Module 6: Two Dimensional Problems in Polar Coordinate System

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1 odule 6: Two Dmenonl olem n ol oodnte Sytem 6.3. RS WITH LRGE INITIL URVTURE Thee e ctcl ce of, uch hook, lnk nd ng, etc. whch hve lge ntl cuvtue. In uch ce, the dmenon of the coecton e not vey mll n comon wth ethe the du of cuvtue o wth the length of the. The tetment tht follow ed on the theoy due to Wnkle nd ch WINKLER S H THEORY umton. Tnvee ecton whch e lne efoe endng emn lne even fte endng.. Longtudnl fe of the, llel to the centl x exet no eue on ech othe. 3. ll coecton oe vetcl x of ymmety lyng n the lne of the centodl x ng though (Fgue 6.) 4. The em ujected to end coule. The endng moment vecto noml thoughout the lne of ymmety of the em. Wnklech Fomul to Detemne endng Ste o Noml Ste (lo known cumfeentl Ste) () () Fgue 6. em wth lge ntl cuvtue led Eltcty fo Engnee T.G.Sthm & L.GovndRju

2 onde cuved em of contnt coecton, ujected to ue endng oduced y coule led t the end. On the of lne ecton emnng lne, we cn tte tht the totl defomton of em fe oey lne lw, the em element otte though mll ngle Dd. ut the tngentl tn e doe not follow lne eltonh. The defomton of n ty fe, gh e c Rd ydd whee e c denote the tn of the centodl fe ut the ognl length of the fe gh (R y) d Theefoe, the tngentl tn n the fe gh e [ e Rd ydd ] c ( R y) d Ung Hooke Lw, the tngentl te ctng on e d gven y e c R y( Dd / d ) E ( R y) Dd Let ngul tn l d Hence, Euton (6.6) ecome e R yl c ( R y) E ddng nd utctng e c y n the numeto of Euton (6.6), we get, e c R yl e c y e c y E ( R y) (6.6) (6.6) Smlfyng, we get y êe ( R y) E c ( l e c ) (6.63) ë The em ecton mut tfy the condton of ttc eulum, F z nd x, eectvely: \ò d nd ò yd (6.63) Suttutng the ove oundy condton (6.63) n (6.63), we get led Eltcty fo Engnee T.G.Sthm & L.GovndRju

3 ò y êe c ( l e c ) ( ) d ë R y odule6/leon3 y o ò e c d ( l e c ) ò d ( R y) y o e c ò d ( l e c ) ò d (6.63c) R y lo, ( ) y êe cò yd ( l e c ) ò d E ( R y) (6.63d) ë Hee ò d, nd nce y meued fom the centodl x, ò yd. Let y ò d m O m ( R y) Theefoe, y ò ( R y) d y ò Ry d ò y ( R y) ( R y) \ ò yd ò R [m] y ( R y) ( R y) d Ry d ò d mr Suttutng the ove vlue n (6.63c) nd (6.63d), we get, e c (l e c )m nd E (le c ) mr Fom the ove, we get e c nd l ER E R mr (6.63e) Suttuton of the vlue of Euton (6.63e) nto Euton (6.63) gve n exeon fo the tngentl te n cuved em uject to ue endng. led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

4 Theefoe, R y ê ë m( R y) odule6/leon3 (6.64) The ove exeon fo genelly known the "Wnklech fomul". The dtuton of te gven y the hyeolc (nd not lne n the ce of tght em) hown n the Fgue 6. (). In the ove exeon, the untty m ue nume, nd the oety of ech tcul he of the coecton. Tle 6. gve the fomul fo m fo vou he of the coecton. led Eltcty fo Engnee 4 T.G.Sthm & L.GovndRju

5 Tle 6.. Vlue m fo vou he of coecton oecton Fomul fo m R R R m R R R m R ë R R m R/ h {[ h (R )( )] R h n R ( ) h} R Fo Rectngul Secton: ; Fo Tngul Secton: D R t t t h m R [ t.n (R ) ( t).n (R ).n (R )] E t R t t 3 m R[.n (R ) ( t ). n (R ) ( t). n (R ).n ( R )] 3 led Eltcty fo Engnee 5 T.G.Sthm & L.GovndRju

