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7/30/2019 WRC INFORMATION

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16426/16587 - Pressurised Systems

141

11 THE ANALYSIS OF LOCAL LOADS AND SUPPORTS

This section deals with the influence of local forces and moments on cylindrical and

spherical vessels. These actions generally arise in the vessel support regions, at

brackets used for ancillary equipment or at lifting lugs. When the support, or bracket,contact area is relatively small it is permissible to assume a simplified form for the

interface force distribution between the vessel and its support (or bracket). This

approach is used in BS 5500 Appendix G.1 to G.2.4 and in the Welding Research

Council Bulletin WRC 107, used widely when designing to the ASME Code.

During the life of a pressure vessel it is subjected to a wide variety of loading

conditions, all of which must be considered during design. In some instances the

internal pressure is not the dominant form of loading and special attention has to be

given to other load cases which combined together could cause premature failure of the

vessel.

The local loading at the supports and lifting brackets which are welded to the vessel are

such a case and although the resulting stresses are generally not excessive, checks must

be made to establish their value. This is particularly the case for the horizontal vessel

which is supported on two saddles where high stresses occur at the uppermost point of

the saddle (known as the horn), during the hydraulic test.

The approach given in BS 5500 and WRC 107 for the local loading of the patch is

based on elastic, small displacement linear analysis. The stresses for the various load

cases are, therefore, superimposed to provide a value for the maximum stress in the

vessel. The approach is essentially a design by analysis method, in which the valuesfor the stress intensity are limited to prescribed values given in the Standard (BS 5500,

Appendix A).

11.1 THE CYLINDRICAL VESSEL WITH LOCAL LOADS ON

RECTANGULAR ATTACHMENT

Local loads arise in the vessel support regions, at brackets used for ancillary equipment

or at lifting lugs.

In this case it is assumed that the attachments are rectangular, or square, with

boundaries coinciding with the parallel circle profile, associated with the co-ordinate

, and the axial generatorx - as shown in Figure 11.1

When a radial force or a bending moment is applied to the attachment the interface

forces between the attachment and the vessel are rather complicated. Their distribution

depends upon the relative rigidities of the vessel and the attachment. For example, if

the attachment is very rigid compared with the vessel, one would expect the interface

forces to be concentrated round the attachment. A further complication is the fact that

the attachment is only fixed to the vessel round its periphery.


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Figure 11.1 Rectangular attachment on cylindrical vessel

The BS 5500 and the WRC 197, do not attempt to handle these and other intractable

modelling problems but make a series of assumptions regarding the interface pressure

loading. These are as follows

(1) When a radial force is applied it is assumed that the interface reactive force is

uniformly distributed over the attachment area - see Figure 11.3.

(2) When a moment loading is applied, either in the circumferential or

longitudinal directions, the interface pressure distribution is assumed to betriangular. In BS 5500 this is considered to be equivalent to two patches of

uniformly distributed radial loading of length equal to one-third the patch

length. The forces on the two patches are equal and opposite and consistent

with the direction of the applied moment - see Figures 11.6 and 11.7. In WRC

107 a triangular loading is considered without the assumptions made in BS

5500. A large number of charts are thus required in the WRC 107 to provide

all the information.

11.1.1 Analytical Methods

Two analytical approaches have been developed to handle the above radial loading

problems. These are briefly outlined as follows.

(a) In the first, the initial step is to solve the case of a radial line load applied at the

zenith (the top of the cylinder) and acting along part of the generator, (i.e. in the x

direction), at the mid-length of the end supported cylinder - see figure 11.2.

This can be expressed in Fourier series form:

P Pm x

Lr m

m

cos

, , ..1 3

(11.1)

wherePmis the loading term.


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Figure 11.2 Line load on the generator

Using the shell equations, the displacements, rotations and stress resultants for this

case were obtained. These line load results were later used to obtain the stresses

and deflections in the region of rectangular areas of loading by direct integration.

The procedure was adopted by ICI and later incorporated into the British

Standards. In 1976 it became part of BS 5500 Appendix G. Although information

was available for obtaining the longitudinal and circumferential moment cases

precisely (i.e. using the triangular distribution), the procedure referred to earlier

was adopted where two equal and opposite uniformly distributed radial loads were

used to represent the moments.

