231
kwu, Celestine yKBy CenecrnH Nkemakolam HxenaarcoJraM Solved Problems In 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- + nG S- () do cO + . 6ft,r,wru"*. ll l*, i :'l'. ' di' jr. ,'\' . .; (l.t,i \ r.ir.

Mechanics of materials

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Page 1: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

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{

i:i! il1;.

Page 5: Mechanics of materials

,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.,- .

-{.,'wv'. v .*.,/\tz d*r fe^C-a. bd-*<t-^ K ha-Xn,r1 ",^A K4,e4 +) be,n?* Nl*b ansp^,/ r<,"r1r)1 )n odc-nu ;, j

'?4< f*YU Jza-"lt fo$ K t4tuKi,t f+mq.inA S.tazlrc

d414 Pur,+in

I n *hL Sa/vn-e-

(e-ri"l-rt; A tfd^lt*.r r- +o

J41,e

44u\

l.

'li

.-- -v_

Tl4R- los;-+-on of a a"*X/g+a<, Sr.tc.,tt a4 Lq*aa, )<lta,-ru,,t,fitlu,rrtl"a* v Kan7a1_"

' ,/_0.0, rhzi* 4 "

'h"*f ' b''(* J-h-a*rzfu<^-u "ta^-'u +o #Lo4< a',,'42 co-Alc! ,/r: ";.,-.

J.. - :

w'j ll14!,t4- 6v< /!-a4 ' . - d

ftrp-atq-< ,",*'J;r**rn :3,Y; , * -'Y t:

Yre;

lhA/l4/U.{ ) J-h-L

brrrA,'npul7ui*w^ q

See+1'6r:.-a

'{'t''a':.I'S ($-4rq.c N lr'La_

.J

futtV;l.--, Ct"il \/*y /"e&r,"/;"fl wr-

Page 6: Mechanics of materials

14p r44 beJrn'

(ir (ir;1

dA $Iu( Lr,a,^Lo epr^/ rlu,n-orr.r,o-'t tl o/-'+<-a * d-t"n (<t'-/.'1,.,"1

Wh,'\4 q'+4W of Vlt-e+e W$D-r44 ) ktrl- ,WLc. fo\tVte-nz . Nt'Ll tt #\-L0Scwt\i-' Fff J.Lvo Sq','A OF Cgtxtl<,ru'ent Ce, Le* ua Clwu< e-p#un

(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>

J+vv a^^nl- (..y,Qp^'o,--4e *! Irur^r: n4\a* i+ A n-! a?€,4 jn l4\;f ft,'l-^ u(f a4( Lwl*;, fr* -

OrTeA 04rNsrr o! of CA/ Tfrat+ S+ Secnon/ ( l. nJ ,. )Fvr \a"

r/_r/c

r\zL=l

SrJ

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.

,*-l 2 * awo UF 4,^-A- re-fe.."/^to. qs4 - NZ.+-1^,r* .Y^e* fi;a<,'*cn- bqfursyrL * "^) Z; i

tr

L W6 e-4w-ootttYI

+o -{4^n- t -qt'.$e.c*'in" *^J

Sec+rrn

:'e-A-

.y^-Q-

Cfr"LA 1r. f

=::-t-i'::9.-

Page 7: Mechanics of materials

qUc

I*

Ln YL\t

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

L

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; *.'

ZA;Lc I

L

E< J*l*^lg' be*'<s'++-

Yi-qocn ,-A *414e-/J

qx; t

= - la 51"m

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",-

.- ,i.: .,S, , -,,i1i,, ;" j. :! r. : u,

6t

Page 8: Mechanics of materials

W',uA 'd.r ,l ::: t :!iii r,:

q flrA 4*,t^sri ,i

qrr; of {L\a >4a4-*e }€

Lv,r*;-L (-<^f*'J q)c-ur

I

d, 14, {1 t tY1 4A n4.^L

?d" \ Oe ^*4 e*,J-1*r*/1" t+, ' tWzo{,trr*" df s{*t+k) b-l-"ri

Ud"^"''a 9'-L, t '1r-z-

^A -tr4A* $ n-da

h:',;'A,^* ru'"tur 4

.U

tlI

,,

) !./t)

u?-

6N;u*-* be*-<t r t,aJ s. = i= *or+?c: 3"++l.sct= /t-"teczn''

= h_f.a

= +-l "S? : t.4lett,

r.r , , l* o^-l t1" = n

\r ', ', Zt cA Zc- tfL-* $ * f .a

= j- -o"&f = l.i6c",a. - --u

*,d?c =,<= !o,-rfc\. = lr5&+ 0.6r/-

7, *8.Un'

IDFTEdltrn,t+t'0.,O OF Ce+,sr &A4* fi\,Otvt €-^/Tt Sf tN,,L-I

e-jgo

\..; T._L

l-+$n*s

f-.LI,,

ET

T,,

rn

T5

+ Au)*+- At' Gn)t'

nt)_

., A*;a

+lr"'l r'-

{L is (-V+) b+.cr.,u,te i* LiU

nf1^-4,< T'*t

TB*.*Uo -

tI

,

I.-!

L .,-.n -rQ.-- Trl * frWt To] *fro&n)o^lta te1^h^Le tc qia .

Page 9: Mechanics of materials

l5+ fa76, t{ t ciru

dnqo

r^&r4- Tr = &gi*

Ift, f +".

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<

[.r'aa a]d',\4

,i

b4-'Cu,utQ r?

-V{ \, -.rtLt'7'dv

J)*fef-nrr'J4 Tr ua] $F fRocucr . s+ rnl&Ar,c

\ &6'-'+*, , I s,' ..' t A U,' t.'

L

Ir* + Ir,,lL

= Tt,+, 'f A'[J)(x)

=T:. t ft"Gn)(-r)JD t.O

To" *,.

_Lrro

E

Tr*

l":Jw-l of- in-e"'h-q uf

1e cfi'.rvra to i V'. ?rtt- e'

+1"ro o,r,-y it 1.1*trada

lr#*vr is e1,.oJ-.taZQ-rn. t>cc^,r^+-tt-,s

i" ut ,- nlc re veu-.1 Lt t .

C,r,'rc-LLrl , 4

*trJ5, o- = O S'{ &- 1:{- lt!-' a^1

$rr-*aa, r.tnl^ t cr* Tort*, i s nu+ 3r-{n .r., b^r- rn.t,

a*, ':f \r"at*ax '

C*-J*{ Lale*o.,t< l+ he A*-A +{,r4- ( ov*w,,,t-t+,,< 6i,lq,.

M{- ar4- 0i\r{i4 qns,* '

]ova d, = O, 3&+ t

*ld. = te.-, O,3 f+

= ,-1. l6-

:-t-e, !I ?4-------+.. I

^rd-

= (++., - rs+ ) 4-^ (2xxt.tb)

Page 10: Mechanics of materials

Cht"'u<+1 , "44,.a

::l!i-- ': -

't)ac...t'* kl+t.r*lt

TtLa. aft1A Ocar^fi' -l \ ryru1la$-ia $,et-+i4rr1 i s fn'Bre 4'14e'n

U.r)";* occn'f'<) \ #" fas'+-^t-a

t<r,f',ali. . -fWa Iy, e, - - V€

TLr;^ t J irt aArc-P+oazf w1*+1 \+14-{

Nl*+,t v"< y.Q.U^^!eA ; {-l.La-Co.l C-ta,tA>frcv t - 4 t t 16 r^4,

ryyr)t t1Lan l1{.1 auf*& Oiaa,U

t! LDh-u*..

