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Mechanics of materials
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kwu, Celestine
yKBy CenecrnH
Nkemakolam
HxenaarcoJraM
Solved ProblemsIn
0 ftil.!
Strength of Materials
Peuemne 3aAaq[Io
Con porll BAeH He Marepnanoe
cwott WWW {uiYwt
".' tX )'m4t 't (1'J V )'n^z I a 6
kf /krya-ts fu-"Pr+ l*e4 g s69 sa34 '
00.-L{(-((Ya
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ll
l*, i:'l'.'
di'jr.
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. .; (l.t,i\ r.ir.
Preface
This book is written to assist students studying Strength of Materials in National Aero-
space University, Kharkov Ukraine, towards solving typical home tasks, school examina-
tion questions and All Ukrainian Olympiad questions. This is not a substitute to text-
books and lecture materials provided by lecturers; and on that note lwill advice students
to study the lecture materials provided by their lecturers before using this book,.
lwould like toexpress my gratitude to the Almighty God who directed me towards the
course of writing this material. I owe everything I have acquired to Him. My parents, Chief
and Lolo Bartholomew Chukwu Agwu, and my siblings have been of immense support to
me-both financially and otherwise. My friends George Fidelis lmafido( Presido), Nassir
Babiito, Dami Molu, Ronald Ngobi, Orok Donatus, Shaikh Farhan, Mittul Parikh, Victor
Mukungunugwa, Winrrie Kalyango, Philip Angule, Bhattacharyya Prantosh Pran,
Saikia Hridayananda, Kafeero Derrick, A.shish Gupta, Okrruir Simon, Praveen Jose, Amit
Krishnan, Vikas , Bharat and many others too numerous to rnention provided nre with
moral suppor"ts and encouragement. Pastor Richard Ogbunuju and his wife Marina
Ogbunuju have shown me such love that I cannot quantify. I thank you all for your help'
ing hands.
I want to specially thank my lecturer, Professor Vladislav Demenko, for teaching me this
course. He was very eager to give me every valuable information I needed, and I must
say that God used him to open my eyes to the world of engineering.
I am a Nigerian student studying Aerospace Engineering at Nationat Aerospace Univer-
sity, Kharkiv Ukraine, and I hope to graduate in 2010. Writing poems iSmy pastime.
&butra,X,.to,^Ghukwu, Celestine Nkemakolam
This material MUST not be modified in any form without a prior permission from me.
Forfurther enquiries, comments orsuggestions, contact me through
n k e mo n I i n e200 1 @ia h o o.c o m
-r
Ghapter
t".$rH,ffi;f,ffi, ,;n
Goritentsf$lr'utt'^-t " '
page
Preface
Contents
1 Geometricat Characteristics of Cross-Sections of Rods
2 Construction of Diagrams of Internal Force Factors
2,1 Diagrams of Normal Forces
2.2 Diagrams of Twisting Moments
3. Construction of Shear Force and Bending Moment Diagrams
3.1 Cantilever Beams
3.2 Two-Supported Beams
3.3 tnternal Force Factors for Statically Determinate Frames,.
4. Analysis of Stresses and Strains
4.i Calculaiion of Stresses in Planes of General Position Using Analytical
and Mohr's Graphical Methods
4.2 Determination of Principal Stresses t\.-_
1
2
3-18
19
20 -23 I
J
23-2526-2828-36
36-4243-51
52
53-56
57 --624,3Deter'minationofStressandStrainUsingGeneralize
5. Strength Analysis and Designing of Rods and Rod Systems in Tension and
Compression,
6. Strength Analysis and Designing of Shafts and Beams in Torsion.
7, Strength Analysis and Designing of Beams in Plane Bending.
8. Combined Loading of a Bar System.
L Unsymmetrical/Oblique Bending
10. Eccentricity in Tension and Compression.
11, Generalized Force. Generalized Displacement.
12. Multi-Span/ Continuous Beams
13. Buckling of Columns
14. Pressure Vesiels ( Symmetricat Thin-Walled Shells)
References
. :.1.
63-80
81 - 101
102 - 133
134- 158
159- 164
165 - 172
173 -. 198
199 - 217
218 - 23A
241
,f;r;-iJ.:';' 9
Geometrical Characteristics ofGross-sections of a Rod
Qreatness is not something orle ccuTL infierit nor 7uy; itis rather cLrL fi,onorary titfe which frfn cortfers on har{'w orfring vnerl art{ w omerL
Chukwu, Celestine Nkemakolam.
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r W cAt"*t"&A v^/nPz
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7. zi "ot
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il
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5l A. q5 *"*
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rr"
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TAtIo.
gF Pf-oa q cf W tN*Tt A oF
A
=Lt3L
= 5xn5 o--;7- =AB ?c',ntoE -.-
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= *2- c.,,, /
-
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2z3o t- tl'.y ($n)^ + +2 t- b6 (-s,Tt)QSlo6cot'f
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/
LL h4
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Trgelc
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LJ
Dao =rJ5ri c.e r'fq*rs #
ha-s t?1e
4^*7,-
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q
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0)G*)A
7+toht+Z
l.e*r"rr;.,.r*- *t\L )L2 n
.A 2-
= 6xt?-_ e7t
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4z*-.
tlr\ +ta f/uczJr*,< bi^nt,
f5
I_tJ
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+z
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rwefie^)T s 6f rN&Tt+ 0t TF+e
f* Yh! D n'e..<l-oohlrn'.
han 2 d,p = A h"*.
-t- L1
U.F T FTtr
trt rt+E
56Crr unf
- Ve. ezre-a.
tfuVtt ,fLt-a-rt
g-Lr-r. eyl4w"-/
t S alf ruXt'lha,&lc
*rle - Th-4; htre *,
Ade =
?nnsct/kt*fnwGt f fra-
T, o,^A -f .LV
("A)TfrnrQ*-t-LENTA,OI ? I4.L
;,
.r...ii!t ," ;:",..'.
-T.J
I
*n0- ft+<-olrutqdLl'd
lu = l,'-9. =
T^*t-
*mm
Tr, * f*.---2,
tn4." h-aic:
r ll*" =r*.) + rri*.-VL a /l/ge,?s *slot\4 /
.r/(==-/ + (-w.aO
? 3tt 7. ? /
7.1I *. 5t/?-s?r_,^*7- ?/ : +?7. s4 .^^^ f
= 5l?,15+5loL +/,
= 2307.+B
9-9o7.+p 1-
2Y01.+y
2-
_f
A3o
23o
Cl"-e-ot<";-4 Jaa ye4u44
(i) T*"n
5)t 1.s? 7 5/oL 7'StX,,Qs2
lrt rnotl- Ca:e_t L.rco-.+Az fre!,o'ta %n *f7r,
th.a.f-*, (;) il rtff o_*aASot.* C+,|-e--a
4 ?7-5y
li;;) Lr = 4.*" Cos r,d, f *f --T+5c -La.-_-;--
561s.
5 6t8.
o//b
?s
- 2+t.gt
-?"1L, 39
eos(- 6.r+) #
* z+6. t3
3i^ 74 O
5t2,q5 ^S toL s;. C-G.t*)
fto
Lrie,ntai*,a ry rdp, S lb-u! 4. Jo*, un;./41
+o ,1" e*i s .
$de e,rr-J4
-dp ,, ,,
qnql'r'- c to qa.rt,>e- nflrr*^t-,
ChcKota* rryd;
rqvaa 7-
re +sc
!,is}:i'l
Sin C*
chy< fiV V*^
J*.+4;a *"
^ qX,t
7Tr,,^ q{-+)
^-J
hc"*t'(t-o
i:,,.i:r _r r.
'-1.., ,'
U,r*r qx,a Sb*d) l'e
t{') /""D^ fe'/'^)'Od+-
ii.'_!
trl
{ ..-J.. - -
--r
-:I
I
Gonstruction of Diagrams of lnternalForce factors
Normal Force Diagrams.
Twisting Moment Diagrarns'
fhere ctre two tyyes of yeoyte in the worfd: tftose wfin ntafre
things hayyen ancftfr"ose wfrn watcfi. tfiings hayyen. lf ott, 6LS 611't
n lgirnnr ilTout{ strive to tnafr.e thinq hayyen--:ro! things
ffi"at can {estroy, but tfiings tfr,at witf irnyrove tlie fives oirrtanfr.in{.
Chukw u Cel estine Nkema ko I am .
rg
0NgTF,"uterto,0 Df NU{qiY,{w . h.tcesT Erg Sror'r A_^).s Eq{{Sjl ilSrrue, mef *oa*-vF SEcr t Orr;S ]
+ {Ylaltz q Sec*ton q+
VP a P Phcat-t mt ef q
di +rnce ,tc, Q,fj<,r *Lu
*r. ,tl*,r<- 44^9,4.e ,s a./L
E@,,/
fu*f c';,ut
+lr""az1
A* S-<c./*,rn
-,u- ;t +d-
/s
B1
F,
EAo
Givr."'-Q = J,rn
$: LaC-= )bwl.)_A : Sot*
"ora) - R. =I
boxn '
N&,*-t fo.&
2fi Ki
t-u.ef-a.a t
do{e;9,
LtTJ
J"-^t*-Hn..q-
n-14244
Uhc*qe- i," Co',:>t- Sechnta,L Q+c-a oP ct $e.aaot'
=) fr'-C<..s ,L'r" c4-e I j-oru a,,-Ls .+4t-4-
Cavtl,< s si v€ , clrnrl ; ,ln s;q n ar< ,! a)cLir<c+e-J &alay fr--, #: t< c*ri--
f s -l_enug, Lt- "
JX :", ff* :.:;*tr, n!* &;' ^T; J:ru .Jr'p op Jata bear,,t ..tt l*rt O-i t're,"ufdrr-t- C* Sc c.*-u,-',u ,a (a,<.) Con y)e," J . firyc. (i)Cu-,t,,'e,";1 Cj,* Scc.l-.,.^ a.,>-e- nfr_ kyr.t., olr.* rl q* l-;a
=) flos,'i,tr C*v.;irr [-Lc rwyy*-vrta
j"ofrlr* clh<
T-y1u a+<
pcS'i*..--'. af*
A/>io-. ,.4.cr:
fl+Cn4;-.-;7--
tdlnt*on- cvbo.Le *Lt_q ZL.a L,,tir1Ct,rr4tn'il tX.lr;i -
fr,-caz
SrrrC; ft
rz/r:.i cr
t-4u !o-r/ e
{ttoy, Sec,{-'p-ru f,, n q^a Tff a,f44.p , )..--h, tr %A A.i
Do+*"n) r,4-1,a.t DF f.l* crt/o-4 \<,0*;*\
TI
Nr=e, Fr = 7o x'tJ
:[ -E.lr
= ?o+eo =D-B.'ur
N, tl = F, pFz F3=' ?o*a-o = L-c =
Pos, +"u-q 3o XrJ ,. cl-i,t cl*..["J-tr-az.e {i+e ., .}4te )_(_4 c+?r+}1 fo-&O'[ f un'rzl-e J,^te crl-* +..=.* R,
1o xA
0 A..rc < I
B o x"l,
c.iel,; --+-n-,( .J-hrs ---> l"'I., R, "5 - 3 o q,.i Ja,..in-f
,lI
j
I
I
' I bbile'un'|' , KN , e. nklAtw< +nA*
14,,s- "U+n^(a,IrW .t-l^-L
b4,14"rcn*. pv\*rt
ID,:r,",d- Prv*l-
No I I. rr]]
3"4)dF fu<-z
flrt-ct
eqnA-r<a*J C/,r** cv??L,rt*-l,,
&\A- I'Cr-Cd-qJh &
Ir I
R
I
I
F3 E,2>_--_ tr
r'x
$ec*,.i. ( I, S, D ).
Fr=Fa=
ryft=-
,q');;'3"rtid = lrtg -. li,aa ''
4:'4ni5o xrr]
1o KN
6o t<nl
5ft1^
i
I
;
I
-_
I
I
I
i(
la:
I
I
lI :.
1
si^ c-e Nrl' LA* Po-cs
+4/\-e. Y<gs1*n Fr-Ca ,
t9,o ------> R
fYLatts- fu,cr^.-ot ) - E
e+e,4'; d4r.-; of - AJx ,d"y kc,t<,e,t t
II.rfNz &)
-tr -'r,tr
N, ['r;
xr-r.E
/rjx &, -*F, + F,5o + cl o =: i*O ruu)
JJ._ E.rV
,Ux 8) E +F^,-F-5o +1o *6p
q1*;*q c^t +hj0
f7, sLw^Ld b<
-trlt
- Fr =
-5o K^l
*5O 114
O 4:r ./- ct
CI<.x t 4
I no n-0.,\ F"tt is\ b"* ct t)ta..at in
o ('x"4 c
D<x<J
-ao Ki
*l*??o4 * )d;u**A
"a PL;*,1 )/|t'Ltt /
'4-b -Q.o t
*Lta,ta'
.4ht-,
a
-*it--
-i--Ti
tts'j
rft . ,l s.N,
: .tr F3- <-<
...:r.'I
l
I
-
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DRMdttlu cr -f il/S f t N C.t [YtarYlHJ f Dt.trc, R. MnA
ln oA l,ul'.*- rc -J'l,rc ftllvtts ̂ *16 l"tu
, Wt-Lr"J of clr.*,*.4s{ c$-4'7u?rj L'5 ''fttq s a-nc a4 tr1 't-<.n svvn a-e Caiup*.fl r;u .
' lf bc\t' V'J ert 44u* ,sLaft f-n ,#,a e*urA , c Lccaror..r-€ pwva.dfilL*, S'lu,Lt bq Corx"^f,c,or0 pou-i-,tn u,Iti"Lc c-tvtl-t- ckc*t,,onf\\,s*'R** b *A**_-*
AtrI(W d,,"
tF .J4tq .
#& , +4tn-n-
+-o ec,l- 'tw
Yn* Cqrt c.ttarrV *{r ) 5
\')J"n C*tw+,-x;y, , btct-Auurc- 44v*fu wtr*.u'.t'h:x3*. "A q.dyp+o{,.i-
de.u.r ,*#hLq,Lcdr,t<-8 ftWrn-at* q-C+it
+41Ja fe*c+rt 4 mn a,,*,'r:, ff*OY'T:^-14^{ n e<qtj:. Jr,r" c{-rn a*_J .\r"& Ve<a -
29
-rt
,,.k'*'4
r>ll*5VL
. G;lao':.,:,
d,2 \t*( = ttfi,L = StY\-'
lil, = SO,1,11r1-
fl'l1 = 'lo luo,*
01. : 80 xr/"t
Strl,'Ll;,r^
Ler ul Lt'&<
T*fi1
*lt- p6,l64ra1,a.rd \niln Csy Lew*iatL
Orv-&
[-
, C -r- 4 J-
)e*a-nr.ri tt*t"rn o-F 6x: .1.^l is nf,O"vr-t -fl J-€
TT,,Ilil;, Gl --
lI-*n-
nr} e) *
=
m=- E
[t't,I oq =
- lr4t -: * 5c K'rd rru
o (x z- Q
0,(x { l,
o <)c < c
- ftlr 1_ il'i.1
* sc.r (lO = 4O lv,.l rn
- 0t, {-fl1,A - n,l3
-50 +cto ^to - * tfDt<t) m
1 of- l-.t"J,.i $ tVld-tl,t-t-vr:f
I fvjei Er rtrMg
lIT1
tin@ t x'i'n]
5
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NGrtrd-'t4''rf-{Ar- " rc-a"cr4qJvt fvtsrnr^:F MR, h^1 +,,
s;ca-;1A;;-i*+: +'*U
[11u,, ',{ - +oltt)m, i-tLo*LM$'!\a,"*
b,e + +OK/Jm
tg*r),Q,^*1'rj* .
l
Q*
A=t ,1,
rc
Qi,tea )
q - ?-an'A
= +,nC = ,rn-
il1, :8Dx,urrt01o *- kc xru ruil1o: *oK,v.,*
T'C+ezu ,; .a+.ry oF +w ,l. .f\,q fno.Lrc4./- &tft_r+ E< c+-u."'-l- .iJ_ J
l
Ii
1_
!
Irlic,r
tr-rh1,E&.,
'm -ilrn1rEo.,r
=, fl'1, = &c'ltarr.f <x < r
= f\'1, +fl1r
= E, + a-D = IUO Kdrn
= 44, J- n1d * 1113
: 80 .+ro -- ?r, _: ZC rcnn
S'' c-q r11r[8t --- +3op,l- , {trt-t-,1 (l1a : - 30tr^rrt
7a.411 tF .l--r; rwanat Jsh' ['q-*"-r'-5
[1r^1
E lylg[.
fvl
A An=5 ,1,
2
80
M, I xru, ]
Nkr4^ilA,,*
Gonstruction of Shear Force andBending Moment Diagrams
Cantilever Beams
Two Supported Beams
Statically Determinate Frames
{he cantifeyet wing of one aityfane tofcf 3t4t'. Tttgin.e and Jt4rs. fue["r, a.[ot"Le, sLLyyort tfi"e weight of thefttsefane inffigfi"t. ?fease, I can-not carry yoLL both". 'Br.tt it fiec{ earty c{tte to ittcreasecf 1enditryntoynent acting on it. 3{oy,eyet', tlte whtg of another ahyfane tofdthe couyfe " Britq affyour fanni[y members to tny honte". Attditfivec{ fonget hecantse those acfditiottafx,eights it carried c{ecreasedthe bencf|lxg rnoment acting on it. I 6now tfi-e acadentic stress irti<fLAI is 6ig; frovever, the wtore yoLL hufu yout'feffotv stttdents tot tnder s t and t hittg s, t he rnor e y ou w iff g ain Enotv [e dg e for y ot n's e [f.Don't 6e art is[and!!!!
Chukw u, Celestine Nkemakol am.
" ir' :,;i i ::+'s t ;..' * b
S HE44
g\rsr- W* ia At*"^'-fr q^ qlforc{ , d*taw +o #"4 b++ or +o
W str,,rrr of oil,!: {t"<.tL"l.{4^-a. riqhf, na*nL s;clq- .vF q sec+rrt .
U
"Jt*l*c"'t Ste'rna 0P i4^iL-
L$* w *JG ,}F{-o {ntL.
c
[-,,,J?.
bo4*p bnb"<r+Ir
vy
be'^A;^1 N\'s.*'a'\l i. dap.,*-O 4A +t^2
fWe,rnns,utS o'F crJJ. *l* WA e-;a+r_t",
q St C{^ztrt'
Fh. Slu_A- Wa Csr),Q, Slt-ei,, fu-ce na^t^-? cr,yt w1fiw.A l,r<cli,n-+o -vt\-L Lep+ [tr*,[ EF q Se c*. crru €ry .[r,.tn "ro-.( Ju rec*i:n 'jc. .t-t^rz
fi3lct !ta-^^.{ c+ s.'c{-e of- .{4.{ SeCfi'crr lait[ b€ +ayprL a4 Pi\,firfu;Si,*i \c-h s e. n(Aa.-h{r Str uritt bs e"u -J,i,ur,.-* l.1aut a Jt*l n,y*,JC", r* U{-r*, +r ,f.lnt
--*;-+** LOA- U F .ptz Se C..pr,u,x Vr up;icv,Js
c{,i"ecf*'. & -Pfta "i;3
Ur* ofi ,l.r,a seL..(-,,),re . lrr E{Lrw ,.t.Jr
/ A .CLtr,tut*<- ,lir.r- r.tiut b€ *+at*nt a^ frx^pig4 (+ve) i,L *''J crry\ a^*.d.iol{ur:i+-c -\\tc r-.r- r.d iLL )ae lo,tts^ u ng t*-,<r C- r.l J
+Q -e| -c-lr:
Fo (L._ uzl .
