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2.1
2. Design by Analysis
The aim of this section is to summarise issues related to the current use of design by analysis in
order to put the new European rules in context. The concept of design by analysis was first
formulated in the US ASME Pressure Vessel and Boiler Code in the early 1960’s; with almost forty
years of use various critical problem areas have arisen, most of which have been addressed in the
new European rules. These problem areas are discussed in the following since they highlight
implicit difficulties with an apparently simple and straightforward set of design rules. In the
following the approach devised by ASME is briefly summarised, followed by a description of the
usual methods by which the rules are implemented and a discussion of the problem areas which
arise. After this the differences in implementation of design by analysis rules in the European
Standard are described.
2.1 Design by analysis: the current Stress Categarisation route
The design by analysis procedure is intended to guard against eight possible pressure vessel failure
modes by performing a detailed stress analysis of the vessel. The failure modes considered are:
1. Excessive elastic deformation including elastic instability.
2. Excessive plastic deformation.
3. Brittle fracture.
4. Stress rupture/creep deformation (inelastic).
5. Plastic instability - incremental collapse.
6. High strain - low cycle fatigue.
7. Stress corrosion.
8. Corrosion fatigue.
Most of the design by analysis guidelines given in the codes relates to design based on elastic
analysis – this is the so-called elastic route. Essentially it was recognised when the rules were being
developed that only elastic stress analysis was feasible. In the 1960s, most designers were restricted
to linear elastic stress analysis, and in the case of pressure vessel design most analysis was defined
in terms of elastic shell discontinuity theory (also known as the influence function method). The
nature of elastic shell analysis impinges significantly upon the way the above failure modes are
treated in the Code. Thus, rules were developed to help the designer guard against the various
failure mechanisms using elastic analysis alone. These guidelines guard against three specific
failure modes - gross plastic deformation, incremental plastic collapse (ratchetting) and fatigue.
These failure modes are precluded by failure criteria based on limit theory, shakedown theory and
fatigue theory respectively. It is essential to appreciate at the beginning, the excessive plastic
deformation and incremental plastic collapse cannot be dealt with simply in an elastic analysis, as
the failure mechanism is inelastic. In addition, the type of loading causing the stress can
significantly affect the level of permissible stress. Ideally, these inelastic failure modes should be
assessed by an appropriate analysis which adequately models the mechanism of failure.
In this approach the designer is required to classify the calculated stress into primary, secondary and
peak categories and apply specified allowable stress limits. The magnitude of the allowable values
assigned to the various stress categories reflect the nature of their associated failure mechanisms,
therefore it is essential that the categorisation procedure is performed correctly.
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Stress categorisation (sometimes, classification) is probably the most difficult aspect of the design
by analysis procedure and, paradoxically, the problem has become more difficult as stress analysis
techniques have improved. When the design by analysis procedure was introduced, the dominant
analysis technique in pressure vessel design was thin shell discontinuity analysis or the influence
function method. This is reflected in the definitions of stress categories given in the Codes, which
are based on the assumption of shell theory stress distributions; membrane and bending stress. It is
therefore difficult to equate the calculated stresses and the code categories unless the design is
based on shell analysis. The various stress categories are described first in the following:
2.1.1 Stress Categories
The object of the elastic analysis is to ensure that the vessel has adequate margins of safety against
three failure modes: gross plastic deformation, ratchetting and fatigue. This is done by defining
three classes or categories of stress, which have different significance when the failure modes are
considered. These three stress categories are assigned different maximum allowable stress values in
the code: the designer is required to decompose the elastic stress field into these three categories
and apply the appropriate stress limits.
The total elastic stress which occurs in the vessel shell is considered to be composed of three
different types of stress primary, secondary and peak. In addition, primary stress has three specific
sub-categories. The ASME stress categories and the symbols used to denote them in the code are
given below;
(1) Primary Stress
General Primary Membrane Stress, Pm
Local Primary Membrane Stress, PL
Primary Bending Stress, Pb
(2) Secondary Stress, Q
(3) Peak Stress, F
and depend on location, origin and type. Before we can give a proper definition of these stresses, we
must first give some terminology:
Gross Structural Discontinuity: A gross structural discontinuity is a source of stress or strain
intensification that affects a relatively large portion of a structure and has a significant effect on the
overall stress or strain pattern or on the structure as a whole.
Examples of gross structural discontinuities are:
∗ end to shell junctions,
∗ junctions between shells of different diameters or thickness,
∗ nozzles.
Local Structural Discontinuity: A local structural discontinuity is a source of strain intensification
that affects a relatively small volume of material and does not have a significant effect on the
overall stress or strain pattern or on the structure as a whole.
Examples of local structural discontinuities are:
3
∗
∗
∗
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small fillet radii,
small attachments,
partial penetration welds.
Normal Stress: The normal stress is the component of stress normal to the plane of reference; this is
also referred to as direct stress.
Usually the distribution of normal stress is not uniform through the thickness of a part, so this stress
is considered to be made up in turn of two components one of which is uniformly distributed and
equal to the average value of stress across the thickness of the section under consideration, and the
other of which varies with the location across the thickness.
Shear Stress: The shear stress is the component of stress acting in the plane of reference.
Membrane Stress: The membrane stress is the component of stress that is uniformly distributed and
equal to the average value of stress across the thickness of the section under consideration.
Bending Stress: The bending stress is the component of stress that varies linearly across the
thickness of section under consideration.
With this terminology as background, we now can define primary, secondary and peak stresses
properly.
Primary Stresses: A primary stress is a stress produced by mechanical loading only and is so
distributed in the structure that no redistribution of load occurs as a result of yielding. It is a normal
stress or a shear stress developed by the imposed loading, that is necessary to satisfy the simple
laws of equilibrium of external and internal forces and moments. The basic characteristic of this
stress is that it is not self-limiting. Primary stresses that considerably exceed the yield strength will
result in failure, or at least in gross distortion. A thermal stress is not classified as a primary stress.
Primary stresses are divided into ‘general’ and ‘local’ categories. The local primary stress is defined
hereafter.
Typical examples of general primary stress are:
∗ The average stress in a cylindrical or spherical shell due to internal pressure or to distributed
live loads,
∗ The bending stress of a flat cover without supporting moment at the periphery due to internal
pressure.
Primary Local Membrane Stress: Cases arise in which a membrane stress produced by pressure or
other mechanical loading and associated with a primary together with a discontinuity effect
produces excessive distortion in the transfer of load to other portions of the structure.
Conservatism requires that such a stress be classified as a primary local membrane stress even
though it has some characteristics of a secondary stress. A stressed region may be considered as
local if the distance over which the stress intensity exceeds 110% of the allowable general primary
membrane stress does not extend in the meridional direction more than 0.5 times (according to
BS5500 - 1 time according to ASME and CODAP) the square root of R times e and if it is not
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closer in the meridional direction than 2.5 times the square root of R times e to another region
where the limits of general primary membrane stress are exceeded. R and e are respectively the
radius and thickness of the component.
An example of a primary local stress is the membrane stress in a shell produced by external load
and moment at a permanent support or at a nozzle connection.
Secondary Stresses: Secondary stresses are stresses developed by constraints due to geometric
discontinuities, by the use of materials of different elastic moduli under external loads, or by
constraints due to differential thermal expansion. The basic characteristic of secondary stress is that
it is self-limiting. Local yielding and minor distortions can satisfy the conditions that cause the
stress to occur and failure from one application of the stress is not to be expected.
Examples of secondary stresses are the bending stresses at dished end to shell junctions, general
thermal stresses.
Peak stresses: Peak stress is that increment of stress which is additive to the primary-plussecondary stresses by reason of local discontinuities or local thermal stress including the effects (if
any) of stress concentration.
The basic characteristic of peak stresses is that they do not cause any noticeable distortion and are
only important to fatigue and brittle fracture in conjunction with primary and secondary stresses. A
typical example is the stress at the weld toe.
2.1.2 Stress intensity
Pressure vessels are subject to multiaxial stress states, such that yield is not governed by the
individual components of stress but by some combination of all stress components. Most Design by
Formula rules make use of the Tresca criterion but in the DBA approach a more accurate
representation of multiaxial yield is required. The theories most commonly used to relate multiaxial
stress to uniaxial yield data are the Mises criterion and the Tresca criterion. ASME chose the Tresca
criterion for use in the design rules since it is a little more conservative than Mises and sometimes
easier to apply.
For simplicity we will consider a general three-dimensional stress field described by its principal
stress components, which we will denote σ1, σ2 and σ3, and define the principal shear stresses:
1
τ 1 = (σ 2 − σ 3 )
2
1
τ 2 = (σ 3 − σ 1 )
2
1
τ 3 = (σ 1 − σ 2 )
2
According to the Tresca criterion yielding occurs when
1
τ = max(τ 1, τ 2 , τ 3 ) = σ Y
2
where σY is the uniaxial yield stress obtained from tensile tests.
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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
was defined:
Stress differences, S12, S23 and S31 are equated to twice the principal shear stress given above, such
that:
S12 = (σ 1 − σ 2 ) = τ 3
S23 = (σ 2 − σ 3 ) = τ 1
S31 = (σ 3 − σ 1 ) = τ 2
The Stress Intensity, S is then defined as the maximum absolute value of the stress differences, that
is S = max (|S12|, |S23|, |S31|), so that the Tresca criterion reduces to:
S = σY
Once an analysis has been performed, the Stress Intensity for each stress category is evaluated and
used in the design stress limits.
2.1.3 Stress limits
The primary stress limits are provided to prevent excessive plastic deformation and provide a factor
of safety on the ductile burst pressure (ductile rupture) or plastic instability (collapse). The primaryplus-secondary stress limits are provided to prevent progressive plastic deformation leading to
collapse, and to validate the application of elastic analysis when performing the fatigue analysis.
The allowable stresses in the Codes are expressed in terms of design stress Sm. The tabulated values
of Sm given in the Code are based on consideration of both the yield stress and ultimate tensile
strength of the material. Sm is notionally two-thirds of the "design" yield strength σY. Code
allowable stresses for primary and secondary stress combinations are shown in the following table
in terms of both Sm and σY.
ALLOWABLE STRESS
STRESS INTENSITY
2/3 k
σY
General primary membrane, Pm
k Sm
Local primary membrane, PL
1.5 k Sm
k σY
Primary membrane plus bending
1.5 k Sm
k σY
3 Sm
2 σY
(Pm + PB) or (PL + Pb)
Primary plus secondary
(Pm + PB + Q) or (PL + Pb + Q)
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In addition to these allowables, when fatigue is considered relevant the total sum of (PL+Pb+Q+F)
should be less than an allowable fatigue stress intensity range, Sa. The value of the k factor depends
on the load combinations experienced by the vessel. For load combinations including design
pressure, the dead load of the vessel, the contents of the vessel, the imposed load of the mechanical
equipment and external attachment loads the k factor has a value of 1. When earthquake, wind load
or wave load are added to the above, a k value of 1.2 is used. Special limits are also stipulated for
hydraulic testing. Under design load conditions k = 1 and the maximum value of the primary stress
combinations is yield the yield stress of the material. Primary stress is yield limited to ensure gross
plastic deformation does not occur. The primary plus secondary stress combinations have a much
higher allowable stress: twice the yield stress of the material. Primary plus secondary stress is
limited to ensure shakedown of the vessel.
