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1
CHAPTER – 1
INTRODUCTION
Pressure vessels are leak proof containers. They may be of any shape and
range from beverage bottles to the sophisticated ones encountered in engineering
construction. The ever-increasing use of vessels for storage, industrial processing
and power generation under unusual conditions of pressure, temperature and
environment has given special emphasis to analytical and experimental methods
for determining their operating stresses.
Pressure vessels are probably the most widespread “machines” within the
different industrial sectors. In fact, there is no factory without pressure vessels,
steam boilers, tanks, autoclaves, collectors, heat exchangers, pipes, etc. more
specifically, pressure vessels represent fundamental components in sectors of
enormous industrial importance, such as the nuclear, oil, petro-chemical and
chemical sectors. There are periodic international symposia on the problems
related to the verification of pressure vessels.
Pressure vessels encountered in nuclear, aerospace and other structures are
rotationally symmetric shells subjected to internal pressure. In the design of large
rocket motor cases, the number of individual welded segments (viz., head end
segment, nozzle end segment, cylindrical segments ) will be chosen based on the
feasibility of propellant casting, hardware fabrication limits and ease of
transportation / handling, etc. These segments may be connected to each other
through the tongue and groove type of joints. End domes having central circular
openings will be provided at the head end and nozzle end of the motor case. The
cylindrical portion of the casing for this type of configuration will be stressed
maximum under internal pressure and hence governs the design.
2
In recent decades, various methods have been proposed for strengthening the
vessels. The autofrettage process is possibly the most well known method.
Autofrettage is a process in which the cylinder is subjected to a certain amount of
pre-internal pressure so that its wall becomes partially plastic. The pressure is then
released and the residual stresses lead to a decrease in the maximum von-Mises
stress in the working loading stage. This means an increase in the pressure capacity
of the cylinder.
Since prediction of structural behavior at the regions of autofrettage in
pressure vessels is the ultimate goal of the present study, comparisons are made of
different solutions, viz., finite element solution obtained using ANSYS. The
analytical stress results at various junctions of cylinders like outer radius, inner
radius and autofrettage radius are found to be in good agreement with FEA results.
1.1 Stresses in Pressure vessels:
Structures such as pipes or bottles capable of holding internal pressure have
been very important in the history of science and technology. Although the ancient
Romans had developed municipal engineering to a high order in many ways, the
very need for their impressive system of large aqueducts for carrying water was
due to their not yet having pipes that could maintain internal pressure. Water can
flow uphill when driven by the hydraulic pressure of the reservoir at a higher
elevation, but without a pressure containing pipe an aqueduct must be constructed,
so the water can run downhill all the way from the reservoir to the destination.
Airplane cabins are another familiar example of pressure containing structures.
They illustrate very dramatically the importance of proper design; since the
atmosphere in the cabin has enough energy associated with its relative
pressurization compared to the thin air outside that catastrophic crack growth is a
real possibility.
3
Combined Stresses:
Cylindrical or spherical pressure vessels (e.g. hydraulic cylinders, gun
barrels, pipes, boilers and tanks) are commonly used in industry to carry both
liquids and gases under pressure. When the pressure vessels is exposed to this
pressure, the material comprising the vessel is subjected to pressure loading, and
hence stresses, from all directions. The normal stresses resulting from this pressure
are functions of the radius of the element under consideration, the shape of the
pressure vessel (i.e., open ended cylinders, closed end cylinders, or sphere) as well
as the applied pressure.
THIN WALLED PRESSURE VESSELS:
Cylindrical vessels: A cylindrical pressure with wall thickness t, and inner radius
r, is considered (fig 1.1). A gauge pressure p, exists within the vessel by the
working fluid (gas or liquid). For an element sufficiently removed from the ends of
the cylinder and oriented as shown in fig 1.1, two types of normal stress are
generated: hoop stress σh , and axial stress σa, that both exhibit tension of the
material.
Hoop membrane stress: The average stress in a ring subjected to radial forces
uniformly distributed along its circumference.
FIG.1: Cylindrical thin walled pressure vessel
4
Longitudinal stress: The average stress acting on a cross section area of the
vessel.
For the hoop stress, consider the pressure vessel section by planes sectioned
by planes a, b and c for fig 1. A free body diagram of a half segment along with the
pressurized working fluid is shown in fig.3. Note that only the loading in the x-
direction is shown and that the internal reactions in the material are due to hoop
stress acting on incremental areas A, produced by the pressure acting on the project
area, Ap.
Fig. 2: Cylindrical thin-walled pressure vessel showing co-ordinate axes and
cutting planes (a, b and c)
For equilibrium in the x-direction we sum forces on the incremental segment of
width dy to be equal to zero such that:
𝐹𝑥 = 0
2[σh.A] – p.Ap = 0 = 2[σh.t. dy] – p*2r*dy
5
or solving for σh, we have
σh = 𝑝∗𝑟
𝑡 …..(1)
where dy = incremental length; t = wall thickness; r = inner radius; p = gauge
pressure and σh = hoop stress.
For the axial stress, consider the left portion of section b of the cylindrical
pressure vessels shown in fig.2. A free body diagram of a half segment along with
the pressurized working fluid is shown in fig.4. Note that the axial stress acts
uniformly throughout the wall and the pressure acts on the end cap of the cylinder.
For equilibrium in the y-direction we sum forces such that:
Fig.3: Cylindrical thin-walled pressure vessels showing pressure and internal
hoop stress
𝐹𝑦 = 0
σa.A – p.Ae = 0 = σa*π(ro2 – r
2) – p*πr
2
or solving for σa, we have
σa = 𝑝∗ 𝜋𝑟2
𝜋(𝑟𝑜2− 𝑟2)
Substituting r = ro + t, we get
σa = 𝑝∗ 𝜋𝑟2
𝜋((𝑟+𝑡)2− 𝑟2)
σa = 𝒑∗ 𝒓𝟐
𝟐𝒓𝒕 +𝒕𝟐 …..(2)
6
Since, this is a thin wall with a small t, t2 is smaller and can be neglected such that
after simplification,
σa = 𝒑∗ 𝒓𝟐
𝟐𝒓
where ro = inner radius and σa = axial stress.
Fig. 4: Free body diagram of end section of cylindrical thin-walled pressure
vessels showing pressure and internal axial stress
Note that in equations 1 and 2, the hoop stress is twice as large as the axial
stress. Consequently, when fabricating cylindrical pressure vessels from rolled-
formed plates, the longitudinal joints must be designed to carry twice as much
stress as the circumferential joints.
