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16587 Pressurised Systems

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11.4 STRESSES IN HORIZONTAL CYLINDRICAL VESSELS SUPPORTED ON TWINSADDLES - The PD 5500 Approach

The design of horizontal vessels supported on twin saddles (see Figure 11.12) has beendealt with by several authors over the years. However, the approach given in PD 5500 isessentially the work of one man - L P Zick. He used a modified beam and ring analysis sothat the mathematical model for the vessel predicted values which agreed with theexperimental results he had available. More recent experimental work has indicated thatZick’s treatment for the vessel full of fluid predict stresses which are in reasonableagreement with the experimental values only when a flexible saddle is employed. Whenthe 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 circumferentialdirection. In some cases they have a magnitude which is double that which occurs when aflexible saddle is employed.

When vessels of this type are supported at more than two cross-sections the supportreactions are significantly affected by small variations in the level of the supports, thestraightness and local roundness of the vessel and the relative stiffness of different parts ofthe vessel. Support at two cross-sections is thus to be preferred even if this requiresstiffening of the support region of the vessel.

In this approach one of the supports should be designed at the base, to provide freehorizontal movement, thereby avoiding restraint due to thermal expansion. For very longvessels multi-saddle supports may be required. An approximate approach to this case is toderive the support forces and longitudinal moments assuming the vessels behave like acontinuous beam. These values can then be used in the manner outlined below for thetwin support case.

Figure 11.12 Horizontal vessel with twin saddle supports, hemispherical ends,3.044 m diameter, 24 m tan to tan length, 78 mm thick, design pressure 99.3 bars,design temperature -27oC to +38oC


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11.4.1 Longitudinal Bending Moments

To determine expressions for the longitudinal bending moments the approach adopted is toconsider 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 endsand by the 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).


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

11.4.2 Longitudinal Stresses

Longitudinal Stress at Mid-Span

The stress due to the overall mid-span bending moment M3 is calculated by assuming thatthe full vessel section is available and that the cross section remains circular, i.e. secondarybending in the circumferential direction is small. This assumption will be adequate formost cases. However, for very thin vessels it is found that the cross-section does notremain circular; especially so during filling with liquid. Furthermore, the axial membranecompressive forces in the partially full condition are found to be larger than those when thevessel is full. These vessels, therefore, have a tendency to buckle inwards at the locationof the liquid height during filling. Despite this, experience has shown that for steel andaluminium alloy vessels with diameter to wall thickness ratio up to 1250/1, the methodspresented herein, based upon the full condition and assuming the cross section to remaincircular, 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 subjectto an axial stress due to the hydraulic pressure on the ends of the vessel. This correspondsto p r tm 2 where pm is the internal pressure at the equator (horizontal centre line of thevessel). The total stresses are thus(1) at the highest point of the cross-section the stress f1 is given by

f p rt

M rI

p rt

Mr t

m m1

3 322 2

==== −−−− ==== −−−−ππππ (11.8)

(2) at the lowest point of the cross-section the stress f2 is given by

f p rt

M rI

p rt

Mr t

m m2

3 322 2

==== ++++ ==== ++++ππππ (11.9)

Longitudinal Stress at the Saddles

The stress due to the overall saddle bending moment M4 is calculated on the basis thatonly part of the cross-section of the shell at the saddle profile is effective. The effectivepart is shown in Figure 11.15. The position of the neutral axis, NA, and the secondmoment of area INA about the axis can be found. To this stress, which arises from the


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longitudinal bending moment M4 must be added the longitudinal stress due to the endpressure, as before.

Stress at the highest point, of the effective cross-section, f3 is given by

f p rt

MI

y p rt

MK r t

m

NAT

m3

4 4

122 2

==== ++++ ==== −−−−ππππ (11.10)

The values of K1 are given in PD 5500 in Table G.3.3.2.3, reproduced in these notes asTable 11.1.It should be noted that when the full section is available K1 = 1

Stress at the lowest point of the cross-section, f4 is given by

f p rt

MI

y p rt

MK r t

m

NAC

m4

4 4

222 2

==== ++++ ==== ++++ππππ (11.11)

The value of K2 is also shown in Table 11.1

Allowable Stresses


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The calculated stresses f1 to f4 which are essentially membrane stresses in the axialdirection, that is they are σσσσ z , together with the circumferential membrane stressσσσσ θθθθ ==== p r t have to satisfy two requirements:

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

σσσσ σσσσ σσσσ σσσσθθθθ θθθθ−−−− ++++ ++++z z p p; . ; .0 5 0 5this shall not exceed the design stress, f.(2) to avoid buckling of the vessel the longitudinal compressive membrane stress, σσσσ z ,

shall not exceed ∆∆∆∆ s f , where ∆∆∆∆ is obtained from the section in the Standard (PD5500) dealing with external pressure loading.

