07 GITA
SPRUCE . . . . . . . . .

XII. .'ALLTS OF Ti0 E 3!IT.AF STRESS O':P:, F :i
;I VT POI:TS ALO..""- ''t-i ,." DA7 OF
A '3ID ELLIPTIC DI.S[ (*.
V'IT!' ,F" -. AT T"L OF:n ]" K '"-h '
,P.ATO-SAW . . .

XIII. '.AI.TS OF THE STA-,. STATE 33 .70 CP T-
.,- AT POI "S AL.OC:.U L. '. 1T ': O'
A RIJID ELJ.-P'TIC DiJ (. "
.''ti C '"T .F AT T OT 7 7.'. A
PLAT: -SArF. FLABE OF S77, A SPT . .














LIST OF FIG'RFES


F i r' Pa -

3. Plai-r-savn hoa'r r'elati ,e to th.. lo-
froc-1 t:ilch it :s cut . . . . .

'. Qutrt=--r sa-,l: .,,ari r lat: to ** I c.
t''oi ;,h c. 1 it is cut . . .

Elliptic Dis': i. an Ort-hotrnpir Plt. . 7

4. Plain-sa'.:n woorc-ien lat.- with ''a-e :
ellastic : t . .

'r,'i .tilun.. of the ncr-ial ,t2'frs- .i.j"-
poricr.t X.. at polrit. ilon t :.-i:.
-:etercior t'o a rI 6;u lroulr dic-k :.,
radio's a :. th c- rt-r at. t.h-: c "r
for a l.ir-s.'- l'- latt- o Sit.:
s r,' a-,r fPr- a:, ilEtropic 'late

SVariatl io of the normal str,-s -' -
ponent Y.. at ~po:it al,-o th'.e -a:
tr,: 2 or) .o a r i(i C-1 c7cular s1": of
'aacius a itil centcr t. '- : r. ..
f'r a 'lain-savrn -liatL. of S: L:9
spruce and for an Isotr l c p-]'t? . .

a. 'aratlo i of the r.nr:-al tr-:C. 3 co1-
por:-.i; Xy, at. poir.tc aloi- tl.c : -1?xls
-::tei -or to a ri d Cirr'c.lar Cilci; of
racilus a. V ith c-nt-: a- cr' *in

ruco L ar. S. D,- .g 7 Ll tr


r.onnnt Y., at oolnts aloof: tir -ax s
:.te'.or' to a rl id -ir.ular C If;': of
radius a ,Uvith ceCnt-er at tli- or-i A
forr a plair-sa,!n plate of Sit':a
spruce and for an isot-ronic ,T!. . "


- ii







LIST OF FIGCURES--Continued


Figure Pace

9. Variation of the shear stress com-
ponent X,, at points along the
boundary of a ri.id circular dis
of radius a with center at the
origin for a plain-sawn plate of
Sitka spruce and for an isotropic
plate . . . . . . .... 7

10. Variation of the shear stress com-
ponent X,, at points alont the line
: 2a ?.tc-rior to a ri-ld circu-
lar disk of radius a vith center at
the origin for a ulain-saun plate of
Sitka spruce and for an isotropic
plate . . . . . . . . . 7

11. Variation of the nor-al stress com-
ponent Xx at points alonj the x-axis
exterior to a riid elliptic disk
(b = 5a) with center at the ori-in
for a plain-sawn plate of Sitl:a
spruce . . . . . . . . 4

12. Variation of the normal stress co'i-
ponent Yy at points alon- the x-axis
exterior to a r:icid elliptic dis!:
(b -a) with c-nter at the ori i.n
for a olain-s.'w-n .late of Sitl:a


153. Va;i atlio of tha ncrral ctr'. co -
:o..on- t X.. at poli ts alor.- the --a:,.i~
.:*:trior. to a ri id e ll.tlc d i
(b -= a) wlth cet-r at the orilir.
for a -niain-snwn plate of 3St!:a
ALL%.







LIST OF FTI'3:ES--Co:ntnuud


F 1 Le Pa e

14. 'ar'iation of tie normal str 'ss co!m-
ponent X.. at points Zlon- the
'-a::ls exterior to a r1i d6 eliotic
disl: (b : .a) with center at the
orl in for a plain-savn plate of
Sltl:a pruc . . . .. . . .

Vcariat'on: of tlic S'l-ar stre.s co--.-
on;ent ,, at Doints alone th,,
bour.daru- of a ri Id lll'.tie disl:
(b 7- a) wvth center at the origin
for a plain-sawun plat, of Sitl:a
S r ucO . . . . . . . .

1-. Varlatlon of thn n1or1al stress c .'-
nonc.t X at point. al3.-: te :-a.::s
e::tcrlor tI a r' 'Id 011l tic CiZ s:
(a :r ') with center at the ori in
for a plal: -sc. plate of Sitl-.
S7r'Uce . . . .. . . . .

1". "'arlatio: of th- shcar stress crj--
.oier.t X, at c'in.ts alt.ij tic
jounidar' of a ri-id eliticl d;sl:
(a = _b) wilh c'entrr at t:ih or2 in
for a ilain-savn --late Af Sit.:a
S r ce . . . . . . . . ."

1, Yaoil .atln, of tli- "e,:.r st-. -r s co,:-
pon':t X.. at points ul n t.-
.o'l ar.,- of a .i id clll il"-? .i sl:.
(a l',ti) wv th co,'-r ,t th ori in
for a ,ali.-22.v;- late of Sit.:a















CHAPTER I


INTRODUCTI OI


A substance having three mutually perpendicular planes

of elastic s'ry'-.etry is said to be orthotropic. A thin plate

of uniform thickness taken from such a material with its

plane parallel to a plane of elastic symmetry will possess

two perpendicular axes of elastic symruntry in the plane of

the plate. Such a plate is said to be an orthotropic plate.

In this dissertation, a large rectangular plate of this type

with its ed es parallel to the axes o:'f s'-rnc'-try and vith a

small rigid elliptic disk at its center is discussed. A uni-

form tension is assumed to act along two opposite c(dges of

the plate and a mathematical analysis of the stress distribu-

tion near the elliptic disk is given. It is assumed that

the strains are small and remain within the limits of perfect

elasticity; that is, the material returns to its initial

shape after the external load is removed.

It is well established that a plain-savn or a quarter-

sawn board can Le considered as an orthotropic plate. The

position of a plain-sawn board relative to the log fror which

it is cut is illustrated in Fli nre 1. FiRure 2 illustrates

the position of a quarter-sawn board relative to the lo- from
































Figure l.--Plaln-sawn board relative to the log
from which it is cut.


Figure 2.--Quarter-sawn board relative to the log
from which it is cut.











vhich it is cut. The problem presented here is the inathe-

natical idealization of the probile-? of a wooden plate pierced

by an elliptic steel rcc: or, say, a wooden plate containing

an elliptic ir.ot.

It is desired to locate the points of stress concentra-

tion; that is, those points at which the ratio of the stress

at the point to the normal stress at infinity is large con-

pared to the ratio for other points in the plane. For a

wooden plate, shearing stresses are probably n-ost important

in producing failure in the plate. Formulas for these ratios

at any point on the boundary of and exterior to the rilid

elliptic disk are obtained.

For the problem of a hom'oajeneous orthotropic plate, in

other words, a plate containing no holes or inserted dlsi3s,

with the exterior boundary conditions ,given above the ratio

of the shear stresses (in the directions perpendicular and

parallel to the direction of applied tension) at any point

to the normal stress at infinity equals zero. The ratio of

the normal stress (in the direction perpendicular to the di-

rection of the appiped tension) at any point to the normal

stress at infinity equals zero. The ratio of the normal

stress (in the direction of the applied tension) at any point

to the nor-al stress at infinity equals one. These ratios

are constant for all points and there are no points of stress

concentration.









A plate containing a hole or a ri id elliptic dici: Is

non-homo ljneous and points of stress co;ncent.ratilon occur.

Professor C. P. Smith1 has solved the ; roblen it!:; t.e Py-

te.rior boundary conditi:rns ci.-;'r above for a~r infi:-;lt r-c-

tangularr orthot-ropic plate cor.ta inin at Its cer.t- .h. : -

tic hol,'. By taking the major and minor r axLYs equal, t;- s.~ia-

t1on for a circular hole was obtaineci. For a plain-s.~vn

plate of Slti:a o r'uce coritainLn_ a c'rrL.lar '.oile, a -.a: :--uJ

tensil] stress of 4 4S occurred on the Fd 'e of t..- 0ioi 5t,

the eindL of the dian-etr perp.-ndicularx tj the directi ? of

uniform ter.slcon. S is tr.e uniform tepnrs.i applied ii t.:e r:-

rection of the ,rain at two ocp.osite ced e The -i-j.

shoar .tr.-ss obtainea 'vas '."71.

