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Stress reduction factor associated with saddle support with extended top plate
hr. ./. Pm. Printed ELSEVIER
0308-0161(94)00010-7
TECHNICAL
Ves. & Pipmg 62 (1995) 205-208 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030x-0161/95/$09.50
NOTE
Stress reduction factor associated with saddle support with extended top plate L’. S. Ong & G. Lu Natzyong
Technological
University,
School of Mechanical
& Production
Engineering,
Nanyang
Avenue,
Sitzgapore
2263
(Received 21 February 1994:accepted 3 March 1994)
A chart, which consists of a series of parametric curves, is provided to determine the stress reduction factor associatedwith the use of a saddle support with extended top plate. The theoretical basis for the parametric curves and dimensional parameters is briefly described. It is found that a saddlesupport with an extended top plate could reduce the peak stressat the support by as much as50% or more. An example is shown on the use of the chart. The chart will be a useful designaid to vesseldesignersin specifying suitable extended plate dimensions.
1 INTRODUCTION
have a thickness equal to that of the vessel and should subtend an angle between 6” and 12” above the tip of the saddle web-see Fig. 1. The Code does not provide any technical reference to its recommendation. The author has investigated this problem and published a table of results for use in the determination of the stress reduction factor. However, the tabulated results are restrictive in their ranges of application as only a few selected ranges of saddle top plate dimensions are considered. The present paper reexamines the problem and establishes a set of geometric parameters which are then used to reexpress the results by parametric curves. The use of geometric parameters greatly reduces the number of results to be represented. For this reason, additional data are generated to extend the range of application. All parametric curves are plotted in one single chart. This is considered to be particularly convenient to use so that it will be useful to vessel designers in specifying saddle support design details.
Cylindrical vessels and liquid containers are usually supported horizontally by means of saddle supports. One of the design considerations is to limit the peak stress developed at the support to avoid local or fatigue failure. As all forces acting on the cylindrical vessel are ultimately transferred to the support, stress concentration at the support must be checked so that it will not become the main design factor. Past studies’-’ have shown that the severity of stress concentration at the support depends very much on the construction details of the support. A rigid support will give rise to greater stress concentration compared to a flexible one. The main cause of stress concentration is the abrupt transition of structural rigidity between the support and the vessel. As the vessel is far less rigid than the support some stress concentrations must exist. One of the best ways of moderating and cushioning the stress concentration at the support-vessel junction is to use a support with an extended saddle top plate. Indeed, this is a recommended practice in Appendix G of British Pressure Vessel Code BS 5500.’ The Code recommends that the saddle top plate should
2 ESTABLISHMENT PARAMETERS
OF GEOMETRIC
The advantage of incorporating an extended saddle top plate in the design of a saddle support 205
206
L. S. Ong, G. Lu
stiffeners
Fig. 1. Saddle support with extended plate.
can be quantified via a stress reduction factor as follows: Peak stress at the support (with extended top plate) k, = Peak stress at the support (with no extended top plate)
vessel is governed by the flexibility of the vessel and the support as follows:
PI cxWlv + FIJ’ (1)
References l-3 outline the theory for finding the stress distribution in a cylindrical shell at its support. Basically, the theoretical approach consists of finding the contact pressure distribution between the saddle support and the vessel and then applying the pressure distribution to the cylindrical shell so as to obtain the stress distribution. The first step involves contact stress formulation and the second step is merely an analysis of a loaded cylindrical shell. As the support embraces only part of the cylindrical shell surface, high magnitudes of stress are expected to concentrate along the top edges of the support. This loading action is analogous to a rigid die acting on an elastic medium. As the support is structurally more rigid compared to the vessel shell, an extended top saddle plate will be an effective means in moderating and cushioning the abrupt transition of rigidity between the support and the vessel. The stress distribution at the support will be ultimately governed by the relative rigidity (or flexibility) of the saddle support and the vessel. From Refs 1 and 2, it is known that the contact force distribution between the support and the
X
(applied displacement components)
(2)
In eqn G?, PI is the contact force vector and Plv and [Fls are the flexibility matrices associated with the vessel and the support, respectively. The flexibility matrix consists mainly of displacement coefficients. The elements in each column of the flexibility matrix are actually a set of displacements due to a unit load applied somewhere at the contact boundary. The dimensional groups associated with the two matrices, [Fly and [F],, will be established in the following. The displacement of the vessel due to a normal force has the following form:
(3) For the saddle top plate, the theory described in Refs 1 and 2 has assumed that only the extended portion of the plate is deformable and it can be idealised as a cantilever curved beam. A simple curved beam formula is, therefore, used to generate the displacement elements in matrix [F],. For a cantilever curved beam subject to a normal force, the displacement has the following form: r3 wsxz=
12r3 Ebt:
(4)
Stress reduction factor associated
with saddle support
parameter (CLJ is associated with the rigidity of the extended plate.
