scad engineering college-ce6402 strength of materials

EGELLOC GNIREENIGNE DACS - SLAIRETAM FO HTGNERTS 2046EC

1

2046EC - SETON SLAIRETAM FO HTGNERTS UN TI – I

YGRENE P ICNIR P SEL

RENE NIARTS GY

revenehW a ob dy si ,deniarts eht ene gr y is ba s debro ni eht ob dy. ehT ygrene hcihw si dob eht ni debrosba y grene niarts sa nwonk si tceffe gniniarts ot eud y. ygrene niarts ehT eht ni derots

b enod krow eht ot lauqe si ydob y dob eht gnihcterts ni daol deilppa eht y.

P OOR F SSERTS

dob citsale na ni decudni sserts ehT y grene niarts mumixam sessessop ti nehw y mret si e sa d oorp sti f rts e ss .

rene niarts rof noisserpxe eht evireD etsyS citsalE raeniL ni yg m llof eht rof s o gniw c .sesa ( )i )elpuoc )ro( tnemom( gnidaoL laruxelF )ii( gnidaol laixA

)i( idaoL laixA ng

L fo rab thgiarts a redisnoc su te L htgne L c mrofinu gnivah , r sso - .A aera lanoitces I laixa na f audarg deilppa si P daol noitamrofed a seogrednu rab eht fi dna ,yll nod krow eht ,∆ e rots , e d

grene niarts sa y dob eht ni )U( y b deilpitlum )P 2/1( ecrof egareva ot lauqe eb lliw , y eht de of r noitam .∆

uhT s U = P ½ . ∆ B tu ∆ = PL EA /

U = P ½ . PL EA/ = P2 L EA2 / -- ------- - )i(

I ht revewoh ,f e h rab a nel fo llams a redisnoc ,noitces ssorc fo aera elbairav s g aera dna xd ht es ssorc fo c xA noit . grene niarts ehT y nel fo tnemele llams siht ni derots Ud g ,eb lliw xd ht

fr uqe mo a )i( noit

P2 d x Ud = ----- - ---

2Ax E niarts latot ehT grene y b deniatbo eb nac U y nitargetni g htgnel eht revo noisserpxe evoba eht

of b eht ra .

P 2dx 2AxE

L

ò 0

U =

www.vidyarthiplus.com



2

( )ii F ( gnidaoL laruxel M ) elpuoc ro tnemo

L nel fo rebmem a redisnoc won su te g ht L nidneb mrofinu ot detcejbus g M tnemom . nel fo tnemele na redisnoC g d tel dna xd ht i nahc eht eb g ht ni e e ht fo epols e e mel e ot eud tn

a ilpp e mom d e .M tn llaudarg deilppa si M fI y grene niarts eht , y lliw tnemele llams eht ni derots be

Ud = ½ dM i B tu

di d -- --- - = -- -- - d( y d = )xd/ 2y d/ 2 IE/M = x

dx dx M

di = ---- --- dx EI

ecneH M( M ½ = Ud / xd )IE

M( = 2 E2/ I xd )

I gnitargetn

2

U =

CNEILISER E ehT ecneiliser si denifed sa eht icapac ty fo a ts r denia ob dy rof gniod row k no eht omer v la fo niniarts eht g f cro e . grene niarts latot ehT y dob a ni derots y lnommoc si y ecneiliser sa nwonk .

OORP F ECNEILISER

titnauq eht sa denifed si ecneiliser foorp ehT y grene niarts fo y dob a ni derots y deniarts nehw le ot pu a its c grene niarts mumixam ehT .timil y rots e dob a ni d y ecneiliser foorp sa nwonk si .

O SULUDOM F ECNEILISER

I ed si t f r foorp eht sa deni e ep lairetam a fo ecneilis r .emulov tinu

oorP f er neilis ce fo suludoM er neilis ce = ----- - -------------

o emuloV f dob eht y

owT m lana rof sdohte y tedni yllacitats eht gniz erm .serutcurts etani

L

ò 0

M xd 2EI




3

a. muirbiliuqe( dohtem tnemecalpsiD fits )ro( dohtem fne feoc ss f ici e tn dohtem

tilibitapmoc( dohtem ecroF.b y tilibixelf )ro( dohtem y )dohtem tneiciffeoc

eroeht tsrif s’onailgitsaC m eroehT dnoces m. F oehT tsri rem.

I b desuac noitcelfed eht taht setats t y na y ap eht ot lauqe si ecrof lanretxe laitr evitavired o eht f ne niarts e gr y r htiw e ecrof taht ot tceps .

oehT dnoceS er m I “ taht setats t I grene niarts latot eht si U f y na rednu muirbiliuqe ni krow emarf a ni pu derots

am sti ;ecrof lanretxe g awla si edutin y muminim a s .

tsrif s’onailgitsaC eroeht m: tI setats taht eht tcelfed i no desuac by any ex ret n la ecrof si qe u la ot eht laitrap evitavired

fo eht niarts rene gy tiw h pser e tc ot taht ecrof . A eg n dezilare etats m tne of eht meroeht si sa :swollof

“ fI ereht si any citsale s sy t me ni rbiliuqe ium rednu eht noitca fo a tes fo a of r ec s

W1 , W2 , W 3 ………… .Wn stnemecalpsid gnidnopserroc dna δ1 , δ ,2 δ3… .……… δn a dna tes fo stnemom M 1 M , 2 M , 3 ……… Mn rroc dna e nidnops g Φ snoitator 1 Φ , 2 , Φ3 …, … .. Φn , neht eht laitrap evitavired of eht latot niarts ene gr y U htiw tcepser ot any eno fo eht rof c se or

stnemom nekat laudividni ly dluow dleiy sti co rr gnidnopse tnemecalpsid s ni sti noitcerid fo ac ”.snoit

mehtam desserpxE a it ac ll y,

------------- i( )

-- --- - --- - --- )ii(

redisnoC na citsale ydob sa wohs ni gif detcejbus ot sdaol W ,1 W2, W3 … …… e ct . hcae deilppa tnednepedni l .y teL eht ydob eb pus p detro ta ,A B .cte ehT snoitcaer RA R, B cte

od ton krow elihw eht body d smrofe suaceb e eht egnih noitcaer si if x de dna tonnac om ve dna( erofereht eht krow enod si )orez dna eht rellor tcaer i no si ep r ralucidnep ot eht

stnemecalpsid fo eht .rellor imussA gn taht eht lairetam swollof eht ekooH s‟ ,wal eht stnemecalpsid fo eht stniop fo gnidaol lliw eb il n rae tcnuf i sno of ht e aol ds dna eht irp n pic l se

of pus e .dloh lliw noitisopr L δ te 1 d eht eb cte ………3δ ,2δ , e noitcerid eht ni cte ,3 ,2 ,1 stniop fo snoitcelf eht fo

sdaol a ht t ese stniop . ig neht si U ygrene niarts latot ehT v ne b y

¶U ¶W1

¶U ¶M 1

Pr :foo

= d1

= f1




4

U = W( ½ 1δ1 + W2 δ2 + ).……… -- --- - --- )iii(

L W daol eht te 1 cni eb r b desae y Wd tnuoma na ,1 etfa r eb evah sdaol eht e ilppa n ed . uD e ot ,siht ereht lliw eb llams segnahc ni eht d ofe itamr on fo eht ob d ,y dna eht ts r nia ene rgy liw l eb

desaercni thgils ly by na tnuoma .Ud isserpxe ng t sih llams aercni se as eht etar of chan eg of U W ot tcepser htiw 1 semit Wd 1 rts wen eht , a ne ni e gr y eb lliw

-- --- - --- )vi(

nO eht noitpmussa taht eht elpicnirp fo noitisoprepus ,seilppa eht lanif niarts rene gy eod s ton dneped nopu eht redro ni hcihw eht rof ces era .deilppa ecneH nimussa g taht Wd 1 si

gnitca no eht ob d ,y roirp ot eht noitacilppa fo 1W , W2, W3 ,cte……… eht snoitcelfed lliw eb yletinifni llams na d eht idnopserroc ng ts r nia e en rgy fo ht e se dnoc ro d re nac be ne .detcelg

woN nehw 1W , W2, W3 ,cte……… era deilppa ( htiw Wd 1 llits gnitca laitini l ,)y eht stniop ,1 ,2 3 te c liw l vom e orht ug δ h 1, ,2δ ……3δ … .cte ni eht id r noitce fo seht e f secro dna ht e iarts n

ygrene lliw eb ig v ne sa .evoba euD ot eht lppa ci noita fo W1, sedir orht u hg a sid tan ec δ1 dna lanretxe eht secudorp Ud tnemercni krow = dW1 . δ1 niarts eht ecneH . ene gr ,y wh ne eht sdaol

era deilppa si

U+ Wd 1.δ1 ----- - --- -- )v(

lanif eht ecniS grene niarts y b si y .)v( & )vi( gnitauqe

U+ Wd 1.δ1= U + xdW1 ¶W1

¶U ¶W1

p eht sevorp hcihW r noitropo . lralimiS y Φ taht devorp eb nac ti 1= .

¶M 1

aeb fo noitcelfeD m ht tsrif s’onailgitsac yb s eorem:

I rebmem a f c ra r rof laixa na sei c rene eht e g erots sei b nevig si d y P 2dx 2AxE

fo noitcnuf eht si dna rebmem eht ni ecrof laixa eht si P ,noisserpxe evoba eht nI lanretxe daol W1, W2 W, 3 e ct . d eht etupmoc oT e elf c δ noit 1 W fo noitcerid eht ni 1

δ1= = dx

¶W1 0 AE ¶W1

grene niarts eht fI y neb ot eud si daol laixa ot eud ton dna gnid M xd 2EI

¶U ¶W1

xdW1 U +

¶U

δ1=

¶U

L

ò 0

= U

¶U P ¶p L

ò

L

ò 0

2

U =




5

δ1= = M

¶W1 0 ¶W1 EI

I nitca si daol on f g if ,derised si noitcelfed erehw tniop eht ta c ilppa si W daol suoitit e d a ht t e hw noitcerid eht ni tniop ere elfed eht c r si noit e riuq ed . tfa nehT e ffid r e nitaitner g tub rofeb e

etni g nitar g orez ot tes si daol suoititcif eht . sihT m suoititcif eht sa nwonk semitemos si dohte daol dohtem . I eht f r Φ noitato 1 r si e riuq e ht ni d e rid ec o noit f M1 .

¶M xd ¶M1 EI

sdne ta epols eht dna noitcelfed lartnec eht etaluclaC ppus ylpmis a fo o gniyrrac maeb detr htgnel tinu /w LDU a o ht rev e naps elohw .

loS u :noit

elfed lartneC )a c :noit

nitca si daol tniop on ecniS g lppa ,deriuqer si noitcelfed eht erehw retnec eht ta y eht t neht ,W daol suoititcif h liw B dna A ta noitcaer e W( l L )2/W + 2/

↑ hcae .

δc= = ¶ W ¶W EI

es a redisnoC c cnatsid a ta noit e x f mor A. B nidne g mom e ,x ta tn

M= æ + - è ø

¶M x

2

d =

nittuP g W= ,0

2

d =

4

= ø0

= M 0

L

ò Φ1= ¶U ¶M 1

¶U ¶M xd L

ò 0

wL W xw ç ÷

2 2 2 öx

2

= ¶x 2

l

ç

2 EI 2

EI

æ æ wL W 2

ò 0

l 2

ò 0

ö xw x ÷ öx -

ø

ç ç è

æ ç æ Lw x 3

ç ç è æ æ

ç ç 21 è

ç è 2

÷ ç è 2 +

2 2 2 ÷

è

wL

ø c xd

wx x ÷ ÷ 2 2 ø

ö ÷ xd xö - ø c

l

ö÷ 2

ø ÷÷ ÷

ö 2 EI

- 61

wx

¶U ¶M xd L

ò




6

5 83 4

polS )b e ta e sdn

o oT as ,A dne eht ta epols eht niatb y lppa y gif ni nwohs sa A tnemom snoitcirf a . ehT

snoitcaer a A t a dn B b lliw e - ÷ ç ÷ è ø è ø

nirusaeM g w ,b morf x e ge t

. xD --------------------- - --- - --- - -- 2

0

nemom eht si xM erehW t tsid tniop a ta morf x tna iro eht g o noitcnuf a si )B ,ei( ni f .M

Mx = æ + x -

è ø

= in 2

wl m Wx 2 ç ÷

2 l 2

nittuP g M= 0

fa =

L

f A = û 0

wL3 24EI

oehT dnoces s’onailgitsaC eht evorp dna etatS rem.

noces s’onailgitsaC d the ero m:

tI setats taht eht niarts ygrene fo a ylraenil el citsa metsys taht si laitini ly deniartsnu lliw ah v e derots ygrene niarts ssel i jbus nehw ti n e ot detc a t daol lato s aht metsy n evah dluow ti i erew ti f

se fl - rts a .deni

d = c wl EI

wl m wl m ç na d +

2 l 2 l æ ö æ ö

1 l

EI Mx

¶Mx ¶M

ò

¶u ¶m

= f = A

wl m Wx 2 ç ÷

2 l 2 ö

¶Mx x ¶m l

æ ö l

ò 0

+ x - è ø

1 A =

EI 2/X xD f

WX 2 x 2 l

1 Ei

1

l

ò 0

wl 2

x - dx

xw é xw 6

4 ù 3

- ú ê ë 8L EI

fA =

4




7

¶u ¶t

roF ex ,elpma fi l si llams niarts )ro( alpsid c tneme , htiw in eht el tsa ic limit ni the rid e noitc fo crof tnadnuder eht e ,T

= l

¶t l 0= nehw eht nuder d tna stroppus od ton dleiy )ro( wh ne ht e er is on ni iti la cal k fo tif ni eht re dnud a m tn e srebm . Pr :foo

redisnoC a er dund tna emarf sa nwohs ni fi i.g n hcihw Fc si a r dnude a tn rebmem of oeg m lacirte htgnel L. eL t eht lautca htgnel fo eht rebmem cF eb (L- l l

fo 2F.tif C stneserper suht eht lautca htgnel L( - l ) fo eht .rebmem nehW ti si dettif ot eht ,ssurt liw rebmem eht taht hcus dellup eb ot evah l F 2 a dn F c dicnio e .

Ac nidroc g ekooH ot wal s‟

F2 F1 = De of r = noitam = ( orppa x)

AE AE cudni )elisnet( ecrof eht si T erehW e ht ni d e bmem er.

Henc F e F1=FF2-F1 F2

-------------------------------- - --- ) i (

L ebmem eht te r F devomer eb c a t ta deilppa T ecrof elisnet a redisnoc dn oc eh rne sr F a .C dn

FF 1 r = ela vit e de elf c o noit f F a C dn

= -------------------------------------- - --- ) ii ( ¶T

Accor nid g c ot a its g il a on oeht tsrif s‟ er rehw m e U1 grene niarts eht si y rf elohw eht fo a tpecxe em

m eht fo taht e ebm r cF .

nitauqE g a )i( teg ew )ii( dn ¶u1 TL ¶T AE

o( r)

= 0

¶u

,) gnieb eht laitini kcal

T (l - l) TL

TL AE

l =

¶u1

l = --




8

+ = l ---------------------- - ) iii (

AE

grene niarts oT y si T ecrof a ot eud cF rebmem eht ni derots

UFC = T ½ . =

AE 2AE

¶U TL

v eht etutitsbuS a o eul f

+ = l o( r) = l

U =U nehW 1 .cF U + ,tif fo kcal laitini on si ereht fI l ecneh dna 0=

oN te:

)i onailgitsaC s‟ meroeht fo muminim niarts rene gy si desu for eht rof anal sisy fo lacitats l y ed eht fi,senart latrop sdna maeb etanimretedni g fo eer r cnadnude y owt naht erom ton si .

)ii fI the de eerg of nadnuder cy si om re aht n t ,ow eht pols e fed l tce i no ohtem d ro ht e om m tne si dohtem noitubirtsid m tneinevnoc ero .

A maeb BA fo naps sim3 f dexi ta htob eht sdne dna seirrac a tniop daol of 9 NK ta C d tnatsi m1

morf .A ehT .I.O.M fo eht noitrop CA fo eht eb am si I2 dna taht fo noitrop BC si .I etaluclac eht exif dne d m mo e r dna stn eac snoit .

noituloS :

ra erehT e uof r M snwonknu a aR , , M b dna Rb nO. ly owt e uq a itats fo snoit c are ava li a elb )ei( = 0 dna = 0

nimretedni eerged dnoces fo si melborp sihT acy.

F soohc tsri e MA a M dn B a r s e dnud a .tn

¶u1 ¶T

TL T 2L

AE

)iii( ni ew get

CF = ¶T TL AE

¶u' ¶UFC ¶U ¶T ¶T ¶T

¶U ¶T

= 0

åv åM

TL




9

Mx ¶M ----- - --- -- )1(

----- - --- - --- )2( A

)1 For op r :CA noit

nikaT g o eht sa A rig ni

Mx = -MA + RA x

¶M ¶M ¶RA ¶M

M .O. I = 2I L m1 ot 0 :x fo stimi C 1

Hen ec xd = dx

2 3

ç ÷ 2 3

æ ö è ø

M ¶M (- MA + R x () -1)dx

A

æ ö è ø

nikaT ,BC noitrop roF g A a iro eht s g evah ew ni

- M + R A X - 9(X -1)

= x; = -1 A

.O.M I = I L m 3 ot 1 : x fo stimi

ò x xd ¶U BA δA= = 0 =

¶RA EI ¶RA

B

ò A

¶U AB M ¶M IE ¶M

θA= = 0 = ¶M

x x xd A

x x = x; = -1 A

ò ò A

x x M ¶M (- MA + RA x)x EI ¶RA 0 2EI

+ A æ ö ç ÷ è ø

RA M ç - ÷

3 2 A

- M (1) RA(1) 1 2EI

1 2EI

C 1 x x A

nA d EI ¶RA 0 2EI

ò ò

=

=

xd =

= RA(1) 2

ç ÷ M (1) - = è ø

A 2 æ ö ç ÷

RA ç - ÷ 2

M 1

2EI 1

2EI A

A M = x

¶M ¶M x x

¶RA ¶M




01

Hence

M ¶M (- MA + RA x - x(9 - )1 )x EI ¶RA 1 EI

RA - 24 ù û

dnA

M ¶M (- MA + R x - x(9 - )1 )-1 EI ¶M EI

= [2M - 4RA +18]

)1( ni seulav eseht sbuS & 2( ) we get

= 0

A ê ú ê- A

3 ú

80.2 – MA = 88.9

= 0

A

1 M RA 1 2EI 1 2 EI

MA – R7.1 A = - 2.7

nivloS g 3( ) & ( )4

MA = K 8.4 N – M (a i noitcerid demuss )tcerroc s RA NK 50.7 =

B 3

ò ò C

x x xd = dx

= 1 EI

62 3 ê- 4M A +

é ë ú

ò ò C 1

B 3 x x A

A

xd = dx

1 EI A

¶U BA

¶RA

é ù é EI 3 2 EI 1 RA M 1 26

ë û ë û Þ - + 4M + RA - 42ù = 0

)3(

----- - --- - ---- )4(

¶U BA

A A

ê ú Þ - + [2M - 4RA + 81 ] = 0

¶M

é ù ë û




11

oT dnif MB, ekat emom n st ta B, na d ppa ly ht e co noitidn = 0 .ereht gnikaT esiwkcolc evitisop sa tnemom a vitagen sa tnemom esiwkcolcitna dn e . nikaT g MB w ,esiwkcolc e ha ev

MB – MA R= A 3( ) – 0 = 2x9

MB – 8.4 + ( )3 x50.7 - 81 = 0

MB = 56.1 NK – oc si noitcerid demussa( m rr ce )t

åV = 0 rof eht elohw .emarf R B = 9 – RA = 9- 50.7 = NK 59.1

dna noitcelfed eht enimreted ,meroehT tsriF s’onailgitsaC gnisU fo noitator t eh

fo A dne gnignahrevo t aeb eh m daol ed.

:loS atoR t A fo noi :

RB x L = -M

RB = - L/M

RB = /M L ( ̄ )

& RC = /M L ( � )

B

ò M . ¶M

. xd ____________ )1( C

na roF y = x ot 0 = x ).e.i( B dna A neewteb ,A morf x tnatsid tniop L 3/

Mx = M ; dna = 1 ________ ( )2

¶M

na roF y = x ot 0 = x ).e.i( B dna C neewteb ,C morf x tnatsid tniop L

Mx = ( /M L x ) ; dna

)2( sbuS & ( ni )3 ( )1

R dnif oT B pA lp y

¶U 1 B

ò A

¶M 1 EI

q = = ¶M EI A

¶M x M . dx +

¶M x x

x

¶M x

x x L

¶M ________ )3( =

¶M

åM




21

L / 3 L

ò ò ç

ML ML 3EI 3EI

= ( siwkcolc e) 3EI

elfeD )b c A fo noit : lppa ,A ta noitcelfed eht dnif oT y sa noitcerid drawpu ni ,A ta W daol suoititcif a

if ni nwohs g .

RB Lx = -(M + WL) 3

RB = -(M + WL) 3

RC = (M + WL)

d = =

¶W EI

= x dna A ta 0 = x ,BA noitrop eht roF L B ta 3/

Mx = M + Wx

= x

= x dna C ta 0 = x ,BC noitrop eht roF L B ta

æ ö

x

x 3

L / 3 L

ò (M +Wx)x + IE ò ç M +

3LW ö

L .

3xd

0 0

M A

ö x÷ dx

ø

x L

æ q = = M 1( ).dx + ¶U 1 1 ÷

0 0 è ¶M IE EI L ç

= +

2ML

4

4 1 L

1 L

RB = (M + WL) 3

4 (¯)

(�) ¶M

x

1 3

B

ò M ¶W

+ A

x

1 L

¶M x ¶U 1 1 EI

M . xd ¶W x

B

ò C

A

¶M x

¶W

1 1 ç ÷

8 L M = M + WL .x

è ø ¶M x =

A

¶W

d = 1 1 1 x x

÷ è ø æ

EI




31

L / 3 L

ò (Mx) xd + IE ò ç

3L ÷

0 0

M x2 L M x3

L EI 2 3EI 3

ML2 ML2 18EI 9EI

ML2 6EI

KROW LAUTRIV FO ELPICNIRP

I b erutcurts a no enodkrow eht taht setats t y grene lanretni eht ot lauqe si sdaol lanretxe y derots urts a ni c ( erut eU = Ui)

sdaol lanretni fo krow = sdaol lanretxe fo kroW

RENE NIARTS G O DOR A NI DEROTS Y F NEL G YTIDIGIR LAIXA DNA L HT A OT E

LAIXA NA F P ECRO grene niartS y derots

P2 L =U --- - ----

EA2 TNIOJ EHT GNITUPMOC ROF SDOHTEM SUOIRAV EHT ETATS ED F FO NOITCEL

A P RE F TCE F EMAR

1. dohtem daoL tinU ehT 2. De lf ec b noit y Ca its g il a on ‟ s F roehT tsri em 3. tolliW : dohtem lacihparG – rhoM D gai ram

ED EHT ETATS F O NOITCEL F TNIOJ EHT NOITAMROFED RAENIL OT EUD

n δv Σ = U x ∆

1

n δH = Σ U‟ x ∆

1

A

nittuP g W = 0

d =

æ ö ç ÷ xd

è ø

1 1 Mx 2

EI

d = ( )0 / 3 + ( )0 A

d = + A

A d =




41

PL ∆ = -- - --- - --

eA

U= noitcelfed lacitrev U =‟ latnoziroh noitcelfed

O NOITCELFED EHT F MET OT EUD TNIOJ P NOITAIRAV ERUTARE

n δ = Σ U A X

1 = U1∆1 + U2 ∆2 + …… ……+ Un ∆n

( htgnel ni egnahc eht fI esoht rof ∆.U tcudorp eht ,orez si rebmem niatrec fo )∆ me ebm rs detutitsbus eb lliw a eht ni orez s a uqe evob a noit .

O NOITCELFED EHT ETATS F CAL OT EUD TNIOJ A K OF F TI

n = δ U Σ ∆

1 = U1∆1 + U2 ∆2 + …… ……+ Un ∆n

I lno si ereht f y kcal gnivah rebmem eno o tif f ∆1 d eht , e lf ce o noit f a lucitrap a eb lliw tnioj r e uq a t l o U1∆1.

nip eht fo C tniop eht fo stnemecalpsid latnoziroh dna lacitrev eht enimreteD - arf detnioj m e gif ni nwohs . mqs 001 si BA fo aera lanoitces ssorc ehT m A fo dna C 051 CB dna mm2 each .

01 x 2 =E 5 m/N m2 . daol tinu yB( m )dohte :loS

a lacitrev ehT nevig era C tnioj eht fo snoitcelfed latnoziroh dn b y

d V =

d =

tS )A r daoL lanretxE ot eud sesse i :gn 32 + 42 = 5m

R cae :noit RA = - 4/3 RB 4/3 =

θ niS = ;6.0 = 5/3 soC θ = 8.0 = 5/4

nivloseR g llacitrev y ew ,C tnioj eht ta g te P = 6 A C c + θ so P CB nis θ

llatnoziroh gnivloseR y teg ew ,C tnioj eht ta

P CA soc θ = P CB nis θ; PA C P = CB

P Lu å AE

H å uP ' L AE

CA =




51

P CA P + θ nis BC θ nis = 6 P 2 A C θ nis = 6

P CA 6.0 x 2/6 = θ nis/6 = = K 5 N )noisnet(

P CA = P CB )noisnet( NK 5 =

llatnoziroh gnivloseR y w ,C tnioj eht ta e g te

PA B = PAC θ soc PAB = 5 soc θ ; PAB .0 x 5 = 8

P BA )pmoc( NK 4 =

v tinu ot eud sessertS )B e itr c :C ta daol la lppA y C ta daol lacitrev tinu . rtS ehT e e ni sess a o naht 6/1 eb lliw rebmem hc f esoht

ud deniatbo e etxe ot rna ol l a .d u CA = uBC 5 = / 6 u AB = -4 / 6 = -2 / 3

:C ta daol latnoziroh tinu ot eud sessertS )C

wot daol latnoziroh eht emussA a if ni nwohs sa tfel sdr g.

nivloseR g llacitrev y ew ,C tnioj eht ta g te (u AC )' is n q = (u BC )' is nq \

(u AC )'= (u BC )' nivloseR g llatnoziroh y w ,C tnioj eht ta e g te

(u BC )'c so q + (u AC )'c so q = 1 (u BC )'c so q + (u BC )'c so q = 1 2uCB 'c so q = 1

= 5 8 / KN ( oisnet n)

\u AC '= -5 / 8KN u AC '= 5 8 / KN ( moc p)

llatnoziroh gnivloseR y tnioj eht ta B ew , g te

u BA '= -u CB 'c so q u BA '= -5 / 8x0 8. = - .0 5KN u BA '= .0 5KN ( moc p)

1 2c so q

1 2x 8.0

uCB '= =




61

ebmeM r L ne g (ht L) Area (P K )N U )Nk( UP L A/ U NK(‟ ) UP /L‟ A

mm )mm( 2 BA 008 0 01 0 -4 - 3/2 /046 3 - 2/1 061 CB 005 0 51 0 5 /5 6 1/0052 8 8/5 42/0052 AC 005 0 51 0 5 /5 6 1/0052 8 - 8/5 42/0052

E = 2 X 01 5 mm/n 2= 002 m/NK 2

= 2.45mm

dh = = = 0.8mm AE 200

ehT marf e ohs wn ni .gif stsisnoC fo ruof slenap hcae 25m wid ,e and eht sorc s es c noit al

saera fo eht mem b re era hcus taht , nehw eht rf a em ca rr sei lauqe sdaol ta eht pan le iop n st fo eht ol w re ,drohc eht sserts ni lla eht t oisne n memb re s si f n m/ m2 dna eht sserts in la l eht

nosirapmoc srebmem fo 8.0 f mm/N 2 imreteD. ne eht seulav of f if eht oitar of eht xam imum /1 si naps ot noitcelfed 9 kaT 00 e 2 =E 01 x 0. 5 N/mm2.

:loS ehT pot drohc mem b sre lliw eb ni moc p noisser a dn eht mottob hc o dr me ,srebm ve ,slacitr

dna slanogaid lliw eb ni .noisnet euD ot s rtemmy i lac idaol n ,g eht mumixam noitcelfed cco u sr ta C. ppA ly tinu lo da at C ot fi dn u ni la l eht mem be .sr llA eht rebmem s ah ve be ne ebmun r de 1, ,2 3… .. b ,.cte y eht r u elu 8 01u = u = 21 0 = .