Because of the different allowable values for primary and primary plus secondary stress, it is
essential that the calculated elastic stress is correctly decomposed into the various categories. This
is one of the most difficult problems encountered in DBA and has potentially critical effect on the
final design. If primary stresses are classified as secondary the design may be unsafe, whilst if
secondary stresses are classified as primary the design will be over-conservative. The code provides
explicit classification guidance for certain typical vessel geometries and load through Table 4.120.1
Classification of stresses for some typical cases. In situations other than these cases the designer
must rely on the basic code definitions of primary, secondary and peak stress and his own
judgement to properly classify the elastic stress. In fact some of the stress classifications
recommended in Table 4.120.1 have been in doubt for some time, and must be used with care.
2.2 Design by Analysis: the ASME inelastic route
The ASME VIII Division 2 rules for inelastic analysis are given in Appendix 4-136 Applications of
Plastic Analysis. These rules “provide guidance in the application of plastic analysis and some
relaxation of the basic stress limits which are allowed if plastic analysis is used.”
The rules for inelastic analysis considered here pertain to calculation of permissible loads for gross
plastic deformation only. Rules are given in the Code for shakedown analysis but in practice
shakedown analysis is difficult and it is simpler to apply the 3Sm limit to an elastic analysis.
Two types of analysis may be used to calculate allowable loads for gross plastic deformation: limit
analysis and plastic analysis.
Limit analysis is used to calculate the limit load of a vessel. By definition, the analysis is based on
small deformation theory and an elastic-perfectly plastic (or rigid-perfectly plastic) material model.
Plastic analysis is used to determine the plastic collapse load of a vessel. The analysis is based on a
model of the actual material stress-strain relationship and may assume small or large deformation
theory as required. Material models can vary in complexity (or degree of approximation) from
simple bilinear kinematic hardening models to more complex curves defining the actual stress-strain
curve in a piecewise continuous manner.
Including strain hardening in the analysis may give a higher plastic collapse load than the limit load
but in the design by analysis procedure the allowable load is dependent on the criterion of plastic
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collapse used. Including large deformation effects in the analysis may increase or decrease the
calculated allowable load depending on the geometry of the vessel. Some structural configurations
exhibit geometrical strengthening when non-linear geometry is considered whilst others exhibit
geometric weakening.
The expression ‘plastic collapse load’ is to some extent a misnomer, as a real vessel may not
physically collapse at this load level, hence the ‘plastic collapse load’ is often referred to simply as
the ‘plastic load’.
2.2.1 Limit analysis
ASME VIII Division 2 Appendix 4-136.3 Limit Analysis states:
“The limits on general membrane stress intensity ...local membrane stress intensity ... and primary
membrane plus primary bending stress intensity ... need not be satisfied at a specific location if it
can be shown by limit analysis that the specified loadings do not exceed two-thirds of the lower
bound collapse load. The yield strength to be used in these calculations is 1.5Sm.”
Thus allowable load Pa is Pa =
2
PLim where PLim is the limit load of the vessel.
3
Clearly, if the limit load can be calculated this procedure is much simpler to apply than the elastic
analysis stress categorisation procedure. However, there are two additional requirements that must
be satisfied when applying this approach. Firstly, the effects of plastic strain concentrations in
localised areas of the structure such as points where plastic hinges form must be assessed in light of
possible fatigue, ratchetting and buckling failure. Secondly, the design must satisfy the minimum
wall thickness requirements given in the design by rule section of the Code. In effect, the design by
rule formulae for wall thickness have priority over design by analysis calculations.
2.2.2 Plastic analysis
ASME VIII Division 2 Appendix 4-136.5 Plastic Analysis states:
“The limits of general membrane stress intensity ...local membrane stress intensity ... and primary
membrane plus primary bending stress intensity ... need not be satisfied at a specific location if it
can be shown by limit analysis that the specified loadings do not exceed two-thirds of the plastic
analysis collapse load determined by application of 6-153, Criterion of Collapse Load (Appendix 6)
[Mandatory Experimental Stress Analysis], to a load deflection or load strain relationship obtained
by plastic analysis.”
Thus allowable load Pa is Pa =
2
PP , where PP is the plastic load of the vessel.
3
Calculating plastic loads is more problematic than calculating limit loads as no rigorous definition
of what constitutes a plastic load is given. Instead, the twice elastic slope criterion as used in
experimental analysis is prescribed.
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2.3 Analysis methods for the ASME approach
Design by analysis procedures do not specify particular implementation tools: it has been left to the
analysts to choose the technique they feel most appropriate. Shell discontinuity analysis was the
primary tool in the early days of design by analysis, where stresses could easily be categorised in
terms of shell-type membrane and bending stress. By now analysis techniques have developed
significantly and although shell discontinuity analysis is still used very often in structural analysis,
it is replaced more and more by computer based numerical methods. The most widely used
technique in contemporary pressure vessel design is the finite element method, a powerful
technique allowing the detailed modelling of complex vessels. Shell discontinuity analysis and the
finite element method are discussed in relation to pressure vessel design by analysis in the
following sections.
2.3.1 Shell discontinuity analysis
Shell discontinuity analysis was the primary means of stress analysis in the early days of design by
analysis procedures. Although largely replaced by finite element analysis, shell discontinuity
analysis remains a useful tool for simple geometries, and indeed many engineering software
companies still supply programs for discontinuity analysis.
V
Forces
H
u
q
Semi-infinite cylinder
Cone
Displacements
v
Hemisphere
Flat end
Figure 2.1.: Shell discontinuity forces and moments.
Shell discontinuity analysis is primarily used to evaluate shell membrane and bending stresses for
axisymmetric vessels under internal pressure. It makes use of the fact that typical vessel
configurations are composed of regular parts - spheres, cylinders, cones and flat ends in particular.
For pressure loading, simple regular shapes exhibit mainly membrane stress. However, at junctions
local bending (and additional membrane) stresses are generated. These stresses are called
discontinuity stresses for obvious reasons. Shell discontinuity analysis allows these junction
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2.9
stresses, and their effect in the vessel, to be readily calculated using a simple engineering force
method.
This force method uses analytical solutions for the local bending and shear stress close to junctions
which allow so-called edge forces and moments to be related to edge displacements and rotations,
Figure 2.1. These edge relations are evaluated for each part of the vessel and then assembled at
junctions. Continuity of displacement and rotation between parts then allows the edge forces and
moments at the junction to be derived and finally the stresses in the various parts can be calculated.
2.3.2 Finite Elements for Pressure Vessel Analysis
In creating a model, element selection and mesh definition are crucial aspects of finite element
analysis. The type of element used in a finite element analysis for pressure vessel design can greatly
influence the design procedure, so a brief overview is given here. Most commercial programs
include large finite element libraries, however, in pressure vessel design the most common element
types are 3-D solid, axisymmetric and shell elements.
2.3.2.1 3-D Solid Elements
Solid (or continuum) elements are based on the mathematical theory of elasticity, which describes
the behaviour of a deformable component under load assuming small deformation and strain. The
most general theory is three dimensional, but under specific circumstances certain two dimensional
reductions are possible. 3-D solid elements are used to model real three-dimensional structures
such as the part model of a nozzle-vessel intersection shown in Figure 2.2.
Figure 2.2: 3-D solid model of a nozzle-vessel intersection.
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Elastic 3-D solid elements are based on 3-D elasticity theory. A general system of forces acting on a
three dimensional elastic body sets up internal forces within the body, which vary with position
throughout the body. The state of stress at a point in the body is fully defined by six components:
Direct stresses: σx, σy, σz
Shear stresses: τxy, τyz, τzx
σY
as illustrated in Figure 2.3.
Three degrees of freedom are defined at each node
of a 3-D solid element: orthogonal displacements
ux, uy and uz. Displacement throughout the domain
of the element is defined in terms of these nodal
displacements by the interpolation functions used in
the element formulation. Most commercial finite
element packages offer solid elements based on two
different orders of interpolation:
•
τ XY
τYZ
τ XY
σX
τYZ
σZ
Y
Z
τ ZX
τZX
X
Figure 2.3: Stresses acting on a differential cube
of 3-D elastic continua.
8 node linear element: Figure 2.4.
Each element has 24 (8 node x 3) associated degrees of freedom.
uy
8 NODE LINEAR
ux
uz
ISOPARAMETRIC
3-D SOLID BRICK
ELEMENT
Deformed geometry
Y
X
Z
Original
geometry
ELEMENT DEFORMATION
Figure 2.4: Linear 3-D solid brick element
Edges/sides must be straight/ plane before deformation, as the geometry is defined by linear
equations.
When the element is loaded it deforms such that the sides remain straight, as the displacements of
the element are also defined by linear equations.
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2.11
20 node quadratic element: Figure 2.5.
ORIGINAL GEOMETRY
STRAIGHT SIDES
Deformed geometry
20 NODE QUADRATIC
ISOPARAMETRIC
3-D SOLID BRICK
ELEMENT
Original
geometry
ELEMENT DEFORMATION
ORIGINAL GEOMETRY
QUADRATICALLY CURVED
SIDES
Figure 2.5: Quadratic 3-D solid brick element.
Each element has 60 (20 nodes x 3) associated degrees of freedom.
Edges/sides may be defined as quadratic curves/surfaces, as the geometry of the element is defined
by quadratic interpolation. This means that the 20 node brick can more closely model the true
shape of a curved body than the simpler 8 node element.
When the 20 node element is loaded the sides may deform quadratically, as the displacements of the
element are also defined by quadratic interpolation.
Both 8 node and 20 node brick elements may be
degenerated to give elements of wedge and
tetrahedron shape by defining two or more nodes
at the same position, as shown for the 8 node
element in Figure 2.6. In general, degenerate
elements do not perform as well as the brick
elements, however they can be used to model areas
of a structure which cannot be meshed using brickshaped elements.
Solid elements based on tetrahedral geometry (as
opposed to being degenerate bricks) are also
available in many finite element programs.
4 NODES
BRICK
2 NODES
2 NODES
TETRAHEDRON
WEDGE
OR
PRISM
Figure 2.6: Degenerating brick geometry to give
wedge and tetrahedron geometry
3-D solid elements can, hypothetically, be used to
model any type of structure but well-designed 3-D solid models are usually large in terms of
computing requirements. Hardware limitations etc. tend to restrict their use to situations where
simplified models (as discussed below) are not viable; for example, thick non-axisymmetric vessels,
thick axisymmetric vessels under non-axisymmetric boundary conditions (loading and support) or
perhaps thinner vessels with unusual or significant geometric details.
Modern finite element programs make it relatively simple (although perhaps time-consuming) to
create complex 3-D models of pressure vessels. The most significant problem in practical design by
analysis using solid models is not in model creation but interpretation of the results of the analysis
in the light of code requirements.
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As stated above, solid elements are based on elasticity theory, in which stress at a point is defined in
terms of six stress components: σx, σy, σz, τxy, τyz, τzx. These stresses vary continuously throughout
the body and in thick pressure components the through-wall distribution is non-linear. This form of
stress distribution is significantly different from that envisaged when design by analysis was first
implemented, which implicitly assumed a linear through thickness, shell-type stress distribution that
could be decomposed into membrane and bending constituents. This incompatibility in format
between the stresses calculated in the solid model and those required for design by analysis
procedures often makes it extremely difficult for the designer to classify the calculated stresses as
primary, secondary and peak and apply the appropriate category stress limits.