Fe' "d\
ruoYh:,.^*
*Mo
rl, v, r J,J,,
L.
I Pq.I,
\/l4^ >ov2.+
x
= k,r.q vf *tlr.
f (-cla,nA"diq'rU
dis'triuw+v"t
llrqq W l+t-s-fticwqu/a.rcl,i s+r,! t
"r-l-r' o:rt-
P"r= 7 t
is trrrrun be La.,t Zen-d [ .uc
lno;n cvLre^t-t Zo,.t, L; os
,rb S<f cu-,,*e
b"t"lr fl n1frr\\t^,* Wf1^
.|Ve Sl,tar..pr..C( fl*f\ iS
C Aal f , \ qJ. E****F-"ftt1i3--(go*S )I nJir( +A +D git\L,fu_ *Leo_ fi.rs.l- fnrblon.t(gtvti,leo-,t*imt ffi *t,a Aclin-y firtos
+\fa_
v,Sitt.
',- ,AL -\ ]bL
_____/O___-=
C1
Qrienux-I'l--) Q: a,rn
0 c- trw' cl ^ t[.r^4
? = ?.0 r<ruil
M = fic KtlrttI, = ts kN ln
Q- "J nL,"*'gl*\r+u*n .
I,X a^i
w &i",tM'
be*+tet^n
4,,f Ph'ctuh\n_
\
TIrf_tr
/+-B,(!-c.r,^ul c-*oU''F- L.x,.J. ,
qr( -&'s|*nco r, x ,
I
6;id[i"''iicdo" f .-f.t'
&s+- ,'b^j<A
&;s+or"cp L,)
?rt
Fs'r .ta,u S\n-a-or f$-@ rv,Q-
Pa ro$+rJea +'t,ts bzqn*
P lo}-ar,{a.A "Yt-q leu"*t
e4/.1; \J'J-a,^X-, ?u2. =t'
SXLatl Lux,tLe
'}+rnl.\ +" rg-{q,,Fe \l.hi- bea,rtt- "*^J"^J Sec.{rz'n f \t cuA"cttxuo;pd^ ,a c.lau/t v0l^^ d^ i s Ao{
^rt r]j W si5 n, b.^;,1,.\*^i1,t
. . r tp+€"" & {-o ber^A -H,.!. bea,r,+ q s s h,fi,.rr..-
f pL,t cl.a..t Ol pn^^ rl c.a^ r l
+q,^) seL'hrn L is Dt
ftltt,= P.;c : e.xr,ri.on ,}= O, px- = so.xO .= Di,ul,u,n rc=cl=4-, ?;c= Q.r;t+: EOxrtm:
f'-E0 (:c< e
O/-d
IiL-
-PP I: x(o)-p l=*, -olr=) =-?.Drcil
l"==.=, =
b,-*,1 t", i+s
F*" 13 , u.rt
l
[uo*t- 1r1,\e,*
C6ox-t,bn\ = _F,Je
arv\*idn;u<"ou-<\ = - Vl<.'
_'. eF, - ?*= /x-
l5 *'2D= - SKA)
1e,pla{-^j 4^A
AC.h'r-1 ct:F q.
P lPe+B
r.l\Y Vr.ar: fvw,rwr/"\f' q/Ym il-b$rn q4 SLG"^m i
P"l. h-a4 t r*r*r** @rt^
q SLwtrn'
f*,r1,t
= *V€
III ,--l br."J s *.t e beA,^
I= -ve.
gunll*r*r- Exn*;*90 1'144 .7."4U mu,Yt Lzt1 LctrJl .
T[Lo4-{ is no lcLrictA,cwe.P rlgn in VaJlrer cF
Qz. Tkcreprs ,,iJ" tShr,c( rtr-) l)e ColC*rUU*
Itr'
lP
-'- nlrfro : ?(A+*)- pn1-
= pC.t+x)* 7r.*p(,0"u) - 1.0.4
* f,wr1 St c*rai,rr<
: 7D G) =r D = &) r-t"^,*..
"o(++t) * is./
-,)-"
:- c1?" s.r,ln+
O< >c
I
:E -*trr
rrh%, =
P*r- P
1,c-P =
l?. 5 1<dq
=
=Jc=o
P ( ,0+e+x)
xo Cs+*)
v r.-lu , 1Jf - lrq r---'->7"/{+
o l4;,t ,f'^,nn1
lo+Ai1 IVren,u,.J ty^,f1\ W-\ ,.r^ nub ,rup * ( t *+ ')
;!,;
:
l/+.s
ltlltL
Q,trl ,[r, ]2C^u
*\-o* Jaq/,\ Ltua V^rtl\ e? *,tiLl S lL ilos i-gt"S - ltL- : M
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I.II
/I
M
G"ue- '
Q= 5ot6 - 3,.tCs 4m-
P - ;DK/J01 6o X.rnrtu
hu = ?af,N/m
r 7,.vb
- iD QP [r<nri
6O
I
1t.s
43,3
T-rQ*"n,
M,,,
X--t . ,0<>c
T
Qov = -P + 7,*^'kl*__*la-LbJ= -so t- tD>< C r-{ l
-nl r- P*-ry_[q+] L*il+
I
it
,
l
)
l
!
t)
= *M = -6ox,'srrt
q,rn$-'
W'e {""}-\{* .{t,j
l-' thY),,./
o *U*'"t) '''>lw)
I = -soK^, / =- -so+3ol:'=-J I * = g=.
"l -= -R[)Kd'
b si ^a.,
t r,r
/ rtr*t1d
oi-'t =-
'{n'r"lLoi r*Le
llLxZ;
7*.^o ('8 -, )----7-<)
P"l.= fhto.S, ctr
ff- -i4a -t;',.|s 7s;.\.. te J *=t:/ '*bo
tt,;
r 5Dx
6ro rr{rt
._ aor'fgc.)-x]-; L _?-Ji_/ -- Cox'ttnt/ x:3
ffin<.---__a_--
Ir
Qrvt =
-D
-P'r 7*4 .t-
0 (>c ( q--
*T5XL
^ /tzLLx + ,/rno-\--*::r, .lc
= /r,- ('L^^^ r Z*^ ('t-r))\ r
./)
b /u'
6\
:O T-Z,o^X I i* t'r ,.'-la- L bJ
= .X^* x- [ar- x- 'le L b.-J
7, )c
+^3
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l5x
=4:
)
.f,
1,. ,
+*
= _5O 1_?O
I
I : -?fi x,,t,(r.=,
: -fi1 +p (Se-
0**) - 3D
l=I
Ix=
S K.ti *.
f\tr,ri.,. c***.1& sf #\*
'To q.Di d ++.rs
l>nrLtll'*, $O,lt Ce4 St,"4t'tlrh,e f r; a6wtt+ l--rl {"'lai sh, q3
It
H*,fa+, 1
l at \ -l-o+tz.'"nn4 eLzcxl =fzu'-x1x^ I yzu -x, JT
= -eD fsoI/- ?ox,,t*lx =o ?;3
2 e. SKNtl
s
{
l
e-
fn*
'-riw;i;;;atr; dA rti,lr,n *,
tOhUn.tur *hore is uru fur,r,f, (rnrto,,g.,A* ) /;s*;'t,*"!U"*r-l;- ntfin-o^..st
"u^r*;"d f fuY l1+e,'2|' S<c*'an S4\4t( bQ 7tu*1""nc
h"l, -flcz
i < h^4 rC rai *e,l ,t,s ftw+ I f *rl . ln 14A;4f G4< ) we SW*rlca{fu.tal-e ,H\a /hA,4rw,?, b*^l;fl w.t il(r*^ uL ,n-a E4cr/-lir, )funl i'.ls /ci* W lica*t4,rr, Au . tF Crvtcule*el valra. of Xe/'s rlLfiR r-raa.n ,t lro Ju's{€v;4t:!- rF *a-e. /_or, il 5/194-r;72air,< , i4t-!_ra$rs<l b< lLo npnl -i-o (e,
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Tl.tn_ b444- a^n eol)e-, d^liR A 4-Lrq .
f ,: !"r, , s ct c.hr,^fl< vh 5,.,t4 66
LC"x aun +e{, a*"-1ff-fi1 ii
JSLLq,L(
ct = 5.*t
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lA/ eo
ln
%;
8/t*.r-J, &wep,/* frhJ /t$*^ b 6A4hry )( = X<eJn 2-
CL+ llLl^o^,
T lt-ao<f*e, (t :a. ta*,a'd
@x:
n -D\-/ l.
''J<. =
J4une /'s al,sa Clw_/x ,,* 2^0 r-ri ^-1 * e S
eyo+'u:"
au ; ,n,
/-) ]ri(1 1- (xi
t't
e1r".*l ./b
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+ {r-Y":a
F 1-,** 4
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qt-t
7-^^ 4 l
u uJu<t v/: 0* ?Btr ,ur"oia j
Zont) a,^-r;l ttan j4cfiazal
= I [-So-si(il1rs L --*- .J
: lnT3m d/fiq
{dwa 6f QJ,,, C^lc^tr-r-,/
II
EH
:1
f,.1
.:
Ei
"]I
I:rl
i
4 T!: ,o ),o,',
Gi'l t-, '.
" q^ 2.tfl6- 4m
I ,\tL P, = zst<^)7-' Pe= fuxa.)
q'; = 15 r'f't4"L
7_** = Zow/^ll,l =_ +orrlr^'
lr- .+*r{ r[ al i{"l,ra>t
Sr", Ce rl CgurCn.*r.4-oL
O&), l- xultt L
t\u En':
3 61--*'b"'
h,le shp.l fiww4- *r" *u"r,*.{4^r- rQ-c}+,^awL{}r Jr'sfr,'br+e c0 t0-4.,0
o
fsrca , ?,', i^ o4plid uol*{.r,irr- i+.
'rn -'l/ z-
+-O
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I r i --rL-=!--1. 'r--{'q -to-!/l-.-,=-<---*---_--. - .-,
.
iI
I
!, -.-c:.^ffllbs
7-L= 7r-^ *Gn;,' rlgrr'"a +r\e +tt>^A,
,,rr,t JiSkK
z,DTmrl--
0(x4gfi^,1,,,^;fund;r",, ft^z
/ta/h rwq+ b-<'q Aq-Lle
I/ - 3oKd'I
f )c - c_-_3
':I- fQu"n, q x'-
'* T.
= tp)L" i :,3 lio
-r
- - 20 Y/nn'\
,r= d'.r I z
rlI = -3o Kilrn-
/ra - c--'\
"o, = -L*t(+)ac L5/
I /,=,
l ;j,: )
,.] +M
o 'f- 7"
[t"'f Ll-
Aaic,r =
lL\i
3- -u
ffet =
>c= t/t=t
c'r X,{,).t
s-4 l= l0K.,r;^
I rc:)
4CkthJDiJ
-r--30 (r+x)
o <r< E^I[ -8.
Qt, .= 3c>- 3o* B5
.ilI\4ic.,= *"J.^,^C f gr!
a L3 a
= -3ofz.t, ] + 4D
+-7,t2'
.+- () '/ )C' y i L,r-
15^ I=as / = 5rc*[2c =o l r= 6/^ ="_-t
"xJ+ M r/4t#-, _f;x
.f :o[l*>cJ -_ 3sr( + ], s**] = -6orct{rv\lx= 8/r=a
a]lnart t- ,1, g
-1
)c:o
E_Q,*
U@)= w
e
---. ,./-"::'-/- -'----<
' ''I[v&l : *fu: [*+ tur] * n'ir,b(*i) p,(++x) + n;
*3o f s+. ] +- 4o + Go(a+x) -aE (z r x) + 2oxl_ ,,, t
| - -6roxnrrr,f : - -3.OKAl rn @llu_')L=ro f !-:e=2-
GRrt++s sr Q?fi1iu,)6"' lYWfYlQJr ff^ P. . .'S peA/&' :Frfl' Ce
Tr,rfo swrtFb*reo '.. eq+lqI
G,,it-:
F= tor<.r0
7, = l-O*rtlrrt
lYl = rto KNzLQ= a-m (: lm(- s Zl^
.(t{,t +rrry
l.'-
(,i,1 C [r'ir're *oyru szh,. c*twtr-t^vrt*,t f-r.. *au at*e-a-nu.1a_fi\.rr_Df- N!r,!La^dt ( tr.ot br.r.,. J,_g n v*r^^.t ) ,. Ls+ ri-s ur us, por .,r 51*v,. &,
Gn ^ c'*o *D
(irtS t [**: 61,a1,,'.7,-c,t-r7. siA,^ .Crmtt+m:t_.o,_
Lcl- t La $tt.c J; .*i 1 +,
(;)r(ii),.^^A Q,,) Slr,r,^ra/ bQ- ,t+,,1 !{!y Jr*'^X ,nu ch.d.s,ryta,,LA:r,{-
Df Y{o\c4-"on fcr(u.
(f rl LLDI" efla-a4za-t3 e,f e7tt'L Lr,'u,ryL-
fttct't ZFy-a
(v) tf b.,ifxu-t ,
f,Z lv =- o2tvL =- o
a.*g ef0r+ q n?1A*iqe- S,yn jru
^q4ei+ 1'^4{r"es l-hA* W AVgwgf- 4tu qrb,'+*^X qnm) Csrr.c-yF,n),^rt
(R6
Q*('' , Mrr- 1
tc,) ?r.o"o yeqc4ir_ Rrc<s k" ^"4(e cut \,*f l&{+4 * *) SIn ArN Y o, r";ya,rp- dt re t+i.on
+-) At /o"irrr.- Jrloa< u*u&rata1,1_
h - lrt-;zo,^-l-'-!
V - n+-rt tnJ-'
Lqiad-^rJ peaa;d4a,rw 4.h4 dr,.ec./r+n
no *tu ftrA (s ) ,
-
-l
-
H.
)
= -ID (^* t.r-a)'F aox2.(,'*,o*) t+D - 4Cm)
:- ?, s rcnl
Zn4R = ,3
: -c, .+ |ct.grq '2'
R* = F..- rlt*7
- a" fl,i + Rn*.
Si"& 4,ha-
t^e+a.;a fi4q a-r b i.l-rcvry dr' r< A^),-'t
o.<,>c c
S;3n is 6'S,+;Je- ) ute St*".fJ
+ 04- afi1 .t R*(b.+c) =C
frA' .+- X./t'l
b+L.
= ?.o *+o - 4 a-t- z(+c)/l-
(L*ox-^a -i-.[,ta re-yw(*
Ztrrt -- o
= Rr+* fle.f F Xrzs'+2s '+-,o -ao(a)= O ,..
D%+,rn*tnc\,h',^ c+ SWp,l fc,A a, _J trcr*).,,\1 q@hde-€"^f:-J."-
-
IAk k)^ ,br*r Se0hi,<a SfrrH?,A e),.t41a4- .feu ** h+_{ 01_Yi A.^*.
J_ I
Qle,Mfe.r
= 5K0,
-
-e(+o) +si( 6 s x^i,n
-t,:' Qt,r [XruJ5
T'T-Qin, =IIf o.r :
5
5a
*Ro
-214
-Bo
f|rrrt Lx nnl
- - Sx")
t (n(ctr) t n1
t' s (?*>c) 'i-^lo
o<>< 4
A /-x. < ci
l=-*5xNn l= -eDld.,Pt'l:-;D lsc=b. r
.--
I
trr-F^'flU*w) = -Ro-(o+?,r,
: *S-?-s.t- QDrr l= -3or.n J= lOi(^/lx=a l*="t=^
t,rh t^-.t,
0+ u",t^t-lq. drf+"u-rJ ;)qJ c^,u.A c
Co,Lca,taet\. T.n-o.u.eq-er< r rV"LW Vrr;;ffiNtflo" = -trn + (n (.o 4+rr) + rv1+ fls,. xt
= *8o +5 (++x) ++D +asx-tD.:cz a'
I x:o lx=q=e,38
ir'rrt .rl
"4/-^+< Q27oo, = ''o " "' " BaJrSh*h/tp .2q-:"ae'
(,1+]fu = S-las = 1,5
hl
, 6i*.o\ * fc,*+ A,\ n -1-va '(\t,
"k-",f\rtl.,.wp* 4,t.fl.i,'t K v-tfl.ct+s n1 = *aS-(_65)_-- /*Oj<rur't'
i
o/.mq \
6
^*.4 ka
Fot;r-r'(-a
(r\)
Cti,i iln< .
Q:.t-n1.
I =ZmC = lrrt
F = l5/<N'
|u,rx = ).0 fW ft,t
L = lap.t/,,n
fl4 = tO Kilm'h^
toa< I*J I'R* Ra
**( d.o*nu$,
*9* *.'(c * 1.x.
1.
8U?*1?"\R .Q !*=XeTTnA (rt)trlnnax = --8u+s (**
= 9." 5 i(4 ,rt
Clwra;, f* o*.**,ln + e .)4-...Ph crp Q.z
"i4r.{- aLr",'14 q* 1,i,n* 6-H^t t$+ e^^,{ refle c*g
a_-D
Mfe.l eV,n.-*a}w
l.s ) 1- /+o r- ?-5 ( t,s)'- t o ( 4 'tJ a
(+
retLa-s Re i< - 3tl .+ro: - eA'KiF = l0r<al
t4 rt F{uu+ard = '5 iqv '
*)
31
r<qc'(<iq Fr.U-4 Rft a--A ft.4
Zffi+ = t-)
-Fq +
Fq2.
+b)fl{ +Mr(u[u+c)- 2d(+,a a. \5R,- qr -Mt W(+.nr)
Ic.r c
tsCe) .*tod _ ,o + Q,Dr.Ia ( t ..)2.
3'1- *
ClAo*r^, 4+\L ?eu.rllli
ZFrn - o
PoS' t ^U <ttDJ.^/ez. +-n |a-<'Ll -yLr-e.J*Utt re I LaC ra(_C_J ,D (lyc+w1q *t4tlArL,',V*n J ,,.e c/r.r..
^!Xq'l-nL, ct+tst,,}K p,nl-a-,'lA
4-1^e,;4. r,.,rt fUdST ch**6< +1^AIna.h-ql cl"e"*r- .+ n; Fnw\_f *o t,
? .CI? KT
I zl\tu .:Q
= '-F ( a+++t)- Rn (urc) * xa(+ rr)in, * ,/,*cR,r = *F(,ur,+c) *rl,b(*n.) *,u, *
3l- I
* 3+, 1X t<,rl
(r.,
I
I
Il
1,.
