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E
byJa Hartley DanielsLe- II Lu
ELIla TO YfE
Plastic esign ~F Multi-Story Frames
E
FRITZ E~~GINEEFHf\IG
Lt\80HI\T()HY L:BliARY
u
Frif:z Engineering Laboratory Report No. 273&54
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273.54
SYNOPSIS
This report presents 78 design charts which are to be used in con
junction with an earlier report describing the Sway Subassemblage method
of designing unbraced multi-story frames. l These charts relate the
lateral load versus sway deflection behavior of restrained columns per- '
mitted to sway. They have been prepared to cover the range of axial
load ratios pip from 0.30 to 0.90, and the range of slenderness. yratios hlr from 20 to 30.
The composition of the design charts is briefly discussed. The
scope of application of the charts is also given along with two worked
examples to illustrate their use in solving restrained column problems
similar to those encountered in the design of multi-story frames.
i
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273.54
TABLE OF CONTENTS
SYNOPSIS
1. INTRODUCTION
2 • SCOPE OF APPLICATION OF THE CHARTS AND EXA:MPLES
3. ACKNOWLEDGMENTS
4. NOTATION
5. REFERENCES
6. CHARTS
ii
i
1
3
7
8
9
10
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273.54
1. INTRODUCTION
A new semi-graphical method of designing columns and girders
in rigidly jointed, unbraced multi-story frames was presented in
Reference 1. The design method discussed there, starts with columns
and girders which have been selected initially from a previous pre-
liminary design of the story under consideration. Then, through
application of an analysis procedure which makes use of the charts
presented in this report, the lateral load versus sway deflection~
behavior of that story can be determined.
The design charts give the lateral load versus sway deflection
relationships for restrained columns which are permitted to sway, and
which are loaded by a constant axial load P and a gradually increasing
lateral load Q. In addition to the above loads, the columns are also
subjected to a bending moment M defined byn
Mn
(1)
in which h/2 represents the column height and A/2 the sway deflection.
The reasons for choosing such a moment is discussed in Reference 1.
Each chart is prepared for a given axial load ratio pip andy
a given slenderness ratio h/r for the restrained column. The lateral
load Q is nondimensionalized as Qh/2M with M being the plasticpc pc
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-2-
moment of the column corresponding to the given pip ratio. M valuesy pc
for commonly used rolled wide-flange column sections in both A36 and
A44t steels are listed in Part II of Reference 3. Each chart contains
two sets of curves which define the relationship between Qh/2M andpc
A/h for certain values of the restraining function. The derivation
of these curves and the procedure for preparing the charts, is presented
in Appendix I of Reference I. Use is also made of the moment-rotation
curves given in Part III of Reference 3 for the preparation of the charts.
The application of these charts to the design of columns in unbraced
frames, is explained in Reference I and also in lectures 18 and 19 of
Reference 2.
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273.54
2. SCOPE OF APPLICATION OF THE CHARTS AND EXAMPLES
-3-
The charti presented in this report give a precise in-plane load
deflection curve for any initially straight steel wide-flange column'
which is pinned at one end and elastically restrained at the other end,
while free to sway. In addition, the column must be adequately braced
against out of plane movement, and subjected to the loading condition
outlined in Section I of this report. The load-deflection curves take
into account residual stresses in the column, plastification of the
column, reduction of plastic moment due to axial force in the column
and the effect of column axial force, P, acting through the sidesway
displacement, ~, commonly referred to as the P - 6 moment. Although
the columns in an unbraced multi-story frame do not comply exactly
with the above requirements, they can be reduced to this case by making
certain simplifying assumptions which are stated in Reference I. Thus
the charts given in this report can be used to give reasonably accurate
designs of such frames.
The following two examples will illustrate the use of these charts
in solving restrained column problems similar to those encountered in
the design of multi-story frames.
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webs is permitted. The column is subjected ,to a constant axial force
P = 1960 kips and a varying lateral force Q. In addition, a moment Mn
is applied at the top of the column which is related to the forces P
and Q according to Eq. 1. Both the column and the beam are made of A36
"ksteel. Determine the maximum lateral force that can be resisted by
the column, and the corresponding sway and joint rotation.
