c.alifoii!iil,'.,. n.ts'ri'l'lj'.fl of t~:;hnology · 2015. 6. 18. ·...
TRANSCRIPT
C.ALI F Oii!IIl,'.,. n.TS'rI'l'lJ'.fl OF T~:;HNOLOGY
,.,.. ' i -- \ --·-·
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D..!}V2:10P·I) BY
CH.A..-LBS 3 H\J;/il.:.W ' 23.
1
1 . :;)3V'J LOP:J.:.!iH OF FJX.1lJLAS
A. Assu~npt ions
1. 'i'lrn union b0 twe on th) stoGl and th'~ coEcret:= is s :.1ff lcL:mt
to co.us ':! t.t12 t;;;o :.1at·.:?ri[lls to act a s on·:! ::10.t<:n·io.l.
2. ~o initi a l str8sse s ar0 considarol in ~i th~r the concret0
or th-9 sV~:-1 du._ to t .3,np2ra.tur .; or sI1rinl::2.g _.
3 ~ '.C'lB ::ipp l iocl forc)s ar'.:J :;;inr:ilL:l to 3a ch oth3r and. por-
_p::mrJ_ic v.lar t o tn-2 n-'!1.1tr2l surf::i.c= of tne b -.:?.'.l!'l b0for·3 bendi ilf;.
b ?ildiug '.'ii t.rlin th.; e l astic J.i::ii t o:t' th·:; s teel.
B. ::'!1.a follow i :1,3 notat io:~s wi ll b 3 ·3::1ploy2cl L1 r·::f~rr .i.:1'.:,,' to .'.!'ig. 1.
i.ot f c
crotc nr..d.er a giv:m lo o.ti .
f :.1c;xL1v..::1 ill t •.nls i ty of t :m si l ! st ro s s in t h-J ::1'..) t a l s
l s as s u:::L:d t o b'.: so small -,'ii t h r·:: f:::r-2nc2 to t.u
totn l ar:; ~ o f cross- s~ction of th~ b2~~ t na t t n 0
str;; ss in thJ m0tal is pract ica lly LLYlifJI':il) .
C - ro tul CO!'lpre s s ion in concra tc u t o. section of t.h·~ ben.:i.
'i' Tot:J. l t::msion of tin 3tJol ::.t t 2 s ~ction o f th,, b~a:·n.
";\
·-'c 3apro s 2nt s tha m0Qulu 3 of al~sticity of co~crctJ i n
compr.ssion.
·~ J.epr :Js ~11ts th·J :noclulus o f ~lastici t y of s t e,~ 1 in t8 .?.1s i on. s
n rhc rat io o:f ~ 1~ . s c
J. i'he di stance fro:ri CO!n}_) r ,3ssi'.)ll surf ace to .:ixi 3 of
rei£1for c:::.aent.
2
k 'l':h::! prorJort i :;eat ·~ d ep th of il(3utra l a;:. i s fro:I'-
b::!lO\'I th·3 C:l!ripres s ion surf ::tce.
a '.c'he ::i.r00. cro s s- s oct ion •)f sto _1 . s
st :iol to th'.) area of concr::::t _.
b
bl = The widtn of be~~ at axi s of tension stc :::l .
.. -··c
c ret0.
i '1D r.)s i stL1~; mo.::.cnt a s cl J ter::1incU. by the ste .. ~l.
l1 '.i.'iH~ b :m.d i ng ;no:n:::nt or resi :3ting .nom~:mt in een ~ral •
•
3
c
jd
Fiq.I
4
To h;,rl Xo g,,d' C
b - b -rzy - b r(/7-baJ¥ 7 - ~ - ;? xct
Let re'=- ro~,,, on elemen*17 r::rrt?« dy el,r, f/- ~,r c - ka"
Lef /V7, =Moment" o-r allfdr<es abovPlV:A/ ;VI,: ffc'by-rdA"
=f Tc.K[ b +- {" -b~) ~] x d,,r kd '2 kn'
=[ o fc ,,tl.J;,R dA" + -rr E3' ( f7 -be) d.-r ~ ~qi kd' { kt:TJl.
= fC b,2 { kd)Z.r rt: l b-b2){kel)2. 3 'f-
= ~c ( Kd'J'(.bg ~3b) /~
c = ff c f?y c//}"
- ~[fcxPa. r ~K2(b-J:>2)Jd~ -;j kif/ (/(t/)2.
k~ ~ ( -fc;rbg clK +- f;:K2. (b-b
12)t:IA.
