PIPES, TUBES, AND CYLINDERS 7.20 CHAPTER SEVEN Particular Formula PROBLEM A closed end cylinder made of ductile material has inner diameter of 10 in (250 mm) and outside diameter of cylinder is 25 in (625 mm) The pressure inside the cylinder is 5000 psi Use Clavarino’s equation from Table 7-8 R¼ 25 ¼ ¼ 2:5 di 10 Mark on scale b at 2.5 Draw a perpendicular from x and this perpendicular meets scale d at y Join y and (5000 psi) on scale e Produce y–5 to meet scale f at z y–5–z meets scale f at 8.25 Stress ¼ 8:25 ¼ 8250 psi Stress in SI units ¼ 8250  6:894  103 ¼ 56:88 MPa Check by using Clavarino’s equation from Table 7-8 0:4 ỵ 1:3R2 0:4 ỵ 1:32:5ị2 ẳ 5000 ¼ p1 R2 À ð2:5Þ2 À 0:4 ỵ 8:125 4:2625 ẳ 5000 ẳ 104 6:25 5:25 ẳ 8120 psi 56 MPaị The stress obtained from nomogram 8250 psi (56.88 MPa) is very close to stress value found from Clavarino’s equation REFERENCES ‘‘Rules for Construction of Power Boilers,’’ Section I, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, 1983 ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 1, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, July 1, 1986 ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 2—Alternative Rules, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, July 1, 1986 Nicholas, R W., Pressure Vessel Codes and Standards, Elsevier Applied Science Publications, Crown House, Linton Road, Barking, Essex, England Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 Courtesy: Durham, H M., Stress Chart for Thick Cylinders Greenwood, D C., Editor, Engineering Data for Product Design, McGraw-Hill Book Company, New York, 1961 Lingaiah, K., Machine Design Data Handbook (SI and U.S Customary Systems Units), McGraw-Hill Book Company, New York, 1994 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Source: MACHINE DESIGN DATABOOK CHAPTER DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS SYMBOLS13;14;15 a A A A A1 A2 A3 A41 , A42 , A43 A5 Ab Am Am1 ¼ Wm1 =sb Am2 ¼ Wm2 =sa length of the long side of a rectangular plate, m (in) pitch or distance between stays, m (in) major axis of elliptical plate, m (in) long span of noncircular heads or covers measured at perpendicular distance to short span, m (in) (see Fig 8-10) factor determined from Fig 8-3 total cross-sectional area of reinforcement required in the plane under consideration, m2 (in2 ) (see Fig 8-17) (includes consideration of nozzle area through shell for sna =sva < 1:0) outside diameter of flange or, where slotted holes extend to the outside of the flange, the diameter to the bottom of the slots, m (in) area in excess thickness in the vessel wall available for reinforcement, m2 (in2 ) (see Fig 8-17) (includes consideration of nozzle area through shell if sna =sva < 1:0) area in excess thickness in the nozzle wall available for reinforcement, m2 (in2 ) (see Fig 8-17) area available for reinforcement when the nozzle extends inside the vessel wall, m2 (in2 ) (see Fig 8-17) cross-sectional area of various welds available for reinforcement (see Fig 8-17), m2 (in2 ) cross-sectional area of material added as reinforcement (see Fig 8-17), m2 (in2 ) cross-sectional area of the bolts using the root diameter of the thread or least diameter of unthreaded portion, if less, Eq (8-111), m (in) total required cross-sectional area of bolts taken as the greater of Am1 and Am2 , m2 (in2 ) total cross-sectional area of bolts at root of thread or section of least diameter under stress, required for the operating condition, m2 (in2 ) total cross-sectional area of bolts at root of thread or section of least diameter under stress, required for gasket seating, m2 (in2 ) 8.