Reac R noit A = R B = 2/1

θ = soc ; º54 θ nis = θ =

( moc p)

u3 = u7 cosq = . = = u4 ( oisnet n)

2

u9 = =

,oslA u7 c so q + u9 c so q = u1

= 0.1 ( moc p)

å å

194 002

dv = = AE P lu

pu 'l 061

1 2

RA \u7 = = is nq

2

2

2 1 1 2 2 ( oisnet n) 2 u4

c so q 2

2 1 x 2 1 2

x 2 2 2

+ u1 =




71

U PUL - .1 0 + F0002 + 2/1 + F0521 + 2/1 + F0521

- 2( ) 5.0 /2 + F0002 0 0 )2(+ 5.0 2/ + F0052

muS : + F0009

= = 0.0 9 F mm

x 00001 = mm 9

He )ro( 9/001 = F 90.0 ecn F mm/N 5.321 = )90.0 x 9(/001 = 2.

teD enimre eht v tre i lac noitcelfed of eht tnioj C of eht f emar sho nw in f .gi eud ot repmet a erut esir fo º06 F ni eht reppu ch ro ds o .yln ehT tneiciffeoc fo noisnapxe = 0.6 x 10-6

p re º1 F 01 x 2 = E dna 6 c/ gk m2 .

loS : I fo rebmem hcae fo htgnel ni esaercn = drohc reppu eht L t α

x 004 = 6 01 x -6 06 x mc 441.0 =

b nevig si C fo noitcelfed lacitrev ehT y

åuD lppa ,u dnif oT y C ta daol lacitrev tinu . cniS e nel ni egnahc eht g ( ht lno srucco )∆ y eht ni

lno srebmem eseht ni sesserts ,srebmem drohc pot eerht y en e b d e tuo dnuof .

21/4 = A ta noitcaeR = 3/1 3/2 = 21/8 = B ta noitcaeR

nissaP g nittuc noitces a g nikat dna ,4 dna 1 srebmem g g ew ,D ta stnemom et

U1 = ( 3/1 )4 x 3/1 = ( 9/4 c )pmo

nemom gnikat dna 9 dna 3 srebmem gnittuc noitces a gnissap ,ylralimiS eg ew ,C ta st t

m )L( htgneL m mm/N( P 2) rebmeM 1 3 4 7 8 9

δC =

- 8.0 F F+ F+

- F8.0 F+ F+

0052 0052 0052

0052 )2( 5.0

0052 )2(0052 .0 5

PUL 009 0 + 2 E 2x105

n

å 1

1 1 09 0 009

001 d C = napsx =

d =




81

u3 = x4ö = ( moc p)

è ø

oslA u2 = u1 = ( moc p)

d C = u1D1 + u 2 D 2 + u3D 3

æ ö æ ö æ ö

î þ d C = - 652.0 mc

fo elpicnirp eht gnisU ni nwohs emarf latrop eht ezylana ,krow tsael F gi . tolp oslA t eh .B M D. .

:loS

ehT troppus si ih n .deg ecniS ht e er are owt itauqe o sn ta e hca ppus o .str yehT era HA, VA, HD , a dn VD . eerht si noitauqe muirbiliuqe elbaliava ehT . .e.i( ) = 0, = 0, = 0 . \ yllacitats si erutcurts ehT animretedni t ot e tsrif .eerged ( H latnoziroh eht taert su teL ¬ ta ) A

as .tnadnuder ehT latnoziroh noitcaer ta D lliw yltnedive eb = 3( - )H ( ¬). yB gnikat stnemom ta ew ,D eg t

V( A )3 x + H (3- )2 + ( )1 x 3 ( 2 – )5.1 – = )2 x 6( 0 VA = 5.3 – /H 3 VD = 6 – VA = 5.2 + 3/H

By grene niarts muminim fo meroeht eht y,

¶U ¶H ¶U ¶U ¶U CE ¶U ¶H ¶H ¶H ¶H

em roF)1( m :BA reb nikaT g o eht sa A rig .ni

ç ÷ 2 1 8 3 3 9

æ

4 9

ç ÷ ç ÷ ç ÷ 9 9 9 í ýx(+ )441.0 d C = + + ì - 4 - 4 - 8 ü

è ø è ø è ø

åM åH åV

AB EB DC

= 0

+ + + = 0




91

M = + H .x

2 ¶M ¶H ¶U 1 ¶M ¶H EI ¶H

+ Hx ÷x xd ø

Hx 3

3

= [9H - 1.01 2]

em eht roF )2( m :EB reb nikaT g B iro eht sa g .ni

M = (H x 3) - (3 x 1 .1 5) + ç .3 5 è

Hx M = 3H - 5.4 + .3 5x -

3 ¶M ¶H ¶U

æ3H - 5.4 + .3 5x -

ç - 5.31 + .01 5x - Hx - Hx + .1 5x - 6.1 7x 2 + è

ç -1 5.3 +12x - 2Hx - 6.1 7x 2 + è

-1.x 2

= x

= M dx 3

ò 0

BA

ö 1 EI 1

æ - x 2 ç

3

ò ç ÷

= EI

1 EI

0

é ê ë

è 2

-

=

ú û 0

3

ù x 4

8

ö æ ÷x ø

H 3

3 -

1 EI

=

EB =

¶H

x 3 1

ò 0

Hx 3

xd M ¶M ¶H

1 EI

1

EI

1

EI

1

ò 0

1

x 3

ödx

ø = = =

ç è

æ

÷ ÷ç öæ3 -

øè

ò ç9H

0

1

Hx 2 ÷

÷ xd ø

ö 9

ö Hx 2 ÷ æ

÷ xd

ø ò ç9H

0 9




02

1

ç -13.5x + 6x 2 - Hx 2 - 83.0 9 x 3 + è

- 7.9]

em eht roF )3( m :EC reb

nikaT g iro eht sa C g ni

M = - 3( - H )x2 + 5.2( + )x 3

M = -6 + 2H + 2.5x +

¶UCE ¶H

2

ò ëç - 6 + 2H + 2.5x +

ê- 21 + 4H + 5x + 6.6 7 xH - 2x + 6.6 7 xH + 0.833x2 +

ê- 21 + 4H + 3x + 43.31 xH - 2x + .0 38 3x2 +

= H69.01( - 87.51 )

em eht roF )4( m :CD reb nikaT g o eht sa D rig ni

M = -(3 - H )x = -3x + Hx ¶M ¶x

¶U

(- 3x + Hx)(x)dx

æ9H - 5.31 + 6 2 - H - 983.0 +

è 1

EI

=

æ ç9Hx

ö ø

H 72

ö 1 EI

Hx 3

72 ÷ ç ÷ = ÷ ø0

=

1 EI

[9H

H

Hx 3

3 2

ò M 0

= ¶M ¶H

1 EI

öæ 2 +

øè æ x ù ö

ø xH

3 é 1

EI

1 EI

1 EI

1 EI

êè 0

÷ç ÷ú û 3

=

=

=

é 2

ò 0

2

ò 0

ù údx

û

Hx 2

9 ë é ù

údx

û

Hx 2

9 ë

= x

2

ò 0

= dx 1 EI

M ¶M ¶H

CD

¶H

1

EI (- 3x 2 + Hx2 )dx 1

EI

2

ò 0

2

ò 0

= =




12

ç- x 3 + è

= H76.2( -8)

ht sbuS e va seul ¶U ¶H

E/1 I 9( - .8( + )2.01 H40 - H69.01( + )9.7 - ( + )87.51 - 0 = )H76.2+8 08.14 = H76.03

H = NK 63.1 Hence

VA 5.3 = - 3/H = 5.3 - NK 50.3 = 3/63.1 VD 5.2 = + 3/H = 5.2 + 3/63.1 = 59.2 K N

MA= MD 0= M B = (- x 1 32 2/) + )3 x 63.1( = - K 24.0 N –m MC = - 3( - 2 )H = - 3( - 63.1 ) 2 =- K82.3 Nm

:margaiD tnemom gnidneB

2 ta NK 54 fo daol detartnecnoc a ot detcejbus si m6 naps fo maeb detroppus ylpmis A m orf m oppus tfel eht rt . noitcelfed eht etaluclaC u tniop daol eht redn . 01 x 002 = E ekaT 6

K /N m2 01 x 41 = I dna -6 m4 .

noituloS : nikaT g m mo e .B tuoba stn

VA 6 x – 0=4 x 54 VA 6 x - 0 = 081

VA NK 03 = VB daoL latoT = – VA = K 51 N

uqe krow lautriV a :noit

(dc ) =

lppA y etsni c ta daol lacitrev tinu a NK 54 fo d

RA 6 x - 4 x 1 =0 RA = NK 3/2

÷ xd ø0 ÷ xd = + = ÷ ç ÷

ø0

ç ç 3 è

= 0

m xdM EI

L

ò 0

V

2 ö Hx 3

3 æ - 3x 3 æ

2 ö Hx 3

3 1

EI

1

EI

1 EI




22

RB = toT a ol l a d –RA = K 3/1 N

lautriV oM me :tn

es redisnoC c b noit e CA neewt

M1 X 3/2 = 1 ]2 ot 0 timil[ BC neewteb noitceS

M2 X 3/2 = 2- X( 1 2- ) 2 ] 6 ot 2 timil[

Re oM la m :tne

nretni ehT a ot eud tnemom l g gnidaol nevi

M1 X x 03 = 1 M2 = 03 x X2 - 54 X( 2 - )2

m1M1dx1 6 m2M dx2

V

ç - (x2 - 2)ö(30x2 - 45(x

2 6

ò 20x + ò ç 3

x2 - x2 + 2ö(30x2 - 45x2 +

90)dx2

= + 2ö(-15x2 + 90)dx2 EI 3

= 5x2 - 30x2 - 30x2 + 81 0dx2

20x1 5x2 60x2 ë û 0 ë

3 2

ç - 23 )- 30(62 - 22 )+ 180(6 - 21)ö

2

ò 0

(30x1 )

(dc ) = 2 +

3 3

EI EI

æ ö 2

2 6

= dx1 + 0 2

ò 2

ç ÷ x2

ò ò ÷

è ø è ø EI EI

æ 2x1 - 2)) dx2

æ 1 EI

2 2 1

0 2

2 6

ò 20x +

ò 0 2 2 6

ò 20x +

ò 0 2

= ÷

ç ÷

è ø

æ - x2

è ø 2 1

1 2 2

1

1 EI

1 EI

02

EI

6 é ù

+ - + 81 0x 2 ú û 2

3

ê 3 3

3 =

=

ê ú é ù

1 EI è ø

÷ (63 5 3 è

8 ç ÷ æ ö +

3 ø

æ




32

[53 3. + 3 43 6 76. - 069 + 27 0]

= 1750.0 m (or ) 57 1. mm

dnu noitcelfed ehT e ht r e daol = mm 1.75

De s’llewxaM eht evorp dna enif r oeht lacorpice rem.

hT e wxaM e ll s‟ er c orpi c ht la eor ts me a det a s “ hT e ow r nod k e by eht f s tsri y ol mets a sd b desuac stnemecalpsid ot eud y s dnoces a y b enod krow eht slauqe sdaol fo mets y dnoces eht

sy b desuac stnemecalpsid ot eud sdaol fo mets y s tsrif eht y ”sdaol fo mets .

llewxaM s‟ meroeht fo lacorpicer snoitcelfed isrev eerht gniwollof eht sah o :sn

1. ta ecrof tinu ot eud A ta noitcelfed ehT B ed ot lauqe si f ta noitcel B of tinu ot eud rce ta A .

δ BA = δB A

2. A ta epols ehT t eud B ta epols eht ot lauqe si B ta elpuoc tinu ot eud elpuoc tinu o A

Φ BA Φ = BA

3. ta noitcelfed ot lauqe si B ta daol tinu ot eud A ta epols ehT B ud e elpuoc tinu ot . f 'AB = d

Pr :foo

By ,dohtem daol tinu

d =

hW ere,

na ta tnemom gnidneb =M y .daol lanretxe ot eud x tniop

a ta tnemom gnidneb =m ny rehw tniop eht ta deilppa daol tinu ot eud x tniop e lfed ce r si noit e riuq ed.

L m te XA=be nidn g mom e ta tn any ud x tniop e ol tinu ot a d a A t L m te X = B be nidn g ta tnemom any ud x tniop e ol tinu ot a d a t B.

,A ta deilppa si )daol lanretxe( daol tinu nehW

=

061 02 0x106 x14x10-6 = =

EI 061

' BA

M xdm EI ò

1 EI




42

m=M AX

ta noitcelfed dnif oT B lppa ,A ta daol tinu ot eud y daol tinu a t B hT. e m =m n BX

Mmdx mXA.m BX EI IE

lralimiS y,

hW e daol tinu n (e etx rna ol l a ilppa si )d e ta d B m=M , XB

d eht dnif oT e elf c noit a ud A t e ol tinu ot a d a t B lppa , y daol tinu a t m =m neht.A AX

___________ _ )ii(

nirapmoC g )i( & w )ii( e get

δ BA = δB A

eroeht s’onailgitsaC gnisU m reted , m elfed eht eni c velitnac eht fo dne eerf eht fo noit e aeb r m eht ni nwohs gif . ekaT /NM 9.4 = IE m2 .

noituloS :

lppA y mmud y B ta W daol . S o noitcelfed eht enimreted ot evah ew ecni f ht e f eer e .dn redisnoC a cnatsid a ta xx noitces e x f mor B . hT en

M = W x + 30(x -1) + 02 1* *(x -1.5)+16(x - 2) M ¶M EI ¶W

2 3

ò ò íí xW * x + 03 (x -1)x + 02 (x - 1 () 2

)xü xd + ò xW * x + 03 (x -1)x{ 02 *1(x - )5.1 * x + 61 (x - 2)}ü 0 1 îî

2 þ

ò ò BA

Hen ec ,

d = = dx ___________ _ )i(

Mmdx mB.m AX

EI IE d = = xd AB ò ò

x

xd d = ìì

ò

ê ê ë

é 1 x -1 Wx * x xd + 1

EI ý ý þ

d




52

æ ö æ

û1 ú

3

- 7.0 5x 2 ÷ +16ç

ø è

5 19 19 ù ÷ ç - 7.3 5 ö +16æ - 5ö

ú

+ 01 + 03 + 20 * .2 58 + 61 6 2 6 3 ú

d = (25 + .5 83 + 11 5 + 51.6 + .12 33) d 644.0 = m (or) 46.44 mm

,revelitnac A 8m ,gnol iyrrac ng a tniop sdaol ta NK 5 eht retnec dna an ldu fo 2 m/NK f ro a

htgnel m4 rf om eht dne .B fI IE si eht laruxelf ytidigir of eht ac n revelit find eht noitcaer ta eht porp . (N CED/VO – 4002 )

loS u :noit

)NK ni( R ,porp eht ta noitcaeR dnif oT

iro ( :CA noitroP g ) A ta ni (Rx) dx

2EI

iro ( :BC noitroP g ) C ta ni B nidne g M tnemom x x( R = +4) – x5 – 2x2 2/

x( R = +4) – 5x –x2 4 2

U 2 = 2EI

2x3 x 2

3 2 W + + 30ç ÷ +10ç - ÷ ç - +

ø è

x 2 x 4

2 4

ö æ

ö æ x3

3

1 x3 éWx3 3

ç

è ø0 3

è

ê ç ÷ 3 ÷ ê ç ÷ ø ê

= 1 EI

ö ú ÷ú

÷ ç

ë ë û

ö æ x3

3 æ x3

2 ç + 30ç -

ö ÷ éWx

3

+ ê ù x 2

÷ ç x3

2 3

+ 20ç - x 2 ÷ ú

ú ÷ ø è

é 7 ê30ç

3 -

ê è ø 2 û

ë æ è

ë 3

nittuP g W =0

d =

d =

14 3 19 ÷ ç

3 2 3 ö 3 15

ø è ö 1

EI

1

EI

2 4 ö20æ +10æ - + + 30æ - ÷ ç

ø è ø è ø è ø ÷ ç ÷

2 3 3 û

é30

ë ù û ê

5 7 32 4

1x103

.4 9x106

û 0

6EI

ú ù

4

= U1 = 64R 2 32R 2

= 3EI

R 2 x 3

6EI é

= ê 4

ò 0

2

ë

x (M ) xd ò 0

é 2 ù ù




62

toT a ts l ra ne ni e gr y = U1 U+ 2

p eht tA r ppo e d e dn

64R æ ö xd

è ø

[R(x - 4)- 5x 2 - x2](x + 4) xd

[R(x - 4) - 5x (x + 4)- x2 (x + 4)]dx

[R(x 2 + 8x +16)- 5(x 2 + 4x) - (x3 + 4x 2 )]dx

+ 4x 2 +16x÷ - 5( + 2x 2 ) - ( - )ú

3 4 3

+ 64 + 64÷ - 5( ø

R 33.12 = + ( R33.941 – 76.662 – )33.941 R 33.12 = + ( R 33.941 – )614

R 33.941+ R 33.12 – 614 0=

R = NK 743.2

lpmis A y rrac si L naps fo maeb detroppus y ni g lmrofinu a dna ertnec eht ta W daol detartnecnoc a y d tisnetni fo daol detubirtsi y nel tinu rep w fo g ht . ht wohS a wxaM t e ll s‟ re orpic ac ht l e ro e sdloh m g doo a t

c eht e rtn e ht fo e eb am . noituloS :

L lmrofinu eht neht dna tsrif deilppa si W daol eht te y w daol detubirtsid .

ertnec eht ta W daol ot eud noitcelfeD nevig si maeb eht fo b y

4

d W = 83 4EI

:yb nevig si w ot eud W yb enod krow ecneH

¶U ¶R

= 0

3EI IE dR = + x M Md

ç ÷ x x 4

ò 0

¶U ¶R

0

4

ò 0

1

= + 3EI EI

64R = + 3EI EI

64R 1 = + 3EI EI

64R 1 = +

3EI EI

64R 1

è

4

ò 0 4

ò 0

é

ç

êRç

ë

æ

è

êRç

3

æ

64R 46 = +

3 é

ë

ç

2

4 x 3 x 4 4x3 ù ÷ x 3

3 ö

û 0 ø

+ 3 )2 - ( + )ú 256 52 6 4 3

ù 64 3

÷ ö

û

5 lW




72

4

U = xW

De lf e yb nevig si W ot eud dne tfel eht morf x ecnatsid a ta noitc

d W (x ) = (3l x - 4x 2 ) W ot eud htgnel tinu rep w yb enod kroW ,

l / 2

UB,A = 2 xw 0

wW 24EI

æ ö ç16 ÷

5 Wwl

384 EI Henc vorp e e .d

ETATS T ID EH FF RENE NIARTS DNA DAOL TINU NEEWTEB ECNERE G Y

FO NOITANIMRETED EHT NI DOHTEM S ERUTCURT S . nI niarts ene gr y ohtem d, na igami nary daol P si lppa dei ta eht tniop wh ere eht d tcelfe i no si

d eb ot derised ete nimr e eht ni orez ot detauqe si P .d f niatbo si noitcelfed eht dna pets lani ed. nI the tinU daoL ohtem d, a tinu daol daetsni( fo )P si deilppa ta eht tniop erehw the fed l tce i no

si derised .

ETATS HT NI EDAM SNOITPMUSSA EHT E DAOL TINU M DOHTE

2. ra secrof lanretni dna lanretxe ehT e muirbiliuqe ni 3. elbissop si tnemevom on dna digir era stroppuS 4. rts si lairetam ehT a w deni e eht nihtiw ll e timil citsal .

MOC EHT ETATS P O NOSIRA F ITSAC G S’ONAIL F RI TINU DNA MEROEHT TS

DOHTEM DAOL nevig si dohtem daol tinu eht yb noitcelfed ehT b y

n PUL

5 lw 83 4EI A,B

W 48EI

2

2 W 48EI

(3l x - 4x2 ) xd ò

÷ ú 4 ù ö 2 é3l

2

÷ ê æ

- ç ç ê ë è ú

UB, A =

ø û

l UB, A = l 2 2 2

ë

é3l 4

- ç ÷ ú ê ê

ù

ú è ø û

l wW 24EI

4

8 4

U = A,B




82

= δ Σ -- --- -- 1 EA

n PL

= δ Σ ------- x U 1 EA

n

= Σ U x ∆ -- -- - )i( 1

hT e de elf c b noit y ca its g il a on ht s‟ e ro e si m g yb nevi

-- --- - --- )ii(

By pmoc )i( gnira & )ii(

¶P ¶W

S’LLEWXAM ETATS ICER P MEROEHT LACOR

hT e wxaM e ll R s‟ ce orpi ca ht l e ro e ts m a sa set “ hT e d krow one by eht f s tsri y ol fo mets a sd b desuac stnemecalpsid ot eud y s dnoces a y b enod krow eht slauqe sdaol fo mets y s dnoces eht y mets

aol fo b desuac stnemecalpsid ot eud sd y s tsrif eht y sdaol fo mets .

YCNADNUDER FO EERGED llacitats eb ot dias si emarf A y rts ro snoitcaer nwonknu fo on eht nehw etanimretedni e ss ebmun latot eht deecxe stnenopmoc r fo c muirbiliuqe fo snoitauqe noitidno .

P RE F TCE F EMAR

I uqe si snwonknu fo rebmun eht f a o rebmun eht ot l f va snoitauqe snoitidnoc a ht ,elbali e fra em

p a eb ot dias si e fr ce emarf t .

O SEPYT OWT EHT ETATS F SEIGRENE NIARTS c. )ygrene niarts raehs( noitrotsid fo ygrene niarts

grene niarts.d y inu fo )ygrene niarts cirtemulov( noisnet )ro( noisserpmoc mrof

ESU EB NAC MEROEHT S’ONAILGITSAC ,SESAC HCIHW NI ETATS D .1 cilpmoc fo stnemecalpsid eht enimreted oT a urts det c rut es.

2. niraehs ot eud smaeb fo noitcelfed eht dnif oT g ro( ) nidneb g )ro( secrof neb nid g ra stnemom e nwonknu .

.3 fo snoitcelfed eht dnif oT c nirps smaeb devru g te s c.

PL ¶P AE ¶W

n

å 1

d =

= U




92

UN TI – II

RETEDNI M NI A AEB ET MS

SMAEB ETANIMRETEDNI YLLACITATS fI eht srebmun fo er a noitc stnenopmoc era om re naht eht snoitidnoc tauqe i ,sno eht erutcurts si

s sa denifed llacitat y eb etanimretedni a sm . E = R – r

eD = E g ycnadnuder lanretxe fo eer o rebmun latoT = R f stnenopmoc noitcaer

r = liava snoitauqe noitidnoc fo rebmun latoT a elb . t a si maeb suounitnoc A y llanretxe fo elpmaxe lacip y erutcurts etanimretedni .

ETATS REVELITNAC DEPPORP NI YCANIMRETEDNI FO EERGED EHT

nidaol lareneg a roF g )R( stnenopmoc noitcaer latot eht , are e uq a 3( ot l + )2 = ,5

uqe noitidnoc fo rebmun latot eht elihW a ( snoit r ra ) e 3 ot lauqe . llacitats si maeb ehT y ,etanimretedni e etx r llan y ces ot ed dno g .eer citrev roF a nidaol l g llacitats si maeb eht , y

nis ot etanimreted g ged el eer . E = R – r

5 = – 2 = 3

EHT ETATS D A NI YCANIMRETEDNI FO EERGE F MAEB DEXI

s lareneg a roF y nidaol fo mets g llacitats si maeb dexif a , y .eerged driht ot etanimretedni F citrev ro a l llacitats si maeb dexif a ,gnidaol y es ot etanimretedni c ed dno g eer .

E = R – r s lareneg roF y :gnidaol fo mets

3 = r dna 3 + 3 = R 6 = E - 3 = 3

:gnidaol lacitrev roF 2 = r dna 2+2 = R

E = 4 – 2 = 2 HT NI YCANIMRETEDNI FO EERGED EHT ETATS AEB NEVIG E M

llacitats si maeb ehT y s lareneg fo eerged driht ot etanimretedni y nidaol fo mets g. 6 = 1+1+1+3 = R

R = E - r 6 = - 3 = 3




03

EHT NI YCANIMRETEDNI FO EERGED EHT ETATS G MAEB NEVI

llacitats si maeb ehT y nimreted a et . e noitidnoc fo srebmun latot ehT era snoitauq e uq a 3 ot l + = 2 .5 reht ,ecniS e knil a si a t B . ra snoitauqe noitidnoc lanoitidda owt ehT e knil ta .

E = R- r

2+1+2 = - 5 5 = - 5

E = 0

YLLACITATS GNIZYLANA ROF ELBALIAVA SDOHTEM EHT ETATS RETEDNI M S ETANI T SERUTCUR

i. dohtem ytilibitapmoC ii . ohtem muirbiliuqE d

XE EHT ETIRW P NOISSER F ED DNA STNEMOM DNE DEXI F NOITCEL F A RO

A DAOL TNIOP GNIYRRAC MAEB DEXIF T ERTNEC

M = M =

WL3 91 2EI

XE EHT ETIRW P NOISSER F ED DNA STNEMOM DNE DEXI F NOITCEL F A RO DAOL TNIOP CIRTNECCE GNIYRRAC MAEB DEXIF

Wab

A

2

M =

3

ymax = ( rednu eht aol d )

XE EHT ETIRW P NOISSER F STNEMOM DNE DEXI F A RO F OT EUD DEXI PUS FO GNIKNIS P TRO

M = M =

WL 8 A B

ymax =

2

M = L2

aW b L2

aW b3

3 IE L3

B

6EId L2 A B




13

aeb dexif A m 6 htgnel fo BA m Nk 061 fo daol tniop seirrac a k 021 dn N 2 fo ecnatsid a ta m m4 dna orf m ht e tfel e A dn . dne dexif eht dniF m stroppus eht ta snoitcaer eht dna stnemo . .B warD M dna .S F argaid ms .

noituloS : G :nevi

L = m6 Loa d a W ,C t C Nk 061 = Loa d a W ,D t C Nk 021 =

natsiD ec CA = m2 natsiD ec DA = m4

C ta sdaol ot eud stnemom dne dexif eht etaluclac tsriF letarapes D dna y neht dna a stnemom eht pu dd .