T'. ai-naly, s of th:ls c. ss rta t)n aupli o to t:. st:'- s

distribution in L- plain-s.--n ..'lat -~f Si t.:' sruce c tt. t ir

at its center a rixl c ci-crcui la' dis,: -lves --;a: u .: L i"' cil

stress and a may. mur sheer stress less th-nii t.hss- fou2:d co-

Professor Srith. These results do not a:iLuar un.:ra .Fna:tle.

We would c r ::oect t. w.al:er: a w- odc'-n ;.iat. riL 'n 1 1.oI P

i' it. Also It ser.ms r a-sonable thlat. a c' s,-- t' t : st.-l

dish: n t'h L-0 oul' stre:. th n the plat alth',u 1h tht- r lat

would be veask~r than it was .before th-. !;lo>, V-s L rill .'.


-C. B. Smith. Eff ct of Fllitic or Circular Holes on
the Strnss Distribution in Plates of Wood or P1'-voc.i Corsid-
ered as Orthotr-opic Mfatorials, For-s Products Laboratory
Report 10. 1 .' 1'-44.











T.1 iscrttcn s ".: i :. .r t 'n n

S .--:." .' :r .t ..- : :. .. .

1: t ? *r.r t -: : r .' it -

i t r x.. :: ::2 : : 3*.--- it-



tion of the stress d:rtribution in :: infinite rect: ;ular

2rt'-:otr"t 'ic ."l t- cort t ir _- ,. at :".: -. -':. . ... .. . ".

d ..... .? - r. : = t . t ... '. .. .. : 1-.

t roth caJ-. : '.--. str .. fm^ ttlar, t.0: -:' "C.-C t :,V: a.-.:t -

-,lu_ r, S ; :, :'. c . 1 .. r,- t 't -

t.lr. ;r' the- r .o :- c -; a' 1' "' .. l ..'"
_d. :P ,* ., .-..



a- -3 ,- a



." Cr:. t "r ;I is apy lie .. a ~ :.-...-:. 3:t_;, *L,: 'i .

o. tt r 3 C:.(i' "- i "', '- Vcr'u. -" L'.-['_" :" .... t.
i a >, 7 "" ..- "C ... ,- .a r
. . -. . ? '* .D :- 't_ ._* -. .
'.C-i; 'r 1 i, t .: -t 1 : I.. .Z c - . .- . c .i



to t..e r i. c r3..; : s. j:a '.-.Xe Lcu:.' ar- .': .t-. cls u:. Pr 'oabo


of --ore- practical 1i-. rtance *i t:e plat.- ,c.rta Inin_ 3 11 1_,

c'ic_ -. i .c: :-'.t.cr. t'., n a r' .:r t>- il?.-. .1U. t ,-.

fr"- .r a e, t;r..- str zsses fjr: t:-. car'r c ir, ;: -r .-

. n a. ; t - *.:_. -ar r .,. -L :. -,:'. f - -

c: ar 3cn.













CHAP'rRi II


THE EXACT SOLUTION FOR PLAI-E STRESS

Th" mathematicalal ?crmulation of the
Boundary Cornditions
Clhoose thp center of the ellipse a3 the oricin of tnI

coor'inatr system as show ini Fiiul'e j. The axes of the rl-

lipse are taken to colnrlde with the coordinate a:xes.

The boundary- conditions mray b' stated :-athe-naticall-

usirinU Lhe_ n3t.ltion of A. F.. H. Lovely as


X c 0 57 ,


-T / s r =| O C)

and u(.,y) 0 v(,,7) 0 on thn circunforence of the dls:

.i-ein by the paraa;ietr!c equationss a o y = b sin-.

The displacements in the x and y directions are u(x,y) a.nd

v(x,-), respectively.


1A. E. H. Love, A Treatise on the Matlhematical Theory
of Elastlcit:- (:Iew York), 19';4.













































Figure '.--Elliptic Disk in an
Orthotropic Plate.








Tie Derivation of the Differential Equation
for an Orthotropic Plate in a
State of Plane Stress
If the a.:e of elastic sy :etry are taken as the -, y
and z axes, the components of stress and strain are related
by th" Follovin- enuations:2





(A. XV -A _t _- (2)
E, E





vt _-LJ _




Sy X
( -- = -- ( ,







In th-Sce -cquations E.., B.,, and E_ are You~2's ,ioduli
in the :., and z c:l"ctis. respectivel_ -. Tho s-i.bol -
repr'esents Po'sson's ratio; in particular, o-. is thle ra'io
of the contraction :arallel to the "-ax.s to the extension

2H. W. .larch, Stress-Strain Relatio;ns i..n Wood and Pl--
;:ood Cron3rsidrred as Orthotropic I.ater!als, Forest Products
Lab:or;ator-y report 1No. R1J February,, 1;1, ?.









pa Ua l cl to the _-EL .-^ ass:.cl ". : th a t r- "cr. _'.



.r c ,--, P .- ,_. .
th :.-a ; _.I T ., .iu a -t it: i ? .-- .* .. .1.c t . . 1--

particu : ,2 C .... is the r..> . * c, '4 i






ZT h e .r -- : cS. 4' ..). C, rr . L
anc. -no






-..io. : i, .r. C : b" ",.'" : .'>r ( ,"-
r ; s. u()

Z U ." _












and (i ) r'f ..ca? co:-'



S= =v --- t )


Y -E E








Iv ru., ,,
"PA2y
.u'-~~~~~~ t i lt PL








The equations of equilibrium for plane stress are

STx +- 3 x = 0, (n)
x Y

aY +- r = 2o. (11)
)x /

The equations of equilibrium are deduced from the conditions
of static equilibrium with no reference to the laws relating
stresses and strains. They will be satisfied by a stress func-
tion F(x,y) such that

~- 'F I F ( 1 F
^ y ax- y (1?)

The compatibility equation for plane stress,


Y x' 2x y
is derived using the concept that u(x,y) and v(x,y), the dis-
placemcnt in the x and y direction, are single-valued and
continuous (class three).
Equations (12) substituted into equations (7), (8), and
(9) live

/ X FF -r 2 Z E
Ex YZ E, Y X

4G. B. Airy, Report of the British Association for the
Advancement of Science, 1862.




11




e + -- E


i. F. (15)
/xy X XY
Comibininc .-uatcns (1'), (14), (1), and (16) and using
G5
S-X(17)
Ey Ex
leads to the diff-re-tial eqoution,

E- "+F (/ -2 + I_- (F0. (-0)
Ey X "I [ Ex ) W- E, f -
Equation (I1) may be rewritten as

SF +X aXF aF =0 (1.)

%where




by the cubstltut!on

= Y,6 (21)

5garch, op. cit., 7.
CH. W. !iarch, Plat PFl.tes of Pl.-v.ood urndcr Unlfcr-i or Con-
ccnt:rated Loads, Forest ?r-.:cucts La 'ato:-y Report ... 1712,
iarcl., 19 2, i. -' .








whcre


1



The General Solution of the
Differential Equation
Sub'titutin- a solution of the form F F(.-"+h) in
equation (19) leads to the characteristic equation

VL +2 rV+/=o (2)

a quadratic in U2 or

(24)

For ijod K as defined here is probably always creator
than 1.'

Let = COSh (5)

then

L= costh Vcosh -;

=-cos h sinh0


7Smith, op. cit., p. -.









e2
2


The roots of equation (23) are


here 1
where i


e


-^


V-1. The differential equation (19) can nov be


written


(2 )


where


(27)


since


= 2 K
It is important to note that


L= #-= x k- -V7- -,


V3= -A


Qcosh b


(23)


(29)





(32)


- e+ e-0
2


V3 =-i


'= e




K;'+2= e+e -


S+ Kt' +q: F=
f~ t ^ i-JL^ L-f^-







The general solution is

F= F,(x+ ix ) + Fa(x-Axr

+F3 (x +A ?PP) + F4 (x-A'PR). (:

The stresses obtained from this stress function rust be real.
It will bc shown that a real solution can be obtained from
the complex solution. Consider

R C F (x+i rK)l
which substituted in equation (19) Gives

( -2K+p(x ) R { F "} = 0, )

where the symbol

R{ }
denotes the real part of the quantit:; enclosed in the bracket.
Equation (34) is satisfied b: virtue of equations (27) and
(29). Similar results are obtained with F2, F.. and F4.
Since


it follows that a real solution of the differential equa-
tion (19) is

R[ F = R (, (X+ r)+F (x+i# )}, (
where










F, = F,+E d F, = F3+ E.



The Stresses Obtained from the
Stress Function
0
Some wor-: of Professor I. S. Sokolnikoff-on problems
using isotropic material suggested the general for: of the
stress function. In the solution of the Isotropic prob-
lems, a function of one complex variable is used. How-
ever, in the orthotropic case, a function of two complex
variables is used. Several functions of two complex
variables which satisfied the differential equation but
did not satisfy the boundary conditions were tr:ie. A
suitable function given by Professor C. B. Smith9, except
here the constants are complex, is


f _A + (


F = R ( 2, -^3 w)'+ Y+(V 2 2 +


where

A=A, iA B=B-1B+ (37)


'I. S. Sokolnikoff, Mathematical Theory of Elasticity
(Providence, Rhode Island), 1941, chap. v.
9Smnith, op. cit., p. 6.









