Design_piping_systems_strathclyde

Pr^vl

r

Design & Analysis

of

PIPING SYSTEMS

A Short Course

University of Strathclyde

DESIGN & ANALYSISOF

PIPING SYSTEMS

James T BOYLE

Department of Mechanical EngineeringUniversity of Strathdydelasgow, Scotland, UK

Design & Analysis ofPiping Systems

Table of Contents

1 INTRODUCTION 11

1.1 BASIC DESIGN PHILOSOPHY 1.11.2 OVERVIEW OF COURSE CONTENTS 1.31.3 SUGGESTED READING 1.4

2 BASIC CONCEPTS 9 i

2.1 MATERIAL PROPERTIES"i^'i&LOWABLE^ 2!l2.1.1 Allowable stresses 2 12.1.2 Plasticity 2*4

2.2 MECHANICS OF PIPE BEHAVIOUR ".' 2.12.2.1 Pressure stress 2.72.2.2 Torsion stress 2!lO2.2.3 Bending stress 2.122.2.4 Combined stress 2.132.2.5 Component flexibility 2 14

2.3 PLASTIC DESIGN CONCEPTS .'. 2!l72.3.1 Limit loads 2.172.3.2 Shakedown & ratchetting . 2 20

2.4 FATIGUE - i-FACTORS Z 2.22

3 OVERVIEW OF PIPING CODES 3 l3.1 BS806: FERROUS PIPES AND PIPING 3*13.2 ANSI/ASME B31.3: REFINERY PIPING CODE 3*63.3 ANSI/ASME B31.1: POWER PIPING CODE 3*93.4 ASME BOILER & PRESSURE VESSEL CODE Z'Z 3.'lO

4 FLEXIBILITY ANALYSIS OF PIPING SYSTEMS 4 14.1 THE NEED FOR FLEXIBILITY ANALYSIS 4*14.2 FLEXIBILITY ANALYSIS OF PIPING SYSTEMS FOR THERMALEXPANSION 4.4

4.2.1 Energy Methods ""!"."."""!!!! 4!44.2.2 Matrix Displacement & Finite Element Methods 4 8

4.3 COMPUTER ANALYSIS OF PIPING SYSTEMS 4*134.4 IS FLEXIBILITY ANALYSIS RELIABLE? ZZ 4.*35

4.4.1 Is Code flexibility analysis conservative? 4^354.4.2 Does flexibility analysis represent real behaviour? 4^38

Tabl&ofContents

5 BEHAVIOUR OF COMPONENTS: PIPING ELBOWS 5.1

5.1 INTRODUCTION 5.15.1.1 In-plane bending of a pipe bend - von-Karman's analysis 5.15.1.2 Behaviour of piping elbows 5.5

5.2 CURRENT DESIGN PHILOSOPHY 5.105.2.1 BS806 5.105.2.2 ANSI B31.1 5.145.2.3 ANSI B31.3 5.165.2.4 ASME III Class 1 5.17

5.3 FUTURE PROSPECTS 5.205.3.1 ANSI B31 i-factors & ASME III C-factors 5.205.3.2 ASME III B-factors 5.245.3.3 Summary 5.26

6 BEHAVIOUR OF COMPONENTS: BRANCHES 6.1

6.1 INTRODUCTION 6.16.2 STRAIGHT PIPE 6.16.3 BRANCHES - PRESSURE LOADING 6.26.4 BRANCHES - MOMENT LOADING 6.96 5 BRANCHES - COMBINED PRESSURE & MOMENT LOADING 6.116!6 ASME IH PROCEDURE FOR BRANCH DESIGN 6.136.7 BRANCH JUNCTION FLEXIBILITY 6.15

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Design & Analysis of Piping Systems . -

1 INTRODUCTION

The tradition of pipework design has a long and established history, but has undergone a

significant change in the past few decades. The increasing public need for structural safety,

together with the additional requirement of economy has required the development and

adoption of new design methods and associated analytical tools, for the most part based

oncomputermethods. Indeedmost piping design and analysis is done todayusingcomputer

pipe stress analysis packages. Thus this course is primarily concerned with piping stress

analysis rather than problems of detailed design. The reason for this is straightforward:

for a 'safe' piping system design it is necessary to avoid any overstressing of the piping

components whichmaylead to structural failure or overloading ofthe connectedequipment.

It has become common practice to design and fabricate pipework to some particular design

code or specification: design by analysis and associated criteria for pipework has been in

existence since the issue of the American National Standards Institute ANSI B31.1 Code

for Power Piping - most national codes have followed this approach since then. Thus the

common basis for 'safe' piping design is analysis.

In the writer's experience piping design and analysis has become rather routine: either

some standard design procedure is adopted, which may avoid analysis altogether, or the

whole design assessment is handed over to some prescribed analysis package. In the latter,

if the design criteria are not met then the pipework designer may use his experience to

adjust the design ifthis is possible. In either case it may be argued that the routine nature

of the design process leads to a lack of understanding as to what we are calculating. This

problem exists in all areas ofthe piping industry and is evident when design requirements

are specified without a basic understanding of the background to the code. It is fortunate

that the people who devised the code rules, and apparently the pipe itself, understand so

that the design procedure works. However it is equally wrong to presume that the loads

and stresses we are calculating are exact when in fact we are only calculating certain

theoretical results required by the design procedure in a way which the code allows and

expects. The code procedure does make an attempt to develop a conservative estimate; butas we shall see sometimes this is in error.

The aim ofthis course then is to provide the necessarybackground to the design and analysis

sections ofthe various codes in the hope ofproducing a better educated and aware piping

designer. Traditional manual methods of calculation are not covered; it is felt that the

designer who is using such techniques possibly has no need for a course like this - the

problem is with the piping designer who adopts computer assisted methods! To start with

we will go back to fundamentals:

1.1BASIC DESIGN PHILOSOPHY

Loads on piping systems are many and varied, but fortunately from the point of view of

the writer ofa design code can be broadly grouped according to their effect:

• internal (and external) pressure,

• dead weight effects of piping together with insulation and contained fluid,

LI

INTRODUCTION

• thermal expansion and possibly through wall thermal gradients, and /

• dynamic loadings for example due to wind, earthquake or blast loadings.

At their most basic level the various design codes and associated standards aim to provide

protection against two kinds of failure;

Firstly, recognizing that pipework is principally a means by which fluids or gases may be

transported between different plant items there must be some basic protection against a

catastrophic or "burst' type failure. This is usually provided by requiring, as an absolute

minimum: the use of standard fittings for which prototypes have been demonstrated to

meet a simple pressure burst test, that certain fabricated branch connections meet

established reinforcement rules, that the pipe wall is thick enough to prevent pressure

bursting, and the stresses arising from other sustained or occasional loads to which the

system will be subjected are kept within certain limits.

The last requirement is necessary since the basic code procedure to protect against bursting

is rather simple: in terms of stress analysis the averaged, or 'membrane', calculated stress

due to pressure must be kept below some fraction ofyield on the argument that this prevents

gross plastic yielding through the pipe wall and thus obviates bursting. The sustained

stresses must then be limited also to keep to the spirit of this requirement. However it

must also be recognized that failure of the pipe wall can also occur through mechanisms

other than gross yielding, for example creep damage at elevated temperature. This must

also be taken into account.

Secondly thermal expansion between different plant items will induce internal stress and

deformation in the pipe and end reactions on connecting equipment. Piping systems should

then have sufficient flexibility so that these stresses, deformations and forces are limited.

It seems fairly obvious that distortion and end reactions should be limited to avoid leakage

or service failure at joints or in connected equipment. But it is not clear how or why the

stress levels should be limited. Operating stresses due to thermal expansion will be cyclic

and there is then the need to protect against potential failure due to the repeated application

of stress. The design procedures should thus provide some protection against material

fatigue cracking, a leak type failure. The pioneering ANSI B31.1 provided this protection

against fatigue in a rather obscure manner, through the concept of a stress intensification

factor. However it must also be recognized that other failure mechanisms due to cyclic

stress are possible; in particular that of excessive repeated deformation due to ratchetting.

The design requirement in this case is that of shakedown; BS806 is based upon design for

shakedown.

With very few exceptions, given a minimum pipe wall thickness, the basis for piping design

is a flexibility analysis for thermal expansion and sustained loads. This allows forces, in

particular bending moments, to be evaluated on each component and resultant stresses to

be calculated and compared to code allowables. Again it must be emphasised that we are

not calculating real loadings. In the process ofcarrying out a flexibility analysis the analyst

must make certain assumptions concerning the modelling of restraints, supports, anchors

and nozzles. In the evaluation of expansion loads it is generally assumed conservative to ■

assume an infinite stiffness for anchor points; experience has demonstrated that this

assumption is acceptable and does provide a margin of safety. But our restraints are not

I

[: 1.2

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: Design: & Analysis ofPipingSystems :

rigid, so we are not calculating real loads. It must also be remembered that this assumption

is only conservative for static loads; for dynamic analysis it artificially increases the fun

damental frequency of the system. Similarly, assumptions as to the direction and stiffness

of supports and hangers are also usually less than realistic, and there is less evidence that

the usual assumptions are conservative. This will be discussed further in the course.

If we recognize that we are not representing the actual piping system behaviour and that

the calculated loads and stresses are not realistic, why should we respect the design codes?

Simply, because we have done quite well with them in the past! The problem is that many

people who use the piping design codes and analysis packages may believe that the numbers

being generated are representative of real pipe behaviour. While this may be expected inmany other areas of engineering design and stress analysis, it should not be for piping.

Perhaps what this course is trying to achieve is an educated and sceptical piping designeras an additional margin of safety.

■

1.2 OVERVIEW OF COURSE CONTENTS

Following the present lecture, the course is broadly divided into six summary lectures overtwo days:

The first day deals with the fundamental ideas of piping system design and analysis:

Lecture 2: Basic concepts, such as material allowables, behaviour of pipes under variousloading conditions and design criteria - limit loads, shakedown and fatigue - are recalled.

Lecture 3: The requirements of a representative sample of piping Codes, BS806 ANSIB31.1 & B31.1 and ASME III & VIII are summarised

Lecture 4: This covers basic concepts ofstatic piping flexibility analysis, the basic analysisassumptions which are made and the methods of analysis which are usually employed

together with a sample computer analysis and a discussion of the reliability of solution.

The second day covers the background to the design codes and the design procedures forspecific components:

Lecture 5: This covers the mechanical behaviour of piping elbows, the main source offlexibility in design, covering the state of current knowledge and stress analysis togetherwith an attempt at a rational explanation of their treatment in the Codes.

Lecture 6: (DrDMoffat, Department ofMechanical Engineering, University ofLiverpool):The difficult and sometimes mysterious behaviour of branch connections are introducedhere for the strong of heart.

1.3

INTRODUCTION ^

1.3 SUGGESTED READING ?

Included in the course notes are a collection ofresearch papers (and other documents) whichthe authors consider essential background reading. Some will be examined during thecourse. The following handbooks should also be in the library of responsible piping

designers:

MW Kellog & Co.: "Design ofPiping Systems" 2nd Ed, Wiley, 1965

S Kannappan: "Introduction to Pipe Stress Analysis" Wiley, 1986

PR Smith & TJ van-Laan: "Piping & Pipe Support Systems", McGraw-Hill, 1987

1

1.4

Design &■ Analysis of Piping. Systems"

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2 BASIC CONCEPTS

The aim of this Lecture is to summarize several basic concepts from mechanics which are

employed in the various Codes and which will be used in the following lectures. Following

a brief review of basis of a design stress and the mechanics of pipe behaviour underload,

important concepts from the theory ofplasticity - the limit load and the so-called shakedown

load - will be defined. We will also briefly summarise some basic concepts from fatigue

which form the basis of the stress intensification factor which is used in the US Codes.

2.1 MATERIAL PROPERTIES AND ALLOWABLE STRESS

l.lAllowable stresses

Allowable stresses as specified in the various codes are generally given in terms of certain

characteristic material properties and are typically classified as being either time independent or time dependent.

'Time independent allowables are related either to the (initial) yield stress or the tensile

. strength as measured in a simple tensile test, Figure 1. The yield stress is the elastic limit,

that is stresses below this value are proportional to strain and when the stresses areremoved there is no permanent distortion of the tensile specimen.

a

r

0.2%

Figure 1: Tensile test in a ductile material

2.1 ifK

BASIC CONCEPTS

The elastic limit is often difficult to determine, especially for ductile materials as shown

in Figure 1, and instead the so-called 0.2% proof stress may be used. The tensile strength

is the highest stress which the specimen can accommodate without failure, Figure 1. Care

is often needed in defining a suitable stress value since at strain levels close to failure the

specimen is either necking or suffering damage so that the simple definition of stress as

load over area needs to be modified.

As we will see later in Lecture 3, ANSI B31.1 uses allowables Sc and Sh which are the

smallest of 1/4 the tensile strength or 5/8 of the yield strength whereas B31.3 uses 1/3 the

tensile strength and 2/3 (and as high as 0.9 for austenitic stainless steels) the yield strength.

BS806 uses a factor of 0.9 or 0.8 on the 0.2% proof stress. However BS806 and ANSI B31.3

(but significantly not ANSI B31.1 and related ASME codes) both also use time dependent

allowables at higher temperatures:

The time dependent allowable is usually related to the creep rupture strength at high

temperature. At temperatures above about 1/3 of the melting temperature most metals

will exhibit creep - that is in a standard tensile test, ifthe load is kept constant the specimen

will continue to deform with time, as shown in Figure 2.

Time

Figure 2: Standard creep curve

Under constant load the rate of creep strain will decrease initially to a steady state andlater will increase rapidly until the specimen ultimately fails due to creep rupture. Thesethree phases of creep are usually termed primary, secondary and tertiary. The importantpoint here is that if creep is present the specimen will fail at most stress levels, but as thestress level decreases the time to rupture will decrease. Results from many creep tests atthe same temperature but at different (initial) stress levels can be cross plotted as creeprupture curves giving time to rupture for a given initial stress, Figure 3. However such

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-

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2.2

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Design & Analysis of Piping Systems

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jross plots invariably give rise to a high degree of scatter and it is more appropriate to

define scatter bands for a specified rupture time and to use the minimum and average

stresses from this band.

log a,

wV)

CD

V)

la

'c

Jog t R

Time to rupture

2B H 16 «Q

Log^iime in hoofi

Stress rupture data for S'imonic 80. \, log-iog plot.

Figure 3: Creep rupture curves

ANSI B31.3 thus uses an allowable which is the smaller of the time independent and the

time dependent allowable stress. The time dependent allowable stress is then the smallest

, of 67% of the average stress to cause creep rupture in 100,000 hr, 80% the minimum stress

to cause rupture in 100,000 hr or 100% of the stress to give 0.01% creep rate per hour (that

is, the rate of deformation must also be kept within bounds). BS806 uses a factor of 0.9 on

the mean stress to cause rupture in the design life at the design temperature (or the

minimum stress for sustained loads).

For cyclic loading there is of course another 'time dependent' allowable - related to the

fatigue life. We will leave discussion of this aspect until later since it is the basis of the

oncept of a stress intensification factor.

2.3

BASIC CONCEPTS

2.1.2 Plasticity

We have briefly described the fundamental idea of plasticity in the above, deriving the

concept ofinitial yield, the elastic limit, from a simple tensile test. The concept ofplasticity

is not simply that beyond yield the stress and strain are no longer proportional and exhibit

hardening (equal increments of stress give progressively greater increments of plastic

strain) but also the behaviour on unloading. A material is elastic if there is no permanent

deformation (residual strain) on unloading; rubbers are elastic, but the stress and strain

are not proportional (nonlinear), Figure 4.

LINEAR ELASTICITY NONLINEAR ELASTICITY

UNLOADING

LOADING

UNLOADING

'LOADING

Figure 4: Nonlinear elasticity

In a material exhibiting plastic behaviour, if a tensile specimen is loaded beyond yield,unloaded and subsequentlyreloaded it remains more or less linear elastic up to the previous

highest stress which was reached, Figure 5.

Hence we must be careful to denote the elastic limit as initial yield only.

For engineeringpurposes, althoughit is possible to develop a tensile (uniaxial) stress-strainrelation, usually called nonlinear hardening, to describe the tensile curve one of two simplifications is more usually adopted - either bilinear hardening or the hypothetical perfectplasticity, Figure 6. Perfect plasticity is important in the definition of possible failuremechanisms for components, in particular it is the basis for the development ofa limit load.

2.4

Design & Analysis ofPiling Systems

ELASTIC

UNLOADING

Permanent deformation

Figure 5: Plasticity - behaviour on unloading

NONLINEAR

HARDENING

BILINEAR

HARDENING

PERFECT

PLASTICITY

Figure 6: Models of material behaviour

2.5

BASIC CONCEPTS

An engineering description ofgeneral plastic behaviour is quite complex: as well as initialyield it is necessary to develop a suitable multiaxial yield criterion. That is, in a material ]

subject to multiaxial stress, what combinations ofstress cause yield to occur ? The simplest

is the maximumprincipal stress criterion which assumes that yield occurs when the largest «n

principal stress component reaches the experimentally determined yield in tension. jAlthough this is largely unrealistic, it is used in some parts of the design codes since it is

simple. The two criteria most found in practice are the Tresca criterion and the von-Mises **

criterion. 1

In the Tresca it is assumed that the value ofthe maximum principal shear stress governs «i

yield, in the latter it is assumed that it is the value ofthe root mean square ofthe principal jshears. If a!,a2,a3 are the principal stresses, then the principal shears are defined as

G>2 — G3 CT3 —CTi

and the maximum principal shear is

According to the Tresca criterion, yield under multiaxial stress occurs when the maximum

principal shear reaches a critical value. Assuming a uniaxial stress field this implies the

criterion

where oy is yield in tension (obtained from tensile tests). The Tresca criterion is inherent

in most ofthe piping Codes, as we will see. Sometimes the stress intensity, S = 2tmix is used,

so that the Tresca criterion is simply S =Gy. ,

The Tresca and von Mises criteria are shown plotted against some typical biaxial tests in

Figure 7.

Although the von Mises criterion is more accurate, the Tresca criterion is generally con

servative and is thus preferred in design. It is also essentially easier to use when applied

to specific stress sytems.

This of course is only part of the problem; the Tresca and von-Mises criteria are only used

for initial yield - we still face the problem ofdescribing multiaxial plastic behaviour beyond

yield, and for subsequent yield. Thankfully such problems need not be addressed in design.

2.6

Design & AnalysisofPiping Systems

Figure 7: Multiaxial yield criteria

2.2 MECHANICS OF PIPE BEHAVIOUR

It is worthwhile at this stage recalling some basic features of the mechanics of pipe

behaviour under pressure, bending and torsion. These will feature in later discussions of

the Codes:

2.2.1 Pressure stress

With reference to Figure 8, it is well known from elementary strength of materials texts,

that the pressure stresses in a long thin pipe under internal pressure are given by

Hoop (circumferential or transverse) stress: <** = 7"

Axial (longitudinal) stress: EL

where p is internal pressure, r mean radius and t wall thickness. The longitudinal stress

assumes that remotely the pressure gives rise to an axial force (such as with closed ends).

•

. 2.7.