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.2 CHAPTER EIGHT length of short side or breadth of a rectangular plate, m (in) short span of noncircular head, m (in) (see Fig 8-10 and Eq 8-86a) effective gasket or joint-contact-surface seating width, m (in) basic gasket seating width, m (in) (see Table 8-21 and Fig 8-13) factor determined from the application material–temperature chart for maximum temperature, psi inside diameter of flange, m (in) corrosion allowance, m (in) basic dimension used for the minimum sizes of welds, mm (in), equal to tn or tx , whichever is less empirical coefficient taking into account the stress in the knuckle [Eq (8-68)] empirical coefficient depending on the method of attachment to shell [Eqs (8-82) and (8-85)] empirical coefficients depending on the mode of support [(Eqs (8-92) to (8-94)] bolt-circle diameter, mm (in) finished diameter of circular opening or finished dimension (chord length at midsurface of thickness excluding excess thickness available for reinforcement) of nonradial opening in the plane under consideration in its corroded condition, m (in) (see Fig 8-17) diameter or short span, m (in) diameter of the largest circle which may be inscribed between the supporting points of the plate (Fig 8-11), m (in) diameter as shown in Fig 8-9, m (in) factor, m3 (in3 ) b b bo B B c c c1 c2 c4 , c5 C d d d U h g2 V o o U h g2 d¼ VL o o d0 d¼ de di , Di , Do dk D Dp e for integral-type flanges for loose-type flanges diameter through the center of gravity of the section of an externally located stiffening ring, m (in); inner diameter of the shell in the case of an internally located stiffening ring, m (in) [Eq (8-55)] outside diameter of conical section or end (Fig 8-8(A)d), m (in) inside diameter of shell, m (in) outside diameter of shell, m (in) inside diameter of conical section or end at the position under consideration (Fig 8-8(A)d), m (in) inside shell diameter before corrosion allowance is added, m (in) outside diameter of reinforcing element, m (in) (actual size of reinforcing element may exceed the limits of available reinforcement) factor, mÀ1 (inÀ1 ) F ho F e¼ L ho for integral-type flanges E Eam modulus of elasticity at the operating temperature, GPa (Mpsi) modulus of elasticity at the ambient temperature, GPa (Mpsi) e¼ for loose-type flanges Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS hub stress correction factor for integral flanges from Fig 8-25 (When greater than one, this is the ratio of the stress in the small end of the hub to the stress in the large end For values below limit of figure, use f ¼ 1.) fr strength reduction factor, not greater than 1.0 fr1 sna =sva fr2 (lesser of sna or spa Þ=sva fr3 spa =sva F total load supported, kN (lbf ) total bolt load, kN (lbf ) F correction factor which compensates for the variation in pressure stresses on different planes with respect to the axis of a vessel (a value of 1.00 shall be used for all configurations, except for integrally reinforced openings in cylindrical shells and cones) F factor for integral-type flanges (from Fig 8-21) FL factor for loose-type flanges (from Fig 8-23) ga thickness of hub at small end, m (in) thickness of hub at back of flange, m (in) g1 G diameter, m (in), at location of gasket load reaction; except as noted in Fig 8-13, G is defined as follows (see Table 8-22): When bo 6:3 mm (l/4 in), G ¼ mean diameter of gasket contact face, m (in) When bo > 6:3 mm (1/4 in), G ¼ outside diameter of gasket contact face less 2b, m (in) h distance nozzle projects beyond the inner or outer surface of the vessel wall, before corrosion allowance is added, m (in) (Extension of the nozzle beyond the inside or outside surface of the vessel wall is not limited; however, for reinforcement calculations the dimension shall not exceed the smaller of 2.5t or 2.5tn without a reinforcing element and the smaller of 2.5t or 2.5tn þ te with a reinforcing element or integral compensation.) h hub length, m (in) h, t minimum required thickness of cylindrical or spherical shell or tube or pipe, m (in) thickness of plate, m (in) thickness of dished head or flat head, m (in) actual thickness of shell at the time of test including corrosion allowance, m (in) hc thickness for corrosion allowance, m (in) hD radial distance from the bolt circle, to the circle on which HD acts, m (in) hG ẳ C Gị=2 radial distance from gasket load reaction to the bolt circle, m (in) pffiffiffiffiffiffiffiffi ho ¼ Bgo factor, m (in) hT radial distance from the bolt circle to the circle on which HT acts as prescribed, m (in) H ¼ G2 P=4 total hydrostatic end force, kN (lbf ) HD ¼ B2 P=4 hydrostatic end force on area inside of flange, kN (lbf ) HG ¼ W À H gasket load (difference between