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Stresses in ellipsoidal pressure vessel heads with noncentral nozzle
Nuclear Engineering and Design 198 (2000) 317 – 323 www.elsevier.com/locate/nucengdes
Technical note
Stresses in ellipsoidal pressure vessel heads with noncentral nozzle V.N. Skopinsky * Moscow State Industrial Uni6ersity, Moscow, 109280, Russia Received 28 September 1999; accepted 8 December 1999
Abstract The objective of this paper is further investigation of the shell intersection problem. The shell theory and finite element method are used for stress analysis of nozzle connections in ellipsoidal heads of the pressure vessels. Ellipsoidal heads having attached nozzles considerably displaced from the head axis are mainly considered. The features of the numerical procedure, structural modeling of nozzle-head shell intersections and SAIS special-purpose computer program are discussed. The results of stress analysis and parametric study of an ellipsoidal vessel head with a noncentral nozzle under internal pressure loading are presented. © 2000 Published by Elsevier Science S.A. All rights reserved.
1. Introduction Nozzle connections in pressure vessel heads that could be ellipsoidal, hemispherical or torispherical are widely used in many industries, for example, in the chemical, petrochemical, petroleum-processing, power and nuclear engineering industries. For these structures often operating in diverse and complex conditions (in cases of possible departure from the technological standards of operating regime, unusual or complex vessel design, large vessel diameter, vessels of a high pressure state, at low or high temperature, etc.), the local increased stresses and stress distri* Tel./fax: +7-095-275-2256. E-mail address: [email protected] (V.N. Skopinsky)
bution features must be taken into consideration, particularly for the proper design of the nozzle connections. Increasing requirements in various industries make it necessary to conduct the detailed stress analysis under primary loads for analyzed structure configurations. In particular, known Codes such as ASME (1983), BS 5500 (1976), and Russian GOST (1989) do not contain enough information about nozzle connections on pressure vessel heads. Therefore, it is necessary to carry out a systematic investigation of the nozzle connections in various structural applications in order to contribute to an overall understanding of this problem. This paper presents a numerical stress analysis of the ellipsoidal vessel heads with cylindrical nozzles, in the general case, nonradial and
0029-5493/00/$ - see front matter © 2000 Published by Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 3 4 2 - 8
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V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
noncentral, subjected to uniform internal pressure. In the study, special attention is paid to stress concentration in the head and attached nozzle, and to the features of stress distributions along the intersection region.
2. Structural modeling and numerical procedure The geometry of a standard ellipsoidal head with noncentral nozzle being analyzed is shown in Fig. 1. In the present analysis this nozzle-head connection is considered as the shell intersection, where the ellipsoidal head is defined as the basic shell (Skopinsky and Berkov, 1994). The important non-dimensional geometric parameters of the nozzle-ellipsoidal shell intersection are as follows: b/a, r/a, a/H (or r/h), h/H, a, x¯0
(1)
Parameter x¯0 defines the relative displacement of the nozzle from the central position at the ellipsoidal shell (head). The angle a defines the angular deflection of the nozzle axis from the
Fig. 1. Geometry of ellipsoidal head with nozzle.
Fig. 2. Finite element model of ellipsoidal head with nozzle.
normal n0 in the main plane of the shell intersection. (According to definitions introduced in previous work (Skopinsky, 1993), the main plane passes through the axis of the basic shell and the normal n0 to the basic shell surface at the point of the intersection of this surface by the nozzle axis.) For elastic analysis of shell intersections, the finite element method based on a modified mixed formulation and thin shell theory were employed. The finite element modeling of the shell intersection was carried out using the four-node arbitrary double curved shell quadrilateral element with 20 d.o.f., five at each node. The nodal d.o.f. are three purely displacements and two out of plane rotation components in the curvilinear coordinate system of the middle shell surface. For the approximation of the spatial intersection region the mixed model of the beam curved element was used. The special scheme for the independent assumptions of the displacement and strain components for the mixed element models gives the improved characteristics of an element stiffness matrix in comparison with the usual displacement method. The solution procedure used in this study involves the application of the curvilinear coordinate systems (s, 8, z) and (s%, 8%, z%) at the middle surfaces of the ellipsoidal shell and nozzle (cylindrical shell), respectively (Skopinsky and Berkov, 1994). An irregular finite element mesh of the shell intersection was applied using an automatic mesh generation which used nonlinear geometric relationships describing the intersection curve. These relations are given completely in the abovementioned paper. From previous studies, it is known that the stresses in the intersecting shells change sharply reaching the membrane stress levels away from the intersection region. Therefore, the mesh with the least element sizes is employed in the vicinity of the intersection curve to achieve accuracy in results, and the procedure of grading the mesh from fine to coarse is realized with the mesh generation to economize the computational effort. Due to symmetry, only half of the shell intersection was used for analysis. For example, the finite element model of half of the ellipsoidal head with noncentral nonradial nozzle is shown in Fig. 2.
