wrc information
Post on 04-Apr-2018
217 Views
Preview:
TRANSCRIPT
-
7/30/2019 WRC INFORMATION
1/14
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.
-
7/30/2019 WRC INFORMATION
2/14
16426/16587 - Pressurised Systems
142
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.
-
7/30/2019 WRC INFORMATION
3/14
16426/16587 - Pressurised Systems
143
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
-
7/30/2019 WRC INFORMATION
4/14
16426/16587 - Pressurised Systems
144
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.
-
7/30/2019 WRC INFORMATION
5/14
16426/16587 - Pressurised Systems
145
Figure 11.4 Cylindrical vessel with radial load: circumferential stress resultant permillimetre width.
-
7/30/2019 WRC INFORMATION
6/14
16426/16587 - Pressurised Systems
146
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
-
7/30/2019 WRC INFORMATION
7/14
16426/16587 - Pressurised Systems
147
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].
-
7/30/2019 WRC INFORMATION
8/14
16426/16587 - Pressurised Systems
148
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.
-
7/30/2019 WRC INFORMATION
9/14
16426/16587 - Pressurised Systems
149
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.
-
7/30/2019 WRC INFORMATION
10/14
16426/16587 - Pressurised Systems
150
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)
-
7/30/2019 WRC INFORMATION
11/14
16426/16587 - Pressurised Systems
151
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.
-
7/30/2019 WRC INFORMATION
12/14
16426/16587 - Pressurised Systems
152
Figure 11.10 Moments and membrane forces in a spherical vessel subjected to a radialload W - BS 5500
-
7/30/2019 WRC INFORMATION
13/14
16426/16587 - Pressurised Systems
153
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 )
-
7/30/2019 WRC INFORMATION
14/14
16426/16587 - Pressurised Systems
154
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.
top related