AN EXPERIMENTAL INVESTIGATION OF THE EFFECT OF LIP OW THE COLUMN"" STRENGTH OF
ROLLED SHEET ZEE AND AJ.~GLE SECTIONS
Thesis submitt ed by
Lt. Jeremiah C. Hege and
Thomas Goebel
In Partial Fulfillment of the Requirements for t he Degree of Master of Science
California Institute of Technology
The authors ~Iish to express their thanks to the Consolidated Vultee Aircraft Corporation for supplying the test specimens used in this program. Further thanks are due to Dr. E.E. Sechler
of the Guggenhiem Aeronautical Laboratory for the suggestion
of the topic, and to both Dr. Sechler and Mr. Roy A. l-filler of
TABLE OF OONTENTS
Introduction •
Preparation of Test Specimens •
Test Proceedure and Equipment
T est Spec imens •
Angle Section Test Data .
Angle Section Ourves •
Zee Section Test Data
Zee Section Ourves •
Typical Strength-Weight Ratio versus Lip Ourves •
Results and Oonclusions •
1
2
2
)
7
.10
)1
)6
• 66
[image:3.551.58.425.97.332.2]llITRODUCl' ION
The hat section used by certain of the aircraft companies
has been quite l'iidely accepted as the most e.:fficient compression
carrying member when stiffened by a sheet. However, due to the
difficulty of inspection and corrosion control, which is particularly
important in naval aircraft, the Consolidated Vultee Aircraft
Corporation has for some time been using open angle and zee sections
formed from dur~l sheet. For this reason the members of the
engineering staff of Consolidated VulteE became interested in the
effect of lips on zee and angle sections such as are frequently
used to stiffen panels occuring in the compression side of sem
i-monocoque structures subjected to bending loads.
This paper is an attempt to det ermine experimentally the optimum
amount of lip for the sections under consideration. Sufficient tests
are made to locate the column curves for these sections, and from
2
-PREPARATI0r OF TEST SPECI lENS
The test specimens were formed of commercial Alclad 2480
(AN-A-l) sheet on Yoder rolls and treated in the following
manner:
1. Heat treated and stretched ~~ to 24RT condition.
Specimens in this group are given plain numbers for
identification.
2. Heat treated and stretched l~ and artificially aged
to 24s-T8l condition. Specimens in this group are
given numbers followed by the letter "A" for identi
-fication.
). Heat treated and stretched 4% and artificially aged
to 24s-T84 condition. These specimens are given numbers
followed by the letter "B" for identification.
Various shapes and thicknesses were rolled (see table on
page~) and then cut to approximately the desired length. The
ends were then cast in ~ood' s metal and milled flat to final
length.
TEST PROCEEDURE AND EQUIPMENT
The specimens were tested in compression between flat plat es
mounted on both the stat ionary and moving heads of a )00,000
pound Southwark-Emory hydraulic testing machine located in the
TEST SPECIMENS
Since the test specimens were formed in a production shop
from commercial sheet, the tolerances were of the same order
of magnitude as is customarily encountered in actw,l aircr8 t
nroduction. The overall dimensions of the specimens were found
to be as close to nominal as could be measured, but variations
from nominal in sheet t ickness of ±O.002 of an inch, and in
length of lip of ±O.OI of an inch \-Iere not uncommon. The bend
radius \-Ias one-eighth of an inch for all the test specimens.
The section properties presented in the following tables
4
-S
EC
TI
ON
PROPE
R
TI
E
S
O
F ANGLE T
EST
S
P
E
CI
MEN
S
~
t:C
T
Spec W t C Area t>Min
1 2.00 .053 0 .209 .413
2 2.00 .052 .20 .223 .444
3 2.00 .052 .30 .236 .454
4 2.00 .050 .40 0233 .474
5 2.00 .072 0 .283 .401
6 2.00 .072 .18 .299 .423
7 2.00 .072 .25 .309 .438
8 2.00 .072 .40 .330 .469
9 2.00 .087 0 .341 .400
10 2.00 .089 .15 .359 .414
11 2.00 .088 .25 .373 .433
12 2.00 .1213 0 .4706 .399
13 2.00 .1215 .25 .503 .424
14 .75 .031 0 .03403 .182
15 1.00 .038 0 .07976 .204
16 1.00 .063 0 .:J2~J .239
17 1.00 .030 .1~G .0624 .249
18 1.00 .062
.
a,
O
.1264 .25419 .625 .031
.
r,a
.0405 .162ZEE SECTION with LIP
Section properties are tabulated on the following page.
6
-SECTION PROPERTIES OF ZEE TEST SPECIMENS
Speo B C t Area .f'Min
1 1.25 0 .049 .209 .288
2 1.25 0 .073 .308 .279
3 1.25 0 .085 .356 .284
4 1.25 0 .122 .499 .276
5 1.75 0 .050 .265 .364
6 1.75 0 .071 .370 .356
7 1.75 0 .089 .461 .356
8 1.75 0 .122 .621 .359
9 1.25 .15 .048 .199 .294
10 1.25 .25 .049 .211 .314
11 1.25 .30 .071 .304 .370
12 1.25 .25 .09l .375 .335
13 1.75 .12 .049 .264 .391
14 1.75 .25 .048 .267 .377
15 1.75 .35 .049 .281 .473
16 1.75 .10 .071 .379 0366
17 1.75 .25 .072 0392 .366
18 1.75 .40 .071 .408 .462
19 1.75 .10 .089 .470 .370
20 1.75 .25 .088 .472 .369
21 1.75 .25 .125 .649 .353
22 .75 0 .050 .164 .180
23 .75 .20 .050 .162 .193
24 .75 0 .072 .232 .174
25 .75 0 .088 .280 .173
ANGLE SECTION TEST DATA
Length .. 16" Length • 32" Length • 48" Length • 64"
Spec. p U P ( ) P ( J P
cr
1 2,360 11,300 1,695 8,110 1,130 5,400 790 3,780
2 2,790 12,500 2,160 9,700 1,390 6,250 1,180 5,300
3 3,340 14,200 2,680 11,400 1,650 7,020 1,080 4,590
4 3,630 15,600 2,320 9,975 1,940 8,350 1,310 5,630
5 4,870 17,220 3,130 11,060 2,120 7.500 1,500 5,300
6 4,930 16,500 4,260 14,260 2,410 8,060 1,910 6,400
7 4,950 16,020 3,590 11,600 2,765 8,960 1,810 5,860
8 5,750 17,400 4,490 13,600 2,930 8,880 2,110 6,385
9 6,570 19,300 5,055 14,850 3,100 9,100 2,520 7,400
10 6,390 17,800 5,000 13,920 3,380 9,430 2,330 6,500
11
.