F oM dnE dexi m :stne

F a ,C ta daol eht ro = b dna m2 = m4

M =

M = = 41 2. 22 k mN

(6)2

W C a2b L2

61 0x22 x(4) (6)2

m2 = b dna m4 = a ,D ta daol eht roF

A2

= 33.35 kNm

B 2

M = = 66.601 k mN

(6)2

nixif latoT g ,A ta tnemom MA = MA1 + MA2

= 22.241 + 33.35 MA = 55.591 mNk

nixif latoT g ,B ta tnemom MB M= 1B + M 2B

= 11.17 + 66.601

W C ab2

L2

61 0x2x(4)2

A1

A1

B1

B1

M =

M = = 71 11. k mN

WDa b 2

L2

21 0x22 x(4) (6)2

WDa 2b L2

61 0x2 x(4)2

A2

M =

M =

M =

B 2




23

= 77.771 Nk m :sdaol lacitrev ot eud margaid M.B

lpmis sa BA maeb eht redisnoC y .detroppus Le R t A* a R dn B

* are ht e reac ta snoit lpmis ot eud B dna A y maeb detroppus . eg ew ,A tuoba stnemom gnikaT t

RB*x6 = 61 0x2 + 21 0x4

RB

* = = 33.331 kN

RA* = toT a ol l a d - RB

*= 061( + )021 – Nk 76.641 = 33.331 B A ta M. = 0 B R = C ta M. A

* 2 x = 2 x 76.641 = Nk 43.392 m 31 = D ta M.B m Nk 66.662 = 2 x 33.3

B ta M. B 0 = .S F :margaiD

L R te A v dna stnemom dne dexif ot eud A ta noitcaer tnatluseR = e itr ac sdaol l RB noitcaer tnatluseR = a B t

nitauqE g itna dna stnemom esiwkcolc eht - ,A tuoba stnemom esiwkcolc

RB 6 x + MA = 2 x 061 + 021 M + 4 x B

RB= 73.031 Nk RA = tot a ol l a d – RB Nk 36.941 =

.S F a A t = RA = Nk 36.941 = C ta F.S 36.941 - = 061 - k 73.01 N

.S F a D t = - 73.01 – = 021 - k 73.031 N .S F B ta = NK 73.031

6 htgnel fo BA maeb dexif A m d a ta hcae Nk 03 fo sdaol tniop owt seirrac m2 fo ecnatsi

orf m ht e ob t h e sdn . reteD m B eht ward dna stnemom dne dexif eht eni . argaid M m.

noituolS : :neviG

m6 = L htgneL daol tnioP a C t = W1 = Nk 03 daol tnioP a D t = W2= 30 Nk

:stnemom dne dexiF MA = F nixi g nixiF + C ta daol ot eud tnemom g mom daol ot eud tne a D t

W1a1b12 W2a2b2

2 L2 L2

30x2x42 30x4x22 62 62

s si maeb eht ecniS y ,lacirtemm

08 0 6

= +

= 40kN m

MA = MB mNk 04 =

= +




33

argaiD M.B m : aid M.B eht ward oT g lpmis sa BA maeb eht redisnoc ,sdaol lacitrev ot eud mar y .detroppus

dna A ta snoitcaer ehT B uqe si a Nk03 ot l . a M.B 0 = B dna A t

C ta M.B 03= x mNk 06 = 2 mNk 06 = 2 x 03 = D ta M.B

gnixif eht dniF mom rofinu a gniyrrac ,m6 htgnel fo BA maeb dexif a fo snoitcaer troppus dna stne m yl /Nk4 fo daol detubirtsid m flah tfel eht revo o naps eht f .

noituloS :

aM ac alu y s‟ me c doht a b n e esu d a rid dn ce lt y nixif eht g mom e dna stn e r dn e ca snoit ac b n e ac lucl a det . aid M.B fo saera eht erehw desu si dohtem sihT g ltneinevnoc denimreted eb tonnac smar y.

en si ti dohtem siht roF c rasse y LDU taht oc neht dna B ot pu dednetxe eb dluohs of detasnepm r drawpu nel rof LDU g CB ht

yb nevig si A morf x ecnatsid a ta noitces yna ta gnidneb ehT , 2

EI = RA x - M - wx + *w (x- )3

R= A x – MA- ( ) + (4 )

= RA x – MA- x2 2 + x(2 -3)2

teg ew ,gnitargetnI dy x2 x3 2(x - 3)3

dy

EI R= A -MA x - 2 + dx 2 3 3

I rgetn a nit g we ge t x3

A 2x 4 2(x - 3)4 6 2 12 12 0 = y , 0 = x nehW

radnuob eseht gnitutitsbus yB y ,enil dettod eht otpu snoitidnoc C2 = 0

EI y = - - + ________ ( )ii

(x - 3) 2 A

d y x dx 2 2

4x2 x - 3)2

2 2

EI R= A -MA x - 2 C+ 1 + -- --- -- )1( dx 2 3 3

,0=x nehW =0. dx

utitsbuS t ni g voba eht ni eulav siht e e uq a ,enil dettod ot pu noit C1 = 0

rehT e rof e e uq a b )1( noit ce semo dy x2 x3 2(x - 3)3

EI y = RA - - + C2 + M x 2

A RA x 3 M x 2 4 - 3)4

6 2 6 6

x 1(x




43

By & 6= x sbus y )ii( noitauqe ni 0 =

R A 63 M 62 6 2

= 36RA -18M - 12 6 + .31 5

R81 A – M 9 A = 52.101 -- ----- - --- -- )iii(

,6= x tA 0 = tauqe ni i )i( no xd

62 2 2 2 3 3

18RA - M x6 -144 + 81 = 0 18RA - 6M = 126

By nivlos g )iii( & )vi(

MA mNk 52.8 = By nitutitsbus g MA )vi( ni

621 = 81 RA – 52.8( 6 ) RA = Nk 57.9 RB daol latoT = – RA RB Nk 52.2 =

By B tuoba stnemom esiwkcolcitna dna stnemom esiwkcolc eht gnitauqe MB + RA 6 x = MA + .4( 3x4 5 )

MB = 57.3 mNk Re :tlus

MA = mNk 52.8 MB = 57.3 mNk RA = Nk 57.9 RB NK 52.2 =

EHT ETATS ROEHT E O M F STNEMOM EERHT

eroehT m stnemom eerht fo

I “ taht setats t I lno era DC dna CB f y ejbus maeb suounitnoc a fo naps evitucesnoc owt c ot det etxe na r nidaol lan g ht , e M stnemom eht n B M , C an M d D a ht t e stroppus B C , a D dn are g vi e n

by

6 1 6( - 3)4

+ 6 6

4 A 0 = - -

A

yd

A 3 3 0 = RAx - M x6 - x(6) + (6 - 3)

A

A




53

M L1 + 2M C (L1 + L2 ) = M .L2 = +

hW ere

MB = B nidne g etxe ot eud B ta tnemoM rna ol l a gnid MC = B nidne g etxe ot eud C ta tnemoM rna ol l a gnid M D = Be nidn g moM e ud D ta tn e etxe ot rna ol l a gnid L1 nel = g BA naps fo ht L2 CB naps fo htgnel = a1 ra = e fo a B ot eud D.M. naps no sdaol lacitrev B C a2 ra = e fo a B citrev ot eud D.M. a ol l a DC naps no sd _ x B morf CB no sdaol lacitrev ot eud D.M.B eht fo G.C fo ecnatsiD = _ x ol lacitrev ot eud D.M.B eht fo G.C fo ecnatsiD = a DC no sd f .D mor

dne dexif eht era tahW m tnemo aeb dexif a rof s m jbus ’L‘ htgnel fo e tartnecnoc a ot detc e d a ’w‘ daol t natsid a ec a‘ ’ from tfel e ?dn

xiF e :tnemoM dnE d aW b2 L2

aW b2 L2

FE EHT NIALPXE F O TCE F O TNEMELTTES F PUS P SUOUNITNOC A NI STRO

AEB M uD e oc a ni stroppus fo tnemelttes eht ot nidneb eht ,maeb suounitn g sretla lliw sesserts

rppa e lbaic y nidneb mumixam ehT . g fo esac ni tnemom c hw ssel si maeb suounitno e eht ot erapmoc n lpmis y maeb detroppus .

O SEGATNAVDA F SMAEB DETROPPUS YLPMIS REVO SMAEB SUOUNITNOC

ehT)i( am x mumi gnidneb m mo tne ni esac fo a suounitnoc maeb si hcum ssel naht ni esac fo a lpmis y rrac naps emas fo maeb detroppus y sdaol emas gni .

)ii( nI esac of a itnoc n suou ,maeb eht eva r ega M.B si ressel dna eh n ec rethgil slairetam fo dneb eht tsiser ti desu eb nac noitcurtsnoc tnemom gni .

aeb dexif A m 5 htgnel fo m rofinu a seirrac m daol detubirtsid yl o k 9 f N/m eht revo nur itne er naps . x5.4 = I fI 10-4 m4 1x1 = E dna 07 kN/m2 dnif , the gnixif m mo e a stn t the dna sdne

ta noitcelfed the ec rtn e.

1 B D

1

2

A

B

M =

M =

_ _

6a1 x 6a2 x2

L1 L2




63

loS u :noit

:neviG L m5 =

W = k 9 m/N 2 , I = 01x5.4 -4 m4 a E dn = 1x1 07 k m/N 2

ldu gniyrrac maeb eht rof tnemom dne dexif ehT )i( :

MA = MB =

= = 57.81 NK m 12

ud ertnec eht ta noitcelfed ehT )ii( e :ldu ot WL4 483 EI

yc = = 452.3 mm

83 4x1x107 x .4 5x10-4

De lf e drawnwod ni si noitc erid c .noit

l m6 ,BA maeb dexif A o etnec sti ta Nk 04 fo daol tniop a gniyrrac si gn r . ehT M ht fo I.O. e aeb m 01 x 87 si 6 mm4 eulav dna o E f b rof eam m ta er 01x1.2 si lai 5 m/N m2 . teD erm )i( eni

B dna A ta stnemom dne dexiF . noituloS :

dexiF e mom dn e :stn

M = M =

M = M =

aeb dexif A m 3 htgnel fo BA m 01 x 3 = I I.O.M gnivah si 6 mm4 av dna l E fo eu aeb rof m m ta re 01x2 si lai 5 mm/N 2 . s ehT u nwod sknis B tropp m3 yb m . )i( enimreteD f dne dexi m a stnemo t B dna A .

noituloS : G :nevi

L m3 = = mm0003 I = 3 01 x 6 mm4

E = 01x2 5 /N mm2 d mm3 =

WL2

21

9x(5)2

yc =

9x 5( )4

WL 8

50x6

8

A B

A B = 5.73 k mN




73

M = M =

= 003( 0) 2

= 01x21 5 Nk 21 = mm N .m

3 ,BA maeb dexif A m orf m2 fo ecnatsid a ta Nk 54 fo daol tniop a gniyrrac si gnol m A . fI ht fo IE )e.i( ytidigir laruxelf eht e beam 01x1 si 4 Nk m2 . De ret m tcelfeD )i( eni eht rednu noi

daoL . noituloS :

G :nevi L m3 =

W = Nk 54 EI = 01x1 4 mNk 2 lfeD e :daol eht rednu noitc

I daol cirtnecce ot eud daol eht rednu noitcelfed ,maeb dexif n Wa3b3 3 IE L3

45x(2)3 x(1)3 3x1x104 x 3( )2

yC = 444000.0 m

yC = 444.0 mm oitcelfed ehT wnwod ni si n a id dr rec .noit

5 fo maeb dexif A m orf daol gniyrav yllaudarg a seirrac naps m /Nk 01 ot A dne ta orez m ta

e B dn . F xif eht dni i gn m sdne dexif eht ta noitcaer dna tnemo . noituloS :

:neviG

L m5 = W = m/Nk 01

( )i :tnemoM gnixiF

dna MB =

MA = = = 33.8 k mN 30 03

A B

6x2x105 x3x106 x3

yC = yC =

WL2

03 WL2

02 M = A

10(5)2 2 05

6EId L2




83

MB = = = 5.21 k mN

20 20

(ii) noitcaeR a oppus t r :t 7 LW

02

RA = = = 5.7 kN

RB = = = 17 5. kN

aeb suounitnoc A m dna BA naps evitucesnoc owt srevoc CBA l fo CB e 4 shtgn m 6 dna m , rofinu gniyrrac m detubirtsid yl sdaol o /Nk6 f m /Nk01 dna m ylevitcepser . dna A sdne eht fI

is era C m troppus eht dnif ,detroppus ylp m C dna B,A ta stnemo . na D.M.B osla ward d .S F .D.

noituloS : G aD nevi t :a L ne g A ht B , L1= m4 . L ne g ht B ,C L2 m6=

LDU w ,BA no 1 m/Nk6= DU L no B w ,C 2= m/Nk01

( )i uS p oM trop me stn : lpmis era C dna A sdne eht ecniS y eb lliw C dna A ta stnemom troppus eht ,detroppus

ze or . By nisu g c el y norep ‟ uqe s a o noit f rht ee mom e ta stnemom troppus eht dnif ot ,stn B ei( )

MB.

MAL1 + M2 B(L1+L2 ) + MCL2 = - 0 + M2 B 4( + )6 + 0 = -

M02 B = - a2x2

2 lpmis a no D.M.B ehT y rrac si maeb detroppus y nivah alobarap a si LDU gni g edutitta na

of

8.

3WL RA = and RB =

20

3 01* * 5 051 20 02

7 01* 5* 053 02 02

- 6a1x1 6a2x2

4 6 - 6a1x1 6a2x2

4 6 - 3a1x1

wL2

10(5)2 250




93

erA a fo B D.M. = *L h*

3 2 wL2 3 8

a siht fo G.C fo ecnatsid ehT r no morf ae e ,dne =

. a1 A= r fo ae B DU ot eud D.M. L A no B,

= *4* 3 8 23=

x1 =

= 2/4 = 2 .m

a2 A = r fo ae B DU ot eud D.M. L no B ,C

= *6* 3 8

= .m081 x2=L2 2 / = 2 / 6 = m3

qe ni seulav eseht etutitsbuS au i(noit ). eW eg ,t

M02 B = + ( 081 )3* 2

045+69 = MB 8.13= mNk .

(ii) .B M D.

ud D.M.B ehT ev ot e r DU( sdaol lacit L A naps no ) B naps dna B .C A napS B :

w1L12

= 8

mNk21=

pS a n B :C =

2

= * napS *

ps an 2

2 6(42 )

L1

2

2 1 (0 62 )

*3 23 * 2

= 8

6 * 42

w2L2 2

8




04

01 * 62 8

mNk54= iii( ) .S F :D.

,snoitcaeR etaluclac oT eg ew ,B tuoba stnemom gnikat ,BA naps roF t

R( A )4* - 2*4*6( ) – MB 0= R4 A – M( 8.13 = 84 B ,8.13= - is ev ng si .tnemom gniggoh ot eud

RA= Nk50.4 lralimiS y ,

nikat ,CB naps roF g ,B tuoba tnemom R( c )6* - )3*01*6( – MB 0= R6 C – =081 - 8.13

RC k7.42= N . RB= toT a ol l a A no d B C – R( A R+ B)

))6*01(*4*6(= - 50.4( + )7.42 = .Nk52.55

TLUSER :

MA M= C 0= MB mNk8.13= RA= Nk50.4 RB= Nk52.55 RC Nk7.42=

aeb suounitnoc A m 51 htgnel fo DCBA m ruof no stser s stroppu c ps lauqe 3 gnirevo a seirrac dna sn rofinu a m eht etaluclaC. htgnel m/Nk 5.1 fo daol detubirtsid yl m eht ta snoitcaer dna stnemo

oppus r st . .S ehT warD F .B dna D. M .D.

ituloS no : G :nevi

L ne g A ht B = L1 m5 = L ne g ht B = C L2 = m5 L ne g = DC ht L3 m5 =

l.d.u w1 w = 2 w = 3 m/Nk 5.1 =

lpmis era D dna A sdne eht ecniS y oreZ eb lliw D dna A ta stnemom troppus eht ,detroppus . MA= 0 a M dn D 0=

s roF y rtemm y MB 0=

troppus etaluclac oT)i( m :stnemo oT f ht dni e mom troppus e ta stn B a b ,C dn y gnisu c pal e nor s‟ e uq a rht fo snoit ee mom e of stn r

AB dna C B .DC

=




14

- 6a1x1 6a2 x2

L1 L2

- 6a1x1 6a2x2 5 5

M02 B M5+ C= (a1x1 + a2 x2 ) ----------------------------------- - -- )i( 5

a1 fo aerA= B ot eud DM DU L A no B A nehw B lpmis sa deredisnoc si y oppus r b det ae m. w1L1

8

= 5* * 3 8

526.51= x1=L1 2/

m5.2=2/5= uD e s ot y mm e rt y

.a2 a= 1 526.51= x2 x= 1 5.2=

v eseht sbus a nqe ni seul ( )i

M02 B M5+ C = [( *526.51 )5.2 + *526.51( ])5.2 5

57.39= uD e s ot y mm e rt y MB M= C M02 B M5+ B 57.39=

MB Nk57.3= m . MB M= C Nk57.3= m .

itrev ot eud MB etaluclac oT )ii( c :sdaol la

DU ereh(sdaol lacitrev ot eud DMB ehT L ( DC dna CB ,BA naps no ) c niredisno g naps hcae lpmis sa y b nwohs era ) detroppus y rap a edutitla fo salob

w1L12 .1 5 * .1 52

8 8

:snoitcaeR troppus etaluclac oT)iii(

L R te A R, B R, C R dna D a oppus eht er r r t eac snoit a ,A t B C, a .D dn uD e s ot y rtemm y

RA R= D RB R= C

,B tuoba stnemom gnikaT ,BA naps roF eW eg t

MB R(= A )5* - )5.2*5*5.1( - 57.3 = R( A 5* )- 57.81 RA= .Nk0.3

,CBA roF

MAL1+ M2[ B(L1+L2 ]) +MCL2= -

0+ M2[ B 5( +5)]+ M[ C(5)]= -

6

2 = * BA * alobarap fo edutitlA rap fo edutitlA( a lob a= )

3 2 .1 5* 5( )2

6

= 4. 786 5 mNk ae c .h =




24

uD e s ot y rtemm y RA R= D= Nk0.3

RB R= C RA R+ B R+ C R+ D latoT= daol no DCBA

R+3 B R+ B 51*5.1=3+ RB Nk52.8= RC k52.8= N .

Re :tlus

MA = MD = 0 MB M= C .mNk57.3= RA R= D= Nk0.3 RB Nk52.8= RC k52.8= N .

aeb suounitnoc A m is ,DCBA m troppus ylp e daol si D dna C ,B,A ta d e hs sa d o .gif ni nw iF eht dn mom aeb eht revo stne m .S dna D.M.B ward dna F D. . 3002 ceD /voN( )

noituloS :

G :nevi L ne g A ht B = L1 m6 = L ne g ht B = C L2 m5 = L ne g = DC ht L3 m4 =

daol tnioP W1 = Nk9 daol tnioP W2 = Nk8

,DC no l.d.u w = m/Nk 3

kat sdaol lacitrev ot eud D.M.B )i( i is sa naps hcae gn m troppus ylp e :d

,BA maeb redisnoC B = E ta daol tniop ta M. =

lralimiS y = F ta M.B = = 6.9 kNm

L2 6 lpmis a fo ertnec eht ta M.B y rrac ,DC maeb detroppus y L.D.U gni

wL32 3 * 42

:stnemom troppus ot eud D.M.B )ii( maeb eht ecniS detroppus ylpmis si MA M= D 0 =

lC gnisu yB a noitauqE s’noryep o moM eerhT f e :stn

a) dna BA snaps roF B C

W1 ba L1

9 * 2 * 4 6

= 12k mN

W 2ab 8 * 2 *3

= = = 6kNm 8 8




34

MAL1 + M2 B(L1+L2 ) + MCL2 = -

0 + 2M 6( + 5) + M c (5 ) = - -

22M + 5M = a1x1 a2 x2 --------- - -- )i(

5 a1x1 *21*6*½ = L+a 3/ = *21*6*½ (6+ 3/)2 69 = a2x2 = *6.9*5*½ L+ *6.9*5*½ = 3/b (6+4) 46 = 3/

qe ni seulav eht etutitsbuS au )i( noit M22 B + M5 C = 46*5/6+69 M22 B + M5 C = 8.271 -- --- - --- - -- )ii(

)b For ps a sn B C a DC dn

MBL2 M2 + C(L2+L3 ) + MDL3 = -

MB 5* + M2 C 5( +4) + 0 = -

5M +18MC = + -- -- -- --- -- )iii(

a2x2 (* 6.9 * 5 * ½ = L+a 5(* 6.9 * 5 * 2/1= 3/) +2) 3/ = 65 a3x3 = 2/4*6*4 * 3/2 =32

uqe ni seulav eseht etutitsbuS a )iii( noit

5M +18M = + 5 4

5M +18M = 11 5 2.

By )vi(& )ii( snoitauqe gnivlos

MB = 48.6 mNk dna MC = 84.4 mNk

S )iii( up op r t eR ac :snoit

A naps eht roF B nikaT , g tuoba tnemom B, MB R = A 6 * – 4*9

= 6RA -36 RA = = 4.86KN

6 nikat ,DC naps eht roF g C tuoba stnemom

MC = RD ´ 4 - 3´ 4´ (MC = - )84.4 2

- 6a1x1 6a2x2

4 6 6a1x1 6a2x2

6 5 B

6 B C

- 6a2 x2 6a3 x3

L2 L3

- 6a2 x2 6a3 x3

5 4

6ax2 6a3 x3

5 4 B

6 * 65 6 * 23 B C

B C

63 - .6 48

4




44

RD = NK88.4 nikat CBA roF g C tuoba tnemom

Mc = RA *(6 + 5)- 9(5 + 4)+ RB * 5 - 8*3 5RB 8 = 1+ 24 - 8.4 6*11

RB = 14.9 Nk RC = toT a ol l a A no d B DC – R( A R+ B R+ D) RC = (9+8+ )3*4 – 68.4( + 88.4+14.9 ) RC = 58.9 Nk Re :tlus

MA = MD = 0 MB = 48.6 mNk dna MC = 84.4 mNk RA = Nk68.4 RB = Nk14.9 RC = 58.9 Nk RD = NK88.4

t gnisU eroeht eh m eerht fo m gnidneb dna ecrof raehs eht ward stnemo m gaid tnemo ram s tnoc gniwollof eht rof in aeb suou m . M / lirpA( a 3002 y )

loS u :noit G :nevi

,BA htgneL L1 m4= . L ne g ,CB ht L2 m3= .

htgneL L ,DC 3 m4= . m/Nk 4=w ,BA no LDU

ni daol tnioP B W ,C 1= Nk4 m/ DC ni daol tnioP , W1 k6= N

( )i oM gnidneB m tne t :sdaoL lacitreV o

A maeb redisnoC B , B =M. = 8 8

Nk8= m . lralimiS y ,CB maeb rof

6 * 2 1* 3

mNk4= lralimiS y ,DC maeb rof

wL2 4 4 * 2

B =M. = W1 ba

L2




54

W2 ba L3

mNk6=

(ii) oM gnidneB m troppus ot tne m :stnemo

L M te A M, B M, C dnA MD be oppus eht r ,A ta stnemom t B C, a D dn . eht ecniS e si sdn lpmis y troppus e M ,d A M= D=0 .

By nisu g alC ype nor ‟ s e uq a rht fo noit ee mom e of stn r A naps B a dn B ,C

MAL1+ M2[ B(L1+L2 ] ) + MCL2 = +

0+ M2[ B 4( +3) M ] C 3( ) = +

M41 B M3 + C a5.1 = 1x1 a2 + 2x2 ---------------------------- )i(

a1x1 moM = e fo tn area ud DMB e LDU ot

= * * ( aB se * Altit du e) 3 2

= * * 4( )8*

3 2 = 33.24

a2x2 moM = e fo tn area ud DMB e ol tniop ot a d a B tniop tuob

= * * 2( * )4 2 3

= 33.5

nisU g v eseht a ,)i( nqe ni seul M41 B + M3 C = (5.1 4 33.2 ) + 5*2( . )33

-- --- - --- - --------------- )ii(

6a2 x2 6a3x3 L2 L3

6a2 x2 6a3x3

3 3 M3 B M21+ C = a2 2x2 + a2 3x3 ------------------------ )iii(

a2x2 fo tnemoM = area ud DMB e ol tniop ot a d a C tniop tuob

= 2/1( ) *4*2* 3

B =M. = 8 *1* 3 4

6a1x1 6a2 x2

L1 L2

6a1x1 6a2x2

4 3

2 B esa

2 4

1 2 * 2

M41 B + M3 C 66.01+594.36=

oF r aps n B C a ,DC dn

MBL1+ M2[ C(L2+L3 ] ) + MDL3 = +

MB 3( )+ M2[ C 3( +3) ]+ MDL3 = +

2 1*




64

66.2= a3x3= moM e fo tn area ud DMB e ol tniop ot a d a D tniop tuob

1 2 * 3 2 3 6=

nisU g ,)iii(nqE ni seulav eseht

M3 B+ M21 C =2( 66.2 ) + ( )6*2 M3 B + M21 C = 23.71 ----- - --- - --------- )vi(

nisU g ,)iii( dna )ii( nqe MB = 962.5 Nk m

MC = 921.0 Nk m iii( ) :noitcaeR troppuS

nikat ,BA naps roF g B tuoba tnemom M = RA * 4 - 4* 4 * 2 - 962.5 = RA 4* – 23

RA 4* = 137.62 RA = Nk 86.6

,DC naps roF C tuoba tnemom gnikat

M = RD * 4 - 8 *1 - 921.0 = RD 4* - 8 RD = 769.1 Nk

BA rof C tuoba tnemom gnikat woN C M C = RA )7( - 4 * (4 * 5)+ RB *3 - 6 *1 M C = 7RA - (4 2 )0 + 3RB - 6 0. 21 9 = (7 6.68) - 80 + 3RB - 6 RB = 730.31 Nk RC = toT a ol l a d – R( A R+ B R + C) =[(4 * 4)+ 6 + 8]- ( .6 68 +1. 69 7 +13.037) RC = 613.8 Nk

Re :tlus MA = MD = 0 MB = 962.5 Nk m MC = 921.0 Nk m RA = Nk 86.6 RB = 730.31 Nk RC = 613.8 Nk RD = 769.1 Nk

= *1* 6 *

B

C




74

aeb A m 4 fo BA m a dedaol si dna sdne eht ta detroppus ylpmis si naps s ni nwohs .gif enimreteD ixaM )ii( C ta noitcelfeD )i( mum A dne eht ta epolS )iii( noitcelfed . 01 x 002 =E 6 /Nk m2 na d 01 x 02 = I -6 m4

noituloS : G :nevi L = m4

01 x 002 =E 6 /Nk m2 na d 01 x 02 = I -6 m 4

R etaluclac oT e :noitca nikaT g mom e A tuoba tn

RB = 4 * 02 *1+ 01 * (2 +1 +1) 2

RB 4* = 02 + )3(02 RB = 4/08 = 02 Nk RA = toT a daol l - RB

)02+2*01( = - 02 RA = Nk 02

s’yaluacaM gnisu yB m :dohte

d y 01 (x - 2)2 X

EI = 10x 2 + C1 - 01 (x -1)2

IE y = + C1x + C2 - - -- --- - --- - )ii( 3 3 21

,0 = x nehW y itauqe ni 0 = ew )ii( no g C te 2 0 = ,m4 = x nehW y )ii( noitauqe ni 0 =

0 = (4)3 4 + C1 - ( 4 -1)3 - (4 - 2) = 33.312 + C4 1 – 09 - 76.6 C1 = - 61.92

He epols eht ecn a lfed dn ce noit e uq a ra snoit e

auqE epolS t noi :

EI = 10x 2 - 61.92 - 01 (x -1)2

elfeD c itauqE noit no :

10x3 01 (x -1)3 5(x - 2)4

2

2

M = EI = 20x - 02 (x -1) - d x 2 2

teg ew gnitargetnI dy - 5(x - 2)3

dx 3 teg ew gnitargetnI 10x3 01 (x -1)3 5(x - 2)4

01 01 5 3 3 21

4

dy dx

- 5(x - 2)3

3

EIy = - 1.92 6x - - 3 3 12




84

( )i De elf c ta noit C y , C :

nittuP g = x w ,noitauqe noitcelfed eht ni m2 e get

2(01 )3 2(01 -1)3 3 3

76.62 = - 23.85 - 33.3 = - 89.43

yc )drawnwod( 47.8 = (ii) ixaM mum De elf c y , noit m xa :

liw noitcelfed mumixam ehT dim ot raen yrev eb l - C tniop . Le su t es eht ni srucco ti taht emussa c D neewteb snoit a C dn . elfed mumixam roF c nitauqe noit g ht e

eg ew ,orez ot noitces eht ta epols t

EI = 10x 2 - 29. 61 -10(x -1)2

x01 2 - 61.92 - x(01 - )1 2 = 0 x01 2 - 61.92 - x( 01 2 -2x+1) = 0

x = 859.1= 02/61.93 m 01 ( .1 59 8)3 01 (1. 59 8 -1)3

yd xd

EIy = - )2(61.92 -

yd xd

EI y = - 2 .9 16( .1 958) - 3 3

y am x = - E/53 I y am x = mm 57.8 ( wnwod a )dr

iii( ) ta epolS t eh e θ ,A dn A :

nittuP g pols eht ni 0 = x e e uq a ,noit

EI = - 61.92

θA d = y = xd/ - IE/61.92 θA = - r 92700.0 a id a sn θA = - º714.0

Re :tlus ( )i mm 47.8 = C ta noitcelfeD (ii) mm 57.8 = noitcelfed mumixaM

iii( ) θ ,A dne eht ta epolS A = - º714.0




94

.9 aeb suounitnoc A m ni nwohs si gif . eht warD tneilas gnitacidni DMB stniop .

noituloS : G :nevi L ne g ht L1 m4 = L ne g ht L2 m8 = L ne g ht L3 m6 =

no ldU B w C = m/Nk 01 daol tnioP W1 = Nk 04 daol tnioP W2 = Nk 04

( )i :sdaol lacitrev ot eud M.B

b redisnoC e A ma B , B M. = =

For eb a m B ,C

B M. = = 8 8

oF r eb a ,DC m

B M. = = 4

:stnemom troppus ot eud M.B )ii(

teL MA, MB, MC, MD eb eht troppus stnemom ta ,A ,B ,C D . cniS e eht e lpmis era D dna A dn y troppus e M d A = MD = 0

By nisu g epalC y nor ‟ uqE s a o noit f hT r ee mom e .stn

F :CB dna BA napS ro

M L1 + 2M (L1 + L 2 ) + M L2 = - -

0 + 2M 4( + 8 + ) M C (8 ) = - -

M2 B )21( 8+ MC = - a5.1 1x1 – 57.0 a2 x2 42 MB 8+ MC = - a5.1 1x1 – a 57.0 2 x2 ----------- )i(

a1x1 moM = e fo tn area fo B ud D.M. e ol tniop ot a d 021 = 3*3/2*03*4*½ =

a2x2 moM = e fo tn area fo B ud D.M. e ldu ot saB( 3/2 = e utitlA x d 2/esaB x )e

76.6071 = 2/8*)08*8( 3/2 =

04 *3 1* 4

W1ab L1

= 03 k mN

= 08 k mN

06 kNm

wL2 01 (8)2

40* 6 4

W2 L3

6a1x1 6a2 x2

L1 L2

6a1x1 6a2x2

4 8

A B C

B




05

nisU g uqe ni seulav eseht a )i( noit 42 MB 8+ MC = - )021(5.1 – 57.0 )76.6071( 42 MB 8+ MC = - 5200.0641 -- --- - --- - ------ )ii(

F :DC dna CB napS ro

M L2 + 2M (L2 + L3 ) + M L3 = - -

M (8) + 2M C (8 + 6) + 0 = - -

8 MB + 82 MC = - 57.0 a2x2 - a3x3 --------- - ---- )iii( a2x2 = moM e fo tn a er a fo B .M. ud D e ldu ot

saB( 3/2 = e x )edutitlA x B 2/esa 76.6071 = 2/8*)08*8( 3/2 =

a3 x3 D.M.B fo aera fo tnemoM = daol tniop ot eud *h*b * ½ = L 3/

= 3/6*06*6 * ½ = 063

nisU g )iii( noitauqe ni seulav eseht 8 MB + 82 MC = - 57.0 )76.6071( – 063 8 MB + 82 MC = - 5200.0461 --------- - -------- )vi(

F mor ( )ii & ( )vi MC = 625.54 mNk MB = 756.54 mNk

Re tlus : MA = MD = 0 MC = 625.54 mNk MB = 756.54 mNk

BA maeb revelitnac A o ta depporp dna A ta dexif si m6 naps f B . ehT aeb m fo ldu a seirrac /Nk2 m hw sti revo htgnel elo . F pporp ta noitcaer eht dni e d e .dn

noituloS : G :nevi

L= ,m6 m/Nk 2= w

en ldu eht ot eud B ta noitcelfed drawnwoD g nitcel g r porp eac ,P noit

4

yB = 8EI ta noitcelfed drawpU B ta P noitcaer porp eht ot eud B en g nitcel g ht e ,ldu

6a2 x2 6a3x3

L2 L3 B C D

6a2x2 6a3 x3

8 6 B

lw




15

Pl3 3EI

wnwoD = noitcelfed drawpU a d dr e lf ce noit

Pl3 wl 4

yB =

= 3EI 8EI

W3 = P L nK 5.4= 8/6*2*3 = 8/




25

UN III TI

ULOC M SN

ULOC M SN

fI eht rebmem fo ht e st erutcur si lacitrev na d b hto fo its sdne are if x de digir ly elihw rpmoc laixa ot detcejbus e viss e ht ,daol e bmem e nwonk si r a .nmuloc s

:elpmaxE itrev A .roolf dna foor eht neewteb rallip lac

RTS UTS

fI eht rebmem of eht ts r erutcu si ton lacitrev na d eno )ro( htob fo sti dne s si degniL o( r) b eht ,detnioj nip a nwonk si r a turts s .

nitcennoC :elpmaxE g sdor notsip ,sdor e ,ct

SESSERTS EHT NOITNEM W RA HCIH SER E P ELBISNO F ULOC RO M F N A .ERULI

i. rts evisserpmoc tceriD e ss e s ii . B nilkcu g sesserts

.iii rid fo denibmoC e nilkcub dna evisserpmoc tc g rts e sess .

