Pipe Support Attachment Wrc Analysis

Proceedings of the ASME 2013 Pressure Vessels & Piping Division Conference PVP2013 July 14-18, 2013, Paris, France

PVP2013-97622 STRESS ANALYSIS ANALYSIS OF PIPE P IPE SUPPORT ATTACHMENTS: ATTACHMENTS: A COMPARISON COMPARISON OF ANALYTICAL ANALYTICAL METHODS AND FINITE ELEMENT ANALYSIS ANALYSIS FOR CIRCULAR AND NON-CIRCULAR ATTACHMENTS TTACHMENTS Anindya Bhattacharya Technical Head, H ead, Stress Analysis CB&I, 40 East Bourne Terrace, Terrace, London, W 2 6LG, 6LG, United Kingdom. Phone: +442070535668

ABSTRACT 4. Comparison of FEA, WRC 107, WRC297 and “Kellogg” methods w.r.t the following parameters: • Type of loading (Radial, Longitudinal, Circumferential) applied in a “ stand-alone manner” is absence of pressure • D T , t T and d D ratios • Combined loading including pressure • Different element types

Despite the availability availability of special purpose purpose FE codes with post processing facilities as per rules of ASME SEC VIII Division 2, use of simple analytical methods like ring loading around a circumference or more complex methods like Welding Research council bulletins 107 and 297, will continue to be used in the industry industry for a significant significant period of time for for stress analysis of pipe support attachments. The reasons are few: not all engineering companies have such custom made FE codes, lack of trained personnel to work with general purpose FE codes, ease of implementation of the available methods and their successful design history, cost and time issues with FE analysis etc. In this paper these available methods will be reviewed based on their theoretical background, their range of appli cability w.r.t w.r.t the typical t ypical design parameters and their comparison with FE analysis. More recent recent analytical methods methods based on mathematically mathematically accurate accurate space curves of intersections for circular attachments will also be discussed. This study will include both circular as well as non-circular attachments. This paper will highlight the strengths and weaknesses of the conventionally used methods especially especially with respect respect to their mathematical mathematical limitations limitations to make an analyst aware of the potential over conservatism and under conservatism of these analytical methods. Finite element analysis models will be discussed in detail specifically in relation to elements used, element parameters, boundary conditions and post processing.

NOMENCLATURE compone nents nts in the the ( ξ ,ϕ ) coordinate of the main ξ , ϕ - compo shell surface radii radii of the branch branch pipe pipe and main main shell r , R - mid surface oung’s modulus and Poisson Poisson ratio respectiv respectively ely E ,ν - Young’s radial displac displaceme ement nt un - radial diamete eterr ratio ratio = d ρ 0 - diam

D T - thickn thickness ess of main main shel shelll φ - Airy Airy stre stress ss func functi tion on - inte interna rnall press pressure ure cylindrical drical coordin coordinates ates in 3D space space ρ θ , z - global cylin surface force components components in the pξ , pϕ - surface

directions directions

verticall disp displac laceme ement nt w - vertica flexural ral rigidi rigidity ty of shel shelll = H - flexu

INTRODUCTION

ET 3 12( 1 − µ 2 )

loading g in vertica verticall directio direction n Z - loadin t - thickn thickness ess of of attache attached d shell shell ET K - foun founda datio tions ns stif stiffn fnes esss = 2 R direction directio n of longitudinal longitu dinal axis of of cylinder cylinder x section modul modulus us of the trunn trunnion ion pipe pipe S - section A - area of cross cross section section of trunnion trunnion pipe pipe

In this paper, the subject matter has been structured in the following manner: 1. Discussion of the available theoretical methods, from the simplest to the advanced. 2. Brief overview of basic shell mathematical model. 3. Brief overview of available finite element options.

1

Copyright © 2013 by ASME

2. AVAILABLE THEORETICAL APPROACHES TO THE PROBLEM PROBLEM OF ANALYSING ANALYSING A CYLINDRICAL CYLINDRICAL SHELL SHELL WITH CYLI CYLINDR NDRICAL ICAL OR NON-CY NON-CYLIN LINDRIC DRICAL AL ATTACHMENTS.

- resultant resultant applied applied bending bending moment moment on trunnion trunnion P - load per per unit of of circumferenc circumferencee (applied (applied as a ring load) radiuss of trunn trunnion ion r t - radiu

2.1 Approach 1:

F - forc forcee on trunn trunnio ion n attachment parameters parameters for rectangular rectangular attachments attachments = β 1 , β 2 - attachment c1

&

R c1 , c2 - half

This approach approach is popularly popularly known as “Kellogg “Kellogg”” method in the piping industry industry.. This approach approach has been so named as it appeared appeared for the first time in [4] and is based on ring loading loading around a circular cylinder.

c2 R dimensio dimensions ns

of

the

rectang rectangle le

circumferential and meridional respectively 2 Laplacian ian Operat Operator, or, ∇ .∇ ( ) ∇ - Laplac

along along

Governing Governing differential differential equation [1]:

directions

For an axi-symmetric loading on a circular cylinder, the governing differential equation is the well known beam on elastic foundation equation:

1. SHELL THEORIES: THEORIES:

H

There are various shell theories and each one has its own protagonist. Any shell theory has to be evaluated within the postulates of Sanders-Koiter’s approach [12, 21] which can be summarised as follows: 1. The equations can be written in general tensor form. 2. The deformations deformations are are described described by six strain measures measures,, three of which are components of the usual membrane strain tensor and the other three deviate from the components of the geometrical geometrical curvature curvature change change tensor only by terms that are bilinear in the components of the curvature and membrane strain tensor. 3. The stresses are described by six stress measures that satisfy the equations of equilibrium without approximation. 4. The theory has a principle principle of virtual work that that is exact for displacements obeying the Kirchoff hypothesis; hypothesis; in conjunction conjunction with approximate constitutive relations between the stress and strain measures. Well-set boundary value problems can be formulated, and the usual minimum and reciprocal relations of structural mechanics hold good. 5. The theory contains an exact static-geometric analogy. analogy. This analogy can be formulated by replacing the static quantities by corresponding geometrical quantities in homogeneous equations of equilibrium and the resulting equations become identical with the compatibility conditions. 6. When applied to the symmetrical bending of shells of revolution, the stress and strain measures agree with those generally generally used. used. They are consistent consistent with those those of the most simple curved beam theory. theory. For the present purpose, we will discuss the issue of cylindrical pipes with circular (referred to as trunnion) as well as non-circular (referred to as pipe shoes) attachments. Hence there is no “puncture” in the header pipe. The mathematical problem of the main shell with cut-out is a boundary value problem of partial differential equation. It means that the cylindrical shell equation, whose general solutions have many unknown constants, is suitable on the shell surface with or without cut-out. In order to determine the unknown constants the boundary boundary conditions conditions have to be used.

d 4w dx 4

d 4w dx

4

+

+ Kw = Z ETw HR

2

=

(1)

Z

(2)

H

Introducing β 4

=

3( 1 − µ 2 ) 2

2

R T

= , i.e. β =

1.28 RT

considering

µ = 0.3 . we therefore get

d 4 w dx

4

+ 4β 4 w = Z

(3)

H

The solution of this differential equation and boundary conditions are detailed in [1] Extending the above analysis to a case of bending of a cylindrical shell by a load uniformly distributed along a circular section [1], we get: P Maximum Bending Moment = , where P= load per unit 4 β length of circumference. Bending stress, σ bending

P = 1.17 1.5

R

(4)

T

P, can be defined in terms of a local radial load, P r and local moment, M r . This This is necess necessary ary because because P is a line line load load distributed around the circumference of the shell. If a load P r is divided by the attachment attachment perimeter it becomes becomes

P r 2π r t

for a nozzle of radius, r t . or a moment over section

modulus of the attachment becomes,

M r

π r t 2

.

