BS EN 13445-3 (2009)

BS EN 13445-3:2009 EN 13445-3:2009 (E) Issue 1 (2009-07)

15 15.1

Pressure vessels of rectangular section Purpose

This clause specifies requirements for the design of unreinforced and reinforced pressure vessels of rectangular cross-section. For fatigue, designs shall be checked against either clause 17 or clause 18.

15.2

Specific definitions

The following terms and definitions apply in addition to those in clause 3. 15.2.1 membrane stress equivalent uniform stress through the wall of the vessel, see also C.4.4.2 15.2.2 bending stress equivalent linear distributed stress through the wall of the vessel, see also C.4.4.3

15.3

Specific symbols and abbreviations

The following symbols and abbreviations apply in addition to those in clause 4: a

is the inside corner radius;

A1

is the cross-sectional area of a reinforcing member which is attached to the short side of a vessel;

A2

is the cross-sectional area of a reinforcing member which is attached to the long side of the vessel;

b

is the unsupported width of a flat plate between reinforcing elements, see Figure 15.6-1;

be

is the effective width of a plate in combination with a reinforcing member, see Figure 15.6-1;

bR

is the pitch between centrelines of reinforcing members on a vessel;

c

is the distance from the neutral axis of a section to the outer fibre of a section and is positive when inwards;

C

is a shape factor determined from the long and short sides of an unsupported plate between stiffeners, see Table 15.6-2;

d

is either the diameter of an opening or the inside diameter of a welded connection if attached by a full penetration weld;

g

is the length of an unsupported span;

h

is the inside length of the long side;

h1

is the distance between the neutral axes of reinforcing members on the long side;

H

is the inside length of the short side;

H1

is the distance between the neutral axes of reinforcing members on the short side;

I1, I2

is the second moment of area per unit width of a strip of thickness e;

I11

is the second moment of area of the combined reinforcing member and plate of on the short side of the vessel;

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I21

is the second moment of area of the combined reinforcing member and plate on the long side of the vessel;

k

is a factor, see equation (15.5.2-4);

K3

is a factor for unreinforced vessel to Figure 15.5-1;

l1, lx, L, Ly

are the dimensions of the vessel;

MA

is the bending moment at the middle of the long side, it is positive when the outside of the vessel is put into compression. It is expressed as bending moment per unit length (in N.mm/mm);

p

is the hole pitch along the plate length, see Figure 15.5-2;

ps

is the diagonal hole pitch, see Figure 15.5-2;

D

is a factor, see equation (15.5.2-5);

D1

is a factor, see equation (15.5.1.2-13);

D3

is a factor, see equation (15.5.1.2-14);

E

is the angle between the line of the holes and the long axis, see Figure 15.5-2.

T

is an angle indicating position at the corner of a vessel, see Figure 15.5-2;

P

is the ligament efficiency;

Vb

is the bending stress;

Vm

is the membrane stress;

I

15.4

is a factor, see equation (15.5.1.2-15).

General

The equations given in this subclause shall be used for calculation of the membrane and bending stresses in unreinforced and reinforced rectangular pressure vessels. The maximum stress at a given location shall be taken as the sum of the membrane stress and the bending stress at that location. For vessels operating with extensive fatigue loads (for example sterilizers) the longitudinal corners of the vessel shall be provided with an inside radius not less than three times the wall thickness. For pressure vessels provided with doors a special analysis shall be performed to detect any deformation in the door and the edge of the vessel. NOTE

15.5

Special care should be taken in the choice of gasket for the door.

Unreinforced vessels

15.5.1 15.5.1.1

Unreinforced vessels without a stay General

This method applies to vessels of the type shown in Figure 15.5-1. It is assumed that the thicknesses of the short and long sides are equal. When they are not, the method in 15.6 shall be used.

