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Gerry C. Slagis e-mail: [email protected] G C Slagis Associates, 258 Hillcrest Place, Pleasant Hill CA 94523-2184

ASME Section III Design-ByAnalysis Criteria Concepts and Stress Limits1 The ASME Section III design-by-analysis approach provides stress criteria for the design of nuclear components. Stresses are calculated elastically for the most part, although plastic analysis is recognized. Limits are specified for primary, secondary, and peak stresses. Inherent in these limits are factors of safety against several modes of failure. The purpose of this paper is to explain the design-by-analysis criteria and fundamental concepts behind the approach. Topics covered include the bases for the primary stress limits, shakedown to elastic action, fatigue, simplified elastic-plastic analysis, and thermal stress ratchet. Issues that are explored are separating primary and secondary stresses in finite element analyses, material ductility requirements, and the meaning of the fatigue penalty factor. 关DOI: 10.1115/1.2140797兴

Introduction The design-by-analysis concept was first introduced in 1963 with the publication of the nuclear vessels code 关1兴. In comparison to the nonnuclear vessels code, a lower factor of safety on pressure design is incorporated. To justify the lower factor of safety, detailed stress analysis and an evaluation of fatigue, including explicit consideration of thermal stresses, are required. Different categories of stress are assigned different allowable values. A criteria document 关2兴 to explain the design-by-analysis approach was published by ASME in 1969. For nuclear piping, a simplified design-by-analysis approach was first published in 1969 as USAS B31.7 关3兴. The Foreword section of B31.7 gives a description of the design philosophy for nuclear piping. The piping rules were incorporated with the vessel rules in 1971 when Section III was revised to include rules for all nuclear components. Some basic questions regarding interpretation of the design-byanalysis rules have come up in recent years. Some of these questions result from extensive use of finite element methods to determine stresses. For example, how are primary stresses extracted from a finite element analysis? The purpose of this document is to review the design-by-analysis criteria, discuss the fundamental concepts behind the criteria, and provide insight into some of the technical issues. The fragmented nature of code developments and the related literature makes it difficult to fully understand all aspects of the concepts involved. Discussions of the design-by-analysis criteria are based mainly on the 1974 Edition of the Section III Code although the stress limits are taken from the 2001 Code. The Code rules given in NB-3200 apply to any pressure retaining component. The piping rules given in NB-3600 are a simplified version of the NB-3200 rules. Some piping terms and criteria will be used to illustrate certain aspects of design-by-analysis.

Criteria There are two basic concepts underlying the design-by-analysis criteria. First, stresses are categorized into three types with differ1 This is a minor revision of a paper 共PVP2004-2614兲 of the same title that was presented at the 2004 PVP Conference. Contributed by the Pressure Vessels and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received August 26, 2005; final manuscript received October 24, 2005. Review conducted by G. E. Otto Widera. Paper presented at the 2004 ASME Pressure Vessels and Piping Division Conference 共PVP2004兲, July 25, 2004–July 29, 2004, San Diego, California, USA.

Journal of Pressure Vessel Technology

ent stress limits. Second, the stress limits change for different service levels. The three categories of stress are primary, secondary, and peak. Primary stresses are load controlled; secondary stresses are displacement controlled; and peak stresses are local in nature. Primary and secondary stresses can be membrane or bending. Stress limits are established for Design, Level A, Level B, Level C, and Level D loadings. Design conditions 共design pressure, design temperature, and design mechanical loads兲 establish the required wall thickness of the vessel. Level A conditions are those originally referred to as normal conditions 共1971 edition兲 and Level B as upset conditions. Level A and B loadings are expected to occur in the operation of the component. Stress limits for Level A and B are selected so that there is no damage to the component that requires repair. Level C stress limits permit large deformations in areas of structural discontinuity which may necessitate the removal of the component or support from service for inspection or repair of damage.2 Level D stress limits permit gross general deformations with some subsequent loss of dimensional stability and damage requiring repair, which may require removal of the component or support from service. The allowable limits of stress intensity from NB-3200 关4兴 are shown in Fig. 1 as given in the 2001 Edition. Secondary and peak stresses are not limited for Levels C and D on the basis that fatigue analysis is not required since only one such event is anticipated, followed by shutdown for inspection or repair 关5兴. NB3200 Level D limits are given in Appendix F of Section III. Sm, the allowable material stress intensity, is based on a fraction of the material yield stress and the ultimate stress. For ferritic steels, Sm 1 2 is the lower of 3 minimum tensile strength or 3 minimum yield 1 strength. For austenitic steels, Sm is the lower of 3 minimum tensile strength or 90% of the minimum yield strength. The increase to 90% of yield strength is to allow for the strain-hardening characteristics of austenitic steel. When the emergency 共Level C兲 and faulted 共Level D兲 conditions and stress limits were first identified in 1971, the probability of the condition occurring was discussed. For emergency—The conditions have a low probability of occurrence …; for faulted— Those combinations of conditions associated with extremely-lowprobability events. Elastic stress limits for piping for emergency and faulted were first defined in 1974. The probability of occurrence of the loads 2 Italics indicate wording taken from the Section III code document 共if no reference given兲 or the referenced document.