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
Application of the curved shell elements allows use of a rational calculation algorithm of the numerical procedure developed for analysis of the shell intersections. The resulting equations with respect to nodal displacements for the finite-element model are obtained in the form Kd = F,
K =%Ke,
F =%Fe
e
(2)
e
where the stiffness matrix Ke and the load vector Fe of the element are obtained in the curvilinear coordinate system of every shell. The element characteristics (Ke, Fe) of the nozzle elements having nodes at the intersection curve are transformed to the coordinate system of the basic shell. These transformations are represented in block form corresponding to a nodal scheme of the element characteristics Kij =LTi K%ij Lj,
Fi =LTi F%i
Lk = Ælk, lk É,
k =i, j
(3)
where Kij, Fi are blocks of the element stiffness matrix Ke and the load vector Fe corresponding to i-node at the intersection curve (blocks in prime correspond to the nozzle coordinate system); Lk is the transformation matrix for node k of element (in case this node does not lie at the intersection curve, the matrix Lk is a unit diagonal matrix). The submatrix lk for nozzle-ellipsoid shell intersection was given in Skopinsky and Berkov (1994). The frontal method is used to form and solve the resulting Eq. (2). The nodal displacement vector of the nozzle elements having nodes at the intersection curve is obtained in the curvilinear coordinate system of the nozzle using the following transformation: d%i =Lidi
(4)
where di is block of the element displacement vector corresponding to the ith node. The stress components (ss, s8, ts8 ) are determined at the nodal points of an element for the outside and inside shell surfaces. Moreover, the stress components of the basic shell and nozzle are divided for nodal points of the intersection curve.
319
Also, using the obtained stress components, the principal stresses and effective stress on the basis of the Tresca criterion are defined as: se = s1 − s3
(5)
where s1 and s3 are the maximum and minimum principal stresses. In general, application of the curvilinear coordinate systems makes it possible to use minimum coordinate transformations in order to obtain the resulting equations of the element model, and also displacement, strain and stress components in elements. In combination with using effective shell element models, these factors allow significant reduction of the cost–time numerical procedure.
3. Stress analysis Numerical stress analysis of the various nozzlehead shell intersections was carried out using the SAIS special-purpose computer program (Skopinsky, 1993; Skopinsky and Berkov, 1994). SAIS (S6 tress A6 nalysis in I6 ntersecting S6 hells) is an object-oriented program providing a complete engineering environment (pre-processing, modeling, analysis and post-processing), and practical efficiency at relatively modest cost–time. Numerous comparisons between experimental and numerical results carried out by the author for various test models of the intersecting shells demonstrated good accuracy in results, and a high efficiency and capability of the program to analyze the different shell intersections including both thin and thick shells, in particular for problems with regions of high stress gradients. Some results of these comparisons are given in Skopinsky (1993, 1996, 1997). Stress analysis and examination of stress concentrations were conducted for nozzle connections on the standard ellipsoidal head of the reactor vessel with non-dimensional geometric parameters a/H= 50, b/a= 0.5, r/a= 0.11, h/H= 1.0
(6)
The load is the working internal pressure p. Elastic stress analysis was carried out for different structural configurations of the nozzle connection: central and noncentral, radial and nonradial posi-
320
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
tions of the nozzle in the ellipsoidal pressure vessel head. It is assumed that materials of the nozzle and head are the same. In the finite element analysis, symmetric boundary conditions were imposed on the symmetry plane; the boundary conditions at the end section of the nozzle corresponded to a flange fixing (axial forces from the internal pressure were applied to the nozzle), and at the edge section of the ellipsoidal shell corresponded to a simply supported shell (meridional and circumferential displacements, and two rotations are zero). From the stress analysis of the nozzle-ellipsoidal shell intersections, the following observations can be made: 1. The stress state in the nozzle and head is of a local nature, and the stress concentration is caused by the geometric discontinuity. The stress concentration of the nozzle is the most significant (so, for shell intersections with parameters (Eq. (6)), a stress concentration factor for the nozzle (SCFn =s ne,max/s n0, s n0 = pr/h) comes to a value of approx. 