Elastic Design Chapter 4

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ELASTIC DESIGNELASTIC DESIGN

&&

STRESSSTRESSCLASSIFICATIONCLASSIFICATION



Principal Failure Mechanisms

• Gross plastic deformation• Incremental Plastic collapse• Fatigue

– The first two are plastic failure modes – Introduces the problem of relating elastically

calculated stresses to plastic behaviour • The post-yield behaviour is not simulated by elastic

analysis



Stress Categorisation

• Technique used to relate elastic stress and plastic failuremechanisms

• Elastic stress is partitioned into three constituent

stresses – Primary stress

• associated with gross plastic deformation under

static load – Primary plus secondary stress

• associated with incremental plastic collapse under

repeated load – Primary plus secondary plus peak stress

• associated with fatigue under cyclic load



Elastic DBA procedure

• Perform linear elastic stress analysis using anappropriate stress analysis technique

– When considering gross plastic deformation• Consider only mechanical design loads

– When considering incremental plastic collapse andfatigue

• Consider mechanical and thermal operating loads



• Calculate the elastic stress distribution• Partition the elastic stress into stress categories

– Primary stress – Secondary stress – Peak stress

• Compare calculated stress with allowable stress – Specified in terms of design stress intensity

• S m in ASME Section II, Part D, Subpart 1, Tables2A, 2B and 4

– If less than the allowable, the vessel is acceptable. – If allowable exceeded, redesign is required



Gross Plastic Deformation & Primary Stress

• Gross plastic deformation or ductile collapse• Occurs when the mechanical forces and pressures are

great enough to cause a plastic failure mechanism – Yielding through the entire vessel – Formation of plastic hinge mechanism

• Elastic analysis – Can determine when first yield occurs – Does not model the post yield behaviour

• Elastic analysis cannot determine how close adesign is to the collapse state



Gross Plastic Deformation Criterion

• Collapse behaviourof a simple

rectangular beamunder combinedmembrane plusbending action isadopted as ageneral model ofplastic collapse

– Represents asection ofpressure vesselwall

InternalPressure

Bending Moment



• Little plastic stressredistributionbetween the yieldload and the limitload – If the stress

approaches yieldthere is little orno margin ofsafety againstcollapse

ybh N

σ 2

2bh M

yσ

1

ElasticDeformation

PlasticDeformation

I n i t i a l Y i e l d

L i m i t S u r f a c e

1

2/3

Membrane Dominated Response



Bending Dominated Response

• Considerable plasticstress redistribution

occurs between theyield load and thelimit load – The stress can

approach yieldand retain asignificant margin

of safety againstlimit collapse ybh

N σ 2

2bh M

yσ

1

ElasticDeformation

PlasticDeformation

I n i t i a l Y i e l d

L i m i t S u r f a c e

1

2/3



Primary Stress

• Two classes of elastic stress are associated with theequilibrium response of a vessel – Primary membrane stress – Primary bending stress



Primary Membrane Stress

• The stress associated with membrane action :

– A = 2bh is the cross sectional area of the beam

• Thus

A N

z =)(σ

bh N

m 2=σ



Primary Bending Stress

• The stress associated with bending action :

– I = 2/3 bh 3 is the second moment of area of the

beam cross section• The bending stress at the outer fibre of the beam is:

I Mz

z =)(σ

223

bh M b =σ



Yield & Limit Surfaces

• Determined by – Elastic analysis

– Limit analysis• The definitions of membrane and bending stress are

used to redraw the diagram in terms of parameters that

can be determined by elastic analysis only



Yield Condition

• Maximum stress – Sum of the membrane plus bending stress

• Initial yield occurs when the maximum stress equals theyield stress of the material.