UE EHT NI EDAM SNOITPMUSSA L ULOC S’RE M YROEHT N

1. llaitini si nmuloc ehT y aol eht dna thgiarts yltcefrep yllaixa deilppa si d . 2. ssorc ehT - uorht mrofinu si nmuloc eht fo noitces g l sti tuoh eng ht . 3. ehT nmuloc lairetam si tcefrep ly ,citsale egomoh n suoe dna ciportosi dna syebo

ekooH ‟s al w. 4. iew fles ehT g en si nmuloc fo th g il g lbi e.

O SNOITIDNOC DNE F LOC SNMU

1. B o( degnil era nmuloc eht fo sdne eht hto r nnip e )d 2. ehto eht dna dexif si dne enO r e rf si dn ee . 3. a nmuloc eht fo sdne eht htoB r xif e e .d 4. ehto eht dna dexif si dne enO r nnip si e .d




35

O ERULIAF EHT NIALPXE F NMULOC GNOL noituloS :

A gnol muloc n fo mrofinu sorc s- lanoitces ar ae A dna fo htgnel ,l cejbus t de ot na ax lai evisserpmoc daol ,P sa nwohs ni .gif A nmuloc si nwonk sa gnol nmuloc fi eht htgnel fo eht

nmuloc ni pmoc a nosir ot sti aretal l snoisnemid si ev ry al r .eg cuS h c ulo m sn od ton iaf l y c )gnilkcub nwonk osla( gnidneb yb osla tub ,enola gnihsur

ehT daol , ta hw i hc ht e c nmulo suj t ,selkcub si kn nwo sa bu lkc i gn daol and ti si ssel naht nihsurc eht g nihsurc eht naht ssel si daol g nol a rof daol g nmuloc .

gnilkcuB daol si osla nwonk sa irc t ci la tsuj )ro( gnilppirc .daol ehT lav ue fo lkcub i gn

daol rof ol ng snmuloc are gnol snmuloc si wol w aereh s of r trohs uloc mns eht ulav e fo lkcub ing ih si daol gh.

L te

l = nel g nol eht fo ht g nmuloc p = L ( dao c ht hcihw ta )evisserpmo e c h nmulo a suj s

elkcub d . A = ssorC - era lanoitces a h fo e c nmulo e = nidneb mumixaM g ht ta nmuloc eht fo e ec rtn e.

= daol tcerid ot eud ssertS =

nidneb ot eud ssertS g eht fo ertnec eht ta c nmulo P ́ e

Z hW e er

Z b fo sixa eht tuoba suludom noitceS = e nidn g.

ehT no sesserts emertxe t dim eh - nevig era noitces by

mumixaM s rt e = ss

rts muminiM e = ss

ehT nmuloc lliw liaf wh ne am x mumi sserts )e.i( s + si erom eht gnihsurc st sser fc .

nI esac of ol ng ,nmuloc eht tcerid pmoc r evisse sesserts era n lge i elbig sa derapmoc ot ilkcub n g esserts s. ecneH rev y l no g snmuloc are ejbus c det to ub c nilk g ts r esse s .

P A 0 s

b s =

=

s 0 b s +

s - 0 b s

0 b s




45

ETATS EHT SNOITPMUSSA MA ED NI EHT S’RELUE MULOC N HT E .YRO AND NMULOC NI DEREDISNOC SNOITNEVNOC NGIS EHT NIALPXE S .

llof ehT o ussa eht era gniw m snoitp m eht nmuloc s’reluE eht ni eda o yr :

1. a deilppa si daol eht dna thgiarts yltcefrep yllaitini si nmuloc ehT x lai l y 2. ssorc ehT - ht mrofinu si nmuloc eht fo noitces r uo g l sti tuoh eng ht . 3. ehT nmuloc lairetam si tcefrep ly el ,citsa suoenegomoh dna ciportosi dna

syebo ekooH s‟ wal . 4. ehT htgnel fo eht oc l nmu si yrev l egra sa co derapm ot its tal e lar

snoisnemid 5. sserts gnidneb eht ot derapmoc sa llams yrev si sserts tcerid ehT 6. b liaf lliw nmuloc ehT y nilkcub g enola . 7. fles ehT - en si nmuloc fo thgiew g il g elbi .

uloc ni deredisnoc snoitnevnoc ngis eht era gniwollof ehT m :sn

1. A tnemom hcihw lliw dnet ot dneb eht nmuloc htiw sti evnoc xity ot w sdra its

c laitini e vitisop sa nekat si enil ertn e . 2. A tnemom hcihw lliw t dne ot dneb ht e muloc n htiw its vacnoc ity ot w sdra its

c laitini e en sa nekat si enil retn ga vit e.

EVIRED EHT RPXE E NOISS ROF IRC P LP I GN DAOL NEHW EHT HTOB DNE S FO DEGNIH ERA NMULOC EHT

noituloS : redisnoC a nmuloc BA fo htgnel L degnih ta htob sti sdne A dna B seirrac na ax lai

c nilppir g daol a A t .

na redisnoC y es c X noit - morf x fo ecnatsid a ta X B.

L X ta noitcelfed eht te - si X y.

\ nidneb ehT g X ta tnemom - = M ,P daol eht ot eud X - P.y

2 = = -k 2 y

IE

p EI

2

+ k 2 y 0 = dx2

reffid siht fo noituloS e uqe laitn a si noit

d y - yP dx2

erehW k = 2

d y \ `




55

y = Ac so xk + B nis kx

\ y = A soc xç ø è

By radnuoB gnisu y ,snoitidnoc

,0 = x 0 = y Þ 0 = A x = ,l y = 0

p

EI

= 0

= 0,p ,2p ,3p .. ... .

t gnikat woN h tsel e )e.i( eulav tnacifingis p

p EI

p EI l 2

\` hT e reluE ‟ nilppirc s g daol for nol g c htob htiw nmulo e nih sdn ge .d

p EI l 2

EVIRED NOISSERPXE EHT ROF LKCUB I GN DAOL )RO( CRI ILPP NG DAOL NEHW DEXIF ERA NMULOC EHT FO SDNE HTOB

noituloS : redisnoC a nmuloc BA fo htgnel l if x de ta htob eht sdne A dna B dna seirac na ax lai

gnilppirc aol d P ta A due ot hcihw bu lkc i gn occ .sru rednU eht tca i no of eht daol P eht uloc mn d lliw e lf ce if ni nwohs sa t g.

na redisnoC y X noitces - X ta noitcelfed eht teL.B morf x ecnatsid a ta X - si X y.

ö ÷ p EI

æ p EI

÷ è

+ B si n xç ÷

ö æ ÷ ç ç ø

tA B , tA ,A

\ = 0 B sin l

lniS

l ´

p EI

p

EI

p EI

l æ è

; = p ÷ ç l = p ö ø

2 2

p = 2

p = 2




65

tixif ot euD y .M si B ro A ta tnemom eht tel ,sdne eht ta

\ M = XX ta tnemom latoT – P.y fiD f itnere a uqe l a o noit f ht e e citsal c vru e si

2

EI = M - Py

d y py M dx 2 EI IE

d y py M p dx 2 EI IE p

d y py P M dx 2 EI EI P

reneg ehT a oitulos l itnereffid evoba eht fo n a uqe l a si noit

y = A soc x( P / EI )+ B s ni x( P / EI )+

rehW e dna A B etni eht era g tsnoc noitar a tn

,tA N . dna 0 = x y = 0 \ F mor ( )i

0 = A ́ 1 + B ́ 0 +

A = -

nitaitnereffiD g x ot tcepser htiw )i( noitauqe eht ,

iS n(x. P / EI )+ B

dne dexif eht tA B x , = 0 a dn = 0

xd

P EI

d y dx 2

+ = 2

+ = ´ 2

+ = ´ 2

M P

)i( + 0

M p

M p

P EI

= -A P EI ø

æ oC sç x.

è ç yd

dx ÷ ÷ ö P EI

yd

\ B = 0




75

rehtiE B = o( 0 r)

¹ 0 sa p ¹ 0

B = 0

dna B = uqe ni 0 a )i( noit

æ y = - oc sç x.

è

ê1 - c so ç x.. ë

0 = y ,l = x ,A dne dexif eht ta niagA

0 = [1 - oC s (l. P / EI )]

l. P / EI = 0,2p ,4p ,6p ........ is tsael eht ekat woN g ulav tnacifin e 2p

= 2p

l.2 ´

2

P =

\ nol rof daol gnilppirc ehT g tob nehw nmuloc dexif era nmuloc eht fo sdne eht h

4p EI L2

P EI

ecniS

M p

S sbu A = -

÷ EI P P M

ø ç ö M

P + ÷

ù ç

û ÷ú ê è ø

é æ P EI

M P

y = ö ÷ú

M P

l. P EI

P EI

= 4p 2

4p EI 2 l

P = 2

P EI

= 0




85

EVIRED EHT NOISSERPXE ROF RC I LPP I GN AOL D HW EN COL NMU IW TH ENO NE D F A DEXI N NE REHTO D D NIH G DE

loS u :noit

fo BA nmuloc a redisnoC htgnel l if x de ta B degnih dna ta .A na seirrac tI ax lai gnilppirc

daol eht hcihw rof A ta P c ub tsuj nmulo c selk . ht ereh sA e nmuloc A if si B x ta de ereht ,B if emos eb lliw x ne de d tnemom B ta . L eb ti te

M. nixif siht ecnalab oT g eb lliw H hsup latnoziroh a ,M tnemom e .A ta detrex X noitces yna redisnoC - ta X a ecnatsid x orf eht m if x de dne .B noitcelfed eht teL ta xx si

y.

l( H = xx ta tnemom gnidneB - )x - Py

\ eht fo noitauqe laitnereffiD e citsal c vru e ,si

2

EI = H (l - x) - Py

d y P 14(l - x) dx 2 EI EI

d y P H (l - x) p dx 2 EI EI P

d y P H (l - x) ´ p dx 2 EI EI EI reneg ehT a uqe tnereffid evoba eht fo noitulos l a si noit

æ

y = A oc sç x. è

dna A erehW B etni fo stnatsnoc eht era g .noitar )i(

,0 = x ,B tA y 0 =

\F mor ( )i

P EI

d y dx 2

+ y = 2

2

+ y = ´

2

+ y =

ç p H (l - x) ö æ

÷ ç ÷ ø ÷ p

EI ø è +

ö EI P

+ B is nç x. ÷

A = - lH P

= B H P




95

B = ´

P dne eht ta niagA ,l = x ,A .0=y \ B dna A ,y ,x fo seulav eseht etutitsbus i i( noitauqe n )

oC s(l. P / EI ) +

Sin.(l. P / EI ) ö= oC s(l. P / EI )

ø

at n(l. P / IE .l)= P / IE .l

at n( P / IE .l) lno y nat esohw elgna g tne uqe si a snaidar 94.4 tuoba si ,flesti ot l .

P / IE . l = 4. 94

l = ( .4 49)

(a xorpp )

2p EI l 2

\ nilppirc ehT g daol ( nilkcub )ro g daol f nih dne eno dna dexif dne eno htiw nmuloc eht ro g .de

2p EI 2

EVIRED EHT NOISSERPXE ROF GNILKCUB DAOL ROF EHT NMULOC HTIW DNE REHTO DNA DEXIF DNE ENO F EER

noituloS :

redisnoC a nmuloc BA fo el n htg ,l if x de ta B dna eerf at ,A rac r iy ng an ax lai lppir i gn daol P ta D de ot hw i hc it suj t .selkcub ehT fed l detce mrof fo eht uloc mn BA si nwohs ni fi .g

L A si A fo noitisop wen eht te 1.

a teL .dne eerf eht ta noitcelfed eht eb na redisnoC y X noitces - .B morf x ecnatsid a ta X

IE p

0 = - EI P

iS n(l. P / EI ) Hl P

EI p

H P

lH

P ç H

P ÷ ç

æ

è ÷

ehT eulav of ni snaidar sah ot eb hcus taht sti tnegnat si lauqe ot .flesti ehT

2 2 P EI P EI

2 2 l = 2p

P = 2

l P =

2

H




06

L si xx ta noitcelfed eht te y.

x ta P daol lacitirc ot eud tnemom gnidneB x ,

2 M = EI = P(a - y)

2 EI = aP - py

d y py pq dx2

reffid evoba eht fo noitulos ehT e ,si noitauqe laitn

+ a dna A erehW B etni fo stnatsnoc era gra .noit ø è

,0 = x ,B tA 0 = y

\ F mor ( A ,)i = 0

nitaitnereffiD g ( noitauqe eht I .w r . ot x

æ

niS ç x. è

dne dexif eht tA B x , = 0 a dn = 0 xd

= 0 B

(\ p ¹ 0)

= A etutitsbuS - ew )i( noitauqe ni 0 = B dna a g ,te

æ y = -a c so ç x.

è

y = aê1- c so ç x..

è

d y dx2

d y dx2

2 + =

EI EI

ö ÷ P EI

æ y = A oc sç x.

è

P EI

÷ ç ö æ ÷ ç + B is nç x. ÷ ø

ç ö ÷ ÷ P

EI æ

oC sç x. è

P EI

P EI

yd ø ø

)ii(

+ B = -A ÷ ç ö yd

dx ÷ P

EI

P EI ¹ 0 As P

EI

ç ÷ ÷ ø

ö P EI

+ a

÷ú ÷ú

P EI

ù æ ö

ê ç é

ø û ë




16

,l = x ,A dne eerf eht tA y ulav eseht etutitsbus ,a = e uqe ni s a )ii( noit

a = aê1- c so ç1 ..

ë

æ cosç1 ..

è

= , , 2 2 2

nikat woN g is tsael eht g ulav tnacifin e,

P EI

P EI

p EI

4l \ nilppirc ehT g daol f ht ro e tiw snmuloc rf dne rehto dna dexif dne eno h ee .

p EI

4l

ETIRW EHT NOISSERPXE ROF RC I ILPP NG DAOL HW EN HT E OB TH DNE S O EHT

ULOC M N A DEGNIH ER

p IE 2

daol gnilppirC = P E = gnuoY s‟ suludoM I moM = e eni fo tn r it a

nmuloc fo htgneL = l

ç ø û ê è = 0

÷ú

é æ ù ö ÷ ú P IE

÷ P EI ø ÷ ç

ö

p 3p 5p P EI

1

p = 2

1 12 p =

2

4

2

2 P =

P = 2

2

l P =

2




26

NOISSERPXE ROF BU GNILKC DAOL O( R) RC I GNILPP DAOL NEHW HTOB EN SD DEXIF ERA NMULOC EHT FO

4p IE

L2



NOISSERPXE ROF ILPPIRC NG DAOL W NEH NMULOC HTIW NO E E DN DEXIF DEGNIL DNE REHTO DNA

2p EI

l 2



NOISSERPXE ROF B ILKCU NG DAOL FOR EHT NMULOC HTIW ENO DEXIF DNA DNE REHTO F EER

p EI

4l 2

daol gnilppirC = P

E = gnuoY s‟ suludoM I eni fo tnemoM = r it a


XE P NEL TNELAVIUQE NIAL G FE )RO( HT F NEL EVITCE G HT

fI l si lautca htgnel fo a ,nmuloc neht sti tnelaviuqe htgnel )ro( evitceffe htgnel L may be deniatbo by pitlum l iy ng it htiw emos tnatsnoc caf t ro ,C hcihw sdneped no t eh ne d if xati no fo eht

)ei( nmuloc L l x C = .

ETIRW EHT TNELAVIUQE HTGNEL )L( FO EHT NMULOC NI HCIHW BOTH DAOL GNILPPIRC EHT ETIRW DNA DEGNIH SDNE

p EI nilppirC g L dao

( htgnel tnelaviuqE L )l( htgnel lautcA = )

P = 2

P = 2

P = 2

L2 P = 2




36

daol gnilppirC = P E = oY ung suludoM s‟

I moM = e eni fo tn r it a neL =L g nmuloc fo ht

ETIRW HT E LER A NOIT TEB W NEE E VIUQ A TNEL NEL G HT NA D AC UT AL

HTGNEL F O SNOITIDNOC DNE LLA RO F ULOC MN.

B gnil sdne hto ed

B xif sdne hto ed

nO e e xif dn e d a ehto dn r degnih dne

nO e e xif dn e d a ehto dn r f dne r ee

A dlim leets ebut 4m ol n ,g mc3 lanretni retemaid dna mm4 kciht si desu sa a turts w hti

h sdne htob i .degn gnilppirc eht eb lliw tahw ,daol gnispalloc eht dniF l fi dao

.i tliub era sdne htoB i ?n .ii tliub si dne enO – ni and erf si dne eno e?

loS u :noit

G :nevi

m4 = l ,ebut leets dlim eht fo htgnel lautcA = 004 c m I nretn a o retemaid l f ,ebut eht mc 3 = d

nkcihT e but eht fo ss e mm4 = t , = 4.0 c .m

\ t eht fo retemaid lanretxE t2 + d = D ,ebu = 3+2( )4.0 = 8.3 mc .

nimussA g E for ets e 1 x 2 = l 06 Kg mc/ 2

.O.M I ,noitces nmuloc eht fo

[D 4 - d ]

= [(3.8) - (3) ] 46

I = 62.6 c m 4

l 2 l

L = l tnatsnoC = 1 1 2

1 2

L = L =

2

L = 2l

tnatsnoC =

tsnoC a = tn

tnatsnoC = 2

I = 4 p 46

4 2 p




46

i. eht fo htgnel evitceffe eht ,degnih era ebut eht fo sdne htob eht ecniS nmuloc nehw ob sdne ht are nih ged.

L = l = 004 cm

2

\ reluE s‟ gnilppirc daol Þ

p ´ 2´106 ´ 6. 62 2

Prc = 77 .2 30Kg. \ daol despalloc deriuqer ehT = K 03.277 g. ii . tliub era nmuloc eht fo sdne htob nehW – ni ,

nel evitceffe neht g ,nmuloc eht fo ht

L = = = 02 0 mc 2 2

\ reluE s‟ gnilppirc ,daol

p EI L2

2

= ( 02 0)

Pcr = K 91.9803 g. ii i. o eht dna ni tliub si nmuloc eht fo dne eno nehW the r e rf si dn ee ,

nel evitceffe g ,nmuloc eht fo ht L l2 = = 04 x 2 0

mc 008 =

\ reluE s‟ gnilppirc ,daol

2

Prc =

2

= ( 08 0)

Pcr = K 70.391 g.

p EI L2 Prc =

= ( 04 0)

2

l 004

Pcr = 2

p ´ 2´106 ´ 62.6 2

p EI L2

p ´ 2´106 ´ 62.6 2




56

A nmuloc gnivah a T noitces htiw a f egnal 021 mm x 61 mm dna w be 051 mm x 61 mm si

m3 .gnol ssA u gnim eht nmuloc ot eb degnih at tob h en ,sd nif d eht irc p ilp ng aol d yb gnisu E of s’relu rm .alu E 1 x 2 = 06 K c/g m2.

loS u :noit

G :nevi

nalF ge htdiw = mm 021 = 21 c m nalF g iht e c nk e ss = mm 61 = 6.1 c m

L ne g w eht fo ht eb = mm 051 = 51 c m w eht fo htdiW eb = m61 m = 6.1 c m

E = 2 01 6 mc/gK 2

L ne g 003 = m3 = l ,nmuloc eht fo ht c .m

degnih si nmuloc eht ecniS a nel evitceffe ,sdne htob t g eht fo ht c .nmulo

L = l = 003 c .m

morF ht e .gif Y-Y si eht sixa fo temmys r .y \ T eh G.C fo eht elohw noitces seil no Y- Y

sixa .

Y = noitces eht fo rebif tsompot mm 61 eht morf G.C eht fo ecnatsid eht teL

.1 6 15 2 2

12 ́ 6.1 + 15 ́ 6.1

Y = 4.5 1 mc

rbif mottob morf G.C fo ecnatsiD e 51( = + )6.1 - 14.5 = mc91.11

X tuoba noitces elohw eht fo I.O.M woN - sixa X .

3

+ (12 ́ .1 6)æ 14.5 - ÷ ú ê

I XX = 1188.92 mc 4

12 ́ 6.1 ́ + 15 ́ .1 6æ 6.1 + = \Y è ø

ç ÷ ö

ê é 21

2

+ ( 6.1 ́ 15)æ 91.11 - ÷ ú ù ´( .1 6)

21

2

2 12 è ø ç

2 ë û ë I =

ù é1 6. ´(15)3

+ XX ê û

ç ú ê è ø ú

15 ö 1 6. ö




66

.M I Y tuoba noitces elohw eht fo - sixa Y

.1 6´(12)3 + 15´( 01 6)3 = 32 5.52cm4

12 21

= 32 5.52cm4

\ reluE ‟ nilppirC s g ol a ,d

p EI L2

p ´ 2´106 ´ 32 5 25. 2

A leets rab fo dilos ralucric sorc s- noitces si 05 mm in d mai e ret . ehT rab si pi nn de ta htob

sdne dna detcejbus ot laixa .noisserpmoc fI eht timil of ytilanoitroporp fo eht airetam l si G 002 = E dna aPM 012 P reted ,a m eht eni m m ini mum ot htgnel hcihw s’reluE

alumrof si .dilav oslA enimreted eht lav ue fo s’reluE gnilkcub daol fi eht nmuloc sah siht m ini mum .htgnel

loS u :noit

,neviG ssorc ralucric dilos fo aiD - ,noitces mm 05 = d

pM 012 = f ,timil lanoitroporp ta ssertS a

= 012 mm/N 2

gnuoY ,suludoM s‟ 01 x 002 = aPG 002 = E 3 /N mm2

A = ´ (50)2 = 1963 4. 9mm2

4 L ,noitces nmuloc eht fo aitreni fo tnemom tsae

´ (50)4 = 7.6.3 9´ 01 3 mm 4

noitaryg fo suidar tsaeL ,

k 2 = = ´ (50)4 = 2.651 5mm2

� b ehT a dennip si r a htob t e ,sdn \ ,htgnel evitceffE L A = c nel laut g l ,ht

I yy =

\ I min

Prc = 2

= ( 03 0)

2

Prc = 15 56 .5 32Kg. ;

p rA ae fo c ssor – es c ,noit

I = p 46

I 97.603 ́ 103

A .3691 49




76

\ reluE s‟ gnilkcub ,daol

2

Prc =

2 =

(L / K )2

roF reluE s‟ alumrof ot eb ,dilav eulav fo sti muminim evitceffe htgnel L yam eb dnuof ot sserts gnilkcub eht gnitauqe yb tuo f

p E 2

÷

p ´ 2 ́ 105 ´15 .6 25 210

m 212.1 = mm 2121 = mm 98.1121 = L

\ = l htgnel lautca muminim deriuqer ehT L m 212.1 =

oF r ulav siht e nel muminim fo g ,ht

p EI L2

p ´ 2´105 ´ 03 6.75´103

( 121 2) = N 452214 = NK 452.214

12.1 = L = l htgnel lautca muminiM 2 m reluE ‟ nilkcub s g Load = NK 452.214

O LENREK )RO( EROC F NOITCES A

W neh a daol stca ni such a way no a re noig a or u dn eht GC of eht s oitce n oS taht ni taht noiger sserts eve ryw eh er si evisserpmoc dna on t noisne si epoleved d an erehwy , neht taht ar ae si

ht dellac eroc e )ro( lanreK fo a .noitces ehT lenrek fo eht noitces eht si aera nihtiw hcihw eht enil ssorc eht tuc tsum P daol cirtnecce eht fo noitca fo - rts eht fi noitces e t emoceb ot ton si ss e elisn .

p EI L2

A P rc p E

æ è

2

= 210

L2 =

ç

2

L K

ö ø

p E ́ k 2

12 0 L2 =

2

= =

2

reluE s‟ gnilkcub daol Re :tlus

2

2




86

NOITCES RALUGNATCER A FO EROC ROF NOISSERPXE EHT EVIRED ehT timil fo ce c icirtne ty fo a r atce n lug ar noitces b x d no rehtie edis fo XX ax si )ro( YY

6/d si sixa isnet diova ot o natcer eht fo eroc esab eht ta n g alu r s ce noit .

natcer eht fo eroC g dahs eht fo aerA = noitces ralu e noitrop d

= 2 ́ ´ 2

bd 18

EVIRED EHT PXE R NOISSE ROF EROC FO A SOL DI CRIC U RAL SE ITC ON FO ID MA D RETE

ehT timil fo icirtnecce ty no rehtie edis fo htob XX )ro( YY ax si = 8/D ot diova noisnet fo

b eht a .es noitces ralucric eht fo eroC noitrop dedahs eht fo aerA =

= p(D / 8)2 pD 2

(d ) = ( 06 0) 63.6 = ´109 mm 4

cniS e eht c degnih si nmulo a ht t e ,sdne htob

\ htgnel tnelaviuqE L l =

p IE

L2 p ´ .2 1´10 5 ´ 6 63. ́ 10 9

( 0008 )2 = 0.2 6´108 N

d 6

1 b 3 ´

=

p p 46 46

= 46

A LEETS NMULOC SI FO HTGNEL 8M DNA ID AME ET R 006 MM IW TH BOT H

NE DS EGNIH D. ENIMRETED EHT PIRC P NIL G AOL D BY E RELU ’S F ALUMRO . kaT e /N mm2. E = 2.1´105

4 4 I =

P rc = 2

= 2




96

ITAR SSENREDNELS O

tI si d denife sa eht itar o fo eht evitceffe el n htg fo eht c nmulo ( )L ot ht e tsael suidar of

gy ssorc sti fo noitar – )K( noitces ( r eht )e.i a fo oit nwonk si a .oitar ssenrednels s K

dnelS e nr e r ss a oit =

ETATS T TIMIL EH A O SNOIT F ROF S’RELUE M ALU

a. reluE ‟ of s r lum a acilppa si r ssenrednels eht nehw elb a si oit g 08 ot lauqe ro naht retaer b. luE er‟ of s r lum a cilppa si a lno elb y of r nol g c nmulo c. reluE s‟ alumrof si suht elbatiusnu nehw eht ssenrednels oitar si ssel th na a ce niatr

ulav e .

NAR EHT ETIRW K S’ENI F ALUMRO F ULOC RO M SN

f c ´ A 2

1 + aæ ö

è ø

L g fo suidar tsae y noitar

daol gnilppirC nmuloc eht fo aerA

ped eulav tnatsnoC e m eht nopu sdn a et r .lai

= eniknaR s‟ tnatsnoc

NAR EHT ETIRW K ALUMROF S’ENI F CE RO C CIRTNE C ULO MN

f c ´ A

æ1 + öé1 + a æ ö

è øë è ø ú

L tsae g fo suidar y noitar

daol gnilppirC nmuloc eht fo aerA

ped eulav tnatsnoC e m eht nopu sdn a et r .lai

L

L K

P = =

= = =

L ç ÷ K

= I A

K

P A fc a =

2 fc

p E

2 ê ú û ê

eyc L ù 2

ç ÷ ç ÷ k k

P =

K =

P = A = fc =

= I A




07

a = eniknaR s‟ tnatsnoc

nialpxE aR n s’enik Fo alumr dna evireD eht s’eniknaR alumrof rof htob trohs dna gnol c ulo mn.

loS u :noit

s’eniknaR F ro m :alu

reluE s‟ alumrof sevig tcerroc stluser ylno rof gnol ,snmuloc hcihw f lia niam ly eud to ub c nilk g. saerehW eniknaR s‟ desived na laciripme alumrof esab nod pr lacitca ex stnemirep rof

gninimreted the ilppirc ng ro lacitirc daol cihw h si elbacilppa ot lla oc l nmu s tcepserri i ev of eht rehtehw y nol ro trohs a g.

