Horizontal Vessel 2 Saddle design as per BS5500

16426/16587 - Pressurised Systems

11.4 STRESSES IN HORIZONTAL HORIZONTAL CYLINDRICAL CYLINDRICAL VESSELS VESSELS SUPPORTED SUPPORTED ON TWIN SADDLES - The BS 5500 Approach

The design of horizontal vessels supported on twin saddles (see Figure 11.12) has been dealt with by several several authors over over the years. years. However, the approach approach given in BS 5500 is essentially essential ly the work of one man - L P Zick. He used a modified beam and ring analysis so that the mathematical model for the vessel predicted values which agreed with the experimental results he had available. More recent experimental experimental work has indicated that Zick’s treatment for the vessel full of fluid predict stresses which are in reasonable agreement with the experimental values only when a flexible saddle is employed. employed. When the saddle is rigid the treatment under-estimates the maximum stresses in the vessel. These stresses occur at the horn (the highest point on the support) in the circumferential direction. In some cases cases they have have a magnitude which is double that which occurs occurs when a flexible saddle is employed. employed.

Figure 11.12 Horizontal vessel with twin saddle supports, supports, hemispherical ends, 3.044 3.044 m diameter, 24 m tan to tan length, 78 mm thick, design pressure 99.3 bars, design o o temperature -27 C to +38 C When vessels of this type are supported at more than two cross-sections the support reactions are significantly affected by small variations in the level of the supports, the straightness and local roundness of the vessel and the relative stiffness of different parts of the vessel. Support at two cross-sections is thus to be preferred preferred even if this requires requires stiffening of the support region of the vessel. In this approach one of the supports should be designed at the base, to provide free horizontal movement, movement, thereby avoiding avoiding restraint due to thermal expansion. expansion. For very long

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vessels multi-saddle multi-saddle supports may be required. An approximate approach to this case is to derive the support forces and longitudinal moments assuming the vessels behave like a continuous beam. beam. These values values can then be used in the manner outlined outlined below for the twin support case. 11.4.1 Longitudinal Bending Moments

To determine expressions for the longitudinal bending moments the approach adopted is to consider that the vessel behaves like a beam supported at the saddles (see Figure 11.13). Consideration is given to the additional moment caused by the weight of the dished ends and by the hydraulic hydraulic pressure on the ends. The result is shown in Figure 11.14(a). The distribution of the bending moments and shear forces are shown in Figure 11.14(b) and (c).

Figure 11.13 11.13 Cylindrical vessel on saddles. Bending Moment at the Support Sup port M 4

From Figure 11.14 (c) the bending moment at the support is given by 1

W1 A 1

M4

r2

A L 1

b2

2AL

(11.6)

4 b 3 L

Bending Moment at Mid-span Mid-spa n M 3

Again from Figure 11.14 (c) the bending moment at the mid-span position is given by

M 3

W1 L 1 4

2 r2 1

b2

4 b 3 L

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L2

4 A

L

(11.7)


Figure 11.14 Cylindrical vessel vessel acting as a beam beam over support - BS 5500

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11.4.2 Longitudinal Stresses Longitudinal Stress Stre ss at Mid-Span

The stress due to the overall mid-span bending moment M 3 is calculated by assuming that the full vessel section is available and that the cross section remains circular, i.e. secondary bending in the circumferential direction direction is small. This assumption will be adequate adequate for most cases. However, for very thin vessels it is found found that the cross-section does does not remain circular; especially especially so during filling with liquid. Furthermore, the axial membrane membrane compressive forces in the partially full condition are found to be larger than those when the vessel is full. These vessels, vessels, therefore, have have a tendency tendency to buckle inwards at at the location of the liquid height during during filling. Despite this, experience experience has shown shown that for steel and aluminium alloy vessels with diameter to wall thickness ratio up to 1250/1, the methods presented herein, based upon the full condition and assuming the cross section to remain circular, produce designs which are satisfactory for the partially full condition. In addition to the stress due to the bending moments the vessel cross section is also subject to an axial stress due to the hydraulic hydraulic pressure on the ends of the vessel. This corresponds to p m r 2 t where p m is the internal pressure at the equator (horizontal centre line of the vessel). The total total stresses are thus (1) at the highest point of the cross-section the stress f 1 is given by f 1 (2)

p m r

M 3 r

pm r

2 t

I

2 t

M 3 r 2 t

(11.8)

at the lowest point of the cross-section the stress f 2 is given by f 2

p m r

M 3 r

pm r

2 t

I

2 t

M 3 r 2 t

(11.9)

Longitudinal Stress Stre ss at the Saddles

The stress due to the overall saddle bending moment M 4 is calculated on the basis that only part of the cross-section of the shell at the saddle saddle profile is effective. The effective part is shown in Figure 11.15. 11.15. The position of of the neutral axis, NA, and and the second second moment of area I area I NA about the axis can be found. To this stress, which arises from the longitudinal bending moment M moment M 4 must be added the longitudinal stress due to the end pressure, as before. Stress at the highest point, of the effective cross-section, f cross-section, f 3 is given by p m r M 4 p m r M 4 f 3 yT 2 t 2 t I NA K1 r 2 t

(11.10)

The values of K of K 1 are given in BS 5500 in Table G.3.3.2.3, reproduced in these notes as Table 11.1.

