design by rule

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16587 Pressurised Systems

76

6. DESIGN BY RULE6. DESIGN BY RULE

The philosophy of many national pressure vessel design standards embraces the concept of

Design by Rule. Essentially this involves simple calculations to arrive at basic scantlings

via an allowable standardised design stress followed by strict adherence to specific rulesdelineated in the Standard for the component detail.

The method may not be apparent to the designer without further explanation. The

approach has the benefit of simplicity and to some extent clarity but the approach mitigates

against the use of rational extension of the Standard to deal with more complex situations.

These may occur when the geometry of the component is outwith the Standard or the

loading is slightly different from that set out in the Standard.

6.16.1 THE BASIC CONCEPT OF DESIGN BY RULE.THE BASIC CONCEPT OF DESIGN BY RULE.

Simple formulae are given in the Pressure Vessel Standards, for example section 3 of BS

5500, which is used in the UK, gives equations to derive the wall thickness of a range of

standard components, such as, spheres, cylinders, cones and dished heads, etc. When

these are used with the design stress,f, the basic minimum thickness of the component can

be found. The basic idea of Design by Rule is that once the leading scantlings are fixed in

this way the designer simply obeys the rules laid down in the procedures for specified

components such as nozzles, flanges, local supports, etc. This is the most common

approach used in national design standards.

The approach, of course, does not provide the designer with a value of the stress in the

component, since the aim is to lead to a value for the wall thickness, or the plate thickness.

However, the information obtained from the approach can be used to assess likely vessel

weight and from this, by certain complex financial formulae, give an estimate of the cost of

the vessel. This may be of advantage in giving a budget cost for the vessel. On the other

hand it can leave the designer with the head-ache of producing a detailed design which

can be built within the budget cost; sometimes this leads to a vessel which may not be

commercially viable.

The other concern in using a Design by Rule is that there is a lack of consistency in the

design criteria used throughout the Standard. Some parts are based on elastic analysis with

some limitation on the maximum stress (although the limit is different in different cases),

some are based on shakedown concepts without regard to the actual stress range, whilesome are based on limit load concepts with suitable (unknown) safety factors. It is true to

say that the criteria are HIDDEN.

However, the design by rule approach has the great advantage of simplicity and, having

been used for many years, is backed up by long experience of users who have found the

approach works well. In other words it reflects the voice of the industry who by using the

methods have not had too many, if any, failures. It also had the advantage that every body

is using the same method and so one assumes there is a level playing field. Actually there

seldom is, since other subsidies move the goal posts !!!!


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As indicated above the greatest disadvantage of the method is that it cannot be easily

extended either to different geometries or to additional loadings beyond the normal

pressure case. Unless such additional loads are rather small it is unclear how the combined

situation can be tackled logically especially where the criteria which have been used are

uncertain. Similarly, because the stress and strain levels are, in general, unknown there aredifficulties in conducting a fatigue analysis.

7. MATERIAL PROPERTIES AND ALLOWABLE DESIGN STRESSES7. MATERIAL PROPERTIES AND ALLOWABLE DESIGN STRESSES

The Standard (BS 5500) gives basic information regarding allowable design stresses.

Normally these can be selected from tabular information for specific recommended grades

of steel. The allowables are based on actual material data for tensile tests and information

on how these are arrived at is given in BS 5500, Appendix K, Stated simply the design

stressfis typically:-

eR mR rtS

1 5 2 3 5 1 3. . .o r or

where,

Re is the minimum value of the specified yield strength for the grade of steel concerned

at room temperature or at temperature Re(T)

Rm is the minimum tensile strength at room temperature.

Srt is the mean value of the stress required to produce rupture in time t at temperature T.

An example of the way the design stress values, f, are provided is given on p.25 of the

notes. The values given in Table 2.3(a) are for Plates.

8. DESIGN BY ANALYSIS8. DESIGN BY ANALYSIS

The concept of Design by Analysis originated in the ASME code and has been adopted byother including BSI in BS 5500. The approach assumes that a sufficient elastic stress

analysis can be conducted. Particular stress categories are defined and then identified with

elements of the stress analysis. Thereafter via a Tresca criteria framework the different

categories of stress are compared with recommended limits. The limits are also dependent

on the category.