where 1, = &f/l2 is the second moment of area of the reinforcing plate cross-section. In view of eqns (3) and (4), flexibility matrices [Flv and [Fls can be alternatively expressed by
3 DERIVATION CURVES
The contact force vector [PI, given by eqn (2), can now be expressed by (j51’[F,”
+ g
,F],]-’
-Yr 1 -YL:’ r
-I
(7)
Equation (7) suggests the following geometric parameters to be defined:
two
[F], + $-(
fi3[Qs
(9) The first parameter (Q) is associated with the width of the saddle top plate and the second
lx,= g
0
0.2
0.4
0.6
0.8
1
1.2
OF PARAMETRIC
The stress reduction factor, defined in eqn (l), will be dependent on various geometric variables such as the mean radius (Y) and thickness (t) of the vessel, and the width (b), thickness (tS) and angular extension (cy) of the extended saddle top plate. All these geometric variables can be considered through the use of the two dimensionless parameters, ab and (Y,, established in Section 2. Figure 2 shows a chart for finding the stress reduction factor (k,) associated with the use of an extended saddle top plate. The applicable ranges of the parametric curves in the chart are stated in the following. The angular extension (cy) of the extended plate ranges from 3” to 15”. The extended plate ‘rigidity’ parameter, (Y,, ranges from zero to 2.8, the higher value of a, denotes thicker plate thickness and, therefore, has a higher rigidity. The ‘width’ parameter q, varies from O-01 to O-1, and covers the width of the plate to as large as the shell radius. The ranges of
(6)
[P] rx [-$
207
with extended top plate
1.4
1.6
1.8
60<20<150
2
Fig. 2. Stress reduction factor at the support.
2.2
deg.
2.4
2.6
2.8
3
208
L. S. Ong, G. Lu
the geometric parameters thus cover adequately all practical extended plate dimensions. The chart (Fig. 2) also shows the location of the peak stress, The peak stress can occur within the extended plate region, which is likely to happen when the extended plate is very thin compared to the shell thickness or when it has a large angular extension. Otherwise, the peak stress would occur at the edge of the plate, expecially when using a thicker extended plate. From Fig. 2, it can be observed that when the peak stress is located within the extended plate, the ‘width’ parameter ((Ye) dictates the value of k,. On the other hand, when the peak stress is located at the edge of the extended plate, the angular extension (a) is the controlling factor. An extended plate having a larger angular extension will always have a lower stress reduction factor. The parametric curves are applicable to a range of support angles (2p = 60-150”). It is merely because stress reduction is of a local character and it is insensitive to angular position. It can also be observed that, with properly selected extended plate dimensions, the peak stress can be reduced by more than half. Current design practice of recommending an extended plate having the same thickness as the pipe wall and extending not more than 6” away from the edge of support may not always provide the optimum solution.