( ) bm σ σ σ

+=max

ybm σ σ σ =+ – Thus, the yield condition can be written:

1=+ y

bm

σ

σ σ



Limit State

23

12

σ

σ

σ

σ b

Y

m

Y

⎛ ⎝ ⎜

⎞ ⎠⎟

+ ⎛ ⎝ ⎜

⎞ ⎠⎟

=

• Derived by a limit analysis based on the post-yield stressdistribution

• The terms in the equation are similar to the definitions ofprimary bending and primary membrane stress

– Substituting

M

bh

N

bhY Y σ σ 2

2

21+ ⎛

⎝ ⎜

⎞

⎠⎟ =

bh N

m 2=σ

223

bh M

b =σ



NB!• This substitution is a ”trick”

– It is not mechanically or mathematically coherent• The limit surface cannot actually be derived from the

elastic distribution – A limit analysis with post-yield stress redistribution is

required

N N

z

M M

σ y

N N

z

M M

σ y

−σy(a) Partial Plasticity (b) Fully Plastic



Primary Stress Limit

• Overall, the totalprimary stress must beless than yield underthe specified designloads – Different factors of

safety are applied,depending onwhether the stress ismembrane stress ormembrane plusbending stress

0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.2 0.4 0.6 0.8 1

y

bmσ

σ σ +

y

mσ

σ

ElasticDeformation

PlasticDeformation

L i m i t S

u r f a c e Initial Yield



Primary Membrane Stress

• Pure membrane action – Yield and limit

surfaces arecoincident – It is unsafe to

approach yield

• Primary membrane stressto 2/3 of the yield stress

– 50% margin againstlimit load collapse

σ σ m Y ≤

23

0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.2 0.4 0.6 0.8 1

y

bmσ

σ σ +

y

mσ

σ

ElasticDeformation

PlasticDeformation

L i m

i t S u r f a c e Initial Yield



Primary Bending Stress

• Pure bending action• Limit load is 50% higher

than the yield load – If the bending stress

reaches yield it hasthe same margin ofsafety against limitcollapse as themaximum membranestress

0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.2 0.4 0.6 0.8 1

y

bmσ

σ σ +

y

mσ

σ

ElasticDeformation

PlasticDeformation

L i m i t S




Primary Membrane Plus Bending Stress

• There is a significantmargin against limitcollapse right up to themaximum permissiblemembrane stress (2/3yield)

– The codes limit thetotal primarymembrane plusprimary bending stress

to yield0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.2 0.4 0.6 0.8 1

y

bmσ

σ σ +

y

mσ

σ

ElasticDeformation

PlasticDeformation

L i m i t S


( )σ σ σ m b Y + ≤



ASME Elastic Design Region

• The safetymargin is notuniform for allcombinations ofmembrane plusbending stress – At its lowest

for a highmembranestress

• Proven to be

effective indesign0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.2 0.4 0.6 0.8 1

y

bmσ

σ σ +

y

mσ

σ

ASMEDesignRegion



Primary Stress

• Gross plastic deformation is prevented by ensuring thatthe primary stress intensity does not exceed the

specified allowable value• The ASME interaction diagram and Code limits on

primary stress are based on

– Limit analysis of a rectangular beam• Rewritten in terms of elastic stress distributions• It is assumed all pressure vessel components behave in

a similar manner if – the primary membrane and primary bending stresses

are correctly identified according to their Codedefinitions



ASME III Primary Stress

• NB-3213.8 – (PD 5500 and EN 13445 have similar statements)

– “Primary stress is any normal stress or shear stressdeveloped by an imposed loading which is necessary

to satisfy the laws of equilibrium of external andinternal forces and moments.”



Primary Stress

• The primary stress is a load controlled stress• The only requirement of the (internal) primary stress

distribution is that it satisfies equilibrium with the(external) applied load – It does not have to satisfy the two other general

conditions for an elastic stress field• Compatibility of strain and deformation• Linear elastic relationship between stress and

strain.



Not Self Limiting – Primary Stress

• Unlimited plastic deformationoccurs when a load controlledstress exceeds yield

– The plastic deformation doesnot lead to a reduction instress or external load.

