5. Pressure Vessels and

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1 5. Pessue Vessels and Axial Loading Applications

2 5.1 Intoduction Mechanics of mateials appoach (analysis) - analyze eal stuctual elements as idealized models subjected simplified loadings and estaints.

3 5.2 Defomation of axially loaded membes - Unifom membe: - Multiple loads/sizes: - Nonunifom defomation: s L d = e L =, d = E PL d = = d = n n i i ådi å i= 1 i= 1 Ei Ai L Px ò dx 0 EAx PL EA

4 5.2 Defomation of axially loaded membes Example Poblem 5-1 D A = 100 mm, D B_in = 100 mm, D B_out = 150 mm, D C_in = 125 mm, D C_out = 200 mm, E A = 73 GPa, E B = 100 GPa, E C = 210 GPa - Detemine the oveall shotening of the membe

5 5.2 Defomation of axially loaded membes Example Poblem 5-3 A homogeneous ba of unifom coss section A hangs vetically - Elongation of the ba due to its own weight W in tems of W, L, A, and E - Elongation of the ba if it is also subjected to an axial tensile foce P at its lowe end

6 5.3 Defomations in a system of axially loaded bas d d ( ) 2 2 = L - L = L + v + u - L AB f i B B + 2Ld + L = L + 2Lv + v + u AB AB B B B v AB In a simila manne, d u B B In case of L = R, BC ( cos ) ( sin ) 2 2 d = R q - u + R q + v - R BC B B d + 2Rd + R = R cos q - 2Ru cosq + u + R sin q + 2Rv sinq + v BC BC B B B B v sinq - u cosq = d sinq -d cosq BC B B Fo small displacements, the axial defomation in any ba may be assumed equal to the component of the displacement of one end of the ba taken in the diection of the unstained oientation of the ba. AB BE

7 5.3 Defomations in a system of axially loaded bas Example Poblem 5-5 Coss-sectional aeas of tie od AB and pipe stut BC ae 650 mm 2 and 925 mm 2, espectively. E = 200 GPa - Nomal stess in AB and BC - Lengthening o shotening of AB and BC - Hoizontal and vetical components of the displacement of B - Angles though which membes AB and BC otate

8 5.4 Statistically indeteminate axially loaded membes - The numbe of unknowns should be the same as the numbe of equations. - Fo many mechanical systems, the equations of equilibium ae not sufficient fo the detemination of axial foces in the membes and eactions at the suppots. - Additional equations involving the geomety of the defomations of the system can be helpful to solve the poblems. - Hooke s law and the definition of stess and stain can be used to elate defomations and foces when all stesses ae unde the popotional limit of the mateials.

9 5.4 Statistically indeteminate axially loaded membes - Calculation of the tension of the cable 1) Rigid cable Wa å M B = 0 : TR - Wa = 0 T = R 2) Defomable cable Wa å M B = 0 : TR - W ( a cosq ) = 0 T = cosq R TL d AE Wa d = = cosq AE L R 2 d = Rq R EAq = WaL cosq

10 5.4 Statistically indeteminate axially loaded membes - Compaison of tensions with vaious elastic moduli of the cable

11 5.4 Statistically indeteminate axially loaded membes Example Poblem 5-7 A pie has nine 25-mm-diamete steel einfocing bas (E = 200 GPa) in the concete (E = 30 GPa). P = 650 kn - Stesses in the concete and the steel bas - Shotening of the pie

12 5.4 Statistically indeteminate axially loaded membes Example Poblem 5-10 A 0.5 in.-diamete bolt (E = 30,000 ksi) & a sleeve (E = 15,000 ksi) of in 2 coss-sectional aea ae defomed by a nut (0.02 in.). - Stesses in the bolt and the sleeve

13 5.5 Themal Effects - When the defomation of a ba is pevented the themal stain is offset by the mechanical stain in opposite diection. L L TL s dtotal = d L T + ds = et + es = ad + = 0 E

14 5.5 Themal Effects Example Poblem 5-11 A 10-m section steel ail (E = 200 GPa, α = 11.9(10-6 )/ C) has a coss-sectional aea of 7,500 mm 2. Defomations in all diections ae esticted. Fo an incease in tempeatue of 50 C, detemine - Nomal stess in the ail - Intenal foce on a coss section of the ail

15 5.5 Themal Effects Example Poblem 5-12 The tempeatue inceases 100 C. The themal coefficients of expansion and the modulus of elasticity ae 22(10-6 )/ C and 75 GPa fo the od A, and 12(10-6 )/ C and 200 GPa fo the od B. The coss sectional aea of A and B ae 1000 mm 2 and 500 mm 2, espectively. The od CD is igid. - Nomal stess in bas A and B - Vetical component of the displacement of point D

16 5.6 Stess concentations - The stess is concentated aound discontinuities that inteupt the stess path.