20). 2. Locations of points of maximum stresses in shells along the intersection curve have a certain interest. The end points A and B of the intersection curve at 8% =0 and 180° are shown in Fig. 1. The angular coordinate 8% along the intersection curve AB is the circumferential coordinate in cross-section of the nozzle. For the shell intersections with a noncentral nozzle and for relatively small values of parameter x¯0 (at x¯0 50.2, including the case of a central nozzle), the location of the maximum stresses were at 8% =180 or 0° (at the acute or obtuse corner regions). For other shell intersections with a noncentral nozzle (x¯0 \0.2), the maximum stresses are located at 8%:90°. 3. The meridional stress is primarily a bending stress changing sharply with distance away from the intersection region. This stress is characterized by high stress gradients adjacent to the juncture. It can be noted that the meridional stress in the nozzle and ellipsoidal shell may vary significantly along the intersection region, not only in value, but also in sign. For the shell intersections having a noncentral noz-
zle with a relatively small value of parameter x¯0, tension membrane components are primary for the circumferential stress. For shell intersections with a noncentral nozzle and for a relatively large value of x¯0, the circumferential stress in the nozzle is primarily a bending stress at the acute and obtuse corner regions. As it is distinct from the nozzle, the ellipsoidal head is a shell of double curvature having greater tangential stiffness than that of zero curvature. As a consequence of the membrane stress is the prevalent component for the circumferential stress in the head. The position of the nozzle on the head, central or noncentral, appreciably influences the stresses in shells. It is known that for a ‘smooth’ ellipsoidal shell (without hole) under internal pressure loading, the membrane circumferential stress away from the pole to edge section (perpendicular to ellipsoid axis) is changed from positive (tensile) to negative (compressive). Therefore, for the head with a noncentral nozzle and for x¯0 \ 0.5, the bending component of the circumferential stress in the ellipsoidal shell at the edges of the juncture is increased. Distributions of the effective stresses in the nozzle (sen) and ellipsoidal head (see) on the outside and inside shell surfaces along the intersection curve are shown in Fig. 3 for shell intersection (Eq. (6)) with a radial nozzle (a= 0) and for x¯0 = 0.8. These stresses are presented in the non-dimensional form, as the stress ratios s¯ e = se/s0,
s0 = pa 2/2bH
(7)
For comparison, the maximum effective stresses seo in the nozzle and head for a radial central counterpart (a= 0, x¯0 = 0) are shown by dashed straight lines. The maximum effective stresses occur at the inside surfaces of the nozzle and head. Moreover, the increase in the effective stress in the nozzle for a noncentral nozzle connection in comparison with its central counterpart is due to a significant increase in the circumferential stress at the inside surface of the nozzle within the juncture range 60 B 8%B 120°. The local increase in the effective stress in the ellipsoidal head in this juncture region is caused by significant shear stress at the inside surface (ts8 : 0.62s0).
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
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role of separate stress components. In the present study, for the standard ellipsoidal head with parameters (Eq. (6)), primary emphasis is given to the influence of parameters a, x¯0 and h/H, in particular for nozzle connections with a relatively large value of x¯0. Some stress results for the parametric study are shown in Figs. 4–6. Fig. 4 represents the effect of the parameter x¯0 on maximum effective stresses (Eq. (7)) in the nozzle and head for radial nozzle connections. Indexes (+) and ( −) for stress ratios correspond to the outside and inside shell surfaces, respectively. The numerical analysis indicates that the variation of maximum effective stresses in shells is negligible in nozzle-head connections with x¯0 B 0.6. For nozzle connections with x¯0 \ 0.6, there is an appreciable increase of the maximum effective stress level at the inside surface of the nozzle caused by the above-mentioned reasons. Fig. 3. Effective stress distributions along the intersection curve (a=0, x¯0 = 0.8).
