C100 Cryomodule Vacuum Vessel Structural Analysis Gary G. Cheng, William R. Hicks, and Edward F. Daly

Transcription

1 Introduction C100 Cryomodule Vacuum Vessel Structural Analysis Gary G. Cheng, William R. Hicks, and Edward F. Daly Cryomodule (CM) prototypes for CEBAF 12GeV upgrade project have been built in the past and the design has been improved since then. The vacuum vessel, as an essential component of the CM, has adopted a few changes. A previous JLab technical note [1] has described the structural analysis procedures per requirements in ASME Boiler & Pressure Vessel Code, Section VIII, Division I for the vacuum vessel used in the prototype CMs. This technical note serves as an amendment to JLAB-TN to reflect changes in design and the associated analyses. Mechanical loads include those occur at normal work condition and the transportation loads (termed as abnormal loading in JLAB-TN ). Formulas and rules are cited from the 2007 ASME BPV code [2] and mechanical engineering design reference books. I. Vacuum Vessel Design A brief introduction of the 12GeV upgrade project CM vacuum vessel was given in JLAB- TN Changes have been made since then include: 1. In each vacuum vessel, four instrumentation ports accommodating NW200 half nipples (8.0" tube OD) are added. 2. Space frame is anchored to the vacuum vessel through 6 sets of studs distributed along the length of the vessel. Each set contains three studs located 120 o apart from one another along the circumference of the vacuum vessel. The space frame wheels are suspended while studs are functioning so that they don t provide any form of support after the assembly process is completed. 3. Openings for waveguide attachments are smaller. Currently, there are six 11.72" diameter openings and one 14.9" diameter openings. The half nipples and tubes are all welded to, instead of being pulled out from, the openings on vacuum vessel. 4. The two ground supports are relocated toward ends of the vacuum vessel. The code design analyses in this report will take into account all above changes. II. Mechanical Loads 1. Structural Loads During Normal Operation Weight of end cans: 750 lbf/each Weight of cavities and tuners: 3,000 lbf Weight of space frame: 850 lbf Internal design pressure: 29.6 psi (2.0 atm) External design pressure: 29.6 psi (2.0 atm) 2. Transportation Loads (cited from JLAB-TN ) 1/27

2 4.0g vertically (which includes the initial weight) 2.0g axially 0.5g horizontally III. Analyses per ASME BPV Code 1. Determine Wall Thickness of Vacuum Vessel (UG-22 and UG-23) The BPV code section VIII, Division 1, para. UG-23 (c) requires that the wall thickness of a vessel computed by these rules shall be determined such that, for any combination of loadings listed in UG-22 that induce primary stress and are expected to occur simultaneously during normal operation of the vessel, the induced maximum general primary membrane stress does not exceed the maximum allowable stress in tension JLAB-TN addressed this provision by conducting stress analyses via shear-moment diagram. Herein a simplified model for finite element stress analysis is used to estimate the membrane and bending stresses in the vacuum vessel subjected to both normal loads and transportation loads. 1.1 Assumptions for Model Simplification 1) Small openings, including the eight tuner openings and other openings with diameters smaller than 4.5", are omitted from the model for brevity. 2) With the 18 studs, the space frame and cavities are tied tightly to the vacuum vessel. It is thus assumed that the weights of all parties are evenly distributed along the length of the vacuum vessel. This assumption allows a beam-like finite element model. 3) Flanges are omitted from the model. 4) Ground supports are assumed to be fixed to the vacuum vessel by welding. 1.2 Finite Element Model The model is created in ANSYS by use of PIPE16 element. The element has three translational and three rotational degree of freedoms (DOF) at each of its two nodes. Pipe outer diameter and wall thickness are input as real constants. Figure 1 is a snapshot of the finite element model. 1.3 Applied Loads and Boundary Conditions 1) Two ground supports are added at nodes that are (100.58"+96.08")/2 = 98.33" away from the center of the vessel. They are welded (all DOFs fixed) to the vessel. 2) End can weights of 750 lbf are applied at both ends of the vacuum vessel. A bending moment of /2 = 13, lbf-in is also applied at each end of the vessel. The 37.23" is the estimated overall length of the end cans. 3) Normal load is applied by specifying 1.0g acceleration in the y coordinate direction. 4) Transportation loads are applied by specifying accelerations in all three axes. End can inertias are included by scaling the weight and bending moments. 5) Equivalent axial forces to simulate pressures on capped ends are applied. 2/27

