stress categorization derivitions

16587 Pressurised Systems

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6. DESIGN BY RULE6. DESIGN BY RULE

The philosophy of many national pressure vessel design standards embraces the concept ofDesign by Rule. Essentially this involves simple calculations to arrive at basic scantlingsvia 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. Theapproach has the benefit of simplicity and to some extent clarity but the approach mitigatesagainst 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 theloading 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 BS5500, which is used in the UK, gives equations to derive the wall thickness of a range ofstandard components, such as, spheres, cylinders, cones and dished heads, etc. Whenthese are used with the design stress, f, the basic minimum thickness of the component canbe found. The basic idea of Design by Rule is that once the leading scantlings are fixed inthis way the designer simply obeys the rules laid down in the procedures for specifiedcomponents such as nozzles, flanges, local supports, etc. This is the most commonapproach used in national design standards.

The approach, of course, does not provide the designer with a value of the stress in thecomponent, 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 vesselweight and from this, by certain complex financial formulae, give an estimate of the cost ofthe vessel. This may be of advantage in giving a budget cost for the vessel. On the otherhand it can leave the designer with the head-ache of producing a detailed design whichcan be built within the budget cost; sometimes this leads to a vessel which may not becommercially viable.

The other concern in using a Design by Rule is that there is a lack of consistency in thedesign criteria used throughout the Standard. Some parts are based on elastic analysis withsome 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 tosay that the criteria are HIDDEN.

However, the design by rule approach has the great advantage of simplicity and, havingbeen used for many years, is backed up by long experience of users who have found theapproach works well. In other words it reflects the voice of the industry who by using themethods have not had too many, if any, failures. It also had the advantage that every bodyis using the same method and so one assumes there is a level playing field. Actually thereseldom 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 easilyextended either to different geometries or to additional loadings beyond the normalpressure case. Unless such additional loads are rather small it is unclear how the combinedsituation can be tackled logically especially where the criteria which have been used areuncertain. 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 gradesof steel. The allowables are based on actual material data for tensile tests and informationon how these are arrived at is given in BS 5500, Appendix K, Stated simply the designstress f is typically:-

eR mR rtS15 2 35 13. . .

or or

where,

Re is the minimum value of the specified yield strength for the grade of steel concernedat 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 thenotes. 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 stressanalysis can be conducted. Particular stress categories are defined and then identified withelements of the stress analysis. Thereafter via a Tresca criteria framework the differentcategories of stress are compared with recommended limits. The limits are also dependenton the category.

8.1 INTRODUCTION8.1 INTRODUCTION

The basic philosophy of Design by Analysis originated in the ASME Pressure Vessel andBoiler 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 stressintensities and then compared to different allowables depending on the categorisation.The philosophy has been adopted in BS 5500 Appendix A with some simplification inwhich some of the detail has been omitted. What follows is based largely on the ASMEexplanation.

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

== 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 LIMITSTHE ASME STORY FOR STRESS LIMITS - 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 vesselbecoming plastic. The Fig 8.1 shows a simple case with an element of the wall stressed inone 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 s(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 externallyapplied 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 zdzh

h

s .-- ; and N = b dz

h

h

s--

Provided the behaviour is purely elastic, these two equations produce the simple beambending theory, which is given as follows:-

s ( )zNA

M zI

== ++

where A = 2bh is the area and I = 23

3bh is the second moment of area (bt3

12) of the beam

element cross section.

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

Nbh

M

bhY2

32 2

++ == s

This equation can be plotted as follows:-

Following yield, if the load is further increased, plasticity will spread through the vesselwall (i.e. the beam cross section) as in Fig 8.2. For perfect plasticity, the fully plastic statecorresponds to the LIMIT STATE and the postulated (linear) distribution of plastic flowstrain is shown. The exact nature of the strain does not need to be specified except thatthe neutral axis is off-set by an amount ho below the centre line. Above the neutral axis inthe fully plastic state the stress must equal sY, while below this axis it must be equal to -sY.


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Figure 8.2 The Progessive Development of Plasticity in the Beam

Mathematically, the stress distribution is expressed as:-

ss

s( )z

z h

z hY 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

== ++ --

== ++ -- -- ++--

--

-- s s s

This simplifies to; N bho Y== 2 s

\\ ==hN

bo Y2 s

\\ == --

M b hN

bY Ys

s2

2

2

Mb

hN

b hY Ys s== --

22

2 2 21

4

\\ ++

==

M

bh

NbhY Ys s

2

2

21 THE LIMIT CONDITION

Owing to the nature of the stress in the fully plastic state the stress cannot increase abovethe yield stress, sY. 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 limitcondition.

