T H I R D E D I T I O N
L
I L'lll!lI
I.1
THIRD EDITION
PRESSUREVESSEL
DESIGNMANUAL
THIRD EDITION
PRESSUREVESSEL
DESIGNMANUAL
Illustrated
procedures
for solving
majorpressure
vessel design
problems
DENNISR....
Gulf Professional Publishing is an imprint of EIsevier
200 Wheeler Road, Burlington, MA 01803, USA
Linacre House, Jordan H...
Contents
PREFACE, ix
CHAPTER 1
STRESSES IN PRESSURE VESSELS, 1
Design Philosophy, 1
Stress Analysis, 1
Stress/Failure Theo...
vi Pressure Vessel Design Manual
CHAPTER 3
DESIGN OF VESSEL SUPPORTS, 109
Support Structures, 109
Procedure 3-1: Wind Desi...
Contents vii
Procedure 6-3: Design of Square and Rectangular Platforms, 304
Procedure 6-4: Design of Pipe Supports, 309
Pr...
Preface
Designers of pressure vessels and related equipment frequently have design infor-
mation scattered among numerous ...
Cover Photo: Photo courtesy of Irving Oil Ltd., Saint John, New Brunswick,
Canada and Stone and Webster, Inc., A Shaw Grou...
1
Stresses in PressureVessels
DESIGN PHILOSOPHY
In general, pressure vessels designed in accordance with
the ASME Code, Se...
2 Pressure Vessel Design Manual
effects long or short term? Do they apply to a localized
portion of the vessel or are they...
Stresses in Pressure Vessels 3
material depends only on the numerical magnitude of the
maximum principal or normal stress....
4 Pressure Vessel Design Manual
,-Failuresurface (yield surface boundary)
t O1
P
Figure 1-2. Graph of maximum shear stress...
Stresses in Pressure Vessels 5
For thick-walled vessels (R,,,/t < lo), the radial stress
becomes significant in defining t...
6 Pressure Vessel Design Manual
LOADINGS
Loadings or forces are the “causes” of stresses in pres-
sure vessels. These forc...
Stresses in Pressure Vessels 7
Loadings can be outlined as follows:
IA. Categories of loadings
1. General loads-Applied mo...
8 Pressure Vessel Design Manual
Primary plus secondary stresses are allowed to be higher
and so on. Before considering the...
Stresses in Pressure Vessels 9
stresses have some self-limiting characteristics like secondary
stresses. Since they are lo...
10 Pressure Vessel Design Manual
allowable stress (see Table 1-1). The combined classes of
stress due to a combination of ...
Stresses in Pressure Vessels 11
and then the core is expanded hydraulically. The
core is stressed into plastic range but b...
12 Pressure Vessel Design Manual
Thermal stresses are “secondary stresses” because they
are self-limiting. That is, yieldi...
Stresses in Pressure Vessels 13
very high magnitude. Discontinuity stresses are “secondary
stresses” and are self-limiting...
14 Pressure Vessel Design Manual
REFERENCES-
1.
2.
3.
4.
5.
6.
ASME Boiler and Pressure Vessel Code, Section VIII,
Divisio...
4
General Design
PROCEDURE 2-1
GENERALVESSEL FORMULAS 11, 21
Notation
P =internal pressure, psi
D,, D, =insidehtside diame...
9
5
$Table 2-1 v)
tD
v)
v)
0
<
GeneralVessel Formulas 2
0
tD
E.
(0
3
zm
Stress, SThickness, t Pressure, P
I.D. OB. I.D. 0....
General Design 17
1
3
16
7
8
-
1_3
16
-3
4
1!
16
-5
8
9
1%
1
2
-
7
1%
3
8
-
5
i6
1
4
3
1%
1
8
-
1.219
1.156
1.094
1.031
0....
18 PressureVessel Design Manual
WWm 0 N d W
0
2 9 2 ? 7
W
7
m
Vessel Diameter, Inches
Figure 2-la. (Continued)
dWNb0WWdWmdN
General Design 19
PROCEDURE 2-2
EXTERNAL PRESSURE DESIGN
Notation
~~ ~
A =factor “A,” strain, from ASME Section TI, Part
A...
20 Pressure Vessel Design Manual
possible, stiffeners should always clear shell nozzles. If una-
voidable, special attenti...
Case A
-4-
L
Portionpf a cone
f
General Design 21
I
1
.kj$r2
Dss
I
1
Case B Case c I
Case D Case E
Figure 2-1b. External p...
22 Pressure Vessel Design Manual
Large End Small End
Figure 2-lc. Combined shelkone
Design stiffener for large end of cone...
General Design 23
Figure 2-le. Geometric chart for componentsunder external or compressiveloadings (for all materials). (R...
Design Procedure For Spheres and Heads
Step 1:Assume a thickness and calculate Factor “A.”
0.125
A = -
Ro
Step 2: Find Fac...
General Design 25
0.0625E
P, = ~A to left of material line
Bt
P. - -
- R,,
A to right of material line
Notes
1. As an alte...
26 Pressure Vessel Design Manual
Table 2-1b
Moment of Inertia of Bar Stiffeners
Thk Max, Height, h, in.
t, in. ht, in.
1 1...
General Design 27
Table 2-lc
Moment of Inertia of Composite Stiffeners
-I
- f2
H
1
tiH3
11 =-
12
Iz ==-
12
1
1
2
3
4
5
6
7...
28 Pressure Vessel Design Manual
As
E=modulus of
elasticity
STIFFENING RING CHECK FOR EXTERNAL PRESSURE
If B> 2,500 psi,
d...
General Design 29
stress value at each temperature would be used. When more
than one set of conditions is specified for a ...
30 PressureVessel Design Manual
PROCEDURE 2-4
STRESSES IN HEADS DUE TO INTERNAL PRESSURE [e,31
Notation
L =crown radius, i...
General Design 31
U X
I TORISPHERICAL HEADS I
U
&
K
In Crown I
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
.5 .57 .65 .73 ...
32 Pressure Vessel Design Manual
0
-k
Figure24. Dimensional data for an intermediate head.
Seal OF
A, = t, + lesser of t2 ...
General Design 33
0 External pressure, P,. Assume corroded head thickness, th
0.125th
Factor A = ____
L
Factor B can be ta...
34 Pressure Vessel Design Manual
Calculating Angle, cc
T.L.
T.L.
tI
$
I
B
I
I p& - 0'
L o>
~~
Case 1
0 > 0'
Step 1:
Step 2...
General Design 35
Dimensional Formulas
DI D - 2(R - R COS^)
DI
L, =-
2 cos a
Large End (Figure2-7)
Figure 2-7. Dimensional...
36 Pressure Vessel Design Manual
Figure 2-8. Dimensional data for the small end of a conical transition.
Circumferential s...
General Design 37
ASME Code [l]. This paragraph requires a discontinu-
ity analysis of the cone-shell juncture.
3. No rein...
38 Pressure Vessel Design Manual
_ _ _ _ _ _ _ _ _ _ _ _ ~
Formulas
C - dia. HD
2
hD =
C - dia. HT
2
hT =
C - G
hc =-
2
ho...
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Transcripts - Pressure vessel design manual d moss - 3 ed

  • 1. T H I R D E D I T I O N L I L'lll!lI I.1
  • 2. THIRD EDITION PRESSUREVESSEL DESIGNMANUAL
  • 3. THIRD EDITION PRESSUREVESSEL DESIGNMANUAL Illustrated procedures for solving majorpressure vessel design problems DENNISR. MOSS AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK *OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO G p Gulf Professional +P @ Publishing ELSEVIER Gulf Professional Publishingis an imprintof Elsevier
  • 4. Gulf Professional Publishing is an imprint of EIsevier 200 Wheeler Road, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright 02004, Elsevier, Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier,corn.uk. You may also complete your request online via the Elsevier Science homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” 00 Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible. Library of CongressCataloging-in-PublicationData Moss, Dennis R. vessel design problems/Dennis R. Moss.-3rd ed. ISBN 0-7506-7740-6(hardcover: alk. paper) Pressure vessel design manual: illustrated procedures for solving major pressure p. cm. 1. Pressure vessels-Design and construction-Handbooks, manuals, etc. I. Title. TA660.T34M68 2003 681’.76041 4 ~ 2 2 2003022552 British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library. ISBN: 0-7506-7740-6 For information on all Gulf Professional Publishing publications visit our website at www.gulfpp.com 0405060708 10 11 9 8 7 6 5 4 3 2 1 Printed in the United States of America
  • 5. Contents PREFACE, ix CHAPTER 1 STRESSES IN PRESSURE VESSELS, 1 Design Philosophy, 1 Stress Analysis, 1 Stress/Failure Theories, 2 Failures in Pressure Vessels, 5 L,oadings, 6 Stress, 7 Special Problems, 10 References, 14 CHAPTER 2 GENERAL DESIGN, 15 Procedure 2-1: General Vessel Formulas, 15 Procedure 2-2: External Pressure Design, 19 Procedure 2-3: Calculate MAP, MAWP, and Test Pressures, 28 Procediire 2-4: Stresses in Heads Due to Internal Pressure, 30 Procedure 2-5: Design of Intermediate Heads, 31 Procedure 2-6: Design of Toriconical Transitions, 33 Procedure 2-7: Design of Flanges, 37 Procedure 2-8: Design of Spherically Dished Covers, 57 Procediire 2-9: Design of Blind Flanges with Openings, 58 Procedure 2-10: Bolt Torque Required for Sealing Flanges, 59 Procedure 2-11: Design of Flat Heads, 62 Procedure 2-12: Reinforcement for Studding Outlets, 68 Procedure 2-13: Design of Internal Support Beds, 69 Procedure 2-14: Nozzle Reinforcement, 74 Procedure 2-15: Design of Large Openings in Flat Heads, 78 Procedure 2-16: Find or Revise the Center of Gravity of a Vessel, 80 Procedure 2-17: Minimum Design Metal Temperature (MDMT),81 Procedure 2-18: Buckling of Thin-Walled Cylindrical Shells, 8.5 Procedure 2-19: Optimum Vessel Proportions, 89 Procedure 2-20: Estimating Weights of Vessels and Vessel Components, 95 References. 106 V
  • 6. vi Pressure Vessel Design Manual CHAPTER 3 DESIGN OF VESSEL SUPPORTS, 109 Support Structures, 109 Procedure 3-1: Wind Design per ASCE, 112 Procedure 3-2: Wind Design per UBC-97, 118 Procedure 3-3: Seismic Design for Vessels, 120 Procedure 3-4: Seismic Design-Vessel on Unbraced Legs, 125 Procedure 3-5: Seismic Design-Vessel on Braced Legs, 132 Procedure 3-6: Seismic Design-Vessel on Rings, 140 Procedure 3-7: Seismic Design-Vessel on Lugs #1, 145 Procedure 3-8: Seismic Design-Vessel on Lugs #2, 151 Procedure 3-9: Seismic Design-Vessel on Skirt, 157 Procedure 3-10: Design of Horizontal Vessel on Saddles, 166 Procedure 3-11: Design of Saddle Supports for Large Vessels, 177 Procedure 3-12: Design of Base Plates for Legs, 184 Procedure 3-13: Design of Lug Supports, 188 Procedure 3-14: Design of Base Details for Vertical Vessels #1, 192 Procedure 3-15: Design of Base Details for Vertical Vessels #2, 200 References, 202 CHAPTER 4 SPECIAL DESIGNS, 203 Procedure 4-1: Design of Large-Diameter Nozzle Openings, 203 Procedure 4-2: Design of Cone-Cylinder Intersections, 208 Procedure 4-3: Stresses at Circumferential Ring Stiffeners, 216 Procedure 4-4: Tower Deflection, 219 Procedure 4-5: Design of Ring Girders, 222 Procedure 4-6: Design of Baffles, 227 Procedure 4-7: Design of Vessels with Refractory Linings, 237 Procedure 4-8: Vibration of Tall Towers and Stacks, 244 References, 254 CHAPTER 5 LOCAL LOADS, 255 Procedure 5-1: Stresses in Circular Rings, 256 Procedure 5-2: Design of Partial Ring Stiffeners, 265 Procedure 5-3:Attachment Parameters, 267 Procedure 5-4: Stresses in Cylindrical Shells from External Local Loads, 269 Procedure 5-5: Stresses in Spherical Shells from External Local Loads, 283 References, 290 CHAPTER 6 RELATED EQUIPMENT, 291 Procedure 6-1: Design of Davits, 291 Procedure 6-2: Design of Circular Platforms, 296
  • 7. Contents vii Procedure 6-3: Design of Square and Rectangular Platforms, 304 Procedure 6-4: Design of Pipe Supports, 309 Procedure 6-5: Shear Loads in Bolted Connections, 317 Procedure 6-6: Design of Bins and Elevated Tanks, 318 Procedure 6-7: AgitatordMixers for Vessels and Tanks, 328 Procedure 6-8: Design of Pipe Coils for Heat Transfer, 335 Procedure 6-9: Field-Fabricated Spheres, 355 References, 364 CHAPTER 7 TRANSPORTATION AND ERECTION OF PRESSURE VESSELS, 365 Procedure 7-1: Transportation of Pressure Vessels, 365 Procedure 7-2: Erection of Pressure Vessels, 387 Procedure 7-3: Lifting Attachments and Terminology, 391 Procedure 7-4: Lifting Loads and Forces, 400 Procedure 7-5: Design of Tail Beams, Lugs, and Base Ring Details, 406 Procedure 7-6: Design of Top Head and Cone Lifting Lugs, 416 Procedure 7-7: Design of Flange Lugs, 420 Procedure 7-8: Design of Trunnions, 431 Procedure 7-9: Local Loads in Shell Due to Erection Forces, 434 Procedure 7-10: Miscellaneous,437 APPENDICES, 443 Appendix A: Appendix B: Appendix C: Appendix D: Appendix E: Appendix F: Appendix G: Appendix H: Appendix I: Appendix J: Appendix K: Appendix L: Appendix M: Appendix N: Appendix 0: Appendix P: Appendix Q: Appendix R: Guide to ASME Section VIII, Division 1,443 Design Data Sheet for Vessels, 444 Joint Efficiencies (ASME Code), 445 Properties of Heads, 447 Volumes and Surface Areas of Vessel Sections, 448 Vessel Nomenclature, 455 Useful Formulas for Vessels, 459 Material Selection Guide, 464 Summary of Requirements for 100% X-Ray and PWHT, 465 Material Properties, 466 Metric Conversions, 474 Allowable Compressive Stress for Columns, FA,475 Design of Flat Plates, 478 External Insulation for Vertical Vessels, 480 Flow over Weirs, 482 Time Required to Drain Vessels, 483 Vessel Surge Capacities and Hold-Up Times, 485 Minor Defect Evaluation Procedure, 486 References, 487 Index, 489