8.1 INTRODUCTION8.1 INTRODUCTION

The basic philosophy of Design by Analysis originated in the ASME Pressure Vessel and

Boiler Code Section III and Section VIII Division 2. The concept is that a designer canperform his own analysis to obtain the stress levels in a component under any load


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condition. The stresses are assigned to certain categories before being arranged as stress

intensities and then compared to different allowables depending on the categorisation.

The philosophy has been adopted in BS 5500 Appendix A with some simplification in

which some of the detail has been omitted. What follows is based largely on the ASME

explanation.

Appendix A (of BS 5500) is intended to cater for situations not covered by Section 3 and is

supposed to ensure that in such situations the design basis is consistent with Section 3.

The aim is laudable but having seen how variable Section 3 can be, this consistency will not

be easily forthcoming. For example it is stated that the margin on gross plastic deformation

should be the same as that in the membrane region (i.e. Y

f== 1.5 ). This is rather difficult

for the designer to check. But it is a useful statement if limit analysis is actually employed.

8.2 THE ASME STORY FOR STRESS LIM ITSTHE ASME STORY FOR STRESS LIM ITS - Two aspects of Design areconsidered:-

(1) Avoidance of Gross Distortion or Bursting

To avoid gross distortion or bursting it is necessary to avoid the full wall section of a vessel

becoming plastic. The Fig 8.1 shows a simple case with an element of the wall stressed in

one direction.. The vessel wall is idealised as a beam, of width b and thickness 2h (or t),

subject to an end force N and a bending moment M.

Figure 8.1 The vessel wall analysed as a beam


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Let (z) be the circumferential stress at any position z through the wall.

At section z it is possible to write down the equations of equilibrium in which the externally

applied loads M and N are equated to the internal forces, that is the stresses in the beam,

the result is the following two expressions:-

M = b zdz

h

h

.

; and N = b dz

h

h

Provided the behaviour is purely elastic, these two equations produce the simple beam

bending theory, which is given as follows:-

( )zN

A

M z

I== ++

where A = 2bh is the area and I =2

3

3b h is the second moment of area (b t3

12) of the beam

element cross section.

Suppose the material is elastic, perfectly plastic with yield stress Y, then with N tensile,yield first occurs in tension in the outer fibre (z = +h) when

N

bh

M

bhY

2

3

2 2++ ==

This equation can be plotted as follows:-

Following yield, if the load is further increased, plasticity will spread through the vessel

wall (i.e. the beam cross section) as in Fig 8.2. For perfect plasticity, the fully plastic state

corresponds to the LIMIT STATE and the postulated (linear) distribution of plastic flow

strain is shown. The exact nature of the strain does not need to be specified except that

the neutral axis is off-set by an amount ho below the centre line. Above the neutral axis in

the fully plastic state the stress must equal Y, while below this axis it must be equal to -Y.


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Figure 8.2 The Progessive Development of Plasticity in the Beam

Mathematically, the stress distribution is expressed as:-

( )z

z h

z h

Y o

Y o

==>>


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(( )) (( )){{ }} N b dz dz b h h h hY Yh

h

h

h

Y o o

o

o

== ++

== ++ ++

This simplifies to; N bho Y== 2

==hN

bo

Y2

==

M b hN

bY

Y

22

2

M

b h

N

b hY Y ==

2

2

2 2 21 4

++

==

M

bh

N

b hY Y

2

2

21 THE LIMIT CONDITION

Owing to the nature of the stress in the fully plastic state the stress cannot increase above

the yield stress, Y. If the combinations of the two loads M and N change in such a waythat the above equation is always satisfied then the vessel will always be within the limit

condition.

The IMPORTANT POINT here is that there is not a single limit load; rather, for multiple

loading, there are certain combinations of load which put the structure in the limit state

where flow occurs. It is usual to describe the initial yield conditions, given earlier, and

the limit loadcondition given in the above equation, in an INTERACTION DIAGRAM

(in Load Space) as shown below:-

The limit condition, given above, is commonly

referred to as a LIMIT SURFACE on this

interaction diagram.