• NB-3213.8 – “The basic characteristic of a

primary stress is that it is notself-limiting”

– “Primary stresses whichconsiderably exceed theyield strength will result infailure or, at least, in gross

distortion.”

m

A

σ

σ

σ

εE

σy

unlimited plastic strain



Thermal Stress

• “A thermal stress is not classified as a primary stress.” – Thermal stress and strain are deformation controlled,

not load controlled• Deformation controlled stresses are intrinsically self

limiting and do not lead to unlimited gross plasticdeformation and collapse. – They are not considered when considering gross

plastic deformation



Primary Membrane Stress• “Primary membrane stress is divided into general and

local categories.” – “A general primary membrane stress is one which is

so distributed in the structure that no redistribution ofload occurs as a result of yielding.”

• NB-3213.10 Local Primary Membrane Stress :

– “… a membrane stress … associated with a[structural] discontinuity [that] would, if not limited,produce excessive distortion in the transfer of load toother portions of structure.

• “Conservatism requires that such a stress beclassified as a local primary membrane stresseven though it has some characteristics of a

secondary stress.”



Local Primary Membrane Stress

• Associated with a degree of self-limiting behaviour – Incorporates an intrinsic safety margin between first

yield and limit collapse of the component.• Permitted to exceed the 2/3 σy limit applied to general

primary membrane stress

– Allowed to approach the primary bending stress limitof σy).



Intensity of Primary Stress

• Pressure vessels experience 3-D stress distributions – In some cases simplified to 2-D

• The primary stress used in the ASME DBA procedure isthe “intensity of primary stress”

• ASME stress intensity S is defined according to theTresca yield criterion:

– PD 5500 has a similar definition

[ ]133221 ,,max σ σ σ σ σ σ −−−=S



Example: Thin Cylinder Under Internal Pressure

• Stress distribution is purely membrane – Constant through thickness

• Hoop, axial and radial directions arealso the principal directions

• The primary membrane stress intensity is

σθ

σ a

σθ

σ a

P

t Pr =θ σ

t a 2Pr =σ σ r =0

t Pr

1 =σ t 2

Pr 2 =σ σ 3=0

t t S

Pr Pr 013 =−=−= σ σ



Design for Shakedown

• The ASME Codes requires the vessel to exhibitshakedown under repeated thermal and mechanicaloperating loads – Elastic shakedown and plastic shakedown

(alternating plasticity) are permitted – Ratchetting is not permitted

• The ASME shakedown criterion is derived from a modelsimilar to that used for gross plastic deformation

– The load is an applied thermal strain range, ε R

– The response is deformation controlled



• Outer fibre of wall thermallycycled

– ε

=0 toε

=ε

R toε

=0• Loading

– Yield at A

– Plastic deformation:perfectly plastic material• At the end of the half cycle,

B, the total strain is ε R

E

σ

ε

σ y

σ y

E

σr

O

A B

C

εRStrainRange

ε=εRε=0



• Unloading – Strain reduced to

zero – Material elastically

unloads from B• At the end of the load

cycle, C , a residualcompressive stress σ r isestablished

E

σ

ε

σ y

σ y

E

σr

O

A B

C

εRStrainRange

ε=εRε=0

R yr E ε σ σ −=



Shakedown Elastic Stress Range

• Maximum strain rangewith no unloading yield

• E ε R is treated as an

elastic stress range ,σ

R• The condition for

shakedown is

Y R E σ ε 2=

σ

ε

σy

σy

r

o

A B

C

StrainRange

εR

E

Y R σ σ 2≤



Shakedown & Secondary Stress

• The shakedown analysis includes – Compatibility condition

– Elastic material model• The shakedown elastic stress range must include

– All load controlled or primary stress

– All displacement controlled or secondary stress



Secondary Stress

• NB-3213.9 – PD 5500 and EN 13445 have similar statements

• “Secondary stress is a normal stress or a shear stressdeveloped by the constraint of adjacent material or byself-constraint of the structure."