2.3.2.2 Axisymmetric Elements
Whilst, ideally, all three dimensional structures can be
modelled using 3-D solid elements to a greater or lesser
degree of accuracy, it is not always necessary to perform a
complete three dimensional analysis. In certain classes of
structure, advantage can be taken of the structure geometry
and loading to reduce the problem to two dimensions. Such
structures can be analysed using much simpler and smaller
finite element models.
Y
r
The most useful class of three-dimensional vessel which can
be analysed using two-dimensional elements is the body of
revolution. Here geometric and material properties are
symmetric about the symmetry axis subject to loading
symmetric about the symmetry axis, as illustrated for an
example structure in Figure 2.7.
X
Z
Figure 2.7: Axisymmetric structure.
This type of structure is called an axisymmetric structure. In finite element practice the component
geometry is defined so that the Y axis is the axis of rotational symmetry. For convenience, the
geometry, loading, stresses and strains of the component are defined in polar co-ordinates: distance
r in the radial direction R from the origin, circumferential or hoop position θ in the circumferential
direction Θ and meridional or height position y in the axial direction Y.
When a rotationally symmetric body is loaded
symmetrically about the symmetry axis - that is,
loads are applied radially or vertically and
uniformly with respect to circumferential position,
as illustrated in Figure 2.8, points in the body can
move radially in or out and vertically up or down.
The material does not move sideways, in the Θ
(hoop) direction, nor rotate as there are no loads
causing the body to deform in this manner.
ELEVATION
PLAN
Figure 2.8: Axisymmetric loading
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As the points on any R-Y plane can move only in that
plane, they have two degrees of freedom: radial
displacement ur and vertical displacement uy. As the
behaviour will be the same at all R-Y planes in the
body, the behaviour of the entire component is fully
defined if the behaviour of any such plane is defined.
Consequently, in an axisymmetric analysis only a
single 2-D section of body need be analysed. In finite
element practice, axisymmetric models are created in
the global X-Y plane, as shown in Figure 2.9.
Y
uy
ux
X
Figure 2.9: Axisymmetric degrees of freedom
The state of strain at a point in an axisymmetric body
may be defined by considering the deformation defined
above. As deformation is symmetric about the Y axis,
no strains are present which would give rise to nonsymmetrical deformation. Thus, no shear strains can
arise perpendicular to the X-Y plane. Under this
condition, the number of stresses at a point reduces
from six (3-D) to four: σx, σy, σθ, τry, as illustrated in
Figure 2.10.
Clearly, the form of stress distribution calculated in an
axisymmetric solid analysis is similar to that calculated in 3-D analysis. Consequently, the same
incompatibility exists between the form of calculated stress results and the form required by design
by analysis rules as discussed for 3-D analysis above.
Y
u2
u1
σθ
σy
τ ry
X
σr
σθ
σr
σy
Figure 2.10: Axisymmetric element stresses.
A range of axisymmetric solid elements are available in most commercial finite element programs,
as illustrated in Figure 2.11.
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2.14
The linear quadrilateral element has four nodes, each with two degrees of freedom. Thus, each
linear element has 8 associated degrees of freedom, compared with 24 for a linear 3-D solid. The
quadratic element has eight nodes with two degrees of freedom. Thus, each quadratic element has
16 associated degrees of freedom, compared with 60 for a linear 3-D solid. Clearly, the use of
Y
LINEAR TRIANGLE
(CONSTANT STRAIN TRIANGLE)
r
X
Z
QUADRATIC TRIANGLE
LINEAR QUAD
QUADRATIC QUAD
Figure 2.11: Typical elements
for axisymmetric
analysis.
axisymmetric elements leads to smaller models in terms of degrees or freedom or, if preferred,
permits a finer mesh for the same model size.
An axisymmetric model of the 3-D nozzle
intersection shown in Figure 2.2 is shown in
Figure 2.12.
The use of axisymmetric
elements allows the analyst to produce a finer
mesh through the thickness of the vessel wall
without creating an excessively large model.
Care must be taken when defining loads for
axisymmetric models. Forces may be defined
as a total applied force or on a per radian
basis. The program user’s manual should be
consulted to check the situation for the
software in use.
Y
It is worth noting that many pressure vessel
problems relate to axisymmetric structures
Figure 2.12: Axisymmetric model of nozzle intersection.
under non-axisymmetric loading. For linear
elastic analysis, it is possible to treat this as an
axisymmetric problem, and model the loading
using Fourier series around the circumference. Some commercial finite element software offers this
capability through modifications to the basic axisymmetric element, known as an harmonic element.
X
Solid elements are extremely versatile, powerful, and suitable for a wide range of applications.
However, there are two significant factors which limit the use of solid elements in practical finite
element analysis, particularly for pressure vessel problems. These are:
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2.15
•
The aspect ratio of solid elements should ideally be 1 - that is, the element is a cube - but in
practice limited to 2 in the case of linear elements and 5 in the case of quadratic elements.
•
Solid elements do not respond well to bending loads, and at least three linear or two quadratic
elements must be used through thickness when bending is present.
Taken together, it becomes clear that these factors often make it impracticable to model thin shell
structures such as the longitudinally supported vessel shown in Figure 2.13 using solid elements.
Figure 2.13: Shell model of vessel.
In general, shell structures are thin in one direction and carry both membrane (in-plane) and
bending (out-of-plane) loads. The load-carrying capacity of a shell mainly is derived from its
membrane strength, but it is impossible to construct real shell structures, such as pressure vessels,
without inherent bending during loading - for example, the junction between a cylinder and a
spherical end cap gives a discontinuity in curvature which induces a bending stress (called a
discontinuity stress). Bending stresses can also result from mechanical and thermal loading - for
example, piping forces on a nozzle.
Several solid elements are required through the shell thickness to adequately represent bending
behaviour but these elements cannot themselves be thin or they will violate aspect ratio
requirements of the formulation. Consequently, a large number of solid elements are required in
order to model even simple shell structures. These shell analysis problems are avoided by using
special shell elements, which incorporate assumptions about the nature of the bending in the
formulation.
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2.16
2.3.2.3 Thin Shell Elements
The traditional method of analysing shell structures is to simplify the behaviour of the structure by
assuming an appropriate thin shell theory in which the behaviour of the three dimensional structure
is described in terms of the deformation of a doubly curved reference surface. This reduces the
number of degrees of freedom required to model the real behaviour of the structure as only one
element is used through thickness. In addition, the definition of the finite element model is also
considerably simplified, as only the mid-surface has to be defined and meshed with surface or area
elements. However, the reduction of a real three dimensional shell structure to a reference surface
model is considerably more complicated than reducing three dimensional elasticity to axisymmetry
or plane strain, as discussed earlier.
In general, shell structures are doubly curved
and the radii of curvature are not constant
throughout the body. Shell structures support
both membrane forces, which act in the plane
of the shell, and bending forces or moments
which arise due to out-of-plane loading.
These are shown diagrammatically in Figure
2.14.
MEMBRANE MODES
BENDING MODES
The membrane and bending forces are
Figure 2.14: Thin shell bending & membrane modes.
coupled throughout the shell, but their relative
magnitude differs with position in the
structure. In certain locations membrane action may predominate, whilst in other locations, most
notably at structural supports and discontinuities, bending becomes more significant. In flat plates
and in the theory of shallow shells, membrane and bending actions can be sensibly de-coupled - this
simplification allows a simpler element formulation, and for this reason flat plate elements are
commonly used in the analysis of general shell structures.
It is possible to simplify the real three-dimensional problem if certain assumptions are made about
how the thin plate or shell deforms, particularly during bending. The most significant assumption
of thin plate or shell theory is that straight lines initially perpendicular to the mid-surface remain
straight during deformation, and follow either the Kirchhoff hypothesis or the Mindlin hypothesis.
In the Kirchhoff hypothesis, when the shell deforms, straight lines normal to the mid-surface rotate,
but so that they remain straight and normal to the deformed mid-surface, as illustrated in Figure
2.15. In the less restrictive (but more complex) Mindlin Hypothesis these lines are not required to
remain normal during deformation, such that the influence of shear strain can be better represented.
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2.17
PLANE SECTIONS REMAIN PLANE
AND NORMAL TO THE NEUTRAL AXIS
KIRCHHOFF HYPOTHESIS
PLANE SECTIONS REMAIN PLANE
BUT NOT NECESSARILY NORMAL
TO THE NEUTRAL AXIS
MINDLIN HYPOTHESIS
Figure 2.15: Kirchhoff and Mindlin Hypotheses
The Kirchhoff hypothesis is common in shell theory, and most published results for thin pressure
vessels are based on this. However, some modern finite element formulations for thin shells use the
Mindlin hypothesis, (essentially since numerical analysis is used and there is no need to be so
restrictive). In addition, the Mindlin hypothesis represents shell behaviour more accurately in the
vicinity of discontinuities and restraints, where transverse shear effects are more significant.
The effect of using one of these simplifying hypotheses in the finite element method is that the
deformation at any point can be defined if the displacement and rotation of the mid-surface are
known. The rotation at any point on the mid-surface is defined by interpolating between rotational
degrees of freedom defined at the nodes. Therefore beam, plate and shell elements have
translational and rotational degrees of freedom. Only these bending elements have rotational
degrees of freedom: solid elements do not need rotational degrees of freedom to define the element
deformation (although it is possible to formulate solid elements which include rotational degrees of
freedom to enhance performance but at the cost of increasing the number of degrees of freedom per
element).
A great deal of work has been undertaken on formulating shell elements, and it is an indication of
the complexity of the problem that no single type of formulation has been universally accepted as
being the best. Classical shell theory produces equations which are difficult to solve and which are
remarkably sensitive to slight variations in shape (which are common in the approximate finite
element method). A large number of different approaches have been developed over the years but
basically only three types of shell element are used in practice:
•
Facet (flat) shell elements, formed by combining membrane and plate bending
elements
•
Curved shell elements, based on classical shell theory
•
Reduced (or degenerate) solid (continuum) iso-parametric elements which directly
take account of thinness and the Mindlin hypothesis in their formulation
The most popular are flat elements and reduced solid elements, both of which appear in commercial
software.
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2.18
Flat (Plate) Elements
The geometry of a doubly curved shell surface can be approximated by a faceted surface formed by
connecting flat triangular elements together at their vertices. A flat three-noded triangular shell
element would have six degrees of freedom per node: in an element co-ordinate system (x,y,z), as
shown in Figure 2.16, there would be three translational degrees of freedom, (ux, uy, uz), and three
rotational degrees of freedom, (φx, φy, φz), giving a total of eighteen degrees of freedom per
element.
This element can be used to represent a shell by including both membrane, (ux, uy, φz), and bending,
(uz, φx, φy), degrees of freedom. Membrane stiffness is derived from simple plane stress conditions,
with the added drilling degree of freedom, φz. The most significant aspect of the derived shell
element is that the membrane and bending stiffness are uncoupled, although there is a degree of
coupling when the elements are assembled.
uz
Y
Z
uy
ux
K
Z
Y
X
I
X
J
Figure 2.16: Triangular flat plate element
Used on their own, it has been found that triangular shell elements based on plate bending elements
do not perform very well, having an artificially high bending stiffness and spurious torsion modes.
Other flat shell elements have been formulated, among the most common of which is the BatozRazzaque element. This is a quadrilateral element formed from four flat shell elements such that the
diagonals are continuous, Figure 2.17. This formulation works well and can be found in many
commercial programs.