>+t

=

"*n \+ u/ ^^-t ''6*.* * fl'v^ Ywt-t

'

rt 4r

ln Po\,hrrp- At<c, ,,

i rll- - i..!L.:t.t-r1j Tlvu.t, g.1n. crra4\^e-,.1.9:o1il-'---- -.Witfl--- - ----*)Ye,'t.Q, 'i L',

L/-L)'-J.;

-?l

D*r**nr^J*T,trzi 0f g4,t6 ?FTuuru s crF TH€ P*n't wrn-t cetlTg6t ) rl-L

, i*1n1=f)

F)xe) =(') tr)x&): f)

W-) hre s1^,,r.1 6bta,.;

= Tr,,, + r+'@(K) r r;., + i1'6n)(-^)

- 4s. oL + ts,s(+.n)(a,+e) + C + z7(-e,+,)(*1" ib)?).+,1,To.-t

Tg. *.

r \",1c&)^-z'

l$fsN rfq lGSC6'OrfOrpX-r 0LsflrHJ"S O+ lr.J6,<Tr A s fiF T**e (r-, ,-.. - vl lrwst-!.ns *

*on ?.dp 2Is.u-

T*"

I A.o.Ad\p :

l--L+c

t.i

Page 11: Mechanics of materials

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

87t,$ i-aqt.C tz r{t . / +aQ+'?+

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

Ir ao,o g

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

e,

'la:t-

--l

i

-

I

I

Page 12: Mechanics of materials

34. n'3,

\ cay'c

3"o44, fq.f,a-

Tha by* aL.rrt! St4o..t-l

l.6w'a1 1i't"a* ^") Cat Crlt* oi

v/ A+ P"P?-'

bE J,r^*n

P A4ai11LlzA Vn

U*0,

+o

C1

I*;

Aruv") t4\-e- .prA c"6>al qX,,a +O bzI

t\ i^7, ri &-c"s--,-n +h, 0r; e*euyi-lt .

C{l'rec4usn yf hytel*;4ru

2""/4!

b"tl no f,ou.,Uk,

^t( f a*J tls

ll- q Ml [onvful4s7 "t'a Stwi ; 1+rb!4, ,i /.,,*,)n-^ll^Drr^,a wo>4+ .t+u " [r* * ,xr; " *rLu," ;* fitr LLt!(ry { !Sixn or ptJ'* or tn'e'rtua. yo- sr"a,,Lil we rt a) q,nff C)wu< +o /ttu*,,r* ilh!i-t4,o_ L

r*-tu* L ;, frA o,,ta.. ,;*:JZTT*U,W ov4vor wt?+L

f* bcrloru- St4,a.erl. U/:i (. U ru-t ) PLe q.r4 ltrd

7/4!44rt*t . .

Page 13: Mechanics of materials

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: Mechanics of materials

w l*ry Cs^tl be s,i'*vL ,eJSticz

, * qxi **,"tT /;v;/t s<c:hvra U

.*o /fu+.,m *-Ju *,* r/.a' *dcrep^l #4b* r* 6p.t_wr.,4 **lT

q t:o (.u",ol tU b 4rrt.oe a* ,,f-ra 4alry h*; t N€ -

, U*,dr'tc"--/ gfi;,'rt 0') . /ut- ,+t tu< or./-,^r4 (iJ +-..r/+zt- .ft^a..-,h t*a-m;71a* inr/.,stu* &;.{b.? L*^1 e474_4,h?r44.,r',

? 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 (*,^)

A"+4Lls.Y (*+e) * zs(+),

,-8.v + zL

IhLA

tAxL -., IA - COd^

Ut.Jor = A+ t5-

^- L* + =)?*.?-

Zo, = L-L = Zt^33

=3r(63

4uYt

Ao=

It. --__-_U_o__r=..-

-t'--2

?s-r (,o) + z6(+)A&,8 + B6

1'*t r,*'

at-a

5'ge orr't '

h" + r4L

Page 15: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

!,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: Mechanics of materials

{ ..-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: Mechanics of materials

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: Mechanics of materials

{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: Mechanics of materials

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: Mechanics of materials

-

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: Mechanics of materials

-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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

- - 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: Mechanics of materials

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: Mechanics of materials

-

-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: Mechanics of materials

-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: Mechanics of materials

"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: Mechanics of materials

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: Mechanics of materials

^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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

;

'|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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

/(}{-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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-

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: Mechanics of materials

"'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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

/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: Mechanics of materials

,-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: Mechanics of materials

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

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

F^.{ a) d"o

b) Uwr,fYtta

D

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

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

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

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

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

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

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

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

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

0-

tlc ova-el O-p

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

t€ CoirJr Sl^r,t{

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

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

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

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

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

ld.o

\-*t

4fi4^l"y ,.

6

-;--ttu=!nrd,^ ,

,

6p (t,*\ - K

\tA\ 6/14fu+ c^{

5B I

Page 60: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

'., ..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: Mechanics of materials

-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: Mechanics of materials

-: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: Mechanics of materials

-,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: Mechanics of materials

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: Mechanics of materials

/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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

-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: Mechanics of materials

.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: Mechanics of materials

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: Mechanics of materials

': .-.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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

!

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

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: Mechanics of materials

'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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

. ';;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: Mechanics of materials

[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: Mechanics of materials

<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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

[\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: Mechanics of materials

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: Mechanics of materials

: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: Mechanics of materials

-

--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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

( "', 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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

,* 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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

,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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

..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: Mechanics of materials

= 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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

)! >.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: Mechanics of materials

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: Mechanics of materials

\

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

!_-."._...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: Mechanics of materials

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: Mechanics of materials

-

I

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

frn C*)r"i*' et Yt.,'T 'Ytn

r&,.-N^;'

^r^ir= \1,,- =

. f ) ^/i' W, 7t .tttYm*qJ-CFJ

2

=- trxr-o' . :. 4."f{xto*t rn5tWxto6 a ++.4r_nS. .a)

^?, )-c-tr1 '

F** tutaol-+tuea* t iLe rv-a.ra{ til^4 i s ?9., y c,rn

: . Otr,. b ^qzis C*,r,.<-4f a^As *o x- e xz s ind-{a,u,r, UDr. in q s S e?.r*rr€^i* t)l fo. gr_n_

0^-" =I4*=N. 12

,J ^.t

)gx to'

-- -/,('^ 2Q" &X {rJ "

3 ,', rug s.d-4^a q-vy***-Il,L.q( G,La-*'st I

:)

l*5 " + fti.pq.._-

-

'.(ytr^*

2 h]*f-J Lq

0;,o" ' Tt^.l

I

WNW)*l_,

Sft-+r^rl.

)4/+rmvma*un dF ,^l lt /-'r+",'r,q>l.^-

f.-rQ.1",

= O,Ts t<)\= nq = 6")s

Gi,.e-'. Ct= O,\5m

1 rn: jo r<-rum

/ I o/- = zKN /rtIV b= sxro-am

\=ab

i

i

i

:

iiI

I

I

I

tf

i

,]I

tr

= [x- ?x^z(zq")

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

[r: = ru*, n^,r4

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

O(. rc ( 3q

t*ot*s ;nQn^u,

.'L.

e./\\7

r ) rr1,',, j'Y J

i/. t"LI

I

=b l--X=D l*.

* O,3?s tcdm0.}s

= -x2+- a x34's

La+ ;

Page 126: Mechanics of materials

,.5

9.rs

fI *urrJrLf'z E.r

Il-rt

QI",=

.E'fVt u, =

0(x1 3c1

X,3

?x'T(zqS

A,vs *,x x

o'7s

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

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

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

e

I

I

*)

x)

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

+- ,ryf+u-l +-

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

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

3rc

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

{s+ao **]

6(zq)

2 -3x+.5

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

a-

I

-2x I :o. lx=D

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

rn + L-xo'

sK)

* xY L#*,.]