3 Support ring Instrumentation port Waveguide ports with (11.72" diameter) Waveguide ports with 14.9" diameter Fig. 1 Finite element model of the vacuum vessel 1.4 Results: Internal Pressure and Normal Load A picture of the deformed vacuum vessel under internal pressure and normal load is shown in Fig. 2. The 1 st principal stress in the vacuum vessel is plotted in Fig. 3. The moment diagram in vacuum vessel is plotted in Fig. 4. The peak bending moment is found to be 69,555 lbf-in occurring at support points. Stresses of interest to know are: Peak stress induced by bending is 354 psi. Peak axial stress is 1,850 psi. Peak shear stress is 112 psi. Peak hoop stress is 1,850 psi. 1.5 Results: Internal Pressure and Transportation Load The deformed vacuum vessel under internal pressure and transportation load is shown in Fig. 5. More stress results are summarized in a subsequent table. 3/27

4 Fig. 2 Deformed vacuum vessel under internal pressure and normal load Fig. 3 1 st Principal stress in vacuum vessel under internal pressure and normal load 4/27

5 Fig. 4 Moment diagram for vacuum vessel under internal pressure and normal loading Fig. 5 Deformed vacuum vessel under internal pressure and transportation load 5/27

6 1.6 Stress Results Summary Table 1. A summary of peak stresses in vacuum vessel subjected to various loadings Peak Stresses internal pressure Internal pressure & External pressure External pressure & (psi) & normal load transportation load & normal load transportation load Bending stress 354 1, ,429 Axial stress 1,850 1,960 1,909 2,020 Hoop stress 1,850 1,850 1,880 1,880 Shear stress Max. Principal Stress 2,204 3,390 2,264 3,446 None of the peak membrane stresses (axial, hoop, and shear) in Table 1 exceed the 20,000 psi allowable stress for 304 stainless steel (cited from Table 1A of reference [3]), neither does the sum of the membrane stress and bending stress. 2. Thickness of Shells under Internal Pressure (UG-27) In JLAB-TN-00-03, the minimum required shell thickness was calculated per UG-27 c-1 and c-2. The greater of the two calculated minimum thicknesses is found to be t = 0.038". Therefore the actual vacuum vessel wall thickness of 0.25" is sufficient. 3. Thickness of Shells under External Pressure (UG-28) Due to the fact that the supporting scheme for the vacuum vessel has been changed since the prototype designs, the shell thickness calculations per UG-28 need to be revisited. Fig. 6 illustrates how the vacuum vessel is supported by flanges, ground supports, and support rings. The length of the end can is roughly 37.23". It can be treated as the head of the vacuum vessel " 53.22" Ring 1 Ring 2 Ring 3 Ring 4 Ring 5 Ring " 5.57" Symmetry plane Fig. 6 Vacuum vessel supports and stiffening rings layout The calculation of allowable pressures is performed for each segment of the vacuum vessel. A few parameters are common in all calculations: vessel outer diameter is D o = 32 in, design 6/27