--:: I
]
'l1
-*! j
)(ra7Lrur/ I
=O D-
^r4.-'.* 0e**Xt".ejI..fi;*IL
0*{'' I rr3
i
l
O /'.X- t- cl
7.o8
f-;.
lel := F .- l5i<nl
F-)t
lr';c l:a I = 3ot<"1^tr!:-D / :c = cl -- a
[1r'0,) =
0,<.x < b
{ Tlaa<. A qI
I ?*v. ,n s,-r,,l'ff{;,*:;-
-tr -'n^'EQu6*) - F -. R+' + t, >t
Mf0,) = FCq+*)_[Q*x+ X,*:L
= ls(a+ac) - 3?. 1e,x -,- s)Cl'Pdetwiql,,^r\
KNm'.I = 6.a+/z- b- g
/ = 3ol*
=.o
97 1e *Pto
6= O- F- A* r7-,x.A<= R**f = = 2,21 m.at-)
Jff'*h^ l[.
Ui bL) F*R*+7,1 -futAC
l5-Z+.clA.rBo_lDx*
-------- *r o O- _IthS t 5 'l7l-,t r).^j-r-L e4^*-riv .
.'fU-o,r<1-cx,-{)yC fuqh /yLu,r+ Aq (*+rle '/
X,L.C* o"). r11 - '/#'ttlr* 3o (t.s +x)*iD_
€=,
f"u-\..t'|tE €-f,,"t:w,(rt
= I $,&eFr.,
S,r,t's.h.l^'.fe r.F ' i""*"r
lo (t,5 + D, *) -to -
m.frLr,!,)^) = F Cq++,+*)
tE(srr()A* 5.'t 6r:C
|} c:,J e, rwi trq.lvt;.t fi-
b +x) .r
( a 'x)c
- 4.?, 1 2 * C, ") s -3 5. qtrrtA,1q t<)
*Ro(
3t'12t:l* =,
l
i
I
/\n_
Xea = t t--0'*r,{ }t':Luf- uA {-{r"*o -14^{ ;aos,{-..,^r_v VruL,^q # X" o*rcAe1^-r.rl-a, + f,lil
rtIttr_#.., = r s (s r s,9+) .- z+,1? (z +_ o, r +) +
= 9,38H- \ j,i,r
LLaw.Lra Jht r<a<u*i N.I t "
l* -t tu a- t.,4t1*M',".f+ o,* pt ft = Q* =.)r $ tt B = RB=
tn+'tr,e. hed'^x *!f4aL*a o* ptr K = tvl =
! ('.;
O A.x< c
6. Lrf - G+L) .- I o x,r/r.t
#*i
9"wf fvr++.
x,)
Sitc/t q \urlftn* |aa'lr;cl+ r,te*r'r:vu 4 *& wte<rrrn i,n rlo,r+ica.L o*r-rl, hfr,'ri zna-*-n'!
I
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fo.cc'
'TluLs Suuff*rt res.f,n-o{-s l'T;;;-^ yLJ i,, ..,ur-o
V4r1lta.0 al, anc-{-,o^ , }arut c11lg+u\\ l.uryr,-centr-\J
flA,O+.f:vu cr,,. c[ v'&,fa"t,'On . 'fLuctc .fel- e , ex.L,r-
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x.l
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)'Q. q c'tri-', {:ur Ce:.+ a^n ol
Se c+,ea"a o,r^-l ? -cix4 \ ,l; r*c*".eaar
la a-t-au
n.'t'f ct tlri ftvo.r- t'rrr,\
61,.r re(li,.c-';r. . a'lun< f-6r< ,
tJ cr-,h'tr-J ca.,^) ll0z,' Lc-rc*-,-i
fCr<Clra-'t fi++t-rwc."-J. t .
Qr
(ul
=>
+
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P'O i t -t -l-0,r":a-'c0-t
? ro-i -1,4^A
hfl. tr. -l,L\-a-
bL lnoten b,.c,lt^
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a.J>luv
tij:
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&rr.c+** oF Z -qx.1,r *o goi^;t-Jttn^ @-;-t f n +hP Lu>Vrowntut roJA
*o*'c,,3- f6.{r, \1-0.*rd7 - ax,A o'F +4"q
{4^Ad- *r? r.,J, lL faU ea.d^ vw,o,, C,"
6i,tc^ , q= un"-*- ':. b; dm
'tf
C= 3m,l= € = lrrt
F - 6ox,r)
Nl = 46y:^Jmv
I
7*u = n, = *o t<il/n
Nlo , Q*', fvlr""
(cLtn*-"o-,t
rd,'L[ lt ovLo {r.t'0 f.2ctti-}e-n
,*\ cta-b,\lro-f .Lit* c.hz:,r
^ 'n l't
f€..Ce.s Ari a^-ri Ko{ ca,ut\ t,,te
fe.r- '.J-tr.c.+'r
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.
ZF"
6or-*)
tt (+
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= F - (^it'
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i
)
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:r-rN,It'; - rc
Qj*, = fur:?q
I
= lox{ J= o
lr-
0/.x < q,
+O KN
Mr'e'r =
Ilr=o=,
o'<'i,'{"?[
?+Dt(N,"'
D{:c- < c.
, ad rn)
o /:) / r"- J=,r-
- tii.i- ,.-l 6'j'
.^t: -ffiKN
= &a^ h
NIu',
QJ*
rfe
E.N r.f,
GI-'Mi,
-- Rf
-FFrq
'-+o K/orn
*'trr-'
:-o:-D
= -* /'l'l :
Dd:c < Q.rV-lV :_
rrf "'* =
f,)r{c,l =
n{j'u =
l
I
i
lI
= - 3t *gxil6o xnl
6t,r< 2+01!-r-rrrt
_Ri,111.l\+
R,l ,. :A-:<,.ti-
rrtr', =
V-_
/riJ er
Q*u t,.l
^n:P%-w,wn a"hoa .t_}gg_
QL,=o = R^u - ?,xnxe= &': gT:y=
o,?Lrtz, +o
^, vHr#l = 3?.le (o.r+) * x+o +&D - ard 0,1,t)l *o= Aq+, g+,Frtn
o(x< b
*sr,AQ xN
l
l1n '[, )L
Zar.+s - 4f D2c
(ourc .+- (onq +rlt
/:r* l= 2tz'3+12c:d I 2(:3
I
I.l46
1
I-1
*t
q-
0 rri.{-J
,/Z FLt = G-D.-b.o = @
Z Fv = t \?.2-?,- tlo - 82.'e4
=oZ M = 2+ o -a. r3 .3a -\t,ts
-{B
37.+s
L\.ti.r.'i<^^', { 4ues_e&_"!
p ,^,{ i<
Jo'+-L P
iI
--l----+ 6 oI
tl,"3+.I8
6v i+.+e:.+o st.- " ',-7
I X2. , r.2 6o'
ZFy,- 6o-6o=-oZFv :3?.a.s -3+,+&:-O
ZM = ,.8r, ++o-?+D = EO
Y.,, c@r)
8e.fi\wvt*n*,
O,b+4r4f- {4^* ,J4^{- \ u,u {p,,',
J oi^l- f i.} eys,l *ebec**, r? lul
Vl 16 ix,,t r,-jJ.
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J o,n/.- l(2.1b.3,t
7jt/__-- i
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Gi.l'e^:{\1= 30Kd/n
[ : 5 Or<ritt - aQrrtt/nQ= Alnb^ vffic. = lrvte- = lfrYL
d = *rrt-
[{e a c*.',"'', Fu'u,s
r/ rtt
d,'4&'
A) a\?-rn^,;1at-4-rrvu c-r fln'
(:
*o= (,]'
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llg.t-1-:-
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?tr- I u.r; xrrj
z $11; =-o
F.r *ovIt p,
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t-lbl,f
J- 2Fc t
b+ ('* rr4 - e: (e-c0 )- n! r,RFC
F.r
I\i
Z-M e
Ro'
x * *Mt: d' ( u-,1)
= luD + tu.o .1- toCs)q* 3D *.lr:o ( +-a)
F Cq.+a) - ar ( e-c *J )*F (q+-tr) r-eF C e-c-.J)
*Eo(=) +too(r) + ,D (3)i+-3o
I
+
=-O
=
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st +M^
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4^e
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+- If,
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=
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= -tUOrcnl
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T -]L
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J"'
:
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F,t1,ra"
4
i
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=
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"'
il
Q**ocJ=o = R:-7,xnJ
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/\vf.- Rn'_ -W =11e4
n I --- -totl{r\)ntL' I
z=J Ir=4- *
fII- -1trN,Tc.,=
QJbc) =
O
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5Ox
.n tlI(n
tt tl= fin=-tOK^l
= lrl0x \- olx-o
Gr^dl^.\ u!
t 0 K,rt
=- 3 ut KnJ"-2c=3
r}.+e",*J W! f-4cfir"d'
D(rc /- ct
g "1::c < C.
0{rc< e-ce -c: J
til * r./
Nl't ,
Qrt*
:=
t
r\.ll\Jruun,,
V- V
xll o.r
r-r rJ
Vu Co'l
fitie,
2.')
K.
_ I NIE:J
It
Wn' '["*)
J0ua,6-, P cv^,.-.l K
to;ru1. PJ0i4J K
'Y ,IaoI I i*.--1'or(:-H -Klii-il".rl
,i...,..1
- *j--***-+ ro't)
.- txo ,
--l / *':<'r-Z- FX'r1 =
Z- l-y =
I d1) -- luF0 "=. O
tA++C*5O-:()
Ruu .+-'iu\ .*3-0rJ
ZFrt = lrr0 -tr,'\) =Q
Z Fu = ?,<: '- 2*4, = C2111..= ilo -,3-o = O'K
Z'f\4p =-
st
-
T'
,.:i
:
4-L\d
l
Analysis of Stresses and Strains
Calculation of Stresses in Planes of General PositionUsing Analytical and Mohr's Graphical Methods
Determination of Principal Stresses
Determination of Stresses and Strain Using GeneralizedHooke's Law.
foofs andfaiftn es syelu{tiyne cfiscttssing the vealittesses of tfteh'teacher. A teacher's yrimary ylLvyose is to gtdcfe yott totuan'ds tfteacqttisition of knowfedge, 6ttt no.t to give yott ttn{erstandhq. IfyoLL cantnot understancf things, stttc{y liar[er!!!
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EIED
Sl#4/r,tt4/Ag"r,L
Strength Analysis and Designing ofRods and Rod Systems in Tension
and Gompression
kententber tfr"at yolryarettts o1' syol'Lsovs syend so ntttcft tnoneyorL yoltt ecfttcation 1ecattse they tvantt yoLL to 1ecorne slLccestfrrttontot'row. Do not fet tfievn down!!!
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1
-: Nr..rr*,. - Rn'= -+t'KJ., INlro[,c, -- -Ro'+F= -+t i,o
T)t*g./hrh 6,r-r,tr:h Df Sfr,e.g{", J.,lna \{ r+..r!d-
6ro := D
[ro = aln:p = t&,E-A^
* *rxtJ3x z
.\'= * 6gt<N
= -l,sLxrD*3,,r-l
i
i
I
i
{r 8 *"1''",a-A eY 5X tur'-f
ft,o,t ,+ r s c.Lir.:err cL\ Or,'.i,r'
(o?B -
=
ax t)"x sx ,6*fA t-n*
ALne + ALsB 0j, d.eJ
- [,5 6xto-3+ ,\r= b , dt AtbA?ft
= -l'56xro*3+ 6gxro3xs + l,bxtu'Snbpxs2.Xxxtottx Sxta**
l,P6xto-3 rr- t, t6Xlo*3
Sxro-t,r : ALr',
II
I
'v)
i iii :, rli:ti,,:-.':rrrl
f;ffrt.,:jjrr l:1.;::t',.,, :: r.i rr :'': ,:.tl't:
qe
Gi..l<,., I Ir: trn , p= 2.gp^)
A = Boo i,I] , I \roftut24 , S= 7,rxto\t1/n3)e"l+-m'rw'' $ Areat F, ^l-!
'F- ft.r*t tlari:an.A ct
LJ,) fhe- lyt a,y fi* u+t* t *,X,m df fiq ns_&:
zFa
/^
:\
a"*rl b
, i €.au/, \lctr|r",nX
SS1^**fil- Vrr r i ca4;[- 1y
ZI-z = U- P, gad - % S-. c.= r]
"'P,= p,
o
P, [tt d
"' Pt =p,,=$"* Lo^ J,-h,,a,
o+ ( ru-lf-1
mAr\ Pr- *t (r",fht , eq= Sp
'l* P,r Co.i - P
a cDr d' ;;3Lr [$i'^:of Sfvc""Jr4;
F, 2ro -3lr-- = ll"ssxl0-
[tlr] - ta- ^,"
:
*l
l*j
I*l
i
\'lqri",ut- b?#
?r
5 ?,trxn 3x ],2q x /o^s .= O.s 6z tg /^
ZL= Q
, PrSr',4 ^P,t;aZT
1'-r:.. ?,) 'rb), 1Ai\
': .-.k'.--e
q a{-. y', C.os d *f z _fl =Tti ),
Thr \are,n ir rfril\ s+"*til, i"rLa*r-r"*,^+ .
dtat*e.fisrc ) tn ch-& kx*p^1.\,LirJ ely.aoh'trr-.
t'^+D eJa.-0,).-rr;,^ I
.tft: ?
2'o xLo 3
8, + rcr,)
6. st i<.'v '
5{r+^1h,1 ,
= 8,+x tD3 : 5,4+ x LD 'rrL *
l't = fz
Are n\a4
&tsuurrnr ,,7 44t-*+ a,?*1/ Ar{*-^^*;*. uF +r.s ^J, *ttp Ct,^*pz ;T#, Ad is n"Jt$;bb tM!il)hre- ch,'^ s+47.
Ilxc. v
*sr$\tCw*tra*- t,,L;.tH .!n ofr or<-- NtWA L, Cus3. : A Lr
P"h crxc - P, ltEFa
P, = /r- Cr:sL d
S"thfi.r*hi? +4r qtr
), ( l, Lri'd ) c^ ,x
A= -L.- :a_ Ccrjd + r
R = Pt &;sta
= &. * c-u.o 3. =
F,.*n tm ir'+-'a". Df
err.tllD +,
Fr) WLD.I ttoxn 6
3F=
Vo & xto9x S,+rr xro-5 : O ,4 "+
tg /rrt
fh an P"- u,u+ b-Vt'r G b
A,5b3 = l"Z3O,+L+
('r;l
P^^-*
F:.
9q_ -=Gu
C*^ct u\,\'r's (a-n
ro-l,t ,
YIb+1 Pc.' c,,nal b-^J.+k
,J'he"4 +h4- fi'rr4 t!.or;an*of J4-14 kbi./.Alan l4n4/A i+
t/aa-th,a*14e.4 ruer<
E Fl,
i
. -;id.*- t-
Q;,t*., b= l,JXto-arn, f,= O.oarrol
l=r.F.=Fl= Zxrr-,-f ^u
Er=E:-:€3 r ?xtosm,f.q
F ^-/ +4'\a nv'''^"l fi'YU "a'h\1 u'\ J4'\-(
ro J.,t a++e4 Q)4en bh*fl'
Jda4-q,A>,n bliy J4ts io.Lt, 'YL\a f+t YQc,t.l-tc,< fvrtct SLrctz./ Lvix
U= Br)
k?+wJ4uu",r l
1'IL__)
x
N,\ /,tt
J'l:
Z?*=a= N, Sr'n (.r - Al, Sri, 15'
N': Nr
= 2 fi, CorW- Nr.:0 - -- (iJta.rl r..n.0 is a_l ,i+,_q F 1,__n rx-/.ci
l\.TLc &' Y,o*, ,I: Aa{+"r,n a"d1c\
',.tt-. - ft-.li
\ t,-t,
Csnaot\,\ eynnr(,1-Air) cosre =- ALr
f * ryu(i-r) 1^,p =L Ea.Fz J
i+' trrC{,, ctr+-:.r Jrrr f c., *,,;-ruef-++t.c ro,L5 il*,, ^4QSSenlbt,'"-5
fr" i,, &-t^r.o {,r\a f.,\i-ir".,.rcL|4er, CLt\.<.rn"lt t;a7
'.l,,
Il{| . (,i
AI 't-l
't, ,
\n L -'/ -*'
":'r::-i,.
(
N, IGq E, F,
( -)
\
r-l\o\
-t
k?+* rzav(a"'er*i q^,t mffy t1^ed tr = 6e a-J F. : F r,
[ *'rosxtocxax to-1.0'or x,, "- /r]. ( ,o*,0-1 0.oz^,".)-].&x q -
-'o.\
tit
= 45,1s xl
t1 :3ao
C.t,rl<,rl Q: tnt
P: itroxnlJ,/rJ^ = 0. &
l, = Lt, Xru /,nc/z . -i- i, Xrl /'rn
U = tu\ w1't.q
..**.:.,, :, .: ,,:
" (* xuo** o. 1qr nJ") C,ls
s w - AJ r- = \>
A)r= lD3f 2z,os
15,2s ?6.r 3 Ki
),q @1",^
I
rI
4l.e
j
*t. ,o )t l' 5q x_-- l'5 4-__,r*--g_)
Solrr,/r.o,r
Agoc+"rr,'' f-vrCe> r.^-&
F,; A
{-trLa-,> or1 tr.,*
----' ---- ---l---- ---- -dl ad ,ll
*"J cnl- j Lr,uur.Y\ -flk{.lQxlan-^-l-
N1Rq
k
6L7
zrt1fr
loSQ q.--_:_---)<-.-.--.---_)
O= R+r1,,(err.1r,scr)_AJ,J_p*4 (,"un *?)^ Nc.
:o1,, (u .rt" sa)
6
-N*Sq
2q. t- ?(s-q it,5ct).1-
= tr21.5 &.N, +3so +ass,3 _5drAln I t46. tL - o. aN, [,0
\tv-n vf trd is 3il4*f cat! ;cl,.#u Mr',.'-*e..
+ (, s ct +q)pk'ur) + s,s q)
Tha
a C4rU ?-LasJo
!
i
j
I
.{
i
iI
-
I
-l .)
riF , 'r ro* d,n.fl,r r^q*,x
t .r^ f-14-Q Lr1y.c\,ttt )
gL.:- !c' I un' = aLy
l.r':ALzi 1or-_***+:;*=rc,@r
w€ C ct+\ Co.a.1r^ * Cvn F r*,tr, tiy e-/uA!-,.-,
Act 5q,
bBB
5AL, := ?CC'
2ALz
sNgFr,
N'.Nr,
= lN:_a. E.F,
r- / ,=r\5l r,/
F,
E n d"7r= Trlil
e / ',estu't'i = c'lsA
$uvLcshhuh^X Nr= 0, zs[Nt r,u,fo elwArtac.l i,'
i\1.* = tkl. L * o.+ (a"rsa No)': i33Kd
Nl, = D.ase 0tl)= ZLtxrl
c [:o-.o --j:*- -ry!-Y:tt* to',1"1 rA= N,1\-Ur i4 xrJ3:_-zt =
TI- d, *.\
l3 3xr03 ,4TFJ,"
ljLxttS,Tr d ,''
= Z+0.s x tD3.-12-ll dt
ii
W t^r- tYtArt+ eary ""^-tt^{ A)*r*no gp.,4^!
J.= t0.3:
= t,zs Jt
-11norTwr.t . ruA e is
tJ)t i+ tu Aul+'n";*
g;a.[([rJ",^J.