Page 127: Mechanics of materials

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

t ( tu,o"'tJ 3

lO* 2x.

- S<t*r.un r'j

Ar= ttL: z-be

S1^ o..^X

j

= 1"\sxtD-

gir.

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

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

f\\ = l,rSx-dnt.

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

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

lrc =o I n=o.+s

:\\

li= .trl,

t6

72*1.)35

\

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

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

f0"o D,.o

= I ll nr,ro*i

4..=.An

- L'r"s xtO3

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

FrF,, o"r,

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

ry:&lxrro =t0-J

:. {-t^e

0-*er

P orcr^"t4y^,

Af,= lll-ttry xtuD:t iJD

% 7 s% qAwa6L!',

,rlj, turt

Page 128: Mechanics of materials

4*; os/o-6

hluni ni,^,rtr,

stce-L

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

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

U = tr (r,.r')rbp

L<-+

(',0 =

Gs+'

) Vn,

\lt

h/a

: C"jS

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

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

S r:[ut*Luh

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

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

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

DlLr

Fsr

Ge

eF

=l

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

*, = [{,* Srr, +-

poi* q is

Q^ Ss*

*t'l{ n*+t-4

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

ftt**u+7- 6fkra-a

/+ - frae4 {

b.L

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

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

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

.:

A,* = hn=

I.' : O,3f h

-

%___-r'

Page 129: Mechanics of materials

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

O' 6 q 1.,,,j

f lexu- ^l

G. = f\l z, Fs.r

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

^*\

f-+vit+La,r

I

d*rrwtu-t,*-q>bv

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

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

'2 Er = fr*

= fil z, t0rr--- T*V

: 58" SottPq

-r

q

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

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

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

T3

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

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

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

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

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

Fw ru,p s.l-eel

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

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

- qffax

tv,uw.^* W in<

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

z1 f'l *q^l !JL

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

^\,-^J ^

)JI/

q

Y.

"-Yk

!)t)

[f,*fr., *

.q r.i r.J,r1

"vUe-l rirc,'\

rArr(L^-

rr-'(,0-i] 3

ll-

tlrv'-l'r'4

--3 '1:0-,,

b(+J i7-

l?-

//

- o,tsl t

t

C-rl-

,L 14\,

f<c<71C7{-

11

t+L{*{

y_Ls

-lt tcrr2

I?

oa*

[*,

' r'J

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

= rn(+- z,)

;}o Bx w-1

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

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

2a

Page 130: Mechanics of materials

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

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

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

I

q *,8X tO*1

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

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

i*. B xr-o -{

: [1,"12.wrr-q.

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

:'ti' d

f,,u' , o'no

7a) -rl€ ba-,*\ , s

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

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

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

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

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

, e k+=flB

ArQ

6-a

o tft* = l,"\ot Jx,

e?"^/.

Page 131: Mechanics of materials

I

' '- 5) tt','

%x3 1zr,

6Lu Jo

0' 6+ t" t

OLL .?5 \ %+(Jo v

('o

= lZ,tx'l-

LLL

R*=P4=

= 2XoL

: O"z3

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

o.or+?ot\I

Io.o3+7o1t- iA;

/Fr- i

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

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

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

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

=-lilft\r br iF\;-

+) : o"d+ z,t

Qrt"' ' N

Lo**(, o ,

/ l-L

%t

f\4,a,

ulr{'' , /Uoi

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

Stn Cr q* 44^\

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

I -L

Q3,=n'l^

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

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

01 cc . q..

!

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

R*\As\r\--

'. il.il

0,lo4Zot

Page 132: Mechanics of materials

I

I

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

.. 01j (r',

- 0" og+ f" t a*'

,/,

1x<- L,./2

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

J "=o

x

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

llx^J

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

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

l_I _-Oll,= r/t

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

Jo

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

I

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

i__

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

l

1

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

\ ftrce P,

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

m'J ll"<h I

fYLqx.IFt" ,)

Page 133: Mechanics of materials

s,

T

iJ

T

.T

T

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

,1 :

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

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

ltt11 =

h1 -- 4D

)C

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

S.fre-t: tr=fd^^

sic"tl" 1r.ae

lLt = * Ptt

\"1

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

jr* : lno +

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

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

E^- ff I ,:. \

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

-"; -'(,/

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

/ ,.

l

rv1

I,r) '

r")JlL., L-, olc

(t .

(t.'

2tx

zLtT

h

,,I

+

*)u-

\,1. ,c

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

d'x

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

I.^,r*-. e

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

2 h'xaLL\

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

12

--uyz

hr( Et"jut( ULr^nn<

f uti +"vt \LsL,"!_

2C= t L,L-Tr

I

'_tr-j

Ir-i

-L

Page 134: Mechanics of materials

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

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

iLhh,.#.

Page 135: Mechanics of materials

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: Mechanics of materials

->{ 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: Mechanics of materials

It.t]4,..d

I

I*-1

i

.l

I

I.--,

l

)

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

ci +ru < [u-]

Ar^)

' trl

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

),lr^e,r\

,:r",.-

H JII'0oo =-P

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

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

rh;l /01-J- s4

0-f

.1_( t\4x

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

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

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

"^-J

Si..@ {-4,ro'

ta.rr,,/t1 qyi Le

M6=

ha t f D1"atr

cle fg"rrnp1'{.4r1l

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

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

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

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

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

(st

frt;rLF o, *

W=t

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

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

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

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

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

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

Page 138: Mechanics of materials

[{t,,1*.f

:"t-4r T @

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

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

nl3 ?il* ,

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

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

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

/r,rtto*u;

[u

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

N p. \lLa^l\

d4te*\

(Fl -b)IN no,

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

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

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

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

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

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

i*-e Shc*7{+,

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

SWJ+L)

b"-*"

f - a-6,<n)t-

7'- lo r-rn lnrE.t-

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

L= Q = ar4'd = LFcL

[O-: = tbowq'

,^"1f", **

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

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

[ar* | wtivrt-

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

FE

Page 139: Mechanics of materials

Lat

Note :

S6l"'l't"t^'

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

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

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

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

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

* -4.A

\ tt<n" '

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

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

C l,v>-rr1< fr"-

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

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

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

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

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

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

,;ll Ji: x' ; .,

n *|:7

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

t/

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

3 i,t_e fyty

(6\

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

* 7

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

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

t

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

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

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

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

Page 140: Mechanics of materials

fYlroc*, ^ 'sj0*

F/irh) 4L eff.c/+

€.?lti,rqU-* ,l-o

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

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

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

l.17et t- ttp ZF7zFA

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

/Vlr: l.F b

= Nlo,=F G+*)

F

t+ Co nrufl' Tzcd< C*,

Pci* c w;k14 i+s

iL+p

O4,x + e.

{F rn S.ec*ui h.

-D2Cs 2-

l?"")

2F

e.t

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

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

,

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

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

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

--\t--)/

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

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

(**qltJet+

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

Qfl- = 3F. 6oxrJ

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

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

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

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

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

eoc

Page 141: Mechanics of materials

flAfle,

filfo. '=

eFbl?'o t<"t "'

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

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

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

:. . l

6o xnt^'

(r+ (+ -)

fil,e+na,r*< ei1

TLa *wish.^X

17. Ba"),;7

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

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

v" D.

flWra-o^^A , frl *

qc

D

lc

I'tli:2F b

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

3'q-)

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

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

Nl'* = *3F

Qr'u* = F s

Mu]'r*, = 3 Fc *

* tr: 'ryh uuux;

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

* - 6or)

?,o xrt)

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

Page 142: Mechanics of materials

I

-TI

--1I

tlAoi.t"cu'

I

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

[x, 'i

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

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

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

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

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

{vly'tv, .6

7

{v1)uo, =

V-

L

\i

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

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

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

0{>c 1 e.

e/#c*sCon nTcur.e

N.lt = F

Q,O, :

I!lr'ft, =

QJn,

fYUo, s

=

: OoKA

3F . 6o yrl

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

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

t-/-lr.. e__.

