Mechanics of materials

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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

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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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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

-

N,&) Ixnr]

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

! *^f ar(tco;,-? ,to" r;,n L,f*

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

'?^o,,

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

?*

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,

.J.4etue. J-Lua,a /,otxx-,r ^r;'ff^k;"*' L''n* i'F ae r'-s b'.4.

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

*7n *

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

It ccg?

+ {r-Y":a

F 1-,** 4

I

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

-o+

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

&rqc{"Vna r b1^.* c;"Ll*^ls r vtr*t'uru - -Th-a're{gaq ,J,la-o,rz ud irt 1-*t- cl h,r,'zrc*.**.g 0il.V*-l.. CiL( f{qcfi'urr

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-

\+r-h-c,J f{clc-l"t.n f.srte uti,'LL al pe.c,o

x.l

.J) ro'i rh-,

*L1--)

\0-l

'Tlr, s \wf f vt't A-rorn ci. Pg'1-a>{el'r'r- i', ,."1[{a SLur.r.t-L l,..a,.ee bcrl<

)'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

=>

+

? t-i;i^ur ,4+,,k

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^

VY f 6l-tt esch e<a"w)

a.J>luv

tij:

,/,,rur+*- D+t.,

eiVhl4'., -.'g,F

:c sd 4,^_e* 4*1$t^, JD;n i A sy ,3 ,

&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

ID t':{-rl,-rrurn.a:J r'u-, 8f A'q..*-a 9,utxl, t"'';t t:'F Q 7 tti L'' bn *rrt

.

ZF"

6or-*)

tt (+

.,?*.*(**q-da =o

= F - (^it'

K*n= [ -

i

)

Swf ytw'+ A

!Iucut{ CLtturi

$w1of crt- B u,^t;t( f,.A"z-s 04\l.to.tt cr.,t4r: Czt,tcrV ,r'*d a{b' Lc+.-A

Z {\,111 : D

,\l '- ?rv1 -(',

7,b: *^

fvt*uvt - q,+ -ry(*.r)

Ao * 8o * e-DCz)^ *40(* + 3)

-\./

fe p,,:lutz--.I rta rv,*nqe^'iI wu^*,I d-=n{._ _ .. ___-./

r

3

I,.A Kru

C

- ,et,

,o#&

;

'|k{

r.

I

1

I

I

I

I

I

II

ItI

1

1I

II

I

II

l

I

.

I

ri

:

I ,:

I

*3+^*8 rcnt'

= * +D ('3) * a.o (a) .F 3?, lB -F r r. ?., a x,

. ,D<-lU.trvrr cul-vo.r.r ef 1ryfi?,r-tt-l.,t fOr{t: fqg.k,.1

: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.

e-l__e v.?, S mcot.t 8)<.,.lrtc.'.tf 0-|

J o,n/.- l(2.1b.3,t

7jt/__-- i

4r+H1--r ---l J+o,\ l,8it,ra L,)

/l\! >hI

- l??'l'7,--.-___---l a+D

, a4 p6<4.

4+

-T'.

TJ

..1

I

l

l

1

I

I

I

l

i

Ca+<. .,t,.. t. :::.: t,,

K

i{^94:' i1a. lrlri ,' La.gA a.,,t @ ,"1A-\-4i:i.r r' I .. rr I . :j I .

_"_;.,--.

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= (,]'

R,,h --

llg.t-1-:-

-o ,c

?tr- I u.r; xrrj

z $11; =-o

F.r *ovIt p,

,,)

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

=

Ro''- - Zt *M2

st +M^

3 - , O ,< )'%

4^e

/(}{-5o-2-rjt4*tQ :o t/

Ira*e.d Feya Fq c-*ry,-s

+- If,

Nl" [n) =/-. X.

tl'- 8l =

ATTlvL,r 6c) =U

=

(oo'

- Rn\

(*h r.t 0O>c

= ?0 K,tl:

= -tUOrcnl

rltI?D /= -3otz=-D

0<x<l

I *O J<aJ ,ru'

0{>c < b

T -]L

Nlc^l = A) = rDD K^r

Qlnt = Rl* q,*

20-- 2D,c 1; ror^r

t--I-lx=a

I :- -'to Kr't

I x= b =3

J"'

:

-R"'x*n4+ BIJ r- lox - 3o + rDDCe)

$t.2

+ lDr, J=*lx=3

F,t1,ra"

4

i

.j

=

)

. iDr-,4ortnrrutlula #'. : lst,.^rr.'-.n -. ' t t't

"'

il

Q**ocJ=o = R:-7,xnJ

Mfu? = *rr(,)-ao+- en,'+tD(')o: 149

/\vf.- Rn'_ -W =11e4

n I --- -totl{r\)ntL' I

z=J Ir=4- *

fII- -1trN,Tc.,=

QJbc) =

O

*F

-Fx

= - sDKrr)

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!!!

Chukw u, Cel estin e N kem akol am .

5X

"'AAJfl-LYsrsOf' , Sr RFss e* d+l -D ,gr1 rl-rrrl

Ca[ Cur-a+ion' sf s'lrsl.{ed i. pt@r?. . ea posi+ .rn usr*1;-,-_-.-

rl,

, Si:lru CEtrteri+i u*t

a '-1qn i, l<- S*,'ea1g-l cL.y-. cLe t;,j",r#u.0S,bt t.r-l_ S a"< ru*4o;h tl_tt

) fu+, c Ltu*r,t-e wnlwia* r-u<lqj-,.eh , d,11246t'l-i'J"e

,s fcs,.t-iJe ,

fe.,-,.1 \ -li) ;-s:/-c2:lq *l/,ti.

s dJt t,'q rr,afu cf pir,,r'.{r'r/rJ

Sl.u,e" 1-|re-1.1 " 1', .*{er"J-

C. [^., c+1,'O,'* .t,r' 'e a'l-':rt' i\i3n ir .l-r"+"-J f.,rr ou4#, C LOr,<

",.t i s< Lr rtt.la.i1

J

-|i"ygL*_Zy

*:pll-.

^\--O\r-- Cz Sinad.

ac\-(rrt'

Gi .l*n'

*d.

tdth

G]i'l"it

O-

md

- tO rrur^

= 2gnf4

= Zoo

=fr(D

-Ji

il z,JI >tr

p"'S,'.f, u(o w1,",Lr LDn4--a, n-,,.d

C Lc, f +.. rr),'ge ,S'

?l<.t':-,.t* .i "

,J}r-. i-q (tltq.{t'1,.-/

S Lr-e_.i.r- t.trUt .

n") q, Ta, fr, '79 1

U; = f,c,x'a t-E sin^d

: Bo caa3o +. 2o€rnn*r(.

= 6srw*,

ftr = t[5,i",* t- D-^ Cu,s'.i'

= $Ds,-t JD'r l-c cr,'ra1Tf

= 3s rvP4'

CL,,ecad^-+

Dlr[.: CI +rleg.CtaO = 6S.+-as

( C.L"tu.t,. tt )

Td=

=

Srn?-d

:j 'i 6o .: A 6Wer

?p *t{rWq

Uq = 6s*ea

= 2.5 rn,e4

G= lorrrPr

Gta,uh'ca.( !O1zt*.oa. ugi,"-? fnOhr's _C,; ch .

?o (f, o) ,o( =. Boo

f""J pc C rX,

Po ( tre,

?n (tr, c)

7")Tr)

II-E

II

:r1

s(4, %) - _

* (q, o)

[=

Il =-",,r

E

[,i =

e(q,o

tftj,f)

0l+ rle l-'.o, - 4 (rs2ot

e^

[1[^ + CI-DCurSr-,r2 .:-

Eqt-P r -E.'-, Crr c-c -=- krurae2

ry] -- 4=*- o:sad

5C SOCos bt> : TSrtvfq

q - C. 9r,, L,r. = :.. :6.tvt+r't

til = *?*

-tt t,q. \^{_}ur-{

Si{,uU.^;.",I-? ce( ,J{ J ru J,.r, ..r.olg l-,-.,r(

::- - ?- Ln1"{r4'

Oi* *-tuz Scr-.trta cV, .tl.Lq g.1.t( \

I

I

I

Qu<r"

tr"")0: ,F Oo n fq ) f= - 16t rvv?q ,

trd, aa ,, 0*9, T:,

d = +s o

U,7ii.r-2 Fr

O = lrnranOa:.Oq = *+rr3 Nei-

/ft.soicr.n" C-.4m) t (*+,st) bf+s : '

i::'r

Lfr-

(d +.,o - G+nl) ,,,n C-l.o). = - 46o rwpta

'Ty * 4,sotwql :

fr" =-O

Qn"fl;t ij- !rjuq+"p

();,.0)\

0j,D )

-+>

q

?ft({',J (d,--

t-

i

I

I a

qT,<\

t'= *'rol : ofsa,a.pa '

c(U,7o)

,-r

l

I

q^iNt-r v,, a(i) Or^A 0( '

A'.rn4.1"1.1-,-c^t Suri'n*'''b'',r

.l-an xdt =Q_-h ,

G-o; )' d= r -tf'- 2xruc--1t+4, / =#_i:_J: 6- a-[D

.+,)+.+ (^

Lhe ut; g .

fr*q = ffi+%Jll-6rt = 3oo- &-r.d;

mfdO,,

a1/[d. :(B

7y,= 7m rwfq

: * 6ntw<

- t uO fi/P<'

'lvt'rvr'q

[=

E=

3u, '- 6"o

l5D + L{bi = ?tlrn4q

- tF) * 161 * 6il wvpr.

5r= bttr'4a

A, = jtl wt+

d= 6.t t'

/q-Dr. ltL a/

Gri,fl;c^l IM;; , C-it ;;- &,&Ji

Gi,te^ -*A-( [^ , tn)B(G,T^)

F^.{ a) d"o

b) Uwr,fYtta

D

l.lrvl .hc &"g; 'lztr-r.1 T,^&'I.e.f,,f 't-L..C 7p*C re{e,re,.a ct t\\ta .,(: -(, n, Jnr*

P' ( dr= , Tr) [-04{".r" .t'r a'^ i w2 "ttr' &:

k ( fi , Y^ ) 'r , 3lr,r!.r..d-{

a*^ .t .fr.rri n ,1cr-'t0l

JO,. -i4u *u{o., f O,',,,+S A c.uw) [\ tt ( !,1ra....1;f{4ll i',*e- LuP.*,*

^tr .1.r, f- q;t. s r..,1.n c.l^ Sut.cu-l k

f\W1,.,,, 'l A"- c.i-t -Drrr,v.) _E., c\ ( r. Cb ,,* pJ ,.jrq c{, * rt [7;:1-.----l( = \/d;-d;\'-{L==, '' ro'

'Fi ".!. -l-.u.o i",'l-.-clr+s ef #^-. ow cts a-+ .J 4ui fr- ,, r."' , o"....-i.L.'Vrot *lt,-,tt ct-t L r,,.^J ? . Tl4r,-a f o, ^j; SLuut( Grrra g-;.t t': .-l.o

C*,; ,^-i m* rt+y)e c+oe-!-.6 - (\\.La4u",e -ptu V.ntz,rr,, u-p

0-

tlc ova-el O-p

WIU ryt +k L'*U/4 o-\ Ac ca-a{a

t€ CoirJr Sl^r,t{

- LUiY q P,ot"" c*tr, (vv.-a-s,^,< *W

ft e ftqlu) wtyh^ +rq 6- AxL. I F

s cA& tu-A J"*";;3, vw a,na,,re.,-s.b{" ?y^t .{-o {4^s r<A(lLa4. *

'''t'&lt ru cs" t/e J

(,',.L r.^i' i..,J-..|4rl Ce,^ d >< tFl< Lij r r.1 .; .)..._L

ld.o

\-*t

4fi4^l"y ,.

6

-;--ttu=!nrd,^ ,

,

6p (t,*\ - K

\tA\ 6/14fu+ c^{

5B I

lfisof, 'r La") Frr. Ae'^,\a S!c7y,ur' .a*) ..Gen - o,)

'i,,

t*tJr IJIlhi.tK{

€*=

whBy-e

C.Lx

Lt=

tv=

L,l =

?

0;=%=0-r =

/\*- -! / mtr2 )a E\ w ,/

*- u (c*+G)e T t+ _ + (u,+4)

€,' - Arnw+., S*rruia.s in Lr*c*.,r^ ef c

E You.*3's fturlJur.Lrcr et eLa-.sh ci.l-3

l,,l = f Di sr<:z rq+.,.b

€-r + (-,, + [r.

ry[mr{r d;JrltLnqlr/: t, - lgh,rr"c+".:'^

,9..SLt *rt-q,

t,c* fr+ L7'

,=+* [O;*Fv "d;l

,Lt = o'2 5j

fr,; rl t u

rl.!:

ilr-\\r

4Vrr*q'

- &enm4o<t'

Sowq

-,.€v = t - z Co,zt)

axros

8. 8 x,D:s:===

foo- &o + *DJL

h$),s,-ry .

-ll

--1-

E - 2x r0' ffv74.'

l'1 + oo3

d =80

m=.:

=

vr)

=

'12(n 1::

: Stil'unr'l-,

ill c-'a + F, s; "'. 'l / \ '2-t t.o G) ' i D t-(- 5D) Sr L,--3 D

6 ^,

s m,{t-q

CII S,i,'d 1- Fu C*tx

i du g'i"." :;. -t ( -sc) C-rx'} D

!:l-9-e s,"&* '

,_'

F-^l '& +l^{ s&rua ^* '}'r\r' t } c L'" "'riwI

A,--d € * .

= - lt"5*r'P'q'

= ltlu*sD g,_ &o = al,bSrzrf")--lp .= -'YF = - 7l . Lt rnlt't

:= t- e(o2r los

fAa

L (,

t

L

'-, )g1o.-'

Q-pzr-L--

l{=

i0xr o')^(mrwpa

&.Xro s fwPn,

o,3 0o//q\

7ut Fr^; t,, , ?n,

,^X Zv

,.{d0

to'G: rd,'d;

L,0\#' f-)

[..- , J i

- I I /1,I -Fr'\J

60

X5tnry:

A* =o*,'' q -o ( Canwu* ery*^J oun1 1^'), * Cs = - P = - lo W

IJ

i,s6*ro-4-otlo7.@ aLvPiad

.L__-

/f

lru,4z.

q

i, =- o,B.14-

ln Nhioh o?vts frJ* q/

flMYn'mt*rn ClM"Aa

( a.sguttt J'14A* frb,*o c ,'s

tlo "

A o,,.-nl C SLwt( Nt

eu 3,

rf g,'l ) 6tI

-i

f* f-ly: ?. * '*D;=O, DS=O, A*=-dE

E o,.rt* b

0i=o, tc:DLcu= c= -t'- 4(c.+4)GE

* 0"46;E'v

i

L-q"r,n c

,\-€, -- o'= L) 'cE

c= c1 .-- tt )

#(w-m)-L( s;, - fi)(-L

.Gl =N-t

tr

0* 6,,E

L=d, = 2 o-;-f

b"btl,"\ i,,ro el ,f

ev: ry [0;+e+o]l

I %- ,'. €u *^n C.gy,'e*?,*rj" -r-.3 f-j*"

'., ..t -irl ji r.,i'.

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!!!

Chukw u, Cel estine Nkemakol am .

6s

-TJ S'i-Reroc"I,i;: i4tniI,:$ri

"

qt 8oa, 4l]a,,&q-L Y-lr-e/.---LN*-T E,.t) sr q."l ' Ad' Cg/n'lRe ssr urJ a - .

AL.- {

{wrrr,"*,- F"- l.hrtr1ra 1r t 'J "}4L":r T--}--

D;- €F Q= s.fra.r^=4} . Lf=L/

N = AL L- , L* ori3i..n( La^5+,. or JA L aL - d^'&^4e 'L Va".,YL'''d' ,-l '

lrrhLo,"''t

)-A7> a,!-c', . cl = jn'r

\ = H-nt

C= lm

;|: 5 c'*' '

F, = 6t, x-n

F:- : !o x'rv

Fl .= irr: i.:nl

F.,."n /ir.(r) ., C r,r,, /] i-4i r

NT

N,,6'1[ta))

tr"@')

At- =e+*A )<4*-***^^U Cowp:u+"-\, LI u"l,*""*^

t" .1-e-v\a"\r.^ ar,"=a| 'CUwfr<s4,-. ti

!d;1^a.^1.

1,63

r)1. (mM)

6+

-:r'J

l

'"f.J

!J _TI

N, ['r =-IL' _ TE

!

:

u" -urr TN.ri e) =

-

-dU, c*r =

,-- filUr (r/ --

l-r = 60 lu.,i

F,-F. tFa6-0 - &o +la-D

D 4.:c .q q.

Cot { a^ ta*tirn of A C'h'nX S.k4-n<,.t oJ-a.^,,, !< c.l,rnl,i1,.) = Atf

= .*-.-?fr

*,UJ:3A

N,t, =

bo n to3axsxro-f

*2rx t03

3'* sxro-*?

luD r tO -'

=5xro-f

lu"ro K;

- l3 .3 m,p4:.4-

2emrxf4'---:-*

I

I

I

I

fr

Ctutqr{rarr

.:!

(-.

dpg = ALle

=

Cc\t

-tUPJ

tr"

ef Ois PL". k mt--,:l\ c+l,s.'u. Se c*,a4a ( d)

; < dAl*^* 'o a,* S4^4 \ uvl, f ".r{. r'J La.,-o

ALr. = f\J" = lrt$xto5FA- ax lDilx 5x Lo*t' '= l,Oxlo-; = lnrn-

*"31x to*?m s 0. Vlt rnrn

: Alr.'* al.r.

[,D x,r-3 r N ]w b3*r+

t.oxto-3+ (-al^,rt) +Zx qx,ot'x sx , df

A Lra * NJr*e?-ffi

*rzz xto-t + &ox r 03x3

ill*J'rl-"-+l, Lg xr o-3 n : 1,6 g ft.fti .

"tIIIAL+, F

-

Er+- -

-,EF!5tI

I

- 511,'h,;

Ft = BK/0

[-" |ruJFc= rfi

Q =nQ"l m

lt = O"3r?a

c = o'85 zr

f -*-1LU- l" = 1sD ft,f,i

\f(I]o = i&6"tu?'