BASIC CONCEPTS

Figure 8: Pressure stress in thin cylinder

As a simple rule in establishing the thickness of a pipe which must carry a given volume

of fluid at a given pressure (so that and p are prescribed) the maximum stress, a+ in this

case, should be less than the design stress, f,

ie

In fact most Codes are not quite as simple as this:

For a thick pressurised cylinder, Figure 9, Lame's equations give the internal and external

surface hoop stresses as

p{D2+d2)

°*L~ D2-d2

2pd

where d is the internal diameter, D the external diameter (D = d + It).

2.8

Design & Analysis ofPiping:Systems

Figure 9: Pressure stress in thick cylinder

There is also a radial stress such that ar = -p at d, and or = 0 at D (and an axial stress can

be developed as above). The maximum hoop stress is at the inside.

In order to establish the pipe thickness these relations, simplified for thin to moderatelythick pipes, are used together with a Tresca combined stress such that the stress intensity($ = Tm«x) is limited by the design stress, S </:

The maximum stress intensity occurs at the inside surface

c__, 2pD2

The right hand side may be written as

2pD

D2-d2 2 - +1

which can be approximated as

■• ■■

^D/t)« 1, since the second term is negligible. Hence for thin to moderate thickness, the

2.9

BASIC CONCEPTS . -

stress limit is

which may be rearranged as

pD

It is this form which is adopted in BS806 and ANSI B31 (with additional e, or E &Y factors

on design stress and pressure - additional factors of safety, or otherwise, depending on

material or pipe manufacture).

It is to be noted that a Tresca criterion for multiaxial stress is used in the development of

these formulae. *

It may be shown that the maximum hoop stress at the inside can be written as

can be similarly simplified for moderately thick pipes as

which also appears in BS806 and ANSI B31

Without going into too much detail, the longitudinal (axial) stress may be derived as

x

which is not simplified further.

2.2.2 Torsion stress

The shear stress in a (thick) cylinder under torsion, Mt, Figure 10, can be found from

elementary texts as

M,D M,(d + 2t)

t-r'?\

2.10

Design & Analysis1 ofPiping Systems :

./here

rFigure 10: Torsion stress

For thin tubes this may be simplified to

M,r

where r is mean radius, or as

v 2Z

where Z = A t, with A the enclosed area ofthe pipe centre line (A = wr2), the section modulus.

2.11

BASIC CONCEPTS

2.2.3 Bending stress

Engineer's Theory of Bending, or more correctly Euler's Theory of Bending, makes two

simplifying assumptions which capture the essential behaviour of long slender straight or

(solid) curved beams under bending:

• Plane sections remain plane during bending

• The cross section of the beam does not deform during bending

With these assumptions only longitudinal (axial) stress and strain are induced due to

bending. For a long straight pipe under a bending moment M, Figure 11, the longitudinal

stress is given for thin pipes as

Mr

tSSft

while for thick pipes at the outside

M(d+2t)

21

where*W|

or £r64

respectively.

M

Figure 11: Bending stress

• 2.12 •


Jnder combined bending M{ and Mo, Figure 12, the maximum stress at the outside of a

thick pipe is

D

M

M.

Figure 12: Combined bending

2.2.4 Combined stress

A pipe subject to bending moments (in plane and out ofplane), internal pressure and torsion

can result in a fairly complex (but tractable) stress system. The bending gives rise to

.ongitudinal stress, pressure to hoop (transverse) and longitudinal stress and the torsion

to shear stress.

Either ofthe normal stresses (transverse or longitudinal), denoted by a, must be combined

with the shear stress, x so that the maximum shear can be evaluated

(for example using a classical Mohr circle, Figure 13).

2.13

BASIC CONCEPTS

Shear stress

Figure 13: Evaluation ofcombined stress using Mohr circle

It is this maximum shear which must be limited by the design stress in the piping Codes;

again a Tresca criterion is adopted

vmax — V

or, in terms of combined stress

where the design stress may be derived from yield or proof stress, or tensile strength etc.

Most piping Codes will take as the normal stress the largest of the hoop or longitudinalstress. Strictly this is an approximation for combined loads, but apparently sufficient fordesign. (Most piping analysis software could evaluate combined stress more accurately if

the Codes allowed).

2.2.5 Component flexibility

Unfortunately some important piping components cannot be directly modelled in this wayusing simple beam bending theory. In particular a pipe bend is more flexible than anequivalent curved beam since the cross section does ovalise, Figure 14. ,

2.14

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Design ■& Analysis of Piping Systems

vcn-Karman Effect

In-plane benaing

Undeformed Deformed

Out of plane bending

Figure 14: Deformation of a pipe bend

This is known as the von-Karman effect - the ovalisation of the cross section leads to

increased flexibility and induces higher, and more complex, longitudinal and hoop stresses.

The additional flexibility ofa curved pipe is takeninto accountin the Codes using a flexibility

'xctor which is derived from more complex shell analysis. This, and other aspects of the

behaviour of pipe bends, will be examined in more detail in Lecture 5. Of course the stress

levels must also be modified and a stress intensification factor is introduced to factor the

maximum stress to basic beam (bending) stress. Other components also need some special

treatment, such as mitres, branches and expansion joints.

The flexibility factor for a pipe bend is necessary because, as we will see in Lecture 4, the

piping system is being analysed using beam bending theory. The concept of a flexibility

factor is not as simple as it may at first seem; we will examine this later in Lecture 5. The

Codes adopt a fairly simple approach: the essential behaviour ofa pipe bend can be described

using a single parameter - the pipe bend parameter or pipe factor. This is usually defined

as, Figure 15,

.■/here R is the bend radius.

2.15

BASIC CONCEPTS

Figure 15: Gemetry of a pipe bend

BS806 provides a chart, Figure 16, for the value ofthe flexibility factor in terms of the pipefactor;

2CO

O.OI Q02 0.03 0.05 0.! Q2 0.3 0.5 I

0.04 0.4

Pipe factor A

Figure 16: BS806 Flexibility factor

2 3 4 5 10

1

2.16 ■ -

rDesign & Analysis:ofPipingSystems :

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r

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The ANSI/ASME Codes use a different notation, usually denoting the pipe factor by h, and

have a simple formula for the flexibility factor, k,

1.65K — .

Stresses are also modified using a stress intensification factor, usually denoted by i, either

from a chart as in BS806, or from a simple formula, as in ANSI B31.1

. 0.9

2/3

In fact, as will be discussed in Lecture 5, this is rather misleading since in the ANSI Codes

it represents a design factor rather than the true stress in the component.

ANSI B31.3 (and BS806) uses different stress factors (sometimes called i-factors) for in-

plane, ii} and out-of-plane bending, i0 ; then the maximum stress for combined bending in

a pipe bend is usually taken from the above as

2.3 PLASTIC DESIGN CONCEPTS

There are essentially three distinct design concepts used in the various piping Codes - the

concepts of a limit load and that of shakedown for cyclic load and fatigue. These concepts

appear either directly in the formulation of the design rules themselves, or in the choice of

various stress multiplication factors for various components. Here the plastic design con

cepts will be developed:

2.3.1 Limit loads

If some suitable elastic-plastic stress strain relation is specified, such as nonlinear

hardening, itis quite possible to analyse the behaviour ofeven the most complex engineering

component (in particular if finite element techniques are used). However the nonlinear, or

even the bilinear, hardening rules are somewhat unrealistic since they contain no mech

anism of failure - the stress can increase indefinitely, it merely causes larger plastic strain

and distortions. A simple plastic failure mechanism -plastic collapse - is however provided

by the simple perfect plasticity model: stresses above yield are not possible. If a component

is assumed to be made of a perfectly plastic material then it is generally not possible to

continue increasing the load - a limit must be reached when no more stress can be

accommodated and the component collapses. The maximum load which the component can

■2.17

BASIC CONCEPTS

take is then called the limit load. Limit loads have been calculated for many engineering ,-

components and pressure vessel & piping components; an example is shown in Figure 17

for a nozzle in a cylinder under internal pressure(1).

30.

Figure 17: Example of nozzle limit load

The limit load can thus be used as an indication ofgross plastic deformation, and design

should ensure loads significantly less than the theoretical limit load. If the component is

subject to multiple loads then the combination of loads which cause collapse are called a

limit surface:

An important example of a limit surface is given by a simple beam under tension and

bending, which can serve as an elementary model of the bending of a shell wall under

pressure (and other sustained loads). The appropriate limit surface is easily evaluated and

is shown on an interaction diagram, Figure 18.

This interaction diagram can be interpreted in terms ofmembrane stress and bending stress

and is then replotted as in Figure 19. This diagram provides a simple interpretation of the

ASME Sec.III design by analysis criteria, as shown.

As we will see, limit loads form the basis for design of Class 1 piping in the ASME codes.

(1) - RL Cloud & EC Rodabaugh: Approximate analysis of the plastic limit pressure in nozzles in cylindrica|shells. Trans ASME, Journ Engng for Power, 171-176, 1968

2.18

1

1

Design & Analysis of Piping Systems•

Direct

Shell wa I

Sensing

PARTIALLY

PLASTIC

M

A

-a

ELASTIC

FULLY

PLASTIC

Figure 18: Interaction diagram for beam in bending & tension

BTI 2.19

BASIC CONCEPTS.

A °r

1.5

INITIAL YIELD

■ASME

DESIGN

REGION

UMIT CONDITION

m

f

o

am

o 2/3

r

i

i

Figure 19: ASME Design by analysis criteria for primary stress

■

2.3.2 Shakedown & ratchetting

The concept of a limit load is important when limiting the stresses obtained from sustainedloads such as pressure (and deadweight) - the so-called primary stresses. However anotherpossible failure mechanism associated with the plasticity ofcomponents is more significantin the case of thermal loads (the so-called secondary stresses). Such stresses are usuallycyclic in nature. Ifwe recall that an important aspect of material plasticity is the behaviouron unloading, we should realise that cyclic loading can be quite complex. Again our

description of component behaviour for cyclic loads is mostly limited to the assumption ofperfect plasticity. Two concepts are important - that of shakedown and that of ratenetting(which is what happens if a condition of shakedown is not achieved). In general for cyclicloading we design for shakedown in order to avoid ratchetting which can lead to incremental

collapse.

2.20

1

'

1

1

1

1

J or cyclic loading shakedown is the condition that after the first cycle ofload, the componentbehaviour is purely elastic; some plastic strain does take place in the first cycle but not inthe second or subsequent cycles. The highest load for which we can assure shakedown iscalled the shakedown load. This is shown in Figure 20 which represents a load against(maximum) strain plot for a hypothetical pressurised (or mechanically loaded) component

i LOAD

Shakedown load

Below

shakedown

load

Above

shakedown

load

Residual strain Ratchetting

Figure 20: Shakedown load

IttfTt^ °btainef,then in each Cycle there is additional plastic strain accumulated -tlus behaviour is called ratchetting and should be avoided in design. (It is possible

but eattheaen°d T^T t "f^°T * **** * Zem'pla8tic ****** does ^keJw £ ?«»d of the cycle it is reduced to zero - this behaviour is known as averse

S £??? ^^&tigUe g0VemS the design)- Shakedown loads have alsofor nozzles) mEny lmpOrtant engineering components (as in BS5500 Appendix G

Shakedown is the basis for the ASME design rules for secondary stress and directly givesnse to the well known 3Sm limit (which is equivalent to 2a,). An elementary justificationfor this is given in Figure 21. The aim is to keep the elastically calculated stress range tolfcXwn, * d ttenSUre shakedown=the »ubt»e point about this is the elasticitycalculated stress range - these are fictitious stresses and would not be obtained in practice^but allow the use of elastically calculated stress as a basis for design

BASIC CONCEPTS

RESIDUAL

STRESS

V y £. C R _ U y

Eeo <

ao <2a

G^l

Figure 21:ASME Shakedown (secondary stress) criterion

We will see that shakedown considerations have been important in the development of

BS806 and ASME Sec.III Class 1 piping rules.

c

1

2.4 FATIGUE ■ i-FACTORS

We have already discussed the principal failure mechanisms which the design codesattempt to avoid - bursting due to primary stresses and ratchetting due to (cyclic) secondarystress Another failure mechanism is also dominant - under cyclic load the possibility oflocal structural failure due to low cycle fatigue cracking. Low cycle fatigue requires a different treatment from the more well established high cycle fatigue which is associated withrotating equipment; in this case it is the familiar endurance limit which is of significancein design. Low cycle fatigue is governed by the pea* stress in the component which may beexpected to be in the plastic range. In this case a fatigue life should be specified and designbased upon keeping stress levels below the minimum cyclic stress range required to causefatigue in a standard fatigue specimen with a prescribed number of cycles to failure. Thisinformation may be obtained from standard S-N curves, as shown in Figure 22, where thealternating stress SA is usually one half the peak stress.

1

"2 22 -


logAS

AS[

. L- Alternating stress

—a

Endurance limit

No. cycles to failure

22: Fatigue S-N Diagram

Standard S-N curves assume a constant amplitude, while in practice the stress range maybe expected to vary during the service life. This is dealt with in design by defining usefraction sums or fatigue damage sums according to the well known Miner's Rule, whichhas proven with experience to be acceptable for design purposes.

It is fatigue which is the basis of the stress intensification factors which appear throughoutthe US ANSI B31.1 and ASME Sec.III Class 2 & 3 piping rules. Stress intensificationfactors for pipework components were introduced in the pioneering paper by ARC Marklin l7O^in

r aS^/aS^1^*1011 faCt°rS (SIF °r i-factors) were first introduced into the ASA (now[ ANbi/ASME) Code for pressure piping in 1955. These factors were based almost entirely

°?f^fr?fl fatiffUe t6StS by a sma11 team ofTubeTurns Co' engineers under the directionP ofARC Markl, HH George and EC Rodabaugh; they represent the fatigue strength ofpipingt components relative to the fatigue strength of a typical girth butt weld in a straight pipe

Fewer than 500 full sized components were tested by the Markl team; these components -welding elbows, mitre bends, reducers, flanged connections and various types of branchconnections - were nearly all 4 inch IPS (NPS) standard weight (4.5in O.D and 0 237innominal wall thickness, A106 Grade B material) from the stocks of one supplier. The

[ experimental results for these components were compared to analytical results andextrapolated to-provide SIF's for a range of geometries ! In fact it has recently been 'dis-

2.23

BASIC CONCEPTS

J

covered' that these extrapolations may not be conservative, particularly for branch con- f

nections. Unfortunately little additional testing has been performed since Markl's tests,

and a new programme has been initiated by ASME.

In the Markl tests, specimens were usually bolted at one end and subject to a fatigue test "1through loads at the Tree' end in which the displacements were measured; the specimens

were pressurised with water and subject to cycles of load - failure was considered to haveoccurred when leakage from a through wall crack was observed. The tests were thus jessentially ones with 'controlled displacement'. Displacement was converted to an equiv

alent elastic stress and stress amplitude was plotted against number of cycles to failure. «■«

Test data conformed reasonably to the relationship, I

1where S is the nominal cyclic stress and N is the number of stress reversals to failure; Cis a material constant (245,000 for carbon steel ASTM A-106 Grade B). An S-N curve for (

the base test on a butt welded straight pipe, where

is shown in Figure 23.

§

A Controlleddisplacement

Girth butt weld

150,030

4" Standard Wjiqhf Ptoe:

• Wtthvcnc-1! bccfcinq rm;s

o No bac'*i.--q nnqs

//'

1

The stress intensification factor i was thus introduced; it was derived for each component -iusing the following procedure: Figure 24 shows the results from a nominal test on a 90deg jpiping elbow subject to a cyclically applied in-plane end displacement. The cyclic strain

Figure 23: Markl's base test on butt welded pipe

2.24


isplacement range caused stress to exceed yield in nearly all tests (and was in fact anecessary element). An equivalent elastic load/stress range, SA, could then be derived - thisis of course fictitious.

Load

Equivalent

elastic

load range

SA

Displacement 5

Cyclic displacement

Figure 24: Markl's fatigue tests

^he elastic stress range could then be used to derive an S-N curve, Figure 25 for thecomponent, which could be plotted together with that for the nominal butt welded straightpipe. The i-factor is then obtained as

i =

OB

OA

The i-factor thus compares the fatigue strength of the component to that of a (similar)welded straight pipe as the stress which gives a fatigue failure in the same number ofcycles.

2.25

BASIG CONCEPTS

tlog S,

r

Reference test

Component test

log Nf

Figure 25: Derivation ofi-factor

Commercial straight pipe without a girth butt weld was also tested as well as polished

bars, but the results were not accurate. Thus it was decided to use the fatigue life ofa girth

butt weld as a reference point for all data. Hence an i-factor of 1.0 means that a component

has a fatigue life equivalent to that of a girth butt weld in a straight pipe and it is these

results on which the ANSI codes are based.

It is thus important to emphasis that the i-factors which appear in some Codes are based

on a prediction of fatigue stresses for use with a matching fatigue design curve.

2.26

i


3 OVERVIEW OF PIPING CODES

The various codes necessarily contain all the information required in the design including

material specifications, acceptable dimensional standards, rules for the basic pressure

design and rules for the evaluation and limitation of internal stresses as well as end

reactions and movements due to thermal expansion, external forces and pressure, rules for

manufacture (fabrication, assembly and erection) and rules for examination, inspection

and testing. Once the various flexibility analyses are available, the results are applied to

most codes in a selective manner. Here we will briefly summarise the design rules given

in several familiar codes: BS806, ANSI/ASME B31.3, B31.1 and ASME Section III, Sub

section NB. A study ofthe different approaches to design used in these Codes form a useful

background to piping design and analysis.

3.1 BS806: FERROUS PIPES AND PIPING

The loadings requiring consideration in this Code are sustained loadings (pressure and

deadweight), flexural (thermal expansion) stress and bending stresses caused by external

loads. (BS806 makes no provision for occasional loads such as earthquake, wind etc.; in

practice analysis is carried out using ANSI/ASME B31.1 (Enquiry Case 806/2)). The stress

levels must be calculated for all the various operating conditions both 'cold' and 'hot'.

BS806 is an elegant Code, being fairly straightforward. The principle section of interesthere is Section 4: Design:

A general background to the purpose of the Code is given in Section 4.1 where it is stated

that"... piping installations shall be designed ... to withstand the design pressure at the

design temperature sustained, where relevant, for the design lifetime ...". Interestingly

that "... this section also covers the assessment of stresses arising from the thermal

expansion and deadweight loading of piping systems ..." appears as an added Note.

The following Sections 4.2-4.9 mostly deal with basicpressure design ofvarious components

- straights, bends, branches (and joints and valves). In most cases this is straightforwardbeing a calculation ofminimum thickness:

._ PD3 2fe+p

where,

3.1

OVERVIEW OF PIPING CODES

p - design pressure v

D - mean outside diameter

Bs?)

f-design stress |

e - a factor «i

The derivation of this formula was discussed in Lecture 2.

