Page 1
kwu, Celestine
yKBy CenecrnH
Nkemakolam
HxenaarcoJraM
Solved ProblemsIn
0 ftil.!
Strength of Materials
Peuemne 3aAaq[Io
Con porll BAeH He Marepnanoe
cwott WWW {uiYwt
".' tX )'m4t 't (1'J V )'n^z I a 6
kf /krya-ts fu-"Pr+ l*e4 g s69 sa34 '
00.-L{(-((Ya
s-+nGS-()docO
+
. 6ft,r,wru"*.
ll
l*, i:'l'.'
di'jr.
,'\'
. .; (l.t,i\ r.ir.
Page 2
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
Page 3
-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
Page 4
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.
ta
l*l t,
. . .. i:..,;;r!:Jf't::., : ..rtt:.;::{,ii1ji.:.:: t .."r\ -1.:.,::;..!;/att{
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Page 5
,A,T POr***t ut r<cta'*1il.e''. S€cr{-"4m
h = tur"{ = }tun-
P**^tl*, of "L" sec+isn
fin^ "L" ,",*;n NQ tok''lt'' 'ft' = 15' |c'n*
to, = l, 5K a'r'
fuot = Z, +0 arn
$ = toonror CrOr^)B = 6zn^ (&1r.^)d - 1o ntrr Ct un)
fr, = t54c,m4
Jzr = ++-'lc+n+
{<tnd, = O,t&+'Tirt-
{h.rb\<rn iWor^r* *-vu.o (e,^L;ne_rl Sac*,rrn: ur; r..i/,r-
Se.ya^-.*+- Cera,t-,u.'' J-s ,r"-l rnan-oal< yf t n-o-t,-, .a s . &r. +ei^
r^ rc ,-he (,nAt*p*cr co^x-,t r !- , /y,offi'^*t ufinp<r-a , fu>l,nut af tn>zr4.-r) s , fr,,) orp^-!-. ryh/^!-1-\>ls gf i^r"l;,a*.1- Jazir Orie-a-*a*io.,- .
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14p r44 beJrn'
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dA $Iu( Lr,a,^Lo epr^/ rlu,n-orr.r,o-'t tl o/-'+<-a * d-t"n (<t'-/.'1,.,"1
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(i;) S,,i c< 'L+ ,jan-c(-5 !-o c/,'v.rb l+to 'f,on, t,i,.,t-t-| St c*i,-, i.ti-.r>
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r/_r/c
r\zL=l
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k=n7l=l
2 A,.L=l
* A,$;r=t
[)= nunvbtl 6 Stc*"t-a : Q'
Sz" = S.l-e*.c m,or^,'7'';\ t'rJ'*h
Sz = Ad"
A.
feAYzL+ tP /- o.lAzt
Ar.-
k1<A 6f C-,nwt-\<c*..tm
S+-4*€^ *,{,r.a.j- t4^-4. S+4r{i4 iywtyu,14* rdr'{t y<4fLc+
; ef^l *o {a"o, f*d^e Vf h<-c, frr' , fr 141R.
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tr
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+o -{4^n- t -qt'.$e.c*'in" *^J
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qUc
I*
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Fo, "ac. ,
1
LSz + Sz
fr'+fttr
= A' Cy,,) r ,+ o (-t, J
A"tt- Ao :
lE.s (t,sr) +- 3A (-t)15.5 + 3 a'
: lS " &'t c*a-
fro : A^Rq e+ r,***''da-saSt.c+ur-\
=ht= $i{.rr a Z"-uy1
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tr^ = *. t-ve)S ui Ca it* L; u Jr"nrut"ru. ye4_44a^*e 3*4r.
tl< O' &4 r-,ra-
$- 4 >c'a '
*q rV€ r<-f<a<^^*
-t_LC So,
A.'
A; *.'
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L
E< J*l*^lg' be*'<s'++-
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n-
L=tl'\2e
r3l
rL<.,.i=t
Sb+ Sr
ft"+ A"
= 'A'fao,) ,-{tG *rJA'+ AO
rEas (z'+) t 3e G s/*)
l"slortl fu0 2a@ral!
l5.s -|-3a'
,t.- t -,^b",-
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Page 8
W',uA 'd.r ,l ::: t :!iii r,:
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7, *8.Un'
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Page 9
l5+ fa76, t{ t ciru
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LtrI+o, = Tr + I"
Tu'= f*i* A'(r)*Ino = f*: +- Ao 6r)"'
I+. = I*i * A.E) ** f;
= H..+"r+ tB.S(*,nn)"
= 223.6O c."^T
+ Atr [-r)o
+ 3{t3 f 3x (-t,tr,)
m 'b '-v<
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E
Tr*
l":Jw-l of- in-e"'h-q uf
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i" ut ,- nlc re veu-.1 Lt t .
C,r,'rc-LLrl , 4
*trJ5, o- = O S'{ &- 1:{- lt!-' a^1
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a*, ':f \r"at*ax '
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M{- ar4- 0i\r{i4 qns,* '
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*ld. = te.-, O,3 f+
= ,-1. l6-
:-t-e, !I ?4-------+.. I
^rd-
= (++., - rs+ ) 4-^ (2xxt.tb)
Page 10
Cht"'u<+1 , "44,.a
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't)ac...t'* kl+t.r*lt
TtLa. aft1A Ocar^fi' -l \ ryru1la$-ia $,et-+i4rr1 i s fn'Bre 4'14e'n
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ryyr)t t1Lan l1{.1 auf*& Oiaa,U
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=
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'
rt 4r
ln Po\,hrrp- At<c, ,,
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= Tr,,, + r+'@(K) r r;., + i1'6n)(-^)
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Page 11
dp = = 3trl2'a
t 1't. +ei
n" noluwl'"l of- L "^J I,
C*eCrct"t C1 Ttt-e ll.ettur.t
(,:)
/, \LL) J,,-.7Tr" 7 L 7 T*n*
= t !-+.-qe '
= l:+To' t
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560,o+ t #o6.4o
5 btt.o+ + 406-+ c
7 6 [' 14 c,*', 4
S rrc . a+ - +0 g.,+
[53' 64 a^.+
I I a,o.o k
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f,t = {.*. b* M, + I.;_ Ta" sn ed4e-' = Ae+.1+ tu Gt+, ta) + y1i:q? *att,6
I yt.+3 - /&&.*2
[,li')
=Q
sl^ 6-, +.r* )
o
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--l
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Page 12
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b"tl no f,ou.,Uk,
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f* bcrloru- St4,a.erl. U/:i (. U ru-t ) PLe q.r4 ltrd
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Page 13
O,, D( O, fiil'/* h'< q^Lr4 {144l/tna
DI warhp,toyi eF 4+\L
h : lQ-c*,,.