Rl r ^.'N*(,?-, k, Dlt..,, Ak.,.I

\r -/^/.y'-I(ri-

'- - q,t sn b,a j.dr're cJ.i*,t

A-tttC4tt-q_{& te c'/-^r'"

l\ *0,,, xil

4.zw

lx0,3,rjI

f)a,, Mpa

'ALua.t xtq-t (fi)

)ef4rr'ftv;na.:J-rnrr, 0f U)rW\*1,1r1^ryL p_l*in ftrrlq (l* ^n A"

ZF;:e = -(u+ R*+ q4-Fr -Ft-[-[tn's is Si,"gt? S]q*,ce,*b \n &.rt*ni^a#€

, e, uNaae,nn I e41tql_r,r) : l,

l,lta

i

-cr1 61,

IIl

.:

Ilt.;

iI

.rf(. rT!,Y4a) ALnu - o

IA w Nr,.J4 Ll+q*; rn,'Nr"(*t-',--Ra

Nroo., = *Rn *Fr

L<+ uA t'< ,d., trp e-?r,^r.ti

I

l

= ALn. + AL.o + ALre -l- Alc_a >^- -.t-0,

ots rtvm^l fr-a"AJ fg-, s - Rn.*E .r f,,.. t'JAjr ()r): _R* _h.*Fa_

t in .{-?,rm, . rF fLo-lu.^!. -fuun

: t?lo , (- fto * F,)aqO.+AE L) "^f l1E

+ ('- R* -p, +r.r)b

F. l='q

Atno = C=

€-4-

\OtJr= i+ue e.iflia>l.rw f.,r _,.-t.r< L/r)K,,ur.,.rz d_,:r(c i?,+f_a -r- R*aq+- Rnb., Roc=_Fe,o'' 4e T-on* ' ff '. -=, - -,-:-, -+ r_E]= F, c .r r*c

,l w. A e c,.^n ,0 T4_(1 v..,n -^ - _. c+ [-A Er+

R^[f" rk "r+c] = -*#:l-

"L- sc r-8. _F.

"DTK6- i_ t -F,eeL ;; - l-; u + LL - F, (+A. -F..1+ O-^A iil

.r- C-r"--l-,_lB:FJ_.E; /+

t=#3 -8(o'r)'r-5 6,)-s( o.xs)rs(

_-!.6l rt

o"gsl *o",!-s

f o.iL; - **'Fo'-r -i-

: 4,26t xN

D" es ]

f urle Sf',^ro.{

l- &;eu".- "rL:-*-

^ t .-agsq ara-{w4etV -tttc +ir.61*(,RA frs*

' RA- > )D

Lnse-#rl ,(e= -4.1 *xa11lje", = - RA .* - (-+. ztr)N ,1., = - R.* -Fr - +,36 &-

lrrrfo f,n 2-. :\

'c)F 4,36tKN'

g : -3,(Zx)

/UI(t,, = ^(.,*-F, +l, = l' zbern

tttr'* = '-Ao -F'+ F.'E = o-ges Kr{'

a ) (xn .,! ) eo,^rtr 44\a- lytqti tuwnn . Al * rlJ*r i, Cwrupr*_^a, in G r,*h wg rhp'{ 6h+4^/1 ,'d1,.A* i+occrne ArF. ru_3 = *2, 63xas-

,'. \tt CVtv,frcatr,i),

\m"*l =d = /UJo.+ 4..t LLm?

trI t cwr-fo.+ [tr[ = 3. 63xt o

l.D.+ A (,oXrO6

fu_c" Al,,ill"ri Y" t\t*ctr ten*", tt

ln =k,rt l,irn '

[c-,*- ,.-,,;,1

= 4. Zlrt *lD 3

o,/lx tirovto6

feme,'n SJ-o,rtto

9ha l( C Lrao s <

S,icp /+** > A. _,, fr

f'+ft O"4 xI, tslsx lo-5

= N*, 2_

.= l'3Lgx tD'

0f* -- , N,fr, : -3'd 3xro 3

# sh-t^"h

= '1 -c'*5r,Jt m'2-^

,rr JN', : 4. z (,s rcrt,

[D- l.= {'s25a ,u-'rr'

', n Lt+h .j-<,n_t,- orw 4,q_&-l%a /l1a*'mu,rn Vilw * 4r<r,

A.>t--(L-: l1'c*.,.- = 1, O+g X I o' nz

15. 0V fiAa, .

,nN*

= N'" i"' /6\

O'rt- A \

Frt 14tp ro,i +-tl

(j,)nw/N--tat c- I tl {

= -'/o-o twq

i=-IU* oc)

1,0+sxto-5

F l\)

6, G':i

-tF

0;,36 ax"t

7'o?5x t o-5i

Do{+-rm;n f; s* Ort plabrl4er.d

ln

6pn

Al* xert

AL6,: = Kc<

I

l

= AL^. = 4.s6ii r rd 3n o. I

D"qK ?x lO"rrl.O*S xt[S

A[n.o = dlu. I AL.,r'-rE= 6 DfXtr)->'* Nr '?c(

O"+ EA-

3"6z>iIo3xo-a0'-l X lx t1'i^ r/. rr.7, n 1r*s

'r

.= N;<,;64*

/ ^- --q= b.U).X(D -m

rL'Po =

3,,/yxtisr4

lLau: *

= -)"*e.^

to*5 lyl

-

Cdru=

2

*

\

.- /. T\xto-S +- t, /zx,o^s --: t-/

(^dpe = Alnu :

- 3. g& xto-S

-3- q &x rd*s

A Lrto

A Lne 'l

* l,TLxta-5

AL,,o+ @p AL,u

+./')}lEt+

+- l.l6s * loix D"3

^x ro" x I. ol5 v 1,1*s

- Nr'u "

AL; U:; o"Ed& xto3^o'ss

axt1tt x 1. O+S xt}t.:

-s tt/R

?m,

1in )4/1antin/e

*4 {4ue- ?nr+;iniq h<""*eJ..

Sol.u,l*m

Q:etn , b=

E= 2-xta,t fqF-- I o K^)

A c $t!nw/|: scma

{.^-e A)*rr., , Er , AlaiI

AtS,"*q d4^r**t-q. ro.0 ] c

l'^-c

rJ

vOl+LS4r"n

c"f -l--fr-e. )

F .lr'c Xq ,

r'tt ri".r/-o 3

I zt, .d,

/ Rn-=.

I ut: -/in:-f =-toEtrI AJ^" -- *Rn f F = ru_1r,, =!

tP. = A L,tc: ALo, .+ ALs.

x= ru+g' " + AL). (,rt) +4

= -+++drArr*4

?: -lDXrO'X?

ALr. ( ot)

f l,6xto'Sx /roxg

= 2,6 Emm

!-e d-ra:'r :) e'i CLo-re.l,

,{ ri n-1.-)

s+q:fita,t /f ,n)ndn-rru^4e-JQ..d

{\ e|we yU € f ro cea.( ., r^.t { sV,pt L( J,o-rt-- ^

i ^,W,f Pt ccu*r,um (rF +4\,a. $e,--Cp F a,^..& lrr;_

^1.,^r7iAlOLd d C(t,^AC -ltrc ct h./) n

*.rre "oJ *r.,1 n'";.'m o -1" be , c t,r"-'f ' r

[a q c.i-,i,." trr, rr €-rvie+X.e .*t ]i.f :]r": 'Lp"t*'

f/ffi!i;' 1L:,-a_r#li taK

CqlGuLa;trure rF J;tpttlLa-m_o*ci c-f goi.^_t c

ax to'l x 5lro-4

<' ^'5tx1s-3rnC

Y*f*, {,ir€ foJ 35 s$ta4 b< (Dlrtta

-rl

'Y

,&*

(xt, Kl

, MPq

o dL,, t f,irtt.

1,56I

ZI]= o = *Ru -F +Rrr, 2- | : t, nr

?*i"' *+ tu .1*_:r _ __Qlf:_l :.!_!-h - -

AL*a - A

=- ALA,(nt,t)rAL*(,U,o. A*): A

' = -Nlgo r4J,"" ,_d*a+.t =A*A e .Ey+

j -& * GR+'+F) b*

- n-^

ad+ n+b = A

#'[^-]J = dr ar b - a+ *,

N,*"u,, .N,lr.

* Ro'

- (r'+ F'

2eA

qr Cc)

Rn = r*fdrafb*A+%l

.eiiol'x-sxr o'P./,'1, 6'x t t) i b x g,r.::0, 3 X.l Ol.i.t2,x a.xr 6'rx 5x ic-f

,

1-,rl*J

_d

,r

"_]4

:

fi *'? I''''

''t"

l

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

.,-_-3-*..->

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%'-

( "', 0;4'4 6 s 't,I

=r l,++KN'"1

^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

i+**, l-o*ten v il^n Gf tln-)' Yt**A b< Cr*r*

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'

|f (16 = lYl e

?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

":''/r4r')i.......-.-.:.":-{fc.

i

,a:.@i::l:n).a

rs* -=o

&L

I

L-

i

Il

I

l

i-

l

I

lI

IL-

\tI

+ :Nl -{ylaL-Lsr J g*

[4*(t, - t, "g,ri,Ce [l4*..(Ae

t tUllTJo

q )-Ts.J *

; {Y1 u * fi4t

: LM* = {W'

Ir,. 'IJ,

M(L - r- )\ Tr. 'Ts,J

L)I

rs, )fv1 (* =zm(g.

LE^_

.! eLrf. r;-o +-'--

.lS,

-QIj,

*3_1 :7-I-lr

_aI'f.

=! r3-')Js" Tr,)

c7tL-

Jfe lf,

ll +''rx)

-lLIy, +Jr.

Tr (' J. f- l, r)f s^ =3^

lu-q- d, I

-.Ld,r+

I

F,^-l L;)

(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-

t\W4n!*d e?l

T1^L fi,\La4 S4v<ta

SC""+"uu..

L+* I rz,p<a,o--.d 5,t<!-(, .2 - b,o,:t ,'f l,ua Canwy owtn SLv,pa i s q ut"rl LLftn

n1'< , r4,n I urJ, L[ \e Te))t s+-rl b.J

a^--J frl* W d4L0 {,v.e-t tn-p;{e+-,-qlt

frvr* br", ,L +""',t 6g €-/,tcr ln)r,,t^.

3- */*vlTl,tntLtt1.1

% q 4\.*"vi, .**n)rn /yyg.1t17,-i

++r4 fi,LUva,pyr13 1,11, , {U Z

t'r

S,,, c< il,t- f 14,44 a,N r ) g, r!-LS ln *ri* .0 *t,e-t4vt, 4nyyyy-ed-z ,"b l4tp Sqrnp cLtuaa2rr+ , , "

.fLue #u"t i y{b< S+r-t \cunp . 1J<,r^ k, .fn^rrn C*ryuf ,r-+i L, bi3 "r:#

th uA?

lltl<,r1

i.u,t /.taaiQ

fi4,

W, . W": lQ,

M"

1

J!) =* [(+xro-")l (s^,.-"J *] = b*4+xto

G u, TJ. G* Ir.

lI (,^,o-")t - L.tZLxto-"n{.\j2-

/J,,=

[?rrr =

ftr.=

-l t* !4X */ne dtlis*-^X trryrary^t. 4*&(,iti.): i,

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"

Fer' *lan; ni ',t *t

l,+++ Xto- L

l"T4+X,to-6

boW<*,u ftl*uniniu,r,t

: 0.4lrn4q

O, 5 & fi,aq

,,,^l- br*U

-150.83x 3o5xto

tf, o 8*xw*6- 0'4*rruP?'

,* o$4 xro-6

.rn,riifi.ii:

:::'ririirli.

':., 'i:,.; ,:

I

I

il

i

l1

l

I

I

l

1

I

{

l

I

I

!-

t_

lL

l.llIi*;.,,']

ltitti-i

i,i,

Y/6b,1r"tL!s,"*'

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

1 srReil6TH ff beqnt 'irt

2ot441 A-,+ ?a.iu .' s*+,ry*A ff rcu*^nT ht beq"'t g< c{r^s^ s h*"1il ruA^/t. wfi, a"*..r s#+*Y.{t<-

+a r.'vl*t l^e- ex++on-J b,on l|n , rna-rnut. -h €mAu,r--e_s'lt**'1tu gP q b<arn, r'f ri nno*o*a +4LA* d44!- \r<-eLfa,rt+eA4,'"1!. ^J Xru,+a* Cwr?ne-l,s;,1<, Sly-ea_za 4t 44LLC,r,""t-,-co,l- )Qc-t1,s1 (t" a*l-Le Sec.h.':a.n

^t"*-* L. tt .

is rrt qx-trvutm ) sr,wut,l nr><* kcru-o( {Lrpn,. * nil ttlofat'tt

qLLa*aJoLq \-f*e.rla f* n* ttq:fet^,^J. r<4/t'ncL:'-'

Le* Lc o.^-l Z. be .,p a J\#,^ t:u,, st /-Le Fn,thno+,|-o,t 5i (-c *,--,tr.

. Grrtuyros t', ,)e f, b, e.s .r<,s f< c +_, ,/dJ F*n J-tte

fi-ea,t -J G,tut (; , c('x.t) rri)te"< Shr-+t ,-s ,r-^ ) ,ie stro:tt lsut!.e,

ln f"*-J

T ,h-),,[2, L ]k,, tt

sf duti j n

En^rr s [4J qttauraatp

, 64.a/t

C*"rL'+,)-,t ul_ Sl)-.,^n1yh *r.

14)no, --

.j4u

r'S

,^) oLlvL44, < [q ) ^,,u--*to

r,-L/_ lvrtxIr\)no,

F* ,*r^1 sl'c*-t*ru

-ffa3

I

;

Ia

to3;

,n0 3a'

,t

*-,

,l

*!

& '''i$ffi s*,qr+'

tlfrl-l . /h'r-q ot-'

YtL+{+y wLaa-S-€ c{tn7r'

Eit lDnu =t.r0

{-hqx

3^^= !

F* Sh!-a4, g$*ae.r,

m>Y+wJJw

Tr,, - A S.r*

T- Ltzr

fur r < c,l-Oftr1"1*. Se c**.u

bL3t1- - h/L

bx,'6

hre Lt+e f,tu 7znt-</ Z lnt*^ rt s \, s

' S

* = S*qrh-c /tx*rn->,,,* ,, ,i,, r!4 p<*

+-o !*qv_ro aF a yer,fua- ,JfQt'<q hsu-,," ,l

^.k14X *1"!Z .* ,l:.te

Fr<.-_5_>

1- rt

l- mnx e*"A^

1-)a

Cir c,il-s"r \< ch):n)

Qn**N-rrr.A4 3

lo Atur"-J, Nh,'m^l Slran ,0- qre ztnc N*l+< @,

nu1-{ q?L+r *"1 ,r h }.Q"M<A ,tt ,J. a-cla. i-ht Dr,t f* . fi'br<-l .

Sl,Lze* Slv<rYS , T qi<. l)'LA*'r?ur,rrL a* n4,1-e ne.Llrfta( Ayj;

"") &q" * +r> ?-pna .t-oao^*la U4a guf-or fi:,br*<. ,

tx

'-{

Q, , ''')

*;w,ho 'S l,vtnr-l' lil p'{, lrr^ ,u-a-1 L-trr J< *!, f re.tr,t etert3 Lr-, p,

3 .es 3T - g s

A'^.e(- l'Lrz 1-,.f h r'S Jr-n*,, a",Lo,stlL .

J

J-te-t,.X-n frfbLtn t Nr L( b< s0lJtc0 tUin6 c*rni,+;un rf.g,lr u^t;+h i rt WrrrY2',1 S*'re*lea,

: ur*[4Jy e*

ov.-.L

lMr^*. )

T-hL

UVr*o* I

N iLL b4,

.-

3,+E.x lo-*mv':: t''

'g75onT

-*"Q.- SJtL

[ - Lt-tyl-

C ^ brn-

P, = aSrcn)

Pt = anr<,,O

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

F;l t )ru , L^ (,,i, [=,f - /V 9 (I -)i,(t:.tu niu,*'+t")

[ 6-,', q.\Serh11-L' X'f

]>+n.t!*^., ltlgn-t^< l-J

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

TUt-t CAr rc't.e n^ L,<l rtJrtr..r t &h C*.^,** r.s inn a I =

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*.^

@, .,, ^ e'= /l .,+

n

: b,t+x 0.t86'+

r Q*t- nn-ar )rua

All+

't

bE) 'I.,-a

Q. ^*{+o

=L3

);1--t;'"+;"

: %X*/Anpq.<< tr^_

tn{, a,

@.0,olQl...- :

/vis52"3:11? '

i

"l' !

b s4mL

?1" l, E3:Sfi.ry m'roq Tr,@ ,

Lal.a,tb*,+r",A,

O"tr dh rn-on rt,,s", Uf l e

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,

,)n2- Vt t...-a^a *na*

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

+l

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

..,1t1

Q**o,o

l.[o+el

..in.!!--"r,--nrxail!,r.F-r{+t'+dirdsi!;i;--i}-.-!;r l.'*i;*i@#'r4""'

)<4-s( n4 fi afus)1 OF f . b<-".rn St c-{r,tvrrI

\'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

r IF Our f w ca'nloj< ov"er'5{re+\ , a fl vr*:rq (-ut -r+t aat s o/-| '^r ^-., [] :, r (J L, t l'r*-r< GUI -1+t aat 5 -)

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/,

@, l.VL )?LVtltL-L4 L,U.O1d)4 Qr>o(teor, J_ ta-e,arwt

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-|

, I .-J eL r rv\e Urrrarr ir\r,i( eil+t_a_r.

[O " L+,nr {tto,n [U ] R, * jiN-o;1-sn ,Jltaq Lf: tl S %

Sin ce DI*^ ( f, I N 0 27 r-S Ll,,.aten a-,s a"ry<-iecl-,,.;,.e+ gl'/r be-ar,'/l

Geern-e{-.;cAl A^f- ,$- T- ruo 27 f *U+4,-ed )(-Q)u'1s i'n *Ytua-

(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

=q* [-,,'(>T LsJ{

-(l^ \Q'l

F,of -G.*a{ o,uo?s ")I

,]

-LLr

T'x(r)

alt-I n"nx

'.nq-* qr'

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

'o-(ps

G-"4 of S.'l-n4rt dl f-.r**;

E&t,,tnpc T*1't , h pq.':

t-T-2<---.\

o+ S+rl.,tr

,54

-kt a Cautrj -ie c,{-'-, ---'.iC'n,lr?r3.a-t'{-l.ut (*+5.,,.1s*nt

f o i"*s c,LN

tr, Pu,ntj q!Th-a s-i,.ezr e.s , d, ?;3 I i n ,tas 3*ft

.l-t4rL\ .l_,t^.u s*e+e ef S+rea_l .'s u,rur)r,ocl-q_(

rAt;-rlo. y'lvr;.r.us

S.,u Ct b.e.^ ,Ln, rn,o,,***-\It, Uf f en f , Urc.i .1*he b€a*t 4r< .fe.nt,; p&).3t" *"g Ltwc,r s,_,^ctdu,,t

no.I - ir,*a t c.*f7 dra gdaA.fi\a oa-u*_q_{q f,-{' ! , ,r4vL{., 4-\-L1 ,'r<

,1-per,r7.r r.* { "(_V" S+{41)

QG\ xwr" gfr<,nYo ^nn,

- t, ) = r,o x.o= (}} - o, oolr )Irq.