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Fig. 1 NB-3200 design-by-analysis stress limits „2001…

and an equal reliability approach was used as a technical basis for the stress limits. For piping, the Design limit of 1.5Sm for primary membrane-plus-bending was increased to 2.25Sm for emergency and 3Sm for faulted 关6兴.

Stress Definitions The definitions given in NB-3213 for primary, secondary, and peak stresses are given below. Primary stress is any normal stress or a shear stress developed by an imposed loading which is necessary to satisfy the laws of equilibrium of external and internal forces and moments. The basic characteristic of a primary stress is that it is not self-limiting. Primary stresses which considerably exceed the yield strength will result in failure or, at least, in gross distortion. A thermal stress is not classified as a primary stress. Secondary stress is a normal stress or a shear stress developed by the constraint of adjacent material or by self-constraint of the structure. The basic characteristic of a secondary stress is that it is self-limiting. Local yielding and minor distortions can satisfy the conditions which cause the stress to occur and failure from one application of the stress is not to be expected. Peak stress is that increment of stress which is additive to the primary plus secondary stresses by reason of local discontinuities or local thermal stress including the effects, if any, of stress concentrations. The basic characteristic of a peak stress is that it does not cause any noticeable distortion and is objectionable only as a possible source of a fatigue crack or a brittle fracture. The key to understanding the difference between primary and secondary stresses is that primary stresses are required for equilibrium with an applied “mechanical” load. Pressure is an applied mechanical load. The hoop stress in a cylinder to react the pressure load is a primary membrane stress. An applied moment to a horizontal cylinder from self-weight produces a primary bending stress. If the cylinder includes a gross structural discontinuity, secondary stresses will also be created by a mechanical load. If the cylinder includes a local structural discontinuity, peak stresses will be created. If a piping system is subjected to a fluid temperature increase, thermal expansion stresses are created. These thermal expansion stresses 共restraint of free end displacement兲 are secondary stresses. Thermal expansion also causes peak stress at a local structural discontinuity 共a girth butt weld, for example兲. A through-wall temperature gradient in a cylinder can cause a 26 / Vol. 128, FEBRUARY 2006

secondary stress 共general thermal stress兲 and a peak stress 共local thermal stress兲. The secondary stress is the equivalent linear stress produced by the radial temperature distribution in a cylindrical shell. The peak stress is the difference between the actual stress and the equivalent linear stress resulting from a radial temperature distribution. In piping terminology, the secondary stress in straight pipe is E␣⌬T1 / 2共1 − ␯兲, and the peak stress is E␣⌬T2 / 共1 − ␯兲. In 1982, the piping rules 共NB-3600兲 were changed to reclassify the ⌬T1 stress as a peak stress. An axial temperature distribution in a cylindrical shell or a temperature difference between a nozzle and the shell to which it is attached can cause a secondary stress. In piping terminology, the secondary stress is C3Eab共␣aTa-␣bTb兲. The peak stress is K3C3Eab共␣aTa-␣bTb兲.

Failure Modes The fundamental failure mode of concern for a pressureretaining component is burst. Another failure mode that is considered is plastic deformation. Plastic deformation 共yielding兲 is a functional concern more than a pressure boundary concern. An owner of a vessel will not be pleased if a Level A or B loading results in observable deformation of the vessel. From the criteria document 关2兴 … The primary stress limits are intended to prevent plastic deformation and to provide a nominal factor of safety on the ductile burst pressure. The primary plus secondary stress limits are intended to prevent excessive plastic deformation leading to incremental collapse, and to validate the application of elastic analysis when performing the fatigue evaluation. The peak stress limit is intended to prevent fatigue failure as a result of cyclic loadings.

Basis for Stress Limits The code stress limits are derived from … application of limit design theory tempered by some engineering judgement and some conservative simplifications 关2兴. Section III 共NB-3213兲 defines limit analysis and allows the use of limit analysis to establish a lower bound to the collapse load. This use of an ideally plastic material without strain hardening has led some to conclude that limit load is the failure load. But, limit load is not the failure load. The fundamental failure load of concern is the ultimate load to burst or plastic instability in the case of primary membrane stress. In the case of primary bending stress, the failure mode of concern is ultimate collapse. Transactions of the ASME


Fig. 2 Rectangular cross-section bar in tension

Pressure Design. In design-by-rule, the minimum required wall thickness for a cylinder is specified for pressure design. This minimum wall equation is not contained in NB-3200, but is contained in NB-3300 for vessels and NB-3600 for piping. The primary membrane stress limit for straight pipe 共NB-3640兲 for design pressure is met by meeting the design-by-rule minimum required wall thickness equation. tm = PDo/2共Sm + 0.4P兲

where P is the design pressure 共1兲

The piping equation is similar to the cylindrical vessel equation given in NB-3324. t = PRo/共Sm + 0.5P兲

