Design of Superconducting Cavities for LinearParticle Accelerators using ASME Section VIII,Division 2, Design-by-Analysis Requirements

Donato Passarelli†,‡,∗, Robert H. Wands†, Margherita Merio†, Leonardo Ristori††Fermi National Accelerator Laboratory, 60510 Batavia, IL, USA

‡Department of Civil and Mechanical Engineering, University of Pisa, 56122 Pisa, Italy

This paper presents the procedures to design a supercon-ducting cavity using the rules of the American Society of Me-chanical Engineers (ASME) Boiler and Pressure Vessel Code(BPVC). Several factors such as exotic materials, unquali-fied brazing procedures, limited nondestructive examination,and the general R&D nature of these early generations ofcavity design, conspire to make it impractical to achieve fullcompliance with all ASME BPVC requirements. However, itis the intent of the cavity developers that these devices meetthose requirements to the greatest extent possible. The proce-dure adopted to design a superconducting cavity, the SingleSpoke Resonator of type 1 (SSR1), is reported in this article.

1 IntroductionSuperconducting radio-frequency (SRF) cavities are

crucial components of modern high-performance particle ac-celerators to impart energy to the charged particles. Theirdramatically lower electrical losses allow operation at sub-stantially higher duty cycles than conventional copper cav-ities. Accelerators are used in a large number of areas forhigh-energy physics, low-energy to medium-energy nuclearphysics research and free-electron lasers. Nowadays, theyare also essential tools in industry for industrial processes, inmedicine for cancer therapy, for national security, and manyadditional future accelerator applications are envisioned andunder studies.

From a mechanical engineering standpoint, a dressedSRF cavity is typically comprised of an inner niobium vessel(or SRF cavity) surrounded by a liquid helium containmentvessel made of stainless steel or titanium. The helium bathmay reach pressures exceeding 15 psi and generally it has avolume greater than five cubic feet. All this leads to considera dressed SRF cavity such as a system of pressure vessels.

Based on the Department of Energy (DOE) directive 10CFR 851, it is mandatory for safety reasons that all pressuresystems designed, fabricated and tested by U.S. NationalLaboratories conform to ASME Codes. As a consequence,dressed SRF cavities fall within the scope of the following

∗Address all correspondence to this author: donato@fnal.gov

sections of the ASME Boiler and Pressure Vessel Code:

Section VIII - Pressure Vessels, Division 1 and 2Section II - Materials, Parts A through DSection V - Nondestructive ExaminationSection IX - Welding and Brazing Qualifications

It is acknowledged that a true Code design is not cur-rently possible, primarily due to the use of non-Code mate-rials, the unfeasibility of Code-required nondestructive ex-aminations of welded joints and the use of unqualified pro-cedures for welding and brazing. A set of rules have beendeveloped by engineers at Fermi National Accelerator Labo-ratory [1] based on their current understanding of best prac-tice in the design, fabrication, examination, testing, and op-eration of the dressed SRF cavities. These guidelines com-ply with Code requirements wherever possible, and for non-Code features, procedures were established to produce alevel of safety consistent with that of the Code design.

2 Design Methods for Dressed SRF CavitiesThe mechanical design of dressed SRF cavities can

be approached starting with the ASME BPVC, SectionVIII [2], [3]. This section provides detailed requirements forthe design, fabrication, testing, inspection, and certificationof both fired and unfired pressure vessels. Section VIII con-tains three divisions, each of which covers different vesselspecifications. Division 3 provides rules to pressure vesselsthat operate at pressures exceeding 10,000 psi. It is not ap-plicable to the design of SRF cavities since they are designedfor much smaller working pressures.

Division 1 (Div. 1) is directed at the economical designof basic pressure vessels, intending to provide functional-ity and safety with a minimum of analysis and inspection.Common components’ geometries can be designed for pres-sure entirely by these rules but they may be not enough forthe design of all cavity components under all loading cases(cooldown, additional forces, etc.). Nondestructive exami-nations (NDE) of welds can typically be avoided by taking apenalty in overall thickness of a component.

Division 2 (Div. 2) is directed at engineered pressurevessels, which can be thought of as vessels whose perfor-mance specifications justify the more extensive analysis andstricter material and fabrication controls and NDE requiredby this Division. The design is governed by two loosely-coupled provisions: Part 4 (Design by Rule), and Part 5(Design by Analysis). A device may be designed by eitherPart; regardless of the Part used, the provisions of Part 3, 6,and 7 (materials, fabrication, and inspection) must be met.The rules of Part 4 are very thorough, duplicating manyof the rules of Div. 1, while expanding them to cover awider range of geometries. The rules of Part 5 provide fora strictly analytical approach to the vessel design. A numeri-cal analysis technique is assumed, and either an elastic or anelastic-plastic analysis is permitted. The mandatory NDE forwelded joints in this Division is extensive.

