52357113 Final Report Rev1

BEng Mechanical Engineering Design

Flexibility Analysis of the Effects of Trunnion Supports on Piping Systems

of 66

Flexibility Analysis of the Effects of Trunnion

Supports on Piping Systems

With

Particular Focus on Compressor Piping Systems

Project Dissertation

________________________________________________________________

Date: 21st May 2010



of 66 Project Dissertation (PD399B)

Jonathan de Jong (20640776)

ABSTRACT

The flexibility characteristics of bends with trunnion attachments and straight pipes

with trunnion attachments are analysed using Finite Element Analysis (FEA), and

equations are developed which can be used to predict flexibility factors. The range of

applicability of these equations is based on a survey of 15 different compressor

piping systems. This survey means that the equations created should be widely

applicable to all compressor piping systems. The analyses are performed in FE/PIPE,

FEA software customized for the analysis of local stresses in vessel and piping

attachments. Regression analysis is then performed on the FEA results to create

equations that predict flexibility factors. These are compared against available test

data and previous analyses. The equations created can then be used to calculate

flexibility factors for the bends with trunnion attachments and straight pipes with

trunnion attachments in piping system stress analyses, which use Engineer's beam

theory. The comprehensive nature of the equations developed in this report enable

the easy and accurate incorporation of trunnion flexibilities into all future piping

system stress analyses.





Contents

ABSTRACT ................................................................................................... 2

1.0 Glossary .............................................................................................. 6

2.0 Introduction ......................................................................................... 8

2.1 An Introduction to Compressor Piping Systems ....................................................... 8

2.2 The Behaviour of Trunnions in Piping Systems ........................................................ 9

2.3 The Project’s Aim and Objectives ........................................................................... 11

2.3.1 The Project Aim ...................................................................................................... 11

2.3.2 The Project Objectives ............................................................................................ 12

3.0 Current Literature Survey ................................................................. 13

3.1 Formats Used to Represent Flexibilities in a Piping System....................................13

3.1.1 Flexibility Factors for Bends .................................................................................... 13

3.1.2 Flexibility Factors for Branch and Trunnion Attachments ......................................... 13

3.2 Reports on Bends with Trunnion Attachments .........................................................15

3.2.1 The Flexibilities of Bends with Trunnions ................................................................ 15

3.2.2 The Flexibility of the Trunnion-Bend Connection ..................................................... 17

3.3 Reports on Trunnion Attachments on Straight Pipe .................................................19

3.4 Reports on Similar Systems ....................................................................................20

3.4.1 Reports on Bends (Without Trunnions Attachments) ............................................... 20

3.4.2 Reports on Branch Connections on Straight Pipe ................................................... 24

4.0 Methodology...................................................................................... 26

5.0 Results ............................................................................................... 28

5.1 Generally Applicable Decisions Made .....................................................................28

5.2 Trunnion Attachments on Straight Pipe ...................................................................29

5.3 Trunnions Attachments on Bends ...........................................................................34

5.4 Bends with Trunnion Attachments ...........................................................................38





6.0 Discussion ......................................................................................... 45

6.1 Benefits of the Report’s Completion ........................................................................45

6.2 The Practical Use of the Equations Created ...........................................................46

6.3 Trunnion Attachments on Straight Pipe ...................................................................47

6.3.1 Accuracy of the FEA Results and Equations Developed ......................................... 47

6.3.2 Advantages and Disadvantages of the Equations and Beam Model Selected ......... 50

6.4 Trunnion Attachments on Bends and Bends with Trunnion Attachments .................51

6.4.1 Accuracy of the FEA Results and Equations Developed for Bends with Trunnion Attachments ........................................................................................................... 51

6.4.2 Accuracy of the FEA Results and Equations Developed for Trunnion attachments on Bends .......................................................................................................... 53

6.4.3 Advantages and Disadvantages of the Representation Methods Selected .............. 54

6.5 A Critique of the Tools Used During the Project.......................................................56

6.5.1 Current Literature Review Tools .............................................................................. 56

6.5.2 FE/PIPE .................................................................................................................. 57

6.5.3 Stat-Ease Design Expert 8.0 ................................................................................... 58

6.5.4 Project Planning Tools ............................................................................................ 58

6.6 A Critique of the Project’s Running..........................................................................59

7.0 Recommendations for Further Work ............................................... 60

8.0 Conclusions ...................................................................................... 62

8.1 A Summary of the Report Results and Recommendations ......................................62

8.2 A Summary of Personal Lessons Learnt .................................................................64

9.0 References ........................................................................................ 65





List of Appendices

Appendix A: A Study on the Effects of Trunnions in Piping Systems

Appendix B: Selection of an Acceptable Tolerance Level for the Report

Equations Developed

Appendix C: Advice on the Creation of Spreadsheets Which Include the

Report Equations

Appendix D: A Summary of the Comparison of the Report Results Against

Experimental Data

Appendix E: A Summary of the Meshes and Finite Elements used in the

Project

Appendix F: Final Project Gantt Chart

Appendix G: Electronic Copies of the FEA and Reports Created During the

Project





1.0 Glossary

CAESAR II: Beam type analysis software widely used in the in the

stress analysis of piping systems

Design Expert (8.0): Software which is able to perform statistical Design of

Experiments, and also use linear regression analysis to

create response surface models.

Do, D, T: The outer diameter, mean diameter, and wall thickness

respectively of a pipe. Do = D + T.

do, d, t: The outer diameter, mean diameter, and wall thickness

respectively of a trunnion. do = d + t.

E: Young’s Modulus

FEA : Finite Element Analysis.

FE/PIPE: FEA software customized for the analysis of local stresses

in vessel and piping attachments.

h : The flexibility characteristic of a bend. Defined as 4TR/D2

R : Bend radius of a Long Radius bend = 1.5Do

Flexibility Factor: A dimensionless ratio used to quantify the flexibility of a

component. It is the ratio between the flexibility of a

component, and the flexibility of a reference component

whose flexibility is well defined.

kd Flexibility factor for moments in the d direction ( i = in-

plane moment, o = out-of plane moment, t =torsion).

Piping System: Interconnected piping subject to the same set or sets of

design conditions (Nayyar et Al, 2000)





Piping stress analysis: Any of a number of methods which can be used to

determine whether a piping system meets code

requirements. In this report it refers to a beam type

analysis, typically completed using a suitable computer

program.

Regression Analyses: Regression analysis is a statistical tool for the

investigation of relationships between variables and

responses. It can be used to create predictive equations

based on a data set.

Trunnion: Trunnions are lengths of pipe welded to a piping system

that are used as points of support or restraint in that

system.





2.0 Introduction

2.1 An Introduction to Compressor Piping Systems

Compressor piping systems require a high level of design effort to complete a good

piping system design. This is because of the strict requirements placed on maximum

compressor nozzle loads by design standards such as API 617 (2002).

The reason for the strict design

standards and extra design effort

required is that high loads applied to

compressor nozzles can lead to early

equipment failure. Compressors are

expensive, long lead time items of

equipment, and this is not desirable!

The cause of failure is that the forces

and moments applied by a piping

system on the nozzle can distort the

compressor casing, resulting in a

misalignment between it and the

rotating shaft,

To maintain low loads on a compressor’s nozzle, the connected piping system has to

be designed so that it doesn’t transfer its operating loads on to the nozzle. As the

piping becomes larger and stiffer this becomes more and more difficult to achieve.

Because of these requirements to keep nozzle loadings low, it becomes important to

correctly model the flexibility of the piping system in its stress analysis. Doing this

ensures that the transfer of forces and moments through the piping system is

correctly modelled and, consequently, that the locations of these forces and moments

in the stress analysis is representative of reality.

Figure 2-1: Internals of a compressor: its

casing and rotating shaft





2.2 The Behaviour of Trunnions in Piping Systems

This report focuses on the effect that trunnions have on the flexibility of piping

systems, and specifically compressor piping systems (these are referred to

generically as piping systems from this point forward).

Trunnions can affect the flexibility of a piping system

in 2 ways:

1. The local deformation that occurs at a

trunnion-pipe connection when a moment is

applied to a trunnion increases the flexibility of

a piping support. This increases the

deformation which can occur at the support,

which would normally be assumed to be

infinitely rigid.

2. The stiffening effect of a trunnion on the shell

that it is attached to, where it acts as a

webbing, decreases deformation of that part of

the shell under loads applied through the shell,

increasing the stiffness of the piping system

The effect of a trunnion on the flexibility of a piping

system is generally modelled in a stress analysis in

one of 2 ways:

- The trunnion-shell connection and the stiffening effect of a trunnion on a bend

are assumed to be unimportant, and are ignored.

This reduces the amount of time required to complete a stress analysis. However, as

shown in Appendix A, which is a study on the effect of considering a trunnion’s

flexibility in a stress analysis, this can lead to significant errors. Realistic piping

systems are considered, and it was highlighted that including the trunnion’s flexibility

in a stress analysis could change the calculated loads at a point within a piping

system by up to 29%.

Figure 2-2: Trunnion on a

bend with guide and anchor

steel stops.





- Using FEA, the effects of trunnions on the flexibilities on a piping system are

determined, and then included in a stress analysis.

There are a number of disadvantages in this approach too. FEA software is

expensive, and there are a limited number of experienced users within the industry.

As well as this, using FEA increases the amount of time required to complete a stress

analysis. The continuous use of FEA is also very repetitious, considering the fact that

there are only a small number of parameters which need to be varied to fully describe

a trunnion’s configuration within a piping system.

There are a few equations within the current literature which can be used to calculate

flexibility factors for various trunnion configurations. These have a limited range of

applicability, and also do not consider all of the parameters which affect a trunnion’s

flexibility within a given situation, which further decreases the situations in which they

can be used.

Considering all of the above, the usefulness of a set of equations which include all of

the parameters which describe a trunnion’s flexibility, and with a range of applicability

wide enough to include most piping system can be seen. The result of such a set of

equations would be the quick and easy calculation of trunnion flexibilities, and a

resultant increase in the accuracy of piping system stress analyses.

This project set about to create such a set of equations. This is captured in the aims

and objectives.





2.3 The Project’s Aim and Objectives

2.3.1 The Project Aim

This project aims to create equations which can predict relevant flexibility factors for

bends with trunnion attachments and straight pipe with trunnion attachments. These

equations should be applicable to varying sizes of piping systems, with a wide

enough range of applicability to be able to be used on the extra large bore thin or

thick walled pipe that can be found in compressor inlet and outlet piping systems.

The equations developed should also be widely applicable to other similar sized

piping systems to allow for their future use.





2.3.2 The Project Objectives

The project objectives, which are set as goals required to be met to achieve the

project aim are:

1. Develop a range of dimensional parameters which encompass compressor

piping systems.

2. Define the flexibility factors that are required to be calculated for compressor

piping systems.

3. Review current literature applicable to trunnions flexibilities.

4. Using FE/PIPE, perform FEA on the typical models developed.

5. Using ANSYS, perform FEA on the typical models developed.

6. Compare FEA results with available test data and previous analyses.

7. Select equations to be used to calculate flexibility factors for stress analyses.

8. Discuss the effects of adjacent piping components on the flexibility of

trunnions.

9. Discuss the variation in flexibility of the Support Systems caused by internal

pressures

10. Support the master project schedule by providing timely release of trunnion

flexibility factors at a level of detail relevant to the current analysis phase.





3.0 Current Literature Survey

3.1 Formats Used to Represent Flexibilities in a Piping System

For general design purposes, simple, conservative and reasonably accurate analysis

formulae are often more desirable than long and more accurate equations. It is noted

that in general, design codes provide simplified stress analysis formulae in terms of

flexibility factors (Dodge and Moore, 1972).

3.1.1 Flexibility Factors for Bends

Throughout the current literature, the flexibility of a bend is defined using flexibility

factors in the same way. The bend flexibility factors can be calculated as:

nalnomi

actualk

(Dodge and Moore, 1972) (Eq. 3.1)

Where θactual = the actual angle of deflection across the bend under a moment

loading and, θnominal = the deflection under the same moment loading of a straight

pipe whose length L is equivalent to the centreline length of the bend. For example,

for in plane bending, θnominal can be calculated from standard engineers beam

theory by equation 1.2:

L

dxMEI0

1

1 (Rodabaugh and Moore, 1979) (Eq. 3.2)

A word definition of equation 4.1 is: The flexibility factor of a bend can be defined as

the number of times more flexible a bend is than a straight pipe with a length equal to

the bend’s centre line length.

3.1.2 Flexibility Factors for Branch and Trunnion Attachments

Branches and Trunnion attachments on pipe can be represented using the same

flexibility factor methods, as they are similar. The only difference between them is the

presence of a plug at their connection point to the pipe, which is present in trunnion

attachments and absent in branches.





Rodabaugh and Moore (1979) provide a way of representing the added flexibility

present at a branch-pipe connection due to the local deformations in the pipe shell.

The rotation due to local deformation at the branch-pipe connection is considered in

terms of the length of straight branch pipe (measured in branch diameters) required

to produce that same deformation. Incorporating this into equation 3.2 we get:

EI

kMdo

local (Eq. 3.3)

Or, rearranging:

o

localMd

EIk (Eq. 3.4)

This production of the flexibility factor in this manner has a number of advantages. It

makes k dimensionless, and it also allows for the evaluation of the significance of the

value of k. This can be seen considering a word definition of equation 3.4:

The flexibility factor can be defined as the length, measured in branch pipe diameters

do, of branch pipe required to provide the same amount of flexibility as is provided by

the local deformation at the branch-pipe connection.

Another method for representing branch flexibilities is by the use of a point spring,

whose stiffness represents the stiffness of the branch-pipe connection. The stiffness

of the point spring is defined by:

/MS (Eq. 3.5)

Where M is the moment applied at the connection, and θ is the resultant local

deformation rotation of the connection due to the moment. This method of

representing the local deformation flexibility is the way in which the branch-pipe

connection flexibilities are input into Caesar II. The significance of these stiffness

values can be difficult to appreciate as they vary significantly based on the moment of

inertia of the branch pipe.





3.2 Reports on Bends with Trunnion Attachments

3.2.1 The Flexibilities of Bends with Trunnions

2 reports have been found which consider the flexibility of bends with trunnions, and

which can be used to compare against the results from this report. These are

summarised below:

A report by Hankinson and Albano (1989) considers the effect of trunnion

attachments on bend flexibility factors, considering bends with flanged ends, and

also with lengths of attached pipe for the removal of bend end effects. The results

appear reasonable and are compared against the results of this paper.