The procedure adopted in BS 5500 has the advantage of an economy in the number

of charts required since only the uniformly distributed radial loading needs to be

considered. However, using this approach the question of the interaction between

the two equal and opposite radial loads has to be addressed. This invariably

complicates the computation when the analysis is carried out by hand

(b) The secondmethod, is a more general approach to the local loading of shells and

vessels. In this the load is expressed as a double Fourier series, that is, the series is

capable of representing a load with dimensions in both the andx directions:

P P nm x

Lr n m

mn

cos cos

, ,.., , ,.. 1 30 1 2

(11.2)

As in the single series, equation (11.1), the term Pn m is the loading term. Using

this technique the direct and moment loading can be represented as double Fourier

series expressions and introduced into the shell equations to provide values for the

stress resultants and displacements. In order to represent the patch load it is

necessary to have a large number of terms in both the axial and circumferential

directions. The ideal number depends on the size of the patch compared with the

dimensions of the cylinder. In general, 200 terms in each series will givesatisfactory results for most problems. This approach has been used to draw up the


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curves presented in WRC 107. They are widely used in the USA and by users of

the ASME Standards. In this course we will confine our attention to the BS 5500

approach and the use of the charts provided in that Standard.

11.1.2 Uniformly Distributed Radial Loading over the Attachments

The case of a uniformly distributed radial loading of total magnitude W acting over

rectangular areas 2 2C Cx is shown in Figure 11.3. The derived stress resultants are

non-dimensionalised and presented in chart form in both BS 5500 and in WRC 107. In

BS 5500 the form chosen is M W M W N t W N t Wx x, , , plotted as functions

of 64 2r t C rx for four values of C Cx and four values of 2C Lx . To

illustrate the form of plot, the values ofM W are shown in Figure 11.4. The curves

presented in BS 5500 are for the case when L r 8 . The rest of the curves areprovided in the pages from BS 5500, G/6, G/7, G/8 and G/9

Figure 11.3 Cylindrical vessel with radial load

Once the stress resultants have been found the stresses can be found in the usual way

from the following:

W

t

N t

W

M

W2

6

(11.3)

xx xW

t

N t

W

M

W2

6

N Nxand are positive for tensile membrane stress. M Mxand are positive when

they cause compression at the outer surface of the vessel.


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Figure 11.4 Cylindrical vessel with radial load: circumferential stress resultant permillimetre width.


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Position of the Maximum Stress

For the uniformly distributed radial load the maximum value of these stresses, equation

(11.3), will occur at the centre of the attachment. However, as mentioned earlier in the

case of the actual attachment, which could well be rigid compared to the vessel andwelded round the edge, the maximum stress is likely to occur at the edge of the

attachment. In the Standards it is assumed that the maximum stress value obtained

from the idealised distribution is the same as the actual case but occurs at the edge of

the loaded area.

Circular and Elliptical Attachments

When the attachment is of circular cross-section of radius ro the same charts may be

used, by assuming the force is distributed over an equivalent square area. In BS 5500

the half sides of the square is equal; to C C rx o0.85 . This value is less than theactual equivalent square area of the circular attachment of radius ro. The Standard,

therefore, builds in a factor of safety by assuming the total load is distributed over a

smaller area than is actually the case. If the loaded area is elliptical the values

C Cxand should be taken as 0.42 multiplied by the major & minor axis of the ellipse.

Off-Centre Loading

The data presented in Appendix G of BS 5500, refers to attachments at mid-length of

the cylinder. When the load is offset by a distance d from the centre, as shown in

Figure 11.5, the stress resultants are assumed to be equal to those in a vessel of length

Le loaded at its mid-length.Leis called the equivalent length and can be found from:

L Ld

Le

4 2

(11.4)

This equation encapsulates the concept that the bending moment under the offset load

of the actual end supported vessel of lengthL is the same as under the central load of a

vessel of equivalent lengthLe.

Deflections Due to Radial Loads

The deflections due to a local load are required in order to

(1) find the displacement of a vessel due to a unit pipe thrust; and

(2) find the rotation of a branch due to a unit moment.

These flexibilities can be used to provide data to analyse the pipework in an overall

plant layout. The Standard (BS 5500) provides information by way of a nomogram

involving the non-dimensional parameters r t E r W L r, and for a square patch.