Flexural stresses are added to membrane longitudinal and hoop stresses to get total stress = membrane stress in direction i + flexural stresses in direction i computed computed by the expression expression in eq-(4)

2

Copyright © 2013 by AS A SM E

To compute compute P, these were the steps followed: Computation of loads in longitudinal and circumferential directions directions by use of the following expressions: expressions:

•

longitudinal force = (longitudinal force x moment arm)/ π r t 2

•

circumferential force = (circumferential force x moment 2

arm)/ π r t

• • •

radial force = radial load/ 2π r t equivalent equivalent circumferentia circumferentiall force force = 2 x circ. circ. force + 1.5 x radial force equivalent longitudinal force = 1.5 x radial force + longitudinal force

The above forces are used as P in eq-(4) The reason behind the use of the factors 1.5 and 2.0 is attributed attributed to higher flexibilities flexibilities in these directions. directions. The flexural stresses in longitudinal and circumferential directions are then computed using these “equivalent” forces and the membrane pressure stresses are then added to compute the total stresses. Stress in the trunnion attachment is F M + . computed as A S

2.1.1. The case case of Pipe shoes: shoes: Schematic arrangements for some pipe shoes are shown in fig-(1). Dimension B stands stands for shoe/gusset shoe/gusset width, width, G = number of gussets, L = gusset spacing (this depends depends on the design design), ), S = number number of spines spines,, M = spine spac spacing ing,, and A= shoe length. The approach taken for analysis of pipe shoes is similar to that of trunnion trunnion type attachments. attachments. The computatio computations ns of section section properties (few examples) are cited. 1. Pipe shoe with no gusset: 3 longitudinal longitudinal moment moment of inertia = A

12 distance distance to centroid, longitudinally longitudinally = A moment of inertia, circumferential = A

distance to centroid = B

2

12

2

2. Pipe shoe shoe with 4 gussets: gussets: 3 longitudinal moment of inertia = A

distance distance to centroid centroid = A

12

2

moment of inertia, inertia, circumferentia circumferentiall = B distance distance to centroid centroid = B

+ 5 9 BL2 3

3

2

3


Fig-(2) [31, 25] 25]

Bijlaard applied the radial force system, qn instead of vertical force system, q z . When subjected to a radial force system the resultants not only include moments,

xb

or

yb

but also force, F yb . This This will will be stat statica ically lly equi equiva valen lentt to exte extern rnal al load load Z invo involv lvin ing g force force,, F zb , transverse bending

Fig-(1). Schem Schematic de design of of so some com common pipe shoe shoe typ types.

moment,

3. Saddle: 3 moment of inertia, longitudinal = A

distance to centroid = A

12

yb

statically statically equivalent equivalent to the the transvers transversee bending bending moment, moment, 3

6

+ AB

. The

In figfig-3(a 3(a), ), the linea linearly rly distri distribu buted ted force force sy system, stem, q z is

2

moment of inertia, circumferential = B distance distance to centroid centroid = B

and longitudinal longitudinal bending bending moment, moment,

figure below shows the transverse bending moment, M xb case.

2

+ BL 6

xb

2

xb

but in fig-3(b), the linearly distributed force system, qn is

2

statically statically equivalent equivalent to the transverse transverse bending bending moment,

2

xb

and force, F as in fig-3(d).

2.2 Appro Approach ach 2: 2: The The WRC-107 WRC-107 appr approac oach h based based on the work of of Bijlaard [34, 26, 2] Bijlaard derived a theoretical solution based on Timoshenko Timoshenko equation equationss [1, 26] for a cylindrical cylindrical shell on end supports under a force system, qn linearly distributed over a square region defined by ξ

≤ c , ϕ ≤ c ,

where c =

ρ 0

in 2 the developed developed surface. surface. In deriving the equation equationss it has been assumed that ε 0 = 0 (circumferential strain). The force/moment force/moment system system is shown in fig-(2) below

4


Fig-(3) [25]

= qn cos ϕ = qn 0

q z

ϕ0

x

= 2c ∫−ϕ

0

ϕ cos ϕ , q y ϕ 0

qz Rc sin ϕ d ϕ

= qn sin ϕ = qn 0 ϕ 0

= 2c 2 R ∫ −ϕ qn 0 0

ϕ sin ϕ ϕ 0

ϕ cos ϕ sin ϕ d ϕ ϕ 0

(6)

F y

ϕ0

= 2c ∫−ϕ

0

q yc d ϕ

ϕ 0

= 2c 2 ∫ −ϕ qn 0 0

ϕ sin ϕ d ϕ ϕ 0

•

(5)

(7)

Bijlaard used Double Fourier Series to represent loads and displacements. The limitations of Bijlaard’s approach are the d ≤ 0.3 D and the the usage usage of Double Double Fourier Fourier series method which may may not show converge convergence nce for certain boundary boundary conditions conditions [32, 26]. The limitation of d is due to the use of of radial force instead instead D of vertical force. This results in significant error outside aforementioned d limit. The stresses are computed in 8 D specific locations around the intersection. The maximum stresses need not be at these locations! Additionally stresses in Trunnion/Shoes cannot be computed. For rectangular attachments, the limitation is β 1 & β 2 < 0.5 .

Circumferential Moment ( taken from A.3.3.2 and Table A-4 of WRC 107) For the thin walled model, the measured circumferential and longitudinal stresses were both higher than the computed values. Modifications were then done to Bijlaard's original work for both longitudinal and circumferential stresses for the bending components (and for the circumferential stress, for the membrane component also) but for longitudinal stresses there was minimal requirement for correction of the membrane component. Correction factor used was around 2.7 for bending component of circumferential stress + 20-25% for membrane component and correction factor of 2.72 was used for bending component of longitudinal stress (no correction for membrane component). component). It is stated in WRC-107 that the modified curves curves may be more conservative than the original work.

•

Longitudinal Moment ( taken from A.3.3.2 and Table A-4 of WRC 107) For the thin walled model, the measured circumferential and longitudinal stresses were both higher than the computed values. Corrections were made to both the Membrane and Bending components. For the Longitudinal stress, no correction was required for the Membrane component. Higher modification was required for membrane component (30%) compared to bending component (18%) for circumferential stress. Correction factor used was 6.75 in bending component of longitudinal stress (no correction for membrane component)

It is to be noted that that WRC-107 WRC-107 is not only only based on Bijlaard’s theoretical work but also experimental works by Mehrson, Mehrson, Wichman Wichman and Hopper Hopper [34]. The bulletin bulletin shows a comparison comparison of the calculated and measured measured stresses stresses for both thick walled models and thin walled models. Following were the main main issues issues between between the Experimental Experimental works and Bijlaard’s work for thin shells.