The equation provided by Hankinson and Albano (1989) for bends with flanged ends

is:

- ki = 0.891(4TR/D2)-0.67(d/D)-0.34(t/T)-0.03 Eq. (3.6)

And for bends with attached pipe lengths is:

- ki = 0.937(4TR/D2)-1.02(d/D)-0.41(t/T)-0.03 Eq. (3.7)

The range the equations provided are applicable for are:

20<= D/T <= 60, 0.4<= d/D <=0.92 & 0.2 <= t/T <= 3.2.

A report produced by EPRI (1998a) provides equations that can be used to calculate

flexibility factors for bends with lengths of attached pipe. The results appear

reasonable and are compared against the results of this paper.

The equations provided by the EPRI (1998a) are:

- ki(elbo) = 1.01 (d/D)-.372(t/T)-0.12/h Eq. (3.8)

- ko(elbo) = 1.3/h Eq. (3.9)

The range the equations provided are applicable over are:

19<= D/T <= 49, 0.34 <= d/D <= 0.85 & 9<=d/t<=49.





Both of the reports mentioned in this section have a range of applicability that does

not cover the full range of pipe dimensions found in compressor piping. In addition to

this limited range of applicability, the equations created do not account for the effects

of pressure stiffening or fully define the effect of adjacent piping components on the

bend flexibility factors.





3.2.2 The Flexibility of the Trunnion-Bend Connection

The EPRI (1998a) report mentioned in Section 3.2 also considers the flexibility of the

trunnion-bend connection. Considering the flexibility of these trunnions consists of 2

parts: a suitable beam model that

can be used to represent the

behaviour of the trunnion and

bend in a piping system stress

analysis has to be created, and

then appropriate bend flexibility

factors and trunnion-bend

connection flexibility factors have

to be created from the FEA

results.

When performing FEA, the

deformations at the ends of the

bend and trunnion can be

measured, however, a beam

model has to be assumed which

can be used to distil from these rotations the rotation due to the trunnion-bend

connection local deformations.

Figure 3.1 is adapted from the EPRI (1998a) report, and shows the different beam

models that were experimented with in that report to remove beam type rotations

from the FEA results. The simpler of the 2 boundary conditions considered is looked

at: The FEA model is fixed at boundary 1, and a moment is applied at boundary 3.

The length of bend which is considered as bending under the trunnion moment is

first considered as Length 1, then Length 2, and finally Length 3. As the moment

applied is considered to operate across shorter lengths of the bend, more of the

rotation in the FEA results becomes attributed to the trunnion-bend local

deformation. In the report, the model decided upon as correct is that represented by

Length 3, as all calculated flexibility factors are positive. The report does not provide

a more logical reason for selecting this beam model, and because of this the EPRI

A B

C

D Length 1

Length 2

Length 3

Boundary 1

Boundary 2

Boundary 3

Figure 3-1: Different EPRI (1998a) report beam

models for calculating trunnion flexibility factors.

(Adapted from EPRI (1998a)





(1998a) results for trunnion attachments on bends are best considered only as

comparative, rather than absolute.

Another problem with the EPRI (1998a) report is the discrepancy between their

results for different boundary conditions. FEA is run first with Boundary 1 fixed, and a

moment applied at Boundary 3, and then with Boundary 1 and 2 fixed, and a

moment applied at Boundary 3. The final flexibility factors calculated for these 2

different boundary conditions are significantly different, by as much as an order of

magnitude.

The report considers this error as representative of reality, rather than because of

errors in their beam models which they used to remove beam type rotations from the

FEA results, which is the correct reason for the discrepancy. Considering the

geometry and loads moments, the effect of boundary conditions on the trunnion-

bend connection flexibility is not that significant, and their method of considering the

average of the 2 end condition results as correct is wrong.

Because of the points raised in the discussion above, the equations provided in the

EPRI (1998a) report should be treated with caution.

The equations provided by the EPRI (1998a) report are:

- ki = .142 (D/T)1.11 (d/D)-0.22 (d/t)-0.55 Eq. (3.10)

- ko = .146 (D/T)1.41 (d/D)0.36 (d/t)-0.61 Eq. (3.11)


19<= D/T <= 49, 0.34 <= d/D <= 0.85 & 9<=d/t<=49.





3.3 Reports on Trunnion Attachments on Straight Pipe

A report produced by EPRI (1998b) has been found which considers the flexibility of

trunnion attachments on straight pipe. The results appear reasonable and are

compared against the results of this report.

FEA is performed on trunnions under in-plane and out-of-plane moments, and

torsional loading. The flexibility factors derived for trunnions under torsional loading

are low, which is in agreement with Rodabaugh and Moore (1979), who say that the

torsional flexibility factor can be assumed to be zero.

Once more, the range of applicability of these equations is not wide enough to cover

the full range of pipe dimensions found in compressor piping. Also, the equations

created do not account for the effects of pressure stiffening and adjacent piping

components on the trunnion attachment on straight pipe flexibility factors.

The equations provided by the EPRI (1998b) report are:

- Ki = 0.34 (D/T)1.351 (d/D)0.058 (d/t)-0.76 Eq. (3.12)

- Ko = 0.321 (1.47(d/D)-(d/D)2.45) (D/T)1.86 (d/D)-0.14 (d/t)-0.75 Eq. (3.13)

- Kt = 0.56 (D/T)0.75 (d/D)0.998 (d/t)-0.61 Eq. (3.14)


19<= D/T <= 49, 0.34 <= d/D <= 0.85 & 9<=d/t<=49.

The report also provides equations which can be used to calculate flexibility factors

from an FEA model, or experimental data. These are a collection of simple beam

equations which can be used to remove the rotations in the FEA Model due to

bending or torsion on the pipe and trunnion.





3.4 Reports on Similar Systems

A number of reports have been made on systems similar to those that are considered

in this report. The relevant points drawn from these reports are summarised in this

section.

3.4.1 Reports on Bends (Without Trunnions Attachments)

Four properties of a bend in a piping system which have applicability to this report

have been discussed in the current literature, and are summarised here. These are:

the linearity of the relationship between the applied moment and observed rotation for

a bend, a bend’s flexibility factor, the

effects of flanged ends on a bend’s

flexibility factor and the effects of

internal pressure on a bend’s flexibility

factor.

Reports produced by Taupin et Al

(1983) and Greenstreet (1978)

investigate the plastic response of

bends under moment loading, and

both show as a product of their work

that the relevant portion of an bends

response to moment loading is linear

in nature. This can be seen in Figure

3-2, which is a sample curve taken from Taupin et Al’s report.

Thomson and Spence (1988) add to this by highlighting that under combined

moments and pressure, the ratio of bending moment to pressure influences the

flexibility factor. However, they also point out that this interaction effect can be

ignored for relatively lower pressures.

Figure 3-2: Graph of Moment against

rotation for a 90 degree elbow without

internal pressure from Taupin et Al (1983)





The flexibility factor of a bend is defined in section 3.1. With regards to the value of a

flexibility factor to be used to represent a bend in a stress analysis, ASME B31.3

(2006) provides the basic design rule below.

The flexibility factor of a bend with sufficient attached pipe length can be

approximated by the equation:

k = 1.65/h Eq (3.15)

Rodabaugh et Al (1978) suggests corrections to this flexibility factor based on FEA,

suggesting that the flexibility factors of a 90 degree bend with sufficient attached pipe

lengths should be approximated by the formulae:

ki = 1.3/h Eq (3.16)

ko = 1.25/h Eq (3.17)

The reason for the decrease in flexibility factor between Equations 3.16 and 3.17 and

Equation 3.15 is that Equations 3.15 is based upon the no end effects theory, where

it is assumed that all cross sections along the arc of the bend deform identically.

However, even with sufficient attached pipe lengths, an end effect exists on the bend,

with decreased flexibility present at the straight pipe bend connection. Touboul et Al

(1989) point out that experimentally calculated flexibility factors are scattered, with an

upper limit value of k = 2.1/h, and Equation 3.15 being an approximately an average

value, though many data points also fit Equation 3.16. These differences can thus be

ignored as within the range caused by standard manufacturing tolerances.

If a bend is stiffened by the introduction of flanged on its ends, then, according to

ASME B31.3 (2006) the bend flexibility should be multiplied by the correction factors:

C1 = h1/6 Eq (3.18)

C2 = h1/3 Eq (3.19)

Where C1 is the correction factor for a bend with one end flanged, and C2 the

correction factor for a bend with both ends flanged.





A report by Whatham and

Thompson (1979) highlights

that for bends with flanged

ends and large D/T ratios,

the ASME B31.3 (2006)

code guidance significantly

overestimates the flexibility

factor. Based on the

Novozhilov thin shell

theory, and making suitable

assumptions to account for

the effect of flanged ends,

they create a number of

graphs which can be used

to calculate the bend flexibility

factor based on certain

dimensional parameters. Figure

3.3 shows the results from their

report contrasted against the

standard ASME B31.3 (2006)

correction factors.

In the same report, Whatham

and Thompson (1979) consider

the effect of flanges a short

distance away from the ends of

the bends, defining the distance

by the dimensionless parameter

L/R. Graphs are created once

more, an example of which is

shown in Figure 3.4. This

approach is more representative of reality than the ASME B31.3 (2006) approach in

that rather than considering a bend as either being flanged or unflanged, the distance

Figure 3-4: Graph of the effects of the bend

characteristic on the flexibility factors of bends with

flanged ends. From Whatham and Thompson (1979)

Figure 3-3: Graph of the effects of the bend

characteristic on the flexibility factors of bends with

flanged ends a distance of L/R = 2 away from the

end of the bend. From Whatham and Thompson

(1979)





of the flange from the end of the bend can now be taken into account by presenting a

graduating scale of flexibility factors that are dependant of the value of L/R for the

bend configuration.

The final bend property that has a major effect on the flexibility factor of a bend is its

internal pressure. This is generally considered as a second order term with the

stiffening effect due to pressure resisting the change in area caused by the

ovalisation of the pipe (Thomson and Spence, 1983). ASME B31.3 (2006) provides a

pressure correction factor to be applied to bends with and without stiffening rings

which can be calculated by the equation

C = 1 + 6(P/E) (D/2T)7/3(2R/D)1/3 Eq (3.20)

Lubis and Boyle (2003) perform a parametric analysis using non-linear FEA, and

based on that suggest the use of the pressure correction factor below, saying that it

provides better correction factors for bends with lower internal pressures:

C = 1 + 33(P/E) (D/2T)2(2R/D)1/4 Eq (3.21)

Thomson and Spence (1983)

consider the pressure stiffening effect

on bends with flanges on their ends,

suggesting that the effect of pressure

on the bend flexibility factor is

reduced significantly – though it is

worth noting that the flexibility factor

is already significantly lower than for

a bend without flanges on its ends.

They create a graph which can be

used to calculate the bend flexibility

factor based on certain dimensional

parameters. This is shown in figure 3.5. For operating pressure in a practical stress

analysis for steel pipe, the pressure term provided varies between 0 and 0.004.

Looking at the graph it can be seen that for flanged bends, the effects of pressure on

flexibility factors are small.

Figure 3-5: Graph of the effect of pressure on

the flexibility factor of bends with flanged

ends. From Thomson and Spence (1983)





For the calculation of flexibility factors from an FEA model, or experimental data,

Rodabaugh and Moore (1978) provide the 2 equations below:

)2

/()2(R

LM

EIK

efea

i

For bends under in-plane moments Eq (3.22)

)4

/()021.13.2(R

RLM

EIK

efea

o

For bends under out-of-plane moments Eq (3.23)

As a final point, Rodabaugh and Moore (1979) highlight the reason why a torsional

flexibility factor does not exist for a bend. Considering the geometry, it can be noted

that an out-of-plane bending effect on one end of the bend becomes a torsional effect

on the other as the pipe rotates through 90 degrees, and vice versa.

3.4.2 Reports on Branch Connections on Straight Pipe

As mentioned earlier, branches and trunnion attachments on straight pipe are similar,

the only difference between them being the presence of a plug at their connection

point to the pipe, which is present in trunnion attachments and absent in branches. A

number of reports are available in the current literature which discuss the flexibility of

branch-pipe connections.

A report by Rodabaugh and Moore (1979) provide 2 equations for the evaluation of

nozzles (branches) in pressure vessels and piping.

The equations provided by Rodabaugh and Moore (1979) are:

Ki(bran) = 0.1(Do/T)1.5[(T/t)(do/Do)]0.5(t/T) Eq (3.24)

Ko(bran) = 0.2(Do/T)[(T/t)(do/Do)]0.5(t/T) Eq (3.25)


do/Do < 0.5 and D/T < 100





Rodabaugh and Moore (1979) also mention that as the rotation of the straight length

of pipe becomes of a comparable size to the trunnion-pipe connection deformations,

the calculations become more sensitive to inaccuracies in the rotations measured.

Because of this, the lengths of straight pipe attached to the model should only be as

long as is required to remove the effect of end conditions, and no longer. They

suggest that the ratio (θFEA – θbeam)/ θbeam should be greater than 0.1 (an arbitrary cut

off point) to prevent flexibility factors calculated becoming overly sensitive to errors in

the rotations measured from the FEA model.

A report by Xue and Sang (2006) provide the results of a parametric study of

flexibility factors for branch pipe connections. The report is interesting in that its range

of applicability is created to cover all practical piping system dimensions, providing a

single set of equations that can be referred to when considering branch-pipe

connections.

Unfortunately, the equations created by the report significantly overestimate the

flexibility of branch connections. The student was warned during correspondence

with FE/PIPE (2010), that when lengths of attached pipe become excessive, flexibility

is over estimated in FEA. Xue and Sang (2006), before performing their parametric

study, performed a number of FEA runs with different lengths of attached piping to

allow them to define when end effects become insignificant. They decided that,

depending on the value of d/D, this happens between an L/D ratio of 6 and 9. These

excessively long attached pipe lengths affect their results, especially the out-of-plane

flexibility factors calculated.

The equations provided by Xue and Sang (2006) are:

Ki(bran) = 0.680(d/D)-0.242(D/T)0.802(t/T)0.622[3.437(d/D) – 7.414(d/D)2 + 4.766(d/D)3]

Eq (3.26)

Ko(bran) = 0.172(d/D)0.538(D/T)1.515(t/T)0.862[5.935(d/D) - 10.454(d/D)2+ 4.797(d/D)3]

Eq (3.27)


0.333 <= d/D < 1, 20 <= D/T <= 250, d/D <= t/T <= 3





4.0 Methodology

This aim of this report was to define a way of modelling the effects of trunnions on a

piping system’s flexibility such that these effects could be represented in a simple

beam type stress analysis performed on the piping system.