=&D,(: o

7o / .|,c=r-

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

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

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

-:

Page 143: Mechanics of materials

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: Mechanics of materials

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

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

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

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

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

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

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

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

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

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

r) ln

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

q) tile 5r,tot{ CLrrx

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

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

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

l++

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

Page 145: Mechanics of materials

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

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

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

' ' J

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

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

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

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

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

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

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

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

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

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

L* uA

S<c;or^

\arr*-a,

(o;";<

",pP\- t/^!,re

" S ti cz ,rtto

h/.e_ J h_all q;<-

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

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

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

4ru4< *-b

L+5

Page 146: Mechanics of materials

N"'Nln

Nx

{Wz

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

4Ft))adrr /*OX4A

l? O x,,r^ €0 x,,rr."I

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

Sri" ce t-h4

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

./laotnz*v,H1". fdL*

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

d+rc. /usih j_e

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

f\r\+ rAJ

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

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

Nno

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

=I

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

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

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

N,.ft ., ffi Sha((

de"*t^, + ewtiLtl

fi\r^ r tl.'

lTL4 Lru*,rn l,f crLot<a

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

La-en"\,\j

U rtt'f.u<.\

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

,0oo

r+ (0"',t<

t'> tt,t,

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

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

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

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

t r\L/ ^^ ^

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

AJ^+'

Page 147: Mechanics of materials

PYft

Tfr,1Y

f(a4,1

0Cn1

.t.

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

:

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

u;^ Ffr ( [r-)

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

0*t+ +7" < td-J

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

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

_ -iD/\ I

u*t Fn \

-.: I:Nnq

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

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

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

32-x lqn,qsx ros

-

a,trt X I box to 6

tL)n a

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

-

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

->'tUa*fttttn , *

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

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

I S+. t y.atu'

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

DF 7t = g'?+rm

t4?

Page 148: Mechanics of materials

-T ln u*

d*rASe Ce,^rl

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

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

t+ x D. 2+13 ),/

+tZrSxtJ3\{riJ

f3oxto)"+ G^,

ffi

,-z.i+x 0.2+z

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

: l{9.x6tu+q

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

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

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

A D-< s %

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

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

poi,,.+ F or F, EFAla

". * z-cr,)

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

lt4d̂

flult

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

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

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

l7ct-;- a0fa^

ffi wm ,-,fi) Ktirn

I r|8

Page 149: Mechanics of materials

.al

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

Po,* *,

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

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

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

T=D

,b tl tia-o,U

6(rt*) =_A&++

r 4{ry,{n^J

ULNt

N{t+

-t P ..ft-J., rr) = f',b(_ - 11e2 _€

4,lP,tc"l, ,, rV€. 's t.c,L[

S.l>cy+tr lazLvtne-t

bt\L"ul2hL "/ =^

D!

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

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

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

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

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

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

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

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

rb; 7t

,* h's"a,

lt,"l fu+,4JJ

n-Ulltar

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

\ = '2,b

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

+ 3(roJ = T3Spil,n

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

f'{,r*eQ

6Qax)

?3s x ro3

l(*xtoL

l+q

Page 150: Mechanics of materials

),

-t

;.i

'-';.1

l

!

l) oin* f\-

--\_2

)=lo;*

LPu

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

Tuax - . {il*"B 53

Tla S*rer sd-are 13 bl qtid .

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

Sl"o^na'u ,

d=

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

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

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

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

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

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

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

A(t,)

6^+ ,tT" < [D-]

150

Page 151: Mechanics of materials

):.)*: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.

<_--_

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

Page 152: Mechanics of materials

do,=

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

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

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

d [e]-- ,-

( o,+ts

("-"

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

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

I5- 7*,t ,

LaX u+

LF-3

V a&,r4

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

Gn, ,o,,,.X

wuwt& bca,-l F,

l,rt, ^,* 1

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

-3

[^* =

._ l6q,Tnvs

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

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

C*nn dil s, uu

/rro 4tvv = l.tll 5uA

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

= ll. Z ,,*,

trb

t'\

,)ail.7

1o

0.

1

x

q\l,q L;vUU

: ll-Zazt t

b,u= /l,6cn"

1^tA \Wt' I .^o

1=

rF,l) D2 a

Llunne fi^-a.

I 6,6 wL-

7,f.9.*n \

,rl 5r.

Page 153: Mechanics of materials

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: Mechanics of materials

fu\l\'t,tqtb Q*

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

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

lo;n*t C) @,La!

(o S a*-,1 T,

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

t^ry

Lt

4I

-fiut slr<,+t

S4"e^y*<

Trf--

Si-1r(e IJ

+D

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

,"rt'.q ,kqa ffffi;

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

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

,/

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

-- Lo.T-

0, zr

{criir. < rirl

' lNna ( co-l

d^o = 1Ta 33z*.

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

= /?. Lol.t ''

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

|o,; * c t,'r

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

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

ll*-1, I c ,3

7r t:F7

t^r"JrJ

{vlrftt ,,"J

oLc- + h o.on

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

fYtz'+-th.

Page 155: Mechanics of materials

\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: Mechanics of materials

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: Mechanics of materials

/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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

+

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: Mechanics of materials

-

.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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

'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: Mechanics of materials

,..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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

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: Mechanics of materials

-.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: Mechanics of materials

^--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: Mechanics of materials

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: Mechanics of materials

-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: Mechanics of materials

lyl^l*'s hu-t*-l 'cw Ca(atlo+'i.l Ge,^t*-*L'rntr O', f Lott n -iI

TkL Jr^t 1Lrb,r*.ct g- q b-en-,2 Carq*-"( L) 4a,.e CgrorA,,,*r('

[narl.,,-? N ,r, {4r,r' |il", a-,-"{ ill+ c o,^ b< . h) lJL,l uan^4td

WL,'s i,l+.or-.,1 1W4-L*JU

A = +-[1 N"," 'u'* u.

''=' LJ ,-' E+r | !r,k'frru 4,.

J', 5t rt

Ggru e tr-$t-tzcCI FoACe G G^J e&*LrLe.p ors ?+nte nel

fvl*rd il,*,4,r.

L,

l

)r o'i qno&lo". b<e,n of eVr_a/ (arqJ4^ a*-J s izo br :, J<.14\e a.(+1^-^l d"* beaa eA &r_riJnnn Jlnt J;cole J4.Le th.rc,^^l ferca facl*-s

^ cl_,nd 0n tu4qc,tu-\l /w;1,r.. u[ b*o,n -^_J C)r* .14n4 ,u I tnjD .f or*,,e.o orS'eul..;onu a4 N< u.te-d +CI li

(*t,;c/^is rt41-* h-t! y), n4qK4 Fo*.t,4Sa'*'c wa,g ry A" I "in *qe gr,.j;n|d rr,! i.

fi'ul n-ht 'Jr', fLqhi,L;^-A- *f- *ho ,.,.1

lrt t-qr gr,J/\,r-.! ru{, /r*) q krt t.-/* f*A41* a br.a /n"J,*l /ot^* 0n n44L ne^)

-JIs

t- | na*f'M.-'),,tLt

= T-{t

NI,,ob ,l\4,r{n ,Mrr^fa c+e^ *4s a.c*i^tX

N, o' , frr"'' , ilrr., ,

+4"e;. M\n?^uh

, ll4^r r, er<

nb.I *4tq

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

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

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

l\4, t t

fs

cUr<

L,

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

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

fl- A^ns* L<. c*'o,.-,-Q /qr-€-^

t-Lq-a'hii*g

[t ++'o "uJ

;)

,\)

,l;)

,r/ J

\l)

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

" tp lnu* ^) cl4* lo

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

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

l+f,i

Page 175: Mechanics of materials

Its

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

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

qlpf Zac'h Cu*

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

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

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

'I

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

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

;Il(_

bl-K-o,

I

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

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

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

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

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

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

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

f:'''cn

/t

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

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

f *--a--rl'

// // /

l,r! ,F Po,+ *

a -T

L*1!,^

CI {"?c ( e.

