Cylindrical Shell of Pressure Vessel Calculation

1

CHAPTER CHAPTER CHAPTER CHAPTER 7777

cylindrical & spherical shell cylindrical & spherical shell cylindrical & spherical shell cylindrical & spherical shell

Stress element

Stresses in Pressurized CylindersStresses in Pressurized CylindersStresses in Pressurized CylindersStresses in Pressurized Cylinders Cylindrical pressure vessels, hydraulic cylinders, shafts with components mounted on (gears, pulleys, and bearings), gun barrels, pipes carrying fluids at high pressure,….. develop tangential, longitudinal, and radial stresses.

Wall thickness

t

Radial stressσr

Longitudinal stressσl (closed ends)

A pressurized cylinder is considered a thin-walled vessel if the wall thickness is less than one-twentieth of the radius.

< 1/20tr Thin-walled pressure

vessel

Tangential stress σθ

Hoop stress

Stresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized CylindersIn a thin-walled pressurized cylinder the radial stress is much smaller than the tangential stress and can be neglected.

Longitudinal stress, σl

(σl) π/4 [ (do)2 – (di)2] = ( p ) π/4 (di)

2

Internal pressure, p

Σ Fy = 0

Longitudinal stress

(σl) [ (di + 2t)2 – (di)2] = ( p ) (di)

2

4t2 is very small,

(σl) (4di t) = ( p ) (di)2

σl = p di2 t

σl = p (di + t)

4 tMax. longitudinal stress

Pressure area

Stresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized CylindersStresses in a Thin-Walled Pressurized Cylinders

Projected area

Hoop stress

σθ

Tangential (hoop) stress

2(σθ) t (length) = ( p ) (di) (length)Σ Fx = 0

σθ = p di 2 t

Max. Hoop stress

σl = ½ σθ

Procedure of calculation of Cylindrical Procedure of calculation of Cylindrical Procedure of calculation of Cylindrical Procedure of calculation of Cylindrical Shell under internal pressureShell under internal pressureShell under internal pressureShell under internal pressure

Minimal thicknessS’

Real thicknessS = S’+C

Verification allowable pressurep ≤ [p]

Material selection(mechanical properties)

Requiremnetsp,t ca…

6

[[[[ ]]]]

[[[[ ]]]] pppp2222pDpDpDpDssss

pppp2222pDpDpDpDssss

hhhh

nnnn

hhhh

tttt

++++ϕϕϕϕσσσσ====′′′′

−−−−ϕϕϕϕσσσσ====′′′′

Calculation formula for minimum thicknessCalculation formula for minimum thicknessCalculation formula for minimum thicknessCalculation formula for minimum thickness

Material Material Material Material< 5,5< 5,5< 5,5< 5,5 5,5 5,5 5,5 5,5 ÷÷÷÷ 25 25 25 25 ≥≥≥≥ 25 25 25 25

Brittle metal, non-Brittle metal, non-Brittle metal, non-Brittle metal, non-metallicmetallicmetallicmetallic

Ductile metalDuctile metalDuctile metalDuctile metal

[[[[ ]]]]hhhhpppp

ϕϕϕϕσσσσ

[[[[ ]]]]

[[[[ ]]]] hhhh

nnnn

hhhh

tttt

2222pDpDpDpDssss

2222pDpDpDpDssss

ϕϕϕϕσσσσ====′′′′

ϕϕϕϕσσσσ====′′′′

[[[[ ]]]][[[[ ]]]]

[[[[ ]]]][[[[ ]]]] ⎟⎟⎟⎟⎟

⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛++++ϕϕϕϕσσσσ−−−−ϕϕϕϕσσσσ

−−−−====′′′′

⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛−−−−

−−−−ϕϕϕϕσσσσ++++ϕϕϕϕσσσσ

====′′′′

pppppppp1111DDDD5555,,,,0000ssss

1111ppppppppDDDD5555,,,,0000ssss

hhhh

hhhhnnnn

hhhh

hhhhtttt

(((( ))))(((( ))))

⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛ββββ−−−−ββββ

====′′′′

−−−−ββββ++++====′′′′

1111DDDD5555,,,,0000ssss

1111CCCCDDDD5555,,,,0000ssss

nnnn

aaaatttt

1 1,5 2 2,5 3 3,5 4 5

ββββ 2,8 2 1,68 1,5 1,4 1,35 1,3 1,22

[[[[ ]]]]hhhhpppp

ϕϕϕϕσσσσ

7

[[[[ ]]]] [[[[ ]]]] (((( ))))(((( ))))aaaatttt

aaaahhhh

CCCCssssDDDDCCCCssss2222pppp

−−−−++++−−−−ϕϕϕϕσσσσ

====

tttt

aaaa

DDDDCCCCssss −−−−

[[[[ ]]]] [[[[ ]]]] ⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛++++

ϕϕϕϕσσσσ====aaaatttt

nnnnhhhh CCCC2222DDDD

DDDDlglglglg3333,,,,2222pppp

[[[[ ]]]][[[[ ]]]]

11111111DDDD5555,,,,0000CCCCssss

11111111DDDD5555,,,,0000CCCCssss

pppp 2222

tttt

aaaa

2222

tttt

aaaahhhh

++++⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛++++

−−−−

⎥⎥⎥⎥⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡−−−−⎟⎟⎟⎟⎟⎟⎟⎟

⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛++++

−−−−σσσσϕϕϕϕ

====

verification allowable pressure, [p]verification allowable pressure, [p]verification allowable pressure, [p]verification allowable pressure, [p]

verification allowable pressureverification allowable pressureverification allowable pressureverification allowable pressure

VAVAVAVAÄÄÄÄT LIET LIET LIET LIEÄÄÄÄUUUU

≤≤≤≤ 0,1 0,1 0,1 0,1 > 0,1> 0,1> 0,1> 0,1

Brittle metal, non-Brittle metal, non-Brittle metal, non-Brittle metal, non-metallicmetallicmetallicmetallic

Ductile metalDuctile metalDuctile metalDuctile metal

p p p p ≤≤≤≤ [p] [p] [p] [p]

8

Cylindrical Shell with internal pressureCylindrical Shell with internal pressureCylindrical Shell with internal pressureCylindrical Shell with internal pressure AD-B1 (Germany) AD-B1 (Germany) AD-B1 (Germany) AD-B1 (Germany)

• sB : shell thickness [mm]• Da : outer Diameter [mm]• P : Design pressure [bar]• K . Limiting yield stress [N/mm²]• S: Safety coefficient• V : Weld joint efficiency• C1-2 : Additional value of wall thickness … [mm]

Cylindrical

Spherical

9

ASME Code

P = design pressure , psig. P = design pressure , psig. P = design pressure , psig. P = design pressure , psig. r = Inner radius, in.r = Inner radius, in.r = Inner radius, in.r = Inner radius, in.S = Limiting yield stress , S = Limiting yield stress , S = Limiting yield stress , S = Limiting yield stress , psipsipsipsi....E1, E = Weld joint efficiencyE1, E = Weld joint efficiencyE1, E = Weld joint efficiencyE1, E = Weld joint efficiencytptptptp = Shell thickness, in. = Shell thickness, in. = Shell thickness, in. = Shell thickness, in. t = real thickness (exclusive value of additional thickness of corrosion), in. t = real thickness (exclusive value of additional thickness of corrosion), in. t = real thickness (exclusive value of additional thickness of corrosion), in. t = real thickness (exclusive value of additional thickness of corrosion), in.

10

� Fabricated from welded roll Fabricated from welded roll Fabricated from welded roll Fabricated from welded roll bending steel plates.bending steel plates.bending steel plates.bending steel plates.

�Welding Welding Welding Welding � Dimension of steel plateDimension of steel plateDimension of steel plateDimension of steel plate� A : longitudinal seamA : longitudinal seamA : longitudinal seamA : longitudinal seam� B : round seamB : round seamB : round seamB : round seam

Raw material

Processing steps

Ready for welding

Machine welding

Cylindrical shell manufacturingCylindrical shell manufacturingCylindrical shell manufacturingCylindrical shell manufacturing

16

Round seamRound seamRound seamRound seam

17

Cross seamCross seamCross seamCross seam

18

19

false correct

Procedure of calculation of Cylindrical Shell Procedure of calculation of Cylindrical Shell Procedure of calculation of Cylindrical Shell Procedure of calculation of Cylindrical Shell under external pressureunder external pressureunder external pressureunder external pressure

(ductile metal, safety coefficient)(ductile metal, safety coefficient)(ductile metal, safety coefficient)(ductile metal, safety coefficient)



Verification allowable pressurep p ≤≤ [p] [p]