From the interaction diagram we must also

have the conditions:-

M

bh

N

b hY Y

21

21 ,


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From the LIMIT CONDITION equation, on page 81, we can find the LIMIT LOAD

for a beam in bending. We do this by putting N = 0, which gives a value for the Limit

Moment of, M b hL Y== 2 .

Recall from p. 79, that the first yield moment MYcan be obtained by putting N = 0. Thisresults in:-

M bhY Y==2

3

2

From these two expressions we can write;M

M

L

Y

== 1.5.

That is, the complete wall thickness is plastic at a value of moment 1.5 x First Yield

Moment, i.e. there is 50% reserve at first yield.

If we now put the moment, M = 0, in the Limit Condition equation; we find that the

section is fully plastic when N b hL Y== 2

Note from p79 that the first yield value for the direct force is given by N b hY Y== 2

That isN

N

L

Y

== 1, this means that there is no reserve at first yield, for this condition.

The Interaction Diagram, shown on page 81, can be redrawn in an alternative form where

the quantityN

bh2

may be interpreted as:- The Elastic Membrane Stress m

and, the quantity3

2 2

M

bh

may be interpreted as:- The Elastic Bending Stress b at the outer fibre

These two expressions, involving N and M, may be identified from the equation on p.79.

They are clearly only appropriate, if elastic behaviour is assumed.

The maximum stress can then be written as:- max == ++m b

The initial yield is given by:- m b Y++ == .

The limit condition, from p 81, is written :-M

b h

N

b hY Y

2

2

21++

==

From above:- N bh M bhm b== ==2 2 3

2

;


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Substituting these in the limit equation gives,2

3

b

Y

m

Y

bh

b h

bh

b h

2

2

22

21++

== .

This equation is simplified to

2

3 1

2

b

Y

m

Y

++

== .

These equations are replotted in terms of max and m and shown below in Figure 8.3.

Figure 8.3 Design Limits to Avoid Gross Distortion

It is this form of the design limit diagram which issued in ASME. To avoid distortion

and subsequent bursting, it is recommended that the stresses are kept below yield at all

times. However different factors of safety are applied; limits on m and ( ) m b++are imposed as fractions of the yield strength. These are:

(( ))

m Y

m b Y

++

2

3 LIMITS OF STRESS


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An accurate plot of the various equations given above is shown below in Figure 8.4. A

development of the equation to determine the highest value of the maximum stress, which

is of course associated with the turning value, has been derived. The details are provided

on p. 85.

Figure 8.4 Accurate plot of the Design Limits to Avoid Gross Distortion.


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To plot the LIMIT CONDITION on

max

Y

m

Y

~ Graph

we start with the equation given above:-2

3 1

2

bYmY++

==

This can be modified by the addition and subtraction of2

3

m

Y

, as shown below:-

(( ))23

2

31

2

++

++

==

b m

Y

m

Y

m

Y

To find the turning value - 23

2 23

0

++

++ ==

d

d

b m

Y

m

Y

m

Y

++

== == ==d

d

b m

Y

m

Y

m

Y

m

Y

1 3 01

3the turning value is at

as shown on Figure 8.4.

It will be noted from the figure that the factor of safety on m is greater than that on

( ) m b++ since overloads into the plastic region would result in the fully plastic state if m Y . But in the case of ( ) m b Y++ overloads would only cause partialyielding through the thickness and may be acceptable. The above limits are intended to

guard against gross/plasticity or bursting.

(2) Avoidance of Rachetting or Repeated Plastic Straining

On page 78 it was suggested that there was two aspects of design to be considered. The

first was the avoidance ofGross Distortion or Bursting. The second is the avoidance of

Rachetting. The ASME approach is to consider the avoidance of the possibility of

repeated plastic cycling or ratchetting. A simple example which is used, is the case of

thermal cycling applied to the beam element of the vessel wall. Consider the outer fibre of

the vessel wall which is strained (unaxially) to some value R as shown in Figure 8.5 on

page 86, over the cycle OAB, somewhat beyond the yield strain. When we cycle from 0 to

R and back to zero, R is Strain Range.

On unloading at point C, the outer fibre has a residual compressive stress, Y RE .On subsequent reloading this stress must be removed before the stress goes into tension.