""J
a,J < SLto*(
\"a slrrn ,
<-J
lao'-s xt03 < [,*x106]-rr d, q*
d.v 24,,0.5 x ro3 O'0 33'''
O"o4l r*d,-T"
z,l4 x
rOoB -'
Lot) xtpb , '
J.= l"25J, =
Gir,al c( - trn
P = /tro K-n)
+ = o"s C=&,,,dt
?.= 4c xrJ/,n
Lr) = lrjDwra
{ 2cro,r",-a-[
. SOluu+tr,"Y'
*.-rl ya tr r*l+n, Wcr.a
(
A
e,n *l-L( ru._d
-rlh,.. d Ll, , cj a.
Cb-< 5 i"t*rrn b"€{"J
,a.'J
*---) st4nr- -l------------Zt
I
Q*3
Z?*
2U=oRh .r P - Nt' - Nr Si;, ,r,
Zfnf : o .n
= XQt)'- AJ, ('t,6rt)-N, 9;^a(c.,5,t) .+- P(zo)2
= Sl, ?. - l,6^l, - X, as d* + 3 uo
'rtr'f
t ! v-r."l K -i--V\:)J -\(L\/
1.,
qr.
$/\nuu
f$'!.' ,,'
i' :.- .:" i r-
^lllLL _
Auh3'crrn ,p hrd J,l+e",.r,'- \,t-ru*lI
rl:Vv
t4
up! N
Nr
+8
-Y C-*?qh bt tr'$ ,#fu
t , )4"bs' ac'l,6Q 7"6e
,)L,6A$'= l"b cc
?, (, r}t;
&= I4Nr 1,.(,
Cosot.
?o6[rU,l t. L f N7L, IL er,J Lco114 EF.J
Bt|'= AL,c'(": ALr= a6'c-,xd.
Ql\u-minl +44 br^_rJi,k uprol l'= L,
*d*tn ^3*,
op ,,Sj <.
Lr = Lt,/,/Co!4.
Ii= !):4 = ,!,,l-t lT,lr^h W= O.5?
t'-tl IL*""J = #t"H-]
: O,6iS
I
LO.ois+t.+r)+ Krl
N, = 0.,Gts(toV,d = 6["*rci(l'te C Mn--cS rltQ ,U /
,2
"64-8xro'
St,tbshJq*lH
0. 6 r5 lvl* :N^:
NI, = 0, 6,1 5 N, ,^-+o
Ltq,5 - 1,4/ l\)^
7tq"s
e-L (i;, )
T/, ''
108. +xto3-rar-
lu8.
Lf = N' =
E
(L= N.=F,
66,+x,oJ +I
ff d,'a
l0 &'+ ^tl'
, ,t =.I
Tr.fi
lo8,+x to j x4
*=''t,J.= -.d,
1- x(a J,) a
a>o-,'fh-w< f{e, . ,r-J 1 R
i+- +o et&* ^inn ' d4a
*tq t\lunt J*7rr""* ;Jqp #^{ ,"14'
r^^( lL\eil tts<
r.m;ldirr?n?ri,:,1-'-r--, .: r:i, :.\:,; : | ).: :.
,..... .,i, . .r l
d{t';;;-
1
II
L--
d*= -1, @t* 0"058*'
I
I
t!l 'l
l
11-,i
iirl
.:.J
!ri
&:f ,t'D'1
's}!+ffins-:4*].: .l::; .. ..::: .,: ::::: . :l
1:iir'llrlrj:i. I ri. :i'
:j j.- ,.-'! '
.,: '
GB. :
Strength Analysis and Designing ofShafts and Beams in Torsion
I
1*l
Do not stttdy jttst to yass exa?ninatiotts for you vi[[ soortforget vhatyotr [earnt. Stttcfl, ratfter, in order to Enow h-ov' to ayyfy yoln'linowfedge into rea[ fife situations. It is not the Cet'tificate thatnta,[<"es ybrt on ettgineer, br,tt tfie yrochtctit,e Frrowte{ge yoLL a.cqttired.
Chukw u, Celestine Nkemakol am,
rtJ
i--I- '
8t
St rpla#^i^4 A^4
o
,l,f,
trL ,l .. tt
;l.t
:i:ii
tl
.i
A rl\l ,qT ,+ A-^, AJJSTS sr s H74?r s &ru ? Be*nt t^r To4. S r o,'d
Qr< *:d W,lratuna\4"4\1 tru"tu , ;,-ol in' {, .o* Pntp*,
^ \-Ld<.+"A -Fo -{4tq f-o rt"d*? 4".W
A $';rt,",1 +y.w art- qz^-. in f,*f un-J Lr" {o prwe4-4--ratttn,il+"J
r ,r rvvr '1'--
tiit t fe\Lr +" ,,a* V^L in g,a;t ,^A exen<<_J \ f""Adr'r't*^ lvuq c[r,r rr er,1
T€n{,s-^ r<,fen +n {4La -J<rl,.t--^a crf q Sjzu-egr,r-J tvwmb<a-'\^h!,a i+ , s
^* -rt"e bT '
conLp {< s {4^p;l puJ.ou \g-4a*r.ot<- a,t'e,^*
it r L*^1i+*$-"^f cu,i^ - T\uir (-!rr.Ft{ g,r- (vrE.Mr*,J lr; ,'L ( t^a,Lg<o*
+/"-a. yxhwv\p,r ( S!,+pt w Leaw,,.) .J-o a. *Or1.*n Cau,t;^_.,5 *.trr,s{t,r {-g-Sll:-rr ir't '{-{W Veuhp-. Tf*t CO,,r,ptr,-^ }aa-o;-t pndrCc=l*v0,t{-,^{ Uf ct Sd-,r.r-r!t,,-,ui W.mku, ctwe Ccr-.& d i.OrLuve-t ,-l*vOis{--..-3 CU",tr1.ur gy -l+r,,s.E.-q fltt4ru!,,;i< (\1
lYLnna
=f = fi4
TrR
gg=L
L^na.
L-
A
eql_FT
h4
1-r
mALrru*yn Shtaa, S,lve*t A* 3l.t<-pt' Sur- fo C<
Slw-ctr si-re.A4 q.+ anJ "r_kot clc+a,nct_ r€*+^u-!- Ta lt'rta 61 St4q+t
An6t< Uf =l+'t;st
fil^:,l*tlr^ oP r4;L+7 6f tfuL ShA{+ thA*or"JIufil,h of J-lq.' ll^a,p.t--bwya *ra,ruiul+"fi b ** sL*+t
TLw- rgLq-*t,yr,t SJ-arn*e-^ l4,A+ -],lrp in k+t ti{3 gf S.Lq*_ -lA* a,rwS PL\;,^i 'tn l-Lt-e
*.) {t,t t < *grsf o-rt rs
14\-a Co-t'|,-e l;**1+v> CAtr rlr,'q^h^o fr,
14t<-l< fe(a-t",orrz , **.€ g 1,, a.Ll hdtacJ!<-t,Jri",l 4 SLtr-ft gv' r,: r|t un<.Jaa-'
Ts
t'\'ir
0(r< R
C,yrgs * |-ec-+w*, ef cL g t.rre.p t gMJ <4;.f rtt'u-+en-.o-( +r> i+s clra a^" u fr"_r.
N4 =Tr: l- f\\
Liltfe
q/F{4,/
lrtg-ll, .
rf,J,*.
uthtq-< ^lf =
.:U+€V b li5 h t'n-g {tw
--l-- -
J_J'r
'++4, u- "-t 9 < c,h 'on
nW aLolu<'C,urrlr'h; rF SCv<zr , a.Ftovq,UDdqALa-
S4-,-U,t til ,S
tNs \ L-\
Cpn d;l;*- W {1rr; A.,'"1-.
4r _G,*&= GqTsL Ullra< V = * = lrrilt QnAl-o
.pe,r i.*'t t"^X*€->da,bL's!r;, 44L ALtuNaALe 4"^)%+ tudu tWS!ffi\t),
Grs
. ';;i:i'.i,J;:'' ,,l'i;'l:
,:, WT,e -
F* q Oirch,
Ic + fo,
!=Ie =
-t. -rJf,=
.T.i
q--
o.-ra
+l r-,L/ \J
l
*I
)
-:.--'
4
)
Y ntbbn-s
Fi" l: 61 C'^**- ft:, Vt t ,;1.,1 SLpq+ -Dd
,l,on*n' f", lroilA 5iv.ft ,J€: J @
G*,\t, oT ,tu i+rq c+nrlLtr ch s.h", L*l.,rn LQ *
4s O,5m
Mr. l'.)Y-drA
II^.' ?';oY-ir^
IAr: ,:oK,rJrr'
:: | (i^) rt4c ct.'
ri Ii lrii'
) t,J-
N1r,''.' I )(ilr't
G r,J *-zr '
I
d=-L =o.F,Ti.,
,5,,?,
[f ?e4errmtnfu * '*wit*ir,I
/r1,w1!,4* d) S-tr,' buj+ir"4.-*.:_..,-. ,
fia'rw =- - {Yl, = -4oxr$ "n'
fv1ic., = '-0'll *.i11, = -rFo *e5 s - e.r, v.rnm-
: - | OKNnt'
S,r Ce {4^r- CEu\,r- S€C+-',ryr- i S Ce.q+*,^l , f)*l<,sn I- I i-r tVW,<I
dC^ 4t?-v1r+1/\
Ifq*"/= tt-orA,"'
SLa"lt ,!;o ^rr,ro,
1; iDeS<om'urtc<-)ioYt g{ St"fl-./
0+ Str.e.yllltfvtn Cen il,'t;srL
-att)g: J-F = 'T:pr
A 3r+1
= 7lll€,
8.,1 T^n* = 114"..*^ < t - }tus
trA g 77
rTo 3
)')1x,,u.at6
:D )7
Crllb fi|,n,.n<'r Lrl
: o't?*m
,lr',, ^rp,- fr-*, Cr^l"'+wn
llO n",[-r]
fu*)'3) ie+ en-m 'i4li'YL at ShAp+
tr? r;3;J"1
( fti:Gfs
q tqt1T oi ,. 'fi4r*
6^'//s< q filJ^Yta
bl'rYl*,* x I F
W, is , ', '^'l/^!,.1 i" ordr- 't<t Cs*r)'n't
LQ n"". lu/m + rnl /'< ,N(- slut tJ n,rl+'/Q bl'%w.
Pla,a we bn ?*'q.tl> **e,-tle Nl+l*hl,r LWSb Skcn i^ "d/" f',^ ,W"t,
= (ir,n,^n
/)o Tra tl=Jt !.2--.
Coaz 2w'lm*fttet ln M,Ll+;Plt
b % ] sr in t'/* (,,,t1',-."
W "p"rl- $ (*r,te*. n'o *)/*L
t 6.'x ,roxr o3
J.14- xi tultr'r6
)>/ 4Trx Ca [el T
<3a x+o (LD-X t tD
3.,+t ^
grn.*x,,:txl,I C" lzl ft't
lrr- ' W y<-gu)+l vf- Ca tu"l"*Y , 4\A /a3r- Va,lru< *l)'tZtrn i S drr,rSrn, beCautt+ i* eourt r'<.: k+r47 14,a.
-9hrc{-t fo,tnain-t bs*h S+}ryr, artl r;3, J
+) Ca,lalq+-,-n t{- J;q,w}a..t uf- Ja! L^the S. u,+(' ,
A+-i) fr-r, Ca"J,+""rn vf- SWt4"d,
liilt 27 lLl^ ^'^^^
ft'l
A
fto',n (a.,-La;,nL': Tf :D'f -TrJ+
32 32_
sf
N.f @= &-'
{?no
-"()
r ) 3, rlr.g
Tr t} ( i3' I
oa ,, \/, lgax M** nlgD\/r\ n- " e LV.l (r-x*)n-
2z glx lAxrtl'x I &O :- 0,r1?mZ.H? x, gxtotxto6xr x (t-o'e+)
d= [9,&xo't+1 = (9.t11^ x.$.tqm'.fr"*, 44,q r-eau/,t'l W CnltuLaA\o,t, wQ- Shil/' u)-Q- L+r6t^-
dh,*r". ,, t), I S,ltr1 w,lh unp Co"ruV""l,4 inr-,-pn J;o,r*r*-= O'12,m
+,4)
IJ
?-( t*d") zr {Yl* ^ooQ L,t I ^rlso
= 0,i5i'r1
'3/z'= ry'/,- d*)16 t' \
iv h-r4 ,{ = ,1/O
t.'GA 4th€4e.,
sol I
I
! 5)' GnAt, vl T /r's1.,^uu*r^s* ia W I "^1 'h,otlii SLye4.g
&t 444,e
lrnax
bh'J shqt, ',,+ o = A. tz,.
= ry-*r -- / L /14,n^nLffs T, Dt'
Srwyh grtc-A r'tutt T C s#,b u<j-un.t-
+<*rA -t.o b< YD l.r.iyuj ct yv{_l' CLc,C t<-t ,) .1 g L,
= 10,/,ty|t
3,i+ * a. tl ls
7 0.6 n4,
Atu+< "H4A* ilr,-* = (l,lr,trl is ro4_q_frirfl
Orn +-r- ob c,,,<,a,\ d^A " I t^*.+rrr, dro^n n L^-a-
w swt ol
tuu
l-ar *tn
t_Ltv\+/.
gQ,,,,,,\
hot L,;*J S'Lu;vk( ,
'(14, ^^^
c; = D'tl rrt
-_:-*o'( ' 'au)
J) @. tl ..t 5ifti ,
4oxtD3x lL?.i+xD-ts)'(,-,t,U")
r0] "--
de.O, t56 n1
O. I P. rYt-
st -l+te h o t le.) tu^J So t""ct
0 - tAcfl ;^/aAk-t
lrt- V ol,*.nt'
f, * ft{'4'
= loDrql
= 4*[ft"L
u{e,r.q1n4*
l-
l
fu chr, o^ .' S1;L^ l" Sl^a,f+ '4 44,q) ;U hea,Vt +-r l-ko,z l*o tLr,)
Loo{u/. Th, s is t.qa reAyvn dLV lutLoa . St apaa a,r<W4-l in A-tln ,t*.oi in g,r)no +o r<J,r+Ca v1qrA.
+) Gr*fl, st -t^;+',^t ,*Vq J,tb;,,o+*-, (ftf r^o t:r;l \*;tfn, 1 i t-t {
6;fs ! ::,-
f tf r,t* /+- be C Ltc s{-,,r a)\ 0 r," gi,ulu-+
V,r = Wr,, - n4,I(r ct
G],= :Prg-il9-"r i t
-rrr,.^;4:{ay# = *e"/LtLt' -r<'{t
= -?'l6xro-3 - 2oxLD3xo's ./ -g t
g xrctxr.6r.rrx D:Elf ' : - b' LtK x tO rq"dj2-
Wr, [0o, = U*" '+- (0.,
: -6.48x,0*3+-Mfn," ,_ ? ,
tl rr
88
1Il4(i,,r., : q= r).Lm
d = +xto-\nL
[tl = eo rn ?ct
[tA3 = o,E^J/*;lLr = 6x tO ' t^4q
F,^J [Pr_] ., V(o,
'/r--uVJt
<4,<_41<__>
5tA
S-u{rd-;
a. re3H
i)i I
--:_>l
0,t3t,t4
[\/t'o t ' x-)i
0. s61 f\;
I
o..o 9_1+
0.0A+,
Q,trt , rqJ
)e-iterm^o J.'
o?2.+
I
I
0.o
il{jc-' = IVIn
t\^Jn, = M+-50t
JrtM, ['r - lYln * Sr4 -
"fA= fVlo - 3fl4
Mjro,, -- fiLn -,s'fi1 1,/rn + M= {Y1* - 2.(w'
B1
fr*t!,r
G-rup.t1-n;r.1A e4ya*'e-ns ir^, *ors, iy,.,'vl/
e:o: Q*+Qur*L(rr+(t*
ri:,,.,r- !!rier+ : 1,! :t+-,r-
= lllt*q LJL
3',-
['l-il 4 .-
G rr, h fr" G Ifsfit]*o r W
* (ttta -sn) q
qn@):3"-
+ {na }Ds + ( t}to - 2n) o's qGr qsil* Cr:i- C o"g'd \+/
'/3?Ca-. ted,;, r-k{ Gn^n rq f cw+n^t ++,< ,^ & {<. q vr d
'. -
"Y'*3 ,
01,n {- 0,0'*(fi*_gr) + a,o*(tto_,ft,),p t,? l(.*o_rrr ) :g,'ln + a'0 61s 11* 'r Q-+'i'filrt * I'a qrila = a,3tl5r14 + 2,+.t (w1.rr"21(ur[l1o -= 1$,orlf ,, = 2, r 'i A4
-[ grs,.r-,. M,w\x-^/\-* J, r+- b rt*,-o-
nli,it = l)46-sM - -2,g[+fi1
TlrP ret:ut* SL,o-r r^\-* [lrlt*, =
)da-n\^ q{ivrr vt Ltfr)
= lLln -ilvi *.- r, K 6 Vltt
ffln-rB1 .-- D.tzznl
: t,k 66n1
f*^ Grtd,;+. -^
ws^fT7
3klilg,= TGe )3
t6
l,' t 6+rrl
ta xu) (
vf Skulru
h/.f
l6
g,t+ (tu( +xro-") B*
- -i'i-
fi'\, v,
{A )' ,*, .
I [ri, ^,* ;
r lVUu t*, o,sq.(, J3.
lb x2,96* = 2, I rcNrYt
[\4"* < t.z tG Is,
tl
- / ,\ III LLd J3 a'
,S
J-e^ -
nl < 8x ro&
Is" [.r ]iD
6x 3.r^F (^***r.r'J f
j?* a" tr6 + = lllX,{rrt
*^.-[ r;3;l lP +/^rTf.t 'r, J ,litt rurnairu
l.tsl* e* na..o 0'tofl're'L+s
Lm] = 2,8K^),YL
bs-a+t S$.e^7
I s q,f ?1, .J
Ut0o =
k*=a t ct/^
L0* : L0+o + L00.
.+D.or{++ f/lr,,t4
Z if-:. 0.?tct+ -
= 0"d rq1
iilfl4*at q
Bar.lar tobx 3,t4 ('r^n^ ,o ") *
0.or. eTo - O- ooz+ rl = O.02 +a-{ -*J
Q,& (, + [Y1ct
Grr 6,1)/.
,,,s67 (e.,8x tD3* 3 tx ri'l
. gx w t*,0'* s"t I (,1rg *+x i)t
0 - o 1-7 +& = - O.oo ??1*r"l
t(*,
= O.oL}a,t -
M,J qqi,.