•

Radial Load ( taken from A.3.3.4 and Table A-6 of WRC107) Results agreed well on the transverse axis but the theoretical results were conservative by factors as high as 2.0 on the longitudinal axis.

5


on the longitudinal axis. Results for transverse axis agreed well for cases that are restricted by

dm

Dm

Dm

T

Based on this line of reasoning in-order to extend the applicability of the thin shell theoretical solution for cylindrical cylindrical shells with cut-out, cut-out, Xue et al al adopted adopted Morley’s Morley’s equation [9], which has the same order of magnitude of

≤2

(

Theory Theory of thin thin ela elast stic ic she shells lls,, in whi which ch T/R< T/R<<1 <1 is is insignificant in magnitude is derived on the basis of LoveKirchhoff assumptions. A generally accepted fact is this

(

 4 2 ∂ 2  2 ∇ + ∇ − 4 i µ χ = P ( pξ , pϕ , p )  ∂ξ 2   

)

approach approach has an an error of of order of of magnitude magnitude O T . R When a solution is derived by omitting some terms, which has order of magnitude larger than O

(

T R

)

( )

O T R

(

)

T ) from R

Flügge’s equation [13]. This equation is quite simple and can be expressed in complex-valued displacement-stress function form (Lekkerkerk (Lekkerkerker er [15] and Steele Steele [3]) as follows: follows:

 4 ∂ 2  2 4 i µ χ = 0 ∇ −  ∂ξ 2    4µ 2

χ

=

12 (1 −ν 2 )

= un + i

hand side of eq-(11) is a load function dependent on the surface force components acting on the shell. The cylindrical thin shell equations derived by Goldenveizer, Morley, Simmonds and Timoshenko (which was used by Bijlaard) have the same inherent error in order of

for shallow shallow shell equations equations), ), the accuracy accuracy of the

(omitting terms of order of magnitude O

4 µ 2 ETR

(8)

R

(9)

T

φ

(10)

(11)

where, χ and µ are the same as in eq-(10) and (9). The right

(such as

solution is bound to be lower. The detailed analysis of the above-mentioned concept can be found in well-known literature literature and textbooks of thin shell shell theory [6]. The “exact” equations for thin elastic cylindrical shells are very complicated. complicated. For a problem of cylindrical cylindrical shell with cutout out [25], [25], Donn Donnel elll [8] prese present nted ed an app approx roxim imat atee equ equat ation ion

)

accuracy accuracy as the general thin shell theory, theory, i.e. O T , instead R of Donnell’s. Donnell’s. Morley’s Morley’s equation equation is expressed expressed in complexcomplexvalued form by Simmonds [11] as follows: follows:

2.3 2.3 Appr Approa oach ch 3: Post Post WRC-1 WRC-107 07 appro approac ache hes s – WRCWRC297 and works of Morley, Simmonds and Hwang et al. [5, 33, 33, 11, 11, 16, 25]

(

)

magnitude O T . The solution has the order of accuracy R

(

)

O T . WRC-297, which is based on Steele’s work on R shallow shell equations covers a range of only

( )

r ≤ sin π < 0.5 . R 6 For detailed analys analysis is of the approach taken by Xue et al refer [17, 31]. In essence, the approach taken is to use compatibility conditions enforced on the geometrically correct curve of intersection as opposed to an assumed curve of intersection and using theories which are of order O T whic which h may or R may not involve using different different shell theories theories for intersecting cylinders. To summarize, summarize, different different cylindrical cylindrical shell equations equations are suitable to different ranges of the developed surface. Fig-(4) below shows the different ranges of developed surface [25].

(

)

where, u n , is radial displacement and φ , Airy stress function function.. Eq-(8) can be decomposed into two second-order partial differential equations and is easy to solve in polar coordinate system for the problem of cylindrical shell with cut-out [25]. However, However, as pointed pointed out by Koiter, Koiter, eq-(8) can only only be applied for shallow shallow shells. shells. Koiter Koiter [7] had had written, written, “It has has been been noted noted [9,14] that Donnell’s approximation is sometimes inaccurate” and “the generalization of Donnell’s approximation is applicable in the case of shallow shells in which the wave length L of the deformatio deformation n pattern pattern on the middle surface is always small compared with the minimum principal radius of curvature curvature R”. Based on fig-6.14 fig-6.14 in Donnell’s Donnell’s book [10], the the applicable range of shallow shell equation for the problem of

( )

cylindrical shell with opening is only r ≤ sin π = 0. 5 . R 6 The edge effects of general cylindrical shells and shallow shells mathematically mathematically differ. differ.

6


methods like FEM as currently an engineer in an industry does not have an easy tool to compare the FE results against some published benchmarks for d >0.5. In other words, as long as D we do not have analytical tools which are easily implementable and which will address the problems to be analyzed without having significant restrictions on geometry and loading conditions, FE analysis should be the preferred tool for analysis. The objective of this paper is to make an analyst aware of the potential over conservatism and under conservatism conservatism in the available available and and widely widely used methods methods if an analyst is constrained to use them.

4.0 FINITE ELEMENT ANALYSIS ANALYSIS APPROACH TO THE PROBLEM [33] provides an excellent discussion on the issues involving involving conflicts between between shell shell theory and finite element element analysis of shells. To To briefly summarize them:

Fig-(4) [25]

•

Ill conditioning due to significantly different different strain energies between membrane and bending modes.

•

Use of low degree polynomial trial functions in the displacement displacement finite element method generally generally leads to overstiffness in the response to bending actions.

•

Difficulty Difficulty in deriving trial functions for in-extensiona in-extensionall bending.

Donnel [10] showed that his shallow shell equations could be suitable to the range of

− π 6 < ϕ < π 6

i.e. ρ 0

< 0.5 .

In

[15] Lekkerkerker showed that the shallow shell equations could be applied to the range of ρ 0 ≤ 0.25 . The different

Many authors [33, [33, 18, 30] have have recommende recommended d use of hybrid elements. In this paper, however, we have used only displacement based finite element method. Finite elements available for shell analysis can be broadly classified into the following groups:

applicable ranges adopted by different authors are dependant on different allowable intrinsic errors.

3.0 DIFFICULTIES IN IMPLEMENTATION IMPLEMENTATION OF ANALYTICAL ANALYTICAL SOLUTIONS: SO LUTIONS:

1. 2. 3.

Analytical solutions (rather analytical solutions backed by experimental findings like WRC 107/297 methods) are extremely useful in addressing stress stress analysis issues of pipe support attachments as they are available in almost all commercial pipe stress codes and methods like “Kellogg method” can easily be developed into spreadsheets. The difficulty is of course course the limited range of applicability of

Dege Degene nera rated ted solid solid elemen elements ts.. Element Elementss based based on on basic basic shell shell mathe mathemati matical cal mode model. l. Elemen Elements ts based based on on combin combinatio ation n of plate plate and and membran membranee elements.