flange design bolt load and total hydrostatic end force), kN (lbf ) HP ¼ total joint-contact-surface compression load, kN (lbf ) 2b  GmP f Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 8.3 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.4 CHAPTER EIGHT HT ¼ H À HD Is Is0 I I0 k , k2 , k3 , k4 , k k6 K ¼ A=B K K1 l L difference between total hydrostatic end force and the hydrostatic end force on area inside of flange, kN (lbf ) required moment of inertia of the stiffening ring cross-section around an axis extending through the center of gravity and parallel to the axis of the shell, m4 or cm4 (in4 ) required moment of inertia of the combined ring-shell crosssection about its neutral axis parallel to the axis of the shell, m4 (in4 ) available moment of inertia of the stiffening ring cross-section about its neutral axis parallel to the axis of the shell, m4 (in4 ) available moment of inertia of combined ring shell cross-section about its neutral axis parallel to the axis of the shell, m4 or cm4 (in4 ) coefficients factor for noncircular heads depending on the ratio of short span to long span b=a (Fig 8-10) ratio of outside diameter of flange to inside diameter of flange (Fig 8-20) ratio of the elastic modulus E of the material at the design material temperature to the room temperature elastic modulus, Eam , [Eqs (8-26) to (8-31), (8-55)] spherical radius factor (Table 8-18) length of flange of flanged head, m (in) effective length, m (in) distance from knuckle or junction within which meridional stresses determine the required thickness, m (in) perimeter of noncircular bolted heads measured along the centers of the bolt holes, m (in) distance between centers of any two adjacent openings, m (in) length between the centers of two adjacent stiffening rings, m (in) (Fig 8-1) te ỵ t3 factor ỵ T d m gasket factor, obtained from Table 8-20 m ¼ 1= reciprocal of Poisson’s ratio Mb longitudinal bending moment, N m (lbf in) L¼ FIGURE 8-1 Cylindrical pressure vessels under external pressure Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS torque about the vessel axis, N m (lbf in) component of moment due to HD , m N (in-lbf ) component of moment due to HG , m N (in-lbf ) total moment acting on the flange, for the operating conditions or gasket seating as may apply, m N (in-lbf ) MT ¼ HT h component of moment due to HT , m N (in-lbf ) N width, m (in), used to determine the basic gasket seating with bo , based on the possible contact width of the gasket (see Table 8-21) pi internal design pressure, MPa (psi) p maximum allowable working pressure or design pressure, MPa (psi) po load per unit area, MPa (psi) external design pressure, MPa (psi) P total pressure on an area bounded by the outside diameter of gasket, kN (lbf ) design pressure (or maximum allowable working pressure for existing vessels), MPa (psi) Pa calculated value of allowable external working pressure for assumed value of t or h, MPa (psi) r radius of circle over which the load is distributed, m (in) ri inner radius of a circular plate, m (in) inside radius of transition knuckle which shall be taken as 0:01dk in the case of conical sections without knuckle transition, m (in) R inner radius of curvature of dished head, m (in) Ri inner radius of shell or pipe, m (in) ro , Ro outer radius of a circular plate, m (in) outer radius of shell, m (in) R ẳ ẵC Bị=2 radial distance from bolt circle to point of intersection of hub Àg1 and back of flange, m (in) (for integral and hub flanges) R inside radius of the shell course under consideration, before corrosion allowance is added, m (in) Rn inside radius of the nozzle under consideration, before corrosion allowance is added, m (in) t or h minimum required thickness of spherical or cylindrical shell, or pipe or tube, m (in) t flange thickness, m (in) t nominal thickness of the vessel wall, less corrosion allowance, m (in) tc weld dimensions thickness or height of reinforcing element, m (in) te tn nominal thickness of shell or nozzle wall to which flange or lap is attached, irrespective of product form less corrosion allowance, m (in) tr required thickness of a seamless shell based on the circumferential