L = C*.t'
I Or^.,^"a--t "S- W$ = ALo"-6 = K' +*'rg = O '5+o*,
( = 1.O2-o,*
Tl,s,,< qf< lY1a,n'vt'4). f
E,l- ,,
Lho, n<] \€ c*r^cyr- frrr.rx
1-
= 2r 'tor.n
= lgTun"*
= 2-LA'c en-
= ?,rf 6 c-r-t-
(roco4-'*ra
l/g z?q
ft"t-)
T5,
-tJre,3-,-O t
(xs,bb f* {1aLsuch
Page 14
w l*ry Cs^tl be s,i'*vL ,eJSticz
, * qxi **,"tT /;v;/t s<c:hvra U
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? ETtRatNkrtoil of CgilTfrnte tl Tt*e Surcoil T-*J7
ut-0c 7 k,Y,
2 A-;(=,
A" (yo,) + frb(toJ
ftz fr,-L,i=t
2Ar'i,hu (2,,) + fto (*,^)
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,-8.v + zL
IhLA
tAxL -., IA - COd^
Ut.Jor = A+ t5-
^- L* + =)?*.?-
Zo, = L-L = Zt^33
=3r(63
4uYt
Ao=
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-t'--2
?s-r (,o) + z6(+)A&,8 + B6
1'*t r,*'
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Page 15
p.ur1,1inLu
(i,t) , Nef'rla*C"**1 rc C,ert\U-J
aic( be
0{,1!4' +/h'e"ta Srr*Aaa &),{iit) art
d-t'fq+,r<^* V4p of 3" ^J 2.,b* 1442r, [oc*,>ns
C l.U"t<^.^i f"r fl,* C,e-",:t-t;o/'
?ra,,.t q Str,^bh't C-"-a
S+'ra"i,,* 6^2 D^. O,
,htr*rro" b*l.vtu,/L 1, ^J y" =
,d.r>l"^^C-e be+neo-, lna-l y. =
r W cAt"*t"&A v^/nPz
44,12 ar-J;"* h cr*'*, ':
bel4^)<244- /a,'n*sof 0" *d io qr*e.
mtu\* ba q 7oi,.t
f42 $a-,nt -.
D, or^J 0q
hn"e-$ lho,n
q.(e^fl. {e
J = 4,- Lo,! z.gL- A.+L= O'?Gc*r'
(1 = +-- *5-ua. ac
= +- 7'sz= O'G'e oru
&sw-'o Y* z^ ^J2. = (tL = WF Ht I._1.+e - ** (
.o
=K- L+4. -n4 ro
= 6* * - ?.-*s..3
Y, ^.
5'7r u,<
7. zi "ot
Jrr+-o s" baurr"* Z, ^-J 2.
Page 16
3 Er*n 11ttfr'TloA)
Tlt-P ' c.,f/urfrlrrlg,
m.ofieilr 3 64wsq
*SGcTron) USrNq Pw<t*+rct
il
1= LJ f\=lr-f *9c , Jg.
-LJT- - T- 1_ Au (_J)"*r. ; "Uo,
Ta -aIr" = T;" + no(n)^
rL * AUGJ)^r rul* n%.r'
I 8+ r e.&.tr (^ o.ta)t+ ,.8r + gL (o.ss)L
5l A. q5 *"*
lNffiTn T 'r-".r, I?
**t E'-Tttt*e,qnBtr
rr"
Ir" =
TAtIo.
gF Pf-oa q cf W tN*Tt A oF
A
=Lt3L
= 5xn5 o--;7- =AB ?c',ntoE -.-
tt-
= *2- c.,,, /
-
N= J*c -l- T*.-r LJ .LJr*." = r;, + *" G), / r_l^T*: = ri, + ffo G^)^
Id .rT.; n h" (y)'o f ul, * fro G,,^
2z3o t- tl'.y ($n)^ + +2 t- b6 (-s,Tt)QSlo6cot'f
-r,L,
/
LL h4
l*.
T*" =
?---_-&-
-DETgfr, fYrtru*norl
Trgelc
LrLsBL
= t?x (33L
-l-Fl-€ Becur u^J
!" + Ir*-r t-J LJJr* = I*" * *" (-J) ( r)
LJ
Dao =rJ5ri c.e r'fq*rs #
ha-s t?1e
4^*7,-
?
q
.TA aJu* = rio*o + fta' t1loX*:+.,*k ot
0)G*)A
7+toht+Z
l.e*r"rr;.,.r*- *t\L )L2 n
.A 2-
= 6xt?-_ e7t
b d.D,i1 ",{*- -qhsl,ie CanOl.tto4 ef
4z*-.
tlr\ +ta f/uczJr*,< bi^nt,
f5
Page 17
I_tJ
1
I
-l
G) 6r.+ft*&t d,rtlfu Oriud4-l-,',tt tF oru4/
|ec*an. Tlau, +c*St1{t Shntl b€ * r/e
0 + 2t.r 6-o,sr1(up,)
- 24?. 7L or- *+ +A+ a [ (o.s r) C s. *01
<8c ..
-
Cl^trt<. 1 .141e. aCauaa oF ErTn 6f 4crc *^rAt( ,lA'r'cA -*a; y-lt-
+z
?ttmrntDftTturn ff wate"Jrn rwr)
AxLr M{A Tbr€_ IYLA.GNtrvt}C3
rwefie^)T s 6f rN&Tt+ 0t TF+e
f* Yh! D n'e..<l-oohlrn'.
han 2 d,p = A h"*.
-t- L1
U.F T FTtr
trt rt+E
56Crr unf
- Ve. ezre-a.
tfuVtt ,fLt-a-rt
g-Lr-r. eyl4w"-/
t S alf ruXt'lha,&lc
*rle - Th-4; htre *,
Ade =
?nnsct/kt*fnwGt f fra-
T, o,^A -f .LV
("A)TfrnrQ*-t-LENTA,OI ? I4.L
;,
.r...ii!t ," ;:",..'.
Page 18
-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
Page 19
!,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
Page 20
{ ..-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
Page 21
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
Page 22
{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
$ec*,.i. ( I, S, D ).