Jo K~ k~~ = fc X"
2b2Jk"' + fcx 3 ]kd(b-bz) 2 kt?! () J(~d,. 0
: fc kd ( P11. ; 20)
re ( ktT/ { /?~ 7' 12.b.J 6
= Kd(b3 ~J.6) 2 ( b,z r-26.J
5
EJul' /?fl :: b/ r- ( 6-h/ }{ 1-,1c J = br *fb/-h/ :: b -k( b-b,)
= Kcl[br- k{b/-,b}~3b J :. Xa y J?~k(b/-6.J r--<.6
k d [-4 b r k.. ( b/ -b) ] = z 3 b 'I-/(_ { "/ -bl
llv"hen b, = b (lf:pcf-gn911/t:rr Pt'e?m) yo : k d [-4 b 'I- /( ..t o ] = k ti _:J_ - 2 k /
/'• 2.. 9bro TAJ-yo
cT;j . ,,.....,::: b;b _/ p=. f1 : IT £:r
p { Jn•-i;J/} d a'J ::: 2.
Z f-1= 0 or C: T
C::: f~kt/ ( 6-l r2b) = T:: a's Ts fc kd(b. -1- 26 1 = f'sec/( b~6,J
6 ~ / ~ .
l3ur b2 = '7 "k{.6, -b.J ,·. f~A:d' (bi-k{6/-bJ) = f'sp~(P?-6,,.)
or ~~kd [J b r* ( b, -6)] = ~~ e'{ (bf-~} Z3vf ~:: -h/1(/;~J
, '. .k ::: -f=c n
IS -1- h ii: . 1'1~/·e/ [ b- Q/'c:(~-b)] - rs pd( br~}
• • b (J51'-hTc) J -61'H* - 2.
. ·. p
6
(nf;/el)(Jb;'st-3bn'lf)-nb, -f'c -1-n b-h:. or 6(rs -r/7 rc )2. =
2. (.J'n b ~sTc ~-1- ~ nfb rc3- /1 ~b, rc 3
= 3(.,CS -/-/1 rc} ~rs ( b 7'-b/)
= 11 Tc l.( 3 brs 7'-""f '1 b re - /7 ~ nc) ~75- ( 6 rHTc) 2
( b-rh,J
When b, = b (7f'pc/-~n9v/n/ bt"nm)
11/'f i.(!b;!sr- ~nbrc -nbrc} p ~ Jfs (~ ;1-H}f)~ { b ~/,.)
11 re~ 3( i:J 15- ~1?6 re) = 6 .b rs ( T:s r "'rrc} z
,., .J;c 'L /7 7f z. ,,/£ - OK. .
- 26(61-nf'eJ 2(-fs~~nT.sfcJ
Jd::: d-(lat'-ro)= a'-kd-1'-/;, ~ cl - I< p/ ~ tr.L [ '</ b i- .k ( b/ -6) 1
2 3.brk(b,-b}
= d-kd [1-..L[~bt--l<(h/-b) J} Z Jb r-~ (6/ -bJ
.. • [ / [ 4 b i- k. ( b, -b) ] t •• J = /-/(, 1-2 5/,J-1-f({p/-b) J
· When b, =b (leecrnn9ulu;-- P~dm) j::. /-I< (-~ xy +I} = 1-f k. O.~.
7
.:J~/V//19 for /( rrom C = T
fc Kd f:rbrk(b,-b}] = '1fc(l-k_J pd(b-rb,) 6 L::: 2 k..
or 5k6 r kz ( b,-b) = np{l-k.){ brb,) 6 6 £k
or kP ~ ~ ( b, -b J ::. pn{/-/c)( b-/-b,} 3 k
or f ( b,-h) r kz.b -pn (1-1<) {b-i--b,) = o
When b,=b § 3 (b,-b) =O
K~b -pn(l-k.} 2b=O or- k 2 b + 21<.pn b -2pnb =O
I<.=- -2Pnl7z fC2pnJ2.~ "f bt2pnJb 2b
= - pn r 1(,cn J2 -1-2p11 0.1(. .
