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Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 139
5. Finite Element Analysis of Bellows
5.1 Introduction:
Traditional design process and stress analysis techniques are very specific for
each individual case based on fundamental principles. It can only be
satisfactorily applied to a range of conventional component shapes and
specific loading conditions using sound theories. Also this design process
needs continuous improvement till the product becomes matured and proven
successful by customers. After that the product becomes standardized. This
methodology is followed by majority industries for their products.
But in case of customized products, every individual product has unique
design features. Specific geometric parameters are altered in order to achieve
desirable function from the product. Hence, traditional design technique is not
much useful because of very frequent changes in the design calculations.
Expansion joints are such customized products, which needs to be treated
individually for varieties of applications. Every time design procedure is carried
out carefully and minor modifications are also required. During traditional
design process many ambiguities remains in the mind of designers because of
varieties of application areas of expansion joints. Thus, designers normally use
higher safety factors in order to minimize risk. This leads to over design
components by specifying either unnecessarily bulky cross sections or high
quality materials. Inevitably the cost of the product is adversely affected. Finite
Element Analysis (FEA) provides a better solution for design and stress
analysis in the virtual environment.
Finite Element Analysis (FEA) is a computer-based numerical technique for
calculating the strength and behavior of engineering structural components. It
can be used to calculate deflection, stress, vibration, buckling behavior and
many other phenomena. It can be used to analyze either small or large scale
deflection under loading or applied displacement. It can analyze elastic
deformation, as well as plastic deformation. Finite element analysis makes it
possible to evaluate in detail the complex structures, in a computer, during the
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 140
planning of the structure. The demonstration of adequate strength of the
structure and the possibility of improving the design during planning can justify
the need of this analysis work.
The first issue to understand in FEA is that it is fundamentally an
approximation. The underlying mathematical model may be an approximation
of physical system. The finite element itself approximates what happens in its
interior with the help of interpolating formulas.
In the finite element analysis, first step is modeling. Using any special CAD
software, model can be generated using the construction and editing features
of the software. In finite element method the structure is broken down into
many small simple blocks called elements. The material properties and the
governing relationships are considered over these elements. The behavior of
an individual element can be described with a relatively simple set of
equations. Just as the set of elements would be jointed together to build the
whole structure, the equations describing the behavior of the individual
elements are also joined into an extremely large set of equations that describe
the behavior of whole structure.
The computer can solve large set of simultaneous equations. From the
solutions, the computer extracts the behavior of the individual elements. From
this, it can get the stress and deflection of all parts of a structure. The stresses
will be compared to permissible values of stress for the materials to be used,
to see if the structures are strong enough.
Interpretation of the results requires knowing what is an acceptable
approximation, development of a complete list of what should be evaluated;
appreciation of the need of margin of safety, and comprehension of what
remains unknown after an analysis.
There are many softwares available for finite element analysis, which can be
utilized for the engineering applications. They are ANSYS, Pro/Engineer,
CATIA, NASTRAN, Hyper Works, I-DEAS etc.
5.2 Overview of FEA Procedure:
1. Modeling of the component
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 141
A model is required to be generated for the component which is to be
analyzed. Designer has to choose proper type of element for the analysis.
Actually the kind of component behavior is required to be considered at this
stage. The model can be either one dimensional, two dimensional or three
dimensional. One dimensional model can be generated by using 2 D spar
element; two dimensional models can be generated by key points or directly
generating two dimensional shapes like rectangle, circle, etc. Here union,
intersection and subtraction of area kind of commands are very useful. In case
of three dimensional modeling two dimensional shape can be extruded to third
direction, or revolve command is useful. Every designer can have own idea for
generating model. Many times component is consist of many small parts,
hence all parts are required to be modeled, than assembly function is required.
Here type of fit can also be selected as per requirements. Pro/Engineer
software is facilitating this kind of features. While earlier ANSYS software does
not provide assembly feature. This feature is added in workbench module. All
softwares are having their distinct features as well as limitations. User has to
make proper choice for applications.
2. Descritization of the continuum:
The continuum is the physical body, structure, or solid being analyzed.
Descritization may be simply described as the process in which the given body
is subdivided into an equivalent system of finite elements. Elements are
nothing but a small portion of the continuum which represents the whole
continuum that is being analyzed. The finite elements may be triangles, groups
of triangles or quadrilaterals for a two dimensional continuum. For three
dimensional analysis, the finite elements may be tetrahedral, rectangular
prisms or hexahedral.
In some cases the extent of the continuum to be modeled may not be clearly
defined. Only a significant portion of such a continuum needs to be considered
and descritized. Indeed, practical limitations require that on should include only
the significant portion of any large continuum in the finite element analysis.
3. Selection of the displacement models
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 142
The assumed displacement functions or models represent only approximately
the actual or exact distribution of the displacements. A displacement function is
commonly assumed to be a polynomial and practical considerations limit the
number of terms that can be retained in the polynomial. The simplest
displacement model that is commonly employed is a linear polynomial.
Obviously, it is generally not possible to select a displacement function that
can represent exactly the actual variation of displacement in the element.
Hence, the basic approximation of the finite element method is introduced at
this stage.
There are interrelated factors which influence the selection of a displacement
model. Usually, since a polynomial is chosen, only the degree of the
polynomial is open to decision. The particular displacement magnitudes that
describe the model must also be selected. These are usually the
displacements of the nodal points.
4. Derivation of the finite element stiffness matrix:
The stiffness matrix consists of the coefficients of the equilibrium equations
derived from the material and geometric properties of an element and obtained
by use of the principle of minimum potential energy. The stiffness relates the
displacements at the nodal points (the nodal displacement) to the applied
forces at the nodal points (the nodal forces). The distributed forces applied to
the structure are converted into equivalent concentrated forces at the nodes.
The equilibrium relation between the stiffness matrix [k], nodal force vector {Q},
and nodal displacement vector {q} is expresses as a set of simultaneous
algebraic equations,
[k] {q} = {Q} (5.1)
The elements of the stiffness matrix are the influence coefficients. A stiffness
of a structure is an influence coefficient that gives the force at one point on a
structure associated with a unit displacement of the same or a different point.
The stiffness matrix for an element depends upon (1) the displacement model,
(2) the geometry of the element, and (3) the local material properties. For an
elastic isotropic body, a pair of parameters such as the young’s modulus E and
the Poisson’s ratio define the local material properties. Since material
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 143
properties are assigned to a particular finite element, it is possible to account
for non-homogeneity by assigning different material properties to different finite
elements in the assemblage.
5. Assembly of the algebraic equations for the overall descritized continuum.
This process includes the assembly of the overall or global stiffness matrix for
the entire body from the individual element stiffness matrices, and the overall
global force or load vector from the element nodal force vectors. The most
common assembly technique is known as the direct stiffness method. In
general, the basis for an assembly method is that the nodal interconnections
require the displacements at a node to be the same for all elements adjacent
to that node. The overall equilibrium relations between the total stiffness matrix
[K], the total load vector {R}, and the nodal displacement vector for the entire
body {r} will again be expresses as a set of simultaneous equations.
[K] {r} = {R} (5.2)
These equations can not be solved until the geometric boundary conditions are
taken into account by appropriate modification of the equations. A geometric
boundary condition arises from the fact that displacements may be prescribed
at the boundaries or edges of the body or structure.