共2兲

The piping equation was first adopted by the B31 Code in 1955. Reference 关7兴 discusses the derivation of this formula. Over 31 different formulas were considered; these formulas included elastic and plastic calculations. The final equation was selected since it … approximates satisfactorily the available room-temperature tubular bursting data 关7兴. The ASME B16.9 standard describes burst test procedures for pipe fittings. Calculated burst pressure for straight pipe is given as P = 2St / Do where S is the specified minimum tensile strength. Langer 关8,9兴 discusses pressure design of vessels for burst and provides failure pressure data on PVRC disk tests for a range of materials. The intent of the cylindrical vessel equation, as given above, is to provide a nominal factor of safety on burst of 3. But, the theoretical burst pressure is dependent on the strain hardening exponent of the material. For five different materials, the theoretical safety factor was found to vary from 2.75 to 3.34. In NB-3200, a minimum required wall thickness is not specified for pressure design. Instead, the primary membrane stress is calculated by elastic analysis and compared to a stress limit of Sm 共for Design conditions as given in Fig. 1兲. Consider the limit analysis of a simple, straight, rectangular cross-section, bar in tension as shown in Fig. 2. Assume an elastic-perfectly-plastic material model with a yield stress of Sy. Once the cross-sectional stress reaches the yield stress, the maximum load carrying capability of the bar is achieved. Applying any additional load causes the bar to deform until the failure strain is reached, and the bar ruptures. The normal stress in the bar is a primary membrane stress that is required for equilibrium with the applied external load. Instead 2 1 of using Sy as the failure criterion, the code uses Sm 共 3 yield or 3 ultimate兲. Hence, the design-by-analysis criterion for primary membrane stress provides a factor of safety of 1.5 on excessive plastic deformation 共yielding兲. The use of ultimate tensile strength in addition to yield strength to specify Sm accounts for the strain hardening characteristics of the material. Ultimate failure 共plastic in stability/rupture兲 occurs when the primary membrane stress 1 reaches the ultimate strength of the material. Hence, using Sm of 3 ultimate strength provides for a factor of safety of 3 on plastic instability or rupture failure. By the same reasoning, the code primary membrane stress criterion provides a factor of safety of 1.5 on excessive deformation Journal of Pressure Vessel Technology

共yield兲 and a factor of safety of 3 on ultimate failure 共plastic instability or burst兲 for design pressure. For Level D, the elastically calculated membrane stress is limited to 0.7Su. This means that the nominal factor of safety on burst is 1 / 0.7 or 1.43 for Level D. For piping for Level D, the elastically calculated allowable pressure is double that of the allowable pressure for Design conditions. This means that the nominal factor of safety on burst 3 for piping for Level D is 2 or 1.5. Primary Bending Stress. Primary bending stresses are limited to ␣Sm 共or less if there is pressure stress兲. The limit is given in NB-3221.3 and the Code requirement is in terms of the “Primary Membrane Plus Primary Bending Stress Intensity. The Code words are … This stress intensity is derived from the highest value across the thickness of a section of the general or local primary membrane stresses plus primary bending stresses produced by Design Pressure and other specified Design Mechanical Loads, but excluding all secondary and peak stresses. For solid rectangular sections, the allowable value of this stress intensity is 1.5Sm. For other than solid rectangular sections, a value of ␣ times the limit established in NB-3221.1 may be used, where the factor ␣ is defined as the ratio of the load set producing a fully plastic section to the load set producing initial yielding in the extreme fibers of the section. Limit analysis is used to establish the stress limit for primary bending stress. Consider the same rectangular cross-section bar with the added weight applied to cause bending as shown in Fig. 3. As weight is increased, the bending stress increases until the outer fiber stress is at the yield stress of the material 共a weight of Wy in Fig. 3兲. Since the material is ductile, the bar can withstand additional load. Yielding is spread across the section. Maximum load carrying capacity occurs when the cross-section is fully plastic. A plastic hinge forms and the bar collapses—unlimited deformation occurs. For a rectangular cross-section, the moment at collapse is 50% higher than the moment at first yield. Hence, the primary bending stress limit is 50% higher than that for primary membrane. The allowable 共from Fig. 1兲 is 1.5Sm.3 The bending stress in the bar is a primary stress that is required for equilibrium with the applied 2 1 external load. Sm is 3 yield or 3 ultimate. Therefore, the elastically predicted bending stress allowable is 1Sy or 0.5Su. The design-byanalysis criterion for primary bending permits no significant yielding for Design conditions. Use of ultimate tensile strength, as a parameter in Sm, accounts for the strain hardening characteristics of the material. Ultimate collapse for a ductile material occurs when a fully plastic hinge 3 The 1.5 factor is for a rectangular beam. NB-3221.3 refers to ␣Sm based on the shape factor for the section. NB-3600 for piping uses 1.5 for all tubular piping products.