Requiremnetsp,t ca…

calculation stiffening ring

21

Minimum thickness, sMinimum thickness, sMinimum thickness, sMinimum thickness, s′′′′

4444,,,,0000

tttttttttttt DDDD

LLLLEEEEppppDDDD18181818,,,,1111ssss ⎟⎟⎟⎟⎟⎟⎟⎟

⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛⋅⋅⋅⋅====′′′′

�Length of shell, LLength of shell, LLength of shell, LLength of shell, L

� L = H (cylï) (cyl. Shell with flanges, flat head)

( ) ( )3

headcyl hHL +=� (cyl. Shell with 2 ellipsoid heads)

� (shell with stiffening ring)( )( )⎩

⎨⎧

≥−=<=

LbbLLbL

kk

k

1,01,0

ℓℓ

⎩⎨⎧

shell contact to ring stiffening of width:rings stiffeningnearby 2 between distance :

:whilekbℓ

22

[[[[ ]]]] [[[[ ]]]](((( ))))

(((( )))) ⎥⎥⎥⎥⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛ σσσσ⋅⋅⋅⋅−−−−

++++

−−−−σσσσ==== 2222

tttt

ttttcccc

3333aaaa

tttt2222

tttt

aaaannnn

EEEECCCCssssDDDDLLLL02020202,,,,11111111DDDD

CCCCssss2222pppp

Verification allowable pressure , [p]Verification allowable pressure , [p]Verification allowable pressure , [p]Verification allowable pressure , [p]

p p p p ≤≤≤≤ [p] [p] [p] [p]

(((( ))))(((( ))))aaaa

tttt

tttttttt

aaaa

CCCCssss2222DDDD

DDDDLLLL

DDDDCCCCssss22225555,,,,1111

−−−−≤≤≤≤≤≤≤≤

−−−−

(((( )))) 5555,,,,1111

tttt

aaaattttcccc

tttt

tttt DDDDCCCCssss2222EEEE3333,,,,0000

DDDDLLLL

⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡ −−−−σσσσ

≥≥≥≥

(((( )))) 5555,,,,1111

tttt

aaaattttcccc

tttt

tttt DDDDCCCCssss2222EEEE3333,,,,0000

DDDDLLLL

⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡ −−−−σσσσ

<<<<

[[[[ ]]]]5555,,,,2222

tttt

aaaatttttttt

DDDDCCCCssss

LLLLDDDDEEEE649649649649,,,,0000pppp ⎟⎟⎟⎟⎟⎟⎟⎟

⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛ −−−−====

( )

pressure externalunder ncalculatiosimilar

5VAØ2

⎯→⎯>−

>ta

t

t DL

CsD

DL

23

Step 4: stiffening ringStep 4: stiffening ringStep 4: stiffening ringStep 4: stiffening ringMoment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of

axis (passing through the center of cross section), Jaxis (passing through the center of cross section), Jaxis (passing through the center of cross section), Jaxis (passing through the center of cross section), J’’’’ (mm (mm (mm (mm4444))))

(((( )))) ⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡−−−−−−−−====′′′′ 3333

aaaatttt

3333nnnn CCCCssss

EEEEppppDDDD18181818,,,,1111

12121212LLLLJJJJ

Selection stiffening ring suitable to cross section calculated, Selection stiffening ring suitable to cross section calculated, Selection stiffening ring suitable to cross section calculated, Selection stiffening ring suitable to cross section calculated, FFFFkkkk (mm (mm (mm (mm2222))))

(((( ))))aaaattttcccc

nnnnkkkk CCCCssssppppDDDD3333,,,,1111FFFF −−−−−−−−

σσσσ>>>> ℓℓ

24

Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of

axis (passing through the center of cross section), parallel to axis of axis (passing through the center of cross section), parallel to axis of axis (passing through the center of cross section), parallel to axis of axis (passing through the center of cross section), parallel to axis of cylcylcylcyl. shell, . shell, . shell, . shell,

JJJJkkkk (mm (mm (mm (mm4444)))) JJJJkkkk = = = = f(Df(Df(Df(Dnnnn,Lp,Lp,Lp,Lp))))