6 Sgn onventon The followng gn conventon wll e followed: odule6/leon3. endng moment wll e tken otve when t dected towd the concve de of the em (o t decee the du of cuvtue), nd negtve f t ncee the du of cuvtue.. 'y' otve when meued towd the convex de of the em, nd negtve when meued towd the concve de (o towd the cente of cuvtue). 3. Wth the ove gn conventon, f otve, t denote tenle te whle negtve gn men comeve te. The dtnce etween the centodl x (y ) nd the neutl x found y ettng the tngentl te to zeo n Euton (5.5) y \ ê R ë m( R y) yn o m( R yn ) whee y n denote the dtnce etween xe ndcted n Fgue 5.. Fom the ove, mr y n ( m ) Th exeon vld fo ue endng only. Howeve, when the em cted uon y noml lod ctng though the centod of coectonl e, the tngentl te gven y Euton (5.5) dded to the te oduced y th noml lod. Theefoe, fo th mle ce of ueoton, we hve R y ê ë m( R y) efoe, negtve gn octed wth comeve lod STRESSES IN LOSED RINGS (6.65) ne hook, lt ng e the cuved em tht e untned t one end o oth end. Fo uch em, the endng moment t ny ecton cn e clculted y lyng the euton of ttc dectly. ut fo the em hvng etned o fxed end uch cloe ng, euton of eulum e not uffcent to otn the oluton, thee em e ttclly ndetemnte. In uch em, eltc ehvou of the em condeed nd n ddtonl condton y condeng the defomton of the meme unde gven lod develoed n the ce of ttclly ndetemnte tght em. led Eltcty fo Engnee 6 T.G.Sthm & L.GovndRju

7 Now, conde cloed ng hown n fgue 6. (), whch ujected to concentted lod long vetcl dmetcl lne. () () (c) (d) Fgue 6. loed ng ujected to lod The dtuton of te n ue hlf of the ng wll e me tht n the lowe hlf due to the ymmety of the ng. lo, the te dtuton n ny one udnt wll e me n nothe. Howeve, fo the uoe of nly, let u conde udnt of the ccul ng hown n the Fgue 6. (c), whch my e condeed to e fxed t the ecton led Eltcty fo Engnee 7 T.G.Sthm & L.GovndRju

8 nd t ecton ujected to n xl lod nd endng moment mgntude nd the gn of the moment e unknown.. Hee the Now, tkng the moment of the foce tht le to the one de of the ecton, then we get, mn ( R x) ut fom Fgue, x R co \ mn ( R R co ) R \ mn ( co ) () The moment mn t the ecton N cnnot e detemned unle the mgntude of known. Reolvng nto noml nd tngentl comonent, we get Noml omonent, oducng unfom tenle te N co Tngentl comonent, oducng heng te T n led Eltcty fo Engnee 8 T.G.Sthm & L.GovndRju

9 Detemnton of Fgue 6.3 Secton QN onde the eltc ehvo of the two noml ecton N nd Q, dffeentl dtnce t. Let the ntl ngle d etween the lne of thee two ecton chnge y n mount Dd when lod e led. Dd Theefoe, the ngul tn w d.e., D d w d. Theefoe, f we e nteeted n fndng the totl chnge n ngle etween the ecton, tht mke n ngle nd wth the ecton, the exeon ò w d wll gve tht ngle. ut n the ce of ng, ecton nd emn t ght ngle to ech othe efoe nd fte lodng. Thu, the chnge n the ngle etween thee lne eul to zeo. Hence ò w d () o led Eltcty fo Engnee 9 T.G.Sthm & L.GovndRju

10 In tght em the te of chnge of loe of the eltc cuve gven y odule6/leon3 d y dx Whee n ntlly cuved em, the te of chnge of loe of the eltc cuve whch the ngle chnge e unt of c length. Dd w Now, Rd R R mn O w EI EI EI mn fo cuved em Suttutng the ove n euton (), we get R. mn ò d o EI nce R, E nd I e contnt, \ ò d mn o Fom Euton (), uttutng the vlue of mn, we otn ò d R ò d R ò co d Integtng, we get [ ] R[ ] R[ n ] R Rn R Thu EI Dd Rd Theefoe, knowng, the moment t ny ecton uch N cn e comuted nd then the noml te cn e clculted y cuved em fomul t ny deed ecton.., led Eltcty fo Engnee T.G.Sthm & L.GovndRju