5*?^

t)1"Pq

( (zT

-i ffx *r,5:AM?q

II--l

]tj'.rl

\=-=---

(@* =D+ +'- >[h-E=-

,?oi*t 1.

(* 'u-> cgou-de- pc,,..-t

of= q^ = t.tlir (+

E? =Tp^,,,

-P':t =.1 grgWq

;

t*)it?3|'om hit[ b{ ,",'

a

:5o2.

- 2_5 o;0i

-i I

=-25.+blil: -'2S * 7l"l

..1

= 6"lwqr.+ FA, ['fiip4= ':j

r.

l< lris Sj-re,t0,

d"o =r -lt ld.-' / 2T*, \l- |tq c* J

l' ^* lg"5\tTa-u.7

frfo

5s='96'l Wq,

,t8"be

0,=6"! f@qI

er=o '

<+

n t '.prro 1'

I

i*-

L

1 f--l l,^.E

f!:! 1

N\W

et mq*

_ a\_tX.& ( tu-cd

a!- LizQ"-+

1fflireid

- T*-x .- 7+, En*.' I , ,. ,.. :-rt

0-= t*r,,= ? 1:5u ?*

' ,"--q= T^* =2.tp,S twq

C, = -y= - Qtlcl nuPq

i^* \ G=dL=od,= **-'1E*) tzt= lr,..^x.

= +E o

(,anl;c)x,t<xr< rn +*ir.)

lcn I <tD. l<lr t it po,.* *twee is s-r"1rn,-,.r,- '.',J:,'-- ).\ .:

I

I

i '' .:,'i--'t

.rl

1

to-),

Q-= lt\

? = zxat/m

P - 6xnM - L+rd rn

r' --7= Lo,J.= RaDnf',

lA j"', Y'^.t rn'

F^/ lr.

Altc -qqJ_J-

Q.1,.' =

f(,?''

fr -IrQT",

MF.

*Xx

1,Lt Ie-l

I = o I -- - exllr.=, I *=o =,

=o l= lKN,tx:0' /*. o =,

*7.q. s *tKAJ'U(:c< q

ll

- I

t

L

I

I

l

fi

l

*l

1l

'..,1

O (x.5..,(.;#:"-':u:,.. :

/=lr=o= ,

3 nnlr'"

I\1r,* l^*'Ir*

o

4-Lt-

0'q h/-_->

0'+ h.{ L)(o,rh xo,,rh)(q*.*r)

1.,

3I

t-

?l

+t

rh

0\

o8

?,

0"

)r'

T

t*

0

0,a

h

h

:e

)

(o"Bl,,.3.3&4h

0"+

(0.+ + r

\.,..::7t

:=0

'J_

:. Z^^x

) ,(0",r-h * t,)

: 0"tt )r-

elh- : 0'51|:Lh

t\t\'

C*

- 0,sh ( o,th)'r-, (o,B hxo.+r.,)(o.s1h - ry,,) *

la l

+ .o.t=hL' * Co.,Fhxr.)(o"alh : h) *

' le

D. r*++h t

I.,.o ?l &*, *,**[D-]

F*-*,, ,'

::]:.

T :...:

:Il

I

\

l"-i

1

iTl

+x,ro3{ o.g l

0,lt+7 xz,otxd6

,, t:XiO3^ 6"9 g'i}1

ile^tD X. LO oI

'.h

Zlfro= M-:

R*=

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, ;

\lt

al.

l

Sdtuu*ro"

) e{en mi.n",t*n of- r€a cfi;r^ FwC et R n- a.--t, R r,14

ZFn - o: (lo\ - p = D (oh= ?= ax"i

+ Raq + P+ =-o

q

*n P4

xte

xq-^,.

d"

u4

-(i

tr/r

f! \'

= l, KA,

i.:tii,:lr:i,..rii.ri.:': I:.fl.i: :.':ri-ii: ,', , i ,,',, :l'I

Z\AG - oo= fltr- 1{ t R,*{o +

t {+ e4

?-

M-t-nvl(n --

I_TN*"{r, = Ru = ll Kr)

frt'ou : fl

n--EN*oer, = R,r = llK,

Nf o, = P,,

= _4rc l:o[)L=5

E -II.mN,.c"l- -'f = -4-'<-d'

I 3 l(^)

0z-lc<h/t

0{.x.t',ll

P = +x'l

Ch+ci<^^ q .'.r

ZFv = (^u* flu- ?a= l3-ll-2- =0\

OUa,<wi" ofi:

4

6I' Vre.,= o

QT,,=

\0.< "c< 6t

^E'Q 0, = 7.x *(G

= ?.>c+u /: llI }'=D

l = { 31#)'Il9cs4:,

*r

[v1,rQ.., = - (ux - p(+) - X *

ll., t 2:- -t17q -+-x

lU tU

A),,'{c*, = o ^ [r)

t1)*e'= o

8rctn"'

I = -4.KNnI *:D

I

V

:

Q. {",, r..nlE

[t\'o)= -M :-V--

N^'t*.,

[1ru t,.,

-(o'=-ttrcnl

- Ro^rc

* /irt l=-o I = *l6KNrn'lx=D lrr-'h=t

o< x<h

Qlc,.i = -Ro\=-4x^)

D(>c < q

[t/ir7', rtrr rrr

,. ('') , l(,r)

") e*err-mrn a fi.tyn e.A A; ^n^u

r.rn .,ef Lra: r I

C'{-r'.rfULU- uA tenovo'J'w'< T i,*"-(, ?<<(.(f]

l-h.o {yte'ta S*re_ra gat _t44e ldar.rh q*< Cor4*rl L_,,o[ ( N,.) "^-) [ ( 1rr, )

fit. G^J".f,u.n of slre^,q{ao ,+<r''"Ai r- a^^_A Cwre,y,.tnttl

rt+ tn bm4r€sr ,*u I = ry'( H')

N)*"*= /\)r&r= *lBK^r' / =rI +"0- o-r. )

A.

)! >.D.D I sl-m

[n +€,fl+;wt

77 - lltxro3lcroxrDdx0.Dgxti- = c-ot4Tn

6on c&,rrir,., B.F I3rtI :: r,y,l;,

< [r]

&-"-r-lrl

fu +"""

lDr^'

fI\."*l\""

l,Nn^ 7r

A),, ^* =

Ao\

.'- lL/Ulc", =* l0 Kd'

Ae ), Ia"Lu_Jp

-, ","

j

A),'c,, :.

trlr)x

(,- u ,r) xn x trux ro 6

for .Yt"r.

3:>(r -o. r*/x1 x xrux ro 6

C t^^of i*t -1-'{^-e tolea+ rl ",1-,t

3,*^= Di = A,t4rn

Lt)^,,@=ryt(,^, *)lnr-"^) = )Lx/urt

.+*r-, tu-,l F b"u

C*,w?,uacJ F'b'*

= O"llra

oF?

. Cl; 0' f(A.,,,r) = C,tt LrL

[f : =6trJn r rru o,r,,

0-t4m

CO-l - [o- l. = ?,.-o,,.^t

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

Zlll*=e= - a( +)(trr)t rr1 + 9tq(** 6i . (a ( s,?,)

R*= Y*)GfuoG,)84'

t6

z fi|r\ =. D

= *(Q* (sq) u ? (*) (t 6, +*,) r ol

R,*= x (*)(9-^^)+ ?'/q1*?2,?.1

"*- =

Chewil

ZFv =- o

ZGq) - {4n - R* :1-9'-q

61.+1,-"1 *+? -e t/

I

\

b

Ro

arnV,-. )

, ,r,.,

M,--, /,)

Ma-

0 <v-< +q

VlAt ?Vt 3

?'. -.z.:

t L.s

D<x< 2q

1_I

i*. *.,

/=o l= td,t,t ?a

[r=, l*.oq=B

/= o,zz?. /= -sZI>c:D lx=Lq=4

Na4e *+rt Oha,-,e

w sf,trL u-

t0.5?.

:y 4*y: "fu uf Svrr^" ft-u *",A U"^A rt.,;rrnpaa* J^.+l bq,,hL^a

I_I.

Q.'o'

flrft, -

I -rtnJlVtet=

=

=R*-

3q--

R*x

Z?-rc - -N*t-

fu * 7(+q**)1---_-__7-

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

/6Xe - Ae *=- O

+ ir"l

= -/L"sn

7L-

*6+ ^

t6 t

[{r1., = f{* (uo +*) -

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

ZlYla =o= -rye) + {yt * Res.q ?q lu.r-aL3

Stitaa*r'rn

+")

!_-."._...I

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

< m]S'ec'fi'on Y,.\\tAnub-e4 iS A;!S ,,!,.r1.4r t,,trc= A?,& ca,r1'j

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.._-

-

'.(ytr^*

2 h]*f-J Lq

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")

= 2,x- xt /=O. o-?j /*=;

[r: = ru*, n^,r4

tr'*J tr*C, *,,."1.+-tLt Sec+'dr1- r,nl).y*t

O(. rc ( 3q

t*ot*s ;nQn^u,

.'L.

e./\\7

r ) rr1,',, j'Y J

i/. t"LI

I

=b l--X=D l*.

* O,3?s tcdm0.}s

= -x2+- a x34's

La+ ;

,.5

9.rs

fI *urrJrLf'z E.r

Il-rt

QI",=

.E'fVt u, =

0(x1 3c1

X,3

?x'T(zqS

A,vs *,x x

o'7s

F, -- _ 1(sq) (+ ,

= ,-/_/.5 L0"a?S .f

= * 0,3+g-:(:D

e

I

I

*)

x)

= o,*s / : c)L:o / ec. 6 ", .

+- ,ryf+u-l +-

'l* o.*s ( o,e s .r ,c ) .i-

/ = -C-Tsr.tnlr. o.o.

3rc

I = -1,l*= o"?s

{s+ao **]

6(zq)

2 -3x+.5

D{->c < 3q.g(sq) * Xg * ?_ (zq) _2

a-

I

-2x I :o. lx=D

- Tsq[* +24 +xl* s-# IJ Z L

rn + L-xo'

sK)

* xY L#*,.]

(cd4{ oF \ q+ Wg+ C+ut

t ( tu,o"'tJ 3

lO* 2x.

- S<t*r.un r'j

Ar= ttL: z-be

S1^ o..^X

j

= 1"\sxtD-

gir.

Tlu p*.u* ,rF c-al,e,o-lq+i ozr- 44.\g*'

Q**-I =.['SKN-fl,rt Cg+r*df s'. Jl*t

f\\ = l,rSx-dnt.

/\-/\/\- 1"5 ( l.tts *x) i- 0,+s ( t tx) f 0.+5 ( o"zst-cc)

'lli"'= I = -10,+s I -.- 7,2_5y4trrt,

lrc =o I n=o.+s

:\\

li= .trl,

t6

72*1.)35

\

Fornn CcwJr\^^.. 9F f fr*,,.q +4",

^l v( )-ra ^ IVL.,vx 1

f0"o D,.o

= I ll nr,ro*i

4..=.An

- L'r"s xtO3

Qr^.tlt _ cs 6{_.,6,a

FrF,, o"r,

; Lo.Jo{gr-S{'<-a

ry:&lxrro =t0-J

:. {-t^e

0-*er

P orcr^"t4y^,

Af,= lll-ttry xtuD:t iJD

% 7 s% qAwa6L!',

,rlj, turt

4*; os/o-6

hluni ni,^,rtr,

stce-L

Fi" J Li) fl* [e'r-{,-ar ar:.i s

1ii) 'flu sfr<-r t J s*,, rr..h.o-,Lq/"'^J *" C*w .k c1...:,^ , [h I

U = tr (r,.r')rbp

L<-+

(',0 =

Gs+'

) Vn,

\lt

h/a

: C"jS

r -')h =rxroml b= rxrD-?L

flL = t\oyn .Fa+ = 2,xdsr*trl, F*= 0, Txtosnp.

S r:[ut*Luh

Lo cct+ie^ e-f- +-a^{ rrauf'"r'( '10\ -uf *t.c c-aovvpoti.l-e cruuts<c*,,r.

ru lw,-e FF - . Fi. t-i+is."a wa fr,a;*s n)vr. (A

rh e{q. Ca s< C0ut.t bz Fsi LyE+t

DlLr

Fsr

Ge

eF

=l

fl-SSu rvtttuX $-a*G,x;, nr( (tco.-ll

*, = [{,* Srr, +-

poi* q is

Q^ Ss*

*t'l{ n*+t-4

e)y - Sla,h'c e-r

ftt**u+7- 6fkra-a

/+ - frae4 {

b.L

Q,* A,* + V* A-o,

= &. U'.{*] + ANQ, ({ -+)A,+t L0at + Ar* er*

= (rr"t*to^+)t[.00. * Wr*

.:

A,* = hn=

I.' : O,3f h

-

%___-r'

! Lr+ h-Z, =t/9n'.,3 d-l^o Xann*al

O' 6 q 1.,,,j

f lexu- ^l

G. = f\l z, Fs.r

E*f,n, f f,, f*,/trt-.. [,.*X Tlvtv\-a4-q-M

^*\

f-+vit+La,r

I

d*rrwtu-t,*-q>bv

W Corwp*i,t< th_nfu,)qt-t,

fvl ]^ E *,6* f,* t tE* I*=

'2 Er = fr*

= fil z, t0rr--- T*V

: 58" SottPq

-r

q

0= fvle,* = fi4 z' L{rt

(,. I,*, + QnI*t*T,-, r- -E&J,,,

t X..*'rl,\.,1)* = f\'l z' l! *'

T3

\ruie Sf,cuu( Cq Ic,uLcte =lrq.l-,f t+.-o- Cq I C..tr+< J fiU*r.*

-hrh\1 /J, = (,* t+P * A,* (* -D

= ro-*(,u-l) 3.,_ /o?,o_2 [So.rr_i\L

+ t,t-1,0-^ (o'6r - D,rs) z ^,,

= l,5X,,o-1 + 6,go,o'1 = ?,gxto*,| ^f

Fw ru,p s.l-eel

o OU*t r Fil"l-( rIF SrcJ

el;, = lYl +, Qt{ = *q xo.3 &^ ax ro-z

- qffax

tv,uw.^* W in<

f u\ r fi:e-'1 i;uf ir-

z1 f'l *q^l !JL

.0.*)^*,;j',

^\,-^J ^

)JI/

q

Y.

"-Yk

!)t)

[f,*fr., *

.q r.i r.J,r1

"vUe-l rirc,'\

rArr(L^-

rr-'(,0-i] 3

ll-

tlrv'-l'r'4

--3 '1:0-,,

b(+J i7-

l?-

//

- o,tsl t

t

C-rl-

,L 14\,

f<c<71C7{-

11

t+L{*{

y_Ls

-lt tcrr2

I?

oa*

[*,

' r'J

?oir.l:'.tf 'c.sn*qct-

= rn(+- z,)

;}o Bx w-1

Ni.l44 ftLt,n;*.,r,*t

!021 = ,y&lo.3s)^*,, 1*.,3 6,12, Wq.ry t,8 lqrc-.l

2a

,',,1!qr\;4^Vi l

9u*en ?'br. uf ku*uini,r*rI

0* , = II1 +. L0nt = r.e.o ,( 0, B t-x t-xto*^

I

q *,8X tO*1

o 4* ?ai* W c-&l*nc*,, vui+at S*a,t

G{" = U ( 4- z,) Qa= lta eafg.rr)xrx,o-'x o.3s

i*. B xr-o -{

: [1,"12.wrr-q.

G.aft g'F- S{-rr+l .ha.tr,' brttlcrh

:'ti' d

f,,u' , o'no

7a) -rl€ ba-,*\ , s

P or*,Ao t^-ce,il*J Lv^"tu-J 4?.ul = 7,+ (, ;)

F;f 0*{.,,, fll rA ,'

Sr)-,re lr>"1 cl;tfri"bu*,'ern_ a^il tu,://w#.,8\rrrn e"t-,'.u'l , 14L f<eL crhon fut+-l R * ^^,1 Ra SLq,Q

F** tt<.A','c efp'/,'$ri uut<

{l,y r kr, I }' ru, ,l r, _

, e k+=flB

ArQ

6-a

o tft* = l,"\ot Jx,

e?"^/.

I

' '- 5) tt','

%x3 1zr,

6Lu Jo

0' 6+ t" t

OLL .?5 \ %+(Jo v

('o

= lZ,tx'l-

LLL

R*=P4=

= 2XoL

: O"z3

)Q.,fgnmri a:trnm tr Q*b t

o.or+?ot\I

Io.o3+7o1t- iA;

/Fr- i

Le* tu arr,rJlz< Qu ^A ftaegt Y*re+1 yt Jn's*,i,,** J

h), Jr"-X,-^*, r^Jr'tl b< mv*s*-\

b{ Ar""cd\ h ** e*rical

o'x\ct rrtlT ,-,--- .o,lo,f ? o L

=-lilft\r br iF\;-

+) : o"d+ z,t

Qrt"' ' N

Lo**(, o ,

/ l-L

%t

f\4,a,

ulr{'' , /Uoi

fr* -il^-( k+* tip +-o

Stn Cr q* 44^\

441A 7a') t:goi*&ty ^t it(\ n^ u e-[n;co.l vu L.k

I -L

Q3,=n'l^

- J" % r(,-x) J"

f Lz^- %x'ls"L ?,L Zt" J,

01 cc . q..

!

l1_I= o ,-/r=, l*='r,

R*\As\r\--

'. il.il

0,lo4Zot

I

I

iii:-lf:,''rI i;r..

.. 01j (r',

- 0" og+ f" t a*'

,/,

1x<- L,./2

,--t/rll= o'Z?q|,alttJI

J "=o

x

l)+ l= -o,oi+%/'

llx^J

= -TrtnLo)ct l= D 1-3L s"aa' lr,=o I;fr -[.^a0Q. u" = Ro l^.; * / t-,c ) -,ro t-o T (' -#) d,,

= A,fi'flt -c1, f ( h +") {- Z. ( t +n)7L 6az.-

l_I _-Oll,= r/t

Iur**,= Ror( - l'o-';r+(i _r!)*o

Jo

: o,zg%tc %(i**)3, q (L .*-;= +- '':=:8LL

I

/ - 0 rq.sZ,L\Ily^Ltn_ /L

i__

O Qrrry; *.tr fu4Ar.;- z rlr s /urrt{

l

1

Tlre *a,fot.lb<srn tS,, Lood.l

\ ftrce P,

ThA bue, b isbtula,* t,Jla'i l-g'

m'J ll"<h I

fYLqx.IFt" ,)

s,

T

iJ

T

.T

T

r' 1.;'. ':;,.l.,ilir:r:i .