Figure 2.17: Batoz Razzaque quadrilateral flat shell element
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2.19
Reduced Continuum Elements
The reduced continuum shell elements are similar to the 3-D solid elements discussed above but
with the Mindlin hypothesis (plane sections remain plane or linear interpolation) applied through
the thickness. Under this condition, if the deformation of the element mid-surface is known, the
deformation at any other point can be defined through the element shape functions. In this way, the
3-D element is reduced to a shell-type area element. Such elements are usually called reduced
continuum, or degenerate solid, or sometimes thick shell, elements. In practice, the element’s
curved mid-surface is approximated from the given nodal co-ordinates and this can affect the
performance and accuracy of this type of element if it has a poor shape.
2.3.2.4 Discussion
At first sight the most appropriate choice of finite element may seem obvious for a given vessel
under consideration. However when the choice is examined in the light of the pressure vessel design
by analysis elastic route, where limits are placed on membrane and bending stress and stresses, or
indeed parts of stresses, must be categorised, various well-known difficulties arise. These problem
areas lie at the heart of criticism of the ASME design by analysis rules and consequently are
discussed in more detail in the following:
2.4 Implementation Problems of the Stress Categorisation Route
2.4.1 Overview of Problems
Once the linear elastic analysis of a part is complete and the immediate results for stresses and
strains obtained, there is the need to satisfy the design by analysis rules. As mentioned previously
this is not necessarily as straightforward as it may at first seem. Specifically, there is a requirement
to obtain membrane and bending components of primary stress and the calculated stresses must be
categorised. This does not present a problem in cases where the analysis utilises thin shells.
However, for analysis (in particular finite element analysis) utilising solid models (2 or 3
dimensional) where the calculated stress can not be easily identified as membrane, bending or peak
the problems of linearisation and categorisation become apparent. Difficulties implementing this
area of the design by analysis rules have become increasingly evident to both designers and
analysts[1]. This section examines the practical problems associated with the implementation of
these design by analysis rules.
2.4.2 Linearisation
The design by analysis criteria, as formulated nearly thirty years ago, is based on the behaviour of
thin shells and includes the notion of membrane and bending stress. Inherent in this understanding
is the assumption that membrane and bending stress act on a plane under the Kirchhoff hypothesis
that plane sections remain plane during bending. The shell type membrane and bending stresses
cause gross distortions under primary loads and strain enhancement under secondary loads. Most of
our understanding of basic pressure vessel geometry and components come from our knowledge of
their behaviour as thin shells. A consequence of this understanding is the possibility that portions
of total stress, identified as membrane or bending (or peak) can be categorised as primary or
secondary.
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For example, in the case of a nozzle in a
spherical shell (Figure 2.18) with area
compensated reinforcement, only the
membrane stress is primary, despite
significant bending in the shell close to the
nozzle. The bending stress is secondary
(since only the membrane stress and the hoop
stress in the nozzle are required to satisfy
equilibrium with the internal pressure). In
this case, it is essential to consider membrane
and bending stress for the correct
categorisation.
2.20
Reinforcement
M R BENDING
t
p
Pm = pr
t
MEMBRANE
r
Pm is PRIMARY
M R is SECONDARY
Figure 2.18: Stress categories for a nozzle.
l
z
o
p
If the analysis is based on thin shell finite
elements, then there is no difficulty in identifying
membrane and bending stress, as they are part of
the underlying theory, Figure 2.19.
TOP
y
x
k
m
n
i
TOP
TOP
MID
BOT MID
BOT
x
y
Difficulties arise when thin shell analysis is not
used and the finite element analysis is based on
axisymmetric or three-dimensional solid elements.
Z
X
In general, unless the section is indeed thin, the
Figure 2.19: Shell membrane and bending stress
stresses on a through thickness line are not linear,
and further plane sections do not remain plane
during bending. Over the years it has become common practice to linearise the calculated through
thickness stresses in order to separate membrane and bending components.
j
Y
BOTTOM
A technique for linearising stress was first suggested by Kroenke[2,3], and has been adopted in
several finite element postprocessors. A stress classification line (or plane) or supporting line
segment is chosen and the stresses are linearised along this line. The supporting line segment (SLS)
or classification line is the smallest segment joining the two sides of the wall where the stress is to
be linearised. Outside of gross structural discontinuity regions, the SLS is normal to the wall mean
surface, i.e. its length is equal to the thickness of the wall in the analysis. There are difficulties with
this procedure – which seems straightforward, but again is a fundamental difficulty - this will be
discussed in more detail in the following.
Pressure vessel design codes are not particularly helpful on the problem of linearisation. ASME III
& VIII admit a non-linear bending stress, but also contains some ambiguities: bending stress is
described as a normal stress - and it is bending stress that may need linearisation. In Paragraph
NB-3215 a note is provided to the effect that “.. membrane stress intensity is derived from the stress
components averaged across the thickness of the section. The averaging shall be performed at the
component level ...”. This implies that only stress components may be linearised (by definition this
could include shear stress), and not derived principal values. However, through omission from the
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2.21
code, it may be argued that shear stress should not be linearised. Inclusion of shear stress
linearisation will mostly affect the surface stress: in practice linearisation of the normal stress only
is adopted to modify the surface stress in application of the design criteria.
The basic procedure for stress linearisation is the
selection of a stress classification plane or
supporting line segment on which shell type
stress will be evaluated.
A stress classification line (or plane of referencesee page 31) is identified through a section and
the non-linear stress distribution along this line is
linearised in order to extract membrane and
bending stress, as shown in Figure 2.20. The
stress classification line (CL) lies along a local
axis X3; the origin is located at the mid-point of
the CL (i.e. at radius Rc); the abscissa of a point
on the supporting line segment is designated x3.
R
X
c
3
Stress classification line or
supporting line segment of
length h.
Figure 2.20: Stress classification line
In practice, the linearisation procedure is performed automatically by special postprocessors. For
simplicity, some basic postprocessors (in particular self written ones) may require the finite element
mesh to be created so that a line of nodes lie along the chosen classification line, making it
relatively simple to extract stress results. In
postprocessors that are more complex, the
classification line need not pass through a line
STRESS
CLASSIFICATION
of nodes.
LINE
The classification line in Figure 2.21 is defined
from node Ni at the inner surface to No at the
No
Ni
outer. The path of the classification line does
not pass through a line of nodes: it cuts
through the elements. Advanced linearisation
postprocessors use the location of the surface
nodes to define the path through the elements
and then apply interpolation functions to the
appropriate nodal stresses to calculate the
stress along the path. Two possible procedures
Figure 2.21: Interpolation of a classification line
for linearisation have been suggested, Kroenke
which has been discussed by numerous
analysts[4] and a more refined version by
Gordon[5].
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2.22
2.4.2.1 Kroenke’s Procedure
Kroenke’s procedure makes reference to familiar beam bending stress - a uniaxial stress - and
attempts to define an equivalent linear stress distribution on the classification line (CL). Consider a
typical stress distribution along a classification line as in Figure 2.22.
If x3 measures local distance along the classification line then the equivalent linearised stress is,
(σ )
ij L
= ax3 + b .
The membrane stress component is given by the formula
nonlinear stress
σ
distribution
σp
linearised
stress
b
σb
a
σm
classification
line
X3
x3
e/2
e/2
Figure 2.22: Typical Stress Distribution
(σ )
ij
m= b =
1
e
e
2
e
−
2
∫
σ ij dx3 .
The membrane force per unit length of the membrane stress component is equal to that from the
calculated FE stress component.
The bending stress component is given by
(σ )
ij b
= a ⋅ x3 =
12 x3
e3
e
2
e
−
2
∫
σ ij x3 dx3 .
The maximum and minimum bending stresses can then be evaluated (for x3 = ± e / 2 )
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(σ )
ij b , s
6
=± 2
e
e
2
e
−
2
∫
Page
2.23
σ ij x3 dx3
The bending moment per unit length of the calculated FE stress component is equal to
(σ )
ij b , s
⋅
e
e2
= ∫ 2e σ ij x3 dx3
−
6
2
The linearised stress (σ ij )L is found by adding membrane and bending stresses
(σ ) = (σ ) + (σ )
ij L
ij m
ij b
The bending stress of this equivalent linear stress distribution vanishes at x3 = 0 .
The peak value of stress at a point is the difference between the total stress and the sum of the
membrane and bending stresses
(σ ) = (σ ) − (σ ) = (σ ) − [(σ ) + (σ ) ] .
ij
p
ij
ij L
ij
ij m
ij b
2.4.2.2 Gordon’s Procedure– for axisymmetric problems
In Kroenke’s procedure for axisymmetric problems, the shell wall is assumed (locally) straight in
the meridional direction. In some circumstances the meridional curvature is finite. Gordon
suggested a modification to Kroenke’s procedure to allow for this.
Gordon’s procedure for an axisymmetric case is the same, in principle, as the case above, except for
the fact that there is more material at a greater radius than at a smaller radius. The neutral axis is
shifted radially outward to accommodate for this.
Consider the axisymmetric section of a vessel wall as shown in Figure 2.23. ρ is defined as the
radius of curvature of the mid-surface of the shell. In the case of an axisymmetric straight section
such as a cylinder or cone, ρ = ∞.
Adopting the following notation:
ρ − radius of meridional curvature
R1 – radius of circumferential curvature
z – axial co-ordinate
r – radial co-ordinate
θ – angle in hoop direction
φ – angle in meridional direction
X3 – local co-ordinate containing the classification line
X2 – local co-ordinate normal to classification line
x3 – co-ordinate along classification line
e – shell thickness
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2.24
R – radial co-ordinate of a point of the classification line
Rc – radial co-ordinate of mid-surface point
z
R
Rc
X3
Axis of
symm.
φ
X2
ρ
e
z
R1
r
centreline
θ
r
Figure 2.23: Geometry for finite curvature
From an axisymmetric analysis the following stresses would be obtained in the local classification
line co-ordinates:
σX2 - (local) meridional stress
σX3 - (local) radial stress
σθ - hoop stress
σX2X3 - (local) shear stress
The other shear stresses would be zero in an torsion-free axisymmetric analysis.
The aim is to obtain membrane and bending components of these stresses, denoted by subscripts m
and b respectively, evaluated from the average stress across the section and the beam type bending
stress.
The membrane component of the (local) meridional stress on the classification line is given by
(σ )
X2
m
=
FX 2
AX 2
=
e
2
e
−
2
∫
σ X 2 ⋅ (R1 + x 3 ) ⋅ ∆θ ⋅ dx3
R1 ⋅ ∆θ ⋅ e
=
e
2
e
−
2
∫
σ X 2 R ⋅ ∆θ ⋅ dx3
Rc ⋅ e ⋅ ∆θ
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2.25
where the area AX 2 of a small segment extending over the angle ∆θ in hoop direction is given by
A X 2 = Rc ⋅ e ⋅ ∆θ .