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15.5.1.2

Unperforated plates

Where the thickness of the smaller side is not the same as the thickness of the longer side, the calculation method for reinforced vessels in 15.6 shall be used. For unreinforced vessels conforming to Figure 15.5-1, the membrane stresses are determined from the following equations: at C,

V m c at D,

P a  L e

(15.5.1.2-1)

V m D V m C

at B,

V m B

P a  l 1 e

(V m )A

(V m )B

(15.5.1.2-2)

at A,

L



at a corner, e.g. between B and C, it is given by: (V m )B C

P­ ®a  e¯

2

½  l12 ¾ ¿

(15.5.1.2-3)

The second moment of area is given by: I1

I2

e 3 / 12

(15.5.1.2-4)

Figure 15.5-1 — Unreinforced vessels 320

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The bending stresses shall be determined from the following equations: at C, (V b ) C

r

e ª 2M A  P(2a ˜ L  2a ˜ l1  L2 )º »¼ 4 I1 «¬

(15.5.1.2-5)

e ª 2M A  4 I1 «¬

(15.5.1.2-6)

at D, (V b )D

r

P §¨ 2a ˜ L  2a ˜ l1  L2  l12 ·¸º» ¹¼ ©

at A, (V b ) A

at B,

V b B

r

r

M Ae 2I 1

(15.5.1.2-7)

>

e 2M A  P L2 4 I1

at the corner,

V b BC

r

>

@

(15.5.1.2-8)

^

e 2M A  P 2a( L cos T  l1(1  sinT ))  L2 4I1

For these equations the following shall apply: a)

the maximum value of V b BC is given where T

`@

(15.5.1.2-9)

arctan l1 / L

(15.5.1.2-10)

and b)

the bending moment MA per unit length, is given by: MA

P ˜ ( K 3 )

where



l 1 6ij 2 ˜ Į 3  3ʌij 2  6ij 2  Į 3  3Į 3  6ij  2  1.5ʌĮ 3 ˜ ij  6ij ˜ Į 3 32Į 3  ʌij  2 2

K3

D1 = H1 / h1

D3 I

(15.5.1.2-11)

L / l1 a / l1

3

2

2



(15.5.1.2-12) (15.5.1.2-13) (15.5.1.2-14) (15.5.1.2-15)

At a location, the maximum stress shall be obtained as stated in 15.4 by summing the membrane and bending stresses.

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15.5.1.3

Perforated plates

The ligament efficiency of a perforated plate is given by :

P

ªpd 1 min « ; p cos E ¬«

§ p s  d ·º ¸» ¨ ¸ ¨ p s ¹ ¼» ©

(15.5.1.3-1)

where E is the angle defined in Figure 15.5-2.

When P is less than the joint coefficient z, the membrane and bending stresses calculated on the gross area of the section shall be divided by P to obtain the stresses based on the net area of the section.

Figure 15.5-2 — Unreinforced vessels with perforated sides

If the ligament efficiency P is at least 0.2, the membrane stresses shall be determined from the following equations:

V m y V m x

322

V m B P

(15.5.1.3-2)

P

(15.5.1.3-3)

V m C

BS EN 13445-3:2009 EN 13445-3:2009 (E) Issue 1 (2009-07)

> 2M 4I P

@

The bending stresses shall be determined from the following equations:

V b y

r

(V b ) x

r

e

1

>

A

 PL y 2

^

e 2M A  P 2a ˜ L  2a ˜ l1  L2  (l1  l x )2 4 I1P

`@

(15.5.1.3-4)

(15.5.1.3-5)

The allowable values for membrane and bending stresses are given in 15.5.3. The sum of stresses shall fulfil that requirement at all points with no hole circle closer to the other vessel wall than the distance a or 0,5d, whichever is the largest. For holes closer to the wall and for P  0,2, a stress analyses shall be performed. 15.5.2

Unreinforced vessels with a central partition plate

Figure 15.5-3 — Unreinforced vessel with a central partition plate For unreinforced vessels with a central partition plate, as shown in Figure 15.5-3 the membrane stresses shall be determined from the following equations. at C,

V m c at D,



ª2  k 5  D 2 P ˜ h ­° ®4  « 4e1 ° 1  2k «¬ ¯

º»½°¾ »¼ °¿

(15.5.2-1)