7,= x-,-A r, C= X + /?, /?=E (3')

w, 2 Vi- W = ( z)

cab"'-L 42 a-1 b (C4)

The boundary of the disk is given by

X= acosLe)- r= E y= Ebsm-. (1)
S is the uniform tension applied at the ends of the plate.
In order that the stresses may be single-valued, the
square roots in equation (39) are to be chosen so that the
inequalities

Wl mI Y I and I42+VA41I | (L)
are satisfied. These inequalities will be discussed fur-
ther after the stresses are obtained.
The method used in deriving equation (19) establishes
that any function satisfying equation (19) satisfies the
equilibrium equations (10) and (11) and the compatibility
equation (13). A solution for the problem must
(a) satisfy the differential equation (19);
(b) satisfy the exterior boundary conditions


I XS,, =Oy I y _=0;


~y ~ X --P t Y Y V -= CO 0





17

(c) satisfy the interior boundary conditions; that is, the
displacements u and v are zero along the boundary of the disk

X= QCOS e- / =- 6 = ebsim-e-
The following formulas will be found useful throughout
this dissertation.


2-

aI.=


(43)


2;


' W2 .
AWL^^.


It follows that


R L A r


WJ JEL


+ _2 I -
Z2t- W2. j


R(8 I
+(?- )i
/.z \


( ,.h )


J-H Z,)
ix

^x


~-f
~ t,
af
$ t,


~f(~)
~II
af(Tif.~)
~r7


aZii=
3q


A~~f


= AB at


_ af


SZ-,


21-W)l


F


W2







A 2 (ZW W++


B
+ -
2 Y22


2, 21l


From equations (45) and (39)


RL L(+-
-W., +\A

SB -(,- WZ)
+1 W2(+ I)


KA
= R [, + W


,)+ .+w,
Z, + W ,2_l


SW2(L2+W il)


++ W


(46)


Equations (46) and (12) lead to


3XF F
T x


By equations (12),


f-A
W, ( +1--W)
(21), and (41)


xx = E1


Kw(1 A
w,( ,+w,)


+S


and


F p A'+A
a^Wx 7(z,+W,)a


(49)


(415)


1y=


S-B
+ -3
W-2 (7-2 + A


(47)


W (-2+WDI


(:.' )


aFx
Sx


eFF


AB~8
L~l(;t,+IJ~z)


xy= 6









may be written immediately.
The stress equations (47), (43), and (49) contain mul-
tiple valued expressions of the form

I- and
W,Z + W) W2( i2 + WO)

1Y" and Y2 defined by equation (40) may be positive or
negative, depending upon the orthotropic material, orienta-
tion of the axes, and the values of a and b. Assume 2,
is less than zero, say equals -c Define

W-Z, +W, = ,-:V:'+C. (3o)

The branch points of the function W are ic. A straight
line connecting the points +lc and -ic may be ta:kn as the
branch-cut. The branch-cut in the Z1-plane can be napped
onto the V-plane in the follovinC manner.
Define

= =X 0 and
Then

W- u*,Av x, V CA-y,
v here

Y,l a Icl.
Hence
u= .c- v=y,

and u2 + v2 : c2 is a circle vith center at the origin of
the W-plane and radius Ic.









The inequality


Iw =lE,+wZ, l IclI= IY,|

relation (42) given before, restricts the variable Z1 from
following a path crossing the branch-cut and determines which
value to assign W. Hence, by means of the function W, subject
to the relation (42),the entire Z1-plane with the exception of
the branch-cut maps into the region of the W-plane exterior to
the circle

w- 141 e .

If 21 is greater than zero, it can be shown that the
function W has the branch points :Y"l. A straight line con-
necting the points '-Y1 and 1 in the Z1-plane may be taken
as the branch-cut. Here again, by means of the function W,
subject to the relation (42), the entire Zl-plane with the ex-
ception of the branch-cut maps into the region of the W-plane
exterior to the circle


w I Yle .

Similarly for the expression with the subscript 2.
The stresses at any point of the plate except those in-
terior points of the elliptic disk are given by the equations

(47), (48), and (49). The values of the radicals defined itn
equations (39) are determined for thp point by the Inequalities
(42).






21



The Exterior Boundary
Conditions Satisfied
It will be shown that the exterior bcurndary condltloA.s
arc satisfied for any finite value of the co:mple:: constants
A and B. These constants are uniquely determine by the cor.-
dition that the displacements are zero on the bound.ar of th?
rl-id elliptic disl:.
In the previous sector. :t was shown t-hat the function
W Z1 + W1, subject to the relation (42), ma,)s the entire Z1-

plane with the exception of the branch-cut into thc --prlan.
e;:terior to the circle


w= e^."

It also follow:z that t..e point Z-1 oo ias into t:ie ;-ont
W = Z I+ '1 00 o Z oo f'or the c ;c of ': late :: := o .

Si:iilar results wre obtained :ith th- terms of subscri t .

It follows from the abovr remar:-s and rqii~~tior.s (i.) aind

(49) that o; the tvi 3 ed e :: = co








IK'x c r" R o+o + S0.
Sx,- = R o+ 0 + S S





T:y Zi -CO = R[ 0 + 01 0.

These last t': ;:r s-,s ons ar-- '"h c::tr l- bo nund .''.
for the edges :s: =0oo.








Also Z1 = oo for the edges of the plate y =: o0 Again
W = Zl+ W1 = oo evaluated along the edges of the plate y =:oo.
Similar results are obtained vith the terms of subscript 2.
It follows from these remarks and equations (47) and (49)
that along the edge of the plate perpendicular to the y-asxis


Y o-, = R 0 =o

and


y .c00 = R +0 = 0.


These last two expressions are the exterior boundary condi-
tions for the edcs y = oo.

The Derivation of the Displace-
ments u and v
Substitution of the stresses given by equations (47) and
(48) into equation (7) leads to



ax L Ex Ey (x.+W)






Ex







Integrating,

u= R Ai B + L-
,Ex E ] L-+v- Ex Ey+

I US(Y) (52)

where uo(y) is an arbitrary function and will be phown to be
identically zero.
Substitution of the stresses given by equations (47) and
(48) into equation (3) leads to



R+6 + \E ( -+
Ey Ex W-2 + W2

*S~ (5: )
E,
Integrating,

v= r4Aj + cx' [Y I + +1 ]


[__ I ] SY y V X) (51)
ee ) i an abitrr Euntion an will be shown to bex
where vo(x) is an arbitrary function and will be shown to be


Ider.ticall.:, zero.








It follows fr'c-i equations ([2) and (17) that


u=R[-


where


and


iZ =E


Fro:m equations (53) and (17), wh. have


v, R~j A
E [U(:?+w,) +


9a2 8 SyY Crx
A(Se(2+W) EY,


+ Vo(X)


where


and


2. E -
Ey


Substitution of the derivatives obtained from equations
(55) and (57) as required by equation ()) leads to


Y W(,1+w.)

A ( Z, A+W,)


9, a E3 fe
VW (Z1+Wa)

W2 (: + W -
AflE W2(-2+Wjj


+ (Y) + V,(X))


9,,B 5x U )
72+ w+ Ex +


(-,-)


(57)


(5: )


,,A


S Ex


E,


(59)














+u0 +


921
9A2
Ao7


A
W(z, +w,)

B
W ( +W,) S


dx


( --)


('a )


S+ Tj EW

It follows ,'rom equation (22) that

Ex = E' Ey)


92,
A6 ^


S/A y e +A I K
Ey EI

AKE


The relation


S= E- =7


hence

XVY


(62)


A \ 6 +1 E x
= rixr\ Ex


921
k K E


P(E Q,,-


xyK ,- 9"







is obtairncd fror-. equations (17) anc (22).
ironcr,


S,1 ). -


.I /x 3 C. 3 t -t X y
Ev 64

1+ ^-fc


^
~ Ey D< (
From eouatlon (27), ve have

x


c- E-2Iry


2 6 )


T h en


9IT1
A1

6 EyitX


2 2 /


fHxy 2 + 2 +
,1 E, L K E ftP3-!
Equist.on1? (2? ) anc (22) conObinc to "i'.:

2C= VE- _G-
rJxyY C


2QTI ~
ElI


Py(A


(C.: )


20^).