7 pressure is P = 29.6 psi, actual wall thickness is t = 0.25", D o /t = 128, and the young s modulus of 304 stainless steel is E = 2.8e7 psi [1]. Table 2 presents the steps (refer to UG-28c) and results. Segments of vessel are numbered from left to right. Note that the length of the leftmost segment, which is connected to the end can that is treated as the vessel head, is added by 1/3 of the end can length. The formula used for calculating the allowable pressure P a is: 4B Pa = (1) 3( D / t) o Table 2. Calculation of allowable pressure in vacuum vessel segments Segment 1 Segment 2 Segment 3 Segment 4 Length, L L/D o Factor A Factor B 6,230 11,350 6,230 7,000 P a, psi The allowable pressures in Table 2 for all four segments are much higher than the design pressure of 29.6 psi. Therefore the wall thickness of 0.25" is sufficient to resist 2 atm of external pressure. 4. Stiffening Rings for Cylindrical Shells under External Pressure (UG-29) The ASME BPV Sec. VIII Div. I para. UG-29 requires that the available moment of inertia (MOI) of a circumferential stiffening ring shall be not less than that determined by the following formula (pick one from two candidate equations suggested in the code): where Ls As I 2 s o s + = [ D L ( t A / L ) A]/14 (2) s s = one-half of the distance from the centerline of the stiffening ring to the next line of support on one side, plus one-half of the centerline distance to the next line of support on the other side of the stiffening ring. See Fig. 6 for stiffening ring arrangement. = cross-sectional area of the stiffening ring = ( )/ /2 ( ) = in 2 Similar to what is done in Table 2, the required MOI for two representative stiffening rings (see Fig. 6) are evaluated following steps in Table 3. Note that the formula used to determine the factor B is: 3 P D o B = (3) 4 t + As / Ls Once factor B is calculated, value for factor A can be looked up in Fig. HA-1 of section II, part D, subpart 3. 7/27

8 Table 3. Calculation of allowable pressure in vacuum vessel segments Ring 1 Ring 2 Ring 3 Length, L s 52.67"/2=26.335" (5.57"+66.15")/2=35.86" (66.15"+53.22")/2=59.685" Factor B ,437 2,584 Factor A I s, in The actual rings have a MOI > /12 = in 4. Therefore all six rings meet the ASME BPV requirements. 5. Openings and Reinforcements (UG-36, UG-37, UG-40, UG-41, UW-15, UW-16) According to ASME BPV VIII D1 para. UG-36 (b), all openings in the vacuum vessel can be examined for required reinforcement areas by applying rules and formulas given in para. UG-37. There are five sizes of openings in the vacuum vessel: 1) 8 NW40 (1.5" OD and 0.065" wall) weld-on half nipples for cavity/space frame/ vacuum vessel alignments. 2) one 14.9" OD and 0.25" wall waveguide port locating in the center of the vessel. 3) " OD and 0.25" wall waveguide ports scattered along the length of the vessel. 4) 4 NW200 (8.0" OD and 0.12" wall) weld-on half nipples as instrumentation ports. 5) 8 4.5" OD and 0.12" wall tuner ports. Per UG-36(c)(3), the NW40 openings, which have an OD = 1.5" < 3.5", are exempt from the reinforcement requirements. Also according to UW-15(b)(2), no detailed weld strength calculations are required for such small openings. These ports are attached to the vacuum vessel by full penetration bevel welds (refer to Fig. UW-16.1(a)). An outward fillet weld is omitted since no mechanical loads are transferred to these ports and the design pressure is comparatively low. The other openings are evaluated using reinforcement and weld strength calculations per UG- 37, UW-15, and UW-16. The procedures are demonstrated in numerical examples given in the code s nonmandatory appendix L, part L-7. The waveguide ports having a small amount of eccentricity and the instrumentation ports are abutting the vessel wall so that the procedures in example 4 in L-7.4 can be adapted for these openings. The 8 tuner ports have hill-side nozzles so that example 7 in L-7.7 better resembles these ports. Following the two mentioned examples, calculations are carried out as follows. 5.1 Openings Abutting Vessel Wall These openings include: one 14.9" OD and 0.25" wall waveguide port, " OD and 0.25" wall waveguide ports, and 4 NW200 instrumentation ports Wall Thickness Required Shell wall thicknesses required have been determined earlier. The required nozzle neck thicknesses are calculated in Table 4: 8/27

