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Pipe Stress Analysis Institute of Piping Engineering & Building Services www.ipebs.in

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Topic Page No.

Introduction 2

Stress Strain Relationship 6

WHAT IS STRESS ANALYSIS? 8

STRESS CATEGORIES 9

METHODOLOGY - CLASSIFICATIONOF LOADS AND FAILURE MODES 10

THEORIES OF FAILURE 22

REQUIRMENTS OF ASME B31.3 26

FLEXIBILITY ANALYSIS 28

SUGGESTIONS FOR REDUCINGEUQIPMENT LOADING

31

Location of Supports and Restraints 32

Tips for Flexible Layouts 33

What A Piping Designer Should Know

to deal with Piping Stress

35

Stress Critical Lines 36

Information Required For StressAnalysis

37

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I . INTRODUCTION

Pipes are the most delicate components in any process plant. They are also the

busiest entities. They are subjected to almost all kinds of loads, intentional or

unintentional. It is very important to take note of all potential loads that a piping

system would encounter during operation as well as during other stages in the life

cycle of a process plant. Ignoring any such load while designing, erecting, hydro-

testing, start-up shut-down, normal operation, maintenance etc. can lead to

inadequate design and engineering of a piping system. The system may fail on the

first occurrence of this overlooked load. Failure of a piping system may trigger a

Domino effect and cause a major disaster.

Stress analysis and safe design normally require appreciation of several related

concepts.

An approximate list of the steps that would be involved is as follows.

1. Identify potential loads that would come on to the pipe or piping system during its

entire life.

2. Relate each one of these loads to the stresses and strains that would be

developed in the crystals/grains of the Material of Construction (MoC) of the piping

system.

3. Decide the worst three dimensional stress state that the MoC can withstand

without failure

4. Get the cumulative effect of all the potential, loads on the 3-D stress scenario in

the piping system under consideration.

5. Alter piping system design to ensure that the stress pattern is within failure limits.

The goal of quantification and analysis of pipe stresses is to provide safe design

through the above steps. There could be several designs that could be safe. A piping

engineer would have a lot of scope to choose from such alternatives, the one which

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is most economical, or most suitable etc. Good piping system design is always a

mixture of sound knowledge base in the basics and a lot of ingenuity.

Piping Stress Introduction:

There are five basic factors that influence piping and therefore piping stress in the

process plant. There is temperature, pressure, weight, force and vibration. These

factors will come in many forms and at different times. Stress problems become all

the more complex because two or more of these will exist at the same time in the

same piping system. The main objective of the focus when dealing with problems

related to piping systems is not normally the pipe itself. In a very high percentage

of the time it is not the pipe that is the weakest link. Note this: the pipe is normally

stronger and/or less vulnerable to damage than what the pipe is connected to.

Pumps are just one examples of equipment to which pipes are routinely connected.

Misalignment problems caused by expansion (or contraction) in a poorly designed

system can result in major equipment failure. Equipment failures can lead to the

potential for fire, plant shutdown and loss of revenue. At this point it should be

emphasized that the success (or failure) of the plants operation, years down the

road can and will depend on what is done up front by all the members of the design

team during the design stage.

Stress Related Design Factors

Temperatures in piping systems may range from well over 1000o F (537.8 C) on

the high side to below -200 o F (-128.8 C) on the low side. Each extreme on the

temperature scale and everything in between brings its own problems. There will

also be times when both high and low temperatures can occur in the same piping

system. An example of this would be in piping that is installed in an arctic

environment. The piping is installed outdoors where it is subjected to -100 o F (-

73.3 C) over the arctic winter. Six to nine months later it is finally commissioned

started up and may operate at five or six hundred degrees.

The problems that temperature causes is expansion (or contraction) in the piping

system. Expansion or contraction in a piping system is an absolute. No matter

what the designer or the stress engineer does they cannot prevent the action

caused by heat or cold. Expansion or contraction in a piping system it self is not

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so much a problem. As we all know if a bare pipe was just lying on the ground in

the middle of a dry barren desert it will absorb a lot of heat from just solar

radiation. In the hot sun piece of pipe can reached 150 o F (65.5 C). The pipe will

expand and with both ends loose it would not be a problem. However, when you

connect the pipe to something, even if only one end is connected you may begin

to have expansion related problems. When the pipe is anchored or connected to

something at both ends you absolutely will have expansion induced problems.

Expansion induced problems in a piping system is stress. There are a number of

ways to handle expansion in piping systems. Flexible routing is the first and by

far the cheapest and safest method for handling expansion in piping systems. The

other way is the use of higher cost and less reliable flexible elements such as

expansion joints.

Stress will exist in every piping system. If not identified and the proper action

taken, stress will cause failure to equipment or elements in the piping system

itself. Stress results in forces at equipment nozzles and at anchor pipe supports.

Two piping configurations with the same pipe size, shape, dimensions,

temperature and material but with different wall schedules (sch. 40 vs. sch. 160)

will not generate the same stress.

Force in piping systems is not independent of the other factors. Primarily, force

(as related to piping systems) is the result of expansion (temperature) and/or

pressure acting on a piping configuration that is too stiff. This may cause the

failure of a pipe support system or it may cause the damage or failure of a piece

of equipment. Force, and the expansion that causes it, is best handled by a more

flexible routing of the piping. Some people suggest that force can be reduced by

the use of expansion joints. However we must remember that for an expansion

joint to work there must be an opposite and equal force at both ends to make the

element work. This tends to compound the problem rather than lessen it.