Nn, , rcrrl N,,

-F,- 4t1 o

r *1tr'

--:--

nI-u

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

-tr= -l

-F=-f

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

= -?O

t+5

Page 176: Mechanics of materials

-.

t

As=

=

=rCALtl

i

22--ow g<t/v'

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

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

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

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

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

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

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

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

111, . luo F,v,,.

[t* = l5 or-dnt

q - e_fi1 , \)= 4,t1-

6', Ir = Ce,uuFa'ct '

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

Stl[utfi's'a

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

Afo k;

=>

+=)

=)

Cgtaf^rci-

0lmnc^tt

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

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

JomilofrJoU

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

(-D Jx r Jre

+o) (-D lx

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

l+6

Page 177: Mechanics of materials

'F*a *Wkrr1"

Wi+ WtelrL-

l-,r (+ ,

V-=

=

-LG:

_r*GTs

t

I nr/ \ (roo) (o) Jr

s f Jo

f r[)luuxl -sorL lo

Uo -- a-o-D

c&

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

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

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

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

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

D<x< I

o(x<l [Yi' * t ub

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

:-50

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

4= \r.ull; b=*u

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

fluia* ft,

ft

Page 178: Mechanics of materials

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

t^l,€ ir"o[ BoLd( -J'{^-- ,, ( *b;ert

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

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

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

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

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

L'na-o,r.

u n r7(-

FuCa *pr*ezu.

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

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

rr -T

lllg8, i Kd*r

..- F>c: - l0x'

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

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

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

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

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

N! ef Por--h,r,r

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

Page 179: Mechanics of materials

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

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

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

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

(,

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

,)"*u,4 dau f-4, h^( skoilb-pr^ Ji.^-, rwenaL^* ffirb(9 ' a--'l

.dha.w'

,''. \, *'c/a'

[Y\t, , t,, ,^

Ut/'1 :7;ll-e-rrt

Lel- u,l d,wn<

g^hL;u-dt .kt-e

C, -- t1^ Gto)xt

4\JL

W4

I

fo,uttutl

*e0*trD

-+o1,0*. * - ku-+r)+

,0, = *tox4

rLi) s,*3

.rr'. u]&+-

/"s [aru-J

Sirrrilr*bo)'"X A"4 ,< 42-i.J-j

4+ v44

t?q

ryt-,, ffm-ph at *\4 L^JI-V e,ulf4hp_a p w+vna,

o*-q /ot',t 2/ro

4-i *fl\" ru,etto J.,

0*tJe

{t4a

(a.n

W6W,t @t' fottrrl r .

fi'J $4{ { ^)-^}

Page 180: Mechanics of materials

-

' i '-e = fLbffi t/s

L'p ilr )e'

, ZLI .. /

-' h * -bX t/S - A-/atQ '?'oirlr

50 r--...-/ "-l'o

;

,r^-fr' -L = k, fl = -i(z+z):q+b |+oh :lLs = t "

Sf^t L'ftil V<-< s@^,;'s w**JThR. ruLo r'S 44A

o1yffi, illa.D-h"r

L-a__--*J

t!4- ur q{A *dt *'-*-o*S

=J-t3

-+

:- 5k)TT,

O.tJ4_ ,yra.{hU

Lu uA spL'+ *a,,-L

Xr^4Un i, 11 +-w-0'

i" ,f M, J,,.: ?b (t,+"1)*,, (.*"r)1.-,t u

.

JIET otu*s)J +

=SfuFI

fi1r)

(rt ,

r '12=* -'1---------------

(r

(*Lu'(LrtaJ + t

*,(ry,-€)

L .HO= o

f\

SPkn)no]b-'--

A*=

Page 181: Mechanics of materials

er------+ z

Ju

"'1-'- -rlQ, = z-r*b= t*+rt

1-= tott-dlm

flfi= ,fot<fi il

F,^f, vttP. L+",-&;q n"y&w SLey< a..* g ,f Wu.

LvtLi+. b<n),*1 tygt,rc,cl *y pL,'*-,! s-t .

!o1"+,";.tu^-r\ Lulu*

I'_-_l

L_=--

@

(,

:tri , [i=A____r___ 'r1l , --rl-4 I a L--->

('- '

"rlf ffho-,ro*)o)i,,J

Nq cl- /o,'ri"*''

A,*=h

l*.'.qfLr ol>

[/Or+'Ur,IIr t, lcrr rr

- I LU)(-tD)cLJ L -E-

= *8o "oJE-T

\or.ttud'aXin's w*t^,,A Cs,uJ b( .,a{-a( W *} &,1k* T*Ut^ -'l-.1'', e)"'&*"u1 aa, for-a.bo1^i ,^go . ftrcsa q u/^dt* {14a

i

T-rt t C,vr*-J b< r{.orr',.{ L*b ir-f-p*r*....,r.J "i. ,.*l*-;;

eTuorJi:rn e"+ d4L!- F rf'* ill , (,vr^^';Jo*.!.. TrM c-o^.+r.r,rJ

qf^Aw is [4s. &'o"J @,"*n;4is ,*Lo^ - I ImrxJr

t, <x (l. - fx'l= *5X^

D (x,(+ *flq.(*rr) + M

= ^O

- A-0:<

.)14 = r

Page 182: Mechanics of materials

l} \,'*('b{g"' f)ti ' fn'bLamt ar'( at{'at.,( ,l^'rllallabtvt-tor Ue .,e.tl,L€' zt*(er ry'- b;"ei"" V*fu,i. dfu!." rc1uJ,l4 e+ A+eN *,"! t-ent'*t'J)

/n!{ Lh "4u o+ ?,-,flu'a

#,ffil.trl

*|ru:Orc fn-4e,r, oat fa*a

t - l'a.,Ll #vf fr/,r^k1

?t-Eh)=hL' :

7['

hJ d= hL2.

h

li = +\L

D

o

D

uw'{.ter1e f+dnr X;-l-1^e* Ca).bat',14q44

u) = *hL

bz o.

uO = @'t-t') t

7

C _ (zLm)(rrra 1

L3

S+a,fr'cA/t lp d LW nr^*+ Wotu,n2

l*UU^" "f *lru's 4f< q/r< Sotde J N r.Jh ut<

4,, =

Asp :c**A 4tvl/€/v\t\*-

La/AA,lic^9 e.rys*iw

t 6,x, +6.x" [x.+{ d, ,, 'r {,*X. + 6.,k. +\. 6r,x, + 5r-\^ r &. x, +

Si rrr4+ Jt@mq'fint in i d;"ic*e,t

X; 4,:,-" 4 /r--*ra!# b&,),^xN4eaUA_r*tZ

Page 183: Mechanics of materials

F

-* ?*erfiu *'Tavrl' . C*u++J.

nt,- lrrtbl,uht

, .: ,.,'

h4 44^-A-o

i*riI

@._;