When the patch is rectangular, BS 5500 provides a method for the analysis of anequivalent square patch. It should be noted that the nomogram only provides a value


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for the deflection which occurs at the centre of the patch. It does not provide deflection

values away from the loaded area. These nomograms are given in BS 5500 on pages

G/20 and G/21.

Figure 11.5 Cylindrical vessel with radial load out of centre.

11.1.3 Stresses Away from the Edge of the Loaded Area.

A series of curves are provided in BS 5500 to obtain information on the decay or die-out of the stress resultants away from the immediate loaded area. This information is

required for the moment loadings which, in BS 5500, are considered as two equal and

opposite radial loads and also when radial loads, in their own right, are applied close

together. The Standard, indicates that the effect of one load at the position of another

can be disregarded when the distance between the centres is greater than K C1 for

loads separated circumferentially or, K Cx2 for loads separated axially.

K K1 2and values are given in Table G.2.2.2 of BS 5500. This Table only gives

approximate values. If a more accurate interaction assessment is required, or when the

distance between the loads is less than the above, BS 5500 details the method to be

used. This is briefly mentioned below.

It should be appreciated that the curves presented in BS 5500, to derive the extent of the

die-out, are only for a radial line loadalong a generator of the vessel. A procedure is

however, provided in BS 5500, in which these line load curves can be used to estimate

the stress resultants for a patch load at points which lie on the load centre profile (i.e.

round the vessel in the circumferential direction ) and also lie on the load centre

generator (i.e. along x). The appropriate curves are given from BS 5500 along with

these notes, and the details of the method are set out in the Standard. Further

explanations of the method and independent validation of the approach can be found in

the book by Spence and Tooth [Pressure Vessel Design - Concepts and Principles].


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11.1.4 The Application of Circumferential Moments - BS 5500 Approach

A circumferential moment applied to a rectangular area of circumferential length C

and axial length 2Cx (see Figure 11.6) is resolved into two equal and opposite forces :

W M C1 5.

acting on rectangles of sides 2 2C Cx where C C 6 which are separated by a

distance of 2 3C between centres.

Figure 11.6 Analysis of a circumferential moment - BS 5500

The maximum stresses due to the moment occur at the outer edges of the actual loaded

area. The resultants are thus a combination of the stress resultants from both loaded

areas shown in Figure 11.6:-

M M M1 2

M M Mx x x1 2

N N N1 2

N N Nx x x1 2 (11.5)

The quantities with the subscript 1 are the stress resultants for an inward facing load W

distributed over a loaded area 1 which has an area of 2 2C Cx , see the right-hand

side of Figure 11.6. Quantities with subscript 2, refer to loaded area 2 - see left-hand

side of Figure 11.6. They correspond to the stress resultants due to an outward facing

load W at a surface distance 5C , that is an angle equal to 1 5C r , from the

centre of the second loaded area to the outer edge of the first.

The detailed steps for the above are given in BS 5500 (see hand out copies of theStandard). A worksheet, or working form is provided in the Standard.


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11.1.5 The Application of Longitudinal Moments - BS 5500 Approach

A longitudinal moment applied to a rectangular area of circumferential length 2C and

axial length Cz (see Figure 11.7) is resolved into two equal and opposite forces:-W M Cz1 5.

acting on rectangles of circumferential length 2C and axial length 2Cx , where

C Cx z 6 , which are separated by a distance of 2 3Cz between centres. Equations

(11.5) are again appropriate for this case. The steps for this are given in BS 5500,

found in the hand-out copies of the Standard.

Figure 11.7 Analysis of longitudinal moment.

11.1.6 The Effect of Internal and External Pressure

When internal pressure is applied at the same time as the local loads, the total stresses

(and strains) are assumed to be obtained by the superposition of those due to each

system of loading considered separately. This is a conservative approach for internal

pressure.

This method cannot be used when a vessel is subject to external pressure, or a high

axial load, since the deflection due to the radial or moment loading always increase the

out-of-roundness of the shell and therefore, the tendency to buckle. A procedure is

suggested in the Standard, BS 5500 for examining this problem.