The ASME BPVC Section II is a “Service Section” forreference by the BPVC construction Sections providing ta-bles of material properties including allowable, design, ten-sile and yield stress values, physical properties and externalpressure charts and tables. Part D contains appendices whichinclude criteria for establishing allowable stresses. Some ofthe materials for the construction of SRF cavities are notaccepted by the Code, as being suitable for construction ofpressure vessels. As a result, the mechanical properties ofthese materials are not available in Section II, Part D of theCode. Therefore, experimental tests have to be used in thedetermination of the material properties for non-Code recog-nized materials. Subsequently, the material properties deter-mined have to be utilized to calculate the maximum allow-able stress values using the Code methodology.

The ASME BPVC - Section V contains requirementsand methods for nondestructive examination which are ref-erenced and required by other BPVC Sections (i.e. Sec-tion VIII). Examination methods are intended to detect sur-face and internal discontinuities in materials, welds, and fab-ricated parts and components. Examination per the ASMEBPVC is not practical because SRF cavities are constructedof non-Code materials. The ASME Process Piping Code,B31.3, does allow for construction with non-Code materialsand is deemed more applicable to the SRF cavity.

The ASME BPVC - Section IX contains rules relatedto the qualification of welding, brazing, and fusing proce-dures as required by BPVC Section VIII for component man-ufacture. It also covers rules relating to the qualificationof welders, brazers, and welding, brazing and fusing ma-chine operators. The manufacturing of dressed SRF cavi-ties implies the use of electron-beam welding, gas tungstenarc welding (also known as tungsten inert gas welding), andbrazing. Procedures that will guarantee a reasonable level ofcertainty that the SRF accelerating structure to be fabricatedwill be in compliance with ASME BPVC must be developed.In each case, if the welded joint or brazed joint is not a stan-dard ASME Code joint, the development must also includesufficient analysis and mock-up testing to support the conclu-sion of equivalent safety. A base set of acceptable weld andbraze parameters has to be established for each non-standardjoint to assure their integrity examining with a microscope,

metallograph or SEM weld and braze samples made by usingthe contractor’s welding/brazing machine. Weld and brazesamples for each joint must be as representative as possible:mass, geometry and material thickness, of the actual joint onthe structure. Moreover, a sufficient number of samples perweld and braze should be produced to allow tensile tests andbend tests (face and root) at 300 K, 77 K and 4 K. Samplesmust also be radio-graphed or ultrasonically examined.

3 SSR1 CaseSRF cavities called Single Spoke Resonators of type

1 (SSR1) are fundamental to the realization of a linearparticle accelerator at Fermi National Accelerator Labora-tory [6], [7]. The dressed SSR1 cavity consists of two nestedcryogenic pressure vessels: the inner vessel is the supercon-ducting SSR1 cavity, see Fig 1, and the outermost vessel isthe helium containment (or helium) vessel, see Fig 2.

The parts of the cavity are formed and machined of high-purity niobium (Nb), and joined by electron-beam welding.Four flanges on the cavity, through which it interfaces withthe helium vessel, are made of stainless steel connected tothe niobium by means of copper-braze joints. The heliumvessel is entirely made of 316L stainless and it is assembledaround the cavity by full penetration gas tungsten inert gas(TIG) welds.

SHELL

CIRCUMFERENTIALnRIBn(x4)

ENDWALL(x2)

DONUTnRIB(x2)

DAISYnRIB(x12)

BEAMnPIPE(x2)

BPnFLANGE(x2)

SPOKEnCOLLAR(x2)

HALFnSPOKE(x2)

SPOKEnBEAMnPIPE(x2)

VACUUMnPORTnFLANGE

TUBEnINLET(x2)

COUPLERnPORTFLANGE

TRANSITIONnRINGNb-side

TRANSITIONnRINGStainless-side

Fig. 1. Exploded view of the niobium SSR1 cavity

The typical operating temperature of an SRF cavity isin the range from 1.8 K to 2.1 K. A bath of liquid helium,confined by the helium vessel, surrounds the cavity exertinga pressure on both vessels. The cavity is internally pumpeddown to ultra-high vacuum (for the particle beam), and theentire dressed cavity is placed in a cryostat under insulatingvacuum.

The greatest risk with vessels containing superfluid he-lium is that an accidental loss of vacuum results in very rapidboiling of the helium, causing a consequent pressurization ofthe system. Moreover, differential pressure can be detectedbetween the volumes defined by cavity and helium vesselduring the first phases of operation before the cooldown.The authors were charged to design and develop a dressed

SSR1 cavity that has a maximum allowable working pres-sure (MAWP) greater than 0.2 MPa at 293 K and 0.4 MPa at2 K.