  • 8. Preface Designers of pressure vessels and related equipment frequently have design infor- mation scattered among numerous books, periodicals, journals, and old notes. Then, when faced with a particular problem, they spend hours researching its solution only to discover the execution may have been rather simple. This book can eliminate those hours of research by probiding a step-by-step approach to the problems most fre- quently encountered in the design of pressure vessels. This book makes no claim to originality other than that of format. The material is organized in the most concise and functionally useful manner. Whenever possible, credit has been given to the original sources. Although eve^ effort has been made to obtain the most accurate data and solutions, it is the nature of engineering that certain simplifying assumptions be made. Solutions achie7ed should be viewed in this light, and where judgments are required, they should be made with due consideration. Many experienced designers will have already performed many of the calculations outlined in this book, but will find the approach slightly different. All procedures have been developed and proven, using actual design problems. The procedures are easily repeatable to ensure consistency of execution. They also can be modified to incorpo- rate changes in codes, standards, contracts, or local requirements. Everything required for the solution of an individual problem is contained in the procedure. This book may be used directly to solve problems, as a guideline, as a logical approach to problems, or as a check to alternative design methods. If more detailed solutions are required, the approach shown can be amplified where required. The user of this book should be advised that any code formulas or references should always be checked against the latest editions of codes, Le., ASME Section VIII, Division 1, Uniform Building Code, arid ASCE 7-95. These codes are continually updated and revised to incorporate the latest available data. 1am grateful to all those who have contributed information and advice to make this book possible, and invite any suggestions readers may make concerning corrections or additions. Dennis H. Moss ix
  • 9. Cover Photo: Photo courtesy of Irving Oil Ltd., Saint John, New Brunswick, Canada and Stone and Webster, Inc., A Shaw Group Company, Houston, Texas. The photo shows the Reactor-Regenerator Structure of the Converter Section of the RFCC (Resid Fluid Catalytic Cracking) Unit. This “world class” unit operates at the Irving Refinery Complex in Saint John, New Brunswick, Canada, and is a proprietary process of Stone and Webster.
  • 10. 1 Stresses in PressureVessels DESIGN PHILOSOPHY In general, pressure vessels designed in accordance with the ASME Code, Section VIII, Division 1,are designed by rules and do not require a detailed evaluation of all stresses. It is recognized that high localized and secondary bending stresses may exist but are allowed for by use of a higher safety factor and design rules for details. It is required, how- ever, that all loadings (the forces applied to a vessel or its structural attachments)must be considered. (SeeReference 1, Para. UG-22.) While the Code gives formulas for thickness and stress of basic components, it is up to the designer to select appro- priate analytical procedures for determining stress due to other loadings. The designer must also select the most prob- able combination of simultaneous loads for an economical and safe design. The Code establishes allowable stresses by stating in Para. UG-23(c) that the maximum general primary membrane stress must be less than allowablestresses outlined in material sections. Further, it states that the maximum primary mem- brane stress plus primary bending stress may not exceed 1.5 times the allowable stress of the material sections. In other sections, specifically Paras. 1-5(e) and 2-8, higher allowable stresses are permitted if appropriate analysis is made. These higher allowable stresses clearly indicate that different stress levels for different stress categories are acceptable. It is general practice when doing more detailed stress analysis to apply higher allowable stresses. In effect, the detailed evaluation of stresses permits substituting knowl- edge of localized stresses and the use of higher allowables in place of the larger factor of safety used by the Code. This higher safety factor really reflected lack of knowledge about actual stresses. A calculated value of stress means little until it is associ- ated with its location and distribution in the vessel and with the type of loading that produced it. Different types of stress have different degrees of significance. The designer must familiarize himself with the various types of stress and loadings in order to accurately apply the results of analysis. The designer must also consider some adequate stress or failure theory in order to combine stresses and set allowable stress limits. It is against this fail- ure mode that he must compare and interpret stress values, and define how the stresses in a component react and con- tribute to the strength of that part. The following sections will provide the fundamental knowledge for applying the results of analysis. The topics covered in Chapter 1 form the basis by which the rest of the book is to be used. A section on special problems and considerations is included to alert the designer to more com- plex problems that exist. STRESS ANALYSIS Stress analysis is the determination of the relationship between external forces applied to a vessel and the corre- sponding stress. The emphasis of this book is not how to do stress analysis in particular, but rather how to analyze vessels and their component parts in an effort to arrive at an economical and safe design-the rllfference being that we analyze stresses where necessary to determine thickness of material and sizes of members. We are not so concerned with building mathematical models as with providing a step-by-step approach to the design of ASME Code vessels. It is not necessary to find every stress but rather to know the governing stresses and how they relate to the vessel or its respective parts, attachments, and supports. The starting place for stress analysis is to determine all the design conditions for a gven problem and then deter- mine all the related external forces. We must then relate these external forces to the vessel parts which must resist them to find the corresponding stresses. By isolating the causes (loadings),the effects (stress)can be more accurately determined. The designer must also be keenly aware of the types of loads and how they relate to the vessel as a whole. Are the 1
  • 11. 2 Pressure Vessel Design Manual effects long or short term? Do they apply to a localized portion of the vessel or are they uniform throughout? How these stresses are interpreted and combined, what significance they have to the overall safety of the vessel, and what allowable stresses are applied will be determined by three things: 1. The strengtwfailure theory utilized. 2. The types and categories of loadings. 3. The hazard the stress represents to the vessel. Membrane Stress Analysis Pressure vessels commonly have the form of spheres, cylinders, cones, ellipsoids, tori, or composites of these. When the thickness is small in comparison with other &men- sions (RJt > lo),vessels are referred to as membranes and the associated stresses resulting from the contained pressure are called membrane stresses. These membrane stresses are average tension or compression stresses. They are assumed to be uniform across the vessel wall and act tangentially to its surface. The membrane or wall is assumed to offer no resis- tance to bending. When the wall offers resistance to bend- ing, bending stresses occur in addtion to membrane stresses. In a vessel of complicated shape subjected to internal pressure, the simple membrane-stress concepts do not suf- fice to give an adequate idea of the true stress situation. The types of heads closing the vessel, effects of supports, varia- tions in thickness and cross section, nozzles, external at- tachments, and overall bending due to weight, wind, and seismic activity all cause varying stress distributions in the vessel. Deviations from a true membrane shape set up bend- ing in the vessel wall and cause the direct loading to vary from point to point. The direct loading is diverted from the more flexible to the more rigid portions of the vessel. This effect is called “stress redistribution.” In any pressure vessel subjected to internal or external pressure, stresses are set up in the shell wall. The state of stress is triaxial and the three principal stresses are: ox= 1ongitudmaVmeridionalstress 04 = circumferentialAatitudina1 stress or= radial stress In addition, there may be bending and shear stresses. The radial stress is a direct stress, which is a result of the pressure acting directly on the wall, and causes a compressive stress equal to the pressure. In thin-walled vessels this stress is so small compared to the other “principal” stresses that it is generally ignored. Thus we assume for purposes of analysis that the state of stress is biaxial. This greatly simplifies the method of combining stresses in comparison to triaxial stress states. For thickwalled vessels (RJt < lo), the radial stress cannot be ignored and formulas are quite different from those used in finding “membrane stresses” in thin shells. Since ASME Code, Section VIII, Division 1,is basically for design by rules, a higher factor of safety is used to allow for the “unknown” stresses in the vessel. This higher safety factor, which allows for these unknown stresses, can impose a penalty on design but requires much less analysis. The design techniques outlined in this text are a compro- mise between finding all stresses and utilizing minimum code formulas. This additional knowledge of stresses warrants the use of higher allowablestresses in some cases,while meet- ing the requirements that all loadings be considered. In conclusion, “membrane stress analysis’’is not completely accurate but allows certain simplifymg assumptions to be made while maintaining a fair degree of accuracy. The main simplifying assumptions are that the stress is biaxial and that the stresses are uniform across the shell wall. For thin-walled vessels these assumptions have proven themselves to be reliable. No vessel meets the criteria of being a true membrane, but we can use this tool with a reasonable degree of accuracy. STRESS/FAILURETHEORIES As stated previously, stresses are meaningless until com- pared to some stresdfailure theory. The significance of a given stress must be related to its location in the vessel and its bearing on the ultimate failure of that vessel. Historically, various ‘‘theories” have been derived to com- bine and measure stresses against the potential failure mode. A number of stress theories, also called “yield cri- teria,” are available for describing the effects of combined stresses. For purposes of this book, as these failure theories apply to pressure vessels, only two theories will be discussed. They are the “maximum stress theory” and the “maximum shear stress theory.” Maximum Stress Theory This theory is the oldest, most widely used and simplest to apply. Both ASME Code, Section VIII, Division 1, and Section I use the maximum stress theory as a basis for design. This theory simply asserts that the breakdown of
  • 12. Stresses in Pressure Vessels 3 material depends only on the numerical magnitude of the maximum principal or normal stress. Stresses in the other directions are disregarded. Only the maximum principal stress must be determined to apply this criterion. This theory is used for biaxial states of stress assumed in a thin- walled pressure vessel. As will be shown later it is unconser- vative in some instances and requires a higher safety factor for its use. While the maximum stress theory does accurately predict failure in brittle materials, it is not always accurate for ductile materials. Ductile materials often fail along lines 4 5 to the applied force by shearing, long before the tensile or compressive stresses are maximum. This theory can be illustrated graphically for the four states of biaxial stress shown in Figure 1-1. It can be seen that uniaxial tension or compression lies on tlir two axes. Inside the box (outer boundaries) is the elastic range of the material. Yielding is predicted for stress combinations by the outer line. Maximum Shear Stress Theory This theory asserts that the breakdown of material de- pends only on the mdximum shear stress attained in an ele- ment. It assumes that yielding starts in planes of maximum shear stress. According to this theory, yielding will start at a point when the maximum shear stress at that point reaches one-half of the the uniaxial yield strength, F,. Thus for a 9 -1.0 0 1 l+l.o biaxial state of stress where 01 > ( ~ 2 ,the maximum shear stress will be (al- (s2)/2. Yielding will occur when Both ASME Code, Section 1'111, Division 2 and ASME Code, Section 111, utilize the maximum shear stress criterion. This theory closely approximates experimental results and is also easy to use. This theory also applies to triaxial states of stress. In a triaxial stress state, this theory predicts that yielding will occur whenever one-half the algebraic differ- ence between the maximum and minimum 5tress is equal to one-half the yield stress. Where c1> a2> 03,the maximum shear stress is (ul- Yielding will begin when 01 - 0 3 - F, 2 2 This theory is illustrated graphically for the four states of biaxial stress in Figure 1-2. A comparison of Figure 1-1 and Figure 1-2 will quickly illustrate the major differences between the two theories. Figure 1-2 predicts yielding at earlier points in Quadrants I1 and IV. For example, consider point B of Figure 1-2. It shows ~ 2 = ( - ) ( ~ 1 ; therefore the shear stress is equal to c2- (-a1)/2, which equals o2+a1/2 or one-half the stress r Safety factor boundary imposed by ASME Code I / I I _ _ _ _ I IV111 I-'.O tO1 + l . O I Failure surface (yield surface) boundary Figure 1-1. Graph of maximum stress theory. Quadrant I: biaxial tension; Quadrant II: tension: Quadrant Ill: biaxial compression; Quadrant IV: compression.