11 led Eltcty fo Engnee T.G.Sthm & L.GovndRju NUERIL EXLES Exmle 6. Gven the followng te functon f co Detemne the te comonent t nd, Soluton: The te comonent, y defnton of f, e gven follow f f () f () f f t () The vou devtve e follow: f co f ( ) f n co ( ) f n co ( ) f co n Suttutng the ove vlue n euton (), () nd (), we get ( ) n co co n co co n \ f

12 led Eltcty fo Engnee T.G.Sthm & L.GovndRju ( ) ( ) t co n co n \ t Theefoe, the te comonent e n t Exmle 6. thck cylnde of nne du cm nd oute du 5cm ujected to n ntenl eue of. Detemne the dl nd hoo tee n the cylnde t the nne nd oute ufce. Soluton: The dl te n the cylnde gven y o o The hoo te n the cylnde gven y o o the cylnde ujected to ntenl eue only, the ove exeon educe to nd Stee t nne fce of the cylnde (.e., t cm): Rdl te ê ë ê ë ê ë (.) (.5) (.) (.) (.5) (.) (.5) (.) o

13 (.) Hoo te ê ë(.5) (.) o 3. ê ë(.5) (.) Stee t outefce of the cylnde (.e., t 5 cm): Rdl te (.) ê ë(.5) (.) Hoo te (.) ê ë(.5) (.) o ê ë(.5) (.) (.) (.5) ê ë (.5) (.5) (.) ê ë (.) (.) ê ë (.5) (.5) ê ë(.5) (.) Exmle 6.3 teel tue, whch h n outde dmete of cm nd nde dmete of 5cm, ujected to n ntenl eue of 4 nd n extenl eue of 5.5. lculte the mxmum hoo te n the tue. Soluton: The mxmum hoo te occu t. Theefoe, xmum hoo te ( ) mx ê ë ê ë ê ë ê ë ( ) mx ( ) 4[(.5) (.) ] 5.5 (.) Theefoe, ( ) mx (.) (.5) O ( ) mx 8.67 ê ë led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

14 4 led Eltcty fo Engnee T.G.Sthm & L.GovndRju Exmle 6.4 teel cylnde whch h n nde dmete of m ujected to n ntenl eue of 8. lculte the wll thckne f the mxmum heng te not to exceed 35. Soluton: The ctcl ont le on the nne ufce of the cylnde,.e., t. We hve, Rdl te ê ë ê ë t nd, ê ë ê ë ) ( ) ( Theefoe, Smlly, Hoo te ê ë ê ë t nd, ê ë ê ë ) ( ) ( Hee the mxmum nd mnmum tee e 3 nd ut the mxmum he te t mx ( ) 3.e. t mx ( ) ( ) ê ë

15 ê ë 35 ( ) 8.e., 35 ( (.5) Theefoe,.5693 ) ( ) If t the thckne of the cylnde, then.5 t.5693 \t.693 m o 69.3 mm. Exmle 6.5 The ccul lnk hown n Fgue 6.4 h ccul coecton 3cm n dmete. The nde dmete of the ng 4cm. The lod kg. lculte the te t nd. ome the vlue wth thoe found y the tght em fomul. ume tht the mtel not teed ove t eltc tength. Soluton: oectonl e 4 (3) 7.6 cm. Fo ccul coecton m gven y m R c R c Hee R cm R c c.5 cm. (Refe Tle 6.) Theefoe, m R Fgue 6.4 Loded ccul lnk m.5 led Eltcty fo Engnee 5 T.G.Sthm & L.GovndRju