DOME1BS

BELLOWS1ASSEMBLY

PLATE1BS

ADAPTER1RING

CBSC:1Bellows1SideCRSC:1Ring1Side

PLATE1RS

DOME1RS

TRANSITION1RING

SHELL1-1LOWER

TUNER1SUPPORT(x2)

SUPPORT1BASE(x2)

SSR1-G31CAVITY

SHELL1-1UPPER

TUBE1INLETFLANGE

LUG1-1LIFTING(x8)

Fig. 2. Exploded view of the stainless helium vessel surroundingthe SSR1 cavity

4 Design and Analysis of Dressed SSR1 CavityThe need to optimize the complex shape of the dressed

SSR1 cavity under several loading conditions led to ap-proach the design using Div. 2 of the ASME BPVC Sec-tion VIII. The design procedure principally consists of twomethods: design-by-rules requirements, described in Part 4,and design-by-analysis requirements, described in Part 5.

The rules of Part 4 were used to provide useful checks ofnumerical simulations carried out in accordance with Part 5.Part 4.3 was used to set minimum thickness for elementarygeometries such as the cylindrical shell and conical platesof the helium vessel, which are subject to internal pressure1.The resulting minimum thickness of the stainless vessel was2 mm. Part 4.4 was used to set minimum thickness of thecylindrical shell of the niobium cavity, resulting in a wallthickness of 1.5 mm. These minimum thickness were usedas a starting point for the material optimization, and laterincreased to assure stiffness and strength under all loadingconditions. Part 4.5 was followed to design nozzles in theshell and heads of the helium vessel subject to internal pres-sure. Part 4.2 addresses the design of welded joints and wasused extensively in the design of the SSR1 system (see Sec-tion 5). The rules in paragraph 4.19 were applied to designa U-shaped unreinforced bellows expansion joint having hy-droformed convolutions and two collars welded at the endtangents. The bellows is considered being part of the heliumvessel and is made entirely of 316L stainless steel.

The geometries of the structures and loading conditionsof SRF cavities, and therefore the stresses distributions, are

1In this context, internal pressure is defined as pressure acting on theconcave side of the shell

often complicated and do not lend themselves entirely todesign by design-by-rules method. The design-by-analysismethod can be used to optimize those features not amenableto design-by-rules method. Design-by-analysis method as-sumes a numerical analysis technique will be used, and eitherelastic or elastic-plastic analysis is permitted. In the case ofthe SSR1, ANSYS structural analysis software [5] was usedto perform the finite element analyses and provide protec-tion against four modes of failure: plastic collapse, buckling,cyclic loading and local fracture.

4.1 Material PropertiesThe main structural components of the system are con-

structed from type 316L stainless steel and ultra-pure nio-bium. For the analysis, thermal and mechanical properties aswell as allowable stresses for both materials are required, atboth room temperature and 2 K. While the ASME Code pro-vides Young’s modulus and allowable stresses for type 316Lstainless at room temperature, it contains no cryogenic prop-erties for this stainless steel, and makes no mention at all ofniobium.

4.1.1 316L Stainless SteelType 316L stainless steel is an ASME Code material

and has linear properties as well as allowable stresses (S)for elastic analysis at room temperature are provided in Sec-tion II, Part D [4]. However, mechanical properties at cryo-genic temperature are not given by the Code. Since the yieldand ultimate stress of austenitic stainless steels increases sub-stantially at cryogenic temperatures, a conservative approachwas chosen and room temperature properties were used. Ad-ditionally, the coefficient of thermal expansion (CTE) overthe wide temperature range from 293 K to 2 K is not avail-able from the Code, but is well known in literature [8], [9].

The elastic-plastic stress strain curve required for theplastic collapse analyses was derived from the Ramberg-Osgood correlations, which use the Code yield and ultimatestresses in combination with other parameters provided byDiv. 2, Part 3, Annex 3.D [3].

Table 1. Mechanical properties of type 316L stainless steel

T[K] E[GPa] ν Sy[MPa] Su[MPa] S[MPa]

293 195 0.30 172 517 114

2 Conservatively assumed equal to 293 K properties

4.1.2 NiobiumThe ultra-pure niobium (300 RRR) is a non-Code ma-

terial. Experimental tests and literature data were used todefine the mechanical properties of pure niobium at roomtemperature and cryogenic temperature, see Table 2.

0 0.1 0.2 0.3 0.4 0.50

100

200

300

400

500

600

700

800

900

Plastic strain

Stress [MPa]

Fig. 3. Stress strain curve for 316L stainless steel derived from theRamberg-Osgood correlation using the following fitting parametersm2 = 0.75(1.00−R) and εp = 2×10−5

The elastic-plastic stress strain curves at both 293 K and2 K were derived from the same Ramberg-Osgood corre-lations mentioned above. However, since niobium is not aCode material, parameters specific to it were not available.Instead, the parameters for titanium were used. The possibil-ity of using copper parameters was explored, but it was foundthat even though the difference between the two curves wasvery small, the behavior near the yield point was slightly lessconservative with copper parameters.