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Fig. 3 Rectangular cross-section bar in bending

with the entire cross-section at Su exists. The elastically predicted moment stress corresponding to a fully plastic hinge is 1.5Su. Hence, an elastic bending stress limit of 0.5Su provides a factor of safety on ultimate collapse of 3 共for a design mechanical load兲. For Level D, the elastically calculated bending stress limit is 1.5 times 0.7Su. This means that the nominal factor of safety on ultimate collapse for Level D is 1 / 0.7 or 1.43. For piping for Level D, the elastically calculated bending stress is double that for Design 共3Sm versus 1.5Sm兲. This means that the nominal factor of 3 safety on collapse for piping for Level D is 2 or 1.5. Secondary Stress. Two secondary stress limits are provided as shown in Fig. 1. The quantity PL + Pb + Q and the quantity Pe are both required to be less than 3Sm for Level A and B conditions. Meeting the 3Sm limit is a precondition for the fatigue analysis. The failure mode of concern is fatigue. Therefore, the stress range, not the amplitude, is evaluated. The 3Sm stress limit was developed by considering a cyclic secondary stress range. Again, an elastic-perfectly-plastic material model is used to develop this stress limit. Consider a rectangular beam with a rotation that is applied, released, and then applied again. This is a displacement-controlled condition. The magnitude of the rotation is such as to produce an elastically predicted bending stress of 2Sy or a strain of 2Sy / E. The loading/unloading diagram is shown in Fig. 4. On the first half-cycle of rotation, the beam outer fiber will yield. But, on subsequent half-cycles, the outer fiber will not yield. This behavior is called shakedown to elastic action. Since 3Sm is equivalent to 2Sy the design-by-analysis criterion does not impose a factor of safety on shakedown to elastic action. This is reasonable since exceeding the limit does not cause failure. Limiting the thermal expansion stress range 共Pe兲 to 3Sm will ensure that the cyclic thermal expansion stresses by themselves will shakedown to elastic action. This is a basic piping design require-

ment. For other secondary stresses 共Q兲, combining primary stress range with secondary stress range will ensure that the combined primary-plus-secondary stress range 共PL + Pb + Q兲 will shakedown to elastic action. The purpose of the limit as stated in the criteria document 关2兴 is … to validate the application of elastic analysis when performing the fatigue evaluation. The criteria document also states the purpose is intended to prevent excessive plastic deformation leading to incremental collapse. The need to validate the fatigue evaluation will be discussed in the section on “Peak Stress.” Incremental collapse will be discussed in the section on “Thermal Stress Ratcheting.” As shown in that section, the primary-plus-secondary stress range limit does not provide complete protection against ratcheting. Peak Stress. Peak stresses are a concern for fatigue. The total 共primary-plus-secondary-plus-peak兲 stress range for a stress cycle is calculated. The stress evaluation must include consideration of local structural discontinuities 共stress concentration兲. One-half of the total stress range 共stress amplitude兲 is calculated and referred to as Salt. Design fatigue curves are given for various materials. Entering the fatigue curve at Salt gives an allowable number of cycles N. From the specified number of cycles, n, the fatigue damage is calculated as n / N. The damage from all stress cycles are added together, and the accumulated damage must be less than or equal to one 共兺n / N 艋 1兲. A detailed procedure is specified in NB-3222.4共5兲 for determining the effect of superposition of different stress cycles. The criteria document 关2兴 gives an excellent discussion on the generation of the design fatigue curves. The best-fit data from small polished bar specimens are provided. A factor of 2 on stress and 20 on cycles, whichever is largest, was used to establish the design curve from the best-fit curve. In the low cycle region, the factor on cycles governs. The inherent factor on stress is signifi-

Fig. 4 Shakedown to elastic action

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cantly larger than 2 in the low cycle region. The fatigue design method is straightforward. Elastically predicted total stress amplitude, including peak stress from local structural discontinuities, are compared to a design curve to determine the allowable number of stress cycles. The inherent assumption is that the net section stresses and strains are elastic. Only the peak stresses at the local structural discontinuity are in the plastic regime. Therefore, the precondition for the fatigue analysis is that the primary-plus-secondary stresses shakedown to elastic action. If the primary-plus-secondary stress range exceeds 3Sm, there is plastic cycling at the local structural discontinuity, which is very detrimental to fatigue life. The original 共1963兲 fatigue design curves were based on small polished bar specimen test data. These tests were run to the point of separation of the specimen. A common question is—“Does the fatigue design curve represent crack initiation or crack propagation through the wall thickness?” My answer is—The objective of the fatigue design method is to prevent a leakage failure of the pressure boundary. A cumulative usage factor of 1 does not mean that a crack has initiated or that a crack has propagated through the wall. A cumulative usage factor of 1 implies reasonable assurance that leakage will not occur in the design life. Simplified Elastic-Plastic Analysis. The primary-plussecondary stress range limit of 3Sm may be exceeded for a stress cycle including thermal bending if a penalty is taken on the fatigue evaluation 关NB-3228.5 Simplified Elastic-Plastic Analysis兴. The primary-plus-secondary-plus-peak stress range amplitude, Salt, is multiplied by Ke, a plastic strain correction factor. The Ke factor can be substantial. for Sn 艌 3mSm

Fig. 5 Bree diagram

Ke = 1/n = 5 for carbon steel

The maximum value for Ke is a significant design problem for severe thermal transients. Questions concerning Ke are discussed in a later section 共Meaning of Ke兲. “Thermal bending” is not specifically defined in NB-3228.5. Examples of thermal bending as given in NB-3213.13 are 共1兲 the equivalent linear stress produced by the radial temperature distribution in a cylindrical shell, 共2兲 the bending stress produced by an axial temperature distribution in a cylindrical shell, and 共3兲 stress produced by the temperature difference between a nozzle and the shell to which it is attached. By comparison with the piping rules prior to 1982, thermal bending in piping terms are E␣⌬T1 / 2共1 − ␯兲 and C3Eab共␣aTa − ␣bTb兲.