9 5.1.2 Size of Weld Required Table 4. Nozzle wall thicknesses calculation Nozzle OD Actual nozzle wall, t n Nozzle inner radius, R n Req d nozzle wall, t rn Is t n > t rn? yes yes yes Full penetration bevel welds are adopted for mounting the waveguide and instrumentation ports to the vacuum vessel. The Fig. UW-16.1 (a) depicts a code acceptable weld pattern. Herein the outward fillet weld is omitted. The weld strength verifications conducted later will prove the weld size is sufficient Area of Reinforcement Required and Available UG-37 shows how to calculate the required reinforcement area, A, corresponding to an opening with diameter of d by use of the following formula: A = d t F + t t F (1 f ) (4) r 2 n r r1 For calculation of the area available in vessel shell, A 1, two formulas are suggested: A = d ( E1 t F tr ) 2tn ( E1 t F tr )(1 f 1) (5a) A = ( t + t )( E t F t ) 2t ( E t F t )(1 f ) (5b) 1 r 1 2 n 1 r n 1 r r1 The larger value from Eqs. (5a) and (5b) shall be used for A 1. The area available in nozzle projecting outward should take the smaller value from the following two equations: A = 5( tn trn fr2 t (6a) A2 = 5( tn trn ) fr2 tn (6b) 2 ) The sum of areas available is compared to the required reinforcement area A calculated from Eq. (4). If A 1 +A 2 >A, no additional reinforcement is needed. Otherwise, extra reinforcement material is necessary. Refer to UG-37 for a detailed explanation of all dimensional variables. There are a few parameters that are common in calculations for all pertinent openings: t r = PR /( SE.6P) = /(18, ) = 0.025" S n = S v = S p =18800 psi f r1 = f r2 = f r3 = f r4 = 1.0 F = 1 E 1 = 1 for opening in solid plate t = 0.25" is the specified vessel wall thickness The calculations and results are presented in Table 5. It can be easily concluded that none of the three types of openings need additional reinforcement. 9/27

10 Table 5. Required reinforcement areas for openings on vacuum vessel center waveguide port side waveguide port instrumentation port diameter of opening d= wall thickness of nozzle t n = required nozzle thickness t rn = From Eq. (4), A= From Eq. (5a), A 1 = From Eq. (5b), A 1 = larger A1 from above two = From Eq. (6a), A 2 = From Eq. (6b), A 2 = smaller A2 from above two = A 1 +A 2 = A 1 +A 2 >A? Yes Yes Yes Verifications of Weld Strength Figure 7 shows the waveguides and their supports. The waveguide ports carry the weights from waveguides and some accessories. Rough estimation of the weights and bending moments can be obtained using dimensions in Fig. 7. The total weight of components in Fig. 8 is approximately 380 lbs (based on 1,300 in 3 volume and lbf/in 3 density). To be conservative, the side waveguide ports, holding one waveguide each, are assumed to carry 190 lbs and the center waveguide port holding two waveguides supports 380 lbs. In reality, the total weight of components in Fig. 8 is also carried by two other supports in the system. The instrumentation port is not subjected to noticeable mechanical loads besides the pressure load. Stresses due to inner pressure are considered in evaluation of the weld strength. Table 6 presents the weld strength verification (some of the formulas are quoted from reference [4]) for the side and center waveguide ports, as well as the instrumentation port. It is seen that all welds are sufficiently strong to withstand the mechanical loads applied. 10/27

11 Average weld boundary on waveguide port tube 26.0" Support bracket Waveguides 13.5" 13.2" 9.989" z y x Fig. 7 Waveguides and supports Fig. 8 Components used in weight estimation 11/27