6. Solutions for the unknown displacements
The algebraic equations assembled in above step are solved for the unknown
displacements. For liner equilibrium problems, this is a straightforward
application of matrix algebra techniques. For non-linear problems, the desired
solutions are obtained by a sequence of steps, each step involving the
modification of the stiffness matrix and/or load vector.
7. Computation of the element strains and stresses from the nodal displacement
In certain cases the magnitudes of the primary unknowns, that is the nodal
displacements, will be all that are required for an engineering solution,. More
often, other quantities derived from the primary unknowns, such as strains
and/or stresses, must be computed.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 144
The stresses and strains are proportional to the derivatives of the
displacements and in the domain of each element meaningful values of the
required quantities are calculated. These “meaningful values” are usually taken
as some average value of the stress or strain at the center of the element.
5.3 The basic Element Geometry:
When modeling any structural problem, the geometry must be split in to a
variety of element. To do this, elements essentially have one of the five basic
forms shown in table 5.1.
Table 5.1: Basic forms of Elements
Dimensionality Type Geometry
Point Mass
Line
Spring, beam,
bar, spar, gap,
torsion
Area
2D continuum,
axi-symmetric
continuum, plate
or flat shell
Curved Area
Generalized
shell
Volume 3D continuum
5.4 Typical range of elements:
Finite element solutions program has a library of element types it understands.
The model must be built using only these supported elements if we want to
solve the model in solution. The elements supported for the different analysis
types are presented in the table 5.2.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 145
Table 5.2: Types of Elements
Element Type Degrees of Freedom Representations
Mass -
2D bar u, v
2D beam v, Oz
2D continuum
plane stress
plane strain
axisymmetric
u,v
2D interface u,v
Axisymmetric
shell u,v, Oz
3D bar u, v, w
3D beam u, v, w, Ox,
Oy, Oz
3D solid u, v, w
3D Shell u, v, w, Ox,
Oy, Oz
3D interface u, v, w
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 146
5.5 The Rules for Compatibility:
If elements are compatible internally and across their boundaries then, as the
mesh is refined, the solution will coverage to the exact solution of the finite
element method.
Element must have the same order, although one can mix three sided and
four sided elements.
There must be connection between the nodes of adjacent elements if the
element is 1 D, between the edges of adjacent elements if the element is 2 D and
between faces of adjacent elements if the element is 3 D.
5.6 Structure Material Property:
To carry out a successful stress analysis for the purpose of design, analyst must
provide the material properties, in particular the elastic constants (Young’s’
modulus and Poisson’s ratio) and strengths. Other properties such as thermal
conductivity, wear resistance and corrosion allowance may be relevant to the
product function.
A body is homogenous if it has identical properties at all points, and
It is considered as isotropic when its properties do not vary with direction or
orientation.
The property, which varies with orientation, is said to be an isotropic
property. Metal becomes anisotropic when they are deformed severely in a
particular direction, as happens in rolling and forging.
5.7 Meshing:
The arrangement of the elements through the continuum is known as the form or
topology of the mesh. The elements can be arranged in any manner, provided that
the faces of the elements are positioned correctly. This means that to ensure
compatibility of the mesh, the edges of two-dimensional and the faces of three
dimensional elements, which are touching, must be in contact, with edge exactly
matching edge or face exactly matching face and with node matching node.There
are two ways in which the mesh structure can be arranged. The first is the regular
form (topology) or irregular form.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 147
5.7.1 Free and Mapped Meshing:
Nodes and elements are generated by one of the two methods, mapped or free
mesh. Mapped meshing requires the same number of elements on opposite sides
of the mesh area and requires that mesh areas are bounded by three or four
edges. If one defines a mapped mesh area with more than four edges, one must
define which vertices are the topological corners of the mesh. Mapped mesh
boundaries with three corners will generate triangular elements.
5.7.2 Mesh Refinement:
Once the mesh has been generated, it is possible to modify it in such a way that
the better solution can be produced. On many occasions the solution will be good
over most of the model but will need refinement / enrichment in one or two
regions. Mesh modification technique can be applied after a solution has been
produced on an initial mesh. There are three ways to refine mesh, H refinement, P
refinement and HP refinement.
5.7.3 Mesh Enrichment:
The original mesh has regular spacing but enrichment is required near to stress
concentration area. Here more number of nodes and elements are generated in
the mesh enrichment.
5.7.4 Quality Criteria of meshing:
New generation Computer Aided Engineering (CAE) softwares have very useful
meshing features. Modern software consists of wide range of meshing
characteristics. The main objective for the quality mesh are control on size of
elements, coarse or fine mesh, addition and removal of node points depending on
surface, and editing features of elements. They define all these features through
measure of certain quality criteria of meshing. They are described bellow.
5.7.4.1 Aspect ratio: Quadrilateral area will have four side characteristics. Other
shape may be divided into triangles. An aspect ratio is defined as the ratio of
maximum to minimum characteristic of dimensions. Equilateral triangular
configurations are best elements. But it is very difficult to achieve all elements of
similar size. Therefore practically at least 70% elements should have similar size.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 148
5.7.4.2 Maximum angle: The corner angles of an element will have variable
values in terms of degree. The maximum angle should be up to 1200.
5.7.4.3 Minimum angle: The corner angles of an element will have variable
values in terms of degree. The minimum angle should be at least 300.
5.7.4.4 Biasing
The biasing sub-panel allows the user to control the distribution of nodes during
the nodes seeding by selecting biasing in the form of linear, exponential or bell
curve distributions. Figure 5.1 shows basing node arrangement.
Figure 5.1 : Biasing of nodes along length
5.7.4.5 Skew: The skew angle is the difference between right angle and angle of
a parallelogram. For homogeneous element arrangement the minimum and
maximum skew angles are observed. The minimum difference is always
preferable in skew angle.
5.7.4.6 Morphing:
This is a mesh morphing tool that allows user to alter finite element models while
keeping mesh distortions to a minimum.
Morphing tool can be also useful to:
• Change the profile and the dimensions of mesh
• Map an existing mesh onto a new geometry, and
• Create shape variables that can be used for optimization
5.7.4.7 Masking:
The masking tools allow the user to show and hide select entities that might
interfere with desired visualization.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 149
5.8 Restraints:
Restraints are used to restrain the model to ground. Restraints also have six
values at nodes; three translations and three rotations. Each entry can either have
a value for the fixed displacement or is left free to move.
5.9 Constraints:
Constraints are used to constrain nodes to other nodes, not to ground. They can
be used to impose special cases of symmetry boundary conditions, or special
relationships between nodes.
5.10 Structural Loads:
Structural loads can be nodal forces or pressures on the face or edge of an
element. A nodal force has six values for three forces and the three moments.
5.11 Boundary Conditions:
Any analysis case consists of model subjected to constraints, restraints, structural
loads and heat transfer or other similar scalar field loads. The boundary conditions
are applied to build analysis cases containing loads and restraints of the model. In
finite element analysis, the model is considered to be in equilibrium. So the loads
and the moments should be such that the equilibrium condition satisfied.
ΣF = 0 & ΣM = 0 (5.3)
If the end condition of the model is not applied to the model then the reaction at
that point or edge or surface should be applied to make it in equilibrium.
Boundary conditions can be applied to the part geometry before meshing or the
resulting nodes and elements after meshing. Applying boundary conditions to the
part geometry will mean that if the part is changed and the model is updated, the
boundary conditions will also be updated.
There are two special cases of boundary conditions; symmetry and anti-
symmetry, which can be utilized for as per requirement.