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Effectively the elastic range is increased from OA to CD. Provided this residual stress at C

in the outer fibre is less than yield, the subsequent behaviour is purely elastic - i.e. the vessel

wall exhibits SHAKEDOWN. The limit of Shakedown in this simplified case is obviously

Y R Y

E

== .

That is , and the max strain range for shakedown is given byREE

R Y RY

== ==2

2,

Figure 8.5 Design Limit to Avoid Ratchetting

In a Design context, we may interpret E R as the elastically calculated maximum STRESS

RANGE, R . Thus for Shakedown the elastic stress range is TWICE YIELD i.e.

R Y 2

The two equations, which give the limits of stress on p. 83:-

(( ))

m Y

m b Y

++

23

and the above equation:- R Y 2 , together define the three main limits of stress inthe ASME code and in BS5500 Appendix A.

However, IT REMAINS TO DECIDE IN WHAT CIRCUMSTANCES THEY

SHOULD BE APPLIED. The answer to this question is addressed in the following

discussion.


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8.3 MULTI AXIAL STRESS STATES AND CATEGORISATION8.3 MULTI AXIAL STRESS STATES AND CATEGORISATION

In the above, the stress limits were derived for the simple situation where only

circumferential effects were considered, that is it was limited to a uniaxial condition. In

real vessels there will also be longitudinal effects. The same type of philosophy can be

applied, but consideration must be given to the multiaxial stress state. In the presence of

multiaxial stress states, yield is not governed by the individual components of stress, but by

some combination of all stress components, as we saw in the lectures on Plastic Design,

p.67.

The theories most commonly used are the von Mises criterion (also known as the

octahedral shear theory or the distortion energy theory) and the Tresca criterion (also

known as the maximum shear stress theory). In fact, many Design by Rule codes make

use of the maximum stress criterion but in the Design by Analysis approach a moreaccurate representation of multiaxial yield is required. Although it is generally accepted

that the Mises criterion is more accurate for common pressure vessel steels, the ASME

Code uses the Tresca criterion since it is a little more conservative and sometimes easier to

apply. BS 5500 follows the same procedure.

Let 1 2 3, and be the principal stresses at some point in a component. Then the shear

stresses are:-

(( )) (( )) (( )) 1 2 3 2 3 1 3 1 21

2

1

2

1

2== == == , , ; (see p.67 of these notes)

Yielding occurs according to the Tresca criterion (see Section 5.1), if :-

(( )) == ==max 1 2 31

2, , Y ; (see p.67 of these notes)

In order to avoid the unfamiliar (and unnecessary) operation of dividing both calculated and

yield stress by two, a new term called "equivalent intensity of combined stress" or simply

STRESS INTENSITY is defined. The STRESS DIFFERENCES, denoted by

S S and S12 23 31, are equated to twice the shear stress. We can then write;

S S S12 1 2 23 2 3 31 3 1== == == ; ;

The STRESS INTENSITY, S is the maximum absolute value of the stress difference

i.e. S = max (S12,S23,S31), so that the Tresca criterion reduces to:-

S Y== .

This caters for the multiaxial aspects.


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It now remains to deal with different types of categories of stress. It is assumed (sensibly)

that different types of loading, or different types of stress, require different allowable stress

limits. Since terms like membrane and bending, are often used rather loosely, ASME

chose to strictly define different STRESS CATEGORIES to which different limits were to

be applied.

Basically the stresses which occur in vessel shells are divided into two categories

PRIMARY and SECONDARY together with subcategories.

(1) PRIMARY STRESS

General Primary Membrane Stresses, fm(BS 5500) andPm (ASME).

This is the stress produced by mechanical loads - like internal pressure; it excludesthe stress due to discontinuities and concentrations. It is derived and required by

equilibrium of the component.

Local Primary Membrane Stress,fL (BS 5500) andPL (ASME).

This is again produced only by mechanical loads; it considers discontinuities, but

not concentrations. The termpa

o2

cos , given in Figure 4.22, p. 58, for the nozzle

in the sphere is a good example of Local Primary Membrane Stress.

Primary Bending Stress,fb (BS 5500) andPb (ASME).