( ")


( )


(c,)


Ey 2K+


Hyk VE 9








and substituting from equation (28), equation ((6) is


rx

K 62 2 6~ + 2 62 ~ .
^^( g 6^+2uOy
(-(7)

Equation (E ) by use of the result. :'f Pquation ('C) --educes
to

F i 22)j @ (SCZ)


In an analo.ous manrneIr, it follows that

y( 9A)= e. (69)

Equations (CS) and (c9) substitute into equations (Cs.) reduces
it to


W ( "I W .IO) W2 (Z2+wa)i

+ dY + dVx a (70)

It is seen b:; direct comparison of equ:tion3 (4.-) arid
(7)) that tlh above value for the shear snd the siear ribtain-
edc-fro;r the stress fiuLction are -iqual if anr. only if

HX = dx ) ( )








but


hence
du. do o
dY dx -

Ilow dUo/dy is a function of y alone, and dvoc/dx is a function
of x alone. It foll.v,'s that

d U. and dv.
dY dx

where is an arbitrary constant.
Inte ratin-,

Uo= /y-t-rm and Vo=-J X +

~iere m ancr n are arbitrar: constants. But m : n = 0 since
the displacenents are zero alone; the boundary of the ri id
elliptic disk for all values of -e-. Further, the plate is
'f-Im-etrical about, the x-axis, and the stresses ar- applied
sy rmtrically; er -rfor-., the dlsplacenents must be sy.netri-
cal with respect to the ).-axis. This is expressed matheratical-
ly by requiring u,(y') to be an even function. It follows then
that Aj must c- zero. Tho arbitrary function uo(y) Is identlcal-
ly zero.
By a similar arjuent, it can be shown that the arbitrary
function Vo(y) is also identically zero.







A and B Determined from the Interior
Boundary Cond tions
The follovlng quantities ar- ev'aluated on the boundary
of the ridld elliptic d-sl.:.10

W, = pI e b cos + i a. sn -e- ( ,)

W2 = b cos'e-st- La si (7

Z, +W, =(cL + b)e (7)

,I+W,=(aC+P6b) e^ -)
Suostitutlon of the -quatiars (75) ar,, (76:) into ccuatlon
(52), recallin that Uo(:,) is Identic.al]y zero and that the
d isplacPm onts on the boundary of ti.e ri'd cdsl: 'trE zero, 'iv.s

u- Rf A [ E 1-e
L L Ex EyJ a+K Eb
+B[11 _1 -+""


+ S os- 0e (77)
Ex

A A 2 + B I 11 x 72
-a+KEbLEx E j a+pEb LEx E y


1]Smith, op. cit., 7.








E,
FtE)-cos-e- +


+ 2 F
Q +-P6tb LEx


Since u =- 0f for 0 -- 27T, it follows that

A, l .' 71 B. & W#+ a7x
ci+xcb Ex E a+peb Ex Ey1

-- =0
Fx


QA.4- (bL Ex FyJ


+4 B2.
*+0&b


2l t]0


Rew itin.i equatic n (7-) with the use of equation (17),
becomes


a+Kp 4 E bA, +
a + X 6. b


en+oy B -5. = o.
4.+ P b
-cL 1-Sz.=O


Substitution of the equations (75) and (76) into equa-
tion (54), recalling that vo(x) is identically zero and
that the displacements on thp boundary of the riic disk: are
zero, 1:'.'Ps

v= R4A[ -, +J ~
VE E E(CL+K6h)


aL-teL Ex E,


SIn-e-= 0.


(7s)


(7:.)


(o0)


(Si)








E ]r-1 -A e
Ex _PE (a.+3p b)


_ L Sb sin-= 0
EX


(2)


( itp C-EI-y E +[I'


Sin --


-A6 xI +i -
.b)(-+ [r ) + + -LB
L6 (.KeLb)yLE EEx -I L+st- E b)


os-e- = 0.


(i-=)


Since = f.r -e- TT. it follows that

A, E- x +- + f + AY
Y Ex ] |, +vb) *' + Fx _


(34)


(1L -b) ExSb=0


arnd


Ex


A2 I-Fy Ex. I Q+R+Kb) 7


P (CL +(86 b) -I


( 35)


E< J


f[ t


[Y +


+ telFe
-y' F. J;


B2 +
L y








With the use of equation (17), equation (S4) may be rev'rltten
as


I (tx2 e A b),
;

l+~ : + rTSb= 0
PE(a+pBE b)'


Solving equations (80) and ('3) simultaneously,

A2= B3= 0.
Solving equations (,1) and (?6) simultaneously,


0S -(


a-, Sbj e ( z+ 7)+
S (+E XVI '+ ~)


or, lith the use of l I = 1 obtained from equations (27),


and,


Bj




B,-


+ Sbo (f6)+ -)


aSC-,b(+ aO) + SQl.(I+62' OX)
-( -:




SE (CLt+ E + b)


SB (I+ EXpx) +Sb tP-x (Eo; )


(C.+ Eb) ,


(so)


(,7 )


E(a + P6b)


I+c' X)


_ (fl+-tKJYx)( C4 + Y


S v(I+ e )








The equations (7S), (S8), (89), and (36,) 1l-ad to the
stress function


F=P
(f K a(p6+G l+e2 C) 2( aeI(l+xE+6Ly)
S(a+ K b) E (Ca 4 : b)






+ + ( i+ eV ) 6+2 SbK _+ _-Ea)
-6
2"r c xP'+6a--v)(l+gS ) p(d+t'P(rd)(r +qG
e(a-, t b) E(cL+ P b)


W()++W.J) t C



The Isotro.,ic Stress Function as the Lir-tl.n:
Case of the Ortiiotropic Stress Function
It is desira-le tc shove that the c.rthotrooic ctrcss func-
t-o., contains as a special ?ase the isotropic stress function'
for a : b. First, set 6 = i. 3:' virte of qcuation (-2),
E. = E,; but, Lhis is not necessarlly so f.r everr' orientation
of the a.:es. E., E,. isa a necessary, condition for the iso-
t.ro-.ic solution, but it is ;.ot a sufficient co,- ition.






It follo-'s fro.r equation (17) that wve can write

^ = (7y- = c-
By virtut- of the aboi--, ro,-' r!:s and equation (4c), the actress
function (93) si- lif'p e to th- exr, rns io:.

Sa x (I~Tr)+-S bo-+o) b)
.x- (l+_0P(+ j F)(P-A+ 0-) (a- K b)



--W+ (a ()+w
S( o-)(l+r)- B(+ I -)( 0+ 21(c- P b)



v er e Z- -i- I + antd ., x -+- i p ;,
Rewrtin_,

F=R (((2+)x(l+ b))-(lt r)( f+ (I. -b)

Sa x(l+0')+5 bWY(P'+ (t-wF-+',L (z+w)


-B(Sa Pgl+P o -((T<+5 (+ ))

(2(- WV + y+a (,+w) + -+w (+







For the isotropic caS--, ILt s n3-. sufficient tiat :;
since, Dy equation (2-") 1---i and .-e.atLicn (1 ) reduC.LLc t tih
biharn:ornic equation for' the 1 'i 'o,:rL.r as


The followir.nc

lim ,


I r ? '


m r
tQ


21 =


Vc -U' = "
v0Q-^=w


I,, k = llm P=j
(--- 0 0--y 0

If = the oracl:ct.-d 'xp:'r ssio: ;f c uat: :. ( ) s i id.'-
t :,iirte. In order to e'valuat- the .nd- ter-iL--t expr.eC3ion,
first re:.'rite oqu:atl.on ('2) wvih th- use of equIatio.:i (2?).
Thzis ads to


FH


-;r /


In.ts w il ber founc us ful.

Ilm Z o X +.y-
-o


Ilm W. = rlm W, =
-0 0-- 0


a.e( e- ~ +S b -( e-















I


(93)


vh7 re


, = x+e e y,
r =x+ ke *, Y


Evaluatln by iffrerer.tiatir,. both numerator and Cdnolinator
with respect to it foJ Lcvs that


e 2I+e o-)


+Sbcr(e-


(I-+ -) +S e/)


Se a. (liea-)+sb-(e +r-)


F= R


ea- e-%,
zzx+ e- /


-w,) + Y,-


La ( t+ )


/ s a. e
+
S(a -- ebS)











Sa e(1, + e r) + Sb6-(e +)


(I
2t~~(cL-e~


w, ey+ eb'
W,


Sae (i+-


e"ci-)


cii))


L b
\ ) ^s-e~b


-whf+Y (i(,+n


- e-)-sb c-


(2, +W,) +


-e~b0a


+Sbr-(e +








+e-) So. e-


Se
'2


O + Sbr e0


(i+ecr-)


+Sb -c(e i-


e- y_
e y/-


ar)


+eb)
w2
v^--7


e Y+ e


b-


--- + -W2


2


Sae


C-


(94)


__ __


+e~Rrat


SA I








ee (1+ e- cr)


v=-i


e e cr


+e-2 I+


Le, d 0 eand qua:l.o. (.) ta.eos the forn


F2= R I (-- b
1(0 +)(3 F [2(aL-b


- Sb r- (--w)+


I ++ Sb (i+0-))


tji~b


-+ r) +5b c-(l +))


y-
wY+'


SiY- + b3
4 w


-) b' m(- +W)


where


U= I


(I
2


(Sa(


e +ot-(+ e-,T +


ce*+ -) (-e- -)


+ ek


ek,+,


e


e u) ( e


e. a) (- ).