The methodology employed in producing the results for this report had 2 aims:

1. To be able to produce equations and beam models that provide an

acceptable level of accuracy in their use, even though they are simplified

models of the real piping system.

2. For the end results of the report to be simple enough to be easily used

during a practical piping system stress analysis.

Considering the above points, a number of questions had to be answered.

One of the key questions was what is an acceptable tolerance for the deviation of the

results from the correct value? This was considered in Appendix B, and a reasonable

answer was provided which enables the usefulness of the final equations created to

be considered.

With regard to the FEA performed, the validity of the results had to be considered.

Validation of the results against the current literature, and experimental data where

possible, helped answer these questions. In addition, the use of FE/PIPE, which is an

industry specific and largely template driven tool also provided extra confidence in

the mesh and elements used in the finite element models.

Validating the results also includes confirming that the selection of boundary

conditions and seemingly unimportant model parameters was correct. This was

achieved through a current literature survey and creating a list of lessons learnt from

the papers. Advice provided in discussion with FE/PIPE on some of these topic areas

was also invaluable.





The FEA results were not the only thing that needed to be validated. The usefulness

of the beam models created to represent the trunnion systems also had to be

considered. The use of established techniques where they were available, and the

validation against the current literature and experimental data where they were not,

helped confirm the validity of the beam models created

Before designing an experiment, a suitable range of applicability had to be decided.

This was done by surveying a number different compressor piping systems. This step

ensured the usefulness of these equations for their use in future stress analyses.

FEA results could then be performed within this range of applicability. For results to

be useful for the development of equations, a suitable number of FEA runs had to be

performed. To ensure that the equations are capable of predicting the response

variable away from the FEA run factor levels, these runs also had to be evenly

spaced throughout the design space. This was ensured by the use of the software

tool Design Expert.

After the FEA runs were complete, suitable equations had to be developed which

fitted up to 5 model parameters. Once more, Design Expert has a response surface

modelling tools which can be used to help create these equations

Making these decisions as part of the project provided worth to the final report

results.





5.0 Results

5.1 Generally Applicable Decisions Made

The following points are results of general considerations made during this project,

which should be noted:

- An acceptable tolerance level of 50%, with an aim of keeping the actual tolerance

level below 25% was selected for the final report results. Refer to Appendix B for

a discussion of this.

- The effects that trunnion attachments have on piping systems have been split

into three generic models which are discussed in Section 5.2, 5.3 and 5.4. These

are: the flexibility of the trunnion-pipe connection on straight pipe, the flexibility of

the trunnion-bend connection and the flexibility of a bend with a trunnion

- The range of dimensional parameters which cover normal compressor piping

systems are summarised in table 5.1 below:

D/T d/D t/T σhoop L/D R L/do w/D

Maximum Value: 20 0.3 0.4 0 0 1.5 0 0Minimum Value: 100 0.9 1 90 3 1.5 2.5 0.4

Note: The maximum and minimum values stated in this table are not the actual limits of

the piping system geometries surveyed, but logical boundary points which incorporate

Parameter

Broadly Applicable Parameters Bend Only Straight Pipe Only

Table 5-1: Limit of relevant piping system dimensional parameters taken from the

piping survey





5.2 Trunnion Attachments on Straight Pipe

The method of representation of the trunnion-pipe connection flexibility selected is

the flexibility factor defined by Equation 3.4. The beam model used to represent the

trunnion-pipe connection flexibility is therefore (with reference to Figure 5-1):

From 1-2: Flexibility

across length is same

as for straight pipe of

length L1.

From 2-3: Flexibility

across length is same

as for straight pipe of

length L2.

From 2-4: This is a

rigid link from the pipe

centreline to its outside

diameter which has no

flexibility.

From 4-5: This is the location of a point spring which is used to represent the

flexibility factor defined for the trunnion-pipe connection flexibility. The flexibility factor

often has to be changed into a stiffness in Nm/deg before being used in stress

analysis software such as Caesar II.

From 5-6: Flexibility across length is same as for a trunnion of length l1.

The simplified model parameters selected

as affecting the trunnion-pipe connection

flexibility, and their range, are shown in

Table 5-2. These were used to define the

experimental runs and were modified

slightly during the creation of predictive

equations.

Parameter Parameter Range

D/T 20 < D/T < 100

t/T 0.4 < t/T < 1.0

d/D 0.3 < d/D < 0.9

L/do 0 < L/do < 2.5

w/D 0 < w/D < 0.4

Direction of Moment In-plane or Out-of-plane

Minpl

Moutpl

Do

T do t

L2 L1

1 2 3

4

6

5

l 1

w

Figure 5-1: Beam model, and selected model dimensions

for a trunnion attachment on a straight pipe

Table 5-2: Simplified model parameters

for a bend with a trunnion attachment





Tables 5-5 and 5-6 shows the experimental runs and the factor levels set for each

experiment, as well as the results. Also included when available are comparative

results which are available in the current literature.

Linear Regression Analysis was then performed on the experimental runs and

equations were created. Statistical measures of the goodness of fit and the number

of terms in each equation are summarised in Tables 5-3 and 5-4.

The most suitable equations developed to predict Ki and Ko are:

(Ki/F)0.45 = -0.978 + 0.0822A + 1.44B + 0.591F + 0.523D - 2.32C - 0.0493AB -

0.0133AF + 9.24E-04AD - 0.00612AC - 1.06BF + 1.55BC - 0.296DC – 6.04E-04A2 -

0.108D2 + 2.46C2 + 0.0175ABF + 1.39E-04A2B + 3.85E-03AC2 + 0.254DC2 + 2.07E-

06A3 -1.20C3 Eq. (5.1)

(Ko/F)0.2 = 0.0886 + 0.0357A + 1.49B - 0.149F + 0.698D - 1.44C - 0.0112AB -3.65E-04

AD -0.083BD + 0.553BC – 3.28E-04A2 - 0.89B2 -0.344D2 + 1.40C2 + 2.70E-03ABD +

1.36E-06A3 + 0.0601D3 - 0.522C3 Eq. (5.2)

Where A = D/T, B = do/D, C = w/do, D = L/do, and F = t/T.

These are applicable when the parameters in Table 5-2 fall in the ranges specified.

Linear 2FI Quadratic Cubic Cubic of Ki/(t/T)

D/T 1 1 2 3 3

do/D 1 1 1 1 1

t/T 1 1 1 2 1

L/do 1 1 2 3 2

w/do 1 1 2 3 3

Interaction Effects 0 2 4 9 11

Total (Including Intercept) 6 8 13 22 22

112 81 66 37 28

38 29 17 9 8

0.818 0.874 0.966 0.991 0.991

Maximum Polynomial level term Included in Equation

Summary of Predictive Equations developed for K i

No of

equation

terms

for:

Maximum Diference (%):

C.V. (%):

Adjusted R squared:

Table 5-3: Summary of the various predictive equations developed for Ki





Table 5-4: Summary of the various predictive equations developed for Ko

Linear 2FI Quadratic Cubic Cubic of Ko/(t/T)

D/T 1 1 2 3 3

do/D 1 1 2 2 2

t/T 1 1 1 1 1

L/do 1 1 2 3 3

w/do 1 1 2 3 3

Interaction Effects 0 3 4 8 5

Total (Including Intercept) 6 9 14 21 18

116 95 38 26 25

37 32 13 7 8

0.850 0.895 0.981 0.994 0.995

0.831 0.873 0.975 0.990 0.989


C.V. (%):

Adjusted R squared:

Predicted R squared:

Summary of Predictive Equations developed for Ko


No of

equation

terms

for:





Table 5-5: FEA runs, their factor levels, and results, for trunnion-pipe connections (Table 1 of 2).

In-plane

displacement

Out-of-plane

displacement

1 80 0.8 0.7 0.85 0.26 1000 2.00E+05 4.85E+05 4.63E+00 1.10E+01 0.40 0.73 3 2.06 8.94

2 60 1 0.7 2.55 0.26 1000 2.00E+05 6.44E+05 4.51E+00 1.48E+01 0.49 0.82 3 3.00 16.44

3 60 0.4 0.9 0 0 1000 2.00E+05 6.44E+05 2.32E+00 3.67E+00 0.39 0.58 1 0.65 1.53

4 40 0.6 0.5 1.7 0.26 1000 2.00E+05 9.58E+05 7.47E+00 1.32E+01 0.30 0.59 3 1.31 4.56

5 100 1 0.5 2.55 0.39 1000 2.00E+05 3.89E+05 7.69E+00 2.79E+01 0.59 0.88 3 4.42 23.82

6 100 1 0.5 0 0.26 1000 2.00E+05 3.89E+05 6.35E+00 1.58E+01 0.51 0.80 3 3.15 12.22

7 20 0.4 0.3 0 0 1000 2.00E+05 1.87E+06 5.53E+00 8.07E+00 0.18 0.44 1 0.23 0.78

8 40 0.8 0.9 2.55 0.39 1000 2.00E+05 9.58E+05 2.03E+00 6.35E+00 0.57 0.76 1 1.64 6.81

9 60 0.8 0.7 2.55 0 1000 2.00E+05 6.44E+05 6.51E+00 2.75E+01 0.57 0.88 3 4.04 4.14 26.66 21.48

10 100 0.8 0.5 0.85 0.13 1000 2.00E+05 3.89E+05 8.13E+00 2.13E+01 0.53 0.82 3 3.35 13.61

11 20 0.8 0.3 1.7 0 1000 2.00E+05 1.87E+06 6.03E+00 1.03E+01 0.58 0.75 1 1.40 3.07

12 80 0.6 0.5 0 0 1000 2.00E+05 4.85E+05 8.88E+00 1.80E+01 0.43 0.71 3 2.23 7.60

13 100 0.8 0.9 0 0.39 1000 2.00E+05 3.89E+05 3.14E+00 5.56E+00 0.31 0.57 3 1.41 4.56

14 100 0.8 0.9 0.85 0 1000 2.00E+05 3.89E+05 4.14E+00 1.07E+01 0.48 0.77 3 2.82 11.84

15 20 0.4 0.9 0 0.26 1000 2.00E+05 1.87E+06 1.77E+00 2.65E+00 0.19 0.39 1 0.24 0.75

16 100 1 0.7 1.7 0 1000 2.00E+05 3.89E+05 6.63E+00 3.27E+01 0.66 0.92 3 6.02 41.42

17 20 0.6 0.9 1.7 0 1000 2.00E+05 1.87E+06 3.94E+00 6.76E+00 0.25 0.49 3 1.06 3.54

18 100 0.6 0.3 0.85 0.39 1000 2.00E+05 3.89E+05 1.74E+01 3.75E+01 0.51 0.77 3 3.08 10.13

19 40 1 0.3 0.85 0.26 1000 2.00E+05 9.58E+05 4.50E+00 7.56E+00 0.58 0.74 1 1.37 2.96

20 100 0.8 0.3 2.55 0 1000 2.00E+05 3.89E+05 2.72E+01 1.05E+02 0.76 0.94 3 9.53 10.16 46.36 51.14

21 100 1 0.9 1.7 0.39 1000 2.00E+05 3.89E+05 3.30E+00 1.08E+01 0.45 0.79 3 2.67 15.23

22 100 0.8 0.9 2.55 0.26 1000 2.00E+05 3.89E+05 4.31E+00 1.79E+01 0.48 0.84 3 2.93 21.35

23 20 0.4 0.7 1.7 0.39 1000 2.00E+05 1.87E+06 2.71E+00 4.93E+00 0.31 0.56 1 0.47 1.54

24 60 0.4 0.3 0.85 0.26 1000 2.00E+05 6.44E+05 1.81E+01 2.74E+01 0.29 0.53 3 1.24 3.40

25 60 1 0.3 0 0.39 1000 2.00E+05 6.44E+05 9.71E+00 1.68E+01 0.44 0.67 3 2.33 6.18

26 20 1 0.5 0 0.13 1000 2.00E+05 1.87E+06 1.56E+00 2.37E+00 0.25 0.48 1 0.35 1.00

27 20 0.4 0.3 2.55 0.13 1000 2.00E+05 1.87E+06 5.06E+00 7.50E+00 0.10 0.39 1 0.12 0.64

28 20 0.6 0.7 2.55 0.26 1000 2.00E+05 1.87E+06 4.58E+00 7.07E+00 0.17 0.40 3 0.63 2.32

29 40 0.6 0.7 0 0.39 1000 2.00E+05 9.58E+05 4.62E+00 7.26E+00 0.20 0.47 3 0.78 2.83

30 80 0.8 0.3 2.55 0.26 1000 2.00E+05 4.85E+05 1.41E+01 3.16E+01 0.53 0.79 3 3.41 11.37

31 100 0.4 0.5 1.7 0.26 1000 2.00E+05 3.89E+05 1.29E+01 3.59E+01 0.41 0.78 3 2.11 11.20

l/do

Ki

Calculated

from FEA

Ki

(EPRI

1998b)

Ko

Calculated

from FEA

Ko

(EPRI

1998b)D

Youngs

Modulus

(Mpa)

Moment through

header producing

50Mpa stress

In-plane

Displacement

(mm)

Out-of-plane

Displacement

(mm)