5.12 Computer Aided Engineering softwares:
Various softwares are available for the finite element analysis. All are having
different area of specialization. Some softwares like Pro/Engineer, I-DEAS,
Mechanical Desk Top etc. are having very wide range of modeling features.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 150
Hyper-Mesh software is good at meshing or descrtization features. CosMos, LS-
Dyna, ANSYS etc. are good at engineering analysis. Effluent is specialized for
Computational Fluid Dynamics (CFD).
5.13 Linear Analysis and Non-linear Analysis:
When one could not achieve accuracy in the solution from linear finite element
analysis, non-linear methodology should be utilized. A nonlinear solution is a
series of successive linear steps (iterations) along a path that is not straight. But
nonlinear solutions require more data and it takes more time to setup and solve.
Most of the world is nonlinear. In many cases, simply understanding the
effects of the nonlinearity can enable a design engineer to make sound
design decisions on linear results. All problems could be run as nonlinear
analyses, but it should be used when only it is necessary.
5.13.1 Types of nonlinear behavior:
Following are the types of non-linear solutions.
Yielding/plasticity (beyond Hooke's law: s = Ee)
Changing contact or interference
Large displacement, large rotation, large strain, stress stiffening
Manufacturing processes (mold filling, forging, rolling, stamping, welding)
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 151
5.14 Stress Analysis using FEA:
The bellows are designed by customized approach for individual application. The
prototype testing is highly time consuming and costly task. Also, the measurement
of stresses is very difficult part during the testing bellows. Hence, computer based
Finite Element Analysis will be very much useful for the designers to estimate the
stresses for newer geometry bellows. This exercise is carried out with the
objective that the stresses can be estimated of bellows with FEA methodology.
This technique will be beneficial to designers and manufacturers for faster design
and analysis.
The primary function of expansion joint is to absorb axial (longitudinal),
perpendicular (lateral) and angular motions in the long piping and ducting. The
bellows are most critical part of expansion joint assembly, which takes of all these
movements of piping. Figure 5.2 shows axial movement of metallic bellows.
Figure 5.2: Axial Motion of bellow
The motions or movements are developed because of differential variation in
pressure and temperature inside the long piping. Many times shocks are also
developed because of sudden stop and start of fluid flow in the piping. The
pressure fluctuations and temperature variations creates unpredictable stresses in
the piping. Since the bellows are formed from very thin sheet metals, the
movement creates deformation in the elastic as well as plastic region. Hence, It is
very difficult to estimate the stresses developed in the expansion joint assembly.
The induced membrane stresses in the bellow material must be less than the
allowable stress of the materials at the design temperatures. The bellow should be
flexible in order to get flexibility and tough to resist pressure fluctuations. This
conflicting need for thickness for pressure capacity and thinness for flexibility is
the unique design problem faced by the expansion joint designers.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 152
Bellow is made from SS 304 stainless steel sheets. Other properties are
mentioned in material properties.
5.14.1 Geometry of bellows:
Figure 5.3: Geometry of a bellow
Figure 5.3 shows a bellow with two convolution and single ply material with
reference to following geometrical nomenclature.
Db = Inside diameter of the pipe / bellow = 30 cm
N = Number of convolution = 2
w = Height of convolution = 3.5 cm
q = Pitch of convolution = 4.0 cm
Lt = Lc = Tangent length and Collar length = 2.5 cm
Thickness of materials = 0.05 cm
Number of ply of material = 1
1
X
Y
Z
AUG 17 201009:39:51
ELEMENTS
Figure 5.4: Bellow model
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 153
5.14.2 Stresses in metallic bellows:
The expansion joints are loaded with internal pressure due to flowing fluid at inner
surfaces. In order to get higher flexibility, bellow is made from thin sheet metal.
Since the thickness of the material is very less compare to other two dimensions,
membrane stress are produced.
The stresses are developed in the radial / circumferential direction as cylindrical
shape of bellow. The outer cylindrical surface of bellow, undergo maximum stress
value, called hoop stress or circumferential stress. The approximate value of this
stress can evaluate by following equation number.
Circumferential stress = tDP
2 (5.4)
The stress produced in the longitudinal direction, along the flow of liquid is
longitudinal stress or meredional stress. In case of hollow pipes, the longitudinal
stress is approximately half the circumferential stress.
Longitudinal membrane stress = tpnwP
2 (5.5)
Longitudinal bending stress = Cptpw
nP
2
2
(5.6)
These relationships are based on shape of convolution; they may not give true
stress value for all types of bellows. The bellows consists of some number of
convolutions, and hence a stress due to bending is produced, which is very high
compare to its membrane stress due to fluid pressure. The total longitudinal
stress will be combined effect of stress due to membrane and bending.
Estimating the stresses produced is depending upon number of parameters. They
are internal pressure fluctuations, inside temperature and its variation, material
properties, geometrical parameters, convolution shapes, material thickness,
number of plies, heat treatment of the material etc. Considering the complexity of
the case, accurate prediction of stress is difficult. Many researchers’ have
contributed to develop mathematical models, but the results are varied because of
change in the geometry of the convolutions. Hence there is no general purpose
solution available to this.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 154
The EJMA has developed the codes for evaluation of stresses. This includes finer
details of the shape of bellow and estimates precise stresses. Actual
experimentation is possible but it is very difficult to measure the various stresses
developed. Therefore computerized technique is more convenient for the stress
analysis2. Even one researcher has suggested that consideration of strain
concentration can be also a useful approach for the greater accuracy in design of
bellows.
In the present study ideal geometry U - shape of convolutions are selected. Finite
Element Analysis carried out using ANSYS software.
Assumptions for the analysis:
1. The material used is homogeneous and isotropic.
2. The material thickness is uniform throughout its cross section.
3. The inside temperature is room temperature and it is constant.
4. The deformation taking place is within elastic limit. Material obeys Hook’s
law of elasticity.
5. All convolutions are equal in size at pitch distance.
Element selection:
The important task at the beginning of Finite Element Analysis is selection of type
of element. Bellow material have very less thickness, hence shell element should
be selected. There are various shell elements which can be used for the analysis.
They are Shell 28, Shell 41, Shell 43, Shell 63, Shell 93, Shell 143, Shell 150, and
Shell 181. Here shell 181 element is chosen for the analysis which is having
following features.
SHELL181 is suitable for analyzing thin to moderately-thick shell structures. It is a
4-node element with six degrees of freedom at each node: translations in the x, y,
and z directions, and rotations about the x, y, and z-axes. In case of membrane
option used, the element has only translational degrees of freedom. The
degenerate triangular option should only be used as filler elements in mesh
generation.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 155
SHELL181 is well-suited for linear, large rotation, and/or large strain nonlinear
applications. Change in shell thickness is accounted for in nonlinear analyses. In
the element domain, both full and reduced integration schemes are supported.
SHELL181 accounts for follower (load stiffness) effects of distributed pressures.
SHELL181 may be used for layered applications for modeling laminated
composite shells or sandwich construction.
Material properties:
The bellow material is SS 304 sheets. It possess following properties.
Modulus of elasticity, E = 19728607 N/cm2.
Poisson’s ratio = 0.3
Yield stress of the material, Sy = 20310 N/cm2.
Allowable stresses of the material, Sab = 12738 N/cm2.
Coefficient of Thermal Expansion, α = 17.3 x 10-6 m/m K.
Reference temperature, T = 273 K.