.
.
. .
...
5,350 14,340 3,640 9,750 2,790 7,49012 11,600 24,600 7,870 16,750 6,485 13,700 5,520 11,720
13 11,000 21,900 9,820 19,750 7,360 14,660 5,400 10,750 .
Length
=
7.5" Length=
15" Length=
20" Length a 30"14 660 19,390 400 11,760 310 9,110 150 4,405
15 1,070 13,420 770 9,660 550 6,900 430 5,400
16 2,940 ~~-,SOO 2,200
11
,
a
OO
1,140 1.,2.:mO 1,110 1!.~017 910 14,600 770 12,350 590 9,460 390 6,250
18 3,090 24,450 2,600 20,590 2,160 17,100 1,430 11,300
-~-ANGLE SECTION TEST DATA
Length .. 16" Length .. 32" Length • 48" Length I I 64"
Spec. p <r p c:r p c:r p (j
lA 2,780 13,800 1,710 8,500 1,280 6,355 870. 4,320
2A 3,160 14,860 2,160 10,150 1,290 5,800 930 4,175
3A 3,800 16,150 2,710 11,510 1,640 6,980 1,080 4,595
4A 4,000 17,200 3,190 13,700 2,080 8,950 1,240 5,330
5A 5,070 17,920 3,810 13,470 2,100 7,430 1,520 5,380
6A 6,130 20,500 4,110 13,750 2,380 7,970 2,200 7,375
7A
.
.
.
.
.
.
...
4,130 13,060 2,890 9,250 1,680 5,3708A 7,390 22,360 4,904 14,880 3,070 9,300 2,240 6,810
9A 7,530 22,100 5,550 16,300 3,050 8,950 2,640 7,750
lOA 8,360 23,300 5,890 16,400 3,620 10,080 2,780 7,750
11A 8,480 22,300 7,260 19,050 3,810 10,000 3,020 7,930
12A 14,000 29,000 16,000 33,000 7,180 14,820 4,920 10,170
13A 14,200 27,750 11,800 23,000 7,650 14,920 6,400 12,500
Length ... 7.5" Length • 15" Length .. 20" Length
=
30"14A 710 20,850 410 12,040 280 8,230 150 4,400
15A 1,500 18,800 920 U,530 710 8,900 390 4,900
16A 3,400
2'1
,,
00
2,620 21,200 1,8201
4
.
6
00
1,330 le;~17A 560 8,990 920 14,750 640 10,260 240 3,890
18A 2,850 22,500 2,360 18,690 1,930 15,280 1,650 13,050
ANGLE SECTION TEST DATA
Length • 16" Length • 32tt
Length
=
48" Length=
64"Speo. P 0 - P 0- P
cr
P ( JI-B 2,920 14,000 1,640 7,850 1,120 5,350 900 4,300
2-B 3,160 14,580 2,460 11,350 1,370 6,320 870 4,015
3-B 3,530 15,800 2,220 9,980 1,530 6,500 990 4,450
4-B 3,820 16,400 2,750 11,800 1,710 7,350 910* 3,910
5-B 5,400 19,100 3,480 12,300 2,020 7,140 1,540 5,450
6-B 5,900 17,650 3,740 12,500 2,280 7,650 1,850 6,200
7-B 6,470 21,000 4,010 13,000 2,570 8,325 1,900 6,150
8-B 7,530 22,800 4,970 15,040 3,000 9,080 2,020 6,110
9-B 7,730 22,700 5,100 14,970 3,230 9,480 2,490 7,300
10-B 7,640 21,250 5,480 15,290 3,490 9,730 2,600 7,250
11-B •••••
.
...
5,460 14,650 4,190 11,330 3,030 8,13012-B 13,800 29,100 9,960 21,000 6,090 12,820 5,600 11,800
13-B 14,400 28,400 11,450 22,550 7,460 14,700 5,790 11,400
Length - 7.5" Length - 15" Length - 20" Length - 30"
14-B 850 25,000 400 11,760 •••••
.
...
190 5,58515-B 1,480 18,570 910 11,400 690 8,650 380 4,760
16-B 3,640 ~9,400 2,610 20,.,00 1,340 10.~ 1,370 U,,~
17-B 1,380 22,100 990 15,880 690 11,060 420 6,740
18-B 3,030 24,000 2,540 20,100 2,160 17,100 1,410 11,250
1 -1
0-.: .'
J'j.
~"
.
·:·
·
.~r---
.
-.
-.--
, l -r--·:·
'1'·
:-
TI'~
- " ,,
.
' t
t
,
I , t~
---I····
;
r -1 ...• -•.
I"--1-l..
-
'
··1·_·-
",
-t--"-' 'j
.: ;
~:
j • .'~
" .: ( . :I::'",
'.:~:
: ... ,.
·1·
: \.
J
I
: .
t
'"
~..
.
;~.:~.
F
1'
~
~
i· ..~·:
~
. ' . . :r'
T
.
r.;
.
"
t
l 'i
1-. -.
t
':.'. .
..
.
t
;:-T
~-:-r-
'-.:
j:
! . . . --:-'17-~
--
1
[:
l':
r~:;:jJ~
:
o:~
'
4
~1~
:
n
-
.CLf~~r' '.~ .~. r
~
"
-·I,~·
I
-:.
, !t
,
~
. <-~
~
~I
.;
I .. ·· ..
j
...
1·
-~.
~;.;
.
..
~+:
.,.::~:.
' !
I
:-
+-j-
.---.:-.
-
.---...:..---
-
.
~--
--
~
.
'f .' . . .-~
'---1-'
--+-, _ ~ ~ . . ; , , : . . . ' \~ ~; ,"'; ~·'t~~f : . ' .t~·:.~.