Spherical Vessels:
A spherical vessel can be analyzed in a similar manner as for the cylindrical
pressure vessel. As shown in fig.5, the “axial stress” results from the action of the
pressure acting on the projected area of the sphere such that,
𝐹𝑦 = 0
7
σa.A – p.Ae = 0 = σa*π(ro2 – r
2) – p*πr
2
or solving for σa, we have
σa = 𝑝∗ 𝜋𝑟2
𝜋(𝑟𝑜2− 𝑟2)
Substituting r = ro + t, we get
σa = 𝑝∗ 𝜋𝑟2
𝜋((𝑟+𝑡)2− 𝑟2)
σa = 𝒑∗ 𝒓𝟐
𝟐𝒓𝒕 +𝒕𝟐 …..(3)
Since, this is a thin wall with a small t, t2 is smaller and can be neglected such that
after simplification,
σa = 𝒑∗ 𝒓𝟐
𝟐𝒓
Note that for the spherical pressure vessel, the hoop and axial stresses are
equal and are one half of the hoop stress in the cylindrical pressure vessel. This
makes the spherical pressure vessel more “efficient” pressure vessel geometry.
Fig. 5: Free body diagram of end section of spherical thin-walled pressure
vessel
8
The analyses of equation 1 to 3 indicate that an element in either a
cylindrical or a spherical pressure vessel is subjected to biaxial stress (i.e., a
normal stress existing in only two directions).
Fig. 6: Spherical thin-walled pressure vessels showing pressure and internal
hoop stress
Thick walled pressure vessels:
Closed form, analytical solutions of stress states can be derived using
methods developed in a special branch of engineering mechanics called elasticity.
Elasticity methods are beyond the scope of the course although elasticity solutions
are mathematically exact for the specified boundary conditions are particular
problems. For cylindrical pressure vessels subjected to an internal gauge pressure
only the following relations result:
σh = 𝑝∗ 𝑟𝑜
2
𝑟𝑜2− 𝑟𝑖
2 1 + 𝑟𝑜
2
𝑟𝑖2
σa = 𝑝∗ 𝑟𝑖
2
𝑟𝑜2− 𝑟𝑖
2 …..(4)
9
σr = 𝑝∗ 𝑟𝑖
2
𝑟𝑜2− 𝑟𝑖
2 1 − 𝑟𝑜
2
𝑟𝑖2
where ro = outer radius; ri = inner radius and r is the radial variable. Equation 4
applies for any wall thickness and is not restricted to particular 𝑟 𝑡 ratio as are the
equations 1 and 2. Note that the hoop and radial stress (σh and σr) are functions of
r. (i.e. vary through the thick wall) and that the axial stress σa, is independent of r
(i.e. is constant through the wall thickness). Fig.6 shows the stress distributions
through the wall thickness for the hoop and radial stresses. Note that for the radial
stress distributions, the maximum and minimum values occur respectively, at the
outer wall (σr = 0) and at σr = -p as noted already for the thin walled pressure
vessel.
Fig. 7: Stress distribution of hoop and radial stress
10
Equation 4 can be generalized for the case of internal and external pressures
such that,
𝜍ℎ =
𝑟𝑖2∗ 𝑝𝑖 − 𝑟𝑜
2∗ 𝑝𝑜 − 𝑟𝑖2∗𝑟𝑜
2 𝑝𝑜2− 𝑝𝑖
2 𝑟2
𝑟𝑜2− 𝑟𝑖
2
𝜍ℎ = 𝑟𝑖
2∗ 𝑝𝑖 − 𝑟𝑜2∗ 𝑝𝑜
𝑟𝑜2− 𝑟𝑖
2 …..(5)
𝜍ℎ = −
𝑟𝑖2∗ 𝑝𝑖 − 𝑟𝑜
2∗ 𝑝𝑜 + 𝑟𝑖2∗𝑟𝑜
2 𝑝𝑜2− 𝑝𝑖
2 𝑟2
𝑟𝑜2− 𝑟𝑖
2
11
1.2 CONCEPT OF AUTOFRETTAGE:
Autofrettage is a metal fabrication technique in which a pressure vessel is
subjected to enormous pressure, causing internal portions of the part to yield and
resulting in internal compressive residual stresses. The goal of autofrettage is to
increase the durability of the final product. Inducing residual compressive stresses
into materials can also increase their resistance to stress corrosion cracking; that is,
non-mechanically-assisted cracking that occurs when a material is placed in a
suitable environment in the presence of residual tensile stress. The technique is
commonly used in manufacturing high-pressure pump cylinders, battleship and
tank cannon barrels, and fuel injection systems for diesel engines. While some
work hardening will occur, that is not the primary mechanism of strengthening.
Fig. 8: Highly strengthened Pressure vessels
The start point is a single steel tube of internal diameter slightly less than the
desired calibre. The tube is subjected to internal pressure of sufficient magnitude to
enlarge the bore and in the process the inner layers of the metal are stretched
12
beyond their elastic limit. This means that the inner layers have been stretched to a
point where the steel is no longer able to return to its original shape once the
internal pressure in the bore has been removed. Although the outer layers of the
tube are also stretched the degree of internal pressure applied during the process is
such that they are not stretched beyond their elastic limit. The reason why this is
possible is that the stress distribution through the walls of the tube is non-uniform.
Its maximum value occurs in the metal adjacent to the source of pressure,
decreasing markedly towards the outer layers of the tube. The strain is proportional
to the stress applied within elastic limit; therefore the expansion at the outer layers
is less than at the bore. Because the outer layers remain elastic they attempt to
return to their original shape; however, they are prevented from doing so
completely by the now permanently stretched inner layers. The effect is that the
inner layers of the metal are put under compression by the outer layers in much the
same way as though an outer layer of metal had been shrunk on. The next step is to
subject the strained inner layers to low temperature heat treatment which results in
the elastic limit being raised to at least the autofrettage pressure employed in the
first stage of the process. Finally the elasticity of the barrel can be tested by
applying internal pressure once more, but this time care is taken to ensure that the
inner layers are not stretched beyond their new elastic limit.[1]
When autofrettage is used for strengthening cannon barrels, the barrel is prebored
to a slightly undersized inside diameter, and then a slightly oversized die is pushed
through the barrel. The amount of initial underbore and size of the die are
calculated to strain the material past its elastic limit into plastic deformation,
sufficiently far that the final strained diameter is the final desired bore.