Uniform temperature = 300 K.
Constraints:
As the expansion joints are fixed through the collar at both ends. The
displacement constraint is made fixed at tangent length on both sides. The inside
fluid pressure will be acting on inner wall of convolution as well as tangent area.
Collars are not included in the model; hence its effect should be neglected for
stress evaluation. It is also assumed that the fluid pressure is to be born by
convolutions only.
Tangent length of bellow = Ux = Uy = 0
Loading conditions:
The material properties are given for the analysis as following. The element ‘shell
181’ is suitable for analyzing thin to moderately-thick shell structures. It is a 4-
node element with six degrees of freedom at each node: translations in the x, y,
and z directions, and rotations about the x, y, and z-axes. Change in shell
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 156
thickness is accounted for in nonlinear analyses. It is may be used for layered
applications for modeling laminated composite shells or sandwich construction.
In actual practice the bellows are pressurized by high pressure fluid flow. In the
present study, this is simplified by applying uniform pressure at inside surfaces of
bellow. The pressures (gauge) at inside surface are taken as 2.5, 5, 7.5 and 10
N/cm2.
5.14.3 FEA Results:
Table 5.3: FEA Results from ANSYS
Gauge Pressure
N/cm2
Deflection,
cm
Circumferential stress
N/cm2
Longitudinal stress
N/cm2
2.5 0.00415 883 2883
5 0.0083 1766 5766
7.5 0.0125 2649 8649
10 0.0166 3532 11532
FEA result images are shown in figure 5.5 and 5.6. It shows that the maximum
stresses are developed near to root area of the convolutions. This is because of
stress concentration effect. The maximum stresses are surrounding the root
diameter because of its symmetrical shape.
5.14.4 Analytical Results:
Table 5.4: Analytical Results Gauge Pressure
N/cm2
Circumferential stress
N/cm2
Longitudinal stress
N/cm2
2.5 750 3039
5 1500 6078
7.5 2250 9117
10 3000 12156
Table 5.4 shows analytical results of stresses of bellows calculated using EJMA
codes. Results are compared at gauge pressure of 10 N/cm2. Analytical results
are calculated using equations 5.1, 5.2 and 5.3.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 157
1
MNMX
XY
Z
-1016
-797.902-579.785
-361.668-143.551
74.566292.683
510.8728.917
947.034
SEP 4 201014:18:23
ELEMENT SOLUTION
STEP=1SUB =1TIME=1SX (NOAVG)RSYS=0DMX =.019429SMN =-1016SMX =947.034
Figure 5.5 : Results from FEA
Figure 5.6 : Stress distribution
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 158
Circumferential stress = tDP
2 =
05.023010
xx = 3000 N/cm2.
Longitudinal membrane stress = tpnwP
2 =
047.0125.310
xxx = 372 N/cm2.
Longitudinal bending stress = Cptpw
nP
2
2
=2
047.05.3
1210
xx0.425=11784 N/cm2.
Total longitudinal stress = 372 + 11784 = 12156 N/cm2.
Longitudinal membrane & bending stress ≤ Sab x Factor for formed bellow
12156 ≤ 12738 x 3 = 38214; stresses are within safe limit.
5.14.5 Graphs:
00.0020.0040.0060.0080.01
0.0120.0140.0160.018
2.5 5 7.5 10
Pressure, N/cm2
Disp
lace
men
t, cm
Displacement
Figure 5.7: Nodal Displacement
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 159
0
2000
4000
6000
8000
10000
12000
14000
2.5 5 7.5 10
Pressure, N/cm2
Stre
sses
, N/c
m2
Circumferential stress Longitudinal stress
Figure 5.8: Stresses developed in bellows vs pressure (FEA)
0500
1000150020002500300035004000
2.5 5 7.5 10
Pressure, N/cm2
Circ
umfe
rent
ial s
tres
s, N
/cm
2
FEA Analytical
Figure 5.9 : Comparison of Circumferential stress
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 160
0
2000
4000
6000
8000
10000
12000
14000
2.5 5 7.5 10
Pressure, N/cm2
Long
itudi
nal s
tress
, N/c
m2
FEA Analytical
Figure 5.10 : Comparison of Stress intensity (longitudinal) 5.14.6 Stress distribution in the bellow:
The stresses developed due to loading can be visualized as per their location.
Some node locations are selected as showing in figure 5.11. The values of
resultant stresses are shown in appendix C.
Figure 5.11 : Selected nodal point locations for stress analysis
The results of stresses are plotted as per node location are shown in figure 5.12
and figure 5.13. They show distribution of longitudinal stresses and distribution of
circumferential stresses at selected nodes.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 161
Maximum stress intensity (Longitudinal)
-2000
0
2000
4000
6000
8000
10000
12000
0 5 10 15 20
Location Number
Max
imum
str
ess
inte
nsity
, N/
cm2
Figure 5.12: Longitudinal stress distribution
Figure 5.10 shows that the maximum longitudinal tensile stresses are developed
at location number 5, 6, 11, 12, & 13. The longitudinal stress at tangent length is
tends to zero. The longitudinal stress is maximum at convolution faces and root
area, while it is minimum at crest of convolution.
Circumferential stress distribution
-4000
-3000
-2000
-1000
0
1000
2000
3000
0 5 10 15 20
Location number
Circ
umfe
rent
ial s
tress
, N
/cm
2
Figure 5.13: Circumferential stress distribution
Figure 5.13 shows that the circumferential compressive stresses are developed at
location number 3, 4, 7, and 8. The maximum tensile circumferential stresses are
developed at 12. This is root area of bellow, which undergoes very high
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 162
compressive stress. The bellow convolution will be deflected due to loading. The
displacement of each node is listed in table 5.5. Table 5.5: Displacement at various nodes
Location No. Node No. δx δy
1 1 0 0
2 37 0 0
3 120 -0.00135 0.000180
4 103 -0.000298 0.000398
5 215 -0.0170 0.00783
6 218 -0.0170 0.00621
7 312 -0.00377 0.00189
8 295 -0.00571 0.00377
9 408 0.00364 0.00786
10 391 0.000868 0.00669
11 1841 -0.0000204 -0.00206
12 1843 -0.0000209 -0.00223
13 632 -0.000150 0.00304
14 615 -0.0000214 0.000121
The displacement curve as per absolute co-ordinates is plotted in figure 5.12. The
deformation curve shows that the deflection is uniform and maximum is near to
convolution flank, which is developing longitudinal stress.
Figure 5.14 : Displacement curve of convolution surface
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 163
5.14.7 Observations:
1. Nodal displacement (axially) increases with increase in internal pressure of
bellow. (Figure 5.7) Circumferential stresses and longitudinal stresses
increase with increase in pressure. (Figure 5.8) The stresses developed
are well within the permissible limit of the material.
2. Longitudinal stresses are higher than circumferential stresses. This is
because of bending effect at convolution faces. As stresses because of
bending is always higher than direct stresses causes due to fluid pressure.
This is agreeable to the analysis of EJMA.
3. Maximum stresses produced at the root area of bellow. This is due to
stress concentration effect. The remedial action can be taken to control the
stresses as convolution rings can be used at root area. Infect, U shape
convolution geometry produces minimum stress concentration effect
compare to any other shape of convolutions.
4. Stresses calculated by FEA are near to analytical values. This validates the
results derived from FEA. The variations are up to 13%.