Goodness of Model for

creating K measures

Run D/T t/T d/D L/do w/D





In-plane

displacement

Out-of-plane

displacement

32 40 0.8 0.7 0 0 1000 2.00E+05 9.58E+05 1.97E+00 3.99E+00 0.52 0.74 1 1.11 3.22

33 60 1 0.5 0.85 0 1000 2.00E+05 6.44E+05 7.95E+00 2.22E+01 0.60 0.85 3 4.56 18.06

34 60 0.4 0.7 0.85 0.13 1000 2.00E+05 6.44E+05 7.39E+00 1.38E+01 0.27 0.60 3 1.11 4.59

35 80 0.6 0.7 2.55 0.39 1000 2.00E+05 4.85E+05 4.11E+00 1.88E+01 0.68 0.91 1 2.34 14.24

36 20 1 0.5 2.55 0.39 1000 2.00E+05 1.87E+06 2.05E+00 3.84E+00 0.41 0.63 1 0.74 2.11

37 60 0.8 0.3 0 0.13 1000 2.00E+05 6.44E+05 1.10E+01 1.74E+01 0.39 0.61 3 1.93 4.78

38 40 0.4 0.5 2.55 0 1000 2.00E+05 9.58E+05 1.28E+01 3.16E+01 0.40 0.75 3 2.01 2.44 9.36 9.91

39 40 0.8 0.9 1.7 0.13 1000 2.00E+05 9.58E+05 3.46E+00 7.92E+00 0.35 0.66 3 1.73 7.36

40 40 1 0.3 2.55 0.13 1000 2.00E+05 9.58E+05 5.41E+00 1.22E+01 0.65 0.84 1 1.84 5.41

41 100 0.4 0.7 1.7 0 1000 2.00E+05 3.89E+05 1.04E+01 4.04E+01 0.48 0.86 3 2.76 19.39

42 100 0.6 0.7 0 0.13 1000 2.00E+05 3.89E+05 3.12E+00 7.79E+00 0.61 0.83 1 1.58 5.39

43 80 0.6 0.9 0.85 0.26 1000 2.00E+05 4.85E+05 4.08E+00 8.10E+00 0.30 0.61 3 1.32 5.32

44 20 0.6 0.3 0 0.39 1000 2.00E+05 1.87E+06 3.62E+00 4.62E+00 0.12 0.30 1 0.13 0.43

45 20 0.6 0.5 0.85 0.13 1000 2.00E+05 1.87E+06 2.52E+00 4.05E+00 0.28 0.53 1 0.40 1.21

46 60 0.4 0.9 2.55 0.13 1000 2.00E+05 6.44E+05 6.35E+00 1.73E+01 0.32 0.71 3 1.47 8.91

47 60 0.6 0.3 1.7 0 1000 2.00E+05 6.44E+05 2.20E+01 5.35E+01 0.60 0.84 3 4.53 15.41

48 60 0.4 0.3 2.55 0.39 1000 2.00E+05 6.44E+05 1.89E+01 3.08E+01 0.32 0.58 3 1.41 4.19

49 80 1 0.3 1.7 0.13 1000 2.00E+05 4.85E+05 1.25E+01 2.86E+01 0.57 0.81 3 4.03 13.01

50 100 0.4 0.9 0.85 0.39 1000 2.00E+05 3.89E+05 5.54E+00 9.99E+00 0.24 0.55 3 0.94 3.94

51 100 1 0.3 0 0 1000 2.00E+05 3.89E+05 1.69E+01 3.94E+01 0.69 0.87 3 6.59 19.40

52 40 1 0.9 0 0.26 1000 2.00E+05 9.58E+05 2.25E+00 3.31E+00 0.21 0.41 3 0.84 2.37

53 100 0.4 0.3 0 0.13 1000 2.00E+05 3.89E+05 2.19E+01 4.07E+01 0.42 0.69 3 2.17 6.63

54 20 0.8 0.7 0.85 0.39 1000 2.00E+05 1.87E+06 1.47E+00 2.63E+00 0.32 0.55 1 0.50 1.53

55 20 1 0.9 2.55 0 1000 2.00E+05 1.87E+06 2.89E+00 5.48E+00 0.33 0.52 3 1.63 2.15 4.86 5.40

56 60 0.8 0.5 1.7 0.39 1000 2.00E+05 6.44E+05 6.77E+00 1.54E+01 0.42 0.74 3 2.20 8.75

57 100 0.6 0.5 2.55 0.13 1000 2.00E+05 3.89E+05 1.16E+01 4.44E+01 0.56 0.88 3 3.84 23.14

58 100 0.8 0.9 0 0.39 1000 2.00E+05 3.89E+05 3.14E+00 5.56E+00 0.31 0.57 3 1.41 4.56

59 20 1 0.7 1.7 0.13 1000 2.00E+05 1.87E+06 3.10E+00 5.31E+00 0.22 0.49 3 0.88 3.33

60 80 0.4 0.5 0 0.39 1000 2.00E+05 4.85E+05 6.23E+00 1.72E+01 0.59 0.85 1 1.46 5.76

61 80 1 0.9 0 0.13 1000 2.00E+05 4.85E+05 1.39E+00 2.82E+00 0.57 0.73 1 1.40 3.65

62 20 1 0.9 0 0 1000 2.00E+05 1.87E+06 8.71E-01 1.40E+00 0.29 0.46 1 0.42 1.10

l/do

Ki

Calculated

from FEA

Ki

(EPRI

1998b)

Ko

Calculated

from FEA

Ko

(EPRI

1998b)D

Youngs

Modulus

(Mpa)

Moment through

header producing

50Mpa stress

In-plane

Displacement

(mm)

Out-of-plane

Displacement

(mm)

Goodness of Model for

creating K measures

Run D/T t/T d/D L/do w/D

Table 5-6: FEA runs, their factor levels, and results, for trunnion-pipe connections (Table 2 of 2).





5.3 Trunnions Attachments on Bends

The method of representation of the

trunnion-bend connection flexibility

selected has the same format as the

flexibility factor defined by Equation

3.4. The beam model used to

represent the trunnion-bend

connection flexibility is therefore

(with reference to Figure 5-2):

From 1-2: Flexibility across length is

same as for straight pipe of length

L1.

From 2-3: Flexibility across this

length is defined as that of a straight

pipe with the same centre line length

multiplied by the bend flexibility

factor for a bend with a length L1 of

straight pipe attached on both of its

ends.

From 3-4: Flexibility across this length is defined as that of a straight pipe with the

same centre line length multiplied by the bend flexibility factor for a bend with a

length L2 of straight pipe attached on both of its ends.

From 4-5: Flexibility across length is same as for straight pipe of length L2.

From 3-6: This is a rigid link from the pipe centreline to its outside diameter which has

no flexibility.

From 6-7: This is the location of a point spring which is used to represent the

flexibility factor defined for the trunnion-bend connection flexibility. The flexibility

factor often has to be changed into a stiffness in Nm/deg before being used in stress

analysis software such as Caesar II.

Direction of Moment

t/T

L1/D

L2/D

Minpl

Moutpl

Do

T

do

t

L1

L2

θ = 41.4° R = 1.5Do

1 2

3

4

5

6

7

8

l 1

Figure 5-2: Beam model, and selected model

dimensions for a trunnion attachment on a

bend





From 7-8: Flexibility across length is same

as for a trunnion of length l1.

The simplified model parameters selected

as affecting the bend flexibility, and their

range, are shown in Table 5-7. These were

used to define the experimental runs, and

were modified slightly during the creation of the predictive equations.

When performing FEA, the deformations at the ends of the bend and trunnion can be

measured, however, a beam model has to be

assumed which can be used to remove those

rotations not due to the trunnion-bend

connection local deformations from the FEA

results.

For this report, the rotations due to the trunnion-

bend connection local deformations are

calculated as (with reference to Figure 5-3):

θtrun-bend = θtotal – (θelb-beam + θCD) Eq. (5.3)

The same basic equation can also be used to

calculate the rotation due to the trunnion-bend

connection for out-of-plane moments.

θtrun-bend can then be used in Equation 3.4 to

calculate flexibility factors to represent the trunnion-bend connection flexibility.

Table 5-10 shows the experimental runs and the factor levels set for each

experiment, as well as the results. Also included when available are comparative

results which are available in the current literature.

Linear Regression Analysis was then performed on the experimental runs and




D/T 20 < D/T < 100

t/T 0.4 < t/T <1.0

d/D 0.3 < d/D < 0.9

Direction of Moment In plane or Out of plane

Table 5-7: Simplified Model Parameters


Minpl

θtotal

θelb-beam

θtrun-bend

A

B C

D

E

Figure 5-3: Rotations in a bend

under a in-plane moment loading

on the trunnion





The most suitable equations developed to predict Ki and Ko are:

Ki-0.3 = 1.12- 3.16E-03B + 0.0183C - 0.556A – 3.27E-03BC + 2.60E-03BA + 0.700CA

Eq. (5.4)

Log10(Ko) = -0.243 + 0.00861B + 0.155C + 0.915A – 3.74E-03BA - 0.990CA

Eq. (5.5)

Where A = t/T, B = D/T and C = d/D.

These equations are applicable within the relevant parameter ranges specified in

Table 5-7.

Linear 2FI Quadratic Cubic

D/T 1 1 2 3

do/D 1 1 1 2

t/T 0 1 1 1

Interaction Effects 0 3 3 6

Total (Including Intercept) 3 7 8 13

117 20 18 4

28 10 8 2

0.824 0.962 0.983 0.997

0.785 0.928 0.964 0.993Predicted R squared:

Summary of Predictive Equations developed for Ki


No of

equation

terms for:


C.V. (%):

Adjusted R squared:


D/T 1 1 2 3

do/D 1 1 1 2

t/T 0 1 1 1



59 23 30 17

20 11 10 6

0.886 0.946 0.962 0.994


Summary of Predictive Equations developed for Ko


No of

equation

terms for:


C.V. (%):

Adjusted R squared:







In-plane

rotation

Out-of-plane

Rotation

1 60 0.4 0.7 1000 2.00E+05 6.44E+05 1.09E-02 1.56E-03 1.11E-02 3.38E-04 0.46 0.59 1.97 1.12 2.53

2 100 0.4 0.3 1000 2.00E+05 3.89E+05 8.48E-02 1.74E-03 1.08E-01 5.35E-04 0.54 0.83 3.30 5.08

3 40 1 0.9 1000 2.00E+05 9.58E+05 2.69E-03 1.22E-03 2.14E-03 2.97E-04 0.19 0.33 0.83 1.42

4 40 0.6 0.5 1000 2.00E+05 9.58E+05 1.37E-02 1.49E-03 1.54E-02 2.99E-04 0.40 0.61 1.65 1.44 2.51

5 80 0.6 0.3 1000 2.00E+05 4.85E+05 5.80E-02 1.68E-03 7.46E-02 5.19E-04 0.54 0.85 3.37 5.26

6 40 0.8 0.7 1000 2.00E+05 9.58E+05 5.26E-03 1.43E-03 5.15E-03 2.73E-04 0.31 0.51 1.28 1.30 2.10

7 100 1 0.3 1000 2.00E+05 3.89E+05 4.58E-02 1.75E-03 6.28E-02 5.43E-04 0.64 1.03 5.13 8.33

8 20 1 0.5 1000 2.00E+05 1.87E+06 8.30E-03 1.23E-03 8.52E-03 2.08E-04 0.36 0.51 1.44 1.30 2.04

9 80 0.4 0.5 1000 2.00E+05 4.85E+05 2.41E-02 1.66E-03 2.79E-02 4.61E-04 0.52 0.72 2.49 3.49

10 20 0.8 0.9 1000 2.00E+05 1.87E+06 2.73E-03 9.94E-04 2.29E-03 3.12E-04 0.20 0.29 0.70 1.02

11 40 0.4 0.3 1000 2.00E+05 9.58E+05 6.55E-02 1.51E-03 7.27E-02 3.36E-04 0.41 0.54 1.92 1.71 2.52

12 40 0.8 0.5 1000 2.00E+05 9.58E+05 1.12E-02 1.48E-03 1.26E-02 3.01E-04 0.42 0.65 1.86 1.69 2.88

13 100 0.4 0.9 1000 2.00E+05 3.89E+05 1.11E-02 1.58E-03 1.05E-02 3.43E-04 0.64 0.70 4.66 5.08

14 20 0.8 0.3 1000 2.00E+05 1.87E+06 3.27E-02 1.28E-03 3.37E-02 1.66E-04 0.39 0.45 1.71 1.70 2.00

15 100 1 0.9 1000 2.00E+05 3.89E+05 3.41E-03 1.49E-03 2.76E-03 3.23E-04 0.28 0.43 1.55 2.38

16 60 1 0.3 1000 2.00E+05 6.44E+05 3.84E-02 1.62E-03 4.77E-02 4.71E-04 0.57 0.84 3.79 3.55 5.60

17 60 0.6 0.9 1000 2.00E+05 6.44E+05 4.50E-03 1.40E-03 3.89E-03 3.08E-04 0.33 0.44 1.47 1.94

18 80 1 0.7 1000 2.00E+05 4.85E+05 5.32E-03 1.61E-03 5.40E-03 3.73E-04 0.37 0.62 1.90 3.19

19 80 0.8 0.9 1000 2.00E+05 4.85E+05 3.82E-03 1.45E-03 3.19E-03 3.17E-04 0.31 0.44 1.52 2.18

20 20 0.4 0.7 1000 2.00E+05 1.87E+06 7.69E-03 1.21E-03 7.53E-03 2.23E-04 0.28 0.39 0.86 0.60 1.19

21 20 0.4 0.9 1000 2.00E+05 1.87E+06 4.61E-03 1.09E-03 4.16E-03 3.04E-04 0.25 0.32 0.77 0.99

22 80 0.8 0.5 1000 2.00E+05 4.85E+05 1.33E-02 1.66E-03 1.63E-02 4.56E-04 0.50 0.81 2.65 4.31

23 20 0.6 0.5 1000 2.00E+05 1.87E+06 1.19E-02 1.25E-03 1.24E-02 1.88E-04 0.33 0.46 1.18 0.98 1.64

24 100 0.6 0.7 1000 2.00E+05 3.89E+05 8.99E-03 1.68E-03 9.33E-03 4.12E-04 0.49 0.67 2.59 3.5425 100 0.8 0.5 1000 2.00E+05 3.89E+05 1.43E-02 1.72E-03 1.78E-02 4.86E-04 0.53 0.86 3.01 4.90

Run D/T t/T d/D

In-plane

Rotation: Bend

end (Radians)D

Youngs

Modulus

(MPa)

Moment through

pipe producing

50Mpa stress

In-plane

Rotation:

Trunnion end

(Radians)

Ko Calculated

from FEA

Out-of-plane

Rotation:

Trunnion end

(Radians)

Out-of-plane

Rotation: Bend

end (Radians)

Ki

Calculated

from FEA

Ki (EPRI

1998a)

Goodness of Model creating

K measures:

Table 5-10: FEA runs, their factor levels, and results, for bends with trunnion attachments and excluding pressure





5.4 Bends with Trunnion Attachments

The method of representation of the bend flexibility selected is the bend flexibility

factor which is defined by equation 3.1. This is improved by the splitting of the bend

into two sections, each with a potentially different flexibility factor. Thus the beam

model selected to represent the bend flexibility in a stress analysis is (with reference

to Figure 5-4):

From 1-2: Flexibility across length is same as for straight pipe of length L1.


defined as that of a straight pipe

with the same centre line length



straight pipe attached to both ends.


defined as that of a straight pipe

with the same centre line length



straight pipe attached to both ends.


same as for straight pipe of length

L2.