Pressure in piping systems also range from the very high to the very low. Piping

systems with pressure as high as 35,000 psi in some plants are not unusual. On

the other hand piping systems with pressures approaching full vacuum are also

not unusual. The pressure (or lack of) in a piping system effects the wall

thickness of the pipe. When you increase the wall thickness of the pipe you do

two things. First, you increase the weight of the pipe. Second, you increase the

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stiffness of the pipe thus the stress intensification affecting forces. Increasing the

wall thickness of the pipe is the primary method of compensating for increases in

pressure. Other ways, depending on many factors include changing to a different

material. With low or vacuum systems there are also other ways to prevent the

collapse of the pipe wall. Among these the primary method is the addition of

stiffening rings. Stiffing rings may be added internally or externally depending on

the commodity type and the conditions.

Weight in a piping system is expressed normally as dead load. The weight of a

piping system at any given point is made up of many elements. These include the

weight of the pipe, the fittings, the valves, any attachments, and the insulation.

There is also the test media (e. g. hydrotest water) or the process commodity

whichever has the greater specific gravity. Piping systems are heavy, period.Everybody involved in the project needs to understand this and be aware that this

weight exists and it needs to be supported. Ninety-nine times out of a hundred this

weight will be supported from a structural pipe support (primary pipe support

system) of some kind. However there are times when the piping (weight) is

supported from a vessel or other type of equipment.

Vibrations will also occur in piping systems and come in two types. There is the

basic mechanical vibration caused by the machines that the piping is connected

to. Then, there is acoustic (or harmonic) vibration caused by the characteristics of

the system itself. Typically the only place severe vibrations will be found is in

piping connected to equipment such as positive displacement reciprocating pumps

or high pressure multi-stage reciprocating compressors and where there is very

high velocity gas flows.

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II. STRESS - STRAIN RELATIONSHIP

STRESS: Stress of a material is the internal resistance per unit area to the

deformation caused by applied load.

STRAIN: Strain is unit deformation under applied load.

STRESS STRAIN CURVE: It is a curve in which unit load or stress is plotted

against unit elongation, technically known as strain.

1. O A represents the stress is directly proportional to strain, and point A isknownproportional limit.

2. Point B represents elastic limitbeyond which the material will not return toits original shape when unloaded but will retain a permanent deformation

called permanent set.

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3. Point C is called yield pointand is the point at which there is an appreciableelongation or yielding of the material without any corresponding increases of

load.

4. Point D is ultimate stress or ultimate strength of material.5. Point E is the stress at failure known as rupture strength.

OBJECTIVE AND SCOPE

With piping, as with other structures, the analysis of stresses may be carried to

varying degrees of refinement. Manual systems allow for the analysis of simple

systems, whereas there are methods like chart solutions (for three-dimensional

routings) and rules of thumb (for number and placement of supports) etc. involving

long and tedious computations and high expense. But these methods have a scope

and value that cannot be defined as their accuracy and reliability depends upon the

experience and skill of the user. All such methods may be classified as follows:

1. Approximate methods dealing only with special piping configurations of two-,

threeor four-member systems having two terminals with complete fixity and the

piping layout usually restricted to square corners. Solutions are usually obtained

from charts or tables. The approximate methods falling into this category are limited

in scope of direct application, but they are sometimes usable as a rough guide on

more complex problems by assuming subdivisions of the model into anchored

sections fitting the contours of the previously solved cases.

2. Methods restricted to square-corner, single-plane systems with two fixed ends,

but without limit as to the number of members.

3. Methods adaptable to space configurations with square corners and two fixed

ends.

4. Extensions of the previous methods to provide for the special properties of curved

pipe by indirect means, usually a virtual length correction factor.

The objective of this project is to perform the static analysis of a piping model.

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III. WHAT IS STRESS ANALYSIS?

Piping Stress analysis is a term applied to calculations, which address the static and

dynamic loading resulting from the effects of gravity, temperature changes, internal

and external pressures, changes in fluid flow rate and seismic activity. Codes and

standards establish the minimum requirements of stress analysis.

PURPOSE OF PIPING STRESS ANALYSIS

Purpose of piping stress analysis is to ensure:

a) Safety of piping and piping components.b) Safety of connected equipment and supporting structure.c)

Piping deflections are within the limits.

HOW PIPING AND COMPONENTS FAIL (MODES OF FAILURES)

There are various failure modes, which could affect a piping system. The piping

engineers can provide protection against some of these failure modes by performing

stress analysis according to piping codes.

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FAILURE BY GERNRAL YIELDING: Failure is due to excessive plastic deformation.

1. Yielding at Sub Elevated temperature: Body undergoes plasticdeformation under slip action of grains.

2. Yielding at Elevated temperature: After slippage, material re-crystallizesand hence yielding continues without increasing load. This phenomenon is

known as creep.

FAILURE BY FRACTURE: Body fails without undergoing yielding.

1. Brittle fracture: Occurs in brittle materials.2. Fatigue: Due to cyclic loading initially a small crack is developed which grows

after each cycle and results in sudden failure.

IV. STRESS CATEGORIES

The major stress categories are primary, Secondary and peak.

PRIMARY STRESSES:

These are developed by the imposed loading and are necessary to satisfy the

equilibrium between external and internal forces and moments of the piping system.

Primary stresses are not self-limiting.

SECONDARY STRESSES:

These are developed by the constraint of displacements of a structure. These

displacements can be caused either by thermal expansion or by outwardly imposed

restraint and anchor point movements. Secondary stresses are self-limiting.

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PEAK STRESSES:

Unlike loading condition of secondary stress which cause distortion, peak stresses

cause no significant distortion. Peak stresses are the highest stresses in the region

under consideration and are responsible for causing fatigue failure.

CLASSCIFICATION OF LOADS

Primary loads:These can be divided into two categories based on the duration of loading.

Sustained loads

These loads are expected to be present through out the plant operation. e,g.

pressure and weight.

Occasional loads.

These loads are present at infrequent intervals during plant operation. e,g.

earthquake, wind, etc.