This formula is used for straights and bends, but is modified for mitre bends and branches )with additional factors. The background to mitre bends will not be discussed in this course

but can be found in the commentary by Battle et al(1). Branches will be discussed in Lecture q

6. I

The design stress is outlined in Appendix B of the Code, derived from one of the following: <m

Rn, - the tensile strength at room temperature

Re - the yield (or proof) stress at room temperature

R^t) - the yield stress at temperature T

Siu - the mean stress required to produce rupture in time t at temperature T i

depending on the material and temperature. Essentially the time independent design stress «i

is given as the lower of J

15 °r 235JE~~ 1 e Or -ioc

for temperatures up to 50degC, substituting R^ for temperatures above 150degC with

linear interpolation between in the intermediate range. If specified elevated temperature

values are not available then those for similar materials may be used, except that an(

additional factor of safety, replacing 1.5 by 1.6 should be adopted. For austenitic steels, the

enhanced ductility at elevated temperature allows reduced factors of safety, replacing 1.5

(or 1.6) in the above by 1.35 (or 1.45) and 2.35 by 2.5. Finally the time dependent design

stress is specified as

*

Once the basic component thicknesses have been evaluated, the pipe run needs to be jassessed for expansion and deadweight: rules for the calculation and limitation of system

stresses are given in Section 4.11. Flexibility: "j

1(1) - K Battle et alThe design ofmitred bends - a background to BS806.1975 ammendmentNo.3. Proc IMechEConf on "Pipework Design & Operation", p9, Vol.C22, 1985

3.2


BS806 is remarkably clear in its aims with regard to flexibility (Section 4.11.1). "... the

pipes shall be arranged so that the system is sufficiently flexible to ensure that the end

reactions, under any operating conditions either hot or cold ... do not exceed ... maximum

values ..." which have been agreed between purchaser and manufacturer (say through

design of attached vessels). "... the pipes shall also be sufficiently flexible to absorb the

whole of its own expansion and that of the connecting equipment without exceeding the...

maximum permitted stresses specified in Section.4.11.2. "... where practicable, the

requisite flexibility shall be provided in the layout of the pipes ...".

There is some subtlety in the above: by inference the connecting equipment are treated as

(rigid) anchors since it should be demonstrated that the whole thermal expansion (modifiedby deadweight) be absorbed. This is achieved through flexibility of the pipe layout, but at

the expense ofstress intensification, whosemagnitude should be limited. That conventionalflexibility analysis also allows for the expansion of the connecting equipment is a subjectof debate.

It is then stated that"... a flexibility analysis shall be carried out if there is any doubt as

to the ability of the system to satisfy the specified requirements ...". Circumstances wherethere is no doubt in this respect have never been clear to the writer in the context ofresponsible engineering practice. Requirements for calculating the flexibility are thenlisted. Two points are of interest:

Firstly it is required that"... linear and rotational behaviour ofconnecting equipment shall

be taken into account...". It is not clear to the writer to what extent this requirement is

followed in practice; as will be discussed in Lecture 4 the intent of such a modellingassumption is clear in that the inclusion oflinear and rotational behaviour can significantlyaffect system flexibility and stress levels.

Secondly, it is required that"... flexibility and stress intensification factors for bends andbranches shall be utilised..." and refers to BS806 Figures 4.11.1(1) - 4.11.1(8) (see Lectures5 & 6). It is not clear here if the Code values have to be used, or if alternative improvedfactors can be utilised.

Finally, Sections 4.11.2, 4.11.3 specify the maximum permitted stresses and the methodof calculation of stress levels:

Design is based onthe limitation ofan 'equivalent combined stress' from eitherfor straightsand bends,

or for branches,

where,

3.3


F - the greater of the transverse stress fT and the longitudinal stress fL being thef

sum of the relevant maximum stresses for pressure and bending,

fB - the (maximum) torsional stress

fB - the transverse pressure stress at a branch junction plus the non-directional

bending stress,

fsB - the torsional stress at the branch,

q - a relaxation factor for hot stress evaluation

For example for straights and bends the pressure stresses are evaluated as,

Jt It

f ~4t(d + t)

and the bending stress for straights is

while for bends

where,

d - mean inside diameter

t - mean thickness

r - mean radius of pipe

I - second moment of area

p - design pressure

M4 - maximum in-plane moment

Mo - maximum out of plane moment «~

and FrJFTb,FIilFLo are in plane and out of plane stress intensification factors for bends as Ispecified in charts.

3.4


Additional complex formulae are included for mitre bends but are not discussed here.Branches will be discussed further in Lecture 6.

In general (for straights and smooth bends with modifications for mitres and branches) the

bending stresses are derived from simple beam bending theory for combined load (in and

out ofplane bending and torsion) with factors to account for the elbows (smooth and mitred)

and branches. The effect of loads other than bending and torsion are not considered. It is

to be noted that the maximum pressure and bending stresses are added, and that these

are derived on the basis of an equivalent stress also based on the summation ofmaximum

stresses. These maxima would not be expected to occur at the same location, so the 'actual'

stress is not being calculated, rather a 'worst case' is derived. Moreover in the calculation

of stress range, it is the range of bending moment from cold to hot condition which must

be used; indeed this calculation can be quite tedious; the words ofAppendix F are significanthere:

" ... the calculation of bending moments and the identification of maxima ... becomes a

lengthy operation.... it is expected that where such systems are required to be analysed

recourse will have to be made to a computer analysis using a program developed to complywith this standard ..."!!

Three different limits are specified for maximum stress range , maximum hot stress,

ifapplicable, and sustained stress. The allowables are based on selected material failure

stresses for both the hot and cold conditions and applied to selected loadings - pressure,

thermal expansion, deadweight and cold pull. Ifapplicable the hot stress limits are appliedto all loadings; the stress range limits apply only to pressure and thermal expansion, the

sustained stress limits only to pressure and deadweight. The following permitted stresslevels are given:

(a) Maximum stress range is the lower of

(1) H times the proof stress at room temperature plus H times the proof stress atdesign temperature

(2) H times the proof stress at room temperature plus the average stress to rupturein the design life at the design temperature

where H=0.9 (except for some branch geometries where H=1.0)

(b) Maximum hot stress is the average stress to rupture if (2) above is used

(c) Sustained stress is the lower of 0.8 times the proof stress or the creep rupturestress (time dependent allowable), with modifications for branches.

Simplifying, except at elevated temperature, the stress range for pressure and thermalexpansion must be limited to 80% of twice yield (recall the shakedown criteria of Lecture

2) and the sustained stress for pressure and deadweight to 80% of yield. At elevatedtemperature the creep rupture strength should be taken into account.

3.5

OVERVIEW OFPIPING CODES

3.2 ANSI/ASME B31.3:REFINERY PIPING CODE <

Like BS806, ANSI B31.3 Chemical Plant and Petroleum Refinery Piping Code is fairly

straightforward. It covers pipework in petroleum refineries, chemical plants and natural «

gas plants among others. The loadings considered are pressure, deadweight and thermal jexpansion as well as wind and earthquake loadings at various service levels. The Section

which will be examined here is Chapter II: Design which consists of several parts, in par- -»

ticular: J

• Part 1: Conditions and Criteria which covers Design Conditions {Paragraph 301) «*

and Design Criteria {Paragraph 302). j

• Part 2: Pressure Design ofPiping Components which covers, in Paragraph 304 the m,

basic pressure design and calculation ofcomponentminimumthickness for straights, ]bends, branches etc.

• Part 5: Expansion, Flexibility and Support which covers in particular in Paragraph( J319 the evaluation of expansion and flexibility stresses.

These will be discussed further below: ]

Paragraph 301 covers the design conditions, defining the temperatures, pressures andloadings applicable to the design of piping systems. Of particular interest is Paragraph j301.8 which, like BS806, requires that"... the effects of movements of piping supports,

anchors and connected equipment shall be taken into account ... these movements may

result from the flexibility and/or thermal expansion ofequipment, supports or anchors ...". jAgain the extent to which this is done in practice, with the consequent effect on calculatedstresses (as discussed later in Lecture 4 with various assumptions of conservatism) is not «i

clear. J

Paragraph 302 lists the general design criteria, specifically allowable stress levels in 302.3: „Allowable stresses and other stress limits for metallic piping. With the exception ofbolting ]materials, cast and malleable iron, the basis for the design stress is the lowest of:

1/3 the minimum tensile strength at room temperature J

1/3 the tensile strength at temperature «

2/3 the yield at room temperature

2/3 the yield at temperature "1

100% the average stress for creep rate of 0.01% per lOOOhr

67% the average stress for rupture in 100,000hr ^ I

80% minimum stress for rupture in 100,000hr ~

Again for austenitics, enhanced ductility allows the factor on yield at temperature to rise Ito 90%. For structural grade materials the design stress is reduced further by a factor of

0.92. ]

3.6


The limits for calculated stresses are then prescribed in Paragraph 302.3.5:

• Pressure stresses are limited through a minimum wall thickness using the appro

priate design stress

• Longitudinal stress, SL, for pressure, deadweight and other sustained loadings shall

not exceed Sj,

• Stress Range, SE, for displacement stress (thermal expansion etc) shall not exceed

the allowable, SA, where

SA=f(l.25Se+0.25Sh)

This limit is modified if Sh > SL.

In the above,

Sc = basic allowable stress at minimum (cold) temperature

Sh = basic allowable stress at maximum (hot) temperature

f = stress range reduction factor

The stress range reduction factor is related to the design philosophy ofB31.3 - that ofyield

(or creep rupture) limited design for pressure and deadweight, and fatigue design for

expansion stresses, as we will see.

Pressure design requirements for various components are given in Paragraph 304. Similar

to BS806, the pressure stress is limited through a minimum wall thickness (with corrosion

allowance) - the pressure design thickness for a straight pipe is

PD

I 2(SE+PY)

P where

S - design stress

r E - quality factor (tabulated in Code)

*• P - pressure

p although the simpler and more obvious formt

PD

2SE

may also be used (or indeed Lame's equations, Lecture 2). The factor Y is tabulated

depending on material, geometry and temperature varying from 0.4 to 0.7 over a tem

perature range for ferritic and austenitic steels (it is discussed in Lecture 2).

The minimum thickness for pipe bends is taken to be the same as for straights, while

additional rules are given for mitre bends and branches.

3.7


The evaluation oflongitudinal stress for sustained loads, and stress range, SE, for thermal

expansion is developed in Part 5, in particular Paragraph 319 which provides "... concepts,

data and methods for determining the requirements for flexibility in a metallic piping

system..." These requirements are that"... the computed stress range shall not exceed the

allowable stress range ..., that reaction forces ... shall not be detrimental to ... connected

equipment...". The aim of the design rules is that"... piping systems shall have sufficient

flexibility to prevent thermal expansion ... from causing ... failure of piping ... leakage, or

... detrimental stresses or distortion ...".

Various terms are defined in Paragraphs 319.2: Concepts and Paragraph 319.3: Properties

for Flexibility Analysis. The most interesting is Paragraph 319.3.6: Flexibility and Stress

Intensification Factors:"... in the absence of more directly applicable data, the flexibility

factor k and stress intensification factor shown in Appendix D, shall be used in flexibility

calculations ...". That is, apparently unlike BS806, better values may be used if available

(and they are!).

Paragraph 319.4 covers analysis methods. Formal flexibility analysis can be avoided underCcertain conditions, otherwise can be carried out using "... simplified, approximate or

comprehensive ..." methods; more of this in Lecture 4. The flexibility analysis provides

in-plane, Mj, .and out-of-plane, Mo, moments which are used to evaluate the "... computed

displacement stress range, SE ..."

where Sb is the bending stress

and St is the torsional stress

where Mt is the torsional moment, Z the section modulus of the pipe cross section and ij, iothe in-plane and out-of-plane stress intensification factors for elbows, mitres etc. Again

branches are treated slightly differently.

The stress combination is similar to BS806. However the most interesting feature is thatthe evaluation of longitudinal stress for sustained loads is not detailed. Only bending

moments are used in the stress calculations.

3.8


3.3 ANSI/ASME B31.1:POWER PIPING CODE

The Power Piping Code B3.1.1 is similar to B31.3 in the use of i-factors etc, but does not

use combined stress or time dependent allowables but more specific rules are given for

sustained (and occasional) loads:

The loadings requiring consideration in this Code are sustained loadings (pressure and

weight), occasional loadings (wind and earthquake when applicable), vibration and thermalexpansion:

Stresses due to sustained loadings:

PD0 Q.75iMA

~4/~~+ Z " ' *

Stresses due to occasional loadings:

. 0J5iMB ^ , „

Stresses range due to expansion loadings:

iMc

where, in the above,

P = design pressure

Do = outside pipe diameter

tn = nominal wall thickness

Z = section modulus

i = stress intensification factor for component

MA = section bending moment due to sustained loads

MB = section bending moment due to occasional loads

Me = range ofbending moments due to thermal expansion

SL = calculated sustained stress

k = occasional load operating factor

f = stress range reduction factor for cyclic conditions

together with the allowables

Sh = basic material allowable at maximum temperatureSA = allowable stress range for expansion stress

Sh=f(\25Sc+O25Sh)

Sc = basic material allowable for minimum (cold) temperature

3.9


The Code allowables are derived from selected material failure stresses depending on theftemperature (Lecture 2). The main point again, as in BS806 and B31.3, is that the design

stresses used are essentially based onsimple beam bending stresses alone, with modification

factors for certain components (such as pipe bends, mitres, branches etc); the basis for thesemodifications will be examined in Lectures 5 and 6. There is no combination of stresses fordifferent sectional loading, and no consideration of stress other than longitudinal stress !For bends, no distinction is made between in-plane and out-of-plane moments (the same

stress intensification factors are used).

3.4 ASME BOILER & PRESSURE VESSEL CODE

Finally it is instructive to examine the rules developed from the ASME III nuclear designCode, as this represents a fairly significant departure in philosophy from the conventionalUS B31 approach. As will be seen later, the philosophy is similar to BS806 in design intent. ^ • j

The ASME nuclear design by analysis code provides different design rules for what it callsClass 1 and Class 2 and 3 components. The rules for Class 2 are essentially modifications(based on the approach used for Class 1 components) to those of ANSI/ASME B31.1 andmost Class 3 components are treated as Class 2 for design purposes. The rules for Class 1components are a special modification of the ASME design by analysis philosophy forpipework. This philosophy separates component stresses into two types - primary andsecondary. Primary stresses arise from pressure or sustained loading and can directly lead -to catastrophic or burst type loading; secondary stresses arise from thermal and many ]other loadings and can lead to ratchetting, distortion or fatigue failure. A specific procedureis laid down in the ASME Code, with different limits for primary and secondary stresses -i(since they are dealingwith different failure mechanisms) requiring the calculation ofstress |intensities, which are related to maximum principal shear stress, for each stress category.

This procedure has been modified for piping to account for the traditional techniques of "1

flexibility analysis. The stress rules which must be satisfied are:

Primary stress intensity:

BXPDO

Primary plus secondary stress intensity:

where

3.10

1

1


P = design pressure

Po = range of service pressure

t = nominal wall thickness

I = second moment of area (moment of inertia)

Ta,Tb = range of average temperature for gross structural discontinuity

aab = coefficients of thermal expansion at gross structural discontinuity

Eab = average modulus of elasticity at gross structural discontinuity

Mj = section bending moment or range (for secondary rule)

BltB2 = primary component stress indices

C1,C2,C3 = secondary component stress indices

with allowables

Sm = allowable for primary stress

k = factor for load service levels

with additional rules if the latter rule is not met for all load cases at a gross structural

discontinuity. Rules for separate fatigue design are also specified.

The stress indices are specified for each component (elbow, branch etc) and, it must be noted

at this stage, are essentially different from those of the related ANSI B31.1 being based

on an entirely different design philosophy (ANSI B31.1 being based on fatigue, ASME Sec

III NB being based on maximum elastic stress or limit stress). Again only longitudinal

bending stresses are considered, even though the main ASME rules do allow for more

complex calculated stress. It will become clear later that here special modification to the

design by analysis rules has been made for the information which can be obtained from

conventional piping flexibility analysis !

3.11

\- ' >* \ OVERVIEW OF PIPING QbDESt:;;^i;;/:-;

1

1

1

3.12

Design & Analysis ofPiping Systems .

psi

4 FLEXIBILITY ANALYSIS OF PIPING SYSTEMS

It is the purpose ofthis Lecture to describe the basis offlexibility analysis ofpiping systems

for static loads. To the experienced pipework designer this may seem rather routine;

however in almost all such analyses several simplifying assumptions are introduced. These

assumptions not only make the analysis more straightforward, but also have a significant

effect on the treatment of many piping components, such as bends and branches, both in

analysis and in design.

Following a brief summary of the need for a flexibility analysis, the mechanical basis of

the various simplifying assumptions are described. The two most common analysis pro

cedures are then described - one based on the use ofenergy methods ofstructural analysis

and the other based on matrix displacement or finite element techniques. Following this

modern computer analysis of piping systems is discussed.

4.1 THE NEED FOR FLEXIBILITY ANALYSIS

To begin with, it is worthwhile reminding ourselves ofthe need for flexibility in a pipeworksystem:

For the most part in this lecture we will be dealing with the analysis of piping expansion

stresses. Traditionally stress analysis for other loads, in particular sustained stresses such

as deadweight, could be quite adequately carried out using manual methods and the

emphasis was on support design. The problem was with the calculation of thermal

expansion stresses and, as we shall see later, many ingenious techniques were developed

to allow this calculation. However, the availability of computer based structural analysis

rendered this analysis rather straightforward and also allowed analysis for sustainedstress. Although we will describe here the basis for such structural analysis, we will spe

cifically only look at the flexibility analysis for thermal expansion, without going into anydetail on the inclusion of sustained stress in the analysis.

So, why is flexibility analysis necessary ? It is to be recalled that a good design aims atkeepingthe pipe stresses and end reactions within certain specified allowables. As pipeworkundergoes a temperature rise, thermal strains are induced according to the well knownformula:

If the ends of the pipe are restrained then a thermal stress is induced, giving rise to endreactions on the restraints. As an example of a stiff system, Figure 1, the force developedin a lOin. sch 40 carbon steel pipe A53 Grade B subjected to 200degF from an installationtemperature of 70degF is 273,9081b !

4.1

Flexible piping

Figure 1: Thermal expansion ofa stiff system

In order to reduce these endreactions (andinternal pipe stresses) it is necessary to introduce

some flexibility into the system to absorb the thermal expansion strain. Basically this is

achieved through changes in direction ofthe pipe run either using expansionloops or simply

by re-routing (or by using expansion or other joints), Figure 2.

All design codes have recognised this need and specifically require the reduction ofthermalexpansion stresses, although they differ as to the mechanical basis for the design stress,

as we will see later. Codes also differ on the need for flexibility analysis. For example BS806

specifies that:

"... a flexibility analysis is required if there is any doubt as to the ability of the system to

satisfy the specified requirements..."

This helpful statement is similar to early versions of B31.1 (1942):

"... formal calculation shall be required only where reasonable doubt exists as to the

adequate flexibility of the system..."

The current version of B31.1 specifies a flexibility analysis unless certain conditions are

met:

"... it shall be the designer's responsibility to perform an analysis unless the system meetsone ofthe following... all systems not meeting the above criteria or where reasonable doubtexists as to the adequate flexibility of the system, shall be analysed by simplified,

approximate or comprehensive methods of analysis ..."1

4.2 •


Loop

4f

Expansion joint

—m—

Figure 2: Flexible piping systems

4.3

FLEXIBILITY ANALYSIS OF PIPING SYSTEMS

The basic design philosophy is to calculate stresses in each component and compare these^

and anchor reactions to given allowables for given materials and temperatures. The stress

levels in each component are derived from the forces and bending moments calculated to

be acting on it from the flexibility analysis according to procedures specified in the Code.