Page 23
Fr=Fa=
ryft=-
,q');;'3"rtid = lrtg -. li,aa ''
4:'4ni5o xrr]
1o KN
6o t<nl
5ft1^
i
I
;
I
-_
I
I
I
i(
la:
I
I
lI :.
1
si^ c-e Nrl' LA* Po-cs
+4/\-e. Y<gs1*n Fr-Ca ,
t9,o ------> R
fYLatts- fu,cr^.-ot ) - E
e+e,4'; d4r.-; of - AJx ,d"y kc,t<,e,t t
II.rfNz &)
-tr -'r,tr
N, ['r;
xr-r.E
/rjx &, -*F, + F,5o + cl o =: i*O ruu)
JJ._ E.rV
,Ux 8) E +F^,-F-5o +1o *6p
q1*;*q c^t +hj0
f7, sLw^Ld b<
-trlt
- Fr =
-5o K^l
*5O 114
O 4:r ./- ct
CI<.x t 4
I no n-0.,\ F"tt is\ b"* ct t)ta..at in
o ('x"4 c
D<x<J
-ao Ki
*l*??o4 * )d;u**A
"a PL;*,1 )/|t'Ltt /
'4-b -Q.o t
*Lta,ta'
.4ht-,
a
-*it--
-i--Ti
tts'j
rft . ,l s.N,
: .tr F3- <-<
...:r.'I
l
I
Page 24
-
N,&) Ixnr]
DRMdttlu cr -f il/S f t N C.t [YtarYlHJ f Dt.trc, R. MnA
ln oA l,ul'.*- rc -J'l,rc ftllvtts ̂ *16 l"tu
, Wt-Lr"J of clr.*,*.4s{ c$-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
Page 25
-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]
Page 26
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$'!\a,"*
b,e + +OK/Jm
tg*r),Q,^*1'rj* .
l
Q*
A=t ,1,
rc
Qi,tea )
q - ?-an'A
= +,nC = ,rn-
il1, :8Dx,urrt01o *- kc xru ruil1o: *oK,v.,*
T'C+ezu ,; .a+.ry oF +w ,l. .f\,q fno.Lrc4./- &tft_r+ E< c+-u."'-l- .iJ_ J
l
Ii
1_
!
Irlic,r
tr-rh1,E&.,
'm -ilrn1rEo.,r
=, fl'1, = &c'ltarr.f <x < r
= f\'1, +fl1r
= E, + a-D = IUO Kdrn
= 44, J- n1d * 1113
: 80 .+ro -- ?r, _: ZC rcnn
S'' c-q r11r[8t --- +3op,l- , {trt-t-,1 (l1a : - 30tr^rrt
7a.411 tF .l--r; rwanat Jsh' ['q-*"-r'-5
[1r^1
E lylg[.
fvl
A An=5 ,1,
2
80
M, I xru, ]
Page 27
Nkr4^ilA,,*
Gonstruction of Shear Force andBending Moment Diagrams
Cantilever Beams
Two Supported Beams
Statically Determinate Frames
{he cantifeyet wing of one aityfane tofcf 3t4t'. Tttgin.e and Jt4rs. fue["r, a.[ot"Le, sLLyyort tfi"e weight of thefttsefane inffigfi"t. ?fease, I can-not carry yoLL both". 'Br.tt it fiec{ earty c{tte to ittcreasecf 1enditryntoynent acting on it. 3{oy,eyet', tlte whtg of another ahyfane tofdthe couyfe " Britq affyour fanni[y members to tny honte". Attditfivec{ fonget hecantse those acfditiottafx,eights it carried c{ecreasedthe bencf|lxg rnoment acting on it. I 6now tfi-e acadentic stress irti<fLAI is 6ig; frovever, the wtore yoLL hufu yout'feffotv stttdents tot tnder s t and t hittg s, t he rnor e y ou w iff g ain Enotv [e dg e for y ot n's e [f.Don't 6e art is[and!!!!
Chukw u, Celestine Nkemakol am.
" ir' :,;i i ::+'s t ;..' * b
Page 28
S HE44
g\rsr- W* ia At*"^'-fr q^ qlforc{ , d*taw +o #"4 b++ or +o
W str,,rrr of oil,!: {t"<.tL"l.{4^-a. riqhf, na*nL s;clq- .vF q sec+rrt .
U
"Jt*l*c"'t Ste'rna 0P i4^iL-
L$* 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
Page 29
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
Page 30
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
Page 31
r.l\Y Vr.ar: fvw,rwr/"\f' q/Ym il-b$rn q4 SLG"^m i
P"l. h-a4 t r*r*r** @rt^
q SLwtrn'
f*,r1,t
= *V€
III ,--l br."J s *.t e beA,^
I= -ve.
gunll*r*r- Exn*;*90 1'144 .7."4U mu,Yt Lzt1 LctrJl .
T[Lo4-{ is no lcLrictA,cwe.P rlgn in VaJlrer cF
Qz. Tkcreprs ,,iJ" tShr,c( rtr-) l)e ColC*rUU*
Itr'
lP
-'- nlrfro : ?(A+*)- pn1-
= pC.t+x)* 7r.*p(,0"u) - 1.0.4
* f,wr1 St c*rai,rr<
: 7D G) =r D = &) r-t"^,*..
"o(++t) * is./
-,)-"
:- c1?" s.r,ln+
O< >c
I
:E -*trr
rrh%, =
P*r- P
1,c-P =
l?. 5 1<dq
=
=Jc=o
P ( ,0+e+x)
xo Cs+*)
Page 32
v r.-lu , 1Jf - lrq r---'->7"/{+
o l4;,t ,f'^,nn1
lo+Ai1 IVren,u,.J ty^,f1\ W-\ ,.r^ nub ,rup * ( t *+ ')
;!,;
:
l/+.s
ltlltL
Q,trl ,[r, ]2C^u
*\-o* Jaq/,\ Ltua V^rtl\ e? *,tiLl S lL ilos i-gt"S - ltL- : M
'?^o,,
I.II
/I
M
G"ue- '
Q= 5ot6 - 3,.tCs 4m-
P - ;DK/J01 6o X.rnrtu
hu = ?af,N/m
r 7,.vb
- iD QP [r<nri
6O
I
1t.s
43,3
Page 33
T-rQ*"n,
M,,,
X--t . ,0<>c
T
Qov = -P + 7,*^'kl*__*la-LbJ= -so t- tD>< C r-{ l
-nl r- P*-ry_[q+] L*il+
I
it
,
l
)
l
!
