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A comparative study of the stress field around a reinforced and an unreinforced normal intersection of two cylindrical shells
Int. J. Pres. Ves. & Piping 15 (1984) 79-92
A Comparative Study of the Stress Field Around a Reinforced and an Unreinforced Normal Intersection of Two Cylindrical Shells*
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, Norman, OK 73019, USA
and Glynn Woods WFI International, Inc., Houston, TX, USA (Received: 7 February, 1983)
ABSTRACT The results of a comparative study of the effects of reinforcement on the stress fields in the critical intersection region of two normally intersecting cylindrical shells are presented. The results for in-plane and out-of-plane moments have been obtained experimentally by the use of electrical resistance joil gages, and numerically by the use of a three-dimensional fnite element program. The finite element analysis also includes internal pressure loading. A comparison of results obtained by using a thin shell element with six degrees of freedom at each of the four nodes (three displacements and three rotations) and a three-dimensional element with three displacement degrees of freedom at each of the eight nodes, is also given. * Paper presented at the 1982 ASME-PVP meeting, held in Orlando, Florida, USA 27 June 2 July. 79
Int. J. Pres. Ves. & Piping 0308-0161/84/$03.00 © Elsevier Applied Science Publishers Ltd, 1984. Printed in Great Britain
80
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
INTRODUCTION The wide use of normally intersecting cylindrical shells in nuclear and fossil power plants and petrochemical industries makes it an important, necessary and interesting topic for study. Since the early analytical investigation of Bijlaard,l there have been numerous other studies on this subject, including the analytical work of Eringen and Suhubi, 2 the experimental and numerical analyses of Gwaltney et al. a and the finite element parametric investigation of Bryson et al. 4 All the abovementioned analyses, as well as most of the other studies referred to in these analyses (with one exception3), were limited to configurations where the ratio of the diameter of the branch to that of the main shell was less than 0.5 and where the diameter to thickness ratio of the main shell ranged from 10 to 100. In the study of Gwaltney et al. 3 one model with d i D equal to 1.0 was also included. Thus experimental or numerical studies on intersections with diameter ratio larger than 0.5 are generally lacking. We have therefore embarked on a comprehensive project focusing on diameter ratios greater than 0.5; the present paper contains the results of the initial phase of this project. A comparative study of the stress field in the intersection region of two cylindrical shells, with no reinforcement and with full reinforcement, is performed and reported here using experimental and finite element techniques. In the case of experimental investigation, in-plane and out-ofplane loads were applied to the end of the attached or branch shell. The numerical analyses include these loading situations as well as internal pressure. The outside diameter and thickness of the main shell were 168.3 mm (6.625 in) and 7.1 mm (0.28 in), respectively, while the same parameters for the attached or main shell were 114.3 mm (4.5 in) and 6 mm (0.237 in). Therefore, the diameter ratio and diameter to thickness ratio for the main shell were 0.68 and 23.77, respectively. The full reinforced intersection used in this study was of the 'Vesselet' type, manufactured by WFI International, Inc. The unreinforced intersection is what is generally known as 'stub-in'. The experimental analyses were performed on the as-welded condition in both cases. EXPERIMENTAL DETAILS The loading frame used in the experimental investigation has already been described in detail;5'6 only a brief description is provided here. One
Stress field oj normally intersecting cylindrical shells
81
end of the main shell was clamped in 12.7 mm (0.5 in) thick split plate, and the in-plane and out-of-plane forces were applied to the end of the branch or the attached shell. The stresses in the intersection region were determined for the b o u n d a r y condition, of one end clamped and one end free. K y o w a electrical resistance foil three-element rosette gages of type KFD-2-D17-11, and A u t o m a t i o n Industries Metafilm two-element rectangular rosettes of type C6-124-R2TC were used for the determination of strains. The resistance, gage length and gage factors for the K y o w a and AI gages were respectively 120 ~), 2 mm and 2.08, and 350 fl, 6.3 mm and 2.06, respectively. Figures 1 and 2 give the location of these
k,
Y
Z
~
-- I i - 5
I
II-4
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II-31 1
Z
X'x~" 11-6 __~I I-7
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II-6 / / I - I I
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Fig. 1. The unreinforced intersection; three-element rosettes are markedas 111-1 to llI-6, while two-element rosettes are shown by I I-1 to ll-l I. I n the results presented in later figures, the distance is measured from the intersection line shown above. All dimensions are in ram.