25

Distance from the center of cross section ( stiffening ring) to mid-Distance from the center of cross section ( stiffening ring) to mid-Distance from the center of cross section ( stiffening ring) to mid-Distance from the center of cross section ( stiffening ring) to mid-

surface of shellsurface of shellsurface of shellsurface of shell, e (mm), e (mm), e (mm), e (mm)

kkkk

kkkk

FFFFJJJJeeee ====

26

� Distance from the center of cross section ( stiffening ring) to mid-surface of Distance from the center of cross section ( stiffening ring) to mid-surface of Distance from the center of cross section ( stiffening ring) to mid-surface of Distance from the center of cross section ( stiffening ring) to mid-surface of

shell, shell, shell, shell, eeeeoooo (mm) (mm) (mm) (mm)

� effective length of shell (for the calculation the total cross section of effective length of shell (for the calculation the total cross section of effective length of shell (for the calculation the total cross section of effective length of shell (for the calculation the total cross section of

stiffening ring and shell ), stiffening ring and shell ), stiffening ring and shell ), stiffening ring and shell ), llllhhhh (mm) (mm) (mm) (mm)

� sectional area of the shell, sectional area of the shell, sectional area of the shell, sectional area of the shell, FFFFoooo (mm (mm (mm (mm2222))))

�Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of Moment of inertia of the cross section (of the stiffening ring and the body) of

axis (passing through the center of cross section) x axis (passing through the center of cross section) x axis (passing through the center of cross section) x axis (passing through the center of cross section) x –––– x, x, x, x, JJJJxxxx (mm (mm (mm (mm4444))))

� Verification value of pressure for stiffening ring, [p] Verification value of pressure for stiffening ring, [p] Verification value of pressure for stiffening ring, [p] Verification value of pressure for stiffening ring, [p]

[[[[ ]]]]⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧

≤≤≤≤

′′′′≥≥≥≥ppppppppJJJJJJJJxxxx

(((( )))) (((( ))))⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡ −−−−′′′′====

xxxx

aaaa

tttt

2222

JJJJ12121212CCCCssss

DDDDLLLL2222ffffKKKK ℓ

� specific parameters to the critical pressure , K specific parameters to the critical pressure , K specific parameters to the critical pressure , K specific parameters to the critical pressure , K

Value KValue KValue KValue K

5 10 20 50 100 200 500 1000 2000 5000100002000050000

KKKK 0,05 0,1 0,24 0,7 1,0 2,0 2,6 5,0 7,5 12,5 15 23 30

(((( )))) (((( ))))xxxx

aaaa

tttt

2222

JJJJ12121212CCCCssss

DDDDLLLL2222 −−−−′′′′ ℓ

27

[[[[ ]]]] (((( ))))3333tttt

xxxx3333

aaaatttt

DDDDJJJJ12121212CCCCssssEEEE85858585,,,,0000pppp

ℓℓ ++++−−−−

====

Calculation and verification sequence of stiffening ringCalculation and verification sequence of stiffening ringCalculation and verification sequence of stiffening ringCalculation and verification sequence of stiffening ring

≤≤≤≤ 0,5 0,5 0,5 0,5((((andandandand¬¬¬¬ L L L L′′′′ ≤≤≤≤ 8D) 8D) 8D) 8D)

> 0,5> 0,5> 0,5> 0,5

llllhhhh

FFFFooooeeeeoooo

FFFFoooo = = = = llllhhhh (s (s (s (s –––– C C C Caaaa))))

JJJJxxxx

[p][p][p][p]

ttttDDDDℓ

[[[[ ]]]](((( ))))

(((( ))))⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡ −−−−++++

′′′′ππππ

====9999,,,,10101010

CCCCssssJJJJKKKKLLLLDDDDEEEEpppp

3333aaaaxxxx

2222tttt

tttt2222

ℓ

(((( ))))aaaannnnkkkkhhhh CCCCssssDDDD1111,,,,1111bbbb −−−−++++====ℓ

If selected: (((( ))))⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧

−−−−++++≤≤≤≤<<<<

aaaakkkkhhhh

hhhh

CCCCssss30303030bbbbℓℓℓ(((( )))) (((( ))))[[[[ ]]]]aaaannnnkkkk3333

aaaahhhh CCCCssssDDDD1111,,,,1111bbbb

JJJJCCCCssss0278027802780278,,,,0000JJJJ

−−−−−−−−−−−−⋅⋅⋅⋅′′′′++++−−−−

′′′′−−−−==== ℓ

ℓℓℓ

kkkkoooo

kkkkoooo FFFFFFFF

eeeeFFFFeeee++++

====

(((( )))) (((( ))))⎥⎥⎥⎥⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛−−−−