stress, or of a formed head, computed by the rules of this chapter for the designated pressure, m (in) trn required thickness of a seamless nozzle wall, m (in) nominal thickness of cylindrical shell or tube exclusive of ts corrosion allowance, m (in) tw weld dimensions tx two times the thickness go , when the design is calculated as an integral flange, m (in), or two times the thickness, m (in), of shell nozzle wall required for internal pressure, when the design is calculated as a loose flange, but not less than 6.3 mm Mt MD ¼ HD hD MG ¼ HG hG Mo Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 8.5 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.6 CHAPTER EIGHT (1/4 in) T U V VL w W W W Wm1 Wm2 y y ymax Y Z , , sy sa e sam sd sa sna sva spa sbat sbd sfd snd H R 0 r s su z or l factor involving K (from Fig 8-20) factor involving K (from Fig 8-20) factor for integral-type flanges (from Fig 8-22) factor for loose-type flanges (from Fig 8-24) width, m (in), used to determine the basic gasket seating width bo , based on the contact width between the flange facing and the gasket (see Table 8-21) weight, kN (lbf ) total load to be carried by attachment welds, kN (lbf ) flange design bolt load, for operating conditions or gasket seating, as may apply, kN (lbf ) minimum required bolt load for the operating conditions, kN (lbf ) (For flange pairs used to contain a tubesheet for a floating head for a U-tube type of heat exchanger, or for any other similar design, Wm1 shall be the larger of the values as individually calculated for each flange, and that value shall be used for both flanges.) minimum required bolt load for gasket seating, kN (lbf ) gasket or joint-contact-surface unit seating load, MPa (psi) deflection of the plate, m (in) maximum deflection of the plate, m (in) factor involving K (from Fig 8-20) factor involving K (from Fig 8-20) a factor for non-circular heads [Eq (8-86b)] angles of conical section to the vessel axis, deg (Fig 8-8(A)d) difference between angle of slope of two adjoining conical sections, deg (Fig 8-8(A)d) normal or direct stress, MPa (psi) 0.2 percent proof stress, MPa (psi) maximum allowable stress value, MPa (psi) equivalent stress (based on shear strain energy), MPa (psi) allowable stress at ambient temperature, MPa (psi) design stress value, MPa (psi) allowable stress value as given in Tables 8-9 to 8-12, MPa (psi) allowable stress in nozzle, MPa (psi) allowable stress in vessel, MPa (psi) allowable stress in reinforcing element (plate), MPa (psi) allowable bolt stress at atmospheric temperature, MPa (psi) allowable bolt stress at design temperature, MPa (psi) allowable design stress for material of flange at design temperature (operating condition) or atmospheric temperature (gasket seating), as may apply, MPa (psi) allowable design stress for material of nozzle neck, vessel or pipe wall, at design temperature (operating condition) or atmospheric temperature (gasket seating), as may apply, MPa (psi) calculated longitudinal stress in hub, MPa (psi) calculated radial stress in flange, MPa (psi) calculated tangential stress in flange, MPa (psi) hoop stress, MPa (psi) radial stress, MPa (psi) strength, MPa (psi) ultimate strength, MPa (psi) longitudinal stress, MPa (psi) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.7 zt tensile longitudinal stress, MPa (psi) zc compressive longitudinal stress, MPa (psi) shear stress (also with subscripts), MPa (psi) Poisson’s ratio joint factor (Table 8-3) or efficiency ¼1 (see definitions for tr and trn ) when an opening is in the solid plate or joint efficiency obtained 1 ¼ from Table 8-3 when any part of the opening passes through any other welded joint Note: and with initial subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook Other factors in performance or in special aspect are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage Particular PLATES13;14;15 Formula Refer to Table 8-1 For maximum stresses and deflections in flat plates Plates loaded uniformly The thickness of a plate with a diameter d supported at the circumference and subjected to a pressure p distributed uniformly over the total area The maximum deflection