Expansion loads:These are loads due to displacements of piping. e,g .thermal expansion, seismic

anchor movements, and building settlement.

V. METHODOLOGY - CLASSIFICATION OF LOADS AND FAILURE MODES

Pressure design of piping or equipment uses one criterion for design. Under a steady

application of load (e.g.. pressure), it ensures against failure of the system as

perceived by one of the failure theories. If a pipe designed for a certain pressure

experiences a much higher pressure, the pipe would rupture even if such load

(pressure) is applied only once. The failure or rupture is sudden and complete. Sucha failure is called catastrophic failure. It takes place only when the load exceeds far

beyond the load for which design was carried out. Over the years, it has been

realised that systems, especially piping, systems can fail even when the loads are

always under the limits considered safe, but the load application is cyclic (e.g. high

pressure, low pressure, high pressure, ..). Such a failure is not guarded against by

conventional pressure design formula or compliance with failure theories. For piping

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system design, it is well established that these two types of loads must be treated

separately and together guard against catastrophic and fatigue failure.

The loads the piping system (or for that matter any structural part) faces are broadly

classified as primary loads and secondary loads.

Primary Loads

These are typically steady or sustained types of loads such as internal fluid pressure,

external pressure, gravitational forces acting on the pipe such as weight of pipe and

fluid, forces due to relief or blow down pressure waves generated due to water

hammer effects. The last two loads are not necessarily sustained loads. All these

loads occur because of forces created and acting on the pipe. In fact, primary loads

have their origin in some force acting on the pipe causing tension, compression,

torsion etc leading to normal and shear stresses. A large load of this type often leads

to plastic deformation. The deformation is limited only if the material shows strain

hardening characteristics. If it has no strain hardening property or if the load is so

excessive that the plastic instability sets in, the system would continue to deform till

rupture. Primary loads are not self-limiting. It means that the stresses continue to

exist as long as the load persists and deformation does not stop because the system

has deformed into a no-stress condition but because strain hardening has come into

play.

Secondary Loads

Just as the primary loads have their origin in some force, secondary loads are caused

by displacement of some kind. For example, the pipe connected to a storage tank

may be under load if the tank nozzle to which it is connected moves down due to

tank settlement. Similarly, pipe connected to a vessel is pulled upwards because the

vessel nozzle moves up due to vessel expansion. Also, a pipe may vibrate due to

vibrations in the rotating equipment it is attached to. A pipe may experience

expansion or contraction once it is subjected to temperatures higher or lower

respectively as compared to temperature at which it was assembled.

The secondary loads are often cyclic but not always. For example load due to tank

settlement is not cyclic. The load due to vessel nozzle movement during operation is

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cyclic because the displacement is withdrawn during shut-down and resurfaces again

after fresh start-up. A pipe subjected to a cycle of hot and cold fluid similarly

undergoes cyclic loads and deformation. Failure under such loads is often due to

fatigue and not catastrophic in nature.

Broadly speaking, catastrophic failure is because individual crystals or grains were

subjected to stresses which the chemistry and the physics of the solid could not

withstand. Fatigue failure is often because the grains collectively failed because their

collective characteristics (for example entanglement with each other etc.) changed

due to cyclic load. Incremental damage done by each cycle to their collective texture

accumulated to such levels that the system failed. In other words, catastrophic

failure is more at microscopic level, whereas fatigue failure is at mesoscopic level if

not at macroscopic level.

The Stresses

The MoC of any piping system is the most tortured non-living being right from its

birth. Leaving the furnace in the molten state, the metal solidifies within seconds. It

is a very hurried crystallization process. The grains, crystals of the material have no

time or chance to orient themselves in any particular fashion. They are thus frozen in

all random orientations in the cold harmless pipe or structural member that we see.

When we calculate stresses, we choose a set of orthogonal directions and define the

stresses in this co-ordinate system. For example, in a pipe subjected to internal

pressure or any other load, the most used choice of co-ordinate system is the one

comprising of axial or longitudinal direction (L), circumferential (or Hoope's) direction

(H) and radial direction (R) as shown in figure. Stresses in the pipe wall are

expressed as axial (SL), Hoope's (SH) and radial (SR). These stresses which stretch

or compress a grain/crystal are called normal stresses because they are normal tothe surface of the crystal.

But, all grains are not oriented as the grain in the figure. In fact the grains would

have been oriented in the pipe wall in all possible orientations. The above stresses

would also have stress components in direction normal to the faces of such randomly

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oriented crystal. Each crystal thus does face normal stresses. One of these

orientations must be such that it maximizes one of the normal stresses.

The mechanics of solids state that it would also be orientation which minimizes some

other normal stress. Normal stresses for such orientation (maximum normal stress

orientation) are called principal stresses, and are designated S1 (maximum), S2 and

S3 (minimum). Solid mechanics also states that the sum of the three normal

stresses for all orientation is always the same for any given external load. That is

SL + SH + SR = S1 + S2 + S3

In addition to the normal stresses, a grain can be subjected to shear stresses as

well. These act parallel to the crystal surfaces as against perpendicular direction

applicable for normal stresses. Shear stresses occur if the pipe is subjected to

torsion, bending etc. Just as there is an orientation for which normal stresses are

maximum, there is an orientation which maximizes shear stress. The maximum

shear stress in a 3-D state of stress can be shown to be

Tmax = (S1- S3) / 2

i.e. half of the difference between the maximum and minimum principal stresses.

The maximum shear stress is important to calculate because failure may occur or

may be deemed to occur due to shear stress also. A failure perception may stipulate

that maximum shear stress should not cross certain threshold value. It is therefore

necessary to take the worst-case scenario for shear stresses also as above and

ensure against failure.