82
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao. Glynn Wood~"
,m_,,~..L I I - 1
IntersectiOno ° Line~L~
L 11-5
~Jl
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51
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±
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_ _ ~
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Fig. 2. The reinforced intersection; three-element rosettes are III-I to 111-11, while twoelement rosettes are shown by lI-1 to II-8. In the results presented in later figures, the distance is measured from the intersection line shown above. All dimensions are in mm. gages on the two shell intersection structures included in the present study. The global coordinate system is also shown on these figures. Due to symmetries of the structure, the measurement is made in one quadrant only. Three-element rosettes are used where appreciable shear stresses are expected, while away from the critical area two-element rosettes are used.
FINITE E L E M E N T A N A L Y S I S DETAILS The finite element analysis has been performed using the Structural Analysis Program (SAP) developed at the University of California,
Stress field of normally intersecting cylindrical shells
83
Berkeley. 7 In all analyses, the eight-node brick element Z1B8R9 originated by Irons and Zienkiewiczs is used. The element has three degrees of freedom (displacements) at each of the eight nodes. In the case of the unreinforced intersection, analysis was also performed using a thin shell element which is an assembly of four compatible triangular fiat-plate elements; each triangular element is a combination of constant Strain triangle (LCT) and linear curvature compatible triangle (LCCT 9).The displacement expansion for this LCCT 9 element is the same as that designated HCT element and described in Ref. 9. The element has six degrees of freedom (three displacements and three rotations) at each node.
Fig. 3.
The developed (unfolded) view of the finite element mesh for the main shell of the unreinforced intersection.
The developed finite element meshes for the attached shell and the main shell are shown in Figs 3 and 4 for the unreinforced or 'stub-in' intersection. Due to symmetry of the structure and applied loads, only half of the model was used for analysis. The complete model has 1110 nodes and 521 elements. Only one element through the thickness was used for both the main and branch shells, to compare the results obtained by the use of eight-node brick and thin shell elements in the SAP program. The developed finite element meshes for the reinforced intersection (WFI's Vesselet) are shown in Figs 5 and 6. In this case, the total numbers of nodes and elements were 891 and 562, respectively. This time two elements through the thickness are used in the main shell, while in the reinforced region of the branch shell three elements through the thickness are used. In the remaining portion of the branch shell one element through the thickness is included in the model. In both cases, the weld itself is not
84
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glvnn Woods
J
Fig. 4.
The developed (unfolded) view of the finite element mesh for the attached shell ot the unreinforced intersection.
included in the finite element model. The boundary conditions used in the analyses are as follows. All the nodes on the top and bottom horizontal lines in Figs 3 and 5, and all the nodes on the two extreme vertical lines in Figs 4 and 6, are constrained to move in the direction perpendicular to the plane of the figures. Further, all the nodes on the extreme left hand vertical lines in
m
Fig. 5.
_
m
The developed (unfolded) view of the finite element mesh for the main shell of the reinforced intersection.
Stress field of normally intersecting cylindrical shells
85
E
i i
Fig. 6.
The developed (unfolded) view of the finite element mesh for the attached shell of the reinforced intersection.
Figs 3 and 5 are fixed with respect to all three displacements (if threedimensional elements are used), and in the case of thin shell element, the nodes are constrained also with respect to the three rotations. Horizontal and vertical nodal forces are applied on all the nodes on the topmost horizontal line in Figs 4 and 6 to produce in-plane and out-of-plane moments in the intersection region.
RESULTS A N D DISCUSSION The experimental and finite element results are given in Figs 7 to 9 for the unreinforced intersection (stub-in). The stress components are normalized by dividing by 'beam type stress' for moment leadings ( = M/z; M and z are the m o m e n t and section modulus of the main shell, respectively); in the case of internal pressure loading, the normalization has been achieved by dividing the stress components by 'hoop stress' in the main shell. The solid and dashed lines give the axx/ao, and ayy/a o, obtained from finite element analyses, respectively; the respective
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
86
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Distance From The Intersection (mm).