16 t ecton, the lod eolved nto lod nd endng coule whoe moment otve. The te t nd condeed to e the um of the te due to xl lod, nd the te due to the endng moment. Theefoe, Ste t ont R y ê ë m( R y ) (3.5 ) (.5) ê ë.5(3.5.5) o 4.65 kg/cm (comeve). The te t ont gven y y ê R ë m( R y 35 ê ë \ kg/cm (Tenle).5.5(3.5.5) omon y Stght em Fomul The moment of net of the ng coecton out the centodl x I d (3) 3.976cm If the lnk condeed to e tght em, the coeondng vlue e y I ( 35)(.5) \ 46.6 kg/cm (comeve) & kg/cm (tenle) led Eltcty fo Engnee 6 T.G.Sthm & L.GovndRju

17 cm odule6/leon Stght em d. entod x Neutl x 46.6 uved em Exmle Fgue 6.5 Stee long the coecton n oen ng hvng TSecton hown n the Fgue 6.6 ujected to comeve lod of, kg. omute the tee t nd y cuved em fomul.,kg cm.. 8cm cm 4cm..34cm R Fgue 6.6 Loded oen ng led Eltcty fo Engnee 7 T.G.Sthm & L.GovndRju

18 Soluton: e of the Secton 4 48 cm odule6/leon3 The vlue of m cn e clculted fom Tle 6. y uttutng fo the unymmetc Iecton. Fom Fgue, R cm c c 3.34 cm c 3.66 cm, c 5.66 cm t cm, cm m gven y R m 3 ln Theefoe, m.4 Now, te t, 48 [.ln( R c ) ( t ).ln( R c ) ( t).ln( R c ). ( R c) ] R [ ( ) ln( ) ( ) ln( ) ln( ) ] y ê ë m( R y ) (x3.66) (5.66) ê 48x3.66 ë.4( ) \ kg/cm (comeve) Smlly, Ste t gven y R y ê ë m( R y ) ê ë.4( ) \ 58.5 kg/cm (tenle) led Eltcty fo Engnee 8 T.G.Sthm & L.GovndRju

19 Exmle 6.7 ng hown n the Fgue 6.7, wth ectngul ecton 4cm wde nd cm thck. It ujected to lod of, kg. omute the tee t nd nd t nd D y cuved em fomul. kg 4cm 3 D. 4cm o ecton cm Fgue 6.7 Loded ng wth ectngul coecton Soluton: e of the ecton 4 8cm The Rdu of cuvtue of the centodl x R 4 6cm. Fom Tle 6., the m vlue fo tezodl ecton gven y, R ì R c ü m í[ h ( R c )( )] ln ( ) hý h î R c þ ut fo ectngul ecton, c c,, R ì R c Theefoe m [ h ( R c)( ) ] ln ( ) ý ü þ í h î 6 ì 6 ü m í[ 4 ( 6 )( ) ] ln ý 8 4 î 6 þ Theefoe m. 397 y Now, te t ê R ë m R R c ( y ) 6 ê ë.397 \ 348.6kg / cm (omeon) ( ) ( ) 6 led Eltcty fo Engnee 9 T.G.Sthm & L.GovndRju

20 y Ste t ( ) ê R ë m R y 6 ( ) ê ë Theefoe, 574.3kg / cm (Tenon) To comute the tee t nd D. D 3 kg 3 O.. 6cm Fgue 6.8 t ecton D, the endng moment,.e., R co3 6 co3 39 kg cm omonent of noml to D gven y, N co3 co3 73kg. N y Theefoe, te t c ê R ë m R y ( ) ê ë.397 ( ) ( ) 6 \ c 76.7kg / cm (omeon) Ste t N y D D ê R ë m R ( y ) led Eltcty fo Engnee T.G.Sthm & L.GovndRju

21 73 39 ê ë.397 ( 6 ) Theefoe, 363.4kg / cm (Tenon) D Exmle 6.8 The dmenon of tonne cne hook e hown n the Fgue 6.9. Fnd the ccumfeentl tee nd on the nde nd outde fe eectvely t the ecton. Y Y. R t 3cm 7cm 9cm cm Secton Fgue 6.9 Loded cne hook 9 3 Soluton: e of the ecton 7cm 9 3 Now, y 5cm. 3 ê ë 9 3 Theefoe y ( 5) 7cm. Rdu of cuvtue of the centodl x R 7 5 cm. Fo Tezodl co ecton, m gven y the Tle 6., ì 7 ü m í[ ( 3 ) ( 7)( 9 3) ].ln ( 9 3) ý 7 î 5 þ \m.8 oment Now, Ste t R, kg cm y ê R ë m R ( y ) led Eltcty fo Engnee T.G.Sthm & L.GovndRju