It is easy to define stresses in the co-ordinate system such as axial-Hoope s-radial

(L-H-R) that are defined for a pipe. The load bearing cross-section is then well

defined and stress components are calculated as ratio of load to load bearing cross-

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section. Similarly, it is possible to calculate shear stress in a particular plane given

the torsional or bending load. What are required for testing failure - safe nature of

design are, however, principal stresses and maximum shear stress. These can be

calculated from the normal stresses and shear stresses available in any convenient

orthogonal co-ordinate system. In most pipe design cases of interest, the radial

component of normal stresses (SR) is negligible as compared to the other two

components (SH and SL). The 3-D state of stress thus can be simplified to 2-D state

of stress. Use of Mohr's circle then allows to calculate the two principle stresses and

maximum shear stress as follows.

The third principle stress (minimum i.e. S3) is zero.

All failure theories state that these principle or maximum shear stresses or some

combination of them should be within allowable limits for the MoC under

consideration. To check for compliance of the design would then involve relating the

applied load to get the net SH, SL, and then calculate S1, S2 and Tmax and some

combination of them.

Normal And Shear Stresses From Applied Load

As said earlier, a pipe is subjected to all kinds of loads. These need to be identified.

Eachsuch load would induce in the pipe wall, normal and shear stresses. These need

to be calculated from standard relations. The net normal and shear stresses resulting

in actual and potential loads are then arrived at and principle and maximum shear

stresses calculated. Some potential loads faced by a pipe and their relationships to

stresses are summarized here in brief

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Axial Load

A pipe may face an axial force (FL) as shown in Figure. It could be tensile or

compressive.

What is shown is a tensile load. It would lead to normal stress in the axial direction

(SL). The load bearing cross-section is the cross-sectional area of the pipe wall

normal to the load direction, Am. The stress can then be calculated as

SL = FL / Am

The load bearing cross-section may be calculated rigorously or approximately as follows.

The axial load may be caused due to several reasons. The simplest case is a tall

column. The metal cross-section at the base of the column is under the weight of the

column section above it including the weight of other column accessories such as

insulation, trays, ladders etc. Another example is that of cold spring. Many times a

pipeline is intentionally cut a little short than the end-to-end length required. It is

then connected to the end nozzles by forcibly stretching it. The pipe, as assembled,

is under axial tension. When the hot fluid starts moving through the pipe, the pipe

expands and compressive stresses are generated. The cold tensile stresses are thus

nullified. The thermal expansion stresses are thus taken care of through appropriate

assembly-time measures.

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Internal / External Pressure

A pipe used for transporting fluid would be under internal pressure load. A pipe such

as a jacketed pipe core or tubes in a Shell & Tube exchanger etc. may be under net

external pressure. Internal or external pressure induces stresses in the axial as well

as circumferential (Hoope s) directions. The pressure also induces stresses in the

radial direction, but as argued earlier, these are often neglected.

The internal pressure exerts an axial force equal to pressure times the internal

cross-section of pipe

This then induces axial stress calculated as earlier. If outer pipe diameter is used for

calculating approximate metal crossection as well as pipe cross- section, the axial

stress can often be approximated as follows.

The internal pressure also induces stresses in the circumferential direction as shownin figure

The stresses are maximum for grains situated at the inner radius and minimum for

those situated at the outer radius. The Hoope's stress at any in between radial

position ( r ) is given as follows (Lame's equation)

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For thin walled pipes, the radial stress variation can be neglected. From membrane

theory, SH may then be approximated as follows.

Radial stresses are also induced due to internal pressure as can be seen in figure

At the outer skin, the radial stress is compressive and equal to atmospheric pressure

(Patm ) or external pressure (Pext) on the pipe. At inner radius, it is also

compressive but equal to absolute fluid pressure (Pabs). In between, it varies. As

mentioned earlier, the radial component is often neglected.

Bending Load

A pipe can face sustained loads causing bending. The bending moment can be

related to normal and shear stresses. Pipe bending is caused mainly due to two

reasons: Uniform weight load and concentrated weight load. A pipe span supported

at two ends would sag between these supports due to its own weight and the weight

of insulation (if any) when not in operation. It may sag due to its weight and weight

of hydrostatic test fluid it contains during hydrostatic test. It may sag due to its own

weight, insulation weight and the weight of fluid it is carrying during operation

All these weights are distributed uniformly across the unsupported span, and lead to

maximum bending moment either at the centre of the span or at the end points of

the span (support location) depending upon the type of the support used.

Let the total weight of the pipe, insulation and fluid be W and the length of the

unsupported span be L (see Figure).

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The weight per unit length, w, is then calculated (w = W/L). The maximum bending

moment, Mmax, which occurs at the centre for the pinned support is then given by

the beam theory as follows.

For Fixed Supports, the maximum bending moment occurs at the ends and is given

by beam theory as follows

The pipe configuration and support types used in process industry do not confirm to

any of these ideal support types and can be best considered as somewhere in

between. As a result, a common practice is to use the following average formula to

calculate bending moment for practical pipe configurations, as follows.

Also, the maximum bending moment in the case of actual supports would occur

somewhere between the ends and the middle of the span.

Another load that the pipe span would face is the concentrated load. A good example

is a valve on a pipe run (see figure ).

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The load is then approximated as acting at the centre of gravity of the valve and the

maximum bending moment occurs at the point of loading for pinned supports and is

given as

For rigid supports, the maximum bending moment occurs at the end nearer to the

pointed load and is given as

a is to be taken as the longer of the two arms (a and b) in using the above formula.

As can be seen, the bending moment can be reduced to zero by making either a or b

zero, i.e. by locating one of the supports right at the point where the load is acting.

In actual practice, it would mean supporting the valve itself. As that is difficult, it is a

common practice to locate one support as close to the valve (or any other pointedand significant load) as possible. With that done, the bending moment due to pointed

load is minimal and can be neglected.