t)
= *M = -6ox,'srrt
q,rn$-'
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'*, $O,lt Ce4 St,"4t'tlrh,e f r; a6wtt+ l--rl {"'lai sh, q3
It
H*,fa+, 1
l at \ -l-o+tz.'"nn4 eLzcxl =fzu'-x1x^ I yzu -x, JT
= -eD fsoI/- ?ox,,t*lx =o ?;3
2 e. SKNtl
Page 34
s
{
l
e-
fn*
'-riw;i;;;atr; dA rti,lr,n *,
tOhUn.tur *hore is uru fur,r,f, (rnrto,,g.,A* ) /;s*;'t,*"!U"*r-l;- ntfin-o^..st
"u^r*;"d f fuY l1+e,'2|' S<c*'an S4\4t( bQ 7tu*1""nc
h"l, -flcz
i < h^4 rC rai *e,l ,t,s ftw+ I f *rl . ln 14A;4f G4< ) we SW*rlca{fu.tal-e ,H\a /hA,4rw,?, b*^l;fl w.t il(r*^ uL ,n-a E4cr/-lir, )funl i'.ls /ci* W lica*t4,rr, Au . tF Crvtcule*el valra. of Xe/'s rlLfiR r-raa.n ,t lro Ju's{€v;4t:!- rF *a-e. /_or, il 5/194-r;72air,< , i4t-!_ra$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-,/
Page 35
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
Page 36
- - 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-
Page 37
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 ) ,
Page 38
-
-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
Page 39
-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
Page 40
"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
Page 41
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-
Page 42
^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-)
Page 43
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
Page 44
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,"
Page 45
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#&
Page 46
;
'|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=,
Page 47
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
Page 48
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+
Page 49
-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
Page 50
/(}{-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
Page 51
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
Page 52
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
Page 53
-
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
Page 54
"'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
Page 55
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)
Page 56
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-
Page 57
/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)
Page 58
,-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/
Page 59
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'< 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
Page 60
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 .
Page 61
-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
Page 62
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
Page 63
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*"
Page 64
'., ..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
Page 65
-TJ S'i-Reroc"I,i;: i4tniI,:$ri
"
qt 8oa, 4l]a,,&q-L Y-lr-e/.---LN*-T E,.t) sr q."l ' Ad' Cg/n'lRe ssr urJ a - .
AL.- {
{wrrr,"*,- F"- l.hrtr1ra 1r t 'J "}4L":r T--}--
D;- €F Q= s.fra.r^=4} . Lf=L/
N = AL L- , L* ori3i..n( La^5+,. or JA L aL - d^'&^4e 'L Va".,YL'''d' ,-l '
lrrhLo,"''t
)-A7> a,!-c', . cl = jn'r
\ = H-nt
C= lm
;|: 5 c'*' '
F, = 6t, x-n
F:- : !o x'rv
Fl .= irr: i.:nl
F.,."n /ir.(r) ., C r,r,, /] i-4i r
NT
N,,6'1[ta))
tr"@')
At- =e+*A )<4*-***^^U Cowp:u+"-\, LI u"l,*""*^
t" .1-e-v\a"\r.^ ar,"=a| 'CUwfr<s4,-. ti
!d;1^a.^1.
1,63
r)1. (mM)
6+
Page 66
-:r'J
l
'"f.J
!J _TI
N, ['r =-IL' _ TE
!
:
u" -urr TN.ri e) =
-
-dU, c*r =
,-- filUr (r/ --
l-r = 60 lu.,i
F,-F. tFa6-0 - &o +la-D
D 4.:c .q q.
Cot { a^ ta*tirn of A C'h'nX S.k4-n<,.t oJ-a.^,,, !< c.l,rnl,i1,.) = Atf
= .*-.-?fr
*,UJ:3A
N,t, =
bo n to3axsxro-f
*2rx t03
3'* sxro-*?
luD r tO -'
=5xro-f
lu"ro K;
- l3 .3 m,p4:.4-
2emrxf4'---:-*
I
I
I
I
fr
Ctutqr{rarr
.:!
(-.
dpg = ALle
=
Cc\t
-tUPJ
tr"
ef Ois PL". k mt--,:l\ c+l,s.'u. Se c*,a4a ( d)
; < dAl*^* 'o a,* S4^4 \ uvl, f ".r{. r'J La.,-o
ALr. = f\J" = lrt$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+- -
Page 67
-,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,
Page 68
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)
Page 69
/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
Page 70
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.:
Page 71
-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
Page 72
-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
Page 73
.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',
Page 74
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\
Page 75
': .-.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,
Page 76
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.\
Page 77
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
Page 78
!
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*
Page 79
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 -\(L\/
1.,
qr.
$/\nuu
f$'!.' ,,'
i' :.- .:" i r-
^lllLL _
Auh3'crrn ,p hrd J,l+e",.r,'- \,t-ru*lI
rl:Vv
t4
up! N
Nr
+8
Page 80
-Y C-*?qh bt tr'$ ,#fu
t , )4"bs' ac'l,6Q 7"6e
,)L,6A$'= l"b cc
?, (, r}t;
&= I4Nr 1,.(,
Cosot.
?o6[rU,l t. L f N7L, IL er,J Lco114 EF.J
Bt|'= AL,c'(": ALr= a6'c-,xd.
Ql\u-minl +44 br^_rJi,k uprol l'= L,
*d*tn ^3*,
op ,,Sj <.
Lr = Lt,/,/Co!4.
Ii= !):4 = ,!,,l-t lT,lr^h W= O.5?
t'-tl IL*""J = #t"H-]
: O,6iS
I
LO.ois+t.+r)+ Krl
N, = 0.,Gts(toV,d = 6["*rci(l'te C Mn--cS rltQ ,U /
,2
"64-8xro'
St,tbshJq*lH
0. 6 r5 lvl* :N^:
NI, = 0, 6,1 5 N, ,^-+o
Ltq,5 - 1,4/ l\)^
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<
Page 81
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
Page 82
'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
Page 83
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
Page 84
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
Page 85
. ';;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,,?,
Page 86
[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
Page 87
<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
Page 88
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*
Page 89
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
Page 90
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
Page 91
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
Page 92
[\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$.e^7
I s q,f ?1, .J
Ut0o =
k*=a t ct/^
L0* : L0+o + L00.