F= 2-o r*A)lvl = lo KArt

Q = zJYl

[ = 2-tY\

h = +rrt

/ : 9-l l<r': /,rt

A*o

JG_-n*,9Y<-/-i

fl"!

r---T-^I l.mA,'

{e)

I,ar uA cJ-rlrlr

rrbt

sfh'an @)'

r--r---zI I

f77'l

tA77T r7r',

(c)w

(J)

fr-*l Nra' , }rkt, llfrt't a,-1'

drrr,\ *Lai,' f r*/ lu> '

\5gil'w4t'r:l

r) ]<4 t-<s- oF- .t*."-( r,, c[t-'l-+-, rvtu,l n c., (,<)J*J'-J

, Nwu.be,r OF Urlt<tuswrt y4qLh',tttt e.r C&rUtrain*S '/14 =5o IVLa l)rnu.n flunb+, sf L+nl4,cui^# +er Slet*,rC. -/o,(r.t>,)q,rn ft - 3

K = fyl */t'= 5-3 =

a) (Lms,"1 {^e bue. Si"', C-o ltt r- , de Uod

q N c,1 taa_a* +&4 fuite.^Hea-q ar< Varrio,.^.*{'

+"/D C0"t47-o,'Ot in suc)<,r"*.1 b< c,er^p S+qrh'cr,"L{J S+-,\.bL€.

*Xa<u^

Jn r<Jvt^Ga

BS,

Page 184: Mechanics of materials

I-

I-_Jt#

j.Nl::sj

ti1trTi

fr n4- ru

n ?<;ijrtinl {-ha. e/,u,.t'Jate,4l- fl1rcn. -:- "

TW s in uolure.r l\uh'^A q.L( *lLt f6le-n*( ferAs. r,u.r.D *trt.b+tx 9,14a<,r,t tuJ -a

rl lr'*X |-wD fisrtot X; ^-l X^ crt- lt*-</

f Oiw,rs $f y'44v6t/-2.0 C,go', 5.7a7'r'ul::i r4-r7 cl-, *fo .

"fA

ES

+) N< |L'c<'L( w)e C.artue'tu-c-ql u/.uotlt'ez-t er fe,<L 1)L{-{{,,-,' +of 'r-l U rl tcrLaw-'t'L f-ea cn'-,"a i,. ti_ rl-r.,r, )cr-d- L,uus+-c, zry_ S

&, X, +ct*X.+A,r -c&, x, ,- [. x^ +- A.o -D

irr o.Jc"- +D fi.,.. .(1 11,;Kl Lg1unr Cc tq4, C,,-e,ti-s t[, a[ = 6f,, ,, A , , *"- l,l\,-r ,

a ierufA

jt L,Jtruu[ c( l.6a ))€ c€ s sq.,.U fu Csrr s , lou *r^t trnt.]- e_J

fil

VV;i-e,,-r

7tlt

@i,:r

\sl-Jnl:,s-lcr %tW

A.r,

vni* ls+en

I8{

Page 185: Mechanics of materials

i}t*enw\qty '*','$-* fsra i^ ,f*.& Xifi+qy_

' 'L = - Xh+ Rrh= o k*n - ?-lrt = 8>r)

Z lvl* =- o= M+ RJC"+) -F(q*)-I-fi + LhZ"

Ri = -trLq*) - tvr +tfi - ?-h%qyb

r-o(+) -ro +eo - r,r (+"J

2M,+

h+rn

7,h^/^q"+b +

- 3+. s rc,..)cB*.n

=-D

,1 tll\n =

* ?h"9-

+ ?Jn

il

G (q,-a) * (J*R"th-ru+

- ,+" s l(d"

- 7fn = O

= - eo (+)*to -F., tJ^+?+eD

Z Fr : D

= A.'-F- R*'= l?. 5 _ 2, * (*:+,S ) :o

-t.,l+a,-m; ^ q.+r"tln op p.

"*C*r^{nL utu{

Rl, = D

Cq-+', ) = o Rrr = o

t-

Ro,' = i.,

nJ - o

:l

pg+€' rhi'r\a+ie.r\ SP Rco.Cl-u.tn -{lrrCea in um;,- SrS+o* (,

Z{fro =o[.o(a+a1-X,b=D -. n,'"

=

X"4 =o2-

-&jeT

q,+4

= 0--S

g i.aYRL= X,,q

l8s

a"tb = 0-5

Page 186: Mechanics of materials

-

&l=.o ; fuv" l#-s ,, 6.d: t?'5

I

IJ,

Rf"= Qou=r.

*f*bln sl Q1^ -l-''*4 NW*\a-c*

lrlSi"l l4tes< €l/4Aitpttr, uJ(

bl^lrq rrwmp-q* 444q* Wit(

fna-+4^-l {a Cq t c-*LAr-n E, ,

Sha.L( des,jrt -t-h-Q. ruu Ebe A"f . d it+1 l+re shclo<X ir.t , s

6,., [,,.= 4.,.

Nt) vfPor+i ovt

Loafh ['10.(r , K,tnr li1s,[r, r14 l1lrJ"', /11

r-r 0{r<a -Rl * Eol ,.

=?ct)

x*T

]Ir-E

0<x{2-

94x(t

R;x-Ro'h- rr4

= 31"5X *go(+) -LO= i+.sx - 33o

- lxYa

= *10x1-

-hRo, h

=+)C, x:x

A;^ )c

= o,5x

,

lV-trI 0(?c( 9.

*RXx-F'x-1144

: *3?-.sx -rD o R,, .:(

= o,5x

i

lss

Page 187: Mechanics of materials

[4yr , ,<a) /

I

l'v*< i s t1\LlF ultq )L,,or"tl 1$a'*t' r-h C-nLcqLq-*

fu,r,nd't* 'F LUtr\o.-

d, = =,*.').,

/Jl,' t)f l"

a fttoJr, , rc-t; j uL: riui +.,"re

- ,rur,1n*l ,

l(:ler-rlr,),o--lt*, eP t, "t. {, = 6f ut-,1f,

?t*t rrvrinq*r'a',a Rt A,F a^.d A.-rr,rrninqfl'an B an d) l\* Wh iyl,tl*,t t.t

ne,u^A ir: ffiA,p = lt [t:(-ra,.)

(u),)x r ! O sx -zto)(+) dn,-J"C,,,la

= & FTt lit..#*^-{3ro"/^- Vll: lkJc L z a - -rtr- J 3&Ly

)1"1 \

J

tBr

Page 188: Mechanics of materials

:T

f

-

a*, ; *. tJ:(r+s*-aao)9nf-,A.i

= I I lg.+5x ^tLx-- 18.*srt-.ET I a t b

r_- .+.9 L

= r ftr.+t(*) -lbsx*:. tB.+s*rjl%EIuL 3 e z

-t\/ \ t I

sx-roJ p.sx) dx I

.:-J

al( Itt/, J

.,-l -Bso,- =-=I bJ--

-l \r

*a-*^co-Q- e-4qLrch-uv\l/.

(0.s") J* +

S *,t sl,'*.++v7 l-|^iae Co "+f; c e-**s

D

lbP x, + Bx* 1t9L' =- rrq3 3I

Lgx, +- +X" 3sD =-D

V -LAz= Ept = *e"osr)

; = & : ++tKr/\l rt

Lo+ u4 yt-.Hta-n +o 44,_a. -e14.r.,1+tt ,,,* *.ytw"n

Fo..u X.,tu0 X.^-A orfpA 1,..L

(c!7rY1

c

F

zF^ _ A1_

, ol:

qt- P!-sd'w-, 6tyt aa fl^' , K^\ *-A

7h + R,*h =o

7h- X,go - ++.t =

=D= fvl +Xot nL4'*^n1

9s" 1 xl

+ Rou hw) --'F G,w) = oZMn

Page 189: Mechanics of materials

--r

R.'- F (*4'i - M'' X* t '- '7L\q+4

s ao( +)

-

4I

=. -18,5 asr)

zrh.