I ht si P f e c nilppir g b daol y Ra enikn s‟ for lum a .

Pc nihsurc eht si g eht fo daol c iretam nmulo a l PE si eht gnilppirc reluE yb daol alumrof s‟ .

hT e eht n E iripm ca d alumrof l e b desiv y Ra nwonk enikn a R s a nikn e‟ lumrof s a dnats a :s

=

P Pe

roF a trohs ,nmuloc fi eht evitceffe htgnel si ,llams eht eulav fo PE lliw eb ev ry hi hg dna

1 1 v eht a o eul f rev eb lliw y rapmoc sa llams e ot d en si dna g il g lbi e.

trohs eht roF c )e.i( ,nmulo c

hT su rof the trohs ,nmuloc eulav o gnilppirc f load by R enikna si mo er or le ss lauqe ot eht

nihsurc fo eulav g :daol

1 gnivah nmuloc gnol roF rehgih ,htgnel evitceffe eht eulav fo PE si llams dna

ral eb g uone e g ot nosirapmoc ni h

1 PE

+ 1 1

PC PE

1 P

= P 1 P = PC

PE iw ll

1 PC

1 PC

. oS i si g ron e .d

= 2 fc

p E




17

\ nol eht roF g ,nmuloc

suhT rof eht ol ng nmuloc eht eulav fo ilppirc ng l dao by R enikna si rom e ro ssel lauqe ot nilppirc fo eulav eht g b daol y reluE .

1 PE

PE ´ P P ´ PE

Pc ´ PE PE ´ Pc

2

ht etutitsbuS e va o eul f Pc = f c A an d

fc ´ A

fc ´ A p EI / L2

eht fo sserts gnihsurc etamitlU .lairetam nmuloc

ssorC - nmuloc eht fo era lanoitces

nel evitceffE g nmuloc eht fo ht

kA 2

g fo suidar tsaeL = k erehW y .noitar

fc ´ A

fc ´ A p EI / L2 fc ´ A

æ ö è ø

hw ere a = eniknaR s‟ tnatsnoc

» 1

PC .i( e p ) » PE

+ c

=

=

1 P

1 P 1 P

c

c

1 +

PE =

Pc

PE

p EI L2

p = Pc p =

;

ht ni e a vob e e uq a ,noit

p = 1 +

fc = A = L = I =

2

hW ere, p = =

1+ 2 2 1+

p =

fc ´ A fc ´ A´ L2

p AE k2

L K

1 + a 2

ç ÷

2

fc = p E

1 PE




27

gnihsurC daoL 1 + a(L / k )

nehW eniknaR s‟ tnatsnoc si ton nevig neht nif d

fc a = p E

niwollof ehT g f fo eulav eht swohs elbat c a dn a m tnereffid rof a et r .slai

taM e ir al fc mm/N 2

uorW g nori th 052

Ca nori ts 055

ets dliM el 023

bmiT er 05

dellor A leets BMSI tsioj 003 si ot desu eb a loc u nm fo 3 sretem htgnel w hti b hto sdne f .dexi dniF eht as fe aixa l aol d on eht loc u .nm ekaT rotcaf fo s ytefa ,3 fc = 023 mm/N 2

anda = . uloc eht fo seitreporP m es n c noit . 57 00

Are 6265 = a mm2, I XX 01 x 306.8 = 7 mm 4

I yy 935.4= 01 x 7 mm 4

noituloS : G :nevi

m3 = l ,nmuloc eht fo htgneL mm 0003 = tefas fo rotcaF y 3 =

fc mm/N 023 = 2, a =

A aer A , = 65 mm 62 2

I XX 01 x 306.8 = 7 mm 4

I yy 935.4= x 01 7 mm 4

xif si nmuloc ehT e d a eht htob t e ,sdn

2 P =

2

a = f c

p E 2

1 009 0 1 061 0

1 057 0

1 57 0

1

1 57 00




37

\ ,htgnel evitceffE L = =

2

I ecniS yy I neht ssel si xx , \ hT e es nmuloc c ,noit

I = I min = I = 4. 35 9 ́ 10 7 mm4 \ g fo suidar tsaeL y es nmuloc eht fo noitar ,noitc

935.4 ́ 107 6265

nilppirC g daol a s g b nevi y Ra enik s‟ for lum a,

320 ́ 6265 1 0051

7 05 0 8 .9 82

P rc = 83.2253431 N

fas fo rotcaf gniwollA ety ,3

rC gnilppi daoL rotcaF fo as f yte

= = 48744 0.79 N 3

Re tlus :

i. nilppirC g Loa P( d cr ) = N 83.2253431 ii . daol efaS N97.048744=

A uloc pu tliub m aeb leets dellor fo gnitsisnoc n m owt htiw 003 BWSI p m 002 setal m 01 x dna pot eht ta detcennoc mm ottob m gnalf e .s ht daol efas eht etaluclaC e c ulo m n

3 si htgnel eht fi ,yrrac m ra sdne htob dna e .dexif fo rotcaf ekaT 3 ytefas fc = 023

/N mm2 na d 57 00

kaT e pr po re it e m 3316 = A :tsioj fo s m 2

I XX 01 x 6.1289 = 4 mm 4 I ; yy 01 x 1.099 = 4 mm 4

l 03 00 2

= 051 0 mm

yy

= 89 8. 2 mm = I A

K =

æ ö è ø

æ ö è ø

1 + a L K

fc ´ A 2

ç ÷ 2

ç ÷ pcr = =

1 +

Safe daol =

253431 2 83.

1 a =




47

loS u :noit

G :nevi

L ne g ,nmuloc pu tliub eht fo ht m3 = l mm 0003 =

ytefas fo rotcaF = 3

fc 023= mm/N 2

1

57 00 iub eht fo aera lanoitceS l uloc pu t ,nm

A = 6133 + 2(200 ́ 10)=10 31 3mm 2

xx tuba noitces nmuloc pu tliub eht fo aitreni fo tnemoM a ,six

é ù XX ú

ë û = 49.1 x 01 8 mm 4

YY tuba noitces nmuloc pu tliub eht fo aitreni fo tnemoM a ,six

IYY = 99 .0 1´104 + 2ç è ø

= 32.0 x 01 8 mm 4

iS cn e I yy l si e ht ss a n Ixx hT , e c ub ot dnet lliw nmulo c elk a Y tuob - Y a .six

L eni fo tnemom tsae r eht fo ait c es nmulo c ,noit

I = I min = IYY 2.0 = 3´108 mm 4

dexif si nmuloc ehT a htob t e sdn .

\ htgnel evitceffE ,

= 51 00 mm

\ o noitaryg fo suidar tsaeL noitces nmuloc eht ,

32.0 ́ 108 01 1 33

a =

02 0 ́ 103 2 I = 9 28 1.6 ́ 104 + 2ê + ( 02 0 ́ 10)(155)

12

æ 10´ 3002 ö ç ÷ 12

÷

l L = =

2 03 00 2

J A

= 47.64 mm = K =




57

nilppirC g daol a s g b nevi y Ra enikn s‟ for lum a,

320 ́ 01 133

æ ö è ø

= N 3.3204682

nilppirC g daol ytefas fo rotcaF

Re :tlus

.i nilppirC g daol = 3.3204682 N ii . Safe daol = 4.476459 3 N

vireD e dna s’eniknaR reluE f alumro f ro gnol snmuloc rednu gnol oc l snmu dnu re Eccen irt c gnidaoL ?

i. rof s’eniknaR m :alu

redisnoC a hs o tr muloc n detcejbus ot na ec rtnec ic daol P htiw an ec rtnec i ic ty e mrof eht

sixa .

ssertS tceriD = sserts mumixaM sserts gnidneB +

fc = +

I = Ak 2

k =

eS c noit a a l er ht fo e c nmulo nmuloc eht fo suludom lanoitceS

rtxe fo ecnatsiD e em f morf erbi N A. L g fo suidar tsae y noitar .

pcr = fc ´ A 1 1500

7 05 0 46.74

204682 3.3

æ ö è ø

1 + a 2

ç ÷

= = 76459 4.43N 3

= 1 +

2

ç ÷ L K

Safe daol =

I y

Z = M Z

A Ak

= = = =

P A +

p.e.yc 2

P =

hw e er

A Z yc k

I A




67

P eyc è ø

fc ´ A cE ec irtn c ,daol eyc

k 2

æ ö hW ere ticirtnecce rof rotcaf noitcuder eht si y .gnidaol fo

è ø

,daol gnilppirc eht ,gnidaol laixa htiw dedaol ,nmuloc gnol roF

fc ´ A 2

ø

nilkcub rof rotcaf noitcuder eht si g nol fo g .nmuloc

è

He of ecn r nol a g aol nmuloc nidaol cirtnecce htiw ded g fas eht , e ,daol

fc ´ A

æ1+ öé1+a

è øë è ii . reluE ‟ alumrof s

mumixaM s rts tceriD = nmuloc eht n ssert e B + ss e nidn g rts e ss

P ́ e s ce P / EI

= + A Z

,ecneH eht mumixam sserts decudni ni eht oc l nmu gnivah htob sdne degnih dna na æ ö

ticirtnecce y fo e si + secç P / EI è ø

fc =

\

P = 1+

1 + 2

eyc ç ÷ k

ç è

ö ø

2

÷

1+ a æ

P = ö ÷

L K

ç1

ç L

ö ÷ æ hW ere + aæ

è ø ÷ ç K

ç ÷ ç L K

ù 2

ú ê ê

P = æ eyc ÷ ú

û 2 K

ö ø

P l 2

÷ P Pe l A Z 2 ç ÷

æ1+ ö A k

ç ÷ 2




77

ehT umixam m erts ss decudni ni eht nmuloc htiw rehto dne snoitidnoc era denimreted by nahc g ni g eht nel g ht ni et r sm of eff ce vit e nel g .ht

A nmuloc of ralucric se noitc sah 051 mm aid na d m3 .htgnel htoB sdne of eht loc u nm era

.dexif ehT nmuloc seirrac a daol fo 001 NK ta an rtnecce i ytic fo 51 mm rf om eht lacirtemoeg sixa fo eht .nmuloc dniF eht xam imum comp evisser sserts ni eht

nmuloc s .noitce dniF osla eht mixam um rep m elbissi necce t yticir ot a iov d et n ois n ni uloc eht m es n c .noit 01 x 1 = E 5 m/N m2

noituloS : G ,nevi

D etemai r ht fo e c ,nmulo D = 051 mm

l ,nmuloc eht fo htgnel lautcA = m3 = 0003 mm L ht no dao e c ,nmulo P = x 0001 = NK 001 01 3 N

E = 1 x 01 5 mm/N 2

ticirtneccE y, e = 51 mm

tces nmuloc eht fo aerA i no A =

= ( 51 0 )

4 = 17671 mm 2

aitreni fo tnemoM .A.N noitces nmuloc eht fo ,

I =

= 58.42 x 01 6 mm 4

eS c ,suludom noit

I D / 2 2 .4 85 ́ 106

51 0 2

B a 2 nmuloc eht fo sdne eht hto r .dexif e

evitceffE l ,nmuloc eht fo htgne L = = = 51 00 mm

2 2 na eht ,woN g el

p ´ D 2

4 p 2

4 ´ ( 51 0) p 46

´ D 4 = p 46

I Z = =

y

= = 133 3 93 mm3

l 03 00




87

01 0 ́ 103 0051 1´105 ´ 2 .4 85 ́ 106 2

r 4051.0 a d = 16.8 o

visserpmoc mumixaM e s rt e ,ss

P ́ e Z

01 0 ́ 103 01 0 ́ 103 ´15 ́ es c 16.8 o 1 67 71 331339

= mm/N 22.01 2

t diova oT e w noisn e ,wonk

P M

= A Z

100 ´103 100 ´ 01 3 ´ e ´ es c 6.8. 1o

e = 05.81 mm

sserts evisserpmoc mumixaM mm/N 22.01 = 2 ticirtnecce mumixaM y = mm 05.81

ussa eht etatS m aL evired dna snoitp m oehT s’e r ?y

1. ussa ehT m aL ni devlovni snoitp m hT s’e eor .y

i. llehs eht fo lairetam ehT ciportosi dna suonegomoh si ii . enalP ces snoit lamron ot eht lanidutignol ax si fo eht c rednily niamer enalp

etni fo noitacilppa eht retfa rna rp l e russ e. .iii fo serbif eht llA lairetam eht dnapxe )ro( tcartnoc tnednepedni ly uohtiw t gnieb

rtsnoc a b deni y serbif tnecajda rieht .

2 De L fo noitavir am oehT s’e ry

redisnoC a c kciht y dnil er

ö ø

ç ÷ P / EI P L 2

æ ces

è = +

A

= +

= A Z

P p ́ e´ es c 6.8. 1o

Þ

Re tlus :

= 71 671 3 13 339

.i ii .

L P / IE ´ =

2 =

´




97

L te rc = rednilyc eht fo suidar rennI r0 = c eht fo suidar retuO y rednil Pi = I erusserp laidar lanretn Po = erusserp laidar lanretxE

ht fo htgneL e cy rednil

L no g rts laniduti e ss .

x

b x 2

- a + 2a

+ a

rts pooh e udni ss ec ht ni d e r ni g.

I if eht ni erusserp laidar lanretn g . r lanretxE a p laid er russ e ht ni e nir g.

ehT seulav fo eht owt stnatsnoc a dna ot b era dnuof tuo gnisu eht gniwollof adnuob ry

c snoitidno :

.i etni eht ecniS rna r l a p laid re russ e P si ,i tA r = x i, Px P = i

ii . cniS e eht e etx rna r l a p laid er russ e P si 0

, tA r = x 0, Px P = 0

L = f2 =

emaL s‟ :noitauqE

f = px + 2 a

Px = \

- a

x

x f =

hw e er fx px

Px + Pd x

f = b x 2

b x 2

= = =




08

CIHT K REDNILYC fI eht oitar fo ssenkciht of eht lanretni di retema of a c lacirdnily or rehps ical llehs ex sdeec

demret si ti ,02/1 a hs kciht a s e ll . ehT pooh sserts ed v ole p de ni a kciht llehs rav i se morf a am xim mu eulav ta eht renni

eulav muminim a ot ecnerefmucric a retuo eht t c ri c fmu ere cn e. 02/1 > ssenkcihT

MUSSA EHT ETATS P I DEVLOVNI SNOIT N AL M YROEHT S’E

j. omoH si llehs eht fo lairetam ehT g ne e suo a ciportosi dn . ii . enalP noitces lamron ot eht anidutignol l ax si fo eht c nily d re er m snia enalp retfa

eht a cilpp a o noit f tni e rp lanr e russ e. .iii llA eht srebif fo eht lairetam ex dnap )ro( tcatnoc tnednepedni ly ohtiw ut gnieb

rtsnoc a b deni y f tnecajda ereht bi e sr .

DIM D ELUR DRIHT EL nI atcer n ralug ,snoitces t eh rtnecce ici yt e„ ‟ tsum eb ssel naht ro lauqe ot .6/b ecneH eht

tsetaerg necce t icir ty of eht daol si 6/b mrof eht ax si Y-Y dna htiw tcepser ot ax si X –X1 eht icirtnecce ty seod ton deecxe .6/d neH ce t eh daol may be lppa i de htiw ni t eh mi eldd ht i dr fo th e

sab e ( 3/d elddiM )ro . A kciht eets l rednilyc gnivah na lanretni retemaid fo 001 mm an txe e lanr emaid ter of 002

mm si detcejbus ot na in lanret erusserp fo 55 M ap dna na extern la p sser u er of 7 pM a. F eht dni m ixa mum sserts pooh .

noituloS :

,neviG

c eht fo suidar rennI y ,rednil

c eht fo suidar retuO y ,rednil

,erusserp lanretnI Pi = 55 apM

,erusserp lanretxE P0 = 7 apM

nI eht pooh serts s dna laidar sserts ni the c rednily ta a sid t na ce of x morf t eh c rtne e si fx p dna x re ps ec vit ely nisu , g Lame s‟ e uq a ,snoit

i( )

)ii(

ri = = 50 mm 2 001

200 ro = = 100 mm 2

f = Px =

+ a x b x 2

- a b x2




18

b dna a erehw era stnatsnoc ,

b woN y P ,mm 05 = x ta ,noitauqe x radnuoB( aPM 55 = y )noitidnoc

adnuob eseht gnisU r )ii( noitauqe ni noitidnoc y

Px =

55 =

,mm 001 = x nehT px apM 7 =

radnuob eseht gnisU y )ii( noitauqe si noitidnoc

b 01 02

nivloS g ( )iii & )vi(

b /( 01 0)2 - a = 7

b /(50)2 - a = 55 (- ) (+)

3b 10000

b = 000061

a = 9

etutitsbuS a & uqe ni b a )i( noit

f = + 9

x 2

f fo eulav ehT x m si a ix m hw mu e im si x n n mumi f suhT x m si a of mumix r x = ri = m 05 m

\ sserts pooh mumixaM = + 9

= pM 37 a t( e lisn e) Re tlus :

- a b x2

- a b (50)2 )iii(

7 = - a vi( )

- = - 84

x 061 00 0

2 (50) 160000




28

pooh mumixaM s )elisnet( aPM 37 = ssert

tsac A i nor epip sah 002 mm lanretni etemaid r a dn 05 mm em lat ciht kn .sse tI seirrac retaw rednu a erusserp fo 5 N/ mm 2. dniF eht xam imum dna im n mi um seitisnetni fo

ucric mf re laitne ts re .ss oslA hcteks eht noitubirtsid of mucric f laitnere sserts dna ts laidar r ca sse r ht sso e es c noit .

loS u :noit

G :nevi

I nretn a maid l ,rete di = 002 mm

nkciht llaW e ,ss t = 05 mm I ,erusserp lanretn Pi = 5 mm/N 2

etxE rna rp l e russ e, P0 = .0 \ I nretn a suidar l

r0 = ri + t =100 + 50 = 51 0 mm teL fx dna Px eb eht laitnerefmucric sserts dna laidar sserts ta a ecnatsid fo x morf eht ertnec fo

levitcepser epip eht y.

\ nisU g L ema s‟ e uq a ,snoit

f =

px = - a

yrartibra era b & a ,erehw .stnatsnoc

oN w a x t = P ,mm 001 x = mm/N 5 2

x tA = P ,mm 051 x = 0

radnuob gnisU y )ii( si noitidnoc

b ( 01 0)

b ( 51 0)

)iii( gnivlos yB a )vi( & ; 4 = 0009 = b 0

id ri = = 2

= 01 0 mm 002 2

etxE rna r l a suid

+ a x b x 2

b x 2

i( )

)ii(

)ii(

vi( )

5 =

0 =

2

2

- a

- a




38

fx = + 4,

x 2

nittuP g m ,mm 001 = x a itnerefmucric ix a rts l e .ss

f x = + 4 = 13N / mm 2 ( lisnet e)

nittuP g im ,mm 051 = x n ric i c fmu e rts laitner e .ss

f x = + 4 = 8N / mm 2 ( net s eli )

dnuopmoc ni sesserts eht nialpxE ednilyc kciht rs .

noituloS : c kciht dnuopmoc a redisnoC y rednil a if ni nwohs s g.

L ,te r1 = c dnuopmoc eht fo suidar rennI y rednil

r2 = c owt eht fo noitcnuj eht ta suidaR y srednil

r3 = rednilyc dnuopmoc eht fo suidar retuO

W neh eno rednilyc si sh knur revo eht ,rehto renniht c rednily si rednu co sserpm i no dna eht retuo c rednily si nu d re .noisnet euD ot iulf d erusserp edisni eht c ly i ,redn pooh sserts lliw

deve .pol ehT tnatluser pooh sserts ni eht dnuopmoc sserts si taht egla b ciar mus fo eht ho po sserts gaknirhs laitini ot eud e a ud taht dn e rp diulf ot e russ e.

a. knirhs laitini ot eud sessertS age :

lppA y ni g La em E s‟ qua of snoit r tuo eht e c r y dnil e ,r

Px = - a1

fx = + a1

r = x tA 3, Px 0 = dna r = x ta 2, px = p

lppA y ni g La em E s‟ qua of snoit r nni eht e c r y dnil er

\

2 ( 01 0) 90000

2 ( 51 0) 90000

b1

x2

b1

x 2

00009 Px = - ,4 x 2

0009 0




48

Px = - a2

fx = + a2

r = x tA 2, Px p = dna r = x ta 3, px = 0

b. ot eud sessertS I etn rp diulf lanr e russ e.

oT dnif eht sserts ni eht dnuopmoc rednilyc eud ot anretni l diulf erusserp ,enola eht renni dna retuo c dnily e sr lliw eb deredisnoc to rehteg sa eno ht i kc .llehs woN gniylppa emaL s‟ uqE a ,noit

Px =

f =

r = x tA 1, Px p = f ( Pf )erusserp diulf lanretni eht gnieb

r = x tA 3, px = 0

ehT r tnatluse pooh ts r se s si eht ciarbegla mus of t eh pooh ts r sse ud e ot hs r kni i gn na d eud rp diulf lanretni e russ e.

A co dnuopm rednilyc si co desopm of a ebut of 052 mm in ret n la d temai er at 52 mm w lla

nkciht e .ss tI si knurhs no ot a ebut fo 002 mm in lanret dia em et .r ehT laidar erusserp ta eht noitcnuj si 8 N/ mm 2. dniF eht noitairav of h oo p ts r sse sorca s eht

oc eht fo llaw m fi ,rednilyc dnuop tni na rednu si ti e mm/N 06 fo erusserp diulf lanr 2 oituloS n: G :nevi I nretn a id l a o retem f d ,ebut retuo eht 1 mm 052 =

nkciht llaW e eht fo ss o etu r ebut r , t = 52 mm I nretn a o retemaid l f nni eht e but r e d , 2 = 002 mm

usserp laidaR er a ht t e noitcnuj P = 8 mm/N 2 P rednilyc eht nihtiw erusserp diulf lanretnI f = 06 mm/N 2

\ c dnuopmoc eht fo suidar lanretxE y ,rednil

d1 + 2t 2

- A

+ A

B x 2

B x 2 x

r2 =

b2

x 2

b2

x 2




58

= (250 + 2´ 25) = 51 0mm

c dnuopmoc eht fo suidar lanretnI y ,rednil

r1 = =

Ra ht ta suid e ,noitcnuj

teL eht laidar sserts na d pooh sserts ta a ecnatsid fo x morf eht rtnec e of eht rednilyc eb

px dna fx re eps c vit ely. i. pooH sesserts eud ot niknirhs g of ht e etuo r dna renni c rednily s be of re diulf

p er russ e ttimda si e .d a. F dnilyc retuo ruo re :

gniylppA emaL noitauqE s‟

Px = - a1

f = + a1 )ii(

a erehW 1 b dna 1 era .rednilyc retuo eht rof stnatsnoc yrartibra

oN w a x t = P ,mm 051 x = 0 X = P ,mm 521 x = mm/N 8 2

o = - a1

8 = + a1

a )vi( & )iii( noitauqe gnivloS 1 ; 81 = b1 190904 =

f = + 18 v( ) x 2

a sserts noitauqe evoba eht ni mm 051 = x gnittuP ,ecafrus retuo eht t

f = + 81 = 36N / mm 2

( 51 0)2

= x gnittup niagA ,noitcnuj ta sserts ,)v( noitauqe ni mm 521

200 2

= 1 00 mm

r1 = =

d1

2

d2

2 52 0 2

= 125mm

b1

x2

b1

x 2

)i(

x

2

b1

( 51 0) \ )iii(

)vi( 2

b1

( 21 5)

x 4 90 091

90904 1 x t( e lisn e )

1 2




68

f x = +18 = 44N / mm 2

)elisnet(

b) . F :rednilyc renni ro

lppA y ni g La em E s‟ qua usu htiw noit a .snoitatoN l

Px = - a 2

fx = + a2

oN w a x t = P ,mm 521 x = mm/N 8 2

x = P ,mm 001 x = 0

8 = - a2

o = - a2

By nivlos g a )iiv( & )iv( 2 = - 22 b2 = - 222222

f = - 22 = -4 .4 2N / mm2

f = - 22 = -3 .6 2N / mm2

(125)2

ii i . oc eht rof enola erusserp diulf lanretni ot eud sesserts pooH m nuop d :rednilyc

In siht esac , eht wt o sebut lliw eb at k ne as a si elgn ht ick c rednily . App gniyl emaL s‟ uqe a usu htiw snoit a ton l a snoit .

Px =

f = + A

x 2

051 = x tA mm, Px 0 =

2 ( 21 5) 04 9091

b2

x 2

b2

x 2

)vi(

)v(

)iv(

)iiv(

2

b2

( 521 ) \

2

b2

( 01 0)

(100)2

- 222 222 x (c )pmo

- 222 222 \ x )pmoc(

2 x - A B

iiv( )i

xi( )

B x




78

= x m 001 m, Px p = f mm/N 06 = 2

\ F uqE mor a ( noit v )iii

)x(

ix( )

By )ix( & )x( gnivlos

A = ,331 B = 01 x 3 6

fx = + 31 3 x 2

oh ,mm 051 = x gnittuP o afrus retuo eht ta sserts p ec

f = + 133 = 62 6N / mm2 )elisneT(

A oitcnuj eht ta sserts pooh ,mm 521 = x gnittup niag n

f = + 331 = 23 5N / mm2 ( sneT li e)

oh ,mm 001 = x gnittuP o afrus renni eht ta sserts p ce

f = + 331 = 34 3N / mm2 (Te sn ile)

.iii Re + egaknirhs( sserts pooh tnatlus F :)erusserp diul a . c retuO y nil red

63 = ecafrus retuo eht ta sserts pooh tnatluseR + 662 = mm /N 203 2 ( eT lisn e)

523 + 44 = noitcnuj eht ta sserts pooh tnatluseR = mm/N 963 2 t( e lisn e )

b . I c renn y ;rednil

= ecaf renni eht ta sserts pooh tnatluseR - 2.44 + 334

- A B ( 51 0)2 O = \

B

(100)2 60 = - A

3´106

\

x 3´10 6

( 51 0)2

3´10 6

( 21 5)2

3´10 6

( 01 0)2

x

x




88

= mm/N 8.883 2 ( eT lisn e)

noitcnuj eht ta sserts pooh tnatluseR = - 23 + 2.63 5 = mm/N 8.882 2 (Te lisn e)

nmuloc A w ih dne enola hti n a sah dexif dne rehto eht dna deg htgnel fo m5 na d a h lo l wo es ssorc ralucric c tuo fo noit er m 01 ssenkciht llaw dna mm 001 retemaid m. 06.1 = E fI

x 01 5 mm/N 2 htgnerts gnihsurc dna s = 53 0N / mm daol eht dniF , taht eht loc u nm yam yrrac w a hti f 5.2 fo ytefas fo rotca gnidrocca ot reluE ht yroe na d naR kine – odroG n

roeht y. ot gnidrocca daol efas eht dnif ,sdne htob no degnih si nmuloc eht fI eht owt eiroeht s. )3002 yaM/lirpA( noituloS : G :nevi

m 5 = L mm 0005 = id retuO a retem D = 001 mm

I id renn a retem d = D- t2 = 001 – m 08 = )01( 2 m ssenkcihT = 01 mm

I = 1 x 06.1 05 m/N m2 s = 53 0 N / mm

f = 5.2

.i b daol fo noitaluclaC y yroehT s’reluE :

dexif dne eno htiw nmuloC a ehto dn r e degnih dn .

2

L = = = 60.6353 mm 2 2

2 ́ ( .3 14) ´1.60 ́ 105 ´ I

(3 35 .6 06)

(D 4 - d )

(01 04 - 804 ) ( 00000001 0- 0006904 0)

I = 69.82 01 x 5 mm 4

0 2

0 2

P = 2p EI

L2

l 5000

2

2 P =

I = 4 p

46

p =

46 p

= 46




98

2 ́ ( 1.3 4) ´ .1 60 ́ 105 ´ 2 .8 96 ́ 105 21 05 3 17 6. 41

p = 01 x 470.37 3 N ii . eniknaR yb daol fo noitaluclaC - :yroehT nodroG

eniknaR tnatsnoC s‟ a = ).leets dlim si lairetam nmuloc eht emussa( 7500

f c ´ A

2

1 + aæ ö

è ø noitaryG fo suidar tsel = K

69.82 ́ 105

2826

A = (01 0 2 - 802 )

4

= ( 00001 - 046 0) 4

= 6282 mm 2

053 ́ 28 62.

353 6 60. è ø

1989 00

3.1 3´10-4 ´12 02 3. 30 6

P = 60.94 ́ 104 N

.iii ra sdne htoB e nih ged reluE roeht s‟ y

p EI

L2

(3.14) ´ 6.1 0´105 ´ 9.82 6´105 ( 005 0)

1

p = L

ç ÷ K

\

I A

=

c

= = 23 10.

fc = s

p

p

2

ç ÷ 1

0057 1 + æ ö

32 10.