.+D.or{++ f/lr,,t4
Z if-:. 0.?tct+ -
= 0"d rq1
iilfl4*at q
Bar.lar tobx 3,t4 ('r^n^ ,o ") *
0.or. eTo - O- ooz+ rl = O.02 +a-{ -*J
Q,& (, + [Y1ct
Grr 6,1)/.
,,,s67 (e.,8x tD3* 3 tx ri'l
. gx w t*,0'* s"t I (,1rg *+x i)t
0 - o 1-7 +& = - O.oo ??1*r"l
t(*,
= O.oL}a,t -
M,J qqi,.
L0* | Q"*
D" tle111 q-
UEU, *
l-4. A uc olr, Itri3r'1
3? x2..r33x ?"txrD3 xO " I
rrrcta toL Z,t4 ( +x,, :J * = 0, o 2c1 7D r,,t
Q rr:
Page 93
Q**i r.;::,.t,: :i.4 ... i- ,i r t., . -
t(+, t t8>e
o, .lDqa+ + 04r,tJ*, 0'5q
i
:
-#_--'
2-
= -0, OOLL+ + O'l$x?o8xr0'xo.5xo,ixl.
tr^ ,+o ltl 6x 3. * (' O:$ztrfx,o- "J '#
r:<- C
Gl,ra.: cl.= P'snn., b= l'2'm
' fV = 30 t{rimCt = Sxrun rnPo
[tl,] = lo/nt = *" ,1d =o",'T)
-f
[f I : t,ut tYtf €L
F*l ),rl,t!*
[V*b' , l<rl rn
= 'u Oo {9 dAt+
Ca rrs
"l- 0-o,ozr--6
flin
.-l
rF3.+lt
V*r'n,o 1' .o"t'
qx
Page 94
:Dedern'tnqh)e* Vt fuqC1..srt ftLbtv2n*s fi|n a*-A ,lt+l*J
ft rtcr, - fY1"i
.}^A
il1, e = lll+-1t4
JU ^A
{Yln t"t = ll'la -Q1Y1
fl1fo = {lLa -ltn +NL
= fll,v * fvl
(brwfa*; L;U'h €-yAhnt rrr( h_o,.,tp.
= fl4,, L
Gr ry/ l)-
!rt\,bsf ' +,r+<
[1^ r, t' ( *to
I11a = Rlt't t,
ab1-3-Iq
= ? ( go,. t,r) r +s.( gox o.o)
?( i.r) t-3?.(o.r)= +3, +t 4 lrrr;,t1
[gr-S, r'r, fn,- tu\L-t &"s.F", b't+ia..
Mjo.r.]f,
nl,. 6,
t\1o : +Z " .+t+ ten m
fi1o - 2-fi1'
+3.il+- h = -/(r-Zli6t<rtm
J{
iJi* t";
[t'1,)r ttu =
fivrn 44t< f<.5ul,H f<C.eirttil,44^-e .f,,.r+ forfuou, T_T, ,.S
+l^a tui{+ ,["^1 *r-: r,tri+zr / [,[**^^ f'= ,fil;i, = + Z,*t 1Kilrt.\ r l.
pf*+rtntn a+,ry^ qt- dr.A(vv.144 qF W St<aa-6. tSka4-E '
frwn tvn&rll,sn g+ St^r,ft'
[Vtr*^^
\ *4 Cn-J,'+;o^ s? r;S; l;n
D*, -l4t< f<,5u1,H r<c.eiunl ,
I n UVst+ txa,rqi^A*'c.14-
lU<a+;e*a, W n1 nnrt Ee
grricaa. LtJ. tn t,l^El CA-J? ,
+ttz drrtwrel+-n is Cq. tc,ltto**"A
Lfn. : (0* | qPB. + (,, .+ lp"r, :-)
+ (rn^ *t,nt)t, .r
Qir "2'3t
J = 0.5?
-tmJb +- tL
+ 3a /ltq +
( n'l,q - zfi)cr
ctn d +/s,
qncl, Ca,i C<L(
(l''to*^rn)q,+ rl,
t 6vtt q
+ C fi\,, - ttt) qq"l%:
Co u ruru.r *e, rn-l
( (Ylrt -fi1) ( : D
fil+ * I l\- 16.'r e6 Krvnr
fi4* - lttI 3, +t/t- /0nm
-()
Ns(kl
-q*
Page 95
-
--r I
+g,+t+x firxtL
''ld, z @:.tF 7t
-)Vf *xr0 6
?vr,1 Cs^l,^+',on u? riqi Lfl,
0"11Irru
.l = O"5.D: C"tl 1\sm
L{3.+t+7,1s3
l6oXtD 6
GTs
a )7 ilJ,nyq u{l* 3;*t+ XrO3
grr.,,rrt oL lo/ro]f_-_--_-_-_----=;\ +3. Tt f xlcr' ltO I 3(+ t-----_-==- = D,lZ+rrtDv --lJ Tt xsx,pa*,06
iL/n ,6 'xrDe
J. o"s:? - 0"o 6*n
Ft* l4tt-
&;u^** ,
1^+ et't 5,t -e'5
red+t(* ut- Cct t Adq$'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*
Page 96
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
Page 97
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%'-
Page 98
( "', 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
Page 99
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+
Page 100
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.=
Page 101
-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?'
Page 102
,* 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,
Page 103
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
Page 104
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'
Page 105
,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*<. ,
Page 106
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
Page 107
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,@ ,
Page 108
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\ $*,^ is Jaaan ..ro- \%A*.Jr^a,. b%,,w
n-ef qiJ *\ " Lrxer,o* , v1^!- &rrur*in 0f 0.r^_* gL,,,,.r/.yfut* ils ,..3i3r", in .+.kq S€c-frun.
l'0t.'l
..,1t1
Q**o,o
l.[o+el
Page 109
..in.!!--"r,--nrxail!,r.F-r{+t'+dirdsi!;i;--i}-.-!;r l.'*i;*i@#'r4""'
)<4-s( n4 fi afus)1 OF f . b<-".rn St c-{r,tvrrI
\'lr+*-prdaa i rt V'l.z^-&"^e!