To meet this need a whole piping design and analysis industry has developed. Many piping

analysis packages are available, varying in degrees ofsophistication, ranging from the well

established mainframe/mini computerbased systems to the newer pc based systems. While

the different packages have different capabilities, such as CADD, bills of materials data

bases etc., most will aim to do the basic flexibility and sustained load calculations with a

code assessment. In this respect they are all similar, adopting one oftwo related structural

analysis techniques. Essentially for the purposes of a force analysis, a complex piping

system is considered as an assembly of simple beams. The piping system is then treated

as a framework in order to calculate the forces on each component. Once the forces on each

component are found engineer's theory of bending is used to calculate direct and bending

stresses. f

4.2 FLEXIBILITYANALYSIS OFPIPINGSYSTEMSFORTHERMALEXPANSION

How do we do the flexibility analysis ? In principle any convenient structural analysis

method for space frames may be used - the Theorem of Minimum Potential Energy,

Castigliano's Theorem, the Unit Load Method, the Matrix Displacement or Force Methods

and so on. Here we will look at two specific methods, on which most commercial piping

analysis packages are usually based:

• Classical energy methods

• Matrix displacement, or finite element, methods

The aim is to clarify certain important similarities and differences between the two tech

niques.

4.2.1 Energy Methods '

tSPi

Consider the simple planar piping elbow shown in Figure 3 consisting oftwo straight pipes

of lengths Lx and L2 attached to a 90deg pipe bend of radius R. The elbow is fixed at one

end, A, as shown and can have applied in-plane deflections 8x,8y and rotation yg. «

1

i

4.4

1


Figure 3: Example piping system

To analyse this we assume linear elastic material behaviour and use an energy method to

evaluate the in-plane forces Fx , Fy and ^ at B.

For simplicity, assuming only bending to be significant (this is a reasonable assumption

here), the strain energy is given by

where M(s) is the in-plane bending moment acting at some point along the elbow axis and

s represents a measure of distance along the pipe axis.

4.5

FLEXIBILITY ANALYSIS OF PIPING SYSTEMS -

This expression for strain energy requires some modification for the pipe bend; since it is (

more flexible its strain energy is higher than would be expected from simple engineer's

theory of bending. This additional strain energy is given by the flexibility factor defined

in Lecture 2. Ifwe introduce the flexibility factor as the ratio ofthe end rotation ofthe bend

to that of an equivalent straight pipe, then the strain energy due to bending of the bend

should be multiplied by the flexibility factor.

With reference to Figure 3, the strain energy can be written as,

where u, v and 0 measure distance along the straights and angle around the bend

respectively.

In the straight pipe of length Ll we have,

whereas in the straight of length L2,

M=Mt-Fxu

M =-

S 5 y

"~dF, y dFy '' dM,

If all the appropriate substitutions, integrations and differentiations are carried through

then the following relation is obtained in matrix form

where we have defined, in matrix notation

I

and in the bend

I

We may then use Castigliano's theorem (or indeed Unit Load) to relate the forces and

displacements

1

"I


and where the 3x3 symmetric flexibility matrix [K] = (l/EI){Ky} is given by:

Thus ifthe thermal expansion displacements are given at end B then the resulting reactions

can be found by solving the above matrix equation. Then by equilibrium the forces acting

on each component (straight, bend or straight) can be evaluated.

The above method can be easily extended to deal with any number ofbranches and anchors

and to include thermal strains and even deadweight. It is in fact the basis ofthe well known

General Analytical Method of the MW Kellogg Co. It was first derived in full by HV

Wallstrom, DB Rossheim, ARC Markl and E Slezak in 1941; this paper is generally

considered to be the first comprehensive treatment of the flexibility analysis of a piping

system and indeed is still valuable for piping engineers. This approach was later formulated

for computer solution by JE Brock in 1952, although as in most of Brock's work, it is fairly

hard going.

Historically specialised simplified techniques were developed to analyse, or simply size,

piping systems. Indeed some of the very simple methods are still used today, such as the

guided cantilever method and in particular the ANSI criteria for need for analysis:

DY <0.03L-W'

where D is the nominal pipe size, Y the resultant of (thermal and anchor) movements, L

the developed (total) length and U the distance between anchors, assuming no more than

two anchors.

4.7

- FLEXIBILITYANALYSIS OF PIPING SYSTEMS••

To a certain extent tabulated solutions such as the Kellog expansion loop formulae or ITTf

Grinnell's Piping Design and Engineering charts are still valuable in appropriate cir

cumstances; for example a complete worked example using the Grinnell charts is given in

Problem 5.7 of Smith and van Laan's book.

However most of the manual solution methods which were developed in the years from

1920 to 1956 have largely disappeared. They were mostly routine procedures developed

using charts and tables which could be easily documented and checked; nevertheless they

were based upon the simple but sound mechanical principles described above. Worth

mentioning are: Flex-Anal Charts ("Design ofPiping for Flexibility with Flex-Anal Charts"

by A Wert & S Smith, Blaw-Knox Co. 1940), the Grapho-Analytical Method ("Methods of

making piping flexibility Analyses" in "Heating, Piping & Air-Conditioning", 1946 - an

example is given in Chap 12 of E Holmes textbook), SpielvogaVs (Elastic Centre) Method

("Piping Stress Calculations Simplified" SW Spielvogal, published by the author, 1951)

and the GeneralAnalytical Method ("Design ofPiping Systems"MWKellogg Co. 1956,1964)

amongnumerous others. The professional computeroriented piping designer/analyst would!

do well to re-examine some of these techniques.

4.2.2 Matrix Displacement & Finite Element Methods

Many major piping analysis packages have changed over to a finite element formulation

since it offers several advantages; the main advantage however lies in future procedures

for piping analysis - these are discussed later.

The formulation which we will give here is not a true finite element formulation since we

will avoid a discussion of the main feature of the finite element method - displacement

interpolation. Instead we are more properly describing the matrix displacement method

for structures. For beam structures the two formulations - finite element or matrix dis

placement - give identical results. Finite element theory goes further. I

In the present case the piping elbow of Figure 3 is modelled using three simple elements -

two straight pipe elements and one curved pipe element. Each ofthese elements has a node

at each end, as shown in Figure 4. Each straight pipe element has a node in common with

the curved pipe element; the complete piping elbow thus has four nodes as shown.

4.8


Node

Element

©Figure 4: Finite element model ofexample system

The basis of the finite element method is that the behaviour of each element is entirelygiven in terms of the displacments and forces at its nodes. Thus if some element is ter

minated by nodes i and j then it is possible to write a global element stiffness matrix in thepartitioned form

F:

KJ Kji

A.

where the displacements and forces at node i are denoted by,

F =(F- F M )

in the global coordinate system (x,y).

The global element stiffness matrixmay be derived using several different techniques, suchas an energy method or unit load. In the finite element method displacement interpolation

is used: the displacment within the element is interpolated in terms of the displacements

at the nodes. In the theory ofcontinuous beams using Euler's Theory the deformed geometry

4.9

is completely described in terms ofthe displacementfrom the beam axis, v(x), which is related

to the bending moment at any point on the axis, Figure 5 for a straight pipe, by the familiarequation

d2v _ M{x)

dx2~ El

Undeformed

Figure 5: Displacement interpolation ofa continous beam

The displacement v(x) may then be conveniently represented by a cubic polynomial

The coefficients a,b,c and d can be directly related to the nodal displacements. Finally the

inverse form ofCastigliano's Theorem can be used to derive the preceding relations between

nodal forces and displacements.

The stiffness matrix for straight or curved beams is normally given in terms of a local

coordinate system, Figure 6, which is the same for all components ofthe same material and

geometry.

i

4.10

Design ^Analysis ofPiping Systems

Local

coordinate

system

Global coordinate system

x

Figure 6: Global & Local pipe coordinate systems

be related global element

— rxi7*

tranformation ™*i* at node i. Transformation matrices are

i tStraight and cur7ed P^6 are &™in ^e Paper bySn, . Agamm the case ofthe curved pipe a flexibility factor has beenused in the energy expressions in a manner similar to the above.

The basic procedure is then as follows:

• The system geometry is specified: number of nodes and elements, location of eachnode and which nodes are attached to which elements. Section properties (thickness,radius, bend radius etc) and material properties are also specified.

• The local element stiffness matrix is formed for each element, Mowed by theappropriate transformation matrices. These are combined as in the above to formthe global element stiffness matrix.

FLEXIBILITYANALYSIS OF PIPING SYSTEMS

The next stage is to assemble eachglobal element stiffness matrix as developed above {

into a global stiffness matrix for the complete structure - the piping elbow in this

case. This is done by requiring that at nodes common to two elements the dis-

placements for each element at thatcommon node are the same and also by requiring

that the nodal forces from each element at that node are in equilibrium with each

other (and any externally applied loads). The assembly process leads to the global

stiffness relation,

where in the present example A contains the twelve nodal displacements and F I

contains the twelve possible nodal forces.

• The boundary conditions are specified. Usually the anchors are fixed, as at Node 1 Iin the present example.

• Finally any applied forces or moments are specified; in the present example forces { Iare specified at Node 4.

• There results a matrix equation for the unknown nodal displacements, which may |be solved using the usual matrix reduction techniques.

• Given the nodal displacements for each element the element stiffness matrix can be |used to evaluate the nodal forces and moments for each component.

• Then using Code specified stress indices or factors, and engineer's theory ofbending, |the appropriate component stresses may be found and compared to the Code allow

ables. ""I

This method of analysis, in the form of the matrix displacement method using stiffness

matrices, first seems to have been suggested by LH Chen in 1959. It is now possible to «

contrast the general analytical method with the finite element method. Forbeam structures |both should yield identical results since both are wholly based on applications of energy

methods to engineer's theory ofbending. However the general analytical method solves for ( *j

the unknown anchor reactions while the finite element method solves for the unknown jnodal displacements. Obviously there can be a much greater number ofnodes than anchors

in a piping system so that the finite element method leads to larger equations with more "1unknowns. So why is it to be preferred ? Simply, it is a more general technique which can Jbe easily (!) extended to different analysis types (for example dynamic) and, as we shall

see, to include different element types. In particular special ovalising pipe bend elements, |

which are more general and which avoid the need for simplified flexibility factors, can be

formulated and included in the same solution procedure. ^

The reader should appreciate that the procedure for flexibility analysis described above is

quite straightforward. Even for a complex three dimensional system the same procedure

is used. In addition to straights and bends, it is also necessary to include tees, hangers,

supports etc. These are simply modelled using either simple spring or bar elements (as in «

the case of hangers and supports), rigid elements (as for valves) or as three noded rigid )connections (as for tees, but some flexibility can be included). Appropriate local element y


stiffness matrices and transformation matrices are formed and the whole problem

assembled. Boundary conditions are specified. Piping forces, such as thermal expansion,

deadweight etc, are transformed to element nodal forces using equilibrium considerations

and so on. The basic mechanics involved in this is really very simple, requiring no advanced

concepts. In fact computer coding ofthe basic procedure: formation (from anelementlibrary)

of element stiffness matrices, formation of transformation matrices and global element

stiffness matrices, assembly, application of boundary conditions and forces and solution

for unkown nodal displacements and finally element forces and stresses is very simple.

Commercial piping analysis software use no more complex procedures. The main program

design relates to problem specification (pre-processing) and post-processing to variousCodes.

4.3 COMPUTER ANALYSIS OF PIPING SYSTEMS

Computer technology has had no less of an influence on piping analysis than elsewhere.

Indeed piping stress analysis was one ofthe very first non-militaryengineering applications

to be programmed for the new digital computers in the early Fifties ("The solution of pipe

expansion problems by punched card machines" by LH Johnson ASME Paper 53-F-23) and

represented some of the first commercially available engineering software outside the

proprietory IBM business database applications. Particular mention is given to the MW

Kellogg Piping Program, 1955, the Blaw-Knox/AD Little program, 1956 and the MEC-21,

Marc Island Naval Shipyard program, 1959. During the 1970's many familiar commercial

piping analysis programs were developed for mainframe computers which are still in use

today - PSA5, PIPESTRESS, ADLPIPE, DYNAFLEX and so on. These programs largelydeveloped as special purpose software separate from the main stream of finite elementsoftware as they offered an understanding ofpiping Code peculiarities providing a complete

design, analysis and Code assessment. The analysis could of course also be done (withessentially the same results) using conventional commercial finite element software which

includes curved pipe elements (that is with flexibility factors), such as ANSYS (which in

fact includes a special piping pre and post processing module), NASTRAN and several

others. Finite element software has usually been more versatile and efficient using state

ofthe art analysis and graphics techniques. In order to compete most specialised pipeworksoftware has had to include more specialised piping specific pre and post processing - atthe risk of much protest, it is fair to say that most of this is purely cosmetic. To develop a

(static) piping analysis program itis onlynecessary to adapt well documented public domainfinite element matrix handling and solution algorithms to include straight pipe elements,curved beam elements, three noded branch elements and a variety of spring and beam

elements for supports, hangers, valves etc, the stiffness matrices for which are readilyavailable. The evidence for this can be seen in the large amount of software appearing forpersonal computers !

It is common now for many routine piping analysis and design to be made using PC based

software. There are several of these available, the most common being CAESAR, PSA5CAEPIPE, SUPERPIPE, AUTOPIPE and about a dozen more.

It is useful to look at an example:

4.13

^^EEXIBILITY ANALYSIS OF PIPING SYSTEMS

The software system which will be used here is CAEPIPE from SST Systems, Sunnyvale^

California. CAEPIPE, as well as being a fairly complete static and dynamic piping analysis

system which can handle a wide variety of worldwide Codes, is a useful system to learn

and demonstrate piping analysis and design (a full feature static demo version, with limited

problem size is available). The problem to be examined here is taken from the user's manual.

The sample problem is shown in Figure 7.

6" Std pipe

Carbon steel

Calcium Silicate insulation. 2" Ihk

200 psi. 6C0 F

Contents specific gravity = 0.3

6'0"

50 Specified displacement : Y = 0.5"

Figure 7: Sample piping system

This system consists of a simple three branch three dimensional system with a common,

junction supported by a single hanger with three anchors, which are assumed rigid. There

is one standard long radius bend, one non standard 18" bend and a valve. The main run is

6" standard pipe, with one branch of 8" Sch 80 pipe; there is 2" thick Calcium Silicate

insulation. The material is carbon steel. The fluid contents is at 200psi at 600degF, with

a specific gravity of 0.8. A specified displacement at one anchor of 0.5" is also specified.

Wind loading is specified. Finally the system is to be assessed to B31.3.

The CAEPIPE system will run on the most basic IBM compatible PC, preferably with a

hard disc and is fairly simple to use, having an excellent user interface which uses a series

of input screens. Instant graphics are also included. The input screens and graphics can

be navigated using the PC keyboard function keys, cursor keys and numeric keypad; the

(Eac] key is used to toggle between graphics and input screen while the numeric keypad can

be used to move the model and zoom. Sufficient information is given on screen to follow

this without a manual; much use is made of the !S9 key combinations.

4.14;

1

1

s


Once CAEPIPE has started the user is presented with the Main Menu:

CAEPIPE(EUROPIPE)

Version 3.25

Fl

F2 - Analyze

F3 - Output

F4 - Directory

F5 - Databases

F6 - Setup

Model SAMPLE

Alt X - Exit

4.15

FLEXIBILITY ANALYSIS OF PIPINGSYSTEMS

This gives various options, but here the function key ® is used to generate a new model, (which gives an input screen to specify the problem title and select various options. In this

case the title "Sample problem" can be used.

SAMPLE

Title

Sample Problem_

F2 -

F3 -

F4 -

F5 -

Alt X -

From most screens

s^_

Piping layout

Piping code = B31.1

Reference temperature = 70 (F)

Options

Q A block

Exit

use : Alt U for units menu

Alt P for graphics menu

Esc for graphics screen

<3^

4.16

1

1

1


The piping code is selected using the @D key, then 2D to select B31.3. Then (fio) is used to

return to the Title Screen.

Fl -

F2 -

F3 -

F4 -

F5 -

F6 -

F7 -

ANSI/ASME

USAS B31.]

ANSI/ASME

ANSI/ASME

ANSI/ASME

B31.1

Piping Code

(1986)

L (1967)

B31.3

B31.4

B31.8

ASME Section III

ASME Section III

(1987)

(1986)

(1982)

(1980)

(1986)

FIO -

Alt

Alt

Alt

Alt

Alt

Return

Fl

F2

F3

F4

F5

- RCC-M (1985)

- Swedish (1978)

- STOOMWEZEN (1978)

- Norwegian (1983)

- BS 806 (1986)

4.17

I


At this stage the model needs to be developed. This is done on the main Layout Screen, /

which is entered using the r~>*

SAMPLE

Number of elements = 0 # 1 Units :

From _ To Type DX DY DZ

Material =

Nominal OD

Corrosion

Insulation

Pressure =

=

allowance =

type =

Schedule =

Insulation

Temperature

OD =

Mill tolerance (1

density =

= 70

Thickness =

i) =

Insulation thk =

Specific gravity = Additional weight Wind load(y/n) =

Fl-Help F2-Material F3-Supports F4-SIF F5-Loads F6-Cmass F9-Flange Alt O-Other

Enter Alt M-Modify(l)

Alt E-Edit Alt A-Modify{*)

Alt D-Delete

Alt S-Split

F7-Previous Alt J-Go to

F8-Next FlO-Return

4.18

1

1


Before proceeding it is necessary to select the material to be used: form the Layout Screen

use ED to access the Material Properties screen.

r

r

(SI

Number of materials = 0 Material Properties Units :

Mat. number = _

Density =

Temp E

Description :

Nu =

Alfa

Joint factor =

Allow.str Yield str

Type =

Tensile str =

Rupt.str

«—' Enter

Alt M-Modify

Alt D-Delete

Fl-Database

F7-Previous

F8-Next

FlO-Return

4.19 : ..


In CAEPIPE a material database for B31.3 can be used; this is selected using the (fi) key. {The cursor keys select the material. Here the first material, Carbon steel, Carbon < 0.3%,

is highlighted and may be selected using the [Enter] key.

Name : B313

Use

cursor keys,

PgUp, PgDn,

Home and End

to select

material.

Material Database

Carbon steel, Carbon <= 0.3 %

Carbon steel, Carbon > 0.3 %

Carbon Molybdenum steel

Low chrome moly steel

Intermediate Chrome Moly Steel

Austenitic Stainless steel

Straight Chromium Steel

25% Chrome, 20% Ni (Type 310)

<—' Retrieve F10 - Return

-1

#*^1

era

i

4.20

1


(Other material databases are available and can be specified at the Main Menu). The

material properties are retrieved:

(am

r

r

r

Number of materials = 0 Material Properties Units :

Mat. number = _

Density =

Temp

70

200

300

400

500

600

700

800

900

1000

1100

0.2841

E

27.90E+6

27.70E+6

27.40E+6

27.00E+6

26.40E+6

25.70E+6

24.80E+6

23.40E+6

18.50E+6

15.40E+6

13.00E+6

Descriptior

Nu = 0.292

Alfa

6.070E-6

6.380E-6

6.600E-6

6.820E-6

7.020E-6

7.230E-6

7.440E-6

7.650E-6

7.840E-6

7.970E-6

8.120E-6

i : Carbon steel, Carbon <= 0.3 %

Joint factor =1.00

Allow.str

20000

20000

20000

20000

18900

17300

16500

10800

6500

2500

1000

Yield str

Type = CS

Tensile str = 60000

Rupt.str

«—1 Enter

Alt M-Modify

Alt D-Delete

Fl-Database

F7-Previous

F8-Next

FlO-Return

4.21


This material properties table will be denoted Mat. number 1; using the cursor keys (

highlight this field and type 1 and press [Enter]; finally press (fw) to return to the Layout

screen shown above.