– Deformation controlled stress – Determined by the requirements of

• Compatibility of strain and deformation

• Linear elastic material – All general thermal stresses are secondary stresses.



Self Limiting – Secondary Stress

• Deformation controlled – The plastic deformation reduces the load acting on the

bar and ductile rupture is not expected to occur directlyfrom application of a secondary stress

d

L

σ

σ A

σ

ε

E

σy

d

L

• NB-3213.8 – “The basic characteristic

of a secondary stress isthat it is self-limiting.Local yielding and minordistortions can satisfy theconditions which causethe stress to occur andfailure from oneapplication of the stress is

not to be expected.

I i f P i l S d S



Intensity of Primary plus Secondary Stress

• Multiaxial stress systems – Elastic stress range used in the shakedown

assessment is the intensity of primary plus secondarystress

• Primary plus secondary stress intensity



Fatigue & Peak Stress

• The constituent of the elastic stress due to local stressconcentration effects – Local structural discontinuities

• Small holes• Fillets• Welds

• Very localised – Does not cause any noticeable distortion of the vessel – Does not affect the global response

• Does not contribute to the limit or shakedownresponse

• Objectionable only as a possible source of a fatiguecrack or a brittle fracture



Peak Stress

• NB-3213.9 – (PD 5500 and EN 13445 have similar statements).

This states:• “Peak stress is that increment of stress which is additive

to the primary plus secondary stresses by reason of localdiscontinuities or local thermal stresses …”

• “The basic characteristic of a peak stress is that it doesnot cause any noticeable distortion and is objectionableonly as a possible source of a fatigue crack or a brittle

fracture.” – Peak stress is not considered when assessing global

failure mechanisms such a gross plastic collapse or

incremental plastic collapse.

Summary of Code Stress Categories



Summary of Code Stress Categories

• Primary – General primary membrane

– Local primary membrane – Primary bending

• Secondary

– Secondary membrane and bending stress not definedin ASME and PD

– EN13445 defines membrane and bending but states

secondary membrane plus bending is used in DBA• Peak stress

Stress Notation



Stress Notation

FMem + bend: Q

Mem: Q m

Bend: Q b

P bP LP mEN

f pf gf bf Lf mPD5500

FQP bP LP mASME

PeakStress

SecondaryStress

PrimaryBending

Stress

LocalPrimary

MembraneStress

GeneralPrimary

Membrane

Stress

Allowable Stress



Allowable Stress

• Defined in terms of a specified design stress for thegiven material and design or operating temperature

– ASME design stress intensity, S m, is tabulated inSection II, Part D, Subpart 1, Tables 2A, 2B and 4 . – The tabulated value of S m for most pressure vessel

steels has a value of around S m=2/3 σy.



ASME III Stress Limits

• NB-3220 Stress Limits for Other Than Bolts – Stress limits under Design Loadings are specified in

NB-3221 – Special Stress Limits are given in NB-3227 – Level A Service Limits are defined in NB-3222



ASME III Stress Limits

s y3 S mNB-3222.2Primary plus secondary

(P m + P b + Q ) or ( P L + P b + Q )

s y1.5 S mNB-3221.3Primary membrane plus bending

(P m + P b ) or ( P L + P b )

s y1.5 S mNB-3221.2Local primary membrane P L

2/3 s yS mNB-3221.1General primary membrane P m

Allowable(wrt Yield Stress s y)

Allowable(wrt Design Stress S m )

Stress Intensity

Fig NB-3221-1 Hopper Diagram: Primary Stress



Fig. NB 3221 1 Hopper Diagram: Primary StressPrimaryStress

Category General Membrane Local Membrane Bending

Description(For ex-

amples, seeTable

NB-3217-11)

Average primarystress across solid

section.Excludes

discontinuities andconcentrations.

Produced only bymechanical loads.

Average stressacross any solid

section.Considers

discontinuities butnot concentrations.Produced only bymechanical loads.