::.. ..
~.:.' !t:- ..
:.. "
.~..
'
.
.:
!
-..l "
.
'",
1
..•
J.' .~ ~';.: '/:;:":--l:'''"i
~"i:'"I
L ,:
.
~~
.-+
.~
~
t_~~
'
~-:
:;~.::
:~>f
.
~::r~
:~ :
~
~·
-·
w:
:,·;II.··.~··'.t---
..
'
fi./~-
l
"
. 1 t. :
-s
'~:'...
,',! -·II·.~:::.1
\ '
.. :.':
r'" "
.
iti
'" ..
1 ' ... -. -- ,. -.. . .. -: . .... I • - I." ..t
Lf
_.
+
l.
:._'
v:!t
_.
J
.
.~~~ :~J..~:...1-_.
: . - ; - : . -J ...
_.
"l
i
-
.
-~
..
~. .!~
rl" I·
f ' · - : - 1 ' ,
t
.
J,.I
:.
~
.(
·:
·:t··
'I
'
;
j.:.;
i--
t
;
II
.~_.~_:.
~
.
'-:~"-il
_ ...~
__-i,.:'-: ..
-k--
I ; . . / :;, .i :
t:t
I . :. . I - I' .; i. /' • ,i: .
,
~
,
: !.
·'1 .
.,
i '
! /I
'
;~r.
;-
,
0
·
.
•
~·-i--:I~-r·
~
..
r---I
.
yi
r
i
..
,
-·
i~
·
~t.-
...
-
·
'
...
:~-t"""':':"~·
.~_I
I . . /. ' ).'.~--: ~.
--- ' .. _--... ---' . /~
)-..
..l:.\'
··
l
.
;,
" I - , : . ' I ' .I .
.' :'
l
'
..
:
/
.
I1 I , . ) ,
I - . C . - • I ' . ., I . : ... .' • j
I
. :.
'!
.
-:
1:-
.,
.
I"
"
.
I . / I7
+
t . . . . , ' i ," I .J
I
:. -! , ' :
~~' I
·
~t--
'r---'hT
~h
--;'-~-~'~f-'--! -~l"-
.
;:
/
·
i · ) - ·
.:-
+
\
";
;.
i 1 • ' " : ; " ; ; :1
/
'- . -.
~L
~..
- - . ..-1.-
'
-
.
...j.-~.
1.
-'~-f
-:
~-
..
! .. -....~
.. - .. -- .!.~.-.-. ' \ ; I • • • t 1 / j
, i . . ·
!
.
I ' ! I I , · · ·I
'
I..
I,":
! . ',' "r:
i
..
4 :.I·'
I i :
> 0V
/r
"
.
,
J
/
~ --'i~'
· ..
--r-:-·--~-·-~-l·
~
I
.
·-r-·~
"1-'
-"
t,
rr~-'
-... -:
/
J0
,
..
~I":
'.: ','"
'1
-=t....
,~...
; :
.
l
--
1 .J'"
I
...
' . • I I I , I . .,;.a; :
. '1 . . . - . ' I . I
;-
...:.;,.
'
.
.
j----+
'-:-T
.
--,-
-
-1-- : ; - t · __ ·-t· -.. . ., . . I . I ; .. _ '. '
I ' . '
:
.
_ . . • l : . I • . • - " , ... • . . . . I
..
,I I
.j
.
I
I · I ·i. · · : ., -' ~ , .;. ... ' .~,) , !
~- ~-l.!
j--
~
r-
c
.
. '.
.
.
-.-
'.
r-'''--
0
! • . -,'.1 :
·.:
rJ
.
\
~
.~
.,.
-.,
,,'''j .~
.; i
Hr----
: :'
!' .-L:
L:
'
c
l
' ! ' . _ (>I ' . t..:.i' . . .. 1 , - • I ' ,
.. ---.jC..~.- ;...-- ... - - f-- .-. ,--
---,--r--
--:.:....:--c--
.. ·
1 - ---1"""--7,""
-
.
I , , . .. ' I , . j. i
.
~
-
;
"
r
i'
iL :
oj "
I·
,
i ' :
:
.
I
.... '
!-,-.~-
-
~_"
.
..J--t-.--.-+
..
-:
.-.
+--
--~-~-.-
..
_t ..
,
~
.l
I
--r-""
" .
:
'''!I
:~:
If'+1
'(-1
1
,
- t o
~
. . . 1" . 1~
~
"
•• J • I . t . . .
i
-"T--:- : - ...
-
;--'
--!--:-r"'"
.
:
:,'
----.-,
-~ ~-.1--; .
.j
..
1 .I
I ' 'j -. , i · , - ' . -I
"
.
· ! ' , t 1 ' ·, I : .' . I
f . l I
'- --r
-:
-
1 - -... ,- -;-+
,
-
,
--+
.
---'
.
l-. ~-. -;,.---,;--.1.--. -4-1----,;~--.;..\ - . --f()- ---'<tI, .! . ...: . . . ; . j . . . --, ' \ ' " . . , .. ' . - . . . . . .' ,.
l'
·
.I
~ .. -.
1
I
-T---.--~
,
_ .
-
.
..
--j -~.;- . ~-- ; .. __ 1. . . . :" , I ' . '
I
.
.
-'-
-~
---r -_.
~-.-
..
-
'-!
~
-
~11
1
'
:
1
JJ
+-+'-I--I-,-i-'
j
i'
"
~
u
"
-'.
,.
1
--
'1:
.:
I
r
.,
I
'
:
I.
':
1, i i ' u :
.
.J
'
t
.
.
.
t j I ,·
-
-;
- --.,
-
--
r
I ; -I --. j-''"7- -~-
t
--··-
-
-~--'''--l''----I'
-..
- k ' - -1/"
)1
. ' : I }e-~ ~ i 'I : ,'. I ' , ' : ; . !
,
!
-
I '
.
~- . -
I' - " . .I
.
..: -.
:
I iI
. , I , , :
til
i I;--._j---+---..-c
....
, ._-:_-f--:-
~- -~I;:
j---
- --
- --
...
.