A
N*J"J Scc+iu^"/ mo ) 'r l"r N* = 3, +E xto- f ,rLu
v -.1= 9,7,8-cmtt--'
fltz c'La*a \/ila-1 crA .bO r, fr"ern uasrn-ryLa*,x cg-r,44 f, e-^A
ttsi*h T- l)4r.r, flu,ru,.tbtr '@r12T ___2 ,Ono,.: ZTI A;3
9^Q o
Mlt: 'W
5** t& 'fi.sy< la,ta,r't t[J = ,b,c.yvf,,, d <
\tugur,L J Catl, cwL^'+e *ut- f tn' (e"*ag< Otl.er s*rz+< ay d,t4!
fwru'eLw'
A 6 .:: C);^ __ft[--] , -- 2 r- o1. =1.6t'r:!Ltxt,pl]L]"- ^lur' < E -h' - {e''>
='' ,-, qr
r IF Our f w ca'nloj< ov"er'5{re+\ , a fl vr*:rq (-ut -r+t aat s o/-| '^r ^-., [] :, r (J L, t l'r*-r< GUI -1+t aat 5 -)
lwto" ^)t sL"euJ u)< 4ue T-Lt-ec.,vn flurntbp,y-) tr hjLuo..-) ,l4y onrt 5 "L , ,v e sLu,tll C.b.ooto e n tt(r.e0.,. Tr/,
@, l.VL )?LVtltL-L4 L,U.O1d)4 Qr>o(teor, J_ ta-e,arwt
i f, tt tvttntt-on' P**, 1,1.,-t cUi.W4<ul ,d- trt i',)-L-, .n f ar€..,_ S< f #, errr^l
I U^c[! L".l Sutch ,lta-a-t *tu. r<-t L{1+1,\o, f-|
, I .-J eL r rv\e Urrrarr ir\r,i( eil+t_a_r.
[O " L+,nr {tto,n [U ] R, * jiN-o;1-sn ,Jltaq Lf: tl S %
Sin ce DI*^ ( f, I N 0 27 r-S Ll,,.aten a-,s a"ry<-iecl-,,.;,.e+ gl'/r be-ar,'/l
Geern-e{-.;cAl A^f- ,$- 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'
Page 110
= 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
Page 111
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
;
Page 112
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
Page 113
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
Page 114
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-]
Page 115
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,
Page 116
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 *
Page 117
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.
Page 118
)! >.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 +
Page 119
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
Page 120
\
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
Page 121
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';
Page 122
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
+")
Page 123
!_-."._...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.
Page 124
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.
Page 125
-
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
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I
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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
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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+ ;
Page 126
,.5
9.rs
fI *urrJrLf'z E.r
Il-rt
QI",=
.E'fVt u, =
0(x1 3c1
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?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
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Page 127
(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!',
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Page 128
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*
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e)y - Sla,h'c e-r
ftt**u+7- 6fkra-a
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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
-
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Page 129
! 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
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2a
Page 130
,',,1!qr\;4^Vi l
9u*en ?'br. uf ku*uini,r*rI
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I
q *,8X tO*1
o 4* ?ai* W c-&l*nc*,, vui+at S*a,t
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Page 131
I
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%x3 1zr,
6Lu Jo
0' 6+ t" t
OLL .?5 \ %+(Jo v
('o
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o.or+?ot\I
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f Lz^- %x'ls"L ?,L Zt" J,
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Page 132
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I
iii:-lf:,''rI i;r..
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- 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,
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m'J ll"<h I
fYLqx.IFt" ,)
Page 133
s,
T
iJ
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T
r' 1.;'. ':;,.l.,ilir:r:i .
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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
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+
*)u-
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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
Page 134
-- 6?x-:-----*--b ( t.*) o
b (au,Jem,* = : :-('P'P, ^ ZPL
iLhh,.#.
Page 135
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+
Page 136
->{ 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;*
(-=
Page 137
It.t]4,..d
I
I*-1
i
.l
I
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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 - ,
Page 138
[{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
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Page 139
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
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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-'?*'{
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Page 140
fYlroc*, ^ 'sj0*
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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
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fhJ*= ?r'x-P(a;+)= Lo*-
"o(b) J= -l?olx= o
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Page 141
flAfle,
filfo. '=
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6o xnt^'
(r+ (+ -)
fil,e+na,r*< ei1
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D(rc ,- 4'ro I pc (s..** E )
v" D.
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7,<.,r,- h""r* doa rc er
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* tr: 'ryh uuux;
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Page 142
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
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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,, .
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Page 143
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
Page 144
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 , ,, /
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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^
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u.l Lr^/. '\ t*"d.A.
l++
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Page 145
?'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
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;; *ffi: (w**^ is rwt- e-at.J,*.hg,<. Thro<*or<" u.,<
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,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
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Page 146
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
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N,.ft ., ffi Sha((
de"*t^, + ewtiLtl
fi\r^ r tl.'
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U rtt'f.u<.\
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r+ (0"',t<
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^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^+'
Page 147
PYft
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f(a4,1
0Cn1
.t.
''f l4t }fucu 5]a"a<
:
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u;^ Ffr ( [r-)
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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
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r [F)Th!- Se Lt1*, rr; rLr llLhtu.tttat,c llle . $,Lw,tL
fl,r l."c<-utr. F, /i?.6= ffi =P* 9<uf"in P, /hr' = olD"trro"f +0. .:Tltn*6,r<, F. Bl4ee( b< Cun h.Jc-r*_J
32-x lqn,qsx ros
-
a,trt X I box to 6
tL)n a
f,^.r It ,-e (Vlc '=
-
i /14.'o frr' + /Ltt'
->'tUa*fttttn , *
I frtr^r *r.*,rt.^
b+ (.&Ls;"t€r.J 4s t)tL\i- 6{-e,,ye,e-Lx-
I S+. t y.atu'
9-Z|l,13l<dtl't
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Page 148
-T ln u*
d*rASe Ce,^rl
btd e++//rad" ,, LQ* ,u
O t/p.a-s*"eN'*F-:!:3'9*t' sf p**
t+ x D. 2+13 ),/
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ffi
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+ r['tr"::'ot ) "(1.,+ 4 s.z+23 )
: l{9.x6tu+q
C q t C,/a*; vf- LwzL * o!,pt-s1,-<za
A F= er,r-f tr.f x Llo _:_ /6.s.zt _r lotb-j t&o
x ! lo .:- ? - t>rt- d/.,
A D-< s %
ll.u.o{rvc, Jn-r.,rr*-du, o-t rowntl S.ec,/ru," n ?=D "LL{.J-
=?{:ro{ , &F. Rec aA1i(1L<L,4-R
poi,,.+ F or F, EFAla
". * z-cr,)
frlr, 4-0f.,:. lot<,," d^
lt4d̂
flult
Po;n+* A, 13 **A e alre pd<,41;* ,t ry-"-,f+ n- ia +o,rdrb) 4\rr+ fi'nooa t,2 N*, Mo il tvl"I
r+ .c is {-,D +i,\r-& *e*$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*_
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Page 149
.al
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Po,* *,
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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
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-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+.