13
1.3 MATERIAL STRESS-STRAIN CURVES:
The relationship between the stress and strain that a material displays is
known as Stress-Strain curve. It is unique for each material and is found by
recording the amount of deformation (Strain) at distinct intervals of tensile or
compressive loading. These curves reveal many of the properties of a material
(including data to establish the Modulus of Elasticity, E). The following figure
shows the stress-strain curves for mild steel, cast iron and concrete.
Fig. 9: Stress-strain curve for Mild steel, Cast iron and concrete
It can be seen that the concrete curve is almost a straight line. There is an
abrupt end to the curve. This, and the fact that it is a very steep line, indicates that
it is a brittle material. The curve for cast iron has a slight curve to it. It is also a
brittle material. Both of these materials will fail with little warning once their
limits are surpassed. Notice that the curve for mild steel seems to have a long
gently curving “tail”. This indicates a behavior that is distinctly different than
either concrete or cast iron. The graph shows that after certain point mild steel will
continue to strain (in the case of tension, to stretch) as the stress (the loading)
remains more or less constant. The steel will actually stretch life taffy. This is a
material property which indicates a high ductility.
14
1.4 FEA – A SIMULATION TOOL
Finite element analysis (FEA) is a discipline crossing the boundaries of
mathematics, physics, and engineering and computer science. The method has
wide application and has a extensive utilization in the structural, thermal and fluid
analysis areas. The finite element method is comprised of three major phases:
Phase – 1 Pre-processing, in which the analyst develops a finite element mesh to
divide the subject geometry into sub domains for mathematical analysis, and
applies material properties and boundary conditions,
Phase – 2 Solution, during which the program derives the governing matrix
equations from the model and solves for the primary quantities, and
Phase – 3 Post-processing, in which the analyst checks the validity of the solution,
examines the values of primary quantities (such as displacements and stresses), and
derives and examines additional quantities (such as specialized stresses and error
indicators).
The advantages of FEA are numerous and important. Once a detailed CAD
model has been developed, FEA can analyze the design in detail, saving time and
money by reducing the number of prototypes required. An existing product which
is experiencing a field problem, or is simply being improved, can be analyzed to
speed an engineering change and reduce its cost.
The finite element method was first employed on truss-like structures and
over time expanded to include most physical/mechanical phenomena.
15
CAPABILITIES
Pre-processing
The goals of pre-processing are to develop an appropriate finite element
mesh, assign suitable material properties, and apply boundary conditions in the
form of restraints and loads.
The finite element mesh subdivides the geometry into elements, upon which
are found to be nodes. The nodes, which are really just point locations in space, are
generally located at the element corners and perhaps near each midside. For a two-
dimensional (2D) analysis, or a three-dimensional (3D) thin shell analysis, the
elements are essentially 2D, but may be “warped” slightly to conform to a 3D
surface. An example is the thin shell linear quadrilateral; thin shell implies
essentially classical shell theory, linear defines the interpolation of mathematical
quantities across the element, and quadrilateral describes the geometry. For a 3D
solid analysis, the elements have physical thickness in all three dimensions.
Common examples include solid linear brick and solid parabolic tetrahedral
elements. In addition, there are many special elements, such as axi-symmetric
elements for situations in which the geometry, material and boundary conditions
are all symmetric about an axis.
The model’s degrees of freedom (dof) are assigned at the nodes. Solid
elements generally have three translational dof per node. Rotations are
accomplished through translations of groups of nodes relative to other nodes. Thin
shell elements, on the other hand, have six dof per node; three translations and
three rotations. The addition of rotational dof allows for evaluation of quantities
through the shell, such as bending stresses due to rotation of one node relative to
another. Thus, for structures in which classical thin shell theory is a valid
approximation, carrying extra dof at each node bypasses the necessity of modeling
the physical thickness. The assignment of nodal dof also depends on the class of
16
analysis. For a thermal analysis, for example, only one temperature dof exists at
each node.
Developing the mesh is usually the most time-consuming task in FEA. In the
past, node locations were keyed in manually to approximate the geometry. The
more modern approach is to develop the mesh directly on the CAD geometry,
which will be (1) wireframe, with points and curves representing edges, (2)
surfaced, with surfaces defining boundaries, or (3) solid, defining where the
material is.
The geometry is meshed with a mapping algorithm or an automatic free
meshing algorithm. The first maps a rectangular grid onto a geometric region,
which must therefore have the correct number of sides. Mapped meshes can use
the accurate and cheap solid linear brick 3D element, but can be very time-
consuming, if not possible, to apply to complex geometries. Free-meshing
automatically subdivides meshing regions into elements, within the advantages of
fast meshing, easy mesh-size transitioning (for a denser mesh in regions of large
gradient), and adaptive capabilities. Disadvantages include generation of huge
models, generation of distorted elements, and, in 3D, the use of the rather
expensive solid parabolic tetrahedral element. It is always important to check
elemental distortion prior to solution. A badly distorted element will cause a matrix
angularity, killing the solution. A less distorted element may solve, but can deliver
very poor answers. Acceptable levels of distortion are dependent upon the solver
being used.
Material properties required vary with the type of solution. A linear statics
analysis, for example, will require an elastic modulus, poisson’s ratio and perhaps
a density for each material. Thermal properties are required for a thermal analysis.
Examples of restraints are declaring anodal translation or temperature. Loads
include forces, pressures and heat flux. It is preferable to apply boundary
17
conditions to the CAD geometry, with the FEA package transferring them to the
underlying model, to allow for simpler application of adaptive and optimization
algorithms.
Solution:
While the pre-processing and post-processing phases of the finite element
method are interactive and time-consuming for the analyst, the solution is often a
batch process, and is demanding of computer resource. The governing equations
are assembled into matrix form and are solved numerically. The assembly process
depends not only on the type of analysis (e.g. static or dynamic), but also on the
model’s element types and properties, material properties and boundary conditions.
In the case of a linear static structural analysis, the assembled equation is of
the form K.d = r, where K is the system stiffness matrix, d is the nodal degree of
freedom (dof) displacement vector, and r is the applied nodal load vector. The
strain-displacement relation may be introduced into the stress-strain relation to
express stress in terms of displacement. Under the assumption of compatibility, the
differential equations of equilibrium in concert with the boundary conditions then
determine a unique displacement field solution, which in turn determines the strain
and stress fields. The chances of directly solving these equations are slim to none
for anything but the most trivial geometries, hence the need for approximate
numerical techniques presents itself.