  • 13. 4 Pressure Vessel Design Manual ,-Failuresurface (yield surface boundary) t O1 P Figure 1-2. Graph of maximum shear stress theory. which would cause yielding as predcted by the maximum stress theory! Comparison of the TwoTheories Both theories are in agreement for uniaxial stress or when one of the principal stresses is large in comparison to the others. The discrepancy between the theories is greatest when both principal stresses are numerically equal. For simple analysis upon which the thickness formulas for ASME Code, Section I or Section VIII, Division 1,are based, it makes little difference whether the maximum stress theory or maximum shear stress theory is used. For example, according to the maximum stress theory, the controlling stress governing the thickness of a cylinder is 04,circumfer- ential stress, since it is the largest of the three principal stresses. Accordmg to the maximum shear stress theory, the controlling stress would be one-half the algebraic differ- ence between the maximum and minimum stress: The maximum stress is the circumferential stress, a4 04 = PR/t 0 The minimum stress is the radial stress, a, a, = -P Therefore, the maximum shear stress is: ASME Code, Section VIII, Division 2, and Section I11 use the term “stress intensity,” which is defined as twice the maximum shear stress. Since the shear stress is compared to one-half the yield stress only, “stress intensity” is used for comparison to allowable stresses or ultimate stresses. To define it another way, yieldmg begins when the “stress in- tensity” exceeds the yield strength of the material. In the preceding example, the “stress intensity” would be equal to 04 - a,.And For a cylinder where P =300 psi, R =30 in., and t =.5 in., the two theories would compare as follows: Maximum stress theory o = a4 = PR/t = 300(30)/.5 = 18,000psi Maximum shear stress the0y a = PR/t +P = 300(30)/.5 +300 = 18,300 psi Two points are obvious from the foregoing: 1. For thin-walled pressure vessels, both theories yield approximately the same results. 2. For thin-walled pressure vessels the radial stress is so small in comparison to the other principal stresses that it can be ignored and a state of biaxial stress is assumed to exist.
  • 14. Stresses in Pressure Vessels 5 For thick-walled vessels (R,,,/t < lo), the radial stress becomes significant in defining the ultimate failure of the vessel. The maximum stress theory is unconservative for designing these vessels. For this reason, this text has limited its application to thin-walled vessels where a biaxial state of stress is assumed to exist. FAILURES IN PRESSURE VESSELS Vessel failures can be grouped into four major categories, which describe why a vessel failure occurs. Failures can also be grouped into types of failures, which describe how the failure occurs. Each failure has a why and how to its history. It may have failed through corrosion fatigue because the wrong material was selected! The designer must be as familiar with categories and types of failure as with cate- gories and types of stress and loadings. Ultimately they are all related. Categories of Failures 1. Material-Improper selection of material; defects in material. 2. Design-Incorrect design data; inaccurate or incor- rect design methods; inadequate shop testing. 3. Fabrication-Poor quality control; improper or insuf- ficient fabrication procedures including welding; heat treatment or forming methods. 4. Seruice-Change of service condition by the user; inexperienced operations or maintenance personnel; upset conditions. Some types of service which require special attention both for selection of material, design details, and fabrication methods are as follows: a. Lethal b. Fatigue (cyclic) c. Brittle (low temperature) d. High temperature e. High shock or vibration f. Vessel contents 0 Hydrogen 0 Ammonia 0 Compressed air 0 Caustic 0 Chlorides Types of Failures 1. Elastic defi,rmation-Elastic instability or elastic buck- ling, vessel geometry, and stiffness as well as properties of materials are protection against buckling. 2. Brittlefracture-Can occur at low or intermediate tem- peratures. Brittle fractures have occurred in vessels made of low carbon steel in the 40’50°F range during hydrotest where minor flaws exist. 3. Excessive plastic deformation-The primary and sec- ondary stress limits as outlined in ASME Section VIII, Division 2, are intended to prevent excessive plas- tic deformation and incremental collapse. 4. Stress rupture-Creep deformation as a result of fa- tigue or cyclic loading, i.e., progressive fracture. Creep is a time-dependent phenomenon, whereas fa- tigue is a cycle-dependent phenomenon. 5. Plastic instability-Incremental collapse; incremental collapse is cyclic strain accumulation or cumulative cyclic deformation. Cumulative damage leads to insta- bility of vessel by plastic deformation. 6. High strain-Low cycle fatigue is strain-governed and occurs mainly in lower-strengthhigh-ductile materials. 7. Stress corrosion-It is well known that chlorides cause stress corrosion cracking in stainless steels; likewise caustic service can cause stress corrosion cracking in carbon steels. Material selection is critical in these services. 8. Corrosion fatigue-Occurs when corrosive and fatigue effects occur simultaneously. Corrosion can reduce fa- tigue life by pitting the surface and propagating cracks. Material selection and fatigue properties are the major considerations. In dealing with these various modes of failure, the de- signer must have at his disposal a picture of the state of stress in the various parts. It is against these failure modes that the designer must compare and interpret stress values. But setting allowable stresses is not enough! For elastic instability one must consider geometry, stiffness, and the properties of the material. Material selection is a major con- sideration when related to the type of service. Design details and fabrication methods are as important as “allowable stress” in design of vessels for cyclic service. The designer and all those persons who ultimately affect the design must have a clear picture of the conditions under which the vessel will operate.
  • 15. 6 Pressure Vessel Design Manual LOADINGS Loadings or forces are the “causes” of stresses in pres- sure vessels. These forces and moments must be isolated both to determine where they apply to the vessel and when they apply to a vessel. Categories of loadings define where these forces are applied. Loadings may be applied over a large portion (general area) of the vessel or over a local area of the vessel. Remember both general and local loads can produce membrane and bending stresses. These stresses are additive and define the overall state of stress in the vessel or component. Stresses from local loads must be added to stresses from general load- ings. These combined stresses are then compared to an allowable stress. Consider a pressurized, vertical vessel bending due to wind, which has an inward radial force applied locally. The effects of the pressure loading are longitudinal and circumferential tension. The effects of the wind loading are longitudinal tension on the windward side and lon- gitudinal compression on the leeward side. The effects of the local inward radial load are some local membrane stres- ses and local bending stresses. The local stresses would be both circumferential and longitudinal, tension on the inside surface of the vessel, and compressive on the outside. Of course the steel at any given point only sees a certain level of stress or the combined effect. It is the designer’s job to combine the stresses from the various loadings to arrive at the worst probable combination of stresses, combine them using some failure theory, and compare the results to an acceptable stress level to obtain an economical and safe design. This hypothetical problem serves to illustrate how cate- gories and types of loadings are related to the stresses they produce. The stresses applied more or less continuously and unqomly across an entire section of the vessel are primary stresses. The stresses due to pressure and wind are primary mem- brane stresses. These stresses should be limited to the code allowable. These stresses would cause the bursting or collapse of the vessel if allowed to reach an unacceptably high level. On the other hand, the stresses from the inward radial load could be either a primary local stress or secondary stress. It is a primary local stress if it is produced from an unrelenting load or a secondary stress if produced by a relenting load. Either stress may cause local deformation but will not in and of itself cause the vessel to fail. If it is a primary stress, the stress will be redistributed; if it is a secondary stress, the load will relax once slight deforma- tion occurs. Also be aware that this is only true for ductile materials. In brittle materials, there would be no difference between primary and secondary stresses. If the material cannot yield to reduce the load, then the definition of secondary stress does not apply! Fortunately current pressure vessel codes require the use of ductile materials. This should make it obvious that the type and category of loading will determine the type and category of stress. This will be expanded upon later, but basically each combina- tion of stresses (stress categories) will have different allow- ables, i.e.: 0 Primary stress: P, < SE 0 Primary membrane local (PL): PL = P, +PL < 1.5SE PL = P,, +Q, < 1.5 SE 0 Primary membrane + secondary (Q): Pm +Q < 3 SE But what if the loading was of relatively short duration? This describes the “type”of loading. Whether a loading is steady, more or less continuous, or nonsteady, variable, or tempo- rary will also have an effect on what level of stress will be acceptable. If in our hypothetical problem the loading had been pressure + seismic + local load, we would have a different case. Due to the relatively short duration of seismic loading, a higher “temporary” allowable stress would be ac- ceptable. The vessel doesn’t have to operate in an earth- quake all the time. On the other hand, it also shouldn’t fall down in the event of an earthquake! Structural designs allow a one-third increase in allowable stress for seismic loadings for this reason. For steady loads, the vessel must support these loads more or less continuously during its useful life. As a result, the stresses produced from these loads must be maintained to an acceptable level. For nonsteady loads, the vessel may experience some or all of these loadings at various times but not all at once and not more or less continuously. Therefore a temporarily higher stress is acceptable. For general loads that apply more or less uniformly across an entire section, the corresponding stresses must be lower, since the entire vessel must support that loading. For local loads, the corresponding stresses are confined to a small portion of the vessel and normally fall off rapidly in distance from the applied load. As discussed previously, pressurizing a vessel causes bending in certain components. But it doesn’t cause the entire vessel to bend. The results are not as significant (except in cyclic service) as those caused by general loadings. Therefore a slightly higher allowable stress would be in order.