In this linearisation the bending stress component of the (local) meridional stress on the
classification line vanishes at x3 = x f , where x f is the x3 - co-ordinate of the resultant of a constant
stress distribution σ X 2 and of the centroid of the considered area.
x f is given by
xf =
e2
e 2 cosφ
,
=
12 R1
12 Rc
thus the bending stress component is given by
(σ )
X2 b
=
M X 2 ( x3 − x f )
Im
,
where
M X 2 = ∫ 2e (x 3 − x f )σ X 2 R ⋅ ∆θ ⋅ dx3
e
−
2
and
 e2

I m = Rc e ⋅ ∆θ ⋅  − x 2f  ,
 12

which leads to
(σ )
X2 b
=
x3 − x f
 e2
2
Rc ⋅ e − x f 
 12

e
2
∫ (x
3
− x f )σ X 2 R ⋅ dx3 .
e
−
2
The hoop stress is evaluated in a similar manner to the above; however in this case the meridional
curvature, ρ must be taken into account:
The membrane component of the hoop stress is given by
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(σ )
θ m
=
Fθ
=
Aθ
Design by Analysis
e
2
e
−
2
∫
σ θ ( ρ + x3 ) ⋅ ∆φ ⋅ dx3
=
ρ ⋅ e ⋅ ∆φ
e
2
1
σθ
e ∫e
−
Page
2.26

x 
⋅ 1 + 3  dx3
ρ

2
where the area Aθ of a small segment extending over die angle ∆φ in meridional direction is given
by
Aθ = ρ ⋅ e ⋅ ∆φ .
In this linearisation the bending stress component of hoop stress on the classification line vanishes
at x3 = xh , where x h is the x3 - co-ordinate of the resultant of a constant stress distribution σ θ and
of the centroid of the considered area, where x h is given by
xh =
e2
.
12 ⋅ ρ
Thus the bending stress component is given by
(σ θ )b
M θ ( x3 − x h )
,
Ih
=
where
e
M θ = ∫ 2e ( x3 − x h ) ⋅ σ θ ⋅ ( ρ + x 3 ) ⋅ ∆φ dx3 ,
−
2
and
 e2

I h = ρ e ⋅ ∆ϕ ⋅  − x h2  ,
 12

which leads to
(σ θ )b
=
x3 − x h
e
2
e − x h 
 12

2
e
2
∫ (x
−
e
2
3

x 
− x h )σ θ ⋅ 1 + 3  ⋅ dx3 .
ρ

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2.27
(Local) radial stress on the classification line is treated in a special way: in most situations the radial
stress will equal the applied pressure at the internal surface and be free (zero) at the outer surface.
Therefore, membrane stress may be evaluated,
(σ )
1 e2
X3
e σ X dx 3
e ∫− 2 3
but it is questionable whether a bending stress should be evaluated. Either this should be taken as
zero
=
m
(σ )
= 0
X3 b
or as the simple difference between actual and averaged value
(σ ) =σ
X3
b
X3
( )
− σ X3
m
which may not be linear.
Similarly, an average membrane shear stress can be determined along the classification line
(σ
X 2X3
)
=
m
1 2e
e σ X X Rdx3
Rc e ∫− 2 2 3
Since the shear stress would be expected to be nearly parabolic (from basic elasticity theory), and
zero at the surface, the bending stress should be taken as zero
(σ
X2X3
) =0
b
The surface values of shear therefore only contribute to peak stress.
The development of Gordon’s procedure given here is in terms of stress components in a local coordinate system (X3, θ, X2). In practice these would be transformed into the global (r, θ, z) coordinate system, to give global linearised stress components. Once the global stress components
have been linearised, the principle stresses and stress intensity can then be evaluated, as total values
and as averaged membrane and surface bending stresses.
2.4.2.3 Discussion
As mentioned previously, several finite element programs contain post processing options to
directly calculate the equivalent linearised stresses on any prescribed classification line. Usually all
stress components are linearised. A short consideration of the linearisation procedure immediately
brings several possible problem areas to mind.
.
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2.28
• Selecting the stress classification line. This should be a line through the vessel wall, which
would be expected to yield shell type deformations, namely straight lines remaining straight.
However close to discontinuities some
warping and shear would be expected (and
indeed observed in the finite element
calculations) and the concept of averaged
membrane and linearised bending stress is
tenuous. Ideally, inner, outer surfaces,
and, thus, the mid-surface should be
Normal to
LINE 1
Normal to
Inner Surface
parallel, with the stress classification line
Outer Surface
perpendicular to these surfaces. Of course
LINE 2
in some situations ambiguities can arise, as
illustrated in Figure 2.24.
• Selecting which stress components
should be linearised. The global stress
Figure 2.24: Ambiguous classification lines
components, including shear, may be
linearised - but what about the principal
stresses? With principal stresses, there is the obvious problem that principal directions can alter
from point to point through the thickness unless the classification line is far from a discontinuity.
As mentioned previously, the ASME Code itself implies that the linearisation should be performed
on the global component stresses: “... membrane stress intensity is derived from the stress
components averaged across the thickness of the section. The averaging shall be performed at the
component level ...”. There is the related question as to whether linearising the component stresses
then calculating the principal stresses from the linearised components is consistent with linearising
the principal stresses - plane sections remaining plane should provide this consistency. BS5500
implies that all stress components should be linearised.
Selection of the classification line has received little attention in the literature. However, selection
of which stress components to linearise has been examined, on behalf of the ASME Code
committee, by G L Hollinger and J L Hechmer[6].
Hechmer & Hollinger analysed a representative axisymmetric vessel problem and examined several
different methods of stress linearisation. Two methods appear which were identified as being both
conservative and consistent: either linearise the two normal stress components on a line (in the hoop
and meridional directions) and use the total normal radial and total shear stress at the surface, or
linearise the meridional principal stress and use the total stresses for the other principal direction
(the exact technique is not wholly clear from the paper). Neither of these would appear to be
common practice.
2.4.2.4 Three Dimensional Problems
Three-dimensional solid finite element analysis poses a significant problem for stress linearisation.
In 3-D analysis it is necessary to find a consistent stress classification plane, which again could
cause problems near the very features the designer is concerned with (fillets and gross structural
discontinuities). There is the added problem of defining exactly what should be meant by plane
sections remaining plane in this case. Three possibilities arise: firstly the stress components at a
point are directly used to evaluate the stress differences and stress intensity; this is easy, but the
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2.29
subsequent categorisation of these stress intensities is not. Secondly, stresses are linearised along
radial lines to obtain beam type membrane and bending stresses; this suffers from the same
problems mentioned above. Thirdly, selected planes are specified and two sets of stresses on
distinct lines (on the plane) are used to evaluate shell type direct and bending stress on a plane. To
the writers knowledge, no commercial post processors offer three dimensional stress linearisation
capability over a plane, only along a line, only one by a vessel manufacturer[7].
In the case of the three dimensional problem, Hechmer & Hollinger[8] analysed a complex nozzle
shell assembly using brick elements and examined the consequences of the three different
assessment methods described above to calculate the stress intensity. As expected their study
demonstrated a wide variation in the calculated results for the various methods (mainly because
there are many possibilities open to the analyst) - a variation in stress intensity of over 35% was
noted in this example. The results are indeed inconclusive: stress at a point calculation is easiest to
apply but the results are not always conservative while stress along a line calculation is more
advantageous with respect to Code rules.
2.4.2.5 Linearisation Guidelines
It has been apparent for some time that there are deficiencies in the rules for design by analysis
when the finite element method is used. In particular this has highlighted problems with the design
criteria and the underlying philosophy of assessment. Over the past few years, the US Pressure
Vessel Research Council (PVRC) has funded a project to consider recommendations for updating
the ASME Code. It is worthwhile reviewing some of these recommendations; a summary has been
given by Hechmer & Hollinger[9].
The short term recommendations consisted of six sections. The second, fourth, fifth and sixth
recommendations are related to linearisation problems for primary stress and three-dimensional
problems. The first and third recommendations are of a more fundamental implication since they
relate to the use of finite element methods for design by analysis using the existing ASME - Code
criteria. The project members have been very careful with the wording of the recommendations, and
some interpretation is required. These recommendations consider essential pressure vessel
components, which are basic structural
elements:
•
•
•
Shells of revolution and circular plates
with either constant or variable thickness
(transition
elements)
normally
connecting one structural element to
another.
Smooth junctions - where the model
represents the actual geometry for
example connecting fillet or blend
radius.
Sharp junctions - where the model does
not represent actual geometry, such as
sharp corners or notches, as shown in
Figure 2.25.
Basic
Structural Element
Smooth Junctions
Sharp Junction
Fillet
Transition Element
Sharp Junction
Basic
Structural Element
Figure 2.25: Pressure vessel ‘elements’
Blend
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The first and third recommendations are summarised below:
First recommendation: This relates to the use of finite element analysis (FEA) in pressure vessel
design by analysis. It is recommended that for the majority of pressure vessel components, which
are basic structural elements, FEA is inappropriate. Pm stresses should be calculated using general
equilibrium considerations, with Pm+Pb evaluated by hand calculations for conditions where Pm is
small (for example in flat plates). FEA is appropriate for calculating PL+Pb stresses near
discontinuities (see third recommendation below) and for the calculation of P+Q stresses in general.
Notably it is only in complex components where basic structural analysis does not exist that FEA is
recommended as appropriate for Pm and Pm+Pb stress evaluation. “... the thrust is that the designer
should be applying his ingenuity to calculating equilibrium stresses, not to extracting stresses from
a general finite element model ...”.
Third recommendation: This relates to the locations in a pressure vessel where stress evaluations for
Code compliance should be considered. It is recommended that it is appropriate to perform Pm+Pb
(PL+Pb) and P+Q evaluations in basic structural elements, but inappropriate in discontinuity type
transition regions. If there is a smooth junction then the stresses should be evaluated in the row of
elements adjacent to the junction (or the line of nodes at the junction). When there is a sharp
junction, the evaluation must be far enough from the junction so that the stresses are not affected by
the notch behaviour. This recommendation should eliminate the need to linearise erratic stress
distributions; “... the thrust ... is that plastic collapse and gross strain concentration will not occur in
stiff transition regions; they will occur in the more flexible shell elements ... the purpose of the
P+Q limits is to validate the fatigue analysis by precluding strain concentration and ratchet. It is
highly unlikely that ratchet could occur in a transition element ...”
The first recommendation is rather subtle. In the light of the ASME Code (as it stands), finite
element analysis is only appropriate in certain special cases in primary stress calculation - in
general, equilibrium and shell discontinuity analysis are to be preferred. However, FEA is
appropriate for secondary (and peak) stress evaluation. In the context of the discussion given this
may be interpreted further as follows: finite element analysis may be used to evaluate the overall
stress distribution for shakedown and fatigue assessment but the analyst should use simple
calculations and strength of materials arguments to extract the primary stress components. In other
words, elastic finite element analysis should not be used as the basis for categorisation or evaluation
of primary stress.
The third recommendation also needs careful interpretation and is the most intriguing of all those
provided by the PVRC project. The implication to the writers is clear - ignore the calculated stresses
in sharp transition regions, since they will not affect the post yield failure mechanisms.
The mid term recommendations aim to provide additional tools and procedures to assist the
designer in making better use of the existing ASME Code rules, specifically to address the problems
of categorisation and linearisation directly through finite element analysis.
Finally the long-term recommendations aim for a more fundamental assessment of the ASME Code
philosophy and criteria and require extensive new research. It is felt that new rules should be based
on specific quantities required to prevent a failure mechanism, perhaps moving away from simple
elastic analysis and stress evaluation. For example, the limits based on shell type membrane and
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bending stress are difficult to understand and often misinterpreted, while the secondary limits are
probably oversimplified and over-conservative, particularly in the presence of combined load.
Considerable research on shakedown and ratchetting over the past twenty five years has confirmed
this.