5. Since, experimentation and actual prediction of stresses developed in the
metallic bellows are difficult to predict and measurement incase of
experimentation hence, this methodology can be very much helpful in
practical applications.
5.15 Axi-symmetry Approach of FEA:
Many objects have some kind of symmetry like axi-symmetry, repetitive (cyclic)
symmetry or reflective (mirror image) symmetry. An axi-symmetry is observed in
many engineering components like metallic bellows, flywheel, arms of flywheels,
coupling, light bulb etc. Repetitive symmetry can be visualized in evenly spaced
cooling fins on a long pipe, teeth of gears along pitch circle diameter etc. The
reflective symmetry can be visualized in connecting rod, moulded plastic
containers.
When an object is symmetric about center line, one can often take advantage of
that fact to reduce the size and scope of the model in Finite Element Analysis.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 164
Symmetric object means similarity in geometry, loads, constraints, and material
properties.
5.15.1 Axi-symmtery Structures:
Any structure that displays geometric symmetry about a central axis in case of
shell or solid of revolution in any object is an axi-symmetric structure. Examples
would include straight pipes, cones, cylindrical vessels, circular plates, domes,
flywheels, couplings and so forth.
ANSYS software suggests that, models of axi-symmetric 3-D structures may be
represented in equivalent 2-D form. One can expect that results from a 2-D axi-
symmetric analysis will be more accurate than those from an equivalent 3-D
analysis[3].
By definition, a fully axi-symmetric model can only be subjected to axi-symmetric
loads. In many situations, however, axi-symmetric structures will experience non-
axisymmetric loads. In this case one must use a special type of element, known
as an axi-symmetric harmonic element, to create a 2-D model of an axi-symmetric
structure with non-axisymmetric loads.
5.15.2 Requirements for Axi-symmetric Models
Special requirements for axi-symmetric models include:
1. The axis of symmetry must coincide with the global Cartesian Y-axis.
2. Negative nodal X-coordinates are not permitted.
3. The global Cartesian Y-direction represents the axial direction, the global
Cartesian X-direction represents the radial direction, and the global Cartesian
Z-direction corresponds to the circumferential direction.
4. Unless otherwise stated, the model must be defined in the Z = 0.0 plane. The
global Cartesian Y-axis is assumed to be the axis of symmetry. Further, the
model is developed only in the +X quadrants. Hence, the radial direction is in
the +X direction.
5. Model should be assembled using appropriate element types.
For axi-symmetric models, use applicable 2-D solids with plane stress, plane
stress with thickness or axi-symmetry option. The model can be created using
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 165
3-D axi-symmetric shells also. In addition, various link, contact, combination,
and surface elements can be included in a model that also contains axi-
symmetric solids or shells. The program will not realize that these "other"
elements are axi-symmetric unless axi-symmetric solids or shells are present.
5.15.3 Guidelines for Modeling:
Small details that are unimportant to the analysis should not be included in the
solid model, since they will only make your model more complicated than
necessary. However, for some structures, "small" details such as fillets or holes
can be locations of maximum stress, and might be quite important, depending on
your analysis objectives. One must have an adequate understanding of the
structure's expected behavior in order to make competent decisions concerning
how much detail to include in the model.
In some cases, only a few minor details will disrupt a structure's symmetry. One
can sometimes ignore these details, in order to gain the benefits of using a
smaller symmetric model. Designer must weigh the gain in model simplification
against the cost in reduced accuracy when deciding whether or not to deliberately
ignore unsymmetrical features of an otherwise symmetric structure.
If the structure contains a hole along the axis of symmetry, one has to provide the
proper spacing between the Y-axis and the 2-D axisymmetric model.
Figure 5.15 shows a metallic bellow, which is formed type and made from thin
sheets. Its geometry is symmetric about the axis. The metallic bellows are used in
piping as a flexible element to take the axial, lateral and angular variations
occurring in the piping. The variations are because of fluctuation in pressure and
temperature.
Figure 5.15: Shape of a metallic bellow
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 166
To analyze this component, axi-symmetry option can be utilized as shown in other
sketch. This analysis gives nearby results of the stresses. Here Y axis is axis of
symmetry and X axis is the radial direction.
Actual Metallic Bellow
Axi-symmetry model
Figure 5.16: FEA model – Metallic Bellow
Notations and Dimensions:
Number of convolutions = 3 Inside diameter of Bellow = 30 cm Pitch of the bellow = 4 cm Height of convolutions = 3.5 cm Tangent length = 2.5 cm Material properties:
Modulus of elasticity = 19897349 N/cm2 Poisson’s ratio = 0.3 Boundary conditions:
Tangent lengths of both sides are locked with zero degree of freedom. Loading conditions:
Surface pressure selected at inner wall = 10 N/cm2 5.14.4 Results from ANSYS:
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 167
Figure 5.17 : Circumferential stress of a bellow using 3 D shell element
1
MN
MX
-540.518
-420.286-300.054
-179.822-59.589
60.643180.875
301.107421.34
541.572
DEC 18 200910:31:44
ELEMENT SOLUTION
STEP=1SUB =1TIME=1SX (NOAVG)RSYS=0DMX =.092399SMN =-540.518SMX =541.572
Figure 5.18: Axi-symmetry analysis of a bellow
1
MN MX
XY Z
-498.444
-390.616 -282.789 -174.962 -67.135 40.693 148.52 256.347 364.175 472.002
DEC 18 2009 10:29:05
ELEMENT SOLUTION STEP=1 SUB =1 TIME=1 SX (NOAVG) RSYS=0 DMX =.456713 SMN =-498.444 SMX =472.002
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 168
5.15.5 FEA Results:
Table 5.6: FEA Results
Type of Analysis 2 D with axi-symmetry option
(solid element) N/cm2
3 D model Analysis (shell element)
N/cm2 Circumferential stress 3690 3532
Longitudinal stress 11610 11532
3690
11610
3460
11532
0
2000
4000
6000
8000
10000
12000
14000
Circumferential stress Longitudinal stress
Str
esse
s, N
/cm
2
Axi-symmetric 3 D shell
Figure 5.19: Graph showing comparison of Axi-symmetric and 3D approaches
5.15.6 Observations:
1. The use of symmetry allows us to consider a reduced problem instead of
actual full size problem.
2. Modeling time is greatly reduced as geometry is simplified. The modeling of
axi-symmetry is in 2D plane.
3. For the axi-symmetry geometry model, number of nodes, number of elements
and number of equations are reduced compare to actual 3D analysis.
4. For the analytical solution, the order of the total stiffness matrix and total set
of stiffness equations are reduced considerably.
5. By taking advantage of symmetric geometry of the components, finite element
analysis becomes simple and fast.
6. 2D axi-symmetry analysis may be proven more accurate than an equivalent
3D analysis.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 169
5.16 Practical considerations in FEA:
This exercise is carried out with an objective of considering various practical
aspects while performing Finite Element Analysis. A metallic bellow is considered
as a case study for the finite element analysis to study above stated objective. For
finite element analysis actual and full size component should not be considered
for the analysis, but various practical aspects should be taken in to account. There
are many practical aspects for FEA. They are planning of the analysis, choosing
type of model, use of symmetry, selecting critical area for maximum stresses,
meshing quality parameters, aspect ratio of elements, etc. A case study of bellow
is considered for Finite Element Analysis for validation of practical considerations.
The results are obtained using ANSYS software.