  • 16. Stresses in Pressure Vessels 7 Loadings can be outlined as follows: IA. Categories of loadings 1. General loads-Applied more or less continuously across a vessel section. a. Pressure loads-Internal or external pressure (design, operating, hydrotest. and hydrostatic head of liquid). b. Moment loads-Due to wind, seismic, erection, transportation. c. Compressive/tensile loads-Due to dead weight, installed equipment, ladders, platforms, piping, and vessel contents. attachment. d. Thermal loads-Hot box design of skirthead 2. Local loads-Due to reactions from supports, internals, attached piping, attached equipment, Le., platforms, mixers, etc. a. Radial load-Inward or outward. b. Shear load-Longitudinal or circumferential. c. Torsional load. d. Tangential load. e. Moment load-Longitudinal or circumferential. f. Thermal load. B. Typey of loadings 1. Steady load-Long-term duration, continuous. a. InternaVexternal pressure. b Dead weight. c. Vessel contents. d. Loadings due to attached piping and equipment. e. Loadings to and from vessel supports. f. Thermal loads. g. Wind loads. a. Shop and field hydrotests. b. Earthquake. c. Erection. d. Transportation. e. Upset, emergency. f. Thermal loads. g. Start up, shut down. 2. Nonsteady loads-Short-term duration; variable. STRESS ASME Code, SectionVIII, Division 1 vs. Division 2 ~~ ASME Code, Section VIII, Division 1 does not explicitly consider the effects of combined stress. Neither does it give detailed methods on how stresses are combined. ASME Code, Section VIII, Division 2, on the other hand, provides specific guidelines for stresses, how they are combined, and allowable stresses for categories of combined stresses. Division 2 is design by analysis whereas Division 1 is design by rules. Although stress analysis as utilized by Division 2 is beyond the scope of this text, the use of stress categories, definitions of stress, and allowable stresses is applicable. Division 2 stress analysis considers all stresses in a triaxial state combined in accordance with the maximum shear stress theory. Division 1 and the procedures outlined in this book consider a biaxial state of stress combined in accordance with the maximum stress theory. Just as you would not design a nuclear reactor to the niles of Division 1, you would not design an air receiver by the techniques of Division 2. Each has its place and applications. The following discussion on categories of stress and allowables will utilize informa- tion from Division 2, which can be applied in general to all vessels. Types, Classes, and Categories of Stress The shell thickness as computed by Code formulas for internal or external pressure alone is often not sufficient to withstand the combined effects of all other loadings. Detailed calculations consider the effects of each loading separately and then must be combined to give the total state of stress in that part. The stresses that are present in pressure vessels are separated into various cla.~.sr~sin accor- dance with the types of loads that produced them, and the hazard they represent to the vessel. Each class of stress must be maintained at an acceptable leL7eland the combined total stress must be kept at another acceptable level. The combined stresses due to a combination of loads acting simultaneously are called stress categories. Please note that this terminology differs from that given in Dikision 2, but is clearer for the purposes intended herc,. Classes of stress, categories of stress, and allowable stresses are based on the type of loading that produced them and on the hazard they represent to the structure. Unrelenting loads produce primary stresses. Relenting loads (self-limiting) produce secondary stresses. General loadings produce primary membrane and bending stresses. Local loads produce local membrane and bending stresses. Primary stresses must be kept l o ~ e rthan secondary stresses.
  • 17. 8 Pressure Vessel Design Manual Primary plus secondary stresses are allowed to be higher and so on. Before considering the combination of stresses (categories), we must first define the various types and classes of stress. Types of Stress There are many names to describe types of stress. Enough in fact to provide a confusing picture even to the experienced designer. As these stresses apply to pressure vessels, we group all types of stress into three major classes of stress, and subdivision of each of the groups is arranged according to their effect on the vessel. The following list of stresses describes types of stress without regard to their effect on the vessel or component. They define a direction of stress or relate to the application of the load. 1. Tensile 2. Compressive 3. Shear 4. Bending 5. Bearing 6. Axial 7. Discontinuity 8. Membrane 9. Principal 10. Thermal 11. Tangential 12. Load induced 13. Strain induced 14. Circumferential 15. Longitudinal 16. Radial 17. Normal Classes of Stress The foregoing list provides examples of types of stress. It is, however, too general to provide a basis with which to combine stresses or apply allowable stresses. For this purpose, new groupings called classes of stress must be used. Classes of stress are defined by the type of loading which produces them and the hazard they represent to the vessel. 1. Primay stress a. General: 0 Primary general membrane stress, P, 0 Primary general bending stress, Pb b. Primary local stress, PL a. Secondary membrane stress, Q, b. Secondary bending stress, Q b 2. Seconday stress 3. Peak stress, F Definitions and examples of these stresses follow. Primary general stress. These stresses act over a full cross section of the vessel. They are produced by mechanical loads (load induced) and are the most hazardous of all types of stress. The basic characteristic of a primary stress is that it is not self-limiting. Primary stresses are generally due to in- ternal or external pressure or produced by sustained external forces and moments. Thermal stresses are never classified as primary stresses. Primary general stresses are divided into membrane and bending stresses. The need for divilng primary general stress into membrane and bending is that the calculated value of a primary bending stress may be allowed to go higher than that of a primary membrane stress. Primary stresses that exceed the yield strength of the material can cause failure or gross distortion. Typical calculations of primary stress are: TC and - PR F MC Jt ’ A ’ I ’ Primary general membranestress, P,. This stress occurs across the entire cross section of the vessel. It is remote from dis- continuities such as head-shell intersections, cone-cylinder intersections, nozzles, and supports. Examples are: a. Circumferential and longitudmal stress due to pressure. b. Compressive and tensile axial stresses due to wind. c. Longitudinal stress due to the bending of the horizontal vessel over the saddles. d. Membrane stress in the center of the flat head. e. Membrane stress in the nozzle wall within the area of reinforcement due to pressure or external loads. f. Axial compression due to weight. Primary general bending stress, Pb. Primary bending stresses are due to sustained loads and are capable of causing collapse of the vessel. There are relatively few areas where primary bending occurs: a. Bending stress in the center of a flat head or crown of a dished head. b. Bending stress in a shallow conical head. c. Bending stress in the ligaments of closely spaced openings. Local primary membrane stress, PL. Local primary membrane stress is not technically a classificationof stress but a stress category, since it is a combination of two stresses. The combination it represents is primary membrane stress, P,, plus secondary membrane stress, Q,, produced from sus- tained loads. These have been grouped together in order to limit the allowable stress for this particular combination to a level lower than allowed for other primary and secondary stress applications. It was felt that local stress from sustained (unrelenting) loads presented a great enough hazard for the combination to be “classified” as a primary stress. A local primary stress is produced either by design pressure alone or by other mechanical loads. Local primary
  • 18. Stresses in Pressure Vessels 9 stresses have some self-limiting characteristics like secondary stresses. Since they are localized, once the yield strength of the material is reached, the load is redistributed to stiffer portions of the vessel. However, since any deformation associated with yielding would be unacceptable, an allowable stress lower than secondary stresses is assigned. The basic difference between a primary local stress and a secondary stress is that a primary local stress is produced by a load that is unrelenting; the stress is just redistributed. In a secondary stress, yielding relaxes the load and is truly self-limiting. The ability of primary local stresses to redistribute themselves after the yield strength is attained locally provides a safety- valve effect. Thus, the higher allowable stress applies only to a local area. Primary local membrane stresses are a combination of membrane stresses only. Thus only the “membrane” stresses from a local load are combined with primary general membrane stresses, not the bending stresses. The bending stresses associated with a local loading are secondary stresses. Therefore, the membrane stresses from a WRC- 107-type analysis must be broken out separately and com- bined with primary general stresses. The same is true for discontinuity membrane stresses at head-shell junctures, cone-cylinder junctures, and nozzle-shell junctures. The bending stresses would be secondary stresses. Therefore, PL=P, +Qlllrwhere Q,, is a local stress from a sustained or unrelenting load. Examples of primary local membrane stresses are: a. PI,,+membrane stresses at local discontinuities: 1. Head-shell juncture 2. Cone-cylinder juncture 3. Nozzle-shell juncture 4.Shell-flange juncture 5. Head-slurt juncture 6. Shell-stiffening ring juncture b. P,, +membrane stresses from local sustained loads: 1. support lugs 2. Nozzle loads 3. Beam supports 4. Major attachments Secondarystress. The basic characteristic of a second- ary stress is that it is self-limiting. As defined earlier, this means that local yielding and minor distortions can satisfy the conditions which caused the stress to occur. Application of a secondary stress cannot cause structural failure due to the restraints offered by the body to which the part is attached. Secondary mean stresses are developed at the junc- tions of major components of a pressure vessel. Secondary mean stresses are also produced by sustained loads other than internal or external pressure. Radial loads on nozzles produce secondary mean stresses in the shell at the junction of the nozzle. Secondary stresses are strain-induced stresses. Discontinuity stresses are only considered as secondary stresses if their extent along the length of the shell is limited. Division 2 imposes the restriction that the length over which the stress is secondary is m.Beyond this distance, the stresses are considered as primary mean stresses. In a cylin- drical vessel, the length arepresents the length over which the shell behaves as a ring. A further restriction on secondary stresses is that they may not be closer to another gross structural Qscontinuity than a distance of 2 . 5 m . This restriction is to eliminate the additive effects of edge moments and forces. Secondary stresses are divided into two additional groups, membrane and bending. Examples of each are as follows: Seconday membrane stress, Q,,,. a. Axial stress at the juncture of a flange and the hub of the flange. b. Thermal stresses. c. Membrane stress in the knuckle area of the head. d. Membrane stress due to local relenting loads. Secondary bending stress, QL. a. Bending stress at a gross structural discontinuity: b. The nonuniform portion of the stress distribution in a c. The stress variation of the radial stress due to internal d. Discontinuity stresses at stiffening or support rings. nozzles, lugs, etc. (relenting loadings only). thick-walled vessel due to internal pressure. pressure in thick-walled vessels. Note: For b and c it is necessary to subtract out the average stress which is the primary stress. Only the varymg part of the stress distribution is a secondary stress. Peak stress, E Peak stresses are the additional stresses due to stress intensification in highly localized areas. They apply to both sustained loads and self-limiting loads. There are no significant distortions associated with peak stresses. Peak stresses are additive to primary and secondary stresses pre- sent at the point of the stress concentration. Peak stresses are only significant in fatigue conditions or brittle materials. Peak stresses are sources of fatigue cracks and apply to membrane, bending, and shear stresses. Examples are: a. Stress at the corner of a discontinuity. b. Thermal stresses in a wall caused by a sudden change c. Thermal stresses in cladding or weld overlay. d. Stress due to notch effect (stress concentration). in the surface temperature. Categories of Stress Once the various stresses of a component are calculated, they must be combined and this final result compared to an