Whenever the pipe bends, the skin of the pipe wall experiences both tensile and

compressive stresses in the axial direction as shown in Figure .

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The axial stress changes from maximum tensile on one side of the pipe to maximum

compressive on the other side. Obviously, there is a neutral axis along which the

bending moment does not induce any axial stresses. This is also the axis of the pipe.

The axial tensile stress for a bending moment of M, at any location c as measured

from the neutral axis is given as follows.

I is the moment of inertia of the pipe cross-section. For a circular cross-section pipe,

I is given as

The maximum. tensile stress occurs where c is equal to the outer radius of the pipe

and is given as follows.

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Shear Load

Shear load causes shear stresses. Shear load may be of different types. One

common load is the shear force (V) acting on the cross-section of the pipe as shown

in figure

It causes shear stresses which are maximum along the pipe axis and minimum along

the outer skin of the pipe. This being exactly opposite of the axial stress pattern

caused by bending moment and also because these stresses are small in magnitude,

these are often not taken in account in pipe stress analysis. If necessary, these are

calculated as

where Q is the shear form factor and Am is the metal cross-section.

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Torsional Load

This load (see figure ) also causes shear stresses.

The shear stress caused due to torsion is maximum at outer pipe radius. And is giventhere in terms of the torsional moment and pipe dimensions as follows.

RT is the torsional resistance (= twice the moment of inertia).

All known loads on the pipe should be used to calculate contributions to SL, SH and

t. These then are used to calculate the principal stresses and maximum shear stress.

These derived quantities are then used to check whether the pipe system design is

adequate based on one or more theories of failure.

VI . WHEN PIPING AND COMPONENTS FAIL (THEORIES OF FAILURE)

A piping system in particular or a structural part in general is deemed to fail when a

stipulated function of various stresses and strains in the system or structural part

crosses a certain threshold value. It is a normal practice to define failure as occurring

when this function in the actual system crosses the value of a similar function in a

solid rod specimen at the point of yield. There are various theories of failure that

have been put forth. These theories differ only in the way the above mentioned

function is defined. Important theories in common use are considered here.

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Various theories of failure have been proposed, their purpose being to establish the

point at which failure will occur under any type of combined loading.

The failure theories most commonly used in describing the strength of piping

systems are:

Maximum principal stress theory

This theory states that yielding in a piping component occurs when the magnitude of

any of the three mutually perpendicular principle stresses exceeds the yield point

strength of the material.

This is also called Rankine Theory. According to this theory, failure occurs when the

maximum principle stress in a system (S1) is greater than the maximum tensile

principle stress at yield in a specimen subjected to uni-axial tension test.

Uniaxial tension test is the most common test carried out for any MoC. The tensile

stress in a constant cross-section specimen at yield is what is reported as yield stress

(Sy) for any material and is normally available. In uni-axial test, the applied load

gives rise only to axial stress (SL) and SH and SR as well as shear stresses are

absent. SL is thus also the principle normal stress (i.e. S1). That is, in a specimen

under uni-axial tension test, at yield, the following holds.

SL = SY, SH = 0, SR = 0

S1 = SY, S2 = 0 and S3 = 0.

The maximum tensile principle stress at yield is thus equal to the conventionally

reported yield stress (load at yield/ cross-sectional area of specimen).

The Rankine theory thus just says that failure occurs when the maximum principlestress in a system (S1) is more than the yield stress of the material (Sy).

The maximum principle stress in the system should be calculated as earlier. It is

interesting to check the implication of this theory on the case when a cylinder (or

pipe) is subjected to internal pressure.

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As per the membrane theory for pressure design of cylinder, as long as the Hoope's

stress is less than the yield stress of the MoC, the design is safe. It is also known

that Hoope's stress (SH) induced by external pressure is twice the axial stress (SL).

The stresses in the cylinder as per the earlier given formula would be

The maximum principle stress in this case is S2 (=SH). The Rankine theory and the

design criterion used in the membrane theory are thus compatible.

This theory is widely used for pressure thickness calculation for pressure vessels and

piping design uses Rankine theory as a criterion for failure.

Maximum shear stress theory

This theory states that failure of a piping component occurs when the maximum

shear stress exceeds the shear stress at the yield point in a tensile test.

In the tensile test, at yield, S1=Sy (yield stress), S2=S3=0.So yielding in the

components occurs When

Maximum Shear stress =max=S1-S2 / 2=Sy / 2

The maximum principal stress theory forms the basis for piping systems governed by

ASME B31.3.

Note: maximum or minimum normal stress is called principal stress.

This is also called Tresca theory. According to this theory, failure occurs when the

maximum shear stress in a system max is greater than the maximum shear stress

at yield in a specimen subjected to uni-axial tension test. Note that it is similar in

wording, to the statement of the earlier theory except that maximum shear stress is

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used as criterion for comparison as against maximum principle stress used in the

Rankine theory.

In uniaxial test, the maximum shear stress at yield condition of maximum shear testgiven earlier is

The Tresca theory thus just says that failure occurs when the maximum shear stress

in a system is more than half the yield stress of the material (Sy). The maximum

shear stress in the system should be calculated as earlier.

It should also be interesting to check the implication of this theory on the case when

a cylinder (or pipe) is subjected to internal pressure.

As the Hoope's stress induced by internal pressure (SH) is twice the axial stress (SL)

and the shear stress is not induced directly ( = 0) the maximum shear stress in

the cylinder as per the earlier given formula would be

This should be less than 0.5Sy, as per Tresca theory for safe design. This leads to a

different criterion that Hoope's stress in a cylinder should be less than twice the yield

stress. The Tresca theory and the design criterion used in the membrane theory for

cylinder are thus incompatible.