L0* | Q"*
D" tle111 q-
UEU, *
l-4. A uc olr, Itri3r'1
3? x2..r33x ?"txrD3 xO " I
rrrcta toL Z,t4 ( +x,, :J * = 0, o 2c1 7D r,,t
Q rr:
Q**i r.;::,.t,: :i.4 ... i- ,i r t., . -
t(+, t t8>e
o, .lDqa+ + 04r,tJ*, 0'5q
i
:
-#_--'
2-
= -0, OOLL+ + O'l$x?o8xr0'xo.5xo,ixl.
tr^ ,+o ltl 6x 3. * (' O:$ztrfx,o- "J '#
r:<- C
Gl,ra.: cl.= P'snn., b= l'2'm
' fV = 30 t{rimCt = Sxrun rnPo
[tl,] = lo/nt = *" ,1d =o",'T)
-f
[f I : t,ut tYtf €L
F*l ),rl,t!*
[V*b' , l<rl rn
= 'u Oo {9 dAt+
Ca rrs
"l- 0-o,ozr--6
flin
.-l
rF3.+lt
V*r'n,o 1' .o"t'
qx
:Dedern'tnqh)e* Vt fuqC1..srt ftLbtv2n*s fi|n a*-A ,lt+l*J
ft rtcr, - fY1"i
.}^A
il1, e = lll+-1t4
JU ^A
{Yln t"t = ll'la -Q1Y1
fl1fo = {lLa -ltn +NL
= fll,v * fvl
(brwfa*; L;U'h €-yAhnt rrr( h_o,.,tp.
= fl4,, L
Gr ry/ l)-
!rt\,bsf ' +,r+<
[1^ r, t' ( *to
I11a = Rlt't t,
ab1-3-Iq
= ? ( go,. t,r) r +s.( gox o.o)
?( i.r) t-3?.(o.r)= +3, +t 4 lrrr;,t1
[gr-S, r'r, fn,- tu\L-t &"s.F", b't+ia..
Mjo.r.]f,
nl,. 6,
t\1o : +Z " .+t+ ten m
fi1o - 2-fi1'
+3.il+- h = -/(r-Zli6t<rtm
J{
iJi* t";
[t'1,)r ttu =
fivrn 44t< f<.5ul,H f<C.eirttil,44^-e .f,,.r+ forfuou, T_T, ,.S
+l^a tui{+ ,["^1 *r-: r,tri+zr / [,[**^^ f'= ,fil;i, = + Z,*t 1Kilrt.\ r l.
pf*+rtntn a+,ry^ qt- dr.A(vv.144 qF W St<aa-6. tSka4-E '
frwn tvn&rll,sn g+ St^r,ft'
[Vtr*^^
\ *4 Cn-J,'+;o^ s? r;S; l;n
D*, -l4t< f<,5u1,H r<c.eiunl ,
I n UVst+ txa,rqi^A*'c.14-
lU<a+;e*a, W n1 nnrt Ee
grricaa. LtJ. tn t,l^El CA-J? ,
+ttz drrtwrel+-n is Cq. tc,ltto**"A
Lfn. : (0* | qPB. + (,, .+ lp"r, :-)
+ (rn^ *t,nt)t, .r
Qir "2'3t
J = 0.5?
-tmJb +- tL
+ 3a /ltq +
( n'l,q - zfi)cr
ctn d +/s,
qncl, Ca,i C<L(
(l''to*^rn)q,+ rl,
t 6vtt q
+ C fi\,, - ttt) qq"l%:
Co u ruru.r *e, rn-l
( (Ylrt -fi1) ( : D
fil+ * I l\- 16.'r e6 Krvnr
fi4* - lttI 3, +t/t- /0nm
-()
Ns(kl
-q*
-
--r I
+g,+t+x firxtL
''ld, z @:.tF 7t
-)Vf *xr0 6
?vr,1 Cs^l,^+',on u? riqi Lfl,
0"11Irru
.l = O"5.D: C"tl 1\sm
L{3.+t+7,1s3
l6oXtD 6
GTs
a )7 ilJ,nyq u{l* 3;*t+ XrO3
grr.,,rrt oL lo/ro]f_-_--_-_-_----=;\ +3. Tt f xlcr' ltO I 3(+ t-----_-==- = D,lZ+rrtDv --lJ Tt xsx,pa*,06
iL/n ,6 'xrDe
J. o"s:? - 0"o 6*n
Ft* l4tt-
&;u^** ,
1^+ et't 5,t -e'5
red+t(* ut- Cct t Adq$'an , i^)L [o-Vu yc-iorp_ uF
0. t 1 t rtl rail\ c{, = O" CIq S,tft r' s L-Lr,n<,rt b< Ca,,.,-t<
44^at-#h{ ,-r:ui 1'c,,rv.aint bo+h *r*-fl ,"uA r;1; &.
= D'o ts ffi '-"J' "
PO,^rt A is Ch'ut+'n ct,r DriXin
l.!p:o[0ns = Mi*L =
q TI bt/g^
at[-3,7-t4xl0-x3tx I, Affi
'\0*. = LI,++i ..Aae'
= O.or s++ +- [4,] b
grrq*
x l,xY:eyx roTIro6X
.1
L0r. + l.-0o,Ttr
0,()otrq + (\1*aq"ttL I
G xfi l,/g-_-
0.0rJ1&1 . l6"R16xrtrsxtexo,tr'8x ro{x to 6n
0. o'lrs'f - O "D 5 7+6 ,.-r!
LJ.r
o"tq t+0;gs.-{rt ,n!'
Uo, =
L(*t : U^, * Qa.3 - 0'o5a-+6 +-
- D. os2+6
t'J
il'l' bct Q
G ri J o/r,g* 0.0gr+6 ..1- 13. +r4 xro'X 0,8X32
8x'ro+xturlx o.o'1gS+
i-0.oSi-*L -_O
S o Lr4fur-
G 've''
'
h=1" s Lf 4 -tl, b'= Cl - Lxto '[rttnIv [t') = gDrh4c(0
'r=J
E^/ [rn I
Wlo'o,= {/'b't= {v1
Cat o-r[nr 1 ,I
[D r{-ru3
dACA.l^&'4-te-t [f- Se:.^"lth VF rcn ,n-A
a'
= T'0 3 L-rJ" tb
': / -r'l 3 LZ"i4 x, Lq.x'to-'J x foxr0"T-'= l.o I KN/Y1
W
It4:Its
,rr,lu,
^)\.n\r.-J-,
%., = tfl
NN\\/r
- -\ ,/, \\/1.fl\', v')
z) C"<l tcrrl.q-{rty" of [nl l. [-^"ry-4 cfaaar, v*ay f oi-.+#-
Cor*r{r*r.yL g.p- llrc,ng#i rF
'[4,]]t'' ( k)tArK
rZc,*ran,rO u)_h.. Se c]"yru,
tv luoQ f Jny*^rla y1
h/u ",a "i da*n,4rl
&,,t**;^ur( erl'*; rt a^"*<t$
Ftl0x = "f b'
S-er*i rl iln+,t ,IF fb<,1^)
1.,,.,}
..,\,4 l
i1, :.ii"r ',\\
-I-LLL y-o;h.o tf
€x.f er, n ,^l*ll
e< Sl"eStt
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tu,*#*'\a:y-v\6
gu,r yr%lan,
Cwreo P*,fuhrute 44d
f*n *w JAl{b+ = l'5 , Thu.-fn< ,
=.13. O.3*bS
l,V<
f
hlu I l'S n4 3 f 6 g t0 >top
0,n08 0,r*bt o,*4t o.go\ \)n8 {q*'avqb
a-' b.9 "r-3%'-
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^ef -a)' /( +x Lo ^J ( YDrro 6.
{
rll: - ll ^^^-t^ .l^ o^^t.-fi,, U"A f W**! a^"1 TACJa,^-r) tule- S+ C*ta"" 1-,J (An ar*L
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tr" ,t 44&. rcr-r\o * Sr.tcl1
.W\^* +4A *6 i 5.fra,t3 tWat^lsa,1- forr.uFs '* *,^,4 C t..,r [(
be *r,"r S ct ,,.o .
Frt* <Jn"*rv-' of
-fi4++tt1 -{nc =C]4rwt
II*'8, = *tl4n
^LAn1t-rt't'
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?lL4/+ ^ /14
F-^ Wq+-teituq Q?uA*,ry\
[ro.=p = 18*pQn"
= - fil*q +- (-tuonrn) ( L* q)
O$,tvfi,"l ot/07,
6l Irr G lr* =O
i. 1.. ,l
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i
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rs* -=o
&L
I
L-
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[4*(t, - t, "g,ri,Ce [l4*..(Ae
t tUllTJo
q )-Ts.J *
; {Y1 u * fi4t
: LM* = {W'
Ir,. 'IJ,
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L)I
rs, )fv1 (* =zm(g.
LE^_
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ll +''rx)
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lu-q- d, I
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I
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(t"t
oq'6): o;;l;; t -1d, = 5xto ^trL
d^= Txto-2 rrt
dg= lXro:Arn'
Gr* = D'Sxrotr^'Po
9 r,n - 0'4 x t,:' rwf q
Qnt : O,3x,otwf'u
ff4u = lso dnn-
eAl" (p.rnfirs-vla SX,.oi+.
A +,r, veu.c,rt T(st
-tt tLq -l+tJ r 5t-*ug
Po-+w-
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3- */*vlTl,tntLtt1.1
% q 4\.*"vi, .**n)rn /yyg.1t17,-i
++r4 fi,LUva,pyr13 1,11, , {U Z
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th uA?
lltl<,r1
i.u,t /.taaiQ
fi4,
W, . W": lQ,
M"
1
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G u, TJ. G* Ir.
lI (,^,o-")t - L.tZLxto-"n{.\j2-
/J,,=
[?rrr =
ftr.=
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ft,'lo= 9& f/1,gs+ Tf,
/tr4. = Gn I3o
G,n Ir,
i
= 0'#xtosx l'+++^,r-4 ltl,
0.8x tosg{ ^
[,,16x,j,i?
o, &x ro5x 6. tsL xro-?
iF
= l.+t il/],
(t4, = 2"s fil,
S-A*t''t*1"^2 +htv< ra-Lq-t-f trt s 't^;n e y ,r*-i- L
il{,.= ,fl4, * l.+2.fv1t -F a"s.U\'ff{, - IMS = I oD = a'0, Aa r1ryrefu.lr" '+"q2
. M" :' l.qc, ( ^o,at )
: 2 &. &? dnr
ffr= a.s (,ro 3jJ = 5A.gsNn,.
lDt-*e"M'. ^J-''c'1 ef s1^.-q-rl S frre,n k, wb^*r*- Tts s
fur s+epl
70.t3 x 2.-s xto-a6. tZ6 ,i r,.l-?
0"8'awrt
't;, : /ag -Te91
k+ Graa-* **w-?!4 br.r-,yt ",_t fll,u,rni ruurn.
.1-[** = Mr*
= 28.&?x g,Sx,o-t =
W bnf<c*
,Tzr= W"T. *Ss
W C*"/Lla$ tazrt-tt-r L2ratr) o*A SX".L
,.8,8?. x l,sxro*1
rt"
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Strength Analysis and Designingof Rods and Beams in Plane
Bending.
I
!/ottr c{estitty is the nta-stet'yfan of yoLLl' cx/eatt@V-Qoc[. yott ccll'L-
not discoyeT yoursef outsicfe 3{irn, antdyoLL caluLot stLcceed
without t{iyn. htrny o1vll ca.se, I c{iscovet'edthe tordsfiiy of JentsCftrist, and 5{e ftas 6een cfirectilxg nxe ever shrce.
Chukwu, Celestine Nkem akol am .
t01
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^t"*-* L. tt .
is rrt qx-trvutm ) sr,wut,l nr><* kcru-o( {Lrpn,. * nil ttlofat'tt
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fi-ea,t -J G,tut (; , c('x.t) rri)te"< Shr-+t ,-s ,r-^ ) ,ie stro:tt lsut!.e,
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L, = rsXN/'n
L^^n: X-ri t"ttl/'"t
[[\: t4}ruM4
ltri = t&DrYvtrl
hd,^1 fi*n* fl"",ftr
( tol
rurl"l Sect*^n-l
frr" {-t4qqlvr\!4 '
lwD (Lt k4
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[ 6-,', q.\Serh11-L' X'f
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trJ t[^>x toL
105
C" ( a,,r/ q+t -l' + rsl.,,-& \t c{r'.v'a d"; ^^l** 3 a
f Z I 32-xvr)no,\ tH = 3 I 31]:n:11_xr,-f
= W,'o,tg6rtG,4, * 3-l-7oa c0, s+ib,,*;- ," lfu Ct vsleu(-ru,, wi*t<Qo^^^'
\0',^*.
T Lz) -=- aL
/->_. @t._t n"ax
= 3 SK^i
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I tn, = IL_--)L'
t,,0 r,-a b,TE x ro-# ;= EY"g;rnfq
"F ft43 q* #^lQ-O yI r,,l'
bxro 3
L\' (.e c+rr*.^
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n
: b,t+x 0.t86'+
r Q*t- nn-ar )rua
All+
't
bE) 'I.,-a
Q. ^*{+o
=L3
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: %X*/Anpq.<< tr^_
tn{, a,
@.0,olQl...- :
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i
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I
N*, [:'t
W,N
0"l6smtn A'oy- \.-ec*ic.,.
n t14-F\nvt-
6*41, ot
lilnut4 -ab
h=z-L6
2
3
0" 0 9zE- m\1.. e-h=
S4ryry) cl-t' st-, \u*'e,^ o-tr Qu*
Z "?9 x,o-*
{\1,i,\ AUU o.q
LT (t) : v,Cnn
2-.r; r t.0 3
/'v- nt*L rvra,4,
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3 a 3sx to3,- 0,0136
Z,75xtD*'i
AD=
= 3"frn"rg< L<
l. '
O"og:zs xor/69o,ot 36 n2
3, I
T-r{r" m Pq
bh*
[[c*,
, SE*
Qetrnr o
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sec'l-i '^r"+nn Q*** f s nalah,oa , .rwsfuu, 14\L Jir"c*i"$ fi\ $*,^ is Jaaan ..ro- \%A*.Jr^a,. b%,,w
n-ef qiJ *\ " Lrxer,o* , v1^!- &rrur*in 0f 0.r^_* gL,,,,.r/.yfut* ils ,..3i3r", in .+.kq S€c-frun.
l'0t.'l
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Q**o,o
l.[o+el
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\'lr+*-prdaa i rt V'l.z^-&"^e!
A
N*J"J Scc+iu^"/ mo ) 'r l"r N* = 3, +E xto- f ,rLu
v -.1= 9,7,8-cmtt--'
fltz c'La*a \/ila-1 crA .bO r, fr"ern uasrn-ryLa*,x cg-r,44 f, e-^A
ttsi*h T- l)4r.r, flu,ru,.tbtr '@r12T ___2 ,Ono,.: ZTI A;3
9^Q o
Mlt: 'W
5** t& 'fi.sy< la,ta,r't t[J = ,b,c.yvf,,, d <
\tugur,L J Catl, cwL^'+e *ut- f tn' (e"*ag< Otl.er s*rz+< ay d,t4!
fwru'eLw'
A 6 .:: C);^ __ft[--] , -- 2 r- o1. =1.6t'r:!Ltxt,pl]L]"- ^lur' < E -h' - {e''>
='' ,-, qr
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lwto" ^)t sL"euJ u)< 4ue T-Lt-ec.,vn flurntbp,y-) tr hjLuo..-) ,l4y onrt 5 "L , ,v e sLu,tll C.b.ooto e n tt(r.e0.,. Tr/,
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i f, tt tvttntt-on' P**, 1,1.,-t cUi.W4<ul ,d- trt i',)-L-, .n f ar€..,_ S< f #, errr^l
I U^c[! L".l Sutch ,lta-a-t *tu. r<-t L{1+1,\o, f-|
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(a*l*"+rn<.+* CoYr<a yv^rh -t:)t9u+. \_ AtA
:'h= 2?O: lnm = O'77-m
f,= |1-$mn =0'i'75rug=t, = \dh,, * O"O,o 6rnt =tr = ni6 r,r,,r = O,OOcl I m
h = 40,2'cf =? = 4to, -l+'.lxl7 '/v7
S* + ss = S,,q =* il3 lTr\u * Z
J* + I.o= I^o = 5010c^$ = 5 otO xlo-'8i'r6f
:-
r 08'
= z,cxro96 ?5 Dl CItuo-s
hl+ 3Ltal,t Ca,tc.tlrr+e -j4(. Skaa,r. g.hrqj ,Lt foirr-ts l, \,3q^ i^ .l,'Co.fed in .t-t^-q- &-t o*t',:e- uJ;^.L"t Zlru,nov-r t<3,_l
fur n*rla-r P g int g. i s Fu",,-t
va- J^ o. .[. .0 i .\.h) R(F] f-.- Ft r--. *^_l9.tY) F- rib
Q*,* Sr"
b et 'rY'Y L*'t =\x?
(e- = C)
(
^[rt
:= Q" *^ bt+I: Its
= '35xro*]
g X 5ot}x,.t0,- 8
= 0,87Wq
I='2.(r) Qo*^* Lt, ( -\ +o)_/c-/
+" T1
(v-LJ. -4.)'- 1
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-(l^ \Q'l
F,of -G.*a{ o,uo?s ")I
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T'x(r)
alt-I n"nx
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i--_
+, 'f,-t
= o'175 x o.gq0,0o,'b
= 'l8.bWq
BExr os x""21o ot:o,orr6 x SDto(lD*I
*"fl-,8 fif q
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G-"4 of S.'l-n4rt dl f-.r**;
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t-T-2<---.\
o+ S+rl.,tr
,54
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f o i"*s c,LN
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rAt;-rlo. y'lvr;.r.us
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5*?^
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;
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a
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r.
l< lris Sj-re,t0,
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= +E o
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to-),
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ll
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t
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t\t\'
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la l
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'.h
Zlfro= M-:
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0 "'o 6ln= 6lnm
Q- t/4
fu= 2nA - tpu /rn
P = +Y.rr.,
['] = 8,qJ /rl
f* ''r
LU. J = &tsarwPq
f- -r
LD Jr. Luotwfn
')/, * o'x
AP: rV,,, 0*, fi\, , J, ;
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al.
l
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+ Raq + P+ =-o
q
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d"
u4
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tr/r
f! \'
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t {+ e4
?-
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frt'ou : fl
n--EN*oer, = R,r = llK,
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= _4rc l:o[)L=5
E -II.mN,.c"l- -'f = -4-'<-d'
I 3 l(^)
0z-lc<h/t
0{.x.t',ll
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Ch+ci<^^ q .'.r
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4
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= ?.>c+u /: llI }'=D
l = { 31#)'Il9cs4:,
*r
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ll., t 2:- -t17q -+-x
lU tU
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t1)*e'= o
8rctn"'
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I
V
:
Q. {",, r..nlE
[t\'o)= -M :-V--
N^'t*.,
[1ru t,.,
-(o'=-ttrcnl
- Ro^rc
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o< x<h
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D(>c < q
[t/ir7', rtrr rrr
,. ('') , l(,r)
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r.rn .,ef Lra: r I
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A.
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< [r]
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fu +"""
lDr^'
fI\."*l\""
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Ae ), Ia"Lu_Jp
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trlr)x
(,- u ,r) xn x trux ro 6
for .Yt"r.
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3,*^= Di = A,t4rn
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.+*r-, tu-,l F b"u
C*,w?,uacJ F'b'*
= O"llra
oF?
