Mechanical Principles

Transcription

1 Unt 4: Unt de: QCF level: 5 Credt value: 5 Mehanal Prnples F/60/450 OUTCOM TUTOIAL 4 - THIN WALLD VSSLS and THICK WALLD CYLINDS Laded eams and ylnders elatnshps: slpe: Defletn y I I Mdx dx Mdx Laded eams: slpe and defletn fr laded eams (eg antlever eams arryng a nentrated lad at the free end r a unfrmly dstruted lad ver the entre length, smply supprted eams arryng a entral nentrated lad r a unfrmly dstruted lad ver the entre length) Stresses n thn-walled pressure vessels: rumferental hp stress and lngtudnal stress n ylndral and spheral pressure vessels sujeted t nternal and external pressure (eg mpressed-ar reevers, ler steam drums, sumarne hulls, ndenser asngs); fatr f safety; jnt effeny Stresses n thk-walled ylnders: rumferental hp stress, lngtudnal stress and radal stress n thk-walled ylnders sujeted t pressure (eg hydraul ylnders, extrusn des, gun arrels); Lame s thery; use f undary ndtns and dstrutn f stress n the ylnder walls Yu shuld judge yur prgress y mpletng the self assessment exerses. These may e sent fr markng at a st (see hme page). When yu have mpleted ths tutral yu shuld e ale t d the fllwng. Defne a thn walled ylnder. Slve rumferental and lngtudnal stresses n thn walled ylnders. Slve rumferental and lngtudnal stresses n thn walled spheres. Calulate the urstng pressure f thn walled ylnders and spheres. Defne a thk walled ylnder. Slve rumferental, radal and lngtudnal stresses n thk walled ylnders. Calulate hanges n dameter and vlume due t pressure. Slve prlems nvlvng the mpressn f fluds nt pressure vessels. Slve prlems nvlvng nterferene fts etween shafts and sleeves.

2 CONTNTS. Thn Walled Cylnders. Thn Walled Spheres 3. Vlume Changes 4. Thk Cylnders 4. Lame s Thery 5. Interferene Between Shafts and Sleeves D.J.DUNN

3 . THIN WALLD CYLIND. A ylnder s regarded as thn walled when the wall thkness t s less than /0 f the dameter D. When the wall s thker than ths, t s regarded as a thk wall and t s treated dfferently as desred later. Cnsder a ylnder f mean dameter D, wall thkness t and length L. When the pressure nsde s larger than the pressure utsde y p, the ylnder wll tend t splt alng a length and alng a rumferene as shwn n fgures and. Fgure Fgure The stress prdued n the lngtudnal dretn s Land n the rumferental dretn s. These are alled the lngtudnal and rumferental stresses respetvely. The latter s als alled the hp stress. Cnsder the fres tryng t splt the ylnder aut a rumferene (fg.). S lng as the wall thkness s small mpared t the dameter then the fre tryng t splt t due t the pressure s πd F pa p...(.) 4 S lng as the materal hlds then the fre s alaned y the stress n the wall. The fre due t the stress s F multpled y thearea f the metal πdt...(.) L L D.J.DUNN 3

4 quatng. and. we have pd L...(.3) 4t Nw nsder the fres tryng t splt the ylnder alng a length. The fre due t the pressure s F pa pld...(.4) S lng as the materal hlds ths s alaned y the stress n the materal. The fre due t the stress s F multpled y thearea f the metal Lt...(.5) C quatng.4 and.5 we have pd C...(.6) t C It fllws that fr a gven pressure the rumferental stress s twe the lngtudnal stress. WOKD XAMPL N. A ylnder s 300 mm mean dameter wth a wall mm thk. Calulate the maxmum pressure dfferene allwed etween the nsde and utsde f the stress n the wall must nt exeed 50 MPa. SOLUTION The slutn must e ased n the rumferental stress sne ths s the largest. = pd/t = 50 MPa p = 50 MPa x t/d = 50 x x 0.00/0.3 p = MPa D.J.DUNN 4

5 . THIN WALLD SPH A sphere wll tend t splt aut a dameter as shwn n fg.3 Fgure 3 The stress prdued n the materal s equvalent t the lngtudnal stress n the ylnder s pd C...(.) 4t WOKD XAMPL N. Calulate the maxmum allwale pressure dfferene etween the nsde and utsde f a sphere 50 mm mean dameter wth a wall 0.6 mm thk f the maxmum allwale stress s 50 MPa. SOLUTION Usng equatn G we have = pd/4t = 50 MPa p =.5x0 6 x 4t/D =.5x0 6 x 4 x /0.05 = 7 kpa SLF ASSSSMNT XCIS N.. A thn walled ylnder s 80 mm mean dameter wth a wall mm thk. Calulate the lngtudnal and rumferental stresses when the nsde pressure s 500 kpa larger than n the utsde. (Answers 0 MPa and 0 MPa).. Calulate the wall thkness requred fr a thn walled ylnder whh must wthstand a pressure dfferene f.5 MPa etween the nsde and utsde. The mean dameter s 00 mm and the stress must nt exeed 60 MPa. (Answer.5 mm) 3. Calulate the stress n a thn walled sphere 00 mm mean dameter wth a wall mm thk when the utsde pressure s MPa greater than the nsde. (Answer -5 MPa). D.J.DUNN 5