The IMPORTANT POINT here is that there is not a single limit load; rather, for multipleloading, there are certain combinations of load which put the structure in the limit statewhere flow occurs. It is usual to describe the initial yield conditions, given earlier, andthe limit load condition given in the above equation, in an INTERACTION DIAGRAM(in Load Space) as shown below:-

The limit condition, given above, is commonlyreferred to as a LIMIT SURFACE on thisinteraction diagram.

From the interaction diagram we must alsohave the conditions:-

M

bh

NbhY Ys s

21

21 ,


82

From the LIMIT CONDITION equation, on page 81, we can find the LIMIT LOADfor a beam in bending. We do this by putting N = 0, which gives a value for the Limit

Moment of, M bhL Y== s2.

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

M bhY Y==23

2s

From these two expressions we can write; MM

L

Y== 1.5.

That is, the complete wall thickness is plastic at a value of moment 1.5 x First YieldMoment, 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 thesection is fully plastic when N bhL Y== 2 s

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

That is NN

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 quantity Nbh2

may be interpreted as:- The Elastic Membrane Stress s m

and, the quantity 3

2 2M

bhmay be interpreted as:- The Elastic Bending Stress s bat 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:- s s smax== ++m b

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

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

bh

NbhY Ys s

2

2

21++

==

From above:- N bh Mbh

m b== ==22

3

2s s;


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Substituting these in the limit equation gives, 23

s

s

ss

b

Y

m

Y

bh

bh

bhbh

2

2

222

1++

== .

This equation is simplified to 23

12

ss

ss

b

Y

m

Y

++

== .

These equations are replotted in terms ofs maxand s 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 distortionand subsequent bursting, it is recommended that the stresses are kept below yield at alltimes. However different factors of safety are applied; limits on s m and ( )s sm b++are imposed as fractions of the yield strength. These are:

(( ))

s s

s s s

m Y

m b Y

++

23 LIMITS OF STRESS


84

An accurate plot of the various equations given above is shown below in Figure 8.4. Adevelopment of the equation to determine the highest value of the maximum stress, whichis of course associated with the turning value, has been derived. The details are providedon p. 85.

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


85

To plot the LIMIT CONDITION on ss

ss

max

Y

m

Y~ Graph

we start with the equation given above:- 23

12

ss

ss

b

Y

m

Y++

==

This can be modified by the addition and subtraction of 23

ss

m

Y , as shown below:-

(( ))23

23

12

++

++

-- ==

s ss

ss

ss

b m

Y

m

Y

m

Y

To find the turning value -23

223

0

++

++ -- ==d

d

b m

Y

m

Y

m

Y

s ss

ss

ss

\\

++

== -- == \\ ==d

d

b m

Y

m

Y

m

Y

m

Y

s ss s

ssss

s

1 3 013

the turning value is at

as shown on Figure 8.4.

It will be noted from the figure that the factor of safety on s mis greater than that on( )s sm b++ since overloads into the plastic region would result in the fully plastic state ifs sm Y . But in the case of ( )s s sm b Y++ overloads would only cause partialyielding through the thickness and may be acceptable. The above limits are intended toguard 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. Thefirst was the avoidance of Gross Distortion or Bursting. The second is the avoidance ofRachetting. The ASME approach is to consider the avoidance of the possibility ofrepeated plastic cycling or ratchetting. A simple example which is used, is the case ofthermal cycling applied to the beam element of the vessel wall. Consider the outer fibre ofthe vessel wall which is strained (unaxially) to some value e Ras shown in Figure 8.5 onpage 86, over the cycle OAB, somewhat beyond the yield strain. When we cycle from 0 toe R and back to zero, e R is Strain Range.

On unloading at point C, the outer fibre has a residual compressive stress, s eY 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 Cin the outer fibre is less than yield, the subsequent behaviour is purely elastic - i.e. the vesselwall exhibits SHAKEDOWN. The limit of Shakedown in this simplified case is obviously

s e sY R YE-- == -- .

That is , and the max strain range for shakedown is given by RE ER Y RYe s e e

s== ==2

2,

Figure 8.5 Design Limit to Avoid Ratchetting

In a Design context, we may interpret E Re as the elastically calculated maximum STRESSRANGE, s R . Thus for Shakedown the elastic stress range is TWICE YIELD i.e.

s sR Y 2

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

(( ))s s

s s s

m Y

m b Y

++

23

and the above equation:- s sR 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 THEYSHOULD BE APPLIED. The answer to this question is addressed in the followingdiscussion.