++++−−−−

++++−−−−++++====2222

aaaa

oooo2222

aaaaoooo2222ooookkkkkkkkxxxx CCCCssss

eeee1212121211119999,,,,10101010CCCCssssFFFFeeeeeeeeFFFFJJJJJJJJ

� for brittle metal, non-metallic: similar sequence, change equal for brittle metal, non-metallic: similar sequence, change equal for brittle metal, non-metallic: similar sequence, change equal for brittle metal, non-metallic: similar sequence, change equalt

B

t

c σσ 5,0

28

Cylindrical shell under external pressure Cylindrical shell under external pressure Cylindrical shell under external pressure Cylindrical shell under external pressure

29

Shell under external pressure Shell under external pressure Shell under external pressure Shell under external pressure AD-B6 (Germany)AD-B6 (Germany)AD-B6 (Germany)AD-B6 (Germany)

Calculate the pressure under elastic deformation

Pressure of deformation

while

30


31

Compare stress after stiffened and limited allowable stress

Deformation pressure after stiffened

Shell length after stiffened resist deformation


Cylindrical Shell under external Cylindrical Shell under external Cylindrical Shell under external Cylindrical Shell under external pressurepressurepressurepressure

32

External pressure generated from an internal vacuum

33

Cylindrical Shell under external Cylindrical Shell under external Cylindrical Shell under external Cylindrical Shell under external pressure with stiffening ringpressure with stiffening ringpressure with stiffening ringpressure with stiffening ring


34External pressure exist for the length of the jacket

35


Half pipe jacketsHalf pipe jacketsHalf pipe jacketsHalf pipe jackets

36


Stayed surface jacketsStayed surface jacketsStayed surface jacketsStayed surface jackets

37


A dimple jacket in form of stayed surface

38

Cylindrical Shell under external pressureCylindrical Shell under external pressureCylindrical Shell under external pressureCylindrical Shell under external pressure

Half pipe jacket reactor

Stiffening ringStiffening ringStiffening ringStiffening ring

39

40


41


42


Dimple Jacket stiffened reactorDimple Jacket stiffened reactorDimple Jacket stiffened reactorDimple Jacket stiffened reactor

43

Jacket ShellJacket ShellJacket ShellJacket Shell

44

Jacket ShellJacket ShellJacket ShellJacket Shell

45

Procedure of calculation of cylindrical Procedure of calculation of cylindrical Procedure of calculation of cylindrical Procedure of calculation of cylindrical shell under axial loadshell under axial loadshell under axial loadshell under axial load



Stress under axial loadMaterial selection

(mechanical properties)

Responding requirementsp,t ca…

Dependent parameters and coefficients

kc, Kc

(((( ))))(((( ))))aaaattttnPnPnPnP CCCCssssssssDDDD

PPPP−−−−++++ππππ

====σσσσ

Verification stress

47


[[[[ ]]]]nnnnttttDDDDPPPPssssσσσσππππ

====′′′′

≤ 5 > 5

ssss′′′′ [[[[ ]]]]ϕϕϕϕσσσσππππ====′′′′

nnnnttttDDDDPPPPssss

ttttDDDDLLLL ′′′′′′′′

with with with with ϕϕϕϕ = f ( = f ( = f ( = f (λλλλ)))) ((((DiagrammDiagrammDiagrammDiagramm 5.3) 5.3) 5.3) 5.3)tttt

nnnn

DDDD82828282,,,,2222 ℓ

====λλλλ

llll

Free end

clamped

llllnnnn = 2L = 2L = 2L = 2L″″″″

PPPP

llll

hinged õ

llllnnnn = L = L = L = L″″″″

hinged õ

PPPP

llll

hinged õ

llllnnnn = 0,7L = 0,7L = 0,7L = 0,7L″″″″

clamped

PPPP

llll

llllnnnn = 0,5L = 0,5L = 0,5L = 0,5L″″″″

clamped

PPPP clamped

48

25 25 25 25 ÷÷÷÷ 250 250 250 250 > 250> 250> 250> 250

KKKKcccc KKKKcccc = = = = kkkkcccc

Value Value Value Value KKKKcccc

(((( ))))aaaa

tttt

CCCCssss2222DDDD−−−−

cccctttt

ttttcccc

cccc kkkkEEEE

875875875875KKKK σσσσ====

Value Value Value Value kkkkcccc

50 100 150 200 250 500 1000 2000 2500

kkkkcccc 0,05 0,098 0,14 0,15 0,14 0,118 0,08 0,06 0,055

(((( ))))aaaa

tttt


Factor Factor Factor Factor kkkkcccc

49

Verification allowable stress under Verification allowable stress under Verification allowable stress under Verification allowable stress under axially load, [axially load, [axially load, [axially load, [σσσσnnnn]]]]TBTBTBTB