Rou

T-rNl*tc*r =*(*'=Qrto' = -)Qon=

I%tt.r = -R*hro

- 26.++s K^)

- ?5,1*'S

- 3s.q rt

iC" fa^'(a++) "F d t^ + M - X,4 t X, h * /L" *2-tY1 =-a

- R^h h - rvt +-x* q - Y,h +' 1,h2 + o-A

= - 3L. ++Sk r

Chec,rJ- 1.

ZF'l -D(^'* F -Xr- *. &"'* 3 6"+ts +-2,c * t; o S

\D <r4<*rn'n q h' ea

T-IItlJn, e *(*h * *ts.rKt)

,Qlr, = [1r' = 86.++s xd'

tl&,azs

tn*<"-nnl

tO 1xz-f

t+3 .6 r',tl n

*BS.q(+) * lo t e.os (^) ^4+-t (+) r tbo

O.'\cc\ 2

Page 190: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

!.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: Mechanics of materials

!

-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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

-

[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: Mechanics of materials

--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: Mechanics of materials

-,

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

(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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

= 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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

--;\(,

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: Mechanics of materials

)

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: Mechanics of materials

-. 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: Mechanics of materials

-

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: Mechanics of materials

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: Mechanics of materials

!

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: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

--T

I

^r)

T)

T

{'

(

t

T

r)

7

r

/ -D's6) (et-s6"s)$,E6+ (o'rq

=6r> - so

a= tot-t/"r* = o.&+r:g'srLgoL2" "T -_-o"r- t

/:>S% T1^l44't-€"< hrc ft^o-t-A

1n t iA t<- Ce"-d arl P*n&

v L ----E- a-

l.f"^fi,"? d4^-( la/rPL€ \4y<,4c< eF

Svw +(frt^ c't

13oxr03 :iiii,,o*,ot

O- 6s=

CaLc6tct*fcrn fef fe*, t\ ti +t-a.

o.s+r

= 74.t "A

C*.,ti )L, q

v,'' z w'

X C-) n4 fw< cl',

2)

A-: ? :-, L2-LU-I

TL.,r- 1 V[ :/tLra t.*

tti- Ce

3) l-?'

A=

'fl"L Y1ea,rult At-Lq

T l* n 4 a re-at V il*e t* a+<-q .fwn

(q: Ntg lD[B) ru,'+r" $ = /5. 6 c,n?,

(ut Ng 1(1) t,,ti't't, A-:t5"6c,n\,k* tt+ @)

1.6+,n <

= I [.+e"*.'

Q S S or-*rr-ue--;t ar< fe-rr^-! ' jl

f*r"= 6o.7r.,,"*, Ln,,n= l'1Pr*^

I],t - +3. rc,nf , ('n ,; : 1" ?7c,.

0,?-x e : +0.)-1,1 g x'1o-z

flt r-S g ql*,e L'U bc).,t.e-o.,rr- ), = 7-,c tt t..t41

7 y = 8o d;1-t'L U" = 0, +str-f" ^4{t a *.'u,. 2;u Ju'

Vo'- 0.+-s + &.u,-D.)s)(so-?D.+)6-0 - k)

st. ^lJ-' :[

A: n -U *,$ = O.&oL-O.6ts100 ' E I

Q,=O'e/ ^--"!

O. yot

v*' , ?at+.66 q"

'A 7 s{L, o,e \att.tl Cgn t,' ) t4- 4 +h;.il aV prtocL

rl: O.6gs *o.soL = O-+ +LA

2lKoxto-

t

_!_(!=[tr]

-rl,{-+u

=_;-?\

g"+,+6 xlGorrob ;fr"*, A |L-vrl-tnp**

l-tStxto 3.*^

tS-l,o,L#/'S fer-*l ,t, N!

)

tt (*)2\7

Page 223: Mechanics of materials

-

276"atta*i" p;eth'

+ c,c"ta- o'bt) cqi' 63.q)

40- L-= 0-vLtl'

Q-' 2 to*'

7 5'ao^ +46

F%rrrh qrfit" rh

V'u = lT*-t!t'=P.

y tfr t(.-T t ), ?3

il'= CI"k/

fi= o"v+l^ o"+4LL

A= I : teoxro1- l"+eyLor*oL/''L (fJ D'T1,t x rLo xto L

: t4. q c* \

X, *1"+ ( o . x t- D"lr) Cs-o- +q.. i

go - ?-o

O'7+6 t- O"s,fl i 0"+1+

: +1 "l,..t iy4. L0, = o.& I ^-lut;++. Q= O,+S

= O.+ss.

- \lr-a,"u+ f"tL^s OF # frt,'t aatuy*m^t- i's feu.^l;n N:1&)wi*t A= lg,qa,^o, ]^,". 43.yr.^* , (;^ -- l-++

) -Y-L = Axo.v/t _._Lm,tt /,??X /O-A

This L,nz 2, = T-o

2Y= &o

| 2sar.<1zJl cuaroa

0'''= o'+s

Sr ,i Ce' Lr'u' < '.-e

tu

{t,t*, Lq] = V t'= 0.}ts ^A sLw c-t+tg.. seL+1},, ,. i

44\e,1-oirl< A,l-3 1(g) w f Yh $ = B, "t u-4

N0.lc: lF V'u' *rnu *rn 9tucr,-r q*, CorLc,*u.r< A 4,J C\.ec-t< v,sL,u-t,r*A zsI" uv A<50/". rF A<5'/., d-4r-* €Jte6futrvt-e +"rs+a";-z**fg"-Lc dF q o"r*l {.v.{ C*a-a e**1,^-r1 /t"-r-r. f-n a+t_t-nu^:(_ fs- U-.c(ll""*"n

Page 224: Mechanics of materials

II

t*

I

-l

I

-l

1

I

I

WW^4ttn^", \)U4J4T1 r\U ,v '

I rt Ca,rct v,lltp,r-e- e*t Lotanfz- *tvo ir'pQ*.^ l' S< L4-,r", lU g w tdz't-

Sa,rht , t'*'*o^r) b{ .wis< +, Ll.t'^rl1' to*,no up

J/l^-e. ntLt1,\,UrA' {n,44nA-/' CMz' tr^ t'\);LL qrr';{'e a* +L\P- n<A-l<-!'