A finite element mesh is actually a displacement-nodal displacement
relation, which through the element interpolation scheme, determines the
displacement anywhere in an element given the values of its nodal dof. Introducing
this relation into the strain-displacement relation, we may express strain in terms of
the nodal displacement, element interpolation scheme and differential operator
matrix. Recalling that the expression for the potential energy of an elastic body
includes an integral for strain energy stored (dependent upon the strain field) and
18
integrals for work done by external forces (dependent upon the displacement field),
we can therefore express system potential energy in terms of nodal displacement.
Applying the principle of minimum potential energy, we may set the partial
derivative of potential energy with respect to the nodal dof vector to zero, resulting
in a summation of element stiffness integrals, multiplied by the nodal displacement
vector, equals a summation of load integrals. Each stiffness integral results in an
element stiffness matrix, which sum to produce the system stiffness matrix, and the
summation of load integrals yields the applied load vector, resulting in K.d = r. In
practice, integration rules are applied to elements, loads appear in the r vector and
nodal dof boundary conditions may appear in the d vector or may be partitioned
out of the equation.
Solution methods for finite element matrix equations are plentiful. In the
case of the linear static K.d = r, inverting K is computationally expensive and
numerically unstable. A better technique is Cholesky factorization, a form of Gauss
elimination, and a minor variation on the “LDU” factorization theme. The K
matrix may be efficiently factored into LDU, where L is the lower triangular, D is
the diagonal, and U is the upper triangular, resulting in LDU.d = r. Since L and D
are easily inverted, U is the upper triangular; d may be determined by back-
substitution.
In case of h-code elements, the order of the interpolation polynomials is
fixed. Another technique, p-code, increases the order iteratively until convergence,
with error estimates available after one analysis. Finally, the boundary element
method places elements only along the geometrical boundary.
19
Post-processing:
After a finite element model has been prepared and checked, boundary
conditions have been applied, and the model has been solved, it is time to
investigate the results of the analysis. This activity is known as the post-processing
phase of the finite element method.
Post-processing begins with a thorough check for problems that may have
occurred during solution. Next, reaction loads at restrained nodes should be
summed and examined. Reaction loads that do not closely balance the applied load
resultant for a linear static analysis should cast doubt on the validity of other
results. Error norms such as strain energy density and stress deviation among
adjacent elements might be looked at next, but for h-code analyses these quantities
are best used to target subsequent adaptive remeshing.
Once the solution is verified to be free of numerical problems, the quantities
of interest may be examined. Many display options are available, the choice of
which depends on the mathematical form of the quantity as well as its physical
meaning. For example, the displacement of solid linear brick elements is a
3component spatial vector, and the model’s overall displacement is often displayed
by superposing the deformed shape over undeformed shape.
Dynamic viewing and animation capabilities aid greatly in obtaining an
understanding of the deformed pattern stresses, being tensor quantities, currently
lack a good single visualization technique, and thus derived stress quantities are
extracted and displayed.
20
1.5 ANSYS:
ANSYS is general-purpose finite element analysis (FEA) software package.
Finite element analysis is a numerical method of deconstructing a complex system
into very small pieces (of user-designated size) called elements. The software
implements equations that govern the behavior of these elements and solves them
all, creating a comprehensive explanation of how the system acts as a whole. These
results then can be presented in tabulated or graphical forms. This type of analysis
is typically used for the design and optimization of a system far too complex to
analyze by hand. Systems that may fit into this category are too complex due to
their geometry, scale, or governing equations.
ANSYS is the standard FEA teaching tool within the Mechanical
Engineering Department at many colleges. ANSYS is also used in Civil and
Electrical Engineering, as well as the Physics and Chemistry departments.
Use of ANSYS:
ANSYS provides a cost-effective way to explore the performance of
products or processes in a virtual environment. This type of product development
is termed virtual prototyping.
With virtual prototyping techniques, users can iterate various scenarios to
optimize the product long before the manufacturing is started. This enables a
reduction in the level of risk, and in the cost of ineffective designs. The
multifaceted nature of ANSYS also provides a means to ensure that users are able
to see the effect of a design on the whole behavior of the product, be it
electromagnetic, thermal, mechanical etc.
General Steps to solving any problem in ANSYS:
Like solving any problem analytically, you need to define
(1) Your solution domain,
(2) The physical model,
21
(3) Boundary conditions and
(4) The physical properties.
You then solve the problem and present the results. In numerical methods, the
main difference is an extra step called mesh generation. This is the step that
divides the complex model into small elements that become solvable in an
otherwise too complex situation. Below describes the processes in terminology
slightly more attune to the software.
Build Geometry
Construct a two or three dimensional representation of the object to be
modeled and tested using the work plane co-ordinate system within ANSYS.
Define material properties:
Now that the part exists, define a library of the necessary materials that
compose the object (or project) being modeled. This includes thermal and
mechanical properties.
Generate mesh:
At this point ANSYS understands the makeup of the part. Now define how
the modeled system should be broken down into finite pieces.
Apply loads:
Once the system is fully designed, the last task is to burden the system with
constraints, such as physical loadings or boundary conditions.
Obtain solution:
This is actually a step, because ANSYS needs to understand within what
state (steady state, transient state… etc) the problem must be solved.
Present the results:
After the solution has been obtained, there are many ways to present
ANSYS’ results, choose from many options such as tables, graphs, and contour
plots.
22
INTRODUCTION:
ANSYS has a large-scale general purpose Finite Element program with
diversified capabilities for analysis of a wide range of engineering applications.
The major capabilities in ANSYS include linear and non-linear static, dynamic,
buckling and heat transfer analyses. A comprehensive library is also available. It
also includes 2D and 3D point masses, springs, spars and beams, 2D plane and
axisymmetric solid elements, 3D solid elements, axisymmetric 3D shell elements,
3D layered composite solid and sandwich shell elements, gap elements, and axi-
symmetric shell and solid elements with non-symmetric loading. The available
material models in ANSYS include linear isotropic and orthotropic elastic
materials, elastoplastic material with various forms of consecutive relationships.
Other capabilities includes extensive data checking with explicit diagnostic
messages, an efficient wave front solution technique and element resequencing
algorithm, multiple load cases with different boundary conditions and numerous
output options. ANSYS completely interface with the DISPLAY for model and
input data generation and post processing of analysis results. Highlights of the
main features of the pre and post processor DISPLAY program are given in
subsequent topics.
DISPLAY:
The primary goal is to provide an easy user interface for creating input files
for the finite element solver. The first step in the modeling process is to create a
geometric model which is the representation of the system or component. After
completing this step successfully, finite elements should generate and load and
boundary conditions may be imposed to obtain the finite element model. This
FEM model then creates input files for different solvers offered within ANSYS.
