Piping and Pmssure Vessels 215 Procedure 2 Stresses in Heads Due to Internal Pressure [3,4] Notation L = crown radius, in. r = knuckle radius, in. h = depth of head, in. RL = latitudinal radius of curvature, in. R, = meridional radius of curvature, in. q, = latitudinal stress, psi ox = meridional stress, psi P = internal pressure, psi Fonnulas Lengths of RL and R, for ellipsoidal heads: At equator: hZ R R,=- RL=R At center of head: R2 Rrn = RL = h 9 At any point X: R: h2 R4 R, =- Notes 1. Latitudinal (hoop) stresses in the knuckle become compressive when the R/h ratio exceeds 1.42. These heads will fail by either elastic or plastic buckling, de- pending on the R/t ratio. 2. Head types fall into one of three general categories: hemispherical, torispherical, and ellipsoidal. Hemi- spherical heads are analyzed as spheres and were cov- ered in the previous section. Tonspherical (also known as flanged and dished heads) and ellipsoidal head for- mulas for stress are outlined in the following form. Figure I. Direction of stresses in a vessel head. 21 6 ax Rules of Thumb for Mechanical Engineers 9 ux=2t PL PL ax = - 2t a+ = y3 - i) In Crown q = LTx ELLIPSOIDAL HEADS PR ux=n PR a+ = - t PR 2t ox = L At Center PR2 ax = - 2th At Tgngent PR ax=Ti of Head u+ = ox Line Piping and Pressure Vessels 217 Joint Efficiencies (ASME Code) [5] Miter Elbow or - L@ A c If a > 30° Flat head Hemi-head only See Note 3 S.O. flange Figure 1. Categories of welded joints in a pressure vessel. Table 1 Values of Joint Efficiency, E, and Allowable Stress, S* Case 1 Case 2 Case 3 Case 4 Extent of Seamless Seamless Seamless Welded Welded Seamless Welded welded E S E S E S E S E S ES E S ES Radiography Head Shell Head Shell Head Shell Head Shell ~~~ ~ Full(RT-1) 1.0 100% 1.0 100% 1.0 100% 1.0 1000/, 1.0 100% 1.0 100% 1.0 100% 1.0 100% Spot (RT-3) 1.0 85% 1.0 85% 1.0 85% .85 100% .85 100% 1.0 85% .85 100% .85 100% Combinationt 1.0 100% 1.0 100% 1.0 100% .85 100% 1.0 100% 1.0 100940 1.0 100% .85 100% None 1.0 80% 1.0 80% 1.0 80% .7 100% .7 100% 1.0 80% .7 100% .7 100% *See Note 1. jSee Note 2. 218 Rules of Thumb for Mechanical Engineers Notes Table 2 Joint Efficiencies X-Ray TYPes of Joints Full Spot None Single and joints 1.0 .85 .7 Single butt ing strip joint with back- .9 .8 .65 Single butt joint witho;; backing - - .6 Double full joint Single full with plugs Single full joint fillet lap - - .55 fillet lap - - .5 fillet lap - - .45 te3. 1. In Table 1 joint efficiencies and allowable stresses for shells are for longitudinal seams only! All joints are assumed as Type 1 only! Where combination radiog- raphy is shown it is assumed that all requirements for full radiography have been met for head, and shell is spot R. T. 2. Combination radiography: Applies to vessels not fully radiographed where the designer wishes to apply a joint efficiency of 1.0 per ASME Code, Table UW-12, for only a specific part of a vessel. Specifically for any part to meet this requirement, you must perform the fol- lowing: (ASME Code, Section UW-ll(5)): Fully x-ray any Cat. A or D butt welds (ASME Code, Section UW-l1(5)(b)): Spot x-ray any Category B or C butt welds attaching the part (ASME Code, Section UW-l1(5)(a)): All butt joints must be Type 1 3. Any Category B or C butt weld in a nozzle or com- municating chamber of a vessel or vessel part which is to have a joint efficiency of 1 .O and exceeds either 10 in. nominal pipe size or 1% in. in wall thickness shall be fully radiographed. See ASME Code, Section UW- 1 l(aI(4). ~ ______ ~ Properties of Heads L. h 4- Figure 1. Dimensions of heads. Piping and Pressure Vessels 219 D-2r 2 a=- p =go-a b =cosar c =L-cosaL e = sin a L (9=- P 2 Volume VI = (frustum) = .333b n(e2 + ea + a2) V2 = (spherical segment) = nc2(L - c/3) V3 = (solid of revolution) 120r~nsin 6 cos 6 + a6n2r2 - 90 Total volume: VI + V2 + V3 Table 1 Partial Volumes Type Volume to Ht Volume to Hb Volume to h u D2Ht u DHE 'K h2(l .5D - h) 6 Hemi - 9 -[ 4 3D4 2 2:l S.E. 100%-6% F & D rDH2 1 [ 21 'K h2(l .5D - h) 12 D is in it. Table 2 General Data Depth of Points on Heads C.G. - m Type Surface Area Volume Empty Full Head, d X- Y= m Hemi u D2/2 u D3/12 .2878D .375D .5D 2:l S.E. 1.084 D! R DV24 .1439D .1875D .25D .5 ,/D2 - 16Y2 .25 100%-6% F&D .9286 D2 .0847D3 .100D .169 D D is in ft. Notes 1. Developed length of flat plate (diameter) D.L.=2 - rr+2 - nL+2f Go) (1;o) 2. For 2: 1 S.E. heads the crown and knuckle radius may be approximated as follows: L = .9045 D r = .1727 D 220 Rules of Thumb for Mechanical Engineers 3. Conversion factors Multiply ft3 x 7.48 to get gallons Multiply ft3 x 62.39 to get lb-water Multiply gallons x 8.33 to get lb-water 4. Depth of head A=L-r B=R-r d=L-dm Volumes and Surface Areas of Vessel Sections Motation 1 = height of cone, depth of head, or length of cylinder a = one-half apex angle of cone D = large diameter of cone, diameter of head or cylinder R = radius r = knuckle radius of F & D head L = crown radius of F & D head h = partial depth of horizontal cylinder K, C = coefficients d = small diameter of truncated cone V = volume e= I /: Table 1 Volumes and Surface Areas of Vessel Sections ~~ Section Volume Surface Area r D3 r D' - Sphere 6 Hemi-head u D3 12 - r D3 24 - 2:l S.E. head Ellipsoidal head r D2e 6 - u D2 2 - 1.084 D2 RP l+e 2 r R2 + - In - e 1-e 100-6% .OB467 D3 .9286 D2 F & D head F & D head 2 R R3K 3 hlD C Cone u D'e - 12 r De 2coscu Truncated cone r e(D' + Dd + d2) 12 30° Truncated .227(D3 - d3) 1.57 (D' - d2) cone Table 2 Values of c for Partial Volumes of a Horizontal Cylinder R-h R 8 = arc cos - or V = uR2t'c Figure 1. Formulas for partial volumes of a horizontal cylinder. .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65 .7 .75 .8 .85 .9 .95 .0524 0941 . 1 424 .1955 .2523 .3119 3735 A364 .5 5636 .6265 .6881 .7477 b045 .8576 .9059 .9480 .9813 Piping and Pressure Vessels 221 Maximum length of Unstiffened Shells Thickness (in.) Diameter 1/4 5/16 3/83 7/16 '12 9/16 5/s 11h6 3/4 13/16 7/83 15/36 1 Ilh6 1118 13/16 fin.) 36 42 48 54 60 66 72 78 84 90 96 102 108 114 1 20 126 132 138 144 150 156 162 204 OD 168 31 3 1 42 264 122 228 104 200 91 1 74 79 152 70 136 63 123 57 112 52 103 48 94 44 87 42 79 39 74 37 69 35 65 33 62 31 59 280 00 235 437 203 377 178 330 157 293 138 263 124 237 110 212 99 190 90 173 82 160 76 148 70 138 65 1 28 61 120 57 113 54 106 51 98 49 92 46 87 44 358 00 306 W 268 499 238 442 21 3 396 193 359 1 75 327 157 300 143 274 130 249 118 228 109 21 1 101 197 95 184 88 173 83 163 78 154 74 146 70 138 67 437 38 1 OD 336 626 302 561 273 508 249 462 228 424 210 391 190 363 1 76 337 162 31 1 1 49 287 138 266 1 29 248 121 234 114 221 107 209 101 199 96 458 00 408 03 369 686 336 625 308 573 284 528 263 490 245 456 223 426 209 400 195 374 181 348 169 325 158 304 148 286 1 40 271 133 537 483 03 438 81 6 402 748 370 689 343 639 320 594 299 555 280 521 263 490 242 462 228 437 214 41 1 201 385 189 363 1 78 616 559 51 0 01 00 470 875 435 81 0 405 754 379 705 355 660 334 621 31 5 586 297 555 275 526 261 499 248 475 233 Motes: 1. All values are in in. 2. Values are for temperatures up to 500OF. 3. Top value is for full vacuum, lower value is half vacuum. 