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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.

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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.

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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

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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

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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)

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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

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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

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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)

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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

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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

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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

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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.

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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.

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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.

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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

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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

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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

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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.

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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.

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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,

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(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.

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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.

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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

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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

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3. Brownell LE, Young EH, “Process equipment design”, New

York:Wiley;1959.

4. Farnklin GJ, Morrison JIM, “Autofrettage of cylinders; prediction of

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5. Blazinski TZ, “Applied elasto-plasticity of solids”, Hong-

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6. Chen PCT, “Stress and deformation analysis of autofrettaged high pressure

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9. Avitizur B, “Autofrettage stress distribution under load and retained stresses

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