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0-t4m
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t3'iro3?,rr 2i1o6x o"o g Tf
t'l +
G,ic-', 4=fun
'''%lnL-
#
M= 27q Lr -2n= I Oxro lrlh._nb
f*-L0 J, = 0-.r,nc rrvta<
LC -Jc= 3 ffirw4a'el
I
^\r.(N.,
i;
7Lt-q)2
z7
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R*= Y*)GfuoG,)84'
t6
z fi|r\ =. D
= *(Q* (sq) u ? (*) (t 6, +*,) r ol
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"*- =
Chewil
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I
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w sf,trL u-
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s, ( oq,)
zf. - ?( *+x)"Ltf
X!:z( to)
2z: l: z?L+l lz:u
A, \z 'L-8
P.si.ffl
0= Ro
0=
.f2
Xe=
L. f e*+ t6XeT+La
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+ ir"l
= -/L"sn
7L-
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t6 t
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ule fu,qt-( posi*\e \f aJ,^a
7 Gq+x) ", - -6('6q)
"*], 0, 1?^f(< ^ 6.rtrl^
Y7- h: e7?- l:=^::_
37(6"+{) -,t(r*{J3__,/7 7k
a,*L:[r_E
Q.%.,- R,,- ^G!
F ?x :'"'. e-re.
?X -67f 1x i= -37, /- a- lx=-g I *: ^ ' Ck"7t r)i s'i,1,^r x*- 2q = +
6,1 ._3L= 3.L
{4* ( G,,x) - ry t_+ **_J - n r ?+,:-
zt ( trr*) * 6x (^^+x) _"f q2+ Z+1 [.=:
, I _:,
I = -t0's7'/>c: Xe *- 3
Xe=
^ 'iil
rr1, o, :
=C4.+
Frr l-r* yzmur4 .sF c*Lul,a*-r,r
Oil+A4ni^ art-,;* ap L.tl,lMr^^_f =lttlr'l= t(,t7 7
*i
-[tt*- ([r]L\)nu
lr0..o =
/
Ll',* I I ?t-b
t . Ib
Z != Y^h>-i;
le0ll
@L
.r1.,:r';
F
it Fq, J^e {ouraitzl , .f;b^e'
fi'l;^^+ ,'d---:' ( LU lo iNna ffJ, i
tt,*lL s ht [r],l2-
fu e.rz C-rr4,"rg*l F'b*'
_fu^^* { [l]" -'
Lt)na.
N < sL,a-* J C),,xtz J4^a (u,*l
= O" 't *6 r,,t/^ '* I xd /m
I lx i6" x,l
i (r,:x,d-r'* ,,., ,,oL -
'z
bb'b
7"
i 2.X tL. Xcl
i* bfr*h.feru,'u' ,rJl$il^a Lr{ ?-
/" +E ,r,^t/,n,
brtt-P,.€si,;n ,
Giru. : Q.: vrY1.
01 = t}l<r\) ftL
7= ^st irn
/--'rL0 ):16rrm44'
2e4tl-fi t n-!- 44tr'- Secl1 ttr4
ntt-ttbzr (U!) F,ornq/Yyorhyv4* .
t
7 e,t&rmina*',tn ar: 9a"c*t^wt fsrc,e.L R* a^l RetII
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Stitaa*r'rn
+")
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J
l
l
kl= U-^-m+ ?.a(mnr)a _" -'-lE '''aA!
74
3+ io + zG) JA x).Z6
g lvi c*ia
ZFv = D
= q?"|rt Rn---J-a
' = 6'- S-: = o33
Ru
2 e#4 rtt ry ,+-t-vu-r, e(L S hra_a- .[g.U
-!^/tLt--)I
i
l
,fu;:i s eQ.at'v aJ-o-,,u,+- :-9t
Illll*[1 ,"==riljlj_ 20*t, n'
+/3
wa,, KrJffL.
o<1< g
.i 1.a
- l,x + 2x., -tx4'2,x * -t+ 4-t3
fvlr*Cq o -,L e.' J - = -Tja + Llt _.2 =*, l-
= -xL*i+-j^x13T -Ir
nIfQ* ft'=
*
R+
t-l-/x=o
R*x
/- cl-I >L =L)
tt I = 5/" rc'r)IO3t,I >c 'Q.:-q
/;"=8 t<'o'n
p1r .x<q
U*0,, - -7rt+ 14_ ?t r R,*
= -3 -;1] r t! l= '/= I = -€,J L\a^-'y,u S,eh3 l*=a lx-q"3P.ur;+.-. .iF Q * -- 0, S.* ^J lVq,,-^,<
C = -3- 1,?'+:ts3
:(< =t JE :-- t 7,L4-m'r^Ie CL^t'tz 14^g- f oS, + U<
fulfc'o = *?ql"Tr"J* ry(*r,) , Ro(<o.) _ It _ft1
6ct
= *L(+n*) r3(r**)*#(r"*) -# _/o
)=-t l= D l= O,+Bt<,\),^[x=p /x:q=s /r=r" J^,r,F*. d4\s ftr.rL.r+ sp csLlc,{q*{_ityn,
In4a *, l=lr, 4 = ts x,Op
2<f+rdr n 4+urh ee LL,,o_nneL S e C+i u^ fivuttbt r
Tr'E
JJ
I -'- EJ,
a'
J
()
t:f-r t:.Llj*e-
L.,
rd,;L)
= dr^,e.
-
I
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frn C*)r"i*' et Yt.,'T 'Ytn
r&,.-N^;'
^r^ir= \1,,- =
. f ) ^/i' W, 7t .tttYm*qJ-CFJ
2
=- trxr-o' . :. 4."f{xto*t rn5tWxto6 a ++.4r_nS. .a)
^?, )-c-tr1 '
F** tutaol-+tuea* t iLe rv-a.ra{ til^4 i s ?9., y c,rn
: . Otr,. b ^qzis C*,r,.<-4f a^As *o x- e xz s ind-{a,u,r, UDr. in q s S e?.r*rr€^i* t)l fo. gr_n_
0^-" =I4*=N. 12
,J ^.t
)gx to'
-- -/,('^ 2Q" &X {rJ "
3 ,', rug s.d-4^a q-vy***-Il,L.q( G,La-*'st I
:)
l*5 " + fti.pq.._-
-
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0;,o" ' Tt^.l
I
WNW)*l_,
Sft-+r^rl.
)4/+rmvma*un dF ,^l lt /-'r+",'r,q>l.^-
f.-rQ.1",
= O,Ts t<)\= nq = 6")s
Gi,.e-'. Ct= O,\5m
1 rn: jo r<-rum
/ I o/- = zKN /rtIV b= sxro-am
\=ab
i
i
i
:
iiI
I
I
I
tf
i
,]I
tr
= [x- ?x^z(zq")
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tr'*J tr*C, *,,."1.+-tLt Sec+'dr1- r,nl).y*t
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.'L.
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I
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{s+ao **]
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a-
I
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/\-/\/\- 1"5 ( l.tts *x) i- 0,+s ( t tx) f 0.+5 ( o"zst-cc)
'lli"'= I = -10,+s I -.- 7,2_5y4trrt,
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:\\
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t6
72*1.)35
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= I ll nr,ro*i
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U = tr (r,.r')rbp
L<-+
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) Vn,
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S r:[ut*Luh
Lo cct+ie^ e-f- +-a^{ rrauf'"r'( '10\ -uf *t.c c-aovvpoti.l-e cruuts<c*,,r.
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= ro-*(,u-l) 3.,_ /o?,o_2 [So.rr_i\L
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tse rernitt{ecf tfi.at the sword of inteffigent an[ successf'uf stuc{ents rfi,ostay a-6ot,e tfi"eir conte,rnyora.ries is mafe of tfrree comyosite matet"iak:regutfcw attencfctyrce to fectut'es y,itfiin tfi.e umbreffa of attentiteness,yiivate stucfying 6eyon{ cfass vorks, ctn{ consuftation vitli fectut'ersto asT ctn{receiye anstue't's to questtons vfi.ich has yroven frcr[ to
tacftfe.
Chukw u, Celestine Nkemakol am .
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q
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I
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7
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QJn,
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=
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t-/-lr.. e__.
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Cs^o5i J--J {\A.er? tr*fi-".,^" .
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d=
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il r ,'rl4 ,|l4i b,y\ Jr'+-r"o-n
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= ll. Z ,,*,
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g"t+ x o,lot3
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rr f o-J
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lJ
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U^ q-mB:ttz .
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[aJ = tboUeo< #\-a |f-re,fllh qL frrp q./hS<ctu; ( W* s+c+.a ) .
r5s
I
II-r
.tf.r-:
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a -T.N.fl,. CrI'' : /I4*o = D/I4*%. -1rZ .:- 4s;at*t.
,'4. T
,t frf\q ut : -1-qrc= *3ox / =o/ X-3
tl{}ou : Q},, =
7-r, - Zx?/zq.l0sc * Vr*^
lr/ - rrl
N r'L, 'Qr"u' =
o)'n.,
(w,4'ou,
Wlu'u (*, =
- ?-+" u _?-o:6
= bO t<tt7- q .= Zo rrn.
- 7.q. Co*" )-zo C z+x ) l,:, * 1o t<,t^
I ,,= - t eo Rrvn
qoLk+ = -45ra"t
1.. . - 1 o pru,rt
lr -mr -[0
Nl^ [,t) --
QF,=
01X < q.
naf r, :c
0 (.rc < q..
ls6
/4u6',, .*n
' llt) rr
Na Ln*,*,,, 0, ""_l 0*11^A .4t-h pw*ln n Lou/-!)A/, = * ts F.r, //1t-.- ^3 O r,nt,
qf* *LLa S */ pr* +/Ylb*- *ttu@-, /14"t: fsra),,.
potn t-r A ., t< . a,^ l c a4_L
b1r^i-,z/+ Ja.y**:)
L=>
=>
4
L-,.-u
rk4"L(Ir* c*l
Pt- f ;: c{4,/?resta .liq,ee *omesF0v*1, tr ( ttr1, Lr( rttEl
t4- * fs ilo *r,,,-s Cer./,el.*)U}t") , o{/hsy a^) *u*) 7rlt C ,t {tn 4.^o3 Cy^-fre.ad0- C tu{ )., F( ry) "^J suwef U,j
Pg i*ri--r-J
lr,") = * 9b + 6rutbht T;o
Nr.b6[*)tr(,,rv)
ls ^to
3 "'
-+:
px ro)xre-f
Tl4a .,>4 i s
Tha sD hd\.c^
S< bh;\
No+e .' tF lD- | r*
-6
lo tztv , Q, : 2i y4,
= / bg+t-q 1_4tS.J/^r^.r< .!e c-*r-utn C So
,1t6xtrono' , / ?;-----"-:.- t OfSOXtd-
(toxs") xto
I.ftSnl,lLr >22[O-)
P_Lr)
'L - 2-m'
It'>'(crtrvt I i O-p
S a+t) ,
F;"1 {4.\l
t'ln,'rJ ,-,rJ
/*& trqLl<./ .1 t, t<'u t f'Orfh-L( h,tt ,rJ ) .
,*7 S*"_2
N or^Ll 6<
l"^1 b<e"1' l2o,"a* a . a^^_d (co*7*.1 r-LLt )
txc+-ct-c-e'=----
S,-h fn-1 :
. L" ' J ru ,:
1 ?<^4te- /-1"i, tF Jfd ," asa* *^rr_,r;. . uf /4<a.
.u ,"*3 *,, ! sl>ayru tl ha^{e cao'4,,t d:
t/ q/"444 6f (O- _1 , /r*2 q."_rl i
fi # -"* C he c4 szu Slr"^,;;;d4*t+-t-h ro c[ ( (.^* *A S
C(= 2tu, o- t. L_ ,OKrl/rn.f/] . toKilt,, tf::tL,)
Fi d tu J,-r,,.:*,,*_ i,'#il6'; :f"*
,. / f9u"ql ;" S L."*e -
Q= lntcrl = t/r.,2pata
P = totci/ = /ttt<tt/rn
-l
U nsym m etri ca I/O b I i q u eBending
Tlte conLycLtay you lieey is a yar{stict to yom'y.rogress in fife.Isofate yottrseff front those wfr.o are ricfr. in evif deeds, for tlieyviffn"ot onfy tnalie yottfat[ 6ut a[so fr.eey youfaftbn- Tmbracethe hosotn of tnature, htteffigent anrdsuccessfitfyeoyk,for youwiff feartt to be fifr.e tliem andtftey wiff c{rav yoLL tty sfiouf{yottstuvnife anc{faff.
Chukw u, Celestine Nkem akol am.
,v
I 51{,
+
5UN SYrnftlEKRtckL g 6Li,Qi e '9,6--711 o rfu
T )t,L^;y4plaan sF
{),,; a f ,,!
ba"ln 4 li +-t^"dbc,^l;^i1 ltto.nqnt
un Lu"clt *4"a,
tr/ t't'h la<-A'
f,reJtc, rw+
le.*, ),L3 iu
Cg,; c,"J-e
-1
q'Ltrs 0F J-ht Sr a*Uh _
lo K.il /,n
anv)A/A
[D-] " t6on'rq
T- /V3 la'
0/-2
p=L=/E-.
alJ
S.,tLul r" ot. finrl,l4to
tl4r' th u a'"-/
|/l1, r,,a,n
Srr.ao
Ar*c(
I ltu',
9.7L')*
lth = */^
1,:: 1,,: '/,,:.u
b,-l'
Krn =
tfrrt, =
Roh =8= top,J'
.* ,(6 h.rc
- /Dry 1.=,0 l . =l)r =5 lX =* Roh,, (* a) t Fx.
- lo pt^IE
f4* h' =-
c =^tD/= O=o /,'= t
Sf"'uyf^ atr *re ,oJ ,
j-tq {^* I J)i s#,\,ur_r.c,vtS a?t /wl's 6a,,-a lX rn nu-/1 ,-c -.1,R4v n^-rl Ror 5l,co.tt bc /-Ls
h'r, - ?zc
ll)x - sxL
lA2tKilr4 ?)'^a 4 l,e,i,-*--t
Y!2 Lu twhcq-o (^'^)
S 41\t ct,.-ci- eyto-( I oj--
{) = tox-''")
*to (*x)*e_a* l,
-
.T
W^^,1 = sKl* lN*^^*)- lDKt't"'*
iir-
{\r;'n^x^n* s* 4-1^-a- T -bear S h alL be L|n r<l fi i'tL **u 'rF\-/
3,n ce t,/ln ? fVl,I, 44n< T* beo^ \kat( b</Vtt ^0 Mt - 3,n ce lYh
o L"a'.r.-& l,r*";r*^"*^r1"1 80 +k * (frt w r'tt a ct- a* f4\4 S'e c'/-.,1'l-L
rnr , d {n-al*.vrr,.r*n \ecti*^-r.lL n'*A"h^ ' t'fi n^ ,
qt
du= r't0e- ' 58
h= t?,&n,
T- ruo '2-
tt\-l\
UUr; = eyld'a
O-(ria*; = ffry^udu
htl+a ux s.f-i.
Cfusl?*fnu,-e {-tu
e-+' q c'n *'4-
ZSo c--,'.
?lpA-o* aala la-av( {\e*v-4- - y-ax.tz W<J herais Yr* &'ff'o""***'frr""4 Ja.< A^e ,/&Ain lra,o,^X t 4. b+,_),j.
m,oa^4^;*. tU4 *t-o fo-.^o",Y- "r!r +D ;^ L'c,* #<e.b4"" ),...-) fhaat-rj-t l-.ar_e .
:- 5?3.+ (tL1'4,'
= ltgxLo-t*.ro*,r^t : lVt' e' r11-{Ln'
d tl'n )dturtr)
da.a. Sfie+a ia),w< /ratLlru-+*-l a)u,tUa,
P tn, +*,,, $F n {.u{--4{ ae La ,
tNY Cs.-'r,
G.*pL;d Jal*m; qt-ttA ,ye nlurf,-,t q"d-i.l P*wu1:*
J1.
ftnil1+;.r,1 O c+e,rrn,',. qfi",r," gf A,,errn_
q) CLr"'r. n4d stdb+e-n^ op CBo-J,.*to, (n,,0, !,). 1,..d,.c-#€,ar\Y flortt . K u^rfricl. hq b<,h^.,t-e,t- {_4{e fos;+.,re ?nu!-ro-^zf0f-+4u na) ercr-s, *,J sL.,ri i+* cs@tJ_l^*__ Z,'rr^_"(g,.Po,i.+ K is ct \Sta,.,?a. J {-r 4 +o J4.o nt.r{shd Gr i s
[t,r"" 0+w C Ltus"-" Cro-- J, )-.,,-*4-S;
l'ol-atiarr fi,* Fdsi-fi.Jr 3 *.oPol''h',uq L qxL^ Sh,,-t br.Px,-.t'''l* D Cldc<,rix . -tl**<$na,a,ne2lt' ol,t ctu-xlr' !.( sh-rrr( b< fu% cLt\ i.\ t
[ = D(0, r,) =-r) S*rcts is Ze4-o a-tn.e^.cf.a._L a:ctf.
= llt,t.v+ M*9,
)1,
=-OT"
Z, = -fL I, r^
. |}h f,* \-r r "
TL<. nott*ral cr xL) ir q
fe.(*a.+t t'.1-
i, . K{,
sl"rcule.ct L'n* , q^.1 n( (c-.tWt e-y,\o\*-io( trp S +ro"7*r1- [,.^< ,
(s +€,hq c MtL"lrlh T*
hn^r"4
-l= +€vn '
.:-' .
/.:\- I r)
b ( av.*; akc*o,a )lv t -' \ n?a:*z'$t ,/
Aw ru&{++4^{ f<Altti
Whic<tfu-
dn,i c,-J<-a r,,l r\-{rf<cul1<-il
e=t*=o,B[[] = t6ctweo
cl= lok:/nt, L=Jr'1
lVl: torerJ^
F,".J J "t'J D,
5r:14+r-orr
[t4ru,=
r14r@t =
^/Yl .
u^-a
sx*-
- Io l<t';'"t'
.7cc36t:
toxqf.ul= ro-/gl I/ z:; I x: ,,
kin,
Itb,,oltt
[W"*J = lo r'v^
lYlr, F^^ IfVl.,-^l = 1ox,,tm-
fiL" + fVL*
Lt)t
ailr=
=Ws(r3t- \
Eilz*c
t(I;7
ic;l=t/,)g
'fu Ufi&ai^*
Itl" + lYl* ,\
t'l,r-o
3a-
gf' 9+ve-44 /4. ../,l
erl ) ul* 27 l\:'-L[hLrl
ry'('*d*) z7 fi\+il2uJna'
0 zt \3 lYfutuu")\ -n tO-]
-
= ? I g"( torroJxro3{;-r^;z,1-= D.&.p : C.toIm
L,o, cr+e
:r-fiu
to-l
F^ I 4+\a (c\t**'m uF llez'q-t'oJ a X i3
Fr^ 4 da,..-q !r'o*rr* t*+t-.r,a-.1 rA t+ i: {^r*_.I_rJ 4Mr, = L0 ynu , !il, = Q-obJra,
M*= 3ot<.^l *, IEJ = r btroe*
N! IL
= 9 0 )ttu,n ., (Vlt : / 0 l'.-.nrn'
=- t b'o
NLc = +0,<"\.1 ,
t
Fir-.Q *[u. Pu]i+, r]1 rrF- nud"-41 i
Q xr-s , t+ tfJ = r h)nfq. f\.**,'ifVL * *O*rt,', h + tf'n, b= Sr.,O-\rni
:
i
t64 i
Ii
^iI
,J
,..1
t-]
I
+i
Eccentric Tension andGompression
?ractice nta.fr.es yetfect. This 6ooft gtddes yoLL tht'or.tgft tfi"e yro6'[ern-so6,ing yrocesses. 3{o^vveve?', y.o1.L can onfy yerfect yoLLl'
yro6[ent-so[ving sfr.iffs 6y sofvilxg as nxcLtly yt'o1fents as yossi6{e
Chukw u, Cel estine Nkemakol am,
'lrlir.rli*::-:.:,'+rl, l6s
CPdrRrc T€ffsrod A^ra Cornfk-crtro,n){
l1lrl € CU*r*r;1 44t4i'; AJ 6ruzyre {S t oa.,
exle"-^! Wb)AXis o-F q ,'oJ
J-r:A rr{+ OS;^c,-J<
L"* ls ghr'f4-znl..
