Unfired pressure vessels- Part 3: Design

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Kristin Caldwell
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1 Unfird prssur vssls- Part 3: Dsign Analysis prformd by: Analysis prformd by: Analysis vrsion: According to procdur: Calculation cas: Unfird prssur vssls EDMS Rfrnc: EF EN V1 Introduction: This Mathcad tmplat is mad for th purpos of aiding, automating and simplifiying th calculations of various paramtrs concrning unfird prssur vssls, according to th standard NF EN Plas not that th worksht is intndd as a supplmntary tool to th standard, and should thrfor always b usd alongsid th standard, and nvr alon. To nsur minimal risk of rror, it is rcommndd (although not ncssary) to opn a nw vrsion of th worksht for ach nw calculation. Whn printing th worksht, it is rcommndd to sav as pdf and print in A4 siz. Du to th natur of hiddn aras in Mathcad, it may b ncssary to add a fw lins to nsur no txt or calcultations gts split in two btwn pags. To us th worksht, all rlvant input paramtrs should b ntrd first. Th worksht uss two sts of input paramtrs: Global input paramtrs and local input paramtrs. Th global input variabls ar valus concrning th matrials of th vssl and th intdd prssur, and ar found undr this introduction. Th local input variabls concrn th spcific gomtry of th vssl. Each sction contains its own rlvant input paramtrs at th top of its hiddn ara. Evry input paramtr is colourd blu. Aftr all rlvant input paramtrs hav bn ntrd, th worksht will automatically calculat th output valus furthr down. Th most rlvant output paramtrs ar grn. An orang valu might b of intrst to th radr, but it is not th nd rsult. Global input paramtrs: 0.2% proof strnght: R p0.2;t 200 [MPa] 1.0% proof strnght at tmpratur T: R p1.0;t 98 [MPa] 1.0% proof strnght at tst tmpratur: R p1.0;ttst 110 [MPa] 0.2% proof strnght at tst tmpratur: R p0.2;ttst 211 [MPa] Tnsil strnght at tmpratur T: Tnsil strnght at tst tmpratur: Tnsil strnght at tmpratur 20C: R m;t 100 [MPa] R m;ttst 100 [MPa] R m; [MPa] Calculation of dsign strss as shown in 6.6: Th valu of f dn abov, whr n is th dsird stl dsignation, should b ntrd hr as f f 154 [MPa] Joint cofficint: Calculation prssur: z 1 [unitlss] P 4 [MPa] Claus 7 - Shlls undr intrnal prssur

2 7.4.2 Cylindrical shlls Local input paramtrs: Intrnal diamtr of shll: Extrnal diamtr of shll: 25 [mm] D [mm] Calculation valus: Man diamtr of cylindr part: D m 1 = 2 D Analysis thicknss: a 1 = 2 D Output paramtrs: Minimum rquird thicknss of shll: = z- P OR D = z+ P For a givn gomtry, th maximum intrnal prssur is: P max z a = D m Conditions of applicability: status_r1 if 0.16 D rturn OK Not Valid if only th intrnal diamtr is known: status_r1 i if rturn OK Not Valid Rquirmnts Rsults R1: 0.16 status_r1 = OK D status_r1 i = OK Sphrical shlls Local input paramtrs: Intrnal diamtr of shll: Extrnal diamtr of shll: 45 [mm] D [mm] Calculation valus: Man diamtr of cylindr part: D m 1 = 2 D

3 Analysis thicknss: a 1 = 2 D Output paramtrs: Minimum rquird thicknss of shll: = f z- P OR D = f z+ P For a givn gomtry, th maximum intrnal prssur is: P max 4 f z a = D m Conditions of applicability: status_r1 if 0.16 D rturn OK Not Valid Rquirmnts if only th intrnal diamtr is known: Rsults status_r1 i if rturn OK Not Valid R1: 0.16 status_r1 = OK status_r1 i = OK D Conical shlls Local input paramtrs: Intrnal diamtr of con at point undr considration: Extrnal diamtr of con at point undr considration: Man diamtr of con at point undr considration: Man diamtr of cylindr at junction with con: Analysis thicknss of con at point undr considration: Smi angl of con at apx: 20 [mm] D 22 [mm] D m [mm] D c 300 [mm] con.a 1.36 [mm] α 30 [dgrs]