For the 14W264 section, P = 2795 kips and r = 6.74 in. Sincey x
h/2 = 81 in. , h/rx
= 24. Pp
y
1960=--
2795 == 0.7,"k "k
M = 551 kip-ft.pc
Ix
Fur the 30Wl16 section, Z
4919.1 in.4
377.6 in.3
MP
1132 kip-ft, and
Note that the plastic moment of the beam is larger than two times
that of the column. A plastic hinge will therefore form at the top of
the column. The restraining function M , (Ref. 1) is given byr
3EI(3 ) (29500) (4919.1)x
M 8M 8Mr
LM,pc pc
pc (12) (23 .4) (12) (551)(2)
234 e Mpc
where L is the clear girder span plus half the column depth. (Ref. 1)
By entering Chart 51 (P = 0.70' PY
h = 24 r) and locating the curve for
M = 234 eM, the peak value of Qh/ZM is found to be 0.680. Thusr pc pc
the maximum lateral force-Q which can be resisted by the column ismax
(0.680) (551)
6.75
0;'(
For A44l Steel see p. 50 REF •. .1
Part II. REF. 3
55.6 kips (3)
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273.54 -5-
The sway deflection index 6/h corresponding to Q is also foundmax
from Chart 51 to be equal to 0.0115. Thus the sway deflection at Q ismax
(0.0115) (81) 0.932 inches. (4)
Referring again to Chart 51, the value of M' corresponding to Qr max
is 1.94 M The joint rotation corresponding to Q is determined bypc max
equating M and M'r r
1.94 Me = pc
234 Mpc
i'(
= 0.0083 radians. (5 )
Example 2: Determine the maximum lateral force that can be re-
sisted by the column of Example I, and the corresponding sway and joint
rotation, if the beam is reduced in size to a 24Vf76.
For the l4\f264 section, P = 2795 kips, r = 6.74 in.y x
and h/4 24.
p0.7, M 551 kip-ft.p pc
y
For 'the 24Vf76 section Z = 200.1 in.3
M 600 kip-ft. an-d I 2096 in. 4p x
Note that two times the plastic moment of the column is larger than
that of the beam. A plastic hinge will therefore form in the beam at
the column face.
i'(
Note that the numerical value of e will always be equal to or less
than the numerical value of 6/h for positive values of M .r
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The restraining function M is calculated as shown in Example I.r
Mr
100 8Mpc
(6 )
Assuming an idealized elastic-plastic moment-rotation relationship
for the beam, the joint rotation e corresponding to M at the columnp
face is given by
M .{.e = --E.
3ET
(12) (600) (12) (22.75)
(3) (29500) (2096)
= 0.0106 rad. (7)
where t is the clear girder span. The maximum value of the restraining
function M' is obtained by substituting Eq. 7 into Eq. 6.r
M'r
1.06 Mpc
(8)
h = 24 r)Entering Chart 51 (P = 0.70 PY
and locating the curve for M = 100 8M we find that its intersectionr pc
with the curve M' = 1.06 M gives a value of Qh/2M equal to 0.240.r pc pc
Thus the maximum lateral force Q which can be resisted by the column ismax
(0.240) (551)
6.75= 19.6 kips (9)
The away deflection index cO,rresponding to Qmax
is o. 0115. Thus the
sway deflection at Q ismax
(0.0115) (8l) '0.932 inches (10)
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273.54
3. ACKNOWLEDGMENTS
-7-
This study is part of a general investigation "Plastic Design of
Multi-Story Frames" currently being carried out at Fritz Engineering
Laboratory, Department of Civil Engineering, Lehigh University.
Professor L. S. Beedle is acting Head of the Civil Engineering Depart
ment and Director of the Laboratory. This investigation is sponsored
jointly by the Welding Research Council, and the Department of the Navy,
with funds furnished by the American Insittute of Steel Construction,
American Iron and Steel Institute, Naval Ships Systems Command and Naval
Facilities of Engineering Command. Technical guidance is provided by
the Lehigh Project Subcommittee of the Structural Steel Committee of
the Welding Research Council. Dr. T. R. Higgins is Chairman of the
Lehigh Project Subcommittee.