Plates loaded centrally The thickness of a flat cast-iron plate supported freely at the circumference with diameter d and subjected to a load F distributed uniformly over an area (do =4) The deflection Grashof’s formula for the thickness of a plate rigidly fixed around the circumference with the above given type of loading h ¼ k1 d p sd 1=2 ð8-1Þ Refer to Table 8-2 for values of k1 p y ¼ k2 d Eh3 Refer to Table 8-2 for values of k2 ð8-2Þ 0:67do F 1=2 h ¼ 1:2 À d sd ð8-3Þ 0:12d F Eh3 F d 1=2 ln h ¼ 0:65 sd y¼ y¼ 0:055d F Eh3 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð8-4Þ ð8-5Þ ð8-6Þ Form of plate Distributed on circumference of a concentric circle of radius r 8-132 8-134 Edge supported Edge fixed 8-133 Edge fixed 8-131 Edge supported Distributed over a concentric circular area of radius r 8-130 Edge fixed Eq 8-129 Type of support Distributed Edge over the entire supported surface Type of loading TABLE 8-1 Maximum stresses and deflections in flat plates 2rp 2rp r2 p r2 p r2 p o r2 p o Total load, F 3F 4h2 3F3m ỵ 1ị 8mh2 Center Edge Center Edge Center Location of max 8.8 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website r ¼ 3F r2 1À 2 2h ro All points inside the circle of r r radius r ỵ m ỵ 1ị loge o m 1ị r ro 3F Center when r ẳ ẳ m ỵ 1ị 4mh2 r < 0:31ro Edge when ro r2 r > 0:31ro  loge ỵ r ro r ẳ ẳ 3F r m ỵ 1ị loge ro 2mh2 r2 ỵ m ỵ 1ị 4ro 3F m À r ¼ ¼ 2mh2 r ẳ ẳ 3F r m ỵ 1ị loge o r 2mh r À ðm À 1ị ỵ m 4ro 3F r2 r ¼ 1À 2 2ro 2h r ¼ r ¼ ẳ Maximum stress, max ro 7m ỵ 3ị r mỵ1 r 3Fm2 1ị 2Em2 h3 3m ỵ 1ịr2 r2 ị r o r2 loge o 2m ỵ 1ị r 3Fðm À 1Þ ro 2 ðro À r Þ À r loge r h3 2Em 3Fðm2 À 1Þr2 o 4Em2 h3 when r is very small (concentrated load) 3Fðm2 À 1Þ r 4r2 À 4r2 loge o À 3r2 o h3 r 16Em À4r2 loge 3Fðm2 À 1Þr2 o 16Em2 h3 3Fm2 1ị 12m ỵ 4ịr2 o mỵ1 16Em2 h3 3Fm 1ị5m ỵ 1ịr2 o 16Em2 h3 Maximum deflection, ymax DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Form of plate 8-137 8-138 Distributed Outer edge over the entire fixed and surface supported Distributed Outer edge over the entire fixed and surface supported, inner edge fixed 8-135 8-136 Uniform pressure over entire lower surface Distributed over a concentric circular area of radius r Eq Distributed Outer edge over the entire supported surface Type of support Type of loading F ¼ ðr2 À r2 Þp o i F ¼ ðr2 r2 ịp o i F ẳ r2 r2 Þp o i r2 p Total load, F TABLE 8-1 Maximum stresses and deflections in flat plates (Cont.) r ¼ ¼ ¼ ro r 3p ðr2 þ r2 Þ o 4h2 4r2 r2 r À o loge o r ro À r À3pðm2 À 1Þ 4mh2 r r2 À r4 À r2 r2 loge o o i o i ri  ro ðm À 1ị ỵ r2 m ỵ 1ị i 4m ỵ 1ịr2 r2 loge o i ỵr4 m 1ị 4mr2 r2 i o i 3P r4 3m ỵ 1ị o 4mh2 ðr2 À r2 Þ o i Maximum stress, max 3F r m ỵ 1ị loge o r ¼ ¼ r 2mh mÀ1 r ỵ ro Inner edge Inner edge Inner edge Center Location of max 3pðm2 À 1Þ ro ỵ 3r4 4r2 r2 i o i 16Em2 h3 r 16r r r 4r2 r2 loge o ỵ o i2 loge o o i ri ro À ri ri r4 m ỵ 3ị r2 r2 3m ỵ 1ị i o i 8m ỵ 1ị 2m ỵ 1ị r2 r2 3m ỵ 1ị r o i loge o 2m 1ị ri 2r r m ỵ 1Þ r À o i2 loge o r ro ri ịm 1ị ỵ ỵ 3Fm2 1ị r4 5m ỵ 1ị o h3 8m þ 1Þ 2Em where r is very small (concentrated load) 3Fm 1ị7m ỵ 3ịr2 =16Em2 h3 o 3Fm2 1ị r 3m ỵ 4r2 loge o ỵ 2r2 mỵ1 r 16Em2 h3 2 7m ỵ r r ịr r4 ỵ r2 ỵ o 2 o mỵ1 r ro ro Maximum deflection, ymax DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 8.9 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Particular 8.25 Formula Ellipsoidal heads The required thickness of a dished head of semiellipsoidal form in which half the minor axis (inside depth of the head minus the skirt) equals one-fourth of the inside diameter of the head skirt, shall be determined by Eq (8-62) as per ASME Boiler and Pressure Vessel Code tẳ PD 2sa 0:2P 8-62ị The maximum allowable working pressure or design pressure as per ASME Boiler and Pressure Vessel Code pẳ 2sa t D ỵ 0:2t 8-63ị tẳ 0:885PL sa 0:1P 8-64ị Torispherical heads The required thickness of a torispherical head for the case in which the knuckle radius is percent of the inside crown radius and the inside crown radius equals the outside diameter of the skirt, shall be determined by Eq (8-64) as per ASME Boiler and Pressure Vessel Code The maximum allowable working pressure as per ASME Boiler and Pressure Vessel Code