*1"-a fQ Sq Ha-* e-F
('
Soiu-t-^u,
CL^rr r-* arn ;;.,
.-.--.--.-_.*\,5
Aw- src+,.(m
Qt^s 6-<r^*<J
F - twrdh :' Q-Dc,rt
b: too+t
F,'*J
a"J5 ^^"
nfuifr-a,L ax,,-r f,\i+'lt'
A Ac+ 64-l F,^J _!4t_d
'? -!4\+ g1.1e-,t_{ ie-u , F _
,,DL
ln *€"-ru q-L fwU.t ^ I'."c,L
l.-*----b -----rl
Nz='-Fl\= F*
t4* =F
I
= - / UDI<j)
= i roxo",a-'
= 5K'dn
4 = IfD x o'2-A
= lofdm.
tnl<. 51,.-at) CpuiJ+- A,'f9,c ? \
S.ir^C.o {*f qn< fo++,^f-,-a-LLX
dr""A""*ra,
| .',
i)-t; I)l:o-r-r- -.--.)- 7- ? -7 - z r r /
F L/"'// t
b
I
_2- - /L-, )C
A/ / -l---'e
t4JN
D=t
I
tet
e .Al,CjilA*i on sf- kLari ^_,q S+rUaA,
' t * 4+L"* fdr* ft urlr't( bcl# rs Qtliclt'-
,,1^ * t+ is +l*'<-Q -h"r'4 t''o'rwf*ne el'
\ o-c^'"1 J
\ctc,t)/ =
bh"
a
lcox tD )_ zJ-
Q,DX tox lO t
36x sxlo-
lo^20^xt,o-b
,3: (" XtOXtO
ao* ,.-r ,-a
6{vl^bh.L
6til"\ o- cu"l,)
I' d+ =
C
f,;=
o* (,rt"J +- O; ( rUl5 + )-,s _r_ 3D
C(N,,) -f (frr) +5 -- ?-.s +-3D
G^4^ of S+r,en flrl+, t ^l'r'rsr-. ^-) 3e,t<r-rtn;^^+; on of Neu*A *xr'.,
{gA Qe*erm,)vti-,.o,t *F Afeut .-,(, r*X t.s fOS ilr'mt p*,-7,/n4-e.€-|rua+lr^ Yf- Z!*D S*-e,ra'
C*o-),'nat<l (Zrt 0., j, )/< (*h"ch isassq,,il
1il4 SUA"// tt)4 q na,; Xyrltc.n 4qncl. in-rL^rr* a4 q,r b;+,a,n poi*+-o bc-l+na $ <-at- . rtetb-at!
^ rri )
hbA
un;Vuo)2' J*auze4 4
S lvLpq'
= AOf+{tq'
- q., (tvt*)
=- *+?"5fifet--.l:-...r-.-.-;l-
ff(rttv)
+ 5- (rwr1
.-: iA.s rwfq.
tr (u*1
= l?_- s M4t
-r{
Ck=
= ./tL + te[4.yt l,bh hbs
t\l= tvrr + 6 lf1^, iab^
'r'
tG utlVp-, F, - o
3, = *a[- ,Q.bT? ba E7;,
N'Lt^ e =..D
l? ri,fy,
+ -W Y:aJt, r*,
'^,Y-# =- 6bh3
* !n:,i, =- ob+
= l/uDxrd3x o.,'vxtoX ro3
-- _4.L, 'O i7 14
I- z:" t C,*1 ,
h=zo
= /Vn L"j tyl *,
..-luoxrDJx o-l q
12,x. SXro3= -O.ot6?r<
= I . 6lu"t
This re4w+ CD,,i a,"!<-alt< Co,.,v.q_/
ru h i c+tl,X,
*^z Pz c
P= ?opt?_ to1<rt/ma
b=- 4 r,rtt
Y ez''
2e4+rltunt1S+.*4dh Jae
+h1- tt4-ctytout*t
b<.*r4-
r
)_i
:
P - 74' = 2oxrrrs ^(tor+rr*L! -n^ , o)
xb4--T-
fu rileo,^2 qt4t- g< t*t-o-,*Aco a.^ Fd*-f-,-qlhI
Clct"le"-o.ta , ul€. Sltal l@"k
_\ 4-l^4 fa/^,^f tq:!a.rt4 fs-cp
A,I"
fiL\
{Vir
=
.- Pbla./ =
TLh.^ o lL/6 ?!
OJu.) = I/. =l]-
0-(oq ) = 6 lvl" {bh(
/r lar,a+ 6.zs =
lclrt4 X t Au>-t
+ 4 s^ x to -4
t g '* S ttt'12 /1'
lI
4z' ?'t rn41-1e*--c
= 6.lS/t'4 q--.-.-..1-
-/ (,
'n 2-,c xtd-xY71o-r)
0-(,uo) 3 6!3=hva
cE = lfrr-lln I trout
= 18,+s-r15,?5
I o-(ntr) I -18.*s
/ &" +9 lrtPq
7 /rox ro3*+tr rlg t- Q-^>xt,:5* < --)- L I '-' '-..'' 'l " dr\to f Q-ttxtnk <xro
/\ /6gx +^xLo-(
6 ( 7 t^/,l..#\ /6
-nF
CI; =
=
lrCuSl + Iol^,,,.t I
)
/,'l. r/..' I,f.L_
.t/
= 6"^spt*4
'loxr03^E?+xro
lS,+s +
t6q
-.J
--i
.)
I-i
fT-:uc=
\JP
* lo.r.bl = lo'-( ,!,e)l-L' I rtu,)l.i
-:-l8,ts * l8-"75'f 6,zs =. *3I.s,srwfq
lr lu")l- lrCur) / r- IrrC,u,, I
lB"+s l8'-+s +*(""2s =Grcvt)h sP S*relS O is*,'i brt*r'svr-
("., S /w.?i
+L, *5 rwp,?--------*_
+5.+5
6.2>
a'n! tL, 7=
P= l,tYn\, L=
Lf J =- / Lo t'F-'r
5fl=to
/o4Ao i11'
Nr, G)
Wa,
. SoLx.?i,,n
P 8',; a'.Fios,ir3o = sPl
a*"* z,t o pxe- 6L
sx*+* J-Q?ct* r.r,IL
Ch-e-u< l-t"p Srr<*l++t qedfu b<-c,,r,a *
3tr'6sv^t*,l=z
cot a,
t: /..>C
^--l
-
,l
6
bo.b5
iAt, ^*AJ,,{" /tn}
9,. s tvvq-q
= 4.81,Lfi44
: Sd
ln'fr^*1= 3o'brK.,nft1
4,L-j
k[v\v-il;
lI-lt tL cLot<'
/t : lbc',,Ja: V-[c.'r''-T- q1-'2. 4I? = 6f5Otr<-T- ,;A"J = 5Y- 6c,vt'ta.,\ 3/.L'", : lo I c"t
J
AL : 'L0 . Lc,,,, z
lt
'>9
lrtu,.ri
lrcwt)
?.
5X to-_tL
zD-?. xro t
-t3o-65xto--'------------=7loq xLo o
ct-a . 0;
fi .e D-p
= lCr(n,,.)+ lfUrL;}= t-"S+ s,&l-2-= e&L.Trwf-q
= Intru") l- fiO-cottf
l[^-^t' = c'Y}"+ /vuf1 /
Afi= UY3"+-- lGD xP ^ ?*'s/o"tGc t
$tn/u- AF77 5oA
T1^a ro 4 is ru,r+ Sdrr.^1. '
l* is rl*. &,tt a*;1 t'D in b<-qt* r"*.s
d".^a-i vtts
-7y
I
l
;
1,1I
/)"
Jr-=;-i.ox-at n
P= loPlfl = 7oo
Fm& tr"'t td3T4 .ft ufr, m" s a^]. dp*ra" t )-e. #&-
flell,h'"l qXA P*tnt ^^tl"
U ( A ps,,;a) '-, lvlt ( p c," *)
Q =-er:f-,t' /^, f . Z,C t<.,1'
lq: ?c,'r, b=+c.,n'>1 Ia F,^ u( tta ll'Lctx,'ttLl'.ttl Si-r-<-t) a'
rt4t- k-a,'t- ^J "L'""t J'o7o'nn
'Cf ,*r^ ,L-r*, br'tftit''r
/
q.
Lv&
Lrz ^, 7= loXN //u
P = tottrn
lNo_ +o
tr.l+r*t"l fiNlfi;t il*afuc+o.r, uvl
ft4+ga' u,*L
&^qf,a-,a tf
,y1\4 fut d4"4-
9l"tm a^*l|lr<-tt Jr'slr, bqh rt-
( h+u< ++tq \*r<,,.,'"fi+t ry1412 b4e",r7. ( t- N! lg )tF i+ i s [o* d<. d b,X.
f\= torN,"r , fi4y' {.0 t< )A
N,. = t DFd, ff ] = t|l wf c.
-l
'u,oGW""^u*
Generalized Force, GeneralizedDisplacement,
1|rfi.en tfte teeth of sfeey beghts to 6it yott c{tn hry yrittate stuc{ies,cfose yollt' booli and sfeey. ^lilfren confusion beplitts to irtterferewith yoLLl" tntderstanding cfi.re to fong hours of studyhtg, go orrtandyfay. 3-[otvet,et', if tlie ntnttbet' of ftolLrs yoLL speltd'ott sfeey-ing andy[aying 1econtes nLore th"an those syent on shtc{yhtg,then htcrease the frmettsiorts of tfi"e rod of disciyfttLe olL yotn se{f,
for tfte yirus of faziness has caugfrt yortlll
Chukw u, Cel estine Nkemakol am .
l+3
lyl^l*'s hu-t*-l 'cw Ca(atlo+'i.l Ge,^t*-*L'rntr O', f Lott n -iI
TkL Jr^t 1Lrb,r*.ct g- q b-en-,2 Carq*-"( L) 4a,.e CgrorA,,,*r('
[narl.,,-? N ,r, {4r,r' |il", a-,-"{ ill+ c o,^ b< . h) lJL,l uan^4td
WL,'s i,l+.or-.,1 1W4-L*JU
A = +-[1 N"," 'u'* u.
''=' LJ ,-' E+r | !r,k'frru 4,.
J', 5t rt
Ggru e tr-$t-tzcCI FoACe G G^J e&*LrLe.p ors ?+nte nel
fvl*rd il,*,4,r.
L,
l
)r o'i qno&lo". b<e,n of eVr_a/ (arqJ4^ a*-J s izo br :, J<.14\e a.(+1^-^l d"* beaa eA &r_riJnnn Jlnt J;cole J4.Le th.rc,^^l ferca facl*-s
^ cl_,nd 0n tu4qc,tu-\l /w;1,r.. u[ b*o,n -^_J C)r* .14n4 ,u I tnjD .f or*,,e.o orS'eul..;onu a4 N< u.te-d +CI li
(*t,;c/^is rt41-* h-t! y), n4qK4 Fo*.t,4Sa'*'c wa,g ry A" I "in *qe gr,.j;n|d rr,! i.
fi'ul n-ht 'Jr', fLqhi,L;^-A- *f- *ho ,.,.1
lrt t-qr gr,J/\,r-.! ru{, /r*) q krt t.-/* f*A41* a br.a /n"J,*l /ot^* 0n n44L ne^)
-JIs
t- | na*f'M.-'),,tLt
= T-{t
NI,,ob ,l\4,r{n ,Mrr^fa c+e^ *4s a.c*i^tX
N, o' , frr"'' , ilrr., ,
+4"e;. M\n?^uh
, ll4^r r, er<
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tT"otl, du = o'9
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e [J:4. + J^(r-o.sx)
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I
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NI ol |le,'rlo,, wYu MIO, tLNrt fr,,
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f,,,-'flnt= 'l - 0"5x
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)
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fo F^0q ilru+
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A
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1
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6
Zfi'ln-p : *M + 4@-/1;'q ---
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Z Ft =o *,{{lo fl* =D
A .= tSPi.a
t14,+ = o
tD- co +,5: tordot
JI
A
114.
l-.l
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7 l 0w /rt't'
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o
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zfn:RI
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o
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+'
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= - 10,+t
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tfr, h- 2-ryt
l7lul, M:40KrJt^
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?a 4t+dj3o -f ,) .-D
lA r:. fl1"
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Zf4,'
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{s +*^ * iln - 0{s * /t4r
= 'l A"+5 r..,.)
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\+- 9.q (* +b) + tAa * {]rts + Nt,<
b
.crl
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.u
&a-2o?,
F
l
I!
I 4q -,rr:(+.r}+ro-to iao
'' ct""g.qj.
Z F{ =-o: ?q+Rn-^Ru' s
F+" t'ha Yig \-16 L4alnU
Zfl,la: o= -7-t+
a
ttn2_
s ,Z'+5 r<,i
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R" c - e,*(*J..) + P (c+J) +
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[1x -o
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Z/Vle :o= Mx+
,)
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C-
xq - ?4 (?e)
" z:! (:r) -,,l :
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+, Pd
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c,
Clrtil.J,^4
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= -2Stt_r-r
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gf 'l n+e,-rurut th-Ct- {..t t --t
l,
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T-T
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o.1x< t
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n{ - rrl
alV
t) o',, =rt
:
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IYlc
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1
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o f = /orcdl2(=t
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A 1:r<ry
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l"tZm
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o{- 'Y<e'c*'uJ'l
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Ru= 0
(n' = 2's+,3.+s
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a.ck*o,! V ql/*
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Z /vl,s = o
= X,Lr- ['c=o: E. = -j* =-_/rc-;?
D
X, I t,tcJ _- Ro c
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*oA
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D
Zfic =
o
o
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D<U.(t
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: 40 .-ro(o,s+x) +.lo
= rFS .- lOx
P* - ?L^a
: l0x * sxa+ rg xs
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tV-tv 0<r(e,
Zsb
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FT
Ar.p *
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lJl x\t* ...f +*\ J"f
t+'l: .
cL\.Lu . rAJ i.lar guw
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trI
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3,+sxr,)----
gll I\sl I3 lo J
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_a3ET
2.
(e,sx - s
L
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tQo
a!/
Q: ztn/: qyN lrnp = lo rcd'
C.o+tstruttl- dr-<-X"o,*r, st
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f4lLa.(q +4^!.
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s$q-+ic4th ,t, Aa*+^.ur n e/e
N i{44 a p ic{, +io"cr Ze",o
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ffio (., + rM, ([,* t.) + fio L.
'2M,q. r M^a * o
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d e = ar(t)
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/Vl. . + t{Nm'-.t.
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sl^o.lt px-ctud-p 44^.0- Fic*,'c,eq,l \p*rr)
Oe +*, > (r-.^'
= - 6Er (d,*d,)
N'= -7*- - g Kr)'h
/h6= [, = d, = 4z_= olz = q.
,-;
fu" *"1"e
Zfvln =.a
= Mr+
of rvd- Va*t'
Mr* {rtr-e-
*d=
d,)r11ur-.$'Cr-
= PqeI6 EI
o,to? pN
-D
LeJ-
h^J
[Y\ r
*/\
Rn 'l
J-rLt
(*.
tvlrG
)lt1q - Re -'o (^] =
Rq=
{ =O
, ;.-..'::'?-
W vtP
'Z.fYlu =-
S'e cor. d Lean'
* fi,^ - P* * R"q
fle= -M. ^Po/^
1- P+
7L.Lx-
=o*r/o - to *g( wz)
q)-r zq(*"7q (4 +q)
q- -)4
= * lT,0+rlZ l1'lc = o
= -0'la+ R"*o
R,i' = ll'l^ - P* + '/a *to + &2
A-o'82i t
R*= o.tc+ I ,
l rr i.tr q-( SVlte*...
q q-'4
a-
Ru = (,, * Ru"
= D,loT- 0"qs =V\l.I{c - l?.0} rn; ! ',
L.e+ A) nc+j y.g,(t&rl -l-o +h-a
Forcc Fq.c-|e'-o '
: - O"qg
/)1, = ll1,= **ctu <L fi',. c(
Mrrr" ')
= 0,cll xn:rr,.
in*'-n-a,L
:(n-
i tv In iri- jIi l^^', to i lA"
iI I I /' i t-j=-7F__r_ u r ;Iilq:LrrJ,f--I Lt- |._i l,= ,,--*l,ri*
Q et ,,-)lo
7.x
- ?Z:a
* 2.*2
=OL:D
+
)
+-
g r.l$
8 t<d
o <)a< 2
o<.r< I
T-A
Qle,
rvL1,
:tr -ir
QIN,
[f e-,
llT- -iir
C*'n,,
Pf*
ftj'0,,
ir
/;=;
:( =o
cI
r,1v'r I(c^
l'?: Olx
- 1'o+ r"l
rc: D
7,q" - R"
I - l+.0
- Z.q (* ***8( r+r)
a+x) *p(***) rRunro ( r+*) # o.8as * /= o,t+ /= - o,o+
/r.o /r=*
7-a" *(u +p8 - t+. o.+ *- ro = O.gg r<,)
*?q G.* *x) + {?. (+ **) - ?u
-s ( r"+x) + /*. o+ (+x)- rox 0 ' 11 r,",,ut,;
-& l.0t rrnr,n
= /.oI2C:D
U<ac< ?.
-. D.lO+ t<N,a
Qo
0.8?.3
+(c C
R+;c) *
1.q. - (. r- p
8-l+"0++ l0-
- 7q. (* *+,.)- r( a+x) + r+. o+ (
= fii.c
*o)
d, =D 4^= d^(1.)
+ rDq : - (,q ( *Zot l\-6tr/_AD" *srstrn-
iI
+ d. )t"= ?7 q.3.
Z6-o e t
We. Gv+ *t<-o V*nA- cL/- + cz^-nl dr-sco,dn ,j.<-
*,tua. Fiil.,+.or.,l s( a,+t
fr= t.-i".1
?ul, 4
M'a
Rn
(u c.o
Q=Afi,, ?= Zov^r,hr
M= lot<,lJrn
ibl = t6a Nv?q
Cgyv+-r.*ut Jr;y,^*r, vF Q* d[Vl,x , ^d F'^d +a.s Jiqft,-a-t* o-F{4^e b.eqcrt
ft/lz
fi^r O
Lr=
9-o.7s-1O+tO3
= I t) K.N r/.