.Mc = Cjd= fc6kd[_Jb1-~(b1 -6J]/d
= f'c :jdZ [! f7 t-A: { b, -b)]
8
Resl/LT.5 and roR.?JUlRS
p = zcts _ n'k&.{5brs+4nb"fc:-n'7,-fc) (brb,Jd - 3fj (-& r/?Tc ) 2
( b~b,)
- p(bt-b,Jd as - 2.
K. JC( b,-b) 1-f(~{b)-pn(l-l<J( b-rk) = o 3
JV'/c = .ri:;jdz.[.;b ""* ( b, -b) J /Vis = p6jd2.( b'l"b,)
2.
9
.:50/v/119 for values of p 17=15
b, = o.2b nfc2.(3b'/s t--4xl5brc - /5)( o.zbfc
J'= Yfs(6-rnrc)Z(b-ro.2bJ
= /,,fc2.{3brs f-.7/Tc) .:Yfs (f"sT--/frc)Z,-r/.2b
_ 1f"fc2.{fi.r/9Tc)
- r., (Ts f- / .rfc) 2~ /. .<
61 = o.~b 1fhZ-frs.;-17hJ P = rs('h-r/.rrcJ~xJ.6
b,-= 0 .6 b ;.r/c~(4t-16~) f> = f.s ( f; rl.J7i:)2.x /.8
b, = 0 . ab ;3-/;2(fsr-!6rc)
/> = & r fs -r1.rrc)<x 1.8
For Vt1/ves .5ee ,Fo//ow1119 Table.
TR8LEI
f c f s fl- rh l§flJk % 1'1Tc !Bf'c 17k 16'/C /~~ f!j.fs1 /81. !.E(f.m'f:. ~{~t'''
Valves of p /,~ I,~ J
ih,:.2t b,=fb /h:.b6 /;/.-.86 r1 B c D E ;:: 6 If / .J I< t.. /VT IV 0 p (µ R .s
I 12Xlt>3 /J..J.3 18.U~J JH;/J -f,//Klia 2:1.fxttl 'J()o.tl~'J '71:f,r/tT 1.JJ-t!i3 l.dtJ6 .8-f? ' 7,;.; I 6,<..9 2 11 " /6,/'/(i'I 11._,.Z 20 f()t> l.?,86 2'70 221 I.IS' ''" 1'TTI . 61'~ , f.76 . 1'~6
3 100 16 II /O.tJO 22 18tl l2.b6 7otX' 7200 &BOO i6"f()O 2.9.r '1:ft9 tl"I 186.f . 6tJ~ '.f"/.2 , "f"f .t ,Jtf'f' 'f 18 ,, 8.!J() 2.1 )/6 l.f'f :120 26~ 2Jt.!" ?()3.J" I "f .9"f ,f/6 '3.7.9 .3 /"! .1' 20 ti 8.0d .26 ~76 /.28 lt3"fr l~~(J 1.n.r 1.22" -(~/ ,37/ , .U:t.. .2YI
6 12" 2(), fc ;9,.,~ 380 .r.-ft9 2~f .12S- /?2 /"lf.6 /,'1/f /, ..l .1.f" /, tJ.7,Z . 9/0
7 1-f,, 2fr10"" 178'J 21. .r ii'!~Z J .8.r .29~ ~f~ • .r 2/~ /tJ'J..1 /,/3~ . 9 $"() .~32. ,?a:,
8 ,;oo 16 II VJ:60 .tJ . .f' r.u ~.83 9.1'1J() 9000 ~.roo 8000 3Z9 1,,2~8 ,2./() ~.:1-" ''9''1~ ,7f8 ,6ft:J ,.;6.r 9 /8" 1/3.88 :t .r._, .. (,}d ~. 14 l,J'# '2~~J- fH4.f' ~N.6 I 73.3 .6/J .f2.9 ,"f6/
/(J 2~" V2-fd J'!.f 7f6 /.6.f' ~69 311 ?~Zf' ?.1~3 ,6// • .r1.r .-f'ft ,JS<;
II V-? ,, 3'1. "" ,21 ff/ 6.80 ~n .. r· .2-H..1" .2t:'8 / 4'0 /, 99'0 /.660 1.1'/f /.2.2<J 12 ;.,. If 36K/(7) 2f-lt:' .23 S-~9 'f. s-~ ~/7,,)- ~'6 2.<'7 /?~,.r v . .r~.:r /.2?~ /,/O'I ,9.r.r 14 600 V6" 'i:Z..S"" 2f' 62.f" 3.6" l/'100 10800 /(J.200 9600 ,ff./..,- .tf;u- ,2~£ .2 / -,! /,2.J~ /.t:73.!" , 886 . 7"71 1-1- IB" 2"·"" 1.17 72~ 2.7"1 3•?: .>- .)'Of ~~.r .2J~ /.de:!?' .'577 .7,26 . ~jO