Pẳ sa t 0:885L ỵ 0:1t 8-65ị Hemispherical heads The required thickness of a hemispherical head when its thickness does not exceed 0.36L or P does not exceed 0.665sa , shall be determined by Eq (8-66) as per ASME Boiler and Pressure Vessel Code The design pressure tẳ PL 2sa 0:2P 8-66ị Pẳ 2sa t L ỵ 0:2t 8-67ị hẳ pde c1 2sa 8-68ị Conical ends subject to internal pressure (Fig 8-8d) as per Indian Standards KNUCKLE OR CONICAL SECTION The thickness of cylinder and conical section (frustrum) within the distance L from the junction pffiffiffiffiffi (L ¼ 0:5 de h=cos ) The thickness of those parts of conical sections not less than a distance L away from the junction with the cylinder or other conical section Refer to Table 8-4 for values of c1 h¼ pdk 2sa À p cos Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð8-69Þ DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.26 CHAPTER EIGHT FIGURE 8-8(A) (a) Ellipsoidal; (b) spherically dished (torispherical); (c) hemispherical; (d) typical conical shell connection; and (e) toriconical (cone head with knuckle) Particular Formula SHALLOW CONICAL SECTIONS The thickness of conical sections having an angle of inclination to the vessel axis more than 708 pffiffiffiffiffiffiffiffiffiffiffi p=sa ð8-70Þ 90 The lower of values given by Eqs (8-69) and (8-70) shall be used h ¼ 0:5ðde À ri Þ Conical heads (without transition knuckle) as per ASME Boiler and Pressure Vessel Code The required thickness of conical heads or conical shell sections that have a half-apex angle not greater than 308 shall be determined by Eq (8-71) t¼ PD cos ðsa À 0:6PÞ ð8-71Þ where D ¼ inside diameter ¼ minimum joint efficiency, percent TABLE 8-4 Values of c1 for use in Eq (18-68) (as function of É and ri =de ) ri =de É 0.01 0.02 0.03 0.04 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 108 208 308 458 608 758 0.70 1.00 1.35 2.05 3.20 6.80 0.65 0.90 1.20 1.85 2.85 5.85 0.60 0.85 1.10 1.65 2.55 5.35 0.60 0.80 1.00 1.50 2.35 4.75 0.55 0.70 0.90 1.30 2.00 3.85 0.55 0.65 0.85 1.20 1.75 3.50 0.55 0.60 0.80 1.10 1.60 3.15 0.55 0.55 0.70 0.95 1.40 2.70 0.55 0.55 0.55 0.90 1.25 2.40 0.55 0.55 0.55 0.70 1.00 1.55 0.55 0.55 0.55 0.55 0.70 1.00 0.55 0.55 0.55 0.55 0.55 0.55 Source: IS 2825, 1969 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.27 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Particular Formula Pẳ 2sa t cos D ỵ 1:2t cos ð8-72Þ The required thickness of the conical portion of a toriconical head, in which the knuckle radius is neither less than percent of the outside diameter of the head skirt nor less than three times the knuckle thickness and pressure shall be determined by Eqs (8-73) and (8-74) tc ẳ PDi cos sa 0:6Pị 8-73ị Pẳ 2sa t cos Di ỵ 1:2t cos 8-74ị The required thickness of the knuckle and pressure shall be determined by Eqs (8-75) and (8-76) tk ¼ The design pressure Toriconical heads where Di ¼ inside cone diameter at point of tangency to knuckle PLM 2sa À 0:2P ð8-75Þ or refer to Eqs (8-66) and (8-67) where M ¼ factor depending on head proportion, L=r L ¼ Di =2 cos The design pressure Toriconical heads may be used when the angle 308 Pẳ 2sa tk ML ỵ 0:2tk 8-76ị FORMED HEADS UNDER PRESSURE ON CONVEX SIDE The thickness of heads and ends under pressure on convex side (minus heads) as per Indian Standards (a) Spherically dished ends and heads The thickness of the spherically dished heads and ends shall be the greater of the following thicknesses: (1) The thickness of an equivalent sphere, having a radius ro or Ro equal to the outside crown radius of the end as determined from Eq (8-39) (2) The thickness of the end under an internal pressure equal to 1.2 times the external pressure (b) Ellipsoidal heads The thickness of ends of a semiellipsoidal shape shall be the greater of the following: (1) The thickness of an equivalent sphere, having a radius ro or Ro calculated from the values of ro =do or Ro =Do in Table 8-5, determined as per Eq (8-39) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.28 CHAPTER EIGHT Particular Formula (2) The thickness of the end under an internal pressure equal to 1.2 times the external pressure (c) Conical heads under external pressure: For a conical