(c- -W + Y2, ( +w;)


+ a -( b t (2g S a. -
a-b 2


+ C2(


+ 4-) +4








5fy
t ,/ (9:.)
+2
Equation (95) is the strce-, function for the effect of a
rlIid elliptic disl: on the stress distribution. in nn isotrro ic
plate. The pro.-le of a circular disk Is the special case
w.heL a = b. On settinC a : b, parts of equation (9') teco-e in-
determii.ate. However, equation (95) :,s;- be evaluated b:. succce-
sive applications of L'Hospital's rule, that is, by differentiat-
Ing both numrierator and denominatz.r with respect to b, remembering
that

W = \ f- c + b-.
It 13 susceu:ted that the underlined tc'rmr of equation (p") Le
combined before using L'Hospital's rule. Thie -ethod is then
strain htforward and for b = a gives


F= R .(1++(3-+) 4- 2


+S 'l -3

--')- -S "(I + ) "
+S a? (+r/ ^ Z A+ Sal of

S.nce constant terns Co not appear rn the stresses In 2Z
Le written as In Z. Also,

Ay= ()
2










a,: r.ot' i t ;: t ,. i, Zt-r- c- -,- 1 t.'L1..1 (


S) .! ,.


( ,L) rr-cucezs tj

SS(1+9-) S(--o-) 0- S(-+)fa-J
F=R +( ,- t (l-) 23-
B(3-a-) z+ 2(+ 2(3-9 1


SI2
I-i-


VIP (. <. ) L .I-


r ic *. ir''.: i_:' &i ".: ;: '.-- :f'I.:.' s ,: _': .. t,'-


--I -J.


* !- -.h -. '- .


JC ~ i. 03 s'wv in'. -..


- n :1:t.I


C. I*C t~. ~S:' r -


I I


c" ; r ;:'i*'.












CHAP:TR II


THE FXACT SOL.I'JOi POAR PAIT STF.AI:'

The Derivutoin -o t.te Difi' '... ial EuLFatJ. n ."fr ar.
Or'thctopic Pla:. ir. a Stat.r of Plane Strai,
For: the ?r obln of pl~ i v- trair the ,,-. -r._.;. of th.- hocvr
in thLe Z c.r'c-" r. _i c'.uld-'.:. ]L Je, a.r.!d t.e abo y is Ic.l c ed
uri _,r.-.i "r. tile diz .lt o 1,oe s v:nic.i act er' p di,.* .,a
t u tih r-a:; s Th't fu'Lnc I. s "'.c ..lesc .':..-cOn .r ,ut
f n; ctjons ,f -. Tle. Fr;:.r: :' D f' thi- L' ..,* Gt ." L stL r.ce f.o-
.nh--, er .n hr.L: a .a;)n c 1:c '- . l h' c ;G,lace.... "'ts u ,d ';
a:': no' funct f r- .. = A .
F,,o t;'- r ".-'pt: -p it I l ',:r .::i .- -r .... (I),




(2), .-: ) :-( )





+ a-,,



Efx ( --?-)
+t~i-ny~ii~x ,(130)








+ -





E


S _.y


(1i 1)


(132)


It fol2oCs from equations (12), (i,),


ar.c (1:2) that


oax.'


+ E / I
(- Oi0tx


Sa"F


+ Ey(I- OTx G- F 0.
Ex-(y-) ay


Let n 6 Fa


F,(lquat)n ( )
Equatilc.n (1-) -ay nc oc vvitter,


c'F
JX* t


E, E,


(14)


( 5)


\( I Y EE. 9


i- rch, Str-'ss-Strair, Re -iati cns In Ilooc aid P i' :o. : '.


Ex


~ x


-E,


Uxt
Ex


Ey


Ex


il~ ),
J.


2 K' a+F
34F~~


4 F
a =0
^ ~r








The general soluticn of equation (105) Is equat:o- (75)
where C and r7 are defl.ed by:,- equation (134) and % and 9
are defined by equations (29) anr (j3) wthl K' replacing .

A' and B' Dfter.,ir:eL fro-- the
Intericr 3Bo',-dar5, Conditions
The boundary conditions for the problem in lane strain
are the sam- as those of the orobicr solved in Cniater IiJ fcr
rlanp strE.s.
A suitable str-ss Cuc-ctlon is a-ain 'nrlution (~j) :.'Iere
E f n P D-.-. "T-. d f'ned in the oI ncePdin sectLor,
and t;he co;tonts A' = A+' + ,A-.' 1"c D' = 3-' -- -.2' are
printed to dcstin .-ush they .'.'." t'.? con'.tL.nt: of C'iaacFCr, II.
This new stress function is depi.-nat:c' equation (7' ). Fqua-
tions (47), (''). and (49) vith A n-6 B -r.e-d are de'si inated
equations (4 ), (4,'). a,- (i. ;' ), the stresses Y, X;. and
X.,, respective :,.
The result of subst.ttir.n e-at, Ins (47') anc (hb') into
equation (o9) and inte Tratin- wth respect to x is

U= R -A I ( Ix ) t G
Ex EJ

6- -1 x t -4- K
+ w, E E J


F+ ES ?-)( + U Y).







Uo(y), an arbitrary function of 7, is identicall;, zero. This
can be shown in a manner analogous to the method used in estab-
lishing uo(y) 0.
Since u = 0 on the bounda:'r of the rigid elliptic dis:,
the result of substituting from equations (75) and (76) ( Q, ,
and E defined as in Chapter III) is

U= -Ai a C 2) ( I- -G \
LQ[+DIb Ex Ev

B: 1 '(--, x + -,o ,,
a+9E b Ex E,

+0-ZY A Ib)
+ Sc (i-oYx a )lcos- _
2E x 2 1I- Tax + + + b

F, E, / a+ eb/
'( I- 'T;, A ) x +-0x sIn 0. (102)
E, + Ey /J
Since u 0 for' 0 2TT, it follows that the t-ers in
the square brackets may: be equated to zero. Equating thcse to
zero and simplifying : by virtue of equation (132) loads to

A' 2"6"( '-- x0+x)-+ -y--~~ +(



C. + PE 6 L








?a.Cl- 0 x O')=0 (1E)
A, -,
and




+a.+P b Ex + E,0 (=o)
The result )f sulstitut.ng equatiors (47') and (4'.')
into Peuation (130) anc ir.tc ratlni_ with respect to y il


E- L E: E, J A- E(,+w,)

+ B(G V+ -T.J y+ (S Y I --
EK Ey JAPE(2&+ vVW)

y ((07, + 7iaT- ,) -V- Vo (x))

Vo(x), an arbitrary fnctli of x, is Identically zcro. This
can be shown in a. rlarncr apalo joi3 to the 7etho( used in estab-
lishin_; vY(::) identically: zero.
Since v : o;- thei boundary of the ri'd cll ptic cisk,
the result t of substituting from equations (75) and (76) ( ,
and E defined as in Chapter III) is

-Al/ y +Gy+IyW G) i-\G )
-E(aL ->c+- b) Ex E,







B,' V6(\



Bo7 I f 0eeTo.. + e (b12)
s + A,

E, X* & (0-+K c b)


Ex E, lpe(_+o b)/

(qv+Q ) v(7 cos o. (11)
E, Ey CJ
Since v 3 for 0 21T it follows that the ter.-is in
t;"I square b:'ac::ets :na-, el equated to z,-ro. Equa-tin these
to zero and implhif.'fin- b,' -.irtue of equations (1 ) leads t.c

A: x -x( +y-y (0x -*)+ (FX I^)
I PIt (a+ N b)

+B,' x( t +v t)+ r(1- G6o ux)
1 (cL+ P 6 b)

+ Sb IX, (7x, C a ;v ) = o (O7)
a nrc


te(a.+ 6-b) Ex Ey

B+ BxEc +J rJ-y 1-E y = 0. (1c4)
W t(a.+P6b) Ex Ev






Sol', i. ,:,qua ionr~, (11 ) and (11.) i-.ultane.Jusl- 1v\:'

A = B = (11,)
Solvi:-n equations (i10) and (12J) 0.:ul tar e usl.- iv.

A A., +A,
A A22


B B,,- + B.
S= --( v )
M '., n l'.I


A,,= (cl-Gr) Yx(y^-^y)+(a

A,2=S bo- (N(y+F ) -xJ) Ez16207-7 GJ+ ) + 0GEy]
A i F x ( O- x [ ( a Y+(JI+(Y1


[ 2 ( x (G -. 0y +i Oy) + J7 ( I y J G ) ,


( Lc + 37eb)-
A0 ( X ( y7) + G Y+ Ty- a y-Z j -Iy]
B" .F-i x ?x ( y- rc CY(I-Exa-0+Gy+G;to








BI= Sbo 0(Jay+( ra )[, Ia-Gx ;) + 7+c-r oR j,


E (I-. T)+ AY + TA
B,,- ~(a Pe b)






B -_Y- rx ( 7 y + 0 1Y a) + avy(l g iy 671
BVI- F (c- + fe b)


pV (Y- -kx C) + Y + (6 + GY -

Equation (w( ') with -.I' B' A2 A, b',
recu tions (11-), (ic'), a ;d (17) o3 t.O stres- full ctirn 1.'-
In t th ir effect of' a ri.l ell p- c c::1iir.d o:' t". st:---s d s-
trlout lo: In a tod, of ort.-otr.o ic atrialil in a state- of
plane strair..