Table 2. Mechanical properties of Niobium

T[K] E[GPa] ν Sy[MPa] Su[MPa] S[MPa]

293 105 0.38 74 160 46

2 105 0.38 317 600 171

4.1.3 Brazed JointThe connections between Niobium and Stainless Steel

were made adopting a copper-brazed transition tech-nique [10]. The filler metal is Oxygen Free Electronic Cop-per (CDA-101). Full scale samples were constructed andqualified through a series of leak tests, load tests, thermalcycling, visual and microscopy (SEM) inspection [11]. Re-sults from tensile tests on brazed specimens were taken intoaccount to define the allowable stress (S), see Table 3. Loadtests at cryogenic temperature were not performed but it isreasonable to assume that the static mechanical properties arehigher than at room temperature. Thus, to be conservative,the allowable stress at cryogenic temperature was assumedequal to the value at room temperature.

0 0.05 0.1 0.15 0.2 0.25 0.30

100

200

300

400

500

600

700

800

900

Plastic strain

Stress [MPa]

@T=2K @T=293K

Fig. 4. Stress strain curves for pure niobium derived from theRamberg-Osgood correlation using the following fitting parametersm2 = 0.50(0.98−R) and εp = 2×10−5

Table 3. Allowable stress of Cu-brazed joint defined according toTable 1-100 of Section II [4]

T[K] E[GPa] ν Sy[MPa] Su[MPa] S[MPa]

293 – – – 120 34

2 Conservatively assumed equal to 293 K properties

4.2 Loads and Load Case CombinationsAccording to Table 5.5 of Div. 2, the loads applied to

SSR1 system have been identified (see Fig 5) and they are:

P - Pressure into the helium space to satisfy the condi-tions of MAWP = 0.2MPa before the cooldown at 293 Kand MAWP = 0.4MPa at 2 KpHe - Static head from liquid helium (considered as neg-ligible)D - Dead weight of the vessel and cavity: 1250 NT1 - Elastic tuning, maximum displacement of 0.26 mmof the beam pipeT2 - Cooldown from 293 K to 2 K

Four load case combinations have been defined and reportedin Table 4 to study all critical conditions that might occurto the SSR1 system. Load cases 1 and 2 are representativeof the critical scenarios at room temperature (293 K), wherethe system is stressed by dead weight (D), a maximum of0.2 MPa of differential pressure (P) into the helium space,and maximum displacement of the beam pipe (T1) to allowresonant frequency adjustment by a device named “tuner”.The worst scenarios of loading at 2 K are described by loadcases 3 and 4, where the system is subject to thermal stressesdue to the cooldown (T2), the dead weight (D), a maximumof 0.4 MPa of differential pressure (P) into the helium space,and maximum action of the tuner (T1).

RF volume

Helium space

Insulating volume

T1

293 K

2 K

T2 gravity

D = mg

P (MAWP)

Nb cavity

Helium Vessel

Fig. 5. Dressed SSR1 system with schematic loads applied. Thepressure (P) is applied normal to the surfaces defining the heliumspace, the frequency tuning displacement (T1) is applied normal tothe beam pipe (along the arrows), the gravitational force (D), andthe cooldown (T2) applied to the entire system

Table 4. Load cases and design temperature used to study the be-havior of the SSR1 system. Load combination factor will be appliedfor specific analyses

Case 1 Case 2 Case 3 Case 4

P+D P+D+T1 P+D+T2 P+D+T1 +T2

293 K 293 K 2 K 2 K

4.3 Protection Against Plastic CollapseThe analysis for this failure mode focuses on the inter-

nal pressure of the vessel and prevents plastic instability, en-suring that the pressure vessel does not experience plasticdeformation that may lead to collapse. Moreover, the analy-sis avoids unbound displacement in each cross-section of theSSR1 system.

Three separate approaches are permitted by Div. 2,Part 5.2, for protection against plastic collapse: elastic stressanalysis, limit load analysis, and elastic-plastic analysis. Theelastic-plastic stress analysis method was chosen, as it per-mits the most sophisticated material property characteriza-tion, and is likely to produce the highest MAWP.

4.3.1 Elastic-Plastic Stress Analysis MethodThe Load Combination Factor (LCF) for each load case

is defined using Table 5.5 of Div. 2, having LCF = 2.4 forload cases 1 and 2, and LCF = 2.1 for load cases 3 and 4.