/S" 20 ,, /s.aa IJ9 Ml 12./4- .19~ • .r l33o • .r .tdJ ~'f~.1" ,tJ"/O ,Toa . ovo ,J..t8
/~ 1.2 ,, "'"· f
,2,2,,J- .1(}6 dt:J6 .3/• ~•.l..f 2.2-f / $3 1.1..r.r" .?.12$" /,6'd6 J.r.r.r /7 v~ I/ ff9,r/"1 ~.r.() .t'f • .,- oao f.8$ 133(}() 1260() ll!JOtJ 11200 ff/ .te.r .2"13 J/~ v. 99t:> /,lf6t:J /,'fl'/ l/,,.l,:?f
/8 700 V6 If Jd.6 .2~ . .r 701 .f,:J, f6o JtJ • . ,)- -<''-< .127 /,fP6 /,3Jf /./-IR • 99'0 1,g llA 11
~ 7..2 Je,.,- 812 5. :.r4 39/ ~./B A'9/ 4~.Jf /,.JOO /,t:JPO ,fJ3 .808 :/(} 2t:> I/ .t"f.f Jo . .r 9ZO ,2.63' 1'/6 .J'f9..r jm:J .24"0 /. 0T7' , 7'..?0 • 76.:I 16b"..,.
.1/ 1.a ,, .r..1,J .?1.o .1"76 9.26 wo .2C'3 .2"'10 .,//~.~ J./ft:J IJ,6.2() .l.122 .1.d<Jb
2a IA/- ,, 6-f,r/~ '1'9,). 7 .1' 676 •.76 lf.t'()(J /#tJO IJot:J() V.2800 ~6,r .30-f:~· .;.r.9 2.23..3 ,?:??P "R.t:Jlf() /,7-'" /, .,1""/0
:J~ 800 /~ t" ~(). () .28 7c9"f r./o 'i,J.$'d ;J,/~ .27B '2'1" /, 9?&' /,l{b'I /, f/i /,:/,<~
-21' 18 ,, Jf.f .Jo !loo 3.9.f' ~/.r ~</?.f ~9.7 .1$"'6.6 /,b'..,CJ /,.771' /,17-1 1,/.()/,7 .t~ ~" ,, ;,t.() 1.:1.Z /(J2~ :f.l.Z .,,~&j "./('f .3/6 ;.4'/J,J /,37.R /. /f'.? •. 4/16 · .e.r~
TRBleI
Vo/ue~ of" Volu'7s of I< a= tJ.Zb b~ =o.ttlb h::o.6b P,:G 8/:J
o.2. o.4 O.d o.8 a2 o.4 a6 o.B /-<):'!/(, 1.7-aB/'\ J /-(}.~/( 3-()4; j 1-a21c J~o . .Y.it: j 1-0./1( 3-().2~
/ ,67'1 881 • 'J-f3 ~f'1T ,JftJ ,3f2 ,3oz ,3"8 .8.r7 ;'.'!/'I 88'7 ,8-f"'I 2.789 .8.94 I ,9.1,1() .?.87.9 .~~ . .966 2..932 2 ,,,-,zo .6~.Z. ,723 1.968 ,521 ,'3'/6 .270 ,304 8?~ 12 7.,.3 .8.!14 .9()S- 2.A/0 ,A•A ,q,Y-6 l~.B.92 • .9/2 ,97"0 .(' • .95.9
3 ,111 • .r.1'8 ,.f'7S- /.ff'T .290 ,2'!]6 1R?6 .88'1 1 .?.76~ , ,5/(J? • 91"1- 2.8~ .9tJ8 • .9?2 .f . .9~.f
~ ,333 ,..tf.J?' .~ft3 /,272 .J . ·~YO ,3f() 1"'1'18 /J.$8
6 .99~ /..t9'l /,.J6$- 3.~86 .-ti?' ,1()8 , Tf5.2 • .7.9J! .BJ'f ~~66 .6'?/ .B?lJ 2.1~.r .870 .9:1C .?.6fR .876 . .161 .?..922 7 '>'•11 .ffl /,tJ&Z .R.8..79 ,.?76 • .Jo8 ,32Q ,Jf4 .e.ro ?.699 ·882. ,8!JC :/. 77.!J .88~ .!)56 .?. 872. .88? ,.!J6S ·2 . .9M 8 .btJ9 ,7P6 .>t.J" ifl,:10'" ,3-f~ ,336 .288 ,J,2~ 863 :/.726 1 892 .89.9 2.798 ~ss:z • .91/2 ,?.88f -~6 ,968 .?..95~ 8 ,1f.1' ·'~ .~68 /.867 ,JI"/ ,,30.!J .,;1,98 87~ 2.7~.!J ,!JQO ,.!