L.= q-
z lfro
Z lVa
L.+ trn
Rqa
+ aq^*6-q_
Ye*urn fi5 {-rLo
+- ?,q.a.
s=O
= fvl,
(+ = f|4, +tfia-U
o,r;3i,.J lr<ar,^
Rs
?5, 25 KA
1.5.1s
l') , tzt'lY?
t
t4
)(q =
11,+s :
-11.?5 l*30r -l=-tT"+sl-/rt=D l-
/ X-*a
t 3O1. -SX€e
14e,1.,'t,n
wrd S"l5
vo,Lue.,r" uvg
il Lc+" *t *,t
11. o I x,,,rr,
ri,d8
)e,+,?-fn;nqt.yr- dF C ta"Me.+u ,Jl
I
{4^e\8€ cvry) 1-
.s
7x)
x
o.
p
+
D
-7x6q
-19 >c'b
t--t.._
q.
j
+ftr
+ /oF'sr1 h J,+- eF gi-.".,014
- tq.
l:"':)c
I IVt, -* ,l- tJ,*
Tr o'32.
he10. o" = lT-D'
n( to=l
z/ e*tcrl'
I{t|-,^^ }= ?,0.+s t<-a)n,
0,t01?m
= llom
^"-A 6,*S 0* ^^A {U
Quu,. -(6 t?x *ul*
Lct?5X
O = - r1""s
-r,t J;o{,r+ (sxrg.ts]*zCs)
-Jp. :y : o,+s^lo
S ri6q W( hc"tte -l-ru-o p,;t,.,h.v<! hruC,( 4'cv+tp *ttJ_ Va,[.ze ru hicln
*{.o t-r^Xl+. q..
a< { q..
Xe ;. O,?S
lVlor,,t = x-
25"2sx,J
+
3t x to.+s xros
B.t+XtGoXtolD z ..zl vql%-- :
V rr [nt
t lr,-
q ^ l/u, P= lokrl
7- = 2-o y-rn lh
WQ T Ltye-o rnen4-o-J *UA"3
Iil o
*
NrP
-6rt (d,o
/',= d,(f) :
d"=d.(z)=
(,grr,>r )e"i,-X +ftq ll t+ Po,+'
11,J. tq + ti,l, ( a-n +*.) + [1, ta
= 4fY\, +?rvt* =-f,(^.trS/ylrt an,)r_= -SSU1. = -1T.5 ^ 2.rl1,
Co"rsiJo-\ .J-h{ ?.^J f *.
M, tq + M^6ta-r-to) + M" Aq
,{Jp (xil\I6 EI
7.Go)'z+ EJr-p)
3l
= *6E d,* + du)
o, d3t,D
0
r(lfrt=
ri;J
Cu"Atntct- J.,-y*, ur
0*t* o^l' l4rt t -
yr* e-,*nS+a,lr'rqtb ti le+a*';*a-otp b< q"r,.
R..t\
)n' '-z!a'
llt
= X"5EI
=?n3trr
lA3' r/'o
er3
--;\(,
Zm*=
F--Mc = olv1,
R;,
^l*Pa.* Rue4-fi4t
Au' = _{U p.. =AA,
-Knxqf p*-M,-'.(* = ?q-fi4, =9-a
1(7t\* -M*+ RJ
2 bq)" * {U1,, * tfrz21
LA
^ ^A- 1)
1_W) _ Ru e,q _
M, t z@q*-m.e,A
* eyn*.,e"a
^AfVi,) = - u,
., /Y1., =
b<a.,n: a*
-TSOtrt'", V-t^e t;o*+
?fvl, + + (-a+.S
l\, = ^1%
Lc+ u^ sf tt'+ ,t1"2
r:0 q c-haDL wtt t
= * 4--b"K^)'
(l d 0r
_E6
fL e'-1 c
t! ('6
/tr1,o
2M,o=D
l0,s+. xa)'
Z Mr,, = c)
O
[,\, *')
Rc:
l0^ ll" 6T = -- O. &+/<J
= t_1. 6l-+ +o ^+, S* _.- I q. s ,)7
II
=
=- 3.67 =
-Kq =
- e, 1+Fd
".5+ K)
^^A clu-ia-rn *j
@ZfYl c ='o
iu4
J,,,/^
:- IG=:n43AA
nn -T(C -
)1, 6* + rD
' [Y1** R, k
)
R.
R"
A* = 0,t +^ I Rs= q+ Ro' 6: 10.8+ + a6,5 = B+.2+ I
tL.S + a. Z,l- = t g" t+ r)
J
.l4uPP;u,ni^6 +o Qvil,.,'-a! hd S24T<.,,, qo I fgrp5.f,ac*.,uc3 dr-aX,u-,^Bt Q* "*i IVL.
Ro
9
(J+ tt."
1.3+xl1
o,Bq-
Ir '' 3':'
iN
/iilti,I;tl,6e
,_,31t'\uv ' Q;''' KJ
Nt r' , yt'Jtn
I_,^, T-
U* 8'
MJ o,
T-n
Ql o,
{7, =
-&x
= Ro -("= ?"b+ -7( = t&.s.f
0 (.rc4 q
-4.6txa,^
I
LDx f =I x=J
= O"Bxsrn
a, a+ rd
'l=o l=/ x= D /r."
0 (ec <a
l:.r ?x'
I 8,84- f
- e.R++-
2-,1 3
*t6.5 q3,s
9.t s
-. t4ia, = *l?r(ao+r) + 4x -Xx?"
= - 9,i ?+ (a+x) * t&,8+x - I 0*o
I =- ?,1\ Y,,t*,Ir.t=r(€: O.8.lS
:
I =-. +,(,slx=o
I = -ll.(r8 ru;m'Ir.^
G-m
OJ,q**
:[i--D
Q."o'
[4y"r,,,
0 rc<, 1
s -A+ E -g.g4*,,J
= .-(?nx
= -0.84x h,o /:,- 0.t4w^
-f H rStp sra ,\LO?e* A4' vr ber+ans
D(r<-t
= -Ro-P =-to.g+Ki I
= ._ (o C<o") Pr.
= *0. r+(r+*) -_rox l;;c)'&+ I =,)(--l
* tt,t!xnrrrt
o;)L:e+!J
al-
p,dn= dn=
Am__"Am$ do= Ll L3 +7 L'I29EI
.t' 40=
b84 f=IL
d*
d,
= 7q^2+LET
0
= 7q^
";A
z ..4( 2L *q)
2- r.\-q /
W'"F"rfr(^,
-
do = do.= Pt^
dt=
du'
tbEr
pau (Ltb)6 L EI
Pqb(*+)6 trt
)tu\rr= Qg =
dn = /},1oL-:.=-f
2t_,J! +
d,n = l"].!,ii.Lt E 7..
d*= U"- ( g"t6LLL \
dn = ,!" (Zo^6r e r \
P,. ( t -o)o tr-T
+) fi10l-r2
:h ---- -.LLI 7-/71
.ldn=
l4o L
6trI
o) N1
&!?LI EI
()2-Jq
a\-L )
^\^ 1K I*AL
/
lal,-'?7F- - ---.-:1----+S__ "__>,l-l
d, = U-L:Bko e t ) ,l =. 7, L3"rB i5 Er
7u9,;n*r_
A
Nn= do= 5?" L3112, Er
d,5= %tsIT, EJ
d*=
F
g g;
Buckling of Golumns
I
I
I
!.*
^lUe are bont to intyrove tftet o a.c crntttttfat e ., no[t h. !{owyoLL?' tife?
fives of yeoyfe an'otnuf us, antdnotmcLny yeoyfe fi.ave you fietpedht
Chukw u, Celestine Nkem akol am,
?.";l i?':::;
!
bucKtrd cr W COtuMrus
C-O\".*, 'il a CanVft,,rtto'rt therMa J-ha* r'.9 .8-,, S le,--)p--
(p,oyo..J +o i +g @An, J44^a uJe" fj,-"4 inoreu.*q
b-"!, r { f ai ts t X brau<L4 c* Lools CDns, )u-'*t7" (t rs
l-l+a,,.,- {l+ow r<Xu;r"l h ca-'q-+a foi(ur-e 7 hqsh,'^l.. 9u<,l\t a-laa' s ( h-e,rw-rno, on , a J-r;r7 4-o,r'J in i a-,-q-!- @t rcis +.c> t,'^ rl Suul, lhirtt'rt"r,r-rrt k,'{;cJ [ool, lor. ,Nh,'do u)Ord el Cau 52 cz SLp*r)e," nc,ruzbz- /p bot tt</<
, ( la.fo.^ h,lo- rb * u*J /r," ^ tu"j ,'/r.J, ^.Jl L" l ) .\
T*o fu-r"4lq s a4< u,r..r! fry, /e,p,-u,n,^l PuI
I
I
Il
I t) Ct le*'s
l')la =
2 > 7.u'^
fo-*"{*T'E I^,'^
Cu L)"
V- h^X+h reJrrc+-i-Coe44,C.e,ru-t
L _ 0ri1""_>_.1, k^l{4 tt:CoLtu.a
YL =Lo = t4p<c*.,tz b^"rtt
E - 'le,*a't rnr'5"1""4
-) Yo , in s rt'S farn,^tt -
p* = k_b^)frA, <A<7u^
tLn< SLQ^. lo-rnu-l rcll-1.D i s
rJ l+c)-r-l lLo-,)*nor^ t\*;o qF
q', l, Ctrl Ccte.{ f1L-e^-* \dr,fr-"r1..-l gn *r.c n ctr,rt ()Ff1 a*.reri^-( a-.c{ r.j Jiu,r.u ,,,
+-"/6U
fr * Aun- s<c*'ar,-+L ft:-<4.
2 - 3(t--tl+-naa", rt-l.oCc<( c+,. te<rt-"-rl % *W f,s-r,.,,*ts*.
+-t+z na.,t<*r*q-t
L',n,i /?lin,'rt<w,t fa Jr'ns fp3 ?'-*v^Ihi" i /tta,'4 t14a-r.Lt ^t- ftrlnirh\.
0; - 'f,*r",+,J { "/N ,,f3+re+4 vlLo^1 H-ot l<< )s
[,i-J is v q U-0 .
I
I
L:J L'a,'d',-J,d
ALi,r T"EU?,
3b^)+nnrt ruri.W.
'-r1
tt,< A,t-Lon's fe'-,'^19"
Lt^4' fa s, :3 t?1' \ {sv/t4^1
ae,ffi'de^:t ; V, i s &4.h/'""*J,'*t-. gP I oa lirn
l?, ?*1...fflN 'fv
ttIIIllL)-
!=O's V=0'*
lF, 2 z, 2u.,^
2( )u.,n
T1^e k "4?h rulooh*vF "-^i C s* t/sr+) C/'"
'-tfv t"lrtlllA)-V: I Y=q
ar
VrL <4"{ frP"
' .:l
,.2. O
F*, Yh*rt Cntumu ( s'n-u 2\
C\ = [* = rl. LoJF.t- L afi3 C-oLt' n*t
6*= n" f trlyF ,r< J-",ru+. _W = Q6A
fr,'{'L
f'fi " - f"r+,y cf \cwft'*7
t)s S+.n-\ ilu,rg h.++- of s,,p,,*r.
r(_ atlu*r.=-Ek
q [r] " ,.:I;:.^_f-qc;--..
yaln,es.f W. (Z) e.v<- 3,*. ln-+-4"L14 q,+ .l4q e^*{ r)- +L"vr ct-*(?+- .
-1t^o- [.*.= \-QD-,t
Co"^.{^'+,"r" vT- g+",,ti tj},.j
A- F ,, t{ LU1 ,uAs
et r-e !9.-Jg-'^,^ Zta,
'a ^-A L;r il I,rL,
*J [0]* =
i f "Yl\Lye,+Or,e.
S%
l,t gn6V,tw ri, v,ltO,^2 JtJ+<r-tv,'r.-..,.{.,s", g+ n.g c€-) \*aqlu'fla-^at*,,n vF q C,6Lur^A, taJ.e vA4 a,n lyrrhq.f ,r.L*l u€ U.'I=C.S.lF a,We^' Ct^r.resr,tg rl-k-Q J-n'r*l-r,'u-- , Fr1^.^ cvx<+--q-t,tc-* a.Jl4I ' ( tac",^c"aA wri-tr fr'.n- clnot+n );,'.y*r,'* ) rn, f^- (yLtr-t )w..o-a
QO:
=t4^-!,^ rn lo
,Q'o Q|'t<
C a (C,Aa*>l- tJ t,l/La
I hWKr",M t €le S*i
J
L=g-rri , P= lsoxrt)
ttr) = tbornf q
{n cdu-,-^!. - S+€€l I (Cttotb 3)
Fi- &A/ftA
./-t"-tz
of {44-a-
nL &/t4A,*Jon&
A"J,A- yec*t*^-^l
9<c+ierr, tv{*Lt-
l) ln hp fi.,y+
V.' = o'-s
4^N^ A_ :_' \Ar I o-]
-Tt.t- /14ar<*t @4 e.<cr fr*^ esSo'*rn-o<t- r.s. for,.^-r( f rtNg lq"E (l) wi*t en-e-a, = g.^"Dr*", T^,n= lzsc,_fFu, +t.', c.Lwtn^ J.'*rr',,, tl < S Lt,tt( ;:;'" *'
-;1,o'*'io' ,:*":-',kt* \{4";l"r^0 {-et ut-*r of- S^4?-t7 L0,' ^, {-o lk^,s
1- V(. = D.+ x0-/t - L',,i^ a4-x.r-1 = 56r S
SolvL+;t+
U/|7rvauh, LA-+
= I kr:xt030.sxli'-oxto[
u44
: R.isi,o-s lrt o
= A-C.5 C-,r*t 4
V:- O" ?-
Nl^!-. r.t)? c hecz< i^ *ts d-4hu 0f , Q 0) f,rr stu-( s, wt \h_*L(9'6.s+,,-l< +14-^rF Q,t-ta- \f qlL,< (,.e< be,.l-,_e_p^
2, = 5o u.r i+4,t 14, = O, trl o^_[
k* ua J" q C'n*^, €x*rq4 o Lqfigly b<,+1-u^ ih!41 beunJora,V ilR,L +o fi'^*l a,vt a /t)r,o t,t'i.4r,+<- V>il^^r- op L!.' ''
q
Q.'
1pr'
r{L
v, - 10..
),-),: Q''- q^
).-)
)^ ) --6'v._7,
k
B{--l
I
" l0''= Q*n (rp, U")(),-))
'9.a.1
--T
I
^r)
T)
T
{'
(
t
T
r)
7
r
/ -D's6) (et-s6"s)$,E6+ (o'rq
=6r> - so
a= tot-t/"r* = o.&+r:g'srLgoL2" "T -_-o"r- t
/:>S% T1^l44't-€"< hrc ft^o-t-A
1n t iA t<- Ce"-d arl P*n&
v L ----E- a-
l.f"^fi,"? d4^-( la/rPL€ \4y<,4c< eF
Svw +(frt^ c't
13oxr03 :iiii,,o*,ot
O- 6s=
CaLc6tct*fcrn fef fe*, t\ ti +t-a.
o.s+r
= 74.t "A
C*.,ti )L, q
v,'' z w'
X C-) n4 fw< cl',
2)
A-: ? :-, L2-LU-I
TL.,r- 1 V[ :/tLra t.*
tti- Ce
3) l-?'
A=
'fl"L Y1ea,rult At-Lq
T l* n 4 a re-at V il*e t* a+<-q .fwn
(q: Ntg lD[B) ru,'+r" $ = /5. 6 c,n?,
(ut Ng 1(1) t,,ti't't, A-:t5"6c,n\,k* tt+ @)
1.6+,n <
= I [.+e"*.'
Q S S or-*rr-ue--;t ar< fe-rr^-! ' jl
f*r"= 6o.7r.,,"*, Ln,,n= l'1Pr*^
I],t - +3. rc,nf , ('n ,; : 1" ?7c,.
0,?-x e : +0.)-1,1 g x'1o-z
flt r-S g ql*,e L'U bc).,t.e-o.,rr- ), = 7-,c tt t..t41
7 y = 8o d;1-t'L U" = 0, +str-f" ^4{t a *.'u,. 2;u Ju'
Vo'- 0.+-s + &.u,-D.)s)(so-?D.+)6-0 - k)
st. ^lJ-' :[
A: n -U *,$ = O.&oL-O.6ts100 ' E I
Q,=O'e/ ^--"!
O. yot
v*' , ?at+.66 q"
'A 7 s{L, o,e \att.tl Cgn t,' ) t4- 4 +h;.il aV prtocL
rl: O.6gs *o.soL = O-+ +LA
2lKoxto-
t
_!_(!=[tr]
-rl,{-+u
=_;-?\
g"+,+6 xlGorrob ;fr"*, A |L-vrl-tnp**
l-tStxto 3.*^
tS-l,o,L#/'S fer-*l ,t, N!
)
tt (*)2\7
-
276"atta*i" p;eth'
+ c,c"ta- o'bt) cqi' 63.q)
40- L-= 0-vLtl'
Q-' 2 to*'
7 5'ao^ +46
F%rrrh qrfit" rh
V'u = lT*-t!t'=P.
y tfr t(.-T t ), ?3
il'= CI"k/
fi= o"v+l^ o"+4LL
A= I : teoxro1- l"+eyLor*oL/''L (fJ D'T1,t x rLo xto L
: t4. q c* \
X, *1"+ ( o . x t- D"lr) Cs-o- +q.. i
go - ?-o
O'7+6 t- O"s,fl i 0"+1+
: +1 "l,..t iy4. L0, = o.& I ^-lut;++. Q= O,+S
= O.+ss.
- \lr-a,"u+ f"tL^s OF # frt,'t aatuy*m^t- i's feu.^l;n N:1&)wi*t A= lg,qa,^o, ]^,". 43.yr.^* , (;^ -- l-++
) -Y-L = Axo.v/t _._Lm,tt /,??X /O-A
This L,nz 2, = T-o
2Y= &o
| 2sar.<1zJl cuaroa
0'''= o'+s
Sr ,i Ce' Lr'u' < '.-e
tu
{t,t*, Lq] = V t'= 0.}ts ^A sLw c-t+tg.. seL+1},, ,. i
44\e,1-oirl< A,l-3 1(g) w f Yh $ = B, "t u-4
N0.lc: lF V'u' *rnu *rn 9tucr,-r q*, CorLc,*u.r< A 4,J C\.ec-t< v,sL,u-t,r*A zsI" uv A<50/". rF A<5'/., d-4r-* €Jte6futrvt-e +"rs+a";-z**fg"-Lc dF q o"r*l {.v.{ C*a-a e**1,^-r1 /t"-r-r. f-n a+t_t-nu^:(_ fs- U-.c(ll""*"n
II
t*
I
-l
I
-l
1
I
I
WW^4ttn^", \)U4J4T1 r\U ,v '
I rt Ca,rct v,lltp,r-e- e*t Lotanfz- *tvo ir'pQ*.^ l' S< L4-,r", lU g w tdz't-
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= D,\3 ,nn '
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