Fig. 7.
The stress ratios for the in-plane moment load (unreinforced intersection).
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Fig. 8.
The stress ratios for out-of-plane m o m e n t load (unreinforced intersection).
Stress field of normally intersecting cylindrical shells
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127
The stress ratios for the internal pressure (unreinforced intersection).
experimental values are given by closed and open circles. The stresses are given with respect to the local coordinate system; x and y are axial and circumferential directions of the main shell, respectively. The experimental determination of stresses was only performed for in-plane and out-ofplane loadings, thus Fig. 9 contains finite element results only. Considering the fact that the finite element results do not include the stress intensification effects due to the presence of the weld, the agreement between the finite element and experimental results is really remarkable. %×/'0~yy/% --] _ - _ •
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The stress ratios for the in-plane moment load (reinforced intersection).
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
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Fig. 11.
The stress ratios for the out-of-plane m o m e n t load (reinforced intersection).
The results for the reinforced 'Vesselet' intersection are given in Figs 10 to 13. Once again the stress ratios are obtained by dividing the stress components by 'beam type stress', a o, for moment loads and by 'hoop stress' for the internal pressure case. In Fig. 12, a plot of stresses is given at 1 12.5 ° (22.5 ° in the counterclockwise direction from the global z axis). The maximum stress ratio for out-of-plane loading occurs along this line.
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The stress ratios for the out-of-plane moment load at 112.5 ° line (reinforced intersection).
89
Stress field of normally intersecting cylindrical shells
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Fig. 13.
The stress ratios for the internal pressure at 180° line (reinforced intersection).
Once again, except in the region where the stress intensification effect of the weld is prominent, the agreement between the stress ratios obtained by the use of finite element and experimental techniques is good. The stress ratios for both intersections are given in Table 1 for comparison, showing significantly lower values for the reinforced intersection. In order to see the effect of different choices of elements on finite element analysis results, a comparative study is made for a thin shell element used in previous studies (Refs 5 and 6) and a three-dimensional eight-node brick element used in the present study as well as that of Bryson et al. 4 The thin shell element has six degrees of freedom at each of TABLE 1
Magnitude of Maximum Stress Ratios for the Two Intersections; the Location is the Distance from the Intersection Line Along the Given Direction Type oJ loading
In-plane moment Out-of-plane moment Internal pressure
UnreinJorced intersection
ReinJorced intersection
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Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods"
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Fig. 14.
Comparison of results obtained by using thin shell and three-dimensional elements for in-plane moment load (unreinforced intersection).
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Fig. 15.
Comparison of results obtained by using thin shell and three-dimensional elements for out-of-plane moment load (unreinforced intersection).
Stress field of normally intersecting cylindrical shells
91
the four nodes (three displacements and three rotations), while the eightnode brick element has three displacement degrees of freedom at each node; both elements have a total of 24 nodes. These results are given in Figs 14 to 16 for the unreinforced intersection. As can be seen from these graphs, the results are fairly close except for internal pressure loading (Fig. 16), where the agreement is not good. In the case of internal pressure, the peak stresses are fairly close but the two curves are somewhat displaced by approximately 1/8 in (or 3 mm). This is due to geometrical discontinuity in the model if a thin shell element is used. It must also be mentioned here that finite element analysis was also performed for in-plane and out-of-plane moments, rather than forces, and the results for moments and forces were not appreciably different. o:*,*,~00
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127
Fig. 16. Comparison of results obtained by using thin shell and three-dimensional elements for internal pressure load (unreinforced intersection).
CONCLUSIONS It can be concluded from this study that, for moment loading, the thin shell element as well as the eight-node brick element of the Structural Analysis Program (SAP) is effective in providing stress fields in the intersection region of normally intersecting cylindrical shells and these results compare well with experimentally determined stress fields. In the case of internal pressure loading, the peak stresses in the two cases are fairly close but the stress distribution curve obtained by the use of the thin shell element is higher than the one obtained by the use of the threedimensional element.
92
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
The stress ratios given in. Table 1 show that the reinforced 'Vesselet' type intersection is effective in reducing high stresses in the critical region for in-plane and out-of-plane m o m e n t loads as well as for internal pressure loading.