[[[[ ]]]] (((( ))))ssssDDDDCCCCssssEEEEKKKK

tttt

aaaattttccccTBTBTBTBnnnn ++++

−−−−====σσσσ

ttttcccc

aaaa EEEEKKKKPPPPCCCCssssHayHayHayHay

ππππ≥≥≥≥−−−−

[[[[ ]]]]⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧

σσσσ≤≤≤≤σσσσ≤≤≤≤

TBTBTBTBnnnnnPnPnPnP

cccc 155155155155,,,,0000KKKK

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cylindrical shell under axial loadcylindrical shell under axial loadcylindrical shell under axial loadcylindrical shell under axial load

Procedure of calculation of cylindrical shell Procedure of calculation of cylindrical shell Procedure of calculation of cylindrical shell Procedure of calculation of cylindrical shell under Moment bendingunder Moment bendingunder Moment bendingunder Moment bending




Dependent parameters and coefficients

ku, Ku

verification stress


[[[[ ]]]]uuuu2222tttt

uuuu

DDDDMMMM4444ssssσσσσππππ

====′′′′

Bending stress

(((( )))) (((( ))))aaaa2222

tttt

uuuuuuuuuuuu CCCCssssssssDDDD

MMMM4444WWWWMMMM

−−−−++++ππππ========σσσσ

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Value Value Value Value kkkkuuuu

50 100 150 200 250 500 1000 2000 2500

kkkkuuuu 0,06 0,12 0,15 0,17 0,17 0,14 0,10 0,08 0,07

(((( ))))aaaa

tttt


Factor Factor Factor Factor kkkkuuuu

25 25 25 25 ÷÷÷÷ 250 250 250 250 > 250> 250> 250> 250

KKKKuuuu KKKKuuuu = = = = kkkkuuuu

Factor KFactor KFactor KFactor Kuuuu

(((( ))))aaaa

tttt


uuuutttt

ttttcccc

uuuu kkkkEEEE

875875875875KKKK σσσσ====

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Verification allowable stress of shell while bending, Verification allowable stress of shell while bending, Verification allowable stress of shell while bending, Verification allowable stress of shell while bending, [[[[σσσσuuuu]]]]TBTBTBTB

[ ]( ) t

tu

uat

tu

ua

t

at

uTBu EDKMCsor

EsDKMCsor

sDCsEK 27,127,1

≥−+

≥−+−

=σ

(((( ))))

[[[[ ]]]]⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩⎩⎩⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨⎨⎨⎨

⎧⎧⎧⎧

σσσσ≤≤≤≤σσσσ≤≤≤≤

⎥⎥⎥⎥⎦⎦⎦⎦

⎤⎤⎤⎤⎢⎢⎢⎢⎣⎣⎣⎣

⎡⎡⎡⎡ −−−−≤≤≤≤ττττ

TBTBTBTBuuuuuuuu

uuuu

5555,,,,1111

tttt

aaaatttt

185185185185,,,,0000KKKKDDDD

CCCCssss2222EEEE07070707,,,,0000

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Shell under combine load ( external pressure Shell under combine load ( external pressure Shell under combine load ( external pressure Shell under combine load ( external pressure ––––

axially load axially load axially load axially load –––– Moment bending) Moment bending) Moment bending) Moment bending)

step 1: calculate the thickness separately of each case, choose then step 1: calculate the thickness separately of each case, choose then step 1: calculate the thickness separately of each case, choose then step 1: calculate the thickness separately of each case, choose then

the absolute maximal value the absolute maximal value the absolute maximal value the absolute maximal value

Step 2: stable condition of shell for any cross sectionStep 2: stable condition of shell for any cross sectionStep 2: stable condition of shell for any cross sectionStep 2: stable condition of shell for any cross section

[[[[ ]]]] [[[[ ]]]] [[[[ ]]]] 1111pppppppp

TBTBTBTBuuuu

uuuu

TBTBTBTBnnnn

nPnPnPnP ≤≤≤≤σσσσσσσσ

++++σσσσσσσσ

++++

Note: if any part missed , value equal 0Note: if any part missed , value equal 0Note: if any part missed , value equal 0Note: if any part missed , value equal 0