-
~-
-
!-:-:':"I
-:--1'-'-
'-
-~!l-
-
..
~
..
, I ->' ' . I -. . ' . . I ,
I • ' , ' _ .
I
. I.. . " ' , :
• ._. _. __ _ • . _. _ . : 1 . _ " .. ~_.. t · . , i ./I . ,
!
. '
, ;
I"1
: . .
II
I i ,.-- - -_ ... ,--l--~ . .. ' -_·-t----I --_... --'-:1 --. - .
-
r
/"j-",.J
-!
·
t
:::-<~
.
-
~
.. ;':
.
:.: ..
~
.
:
i
'.
,I.i
';
I
I . . : ' ,
J
'
! '
, } ,
I
I- -- ; - • i - ' - - - , -- ' - .. --.-~ .~ - . _ . . -.
'0r-
-
I
"
I t.~
o~ ~
~
"..
., .~
~
-
3
I~
~
...
'"
~ ::!~
iJ
r-~
0 ~
.-~
~
,...
-. !
i ;
I
'
!
I
t'
:-~~~-
I
I--\ ··~I~-
-_
~:/:!
i
I ' I "
I
[
-
~:
.
i i '
.
-'L
·
~;r
-I-
-l-I--l·-:--··
-
--
'-r---; '-
-
-
-,--'
-, -
- -
i
._-
.
'-
_J --
,--t
... --
..
-t---.
--t--"!':---j'-""':- -_.j,
,
.
' !
..',
!
I I . " , II
_
~
I I Ir
<--i~
0
[J ~p
:---1 -~~.-
-~-
.
'::1-
..
,.- ; ..
...
-
i":~t--
'.~.
: -,-,'
,'--t-~-'
!-'
-
-
'
71.-
:
/
I
/
.
J. I I . " M '
)' .. . I . ; . !
J
'
lo!J II ' ' ! ' i
_ •. J .. ___
.!...._ . .
_.~I t .
t • ;-'1"---
-..
-
-
,--'
--;
.
.
,.-
r-: -
-I
.-.-
i .---
;
.
i
-~/-;
-" . I • ,
l . .
V·
.,
.
,.
---..
-I
-~r!1
-~~.
'-'
-i--
·-
b
:Al'-:~~·
(':
'-:
-
I"
I / ./
~
:, ..
---
: ---
-
-
..
!~
--.
-
~-.r__>
f~
!
;
~
-. :
-,
~
-- I1·,...---:
,
(),
'
.
...--t:=
-
I •J; ' . "
~:----~-:~~.
r-:--';
'
/
"J
~
--
:
---!
..
- - - - .-
--1--I .
J'~ ' ~
' -
----
... _ .. '-
,
- -~---- .-..
"
"
I
,
L . .:..
.1_
';t - -
~"---
,.--u
·
.j
'-1
.. _1_
:. .... . . __ .1 __ ._ !
,
!
'T
I
,
--
-
;
..---
-~-..
- ..,
-+- I ..
I
~r-
---, r
I ..; -I
__
~_,I
-
Ii'---
i
!
' k ,
- -t
I '
~
-
-:----,
,-, "
,
!
.
I ! I
• ____ J_
-
.-.
--:- --"
;
,
-:--,
-
Ie.-.
.
--I
I
I-i
.
,
---" -,
~-,
--
-~
!
,
-'j' -_.
-I
" .. I
I !
..
,
,
.
'
~ j ~
I
" I
.
I
J •
, .
_ iT" __ _ I _ • _ . ,
,
.!
, '
,
..
....
i ." ' I " ..
! _. ,
,
"
,
\'
.
;
.. "
'r"
" , , "
0'
~
.-: -
- J-
-
-
~'
:nL
""
T
'
i
jl·:
·.i
: ' !
r
~
.
:.- 1--
':-
I
.1
i
j
i
.
1-"
"
....
-+.".
~
.-;
T
i.
:
:
!:
1
U ' _j-'
~"',
'
.
I
.
;.,
+
.
I.~
_~
, .: I '
7':-+-
T!
,
~
!
_
I
_.
"
I " "!
.
_._.
'
__
~_._
I I...
..
!.
i:"l
!
i
~
.
r -.!,
.
I '"-l'
':'---
.
-
~
.
~
:
/
r
'
,
L.
1
'
" .
'
.
..
..
.
-
I,.
-:-:-:--.
,..
-,---r-.; :,
l
-'.
...
.
'
':
, :
.
..
I-I. , .. , .:r
l-
'II
_
_
. . . , . .
'
I.
1,
~..:
-
.
-+-
~
" - 1i..
,
.
;_.:
"~;::'
t "-,':
~_
.
t~
·-
i-"~·',
-':
!
,
'
I I.
L
I .. ... .'.
1
.
'
.
1 " . .. • II
:
.' i"
~L_L
...•
!
'/-"'" ..
,
j"
".1
' ....
,,;:.
II
.
.,
1
"·c· ,\--C.
i
/-
•.
~'-_'
.
:-"
"··
r
··
"
"
.:..tc+-"
"-,·"
r:-I
.
l
' I
'
+"..
I ' 1 I " " , : 1 '~
- .
j
-
1J
....
~~
:'
·
i..-+:~
.
.
'
1-1=
..
I
'.-
1
... , ..
!
...
j ... , "I
-t
'
-"!"
.
'
-
1-"
C - , . . .':!':?
:
, : .
I,
.
__
_
_
;
.. __
.J
j
I t.J I ' __ ' "
i
'
!'--,
'
r . .-
L:"'
::,
I.,
:
I.
.:
_L
' ,. ,
_.h---
'
, ., .
r
I
·
<>"
,
I. .'
.. 1.:"_!
'
=t
-:-!
-
_.
_
'
i..
,'.
f
:i,:
'
,
:
~.-,""-
..
~
'.
I
':.
. .
I
'I ..
i
I
~i
I
."
;
mi
....
.;~"
...:~
_
1
-
"j"
~_":I'_
.
."
. . , .
".c
,,
"'
...
~,
i.
I '-
---
!
.
i
' .
\
:
.' I
.
.
I i
.. ",--.;
_~
.!
,
I : .. - .-.. I I . /
:'
f
-1
~
•.