in diameter on each cycle. The ordinate is the thermal stress. With zero pressure, a thermal stress range of 2Sy shakes down to elastic cycling. A thermal stress range of 2Sy shakes down to elastic cycling as long as the pressure stress is less than 0.5Sy. Once the sustained pressure stress exceeds 0.5Sy, a thermal stress range of 2Sy will result in ratcheting for an elastic-perfectly plastic material model. Hence, the primary-plus-secondary stress range limit of 共P + Q ⬍ 3Sm = 2Sy兲 does provide protection against ratcheting as long as the sustained primary membrane hoop stress is less than 0.5Sy. As discussed in the section on “Pressure Design,” the hoop membrane stress is limited to 2 / 3Sy. The primary-plus-secondary stress range limit does not provide complete protection against ratcheting.

Thermal Stress Ratchet. Thermal stress ratchet is discussed in NB-3222.5.

Ductility

3.3 for austenitic steel

共3兲

It should be noted that under certain combinations of steady state and cyclic loadings there is a possibility of large distortions developing as the result of ratchet action; that is, the deformation increases by a nearly equal amount for each cycle. Limits for one particular loading, a through-wall temperature distribution are given. The maximum allowable range of thermal stress, as a function of the steady-state pressure stress, is given for a linear temperature distribution and a parabolic temperature distribution. These limits are based on the work of Miller 关10兴. The ratchet phenomenon can be quantified by the Bree diagram as shown in Fig. 5. The Bree analysis 关11兴 considers a cylinder with a steady-state pressure load 共primary membrane stress兲 and a linear through-the-wall temperature distribution 共secondary bending thermal stress兲 that is applied and then removed. Material properties are elastic-perfectly-plastic. A one-dimensional analysis is performed. Only the hoop direction is considered. The regimes are E for elastic behavior, S1 and S2 for shakedown to elastic action, P for plastic cycling, and R1 and R2 for ratcheting. For R1 and R2, there is an incremental plastic strain on each cycle of loading. If ratcheting occurs, the cylinder will permanently grow Journal of Pressure Vessel Technology

The design-by-analysis criteria presume ductile material behavior. Allowing secondary stresses to exceed the yield strength of the material requires that the material have sufficient ductility to accommodate the required plastic flow without failure. Typical yield strains from secondary stress, and even plastic strains at local discontinuities, are not that large in comparison to elongation to failure of 33% or more for typical carbon steels. Materials acceptable for code use are specified, but minimum ductility is not one of the specified parameters used for material selection. Material characteristics and ductility are discussed in Ref. 关12兴. One quote from this document 共1964兲 is … The amount of ductility required to insure satisfactory performance of a pressure vessel has never been definitively established. There are two other aspects of the design-by-analysis criteria that are only directly applicable if the material has sufficient ductility. For pressure design, the primary membrane stress limit is intended to provide a nominal factor of safety of 3 on burst pressure. In a cylindrical shell with a gross structural discontinuity, there will be significant secondary bending stresses at the discontinuity. The material must have sufficient ductility such that the burst pressure of the cylinder is not significantly reduced. FEBRUARY 2006, Vol. 128 / 29


Fig. 6 Discontinuity analysis

The code also allows the primary stress limits to be exceeded if it can be shown by limit analysis that the specified loadings do not exceed two-thirds of the lower bound collapse load. Limit analysis implicitly assumes that the material possesses sufficient post-yield ductility to ensure that the limit analysis is appropriate for the specified geometry.

Is it Primary or Secondary? A perennial problem in running FEA is determining the primary stress from total stress results. Judgment is definitely required. The fallback position seems to be to consider all stresses as primary, but this is unreasonable. The key to resolving a stress distribution into primary and secondary components is to understand that the primary stress is required for equilibrium with an applied mechanical load. If there is no mechanical load, or if stress is a result of compatibility considerations at a gross structural discontinuity, the stress is secondary in nature. Consider a long cylindrical vessel with a thickness change in the middle 共gross structural discontinuity兲 subjected to internal pressure 共Fig. 6兲. Simplify the structural model by assuming an abrupt change in thickness for the stress analysis. In this example, the thickness change is assumed to be sufficiently removed from the vessel ends so that end conditions do not affect the stress analysis at the discontinuity location. Pressure causes a hoop membrane stress of pR/ t 共thin wall approximation兲 resulting in hoop strain and growth in diameter. The growth is larger in the thinner member with the higher hoop stress. An internal shear and moment are required to restore compatibility at the joint. Pressure is an internal mechanical load. The hoop membrane stress is required for equilibrium with the applied pressure 共primary membrane兲. The internal shear and moment are self-equilibrating and are not required for equilibrium with a mechanical load. Hence, the shear and moment at the discontinuity are secondary in nature. Consider a second geometry—a cylindrical vessel with a flat plate closure 共Fig. 7兲. This geometry is an exception to the rule as

noted in Table NB-3217-1. The internal shear and moment are required for displacement compatibility at the vessel/plate joint. But whether the moment is classified as secondary or primary depends on how the flat plate is evaluated for pressure design. Stresses in the flat plate are dependent on the magnitude of the end moment. If the flat plate is analyzed by itself without the restraining effect of the compatibility end moment, then the moment is classified as secondary. If the flat plate is analyzed as a vessel/plate structure, then the restraining effect of the compatibility moment reduces the bending stress in the plate, and the compatibility moment is classified as primary.