REFERENCES 1. Bijlaard, P. P., Stresses from local loadings in cylindrical pressure vessels, Trans. ASME, 77 (1955) 805-16. 2. Eringen, A. C. and Suhubi, E. S., Stress distribution at two normally intersecting cylindrical shells, Nucl. Struct. Engng, 2 (1965) 253-70. 3. Gwaltney, R. C., Corum, J. M., Bolt, S. E. and Bryson, J. W., Experimental stress analysis of cylinder to cylinder shell models and comparisons with theoretical predictions, J. Pres. Ves. Tech., Trans. ASME, No. 1976, 283-90. 4. Bryson, J. W., Johnson, W. G. and Bass, B. R., Stresses in reinforced nozzlecylinder attachments--a parameter study, A S M E special publication, PVP, 50 (1981) 51-66. 5. Hsiao, C., Finite element and experimental analyses oJ two obliquely inclined cylindrical shells, MS Thesis, School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, 1981. 6. Khan, A. S. and Hsiao, C., Obliquely inclined cylindrical shells; an experimental study, Proc. SESA spring meeting, Dearborn, Michigan, 31 May-4 June, 1981, pp. 379-84. 7. Bathe, K. J., Wilson, E, L. and Peterson, F. E., SAP IV--a structural analysis program for static and dynamic response of linear systems, EERC 73-11, University of California, Berkeley, June 1973 (revised April 1974). 8. Irons, B. M. and Zienkiewicz, O. C., The isoparametric element system--a new concept in finite element analysis, Proc. Conf. Recent Advances in Stress Analysis, Royal Aeronautical Society, London, 1968. 9. Clough, R. W. and Tocher, J. L., Finite element stiffness matrices for the analysis of plate bending, Proc. ConJ] Matrix Methods in Structural Mechanics, Report No. AFFDL-TR-66-80, Wright Patterson AFB, 1966, pp. 515-46.
A Comparative Study of the Stress Field Around a Reinforced and an Unreinforced Normal Intersection of Two Cylindrical Shells*
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, Norman, OK 73019, USA
and Glynn Woods WFI International, Inc., Houston, TX, USA (Received: 7 February, 1983)
ABSTRACT The results of a comparative study of the effects of reinforcement on the stress fields in the critical intersection region of two normally intersecting cylindrical shells are presented. The results for in-plane and out-of-plane moments have been obtained experimentally by the use of electrical resistance joil gages, and numerically by the use of a three-dimensional fnite element program. The finite element analysis also includes internal pressure loading. A comparison of results obtained by using a thin shell element with six degrees of freedom at each of the four nodes (three displacements and three rotations) and a three-dimensional element with three displacement degrees of freedom at each of the eight nodes, is also given. * Paper presented at the 1982 ASME-PVP meeting, held in Orlando, Florida, USA 27 June 2 July. 79
Int. J. Pres. Ves. & Piping 0308-0161/84/$03.00 © Elsevier Applied Science Publishers Ltd, 1984. Printed in Great Britain
80
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
INTRODUCTION The wide use of normally intersecting cylindrical shells in nuclear and fossil power plants and petrochemical industries makes it an important, necessary and interesting topic for study. Since the early analytical investigation of Bijlaard,l there have been numerous other studies on this subject, including the analytical work of Eringen and Suhubi, 2 the experimental and numerical analyses of Gwaltney et al. a and the finite element parametric investigation of Bryson et al. 4 All the abovementioned analyses, as well as most of the other studies referred to in these analyses (with one exception3), were limited to configurations where the ratio of the diameter of the branch to that of the main shell was less than 0.5 and where the diameter to thickness ratio of the main shell ranged from 10 to 100. In the study of Gwaltney et al. 3 one model with d i D equal to 1.0 was also included. Thus experimental or numerical studies on intersections with diameter ratio larger than 0.5 are generally lacking. We have therefore embarked on a comprehensive project focusing on diameter ratios greater than 0.5; the present paper contains the results of the initial phase of this project. A comparative study of the stress field in the intersection region of two cylindrical shells, with no