22 ê 7 7 ë.8 \ 4kg / cm (omeon) ( 5) ( ) 5 Ste t ê R ë m y ( R y ), ê 7 7 ë \ 639.6kg / cm (Tenon) 7.8 ( 7) Exmle 6.9 ccul oen teel ng ujected to comeve foce of 8 kn hown n the Fgue 6.. The coecton of the ng mde u of n unymmetcl Iecton wth n nne du of 5mm. Etmte the ccumfeentl tee develoed t ont nd. W Y 6 Y. O R Fgue 6. Loded ccul ng wth unymmetcl Iecton led Eltcty fo Engnee T.G.Sthm & L.GovndRju

23 Soluton: Fom the Tle 6., the vlue of m fo the ove ecton gven y R m 3 ln Hence R Rdu of cuvtue of the centodl x. Now, [ ln ( R c ) ( t ) ln ( R c ) ( t) ln ( R c ) ( R c) ] 8 6mm ( ) ( 8) ( 8 5) y 75.33mm. 6 ( ) 84.67mm. \ y lo, R ( ) 5.33mm \ m 6 ê ë \m.7. 8ln( ) ( 8) ln( ) ( ) ( ) ( ) ln ln oment R N mm. Now, Ste t ont \ y ê R ë m R ê ë.7 \ 93. N / mm (omeon) ( y ) ( 75.33) ( ) Ste t ont y ê R ë m R ( y ) ê ë.7 \ 5.6 N / mm (Tenon) Hence, the eultnt tee t nd e, 5.6 N / mm (Tenon), 93. N / mm ( ) (omeon) led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

24 Exmle 6. lculte the ccumfeentl te on nde nd outde fe of the ng t nd, hown n Fgue 6.. The men dmete of the ng 5cm nd coecton ccul wth cm dmete. Lodng wthn eltc lmt. kg O n. R.5cm O m co n o Soluton: Fo ccul ecton, fom Tle 6. R m c.5 \m.435 R c.5 R c Fgue 6. Loded cloed ng.5 We hve, R.364 R \. 9 Now, ê R ë m.9 ê.5 ë 5. y ( R y ) ( ) ( ) led Eltcty fo Engnee 4 T.G.Sthm & L.GovndRju

25 \ 6. (omeve) yo ( ) ê R ë m R y.9 ( ) ê.5 ë \. 755 (Tenon) o R Smlly, ( ) (. ) 364 R R.636 R \. 59 Now, ( ) y ê R ë m R y.59 () ( ) ê.5 ë \ 9. (Tenon).59 ( ) nd ( ) o ê.5 ë (omeon) Now, uttutng the vlue of 5kg, ( ) 3.459cm, ove tee cn e clculted elow kg / cm kg / cm kg / cm kg / cm led Eltcty fo Engnee 5 T.G.Sthm & L.GovndRju

26 Exmle 6. ng of mm men dmete h ectngul coecton wth 5mm n the dl decton nd 3mm eendcul to the dl decton hown n Fgue 6.. If the mxmum tenle te lmted to N / mm, detemne the tenle lod tht cn e led on the ng. W D 5mm. mm Y. Y 3mm 5mm R Secton W Fgue 6. loed ng wth ectngul coecton Soluton: R mm, e of coecton mm Fom Tle 6., the vlue of m fo the ectngul ecton gven y ì 5 ü m í[ 3 5 ] ln ý 5 5 î 5 þ \m.7 led Eltcty fo Engnee 6 T.G.Sthm & L.GovndRju