Component ofprimary stressproportional todistance from

centroid of solidsection.Excludes



Note [(1)]

Symbol[Note (2)] Pm PL Pb

Pm

P L

P L Pb

Sm

1.5S m

1.5S m+

Combinationof stresscomponentsand allowablelimits ofstressintensities

• Maximum primarystress intensity isyield limited toprevent gross plastic

deformation ym P σ

32≤

( ) ( ) yb Lbm

P P or P P σ ≤++

Fig NB-3222-1: Hopper Diagram: Primary plus



Fig. NB-3222-1: Hopper Diagram: Primary plusSecondary Stress

• Primary plussecondarystress

intensityunder Level Aservice limitslimited to

twice yield toensureshakedown

PrimaryStressCategory General Membrane Local Membrane Bending

SecondaryMembrane

plus BendingPeak

Description(For ex-

amples, seeTable

NB-3217-1)


section.Excludes

discontinuities andconcentrations.Produced by

pressure andmechanical loads.


section.Considers effectsof discontinuities

but notconcentrations.

Produced bypresure andmechanical loads,

including inertiaearthquake effects


centroid of solidsection.

Excludes effects ofdiscontinuities and

concentrations.Produced bypressure and

mechanical loads,including inertia

earthquake effects.

Self-equilibriatingstress necessary tosatisfy continuity of

structure.Occurs at structuraldiscontinuities. Can

be caused by

presure,mechanical loads,or by differential

thermal expansion.Excludes local

stressconcentrations.

(1) Increment added toprimary or secondary

stress by aconcentration (notch).

(2) Certain thermalstresses which may

cause fatigue but notdistortion.

Symbol

[Note (2)]Pm P L Pb Q F

3S m

S a

Pb QPL + +

Pb QP L + + + F

Combinationof stresscomponentsand allowablelimits ofstressintensities

[Note 3] [Note 3] [Note 3]

( ) ( ) yb Lbm Q P P or Q P P σ 2≤++++

ASME VIII Allowable stress



ASME VIII Allowable stress

2 y 3 S mPrimary plus secondary

( P m + P b + Q ) or ( P L + P b + Q )

k y1.5 k S mPrimary membrane plusbending

( P m + P b) or ( P L + P b)

k y1.5 k S mLocal primary membrane P L

2/3 k y k S mGeneral primary membrane P m

Allowable StressIntensity

(wrt Yield Stress σy)


(wrt Design Stress Sm)

Stress Classification – Design stress denoted S m

• Fatigue: total stress ( P L+P b+Q+F ) should be less thanallowable fatigue stress intensity range, Sa

k Factor



• Depends on the type of load – Standard load combinations

• Design pressure• Dead load• Weight of contents and insulation

• Imposed loads from mechanical equipment• External attachment loads

– k = 1

k Factor



• Depends on the type of load – Exceptional loads

• Earthquake

• Wind load• Wave load

– k = 1.2

• Special limits are also stipulated for hydraulic testing

Normal Design Loads: k = 1



g

2 σ y Primary plus secondary

( P m + P b + Q ) or ( P L + P b +Q)

σ y Primary membraneplus bending

( P m + P b) or ( P L + P b)

σ y Local primarymembrane P L

2/3σ y General primary

membrane P m

AllowableStress

Intensity

Stress Classification

ASME Hopper DiagramS d




SecondaryMembrane

plus BendingPeak

Description(For ex-

amples, seeTable

4-120.1)


section.Excludes




section.Considers

discontinuities butnot concentrations.Produced only bymechanical loads.


centroid of solidsection.

Excludesdiscontinuities and

concentrations.Produced only bymechanical loads.

Self-equilibriatingstress necessary tosatisfy continuity of

structure.Occurs at structural

discontinuities.Can be caused bymechanical load or

by differentialthermal expansion.

Excludes localstress

concentrations.

(1) Increment added toprimary or secondary

stress by aconcentration (notch).

(2) Certain thermalstresses which may

cause fatigue but notdistortion of vessel

shape.