23
The modeling process can be broadly divided into two steps:
GEOMETRIC MODELLING:
The basic purpose of the geometric modeling phase is to represent geometry
of points (grids), lines, surfaces (patches) and volumes (hyper patches). It is an art
to represent the geometry of a body in an idealized form using various entities of
cad systems. It can be classified into three groups.
a. Wire frame modeling: It is the name given to a technology where a structure is
represented in the form of grids and lines in space where no continuum exists
among the grids, enclosed by the envelope of associated lines.
b. Surface modeling: It is the name given to the technology where a structure is
represented in the form of grids and lines in space where continuum exists in
form of surface patches among the grids, enclosed by the envelope of
associated lines.
c. Solid modeling: It is the name given to a technology where a structure is
represented in the form of solid entities viz., sphere, toroid, etc., here the
volume is associated with model.
For better understanding, let us take for example that a cantilever plate is to
be modeled. One approach to this problem will be to generate four corner points
(grids) by defining their co-ordinates or by snapping the cursor on the screen. Now
the grids are joined to create four lines. Now the patch is created. Note that
although the cantilever has a third dimension i.e., the thickness, the geometric
representation ignores this dimension at this stage of modeling. Later the thickness
can be encountered at the time of meshing the elements. However if the third
dimension is of significance, one may decide to model it as a solid structure. In this
case, the existing patch is translated in the third dimension by amount of thickness.
By joining opposite patches in the direction of thickness, volume geometry (hyper
24
patch) is obtained. This representation of geometry in the form of grids, lines,
patches, and hyper patches is referred as geometric modeling.
FINITE ELEMENT MODELING:
It is described as the representation of geometry model in terms of finite
number of nodes and elements; this model also contains information about material
and other properties, loading and boundary conditions. Following steps performs
this phase of geometry:
Finite element generation:
In this step, the user maps geometric entities may be created in geometric
modeling with nodes and elements. The complete geometry is defined as
assemblage of discrete pieces called elements, and is connected together with
nodes. The mapping is achieved by automatic or semi-automatic options available
in DISPLAY. If complete automatic meshing is used then there is no need to check
for verification of model.
Model verification:
The model is verified to ensure that it is correct physically and numerically.
Model verification makes sure that all the elements generated earlier are acceptable
to the finite element solver. Any wrapped, skewed or distorted element will be
highlighted in this process. Special attention is also be paid for discontinuities,
aspect ratio etc by checks. The model is checked by shading and shrinking the
elements to confirm for proper connectivity.
Problem definition:
At this stage, the material and element properties are defined along with
loadings and boundary conditions.
File management:
This is the final step of modeling and is useful in order to customize the
ANSYS input file to be generated as per the needs of the particular analysis type.
25
The user has flexibility to define multiple loadings and boundary conditions at this
stage (combinations of some or all the specified loads, pressures, specific
temperature or boundary conditions).
These steps discussed above are guidelines for easy modeling, though the
modeling for realistic problems is an art and do not follow any rigid rules. The
same problem can be modeled by a variety of approaches depending on user.
PRE-POST PROCESSING:
Pre processing:
Pre-processing module of the DISPLAY program is a 3D interactive color
graphic program with extensive modeling capabilities for finite element model
generation and problem definition.
CAD/CAM interface, directly from geometry database or through IGES
format.
Both command and menu driven modes, with on-line help.
3D geometric modeling including points, lines, arcs, curves, surfaces and
solids as well as surface intersection.
Geometric transformations including translation, rotation, scaling, mirror
imaging and dragging a curve along an arbitrary 3D path.
3D interactive Finite Element mesh generation including automatic node and
element generation.
Mesh grading with uniform or non-uniform spacing.
Merging separate models into a larger one.
Definition of element attributes including material and geometric properties.
Specification of loading and boundary conditions.
Extensive model editing capabilities.
26
Extensive plotting options including boundary line, hidden line removal and
shrink element plots for selected elements or regions.
Color shading and light effects.
Model checking including calculations of elements areas, volumes, normal
and distortion index.
Complete ANSYS data deck generation.
INPUT FEATURES:
Executive command date block: this consists of alphanumeric commands
specifying general control parameters for the analysis, e.g. specifying the
type of analysis to be performed.
Model data block: This block generally represents the bulk of the input data.
It describes the model characteristics in terms of nodes, elements, material
and geometric properties etc.
Analysis data block: This block describes data pertinent to various analysis
types, e.g. loading, boundary conditions, output control etc.
OUTPUT FEATURES:
Various output options is offered to meet individual requirement, with
comprehensive output control for all analysis types.
For static, non-linear, buckling and eigen value analysis the various output features
available are as below:
Displacements in global or local directions.
Reactions and summation of reactions in global directions.
Elements internal forces and strain energy.
Element stresses at centroid, gauss and nodal points as well as principal
stresses. The following stress intensities can be included.
1. Maximum shear stress, Smax
27
2. Von-Mises equivalent stress, Seq
3. Octahedral shear stress, Soct
POST PROCESSING:
Graphical representation of the results is obtained interactively using the
post processing module of DISPLAY following run.
Various output features include:
Various geometry plotting options including hidden line removal, boundary
and feature line plots and view manipulation including rotation, scaling and
zooming.
Deformed geometry plots, separate or super imposed on undeformed
geometry.
Animated deformed shapes.
Deformed history plots for non-linear static analysis.
Contour plots for displacements, mode shapes, stresses and temperatures.
Contour plots for cut sections of 3D models.
XY history plots for various output quantities.
AXISYMMETRIC ELEMENTS:
The axi-symmetric element represents a significant reduction in both model
creation and solution effort when its selection is appropriate. The primary
consideration is that both the geometry and the loading must be symmetrical about
the axis of revolution for this element to be validly employed. The head of a vessel
under pressure loading is an example of a good candidate for the use of axi-
symmetric elements. Often, however, vessels have non axi-symmetric mechanical
loading and/or thermal profiles. One side of the vessels is often at a significantly
different temperature than the other side of the vessel. This is especially true for
horizontal vessels. For these cases, the use of axi-symmetric elements will
28
introduce inaccuracies into the solution; another type of element will likely
produce better results.
VARIOUS TYPES OF ANALYSIS:
Different analysis capabilities of ANSYS include:
Linear static analysis
Non-linear static analysis
Dynamic analysis
Eigen value analysis
Modal dynamic analysis
Transient dynamic analysis
Random vibration analysis
Frequency response analysis
Shock spectrum analysis
Buckling analysis
Heat transfer analysis.