4 Output paramtrs: Minimum rquird thicknss of th conical part is: = z- P cos (α) OR D = z+ P cos (α) For a givn gomtry, th maximum intrnal prssur is: z con.a cos (α) P max = D m NOTE: both th input and output paramtrs in this sction only considrs on spcific point on th con, so multipl calculations may b rquird Conditions of applicability: status_r1 if α 75 rturn OK Not Valid status_r2 con.a cos (α) if > D c rturn OK Not Valid Rquirmnts Rsults R1: r 0.2 status_r1 = OK R2: r Torisphrical nds Input paramtrs: 0.06 status_r2 = OK Insid radius torisphrical nd: Insid radius of curvatur of a knuckl: Intrnal diamtr of cylindrical flang: Analysis thicknss: R 600 [mm] r 40 [mm] 430 [mm] a 2 [mm]

7.5.3.2 Dsign r = 0.093 0.75 + 0.2 D i R P f = 0.023 Th valu of β a can b rad from th chart abov or calculatd through th itrativ procss shown in 3.5.3.5, a) β a [unitlss] Calculation valus: R p0.

5 Dsign r = D i R P f = Th valu of β a can b rad from th chart abov or calculatd through th itrativ procss shown in , a) β a [unitlss] Calculation valus: R p0.2;t f b1 = OR, for cold spun samlss austntic stainlss stl: R p0.2;t f b2 = Th corrct valu for f b should b ntrd with rgard to th matrial usd: f b 133 [MPa] (at tst conditions th valu 1.5 in th fb quations shall b rplacd by 1.05) Th rquird ticknss shall b th gratst of s,y and b, whr: R s = z- 0.5 P R y β a P =? f P b 0.75 R D i = f b r Output paramtrs: max s, y, b =? NOTE 1 For stainlss stl nds that ar not cold spun, fb will b lss than f. NOTE 2 Th 1.6 factor for cold spun nds taks account of strain hardning.

NOTE 3 Th insid hight of a torisphrical nd is givn by: h i R - R - D i R + - 2 r = 68.046 2 2 7.5.3.3 Rating a = R 0.003 r = 0.

5 OR, for cold spun samlss austntic stainlss stl: 7.5-4 1.6 R p0.2;t f b2 = 213.333 7.5-5 1.

6 NOTE 3 Th insid hight of a torisphrical nd is givn by: h i R - R - D i R r = Rating a = R r = Th valu of β b can b rad from th chart abov or calculatd through th procss shown in , b) β b [unitlss] Calculation valus: R p0.2;t f b1 = OR, for cold spun samlss austntic stainlss stl: R p0.2;t f b2 = Th corrct valu for f b should b ntrd with rgard to th matrial usd: f b 133 [MPa] (at tst conditions th valu 1.5 in th fb quations shall b rplacd by 1.05) For a givn gomtry Pmax shall b th last of Ps, Py and Pb, whr: z a P s = R a f a P y =? β b 0.75 R a 1.5 P b 111 f b r = R Output paramtrs: P max min P s, P y, P b =?

7.5.3.4 Excptions 7.5.3.5 Formula for calculation of factor β a) itrativ procss for finding th minimum thicknss NB! Origin has to b 1 Functions Y y min y, R 0.04 Z y log 1 Y y 1 N (Y) 1.006-6.

7 Excptions Formula for calculation of factor β a) itrativ procss for finding th minimum thicknss NB! Origin has to b 1 Functions Y y min y, R 0.04 Z y log 1 Y y 1 N (Y) (90 Y) 4 y (β) 0.75 R β P f f Δ ( x, y) y- x x Calculation quantitis X r = Initial bta ( β ): Convrgnc limit: Bta Cod β β 1.8 (Th initial valu dos not hav any impact on th rsult) Δ lim (Prcntag chang) Outputs β cod Δ 1 i 1 β β i 0 whil Δ> Δ lim i y β i Y Y i Z Z i N N (Y) β i + 1 β 0.06 N Z Z β 0.1 N Z Z Z β 0.2 max Y Y 2, 0.5 β A 25 ( X) β ( X ) β 0.1 β B 10 ( X) β ( X - 0.1) β 0.2 if X=0.06 β 0.06 if 0.06 < X < 0.1 β A if X=0.1 β 0.1 if 0.1 < X < 0.2 β B if X=0.2 β 0.2 Δ f Δ β, β i i + 1 i i + 1 rturn β From all itrations: β cod = Numbr of itrations: Rsulting valus: y β cod = n lngth β cod = 4 β β codn = y y (β) = Th minimum rquird thicknss is thn: max s, y, b =