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= Irg {- W +_ 3rtlrtbo 1b3 ]f
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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
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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
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Page 150
),
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l) oin* f\-
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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 .
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I_)_t3E .1tr-)"**(y)" <[rJ
A(t,)
6^+ ,tT" < [D-]
150
Page 151
):.)*: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 [email protected] 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.
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Page 152
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- +/
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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'\
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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 \
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Page 153
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 $l^
[u-] = t|a,ry'q
Ft,4l {'ue l,"qnrii4. uF #Le ru4
r*f-
l01ct-q y
D<x < a.
(lluk,t ta-,,rnQou, t ta)
ls3
Page 154
fu\l\'t,tqtb Q*
ule Shr lu ry nsre 0l il e, S r.i^ 6,l14wuh l'n ftu'o*t'tt e.'L *L'e" ,-rtJ
fr"A &a r! /r"h) o* itt (aa+ fioit* /)
lo;n*t C) @,La!
(o S a*-,1 T,
to- J", = U-T!vu)no/
t^ry
Lt
4I
-fiut slr<,+t
S4"e^y*<
Trf--
Si-1r(e IJ
+D
b i* ,''r.' ..( T f^_
,"rt'.q ,kqa ffffi;
T,. t1lr = /l*tos l;*o.L'/Y SltaLt 4-7q n4 r; rl
*f S dre"<,1-t^ .
,/
n F-Nnq )r J (lly'ytl*.
-- Lo.T-
0, zr
{criir. < rirl
' lNna ( co-l
d^o = 1Ta 33z*.
3"t+ x t&r>X to ( = () -/zzt<
= /?. Lol.t ''
\ Nr, = -lDt<,i, /14x = tot<,N. (
|o,; * c t,'r
ftrrt*"*ztty d*7-r,*^.b*,;A f*tf ;eA ,Cor,r4ru*;
b tr0,t*1 "*J,^(/rtaJ. j ls+
ll*-1, I c ,3
7r t:F7
t^r"JrJ
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C'.c>*, e#-li,l'-.lno J"-l
fYtz'+-th.
Page 155
\tr""^l - *'TlD "
la0rrt ' i
fya-Tr.} 3
0'/o rnrlo't^
TD \in p L' fy SDL,th'on t, ,!.e*
TI-J 3 \. \
r 22.: I 3J! _ I \zxa.axro3.-5
\l 3-,tr x t6oxto6
0;. -- -$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
Page 156
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
Page 157
/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)
Page 158
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
Page 159
-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{,
Page 160
+
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,
Page 161
-
.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, +*,,, $F n {.u{--4{ ae La ,
tNY Cs.-'r,
Page 162
G.*pL;d Jal*m; qt-ttA ,ye nlurf,-,t q"d-i.l P*wu1:*
J1.
ftnil1+;.r,1 O c+e,rrn,',. qfi",r," gf A,,errn_
q) CLr"'r. n4d stdb+e-n^ op CBo-J,.*to, (n,,0, !,). 1,..d,.c-#€,ar\Y flortt . K u^rfricl. hq b<,h^.,t-e,t- {_4{e fos;+.,re ?nu!-ro-^zf0f-+4u na) ercr-s, *,J sL.,ri i+* cs@tJ_l^*__ Z,'rr^_"(g,.Po,i.+ K is ct \Sta,.,?a. J {-r 4 +o J4.o nt.r{shd Gr i s
[t,r"" 0+w C Ltus"-" Cro-- J, )-.,,-*4-S;
l'ol-atiarr fi,* Fdsi-fi.Jr 3 *.oPol''h',uq L qxL^ Sh,,-t br.Px,-.t'''l* D Cldc<,rix . -tl**<$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
Page 163
-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
Page 164
'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
Page 165
,..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
Page 166
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
Page 167
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
Page 168
-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-
Page 169
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
Page 170
-.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
Page 171
^--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
Page 172
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.
Page 173
-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
Page 174
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('
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Page 179
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Page 187
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Page 188
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Page 189
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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
Page 190
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
Page 191
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
Page 192
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 $q
X. f Ar, :- 1l
h, [ .[] {-,r,) ( r- o'sil') J, Ir [- -,o{* +- stt' )
t I =Er"L *e- '? /o J
e [J:4. + J^(r-o.sx)
(t-0,s,
fun
&.
4,, =
:3!3*Ts
.-.\
)J, I'.)
2l = gt-o J 3FI
C0rz
I
-r
NI ol |le,'rlo,, wYu MIO, tLNrt fr,,
X, = l'
f,,,-'flnt= 'l - 0"5x
T-I 0<x<& o
tr -lr o<x(e, - A+x: - t0:t
E-*ro 0(x{{ - l)'1j-a"o 0
'ltqe,
Page 193
!.I
!-1
*":l
-t-
EI
s,h,s*3 -
lctb
x, = - Alr Toxz....- =
3 X,8-2,.5 t<rtr*' .
lrw lt,
^r?h
zlYlt
)
Ro=
f,,,
fo F^0q ilru+
Ez
t.,
A
W< SU-r^L &
wn i* \7U<,rrt-!4/,?a-^-l
Itr
1
R^= Ru =4
= D's
{)
-o= _{v\
&;=
(i=
* ko zq
0a*Xreq
* ll"lv
-Xs= 2", {- x,5 = lt. r_s x;
.̂a
Kri
Ao= 1
EL
_l*E.IO
,r J.,.lJJ
t':"
t J, [*.r-rt,rsrr) [o.u*) J* - j.'6*r,n, - tt.rcr)/o,s - o.s
,-I l,tsxl-
l.a+,b1
t[-,8+lsx -.["e8rca
2'j
d- 5. 6rs * l
d4^q Ve,rd'co-l- J,a pAUr""^* o* SJ
fe-u a* PtA in
N! sP lw+'v, Lo.'rJfi lffyp I tu)n fr1, , rL
T -T 0( x'1 '1 Xg = a.oS o
tr-lr D.(r( I X3 -R+>(; 2o 5 - ll. t.rx
Rrn ,c'
= 'Q'5x
m .D 0 ()c< I Xr,- f'r' (^+*J
: 8.+s* ll.t5rrP,s$+rt1- X ,o
: 0.S *O.sx,t.V* l,l 0(r<r - f/\
= -c-Do
Page 194
!