$.eCrf.u," ,V g (n, & M,rffi. es" l-i**. h* *norn'tfu' tF t^JP L'nJ u'tQ l t'1^9-

\4r_"- J s<c{r.o,, r1uilazl J-t*i,.rtr w'r s< c*^-J c4rynJ, W JJL'tJ't'c--"

'N1-r-A t4o,^/-e- b4 Q-et'5t'a'v'c'*A Sh'e*-L-r '

frj I ,,b

L* Qrn

L Ng 6,g(s)

l\4ca-et'a! - S.ke^ - Cn 3

til: lhork'P4

F,^--J q1[6s^'ti, tn -( tf]

Cc, L CrtLttt-u.s-,

7:+Sd^^.,tu; .

oF a c'l'r,-,-(- \ Lon..)e"-e'"4

,-L1 vful t>t

w,'yh # ('zyt l*n

A'= tY. q 6 "-' ,

Zo=Jo= 1.6'lnn,

filhtt,^u /o-+'t9/

Tu'= I 8''l u'n+ , k,--;-l't6-) or"'

)-cu{r'u1

(netyt

A): (, th)

{

+uaL

rr=b:

_ --L') | - 2xt&.1 =^*u2LQ'+ fr'Q)")2 Lrr.l ++. 1L(r.tt)

,.fr,*.tr"l (6*4r"*J Re4<+,vtz ,r-e SLrA{l lc-c".t{(

3+-v o- +

6 [. I c--l

-JLI_*J-.--JLl=J+

1=T.J-Ih r n

IJL -lt

l, h t,.,

: J, = 3+' Y 'o^t

2fi'= ?x1-'tb

T*,;

-

5

r+

= 1.It c,rnq._--

2.7+

JO

: l.qSow1,

Page 225: Mechanics of materials

)

,

i

*lI

i

\Jot 95-rr "rQ

Z,t+l xextof l

/b xruL

n vL 2-) ^ - -

/l -4't tfi,\ l'15Y, lO- * I 09,. L

1) ?e*e,r-;,n\,. q.'h\a of L r^i+'^4 SLa"-olo"'.l'l t^,'l"b

= lll,l

Cor( cuut "+c Ot-i+,^c-"-L Lao I ct^-&

[*: q-b)' ;::

P* = (o* ,)) k= 5;k,

Lt.\4 .f as;a'y11'5 fe-,n,-Lo,

S*frr*t,

Stet l 3,

ZlOu,fqI ' lt1- na41

ls'1 "L rcil

l7l 5rci

F".-+--*

0;=tu-l

E f*'st<-(=2-x to'tfq

sF \*felt

fJ = l.?.l6n

f"-n:L>

P*:LPI =

3ro: t-r+(o^6) = l'l gnaq

0; A : lclbxrolx ?-'l 2 xtD-'f -=

bI'16.

11" 5

nJ

I "2-

IL= 2n, t3 =

P= ls rcri

[D-] = Lotvtr4

lDxro-A n4'

,lI

l,

nrr"ri%{ - W,# CP,lel

si_q b,'14'+J W t44aC Le ox U",a-

CDU^r'*(\'

$dLtn,h"ort.

Ja = Jg =J^;IL

:'8t;,

IL

= f3.{!',"-

A-= Ba?\5 ,

D cr"'; =

Page 226: Mechanics of materials

(o r,,frtl'I =-1 ra.x6,

a.) 2= +ILm,i,

= 2,'1x,a-of^ = 2, ey enr,

c O.+xr. tt G\ 6:_: f-r- )r- \,(a (J' v

?.1xrc'L

r)

+)

ln t-?$b ,tt A Q)',g^J* *r,,1, pfu- VA-t{ @ &{-

btl.,{a.}.^ 2r=*O wi*tt Q,=0.61 "J 2"-=Stwi*ttLrb^.foarr;f''n2 +44-aAe U et^aa-a ., r* g1^-c+U f_g CU,le

Q; o"s++ I = o.s;5o -+o.lf,r= q-Lrl

= 0.s9xuO = s"q{vLPq'

s ) Ac+,"-l-t 5 Fe-+r 15\ -l-LL( ^J,T rr-- {): tsxrD}

n (oxrr-)"

2 L"<-z

t(r= 0.s?.

/*'u^l

J,?

N

L= h,EJ" =

1Y1e.1,1-u,a,-t

P. 4orfit &o fi/11-4

s+€/ s ( C"Sl

O<.++"-rtrr^t J4tp &'qn4L/-*

ba*L|** , "l-

,

aF-ke

l,n +h2 ir"t-Ut = o.s

+r,-er\ oIt= r

U,D]

*Slnma

'uPlrrt, t l.'

A_: 40 xto' LLO

= -5xtr3-llr, ^

at

= 5 C.wL^o,> x t(roxn6

12.6

Page 227: Mechanics of materials

,s

+l

0"SxC-

6 -s xro-j

V,,, =I7=,ro

Q'' " '{"

@'Q zr)

/c=-

I

6. l,s xto-5 rrt

O'e4

-|l,v-+ fa-+ t^.q u-g1t {. ) n'*q S< t*'^^-ri cvtt {>ru'"-ch '

l_.,VJn,V*,il \J<*e4+]. ".t ru

= 0'oas.rn

i Td+64

T/"r

V- .:-S

^fc.[ +. Coryt s i ")*+

/1.

q = rfl/\L nin

,ll

. \ I-T ,, L^,\= !7-frE+ =I

! d+vrJ'

4x >xro't

rlil*srr

Tlr.z Bfh'*lu*i rf".lq uP q r S V^etl -r ()

ttrz-rf '-r -' 2Tr-t

re.-lfo'== 2[J*'rA"(*).]

l' &'0,.*r* J r O.ozsn= ); g crrrc

,C.,,+*' ,

L t*: ln l"o. t-L\a f*b,^A d-r,jn, ,

L= 2*fn, 'flitr.l+,n-l . - S+LII jCc.-z)

tN:?tF^rl {444 p/h'nu*,n g +/*"ag{- q @ (li.'-5, Je-u,^.1 la-e_fvve*<-e* t* tnrn-J-^,

/, n qa,(

lu'nr-l-^n czJ C-a.i c^^l-t^l+

J-h! b"h'L,4I PnCZ P^

AL = R$.+*o, Jut= tr.yr*,4ntJs .= lSe) c^^ t,

*44r"* t,v.rr*-"l fiA4<<z-l

,ai<-____-_r_) tX+ i

Page 228: Mechanics of materials

+ (\' * J.')

-l-

+(,sro- rl,r)= lS"6c4

A z'+

Yli4.,(ravt V</L< f\'y\A.

8,D6,*'

r-t -T [tS'.k _Lj-- J2hrg Sl,r-r-U ka-rf-a'

tfyc= 2f I.'* A:(+)"]

LJz

ft'=

L/y'r tn

n/:

+

+

l r1T-* J-entr--

2 x l5e/c

a (a 3"4)

3o +D+6.r

W.

,l/lt'^ -

fn.,l*

T^E :a;

2_il5.t+ x,lxlo

a.x1r' -..-

VL

L ^.,i

fu'-r

= .o'+*N:* -= lT"z68-o6 Xtu-a

Sr-e{/ 3 = e,co lttf c1.

l,- ts.?-

cts t< "0*. L<c{,,"ve"-tD fnr:.l^'cl--t ttjt\ L\);+L\

d-q4 k *tr wtvfe.,-; 6J-)q-"1 W,,a- n.,Zu'^,Q,q. b d,

7-a"<

/ Lo rna.

= 1'1,'l ,

= /urr,

-Tht-<y'-a<, I ccs,,.s/<J 2 J ftg>-n,,,(a,

P* = (a_ ,,)) ft

= 6,o * t.t+(ft.g;)fvte "r

= lZ"I Ki

5 L.c"+l b< ,r r."!,,

€o- vt'-rl g,

4: j tO fu,f sb = t-t+ nn4ra'

Q-= lS"L r-n

6crn

1m

I

. CI-'\_l

, 3=, $,^.

d=.(-=V

^[ F,'^*l *ta

,i'q-J

C"i't zJ V-\L,i uf

Po-

X.lB r

Page 229: Mechanics of materials

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: Mechanics of materials

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: Mechanics of materials

!-

I6Worua,l"la-

Pressure vessels( Symmetrical Thin-Walled Shells)

..!

I

I

l

l

I

I

I

IL-

l

Tfieyeoy[e that getfiue (j in

different from yoLL. lou fiavefttclittt

exanthtatiotts 1yere rtot ct eatedaffit tafr.es to plet tfiat nrarli. Qoocf

Chukw u, Celestine Nkemakol am,

?zr