12 Table 6. Weld strength verifications for waveguide and instrumentation ports Side ports Center port Instrumentation ports Weld leg h, in Weld inner radius, i.e. port tube outer radius r o, in Port tube inner radius r i, in Throat area A=1.414π h r o, in Unit 2nd moment of area, I u = π r o 2, in MOI, I = 0.707h*I u, in stainless steel yield strength S y, psi * 30,000 30,000 30,000 Internal pressure P i, psi Logitudinal force due to pressure F x = P i π r 2 i, lbf 2,927 4,821 1,400 Vertical force F y, lbf Distance from C.G. to weld center, in Moment M z, lbf-in 4,940 4,940 - Shear stress due to F y, τ y =F y /A, psi Shear stress due to F x and M z, τ x = F x /A+M z *r/i, psi Hoop stress due to pressure, τ z = 2 r i 2 P i /(r o 2 -r i 2 ), psi 1,968 1, Total shear stress, τ=(τ x 2 +τ y 2 +τ y 2 ) 1/2, psi 2,073 1,965 1,147 Safety factor = 0.577*S y /τ * From [3], Table Y-1, 18Cr-8Ni, SA-240, 304 stainless steel. This yield strength is used in subsequent calculations. 5.2 Hill-side Openings Only the tuner ports belong to this type. Follow the steps in L-7.7 example 7, a series of calculations regarding the tuner ports are conducted: Wall Thickness Required The required vessel wall thickness has been determined earlier. For the tuner port half nipples, the required nozzle wall thickness is 12/27

13 t P Rn 29.6 ( ) / 2 = = S E 0.6P 18, rn = in. The actual half nipple wall thickness is t n =0.12", which is greater than the required t rn Size of Weld Required Full penetration bevel welds are employed to fix the tuner ports onto the vacuum vessel. The weld size is verified by mechanical strength calculations described in paragraph The Strength Reduction Factors f r1 = 1.0 f r2 = S n /S v = 18,800/18,800 = Non-radial Opening Chord Length at Midsurface Rm = R + tr / 2 = ( ) / / 2 = in. L = 8.0" from drawing. Nozzle inner radius R n = = 4.26 in. 1 L + Rn o α 1 = cos = cos = R m L Rn o α 2 = cos = cos = R m α = α 2 -α 1 = 36 o 2 2 d = 2Rm 1 cos ( α / 2) = cos (37.33/ 2) = in. F = 1.0 because no integral reinforcement is used Area of Reinforcement Required and Available Required area of reinforcement is A = d t r F+2 t n t r F(1-f r1 ) = = in 2 Area available in vacuum vessel shell is A 1 = larger of the following = d (E 1 t-f t r )-2 t n (E 1 t-f t r )(1-f r1 ) = ( )-0.0 = 2.27 in 2 or = 2(t+t n )(E 1 t-f t r )-2 t n (E 1 t-f t r ) (1-f r1 ) = 2( )( )-0.0 = in 2 So, A 1 = 2.27 in 2 Area available in nozzle A 2 = smaller of the following: = 5(t n -t rn )f r2 t = 5( ) = in 2 13/27

14 or = 5(t n -t rn ) f r2 t n = 5( ) = 0.07 in 2 So, A 2 = 0.07 in 2 Total area available = A 1 + A 2 = 2.34 in 2. This is greater than the area required A = in 2. Therefore, no reinforcement is needed Weld Strength Verification At the tuner ports, internal/external pressures are the major mechanical loads. The weld strength is thus checked against shear stresses created by pressures. The elliptical cross-section of the weld is projected as an annulus for simplification. The shear stresses inside weld can be evaluated as follows: Weld length h = 0.12 in. Weld inner radius = tuner tube outer radius = r o = 4.5/2 = 2.25 in. Tuner tube inner radius is r i = ( )/2 = 2.13 in. Throat area A = πhr o = 1.2 in 2 For internal pressure P i = 29.6 psi: The hoop stress at outer surface of tuner tube is: 2 2 2ri Pi σ t = = = 511 psi ro ri The longitudinal force caused by internal pressure is: F L = P i π r 2 i = 29.6 π = 422 psi The shear force in weld due to longitudinal force is: σ L = F L /A = /1.2 = 352 psi The resultant shear stress in weld is: τ = (σ 2 t +σ 2 L ) 1/2 = 663 psi The 304 stainless steel has a typical yield stress of S y = 30,000 psi. The safety factor in weld is: Sy/τ = ,000/663 = This indicates a very conservative weld size. 6. Stresses in Space Frame Lockdown Studs Studs clustered on 6 cross sectional areas perpendicular to the vessel s longitudinal axis are used to anchor the space frame onto the vacuum vessel. On each cross section, there are three studs evenly distributed: one at the top and the other two are 120 o apart from the top stud. Therefore, there are a total of 6 3 = 18 studs. The strength of these studs needs to be verified for the worst loading scenario: the transportation load case. A few conservative assumptions are taken to quantify the loads per stud: 1) Total weight of space frame plus cavities is 3,850 lbf per cryomodule. 2) Vertical acceleration of 4g will be carried all 18 studs. That means each top stud takes a vertical load of F z = /18 = 856 lbf. 3) Horizontal acceleration of 0.5g is carried by all 18 studs. That means the top stud carries 14/27