'
-... "
.
_
.
I i
,,:
..
~
.,
'
,
~
··
~t--
·--,,
-.
I..
i :
.
y
/
'
....
~
i
"
~
I_
.. ;--/ ;! ,
-
..
..
'
!
;---(
.;
-." _.. 1
~_
, .;-- . - ' - ' _j
------ .. ~
L . .. ' .. _.
! ' 0,. ,
~
. ---_. -.. - ' " _ , .----:--r -....,... , ' . : 1 .- ... ,---<'--;-- ,- - - ;
i
I V
:
~
i i ..:
tV _ _
:'
:
. .
:. -, __
;- c __~
"
: ' .-
, , . .:..----.. I...J ...
~-
, .__i
;
.,..;..-
,
..
" .~-
...--~
~
.. _ .... '1'
I
i
I
-. I
/
I
,
. l
, r -
--,
,
- ) '-'-
-!
"
~--.-
.
. +
1- .--- .
-
. -.-: '
I
---.-.. - -
-I
L
I
_
_
. . -'" .- --
_.-i
,
I
t-L
.
--=-I
i :
- . - . - - --.-- - t -. -- -- - •
. j
. J I
I
---,- _ ___ . __ f
-- ! . 1
I
-"-'
-T
,
:
I .
. . _.- -1 - . .
I
.
_
+-r
- 14.
-.
,
..• 1- . _
I
.-
-
.... . • - 0-' _ .' • _ I t ·I I I
I
! -
-.
,
,
l
i:
~
"
-
+--
-'+-
:-'
-~--:
=-l--:
- .:....--
;
.
.
! ' I ..
I
.:
.1': . ! . ,
I
.
t
..
~ ' 1t -·1 .. ·- -· !-_.
-
--f
'·-
.
..
_
.
L
-
.
..
I . !
I : I ' I
1
!
i
,
.
l•
.. I
7
- - -.. - I
, -j
t
-· . -
-I - ... -r-·~.. -
i
-
-·
-
/
.,
I' . j ; I
I
.
!
/
~
.
.
.
--
!
.
-
.
-i-
.
..
.
.
+--
;.
/ . //
l
.
f
.
'
·
,,
/~
·
/
~
-
'/
~
-
, . ,,/ . ... ;....
"
.
;;.
: ...-<-
-I
.,-'
..---_
.
... tf- - - ---
-
-
-
-v
'
I
I '[
. -. "'1' _. -~.- ... - ~- - -t .. L . I
I
/
.____ -J
,
-r
,
~_:
__
.
:L __
:
,
"
,
r
--r
v
I)
~
~
.,;iJl~ q;
~
I
~
I1
01
0
~
-~
....
Ii:
..
~
...
,
0
T
. . ~~I
,--I
-/6-,
. .1. - ~ 1
~.
I
i
~ I
I.
I
I
,.
I
! .
- -- - i ·
.;
'-+---'--1
I
q
~
~
I.lJ
">''-1 ~
~ 1\.1
.. ~
.
Q:. q:
..
. . I
-
,
....
----<".~~
--(}
•• - T :
-I
t • II
,
.
.-~ ; - - - - , . . -• • - - I ...
. I
,
---
tI
i
-
t-
..
-.\ .... .:... :--_:_ .1 .
I •
•
"
r .'
!
I
I
, !)t
'
/
sd
tI
iJ
.
SfJUlJSTlolJI.
~
I
~
. L ...
. t- ...
I
L
.: .. ,
o
o
~.
Q '
Q
---, -
-- 1
(1-- (1--1
i i ,
-
r-~--
r
1-
-
-1
-
--t
--
--1
,----]'
1-
---
IL-~-
_L , ' t" .
----L
IL
'-
' I
.'I
•II'
-t- -
I. : -
I 1--r--
-
--~I ~~~---.-
--·!---- --I--r---:-t---~-I---:~---t·-·- ~--- --~--
-j --
'
+
:
-!
-+J
\
-:-
.,+
:
I ,- 1--'~"-'I
f,i
,
:
--
j--":'-l-
-
--;-
_
.
-
\
,:
ll
t
-
--
-
:--'-1 I .----.JI-
-
~l~+
I I ' ;-\--~;
I ,- ,: .: -
,
.~
- :: :.
i
4
:
j
~
-
~T;
I
'
n
-
;~
-
:'-:--1
-I·
i
0I
~
r-:
+~~+-
--.--rr
~~
:l!:
-
-
'-71Tl-~
:
'-
!
~:-r
-
r-
rn
~
: -.- -
'-
I ;
1"
i :
tJ
' .
J
- -
-
:
I
--
I
-
.
-:
1I
--
iI \
i - --
I - . 'I '
"
i ' ,-
-
1 _
,
---l--r-:~
___~_
-+
_
_
:---L_" ---.--'----f--";"~
!
'
-
..
- - -1- -----~!...
1
--.
---:
r:
~'.'
,
'~' I1
-
,
I : -:
-
,
I''_:_,,::,::1.
-'----~
1
1
(
--'1':":
I:
~
-
';~
t
-
!
;i
j-
t';
'-:----'
i
:
IJ
'
r
,
'
:'
o!
~,
-,-
;,
it
:
!
:
-
.
:
'
-
---
-
.
.,
~---'-' ~ ~~---I---:__--!...-'_
-;---.>---..:...---1--- ---L---l --, :
.
1--1
--
:4
-
':
~
-
~
i
-t::
-
~ ~
:
; .\-..:
-:-
-
-:
!
:
!'
t:":~
.
! --..
I
I
I ' ./. . .~ N . . ' , '1 ... ." I 1" .
r
I E '
'
< " (Ij - : --',- - I ' : - , I _ -, i I
I
J-- --r--- -.- - - - ---~----
--
+
-
-
-
.. --
-
--I :- --r---'
-- - - ,
~
't , - "' ... '" 1 , I l . ! 1_ !',!
I..
;;
~
-1
-
~
m
+I
I
'-_
-"'f
L-:---~
:--{,
: --
/:j
I
'j
t , 'I ''-.-'\'J:/
1I
I--- i___ • ___ : _ ,_; ______ . • _ ! _____
+
__ ..
!-I,
I8
_
: i I I I t\l
- I _ ,
, I
;
i i '
, . I i ; - ; , I I ,
I
-T-
-
-+'~---i--_
..