Finite Element Analysis (FEA) When design-by-analysis was developed, shell analysis was the standard method for determining stresses in a vessel. With shell analysis, membrane stress and bending stress are a direct output of the analysis. And, identification of primary stresses versus secondary stresses is relatively straightforward but still requires judgment. Now, the more common analysis method is FEA. And with FEA comes many questions on interpretation of results. The first problem was linearization. To determine membrane stress or bending stress, common practice is to select a “cut line” on the model and interpolate between discrete stress output points to determine the average 共membrane兲 and linear 共bending兲 stresses across the wall. Many different methods have been tried. The problem is compounded by the fact that the stress intensity referred to as Pb or Q . . . do not represent single quantities, but sets of six quantities representing the six stress components ␴t , ␴l , ␴r , ␶lt , ␶lr , ␶rt 共footnote 2 to Fig. NB-3222-1兲. Some people say that design-by-analysis stress criteria are not applicable for FEA. I disagree. The stress criteria apply. The implementation of FEA and the interpretation of results need to be improved. A PVRC project was established to provide guidance. The project report 关13兴 is very informative with a discussion of

Fig. 7 Moment in flat plate with and without discontinuity moment

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linearization, stress categories, example problems, and recommendations. However, there are major problems with the approach in my opinion. The report is oriented to proper determination of membrane stress and bending stress without categorization as primary or secondary. For primary stresses, a separate “equilibrium” analysis or plastic analysis is recommended. This may be a workable solution. But the report gives the impression that the only other solution is to consider all membrane and bending stresses at a discontinuity as primary. This is unreasonable in my opinion. A knowledgeable engineer is able to separate primary from secondary in FEA results at a discontinuity using the principle that the primary stress is that required for equilibrium with the applied load. If the stress is not required for equilibrium with the applied load, then that stress is secondary. The report also recommends determination of P + Q at a “structural element” not a “transition element.” This is a significant limitation. In my experience, many analyses are performed to determine thermal gradient stresses for the fatigue evaluation. The maximum thermal gradient stress is usually in the transition element, and the P + Q stress is needed in the transition element to determine Ke for the fatigue evaluation. The structural element approach does not seem to be workable for fatigue damage calculations.

Meaning of Ke The maximum value of Ke 共1 / n equal to 5 for carbon steel, 3.3 for austenitic steel兲 has been a concern for design. For many of the high thermal transient situations in nuclear applications, the allowable number of cycles is very low because of the high Ke penalty factor on the fatigue evaluation. Recalling the criteria, the range of primary-plus-secondary stress including thermal bending may exceed the 3Sm shakedown criterion if a penalty factor is taken on fatigue. Reference 关14兴 discusses a different procedure for calculating the fatigue penalty factor as proposed for use in the French code. A Ke factor of 1 / n is not applied to the thermal bending stress; 1 / n is only applied to the mechanical stress. The strain concentration factor applied to thermal bending is based on the Neuber rule. There is one statement in Ref. 关14兴 that I do not agree with— The ASME III NB 3200 rule for Ke definition is clearly devoted to elastic-follow-up effects as stated in Ref. 6.4 The intent of the 1 / n factor is a critical issue. To understand the use of Ke it is necessary to review the development of the simplified elastic-plastic method. This method was originally developed for piping and published in 1969 in B31.7 关3兴. The problem in Class 1 nuclear piping was that secondary thermal gradient stresses were exceeding the 3Sm shakedown to elastic action limit in many cases. There was no technique available to qualify the piping for fatigue without a simplified elasticplastic method. Hence, the B31.7 approach was developed. Secondary thermal gradient stresses 共thermal bending in NB-3200 terms兲 could exceed 3Sm provided a penalty was taken on the fatigue analysis. A full discussion of the development of the simplified elasticplastic rules is given by Slagis 关15兴. The background and technical basis for the B31.7 approach is explained by Tagart in Ref. 关16兴. The B31.7 approach included two penalty factors—a notch factor and a plastic strain redistribution factor. The notch factor accounts for detrimental effects of plastic cycling at a stress concentration. The plastic strain redistribution factor accounts for underestimation of strain by elastic analysis at a gross structural discontinuity when the weaker member yields. The 1968 edition of Section III did not have simplified elasticplastic rules to allow secondary stresses to exceed 3Sm. When the B31.7 rules were incorporated into Section III in 1971, the sim4

The quoted Ref. 6 is Ref. 关9兴 in this document.