The simplified model parameters

selected as affecting the bend

flexibility, and their range, are shown in Table 5-11. These were used to define the

experimental runs, and were modified slightly during the creation of the predictive

equations.

Direction of Moment

t/T

L1/D

L2/D

Minpl

Moutpl

Do

T

do

t

L1

L2

θ = 41.4° R = 1.5Do

P 1 2

3

1 4

5

Figure 5-4: Beam model, and selected

important model dimensions for a bend with a

trunnion attachment





Table 5-16 shows the experimental runs

and the factor levels set for each

experiment, as well as the results. Also

included when available are comparative

results which are available in the current

literature.

Linear Regression Analysis was then

performed on the experimental runs and



The most suitable equations developed to predict Ki and Ko, ignoring the effects of

pressure, are thus:

Log10(Ki) = - 0.0673 + 0.0164A + 0.649C + 0.391B + 0.00731AB - 0.0726BC - 2.19E-

04A2 - 0.578C2 - 0.234B2 - 2.61E-05A2B – 7.08E-04AB2 + 1.025E-06A3 + 0.0392B3

Eq. (5.6)

Ko0.85 = 1.28 + 0.0453A - 0.404C + 0.149B + 0.0703AB – 2.66E-04A2 -0.159B2 -

0.00991AB2 Eq. (5.7)

Where A = D/T, B = L/D and C = do/D.

These are applicable within the relevant parameter ranges specified in Table 5-1.


D/T 20 < D/T < 100

t/T 0.4 < t/T <1.0

d/D 0.3 < d/D < 0.9

σhoop = PD/2T 0 < σhoop < 90

Direction of Moment In plane or Out of plane

L/D 0 < L/D < 3

Table 5-11: Simplified Model Parameters



D/T 1 1 2 3

do/D 1 1 2 2

L/D 1 1 2 3



69 46 26 11

33 23 11 4

0.765 0.914 0.977 0.996


No of equation

terms for:

Summary of Predictive Equations developed for K i



C.V. (%):

Adjusted R squared:






A second set of FEA runs which included pressure as one of the model parameters

was then carried out. Table 5-17 and 5-18 show the experimental runs and the factor

levels set for each experiment, as well as the results. Also included when available

are comparative pressure correction factors, when they are available in the current

literature.

Standard pressure correction factors, C, were developed. They were of the same

format for both in-plane and out-of-plane bending, with the out-of-plane bending one

being of a slightly higher magnitude.

For in-plane bending, analysis of the FEA results produced the equation:

)2.3

2.0)/()(1(1

DLBC Eq. (5-8)

Where B is the pressure correction term recommended by Lubis and Boyle (2003)

and can be calculated using Equation 3.21.

This equation was produced by manual analysis, but is of minimal length and fits the

results well. Equations produced by Linear Regression Analysis have to have a

significantly more terms to produce the same level of accuracy. Statistical measures

of the goodness of fit of this equation to the FEA results are summarised in Table 5-

14.


D/T 1 1 1 2

do/D 0 0 0 1

L/D 1 1 2 2



59 30 28 11

30 16 12 5

0.815 0.978 0.992 0.998

0.791 0.975 0.990 0.997

Summary of Predictive Equations developed for KO

Predicted R squared:

No of equation

terms for:


C.V. (%):

Adjusted R squared:







For out-of-plane bending, analysis of the

FEA results produced the equation:

)2.2

2.0)/()(1(1

DLBC Eq. (5-9)

Where B is the pressure correction term

recommended by Lubis and Boyle (2003)

and can be calculated using Equation 3.21.

Once more, this equation was produced by

manual analysis. Statistical measures of the

goodness of fit of this equation to the FEA

results are summarised in Table 5-15..

C 1

L/D 1

Total 2

13

3

No of equation

terms for:

Summary of Predictive Equations

developed for predicting the pressure

correction factor for in-plane bending

Maximum Difference (%):

C.V. (%):

Table 5-14: Summary of Key

characteristics of the predictive

equations developed for predicting

the pressure correction factor for in-

plane bending

C 1

L/D 1

Total 2

7

1

Summary of Predictive Equations developed

for predicting the pressure correction factor

for out-of-plane bending

No of equation

terms for:


C.V. (%):

Table 5-15: Summary of Key

characteristics of the predictive

equations developed for predicting

the pressure correction factor for out-

of-plane bending





FEA

Run

No D/T t/T d/D L/D d/t D

Youngs

Modules

(Mpa

Moment

Producing 50

MPa in Pipe

In-plane

Rotation

(Radians)

Out-of-plane

Rotation

(Radians)

Ki

Calculated

from FEA

K (Whatham

& Thompson

(19XX))

Ki

(EPRI

1998a)

Ki

(Hankinson

and

Ki

(Rodabaugh

et Al

K (ASME

B31.3

(2006))

Ko

Calculated

from FEA

Ko (EPRI

1998a)

Ko

(Rodabaugh

et Al

1 40 0.6 0.7 1 46.7 1000 2.00E+05 9.58E+05 9.20E-03 5.87E-03 6.98 7.7 6.76

2 60 0.4 0.9 1 135.0 1000 2.00E+05 6.44E+05 1.09E-02 7.14E-03 8.42 8.90

3 60 0.4 0.7 0 105.0 1000 2.00E+05 6.44E+05 3.66E-03 2.56E-03 3.11 2.9 4.79 3.04

4 40 1 0.5 1 20.0 1000 2.00E+05 9.58E+05 9.60E-03 5.90E-03 7.32 7.7 6.81

5 100 0.6 0.9 0 150.0 1000 2.00E+05 3.89E+05 4.10E-03 2.94E-03 3.48 3.70

6 100 1 0.9 1 90.0 1000 2.00E+05 3.89E+05 1.36E-02 9.29E-03 10.71 12.55

7 80 0.4 0.3 0 60.0 1000 2.00E+05 4.85E+05 3.98E-03 2.76E-03 3.38 2.9 3.39

8 40 0.4 0.5 2 50.0 1000 2.00E+05 9.58E+05 1.28E-02 8.07E-03 9.23 8.58

9 20 1 0.7 0 14.0 1000 2.00E+05 1.87E+06 2.53E-03 1.98E-03 2.14 2.2 2.17 2.06

10 20 0.4 0.9 0 45.0 1000 2.00E+05 1.87E+06 2.47E-03 1.97E-03 2.10 2.08 2.04

11 100 0.4 0.9 3 225.0 1000 2.00E+05 3.89E+05 2.40E-02 1.72E-02 17.88 22.12

12 100 0.4 0.3 3 75.0 1000 2.00E+05 3.89E+05 3.17E-02 1.77E-02 24.41 21.45 27.23 22.90 20.63

13 100 0.4 0.5 1 125.0 1000 2.00E+05 3.89E+05 1.71E-02 9.62E-03 13.63 13.0 13.10

14 20 0.4 0.3 3 15.0 1000 2.00E+05 1.87E+06 8.22E-03 6.56E-03 4.55 5.60 4.13 5.24 4.25 4.1 3.97

15 20 0.4 0.3 1 15.0 1000 2.00E+05 1.87E+06 5.84E-03 4.12E-03 4.15 3.8 3.84

16 100 1 0.3 0 30.0 1000 2.00E+05 3.89E+05 4.26E-03 2.98E-03 3.62 3.0 3.76

17 100 1 0.9 3 90.0 1000 2.00E+05 3.89E+05 2.02E-02 1.69E-02 14.63 21.59

18 40 0.8 0.3 0 15.0 1000 2.00E+05 9.58E+05 3.32E-03 2.34E-03 2.82 2.6 2.68

19 80 0.8 0.9 2 90.0 1000 2.00E+05 4.85E+05 1.62E-02 1.23E-02 12.08 15.71

20 40 0.8 0.5 3 25.0 1000 2.00E+05 9.58E+05 1.41E-02 9.39E-03 9.45 8.73 8.41 8.46 10.73 8.93 8.5 8.13

21 80 0.4 0.7 3 140.0 1000 2.00E+05 4.85E+05 2.48E-02 1.49E-02 18.55 17.12 21.73 18.17 16.46

22 20 0.4 0.7 2 35.0 1000 2.00E+05 1.87E+06 6.91E-03 5.41E-03 4.25 4.16

23 40 0.6 0.9 3 60.0 1000 2.00E+05 9.58E+05 1.12E-02 9.11E-03 6.99 6.73 8.45

24 80 0.6 0.5 1 66.7 1000 2.00E+05 4.85E+05 1.50E-02 8.47E-03 11.85 11.0 11.15

25 60 0.6 0.3 2 30.0 1000 2.00E+05 6.44E+05 1.81E-02 1.05E-02 13.71 12.70

26 20 1 0.9 2 18.0 1000 2.00E+05 1.87E+06 5.63E-03 5.19E-03 3.16 3.79

27 100 0.8 0.3 1 37.5 1000 2.00E+05 3.89E+05 1.71E-02 9.62E-03 13.64 13.0 13.10

28 100 0.6 0.5 3 83.3 1000 2.00E+05 3.89E+05 3.15E-02 1.77E-02 24.22 21.45 27.23 22.89 20.63

29 100 1 0.5 2 50.0 1000 2.00E+05 3.89E+05 2.71E-02 1.49E-02 21.36 20.21

30 20 0.8 0.7 3 17.5 1000 2.00E+05 1.87E+06 7.59E-03 6.49E-03 4.02 3.76 3.54 4.13 5.24 4.13 4.1 3.97

31 40 0.8 0.9 1 45.0 1000 2.00E+05 9.58E+05 7.63E-03 5.69E-03 5.65 6.45

32 80 1 0.3 3 24.0 1000 2.00E+05 4.85E+05 2.61E-02 1.50E-02 19.61 17.12 21.73 18.31 16.46

33 20 0.8 0.3 2 7.5 1000 2.00E+05 1.87E+06 7.21E-03 5.44E-03 4.50 4.22

34 100 0.8 0.7 0 87.5 1000 2.00E+05 3.89E+05 4.24E-03 2.97E-03 3.60 3.0 3.75

35 80 0.8 0.5 1 50.0 1000 2.00E+05 4.85E+05 1.49E-02 8.50E-03 11.84 11.0 11.20

36 20 0.6 0.5 0 16.7 1000 2.00E+05 1.87E+06 2.66E-03 2.01E-03 2.26 2.2 2.48 2.12

37 20 1 0.3 3 6.0 1000 2.00E+05 1.87E+06 8.18E-03 6.55E-03 4.52 5.02 4.13 5.24 4.24 4.1 3.97

38 60 1 0.7 2 42.0 1000 2.00E+05 6.44E+05 1.67E-02 1.04E-02 12.46 12.50

39 100 0.6 0.7 2 116.7 1000 2.00E+05 3.89E+05 2.61E-02 1.49E-02 20.43 20.0740 60 1 0.9 0 54.0 1000 2.00E+05 6.44E+05 3.39E-03 2.50E-03 2.88 4.27 2.95

Table 5-16: FEA runs, their factor levels, and results, for bends with trunnion attachments and excluding pressure





Table 5-17: FEA runs, their factor levels, and results, for bends with trunnion attachments and including pressure (Table 1 of 2).

In-plane

Rotation

(Radians)

Out-of-plane

Rotation

(Radians)

In-plane

Rotation

(Radians)

Out-of-plane

Rotation

(Radians)

Excluding

pressure

Including

pressure

Excluding

pressure

Including

pressure

Calculated from

FEA for in-plane

moments

Calculated from

FEA for out-of-

plane moments

Thomson and

Spence

(1983)

Lubis and

Boyle

(2003)

ASME

B31.3

(2006)

1 100 1 0.9 0 0 0 1000 2.00E+05 388850 0.061 4.050E-03 2.938E-03 4.050E-03 2.940E-03 3.43 3.43 3.67 3.67 1.00 1.00 1.00

2 40 0.6 0.7 3 60 3 1000 2.00E+05 958401 0.154 1.330E-02 9.337E-03 1.106E-02 7.918E-03 8.80 6.91 8.84 6.43 1.27 1.37 1.26 1.14

3 40 1 0.3 0 90 4.5 1000 2.00E+05 958401 0.154 3.326E-03 2.345E-03 3.190E-03 2.273E-03 2.81 2.70 2.66 2.54 1.04 1.05 1.07

4 100 0.6 0.5 1 90 1.8 1000 2.00E+05 388850 0.061 1.703E-02 9.623E-03 1.208E-02 6.965E-03 13.62 9.42 13.10 8.59 1.45 1.53

5 20 1 0.9 1 90 9 1000 2.00E+05 1874671 0.315 4.487E-03 3.908E-03 4.238E-03 3.697E-03 3.00 2.79 3.48 3.12 1.08 1.11

6 20 0.4 0.9 3 0 0 1000 2.00E+05 1874671 0.315 7.336E-03 6.441E-03 7.336E-03 6.441E-03 3.80 3.80 4.06 4.06 1.00 1.00 1.00 1.00

7 60 0.4 0.3 3 0 0 1000 2.00E+05 643948 0.102 2.025E-02 1.221E-02 2.026E-02 1.221E-02 14.68 14.69 13.67 13.67 1.00 1.00 1.00 1.00

8 20 0.4 0.9 0 0 0 1000 2.00E+05 1874671 0.315 2.470E-03 1.969E-03 2.471E-03 1.969E-03 2.09 2.09 2.02 2.02 1.00 1.00 1.00

9 60 1 0.9 3 90 3 1000 2.00E+05 643948 0.102 1.368E-02 1.165E-02 1.020E-02 8.285E-03 9.11 6.15 12.72 7.00 1.48 1.82 1.59 1.36

10 80 1 0.7 2 0 0 1000 2.00E+05 484889 0.076 2.130E-02 1.267E-02 2.131E-02 1.267E-02 16.40 16.41 16.35 16.35 1.00 1.00

11 80 0.8 0.9 3 30 0.75 1000 2.00E+05 484889 0.076 1.782E-02 1.438E-02 1.547E-02 1.169E-02 12.61 10.61 17.33 12.76 1.19 1.36 1.26 1.18

12 100 0.4 0.5 3 30 0.6 1000 2.00E+05 388850 0.061 3.156E-02 1.767E-02 2.405E-02 1.321E-02 24.26 17.89 22.90 15.32 1.36 1.49 1.33 1.24

13 40 1 0.7 0 30 1.5 1000 2.00E+05 958401 0.154 3.252E-03 2.328E-03 3.213E-03 2.303E-03 2.75 2.72 2.63 2.59 1.01 1.02 1.02