It should be noted that if the longitudinal stresses in the saddle region exceed the allowablestress then rings may be placed in the saddle centre profile - Table 11.1 show theinfluence of such on the values of K1 and K2

11.4.3 Shearing Stresses

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

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

W L AL b1 2

4 3−−−−

++++

The saddle region of the vessel may be either unstiffened (i.e. left as a plain cylinder) orstiffened with rings. The values of the shearing stresses for both these cases have to beconsidered.

Shell Stiffened with Rings in the Plane of the Saddle or Stiffened by being Located near theEnds i.e. A r≤≤≤≤ 2

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

q V rIoo

==== −−−−2

sinφφφφ

where, V is the shear force and Ioo is the second moment of area of the full cross-section ofthe cylinder. That is,


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(((( ))))q K Wr t

L AL bmax ====

−−−−++++

3 1 2

4 3 (11.12)

where K3 = 1 0 318ππππ ==== ⋅⋅⋅⋅

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 stressacts on a reduced cross-section. As in the case of the longitudinal stresses, the upperportion of the shell is considered as being ineffective in carrying shear. The shears in theeffective portion, that is close to the saddle, will therefore, be increased. The form of theshear stress remains the same - that is equation 11.12 - but the value of the factor K3 isincreased. The values are shown in PD 5500 in Table G.3.3.2.4 for the various saddleangles. This is reproduced in these notes as Table 11.2


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Table 11.2 Design factors K3 and K4 and allowable shearing stresses

The Standard also provides details by which the shear stress in the dished end and also inthe shell may be obtained, when the saddle is located near the head.Allowable Shear Stresses

These values are given above in Table 11.2 (Table G.3.3.2.4 of PD 5500). In this the


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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 shear stress. The latter value has its origins in theavoidance 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 thesaddle centre profile. The first is at the lowest point of the cross-section, known as thenadir. The second, by far the most important, is at the saddle horn (i.e. the highest point ofthe saddle support).

Stress at the Nadir

The circumferential stress, given in the Standard (PD 5500), at this point is obtained bysumming the shear stresses in the saddle region. The width of the shell that resists thisforce was considered by Zick to be the saddle width plus 5t on either side, i.e. (((( ))))b t1 10++++ .Thus, the circumferential stress at the nadir is given as,

(((( ))))f K Wt b t5

5 1

1 10==== −−−−

++++ (11.13)

The values of K5 are given in Table G.3.3.2.5.2 of PD 5500, provided here in Table 11.3.

Table 11.3 Values of Constants.


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It should be noted that when the saddle is welded to the vessel the values of K5 given inthe Table 11.3 (G.3.3.2.5.2 of PD 5500) should be taken as one-tenth of this value. Whenloose saddles are employed the full values from the Table should be used.

Allowable Value for Circumferential Stress at Nadir

When the saddle is welded to the vessel the value of f5 should not exceed the designstress f.

When the saddle is not welded to the vessel the value of f5 should not exceed εεεε E 3 ,where εεεε is the circumferential buckling strain. The value of this is obtained from theequation given in Figure 3.6(2) (of PD 5500), which in turn uses the n value from Figure3.6(1). In this derivation the value of L R2 always equals 0.2, both in Figure 3.6(1) andin equation in Figure 3.6(2). Further explanation of this 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 atthe abutments (that is the horn) and loaded with shear stress (((( ))))q V r==== ππππ φφφφsin . This is aredundant structure which can readily be solved. The resulting distribution of the bendingmoment Mφφφφ is shown in Figure 11.16.

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


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In all cases the maximum value of Mφφφφ occurs at the horn B which is identified in PD 5500at an angle ββββ from the zenith, i.e. M K W rββββ ==== 6 1 . Values of K6 are given in Table 11.4

below, which is taken from Table G.3.3.2.5.1 (PD 5500). Those for A r ≥≥≥≥ ⋅⋅⋅⋅1 0 arederived from the above analysis (the ring loaded with a shear stress). When A r <<<< ⋅⋅⋅⋅0 5the above factors are divided by 4. The variation in the range 0 5 1⋅⋅⋅⋅ <<<< <<<<A r is assumedlinear.

Bending stress at the horn.

Having obtained the bending moment at points round the ‘ring’, i.e. the shell in the regionof the saddle, we now have to determine the stress. This is the same problem we hadearlier for the axial and the shear stresses. In these earlier cases we obtained the bendingmoment and the shear force relatively easily, but had to use a measure of scientific‘cunning’ to find the stresses corresponding to these. We have to do the same here.

Zick made the assumption that a certain width (i.e. axial length) of shell was effective inresisting the moment Mββββ - see Figure 11.17.