  • 19. 10 Pressure Vessel Design Manual allowable stress (see Table 1-1). The combined classes of stress due to a combination of loads acting at the same time are stress categories. Each category has assigned limits of stress based on the hazard it represents to the vessel. The following is derived basically from ASME Code, Section VIII, Division 2, simplified for application to Division 1vessels and allowable stresses. It should be used as a guideline only because Division 1 recognizes only two categories of stress-primary membrane stress and primary bending stress. Since the calculations of most secondary (thermal and discontinuities) and peak stresses are not included in this book, these categories can be considered for reference only. In addition, Division 2 utilizes a factor K multiplied by the allowable stress for increase due to short-term loads due to seismic or upset conditions. It also sets allowable limits of combined stress for fatigue loading where secondary and peak stresses are major considerations. Table 1-1sets allowable stresses for both stress classifications and stress categories. Table 1-1 Allowable Stresses for Stress Classifications and Categories Stress Classification or Cateaorv General primary membrane, P, General primary bending, Pb Local primary membrane, PL Secondary membrane, Q, Secondary bending, Qb Peak, F (PL=P, +QmJ p m f Pb +em+Qb pL+ Pb p m +Pb +Q& +Qb Pm +Pb +Q& +Qb +F Allowable Stress SE 1.5SE < .9Fy 1.5SE 4 .9Fy 1.5SE < .9Fy 3SE < 2Fy UTS 3SE < 2Fy < UTS 1.5SE < .9Fy 3SE < 2Fy < UTS 2Sa 2Sa Notes: Q,, =membrane stresses from sustained loads W, =membrane stresses from relenting, self-limiting loads S=allowable stress per ASME Code, Section VIII, Division 1, at design F,= minimum specified yield strength at design temperature UTS=minimum specified tensile strength S,=allowable stress for any given number of cycles from design fatigue curves. temperature SPECIAL PROBLEMS This book provides detailed methods to cover those areas most frequently encountered in pressure vessel design. The topics chosen for this section, while of the utmost interest to the designer, represent problems of a specialized nature. As such, they are presented here for information purposes, and detailed solutions are not provided. The solutions to these special problems are complicated and normally beyond the expertise or available time of the average designer. The designer should be familiar with these topics in order to recognize when special consideration is warranted. If more detailed information is desired, there is a great deal of reference material available, and special references have been included for this purpose. Whenever solutions to prob- lems in any of these areas are required, the design or analysis should be referred to experts in the field who have proven experience in their solution. ~ ~ ~ ~ Thick-WalledPressureVessels As discussed previously, the equations used for design of thin-walled vessels are inadequate for design or prediction of failure of thick-walled vessels where R,,/t < 10. There are many types of vessels in the thick-walled vessel category as outlined in the following, but for purposes of discussion here only the monobloc type will be discussed. Design of thick- wall vessels or cylinders is beyond the scope of this book, but it is hoped that through the following discussion some insight will be gained. In a thick-walled vessel subjected to internal pressure, both circumferential and radlal stresses are maximum on the inside surface. However, failure of the shell does not begin at the bore but in fibers along the outside surface of the shell. Although the fibers on the inside surface do reach yield first they are incapable of failingbecause they are restricted by the outer portions of the shell. Above the elastic-breakdown pres- sure the region of plastic flow or “overstrain” moves radially outward and causes the circumferential stress to reduce at the inner layers and to increase at the outer layers. Thus the maximum hoop stress is reached first at the outside of the cylinder and eventual failure begins there. The major methods for manufacture of thick-walled pressure vessels are as follows: 1. Monobloc-Solid vessel wall. 2. Multilayer-Begins with a core about ‘/z in. thick and successivelayersareapplied. Each layerisvented (except the core) and welded individually with no overlapping welds. 3. Multiwall-Begins with a core about 1%in. to 2 in. thick. Outer layers about the same thickness are suc- cessively “shrunk fit” over the core. This creates com- pressive stress in the core, which is relaxed during pressurization. The process of compressing layers is called autofrettage from the French word meaning “self-hooping.” 4.Multilayer autofirettage-Begins with a core about ‘/z in. thick. Bands or forged rings are slipped outside
  • 20. Stresses in Pressure Vessels 11 and then the core is expanded hydraulically. The core is stressed into plastic range but below ultimate strength. The outer rings are maintained at a margin below yield strength. The elastic deformation resi- dual in the outer bands induces compressive stress in the core, which is relaxed during pressurization. 5. Wire wrapped z)essels--Begin with inner core of thick- ness less than required for pressure. Core is wrapped with steel cables in tension until the desired auto- frettage is achieved. 6. Coil wrapped cessels-Begin with a core that is subse- quently wrapped or coiled with a thin steel sheet until the desired thickness is obtained. Only two longitudinal welds are used, one attaching the sheet to the core and the final closure weld. Vessels 5 to 6ft in diameter for pressures up to 5,OOOpsi have been made in this manner. Other techniques and variations of the foregoing have been used but these represent the major methods. Obviously these vessels are made for very high pressures and are very expensive. For materials such as mild steel, which fail in shear rather than direct tension, the maximum shear theory of failure should be used. For internal pressure only, the maximum shear stress occurs on the inner surface of the cylinder. At this surface both tensile and compressive stresses are max- imum. In a cylinder, the maximum tensile stress is the cir- cumferential stress, 06. The maximum compressive stress is the radial stress, or.These stresses would be computed as follows: Therefore the maximum shear stress, 5 , is [9]: ASME Code, Section VIII, Division 1, has developed alternate equations for thick-walled monobloc vessels. The equations for thickness of cylindrical shells and spherical shells are as follows: 0 Cylindrical shells (Para. 1-2 (a) (1))where t > .5 Ri or P > ,385 SE: S E + P Z=- SE - P A B Figure 1-3. Comparision of stress distribution between thin-walled (A) and thick-walled (B) vessels. 0 Spherical shells (Para. 1-3)where t > ,356 Rior P >.665SE: 2(SE +P) Y = 2SE - P The stress distribution in the vessel wall of a thick-walled vessel varies across the section. This is also true for thin- walled vessels, but for purposes of analysis the stress is considered uniform since the difference between the inner and outer surface is slight. A visual comparison is offered in Figure 1-3. Thermal Stresses Whenever the expansion or contraction that would occur normally as a result of heating or cooling an object is prevented, thermal stresses are developed. The stress is always caused by some form of mechanical restraint.
  • 21. 12 Pressure Vessel Design Manual Thermal stresses are “secondary stresses” because they are self-limiting. That is, yielding or deformation of the part relaxes the stress (except thermal stress ratcheting). Thermal stresses will not cause failure by rupture in ductile materials except by fatigue over repeated applica- tions. They can, however, cause failure due to excessive deformations. Mechanical restraints are either internal or external. External restraint occurs when an object or component is supported or contained in a manner that restricts thermal movement. An example of external restraint occurs when piping expands into a vessel nozzle creating a radial load on the vessel shell. Internal restraint occurs when the tem- perature through an object is not uniform. Stresses from a “thermal gradient” are due to internal restraint. Stress is caused by a thermal gradient whenever the temperature dis- tribution or variation within a member creates a differential expansion such that the natural growth of one fiber is influenced by the different growth requirements of adjacent fibers. The result is distortion or warpage. A transient thermal gradient occurs during heat-up and cool-down cycles where the thermal gradient is changing with time. Thermal gradients can be logarithmic or linear across a vessel wall. Given a steady heat input inside or outside a tube the heat distribution will be logarithmic if there is a tem- perature difference between the inside and outside of the tube. This effect is significant for thick-walled vessels. A linear temperature distribution occurs if the wall is thin. Stress calculations are much simpler for linear distribution. Thermal stress ratcheting is progressive incremental inelastic deformation or strain that occurs in a component that is subjected to variations of mechanical and thermal stress. Cyclic strain accumulation ultimately can lead to incremental collapse. Thermal stress ratcheting is the result of a sustained load and a cyclically applied temperature distribution. The fundamental difference between mechanical stresses and thermal stresses liesin the nature of the loading. Thermal stresses as previously stated are a result of restraint or tem- perature distribution. The fibers at high temperature are compressed and those at lower temperatures are stretched. The stress pattern must only satisfy the requirements for equilibrium of the internal forces. The result being that yielding will relax the thermal stress. If a part is loaded mechanically beyond its yield strength, the part will continue to yield until it breaks, unless the deflection is limited by strain hardening or stress redistribution. The external load remains constant, thus the internal stresses cannot relax. The basic equations for thermal stress are simple but become increasingly complex when subjected to variables such as thermal gradents, transient thermal gradients, logarithmic gradients, and partial restraint. The basic equa- tions follow. If the temperature of a unit cube is changed TH AT Figure 1-4. Thermal linear gradient across shell wall. from TI to Tz and the growth of the cube is fully restrained: where T1= initial temperature, O F Tz=new temperature, O F (11=mean coefficient of thermal expansion in./in./”F E =modulus of elasticity, psi v =Poisson’s ratio =.3 for steel AT =mean temperature difference, O F Case 1: If the bar is restricted only in one direction but free to expand in the other drection, the resulting uniaxial stress, 0,would be 0 = -Ea(Tz - TI) 0 If Tt > TI, 0 is compressive (expansion). 0 If TI > Tz, 0 is tensile (contraction). Case 2: If restraint is in both directions, x and y, then: 0,= cy= -(~IEAT/1- o Case 3: If restraint is in all three directions, x, y, and z, then 0,= oy= 0,= -aE AT11 - 2~ Case 4: If a thermal linear gradient is across the wall of a thin shell (see Figure 14),then: 0, = O+ = f(11EAT/2(1- V) This is a bending stress and not a membrane stress. The hot side is in tension, the cold side in compression. Note that this is independent of vessel diameter or thickness. The stress is due to internal restraint. Discontinuity Stresses Vessel sections of different thickness, material, dameter, and change in directions would all have different displace- ments if allowed to expand freely. However, since they are connected in a continuous structure, they must deflect and rotate together. The stresses in the respective parts at or near the juncture are called discontinuity stresses. Disconti- nuity stresses are necessary to satisfy compatibility of defor- mation in the region. They are local in extent but can be of
  • 22. Stresses in Pressure Vessels 13 very high magnitude. Discontinuity stresses are “secondary stresses” and are self-limiting. That is, once the structure has yielded, the stresses are reduced. In average application they will not lead to failure. Discontinuity stresses do become an important factor in fatigue design where cyclic loadlng is a consideration. Design of the juncture of the two parts is a major consideration in reducing discontinuity stresses. In order to find the state of stress in a pressure vessel, it is necessary to find both the membrane stresses and the dis- continuity stresses. From superposition of these two states of stress, the total stresses are obtained. Generally when combined, a higher allowable stress is permitted. Due to the complexity of determining dlscontinuity stress, solutions will not be covered in detail here. The designer should be aware that for designs of high pressure (>1,500psi), brittle material or cyclic loading, discontinuity stresses may be a major consideration. Since discontinuity stresses are self-limiting, allowable stresses can be very high. One example specifically addressed by the ASME Code, Section VIII, Division 1, is discontinuity stresses at cone-cylinder intersections where the included angle is greater than 60”. Para. 1-5(e) recommends limiting combined stresses (membrane + dis- continuity) in the longitudinal direction to 4SE and in the circumferential direction to 1.5SE. ASME Code, Section VIII, Division 2, limits the com- bined stress, primary membrane and discontinuity stresses to 3S,,, where S, is the lesser of %FFyor ‘/,U.T.S.,whichever is lower. There are two major methods for determining dis- continuity stresses: 1. Displacement Method-Conditions of equilibrium are 2. Force Method-Conditions of compatibility of dis- See References 2, Article 4-7; 6, Chapter 8; and 7, Chapter 4 for detailed information regarding calculation of discontinuity stresses. expressed in terms of displacement. placements are expressed in terms of forces. Fatigue Analysis ASME Code, Section VIII, Division 1, does not speci- fically provide for design of vessels in cyclic service. Although considered beyond the scope of this text as well, the designer must be aware of conditions that would require a fatigue analysis to be made. When a vessel is subject to repeated loading that could cause failure by the development of a progressive fracture, the vessel is in cyclic service. ASME Code, Section VIII, Division 2, has established specific criteria for determining when a vessel must be designed for fatigue. It is recognized that Code formulas for design of details, such as heads, can result in yielding in localized regions. Thus localized stresses exceeding the yield point may be encountered even though low allowable stresses have been used in the design. These vessels, while safe for relatively static conditions of loading, would develop “progressive frac- ture” after a large number of repeated loadings due to these high localized and secondary bending stresses. It should be noted that vessels in cyclic service require special considera- tion in both design and fabrication. Fatigue failure can also be a result of thermal variations as well as other loadings. Fatigue failure has occurred in boiler drums due to temperature variations in the shell at the feed water inlet. In cases such as this, design details are of extreme importance. Behavior of metal under fatigue conrlltions vanes signifi- cantly from normal stress-strain relationships. Damage accumulates during each cycle of loading and develops at localized regions of high stress until subsequent repetitions finally cause visible cracks to grow, join, and spread. Design details play a major role in eliminating regions of stress raisers and discontinuities. It is not uncommon to have the design strength cut in half by poor design details. Progressive fractures develop from these discontinuities even though the stress is well below the static elastic strength of the material. In fatigue service the localized stresses at abrupt changes in section, such as at a head junction or nozzle opening, misalignment, defects in construction, and thermal gradients are the significant stresses. The determination of the need for a fatigue evaluation is in itself a complex job best left to those experienced in this type of analysis. For specific requirements for determining if a fatigue analysis is required see ASME Code, Section VIII, Division 2, Para. AD-160. For additional information regarding designing pressure vessels for fatigue see Reference 7, Chapter 5.