2.4.3 Problems with categorisation
The process of stress categorisation (or classification) is difficult, as stress may be composed of
both primary and secondary parts as seen in Figure 2.18 for the nozzle reinforcing pad. It is not
sufficient just to categorise a particular stress corresponding to a given load condition, but also to
categorise segments of the stress. This prospect is not inviting, and indeed rarely done in practice
unless specified in Code rules (as in the case of the nozzle).
We have reached a familiar problem - how should finite element (or otherwise) calculated stress be
categorised? This is usually left to experience or strength of materials type arguments if this is
possible. It is usually possible with simple strength of materials analyses or shell discontinuity
analysis to separate primary and secondary stress with the understanding of the fundamental failure
mechanisms that the Code addresses, since the equilibrium calculations were done manually. This is
not obvious with finite element results, and in particular with the results of using continuum
elements. The question is what can be done to ease this problem.
An obvious solution would be to provide additional Code rules. While this is likely to be the case
in the long term, it does not help the designer who must carry out pressure vessel design with the
current rules.
Briefly, the evaluation rules in this route can be summarised. Membrane and other primary
membrane stresses are not allowed to approach yield since beyond yield there is the possibility of a
catastrophic plastic collapse – for example bursting under internal pressure. The total (membrane
plus bending) stress can increase fifty percent above the membrane limit since there is some safety
margin here, but is still yield limited. Discontinuity and thermal stresses (or strain controlled
stresses) must be limited to ensure shakedown under cyclic load; thus the range of secondary stress
is limited to twice yield (or some smaller proportion for particular components). The peak stress
must be limited to ensure a sufficient fatigue life, and certain other failure criteria may need to be
addressed depending on the operating temperature - for example creep rupture at high temperature,
fast fracture at low temperature. At this level categorisation is straightforward: any sustained stress
that, subject to overload, would lead to plastic collapse is primary. The remaining stress (or indeed
proportion of stress) can be classified as secondary and is subject only to the shakedown criterion
(and fatigue limit).
The problem arises because this design by analysis route relies upon elastic analysis. Elastic
analysis on its own cannot characterise the nature of the stress since it is not clear what failure
mechanisms can arise; it is left to the designer to do this. In addition, this approach does not make
use of the ductility of pressure vessel steels, resulting in a wholly inconsistent (conservative) margin
of safety[10]. In the absence of any meaningful information the designer is led to classify all stresses
as primary and base redesign on this.
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It is useful to view the categorisation problem as part of the basic requirement to avoid failure by
the various failure mechanisms: the problem of categorisation should then have a different
interpretation.
The stress system in the component should be such that shakedown is achieved and the fatigue
limits satisfied for all stresses. In fact, these are the basic design requirements. The categorisation
problem can then be interpreted as the need to isolate those stress systems that could cause gross
plastic collapse - that is the primary stresses. The distinction here is subtle - there is no real need to
identify a calculated stress as being primary or secondary; it is only necessary to identify the
primary stresses.
One solution to this difficulty is to calculate the limit load of the vessel by inelastic analysis. Limit
load assessment and calculation of principal stress has been discussed by several authors using a
variety of methods, notably Marriott[11], Kalnins & Updike[12], Mackenzie & Boyle[13], Seshadri[14]
Ponter & Carter[15] and Zeman et. al.[16].
2.5 Implementation problems of the ASME inelastic route
Inelastic finite element analysis[17,18] is more difficult than linear analysis and requires considerably
greater computing resources. Essentially, the non-linear problem is solved in a piecewise manner
using incremental solution techniques. The procedure usually requires the analyst to define an
appropriate number of load steps, equilibrium iterations within load steps and convergence criteria
defining the required accuracy of the solution. Poor choice for any of these parameters can lead to
lack of convergence or indeed “convergence” to the wrong answer. In addition, it is difficult to
make a priori engineering estimates of the inelastic response and to verify results of the analysis
through simple calculations. There is also a shortage of non-linear benchmarks, which the analyst
can use to assess the accuracy of the analysis procedures.
There are two types of inelastic analysis methods, which may be used to guard against gross plastic
deformation: limit analysis and plastic analysis.
Limit analysis is based on an elastic-perfectly plastic material model and small deformation theory.
The assumption of perfect plasticity sometimes causes convergence problems in non-linear analysis
and in practice a bilinear hardening material model with a low value of plastic modulus Ep
(1/10000 of E) is often used. This analysis determines the limit load PL of the vessel. The allowable
load Pa then is defined as a specified fraction of the limit load.
Plastic analysis is based on the ‘actual’ non-linear stress-strain relationship of the vessel material,
including non-linear geometry effects if desired. This analysis determines the plastic collapse load
Pφ. However, determination of the ‘plastic collapse load’ is not straightforward – to understand this,
some basic concepts are required.
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2.5.1 Plastic design loads
The aim with the ASME inelastic route is to estimate limit and shakedown loads directly, which can
then be used to characterise an allowable load (Sec. 2.2). To begin with these are defined:
The First Yield Load Py
The first yield load Py is defined as the load for which the material of the pressure vessel first yields
(from the virgin stress-free state) at the most highly stressed point. Because only one point of the
material is at yield, the surrounding elastic material restrains the vessel from plastic deformation as
a whole.
The Limit Load P0
The classical definition of a limit load P0 according to limit analysis is an idealized one, a
mathematical one. This “theoretical limit load” is the maximum load solution to an analytical model
of the structure which embodies the following conditions:
the strain-displacement relations are those of small displacement theory (first order);
the material response is rigid plastic or elastic-perfectly-plastic (Fig.2.26),
the internal stresses and applied forces are related by the usual linearised equations of
equilibrium which ignore changes in geometry due to deformations.
stress (force)
rigid-perfectly plastic
strain (extension)
elastic-perfectly plastic
stress (force)
•
•
•
strain (extension)
Fig 2.26 : Elastic-perfectly-plastic and rigid-plastic deformation curves
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A (sufficiently) small region in an elastic-perfectly plastic material behaves either elastically (if
stressed below yield), or plastically (for stresses at yield). At loads above the first yield load, P >
Py, but less than the limit load, P < P0, a region of material may have stresses at yield, but this
region is still restrained by the remaining rigid portions of material in the vessel. When the load is
increased to the limit value P0, the plastic region has grown to an extent such that the rigid region
has either disappeared or has become insufficient to restrain the plastic region from motion. The
load for which overall plastic deformation of the vessel occurs is called the limit load. According to
limit analysis theory, it is impossible to have loads greater than the limit load for a perfectly plastic
material.
The Plastic Collapse Load Pc
The plastic collapse load Pc is applied to the actual structure or vessel consisting of an actual strain
hardening material. It includes the effects of geometry change due to large deformations. At this
load, significant plastic deformation occurs for the structure or vessel as a whole (un-contained
plastic flow). The cause is the plastic region in the vessel, who now has grown to a sufficient extent
such that the surrounding elastic regions no longer prevent overall plastic deformation from
occurring. When this occurs, it may constitute a real failure, in the sense that the structure then can
no longer fulfil its intended function. The plastic collapse load can be used as a realistic basis for
design; an efficiently designed structure will be proportioned so that the external (operational)
actions would have to be increased by a specified factor (safety factor) in order to produce failure.
The limit load for an idealised structure then can be an approximation for the plastic collapse load
for the actual vessel, when it is largely plastic at small deflections.
The Ultimate Load Pu
At the plastic collapse load, the vessel does not necessarily collapse. Therefore, the adjective,
“collapse”, is unfortunate. The terminology of plastic deformation load or just plastic load would be
more meaningful. The load at which the vessel actually collapses is the ultimate load Pu. An
example of an ultimate load is the burst pressure for a cylindrical vessel of sufficient ductility.
The Plastic Instability Load Ppi
Plastic instability loads can be of two types:
•
•
of the material instability type, and
of the structural instability type.
Plastic material instability corresponds for example to necking of a tensile specimen at the ultimate
load. The plastic structural instability load, depends upon the yield strength of the material, and is
accompanied by significant changes in shape of the structure or vessel. The plastic instability load is
important because its value is often less than the limit load.
The Shakedown Load PS
All the above load definitions are for monotonic increasing loads. The shakedown load refers to
cyclic loading and is considered briefly because it is important to know the relative margin of safety
on shakedown of a design.
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If upon loading the structure beyond yield into the plastic range to a load value P > Py, and upon
unloading, a residual stress distribution is produced in the structure such that further cycles of load
to value P produce only elastic changes in stress, the structure is said to shake down. The highest
value of P for which shake down occurs is called the shakedown load Ps. Failure to shake down, i.e.
P > Ps, leads to either progressive plastic flow called ratchetting, or to low cycle fatigue failure.
2.5.2 Limit Analysis Theory as Applied to Pressure Vessels
Consider a typical pressure vessel loaded by internal pressure, a perfectly-plastic material, small
deflections and increasing the pressure p.
At small values of p, the vessel material will be elastic and deformation of the vessel will increase
in proportion to p. However, as the pressure is continually increased, a region of the vessel becomes
plastic and the rate of deformation begins to increase, but deformation of the vessel as a whole is
usually still restrained by the surrounding elastic material. Finally, upon further increase in pressure,
a limit pressure or (in this case) a plastic collapse pressure is reached, where the plastic zone has
grown sufficiently large so that the deformation has suddenly begun to increase with little or no
additional increase in pressure. The problem then is this: What will be the magnitude of the limit
pressure of a particular pressure vessel? This is an important question in designing a vessel with a
sufficient margin of safety.
As described above, the problem from the beginning of loading involves initially elastic, then
elastic-plastic, and finally largely plastic behaviour. This is an involved and complicated loading
process. The theory of limit analysis, an idealised theory, enables the limit pressure to be found by
considering:
•
•
a rigid ideal-plastic, or a linear elastic ideal-plastic material, characterised by a sharply defined
yield limit (no strain hardening material),
the small displacement theory (any effect of geometry change of the shell due to deformation is
neglected).
These limitations must be kept in mind when applying limit analysis theory to certain problems
where the effects of strain hardening and geometry change may be important.
2.5.3 Elastic-Plastic Theory as Applied to Pressure Vessels
If effects of strain hardening and geometry change are important, an elastic-plastic analysis is to be
applied. Their influence on the load-deflection curvature is discussed.
Geometry Effects
Fig 2.27 shows a comparison between:
a)
b)
c)
d)
a small-deflection rigid-plastic limit load analysis,
a small-deflection elastic-plastic analysis,
a large-deflection elastic-plastic analysis, with geometrical weakening,
a large-deflection elastic-plastic analysis, with geometrical strengthening,
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e) a large-deflection elastic analysis, with geometrical weakening,
f) a large-deflection elastic analysis, with geometrical strengthening.
2
f
1.8
d
e
1.6
a
b
1.4
a
b
load (bar)
1.2
c
c
1
d
e
0.8
f
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
deflection (mm)
Fig 2.27 : Influence of geometrical effects
The small deflection elastic-plastic solution b) approaches the small-deflection rigid-plastic limit
load solution a) as expected. The large-deflection elastic-plastic solution, with geometrical
strengthening, gives a value higher than the limit load. The large-deflection elastic-plastic solution,
with geometrical weakening, gives a value lower than the limit load.
Effect of Strain Hardening
The effect of strain hardening is to increase the pressure capability above the limit load predicted by
the perfectly-plastic analysis, including the large deflection effect. Fig 2.28 shows that a higher
slope of the plastic part of the load-deflection curvature corresponds to a higher strain hardening
effect.