The Isotropi-c St s F ..c'(tior
'sL the Li:n"-tin; Case of th,,
Orthotropic Stress Funct-or.
To obtain the actress function for th e scotroric cas.-,
first st, E 1, and Pall*.I. ----- It follows tL'.a

F= R [ 7 Sa -1)(+ r-)
'^i~rd-D'O-o-J ^







-ScLa - t S b cr l a-)) (bl+
a-2 ( b (

(t-wf+ (a- b) an(+w) 5 aMI- ,r")(l-^r )


+Sbo- i


&(-+wi)


+- aSr(l- crXI+ r) +5 bo (l+ a(--b)


Z+
vv'] -


2- b
2


In order to obtain the stress function for a rlgid circu-
Lar cylinder, set b = a. Equation (113) evaluated becomes

F = ? ar ( X if? 1 r)1- )m 2
G( I + OfP 0 F]


SSo -(I+r-)
+ --


s~


+ a-a(lnd+ )-y



y= ~- and
2


a- D


0(0+ ri -2 7


2.'
H.


(119)


(a b )- + b


i"i~-
- b2an(i~L


(111^)


_Sa'r(+.-)1


Y~-R


+0-))








Equation (119) may now be written, droppin- a constant term
which does not appear in the stresses,

F= _S(- 0 -2 _-)_m + _ Sa?- Sc -

2 2(32-q) -


2t ( U2 (12c3)


It Is interesting to note that usinC a different. method
Professor I. S. Sok]olnkoff2 obtained this same stress functi, .


2So1:olnikoff, op. ci. chap. v.












CHAP=EEB IV


THE PLAIE STRESS SOLUTION! APPLIED
TO A SITKA SPRUCE PLATE



The results of Chapter II are applied to a large plain-

sawn rectangular plate of Sitka spruce containing a rigid el-

liptic disk at its center. It is possible to apply the results

to a large finite plate since the stress concentration Is lo-

calized near the ri -Jd disl:. The stresses approach the values

of the stresses at infinity at a distance of orly a few multi-

rles of the major axis of the elliptic disk. Figure 4 illus-

trates a plain-sawn wooden plate with its axes of syr-_'netrv,.

Hflre the x-arnd y,-ar):s are parallel to the L and T axes, re-

spectively. The orl jn is at the center of the ellipse with

the axes of the ellipse coincident with the coordinate axes

(see Figure J).

The stresses at any point in the plate are given I:.' equa-

tions (47), (4'), arnd (4,) where A rnual A1 is .iven L-y equa-

tion (SB) and equal B1 is xiven by equation (89).

Shear stresses are probably: most important in producing

failure in a wooden plate. For this reason a formula for the

shear stress alon, the boundary of thu rjiid disk is us.fil.

The shear stress ,iven by equation (49) evaluated on the



















\I,
R
























-T







Figure 4.--Plain-savn wooden plate vith
a::es of elastic symmetry.







boiuo-u-ry of thle ric d dws: rduces :ithl tel use of equations
(73), (y7 ) ),(7:), anc (7.) to

Y, -


SB, sin-e- cose-
P'4' b co2- + ac s.o -e

v:.iEar' Al anc B1 ar. r.: p tr,-inc'. by euationCs (yC) nc ('").
For -1 ratio of a to l equal t. ;:'eatcr t'ln 'en, the
nror-a.l trss X. '.1i-3 a 2'elAt'el '- : h value? at tle point
(a, 0). '-o!- eq nations (!.', ( C), and ( ), we '.cv

N, -N
y =o

i, here




Na S p2e (J+e)(l )+S(E bl (7 p +y,),

D = tbl(' + a--)( + ^PY E ) h6r;+7-0

Th? -'alues of EL ., I r-. .r:c L t.:l.t "- .. E.,
r, ar.d ..,,. :e'esr ct c ., v -, I fc.v Slt..: spruce are tabulnted
In Table 1.1 'e obotnin 0- from eoqatio. (i7). In addition,
S, b, 6 and p air "rcu rcd to obt.al th;e value of

All ta'blec aLrd f urce (except. Fi .urc.S n, and ) nen-tion-
ec6 :1 thi.s cha 'tc:r arc L;ro'uped i. th.e Apenri::.









Al and B1. S is the uniform tension applied in the direction

of the grain, and a and b determine the ellipse. The values of

V1, P and E are computed by means of equations (20), (29),

(30), and (22). Constants evaluated for Sitka spruce occurring

in the expressions for A1 and B1 are tabulated In Table II.

Values of A1 and B1 for several ratios of a to b are given in

Table III.

The ratios of the stresses computed for various points

in a plate containing a rigid circular disk to the normal stress

at infinity are given in Tables IV to VII. These ratios and

the ratios computed for the same problem with an isotropic

plate, where 0, Poisson's ratio, is tal:en to be 0.3, are graph-

ed in Figures 5 to 10.

The ratios of the stresses computed for points in a Sitka

spruce plate containing a rigid elliptic disk (b : 5a) to the

normal stress at infinity are given in Tables VIII to X. These

ratios are graphed in Figures 11 to 15.

The ratios of the stresses computed for various points In

a plate containing a rigid elliptic disk (a = 5b) to the normal

stress at infinity are givenn in Tables XI and XII. These ratios

are graphed in Figures 16 and 17.

The ratio of the shear stress computed for points along

the boundary of a rigid elliptic disk (a : lob) to the normal

stress at infinity are given in Table XIII. These ratios are

graphed in FiGure 13. We find from equation (122) that the









normal stress Xx is .'4435S at the point (x = a = 10b, 0).

It ip interesting to compare the stresses in a Sitka spruce

plate containing a circular hole of radius a obtained by Profes-

sor Smith and the stresses in a Sitka spruce plate containing a

rigid circular disl: of radius a obtained here. Throughout a

plate containing a circular hole Y1,,1 is less than 0.22S and
J
throughout a plate containing a rigid circular disk I, I is

less than 0.03S. The manximun normal stress for a plate contain-

ing a circular hole was found to be X, equal 5.34S at the point

(0,a) while for a plate containing a ri.ld circular disk the

maximum normal stress was X. coual 1.23S at the point (a,0).

The maximum normal stress in the latter plate is smaller and lo-

cated at a different point than in the former plate. In fact,

Xx is practically zero at the point (0,a) for the plate contain-

ing the rigid circular disl:. Comparison of the points at which

the maximum shear stresses occur does not show such a difference

in position. The maximum shear occurs along the boundary of the

disk and the edge of the hole at about 770 and 780, respectively.

At the edge of the hole the maximum shear stress is larger than

the maximum shear stress on the boundary of the disk. The maxi-

mum shear stress on the edge of the hole is 0.71S and on the

boundary of the disk it is 0.1S.

From these results, it is probable that in a plate contain-

ing a circular disk or a plate containing a circular hole, for

S sufficiently large, failure in the plate vill occur appro;:imate-





57



ly alon. the line y : asin 77 o : 0.976awith the crack be-

GinninS at the dis1: or hole. However, higher values of S will

probably be required to produce failure in the plate contaiinin

the 1rlid circular disk than will be required tc produce failure

in the plate containing the circular hole.

It should be noted that the normal stress X.. comnuLed for

the point (a,0) increases as the ratio of a to b increases. For

t!ie ratio of a to b equal one-fifth,X.. at the point (a,3) is

1.03S and for the ratio a to b equal ten, XX at the point (a,0)

is 3.443. The absolute value of the naximu-n shear stress on.

the boundary of the disk increases as the ratio of a to b in-

creases, and for the ratio of a to b equal to or Greater than

one, the ma-imurm shear stress occurs at smaller values of -G- as

the ratio increases. For the ratio a to b equal one-fifth, the

maximum shear stress is approximately -0.D4S at -e- equal 700,

wh'le for the ratio a to b equal ten, the max:iru-7 shear stress

is about 0.44S at -- equal approximately 240.

On the basis of this analysis, we conclude that a large

plain-sawn Sitla spruce plate containlnc a small rigid elliptic

disk, such as a knot or a steel plu., is probably weaker than

a hoimoeneous plate but stronger than a plate containing an el-

liptic hole of the same size. Further, the orientation of the

major a::is is important. We will expect a plate containing a

rigid elliptic disk vith the rlajor axis perpendicular to the

direction of the .rain to withstand higher values of S, where





-,0



S is the uniform tension in the direction of the drain applied

at the edces perpendicular to the grain, without failure than

a plate containing a riJd elliptic dis: of the sanm size with

its major axis parallel to the direction of the -rain. The

ratio of a to b is important. As the ratio increases the

maximum normal stress and the absolute value of the nas.:nmtn

shear stress increase. Therefore, we expect a lar-e late

containing a ri.id elliptic disi: vith the nmaor a:-is parallel

to the grain to withstand higher values of S without failure

than a similar plate where the ratio of a to b is Zreater.

