The Layout screen is used to specify the beginning and end nodes (From and To) of the

elements (at the outset this is shown as element #1 at the centre ofthe top line ofthe screen)

and the distance ofFrom node to the To node with respect to the chosen coordinate system

in terms of distances DX, DY and DZ. If the To node is a bend this may be specified, with

an optional bend radius for non standard components. For this component the material

(Mat. Num 1 in this case), the pipe size, insulation, pressure, contents specific gravity and

temperature are specified. These input fields can be navigated using the cursor keys (or

Tab keys).

Nodes are labelled as in Figure 7; in fact only 8 nodes are required to build the model in

this example - numbered from 10 thro 80. It is assumed that node 10 is at the origin ofthe

coordinate system at one anchor; the orientation ofthe coordinate system is as shown - this

is entirely arbitrary. The From and To nodes should be filled in in the Layout screen. At {

node 20 there is a standard bend; in the Type field, type B for a bend. The distance from '~

node 10 to node 20 is DX=9ft, DY=0, DZ=0. Units are in American standard by default,

and ft-in are taken care ofautomatically; they can be immediately changed to SI using the

(Ah)(u) key combination. Using the cursor keys skip to the Material field, input 1 for material,

then below to the section properties field, input 6 for the nominal OD and std for schedule;

the actual OD and thickness are automatically derived. The rest of the fields are input in

a similar manner and shown below:

SAMPLE


From 10 To 20 Type fiend DX 9'0" DY DZ

Bend radius =

Node at angle =

Node at angle =

Bend thickness = Miter cuts =

Material = 1 Carbon steel, Carbon <= 0.3 %

Nominal OD = 8 Schedule = 80 OD = 8.625 Thickness = 0.500

Corrosion allowance = 0.000 Mill tolerance (%) = 0.0

Insulation type = CS Insulation density =11.0 Insulation thk = 2.000

Pressure 200 Temperature = 600

Specific gravity = 0.800 Additional weight = Wind load(y/n)


«—' Enter

Alt E-Edit

Alt M-Modify(l)

Alt A-Modify(*)

Alt D-Delete

Alt S-Split


F8-Next FlO-Return

1

1

1

1

4.22 . •

1

(IP)

Design & Analysis ofPiping Systems -

Pressing the 0 key inputs the first element to the database. The element number changesto #2 and the From field changes to node 20; the To node automatically changes to 30 based

on past history; a different node number can be used. The screens for element no.2 to

element no.7 are shown:

Number of elements = 7

SAMPLE

# 2 Units :

From 20 To 30 Type __ DX DY DZ 6'0"

Cut short



Corrosion allowance = 0.000 Mill tolerance <%) =0.0


Pressure = 200 Temperature = 600

Specific gravity = 0.800 Additional weight = Wind load(y/n) = y


Enter Alt M-Modify(l) Alt D-Delete F7-Previous Alt J-Go to

Alt E-Edit Alt A-Modify(*) Alt S-Split F8-Next FlO-Return

4.23

-


SAMPLE


From 30 To 40 Type fiend DX DY DZ 6'0"

Bend radius = 18.000

Node at angle =

Node at angle =

Bend thickness Miter cuts

Material = 1 : Carbon steel, Carbon <= 0.3 %





Specific gravity = 0.800 Additional weight = Wind load(y/n) = y


«-J Enter" Alt M-Modify(l) Alt D-Delete F7-Previous Alt J-Go to


Cw^j

SAMPLE


From 40 To 50 Type _ DX DY -6'0' DZ

Cut short =

Material Carbon steel, Carbon <= 0.3 %





Specific gravity = 0.800 Additional weight Wind load(y/n)


Enter

Alt E-Edit

Alt M-Modify(l)

Alt A-Modify{*)

Alt D-Delete

Alt S-Split


F8-Next FlO-Return

4.24

1

1


Number of elements = 7 # 5SAMPLE

Units

From 30 To 60 Type _ DX 6'0" DY DZ

Cut short =

Material : Carbon steel, Carbon <= 0.3 %

Nominal OD " Schedule = STD OD = 6.625 Thickness = 0.280

Corrosion allowance = 0.000 Mill tolerance (%) =0.0



Specific gravity = 0.800 Additional weight = Wind load(y/n)


Enter' Alt M-Modify(l) Alt D-Delete F7-Previous Alt J-Go to


Number of elements = 7 # 6SAMPLE

Units :

From 60 To 70 Type DX 2'0" DY DZ

Weight =200

Thickness X = 3.00

Add. Weight = 50Insulation weight X = 1.75

Offsets : DX = 0.000 DY = 18.000

Alt L-Library

DZ = 0.000


Nominal OD = 6 Schedule = STD OD = 6.625 Thickness = 0.280

Corrosion allowance = 0.000 Mill tolerance (%) =0.0



Specific gravity = 0.800 Additional weight

Fl-Help

Wind load(y/n) = y

F2-Material F3-Supports F4-SIF F5-Loads F6-Cmass F9-Flange Alt O-Other

«-> Enter Alt M-Modify(l) Alt D-Delete F7-Previous Alt J-Go toAlt E-Edit Alt A-Modify(*) Alt S-Split F8-Next FlO-Return

4.25


SAMPLE


From 70 To 80 Type _ DX 6'0" DY DZ

Cut short


Nominal OD Schedule = STD OD = 6.625 Thickness = 0.280




Specific gravity = 0.800 Additional weight Wind load(y/n)


<—I Enter' Alt M-Modify(l)

Alt E-Edit Alt A-Modify(*)

Alt D-Delete

Alt S-Split


F8-Next FlO-Return

1C5S

Toggling graphics using the (US key shows the current state of the model:

Samp1e prob1emSAMPLE

4.26


The supports may now be input by moving to the Supports screen, (F3),

SAMPLE

Supports

Fl - Restraints

F2 - Skewed restraints

F3 - Guides

F4 - Hangers

F5 - Nozzles

F6 - Limit stops

F7 - Snubbers

F10 - Return

. 4,27

FLEXIBILITY ANALYSIS OP PIPING SYSTEMS

then selecting §3 for restraints.

Number of restraints = 3 Restraint SAMPLE

Node number 10

Translational stiffness

Rotational stiffness

XX YY ZZ

R

(lb/inch)

(in-lb/deg)

X Y Z XX YY ZZ

Releases (y/n) n n n n n n

Fl - Anchor

<—' Enter

F7 - Previous

Note : Stiffness = R for a rigid restraint

Alt M - Modify Alt D - Delete

F8 - Next F10 - Return

cs\

1

4.28

1


An anchor is specified using the (H) key so that all stiffnesses are shwon as rigid (R). In theNode Number field, specify node 10 and 0 will input the anchor. In a similar manner therestraints at nodes 50 and 80 are specified.

Press (fio) to return to the Supports screen and press © for hangers,

Number of hangers = 1

Node

Number

Hanger

Number

of hangers

D

Fl - Hanger type : Grinnell

F3 - User defined hangers

«—i Enter Alt M - Modify

F7 - Previous F8 - Next

SAMPLE

Load

Variation

F2 - Mid range

F4 - Options

Alt D - Delete

FIO - Return

FLEXIBILITYANALYSIS OF PIPING SYSTEMS-

followed by ® to design the hanger:

SAMPLE

Hanger Types

Fl - Grinnell

F2 - Basic Engineers

F3 - Bergen-Paterson

F4 - Borrello

F5 - Carpenter & Paterson

F6 - Constant Support

F7 - Corner & Lada

F8 - Elcen

Alt Fl - Fee & Mason

Alt F2 - Flexider (30-60-120)

Alt F3 - Flexider (50-100-200)

Alt F4 - Lisega

Alt F5 - Nordon

Alt F6 - NPS Industries

Alt F7 - Power Piping

Alt F8 - Piping Tech & Products

Alt F9 - SSG

F10 - Return

C^S

4.30

1


Specify 30 for the node number and 0 to input a standard hanger. Press (go) three timesto return to the Layout screen, then gs) for the Loads menu,

SAMPLE

Loads

Fl - Forces & moments

F2 - Specified displacements

F3 - Seismic

F4 - Wind

F10 - Return

4.31

FLEXIBILITYANALYSIS OF PIPING SYSTEMS s

followed by (f2) for specified displacements:

Specifying 50 for Node Number and 0.5 for Y displacement. Press @w) several times to

return to the Layout screen.

Number of displacements = 1SAMPLE

Specified Displacements

Node X

Number

10

F7 -

displacement Y

(inch)

X rotation

(deg)

1 Enter

Previous

Alt M

F8 -

displacement Z

(inch)

0.5000

Y rotation

(deg)

- Modify

Next

displacement

(inch)

Z rotation

(deg)

Alt D - Delete

F10 - Return

1

4.32

1

1

Design. & Analysis ofPiping Systems

At this stage the sample problem has been completely specified. The graphics screen is asshown below:

Sample problemSAMPLE

8

d'8

8.5988

LENGTHCFT'IN") SPECIFIED DISPLACEMENTSCINCH,DEG)


Use the gw) key to return to the Main Menu and press © to analyse the model; this isfdone quite rapidly for this simple problem. Then press © to Output the results. Select theload case, for example Q for sustained, then ® for a Code Check:

SAMPLE

E

1

e

m

1

1

2

3

3

N

o

d

e

10

20A

20A

20B

20B

30

30

40A

40A

40B

Press.

(psig)

Design

Allow.

200

2103

200

2103

200

2103

200

2103

200

2103

ANSI/ASME

Sust

SL

(psi)

1417

912

937

1003

957

1742

1731

886

888

1087

: 302.3

SH

(psi)

17300

17300

17300

17300

17300

17300

17300

17300

17300

17300

B31.3

.5(c)

SL/

SH

0.08

0.05

0.05

0.06

0.06

0.10

0.10

0.05

0.05

0.06

(1987)

Occ :

SL+SO

(psi)

Code Compliance

302.3.6 (a) Exp :

1.33SH SL+SO/ SE

(psi) 1.33SH (psi)

28209

24441

42558

30621

17531

48741

43815

14109

16843

9415

302.3.

SA

(psi)

29325

29325

29325

29325

29325

29325

29325

29325

29325

29325

5(d)

SE/

SA

0.96

0.83

1.45

1.04

0.60

1.66

1.49

0.48

0.57

0.32

Ti PgUp PgDn Home End Fl-Sorted stresses Alt P-Plot stresses F10 - Return

GPi

1

1

4.34


Various other options are available.

The reader should appreciate that this is a relatively simple exercise; more complexproblems only require more input effort. The solution phase, which in this example takesa few minutes, is simply based upon the principles discussed above. The software may look

complex (although not so bad with CAEPIPE), but the mechanics are simple. Howevercomplex the software, the basic analysis engine and Code check are essentially the same

- the rest is cosmetics. CAEPIPE does thejob efficiently, with a well designed user interface.More sophisticated software, such as CAESAR, will do essentially the same, but enhancesthe graphics and ties in to CAD systems; additionally CAESAR will perform a detailedanalysis ofspecific components (bends and tees) using finite element shell analysis amongstmany other enhancements. However, the basic flexibility analysis and Code check remainsthe same as CAEPIPE.

4.4 IS FLEXIBILITY ANALYSIS RELIABLE ?

It is quite common practice for piping designers not to think too much about flexibility

analysis. The sophistication of present software systems make the analysis and Code

assessment straightforward and apparently unambiguous. If the software system predicts

component stresses which do not satisfy Code allowables, then the designer simply reroutes

the pipe, ifpossible - adding flexibility or removing restraint as required. Simply the design

process for piping is established and evidently reliable - it seems to work since there is not

an abundance of piping failures. It is often claimed that this approach to piping design -

flexibility analysis, accepted modelling assumptions (beam behaviour with factors, rigid

anchors, simple hangers and supports and so on) is probably conservative. The designer

and analyst should in fact not believe this claim any more than he should believe that the

results of the analysis are representative of actual pipe behaviour. Two examples shouldsuffice to make this point further:

4.4.1 Is Code flexibility analysis conservative?

Consider the piping system shown in Figure 8, consisting ofa seven bend branchless pipingsystem subject to a uniform temperature loading of 200degC.

4.35


SVS*

/

3868

seaa

= 100°C

V

R = 1000 mm

r= 150 mm

t= 15 mm

E = 191.0 N/mm2v = 0.292

cxr = 11.48£T - 6°C~I

see

38

S308

88

ILEMGTH(MM)

fiSrS

Figure 8: Example piping system

It is anchored at both ends, denoted points 1 and 6, and has rigid translation supports at

points 2,3,4 & 5. The object is to obtain the displacement and maximum stresses at the

mid sections of all seven bends.

This example was first analysed using CAEPIPE to B31.1, using conventional flexibility

analysis. The aim is to compare the results ofthis analysis to a more detailed finite element

analysis which uses special ovalising pipe bend elements(1) which give more accurate andrealistic pipe bend deformations and stresses without the use offlexibility or stress factors;these will be discussed further in Lecture 5. These elements have been included as user

defined elements in the ANSYS finite element system. However, as will become clearer

also in Lecture 5, the stresses calculated in B31.1 (and B31.3) are not the actual elasticstresses in the system where stress factors are used - the stress factors are for Code purposes

(1) - D Mackenzie & JT Boyle: Analyses of piping elbows using two new elbow elements. ASME PressureVessel & Piping Conference, Nashville, 1990

i

1

Design & Analysis ofPiping Systems « .

only. In fact for bends, the B31 stresses correspond to about halfthe elastic stresses. SinceCAEPEPE does not allow stress factors to be altered, the ANSYS piping module was usedwith corrected factors to allow the calculation ofelastic stresses (in fact the Clark & Reissnerformulae to be discussed in Lecture 5).

A comparison of calculated displacments at the middle of all seven bends is given belowwhere Analysis 1 uses flexibility factors and Analysis 2 is the more detailed analysis:

Bend

A

B

C

D

E

F

G

UX(mm)

9.161

15.016

-1.405

-10.863

6.323

27.224

10.473

Analysis 1

UY(mm)

-9.493

-4.646

5.579

-5.112

-11.374

-1.695

8.587

UZ (mm)

-0.613

2.620

-5.270

-0.517

-1.193

-18.840

-9.430

UX(mm)

9.151

15.034

-1.396

-10.530

6.650

28.176

11.028

Analysis 2

UY(mm)

-9.54

-4.672

5.516

-5.162

-11.533

-1.831

8.593

UZ (mm)

-0.647

2.694

-5.161

-0.521

-1.206

-19.089

-9.774

The maximum stresses, at the middle of the bends is

Bend A

BendB

BendC

BendD

BendE

BendF

BendG

Analysis

1

2

1

2

1

2

1

2

1

2

1

2

1

2

atmu (N/mm2)

23.6

20.7

18.9

10.5

53.3

35.5

65.4

53.1

36.8

17.7

35.3

39.8

45.1

18.0

% Difference

+14.0%

+80.0%

+50.1%

+23.2%

+7.9%

-11.3%

+150.5%

It can be seen that the displacements compare favourably. However the stress comparisonsare more interesting:

• In the most highly stressed bends, C, D & G, the conventional flexibility analysis isnot just conservative, but clearly over conservative.

• In bend F the conventional flexibility analysis is not conservative.

4.37

FLEXIBILITY:^A]^ffi:Q&-PIPING SYSTEMS

In fact studies of this sort are very rare, even though it is possible to obtain more detailed^

analysis with current finite element analysis software(2). One obvious conclusion to be madehere is that one possible means of overcoming problems in satisfying the Code allowables

could be to perform more detailed analysis; alternatively, this could make the stresses

worse!

4.4.2 Does flexibility analysis represent real behaviour?

A simple answer - probably not!

One ofthe points beingmade in this course is thatmany simplifications and approximations

are made in piping flexibility analysis, such as the use of beam theory, simple flexibility

and stress factors, simplification of anchor, support and hanger behaviour and so on. More

discussion ofinherent simplifications will be given in the next two Lectures where the real

behaviour of elbows, the main source of flexibility, and branches, will be examined. There £

have been very few studies of the effect of these simplifications. Nevertheless a significant

study was undertaken by Carmichael & Edwards(3):

Carmichael and Edwards took a notional pipe run and examined the effect of modifying

several of the conventional modelling assumptions. Two are particularly interesting:

• With the assumption offree rotation at the anchors, rather than a complete fixing,

the piping deflection did not alter greatly. The predicted maximum stresses were

significantly greater in most of the run with the rigid anchors (as expected) but also

demonstrated that this assumption was unconservative in one part of the run.

• It conventional piping analysis hangers are assumed to be simple (linear) springs.

Carmichael & Edwards looked at the effect of hanger lateral stiffness. Again the

predicted deflections were similar with and without lateral stiffness, while the

maximum showed great variation - in particulr that the conventional assumption

of no lateral stiffness was particularly unconservative!

r ■ fOne implication of these observations is that if in practice the piping system deflections

are monitored to verify the analysis reliability, then any problems will not be apparent.

Further, conventional assumptions are demonstrated to give unconservative results! The

reader is left to ponder the implications.

i(2) - D Mackenzie, T Comlekci & JT Boyle: Comparison of flexibility and finite element analysis of examplepiping systems. Proc 11th Int Conf on Structural Mechanics in Reactor Technology, Tokyo, 1991(3) - GDT Carmichael & G Edwards: Some observations on the analysis of high temperature steam piping jsystems. Proc Inst Mech Engnrs, Vol.193, 149-158, 1979


5 BEHAVIOUR OF COMPONENTS: PIPING ELBOWS

5.1 INTRODUCTION

The simplest, and often most ecomomical, means ofobtaining flexibility in a piping systemis to introduce sufficient smooth bends to absorb the thermal expansion (for large diameterpipes smooth bends can be replaced by mitre bends). We have already mentioned thatsmooth bends are more flexible than an equivalent beam, and thus that in conventionalflexibility calculations an additional factor must be introduced. In addition, this extraflexibility is accompanied by higher stresses than would be expected from simple bendingtheory. The analysis and design of bends thus becomes one of the important tasks of apiping engineer. In this lecture we aim to examine how these additional factors may befound for smooth pipe bends andhowthey are used in the various design Codes. In particularwe will see that the Codes use a very simplified approach, even though our currentknowledge on the mechanical behaviour ofpipe bends is considerably more advanced.

5.1.1 In-plane bending of a pipe bend - von-Karman's analysis

To begin with we will look at the problem of the elastic behaviour of a smooth pipe bendsubject to in-plane bending and introduce the initial solution of this problem which v/asgiven by von-Karman(I) in 1911:

Von-Karman recognised that a curved pipe under in-plane bending would undergoflattening Covalisation') of the cross section, an effect which could normally be ignored forstraight pipes, and developed a relatively simple stress analysis for this problem whichrelied on several assumptions:

(a) As the pipe bends, plane sections remain plane (as in simple engineer's theory ofbending) and the deformations would be small.