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2.37
1.8
6%
4%
2%
0%
1.6
1.4
1.2
load (bar)
0%
1
2%
4%
0.8
6%
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
deflection (mm)
Fig 2.28: Influence of strain hardening effect
2.5.4 Estimation of plastic loads
Using an elastic-plastic analysis including strain hardening and large deflections or equivalently
considering experimental analysis of an actual vessel, one is confronted with the problem of
defining a realistic measure of plastic loads. A number of estimations have been used. These are
reviewed next. The discussion refers to pressure loading, but the same definitions can be applied to
other types of loadings.
The Limit Pressure p0
Characteristic for the limit pressure definition according to the rigid perfectly-plastic theory is (with
p = pressure and δ = deflection)
dp/dδ = ∞ or dδ/dp = 0 for p < p0,
and
dp/dδ = 0 or dδ/dp = ∞ for p = p0.
Characteristic for the limit pressure definition according to the elastic perfectly-plastic theory is
dp/dδ > 0 for p < p0,
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and
dp/dδ = 0 or dδ/dp = ∞ for p = p0.
These definitions only apply for small-deflection analyses.
The Tangent-Intersection Pressure pti (Fig 2.29)
1.6
1.4
Pti = 1.35 bar
1.2
load (bar)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
deflection (mm)
Fig 2.29 : Tangent-Intersection method
The tangent-intersection pressure is the pressure at the intersection of the two tangents, drawn to the
elastic and plastic parts of the pressure-deflection curves. The value of the pressure obtained by this
method is sensitive to the localisation of the tangent-point in the plastic range
The 1% Plastic Strain Pressure p1
The plastic pressure is defined as the pressure with an equivalent plastic strain of 1%. Methods
based upon an absolute maximum strain not only will depend on the material assumed, but more
significantly on the geometry:
•
•
Material: e.g. a 1% plastic strain is ten times the yield point strain if the yield point stress is 150
MPa, but five times the yield point strain if the yield point stress is 300 MPa. Consequently, the
relative size of the elastic and plastic zones will differ and the shape of the pressure-deflection
response curves will differ.
Geometry: Ellipsoidal heads have been found to deform less than torispherical or toriconical
heads. Whereas a torispherical vessel may reach a 1% strain at a certain pressure, the ellipsoidal
vessel may reach the same pressure at a lower strain.
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At a yield hinge location, strains will be larger than at other locations. Consequently, the selection
of a strain gauge location on an experimental vessel presents a variable when yield hinge locations
are not known precisely or a priori.
Thus, in summary, a strain basis for defining a plastic pressure may be subject to error in locating
the exact location of maximum strain. Also, strain is a local phenomenon that is not indicative of
plastic work.
The Twice-Elastic-Deformation Pressure p2y
A plastic pressure is defined to be the pressure at which the deflection or strain reaches twice the
value of the elastic deflection or elastic strain at the first yield pressure py. Thus, p2y depends upon
py. Exact determination of py using a computer analysis should not be a problem. In experiments
however, determining the elastic limit on the load deflection curve may be subject to error.
The Twice-Elastic-Slope Pressure pϕ
A plastic pressure is defined to be the value at the intercept of a line drawn from the origin of a
pressure-deformation curve at a slope of twice the slope of the elastic portion of the curve (see Fig
2.30).
1.6
Pφ = 1.39 bar
1.4
1.2
load (bar)
1
y = 2.2x
0.8
y = 1.1x
0.6
0.4
0.2
0
0
0.5
1
1.5
deflection (mm)
Fig 2.30 : Twice-Elastic-Slope method
2
2.5
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The 0.2% Offset Strain Pressure p.2
The 0.2% offset strain pressure is a test pressure that causes a permanent strain of 0.2%.
The Proportional Limit Definition ppl
Ppl is a test pressure defined as the pressure causing the displacement versus pressure curve to
deviate from linearity. The displacement of the vessel is to be measured at the weakest point, the
most highly stressed point, giving the lowest value of ppl.
Analytical calculations can determine this pressure correctly. It will not necessarily be equal to the
first yield pressure py. Experimental measures are subject to error in determining the point of
deviation from linearity. Values of ppl up to 30% greater than py can be estimated from an
experimental curve.
This method of determining a plastic pressure will generally give a lower bound to the plastic
pressure found by most other methods.
The Plastic-Instability Pressure ppi
This is an actual plastic collapse pressure and not just an estimate of a plastic pressure. It may be
identical to the limit pressure if large deflection effects are small, e.g. when the vessel is relatively
thick. However, the plastic-instability pressure may be less than the small-deflection limit pressure
as in the case of a large-deflection elastic-plastic solution, with geometrical weakening (see Fig
2.27, curve c). The plastic instability is defined by a zero slope on the pressure-deflection curve.
A large-deflection elastic-plastic analysis is required to detect ppi. It will also be detected in
experiments on actual vessels and it is possible to have plastic instability pressures less than lowerbounds to the limit pressure where the latter are based on small-deflection analyses. It may occur
that some of the above estimations of the limit pressures will be non-conservative estimates of the
real plastic collapse pressure, an instability pressure, if the estimates were applied to smalldeflection theoretical results.
2.5.5 Inelastic Progressive Plastic Deformation - Shakedown
Within the DBA approach the determination of the limit load, for a given constitutive law, is one
step. Proving that Progressive Plastic Deformation (PD) will not occur, or – more stringent – that
neither PD nor Accumulating Plasticity (AP) will occur, in other words, proving that the structure
under consideration will shake down to pure elastic behaviour under cyclic varying actions, is
another step. Considering this proof, which is to be obtained through numerical simulation, the
following information may be useful:
- In this proof the constitutive law may be, but needs not to be the same law as used in the
determination of the limit load. Normally the structure shakes down under cyclic actions which are
to be specified as functions of a single parameter. This parameter determines the sequence of the
actions and quite often this parameter is time. The proof of shakedown is easier to perform than the
proof against PD. Being conservative this approach yields the proof that neither PD nor AP occurs.
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- In the inelastic simulation the proof of shakedown can be performed by applying the action cycle
repeatedly.
- The proof may also be performed using Melan's shakedown principle:
•
Using an equivalent linear elastic structure, a given cyclic action results in a corresponding
cyclic stress field. Additionally a time-independent self-equilibrating stress field should be
found, such that, using superposition of both stress fields, the stress intensity does not exceed
the yield limit at any time in the cycle.
•
This approach is especially attractive in those particular cases where an appropriate selfequilibrating stress field is already known. A thermal stress field may serve as an example, as
well as the difference between a purely elastically determined stress field and the corresponding
field using plastic constitutive laws.
•
In many cases the proof may be performed using the check of primary + secondary stresses used
in a linear elastic DBA route, against the so-called 3f-criterion. The fulfilment of this criterion
is a necessary condition for shakedown. It is considered accurate enough for most cases
especially in combination with some other checks. However care should be taken whenever the
cyclic action contains a non-negligible time-invariant part e.g. a large contribution of selfweight.
2.5.6 Discussion
Again it can be seen that apparently simple requirements of the inelastic route can be problematic.
Limit or shakedown analysis could be used to directly estimate the limit and shakedown loads, but
until recently this was difficult if not impossible for complex structures. If elasto-plastic finite
element analysis is used there remains the problem of defining the plastic load – there are various
estimations as described above. The twice-elastic-slope method recommended by ASME has been
shown to give inconsistent results. The European standard aims to remove some of these problem
areas. In the following an overview of the new rules is given:
2.6 Design by Analysis in the European Standard
2.6.1 General
The European Standard has introduced the possibility of satisfying the requirement to avoid various
failure mechanisms directly through the detailed rules embodied in the new Direct Route, while
retaining the ‘conventional’ elastic route which uses stress categorisation. In addition it also
introduces several new concepts to help overcome the known difficulties with the current design by
analysis approach and to assist the formulation of the Direct Route. In particular the notion of an
‘action’ rather than a force, and the inclusion of ‘partial safety factors’ is a novel and welcome
addition to the area of pressure vessel design by analysis.
In the following some background to these new concepts is provided, followed by a summary of the
required design checks, with some explanation if required.
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2.6.2 New concepts
2.6.2.1 Principles and application rules
Like in the Eurocode (for steel structures) distinction is made between principles and application
rules. Principles comprise general statements, definitions and requirements for which there is no
alternative, and requirements and analytical models for which no alternative is permitted (unless
specifically stated). Application Rules are generally recognised rules which follow the principles
and satisfy their requirements; alternatives are allowed provided it is shown that they accord with
the relevant principle.
Typical examples are the primary and the primary & secondary stress criteria of the stress
categorisation approach, which are stated here, in slightly modified forms, as application rules.
2.6.2.2 Actions
This term, which replaces the old term loadings, denotes all thermo-mechanical quantities imposed
on the structure causing stress or strain, like forces (including pressure), temperature changes and
imposed displacements.
Actions are classified by their variation in time:
•
•
•
•
permanent actions (G)
variable actions (Q)
exceptional actions (E)
operating pressures and temperatures ( p, T ) - although these are variable actions, they are
considered separately to reflect their special characteristics (variation in time, random
properties, etc.).
The notion variable actions encompasses actions of quite different characteristics – from those
actions which are deterministically related to pressure and/or temperature, via actions not correlated
with pressure or temperature but with well defined (bounded) extreme values, to actions which can
be described only as stochastic processes not correlated with pressure or temperature, like wind
loads. Actions with a deterministic relationship with pressure and/or temperature shall be combined
in the pressure/temperature action and the relationship, exact or approximate, shall be used.
The characteristic values of actions describe the regime of actions which envelops all the actions
that can occur under reasonably foreseeable conditions. The characteristic values are used in
determining the design values of the actions, and they depend on the actions' (statistical) properties.
The characteristic values of permanent actions are usually their mean values (or credible extreme
values). The characteristic values of variable actions are defined as mean values, or p% percentiles, of extreme values, and values specified in relevant codes for wind, snow, earthquake
may be used; usually they are adapted to Eurocode concepts anyway. The upper characteristic value
of pressure shall not be smaller than the lesser of the set pressure of the protecting device or the
highest credible pressure that can occur under normal and upset conditions (reasonably
foreseeable), and the upper characteristic value of the temperature not smaller than the highest
credible temperature (under the same conditions). Therefore, the (limited) pressure excursion
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(overpressure) that occurs if a safety valve opens need not be included in the (maximum)
characteristic value of pressure; it is taken care of in the partial safety factors.
2.6.2.3 Partial safety factors
To allow for an easy, straightforward combination of pressure action with environmental ones, and,
at the same time, to give the flexibility, expected from a modern code, to adjust safety margins to
differences in action variation, likelihood of action combinations, consequences of failure,
differences of structural behaviour and consequences in different failure modes, uncertainties in
analyses, a multiple safety factor format was introduced, using different partial safety factors for
different actions, different combinations of actions, different failure modes and corresponding
resistances of the structure. Examples of partial safety factors are given in the following Table. The
corresponding combination rules for e.g. Design Check GPD-OC Global Plastic Deformation –
Operating Conditions are:
•
•
•
•
all permanent actions shall be included in each load case
each pressure action shall be combined with the most unfavourable variable action
each pressure action shall be combined with the corresponding sum of variable actions;
stochastic actions may be multiplied by the combination factor.
favourable actions shall not be considered.
The partial safety factors of pressure and resistances are calibrated with respect to the DBF results;
no attempt has been made to justify the partial safety factors by probabilistic investigations or
decision theory under uncertainty; if pressure is the only action the approach can be transformed to
a nominal design stress one.