-
j----+:--
r----
--~-
~
--;
!
--
--i
---
+- -
Y
--I'1-: : - ,'-
~
; - .t .. - ; - - I . 1- 1- 'I
/1 /;
. '
:
i
i
,-
--
:
-
-/
,---T-,
'l
:--
--t-'--:--r -.. ; -
~--r----
! --
-t-T
-
-~----
--
f
- --
j-;
:..
:
f
.,
.r
.--
-• I • • • 1 • - I
I
.
'/
I--+--~--f-+-- :-~~-T
· ,;,
I I )~-l~~-~~--J-
-
..
, ---
~--
-
-1L.~
__
~/
__
~_:
-
1>-
-:
. 'I
! I' ' ! :..-
~'
-
i
/
I,
~
-j --, ' . / " , I
! ' i i ! I _ . . Y " ' , .... '
- -
--l---r:---r-
-
---1-
_..l _____ -- -, - - ;":"-7"-~1:
/
-
r
----
.
-, L ' I . . - Y i '
, . ~-1- - 'j' ; --T-' .! ~ ' ~I
.
/
:
;
-II - - 0,: ~ - : ' /'
---t
--
--_
,---i'----~;l~-r---r-----'=-.::
,;.
\-
'- :
/ , -
--:._-.
t
1_ -' -
l
1
~ : _l~,-
I - ~ ,; - '<>'
1 I _ "t...--
~
:!
1 " I , '- _I '-__ I ' - - -
-.S:.t
, _____
;.. --.-
_
----~-_
~ , r-- -,----r---' , 1 • .:.--- • I
\
-
I
;
t ; - -_- ,L
j ,"
~
-
I · J--
-+.---
.~--
--
-r - - - . - -. -. ,- ..---~--
-
:-i
---
---..
-;
: ,
i
!
.
-
\- -
i ,-:-
- _
t--.. ---...,--.. --. t---i - _ . r:
--_~
~
_
J-
I
__
_
.1,__;
!- . - - - 1- , I , I _ , - --- -,
i
!
_I.,
i
- ..
:----~
o
I
,
i
lO
-I
I
"
. . , ..
,
, - - - .
---J, + ' ... __
,
j ,
I , I
I
,
..
I
!.:
.j"
:
; I
I ·
I ' . ' .. ' ...' .. 1 '" .: .. -..•..•
-,._-+
-.
'
...
.
~~'-.. i - .. .
_
... : ..
.
..
.
..
:
...
I i :
I I "I
.
I
' -tJ" ' 1
I·
'.
I ' '"
"f
Ii · ;
:
,
,_.-
jj
;
.
--r..
t..
.
_.
-
r'
...•
-
-
~
_
....
-
_j.. .I •. _ ' . . j •• __ ._ .... ' I :_". i.,- '" .. ' i' " I
I
:
,
I, " 1 >
I
! : , . !
'Ii
I
L
'
> ,I
I ,I .
r.
.
-L
~
_.
"'~
_
'
;
"
'+--.
+_,
_
-1-;- :
. :
I
I
I
I :
~
~
t
,
.
!
.
J.-~·...J
_
:
_·t
l:
..
~
~.
--
..
~
.
;-
._-
_. ;
/
',1'
··_- · · t · · .. - .. - -
i
i
'
h:"~."
I····
I.
,
',I
I
.
!
';
I!
t , ' " : ' L ...i.
__
...
_i..
,
i ' _ L . . . . . 1_._ __ ..
-+-.-
..
-
,
....,
.,
1,.. , ... -- ... t ....
I
: .
~.,
, . ,
!
! j I I~
()
~
.
i
.,
I i '
t
"
"
i
! : /.W) _ I
-+---
"r
--"-
-
-
--.~.---.-
. .
-:-
:,,
-
r-·-r-
_A, - - - . - . -:- _ /1:I·
I :
'
l··j
i
'
,!
:
'
I1/
~
.
~
'-'-'--+-~-rt--i-"
/
,
':/
I l
.,
" .
_··'t'
-
_.L -'
-
T-' --.
j', !
.. -l . It , ' . . , ' - ' T '
I
' ,
'
i /! ' ,0 ; ,
!
! . ! :
i ' , ' I . :,': 1_ . _ :...
.J
_
_
,
__
.:
....
_.1
.
-
-
-L
_~I
- - t-.- ..I
.
, .... c' -- .. - J • -; ' I I
I'
I
I , !
, !
'
"
,
I ,
I I 'I ,
!' '
..
-
'j' I
\
l " ', , \ , : ' ..
.
.
'
-1
' , I
-
...
-
, ... -
,
1. _____ ... _' I ,J-·_ ....
-
r'
"---
) i
'"
' 0 ,---; . . .
~
" I Ii
,
~:!
l'I
;
'~,.!,
- .
~
....
~-
-:
..
't-+·
-
r:
~
·
~
~
~
:-j
--
..
~
.
i
.
I, /: '
I
'"
I ; ' _: .... _.. . _.. /·-·-r--· ---.-;-- - .
r
-
..
'1'-';"
I i ~ : /..
I'
!
,
.- - - ... - . .
-.--
-
--. -'---
.-~---i
-.
!'
~k
-r----~
~
~... '
ir
..
__
...
)
~/
__
..
1-
-I
, ,.
I
..
~
i
•I
-
...
I5//'
: ' ..
;,
-
,
\J:.:~ ~
~ ___'
-
;..~
"
'
.-.
:
.
~
f\( _ _ _ I
,,
--e..
:
...
1
1----o
'!
~
,
.. L ..
"
J
J~cI;/~
_
.
-;. -.
-~-
_.
-;- .... --
l·
--I
·
-
T - .,...• • • 4 . • 1
I • I I
; :. I ' " t
'''
•'1
.- .
•I
..
... +-
---1 . .~
--.-
- ' - - - '-
~
.. . --
-
.
~-~
. . - ' ...-L
..