(b) Each cross section of the pipe would have the same degree of ovalisation (the purebending assumption).

(c) The ovalisation deformation in the cross section would be inextensible (that is thelength of the circumference of the pipe would not change).

(d) The pipe is of long radius - that is the bend radius R is much larger than the tuberadius, r.

The deformation of the cross section, and the pipe geometry, are shown in Figure 1.

ie Formanderung dunnwandiger Rohre, insbesondere federnder Ausgleichs-

5.1

t! BEffiOTOUR OFCOMPONENTS: PIPING ELBOWS

1

Figure 1: Deformation ofa pipe bend in pure bending

It may be described mathematically using a radial, w, and tangential, v, displacement;however the assumption of inextensibility provides a simple relation between v and w.Von-Karman assumed that the radial displacement could be represented well by a trig

onometric series,

w = a2 cos 2<|> + a4 cos 4$ + a6 cos 6ty...

which is simplified to an even cosine series due to symmetry of deformation of the cross

section. The coefficients a2, a4, etc are unknowns to be determined. Von-Karman chose anenergy method to find these coefficients: specifically he used the theorem of minimum

potential energy which states that,

Amongst all the possible deformed shapes which a structure can assume under a speciftc

loading, that which minimises total potential energy will be the one which occurs.

Thus the coefficients a2, a4, etc must minimise potential energy. The analysis then proceedsby formulating the total potential energy in terms of the strain energy of the deformation

5.2

Design & Analysis ofPiping Systems ^ \ - »

of the cross section, U, and the work done by the bending moment, M, in changing thecurvature of the centre line as represented by the change in subtented angle of the endplanes of the pipe, the 'end rotation*, y, (as in simple engineer's theory of bending):

Tl = U-My

Finally some simple mathematical analysis is done to find the values of coefficients whichprovide the minimum.

Von-Karman did a simple analysis with only a single term in the series expansion for w,

W =

The analysis is not given in detail here but can be readily found in the literature(2>.

A flexibility factort k, as the ratio of end rotation of the pipe bend to that of an equivalent

straightpipe (same material, cross section andload, butwith equivalent length) was definedwith the result that,

12A,2+10

where the pipe bend parameter or pipe factor, is defined as

r2

The variation of the flexibility factor with the pipe bend parameter is given in Figure 2.

Circumferential (hoop) and longitudinal stresses are induced in the cross section and thesemay be calculated as

— = v[k cos <j> - ks cos3 <|>] ± -ksX cos 2$

°e 3— = [k cos <|> - ks cos <|>] ± -v&j

where the positive sign refers to the outside surface, k is the flexibility factor as before and

' 12X2+1

(2) - RKitching: Smooth and mitred pipebends. Chap 7, "The Stress AnalysisofPressureVessels and PressureVessel Components" Ed SS Gill, Pergamon, 1970

5.3

: BEHAVIOIJB DP;COMPONENTS: PIBM

Figure 2: Von Karman's Flexibility Factor

OUTSIDE SURFACE

2 - —

\\

INSIDE SURFACE

Figure 3: Von Karman's stress distributions

5.4

1


The nominal bending stress in a straight pipe under bending is,

Ma. =

ntr2

These stress distributions are plotted in Figure 3 for the case X = 0.5, v = 0.3.

There are three immediate points to notice about the results:

(a) Flexibility and stresses (factored with nominal stress) depend only upon the pipe bend

parameter. This is a result of the long radius assumption. In fact if this assumption was

not made, then they would also be found to depend upon a second parameter, the radiusratio, being the ratio ofbend radius to cross sectional radius,

R

(b) As the pipe bend parameter decreases the flexibility factor increases; thus as a pipebecomes thinner its flexibility will increase, or as it increases in diameter.

(c) The longitudinal bending stress has increased, but its maximum is no longer at the

positions expected by simple bending, being closer to the intrados, or extrados. Further,

the hoop circumferential stress, mainly bending, is larger( this stress is negligible for simplebending) and occurs at the intrados. The maximum stress can be intensified by up to three

or four times over the maximum longitudinal stress predicted by simple bending theory.

5.1.2 Behaviour of piping elbows

The preceding von-Karman analysis is based upon several simplifying assumptions.

Naturally many extensions and alternatives to this theory have appeared in the past

seventy years, together with a large amount of experimental evidence. Principally the

von-Karman analysis as found to be sufficient only for pipe bends with pipe bend parameters

larger than 0.5. The most significant alternative analysis was carried out by Clark &Reissner(3) in 1951 (as part of Clark's doctoral dissertation at MIT). Using Reissner's shelltheory and analytical methods Clark reproduced von-Karman's results using trigonometricseries, but also gave an 'asymptotic' solution for small values of the pipe bend parameter:

(*3)" RA.0181* & E Reissner: Bending of curved tubes. In "Advances in Applied Mechanics" Vol 2 o93Academic Press, 1951 ' H '

5.5

BEHAVIOUR OF COMPONENTS: PIPING ELBOWS

c+ 1.892 0.480 '

which as we will see has formed the basis for the ANSI B31 design Codes.

It is difficult to provide a complete review here, but several milestones should be pointed

out:

(i) In 1952 extensive research was carried out at Imperial College into the behaviour of

pipe bends under a variety of loading conditions. Gross and Ford(4X5> found by experimentthat three terms in the von-Karman series were sufficient for most practical pipe bends,

and further Gross removed the assumption of inextensibility which was found to have a

minor effect on stress (the so-called 'Gross correction'). Turner and Ford(6) in 1957 gave afairly extensive review of the various analytical methods for pipe bends and provided a

detailed numerical analysis of the problem using shell theory (this was later updated by

Blomfield(7) in 1971) which later formed the basis for BS806 design curves for pipe bends. (

An attempt was also by Smith(8) in 1967, extending an earlier analysis by Vigness in 1943,

to look at the problem of out ofplane bending. These analyses were not particularly suc

cessful since the pure bending assumption is not very good in this case, and some exper

imentally determined adjustments had to be made. Nevertheless these analyses for out of

plane bending form the basis of BS806.

(ii) Rodabaugh and George(9) in 1957 addressed the problem ofinternal pressure in a pipe

bend subject to bending. Making similar assumptions to von-Karman they evaluated the

work done by the applied pressure in changing the cross sectional area ofthe pipe and then

followed the standard energy analysis. The flexibility factors are modified according to,

k =k

Thus as the pressure increases, the flexibility decreases, depending upon the exact (

geometry. These results are in fact used in the ANSI B31.3 and ASME codes, but are not

very realistic - we will discuss this further below. The problem is that the deformations are

no longer small, and a much more complex analysis should be used, as was pointed out by

(4) - N Gross: Experiments on short radius pipe bends. Proc IMechE, Vol.IB, p465, 1952

(5) - N Gross & H Ford: The flexibility of short radius pipe bends. Proc IMechE, Vol.IB, p480, 1952

(6) - CE Turner & H Ford: Examination of the theories for calculating the stresses in pipe bends subject toin-plane bending. Proc IMechE, Vol.171, p513, 1957

(7) - JA Blomfield & CE Turner: Theory of thin elastic shells applied to pipe bends subject to bending andinternal pressure. J Strain Anal, Vol.7, p285, 1972

(8) - RT Smith: Theoretical analysis of the stresses in pipe bends subjected to out of plane bending. J MechEngSci, Vol.9, pll5, 1967

(9) - EC Rodabaugh & HH George: Effect of internal pressure on flexibility and stress intensification factorsof curved pipes or welding elbows. Trans ASME, Vol.79, p939, 1957 <

5.6

1


Crandall and Dahl(10> in a 1956 extension to Clark and Reissner's work. It is only fairlyrecently that a correct pure bending analysis for a curved tube has been given by Boyle &

Spence(11), although the results are inconclusive.

(iii) With the advent of modern computing technology the pure bending problem could be

analysed to any degree ofexactness; there is a large literature on this problem using either

the energy approach of von-Karman or the thin shell method of Reissner using modern

numerical techniques, finite differences and of course finite elements. The first finite

element analysis of a piping elbow seems to have been done by Natarajan and Blomfield(12)

at Imperial as part ofNatarajan's doctoral thesis in 1971; an extensive set ofanalyses were

carried out at ORNL<13) in the middle seventies. Indeed in 1974 Hibbitt<14) formulated andimplemented a special pipe bend element for the MARC finite element program for West-

inghouse; numerous other such pipe bend elements have been formulated and these

probablyrepresent the future ofpiping analysis. Studies undertaken at Strathclyde suggest

that a relatively simple elbow element(15), which has been coded as a user element for theANSYS finite element program, gives results comparable to more detailed analysis.

(iv) For in-plane bending the most difficult assumption to remove was that ofpure bending

and plane sections remain plane. In practice a pipe bend will have attached straights, or

even more severe, flanges. In either of these cases it cannot be assumed that every cross

section of the bend will ovalise in the same manner. For example, the attached straights

will suffer some ovalisation themselves and will restrict the ovalisation of the bend. This

will reduce the flexibility of the bend and alter the stress distribution, usually reducing

the maximum stresses. How severe this is depends upon the geometry ofthe bend. Several

analyses were given for the end effects problem. A numerical analysis was given by Kal-

nins(16) in 1969, but detailed results were not available until the work ofThomson & Spence(see later) who extended the classical von-Karman energy analysis and by Whatham(17) in

1978 who carried out a detailed numerical analysis and parameter survey of the shell

equations. We will look at this aspect later; however most ofthese analyses did not include

internal pressure (Thomson & Spence did include this in a manner similar to Rodabaugh

and George) and were mostly restricted to in-plane bending.

(10) - SH Crandall &MC Dahl: The influence ofpressure on the bending ofcurved tubes. Proc 9th Int Congressof Applied Mechanics, ASME, 1956

(11) - JT Boyle & J Spence: The nonlinear analysis of pressurised pipe bends. Proc 3rd Int Conf on PressureVessel Technology, Tokyo, 1977

(12) - R Natarajan & JA Blomfield: Stress analysis of curved pipes with end constraints. Comp & Struct,Vol.5, pl87, 1975

(13) - EC Rodabaugh, SE Moore & SK Iskander: End effects on elbows subject to moment loading ORNLRep 2913-0, 1977 (ASME Special Publ No.h00213,1982)

(14) - HD Hibbit: Special structural elements for piping analysis. ASME Special Publ "Pressure Vessels &Piping: Analysis & Computers" ASME, 1974

(15) - D Mackenzie & JT Boyle: Analyses of piping elbows using two new elbow finite elements. In "Designand Analysis of Piping and Components - 1990" Ed QN Truong et al, ASME PVP Vol.188, 1990(16) -A Kalnins: Stress analysis of curved tubes. Proc 1st Int Conf on Pressure Vessel Technology, Delft1969

(17) - JF Whatham: In-plane bending of flanged pipe elbows. Proc Metal Struc. Conf, Perth, Australia (1978)

5.7. •

BEHAVIOUR OP COMPONENTS: PIPING ELBOWS

The paper by Thomson & Spence provides a detailed summary of the extension of the s

von-Karman energy analysis to the problem ofin-plane bending ofa smooth pipe bend with

tangent straights and gives graphs and formulae for modified flexibility and stress factors.

The paper by Rodabaugh and Moore also summarises new results for pipe bends, but puts

them in the context of the ASME Code and provides the background to the ASME Code

CaseN-319.

The reader could use any ofthese results to provide more realistic design factors ifnecessary.

A fuller discussion, in the context of the Codes, will be given in Section 5.2.

In fact developments in computer and finite element technology make it quite easy to

perform a detailed analysis of any geometry of piping elbow:

A finite element model for a 90deg elbow is shown in Figure 4, loaded under an in-plane

force. Stress contours for outside surface circumferential and longitudinal stress dis

tributions are shown in Figures 5 & 6 respectively. The complete analysis, from geometry <

modelling to meshgeneration and solution, using theANSYS program on a laptop computer

takes no more than fifteen minutes; on a workstation in a few minutes.

ANSVS <*.AUNIU VERSIONAUC 26 1991

*MP3?POSTX ELEMENTSTYPE NUH

SB iZU =1

isT*XF =2.261

III S£*W«PRECISE HIDDEN

Figure 4: Finite element model ofpiping elbow

5.8

1C5T1

1

1

Design & Analysis ofPiping; Systems

ANSVS 4.4

ITER=1

ELEM CS

DMX =0.869378SMN =-2383SMX =3808

XU = 1

2b1

*VF =1

•ZF =29.S74

PRECISE HIDDEN9 =-2228C =-803.06E =612.136G =2032I =3433

Figure 5: Outside surface circumferential stress contours

1

PIPING CLBOU

t

,'

i'4i *

• CU 1

I

\>J

si

4WffwMX

\ v\v

s

7?

3^l\\

.\^«S

A,

SI

S.>N

>^

sc

S

\ * i

ANSVS A.4

POSTX^XRESS

ITER=i<AUG>

ELEM CSDMX =0.069378SMN =-4478SMX =8049

XU =1VV =1ZU =1

►VF =1.165»ZF =29.S74PRECISE HIDDEN

15!=1783=4966=7349

Figure 6: Outside surface longitudinal stress distribution

BEHAVIOUR OF COMPONENTS: PIBTNG ELBOWS

The current state of the art in our knowledge of the behaviour of pipe bends is rather'

curious: on the one hand it is very easy now to carry out a detailed finite element analysis

ofthe linear elastic behaviour ofanypipe bend. On the other hand, useful design information

has not been forthcoming, particularly for out-of-plane bending and for internal pressure.

This should be possible for the former, but, perhaps surprisingly, a detailed analysis for

internal pressure is extremely difficult. The reason for this is the so-called Haigh Effect.

Any pipe, straight or curved, subject to interal pressure departs significantly from simple

engineer's theory ifthe pipe cross section is not circular (say induced by manufacture). The

size of this effect depends upon the geometry and loading, but can intensify membrane

stresses by a factor of six! The problem is that the deformation of the pipe cross section can

no longer be assumed to be small, and subsequently the analysis is much more complex

and nonlinear (although ofcourse still amenable to finite element analysis). In a pipe bend,

even if the cross section is initially circular, any applied bending will ovalise the cross

section, and if pressure is present the Haigh effect will be important. The main result is

that the coupling of bending and pressure in a pipe bend is nonlinear. While the applied

moment tends to flatten the cross section of the bend, the internal pressure tries to work ^

against this - it tries to open up the bend (the Bourdon Effect). Very little information on

this effect is available apart from the crude Rodabaugh and George analysis. Not only is

the stress distribution significantly altered, but also the flexibility is reduced. This

reduction is recognised in some design codes. When this is coupled with end effects, the

behaviour ofpressurised pipe bends becomes very complex - so far too complex to warrent

inclusion in the design Codes

5.2 CURRENT DESIGN PHILOSOPHY

Wewill nowlook at how these results have been assimilated into the various Codes, choosing

here four examples. We will see that really only the simplest results have been used, having

been incorporated into the various Codes some twenty to thirty years ago, even though

more accurate results are available:

5.2.1 BS806

The background to the design philosophy ofBS806 is well documented in the paper by PL

Popplewell and J Hammill. The main aim ofthe Code is to achieve shakedown for the basic

pressure and bending stresses and to keep the mean hot, if applicable, and sustainedstresses below yield and less than the creep rupture stress for the given material at design «temperature in the design life. The sustained and hot stresses thus must lead to the )possibility of creep rupture and should not cause yielding. Shakedown is achieved byessentially limiting the elastically calculated stress range to twice yield (with a 'safety' -»factor, and modifications for branches), although the actual allowable which is applied may Jalso be modified to the creep rupture strength for the mean stress.

5.10


These limits are applied on the basis of elastic stresses (the elastic stress for a given load

condition is not limited, rather the elastic stress range):

For a straight pipe a hoop (transverse) and longitudinal stress are specified for internal

pressure:

JL -4t(d

which are the familiar engineer's theory for thin pressurised tubes in the Code notation

(see Lecture 2); the transverse stress has been modified to include the average ofthe radial

stress through the thickness.

These same equations are also used for pressure stresses in a bend, although they are not

strictly correct they are conservative and easier to use.

For a straight pipe only a longitudinal stress is induced,

A

which again is simple engineer's theory of bending to give the maximum elastic stress at

the outside of the pipe under combined in plane and out of plane bending.

For a smooth bend both hoop (transverse) and longitudinal stress are induced,

which are modifications to simple maximum elastic bending stresses, evaluated at the mid

section ofthe pipe. The stress intensification factors are given in figures 4.11.1 in the Code,

which are reproduced here for reference in Figures 7-10. These factors are based on the

maximum stresses derived in CE Turner & H Ford (1957) for in plane bending and by RT

Smith (1967) for out of plane. The flexibility factor prescribed in the Code for a smoothbend is also derived from these studies.

5.11


ILU /? ■ radius of bend

i;—i—.'-i-: ... See also

r ri-rr.prm ~~r. \:r\:: \ m • i :jz

jferH??tmi|:ti;hbk t-.o

OOl 0.0 2 0.C3 C05 010.04

0 2 0.3 0.50.4

Pipe factor

Figure 7: BS806 In-plane transverse stress factor

30

20

10

5

4

3

k3 2<_

o

u

v/l

<r>

4/i_

5 OS0.4

03

02

LU

t

!

-p

i

■4—

1 * i

1 ! 1

I i I

1 ' '

\ < ' i ;

1

i

=

-

i

i

i—fr

1 i

f A -i

r fl ■ r

■^ See dl

-i A-C

1 i 1 •

R

1

idiui of bend

so Fiq.4.ll.|(o)

■ ■ i -i—i—'-\

11 • •

? .

n>t

i ■ t

m

!!'

1

, „! i !

0 01 002 003 005 01 020305 I

Figure 8: BS806 In-plane longitudinal stress factor

10

GfPi

5.12

Design & Analysis ofPiping Systems ■

50

30

0.3

02-

0.1I I

{■»■

001 QO2O.O3 00S Ol0.04

0.2 0.3 0.50.4

Pipe factor

2 3 4 5 10

Figure 9: BS806 Out-of-plane transverse stress factor

30

10

o

ii 2

!

0.5

0.4

0.3

0.2

t i . . ^ShS

i ! 1;

I'll

V: ■

•

, 1

hi

. :]' '

1 1 1

!i

1

fvl-l 1

h-irrf-

i

i | •

i > i

t=

i

-——h—i—1 i ! ' , 1 : -n

: ' ;

1 ' 1

! " 1 ' ' i ' ■ ■ ■:| I ! ! l.|

• | 1 i !

lit'

lil.i in

nli

. ■ ii

• i •

, !

j :.

11

i

p

■ 1

•

_

i

i

:p

1

• i •

i :

X = .

=1

/-'

P ■ radius of

See also Fiq4.

rtrrr

■ ■ ! 'i i i i

: 1 ! 1

2 -H

A-

Mil

1 • 1 ! i i.,

i:i-

b<

II.

r_J

»nd

1(9)

....