}07 z.s15 ,,.9CJO • .9'70 ,t..9¥0 10 .flZ ,f~/ .f?f /,ftf.5 ,<.90 .21% J7?' .se-; 2,7~ ,!id7 ,!J//f. 12.828 ,907 ,IJ'/'2 !ll~-1/1' II /,3'4/3 /,l"'l.J ;,838 -t.'lJ"7 .%?' :1S"7 ,Jf6 ,"/.J8 .8/3 ,2.~26 8~~ ,86:5 2.726 8$'6 .921 12..~ .87Y .956 7..5JIR 12 /,Of.J' l/,.1f.!J /,"'/.35 J.868 ,f2:1' ,"f /6 ,.J,;.9 , ?.9.9 8.YO 2 .660 .86?' .B'J'f' ?. 74"~ ..tl6.9 09..Q l?-'lj-6 .88/./ .$60 .?.92() 13 ,8.f.:f /,()d7 /, /.')';2.. 3./26 .J~9 .,.9&2 ,.J.2'9 '166 .844. 12.689 • 8'78 .se~ 2.771 .S'18 03.y '7N6!} .8.95 ,.965 ~9P7 14 ,680 ,t!Jt39 .9-f~ 2.ff~ ,.1.r8 ,.1.r"I ,3()Jl ,339 .s~? 2.'7/~ .l!J88 ,R!Jlf 9.7B.t; .988 .91/0 !P87.j • .900 . .966 ~ 0'2'?
If . ..r6? ,)'/.1 ,7&t; ,R,/.J~ ,,J.12 ,32~ ,.,2~0 ,$/~ .867 2.734 ,8.9.r , 9~-ft ,!.845 .8.9.; ...9'1-V 2.~ • .90.9 . .968 2.9.:r7 16 14' 7.Z/ '12.231 12.318 ,,2~8 f/CJ , 'f.5''9 -fJ1 ."'176 ,?.96 :/.f".92 .8"f.J .8.r/ 2.7.rl 8¥'6 .91~ 17 ""~' 860 • .!J.J~ P . .9tJ$'
17 V,.1.,3 /, /'f.3 ;,HR 1.94/ .'167 ,f5''5 ,J'J6 ,'f.J4 .$/3 2.'"26 .BJo 186"1' 2.126 ,8f6 • .921 2.8¥2 .87/!. .95"6 1,.912. /C' 1/.0'?7 IJ,<{(}2 l/,~5 -f.c>/CJ ,.,..?" 1'?.2/ 1.36~ ,~trl .828 2~.17J .866 ,87~ 2.7'1? .R66 .927 ~~j'~ .&e.2 . .960 ~.41~ 1.9 .876 /.If.ti V.2/.3 31272 ,Jf7 ,j.9(j ,.J.J6 ~.17'9 8?1 .76~ .87$"' .es5 2.766 .B?'~ . .9.5~ ~A6'6 ,8.!J/ .$65 I.?. '1~.5 20 • 7.J.t!J ,9b6 V.d..t6 ~. 7?''7 ,.J?d ,36-1 ,..11.3 ,.JJ"() .8$'~ 2.7~ 188~ ,8.91 ~782 .88¥ ,.9.37 .2.47,j .8.9<!1 • .96.:f' .?..93'0 :ll R,/..26 R,7f/ .f,48., l!J.12~ ,.;J$' .-<1&e ,f..20 ,839 ~.~7$ .c9J3 .906 .1,815 ,8.Y.9 ,,9.1/8 12.d9t: l:Z~ /. ""'7 .1,/6:5 !tJ.~?'J .. //6 ,,fd~ ,.,,., t'f28 ."'172 ,7..98 2.P'l .8-f-J' . 8.5'~ 2.701! .81/.Y ,91~ ~82.S .862. .,9Y5 R~ :J..3 /,,H,J ;,7f~ /.d'f.J fJff ,f•? .~3? ,3,16 ,"IJ8 ,8/~ 2. ~26 .8/!-6 .863 2.726 .8f~ 92/ !2.~¥2 ,,872. • .9f'6 :.?..5'12 ~~ l/./e'?" lt,'f'/.J ;,.a't:' l'f.le'S ,f.f"I ,~.i'6 ,.J68 ,'1'78 .826 ~16~~ .86.f .87-2 2.7if't .86S . .926 ..?.8.i':. .880 ,9.;9 ,,?.,9/8 .<'f' ,926 l/,lll /,,,?4~ _ y -41 .f'.9 1"/d6 ,3..9c!J • .H.3 .34~ ,A~ ;.&75 .S'/'3 .es1 ~- '161 .9?'3 .95/ 2.60~ ·881fj .$62 ?..!12'1
TaEJlE I
iti=OllJ, ~ = o.2b li~a'lb b,=o.tb tJ, =0./1b •
0,6',?j f1l2 am lfJ:,_ o-"~ ffF3 o.s]j Tri~
J tJl:l'2. b4:Z.