5.16.1 Practical Considerations in FEA:
1. Planning of the Analysis:
Before beginning the model some important decisions must be made by user. The
accuracy of results will depend on these decisions; hence they should be taken
very carefully.[B1]
a) Objectives of analysis:
b) Whether the full model or only portion of a physical system is sufficient.
c) Details to be included in the model.
d) Selection of elements
e) Meshing density
2. Choosing type of Model:
The finite element model may be categorized as being 2-D or 3-D, and as being
composed of point elements, line elements, area elements, or solid elements. Of
course, these can be used as combined different kinds of elements as required.
3. Use of Symmetry:
The appropriate use of symmetry will often expedite the modeling of a problem.
Three types of symmetry can be considered in the modeling[B3]. They are
axi-symmetry, repetitive (cyclic) symmetry or reflective symmetry. Appropriate use
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 170
of symmetry in the modeling allows designer to minimize the problem size instead
of the actual problem.
An axi-symmetry means object is symmetrical about its axis of revolution. The
object shapes may be either cylindrical or conical. The examples falls into this
category are rotors, cylinders, couplings, pistons, flywheels, electric bulb, bottles
or jar, glass, etc.
A repetitive symmetry means, similar pattern is repeating either on a straight line
or radial line. The examples in this type of symmetry are fins of an engine or worm
gear box, metallic bellow, teeth of a rack, spur gear etc.
A reflective symmetry means object is symmetrical about any one or two axis. It
appears like mirror image on other side of an axis. The examples in this type of
symmetry are rectangle plate with a circular hole, bearing cover, plastic container
etc. Figure 5.20 shows FEA model representation of bellows considering repetitive
symmetry.
Bellow with 3 convolutions
(Actual Problem)
Single convolution model
(FEA model representation)
Figure 5.20: Axi-symmetry and Repetitive Symmetry of Bellows
Notations and Dimensions:
Number of convolutions, N = 3 Inside diameter of Bellow, Db = 30 cm Thickness of material, t = 0.05 cm Pitch of the bellow, q = 4 cm
Height of convolutions, w = 3.5 cm Tangent length, Lt = Lc = 2.5 cm Material properties:
Modulus of elasticity = 19897349 N/cm2 Poisson’s ratio = 0.3
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 171
Boundary conditions:
Tangent lengths of both sides are locked with zero degree of freedom. (Ux = 0, Uy = 0) Loading conditions:
Surface pressure selected at inner wall = 10 N/cm2 5.16.2 Modeling options:
Using axi-symmetric elements either three convolution model is required as
per geometry (figure 5.21). Instead of that, since convolutions are repeating at
regular pitch distance, one convolution model may be considered for the
analysis (figure 5.22). Results are shown in table 5.7.
Figure 5.21:
Axi-symmetric full size model Figure 5.22:
Axi-symmetric one convolution model
1
SEP 2 201007:55:58
ELEMENTS
Figure 5.23: Meshing in solid element
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 172
5.16.3 FEA Results:
Table 5.7: FEA Results Circumferential stresses, N/cm2 Longitudinal Stresses, N/cm2 Pressure
N/cm2 1 convolution model
3 D model 1 convolution model
3 D model
10 3730 3460 11020 11532
1
MN
MX
-185.707
-144.318-102.929
-61.54-20.15
21.23962.628
104.017145.406
186.796
SEP 2 201008:01:18
ELEMENT SOLUTION
STEP=1SUB =1TIME=1SXY (NOAVG)RSYS=0DMX =.006627SMN =-185.707SMX =186.796
Figure 5.24: ANSYS Results, Deformed shape
3730
11020
3460
11532
0
2000
4000
6000
8000
10000
12000
14000
Circumferential stress Longitudinal stress
Stre
sses
, N/c
m2
Repetitive Symmetry 3 D shell
Figure 5.25: Graph showing comparison of full model and single convolution model
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 173
5.16.4 Observations:
1. In FEA analysis of bellow convolution pattern is repeating periodically at
equal distance (pitch). Hence, repetitive geometry concept may be
considered and one convolution sufficient for the stress analysis.
2. Bellow consist of symmetrical geometry, axi-symmetric element may be
used instead of 3 D shell (full size) model.
3. Axi-symmetry and repetitive geometry features can be used in combination
and finite element can be made simpler.
4. Here single convolutions results are compared to three dimensional full
size bellow. So, one convolution is sufficient for FEA. The variations in
results are within 20%.
5. Simple and smaller size of model provides higher accuracy in the results.
6. Practical aspects should be considered in modeling phase of finite element
analysis. Complicated component model can be simplified by use of
symmetry, elimination of least affected features etc. This will save modeling
and analysis time, as well as an accuracy of results will increases.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 174
5.17 Comparison of Convolutions Shapes:
The bellow consists of optional convolution shapes like U, V, S, toroidal etc. as we
desire. Selection of each convolution shape will be based on designers’ choice, its
maximum pressure value and manufacturing facilities available. Each convolution
shape will have different effect on the performance of bellow. This exercise is
carried out with the objective that, study of various performance characteristics of
different shapes of convolution. Generally U shape, V shape, toridal shape are
mostly used in application. Hence these three are considered for the study with
Finite Element Analysis methodology. Figure 5.26 shows the comparative
geometry of these convolutions.
Figure 5.28: Shape and Dimensions of Convolutions
Notations and Dimensions:
Number of convolutions, N = 1 Inside diameter of Bellow, Db = 300 mm = 30 cm Thickness of material, t = 0.05 cm Height of convolutions, w = 45 mm = 4.5 cm Tangent length, Lt = Lc = 25 mm = 2.5 cm Material properties:
Modulus of elasticity = 19728608 N/cm2 Poisson’s ratio = 0.3 Boundary conditions:
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 175
Tangent lengths of both sides are locked with zero degree of freedom. (Ux = 0, Uy = 0) Loading conditions:
Surface pressure selected at inner wall = 10 N/cm2 5.17.1 FEA Results:
U shape
V – shape
Toroidal shape
Figure 5.29 : Meshed model of shapes of convolutions
Table 5.8: FEA Results of U shape Convolutions
Pressure
N/cm2
Circumferential stress
N/cm2
Axial stress
N/cm2
Max. Stress intensity
N/cm2
1.0 5960 7030 13770
Table 5.9: FEA Results of V shape convolutions
Pressure
N/cm2
Circumferential stress
N/cm2
Axial stress
N/cm2
Max. Stress intensity
N/cm2
10 4990 9840 12540
Table 5.10: FEA Results of Toroidal shape convolutions
Pressure
N/cm2
Circumferential stress
N/cm2
Axial stress
N/cm2
Max. Stress intensity
N/cm2
1.0 4770 13450 14340
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 176
Graphs:
0
1000
2000
3000
4000
5000
6000
7000
U shape V shape Toroidal shape
Circ
umfe
rent
ial s
tres
s, N
/cm
2
Figure 5.30: Graph showing circumferential stress
0
2000
4000
6000
8000
10000
12000
14000
16000
U shape V shape Toroidal shape
Long
itudi
nal s
tres
s, N
/cm
2
Figure 5.31: Graph showing Longitudinal Stress
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 177
11500
12000
12500
13000
13500
14000
14500
U shape V shape Toroidal shape
Max
imum
str
ess
inte
nsity
, N/c
m2
Graph 5.32: Graph showing maximum stress intensity
5.17.2 Observations:
1. The geometry of the convolution should have uniform shape, any sharp
change in geometry will create stress concentration effect and which will
leads to higher stress development.