15 horizontal load if F y = /18 = 107 lbf 4) Axial acceleration of 2.0g is shared by all 18 studs. That means each stud carries a F x = /18 = 428 lbf Fig. 9 Stepped stud design concept The resultant of F x and F y is F xy = ( ) 1/2 = lbf. F xy is applied as a bending load and F z as the axial tensile force. Figure 9 shows a schematic of a stepped stud design concept. The bending and tensile stresses are evaluated as follows: The peak stress in stud segment with diameter of d 1 is calculated as follows: d 1 = 0.75 inch A 1 = π d 2 1 /4 = in 2 Moment of inertia, I 1 = π d 4 1 /64 = in 4 c 1 = d 1 / in Force arm to left end, L_arm1 = 2.25 in Bending moment M 1 = 992 lbf-in Maximum bending stress at left end, σ_b1 = M 1 c 1 / I 1 = 23,954 psi Tensile stress at left end, σ_z1= F z /A 1 = 1,937 psi Total stress at left end, σ_tot1= 25,891 psi The peak stress in stud segment with d 2 diameter: d 2 = 0.5 inch A 2 = π d 2 2 /4 = in 2 Moment of inertia, I 2 = in 4 c 2 = 0.25 in σ_z2 = F z /A 2 = 4,357 psi By varying the dimension L 2 in Fig. 9, the peak stress in the 0.5" stud can be adjusted. Table 7 lists the stress magnitudes when L 2 takes various values. Note that L_arm2 = L r_fillet. 15/27

16 Table 7. Peak stress in 0.5" stud as L 2 varies L 2, in. L 1, in. L_arm2, in. M 2, lbf-in σ_b2, psi σ_tot2, psi , , , , , , , , The yield strength of 304 stainless steel is 30,000 psi. The studs are therefore safe when subjected to transportation loads. 7. Support Bracket Structural Analysis There are eight stainless steel square tubes (see Fig. 10) used as support brackets attached to the vacuum vessel. Each bracket supports one ion pump and partial weight of a waveguide. The design of the support bracket is observing a conservative approach. Structural integrity of the tubular bracket and the root weld is analyzed. Two types of loads are identified to be exerted onto the support brackets: 1) partial weight from the waveguide and 2) weight from the ion pump. To be conservative, it is assumed that each bracket carries 1) the total weight of waveguide and its accessories, which is approximately 190 lbf and 2) the weight of ion pump, 34 lbf. A preliminary design required Wall square tubes, refer to Fig. 10. The strength analyses for weld and support bracket tube are performed to verify if tubes with smaller sizes are possible. The following calculations show that square 304 stainless steel tubes can yield sufficient safety factors both in the welds and the tubes. Dimensions of the square tube is: b := 2.0 in d := 2.0 in t := in L := ( ) in Self weight of the bracket is estimated as: cross-section area: A tube := bd ( b 2 t) ( d 2 t) A tube = 1.363in 2 W tube := lbf in 3 A tube L W tube = 7.8lbf Refer to Fig. 7, y-direction force is calculated as: F y := 190 lbf + 34 lbf + W tube F y = 231.8lbf Yield strength of stainless steel is: Sy = 30,000 psi The bending or torsional moments created by F x and F y are: M x := 190 lbf 13.5 in M x = 2,565 lbf in 3.75 ( ) in M z := 190 lbf in + 34lbf ( ) in + W tube 2 2 M z = 3,739.7 lbf in 16/27

