SHELLS

1) BASED ON INTERNAL PRESSURE

a) Cylindrical Shell

𝑡 =𝑃𝐷𝑖

2𝑓𝐽−𝑃=

𝑃𝐷𝑜

2𝑓𝐽+𝑃

Circumferential stress, 𝑓𝑝 =𝑃𝐷

2𝑡

Longitudinal stress, 𝑓𝑎 =𝑃𝐷

4𝑡

Where, p – internal pressure

D – Mean dia. of shell

Under Combined Loadings:

1) Circumferential/tangential stress:

𝑓𝑡 = 𝑃𝐷𝑖+𝑡

2𝑡 (Tensile)

2) Longitudinal stress:

a) Due to the internal pressure-

𝑓1 =𝑃𝐷𝑖

4𝑡 (Tensile)

b) Due to weight of vessel and

its contents-

𝑓2 =𝑊

𝜋𝑡(𝐷𝑖+𝑡) (Compressive)

c) Wind load-

𝑓3 = −+ 𝑀

𝑧 = −

+𝑀/(𝜋𝐷𝑖2𝑡)

(Tensile/Compressive)

M = bending moment

𝑓𝑎 = 𝑓1 + 𝑓2 + 𝑓3

3) Stress due to offset piping and

wind:

𝑓𝑠 = 2𝑇/(𝜋𝑡𝐷𝑖(𝐷𝑖 + 𝑡))

T = torque about vessel axis

Resultant stress –

𝑓𝑅 = (𝑓𝑡2 − 𝑓𝑡𝑓𝑎 + 𝑓𝑎

2 + 3𝑓𝑠2)0.5

fc (permissible)

= 1

12∗

𝐸

√3(1− 𝜇2)∗

𝑡

𝐷𝑜/2

E = modulus of elasticity

µ = Poisson’s ratio

b) Spherical shell

𝑡 =𝑃𝐷𝑖

4𝑓𝐽 − 𝑃=

𝑃𝐷𝑜

4𝑓𝐽 + 𝑃

2) BASED ON EXTERNAL PRESSURE

Cylindrical Shell –

𝑃𝑐 =2.42𝐸 (

𝑡𝐷𝑜

)2.5

(1 − 𝜇2)0.75 ∗ {𝐿

𝐷𝑜− 0.45 (𝑡/𝐷𝑜)1/2}

𝑃𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 =1

4∗ 𝑃𝑐

𝐿 = 𝑢𝑛𝑠𝑠𝑢𝑝𝑜𝑟𝑡𝑒𝑑 𝑙𝑒𝑛𝑔𝑡ℎ

HEADS

1) INTERNAL PRESSURE:

I) CONICAL HEAD:

Circumferential stress: 𝑓 =𝑃𝐷

2𝑡 cos 𝑑

𝑡ℎ =𝑃𝐷

2𝑓𝐽 cos 𝑑+ 𝐶. 𝐴

d = half apex angle of cone

II) TORRISPHERICAL HEAD:

𝑡ℎ =𝑝𝑅𝑐𝑊

2𝑓𝐽+ 𝐶. 𝐴

Rc = crown radius

W = stress intensity factor =

1

4∗ (3 + √

𝑅𝑐

𝑅𝑘)

Rk = knuckle radius = 0.06 Rc

III) ELLIPTICAL HEAD:

𝑡ℎ =𝑃𝐷𝑉

2𝑓𝐽

𝐷 = 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟

𝑉 =1

6(2 + 𝑘2),

𝑘 =(𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠)

𝑚𝑖𝑛𝑜𝑟 𝑎𝑥𝑖𝑠

IV) HEMISPHERICAL HEAD:

𝑡ℎ =𝑃𝐷𝑖

4𝑓𝐽 − 𝑃=

𝑃𝐷𝑜

4𝑓𝐽 + 𝑃

2) EXTERNAL PRESSURE:

I) DOMED HEADS:

𝑡ℎ = 𝑑𝑅𝑐{3(1 − 𝜇2)}1/4 √𝑃/2𝐸

+ 𝐶. 𝐴

Where, d = 4 to 4.4

Rc = crown radius

E= young’s modulus.

