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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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16587 Pressurised Systems
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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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16587 Pressurised Systems
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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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16587 Pressurised Systems
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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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81
(( )) (( )){{ }}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 ,
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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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16587 Pressurised Systems
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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
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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.
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16587 Pressurised Systems
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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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16587 Pressurised Systems
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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.
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16587 Pressurised Systems
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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
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.
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16587 Pressurised Systems
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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.
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16587 Pressurised Systems
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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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16587 Pressurised Systems
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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