P = P =

P = 2

L = l

2

2 =

2

P =




09

472.81 = P 01 x 4 N = daoL efaS ;

N 69037 = eniknaR s‟ yroehT

è ø

053 ́ 2826

1 5000 ç ÷

7500 3 .2 01 ø 89 19 00

3.1 3´10-4 ´ 24398. 18 084.03 = P 01 x 4

Re :tlus

.i luE er s‟ oehT r y 470.37 = P degnih dne eno & dexif dne enO 01 x 3 N

B nih sdne hto ged 472.81 = P 01 x 4 N ii eniknaR . s‟ yroehT

06 = P degnih dne eno & dexif dne enO 49. 01 x 4 N B nih sdne hto ged 084.03 = P 01 x 4 N

ii i . efaS L dao

reluE s‟ yroehT = 7 N 6903 eniknaR roeht s‟ y N 029121 =

A nmuloc si edam pu of owt lennahc CJSI 002 mm dna owt 52 mc x 1 cm f al n eg etalp sa

nwohs in f .gi mreteD i en yb s’eniknaR f alumro eht safe ,daol eht nmuloc of m6 ,htgnel htiw htob sdne ,dexif ac n rrac y w hti a f rotca of saf yte 4. ehT eitreporp s fo

eno lennahc era A = 77.71 mc 2, I xx = 2.161,1 mc 4 dna I yy = 2.48 mc 4. ecnatsiD fo diortnec morf kcab fo w be = 79.1 .mc ekaT fc = 23.0 mm/NK 2 a dn s’eniknaR

tnatsnoC = (Apr )3002 yaM/ li 057 0

noituloS : G nevi :

nmuloc eht fo htgneL l = m 006 = m 6 m s fo rotcaF afety = 4

f ,sserts dleiY c = mm/NK 23.0 2

eniknaR ,tnatsnoc s‟ a = 7500

fo aerA c ,nmulo

.81 72 4 ́ 104

5.2

L ç ÷

K

p = =

f c ´ A 2

1 + aæ ö

æ ö è

1 + 2

= Safe daol = 029121 N 30. 84 0 ́ 104

2.5 =

1

1




19

)1 x 52+77.71( 2 = A A = 45.58 cm2

A = 4558 mm 2

ht fo aitreni fo tnemoM e X tuoba nmuloc - sixa X ö

+ 25 ́ 1´ .01 52 ÷

ø

+ 8.42 +1 .7 77 ́ (5 + 9.1 7)

I yy I < XX \ lkcub ot dnet lliw nmuloc ehT y ni e y- noitcerid

I = Iyy = 0.9944 cm4 dexif si nmuloC a htob t the e sdn

= 003 0mm

= 72.5 mm

.0 32´ 558 4.A

1 3 00 0 ç ÷

7 005 0 7 5.2

= NK 755

Re tlus : faS e daol = NK 755

XX æ

I = 2 ́ 1,16 2.1 + è

25 ́ 13

21 ç ÷ ç = 7 0.938 mc 4

I YY = 2í ü ì

î

1´ 253

12 2 ý þ

= mc 0.994,4 4

l L = =

2 0006

2

4 94 9 ́ 104

58 54

=

2

=

K ç ÷

L

I A

f c .A

1+ aæ ö

è ø

K =

P =

1+

P F .O.S

222 8 4

æ ö è ø

2 8222 = NK

nmuloc fo daol efaS =

=




29

UN VI TI

MID EERHT NI SSERTS FO ETATS E SNOISN

ERTS SS nehW a niatrec metsys fo ex lanret secrof tca no a ob dy neht eht ob dy o sreff re ecnatsis ot

eseht .secrof sihT lanretni ecnatsiser dereffo by eht ob dy p re tinu a aer si dellac eht ts r sse dob eht ni decudni y.

ICNIRP P SENALP LA ehT alp ne ni hw i hc eht s aeh r sserts si z re o si c la l de icnirp p la senalp . ehT enalp cihw h si

pedni e nirp sa nwonk si sserts raehs fo tnedn c nalp lapi e. SP ROSNET LACIREH

é 0 0

ê m

ë s

h sa nwonk osla si tI y ts citatsord r t sse e rosn

s = (s +s + s )

s .sserts naem eht si

SSERTS ROTAIVED T ROSNE

l xy ù ê

ij xy

êt xz

s ê0

ê0

ú

ú t = s 0 ii

ij

ù

ú mû

m

0

m x y z 1 3

m

ù xz yz

ú ú ú

z mû s -s

t t

s x m ú y m ú s -s

ú û

é -s t = êt 1

ë l yz




39

EHT SSERTS STNENOPMOC TA A TNIOP ERA NEVIG BY HT E OF LL GNIWO YARRA é ù

ê ú ë6 01 6 ú

lapicnirp dna sserts lapicnirp eht etaluclaC plane .s

loS u :noit

or eht era sesserts lapicnirp ehT oitauqe cibuc eht fo sto n

s - I1s + I 2s - I3 = 0

hw ere ,

I1 = s +s + s

I2 = s s + s s + s s -t -t -t

I3 = s s s -s t -s t -s t + 2t t t

ra e rts eerht e avni ss ria stn t sserts ehT e rosn

s .

ij yx

ë zx

ht dna rosnet sserts gnirapmoc yB ,yawa nevig e

I1 = s +s +s 01 = + 8 + 6 = 42

I2 = s s + s s + s s -t -t -t

8( + )8 x 01( = )6 x 6( + x )01 - )5( 2 – )01( 2 – )6( 2 84 + 08= 06 + - 52 – 001 - 63

72= 2 2 2

x y z x y z xy yz xz

6 x 8 x 01 = - )01( 01 2 - ) 6( 8 2 - )5( 6 2 )6( )01( )5(2 +

01 5 6 ê5 8 10ú

pM a ê û

3 2 )1(

x y z

2 2 2 x y y z z x x y yz x z

2 2 2 x y z x yz y xz z xy xy yz xz

ê é s

s ù ú ú

t t êt

xy y

xz yz

êt ú z û

t = x

t zy s

x y z

2 2 2 xy yz xz x y y z z x

I3 = s s s -s t -s t -s t + 2t t t yz xz xy




49

4= 08 - 0001 - 882 - 051 + 006 =- 853

v eseht etutitsbuS a 1( ni seul ) e uq a noit

s - 24s + 27 s + 358 = 0

We ht wonk a t

Cos3q = 4 oC s3q - 3 oC sq

F siht mor

4 oC s3q = oC s3q + 3Cosq

soC 3q = C so 3q - 4

,tup

s = rC qso +

= rC qso +

s = oCr sq + 8

\ uqE a ( noit eb )2 c mo e s r 3 oC s3q + 15 2 + 24r oC s2q + 91 2rCosq - 24(r oC s 2q + 64 + 2rCos q ´ 8)+

( 72 r c so q + 8 853 + ) = 0 r3 3soC q 215 + - r 42 2 soC 2 + q 291 + soC r q - 42 r2 soC 2 q - 6351 -

483 soC r q 72 + soC r q 612 + 853 + 0= r 3 oC s3 q - 561 r C so q - 054 = 0

b dediviD y r3

oC s 3q - oC sq - = 0 r r

,teg e w, )4( dna )3( noitauqe gnirapmoC

561 3 r 4 r = 4238.41

a dn

3 2 )2(

)3( 1 3

4 Cosq = 0

I1

3 24 3

2 2

61 5 054 2 3 )4(

= 2




59

054 oC s3q 4

54 0´ 4 (14 238. 4)

soC 3 q = 816155.0 \ q1 48.81 = o

q2 = q 1 21 + 0

q2 48.831 = o

q3 = q2 + 21 0

q3 48.852 = o

s1 = r soC q1 8 +

= 4238.41 48.81( soC o) 8 +

s1 = 40.22 aPM = 4238.41 831 soC . 48 o 8 + = - 71.3 aPM

s3 = r c so q3 8 +

= 4238.41 852 soC . 48 o 8 + = 31.5 aPM

Re :tlus

q1 = 48.81 o s1 = 40.22 aPM

q2 = 48.831 o s2 = - 71.3 aPM

q3 = 48.852 o s3 = 31.5 aPM

TBO NIA EHT NIRP C LAPI SESSERTS DNA HT E ER L TA ED RID E ITC ON OC S SENI SSERTS FO ETATS GNIWOLLOF EHT ROF

ê4 5úM aP

ë6 1 û

loS u :noit

= r3

oC s3q =

3

6 3. 4 ù é 2 5

ú ê ú ê




69

cibuc eht fo stoor eht era sesserts lapicnirp ehT qe noitau .

s - I1s + I2s - I3 = 0 I1 = sx +sy +sz

1 + 2 + 3 = 6 = I 2 = s s +s +s s -tx 2 y -t -t

3( = x ) 2 + 2( x )1 )3 x 1( + - )4( 2 - )5( 2 - )6( 2 = 11 – 61 - 52 - 36

I2 = - 66 2 2 2

x y z x y z xy yz xz

3(= )1 x 2 x - )5(3 2 - )6(2 2 - )4( 1 3 4( 2 + x 6 x )5

6 = - 57 - 27 - 61 2 + 40

I3 = 38

v eseht etutitsbuS a seul qe ni au ( noit 1)

s - 6s - 66s -83 = 0 )2( We ht wonk a t

oC s3q = 4 oC s3 q - 3 oC sq

4 oC s3q = oC s3 q + 3 oC sq

oC s3q = oC s3q + oC sq

oC s3q - oC s3q � oC sq

s = rCosq +

s = oCr sq + 2

emoceb )2( noitauqE s

s - 6s -66s -83- 0

( oCr s q + 2) - 6( oCr s q + 2) - 66( oCr s q + 2) - 38 = 0

r 3 oC s3 q + 8 + 3(r soC q ´ 2) + 3(rCosq + 4)- 6r soC q

3 2 )1(

yz zx 2 2

x y y z x

I3 = s s s -s t -s t -s t + 2t t t yz xz xy

3 2

1 3 4 4 1 3 4 4

)3(

I1

3 tuP

3 2

3 3 3

3 2 2 2 2




79

+ 24r cosq + 42 - 66r c so q - 231 - 38 = 0

r 3 soC q + 27rCos q - 66rCosq -179 = 0

r 3 soC q -39rCosq - 71 9 = 0 b dediviD y r3

soC q -

= r 4

r2 = 651 r = 84.21

71 9 oC s3q r 4

617 = soC 3q x ( 84.21 )3

716

1943.765 soC 3q = 3753863.0

3 q = 56583.86

q1 97.22 = o

q2 = q1 021 +

q2 97.241 = q3 = q2 021+

q3 97.262 =

s 1 = r cosq1 + 2

84.21 = + )097.22( soC 2 s 1 =13. 05 6M aP

s = rCosq 2 + 2

= 84.21 )97.241( soC + 2

3

3

971 3 93 r

)4( oC sq -

3 2 = 0

r

By nirapmoc g 3( ) )4( dna 39 1

2

a dn = 3

Cos3q =

2




89

s = -7.939M aP

s 3 = rC so q 3 + 2

84.21 = )97.262( soC + 2

= PM 086334.0 a

Re :tlus

q1 = 22 97 . o s1 = 605.31 aPM

q2 = 241 97 . o s2 = - aPM 939.7

q3 = 262 97 . o s3 = 086334.0 aPM

TS EHT A RTS FO ET E A TA SS P SI TNIO G YB NEVI

20.

ê- 6

ë10

De ret m lapicnirp dna sesserts lapicnirp eht eni rid ec noit .

noituloS : ehT c cibu e uq a noit

s - I1s + I s - I 3 = 0 I1 = s +s +s

7 + 01 + 02 = 73 = 2 2 2

x y y z z x xy yz zx

)01 x 02(= 01( + x )7 )7( + x 02 )63( + 6( + 4) + )001( 07 + 002= + 041 62 + 46 + 01 + 0

I2= 016

I 3 = s s s -s t -s t -s t + 2t t t

01 x 02(= x )7 - )46( 02 - 01 )001( - 7 )63( 2 + )6( )8( )01(

0041= - 0821 - 0001 – 252 069 + 8031=

)1( noitauqe ni seulav eseht etutitsbuS

2

ù

ú û

-6 10 8

úM aP

ú

10 8 7

ê

ê

é

3 2 )1( 2

x y z

I 2 = s s + s s + s s + t + t + t

2 2 2 x y z x zy y xz z xy xy zy zx




99

s -37s + 610s -1 03 8 = 0 We ht wonk a t

Cos3q = 4 soC q -3Cosq

4C so 3 q = Cos3 + 3Cosq

soC q = oC s3q +

soC q -

s = rCosq +

s = oCr sq + 21 . 33

noitauqE semoceb )2(

s -37s + 610s -1308 = 0

( oCr sq + 33.21 ) - 37(rC so q + 33.21 ) + 610( soCr q + .21 33) -1308 = 0

r 3 soC q +1874. 15 6 + r soC q( 99.63 )+ 54 6. 80 7rC so q -[ 73 (r soC q)+ 2 .4 66rC so q

+ 51 2.0 82 9]+ 061 r soC q 08.2791 + - 8031 0 =

r 3 oC s3q + 615.4781 + 9.63 9r 2 oC s2 q + 80.654 7 oCr sq - 37r 2 oC s 2q - 1.9 2r 2 oC s2 - 3960.5265 +

061 r soC q 08.2791 + - 8031 0 =

r3Cos3q - 92 5- 694 0.2 96 3 = 0 ¸ r 3

C so q -

By )3( gnirapmoc

)2(

)3(

3

3 4 3 4

1 3 oC sq

oC sq

4 1 4

3 oC s3q -

I1

3 tuP

3 2

3 2

3 2 2 2 2

2

694 0 3962. 3 592 r

)4( oC sq -

3 2 = 0 r

& ( )4

3 2




001

1 295 2

r2 = 0811 r = 53.43

a dn

oC s3q 694 0 962. 3 3

= 4 3504 4 133.

3q = 0396.06

q1 = 132.02 o

q2 = q1 021 +

q2 = 041 32.

q3 = 132.62 o

s 1 = rCosq1 +1 .2 33

32.041( soC 53.43 = o) 33.21 +

s 1 = 35.44 0 aPM s = rCosq 2 + 3.21 3

132.041( soC 53.43 = o) 33.21 + s = -1 .4 12 7M aP s = rC so q 3 + 3.21 3

132.062( soC 53.43 = o) 33.21 + s = 6. 105 6

Re :tlus

q 1 132.02 = o q 3 132.062 = o s2 = - 712.41 aPM

q 2 32.041 = o s1 = 035.44 aPM s3 = 6105.6 aPM

= 4 r

= 4 r

oC s3q 694 0 3962.

o

2

2

3

3




101

NIALPXE EHT YGRENE FO NOITROTSID ( RAEHS NIARTS YGRENE ) DNA NOITATALID

grene niarts ehT y niwollof eht no pu tilps eb nac g ne niarts owt erg sei .

i. ygrene niarts raehs( noitrotsid fo ygrene niartS ) ii . niartS ygrene fo tataliD i no niartS( ene gr y fo nu i mrof oc m noisserp ro( )) noisnet

) ygrene niarts cirtemulov )ro(

eL t e1 e2 na d e3 eb eht lapicnirp niarts ni the id r tce i sno fo p lapicnir erts s ses s1, s2 dna s3 .

nehT

e1 = [s - m(s + s )]

e2 = [s - m(s + s )]

e3 = [s - m(s + s )]

ew noitauqe evoba eht gniddA g ,te

e1 + e2 + e3 = [(s1 +s + s ) - 2m(s + s + s )]

= (1- 2m ) B tu e1 + e2 + e3 e = v )niarts cirtemuloV(

ev = (s1 + s + s )

s1 +s 2 + s 3 0 = ,ev = 0 . sihT snaem th ta fi mus fo eht ht r ee p icnir p la sserts si lno tub ,egnahc cirtemulov on si ereht orez y srucco noitrotsid eht .

eht morF a vob e ,noissucsid

1. nehW eht mus fo eerht lapicnirp sesserts si ,orez ereht si on ulov m cirte ahc n eg

lno tub y srucco noitrotsid eht .

1 E 1 E 1 E

1 2 3

2 3 1

3 1 2

1 E 2 3 1 2 3

2 s1 + s + s3

E

1 - 2m E 2 3

I f




201

2. W neh eht eerht lapicnirp sesserts era lauqe ot o en rehtona ht e er si no di noitrots lno tub y cirtemulov c nah g srucco e .

oN te:

seiroeht xis evoba eht nI , s te , s ec = mis ni timil citsale eht ta sserts elisneT p t el e noisn a dn

rpmoc e ;noiss

s1 , s2, s3 = xelpmoc yna ni sesserts lapicnirP metsys e taht hcus( 1 > e2 > e 3 )

It may be ussa m de aht t eht idaol ng si darg u la )ro( itats c dna( ht e er is on c lcy ci )ro( ol tcapmi a ).d

EHT ETATS P O SEIROEHT LAPICNIR F ERULIAF

1. rts lapicnirp mumixaM se roeht s y 2. )ro( sserts raehs mumixaM yroeht ecnereffid sserts 3. grene niartS y yroeht 4. ts raehS grene niar y yroeht 5. rts lapicnirp mumixaM ia roeht n y 6. rhoM s‟ yroehT

O SNOITATIMIL F ICNIRP MUMIXAM P T SSERTS LA H YROE

1. nO a dlim leets nemiceps nehw leips noisnet tset si deirrac tuo gnidils srucco

letamixorppa y 45o iht ;nemiceps eht fo sixa eht ot s eruliaf eht taht swohs i si esac siht n aehs mumixam ot eud s r t rid eht naht rehtar sser ce lisnet t e rts e ss .

2. tI h sa b nee dnuof taht a lairetam cihw h si ev ne th hguo aew k ni elpmis co noisserpm tey

h niatsus nac y .noisserpmoc elpmis ni timil citsale eht fo ssecxe ni rof erusserp citatsord

MUMIXAM P SSERTS LAPICNIR YROEHT gnidroccA ot siht oeht ry eruliaf lliw rucco when eht am x mumi elpicnirp elisnet sserts

(s1) ni eht xelpmoc s sy t me caer h se eht eulav fo eht am x mumi sserts ta eht citsale timil (s te ) ni lpmis eht e noisnet .

YROEHT SSERTS RAEHS MUMIXAM

sihT yroeht i seilpm taht eruliaf lliw rucco nehw eht am x mumi raehs sserts t mumixam ni eht elpmoc x s metsy sehcaer eht eulav of eht max mumi raehs sserts ni elpmis noisnet ta itsale c

e.i( timil )




301

s 1 -s s lmax = = s1 -s3 = s et

2 2 O SNOITATIMIL F UMIXAM M SSERTS RAEHS TH YROE

i. ehT yroeht seod ton evig arucca te stluser rof eht etats fo sserts fo erup raehs ni hcihw

fo tnuoma mumixam eht t noisroT )e.i( depoleved si raehs e ts .

ii. ehT oeht ry seod ton evig su esolc stluser sa dnuof by ex stnemirep no elitcud iretam a .sl H ti ,revewo g fas sevi e re .stlus

NE NIARTS RAEHS ERG YROEHT Y

sesiM noV“ ro ”yroehT ygrene noitrotsiD “ dellac osla si yroeht sihT - kneH y roehT y ”.

idroccA ng ot siht yroeht eht citsale f erulia ucco rs erehw eht raehs niarts rene gy rep tinu

emulov ni eht rts desse lairetam sehcaer a eulav lauqe ot eht raehs niarts rene gy p re tinu emulov tniop timil citsale eht ta i t elpmis eht n e .tset noisn

NOITROTSID FO SNOITATIMIL YROEHT YGRENE

1. ehT oeht ry do se ot eerga eht ex tnemirep stluser rof eht am t laire for ihw ch s at si ffid etiuq ere tn e ct .

2. sihT yroeht si dedrager sa eno ot hcihw mrofnoc tsom fo eht elitcud lairetam rednu eht a t suoirav fo noitc y nidaol fo sep g.

MUMIXAM P ROEHT NIARTS LAPICNIR Y

ehT oeht ry setats taht eht eruliaf fo a lairetam srucco nehw eht icnirp p la elisnet niarts ni eht lairetam ehcaer s eht niarts ta eht citsale limit ni is m elp noisnet ro( ) w neh eht min inim m mu

lapicnirp niarts ei( ) max mumi lapicnirp sserpmoc i ev niarts sehcaer eht citsale li tim ni is m elp c rpmo e .noiss

M NI SNOITATIMIL YROEHT NIARTS LAPICNIRP MUMIXA

i. roeht ehT y iretam elitcud fo ruoivaheb eht setamitserevo a sl . ii . ehT oeht ry seod on tif llew htiw eht ex latnemirep stluser ex tpec rof elttirb

iretam a of sl r noisnet laixaib .

NI ROSNET SSERTS MOC NAISETRAC P STNENO

s . xy

ij xy

ë xz

3 et )ro(

ê êt

êt

é x xz yz

ú t t ù

t y ú

ú z û

t = s '

s t yz




401

NI SSERTS EERHT VAR STNAI

noitauqe cibuc eht fo stoor eht era sesserts lapicnirp ehT ,

s - I 1s + I s - I 3 = 0 hw ere

I1 = s +s +s

I 2 = s sy + s s + s s - t xy -ty 2 - t zx

I3 = s sy sZ - sxt - syt - szt + 2t t t

t = f (s )

ehT gnipolevne evruc tsum tneserper ni ht is assicsba s dna setanidro e, eht niraehs dna lamron g nalp eht ni sesserts e .pils fo

s1 +s s1 3

2 2 2

P = (s1 + s 3 )

tm = (s1 - s )

- p) +t = lm2

tS latoT r hT ygrenE nia eory

latot ehT grene niarts y b nevig si noitamrofed fo y

3 2 2

x y z

y z x z

2 2 2 x xy yz xz

YT OWT P RENE NIARTS FO SE G Y

i. grene niartS y grene niarts raehs( noitrotsid fo y) ii . grene niartS y noitatalid fo .

RHOM ’ YROEHT S

xy xz xy

2 2 z

L te

t = f (s )

ç - ÷ + t = ç ÷ æ ö æ -s ö è ø è ø s

2 3 2

1 2 L te

1 2 3

2 (s 2




501

é ù 2 2 2

ês1 + s 2 +s 3 - 2v(s1s 2 + s 2s 3 + s 3s1

)ú ë û

grene niarts dna y si noisnet elpmis ni

U =

XE P ICNIRP MUMIXAM EHT NIAL P NAR ( :YROEHT SSERTS LA K )YROEHT S’ENI - edlo eht dna tselpmis eht si sihT roeht ts y eruliaf fo

- idroccA ng ot siht eht o yr eruliaf lliw ucco r wh ne eht am xim mu elpicnirp isnet le

sserts (s1) ni eht xelpmoc metsys sehcaer eht va eul fo eht mumixam sserts ta eht citsale timil (s te noisnet elpmis eht ni ) im eht )ro( nim mu lapicnirp ts sser taht( ,si eht

am x mumi lapicnirp evisserpmoc ,)sserts caer h se eht citsale limit sserts (s) ni elpmis c rpmo e noiss .

i( e.) s1 = s te )noisnet elpmis ni(

(I rpmoc elpmis n e )noiss

eM a fo eulav laciremun sn - fI eht am x mumi lapicnirp sserts is eht sed i ng ,noiretirc eht am xim mu lapicnirp sserts

nikrow eht deecxe ton tsum g s ht rof e iretam a .l Hen ec ,

s 1 £ s - sihT yroeht sdragersid eht tceffe fo rehto icnirp ap l rts e sess dna fo eht gniraehs serts s se

no rehto enalp orht u hg t eh .tnemele roF lttirb e slairetam hcihw od ton liaf by idleiy ng tub liaf yb elttirb ,erutcarf eht am x mumi lapicnirp sserts oeht ry si noc s deredi ot eb

lbanosaer y rotcafsitas y.

sihT oeht ry sraeppa ot eb orppa x etami ly co tcerr rof idro nary tsac – snori dna irb ttl e tem a sl .

EHT MUMIXAM LAPICNIRP SSERTS YROEHT SI DETCIDARTNOC NI EHT

NIWOLLOF G :SESAC

1. nO a dlim ts e le nemiceps nehw elpmis t noisne tset si ca deirr tuo gnidils ucco rs yletamixorppa 54 o ot ht e ax si fo eht ;nemiceps siht swohs taht eht iaf l eru ni eht

am ot eud si esac rts raehs mumix e r ss a erid eht naht reht c t t e lisn e sserts .

ê U = 1

2E ê ú ú

s 2 0

2E

s ca 3 = s

s 3 3 s




601

2. tI sah be ne dnuof taht a lairetam hcihw si neve hguoht kaew ni is m elp noisserpmoc tey nac niatsus h citatsordy sserp u er rof ni ex ssec fo eht sale t ci timil

elpmis ni c rpmo e .noiss

XE P EHT NIAL AM ID SSERTS )RO( SSERTS RAEHS MUMIX FF YROEHT ECNERE roeht sihT y itseuG dellac osla si roeht s‟acserT )ro( s‟ y.

sihT oeht ry seilpmi aht t eruliaf lliw rucco ehw n eht am x mumi aehs r ts r sse am x mumi ni eht elpmoc x s sy t me caer h se ht e v eula of ht e am x mumi raehs

ts eht ta noisnet elpmis ni sser e .i timil citsal e.

.noisnet elpmis ni

s 1 -s = s

I ised lautca n g n s te sserts efas eht yb decalper si noitauqe evoba eht ni .

- sihT oeht ry sevig doog noitalerroc htiw stluser fo ex stnemirep no elitcud .slairetam In eht ac se fo owt di lanoisnem elisnet serts s dna th ne ht e am xim mu sserts

ot ti etauqe ot detaluclac ecnereffid s te . :yroeht siht fo snoitatimiL

i. ehT oeht ry seod ton evig ucca r eta stluser rof ht e etats fo sserts fo erup raehs ni

ma mumixam eht hcihw o tnu ved si raehs fo e ei( depol ) oT r t nois e ts . ii . ehT oeht ry si ton elbacilppa ni eht ac se erehw eht etats fo sserts oc n is s st fo

laixairt elisnet sesserts fo en arly eq lau ma eduting icuder n ,g ht e gniraehs s ssert ot a llams ma ,eduting os t tah eruliaf dluow eb by elttirb utcaf re r rehta th na by y nidlei g.

.iii ehT yroeht seod ton gi ev sa esolc stluser sa dnuof by ex stnemirep no elitcud iretam a .sl ti ,revewoH g fas sevi e re .stlus

RENE NIARTS RAEHS EHT NIALPXE G YROEHT Y

sihT yroeht is osla dellac noitrotsiD“ renE gy oehT r :”y )ro( noV“ sesiM – neH k y

roehT y”

- gnidroccA ot siht oeht ry eht citsale eruliaf o cc u sr erehw eht sh rae ts r nia en re gy rep tinu emulov ni eht ts r desse lairetam r cae h se a eulav lauqe ot ht e hs e ra niarts grene y timil citsale eht ta emulov tinu rep tset noisnet elpmis eht ni tniop .

eud ygrene niarts raehS t irp eht o n serts lapic s se s 1, s 2, dna s 3 p tinu re v ulo m eht fo e

m sserts a et r lai .

max

- t =

s 1 -s 2 2

et 3 s =

o( r ) 3 et




701

2 2 2 ê 2 2 3 3 ú

ë û

tuB rof eht elpmis noisnet tset ta eht citsale timil tniop erehw ereht si no ly o en lapicnirp )ei( sserts s te grene niarts raehs eht evah ew y yb nevig si hcihw emulov tinu rep

é ù

(s - 0) + (0 - 0) + (0 - s ) ê ú

�s1 = s s = 0 s 3 0 = ]

(s1 -s2) +(s2 -s3) +(s3 -s1) = 2set

ehT oba ve oeht ry h sa neeb dnuof ot vig e tseb res stlu rof elitcud tam e lair f ro hcihw letamixorppa y.

oeht ygrene noitrotsiD fo snoitatimiL r :y

1. eT oeht ry do se ot ga r ee htiw eht ex latnemirep stluser f ro eht lairetam rof hcihw s te si f tnereffid etiuq r mo sec .

2. ehT oeht ry sevig s = 0 rof h citatsordy erusserp (o )r te ,noisn hcihw snaem taht eht lairetam reven lliw liaf u redn any h citatsordy erp s erus )ro( noisnet . nehW eerht lauqe snoisnet era deilppa ni eerht icnirp p la ,snoitcerid elttirb erutcaf cco u sr a dn sa hcus

erts lapicnirp mumixam s lliw s g r evi sac siht ni stluser elbaile e. 3. sihT yroeht si dedrager sa eno ot hcihw mrofnoc tsom fo eht elitcud lairetam rednu

t suoirav fo noitca eht y nidaol fo sep g.