" i

A-

«'i:

.,

i

i i

-+-

■T

6 j-

1*

Ox

1

I

1

7™

-

■

! ;

0

0.01 Q02 Q03 0.C5 O.I

0.04

0.2 0.3 0-5

0.4

I 2 3 4 5

Pipe factor ^

Figure 10: BS806 Out-of-plnae longitudinal stress factor

10

5.13

BEHAVIOUR OF COMPONENTS: PIPING ELBOWS -

1

It is evident that the Code charts and formulae are based upon numerous simplifying^assumptions, with an attempt to make the procedure conservative. Thus the effects of Jpressure and bending in a smooth bend are not coupled; pure bending theory with end

effects ignored for bothinplane and out ofplane bending have beenused to obtainmaximum «elastic stresses which are then summed as in a straight pipe. The charts provided for jflexibility and stress intensification are quite detailed: Popplewell & Hammill give some

relevantbackground,"... it was finally decided that all known confirmedinformation should «i

be included for the guidance ofthe designer...". In fact the work ofTurner & Ford has been 1considerably superseded, while that of Smith is probably not representative of real pipe

behaviour. There is an argument for simple results, as in the formulae used in the TASME/ANSI codes to follow, which are expected to be conservative from experience. The >

BS806 charts are not simple and for consistency should perhaps be replaced with simpler

formulae based on more recent results as discussed above.

Finally, no specific mention has been given here for mitre bends. However an excellentdescription of the background to the design formulae for mitres is given in the paper by . ^

Battle et al<18).

5.2.2 ANSI B31.1

As discussed in the above, the ANSI B31.1 Code has been based upon the concept of simplebending theory with stress intensification factors for specific components which have beenderived from fatigue data. The fatigue tests of Markl have been used as a basis for theANSI/ASME stress factors. We must be very careful in interpreting the derived stressintensification factor in the light ofthe manner in which it is used in the ANSI Codes (but

thankfully avoided in BS806):

From the flexibility analysis, an elastic bending stress in an equivalent straight pipe iscalculated. For an elbow, this simple bending stress is multiplied by the i-factor, but theresulting 'stress1 is not the maximum elastic stress in the elbow, it is the stress which would,cause fatigue failure in an equivalent straight pipe with a girth butt weld in a specifiednumber of cycles (that is, specified in the stress allowables). In fact the i-factor producesstress values which are about one-half the actual maximum elastic stress (this is pointedout very clearly in the Codes which use this result, particularly in cases where experimentally determined i-factors are not available). In fact, as we will see, the i-factors in theANSI codes (and ASME Sec.III Class 2 & 3) are obtained by comparing the theoreticalelastic stresses as obtained from the Clark & Reissner analysis with the experimental

i-factors.

The Codes notwithstanding, the i-factors do give us some indication ofthe fatigue strengthof piping elbows and are perhaps best viewed in this light.

(18) - K Battle et al: The design of mitred bends - a background to BS806.1975 ammendment No.3. ProcIMechE Conf "Pipework Design & Operation", p9, Vol.C22, 1985

5.14

1


Bearing this in mind, the Code is then rather curious: for example the limit provided to

protect against gross plastic deformation (which may lead to catastrophic burst type failure)

for sustained loads (pressure, deadweight etc.) is,

Z

where Sh is the allowable at operating temperature. Note well: this introduces the bending

stress due to sustained loads multiplied by the i-factor for fatigue ! A similar limit holds

for occasional loads,

taking into the bending moment due to occasional loads MB, where the factor k is equal to

1.15 for 10% of the time, and 1.2 for 1% of the time. The factor of 0.75i on the bending

stresses is related to a limit load - this is discussed in the ASME Sec.III code later.

The limit on expansion stresses is intended to protect against fatigue, and thus properlyuses the i-factors,

M

where Mc is the range of resultant moments between the hot and cold conditions, requiredfor the fatigue assessment in this case. This limit is often confusing since it introduces thesustained stress in the allowable; it may be written in the form,

+ Q.15MA)-<f(\25Sc

where Sc is the allowable stress at minimum (cold) temperature. The factor f is a stressreduction factor for cyclic conditions required since real S-N curves are not being applied.

ANSI B31.1 supplies i-factors for a range of components: these have been based on theoriginal Markl tests, modified by comparison with theoretical and experimental elastic

results, on updated tests - occasionally the only available results, say some elastic tests,have been simply modified in the spirit of Markl to obtain a fictitious i-factor ! We areconcerned here with smooth elbows:

For smooth elbows the flexibility factors are obtained from Clark & Reissner's asymptoticsolutions while the i-factors are developed from Markl's tests with the results 'molded' tothe form of the Clark & Reissner maximum stress factors. In fact the i-factor is roughlyone half the maximum elastic stress: from Clark & Reissner


A/f 1.89

whereas from B31.1,

This is perhaps not surprising - the full Markl test data infers that there was a factor of

two between the fatigue data on girth butt welded tubes and the polished bar specimens !

Separate stresses are not distinguished - the maximum stress component is used; separate

i-factors are provided for in-plane and out ofplane loading (in the ASME III Nuclear Codefor Class 2 piping factors are also modified for internal pressure according to the Rodabaugh& George solution) and also for bends with flanges (although the results are very suspect).

Again the factors used in the Code are very crude in comparison with present knowledge/In the case of smooth elbows these are based on approximate solutions to the pure bendingproblem, with no load coupling and inadequate and limited solutions for out ofplane loads

and for flanged bends. The Code has an air of authority - this is misleading.

5.2.3 ANSI B31.3

The ANSI B31.3 Chemical Plant & Petroleum Refinery Piping Code is based on similar

design concepts to B31.1 using i-factors derived from Markl's tests. In the case of elbows

it differs from B31.1 in two respects:

• Whereas B31.1 uses the same stress intensification factor for both in-plane andout-of-plane loading, B31.3 allows the use of a reduced factor for out-of-plane:

. 0.9

■ _a75

although there is a footnote to the effect that the higher value for in-plane may be

used for both cases "... if desired ..."!

In B31.3 it is stated that"... in large diameter thin-wall elbows and bends, pressure

can significantly affect the magnitudes ofk andi...". Ifthis is the case then correctedvalues of flexibility and stress intensification factors should be used (although it isnot mandatory). The correction factor for flexibility is such that,

ra

5.16


*,=*■

and for stress intensification

1+3.

Thus while the flexibility is reduced by this pressure correction, so also is the stress.

B31.1 does not use this correction.

Why B31.3 should provide a lower i-factor for out-of-plane bending, which can be ignored

"... ifdesired..." is rather curious, since both are derived from the Markl tests. As mentioned

in Sec.5.1.2 the treatment of out-of-plane bending through the use offlexibility and stress

factors is rather approximate. A theoretical analysis, as done by Vigness and Smith (used

in BS806) of the hypothetical case of pure out of plane bending, would imply that a single

factor independent of the bend angle, would not be appropriate (and hence the need for

Smith to adjust the results in comparison with experiments). Thus it should be understood

thatthe reduced factor usedin B31.1 for out-of-plane loading is only valid for the geometries

tested by Markl.

The pressure correction derives from the Rodabaugh and George analysis (Sec.5.1.2). For

90deg elbows under pure in plane bending with long attached straights (such that end

effects can be ignored) where the bending stresses predominate, then the Rodabaugh &

George results are reasonable. Otherwise they may be wholly unrepresentative of actual

flexibility and stress in the bend.

The writer has always treated the pressure correction in B31.3 with considerable appre

hension. It is commonly used to reduce stresses, and thereby satisfy Code requirements.

Kannappan(19> quotes this as a virtue "... this information was used to reduce the stress inpiping in real case analyses. Two large diameter long steam lines were built to supply

saturated steam to heavy water plants at Ontario Hydro's Bruce Nuclear Power Devel

opment. In preliminary analysis, the equations of flexibility and stress intensification

factors in Power Code B31.1 were used. In further analysis ... the pressure reduction

equations were used... and the piping was qualified...". It is debatable whether the reduced

stress was noticed by the steam lines! To this writer's knowledge, the accuracy of the

pressure reduction effect in real systems has never been tested.

5.2.4 ASME III Class 1

The ASME Pressure Vessel & Boiler Code Section III for Nuclear Vessels treats piping in

three Classes. Class 2 & 3 have design rules based upon ANSI B31.1 (in fact B31.7 which

(19) - S Kannappan: Introduction to Pipe Stress Analysis. Wiley, 1986 p72

5.17


has been superseded) with modifications for allowables and other specific areas. The rules »

for Class 1 piping are novel, being based upon an application of the pioneering 'design by

analysis' concept, but modified in para NB3600 to account for conventional piping analysis

practice - namely flexibility analysis using simple beam theory. Rather curiously the

background to the Class 1 piping rules is mysterious, even though it has been extensively

documented0205. There is an extensive literature discussing these rules, appearing yearly

and mostly confusing. The idea to bear in mind with the Class 1 rules is that design is

based on the avoidance of gross plastic deformation and collapse through primary (sus

tained - pressure modified for deadweight) loads, with a shakedown criterion for secondary

(thermal expansion loads); most commentaries on the Code view it in the light of the

primarily fatigue based ANSI B31.1 Code (the ASME III Class 1 rules do have fatigue

protection in the form of limited damage sums for peak stress).

The basis for the ASME Code Section III Class 1 rules is the concept of stress indices, as

summarised in Lecture 3,

BXPDO <

The so-called B-, C- and K-indices are derived from a consideration ofthe dominant failure

mechanisms for primary and secondary stress.

Stress indices were introduced into the ASME Code in its first edition in 1963 for nozzles

in pressure vessels. These were derived from a series of photoelastic and strain quage

tests; the maximum stress was written in the form,

PDs=i—

where I was the 'stress index'. This concept was broadened for Class 1 nuclear piping with

the addition of the B-, C- and K- indices where each was related to a different failure

mechanism,

(20) - SE Moore & EC Rodabaugh: Background for the ASME Nuclear Code simplified method for boundingprimary loads in piping systems. ASME PVP-Vol.50,1981

DF Landers: Application ofASME Criteria to piping design. Chap.6.2 In "Pressure Vessels & Piping Design- a Decade of Progress 1970-1980" ASME, 1980

DF Landers: "Piping Design per ASME Section III" Technical Seminars Inc, 1982 \

5.18


B : resistence to gross plastic deformation (limit load concepts)C : primary plus secondary elastic stress (range)

K: peak (highly localised) stress

These indices were identified with a particular type of load by subscripts: 1 for pressure,2 for bending moments and 3 for thermal gradients.

The B-indices have been derived from a consideration of limit loads: this is fairly obviousfor pressure loading. The B2 factor for elbows was originally derived on the basis of somelimited test data, and was given by 0.75C2 for many years until new test data showed thatit was overly conservative, being subsequently changed to 0.67C2 (the 0.75 remains in theANSI B31.1 rules, but for a slightly different reason). The reasoning behind this limit issimple: the test data indicated that plastic collapse did not occur until the stresses wereabout 1.5 times the maximum elastic stress (essentially C2). Thus, recalling Lecture 2, itwould be safe to allow the bending stresses in this case to increase to yield - whereas thepressure stresses should remain below 2/3 of yield. Then since the primary allowable Sis approximately 2/3 of yield, m

C3

<

that is,

21

and hence B2 should be equated with 0.67C2.

The C-mdices give the maximum elastic stress since the criterion used in the primary plussecondary limit is that of shakedown, and the moments used correspond to the range ofloading. A correlation between the i-factors and the C- and K-factors is given by,

which may be used to calculate i-factors. The factor of two is equivalent to changing thefatigue reference to plain straight pipe rather than a butt welded pipe. Again the flexibilityand stress factors are derived from Clark & Reissner as given above and thus have thesame limitations as discussed for B31.1. However Code Case N-319 provides alternativemore accurate factors taking into account end effects for in plane and out ofplane bending-the background to this is given in the paper by Rodabaugh & Moore.


It is quite important to realise that, although the basis for the ASME Class 1 piping rules(is sound, indeed following the ASME design by analysis philosophy, the development of

the rules based on limit load concepts is very approximate. A detailed discussion of thechoice oflimit loads is available in WRC Bulletin No.254 by EC Rodabaugh(21). The reasonfor this is not clear - WRC Bulletin No.254 is not very helpful. The problem is that B-factors

based on experimental limit loads factored for maximum elastic bending stress are used,based upon simple bending tests on elbows fixed at one end with the load applied at the

other: it is assumed that pressure has no effect on the limit load, an assumption which is

demonstrably false (although it is argued that pressure should increase the limit load,

again from limited experimental evidence alone).

5.3 FUTURE PROSPECTS

The above discussion has been in places quite critical of the Codes, in particular the US^- "ICodes. It is fair to say that the ASME Code committee concerned with B31.1 (and B31.3)are currently reviewing possible alterations to the Code. A panel discussion on this problemwas presented at the 1988 ASME Presssure Vessel & Piping Conference in Pittsburgh. In Jparticular there have been several 'early warnings' about a serious lack of conservatismin the Code, specifically for outlet branch connections, and significant over conservatism «

for occasional loads. It appears that this Code will retain the concept of i-factors with the Jmain design basis being fatigue (rather than move over to a complete rewrite in the designby analysis form ofASME III NB3600 which uses plastic design concepts oflimit load and «*shakedown in addition to fatigue) but is requesting additional testing and analysis to Jprovide more accurate and realistic i-factors for specific components. This programme ofanalysis and testing is currently underway. Also some of the criticisms of the flexibility "Iand stress factors in ASME III & VIII have been addressed, but arguably not completely, J

in Code Case N-319.

Nevertheless various researchers have suggested modifications to the current rules, andit is worthwhile summarising some of these developments further here. ,

5.3.1 ANSI B31 i-factors & ASME III C-factors

The ANSI B31.1/3 i-factors (and the related ASME III & VIII C-factors) have served pipinganalysis and design fairly well over the years and there has not been an overwhelming jdemand for changes. The reason is fairly simple: on the one hand the design approach usingthese factors has proven adequate, while on the other the factors are remarkably valid for -a wide range of common pipelines. J

i

(21) - EC Rodabaugh: Interpretive report on limit analysis and plastic behaviour of piping products. WRCBull. No.254,1980 I

5.20CSj3|


For example, Nataranjan(22), carried out a study of the reliability of flexibility and stressfactors for 90deg piping elbows, with long attached straights under in-plane bending. Thetwo tables given below show comparisons between a detailed finite element analysis ofshort and long radius elbows and the Code formulae for flexibility (k) and stress factors(C)

X

k

Short

Long

0.05

33

32.0

30.6

0.1

16.5

16.4

15.7

0.2

8.25

9.4

8.1

0.5

3.3

6.9

3.35

X

c2

Short

Long

0.05

14.4

14.5

17.48

0.1

9.05

8.85

11.51

0.2

5.70

5.73

6.6

0.5

3.1

3.3

3.14

It can be seen that the Code stress factors are representative ofthe more detailed analysisbut underestimating by about 17% for long radius bends. The flexibility factors are alsowell represented, although the Code formula can underestimate flexibility for highervaluesof the pipe factor by 50%. These would seem to be acceptable to most designers.

However problems with the i- and C-factors arise in less simple geometries:

Fujimoto and Soh(23) examined 90deg elbows with long attached straights (and unreinforcedfabricated tees) subjected to both in-plane and out-of plane bending. The componentsinvestigated specifically had large diameter ratios, Dlt > 100. A comparisonofCode formulaefor flexibility factor, k, and stress index, C2 for long and short radius bends compared todetailed finite element analysis is shown in Figures 11 & 12 respectively.

^;?J!atTar^n^alyjisA°^flexibiHties and stress intensification factors in 90 degree bends with endVd 188,1990 Slgn yS1S ° iping and ComP°nents - 1990" Ed QN Truong et al, ASMEI PVPfn^n"X^UJim0t0,.&J S,°h: FlexibiIifcy factors and stress indices for piping components with D/T > 100 subiectedto m-plane or out-of-plane moments. Trans ASME Vol.110, Jourh Press Vess Techn, 374-386,11^§UDjectea

BEHA^6pR:OFeOMPidNENTS: PIPING ELBOWS

I (APPROXIMATE

gr^Q formula, i 55 y.n 5.Tvoc iFE.'JI ' Ki/i ^ n<YPE

T 9 31

MiTER(m:4ll

1.0

0.9-

0.6

oo-

_3_, i ■ roi

TYPE FE.W

APPROXIMATE

FORMULA ,165

2M

231

11^ 2-3OJ '

ViTEQ'-n-- 311 6

MlTERIm:4l| O

MITER imsail °

•O9iis:ii'6''": :i

— — •O9lll3.33tiOw2.2l

!o.9lll3.35t.bi2.21

1.0

0.6

0.05 0.1 0.2 0.01

..£jft*l-='<ii^

0.05 O.I 0.2

(c) Flexibility factors for M, (d) FloxibllMy factors for /fa

Figure 11: Ratio ofFE & ASME flexibility factors

BENO

TYPEFE.M

ELSOWIworili °

ELBOW(longi| V

MlTERC":3l! &

MITERlm=4:

MlTERIms3J

a

o

APPROXIMATEFORMULA 1 95 .„„»,

LINES) a

'0.638

'0.730

-3-3.7* s

-4-JO.692

-3-|0.677

0

0.077

0.120

0 051

0.0 2 7

0.0 3 7

MlTFBImi4li 0 I -4-1100 l

0010.2

(a) Stress indices for M{ (b) Stress indices lor Mo

Figure 12: Ratio ofFE & ASME stress indices

5.22

xt can be seen that the Code formulae overestimate stress by about 40% for lower values

ofthe pipe factor, resulting in a substantial overdesign. Modifications to the Code formulaewere suggested for in-plane (i) and out-of-plane (o) loading as follows

m\

. 1.65

1.95

1.65

1.95

*\ 2/3

where the modification factors a, (3 take the form

except for the out-of-plane stress factor

where the constants a,b and c are tabulated for each load condition for long and short radiusbends. (The factors may also be multiplied by the factor for pressure reduction as necessary)

A far more dramatic problem with these factors has been given by Glickstein & Schmitz(24)who analysed back to back elbow configurations such as shown in Figure 13.

Fixed End

Figure 13: Back to back elbow geometry

£$ :^ Glickstein & LM Schmitz: Stress factors associated with closely spaced thin-walled elbows InPiping Components Analysis" ASMEPVP-Vol.218, 1991 c u«wa. *u

5.23 :

BEHAVIOUR OF COMPONENTS: PIPINGELBOWS

J

Some geometries "... exhibit negligible flexibility ..." compared to the Code values. Infaddition the Code stress factors "... do not provide a set ofconsistent results and cannot be "1used as a valid design method for predicting stresses in multi-closely spaced elbows ..."! J

5.3.2 ASME III B-factors

Perhaps the most troublesome aspect ofASME III & VIII are the B-factors which relate to

limit load concepts. Indeed the basis for the current factors are really inadequate. Recent

background work carried out for the French nuclear codes has suggested a more consistent

approach:

The limit load for a straight pipe in bending is given by the formula,

where t is the pipe wall thickness, r is the mean radius and oy is the yield stress for perfect

plasticity. The load for initial yield on the other hand is given by,

The ratio of these is,

K

Hence the limit moment for a thin straight pipe in bending is only 30% above that for first

yield and in design it is better therefore to limit the stresses to below yield.