17 Fig. 10 Support bracket dimensions Stresses in the weld Weld leg h := 0.25 in Weld throat area A weld := h ( b + d) A weld = 1.414in 2 17/27

18 ( b + d) 3 Unit polar 2nd monent of area J u := J u = in 3 6 Polar 2nd moment of area J weld := h J u J weld = 1.885in 4 d 2 Unit 2nd moment of area I u := ( 3b + d) I u = 5.333in 3 6 2nd moment of area I weld := h I u I weld = 0.943in 4 M z d Shear stress in x-direction: τ x := 2I τ weld x = 3,967 psi 2 F y M x b + d 2 Shear stress in y-direction: τ y := + A weld 2J τ weld y = 2,088 psi 2 2 Total shear stress τ weld := τ x + τ y τ weld = 4,483 psi S y Safety factor: Sf weld := Sf weld = 3.9 τ weld Stresses in support bracket tube 2nd Moment of area: bd 3 ( b 2 t) ( d 2 t) 3 I tube := I tube = 0.754in Polar 2nd moment of area: bd b 2 + d 2 J tube ( ) ( b 2 t) ( d 2 t) ( b 2 t) 2 + ( d 2 t) 2 := J tube = 1.507in 4 M z d Tensile stress in x-direction: σ x := 2I σ tube x = 4,962 psi 2 F y M x b + d 2 Shear stress in yz plane: τ yz := + A tube 2J tube 2 2 Von Mises stress σ tube := σ x + 3 τ yz τ yz = 2,577 psi σ tube = 6,674 psi S y Safety factor: Sf tube := Sf tube = 4.5 σ tube 18/27

19 8. Stiffening Ring Weld Strength Verification The stiffening rings as shown in Fig. 6 are carrying loadings transferred from cavity, tuners, and end cans in the four cases listed in Table 1: Case 1: Internal pressure and normal load Case 2: Internal pressure and transportation load Case 3: External pressure and normal load Case 4: External pressure and transportation load Thus, the welds applied on these rings have to be robust to withstand not only pressure loads, but also the mechanical loads. A combination of bevel welding on the interior surface of vacuum vessel wall and intermittent fillet welds on the exterior wall are adopted. Figure 11 illustrates the welding schemes. The strength of the bevel welds are checked against loads in all four cases. The finite element model described in section 1.2 is utilized to generate the shear force and bending moment at each stiffening ring s interface with the vessel wall. The throat area of weld and weld cross section second moment of area are evaluated and shear stresses are calculated on basis of these. The procedure and results are presented as follows (see Fig. 6 for numbering of stiffening rings): End ring Fillet welds Intermediate ring Fillet weld Bevel weld Bevel weld Weld properties(bevel welds only) Fig. 11 Stiffening ring weld scheme Weld leg h = 0.25 in. Weld inner radius r = in. Throat area A=1.414π h r in 2 Unit 2nd moment of area, I u = π r in 2 2nd moment of area, I = 0.707h*I u in Stainless steel yield strength, S y = 30,000 psi 19/27