Journal of Pressure Vessel Technology

plified elastic-plastic rules for piping were revised and comparable rules were introduced into NB-3200. The A factor was eliminated, and a single fatigue penalty factor Ke was introduced. Ke = 1.0

for Sn 艋 3Sm

= 1.0关共1 − n兲/n共m − 1兲兴 ⫻关共共Sn/3Sm兲 − 1兲兴 = 1/n

for 3Sm ⬍ Sn ⬍ 3mSm

for Sn 艌 3mSm

共4兲

The single Ke factor with a maximum of 1 / n is based on the work of Langer 关9兴. The n parameter is the strain hardening exponent for the material. Included in Ref. 关15兴 are summaries of test data, discussion of the problems with the 1 / n approach, and recommendation for a two separate factors approach as was done in B31.7.

Summary Stresses are categorized as primary, secondary, or peak. Primary stresses are a concern for deformation, burst, or collapse. Secondary stresses are limited to require shakedown to elastic action to ensure the applicability of the fatigue evaluation. Peak stresses are a concern for fatigue. Only primary stresses are evaluated for Level C and D. The Level C and D stress limits permit large deformations that may require repair or replacement of the component. Implicit in the Level C and D limits are lower factors of safety against failure based on lower probability of occurrence of the load. Primary stresses are required for equilibrium with an internal or external applied mechanical load. Pressure is a mechanical load and causes primary stress. Thermal expansion in a piping system, or a through-wall temperature distribution, is not a mechanical load and, therefore, produces a secondary stress. A secondary stress is displacement controlled and is self-limiting. A mechanical load can also cause secondary stresses. Stresses from internal forces and moments, required for compatibility at a gross structural discontinuity, are secondary. Stress limits are derived from application of limit design theory, but limit load is not the failure criterion for primary stress. Burst and collapse are the fundamental failure modes of concern for primary stress. Primary membrane stresses are limited to 1Sm for Design conditions. Sm is the lesser of 2 / 3Sy or 1 / 3Su. Cylinder burst test data indicate that failure will occur when the hoop membrane stress reaches the ultimate stress of the material. Hence, the primary stress limits for Design provide for a nominal factor of safety of 3 on burst. For Level D, the nominal factor of safety is reduced to 1.43. Primary bending stresses 共actually, membrane plus bending兲 are limited to ␣Sm for Design conditions. For a rectangular section, ␣ is 1.5. The limit for bending is higher than for membrane because of the plastic hinge effect. The elastically predicted moment stress corresponding to a fully plastic hinge is 1.5Su. Hence, an elastic bending stress limit of 0.5Su 共1.5⫻ Su / 3兲 provides a factor of safety on ultimate collapse of 3 for a design mechanical load. For Level D, the nominal factor of safety is reduced to 1.43. The primary-plus-secondary stress range limit of 3Sm is to ensure shakedown to elastic action of the through-wall membrane and bending stresses. If the through-wall membrane and/or bending stresses exceed the limit, plastic cycling rather than elastic cycling will occur. Plastic cycling at a local structural discontinuity, such as a notch, is detrimental to fatigue life. The secondary stress limit also provides protection against ratcheting as long as the hoop membrane pressure stress is less than 0.5Sy. The elastically predicted primary-plus-secondary-plus peak stress range for each unique stress cycle is used in the fatigue evaluation. Acceptable number of cycles is determined from a design fatigue curve. The design fatigue curve is based on best-fit polished bar specimen data. Factors of 2 on stress and 20 on FEBRUARY 2006, Vol. 128 / 31


cycles are used on the best-fit data curve to obtain the design curve. The design curve is not based on crack initiation. A cumulative usage factor of 1 implies reasonable assurance that leakage will not occur in the design life. Simplified elastic-plastic analysis rules are provided in designby-analysis. The primary-plus-secondary stress range limit of 3Sm may be exceeded for thermal bending provided a penalty factor, Ke, is taken on the fatigue analysis. Thermal bending is the secondary bending from a through-wall temperature gradient or a mean temperature difference. In piping terms, these stresses are E␣⌬T1 / 2共1 − ␯兲 and C3Eab共␣aTa − ␣bTb兲. The maximum value for Ke, 1 / n, is extremely conservative and should be revised. The simplified elastic-plastic analysis method was first developed for piping and published in B31.7. Two penalty factors were specified in the B31.7 method—a notch factor and a plastic strain redistribution factor. This is the approach that should be adopted for design-by-analysis. The notch factor accounts for plastic cycling at a local structural discontinuity. The plastic strain redistribution factor accounts for underestimation of plastic strain by elastic analysis at a gross structural discontinuity. NB-3200 contains code rules on “Thermal Stress Ratchet.” To prevent ratcheting, limits are placed on thermal stress from through-wall temperature distributions as a function of the value of sustained pressure membrane stress. Ratcheting is incremental deformation on each cycle of loading. The critical parameters are a sustained primary stress and a cyclic secondary stress. General loading cases can be evaluated by the Bree diagram. This is a one-dimensional analysis based on elastic-perfectly plastic material behavior. The design-by-analysis criteria implicitly assume materials with sufficient ductility to accommodate the required plastic flow without failure. A measure of “sufficient ductility” has not been quantified to date. The burst pressure of a vessel with a gross structural discontinuity could be significantly reduced if the material does not have sufficient ductility. Direct application of the design-byanalysis criteria to high strength materials with low ductility is questionable in my opinion. One major problem in finite element analysis is separating primary stresses from secondary stresses. The analyst must exercise competent engineering judgment. The key to making a decision is that primary stresses are required for equilibrium with an applied mechanical load. In general, shear and local bending moment stresses at a gross structural discontinuity are secondary in nature.