reinforcement and with full reinforcement, is performed and reported here using experimental and finite element techniques. In the case of experimental investigation, in-plane and out-ofplane loads were applied to the end of the attached or branch shell. The numerical analyses include these loading situations as well as internal pressure. The outside diameter and thickness of the main shell were 168.3 mm (6.625 in) and 7.1 mm (0.28 in), respectively, while the same parameters for the attached or main shell were 114.3 mm (4.5 in) and 6 mm (0.237 in). Therefore, the diameter ratio and diameter to thickness ratio for the main shell were 0.68 and 23.77, respectively. The full reinforced intersection used in this study was of the 'Vesselet' type, manufactured by WFI International, Inc. The unreinforced intersection is what is generally known as 'stub-in'. The experimental analyses were performed on the as-welded condition in both cases. EXPERIMENTAL DETAILS The loading frame used in the experimental investigation has already been described in detail;5'6 only a brief description is provided here. One
Stress field oj normally intersecting cylindrical shells
81
end of the main shell was clamped in 12.7 mm (0.5 in) thick split plate, and the in-plane and out-of-plane forces were applied to the end of the branch or the attached shell. The stresses in the intersection region were determined for the b o u n d a r y condition, of one end clamped and one end free. K y o w a electrical resistance foil three-element rosette gages of type KFD-2-D17-11, and A u t o m a t i o n Industries Metafilm two-element rectangular rosettes of type C6-124-R2TC were used for the determination of strains. The resistance, gage length and gage factors for the K y o w a and AI gages were respectively 120 ~), 2 mm and 2.08, and 350 fl, 6.3 mm and 2.06, respectively. Figures 1 and 2 give the location of these
k,
Y
Z
~
-- I i - 5
I
II-4
Intersecti°n Line~L'~ II-1~ l
II-31 1
Z
X'x~" 11-6 __~I I-7
II-9 /II-IO
II-6 / / I - I I
1II-4
!!-8
9O
Fig. 1. The unreinforced intersection; three-element rosettes are markedas 111-1 to llI-6, while two-element rosettes are shown by I I-1 to ll-l I. I n the results presented in later figures, the distance is measured from the intersection line shown above. All dimensions are in ram.
82
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao. Glynn Wood~"
,m_,,~..L I I - 1
IntersectiOno ° Line~L~
L 11-5
~Jl
III-7
111-8
51
Z
±
180
II-7 II-8
I /__
~h-1~ __L_~ mT ~-:II-G Ill-If
y
_k~
!II-5
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Z
_ _ ~
Oo
I
Fig. 2. The reinforced intersection; three-element rosettes are III-I to 111-11, while twoelement rosettes are shown by lI-1 to II-8. In the results presented in later figures, the distance is measured from the intersection line shown above. All dimensions are in mm. gages on the two shell intersection structures included in the present study. The global coordinate system is also shown on these figures. Due to symmetries of the structure, the measurement is made in one quadrant only. Three-element rosettes are used where appreciable shear stresses are expected, while away from the critical area two-element rosettes are used.
FINITE E L E M E N T A N A L Y S I S DETAILS The finite element analysis has been performed using the Structural Analysis Program (SAP) developed at the University of California,
Stress field of normally intersecting cylindrical shells
83
Berkeley. 7 In all analyses, the eight-node brick element Z1B8R9 originated by Irons and Zienkiewiczs is used. The element has three degrees of freedom (displacements) at each of the eight nodes. In the case of the unreinforced intersection, analysis was also performed using a thin shell element which is an assembly of four compatible triangular fiat-plate elements; each triangular element is a combination of constant Strain triangle (LCT) and linear curvature compatible triangle (LCCT 9).The displacement expansion for this LCCT 9 element is the same as that designated HCT element and described in Ref. 9. The element has six degrees of freedom (three displacements and three rotations) at each node.
Fig. 3.
The developed (unfolded) view of the finite element mesh for the main shell of the unreinforced intersection.