Actions
Permanent γ G
Unfavourable
Favourable
Pressure γ P
Variable γ Q
Partial safety factors
Design check
GPD-OC
GPD-HT
Combination factor ψ
(stochastic actions)
Resistance γ R
(Temperature γ T )
1
1.35
1.0
1.2 (1.0)
1.5 (1.0)
1.35
1.0
1.0
-
0.91
1.01
1.25
(1.0)
1.05
(1.0)
If not specified differently in the relevant code of environmental actions.
2.6.2.4 Design checks – effects of actions
Design checks are investigations of the structure's safety under the influence of specified
combinations of actions - the design load cases - with respect to specified limit states (representing
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one or more failure modes). Characteristic values of the actions are multiplied by the corresponding
partial safety factors to obtain their design values and their combined design effect (on the
structure) is evaluated:
E d ( γ G G , γ p p , γ Q Q ,... , a d ,... )
In the design checks these design effects are compared with the corresponding design resistances,
obtained by dividing the resistance of the structure, corresponding to the action's combination, by
the relevant partial safety factor of the resistance:
E d ≤ R d = R ( G , p , Q ,... , a d , ) / γ R
This comparison can, in general, be performed in actions, in stress resultants (generalized stresses)
or in stresses.
The resistances are related to the limit states - states beyond which the part no longer satisfies the
design performance requirements.
2.6.2.5 Design checks – resistances
Design checks are designated by the failure modes they deal with. The following ones are
incorporated in the first issue of the standard:
•
•
•
•
•
gross plastic deformation (GPD), with corresponding failure modes ductile rupture and, for
"normal" designs, also excessive local strains
progressive plastic deformation (PD)
instability (I)
fatigue (F)
static equilibrium (SE).
Checks against gross plastic deformation
The design resistances are given by the lower-bound limit loads for
• proportional increase of all actions
• a linear-elastic ideal-plastic material (or a rigid ideal-plastic one)
• first-order theory
• Tresca's yield criterion and associated flow rule
• specified design strength parameters.
Design strength parameters R M and partial safety factors of the resistances γ R are chosen such that
for the simplest structures and pressure action only DBA and DBF results agree. The only exception
are steels, other than austenitic ones with A 5 ≥ 3 0 % , where the design strength parameter R M is
given by R eH , T or R p 0 . 2 , T and γ R = 1.25 for R eH / R m ≤ 0.8 and γ R = 1.5625 R eH / R m otherwise.
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If the procedure used to determine the limit action does not give an (absolute) maximum in the
region with maximum absolute values of principal strains less than 5%, the boundary maximum, for
which the maximum absolute value of the principal strains equals 5%, shall be used.
As an application rule the "usual" primary stress criterion is given, formulated in stresses and - for
structures where the concept of stress resultants is applicable - in stress resultants and local
(technical) limit loads.
These checks (against GPD) are considered also to encompass Excessive Yielding, provided
"usual" design details (with not too severe strain concentrations) exist.
Checks against progressive plastic deformation
On repeated application of specified action cycles PD shall not occur for
• a linear-elastic ideal-plastic material
• first-order theory
• Mises' yield condition and associated flow rule
• specified design strength parameters RM.
A slight modification of the "usual" 3 f criterion is given as application rule; it is noted that this
application rule, which is derived from shakedown considerations, is only a necessary condition for
the fulfilment of the principle, but is considered, together with all the other checks, to be sufficient
to achieve the principle's goal - avoidance of ratchetting in the structure.
Check against fatigue failure
Reference is made to the Fatigue Assessment section of the Standard.
Instability
Static equilibrium
The usual checks against overturning and (rigid body) displacement are stated explicitly, using the
partial safety factors given in the other checks.
2.6.3 Application remarks
Whether the Direct Route or the Stress Categorisation Route is followed, it is imperative that all
stated checks are considered:
•
•
in the Direct Route:
At least the five given checks – sometimes it may be necessary to include additional ones, like
excessive deformation (leakage). Not all of the checks will require calculations, but all must be
considered – e. g. it may be obvious that instability can be excluded.
in the Stress Categorisation Route:
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Check of Primary Stresses
Check of Primary + Secondary Stresses
Check of Total Stresses (Primary + secondary + peak stresses) - Fatigue
Usually it is required to perform each of these checks for different load cases – for different
combinations of coincident actions, as well as for different characteristic values of actions, e. g.
different pressure – temperature pairs.
The Design by Analysis route may be chosen to prove conformity of a design also for a part of a
component, suitably selected and limited; and with appropriate boundary conditions.
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2.7 REFERENCES
[1] W. C. Kroenke, J. L. Hechmer, G. L. Hollinger & A. J. Pedani, “Component evaluation using
the finite element method. In “Pressure Vessel & Piping Design: A Decade of progress, 19701980”, Chap. 2.11, ASME, 1981.
[2] W. C. Kroenke, “Classification of finite element stresses according to ASME Section III stress
categories,” Proc 94th ASME Winter Annual Meeting, 1973.
[3] W. C. Kroenke et al, “Interpretation of finite element stresses according to ASME III,” ASME
Tech. Paper 75-PVP-63, 1975.
[4] N.V.L.S. Sarma, G. L. Narasaiah & G. Subhash, “A computational approach for the
classification of FEM axisymmetric stresses as per ASME Code,” Proc ASME Pressure Vessel &
Piping Conf, Pittsburgh, 1988.
[5] J. L. Gordon, “OUTCUR: An automated evaluation of two dimensional finite element stresses
according to ASME,” ASME Paper 76-WA/PVP-16, 1976.
[6] J. L. Hechmer & G. L. Hollinger, “Considerations in the calculations of the primary plus
secondary stress intensity range for Code stress classification,” “Codes & Standards and
Applications for Design and Analysis of Pressure Vessel and Piping Components” Ed R. Seshadri,
ASME PVP Vol.136, 1988.
[7] B. W. Leib, “An automatic surface element generator for calculating membrane and bending
stresses from three dimensional finite element results,” Proc 4th Int Conf on “Structural Mechanics
in Reactor Technology”, San Francisco, 1977.
K. H. Hsu & D A McKinley “SOAP - a computer program for classification of three
dimensional finite element stresses on a plane,” Proc ASME Pressure Vessel & Piping Conference,
Nashville, 1990.
[8] J. L. Hechmer & G. L. Hollinger, “Three dimensional stress criteria - a weak link in vessel
design and analysis,” ASME Special Publ. PVP 109 “A Symposium on ASME Codes and Recent
Advances in Pressure Vessel and Valve Technology” Ed J. T. Fong, 1986.
J. L. Hechmer & G. L. Hollinger, “Three dimensional stress criteria -application of Code rules,”
ASME Special Publ. PVP 120 “Design and Analysis of Piping, Pressure Vessels and Components”
Ed W. E. Short, 1987.
J. L. Hechmer & G. L. Hollinger, “Code evaluation 3D stresses on a plane, “ Codes & Standards
and Applications for Design & Analysis of Pressure Vessels & Piping, ASME PVP-Vol.161, 1989.
[9] J.L. Hechmer & G.L. Hollinger, “Three dimensional stress criteria,” ASME PVP-Vol.210-2
Codes and Standards and Applications for Design and Analysis of Pressure Vessel & Piping
Components, Ed R. Seshadri & J.T. Boyle, 1991.
G. Hollinger, “Summary of three dimensional stress classification,” Proc Int Conf on Pressure
Vessel Technology, Dusseldorf, 1992.
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[10] R.L. Roche, “Practical procedures for stress classification,” Int Journ Press Vess & Piping,
Vol.37, 27-44, 1989.
[11] D.L. Marriott, “Evaluation of deformation or load control of stresses under inelastic conditions
using elastic finite element analysis,” Proc ASME Pressure Vessel & Piping Conf, Vol.136,
Pittsburgh, 1988.
[12] A. Kalnins & D.P. Updike, “Role of plastic limit and elastic plastic analyses in design,” ASME
PVP-Vol.210-2 Codes and Standards and Applications for Design and Analysis of Pressure Vessel
& Piping Components, Ed R. Seshadri & J.T. Boyle, 1991.
A. Kalnins & D.P. Updike, “Primary stress limits on the basis of plasticity,” ASME PVPVol.230, Stress Classification, Robust Methods and Elevated Temperature Design, Ed R. Seshadri
& D.L. Marriott, 1992.
[13] D. Mackenzie & J. T. Boyle, “A computational procedure for calculating primary stress for the
ASME B&PV code,” Trans ASME, Jrn Pressure Vessel Tech, Vol. 116, No. 4, 1994.
D. Mackenzie, J. Shi, R. Hamilton & J. T. Boyle, "Simplified lower bound limit analysis of
pressurised cylinder-cylinder intersection Shells using a generalised yield criteria", Int Jrn of
Pressure Vessels & Piping, 67, pp. 219-226, 1996.
J. T. Boyle, R. Hamilton, J. Shi, & D. Mackenzie, "A simple method of calculating Limit Loads
for thin axisymmetric shells", Trans. American Society of Mechanical Engineers (ASME), Jrn
Pressure Vessel Technology, Vol. 119, No.2, pp. 236-242, 1997.
[14] R. Seshadri & C.P.D. Fernando “Limit loads of mechanical components and structures using
the GLOSS r-node method”, Proceedings of ASME PVP, Vol. 210-2, pp. 125-134, 1991.
[15] A.R.S. Ponter, K.F. Carter, “Limit state solutions, based upon linear elastic solutions with a
spatially varying elastic modulus”, Jrn of Computer Methods in Applied Mechanics and
Engineering, Vol.140, No.3-4, pp.237-258, 1997.
A.R.S. Ponter, K.F. Carter, “Shakedown state simulation techniques based on linear elastic
solutions”, Jrn of Computer Methods in Applied Mechanics and Engineering, Vol.140, No.3-4,
pp.259-279, 1997.
[16] T. Seibert, J. L. Zeman, „Analytischer Zulässigkeitsnachweis von Druckgeräten“, Techn.
Überwachung, Bd. 35 (1994) Nr. 5, 222-228.
W. Poth, J. L. Zeman, „Grenztragfähigkeit der
Innendruckeinwirkung“, Konstruktion 48 (1996), 219-223.
Zylinder-Kegel-Verbindung
unter
J. L. Zeman, „Ratcheting limit of flat end cylindrical shell conections under internal pressure“,
Int J Pres Ves & Piping 68 (1996), 293-298.
R. Preiss, F. Rauscher, D. Vazda, J. L. Zeman, „The flat end to cylindrical shell connection –
limit load and creep design“, Int J Pres Ves & Piping, 75 (1998),715-726
[17] D. Mackenzie, J. T. Boyle & R. Hamilton, “Application of Inelastic Finite Element Analysis to
Pressure Vessel Design”, International Conference on Pressure Vessel Technology, Volume 2,
ASME 1996.
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[18] J. C. Gerdeen, “A critical Evaluation of Plastic Behaviour Data and a United Definition of
Plastic Loads for Pressure Components”, WRC Bulletin 254, November 1979, ISSN 0043-2326.
[19] A. Kalnins, D. Updike & J.L. Hechmer, “On Primary Stress in Reducers”, ASME PVP-Vol.
210-2, pp. 117-124
[20] D. Mackenzie, J.T. Boyle, J. Spence, "Some Recent Developments in pressure vessel Design
by Analysis" Proc IMechE, Part E, Journal of Process Mech Eng, 1994, 208, 23-30.