Procedure of calculation of Cylindrical shell under combine load : Procedure of calculation of Cylindrical shell under combine load : Procedure of calculation of Cylindrical shell under combine load : Procedure of calculation of Cylindrical shell under combine load : internal internal internal internal pressurpressurpressurpressur –––– axially load (pressing force or tension) axially load (pressing force or tension) axially load (pressing force or tension) axially load (pressing force or tension) –––– moment moment moment moment

(bending or torsion)(bending or torsion)(bending or torsion)(bending or torsion)


Material selection(mechanical properties


(((( ))))[[[[ ]]]](((( ))))aaaahhhh

aaaattttnpnpnpnp CCCCssss2222

CCCCssssDDDDpppp−−−−ϕϕϕϕ−−−−++++

====σσσσ

Stress from pressure

(((( )))) 2222xxxx

2222uuuunPnPnPnPtdtdtdtd 33338888,,,,0000 ττττ++++σσσσ++++σσσσ====σσσσ

Equivalente stress

Minimum thickness S’(choose the maximal value

of different case )

(((( )))) (((( ))))aaaa2222

tttt

xxxxxxxx CCCCssssssssDDDD

MMMM8888−−−−++++ππππ

====ττττ

Torsion stress

Condition for the stability of shell [[[[ ]]]] [[[[ ]]]]

2222npnpnpnp

tdtdtdtd 2222,,,,111187878787,,,,0000 ⎟⎟⎟⎟⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛σσσσσσσσ

−−−−σσσσ≤≤≤≤σσσσ

Note: if any part missed , value equal 0Note: if any part missed , value equal 0Note: if any part missed , value equal 0Note: if any part missed , value equal 0

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Spherical ShellSpherical ShellSpherical ShellSpherical Shell

� Less material used to be fabricated. Less material used to be fabricated. Less material used to be fabricated. Less material used to be fabricated.

� Mainly used for big volume (Mainly used for big volume (Mainly used for big volume (Mainly used for big volume (≥≥≥≥100m100m100m100m3333). ). ). ).

� Applicable for pressure until 2,5N/mmApplicable for pressure until 2,5N/mmApplicable for pressure until 2,5N/mmApplicable for pressure until 2,5N/mm 2222, or vacuum., or vacuum., or vacuum., or vacuum.

note : designing a spherical shellnote : designing a spherical shellnote : designing a spherical shellnote : designing a spherical shell

� main methods of fabricating: butt weld small pieces prefabricated. main methods of fabricating: butt weld small pieces prefabricated. main methods of fabricating: butt weld small pieces prefabricated. main methods of fabricating: butt weld small pieces prefabricated.

The welds must be easily observed.The welds must be easily observed.The welds must be easily observed.The welds must be easily observed.

�The openings must have suitable distance to weld, stiffened if The openings must have suitable distance to weld, stiffened if The openings must have suitable distance to weld, stiffened if The openings must have suitable distance to weld, stiffened if swhellswhellswhellswhell

under external pressure.under external pressure.under external pressure.under external pressure.

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58

• Inside diameter DInside diameter DInside diameter DInside diameter D• Arrangement of segmentsArrangement of segmentsArrangement of segmentsArrangement of segments• Minimum wall Minimum wall Minimum wall Minimum wall thickness(esthickness(esthickness(esthickness(es) after forming s) after forming s) after forming s) after forming s• Fabrication codeFabrication codeFabrication codeFabrication code

Information requiredInformation requiredInformation requiredInformation required

59


60

Spherical HeadSpherical HeadSpherical HeadSpherical Head

Procedure of calculation of Spherical Shell Procedure of calculation of Spherical Shell Procedure of calculation of Spherical Shell Procedure of calculation of Spherical Shell under internal pressureunder internal pressureunder internal pressureunder internal pressure

Minimum thicknessS’

real thicknessS = S’+C




[[[[ ]]]]2222

tttt

aaaatttt

DDDDCCCCssssEEEE87878787,,,,1111pppp ⎟⎟⎟⎟⎟⎟⎟⎟

⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛ −−−−====

62

Minimum thickness, sMinimum thickness, sMinimum thickness, sMinimum thickness, s′′′′FormulasFormulasFormulasFormulas