Based on the above dimensions: a6 = 0.025 (eqn (8)) as = O-292 (eqn (9)) From Fig. 2, k, = O-61, and the peak stress occurs within the extended plate. If now the extended plate thickness is doubled, i.e. t, = 36.6 mm, then a, = 0.585 (eqn (9)) From Fig. 2, k, = 0.68, and the peak stress occurs at the edge of the extended plate. To find an optimum extended plate thickness, Fig. 2 indicates that for LY= 6” and (Ye= O-025, when cy,= 0.38, k, has the lowest value at 0.54. The value of (Y,= 0.33 corresponds to t,/t = 1.3 or t, = 23.8 mm. For this optimum plate thickness, the magnitude of peak stress located within the extended plate equals that at the edge of the plate. For the above example, it is shown that the peak stress can be reduced by almost 50% when an extended plate of 6” extension is incorporated into the saddle support design. In conclusion, the work has shown that using an extended saddle top plate is effective in reducing the peak stress at the support of cylindrical shells or pipes. The parametric curves, besides being able to provide an immediate value of the stress reduction factor for a particular extended saddle top plate, can also be used as a design aid for deriving an optimum set of plate dimensions. REFERENCES
4 AN EXAMPLE The following is an example to illustrate the use of the chart (Fig. 2). The dimensions used in this example are: Vessel: radius (Y) = 1830 mm; thickness (t) = 18.3 mm Support: support angle (2p) = 120”; width (b) = 457.5 mm; extended plate thickness (tr) = 18.3 mm; angular extension (cz) = 6
1. Ong, L. S., Analysis of twin-saddle supported vessel subjectedto non-symmetricloadings.ht. J. Pres. Ves. & Piping, 35 (1988) 423-37. 2. Ong, L. S., Effectiveness of wear plate at the support. ASME J. Pres. Ves. Technol., 114 (1992) 12-18. 3. Ong. L. S., Parametric study of peak stressat the saddle support. ht. J. Pres. Vex & Piping, 48 (1991) 183-207. 4. Tooth, A. S., Duthie, G., White, G. C. & Carmichael,J., Stressesin horizontal storage vessels-a comparisonof theory and experiment. J. Shin Analysis, 17 (1982) 169-76. 5. British Standards Institution, Unfired fusion welded pressure vessels.BS 5500.BSI, London, 1989.
0308-0161(94)00010-7
TECHNICAL
Ves. & Pipmg 62 (1995) 205-208 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030x-0161/95/$09.50
NOTE
Stress reduction factor associated with saddle support with extended top plate L’. S. Ong & G. Lu Natzyong
Technological
University,
School of Mechanical
& Production
Engineering,
Nanyang
Avenue,
Sitzgapore
2263
(Received 21 February 1994:accepted 3 March 1994)
A chart, which consists of a series of parametric curves, is provided to determine the stress reduction factor associatedwith the use of a saddle support with extended top plate. The theoretical basis for the parametric curves and dimensional parameters is briefly described. It is found that a saddlesupport with an extended top plate could reduce the peak stressat the support by as much as50% or more. An example is shown on the use of the chart. The chart will be a useful designaid to vesseldesignersin specifying suitable extended plate dimensions.
1 INTRODUCTION
have a thickness equal to that of the vessel and should subtend an angle between 6” and 12” above the tip of the saddle web-see Fig. 1. The Code does not provide any technical reference to its recommendation. The author has investigated this problem and published a table of results for use in the determination of the stress reduction factor. However, the tabulated results are restrictive in their ranges of application as only a few selected ranges of saddle top plate dimensions are considered. The present paper reexamines the problem and establishes a set of geometric parameters which are then used to reexpress the results by parametric curves. The use of geometric parameters greatly reduces the number of results to be represented. For this reason, additional data are generated to extend the range of application. All parametric curves are plotted in one single chart. This is considered to be particularly convenient to use so that it will be useful to vessel designers in specifying saddle support design details.