Fig. 5. Effect of angle a on maximum effective stresses in nozzle and head. Fig. 4. Effect of parameter x¯0 on maximum effective stresses in nozzle and head.
4. Parametric study The radial and nonradial nozzle-ellipsoidal shell intersections have the most practical importance. The non-dimensional geometric parameters (Eq. (1)) exert the greatest influence upon the stress state, maximum stresses in the individual shells and the locations of these stresses, determine a
Fig. 6. Effect of thickness ratio on maximum effective stresses in nozzle and head.
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V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
The effect of the angular parameter a on the maximum effective stresses in the nozzle and head for shell intersections with a noncentral nozzle (x?0 = 0.8) is shown in Fig. 5. The calculated results indicate that deflection of the nozzle axis from the radial position to the ellipsoid axis (at a \5°) leads to certain reduction of maximum effective stress s− en in the nozzle (which is maximum stress for shell intersection) due to material reduction of the circumferential stress at the inside surface of the nozzle (although, besides an increase of shear stress in this region, it maintains the maximum stress level). Fig. 6 represents the effect of the thickness ratio h/H on the maximum effective stresses in shells for the radial noncentral nozzle connection with x¯0 =0.8 (h= var, H = const). The numerical analysis indicates that it is rational to use connections with a thickness ratio within the range 1.0 5 h/ H 51.2. For a nozzle connection with h/H\ 1.2, the effective stress at the outside surface of the ellipsoidal head is maximum stress for the connection due to an increase of the maximum stress at the outside surface of the head at 8% 5 20°. From the results of the study and analysis of the stress components in the nozzle and head, it can be suggested to use local reinforcement in juncture in order to reduce the circumferential stress in the nozzle. Some local reinforcement variants of the nozzle connections in reducing the stress level were considered in a recent work (Skopinsky, 1998).
give the possibility of achieving a more reliable design of the nozzle connections on the pressure vessel heads, and allow utilization of rational reinforcement in order to increase the strength of the structure. Also, the SAIS program can be used for design optimization purposes (e.g. nozzle location).
Appendix A. Nomenclature major half-axis of ellipsoidal head (mean radius of vessel) b minor half-axis of ellipsoidal head (depth of ellipsoidal head) H thickness of ellipsoidal head h, r thickness and mean radius of nozzle h/H thickness ratio x0 distance between nozzle and ellipsoid axes x¯0 = x0/a parameter of relative distance a angular parameter for nozzle axis p internal pressure s, 8 meridional and circumferential coordinates for ellipsoidal shell middle surface s%, 8% meridional and circumferential coordinates for nozzle-shell middle surface ss, s8 meridional and circumferential stresses in shell in-plane shear stress in shell ts8 se effective stress a
5. Conclusions References The numerical results of stress analysis are presented for the nozzle connections in the ellipsoidal vessel head under internal pressure loading. In many practical designs, the nozzle is placed at a relatively large distance from the head axis. Special consideration of these cases is given in the present analysis. The results of the parametric study demonstrate the influence of various geometric parameters on the maximum effective stresses in the nozzle and head. The stress analysis results obtained can be useful for a better understanding of this poorly investigated problem and
ASME Boiler and Pressure Vessel Code, 1983. Section III, American Society of Mechanical Engineers, New York. BS 5500, 1976. Unifired Fusion Welded Pressure Vessels, British Standards Institution. GOST 24755-89 (ST SEV 1639-88), 1989. Vessels and Apparatuses, Codes and Methods of Calculation on Strength of Reinforcement of Opening. Skopinsky, V.N., 1993. Numerical stress analysis of intersecting cylindrical shells. Trans. ASME J. Press. Vessel Technol. 115 (3), 275 – 282. Skopinsky, V.N., 1996. Stress analysis of nonradial cylindrical shell intersections subjected to external loading. Int. J. Press. Vessel Piping 67 (2), 145 – 153.
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323 Skopinsky, V.N., 1997. Stress analysis of with torus transition under internal Trans. ASME J. Press. Vessel Technol. Skopinsky, V.N., 1998. Comparative study
shell intersections pressure loading. 119 (3), 288 – 292. of reinforced noz-
.
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zle connections. Nucl. Eng. Des. 180, 175 – 179. Skopinsky, V.N., Berkov, N.A., 1994. Stress analysis of ellipsoidal shell with nozzle under internal pressure loading. Trans. ASME J. Press. Vessel Technol. 116 (4), 431 –436.
.
Technical note
Stresses in ellipsoidal pressure vessel heads with noncentral nozzle V.N. Skopinsky * Moscow State Industrial Uni6ersity, Moscow, 109280, Russia Received 28 September 1999; accepted 8 December 1999
Abstract The objective of this paper is further investigation of the shell intersection problem. The shell theory and finite element method are used for stress analysis of nozzle connections in ellipsoidal heads of the pressure vessels. Ellipsoidal heads having attached nozzles considerably displaced from the head axis are mainly considered. The features of the numerical procedure, structural modeling of nozzle-head shell intersections and SAIS special-purpose computer program are discussed. The results of stress analysis and parametric study of an ellipsoidal vessel head with a noncentral nozzle under internal pressure loading are presented. © 2000 Published by Elsevier Science S.A. All rights reserved.
1. Introduction Nozzle connections in pressure vessel heads that could be ellipsoidal, hemispherical or torispherical are widely used in many industries, for example, in the chemical, petrochemical, petroleum-processing, power and nuclear engineering industries. For these structures often operating in diverse and complex conditions (in cases of possible departure from the technological standards of operating regime, unusual or complex vessel design, large vessel diameter, vessels of a high pressure state, at low or high temperature, etc.), the local increased stresses and stress distri* Tel./fax: +7-095-275-2256. E-mail address: [email protected] (V.N. Skopinsky)
bution features must be taken into consideration, particularly for the proper design of the nozzle connections. Increasing requirements in various industries make it necessary to conduct the detailed stress analysis under primary loads for analyzed structure configurations. In particular, known Codes such as ASME (1983), BS 5500 (1976), and Russian GOST (1989) do not contain enough information about nozzle connections on pressure vessel heads. Therefore, it is necessary to carry out a systematic investigation of the nozzle connections in various structural applications in order to contribute to an overall understanding of this problem. This paper presents a numerical stress analysis of the ellipsoidal vessel heads with cylindrical nozzles, in the general case, nonradial and
0029-5493/00/$ - see front matter © 2000 Published by Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 3 4 2 - 8
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V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
noncentral, subjected to uniform internal pressure. In the study, special attention is paid to stress concentration in the head and attached nozzle, and to the features of stress distributions along the intersection region.
2. Structural modeling and numerical procedure The geometry of a standard ellipsoidal head with noncentral nozzle being analyzed is shown in Fig. 1. In the present analysis this nozzle-head connection is considered as the shell intersection, where the ellipsoidal head is defined as the basic shell (Skopinsky and Berkov, 1994). The important non-dimensional geometric parameters of the nozzle-ellipsoidal shell intersection are as follows: b/a, r/a, a/H (or r/h), h/H, a, x¯0
(1)
Parameter x¯0 defines the relative displacement of the nozzle from the central position at the ellipsoidal shell (head). The angle a defines the angular deflection of the nozzle axis from the
Fig. 1. Geometry of ellipsoidal head with nozzle.
Fig. 2. Finite element model of ellipsoidal head with nozzle.
normal n0 in the main plane of the shell intersection. (According to definitions introduced in previous work (Skopinsky, 1993), the main plane passes through the axis of the basic shell and the normal n0 to the basic shell surface at the point of the intersection of this surface by the nozzle axis.) For elastic analysis of shell intersections, the finite element method based on a modified mixed formulation and thin shell theory were employed. The finite element modeling of the shell intersection was carried out using the four-node arbitrary double curved shell quadrilateral element with 20 d.o.f., five at each node. The nodal d.o.f. are three purely displacements and two out of plane rotation components in the curvilinear coordinate system of the middle shell surface. For the approximation of the spatial intersection region the mixed model of the beam curved element was used. The special scheme for the independent assumptions of the displacement and strain components for the mixed element models gives the improved characteristics of an element stiffness matrix in comparison with the usual displacement method. The solution procedure used in this study involves the application of the curvilinear coordinate systems (s, 8, z) and (s%, 8%, z%) at the middle surfaces of the ellipsoidal shell and nozzle (cylindrical shell), respectively (Skopinsky and Berkov, 1994). An irregular finite element mesh of the shell intersection was applied using an automatic mesh generation which used nonlinear geometric relationships describing the intersection curve. These relations are given completely in the abovementioned paper. From previous studies, it is known that the stresses in the intersecting shells change sharply reaching the membrane stress levels away from the intersection region. Therefore, the mesh with the least element sizes is employed in the vicinity of the intersection curve to achieve accuracy in results, and the procedure of grading the mesh from fine to coarse is realized with the mesh generation to economize the computational effort. Due to symmetry, only half of the shell intersection was used for analysis. For example, the finite element model of half of the ellipsoidal head with noncentral nonradial nozzle is shown in Fig. 2.
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
Application of the curved shell elements allows use of a rational calculation algorithm of the numerical procedure developed for analysis of the shell intersections. The resulting equations with respect to nodal displacements for the finite-element model are obtained in the form Kd = F,
K =%Ke,
F =%Fe
e
(2)
e
where the stiffness matrix Ke and the load vector Fe of the element are obtained in the curvilinear coordinate system of every shell. The element characteristics (Ke, Fe) of the nozzle elements having nodes at the intersection curve are transformed to the coordinate system of the basic shell. These transformations are represented in block form corresponding to a nodal scheme of the element characteristics Kij =LTi K%ij Lj,
Fi =LTi F%i
Lk = Ælk, lk É,
k =i, j
(3)
where Kij, Fi are blocks of the element stiffness matrix Ke and the load vector Fe corresponding to i-node at the intersection curve (blocks in prime correspond to the nozzle coordinate system); Lk is the transformation matrix for node k of element (in case this node does not lie at the intersection curve, the matrix Lk is a unit diagonal matrix). The submatrix lk for nozzle-ellipsoid shell intersection was given in Skopinsky and Berkov (1994). The frontal method is used to form and solve the resulting Eq. (2). The nodal displacement vector of the nozzle elements having nodes at the intersection curve is obtained in the curvilinear coordinate system of the nozzle using the following transformation: d%i =Lidi
(4)
where di is block of the element displacement vector corresponding to the ith node. The stress components (ss, s8, ts8 ) are determined at the nodal points of an element for the outside and inside shell surfaces. Moreover, the stress components of the basic shell and nozzle are divided for nodal points of the intersection curve.
319
Also, using the obtained stress components, the principal stresses and effective stress on the basis of the Tresca criterion are defined as: se = s1 − s3
(5)
where s1 and s3 are the maximum and minimum principal stresses. In general, application of the curvilinear coordinate systems makes it possible to use minimum coordinate transformations in order to obtain the resulting equations of the element model, and also displacement, strain and stress components in elements. In combination with using effective shell element models, these factors allow significant reduction of the cost–time numerical procedure.
3. Stress analysis Numerical stress analysis of the various nozzlehead shell intersections was carried out using the SAIS special-purpose computer program (Skopinsky, 1993; Skopinsky and Berkov, 1994). SAIS (S6 tress A6 nalysis in I6 ntersecting S6 hells) is an object-oriented program providing a complete engineering environment (pre-processing, modeling, analysis and post-processing), and practical efficiency at relatively modest cost–time. Numerous comparisons between experimental and numerical results carried out by the author for various test models of the intersecting shells demonstrated good accuracy in results, and a high efficiency and capability of the program to analyze the different shell intersections including both thin and thick shells, in particular for problems with regions of high stress gradients. Some results of these comparisons are given in Skopinsky (1993, 1996, 1997). Stress analysis and examination of stress concentrations were conducted for nozzle connections on the standard ellipsoidal head of the reactor vessel with non-dimensional geometric parameters a/H= 50, b/a= 0.5, r/a= 0.11, h/H= 1.0
(6)
The load is the working internal pressure p. Elastic stress analysis was carried out for different structural configurations of the nozzle connection: central and noncentral, radial and nonradial posi-
320
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
tions of the nozzle in the ellipsoidal pressure vessel head. It is assumed that materials of the nozzle and head are the same. In the finite element analysis, symmetric boundary conditions were imposed on the symmetry plane; the boundary conditions at the end section of the nozzle corresponded to a flange fixing (axial forces from the internal pressure were applied to the nozzle), and at the edge section of the ellipsoidal shell corresponded to a simply supported shell (meridional and circumferential displacements, and two rotations are zero). From the stress analysis of the nozzle-ellipsoidal shell intersections, the following observations can be made: 1. The stress state in the nozzle and head is of a local nature, and the stress concentration is caused by the geometric discontinuity. The stress concentration of the nozzle is the most significant (so, for shell intersections with parameters (Eq. (6)), a stress concentration factor for the nozzle (SCFn =s ne,max/s n0, s n0 = pr/h) comes to a value of approx. 20). 2. Locations of points of maximum stresses in shells along the intersection curve have a certain interest. The end points A and B of the intersection curve at 8% =0 and 180° are shown in Fig. 1. The angular coordinate 8% along the intersection curve AB is the circumferential coordinate in cross-section of the nozzle. For the shell intersections with a noncentral nozzle and for relatively small values of parameter x¯0 (at x¯0 50.2, including the case of a central nozzle), the location of the maximum stresses were at 8% =180 or 0° (at the acute or obtuse corner regions). For other shell intersections with a noncentral nozzle (x¯0 \0.2), the maximum stresses are located at 8%:90°. 3. The meridional stress is primarily a bending stress changing sharply with distance away from the intersection region. This stress is characterized by high stress gradients adjacent to the juncture. It can be noted that the meridional stress in the nozzle and ellipsoidal shell may vary significantly along the intersection region, not only in value, but also in sign. For the shell intersections having a noncentral noz-
zle with a relatively small value of parameter x¯0, tension membrane components are primary for the circumferential stress. For shell intersections with a noncentral nozzle and for a relatively large value of x¯0, the circumferential stress in the nozzle is primarily a bending stress at the acute and obtuse corner regions. As it is distinct from the nozzle, the ellipsoidal head is a shell of double curvature having greater tangential stiffness than that of zero curvature. As a consequence of the membrane stress is the prevalent component for the circumferential stress in the head. The position of the nozzle on the head, central or noncentral, appreciably influences the stresses in shells. It is known that for a ‘smooth’ ellipsoidal shell (without hole) under internal pressure loading, the membrane circumferential stress away from the pole to edge section (perpendicular to ellipsoid axis) is changed from positive (tensile) to negative (compressive). Therefore, for the head with a noncentral nozzle and for x¯0 \ 0.5, the bending component of the circumferential stress in the ellipsoidal shell at the edges of the juncture is increased. Distributions of the effective stresses in the nozzle (sen) and ellipsoidal head (see) on the outside and inside shell surfaces along the intersection curve are shown in Fig. 3 for shell intersection (Eq. (6)) with a radial nozzle (a= 0) and for x¯0 = 0.8. These stresses are presented in the non-dimensional form, as the stress ratios s¯ e = se/s0,
s0 = pa 2/2bH
(7)
For comparison, the maximum effective stresses seo in the nozzle and head for a radial central counterpart (a= 0, x¯0 = 0) are shown by dashed straight lines. The maximum effective stresses occur at the inside surfaces of the nozzle and head. Moreover, the increase in the effective stress in the nozzle for a noncentral nozzle connection in comparison with its central counterpart is due to a significant increase in the circumferential stress at the inside surface of the nozzle within the juncture range 60 B 8%B 120°. The local increase in the effective stress in the ellipsoidal head in this juncture region is caused by significant shear stress at the inside surface (ts8 : 0.62s0).
V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
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role of separate stress components. In the present study, for the standard ellipsoidal head with parameters (Eq. (6)), primary emphasis is given to the influence of parameters a, x¯0 and h/H, in particular for nozzle connections with a relatively large value of x¯0. Some stress results for the parametric study are shown in Figs. 4–6. Fig. 4 represents the effect of the parameter x¯0 on maximum effective stresses (Eq. (7)) in the nozzle and head for radial nozzle connections. Indexes (+) and ( −) for stress ratios correspond to the outside and inside shell surfaces, respectively. The numerical analysis indicates that the variation of maximum effective stresses in shells is negligible in nozzle-head connections with x¯0 B 0.6. For nozzle connections with x¯0 \ 0.6, there is an appreciable increase of the maximum effective stress level at the inside surface of the nozzle caused by the above-mentioned reasons. Fig. 3. Effective stress distributions along the intersection curve (a=0, x¯0 = 0.8).
Fig. 5. Effect of angle a on maximum effective stresses in nozzle and head. Fig. 4. Effect of parameter x¯0 on maximum effective stresses in nozzle and head.
4. Parametric study The radial and nonradial nozzle-ellipsoidal shell intersections have the most practical importance. The non-dimensional geometric parameters (Eq. (1)) exert the greatest influence upon the stress state, maximum stresses in the individual shells and the locations of these stresses, determine a
Fig. 6. Effect of thickness ratio on maximum effective stresses in nozzle and head.
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V.N. Skopinsky / Nuclear Engineering and Design 198 (2000) 317–323
The effect of the angular parameter a on the maximum effective stresses in the nozzle and head for shell intersections with a noncentral nozzle (x?0 = 0.8) is shown in Fig. 5. The calculated results indicate that deflection of the nozzle axis from the radial position to the ellipsoid axis (at a \5°) leads to certain reduction of maximum effective stress s− en in the nozzle (which is maximum stress for shell intersection) due to material reduction of the circumferential stress at the inside surface of the nozzle (although, besides an increase of shear stress in this region, it maintains the maximum stress level). Fig. 6 represents the effect of the thickness ratio h/H on the maximum effective stresses in shells for the radial noncentral nozzle connection with x¯0 =0.8 (h= var, H = const). The numerical analysis indicates that it is rational to use connections with a thickness ratio within the range 1.0 5 h/ H 51.2. For a nozzle connection with h/H\ 1.2, the effective stress at the outside surface of the ellipsoidal head is maximum stress for the connection due to an increase of the maximum stress at the outside surface of the head at 8% 5 20°. From the results of the study and analysis of the stress components in the nozzle and head, it can be suggested to use local reinforcement in juncture in order to reduce the circumferential stress in the nozzle. Some local reinforcement variants of the nozzle connections in reducing the stress level were considered in a recent work (Skopinsky, 1998).
give the possibility of achieving a more reliable design of the nozzle connections on the pressure vessel heads, and allow utilization of rational reinforcement in order to increase the strength of the structure. Also, the SAIS program can be used for design optimization purposes (e.g. nozzle location).
Appendix A. Nomenclature major half-axis of ellipsoidal head (mean radius of vessel) b minor half-axis of ellipsoidal head (depth of ellipsoidal head) H thickness of ellipsoidal head h, r thickness and mean radius of nozzle h/H thickness ratio x0 distance between nozzle and ellipsoid axes x¯0 = x0/a parameter of relative distance a angular parameter for nozzle axis p internal pressure s, 8 meridional and circumferential coordinates for ellipsoidal shell middle surface s%, 8% meridional and circumferential coordinates for nozzle-shell middle surface ss, s8 meridional and circumferential stresses in shell in-plane shear stress in shell ts8 se effective stress a
5. Conclusions References The numerical results of stress analysis are presented for the nozzle connections in the ellipsoidal vessel head under internal pressure loading. In many practical designs, the nozzle is placed at a relatively large distance from the head axis. Special consideration of these cases is given in the present analysis. The results of the parametric study demonstrate the influence of various geometric parameters on the maximum effective stresses in the nozzle and head. The stress analysis results obtained can be useful for a better understanding of this poorly investigated problem and
ASME Boiler and Pressure Vessel Code, 1983. Section III, American Society of Mechanical Engineers, New York. BS 5500, 1976. Unifired Fusion Welded Pressure Vessels, British Standards Institution. GOST 24755-89 (ST SEV 1639-88), 1989. Vessels and Apparatuses, Codes and Methods of Calculation on Strength of Reinforcement of Opening. Skopinsky, V.N., 1993. Numerical stress analysis of intersecting cylindrical shells. Trans. ASME J. Press. Vessel Technol. 115 (3), 275 – 282. Skopinsky, V.N., 1996. Stress analysis of nonradial cylindrical shell intersections subjected to external loading. Int. J. Press. Vessel Piping 67 (2), 145 – 153.
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shell intersections pressure loading. 119 (3), 288 – 292. of reinforced noz-
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zle connections. Nucl. Eng. Des. 180, 175 – 179. Skopinsky, V.N., Berkov, N.A., 1994. Stress analysis of ellipsoidal shell with nozzle under internal pressure loading. Trans. ASME J. Press. Vessel Technol. 116 (4), 431 –436.
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