:
~
:
l
• t t j
I
' "
I~
'
. ; I'
~
i
."~"I
'~'
,i
:
i
-. _-. -
.
-
:,'
__
_
__
1.
,-'
l-:'
-
I '-
-
-.-
,..
t
' : ,
'
~
11 . ..~- .-" " -. - '- 'T- - " - . -'
-I
'I:
'
' !
I- t . ' - 'i - ;. .
..
1_' _-_. -+J!
j .
:
I :'; " -:'1
; .
,.
.
~
.
..
!
, ;
:
--.-l
.
-
t--
t-'
--.~-
I--
,'
-
• •'
--
'
+-
I
".-
I-+
,,
+:
,-.;...-
J
:-.
r
~
'
;
. '
-;
,
' ; '
1 "':,
~
"
---l
--+
-:....-+-
-!;~l!:=l~
-~
_
___
~
.~
,
,-,
--
-
..
r--
'
'1
i
!,. " --- i
, I
i
.. -._ . .
"'
l
h
~
~
:
I
':
; ,-:-
-'
~
i
.
l
·
, - _ .- 4 - - - 4 - -' ' ..:..:' ____
I
:-t
,
~
:...;.~
. ;.
;
:
,
I:,' . .
I
.
i ''
1
'
· ' ·
·
..
-
. :.
.
i---!--+-+-I-.~-'- .. '. ! . : I -' • . '. ' ':' : 'f' . ' .. .
4
-_....
.,
--
..
..
-.. _
, ..
.
..
_I ..... __ ..:
+
.
' '-:"' _
_
+
_'
..:.t-
_i
_
l
:.',
~:
-'-:
_
'!
:
:
.
;:(
.
'
L~
.
:'
~
~,;:
.. -.
I~
T
1 , ' I )' ; . : ; -. .. -: .
l . ' : .~. .. ~ t f I
-
~.
' ..
, I
.
"
"
.
" . . ,
_
___ L
..
~
'
r
__
+-
! :
'~
I'
· ;._;~
:
7
:
"
.
>
~
'r
:
;
I
'
::
-'1'
-: .. '
:,"
:
/
1
;
f
, !
;
,
.
1
'J-:~
--
~ ~ r
-":--:T
-
-
~! "'
~
I
J 'i.
.
..
!
.
,;
,
.
1
.-
I
';
,
t
.
,
;
I
I
.~ 0
I
!!
.
I
t
"
:()_. _.+--_ .• _1.
_~
t' " IJ
.
I! ., ,~- , l' - - , .- ; ... '''r-''-'-+'''''- -..
"
7-
"
; . ! . ," " I
iJ
a
ifi<!.
~ ,
,
'
I
. . :
l
i'i
r,
I :t
I
i'
~
-
~
t..
, .,«.3 ~ 1<1
.,...
~---:-
-
.
:..
..
'~--
'
-
t' , -'-
'
-
"
.
,
"
v
I
"
~
-J
:~li
~
-
~
-:
4-~
~
.
]
·
j-
-
·-
i
'
!
.tL
,
:~
r
l
-
.
~C
;1
J.
'
i
-i
~
,:
i ,
i ,. " " .!
. '.,"".
\
.:
'
j
' .
L
+
~
---- :
I
]
: ;
,
-
..
~-
:
-
+~--t
.
-71-
'
- !~
''''1' -:~
,
- - -....-: - . ' . ;- I __:
Y
_
. ,
.
/~
" /
,.
/
,
'"
j
,
!: - r
ol
l
/
'
,
.
:
____
~i~·
,
·
-I
"'
~
:
:1
_
_
:
:
",
~
.
~'
;
i
'
I
.
'w'.~,
t'
j;f
'
I
: !
<;J
!
,
,
,
1
U_:
.
'
-
--o
r
'--j~''-- -, .... - ,- ,!• ~ I I I "
. . . . : . , I ; I I I " I i . ;
, • 1 ' ,
i
"r ' ~ . , , ' ,. or ! " t - '! ' o·.
-+-
','
1
.
! : :I
.
~,
'
. :
·
i
~~·r·
I
~
:-
r~
~
:F
~
~
!!
-
t
-·
i
-I··
;
i
-
j
-_
.
r'
-
~r-
'
-i- .
~
,
;
'
_
Ii
~
~~
-
!
~
~
I
~
:--
i7
I __
~
0
.
I - - , j! :
~r---
,
I I ,I ' , ,~
.(
I
'
:
,
-.. --
--I----~..-
=;
--
;--
-
-
t-
,,
·
~~'
r
--
,"
---
1'
--
"
--1' ,,
-I ';;':..-_ '
~
- . r~L
'
.
~
!i
'
V1 I j . ; , . . . j
- .. _ .. - r
.
-
-
~
'-:"-'i .-
-t
,-
-
-
-; ----
.
-
-:+
-~
:--
I
,
;
'-' -.
·
~
i·-
..-
-t
-
.
1-'
-
'
r
! , . !
j . ' : . .
,
:
I,
-
,
, .
• l • I . I
• : I ' I ' , . ' . ;
.. -.
··----T
-
'-,''';-'
-:-
-
---j
--
- -
-
,
..
..
~---.·
+-
-
-1
-·
-
·-
,
',
...
" I.~ t · . t I I . t
I .
1
!l' :
.
.
! .,_ .. ___ 1 ___ ' : I
- - --
'--I
_ L
,
-.~
-_ ....l.
-<
," I
--.---- .
,
.
-.
~--•
I
---t-,
_-L __
I
i
_
-
--t--."---t
--+! ~-
--.--t
,-•
,
--.. t
r
-L--i
I I
I
_ _ _
4-
-zz-I
!
L
-
,
--I
-
"-1
:..
I
i
.
.~
..
.. -- - -,;
!
...
-•
~
L
,
r
!
I
.
,
;
-. ,
,
.
, r
---1
i
L_~
"
,
1 __ --1--, .. - I
,
-!
I
J . ' I I
i
-r- _.-
ht
- I ---.1
I
.
I
',+
._
~
~
..
!..1
• I
j
r , ' _
,
,
' ,---- ::1
.
,, I
: : - .. t
. j
1 ,
~
-, i. - - .... - - r
I i "
i
_ _ -1 _ _
I 1
' i. ,
-.,
I;
!