27 To fnd W N W R W Fgue 6.3 The endng moment t ny ecton N cn e detemned y WR N ( co ) \ t, mn WR ut W \ 8. 7W y Now, ( ) ê R ë m R y W y ê R ë W 5 \. 73W (Tenle) y nd ( ) ê R ë m R y w 8.7W 5 ( ) ê 5 5 ë.7 5 \. 943W (omeon) m( R y ) (8.7W ) ( 5) ( ) led Eltcty fo Engnee 7 T.G.Sthm & L.GovndRju

28 To fnd tee t nd D WR We hve, mn ( co ) WR \ t 9, mn D \ D 8.7W W 3. 83W D y Now, te t ( ) ê R ë m R y 3.83W ( 5) ( ) ê 5 ë W (omeon) D y D nd te t ( ) D D ê R ë m R y D 3.83W ê 5 ë \ D. 7W (Tenle) 5.7 ( 5) y comon, the tenle te mxmum t ont D. \.7W \ W N o 55. 3kN Exmle 6. ng of men dmete mm mde of mld teel wth 5mm dmete. The ng ujected to fou ull n two decton t ght ngle to ech othe ng though the cente of the ng. Detemne the mxmum vlue of the ull f the tenle te hould not exceed 8 N / mm led Eltcty fo Engnee 8 T.G.Sthm & L.GovndRju

29 W D W N W. mm W W W W Soluton: Hee R 5mm Fgue 6.4 loed ng wth ccul coecton Fom Tle 6., the vlue of m fo ccul ecton gven y, m \m.6 R 5.5 R 5.5 R 5.5 e of coecton ( ) mm We hve, Now, WR W W ê R ë m y ( R y ) led Eltcty fo Engnee 9 T.G.Sthm & L.GovndRju

30 \ W 9.85W ê ë.6.84w (Tenle) y ê R ë m R y ( ) (.5) ( ) 5.5 W 9.85W.5 ê ë.6 \. 398W (omeon) W lo, D ( R) \ D 5. 95W Now, D y ê R ë m R ( y ) 5.98W ê ë.6 \. 3W (omeon) nd D 5.98W.5 ê ë.6.88w (Tenon) (.5) ( ) 5.5 ( 5.5) Stee t Secton D due to hozontl Lod We hve, moment t ny ecton N gven y R N ( co ) t ecton D,, W \ D R D 9. 85W ( co ) ( 5.5) led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

31 \ D R W y ê ë m R \. 836W (Tenle) ( y ) ( 9.85W ) (.5) ( ) ê ë nd D D R y ê ë m R D ( y ) D ( 9.85W ) W.5 ê ë.6 ( 5.5) \ D. 398W (omeon) Reultnt tee e (.3W.836W ). 464W (omeon) D (.88W.398W ). 48W (Tenon) In ode to lmt the tenle te to 8 N / mm n the ng, the mxmum vlue of the foce n the ull gven y.48w 8 \ W N o kn EXERISES. I the followng functon te functon? f n If o, fnd the coeondng te. Wht the olem olved y th functon?. Invetgte wht olem of lne te olved y the followng te functon () f n K () f n 3. Deve the eulum euton fo ol coodnte ytem. 4. Deve the exeon fo tn comonent n ol coodnte. led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

32 5. Sttng fom the te functon f log log D, otn the te comonent nd n e ujected to ntenl eue nd extenl eue o. Otn the mxmum vlue of when o nd ndcte whee t occu. 6. heck whethe the followng te functon f c co tn whee contnt. [ ( ) ] (3 m) 3 7. Sttng fom the te functon f, deve exeon fo 8 nd n ce of ottng dk of nne du '' nd oute du ''. Otn the mxmum vlue of nd. 8. Show tht the te functon f log log D olve the olem of xymmetc te dtuton, otn exeon fo nd n ce of e ujected to ntenl eue nd extenl eue. 9. Show tht the followng te functon olve the olem of xymmetc te dtuton n ol coodnte f log log D. Exln xymmetc olem wth exmle.. Deve the genel exeon fo the te functon n the ce of xymmetc te dtuton.. Deve the exeon fo dl nd tngentl te n thck cylnde ujected to ntenl nd extenl flud eue. 3. cuved ent nto c of ccle hvng ntenl du nd extenl du ujected to endng coule t t end. Detemne the tee, nd t. 4. Fo the te functon, f log, whee contnt, comute the te comonent, nd t. 5. thck cylnde of nne du 5mm nd oute du mm ujected to n ntenl eue of 5N/m. Detemne the dl nd hoo tee n the cylnde t nne nd oute ufce. 6. The ntenl nd extenl dmete of thck hollow cylnde e 8mm nd mm eectvely. It ujected to n extenl eue of 4N/m, when the ntenl eue N/m. lculte the ccumfeentl tee t the extenl nd ntenl ufce nd detemne the dl nd ccumfeentl tee t the men du. 7. thckwll cylnde mde of teel (E G nd n. 9 ), h n nde dmete of mm, nd n outde dmete of mm. The cylnde ujected to n led Eltcty fo Engnee 3 T.G.Sthm & L.GovndRju