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It should be noted noted that when the full section section is available K available K 1 = 1

Figure 11.15 Effective vessel vessel at the saddles saddles Stress at the lowest point of the cross-section, f 4 is given by

f 4

p m r

M 4

2 t

I NA

p m r

yC

2 t

M 4 K 2

2

r t

(11.11)

The value of K K 2 is also shown in Table 11.1

Table 11.1 Design factors K factors K 1 and K and K 2 Allowable Stresses

The calculated stresses f 1 to f 4 which are essentially membrane stresses in the axial direction, that is they are z , together with the circumferential membrane stress θ

p r t have to satisfy two requirements:

(1) the general general primary membrane stress intensity, intensity, acting at the various points and for the different fill conditions, shall be taken as the greater of θ

z ;

z

0 .5 p ;

this shall not exceed the design stress, f stress, f .

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θ

0 .5 p


(2) to avoid bucklin buckling g of the vessel vessel the longitudinal longitudinal compres compressive sive membrane membrane stress, stress,

z ,

shall not exceed s f , where is obtained from the section in the Standard (BS 5500) dealing with external pressure loading. It should be noted that if the longitudinal stresses in the saddle region exceed the allowable stress then rings may be placed in the saddle centre profile - Table 11.1 show the influence of such on the values of K of K 1 and K and K 2 11.4.3 Shearing Stresses.

As the bending moment varies along the length of the vessel, so also does the longitudinal stress. The effect effect of this is to introduce longitudinal shear stress together together with complementary shear stress stress which occurs in the plane of the cross-section. cross-section. The distribution of the shear shear force is given earlier in Figure 11.14 (b). The inner saddle shear shear force is invariably the greater since L 4 A 4 3 b . In which case the value is given by W1 L

2A 4b 3

L

The saddle region of the vessel may be either unstiffened (i.e. left as a plain cylinder) or stiffened with rings. The values of of the shearing stresses for both these these cases have have to be considered. Shell Stiffened with Rings in the Plane of the Saddle or Stiffened by being Located near the Ends i.e. A r 2

In this case the full vessel cross-section cross-section is available to carry the shear stress q, thus

q

V r2 I oo

sin

where, V is and I oo is the second moment of area of the t he full cross-section of V is the shear force and I the cylinder. That is,

q max where K 3 = 1

K 3 W 1 r t

L L

2A 4b 3

(11.12)

0 318

Shell in the Saddle Region A

r 2 and Unstiffened by Rings

When the shell is free to deform above the saddles, it is considered that the shear stress acts on a reduced reduced cross-section. cross-section. As in the case of the longitudinal longitudinal stresses, stresses, the upper portion of the shell is considered considered as being ineffective in carrying carrying shear. The shears in the

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effective portion, portion, that is close to the saddle, will therefore, be increased. The form of the shear stress remains the same - that is equation 11.12 - but the value of the factor K 3 is increased. The values are shown shown in BS 5500 in Table Table G.3.3.2.4 for the various various saddle angles. This is reproduced in these these notes as Table 11.2 11.2

Table 11.2 Design factors K factors K 3 and K and K 4 and allowable shearing stresses The Standard also provides details by which the shear stress in the dished end and also in the shell may be obtained, when the saddle is located near the head.

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Allowable Shear Stresses Stre sses

These values are given above in Table 11.2 (Table G.3.3.2.4 of BS 5500). In this the smaller of 0 8 f , which is derived from strain gauge tests by Zick, and 0 06 E t r , should be taken as the allowable allowable shear shear stress. The latter value has has its origins in the the avoidance of shear buckling in the region of the support in vessels with a high r t ratio (up to 625 : 1). 11.4.4 Circumferential Stresses for a Shell not Stiffened by Rings.

Important values of circumferential stress occur at two locations in the vessel, both in the saddle centre profile. The first is at the lowest lowest point of the cross-section, cross-section, known as the nadir. The second, second, by far the most most important, is at the saddle horn (i.e. the highest highest point of the saddle support). Stress at the Nadir

The circumferential stress, given in the Standard (BS 5500), at this point is obtained by summing the shear shear stresses in the saddle region. The width of the shell shell that resists this force was considered by Zick to be the saddle width plus 5t plus 5t on either side, i.e. b1 10 t . Thus, the circumferential stress at the nadir is given as, f 5

K5 W 1 t b1

10 t

(11.13)

The values of K G.3.3.2.5.2 of BS 5500, provided here in Table Table 11.3. K 5 are given in Table G.3.3.2.5.2

Table 11.3 11.3 Values of of Constants. Constants.