Symbol[Note (3)]

Pm P L Pb Q F

Pm

P L

PL

Pb

Pb

QP L

Pb QP L

kS m

1.5kS m

1.5kS m

3S m

S a

+

+ +

+ + + F

Note (1)

Note (2)

Combinationof stresscomponentsandallowablelimits ofstressintensities

Use design loads

Use operating loads

PD5500 Allowable Stress



• Limits similar to ASME – No specific reference to load factor k – Design stress is represented by the symbol f

2 σ y3 f Primary plus secondary

( f m + f B +f qQ) or ( f L + f b +f q )

σ y1.5 f Primary membrane plus bending

( f m + f b) or ( f L + f b)

σ y1.5 f Local primary membrane f L

2/3 σ y f General primary membrane f m




(wrt Design Stress f )

Stress Classification

Exceptional Loads



• Wind and earthquake conditions – “All allowable tensile stresses and stress intensities

(membrane or bending, primary or secondary) maybe increased by a factor of 1.2…”• Load factor applies to both primary and secondary stress

PD 5500 Hopper DiagramPrimaryS Secondary




Secondary

Description(For ex-

amples, seeTable A..1)

Average primary stressacross solid section.Excludesdiscontinuities andconcentrations.Produced only bymechanical loads.

Average stress acrossany solid section.Considersdiscontinuities but notconcentrations.Produced only bymechanical loads, bydefinition includes fm in those cases whereit is present.

Component of primarystress proportional todistance from centroid ofsolid section.Excludes discontinuitiesand concentrations.Produced only bymechanical loads.

Self-equilibriating stress necessary tosatisfy continuity of structure.Occurs at structural discontinuities.Can be caused by mechanical load orby differential thermal expansion.Excludes local stress concentrations.

Symbol[see note (3)]

f m f L f b

f m

f L

f

1.5 f

f L f b 1.5 f +

Note (1)

Combination

of stresscomponentsand allowablelimits ofstressintensities

f g

f bf L 3.0 f + + f g

f b 1.5 f

EN 13445 Allowable Stress



• Stress limits similar to other codes – Design stress denoted f

• Classified stresses defined in terms of equivalent stress – Tresca or von Mises

• Primary membrane plus primary bending stress is

represented by the symbol ( σ eq ) P

( σ eq ) P =[( σ

eq ) P m + ( σ eq ) P b] or [ ( σ

eq ) P L + ( σ eq ) P b]

EN 13445 Allowable Stress



2 σ y3 f Primary plus secondary

( σ

eq )P + Q

kσ

y1.5f Primary membrane plus

bending

( σ eq )P

k σ y1.5 f Local primary membrane

( σ eq )P L

2/3 k σ

y f General primary membrane

( σ eq )P m

Allowable EquivalentStress


Allowable EquivalentStress

(wrt Design Stress S m)

(Equivalent)StressClassification

Exceptional Loads



• “The value of f to be retained shall be that consistentwith the type of loading condition considered (normaloperation, exceptional operation, proof test) … at thecalculation temperature of that condition.”

Stress Categories

EN 13445 Hopper Diagram



Primary stress

Generalmembranestress

Localmembrane

stress

Bendingstress

Secondarymembrane + bending

stressPeak s tress

Description

(For practicalexamples,

seeTable C-2)

Primary meanstress calculatedacross the wall

thickness withouttaking into accountdiscontinuities and

concentrations.

Caused only bymechanical loads.

Primary meanstress calculatedacross the wall

thickness withouttaking into accountdiscontinuities, butnot concentrations.

Caused only bymechanical loads.

Primary stresscomponent

proportional todistance from

centroid of the solidwall section.

Does not include

discontinuities andconcentrations. Caused only by

mechanical loads.

Self-equilibriating stressnecessary to satisfy the

continuity of the structure.Occurs at large

discontinuities but doesnot include stress

concentrations.

Can be caused by bothmechanical loads and

thermal effects.