17 5.6 Stess concentations - Stess concentation facto, K, is defined based on an aea at the educed section (net aea) o on the goss aea: P s = K A

2 2 4 æ ö æ ö s a s 4a 3a s = ç 1-1 cos 2 2 - ç - + q 2 4 2 è ø 2 è ø q 2 4 æ ö æ ö s a s 3 a sq = ç 1+ 1 cos

18 5.6 Stess concentations - Kisch s solution: stess distibution aound a small cicula hole in a wide plate unde unifom unidiectional tension æ ö æ ö s a s 4a 3a s = ç 1-1 cos ç - + q è ø 2 è ø q 2 4 æ ö æ ö s a s 3 a sq = ç 1+ 1 cos ç + q 4 2 è ø 2 è ø t 2 4 s æ 2a 3a ö = ç 1+ + sin 2q è ø ( ) s = 0 ; s = s 1+ 2cos 2 q ; t = 0 at = a q 2 4 s æ a 3a ö s o = 2 s 1.074s q = 0 ç o = q = 0, = 3a 2 è ø q

19 5.6 Stess concentations - Saint Venant s Pinciple: Localized stess concentation disappeas at some distance. The diffeence between the stesses caused by statically equivalent load systems is insignificant at distances geate than the lagest dimension of the aea ove which the loads ae acting.

20 5.6 Stess concentations Example Poblem 5-14 The machine pat is 20 mm thick and the maximum allowable stess is 144 MPa. - The maximum value of P

21 5.7 Inelastic behavio of axially loaded membes - When the stesses in some membes extend into the inelastic ange, stess-stain diagams must be used to elate the loads and the deflections and solve the poblem. Steel (elastoplastic) Aluminum alloy (stain hadening) Magnesium alloy (stain hadening)

22 5.7 Inelastic behavio of axially loaded membes Example Poblem 5-15 A is a 0.5-in.-diamete steel od (elastoplastic) which has a popotional limit of 40 ksi and a modulus of elasticity of 30,000 ksi. Pipe B has a coss-sectional aea of 2 in. 2 and shows stain-hadening as below. Load P is 30 kip. - Nomal stess in A and B - Displacement of plate C

a plane pependicula to the suface is unifom thoughout the thickness.

23 5.8 Thin-walled pessue vessels - A pessue vessel is a thin walled containe whose wall thickness is so small that the nomal (axial, meidional) stess on a plane pependicula to the suface is unifom thoughout the thickness. 1) Spheical pessue vessels: no shea stess (stain) 2 ( ) ( 2 a ) P pp = R p ts s = a p 2t

24 5.8 Thin-walled pessue vessels 2) Cylindical pessue vessels: Hoop (tangential, cicumfeential) stess vs. axial (meidional) stess ( s ) ( ) 2Q Lt = P p2l s = h x h p s = \ s = 2s a 2t h a p t

25 5.8 Thin-walled pessue vessels 3) Thin shells of evolution: Shapes made by otating plane cuves (meidian) about an axis sphee, hemisphee, tous (doughnut), cylinde, cone, and ellipsoid. P - 2F sin dq - 2F sin dq = 0 m m t t ( )( ) = ( ) + ( ) p 2 dq 2 dq 2s 2t dq sin dq 2s 2t dq sin dq t t m m m t t m t m m t dq s s p t m t t» dqm + = m t

26 5.8 Thin-walled pessue vessels Example Poblem 5-16 A cylinde whose diamete is 1.5 m is constucted by 15-mm-thick steel plate, and is unde 1500 kpa of inne pessue. - Nomal and shea pessue at welded edge foming 30 fom hoop diection

27 5.8 Thin-walled pessue vessels Example Poblem 5-17 A pessue vessel of ¼ in. steel plate is closed by a flat plate. The meidian line follows a paabola of y = x 2 /4. P = 250 psi, m = (1+(dy/dx) 2 ) 1.5 /(d 2 y/dx 2 ) - Meidional and tangential stess σ m and σ t at a point 16 in. above the m t bottom

28 5.9 Combined effects axial and pessue loads - A case of an axial load (P) combindedly added to an axial pessue (p) s x = s xp + s xp