NOZZLES

AB = Max. horizontal distance = 2d

AD = Max. vertical distance = 6t or 3.5ts + 2.5tn

H1 = H2 = 2.5ts (by nozzle only)

Else, H1 = 2.5tn (by both ring pad and nozzle)

𝐴 = 𝑑𝑡𝑠,

𝐴𝑠 = 𝑑(𝑡𝑠 – 𝑡𝑠′ − 𝐶. 𝐴) ,

𝐴𝑛 = 2𝐻1(𝑡𝑛 – 𝑡𝑛′ − 𝐶. 𝐴),

𝐴1 = 2𝐻2(𝑡𝑛 – 2𝐶. 𝐴)

𝐴′ = 𝐴𝑠 + 𝐴𝑛 + 𝐴1 =

𝐶𝑜𝑚𝑝𝑒𝑛𝑠𝑎𝑡𝑖𝑜𝑛 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒

𝐴 − 𝐴′ = 𝐴𝑟𝑒𝑎 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑

FLANGES

BOLT LOAD

𝑊𝑚𝑎𝑡𝑚= 𝜋𝑏𝐺𝑌𝑎

Where Ya = gasket seating stress

b = bo if bo < 6.3

2.5√𝑏𝑜 if bo >6.3

bo=(G0-Gi)/4

𝑊𝑚,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛 = 𝜋2𝑏𝐺𝑚𝑃 +𝜋

4𝐺2𝑃

m = gasket factor

𝐺 = (𝐺𝑜 + 𝐺𝑖)/2

𝐴𝑚,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 = 𝑊𝑚,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔

𝑓𝑚,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔

𝐴𝑚,𝑎𝑡𝑚 =𝑊𝑚,𝑎𝑡𝑚

𝑓𝑎𝑡𝑚

Tensile stress,

𝑡𝑙 = 𝐺√𝑃

𝑘𝑓+ 𝐶. 𝐴

𝑘 = 1/(0.3 + 1.5𝑊𝑚ℎ𝐺

𝐻𝐺)

ℎ𝐺 = 𝐵 − 𝐺

2

𝐻 =𝜋

4𝐺2𝑃

f= permissible stress for flanges

Where,

Wm = total bolt load

B = Bolt circle diameter

G = diameter of gasket load

reaction

𝑌 = 0.51𝑋2.09

Where,

Y = bolt area

X = nominal diameter

𝐴𝑏 =2𝜋𝑌𝑎𝐺𝑁

𝑓𝑎

SPHERICAL SHELL

Subjected to external pressure-

𝑃𝑐 = 8𝐸

√3(1 − 𝜇2) (

𝑡

𝐷)

2

D = outside diameter

Pallowable = 1/6*Pc

SHELL WITH EXTERNAL JACKET

𝑡𝑠 = 𝑑√𝑘1𝑃

𝑓1+ 𝐶. 𝐴

𝑡𝑐 = 𝑑√𝑘2𝑃

𝑓2+ 𝐶. 𝐴

Where,

k1 = 0.167

k2 = 0.12

SHELL WITH HALF COIL JACKET

𝑓𝑃𝐶 =𝑝𝑑𝑖

2𝑡𝑐

𝑓𝑎𝑐 =𝑝𝑑𝑖

4𝑡𝑐+2.5𝑡𝑠

𝑓𝑃𝐶 = 𝑓𝑃 + 𝑓𝑎𝑐 (total

circumferential stress)

= 𝑃𝐷𝑖

2𝑡𝑠 +

𝑝𝑑𝑖

4𝑡𝑐+2.5𝑡𝑠

Total longitudinal stress:

𝑓𝑎𝑠 = 𝑓𝑎 + 𝑓𝑎𝑐 + 𝑓𝑏

𝑓𝑎𝑠 = 𝑃𝐷𝑖

4𝑡𝑠 +

𝑝𝑑𝑖

2𝑡𝑠+

2Δ𝑃𝑑𝑜2

3𝑡𝑠2

P= internal pressure of shell

P=internal pressure of coil

Di= shell internal diameter

d= internal diameter of coil

FLAT PLATE HEAD

𝑡 = 𝐶 ∗ 𝐷√𝑃

𝑓

f = shear stress