< 2,75< 2,75< 2,75< 2,75 2,75 2,75 2,75 2,75 ÷÷÷÷ 12,5 12,5 12,5 12,5 ≥≥≥≥ 12,5 12,5 12,5 12,5

ssss′′′′

[ ]hp

ϕσ

[[[[ ]]]][[[[ ]]]] ⎟⎟⎟⎟

⎟⎟⎟⎟⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛−−−−

−−−−ϕϕϕϕσσσσϕϕϕϕσσσσ

====′′′′ 1111pppp33332222

2222DDDD5555,,,,0000sssshhhh

hhhhtttt [[[[ ]]]] pppp4444

ppppDDDDsssshhhh

tttt

−−−−ϕϕϕϕσσσσ====′′′′ [[[[ ]]]] hhhh

tttt

4444ppppDDDDssssϕϕϕϕσσσσ

====′′′′

Procedure of calculation of spherical shell under Procedure of calculation of spherical shell under Procedure of calculation of spherical shell under Procedure of calculation of spherical shell under external pressureexternal pressureexternal pressureexternal pressure

(stable safety coefficient, ny = 2,6)allowable Ovality any means ≤ 0,5%

minimum thicknessS’

real thicknessS = S’+C




[[[[ ]]]]2222

tttt

aaaatttt

DDDDCCCCssssEEEE87878787,,,,1111pppp ⎟⎟⎟⎟⎟⎟⎟⎟

⎠⎠⎠⎠

⎞⎞⎞⎞⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛ −−−−====

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tttttttt EEEEppppDDDD73737373,,,,0000ssss ====′′′′

Ductile metalDuctile metalDuctile metalDuctile metal Brittle metal Brittle metal Brittle metal Brittle metal –––– non metal non metal non metal non metal

tttttttt EEEEppppDDDD73737373,,,,0000ssss ====′′′′

6666,,,,2222nnnn

EEEEppppDDDD73737373,,,,0000ssss BBBB

tttttttt ⋅⋅⋅⋅====′′′′

65

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ExampleExampleExampleExample : : : : thicknessthicknessthicknessthickness accordingaccordingaccordingaccording to to to to diameterdiameterdiameterdiameter and and and and pressurepressurepressurepressure / Material / Material / Material / Material

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exercise

Calculate the thicknesses cylindrical shell of a jacketed reactor, Material used : Alloy stainless steel 304, E150 =1,95.105N/mm2, б150=220N/mm2

, [бn] =140N/mm2, [б] = 140N/mm2, corrosion factor Ca =1,2mm, shells were welded automatically with a weld joint efficiency φh=0,98; correction factor = 1,05 (other parameters may be choosed by yourself)1/ Working pressur inside the reactor: 2 bar2/ use saturated steam for heating about 5 bar3/ Acidic reaction environment with : pH=3

(See picture on next slide)

68

exercise

Calculate thickness of cylindrical shell , material : stainless steel 304, inner diameter Dt =2,5m. Heigth H = 10m, chemical medium KLR = 1600kg/m3, joint efficiency = 0,98; safety factor =1,3; working pressure 3bar, corrosion rate T = 0,06mm/year, allowable stress of stainless steel 304 is 180N/mm 2

Cylindrical Shell in Cylindrical Shell in Cylindrical Shell in Cylindrical Shell in Chemical EquipmentChemical EquipmentChemical EquipmentChemical Equipment

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““““Fix Tube SheetFix Tube SheetFix Tube SheetFix Tube Sheet”””” Heat Exchanger Heat Exchanger Heat Exchanger Heat Exchanger

Split View of PartsSplit View of PartsSplit View of PartsSplit View of Parts

GasketGasket

GasketDistributor

End Flange

Cover

Tube Sheet

Shell

Saddle

Tube Sheet

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““““Fix Tube SheetFix Tube SheetFix Tube SheetFix Tube Sheet”””” Heat Exchanger Heat Exchanger Heat Exchanger Heat ExchangerGeneral Assembly viewGeneral Assembly viewGeneral Assembly viewGeneral Assembly view

Shell side OutletOutletOutletOutlet

Shell side InletInletInletInlet

Tube side InletInletInletInlet

Tube side OutletOutletOutletOutlet

Shell

Distributor

Tube Sheet

Gasket

Saddle

End Flange

Tube SheetGasket

Gasket

Cover

Cylindrical Shell of Pressure Vessel Calculation

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