20 The safety factors calculated in Tables 8 and 9 are greater than 1.0. That means even with the bevel welds only, the stiffening rings welds are sufficiently robust. In reality, there are outside fillet welds, although discontinuous along the circumference of vacuum vessel wall, providing additional strength. Therefore, the overall weld size design is conservative in consideration of all cases of loadings. For reference, the shear diagrams and moment diagrams for the studied cases are exhibited in Figs. 12~22. Table 8. Stiffening ring weld strength verification for vacuum vessel with internal pressure Case 1 Case 2 Ring 1 Ring2 Ring3 Ring 1 Ring2 Ring3 Axial force F x, lbf 23,068 9,688 9,688 24,590 12,052 10,343 Vertical force F y, lbf 761 1, ,044 4,729 1,312 Horizontal force F z, lbf F yz = (F y 2 +F z 2 ) 1/2, lbf 761 1, ,067 4,768 1,325 Moment M y, lbf-in ,361 16,832 8,503 Moment M z, lbf-in 14,722 34,502 16,670 58, ,030 66,805 M yz =(M y 2 +M z 2 ) 1/2, lbf-in 14,722 34,502 16,670 59, ,053 67,344 Shear stress due to F yz, τ 1 =F yz /A, psi Shear stress due to F x and Myz, τ 2 =F x /A+M yz *r/i, psi Total shear stress, τ=(τ 1 2 +τ 2 2 ) 1/2, psi Safety factor = 0.577*S y /τ ,002 4,499 2,460 8,192 16,589 8,292 3,003 4,500 2,460 8,194 16,591 8, /27

21 Table 9. Stiffening ring weld strength verification for vacuum vessel with external pressure Case 3 Case 4 Ring 1 Ring2 Ring3 Ring 1 Ring2 Ring3 Axial force F x, lbf 23,806 10,004 10,004 22,284 7,641 9,349 Vertical force F y, lbf 761 1, ,044 4,729 1,312 Horizontal force F z, lbf F yz = (F y 2 +F z 2 ) 1/2, lbf 761 1, ,067 4,768 1,325 Moment M y, lbf-in ,361 16,832 8,503 Moment M z, lbf-in 14,722 34,502 16,670 58,889 13, ,805 M yz =(M y 2 +M z 2 ) 1/2, lbf-in 14,722 34,502 16,670 59, ,053 67,344 Shear stress due to F yz, τ 1 =F yz /A, psi Shear stress due to F x and Myz, τ 2 =F x /A+M yz *r/i, psi Total shear stress, τ=(τ 1 2 +τ 2 2 ) 1/2, psi Safety factor = 0.577*S y /τ ,044 4,517 2,478 8,060 16,337 8,235 3,045 4,518 2,478 8,062 16,339 8, /27

22 Fig. 12 Y-direction shear force diagram for case 1 Fig. 13 Y-direction shear force diagram for case 2 22/27

23 Fig. 14 Z-direction shear force for case 2 Fig. 15 Y-direction bending moment diagram for case 2 23/27

24 Fig. 16 Z-direction bending moment diagram for case 2 Fig. 17 Y-direction shear force diagram for case 3 24/27

25 Fig. 18 Z-direction bending moment diagram for case 3 Fig. 19 Y-direction shear force diagram for case 4 25/27

26 Fig. 20 Z-direction shear force diagram for case 4 Fig. 21 Y-direction bending moment diagram for case 4 26/27

27 IV. Summary Fig. 22 Z-direction bending moment diagram for case 4 The vacuum vessel design has been reviewed observing provisions in ASME BPV VIII D1. Issues such as the vessel wall thickness, nozzle thickness, weld size and strength, and necessity of reinforcement areas are investigated in details and the current design is found to be satisfactory in accordance with the code requirements. Code calculation results are displayed in data tables for information. Lockdown studs strength and stiffening ring weld strength are specially investigated and found to be conservative. Flanges on the vacuum vessel are off-shelf standard vacuum flanges or previous designs. Such flanges were found to be robust when used in previous cryomodules. Hence, detailed structural analysis on flanges is waived. REFERENCES [1]. T. Whitlatch, Vacuum Vessel Structural Analysis for JLAB Cryo Upgrade, JLAB-TN , Jefferson Lab, Newport News, VA. [2] ASME Boiler & Pressure Vessel Code, Section VIII, Division 1, Rules for Construction of Pressure Vessels, The American Society of Mechanical Engineers. [3] ASME Bolier & Pressure Vessel Code, Section II, Part D, Material Properties (Customary), The American Society of Mechanical Engineers. [4]. J. E. Shigley, C. R. Mischke, and R. G. Budynas, Mechanical Engineering Design 7 th Ed, McGraw-Hill, New York, /27