4. Values are for carbon or low alloy steel (Fy > 30,000 psi) based on Figure UCS 28.2 of ASME Code, Section VIII, Div. 1. I 637 585 715 00 540 661 795 1.005 m 502 613 738 875 935 m 469 874 440 819 571 687 816 1,064 m 536 642 762 894 997 m 414 504 603 715 839 974 770 938 1,124 391 475 569 673 789 916 1,053 727 884 1,060 1,253 m 369 687 350 652 332 619 309 590 294 449 836 426 793 405 753 385 71 7 367 538 1,002 510 950 485 902 462 859 440 636 744 1,185 m 603 705 1,123 1,312 573 669 1,066 1,246 546 637 1,015 1,186 520 608 864 994 817 940 1,073 774 891 1,017 1,152 737 846 966 1,095 703 806 919 1,042 03 1,442 m 1,373 a, 562 684 819 968 1.131 1.309 1,509 m 83 131 189 258 342 448 114 5/16 3/e 7/16 lk? 9/16 % 11h6 3/4 13/16 7/s 1%6 1 11/16 1% 13/16 222 Rules of Thumb for Mechanical Engineers Useful Formulas for Vessels C2,61 1. Properties of a circle. (See Figure 1 .) C. G. of area c3 e, =- 12A, 120c e2 =- ax 3a.i97(~~ -r3)sin$/2 (R2 - r2)$/2 e, = Chord, C. C = 2R sin 8/2 C = 2 42bR - b2 Rise, b. b = .5C tan 8/4 b = R - .5J4R2 - C2 Angle, 6. C 8 = 2 arc sin - 2R Area of sections 8xR2 - 180C(R - b) A, = 360 xR2a 360 (R2 - r2)n$ 360 A,=- A, = 2. Properties of a cylinder. Cross-sectional metal area, A A = 2xRmt Section modulus, Z. Z = xRkt - nDkt 4 - x(D4 - d4) - 32d Circular ring Figure 1. Dimensions and areas of circular sections. Moment of inertia, I. I = nRit xDkt - a - - x(D4 - d4) 64 3. Radial displacements due to internal pressure. Cylinder. (1 - .5v) PR2 6=- Et Cone. (1 - SV) PR2 Et cos a 6= Piping and Pressure Vessels 223 Spherehemisphere. Torisphericdellipsoidal. R E 6 = - (oq) - vox) where P = internal pressure, psi R = inside radius, in. t = thickness, in. v = Poisson’s ratio (.3 for steel) E = modulus of elasticity, psi a = % apex angle of cone, degrees o$ = circumferential stress, psi ox = meridional stress, psi 4. Longitudinal stress in a cylinder due to longitudinal bending moment, ML. Tension. Compression. ML 6, = (-) - zR2t where E =joint efficiency R = inside radius, in. ML = bending moment, in lb t = thickness, in. 5. Thickness required for heads due to external pressure. L t, =- l/g where L = crown radius, in. P, = external pressure, psi E = modulus of elasticity, psi 6. Equivalent pressure of flanged connection under ex- ternal loads. 16M 4F +p P,=-+- nG3 nG2 where P = internal pressure, psi F = radial load, lb M = bending moment, in lb G = gasket reaction diameter, in. 7. Bending ratio of formed plates. % = lOOt [ 1 - 2) R f where Rf = finished radius, in. R,, = starting radius, in. (= for flat plates) 8. Stress in nozzle neck subjected to external loads. t = thickness, in. PRm +-+- F MR, 2t, A I ox =- where R, = nozzle mean radius, in. t,, = nozzle neck thickness, in. A = metal cross-sectional area, in.2 I = moment of inertia, in? F = radial load, lb M = moment, in lb P = internal pressure, psi 9. circumferential bending stress for out of round shells [2]. D, - D2 > l%D,,, R, = D, +DZ 2 R, = Dl +Dz +t 4 2 ob = 1.5PR1t(D1 - D2) where D1 = maximum inside diameter, in. D2 = minimum inside diameter, in. P = internal pressure, psi E = modulus of elasticity, psi t = thickness, in. Figure 2. Typical nozzle configuration with internal baffle. [...]... that: and 234 Rules of Thumb for Mechanical Engineers CJ= 1.2Ra for most surfaces Qpical RMS values for finishingprocesses are given in Table 1 Of course, widely different surfaces could give the same R,and RMS values The type of statistical quantity needed will depend on the application One quantity that is used in practice is the bearing area curve which is a plot of the surh face area of the surface... Equations 9 and 10 Note that for contact of crossed cylinders of radii R1and R ,respectively, the effective radii are given by: 2 1 -=R’ 1 1 +-= R, 1 -=-+-1 R” Figure 7 Subsurface stresses induced by circular point contact (v = 0.3) (18) = 1 R, so that if Rl = R2, the contact patch is circular and the equations of the previous section (“Contact of Spheres”)hold 232 Rules o Thumb for Mechanical Engineers. .. in Figure 5 The load is P, the total approach is 6, and the radius of