I .88S 72()0 '~2) 18¥'10 '.J.~o 96()0 6.?.78 11?800 oa3:z. 2 • .900 811 fB.1'- 98 'n..2~ 112 f(.'7.!J 1.26 .ff.I/
3 ,909 9& 'r5.1.tJ3 112 l.r!.tJ7 128 j/fj' 5'a39 ~ 108 "3' l.Y.Y /b2 s /'})() V.y''1 i/60 118() ~ .872 72. .9/..1? Blf' !J()./() 96 98.~l Vt18 8.Y.90 ? .885 8/f' A?I}' 98 82.tJO 11.Z 82,S /.2~ ?c9..Y3 8 ,89!$ 9, 'l?.J.Y 11.t 7£7.1 128 7f.'J? l.Y.Y ?./.6f 9 .$0/2. /()8 'l/,},f /.26 ,9.9') /¥"-/ 1/,2 67.36 /() .80.9 /2() '7.7P /Y'() 6.£.J~ /6<;7 /BO &3./& /I ,_R_.of"'6 'f!) 12/.IJ 8¥ 11.9..33 96 118..f'tJ 1/t:J8 //J/6 12. .A69 /f /y l/l:/.f2 !J6 //().()6 112 l/d.93, IP' /P-K.f'? 1.3 ,.0120 OL JIJ-Y.d~ //!) M/7/f 128 //1/. '>? /~9 .<J?7A
It' .883 VoB 9~f8 1.2, 9¥,T,j /.YI(' 9¥.0J. ~1"" 9P.7.J 1.r ,81)6 I J'JA .9d.12 llf'O 88.7/ /~ 88/.1/ /08 A..r/"' /6 .. .O'V.V 'J1 /.f~77 Bf' 115"/,~7/ _q~ 11~..P.ll /9~ 1-w.?~ 11 ,BS-6 IJ'I' l,YJ.IJ.9 98 /J9.2J /It /.78~ /¥¥ /.1,1./;2.
1/8 ,867 e' 1.1.f.~.P //.l /?.9.~ rf.f8 /M.9' /62 /2.l,60 /.() ,8,7 l-t:J8 /.2~8J" /~' 12t1.J.1 l.Y-t' //.P,'71 /~2 //~ zo .88.i" /.10 //~.Jt:> /.V() /(~~ /"" //.1.J3 //1() /~.J!/2 21 .8~ Zt 8-t' IA93.1~ q~ /B/.PJ /tJB /.?.9~ zz .8~.f ;Jlf' ;7$.12 M 174.1~ 112 /~A_0'4 /2~ /LA77
23 .a.ro ,9.( 1•.J.fJ 112 /.1.Y.~ 12A /.1"~2i I.Yr /.li?.98 2.1/ .t9~7 l/t.18 nr;J.2/ /.2~ !/f'-f8 I/~~ l/.Yc9. 7~ /62 /1";~7 125 .A~1 12tJ ;.1,a11 I/~ /l./0.92 V6'7 /f.'l/,()J 180 /?,y-u·
.::5olvin9 for Va lue.:s of /f..