,1 :

f':-O Ko',q a* +14< +a4ren'e d

Le+ rl.\ ..|-ar,re ' 'q ver+f cJ1c;9 , o^.d d/^-s l,r+-ilu"* i s

ltt11 =

h1 -- 4D

)C

Sec.+ie-^J tyr,.b Ar"lL, L,O.rq

S.fre-t: tr=fd^^

sic"tl" 1r.ae

lLt = * Ptt

\"1

Stc*l'c.n a* q cb s+*.^(€ :c frn*,1,,r, .

jr* : lno +

= b(1.,,,) ^ =6

6Px.-.b (r""*+ rr)"

E^- ff I ,:. \

re. t'r,'nx t ure- I.o^r, g{ & = O i.

-"; -'(,/

3;ff-e,re-rt,*y €lrr^+^n ii e__A sdl{,y W )c,

/ ,.

l

rv1

I,r) '

r")JlL., L-, olc

(t .

(t.'

2tx

zLtT

h

,,I

+

*)u-

\,1. ,c

du.-----i-- - O

d'x

? [ln, * -t-\_ t-

I.^,r*-. e

: u (t,+ + ")*(*e r) - ( 6 (x) [^r(t"r *, H]

2 h'xaLL\

),,, *- . h^ Jc' ..*,-\

12

--uyz

hr( Et"jut( ULr^nn<

f uti +"vt \LsL,"!_

2C= t L,L-Tr

I

'_tr-j

Ir-i

-L

-- 6?x-:-----*--b ( t.*) o

b (au,Jem,* = : :-('P'P, ^ ZPL

iLhh,.#.

i

:/v'p\

i/*r

..

-'

I

i 'nr

Gombined Loading of a Bar System{&

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 .

!q+

->{ C0rvrBt^16+ ' Llofr+rN( frF '' F ,,'B; i',Va'rfir?

Cg^rt *-l 6*J-,X i yq4n< tr. q c4^A.,'+;;,n' ",,{Ltp,r<

q

Ci^jus- Seo*.,'.rn up a ba- v\- rrJ i s l^"J-A s i,ua{*aa q/\ 4 nD*n".^,Lq^A !uooz,;^-2 lvrc*.,t., l, e^MX q*-A +-4^, is+iyMa <ttca,^4re- i s q heL'cca+e,- \Lr++ vrr hcl^ b<an_c yr'rttv., lPe"-c"-^ ft*r,., { hc Ufthn f* cu, %=J _Fr.r is*,,g (wsrn+.,* Jrno{-o l-o*4-r'uu". , c?rucl L €rrdk (vlone,/* ar^< +D --; &z.,"ra_r1

t'-=_m Pft---Y) v#)M

t7

l[is /leux<,r1 .l-o {,nnl efluilql+*tl S.trr.,4r04,J^" +o-L1no h"^*'^r-,r lrr,ev1 0(Ar*), [(ryra),-[(11401 , Nh.,r,=r;:;'";

(p,r, <r< J rd i{+r- Aarc L' oa i + *g S .l-r.e-x< gr q tlo nl a_,r,rc S*r.err o+*he . S. ha,{4, hoLl<r,^q.-t J 6_tr__u.t- exfe rinor<-*_o tq ^ kfre.

6lu-e,^ini^X J4La- O. , Nr, , ilu "^_d /14r, d,.r*,,.bwrk^-1*a of a. L*, N< st4q(.( cheu< ;'. Tt::-'::'*

eJ nh-e

-l-ha Cr;+-cat\ (^od-J S<c+,s,zs ( ^uul,,-.t tr:r?_: ,';::,:

s+s,f-e- oF s+--.ro ). 'f ( 0r,-) a,rr ,u^a (h fj n **T 'il":4'btetr n-enCgi t\ Srvtat( rhaSru.+u-zl_e

U,= +_w' Wz.L-- 0)

N^*

u,"N*

+Mde

whue tJK ' f* ^r*,r.,r.n ,*^'^- @

h'a*nj. ft,; secowcl ,r*-tr:t;*

(-=

It.t]4,..d

I

I*-1

i

.l

I

I.--,

l

)

-Tl4A fl r, i a-? a,l",1 S,lYe/t|'r' frrr"l

ci +ru < [u-]

Ar^)

' trl

frrrL J;slot+;on A"!"U {r**"a

),lr^e,r\

,:r",.-

H JII'0oo =-P

af Str ea-t ,!, t * t, q*w , t^l ( S l^*d I b< <.Lk- +o

DF f,1\a SSre s-t r-a r' J Z-Q,v-o qf- t44! p tt, n* t r,n /-<--

rh;l /01-J- s4

0-f

.1_( t\4x

t1 -tuS ,i c< ? ,4 ,ttA L; ); 5t J 'S"1Ler [ ( Cu,t,'1:n, * J. nl i't+r q {*, rr^+,{ a,r (t Iif W#^ ur>"X* fi,I .y1,.4. Cn'rr*^,", o'Fq r<€ry,n-Lo+-\<c{.,,,t^

t,r l* lz a'*h't 'tF fiv,,*JA +, 5 ^

qa* I +o chetx *4< si-r<,a-j*h .

"^-J

Si..@ {-4,ro'

ta.rr,,/t1 qyi Le

M6=

ha t f D1"atr

cle fg"rrnp1'{.4r1l

f gr*rrl &e c"lu* loo du)

r, = #.*,"*Cmn- S.ec*.ior-

+wD b+r^ ),'*-,,U

Y*ne\; t^/.e Covyt-

{%*'rni ,^) NJ= d* = u0no

(st

frt;rLF o, *

W=t

6nn d.ia-lr',im,t,,J

Oh44-{1 t'tl L"."oh

Cp+-1,') er-qJr"'t'n '

[ur- a t,{c&"fl':r,,, )c'c(\,/rr; '[-^o4 O CCu,,rs ct* l-tto

d'1to {a"3e rl' c!rir'w,.t, u'., r,tr L;i-e 'f^^: O (o*rs q* \14-\4

d+',-a g14r..o-r, Jr'rrrr.r ri - ,

[{t,,1*.f

:"t-4r T @

F.* ll4aXtruu*t S1^ta/;,4-? 3+rf-r4 f-ltc4)we Shqt( hqv<.

Ne Ccurt de+*r;n. .t4^e. .]r.o^ofu , 6ftLell<ct-^1 N,. ,i * )r- i-re \;tu4L,F? so Lx; sh^"t<

nl3 ?il* ,

'|F A ef tb vtJ hr{ rnD rq. l-l,ta,rt

tn (,t-<-a-a-a U,\-o- J^'^*r;s-r1A O* J4,\-t-

f e-d'u U 44^L \lv <-,t< , -Tt" a Cp,, J-,+-,-oY *-14 ,tt S l4a./4 .

/r,rtto*u;

[u

J-l+e \<c*io,n

N p. \lLa^l\

d4te*\

(Fl -b)IN no,

k?'*,- ,le+r,rtti,,i7 d.luc 5l.n*,-oo,rsroorrJ , v,/t Srtctt(. ChecxtL)b,i {4r r,r,'le f'; t+ ,"s A\,ttt'ottz . lt<rtf l4r( [.e ttL*,lf,,q[r.^, ., Shcvtl b.e ftr,,y,g -i4rct.rt. qlLor,tu,Lt_n ,[0-J . lr, ,J+"-^':

I er"w.*, "-2,-e O''{*r sJt<-,rz S Lre.ttJ. r1o+ b< rno r.e l4ta4\ S

A[ = Cf,*^ * [rJIo-] s o/,X 160 1

5 uh .1 rl-h*r-,< tdt

S< c*'o- tnt gt-cle,

ld ^4f l,'c-r"*tp )(;

i*-e Shc*7{+,

31'r< 4 1 ,ca-4-c, aLq-,S/lc

SWJ+L)

b"-*"

f - a-6,<n)t-

7'- lo r-rn lnrE.t-

ffl =- vo,-, .'" rn4= tc: 3 rn

L= Q = ar4'd = LFcL

[O-: = tbowq'

,^"1f", **

F,i-n! o Nr0u, 0r*, flr{, fr** a^l ltfre,

0'J J>inpa,t,intY,ecie*\ffJsy

[ar* | wtivrt-

gf r\rr-r"!St c{.?rrtu

FE

Lat

Note :

S6l"'l't"t^'

uA Cfuuu-s-e @. C,oe-- ), n ,l n::'

1*'"u rs alnV &',*<*<J(fdV- S<C:hrq

L - ql-is qLna|s (oi^* J^ur^so.-Jat

J-arfs is JLr"a-J e/.'t4^2,r +raa,.L d4^e

in q la,or, z-o'aa Plqnn.e,r . Thi-<ftgo*., lilyJ-wo V e.r,-,t^.+3 o+ Cbo, );n.r-+c s,lr*ern-z ,h)' L,)

* -4.A

\ tt<n" '

n*^? d41e ,o !.in eg c/-"

Y,:J vr' q,Jag fiaa.t;*Cona kai-e {4124_Q

C l,v>-rr1< fr"-

Le+ rro Cltc'1"5. sj-tr€ fi, l*- \J er.,^rt,rr* cL_ lrt -l h,,:t cctrt.c.r Ll.r.. ct I61+,L,11'tr /.c,Ll1 ,'.*1. b."l ,r ssr," *',,);,' )*j,,.,'^', ,':^ ' ",:

(ct'rt'c' t [)cnt sL'o'''kl

w !,* ,u,( ,^'"["r",i! ;:t:::"t,u!i,^"fr,.-4*'0,')1,',!,i r,s,€v:J';:t in/h-"a

y''''u furce s in l-h.4 Z _ q x_*. ef q ,o_d

^^_l /* *1L{ be,, I o

&n a, reJ L ,' '-'t v' t '''<- *n*nc rhgflaa-"*

,;ll Ji: x' ; .,

n *|:7

_TTl:, ;, :,:u u,iv

t/

D- rs fo.,^-l b (*,3,.):,,r0 qrtL- qr-. flrh ,"rri( qJ_4a

3 i,t_e fyty

(6\

fu-rn J',*"*-l' ^h *of '-o 3i'lt^ L*t c*^ s;)er,"*a a t( *h4 fe-t"^ ,!;..**J ,t[o".q44e J-q xa oF +xa _5r c6t. b<f*2 hna,^)a-"l . t*^* rtn)

* 7

f"'u 3r wit(- ji,t< Nx in t444 ,*! /+< \itt cp t+ qc+X-qtu\ , fl-t".ten)en, f^ +** -/-he a e, 3f

Carr*n y*,1,; nLy illr k Ca,q+, i,* / \ o r^,' .'" ''',' O; "3

t

of rw. ,^J, o y"^;jau*,; t^!_ ,tn"',

a to"'ff date y-'?*'{

t)*Je, 44!- *J, *.t l+.,-r,* Z^qx,a u);t(b: * AV

? hat\ffba+,a< t'f- d,r+.a fal-q+is-rL up 44,te- ,ol ,( Zrrff, ,:T'r*'tu+; -c(ox,si sp eA l4e. Ca s< *o( on.

fYlroc*, ^ 'sj0*

F/irh) 4L eff.c/+

€.?lti,rqU-* ,l-o

A* f $* a., wf,, in s<** +-woTL,t C*oss-J. fa,+s N;LL fo,nt*^;;sl .la"z ,tJ. Th& ye,mat)'ti

{+.te o*-l b+^[*, g rne,ue,J-Jeftea of ferce F

A+ poiruF ct l^*+.*'Tt z G-r sJt f ,u*-(rt*', Lr,xle rl. Pr* + N;l(

l.17et t- ttp ZF7zFA

-:.F n ?*.*tt-o * lD"x

/Vlr: l.F b

= Nlo,=F G+*)

F

t+ Co nrufl' Tzcd< C*,

Pci* c w;k14 i+s

iL+p

O4,x + e.

{F rn S.ec*ui h.

-D2Cs 2-

l?"")

2F

e.t

b.,^a- e/yut;+</g Jr r*U4 fu cea ?Fa, Cou,1c1" (Zf *) o^ [ r'* ri,t( i-er-J in .

un o.ots.eJ f or+ ,,,'t( fn-du&. gttio,,-

,

,l--^/D e.pn r,( b,rl- ,&fl2a\;4<J^J J).r"o-rl. fv-{<< F .

rd,J( frwLtLo ct Ce"r'ta (lwn fiv.L,cez A_) ov^-rl +q_e

ko&- * *e*ra, o- .

--\t--)/

9t" c-e 3F /'s eL*in q

q Ce-,^.{U4' ln +h;4 ,*l ,d.i,**

(**qltJet+

Nl*,= F- r-oK,\)

Qfl- = 3F. 6oxrJ

fhJ*= ?r'x-P(a;+)= Lo*-

"o(b) J= -l?olx= o

rf,*ra7*i -v*t ,.a 3F

**J .t-i4-L nl ,ii_ Jl.rec*g q+

QI*= *AfD l=X=o l*--

eoc

flAfle,

filfo. '=

eFbl?'o t<"t "'

F-r{),nao,'**- y^A* q.(( i4^a farce-a a^-rl

! te a l( hcrve etf< t,tt vA Sr c+frn , V .

Sh-,tut( 3i t e U*J; ^1 fn smp,^* , fr4, vntr' wiLL jit. lilx 6u m

:. . l

6o xnt^'

(r+ (+ -)

fil,e+na,r*< ei1

TLa *wish.^X

17. Ba"),;7

.a?F,c |xZttox - Er { l,= olr(-c *'--.).

D(rc ,- 4'ro I pc (s..** E )

v" D.

flWra-o^^A , frl *

qc

D

lc

I'tli:2F b

l4,r: F(qt,a)ZR-

3'q-)

ll4 J44.!+e f*rra arra.! s,\ S twdl 6e. "h",l+ lyl u,r+ rhrt- rofttd i

7,<.,r,- h""r* doa rc er

Nl'* = *3F

Qr'u* = F s

Mu]'r*, = 3 Fc *

* tr: 'ryh uuux;

e{cal.|trl Q*ivt F4-- ,

* - 6or)

?,o xrt)

p ( a+t) f Ftc. tp (aw-xj= 3Fe

I

-TI

--1I

tlAoi.t"cu'

I

-qDCa-x) l-D / = gretl-nt,n.t)L=D lX^d='n -'1-*"

[x, 'i

Al .. nD-l-l - /\ f

?.,O *+o = -e-orJ',.Fb 4- a Fx * ZC>c

- aF ( t,-x) ? c**1o(r-n)^e-o, I =-,eo / = -tfoKtin

I rt=e - I >t. d.q

.-Ar /'-7L-* + e,Fo* e-o f- E(> = 6o x-^ln '

{vly'tv, .6

7

{v1)uo, =

V-

L

\i

?o *rg ?-c a,tt+.l3Zr *r(oa\ r't il Fg a,--rl J+e,nl;t-e-t*-"J

ef oL( -lut fa,-c.es ,r"l /rtB.7y,p.-.6u

/o,* r<..8u I *t {g,.*h _tA! o-4 ( J

0{>c 1 e.

e/#c*sCon nTcur.e

N.lt = F

Q,O, :

I!lr'ft, =

QJn,

fYUo, s

=

: OoKA

3F . 6o yrl

3Fc + 3Fx +- FJ - F (a*): VF(c*x)_F (q*_ d)= 6a ( r.+x) ra, ( 6*1_)

aFb-eFJ + L"JeF Cd-r) +- ?cJ- +D.( +*{ * *o( +) a 4-o Kttrn--

t-/-lr.. e__.

=&D,(: o

7o / .|,c=r-

RafO l.r.tr^--tF * 7c*to+2r> = -?.,orc^)

M+ a.Fx + a.Fc - ?rh f c,c.tYl+aF(c+x)_Zc(5+x),3o t r1-o(<+x-) -e,, ( t**) I =

/,=, l30 r*^Itr tr'r,, .

-:

G"",ttt a* t*e*r"* ,q, A lAia+-<a ?ran Xrqlr, 6f N,, flrl* , /yh, A* wLorc *r' a*),s n4e ,*J4 ,.

' 'y TYt" '*t

Qo ' n" t'hfun* or ueJ,; ! d4! ,il :;f a4a,U.e.l +o JAa y_ q_a

Kn/rrt

[4:e,N,ir.q

[/rr't, trn)

Q*b.' ' rt 'J

/l'l *t*, Ktlt rtQr'? t'l

t+g

p Th!' 'rnn:na'uoLez e-f' o: h @ s brl 6e. *he 5q1,q.Yw ca'n *4 h,t'e J,fuG , ) r* fu*is o4/t," & q* J44'a *nJ ^ /n 1444*- ca-.t e, *ha )r-7**.* ,i rq<tkaSru''r^Je't s t^n"t il k ,w *o #.a n+) fu-r, rn fi)q (.a_J.o Thp- rn*a ntu'at-oa af rE- q.t *ht-< e,^4 a-f q n -r s k"_.t J--tk 74 ," nrVa',',r'.-el< e +n d4L4 Ux t/ al u< qr_ /4LaI eg i n'' "'-1 + rLn /-rc cz< )''"-2 *=t - yera c a.t e,_t, h nt< q .

't:1yn',''i u' tr q /LC+-\ ftrx is n"//L^e4 rtr-44p

) .*.( w l,t.k J Lt_^* cl il /_c,r*., t.c S lnar"(^{ ,r, F t_, ,t, h:jr, l;:!//lx. ln e*r (e-,t<, r1o rtpA N4s *//lr"nl ,

o 'fl',a rtrJ-< ( Un* rnnrlr"?-rb n,^_/l Jr,recl-rr) vf Q* q**-r\L <*e uf* q ,o4 s< L4Lu.1 s t a^Ur L< e 7il dr) #^_a';n-: -# ?':,

()r / a rLtocror /n,ctn,'^a ,ur) .