Nomenclature A C3 Do E Eab K3 Ke

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

notch factor in B31.7 piping secondary thermal stress index outer diameter modulus of elasticity average modulus of two sides of a joint piping peak thermal stress index plastic strain correction factor

32 / Vol. 128, FEBRUARY 2006

K␧ ⫽ m,n⫽ Ro ⫽ Salt ⫽ Sm ⫽ Sn ⫽ Sp ⫽ Su ⫽ Sy ⫽ T ⫽ tm ⫽ ␣ ⫽ ␯ ⫽ ⌬T1 ⫽ ⌬T2 ⫽

plastic strain redistribution factor in B31.7 material parameters outer radius primary-plus-secondary-plus peak stress amplitude material allowable stress range of primary-plus-secondary stress intensity total stress intensity range material ultimate tensile strength material yield stress through-wall mean temperature minimum required wall thickness coefficient of thermal expansion Poisson’s ratio linear portion of through-wall temperature gradient nonlinear portion of through-wall temperature gradient

References 关1兴 ASME Boiler and Pressure Vessel Code Section III, 1963 Edition, “Rules for Construction of Nuclear Vessels.” 关2兴 Criteria of the ASME Boiler and Pressure Vessel Code for Design by Analysis in Sections III and VIII, Division 2, 1969, ASME. 关3兴 USA Standard Code for Pressure Piping, “Nuclear Power Piping,” USAS B31.7–1969, ASME. 关4兴 ASME Boiler and Pressure Vessel Code Section III, 2001 Edition, “Rules for Construction of Nuclear Facility Components.” 关5兴 Bohm, G. J., and Stevenson, J. D., 1982, “Extreme Loads and Their Evaluation With ASME Boiler and Pressure Vessel Code Limits,” Pressure Vessel and Piping: Design Technology—1982—A Decade of Progress, ASME, pp. 415– 418. 关6兴 Slagis, G. C., 1991, “Basis of Current Dynamic Stress Criteria for Piping,” Weld. Res. Counc. Bull., 367, pp. 15–16. 关7兴 Burrows, W. R., Michel, R., and Rankin, A. W., 1952, “A Wall Thickness Formula for High-Pressure, High-Temperature Piping,” ASME Paper No. 52A-151, p. 6. 关8兴 Langer, B. F., 1972, “PVRC Interpretive Report of Pressure Vessel Research, Section I—Design Considerations,” in Pressure Vessel and Piping: Design and Analysis, A Decade of Progress, Volume 1, Analysis, ASME, New York, pp. 8–60 关reprinted from Weld. Res. Counc. Bull., 95, 1964兴. 关9兴 Langer, B. F., 1972, “Design-Stress Basis for Pressure Vessels,” in Pressure Vessel and Piping: Design and Analysis, A Decade of Progress, Volume 1, Analysis, ASME, New York, pp. 84–94 关reprinted from Exp. Mech., 1971兴. 关10兴 Miller, D. R., 1959, “Thermal-Stress Ratchet Mechanism in Pressure Vessels,” J. Basic Eng., 81, pp. 190–196. 关11兴 Bree, J., 1967, “Elastic-Plastic Behavior of Thin Tubes Subjected to Internal Pressure and Intermittent High-Heat Fluxes with Application to Fast-NuclearReactor Fuel Elements,” J. Strain Anal., 2共3兲, pp. 226–238. 关12兴 Gross, J. H., 1972, “PVRC Interpretive Report of Pressure Vessel Research, Section 2—Material Considerations,” in Pressure Vessel and Piping: Design and Analysis, A Decade of Progress, Volume 3, Materials and Fabrication, ASME, New York, pp. 4–34 关reprinted from Weld. Res. Counc. Bull., 95, 1964兴. 关13兴 Hechmer, J. L., and Hollinger, G. L., 1998, “3D Stress Criteria Guidelines for Application,” Weld. Res. Counc. Bull., 429. 关14兴 Grandemange, J. M., Heliot, J., Vagner, J., Morel, A., and Faidy, C., 1991, “Improvements on Fatigue Analysis Methods for the Design of Nuclear Components Subjected to the French RCC-M Code,” Weld. Res. Counc. Bull., 361. 关15兴 Slagis, G. C., 2005, “Meaning of Ke in Design-by-Analysis Fatigue Evaluation,” PVP2005-71420. 关16兴 Tagart, S. W., 1972, “Plastic Fatigue Analysis of Pressure Components,” in Pressure Vessel and Piping: Design and Analysis, A Decade of Progress, Volume 1, Analysis, ASME, New York, pp. 209–226 关reprint of ASME Paper No. 68-PVP-3, 1968兴.



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