Elastic plastic analyses were performed for each loadcase combination by applying the loads in two steps. In thefirst step, the factored dead weight, thermal contraction, andtuner displacement were applied. In the second load steponly the pressure was applied, and increased incrementallyuntil collapse (i.e., failure of the finite element model to con-verge) occurred. The MAWP was then calculated by divid-

Table 5. MAWP for Plastic Collapse analyses

Case 1 Case 2 Case 3 Case 4

P+D P+D+T1 P+D+T2 P+D+T1 +T2

0.244 MPa 0.323 MPa 0.897 MPa 0.897 MPa

ing the value of pressure at the last converged solution by theLCF. Table 5 summarizes the values of MAWP for the SSR1system determined for each load case combination using thisapproach. Figure 6 shows the location where the failure ini-tiates in all case combinations. In all listed cases, the valuesof MAWP are always above the specified requirements.

Fig. 6. The failure due to Plastic Collapse initiates on the highlightedarea of the niobium cavity

4.4 Protection Against Local FailureProtection against local failure is directed at identified

regions of high triaxial stress, which may not produce signif-icant von Mises stress, but could produce tensile strains suf-ficient to fracture the material. By Part 5, paragraph 5.3 pro-tection against local failure may be assessed by either elasticanalysis, or elastic-plastic analysis.

The elastic analysis approach was chosen for this work.Paragraph 5.3.2 states that, for each point in the component,the following condition must be met:

(P1 +P2 +P3)≤ 4S (1)

where P1, P2, and P3 are the principal stresses, and S is theallowable stress. Very little guidance is given for implemen-tation, other than that the primary local membrane plus bend-ing stress is to be used for the evaluation. This implies thatpeak stresses need not be considered.

Table 6. Results from Linear Buckling analyses

Case 1 Case 2 Case 3 Case 4

P+D P+D+T1 P+D+T2 P+D+T1 +T2

ΦB = 16.75 ΦB = 16.5 ΦB = 7 ΦB = 7

For this analysis, it was decided that it would be con-servative to consider each finite element as a “point”, sincethis would include peak stress effects. A macro was writtenin ANSYS to extract the centroidal values of the principalstresses in each element of the model, and used to verify thatthe requirement expressed above by Eq 1 was satisfied. Allshells, formed heads and ribs on the cavity and the heliumvessel passed the criterion as enforced here, for all load casecombinations.

4.5 Protection Against Collapse from BucklingStructural instability, or buckling, can lead to the sud-

den failure of a mechanical component. The load at whichbuckling occurs depends on the stiffness of the component,not on the strength of its materials, and the loss of stabilitycan happen even when component stresses are in the linearelastic range.

For the SSR1 system, the thin-walled niobium cavity(2.8 mm) is under compressive stress from external pressure,and is therefore susceptible to buckling.

The effects of initial imperfections of geometry were notconsidered in the plastic collapse analysis. When such im-perfection are included, Part 5, paragraph 5.4 does not re-quire a separate buckling analysis, since the maximum ef-fects of out-of-roundness, etc., would have been includedin the plastic collapse calculation. However, it is permittedto perform a separate linear buckling (bifurcation, or Eulerbuckling) analysis on the “perfect” geometry, provided therequired design factors are adhered to.

4.5.1 Bifurcation Buckling Analysis - Type 1The design factor is based on the type of buckling anal-

ysis performed and in this case it shall be:

ΦB ≥2

βcr= 2.5 (2)

A capacity reduction factor βcr = 0.8 was chosen because themost probable area of collapse in the SSR1 system is similarto an unstiffened cylinder.

The four load cases shown in Table 4 were applied in thepre-stress load conditions. For each one of them the load de-sign factor was calculated and is reported in Table 6. In all ofthe listed cases, the safety factor on buckling always satisfiesthe design factor of Eq 2, and the first component to buckleis the shell of the niobium cavity, as shown in Figure 7.

Fig. 7. Failure due to buckling initiates on the red marked area ofthe niobium cavity

4.6 Protection Against Failure from Cyclic LoadingParagraph 5.5 of Part 5 provides procedures to ensure

protection against failure under cyclic loading. Two types ofcyclic loading are addressed: fatigue, and ratcheting. In thecase of fatigue, a screening procedure is presented to deter-mine if the number and magnitude of stress cycles is enoughto require a full fatigue analysis.

4.6.1 Fatigue Analysis Screening, Method AIn SRF structures, fatigue cycles result from frequency

adjustments (tuning cycles, T1) and temperature changes(cooldown/warmup, T2). Frequency tuning comprises of“coarse tuning” and “fine tuning”. The contribution of thefine tuning is not considered in the analyses because of itssmall amplitude.