(1) Addition toprimary or

secondary stressbecause of

stressconcentration.

(2) Certainthermal stresseswhich may cause

fatigue but notdistortion.

Symbol Pm PL Pb F

assessmentagainststatic

loading

Q(=Q + Q )m b

(σ )eq P m f

(σ )eq P L 1,5f

(σ )eq P 1,5f

(σ )eq P+Q 3f

fatigueassessment

(only ifrequired)

(σ )eq P+Qor

max (Δσ )i eq P+Q+F (Δσ )

= design loads

= operating loads

Stress Classification for Typical Cases



• Different allowable values are defined for primary stressand primary plus secondary stress

– It is essential that the calculated elastic stress iscorrectly classified – This is one of the most difficult problems encountered

in DBA• Potentially critical effect on the final design

– If primary stresses are classified as secondary• The design may be unsafe

– If secondary stresses are classified as primary• The design will be over-conservative



• The codes provide explicit classification guidance forsome typical vessel geometries and load conditions: – ASME III Table NB-3217-1 – ASME VIII Table 4.120.1 – EN13445 Table C-2 – PD5500 Table A.1

• Other configurations require the designer to define theappropriate stress classes on the basis of the codedefinitions of primary, secondary and peak stress – The codes are in general agreement on stress

classification although there are some differences – Four cases are presented to illustrate stress

categorisation

Cylindrical or spherical shell, internal pressure



• The main shell of a vessel remotes from any structuraldiscontinuities such as heads and nozzles – A thin shell cylinder or sphere exhibits a purely

membrane response• There is no variation in stress through thickness.



– Thicker shells have a stressvariation, or gradient, throughthickness

– Stress Classification• Membrane stress intensity (or equivalent stress)

– General primary membrane stress, Pm or fm• Gradient of stress intensity (or equivalent stress)through plate thickness

– Secondary stress, Q or f g

– EN 13445 specifically defines secondary bendingstress, Q b

Any shell or head under thermal load



• Thermal loads only give rise to secondary stress• ASME and PD5500

– Both the membrane stress and bending stress areclassified as secondary stress Q or f g in• EN13445 differentiates between secondary responses

– Membrane stress = secondary membrane Qm

– Bending stress = secondary bending, Qb

Shell or end near an opening, internal pressure



• The presence of an opening in apressurised shell leads tocomplex stress classification – The shell is locally subject to

both membrane and bendingaction

– A peak stress may arise atlocal stress raisers

Shell or end near an opening, internal pressure



• Membrane stress – Local primary membrane stress, P L or f L

• Bending stress – Secondary stress, Q or f b – EN13445 specifies secondary bending stress, Qb

• Stress at local discontinuity – ASME: peak stress F – PD: peak stress. Refers designer to the definition of

peak stress and fatigue assessment

– EN: No reference to the presence of a localdiscontinuity or peak stress

Flat head, internal pressure



• Respond to a pressure load in a similar manner totransversely loaded circular plate – Large bending stresses usually present

• Classification depends on the extent to which theshell and shell-head transition region influence thestructural response




• Centre of the head – Membrane stress

• General primary membrane, P m

or f m

– Bending stress• Primary bending, P b or f b

• Classification at junction differs between the Codes• ASME and PD:

– Membrane stress• Local primary membrane, P L or f L

– Bending stress• Secondary, Q or f g




• ASME Footnote – Elastic compatibility edge bending moments may

reduce the bending stress at the centre of the plate

• If yielding occurs at the edge, the momentconstraint is lost

• The stress at the centre would be greater than

calculated by elastic analysis• The discontinuity bending stress behaves like a

primary stress and is classified P b

• No such qualification is stated in PD5500 or EN




• EN: – Membrane stress

• General primary membrane, P m – Bending stress – Secondary bending, Qb

• There is a difference in classification of membrane stressbetween EN 13445 and the other codes for this case

• There is also a possible difference between ASME andthe others for the classification of bending stress

Elastic Design Chapter 4

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