XE P MUMIXAM EHT NIAL P ICNIR P YROEHT NIARTS LA

- tS htiw detaicossa yroeht sihT tneneV - ehT oeht ry setats taht eht eruliaf fo a lairetam srucco nehw eht apicnirp l elisnet

niarts ni eht lairetam sehcaer eht niarts ta eht citsale timil ni elpmis te oisn n )ro( nehw eht muminim lapicnirp niarts )ei( am x mumi lapicnirp sserpmoc ive iarts n

tsale eht sehcaer timil ci i lpmis n e c rpmo e noiss .

rts lapicnirP a nirp fo noitcerid eht ni ni c sserts lapi s1 ,

ê ú é ù

(s1 -s ) + (s -s ) + (s -s 1 ) ê ú

1 12C S U =

1 12C

2 2 2 ê e t a t ú

ë û [

ê ú 1 s U =

et

nitauqE g rene owt eht g w ,sei e get 2

s = s

2 2 2

et ec

2

et




801

(s +s )ù

rts lapicnirP a rts lapicnirp eht fo noitcerid eht ni ni e ss s3,

(s1 +s )ù

nidrocca eruliaf esuac ot snoitidnoc ehT g mixam he ot u picnirp m rts la a roeht ni y :era

e( 1 eb tsum )eV+

e( 3 eb tsum - )eV

(s +s )ù >

(s1 + s ) ù >

[s1 + s 3] > s et

[s1 + s ]> s

verp oT e f tn a uli er :

s1 - (s + s ) < s

2 e c

f citsale fo tniop eht tA a ruli e:

s1 - (s + s ) = s

2 e c

(s1 + s ) = s

(s1 + s ) = s

(where , st dna sc )sesserts efas eht era

1 m 2 3 ú û

e1 = ês 1 -

é ë

1 m ê s 3

-

1 E 2 ú û

e3 =

et e1 >

a dn

s

ec

E s

e3 > E

s é ë é ë ê

s 3

-

ês 1 - m

et 1 1 2 3

û E 1 E

s E

1

ú

et

1 m 1

û ú m E 2

3

s1 - s -

m 3 ec

1 m 1 m 1 m

2 3 et

(s1 + s ) < s 3 s -

1 m

2 3 et

(s1 + s ) = s s - 3 dna

3

3

ised roF g up n r sop e ,s

s -

s -

iL m :snoitati

1 m 1 m 2 c

2 t

1 E

é ë




901

i. roeht ehT y iretam elitcud fo roivaheb eht setamitserevo a sl . ii . eT oeht ry seod ton tif llew htiw eht latnemirepxe stluser exce tp rof elttirb slairetam f ro

noisnet laixaib .

XE P RENE NIARTS EHT NIAL G YROEHT Y grene niats latot ehT y si noitamrofed fo g b nevi y

U = [s +s +s = 2v(s s + s s + s s 1 )]

2E grene niarts eht dna y si noisnet elpmis rednu

U =

ot lairetam eht rof ecneH y ,dlei

s 1 + s +s - 2v(s1s + s s + s s 1 )

ehT latot citsale ygrene derots ni a lairetam erofeb ti sehcaer eht citsalp etats nac evah on ecnacifingis sa a gnitimil ,noitidnoc ecnis rednu hi hg h citatsordy p usser r ,e egral tnuoma fo ts r nia

grene y nisuac tuohtiw derots eb am g rf rehtie noitamrofed tnenamrep )ro( erutca .

XE P YROEHT S’RHOM NIAL A lairetam yam liaf rehtie hguorht citsalp pils )ro( by utcarf re nehw tie h re eht iraehs ng

sserts t fo senalp eht ni s desaercni sah pil . teL t = f (s ) ehT ne v gnipole cu evr t = f (s ) tsum erper se tn ni eht ir a assicsb s dna o etanidr s t , eht

niraehs dna lamron g nalp eht ni sesserts e .pils fo woN

æ ö æ ö

è ø è ø

L te

t = (s1 - s ) neht

(s - p)2 + t = tm 2 sihT noitauqe erper s stne eht imaf ly fo jam or rp i lapicn sserts cric l se ni p temara er fo .mr

ot tcepser htiw gnitaitnereffid yllaitrap yb deniatbo si epolevne siht fo noitauqe ehT P

1 2 2 2 1 2 3 1 2 2 3 3

s 2 e

2E

2 2 2 2 3 2 2 3 3

3 2 3 ç ÷ ç ÷

s 1 + s s 1 + s 2 2

P = (s1 + s ) 3 1 2

2 2

çs - ÷ + t = ç ÷

m 3 1 2

2




011

s - 2 ps + P 2 + t = m 2

s = p + t .

2

sihT si ot noitauqe fo rhoM s‟ epolevne fo eht rojam

rts lapicnirp e rap ni ss a tem e r for .m NI A LEETS ,REBMEM TA A TNIOP EHT ROJAM LAPICNIRP SSERTS SI 081

M/NM 2 DNA EHT RONIM LAPICNIRP SESSERTS IS SERPMOC S .EVI FI EHT ELISNET DLEIY TNIOP FO EHT LEETS SI 522 /NM M2, IF ND THE EULAV FO

EHT ONIM R P NIR C LAPI ERTS SS AT CIHW H NIDLEIY G LLIW ,ECNEMMOC GNIDROCCA OT HCAE FO EHT GNIWOLLOF AIRETIRC FO

F .ERULIA

i. sserts gniraehs mumixaM ii . umixaM m ot t ygrene niarts la

.iii ixaM mum ts raehs r ygrene nia ekaT P s’nossio r 62.0 = oita

noituloS :

nirp rojaM c rts lapi e ,ss

rts tniop dleiY e ss

m = = 0 2. 6 m

rts lapicnirp ronim etaluclac oT e ( ss s2 )

ixaM )i( mum ts gniraehs r noiretirc sse

s1 -s 2 = se

s =s1 -se

081 = - 522

s2 = - 54 m/NM 2

s2 = m/NM 54 2 )pmoc(

ii . ixaM mum grene niarts latot etirc y r :noi

dt m

dp

(dt )

dp

m t = t . 1- m

m

2 1 s = 81 0MN / m

2 s = 22 5MN / m2

1

2

2 2 2 ( s - P)2 + t = t m2




111

(s1s + s s + s s1) = s

(s1s ) = s

)081( 2 + s2

2 - 2 x 62.0 x 081 s2 )522( = 2

00423 + s2

2 - 6.39 s 2 52605 =

s22 - 6.39 s2 - 52281 0 =

(93.6) + 4´ 52281

2

= = -96 0. 8MN / m2

2 lnO( y – sa nekat si ngis eV s2 si c )evisserpmo

s2 = 80.69 m /NM 2 )evisserpmoc(

.iii ixaM mum ts raehs r etirc ygrene nia r :noi

(s1 - s )2 + (s - s )2 + (s - s1 )2 = 2s

2 2 e

(s1 ) + (s ) - 2s1s + 2(s1 ) = (s )

2)081( + (s2)2+ - 081 s2 )522( = 2

(s2)2 - 081 s2 - 52281 0 =

( 81 0) + 4´18225

2

2 m

2 2 2 3 3 e

2 2 2 2 3 s 1 + s + s -

\ s 3 0 =

s 1 + s - 2 2

2 e 2 2 2

m

2

2

9.36 - s =

63.9 - 82 5. 67

2 2 2 3 3 e

nittup g s3 = 0

(s1 - s )2 + (s )2 + (s1 )2 = 2s 2

2 2 2 2 2 2 e

2

2

81 0 - s =




211

s = = -72.25MN / m 2 2

s 2 52.27 = m/NM 2 )evisserpmoC(

NI A LAIRETAM EHT LAPICNIRP SESSERTS ERA 06 /NM M2, 84 M/NM 2 DNA - 63

M/NM 2. ETALUCLAC i. TO LAT S RT A RENE NI G Y ii . RENE NIARTS CIRTEMULOV G Y

.iii NIARTS RAEHS YGRENE iv. OTCAF R FO SA YTEF NO EHT LATOT S NIART YGRENE

I AIRETIRC F IRETAM EHT A A SDLEIY L T 021 M M/N 2.

AT K 002 = E E G M/N 2 A + N /1 D M 3.0 =

noituloS : G aD nevi ta:

rts lapicnirP e :sess

s 1 = + 06 m/NM 2

s 2 = + 84 m/NM 2

s 3 = - 63 m/NM 2

rts dleiY e ,ss s e m/ NM 021 = 2

E = G 002 N m/ 2 m/1 , = 3.0

.i p ygrene niarts latoT re ulov tinu me:

(s1) + (s ) + (s ) -

[(60) + (48) + (36) - 2 ́ 0.3(60 ́ 48 - 48´ 36 ́ 60)]

U = 2.5[ 63 00 + 32 04 + 921 6 - 0.6( 882 0 - 71 28 - 2 61 0)]´

U = K 15.91 N m/m 3

ii . uloV m grene niarts cirte p y re ulov tinu em :

081 - 23 4.5 2

2 m

1 2E ê

é ë

2 2 2 2 3 ú (s1s + s s + s s 1 )ù

2 2 3 3 û

U =

U = 2 2 2 1012

2 ́ 02 0 ́ 109




311

1 1- 2 / m 3 2E

1 1- 2´ 3.0 3

e v 827.1 = m/NK 3

ii i . ene niarts raehs r p yg re ulov tinu me

es = [(s1 - s )2 + (s - s )2 + (s -s1 )2 ]

= 7 .6 923 NG / m 2 ç

[(60 - 48) + (48 + 36) +(- 36 - 60) ]

es = .1 083(144 + 7056 + 129 6)´10-3

es = 7.71 8KNm / m3

vi . F ( ytefas fo rotca F )S.O.

laixainu rednu emulov tinu rep ygrene niartS si gnidaol

2

= ´10 -3 3 = 6 mNK / m 3 2´ 02 0 ´109

63

19. 15 NI A LAIRETAM EHT LAPICNIRP SESSERTS ERA 05 M/N M2, 04 MM/N 2 DNA - 03

N M/ M2 C , A CL U :ETAL

i. RTS LATOT A RENE NI G Y ii . ULOV M IRTE C RENE NIARTS GY

ev = (s1 +s + s ) 2 3 é ù ë û

2 ê ú

2 ev = ù û

( 06 + 84 - 36) ´1012 é ú ´10-3

ê 2 ´ 20 0 ´109 ë

2 2 3 3 12c

E

2æ1+ è

1

,erehW C =

es =

002 2(1+ 0 3. ) =

ö ø ÷

1 m

2 2 2 1´10 21

12´ 29.67 3´109

(21 0´106 ) s e2

2E =

= 1. 548 S.O.F =




411

.iii AEHS R RTS A NE NI ERG NA Y D iv. ROTCAF FO YTEFAS NO EHT LATOT NIARTS YGRENE

I NOIRETIRC F DLEIY LAIRETAM EHT A /N 001 T MM2. AT K 01 X 002 = E E 3 MM/N 2 AN D P R NOISSIO A 82. 0 = OIT

ituloS no :

viG e ,n nirP c rts lapi e :sess

s 1 = 5+ 0N / mm 2

s = 4+ 0N / mm 2

s = -30N / mm

rts dleiY e ,ss

.i p ygrene niarts latoT re ulov tinu me:

(s1) +(s ) + (s ) -

[(50) +(40) +(30) - 2(0.3)[50´40 - 40´30 -30´50]]

[ 052 0 + 061 0 + 09 0 - 0.6[ 002 0 - 021 0 -1 05 0]]

[5000 - 0.6(- 700)]

[5420]

uloV)ii m ulov tinu rep ygrene niarts cirte m :e

ev = (s1 + s + s )2 é1 -

2E/ m ù

ë û

2

2

s = 01 0N / mm 3

e 2

é 2 2 2 1 2E ê ë

2 m 2 3

û (s1s + s s + s s 1 )ù

2 2 3 3 ú U =

=

=

=

2 2 2 1 2´ 02 0´103

= 04 0´103

U = 55.31 m/mNK 3

1 04 0´103

1

04 0´103

1

ê ú 1 2

2 2 3




511

ev = ( 05 + 04 - 30)2 (1 - 2 ´ 0.3)

= (60)2 æ 0.4

ev = (3600)æ 0.001 ö

103

ev 2.1 = / m N K m3

ii i . ts raehS r nia en re yg

2 2 2

1 2 2 3 3

= = 29.67 3´103 N / mm 2

[(50 - 40) + (40 + 30) + (-30 - 50) ]

[100 + 094 0 + 46 00]

es 1 = 2 3. 5 nNK / m 3

vi . ( ytefas fo rotcaF F )S.O.

grene niartS y nidaol laixainu rednu emulov tinu rep g si

( 01 0)2

= 52 4 mNK / m 3 2´ 02 0´103

52

13. 55

1 3

è 004 ́ 103 ø ç ÷

2 ́ 02 0 ´103

ö 1

3

1 3 è ø

ç ÷

1 12C

[(s -s ) + (s -s ) + (s -s 1 ) ] es =

C =

E 02 0 ´103

2(1+1 / m) 2(1+ 0 3. ) w ereh

es =

es =

1 12(76. 29 3)´103

2 2 2

1 29 3. 70 6 ́ 103

2 s e = = 2E

F.O.S = = 1 548.




611

NI A LAIRETAM EHT LAPICNIRP SESSERTS ERA 05 M/N M2, 04 MM/N 2 DNA - 03 M/N M2 C , A CL U :ETAL

v. LATOT S RT A RENE NI G Y

iv . ULOV M IRTE C RENE NIARTS G Y iv i. AEHS R RTS A NE NI ERG NA Y D

.iiiv ROTCAF FO YTEFAS NO EHT LATOT RTS NIA YGRENE I NOIRETIRC F DLEIY LAIRETAM EHT A /N 001 T MM2.

AT K 01 X 002 = E E 3 MM/N 2 AN D P R NOISSIO A 82. 0 = OIT noituloS :

viG e ,n nirP c rts lapi e :sess

s 1 = 5+ 0N / mm 2

s = 4+ 0N / mm 2

s = -30N / mm

rts dleiY e ,ss

.i p ygrene niarts latoT re tinu ulov me:

(s1) +(s ) + (s ) -

[(50) +(40) +(30) - 2(0.3)[50´40 - 40´30 -30´50]]

[ 052 0 + 061 0 + 09 0 - 0.6[ 002 0 - 021 0 -1 05 0]]

[5000 - 0.6(- 700)]

[5420]

2

2

s = 01 0N / mm 3

e 2

é 2 2 2 2 m ê ë

1 2E 2 3

û (s1s + s s + s s 1 )ù

2 2 3 3 ú U =

=

=

=

2 2 2 1 2´ 02 0´103

= 04 0´103

U = 55.31 m/mNK 3

1 04 0´103

1

04 0´103

1




711

uloV)ii m ulov tinu rep ygrene niarts cirte m :e

ev = (s1 + s + s )2 é1 -

2E/ m ù

ë û

ev = ( 05 + 04 - 30)2 (1 - 2 ´ 0.3)

= (60)2 æ 0.4

ev = (3600)æ 0.001 ö

103

ev 2.1 = / m N K m 3

ii i . ts raehS r nia en re yg

2 2 2

1 2 2 3 3

= = 29.67 3´103 N / mm 2

[(50 - 40) + (40 + 30) + (-30 - 50) ]

[01 0 + 094 0 + 46 00]

es 1 = 2 3. 5 nNK / m 3

vi . ( ytefas fo rotcaF F )S.O.

grene niartS y dnu emulov tinu rep e nidaol laixainu r g si

( 01 0)2

= 52 4 mNK / m 3 2´ 02 0´103

ê ú 1 2

2 2 3 1 3

è 004 ́ 103 ø ç ÷

2 ́ 02 0 ´103

ö 1

3

1 3 è ø

ç ÷

1 12C

[(s -s ) + (s -s ) + (s -s 1 ) ] es =

C =

E 02 0 ´103

2(1+1 / m) 2(1+ 0 3. ) w ereh

1 12(76. 29 3)´103

2 2 2 es =

es = 1

70.329 6´103

2 s e = = 2E




811

52

13. 55

NE NIARTS RAEHS ERG Y P ULOV TINU RE ME

é ù

(s1 - s )2 + (s - s )2 + (s - s1 )2 ú

E

2æ1+ è

IROEHT E O S F ERULIAF

:era seiroeht lapicnirp ehT 1. rts lapicnirp mumixaM se roeht s y 2. rts raehs mumixaM e rts )ro( ss e reffid ss e yroeht ecn 3. grene niartS y yroeht 4. grene niarts raehS y yroeht 5. rts lapicnirp mumixaM ia roeht n y

= 1 548. F.O.S =

1 12C ê ú

ë û

2 2 3 3 ê ú

ê s s =

hw ere ç

ö ø ÷

1 m

C =




911

UN TI – V

A O GNIDNEB NI SCIPOT DECNAVD F EB MA S

YSNU M NEB LACIRTEM D GNI nidaol fo enalp ehT g nidneb fo taht )ro( g )ro( ni eil ton seod a sniatnoc taht enalp

ssorc eht fo sixa ladiortnec elpicnirp eht - nidneb eht ;noitces g snU dellac si y lacirtemm nidneb g.

EHT ETATS NIDNEB LACIRTEMMYSNU ROF SNOSAER OWT G

s si noitces ehT )i( y ziv( lacirtemm . natceR g ,ralucric ,ralu I daol eht tub )noitces t htob ot denilcni si enil h nirp e c lapi axes.

snu si noitces ehT )ii( y ziv( lacirtemm . nA g v noitces lennahc )ro( noitces el e citr al w na gnola si enil daol eht dna )be y .sexa ladiortnec

AEHS R ERTNEC

na rof( ertnec raehs ehT y evsnart r eb eht fo noitces es a ht si )m e fo tniop sretni e b eht fo noitc e nidn g nalp eht dna sixa e noitces esrevsnart eht fo . raehS c rtne e si

ertnec“ sa nwonk osla fo ”tsiwt

EHT ETIRW RAEHS NOITAUQE ERTNEC F CES LENNAHC RO T NOI 3b

Aw Af

s eht gnola bew eht morf ) CS( ertnec raehs eht fo ecnatsiD = e y XX sixa cirtemm Aw rA = ae ht fo e we b Af ht fo aerA = e nalf g e

SAH NOITCES LENNAHC A F NAL G C 21 SE M 2 X MC NA D BEW C 61 M 1 X

CM . ENNAHC EHT FO ERTNEC RAEHS EHT ENIMRETED L noituloS :

21 =b - 5.11 = 5.0 c m t1 = 2c t ,m 2 = 1c 81 =h ,m c m Af = tb 1 = 5.11 x 2 = 32 mc 2

Aw = th 2 1 x 81 = = 81 cm2

3b Aw Af

e = = .5 086 cm

6 +

e = 6 +

e = 6 +

3(11.5) 18 23




021

NOITAUQE ERTNEC RAEHS EHT ETIRW F MYSNU RO NOITCES I LACIRTEM

t1h 2 (b2 - b1 ) 2 4I

s eht gnola bew eht morf )CS( ertnec raehs eht fo ecnatsiD = e y XX sixa cirtemm

t1 iht = c nk e lf eht fo ss ang e ieh = h g w eht fo th e b

b1 = ht fo htdiw e f nal ge ir ni g noitrop th . b 2 = ht fo htdiw e f gnal e l ni e op tf rt .noi Ixx .O.M = I ht fo e es c noit a XX tuob a six .

LENNAHC ROF ERTNEC RAEHS FO NOITAUQE EHT EVIRED NOITCES

A lennahc noitces :segnalf( b x t1 ; Web h x t2) htiw XX sa ht e latnoziroh s rtemmy ic sixa .

L ehs deilppA = S te a rof r ec . X drawnwod lacitreV( )

T( )noitcerid drawpu eht ni bew eht ni ecrof raehs eht si S neh S1 = raehS ecrof ni eht pot egnalf ereht( lliw eb lauqe dna etisoppo raehs rof ce

nalf mottob eht ni g ).nwohs sa e ,woN ehs ar sserts (t) ni t eh alf n eg ta a ecnatsid of x morf eht thgir h dna edge fo( the pot

t = t erehw( 1 , )egnalf fo ssenkciht

t = . =

I .t1 2 2I ra yratnemele si ecrof raehS e a

(d A = t1.dx) = t .d A =t t1dz rof raehs latoT ec lf pot ni a egn

b

nalf eht fo htdaerb = b erehw( g )e

S ́ h hs t1 2I xx 2I xx 0

hS t1 b2 I xx 4

teL e = ecnatsiD fo eht s raeh ertnec )cs( f mor gnikat stnemom fo raehs secrof tuoba eht eW,bew eht fo O ertnec eg t

e = xx

lf a gn e)

t = h 2

Ay = (t1.x) xa

SAy I t St1.x h S xh

xx xx

= . xd 0

1 òt .t

0

o( r)

S1 = ;t1.dx = b b

ò ò xdx

S1 = .




1 12

S. e = S1.h

Sht1 b2 S.t1h2b2 I 4 4I

b2h2t1 4I xx

éb ´ t1 ù

Ixx = 2 + b.t1æ ö + ê ú

3

, gnieb elbigilgen ni rapmoc nosi ot rehto

I = (t2h + bb t1 )

12 fo eulav eht etutitsbuS I uqe ni xx a ,teg ew )1( noit

3b2t1

(t2h + 6ht1 ) Let = 1tb Af )egnalf eht fo aera(

th 2 = Aw (are ht fo a e )bew nehT

3bAf e = =

Aw + 6Af

e.i

3b Aw A f

UQE EHT EVIRED FO NOITA EC RAEHS N LAUQENU ROF RET I- NOITCES noituloS :

lauqenu nA I – oba lacirtemmys si hcihw noitces u a XX t x si .

= . .h = xx xx

e =

3 = + +

)1( bt1 b.t1h2 t2h3

6 2 12

3 h t2h3

ê ú ë û

ç ÷ è ø 12 2 21

2

,woN

bt1 h2 t2h3

2 12 = + +

bt1

3 gnitcelgen( eht mret

h2

xx et r ()sm o )r

e = ´ = 21

h2 (t2h + 6bt1 ) b2h2t1

4

3b Aw

Af 6 +

e = 6 +




221

na ni sserts raehS y al y ,re -

t =

3

2 (b1 + b2 ) t1 +(b1 + b2 )t1x ë

S eh ra f ro ec S1 : -

Ad = t1dx.A y = t1.x.

b1

ò t Ad = I t1 2

xt1 xd XX

b1 thS 1

é ù thS 1b1 2I 2 4I

lralimiS y S( ecrof raehs eht 2 nalf eht fo trap rehto eht ni ) g ,e

hS t1b2 4I

nikaT g ,O bew eht fo ertnec eht tuoba secrof raehs eht fo stnemom ew get S2 . h = S1 . e. S + h S( 3 )muirbiliuqe rof S =

ht fo ertnec eht morf ertnec raehs fo ecnatsid = e ,erehw( e we )b or , (S2 – S1 h ) = .S e

Sh2t1(b2 - b1 ) 4I

t1h2 (b2 - b1 ) 4I xx

RELKNIW GNISU SRAB DEVRUC NI SESSERTS EHT EVIRED – ROEHT HCAB Y

ehT elpmis gnidneb ,alumrof ,revewoh si ton ap bacilp el rof d pee ly cu devr smaeb erehw eht artuen l dna ladiortnec ax se od ton icnioc d .e To laed htiw hcus esac s W kni l re – hcaB ehT or y

su si ed . A rab CBA D laitini ly; ni sti iartsnu n de .etats Le A t B b ‟DC‟ e ht e rts a eht fo noitisop deni

bar .

AS y tI

é = I erehw I XX = ê

ê ú ú û

h 3 ù

12 12

h 2

S.x.t1 h S1 =

0

b1

ò 0

2 = =

XX XX

S.x.

ë û ê ú 0

x 2

ê ú h 2 I = t1 xd

XX

2

S2 = XX

2 2

= S.e XX

2 2 e =




321

L rutavruc fo suidaR = R te e ht fo e diortnec a GH sixa l . Y = f FE rebif eht fo ecnatsiD r al ladiortnec eht mo y GH re .

R‟ = fo suidaR c GH fo erutavru ‟ M = ( maeb eht ot deilppa tnemom gnidneb mrofinU a demuss

itisop erutavruc eht esaercni ot gnidnet nehw ev ) q = b dednetbus elgna lanigirO y GH sixa ladiortnec eht a sti t

fo ertnec c rutavru e O a dn q‟ = elgnA dednetbus by GH ‟ retfa( )gnidneb a t eht retnec fo erutavruc

q ‟ roF gnidnif eht niarts dna sserts lamron ot ht e ,noitces redisnoc eht erbif FE ta a ecnatsid

y .sixa ladiortnec eht morf

teL σ eb eht sserts ni eht deniarts la rey FE ‟ rednu eht gnidneb tnemom M dna e si niarts ni the al emas ye .r

EF '-EF (R '+ y ' )q '-(R + y)q EF (R + y)q

e0 eht ni niarts = ec diortn a al l y .i re e . nehw y 0 = R ' q ' R '+ y ' q ' R q R + y q

1 dna + e = . -- --- - -- - )2(

nidiviD g )1( noitauqe a w , )2( dn e get y ' y ' y

1+ e R'+ y ' R 1+ e0 R + y R'

nidroccA g orez si niarts laidar , )3( noitpmussa ot e.i . = y y‟

y y y

,niartS e = 1+

niddA g nitcartbus dna g ht e et r e m 0 . y ew ,R/ eg t

e0 . + + e0 - + e0 - e0 .

e = 1+

(1+ e0 () - ) y

e = e0 + 1+

morF eht fi .g eht la ey rs evoba eht iortnec d la reyal si ni noisnet dna eht la srey woleb eht al ladiortnec y re noisserpmoc ni si .

R'+ y ' q ' e = . -1 R + y q

ro ,niartS e = =

= . -1 ro 1+ e = . -- --- - -- - )1(

R' q ' R q

R ' R'

1+

e0 . + + e0 -

y R

R = . ro e =

e0 . + + e0 - R ' R ' R y R

y R

-- --- - --- - --- )3(

y y y y R ' R ' R R

y R

1 1 R' R

y R




421

1

= eE = σ , ssertS E(e0 + 1+

ht no ecrof latoT e es c ,noit F = . Ad

gniredisnoC a llams pirts fo atnemele ry aera ,Ad ta a ecnatsid fo y morf eht ladiortnec layer ,GH we ha ev

1( + e0 () - ) y F = E . Ad + E(1+ e0 )( - ) dA

R, R R

F = E .A + E(1+ e0 )( - ) dA ____________ ( )5 R, R

R b eht fo noitces ssorc = A erehw a r

nevig si tnemom gnitsiser latot ehT g yb nevi

( +1 e0 () - )y M = .y. Ad =E e0 .y Ad + E

1+

M = E .0 + E(1+ e0 )( - ) Ad

R, R

2

ò 1 + y Ad eL t ò 1+ y

Ad = Ah2 R R

h erehW 2 rab eht fo noitces ssorc eht rof tnatsnoc a =

= M E ( +1 e0) - Ah

ë û

yR R + y

ò 1 + y dA = 0 -

R y R R

He semoceb )5( noitauqe ecn

F = eE 0 A. – E (1+e0 ) -

tces enalp esrevsnart ecniS r snoi e nirud enalp niam g gnidneb F = 0

0 = Ee0 A. – E ( +1 e0 ) - ë û

1 R

) y )

1( + e 0 () - R'

y R

___________ )4(

ò s

1 1 R' R 1 1 y ò e ò 1 + y F = E e0 . Ad + E ò ò 0 dA

y R

1+

1 1 y 0 ò e ò 1 + y

2

Ad

1 1 R' R ò s ò ò y

R 1 1 y 2

ò 0 ò e ò 1 + y cnis( e Ady =

0 )

R 1 1 y y 2

ê ú M = E e+1( 0) - ë û é ù

R' R

R y 1 y 2

2

é

. Ad = - ë

1 1 R' R

-- --- - --- -- )6( é ù

ê

ò 1+

ò ê y

ê ú

2

ú Ad

ú û

ù . Ad =

y ò ò = dy A- y 2

R + y . Ad ,woN ò 1 + y ò y

R + y

1 R

-- --- - --- - )7( . Ad = - Ah 2

1 Ah 2 ù û

1 R' R R ú ê

é ë

é ù ê ú

1 1 Ah 2

R' R R




521

E e0 A. = E ( e+1 0 ) -

o( r) = e+1( 0 ) -

R' R

utitsbuS t ni g v eht a fo eul = 1( +e0 ) - ht ni e e uq a ( noit 6 ) R' R

2

nitutitsbus g v eht a o eul f e0 uqe ni a ( noit 4)

s = + E * * * 1+

s = + * * 1+

T( e lisn e )

rpmoC( e viss e)

ë

ehT devruc rebmem ohs wn ni gif . sah a dilos ralucric orc ss – noitces 10.0 m ni .retemaid If eht xam imum elisnet dna comp evisser sesserts ni eht rebmem era n to ot xe c dee 051

aPM dna 002 M .aP D enimrete eht eulav fo aol d P taht nac as f yle eb deirrac by eht em mb re .

noituloS :

G ,nevi

aPM 051 = G ;m 01.0 = R ;m 01.0 = d / NM 051 = m2 ) elisnet(

s2 m / NM 002 = aPM 002 = 2 isserpmoC( )ev

daoL P:

feR e eht ot r fig . rA e ,noitces ssorc fo a 2

A = = ´ ( .0 10)2 .7 = 854´10-3 m2

B nidne g mom e m ,tn = 51.0( P + 01.0 ) = P 52.0

R' -

ú R R ê

1 R'

1 Ah 2 ù û

é ë

e0 +1( = e0 ) é ù e0 R 1 1

h 2 ë û ê ê ú ú R R

é ù

e0 R

h 2

M RAE

ê ú e0 R 1 1

h 2 ë û

M = E Ah = e0 RAE

e0 = rO

M y R M )ro( s = + E * * y R

RA M y e0 R

y R

h 2

AE R RA h 2

1+

M M Ry 1 y R

AR RA h 2

ë é

M R 2 y é ê1+

ê

úú

é ùù

ê h 2 ë

R + y

ûû

ú

ê1-

ê

AR

M AR

s = s =

R 2 y h 2

ë R

-

ùù

úú ê ûû ú

é y

pd p 4 4

1 Ah 2 ù û

1 é ë




621

01 x 130.7 = -4 m2

,sserts tceriD s d

=

tniop ta sserts gnidneB :M ot eud 1

t( e )elisn

rts latoT e ,1 tniop ta ss s1 = s d + s b1

051 = +

A AR

2.0 5P 1.0 02 ê1 +

58.7 4 ´10 -3 ´ .0 01 30.7 1´10 -4

= - P 23.721 + 5 x P 13.813 . 47 = 87.9961 P P = = 52.88 KN

961 9 87.

tniop ta sserts gnidneB ot eud 2 M:

é ù s = ´ -1ú

ê ú

rts latoT e ,2 tniop ta ss s 2 = s d + s b2

002 = +

A

2.0 5P 58.7 4´10-3 ´ 0. 01

= P 23.721 + 813 . 13 22.31 x P = P 83.5334

P = MN

P = = 1.64 3KN )ii(

34 3 .5 38

By as eht )ii( & )i( gnirapmoc f o ressel eb lliw P daol e seulav owt f

d 1 ( 1.0 0)4 2 h2 = + .

