Question

Problem 4 [40 marks] A brass cylinder (E = 75 GPa and v = 0.33) of outer radius 150 mm and inner radius 100 mm is to be press-fit over a steel cylinder (E = 200 GPa and v = 0.28) of outer radius 100.02 mm and inner radius 50 mm. a) Find the interface pressure; [20 marks] Draw the radial and hoop stresses in both cylinders generated due to misfit; [20 marks] c) For Masters Students only: Draw radial and hoop stresses in both cylinders when the compound cylinder is subjected to the internal pressure of 10 MPa. [20 marks] b )

Answer 1

Solution Brass cylinder E = 75 G Pa rai = 150 mm V, 0.33 dows 290,= 300mm - dii= 2011, rin 200mm -100 mm steel cylinder E 200GPa , 0.28 2 200.04 mm 119) 100.02 mm I doz rig loomm dir = 50mm internal pressure is zoro Po=0 > Brass (sylivalen 2) » steel (cylindery 1)

(@) From lame's equation have Pr a + b k² at 6 2 r For Steel at S Omm 1 = kiz = 1 Por & Equation ④ الأبيا becomes a Ы So r = 100 mm Also assumed interface pressure at .. 95 100mm Pr =P o equation ① will becomes p= + b 1100) a (4 from 4 Р 1 loo SO 10 b = px soʻx100 50-100 put value » ьіч we get a = px 100² 50-100

a and 6 in egration Now put hoop stress ong px100 + px 150 x 100 100 so² -100 hoop stress 7 = 100mm = ) ons Рx 1во“ 50²_100 px 502 502 100² ons P (50% +1002) + 502 1002 For brus ki = 9 - 100mm Pay P :. From earnation ③ to 1002 at Pg O 96 = 90150mm From equation o + а. b 1502 from

P= b 1 1 1580- 6x(150-100%) 100²x150² to = px150x100² 150%-100 get 10 7 2 put b in 9 a = P*1002 150²_1000 2 Now part a and b in equation Ppx150²x100² + 150-100² 1 Suo PX100? 150-100 hoop stress at r =100 The PX100 PX150 150²-100² 1502-100² P (1502 +100%) 12 ono (150-100) 1 - 100mm displaelment be the radial C Strain in cylindley of steel Ensure E E at 7 = 100 mm

. ene 0.28 (-p) Wie P (100 +502) 50-100 Ens p(100+502) 100²-50² 0:2016-03] in bress similarity hoop stansin ень p(18+100%) 150²-1002 -0.3266) E, sted and Total radial de formation at junction bross cylindey - roz Hlin = 100.02 -100 =0.02 mer 0-02 (enst Enoxri (106+50) +0.287+ p [iso+100+0.33 100²-50² 150²-100 » P 0.02 100 75 -> -1.386 2.93 0.04 + 200 75 → P=0.00 6 223 Gla p = 6.223 MPa interface pressure

(62 6n2- hoop stress variation radial stress yorieten au (tonsile) Note :- Abone x-ane's stresses are positive and below it stresses negative •е ( отримга ) is subjected to intonal (6) compound eglindor pressure Pi = 10 MPa .. For steal at Pi 41 = so -a + b > at 75 o S 100 Р at b 100 a (to From (P- Pe) 100+* 50 50%-1002 px 100% Pi xso² 502 100² 2 71 100 put * in egnation ows p(loom + 50%)--22: P: (50) סטן

Now og at 9=100 ь + TI 67 (1002 57 = (P-Pi) (100)?(50)2 so² - 1002 Now og at r=100 on = Por -at b (100) Pp50²-P 100² 50²-1002 + (P-P;) 100x50? xt so² -1002 100 67 P; 50²-P1002 + P 150%) Piso² 502 - 100 og = Brass For and 1= 150, at = 0 Ря - Р = 100 b P x 150 x 100² Palool 100²-1002 150²_100² ono P (150%+1002) calculated earlier for same boundary at 92 100 1502 -100² condition also on -P h

Now strain calculeten at 7=100m rom Eins , [ p(100*750)-2pi (só) it 0.289 50-1002 for trans Eno I E P(1502710022 150²_1002 +0.33p] Total strain Enst Eno 0002 (72=100 100) ²1 +0.33p 0.02 100 so² - 106² 150²-100² Bus (v0*50*3- 2P(50") +0:28 7] + e, pero [13866]+, [2.833-002 + 2x10x50% (50%-100%)* 200 0.005187 G Pa 5-187 MPa 10 p= 1.981 MP = 1.98m Pa dupa бар. дъяа,