For a detailed discussion on type-1, refer [19]. The main feature of these elements is the number and variety of adhoc assumptions made to accommodate the standard procedures of finite element formulation. The variation of strain through thickness isn’t linear. Assumptions regarding dependence of determinant of Jacobian Matrix in the direction of thickness can lead to violation of rigid body properties properties [19]. Type-2 Type-2 elements are usually not available in commercial commercial FE codes. codes. They suffer from from rigid body motion problems problems [18]. The element S8R is ABAQUS is however close to these elements as discussed in [18]. To explain the meaning of the term Basic Shell Mathematical mode, we briefly describe the derivation of the governing shell equations using the tensor approach which involves the following steps [22]:

these methods specially in relation to d ratio and for the D Kellog method , its main drawback is its mathematical oversimplification of a problem, an issue which is not negligible negligible when the predominant predominant form of the loading is Radial. More advanced approaches as explained in section 2.3 of this paper have solved the problem up to d =0.8, but D these methods are yet not available in commercial pipe stress codes or as WRC bulletins and it will be a while before they will be available as handy tools for engineering applications. Such methods can be used to validate numerical analysis

7


Fundamental assumption of the shell theory based on LoveKirchoff hypothesis and zero strain in the through thickness direction. Expressing the base vectors of a surface located “off middle surface” i.e. a general surface in terms of the base vectors of the middle surface (both covariant and contra-variant versions).

•

Expressing the metric tensor of a surface located “off middle surface” in terms of the metric tensor (both covariant and contra-variant versions) of the middle surface.

• •

Expressing Expressing the rotation vector. vector.

•

• •

STRI3 - Small Strain Strain Triangular Triangular Element Element with 3 nodes nodes and quadratic variation of rotation (accurate representation of plate bending because of linear curvature variation) and analytical implementation implementation of Kirchoff Kirchoff constraint constraint at locations locations (DKT or Discrete Kirchoff element). STRI65 - Small Strain Triangul Triangular ar Element with 6 nodes nodes and Kirchoff Kirchoff constraint imposed numerically numerically at points.

8-node reduced reduced integration integration element element for for small strain strain S8R - 8-node formulation. formulation. This element element has similarity with the Basic Shell mathematical mathematical model as described described in [18], [18], although they they are not the same, the main difference being the use of Mindlin hypothesis. This element is susceptible to element distortion.

Expressing the Cristoffel symbols and permutation tensors (Levi-Cevita tensors) of the surface located “off “off the middle surface” in terms of the corresponding tensors of the middle surface. Expressing the strain tensors of a surface located “off middle surface” in terms of the strain tensor of the middle surface (both covariant and contra-variant versions). Strain tensors are expressed as the difference between metric tensors and curvature tensors in the deformed and un-deformed states Writing expression for stress and moment resultants. Using appropriate constitutive relations.

In the Basic Shell Mathematical model version of Finite Element implementation, the interpolation of the shell geometry is accomplished using the Iso-parametric procedure. Covariant and Contra-variant base vectors of the interpolated surface are computed using the usual finite element interpolation procedures and the First Fundamental form, the Second Fundamental form and the Christoffel symbols are then computed from these base vectors. In the Type-2 elements as described in [18], the normal vector is calculated normal to the interpolated middle surface, although the normal vectors vectors at the nodal points are are exactly normal to the middle surface. For a discussion discussion on type-3 type-3 elements elements any standard standard text book on FEM can be referred [19]. The FE code used for the analysis is ABAQUS ver. 6.9-1. The ABAQUS element library [20] for shells is divided into three categories consisting of general-purpose, thin, and thick shell elements. Thin shell elements provide solutions to shell problems that are adequately described by classical (Kirchhoff) shell theory; thick shell elements yield solutions for structures that are best modeled by shear flexible (Mindlin) shell theory; and general purpose shell elements can provide solutions to both thin and thick shell problems. All these elements use bending strain measures that are approximations of those of Koiter-Sanders version of shell theory [12]. For stress analysis, the following elements from ABAQUS library library have been been used.

The Hexagonal element used is a 20-node reduced integration element. The method of analysis is Linear Elastic following the Elastic Stress Classification Route of [28]. The issue of classification of the FE computed stresses on the lines of [28] has has been dealt with with in numerous numerous papers papers and will not be repeate repeated d here. In a nutshell, nutshell, local local membrane membrane stresses are designated as Pl , primary primary + secondary secondary stresses stresses as Pl + Pb + Q and peak stresses as Pl + Pb + Q + F in line with [28]. Primary stresses stresses develop to maintain maintain equilibrium equilibrium with external loads, secondary stresses to maintain compatibility of deformation (global) and peak stresses to maintain compati compatibili bility ty of local local deforma deformation tion.. Pl stands for local primary stress, Pb for primary primary bending bending stress, stress, Q for second secondary ary stress stress and F for peak stress. Peak stresses are significant significant only from the standpoint standpoint of fatigue failure. failure. FE convergence theorems are in L2 or H 1 norms which are difficult to implement when the exact solution is not shown and in this presentation no attempt has been made to evaluate the convergence using these norms. For checking the convergence of an FE model percentage change in stress is considered considered from a model model with very fine mesh to gradually gradually becoming cruder. Stresses are checked at Gauss points for accuracy accuracy and un-avera un-averaged. ged. For conver convergence gence,, monotonic monotonic behavior is checked with a maximum m aximum permissible variation in stress taken as 5%.The mesh size around the intersection is taken as 0.3 rt with progressive mesh grading away from it. For continuum four elements have been used through the thickness at and close to intersections. The objective of the FE analysis wasn’t to catch the peak stresses which are used for fatigue evaluation, because once the Pl + Pb + Q stresses are computed, computed, the fatigue stresses stresses can easily be computed computed using Fatigue Strength Reduction Factors (FSRF) [28]. The results of the analysis can then be extended to compute Pl + Pb + Q + F in a straightforward manner. [27] shows that modeling of welds to properly simulate joint stiffness does not have serious impact on the computed stresses and hence, welds are not part of the models. FSRF can be avoided if Dong’s method [28] is used. However, this requires special post processing ability of the FE Code. If welds are modeled,

8


Pl + Pb + Q can be evaluated at the weld toe directly by (even though it is a singularity) linearization at the stress classification classification line line (SCL) as explained explained by Kalnins [29]. [29]. The only issue with this procedure procedure is the through thickness thickness stress component. To avoid end effect, the location of the trunnion has been taken as 5D [24] i.e. five times the Outside Diameter of the Header Pipe with respect to the end of the header. The worst aspect ratio around the intersection (HEX elements) was 6.0, average aspect ratio 2.0. One end of the header was fixed in all six DOFs and the other end is fixed in five DOF’s. The DOF along the longitudinal axis of the header was kept free to generate longitudinal pressure stress (for models where pressure was applied). Linear and full integration elements were not selected in the quadrilateral and brick versions to avoid shear locking. References [35, 24] provide excellent guideline on modeling of Large Diameter Cylinder intersections.