  • 23. 14 Pressure Vessel Design Manual REFERENCES- 1. 2. 3. 4. 5. 6. ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, 1995 Edition, American Society of Mechanical Engineers. ASME Boiler and Pressure Vessel Code, Section VIII, Division 2, 1995 Edition, American Society of Mechanical Engineers. Popov, E. P., Mechanics of Materials, Prentice Hall, Inc., 1952. Bednar, H. H., Pressure Vessel Design Handbook, Van Nostrand Reinhold Co., 1981. Harvey, J. F., Theory and Design of Modern Pressure Vessels,Van Nostrand Reinhold Co., 1974. Hicks, E. J. (Ed.), Pressure Vessels-A Workbook for Engineers, Pressure Vessel Workshop, Enera- Sources Technology Conference and Exhibition, 7. 8. 9. 10. 11. Houston, American Society of Petroleum Engineers, January 19-21, 1981. Pressure Vessel and Piping Design, Collected Papers 1927-1959, American Society of Mechanical Engineers, 1960. Brownell, L. E., and Young, E. H., Process Equipment Design, John Wiley and Sons, 1959. Roark, R. J., and Young, W. C., Formulasfor Stress and Strain, 5th Edition, McGraw Hill Book Co., 1975. Burgreen, D., Design Methods for Power Plant Structures, C. P. Press, 1975. Criteria of the ASME Boiler and Pressure Vessel Code for Design by Analysis in Sections I11and VIII, Division 2, American Society of Mechanical Engineers.
  • 24. 4 General Design PROCEDURE 2-1 GENERALVESSEL FORMULAS 11, 21 Notation P =internal pressure, psi D,, D, =insidehtside diameter, in. S =allowable or calculated stress, psi E =joint efficiency L =crown radius, in. K,, R, =insidehutside radius, in. K, M =coefficients (See Note 3) crx =longitudinal stress, psi crTd=circumferential stress, psi R,,, =mean ra&us of shell, in. t =thickness or thickness required of shell, head, r =knuckle radius, in. or cone, in. Notes 1. Formulas are valid for: a. Pressures < 3,000psi. b. Cylindrical shells where t 5 0.5R, or P 5 0.385 SE. For thicker shells see Referencr 1,Para. 1-2. c. Spherical shells and hemispherical heads where t 5 0.356 R, or P 5 0.665 SE. For thicker shells see Reference 1,Para. 1-3. 2, All ellipsoidal and torispherical heads having a mini- mum specified tensile strength greater than 80,000psi shall be designed using S =20,000psi at ambient tem- perature and reduced by the ratio of the allowable stresses at design temperature and ambient tempera- ture where required. Ellipsoidal or torispherical head - t-4 ,JK Figure 2-1. General configuration and dimensional data for vessel shells and heads. 3. Formulas for factors: 15
  • 25. 9 5 $Table 2-1 v) tD v) v) 0 < GeneralVessel Formulas 2 0 tD E. (0 3 zm Stress, SThickness, t Pressure, P I.D. OB. I.D. 0.0. 1.0. 0.0.Part Stress Formula Shell Longitudinal [l,Section UG-27(~)(2)] Circumferential [l,Section UG-27(c)(l); Section 1-l(a)(l)] Heads Hemi sphere [I, Section 1-1(a)(2); Section UG-27(d)] [ l, Section 1-4(c)] Ellipsoidal 2:l S.E. [l, Section UG-32(d)] 100%-6% Torispherical [l,Section UG-32(e)] -3 C P(Ro- 1.4t) 2Et PR, a, =- 0.2t P(Ri -0.4t) 2Et 2SEt Ro - 1.4t 2SEt Ri -0.4t SEt Ri +0.6t PRO 2SE +1.4P PRi 2SE +0.4P P(RiEt+0.6t) P(R0Et- 0.4t)SEt Ro - 0.4t PRO SE +0.4P PRi SE -0.6P P(R0 - 0.8t) 2Et P(Ri +0.2t) 2Et See Procedure2-2 2SEt Ro - 0.8t PSEt R,+0.2t PRO 2SE +0.8P PRm PRi ax= a+ =- 2t 2SE - 0.2P PDiK 2SE - 0.2P See Procedure 2-2 2SEt KDo -2t(K - 0.1) 2SEt Do - 1.8t 2SEt KDi +0.2t PDoK 2SE +2P(K - 0.1) PDo 2SE +1.8P PSEt Di +0.2t PD, 2SE - 0.2P SEt 0.8851, -0.8t SEt 0.8854 +O.lt 0.885P1, SE +0.8P 0.885PLi SE-0.1P Torispherical Urc 16.66 [l,Section 1-4(d)] PSEt 1,M - t(M - 0.2) 2SEt 4M +0.2t P b M 2SE +P(M- 0.2) PLiM 2SE - 0.2P Cone Longitudinal P(Di - 0.8tCOS a) P(Do-2.8tc0~CX) 4Etcos cx 4Etcos 0: 4SEtcoscx Do - 2.8tCOS O: 4SEtcos0: Di - 0.8tCOS cx PRm PDi PDo a, =- 2tcos 0: 4COS cx (SE +0.4P) 4c0S cx (SE+1.4P) Circumferential [l,Section 1-4(e); Section UG-32(g)] P(Di +1CO COS K) P(Do- 0.8tc0~CX) 2Etcos cx2Etcos cx 2SEtcos0: Do- 0.8tCOS cx 2SEtcoscx D,+1.2tcos 0: PRm PDi PDo 2COS O: (SE -0.6P) 2 ~ 0 sN (SE+0.4P) cT@ =~ tcos 0:
  • 26. General Design 17 1 3 16 7 8 - 1_3 16 -3 4 1! 16 -5 8 9 1% 1 2 - 7 1% 3 8 - 5 i6 1 4 3 1% 1 8 - 1.219 1.156 1.094 1.031 0.969 0.906 0.844 0.781 0.719 0.656 0.594 0.531 0.469 0.406 0.344 0.281 0.219 24 36 48 60 72 84 96 108 120 132 144 156 168 Vessel Diameter,Inches Figure 2-la. Required shell thickness of cylindrical shell.
  • 27. 18 PressureVessel Design Manual WWm 0 N d W 0 2 9 2 ? 7 W 7 m Vessel Diameter, Inches Figure 2-la. (Continued) dWNb0WWdWmdN
  • 28. General Design 19 PROCEDURE 2-2 EXTERNAL PRESSURE DESIGN Notation ~~ ~ A =factor “A,” strain, from ASME Section TI, Part A, =cross-sectional area of stiffener, in.2 D, Subpart 3, dimensionless R =factor “B,” allowable compressive stress, from D =inside diameter of cylinder, in. Do=outside diameter of cylinder, in. ]I1,=outside diameter of the large end of cone, in. D, =outside diameter of small end of cone, in. E =modulus of elasticity, psi I =actual moment of inertia of stiffener, in. I, =required moment of inertia of stiffener, in.4 I: =required moment of inertia of combined shell- ring cross section, in. L, =for cylinders-the design length for external pressure, including k the depth of heads, in. For cones-the design length for external pres- snre (see Figures 2-lb and 2-lc), in. ASME Section 11, Part D, Subpart 3, psi 4 4 L,, =equivalent length of conical section, in. L, =length between stiffeners, in. I,, - T =length of straight portion of shell, tangent to tangent, in. P =design internal pressure, psi P;,=allowable external pressure, psi P, =design external pressure, psi R,, =outside radius of spheres and hemispheres, t =thickness of cylinder, head or conical section, in. crown radius of torispherical heads, in. t,, =equivalent thickness of cone, in. c( =half apex angle of cone, degrees Unlike vessels which are designed for internal pressure alone, there is no single formula, or unique design, which fits the external pressure condition. Instead, there is a range of options a~ail~bleto the designer which can satisfy the solution of the design. The thickness of the cylinder is only one part of the design. Other factors which affect the design are the length of cylinder and the use, size, and spacing of stiffening rings. Designing vessels for external pressure is an iterative procedure. First, a design is selected with all of the variables included, then the design is checked to determine if it is adequate. If inadequate, the procedure is repeated until an acceptable design is reached. Vessels subject to external pressure may fail at well below the yield strength of the material. The geometry of the part is the critical factor rather than material strength. Failures can occur suddenly, by collapse of the component. External pressure can be caused in pressure vessels by a variety of conditions and circumstances. The design pressure may be less than atmospheric due to condensing gas or steam. Often refineries and chemical plants design all of their vessels for some amount of external pressure, regarcl- less of the intended service, to allow fbr steam cleaning and the effects of the condensing steam. Other vessels are in vacuum service by nature of venturi devices or connection to a vacuum pump. Vacuums can be pulled inadvertently by failure to vent a vessel during draining, or from improperly sized vents. External pressure can also be created when vessels are jacketed or when components are within iririltichairibererl vessels. Often these conditions can be many times greater than atmospheric pressure. When vessels are designed for bot11internal arid external pressure, it is common practice to first determine the shell thickness required for the internal pressure condition, then check that thickness for the maximum allowable external pressure. If the design is not adequate then a decision is made to either bump up the shell thickness to the next thickness of plate available, or add stiffening rings to reduce the “L’dimension. If the option of adding stiffening rings is selected, then the spacing can be determined to suit the vessel configuration. Neither increasing the shell thickness to remove stiffening rings nor using the thinnest shell with the Inaximum number of stiffeners is economical. The optimum solution lies some- where between these two extremes. Typically, the utilization of rings with a spacing of 2D for vessel diairictcrs up to abont eight feet in diameter and a ring spacing of approximately “D” for diameters greater than eight feet, provides an eco- nomical solution. The design of the stiffeners themselves is also a trial and error procedure. The first trial will be quite close if the old APT-ASME formula is used. The forinula is its follows: 0.16D~P,LS EIs = Stiffeners should never be located over circurnferentlal weld seams. If properly spaced they may also double as insu- lation support rings. Vacuum stiffeners, if coinbined with other stiffening rings, such as cone reinforcement rings or saddle stiffeners on horizontal vessels, must be designed for the combined condition, not each independently. If at all
  • 29. 20 Pressure Vessel Design Manual possible, stiffeners should always clear shell nozzles. If una- voidable, special attention should be given to the design of a boxed stiffener or connection to the nozzle neck. Design Procedure For Cylindrical Shells Step 1:Assume a thichess if one is not already determined. Step 2: Calculate dimensions “L’and “D.” Dimension “L’ should include one-third the depth of the heads. The over- all length of cylinder would be as follows for the various head types: W/(2) hemi-heads W/(2) 2:1S.E.heads L = LT-T +0.333D L = LT-T +0.16661) W/(2) 100% - 6% heads L = h - ~+0.112D Step 3: Calculate UD, and D,Jt ratios Step 4:Determine Factor “A’from ASME Code, Section 11, Part D, Subpart 3, Fig G: Geometric Chart for Components Under External or Compressive Loadings (see Figure 2-le). Step 5: Using Factor “A’ determined in step 4, enter the applicable material chart from ASME Code, Section 11, Part D, Subpart 3 at the appropriate temperature and determine Factor “B.” Step 6: If Factor “A’falls to the left of the material line, then utilize the following equation to determine the allowable external pressure: Step 7: For values of “A’ falling on the material line of the applicable material chart, the allowable external pressure should be computed as follows: 4B Step 8: If the computed allowable external pressure is less than the design external pressure, then a decision must be made on how to proceed. Either (a)select a new thickness and start the procedure from the beginning or (b) elect to use stiffening rings to reduce the “L’ hmension. If stiffening rings are to be utilized, then proceed with the following steps. Step 9: Select a stiffener spacing based on the maximum length of unstiffened shell (see Table 2-la). The stiffener spacing can vary up to the maximum value allowable for the assumed thickness. Determine the number of stiffen- ers necessary and the correspondmg “L’dimension. Step 10:Assume an approximate ring size based on the fol- lowing equation: O.lGD~P,L, E I = Step 11:Compute Factor “B” from the following equation utilizing the area of the ring selected: 0.75PD0 g = - t+As /Ls Step 12: Utilizing Factor “B” computed in step 11,find the corresponding “A’ Factor from the applicable material curve. Step 13:Determine the required moment of inertia from the following equation. Note that Factor “A” is the one found in step 12. I, = Ls(t+As /Ls 1-41 14 Step 14: Compare the required moment of inertia, I, with the actual moment of inertia of the selected member. If the actual exceeds that which is required, the design is acceptable but may not be optimum. The optimization process is an iterative process in which a new member is selected, and steps 11 through 13 are repeated until the required size and actual size are approximately equal. Notes 1. For conical sections where c( < 22.5 degrees, design the cone as a cylinder where Do=DL and length is equal to L. 2. If a vessel is designed for less than 15psi, and the external pressure condition is not going to be stamped on the nameplate, the vessel does not have to be designed for the external pressure condition.