14 100 1 0.7 0 90 1.8 1000 2.00E+05 388850 0.061 4.239E-03 2.973E-03 3.985E-03 2.839E-03 3.59 3.37 3.73 3.50 1.06 1.06 1.07

15 40 0.8 0.9 1 0 0 1000 2.00E+05 958401 0.154 7.627E-03 5.692E-03 7.630E-03 5.692E-03 5.65 5.65 6.46 6.46 1.00 1.00

16 80 0.6 0.7 2 60 1.5 1000 2.00E+05 484889 0.076 2.174E-02 1.270E-02 1.562E-02 9.213E-03 16.78 11.58 16.40 10.48 1.45 1.56

17 80 0.6 0.9 0 30 0.75 1000 2.00E+05 484889 0.076 3.814E-03 2.739E-03 3.751E-03 2.697E-03 3.23 3.18 3.33 3.26 1.02 1.02 1.02

18 40 0.4 0.5 3 90 4.5 1000 2.00E+05 958401 0.154 1.417E-02 9.402E-03 1.065E-02 7.449E-03 9.54 6.55 8.95 5.63 1.46 1.59 1.39 1.21

19 60 0.4 0.7 1 0 0 1000 2.00E+05 643948 0.102 1.229E-02 7.268E-03 1.230E-02 7.268E-03 9.60 9.61 9.12 9.12 1.00 1.00

20 100 0.8 0.3 2 0 0 1000 2.00E+05 388850 0.061 2.734E-02 1.496E-02 2.734E-02 1.497E-02 21.53 21.53 20.23 20.24 1.00 1.00

21 60 0.6 0.3 1 30 1 1000 2.00E+05 643948 0.102 1.261E-02 7.304E-03 1.171E-02 6.724E-03 9.86 9.11 9.18 8.19 1.08 1.12

22 20 0.8 0.3 0 0 0 1000 2.00E+05 1874671 0.315 2.680E-03 2.017E-03 2.682E-03 2.017E-03 2.27 2.27 2.11 2.11 1.00 1.00 1.00

23 20 0.6 0.3 3 30 3 1000 2.00E+05 1874671 0.315 8.208E-03 6.555E-03 7.854E-03 6.305E-03 4.54 4.24 4.25 3.83 1.07 1.11 1.07 1.03

24 40 0.8 0.9 0 60 3 1000 2.00E+05 958401 0.154 3.020E-03 2.281E-03 2.942E-03 2.233E-03 2.56 2.49 2.55 2.47 1.03 1.03 1.05

25 20 1 0.9 3 30 3 1000 2.00E+05 1874671 0.315 6.596E-03 6.303E-03 6.393E-03 6.079E-03 3.17 3.00 3.82 3.44 1.06 1.11 1.07 1.03

26 100 0.4 0.7 3 90 1.8 1000 2.00E+05 388850 0.061 3.023E-02 1.759E-02 1.547E-02 9.688E-03 23.14 10.61 22.76 9.35 2.18 2.44 1.98 1.72

27 80 0.4 0.3 0 90 2.25 1000 2.00E+05 484889 0.076 3.974E-03 2.770E-03 3.768E-03 2.654E-03 3.36 3.19 3.38 3.19 1.05 1.06 1.07

28 20 0.4 0.5 0 90 9 1000 2.00E+05 1874671 0.315 2.673E-03 2.015E-03 2.600E-03 1.976E-03 2.26 2.20 2.10 2.04 1.03 1.03 1.07

29 60 0.4 0.3 3 0 0 1000 2.00E+05 643948 0.102 2.025E-02 1.221E-02 2.026E-02 1.221E-02 14.68 14.69 13.67 13.67 1.00 1.00 1.00 1.00

30 80 0.4 0.5 2 60 1.5 1000 2.00E+05 484889 0.076 2.280E-02 1.278E-02 1.620E-02 9.226E-03 17.68 12.07 16.54 10.51 1.46 1.57

31 20 0.4 0.9 3 90 9 1000 2.00E+05 1874671 0.315 7.336E-03 6.441E-03 6.545E-03 5.794E-03 3.80 3.13 4.06 2.96 1.21 1.37 1.20 1.09

32 100 0.6 0.9 2 0 0 1000 2.00E+05 388850 0.061 2.030E-02 1.449E-02 2.031E-02 1.449E-02 15.55 15.56 19.43 19.43 1.00 1.00

33 40 0.4 0.5 0 60 3 1000 2.00E+05 958401 0.154 3.325E-03 2.345E-03 3.236E-03 2.296E-03 2.81 2.74 2.66 2.58 1.03 1.03 1.05

Excluding Pressure Including Pressure

Ki Calculated from

FEA

Ko Calculated from

FEA Pressure Effect Correction Factors

Run D/T t/T d/D

Youngs

Modulus

(Mpa)

Moment

through pipe

producing

50Mpa stress

H charact

eristicL/D

Pstrs

(Mpa)

Pressure

(Mpa) D





Table 5-18: FEA runs, their factor levels, and results, for bends with trunnion attachments and including pressure (Table 2 of 2).

In-plane

Rotation

(Radians)

Out-of-plane

Rotation

(Radians)

In-plane

Rotation

(Radians)

Out-of-plane

Rotation

(Radians)

Excluding

pressure

Including

pressure

Excluding

pressure

Including

pressure

Calculated from

FEA for in-plane

moments

Calculated from

FEA for out-of-

plane moments

Thomson and

Spence

(1983)

Lubis and

Boyle

(2003)

ASME

B31.3

(2006)

34 40 0.4 0.9 2 30 1.5 1000 2.00E+05 958401 0.154 1.065E-02 7.880E-03 1.000E-02 7.312E-03 7.38 6.83 8.27 7.30 1.08 1.13

35 20 1 0.9 1 90 9 1000 2.00E+05 1874671 0.315 4.487E-03 3.908E-03 4.237E-03 3.697E-03 3.00 2.79 3.48 3.12 1.08 1.11

36 60 0.4 0.9 1 90 3 1000 2.00E+05 643948 0.102 1.091E-02 7.145E-03 8.815E-03 5.822E-03 8.43 6.65 8.91 6.66 1.27 1.34

37 20 0.4 0.3 1 60 6 1000 2.00E+05 1874671 0.315 5.840E-03 4.125E-03 5.537E-03 3.955E-03 4.15 3.89 3.84 3.55 1.07 1.08

38 80 1 0.3 3 30 0.75 1000 2.00E+05 484889 0.076 2.607E-02 1.496E-02 2.099E-02 1.195E-02 19.61 15.30 18.31 13.20 1.28 1.39 1.26 1.18

39 100 0.4 0.3 0 30 0.6 1000 2.00E+05 388850 0.061 4.257E-03 2.981E-03 4.183E-03 2.932E-03 3.61 3.54 3.74 3.66 1.02 1.02 1.02

40 100 0.8 0.7 1 30 0.6 1000 2.00E+05 388850 0.061 1.665E-02 9.568E-03 1.484E-02 8.372E-03 13.29 11.76 13.01 10.98 1.13 1.18

41 100 1 0.7 3 60 1.2 1000 2.00E+05 388850 0.061 2.919E-02 1.752E-02 1.802E-02 1.101E-02 22.26 12.77 22.64 11.59 1.74 1.95 1.65 1.48

42 100 0.8 0.3 0 60 1.2 1000 2.00E+05 388850 0.061 4.253E-03 2.980E-03 4.093E-03 2.887E-03 3.60 3.47 3.74 3.58 1.04 1.04 1.05

43 40 0.6 0.3 2 90 4.5 1000 2.00E+05 958401 0.154 1.294E-02 8.089E-03 1.012E-02 6.594E-03 9.33 6.93 8.62 6.08 1.35 1.42

44 80 1 0.9 1 60 1.5 1000 2.00E+05 484889 0.076 1.175E-02 8.205E-03 9.911E-03 6.805E-03 9.14 7.57 10.70 8.32 1.21 1.29

45 20 0.4 0.5 2 0 0 1000 2.00E+05 1874671 0.315 7.160E-03 5.434E-03 7.160E-03 5.434E-03 4.46 4.46 4.21 4.21 1.00 1.00

46 60 0.8 0.3 2 30 1 1000 2.00E+05 643948 0.102 1.812E-02 1.051E-02 1.597E-02 9.189E-03 13.71 11.89 12.70 10.46 1.15 1.21

47 20 1 0.3 1 0 0 1000 2.00E+05 1874671 0.315 5.806E-03 4.120E-03 5.806E-03 4.121E-03 4.12 4.12 3.84 3.84 1.00 1.00

48 100 0.4 0.7 0 60 1.2 1000 2.00E+05 388850 0.061 4.251E-03 2.978E-03 4.080E-03 2.884E-03 3.60 3.45 3.74 3.58 1.04 1.04 1.05

49 40 1 0.5 2 60 3 1000 2.00E+05 958401 0.154 1.269E-02 8.070E-03 1.075E-02 6.979E-03 9.11 7.47 8.59 6.74 1.22 1.27

50 100 1 0.5 1 30 0.6 1000 2.00E+05 388850 0.061 1.703E-02 9.620E-03 1.518E-02 8.404E-03 13.62 12.04 13.10 11.03 1.13 1.19

51 100 0.4 0.3 3 60 1.2 1000 2.00E+05 388850 0.061 3.174E-02 1.768E-02 2.033E-02 1.100E-02 24.42 14.74 22.92 11.57 1.66 1.98 1.65 1.48

52 100 0.4 0.5 3 30 0.6 1000 2.00E+05 388850 0.061 3.156E-02 1.768E-02 2.405E-02 1.320E-02 24.27 17.90 22.91 15.32 1.36 1.50 1.33 1.24

53 20 1 0.3 3 90 9 1000 2.00E+05 1874671 0.315 8.179E-03 6.552E-03 7.150E-03 5.873E-03 4.52 3.64 4.25 3.09 1.24 1.37 1.20 1.09

54 100 0.4 0.9 3 60 1.2 1000 2.00E+05 388850 0.061 2.416E-02 1.725E-02 1.611E-02 1.099E-02 17.98 11.15 22.18 11.56 1.61 1.92 1.65 1.48

55 40 0.6 0.7 2 90 4.5 1000 2.00E+05 958401 0.154 1.205E-02 8.020E-03 9.575E-03 6.566E-03 8.57 6.47 8.51 6.04 1.32 1.41

56 60 1 0.3 0 0 0 1000 2.00E+05 643948 0.102 3.687E-03 2.563E-03 3.687E-03 2.564E-03 3.12 3.12 3.03 3.03 1.00 1.00 1.00

57 80 0.6 0.5 0 0 0 1000 2.00E+05 484889 0.076 3.980E-03 2.769E-03 3.981E-03 2.770E-03 3.37 3.37 3.38 3.38 1.00 1.00 1.00

58 20 1 0.9 3 30 3 1000 2.00E+05 1874671 0.315 6.596E-03 6.303E-03 6.393E-03 6.079E-03 3.17 3.00 3.82 3.44 1.06 1.11 1.07 1.03

59 80 0.8 0.5 3 90 2.25 1000 2.00E+05 484889 0.076 2.584E-02 1.495E-02 1.460E-02 9.142E-03 19.42 9.88 18.30 8.44 1.97 2.17 1.78 1.53

60 20 1 0.3 1 0 0 1000 2.00E+05 1874671 0.315 5.806E-03 4.121E-03 5.806E-03 4.121E-03 4.12 4.12 3.84 3.84 1.00 1.00

61 20 0.8 0.5 1 60 6 1000 2.00E+05 1874671 0.315 5.687E-03 4.107E-03 5.401E-03 3.940E-03 4.02 3.78 3.81 3.53 1.06 1.08

62 100 1 0.3 2 90 1.8 1000 2.00E+05 388850 0.061 2.734E-02 1.496E-02 1.539E-02 8.856E-03 21.53 11.38 20.23 9.87 1.89 2.05

63 40 0.8 0.5 3 0 0 1000 2.00E+05 958401 0.154 1.406E-02 9.394E-03 1.406E-02 9.394E-03 9.45 9.45 8.93 8.93 1.00 1.00 1.00 1.00

64 100 0.8 0.9 2 90 1.8 1000 2.00E+05 388850 0.061 1.929E-02 1.440E-02 1.240E-02 8.830E-03 14.69 8.84 19.28 9.82 1.66 1.96

65 20 0.6 0.7 1 30 3 1000 2.00E+05 1874671 0.315 5.458E-03 4.079E-03 5.338E-03 3.995E-03 3.82 3.72 3.77 3.62 1.03 1.04

66 60 0.8 0.7 1 90 3 1000 2.00E+05 643948 0.102 1.211E-02 7.246E-03 9.600E-03 5.880E-03 9.44 7.31 9.08 6.76 1.29 1.34

Moment

through pipe

producing

50Mpa stress

H charact

eristicL/D

Pstrs

(Mpa)

Pressure

(Mpa) D

Excluding Pressure Including Pressure

Ki Calculated from

FEA

Ko Calculated from

FEA Pressure Effect Correction Factors

Run D/T t/T d/D

Youngs

Modulus

(Mpa)





6.0 Discussion

6.1 Benefits of the Report’s Completion

As discussed in the introduction, there are 2 ways in which the effects of a trunnion

are included in a piping system stress analysis: they can be ignored, or derived using

FEA. Both of these approaches have disadvantages associated with them: one is

simplistic and can lead to significant errors, while the other is time consuming and

requires experienced FEA users.

The reports in the current literature which do consider the effect of trunnions on a

piping system do not cover a large enough parameter range to be broadly applicable

in the calculation of flexibility factors, and also ignore important model parameters

which significantly affect system flexibilities. Because of these 2 reasons they are not

a suitable replacement for FEA in a piping system that requires accurate flexibility

factors to be calculated.

The approach taken in this report is unique in that an attempt is made to capture

within a single report and set of equations all of the model parameters that

significantly alter the flexibilities of piping systems of which trunnions are part over

the full range of values in which they are found in a piping system. This is done by

taking full advantage of the power of FEA and Regression analysis, and collecting a

broad range of current literature for the purposes of model validation.

Having a single set of equations which can be used to calculate flexibility factors that

describe trunnions in a piping system increases the ease of their use, and the

inclusion of these equations in a user friendly excel spreadsheet does this even

more. The final culmination of such an approach is the more accurate inclusion of

trunnion effects in future piping system stress analyses.