5.0 RESUL RESULTS TS The stresses shown in the tables below belong to the 2 Pb + Pl + Q category category and are in N/mm . Only maximum Von

Table-1 (Contd) (Contd) Loading Type

Circumferential force

FEA shell element (STRI3) Cylinder

8

6

3

FEA shell element (STRI3) Trunnion

13

4

5


9

6

3


13

4

5

FEA continuum continuum element Cylinder

9

5

3

FEA continuum continuum element Trunnion

12

3

6

36 inch header header,, 30 inch trunnion, trunnion, wall thickness thickness = 9.52 mm for both. Magnitude of Force = 10KN, length of trunnion = 100 mm, d = 0.84 , t = 1 D T Loading Type

Table-1 30 inch header, header, 24 inch trunnion, wall thickness = 9.52 mm for both. Magnitude of Force = 10KN, length of trunnion = 100 mm, d = 0.8 , t = 1 : D T Radial Force

Longitudinal force

Table-2

Mises equivalent stress values are shown. For continuum elements, stresses have been Linearized using [28] as a guideline. For tables 1-5 the applied loadings are at the end of the Trunnion which makes it a Shear Force + Bending Moment at the Shell-Nozzle interface for the Longitudinal and Circumferential Force applications.. Pressure is not a part of the loadings in Tables 1-5. For WRC-107 and WRC-297 computations, code FE-107 has been used.

Loading Type

Radial Force

Radial Force

Longitudinal force


WRC 107 Cylinder

45

2

12

WRC 107 Trunnion

NA

NA

NA

WRC 297 Cylinder

51

5

16

WRC 297 Trunnion

56

4

16

Kellogg Cylinder

6

2

4

Kellogg Trunnion

0.5

0.2

0.2

Longitudinal force


FEA shell element (S8R) Cylinder

21

7

5

WRC 107 Cylinder

45

3

16

FEA shell element (S8R) Trunnion

15

5

4

WRC 107 Trunnion

NA

NA

NA


17

6

5

WRC 297 Cylinder

50

6

22


11

4

3

WRC 297 Trunnion

54

6

20


20

7

5

Kellogg Cylinder

6

3

5


14

4

3

Kellogg Trunnion

0 .6

0.4

0.4


19

6

4


10

6

3


15

5

6


15

5

6

9


Table 3

Table 4 (Contd.) (Contd.)

36 inch header, header, 12 inch trunnion, trunnion, and wall thickness thickness = 9.52 mm for header; header; and 6.35 mm for trunnion. trunnion. Magnitude Magnitude of Force = 10KN 10KN,, length length of of trunn trunnio ion n =100 =100 mm, mm, d = 0.34 , D t = 0.67 : T Loading Type

Radial Force

Radial Force

Longitudinal force


WRC 297 Trunnion

90

13

44

Kellogg Cylinder

7

4

3

Loading Type

Longitudinal force


Kellogg Trunnion

1

1

1

WRC 107 Cylinder

48

10

31


19

10

6

WRC 107 Trunnion

NA

NA

NA


20

9

7

WRC 297 Cylinder

54

30

41


17

8

5

WRC 297 Trunnion

103

30

75


19

7

6

Kellogg Cylinder

15

11

22


19

10

6

Kellogg Trunnion

2

2

2


20

7

7


46

16

29


17

11

6


48

16

31


19

8

6


42

14

26


43

15

27


45

16

28


47

15

30


44

13

27

Loading Type


46

14

29

WRC 107 Cylinder

Table 5 24 inch header, header, 8 inch trunnion, and wall thickness = 9.52 mm for header header and 8.18 8.18 mm mm for trunnion. trunnion. Magnitude Magnitude of of Force Force = 10KN, length length of trunnion = 100 mm, d = 0.36 , t = 0.86 : D T

Table 4 24 inch header, header, 20 inch trunnion, trunnion, and wall thickness thickness = 9.52 mm for header header and 6.35 6.35 mm for trunnion. trunnion. Magnitude Magnitude of Force Force = 10KN 10KN,, len lengt gth h of of tru trunn nnio ion n = 100 100 mm, mm, d = 0.84 , D t = 0.67 : T

Radial Force

Longitudinal force


47

21

53

WRC 107 Trunnion

NA

NA

NA

WRC 297 Cylinder

69

31

77

WRC 297 Trunnion

74

34

78

Kellogg Cylinder

16

20

40

Kellogg Trunnion

2

4

4

Radial Force

Longitudinal force



48

26

46

WRC 107 Cylinder

44

5

20


43

21

43

WRC 107 Trunnion

NA

NA

NA


44

22

40

WRC 297 Cylinder

44

7

23


38

19

36

Loading Type

10

Copyright © 2013 by AS A SME

Table 5 (Contd) (Contd)

Table 6 (Contd) (Contd)

24 inch header, 8 inch trunnion, and wall thickness = 9.52 mm for header header and 8.18 8.18 mm for trunnion. trunnion. Magnitude Magnitude of Force = 10KN, length length of trunnion trunnion = 100 mm, d = 0.36 , t = 0.86 : D T Radial Force

Loading Type

Longitudinal force



44

22

40


39

19

37


46

24

44


41

19

43

Table 6

e c r o F l a i d a R

) l a e i n c d r o t u F i g r a n e o h L S (

WRC 107 Cylinder

48

4

4

WRC 107 Trunnion

NA

NA

NA

WRC 297 Cylinder

54

4

4

13

153

413

WRC 297 Trunnion

103

6

6

19

295

752


46


48


46

6

9

21

106

359


46

7

8

24

103

403

7

10

9

l a n i d t u t i n e g m n o o L m

l a i t n e r e t f m n u e c m r i o C m

13

99

310

NA

NA

NA

21

23

47

6

10


45

6

FEA continuum element Cylinder

44

FEA continuum element Trunnion

47

l a n i d t u t i n g e m n o o L m

l a i t n e r e t f m n u e c m r o i C m

20

106

361

9

23

104

402

6

5

18

106

360

7

9

20

104

398

For the WRC-107 analysis, pressure loading has NOT been added as a radial load at the trunnion attachment.

Table 7

) l a i t n e e r c e r o f m F u r c a r e i h C S (

6


t n e m o M l a n o i s r o T

For tables, 7-9, applied load in longitudinal, circumferential and radial directions = 10KN (applied together), pressure = 18.9Barg.