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11.1.7 Dimensional Limitations on the Vessel and Attachment

The analytical treatment assumes that the ends of the vessel are remote from the

attachment and the vessel length L, should be greater than the vessel radius r. Whenthe load is offset from the centre the distance from the vessel end to the edge of the

attachment should not be less than r/2. Certain restrictions are also placed upon the

attachment dimensions C r for radial loads and longitudinal moments; and C r 2

for circumferential moments. These are shown on Figure 11.8 as a function ofr t . A

further limitation is placed on the axial length of the patch, in that for values of

C rx 0 25. for radial loads and circumferential moments and for values of

C rz 2 0 25. for longitudinal moments, the data should be used with caution.

Figure 11.8 Restrictions of vessel/attachment geometry (BS 5500)


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11.2 THE SPHERICAL VESSEL WITH LOCAL LOADS

This section is related to the radial and moment loading of spherical vessels through a

rigid attachment - as give in Section G.2.4 of BS 5500. The analysis is based on a

shallow shell theory, but can be used for a complete spherical end providing the loadingis of a local nature and dies out within a region contained by a height to base diameter

ratio less than 1/8. A limitation is placed on the ratio r ro of 1/3. The graphs provided

in BS 5500 give a distribution of the stress away from the loaded region and are

therefore, useful in stressing the end loaded with several discrete attachments.

For convenience the loads are considered as acting on a pipe of mean radiusro which is

assumed to be a rigid body fixed to the sphere..

Loads applied through square fittings of side 2Cx can be treated approximately as

distributed over a circle of radius r Co x . If the bracket is rectangular of sides 2Cx

by 2C the loading is approximately over a circle of radius r C Co x .

11.2.1 Stresses and Deflections Due to Radial Loads.

Figure 11.9 shows a radial load applied to a spherical vessel through a rigid insert of

radius ro .

Figure 11.9 Spherical vessel subjected to a radial load.

The deflections, and stress resultants M W M W N t W N t Wx x, , , are given in

Figs G.2(24) and G.2(25) of BS 5500. To illustrate the form of plot Fig G.2(25) is

reproduced here as Figure 11.10.


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Figure 11.10 Moments and membrane forces in a spherical vessel subjected to a radialload W - BS 5500


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The abscissa of Figure 11.10 is the non-dimensional parameter s x r t1 82. . In this

the parameter s defines the position in the shell at which the force, moment or

deflection is required. The full line on the charts provides the values for u s , that is

the force, moment or deflection at the attachment-vessel junction.

11.2.2 Stresses, Deflection and Slopes Due to an External Moment.

Figure 11.11 shows an external moment applied to a spherical shell through a rigid

attachment of radiusro. In this case the deflection and stress resultants depend on the

angle as well as the distancex.

These values can be found from Figs G.2(27) and G.2(28) of BS 5500. As before, the

full line, u s , provides the values at the attachment-vessel junction.

Figure 11.11 Spherical vessel subjected to an external moment

11.3 ALLOWABLE STRESSES FOR LOCAL LOADS.

Section A.3.3 of BS 5500 provides stress limits for elastically calculated stresses

adjacent to attachments and supports (providing the dimension of the attachment in the

circumferential direction is not greater than one-third of the shell circumference and not

less than 2 5. r t from another stress concentrating feature) which are subject to the

combined effects ofpressure and externally applied loads. These stress limits are

(a) the membrane stress intensity should not exceed 1.2f( 0 8. Y);

(b) the stress intensity due to the sum of the membrane and bending stresses should notexceed 2f(1 33. Y )


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The section A.3.3.3 of BS 5500 considers the possibility of buckling due to local loads

when the membrane stresses are compressive due to a radially inward load or moment.

The membrane plus bending stress (total compressive stress) is limited to 0 9. Y . This

limitation is hardly adequate to avoid snap through condition and should be used with

caution. It was introduced to avoid plastic buckling.

The membrane stress may be considered local membrane if the distance over which

the stress intensity exceeds 1.1fdoes not extend in the meridional direction more than

0 5. r t and if not closer in the meridional direction than 2 5. r t to another region

where the limits of general primary membrane stress are exceeded. If such a condition

can be met the primary membrane stress is limited to 1.5f.

Some designers assume (conveniently !!!) that the local load case falls within the

primary plus secondary stress and use a limiting figure of3f. This, however, is not a

valid interpretation of the Standard with regard to the local loading under discussion

here. Although some secondary bending will occur it is small compared with the

primary bending component.

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