29
CHAPTER – 2
LITERATURE REVIEW
Brownwell and Young3 proposed an iterative calculation method to
determine the optimum radius of the elastic-plastic boundary, which was tedious to
use. Franklin and Morrison4 performed an analysis of residual stresses and
deformation of autofrettaged cylinders. Blazinski5 studied the autofrettage of
cylinders made of elastic-perfectly plastic material. Chen6 also studied the stress
analysis and deformation of autofrettaged cylinders. Kong7 applied graphical
method to determine the optimum radius of the elastic-plastic boundary which was
inaccurate and tedious. Gao8 obtained the elasto-plastic solution of a strain-
hardening cylinder subjected to internal pressure in plane stress by assuming an
isotropic hardening material model (with no Bauschinger effect). Avitzur9 studied
the stress distribution after depressurization of a thick-walled cylinder made of
elastic-perfectly plastic material based on the von-Mises and Tresca yield criteria
in plane stress and plane strain. Zhu and Yang10
determined the optimum radius of
the elastic-plastic boundary and the optimum autofrettage pressure of a cylinder
made of elastic-perfectly plastic material in plane strain. Gao11
obtained the stress,
strain and displacement components of a strain-hardening cylinder subjected to
pressures greater than the elastic limit pressure in plane strain. Majzoobi et al.2
investigated the autofrettage process for strain-hardening cylinders in plane stress
using a commercial finite element code. They showed that the number of
autofrettage stages has no influence on optimization of the stress distribution.
Majzoobi et al.12
by using finite-element modeling also showed that the effect of
the autofrettage process on optimization of compound cylinders is negligible. In
this work, an attempt has been made to analyze the autofrettage in thick walled
cylinders using ANSYS.
30
CHAPTER – 3
LINEAR ANALYSIS OF SIMPLE CYLINDERS
(WITHOUT AUTOFRETTAGE)
A linear analysis is conducted if a structure is expected to exhibit linear
behavior. The deformation and load-carrying capability can be determined by
employing one of the analysis types available in ANSYS, static or dynamic,
depending on the nature of the applied loading. If the applied loading is determined
as part of the solution for structural stability, a buckling analysis is conducted. If
the structure is subjected to thermal loading, the analysis is referred to as thermo
mechanical.
In our project, first linear analysis of a simple cylinder is done by using
ANSYS 11.0 and verifies the analytical results with the theoretical calculations.
3.1 THEORETICAL CALCULATIONS:
Thick walled cylinders:
Cylinders are considered to be thick, if t/d >20 and for thin cylinders,
t/d<1/20
where t and d are the thickness and outer diameter of the cylinders.
The dimensions of the cylinder taken for modeling are:
Outer diameter, do = 105 mm
Thickness, t = 33.3 mm
31
Material Specifications:
Material selected for modeling and analyzing in ANSYS is Stainless steel.
This material was tested and their properties are mentioned below:
Yield stress, σy = 710 MPa
Ultimate stress, σult = 855 MPa
Young’s modulus, E = 2.11 x 105 N/mm
2
Poisson ratio, ν = 0.33
Calculation of maximum von-Mises stress (σvon-max)
Step I : Calculation of pressure limits P Y,i and P Y,o
Let PY,i be the internal pressure required at the onset of yielding at the inner surface
of the cylinder,
= (𝑘2−1)
3𝑘2 σy
Here, “k” denotes the ratio of outer radius to inner radius and its value is calculated
to be 2.734375
P Y,i = 355 MPa.
Let P Y,o be the internal pressure required to cause the wall thickness of cylinder to
yield completely,
=(𝑘2−1)
3 σy
P Y,o = 2655 MPa.
32
From the above, we calculated the pressure limits for a yielding to occur inside a
thick walled cylinder. The minimum pressure is considered as the operating as well
as the internal pressure (i.e.) Popr = Pi = 355 MPa.
Step II: Calculation of hoop or circumferential stress (σθ) and radial stress (σr)
Let, Radial stress, σr = 𝑃𝑖
𝑘2−1 1 −
𝑟𝑜2
𝑟2
σr = -355 MPa
Circumferential stress, σθ = 𝑃𝑖
𝑘2−1 1 +
𝑟𝑜2
𝑟2
σθ = 464.6 MPa.
Since, we are modeling a 2D cylinder in ANSYS, third stress i.e. axial stress σz is
not taken into consideration .
Step III: Calculation of maximum von-Mises stress (σvon-max)
Let, σvon-max = 3
2 (𝜍𝜃 − 𝜍𝑟)
σvon-max = 709.79 MPa.
3.2 FEA ANALYSIS:
ANSYS Procedure
In Preferences module, the individual discipline Structural is chosen.
In Pre-processor module, the element type for analysis is chosen by, Pre-
processor → Element type → Add → Solid → Select “Quad 4node 42”→
Ok. Pre-processor → Element type → Options → Element behavior → Axi-
symmetric → Ok.
33
Pre-processor → Define Material Properties → Select Material Models→
Structural → Linear → Elastic → Isotropic → Predict the values of Young’s
Modulus (Ex) and Poisson ratio (Prxy) → Ok.
The next step is to model the cylinder in 2-dimensional view by, Modeling
→ Create → Areas →Enter the dimensions of the cylinder as required →
Ok.
The created model should be meshed as finite as possible by, Meshing →
Mesh tool → Line set → Select the lines and enter the value for number of
element divisions’ → Ok.
The fully designed model is to be solved by applying the constraints and
pressure value as follows.
Solution → Define loads → Apply → Structural → Displacement → On
lines → Select the lines and arrest the cylinder by applying the constraints
with respect to Y axis (UY) → Ok.
Fig. 10: Cylinder with constraint and load applied
34
Solution → Define loads → Apply → Structural → Pressure → On lines →
Select the line and enter the pressure value → Ok.
Solution → Solve → Current LS → Ok.
The result values for the solved cylinder is determined by,
General Post processor → Plot results → Contour plot → Stress → von-
Mises stress → Ok.
General Post processor → List results → Stress → von-Mises stress → Ok.
Fig. 11: Solution for von –Mises stress for a simple cylinder.