A good example of this is the bending stress in the

central portion of a flat head due to pressure, see

the sketch of this across. This behaviour is also

shown in Figure 4.13 of the finite element results.

It excludes discontinuities and concentration and

is produced only by mechanical loads

(2) SECONDARY STRESS,fg (BS 5500) and Q (ASME)

This is a self-equilibrating stress necessary to satisfy continuity of the structure, and

of course, occurs at structural discontinuities. It can be caused by mechanical loads

or by differential thermal expansion. This what we have called edge bending and

is the stresses due to H and to M at intersection regions like the junction of the

nozzle and sphere.


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The symbols fm(Pm ), fL(PL ), fb(Pb)andfg(Q) are used to denote the categories of stress

(they are not stresses, although careless usage has allowed them to become so). The BS

5500 symbols are shown first in the above with the ASME symbols following. It is

unfortunate that different symbols have been used.

The basic difference between a secondary stress and a primary stress is that the secondary is

largely self-equilibrating, or SELF-LIMITING. It is envisaged that local yielding and

some distortion can satisfy or ameliorate the conditions which cause the stress to occur.

Failure direct from a single application of a secondary stress is therefore not expected. On

the other hand a primary stress is not self-limiting and does not redistribute. Primary

stresses which considerably exceed yield will result in failure or gross distortion.

To summarise

the basic Design by Analysis procedure then involves the categorisation of the

calculated stresses associated with each type of loading, evaluation of the appropriatestress intensity in each category and comparison with the basic limits in each

category.

STRESS LIMITS FOR THE VARIOUS CATEGORIES

STRESS INTENSITY ALLOWABLE STRESS EQUIVALENT

YIELD

General primary membrane,fm f (2/3) Y

Local primary membrane,fL 1.5f Y

Primary membrane plus primary

bending, (fL +fb )1.5f Y

Primary plus Secondary, (fL +fb + fg) 3f 2 Y

It should be noted in passing that the above limits are not always directly applicable. For

example in the ASME code they are used in the above form for design conditions. These

are normally higher than the expected operating conditions, i.e. the actual service loadings,

which may be subclassified for example for nuclear vessels into normal, upset, emergency

and faulted conditions. Also the design should be acceptable for any testing conditions

above the design loads. In particular the limit on primary plus secondary stresses

applies only to the operating conditions. Otherwise k-factors are applied to the limits

given above (i.e. the appropriate limit is multiplied by the factor k). For example for

earthquake k = 1.2, for hydraulic test k = 1.25 etc.

In BS 5500 there are similar restrictions. In Section A3.3. there are specific criteria for

limited application. This refers to stresses local to attachments, supports and nozzles

which are subject to applied loading in addition to the pressure in the vessel.


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Forattachments and supports the limits are:

The membrane stress intensity 1.2f (0.8Y

)

Membrane + bending stress intensity 2f (1.33 Y)

For nozzles and opening:

Membrane + bending stress intensity 2.25f (1.5 Y)

These in-between limits recognise the possibility that some secondary stresses will exist.

Therefore some concession has been made for these components.

In this Design Procedure, the elastic stresses due to the various types of loading areobtained. The stresses are assigned to the stress categories fm, fL, fb and fg . The stress

intensities are determined from the principal stresses.

When we require to calculate (fm +fg ) we calculate the stresses in each category. The final

step is to sum the stresses 1 2 3, and in each category to find a final value of the

stress intensity corresponding to (fm+fg).

THIS FINAL VALUE IS COMPARED WITH THE ALLOWABLE:- in this case the

allowable is 1.5f.

8.4 THE HOPPER DIAGRAM8.4 THE HOPPER DIAGRAM

Both the ASME and the BS 5500 Standards provide helpful information to categorise

components when they are subject to different forms of loading. They provide a Hopper

Diagram which summarises the Stress Categories and the limits of stress intensity. The

one from BS 5500 is provided in these notes as Figure 8.6. In addition a range of typical

cases are also given. A copy of these are given in the notes.


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Figure 8.6 Stress Categories and Limits of Stress Intensity - BS 5500


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Classification of Stresses - Table A.3 from BS 5500


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Completion of Table A.3 from BS 5500

design by rule

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