Paragraph 5.5.2 provides two screening methods to de-termine if a fatigue analysis is required. Method A can beused for materials with a specified minimum tensile strengththat is less than or equal to 552 MPa. This is not the casefor niobium at cryogenic temperature, where Su =600 MPa.However, Method B, which does not have restrictions on ma-terial strength, requires the use of a fatigue curve generatedby the procedures of Annex 3.F of Part 3, Div. 2. The pa-rameters for generating this curve are material-specific, andbecause niobium is not a Code material, no parameters areavailable. It was decided, since both methods require step-ping outside the strict boundaries of the Code, that Method Awould be the best choice for determining if a fatigue analysisis required for dressed SSR1 cavities.

STEP 1 - Determine the load history.There are three loads in the load history: pressure (P),tuning cycles (T1) and cool down/warm-up (T2).

STEP 2 - Determine the expected number of full-rangepressure cycles including startup and shutdown, N∆FP.The SSR1 system will undergo a maximum of 3cooldown/warmup cycles during commissioning and an

additional 1 cycle for each year of operation. Assum-ing an operating life of 40 years, the total number ofwarmup-cooldown cycles will be N∆FP = 43.

STEP 3 - Determine the expected number of operating pres-sure cycles in which the range of pressure variation ex-ceeds 20% of the design pressure for integral construc-tion N∆PO.There will be no operation of an SSR1 system at a pres-sure deviating 20% from the design pressure. Therefore,N∆PO = 0.

STEP 4 - Determine the effective number of changes inmetal temperature difference between any two adjacentpoints, ∆TE .The effective number of such changes is determined bymultiplying the number of cooldown by two: N∆T E =43 ·2 = 86.

STEP 5(a) - Determine the number of temperature cyclesfor components involving welds between materials hav-ing different coefficients of thermal expansion (N∆T α)that causes the value of (α1+α2)∆T to exceed 0.00034.For the SSR1 system, N∆T α = 43

STEP 5(b) - Determine the number of course tuner cyclesexpected during the system’s lifetime, N∆T1 .There are assumed to be 10 coarse tuner cycles per yearfor 40 years of operations. Therefore, N∆T1 = 400.

STEP 6 - If the expected number of operating cycles fromsteps 2-5 satisfies the criterion of Table 5.9 of the Div. 2,then no fatigue analysis is required.The criterion of Table 5.9 requires the sum of the oper-ating cycles steps 2-5 to be equal to or less than 1000.The sum for the SSR1 system is 572 cycles. Therefore,per Method A, no fatigue analysis is required.

4.6.2 Ratcheting Assessment - Elastic-Perfectly PlasticAnalysis

Two analysis methods are allowed by Div. 2 for theratcheting assessment: Elastic Stress Analysis (Part 5.5.6)and Elastic-Plastic Stress Analysis (Part 5.5.7). For the SSR1case, an elastic-plastic stress analysis was performed since amodel for the analysis that accurately represents the com-ponent geometry, boundary conditions, applied loads, andmaterial properties was already developed for previous anal-yses.

The analysis for protection against ratcheting is per-formed by application, removal and re-application of the ap-plied loadings to show that the structure eventually shakesdown to elastic action, i.e., that the incremental increases inplastic deformations from each cycle are small and diminish-ing as the number of cycles increases.

The finite element model was subjected to 20 load cy-cles. Each load cycle consisted of the following eight-steps:

1. Warm pressurization, P = 0MPa→ 0.2MPa2. Thermal cooldown, T2 = 293K→ 2K3. Extension of tuner, T1 = 0mm→ 0.25mm4. Cold pressurization, P = 0.2MPa→ 0.4MPa5. Cold depressurization, P = 0.4MPa→ 0.2MPa6. De-extend tuner, T1 = 0.26mm→ 0mm

7. Warm-up to room temperature, T2 = 2K→ 293K8. Warm depressurization, P = 0.2MPa→ 0MPa

Several deformation probes were established on the geome-try in the most stressed and/or deformed areas. The defor-mations at the probe locations were used to examine whetherdistortions were increasing or decreasing during the cycle.Three of them are shown in Fig 8.

It should be noted that there are two very conservativeassumptions in this analysis. The first is the elastic-perfectlyplastic material model, which essentially limits the maxi-mum stress of any element to the yield stress of the material.In reality, both the stainless steel and niobium demonstrateconsiderable post-yield stiffness. The second conservativeassumption is the use of failure-mode maximum pressures atevery cycle, as well as maximum tuner displacements. It isunlikely that the cavity will be subjected to twenty cycles ofthis severity during service.