.is! ( b, -h) rK2.b - pn{ 1-k.J ( b +-kJ = o 3
b, -=0.2 b /(
3 -3.7.Yl<:i.- 67.3p/\. -1-67.3p=:o
b, = 0. tf.b 3
/( - 5/(2. -/O~,t:J/( r/0.1"p =o
b/=0.6b K 3
- 7.f' I<.~ -t3opk. r /3op = o
t>, = 0.8b /(
3 -1.s:o 1<2 - .r;orpk.. t-48l!Jp =o
n=IS
Volves of K J % e;rA4' y 1n IP6/t? ZL w-~,,...,.... To«nd Q'S ~o /~'1r.S.
.all of the ;;qua tions v1 r~ o f the fo nn (k3:-xk2- yk + y == 0 J, t hsre
b e int: but four diff' ::n .. .;n t values of x ( iil.11 gr<J:ater than 1) fo r the who l o
set of probl .J!ns , :lnd th·3 v ;;. l u::s of k being known to ran::;-:1 b ct-_·;,'.!on .270
and .550. The e quat ion was r~written i n tho form k 2 (k-::c) = y(lc - 1 ),
o r since both x and 1 •.•1ere great~r th:rn k, in th~ fonn lC' (.·-kj - y(l- k) ,
w!;enc c dividing t hru by 1-k , y = 1i;:2 (x- k )/ (1- k ). i:h8 values of y g iven
by th~ s mcc siv: v 2 l uc s of k .270 , .280, .2;10, and so on t o k "" .550
wer·3 next co::nput~d and tabul at-:~d . 'l.'h computat io,1 Vi~s easily pr~for:ned
on a 16 place ;.1onroc by sett inz x-k up on th~ left o f the keyboard , 1 -k
or.. tn.e rizht , and buildin:; 1-k up to li;:2
in the middl'~ c!f tho lower J.ia l,
wh~nc0 y app-:::ar::J ~t t .ht: bft of tile lower di -.1. For exampl e, for k =
.43 (x - 1':) = 3 .32, 1-k"' .57, and k 2 = . 184:i'(found i n a t ~-, ble of squa res),
k ·'.)yboard 3320000057 and by forward tur11s of the crank build 57 up to
184 ':1(023 ) i n th~ ,nidO.l ~ of th~ lowe r dial , when y:;:: l.076 'il{ 74~) ap9ears
' t the 1 ~ft of t he lower dia 1. Go py y = l . O 77 , c 10a.!· dia 1 s, and change
keyboard s e tup to 3310000056 for k = .44 (k-k =3~31 , 1-k = . 56, and
k 2 l ··3·· J = .:J o. Build 56 up to l ';.1 35('376 ) in the miclll.l ~ of the lowe r dia l
1i1nd when y = 1.1440 (001) a pp::2 rs a t t !1e left.
Valu·:?s of ;v fo r inter.:iedi a te 3 pla c e v :alue s of k w~re t h en com-
puted by proportiona l i nterpolat ion, th:ii.t is, for example , having fou.nd.
y = 1 . 0 ?7 for k = .430 :md y = 1.144 for k = .4-40 , di 2 l 10770 and. key-
board 67 (1144 - 10 77 obtai m~•l :r1enta. l ly ) , on~ fo1~,.1ard tur n of t hc "crank
give s y = 1.084 f ork = .431, anothGr turn e;ive s y = l.O:JO for k = . 432,
and so on to y 1.144 fo rk = .440 , thus a uto:nat ic ·lly ch :~cking the
correctness of the ;n:?ntal subtra ction, and. dia lling the v :..lv.~ of y for
k = . 440 i n raudi ness for t~e interpol at ion of the next ~ v a l ue s of y.
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2 .79¥ /.088 /.681 Y.+'~.3 ~ 180f /,/t:J3 V. ?PY J.f~ 6 18/6 /.//8 l/,727 /.1.~.Y8 8 .827 /,/J.Y 1.?fo J . .T.95
u.~90 ID.I 3.36 -¥,~/ 7.11 /If,'/ -~/ .8J8 /,/"Y9 I/. 77.J 13.6¥3 2 • .a"Y.:t /,/6'Y 11.797 3_~'13 ~ 3.lf .r.oo ?.fO /.f.t)() .8GO /./B~ l/.821 .J. 7.Y..-i 6 .871 /./$16 V.B'f'.f' 3.7.93 8 .BB~ /,2// /.868 ..!.8~
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