The developed finite element meshes for the attached shell and the main shell are shown in Figs 3 and 4 for the unreinforced or 'stub-in' intersection. Due to symmetry of the structure and applied loads, only half of the model was used for analysis. The complete model has 1110 nodes and 521 elements. Only one element through the thickness was used for both the main and branch shells, to compare the results obtained by the use of eight-node brick and thin shell elements in the SAP program. The developed finite element meshes for the reinforced intersection (WFI's Vesselet) are shown in Figs 5 and 6. In this case, the total numbers of nodes and elements were 891 and 562, respectively. This time two elements through the thickness are used in the main shell, while in the reinforced region of the branch shell three elements through the thickness are used. In the remaining portion of the branch shell one element through the thickness is included in the model. In both cases, the weld itself is not
84
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glvnn Woods
J
Fig. 4.
The developed (unfolded) view of the finite element mesh for the attached shell ot the unreinforced intersection.
included in the finite element model. The boundary conditions used in the analyses are as follows. All the nodes on the top and bottom horizontal lines in Figs 3 and 5, and all the nodes on the two extreme vertical lines in Figs 4 and 6, are constrained to move in the direction perpendicular to the plane of the figures. Further, all the nodes on the extreme left hand vertical lines in
m
Fig. 5.
_
m
The developed (unfolded) view of the finite element mesh for the main shell of the reinforced intersection.
Stress field of normally intersecting cylindrical shells
85
E
i i
Fig. 6.
The developed (unfolded) view of the finite element mesh for the attached shell of the reinforced intersection.
Figs 3 and 5 are fixed with respect to all three displacements (if threedimensional elements are used), and in the case of thin shell element, the nodes are constrained also with respect to the three rotations. Horizontal and vertical nodal forces are applied on all the nodes on the topmost horizontal line in Figs 4 and 6 to produce in-plane and out-of-plane moments in the intersection region.
RESULTS A N D DISCUSSION The experimental and finite element results are given in Figs 7 to 9 for the unreinforced intersection (stub-in). The stress components are normalized by dividing by 'beam type stress' for moment leadings ( = M/z; M and z are the m o m e n t and section modulus of the main shell, respectively); in the case of internal pressure loading, the normalization has been achieved by dividing the stress components by 'hoop stress' in the main shell. The solid and dashed lines give the axx/ao, and ayy/a o, obtained from finite element analyses, respectively; the respective
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
86
~
----
G
..............
U
-
0
-
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I 127
Distance From The Intersection (mm).
Fig. 7.
The stress ratios for the in-plane moment load (unreinforced intersection).
180°
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Fig. 8.
The stress ratios for out-of-plane m o m e n t load (unreinforced intersection).
Stress field of normally intersecting cylindrical shells
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127
The stress ratios for the internal pressure (unreinforced intersection).
experimental values are given by closed and open circles. The stresses are given with respect to the local coordinate system; x and y are axial and circumferential directions of the main shell, respectively. The experimental determination of stresses was only performed for in-plane and out-ofplane loadings, thus Fig. 9 contains finite element results only. Considering the fact that the finite element results do not include the stress intensification effects due to the presence of the weld, the agreement between the finite element and experimental results is really remarkable. %×/'0~yy/% --] _ - _ •
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The stress ratios for the in-plane moment load (reinforced intersection).
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
88
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Fig. 11.
The stress ratios for the out-of-plane m o m e n t load (reinforced intersection).
The results for the reinforced 'Vesselet' intersection are given in Figs 10 to 13. Once again the stress ratios are obtained by dividing the stress components by 'beam type stress', a o, for moment loads and by 'hoop stress' for the internal pressure case. In Fig. 12, a plot of stresses is given at 1 12.5 ° (22.5 ° in the counterclockwise direction from the global z axis). The maximum stress ratio for out-of-plane loading occurs along this line.
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89
Stress field of normally intersecting cylindrical shells
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Fig. 13.
The stress ratios for the internal pressure at 180° line (reinforced intersection).
Once again, except in the region where the stress intensification effect of the weld is prominent, the agreement between the stress ratios obtained by the use of finite element and experimental techniques is good. The stress ratios for both intersections are given in Table 1 for comparison, showing significantly lower values for the reinforced intersection. In order to see the effect of different choices of elements on finite element analysis results, a comparative study is made for a thin shell element used in previous studies (Refs 5 and 6) and a three-dimensional eight-node brick element used in the present study as well as that of Bryson et al. 4 The thin shell element has six degrees of freedom at each of TABLE 1
Magnitude of Maximum Stress Ratios for the Two Intersections; the Location is the Distance from the Intersection Line Along the Given Direction Type oJ loading
In-plane moment Out-of-plane moment Internal pressure
UnreinJorced intersection
ReinJorced intersection
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Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods"
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Fig. 14.