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It should be noted that when the saddle is welded to the vessel the values of K of K 5 given in the Table 11.3 (G.3.3.2.5.2 (G.3.3.2.5.2 of BS 5500) should should be taken as one-tenth one-tenth of this value. When loose saddles are employed the full values from the Table should be used. Allowable Value for Circumferential Circumfer ential Stress at Nadir

When the saddle is welded to the vessel the value of f of f 5 should not exceed the design stress f stress f . When the saddle is not welded to the vessel the value of f of f 5 should not exceed E 3 , where is the circumferential circumferential buckling buckling strain. The value of of this is obtained from the equation given in Figure 3.6(2) (of BS 5500), which in turn uses the n value from Figure 3.6(1). In this derivation the the value of L of L 2 R always equals 0.2, 0.2, both in Figure 3.6(1) and in equation in Figure 3.6(2). 3.6(2). Further explanation explanation of this method method is found in the book ‘Pressure Vessel Design - Concepts and Principles’ by Spence and Tooth. Stress at the Horn of the Saddle

The analysis of Zick assumes the shell in the region of the saddle to be an arch built in at the abutments (that is the horn) and loaded with shear stress q V r sin . This is a redundant structure which which can readily be solved. The resulting distribution distribution of the bending moment M is shown in Figure 11.16.

Figure 11.16 Distribution of circumferential moment moment resulting from the application application of shear stress round the arc and in the plane of the shell.

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In all cases the maximum value of M of M occurs at the horn B which is identified in BS 5500 K6 W1 r . Values of K at an angle from the zenith, i.e. M of K 6 are given in Table 11.4 below, which is taken from Table Table G.3.3.2.5.1 (BS 5500). 5500). Those for A r

1 0 are derived

from the above analysis analysis (the ring loaded loaded with a shear stress). stress). When A r

0 5 the above

factors are divided by 4. The variation variation in the range range 0 5

A r

1 is assumed linear.

Table 11.4 Design factor K 6 Bending stress at the th e horn.

Having obtained the bending moment at points round the ‘ring’, i.e. the shell in the region of the saddle, we now have have to determine the stress. This is the same problem problem we had earlier for the axial and the shear stresses. In these earlier cases we we obtained the bending moment and the shear force relatively easily, but had to use a measure of scientific ‘cunning’ to find the stresses stresses corresponding to these. We have to do the same here. here. Zick made the assumption that a certain width (i.e. axial length) of shell was effective in resisting the moment M - see Figure 11.17.

Figure 11.17 Diagrammatic representation representation of width of vessel resisting resisting M He found that if the effective width was four times the shell radius or equal to one half the length of the vessel, whichever is the smaller, then the resulting stresses agreed conservatively conservatively with the results from strain gauge surveys. surveys.

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That is if L if L

8 r , the bending component of the circumferential stress is given by:

M β 4 r

t 2 t 3 12

[Note this corresponds to the ‘Engineers Bending’ relationship M y I ]. I ]. This expression simplifies to 3 M β r t 2 2

When L

3 2

3

K 6 W1 r r t 2

2

K 6 W1 t 2

(11.14)

8 r , the bending component of the circumferential stress is given by:

M β

t 2

12

12 M β L t 2

L 2 t 3 12

L t 2

K6 W1 r

(11.15)

In the above equation the effective width is taken as L 2 Direct stress at the horn ho rn

The direct component of the circumferential stress at the horns can be obtained in a similar semi-empirical manner by first of all obtaining the direct thrust at the horn, and then by allowing this to be carried over over an effective width of shell. However, in this case case Zick proposed that the direct load at the horns be W 1 4 distributed over the portion of the shell 10 t . Using this approach the direct

stiffened by the contact of the saddle, i.e. b1

component of the circumferential stress is assumed to be: W 1 4 t b1

(11.16)

10 t

Total circumferential stress at the horns L r

8

Combining equations (11.14), (11.15) and (11.16) where appropriate, gives the maximum stress at the horn on the outer surface: For L r

8 ;

f 6

For L r

8 ;

f 6

3

W 1 4 t b1

10 t

12

W 1 4 t b1

2

K 6

10 t

Lt

2

W 1 t 2

K6 W1 r

(11.17)

(11.18)

The stresses may be reduced if necessary by extending the saddle plate as shown in Figure 11.18 (a) to that shown in Figure 11.18 11.18 (b). [These are Fig G.3(14) G.3(14) of BS 5500.].