Satisfying any one of the following three criteria is suf-ficient to demonstrate protection against ratcheting:

1. There is no plastic action in the component2. There is an elastic core in the primary-load-bearing

boundary of the component3. There is not a permanent change in the overall dimen-

sions of the component

Convergence to pure elastic action cannot be obtained intwenty cycles, see Fig 8. However, all trends are smooth, andeventual convergence can be safely assumed. This behaviorconfirms the existence of an elastic core, meeting the require-ments of the Code. The final permanent deformations of theprobe locations at the end of 20 load cycles must be regardedas absolute upper limits. It is highly unlikely that these lim-its will be reached in practice, due to the very conservativeassumptions made in this analysis.

Nb cavity

Helium Vessel

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

2

4

6

8

10

12

14

16

18

20

22

Number of load cycle

Residual deformation [

µm]

Probe 1 Probe 2 Probe 3

Fig. 8. Residual plastic deformation between load cycles in the di-rection of the probes. The displacements are taken from Load Step 8of each cycle, in which the system is entirely unloaded, and representpermanent deformations

5 Design of Permanent JointsThe types of welds joining stainless parts are defined

in paragraph 4.2 of Div. 2. However, since nondestructiveinspections are not performed, joint efficiencies for unradio-graphed welds, given in Div. 1 - Table UW-12, have beenadopted. All TIG welds are classifiable as single-weldedbutt joints with backing strip, and single full fillet lap joints,which have efficiencies of ε = 0.65 or ε = 0.45, respectively.

It was not possible to strictly adhere to Code rules indefining the types of welds used for joining the niobium partsbecause of the stringent requirements for the high-qualitysurfaces in SRF components. Several welded samples weremade to define and achieve the best welding parameters. Thesamples were etched and studied microscopically, their in-tegrity (absence of defects) was verified and the effectiveweld areas, Ae f f , measured. This level of inspections has ledto the conclusion that the weld efficiency ε = 1 is appropriatefor niobium electron-beam joints.

The same approach was used to qualify the copperbrazed joint, for which ε is also taken as 1.

The integrity of the cavity structural connections wasverified using the results of finite element analyses per-formed with models having bonded contact between parts,i.e., welding beads or brazed interfaces were not explicitlymodeled. A linear elastic analysis was performed for all fourload case combinations (see Table 4) and the stress state atconnections assessed by hand calculations based on a levelof criticality defined by the safety factor η =

σeqS . The von

Mises stress, σeq, was calculated from the nodal forces act-ing on the joints, extracted from the finite element analyses,and the effective weld area, Ae f f . The joint was consideredsafe if the condition ηε > 1 was met.

All joints passed the safety requirement ηε > 1. In addi-tion, stress classification lines (SCLs) were created throughthe more highly stressed joints to support the hand calcu-lations. Figure 9 shows the SCL through the most highlystressed niobium electron-beam joint and confirms the veri-fication approach, with an error between the hand calculatedstress and the simulation result that is less than 10%. Thestress distribution, linearized by the ANSYS program in themanner specified by Part 5 of Div. 2 of the Code, satisfies thesafety conditions σM

ε≤ Sε and σM+B

ε≤ 1.5Sε.

The inclusion of weld efficiencies (characteristic ofDiv. 1 vessel design, where radiography is not mandatory)should permit this analysis to stand as qualification of thevessel under the provisions of U-2(g) of Div. 1.

6 Inspection and ExaminationAll examination requirements of the Code (e.g., weld

examination) have not been strictly followed in the construc-tion of the dressed SSR1 cavity. However, a combinationof visual examinations, radio-frequency measurements, andleak checks confirms the integrity and reliability of the nio-bium cavity and stainless steel vessel.

Visual examinations at each critical step was performedin order to prevent visible defects in the parts. All welds forpressure retaining parts were visually examinated by man-

(a) Stress classification line (SCL)

0 4 8 12 16 20 24 2830

40

50

60

70

SCL path [mm]

σ [MPa]

σM

σM+B

(b) Linearized stresses on the SCL: membrane stress σM and membraneplus bending stress σM+B

Fig. 9. SCL through the more stressed electron beam welded joint

ufacturer personnel under the supervision of Fermilab in-spectors to assure the acceptance criteria of Table 7.6 of theASME BPVC - Section VIII, Div. 2, Part 7. The electron-beam welds joining niobium parts are also examined frominside of the cavity (radio-frequency volume) with the aid ofa video scope. TIG welds for stainless parts are visually ex-amined step-by-step during the highly-controlled process ofwelding. Integrity of braze joints is visually checked lookingat both ends of the joint with a glass magnifier to excludethe presence of defects. None of the joints were subject toNDE. Proper efficiency coefficients for joints design weretaken into account to overcome the absence of NDE. Jointshaving a design that does not meet the rules of the ASMEBPVC were qualified by testing samples of the same specificjoint to prove the manufacturing process and material.