contact is a Geometric considerations very similar to those for the contactingcylinders reveal that the sum of the displacements for the two spheres should satisfy: 230 Rules of Thumb for Mechanical Engineers Po=(=) 6PE*2 113 These equations describe Hertz contact for spheres Notice that these results are nonlinear and that the maximum... wear configurations that can be used to obtain wear coefficients and compare material choices for a particular design A primary source for this information is the Wear Contr-olHandbook by Peterson and Winer [121 236 Rules of Thumb for Mechanical Engineers junctions fail just below the surface leading to a wear particle results in: Table 3 Friction and Wear from Pin on Ring Tests Materials 1 Mild steel... presents a brief discussion of mechanical Vibrations and its associated terminology Its mainemphasis is to provide practical rules of thumb to help calculate, measure, and analyze vibration frequencies of mechanical systems Tables are provided with useful formulas for computing the vibration frequencies of common me- 239 chanical systems Additional tables are provided for use with vibration measurements... in the proceedings of the International Conference on Wear of Materials, which is sponsored by the American Society of Mechanical Engineers Lubrication is the effect of a third body on the contacting bodies The third body may be a lubricatingoil ur a chemically formed layer fiom one or both of the contacting bodies (oxides) In general, the coefficient of friction in the presence of lubrication is reduced... function of height If t e surface does not deform during contact, then the bearing area curve is the relationshipbetween actual area of contact and approach of the two surfaces This concept leads to discussion of contact of actual rough surface contacts (See also Figure 10. ) Distancealong surface (mm) Figure 10 A typical rough surface Contact of Rough Surfaces Much can be gleaned from consideration of the... impossible to estimate p for two materials without performing an experiment As a point of reference, p is tabulated for several everyday circumstances in Table 2 As noted in the rough surface contact section, the real area of contact is invariably smaller thanthe apparent area of contact The most common model of friction is based on assuming that the patches of real contact area form junctions in which... One Degree of Freedom System Solving Multiple Degree of Freedom Systems Vibration Measurements and Instrumentation Table A: Spring Stiffness Table B: Natural Frequencies of Simple Systems Table C: Longitudinal and Torsional Vibration of Uniform Beams Table D: Bending (Transverse) Vibration of Uniform Beams , Commercial Engines, Allison Engine Company, Table E: Natural Frequencies of Multiple DOF Systems... adhesion Assuming that some of the real contact area where V is the volume of material removed, W is the normal load, s is the horizontal distance traveled, H is the hardness of the softer material, and k is the dimensionless wear coefficient repsenting the probability that a junction will form a wear particle In this form, wear coefficients vary from - I@ to There is a wealth of information on wear coefficients . Full(RT-1) 1.0 100 % 1.0 100 % 1.0 100 % 1.0 100 0/, 1.0 100 % 1.0 100 % 1.0 100 % 1.0 100 % Spot (RT-3) 1.0 85% 1.0 85% 1.0 85% .85 100 % .85 100 % 1.0 85% .85 100 % .85 100 % Combinationt. Combinationt 1.0 100 % 1.0 100 % 1.0 100 % .85 100 % 1.0 100 % 1.0 100 940 1.0 100 % .85 100 % None 1.0 80% 1.0 80% 1.0 80% .7 100 % .7 100 % 1.0 80% .7 100 % .7 100 % *See Note. ellipsoidal head for- mulas for stress are outlined in the following form. Figure I. Direction of stresses in a vessel head. 21 6 ax Rules of Thumb for Mechanical Engineers 9