An approximate analysis for the limit load ofa pipe bend under pure in-plane bending was

derived by Spence & Findlay(25); here the ratio of limit moment to the limit moment for a

straight pipe was calculated approximately as,

- = 0.8A,3'5MLS

for values ofthe pipe bend parameter less than 1.0; the detailed results are shown in Figure

14.

(25) - J Spence & GE Findlay: Limit loads for pipe bends under in plane bending. Proc 2nd Int Conf onPressure Vessel Technology, San Antonio, 1976

5.24

Hv:>:r.™-£; Design/& Analysis^

10

J a a a Experimentight pipe value.

0-1

74: Spence & Findlay's pipe bend limit loads

Thus the theoretical limit load for a pipe bend is less than that ofa straight pipe (as shouldbe expected). However this analysis assumes pure bending: this implies that the wholebend will reach yield at collapse. In practice end effects will be important here - an analysiswas given by Chan and Boyle(26), but the results were inconclusive - and there will be adifference between opening and closing moments. No theoretical limit loads are availablefor out of plane loading, or for internal pressure.

Modified B2 factors are given in Code Case N319; again these have been widely criticised.Touboul et al(27), based on the Spence & Findlay results and an examination of the FrenchCEA programme of elbow collapse tests have proposed that the modified formula

where a is the bend angle. An additional factor for pressure was given, based on theRodabaugh & George results discussed above. Also they suggested that the allowable betaken as

= min(l.45,, 0.635J

*"** ** end constraints'

5.25

BEHAVIOUR OFCOl^ONENTS: PIPING ELBOWS

in terms of the yield stress, Sy and ultimate stress, Su. For unpressurised elbows this leads

to a much lower allowable bending moment as compared to the recommendations of Code

Case N319. When compared to ASME NB 3680, the allowable moment is higher for small

angle bends, lower for larger angle bends. For pressurised bends, the allowable moment is

considerably increased since, as discussed in Section 5.1.2, the Bourdon effect resists the

moment! A comparison is given in Figure 15 The significance of these results should be

carefully considered by the reader.

15-

10

05

Toubouletal VjM op=2/3Sy

IS

X = 0 IS

10

CC N319

Toubouletal V°cD=2f3Sy

X = 0 3

CC N319

NB 3660

NB 3680DSl-

30° 60° 90° 120° ISO0 180* 30° 60° 90'

i

120° ISO' 180'

Figure 15: Allowable bending loads

5.3.3 Summary

It should be clear that it would be a fairly easy matter to incorporate modified flexibilityand stress factors into the Codes; indeed this process has already been initiated in theASME III Nuclear Code for Class 1 piping and ,as discussed above, it is likely that therewill be a complete update of the ANSI codes i-factors in the near future. The questionremains as to if, and when, BS806 will be modified. In any case it should be clear thatconsiderable skepticism is necessary ofthe present Codes, and the reader should be carefulof accepting the results of flexibility analysis as being precise.

The way forward in the long term, however, is clear. Conventional beam type flexibilityanalysis for static loads will probably be replaced by more detailed analysis using specialpurpose pipe elements. This would avoid the need for separate flexibility and stress factors.Such an approach is possible now, although it is still moderately expensive, yet a detailedanalysis ofa complete system shouldbe possible in minutes on a super-workstation. Further (.

reductions in the price perfromance ratio ofmodern computers should make this probable.

5.26

1


A suitable design approach is also available in the form of the ASME design by analysis

route. Some problems remain to be addressed if this approach is used (in particular the

treatment ofbranches) but do notrepresent a reason for notpursuingthis. The onlyquestion

is how long it will take.

PPI

rpi

rip)

5.27

BEHAVIOUR OPCOMPONENTS: PIPING ELBOWS •

ra

1

C?rl

-j

Kffl

5.28

Design & Analysis of Piping Systems

6 BEHAVIOUR OF COMPONENTS: BRANCHES

6.1 INTRODUCTION

Branch pipe, or nozzle intersections in piping systems are generally recognised as components that require careful attention if they are to operate satisfactorily when loaded in

service. The primary load is ofcourse internal pressure, but in addition branchintersections

are required to withstand a complex set of moments and forces transmitted via the three

connected pipe limbs, as a result of deadweight, thermal expansion, seismic loading etc.The overall stressing situation is a complex one and has resulted in a great deal ofliteraturebeing published on the various aspects ofthe problem. For example Moore et al. publisheda useful review in 1982(1> which contains a list of 158 references on the subject.

Rodabaugh, the ASME code writer, has commented in a recent report(2) that, "We wouldrate the relative complexity of stress intensification factors for plain pipes, elbows andbranch connections by the ratios 1:5:500." While we might dispute the ratios quoted,Rodabaugh's comment serves to emphasise the difficulties involved in arrivingat acceptabledesign procedures for branch connections. Factors that influence branch design includemanufacturing procedure (welded, forged, extruded), reinforcement, weld details, loadinteraction, thermal stress, fatigue, shakedown, collapse, fracture etc. This presentationwill make no pretence at covering the above, but will concentrate on explaining some ofthe background to the BS 806 design rules for branch junctions, while at the same timedrawing attention to some of the limitations in the code rules. Much of what is said willbe based on the unpublished notes produced by Carmichael et al(3) in connection with thework of the PVE/-/5 working party on the topic, and the use of these notes is gratefullyacknowledged (see also Popplewell and Hamil(4)). Brief mention will also be made hereinofASME III design procedures for branch junctions and tees.

jpl

I! 6.2 STRAIGHT PIPE

The BS 806 mean diameter formulae for determining design thickness ofstraight pipe hasalready been presented, i.e.

(1) - Moore, S.E., Greenstreet, W.L. and Mershon, J.L., "The Design of Nozzles and Openings in Pressure

^^^SS^ Ia982 Plng: °eSign Technol°^ " A Deca*e of Progress, ed. zSfdm™eiT^ IntensiflCati°n FactOrS for Branch Connections", Draft Welding

DeSign *nd Assessment of Branch Connections", BSI PVE/-/5 Specialist

^!^SF806: Ferrous Pipes for and i

6.1

BEHAVIOUR OF COMPONENTS: BRANCHES

or

-p

where f is the design stress and e is a factor depending on pipe quality. (Note: The second

equation references are the relevant BS 806 code equation numbers).

For combined bending, torsion and pressure loading, the combined stress fc is given by

4/* (3) (28)

where F is the greater of fT or fL. The transverse (hoop) stress is,

fT = EL+0.5p (4)<29)f

the longitudinal (axial) stress is

/, = ^ +^m^Mf+M; (5) (32)+ (33)4t(d + t) 2/

and the torsional shear stress is

MM 4-7t\

(6) (35)

6.3 BRANCHES - PRESSURE LOADING

The BS 806 terminology for thickness and diameter is shown in Figure 1. Based on the

bores of the main and branch pipes the design formulae for the thicknesses are,

Minimum main thickness

pdx

tml 2fex-p

and the minimum branch thickness

(7)(19)

2fex-p

6.2


The terms f and e are as for plain pipe, and the term x is the code "weakening factor". Thesource of this factor x is as follows:

rd, D,

\

Figure 1: BS806 Notation

In 1968 Money(5) at CEGB Berkeley conducted a parametric survey ofall the experimentalinternal pressure SCF data he could find for branch pipes and nozzles in pressure vessels'He presented the data as in Figure 2 and from this deduced the relationship

SCF = 2.5(Z)0.2042

(9)

where

H'A"' "Designing Flush CyHnder-to-Cylinder Intersections to Withstand Pressure", Proc. ASME,

:;iHlBEHAOTQ §llli;|liflllilfli

«

I'V

d

Lo, SCF

OGNTICAL

(EXCESSIVELY CWT OF

RELIA'BLI STEEL HCSW.T5

F€SSLEB PMOTO€LASTIC

IIUNOiS PMOTOCLASTtC «6SUt.TS

♦ IO% SCATTER ftANO

O2

Figure 2: Original Money data

Adjusting eqn.(9) to a design format gives

zJscfT,4.897

(10)

Following Money's report, additional data came to light from tests on steel vessels atBabcock and Wilcox, and CEGB Berkeley, which suggested that eqn.(lO) underestimatedSCFs for branch junctions with d2/d1 > 0.7. Hence an adjusted Money relationship was

proposed as follows

6.4


= 2.5(Z)0.241

(11)

which gave the design equation

.142

(12)

The Money and adjusted Money SCF relationships are presented in BS 806, Fig 4 8 5 2 (1)

Si X™eSCFiSde^^j

200 400 600800'1000

Figure 3: BS806 Figure 4.8.5.2(1)


At this stage the term J is introduced, where it can be shown that J is defined as

Peak Stress

Design Stress(13)

Values ofJ were chosen to be 2.5 for d^dx < 0.3 and 2.2 for d^dj > 0.3. These values were

arrived at on a shakedown basis and were intended to give a 20% margin over pressure

stress to allow for external load stresses. Combining eqns.(9)-(13) with the stated limits

for J can be shown, after some manipulation, to give the following expressions for the

Weakening Factor x for branches with proportional thickening, i.e,

0.3

d20.3 <-*£ 0.7

d

Weakening Factor x

,-0.17

(14)

i

where fx is the allowable design stress for the main pipe.

Curves representing the above expressions for the weakening factor x are presented in BS

806 Fig.4.8.5.1. (Figure 4 herein). The additional curves for dj^ = 0.2 and 0.1 are basedon SCF data derived by Leckie and Penny(6) using shell analysis techniques. Note that for

cases where d/di < 0.3, the code states that the x factor need only be used to thicken the

main, an x factor of 1.0 being used for the branch.

BS 806 also presents a design procedure, again based on the Money SCF relationships, for

cases ofnon-proportional thickening, i.e. where the thickness ofthe main is predetermined.The basis for this is not presented herein but can be found in Carmichael et al. (reference

(3)).

(6) - Leckie, F. and Penny, R.K., Welding Research Council, Bulletin, No. 90, 1963.

6.6

6S3"|

1


intermediate

between O.I and 0 2

0.2 dnd 3.3 tobe obtained 2y nn e

interpolation

-± aoove 0.3 and up to O 7

-pabove07 and up to 1.16

1000

Figure 4: BS806 Figure 4.8.5.1

iTJ^Tl^s °n branch junctions it has been shown that the location

Bilml (15)

Figure 5 compares eqn.(15) with the Belgian data ofDecock(7) where it can be seen that the

line defining eqn.(15) approximately distinguishes between crotch corner and weld fatigue

failures.

BS 806 0986)

• o CRjj: Fatigue Test Results

• Failure at Crotch Corner

o Failure at Toe of Weld

(Transverse Section)

oo

oo

00

• o

rL ett

C^i

O2 03 04 05 06__L_0.7

__L_08 0.9

Figure 5: CRIF Fatigue test data (Decock 1975) and BS806 (1986)

Relationship between tIT and dID

(7) - J Decock: Determination of stress concentration factors and fatigue assessment of flush and extrudednozzles in welded pressure vessels. Proc 2nd Int Confon Pressure Vessel Technology, Part II, ASME, 821-835,

1973

6.8


6A BRANCHES - MOMENT LOADING

A simple in-plane piping system is indicated in Figure 6. Due to pressure, deadweightthermal expansion effects etc. moments are transmitted to the branchjunction which willcause stresses that are additional to the pressure stresses.

Figure 6: Typical piping network

Figure 7: BS806 Figure 4.11.5.1

BEHAVIOUR OP COMPONENTS; BRANCHES

1In the general 3D case, using a piping flexibiHty analysis, as discussed earlier, a set of 9 Jmoments can be specified, 3 each, in the x, y and z planes, for each of the three connectinglimbs. Fig.4.11.5.1 of BS 806 (Figure 7 herein) illustrates the situation, and shows thateach limb has an in-plane, an out-of-plane and a twisting moment.

In 1952 Markl(8) conducted fatigue tests on a series of equal-diameter welded branchjunctions, loaded by in-plane and out-of-plane bending on either branch or main (run) pipes,using the "cantilever model" shown in Figure 8. From the tests he developed expressions

for fatigue stress intensification factors - symbol i. Comparison with work by Markl and Jothers on straight pipes and elbows showed that reasonable values for stress concentrationfactors (SCFs) could be deduced from Markl's fatigue i-values by simply multiplying by afactor of2. Markl found that, not surprisingly, the i-values were higher for branch momentloadingthanrunmoment loading. The BS 806 SCF curves formomentloadingFig.4.11.1(6)

(Figure 9 herein) are based on Markl's i-values for branch moments multiplied by 2.0. TheSCFs are plotted against the pipe factor X, which is defined in Figure 9 for welded, reinforcedand forged branchjunctions. [Note: the source ofthe statement, "Based on equalbore branch

flexibility factor = 1" in this figure is unknown!] ^

BRANCH

In-plane .5 ;

RUN

"I

Out-of-plane

BRANCH

T

RUN

Figure 8: Markl bending fatigue tests

- Markl, A.R.C., "Fatigue Tests on Piping Components", Trans. ASME, 1952 287-303

^3


6.5 BRANCHES - COMBINED PRESSURE & MOMENT LOADING

Stress levels under combined loading must be determined for all combinations ofpressureand moment loads to ensure that the stress criteria ofSection 4.11.2 are met. For branches,the BS 806 equation for combining pressure and moment loads is similar to that for straightpipes (eqn.(3) herein) but with SCFs introduced. The combined stress fCB is thus given by

(16) (36)

where fB is the sum of the direct stresses due to p, Mj and Mo (i.e. assumed to act at thesame location) as follows:

PWi + O(17)(37)&(40)

The SCF m in eqn.(17) is given by

2.8r18 (i8)(38)

for small branches (i.e. r^ and tjtx < 0.3) or can be obtained from the Money curves inFig.4.11.5.2ofBS806.

The Bj and Bo stress factors are obtained from Fig.4.11.1(6) (Figure 9 herein).

The shear stress fSB in eqn.(17) is simply the non-intensified plain pipe torsion stress asgiven in eqn.(6) herein, i.e. it is assumed that the SCF for torsional loading is 1.0. Morewill be said about this!

Notes:

1. The hot stress relaxation factor q has been omitted from eqn.(16) above for simplicity,but should of course be included in practice.

2. Eqn.(16) above should be applied to each ofthe 3 limbs of the branch junction in turnto determine the maximum combined stress.

3. The second moment of area I has the conventional definition when used with themain pipe, but in order to allow for the fact that the stress factors in Figure 9 hereinare based on equal-diameter junctions, an adjusted I value is used for branchcalculations (see Section 4.11.5.3).

BEHAVIOUR OF COMPONENTSfBRANCHES

40

In-plane curve -075 » — tQ

30

20

\ \

\ A

\l \

ms

10

9

8

o 7

% 6

* 5

-H

\

;i

I

■*• Cut-ot-pldne curve • Li

X based on equal bore branch flexibility factor ■!

N • i ■ 1

\ ;"

■+-H-

I ^

I;;,

1

• I i

-»-*■

Ski!

0.01 QO2QO3 QOS 0.1 Q2 Q3 0

Pipe factor \

2 3 4 5

r

i i

f I

t

i*>l.33f,

Forged tee meeting

Limiting crotch dimensions

r\

(a)10

9: BS806 Figure 4.11.1(6)


^.6 ASME III PROCEDURE FOR BRANCH DESIGN

The ASME III procedure is based on the so-called "Cantilever model" as explained belowIt can be readily shown that the 9 moments associated with a branch junction are notindependent, and can be reduced to 6 independent moments using equilibrium argumentslne 6 moments are usually represented using the cantilever model shown in Figure 10herein. The ASME III design procedure is as follows:

M.

.v*

Moment Vectors

M

■+»

M.

Figure 10: Moment load categories

First the reinforced thicknesses of the branch junction pipes are determined using thewell-established area-replacement" procedure. Acheckon PrimaryStresses is then carriedout, whuch is essentially aimed at ensuring that the junction does not plastically collapseThe relevant equation in ASME III is

PD. M.

(19)

,where Mb and M, are the resultant branch pipe and run pipe moments given by,

+Mtb2 (20)


and M^Mj +Myr2 +Mt/ (21)

Blt B2b and B2r are the Primary Stress Indices defined, for pressure or bending, by

Plane pipe collapse load

~J-allowable load ( 'and Sm is the design stress.

Next a check on combined Primary and Secondary stressing is carried out. This is to ensure

that combined load stress levels due to, a) nominal pressure and moment loads, and b)

discontinuity effects at the branch junction, are within acceptable limits. Again a linear

interaction rule is used, i.e. stresses due to all loads are considered to act at the same

location, as follows:

pn r f m \ ( m Yi-M U 3S_ (23)1-

where the C1} C2b and C2r Secondary Stress Indices are

0.182/ . NO.367/^ \0.382/, N0.148

r - i 4 —11 —

. \0.367 / t N0.382 / t

*7 S u: :

\0.67

I

and

^=1.151^ I 2 1.5 (26)

The various parameters in the above are illustrated in Figure 11 herein.

Finally, a fatigue check is conducted to allow for the local effects of welds etc. Again a

linear interaction rule is used, with the addition, to eqn.(23) above, ofthe peak stress indices

K,, K2b and K2r. The equation is

:J^W (27)p

where K, = 2.0, K2b = 1.0 and K2r = 1.75. j

Note that, for simplicity, the ASME III thermal stress terms have not been included in the

above expressions. Sp is the allowable fatigue stress range. ]

6.14

Hii^!^^^

Branch Pip«

11: ASME III Notation

6.7 BRANCH JUNCTION FLEXIBILITY

The significance of the flexibility of pipe bends in piping systems has already been

emphasised. In BS 806, it is assumed that there is no additional flexibility introduced into

the piping system due to branch junctions. However ASME III*9* now includes a limited

rocedure for including branch junction flexibility: Of the 6 moment categories of Figure

10 it has been decided that the additional flexibilities due to moment categories M^, M^,

MCT and Myb are sufficiently small to be ignored. The definition ofFlexibility Factor KQ used

for pipe bends, i.e.

Rotation between bend ends (28)

Rotation between ends of straight pipe of same length as bend

is not applicable to branch junctions and a different definition has to be used. This is

explained using ASME III Fig.NB-3686.5-1 (Figure 12 herein).

(9) - ASME III - Division 1, Subsection-NB, Boiler and Pressure Vessel Code, 1985 Addenda.

6.15

Md

Point spring-Element of

negligible length.

Rotation across element

Rigid length

igid juncture

A. A.S :. IE Method

Figure 12: ASME III Flexibility

The additional flexibility is assumed to be concentrated at a "Point Spring" located at the

branch/main junction. The rotational stiffness of this spring is such that the rotation at

the spring is

(29)

where the Flexibility Factor kB is now defined as

(30)

The length Le is a fictitious equivalent branch limb length due to junction flexibility. This

definition of K9 is helpful in that its effect is easily envisaged. For example a KQ of 6 say,

means that the extra flexibility due to the branch is equivalent to an additional fictitious

length ofbranch pipe = 6d.

6.16


The parametrix expressions given in ASME III are:

For moment Mxb,

iOJ

■' [r. (31)

and for moment Mzb>

(32)

Note however that the ASME III procediire is limited to branches with d/D < 0.5. Morewill be said on this topic.

6.17

; ^.behawqto^ P3"

fsn

ess

ra

6,18

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