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It is recommended that the thickness of the saddle plate in the case of steel vessels should be equal to the thickness of the vessel shell shell plate. If the width of this plate is not less than 12 o ) , the reduced stresses in the shell b1 10 t and subtends an angle not less than ( at the edge of the saddle can be obtained by substituting the combined thickness of vessel and saddle plate into the relevant equation, using a saddle angle of θ . A second check must also be carried out to determine the stress stress in the vessel at the edge of the top plate. In 12 o ) may be used to derive the K 6 this case a saddle angle of ( 6 value; thereafter the actual vessel thickness must be used in the equations (11.17) and (11.18).

Figure 11.18 (a) Simple saddle support, support, (b) saddle support with extended extended plate. The appropriate pages of the Standard (BS 5500) give further details of the above.

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Allowable circumferential circumferen tial stress in horn region reg ion

The numerical value of f of f 6 6 found from the above calculations should not exceed 1.25 f , where f where f is is the design stress of the vessel material. 11.4.5 Stiffening Rings in the Region of the Saddle

If the circumferential stress, derived as above, exceeds the allowable the designer has a number of options. options. One of these is to weld ring stiffeners to the shell. These may be placed in the plane of the saddle or adjacent to the saddle on either the inside or the outside of the vessel as shown in Figure 11.19 (a), (b) and (c). These figures are are Fig. G.3 (15) of BS 5500. The analysis presented previously where the vessel is treated as an arch in the saddle region loaded with a shear stress is used to analyse this case. The maximum bending moment occurs at the K6 W1 r . In this case horn, i.e. M the moment is assumed to be carried carried by the stiffener and part of the plate equal to 5 t on either side - shown shaded in Figure 11.19. In some ways the case of the ring stiffener is easier to analyse, in that there is less dubiety as to way the bending moment is carried. The direct force is analysed more exactly than is the case for the unstiffened vessel, since again there is less uncertainty concerning the way in which the force is carried. The Standard (BS 5500) presents the design approach in detail; this can be found on page G/63, section G.3.3.2.5.2. The numerical values of the maximum circumferential stresses should not exceed 1 25 f .

Figure 11.19 11.19 Typical ring stiffeners

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When rings are used on the outside of a vessel in the plane of the saddle, it is usual to use the rings as part of an integrated support system, as shown in Figure 11.20. Arrangements of of this type are are referred to as a ‘ring and leg’ support. support. The stresses in the vessel away from the saddle are given by the same relationships obtained obtained earlier. As with the unstiffened vessel a shear stress is applied to the vessel and ring combination, with the support at the intersection of the load and the centroid diameter of the ring.

Figure 11.20 11.20 Ring supported supported vessel vessel A typical result for the bending moment distribution for this case is shown in Figure 11.21. From this type of analysis the maximum values of the bending moment for various supporting angles angles can be found. The resulting stresses stresses are provided in the Standard Standard (BS 5500) in terms of the least least section modulus and effective area of of the ring. The details are given on page G/64. [Note the section modulus = (Second moment of area, I)/(distance to fibre, y)]

Figure 11.21 Variation of circumferential moment for the case of a support at

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1

60 o


11.4.6 Design Modifications to Reduce the Max. Circumferential Stress at at the Horn Horn

When the calculated value of the maximum circumferential stress f 6 6 is greater than the allowable stress a number of options are available to the designer. designer. These are set set down here. (1) (1)

Incr Increa ease se the the sad saddl dlee ang angle le.. In the Standard (BS 5500) a range of preferred saddle angles are given - 120 to o 150 . It is also possible to increase increase the effective saddle saddle angle by increasing increasing the o o angle of the top plate by 12 , that is a saddle angle of 162 . This is an an effective effective method, since K since K 6 6 is influence considerably by the angle of support.

(2) (2)

Incr Increa ease se the the sad saddl dlee wid width th.. Increasing Increasing the width only effects the first term in i n the equations for f for f 6 6 and is not too satisfactory .

(3) (3)

Incr Increa ease se the the she shell ll thi thick ckne ness ss.. This is effective but rather expensive, unless the increased thickness is confined to the region of the saddle. Details of using a ‘thickened ‘thickened strake’ in the saddle region are now given in BS 5500.

(4) (4)

Move Move the the sad saddle dless nea neare rerr to to the the ends. ends. The value of K the A value, so this is a useful approach - it K 6 6 is influenced by the A does not cost any more, although it is necessary to check the axial stresses in the mid-span position, since the distance between the saddles is now increased.

(5)

Weldin Welding g stiff stiffeni ening ng rings rings in the saddle saddle region region.. This method was discussed discussed earlier. It is effective, but costly and could lead to a fatigue problem in the region of the circumferential welding between the ring and the vessel.

The question has to be answered for each case and is often a balance between material cost and the labour cost involved. involved. The designer is at the forefront forefront of such decision making. making.

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Horizontal Vessel 2 Saddle design as per BS5500

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