Thus it can be seen that a direct benefit of the use of this report is an increase in the

accuracy of stress analyses performed. This is because it has been shown that

modelling the flexibility factors associated with trunnions can have a significant effect

on the location of load within a piping system.





6.2 The Practical Use of the Equations Created

Though the final equations created for the calculation of the various flexibility factors

do not lend themselves to the easy understanding of the way in which flexibility

factors vary with the change of the model parameters, this has not been a problem

for the analyst, who was able to

use 3D surface plots. These are

available in Design Expert, and

allow the more accurate

understanding of the equations

produced.

As an end user does not have

access to such plots, his

understanding of the equations

will be limited. This is a

significant disadvantage of the

linear regression analysis

performed. Other equations, created in the current literature do not suffer from this

same problem and the trends they represent are easily understandable. To create

similar equations from the results using non-linear regression analysis has been

recommended as an area for future work

A second limitation of the equations created is their length, with the longest having 23

terms. Once more this is a significant limitation, especially if the equations were used

manually. To overcome this, the author has created an Excel spreadsheet which

contains the various equations created. This result of this is that the equations are

easily and quickly usable.

The spread sheets, together with details of the best way to include the equations in

an Excel spreadsheet is detailed in Appendix C, which discusses how to create User

Defined Functions.

Figure 6-1: Example of 3D surface plot taken from

Design Expert, which highlights the mainly

proportional relationship Ki and t/T





6.3 Trunnion Attachments on Straight Pipe

6.3.1 Accuracy of the FEA Results and Equations Developed

In Table 5.5, the results for this report have comparative values included from the

EPRI (1998b) report where they are applicable. The general trends between that and

this report are discussed in more detail in this section, and also compared against the

equations provided by Rodabaugh and Moore (1979) for branches on straight pipe.

The Rodabaugh and Moore report is selected for comparison as the equations from it

are included in the ASME BPV (2007) code, and are widely used in industry.

The only portion of the results which can be compared against these 2 reports where

the parameters w/do = 0, L/D = 2.5. This is the most significant part of the equation,

as these are the parameters for which the trunnion-pipe connection is most flexible.

In addition to this, the mesh used in the other models tested is similar, and so its

validity can be inferred from the validity of the cases mentioned above.

Figure 6-2: Plot of the flexibility factors against the

variable d/D





Figures 6-2 to 6-4 show the trends

between the different equations, which

are discussed below:

- The branch flexibility factors

calculated should always be

more flexible than the

trunnion flexibility factors as

the branch configuration

does not have a plug at the

branch-pipe connection.

Where the EPRI equations

provide flexibility factors

higher than the Rodabaugh

and Moore equations, they

are probably in error.

- The slope of the current

equations correlate well

with the Rodabaugh and

Moore equations, varying

in a similar manner with

changes in the equation

variables. This correlation

is significantly better than

that between the

Rodabaugh and Moore

and EPRI equations.

Figure 6-3: Plot of the flexibility factors against

the variable D/T with d/D = 0.5

Figure 6-4: Plot of the flexibility factors against the

variable D/T with d/D = 0.3





- The effect of the length of attached pipe is discussed in the current

literature survey. For the attached pipe length, the EPRI report holds the

value of L/D constant for FEA models with different d/D ratios. This is

compared against the current report, which held the value of L/do constant

in its FEA models. The approach taken in FE/Pipe’s “auto length” feature is

also to hold the L/do ratio approximately constant. Because the area over

which the trunnion-pipe connection local deformation occurs increases

significantly with an increase in d/D, it is considered that the holding of the

L/d ratio constant in the different models is a better method of keeping the

effect of the length of straight pipe in the model constant.

A comparison was also perfomed between a beam model using the current

equations, and an experiment provided in the EPRI (1998b) report, the details of

which are provided in Appendix D. The deflections observed in the model based on

the current equations are 24% lower than those from the experiment. This is within

the report tolerance and maximum difference of the equations from FEA results, but

is a poor level of agreement. The EPRI equations faired slightly better, being 17%

lower than those from this experiment. It should be noted that the experimental data

provided is limited, with only one trunnion configuration tested. The results of further

and more comprehensive comparisons would be interesting, but a brief scan of the

current literature provided no relevant experiments.

In addition to this, further notes on the equations created, and their validity are:

- The equations developed based on the FEA results, which include the 2 new

model parameters L/do and w/do, show the trends expected with the changes

in the values of these parameters. These trends are a decrease in flexibility

with a decrease in the parameter L/do, and with an increase of the parameter

w/do.

- The program used for the FEA runs, FE/PIPE, is template driven with a

largely predefined mesh, and appropriate element type. It is widely used in

the piping industry. This improves the confidence in the FE model, and the

results taken from it.





- The FEA results were insensitive to an increase in the mesh density,

suggesting that they had converged

As a conclusion of the above points, the FEA results provided for trunnion flexibilities

are useful for design purposes, providing acceptably accurate flexibilities factors.

However, the validation of the equations against experimental data suggests that the

trunnion flexibility factors may be underestimated by the current equations.

6.3.2 Advantages and Disadvantages of the Equations and Beam Model

Selected

Because of the established nature of the beam model to be used to represent

trunnion-pipe and branch-pipe connection flexibilities in the current literature, there

are no specific improvements in the beam model selected for use in this report.

The selection of the extra model parameters to be used to describe the equations

developed does significantly improve the equations developed. As trunnion pads and

adjacent flanges are often present in a trunnion on straight pipe configuration, the

inclusion of these parameters significantly increases the number of configurations for

which these equations can be used.

Unfortunately, the inclusion of all model parameters that significantly affect the

trunnion-pipe connection flexibilities was not possible. Because of time constraints on

the project, the effect of the least significant model parameter that affects trunnion

flexibilities had to be excluded from the FEA runs performed. The results of a

screening study showed this to be pressure. For thin walled piping, the effect of high

pressures on the trunnion-pipe connection flexibility factors can still be significant,

and this is a limitation on the use of the equations developed.

Further analyses will ideally be performed to create these equations, which will most

probably be in the form of equations 5.8 and 5.9.





6.4 Trunnion Attachments on Bends and Bends with Trunnion

Attachments

The results for the decisions taken in the representation of bends with trunnions

associated are considered together as they complement each other in their use.

6.4.1 Accuracy of the FEA Results and Equations Developed for Bends

with Trunnion Attachments

The results and equations presented in this report for bends with trunnion

attachments are good, with a high level of agreement with the current literature, and

equations that adequately predict the flexibility factors created from the FEA results.

In Table 5.16, the results from

the FEA analyses for bends

without internal pressure had

comparative values included

where they were available in

the current literature. Figure 6.5

is a plot of various flexibility

factors over a range of

parameters. This shows the

level of agreement between the

various equations for predicting

flexibility factors.

As well as this, the results show

all of the expected trends. The flexibility factors show a significant increase with an

increase in the value of D/T, a significant decrease with the value of L/D,

independence of the parameter t/T, and a smaller decrease in flexibility as the value

of d/D increases. The Paulin Research Group (2003) agree with this final point,

saying that trunnions generally only start to significantly affect a bends flexibility at

d/D ratios of around 0.7.

Figure 6-5: A comparison of the equations created

against similar results from the current literature





The FEA results for the analyses including pressure also show a good level of

agreement with the current literature, particularly with equation 3.21 for bends with

long tangent pipes, and with results from the graph created by Thomson and Spence

(1983) for those with flanged ends. These results were in such a good level of

alignment that an equation based on equation 3.21 which predicts the pressure

correction factor for bends at all L/D ratios was easily created using manual analysis.

This can be seen from the tables of results 5-17 and 5-18. Plotting a graph to see the

variations in flexibility factors calculated in this report again those from the Lubis and

Boyle report is thus pointless, as they are the same for the L/D = 3 case.

Once more, all of the trends expected in the results for the pressure analyses are

present: a significant increase in the pressure correction factors with an increase in

the values of L/D, an increase in the pressure correction factors with increasing

internal pressures, and an increase in the pressure correction factor with an increase

in the parameter D/T, with the latter parameter being more significant.

The equations developed for predicting the bend flexibility factors and pressure

correction factors are also well within the accepted report tolerances.





6.4.2 Accuracy of the FEA Results and Equations Developed for

Trunnion attachments on Bends

The analysis of trunnion-bend connections, and the creation of flexibility factors

which can be used to represent these connection’s flexibilities has not been widely

discussed in the current literature.

Significant difficulties arose in the creation of flexibility factors from FEA results.

Initially the report planned to include the model parameter L/D in FEA analyses for

trunnion-pipe connections. It was found during preliminary analyses that as the L/D

ratio is increased from 0, the goodness of model measure suggested by Rodabaugh

and Moore (1979) quickly falls to less than 0.1, making the results of any equations

developed unreliable and subject to small model and measurement error. This is

because of the significant increase in the flexibility of the bend that occurs with an

increase in L/D. Because of the trunnion-bend connections location on the midpoint

of the bend, its closest flange point is still some distance away, and so the fact that

the parameter L/D was not included in the model is not as significant as it is for a

trunnion-pipe connection. This was confirmed in the experimental and beam model

validation, which produced reasonable results even though the parameter L/D was

not included in the trunnion-bend connection flexibility factor equations

The final flexibility factors created appeared reasonable. It was expected that the

trunnion-bend connection would be significantly stiffer than the trunnion-pipe

connection because of the doubly curved shell present on the bend, this was

confirmed. Interestingly, the difference between the magnitude of the in-plane and

out-of-plane flexibility factors was significantly smaller. This can be interpreted as

being due to the fact that the shapes around which the trunnion pivots under in-plane

and out-of-plane bending are more similar to each other on a bend than those around

which a trunnion pivots under the different direction moments on a straight pipe.

Looking at Table 5-10, it is also interesting to note that the report the EPRI (1998a)

report flexibility factors are always smaller than the current equations, but do follow

the same trend. The author stated in the current literature survey that these values

are best treated as relative rather than absolute.





The equations developed are well within the accepted report tolerances. As well as

this, they are significantly shorter than other equations developed in this report,

increasing their ease of understanding.

6.4.3 Advantages and Disadvantages of the Representation Methods

Selected

The final beam model selected is that shown in Figure 5-3. There are a number of

advantages in its use which were not a feature of other beam models suggested in

the literature:

- One of the main advantages of the beam model selected is the selection of

the point of attachment of the trunnion as its tangent point.

Specifying of the trunnion-bend connection point at its tangent point, and near the

centre of the bend is a good approximation of its point of action on the bend, which

differs dependent upon the direction of resultant moment through the bend. This can

be confirmed by looking at table 6-1, which shows the agreement between the

rotations of the created beam model against the rotations of FE/Pipe when different

boundary and load conditions were applied to the 2 models.

- Another significant advantage of the beam model selected is that the bend is

split in half at the trunnion tangent point, and different flexibility factors can be

applied to each half of the bend.

This allows the bend flexibility to be changed on either side of the trunnion to

represent when different end conditions exist on the bend. This allows for the more

accurate representation of the amount of flexibility available in the bend on either side

of the trunnion (the flexibility factors can vary significantly). It also does not cause the

overall flexibility factor of the bend to change. Looking at the format of the ASME

B31.3(2006) equations which consider a single and double flanged bend, it can be

seen that a similar assumption is made, with a single flanged bend having flexibility

factors midway between a double flanged and unflanged bend.





This representation method also has advantages in the input of bend flexibility factors

into beam analysis programs, which sometimes only allow the specifying of a single

flexibility factor. When different end conditions exist on a bend, ki and ko differ

significantly, as ko is not considered as acting throughout the bend (out of plane

bending at one end transfers to torsion at the other). However, when the use of a

separate bend flexibility to represent the bend flexibility of each half of the bend is

used, ki and ko are similar, and taking their average for use as the flexibility factor is a

smaller change to each individual value.

As well as validating the selected beam model selected against further FEA, it was

also compared against experimental data from the EPRI (1998a) report. This

comparison is summarised in Appendix D. The deflections observed in the model

based on the current equations and suggested beam model are 10% higher than

those from the experiment. In comparison, the EPRI equations and beam model,

produced deflection 17.5% lower than those from the experiment.

The agreement of both sets of results with the experimental data is reasonable,

though the current equations were better. Once more, the available experimental

data was not comprehensive, and it would be useful to compare the current

equations and beam model created with further experimental data.

Table 6-1: Ratios of the deflections observed across FE and beam models under

various loading and boundary conditions

Boundary 1 Boundary 2 Boundary 3

ComplexLoad1 n/a In-plane fixed fixed moment 19 19 0.5 0.5 3 3 0.932

ComplexLoad2 n/a In-plane fixed free moment 19 19 0.5 0.5 3 3 1.036 0.922

ComplexLoad3 n/a In-plane moment fixed fixed 19 19 0.5 0.5 3 3 0.875

ComplexLoad4 n/a In-plane free fixed force 19 19 0.5 0.5 3 3 1.172 0.993

ComplexLoad4 n/a In-plane free moment fixed 19 19 0.5 0.5 3 3 0.899 0.986

ComplexLoad5 n/a Out-of-plane fixed fixed moment 19 19 0.5 0.5 3 3 0.989

ComplexLoad6 n/a Out-of-plane fixed free moment 19 19 0.5 0.5 3 3 1.612 * 1.143

ComplexLoad7 n/a Out-of-plane moment fixed fixed 19 19 0.5 0.5 3 3 0.953

ComplexLoad8 n/a Out-of-plane free fixed force 19 19 0.5 0.5 3 3 0.906

ComplexLoad8 n/a Out-of-plane free moment fixed 19 19 0.5 0.5 3 3 1.094 0.977

ComplexLoad9 Bound. 1 In-plane fixed fixed moment 76 76 0.5 0.5 3 0 0.996

ComplexLoad10 Bound. 1 In-plane fixed free moment 76 76 0.5 0.5 3 0 0.604 * 0.739

ComplexLoad11 Bound. 2 In-plane moment fixed fixed 76 76 0.5 0.5 3 0 1.025

ComplexLoad12 Bound. 2 In-plane free fixed force 76 76 0.5 0.5 3 0 0.956 0.755

ComplexLoad13 Bound. 1 Out-of-plane fixed fixed moment 76 76 0.5 0.5 3 0 1.022

ComplexLoad14 Bound. 1 Out-of-plane fixed free moment 76 76 0.5 0.5 3 0 1.226 1.130

ComplexLoad15 Bound. 2 Out-of-plane moment fixed fixed 76 76 0.5 0.5 3 0 1.100

0.188

1.00

18.8

61.2

Standard Deviation of

Average of Ratios

D/T d/t d/D t/T L1/D L2/D

Coefficient of Variation(%)

Maximum Difference (%)

* These ratios are for the same model boundary conditions and are for the rotations observed at the free end of the

bend. These rotations are small in comparison to the trunnion rotations, and their error is perhaps best interpreted

as the results of a difference point of application of the trunnion moment under the different moment loading

configuration

Ratio of Rotations measured FE/Pipe to

those measured in Caesar

Model Name

Location

of pipe

length L2

Moment

Direction

Boundary

1

Boundary

2

Boundary

3





6.5 A Critique of the Tools Used During the Project

6.5.1 Current Literature Review Tools

A number of tools recommended by Sharp et Al (2002) were used in the current

literature review to help identify subject matter for research that is applicable to the

project, to find papers for that subject matter, and to keep track of the subject areas

already researched.