The authors extend their thanks to all those who assisted with the
preparation of the design charts. To Dr. B. P. Parikh, formerly of
Fritz Engineering Laboratory for his assistance with computer programming,
to Mr. J. R. Dawson of Bethlehem Steel Corporation who made available
their computing facilities for the preparation of some of the design
charts, and to Mr. A. Goodman who translated the computer output into the
initial design charts.
Miss S. D. Gubich and Mr. J. M. Gera Jr. prepared the final charts
and Mrs. D. Eversley typed the manuscript.
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-8-
A
E
I
L =
M =n
M =P
Mpc
Mr
M' =r
p =
p =y
P-b, =
Q =
z
h
h/r =
~ =
r =
~/h =
e . =
4. NOTATION
area of member
elastic modulus
strong axis moment of inertia
center to center span length of girder
moment at joint or end of member
plastic moment
reduced plastic moment
restraining moment or restraining function
maximum restraining moment
applied axial force
axial yield force = A x ay
secondary overturning moment
shear resistance of column
plastic modulus
story height
slenderness ratio·
clear span length of girder
radius of gyrarion
static yield stress of material
joint translation' or sway
deflection index or cho~d rotation
joint rotation
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273.54
5. REFERENCES
-9-
1. Daniels, J. H. and Lu, L. W.THE SUBASSEMBLAGE METHOD OF DESIGNING UNBRACED MULTI~TORY FRAMES, Fritz Engineering Laboratory ReportNo·. 273.37, Lehigh University, March 1966.
2. Driscoll, G. C. Jr., Beedle, L. S., Galambos, T. V.,Lu, Le-Wu, Fisher, J. W., Ostapenko, A., and Daniels, J. H.
PLASTIC DESIGN OF MULTI-STORY FRAMES ---LECTURE NOTES,Fritz Engineering Laboratory Report No. 273.20, LehighUniveristy, Summer 1965.
3. Parikh, B. P., Daniels, J. H., and Lu, L. W.
PLASTIC DESIGN OF MULTI-STORY FRAMES ---DESIGNAIDS BOOKLET, Fritz Engineering Laboratory ReportNo. 273.24, Lehigh University, Summer 1965.
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273.54
6. CHARTS
10
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i.O
0.8
--.QL2Mpc
0.2
o
-0.20.03
CHART
to
DB
0.6
'0.4
0.2
o
-0.2
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1.0
0.8
....QJL2Mpc
Oa4
0.2
o
-0.2
CHART 2
1.0
OB
Oa6
0.4
Oa2
o
-0.2
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0.2
0.6
o
0.4
08
-0.20.03
CHART :3
0.03 1.0
P =0.30 FYh = 24r
l.O
0.8
0,.2
o
0.4
-0.2
--.WL2Mpc
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1.0
0~8
--.Q1L2Mpc
0.2
o
-02
CHART 4
1.0
08
0.6
0.4
0.2
o
-0.2
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1.0 .-~~ ....
0.8
--L2Mpc
0.2
o
0.03
CHART 5
1.0
os
0.6
0.4
0.2
o
-0.2
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1.0
0.8
.JilL2Mpc
0.2
o
-0.20.03
CHART 6
1.0
08
0.6
0.4
02
o
-0.2
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0.8
o
0.2
CHART 7
0.03 • 1.0P =0.35 P
h =20r
1 -02- ,,~
1.0
o
0.2
0.4
-0.2
0.8
0.6
--.Q.L2Mpc
0.6
0.4
~-o.,J
~~r;·" +
~ ~~. t
""'.>~::l' ~~ ,"
--~~
c.---
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1.0
O~8
....Q!L2Mpc
0.6
0.4
O~2
o
r;,~ -0.2r:~~
....;
OJ CHART 8
1.0
08
0.6
0.4
0.2
o
-0.2
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O~03
P =o. 3 5 F? • 1.0
h =24r yLO 08
o
0.4
02
0.03
CHART 9
L..J=:E~::::t:4:-L~~~R::::::t-=+-+:::±::=*=-+-+=AH4::t===F~r-r-r-cr t::t:::±=::bJ -0.2
o
0.2
0.6
-0.2
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1.0
0.2
OB
0.6
0.4
1.0
0.8
0.2
o
0.6
-.Q.!L2Mpc
-O.z
![Page 36: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/36.jpg)
to
h =28r
[[[:
0.03
P :: 0.35 FY I0.01 0.02
-- ----- J --...-,~"l--'-"--""-""-""""-'-"""'---""-""
0.6
0.4
02
o ~ o
![Page 37: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/37.jpg)
1.0
o
0.6
0.4
02
OB
1.0
0.8
o
"
0.03 -0.2
CHART 12
0.6
-0.2
~2Mpc
![Page 38: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/38.jpg)
![Page 39: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/39.jpg)
![Page 40: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/40.jpg)
![Page 41: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/41.jpg)
![Page 42: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/42.jpg)
1.0
0.6
0.4
08
02
i.O
0.8
o _----................ -_- --- 0..........- ----u ~
I ------..~ r ~__ 'to ....,--___~- :~~- -~~ - ~-
r----r--- - r-------- ~"""'--- ......... 0~~~ ~~~ .2~~---- -----... ~~------~ __..___ ...........---- ... 0.1~___-------- ~~-
------ - 0 ___-- ~~r---.L.........