AF PT-DIX



















TABLE I
ELASTIC MODULI OF IITIKA SPRUCE*


a-7
LT .44

EL 1,679,330 Ibs. per sq. in.
FT 76,OD" "

OLT 112, It


*The elastic modull of Sitka spruce are aver-
age values obtained from a larlie number of tects
made at the U. S. Forest Products Laboratory.
















TABLI Ii
CO:-STA::TS E'.'ALUATED FOP, SIT;A SPRUCE*


).0213
2.163


l.l9C
1.615



0.35-'-


12.26


KIE
P6


/6/E





/+ +
(-I p I
)+196 6;


K






K6
'9 1


!1.257

1.?31



2. .6
":, ) '.


*Thrr x and y axes are taken alone the L and T ax-s of s;,'n-
.nl try, respcctively.


















T 'ir :I11


THE. *:,' ~ -. J' A :" .. 'L 1 VALU TF i.

":" S:T:- ... *FUC"
,:-,r,5-" '.


- C.





Sb-


- I -



Si


1 ..


-. 7 ,



VA1 Lu a ,? ".. . .",":-h h > -J., -"}.'L;: a 's



J ,. . ". ... ..
.7 . '- ,

-. . *;- .1 4. J ...
F r -.-,
U.: -k".L:A '1 II,*


1 O.2 -



. 7


-a "^


1-.


I-. 1


-. L -






" 1 .-
c --U


I ~_--------I----- --- _


----------- --


me










TABIF V

VALUES OF THE IIORIiAL STRESS COI;.FOITETS AT POI::TS
ALOIIG THE y-AXIS FXTT.RIOP TO A RIGID CIRCULAR
DISK OF RADIUS a ]ITH CEirTER. AT THE ORIGIN
FCf A FLAI::-SA3Ai PLATE OF SIT7:A SPRUCE


X. Y,
yS S
s "S

1.00a 0.31) +0. 216
1.1Oa 0.7L?
1.25a .183
1,50a .939 -0.003
o.S --3.0072
2. 3a 0.961 -0.0072

4.o0a '. 99i -C.?0
5.O33a 0.?994 -0.0011


TABLE Vrl

VALUES OF THE SHIJAR STRESS COMrIPOrEiT X., AT POII:TS
ALOI:G Tlh BOUNDARY OF A RIGID CIRCULAR DISK
OF RADIUS a WITH CEITTER AT THE ORIGIN FOR
A FLAIH-SAi.'i PLATE_ OF SITATi SPRUCE




S


i10
20
0o
400
50

-0
600


8OrO
50
30
050
I~j


0
-0.0004
+3.0310

+0.0198
+0.0442
+3.0341
+0.1 48
+0.1525
+0.1467
+0.0959
'oog3


-- ----










TABIJ VII

VAL-ES OF TIE SHEAR STRESS COI;PO?;ElT Xy; AT
POINTS ALOI;G THE LINE x = 2a EXTERIOR TO
A RIGID CIRCULAR DISK OF RADIUS a
WITH CE'NT'E AT THE ORIGI: FCR A
PLAIAH-SA]i PLATE OF SITKA SFRUCE




S S

0 0
2. 53?93n +93.279
0.7500a +3. .370
n.9659a +0.?34S

1. rO)a
;. C030a +0.0534
1.5000a +-0.:390
2.?000a --.5311
3.0003a --0.331


TABLE VIll

VALUES OF TE HIORKAL STRESS CCMPOiEI'BTS AT POINTS
ALOITG THE :-.AXIS EXTEP.ICR TO A PIGID ELLIPTIC
DIiK (b -: a) WITH CE]ITER AT THE ORIGIN
FOR A PIJiIN-SAWIT PATE OF SITKA SPRUCE



x ----
S S

la 1.0230 0.0216

2a 1.0316 0. 174

1J vk=
-a 1. 374 0.01i7

4a 1.0363 0.3107

5a 1.3368 0.0092









TABLE IX

VALUES OF THE NORMAL STRESS COMPOHFEITS AT POINTS
ALONG THE y-AXIS EXTERIOR TO A RIGID ELLIPTIC
DISK (b = 5a) WITH CENTER AT THE ORIGIN
FOR A PLAIN-SAWII PLATE OF SITKA SPRUCE


xx y
Xv
S S

5.00a .nj5'; -J.3759
5.05a + 0.7113
5.10&a +0.959
5.2) a +O. g 309
c.0a to.9611
5.60a +0.93666

7.]0a +0.)390 -.'7
,. )Oa +0.92 -. )071
9. 0-a *0.91149
13. 00a .?54


TALE X
VALUES OF THE SHEIAR STRESS COf.!POTIEIT X,. AT POINTS
ALOI:G THE BOUNDARY OF A RIGID ELLIPTIC DISK
(b 5a) WITH CENTER AT THE ORi;I:I FOR
A PLAIN-SAWIH PLATE OF SITKA SPRLCF


'0O

hoo

S0


oO~


JD.143
-]. )071
- J.D11J

-,). 21
-' .j~L4:.2
- C .21'.-
J21?1


-- --


X,

S








TABLE XI

VALUES OF THE T;ORMAL STRESS COMPOIIETS AT POINTS
ALOIIG THE x-AXIS EXTERIOR TO A RIGID ELLIPTIC
DISK (a : 5b) WITH CENTER AT THE ORIGIN
FOR A PLAII--SAWVI PATE OF SITKA SPRUCE


X,
xx
s

la 2.21 -

2a 1.2340


'a 1. 3515

5a i~. 33.


TABLE XII

VALUES OF THE SHEAR STRESS COMPONTEIT X,, AT POIITTS
ALOUG THE BOULrDARY OF A RIGID ELLIPTIC DISK
(a = 5b) WITH CEI'TER AT THE ORIGIIN FOR
A PLAITI-SAWT; PLATE OF SITKA SPRUCE


X ,

S





0 0

rO0 3.2124
70 .1c5
930 o7 0
9 0 0_______


















TABLE XIII
VALUES OF THE SHEAR STRESS COfr.POI'iETIT X.. AT POINTS
ALONG THE BOUTIDARY OF A RIGID ELLIPTIC D-IS:
(a = lob) WITH CE-:TER AT THE ORIGIII FOR
A PLAI7;-SAWI, PLATE OF SITKA SPRUCE


X ,
S




15 0.r '"74
c,0 n.^).423
4 0 0. )0l1

c0 : ",
300

DO ? .1.. o
,0 0. 1224
90

















1.6


1.5




U) 0. E-- E--
1.3




0 0 \



I 2a 30 40 5
















disk of radius a with center at the
orligin for a plain-sazn plate of Sitka
spruce and for an isotropic plate.
sprue? and for an 1 so tropic platf-






9








+ 0.5


o0 +0.4

moo
O Tf. 01
m +0.3

0r 0. E- E-1
.4E-_ +02




0 M \
o21 SITKA SPRUCE
L4 ISOTROPIC
0 U Cp

Ia 2a 30 4a 50
(X)











Figure 6.--Variation of the normal
stress component Yy at points along the
x-axis exterior to a rilid circular
disk of radius a vith center at the
origin for a plain-sawn plate of Sitl:a
spruce and for an isotropic plate.






70



1.0



En 0.9
E-4
S 0.8 SITKA SPRUCE



S 0.7

E /--ISOTROPIC
W :E 0.6
00
zp

o r, 0.5
ur..


S0.4
W:
o:


0.3

:E-
. 0.2





0c -. 0.1











Figure 7.--Variation of the normal
stress component XX at points along the
y-axis exterior to a rigid circular
0 C 3 -." 0 .1 ------------------------















disk of radius a with center at the
origin for a plain-sawn plate of Sitka
spruce and for an isotropic plate.

















+0.3


+0.2


LEI +0.1

n SITKA SPRUCE

". 0 -x

o> > _I SOTROPIC___
o -0.1
E- --, E-


co h0 -0.2
Ell rL
< 0 M' E-
-0.3
Ia 20 3a 4a 5a
(Y)











Fiurce S.--Variation :f the normal
stress co-.!onLe.t Yy at points alonZ the
y-aris e::tcrlor to a rigid circular
disk of radius a with center at the
origin for a plain-saun plate of Sitka
spruce and for an isotropic plate.

















+0.5 -


0. 0 +0.4
0 2-
En E-C V- /


En C\
E-AI-E-- +0.2 -
- z - +0.3 "
[ m0 1 TROPIC




co m +0.1
E- E-
o0,--/ /- SITKA SPRUCE
;_1 j 0 __ __
- Oi V- 0
E =-: r. _
0.1
0 100 200 30" 40 500 600 70 80" 900
e (DEGREES)











Figure 9.--Variation of the shear stress
component X at points along the boundary of
a rigid circular disk of radius a with center
at the origin for a plain-sawn plate of Sitlka
spruce and for an isotropic plate.




















r.