Measurements of the resonant radio-frequency are ex-tremely important in obtaining a dressed SRF cavity with theproper frequency at each step. They are also used to mon-itor the deformations of the niobium cavity during machin-ing and welding, preventing plastic deformations which areabove acceptable limits. This is possible because the reso-nant radio-frequency of a given volume shifts as the volume

varies, with sensitivity on a nanometer scale.Several leak checks of the vessels were performed at var-

ious stages of the fabrication in order to prove that all weldsand brazed joints are leak tight with a minimum sensitivityof 10−9 atm·cc

s .Parts and final assemblies were measured by coordinate

measuring machine (CMM) to check the geometrical proper-ties of the objects and their adherence to released drawings.Material certifications are provided by the manufacturers toprove the specified quality and grade of the material.

7 Pressure TestingPressure testing shall be performed per the ASME

BPVC, Section VIII, Div. 2, Part 8 when the vessels arecompletely welded, machined and inspected. The validityof the design approach, used for the dressed SSR1 cavity,in terms of safety, reliability and leak tightness has been en-sured through a pneumatic pressure test. All safety precau-tions were adopted before and during the execution of thetest.

The helium space was pressurized at the test pressure of1.15 times the MAWP at room temperature, PT = 1.15 ·0.2=0.23MPa, according to Part 8.3 of the Code. The absence ofleaks was proved by monitoring the pressure, which must beconstant for the 10 minutes duration of the test. An additionalcheck was done to detect plastic deformations of the niobiumcavity (the weakest vessel) by measuring the shift of the reso-nant radio-frequency during the test. Comparing the value ofthe frequency after two pressurization cycles showed no shiftin frequency, confirming the absence of plastic deformationdue to the pressure test.

A helium leak check with a minimum system back-ground of 10−9 atm·cc

s was performed after the pressure testand verified the absence of leaks at welds, seals, in otherpossible locations.

8 ConclusionThe SSR1 design case is an example of dressed SRF

cavity designed starting from the rules of ASME BPVC,Section VIII, Div. 2. Some of the most common non-compliances have been addressed using “equivalent rules”in terms of safety, when the Code does not apply.

The use of different technologies and materials in thedesign and fabrication of the various styles of SRF cavitiesdoes not allow the definition of a single, unique procedurethat could be extended to all SRF cavities. Nevertheless, thedesign of other dressed SRF cavities may be approached us-ing the set of rules reported in this paper, as a starting pointto meet the Code at the greatest extent possible.

AcknowledgementsThe activities described in this paper are supported by

Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of En-ergy

Nomenclatureβcr capacity reduction factorε permanent joint efficiencyη design safety factor of a permanent jointν Poisson’s ratio of materialsσeq equivalent tensile stress (von Mises yield criterion)σM membrane stressesσM+B membrane and bending stressesΦB design factor for buckling analysisAe f f effective area of a jointD dead weight of the SSR1 systemE Young’s modulus of materialsLCF load combination factorMAWP maximum allowable working pressureP pressure acting into the helium spaceP1, P2, P3 principal stressesS allowable stress of materials at the design temperatureSy yield strength of materialsSu engineering ultimate tensile stressT design temperatureT1 applied displacement to the beam pipe of the cavity

(elastic tuning)T2 thermal load (cooldown, warmup)

References[1] Guidelines for the Design, Fabrication, Testing and In-

stallation of SRF Cavities, FESHM Chapter 5031.6,TD-09-005, Fermi National Accelerator Laboratory

[2] ASME Boiler and Pressure Vessel Code, Section VIII -Rules for Construction of Pressure Vessels Division 1,2010

[3] ASME Boiler and Pressure Vessel Code, Section VIII -Rules for Construction of Pressure Vessels Division 2 -Alternative Rules, 2010

[4] ASME Boiler and Pressure Vessel Code, Section II,Part D, Materials, 2010

[5] ANSYS Academic Research Mechanical, Release16.0, ANSYS, Inc.

[6] L. Ristori et al., “Development and Performance of325 MHz Single Spoke Resonators for Project X”, Pro-ceedings of SRF 2013, Paris, France

[7] T. Nicol et al., “SSR1 Cryomodule Design for PXIE”,Proceedings of PAC2013, Pasadena, USA

[8] National Institute of Standards and Technology, Cryo-genic Technologies Group, 316 Stainless Steel MaterialProperties

[9] J.E. Jensen, et al., “Selected cryogenic data notebook,Volume II, Sections X-XVIII”, Brookhaven NationalLaboratory

[10] J.D. Fuerst et al., “Niobium to Stainless Steel brazetransition development”, Proceeding of SRF 2003,Lubeck, Germany

[11] L. Ristori et al., “Development at ANL of a copper-brazed joint for the coupling of the niobum cavity endwall to the stainless steel helium vessel in the Fermi-lab SSR1 resonator”, Proceeding of IPAC 2012, NewOrleans, USA