33 ntenl eue of 3. Detemne the te comonent odule6/leon3 nd t mm, 5mm nd 5mm. 8. long cloed cylnde h n ntenl du of mm nd n extenl du of 5mm. It ujected to n ntenl eue of 8. Detemne the mxmum dl, ccumfeentl nd xl tee n the cylnde. 9. old dc of du mm ottng t eed of 3 m. Detemne the dl nd hoo tee n the dc f n. 3nd 8kg/m 3. lo detemne the tee n the dc f hole of 3mm oed t the cente of the dc.. dc of 5mm dmete h centl hole of 5mm dmete nd un t 4m. lculte the hoo tee. Tke n. 5nd 78 kg/m 3.. tune oto 4mm extenl dmete nd mm ntenl dmete evolve t m. Fnd the mxmum hoo nd dl tee umng the oto to e thn dc. Tke the weght of the oto 77 kg/m 3 nd oon to.3.. Invetgte wht olem of lne te olved y the followng te functon 3 3F ì xy ü f íxy y ý. heck whethe the followng te functon 4 î 3 þ f D co y y y y 3. Show tht [ e e ye Dye ] nx eeent te functon. 4. The cuved em hown n fgue h ccul coecton 5mm n dmete. The nde dmete of the cuved em 4mm. Detemne the te t when kn. Fgue cne hook ce lod W kn hown n fgue. The coecton mn of the hook tezodl hown n the fgue. Fnd the totl tee t ont m nd n. Ue the dt gven 4mm, mm, 3mm nd c mm led Eltcty fo Engnee 33 T.G.Sthm & L.GovndRju

34 m n W c Fgue emccul cuved loded hown n fgue nd h tezodl coecton. lculte the tenle te t ont f 5kN mm mm 4mm mm mm mm Fgue cuved em wth ccul centelne h Tecton hown n fgue elow. It ujected to ue endng n t lne of ymmety. The du of cuvtue of the concve fce 6mm. ll dmenon of the coecton e fxed hown excet the thckne t of the tem. Fnd the oe vlue of the tem thckne o tht the exteme fe tee e endng wll e numeclly eul. led Eltcty fo Engnee 34 T.G.Sthm & L.GovndRju

35 t 6mm mm 8mm Fgue cloed ng of men dmete mm h ectngul ecton 5mm wde y 3mm thck, loded hown n the fgue. Detemne the ccumfeentl te on the nde nd outde fe of the ng t nd. ume E kn / mm 5KN 5mm mm.. 3mm 5mm 5KN Fgue hook h tngul coecton wth the dmenon hown n fgue elow. The e of the tngle on the nde of the hook. The lod of kn led long lne 5mm fom the nne edge of the hnk. omute the te t the nne nd oute fe. led Eltcty fo Engnee 35 T.G.Sthm & L.GovndRju

36 5mm 65mm Fgue ccul ng of men du 4mm h ccul coecton wth dmete of 5mm. The ng ujected to dmetcl comeve foce of 3kN long the vetcl dmete. lculte the tee develoed n the vetcl ecton unde the lod nd the hozontl ecton t ght ngle to the lne of lodng. led Eltcty fo Engnee 36 T.G.Sthm & L.GovndRju

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