  • 30. Case A -4- L Portionpf a cone f General Design 21 I 1 .kj$r2 Dss I 1 Case B Case c I Case D Case E Figure 2-1b. External pressurecones 22 1/2"< a<60". For Case B, L,.=L For Cases A, C, D, E: t,. =tcos c(
  • 31. 22 Pressure Vessel Design Manual Large End Small End Figure 2-lc. Combined shelkone Design stiffener for large end of cone as cylinder where: Do = DL t = tI> L1 L2 L,=-+- 2 2 I I Design stiffener for small end of cone as cylinder where: Do = Ds t = t, L --+- 1+- " 2 L3 "[2 3 Ro= 0.9 Do I R o w = Do I ' II Sphere/Hemisphere 2:l S.E. Head Figure 2-ld. External pressure -spheres and heads. Torispherical
  • 32. General Design 23 Figure 2-le. Geometric chart for componentsunder external or compressiveloadings (for all materials). (Reprintedby permission from the ASME Code. Section VIII, Div. 1.)
  • 33. Design Procedure For Spheres and Heads Step 1:Assume a thickness and calculate Factor “A.” 0.125 A = - Ro Step 2: Find Factor ‘‘€3” from applicable material chart. B = Step 3: Compute Pa. Figure2-If. Chart for determining shell thickness of components under external pressure when constructed of carbon or low-alloy steels (specified minimum yield strength 24,OOOpsi to, but not including, 30,OOOpsi). (Reprinted by permissionfrom the ASME Code, Section VIII, Div. 1.) 25.000 20.000 18,OM) 16.000 14.000 12.000 lO.Oo0 8.003 50 7.000 (d u. 6.000 5,000 9.000 m 4.000 3,500 3,000 2.500 o.oooo1 a m i 0.001 Factor A 0.01 0 1 Figure2-1g. Chart for determining shell thickness of components under external pressurewhen constructed of carbon or low-alloysteels (specified minimum yield strength 30,OOOpsi and over except materialswithin this range where other specific charts are referenced) and type 405 and type 410 stainless steels. (Reprinted by permission from the ASME Code, Section VIII, Div. 1.)
  • 34. General Design 25 0.0625E P, = ~A to left of material line Bt P. - - - R,, A to right of material line Notes 1. As an alternative, the thickness required for 2:l S.E. heads for external pressure may be computed from the formula for internal pressure where P =1.67P, and E = 1.0. Table 2-1a Maximum Length of Unstiffened Shells 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 204 168 313 142 264 122 228 104 200 91 174 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 00 280 235 437 203 377 178 330 157 293 138 263 124 237 110 212 99 190 90 173 82 160 76 148 70 138 65 128 61 120 57 113 54 106 51 98 49 92 46 87 44 83 00 358 306 268 499 238 442 213 396 193 359 175 327 157 300 143 274 130 249 228 109 211 101 197 95 184 88 173 163 78 154 74 146 70 67 131 00 00 1i a a3 138 437 381 336 626 302 561 273 508 249 462 228 424 210 391 190 363 176 337 162 311 149 287 138 266 129 121 234 114 221 107 209 101 199 96 189 00 248 458 408 369 686 336 625 308 573 528 263 490 245 456 223 426 209 400 195 374 348 169 325 158 304 148 140 271 133 258 00 00 284 1ai 286 537 483 00 438 816 402 748 370 343 639 320 594 299 555 280 521 263 490 242 462 228 437 214 411 201 385 363 178 342 689 1a9 616 559 510 470 435 810 405 754 379 705 355 660 334 621 315 586 297 555 275 526 261 499 475 233 448 00 00 a75 248 637 585 715 540 661 1,005 00 502 613 935 00 469 571 874 1,064 440 536 819 997 414 504 391 475 369 449 687 836 350 426 652 793 332 405 619 753 309 385 590 717 294 367 00 770 938 727 a84 562 684 795 738 687 642 603 1,124 569 1,060 538 1,002 510 950 902 462 859 440 819 00 00 485 a75 816 762 715 673 1,253 636 1,185 603 1,123 573 1,066 546 1,015 520 968 00 894 839 789 744 705 1,312 669 1,246 637 1,186 608 1,131 00 00 974 916 864 817 774 1,442 737 1,373 703 1,309 00 1,053 994 940 1,073 891 1,017 1,152 846 966 1,095 806 919 1,042 00 co 1,509 co Notes. 1. All values are in in. 2. Values are for temperatures up to 500°F. 3. Top value is for full vacuum, lower value is half vacuum. 4. Values are for carbon or low-alloy steel (Fv>3O,O0Opsi) based on Figure 2-19,
  • 35. 26 Pressure Vessel Design Manual Table 2-1b Moment of Inertia of Bar Stiffeners Thk Max, Height, h, in. t, in. ht, in. 1 1% 2 2% 3 3% 4 4% 5 5% 6 6% 7 7% 8 'I4 2 0.020 0.070 0.167 0.250 0.375 0.5 ~ r [ T h S , 5116 2.5 0.026 0.088 0.208 0.407 0.313 0.469 0.625 0.781 "I8 3 0.031 0.105 0.25 0.488 0.844 0.375 0.563 0.75 0.938 1.125 th3I = -12 7/16 3.5 0.123 0.292 0.570 0.984 1.563 0.656 0.875 1.094 1.313 1.531 4 0.141 0.333 0.651 1.125 1.786 2.667 0.75 1.00 1.25 1.50 1.75 2.00 'lI6 4.5 0.375 0.732 1.266 2.00 3.00 4.271 1.125 1.406 1.688 1.969 2.25 2.53 "8 5 0.814 1.41 2.23 3.33 4.75 6.510 1.563 1.875 2.188 2.50 2.813 3.125 116 5.511 1.55 2.46 3.67 5.22 7.16 9.53 2.063 2.406 2.75 3.094 3.438 3.78 'I4 6 1.69 2.68 4.00 5.70 7.81 10.40 13.5 2.25 2.625 3.00 3.375 3.75 4.125 4.50 l3lI6 6.5 2.90 4.33 6.17 8.46 11.26 14.63 18.59 2.844 3.25 3.656 4.063 4.469 4.875 5.281 7 4.67 6.64 9.11 12.13 15.75 20.02 25.01 3.50 3.94 4.375 4.813 5.25 5.688 6.125 1 8 _______ 5.33 7.59 10.42 13.86 18.00 22.89 28.58 35.16 42.67 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 Note: Upper value in table is the moment of inertia. Lower value is the area
  • 36. General Design 27 Table 2-lc Moment of Inertia of Composite Stiffeners -I - f2 H 1 tiH3 11 =- 12 Iz ==- 12 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 3 3 4 4 4.5 5 5.5 6 6 5.5 6.5 7 8 8 3 4 4 5 5 4 4 5 6 6 6 6 6 6 0.375 0.5 0.375 0.5 0.5 0.5 0.5 0.5 0.625 0.875 0.75 0.625 0.75 1 0.5 0.5 0.5 0.625 0.5 0.625 0.5 0.625 0.625 0.875 0.75 0.75 1 1 2.63 3.50 3.50 5.13 4.75 5.00 4.75 6.13 7.50 10.01 9.38 8.88 12.00 14.00 0.87 1.17 2.04 2.77 3.85 5.29 6.97 9.10 11.37 12.47 17.37 18.07 32.50 43.16 2.50 2.50 3.28 3.41 3.57 3.91 4.01 4.69 4.66 4.42 4.99 5.46 6.25 5.93 2.84 6.45 9.28 0.125 15.12 17.39 25.92 31.82 37.98 48.14 51.60 93.25 112.47 -3.80 Moment of Inertia of Stiffening Rings Figure 2-1h. Case 1: Bar-type stiffening ring. Figure 2-li. Case 2: T-type stiffening ring.
  • 37. 28 Pressure Vessel Design Manual As E=modulus of elasticity STIFFENING RING CHECK FOR EXTERNAL PRESSURE If B> 2,500 psi, determine A from applicable material charts I Moment of inerliawlo shell Moment of inertia w/ shell D2LO +A S / W1: = 10.9 From Ref. 1, Section UG-29. PROCEDURE 2-3 CALCULATE MAP. MAWP. AND TEST PRESSURES Notation Sa=allowable stress at ambient temperature, psi SDT =allowable stress at design temperature, psi SCA =allowable stress of clad material at ambient tempera- SCD =allowable stress of clad material at design tempera- SBA =allowable stress of base material at ambient tempera- SBD =allowable stress of base material at design tempera- C.a. =corrosion allowance, in. ture, psi ture, psi ture, psi ture, psi t,, =thickness of shell, corroded, in. t,, =thickness of shell, new, in. thc=thickness of head, corroded, in. thn =thickness of head, new, in. tb =thickness of base portion of clad material, in. t, =thickness of cladding, in, R, =inside radius, new, in. R, =inside radius, corroded, in. R, =outside radius, in. D, =inside diameter, new, in. D, =inside diameter, corroded, in. D, =outside diameter, in. PM=MAP, psi PW=MAWP, psi P =design pressure, psi Ps =shop hydro pressure (new and cold), psi PF =field hydro pressure (hot and corroded), psi E =joint efficiency, see Procedure 2-1 and Appendix C Definitions Maximum Allowable Working Pressure (MAWP):The MAWP for a vessel is the maximum permissible pressure at the top of the vessel in its normal operating position at a specific temperature, usually the design temperature. When calculated, the MAWP should be stamped on the nameplate. The MAWP is the maximum pressure allowable in the “hot and corroded’ condtion. It is the least of the values calculated for the MAWP of any of the essential parts of the vessel, and adjusted for any difference in static head that may exist between the part considered and the top of the vessel. This pressure is based on calculations for every element of the vessel using nominal thicknesses exclusive of corrosion allowance. It is the basis for establishing the set pressures of any pressure-relieving devices protecting the vessel. The design pressure may be substituted if the MAWP is not calculated. The MAWP for any vessel part is the maximum internal or external pressure, including any static head, together with the effect of any combination of loadings listed in UG-22 which are likely to occur, exclusive of corrosion allowance at the designated coincident operating temperature. The MAWP for the vessel will be governed by the MAWP of the weakest part. The MAWP may be determined for more than one de- signated operating temperature. The applicable allowable
  • 38. General Design 29 stress value at each temperature would be used. When more than one set of conditions is specified for a given vessel, the vessel designer and user should decide which set of condi- tions will govern for the setting of the relief valve. Maximum Allowable Pressure (MAP): The term MAP is often used. It refers to the maximum permissible pressure based on the weakest part in the new (uncorroded) and cold condition, and all other loadings are not taken into consid- eration. Design Pressure: The pressure used in the design of a vessel component for the most severe condition of coinci- dent pressure and temperature expected in normal opera- tion. For this condition, and test condition, the maximum difference in pressure between the inside and outside of a vessel, or between any two chambers of a combination unit, shall be considered. Any thichess required for static head or other loadings shall be additional to that required for the design pressure. Design Temperature: For most vessels, it is the tem- perature that corresponds to the design pressure. However, there is a maximum design temperature and a minimum design temperature for any given vessel. The mini- mum design temperature would be the MDMT (see Procedure 2-17). The MDMT shall be the lowest tempera- ture expected in service or the lowest allowable temperature as calculated for the individual parts. Design temperature for vessels under external pressure shall not exceed the maxi- mum temperatures given on the external pressure charts. Operating Pressure: The pressure at the top of the vessel at which it normally operates. It shall be lower than the MAWP, design pressure, or the set pressure of any pres- sure relieving device. Operating Temperature: The temperature that will be maintained in the metal of the part of the vessel being con- sidered for the specified operation of the vessel. Calculations e MAWP, corroded at Design Temperature P,. Shell: 2:l S.E. Head: e MAP, new and cold, P M Shell: SaEtsn R, - 0.4tsn or SaEtsn Rn +0.6tsn PM = 2:l S.E. Head: e Shop test pressure, Ps. P, = 1 . 3 P ~or 1 . 3 P ~- e Field test pressure, PF. PF = 1.3P e For clad vessels where credit is takenfor the clad material, thefollowing thicknesses may be substituted into the equa- tionsfor MAP and MAWP: Notes 1. Also check the pressure-temperature rating of the 2. All nozzles should be reinforced for MAWP. 3. The MAP and MAWP for other components, i.e., cones, flat heads, hemi-heads, torispherical heads, etc., may be checked in the same manner by using the formula for pressure found in Procedure 2-1 and substituting the appropriate terms into the equations. 4. It is not necessary to take credit for the cladding thick- ness. If it is elected not to take credit for the cladding thickness, then base all calculations on the full base metal thickness. This assumes the cladding is equiva- lent to a corrosion allowance, and no credit is taken for the strength of the cladding. flanges for MAWP and MAP.