Comparison of results obtained by using thin shell and three-dimensional elements for in-plane moment load (unreinforced intersection).
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Distance From The Intersection (mm).
Fig. 15.
Comparison of results obtained by using thin shell and three-dimensional elements for out-of-plane moment load (unreinforced intersection).
Stress field of normally intersecting cylindrical shells
91
the four nodes (three displacements and three rotations), while the eightnode brick element has three displacement degrees of freedom at each node; both elements have a total of 24 nodes. These results are given in Figs 14 to 16 for the unreinforced intersection. As can be seen from these graphs, the results are fairly close except for internal pressure loading (Fig. 16), where the agreement is not good. In the case of internal pressure, the peak stresses are fairly close but the two curves are somewhat displaced by approximately 1/8 in (or 3 mm). This is due to geometrical discontinuity in the model if a thin shell element is used. It must also be mentioned here that finite element analysis was also performed for in-plane and out-of-plane moments, rather than forces, and the results for moments and forces were not appreciably different. o:*,*,~00
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Fig. 16. Comparison of results obtained by using thin shell and three-dimensional elements for internal pressure load (unreinforced intersection).
CONCLUSIONS It can be concluded from this study that, for moment loading, the thin shell element as well as the eight-node brick element of the Structural Analysis Program (SAP) is effective in providing stress fields in the intersection region of normally intersecting cylindrical shells and these results compare well with experimentally determined stress fields. In the case of internal pressure loading, the peak stresses in the two cases are fairly close but the stress distribution curve obtained by the use of the thin shell element is higher than the one obtained by the use of the threedimensional element.
92
Akhtar S. Khan, Jian-Cun Chen, Chiuder Hsiao, Glynn Woods
The stress ratios given in. Table 1 show that the reinforced 'Vesselet' type intersection is effective in reducing high stresses in the critical region for in-plane and out-of-plane m o m e n t loads as well as for internal pressure loading.
REFERENCES 1. Bijlaard, P. P., Stresses from local loadings in cylindrical pressure vessels, Trans. ASME, 77 (1955) 805-16. 2. Eringen, A. C. and Suhubi, E. S., Stress distribution at two normally intersecting cylindrical shells, Nucl. Struct. Engng, 2 (1965) 253-70. 3. Gwaltney, R. C., Corum, J. M., Bolt, S. E. and Bryson, J. W., Experimental stress analysis of cylinder to cylinder shell models and comparisons with theoretical predictions, J. Pres. Ves. Tech., Trans. ASME, No. 1976, 283-90. 4. Bryson, J. W., Johnson, W. G. and Bass, B. R., Stresses in reinforced nozzlecylinder attachments--a parameter study, A S M E special publication, PVP, 50 (1981) 51-66. 5. Hsiao, C., Finite element and experimental analyses oJ two obliquely inclined cylindrical shells, MS Thesis, School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, 1981. 6. Khan, A. S. and Hsiao, C., Obliquely inclined cylindrical shells; an experimental study, Proc. SESA spring meeting, Dearborn, Michigan, 31 May-4 June, 1981, pp. 379-84. 7. Bathe, K. J., Wilson, E, L. and Peterson, F. E., SAP IV--a structural analysis program for static and dynamic response of linear systems, EERC 73-11, University of California, Berkeley, June 1973 (revised April 1974). 8. Irons, B. M. and Zienkiewicz, O. C., The isoparametric element system--a new concept in finite element analysis, Proc. Conf. Recent Advances in Stress Analysis, Royal Aeronautical Society, London, 1968. 9. Clough, R. W. and Tocher, J. L., Finite element stiffness matrices for the analysis of plate bending, Proc. ConJ] Matrix Methods in Structural Mechanics, Report No. AFFDL-TR-66-80, Wright Patterson AFB, 1966, pp. 515-46.