35
Fig. 12: Stress results for Radial stress and Circumferential stress
36
3.3 RESULT COMPARISONS:
STRESSES (MPa) FEA RESULTS ANALYTICAL RESULTS
Radial stress, σr -350.167 -355
Circumferential stress, σθ 459.658 464.6
Max von-Mises Stress,
σvon-max
701.777 709.79
Table 1: Results for cylinder without autofrettage
37
CHAPTER – 4
NON-LINEAR ANALYSIS (With Autofrettage)
The Non-linear analysis option enables the determination of the internal
forces and deformations of beam and shell structures made of reinforced concrete
and steel as well as stress states of solid elements according to bilinear material
law while considering geometric and physical nonlinearities. Due to the
considerable numerical complexity, it only makes sense to use this program
module for special problems.
Equilibrium of the deformed system according to the deflection theory, if
this has been activated for the corresponding load case.
Beams and area elements made of steel with bilinear stress-strain curve
under consideration of the Huber von-Mises yield criterion and complete
interaction with all internal forces.
Beams, area and solid elements with bilinear stress-strain curve and
individually definable compressive and tensile strength.
Compression-flexible beams.
Beam and area element bedding with bilinear bedding curve perpendicular
to and alongside the beam.
Solid elements with bilinear bedding curve in the element coordinate system.
The load type Prestressing can be used with area and solid elements to
simulate prestressing without bond (only load).
Analysis Method:
The model used is based on the finite element method. To carry out a non-
linear system analysis (load-bearing capacity and serviceability checks), the
standard beam elements are replaced internally with flexibly connecting, nonlinear
38
beams with an increased displacement function. For the treatment of area and
volume structures nonlinear layer elements and solid elements are used. In contrast
to the linear calculation and in order to account for the nonlinear or even
discontinuous properties of the structure, a finer subdivision is generally necessary.
Requirements:
The beams are assumed to be straight.
The area elements are flat.
Area and beam sections are constant for each element.
The dimensions of the section are small compared with the other
dimensions.
Shear deformations or beams are accounted for with a shear distortion that is
constant across the section, meaning the section remains plane after the
deformation but is no longer perpendicular to the beam axis.
The mathematical curvature is linearized.
The load is slowly increased to its final value and does not undergo
deviation in direction as a result of the system deformation.
4.1 THEORETICAL CALCULATIONS:
The material is taken as Stainless Steel for nonlinear analysis and the type of
cylinder is to be simple thick-walled cylinder. The material specifications and the
dimension parameters are taken as same as that for linear analysis.
Step I : Calculation of pressure limits P Y,i and P Y,o
Let P Y,i be the internal pressure required at the onset of yielding at the inner
surface of the cylinder,
= (𝑘2−1)
3𝑘2 σy
39
Here, “k” denotes the ratio of outer radius to inner radius and its value is calculated
to be 2.734375
P Y,i = 355 MPa.
Let P Y,o be the internal pressure required to cause the wall thickness of cylinder to
yield completely,
=(𝑘2−1)
3 σy
P Y,o = 2655 MPa.
From the above, we calculated the pressure limits for a yielding to occur inside a
thick walled cylinder. The minimum pressure is considered as the operating as well
as the internal pressure (i.e.) Popr = Pi = 355 MPa.
Step II : Calculation of autofrettage pressure (Pa)
Let “n” be denoted as the ratio of operating pressure to yield stress.
n = 𝑃𝑜𝑝𝑟
𝜍𝑦 = 0.5
Let “m” be the ratio of autofrettaged radius to inner radius
m = 𝑟𝑎
𝑟𝑖
Since, the value of autofrettaged radius was not found, an equation which gives a
relation between “n” and “m” was used to found the value of “m” based on von
Misses criterion.
m = exp ( 3
2 n) = 1.542
40
Let the optimum autofrettaged pressure be,
P a,opt = 𝜍𝑦
2 1 −
𝑒 3 𝑛
𝑘2+ 3 𝑛
= 550 MPa.
Step III : Determination of autofrettaged radius (ρ)
The relationship between autofrettage pressure(Pa) and the radius of the
elastic-plastic boundary i.e. autofrettage radius (ρ) is given as follows:
Pa = 𝝈𝒚
𝟑 𝟏 −
𝝆𝟐
𝒓𝒐𝟐 +
𝟏
𝒏 𝝆𝟐𝒏
𝒓𝒊𝟐𝒏 − 𝟏
where n is the strain hardening exponent of a material chosen
ro and ri be the outer and inner radius of the cylinder.
by applying all the known values in the above equation, we get the solution for ρ
as 25.3 mm.
Step IV : Calculation of maximum von-Mises stress (σvon-max)
The pressure limits for a cylinder to yield is obtain between 355 MPa and
2655 MPa (as calculated in Step I). To calculate the maximum von-Mises stress,
the working pressure (Pw) is to be known. For this pressure, as per the study the
working pressure will always less than that of the internal pressure (Pw < Pi).
Hence, the working pressure is taken as 300 MPa. Therefore, the maximum von-
Mises stress is calculated by,
σvon-max = 3𝜍𝑦𝑛 𝜌2− 𝑟𝑖
2 + 3𝜍𝑦𝑟𝑖2 1− 𝜌 𝑟𝑖 2𝑛 + 3𝑟𝑖
2𝑛𝑝𝑤 𝑟𝑜2
3𝜌2𝑛 ( 𝑟𝑜2− 𝑟𝑖
2)
σvon-max = 198.58 MPa
41
Step V : Generation of Material curve
For a nonlinear analysis in ANSYS, a material curve is to be generated for
the material chosen for modeling with all the parameter provided as follows:
Outer diameter, ro = 105 mm
Thickness, t = 33.3 mm
Yield stress, σys = 710 MPa
Ultimate stress, σult = 855 MPa
Young’s Modulus, E = 211 GPa
Poisson ratio, ν = 0.33
Material constant, ε0 = 0.0040
Material constant, n = 2.26
Fig. 13: Material Stress-Strain curve for AISI4340
42
Here, a material stress-strain curve by using the above parameters of the
material in a stress-strain relationship is generated as follows:
σ = E𝜺 𝟏 + Ɛ
Ɛ𝟎 𝒏
−𝟏
𝒏
where ε0 = 𝜍𝑢𝑙𝑡
𝐸 and n is the parameter defining the shape of the non-linear stress-
strain relationship. The above expression represents essentially the inverse of
Romberg-Osgood equation.
43
4.2 ANSYS PROCEDURE:
In Preferences module, the individual discipline Structural is chosen.
In Pre-processor module, the element type for analysis is chosen by, Pre-
processor → Element type → Add → Solid → Select “Quad 4node 42”→
Ok. Pre-processor → Element type → Options → Element behavior → Axi-
symmetric → Ok.