8 b) non-itrativ procss for finding maximum prssur Y min a, = R Z log 1 = Y X r = N = (90 Y) 4 β 0.06 N Z Z β 0.1 N Z Z Z β 0.2 max Y Y 2, For 0.06<X<0.1 For 0.1<X<0.2 β A 25 ( X) β ( X ) β β B 10 ( X) β ( X - 0.1) β β b if X=0.06 β β 0.06 if 0.06 < X < 0.1 β β A if X=0.1 β β 0.1 if 0.1 < X < 0.2 β β B if X=0.2 β β 0.2 f a β b = 1.12 => P y = β b 0.75 R P max min P s, P y, P b = Conditions of applicabillity status_r1 if r 0.2 D i rturn R1 OK R1 Not Valid status_r2 if r 0.06 D i rturn R2 OK R2 Not Valid status_r3 if r 2 rturn R3 OK R3 Not Valid status_r4 if 0.08 D rturn R4 OK R4 Not Valid status_r5 if a D rturn R5 OK R5 Not Valid status_r6 if R D rturn R6 OK R6 Not Valid Rquirmnts Rsults R1: r 0.2 status_r1 = R1 OK R2: r 0.06 status_r2 = R2 OK R3: r 2 status_r3 = R3 OK R4: 0.08 D status_r4 = R4 Not Valid

9 R5: a D status_r5 = R5 OK R6: R D status_r6 = R6 Not Valid Claus 15 - Prssur vssls of rctangular sction 15.5 Unrinforcd vssls Input paramtrs: Th insid cornr radius: Th dimnsions of th vssl: (s figur ) Th thicknss of th vssl: a 50 [mm] l [mm] L 367 [mm] 3 [mm] Calculation valus: Th mmbran strsss at ach ara is dtrmind by: At C: P ( a+ L) σ m.c = At D: σ m.d σ m.c = 556 At B: P a+ l 1 σ m.b = At A: σ m.a σ m.b = 600

10 At a cornr: σ m.bc P + L2 + 2 l 1 = Th scond momnt of ara is givn by: I 1 = I 2 I 1 = Th following quations apply to calculat th bnding strsss (q q ) θ atan l 1 = L NOTE: th valu of θ varis thorughout th cornr, but th valu givn hr will giv th highst bnding strss ϕ a = l 1 α 3 L = l 1 2 l 1 6 ϕ 2 2 α 3-3 ϕ ϕ α 3-6 ϕ α 3 ϕ + 6 ϕ α 3 K 3 = α 3 + ϕ + 2 M A P -K 3 = Th bnding strsss of ach ara is dtrmind by: At C: ± σ b.c 2 M A + P 2 a L- 2 a l 1 + L 2 = I 1 At D: ± σ b.d abs 2 M A + P 2 a L- 2 a l 1 + L l 1 =? 4 I 1 At A: ± σ b.a abs M A =? 2 I At B: ± σ b.b abs 2 M A + L 2 = I At a cornr: ± σ b.bc abs 2 M A + P 2 a L cos (θ) - l 1 ( 1- sin(θ)) + L 2 = I 1 Output paramtrs:: Th maximum strss at any location is qual to th sum of th bnding and mmbran strsss at that location (as statd in 15.4). Th maximum strss is thrfor: At C: σ max.c σ m.c + σ b.c =

11 At D: σ max.d σ m.d + σ b.d =? At A: σ max.a σ m.a + σ b.a =? At B: At th cornrs: σ max.b σ m.b + σ b.b = σ max.bc σ m.bc + σ b.bc = Allowabl strsss for unrinforcd vssls Th mmbran strsss shall b limitd as follows: σ m f z status_σ m.c m.c f z rturn OK Not Valid status_σ m.a m.a f z rturn OK Not Valid status_σ m.bc m.bc f z rturn OK Not Valid status_σ m.d m.d f z rturn OK Not Valid status_σ m.b m.b f z rturn OK Not Valid This givs: status_σ m.c = Not Valid status_σ m.d = Not Valid status_σ m.a = Not Valid status_σ m.b = Not Valid status_σ m.bc = Not Valid Th sum of th mmbran strsss and th bnding strsss shall conform to: σ m + σ b 1.5 f z status_σ C max.c 1.5 f z rturn OK Not Valid status_σ D max.d 1.5 f z rturn OK Not Valid status_σ BC max.bc 1.5 f z rturn OK Not Valid status_σ A max.a 1.5 f z rturn OK Not Valid status_σ B max.b 1.5 f z rturn OK Not Valid This givs: status_σ C = Not Valid status_σ D =? status_σ A =? status_σ B = Not Valid status_σ BC = Not Valid Brows for Imag...

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt