Development of stress indices for nonradial branch connections

Nuclear Engineering and Design 98 (1987) 421-435 North-Holland, Amsterdam

421

DEVELOPMENT OF STRESS INDICES FOR NONRADIAL BRANCH CONNECTIONS Pal P. R A J U Stone & Webster Engineering Corporation, 245 Summer Street, Boston, MA 02107. USA

Received May 1986

This paper presents a brief summary of the technical basis for the recommended stress indices for 45 degree lateral connections under internal pressure and in-plane moment loadings. Starting with the historical background of the Pressure Vessel Research Committee's (PVRC) long range program on lateral connections, this paper highlights the various aspects intrinsic to this program such as model selection, analysis technique, finite element discretization, loading conditions, material properties and boundary conditions. A discussion of the stress index method and its application to the pressure vessel and piping design is offered as a prelude to the recommended code indices. Proposed changes to Par. NB-3338.2(C) (1) of Subsection NB of the ASME Code pertaining to lateral nozzles in cylindrical vessels and Par. NB-3650 relative to branch connections in piping systems are presented and discussed in detail. Besides, this paper discusses the need to develop additional data in the range of geometric parameters, 0.5 <_d / D _<1.0 and 10 <_ D / T < 50, and under other remaining loading conditions to broaden the application of the stress index method.

1. Introduction The recognition that i n t r o d u c t i o n of nozzles and b r a n c h connections into vessels * a n d pipelines creates a region of increased localized stress in the vicinity of the a t t a c h m e n t s a n d that this elevation of the localized stress could cause low cycle fatigue failure resulted in a series of theoretical and experimental studies since the mid-fifties~ u n d e r the auspices of the Pressure Vessel Research C o m m i t t e e (PVRC) of the Welding Research Council (WRC). Traditionally, the P V R C organization has developed a n d provided the A m e r i c a n Society of Mechanical Engineers (ASME) the necessary data base to introduce new Code rules a n d / o r update requirem e n t s that relate to pressure vessel and piping design. Hence, the A S M E accorded the P V R C the task of developing the necessary guidelines and procedures to

* In this paper, the terms, main vessel and run pipe/lateral nozzle and branch connection are used alternatively to remind the reader that the work reported herein is equally applicable to a lateral branch connection in a piping system, as well as a lateral nozzle in a component such as a reactor vessel, provided the limits on the dimensional parameters are met.

simplify the calculation of stresses near openings in pressure vessels a n d piping. A n early investigation was reported in W R C Bulletin N u m b e r 51 [1] **. F o r m u l a s were derived by E.O. Waters for stresses in the n e i g h b o r h o o d of a circular hole in a flat plate, reinforced with a cylindrical outlet such as a pipe or nozzle. Subsequently, in recognition of the complexity involved in developing closed form theoretical solutions, experimental stress analysis methods were extensively used to determine the state of stress on b o t h the inner a n d outer surfaces of different types of nozzles a n d b r a n c h connections of various shapes and sizes. Full scale models of pressure vessels were tested u n d e r internal pressure by D.E. H a r d e n b e r g h [2] of the Pennsylvania State University, to provide experimental data on the general state of stress at b r a n c h connections, a n d to determine the extent of stress intensification in that vicinity. U n d e r the sponsorship of the Bureau of Ships, C.E. Taylor et al. [3], of the University of Illinois performed a three-dimensional photoelastic study of stresses in ** The number in brackets refers to the reference listed at the end of this paper.

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422

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cylindrical nozzles in spherical and cylindrical vessels loaded by internal pressure. Several configurations of reinforcements were tested to study the effects of geometrical and other variables on the stress field and to determine an optimum design. This study confirmed the importance of providing well rounded corner and fillet radii and smooth transitions between sections of different thickness. It also confirmed the benefit of distributing the reinforcement as close to the intersection as possible. In 1962, a summary report [4] on the PVRC research on reinforcement openings in pressure vessels was published. This report covered the scope and the results of photoelastic, hard model and theoretical investigations conducted heretofore. It also identified that the work performed prior to 1962 covered only the simplest configurations of reinforced openings and alerted that there existed a number of special problems such as nonradial or hillside nozzles in both cylinders and spheres and lateral connections in cylinders which might not prove amenable to closed form theoretical solution. The interpretive report made it also apparent that the problem of reinforced openings was complicated because of the large number of geometrical variables and loading conditions involved. As a sequel to the foregoing report, WRC Bulletin 113 [5] was published covering the extensive photoelastic data developed by Prof. C.E. Taylor, Uhiversity of Illinois, and Prof. N.C. Lind, University of Waterloo, and Mr. M.M. Leven at the Westinghouse Research Laboratories. The effect of diameter ratio ( d / D ) , diameter to thickness ratio ( D / T ) , stress ratio ( s / S ) , fillet radius and nonradial connections in spherical heads was studied using numerous photoelastic models. The effect of variations in local reinforcement was also studied and reported by varying: (1) the length of reinforcement in nozzle wall; (2) fillet radius and weld fillets; and (3) percentage of reinforcement. The rest of the studies in spherical shells involved the assessment of reinforcements primarily in the vessel wall, balanced vs. unbalanced reinforcements and the effect of D / T ratio. The effect of variables cited above was also studied extensively in cylindrical vessels. Distributions of principal stresses were developed for the inside and outside surfaces of the model. Stress concentration factors were computed and reported for each model. All of these models except for four spherical models, were subjected to internal pressure. The other four models were subjected to external bending moments. A preliminary evaluation of the photoelastic test data was performed by J.L. Merhson [6]. Four nonradial connections were tested as part of the PVRC pro-

gram. Of particular interest is the one nonradiaI connec tion tested in a cylindrical vessel The detailed evalua-, tion of the test under internal pressure loading identi-fied two critically stressed regions, namely: ~1! the acute inside corner: and (2) the base of the nozzles on the obtuse side. It was also observed that the stresses m these locations increased as the angularity increased. Meaningful conclusions on the structural behavior of lateral connections in cylindrical vessels could not be reached with a single photoelastic test reported. The last of the series of reports [7] on oblique nozzles connections in pressure vessel heads and shells under internal pressure loading was published m 1970 as WRC Bulletin No. 153. This report included an i n t e r pretive study on oblique nozzle connections bv J.L. Mershon and presented four recent photoelastic studies on 45 degree lateral nozzles by M.M, Leven. These models represented a combination of thick and thin walled nozzles and cylindrical vessels, reinforced and unreinforced nozzles and blunt and sharp acute corners. As important and pioneering as these studies were, the interpretive report rightfully concluded that the available data were not sufficient to draw quantitative conclusions regarding the magnitude of stresses over an?' range of geometries or parametric ratios and loading conditions. Nevertheless, certain qualitative conclusions were reached based on the evaluated data in aid of interested designers and researchers concerned with structural behavior of lateral connections. The major conclusions which have been subsequently confirmed by the parametric studies to be reported herein, are that: (i) the critical stress occurs in the acute corner of a lateral connection to a cylindrical vessel: ~2) reinforcement in the nozzle wall is relatively ineffective in controlling the load stresses in the acute inner corner: and (3) the empirical code formula for lateral connections in cylinders provides only an approximate value of the maximum stress under internal pressure. One unequivocal conclusion which provided the impetus for the parametric studies presented and discussed in this paper was that any conclusions concerning the stress concentration effects should await further results from three dimensional finite element analyses which would permit generation of parametric data. Consequently, early in 1974, the Subconnnittec on Reinforced Openings and External Loadings (ROEL) of the Pressure Vessel Research Committee (PVRC) formed a Task Group on Laterals with a long range objective of carrying out parametric studies with the aid of state-ofthe-art techniques and recommending a set of stress indices for inclusion in the appropriate sections of the ASME Code. As a first step toward this objective, the

P.P. Raju / Development of stress indices Task Group carried out a survey of lateral configurations most commonly used in petrochemical, fossil and nuclear plants. After a careful evaluation of the merits in terms of cost, versatility and flexibility of different analytical tools, the Task Group selected the three-dimensional finite element analysis method in consideration of several beneficial factors such as the feasibility to obtain direct displacement results and view the structural behavior through distorted geometry plots and the ability to perform multi-load analysis using a single model. Next, in order to assess the accuracy of the finite element method in predicting the maximum or peak stresses in critical regions, two independent threedimensional finite element analyses of photoelastic model WC-12B2, reported in WRC Bulletin No. 251 were sponsored. Based on the results of these benchmark studies, reported in WRC Bulletin No. 251, the Task Group concluded that reliable and accurate results could be obtained provided: (1) careful attention is paid to element and mesh selection; (2) a viable technique is employed to obtain surface stresses; and (3) convergence to theoretical quantities is achieved in regions away from discontinuities. Subsequently, the Task Group sponsored a parametric study on four reinforced lateral models, selected on the basis of an industry and literature survey. This is a paper on analytically determined stresses, stress indices and stress intensification factors for the entire configuration of the chosen lateral models. These models were loaded by internal pressure and external in-plane moment loading on the run and branch pipes, respectively. The results presented herein will demonstrate that the principles of engineering mechanics can indeed provide the necessary ingredients toward the development of new and improved construction codes and standards. 2. Stress index method The /-factors, also broadly known as stress intensification factors, were first introduced into the ASME Code for the design of pressure piping components in the mid-fifties. These /-factors were based entirely on cyclic-moment fatigue tests conducted by Markl and George [8] and by Markl [9]. The /-factors provide a measure of the fatigue strength of a piping component compared to the fatigue strength of a typical girth butt weld in a straight pipe. The /-factors are used to determine stresses due to applied resultant moments by the equation of the form:

S = iM/Z,

423

where S = calculated bending stress; M = range of resultant moments; and Z = section modulus of pipe. Stress intensification factors are specified in the current piping Codes such as ANSI B31.1, "Power Piping" and ANSI B31.3, "Chemical Plant and Petroleum Refinery Piping". They are also available in the ASME Boiler and Pressure Vessel Code, Section III, "Nuclear Power Plant Components", for Class 2 Class 3 piping. Stress indices were introduced into the first edition of Section III of the ASME Code (1963) for nozzles in pressure vessels subjected to internal pressure loading only. These indices were obtained from limited photoelastic tests a n d / o r from steel model tests referenced earlier. The stress index, as used herein, is defined as the ratio of the peak (local) stress component to nominal reference stress relative to the stress field remote from the branch or nozzle connection or other discontinuities. The stress due to internal pressure for branch connections in pipes is obtained from these indices by the equation:

S=IP(D+ T)/2T, where I = stress index, P = internal pressure, D = pipe diameter, T = nominal wall thickness less corrosion allowance of pipe. The stress index method was extended to Class 1 nuclear power plant piping, subsequent to its inception into the Code, by the addition of the so called B, C and K indices. The general definition of a stress index in the piping design for mechanical loads is: B,C

or

K=o/S,

where o = elastic stress due to applied load, and S = nominal stress due to applied load. For B indices, o represents the stress magnitude corresponding to a limit load. For C or K indices, represents the maximum stress intensity due to applied load. Each index is related to a specific stress characteristics, namely: B = resistance to gross plastic deformation, C = primary plus secondary stresses, K = local or peak stresses. The stress indices for the specific piping component under investigation are identified with a particular type of load by subscripts and incorporated into equations of the form:

PD° B M BLOT--+ 2 Z --
424

P.P. Rqlu / Developmenl o/.~isc~ mUi~e~

PDo M C1 ~ - + C 2 ~ + CsE, I J a<,T, - at, Ti, l < Sa, PD° K M K I C 1 T T + 2 C 2 z + K3C3E"bI°GT', - °~t'Tt' I<- SA. In the equations above, the subscripts 1, 2 and 3 represent stress indices due to pressure, moments and the> mal gradients, respectively. SA represents an allowable appropriate to each equation. The rest of the nomenclature used is defined as follows:

T,(Th) = range of average temperature on side a(b) of gross structural or material discontinuity, a,(c%) = coefficient of thermal expansion on side a ( b ) of a gross structural or material discontinuity. A correlation between /-factors and C2K 2 (peaks stress index for the resultant moment loading) has been established and is given by C2K 2 = 2i. The factor of two in the preceding equation reflects the change of the baseline or reference standard from typical girth butt weld in straight pipe to plain, straight pipe under moment loading. Hence, if C2 and K 2 indices or their product are available for piping components, the i-factor for the Class 2 and Class 3 component design can be calculated using the correlation equation.

3. Design of lateral connections For tee connections, Section III and Division 2 of Section VIII of the Code provide stress indices at certain general locations due to internal pressure for a range of dimensional ratios. However, the Code does not provide stress indices for lateral connections that can be of practical use. The only stress index mentioned in the Code for nonradial connections refers to an estimate of the % (normal stress component) index on the inside under internal pressure loading for d / D ratio of less than or equal to 0.15. The estimate is given as follows: K 2 = K, (1 + (tan ~)4/3),

where K 1 - % inside stress index for a radial connection. K. = estimated % inside stress index for the nonradiai connection, 0 = angle the axis of the nozzle makes with the norreal to the vessel wall. The Code does not provide or address stress indices for other load cases or other locations within the lateral configuration. The above estimate does not cover many valve and pipe fitting configurations of practical interest. The severe limitation and the narrow range of application of this estimated stress index provided the necessary impetus for the PVRC Task Group to embark on the long range program to develop stress indices for 45 deg. lateral connections.

4. Selection and description of analytical models A careful consideration was given to the selection of parametric configurations, in order to derive the maximum benefit out of the limited studies, a survey of the configuration manufactured and used by the industries and found in the published literature was carried out. It revealed that the 45 degree lateral connection was the most prevalent nonradial connection in use. After having selected the type of nonradial connection (45 deg. lateral), the Task Group decided on the geometric parameters of practical interest. The range of parameters was also carefully chosen to represent the commonly used configurations in the petrochemical and nuclear applications. Generic geometric parameters that define a 45 degree lateral connection are depicted in fig. 1. The selected parametric matrix consisted of two diameter ratios (0.08, 0.5) and two diameter-to-thickness ratios (10, 40). A single stress ratio of 1.0 was selected for all the models. The Code provides several details of reinforcement [111 for nozzle and fitting design. The Task Group chose the generally adopted standard reinforcement configuration, Fig. NB-3632-3(a)-2, Section III of the Code. Other relevant parameters such as L,,, :,~, q, r~,

Table 1 Lateral model geometric parameters Model

D/T

diD

L,~/T

tn/T

t/T

r~/T

T

,'~/ 1

"2j T

1 2 3 4

10 10 40 40

0.08 0.50 0.08 0.50

2.0 2.0 2.0 4.0

0.782 1.733 1.033 3.466

0.08 0.50 0.08 0.50

0.50 0.87 0.70 2.10

3.0 3.0 0.75 095

0.5 0.5 0.5 o.5

1.0 1.0 1.0 l .i)

P.P. Raju / Det~elopment of stress radices

~f

425

tn

•,,t-- r 3

i__

~,--- r 2

I- n

D

L1

L2

Fig. 1. Lateral model geometric parameters.

r~, etc., were normalized against the nominal thickness of shell ' T'. These nondimensional values along with the diameter and diameter-to-thickness ratios are presented in table 1. It is to be noted that constant values of 0.5 and 1.0, respectively, were used for normalized comer and fillet radii. The normalized transition radius (r3) varied from 0.5 to 2.10. In order to achieve a stress ratio of 1.0 for all the four models, the thickness ratios were maintained the same as the diameter ratios. The rationale for the selection of different parameters was mainly based on, as stated earlier, the industry input, published data, code requirements, practicality and above all, economics. Actual dimensions of the models resulting from the choice of parameters are furnished in table 2.

5. Finite element

model

three dimensional finite element mesh was generated for all four models. The intersection of the axes of the run pipe and the branch connection was chosen as the global origin. The model boundaries were chosen in such a way that the discontinuity effects attenuated and the pressure membrane stresses reached Lame' values within the chosen boundaries. Based on the benchmark verification studies, the general purpose A N S Y S [12] finite element software package was chosen for the parametric studies. The finite element mesh was developed using the 3-D isoparametric solid element (STIF 45 of ANSYS) which is defined by eight nodal points having three degrees of freedom at each node. The A N S Y S node and element generator PREP7 was used in generating the mesh. This generator is an extended capability version of the node and element generation capability in the A N S Y S program.

In view of the symmetry of the geometry and applied loads about the longitudinal plane, a half-symmetric Table 2 Lateral model dimensions (all dimensions are in inches) Model

D

d

T

t

t,

rI

r2

r~

1 2 3 4

30.0 30.0 30.0 30.0

2.4 15,0 2.4 15.0

3.0 3.0 0.75 0.75

0.24 1.5 0.06 0.375

2.35 5.20 0.77 2.60

1.5 1.5 0.375 0.375

3.0 3.0 0.75 0.75

1.5 2.61 0.525 1.575

426

P.P. Rqiu / Development of.~tre.~ imh~ev

6. Discussion of finite element models Each lateral model was generated in several segments or regions and merged to obtain the half symmetric mesh. The number of solid elements used ranged from 2135 to 2600. As a minimum, three elements through the thickness were used in all regions except in the connecting lateral pipe where two elements through the wall were considered adequate. In the circumferential or hoop direction, the models, in general, consisted of one element every 10 ° in the lateral nozzle and one element every 15 ° in the main vessel. In order to obtain a smooth stress profile, six to seven elements were modeled through the acute crotch region. For the innermost elements around the crotch region (from 0 ° to 180°), a 3 × 3 × 3 lattice of integration points was used for the numerical (Gaussian) integration procedure. In order to optimize the computing cost. a wave-front reduction was implemented through element reordering. A variety of geometry plots was obtained for all models. This included typically: (1) regional plots representing the sequence in which the mesh was generated; (2) isometric view; (3) plot of the symmetry plane; and (4) sectional plots of the lateral connection, partly including the run pipe and lateral nozzle. For the sake of brevity, only plots of isometric view, acute and obtuse cross sections and the symmetry plane of Model 2 are

I/IIII Fig. 3. Acute corner rcgi~;n~,f Model 2

presented here (figs. 2 through 5). The salient features of the different finite element meshes are presented in table 3. This table includes the total number of elements used for each model, number of elements through the thickness of run pipe, nozzles and branch pipe, number of elements through acute and obtuse crotch regions, and the element density in the circumferential (0) direction of the run pipe and the branch. The table also identifies the regions for which the Gaussian point printout was obtained.

Fig. 2. Isometric view of Model 2.

P.P. Raju / Development o/stress indices

427

Table 3 Details of finite element mesh Model

Number of elements

description

Total

Model 1 Model 2 Model 3 Model 4

Acute

2553 2600 2135 2600

Gaussian printout

crotch

Obtuse crotch

Vessel wall

Nozzle wall

Pipe wall

Vessel '0" direction

Nozzle '0' direction

inner crotch

Pipe transition

7 7 6 7

3 4 4 5

3 3 3 3

3 4 4 5

2 2 2 2

12 12 12 15

12 12 12 12

X X X X

X X

In conclusion, each finite element mesh was considered adequately fine for the type of analysis performed.

7. Material

properties

Consistent sets of material properties were used for all the models. Accordingly, for the elastic analysis p e r f o r m e d and reported herein, a modulus of elasticity of 30 )< 106 psi and a Poisson's ratio of 0.3 were used for all load cases considered. [ I I

I 1 I

I I I

/

I I i

/ /

/

1 . f 7 / / / / / / / /*/

/

'

/ !

8. Loads X

Z

Of the possible 13 load cases (internal pressure, three external m o m e n t s and forces on b o t h the lateral b r a n c h

Fig. 4. Obtuse corner region of Model 2.

¥

I

I

I

l

i. . . i . . r

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]

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I

I

I

I

[

I

I

i ~ //."A';-;'- ~.

I

1

t

I

I

I

I

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o,~I ..::.-Z":/:

I

I

I

I

l

I

I

I

I

I

I

I

I I

t

I

I 1

Fig. 5. Symmetry. plane view of Model 2.

I

]

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i

i

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i

¢

¢

i

¢

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¢

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~Z

i ~/.

i Uy = 0

f

f

i

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i

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i

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tt

I

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@ o n e node

Fig. 6, Applied loads and boundar? conditions, pipe a n d the r u n pipe) only three i n d e p e n d e n t load cases, namely: (1) internal pressure; (2) external in-plane m o m e n t on the lateral b r a n c h pipe; and (3) in-plane m o m e n t on the r u n pipe were considered in the parametric evaluation. Fig. 6 depicts the load cases analyzed. 8.1. Internal pressure

A unit internal pressure of 1000 psi or an internal pressure which would result in a h o o p stress value of 1000 psi was applied o n all exposed internal surfaces of the lateral nozzle a n d the r u n pipe. The b o u n d a r y m e m b r a n e forces were applied as negative (or tensile) pressure at the u n c o n s t r a i n e d or free b o u n d a r y of the r u n pipe a n d at the free edge of the b r a n c h pipe. Since the stress ratio s / S was chosen to be 1.0 identical b o u n d a r y m e m b r a n e forces were applied at b o t h free ends described above. 8.2. External m o m e n t

As stated earlier, two in-plane m o m e n t cases were studied. In the first case, the b r a n c h pipe was subjected to a n in-plane m o m e n t a n d in the second case, an in-plane m o m e n t was applied on the r u n pipe. Stress indices were o b t a i n e d for each of these cases i n d e p e n dently. 8.2.1. In-plane m o m e n t on the branch pipe N o d a l forces were used to simulate the in-plane m o m e n t load on the b r a n c h pipe. F o r the three nodes specified t h r o u g h the thickness, the applied m o m e n t can be expressed as:

M = 0.25F,, cos Oi(r o cos Oil + 0 . 5 F o cos Oi(r m cos 0i) +0.25Fo cos Oi(r i cos O,), = (0.25For o + 0.5For~,1 + 0.25Fori) cos20i,

'i

M

/

J/

where 0, = (), 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, and 180 ° for a distribution of twelve elements in the b r a n c h pipe, r i = inside radius of b r a n c h pipe, r., = m e a n radius of b r a n c h pipe, p~, = outside radius of b r a n c h pipe, Fo = force s u m m e d t h r o u g h the wall. Once the force Fo is calculated from a b e a m theory distribution and the above expression, individual nodal forces can be calculated from a 1 : 2 : 1 distribution through the wall. Only one half of the calculated nodal forces was applied at the symmetry, nodes. In order to make a quick evaluation of the calculated stresses and associated stress indices, the applied mom e n t was chosen to provide a b e n d i n g stress of 1000 psi in a straight pipe. W i t h this procedure, it was relatively easy to obtain stress index at any location by simply multiplying the calculated stress at that location by 10 :~

Table 4 Details of applied loads Description

Model Model Model Model

1 2 3 4

Internal pressure (psi/

181.818 1000 48.78 1000

In-plane moment (in lbs) Lateral nozzle

Main vessel

858.6 2371.5 × 10 ~ a 1800×103 2371.5×10 a 278.55 544.0 × 10 ~ 68.0 × 10 "~ 544.0 X 1 0 3

This particular moment load evaluation was not performed by this author. Swanson Analysis Systems, Inc (SASI) under a separate contract from the PVRC analyzed and documented this case in a technical report which is on file at the Welding Research Council (WRC).

P.P. Rq[u / Development of stress indices 8.2.2. In-plane moment on the run pipe As a second moment case, an in-plane moment load was applied at the unconstrained edge of the run pipe. Again, nodal forces were used to simulate the moment. Table 4 provides the type and magnitude of the loads applied on each of the models analyzed.

9. Boundary conditions Consistent boundary constraints were chosen for all models to reflect half symmetry of the finite element mesh and the applied loads. Accordingly, the normal displacement was constrained along the symmetry plane for the entire model. The left end of the run pipe was constrained in the axial direction. In order to prevent rigid body displacement, one node on the symmetry plane at the left end of the run pipe was constrained in all directions. The schematics of the boundary conditions are illustrated in fig. 6.

10. Analysis procedure In view of the complexity and size of the model, every effort was made to optimize the computer cost. As a first step towards achieving this objective, a wave front optimization was carried out through user directed initial and. intermediate waves. Roughly, a 30% reduction in the RMS wave front was achieved through this procedure. Second, the stiffness matrix generated for internal pressure evaluation was saved to be reused for the evaluation of other load cases. This reduced the cost further by 20%.

429

Since enormous data was involved in setting up each load case, a data check run was first made to verify the input. Upon complete verification of the data, the analysis for internal pressure case was performed. The computed results were checked for consistency and accuracy in the key areas. In the case of internal pressure, hoop and axial stresses were checked to make sure that they attenuated to Lame' values away from the discontinuities. At the boundaries, the axial stress was compared with the applied end cap load. Displacement trend and magnitude were checked at key regions. Subsequently, post processing of the results was carried out to obtain nodal displacements and stresses based on extrapolated centroidal values. Distorted geometry and stress contour plots were also obtained in the process. For the moment cases, as part of the verification of the results, the calculated surface bending stress was compared with the applied stress M / Z in the vicinity of the loaded edge. Upon satisfactory comparison of the stress results, post processing of displacements and stresses was performed to obtain the aforementioned plots. The more accurate Gaussian point stresses and displacements were not factored into the post processing because of ANSYS limitations. Consequently, the post processed surface stresses were not used in determining stress indices or intensification factors.

11. Discussion of results Displacement and stress contour plots (% .... Omin and %~,) were obtained for the entire acute and obtuse regions using the three dimensional solid element post

SO0.90

Fig. 7. Max. shear stress plot. Obtuse region. Model 2. Internal pressure.

430

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b 1

Fig. 8. Max. shear stress plot. Acute region, Model 2. Internal pressure.

processor (POST 23) of ANSYS. These plots are intended to exhibit schematically the stress intensification in the critical regions and the deformation undergone by the same regions. In the displacement plots, the dashed lines represent the undeformed or original configuration and the solid lines indicate the deformed shape to an exaggerated scale. The distortion scale used to plot the deformed geometry is indicated on the individual plot. The maxim u m indicated displacement is equal to ½-inch in all cases.

Again, in order to provide a concise data package, only the maximum shear stress plot for the pressure loading and the maximum principal and shear stress plots for one of the in-plane moment loadings are included. These plots, presented in figs. 7 through 12 cover both acute and obtuse regions of Model 2, On these plots the maximum and m i n i m u m stress values

based on extrapolated centroidal values are indicated by the letters ' X ' and 'O', respectively. 1 l. 1. Surface stresses

Using an option available in the ANSYS program, stresses were obtained at the 3 × 3 × 3 Gaussian integration points for selected elements of the four models, as described earlier. Employing a local smoothing technique based on a bilinear cubic relation, the outermost corner (2 × 2 × 2 lattice) Gaussian point stresses were extrapolated to the element surfaces. It was observed that in view of the close proximity of the innermost Gaussian point to the surface, the calculated Gaussian point stresses did not differ more than ± 2 percentage points from the extrapolated Gaussian point stresses. The difference was even narrower when Gaussian point data were converted to surface nodal data

103 3D

,(

Fig. 9. Max. stress plot. Obtuse region. Model 2. Run moment.

P.P. Raju / Development of stress indices

//

431

152,3d

Y

k

Ad/// ,/'X /,

+

Fig. 10. Max. stress plot. Acute region. Model 2. Run moment.

through a weighted least square fit. As mentioned earlier, surface stresses obtained through extrapolated centroidal stresses were totally avoided in computing stress indices. These stresses could differ from the extrapolated or raw Gaussian point stresses by as much as +30%. Hence, great caution should be exercised in obtaining surface stresses from three dimensional analyses.

These regions are clearly identified in fig. 13. Of course, the entire configuration was screened to determine the absolute maximum index under each applied loading. Following the general practice adopted in the Code for reinforced openings and branch connections, the calculated pressure stresses were normalized against the membrane hoop stress corresponding to the unpenetrated run pipe. Per the Code: Membrane hoop stress = 0"o = P ( D + T ) / 2 T ,

12. Calculation of stress indices

Based upon the review and evaluation of published theoretical and experimental results on branch connections and engineering judgment, three critical regions, namely, acute and obtuse crotch regions and branch pipe to nozzle transition (Section A-A), were selected as a minimum for the determination of stress indices.

Y

where D and T are the inside diameter and nominal thickness, respectively, and P is the applied internal pressure. Stress indices for two components of stress (i.e., maximum principal stress 0"1, and stress intensity S~) defined below were computed for the designated regions. S1 ~

0`1 -

03 ,

g 0 . oo

Fig. 11. Max. shear stress plot. Obtuse region. Model 2. Run moment.

432

P.P. RWu /

De~,elopment ~)/ stre.s ~ Osag(c~

i". 3 ;3

---.... Fig. 12. Max. shear stress plot. Acute region. Model 2. Run moment. where a~, 0 2 and o) are the principal stress components with r, l _> 0 2 >_ 0 3. In the case of in-plane moment loading on the run pipe (MR) and the branch pipe (M~0, the calculated stresses were normalized against the bending stresses defined as follows: o R = MR/ZR,

the branch pipe would be located at the branch pipe to nozzle transition and in the case of the in-plane moment loading on the run pipe it would be difficult to locate priori the location of the maximum stress index.

13. Recommended A S M E stress indices for 45 ° lateral connections

o B = MB/ZB,

where o R, o B

= reference bending stresses on the run pipe and the branch pipe, respectively; Z R, Z n = section modulus of the run pipe and the branch pipe, respectively. Preliminary evaluation indicated that the maximum stress and stress index under internal pressure loading would be located at the inner acute comer, the maxim u m index under applied in-plane moment loading on

As stated earlier, the main objective of the PVRC long range program was to develop an adequate data base to propose revisions to the stress index method relative to lateral connections in cylinders. Although the data base developed is not extensive for reasons quoted earlier, a clear trend has been established within the parametric range considered to recommend a set of stress indices for 45 ° lateral connections under internal pressure and in-plane moment loadings. Consistent with the philosophy of the Code, a conservative number is chosen for the stress index to account for manufacturing tolerances and design deficiencies, if any. Accordingly, the following recommendation is offered for consideration by the respective Code committies. 13.1. ASME Code Section I I I Subsection NB, Par. N B 3338.2. Stress index method

Fig. 13. Critical regions of stress intensification.

Proposed changes to Par. NB-3338.2(c) (i) are presented in Appendix A and in the sections below. The opening is for a circular nozzle whose axis is normal to the vessel wall. If the axis of the nozzle makes an angle < 45 ° with the normal to the vessel wall and if d / D < 0.50, the stress indices of Table NB-3338.2(c)-2 may be used for internal pressure and in-plane moment loadings.

P.P. Raju / Development of stress i,ldices

In lieu of using the recommended indices, stress indices for a specific configuration may also be determined by theoretical or experimental stress analysis, as permitted in NB-3338.1(c). Such analysis shall be included in the design report. 13.2. A S M E Code Section IlL Subsection NB-3680

NB, Par.

Appendix B presents the recommendation with respect to the design of lateral branch connections in piping systems per NB-3680 of Section III of the ASME Code. The dimensional parameters are not to exceed the following: Ratio

Allowable value

D/ t d/D

40 0.5 3.0

a/ ~ffi

To minimize the effort needed to incorporate the proposed changes, the revisions are written in terms of the nomenclature of the specific paragraphs under consideration. In line with the recommendation above, the respective code committees may choose to introduce revisions to Article 4-6 of Section VIII, Division 2, and other relevant ASME Code sections and subsections which require determination of stresses in openings for fatigue evaluations. Considering the limited guidelines and procedures currently available in the Code for the design of non-radial connections in cylinders, the proposed revisions are expected to result in an improved design of lateral connections in vessels and piping systems and provide further impetus to employ non-radial connections.

14. Conclusions and recommendations

The limited benchmark studies performed and summarized in WRC Bulletin 301 [5] have yielded parametric data to draw quantitative conclusions with regard to the magnitude of stresses and stress indices in the critical locations of the 45 ° lateral connections. The proposed Code revisions are a direct result of the data generated through these carefully planned and executed studies. These studies have also demonstrated that current 3-D finite element modeling and analysis techniques, if used judiciously, are versatile and cost-effective enough

433

to use in the analysis of complex configurations such as oblique connections in shells. Other advantages of finite element technique in comparison with experimental techniques are that multiple load cases can be handled with the same finite element model and graphic depiction of stresses and displacements is easier to accomplish. Comparison of the current results with the available photoelastic results and limited scale model tests confirms the following conclusions: 1. For the internal pressure loading, the critical stresses occur at the acute inside corner. For a constant d / D ratio, the stress index value decreases as the diameter to thickness ratio increases. Also, for a constant diameter to thickness ratio, the stress index value increases with increasing diameter ratios. 2. For the in-plane moment loading on the branch pipe, the maximum stress index value increases with increasing diameter ratios and a constant diameter-tothickness-ratio. On the contrary, for the case of in-plane moment loading on the run pipe, the maximum stress index value decreases with increasing diameter ratios and constant diameter-to-thicknessratio. 3. For the in-plane moment loading on the run pipe, the maximum stress index occurs away from the symmetry plane. 4. Within the Code specified parametric range, it is not necessary anymore to determine the stress index (or stress concentration factor) for the radial nozzle configuration in order to calculate the stress index for the 45 ° lateral nozzle. The proposed stress indices can be directly applied to calculate peak stresses in critical areas. The recommended indices also cover in-plane moment loadings which have not been addressed in the ASME Code until now. These eliminate the need to apply a single stress index value, as currently stipulated in the Code, for all regions and stress categories. In many nuclear and petrochemical applications, the lateral branch connections fall in the realm of d / D = 1.0 and are subjected to a variety of external loads not considered in the current program. Hence, considerable additional data is required to broaden the range of applicability of the stress index method to lateral connections. To generate this data, the current program may have to be extended to include configurations with 0.5 _
P.P. Raju / Det,ehq)ment o~ ~rre~ m,&ev

434

stress indices for thermal transient loadings. This can best be accomplished through a long range plan under the auspices of the Pressure Vessel Research Committee

with the support and cooperation ol the energy iJ~dustry, utilities and other research and regulatory agencies.

Appendix A. Lateral nozzles in cylindrical shells (Table NB-3338.2(c)-2) Internal pressure loading Stress

o.... S

Acute crotch

Obtuse crotch

Nozzle to pipe transition

Inside

Outside

Inside

Outside

Inside

Outside

5.50 5.75

0.80 0.80

3.20 3.50

0.70 0.75

1.0 1.2

1.0 1.1

In-plane moment loading on the lateral nozzle Stress

Acute crotch

Obtuse crotch

Nozzle to pipe transition

Inside

Outside

Inside

Outside

Inside

Outside

o .... S

0.1 0.1

0.1 0.1

0.5 0.5

0.5 0.5

1.0 1.0

1.60 1.60

The stresses in the main vessel are inconsequential under in-plane moment loading on the lateral nozzle. Hence, vessel stresses should be related to load rather than to bending stresses in the attached pipe, such as is done in a Bijlaard type of analysis. In-plane moment loading on the vessel Nozzle to pipe transition

Stress

Acute crotch

Obtuse crotch

Inside

Outside

Inside

Outside

Inside

Outside

o.... S

2.40 2.70

2.40 2.70

0.6 0.7

1.8 2.0

0.2 0.3

0.2 0.3

The dimensional parameters are not to exceed the following: Ratio

Allowable value

D/t

40 0.5

J/D d/(Dt

3.0

Appendix B. Stress indices for use with equations in NB-3650

Piping products or joints

Internal pressure

In-plane branch moment

In-plane run moment

45 ° lateral connection

C1 K~

(C2 K2 ) B

(C2 K2 ) R

5.75

1.6

2.7

P.P. Raju / Development of stress indices

Nomenclature Unless otherwise stated, the basic n o m e n c l a t u r e used in this p a p e r will b e as follows: d inside diameter of lateral nozzle of b r a n c h pipe (in.), D inside diameter of cylindrical vessel or r u n pipe (in.), d i D diameter ratio, L1 distance in inches as shown in fig. 1, L2 distance in inches as shown in fig. 1, L3 distance in inches as shown in fig. 1, L, distance in inches as shown in fig. 1, MB applied in-plane m o m e n t o n lateral nozzle, MR applied in-plane m o m e n t o n cylindrical vessel or run pipe, 'O' location of m i n i m u m stress value o n c o n t o u r plot, P internal pressure (psi), rI corner radius (in.), r2 fillet radius (in.), r~ transition radius (in.), s n o m i n a l hoop stress due to pressure in lateral nozzle (psi), = P( d + t ) / 2 t , S n o m i n a l hoop stress due to pressure in cylindrical vessel or r u n pipe (psi), = P( D + T ) / 2 T , S~ n o m i n a l stress due to in-plane m o m e n t o n lateral nozzle or b r a n c h pipe (psi), SR n o m i n a l stress due to in-plane m o m e n t o n cylindrical vessel or run pipe (psi), SI stress intensity (psi), s/S stress ratio, t thickness of b r a n c h pipe (in.), gn thickness of lateral nozzle (in.), see fig. 1, T thickness of cylindrical shell or run pipe (in.), 'X' location of m a x i m u m stress value on c o n t o u r plot, 0 circumferential location of various sections, q, angle, the axis of the nozzle makes with the normal to vessel or r u n pipe wall, m a x i m u m principal stress (psi), ~o.x m i n i m u m principal stress (psi), O'mi n m a x i m u m shear stress (psi).

Acknowledgement The work reported herein was supported b y the Pressure Vessel Research C o m m i t t e of the Welding

435

Research Council. The permission granted by the Design Division to present this p a p e r at the P o s t - S M i R T Conference in Paris is gratefully acknowledged.

References [11 E.O. Waters, Theoretical stresses near a circular opening in a flat plate reinforced with a cylindrical outlet, Welding Research Council Bulletin Series 51 (June 1959). [2] D.E. Hardenbergh, Stresses in contoured openings of pressure vessels, Welding Research Council Bulletin Series 51 (June 1959). [3] C.E. Taylor, N.C. Lind and J.W. Schweiker, A three-dimensional photoelastic study of stresses around reinforced outlets in pressure vessels, Welding Research Council Bulletin Series 51 (June 1959). [4] J.L. Mershon, PVRC research on reinforcement of openings in pressure vessels, Welding Research Council Bulletin 77 (May 1962). [5] C.E. Taylor and N.C. Lind, (1) Photoelastic study of the stresses near openings in pressure vessels, (2) M.M. Leven, Photoelastic determination of the stresses in reinforced openings in pressure vessels, Welding Research Council Bulletin 133 (April 1966). [61 J.k. Mershon, Preliminary evaluation of PVRC photoelastic test data on reinforced openings in pressure vessels, Welding Research Council Bulletin 113 (April 1966). [7] J.L. Mershon (1) Interpretive report on oblique nozzle connections in pressure vessel heads and shells under internal pressure loading, (2) M.M. Leven, Photoelastic determination of the stresses on oblique openings in plates and shells, Welding Research Council Bulletin 153 (August 1970). [8] A.R.C. Markl and George, H.H. Fatigue tests on flanged assemblies, Trans. ASME 72 (1950). [9] AR.C. Markl, Fatigue tests of piping components. Trans. ASME 74 (1952). [10] A,C. Eringen, A.K. Naghdi and C.C. Thiel, State of stress in a circular cylindrical shell with a circular hole, Welding Research Council Bulletin 102 (January 1965). [11] ASME Boiler and Pressure Vessel Code, Section III, Rules for construction of nuclear power plant components, Division I Subsection NB. Class 1 Components, Section VIII, Division 2, Rules for construction of presure vessels, Alternative Rules, ASME, New York (July 1977). [12] G.J. DeSalvo and J.A. Swanson, ANSYS Engineering Analysis Systems User's Manual (Swanson Analysis Systems, Inc., Houston, PA, August 1, 1978).