6 3. VOLUM CHANGS We wll nw lk at hw we alulate the hanges n vlume f thn walled vessels when they are pressursed. CYLINDS Cnsder a small retangular area whh s part f the wall n a thn walled ylnder (fgure 4). D.J.DUNN 6 Fgure 4 There are tw dret stresses perpendular t eah ther, and L. Frm as stress and stran thery (tutral ), the rrespndng lngtudnal stran s : εl L νc s the mdulus f elastty and s Pssn's rat. Susttutng L= pd/4t and =pd/t we have ΔL pd pd pd ε L ν ν...(3.) L 4t t 4t The rumferental stran may e defned as fllws. = hange n rumferene/rgnal rumferene πd ΔD πd ΔD ε C πd D The rumferental stran s the same as the stran ased n dameter, n ther wrds the dametr stran. Frm as stress and stran thery, the rrespndng rumferental stran s : εc C νl Susttutng L= pd/4t and =pd/t we have ΔD pd pd pd εc εd ν D t 4t 4t Nw we may dedue the hange n dameter, length and vlume. Orgnal rss setnal area f ylnder = A = D/4 ν...(3.) Orgnal length = L Orgnal vlume =V = A L=(D/4)(L) New rss setnal area = A = (D + D)/4 New length = L = L + L New vlume = V = AL= {(D + D)/4}(L + L) Change n vlume = V = V -V Vlumetr stran = v = V/V

7 ε V π D ΔD 4 L ΔL πd L 4 πd 4 L D ΔD 4 L ΔL D L 4 D L 4 Dvdng ut and learng rakets and gnrng the prdut f tw small terms, ths redues t ΔL ΔD εv εl ε D...(3.3) L D If we susttute equatn 3. and 3. nt ths we fnd pd ε V 5 4ν...(3.4) 4t WOKD XAMPL N.3 A ylnder s 50 mm mean dameter and 750 mm lng wth a wall mm thk. It has an nternal pressure 0.8 MPa greater than the utsde pressure. Calulate the fllwng.. The rumferental stran.. The lngtudnal stran.. The hange n rss setnal area. v. The hange n length. v. The hange n vlume. Take = 00 GPa and = 0.5 SOLUTION = pd/t = 30 MPa L= pd/4t = 5 MPa D= D/D = (pd/4t)( - ) = 3.5 D = 50 x 3.5 x 0-6 = mm D = mm A = x 50/4 = 767. mm A = x /4 = mm Change n area = 4.68 mm L= L/L = (pd/4t)( - ) = 37.5 L= 750 x 37.5 x 0-6 = 0.08 mm Orgnal vlume = AL = mm3 Fnal vlume = AL = mm3 Change n vlume = 4000 mm3 Chek the last answer frm equatn 3.4 v =(pd/4t)(5-4) = 300 x 0-6 Change n vlume = V x v = x 300 x 0-6 = 4000 mm3 D.J.DUNN 7

8 SPHS Cnsder a small retangular setn f the wall f a thn walled sphere. There are tw stresses mutually perpendular smlar t fg. 4 ut n ths ase the rumferental stress s the same as the lngtudnal stress. The lngtudnal stran s the same as the rumferental stran s equatn 3.3 emes v = D + D v = 3D...(3.5) The stran n any dretn resultng frm the tw mutually perpendular equal stresses s D= (/)(-) Hene v = 3(/)(-)...(3.6) WOKD XAMPL N. 4 A sphere s 0 mm mean dameter wth a wall mm thk. The pressure utsde s MPa mre than the pressure nsde. Calulate the hange n vlume. Take = 05 GPa and = 0.6 SOLUTION v = 3(/)(-) = (nte the sphere shrnks hene the negatve sgn) Orgnal vlume = D3/6 = mm3 Change n vlume = x x 0-6 = -94 mm3 D.J.DUNN 8