Conclusion

Stresses in pipe or piping systems are generated due to loads experienced by the

system. These loads can have origin in process requirement, the way pipes are

supported, piping system s static properties such as own weight or simple

transmitted loads due to problems in connecting equipments such as settlement or

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vibrations. Whatever may be the origin of load, these stresses the fabric of the MoC

and failure may occur.

Fatigue failure is an important aspect in flexibility analysis of piping systems. Often

cyclic stresses in piping systems subjected to thermal cycles get transferred to

flexibility providing components such as elbows. These become the components

susceptible to fatigue failure. Thermal stress analysis or flexibility analysis attempts

to guard against such failure through very involved calculations.

VII. REQUIRMENTS OF ASME B31.3 (PROCESS PIPING CODE)

This code governs all piping within the property limits of facilities engaged in the

processing or handling of chemical, petroleum or related products. Examples are a

chemical plant, petroleum refinery, loading terminal, natural gas processing plant,

bulk plant, compounding plant and tank farm.

The loadings required to be considered are pressure, weight (live and dead loads),

impact, wind, earthquake-induced horizontal forces, vibration discharge reactions,

thermal expansion and contraction, temperature gradients, anchor movements.

The governing equations are as follows:

1. Stresses due to sustained loads.

SL < or equal to Sh

SL = (PD/4t) + Sb

Sh = Basic allowable stress at maximum metal temperature.

The thickness of the pipe used in calculating SL shall be the nominal thickness minus

mechanical, corrosion, and erosion allowance.

2.Stresses due to occasional loads.

The sum of the longitudinal loads due pressure, weight and other sustained loads

and of stresses produced by occasional loads such as earthquake or wind shall not

exceed 1.33Sh.

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3.Stress range due to expansion loads.

The displacement stress range SE shall not exceed SA:

SE < or equal to SA

WHERE

Sb = resultant bending stress,psi

= [(IiMi)2 + (IoMo)2] / Z

Mi = in-plane bending moment, in.lb

Mo = out-plane bending moment, in.lb

Ii = in- plane stress intensification factor obtained from appendix of B31.3

Io = out- plane stress intensification factor obtained from appendix of B31.3

St = Torsional stress ,psi

= Mt / (2Z)

Mt = Torsional moment, in.lb

SA = Allowable displacement stress range:

(Allowable stress) cold = Sc = (2 / 3) Syc Syc = (3/2)Sc

(Allowable stress) hot = Sh = (2 / 3) Syh Syh = (3/2) Sh

Syc = yield point stress at cold temperature

Syh = yield point stress at hot temperature

Allowable stress =Syc + Syh

=3/2 (Sc + Sh )

= 1.5 (Sc + Sh )

= 1.25(Sc + Sh )---- after dividing with F.O.S

Final allowable stress = [(1.25(Sc + Sh) SL]

SA = f [(1.25(Sc + Sh) SL]

Sc = basic allowable stress at minimum metal temperature

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f = stress range reduction factor from table 302.2.5 of B31.3

VIII. FLEXIBILITY ANALYSIS

Flexibility analysis is done on the piping system to study its behaviour when its

temperature changes from ambient to operating, so as to arrive at the most

economical layout with adequate safety.

The following are the considerations that decide the minimum acceptable flexibility

on a piping configuration.

1. The maximum allowable stress range in the system.

2. The limiting values of forces and moments that the piping system is permitted to

impose on the equipment to which it is connected.

3. The displacements within the piping system.

4. The maximum allowable load on the supporting structure.

Purpose of Flexibility Analysis

The purpose of performing a flexibility analysis is to determine that, barring

interferences and assuming a supportable geometry, the anchor-to-anchor piping

configuration (layout) is acceptable. Adequate flexibility is required to avoid an

expansion (or contraction) induced fatigue failure and to limit anchor loads on

equipment. A flexibility analysis typically (and traditionally) evaluates the range of

stresses encountered by piping system service startup and shutdown. It is generally

assumed that the startup-shutdown stress-range will bound the other thermal

expansion or displacement stress-ranges. The piping flexibility is evaluated between

equipment and structural anchors without locating any intermediate supports.

Weight stresses, then, would not be known. It is presumed that the intermediate

supports for weight and other loads can be added after determining that a piping

system has adequate flexibility without significantly increasing the flexibility stress-

ranges. This is reflected in the circa. 1955 B31 books having an allowable thermal

expansion stress-range

SA = f(1.25Sc + 0.25Sh)

and permitting an additional thermal expansion stress-range allowance of Sh - SL,

when the stresses due to weight and other sustained loads, SL, were known.

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After the flexibility analysis has determined that the piping has adequate flexibility,

using the allowable thermal expansion or displacement stress-range, SA, then span

tables and/or engineering judgement is used to locate intermediate supports for

weight and other loads. If the thermal displacements at a proposed support point are

negligible (i.e., very small), then a rigid support can be located at that point. If the

vertical thermal displacements are significant at locations where weight supports are

proposed, springs (variable or constant) can be used. If the lateral thermal

displacements are significant at locations where lateral supports are proposed,

gapped supports usually can be used. By use of support types that offer minimal

restraint throughout the startup-shutdown excursion, the flexibility stress-range is

not significantly increased and could be expected to be bounded by the additional

thermal expansion allowance, Sh - SL.

The entire flexibility design and analysis process assures that the effects of fatigue

due to thermal expansion, or more generally the restraint of free-end displacements,

are minimized. However, some caution in performing the flexibility analysis is

necessary to see that other frequently occurring normal and abnormal operating

condition stress-ranges do not envelope the startup-shutdown stress-range or to see

that supports do not unduly restrain the load induced expanding (or contracting)

piping system.

