CHAPTER 10 MEMBRANE THEORY FOR THIN-WALLED CIRCULAR CYLINDERS AND SPHERES

Transcription

1 CAPTER 10 MEMBRANE TEORY FOR TIN-WAED CIRCUAR CYINDERS AND SPERES EXERCISE 44, Page A hin-walled irular ylinder of inernal diameer 10 m is subjeed o a maximum inernal pressure of 50 bar. Deermine is wall hikness, if = 0.8 and = 0.4. Assume ha he maximum permissible sress is 300 MPa. (Noe ha 1 bar = 5 10 Pa.) from whih, wall hikness, = 0.08 m = 08 mm from whih, wall hikness, = 0.05 m = 5 mm and design wall hikness = 08 mm he larger of he wo design hiknesses. Assuming ha a submarine pressure hull of exernal diameer 1 m, and wall hikness 5 m, an be designed using inernal pressure heory, and negleing failure due o bukling, deermine is permissible diving deph for a safey faor of. Assume ha he densiy of sea waer = 100kg/m 3, g = 9.81 m/s, = 0.48, = 0.5 and he yield sress of he maerial of onsruion is 400 MPa. Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

2 from whih, P R 1 / P = 1. MPa P = ρgh from whih, h P g from whih, P R 1 / P = MPa ene, design pressure = 1. MPa Therefore, diving deph, h P g = m 3. Design he wall hikness for an airraf fuselage of inernal diameer m, subjeed o an inernal pressure of 0.5 bar. Assume ha = 0.48, = 0.5 and yp = 00 MPa. (Noe ha 1 bar = 5 10 Pa.) from whih, wall hikness, = m = 1.5 mm Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

3 from whih, wall hikness, = m = 0.75 mm ene, he design wall hikness = 1.5 mm he larger of he wo design hiknesses Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

4 EXERCISE 45, Page A boiler, whih may be assumed o be omposed of a hin-walled ylindrial shell body of inernal diameer 4 m, is bloked off by wo hin-walled hemispherial dome ends. Negleing he effes of disoninuiy a he inerseion beween he dome and ylinder, deermine suiable hiknesses for he ylindrial shell body and he hemispherial dome ends. The following may be assumed: maximum permissible sress = 100 MN/m, design pressure = 1 MPa, longiudinal join effiieny = 75% and irumferenial join effiieny = 50% For he ylinder pr σ = 1 pr from whih, = = 0.07 m =.7 m For he dome pr 110 = = 0.0 m = m. If he vessel of Problem 1 is jus filled wih waer, deermine he addiional waer ha is required o be pumped in, o raise he pressure by 1 MPa. The following may be assumed o apply: 11 engh of ylindrial porion of vessel = m, E = 10 N / m, v = 0.3 and K = 9 10 N / m For he ylinder σ = = MN/m 0.07 σ = MPa ε = 1 E (σ υσ ) = w = m Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

5 δv 1 = πrw = 0.048m 3 σ = MN/m ε = 1 E 7(σ υσ ) = u = δv = πr u = m 3 For he sphere σ = MN/m ε = 1 E 8(σ υσ) = w = δv 3 = 4πR w = m 3 For he waer Volume = πr = πr3 = Volume = m 3 δv 4 = pv K 10 = δv = δv 1 + δv + δv 3 + δv 4 δv = 0.1 m 3 3. A opper pipe of inernal diameer 1.5 m and wall hikness 0.1 m is o ranspor waer from a ank ha is siuaed 30 m above i. Deermine he maximum sress in he pipe, given he following: densiy of waer = 1000 kg/m3, g = 9.81 m/s. p = ρgh = = 0.94 MPa Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

6 ene, maximum sress, σ = pr = MN/m 4. Wha would be he hange in diameer of he pipe of Problem 3 due o he applied head of waer? 11 9 Assume ha for opper: E = 1 10 N / m, v = 0.33 and K = 10 N / m. Now r = w 1 v r E w = 11 r E ( /) 10 w = δ = 1 δ = = 0.1 miromeres 5. A hin-walled spherial pressure vessel of 1 m inernal diameer is fed by a pipe of inernal diameer 3 m and wall hikness 0.1 m. Assuming ha he maerial of onsruion of he spherial pressure vessel has a yield sress of 0.7 of ha of he pipe, deermine he wall hikness of he spherial shell. σ = pr Therefore, σ p = For he spherial shell = =.515 p Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

7 Now, r = p ene,.55 = from whih, wall hikness, = m = 3.81 m. A spherial pressure vessel of inernal diameer m is onsrued by boling ogeher wo hemispherial domes wih flanges. Assuming ha he number of bols used o join he wo hemispheres ogeher is 1, deermine he wall hikness of he dome and he diameer of he bols, given he following: Maximum applied pressure = 0.7 MPa, Permissible sress in spherial shell = 50 MPa Permissible sress in bols = 00 MPa σ = pr from whih, = pr m = 0.70 m 50 oad on bols = πr p =.10 MN d 1 σ b d wall hikness of he dome, d = m = 3.4 m 7. A hin-walled irular ylinder, bloked off by inexensible end plaes, onains a liquid under zero gauge pressure. Show ha he addiional liquid ha is required o be pumped ino he vessel, o raise is inernal gauge pressure by P, is he same under he following wo ondiions: (a) when axial movemen of he ylinder is ompleely free, and Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

8 (b) when he vessel is oally resrained from axial movemen. I may be assumed ha Poisson s raio (v) for he ylinder maerial is 0.5. (a) σ = pr σ = pr w 1 v R E pr E = 1 v / 1 pr E w = and w = pr E u p R 1/ v E p R E u 1/ 0.5 p R u. E 0 5 δv 1 = rl pr E 17 δv 1 = δv = πr u R pr0.5 = E 18 δv = pr E 19 δv 3 = pv K 0 Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

9 δv = pr E pv K 1 (b) σ = pr R w = w = pr E E σ = 0 u = 0 δv 1 = πr w =.83 pr 3 δv = 0 δv 3 = pv K δv = pr E pv K Carl Ross, John Bird & Andrew ile Published by Taylor and Franis

10 EXERCISE 4, Page 4 1. A irular ylinder is o be bloked off by hemispherial ends and he whole is subjeed o an inernal pressure. Given ha he inernal diameer of he ylinder and hemisphere are 5 m, and he inernal pressure is.5 bar, and if he join effiieny is 100%, deermine suiable hiknesses for he ylinder and hemispherial ends, assuming ha he ylinder ends defle he same as he hemispherial ends. Assume ha he yield sress = 00 MPa and v = 0.3. (Noe ha 1 bar = 5 10 Pa.) from whih, m ylinder hikness, yl = 3.15 mm From he formula, (from equaion (10.5) in he exbook) s hemispherial end hikness = 1.88 mm Carl Ross, John Bird & Andrew ile Published by Taylor and Franis