29 5.9 Combined effects axial and pessue loads Example Poblem 5-18 A cylindical pessue tank whose diamete and wall thickness ae 4 ft and ¾ in., espectively is unde 400 psi of pessue and 30,000 lb of axial load. s, s, and t - x y xy - Nomal and sheaing stesses on an inclined plane oiented at +30 fom the x-axis

30 5.10 Thick-walled cylindical pessue vessels - Foce equilibium in adial diection dq t 2 ds + s - s t = 0 d (neglecting highe-ode tems) ( s + ds )( + d ) dqdl -s dqdl - 2s ddlsin = 0

31 5.10 Thick-walled cylindical pessue vessels - Obtaining σ and σ t e 1 ( ( )) 1 a = s a - n s + s t s + s t = ( s a - Ee a ) = 2 C1 fom Hooke's law E n ds ds s s t 0 2s 2C d + - = d + = d 2 ( s ) d ds = + = d 2 2s 2C1 2 ( s ) d Integating = 2 C s = C + C d C C \ s = C +, s = C - s + s = 2C ( Q ) t 1 2 t 1 1

32 5.10 Thick-walled cylindical pessue vessels - Applying bounday conditions s = - p at = a i s = - p at = b o ( - ) a pi - b p a b p o i po 1 C C = = - b - a b - a a pi - b p a b ( p o i - po ) \ s = b - a b - a ( ) ( - ) i - b p a b p o i po a p s t = + b - a b - a ( )

33 5.10 Thick-walled cylindical pessue vessels - Displacements ( ) d = 2pd = e 2p d t t = e t When s = 0 a 1 et = ( s t - ns ) E d = s t -ns E ( )

34 5.10 Thick-walled cylindical pessue vessels - Case 1: Intenal pessue only (p o = 0) ( ) a pi - b p a b pi - p o o a p æ i b ö s = - s = 2 2 ç - 2 b - a - b - a è a ø ( b a ) ( ) a pi - b p a b pi - p o o a p æ i b ö s t = + s t = 2 2 ç + 2 b - a - b - a è a ø ( b a ) 2 a pi d = ( s t ) ( 1 ) ( 1 ) b 2 2 E - ns d = n n b - a E é ë ( ) 2 2 ù û - Case 2: Extenal pessue only (p i = 0) 2 2 b p æ o a ö s = ç - 2 b - a è ø 2 2 b p æ o a ö s t = ç + 2 b - a è ø 2 b po 2 2 ( - ) ( 1 ) ( 1 ) d = - é - n + + n b b a E ë 2 2 ù û -Case 3: Extenal pessue on a solid cicula cylinde (a = 0) s = - p t o s = - p 1-n d = - po E o

5.10 Thick-walled cylindical pessue vessels Example Poblem 5-19 A steel cylinde ( E = 30,000 ksi and n = 0.30) whose inside and outside diametes ae 8 in. and 16 in.

35 5.10 Thick-walled cylindical pessue vessels Example Poblem 5-19 A steel cylinde ( E = 30,000 ksi and n = 0.30) whose inside and outside diametes ae 8 in. and 16 in., espectively is subjected to an intenal pessue of 15,000 psi. The axial load is zeo. - The maximum tensile stess - The maximum sheaing stess - The incease of the inside diamete - The incease of the outside diamete

36 5.11 Design - Failue is defined as the state in which a membe o stuctue no longe functions as intended. - Failue modes ae limited to elastic failue (yielding) hee. - Allowable stess design: Stength ³ Stess - Facto of safety: Stength ³ Facto of safety Stess

5.11 Design Example Poblem 5-20 An axially loaded cicula ba is subjected to a load of 6500 lb with the facto of safety of 1.5. The yield stength of the ba is 36 ksi.

37 5.11 Design Example Poblem 5-20 An axially loaded cicula ba is subjected to a load of 6500 lb with the facto of safety of 1.5. The yield stength of the ba is 36 ksi. - Minimum diamete of the ba Example Poblem 5-21 An axially loaded cicula ba has a yielding stess of 250 MPa. The facto of safety is to be Minimum diamete of the ba

38 5.11 Design Example Poblem 5-22 A 40-lb light is suppoted by a igid wie whose yield stess is 62 ksi. The facto of safety should be 3. - Minimum diamete of the wie Poblems: 5-1, 13, 28, 42, 59, 71, 89, 105, 124, 136 by : May 14

7.2.1 Basic relations for Torsion of Circular Members

7.2.1 Basic relations for Torsion of Circular Members Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,