Cylindrical vessels and liquid containers are usually supported horizontally by means of saddle supports. One of the design considerations is to limit the peak stress developed at the support to avoid local or fatigue failure. As all forces acting on the cylindrical vessel are ultimately transferred to the support, stress concentration at the support must be checked so that it will not become the main design factor. Past studies’-’ have shown that the severity of stress concentration at the support depends very much on the construction details of the support. A rigid support will give rise to greater stress concentration compared to a flexible one. The main cause of stress concentration is the abrupt transition of structural rigidity between the support and the vessel. As the vessel is far less rigid than the support some stress concentrations must exist. One of the best ways of moderating and cushioning the stress concentration at the support-vessel junction is to use a support with an extended saddle top plate. Indeed, this is a recommended practice in Appendix G of British Pressure Vessel Code BS 5500.’ The Code recommends that the saddle top plate should
2 ESTABLISHMENT PARAMETERS
OF GEOMETRIC
The advantage of incorporating an extended saddle top plate in the design of a saddle support 205
206
L. S. Ong, G. Lu
stiffeners
Fig. 1. Saddle support with extended plate.
can be quantified via a stress reduction factor as follows: Peak stress at the support (with extended top plate) k, = Peak stress at the support (with no extended top plate)
vessel is governed by the flexibility of the vessel and the support as follows:
PI cxWlv + FIJ’ (1)
References l-3 outline the theory for finding the stress distribution in a cylindrical shell at its support. Basically, the theoretical approach consists of finding the contact pressure distribution between the saddle support and the vessel and then applying the pressure distribution to the cylindrical shell so as to obtain the stress distribution. The first step involves contact stress formulation and the second step is merely an analysis of a loaded cylindrical shell. As the support embraces only part of the cylindrical shell surface, high magnitudes of stress are expected to concentrate along the top edges of the support. This loading action is analogous to a rigid die acting on an elastic medium. As the support is structurally more rigid compared to the vessel shell, an extended top saddle plate will be an effective means in moderating and cushioning the abrupt transition of rigidity between the support and the vessel. The stress distribution at the support will be ultimately governed by the relative rigidity (or flexibility) of the saddle support and the vessel. From Refs 1 and 2, it is known that the contact force distribution between the support and the
X
(applied displacement components)
(2)
In eqn G?, PI is the contact force vector and Plv and [Fls are the flexibility matrices associated with the vessel and the support, respectively. The flexibility matrix consists mainly of displacement coefficients. The elements in each column of the flexibility matrix are actually a set of displacements due to a unit load applied somewhere at the contact boundary. The dimensional groups associated with the two matrices, [Fly and [F],, will be established in the following. The displacement of the vessel due to a normal force has the following form:
(3) For the saddle top plate, the theory described in Refs 1 and 2 has assumed that only the extended portion of the plate is deformable and it can be idealised as a cantilever curved beam. A simple curved beam formula is, therefore, used to generate the displacement elements in matrix [F],. For a cantilever curved beam subject to a normal force, the displacement has the following form: r3 wsxz=
12r3 Ebt:
(4)
Stress reduction factor associated
with saddle support
parameter (CLJ is associated with the rigidity of the extended plate.
where 1, = &f/l2 is the second moment of area of the reinforcing plate cross-section. In view of eqns (3) and (4), flexibility matrices [Flv and [Fls can be alternatively expressed by
3 DERIVATION CURVES
The contact force vector [PI, given by eqn (2), can now be expressed by (j51’[F,”
+ g
,F],]-’
-Yr 1 -YL:’ r
-I
(7)
Equation (7) suggests the following geometric parameters to be defined:
two
[F], + $-(
fi3[Qs
(9) The first parameter (Q) is associated with the width of the saddle top plate and the second
lx,= g
0
0.2
0.4
0.6
0.8
1
1.2
OF PARAMETRIC
The stress reduction factor, defined in eqn (l), will be dependent on various geometric variables such as the mean radius (Y) and thickness (t) of the vessel, and the width (b), thickness (tS) and angular extension (cy) of the extended saddle top plate. All these geometric variables can be considered through the use of the two dimensionless parameters, ab and (Y,, established in Section 2. Figure 2 shows a chart for finding the stress reduction factor (k,) associated with the use of an extended saddle top plate. The applicable ranges of the parametric curves in the chart are stated in the following. The angular extension (cy) of the extended plate ranges from 3” to 15”. The extended plate ‘rigidity’ parameter, (Y,, ranges from zero to 2.8, the higher value of a, denotes thicker plate thickness and, therefore, has a higher rigidity. The ‘width’ parameter q, varies from O-01 to O-1, and covers the width of the plate to as large as the shell radius. The ranges of
(6)
[P] rx [-$
207
with extended top plate
1.4
1.6
1.8
60<20<150
2
Fig. 2. Stress reduction factor at the support.