36 inch header, header, 12 inch inch trunnion, trunnion, wall thickne thicknesses sses 9.52mm and 6.35mm 6.35mm for header header and trunnion trunnion respectively respectively.. Loads applied at shell nozzle interface, Moment=10KN-m and Force=10KN. d = 0.34 , t = 0.67 D T

e p y T g n i d a o L

) l a e i n c r d o t u F i g r a n e o h L S (

e p y T g n i d a o L

Table 6 is to reflect the effect of applying the Forces and moments at the Shell-Nozzle Interface as opposed to at the end of the Trunnion Trunnion in Tables Tables 1-5. Pressure is not a part of the Loading.

t n e m o M l a n o i s r o T

e c r o F l a i d a R

) l a i t n e e r c e r o f m F u r c a r e i h C S (

108

105

30 inch header, 24 inch trunnion, wall thickness =9.52 mm for both (results shown for maximum Pb + Pl + Q in MPa)

363

401

11

WR C 107 C yl yli nd nder

258

WRC 10 107 Tr Trunnion

NA

Kellogg Cylinder

87

Kellogg Trunnion

1

FEAshe FEAshell ll ele eleme ment nt (S8 (S8R) R) Cyl Cylin inde derr

121 121

FEAshe FEAshell ll elem elemen entt (S8R (S8R)) Trun Trunni nion on

63

FEAshe FEAshell ll elem elemen entt (STR (STRI3 I3)) Cyli Cylind nder er

120 120

FEAshe FEAshell ll elem elemen entt (ST (STRI RI3) 3) Trun Trunni nion on

59

FEA FEA shel shelll eleme element nt (STRI (STRI65 65)) Cylin Cylinde derr

125 125

FEAshe FEAshell ll elem elemen entt (ST (STRI RI65 65)) Tru Trunn nnio ion n

66

FEAconti FEAcontinuu nuum m elem element ent (Shel (Shell) l) Cylin Cylinde derr

126 126

FEA FEA cont contin inuu uum m elem elemen entt Trun Trunni nion on

70


Table 8

Table 10

36 inch header, 30 inch trunnion, wall thickness =9.52 mm for both (results shown for maximum Pb + Pl + Q in MPa) WRC 107 Cyli nd nder

307


NA

Kell og ogg Cy Cyl in inder

100

Kellogg Trunni on on

0.8

FEAshe FEAshell ll ele eleme ment nt (S8R (S8R)) Cyli Cylind nder er

146 146

FEAshe FEAshell ll elem elemen entt (S8R (S8R)) Trun Trunni nion on

75


143 143

FEAshe FEAshell ll elem elemen entt (ST (STRI RI3) 3) Trun Trunni nion on

36” pipe, wall thickness = 9.52 mm. Shoe design corresponds to 3-gusset, 3-gusset, A=450, A=450, B=500, shoe plate thickness thickness = 10 mm, L = 350 mm (refer fig-1); magnitude of load = 40KN. Pressure is not applied. β 1 = 0.56 , β 2 = 0.49 Radial Force

Longitudinal Force

Circumferential Force

WRC 107 Cylinder

181

18

61

Kellogg Cylinder

30

18

40

Ke llogg Shoe

2

9

3

76

FEA Shell element element (S8R) Cylinder

75

20

36

FEA FEA shel shelll elem elemen entt (STRI (STRI65 65)) Cylind Cylinder er

145 145

FEA Shell ele ment ( S8R) Shoe

77

30

32

FEA FEA shel shelll ele eleme ment nt (STR (STRI6 I65) 5) Trun Trunni nion on

76

82

18

35

FEAconti FEAcontinuu nuum m elem element ent (She (Shell ll)) Cylin Cylinde derr

148 148

FEA Shell element (STRI3) (STRI3) Cylinder

FEAcon FEAconti tinu nuum um elem elemen entt Trun Trunni nion on

74

FEA Shell element (STRI3) (STRI3) Shoe

82

18

35

Table 9


75

22

37

36 inch header, 12 inch trunnion, wall thickness = 9.52 mm for header and 6.35 mm for Trunnion (results shown for maximum Pb + Pl + Q in Mpa)


80

30

33

FEA Continuum Continuum element element Cylinder

81

24

33

FEA Continuum element Shoe

78

35

32

Loading Type

WRC 107 Cyli nd nder

321


NA

Kell og ogg Cy Cyl in inder

100

Kellogg Trunni on on

0.8


157 157

FEAshe FEAshell ll ele eleme ment nt (S8 (S8R) R) Tru Trunn nnio ion n

108 108


155 155

FEA FEA shel shelll elem elemen entt (STR (STRI3 I3)) Trunn Trunnio ion n

102 102

Loading Type

FEA FEA shel shelll elem elemen entt (STRI (STRI65 65)) Cylind Cylinder er

159 159

FEA FEA shel shelll ele eleme ment nt (STR (STRI6 I65) 5) Trun Trunni nion on

98

FEAconti FEAcontinuu nuum m elem element ent (She (Shell ll)) Cylin Cylinde derr

154 154

FEAcon FEAconti tinu nuum um ele eleme ment nt Tru Trunn nnio ion n

103 103

Table 11 30” Pipe, wall thickness 9.52 mm, Shoe design corresponds to 3 Gusset, A=450, B=500, Shoe plate thickness=10 mm, L=350 mm (refer fig-1), Magnitude of load=40KN. Pressure is not applied. β 1 = 0.67 , β 2 = 0.60

Results Results for Pipe Pipe Shoes: (Stres (Stresses ses at locations locations of singularities singularities have not been considered) considered) Note: WRC-107 method has been used even though in most cases β 1 , β 2 are above the allowable limit. So far Pipe Shoes are concerned, the typically used dimensions render them unsuitable for use of WRC-107. Despite this fact, the author in his experience has seen its usage for computation of local stresses stresses at Shoe Attachments Attachments and its use use is mostly mostly due to availability availability of this module in in most common common pipe stress stress programmes. For the WRC 107 computation of Pipe Shoes, the geometry geometry of the attachment attachment has been conside considered red as Rectangular solid. Pipe Stress Program CAESAR II Version 5.2 has been used for this purpose. For Tables 10, 11 and 12, 2c1=500 mm and 2c2=450 mm.

12

Radial Force

Longitudinal Force


WRC 107 Cylinder

173

21

63

Kellogg Cylinder

27

17

38

Ke llogg Shoe

2

9

3


60

15

18

FEA Shell ele ment ( S8R) Shoe

80

22

12


62

14

18


75

22

13


60

14

20


82

22

14

FEA Continuum Continuum element element Cylinder

63

18

25

FEA Continuum element Shoe

82

20

16


Table 12

Table 14 (results shown for maximum Pb + Pl + Q in MPa)

24” pipe, wall thickness = 9.52 mm. Shoe design corresponds to 3-gusset, A=450, B=500, shoe plate thickness = 10 mm, L = 350 mm (refer fig-1); magnitude of load = 40KN. Pressure is not applied. β 1 = 0.84 , β 2 = 0.75

30” header, wall thickness = 9.52 mm WR C 107 Cyl in inder

312

WRC 107 Shoe

NA

Kel lo logg C yl ylinder

161

Kellogg Shoe

12

Radial Force

Longitudinal Force


WRC 107 Cylinder

174

23

65


126 126

Kellogg Cylinder

24

15

34

FEAshe FEAshell ll elem elemen entt (S8 (S8R) R) Shoe Shoe

115 115

Ke llogg Shoe

2

9

3

FEA FEA shel shelll elem elemen entt (STR (STRI3 I3)) Cyli Cylind nder er

131 131


35

12

22

FEAshe FEAshell ll elem elemen entt (ST (STRI RI3) 3) Shoe Shoe

118 118

FEAshel FEAshelll eleme element nt (STRI (STRI65 65)) Cylin Cylinde derr

128 128

FEA Shell element ( S8R) Shoe

73

22

10 FEAshe FEAshell ll elem elemen entt (ST (STRI RI65 65)) Sho Shoee

113 113


34

12

22

FEA FEA conti continuu nuum m ele eleme ment nt (Sh (Shell ell)) Cylin Cylinde derr