Methods Of Flexibility Analysis

There are two methods of flexibility analysis which involve manual calculations.

1. Check as per clause 119.7.1/319.4.1 of the piping code

This clause specifies that no formal analysis is required in systems which are of

uniform size, have no more than two points of fixation, no intermediate restraintsand fall within the empirical equation

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2. Guided Cantilever MethodGuided cantilever is based on the simple concept of "minimum length".

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When two vessels are connected by a straight pipe, the pipe may buckle or dent the

sides of the vessel when operating at high temperature due to expansion. To

overcome this difficulty a bend is provided as shown in figure above. So that the

movement due to expansion will be absorbed and stresses are restricted to agiven value. The minimum length for this configuration to absorb movement can be

calculated as-

IX . Suggestions for Reducing Equipment Loading

Piping imposes loads on equipment nozzles. These loads may exceed the allowables

provided by the manufacturer or contained in guidelines such as the API 610.

The following guidelines may be helpful in reducing these piping loads on nozzles

connected to equipment.

1. If the dead loads exceed the allowable,- Ensure the piping system is adequately supported,

- Remove unneeded supports; they may be the cause of the problem.

2. If the thermal loads exceed the allowable,

- Check the design and operating temperatures. Consult the process engineer to

obtain correct or reasonable values for different operating conditions.

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3. Try modifying the piping support system and layout

- Add expansion loops if apt,

- Use expansion joints or other flexible joints,

- Consider spring mounted pumps,

- Modify the layout of piping by rerouting,

- Use guides or anchors at strategic locations,

- Use reinforcing pads on vessel nozzles.

X. Location of Supports and Restraints

Placing Dead Weight SupportsGuidelines for placing deadweight supports

Locate dead weight supports using recommended spacing from the code (B31etc.). Consider existing support points.

Decrease span by half off equipment. Decrease span for concentrated loads. Support concentrated loads. Support offset loads.

Decrease span for extra lagging or insulation. Locate supports at changes in direction (no overhung corners, top or bottom ofrisers). Select type (rigid, spring, or constant support) based on thermal expansion

analysis.

Preferred Attachment to "Structure"

Guidelines for dealing with structures when connected with piping.

Apply loads to columns and beams near main-member intersections to minimizebending effects.

Avoid the introduction of unnecessary torsion or lateral bending effects.

Avoid the introduction of movements or transverse loading to slender members(such as wind bracing) and particularly to compression members where instability

controls the design.

Confine connections to an independent structure or a foundation when dealing withpiping subject to pulsating flow or transmitted mechanical vibration, unless a carefuland comprehensive analysis assures that the structures, buildings, etc., are ofadequate strength with nonresonant frequency and sufficient stiffness to control

amplitude within the bounds required by general comfort level of personnel.

Provide anchors and extremely flexible and nonresonant intervening pipe runs

(e.g., expansion joints) to machinery that introduces mechanical vibrations, in orderto isolate the effect by reducing transmissibility.

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Preferred Points of Attachment to PipeGuidelines for selecting preferred attachment points on piping

On a pipe rather than on piping components such as valves, fittings, or expansionjoints. Under highly localized loading, flanged or threaded joints may leak and valvebodies may distort with resulting seat leakage or binding. Attachments to heavycomponents, however, may be acceptable and even desirable where the effect can

be properly provided for.

On straight runs rather than on sharp radius bends or welding elbows, since theseare already subjected to highly localized stresses on which the local effects of theattachment would be superimposed. Furthermore, attachments on curved pipe whichextend well along the length or circumference of the bend will seriously alter theflexibility of the component.

On pipe runs which do not require frequent removals for cleaning and maintenancework.

As close as practical to heavy load concentrations such as vertical runs, branchlines, motor operated or otherwise heavy valves, and minor vessels such asseparators, strainers, etc.

XI. Tips for Flexible Layouts

A system for determining flexibility on an increasing scale is illustrated above. Each

prism shows pipe running between points A and F. In the one at the far left, the pipe

cuts across the face of the prism with leg CD. For this case, the pipe could cut across

any face or into the body of the prism. If calculation shows such a line to be

overstressed, another route must be chosen.

The center sketch shows the same anchor points A and F, but the pipe now runs

along the edges of the prism. The pipe could be run along any of the edges but not

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across the surface or through the prism. This route is more flexible than that on its

left.

Suppose that the pipe in the center sketch is still overstressed. The sketch at far

right shows the line going outside the prism into space. It runs along the edges and

then into space to form a loop between points C and F. This route is the most flexible

of the three possible routes.

It is important to point out that the first route shown (far left) is the usual one and

piping is not necessarily overstressed because it follows this path. The three

sketches are used only to show the successive paths of increasing flexibility. The

prisms provide a means of visualizing, at a glance, a softer piping system. Even the

path at far right, however, can be overstressed if the loop between points C and F is

not large enough.

In laying out hot piping, one should at least consider the following:

1. The expansion of turbines, towers, heat exchangers etc. must be added to the

pipe expansion.

2. A heat exchanger is generally fixed at one end and free to slide at the other.

3. Long radius elbows are more flexible than five diameter bends. The elbows

produce lower forces but higher local stresses because of the flattening of a

curved member when it flexes. The five diameter bend flattens less therefore

produces higher forces but lower local stresses. These local stresses are of

course, in the bend or elbow itself.

4. Pumps, turbines and compressors must have low forces on them as required by

the manufacturer and in compliance with API 610/617 and NEMA SM-23. If the

stress in the piping adjacent to the equipment is limited to 5,000 psi, the forces

will generally be acceptable.

5. Dead weight of piping must in most cases be carried by independent supports

and not by the pump, turbine or compressor. In the case of heat exchangers and

vessels and other non-rotating equipment, some of the piping dead weight loads

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may be transferred to the nozzles but the designer MUST check with the

equipment designer first.