0.03 -0.2
CHART 13
0.6
().2 ()
~2Mpc
![Page 43: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/43.jpg)
1.00.01 0.02 0.03
P =0.40 JY1.0
08
0.6 ... 0.6
;;;;;
~ 0.4
;;:;:;;
L....-4--:
0.2 0.2
I 1_
~
o o
-----"o.lLIUW UW -0.20.02 ~ ( d)-0.2 0.01 T Ra . CHART I 4
![Page 44: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/44.jpg)
LO r-,0.03 1.0
-L2Mpc
~:::::f"""C
0~6 ~ O~6
0.4 0~4
----:..-+==
o---
CHART 15
0.03
~
----I I
I ---.. -0.2
1:1 (Rod.)h
0
I I-0.2
![Page 45: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/45.jpg)
0.03 1.0
h = 26rTl
p =0.40 FY~III
0.02
1I
0.01
I1.0~t--t-....l..-
0.8
-ltL2Mpc
0.6
OB
0.6
~
~ ~
0.4
02I ---
~
r-t---4--l..-
I_LLJ-HHH-t-i-ilill-' I---r--t--_..q~ I I I I 0:03 -UolL-----J~---t____r 0.02 ~ J
-0.2 h (Rod. CHART I 6
-=a
---~
o
![Page 46: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/46.jpg)
LO
08
0.6
0.4
0.2 0
o
r-r-r---j--r--r--r--r--r---t---+---=P::...t-~=-+---t~F=F---k:::+-+--+=--+==~~~::::;:::~==-+:::::t:==-J I0 ~~--+---I
0.03 -0.2
CHART 17
1.0
0.8
0.4 0.2
--.QJL2Mpc
![Page 47: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/47.jpg)
1.0
0.6
0.4
08
02
1.0
0.8
0.2 0
--.Q..L2Mpc
-O.l
CHART 18
![Page 48: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/48.jpg)
![Page 49: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/49.jpg)
![Page 50: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/50.jpg)
![Page 51: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/51.jpg)
![Page 52: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/52.jpg)
I 1 1 10.01
1 1 I0.02n 1 J 1 1
0.03 1.0
~
0.6
08
-
~
~
.....~
~
-......-mO.4
--..---~O.2
~
.....