2.2
deg.
2.4
2.6
2.8
3
208
L. S. Ong, G. Lu
the geometric parameters thus cover adequately all practical extended plate dimensions. The chart (Fig. 2) also shows the location of the peak stress, The peak stress can occur within the extended plate region, which is likely to happen when the extended plate is very thin compared to the shell thickness or when it has a large angular extension. Otherwise, the peak stress would occur at the edge of the plate, expecially when using a thicker extended plate. From Fig. 2, it can be observed that when the peak stress is located within the extended plate, the ‘width’ parameter ((Ye) dictates the value of k,. On the other hand, when the peak stress is located at the edge of the extended plate, the angular extension (a) is the controlling factor. An extended plate having a larger angular extension will always have a lower stress reduction factor. The parametric curves are applicable to a range of support angles (2p = 60-150”). It is merely because stress reduction is of a local character and it is insensitive to angular position. It can also be observed that, with properly selected extended plate dimensions, the peak stress can be reduced by more than half. Current design practice of recommending an extended plate having the same thickness as the pipe wall and extending not more than 6” away from the edge of support may not always provide the optimum solution.
Based on the above dimensions: a6 = 0.025 (eqn (8)) as = O-292 (eqn (9)) From Fig. 2, k, = O-61, and the peak stress occurs within the extended plate. If now the extended plate thickness is doubled, i.e. t, = 36.6 mm, then a, = 0.585 (eqn (9)) From Fig. 2, k, = 0.68, and the peak stress occurs at the edge of the extended plate. To find an optimum extended plate thickness, Fig. 2 indicates that for LY= 6” and (Ye= O-025, when cy,= 0.38, k, has the lowest value at 0.54. The value of (Y,= 0.33 corresponds to t,/t = 1.3 or t, = 23.8 mm. For this optimum plate thickness, the magnitude of peak stress located within the extended plate equals that at the edge of the plate. For the above example, it is shown that the peak stress can be reduced by almost 50% when an extended plate of 6” extension is incorporated into the saddle support design. In conclusion, the work has shown that using an extended saddle top plate is effective in reducing the peak stress at the support of cylindrical shells or pipes. The parametric curves, besides being able to provide an immediate value of the stress reduction factor for a particular extended saddle top plate, can also be used as a design aid for deriving an optimum set of plate dimensions. REFERENCES
4 AN EXAMPLE The following is an example to illustrate the use of the chart (Fig. 2). The dimensions used in this example are: Vessel: radius (Y) = 1830 mm; thickness (t) = 18.3 mm Support: support angle (2p) = 120”; width (b) = 457.5 mm; extended plate thickness (tr) = 18.3 mm; angular extension (cz) = 6
1. Ong, L. S., Analysis of twin-saddle supported vessel subjectedto non-symmetricloadings.ht. J. Pres. Ves. & Piping, 35 (1988) 423-37. 2. Ong, L. S., Effectiveness of wear plate at the support. ASME J. Pres. Ves. Technol., 114 (1992) 12-18. 3. Ong. L. S., Parametric study of peak stressat the saddle support. ht. J. Pres. Vex & Piping, 48 (1991) 183-207. 4. Tooth, A. S., Duthie, G., White, G. C. & Carmichael,J., Stressesin horizontal storage vessels-a comparisonof theory and experiment. J. Shin Analysis, 17 (1982) 169-76. 5. British Standards Institution, Unfired fusion welded pressure vessels.BS 5500.BSI, London, 1989.