end or conical section under external pressure, whether the end is of seamless or buttwelded construction as per Indian Standards Use Eqs (8-68), (8-69), and (8-70) Equations (8-68) to (8-70) are applicable, except that the shell thickness shall be no less than as prescribed below: (1) The thickness of a conical end or conical section under external pressure, when the angle of inclination of the conical section to the vessel axis is not more than 708, shall be made equal to the required thickness of cylindrical shell, in which the diameter is ðde = cos Þ and the effective length is equal to the slant height of the cone or conical section, or slant height between the effective stiffening rings, whichever is less (2) The thickness of conical ends having an angle of inclination to the vessel axis of more than 708 shall be determined as a flat cover The thickness of formed heads under pressure on convex side (minus heads) as per ASME Boiler and Pressure Vessel Code The required thickness at the thinnest point after forming of ellipsoidal or torispherical heads under pressure on the convex side (minus heads) shall be the greater of the thicknesses given here (1) The thickness as computed by the procedure given for the heads with the pressure on the concave side of the previous section using a design pressure 1.67 times the design pressure of the convex side, assuming the joint efficiency ¼ 1:00 for all cases, or (2) The thickness as determined by the appropriate procedure given in Ellipsoidal heads or Torospherical heads as per ASME Boiler and Pressure Vessel Code HEMISPHERICAL HEADS The required thickness of a hemispherical head having pressure on the convex side shall be determined by Eqs (8-41) to (8-43) Aẳ 0:125 Ro =t 8-41ị B Ro =t 8-42ị Pa ẳ TABLE 8-5 Values of spherical radius factor Ko ¼ Ro =Do for ellipsoidal head with pressure on convex side as a function of ho =Do for use in Eq (8-41) ho =Do Ko ¼ Ro =Do 0.167 1.36 0.178 1.27 0.192 1.182 0.208 1.08 0.227 0.99 0.25 0.90 0.276 0.81 0.313 0.73 0.357 0.65 0.417 0.57 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 0.5 0.5 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Particular 8.29 Formula Pa ẳ 0:0625 Ro =tị2 8-43ị where Ro ẳ outside radius in corroded condition The procedure outlined in this section for finding the thickness of a spherical shell can be used to find the thickness of a hemispherical head by using Eqs (841) to (8-43) ELLIPSOIDAL HEADS The minimum required thickness of ellipsoidal head having pressure on the convex side either seamless or of built-up construction with butt joints shall not be less than that determined by the procedure given here The factor A is given by Eq (8-41) TORISPHERICAL HEADS The required minimum thickness of a torispherical head having pressure on the convex side, either seamless or of built-up construction with butt joint CONICAL HEADS AND SECTIONS The required minimum thickness of a conical head or section under pressure on the convex side, either seamless or the built-up construction with butt joints Assume a value of t and calculate the value of factor A using the following equation: Aẳ 0:125 Ro =t 8-41ị Using the value of A calculated from Eq (8-41) follow the procedure as that given for spherical shells to find the thickness of ellipsoidal heads where Ro ¼ the equivalent outside spherical radius taken as Ko Do in the corroded condition Ko ¼ factor depending on the ellipsoidal head proportion Ro =Do (Table 8-5) The required thickness shall not be less than that determined by the same design procedure as is used for ellipsoidal heads given in the Ellipsoidal heads section, using appropriate values for Ro For torispherical head, the outside radius of the crown portion of the head (Ro ) in corroded condition is taken for design purposes This thickness can be determined following the procedure outlined under cylindrical shells in the Torispherical heads section to find factors A and B by assuming a value for te FIGURE 8-8(B) Typical conical sections for external pressure Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.30 CHAPTER EIGHT Particular The symbols involved in design calculations are (1) When 