V m D V m c

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at B,

V m b

p˜H 2e2

(15.5.2-2)

at A, In partition plate

V m P

k

D





P ˜ h ª2  k 5 D 2 º « » 2e3 ¬ 1  2k ¼

I2 ˜D I1

(15.5.2-3)

(15.5.2-4)

H h

(15.5.2-5)

The bending stresses shall be determined from the following equations. at C,

V b C

P ˜ h 2e1 § 1  2D 2 ˜ k · ¸ ¨ 24 ˜ I1 ¨© 1  2k ¸¹

(15.5.2-6)

§ 1  2D 2 ˜ k · · P ˜ e1 § 2 ¨¨ 3H  2h 2 ¨¨ ¸¸ ¸¸ 48 ˜ I1 © © 1  2k ¹ ¹

(15.5.2-7)

§ 1  2D 2 ˜ k · ¸¸ ¨¨ © 1  2k ¹

(15.5.2-8)

at D,

V b D at B,

V b B

P ˜ h 2 ˜ e2 24 I 2

at A,

V b A





Ph 2 e2 ª1  k 3  D 2 º « » 24 I 2 ¬ 1  2k ¼

(15.5.2-9)

The allowable design stresses for membrane and bending shall be as given in 15.5.3. 15.5.3

Allowable stresses for unreinforced vessel

The membrane stresses shall be limited as follows:

Vm d f ˜ z

(15.5.3-1)

The sum of membrane stresses and bending stresses shall conform to:

V m  V b d 1,5 ˜f ˜ z

324

(15.5.3-2)

BS EN 13445-3:2009 EN 13445-3:2009 (E) Issue 1 (2009-07)

where z = 1 for sides without longitudinal or circumferential welds.

15.6

Reinforced vessels

15.6.1

General

Reinforced vessels, as shown in Figure 15.6-1, have a continuous frame which may either follow the contour of the vessel or form a closed rectangle. The reinforcing members shall be fitted to the outside of the vessel in a plane perpendicular to the long axis of the vessel. This calculation method is applicable if the two opposite sides of the vessel have the same second moment of area. Where they do not, a special analysis shall be performed. b

b

e

b

be

be

be

be

be

be

bR

The effective length be is limited by 10e Figure 15.6-1 — Reinforcing member and associated effective plate width 15.6.2

Shear strength of reinforced section

15.6.2.1

General

The reinforcing members and the attached plate elements of the vessel shall be considered to act as a composite unit when calculating the effective second moment of area of the reinforcing members. In order to ensure this structural behaviour, the shear stress in the reinforcement web and in the weld between reinforcing elements and vessel shall be limited as shown below. 15.6.2.2

Continuously welded reinforcements

For continuously welded reinforcements, the shear stress in the weld joining web to vessel shall be calculated by the following equation.

W

Q ˜ A' ˜ j I ˜ bcw

(15.6.2.2-1)

where Q

is the shear load at the section near the corner

A’

is the area of that part of the composite section above or below the calculation point

j

is the distance from the neutral axis of the centroid of A’ 325

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I

is the second moment of area of the composite cross section

bcw

is the net width of the section measured (thickness of the web or in partial penetration welds, sum of weld throat thicknesses a, see definition 3.23, if less).

REINFORCEMENT SECTIONS

Figure 15.6-2

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Table 15.6-1 WEBS

(Flat elements perpendicular to the bending axis)

Sketch

Type of reinforced rection

i Rolled or cold formed i Welded

(a.1, 2, 3) (b.1, 2, 3)

i Rolled or cold formed

(c.1, 2)

i Welded

Width evaluation dw = hr - 1,5 tf dw = hr - tf

dw = hr -1,5 tf dw = hr

FLANGES

(Flat elements parallel to the bending axis)

Sketch

Type of section

i Rolled or cold formed

(a.1)

Width evaluation bf = b - 3 tf

(a.2, 3)

i Welded

VESSEL WALL

(plate space between two reinforcing elements)

Sketch

Type of section

transversal section of reinforced vessel

dw/tw