P:OOH
E-




0: c i
EO r-, 2




eCn Ern .
S0 0 >





E- E-
?- c4 V
co L


Eo- S:M(


+ 0.4


+0.3


+0.2


+0.1


0


-0.1


-0.2


ISOTROPIC



SITKA SPRUCE






) la 2a 3(
(Y)










FiciLre 10.--Variation of
the shear stress component Xy
at points alonj- the line
x = 2a exterior to a rigid
circular disk of radius a with
center at the ori in for a
plain-sawn plate of Sitka
spruce and for an isotropic
plate.


I

















+1.3


SE- +1.2

[.-p, n
E U +1.1
<, SITKA SPRUCE
OX
E- +1.0
0 C' E- x E-
050 +09 ------------------------
o o 0 +t ..

+08
la 2a 3a 40 5a
(x)















FiCure 11.--Variation of the normal
stress component X, at points along the
x-axis exterior to a rigid elliptic disl:
(b = 5a) with center at the origin for a
plain-savn plate of Sitka spruce.






















+0.2


+0.1


0


-0.1


-rI


E E -n
: t-4 0 <
E- 0 E-



c >-i X
z .
E-4Z>1

ZO
0 0 C;z

0 < E-


Figure 12.--Variation of the normal
stress component y at points along the
x-axis exterior to a rigid elliptic disk
(b = 5a) with center at the origin for a
plain-sawn plate of Sitka spruce.


-0.21 1
la 2a 3a 4a 5
(X)


SITKA SPRUCE


+0.3i----

















CD
0 +0.9


c 4. SITKA SPRUCE
E- I +0.8


+ 0.7
C-i
Z +0.7
EH4-i
EO




Z <
0 +0.6
CrC




-cn



o 0..













50 6a 70 80 9o 100
E(Y)
S+0.2str
0
0



O- t





(Y)

Figure 15.--Variation of the normal stress
component X,. at points alon: the y-axis e;.terior
to a rigid elliptic disk (b 5a) with center at
the origin for a plain-saun plate of Sitl:a spruce.
















+0.2.


SITKA SPRUCE


(Y)


Friurip 14.--'arieation of the
norn:ial stress corQiponnt X,, at
points along the j-azic. c:.torior
to a r-ig d ellil'tic dil.. (b = a)
wit;i center at tFh ocriin for a
Ilain-savn plate of Sit.:a spruce.


1C >


S0 z.



i E-4





O z
<)< t
0 -I 0

[JI E-


+01


0


-01


-02


-0.3


1
















+03



+0.2


+0.1



0


-0.1



-0.2


-03


00 100 200 300 40" 500 600


700 800 90


e (DEGREES)















Figure 15.--Variation of the shear stress
component X,r at points along the boundary of a
rigid elliptic disl: (b = 5a) with center at
the origin for a plaln-savwr plate of Sitka
spruce.


rx] E--
o
M E
3:o
E- E


E-1






0
>X
_O


0 0H
&,L~


SITKA SPRUCE


J














H 2.4
X


S2.2


:F. En .
c0 4 l 2.0

IH



cn
Sr ~ 1.6


0
Lz2 E- 1.4





(n \
rz E n SITKA SPRUCE
o0 7 a
10 0
iO
o o z 1.2
E- W
1.0

Ia 20 30 4 5a
(X)














Figure l..--Variation of th] nor al
stress component X,: at points alun,- the
x-axis exterior to a rijid elliptlc disk
(a ',b) with renters at th,- or. -in for a
plain-sawn plate of Sitka spruc-.






















u
',->

E--1 -

- O


o a-i




. O -OII
<;

Eo'0 o
<< b ir I*
0 PC
>. 11 r


+0.


+0.


+0.


+0.


0


-0.


+0.51 1 1 --I I I


4


3


2

SITKA SPRUCE

I -


00 100 200 300 40 500 600 70 800


e (DEGREES)










PL:juri 17.--Varlatior of the s- ia' s tress
_onponent X, at points alonC the buundar of' a
-lJd clliptic disk (a 5) withi ccrtrr at
the origin for a plain-sain niato of Sit.ka
since.


900


I




















M-4
I 11 E- -- ^
Z 4 >- +0.4
0 W- F
U c1 om

S+0.3

, a) 0 0 SITKA SPRUCE

0n o 2+02




E- 0f E n

0 0 0 _ru 0
02:0 C4 o o 2:
-0.I ----
00 10 200 30 400 50 600 70 800 900

e (DEGREES)











Fu:'1. '.--Vari'ation of the shear strr-ss
c'Jrnir.',.'snt X.- al. points alon;r the boundary of a
r'iiC ell)i:itic cisl: (a lob) with center at
th orl,--Jr foa. v plain-sawn plate of Sitk-
S 'nru,. C, .














EIELIOGRAPIHI


ir:;,, G. B. report of the E:Z'tish A'sooiatioin for the
Ad'.- .cemcnt of Science, 12C'.

Love, A. F. 1. A T-l' atLs on tle :lat.c ;t.lt Theor"- of


"'arvc }:. :. I -:, z .. : Fl; c v d u.ic': c .nifcr.: or Co.:-
co.tra L s. li o!, ._col : > C T Ot SCS
cr~ L.c. : s r.t. t: lI r or, T c 1 L .: "r -D

: '"T .... L c 1-- c 1.12, larc ,



1
r- C C -'

1 0 1,- 1: tr St t ; a
.Jor.s cr'c'c L i"-1.:. ._ ac ::3 r .-a- :; ,a '35 n,
-,-6 o i: 'r.. .--, ca-c .. ..: Prodc'-t L -rato o,
Ezp-urzt :o. El;.,7, I;4';.

S3rith, C. Efect Lf Ei..:t' c r Cilrcuiar lioles onr the
Str.-ssn D:S r.uaL o r PTtTtr OT 'o*^c!r~bYThUWThr~ -
sic-:-rec- : :-o "" ..i rials. :-. i3 oi-. 1'cor.s:..
TI.TC ,Lice .o.s- s~ Plouc s L. or.tor: -, crort 7o.


o:l l :of r, 5. .a,.:ei.tlc .l T"L or-- of EIast ci -t
( i-n 'eojr't1.c eoturo Po0es '.-Cr. at i.ro-:n Ln... ltvr
1-!41 Su'-L':C:r Scnsix. for Ac-vanced Instructicon and R -
scarc'h in ::Cc.Ia. cs). P- i,:V'lci, nlioc- Islanc :
r .... "Jr li.V :i Gra..ua e OfcC, 1_'" .














nThe author ;as born S-tc.:Lt'er 2'., 1:1 at Ca.'rrll,

r!issour- In 195 he was -raduascd from Central IllJh School.

Cape Girardeau, Nissouri. He ;ac racu.atedd in 19.0 from

Southeast Missouri State College with the dccree of rachr-]o:

of Arts. He att.enin the ban.t in the barLt ie tren highest

ranklirn freshimne. anc. also the bane icts for the upper f.i'.? pc:

cent 'jC the JnLcr anrd Seni'r:' Classe-c ver. b' the Aercan

Ac: scczt ion of Uni ':e s t,- F'f, s.o' CU oon r.aduatlon, !.9 re-

ceived '.n .apoiit:'.ent as a _raacate assistant. DE oart.:'..t of

P.I:'-C cs, Sta.te Unriv 'P rs it; of I r a. ThT drc IE' of r!t:..rr! of

Science F Jas coferr: -r. June .,:d. .- t.-'r. accepted a Sil-

t-on at th.e SurnKcf. Ln.'c0ratc'r of E. I. cu Porit Ud ',-r.'"

inc. tIe "oi'1 c t'.- -:' I-- 'r. -n vac .-aitea fr-o

,:'.'. l AerIal ::a.': at : 3 Scho C. :. .2' ...:, ,or c '. .. !' i c
n I t7.,truct' ::' .-'.. l .t: :- : s- : -' .! -. J .r._,_" .. P -c ,:,

anid r" a rTa :'C,,o :.c to the:-t r oaJ: of I eu-tena t. Aft-r t i, .,

r.. retu-.rnc to' s- d-, aL L -. -Sta "-. .' t of Iw '.

h'e : s n3 t-2 ; t..:.7 C.:I-it -r Ii th- ol-- c' L ~

S1 Xi as as1 a soclat. '.. -1' I" ...: 21 he tran-

fr.rr-,u to t... Uni '-er3sIt-- of Floi'"da. i !" il ":1 a :rac l t llv-

s ..I fo: th'-! :." '-_ -'l, 1'-- H- :s r :rlo;e:. of th, '"! : L i' -

c, 1 A.-s c atlon! of A ',, r.ca a:', t'I Anrica' :at c:.-.at caL

S 2 c t--.


LIO'RAPHICA.1L SIT'TCi


















T..: cID rt -n *-.s .*re t.z i w i. r the c lrcct '.o:- of

t.:. C c.'-2..r .' c2..u c t.u's i r'c y;i '0"- C '-.' tt c, an-

; .. ". .'ll .T' '.-0 A .? Co: '. I, t as.
n r ... ..
t..'; ,."r. t .. C .q u .. ,etc -, t. .j.. s C '- ;- t.t r,






P.1 c..I -









J .ne. 5, --.












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