-. i
I
-
.
'.
,
.,
,
1 -, - - -,,-°t • i
,
I
~, -- ,t
I
. . I
I 1 •
I
,
.
I
,
J
I
I1
o
-I
-l
---I
I
I~
-
1
j
-
~
I
-.J'" j
- -
-~
I ,j
I
.
~
_~
I
"
'l
I I
J
-1'
~
.
"
-r
~~
-
j
---.-
'1'--' ;- -: -r
-
~~'-~--I
· - - - .- r-- -..
~
.~
..-
j-.-
~
-
1-
i
··--l-
·
~-
·-.
-+-·-
···
t·
·
··
··;
T'- •·
I i
i ; '
i.
j
.
:t
-
~
"'
f'
:'
;'
.
,
--H"
~
'
i'
.
i·
~
-
-r
-.-_
-:
_._:
_-+
.
.
. _.
_!_
._'+ ...
..
; ___ ,
--1.._:--
.
. _
_
.
_
.
_:...
. , ___
_
._.~~__ ._ \
..
...
: ' i
'
. .'
II
. " , .
i
:
I . . I·
.
1
.
j } :-·
:
1.
\-+', . . . . ! . .•. - , j!
;-
--1
'
--:-:-
+' - -
':
'I
jJt
~.-.
".
!
-
' :
;
_LL
__
"
-
i
':
·r
-.:
~
-
._l 1 _ _ .. 1 . . ...---.
. . ;'
I '
. .:
.:
TI
.
1
..
I j' I·
+.- \.:
.
I
.
'.
I' ~ : . . ' I' j..' . . ' .. .. ~-:-: .. - - I. -;
I·
:
u " ~ !.' iI
j : . : . , -l I : ' I , " .~"'.
: --i----r--
f\._:
l.J
r .-:-- .+-.,--f--; . ".... '--.:.--l_~._ . ~· Ii ::.: ' .
I
j;
~
:!.
i
i
': .
.
.
. .
!--.~
~:~.
.
.
.
-~1:'·
.
"
I!
f]
l
I •
i
;--
.
_
!
~
-~.~,-.-
t~'--Jf'-":--r-
~'
."7 -
~r'-,
.'
-~
::':?-:,
4--
.
-
.
--• --• --• .~
. t~1
. 1 j... . •'
1
, I · i - : :·:':'.1 .
!
tf
~
I
-:
..
i. I .. '." .~ . · ' 1 1 • /'"-'-
-r
~
-~'r--'---'
-
~
;
r~
:.c
,".'::-
.
:
~3~:'1-·
-
-~
I
l
'
i
.
"
.
- . [-
•
.
i
~:-'1':::
'
r
':
1
-:
L-r
-:~
~
-[
~
-
.
·
I.-~
:::
I·
-' .
..~-
.
.
.
~
..
_.1--
~
.
~
~~
--I
I
~~-
.
.
':)-"~-
-
L
:
!._/!
·
1
.
·
~
I
:
'.!
.:
·i.
,
;
;
/
1_
.
__
._ ..
_~
__ .
!
.
I
~
1_~
_._ ...L
_
.
J_
....
. __
-1.
--.:...J ....
~_
:_....
Ia
:
. :
,
. e
~
I )"I
.
!..
;.;.
I ":
1
7.
'I
.
til'-'-'
t
----
T"~~~-+--':
-
t-'
'
-'.'~-
,
~
.~
-+
--
,
,I :o:
+_
y
~/
i
-.
~~
'
I.
;
'!
':' ':'
I"
'~':"
l
'
"
'1 . :
j"~
i
.
y
i
.
/1/
ti~
'-
-
---
-T . ':
fl-~
·
·-T
~J-':-
-
;
'
1
..
·
: \
---
r---'-
-
t-
.~
...
!.
'
yi
I
• • ; . . . . I' . ! : . j • i • ' ; ;
! I ; I . ' ! . I - , 1 I •
. '---+--.
-+---
----:--:
.-.
-
..
-:
:
-
.----l--...+--
-" ','-
r .. - - , . :- . ,: . • _J,' . ' I . , I ; 1
I ,
I . 1 " , I I I• . . , . . . ~ . .,. L ' . / " I .
I " ~ I I '
'1
"
I.
i
.
! . . . . . ~I
I / /· ---'T
-:--,---:--:-'-,-'--
-:
:---
1-
-
l-
--
-:---:
.
-:-
-
,~"./
1 . . . . / "·
.i
:
....
:
...
I . . .• .. !·
/
~
.l
/
'
.~I
,:.
!
. \.
_
.:
__
+_
~_..;
'
j,
'
~
~
:
:
:_ ..
:_~
__
,,
_
j
_:.
:
I" I . . . . . , / . I I
·
.
_
.
',":
1
'
-::
,:
"
;.J
I ;-.
~
-
~:
. . . .! '
/
1 • " ")I ,~ : I :. ; ./
·
-'--:----j . -'
'
k
'"
~
-r"-"
-
"-
-
-
+
-'
-
I~~-
-
l
/
._
-
t
.
-
.. ..
·
.• " j
.
!
.
·
·
·.
~
t
I · · •, • • • . , ' ff - /" I . I i .
• I ~.
-.-.---~--
-r---'---l-
_ .. _
.
..J.-__ .+-
-
---
--~ --~ .--~-;-.. !
.
.
-"
i.
-
~~
-
.
i
I,! . ; ..:
.
~
j 'I
·
:
;
l
-
i
, ' I
I
. . ,
1
.
I . ' , " ;~. ~ i !
-~-1----"-r·---·--t - .- I ' r
-I , . 1
'1'
..,
':
.
-II
'
,
.,
;
.-
,
:
."
·!···I
.
f' l ' .J
i . I . :
!
.
I ' •· - ___ '.L_--r- _l __ • _ _ _ _ '-__ -1--~ i ' I
i
' i
" 1
r
I I I
I' : .
i.
.
.!
-r .--."~--~----~----r---~----+---_1~~~--~~--_T----~---~
! .
"
j
I
~: I ..
I
IS'.
~
.
~~.o
:
\.,X)
i
L __