132 132


50

22

10

FEA FEA cont contiinuum nuum elem elemen entt Shoe Shoe

119 119


35

13

22


53

17

10

FEA Continuum Continuum element Cylinder

39

15

FEA Continuum ele ment Shoe

57

21

Loading Type

Table 15 (results shown for maximum Pb + Pl + Q in MPa) 24” header, wall thickness = 9.52 mm WR C 107 Cyl in inder

298

24

WRC 107 Shoe

NA

13

Kel lo logg C yl ylinder

136

Kellogg Shoe

12

FEAshe FEAshell ll elem elemen entt (S8R (S8R)) Cyli Cylind nder er

80

FE A s he he ll ll el ele me me nt nt (S (S 8R 8R ) S ho hoe

85

FEA FEA shel shelll ele eleme ment nt (STR (STRI3 I3)) Cyl Cylin inde derr

84

F EA EA s he he ll ll el el em em een n t ( ST ST RI RI 3) 3) Sh Sh oe oe

89

FEAshe FEAshell ll elem elemen entt (ST (STRI RI65 65)) Cyl Cylin inde derr

82

FEAshe FEAshell ll elem elemen entt (STR (STRI6 I65) 5) Shoe Shoe

83

For Tables 13-15, applied load in longitudinal, circumferential and radial directions = 40KN( applied together), pressure = 18.9 barg. barg. Pressure has been applied applied but not as radial thrust thrust load.

Table 13 (results shown for maximum Pb + Pl + Q in MPa) 36” header, wall thickness = 9.52 mm WR C 107 C yl ylinder

330

FEAcon FEAconti tinu nuum um elem elemen entt (She (Shell ll)) Cylin Cylinde derr

88

WRC 107 Shoe

NA

F EA EA c o nt nt in in uu uu m e le le me me nt nt Sh Sh oe oe

92

K el ellogg Cy Cyli nd nder

186

Ke llogg Shoe

12

FEAshe FEAshell ll ele eleme ment nt (S8 (S8R) R) Cyl Cylin inde derr

180 180

FEAshe FEAshell ll elem elemen entt (S8 (S8R) R) Shoe Shoe

155 155

FEA FEA shel shelll elem elemen entt (STR (STRI3 I3)) Cyli Cylind nder er

184 184

FEAshe FEAshell ll elem elemen entt (ST (STRI RI3) 3) Shoe Shoe

156 156

FEAshel FEAshelll eleme element nt (STRI (STRI65 65)) Cylin Cylinde derr

182 182

FEAshe FEAshell ll elem elemen entt (ST (STRI RI65 65)) Sho Shoee

153 153

FEAconti FEAcontinu nuum um elem elemen entt (She (Shell ll)) Cylin Cylinde derr

188 188

FEAcon FEAconti tinu nuum um elem elemen entt Shoe Shoe

159 159

13


6.0 DISCUSSION OF RESULTS AND SCOPE FOR FUTURE WORK Tables 1,2,4 1,2,4 show that WRC 107 and WRC 297 results results show significant differences with respect to FE results for the radial load case. This is because of the high d ratio and D radial as opposed to vertical load representation of the same in WRC 107 as explained in section 2.2 of this paper. Tables 3, 5 and 6 show that the results are comparable (even for the Radial load case) case) indicating indicating the criticality of the d factor in D WRC 107/297 107/297 approac approaches. hes. For For the Kellogg Kellogg Method, the significant difference is for the radial load case. This is because of t he basis of the method being axi-symmetri cal ring loading which significantly deviates from the actual mathem mathematic atical al model model in the radial radial load load situat situation. ion. The The Kellogg Kellogg method also underestimates the stresses in the Trunnion. This is due to the use of simple simple beam theory theory as opposed opposed to shell theory theory and the non-co non-consi nsidera deration tion of the compat compatibil ibility ity requirement between the header pipe and the Trunnion in this method. Kellogg Kellogg method also in most most (but not not all) cases cases predicts lower magnitude of stresses in the Longitudinal and Circumferential Force applications. However the allowable stresses stresses in the the Kellogg Kellogg method as as long as as they are specified specified as the [28] allowable allowable for local local primary stress, the the error will not in general make the analysis non-conservative except for the Radial Load scenario. For Tables 7, 8 and 9 which are for the combined load scenario, WRC 107 results show significantly higher magnitudes of Pb + Pl + Q with respect to FEA. Even though the Pressure loading has not been modeled as a Radial Radial loading for these Tables, which would have resulted in even higher magnitudes of Pb + Pl + Q if the direction of this load would have been in the same direction as the additive radial load, but the simplistic way of computing pressure stresses also (as in Tables 7, 8 and 9) induces higher stresses in the WRC 107 type type of analysis. analysis. Pressure Pressure induced induced loading at at a cylinder cylinder to cylinder cylinder interface with or without without other external external loadings is complicated and WRC 107 analysis which considers the loading on the cylindrical surface as a rectangular rectangular loading cannot cannot predict predict the stresses stresses correctly correctly and will err on the conservative side for most cases. WRC 107 /297 analysis has shown lower magnitudes of Stress for Shear Forces and Torsion Torsion moments (Table (Table 6 where the loadings have been applied at the Shell-Nozzle Interface) with respect to FEA. However, these loadings, in general are not the govern governing ing loads loads in piping piping applica application tions. s. When When using using WRC107/297 modules of a Pipe Stress Program, an analyst should review the program document to see how pressure is modeled in these modules.

the pipe shoe can be seen as analogous to this parameter. Significant differences exist for the Circumferential loading case also. For both Trunnion Trunnion and Pipe Shoes, for some cases cases , specially specially for the Radial Load scenario, scenario, stresses stresses in the shoes/Trun shoes/Trunnion nion elements elements exceed exceed stresses stresses in the cylinder cylinder which clearly shows the risk of using using the Kellogg Kellogg method for computing stresses in the Pipe support attachments. A point to note is that, the method of computing stresses in the Pipe supports supports cannot cannot be technicall technically y stated as “Kellogg “Kellogg Method” Method” as [4] only discusses computation of local stresses in the Cylinder. The context of using the term “ Kellogg method” for the method of computing stresses in attachments is due the fact that this computation based based on elementary beam theory is an essent essential ial feature feature ( in author’s author’s experience experience)) of the spreadsheets spreadsheets which use the Kellogg method to compute Local Local stresses in the Cylinder. Hence the caution is using elementary beam theory analysis for computation of local stresses in Attachments. Significant differences in results have not been seen in Finite element approach using different element types. This however should not be taken as a blanket statement as the models had proper proper mesh mesh grading grading with with adequate adequately ly small element size and the element distortion control was well within the recommended limits of the FE code. For improper mesh grading, element size, significantly distorted elements and improper integration methods, significant differences in results can be seen between the elements, especially for Triangular elements which suffer from geometric anisotropy. The stress stress analyst analyst should should carefully carefully study the theory theory manual of the FE code which he/she should be using with respect to applicability, element distortion and integration rules. The present analysis has to be extended for different load combinations combinations with varying varying magnitudes magnitudes of the individual load vectors to quantify the degree of over or under conservatism of the available analytical methods. The present analysis mainly focuses focuses on the stand alone alone effect of individual individual load load vectors (although (although Tables Tables 7-9 as well as Tables Tables 13-15 13-15 does address combinations but more tests need to be done with varying magnitudes of the individual load vectors) . Effect of variance variance in mesh grading and element size should be checked checked to assist an analyst in selection of the “best element” for these applications applications,, if an analyst analyst so desires. desires. In the present scope scope of work, the use of proper mesh grading, element size and integration rules have ironed out significant differences between the individual elements. Hence, the take away message for an analyst with respect to individual element types is, as long as mesh grading , element size, distortion control and integration rules are properly used, there are no preferred element s , although alt hough the analyst should carefully read the Theory manual of the FE code which he/she intends to use.