2. In case of U shaped convolutions, circumferential stresses are at
maximum, while longitudinal stresses are at minimum level. This is
because of straight (perpendicular) convolution faces.
3. U shape convolution permits maximum axial displacement (movement)
because of root and crest flexibility. It is because of the vertical edges of
the convolutions, which permits higher deflection.
4. For toroidal shape convolutions, circumferential stresses are at minimum
level, while longitudinal stresses are at maximum level. This is due to its
spherical shape, and toroidal convolution can withstand higher amount of
stresses compared to other types of convolutions.
5. Toroidal shape do not permit higher axial deflection because of its spherical
shape.
6. V shaped convolution performs the stress level between U shape and
toroidal shape convolutions.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 178
5.18 Structural and Thermal Analysis:
The design of bellows is very complex as it involves structural and thermal
aspects. Structural design point of view, the bellows should be flexible enough to
take up movements or deformations causes by pressure fluctuations. The bellows
are manufactured using minimum metal thickness in order to get higher deflection.
Here many times the bellows are deformed beyond elastic range of material,
hence prediction of stresses are very much critical. As the temperature of the
piping rises, the modulus of elasticity of the material is decreases; hence resulting
in the development of the higher stresses. The combined structural and thermal
aspect makes the design of bellows very much critical. The determination of an
acceptable design is further complicated by the numerous geometrical parameters
involved such as diameter, material thickness, and shape of convolutions, number
of convolutions, pitch, height of convolution, number of plies, etc.
5.18.1 Problem definition:
Notations Db = Inside diameter of the bellow = 30 cm
N = Number of convolution = 2
w = Height of convolution = 3.5 cm
q = Pitch of convolution = 4.0 cm
Lt = Lc = Tangent length = 2.5 cm
Thickness of materials = 0.05 cm
Number of ply of material = 1
Figure 5.33 Geometric Dimensions of a bellow
Bellows are made from sheet metal long tube (seam welded in longitudinal
direction). Then the convolutions are formed by any metal forming process using
dies. Generally U shape of convolutions is preferred. Figure 5.31 shows a bellow
with two convolution and single ply material. It shows the basic geometry of a
bellow.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 179
5.18.2 Loading Conditions:
In actual practice the inside surface will be pressurized by fluid. Here we can
simplify the experiment by applying uniform pressure at inside surfaces of bellow.
Pressure at inside surface is taken as 10 N/cm2.
5.18.3 Boundary conditions:
As the expansion joints are fixed through the collar at both ends. The
displacement constraint is made fixed at both sides. The inside pressure will be
acting on cylindrical surface as well as convolution area. Tangent lengths at both
ends of bellows are covered by collars. Collars are made from comparatively thick
material, considering that the stresses are to be bared by convolutions only.
The tangent lengths at both ends are considered as zero degree of freedom as
these ends are welded to flanges and subsequently to long pipes.
Reference temperature is given at 273 K. Uniform temperature is at 3000K, 3500K,
4000K, 4500K, and 5000K are applied at for the stress analysis.
5.18.4 Results: Reference temperature: 273 K
Table 5.11: FEA Results
Pressure N/cm2
Temperature 0K
Circumferential stress N /cm2
Axial stress N/cm2
Max. stress intensity N /cm2
10 300 3310 9470 12700
10 350 4980 26970 36220
10 400 7070 44460 59730
10 450 9170 61960 83250
10 500 11360 79460 106770
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 180
1
MNMX
XYZ
-1016
-797.902-579.785
-361.668-143.551
74.566292.683
510.8728.917
947.034
AUG 21 201023:12:21
ELEMENT SOLUTION
STEP=1SUB =1TIME=1SX (NOAVG)RSYS=0DMX =.019429SMN =-1016SMX =947.034
Figure 5.34: Longitudinal Stresses at uniform temperature 300 0K
5.18.5 Analytical Approach:
Strength of materials decreases with increase in temperature in case of steel. Its
modulus of elasticity is reduced because of thermal vibrations of the atoms in the
material, and hence to an increase in the average separation distance of adjacent
atoms. To consider this parameter, thermal expansion occurs based on its
coefficient of thermal expansion.
The linear coefficient of thermal expansion (Greek letter alpha) describes by
how much a material will expand for each degree of temperature increase.
The thermal expansion coefficient for a pipe is also a thermodynamic property of
that material.
δ = (ΔT) L
Where, δ is the elongation of pipe,
is the thermal expansion coefficient,
ΔT is the change in temperature, and
L is the initial length of the pipe.
The flexibility of a bellow is an important parameter for the designers. The actual
modulus of elasticity is not applicable for the design procedure, as the flexibility
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 181
increases because of its shell structure. The flexibility parameter is based on shell
parameter of a bellow. Many researchers has made attempts to find flexibility
parameter 'E
E for bellows. It depends on geometric parameters like n, b, h, R and
t of bellow.
Figure 5.35 : U shape geometry of a bellow
Shell parameter of bellow is λ = 2btR = 21
05.075.16 x = 0.83
Shell parameter λ = 'E
E , 0.83 = '
19728608E
, E’ = 23769407 N/cm2.
Formulation of stress relationship with reference to thermal expansion of bellow is
Elongation of pipe δ = (ΔT) L = 12.6 x 10-3 x (300 – 273) x 13 = 4.43 x 10-3 cm
Actual Modulus of elasticity, E’ = strainstress
Stress = E’ x strain = 23769407 x 13
1043.4 3x = 8100 N/cm2.
Using same procedure all values are computed in following table 5.12
Table 5.12: Analytical Results
Pressure N/cm2
Change in Temperature 0K
Axial Stresses N/cm2
10 300 – 273 = 27 8100
10 350 – 273 = 77 22780
10 400 – 273 = 127 37580
10 450 – 273 = 177 52380
10 500 – 273 = 227 67180
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 182
5.18.6 Graphs:
0
20000
40000
60000
80000
100000
120000
300 350 400 450 500
Temperature, K
Axi
al s
tres
ses,
N/c
m2
FEA Analytical
Figure 5. 36 : Comparison of FEA and Analytical Results
5.18.7 Observations:
1. Longitudinal stresses developed in the bellows increases due to increase in
temperature, even though the pressure remains constant.
2. The analysis is based on linear relationship, and non linear mode may give
some variations in results. The elongation is based on thermal expansion of
bellow material.
3. Longitudinal (along axis) stresses and strains are always higher compared
to circumferential stresses because elongation takes place along the length
of pipe and bellow and stresses due to bending.
4. FEA results are validated with analytical approach for longitudinal stresses
developed in bellow. Finite element analysis using ANSYS gives near to
realistic results.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 183
5.19 Stability Analysis:
Structural members, which are considerably long in dimensions compare to their
lateral dimensions, starts bending (buckling), when their compressive loading
reaches to some critical value. Buckling can be defined as the gross lateral
deflection of long columns at center sections. Buckling failure of structures mainly
depends upon slenderness ratio.
Expansion joints are used in the piping to take deviations occurring because of
temperature and pressure variations. These deviations may be axial, lateral and
combined. Bellow is a critical element of an expansion joint assembly. Bellows are
normally loaded with internal pressure along with elevated temperature depending
upon the applications. Design of bellow is very much critical as there are many
geometric parameters and many other affecting factors. The stresses developed
in the bellows are due to pressure and deflection. Some times bellows becomes
unstable because of excessive internal pressure. This kind of failure of bellows is
termed as ‘squirm’.