6.*wa p*,t,d,^_l wil t=T:,,n*,i !:-:: *:r;.yrn44^d- txplai"a w\ we ,{^* ,.1_

U ^u. a,t>,,,-1t_}_s6) Ttrs- t)l Lln,h.cl<_.r eT n , ,, /

-"'"r"l't-\ t\ Qa= 7.*71"

o| +'L\-o /ra:ca.,.),; *T'.:,,*'ii;! in *r'l< 3 /'*-*-;,"*)

, Tlte t/ n-l4L( dF f4o qr_ -/44 .er^a( ,g,f q ruJ SLt,t""to[ b-e.€4/'-^'( f,c *t.p uoJ.rr.e q* {+ra LToA. 1*g*,,g7t <r4s* q: ill q J^",2_ {;;r,Sr:!ZwJ - h s_- cA.t<, ^',,r1*rrJri,, a.t/h.J

r) ln

afi*ruw CNL C,ultgw"r , 0,

q) tile 5r,tot{ CLrrx

Etm,'l1utr'aa+*"atX \CoLo^udfu^

,,1-,r^4 [+r+ *.0rFb,N*, {Vlrr, Ml "."A fnt

u.l Lr^/. '\ t*"d.A.

l++

rz'"t 51to.,t t nrytea*

?'U<,rn t'rt q h.tr/1 Aots *sccx-1ine-t) 3'e,t<rninqh4 oT J^^4"*r- Crt>ts*sec+tln in +-t* (aa4 *J -

f osi+iw uf I *7rr-^ ' h- s- *t*tsw Lt . 'lt^l*rt,,n, A 4 V^1 ,l-"0r c/htuJ'^A f4". f o,'"t-< E a.,-.1 S a,r< efr4 l^"yo,a,in .l-ovat+4 (rP Nx J {Llx sf. tp. i"j hote. c.s^at*,tat *X at^a n)fi.

'il,i,,, lyll ^-; rfi* h"c^tu l*rffi,o,*t't^-xr+" r'-l - ft{ po;'^'i E,u rt4y hoo q, r1,r-axiruu-rtt tf t30t<,,tm ,iL:(!/hv ,-o q n4tniott4n4 LtF got<,uq * /o,u ;,- ;;" ; lltini rhLqt4.W 1o t<,.t,u nl li t-n il4* l+u q. lha ri ,r,,

' ' J

;,.'Tk' th**;^)' F( u,-,,:;:; -#7;".w,,r t,-4

;; *ffi: (w**^ is rwt- e-at.J,*.hg,<. Thro<*or<" u.,<

l<.uhw*l^',,. . *.^:o'L:H'^

'i' sr4' ca{u,7,'1.; of ,

,J)**,s).afls. rrt ,:^^- "^ .,"-,uuu* : ^^fu'n-'ct'rvt ''r re con',/<--rl'

luch /h,,bk",^A' L, J*;:f ffi";:ff ,"*L,,, 3 irtin *,)e, 4r sin yL'fy s-iL"*uo" R.d- , !n^L ru-nun nno_le

u.#'*Xfrhuutn ru:vt- /r'rr,a* b ,-LL^* b/-e ccnn rr-ac<ut

ru /tnhLo,,.t ,i r.o!_ ckOar* l'n-,1+az<-*0 fo.ca fo.+on ob-s.

P,:'^t E pl,it p\_1,. 2 cx,i fc x)114 ,, 4 o p,J,1

L[_o t<,r) *rllt,l 1]x1<,r,a 1cK,lll l* S?Kd,r. ?_at KN ro, ,

L* uA

S<c;or^

\arr*-a,

(o;";<

",pP\- t/^!,re

" S ti cz ,rtto

h/.e_ J h_all q;<-

f*'u fa c,run\'J no o-l' ry *--1 /l4t d-

ot^-( J)i" f",^ * rle,fTue--*

4D q ,""*,-J,^tr a--U l- o-"

4ru4< *-b

L+5

N"'Nln

Nx

{Wz

F.F't O Kil '.r-r-)|aa\

4Ft))adrr /*OX4A

l? O x,,r^ €0 x,,rr."I

$O t<rt ^ I{alo xnt,*

Sri" ce t-h4

*r'"u b-e^)';2OYU\4- ir C.fi"tyn

./laotnz*v,H1". fdL*

s i,., & ry is fNia,t<, i+.V+c+ur Sr. r"A f o;* *u,-ra_J-t

d+rc. /usih j_e

prr,.*t ,. J;;.*-:I",9Y n4 n4a-+-')1 t h,tlt. s lte"t4 \ tt th rt 1 4 , ,- ,o-

f\r\+ rAJ

ln Foi'n*t n- ,n"^l rt' , *a lto.t-a

Shs+e-o ( + ;r *<-,-,).J wl,'(-r

Nno

w l.nu v\) z " "l ,1, F ( ,U,r)

=I

u a*1,+-tvt' f t n ,u L-cA ia

[n+ A' o] o' + atu,L)I a I

-[*" n. 0=*G,_t_D-[,u*)

N,.ft ., ffi Sha((

de"*t^, + ewtiLtl

fi\r^ r tl.'

lTL4 Lru*,rn l,f crLot<a

It ' is C*n /re-n-! ).

La-en"\,\j

U rtt'f.u<.\

-.a,-g\ (t (Nx)

,0oo

r+ (0"',t<

t'> tt,t,

^Jllrra,sr'*rrn oF {+u_.l_ t

lD*"i = lc) = flq +-IA),.a

'r'';ff{1 foi'tt-s ( ar^*. /h4)ai- /t,it< ) an< *t^_r, /aru< ,t

T^^*:f!"=ftJ* =r ,fB"E ffi* l'7' (A )

t r\L/ ^^ ^

fi"t* t,n ,ir*n, ry naad he^t, u,?14^W. f"A:"r_ Voi,^+.*4, e)w 6 'G,,j.r>, t S$tuy. q^44"; , w< ,r( iqraf_6a*e l* hf.VF S+YP44 Bfal€. 4* *.

AJ^+'

PYft

Tfr,1Y

f(a4,1

0Cn1

.t.

''f l4t }fucu 5]a"a<

:

,'.! b,^a./a'q[. T'n-or-eforv, W€ gLtat( ttaa U"* th i,.!

u;^ Ffr ( [r-)

/Ylo" i- ru t' i- ft.r..Nno' *^a- < lr]

0*t+ +7" < td-J

Stve.^1*d n<*,y ?b Nrt{e Cr",J_,'t"* e-p gJ-r*_,q/-q:

lb. S,npDf,X *t t ,\Ul^arhun , wt, r( fleaLLc* ) in 1a*_a f ,-s j_ <q/r_*<ctt,l'1^i (*"nyu*ru--i A-( N,, ),, a,^,J Su,ub s h.h-*e. Ng . \ u_tn.e.

_ -iD/\ I

u*t Fn \

-.: I:Nnq

Nnrn = ll o'-7{,7 7z \3 77 llle

r [F)Th!- Se Lt1*, rr; rLr llLhtu.tttat,c llle . $,Lw,tL

fl,r l."c<-utr. F, /i?.6= ffi =P* 9<uf"in P, /hr' = olD"trro"f +0. .:Tltn*6,r<, F. Bl4ee( b< Cun h.Jc-r*_J

32-x lqn,qsx ros

-

a,trt X I box to 6

tL)n a

f,^.r It ,-e (Vlc '=

-

i /14.'o frr' + /Ltt'

->'tUa*fttttn , *

I frtr^r *r.*,rt.^

b+ (.&Ls;"t€r.J 4s t)tL\i- 6{-e,,ye,e-Lx-

I S+. t y.atu'

9-Z|l,13l<dtl't

DF 7t = g'?+rm

t4?

-T ln u*

d*rASe Ce,^rl

btd e++//rad" ,, LQ* ,u

O t/p.a-s*"eN'*F-:!:3'9*t' sf p**

t+ x D. 2+13 ),/

+tZrSxtJ3\{riJ

f3oxto)"+ G^,

ffi

,-z.i+x 0.2+z

+ r['tr"::'ot ) "(1.,+ 4 s.z+23 )

: l{9.x6tu+q

C q t C,/a*; vf- LwzL * o!,pt-s1,-<za

A F= er,r-f tr.f x Llo _:_ /6.s.zt _r lotb-j t&o

x ! lo .:- ? - t>rt- d/.,

A D-< s %

ll.u.o{rvc, Jn-r.,rr*-du, o-t rowntl S.ec,/ru," n ?=D "LL{.J-

=?{:ro{ , &F. Rec aA1i(1L<L,4-R

poi,,.+ F or F, EFAla

". * z-cr,)

frlr, 4-0f.,:. lot<,," d^

lt4d̂

flult

Po;n+* A, 13 **A e alre pd<,41;* ,t ry-"-,f+ n- ia +o,rdrb) 4\rr+ fi'nooa t,2 N*, Mo il tvl"I

r+ .c is {-,D +i,\r-& *e*$u) 4 Nu il Mv *J *:A h f;tt b it .+.^A

finaoa +dt -J t3 0,c "--l M* *) lt^n^J \ f*_

l7ct-;- a0fa^

ffi wm ,-,fi) Ktirn

I r|8

.al

I nb< F1Yft)Cp, d,*;*',^

Po,* *,

. LrJ pr* : bl.'L\)/b

[,r +ka fir.oCer" L'Y-^ vt-

Ne s;i^u fi.,*J lr"*nori*;,of Sh-e,l*g*h i a2F fr, $ qn d' c .

T=D

,b tl tia-o,U

6(rt*) =_A&++

r 4{ry,{n^J

ULNt

N{t+

-t P ..ft-J., rr) = f',b(_ - 11e2 _€

4,lP,tc"l, ,, rV€. 's t.c,L[

S.l>cy+tr lazLvtne-t

bt\L"ul2hL "/ =^

D!

o'tLL ^=t6.6o,.r+.

[^. = Ii r llr+Gtvltvf r\t< bh -bh" , #

= Irg {- W +_ 3rtlrtbo 1b3 ]f

* (;/11) + rrr/,) < tLT-lru

b zt ..3 [ 3/^M" +. 3 n4+__\ [o--]TLtn 3'<-cirw hr i+-tn t'ltqX-r Mu,/v1 (al^{

Cs^o5i J--J {\A.er? tr*fi-".,^" .

Fw s<c*iryn E, g/^r\+3114+= if,tlo

tt ctrtn p, ,lr i+t^ lvtct{.iu.l,ma rto.r*r, S\ntl

rb; 7t

,* h's"a,

lt,"l fu+,4JJ

n-Ulltar

ruy [,e_c* f ( rtl,,-). T L.s t.ettj L1

\ = '2,b

t{- (, d ftLx + , a r) sL,-xtt ia

+ 3(roJ = T3Spil,n

a(e,uo;. ?3S 1<d,n.

f'{,r*eQ

6Qax)

?3s x ro3

l(*xtoL

l+q

),

-t

;.i

'-';.1

l

!

l) oin* f\-

--\_2

)=lo;*

LPu

N, -" (14+R- ' -cd"

Tuax - . {il*"B 53

Tla S*rer sd-are 13 bl qtid .

Sht^y{41 W,r"^t [Sho"),"1 S*rea

Sl"o^na'u ,

d=

w( \hat( *r* 4_h,; A.

il r ,'rl4 ,|l4i b,y\ Jr'+-r"o-n

*f 1",-o-,t-e$6-p ,

4+aD"-1) +-o v{

1^ .l4,.e i'""* or{ lft"&, wt lt *r{ r1z,1le.c* F(n*1 .

--B[;. = { (#)""*(#,)" sir]wrylt. t=-0.

I_)_t3E .1tr-)"**(y)" <[rJ

A(t,)

6^+ ,tT" < [D-]

150

):.)*:d:t*ir{&daiaia;L

T

Y

o,mq" +,+(#Jb7r

Cpruf q r,na

F S t^oll b<

trr' b, 77

lrfq 51,ail aLtA

+.'1^r- Co."Ji+rnm np

h/z2z

j(+) = 0-+'q s

Ia444i f , S<c*h_

= ?-rrr: K^t 14 , ilr€td

' ls 4-^'o '31+r,r.

Qtnrs*rn

= T 7l,,^*-

'lo t.t r, rc

.fr\a /14g 4@.t Valu<t o-f Itt+,.srt *

Yfh'tehnt*^' I m*,-^- Il-_3 I {p*^*,0')} +(yt4\^\=o

3t+

l{DI

t 'La.lv

: I /t"o^rlr-

\ I {0.,,,V\= t 61n"7"<

: tbt*tblba

TL'- atW l,a

b^F )r. D, ,

f oi* c.

AD-

^t,t lr,a-.d

"*/!Y!- )"lo"+t3x D.r trt)

IJ

O.6z

ge*+*J"oryr

= l5,Vm.

[( Nx)

'- -*" 0(u)

S+ff/-! E +q'r€ n L| ry,' c,t

0-0. I = *'* ryNg lT,) =

t{,1{ *4u *4r,d

(,*^

flteA'"3.

<_--_

S.lre^7flrS$<*,-aS*tr

do,=

Nltzl-.r"? Nr, i,,, i+a fi'ar +1tylruac-lt.

x-rL , IUi,,: # {i (;,r).*,t(Iill\ (&rl

ir.t W V.tf,nra uf-. S.cc./_\.,,ru4 Ffow,. J +o b< A ^6r*._ w r{14

d [e]-- ,-

( o,+ts

("-"

r X \?-+ bX ll,) Y16 \

L'L ;r,, n ,*r') 1- +/

I5- 7*,t ,

LaX u+

LF-3

V a&,r4

[ = O,T?St\/h!a h ^.^,i -<o

Gn, ,o,,,.X

wuwt& bca,-l F,

l,rt, ^,* 1

S'e r./-,',rx F: lZ 0 tutnfklr, -- 4 o t<r,; r-

-3

[^* =

._ l6q,Tnvs

Atr = t{,t.+-rLo ^/ 6'vz

otlt".tos q4 ,',Ebj 7t O.tr3,

C*nn dil s, uu

/rro 4tvv = l.tll 5uA

= 0"/t3n.hz zb -- o ,:.26,,-,. -

= ll. Z ,,*,

trb

t'\

,)ail.7

1o

0.

1

x

q\l,q L;vUU

: ll-Zazt t

b,u= /l,6cn"

1^tA \Wt' I .^o

1=

rF,l) D2 a

Llunne fi^-a.

I 6,6 wL-

7,f.9.*n \

,rl 5r.

I

:,' .

ffl = tox<a,a /^ ) P: lar-il,

Q= z-rtl.

Nr"[", = D

Q3-4, = D.(V,trTk, = Q

Qrto, r * P =-top,J/14a=0, = * Px: -lo.rc l=o l= ^2,0t<r,t^

.n, T lrcr') lx-:_'lvlr,6o- -ffI.>c= -lo>c J:o/=

/,. = r- { ,:;

?-Dxnt*t

Wni

Le,t" (+

TrElr

N,1,,. -P -nT

Q)= c,n = ofYL,T: - rLq.

.-- rD)<l)

= - fl' O t<t.l wlc

^T-,1 \Vykt -: OW{0,,= -?q

: -- 2.D t<tJm

llrl,to,, = D

^S^ ",-NlI $l^

[u-] = t|a,ry'q

Ft,4l {'ue l,"qnrii4. uF #Le ru4

r*f-

l01ct-q y

D<x < a.

(lluk,t ta-,,rnQou, t ta)

ls3

fu\l\'t,tqtb Q*

ule Shr lu ry nsre 0l il e, S r.i^ 6,l14wuh l'n ftu'o*t'tt e.'L *L'e" ,-rtJ

fr"A &a r! /r"h) o* itt (aa+ fioit* /)

lo;n*t C) @,La!

(o S a*-,1 T,

to- J", = U-T!vu)no/

t^ry

Lt

4I

-fiut slr<,+t

S4"e^y*<

Trf--

Si-1r(e IJ

+D

b i* ,''r.' ..( T f^_

,"rt'.q ,kqa ffffi;

T,. t1lr = /l*tos l;*o.L'/Y SltaLt 4-7q n4 r; rl

*f S dre"<,1-t^ .

,/

n F-Nnq )r J (lly'ytl*.

-- Lo.T-

0, zr

{criir. < rirl

' lNna ( co-l

d^o = 1Ta 33z*.

3"t+ x t&r>X to ( = () -/zzt<

= /?. Lol.t ''

\ Nr, = -lDt<,i, /14x = tot<,N. (

|o,; * c t,'r

ftrrt*"*ztty d*7-r,*^.b*,;A f*tf ;eA ,Cor,r4ru*;

b tr0,t*1 "*J,^(/rtaJ. j ls+

ll*-1, I c ,3

7r t:F7

t^r"JrJ

{vlrftt ,,"J

oLc- + h o.on

C'.c>*, e#-li,l'-.lno J"-l

fYtz'+-th.

\tr""^l - *'TlD "

la0rrt ' i

fya-Tr.} 3

0'/o rnrlo't^

TD \in p L' fy SDL,th'on t, ,!.e*

TI-J 3 \. \

r 22.: I 3J! _ I \zxa.axro3.-5

\l 3-,tr x t6oxto6

0;. -- -$NI,.c"r * z\ ttl-fTJ z lr_a .t-- +{t0xrD2 + 32.x6x,n3

g"t+ x o,lot3

u4 ngbr.+ No,t

rr f o-J

A[= lL^'t.-,toI 6--

LtJn cLtut -t^tu

x/4 : l"Vs'/t//

uve, [o,q ),'n",d

/ 6 t.rnp,

t cfi-e-ar ut d4<a

S '/o'

quu'LnLLp.

Ca tc^ta,*r-(

lJ

?.il4 E'LLail Ll,wW l4L{nl/t

U^ q-mB:ttz .

@ aL7 = /DKnt/ra., e -3rrh = /ocot'b^ su4

[aJ = tboUeo< #\-a |f-re,fllh qL frrp q./hS<ctu; ( W* s+c+.a ) .

r5s

I

II-r

.tf.r-:

){i

O 4;(<q

L,= ill*7r,.{14,

/or /=" l=r,/z-_o /x__:

=0,o/-x =

7M, j,

nI'(17 w --

{ilr'r*, .=

/,,:,0 /,:, t 5 t< tq

-l-"2-_Sxt / =o

/ '--o

D (:C 4 q..

/= -/fStcn,rr./.rr.3

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<

nb.I *4tq

*he qcJil4 in.le+.nsJ- {s, C<

War, W,Url rf i"++*,-o| f*Ct Fq+-r--, ,<

o?-r +q-( sj S+<-*t_

l\4, t t

fs

cUr<

L,

n TL4\r+e^^-h "t4\L =l€"+-"{ ylwrv*v+. 1s:r- )< c*r-oltz lr,u..Lt

SA '-J'? {6xrr.j' s ttvtl,,.l,,a oF

fl- A^ns* L<. c*'o,.-,-Q /qr-€-^

t-Lq-a'hii*g

[t ++'o "uJ

;)

,\)

,l;)

,r/ J

\l)

ln *ht ne) n,Jgr t<Lsltzu4 nha

" tp lnu* ^) cl4* lo

ot-cna /();r* Kulgr fv"ct fac+trr-J

fu{r,k t'he e-f/A*vna of- d,ha- for&_ fiuo"< J,sdo,,h,i-,>sn

l+f,i

Its

Vr) (9rr*r* J41a- *^rorttJ Vr* rbnX

a"^-\ ob+a; rey"L+

qlpf Zac'h Cu*

Fy *e' .(wttanpor*i. . Nr,l<. trt J-ha

6" f-kS fLl*) * J 'I

egy-n*.sa tn r-t4y//k1 oh r's ;r,l1rA -

'I

gq rta kt e a n.a,- e4-+d*raru

V,' j,^nl ,r^--l n-b)Scll,lt t44a LI **un

;Il(_

bl-K-o,

I

-l-u,,s,,i d (naryre-al, on

4-- ar^ Fr : a.artJlt = !rt1 F]_ = &c p,l( :- 4/1 FS = qa X-,,J'

f-,| - Cet-t]'q*^l-

'F,^ cl. Jaq .(+, pto1i1tt1--J of /uri '-ts B

!dl"*";'7.q "^*?