9 WOKD XAMPL N. 5 In example N.3 the nternal pressure s reated y pumpng water nt the ylnder. Allwng fr the mpresslty f the water, dedue the vlume f water at the utsde pressure requred t fll and pressurse the ylnder. The ulk mdulus K fr water s. GPa. SOLUTION Intal vlume f ylnder = V = mm3 = vlume f unmpressed water Fnal vlume f ylnder = V = mm3 = vlume f mpressed water. If V was unmpressed t wuld have a larger vlume V3. V3 = V + V (all vlumes refer t water). Frm the relatnshp etween pressure and vlumetr stran we have V = pv3/k = 0.8 x 06 x V3/. x 09 = x 0-6V3 V3 = x 0-6V V3 = V3 = mm3 Ths s the vlume requred t fll and pressurse the ylnder. The answer s nt prese eause the mean dmensns f the ylnder were used nt the nsde dmensns. D.J.DUNN 9

10 SLF ASSSSMNT XCIS N.. A ylnder s 00 mm mean dameter and m lng wth a wall.5 mm thk. It has an nsde pressure MPa greater than the utsde pressure. Calulate the hange n dameter and hange n vlume. Take = 80 GPa and = 0.3 (Answers mm and 6 59 mm3). A sphere s 50 mm mean dameter wth a wall 0.5 mm thk. It has an nsde pressure 0.5 MPa greater than the utsde pressure. Calulate the hange n dameter and hange n vlume. Take = GPa and = 0.5 (Answers 0.00 mm and 8.68mm3) 3a. A thn walled ylnder f mean dameter D and length L has a wall thkness f t. It s sujeted t an nternal pressure f p. Shw that the hange n length L and hange n dameter D are L=(pDL/4t)( -) and D =(pd/4t)( - ). A steel ylnder m lng and 0.5 m mean dameter has a wall 8 mm thk. It s flled and pressursed wth water t a pressure f 3 MPa gauge. The utsde s atmsphere. Fr steel = 0 GPa and =0.3. Fr water K =.9 GPa. Calulate the fllwng.. The maxmum stress. (93.75 MPa). The nrease n vlume f the ylnder. (33309 mm3). The vlume f water at atmspher pressure requred. ( mm3) 4a. A thn walled sphere f mean dameter D has a wall thkness f t. It s sujeted t an nternal pressure f p. Shw that the hange n vlume V and hange n dameter D are V =(3pDV/4t)( - ) where V s the ntal vlume.. A steel sphere m mean dameter has a wall 0 mm thk. It s flled and pressursed wth water s that the stress n steel 00 MPa. The utsde s atmsphere. Fr steel = 06 GPa and = 0.3. Fr water K =. GPa. Calulate the fllwng.. The gauge pressure (8 MPa). The vlume water requred. (4.3 x 09 mm3) D.J.DUNN 0

11 4. THICK CYLINDS The dfferene etween a thn ylnder and a thk ylnder s that a thk ylnder has a stress n the radal dretn as well as a rumferental stress and lngtudnal stress. A rule f thum s that radal stress emes mprtant when the wall thkness exeeds /0 th f the dameter. 4. LAM'S THOY Fgure 5 Cnsder a small setn f the wall. L = Lngtudnal stress = adal stress C = Crumferental stress Fgure 6 We have 3 stresses n mutually perpendular dretns, the rrespndng strans are εl L ν C εc C ν L ε ν C L Next nsder the fres atng n a setn f the wall. D.J.DUNN

12 Fgure 7 D.J.DUNN

13 Balane the fres vertally (assumng m f length). ememer the length f an ar s radus x angle The area f the tp urved surfae s (r + r) x The area f the ttm urved surfae s r x ememer Fre s stress x area. The vertal fre up s ( + )(r + r) The vertal fre dwn s r + C r sn/ ememer fr small angles the sn s the same as the angle n radans. sn/ =/ Balanng the fres we have δθ δ r δrδθ r δθ Cδr δ Ths reslves dwn t C δr d In the lmt ths emes C...(4.) dr Wthut prf, t an e shwn that the lngtudnal stress and stran are the same at all rad. (The prf f ths s a lng pee f wrk and wuld detrat frm the present studes f gven here). The stran s gven y εl L ν C C Sne L and L are nstant then t fllws that ( + C ) = nstant. The slutn s smplfed y makng the nstant a C a...(4.) C a Susttute (4.) nt (4.) and rd a a dr multply all y r and rearrange r r d ar r dr It an eshwn that C d r dr d r dr a r a r r d dr ar ardr ar r where s a nstant f ntegratn. D.J.DUNN 3