5.19.1 Squirm in Expansion Joints:
Expansion Joints Manufacturers Association (EJMA) has established the codes
for design of bellows considering buckling. This analytical approach is based on
Euler’s theory and gives near to realistic estimation of buckling load. This exercise
is an attempt to check the bellow for buckling failure using ANSYS finite element
method. The squirming phenomenon was first demonstrated by Haringx[2], who
showed pressure buckling of bellows was analogous to buckling of Euler strut. He
gave following relationship.
22'
lrIEP
Excessive internal pressure may cause a bellow to become unstable and squirm.
Bellows performance is depending on critical pressure. The pressure capacity is
decided based on squirm by keeping some factor of safety. Fatigue also depends
on squirm pressure. There are two basic types of squirm, column squirm and in-
plane squirm.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 184
Figure 5.37 : Column Squirm Figure 5.38 : In-plane squirm
Column squirm is defined as a gross lateral shift of the middle section of the
bellow. In-plane squirm is defined as deflection occurred in individual
convolutions, parallel to the surface of bellow materials.
Squirm is associated with length to diameter ratio, called slenderness ratio.
According to slenderness ratio, bellows can be categorized in long or short
columns. Failure of column depends on the kind of column. Squirm is similar to
buckling of column under compressive load. Buckling failure consists of an elastic
and in-elastic deformation. Since bellows are made from thin sheet metal,
deformation of bellows exists in elastic and plastic region. Hence determination of
critical pressure is essential to avoid squirm failure.
Buckling analysis is a technique used to determine buckling load or critical load at
which a structure becomes unstable or buckle mode shapes - the characteristic
shape associated with a structure's buckled response. Eigenvalue buckling
analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal
linear elastic structure. An eigenvalue buckling analysis of a column will match the
classical Euler solution[5]. However, imperfections and nonlinearities prevent most
real-world structures from achieving their theoretical elastic buckling strength.
Thus, eigenvalue buckling analysis often yields unconservative results. The non
linear analysis gives much accurate results.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 185
5.19.2 Results:
Table 5.13: FEA Results
Model Lb / Db ratio
No. of convolutions, N
Pitch, q (cm)
Buckling pressure (N/cm2)
1 5 5 8 22.70
ANSYS gives results 22.70 as limiting pressure for buckling to occur. Since
geometry of bellow is made in order to achieve maximum flexibility by shapes
and parameters like minimum thickness, height of convolutions, pitch of
convolutions, and convolutions formulations. These features are having
definite effect on the result. They are discussed below.
5.19.3 Observations:
1. Euler’s equation gives satisfactory results for long columns only. ANSYS
buckling is based on Euler’s theory. Hence bellows geometry preferably in long
column should be considered for analysis.
2. Bellows material thickness is very less and it is having hollow structure, so
moment of inertia is very negligible, and hence results may not be accurate.
3. Bellows are having very high spring rate, because of its geometric features like
height of convolution, thin metal thickness etc. While Euler’s column is rigid.
Actually bellows possess more elasticity due to its geometry. Euler’s relation
includes elasticity based on property only.
4. Since EJMA suggests equation to calculate limiting pressure to avoid buckling.
It is an indirect way to make safe design. Equations suggested by EJMA
definitely give much more precise and safe design parameters.
5. Bellows may fail by squirm, if number of convolutions and pitch of bellow
parameters are selected on higher side. Bellows squirm pressure can be
evaluated and design pressure should be well within limit by keeping factor of
safety.
6. FEA for buckling analysis does not give accurate results because of its typical
shell type structure, hence not recommended for FEA.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 186
5.20 Dynamic Analysis:
A steady flow of fluid is passing through the long pipes and expansion joints. The
fluid is passing at pressure higher than atmospheric, hence dynamic analysis of
expansion joint is necessary. Metallic bellows are supposed to be loaded with high
frequency and low amplitude vibrations. The piping system designer should take
care about vibration loads in the piping system. Designers can use external
damping device for reducing the vibration effects in the bellows. For turbulent flow
applications, inside sleeve arrangement must be provided in order to minimize the
vibration intensities. Piping elements are sturdy and rigid, while bellows tends to vibrate in the
convolution length because of pressurized fluid flow. The vibration intensities
depend on overall stiffness of bellow, type of fluid, and weight of bellow along with
fluid in the convolution region. The vibration intensities will be developed in axial
and lateral direction of bellow.
5.20.1 Harmonic Analysis using FEA:
A harmonic analysis, by definition, assumes that any applied load varies
harmonically (sinusoidally) with time. To completely specify a harmonic load, three
pieces of information are usually required: the amplitude, the phase angle, and the
forcing frequency range. Peak harmonic response occurs at forcing frequencies
that match the natural frequencies of your structure. Before obtaining the
harmonic solution, you should first determine the natural frequencies of your
structure by obtaining a modal solution
The amplitude is the maximum value of the load, according to type of problem.
The phase angle is a measure of the time by which the load lags (or leads) a
frame of reference.
Figure 5.37 shows the model created in the ANSYS environment. In solid type of
elements, axi-symmetric harmonic 4 node element is used to make model.
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 187
Geometric dimensions of bellow:
Inside diameter of bellow, Db = 16.9 cm.
Mean diameter of bellow, Dm = 18.63 cm.
Number of convolutions, N = 7
Number of plies, n = 1
Thickness of material = 0.04 cm.
Height of convolutions = 1.65 cm.
Pitch of convolutions, q = 2.60 cm.
Length of bellow, Lb = N x q = 18.20 cm.
Design pressure, P = 25 N/cm2
Figure 5.39 : Axi-symmetric model of bellow
5.20.2 Boundary conditions:
Bellows are always clamped from its tangent length at both ends with collar. All
(Ux and Uy) degree of freedom is locked at inner surface of tangent length.
5.20.3 Loading conditions:
Bellow is loaded with inside fluid surface pressure of 25 N/cm2. Room temperature
condition is assumed for the analysis.
5.20.4 Results:
Following results are derived from the Finite Element Analysis using ANSYS
platform.
Table 5.14: FEA Results
Number of convolutions
Natural frequency (axial) Uy
Natural frequency (lateral) Ux
7 44 Hz 45 Hz
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 188
Graphs:
Figure 5.40 : Natural frequency in Axial direction
Figure 5.41 : Natural frequency in lateral direction
Ph. D. thesis on “Study of Design Aspects of Expansion Joints with Metallic Bellows and their Performance Evaluation” 189
5.20.5 Analytical Approach: Cross section Area of bellow = 1.34 cm2
Volume of bellow = 79 cm3
Mass of bellow = 79 cm3 x 0.008 kg/cm3 = 0.62 kg.
Weight of bellow = 0.62 x 9.81 = 6.082 kgf. = 60.82 N
Initial Axial spring rate of bellow = 1.7 f
pbm
CwntED
3
3
= 9990 N/ cm/ convolution.
Over all spring rate of bellow = Nf iu =
79990 = 1427.1 N/cm
Natural frequency (Axial) = fn = m
KC sr
n = 47.25 Hertz
Natural frequency (Lateral) = fn = m
KLDC sr
b
mn = 48.37 Hertz.
5.20.6 Observations:
Following observations are derived from the analysis.
1. Number of convolution is taken 7 for the test bellow. As the number of
convolution increases, stiffness reduces, which will increase the natural
frequency of bellow.
2. Natural frequency derived by FEA is near to analytical results.