,Lr4*n^rt",r- g\ .!4+a's ,-,-,J ls 'elf<'uu)it-'. O)' t{=lt=H-*t-iar

(.Utt-,,tf't:rr -f ltt.,t,74.x', tr,'l f,lt-crl"( / r L/( lier^ +-kd /1(. A lt , ' > i', 'k1.*Le-f,ttrvt-r,Jn f-rv /ctr* dttt-r cU-o(s t"'t'J-h O^-17 1)ytttwt'( 't'+, Q

f:'''cn

/t

t^*-l Lt) 'Lu', 4' rtL'-) '-'''-{ (tti""+ SlD+"'). o''c1 cfPl, q

Q:(^ g 1r,,'t 13 ri n'';) 1:t' bt4ttitT 11g1''u:u'( clit'et*, u--'

f *--a--rl'

// // /

l,r! ,F Po,+ *

a -T

L*1!,^

CI {"?c ( e.

Nn, , rcrrl N,,

-F,- 4t1 o

r *1tr'

--:--

nI-u

04)L { } -Fr * Fz: a-C

-tr= -l

-F=-f

O(rc< + --Fr +P. * Fs

= -?O

t+5

-.

t

As=

=

=rCALtl

i

22--ow g<t/v'

"L' ,< c*rr,- cF F-- . / h [r.'c t, i \

CUn |tr<,:,s ,^rS i S Cgn-.tc* TLvt t,\* (L, Jtr ,r- J ,.s Lern4,*u u1

#rr,^Yj ,n dr+v,,.0(+'e{L

ryo. ov+? toorc;*, f*,

r* v**.-r ;:: r,"::. :.' ;,,f;: -tr:"*\* t;&* horl.r.**l A;g^ro*,re, "{rt; rt un* lt*r,-.^-,.*.,t-t fiD-1p

f* +,ist- oryV O(W,) , *tfLd cr uu* +\^,-r+.? yna-ne^*f* t*)b q-fl\n

O , qrr 1ta q, u,t * un.,),i-,, n1^w.o^,J-,

:to-irh^J D+torna:li o?a

111, . luo F,v,,.

[t* = l5 or-dnt

q - e_fi1 , \)= 4,t1-

6', Ir = Ce,uuFa'ct '

FtlrJ {,h! {*.|'rr c,,^gap ir*-+".a pc),.t

Stl[utfi's'a

".dJ,+ie-,*l wit wy-e,*r o,"J q4.?ll a Lr,r* ft/isg.^4llT= r a* 6oi,t* K . '-]' '7

Afo k;

=>

+=)

=)

Cgtaf^rci-

0lmnc^tt

+ I N^tN,"Z- ) .-..i=r J., E*. , i

fq ^r,I il,l; i- 1'b-N,I.,!J.r- + f ' NI15]

JomilofrJoU

[, U-. ) bt Ax + J't^r,

(-D Jx r Jre

+o) (-D lx

5 t *0*[ + *0,, l'o ]

l+6

'F*a *Wkrr1"

Wi+ WtelrL-

l-,r (+ ,

V-=

=

-LG:

_r*GTs

t

I nr/ \ (roo) (o) Jr

s f Jo

f r[)luuxl -sorL lo

Uo -- a-o-D

c&

J r^.)o)rx *I&,x,)J,, IrlDJ

- I uO Tk0 tu3,.{--,,e Si 3 n fiLea,nA

e & d-t^a:'- u* '"' '{ .^* Po '; + K i s

*ru,"5+s d[ lh .1 ,{;ra"1r.u.r. v.Jtlosr,k*o +aa 0.,r4. L.t,.c^+._J L {r{ arr,"+ty+**"*r,

N9 Bf fo,tu, 1s'9f,r1 opf) o'+'u" li'kr, Kd,,. lilr,,

D<x< I

o(x<l [Yi' * t ub

JII -l[ | rJ<ra ,1 It4q* tYt^

:-50

[14= to tovmp. = Io,<n)

4= \r.ull; b=*u

Fi'^J Ve,-+'col J,\ flob,,bC* W

fluia* ft,

ft

Solw+r\"r-!:---:

t^l,€ ir"o[ BoLd( -J'{^-- ,, ( *b;ert

fu ) [V\sf,"s t'"r"2J'

'1. i l: '::. :.:!r', :. i ;. ' 1l

u sjn-g : dlne fu,tara,*g Tt$a

(i.,) gr*fLtL..i y*a3t^ct*a1ta.a $L,{4"*A \t"L'J t{V>, tr-f, W

Lit u-{ (.x'rus+v,-l*- Fa'ca a^.-1, u,ru'* s7d<-1144 *J a< n\ agq..},'t-a-A FsTCq- ^* Pnfu r+ i," -t!4 u.rraT tSSi-J.n

L'na-o,r.

u n r7(-

FuCa *pr*ezu.

Ufi;'+- 36a+<--ut'

&*, L;-, /l4.4yn-o^4J- J-a.io,ig.

rr -T

lllg8, i Kd*r

..- F>c: - l0x'

-F (q+x) + M: -tD (e+x) + to

fllrtCa-w-^+*U si^ 1 Mol,.,' s t nfaq ral_

ao = *, tl:(*ro') (-x)Ju . j,(-,0-rox)!z-*)"{,.]

= r [,1::'/-*&ox+-3o3c(r_rD)c3 lt IEIoL = lr ' qv^ '-a- ,-::-::--

l, l- I f go+sD i-trto +- 61D.lEr.L* trD r1+o t 2" ) = #,

N! ef Por--h,r,r

-- F (e+)c)- - t ( 2.*rr-)=' -r.-.t

i I rsi^t V+n;;.t;a;;'s r!ry*J

A^ = *- etL' L=t tfrh) = A-rer. g+ b+v,, ),^fl lt+*rutt*i

T",f,h unl* q fv-A',sv-Zt*t"^. o+ f"- Jf.a. l|fi,.ffi,tfrJ,'* 3f b"*rS'6*M#w r4!. fo!e{:7 y"fu'a* a, /o'"-t (Xrrc-,tfe*),*1

.|1-\ {4L( C444+-L";) sF qr<4 rf+44-e k-s-J,.{ 3"-"4 ,

(,

W4- *'"t"1+'+l*r, )( Ul*ea t. #<-e

,)"*u,4 dau f-4, h^( skoilb-pr^ Ji.^-, rwenaL^* ffirb(9 ' a--'l

.dha.w'

,''. \, *'c/a'

[Y\t, , t,, ,^

Ut/'1 :7;ll-e-rrt

Lel- u,l d,wn<

g^hL;u-dt .kt-e

C, -- t1^ Gto)xt

4\JL

W4

I

fo,uttutl

*e0*trD

-+o1,0*. * - ku-+r)+

,0, = *tox4

rLi) s,*3

.rr'. u]&+-

/"s [aru-J

Sirrrilr*bo)'"X A"4 ,< 42-i.J-j

4+ v44

t?q

ryt-,, ffm-ph at *\4 L^JI-V e,ulf4hp_a p w+vna,

o*-q /ot',t 2/ro

4-i *fl\" ru,etto J.,

0*tJe

{t4a

(a.n

W6W,t @t' fottrrl r .

fi'J $4{ { ^)-^}

-

' i '-e = fLbffi t/s

L'p ilr )e'

, ZLI .. /

-' h * -bX t/S - A-/atQ '?'oirlr

50 r--...-/ "-l'o

;

,r^-fr' -L = k, fl = -i(z+z):q+b |+oh :lLs = t "

Sf^t L'ftil V<-< s@^,;'s w**JThR. ruLo r'S 44A

o1yffi, illa.D-h"r

L-a__--*J

t!4- ur q{A *dt *'-*-o*S

=J-t3

-+

:- 5k)TT,

O.tJ4_ ,yra.{hU

Lu uA spL'+ *a,,-L

Xr^4Un i, 11 +-w-0'

i" ,f M, J,,.: ?b (t,+"1)*,, (.*"r)1.-,t u

.

JIET otu*s)J +

=SfuFI

fi1r)

(rt ,

r '12=* -'1---------------

(r

(*Lu'(LrtaJ + t

*,(ry,-€)

L .HO= o

f\

SPkn)no]b-'--

A*=

er------+ z

Ju

"'1-'- -rlQ, = z-r*b= t*+rt

1-= tott-dlm

flfi= ,fot<fi il

F,^f, vttP. L+",-&;q n"y&w SLey< a..* g ,f Wu.

LvtLi+. b<n),*1 tygt,rc,cl *y pL,'*-,! s-t .

!o1"+,";.tu^-r\ Lulu*

I'_-_l

L_=--

@

(,

:tri , [i=A____r___ 'r1l , --rl-4 I a L--->

('- '

"rlf ffho-,ro*)o)i,,J

Nq cl- /o,'ri"*''

A,*=h

l*.'.qfLr ol>

[/Or+'Ur,IIr t, lcrr rr

- I LU)(-tD)cLJ L -E-

= *8o "oJE-T

\or.ttud'aXin's w*t^,,A Cs,uJ b( .,a{-a( W *} &,1k* T*Ut^ -'l-.1'', e)"'&*"u1 aa, for-a.bo1^i ,^go . ftrcsa q u/^dt* {14a

i

T-rt t C,vr*-J b< r{.orr',.{ L*b ir-f-p*r*....,r.J "i. ,.*l*-;;

eTuorJi:rn e"+ d4L!- F rf'* ill , (,vr^^';Jo*.!.. TrM c-o^.+r.r,rJ

qf^Aw is [4s. &'o"J @,"*n;4is ,*Lo^ - I ImrxJr

t, <x (l. - fx'l= *5X^

D (x,(+ *flq.(*rr) + M

= ^O

- A-0:<

.)14 = r

l} \,'*('b{g"' f)ti ' fn'bLamt ar'( at{'at.,( ,l^'rllallabtvt-tor Ue .,e.tl,L€' zt*(er ry'- b;"ei"" V*fu,i. dfu!." rc1uJ,l4 e+ A+eN *,"! t-ent'*t'J)

/n!{ Lh "4u o+ ?,-,flu'a

#,ffil.trl

*|ru:Orc fn-4e,r, oat fa*a

t - l'a.,Ll #vf fr/,r^k1

?t-Eh)=hL' :

7['

hJ d= hL2.

h

li = +\L

D

o

D

uw'{.ter1e f+dnr X;-l-1^e* Ca).bat',14q44

u) = *hL

bz o.

uO = @'t-t') t

7

C _ (zLm)(rrra 1

L3

S+a,fr'cA/t lp d LW nr^*+ Wotu,n2

l*UU^" "f *lru's 4f< q/r< Sotde J N r.Jh ut<

4,, =

Asp :c**A 4tvl/€/v\t\*-

La/AA,lic^9 e.rys*iw

t 6,x, +6.x" [x.+{ d, ,, 'r {,*X. + 6.,k. +\. 6r,x, + 5r-\^ r &. x, +

Si rrr4+ Jt@mq'fint in i d;"ic*e,t

X; 4,:,-" 4 /r--*ra!# b&,),^xN4eaUA_r*tZ

F

-* ?*erfiu *'Tavrl' . C*u++J.

nt,- lrrtbl,uht

, .: ,.,'

h4 44^-A-o

i*riI

@._;

F= 2-o r*A)lvl = lo KArt

Q = zJYl

[ = 2-tY\

h = +rrt

/ : 9-l l<r': /,rt

A*o

JG_-n*,9Y<-/-i

fl"!

r---T-^I l.mA,'

{e)

I,ar uA cJ-rlrlr

rrbt

sfh'an @)'

r--r---zI I

f77'l

tA77T r7r',

(c)w

(J)

fr-*l Nra' , }rkt, llfrt't a,-1'

drrr,\ *Lai,' f r*/ lu> '

\5gil'w4t'r:l

r) ]<4 t-<s- oF- .t*."-( r,, c[t-'l-+-, rvtu,l n c., (,<)J*J'-J

, Nwu.be,r OF Urlt<tuswrt y4qLh',tttt e.r C&rUtrain*S '/14 =5o IVLa l)rnu.n flunb+, sf L+nl4,cui^# +er Slet*,rC. -/o,(r.t>,)q,rn ft - 3

K = fyl */t'= 5-3 =

a) (Lms,"1 {^e bue. Si"', C-o ltt r- , de Uod

q N c,1 taa_a* +&4 fuite.^Hea-q ar< Varrio,.^.*{'

+"/D C0"t47-o,'Ot in suc)<,r"*.1 b< c,er^p S+qrh'cr,"L{J S+-,\.bL€.

*Xa<u^

Jn r<Jvt^Ga

BS,

I-

I-_Jt#

j.Nl::sj

ti1trTi

fr n4- ru

n ?<;ijrtinl {-ha. e/,u,.t'Jate,4l- fl1rcn. -:- "

TW s in uolure.r l\uh'^A q.L( *lLt f6le-n*( ferAs. r,u.r.D *trt.b+tx 9,14a<,r,t tuJ -a

rl lr'*X |-wD fisrtot X; ^-l X^ crt- lt*-</

f Oiw,rs $f y'44v6t/-2.0 C,go', 5.7a7'r'ul::i r4-r7 cl-, *fo .

"fA

ES

+) N< |L'c<'L( w)e C.artue'tu-c-ql u/.uotlt'ez-t er fe,<L 1)L{-{{,,-,' +of 'r-l U rl tcrLaw-'t'L f-ea cn'-,"a i,. ti_ rl-r.,r, )cr-d- L,uus+-c, zry_ S

&, X, +ct*X.+A,r -c&, x, ,- [. x^ +- A.o -D

irr o.Jc"- +D fi.,.. .(1 11,;Kl Lg1unr Cc tq4, C,,-e,ti-s t[, a[ = 6f,, ,, A , , *"- l,l\,-r ,

a ierufA

jt L,Jtruu[ c( l.6a ))€ c€ s sq.,.U fu Csrr s , lou *r^t trnt.]- e_J

fil

VV;i-e,,-r

7tlt

@i,:r

\sl-Jnl:,s-lcr %tW

A.r,

vni* ls+en

I8{

i}t*enw\qty '*','$-* fsra i^ ,f*.& Xifi+qy_

' 'L = - Xh+ Rrh= o k*n - ?-lrt = 8>r)

Z lvl* =- o= M+ RJC"+) -F(q*)-I-fi + LhZ"

Ri = -trLq*) - tvr +tfi - ?-h%qyb

r-o(+) -ro +eo - r,r (+"J

2M,+

h+rn

7,h^/^q"+b +

- 3+. s rc,..)cB*.n

=-D

,1 tll\n =

* ?h"9-

+ ?Jn

il

G (q,-a) * (J*R"th-ru+

- ,+" s l(d"

- 7fn = O

= - eo (+)*to -F., tJ^+?+eD

Z Fr : D

= A.'-F- R*'= l?. 5 _ 2, * (*:+,S ) :o

-t.,l+a,-m; ^ q.+r"tln op p.

"*C*r^{nL utu{

Rl, = D

Cq-+', ) = o Rrr = o

t-

Ro,' = i.,

nJ - o

:l

pg+€' rhi'r\a+ie.r\ SP Rco.Cl-u.tn -{lrrCea in um;,- SrS+o* (,

Z{fro =o[.o(a+a1-X,b=D -. n,'"

=

X"4 =o2-

-&jeT

q,+4

= 0--S

g i.aYRL= X,,q

l8s

a"tb = 0-5

-

&l=.o ; fuv" l#-s ,, 6.d: t?'5

I

IJ,

Rf"= Qou=r.

*f*bln sl Q1^ -l-''*4 NW*\a-c*

lrlSi"l l4tes< €l/4Aitpttr, uJ(

bl^lrq rrwmp-q* 444q* Wit(

fna-+4^-l {a Cq t c-*LAr-n E, ,

Sha.L( des,jrt -t-h-Q. ruu Ebe A"f . d it+1 l+re shclo<X ir.t , s

6,., [,,.= 4.,.

Nt) vfPor+i ovt

Loafh ['10.(r , K,tnr li1s,[r, r14 l1lrJ"', /11

r-r 0{r<a -Rl * Eol ,.

=?ct)

x*T

]Ir-E

0<x{2-

94x(t

R;x-Ro'h- rr4

= 31"5X *go(+) -LO= i+.sx - 33o

- lxYa

= *10x1-

-hRo, h

=+)C, x:x

A;^ )c

= o,5x

,

lV-trI 0(?c( 9.

*RXx-F'x-1144

: *3?-.sx -rD o R,, .:(

= o,5x

i

lss

[4yr , ,<a) /

I

l'v*< i s t1\LlF ultq )L,,or"tl 1$a'*t' r-h C-nLcqLq-*

fu,r,nd't* 'F LUtr\o.-

d, = =,*.').,

/Jl,' t)f l"

a fttoJr, , rc-t; j uL: riui +.,"re

- ,rur,1n*l ,

l(:ler-rlr,),o--lt*, eP t, "t. {, = 6f ut-,1f,

?t*t rrvrinq*r'a',a Rt A,F a^.d A.-rr,rrninqfl'an B an d) l\* Wh iyl,tl*,t t.t

ne,u^A ir: ffiA,p = lt [t:(-ra,.)

(u),)x r ! O sx -zto)(+) dn,-J"C,,,la

= & FTt lit..#*^-{3ro"/^- Vll: lkJc L z a - -rtr- J 3&Ly

)1"1 \

J

tBr

:T

f

-

a*, ; *. tJ:(r+s*-aao)9nf-,A.i

= I I lg.+5x ^tLx-- 18.*srt-.ET I a t b

r_- .+.9 L

= r ftr.+t(*) -lbsx*:. tB.+s*rjl%EIuL 3 e z

-t\/ \ t I

sx-roJ p.sx) dx I

.:-J

al( Itt/, J

.,-l -Bso,- =-=I bJ--

-l \r

*a-*^co-Q- e-4qLrch-uv\l/.