Summaries were made of each of the papers reviewed, with particular attention being

paid to those subject areas relevant to the project. This is a handy reference for when

questions are raised on particular subject matters, as these paper summaries can be

referred to see what papers contain relevant information.

A relevance tree was also created and maintained to highlight new subject matters

which can be researched, and also to show which research has been completed.

This is shown in figure 6.6 below

Flexibility Factors

Piping Component

Emperical Formulae

Definition of flexibility

Factors

Flexibility Factors

Required

governing Parameters /

equations & current literature

Similar Systems: Straight pipe,

radial nozzles on vessels

use of FEA in previous

studies


studies

Relevance Tree: Piping Components and their

Flexibility FactorsKey:

Subject Area relevant &

researched

Subject Area relevant & not

yet researched

Subject Area irrelevantStress Intensification

Factors







Generic Formulae for

Calculating Flexibilty

use of FEA in

previous studies


studies

Similar Systems: Mitred

Bends

Straight Pipe with trunnionsElbows with

trunnions

Similar Systems:

Elbows

Figure 6-6: The relevance tree created during the current literature survey





In the literature survey, Electronic databases and internet search engines were used

to great effect to find papers relevant to the project. These include:

- Science Direct

- Industry Specific Websites (WRC, ASME, IASMiRT)

- Knovel

- Internet Search engines e.g. Google

In addition to this, the references from the papers read could often be used as a

suggestion for new papers to review. These papers were then searched for using an

internet search engine, the British library online catalogue, or were requested through

the inter-library loan facility.

These tools helped make the current literature survey the best run part of the project,

producing good and relevant results.

6.5.2 FE/PIPE

FE/PIPE is advertised as a template driven software tool customised for the FEA of

piping systems. During the interim report, the author expressed his dislike of this tool,

and its haphazard user interface and user manual.

After taking a long time to become familiar with the interface, the author now thinks

very highly of FE/PIPE. The accuracy of results and ease of altering the FE models

created in the template forms enables the quick and accurate calculation of results. In

addition, the support team for FE/PIPE responded promptly to any queries that the

author had, with this response being illuminated by an understanding of the piping

system being analysed.





6.5.3 Stat-Ease Design Expert 8.0

Stat-Ease Design Expert is a tool for design and analysis of experiments. It enables

part factorial experiments to be designed that avoid confounding of interaction effects

up to a specified level. As well as this it is a tool for the linear regression analysis of

experimental results. Automatic statistical measures of the importance of factor and

interaction effects are automatically produced, which greatly helps in the analysis of

an experimental runs results.

The biggest disadvantage of the design of experiment software is that the regression

analysis is limited to linear regression analysis. Mason et Al (2003) highlight that high

order polynomial terms may be necessary to model relationships that could be simply

stated using nonlinear terms, and this means that the created equations are a lot

more complicated than if they were created using nonlinear regression analysis

This was perhaps the biggest limitation of the software and for that reason it is not

recommended as a tool for future analyses.

6.5.4 Project Planning Tools

The use of three project planning tools is discussed in this section: the Gantt chart,

Project Aim, and Project Objectives. These are all interconnected and their creation

and maintenance should be iterative to ensure their applicability and feasibility.

In this project, all three were well defined and correctly connected in the downwards

direction: the aim was translated into objectives, which were then used as the basis

of the project schedule and Gantt chart. However, the feedback loop from the Gantt

chart back up the chain was missing. This meant that though the Gantt chart

highlighted the difficulty of the project schedule during the project planning phase,

this was not fed back into creating smaller aims and objectives. Rather, the

aggressive schedule and project aim was accepted as a risk taken by the project.

This was unnecessary, and a better approach would have been to limit the project

scope.





6.6 A Critique of the Project’s Running

One of the significant pit falls of the project is that the initial project schedule was too

aggressive, and thus all of the activities on it were not achieved by the completion of

the project. A poorly timed personal decision also contributed to this, which is

discussed below. The final project progress achieved is included in Appendix F.

One of the main points to note in the final progress achieved is that the results of this

project were required for a master project by a specific date. Because of significant

difficulties in the creation of the comprehensive equations included in this report, they

were not completed by the time they were required on the master project. This

resulted in a break away from the structured project FEA to run FEA for specific

piping systems on that project. To run these analyses for the master project took a

total of 3 weeks, which effectively removed that time from the project schedule. A

lesson has been learnt from that enough time in the project schedule should be given

to perform well planned design work, or else inefficient “fire fighting” may be the

result. In addition, this had an impact on the project’s momentum, with the author

finding it difficult to regain his rhythm.

The author also moved house about a month before the final report was due, which

took another week out of the project schedule. Important lessons were learnt from

this about taking into account how busy one’s professional life is when making

personal decisions, and vice versa.

The author could have also made better use of his time by more actively seeking

advice from the experienced members of his industry by which he was surrounded.

Especially, requesting an official supervisor to be responsible to, and who he would

be able to discuss the project with, would have greatly improved the running of this

project. Rather than seeking personal advice, the author made effective use of the

available current literature to comprehensively understand the project subject area.

Though this comprehensive review of the current literature is good, complimenting it

with personal advice is important in achieving full understanding of the project subject

area.





7.0 Recommendations for Further Work

This report aimed to use the power of FEA and regression analysis to create

predictive equations that fully describe the flexibilities of trunnions in a piping system

and whose variation can be fully described by a small number of parameters.

The predictive equations available in the current literature do not describe the

flexibility factors for trunnions using the complete range of model parameters that

effect it, and for this reason have a very small range of use. This report made an

effort to correct that. A good set of equations was created to describe the flexibility

factor of a bend with a trunnion attachment. However, the following require more

work to be done:

- For a trunnion-pipe connection the effect of internal pressure in the equation

describing the flexibility factor was not considered. Preliminary analysis

suggests that pressure can significantly affect this flexibility factor for piping

with a large D/T ratio or with a high internal pressure. Ideally, further work

would be carried out to include the pressure effect on the trunnion-pipe

flexibility factor. This is understood to be the final parameter that significantly

affects the flexibility factors of this model, and its inclusion in predictive

equations would make the equations broadly applicable to all trunnion-pipe

connections.

- In addition, the validation of the trunnion-pipe connection flexibility factors

created against further experimental data would be of use. The initial and

limited experimental data reviewed was not in as good agreement with the

FEA results as would have been liked. One reason for that could be that

perhaps the author specified too short a length of attached pipe for the

removal of end effects.





- For a trunnion-bend connection, the beam model used to extract the rotations

local to the trunnion-bend connection does not provide reasonable rotations

when the bend has attached lengths of pipe on it. This makes it hard to derive

flexibility factors from the FEA results. For this reason, the equation

developed to describe this flexibility factor does not include the effect of the

parameters L/D or internal pressure. Further work could focus in detail on the

behaviour of a trunnion-bend connection, and once this has been carried out

then further FEA runs could be completed to allow for equations including the

effect of L/D or internal pressure to be created.

Another area where further work could be carried out is the subjecting of the FEA

results from this report to non-linear regression analysis. This would enable shorter

equations which describe the flexibility factors to be developed, making them more

user friendly in hand calculations.

Finally, the approach taken in this report in creating equations that predict the

property of a piping component based on the important model parameters that affect

that property could be applied to other piping components. Any broadly applicable

equations developed can then be easily used in future stress analyses to improve

their accuracy.





8.0 Conclusions

8.1 A Summary of the Report Results and Recommendations

This report has relied on the software tools FE/PIPE and Design Expert 8.0 for the

creation of results and equations. The author would recommend the use of FE/PIPE

in future reports, but would recommend that a statistical tool with non-linear

regression analysis capability is used rather than Design Expert.

For the complete bends with trunnion attachments model the following points can be

noted:

- The Equations 5.6 - 5.9 created for calculating bend flexibility factors were in

good agreement with previously available current literature, and do not

require any further work to be performed on them. The inclusion of the

parameter L/D in these equations is reasonable and significantly increases

their accuracy compared to other equations available in the current literature.

- The equations 5.4 and 5.5 created for calculating trunnion-bend flexibility

factors produced reasonable results. The exclusion of the parameter L/D from

the equation does not appear to significantly affect the equations results,

though this could be the subject matter of a more detailed study. A report on

the effects of pressure on the trunnion-bend connection flexibility, perhaps as

part of the same study, would also further increase the accuracy of the

equations.

- The suggested beam model to be used to add the flexibility factors calculated

into a stress analysis is shown in Figure 5.4. When compared against further

FEA and experimental data, the beam model and flexibility factors created in

the report performed well. Validation of the beam model and flexibility factors

calculated against more comprehensive experimental data would be useful.

In summary the bends with trunnion beam model and flexibility factor equations can

be used in future stress analyses to significantly improve the accuracy of these

analyses, though further work could be performed on the trunnion-bend connection

flexibility.





For straight pipe with trunnion attachments the following points can be noted:

- The Equations 5.1 and 5.2 which were created for calculating trunnion-pipe

connection flexibility factors produced results similar to those available in the

current literature, though they do appear to be slightly lower. The inclusion of

the parameter L/D in these equations is reasonable and removes the need to

make assumptions about when flanges affect the trunnion-pipe connection

flexibility. The parameter that has the least significant effect on the flexibility

factors, pressure, was not included in the equations developed , and a report

on the inclusion of the effects of pressure on the trunnion-bend flexibility

factors would further increase the accuracy of these equations.

In summary, the use of the trunnion-pipe connection flexibility factors developed in

this report to in future stress analyses will significantly improve the accuracy of these

analyses. However, further work could be done to validate the results, and to include

the effects of pressure in the equations developed.





8.2 A Summary of Personal Lessons Learnt

During the writing of this report, and the prior 8 months of project work, there has

been time for reflection on the good and bad decisions made by the author. As this

report is at the conclusion of an individual project, some of the insights gained into

the management of projects are personal, and reflective of the character of the

author.

These were discussed earlier and are summarised below:

- The author has been reminded of the importance of creating realistic

objectives for a task that he is performing

- The importance of learning lessons from the current state of knowledge in a

subject has been highlighted.

- Maintaining a working relationship with, and seeking advice from, a more

experienced work supervisor is important to reduce wasted time in a project,

and to provide insights into the work being performed by the author.

- The importance of maintaining momentum in a task being performed has

been highlighted.

- The importance of taking account of busy periods in the personal and

professional life of the author, and ensuring that they do not clash, has been

highlighted.





9.0 References

1. API(2002). Axial and Centrifugal Compressors and Expander-compressors for

Petroelum, Chemical and Gas Industry Services. API Standard 617.

Washington: API.

2. ASME (2006). Process Piping. ASME Code for Pressure Piping, B31. ASME

B31.3-2006. New York: ASME

3. ASME (2007). Alternative Rules for the Construction of Pressure Vessels.

ASME Boiler and Pressure Vessel Code, Section VIII. ASME Section VIII

Division 2. New York: ASME

4. Dodge W. G. & Moore S. E. (1966) Stress Indices and flexibility factors for

moment loadings on elbows and curved pipe. Report ORNL-TM-3658.

Tennessee: ORNL.

5. EPRI (1998a) Stress Indices for Elbows with Trunnion Attachments. Palo Alto,

CA:1998. Report TR-107453

6. EPRI (1998b) Stress Indices for Elbows with Trunnion Attachments. Palo Alto,

CA:1998. Report TR-107453

7. Greenstreet W. L. (1978). Experimental Study of Plastic Reponses of Pipe

Elbows. ORNL/NUREG-24.

8. Hankinson R.F. & Albano L. D. (1989) An Investigation of Elbow Flexibility for

Elbows With Circular Trunnion Attachments, presented at the ASME/JSME

Pressure Vessels and Piping Conference, July 1989.

9. Mason et Al, (2003). Statistical Design and Analysis of Experiments. 2nd

Edition. Hoboken: New Jersey. pp. 514-516.

10. Paulin Research Group (no date). Pipe Stress Errors – When you’re off by

twenty times! Available from: http://www.paulin.com/WEB_FESIF.aspx [

accessed online 04 April 2010].

11. Paulin Research Group (2003). FE/PIPE 4.111 Program Manual. Houston:

Paulin Research Group.

http://www.paulin.com/WEB_FESIF.aspx





12. Rodabaugh, E.C et Al (1978). End Effects on Elbows Subjected to Moment

Loadings. ORNL/Sub-2913/7.

13. Rodabaugh and Moore (1979). Stress Indices and Flexibility Factors for

Nozzles in Pressure Vessels and Piping. ORNL/Sub-2913/10

14. Sharp J. Et Al(2002) The management of a student research project. 3rd

Edition. Aldershot: Gower Publishing Limited.

15. Taupin et Al(1983). Experimental Study of Stainless Steel Pipes and Elbows

Under Pressure and Moment Loadings. 7th International Conference on

Structural Mechanics in Reactor Technology.

16. Thomson and Spence (1983). Combined Pressure and In-plane bending on

pipe bends with Flanges. 7th International Conference on Structural

Mechanics in Reactor Technology.

17. Touboul et Al (1989). Design Rules for Piping: Experimental Validation of

Flexibility and Elastic Stress Indices for Elbows Under Bending. 10th

International Conference on Structural Mechanics in Reactor Technology..

18. Whatham and Thompson (1979). The Bending and Pressurizing of Pipe Bends

With Flanged Tangents. Nuclear Engineering and Design. 54 (17-28) T

19. Xue and Sang (2006). Flexibility Factors for Branch Pipe Connections

Subjected to in-Plane and Out-of-Plane Moments. Journal of Pressure Vessel

Technology. 128 (89-94).

52357113 Final Report Rev1

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