-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.
Page 195
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
Page 196
eot S$,",.*iqn W Wa' an|A
ZFt = o
Ko'4-? * +q =- o
R*' = 'l''t -P
Z Fh- o.t1 \' t\n
7*lA+ --o
X.= I
2-='l
Z lyln =- o
*X-q a 1)lo-5
o
ZF! =. o
?*u= i
zfn:RI
9_
Pq. - 'l-cl + trr+ Mn --o,-
.lU+= 'Lt- Pq*rn*= 5 -,o -6 = .-llKort'
\!
N[ uP Pe't,'. L*X+^ lLtrlr' KnJ rn fA
T -f 0<)c(, -Px + x*%= -lDx r- sx {
o
lI -[ 0<x(q lVl*
=[l* lYla
:-l
nt-fi t
o<x< | Il,l: 6
*, t.,[* r* + J, *. J,._]5;=
tq 6
Page 197
I,!itl
'i
I
,0"+r xl)!%
<.
elru;la. to^* WS,S+<-n't ^^-J
zl:u'=o:- P+ x^ * Xq - /1*n
-'' /{av = l) +xz - ,-q: l0 | lD.+, *lo = /0 "+t t<,J
Z (iln- =1;Pq *
"*' + lr'r -xoq - t4*
lY]+ =-5 + l0 + 6 :-t0 .+r = O"tLIy^,
+'
_o ; 0J6,
3Er
= *P +-Xx= -lO+ t0r
0 (>c< I
l= -to I = ol, - I [.=,
-6x
/, 1h",,le
)"J*25
E;T
ru=)-
L.+Flr^,a
P
LLj 'tn \er* X. = ID,+l r<,0 ln;lReq C{1 sfl fi:" Cr s qtl fttAruon-t
Tt * '/i
rii1
i
TP <rt1^'fl1),.,r*'l'.*i D+ t \ +<n-ft,r( f*ts
14o
f -rN*t n,
,Ifvl5{:'r =
=
T-E5f],a= -#a*L
Q* 8t =O
E -llr.$
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,**
Page 198
-
[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
Page 199
--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
Page 200
-,
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
Page 201
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
?
Page 202
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?,
Page 203
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
Page 204
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/*
Page 205
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
Page 206
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
Page 207
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
Page 208
(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-.$'Cr-
= PqeI6 EI
o,to? pN
-D
LeJ-
h^J
[Y\ r
*/\
Rn 'l
J-rLt
(*.
tvlrG
)lt1q - Re -'o (^] =
Rq=
{ =O
, ;.-..'::'?-
Page 209
W vtP
'Z.fYlu =-
S'e cor. d Lean'
* fi,^ - P* * R"q
fle= -M. ^Po/^
1- P+
7L.Lx-
=o*r/o - to *g( wz)
q)-r zq(*"7q (4 +q)
q- -)4
= * lT,0+rlZ l1'lc = o
= -0'la+ R"*o
R,i' = ll'l^ - P* + '/a *to + &2
A-o'82i t
R*= o.tc+ I ,
l rr i.tr q-( SVlte*...
q q-'4
a-
Ru = (,, * Ru"
= D,loT- 0"qs =V\l.I{c - l?.0} rn; ! ',
L.e+ A) nc+j y.g,(t&rl -l-o +h-a
Forcc Fq.c-|e'-o '
: - O"qg
/)1, = ll1,= **ctu <L fi',. c(
Mrrr" ')
= 0,cll xn:rr,.
in*'-n-a,L
:(n-
i tv In iri- jIi l^^', to i lA"
iI I I /' i t-j=-7F__r_ u r ;Iilq:LrrJ,f--I Lt- |._i l,= ,,--*l,ri*
Q et ,,-)lo
Page 210
7.x
- ?Z:a
* 2.*2
=OL:D
+
)
+-
g r.l$
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+ (
Page 211
= 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
Page 212
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,-
Page 213
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
Page 214
--;\(,
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
Page 215
)
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
Page 216
-. 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(^,
Page 217
-
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*=
Page 218
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?':::;
Page 219
!
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
Page 220
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
Page 221
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
Page 222
--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
=_;-?\
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l-tStxto 3.*^
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tt (*)2\7
Page 223
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276"atta*i" p;eth'
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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
Page 224
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J/l^-e. ntLt1,\,UrA' {n,44nA-/' CMz' tr^ t'\);LL qrr';{'e a* +L\P- n<A-l<-!'
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frj I ,,b
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w,'yh # ('zyt l*n
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Zo=Jo= 1.6'lnn,
filhtt,^u /o-+'t9/
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)-cu{r'u1
(netyt
A): (, th)
{
+uaL
rr=b:
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3+-v o- +
6 [. I c--l
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1=T.J-Ih r n
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l, h t,.,
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T*,;
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5
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2.7+
JO
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3ro: t-r+(o^6) = l'l gnaq
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bI'16.
11" 5
nJ
I "2-
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P= ls rcri
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Page 226
(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)
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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-
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U,D]
*Slnma
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A_: 40 xto' LLO
= -5xtr3-llr, ^
at
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Page 227
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+l
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6 -s xro-j
V,,, =I7=,ro
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l_.,VJn,V*,il \J<*e4+]. ".t ru
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T/"r
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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
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L= 2*fn, 'flitr.l+,n-l . - S+LII jCc.-z)
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/, n qa,(
lu'nr-l-^n czJ C-a.i c^^l-t^l+
J-h! b"h'L,4I PnCZ P^
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*44r"* t,v.rr*-"l fiA4<<z-l
,ai<-____-_r_) tX+ i
Page 228
+ (\' * J.')
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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.
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fn.,l*
T^E :a;
2_il5.t+ x,lxlo
a.x1r' -..-
VL
L ^.,i
fu'-r
= .o'+*N:* -= lT"z68-o6 Xtu-a
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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,
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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
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Page 229
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)
Page 230
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
Page 231
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Pressure vessels( Symmetrical Thin-Walled Shells)
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l
l
I
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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,
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