(0.s") J* +

S *,t sl,'*.++v7 l-|^iae Co "+f; c e-**s

D

lbP x, + Bx* 1t9L' =- rrq3 3I

Lgx, +- +X" 3sD =-D

V -LAz= Ept = *e"osr)

; = & : ++tKr/\l rt

Lo+ u4 yt-.Hta-n +o 44,_a. -e14.r.,1+tt ,,,* *.ytw"n

Fo..u X.,tu0 X.^-A orfpA 1,..L

(c!7rY1

c

F

zF^ _ A1_

, ol:

qt- P!-sd'w-, 6tyt aa fl^' , K^\ *-A

7h + R,*h =o

7h- X,go - ++.t =

=D= fvl +Xot nL4'*^n1

9s" 1 xl

+ Rou hw) --'F G,w) = oZMn

--r

R.'- F (*4'i - M'' X* t '- '7L\q+4

s ao( +)

-

4I

=. -18,5 asr)

zrh.

Rou

T-rNl*tc*r =*(*'=Qrto' = -)Qon=

I%tt.r = -R*hro

- 26.++s K^)

- ?5,1*'S

- 3s.q rt

iC" fa^'(a++) "F d t^ + M - X,4 t X, h * /L" *2-tY1 =-a

- R^h h - rvt +-x* q - Y,h +' 1,h2 + o-A

= - 3L. ++Sk r

Chec,rJ- 1.

ZF'l -D(^'* F -Xr- *. &"'* 3 6"+ts +-2,c * t; o S

\D <r4<*rn'n q h' ea

T-IItlJn, e *(*h * *ts.rKt)

,Qlr, = [1r' = 86.++s xd'

tl&,azs

tn*<"-nnl

tO 1xz-f

t+3 .6 r',tl n

*BS.q(+) * lo t e.os (^) ^4+-t (+) r tbo

O.'\cc\ 2

fntfl* =

Mrl" =

t4;; * t4:

26.+*s x -h .- rrt,r.4 (4 ^to

I.-mo

Nr. 60 =

Ql,=I = -++.1lr-o

2,?o 5 n?LE

1/- Z5'1Krtlrr- u

l:.'|u hX, rc ?xZt+4.1x loxa /=DI a=D

Q*uou= RJ-*F

+5.62 X,rr)'rt-

:tg s (. 41arj

\-vv

Nr. 8' =D

fvlj'*, -fu*n) x - r.01

.-38. 5a5)( -- 2-D

lDt o- lra,n Sf

I 8.s t5 *en *-

l=-tn l:/==o 1r=

i''tlu'-^A) f+r@-

0 (rc.( x-

\38.sr-(Fr'J

-9 *, osa

Fe<c*sa

'58.52s3L'+ts

25,1

Q er, xJ36.+ts ?..05

rsg,6

,5,q ++,1

tb,+,

(o t tetm

tqo

lT,og T Fn: g5'q -3s1{ =-:s

Z Fv < g6'4rs 1' (;Ds -38.es_U

zV\= 80. 6 s t-tL"+ * 77'os

-l̂ - \-/

Q.=lm , h-c-fii

fi4 = Lo Ntttn

L8 Szs

{

Fr" c0 {4.Q f/++.1-,"c,J

vf poi^* o'di< plo(Mna-Lt

S Ol,"t'lr' Lrvr

*f-l^-e

hor,l-q

t\

Ba 1,.

u/e

Xr; tr}_ uA d-o-t,,j- f+-e

lxa lurl i s I - Wl Sf".,f-'c^rh in Jo+*" M;n qj< b.. tau+<-fetrr Co+U+.a, n-+-* (n, ow-A 44",<j-

= t"h-fl= rt-3 = I

e-/Vo*r^'a ,(- a/n-|. ];ui,ru /1

wy!<.t1Lquiuqk,;r Syll<-,rt

rSx,

lnwd"n +o F'n d {.r"-E

q/"i [ni+ :o7'-1,}?'n4t'

8o

Vnt<,w*tt {vLgrru^*

{v1

at].

.fu t1q Fqca we4 ,

=,ttA : D<*lrlP. --' 'r - rY1 J-,(a flq = o

Fw 44--L utu+ Wy2-(\rs = o

& - R**q'

tT"otl, du = o'9

= L) - roxr0a

.\= -lo1r-A

. /'t fi/1KB= ;;6'l

:R*

R*=:X=2_q

J- ?'(-

tr('----"*- itr

(s

M a

CA,{rup ^t-"b; U' +,A ew't $q

X. f Ar, :- 1l

h, [ .[] {-,r,) ( r- o'sil') J, Ir [- -,o{* +- stt' )

t I =Er"L *e- '? /o J

e [J:4. + J^(r-o.sx)

(t-0,s,

fun

&.

4,, =

:3!3*Ts

.-.\

)J, I'.)

2l = gt-o J 3FI

C0rz

I

-r

NI ol |le,'rlo,, wYu MIO, tLNrt fr,,

X, = l'

f,,,-'flnt= 'l - 0"5x

T-I 0<x<& o

tr -lr o<x(e, - A+x: - t0:t

E-*ro 0(x{{ - l)'1j-a"o 0

'ltqe,

!.I

!-1

*":l

-t-

EI

s,h,s*3 -

lctb

x, = - Alr Toxz....- =

3 X,8-2,.5 t<rtr*' .

lrw lt,

^r?h

zlYlt

)

Ro=

f,,,

fo F^0q ilru+

Ez

t.,

A

W< SU-r^L &

wn i* \7U<,rrt-!4/,?a-^-l

Itr

1

R^= Ru =4

= D's

{)

-o= _{v\

&;=

(i=

* ko zq

0a*Xreq

* ll"lv

-Xs= 2", {- x,5 = lt. r_s x;

.̂a

Kri

Ao= 1

EL

_l*E.IO

,r J.,.lJJ

t':"

t J, [*.r-rt,rsrr) [o.u*) J* - j.'6*r,n, - tt.rcr)/o,s - o.s

,-I l,tsxl-

l.a+,b1

t[-,8+lsx -.["e8rca

2'j

d- 5. 6rs * l

d4^q Ve,rd'co-l- J,a pAUr""^* o* SJ

fe-u a* PtA in

N! sP lw+'v, Lo.'rJfi lffyp I tu)n fr1, , rL

T -T 0( x'1 '1 Xg = a.oS o

tr-lr D.(r( I X3 -R+>(; 2o 5 - ll. t.rx

Rrn ,c'

= 'Q'5x

m .D 0 ()c< I Xr,- f'r' (^+*J

: 8.+s* ll.t5rrP,s$+rt1- X ,o

: 0.S *O.sx,t.V* l,l 0(r<r - f/\

= -c-Do

!

-rI

hLa

W*

) oL,t tr\4r

B\,tcLt( ir, Cl.,r-tcl.-c q uni*

"4, yt"'.l o* f D ;"Ll g

.:O= UYl

1- = Zor'n lnt

lVl= laYilttl

E I * CAn s+u,,1 '

Fin J. l,l*r,'ro^*^t d,,t pt, Cottr-^*g?a f tini B

S:.tll<,rq W t*la0

q h)r ;.a.4r1<,-{ tln^'t-

0

?t+errmin q+,)0.r1 of fuq C{.'r^ +or,Co a^^-[ nVW",tt-tFer {444 Fe"Q S"r{ }+/?l

ZFr= lG\:ozF*= 23-R*'=o ll;=| --;

6

Zfi'ln-p : *M + 4@-/1;'q ---

/fi+= M-23o*k) q =

Z Ft =o *,{{lo fl* =D

A .= tSPi.a

t14,+ = o

tD- co +,5: tordot

JI

A

114.

l-.l

Wt+Fto,lvlt*^

Iilj e+ ler+;on b^l441 {fr1p, N^ fu,, /\:t -r 0<a< I [1a = eD -1"R; I

=2ctr -Tr 0<2c< I flI"+M*-4

= t5x *lo -s: s[]o=-I

u:-.tr' 0(x< I*rn: .-3O X, X

=rC

a

A,B = J*

Th.a.

u) A,,J. 30, J._]

-?D r^l' I-2- /o j

Fq I S; ?ox J'c ,f r'x+,.o 5,c

oJ

t-

0.1. t sr_T(_ J_

Tr_I I e.ox*, 2' +t -

- l5)c +- eox - sx'C--TI^c..'-.{ L 4 a +

So luudr"rvl

!2rlorm is o4\r.*F01"(

Q:tfiat ?= t}Xi, lVl= gKt;m

7 l 0w /rt't'

(g,a5.l-,@L,l &rrl )-€.i)r S o)- 0, /V X@*nt fVL,

\$"..1'a"lh \" ) el"^n"7,y ^,7<

P

lrJ.r^a

l*o(_e

eot S$,",.*iqn W Wa' an|A

ZFt = o

Ko'4-? * +q =- o

R*' = 'l''t -P

Z Fh- o.t1 \' t\n

7*lA+ --o

X.= I

2-='l

Z lyln =- o

*X-q a 1)lo-5

o

ZF! =. o

?*u= i

zfn:RI

9_

Pq. - 'l-cl + trr+ Mn --o,-

.lU+= 'Lt- Pq*rn*= 5 -,o -6 = .-llKort'

\!

N[ uP Pe't,'. L*X+^ lLtrlr' KnJ rn fA

T -f 0<)c(, -Px + x*%= -lDx r- sx {

o

lI -[ 0<x(q lVl*

=[l* lYla

:-l

nt-fi t

o<x< | Il,l: 6

*, t.,[* r* + J, *. J,._]5;=

tq 6

I,!itl

'i

I

,0"+r xl)!%

<.

elru;la. to^* WS,S+<-n't ^^-J

zl:u'=o:- P+ x^ * Xq - /1*n

-'' /{av = l) +xz - ,-q: l0 | lD.+, *lo = /0 "+t t<,J

Z (iln- =1;Pq *

"*' + lr'r -xoq - t4*

lY]+ =-5 + l0 + 6 :-t0 .+r = O"tLIy^,

+'

_o ; 0J6,

3Er

= *P +-Xx= -lO+ t0r

0 (>c< I

l= -to I = ol, - I [.=,

-6x

/, 1h",,le

)"J*25

E;T

ru=)-

L.+Flr^,a

P

LLj 'tn \er* X. = ID,+l r<,0 ln;lReq C{1 sfl fi:" Cr s qtl fttAruon-t

Tt * '/i

rii1

i

TP <rt1^'fl1),.,r*'l'.*i D+ t \ +<n-ft,r( f*ts

14o

f -rN*t n,

,Ifvl5{:'r =

=

T-E5f],a= -#a*L

Q* 8t =O

E -llr.$

NrcE)=?

-Px *x*- t()x + 5x{

= - 10,+t

--i)2C =D

- S tcd,rr

D <.x< I

I D. +' K,i

0(>c(Z

ft)-: il** A,L1 *nlrr

d bc)= Xr=

I q,**

-

[4^ift] = - X* rc +' {V1

* 10,?lx f- 6 &*t t ,rtot'

6"*.fi^ ff b4l+,*r-^1- fedt fqc4+r4

l)*o'Yo,i ,

10,?/

$lrrr,',KtJt.t

Z Fv = tO'l'l - 10,+l = O

Zfilr= s,*D"r.cf * *"*t :D

t xorc^ogt q-*

P=

tfr, h- 2-ryt

l7lul, M:40KrJt^

C^qrut^* L-*7*AI,,o" , Qrn" nJ

ft

\u,

Ctw': Tl"4- ftrt^a,*U*l- Fe Ltr'o+l'n4

U-t+,p,-f f.€r U&t,

P ad4%t eFnrl3r.r+. b_e

--1

i

..t_

I M u lti-spanlGonti n u ousBeams

A stttdent who frnows Strerugth of Svtctteriafs vel'll we[[ catt w-or?irt Ahfil enghteerhq fie[c{.

Chukw u, Celestine Nkemakol am .

tq1

-,

1 uL-r t-sPAd / cDlur I N r/gu s B{A4\A IS rnut+i-sP* becrm,i*, d4^t $rTr' ohic/a,,ht-t

fvtsn.{ *::,**1:oIp*r"';l n*i--i; **" sr'rzrpd+' : ::

*.:., :*':+r,.r17

t n ctUer.m ln-rut* bza"t- s ri cq- {'r'{ nwnfo' of t€aq-fi'o-n G"':'""::*-c ^\l - .1 .- tt C'o nn o,4-1,pa.- \ @r\ Arl.ar lf-LtC'Il'Lulrrt"-''rI'vvr

an< p++T< .J'lr,t.^ 'd'u'-o $l';'/-c' *-17 +l.i "1.@"' @r

fran^ S'l-afic elt''Cb'irtrrt ' Y "l+**;co"(, Ult*T -,,uL "i,l'y --"fv,rl ()'rLat,\

-l/", rr u r ...

?n+tl^o s+ -14-f yr^-a' qa,N b< ea's 'l^-s s 0 I'r<-J rLl r\ g-trr<-e

\\f"S"vt^hJ aJ'wo'*t'su'

1- l,rr+ o t\'L,g<vtP^,J- L1Pnfi'e'n

frnn,

a)

\.

f'fi3rr* q iS 3+.a+,-cr"Lt? v.tt*-nJoLe .' tl-o"rlaJn-r, rF Wq Ji'.fr-.L {t.-l'S Pcvnl c\* n a.,.^-a[ ir+-D A^C,. cT ftrg-yt-L*4i- n , 'trC 5tuer,L.( y',- Cp,,t:<

q :p*+,-r-rru! B{-"\]olo \"<a,rn S5S+ort . Tf.l (r*n<c+11'n b-c-41rLi4 ++,a

fflg+nrct il{^ @'-"\ tct-l-e,*^-oQ tA-@' i s -+i^;

l}1,*,

/1n

flln_,

\,{l^I"-e

L^ t ?tv\n( L^ *t^*,) + M,,*, [n*, = - 6t tr(o(^,^

\ ,Ld,ffi

,-'..X^,"- )dn,., 6I^^ .\ dn, ^*, a/>< CoLl[at *J t-Q-A o.'r'

\-p-(-et*rrrn Cau.-a< & 4 {& e1{-gnn -^.I p" c-o-: €,a.t c {"n- Jvoil-1",. {-.t^!- +/,={ V<++ a^-J ri31^:r 9p*.,s re_8ftc+,r-LJ.TX',t"'r rf qtr,e-\ Cs,"t& L4

Ant,4r-, W ^ npC-*\ ClUr;,J

d-q'tr'g; (or*6 \tt-* rqcL^ot-s lqr44-)" rLnn+'orr*-,, dn.na*a dn,n*-, a4< si*+$ d ri#-<, a) 4 a.^J F s,i nC-U',*f .^ l^^r- l fiw )4^p,'4 *flU^ A4< dg,ferJ"^J- *4 {41( *-r ot 6€ ,d!4{oLht- 0f- {4^4 tpacwn 4 arr?4.,.,,,!- b.cu eJ "J*}.fr*"'_

(

^db- V e orr\qla

U

Aou

Q: tm

b= 4nC - ?-ryr

5l - jrlr'

1,'=tot<ttt/n, p=lOnr.l , /Vlg= &Dt<,un. Mr+= lotcun,Me=.tys(-Nn€grrxt"-.+ J^"X"-" ff Q*e,r *^-A n4{t, o^J Co*t,>rn H-s4*, o,n ht<44

ult".{ FAA*',! i^ry-ar f\rrer-,l-^ a , - .^ ur+1'f>'rn b"*

c>rt't'w<

3 ola,rl->-

Le+Fi.s*af++,r

fUo

M'

Suirn Of {+rr be". La 1 wwyut^ab a Cfr'*J q* q^J k.4e* a $"k{.\t^,!t {t w+ b-e Mo , },^a t ur,rr" oF b€^^_),t rh.g,ax..+< a*- enJ

+-t^{ Loa]- tu"f /0r+ ta< f\4". .

lYIe ?{ + M*-t*o -s.l- lo .=_ *51<-Nrw'

?a 4t+dj3o -f ,) .-D

lA r:. fl1"

(fio

l-o+ UA Wn<t* (Yo 1 {vlx c^-b fY1^ LU,^-1 #^.La rna-mt^* ?^rd".+a*,A f,',J l'Ytr< Vl.t*e^4aa".

l\101, r tt\t, (.,+ (o) r_ Il4* [L = * 6ei" ( dt-y)

l^fr^!^* L, = b, (1 = g'

:l-e* u,t rirs.,) \<N+-al< {4^-o \QA*L h {4.4 S,uTytw{ t3 ,

sriee t'lu f s q+ *+ris et,t* \,wv(s{f) wq h.ltnle {4d. 0f+'o.,r ,r-o

tn cla,rdt r* in e^44.\a,/ t4^q l.*p+ v.y rfll.\t- w^,,r, - W.e gLcr.d,

cl^^$14 +! teut i+ r^i{4\ d-hs \9f4- '\f a4- i+1 Ee?- *ro*.s t

?

I{*{

,)tYis

h =fG) :24 Ea 3Pr

SnVb s+,+r"r,t '^2 4atU< U ryt^^r^ S-F d, CL^^A

.l-4"" d4."-ca lY\,O44rlrt "y@

4s(+) +21\l*(a+'rJ .1_o(t)= -6EI

J)lr< = :fto_e :la

.- 3 O K-rrlrr'

Stln CywJU,/Ur4r*,

wFofi% g. ner4!;;{'1^r,* t^rur.rd ft:".Ju to

O ,uo''*'rn Lc+J i'-atvwut-d- - *oJ,Lend Ot +t"P. bea,r,t

rnrU=L*I. lt vtbr(a- q

PtX,'+,'tl Ar*'61r'

txa".{t.s, *\r- ttt ; fe,nl1drs+;uw<{) Lc*J d

q,CH du*"roq-JA {Sl-rati WJ^tt (v.tlA4 wa--l f-Lctch',:\n ie7(tQ;l gt.; Sq.rpea+ ,...1,,,-LLt

NiL\ fn-*Ja(-c. .?

p*;r,.i-c acnni";,uA

fi1,lgc<i,u;"t-.

lvlu b

3ErL

1.c', :

@?EJ

to

t,t-.lr:

(wt-\ 3ET

+ ro ),'r/

L'c-| u(j 1!e+n J, q

W spa,,.,s ( ri.cL*..t-,.-,1

[\-e v,s^-: C"r( C-ut o,ir-<li

[c^ cfc--s'

14n")

ZM* =o

)oP\s ;

Yuf cr.n-c.4-< C,e"tti ) err<*-rrgr.

+h-s Q-t rd-O a, ct S p crnrl

), f*.d |<<'c+w. fu,-t r

Ro

,J r'1,,,y.

o.a ftu:- Lt l+ a,. J r;. 1\t'

oL,\cr- J" J J,r,-,'.^r, q-t*J

o".*( '1 u.'-"/.t ,,.--aJ Fe-,<r..