61 821 ( 1.0 0)2

(comp) p A

s = ê 1 + ù ú

é R 2

h2 ´ y

R + y M

RA û b1

ê ë ú

- P M t( e lisn e) ú y

R + y

ù ´

R 2

h 2

é ê1 +

ë û ê ú

ù 50.0 - P 58.7 4 ´10-3 ê

ú û .0 01 + 0.0 5 ú

)i(

é

ë ´ 051 = +

051 ́ 103

)pmoc( ë û h 2 ê

M R 2 y AR R - y b2

ù -1ú

û

y R - y

é R 2

h2 ê M

RA P ´

ë ú ê

ê ê ë 30.7 1´ 01

é ù -1ú

û

P 58.7 4´10-3 -4 + ´ 50.0

0. 01 - 50.0 = 1.0 02

ú

002 334 5. 83

02 0´103




721

\ faS e daol = K 31.64 N. A emarf cejbus t de ot a daol fo 4.2 .Nk dniF )i( ehT tnatluser sesserts ta a tniop 1 dna

)ii(;2 P artuen fo noitiso l sixa . )3002 yaM/lirpA(

noituloS : s fo aerA e 1 noitc - ,2

A = 84 * 01*81 -6 = 46.8 * 01 -4m2 gnidneB ,tnemom

= M - 01*4.2 3*( 021 + * )84 = - mN 2.304 kat si M e n a s – uc eht esaerced ot sdnet ti esuaceb ev rva rut e.

iD )i( rec s t tre :ss

riD ce rts t e ss σd =

h 2 = ol g e

æ ö - R 2 D 2R - D

Here mm 84 = R = D ,m 840.0 = mm 84 = m 840.0

40.0 83 ç 40.0(2 8) + 40.0 8 ö

40.0 8 40.0(2 8) - 840.0

= 840.0 2 ol( ge 3 – )1 = 01 * 72.2 -4 m2

ts gnidneB )ii( r :2 tniop ta M ot eud sse

M R y

ê h 2 ë

R - y ûû

s = 1* 0 -6 MN / m 2

6.8 4 1* 0 -4 * 840.0 2.2 7 1* 0 -4 ê 840.0 - 20.0 4 úú

= - 1( 227.9 - )941.01 = m/NM 59.88 2 t( e lisn e)

eud sserts gnidneB )iii( ot M :1 tniop ta

ê1+

ê úú

é

s = 1* 0 -6 NM / m 2 6.8 4 1* 0 -4 * 840.0 2.2 7 1* 0 -4

ë _+ 20.0 4úú

= - .24 m/NM 16 2 = m/NM 16.24 2 (c )pmo

:sserts tnatluseR )vi(

= 1* 0 -6 = 2.77 MN / m 2

8.64 1* 0 -4

P 2.4 1* 0 3

A

R 3 2R + D è ø ç ÷

h 2 = ol g e ç ÷ - 840.0( )2

è ø ÷ æ

é ùù 2

ê1- ê úú

ë ú

é ;

é ùù

s = b2

ë

AR

ê1- ú - 04 3 2. 40.0 82 420.0

ê ë ûû

é

M R 2 y s =

é b1

é ùù

ê h 2 ë

R + y ûû

- 04 3 2. 40.0 82 20.0 4 ùù

ê1+ ú

RA ú

ê

ë

é

ë ûû ê0 840.




821

,2 tniop ta sserts tnatluseR σ 2 = σd + σ 2b = 77.2 + 88 . 59 = m/NM 27.19 2 t( e lisn e)

,1 tniop ta sserts tnatluseR σ 1 = σd + σ 1b = 77.2 - .24 6 1 = m/NM 48.93 2 (c pmo )

v( ) P tuen eht fo noitiso r :sixa la

é ù

R ú

.0 048 * 2.27 *10 -4

ë .0 40 8 + .2 27 *10-4 = - = m 53400.0 - mm 53.4

,ecneH mm 53.4 fo suidar a ta si sixa lartuen

Nk 03 fo daol a gniyrrac gnir A . C talucla e ht e ts er a sess t .2 dna 1

:noituloS fo aerA c ssor -s ce noit

B nidne g mom e M tn

h 2

Here d

h 2

riD ec rtS t e ss σd

B nidne g ,M ot eud 1 tniop ta sserts

é ùù

h 2 ë

R + y ûû

0. 60 ê 531.0 + 0.0 6ú ú

1* 0-6

7 = m/NM 576.71 2 t( e lisn e)

,M ot eud 2 tniop ta sserts gnidneB M R y

ê h 2 ë

R - y ûû

04 50 31.0 52 0. 60

ê 8.9 9 1* 0 -4 ê 531.0 - 0.0 6 úú

47 = m/NM 14.63 2 (c )pmo

Rh y = -ê 2 ú

ê - h 2 û

y = -ê

2 ê

ë é

2

ú ú û

ù

= x122 cm2 1.311 = cm2 13110.0 = m2

4 p

= 01*03 3 * 01*5.31( -2 mN) = 0504 mN d 1 d 16 128 R 2

2 4 = + * + ......

mc 5.31 = 6+ 5.7 = R ,mc 21 =

= + * = 98.9 mc 2 = 01*98.9 -4 m2 16 128 13.5 2

12 2 1 12 4

= = *10 -6 = .2 65 MN / m 2 - P 30 1* 0 3

A .0 3110 1

ê1+

ë ê

é M RA

s = s =

b1

b1

ú

R 2 y ê úú

04 50 31.0 52 é ùù

ë ûû ú ê1+

ê 8.9 9 1* 0 -4 ë

é 6.6*56.2

13110.0 * 531.0

é ùù 2

ê1- ê úú

ë ú

é s = b2 AR

é ùù

ë ûû

s = 1* 0 -6 .31*56.2 ê1- ú ë

b1 13110.0 * 531.0

é




921

Henc σ e 1 = σd + σ 1b = - 56.2 + 576.71 = m/ NM 50.51 2 t( e lisn e)

na d σ2 = σd + σ 2b = - 56.2 – 14.63 = m/NM 60.93 2 (c )pmo

021 fo ebut a fo demrof si rab devruc A mm aid edistuo m m 5.7 dna rete m ssenkciht . Th e tnec r a ralucric a si siht fo enil e r fo c r m 522 suida m . gnidneb ehT m Nk 3 fo tnemo m

ab eht fo erutavruc esaercni ot gnidnet r pa si p il ed . eht etaluclaC m ixa mum te dna elisn oc m rab eht ni pu tes sesserts evisserp .

noituloS :

maid edistuO e et r ht fo e d ,ebut 2 = mm 021 = m 21.0 ihT c nk e but eht fo ss e = mm 5.7

I but eht fo retemaid edisn e d , 1 021 = - mm 501 = 5.7*2 = m501.0 ssorc fo aerA - ,noitces

A = [ 1.0 22 - 1.0 52 ]= 56200.0 m2 4

tnemom gnidneB M = 3 mNk renni fo aerA c ri c ,el

A1 = [ 01.0 52 ]= 66800.0 m2 4 retuo fo aerA c ri c ,el

A2 = [ 1.0 22 ]= 13110.0 m2 4

oF noitces ralucric r , 2 4

h2 = + * + ..... .

oF r enni r ci cr ,el

h2 = + * + ......

h2 = + * = 7.08 *10 -4

ic retuo roF cr ,el d 2

2 d 2 4

61 21 8

Ah 2 = A2 h2 - A1h1 56200.0 h2 = 01*23.9*13110.0 -4 – 01*870.7*66800.0 -4 h2 = 66100.0 m2, dna R2 h/ 2 = 522.0 2 66100.0/ = 94.03

mumixaM s ,A ta ssert

,erehw( y )m 60.0 = mm 06 =

ë

p

p

p

d 1 d R 2 16 128

d12 d1

4 1 61 821 R 2

.0 01 52 1 0 01. 54

0 22. 52 61 21 8

h2 = + * + ..... .; h2= .0 12 2 1 .0 12 4

+ * = 9.32 *10 -4

0. 22 5 2 61 821 2 2

1 R 2

é ùù M R y 2

ê1+ ê

úú ú ê h 2

ë R + y

ûû

é

AR A s =




031

3 1* 03 é 0.0 6 ù A ê1+ 30 94. é

0.0 6úú 1* 0-6 NM / m 2

ë û σA = m/NM 23.73 2 t( e lisn e )

,B ta sserts mumixaM

2

ê-1 + ê úú

ù

B ê-1+ 94.03 é 0.0 6úú

1* 0-6 NM / m2 ë û

σB = m/NM 57.05 2 (comp)

MAEB DEVRUC A H T A SA - NOITCES . TH RENNI E R 003 SI SUIDA MM . TAHW SI TICIRTNECCE EHT Y OF ?NOITCES EHT

loS u :noit

T fo aerA - es c ,noit = b1t1 + b2t2 02*06 = + 02*08 = mm 0082 2

T fo g.c dnif oT - de eht tuoba stnemom gnikat ,noitces g L e L ew , g te A1 x1 + A2 x2

A1 + A2

(60* 20)( + 20) + (80*20)(80*20*1 )0

2 6( 0* + )02 (80*2 )0

oN w R1 = ;mm 003 R2 = 023 R ;mm = 23 ;mm 41.7 R3 = mm 083

:noitaleR eht gnisU

h 2 = . ol g e + t1. ol g e ú - R 2

023 ê80* ol g ( ) + 20* ol g e ( )ú - 41.723( )2

= 8.30521 ( 61.5 + 44.3 ) – 80.215 = 6.020701

2

= y - = = 65.1 mm (-) R 2 + h 2 41.723( )2 + 15 2 80.

erehw y )yticirtnecce( e = .sixa ladiortnec eht morf sixa lartuen eht fo ecnatsid = eN g eht woleb setacol si sixa lartuen taht setacidni ngis evita c sixa ladiortne .

ê 22.0 5 +

û

ù ë

s = 6200.0 5* 522.0

M R y s =

é B

é ùù

ë RA ê h 2 ë

R - y ûû

3 1* 03 é 0.0 6 ù

ú

ë ê 22.0 5 -

û

s = 6200.0 5* 522.0

- x = - x =

60

41.72= mm

é R2 R3 ù êb2 ë A R1 R 2 û

é ù e

ê ú 03 0 023 ë û

083

R 3

h 2 = 41.723( )3

0082

é ù

ê ú ë û

Rh 1.723 4* 80.215 ê ú




131

C- F O DAOL A OT DETCEJBUS EMAR F 021 KN . A TA SESSERTS EHT ENIMRETED DNA B .

noituloS :

L Nk 021 = )P( dao rA ae fo c or ss – b = noitces 1t1 b+ 2t2 b + 3t 3

= 03*021 + 03*051 + 03*081 = mm 5310.0 2 .c dnif oT g L egde eht tuoba noitces eht fo L ,

A1 x1 + A2 x2 A1 + A2

( 021 * 02 * 22 5) + ( 051 * 03 1* 5) + ( 081 * 03 21* 0) y1 =

( 021 * 30) + ( 051 * 30) + ( 081 * 30) y2

= 042 – 311 = 721.0 = mm 721 m R1 = mm 522 = 522.0 m R2 = 522 + mm 552 = 03 = m 552.0 R m 833.0 = mm 833 = 311 + 522 = R3 = 522 + 012 = mm 534 = m 534.0 R4= 522 + mm 564 = 042 = m 564.0

R 3 é ç ÷ R4

ù A R1 R2 R3 û

æ ö h 2 = 1.0 5 ol g e

æ ö + 0.0 3 ol g e æ ö + 1.0 2 ol g e

ç ú÷ - 833.0 2

0. 310 5 22.0 5 ø 52.0 5 ø .0 534

68.2 = ( 77810.0 + 610.0 + )800.0 – 2411.0 = 221800.0 m2

= *10 -6 98.8 = MN / m 2 ( oc mp)

*P = M ,tnemom gnidneB R ,tnemom gnidneb eht ot eud A ta sserts gnidneB

M R y 2 ù

b A ê ú ê ú

P * R 33.0 82 21.0 7

ê 21800.0 22 ê 33.0 8 + 21.0 7 úú

1( 98.8 = + )248.3 = m/NM 40.34 2 t( e lisn e)

B nidne g b eht ot eud B ta sserts e nidn g mom e :tn

M R 2 y1 ù

b A ê ú

311.0

ê 833.0 - 11.0 3úú

= 98.8 ( 1- )460.7

- x =

= mm 311 = m 311.0

è è

æ ö æ R3 ö æ ö h 2 = ol g e ç ÷ + t3 ol g e ç ÷ + b1 ol g e ç

ú÷ - R ê

33.0( 8)3 é 0 52. 5 0 534. ê ç ÷ ç ÷

R2 ç ÷ ç ÷ú

2

0. 64 5 ù ç ÷ú

êb2

ë

ë û è ø ê

è ø è ø è ø

eriD c rts t e σ ,ss d = P 21 0 1* 0 3

A 0.0 31 5

2

ê1+ ú é ù

ë h 2 ë

R + y 2

ûû

é (s ) =

RA

é ê1+ ú ë

é ùù

ë ûû RA (s ) = b A

é ê1+ ú ë h 2

ë R - y1

û û

ê ú

é ù

b A

(s ) = (s ) =

RA

P * R 33.0 82 é ùù

ë ûû ê1+ ú ê 221800.0 ë

é

RA




231

= - m/ NM 9.35 2 = m/NM 9.35 2 (c )pmo rtS e ,A ta ss σA = σd + σ( b)A

= - m/NM 51.43 = 40.34 + 98.8 2 t( e lisn e) ta ssertS B, σB = σd + σ( b) B

= - 98.8 – 9.35 = m/NM 97.26 2 (c )pmo

EHT EVIRED F ALUMRO F AEB FO NOITCELFED EHT RO M OT EUD S YSNU M NEB LACIRTEM D GNI

noituloS :

c htiw maeb eht fo noitces esrevsnart ehtT G diortne . XX a YY dn are owt er c nat g ralu co- ra VV dna UU dna sexa etanidro e nirp eht c na na ta denilcni sexa lapi g eht ot θ el

oc fo tes YX - setanidro sexa . W daol eht si a nitc g nola g maeb eht fo noitces eht no YY enil eht . otni devloser eb nac W daol ehT stnenopmoc owt gniwollof eht :

( )i nola …… θ nis W g GU ( )ii nola …… θ soc W g GV

Le δ ,t u b desuac noitcelfeD = y nola θ nis W tnenopmoc eht g nidneb sti rof UG enil eht g tuoba VV a dna ,six Δv b desuac noitcelfeD = y nola θ soc W tnenopmoc eht g G enil eht nidneb ot eud V g a tdob

UU a six . nidneped nehT g v eht ,maeb eht fo snoitidnoc dne eht nopu a o seul f δu dna δv are g b nevi y

K(W sinq)l 3 EI VV

K(W oc sq )l 3 EI UU

hw ere, A = K c nidneped tnatsno g dne eht no c eht fo snoitidno ola daol eht fo noitisop dna maeb na ,maeb eht gn d

nel = l g maeb eht fo ht δ noitcelfed tnatluser ro latot ehT c :swollof sa dnuof eb neht na

d = (du )2 +(d )2 2 2

çW sinq ÷ ç W oc sq ÷ I VV I UU

æ ö æ ö

è ø è ø

eht htiw ,δ noitcelfed eht fo β noitanilcni ehT b nevig si VG enil y: I UU

dv I VV tanq

d u

d v

v

+ æ ö æ ö ç ÷ ç ÷ è ø è ø

lK E

3

d =

+ I oc s 2 q

ç ÷ UU I

ç ÷ 2 2

s ni q ç ÷

VV

2 ç ÷ d =

Kl 3

E

du nat b =

=




331

A 08 mm x 08 mm x 10 mm an elg se noitc sh nwo in .gif si desu sa a ylpmis etroppus d eb a m revo a naps fo 4.2 .m tI rrac i se a daol fo 004 Nk ola ng eht il ne GY , w ereh G si eht ec n iort d

fo eht oitces n. aluclaC te )i( rtS sesse at eht iop n st ,A B a dn C of eht im d – se noitc of eht maeb )ii( noitcelfeD fo eht maeb ta eht im d-s noitce dna sti d noitceri w hti eht aol d il ne )iii(

P tuen eht fo noitiso r sixa la . kaT e 002 = E G m/N 2

loS u :noit teL )Y,X( eb eht oc - dro i etan fo iortnec d ,G htiw tcepser ot eht atcer n ralug ax se

XB 1 dna YB 1. 80 1* 0* 04 + 70 1* 0 * 5 0023 0+ 35 00

01*08 + 70 01* 008 + 700 eni fo tnemoM r :sixa XX tuoba ait

é80 1* 03 ù é10* 70

3 ù XX

ê ú ê ú

)655872 + 66.6666( = + 33.338582( + )777813 = mm 338988 4 = 01 * 898.8 5 mm4 = I YY nis( ce na si ti e uq a na l g es el c )noit

oC - G fo setanidro 1 04( + = - ,)66.32 - 66.32( - ,43.61( = )5 - )66.81 oC -or tanid e G fo s 2 = - 66.32( -5) . + 54( – ( = )66.32 - 43.12 + ,66.81 )

tcudorP( fo aitreni tuoba eht diortnec ax se si orez esuaceb snoitrop 1 na d 2 era atcer n ralug )spirts

uorht gnissap ,XG htiw sexa lapicnirp fo noitanilcni eht si θ fI g ,neht G h

2I nat 2q = = ¥ = nat 09 o

XY XX

θ2 = º09 e.i . θ1 = º54 dna θ2 = º09 + º54 = º531 are eht in snoitanilc fo eht icnirp p la ax se UG dna VG

levitcepser y. aitreni fo tnemom lapicnirP :

I YY - I ( ) 2 + (I ) 2

2

= 98.8( 5 01* 5 + 98.8 8 1* 05 + ) ( ) 2 + (- 22.5 6 1* 05 ) 2 2 2

= ( 898.8 + 6622.5 ) 01* 5 = 01*5421.41 5 mm4 I UU + I VV = I XX + IYY

IVV = I XX IYY – I UU = 898.8*2 x 01 5 – 21.41 4 1 x 6 05 = 1 x 76.3 05 mm4

( )i ,A stniop eht ta sessertS B :C dna

nidneB g dim eht ta tnemom - ,noitces Wl 04 0 * .2 4 *10 3 4 4

;era stnemom gnidneb eht fo stnenopmoc ehT

M‟ = M nis θ = 4.2 x 01 5 º54 nis = 01 x 796.1 5 Nm m

X woN = Y = = 66.32 mm

I = + 80 1* 0* 66.32( - 5) 2 ú + + 70 1* 0* 54( - 66.32 )2

ú 21 21 ë û ë û ê ê

cnis( e Ixx =Iyy ) YX

I - I

XX YX I UU = (I + I YY ) + XX 1 2

1 98.8 5 1* 05 - 98.8 8 1* 05

= 2.4 *10 5 Nmm M = =




431

M ‟‟ = M soc θ = 4.2 x 01 5 c º54 so = 1 x 796.1 05 mmN oc v,u -or :setanid

:A tnioP = x - 08 = y ,66.32 - mm 43.65 = 66.32 θ nis y + θ soc x = u

= - º54 nis x 43.65 + º54 soc x 66.32 = mm 1.32 = v y x + θsoc nis θ

º54 soc 43.65 = - (- mm 65.65 = )º54 nis x 66.32 tnioP B :

= x - ,66.32 y = - 66.32

θ nis y + θ soc x = u = - ( + º54 soc x 66.32 - 66.32 x = ) º54 nis - mm 54.33

θ nis x + θsoc y = v = - º54 soc 66.32 - (- )º54 nis x 66.32 = 0

; C tnioP x 08 = – 6.32 6 ,43.65 = y = - 66.32 θ nis y + θ soc x = u

º54 soc 43.65 = - mm 1.32 = º54 nis x 66.32 θ nis x + θsoc y = v

= - º54 soc 66.32 - º54 nis 43.65 ) =- mm 65.65 M 'u M "v I VV I UU

96.1 7 1* 0 5 ( )1.32 96.1 7 1* 05 )65.65(

6.3 7x105 421.41 6x105

96.1 7 1* 05 (-3 4.3 5) 0

96.1 7 1* 0 5 )1.32( 65.65 6.3 7x105 421.41 6x105

( )ii lfeD e :δ ,maeb eht fo noitc si δ noitcelfed ehT g b nevi y:

s ni q

ç ÷ VV

erehw K = 84/1 rof a maeb htiw pmis ly detroppus sdne dna rac r iy ng a tniop daol eht ta ec rtn e.

L , dao W = N 004 L ne g ht l = 4.2 m

E = 01 x 002 3 /N mm2 I UU = 6421.41 01 x 5 mm 4

I VV 01 x 76.3 = 5 mm 4 teg ew ,seulav eht gnitutitsbuS

s = + A

s = + = 74.71 N / mm2

s = + = - 74.51 N / mm2

6.3 7x105 421.41 6x105

A

B

ç ÷ UU

s = + B

2 è ø è ø

2 ç ÷ æ ö æ ö

= 887.3 N / mm2

soc q +

2 ç ÷

2 I I d =

KWl 3

E




531

sin 54 o ö

6.3( 7x105 ) 2 ÷

δ = mm 6641.1 ht htiw esiwkcolc β elgna na ta denilcni eb lliw δ noitcelfed ehT ,VG enik e g yb nevi

nat b = tanq = nat 54 o 3.848

β = º34.57 - º54 = º34.03 esiwkcolc htiw eht daol enil YG ‟. :sixa lartuen eht fo noitisoP )iii(

ehT lartuen ax si lliw eb ta º09 - º34.03 = º75.95 a tn i- esiwkcolc htiw eht lo da li ,en a lartuen eht esuaceb d fo enil eht ot ralucidneprep si six e lf ce noit .

ETATS T MUSSA EH P NIW NI EDAM SNOIT K EHT HCAB S’REL O YR 1( ) nirud enalp niamer )esrevsnart( snoitces enalP g nidneb g. 2( ) hT e iretam a ebo l y kooH s e l s‟ a w ( ts timil a orp fo et p noitro a til y ton si

ecxe ede )d aR )3( rts laid a en si ni g .elbigil

na tuohtiw tcartnoc )ro( dnapxe ot eerf era serbif ehT )4( y niniartsnoc g tceffe eht morf a ajd ec rbif tn es.

ITRENI FO TNEMOM LAPICNIRP DNA SEXA LELLARAP EHT ETATS A

I a ,dnuof si aitreni fo tcudorp eht hcihw tuoba sexa owt eht f re eht taht , hcus neht era sexa owt eht ,orez semoceb aitreni fo tcudorp c xa elpicnirp eht della es . ehT

nirp a tuoba aitreni fo tnemom c picnirp eht dellac si sexa lapi a o tnemom l f eni r it a.

NOITARTNECNOC SSERTS cidni ot desu si tneidarg sserts mret ehT sserts a sa sserts fo esaercni fo etar eht eta

ppa si resiar ro ca hed . rts dellac era sesserts dezilacol esehT e rtnecnoc ss a noit .

SSERTS – NOITARTNECNOC F ROTCA I ht fo oitar eht sa denifed si t e rts mumixam e ot ss t rts lanimon eh e .ss

s K t =

s

s mixam = u s m ssert s = nimon a rts l e ss

NOITARTNECNOC SSERTS EUGITAF F ROTCA

itaf ehT g rts eu e ss – rtnecnoc a K( rotcaf noit f nifed si ) e r eht sa d a fo oit f nal ge timil itaf eht ot nemiceps dehctonnu fo g hcton fo timil eu e ro( laixa rednu nemiceps d ) nidneb g

daol s. K = 1+ q(K t -1)

Va o eul f nar q g no ot orez morf se e.

oc s 2 54 o

( 6421.41 x10 5 ) 2

1 04 0x( .2 4 x10 3 ) 3

48 E

2 æ æ ÷

è ø è ø

ö ç + ÷ ç ç ÷

ç d =

I UU

I VV

14 1. 246x105

6.3 7x105

m ax

mon

xam

mon

f




631

RAEHS F WOL nel revo H ecrof raehs latnoziroh fo oitar eht sa denifed si wolf raehS g eht fo ht

eb a x m . nol eht gnola gnitca si wolf raehS g rahcsid ta detacol ecafrus laniduti g e y1 raehS. b denifed si wolf y q.

H Q z

z

H = raehs latnoziroh f ecro

O NOITISOP EHT NIALPXE F SNOITCES SUOIRAV NI ERTNEC RAEHS )i( I s fo sexa owt gnivah maeb a fo esac n y eht htiw sedicnioc ertnec raehs eht ,yrtemm

iortnec d. )ii( I s fo sixa eno gnivah snoitces fo esac n y hs eht ,yrtemm ton seod ertnec rae

s fo sixa eht no seil tub diortnec eht htiw edicnioc y rtemm y.

ERTNEC RAEHS EHT GNITACOL NI DEVLOVNI SELPICNIRP EHT ETATS ssorc a rof ertnec raehs eht gnitacol ni devlovni elpicnirp ehT – maeb a fo noitces

nitca sdaol eht taht si g no nalp a ni eil tsum maeb eht e tluser eht sniatnoc hcihw a ehs tn a r ssorc hcae no ecrof - .sesserts gniraehs eht morf detupmoc sa maeb eht fo noitces

O ERTNEC RAEHS FO NOITISOP EHT ENIMRETED F ES EHT C FO NOIT

AEB EHT M noituloS : t1 = 4 c b ,m 1 = 6 c b ,m 2 = mc 8 h1 03 = – 4 = 62 c m

t1h 2 (b2 - b1 ) 2

4I

I xx = 2ê +14x4(13)3 ú +

ê ú

4x 62 2 8( - 6) 2 4(20 58 2

ETATS T SERTS EH S NEB LACIRTEMMYSNU OT EUD SE D GNI

s = M + ë û

σb nidneb = g eht ni sserts curve rab d ilppa daol eht ot eud tnemom = M e d

IUU = nirP c mom lapi e ni fo tn e eht ni aitr pr picni a xa l e UU s IVV = nirP c mom lapi e o tn f eni r it a ht ni e nirp c lapi axe VV s

F EUGITA

rav rednu lairetam a fo eruliaf eht sa denifed si eugitaF y eht woleb llew ,sdaol gni fo rebmun etinif a retfa ,daol citats etamitlu cy gnidaolnu dna gnidaol fo selc .

q = = V y x I

e = xx

4 é14x43 ù

ë û 21 2x22 3

21 = 20852 mc

e = = 7709.0 cm

é ù b ê

v oc sq u sinq I UU I VV

ú




731

YT P O SE F SSERTS EUGITAF

riD )i( ec rts t e ss gnidneb enalP )ii(

gnidneb gnitatoR )iii( noisroT )vi(

v( ) nibmoC e rts d e sess )a( F nitanretla ro gnitautcul g rts e ss

b( ) veR e sr e rts d e ss .

SERTS ROF SNOSAER EHT ETATS S- ECNOC NOITARTN ral a nehW g co tneidarg sserts e c ra dezilacol ,llams a ni sru e eht ,erutcurts a fo a

ih g rts h e rtnecnoc sserts a sa ot derrefer si ss a noit . noc sserts rof snosaer ehT c era noitartne )ii( muunitnoc ni seitiunitnocsid )i( c atno c rof t ec s.

PEERC

rC ee orp dna wols eht sa denifed eb nac p g htiw lairetam a fo noitamrofed evisser dnu emit e r a tsnoc a rts tn e .ss



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