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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 onlycircumferential effects were considered, that is it was limited to a uniaxial condition. Inreal vessels there will also be longitudinal effects. The same type of philosophy can beapplied, but consideration must be given to the multiaxial stress state. In the presence ofmultiaxial stress states, yield is not governed by the individual components of stress, but bysome 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 theoctahedral shear theory or the distortion energy theory) and the Tresca criterion (alsoknown as the maximum shear stress theory). In fact, many Design by Rule codes makeuse of the maximum stress criterion but in the Design by Analysis approach a moreaccurate representation of multiaxial yield is required. Although it is generally acceptedthat the Mises criterion is more accurate for common pressure vessel steels, the ASMECode uses the Tresca criterion since it is a little more conservative and sometimes easier toapply. BS 5500 follows the same procedure.

Let s s s1 2 3, and be the principal stresses at some point in a component. Then the shearstresses are:-

(( )) (( )) (( ))t s s t s s t s s1 2 3 2 3 1 3 1 212

12

12

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

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

(( ))t t t t s== ==max 1 2 312

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

In order to avoid the unfamiliar (and unnecessary) operation of dividing both calculated andyield stress by two, a new term called "equivalent intensity of combined stress" or simplySTRESS INTENSITY is defined. The STRESS DIFFERENCES, denoted byS 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== -- == -- == --s s s s s s; ;

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== s .

This caters for the multiaxial aspects.


88

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 stresslimits. Since terms like membrane and bending, are often used rather loosely, ASMEchose to strictly define different STRESS CATEGORIES to which different limits were tobe applied.

Basically the stresses which occur in vessel shells are divided into two categoriesPRIMARY and SECONDARY together with subcategories.

(1) PRIMARY STRESS

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

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

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

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

not concentrations. The term pa

o2cosf , 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) and Pb (ASME).

A good example of this is the bending stress in thecentral portion of a flat head due to pressure, seethe sketch of this across. This behaviour is alsoshown in Figure 4.13 of the finite element results.It excludes discontinuities and concentration andis 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 loadsor 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.


89

The symbols fm(Pm), fL(PL), fb(Pb) and fg(Q) are used to denote the categories of stress(they are not stresses, although careless usage has allowed them to become so). The BS5500 symbols are shown first in the above with the ASME symbols following. It isunfortunate that different symbols have been used.

The basic difference between a secondary stress and a primary stress is that the secondary islargely self-equilibrating, or SELF-LIMITING . It is envisaged that local yielding andsome 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. Onthe other hand a primary stress is not self-limiting and does not redistribute. Primarystresses which considerably exceed yield will result in failure or gross distortion.

To summarise

the basic Design by Analysis procedure then involves the categorisation of thecalculated stresses associated with each type of loading, evaluation of the appropriatestress intensity in each category and comparison with the basic limits in eachcategory.

STRESS LIMITS FOR THE VARIOUS CATEGORIES

STRESS INTENSITY ALLOWABLE STRESS EQUIVALENTYIELD

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

Local primary membrane, fL 1.5 f s Y

Primary membrane plus primarybending, (fL + fb )

1.5 f s Y

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

It should be noted in passing that the above limits are not always directly applicable. Forexample in the ASME code they are used in the above form for design conditions. Theseare 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, emergencyand faulted conditions. Also the design should be acceptable for any testing conditionsabove the design loads. In particular the limit on primary plus secondary stressesapplies only to the operating conditions. Otherwise k-factors are applied to the limitsgiven above (i.e. the appropriate limit is multiplied by the factor k). For example forearthquake 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 forlimited application. This refers to stresses local to attachments, supports and nozzleswhich are subject to applied loading in addition to the pressure in the vessel.


90

For attachments and supports the limits are:

The membrane stress intensity 1.2f (0.8s Y )

Membrane + bending stress intensity 2f (1.33s Y )

For nozzles and opening:

Membrane + bending stress intensity 2.25f (1.5s 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 stressintensities are determined from the principal stresses.

When we require to calculate (fm +fg ) we calculate the stresses in each category. The finalstep is to sum the stressess s s1 2 3, and in each category to find a final value of thestress intensity corresponding to (fm+fg).

THIS FINAL VALUE IS COMPARED WITH THE ALLOWABLE :- in this case theallowable is 1.5f.

8.4 THE HOPPER DIAGRAM8.4 THE HOPPER DIAGRAM

Both the ASME and the BS 5500 Standards provide helpful information to categorisecomponents when they are subject to different forms of loading. They provide a HopperDiagram which summarises the Stress Categories and the limits of stress intensity. Theone from BS 5500 is provided in these notes as Figure 8.6. In addition a range of typicalcases 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

stress categorization derivitions

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