~O
~
~
~,o
~
~
~
~o
0.8
~ II, •
2Mpc I --P*......I ........., -..=-10.6
0.4
0.2
-0.20.03
CHART 19
-0.2
![Page 53: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/53.jpg)
-0.2
1.0
0.8
0.2
o
0.03
CHART 20
1.0
08
0.6
0.4
02
o
![Page 54: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/54.jpg)
1.0
0.6
o
0.4
08
02
-0.2
CHART 21
0.03.A. (Rod.)h
0.020.01
1.0 . 0.01 0.02 0.032 , i , , , , , , , , , 0 , , , , i , , i , iii i J i i
P =0.45 FYh = 24r
0.2
o
0.6
0.8
0.4
-0.2
~2Mpc
![Page 55: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/55.jpg)
1.0
o
08
02
0.6
0.4
h = 26r
0.03
P :: O. 45 ~
-0.2~ --0:02 ~ (Rod.) CHART 22GOI
1.0
0.8
0.6
0.2
o
0.4
-0.2
...QL2,Mpc
![Page 56: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/56.jpg)
1.0
0.6
o
0.4
08
02
h =28r
0.03
P 11: 0.45 'YLO
0.2
0.8
o
0.6
...Q..L2Mpc
-0.2 -0.2
CHART 23
![Page 57: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/57.jpg)
1.0
-0.20.03
CHART 24
h =30r
0.03
P =0.45 FY1.0
-0.2
0.8 I-c-::::± I I~ W--1!::::tl v: ~_ I 7'f ::;Pl'5:::ttac:::r= L 1771_IVI;~ 2.n ~6 I I I I I I I I I I I I I OS
~2Mpc
0:6 0.6
0.4 0.4
0.2 I--4tHHW~~<::;t,e ~i>CL: I:::::=+'~ =+==::J::;:::I:-j---r '4=:::::J I r=-=~ I r-- 1--=±:: I P==-I~ I 0.2
0 Iacc::l: EJ?tI I I I E?::t~ I I =p== +=:::::i t I ===t' ack::: I I l::==:at ac:::L:::: I I ?t= -+:::::::: I I F"'-t cc:±:: I 0
![Page 58: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/58.jpg)
![Page 59: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/59.jpg)
![Page 60: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/60.jpg)
![Page 61: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/61.jpg)
![Page 62: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/62.jpg)
1.0
0.6
o
0.4
08
02
~~;;;;::::a...... -0.20.03
CHART 25
LO 0.01 003
P =0.50 FYh =20r
0.8
0.2
o
0.6
0.4
-0.2
....Q!L2Mpc
![Page 63: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/63.jpg)
1.0
0.8
--.Q.L2Mpc
0.6
0.2
o
1.0
OB
0.6
0.4
02
o
-0.2
CHART 26
![Page 64: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/64.jpg)
to
0.6
08
h = 24r
0.03
P =0.50 JY1.0
0.8
0.4 0.4
0.2 02
0 0
-0.2 -0.2- _.~ -- - --
CHART 27
~2Mpc
![Page 65: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/65.jpg)
1.0
~2Mpc
0.2
o
-0.2
1.0
os
0.6
0.4
02
o
Q:-t--:t:::::±:::::I;~-.L..-'---'-..... -0.2
CHART 28
![Page 66: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/66.jpg)
-0.2
LO
0.8
--WL2Mpc
0.4
o
-0.20.03
CHART 29
1.0
08
0.6
0.4
0.2
o
![Page 67: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/67.jpg)
CHART 30
1.0
o
0.2
0.6
0.4
08
-0.20.03
O~03
P = O.50Pyh =30r
A (Rod.)h0.02
O~02
30
0.01
0.01
O~8
1.0
o
0.2
0.6
0.4
-0.2
--IDL2Mpc
![Page 68: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/68.jpg)
![Page 69: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/69.jpg)
![Page 70: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/70.jpg)
![Page 71: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/71.jpg)
![Page 72: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/72.jpg)
1.0
o
0.2
0.6
0.4
08
h=20r
0.03
P =0.55 Py
-....,....;~---.....~-..- -0.20.03
CHART 31
r<:f::::::L I r==r k:J ~20FM~~4:::t:~-L-.J
0.011.0
0.2
o
0.8
0.4
-0.2
....Q.!L2Mpc
![Page 73: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/73.jpg)
1.0
0.8
....Q!L2Mpc
0.2
o
-0.2
1.0
08
0.6
0.4
0.2
o
-0.2
CHART 32
![Page 74: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/74.jpg)
-O~2
1.0
0.8
....QL2Mpc
0.6
0.4
o
-0.20.03
CHART 33
1.0
08
0.6
0.4
0.2
o
![Page 75: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/75.jpg)
1.0
0.8
....QlL2Mpc
0.2
o
-O~2
1.0
OB
0~6
0.4
0.2
o
-O~2
CHART 34
![Page 76: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/76.jpg)
1.0
0.6
o
0.2
0.4
08
-...........L~ -0.20.03
CHART 35
1.0
0.2
o
~2Mpc
![Page 77: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/77.jpg)
1.0
0.8
--.21L2Mpc
0.6
0.4
0.2
o
-0.2
1.0
OB
0.6
0.4
0.2
o
-0.2
CHART 36
![Page 78: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/78.jpg)
![Page 79: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/79.jpg)
![Page 80: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/80.jpg)