  • 39. 30 PressureVessel Design Manual PROCEDURE 2-4 STRESSES IN HEADS DUE TO INTERNAL PRESSURE [e,31 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. r ~ #=latitudinal stress, psi ox=meridional stress, psi P =internal pressure, psi Formulas Lengths of RL and R, for ellipsoidal heads: At equator: h2 R, =- R RL = R At center of head: a At any point X: Notes 1. Latitudinal (hoop) stresses in the knuckle become com- pressive when the R/h ratio exceeds 1.42. These heads will fail by either elastic or plastic buckling, depending 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 Figure 2-2. Direction of stresses in a vessel head. :Figure 2-3. Dimensional data for a vessel head. covered in the previous section. Torisphericz (also known as flanged and dished heads) and ellipsoidal head formulas for stress are outlined in the following form.
  • 40. General Design 31 U X I TORISPHERICAL HEADS I U & K In Crown I 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 .5 .57 .65 .73 .81 .9 .99 1.08 1.18 1.27 1.36 In Knuckle I At Tangent Line I I ELLIPSOIDAL HEADS I I At Any Point X I At Center of HeadI I I At Tangent Line I PROCEDURE 2-5 DESIGN OF INTERMEDIATE HEADS [1, 31 Notation A =factor A for external pressure A, =shear area, in.2 B =allowable compressive stress, psi F =load on weld(s), lb/in. 5 :shear stress, psi E =joint efficiency S =code allowable stress, psi HL)=hydrostatic end force, lb P, =maximum differential pressure on concave side of El =modulus of elasticity at temperature, psi head, psi P, =maximum differential pressure on convex side of K =spherical radius factor (see Table 2-2) L =inside radius of hemi-head, in. =0.9D for 2:l S.E. heads, in. =KD for ellipsoidal heads, in. =crown radius of F & D heads, in. head, psi Table 2-2 Spherical Radius Factor, K
  • 41. 32 Pressure Vessel Design Manual 0 -k Figure24. Dimensional data for an intermediate head. Seal OF A, = t, + lesser of t2 or t, Case 1 Butterto t2 D sin e= 2 ~ A, = lesser of t2or t3 E= 0.7 (butt weld) Case 3 Required HeadThickness, t, 0 Znternal pressure, Pi.Select appropriate head formula based on head geometry. For dished only heads as in Figure 2-5, Case 3: A,= t, Case 2 Reinforcing plate E = 0.7 E = 0.55 Design the weld attaching the head as in Case 3 and the welds attachingthe reinforcingplate to share full load Case 3 Alternate Figure 2-5. Methods of attachment of intermediate heads.
  • 42. General Design 33 0 External pressure, P,. Assume corroded head thickness, th 0.125th Factor A = ____ L Factor B can be taken from applicable material charts in Section 11, Part D, Subpart 3 of Reference 1. Alternatively (or if Factor A lies to the left of the material/ temperature line): AEI B = - 2 peL t'=B The required head thickness shall be the greater of that required for external pressure or that required for an in- ternal pressure equal to 1.67x P,. See Reference 1, Para. UG-33(a). Shear Stress where P = 1.5x greater of Pi or P,. (See Reference 1, Figure UW-13.1.) 0 Shear loads on welds, F. Note: sin8 applies to Figure 2-5, Case 3 head at- tachments only! 0 Shear stress, 5. F AS t=-- 0 Allowable shear stress, SE. 0 Hydrostatic end force, HD. PnD2 Hn=- 4 PROCEDURE 2-6 DESIGN OF TORICONICALTRANSITIONS 11,31 I- d I Notation P =internal pressure, psi S =allowable stress, psi E =joint efficiency PI, P2=equivalent internal pressure, psi ul.uz =circumferential membrane stress, psi fi, f2=longitudinal unit loads, lb/in. a =half apex angle, deg m=code correction factor for thickness of large P, =external pressure, psi MI, M2 =longitudinal bending moment at elevation, in.-lb WI, Wz=dead weight at elevation, lb knuckle Anyplace on cone I D Figure 2-6. Dimensionaldata for a conical transition.
  • 43. 34 Pressure Vessel Design Manual Calculating Angle, cc T.L. T.L. tI $ I B I I p& - 0' L o> ~~ Case 1 0 > 0' Step 1: Step 2: A tan@ = - B @ = Step 3: a = + + @ a = L = COS 4dA2+B2 k 0' Case 2 0' > 0 Step 1: Step 2: A t a n @ = - B Step 3: (Y = 90- 0 - 4 01= L = sin ~ J A ~+BZ
  • 44. General Design 35 Dimensional Formulas DI D - 2(R - R COS^) DI L, =- 2 cos a Large End (Figure2-7) Figure 2-7. Dimensionaldata for the large end of a conical transition. e Maximum longitudinal loady,f i ( + ) tension; ( - ) compression e D&rmine equiualent prmsiire!,PI e Circumferential stress, D1 Compression: PLl '1,"' [;;Ic1=--- - t e Circumferential stress at DI withotit londs, c/1 Compression: e Thickness required knuckle, t,-k [I,section 1-4(d)]. With loads: Without loads: PLlm 2SE - 0.2P trk = e Thickness required cone, t,, [l,section UG-32(g)]. N7ithloads: PiDi 2 cosa(SE - 0.6P1) trc = Without loads: PDl 2 COS u(SE - 0.W) tr, = ~~ Small End (Figure 2-8) e Maximum longitudinal loah,f 2 . ( + ) tension; ( - ) compression e Determine equi.L;alentpressure, P2. 4fz P?.=P+- D2
  • 45. 36 Pressure Vessel Design Manual Figure 2-8. Dimensional data for the small end of a conical transition. Circumferential stress at DZ. Compression: Circumferential stress at 0 2 without loads, 02. Compression: Thickness required cone, at DZ, t,, [l,section UG-32(g)]. With loads: PzDz 2 cosa(SE - O.6P2) t, = Without loads: PDz 2 COS a(SE - 0.6P) trc = Thickness required knuckle. There is no requirement for thickness of the reverse knuckle at the small end of the cone. For convenience of fabrication it should be made the same thickness as the cone. Additional Formulas (Figure2-9) Thickness required of cone at any diameter D', tDl. PD' tD' = 2 cos a(SE - 0.6P) --I I DL *I Figure 2-9. Dimensional data for cones due to external pressure. a Thickness required for external pressure [l,section UG- WOI. te = tcosa D ~ = D 2 + 2 b D, = D1+ 2te L = X - sina(R +t) - sina(r - t) L - -e-:( 1+-::) Using these values, use Figure 2-le to determine Factor A. Allowable external pressure, Pa. where E =modulus of elasticity at design temperature. Notes 1. Allowable stresses. The maximum stress is the com- pressive stress at the tangency of the large knuckle and the cone. Failure would occur in local yielding rather than buckling; therefore the allowable stress should be the same as required for cylinders. Thus the allowable circumferential compressive stress should be the lesser of 2SE or F,. Using a lower allow- able stress would require the knuckle radius to be made very large-well above code requirements. See Reference 3. 2. Toriconid sections are mandatory if angle 01 exceeds 30" unless the design complies with Para. 1-5(e)of the
  • 46. General Design 37 ASME Code [l]. This paragraph requires a discontinu- ity analysis of the cone-shell juncture. 3. No reinforcing rings or added reinforcement is required at the intersections of cones and cylinders, providing a knuckle radius meeting ASME Code requirements is used. The minimum knuckle radius for the large end is not less than the greater of 3t or 0.12(R+t). The knuckle radius of the small end (flare)has no minimum. (See [Reference 1,Figure UG-361). 4. Toriconical transitions are advisable to avoid the high discontinuity stresses at the junctures for the following conditions: a. High pressure-greater than 300pig. b. High temperature-greater than 450 or 500°F. c. Low temperature-less than -20°F. d. Cyclic service (fatigue). PROCEDURE 2-7 DESIGN OF FLANGES [1,41 Notation A =flange O.D., in. 2 AI, =cross-sectional area of bolts, in. A,,, =total required cross-sectional area of bolts, in2 a =nominal bolt diameter, in. B =flange I.D., in. (see Note 6) B1=flange I.D., in. (see Note 6) B, =bolt spacing, in. b =effective gasket width, in. b,, =gasket seating width, in. C =bolt circle diameter, in. d =hub shape factor dl =bolt hole diameter, in. E, hI), hc;, hT, R =radial distances, in. e =hub shape factor F =hub shape factor for integral-type FL,=hub shape factor for loose-type flanges f =hub stress correction factor for integral flanges flanges G =diameter at gasket load reaction, in. %=thickness of hub at small end, in. gl =thickness of hub at back of flange, in. H =hydrostatic end force, lb HD=hydrostatic end force on area inside of HG=gasket load, operating, lb H, =total joint-contact surface compression HT=pressure force on flange face, lb flange, lb load, lb h =hub length, in. h, =hub factor MD=moment due to HD,in.-lb MG=moment due to HG, in.-lb M, =total moment on flange, operating, in.& ML =total moment on flange, seating MT =moment due to HT, in.-lb m, =unit load, operating, lb mg=unit load, gasket seating, lb m =gasket factor (see Table 2-3) N =width of gasket, in. (see Table 2-4) w =width of raised face or gasket contact width, n =number of bolts u =Poisson’s ratio, 0.3 for steel P =design pressure, psi S, =allowable stress, bolt, at ambient temperature, S,, =allowable stress, bolt, at design temperature, Sf,=allowable stress, flange, at ambient tempera- Sf,,=allowable stress, flange, at design temperature, SH=longitudinal hub stress, psi SR=radial stress in flange, psi ST=tangential stress in flange, psi T, U, Y Z =K-factors (see Table 2-5) T,, U,, Y, =K-factors for reverse flanges t =flange thickness, in. in. (see Table 2-4) psi psi ture, psi psi t, =pipe wall thickness, in. V =hub shape factor for integral flanges VL=hub shape factor for loose flanges W =flange design bolt load, lb Wml=required bolt load, operating, lb Wrrr2=required bolt load, gasket seating, Ib y =gasket design seating stress, psi
  • 47. 38 Pressure Vessel Design Manual _ _ _ _ _ _ _ _ _ _ _ _ ~ Formulas C - dia. HD 2 hD = C - dia. HT 2 hT = C - G hc =- 2 ho = nB2P 4 HD=- HT = H - HD HG=operating = Wm, - H gasket seating = W (1 - u2)(K2 - l)U T = (1 - U) +(1+v)K2 K2 +1 Z=- K2 - 1 Y = (1- v2)U K2(I+4.6052 ( 1 + ~ / l-~)log,,K)-l U = 1.0472(K2- 1)(K - 1)(1+u) B1 = loose flanges = B +g, = integral flanges, f < 1 = B +g, = integral flanges,f 2 1= B +g, Uho& d = loose flanges = ~ VL UhO& Urho& = integral flanges = - = reverse flanges = - V V F L e = loose flanges = - h0 F = integral flanges = - h0 G = (if b, 5 0.25 in.) mean diameter of gasket face = (if bo > 0.25 in.) O.D. of gasket contact face - 2b Stress Formula Factors a = t e + l f f f f y = - or - for reverse flanges T Tr

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