608 and for cones having DL =te , values !10: Assume a value of te and determine ratios Le =DL and DL =te The equation for calculating the maximum allowable external pressure Pa for the case of values of factor A falling to the right of the end of the material/temperature line Equation for calculating the maximum allowable external pressure Pa for the case of values of factor A falling to the left of the applicable material/temperature line Formula te ¼ effective thickness of conical section Le ¼ equivalent length of conical section ¼ L=2ị1 ỵ Ds =Lị L ẳ axial length of cone or conical section (Fig 8-8(B)) Ds ¼ outside diameter at small end of conical section under consideration Pa ¼ 4B 3DL =te ị 8-76aị Pa ẳ 2AE 3DL =te ị 8-77ị where DL ẳ outside diameter at large end of conical section under consideration (Fig 8-8(B)) ¼ one-half the apex angle in conical heads and section, deg (Compare the value of Pa with design pressure P If Pa < P, then select a new value for te and repeat the design procedure.) (2) When 608 and for cones having DL =te , values 608 The thickness of the cone shall be the same as the required thickness for a flat head under external pressure, the diameter of which equals the largest diameter of the cone Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS Particular 8.31 Formula Toriconical head having the pressure on the convex side The required thickness of a toriconical head having pressure on the convex side, either seamless or of built-up construction with butt joints within the head The length Le (Fig 8-8B, panel a) The length Le (Fig 8-8B, panel b) The length Le (Fig 8-8B, panel c) The thickness shall not be less than that determined using the procedure followed in the case of a cone having DL =te values 10 for conical heads and sections with exception that Le shall be determined using Eqs (8-81): L DL ỵ Ds Le ẳ r1 sin 1 ỵ 8-81aị DLs Le ẳ r2 Dss L sin 2 ỵ DL Le ẳ r1 sin 1 ỵ r2 Ds ỵ DL DL Dss L sin 2 þ DLs ð8-81bÞ DL þ Ds DLs ð8-81cÞ To find the thickness when lap joints are used in formed head construction or for longitudinal joints in a conical header under external pressure The rules in this section, except the design pressure 2P, shall be used instead of P in the calculations for the design of required thickness UNSTAYED FLAT HEADS AND COVERS (Fig 8-9, Table 8-6) The thickness h of that unstayed circular heads, covers, and blind flanges as per Indian Standards The minimum required thickness t of unstayed circular heads, covers and blind flanges as per ASME Boiler and Pressure Vessel Code The minimum required thickness t of flat unstayed circular heads, covers, and blind flanges which are attached by bolts, causing an edge moment as per ASME Boiler and Pressure Vessel Code h ¼ c2 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð p=sa Þ ð8-82Þ Refer to Table 8-6 for values of c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ d ðCP=sa Þ ð8-83Þ Refer to Table 8-6 for values of C tẳd CP WhG ỵ 1:9 sa sa d 1=2 ð8-84Þ where C is taken from Table 8-6 W ¼ flange design bolt load, lbf W ¼ Wm1 ¼ the minimum bolt load for operating condition, lbf t ẳ 0:785D2 P ỵ 2b 3:14DG mpị G 8-84aị where W ẳ Wm2 ẳ the minimum required bolt load for gasket seating, lbf t ¼ 3:14bDG y Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð8-84bÞ DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.32 CHAPTER EIGHT FIGURE 8-9 Types of unstayed flat heads (K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; K Lingaiah, Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS 8.33 TABLE 8-6 Coefficients c2 and C for determining head thickness for typical unstayed flat heads (Fig 8-9) Coefficient, c2 and C Type of head IS (Fig 8-9)a ASME c2 , IS (Fig 8-9) C, ASME Remarks AðaÞ (b-2) 0.50 0.33 m but ð pK? ?1= 6 ? ?10 0h=do ? ?1= 2 for L 22:4 1: 46 > or > ð pK? ?1= 6 ðh=do ? ?1= 2 or 5 :7 0:58 ? ?10 p=Þ5=2 22:4 > pK ð pK? ?1= 6 or 372 :65  10 3 USCS ? ?10 p=Þ5=2 97: 78 > pK ð pK? ?1= 6 54 :1  10 6... 0 .10 0 .15 0.20 0.30 0.40 0.50 10 8 208 308 458 608 75 8 0 .70 1. 00 1. 35 2.05 3.20 6.80 0.65 0.90 1. 20 1. 85 2.85 5.85 0.60 0.85 1. 10 1. 65 2.55 5.35 0.60 0.80 1. 00 1. 50 2.35 4 .75 0.55 0 .70 0.90 1. 30