![Page 81: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/81.jpg)
![Page 82: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/82.jpg)
-0.2
i.0
0.8
.JlL2Mpc
0.2
o
-0.20.03
CHART 37
to
08
0.6
0.4
0.2
o
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to
0.8
....Q!L2Mpc
0.2
o
-0.2
1.0
os
0.6
0.4
0.2 ,
o
-0.2
CHART 38
![Page 84: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/84.jpg)
1.0
0.6
0.2
o
0.4
OB
0.03
CHART 39
~
-0.2
1.0
0.2
o
0.8
-0.2
--.QL2Mpc
![Page 85: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/85.jpg)
-0.2
1.0
0.8
~2Mpc
0.2
o
-0.20.03
CHART 40
1.0
DB
0.6
0.4
0.2
o
![Page 86: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/86.jpg)
0.6
o
0.4
02
-0.20.03
CHART 41
h = 28 r
0.03 1.0I P = 0.60 Py
0.2
o
-0.2
. _Ul-- 08
...Q!L2Mpc
.:
![Page 87: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/87.jpg)
1.0
0.6
o
0.4
08
02
0.03
CHART 42
"'L ! '::>' ! ~ -0.2
1.0
0.2
0.8
o
-0.2
~2Mpc
![Page 88: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/88.jpg)
![Page 89: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/89.jpg)
![Page 90: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/90.jpg)
![Page 91: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/91.jpg)
![Page 92: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/92.jpg)
0.8
-..WL2Mpc
0.6
0.4
0.2
o
-0.2
CHART 43
1.0
OB
0.6
0.4
02
o
![Page 93: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/93.jpg)
1.0
0.2
o
0.6
0.4
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CHART 44
1.0
0.2
0.8
o
-.QlL2Mpc
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0.8
0.4
0.2
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p z O.65Py, h =24 r
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08
0.6
0.4
02
o
-0.2
45
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1.0
0.2
o
0.6
0.4
08
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CHART 46
0.8
1.0
0.6
0.2
0.4
o
-0.2
.Jlli.2Mpc
![Page 96: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/96.jpg)
1.0
08
0.6
0.2
o
0.4
0.03
CHART 47
0.03---.,..-r-"'-.,-n,-TTTr=D=r=+++=R=t=t=fIEa:p~=~o. 65 FYh =28r
LO.-
0.8
0.2
o
0.6
-0.2
...QL2Mpc
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-0.2
1.0
0.8
--WL2Mpc
0.2
o
-0.20.03
CHART 48
1.0
08
0.6
0.4
02
o
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![Page 100: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/100.jpg)
![Page 101: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/101.jpg)
![Page 102: IG L:BliARY - Digital Scholarshipdigital.lib.lehigh.edu/fritz/pdf/273_54.pdf · Lt\80HI\T()HY L:BliARY u ... A new semi-graphical method of designing columns and girders ... A plastic](https://reader035.vdocuments.net/reader035/viewer/2022070613/5b8883847f8b9a770a8b839a/html5/thumbnails/102.jpg)
0.6
o
0.4
02
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CHART 49
0.03 1.0
I I I P =0. 70 PyI I I I h= 20 r
o
0.2
0.6
-LO OB
-0.2
~2Mpc
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-0.2
1.0
0.8
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0.2
o
-0.03
CHART 50
1.0
OB
0.6
0.4
0.2
o
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o
0.6
0.4
02
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0.03 , 1.0P =0.70 Py
I h=24r
0.2
o
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-0.2
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1.0
0.8
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o
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1.0
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0.6
0.4
0.2
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CHART 52
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,LO- r:::::r-rrlr--t--,-,.....,.......,.~:....-... .......--.....--
0.8
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0.6
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55
1.0
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0.6
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1.0
O~8
....QL2Mpc
0.2
o
1.0
08
0~6
0.4
02
o
-0.2
CHART 56
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