6. Always run a line with a thought as to how it will be supported. Lines should be

grouped whenever possible. If a line needs to be re-routed for the better support,

this should be done.

7. Cold spring is not the answer to lowering stresses in overstressed piping. The

Piping code does not permit this. It allows only a one third reduction in forces and

bending moments if the line is cut short by 50 percent of its total expansion.

8. The stress at flanged connections should be limited to 10,000 psi.

XII. What A Piping Designer Should Know to deal with Piping Stress

Allowable pipe spans All designer need to know and understand the spancapabilities of pipe in the different schedules for a wide variety of common

piping materials. When a new project introduces a new material with severelyreduced span capabilities; supplemental training may be required.

Expansion of pipe All designers must understand that they need to treat apiping system as though it is alive. It has a temperature and that temperaturecauses it to grow and move. That growth and movement must be allowed forand incorporated in the overall design. Not just of that specific line but for all

other lines close by. The process of expansion in a pipe or group of pipes willalso exert frictional forces or anchor forces on the pipe supports they come incontact with.

Routing for flexibility The piping designer must understand how to route pipefor flexibility. Routing for flexibility can normally be achieved in the mostnatural routing of the pipeline from its origin to its terminus. Routing for

flexibility means (a) do not run a pipe in a straight line from origin to terminusand (b) building flexibility into the pipe routing is far cheaper and more reliablethan expansion joints.

Weight and loads (live loads and dead loads) The piping designer needs tounderstand the effects of weight and loading. They need to know andunderstand that everything has a weight. They need to be able recognize

when there is going to be a concentrated load. They need to have access tobasic weight tables for all the standard pipe schedules, pipe fittings, flanges,valves for steel pipe. They also need to have the weight tables for othermaterials or a table of correction factors for these other materials vs. carbon

steel. They need to be able to recognize when downward expansion in apiping system is present and is adding live loads to a support or equipmentnozzle.

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Equipment piping The piping designer needs to know the right and the wrongway to pipe up (connect pipe to) different kinds of equipment. This includespumps, compressors, exchangers, filters or any special equipment to be used

on a specific project.

Vessel piping The piping designer also needs to understand about theconnecting, supporting and guiding of piping attached to vessels (horizontal or

vertical) and tanks. They need to know that nozzle loading is important anddoes have limitations.

Rack piping The designer needs to understand that there is a logicalapproach to the placement of piping in (or on) a pipe rack. It does not matterhow wide or how high the rack or what kind of plant, the logic still applies.Starting from one or both outside edges the largest and hottest lines are

sequenced in such a manner that allows for the nesting of any requiredexpansion loops. The spacing of the lines must also allow for the bowing effectat the loops caused by the expansion.

Expansion loops The designer needs to understand and be able to use simplerules and methods for sizing loops in rack piping. This should include the mostcommon sizes, schedules and materials.

Cold spring/Pre-spring Designers should understand the basics rules of coldspring and pre-spring. They need to understand what each one is along withwhen to and when not to use each.

XIII. Stress Critical Lines

- Liquid lines above 650 F

- All lines above 750 F

- Lines 16" and larger in diameter (e.g. to check for local loads and stresses on pipe

wall at supports)

- Lines having substantial concentrated loads such as heavy valves, fittings,

unsupported vertical risers and branches

- Lines having local reduction in strength due to the installation of special fittings

- Lines connected to vessels or tanks having appreciable settlement or where there

are long vertical runs greater than 30 feet

- Lines with less than standard weight wall thickness

- Lines using non-standard supports and/or having pipe attachments

- All lines connecting to rotating equipment regardless of size

- All lines attached to API 12B bolted tanks larger than 4" NPS

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- Lines where corrosion allowance is greater than 1/16" for lines through 4", or

greater than 1/8" for all line sizes

- All process, regenerating, and decoking lines to and from fired heaters and steam

generators (vibration should be considered for these as well)

- All air-cooled heat exchanger piping (because fin-fans are structurally flimsy)

- All lines with maximum short-term temperature below minus 50 F

- All lines having very long straight runs either horizontally or vertically (definition of

'very long' is in the eys of the beholder)

- All blowdown and flare header systems (forces due to fluid dynamics)

- All multi-phase flow lines (dynamic loads)

- All lines with relief valve with set pressure above 50 psig (thrust load)

- Others: Cast iron lines, FRP, copper, etc.

XIV . Information Required For Stress Analysis

1. Outside diameter of piping, wall thickness (or nominal diameter, schedule

number)

2. Temperature, internal pressure

3. Material of piping. (Expansion coeffcient, Youngs modulus, and

material density will be selected for this material.)

4. Insulation thickness and insulation material. (If not given, standard

thickness for calcium silicate will be selected.)

5. Specifc gravity of contents

6. Any wind load to be considered? If yes, the direction of application is important.

7. Any anchor initial translation. (For towers, exchangers, and so on, nozzle initial

ranslation is important.)

8. Corrosion allowance for piping

9. Flange rating, (ANSI B16.5)

10. Standard valve weight and fange weight will be used. (For special valves markthe weight on pipe stress isometric.)

11. Long radius elbows will be used. (If short radius or any other bend radius, mark

on the isometric.) For short-radius elbow, radius= diameter

12. Any allowable loading from manufacturers on pumps, turbines,

compressors? (From the vendor drawing for equipment.)

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13. Any preference to use expansion loops, expansion joints, and so on,

if needed?

14. Mark type of intersection (reinforced fabricated tee, etc.)

15. Mark support locations (available steel crossing, and so on) on the isometric

16. Is hydraulic testing load condition to be considered to get structural support

loads?

17. Pipe stress isometrics (x-, y-, z-axis) piping plans, and sections are

necessary.

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