7.0 CONCLUSIONS:

Results in tables 10, 11 and 12 again show that the pattern of variance in results between FEA and WRC 107/297 is most significant with respect to radial loads. The reason can still be attributed to the d even though in case of the shoe D dimension dimension “d” is strictly not applicable applicable but the dimension dimension of

1. Use of a particular shell theory requires an understanding of of the order of magnitude magnitude of error inherent in that theory theory and its

14


applicability applicability vis-à-vis vis-à-vis the problem to be analyzed analyzed specially specially with reference to d and D ratios. D T

107/297 based analysis is also not correct as the branch is not pressurized for Pipe support applications where by “branch” we mean the Trunnion.

2. Shell theories should be evaluated on the basis of SanderKoiter postulates.

8. Finite elements for shell analysis have different approaches based on the theoretical considerations that form the basis of their developments, with elements based on basic shell mathematical model being least popular because of the problem of addressing rigid body motion. Commercial FE codes should be having Hybrid elements in the element library for shell applications. applications.

3. The use of an axi-symmetric loading model (which in this paper has been referenced to as “Kellogg method”) has been historically historically the most most popular popular method for analyzin analyzing g both cylindrical and non-cylindrical attachments. 4. WRC-107 method which is based on Timoshenko equations

(

)

has has the the same same err error or O T as Morley, Simmonds and R Goldenveize Goldenveizerr equations. equations. WRC-107 WRC-107 results may be more or less conservative than FE results. Results are are generally overly conservative for d > 0.5 . The analysis results show that in D some cases but not all (generally computation as per Kellogg method has shown lower magnitude of stresses with respect to WRC 107 or FE analys analysis), is), Kellogg Kellogg method significantly significantly underestimates the stresses in Trunnion Trunnion and Pipe Shoes. Shoes. Hence it is recommended recommended that this method method should not not be used and and hence should not be used for evaluating stresses in Pipe supports. supports. A point to note is, the method as at appears appears in [4] addresses addresses only the local stresses stresses at the cylinder cylinder,, so the evaluation of stresses in attachments cannot technically be addressed addressed as “Kellogg “Kellogg method”, method”, rather calculation calculation based based on elementary elementary beam theory theory. It is against this later later which, the author in his his experience experience has seen seen as widely widely used in the Industry as part of of the spreadshe spreadsheets ets based based on “Kellog “Kellogg g method” is what this caution is directed at.

9. Degenerated solid elements have used adhoc assumptions on shell theory to work within the constraint constraint of finite element element formulation. Assumptions regarding the mathematical form of dependence of the determinant of the Jacobian Matrix on the thickness direction coordinate can lead to violation of rigid body properties. 10. Not much difference has been found in results using 8node reduced integration shell element developed on the line of Mindlin hypothesis, hypothesis, triangular triangular elements elements based on discrete discrete Kirchoff constraints (imposed analytically or numerically) and use of solid elements for circular attachments. Stresses at locations locations of singularity singularity have to be carefully addresse addressed d [29]. The pattern of results i.e. relative invariance with respect to element types need not be always correct depending on the D ratio, element distortion, element size and use of T alternate numerical integration rules. In general, as long as thin shell theory is valid and reduced integration rule is used for shear flexible elements, with proper mesh grading and keeping the element size at the intersection region

5. If an analyst analyst is constrained constrained to use Kellogg Kellogg method for for analysis of local stresses on the pipe at support locations, the allowable stress should not be exceeded beyond the allowable for local primary stresses stresses as per [28].

significantly less than rt , type of element is usually not a significant parameter. Stress Analyst should carefully review the Technical Manual(s) of FE Code for the capabilities and limitations of the available elements from the element library.

6. WRC-297 method is based on shallow shell theory and the order of magnitude in error is due to omission of some terms

11. Analytical methods with d as high as 1.0 with ease of D implementation implementation is required required not only because because the available available methods like WRC107/297 etc are inadequate for such applications applications but also as a tool to properly benchmark benchmark the FE results. Till such time, FE models will continue to be benchmarked against WRC 107 type of analysis for similar loading within the limits of the applicable geometry. geometry.

(

)

T and has shown overly R conservative conservative behavio behavior, r, specially specially for the Trunnion Trunnion stresses stresses for most of the cases analyzed. Use of WRC 297 for Pipe support attachments is not recommended.

which are of the order O

7. When comparing results between an analytical and FE approach, it is best to check the model on a component by compon component ent basis basis i.e. the model model is is loade loaded d with with only only one force/moment force/moment component component in the absence absence of pressure. pressure. This check will show show stresses stresses because because of which components components are over/under represented in the final results. Since WRC107/297 107/297 does not not have a provision provision for checking checking pressur pressuree loadin loading, g, simulati simulating ng the same by a modified modified radial radial load load (= (= applied applied radial load load + pressure pressure times area) or superpo superposing sing the results results with the usual membrane membrane stresses stresses in the header header pipe due to to pressur pressuree genera generally lly makes makes the anal analys ysis is overoverconservative conservative.. The modification modification of the radial load in a WRC-

12. Additional tests need to be done for Pipe Shoes for varying effects of D and combined combined loadings. loadings. In author’s author’s opinion opinion it T is futile to expect usability of WRC-107 for shoe attachments, as based on typical dimensions of Pipe Shoes, these geometric parameters will in most cases be not satisf ied. 13. 13. WRC 107 /297 analysis has shown lower magnitudes of Stress for Shear Forces and and Torsion Torsion moments (Table (Table 6 where the loadings have been applied at the Shell-Nozzle Interface) with respect to FEA. However, these loadings, in general are not the governing factors in piping applications. applications.

15


ACKNOWLEDGMENTS

[17]

The author wishes to acknowledge Professor M.D.Xue of Tsinghua University, Department of Engineering Mechanics for providing some valuable suggestions and document references and for answeri ng some questions on her paper. The author also wants to thank Dr.Subrata Saha of Reliance Industries Ltd India, Mr. Suraj Kunder of Costain UK and excolleague and friend Mr.Arijit Chatterjee for providing valuable guideline and suggestions.

[18]

[19]

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[3]

[4] [5]

[6] [7]

[8] [9]

[10] [11]

[12]

[13] [14] [15]

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Pipe Support Attachment Wrc Analysis

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