For nonlinear analysis, the material properties are predicted as follows. Pre-
processor → Material property → Select material models → Structural >
Nonlinear > Inelastic > Rate independent > Kinematic Hardening Plasticity
> Mises Plasticity > Multi linear (General). After this, enter the values of
Young’s Modulus and Poisson ratio and select Ok. Next to this, enter the
values of stress and strain (Maximum 20 values) → Select Ok.
Fig. 14: Material model behavior of a simple cylinder for Nonlinear analysis
44
The next step is to model the cylinder in 2-dimensional view by, Modeling
→ Create → Areas →Enter the dimensions of the cylinder as required →
Ok.
The created model should be meshed as finite as possible by, Meshing →
Mesh tool → Line set → Select the lines and enter the value for number of
element divisions’ → Ok.
Before defining the loads to the meshed model, for a nonlinear analysis to be
done, the below procedure is followed.
Solution → Analysis type → Solution Controls, a dialog box will be
appeared as below.
Fig. 15: Solution control setup for a nonlinear analysis
45
Next, the constraints and the loads are defined as for linear analysis, the only
difference is that the load applied is autofrettage pressure (Pa).
Then, the solution is solved as a nonlinear solution. After the solution was
done, go to Plot → Elements. The following stress image will appear.
Fig. 16: Constraints of a cylinder after applying autofrettage pressure (Pa)
Now, as per the concept of autofrettage, the pressure applied to the cylinder
is removed and a residual compressive stress is setup at the walls of the
cylinder. The loads applied to the cylinder was deleted by, Solution →
Define loads → Delete → Structural → Pressure → on lines → Select the
side, where the autofrettage pressure was applied and select Ok.
46
Solve the solution by, Solution → Solve → Current LS. After the solution
was done, go to Plot > Elements. The constraints of a cylinder with residual
stress will be displayed as follows.
Fig. 17: Constraints of a cylinder with residual stress
Then, go to Solution → Analysis type → Solution Controls → Select
“Calculate prestress effect” → Ok.
Now, apply the operating pressure Popr (Pint) to the side, where the
autofrettage pressure was applied and solve the solution.
Now, the stress values obtained for this nonlinear analysis is to be verified
with the theoretical values and checked.
47
Fig. 18: von-Mises stress results of a cylinder for nonlinear analysis
Determination of Autofrettage radius (ρ) in ANSYS:
As per the ANSYS procedure followed above, the radius of the cylinder is
plotted along X-axis and the maximum von-Mises stress is plotted along Y-axis.
The radius at which the von-Mises stress is minimum is taken as the autofrettage
radius(ρ). The curve is plotted as follows:
48
Fig. 19: Graph between autofrettage radius and max. von-Mises stress
The values are plotted in the table for the above graph as follows:
S.No. Autofrettaged radius, ρ
(in mm)
Maximum von-Mises stress, σvon-max
(in MPa)
1. 19.2 198.77
2. 22.53 89.983
3. 25.86 13.482
4. 29.19 40.974
5. 32.52 48.152
6. 35.85 54.243
Table 2: Autofrettage radius Vs Maximum von-Mises stress
49
4.3 RESULTS AND COMPARISONS:
This table shows the value obtained for maximum von-Mises stress in FEA
results and compares it with analytical solutions.
Nonlinear analysis FEA Results Analytical Results
Maximum von-Mises stress(MPa) 199.027 198.58
Autofrettaged radius ,ρ (in mm) 25.86 25.3
Table 3: Results for cylinder with autofrettage
The below table shows the comparison of von-Mises stress with linear and
nonlinear analysis.
von-Mises stress (in MPa) Linear analysis Nonlinear analysis
Analytical results 709.79 198.58
FEA Results 701.77 199.027
Table 4: Comparison of von-Mises stress for linear and nonlinear analysis
50
CHAPTER – 5
CONCLUSION
The analysis of simple thick-walled cylinder was carried out and the results
were obtained. Then, the cylinder with autofrettage effect was analyzed. The result
shows that the maximum von-Mises stress of autofrettaged cylinder was less than
that of the simple cylinder. Similarly, the cylinder without autofrettage and with
autofrettage was modeled and analyzed using ANSYS. It was found that FEA
results are in good agreement with analytical results.
51
REFERENCES
1. M.H. Hojjati and A. Hassani, “Theoretical and finite-element modeling of
autofrettage process in strain-hardening thick-walled cylinders”,
International Journal of Pressure vessels and Piping, 84 (2007), 310-319.
2. Majzoobi GH, Farrahi GH, Mahmoudi AH, “A finite element simulation and
an experimental study of autofrettage for strain hardened thick-walled
cylinder. Mater Sci Eng 2003;359;326-331.
3. Brownell LE, Young EH, “Process equipment design”, New
York:Wiley;1959.
4. Farnklin GJ, Morrison JIM, “Autofrettage of cylinders; prediction of
pressure, external expansion curves and calculation of residual stresses”,
Proc Inst Mech Eng 1960;174;947-74.
5. Blazinski TZ, “Applied elasto-plasticity of solids”, Hong-
Kong:Macmillan;1983.
6. Chen PCT, “Stress and deformation analysis of autofrettaged high pressure
vessels”, ASME special publication, 110, PVP. New York: ASME United
Engineering Center; 1986.p.61-7.
7. Kong F, “Determining the optimum radius of the elastic-plastic junction, R,
for thick-walled autofrettage cylinder by graphic method”. Petrochem
Equip1986;15;11-2.
8. Gao XL, “An exact elasto-plastic solution for an open ended thick-walled
cylinder of a strain-hardening material”. Int J Pressure Vessel Piping
1992;52;129-44
9. Avitizur B, “Autofrettage stress distribution under load and retained stresses
after depressurization”, Int J Pressure Vessel Piping 1994;57:271-87.
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10. Zhu R, Yang J, “Autofrettage of thick cylinders”, Int J Pressure Vessel
Piping 1998;75:443-6.
11. Gao XL, “Elasto-plastic analysis of an internally pressurized thick-walled
cylinder using a strain gradient plasticity theory”, Int J Pressure vessel
Piping 2003:40:6445-55.
12. Majzoobi GH, Farrahi GH, Pipelzadeh MK, Akbari A, “Experimental and
finite element prediction of bursting pressure in compound cylinders”, Int J
Pressure Vessel Piping 2004;81;889-96.
13. Aseer Brabin T, Christopher T, Nageswara Rao B, “Investigation on failure
behavior on unflawed steel cylindrical pressure vessels using FEA”,
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Recommended