+o -s +to ^2. D+3D

R+

'U'\r' l,f+ bea

Zf4,'

f1'1c 'F X+" - (Yla *{yls * {ylx * (J b = oa

{s +*^ * iln - 0{s * /t4r

= 'l A"+5 r..,.)

o* rflc + 7,q (**) - il1n + lt* t + Ivls *fi1,< = 0

\+- 9.q (* +b) + tAa * {]rts + Nt,<

b

.crl

Rc

.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

R"=

R" c - e,*(*J..) + P (c+J) +

?4(ld*)-e(cat)*rr4r--:..-tJ/1!*

[1x -o

2.0 + &o - ro (s) -3D=-D

Z/Vle :o= Mx+

,)

Ro = * I[,<

C-

xq - ?4 (?e)

" z:! (:r) -,,l :

'))

AoC r* x*'

+, Pd

-3o -?o+ao-:D= * '2s rrn'

c,

Clrtil.J,^4

ZFv =':̂),. (a +-Tc

= -2Stt_r-r

)erft*rur,,,<*r,g,n

*fl. -p+ gd/,

gf 'l n+e,-rurut th-Ct- {..t t --t

l,

"'fl;.I /1. t\

*(

L5t

l)

Ro

Co, , Kt)rt

{VLr,,, w)i4

T-T

QJ*

fl4f4,

o.1x< t

^ftrQoo., =

=

orair" =

n{ - rrl

alV

t) o',, =rt

:

*w-* lorc

IYlc

+o*

1

t*t,;1-*?^

Bra

I

o f = /orcdl2(=t

l=+oIr =o

f: bbrcNnI). =,

fI _E

Qi, * 7-a,* rD

:+s)Cs D

A 1:r<ry

0(.ec<t3

R^

6.?s = *,a,+sxrJfr

MU M. + {il*- Xq(*+r) * R^rc

+o + ro - ro ( o.srrc) -_t,?Sx

]I -Tr

I = -- lDlr.+

x< 20<1)

Rn * 1,x3.5 - tDx

nl* - xr:e

1sx sx2

,)

Ri - l,c 7.x';a2s-.?.4l* EyL

2

g ( as -ar)

-

:C= 1

-torci

3, +s ri'':

+-

| = eslx=o

tr14r,

.- 3o J=-*l l= oIr=o i*=,

1=5l:c -a )c=3

X<:

Mj'0,, =

l"tZm

- M* r RJ' ( c+,, - XcC€ **) I fl. x * 2z:*30 J-e,s (e+x) - a,: ( r+xJ - rox3 I - t"r l,ll;l

.i,, q-4,,4 O" XrA,!"*,.-e1 rep [aur +4qFo,. cr R,p

/ = lL.st xrt,,tI-| >c=:re =[oT3k+ Poi't* R

o{- 'Y<e'c*'uJ'l

RG= R'l +

Ru= 0

(n' = 2's+,3.+s

/ KA=

a.ck*o,! V ql/*

k+ ur u^< ?s,u yp**a (unl*'s t^w< ) n-0 ffe* *t \qr.rqY<-ht*{ r ' '

For Le" St{+{,ryl

?^4,s = i*.

)

i^ )L4<amtn-a:l-{ ) wt044.e v< A",rr) a-^X-

&s

trS

+, 5 Kn)

3 r* f-c *l^q S2t+orn i t &4e FgL"lShpu[ 1a4v6l-a 0?t{ oF {-i^[ \r.t4t ( Wt<C,D*5*'rr;q*, we lir-q.ft Lo,al.s .

3q-al..-c"*,{a

wtrl< "4

.c.h--

11. aA ^

*- SYS*cza

(t: )

;r'B

*

+

={r77)

A.

?t+a

xs! +2l

d.)+ p ft+c)* Ms

-P(J+c) *{As

G.)

-21a

tdo

1

R.C

7_4_

1(*))+r).5 K,

C

tq: lo (+.s) + ,D +-e., t- rs(+ - lo (s) -to

tfl. =O t

= */il. a- ?q. (+ +brc) ^ ffr* * Ms - R; c + 1-c\ -a

=e+?

Rr=*+a f-ro (6.s) * to +aD +zD - Bo {*eo

chec*)nq

ZFv =o i

7-q+qc+ 4-p-R,'= lt)*r-o tls -lo- ?"s

Un'} SYt S+<2"'t

Z /vl,s = o

= X,Lr- ['c=o: E. = -j* =-_/rc-;?

D

X, I t,tcJ _- Ro c

Lr^lY\ ll/tx6, ltl *

- (2'n

- at"s =

*oA

L= &!lr,L

D

Zfic =

o

o

D<-a'93n,"1

Ng oF pn,-rt

D<U.(t

0(3(< rt Me - ?" (+ tx) + 016

: 40 .-ro(o,s+x) +.lo

= rFS .- lOx

P* - ?L^a

: l0x * sxa+ rg xs

?crx) *7d(4 *")*d (4rl+ (.x - 7*Y^

= to (a*x)- :o(r.s+x)+ ls ( r+x)* 9-sx *Sxll_.SX e 5x {

tV-tv 0<r(e,

Zsb

f,n= &=I

FT

Ar.p *

: + f- +sx7+,0*3 lT _prL e.- "i- l"Tt.t Ca,vvl'.tr. cne ul,*_fu

S*X, + A^F --oX^= * A{n : ltD

5,- 3"-fti^ rI .utq_c

dh,ct lYuo*uc,-i-

lJl x\t* ...f +*\ J"f

t+'l: .

cL\.Lu . rAJ i.lar guw

L?L-o'*l"u' , R* =

trI

3. +s x,,,)------------r4

lL"e ,li a"u\ e+s<ltty,ro) I t.r,ctl tLt tz!

3,+sxr,)----

gll I\sl I3 lo J

.,.r,JO

Er'3

* [ j"'Gs-ro^)(-*)J*

_a3ET

2.

(e,sx - s

L

.r- lDr '+

tQo

a!/

Q: ztn/: qyN lrnp = lo rcd'

C.o+tstruttl- dr-<-X"o,*r, st

Q* "J (rL, dut+;,,"-t *

-fka \b S+rrr

f4lLa.(q +4^!.

Sstoth'r.

is .tr.ro -- totdf i.s* \uv({w+

s$q-+ic4th ,t, Aa*+^.ur n e/e

N i{44 a p ic{, +io"cr Ze",o

.W? stucd

Sl aa1

P

ruJ

CD

*gs+<.u rflnl< Ca,r gp("+ +l^4 nJp "fu) D

(4r-

C-g*rs;)e,",'^7 {42 fi*u {la-+

ffio (., + rM, ([,* t.) + fio L.

'2M,q. r M^a * o

fl'l '= *0'5fl1,- -----0)

Cp*";) eni^< -l-{^-a kt.r*-l ()ob,t

Ifl, [* + ,.nlx (t"n t") + flL (3 = - 6€t ( d* *L2=[1 =q ')

d e = ar(t)

= il4, q 1- + ft1^cl * ga. = - (>el (ro.r^ )\rO*JIil'+|FMr- = 0,5 ---(ir.)

Sotrl';-, d-'t\-e STslea.l op eVwrl .w,t (r,1 a.-s[ ;,"

* 0,SM.. + /tM" _ 0,s

/Vl. . + t{Nm'-.t.

fVI,= -J- L,rm.,+'

U) fr,^n) Supo",-"t,1<

f€ac{r o^ WQ)'

b<-o,n M- .lta fOirU-s A at^^) B a,*o{

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-

Sa,rht , t'*'*o^r) b{ .wis< +, Ll.t'^rl1' to*,no up

J/l^-e. ntLt1,\,UrA' {n,44nA-/' CMz' tr^ t'\);LL qrr';{'e a* +L\P- n<A-l<-!'

$.eCrf.u," ,V g (n, & M,rffi. es" l-i**. h* *norn'tfu' tF t^JP L'nJ u'tQ l t'1^9-

\4r_"- J s<c{r.o,, r1uilazl J-t*i,.rtr w'r s< c*^-J c4rynJ, W JJL'tJ't'c--"

'N1-r-A t4o,^/-e- b4 Q-et'5t'a'v'c'*A Sh'e*-L-r '

frj I ,,b

L* Qrn

L Ng 6,g(s)

l\4ca-et'a! - S.ke^ - Cn 3

til: lhork'P4

F,^--J q1[6s^'ti, tn -( tf]

Cc, L CrtLttt-u.s-,

7:+Sd^^.,tu; .

oF a c'l'r,-,-(- \ Lon..)e"-e'"4

,-L1 vful t>t

w,'yh # ('zyt l*n

A'= tY. q 6 "-' ,

Zo=Jo= 1.6'lnn,

filhtt,^u /o-+'t9/

Tu'= I 8''l u'n+ , k,--;-l't6-) or"'

)-cu{r'u1

(netyt

A): (, th)

{

+uaL

rr=b:

_ --L') | - 2xt&.1 =^*u2LQ'+ fr'Q)")2 Lrr.l ++. 1L(r.tt)

,.fr,*.tr"l (6*4r"*J Re4<+,vtz ,r-e SLrA{l lc-c".t{(

3+-v o- +

6 [. I c--l

-JLI_*J-.--JLl=J+

1=T.J-Ih r n

IJL -lt

l, h t,.,

: J, = 3+' Y 'o^t

2fi'= ?x1-'tb

T*,;

-

5

r+

= 1.It c,rnq._--

2.7+

JO

: l.qSow1,

)

,

i

*lI

i

\Jot 95-rr "rQ

Z,t+l xextof l

/b xruL

n vL 2-) ^ - -

/l -4't tfi,\ l'15Y, lO- * I 09,. L

1) ?e*e,r-;,n\,. q.'h\a of L r^i+'^4 SLa"-olo"'.l'l t^,'l"b

= lll,l

Cor( cuut "+c Ot-i+,^c-"-L Lao I ct^-&

[*: q-b)' ;::

P* = (o* ,)) k= 5;k,

Lt.\4 .f as;a'y11'5 fe-,n,-Lo,

S*frr*t,

Stet l 3,

ZlOu,fqI ' lt1- na41

ls'1 "L rcil

l7l 5rci

F".-+--*

0;=tu-l

E f*'st<-(=2-x to'tfq

sF \*felt

fJ = l.?.l6n

f"-n:L>

P*:LPI =

3ro: t-r+(o^6) = l'l gnaq

0; A : lclbxrolx ?-'l 2 xtD-'f -=

bI'16.

11" 5

nJ

I "2-

IL= 2n, t3 =

P= ls rcri

[D-] = Lotvtr4

lDxro-A n4'

,lI

l,

nrr"ri%{ - W,# CP,lel

si_q b,'14'+J W t44aC Le ox U",a-

CDU^r'*(\'

$dLtn,h"ort.

Ja = Jg =J^;IL

:'8t;,

IL

= f3.{!',"-

A-= Ba?\5 ,

D cr"'; =

(o r,,frtl'I =-1 ra.x6,

a.) 2= +ILm,i,

= 2,'1x,a-of^ = 2, ey enr,

c O.+xr. tt G\ 6:_: f-r- )r- \,(a (J' v

?.1xrc'L

r)

+)

ln t-?$b ,tt A Q)',g^J* *r,,1, pfu- VA-t{ @ &{-

btl.,{a.}.^ 2r=*O wi*tt Q,=0.61 "J 2"-=Stwi*ttLrb^.foarr;f''n2 +44-aAe U et^aa-a ., r* g1^-c+U f_g CU,le

Q; o"s++ I = o.s;5o -+o.lf,r= q-Lrl

= 0.s9xuO = s"q{vLPq'

s ) Ac+,"-l-t 5 Fe-+r 15\ -l-LL( ^J,T rr-- {): tsxrD}

n (oxrr-)"

2 L"<-z

t(r= 0.s?.

/*'u^l

J,?

N

L= h,EJ" =

1Y1e.1,1-u,a,-t

P. 4orfit &o fi/11-4

s+€/ s ( C"Sl

O<.++"-rtrr^t J4tp &'qn4L/-*

ba*L|** , "l-

,

aF-ke

l,n +h2 ir"t-Ut = o.s

+r,-er\ oIt= r

U,D]

*Slnma

'uPlrrt, t l.'

A_: 40 xto' LLO

= -5xtr3-llr, ^

at

= 5 C.wL^o,> x t(roxn6

12.6

,s

+l

0"SxC-

6 -s xro-j

V,,, =I7=,ro

Q'' " '{"

@'Q zr)

/c=-

I

6. l,s xto-5 rrt

O'e4

-|l,v-+ fa-+ t^.q u-g1t {. ) n'*q S< t*'^^-ri cvtt {>ru'"-ch '

l_.,VJn,V*,il \J<*e4+]. ".t ru

= 0'oas.rn

i Td+64

T/"r

V- .:-S

^fc.[ +. Coryt s i ")*+

/1.

q = rfl/\L nin

,ll

. \ I-T ,, L^,\= !7-frE+ =I

! d+vrJ'

4x >xro't

rlil*srr

Tlr.z Bfh'*lu*i rf".lq uP q r S V^etl -r ()

ttrz-rf '-r -' 2Tr-t

re.-lfo'== 2[J*'rA"(*).]

l' &'0,.*r* J r O.ozsn= ); g crrrc

,C.,,+*' ,

L t*: ln l"o. t-L\a f*b,^A d-r,jn, ,

L= 2*fn, 'flitr.l+,n-l . - S+LII jCc.-z)

tN:?tF^rl {444 p/h'nu*,n g +/*"ag{- q @ (li.'-5, Je-u,^.1 la-e_fvve*<-e* t* tnrn-J-^,

/, n qa,(

lu'nr-l-^n czJ C-a.i c^^l-t^l+

J-h! b"h'L,4I PnCZ P^

AL = R$.+*o, Jut= tr.yr*,4ntJs .= lSe) c^^ t,

*44r"* t,v.rr*-"l fiA4<<z-l

,ai<-____-_r_) tX+ i

+ (\' * J.')

-l-

+(,sro- rl,r)= lS"6c4

A z'+

Yli4.,(ravt V</L< f\'y\A.

8,D6,*'

r-t -T [tS'.k _Lj-- J2hrg Sl,r-r-U ka-rf-a'

tfyc= 2f I.'* A:(+)"]

LJz

ft'=

L/y'r tn

n/:

+

+

l r1T-* J-entr--

2 x l5e/c

a (a 3"4)

3o +D+6.r

W.

,l/lt'^ -

fn.,l*

T^E :a;

2_il5.t+ x,lxlo

a.x1r' -..-

VL

L ^.,i

fu'-r

= .o'+*N:* -= lT"z68-o6 Xtu-a

Sr-e{/ 3 = e,co lttf c1.

l,- ts.?-

cts t< "0*. L<c{,,"ve"-tD fnr:.l^'cl--t ttjt\ L\);+L\

d-q4 k *tr wtvfe.,-; 6J-)q-"1 W,,a- n.,Zu'^,Q,q. b d,

7-a"<

/ Lo rna.

= 1'1,'l ,

= /urr,

-Tht-<y'-a<, I ccs,,.s/<J 2 J ftg>-n,,,(a,

P* = (a_ ,,)) ft

= 6,o * t.t+(ft.g;)fvte "r

= lZ"I Ki

5 L.c"+l b< ,r r."!,,

€o- vt'-rl g,

4: j tO fu,f sb = t-t+ nn4ra'

Q-= lS"L r-n

6crn

1m

I

. CI-'\_l

, 3=, $,^.

d=.(-=V

^[ F,'^*l *ta

,i'q-J

C"i't zJ V-\L,i uf

Po-

X.lB r

541//*\i.----- ----

:D,) ) =

'I

JTLr" ti.

v= o's

ljl , = lr,)+ *T)+6+#

,[= ,f-lrJo'- 4 '+

/rL4trr

nA=

#r H)rlr++r^ f,-Hr1

ry.f,- (4) ]8.,+x+"[,-lg]") = lO* L lill ]\-'

= D,\3 ,nn '

1"11 *nq

ol2-I c^n

*

E.

L.,+^*.@,? x,ol'

tloxrob

\r= tl.S

'/-\ stt*, i=.fu'a Pyala*r<(1'3 n-11fu w iJh

kLurnin iu'twiwG:o.Txto" Pq'

6oL

6 s"+t] /) 7 )r.^

Th-0".,7* t vlt 5ht tt U-e t^.. Lp-.' J f-orr,,^-L a,

_2.! o- J-lhrn -: I " t+ x o'l x t o'',x l" ll 4 16

0t)' (r,t ^4) {

Lg-p--ati

f

fr'^l P-

= l,'l rcrl

L^ 3r, b= 6*,r*^^'l,t = t oXLD-TrrL, fl = &Xto-^^

-^l* = t? xto''-t4", flj = 3

fUA+<n>4 A ftr^tl. ufSd-e*,( 3

7.o

-v

h/-vt,l

;l

rit

fi^"l. Pr* "*-il A7

J -,vl lQ, L= 2,'1, P= *a r)fD-]. .' 2.,6o r,'f-\

Apox 4aA S[^-lr,'/^*a- At ,*t"t- rXal-Q--"J.

Io-?J

0,?rj Xt".-2

( e)

C -r'tg t+Q= ?c-,

ol":.L LPJ

{)

fr

lrlq<-)

CcLs* r rrrn

hryu

vF 6?.ffi'd4^^*) q tu^rl L

3tou6+3,.t

'fcvbLt af W c<xrr-.1,-c-.u t .

Caef*'ci<nFCrtLrL

.Cr 1---0'xs .9-! q

O'tT'- a: i[-*

[= +m

fuitm,Vq

'5'*.! a,U^r*-rt l'a Lg^rtr [f ]

t .tf3,e,6t.r 6

C tq+

l2.oo

Czs FeP [,o,*!0'/ ootoT -

-0:ii1.

Nl qter;J(cr?, cr3)Ccrs)

S+eei- +o, (lqJh +o

'jif,'c a S*<-c( ( KKe,,ttutf cro ;4.

C'("'Lb / ", -f*.*-txlwil ( l',,e-)

fpe,y.e 4o ( Cocxa) lto' I 2'1"3

Tsa* | Dert'Y-C:-s I t1'Y'x I '- -'-' n

lCo4+iffi| lrlrl->d

20 I 0.a&

'fuzvx

ctq+;Nal

o, I

!-

I6Worua,l"la-

Pressure vessels( Symmetrical Thin-Walled Shells)

..!

I

I

l

l

I

I

I

IL-

l

Tfieyeoy[e that getfiue (j in

different from yoLL. lou fiavefttclittt

exanthtatiotts 1yere rtot ct eatedaffit tafr.es to plet tfiat nrarli. Qoocf

Chukw u, Celestine Nkemakol am,

?zr

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