Figure 11.17 Diagrammatic representation of width of vessel resisting Mββββ


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He found that if the effective width was four times the shell radius or equal to one half thelength of the vessel, whichever is the smaller, then the resulting stresses agreedconservatively with the results from strain gauge surveys.That is if L r≥≥≥≥ 8 , the bending component of the circumferential stress is given by:

(((( )))) (((( ))))(((( ))))M r

t

tββββ 4

2

123

[Note this corresponds to the ‘Engineers Bending’ relationship M y I ].This expression simplifies to

32

32

32

26 1

26 1

2M r t K W r r t K W tββββ ==== ==== (11.14)

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

ML

tt

M L tL t

K W rββββββββ2

212

12 123

22 6 1

==== ==== (11.15)

In the above equation the effective width is taken as L 2

Direct stress at the horn

The direct component of the circumferential stress at the horns can be obtained in a similarsemi-empirical manner by first of all obtaining the direct thrust at the horn, and then byallowing this to be carried over an effective width of shell. However, in this case Zickproposed that the direct load at the horns be W1 4 distributed over the portion of the shellstiffened by the contact of the saddle, i.e. (((( ))))b t1 10++++ . Using this approach the directcomponent of the circumferential stress is assumed to be:

(((( ))))W

t b t1

14 10++++ (11.16)

Total circumferential stress at the horns L r ≥≥≥≥ 8

Combining equations (11.14), (11.15) and (11.16) where appropriate, gives the maximumstress at the horn on the outer surface:

(((( ))))For L r f Wt b t

K Wt

≥≥≥≥ ==== −−−−++++

−−−−84 10

326

1

16

12; (11.17)

(((( ))))For L r f Wt b t L t

K W r<<<< ==== −−−−++++

−−−−84 10

126

1

1 2 6 1; (11.18)


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The stresses may be reduced if necessary by extending the saddle plate as shown in Figure11.18 (a) to that shown in Figure 11.18 (b). [These are Fig G.3(14) of PD 5500.].It is recommended that the thickness of the saddle plate in the case of steel vessels shouldbe equal to the thickness of the vessel shell plate. If the width of this plate is not less thanb t1 10++++ and subtends an angle not less than ( )θθθθ ++++ 12o , the reduced stresses in the shellat the edge of the saddle can be obtained by substituting the combined thickness of vesseland saddle plate into the relevant equation, using a saddle angle of θθθθ . A second checkmust also be carried out to determine the stress in the vessel at the edge of the top plate. Inthis case a saddle angle of ( )θθθθ ++++ 12o may be used to derive the K6 value; thereafter theactual vessel thickness must be used in the equations (11.17) and (11.18).


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

Allowable circumferential stress in horn region

The numerical value of f6 found from the above calculations should not exceed 1.25 f,where f is the design stress of the vessel material.

11.4.5 Stiffening Rings in the Region of the Saddle

If the circumferential stress, derived asabove, exceeds the allowable the


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designer has a number of options. Oneof these is to weld ring stiffeners to theshell. These may be placed in the planeof the saddle or adjacent to the saddle oneither the inside or the outside of thevessel as shown in Figure 11.19 (a), (b)and (c). These figures are Fig. G.3 (15)of PD 5500.

The analysis presented previously wherethe vessel is treated as an arch in thesaddle region loaded with a shear stressis used to analyse this case. Themaximum bending moment occurs at thehorn, i.e. M K W rββββ ==== 6 1 . In this casethe moment is assumed to be carried bythe stiffener and part of the plate equalto 5 t on either side - shown shaded inFigure 11.19. In some ways the case ofthe ring stiffener is easier to analyse, inthat there is less dubiety as to way thebending moment is carried.

The direct force is analysed more exactlythan is the case for the unstiffenedvessel, since again there is lessuncertainty concerning the way in whichthe force is carried.

The Standard (PD 5500) presents thedesign approach in detail; this can befound on page G/63, section G.3.3.2.5.2.The numerical values of the maximumcircumferential stresses should notexceed 1 25⋅⋅⋅⋅ f .

Figure 11.19 Typical ring stiffeners

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


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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 varioussupporting angles can be found. The resulting stresses are provided in the Standard (PD5500) in terms of the least section modulus and effective area of the ring. The details aregiven 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 φφφφ1 60==== o

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

When the calculated value of the maximum circumferential stress f6 is greater than theallowable stress a number of options are available to the designer. These are set downhere.


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(1) Increase the saddle angle.In the Standard (PD 5500) a range of preferred saddle angles are given - 120 to150o. It is also possible to increase the effective saddle angle by increasing theangle of the top plate by 12o, that is a saddle angle of 162o. This is an effectivemethod, since K6 is influence considerably by the angle of support.

(2) Increase the saddle width.Increasing the width only effects the first term in the equations for f6 and is not too satisfactory .

(3) Increase the shell thickness.This is effective but rather expensive, unless the increased thickness is confinedto the region of the saddle. Details of using a ‘thickened strake’ in the saddle region are now given in PD 5500.

(4) Move the saddles nearer to the ends.The value of K6 is influenced by the A value, so this is a useful approach - it 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) Welding stiffening rings in the saddle region.This method was discussed earlier. It is effective, but costly and could lead to afatigue 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 costand the labour cost involved. The designer is at the forefront of such decision making.

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