14 In rder t slve prlems, the nstants a and must e fund frm undary ndtns. ememer: a undary ndtn s a knwn answer suh as knwng what the pressure r stress s at a gven radus. When atmspher pressure ats n ne sde f the wall, t s est t use gauge pressure n the alulatns. Ths makes atmspher pressure zer and all ther pressures are relatve t t. ememer: aslute pressure = gauge pressure + atmspher pressure. WOKD XAMPL N.6 A hydraul ylnder s 00 mm nternal dameter and 40 mm external dameter. It s pressursed nternally t 00 MPa gauge. Determne the radal and rumferental stress at the nner and uter surfaes. Take = 05 GPa and = 0.5 SOLUTION The undary ndtns are Inner surfae r = 50 mm = - 00 MPa (mpressve) Outer surfae r = 70 mm = 0 MPa (mpressve) Susttutng nt Lame's equatn we have = - 00 x 0 6 = a - /r = a - /0.05 = 0 = a - /r = a - /0.07 Slvng smultaneus equatns = 50 kn a = 04 MPa Nw slve the rumferental stress. = a + /r Puttng r = 0.05 = 308 MPa Puttng r = 0.07 = 08 MPa D.J.DUNN 4

15 D.J.DUNN 5 5. SOLID SHAFTS AND SLVS In ths setn we wll examne the stress and stran ndued when a sleeve fts n a shaft wth an nterferene ft. Fgure 8 When the sleeve s ftted we assume here that a pressure p s exerted all ver the surfae f ntat. Fts nsder the shaft. We wll derve the equatns as thugh the shaft was hllw wth n pressure nsde t and then put zer fr the nsde dameter. r a a 0 r At - p p At r a At r r a Put = 0 and = -p The stran n the rumferental dretn = ν p νp p ν ε υ p Δ Δ ε Ths s the hange n the uter dameter f the shaft. s Psn s rat.

16 D.J.DUNN 6 Next nsder the sleeve. r a a 0 r At - p p At r a At r r a The stran n the rumferental dretn = ν - p νp ν ε ν - p Δ Δ ε Ths s the hange n the nner dameter f the sleeve. The derease n radus f the shaft plus the nrease n radus f the sleeve must add up t e the nterferene ft s addng the tw values we get: ν p - ν p δ If the elast nstants are the same fr th materals ths smplfes t : ν - p δ

17 WOKD XAMPL N.7 A shaft has a dameter f mm and s an nterferene ft wth a sleeve 40 mm uter dameter, 30 mm nner dameter and 50 mm lng. Calulate the fre needed t slde the sleeve n the shaft f the effent f frtn s 0.3. The elast prpertes fr th parts are the same wth = 05 GPa and Pssn s rat = 0.5 Calulate the hange n radus f the shaft and sleeve at the nsde. SOLUTION δ p ν - = 0.03 m = 0.0 = m δ p =33.5 MPa ν - The nrmal fre etween the tw surfaes f ntat s N = pa A = L = x 0.05 x 0.05 = 4.7 x 0-3 m N = 33.5 x 0 6 x 4.7 x 0-3 = 69 kn Fre t verme frtn F = = 0.3 x = 88.7 kn Fr the shaft 6 p 33.5 x 0 x 0.05 Δ ν x0 9 05x0 Fr the sleeve at the nsde p Δ 6 ν x0 m Chek y addng = 30m the nterferene ft. 6 m D.J.DUNN 7

18 SLF ASSSSMNT XCIS N.3. A thk ylnder has an uter dameter f 50 mm and an nner dameter f 50 mm. The OUTSID s pressursed t 00 ar greater than the nsde. Calulate the fllwng. The rumferental stress n the nsde layer. (-45 MPa) The rumferental stress n the utsde layer. (-5 MPa). A thk ylnder has an utsde dameter f 00 mm and an nsde dameter f 60 mm. It s pressursed untl nternally untl the uter layer has a rumferental stress f 300 MPa. Calulate the pressure dfferene etween the nsde and utsde. (66.6 MPa) 3. A thk ylnder s 00 mm uter dameter and 50 mm nner dameter. It s pressursed t MPa gauge n the nsde. Calulate the fllwng. The rumferental stress n the utsde layer (74.64 MPa) The rumferental stress n the nsde layer (86.67 MPa) The lngtudnal stress (37.33 MPa) The rumferental stran n the utsde layer (34.9 ) The rumferental stran n the nsde layer (.008 x 0-3 ) The hange n the nner dameter (0.05 mm) The hange n the uter dameter (0.03 mm) Take = 05 GPa and = A shaft has a dameter f mm and s an nterferene ft wth a sleeve 60 mm uter dameter, 45 mm nner dameter and 80 mm lng. Calulate the fre needed t slde the sleeve n the shaft f the effent f frtn s 0.5. The elast prpertes fr th parts are the same wth = 00 GPa and Pssn s rat = 0.3 Calulate the hange n radus f the shaft and sleeve at the nsde. (p =. MPa, F = 36. kn, =48.77 m, =-8.77 m) D.J.DUNN 8